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--- abstract: 'Video image datasets are playing an essential role in design and evaluation of traffic vision algorithms. Nevertheless, a longstanding inconvenience concerning image datasets is that manually collecting and annotating large-scale diversified datasets from real scenes is time-consuming and prone to error. For that virtual datasets have begun to function as a proxy of real datasets. In this paper, we propose to construct large-scale artificial scenes for traffic vision research and generate a new virtual dataset called “ParallelEye”. First of all, the street map data is used to build 3D scene model of Zhongguancun Area, Beijing. Then, the computer graphics, virtual reality, and rule modeling technologies are utilized to synthesize large-scale, realistic virtual urban traffic scenes, in which the fidelity and geography match the real world well. Furthermore, the Unity3D platform is used to render the artificial scenes and generate accurate ground-truth labels, e.g., semantic/instance segmentation, object bounding box, object tracking, optical flow, and depth. The environmental conditions in artificial scenes can be controlled completely. As a result, we present a viable implementation pipeline for constructing large-scale artificial scenes for traffic vision research. The experimental results demonstrate that this pipeline is able to generate photorealistic virtual datasets with low modeling time and high accuracy labeling.' author: - 'Xuan Li, Kunfeng Wang, *Member, IEEE*, Yonglin Tian, Lan Yan, and Fei-Yue Wang, *Fellow, IEEE*[^1][^2][^3][^4][^5][^6]' title: 'The ParallelEye Dataset: Constructing Large-Scale Artificial Scenes for Traffic Vision Research' --- Introduction ============ The publicly available video image datasets have received much attention in recent years, due to its indispensability in design and evaluation of computer vision algorithms [@Geiger2013]. In general, a computer vision algorithm needs a large amount of labeled images for training and evaluation. The datasets can be divided into two types: unlabeled datasets used for unsupervised learning and labeled datasets used for supervised learning. However, manually annotating the images is time-consuming and labor-intensive, and participants often lack professional knowledge, making some annotation tasks difficult to execute. Experts are always sparse and should be properly identified. As we known, the human annotators are subjective, and their annotations should be re-examined if two or more annotators have disagreements about the label of one entity. By contrast, the computer is objective in processing data and particularly good at batch processing, so why not let the computer annotate the images automatically? At present, most publicly available datasets are obtained from real scenes. As the computer vision field enters the big data era, researchers begin to look for better ways to annotate large-scale datasets [@Handa2014]. At the same time, the development of virtual datasets has a long history, starting at least from Bainbridge’s work [@Bainbridge2007]. Bainbridge used Second Life and World of Warcraft as two distinct examples of virtual worlds to predict the scientific research potential of virtual worlds, and introduced the virtual worlds into a lot of research fields that scientists are now exploring, including sociology, computer science, and anthropology. In fact, synthetic data has been used for decades to benchmark the performance of computer vision algorithms. The use of synthetic data has been particularly significant in object detection \[4\], \[5\] and optical flow estimation \[6\]-\[8\], but most virtual data are not photorealistic or akin to the real-world data, and lack sufficient diversity [@Ros2015]. The fidelity of some virtual data is close to the real-world [@Prendinger2013]. However, the synthesized virtual worlds are seldom equivalent to the real world in geographic position, and seldom annotate the virtual images automatically. Richter *et al.* [@Richter2016] used a commercial game engine to extract virtual images, with no access to the source code or the content. The SYNTHIA dataset [@Ros2016] provided a realistic virtual city as well as synthetic images with automatically generated pixel-level annotations, but in that dataset there lacks other annotation information such as object bounding box and object tracking. Gaidon *et al.* [@Gaidon2016] proposed a virtual dataset called “Virtual KITTI" as a proxy for tracking algorithm evaluation. While this dataset was cloned from “KITTI", it cannot extend easily to arbitrary traffic networks. Due to the above limitations, new virtual datasets that match the real world and provide detailed ground truth annotations are still desirable. {width="7in"} Manually annotating pixel-level semantics for images is time-consuming and not accurate enough. For example, annotating high-quality semantics with 10-20 categories in one image usually takes 30-60 minutes [@Kundu2014]. This is known as the “curse of dataset annotation” [@Xie2016]. The more detailed the semantics, the more labor-intensive the annotation process. As a result, many datasets do not provide semantic segmentation annotations. For example, ImageNet [@Karpathy2014],[@Russakovsky2015] has 14 million images, in which more than one million images have definite class and the images are annotated with object bounding box for object recognition. However, ImageNet does not have semantic segmentation annotations. Some datasets provide only limited semantic segmentation annotations. For example, NYU-Depth V2 [@Silberman2012] has 1449 densely labelled images, KITTI [@Geiger2013] has 547 images, CamVid [@Brostow2009],[@Browstow2008] has 600 images, Urban LabelMe [@Russell2008] has 942 images, and Microsoft COCO [@Lin2014] has three hundred thousand images. These datasets play an important role in the study of semantic segmentation. However, these datasets cannot be used directly in intelligent transportation, especially in automobile navigation, because the number of labeled images is insufficient and the segmented semantics have different categories. Currently, computer vision algorithms that exploit context for pattern recognition would benefit from datasets with many annotated categories embedded in images from complex scenes. Such datasets should contain a wide variety of environmental conditions with annotated object instances co-occurring in the same scenes. However, the real scenes are unrepeatable and the captured images are expensive to annotate, making it difficult to obtain large-scale, diversified datasets with precise annotations. In order to solve these problems, this paper proposes a pipeline for constructing artificial scenes and generating virtual images. First of all, we use map data to build the 3D scene model of Zhongguancun Area, Beijing. Then, we use the computer graphics, virtual reality, and rule modeling technologies to create a realistic, large-scale virtual urban traffic scene, in which the fidelity and geographic information can match the real world well. Furthermore, we use the Unity3D development platform for rendering the scene and automatically annotating the ground truth labels including pixel-level semantic/instance segmentation, object bounding box, object tracking, optical flow, and depth. The environmental conditions in artificial scenes can be controlled completely. In consequence, we generate a new virtual image dataset, called “ParallelEye" (see Fig. 1). We will build a website and make this dataset publicly available before the publication of this paper. The experimental results demonstrate that our proposed implementation pipeline is able to generate photorealistic virtual images with low modeling time and high fidelity. ![Basic framework and architecture for parallel vision [@KWang2016].[]{data-label="fig_sim"}](fig/fig2.pdf){width="3.3in"} The rest of this paper is organized as follows. Section II introduces the significance of parallel vision and virtual dataset. Section III presents our approach to constructing artificial scenes and generating virtual images with ground-truth labels. Section IV reports the experimental results and analyzes the performance. Finally, the concluding remarks are made in section V. Parallel Vision and Virtual Dataset =================================== Parallel vision \[23\]-\[25\] is an extension of the ACP (Artificial systems, Computational experiments, and Parallel execution) theory \[26\]-\[30\] into the computer vision field. For parallel vision, photo-realistic artificial scenes are used to model and represent complex real scenes, computational experiments are utilized to learn and evaluate a variety of vision models, and parallel execution is conducted to online optimize the vision system and realize perception and understanding of complex scenes. The basic framework and architecture for parallel vision [@KWang2016] is shown in Fig. 2. Based on the parallel vision theory, this paper constructs a large-scale virtual urban network and synthesizes a large number of realistic images. The first stage of parallel vision is to construct photorealistic artificial scenes by simulating a variety of environmental conditions occurring in real scenes, and accordingly to synthesize large-scale diversified datasets with precise annotations generated automatically. Generally speaking, the construction of artificial scenes can be regarded as “video game design", i.e., using the computer animation-like techniques to model the artificial scenes. The main technologies used in this stage include computer graphics, virtual reality, and micro-simulation. Computer graphics and computer vision, on the whole, can be thought of as a pair of forward and inverse problems. The goal of computer graphics is to synthesize image measurements given the description of world parameters according to physics-based image formation principles (forward inference), while the focus of computer vision is to map the pixel measurements to 3D scene parameters and semantics (inverse inference). Apparently their goals are opposite, but can converge to a common point: parallel vision. From the parallel vision perspective, we design the ParallelEye dataset. ParallelEye is synthesized by referring to the urban network of Zhongguancun Area, Beijing. Using OpenStreetMap (OSM), an urban network with length 3km and width 2km is extracted. Artificial scenes are constructed on this urban network. Unity3D is used to control the environmental conditions in the scene. There are 15 object classes in ParallelEye, reflecting the common elements of traffic scenes, including sky, buildings, cars, roads, sidewalks, vegetation, fence, traffic signs, traffic lights, lamp poles, billboards, trees, cyclists, pedestrians, and chairs. These object classes can be automatically annotated to generate pixel-level semantics. For traffic vision research, we pay special attention to instance segmentation, with each object of interest segmented automatically. In addition, ParallelEye provides accurate ground truth for object detection and tracking, depth, and optical flow. Approach ======== Our pipeline for generating the ParallelEye dataset is shown in Fig. 3. Firstly, the OSM data released by OpenStreetMap is used to achieve the correspondence in geographic location between the virtual and real world. Secondly, CityEngine is used to write CGA (Computer Generated Architecture) rules and design a realistic artificial scene, including roads, buildings, cars, trees, sidewalks, etc. Thirdly, the artificial scene is imported into Unity3D and gets rendered by using the script and the shader. In the dataset, accurate ground truth annotations are generated automatically, and environmental conditions can be controlled completely and flexibly. ![Pipeline for generating the ParallelEye dataset with OpenStreetMap, CityEngine, and Unity3D.[]{data-label="fig_sim"}](fig/fig3.pdf){width="2.35in"} Correspondence of Artificial and Real Scenes -------------------------------------------- In order to increase the fidelity, we choose to import geographic data from OpenStreetMap. Although Google Maps occupies an important position in geographic information, it is not an open-source software. By contrast, OpenStreetMap is an open-source, online map editing program with the goal of creating a world where content is freely accessible to everyone. In OpenStreetMap, the ways denote a directional node sequence. Each node of the network can connect 2-2000 paths, and then arrive at another node. The road information includes direction, lane number, lane width, street name, and speed limit. Each path can form three combinations: non-closed paths, closed paths, and regions. The non-closed paths correspond to the roads, rivers, and railways in the real world. The closed paths correspond to subway, bus routes, residential roads, and so on. The regions correspond to buildings, parks, lakes, and so on. Based on the properties of OSM data, it is easy to relate the real world to the geographic information of the artificial scene. Fig. 4 shows the real Automation Building of CASIA (Institute of Automation, Chinese Academy of Sciences) and its virtual proxy generated by CGA rules. They are similar in appearance. ![The real Automation Building of CASIA (top) and its virtual proxy (bottom).[]{data-label="fig_sim"}](fig/fig4.pdf){width="2.35in"} Generation of Ground-Truth Annotations -------------------------------------- As stated above, ground-truth annotations are essential for vision algorithm design and evaluation. Traditionally, the images were annotated by hand. The manual annotation is time-consuming and prone to error. Taking semantic/instance segmentation as an example, it usually takes 30-60 minutes to annotate an image with 10-20 object categories. Besides, manual annotation is more or less subjective, so that different annotators can make different semantic labels for the same image, especially near the object boundaries. Instead of manual annotation, this paper uses Unity3D to automatically generate accurate ground-truth labels. Fig. 5 shows some examples of ground-truth annotations, including depth, optical flow, object tracking, object detection, instance segmentation, and semantic segmentation. Generating ground truth with Unity3D is accurate and efficient. Semantic segmentation ground truth can be directly generated by using unlit shaders on the materials of the objects, with each category outputting a unique color. Instance segmentation ground truth is generated using the same method, but assigns a unique color tag to each object of interest. The modified shaders output a color which is not affected by the lighting and shading conditions. Depth ground truth is generated using built-in depth buffer information to get depth data for screen coordinates. The depth ranges from 0 to 1 with a nonlinear distribution, with 1 representing “infinitely distant". Optical flow ground truth is generated by calculating the instantaneous velocity of moving objects on the imaging plane and using the pixel changes in the image sequence to find the correspondence between the previous frame and the current frame. Given a pixel point $(x,y)$ in the image, at any time the brightness of that point is $E(x+\triangle x,y+\triangle y,t+\triangle t)$. Let $(u,v)=(\frac{\partial x}{\partial t},\frac{\partial y}{\partial t})$ represent the instantaneous velocity of the point in the horizontal and vertical directions, the brightness change occurs when the point moves. We use the Taylor formula to represent the pixel brightness: $$\label{} \begin{split} & E(x+\triangle x,y+\triangle y,t+\triangle t) \\ & =E(x,y,t)+\frac{\partial E}{\partial x}\triangle x+\frac{\partial E}{\partial y}\triangle y+\frac{\partial E}{\partial t}\triangle t+\varepsilon. \end{split}$$ For any $\triangle t\rightarrow0$, let $\omega=(u,v)$ , the optical flow constraint equation is given by $$\label{} -\frac{\partial E}{\partial t}=\frac{\partial E}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial E}{\partial y}\frac{\partial y}{\partial t}=\nabla E \cdot \omega,$$ where $\omega$ is the optical flow of $E(x,y,t)$. We generate multi-object tracking ground truth based on four rules: 1) when the object appears within the field of view of the camera, the three-dimensional bounding box of the object is converted to a two-dimensional bounding box; 2) when the object appears or disappears from the image boundary, we perform special handling for the bounding box; 3) we do not draw bounding boxes for objects that have less than 15 pixels in width or less than 10 pixels in height; 4) when occlusion occurs and the occlusion rate is higher than a threshold, we do not draw bounding boxes for the occluded object. ![Examples of ground-truth annotations generated automatically by Unity3D. Top: depth (left) and optical flow (right). Middle: object tracking (left) and object detection (right). Bottom: pixel-level instance segmentation (left) and semantic segmentation (right). Best viewed with zooming.[]{data-label="fig_sim"}](fig/fig5.pdf){width="3.5in"} Diversity of Artificial Scenes ------------------------------ In order to increase the diversity and fidelity of artificial scenes, we control the parameters in the script, the material, and the simulated environmental conditions. Specifically, the controllable parameters include: 1) number, type, trajectory, speed, and direction of the vehicles; 2) position and configuration of the camera; 3) weather (sunny, cloudy, rainy, foggy, etc) and illumination (daytime, dawn, dusk, etc). ![Illustration of the diversity of artificial scenes. Top: Virtual images with illumination at 6:00 am (left) and 12:00 pm (right) in a sunny day. Bottom: Virtual images with weather of fog (left) and rain (right).[]{data-label="fig_sim"}](fig/fig6.pdf){width="3.5in"} Traditionally, video image datasets are collected by capturing in the real world or retrieving from the Internet. It is impossible to control the environmental conditions and repeat the scene layout under different environments, and thus difficult to isolate the effects of environmental conditions on the performance of computer vision algorithms. By contrast, it is easy to control the environmental conditions in artificial scenes. In this work, we are able to flexibly control the camera’s location, height, and orientation to capture different contents of the artificial scene. We are also able to dynamically change the illumination (from sunrise to sunset) and weather conditions (sunny, cloudy, and foggy). Although we can change the environmental conditions in artificial scene, the ground-truth annotations are always easy to generate, no matter how adverse the illumination and weather conditions are and how blurred the image details are. This makes it possible to quantitatively analyze the impacts of each environmental condition on algorithm performance, usually called “ceteris paribus analysis". Fig. 6 illustrates the diversity of artificial scenes in terms of illumination and weather conditions. Experiments =========== Based on the proposed approach, we construct the artificial scene and configure virtual cameras to capture images from the scene. The virtual cameras can be moving or stationary. For automobile applications, the virtual cameras are installed on moving vehicles. For visual surveillance applications, the virtual cameras are fixed on the roadside or at intersections. The experiments are conducted to verify that the artificial scenes are repeatable and that the camera’s position, height, and orientation can be configured flexibly. Onboard Camera -------------- In this experiment, an onboard camera is configured at a height of 2 meters, mimicking the camera installed on the vehicle roof. There are totally 67 vehicles on the road, including 52 vehicles parking on the roadside (3 buses, 4 trucks, and 45 cars ) and another 15 vehicles in motion. We turn the camera orientation from left to right and get five orientations (i.e., -30, -15, 0, 15, and 30 degrees with respect to the lane direction). The distance between two cameras on adjacent lanes is 5 meters. These configurations lead to substantial changes in object appearance. Fig. 7 shows three continuous images captured by the onboard camera. ![Continuous images captured by an onboard camera: a sample image (left), another image annotated with object bounding boxes (middle), and the third image annotated with tracking bounding boxes of different colors (right). Best viewed with zooming.[]{data-label="fig_sim"}](fig/fig7.pdf){width="3.5in"} Surveillance Camera ------------------- In this experiment, a surveillance camera is installed at an intersection. We rotate the camera and control the rotation speed at 10 degrees per second, and the rotation range is 180 degrees. We also change the camera height, with the lifting speed of 0.1 meters per second and the lifting range of 2-5 meters. Such settings can fully simulate the role of surveillance cameras. Based on this experiment, the artificial scene provides virtual video images for intersection monitoring. Fig. 8 shows images captured by the surveillance camera. ![Continuous images captured by a surveillance camera: images annotated with object bounding boxes (top row), original images (middle row), and images annotated with tracking bounding boxes of different colors (bottom row). Best viewed with zooming.[]{data-label="fig_sim"}](fig/fig8.pdf){width="3.5in"} In order to increase diversity of virtual images and record the ground truth, we adopt the same operations for both the onboard camera and the surveillance camera. To record the ground truth, we use a green bounding box to record the detection ground truth for each object. We also assign a bound box of unique color to record the tracking ground truth for each object instance. To increase diversity, we dynamically change the illumination (daytime, dawn, and dusk) and weather (sunny, cloudy, rainy, and foggy) conditions in the artificial scenes. These subtle changes simulate different environmental conditions in the virtual world, and would otherwise need the expensive process of re-acquiring and re-labeling images of the real world. The advantage of this setting is that it can increase diversity of the ParallelEye dataset. In the experiments, with image resolution of 500\*375 pixels for ParallelEye, the pipeline for artificial scene construction and ground truth generation runs at 8-12 fps (frames per second) on a workstation computer. We have collected a total of 31,000 image frames, each of which has been annotated with accurate ground truth. We will build a website and make the dataset publicly available before the publication of this paper. Concluding Remarks ================== In this paper, we propose a new virtual image dataset called “ParallelEye". For that we present a dataset generation pipeline that uses street map, computer graphics, virtual reality, and rule modeling technologies to construct a realistic, large-scale virtual urban traffic scene. The artificial scene matches the real world well in terms of fidelity and geographic information. In the artificial scene, we flexibly configure the camera (including its position, height, and orientation) and the environmental conditions, to collect diversified images. Each image has been annotated automatically with ground truth including semantic/instance segmentation, object bounding box, object tracking, optical flow, and depth. In the future, we will improve the diversity of ParallelEye by introducing moving pedestrians and cyclists, which are harder to animate. We will increase the scale of ParallelEye. In addition, we will combine ParallelEye and the existing real datasets (e.g., PASCAL VOC, MS COCO, and KITTI) to learn and evaluate traffic vision models, in order to improve the accuracy and robustness of traffic vision models when applied to complex traffic scenes. [1]{} A. Geiger, P. Lenz, C. Stiller, and R. Urtasun, “Vision meets Robotics: The KITTI dataset," *International Journal of Robotics Research*, vol. 32, no. 11, pp. 1231-1237, 2013. A. Handa, T. Whelan, J. McDonald, and A. J. Davison, “A benchmark for RGB-D visual odometry, 3D reconstruction and SLAM,” *in Proceedings of the International Conference on Robotics and Automation*, IEEE, pp. 1524-1531, 2014. W. S. Bainbridge, “The scientific research potential of virtual worlds,” *Science*, vol. 317, no. 5837, pp. 472-476, 2007. J. Marin, D. Vazquez, D. Geronimo, and A. M. 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Wang, “Parallel vision for perception and understanding of complex scenes: Methods, framework, and perspectives,” *Artificial Intelligence Review*, vol. 48, no. 3, pp. 298-328, 2017. K. Wang, Y. Lu, Y. Wang, Z. Xiong, and F.-Y. Wang, “Parallel imaging: A new theoretical framework for image generation," *Pattern Recognition and Artificial Intelligence*, vol. 30, no. 7, pp. 577-587, 2017. F.-Y. Wang, “Parallel control and management for intelligent transportation systems: Concepts, architectures, and applications,” *IEEE Transactions on Intelligent Transportation Systems*, vol. 11, no. 3, pp. 630-638, 2010. F.-Y. Wang, “Parallel control: A method for data-driven and computational control,” *Acta Automatica Sinica*, vol. 39, no. 4, pp. 293-302, 2014. F.-Y. Wang, X. Wang, L. Li, and L. Li, “Steps toward parallel intelligence," *IEEE/CAA Journal of Automatica Sinica*, vol. 3, no. 4, pp.345-348, 2016. L. Li, Y. Lin, D. Cao, N. Zheng, and F.-Y. Wang, “Parallel learning — A new framework for machine learning," *Acta Automatica Sinica*, vol. 43, no. 1, pp. 1-8, 2017. X. Liu, X. Wang, W. Zhang, J. Wang, and F.-Y. Wang, “Parallel data: From big data to data intelligence,” *Pattern Recognition and Artificial Intelligence*, vol. 30, no. 8, pp. 673-682, 2017. [^1]: This work was partly supported by National Natural Science Foundation of China under Grant 61533019, Grant 71232006, and Grant 91520301. [^2]: Xuan Li is with the School of Automation, Beijing Institute of Technology, Beijing 100081, China, and also with The State Key Laboratory for Management and Control of Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]). [^3]: Kunfeng Wang (*Corresponding author*) is with The State Key Laboratory for Management and Control of Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China, and also with Qingdao Academy of Intelligent Industries, Qingdao 266000, China (e-mail: [email protected]). [^4]: Yonglin Tian is with the Department of Automation, University of Science and Technology of China, Hefei 230027, China, and also with The State Key Laboratory for Management and Control of Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China. [^5]: Lan Yan is with The State Key Laboratory for Management and Control of Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China. [^6]: Fei-Yue Wang is with The State Key Laboratory for Management and Control of Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China, and also with the Research Center for Computational Experiments and Parallel Systems Technology, National University of Defense Technology, Changsha 410073, China (e-mail: [email protected]).

--- abstract: 'Traditional indoor scene synthesis methods often take a two-step approach: object selection and object arrangement. Current state-of-the-art object selection approaches are based on convolutional neural networks (CNNs) and can produce realistic scenes for a single room. However, they cannot be directly extended to synthesize style-compatible scenes for multiple rooms with different functions. To address this issue, we treat the object selection problem as combinatorial optimization based on a Labeled LDA (L-LDA) model. We first calculate occurrence probability distribution of object categories according to a topic model, and then sample objects from each category considering their function diversity along with style compatibility, while regarding not only separate rooms, but also associations among rooms. User study shows that our method outperforms the baselines by incorporating multi-function and multi-room settings with style constraints, and sometimes even produces plausible scenes comparable to those produced by professional designers.' author: - Yu He - Yun Cai - Yuanchen Guo - Zhengning Liu - Shaokui Zhang - Songhai Zhang - Hongbo Fu - Shengyong Chen bibliography: - 'sample-bibliography.bib' title: 'Style-compatible Object Recommendation for Multi-room Indoor Scene Synthesis' --- =1 <ccs2012> <concept> <concept\_id>10010147.10010178.10010187.10010197</concept\_id> <concept\_desc>Computing methodologies Spatial and physical reasoning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010371.10010396</concept\_id> <concept\_desc>Computing methodologies Shape modeling</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012> {width="7in"}

[**DUALITY AND FACTORIZATION THEOREM IN QCD[^1]**]{} I. V. Anikin$^{1 \dag}$, I. O. Cherednikov$^{1, 3}$, N. G. Stefanis$^{2}$ and O. V. Teryaev$^{1}$ [(1) [*Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia* ]{}\ (2) [*Institut für Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany* ]{}\ (3) [*INFN Gruppo collegato di Cosenza, I-87036 Rende, Italy* ]{}\ $\dag$ [*E-mail: [email protected]* ]{}]{} **Abstract** We find that in “two-photon”-like processes in the scalar $\varphi^3_E$ model and also in hadron-pair production arising from the collisions of a real (transversely polarized) and a highly virtual, longitudinally polarized, photon in QCD, there is duality between two distinct nonperturbative mechanisms. These two mechanisms, one involving a twist-$3$ Generalized Distribution Amplitude, the other employing a leading-twist Transition Distribution Amplitude, are associated with different regimes of factorization. In the kinematical region, where the two mechanisms overlap, duality is observed for the scalar $\varphi^3_E$ model, while in the QCD case the appearance of duality turns out to be sensitive to the particular nonperturbative model applied and can, therefore, be used as a tool for selecting the most appropriate one. Introduction {#sec:intro} ============ The only known method today of applying QCD in a rigorous way is based on the factorization of the dynamics and the isolation of a short-distance part that becomes accessible to perturbative techniques of quantum field theory (see, [@Efremov-Radyushkin; @Bro-Lep; @Col-Sop-Ste89] and for a review, for instance, [@Ste99] and references cited therein). Then, the conventional systematic way of dealing with the long-distance part is to parametrize it in terms of matrix elements of quark and gluon operators between hadronic states (or the vacuum). These matrix elements stem from nonperturbative phenomena and have to be either extracted from experiment or be determined on the lattice. In many phenomenological applications they are usually modeled in terms of various nonperturbative methods or models. Generically, the application of QCD to hadronic processes involves the consideration of hard parton subprocesses and (unknown) nonperturbative functions to describe binding effects. Prominent examples are hard exclusive hadronic processes which involve hadron distribution amplitudes (DAs), generalized distribution amplitudes (GDAs), and generalized parton distributions (GPDs) [@Diehl:2003; @Bel-Rad; @NonforRad; @GPV]. Applying such a framework, collisions of a real and a highly-virtual photon provide a useful tool for studying a variety of fundamental aspects of QCD. Recently, nonperturbative quantities of a new kind were introduced—transition distribution amplitudes (TDAs) [@Frank-Pol; @Pire-Szym; @LPS06]—which are closely related to the GPDs. In contrast to the GDAs, the TDAs appear in the factorization procedure when the Mandelstam variable $s$ is of the same order of magnitude as the large photon virtuality $Q^2$, while $t$ is rather small. Remarkably, there exists a reaction where both amplitude types, GDAs and TDAs, can overlap. This can happen in the fusion of a real and transversely polarized photon with a highly-virtual longitudinally polarized photon, giving rise to a final state which comprises a pair of pions. The key feature of this reaction is that it can potentially follow either path: proceed via twist-$3$ GDAs, or go through the leading-twist TDAs, as illustrated in Fig. \[GDAvsTDA\]. Such an antagonism of alternative factorization mechanisms in this reaction seems extremely interesting both theoretically and phenomenologically and deserves to be studied in detail. The intimate relation between these two mechanisms in the production of a vector-meson pair was analyzed in [@PSSW] and it was found that these mechanisms can be selected by means of the different polarizations of the initial-state photon. In contrast, for (pseudo)scalar particles, such as the pions, this effect is absent enabling us to access the overlap region of both mechanisms and their duality as opposed to their additivity. In this talk, we will report on the possibility for duality between these antagonistic mechanisms of factorization, associated either with GDAs or with TDAs, in the regime where *both* Mandelstam variables $s$ and $t$ are rather small compared to the large photon virtuality $Q^2$. ![Two ways of factorization: via the GDA mechanism and via the TDA mechanism.[]{data-label="GDAvsTDA"}](gamma-gamma-pi-new-blobs3.eps){width="40.00000%"} Regimes of Factorization within the $\varphi^3_E$-model {#sec:fact1} ======================================================= Consider first the factorization of the scalar $\varphi^3_E$ model in Euclidean space. To study the four-particle amplitude in detail, it is particularly useful to employ the $\alpha$-representation—see [@NonforRad]. Then, the contribution of the leading “box” diagram can be written as (while details can be found in [@An-Dual]) $$\begin{aligned} \label{Amp1} {\cal A}(s,t,m^2) =-\frac{g^4}{16\pi^2} \int\limits_{0}^{\infty} \frac{\prod\limits_{i=1}^4 d\alpha_i}{D^2} \exp \biggl[ - \frac{1}{D} \left( Q^2 {\alpha_1\alpha_2} + s \alpha_2\alpha_4 + t {\alpha_1\alpha_3} + m^2 D^2 \right)\biggr],\end{aligned}$$ where $m^2$ serves as a infrared (IR) regulator, $s>0$, $t>0$ are the Mandelstam variables in the Euclidean region, and $D=\sum\limits_{i=1}^4 \alpha_i$. Assuming that $q^2=Q^2$ is large compared to the mass scale $m^2$ (which simulates here the typical scale of soft interactions), the amplitude (\[Amp1\]) can indeed be factorized. As regards the other two kinematic variables $s$ and $t$, one can identify three distinct regimes of factorization: (a) $s\ll Q^2$ while $t$ is of order $Q^2$; (b) $t\ll Q^2$ while $s$ is of order $Q^2$; (c) $s,~t\ll Q^2$. **Regime (a)**: The process is going through the s-channel. In this regime, the main contribution in the integral in Eq. (\[Amp1\]) arises from the integration over $\alpha_1$ when $\alpha_1\sim 0$: $$\begin{aligned} \label{GDA-alpha} {\cal A}_{\rm GDA}^{\rm as}(s,t,m^2) =-\frac{g^4}{16\pi^2} \int\limits_{0}^{\infty} \frac{d\alpha_2 \,d\alpha_3 \,d\alpha_4}{D^2_0} \ \exp \left( - s \frac{\alpha_2\alpha_4}{D_0}- m^2 D_0 \right) \left[Q^2 \frac{\alpha_2}{D_0} + t \frac{\alpha_3}{D_0} + m^2 \right]^{-1}\, .\end{aligned}$$ Schematically this means that the propagator, parametrized by $\alpha_1$, can be associated with the partonic (hard) subprocesses, while the remaining propagator constitutes the soft part of the considered amplitude, i.e., the scalar version of the GDA. **Regime (b)**: Here we have to eliminate from the exponential in Eq. (\[Amp1\]) the variables $Q^2$ and $s$, which are large. This can be achieved by integrating over the region $\alpha_2\sim 0$. Performing similar manipulations as in regime (a), we find that the scalar TDA amplitude can be related to the scalar GDA via $ {\cal A}_{\rm TDA}^{\rm as}(s, t, m^2) = {\cal A}_{\rm GDA}^{\rm as}(t, s, m^2) $. **Regime (c)**: The relevant regime to investigate duality is when it happens that both variables $s$ and $t$ are simultaneously small compared to $Q^2$, i.e., when $s,\, t \ll Q^2$. In this case, there are two possibilities to extract the leading $Q^2$-asymptotics, notably, we can either integrate over the region $\alpha_1 \sim 0$, or integrate instead over the region $\alpha_2 \sim 0$. Clearly, these two options can be associated with (i) the GDA mechanism of factorization with the meson pair scattered at a small angle in its center-of-mass system or, alternatively, (ii) with the TDA mechanism of factorization. We stress that we may face double counting when naively adding these two contributions. We interpret such a behavior as a signal of an ingrained tendency for duality between the GDA(s-channel) and the TDA (t-channel) factorization mechanisms. In order to verify the appearance of duality we carry out a numerical investigation of the exact and the asymptotic amplitudes. In doing so, we introduce the following ratios $R_1={\cal A}_{\rm TDA}^{\rm as}/{\cal A}$ and $R_2={\cal A}_{\rm GDA}^{\rm as}/{\cal A}$. ![The ratios $R_1$ and $R_2$ as functions of $s/Q^2$.[]{data-label="fig-rat-1"}](FigN-1.eps){width="50.00000%"} Appealing to the symmetry of these ratios under the exchange of the variables $s\leftrightarrow t$, we take $t/Q^2$ to be $0.01$ and look for the variation of the ratios with $s/Q^2$. This variation is illustrated in Fig. \[fig-rat-1\] from which one sees that in the region where $s/Q^2$ is rather small, i.e., in the range $(0.01, \, 0.05 )$, both asymptotic formulae are describing the exact amplitude with an accuracy of more than $90 \%$. This behavior supports the conclusion that, when both Mandelstam variables $s/Q^2$ and $t/Q^2$ assume values in the wide interval $(0.001, \, 0.7)$, duality between the TDA and the GDA factorization mechanisms emerges. TDA- and GDA-Factorizations for $\gamma\gamma^*\to\pi\pi$ {#sec:fact2} ========================================================= Having discussed the appearance of duality between the GDA and the TDA factorization schemes within a toy model, we now turn attention to real QCD. To analyze duality, we consider the exclusive $\pi^+\pi^-$ production in a $\gamma_{\rm T}\gamma^{*}_{\rm L}$ collision, where the virtual photon with a large virtuality $Q^2$ is longitudinally polarized, whereas the other one is quasi real and transversely polarized. Notice that the GDA and the TDA regimes correspond to the *same* helicity amplitudes. Given that the considered process involves a longitudinally and a transversally polarized photon, we are actually dealing with twist-3 GDAs [@AT-WW]. On the other hand, for the twist-2 contribution, related to the meson DA, we use the standard parametrization of the $\pi^+$-to-vacuum matrix element which involves a bilocal axial-vector quark operator [@Efremov-Radyushkin]. Finally, the $\gamma\to\pi^-$ axial-vector matrix elements can be parametrized in the form, cf. [@Pire-Szym], $$\begin{aligned} \langle \pi^-(p_2)| \bar \psi(-z/2)\gamma_\alpha \gamma_5 [-z/2;z/2]\psi(z/2) |\gamma(q^\prime, \varepsilon^\prime) \rangle \stackrel{{\cal F}}{=} \frac{e}{f_\pi}\varepsilon^\prime_T \cdot\Delta_T P_\alpha A_1(x,\xi,t)\, , \label{eq:gpimeA}\end{aligned}$$ where $P=(p_2+q^\prime)/2$, and $\Delta=p_2-q^\prime$, and noticing that the symbol $\stackrel{{\cal F}}{=}$ means Fourier transformation and that the vector matrix element does not contribute here. To normalize the axial-vector TDA, $A_1$, we express it in terms of the axial-vector form factor measured in the weak decay $\pi\to l\nu_l \gamma$ [@PDG06; @Byc08; @An-Dual]. The helicity amplitude associated with the TDA mechanism reads $ {\cal A}^{\rm TDA}_{(0,j)} = {\cal F}^{\rm TDA} \varepsilon^{\prime\,(j)} \cdot\Delta^T/|\vec{q}\,| $ with $$\begin{aligned} \label{TDAhelam} {\cal F}^{\rm TDA}= [4\,\pi\,\alpha_s(Q^2)]\frac{C_{\rm F}}{2\,N_c} \int\limits_{0}^{1} dy \, \frac{\phi_\pi(y)}{y\bar y} \int\limits_{-1}^{1} dx \, A_1(x,\xi,t)\, \biggl( \frac{e_u}{\xi-x}-\frac{e_d}{\xi+x} \biggr), \label{eq:haTDA}\end{aligned}$$ where we have employed the 1-loop $\alpha_s(Q^2)$ in the $\overline{\rm MS}$-scheme with $\Lambda_{\rm QCD}=0.312$ GeV for $N_f=3$ [@Kataev:2001kk]. \[Note that there is only a mild dependence on $\Lambda_{\rm QCD}$.\] Turning now to the helicity amplitude, which includes the twist-$3$ GDA, we anticipate that it can be written as (see, for example, [@AT-WW]) ${\cal A}_{(0,j)}^{\rm GDA}={\cal F}^{\rm GDA} \varepsilon^{\,\prime\,(j)}\cdot\Delta^T/|\vec{q}\,|$ with $$\begin{aligned} \label{GDAhelam} {\cal F}^{\rm GDA}=2 \frac{W^2+Q^2}{Q^2} (e^2_u+e^2_d) \int\limits_{0}^{1} dy \, \partial_{\zeta} \Phi_1(y,\zeta,W^2) \biggl( \frac{\ln{\bar y}}{y} - \frac{\ln{y}}{\bar y}\biggr)\, , \label{eq:amp_GDA_WW}\end{aligned}$$ where the partial derivative is defined by $\partial_\zeta = \partial/\partial(2\zeta-1)$. In deriving (\[GDAhelam\]), we have used for the twist-$3$ contribution the Wandzura-Wilczek approximation. Duality between expressions (\[eq:haTDA\]) and (\[eq:amp\_GDA\_WW\]) may occur in that regime, where both $s$ and $t$ are simultaneously much smaller in comparison to the large photon virtuality $Q^2$. More insight into the relative weight of the amplitudes with TDA or GDA contributions can be gained once we have modeled these non-perturbative quantities. We commence with the TDAs and, assuming a factorizing ansatz for the $t$-dependence of the TDAs, we write $A_1(x,\xi,t)= 2\frac{f_\pi}{m_\pi} \, F_{A}(t) A_1(x,\xi)$, where the $t$-independent function $A_1(x,\xi)$ is normalized to unity. To satisfy the unity-normalization condition, we introduce a TDA defined by $ A_1(x, 1) = A_1^{\rm non-norm}(x,1)/\int\limits_{-1}^1 dx A_1^{\rm non-norm}(x,1) $ and continue with the discussion of the $t$-independent TDAs. Recalling that we are mainly interested in TDAs in the region $\xi=1$ [@Efremov-Radyushkin; @Bro-Lep], it is useful to adopt the following parametrization $$\begin{aligned} \label{TDAansatz} A_1^{\rm non-norm}(x, 1) = (1-x^2)\biggl( 1+ a_1 C^{(3/2)}_1(x) + a_2 C^{(3/2)}_2(x)+ a_4 C^{(3/2)}_4(x)\biggr) ,\end{aligned}$$ where $a_1, \,a_2, \, a_4$ are free adjustable parameters, encoding nonperturbative input, and the standard notations for Gegenbauer polynomials are used. It is not difficult to show that the TDA expressed by Eq. (\[TDAansatz\]) results from summing a $D$-term, i.e., the term with the coefficient $a_1$, and meson-DA-like contributions. For our analysis, we suppose that $a_1\equiv d_0$ [@GPV], which is equal to $-0.5$ in lattice simulations. With respect to the parameters $a_2$ and $a_4$, we allow them to vary in quite broad intervals, notably, $a_2\in [0.3, \, 0.6]$ and $a_4\in [0.4, \, 0.8]$, that would cover vector-meson DAs with very different profiles at a normalization scale $\mu^2\sim 1\, {GeV^2}$ (see, for example, [@BM-rhomes]). ![Helicity amplitudes ${\cal F}^{\rm TDA}$ and ${\cal F}^{\rm GDA}$ as functions of $Q^2$, using $a_1=-0.5$ found in lattice simulations. The value of $s/Q^2$ varies in the interval $[0.06,\, 0.3]$. []{data-label="fig-TDA-GDA1"}](FigN-2P.eps){width="50.00000%"} The function $\Phi_1(z,\zeta)$ is rather standard and well-known (details in [@Diehl:2003; @An-Dual]). We close this section by summarizing our numerical analysis of [@An-Dual]. We calculated both functions ${\cal F}^{\rm TDA}$ and ${\cal F}^{\rm GDA}$, and show the results in Fig. \[fig-TDA-GDA1\]. The dashed line corresponds to the function ${\cal F}^{\rm TDA}$, where we have adjusted the free parameters to $a_2=0.6,\, a_4=0.8$. The results, obtained for rather small values of these parameters, are displayed by the broken lines in the same figure. The dotted line denotes the function ${\cal F}^{\rm TDA}$ with $a_2=0.5$ and $a_4=0.6$, whereas the dash-dotted line employs $a_2=0.3$ and $a_4=0.4$. For comparison, we also include the results for ${\cal F}^{\rm GDA}$. In that latter case, the dense-dotted line corresponds to the GDA amplitude, where the expression for $\tilde B_{12}$ has been estimated via Eq. (20) of [@An-Dual], while the solid line represents the simplest ansatz for $\tilde B_{12}$ with $R_\pi=0.5$. From this figure one may infer that when the parameter $\tilde B_{12}$, which parametrizes the GDA contribution, is estimated with the aid of the Breit-Wigner formula (provided $s,\,t \ll Q^2$), there is duality between the GDA and the TDA factorization mechanisms. Hence, the model for $\Phi_1(z,\zeta)$, which takes into account the corresponding resonances, can be selected by duality. Conclusions {#sec:concl} =========== We have provided evidence that when both Mandelstam variables $s$ and $t$ turn out to be much less than the large momentum scale $Q^2$, with the variables $s/Q^2$ and $t/Q^2$ varying in the interval $[0.001, \, 0.7]$, then the TDA and the GDA factorization mechanisms are equivalent to each other and operate in parallel. We have also demonstrated that duality may serve as a tool for selecting suitable models for the nonperturbative ingredients of various exclusive amplitudes entering QCD factorization. In this context, we observed that twist-3 GDAs appear to be dual to the convolutions of leading-twist TDAs and DAs, multiplied by a QCD effective coupling. Acknowledgments {#acknowledgments .unnumbered} --------------- We would like to thank A. P. Bakulev, A. V. Efremov, N. Kivel, B. Pire, M. V. Polyakov, M. Prasza[ł]{}owicz, L.  Szymanowski, and S. Wallon for useful discussions and remarks. This investigation was partially supported by the Heisenberg-Landau Programme (Grant 2008), the Alexander von Humboldt Stiftung, the Deutsche Forschungsgemeinschaft under contract 436RUS113/881/0, the EU-A7 Project *Transversity*, the RFBR (Grants 06-02-16215,08-02-00896 and 07-02-91557), the Russian Federation Ministry of Education and Science (Grant MIREA 2.2.2.2.6546), the RF Scientific Schools grant 195.2008.9, and INFN. [99]{} A. V. Efremov and A. V. Radyushkin, Phys. Lett.  B [**94**]{}, 245 (1980). Theor. Math. Phys.  [**42**]{}, 97 (1980) \[Teor. Mat. Fiz.  [**42**]{}, 147 (1980)\]. G. P. Lepage and S. J. Brodsky, Phys. Lett.  B [**87**]{}, 359 (1979); Phys. Rev.  D [**22**]{}, 2157 (1980). J. C. Collins, D. E. Soper and G. Sterman, Adv. Ser. Direct. High Energy Phys.  [**5**]{}, 1 (1988) \[arXiv:hep-ph/0409313\]. N. G. Stefanis, Eur. Phys. J. direct C [**7**]{}, 1 (1999) \[arXiv:hep-ph/9911375\]. M. Diehl, Phys. Rept.  [**388**]{}, 41 (2003) \[arXiv:hep-ph/0307382\]. 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--- abstract: 'Online learning algorithms are designed to perform in non-stationary environments, but generally there is no notion of a dynamic [*state*]{} to model constraints on current and future actions as a function of past actions. State-based models are common in stochastic control settings, but commonly used frameworks such as Markov Decision Processes (MDPs) assume a known stationary environment. In recent years, there has been a growing interest in combining the above two frameworks and considering an MDP setting in which the cost function is allowed to change arbitrarily after each time step. However, most of the work in this area has been algorithmic: given a problem, one would develop an algorithm almost from scratch. Moreover, the presence of the state and the assumption of an arbitrarily varying environment complicate both the theoretical analysis and the development of computationally efficient methods. This paper describes a broad extension of the ideas proposed by Rakhlin et al. to give a general framework for deriving algorithms in an MDP setting with arbitrarily changing costs. This framework leads to a unifying view of existing methods and provides a general procedure for constructing new ones. Several new methods are presented, and one of them is shown to have important advantages over a similar method developed from scratch via an online version of approximate dynamic programming.' author: - 'Peng Guan[^1]' - 'Maxim Raginsky[^2]' - 'Rebecca M. Willett[^3]' title: | Relax but stay in control: from value to algorithms\ for online Markov decision processes[^4] --- Introduction ============ Markov decision processes, or MDPs for short [@AC_MDP_survey; @Puterman; @LermaLasserreMDP], are a popular framework for sequential decision-making in a dynamic environment. In an MDP, we have states and actions. At each time step of the sequential decision-making process, the agent observes the current state and chooses an action, and the system transitions to the next state according to a fixed and known Markov law. The costs incurred by the agent depend both on his action and on the current state. Traditional theory of MDPs deals with the case when both the transition law and the state-action cost function are known in advance. In this case, there are two ways of designing policies [@Bertsekas] – via dynamic programming (where the construction of an optimal policy revolves around the computation of a relative value function), or via the linear programming (LP) approach [@Manne; @Borkar], which reformulates the MDP problem as a “static” linear optimization problem over the so-called state-action polytope [@Puterman]. However, [*a priori*]{} known costs are typically unavailable in practical settings. When neither the transition probability nor the cost functions are known in advance, various reinforcement learning (RL) methods, such as the celebrated $Q$-learning algorithm [@Watkins_Dayan_QLearning; @Tsitsiklis_QLearning] and its variants, can be used to learn an optimal policy in an online regime. However, the key assumptions underlying RL are that the agent is operating in a stochastically stable environment, and that the state-action costs (or at least their expected values with respect to any environmental randomness) do not vary with time. In this paper, instead of considering a fixed or stochastic cost function, we study Markov decision processes where the cost functions are chosen arbitrarily and allowed to change with time. More specifically, we are interested in the [*online MDP*]{} problem: just as in the usual online leaning framework [@Robbins_compound; @Hannan; @PLG], the one-step cost functions form an arbitrarily varying sequence, and the cost function corresponding to each time step is revealed to the agent after an action has been taken. The objective of the agent is to minimize regret relative to the best stationary Markov policy that could have been selected with full knowledge of the cost function sequence over the horizon of interest. The assumption of arbitrary time-varying cost functions makes sense in highly uncertain and complex environments whose temporal evolution may be difficult or costly to model, and it also accounts for collective (and possibly irrational) behavior of any other agents that may be present. The regret minimization viewpoint then ensures that the agent’s [*online*]{} policy is robust against these effects. Online MDP problems can be viewed as [*online control problems*]{}. The online aspect is due to the fact that the cost functions are generated by a dynamic environment under no distributional assumptions, and the agent learns the current state-action cost only after committing to an action. The control aspect comes from the fact that the choice of an action at each time step influences future states and costs. Taking into account the effect of past actions on future costs in a dynamic distribution-free setting makes online MDPs hard to solve. To the best of our knowledge, only a few methods have been developed in this area over the past decade [@McMahan; @EvenDar; @Yu; @onlineMDP_bandits; @AroraTewari; @onlineMDP_full; @Yadkori; @Zimin; @DickCsaba]. Most research in this area has been algorithmic: given a problem, one would present a method and prove a guarantee (i.e., a regret bound) on its performance. There are two distinct lines of methods: the algorithms presented by [@EvenDar; @Yu; @onlineMDP_bandits] require the computation of relative value functions at each time step, while the algorithms in [@Zimin; @DickCsaba] reduce the online MDP problem to an online linear optimization problem and solve it by online learning methods. These two lines of methods correspond to the two above-mentioned different ways of designing polices for MDPs. From a theoretical and conceptual standpoint, it is desirable to provide a unifying view of existing methods and a general procedure for constructing new ones. In this paper, we present such a general framework for online MDP problems that subsumes the above two approaches. This general framework not only enables us to recover known algorithms, but it also gives us a generic toolbox for deriving new algorithms from a more principled perspective rather than from scratch. The online MDP setting we are considering was first defined and studied in the work of [@EvenDar] and [@Yu], which deals with MDPs with arbitrarily varying rewards. Like these authors, we assume a full information feedback model and known stochastic state transition dynamics. (However, it should be pointed out that these assumptions have been relaxed in some recent works — for example, [@onlineMDP_bandits] and [@AroraTewari] assume only bandit-type feedback, while [@Yadkori] prove regret bounds for MDPs with arbitrarily varying transition models and cost functions. An extension of our framework to these settings is an interesting avenue for future research.) Our general approach is motivated by recent work of Rakhlin et al. [@RakhlinRL], which gives a principled way of deriving online learning algorithms (and bounding their regret) from a minimax analysis. Of course, many online learning algorithms have been developed in various settings over the past few decades, but a comprehensive and systematic treatment was still lacking prior to [@RakhlinRL]. Starting from a general formulation of online learning as a (stateless) repeated game between a learner and an adversary, Rakhlin et al. [@RakhlinRL] analyze the minimax regret (value) of this online learning game, which is the regret (relative to a fixed competing strategy) that would be achieved if both the learner and the adversary play optimally. It was known before the work of [@RakhlinOR] that one could derive sublinear upper bounds on the minimax value in a nonconstructive manner. However, algorithm design was done on a case-by-case basis, and custom analysis techniques were needed in each case to derive performance guarantees matching these upper bounds. The work of [@RakhlinRL] bridges this gap between minimax value analysis and algorithm design: They have shown that, by choosing appropriate relaxations of a certain recursive decomposition of the minimax value, one can recover many known online learning algorithms and give a general recipe for developing new ones. In short, the framework proposed by [@RakhlinRL] can be used to convert an upper bound on the value of the game into an algorithm. Our main contribution is an extension of the framework of [@RakhlinRL] to online MDPs. Since online learning problems are studied in a state-free setting, it is not straightforward to generalize the ideas of [@RakhlinRL] to the case when the system has a state, and the technical nature of the arguments involved in online MDPs is significantly heavier than their state-free counterpart. We formulate the online MDP problem as a two-player repeated game with state variables and study its minimax value. We introduce the notion of an online MDP [*relaxation*]{} and show how it can be used to recover existing methods and to construct new algorithms. More specifically, we present two distinct approaches of moving from the original dynamic setting, where the state evolves according to a controlled Markov chain, to simpler static settings and constructing corresponding relaxations. The first approach uses Poisson inequalities for MDPs [@MeynTweedie] to reformulate the original dynamic setting as a static setting, where each possible state is associated with a separate online learning algorithm. We show that the algorithm proposed by [@EvenDar] arises from a particular relaxation, and we also derive a new algorithm in the spirit of [@Yu] which exhibits improved regret bounds. The second approach moves from the dynamic setting to a static setting by reducing the online MDP problem to an online linear optimization problem. After the reduction, we can directly capitalize on the framework of [@RakhlinRL]. We then derive a novel Online Mirror Descent (OMD) algorithm in the spirit of [@Zimin; @DickCsaba] under a carefully designed relaxation over a certain convex set. In short, while the existing methods fall into two major categories, they both can be captured by the above two approaches, and these two approaches arise from the same general idea: move from the original dynamic setting to a static setting, derive the corresponding relaxation, and convert the relaxation into an algorithm. The remainder of the paper is organized as follows. We close this section with a brief summary of our results and frequently used notation. Section \[sec:setup\] contains precise formulation of the online MDP problem and points out the general idea and major challenges. Section \[sec:gfw\] describes our proposed framework and contains the main result. The general framework includes two different methods of recovering and deriving algorithms. Section \[sec:derivealgo\] uses the first method and shows the power of our framework by recovering an existing method proposed in [@EvenDar] and further derives a new algorithm. Section \[sec:derivedick\] uses the second approach to derive a novel online MDP algorithm. Section \[sec:clc\] contains discussion about future research. Proofs of all intermediate results are relegated to the Appendix. Summary of contributions ------------------------ We start by recasting an MDP with arbitrary costs as a one-sided [*stochastic game*]{}, where an agent who wishes to minimize his long-term average cost is facing a Markovian environment, which is also affected by arbitrary actions of an opponent. A stochastic game [@Shapley; @Sorin] is a repeated two-player game, where the state changes at every time step according to a transition law depending on the current state and the moves of both players. Here we are considering a special type of a stochastic game, where the agent controls the state transition alone and the opponent chooses the cost functions. By “one-sided”, we mean that the utility of the opponent is left unspecified. In other words, we do not need to study the strategy and objectives of the opponent, and only assume that the changes in the environment in response to the opponent’s moves occur arbitrarily. As a result, we simply model the opponent as the environment. A popular and common objective in such settings is regret minimization. Regret is defined as the difference between the cost the agent actually incurred, and what could have been incurred if the agent knew the observed sequence of cost functions in advance. We will give the precise definition of this regret notion in Section \[sec:setup\]. We start by studying the minimax regret, i.e., the regret the agent will suffer when both the agent and the environment play optimally. By applying the theory of dynamic programming for stochastic games [@Sorin], we can give the strategy for the agent that achieves minimax regret (called the minimax strategy). It can be interpreted as choosing the best action that takes into account the current cost and the worst case future. Unfortunately, this minimax strategy in general is not computationally feasible due to the fact that the number of possible futures grows exponential with time. The idea is to find a way to approximate the term that represents the “future” and derive near-optimal strategy that is easy to compute using the approximation. Our main contribution is a construction of a general procedure for deriving algorithms in the online MDP setting. More specifically: 1. Just as in the state-free setting considered by [@RakhlinRL], we argue that algorithms can be constructed systematically by first deriving a sequence of upper bounds (relaxations) on the minimax value of the game, and then choosing actions which minimize these upper bounds. 2. Once a relaxation and an algorithm are derived in this way, we give a general regret bound of that algorithm as follows: $$\begin{aligned} \text{Expected regret} \le \text{Relaxation} + \text{Stationarization error}. \end{aligned}$$ The first term on the right-hand side of the above inequality is the expected relaxation, while the second term is an approximation error that results from approximating the Markovian evolution of the underlying process by a simpler stationary process using a procedure we refer to as [*stationarization*]{}. The first term can be analyzed using essentially the same techniques as the ones employed by [@RakhlinRL], with some modifications; by contrast, the second term can be handled using only a novel combination of Markov chain methods. This approach significantly alleviates the technical burden of proving a regret bound as in the literature before our work. 3. Using the above procedure, we recover an existing method proposed by Even-Dar et al. in [@EvenDar], which achieves $O(\sqrt{T})$ expected regret against the best stationary policy. We show that our derived relaxation gives us the same exponentially weighted average forecaster as in [@EvenDar] and leads to the same regret bound. We also derive a new algorithm using our proposed framework and argue that, while this new algorithm is similar in nature to the work of Yu et al. [@Yu], it has several advantages — in particular, better scaling of the regret with the horizon $T$. Both of these algorithms are based on introducing a sequence of appropriately defined relative value functions, and thus can be viewed as instantiations of the first approach to online MDPs — namely, the one rooted in dynamic programming. 4. We also present a different technique for deriving relaxations that implements the second approach to online MDPs — the one rooted in the LP method. This approach allows us to reduce the online MDP problem to an online linear optimization problem over the state-action polytope. This reduction enables us to use the framework of [@RakhlinRL], and the resulting relaxation leads to a novel OMD algorithm that is similar in spirit to the work of Dick et al. [@DickCsaba]. Notation {#ssec:notation} -------- We will denote the underlying finite state space and action space by $\sX$ and $\sU$, respectively. The set of all probability distributions on $\sX$ will be denoted by $\cP(\sX)$, and the same goes for $\sU$ and $\cP(\sU)$. A matrix $P = [P(u|x)]_{x \in \sX, u \in \sU}$ with nonnegative entries, and with the rows and the columns indexed by the elements of $\sX$ and $\sU$ respectively, is called [*Markov*]{} (or [*stochastic*]{}) if its rows sum to one: $\sum_{u \in \sU} P(u|x) = 1, \forall x \in \sX$. We will denote the set of all such Markov matrices (or randomized state feedback laws) by $\cM(\sU|\sX)$. Markov matrices in $\cM(\sU|\sX)$ transform probability distributions on $\sX$ into probability distributions on $\sU$: for any $\mu \in \cP(\sX)$ and any $P \in \cM(\sU|\sX)$, we have $$\begin{aligned} \mu P(u) \deq \sum_{x \in \sX} \mu(x)P(u|x), \qquad \forall u \in \sU.\end{aligned}$$ The same applies to Markov matrices on $\sX$ and to their action on the elements of $\cP(\sX)$. The fixed and known stochastic transition kernel of the MDP will be denoted throughout by $K$ – that is, $K(y|x,u)$ is the probability that the next state is $y$ if the current state is $x$ and the action $u$ is taken. For any Markov matrix (randomized state feedback law) $P \in \cM(\sU|\sX)$, we will denote by $K(y|x,P)$ the Markov kernel $$\begin{aligned} K(y|x,P) \deq \sum_{u \in \sU} K(y|x,u)P(u|x).\end{aligned}$$ Similarly, for any $\nu \in \cP(\sU)$, $$\begin{aligned} K(y|x,\nu) \deq \sum_{u \in \sU} K(y|x,u)\nu(u)\end{aligned}$$ (this can be viewed as a special case of the previous definition if we interpret $\nu$ as a state feedback law that ignores the state and draws a random action according to $\nu$). For any $\mu \in \cP(\sX)$ and $P \in \cM(\sU|\sX)$, $\mu \otimes P$ denotes the induced joint state-action distribution on $\sX \times \sU$: $$\mu \otimes P(x,u) = \mu(x)P(u|x), \qquad \forall (x,u) \in \sX \times \sU.$$ We say that $P$ is [*unichain*]{} [@LermaLasserreMC] if the corresponding Markov chain with transition kernel $K(\cdot|\cdot,P)$ has a single recurrent class of states (plus a possibly empty transient class). This is equivalent to the induced kernel $K(\cdot|P)$ having a unique invariant distribution $\pi_P$ [@Seneta]. The total variation (or $L_1$) distance between $\nu_1, \nu_2 \in \cP(\sU)$ is $$\begin{aligned} \| \nu_1 - \nu_2 \|_1 \deq \sum_{u \in \sU} \lvert \nu_1(u) - \nu_2(u) \rvert.\end{aligned}$$ It admits the following variational representation: $$\begin{aligned} \label{eq:TV_var} \| \nu_1 - \nu_2 \|_1 = \sup_{f:\, \| f \|_\infty \le 1} \left| \ave{\nu_1,f} - \ave{\nu_2,f} \right|,\end{aligned}$$ where the supremum is over all functions $f : \sU \to \Reals$ with absolute value bounded by $1$, and we are using the linear functional notation for expectations: $$\begin{aligned} \ave{\nu,f} = \E_\nu[f] = \sum_{u \in \sU}\nu(u)f(u).\end{aligned}$$ The [*Kullback–Leibler divergence*]{} (or [*relative entropy*]{}) between $\nu_1$ and $\nu_2$ [@CoverThomas] is $$D(\nu_1 \| \nu_2) \deq \begin{cases} \displaystyle\sum_{u \in \sU} \nu_1(u) \log \dfrac{\nu_1(u)}{\nu_2(u)} & \textrm{if ${\rm supp}(\nu_1) \subseteq {\rm supp}(\nu_2$)} \\ + \infty & \textrm{otherwise} \end{cases}$$ where ${\rm supp}(\nu) \deq \{ u \in \sU: \nu(u) > 0 \}$ is the [*support*]{} of $\nu$. Here and in the sequel, we work with natural logarithms. The same applies, [*mutatis mutandis*]{}, to probability distributions on $\sX$. We will also be dealing with binary trees that arise in symmetrization arguments, as in [@RakhlinRL]: Let $\cH$ be an arbitrary set. An $\cH$-valued tree $\mathbf h$ of depth $d$ is defined as a sequence $(\mathbf h_1, \ldots, \mathbf h_d)$ of mappings $\mathbf h_t: \{ \pm 1\}^{t-1} \to \cH$ for $t = 1, 2, \ldots, d$. Given a tuple $\eps = (\eps_1,\ldots,\eps_d) \in \{\pm 1\}^d$, we will often write $\mathbf h_t(\eps)$ instead of $\mathbf h_t(\eps_{1:t-1})$. Problem formulation {#sec:setup} =================== We consider an online MDP with finite state and action spaces $\sX$ and $\sU$ and transition kernel $K(y|x,u)$. Let $\cF$ be a fixed class of functions $f : \sX \times \sU \to \Reals$, and let $x \in \sX$ be a fixed initial state. Consider an agent performing a controlled random walk on $\sX$ in response to signals coming from the environment. The agent is using mixed strategies to choose actions, where a mixed strategy is a probability distribution over the action space. The interaction between the agent and the environment proceeds as follows: --------------------------------------------------------------------------------------------- $X_1=x$ for $t = 1,2,\ldots,T$   The agent observes the state $X_t$, selects a mixed strategy $P_t \in \cP(\sU)$, and then draws an action $U_t$ from $P_t$   The environment simultaneously selects $f_t \in \cF$ and announces it to the agent   The agent incurs one-step cost $f_t(X_t, U_t)$   The system transitions to the next state $X_{t+1} \sim K(\cdot|X_t, U_t)$ end for --------------------------------------------------------------------------------------------- Here, $T$ is a fixed finite horizon. We assume throughout that the environment is [*oblivious*]{} (or [*open-loop*]{}), in the sense that the evolution of the sequence $\{f_t\}$ is not affected by the state and action sequences $\{X_t\}$ and $\{U_t\}$. We view the above process as a two-player repeated game between the agent and the environment. At each $t \ge 1$, the process is at state $X_t = x_t$. The agent observes the current state $x_t$ and selects the mixed strategy $P_t$, where $P_t(u | x_t) = \Pr\{U_{t}=u|X_t=x_t\}$, based on his knowledge of all the previous states and current state $x^t = (x_1,\ldots,x_t)$ and the previous moves of the environment $f^{t-1} = (f_1,\ldots,f_{t-1})$. After drawing the action $U_t$ from $P_t$, the agent incurs the one-step cost $f_{t}(X_{t}, U_t)$. Adopting game-theoretic terminology [@BasarOlsder], we define the agent’s closed-loop [*behavioral strategy*]{} as a tuple $\bd{\gamma} = (\gamma_1,\ldots,\gamma_T)$, where $\gamma_t : \sX^t \times \cF^{t-1} \to \cP(\sU)$. Similarly, the environment’s open-loop behavioral strategy is a tuple $\bd{f} = (f_1,\ldots,f_t)$. Once the initial state $X_1=x$ and the strategy pair $(\bd{\gamma},\bd{f})$ are specified, the joint distribution of the state-action process $(X^T,U^T)$ is well-defined. Let $\cM_0 = \cM_0(\sU|\sX) \subseteq \cM(\sU|\sX)$ denote the subset of all Markov policies $P$, for which the induced state transition kernel $K(\cdot|\cdot,P)$ has a unique invariant distribution $\pi_P \in \cP(\sX)$. The goal of the agent is to minimize the expected [*steady-state regret*]{} $$\begin{aligned} \label{eq:ssregretdef} R^{\bd{\gamma},\bd{f}}_x \deq \E^{\bd{\gamma},\bd{f}}_x\left\{\sum^T_{t=1}f_t(X_t,U_t) - \inf_{P \in \cM_0}\, \E\left[\sum^T_{t=1}f_t(X,U)\right]\right\},\end{aligned}$$ where the outer expectation $\E^{\bd{\gamma},\bd{f}}_x$ is taken w.r.t. both the Markov chain induced by the agent’s behavioral strategy $\bd{\gamma}$ (including randomization of the agent’s actions), the environment’s behavior strategy $\bd{f}$, and the initial state $X_1 = x$. The inner expectation (after the infimum) is w.r.t. the state-action distribution $\pi_P \otimes P(x,u) = \pi_P(x)P(u|x)$, where $\pi_P$ denotes the unique invariant distribution of $K(\cdot|\cdot,P)$. The regret $R^{\bd{\gamma},\bd{f}}_x$ can be interpreted as the gap between the expected cumulative cost of the agent using strategy $\bd{\gamma}$ and the best steady-state cost the agent could have achieved in hindsight by using the best stationary policy $P \in \cM_0$ (with full knowledge of $\bd{f} = f^T$). This gap arises through the agent’s lack of prior knowledge on the sequence of cost functions. Here we consider the steady-state regret, so that the expectation w.r.t. the state evolution in the comparator term $ \E\left[\sum^T_{t=1}f_t(X,U)\right]$ is taken over the invariant distribution $\pi_P$ instead of the Markov transition law $K(\cdot|\cdot,P)$ induced by $P$. Under the additional assumptions that the cost functions $f_t$ are uniformly bounded and the induced Markov chains $K(\cdot|\cdot,P)$ are uniformly exponentially mixing for all $P \in \cM(\sU|\sX)$, the difference we introduce here by considering the steady state is bounded by a constant independent of $T$ [@EvenDar; @Yu], and so is negligible in the long run. In our main results, we only consider baseline policies in $\cM_0$ that are uniformly exponentially mixing, so we restrict our attention to the steady-state regret without any loss of generality. Minimax regret {#ssec:minimaxanalysis} -------------- We start our analysis by studying the value of the game (the minimax regret), which we first write down in [*strategic form*]{} as $$\begin{aligned} \label{eq:strategy1} V(x) \deq \inf_{\bd{\gamma}}\sup_{\bd{f}}\, R^{\bd{\gamma},\bd{f}}_x = \inf_{\bd{\gamma}}\sup_{\bd{f}}\, \E^{\bd{\gamma},\bd{f}}_x\left[\sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right],\end{aligned}$$ where we have introduced the shorthand $\Psi$ for the comparator term: $$\begin{aligned} \Psi(\bd{f}) \deq \inf_{P \in \cM_0} \E\left[\sum^T_{t=1}f_t(X,U)\right]. \end{aligned}$$ In operational terms, $V(x)$ gives the best value of the regret the agent can secure by any closed-loop behavioral strategy against the worst-case choice of an open-loop behavioral strategy of the environment. However, the strategic form of the value hides the [*timing protocol*]{} of the game, which encodes the information available to the agent at each time step. To that end, we give the following equivalent expression of $V(x)$ in [*extensive form*]{}: \[pps:strategyextensive\] The minimax regret is given by $$\begin{aligned} \label{eq:extensive2} V(x) = \inf_{P_1}\sup_{f_1}\ldots\inf_{P_T}\sup_{f_T} \E\left[\sum^T_{t=1}f_t(X_t,U_t)-\Psi(\bd{f})\right]. \end{aligned}$$ See Appendix \[app:strategyextensive\]. From this minimax formulation, we can immediately get an optimal algorithm that attains the minimax regret. To see this, we give an equivalent recursive form for the value of the game. For any $t \in \{0,1,\ldots,T-1\}$, any given prefix $f^{t} = (f_1,\ldots,f_{t})$ (where we let $f^0$ be the empty tuple ${\mathsf e}$), and any state $X_{t+1} = x$, define the conditional value \[eq:condvalue11\] $$\begin{aligned} {V}_t(x,f^t) &\deq \inf_{\nu \in \cP(\sU)} \sup_{f} \left\{ \sum_{u \in \sU}f(x,u)\nu(u) + \E\Big[{V}_{t+1}(Y,f_1,\ldots,f_t,f) \Big| x,\nu\Big]\right\}, \qquad t = T-1,\ldots,0 \\ {V}_T(x,f^T) &\deq - \Psi(\bd{f}).\end{aligned}$$ [*Recursive decompositions of this sort arise frequently in problems involving decision-making in the presence of uncertainty. For instance, we may view as a dynamic program for a finite-horizon minimax control problem [@BertsekasRhodes]. Alternatively, we can think of as applying the [*Shapley operator*]{} [@Sorin] to the conditional value in a two-player stochastic game, where one player controls only the state transitions, while the other player specifies the cost function. A promising direction for future work is to derive some characteristics of the conditional value from analytical properties of the Shapley operator.*]{} From Proposition \[pps:strategyextensive\], we see that $V(x) = V_0(x,\mathsf{e})$. Moreover, we can immediately write down the minimax-optimal behavioral strategy for the agent: $$\begin{aligned} {\gamma}_{t+1}(x,f^{t}) &= \argmin_{\nu \in \cP(\sU)}\sup_{f \in \cF}\left\{ \sum_{u \in \sU}f(x,u)\nu(u) + \E\Big[{V}_{t+1}(Y,f_1,\ldots,f_{t},f) \Big| x,\nu \Big]\right\}, \qquad t = 0,\ldots,T-1.\end{aligned}$$ Note that the expression being minimized is a supremum of affine functions of $\nu$, so it is a lower-semicontinuous function of $\nu$. Any lower-semicontinuous function achieves its infimum on a compact set. Since the probability simplex $\cP(\sU)$ is compact, we are assured that a minimizing $\nu$ always exists. Using the above strategy at each time step, we can secure the minimax regret in the worst-case scenario. Note also that this strategy is very intuitive: it balances the tendency to minimize the present cost against the risk of incurring high future costs. However, with all the future infimum and supremum pairs involved, computing this conditional value is intractable. As a result, the minimax optimal strategy is not computationally feasible. The idea is to give tight bounds of the conditional value, which can be minimized to form a near-optimal strategy. We address this challenge by developing computable bounds for the conditional value functions, choosing a strategy based on these bounds. In general, tighter bounds yield lower regret and looser bounds are easier to compute, and various online MDP methods occupy different points in this domain. In the spirit of [@RakhlinRL], we come up with approximations of the conditional value ${V}_t(x,f^t)$ in . We say that a sequence of functions $\wh{V}_t : \sX \times \cF^t \to \Reals$ is an [*admissible relaxation*]{} if \[eq:condtionvalueshapley\] $$\begin{aligned} \wh{V}_t(x,f^t) &\ge \inf_{\nu \in \cP(\sU)} \sup_{f} \left\{ \sum_{u \in \sU}f(x,u)\nu(u) + \E[\wh{V}_{t+1}(Y,f_1,\ldots,f_t,f)|x,\nu]\right\}, \qquad t = T-1,\ldots,0 \\ \wh{V}_T(x,f^T) &\ge - \Psi(\bd{f}).\end{aligned}$$ We can associate a behavioral strategy $\wh{\bd{\gamma}}$ to any admissible relaxation as follows: $$\begin{aligned} \wh{\gamma}_t(x,f^{t-1}) &= \argmin_{\nu \in \cP(\sU)}\sup_{f \in \cF}\left\{ \sum_{u \in \sU}f(x,u)\nu(u) + \E\Big[\wh{V}_{t}(Y,f_1,\ldots,f_{t-1},f) \Big| x,\nu \Big]\right\}.\end{aligned}$$ \[pps:admissbound1\] Given an admissible relaxation $\{\wh{V}_t\}^T_{t=0}$ and the associated behavioral strategy $\wh{\bd{\gamma}}$, for any open-loop strategy of the environment we have the regret bound $$\begin{aligned} R^{\wh{\bd{\gamma}},\bd{f}}_x = \E^{\wh{\bd{\gamma}},\bd{f}}_x\left[ \sum^T_{t=1}f_t(X_t,U_t)-\Psi(\bd{f})\right] \le \wh{V}_0(x). \end{aligned}$$ See Appendix \[app:admissbound1\]. Based on the above sequential decompositions, it suffices to restrict attention only to Markov strategies for the agent, i.e., sequences of mappings $\gamma_t : \sX \times \cF^{t-1} \to \cP(\sU)$ for all $t$, so that $U_t$ is conditionally independent of $X^{t-1},U^{t-1}$ given $X_t,f^{t-1}$. From now on, we will just say “behavioral strategy” and really mean “Markov behavioral strategy.” In other words, given $X_t,f^{t-1}$, the history of past states and actions is [*irrelevant*]{}, as far as the value of the game is concerned. [*What happens if the environment is nonoblivious? Yu et al. [@Yu] gave a simple counterexample of an aperiodic and recurrent MDP to show that the regret is linear in $T$ regardless of the agent’s policy when the opponent can adapt to the agent’s state trajectory. We can gain additional insight into the challenges associated with an adaptive environment from the perspective of the minimax regret. In particular, an adaptive environment’s [*closed-loop*]{} behavioral strategy is $\bd{\delta} = (\delta_1,\ldots,\delta_T)$ with $\delta_t : \sX^t \times \sU^{t-1} \to \cP(\cF)$, and the corresponding regret will be given by $$\begin{aligned} \E^{\bd{\gamma},\bd{\delta}}_x\left[ \sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right] &\le \E^{\bd{\gamma},\bd{\delta}}_x\left[ \sum^T_{t=1}f_t(X_t,U_t) + \wh{V}_T(X_{T+1},f^T)\right] \\ &= \E^{\bd{\gamma},\bd{\delta}}_x\left[ \sum^{T-1}_{t=1}f_t(X_t,U_t) \right] + \E^{\bd{\gamma},\bd{\delta}}_x\left[ f_T(X_T,U_T)+ \wh{V}_T(X_{T+1},f^T)\right]. \end{aligned}$$ Let’s analyze the last two terms: $$\begin{aligned} & \E^{\bd{\gamma},\bd{\delta}}_x\left[ f_T(X_T,U_T)+ V_T(X_{T+1},f^T)\right]\nonumber\\ &= \int_{\sX^T,\cF^T} {\mathbb P}({\rm d}x^T,{\rm d}\bd{f})\int_\sU P({\rm d}u_T|x_T,f^{T-1})\left\{f_T(x_T,u_T) + \E\Big[\wh{V}_T\big(X_{T+1},f^T\big)\Big|x^T,f^T\Big]\right\}.\end{aligned}$$ In the above conditional expectation, $\bd{f}$ may depend on the entire $x^T$, so we cannot replace this conditional expectation by $\E[\cdot|x_T,\gamma_T(x_T)]$. This implies we cannot get similar results as in Proposition \[pps:admissbound1\] in a fully adaptive environment.*]{} Major challenges ---------------- From Proposition \[pps:admissbound1\], we can see that we can bound the expected steady-state regret in terms of the chosen relaxation. Ideally, if we construct an admissible relaxation by deriving certain upper bounds on the conditional value and implement the associated behavioral strategy, we will obtain an algorithm that achieves the regret bound corresponding to the relaxation. In principle, this gives us a general framework to develop low-regret algorithms for online MDPs. However, with an additional state variable involved, it is difficult to derive admissible relaxations $\wh{V}_t(x,f^t)$ to bound the conditional value. The difficulty stems from the fact that now the current cost depends not only on the current action, but also on past actions. Our plan is to reduce this setting to a simpler setting where there is no Markov dynamics involved. In that setting, we will be able to capitalize on the ideas of [@RakhlinOR; @RakhlinRL] in two different ways. More specifically, using Rademacher complexity tools introduced by [@RakhlinOR; @RakhlinRL], we can derive algorithms in simpler static settings and then transfer them to the original problem. In the same vein, we will also prove a general regret bound for the derived algorithms. Thus we will have a general recipe for developing algorithms and showing performance guarantees for online MDPs. The general framework for constructing algorithms in online MDPs {#sec:gfw} ================================================================ As mentioned in the above section, the main challenge to overcome is the dependence of the conditional value in on the state variable. Our plan is to reduce the original online MDP problem to a simpler one, where there is no Markov dynamics. We proceed with our plan in several steps. First, we introduce a stationarization technique that will allow us to reduce the online MDP setting to a simpler setting without Markov dynamics. This effectively decouples current costs from past actions. Note that this reduction is fundamentally different from just naively applying stateless online learning methods in an online MDP setting, which would amount to a very poor stationarization strategy with larger errors and consequently large regret bounds. In contrast, our proposed stationarization performs the decoupling with minimal loss in accuracy by exploiting the transition kernel, yielding lower regret bounds. Using the stationarization idea, we present two different approaches to construct relaxations, aiming to recover and derive two distinct lines of existing methods. We call the first approach the [*value-function approach*]{}. Making use of Poisson inequalities for MDPs [@MeynTweedie], we state a new admissibility condition for relaxations that differs from the admissibility condition in in that there is no conditioning on the state variable. The advantage of working with this new type of relaxation is that the corresponding admissibility conditions are much easier to verify. The second approach is called the [*convex-analytic approach*]{}. By treating the online MDP problem as an online linear optimization problem, we are able to adopt the idea of [@RakhlinRL] in a more straightforward way, and use the admissibility condition in [@RakhlinRL] to construct relaxations and derive corresponding algorithms. These two approaches can recover different categories of existing methods (it should be pointed out, however, that there is a natural equivalence between these two approaches: the relative-value function arises as a Lagrange multiplier associated with the invariance constraint that defines the state-action polytope; cf. [@MeynCTCN Sec. 9.2] for details). The main result of this section is that we can apply any algorithm derived in the simpler static setting to the original dynamic setting and automatically bound its regret. Stationarization {#ssec:stationarization} ---------------- As before, we let $K$ denote the fixed and known transition law of the MDP. Following [@EvenDar] and [@Yu], we assume the following “uniform mixing condition”: There exists a finite constant $\tau > 0$ such that for all Markov policies $P \in \cM(\sU|\sX)$ and all distributions $\mu_1,\mu_2 \in \cP(\sX)$, $$\begin{aligned} \label{eq:mixingtimeeq} \| \mu_1 K(\cdot|P) - \mu_2 K(\cdot|P) \|_1 \le e^{-1/\tau} \| \mu_1 - \mu_2 \|_1,\end{aligned}$$ where $K(\cdot|P) \in \cM(\sX|\sX)$ is the Markov matrix on the state space induced by $P$. In other words, the collection of all state transition laws induced by all Markov policies $P$ is [*uniformly mixing*]{}. Here we assume, without loss of generality, that $\tau \ge 1$. As pointed out in [@NeuCsaba], this uniform mixing condition is actually stronger than the unichain assumption: $K(\cdot|\cdot,P)$ is unichain for any choice of $P \in \cM(\sU|\sX)$ — see Section \[ssec:notation\] for definitions. The uniform mixing condition implies that the transition kernel of every policy is a scrambling matrix. (A matrix $K \in \cM(\sX|\sX)$ is scrambling if and only if for any pair $x, x' \in \sX$ there exists at least one $y \in \sX$, such that $y$ can be reached from both $x$ and $x'$ in one step with strictly positive probability using $K$ as transition matrix.) Consider now a behavioral strategy $\bd{\gamma} = (\gamma_1,\ldots,\gamma_T)$ for the agent. For a given choice $\bd{f} = (f_1,\ldots,f_T)$ of costs, the following objects are well-defined: - $P^{\bd{\gamma},\bd{f}}_t \in \cM(\sU|\sX)$ — the Markov matrix that governs the conditional distribution of $U_t$ given $X_t$, i.e., $$\begin{aligned} P^{\bd{\gamma},\bd{f}}_t(u|x) = \left[\gamma_t(x,f^{t-1})\right](u); \end{aligned}$$ - $\mu^{\bd{\gamma},\bd{f}}_t \in \cP(\sX)$ — the distribution of $X_t$; - $K^{\bd{\gamma},\bd{f}}_t \in \cM(\sX|\sX)$ — the Markov matrix that describes the state transition from $X_t$ to $X_{t+1}$, i.e., $$\begin{aligned} K^{\bd{\gamma},\bd{f}}_t(y|x) = K(y|x,P^{\bd{\gamma},\bd{f}}_t) \equiv \sum_u K(y|x,u) P^{\bd{\gamma},\bd{f}}_t(u|x); \end{aligned}$$ - $\pi^{\bd{\gamma},\bd{f}}_t \in \cP(\sX)$ — the unique stationary distribution of $K^{\bd{\gamma},\bd{f}}_t$, satisfying $\pi^{\bd{\gamma},\bd{f}}_t = \pi^{\bd{\gamma},\bd{f}}_t K^{\bd{\gamma},\bd{f}}_t$, where existence and uniqueness are guaranteed by virtue of the unichain assumption; - $\eta^{\bd{\gamma},\bd{f}}_t = \ave{\pi^{\bd{\gamma},\bd{f}}_t \otimes P^{\bd{\gamma},\bd{f}}_t, f_t}$ — the steady-state cost at time $t$. Moreover, for any other state feedback law $P \in \cM(\sU|\sX)$, we will denote by $\eta^{P,\bd{f}}_t$ the steady-state cost $\ave{\pi_P \otimes P,f_t}$, where $\pi_P$ is the unique invariant distribution of $K(\cdot|\cdot,P)$. It will be convenient to introduce the regret w.r.t. a fixed $P \in \cM(\sU|\sX)$ with initial state $X_1 = x$: $$\begin{aligned} R^{\bd{\gamma},\bd{f}}_x(P) &\deq \E^{\bd{\gamma},\bd{f}}_x \left[ \sum^T_{t=1} f_t(X_t,U_t) - \sum^T_{t=1}\eta^{P,\bd{f}}_t\right] \\ &= \sum^T_{t=1}\left[\ave{\mu^{\bd{\gamma},\bd{f}}_t \otimes P^{\bd{\gamma},\bd{f}}_t,f_t} - \ave{\pi_P \otimes P,f_t}\right],\end{aligned}$$ as well as the [*stationarized regret*]{} $$\begin{aligned} \bar{R}^{\bd{\gamma},\bd{f}}(P) &\deq \sum^T_{t=1}\left(\eta^{\bd{\gamma},\bd{f}}_t -\eta^{P,\bd{f}}_t\right)\\ &= \sum^T_{t=1}\left[\ave{\pi^{\bd{\gamma},\bd{f}}_t \otimes P^{\bd{\gamma},\bd{f}}_t,f_t} - \ave{\pi_P \otimes P, f_t}\right].\end{aligned}$$ Using , we get the bound $$\begin{aligned} \label{eq:regret_two_terms} R^{\bd{\gamma},\bd{f}}_x(P) &\le \bar{R}^{\bd{\gamma},\bd{f}}(P) + \sum^T_{t=1} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_1.\end{aligned}$$ The key observation here is that the task of analyzing the regret $R^{\bd{\gamma},\bd{f}}_x(P)$ splits into separately upper-bounding the two terms on the right-hand side of : the stationarized regret $\bar{R}^{\bd{\gamma},\bd{f}}(P)$ and the stationarization error $\sum^T_{t=1} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_1$. The latter can be handled using Markov chain techniques. We now present two distinct approaches to tackle the former: the value-function approach and the convex-analytic approach. The value-function approach {#ssec:value_function_approach} --------------------------- The value-function approach relies on the availability of a so-called *reverse Poisson inequality, which can be thought of as a generalization of the Poisson equation from the theory of MDPs [@MeynTweedie].* Fix a Markov matrix $P \in \cM(\sU|\sX)$ and let $\pi_P \in \cP(\sX)$ be the (unique) invariant distribution of $K(\cdot|\cdot,P)$. Then we say that $\wh{Q} : \sX \times \sU \to \Reals$ satisfies the reverse Poisson inequality with [*forcing function*]{} $g : \sX \times \sU \to \Reals$ if $$\begin{aligned} \label{eq:RPI} \E \Big[\wh{Q}(Y,P) \Big| x,u \Big] - \wh{Q}(x,u) &\ge - g(x,u) + \ave{\pi_P \otimes P, g}, \qquad \forall (x,u) \in \sX \times \sU \end{aligned}$$ where $$\begin{aligned} \wh{Q}(y,P) \deq \sum_{u \in \sU}P(u|y)\wh{Q}(y,u) \end{aligned}$$ and $\E[\cdot|x,u]$ is w.r.t. the transition law $K(y|x,u)$. We should think of this as a relaxation of the Poisson equation [@MeynTweedie], i.e., when holds with equality. The Poisson equation arises naturally in the theory of Markov chains and Markov decision processes, where it provides a way to evaluate the long-term average cost along the trajectory of a Markov process. We are using the term “reverse Poisson inequality” to distinguish from the Poisson inequality, which also arises in the theory of Markov chains and is obtained by replacing $\ge$ with $\le$ in [@MeynTweedie]. Here we impose the following assumption that we use throughout the rest of the paper: \[as:existboundQ\] For any $P \in \cM(\sU|\sX)$ and any $f \in \cF$, there exists some $\wh{Q}_{P,f} : \sX \times \sU \to \Reals$ that solves the reverse Poisson inequality for $P$ with forcing function $f$. Moreover, $$\begin{aligned} L(\sX,\sU,\cF) \deq \sup_{P \in \cM(\sU|\sX)} \sup_{f \in \cF} \| \wh{Q}_{P,f} \|_\infty < \infty. \end{aligned}$$ [*In Section \[sec:derivealgo\], we will show this assumption is automatically satisfied when $K$ is a unichain model (or, more generally, when all stationary Markov policies are uniformly mixing, as in Eq. ).*]{} The main consequence of the reverse Poisson inequality is the following: \[lem:comparisonch3\] Suppose that $\wh{Q}$ satisfies the reverse Poisson inequality with forcing function $g$. Then for any other Markov matrix $P'$ we have $$\begin{aligned} \ave{\pi_P \otimes P, g} - \ave{\pi_{P'} \otimes P', g} &\le \sum_x \pi_{P'}(x) \sum_u \left[P(u|x)\wh{Q}(x,u) - P'(u|x)\wh{Q}(x,u)\right] \end{aligned}$$ See Appendix \[app:comparisonch3\]. Armed with this lemma, we can now analyze the stationarized regret $\bar{R}^{\bd{\gamma},\bd{f}}(P)$: suppose that, for each $t$, $\wh{Q}^{\bd{\gamma},\bd{f}}_t$ satisfies reverse Poisson inequality for $P^{\bd{\gamma},\bd{f}}_t$ with forcing function $f_t$. Then we apply the comparison principle to get $$\begin{aligned} \eta^{\bd{\gamma},\bd{f}}_t - \eta^{P,\bd{f}}_t &\le \sum_{x}\pi_P(x) \left( \sum_u P^{\bd{\gamma},\bd{f}}_t(u|x)\wh{Q}^{\bd{\gamma},\bd{f}}_t(x,u) - P(u|x)\wh{Q}^{\bd{\gamma},\bd{f}}_t(x,u)\right).\end{aligned}$$ This in turn yields $$\begin{aligned} R^{\bd{\gamma},\bd{f}}_x(P) &\le \sum_x \pi_P(x) \sum^T_{t=1} \left( \sum_u P^{\bd{\gamma},\bd{f}}_t(u|x)\wh{Q}^{\bd{\gamma},\bd{f}}_t(x,u) - P(u|x)\wh{Q}^{\bd{\gamma},\bd{f}}_t(x,u)\right) \\ &\qquad \qquad + \sum^T_{t=1} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_1.\end{aligned}$$ Note that $\wh{Q}^{\bd{\gamma},\bd{f}}_t$ depends functionally on $P^{\bd{\gamma},\bd{f}}_t$ and on $f_t$, which in turn depend functionally on $f^t$ but not on $f_{t+1},\ldots,f_T$. This ensures that any algorithm using $\wh{Q}^{\bd{\gamma},\bd{f}}_t$ respects the causality constraint that any decision made at time $t$ depends only on information available by time $t$. Focusing on stationarized regret and upper-bounding it in terms of the $\wh Q$-functions is one of the key steps that let us consider a simpler setting without Markov dynamics. The next step is to define a new type of relaxation with an accompanying new admissibility condition for this simpler setting. That is, we will find a relaxation and admissibility condition for the stationarized regret rather than for the expected steady-state regret directly. A new admissibility condition is needed because we have decoupled current costs from past actions, which makes the previous admissibility condition inapplicable. The new admissibility condition is similar to the one in [@RakhlinRL], which was derived in a stateless setting. The difference is that we are still in a [*state-dependent*]{} setting in the sense that the new type of relaxation is indexed by the state variable. Now instead of having a Markov dynamics that depends on the state, we consider all the states in parallel and have a separate algorithm running on each state. The interaction between different states is generated by providing these algorithms with common information that comes from the actual dynamical process. Thus, starting from this new admissibility condition, we further construct algorithms using relaxations and then use Lemma \[lem:comparisonch3\] to bound the regret of these algorithms. For each $x \in \sX$, let $\cH_x$ denote the class of all functions $h_x : \sU \to \Reals$ for which there exist some $P \in \cM(\sU|\sX)$ and $f \in \cF$, such that $$\begin{aligned} h_x(u) = \wh{Q}_{P,f}(x,u), \qquad \forall u \in \sU.\end{aligned}$$ We say that a sequence of functions $\wh{W}_{x,t} : \cH^t_x \to \Reals, t = 0,\ldots,T$, is an admissible relaxation at state $x$ if the following condition holds for any $h_{x,1},\ldots,h_{x,T} \in \cH_x$: \[eq:generalrelaxation1\] $$\begin{aligned} \wh{W}_{x,T}(h^T_x) &\ge - \inf_{\nu \in \cP(\sU)} \E_{U \sim \nu}\left[\sum^T_{t=1} h_{x,t}(U)\right], \\ \wh{W}_{x,t}(h^t_x) &\ge \inf_{\nu \in \cP(\sU)} \sup_{h_x \in \cH_x} \left\{ \E_{U \sim \nu}[h_x(U)] + \wh{W}_{x,t+1}(h^t_x,h_x) \right\}, \qquad t = T-1,\ldots,0.\end{aligned}$$ Given such an admissible relaxation, we can associate to it a behavioral strategy $$\begin{aligned} \wh{\gamma}_t(x,f^{t-1}) &= P^{\wh{\bd{\gamma}},\bd{f}}_t(\cdot|x) = \argmin_{\nu \in \cP(\sU)} \sup_{h_x \in \cH_x} \left\{ \E_{U \sim \nu} [h_x(U)] + \wh{W}_{x,t}(h^{t-1}_x,h_x)\right\} \\ h_{y,t} &= \wh{Q}^{\wh{\bd{\gamma}},\bd{f}}_t(y,\cdot), \quad \forall y \in \sX.\end{aligned}$$ (Even though the above notation suggests the dependence of $h_{y,t}$ on the $T$-tuples $\bd{\gamma}$ and $\bd{f}$, this dependence at time $t$ is only w.r.t. $\gamma^t$ and $f^t$, so the resulting strategy is still causal.) The relaxation $\{ \wh{W}_{x,t}\}^T_{t=1}$ at state $x$ is a sequence of upper bounds on the conditional value of the online learning game associated with that state. In this game, at time step $t$, the agent chooses actions $u_t \in \sU$ and the environment chooses function $h_{x,t} \in \cH_x$. Although this relaxation is still state-dependent, there is no Markov dynamics involved here, which means that now the state-free techniques of [@RakhlinRL] can be brought to bear on the problem of constructing algorithms and bounding their regret. Specifically, we derive a separate relaxation $\{ \wh{W}_{x,t}\}^T_{t=1}$ and the associated behavioral strategy for each state $x \in \sX$. Then we assemble these into an overall algorithm for the MDP as follows: if at time $t$ the state $X_t = x$, the agent will choose actions according to the corresponding behavioral strategy $\wh{\gamma}_t(x,\cdot)$. Note that although the agent’s behavioral strategy switches between different relaxations depending on the current state, the agent still needs to update all the $h$-functions simultaneously for all the states. This is because the computation of the $h$-functions (in terms of the $\wh Q$ functions) requires the knowledge of the behavioral strategy at other states. In other words, the algorithm has to keep updating all the relaxations in parallel for all states. Under the constructed relaxation, the value-function approach amounts to the following: \[thm:main\_ch3\] Suppose that the MDP is unichain, the environment is oblivious, and Assumption \[as:existboundQ\] holds. Then, for any family of admissible relaxations given by and the corresponding behavioral strategy  $\wh{\bd{\gamma}}$, we have $$\begin{aligned} \label{eq:thm2eq1} R^{\wh{\bd{\gamma}},\bd{f}}_x=\E^{\wh{\bd{\gamma}},\bd{f}}_x\left[\sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right] &\le \sup_{P \in \cM(\sU|\sX)}\sum_{x}\pi_P(x) \wh{W}_{x,0} + C_\cF \sum^T_{t=1} \| \mu^{\wh{\bd{\gamma}},\bd{f}}_t - \pi^{\wh{\bd{\gamma}},\bd{f}}_t \|_1 \end{aligned}$$ where $C_\cF = \sup_{f \in \cF} \| f \|_\infty$. See Appendix \[app:main\_ch3\]. This general framework gives us a recipe for deriving algorithms for online MDPs. First, we use stationarization to pass to a simpler setting without Markov dynamics. Here we need to find the $\wh Q_t$ functions satisfying with forcing function $f_t$ at each time $t$. In this simpler setting, we associate each state with a separate online learning game. Next, we derive appropriate relaxations (upper bounds on the conditional values) for each of these online learning games. Then we plug the relaxation into the admissibility condition to derive the associated algorithm. This algorithm in turn gives us a behavioral strategy for the original online MDP problem, and Theorem \[thm:main\_ch3\] automatically gives us a regret bound for this strategy. We emphasize that, in general, multiple different relaxations are possible for a given problem, allowing for a flexible tradeoff between computational costs and regret. We have reduced the original problem to a collection of standard online learning problems, each of which is associated with a particular state. We proceed by constructing a separate relaxation for each of these problems. Because we have removed the Markov dynamics, we may now use available techniques for constructing these relaxations. In particular, as shown by [@RakhlinOR], a particularly versatile method for constructing relaxations relies on the notion of [*sequential Rademacher complexity*]{} (SRC). The convex-analytic approach {#ssec: omd} ---------------------------- In the preceding section, we have developed a procedure for recovering and deriving policies for online MDPs using relative-value functions that arise from revere Poisson inequalities. Now we show a complementary procedure that allows us to use an admissible relaxation with no conditioning on state variables. Specifically, we reduce the online MDP problem to an online linear optimization problem through stationarization, and then directly use the framework of [@RakhlinRL] to derive a relaxation and an algorithm, which is similar in spirit to the algorithm proposed recently by Dick et al. [@DickCsaba]. The idea behind this convex-analytic method is closely related to the well-known fact that the dynamic optimization problem for an MDP can be reformulated as a “static” linear optimization problem a certain polytope, and therefore can be solved using LP methods [@Manne; @Borkar]. Under this reformulation, we are in a state-free setting in the sense that the relaxation is no longer indexed by the state variable, and the policy for the agent is computed from a certain joint distribution on states and actions via Bayes’ rule. As before, we start with the stationarization step. Recall that we decompose the regret $R^{\bd{\gamma},\bd{f}}_x(P)$ into two parts: the stationarized regret $\bar{R}^{\bd{\gamma},\bd{f}}(P)$ and the stationarization error $\sum^T_{t=1} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_1$, that is: $$\begin{aligned} R^{\bd{\gamma},\bd{f}}_x(P) & \le \bar{R}^{\bd{\gamma},\bd{f}}(P) + \sum^T_{t=1} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_1 \nonumber \\ &= \sum^T_{t=1}\left(\eta^{\bd{\gamma},\bd{f}}_t -\eta^{P,\bd{f}}_t\right) + \sum^T_{t=1} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_1 \nonumber\\ &= \sum^T_{t=1}\left[\ave{\pi^{\bd{\gamma},\bd{f}}_t \otimes P^{\bd{\gamma},\bd{f}}_t,f_t} - \ave{\pi_P \otimes P, f_t}\right] + \sum^T_{t=1} \| f_t \|_\infty \| \mu^{\bd{\gamma},\bd{f}}_t - \pi^{\bd{\gamma},\bd{f}}_t \|_1. \label{eq:regretdecomp}\end{aligned}$$ Now, let $\mathcal G \subset \cP(\sX \times \sU)$ denote the set of all *ergodic occupation measures $$\begin{aligned} \label{eq:SAP} \mathcal G \deq \Big\{ \nu \in \cP(\sX \times \sU): \sum_{x,u} K(y|x,u) \nu(x,u) = \sum_u \nu(y,u), \forall y \in \sX \Big\}.\end{aligned}$$ The set $\mathcal G$ is convex, and is defined by a finite collection of linear equality and inequality constraints. Hence, it is a convex polytope in $\Reals^{|\sX \times \sU|}$ (in fact, it is often referred to as the *state-action polytope* of the MDP [@Puterman]). Every element in $\mathcal G$ can be decomposed in the form $$\nu(x,u) = \pi_P(x) \otimes P(u |x), \quad x \in \sX, u \in \sU$$ for some randomized Markov policy $P \in \cM(\sU|\sX)$, where $\pi_P$ is the invariant distribution of the Markov kernel $$\begin{aligned} K_P(x'|x) \deq \sum_{u \in \sU} K(x'|x,u)P(u|x), \qquad \forall x,x' \in \sX\end{aligned}$$ induced by $P$. For this reason, the linear equality and inequality constraints that define $\mathcal G$ are also called the *invariance constraints*. Conversely, any element $\nu \in \mathcal G$ induces a Markov policy $$\begin{aligned} \label{eq:from_ergodic_to_policy} P_\nu(u|x) \deq \frac{\nu(x,u)}{\sum_{v \in \sU}\nu(x,v)}, \qquad \forall u \in \sU(x)\end{aligned}$$ where $\sU(x)$ is the set of all states for which the denominator of is nonzero.* With the definition of the set $\mathcal G$ at hand, now it is easy to see that the first term of is the regret of an online *linear* optimization problem, where, at each time step $t$, the agent is choosing an occupation measure $\nu_t = \pi^{\bd{\gamma},\bd{f}}_t \otimes P^{\bd{\gamma},\bd{f}}_t$ from the set $\mathcal G$ (here we omit the dependence of $\nu_t$ on $\bd{\gamma},\bd{f}$ for simplicity), and the environment is choosing the one-step cost function $f_t$. The one-step linear cost function incurred by the agent is $\ave{\nu_t,f_t}$. Since we can recover a policy from an occupation measure, we just need to find a slowly changing sequence of occupation measures, to ensure simultaneously that the first term of and the stationarization error are both small. Now we have mapped an online MDP problem to an online linear optimization problem. As we mentioned earlier, the idea behind this mapping is simply the fact that average-cost optimal control problem can be cast as a LP over the state-action polytope [@Manne; @Borkar]. For reasons that will become apparent later, it is convenient to consider regret with respect to policies induced by elements of a given subset ${\cal G}'$ of ${\cal G}$. With that in mind, let us denote by $\cM({\cal G}')$ the set of all policies $P \in \cM(\sU|\sX)$ that have the form $P_\nu$ for some $\nu \in {\cal G}'$. For the resulting online linear optimization problem, we can immediately apply the framework of [@RakhlinRL] to derive novel relaxations and online MDP algorithms. For any $t \in \{0,1,\ldots,T-1\}$, any given prefix $f^{t} = (f_1,\ldots,f_{t})$, define the conditional value recursively via $$\begin{aligned} V_T({\mathcal G}' | f_1, \ldots,f_{t}) = \inf_{\nu \in {\mathcal G}'} \sup_{f \in \cF} \left\{ \ave{\nu,f} + V_T({\mathcal G}' | f_1, \ldots,f_{t},f) \right\},\end{aligned}$$ where $V_T({\mathcal G}' | f_1, \ldots,f_{T}) = -\inf_{\nu \in {\mathcal G}'} \sum^T_{t=1} \ave{\nu,f_t}$, and $V_T({\mathcal G}') \equiv V_T({{\mathcal G}'} | {\mathsf e})$ is the minimax regret of the game. Note that we are explicitly indicating the fact that the optimization takes place over the ergodic occupation measures in ${{\mathcal G}'}$. The minimax optimal algorithm specifying the mixed strategy of the player can be written as $$\begin{aligned} \nu_t = \argmin_{\nu \in {\mathcal G}'} \sup_{f \in \cF}\left\{ \ave{\nu,f} + V_T({\mathcal G}' | f_1, \ldots,f_{t-1},f) \right\}\end{aligned}$$ Following the formulation of Rakhlin et al. [@RakhlinRL], we say that a sequence of functions $\wh{V}_T({\mathcal G}' | f_1, \ldots,f_{t})$ is an [*admissible relaxation*]{} if for any $f_1, \ldots, f_t \in \cF$, \[eq: rakhlinadmiss\] $$\begin{aligned} \wh{V}_T({\mathcal G}' | f_1, \ldots,f_{t}) &\ge \inf_{\nu \in {\mathcal G}'} \sup_f \left\{ \ave{\nu,f} + \wh{V}_T({\mathcal G}' | f_1, \ldots,f_{t},f)\right\}, \qquad t = T-1,\ldots,0 \\ \wh{V}_T({\mathcal G}' | f^T) &\ge - \inf_{\nu \in {\mathcal G}'} \sum^T_{t=1} \ave{\nu,f_t}.\end{aligned}$$ We can associate a behavioral strategy to any admissible relaxation as follows: $$\begin{aligned} \wh{\gamma}_t(x,f^{t-1}) &= \nu_t = \argmin_{\nu \in {\mathcal G}'} \sup_{f \in \cF}\left\{ \ave{\nu,f} + \wh{V}_T({\mathcal G}' | f_1, \ldots,f_{t-1},f)\right\}.\end{aligned}$$ In fact, as pointed out by Rakhlin et al. [@RakhlinRL], exact minimization is unnecessary: any choice $\nu_t = \wh{\gamma}_t(x,f^{t-1})$ that satisfies $$\wh{V}_T({{\mathcal G}'}|f_1,\ldots,f_{t-1}) \ge \sup_{f \in \cF}\left\{ \ave{\nu_t,f} + \wh{V}_T({\mathcal G}' | f_1, \ldots,f_{t-1},f)\right\},$$ is admissible. The above admissibility condition of [@RakhlinRL] is different from in the sense that there is no conditioning on the state variable. It is also different from because it is not indexed by the state variable. The following theorem provides the main regret bound for the convex-analytic approach: \[thm:main\_2nd\] Suppose that the MDP is unichain and the environment is oblivious. Then, for any family of admissible relaxations given by and the corresponding behavioral strategy  $\wh{\bd{\gamma}}$, we have $$\begin{aligned} \label{eq:thm2eq2} R^{\wh{\bd{\gamma}},\bd{f}}_x({\cal G'})\deq\E^{\wh{\bd{\gamma}},\bd{f}}_x\left\{\sum^T_{t=1}f_t(X_t,U_t) - \inf_{P \in \cM({\cal G}')}\E\left[\sum^T_{t=1}f_t(X,U)\right]\right\} &\le \wh{V}_T({\mathcal G}' | {\mathsf e}) + C_\cF \sum^T_{t=1} \| \mu^{\wh{\bd{\gamma}},\bd{f}}_t - \pi^{\wh{\bd{\gamma}},\bd{f}}_t \|_1 \end{aligned}$$ where $C_\cF = \sup_{f \in \cF} \| f \|_\infty$. See Appendix \[app:main\_2nd\]. Example derivations of explicit algorithms ========================================== In the preceding section, we have described two different approaches to construct relaxations and algorithms for online MDPs. Specifically, the value-function approach make use of Poisson inequalities for MDPs [@MeynTweedie] to reduce the online MDP problem to a collection of standard online learning problems, each of which is associated with a particular state. We need to construct a separate relaxation for each of these problems. The convex-analytic approach reduces the online MDP problem to an online linear optimization problem, and uses a single relaxation (no longer indexed by the state) to derive algorithms. The common property of these two approaches is that we can apply any algorithm derived in the simpler static setting to the original dynamic setting and automatically bound its regret. The value-function approach {#sec:derivealgo} --------------------------- In this section, we apply the value-function approach to recover and derive a class of online MDP algorithms [@EvenDar; @Yu]. The common thread running through this class of algorithms is that value functions have to be computed in order to get the policy for each time step. The strategies derived in this section using our general framework also belong to a class of algorithms for [*online prediction with expert advice*]{} [@PLG]. In this setting, the agent combines the recommendations of several individual “experts” into an overall strategy for choosing actions in real time in response to causally revealed information. Every expert is assigned a “weight” indicating how much the agent trusts that expert, based on the previous performance of the experts. One of the more popular algorithms for prediction with expert advice is the Randomized Weighted Majority (RWM) algorithm, which updates the expert weights multiplicatively [@LittlestoneWM]. It has an alternative interpretation as a Follow the Regularized Leader (FRL) scheme [@ShaiSinger]: The weights chosen by an RWM algorithm minimize a combination of empirical cost and an entropic regularization term. The entropy term (equal to the divergence between the current weight distribution and the uniform distribution over the experts) penalizes “spiky” weight vectors, thus guaranteeing that every expert has a nonzero probability of being selected at every time step, which in turn provides the algorithm with a degree of stability. The common feature of the strategies we consider in this section is that RWM algorithms are applied in parallel for each state. We start by recovering an expert-based algorithm for online MDPs. Similar to our set-up, Even-Dar et al. [@EvenDar] consider an MDP with arbitrarily varying cost functions. The main idea of their work is to efficiently incorporate existing expert-based algorithms [@LittlestoneWM; @PLG] into the MDP setting. For an MDP with state space $\sX$ and action space $\sU$, there are $\lvert \sU \rvert^{\lvert \sX \rvert}$ deterministic Markov policies (state feedback laws), which renders the obvious approach of associating an expert with each possible deterministic policy computationally infeasible. Instead, they propose an alternative efficient scheme that works by associating a separate expert algorithm to each [*state*]{}, where experts correspond to [*actions*]{} and the feedback to provided each expert algorithm depends on the aggregate policy determined by the action choices of all the individual algorithms. Under a unichain assumption similar to the one we have made above, they show that the expected regret of their algorithm is sublinear in $T$ and independent of the size of the state space. Their algorithm can be summarized as follows: ----------------------------------------------------------------------------------------------------------------------------------------------------------------- Put in every state $x$ an expert algorithm ${\cal A}_x$ for $t = 1, 2, \ldots$ do   Let $P_t(\cdot|x_t)$ be the distribution over actions of ${\cal A}_{x_t}$   Use policy $P_t$ and obtain $f_t$ from the environment   For every $x \in \sX$     Feed ${\cal A}_x$ with loss function $\wh{Q}_{P_t,f_t} (x, \cdot) = \E\left[\sum^{\infty}_{i=0}\left(f_t(X_i, U_i) - \eta^{P_t,{\bd f}}_t \right)\right]$,     where $\E$ is taken w.r.t. the Markov chain induced by $P_t$ from the initial state $x$,     and $\eta^{P_t,\bd{f}}_t$ is the steady-state cost $\ave{\pi_{P_t} \otimes P_t,f_t}$   end for end for ----------------------------------------------------------------------------------------------------------------------------------------------------------------- As we show next, the algorithm proposed by [@EvenDar] arises from a particular relaxation under the value-function approach. For every possible state value $x \in \sX$, we want to construct an admissible relaxation that satisfies . Here we show that the relaxation can be obtained as an upper bound of a quantity called [*conditional sequential Rademacher complexity*]{}, which is defined by Rakhlin et al. [@RakhlinRL] as follows. Let $\eps$ be a vector $(\eps_1, \ldots, \eps_T)$ of i.i.d. Rademacher random variables, i.e., $\Pr(\eps_i = \pm 1) = 1/2$. For a given $x \in \sX$, an $\cH_x$-valued tree $\mathbf h$ of depth $d$ is defined as a sequence $(\mathbf h_1, \ldots, \mathbf h_d)$ of mappings $\mathbf h_t: \{ \pm 1\}^{t-1} \to \cH_x$, where $\cH_x$ is the function class defined in Section \[ssec:value\_function\_approach\]. Then the conditional sequential Rademacher complexity at state $x$ is defined as $$\begin{aligned} \mathcal R_{x,t}(h^t_x) = \sup_{\mathbf h} \E_{\eps_{t+1:T}} \max_{u \in \sU} \left[ 2\sum^T_{s=t+1}\eps_s \left[\mathbf h_{s-t}(\eps_{t+1:s-1})\right] (u) - \sum^t_{s=1} h_{x,s}(u) \right], \qquad \forall h^t_x \in \cH^t_x.\end{aligned}$$ Here the supremum is taken over all $\cH_x$-valued binary trees of depth $T-t$. The term containing the tree $\mathbf h$ can be seen as “future", while the term being subtracted off can be seen as “past". This quantity is conditioned on the already observed $h^t_x$, while for the future we consider the worst possible binary tree. As shown by [@RakhlinRL], this Rademacher complexity is itself an admissible relaxation for standard (state-free) online optimization problems; moreover, one can obtain other relaxations by further upper-bounding the Rademacher complexity. As we will now show, because the action space $\sU$ is finite and the functions in $\cH_x$ are uniformly bounded (Assumption \[as:existboundQ\]), the following upper bound on $\mathcal R_{x,t}(\cdot)$ is an admissible relaxation, i.e., it satisfies condition : $$\begin{aligned} \label{eq:exprelax1} \wh W_{x,t}(h^t_x) = \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^t_{s=1} h_{x,s}(u)\right)\right) + \frac{2}{\rho}(T-t) L(\sX,\sU, \cF)^2,\end{aligned}$$ where the learning rate $\rho > 0$ can be tuned to optimize the resulting regret bound. This relaxation leads to an algorithm that turns out to be exactly the scheme proposed by [@EvenDar]: \[pps:recoverevendar\] The relaxation is admissible and it leads to a recursive exponential weights algorithm, specified recursively as follows: for all $x \in \sX$, $u \in \sU$ $$\begin{aligned} \label{eq:recursiveevendar} P_{t+1} (u|x) = \frac{ P_t(u|x) \exp\left(-\frac{1}{\rho}h_{x,t}(u)\right)}{\Ave{P_t(\cdot|x), \exp\left(-\frac{1}{\rho}h_{x,t}\right)}} = \frac{ \nu_1(u) \exp\left(-\frac{1}{\rho}\sum^t_{s=1}h_{x,s}(u)\right)}{\Ave{\nu_1, \exp\left(-\frac{1}{\rho}\sum^t_{s=1}h_{x,s}\right)}},\qquad t = 0,\ldots,T-1\end{aligned}$$ where $\nu_1$ is the uniform distribution on $\sU$. See Appendix \[app:recoverevendar\]. The above algorithm works with any collection of $\wh{Q}$ functions satisfying the reverse Poisson inequalities determined by the $f_t$’s (recall Assumption \[as:existboundQ\]). Here is one particular example of such a function — the usual $Q$-function that arises in reinforcement learning and that was used by [@EvenDar]. Recall our assumption that every randomized state feedback law $P \in \cM(\sU|\sX)$ has a unique stationary distribution $\pi_P$. For given choices of $P \in \cM(\sU|\sX)$ and $f \in \cF$, consider the function $$\wh{Q}_{P,f} (x, u) = \lim_{T \rightarrow \infty} \E_P \left[ \sum^T_{t=1} f(X_t, U_t) - \ave{\pi_P \otimes P, f}\Bigg | X_1 = x, U_1 = u \right],$$ where $X_t$ and $U_t$ are the state and action at time step $t$ after starting from the initial state $X_1 = x$, applying the immediate action $U_1 = u$, and following $P$ onwards. It is easy to check that $\wh{Q}_{P,f} (x, u)$ satisfies the reverse Poisson inequality for $P$ with forcing function $f$. In fact, it satisfies with equality. We can also derive a bound on the Q-function in terms of the mixing time $\tau$. Let us first bound $\wh{Q}_{P,f}(x, P)$ where $P$ is used on the first step instead of $u$. For all $t$, let $\mu^{P,{f}}_{x,t}$ be the state distribution at time $t$ starting from $x$ and following $P$ onwards. So we have $$\begin{aligned} \wh{Q}_{P,f}(x, P)=\lim_{T \rightarrow \infty} \sum^T_{t=1}\left[\ave{\mu^{P,{f}}_{x,t} \otimes P,f} - \ave{\pi_P \otimes P,f}\right] &\le \| f\|_\infty \sum^T_{t=1} \| \delta_xP^t - \pi_P P^t \|_1 \\ &\le 2 \| f\|_\infty \sum^T_{t=1} e^{-t/\tau} \\ &\le 2\tau \| f\|_\infty, \end{aligned}$$ where $\delta_x \in \cP(\sX)$ is the Dirac distribution centered at $x$, and the first inequality results from repeated application of the uniform mixing bound . Due to the fact that the one-step cost is bounded by $C_\cF = \sup_{f \in \cF} \| f \|_\infty$, we have $$\wh{Q}_{P,f}(x,u) \le \wh{Q}_{P,f}(x,P) + f(x,u) - \ave{\mu^{P,{f}}_{x,1} \otimes P,f} \le 2 \tau C_\cF + C_\cF \le 3 \tau C_\cF.$$ We can now establish the following regret bound for the exponential weights strategy : \[thm:evendarbound\] Let $L \deq L(\sX,\sU, \cF)$. Assume the uniform mixing condition is satisfied. Then for the relaxation and the corresponding behavioral strategy $\wh{\bd{\gamma}}$ given by with $\rho = \sqrt{\frac{2TL^2}{\log|\sU|}}$, we have $$\begin{aligned} \E^{\wh{\bd{\gamma}},\bd{f}}_x \left[\sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right] \le 2 L\sqrt{2 T \log |\sU|} + C_{\cF}(\tau+1)^2 \sqrt{\frac{\log|\sU|T}{2}} + (2\tau+2) C_{\cF}.\end{aligned}$$ See Appendix \[app:evendarbound\]. As we can see, this regret bound is consistent with the bound derived in [@EvenDar]. Therefore, we have shown that our framework, with a specific choice of relaxation, can recover their algorithm. The advantage of our general framework is that we can analyze the part of the corresponding regret bound simply by instantiating our analysis on specific relaxations, without the need of ad-hoc proof techniques applied in [@EvenDar]. The above policy relies on exponential weight updates. We now present a “lazy" version of that policy, wherein time is divided into phases of increasing length, and during each phase the agent applies a fixed state feedback law. The main advantage of lazy strategies is their computational efficiency, which is the result of a looser relaxation and hence suboptimal scaling of the regret with the time horizon. We partition the set of time indices $1,2,\ldots$ into nonoverlapping contiguous phases of (possibly) increasing duration. The phases are indexed by $m \in \Naturals$, where we denote the $m$th phase by $\cT_m$ and its duration by $\tau_m$. We also define $\cT_{1:m} \deq \cT_1 \cup \ldots \cup \cT_m$ (the union of phases $1$ through $m$) and denote its duration by $\tau_{1:m}$. Let $M \le T$ denote the number of complete phases concluded before time $T$. Here we need a describe a generic algorithm that works in phases: ------------------------------------------------------------------------------------------------ Initialize at $t = 0$ and phases $\cT_1, \ldots, \cT_M$ s.t. $\tau_{1:M} = T$ For $t \in \cT_1$, choose $u_t$ uniformly at random over $\sU$ for $m = 2, 3, \ldots$    for $t \in \cT_m$ do     if the process is at state $x$, choose action $u_t$ randomly according to $P_{m} (u|x)$     where $P_{m} (u|x)$ is the state feedback law only using information from phase 1 to $m-1$   end for end for ------------------------------------------------------------------------------------------------ Because now we work in phases instead of time steps, we need to provide an alternative definition of relaxations and admissibility condition. For every state $x \in \sX$, we denote by $h^m_x$ the $\tau_m$-tuple $(h_{x,s} : s \in \cT_m)$, and by $h_{x,1:m}$ the $\tau_{1:m}$-tuple $(h_{x,1}, h_{x,2}, \ldots, h_{x,\tau_{1:m}})$. For each $x \in \sX$, we will say that a sequence of functions $\wh W_{x,m}: \cH_x^{\tau_{1:m}} \rightarrow \Reals, m = 1, \ldots, M,$ is an admissible relaxation if $$\begin{aligned} \wh W_{x,M} (h_{x,1:M}) &\ge - \inf_{\nu \in \cP(\sU)} \E_{U \sim \nu} \left[ \sum^T_{t=1} h_{x,t}(U)\right] \\ \wh W_{x,m} (h_{x,1:m}) &\ge \inf_{\nu \in \cP(\sU)} \sup_{h^m_x \in \cH^{\tau_m}_x} \left\{\E_{U \sim \nu} \left[ \sum_{s \in \cT_m}h_{x,s}(U)\right] + \wh W_{x,m+1} (h_{x,1:m},h^{m+1}_x)\right\}, \quad m = M-1, \ldots, 1\end{aligned}$$ For a given state $x$, we also define the conditional sequential Rademacher complexity in terms of phases: $$\begin{aligned} \mathcal R_{x,m}(h_{x,1:m}) = \sup_{\mathbf h} \E_{\eps_{m+1:M}} \max_{u \in \sU} \left[ 2\sum^M_{j=m+1}\eps_j \sum_{t \in \cT_j}\left[\mathbf h_{x,t}(\eps)\right] (u) - \sum^m_{i=1} \sum_{s \in \cT_i}h_{x,s}(u) \right].\end{aligned}$$ Here the supremum is taken over all $\cH_x$-valued binary trees of depth $M-m$. When recovering the method in [@EvenDar], we replaced the actual future induced by the infimum and supremum pairs in the conditional value by the “worst future" binary tree, which involves expectation over a sequence of coin flips in every time step. By contrast, in the above quantity we replace the real future by the “worst future" binary tree that branches only once per phase. Now we can construct the following relaxation: $$\begin{aligned} \label{eq:modrelax} \wh W_{x,m}(h_{x,1:m}) = \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i}h_{x,s}(u)\right)\right) + \frac{2L(\sX,\sU, \cF)^2}{\rho} \sum^{M}_{j=m+1} \tau_j^2 .\end{aligned}$$ The corresponding algorithm, specified in below, uses a fixed state feedback law throughout each phase: \[pps:recoveryu\] The relaxation is admissible and it leads to the following Markov policy for phase $m$: $$\begin{aligned} \label{eq:recursiveyu} P_{m} (u|x) = \frac{ \nu_1(u) \exp\left(-\frac{1}{\rho}\sum^{m-1}_{i=1} \sum_{s \in \cT_i} h_{x,s}(u)\right)}{\Ave{\nu_1, \exp\left(-\frac{1}{\rho}\sum^{m-1}_{i=1} \sum_{s \in \cT_i} h_{x,s}\right)}},\end{aligned}$$ where $\nu_1$ is the uniform distribution on $\sU$. See Appendix \[app:recoveryu\]. Now we derive the regret bound for : \[thm:yubound\] Let $L \deq L(\sX,\sU, \cF)$. Under the same assumptions as before, the behavioral strategy $\wh{\bd{\gamma}}$ corresponding to enjoys the following regret bound when $\rho = \sqrt{\frac{2\sum^M_{i=1}\tau_i^2L^2}{\log|\sU|}}$: $$\begin{aligned} \label{eq:yuregret} \E^{\wh{\bd{\gamma}},\bd{f}}_x \left[\sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right] \le 2 L\sqrt{2 \log |\sU| \sum^M_{i=1}\tau_i^2} + \frac {2C_{\cF}M}{1-e^{-1/\tau}}.\end{aligned}$$ See Appendix \[app:yubound\]. Our behavioral strategy is a novel randomized weighted majority (RWM) algorithm for online MDPs. Yu et al. [@Yu] also consider a similar model, where the decision-maker has full knowledge of the transition kernel, and the costs are chosen by an oblivious (open-loop) adversary. They propose an algorithm that computes and changes the policy periodically according to a perturbed version of the empirically observed cost functions, and then follows the computed stationary policy for increasingly long time intervals. As a result, their algorithm achieves sublinear regret and has diminishing computational effort per time step; in particular, it is computationally more efficient than that of [@EvenDar]. Although our new algorithm is similar in nature to the algorithm of [@Yu], it has several advantages. First, in the algorithm of [@Yu], the policy computation at the beginning of each phase requires solving a linear program and then adding a carefully tuned random perturbation to the solution. As a result, the performance analysis in [@Yu] is rather lengthy and technical (in particular, it invokes several advanced results from perturbation theory for linear programs). By contrast, our strategy is automatically randomized, and the performance analysis is a lot simpler. Second, the regret bound of Theorem \[thm:yubound\] shows that we can control the scaling of the regret with $T$ by choosing the duration of each phase, whereas the algorithm of [@Yu] relies on a specific choice of phase durations in order to guarantee that the regret is sublinear in $T$ and scales as $O(T^{3/4})$. We show that if the horizon $T$ is known in advance, then it is possible to choose the phase durations to secure $O(T^{2/3})$ regret, which is better than the $O(T^{3/4})$ bound derived by [@Yu]. \[cor:optbound\] Consider the setting of Theorem \[thm:yubound\]. For a given horizon $T$, the optimal choice of phase lengths is $T^{1/3}$, which gives the regret of  $O(T^{2/3})$. See Appendix \[app:optbound\]. The convex-analytic approach {#sec:derivedick} ---------------------------- In this section, we use the convex-analytic approach to derive an algorithm that relies on the reduction of the online MDP problem to an online linear optimization problem over the state-action polytope \[recall the definition in Eq. \]. Structurally, this algorithm is similar to the Online Mirror Descent scheme proposed and analyzed recently by Dick et al. [@DickCsaba]; however, its key ingredients and the resulting performance guarantee on the regret are more closely related to interior-point methods of Abernethy et al. [@AbernethyIPM]. Moreover, we will show that this algorithm arises from an admissible relaxation with respect to . We start by introducing the definition of a *self-concordant barrier*, which is basic to the theory of interior point methods [@Nesterov; @NemTodd]: Let $\cK \subseteq \Reals^n$ be a closed convex set with nonempty interior. A function $F : {\rm int}(\cK) \to \Reals$ is a *barrier* on $\cK$ if $F(x_i) \to +\infty$ along any sequence $\{x_i\}^\infty_{i=1} \subset \cK$ that converges to a boundary point of $\cK$. Moreover, $F$ is *self-concordant* if it is a convex $C^3$ function, such that the inequality $$\nabla^3F(v)[h,h,h] \le 2\left(\nabla^2 F(v)[h,h]\right)^{3/2}$$ holds for all $v \in {\rm int}(\cK)$ and $h \in \Reals^n$. Here, $\nabla^2 F(v)$ and $\nabla^3 F(v)$ are the Hessian and the third-derivative tensor of $F$ at $v$, respectively. We also need some geometric quantities induced by $F$. The first is the *Bregman divergence* $D_F : {\rm int}(\cK) \times {\rm int}(\cK) \to \Reals^+$, defined by $$\begin{aligned} \label{eq:Bregman} D_F(v,w) \deq F(v) - F(w) - \langle \nabla F(w),v-w\rangle, \qquad v,w \in {\rm int}(\cK).\end{aligned}$$ The second is the *local norm* of $h \in \Reals^n$ around a point $v \in {\rm int}(\cK)$ (assuming $\nabla^2 F(v)$ is nondegenerate): $$\| h \|_v \deq \sqrt{\nabla^2 F(v)[h,h]}.$$ Finally, if $F$ is self-concodrant, then so is its *Legendre–Fenchel dual* $F^*(h) \deq \sup_{v \in {\rm int}(\cK)} \left\{ \langle h,v\rangle - F(v)\right\}$. Thus, the definitions of the Bregman divergence and the local norm carry over to $F^*$. Specifically, $$\| f \|^*_h \deq \sqrt{\ave{f, \nabla^2 F^*(h)f}} \equiv \sqrt{\nabla^2 F^*(h)[f,f]}$$ is the local norm of $f$ at $h$ induced by $F^*$ (by the following assumption that $F^*$ is strictly convex, this local norm is well-defined everywhere). Both our algorithm and the relaxation that induces it revolve around a self-concordant barrier for the set $\cK = {\cal G}$, the state-action polytope of our MDP. This set is a compact convex subset of $\Reals^{|\sX| \times |\sU|}$ with nonempty interior. We make the following assumption: \[as:barrier\] The state-action polytope ${\mathcal G}$ associated to the MDP with controlled transition law $K$ admits a self-concordant barrier $F : {\rm int}({\cal G}) \to \Reals$ with the following properties: 1. $F$ is strictly convex on ${\rm int}({\cal G})$, and its dual $F^*$ is strictly convex on $\Reals^{|\sX| \times |\sU|}$. 2. The gradient map $\nu \mapsto \nabla F(\nu)$ is a bijection between ${\rm int}({\cal G})$ and $\Reals^{|\sX| \times |\sU|}$, and admits the map $h \mapsto \nabla F^*(h)$ as inverse. 3. The minimum value of $F$ on ${\rm int}({\cal G})$ is equal to $0$. This assumption is not difficult to meet in practice. For example, the *universal entropic barrier* of Bubeck and Eldan [@BubeckEldan] (which can be constructed for any compact convex polytope) satisfies these requirements. We are now ready to present our algorithm and the associated relaxation. We start by describing the former: ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- For $t = 1, 2, \ldots$ do   If $t=1$, choose $\nu_t = \nu^* \equiv \argmin_{v \in {\rm int}({\cal G})} F(\nu)$; else choose $\nu_t = \nabla F^*\left(\nabla F(\nu_{t-1}) - \rho f_{t-1}\right)$   Construct the policy $P_t = P_{\nu_t}$ according to Eq.    Observe the state $X_t$   Draw the action $U_t \sim P_t(\cdot|X_t)$ and obtain $f_t$ from the environment end for ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- Here, $\rho > 0$ is the tunable learning rate. Note also that, by virtue of Assumption \[as:barrier\], the sequence of measures $\{\nu_t\}$ lies in ${\rm int}({\cal G})$. Next, we describe the relaxation. For reasons that will be spelled out shortly, we focus on the regret with respect to policies induced by elements of a given subset ${\cal G}'$ of ${\rm int}({\cal G})$. For $t=0,\ldots,T$, we let $$\begin{aligned} \label{eq:OMDrelax} \wh{V}_T({\cal G}'|f_1,\ldots,f_t) = \sup_{\mu \in {\cal G}'} \left\{\sum^t_{s=1} \ave{\mu,-f_s} + \frac{1}{\rho} D_{F} (\mu, \nu_{t+1})\right\} + 2\rho (T-t),\end{aligned}$$ Note that $\nu_{t+1}$ is a deterministic function of $f_1,\ldots,f_t$, and therefore the relaxation is well-defined. \[pps:OMDrelaxadmin\] Suppose that the learning rate $\rho$ is such that $\rho \| f_t \|^*_{\nabla F(\nu_t)} \le 1/2$ for all $t = 1,\ldots,T$. Assume that $\| f_t \|^*_{\nabla F(\nu_t)} \le 1$, for all $t = 1,\ldots,T$. The relaxation is admissible, and the algorithm that generates the sequence $\{\nu_t\}$ is also admissible: $$\wh{V}_T({\cal G}'|f_1,\ldots,f_{t-1}) \ge \sup_{f \in \cF}\left\{ \ave{\nu_t,f} + \wh{V}_T({\mathcal G}' | f_1, \ldots,f_{t-1},f)\right\}$$ See Appendix \[app:OMDrelaxadmin\]. Here we impose the assumption that $\rho \| f_t \|^*_{\nabla F(\nu_t)} \le 1/2$ for all $t = 1,\ldots,T$. A restriction of this kind is necessary when using interior-point methods to construct online optimization schemes — see, for example, the condition of Theorem 4.1 and 4.2 in [@AbernethyIPM]. The boundedness of the dual local norm $\| f_t \|^*_{\nabla F(\nu_t)}$ is a reasonable assumption as well. In particular, Abernethy et al. [@AbernethyIPM] points out that if a large number of the points $\nu_t$ are close to the boundary of ${\cal G}'$, then the eigenvalues of the Hessian of $F$ at those points will be large due to the large curvature of the barrier near the boundary of ${\cal G}'$. This will imply, in turn, that the dual local norm $\| f_t \|^*_{\nabla F(\nu_t)}$ is expected to be small. Now we are ready to present the online-learning (i.e., steady-state) part of the regret bound for the above algorithm: \[thm:OMDbound1\] Let $D_F({\cal G'}) \deq\sup_{\nu \in {\cal G}'} D_{F}(\nu,\nu_1)$. Suppose that the learning rate $\rho$ is such that $\rho \| f_t \|^*_{\nabla F(\nu_t)} \le 1/2$ for all $t = 1,\ldots,T$. Assume that $\| f_t \|^*_{\nabla F(\nu_t)} \le 1$, for all $t = 1,\ldots,T$. Then for the relaxation and the corresponding algorithm, we can bound the online learning part of the regret as $$\begin{aligned} \label{eq:OMD_regret} \sum^T_{t=1} \ave{\nu_t,f_t} - \inf_{\nu \in {\cal G}'} \sum^T_{t=1} \ave{\nu,f_t} \le \frac{D_F({\cal G}')}{\rho} + 2\rho T.\end{aligned}$$ See Appendix \[app:OMDbound1\]. [*Since $F$ is a barrier, $D_F({\cal G'})$ will be finite only if all the elements of ${\cal G}'$ are not too close to the boundary of ${\cal G}$. This motivates our restriction of the comparator term to a proper subset ${\cal G}' \subset {\rm int}({\cal G})$.*]{} Finally, we present the total regret bound for the above algorithm, including the stationarization error: \[thm:OMDbound2\] Suppose that all of our earlier assumptions are in place, and also that the uniform mixing condition is satisfied. Then for the relaxation and the corresponding algorithm, we have $$\begin{aligned} &\E^{\wh{\bd{\gamma}},\bd{f}}_x \left[\sum^T_{t=1}f_t(X_t,U_t) - \inf_{P \in \cM({\cal G'})}\right] \nonumber\\ & \qquad \qquad \le \frac{D_F({\cal G}')}{\rho} + 2\rho T + C_{\cF}(\tau+1)^2 T\Delta_T + (2\tau +2) C_{\cF},\label{eq:MD_total_regret}\end{aligned}$$ where $\Delta_T \deq \max_{1 \le t \le T}\max_{x \in \sX} \| P_{t-1}(\cdot|x) - P_t(\cdot|x)\|_1$. See Appendix \[app:OMDbound2\]. The third term on the right-hand side of quantifies the drift of the policies generated by the algorithm. A similar term appears in all of the regret bounds of Dick et al. (see, e.g., the bound of Lemma 1 in [@DickCsaba]). Moreover, just like the Mirror Descent scheme of [@DickCsaba], our algorithm may run into implementation issues, since in general it may be difficult to compute the gradient mappings $\nabla F$ and $\nabla F^*$ associated to the self-concordant barrier $F$. We refer the reader to the discussion in the paper by Bubeck and Eldan [@BubeckEldan] pertaining to computational feasibility of their universal entropic barrier. Conclusions {#sec:clc} =========== We have provided a unified viewpoint on the design and the analysis of online MDPs algorithms, which is an extension of a general relaxation-based approach of [@RakhlinOR] to a certain class of stochastic game models. We have unified two distinct categories of existing methods (those based on the relative-function approach and those based on the convex-analytic approach) under a general framework. We have shown that an algorithm previously proposed by [@EvenDar] naturally arises from our framework via a specific relaxation. Moreover, we have shown that one can obtain lazy strategies (where time is split into phases, and a different stationary policy is followed in each phase) by means of relaxations as well. In particular, we have obtained a new strategy, which is similar in spirit to the one previously proposed by [@Yu], but with several advantages, including better scaling of the regret. The above two algorithms are based on the relative-function approach via reverse Poisson inequalities. Finally, using a different type of a relaxation, we have derived another algorithm for online MDPs, which relies on interior-point methods and belongs to the class of algorithms derived using the convex-analytic approach. The takeaway point is that our general technique of constructing relaxations after a stationarization step brings all of the existing methods under the same umbrella and paves the way toward constructing new algorithms for online MDPs. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported by NSF grant CCF-1017564 and by AFOSR grant FA9550-10-1-0390. The authors are grateful to Profs. Alexander Rakhlin and Karthik Sridharan for helping us construct the relaxation presented in Section \[sec:derivedick\]. Proof of Proposition \[pps:strategyextensive\] {#app:strategyextensive} ============================================== The agent’s closed-loop behavioral strategy $\bd{\gamma}$ is a tuple of mappings $\gamma_t : \cF^{t-1} \to \cP(\sU), 1 \le t \le T$; the environment’s open-loop behavior strategy $\bd{f}$ is a tuple of functions $(f_1,\ldots,f_T)$ in $\cF$. Thus, $$\begin{aligned} V(x) &= \inf_{\bd{\gamma}} \sup_{\bd{f}} \E^{\bd{\gamma},\bd{f}}_x \left[ \sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right] \\ &= \inf_{\gamma_1} \ldots \inf_{\gamma_T} \sup_{f_1} \ldots \sup_{f_T} \E^{\gamma_1,\ldots,\gamma_T,f_1,\ldots,f_T}_x \left[ \sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right].\end{aligned}$$ We start from the final step $T$ and proceed by backward induction. Assuming $\gamma_1, \ldots, \gamma_{T-1}$ were already chosen, we have $$\begin{aligned} & \inf_{\gamma_T}\sup_{f_1, \ldots, f_T} \E^{\gamma^{T-1}, \gamma_T, f^{T-1}, f_T}_x \left\{ \sum^{T-1}_{t=1}\left[f_t(X_t,U_t)\right] + f_T(X_T, U_T)- \Psi(f^T)\right\}\\ &= \inf_{\gamma_T}\sup_{f_1, \ldots, f_{T-1}} \sup_{f_T} \left\{ \E^{\gamma^{T-1}, f^{T-1}}_x \left(\sum^{T-1}_{t=1}\left[f_t(X_t,U_t)\right] \right)+ \E^{\gamma^{T-1}, \gamma_T, f^{T-1}, f_T}_x \left[f_T(X_T, U_T)- \Psi(f^T) \right] \right\}\\ &= \inf_{\gamma_T}\sup_{f_1, \ldots, f_{T-1}} \left\{ \E^{\gamma^{T-1}, f^{T-1}}_x \left(\sum^{T-1}_{t=1}\left[f_t(X_t,U_t)\right] \right)+ \sup_{f_T}\, \E^{\gamma^{T-1}, \gamma_T, f^{T-1}, f_T}_x \left[f_T(X_T, U_T)- \Psi(f^T) \right] \right\} \\ &= \sup_{f_1, \ldots, f_{T-1}} \left\{ \E^{\gamma^{T-1}, f^{T-1}}_x \left(\sum^{T-1}_{t=1}\left[f_t(X_t,U_t)\right] \right)+ \inf_{P_T(U_T | X_T)}\sup_{f_T}\, \E^{\gamma^{T-1}, \gamma_T, f^{T-1}, f_T}_x \left[f_T(X_T, U_T)- \Psi(f^T) \right] \right\}.\end{aligned}$$ The last step is due to the easily proved fact that, for any two sets $A,B$ and bounded functions $g_1 : A \to \Reals$, $g_2 : A \times B \to \Reals$, $$\inf_{\gamma: A \to B} \sup_a \left\{ g_1(a) + g_2(a, \gamma(a))\right\} = \sup_a \left[ g_1(a) + \inf_{b \in B} g_2(a,b)\right]$$ (see, e.g., Lemma 1.6.1 in [@Bertsekas]). Proceeding inductively in this way, we get . Proof of Proposition \[pps:admissbound1\] {#app:admissbound1} ========================================= The proof is by backward induction. Starting at time $T$ and using the admissibility condition , we write $$\begin{aligned} & \E^{\wh{\bd{\gamma}},\bd{f}}_x\left[\sum^T_{t=1}f_t(X_t,U_t)-\Psi(\bd{f})\right] \nonumber\\ & \le \E^{\wh{\bd{\gamma}},\bd{f}}_x\left[\sum^T_{t=1}f_t(X_t,U_t) + \wh{V}_T(X_{T+1},f^T)\right] \\ &= \E^{\wh{\bd{\gamma}},\bd{f}}_x\left[\sum^{T-1}_{t=1} f_t(X_t,U_t)\right] + \E^{\wh{\bd{\gamma}},\bd{f}}_x\left[f_T(X_T,U_T) + \wh{V}_T(X_{T+1},f^T)\right] \\ &=\E^{\wh{\bd{\gamma}},\bd{f}}_x\left[\sum^{T-1}_{t=1} f_t(X_t,U_t)\right] \nonumber\\ &\qquad + \sum_{x_T}\mu_T(x_T)\left\{ \sum_{u \in \sU}f_T(x_T,u)\left[\wh{\gamma}_T\big(x_T,f^{T-1}\big)\right](u) + \E\Big[\wh{V}_T(X_{T+1},f^T)\Big|x_T,\wh{\gamma}_T\big(x_T,f^{T-1}\big)\Big]\right\} \\ &\le \E^{\wh{\bd{\gamma}},\bd{f}}_x\left[\sum^{T-1}_{t=1} f_t(X_t,U_t) + \wh{V}_{T-1}(X_T,f^{T-1})\right], \end{aligned}$$ where $\mu_T \in \cP(\sX)$ denotes the probability distribution of $X_T$. The last inequality is due to the fact that $\wh{\bd{\gamma}}$ is the behavioral strategy associated to the admissible relaxation $\{\wh{V}_t\}^T_{t=0}$. Continuing in this manner, we complete the proof. Proof of Lemma \[lem:comparisonch3\] {#app:comparisonch3} ==================================== Let us take expectations of both sides of w.r.t. $\pi_{P'} \otimes P'$: $$\begin{aligned} &\ave{\pi_P \otimes P, g} - \ave{\pi_{P'} \otimes P', g} \le \E_{\pi_{P'} \otimes P'}\Big\{ \E[\wh{Q}(Y,P)|X,U] - \wh{Q}(X,U) \Big\} \\ &\qquad = \sum_{x,u} \pi_{P'}(x) P'(u|x) \Big\{ \E[\wh{Q}(Y,P)|x,u] - \wh{Q}(x,u) \Big\} \\ &\qquad =\sum_{x,u} \pi_{P'}(x) P'(u|x) \E[\wh{Q}(Y,P)|x,u] - \sum_{x,u} \left(\sum_y \pi_{P'}(y)K(x|y,P')\right) P'(u|x)\wh{Q}(x,u) \end{aligned}$$ where in the third step we have used the fact that $\pi_{P'}$ is invariant w.r.t. $K(\cdot|\cdot,P')$. Then we have $$\begin{aligned} &\sum_{x,u} \pi_{P'}(x) P'(u|x) \E[\wh{Q}(Y,P)|x,u] - \sum_{x,u} \left(\sum_y \pi_{P'}(y)K(x|y,P')\right) P'(u|x)\wh{Q}(x,u) \\ &\qquad = \sum_{x}\pi_{P'}(x) \left\{ \sum_u P'(u|x)\E[\wh{Q}(Y,P)|x,u] - \sum_{u,y} K(y|x,P') P'(u|y)\wh{Q}(y,u) \right\} \\ &\qquad = \sum_x \pi_{P'}(x) \left\{ \sum_u P'(u|x)\E[\wh{Q}(Y,P)|x,u] - \sum_y K(y|x,P') \wh{Q}(y,P') \right\}, \end{aligned}$$ where the second step is by definition of $\wh{Q}(y,P')$. Then we can write $$\begin{aligned} &\sum_x \pi_{P'}(x) \left\{ \sum_u P'(u|x)\E[\wh{Q}(Y,P)|x,u] - \sum_y K(y|x,P') \wh{Q}(y,P') \right\} \\ &\qquad \stackrel{{\rm (a)}}{=} \sum_x \pi_{P'}(x) \left\{ \sum_u P'(u|x)\left(\E[\wh{Q}(Y,P)|x,u] - \sum_y K(y|x,u)\wh{Q}(y,P')\right) \right\} \\ &\qquad = \sum_{x,u} \pi_{P'}(x)P'(u|x) \Big\{\E[\wh{Q}(Y,P)|x,u] - \E[\wh{Q}(Y,P')|x,u]\Big\} \\ &\qquad \stackrel{{\rm (b)}}{=} \sum_{x,u,y} \pi_{P'}(x)P'(u|x)K(y|x,u) \left\{ \sum_{u'} P(u'|y)\wh{Q}(y,u') - \sum_{u'} P'(u'|y)\wh{Q}(y,u')\right\} \\ &\qquad \stackrel{{\rm (c)}}{=} \sum_{x,y} \pi_{P'}(x) K(y|x,P') \left\{ \sum_{u'} P(u'|y)\wh{Q}(y,u') - \sum_{u'}P'(u'|y)\wh{Q}(y,u')\right\} \\ &\qquad \stackrel{{\rm (d)}}{=} \sum_x \pi_{P'}(x) \sum_u \left[P(u|x)\wh{Q}(x,u) - P'(u|x)\wh{Q}(x,u) \right],\end{aligned}$$ where (a) and (c) are by definition of $K(\cdot|\cdot,P')$; (b) is by definition of $\wh{Q}(y,P')$; and in (d) we use the fact that $\pi_{P'}$ is invariant w.r.t. $K(\cdot|\cdot,P')$. Proof of Theorem \[thm:main\_ch3\] {#app:main_ch3} ================================== We have $$\begin{aligned} & \E^{\wh {\bd \gamma}, \bd f}_x \left[ \sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd f)\right]\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum^T_{t=1} \left[ \ave{\pi^{\wh {\bd \gamma}, \bd f}_t \otimes P^{\wh {\bd \gamma}, \bd f}_t,f_t} - \ave{\pi_P \otimes P, f_t}\right] + \sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1 \\ &\le \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \sum^T_{t=1} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_t(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) - P(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u)\right) + \sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1,\end{aligned}$$ where in the first equality we have used , while the second inequality is by Lemma \[lem:comparisonch3\]. Then we write the last term out and get $$\begin{aligned} & \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \left[\sum^{T-1}_{t=1} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_t(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \right) + \sum_u P^{\wh {\bd \gamma}, \bd f}_T(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_T(x,u)- \sum^T_{t=1}P(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \right]\\ &\quad+ \sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \left[\sum^{T-1}_{t=1} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_t(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \right) + \sum_u P^{\wh {\bd \gamma}, \bd f}_T(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_T(x,u)+\wh W_{x,T} (h^T_x) \right] \\ &\quad+ \sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \left[\sum^{T-1}_{t=1} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_t(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \right) + \wh W_{x,T-1} (h^{T-1}_x) \right] +\sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1,\end{aligned}$$ where the two inequalities are by the admissibility condition . Continuing this induction backward, and noting that $\sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1 \le C_{\cF} \sum^T_{t=1} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1$, we arrive at . Proof of Theorem \[thm:main\_2nd\] {#app:main_2nd} ================================== Applying the same backward induction used in the proof of Proposition \[pps:admissbound1\] (also see [@RakhlinRL Prop. 1]), it is easy to show that $$\sum^T_{t=1}\ave{\nu_t,f_t} - \inf_{\nu \in {\mathcal G}'} \sum^T_{t=1} \ave{\nu,f_t} \le \wh{V}_T({\cal G}' | {\mathsf e}).$$ Then it is straightforward to see that $$\begin{aligned} & \E^{\wh {\bd \gamma}, \bd f}_x \left\{ \sum^T_{t=1}f_t(X_t,U_t) - \inf_{P \in \cM({\cal G}')}\E\left[\sum^T_{t=1}f_t(X,U)\right]\right\}\\ &\le \sum^T_{t=1} \left[ \ave{\pi^{\wh {\bd \gamma}, \bd f}_t \otimes P^{\wh {\bd \gamma}, \bd f}_t,f_t} - \inf_{P \in \cM({\cal G}')} \ave{\pi_P \otimes P, f_t}\right] + \sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1 \\ &\le \sum^T_{t=1}\ave{\nu_t,f_t} - \inf_{\nu \in {\cal G}'} \sum^T_{t=1} \ave{\nu,f_t} + \sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1,\end{aligned}$$ where in the first equality we have used . Proof of Proposition \[pps:recoverevendar\] {#app:recoverevendar} =========================================== First we show that the relaxation arises as an upper bound on the conditional sequential Rademacher complexity. The proof of this is similar to the one given by [@RakhlinOR], except that they also optimize over the choice of the learning rate $\rho$. For any $\rho > 0$, $$\begin{aligned} & \E_{\eps} \left[ \max_{u \in \sU} \left\{ 2\sum^{T-t}_{i=1}\eps_i \left[\mathbf h_{i}(\eps)\right](u) - \sum^t_{s=1} h_{x,s}(u) \right\} \right]\\ &\le \rho \log \left( \E_{\eps} \left[ \max_{u \in \sU} \exp\left( \frac{2}{\rho} \sum^{T-t}_{i=1}\eps_i \left[\mathbf h_{i}(\eps)\right](u) - \frac{1}{\rho} \sum^t_{s=1} h_{x,s}(u) \right) \right]\right)\\ &\le \rho \log \left( \E_{\eps} \left[ \sum_{u \in \sU} \exp\left( \frac{2}{\rho} \sum^{T-t}_{i=1}\eps_i \left[\mathbf h_{i}(\eps)\right](u) - \frac{1}{\rho} \sum^t_{s=1} h_{x,s}(u) \right) \right]\right),\end{aligned}$$ where the first inequality is by Jensen’s inequality, while the second inequality is due to the non-negativity of exponential function. Then we pull out the second term inside the expectation $ \E_{\eps}$ and get $$\begin{aligned} & \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^t_{s=1} h_{x,s}(u)\right)\E_{\eps} \left[ \prod^{T-t}_{i=1} \exp\left( \frac{2}{\rho} \eps_i \left[\mathbf h_{i}(\eps)\right](u) \right) \right]\right)\\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^t_{s=1} h_{x,s}(u)\right)\times \exp\left( \frac{2}{\rho^2} \max_{\eps_1, \ldots, \eps_{T-t} \in \{\pm1\}}\sum^{T-t}_{i=1} \big(\left[\mathbf h_{i}(\eps)\right](u) \big)^2 \right) \right)\\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^t_{s=1} h_{x,s}(u)\right)\max_{u} \exp\left( \frac{2}{\rho^2} \max_{\eps_1, \ldots, \eps_{T-t} \in \{\pm1\}}\sum^{T-t}_{i=1} \big(\left[\mathbf h_{i}(\eps)\right](u)\big)^2 \right) \right) \\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^t_{s=1} h_{x,s}(u)\right)\right) +\frac{2}{\rho} \sup_{\mathbf h}\max_{u \in \sU} \max_{\eps_1, \ldots, \eps_{T-t} \in \{\pm1\}}\sum^{T-t}_{i=1} \big([\mathbf h_{i}(\eps)](u) \big)^2,\end{aligned}$$ where the first inequality is due to Hoeffding’s lemma (see, e.g., Lemma A.1 in [@PLG]) applied to the expectation w.r.t. $\eps$. The last term, representing the worst-case future, is upper bounded by $\frac{2}{\rho}(T-t) L(\sX,\sU, \cF)^2$. We thus obtain our exponential weight relaxation from . Next we prove that the relaxation is admissible and leads to the recursive algorithm . To keep the notation simple, we drop the subscript $x$ in the following. In particular, we use $h_t$ for $h_{x,t}$, $\wh{W}_t$ for $\wh{W}_{x,t}$, $\nu_t$ for $P_t(\cdot|x)$, etc. The admissibility condition to be proved is $$\begin{aligned} \sup_{h_{t} \in \cH_x} \left\{\E_{U \sim \nu_t} \left[ h_{t}(U) \right] + \wh W_{t} (h^t)\right\} \le \wh W_{t-1}(h^{t-1}).\end{aligned}$$ Note that $$\Ave{\nu_t,\exp\left(- \frac{1}{\rho} h_t\right)} = \sum_{u \in \sU} \frac{ \nu_1(u) \exp\left(-\frac{1}{\rho}\sum^{t-1}_{s=1}h_s(u)\right)}{\Ave{\nu_1, \exp\left(-\frac{1}{\rho}\sum^{t-1}_{s=1}h_s\right)}} \exp\left(- \frac{1}{\rho} h_t(u)\right) = \frac{\Ave{\nu_1, \exp\left(-\frac{1}{\rho}\sum^{t}_{s=1}h_s\right)}}{\Ave{\nu_1, \exp\left(-\frac{1}{\rho}\sum^{t-1}_{s=1}h_s\right)}}.$$ We have $$\begin{aligned} &\rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^t_{s=1} h_s(u)\right)\right) \\ &= \rho \log \Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^t_{s=1} h_s\right)} + \rho \log | \sU |\\ &= \rho \log \Ave{\nu_t,\exp\left(- \frac{1}{\rho} h_t\right)} + \rho \log \Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s\right)}+ \rho \log | \sU | \\ &\le -\E_{U \sim \nu_t}h_t(U) + \frac{L(\sX,\sU, \cF)^2}{2\rho} + \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s(u)\right)\right),\end{aligned}$$ where the first equality is due to the fact that $\nu_1$ is the uniform distribution on $\sU$, while the inequality is due to Hoeffding’s lemma. Plugging the resulting bound into the admissibility condition, we get $$\begin{aligned} &\sup_{h_t \in \cH_x} \left\{\E_{U \sim \nu_t} \left[ h_t(U) \right] + \wh W_{x,t} (h^t)\right\} \\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s(u)\right)\right) + 2\frac{1}{\rho}(T-t+1) L(\sX,\sU, \cF)^2\\ &=\wh W_{x,t-1}(h^{t-1}).\end{aligned}$$ Thus, the recursive algorithm is admissible for the relaxation . Proof of Theorem \[thm:evendarbound\] {#app:evendarbound} ===================================== Again, we drop the subscript $x$ and write $\nu_t$ for $P_t(\cdot|x)$, etc. We have $$\begin{aligned} \label{eq:evendarregret_repeated} \E^{\wh{\bd{\gamma}},\bd{f}}_x\left[\sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right] &\le \sup_{P \in \cM(\sU|\sX)}\sum_{x}\pi_P(x) \wh{W}_{x,0} + C_\cF \sum^T_{t=1} \| \mu^{\wh{\bd{\gamma}},\bd{f}}_t - \pi^{\wh{\bd{\gamma}},\bd{f}}_t \|_1.\end{aligned}$$ From the relaxation , it is easy to see $\wh{W}_{x,0} \le 2L\sqrt{2 T \log |\sU|}$ for all states $x$ (in fact, the bound is met with equality with the optimal choice of $\rho = \sqrt{\frac{2TL^2}{\log|\sU|}}$). Since we have bounded the first term, now we focus on bounding the second term of the regret bound. The relative entropy between $\nu_t$ and $\nu_{t-1}$ is given by $$\begin{aligned} D(\nu_t \| \nu_{t-1})&= \left\langle\nu_t, \log \frac{\exp\Big(-\frac{1}{\rho}\sum^{t-1}_{s=1} h_s\Big)}{\exp\Big(-\frac{1}{\rho}\sum^{t-2}_{s=1} h_s\Big)}\right\rangle + \log \frac{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-2}_{s=1} h_s\right)}}{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s\right)}} \nonumber \\ &= -\frac{1}{\rho} \Ave{\nu_t, h_{t-1}} +\log \frac{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-2}_{s=1} h_s\right)}}{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s\right)}}, \label{eq:recursivedistance}\end{aligned}$$ where $$\begin{aligned} \frac{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-2}_{s=1} h_s\right)}}{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s\right)}}&= \frac{\displaystyle\sum_{u \in \sU}\nu_1(u) \exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s(u)\right) \exp\left(\frac{1}{\rho}h_{t-1}(u)\right)}{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s\right)}}\\ &= \Ave{\nu_t,\exp\left(\frac{1}{\rho}h_{t-1}\right)}.\end{aligned}$$ Using Hoeffding’s lemma, we can write $$\begin{aligned} \log \frac{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-2}_{s=1} h_s\right)}}{\Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{t-1}_{s=1} h_s\right)}} \le \frac{1}{\rho} \Ave{\nu_t,h_{t-1}} + \frac{L^2}{2\rho^2}\end{aligned}$$ Substituting this bound into , we see that the terms involving the expectation of $h_{t-1}$ w.r.t. $\nu_t$ cancel, and we are left with $$\begin{aligned} D(\nu_t \| \nu_{t-1}) \le \frac{L^2}{2\rho^2}.\end{aligned}$$ Plugging in the optimal value of $\rho$ and using Pinsker’s inequality [@CoverThomas], we find $$\begin{aligned} \| \nu_t - \nu_{t-1}\|_1 \le \sqrt{\frac{\log |\sU|}{2T}}.\end{aligned}$$ So far, we have been working with a fixed state $x \in \sX$, so we had $\nu_t = P^{\wh{\bd{\gamma}},\bd{f}}_t(\cdot|x)$, where $\wh{\bd{\gamma}}$ is the agent’s behavioral strategy induced by the relaxation . Since $x$ was arbitrary, we get the uniform bound $$\begin{aligned} \label{eq:policy_difference} \max_{x \in \sX} \left\| P^{\wh{\bd{\gamma}},\bd{f}}_t(\cdot|x) - P^{\wh{\bd{\gamma}},\bd{f}}_{t-1}(\cdot|x) \right\|_1 \le \sqrt{\frac{\log |\sU|}{2T}}.\end{aligned}$$ Armed with this estimate, we now bound the total variation distance between the actual state distribution at time $t$ and the unique invariant distribution of $K^{\wh{\bd{\gamma}},\bd{f}}_t$. For any time $k \le t$, we have $$\begin{aligned} \left\| \mu^{\wh {\bd \gamma}, \bd f}_k - \pi^{\wh {\bd \gamma}, \bd f}_t \right\|_1&= \left\| \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{k-1} - \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{t} + \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{t}- \pi^{\wh {\bd \gamma}, \bd f}_t\right \|_1 \nonumber\\ &\stackrel{{\rm (a)}}{\le} \left \| \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{t}- \pi^{\wh {\bd \gamma}, \bd f}_t \right\|_1 +\ \left\| \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{k-1} - \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{t} \right\|_1 \nonumber\\ &\stackrel{{\rm (b)}}{=}\left\| \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{t}- \pi^{\wh {\bd \gamma}, \bd f}_t K^{\wh {\bd \gamma}, \bd f}_{t} \right\|_1+ \left\| \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{k-1} - \mu^{\wh {\bd \gamma}, \bd f}_{k-1} K^{\wh {\bd \gamma}, \bd f}_{t} \right\|_1 \nonumber\\ &\stackrel{{\rm (c)}}{\le} e^{-1/\tau} \left\| \mu^{\wh {\bd \gamma}, \bd f}_{k-1} - \pi^{\wh {\bd \gamma}, \bd f}_t \right\|_1+ \max_{x \in \sX}\left\| P^{\wh{\bd{\gamma}},\bd{f}}_{k-1}(\cdot|x) - P^{\wh{\bd{\gamma}}}_t(\cdot|x) \right\|_1 \nonumber\\ &\stackrel{{\rm (d)}}{\le} e^{-1/\tau}\left\| \mu^{\wh {\bd \gamma}, \bd f}_{k-1} - \pi^{\wh {\bd \gamma}, \bd f}_t \right\|_1+ \sum^{t-1}_{i=k-1}\sqrt{\frac{\log |\sU|}{2T}}, \label{eq:k_to_t}\end{aligned}$$ where (a) is by triangle inequality; (b) is by invariance of $\pi^{\wh{\bd\gamma},\bd{f}}_t$ w.r.t. $K^{\wh{\bd\gamma},{\bd f}}_t$; (c) is by the uniform mixing bound ; and (d) follows from repeatedly using together with triangle inequality and the easily proved fact that, for any state distribution $\mu \in \cP(\sX)$ and any two Markov kernels $P,P' \in \cM(\sU|\sX)$, $$\begin{aligned} \left\| \mu K(\cdot|P) - \mu K(\cdot|P') \right\|_1 \le \max_{x \in \sX} \left\| P(\cdot|x) - P'(\cdot|x) \right\|_1.\end{aligned}$$ Letting now the initial state distribution be $\mu_1$, we can apply the bound recursively to obtain $$\begin{aligned} \left\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \right\|_1 &\le e^{-(t-1)/\tau}\left\| \mu_1 - \pi^{\wh {\bd \gamma}, \bd f}_t \right\|_1+ \sum^t_{k=2} e^{-\frac{t-k}{\tau}}\sum^{t}_{i=k-1}\sqrt{\frac{\log |\sU|}{2T}}\\ &\le 2 e^{-(t-1)/\tau} + \sum^t_{k=2} e^{-\frac{t-k}{\tau}}(t-k+1)\sqrt{\frac{\log |\sU|}{2T}} \\ &\le 2 e^{-(t-1)/\tau} + \sqrt{\frac{\log |\sU|}{2T}}\sum^{\infty}_{k=0} (k+1) e^{-\frac{k}{\tau}} \\ &\le 2 e^{-(t-1)/\tau} + (\tau+1)^2\sqrt{\frac{\log |\sU|}{2T}}.\end{aligned}$$ So, the second term on the right-hand side of can bounded by $$\begin{aligned} C_{\cF} \sum^T_{t=1} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1 \le C_{\cF}(\tau+1)^2 \sqrt{\frac{\log|\sU|T}{2}} + (2\tau+2) C_{\cF},\end{aligned}$$ which completes the proof. Proof of Proposition \[pps:recoveryu\] {#app:recoveryu} ====================================== First we show that the relaxation arises as an upper bound on the conditional sequential Rademacher complexity. Once again, we omit the subscript $x$ from $h_{x,t}$ etc. to keep the notation light. Following the same steps as in Appendix \[app:recoverevendar\], we have, for any $\rho > 0$, $$\begin{aligned} & \E_{\eps} \left[ \max_{u \in \sU} \left\{ 2\sum^M_{j=m+1}\eps_j \sum_{t \in \cT_j}\left[\mathbf h_{t}(\eps)\right](u) - \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u) \right\} \right]\\ &\le \rho \log \left( \E_{\eps} \left[ \max_{u \in \sU} \exp\left( \frac{2}{\rho} \sum^M_{j=m+1}\eps_j \sum_{t \in \cT_j}\left[\mathbf h_{t}(\eps)\right](u) - \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u) \right) \right]\right)\\ &\le \rho \log \left( \E_{\eps} \left[ \sum_{u \in \sU} \exp\left( \frac{2}{\rho} \sum^M_{j=m+1}\eps_j \sum_{t \in \cT_j}\left[\mathbf h_{t}(\eps)\right](u) - \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u) \right) \right]\right).\end{aligned}$$ In the same vein, $$\begin{aligned} &\rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho}\sum^m_{i=1} \sum_{s \in \cT_i} h_s(u)\right)\E_{\eps} \left[ \prod^{M}_{j=m+1} \exp\left( \frac{2}{\rho} \eps_j \sum_{t \in \cT_j}\left[\mathbf h_{t}(\eps)\right](u) \right) \right]\right)\\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u)\right)\times \exp\left( \frac{2}{\rho^2} \max_{\eps_{m+1}, \ldots, \eps_{M} \in \{\pm1\}}\sum^{M}_{j=m+1} \left(\tau_j \left[\mathbf h(\eps)\right](u) \right)^2 \right) \right)\\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u)\right)\max_{u} \exp\left( \frac{2}{\rho^2} \max_{\eps_{m+1}, \ldots, \eps_{M} \in \{\pm1\}}\sum^{M}_{j=m+1} \left(\tau_j \left[\mathbf h(\eps\right](u) \right)^2 \right) \right) \\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u)\right)\right) +\frac{2}{\rho} \sup_{\mathbf h}\max_{u \in \sU} \max_{\eps_{m+1}, \ldots, \eps_{M} \in \{\pm1\}}\sum^{M}_{j=m+1} \left(\tau_j \left[\mathbf h(\eps)\right](u) \right)^2 \\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u)\right)\right) +\frac{2}{\rho} \sum^{M}_{j=m+1} \tau_j^2 L(\sX,\sU, \cF)^2,\end{aligned}$$ where the first inequality is due to Hoeffding’s lemma, while the last inequality is by Assumption \[as:existboundQ\]. We thus derive the relaxation in . Now we prove that this relaxation is admissible, and leads to the lazy algorithm The admissibility condition to be proved is $$\begin{aligned} \sup_{h_m \in \cH^{\tau_m}_x} \left\{\E_{U \sim \nu_m} \left[ \sum_{s \in \cT_m}h_s(U) \right] + \wh W_{x,m} (h_{1:m})\right\} \le \wh W_{x,m-1}(h_{1:m-1}),\end{aligned}$$ where $\nu_m = P_{m}(\cdot|x)$ is the Markov policy used in phase $m$. We have $$\begin{aligned} &\rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s(u)\right)\right) \\ &= \rho \log \Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^m_{i=1} \sum_{s \in \cT_i} h_s\right)} + \rho \log | \sU |\\ &= \rho \log \Ave{\nu_m,\exp\left(- \frac{1}{\rho} \sum_{s \in \cT_m}h_s\right)} + \rho \log \Ave{\nu_1, \exp\left(- \frac{1}{\rho} \sum^{m-1}_{i=1}\sum_{s \in \cT_i} h_s\right)}+ \rho \log | \sU | \\ &\le -\E_{U \sim \nu_m} \left[ \sum_{s \in \cT_m}h_s(U) \right] + \frac{\tau_m^2L(\sX,\sU, \cF)^2}{2\rho} + \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^{m-1}_{i=1}\sum_{s \in \cT_i} h_s(u)\right)\right),\end{aligned}$$ Plugging this into the admissibility condition, we have $$\begin{aligned} &\sup_{h_m \in \cH^{\tau_m}_x} \left\{\E_{U \sim \nu_m} \left[ \sum_{s \in \cT_m}h_s(U) \right] + \wh W_{x,m} (h_{1:m})\right\} \\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^{m-1}_{i=1}\sum_{s \in \cT_i} h_s(u)\right)\right) + \frac{2}{\rho} \sum^{M}_{j=m+1} \tau_j^2 L(\sX,\sU, \cF)^2 + \frac{\tau_m^2L(\sX,\sU, \cF)^2}{2\rho}\\ &\le \rho \log \left( \sum_{u \in \sU}\exp\left(- \frac{1}{\rho} \sum^{m-1}_{i=1}\sum_{s \in \cT_i} h_s(u)\right)\right) + \frac{2}{\rho} \sum^{M}_{j=m} \tau_j^2 L(\sX,\sU, \cF)^2\\ &=\wh W_{x,m-1}(h^{m-1}).\end{aligned}$$ So the lazy algorithm is an admissible strategy for the relaxation . Proof of Theorem \[thm:yubound\] {#app:yubound} ================================ The state feedback law $P^{\wh{\bd{\gamma}},\bd{f}}_t(\cdot|x)$ that the agent applies within phase $m$ is the same for all $t \in \cT_m$, and we denote it by $P^{\wh{\bd{\gamma}},\bd{f}}_m(\cdot|x)$. Let $K^{\wh {\bd \gamma},\bd{f}}_m$ denote the Markov matrix that describes the state transition from $X_t$ to $X_{t+1}$ if $t \in \cT_m$. Thus, we can write $$\begin{aligned} K^{\wh {\bd \gamma},\bd{f}}_m(y|x) = \sum_u K(y|x,u) P^{\wh{\bd{\gamma}},\bd{f}}_t(u|x), \qquad \forall x,y \in \sX.\end{aligned}$$ First, we show that $$\begin{aligned} \label{eq:regretboundinphase} \E^{\wh {\bd \gamma}, \bd f}_x \left[ \sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd f)\right] \le \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \wh W_{x,0} + C_{\cF} \sum^M_{m=1} \sum_{t \in \cT_m}\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_m \|_1,\end{aligned}$$ where $\pi^{\wh {\bd \gamma}, \bd f}_m$ is the invariant distribution of $K^{\wh {\bd \gamma},\bd{f}}_m$. To prove , we write $$\begin{aligned} & \E^{\wh {\bd \gamma}, \bd f}_x \left[ \sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd f)\right]\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum^T_{t=1} \left[ \ave{\pi^{\wh {\bd \gamma}, \bd f}_t \otimes P^{\wh {\bd \gamma}, \bd f}_t,f_t} - \ave{\pi_P \otimes P, f_t}\right] + \sum^T_{t=1} \| f_t \|_{\infty} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum^M_{m=1} \sum_{t \in \cT_m}\left[ \ave{\pi^{\wh {\bd \gamma}, \bd f}_t \otimes P^{\wh {\bd \gamma}, \bd f}_t,f_t} - \ave{\pi_P \otimes P, f_t}\right] + C_{\cF} \sum^M_{m=1} \sum_{t \in \cT_m}\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_m \|_1\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \sum^M_{m=1} \sum_{t \in \cT_m} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_m(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) - P(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u)\right) + C_{\cF} \sum^M_{m=1} \sum_{t \in \cT_m}\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_m \|_1,\end{aligned}$$ where the last inequality is by Lemma \[lem:comparisonch3\]. By writing out the first term in the right hand side, we get $$\begin{aligned} & \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \Bigg[\sum^{M-1}_{m=1} \sum_{t \in \cT_m} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_m(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \right) + \sum_u \nu_M(u|x) \sum_{t \in \cT_M}\wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \\ &\quad- \sum^T_{t=1}P(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \Bigg]+ C_{\cF} \sum^M_{m=1} \sum_{t \in \cT_m}\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_m \|_1\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \left[\sum^{M-1}_{m=1} \sum_{t \in \cT_m} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_m(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \right) + \sum_u \nu_M(u|x) \sum_{t \in \cT_M}\wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u)+\wh W_{x,M} (h^M) \right]\\ &\quad+ C_{\cF} \sum^M_{m=1} \sum_{t \in \cT_m}\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_m \|_1\\ &\le \sup_{P \in \cM(\sU | \sX)} \sum_x \pi_P(x) \left[\sum^{M-1}_{m=1} \sum_{t \in \cT_m} \left( \sum_u P^{\wh {\bd \gamma}, \bd f}_m(u|x) \wh Q^{\wh {\bd \gamma}, \bd f}_t(x,u) \right) + \wh W_{x,M-1} (h^{M-1}) \right] \\ & \qquad \qquad \qquad +\sum^T_{t=1} \| f_t \|_{\infty} \left\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \right\|_1.\end{aligned}$$ The last inequality is due to the fact that $\wh{\bd{\gamma}}$ is the behavioral strategy associated to the admissible relaxation $\{\wh W_{x,m}\}^M_{m=1}$. Continuing this induction backwards, we arrive at . Next, we bound the two terms on the right-hand side of . From the form of the relaxation , it is easy to see $\wh{W}_{x,0} \le 2L\sqrt{2 \log |\sU| \sum^M_{i=1}\tau_i^2}$ for all states $x$; in fact, this bound is attained with equality if we use the optimal choice $\rho = \sqrt{\frac{2\sum^M_{i=1}\tau_i^2L^2}{\log|\sU|}}$. Since we have bounded the first term, now we focus on bounding the second term of . From the contraction inequality it follows that, for every $k \in \{0,1,\ldots, \tau_m-1\}$, we have $$\begin{aligned} \left\| \mu^{\wh {\bd \gamma}, \bd f}_{\tau_{1:m-1}+k+1} - \pi^{\wh {\bd \gamma}, \bd f}_m \right\|_1 &= \left\| \mu^{\wh {\bd \gamma}, \bd f}_{\tau_{1:m-1}+1} {(K^{\wh {\bd \gamma},\bd{f}}_m)}^k- \pi^{\wh {\bd \gamma}, \bd f}_m {(K^{\wh {\bd \gamma},\bd{f}}_m)}^k \right\|_1 \\ &\le e^{-k/\tau} \left\| \mu^{\wh {\bd \gamma}, \bd f}_{\tau_{1:m-1}+1} - \pi^{\wh {\bd \gamma}, \bd f}_m \right\|_1 \\ &\le 2 e^{-k/\tau}.\end{aligned}$$ Hence, $$\begin{aligned} \sum_{t \in \cT_m}\left\| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_m \right\|_1 \le 2\displaystyle \sum_{k=0}^{\tau_m -1} e^{-k/\tau} \le \frac {2}{1-e^{-1/\tau}}.\end{aligned}$$ Plugging it in , we have shown that $$\begin{aligned} \E^{\wh{\bd{\gamma}},\bd{f}}_x \left[\sum^T_{t=1}f_t(X_t,U_t) - \Psi(\bd{f})\right] \le 2 L\sqrt{2 \log |\sU| \sum^M_{i=1}\tau_i^2} + \frac {2C_\cF M}{1-e^{-1/\tau}}.\end{aligned}$$ Proof of Corollary \[cor:optbound\] {#app:optbound} =================================== Let us inspect the right-hand side of . We see that both $\sqrt{\sum^M_{j=1}\tau^2_j}$ and $M$ have to be sublinear in $T$. Since $\sum^M_{i=1}\tau_i = T$ and $\sqrt{\sum^M_{i=1}\tau_i^2} < \sqrt{(\sum^M_{i=1}\tau_i)^2}$, at least the first of these terms can be made sublinear, e.g., by having $\tau_j=1$ for all $j$. Of course, this means that $M=T$, so we need longer phases. For example, if we follow [@Yu] and let $\tau_m = \lceil m^{1/3-\eps} \rceil$ for some $\eps \in (0,1/3)$, then a straightforward if tedious algebraic calculation shows that $M = O(T^{3/4})$ and $\sqrt{\sum^M_{j=1}\tau^2_j} = O(T^{5/8})$, which yields the regret of $O(T^{3/4})$. However, if $T$ is known in advance, then we can do better: ignoring the rounding issues, for any constants $A_1,A_2 > 0$, $$\begin{aligned} \label{eq:best_phases} \min_{1 \le M \le T}\min \left\{ A_1 \sqrt{\sum^M_{j=1}\tau^2_j} + A_2 M : \sum^M_{j=1}\tau_j = T\right\} = O(T^{2/3}),\end{aligned}$$ To see this, let us first fix $M$ and optimize the choice of the $\tau_j$’s: $$\min \sum^M_{j=1} \tau_j^2 \quad \text{subject to } \sum^M_{j=1} \tau_j = T.$$ By the Cauchy–Schwarz inequality, we have $$\sum^M_{j=1} \tau_j \le \sqrt{M\sum^M_{j=1} \tau_j^2}.$$ Thus, $\sum^M_{j=1} \tau_j^2$ achieves its minimum when the above bound is met with equality. This will happen only if all the $\tau_j$’s are equal, i.e., $\tau_j = \frac{T}{M}$ for every $j$ (for simplicity, we assume that $M$ divides $T$ — otherwise, the remainder term will be strictly smaller than $M$, and the bound in will still hold, but with a larger multiplicative constant). Therefore, $$\begin{aligned} \min_{1 \le M \le T}\min \left\{ A_1 \sqrt{\sum^M_{j=1}\tau^2_j} + A_2 M : \sum^M_{j=1}\tau_j = T\right\} = \min_{1 \le M \le T} \left( \frac{A_1T}{\sqrt{M}} + A_2 M\right) = O(T^{2/3}),\end{aligned}$$ where the minimum on the right-hand side (again, ignoring rounding issues) is achieved by $M = T^{2/3}$ and $\tau_j = T^{1/3}$ for all $j$. This shows that, for a given horizon $T$, the optimal choice of phase lengths is $T^{1/3}$, which gives the regret of $O(T^{2/3})$, better than the $O(T^{3/4})$ bound derived by [@Yu]. Proof of Proposition \[pps:OMDrelaxadmin\] {#app:OMDrelaxadmin} ========================================== First, we check the admissibility condition at time $t=T$. Since the Bregman divergence is nonnegative, we have $$\begin{aligned} \wh{V}_T({\cal G}'|f_1,\ldots,f_T) = & \sup_{\mu \in {\cal G}'} \left\{\sum^T_{s=1} \ave{\mu,-f_s} + \frac{1}{\rho} D_{F} (\mu, \nu_{T+1})\right\} \\ & \ge -\inf_{\mu \in {\cal G}'} \Ave{\mu, \sum^T_{s=1}f_s}.\end{aligned}$$ Now let us consider an arbitrary $t$. From the construction of our relaxation, we have $$\begin{aligned} &\sup_{f_t \in \cF} \left\{ \ave{\nu_t, f_t} + \wh{V}_T({\cal G}|f_1,\ldots,f_t) \right\} \\ &= \sup_{f_t \in \cF} \sup_{\mu \in {\cal G}'} \left\{ \sum^{t-1}_{s=1} \ave{\mu,-f_s} + \ave{\nu_t - \mu, f_t} + \frac{1}{\rho} D_{F} (\mu, \nu_{t+1}) + \frac{1}{2\rho} (T-t) \right\}\end{aligned}$$ for all $t = 1,\ldots,T$. From the definition of the Bregman divergence, the following equality holds for any three $\mu,\nu,\lambda \in {\mathcal G}$: $$\begin{aligned} \label{eq:Bregman_triple} D_{F}(\mu,\nu) + D_{F}(\nu,\lambda)= D_{F}(\mu,\lambda) + \Ave{\nabla F(\lambda)-\nabla F(\nu),\mu - \nu}.\end{aligned}$$ Since $\nabla F$ and $\nabla F^*$ are inverses of each other, we have $-\rho f_t = \nabla F(\nu_{t+1}) - \nabla F(\nu_t)$. Using this fact together with , for any $\mu \in {\mathcal G}$ we can write $$\begin{aligned} \Ave{\nu_t - \mu, \rho f_t} &= \Ave{ \nabla F(\nu_{t+1}) - \nabla F(\nu_t), \mu- \nu_t} \nonumber\\ &= D_{F}(\mu,\nu_t) - D_{F}(\mu,\nu_{t+1}) + D_{F}(\nu_t, \nu_{t+1}). \label {eq:md1}\end{aligned}$$ Moreover, once again using the fact that $\nabla F$ and $\nabla F^*$ are inverses of one another, we have $$\begin{aligned} D_{F}(\nu_t, \nu_{t+1}) &= D_{F^*}(\nabla F(\nu_{t+1}),\nabla F(\nu_t) ) \nonumber \\ &= F^*(\nabla F(\nu_{t+1})) - F^*(\nabla F(\nu_{t})) - \Ave{\nabla F^*(\nabla F(\nu_{t})), \nabla F(\nu_{t+1}) - \nabla F(\nu_t)} \nonumber \\ &\le \Lambda \left( \rho \| f_t \|^*_{\nabla F(\nu_t)}\right), \label {eq:md2}\end{aligned}$$ where $$\Lambda(r) \deq -\log(1-r)-r = \frac{r^2}{2} + \frac{r^3}{3} + \frac{r^4}{4} + \ldots.$$ Note that, because of the definition of $\Lambda$, the learning rate $\rho$ must be chosen in such a way that $\rho \| f_t \|^*_{\nabla F(\mu_t)} < 1$ for all $t=1,\ldots,T$. By hypothesis, we have $\rho \| f_t \|^*_{\nabla F(\nu_t)} \le 1/2$ for all $t$. The first line of Eq. (\[eq:md2\]) is by Prop. 11.1 in [@PLG], the second is by definition of the Bregman divergence, and the third follows from from a local second-order Taylor formula for a self-concordant function [@NemTodd Eq. (2.5)] (which is applicable because, by hypothesis, $\rho \| f_t \|^*_{\nabla F(\nu_t)} \le 1/2 < 1$ for all $t$) and the fact that the Legendre–Fenchel dual of a self-concodrant function is also self-concordant. Using the inequality $\log r \le r-1$, we can upper-bound $$\Lambda(r) = -\log(1-r)-r \le \frac{1}{1-r}-1-r = \frac{1-(1-r)(1+r)}{1-r} = \frac{r^2}{1-r}.$$ Moreover, since $\rho \| f_t \|^*_{\nabla F(\nu_t)} \le 1/2$ and $\| f_t \|^*_{\nabla F(\nu_t)} \le 1$for all $t \in \{1,\ldots,T\}$ by hypothesis, we can further bound $$\Lambda\left(\rho \| f_t \|^*_{\nabla F(\nu_t)}\right) \le 2\rho^2 \| f_t \|^{*2}_{\nabla F(\nu_t)} \le 2\rho^2 .$$ Applying Eqs.  and , we arrive at $$\begin{aligned} &\sup_{f_t \in \cF} \left\{ \ave{\nu_t, f_t} + \wh{V}_T({\cal G}'|f_1,\ldots,f_t) \right\} \nonumber\\ &= \sup_{f_t \in \cF} \sup_{\mu \in {\cal G}'} \left\{ \sum^{t-1}_{s=1} \ave{\mu,-f_s} + \ave{\nu_t - \mu, f_t} + \frac{1}{\rho} D_{F} (\mu, \nu_{t+1}) + 2\rho (T-t) \right\} \nonumber\\ &\le \sup_{f_t \in \cF} \sup_{\mu \in {\cal G}'} \left\{ \sum^{t-1}_{s=1} \ave{\mu,-f_s} + \frac{1}{\rho} D_{F} (\mu, \nu_{t}) + \frac{1}{\rho} \Lambda \left( \rho \| f_t \|^*_{\nabla F(\nu_t)}\right) + 2\rho (T-t) \right\} \nonumber\\ &\le \sup_{f_t \in \cF} \sup_{\mu \in {\cal G}'} \left\{ \sum^{t-1}_{s=1} \ave{\mu,-f_s} + \frac{1}{\rho} D_{F} (\mu, \nu_{t}) + 2\rho (T-t+1) \right\} \nonumber\\ &=\wh{V}_T({\cal G}'|f_1,\ldots,f_{t-1}) \nonumber.\end{aligned}$$ This shows that the proposed algorithm (behavoiral strategy) is admissible, and the proof is complete. Proof of Theorem \[thm:OMDbound1\] {#app:OMDbound1} ================================== Since the relaxation is admissible by Proposition \[pps:OMDrelaxadmin\], we have $$\begin{aligned} \sum^T_{t=1} \ave{\nu_t,f_t} - \inf_{\nu \in {\cal G}'} \sum^T_{t=1} \ave{\nu,f_t} &\le \wh{V}_T({\cal G}' | {\mathsf e}) \nonumber\\ &= \sup_{\mu \in {\cal G}'} \left\{ \frac{D_F(\mu,\nu_1)}{\rho} + 2\rho T \right\} \nonumber\\ &= \frac{D_F({\cal G}')}{\rho} + 2\rho T \nonumber. \end{aligned}$$ Proof of Theorem \[thm:OMDbound2\] {#app:OMDbound2} ================================== Let us denote by $P_t = P_{\nu_t}$ the policy extracted from $\nu_t$, and the induced marginal distribution of $X_t$ by $\mu_t \in \cP(\sX)$. We also denote by $K_t$ the Markov matrix that describes the state transition from $X_t$ to $X_{t+1}$, and by $\pi_t \in \cP(\sX)$ its the unique invariant distribution. Finally, we denote by $\wh{\bd{\gamma}}$ the behavioral strategy corresponding to our algorithm. Then, for any $\bd{f} \in \cF^T$, we can upper-bound the regret by $$\begin{aligned} \label{eq:regretboundOMD} R^{\wh{\bd{\gamma}},\bd{f}}_x({\cal G'}) &= \sum^T_{t=1} \Ave{\mu_t \otimes P_t, f_t} - \inf_{\nu \in {\cal G}'} \sum^T_{t=1} \ave{\nu,f_t} \nonumber\\ &\le \sum^T_{t=1}\left[\ave{\pi_t \otimes P_t,f_t} - \inf_{P \in \cM({\cal G}')}\, \ave{\pi_P \otimes P, f_t}\right] + \sum^T_{t=1} \| f_t \|_\infty \| \mu_t - \pi_t \|_1 \nonumber\\ &=\sum^T_{t=1} \Ave{\nu_t, f_t} - \inf_{\nu \in {\cal G}'} \sum^T_{t=1} \ave{\nu, f_t} +\sum^T_{t=1} \| f_t \|_\infty \| \mu_t - \pi_t \|_1 \nonumber\\ &\le \frac{D_F({\cal G}')}{\rho} + 2\rho T + \sum^T_{t=1} \| f_t \|_\infty \| \mu_t - \pi_t \|_1. \end{aligned}$$ Now we focus on bounding the third term of the regret bound. For any time $k \le t$, we have $$\begin{aligned} \left\| \mu_k - \pi_t \right\|_1&= \left\| \mu_{k-1} K_{k-1} - \mu_{k-1} K_{t} + \mu_{k-1} K_{t}- \pi_t\right \|_1 \nonumber\\ &\stackrel{{\rm (a)}}{\le} \left \| \mu_{k-1} K_{t}- \pi_t \right\|_1 +\ \left\| \mu_{k-1} K_{k-1} - \mu_{k-1} K_{t} \right\|_1 \nonumber\\ &\stackrel{{\rm (b)}}{=}\left\| \mu_{k-1} K_{t}- \pi_t K_{t} \right\|_1+ \left\| \mu_{k-1} K_{k-1} - \mu_{k-1} K_{t} \right\|_1 \nonumber\\ &\stackrel{{\rm (c)}}{\le} e^{-1/\tau} \left\| \mu_{k-1} - \pi_t \right\|_1+ \max_{x \in \sX}\left\| P_{k-1}(\cdot|x) - P_t(\cdot|x) \right\|_1,\nonumber\\ &\stackrel{{\rm (d)}}{\le} e^{-1/\tau}\left\| \mu_{k-1} - \pi_t \right\|_1+ \sum^{t-1}_{j=k-1} \max_{x \in \sX} \| P_j(\cdot|x) - P_{j+1}(\cdot|x) \|_1, \label{eq:k_to_t_OMD}\end{aligned}$$ where (a) is by triangle inequality; (b) is by invariance of $\pi_t$ w.r.t. $K^{\wh{\bd\gamma},{\bd f}}_t$; and (c) follows from the uniform mixing bound , and (d) follows from the triangle inequality and the easily proved fact that, for any state distribution $\mu \in \cP(\sX)$ and any two Markov kernels $P,P' \in \cM(\sU|\sX)$, $$\begin{aligned} \left\| \mu K(\cdot|P) - \mu K(\cdot|P') \right\|_1 \le \max_{x \in \sX} \left\| P(\cdot|x) - P'(\cdot|x) \right\|_1.\end{aligned}$$ Letting now the initial state distribution be $\mu_1$, we can apply the bound recursively to obtain $$\begin{aligned} \left\| \mu_t - \pi_t \right\|_1 &\le e^{-(t-1)/\tau}\left\| \mu_1 - \pi_t \right\|_1+ \sum^t_{k=2} e^{-\frac{t-k}{\tau}}\sum^{t-1}_{j=k-1} \max_{x \in \sX} \|P_{j}(\cdot|x)-P_{j+1}(\cdot|x)\|_1 \\ &\le 2 e^{-(t-1)/\tau} + \sum^t_{k=2} e^{-\frac{t-k}{\tau}}(t-k+1)\Delta_T \\ &\le 2 e^{-(t-1)/\tau} + B\sum^{\infty}_{k=0} (k+1) e^{-\frac{k}{\tau}} \\ &\le 2 e^{-(t-1)/\tau} + (\tau+1)^2\Delta_T.\end{aligned}$$ So, the second term on the right-hand side of can bounded by $$\begin{aligned} C_{\cF} \sum^T_{t=1} \| \mu^{\wh {\bd \gamma}, \bd f}_t - \pi^{\wh {\bd \gamma}, \bd f}_t \|_1 \le C_{\cF}(\tau+1)^2 T \Delta_T + (2\tau+2) C_{\cF},\end{aligned}$$ which completes the proof. 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--- abstract: 'Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a natural averaging operation of the subdifferentials of the averaged functions emerges. In the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in Hilbert spaces, transforming it into the resolvent average. The theory of resolvent averages contains many desirable properties. In particular, we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average. As a consequence, we recover a significant part of the theory of proximal averages. Furthermore, we shed new light on the proximal average and present novel results and desirable properties the proximal average possesses which have not been previously available.' author: - 'Sedi Bartz[^1], Heinz H. Bauschke[^2], Sarah M. Moffat[^3], and Xianfu Wang[^4]' date: 'May 11, 2015' title: | [The resolvent average of monotone operators:\ dominant and recessive properties]{} --- [**2010 Mathematics Subject Classification:**]{} Primary 47H05, 52A41, 90C25; Secondary 15A09, 26A51, 26B25, 26E60, 47H09, 47A63. [**Keywords:**]{} Convex function, Fenchel conjugate, Legendre function, monotone operator, paramonotone, positive semidefinite operator, proximal average, rectangular, resolvent, resolvent average, strong convexity, strong monotonicity, strong smoothness, subdifferential operator, uniform convexity, uniform smoothness. Introduction {#intro} ============ The proximal average of two convex functions was first considered in [@BMR]. Since then, in a series of papers, the definition of the proximal average was refined and its useful properties were studied and employed in various applications revealing a rich theory with promising potential for further evolution and applications. One of the latest forms of the proximal average we refer to in the present paper is given in Definition \[proximal average def\] below. Some other cornerstones in the study of the proximal average include: [@BGLW] where many useful properties and examples where presented, [@BLT] where it was demonstrated that the proximal average defines a homotopy on the class of convex functions (unlike other, classical averages), and, also, a significant application [@BW] where the proximal average was employed in order to explicitly construct *autoconjugate* representations of monotone operators, also known as *self-dual* Lagrangians, the importance of which in variational analysis is demonstrated in detail in the monograph [@Gho]. A recent application of the proximal average in the theory of *machine learning* is [@Yu]. When subdifferentiating the proximal average, we obtain an averaging operation of the subdifferentials of the underlying functions (see equation  below). Monotone operators are fundamentally important in analysis and optimization [@AusTeb], [@BC2011], [@Borwein], [@BV], [@BurIus], [@RockWets], [@Simons2]. In the present paper, we analyze the resolvent average (see Definition \[resolvent average def\] below), which significantly extends the above averaging operation of subdifferentials to the general framework of monotone operator theory. (See also [@bmow2013], [@Wang], and [@Moffat] for some earlier works on the resolvent average.) We present powerful general properties the resolvent average possesses and then focus on the study of more specific inheritance properties of the resolvent average. Namely, we go through a detailed list of attractive properties of monotone operators and classify them as *dominant* or *recessive* with respect to the resolvent average by employing the following notions: Let $C$ be a set and let $I$ be an index set. Suppose that $\mathcal{AVE}:C^I\to C$. Then a property $(p)$ is said to be 1. **dominant** with respect to $\mathcal{AVE}$ if for each $(c_i)\in C^I$, the existence of $i_0\in I$ such that $c_{i_0}$ has property $(p)$ implies that $\mathcal{AVE}((c_i))$ has property $(p)$; 2. **recessive** with respect to $\mathcal{AVE}$ if $(p)$ is not dominant and for each $(c_i)\in C^I$, for each $i\in I$, $c_i$ having property $(p)$ implies that $\mathcal{AVE}((c_i))$ has property $(p)$. We also provide several examples of the resolvent average (mainly in order to prove the recessive nature of several properties) of mappings which are monotone, however, which are not subdifferential operators. As a consequence, the resolvent average is now seen to be a natural and effective tool for averaging monotone operators which avoids many of the domain and range obstacles standing in front of classical averages such as the arithmetic average. The resolvent average is also seen to be an effective averaging technique when one wishes the average to posses specific properties, especially when the desired properties are dominant. When we restrict our attention to monotone linear relations, our current study extends the one in [@bmw-res] where the resolvent average was considered as an average of positive semidefinite and definite matrices. When we restrict our attention to subdifferential operators, we recover a large part of the theory of the proximal average [@BGLW]. Moreover, we present several novel results regarding the inheritance of desired properties of the proximal average which have not been previously available. In summary, *the resolvent average provides a novel technique for generating new maximally monotone operators with desirable properties*. The remaining of the paper is organized as follows: In the remainder of Section \[intro\] we present the basic definitions, notations and the relations between them which we will employ throughout the paper. We also collect all preliminary facts necessary for our presentation. In Section \[basic\] we present basic properties of the resolvent average. In Section \[dominant\] we study dominant properties while Section \[recessive\] deals with recessive properties. Finally, in Section \[neither\] we consider combinations of properties, properties which are neither dominant nor recessive, and other observations and remarks. Before we start our analysis, let us recall the following concepts and standard notation from monotone operator theory and convex analysis: Throughout this paper, ${\ensuremath{\mathcal H}}$ is a real Hilbert space with inner product ${\langle{{\cdot},{\cdot}}\rangle}$, induced norm $\|\cdot \|$, identity mapping ${\ensuremath{\operatorname{Id}}}$ and we set $q=\frac{1}{2}\|\cdot\|^2$. We denote the interior of a subset $C$ of ${\ensuremath{\mathcal H}}$ by ${\ensuremath{\operatorname{int}}}C$. Let $A:{\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$ be a set-valued mapping. We say that $A$ is *proper* when the *domain* of $A$, the set ${\ensuremath{\operatorname{dom}}}A=\{x\in{\ensuremath{\mathcal H}}{\ensuremath{\;|\;}}Ax\neq\varnothing\}$, is nonempty. The *range* of $A$ is the set ${\ensuremath{\operatorname{ran}}}A = A({\ensuremath{\mathcal H}})=\bigcup_{x \in {\ensuremath{\mathcal H}}} Ax$, the *graph* of $A$ is the set ${\ensuremath{\operatorname{gra}}}A = \{(x,u)\in {\ensuremath{\mathcal H}}\times {\ensuremath{\mathcal H}}{\ensuremath{\;|\;}}u \in Ax\}$ and the inverse of $A$ is the mapping $A^{-1}$ satisfying $x\in A^{-1}u\Leftrightarrow u\in Ax$. $A$ is said to be *monotone* if $$(\forall (x,u) \in {\ensuremath{\operatorname{gra}}}A)(\forall (y,v) \in {\ensuremath{\operatorname{gra}}}A)\quad {\langle{{x-y},{u-v}}\rangle} \geq 0.$$ $A$ is said to be *maximally monotone* if there exists no monotone operator $B$ such that ${\ensuremath{\operatorname{gra}}}A$ is a proper subset of ${\ensuremath{\operatorname{gra}}}B$. The *resolvent* of $A$ is the mapping $J_A=(A+{\ensuremath{\operatorname{Id}}})^{-1}$. We say that $A$ is a linear relation if ${\ensuremath{\operatorname{gra}}}A$ is a linear subspace of ${\ensuremath{\mathcal H}}\times {\ensuremath{\mathcal H}}$. $A$ is said to be a maximally monotone linear relation if $A$ is both maximally monotone and a linear relation. The mapping $T: {\ensuremath{\mathcal H}}\to {\ensuremath{\mathcal H}}$ is said to be *firmly nonexpansive* if $$(\forall x\in {\ensuremath{\mathcal H}})(\forall y\in {\ensuremath{\mathcal H}}) \quad \|Tx-Ty\|^2 + \|({\ensuremath{\operatorname{Id}}}-T)x-({\ensuremath{\operatorname{Id}}}-T)y\|^2 \leq \|x-y\|^2.$$ Obviously, if $T$ is firmly nonexpansive, then it is *nonexpansive*, that is, [Lipschitz continuous]{} with constant $1$, where a Lipschitz continuous mapping with constant $L$ is a mapping $T:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ such that $$(\forall x\in {\ensuremath{\mathcal H}})(\forall y\in {\ensuremath{\mathcal H}}) \quad \|Tx-Ty\|\leq L\|x-y\|.$$ The mapping $T$ is said to be a *Banach contraction* if it is Lipschitz continuous with constant $L<1$. The point $x\in{\ensuremath{\mathcal H}}$ is said to be a fixed point of the mapping $T:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ if $Tx=x$. The set of all fixed points of $T$ is denoted by ${\ensuremath{\operatorname{Fix}}}T$. The function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is said to be *proper* if ${\ensuremath{\operatorname{dom}}}f=\{x\in{\ensuremath{\mathcal H}}{\ensuremath{\;|\;}}f(x)<\infty\}\neq\varnothing$. The *Fenchel conjugate* of the function $f$ is the function $f^*$ which is defined by $f^*(u)=\sup_{x\in{\ensuremath{\mathcal H}}}({\langle{{u},{x}}\rangle}-f(x))$. The subdifferential of a proper function $f$ is the mapping $\partial f:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ which is defined by $$\partial f(x)=\big\{u\in {\ensuremath{\mathcal H}}\ \big|\ f(x)+{\langle{{u},{y-x}}\rangle}\leq f(y),\ \ \forall y\in{\ensuremath{\mathcal H}}\big\}.$$ The *indicator function* of a subset $C$ of ${\ensuremath{\mathcal H}}$ is the function $\iota_C:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ which vanishes on $C$ and equals $\infty$ on ${\ensuremath{\mathcal H}}\smallsetminus C$. The *normal cone* operator of the set $C$ is the mapping $N_C=\partial\iota_C$. We will denote the *nearest point projection* on the set $C$ by $P_C$. We now recall the definition of the proximal average and present the definition of the resolvent average. To this end we will make use of the following additional notations: Throughout the paper we assume that $\mu\in\ ]0,\infty[$, $n\in\{1,2,\dots\}$ and $I=\{1,\ldots,n\}$. For every $i \in I$, let $A_i:{\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$ be a mapping, let $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ be a function and let $\lambda_{i}>0,\ \sum_{i\in I}\lambda_i=1$ . We set: $$\begin{aligned} &{\bf A}=(A_{1},\ldots, A_{n}),\ {\bf A^{-1}}=(A^{-1}_{1},\ldots, A^{-1}_{n}),\ {\bf f}=(f_{1},\ldots, f_{n}),\ {\bf f^*}=(f^*_{1},\ldots, f^*_{n}), \\&\text{and}\ {\boldsymbol \lambda}=(\lambda_{1},\ldots, \lambda_{n}).\end{aligned}$$ \[proximal average def\] The *${\boldsymbol \lambda}$-weighted proximal average* of ${\bf f}$ with parameter $\mu$ is the function $p_\mu({\bf f},{\boldsymbol \lambda}):{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left[-\infty,+\infty\right]}}$ defined by $$\label{proximal average} p_\mu({\bf f},{\boldsymbol \lambda})(x)=\frac{1}{\mu}\Big(-\frac{1}{2}\|x\|^2+\inf_{\sum_{i\in I}x_i=x}\sum_{i\in I}\lambda_i\big(\mu f_i(x_i/ \lambda_i)+\frac{1}{2}\|x_i/ \lambda_i\|^2\big)\Big),\ \ \ \ \ \ x\in{\ensuremath{\mathcal H}}.$$ We will simply write $p({\bf f},{\boldsymbol \lambda})$ when $\mu=1$, $p_\mu({\bf f})$ when all of the $\lambda_i$’s coincide and, finally, $p({\bf f})$ when $\mu=1$ and all of the $\lambda_i$’s coincide. \[resolvents of subs and proximal reformulation\]*[@BGLW Proposition 4.3 and Theorem 6.7]* Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. Then $p_\mu(\bf f,\boldsymbol{\lambda})$ is proper, lower semicontinuous, convex and for every $x\in{\ensuremath{\mathcal H}}$, $$\label{proximal average reformulation} p_\mu({\bf f},\boldsymbol{\lambda})(x)=\inf_{\sum_{i\in I}\lambda_j y_j=x}\sum_{i\in I}\lambda_if_i(y_i)+\frac{1}{\mu}\bigg(\Big(\sum_{i\in I}\lambda_iq(y_i)\Big)-q(x)\bigg),\ \ \ \ \ \ x\in{\ensuremath{\mathcal H}}.$$ Furthermore, $$\label{average of resolvents of sub} J_{\mu\partial p_{\mu}({\bf f},{\boldsymbol \lambda})}=\sum_{i\in I}\lambda_iJ_{\mu\partial f_i}.$$ We recall that $J_{\partial f}=\text{Prox} f$ is Moreau’s proximity operator (see [@Mor]). Thus, we see that $\partial p_{\mu}({\bf f},{\boldsymbol \lambda})$ defines an averaging operation of the $\partial f_i$’s with the the weights $\lambda_i$ and parameter $\mu$ in the following manner: $$\label{resolvent average of sub} \partial p_{\mu}({\bf f},{\boldsymbol \lambda})=\Big(\sum_{i\in I}\lambda_{i}(\partial f_{i}+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}\Big)^{-1}-\mu^{-1}{\ensuremath{\operatorname{Id}}}.$$ We now extend the reach of the averaging operation defined in and which is the subject matter of the present paper: \[resolvent average def\] The *${\boldsymbol \lambda}$-weighted resolvent average* of ${\bf A}$ with parameter $\mu$ is defined by $$\label{ResAvgDef} {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=\Big(\sum_{i\in I}\lambda_{i}(A_{i}+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}\Big)^{-1}-\mu^{-1}{\ensuremath{\operatorname{Id}}}.$$ We will simply write ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ when $\mu=1$, $\mathcal{R}_\mu(\bf A)$ when all of the $\lambda_i$’s coincide and, finally, $\mathcal{R}(\bf A)$ when $\mu=1$ and all of the $\lambda_i$’s coincide. The motivation for naming  the *resolvent average* stems from the equivalence between equation and equation , that is, from the fact that equation is equivalent to the equation $$\label{resolventidentity} J_{\mu {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_{i}J_{\mu A_{i}}.$$ The parameter $\mu$ has been useful in the study of the proximal average; in particular, when taking $\mu\to\infty$ or $\mu\to0^+$ one obtains classical averages of functions (see [@BGLW]). We employ particular choices of the parameter $\mu$ in applications of the proximal average in the present paper as well. Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous, convex and set $A_i=\partial f_i$. By combining Definition \[resolvent average def\] together with equation  we see that $$\label{sub of proximal average} \partial p_\mu(\bold{f},\boldsymbol{\lambda})=\mathcal{R}_\mu(\boldsymbol{\partial}\bold{f},\boldsymbol{\lambda}).$$ We end this introductory section with the following collection of facts which we will employ in the remaining sections of the paper. The next fact from [@RockWets] was originally presented in the setting of finite-dimensional spaces, however, along with its proof from [@RockWets], it holds in any Hilbert space. *[@RockWets Lemma 12.14]* For any mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$, $$\label{must1} J_A = {\ensuremath{\operatorname{Id}}}- J_{A^{-1}}.$$ *[@RockWets Proposition 6.17]* \[NCresolvent\] Let $C$ be a nonempty, closed and convex subset of ${\ensuremath{\mathcal H}}$. Then $$J_{N_C} = ({\ensuremath{\operatorname{Id}}}+ N_C)^{-1} = P_C.$$ *[@BC2011 Proposition 4.2]* \[f:firm\] Let $T\colon {\ensuremath{\mathcal H}}\to {\ensuremath{\mathcal H}}$. Then the following are equivalent: 1. $T$ is firmly nonexpansive. 2. ${\ensuremath{\operatorname{Id}}}-T$ is firmly nonexpansive. 3. \[f:firm-nonexp\]$2T-{\ensuremath{\operatorname{Id}}}$ is nonexpansive. 4. \[f:firm-inequal\]$(\forall x\in {\ensuremath{\mathcal H}})(\forall y\in {\ensuremath{\mathcal H}})$ $\|Tx-Ty\|^2 \leq {\langle{{x-y},{Tx-Ty}}\rangle}$. \[average of firmly\] Suppose that for each $i\in I$, $T_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is firmly nonexpansive. Then $T=\sum_{i\in I}\lambda_i T_i$ is firmly nonexpansive. Employing Fact \[f:firm\], for each $i\in I$, letting $N_i=2T_i-{\ensuremath{\operatorname{Id}}}$, we see that $N_i$ is nonexpansive. Letting $N=\sum_{i\in I}\lambda_iN_i$, we see that $N$ is nonexpansive. Consequently, the mapping $T=\frac{1}{2}(N+{\ensuremath{\operatorname{Id}}})$ is firmly nonexpansive. *[@BC2011 Lemma 2.13]* For each $i\in I$ let $x_i$ and $u_i$ be points in ${\ensuremath{\mathcal H}}$ and $\alpha_i\in{\ensuremath{\mathbb R}}$ be such that $\sum_{i\in I}\alpha_{i}=1$. Then $$\label{f:monoident} {\left\langle{{\sum_{i\in I}\alpha_{i}x_{i}},{\sum_{j\in I}\alpha_{j}u_{j}}}\right\rangle}+\frac{1}{2}\sum_{(i,j)\in I\times I} \alpha_{i}\alpha_{j}{\langle{{x_{i}-x_{j}},{u_{i}-u_{j}}}\rangle}=\sum_{i\in I}\alpha_{i}{\langle{{x_{i}},{u_{i}}}\rangle}.$$ Consequently, $$\label{normstrongconvex} \bigg\|\sum_{i\in I}\alpha_{i}x_{i}\bigg\|^2=\sum_{i\in I}\alpha_{i}\|x_{i}\|^2 -\frac{1}{2}\sum_{(i,j)\in I\times I} \alpha_{i}\alpha_{j}\|x_{i}-x_{j}\|^2.$$ \[equality in convex combination of firmly nonexpansive\] Suppose that for each $i\in I$, $N_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is nonexpansive, $T_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is firmly nonexpansive and set $N=\sum_{i\in I}\lambda_i N_i$ and $T=\sum_{i\in I}\lambda_i T_i$. Let $x$ and $y$ be points in ${\ensuremath{\mathcal H}}$ such that $\|Tx-Ty\|^2={\langle{{x-y},{Tx-Ty}}\rangle}$. Then $T_i x-T_iy=Tx-Ty$ for every $i\in I$. As a consequence, the following assertions hold: 1. If there exits $i_0\in I$ such that $T_{i_0}$ is injective, then $T$ is injective.\[firmly injective\] 2. If $x$ and $y$ are points in ${\ensuremath{\mathcal H}}$ such that $\|Nx-Ny\|=\|x-y\|$, then $N_i x-N_iy=Nx-Ny$ for every $i\in I$.\[equality in convex combination of nonexpansive\] 3. *[@Reich; @1983 Lemma 1.4]* If $\bigcap_{i\in I}{\ensuremath{\operatorname{Fix}}}N_i\neq\varnothing$, then $ {\ensuremath{\operatorname{Fix}}}N=\bigcap_{i\in I}{\ensuremath{\operatorname{Fix}}}N_i. $\[fixed points of average of firmly nonexpansive mappings\] By employing equality  and then the firm nonexpansiveness of each $T_i$ we obtain $$\begin{aligned} {\langle{{x-y},{Tx-Ty}}\rangle}&=\|T x-T y\|^2\\ &=\sum_{i\in I}\lambda_i\|T_ix-T_iy\|^2-\frac{1}{2}\sum_{(i,j)\in I\times I}\lambda_{i}\lambda_{j}\|(T_i x-T_i y)-(T_j x-T_j y)\|^2\\ &\leq\sum_{i\in I}\lambda_i{\langle{{x-y},{T_i x-T_iy}}\rangle}-\frac{1}{2}\sum_{(i,j)\in I\times I}\lambda_{i}\lambda_{j}\|(T_i x-T_i y)-(T_j x-T_j y)\|^2\\ &={\langle{{x-y},{Tx-Ty}}\rangle}-\frac{1}{2}\sum_{(i,j)\in I\times I}\lambda_{i}\lambda_{j}\|(T_i x-T_i y)-(T_j x-T_j y)\|^2.\end{aligned}$$ Thus, we see that $\|(T_i x-T_i y)-(T_j x-T_j y)\|^2=0$ for every $i$ and $j$ in $I$. Hence, $T_i x-T_i y=Tx-Ty$ for every $i\in I$. (i): Follows immediately. (ii): For each $i\in I$ we suppose that $T_i=\frac{1}{2}(N_i+{\ensuremath{\operatorname{Id}}})$ so that $T_i$ as well as the mapping $T=\sum_{i\in I}\lambda_i T_i=\frac{1}{2}(N+{\ensuremath{\operatorname{Id}}})$ are firmly nonexpansive. Then the equality $\|Nx-Ny\|^2=\|x-y\|^2$ implies that $\|Tx-Ty\|^2={\langle{{x-y},{Tx-Ty}}\rangle}$. Consequently, we see that $T_i x-T_iy=Tx-Ty$ for every $i\in I$, which, in turn, implies that $N_i x-N_iy=Nx-Ny$ for every $i\in I$. (iii): The inclusion ${\ensuremath{\operatorname{Fix}}}N\supseteq\bigcap_{i\in I}{\ensuremath{\operatorname{Fix}}}N_i$ is trivial. Now, suppose that $x\in{\ensuremath{\operatorname{Fix}}}N$ and let $y\in\bigcap_{i\in I}{\ensuremath{\operatorname{Fix}}}N_i\subseteq{\ensuremath{\operatorname{Fix}}}N$. Then $Nx-Ny=x-y$, in particular, $\|Nx-Ny\|=\|x-y\|$. Consequently, for each $i\in I$, $ x-y=Nx-Ny=N_i x-N_i y=N_i x-y, $ which implies that $x=N_i x$, as asserted by (iii). *[@Minty] (see also [@BC2011 Theorem 21.1])*\[MintyThm\] Let $A: {\ensuremath{\mathcal H}}\rightrightarrows {\ensuremath{\mathcal H}}$ be monotone. Then $$\label{Minty's parametrization} {\ensuremath{\operatorname{gra}}}A =\big\{ (J_Ax,({\ensuremath{\operatorname{Id}}}-J_A)x)\ \big|\ x\in {\ensuremath{\operatorname{ran}}}({\ensuremath{\operatorname{Id}}}+A)\big\}.$$ Furthermore, $A$ is maximally monotone if and only if ${\ensuremath{\operatorname{ran}}}({\ensuremath{\operatorname{Id}}}+ A) = {\ensuremath{\mathcal H}}$. \[f:Minty\][([@EckBer] and [@Minty].)]{} Let $T\colon {\ensuremath{\mathcal H}}\to {\ensuremath{\mathcal H}}$ and let $A\colon {\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$. Then the following assertions hold: 1. \[f:Mintyi\] If $T$ is firmly nonexpansive, then $B = T^{-1}-{\ensuremath{\operatorname{Id}}}$ is maximally monotone and $J_B=T$. 2. If $A$ is maximally monotone, then $J_A$ has full domain, and is single-valued and firmly nonexpansive, and $A=J_A^{-1}-{\ensuremath{\operatorname{Id}}}$. *[@Fitz]* With the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ we associate the *Fitzpatrick function* $F_A:{\ensuremath{\mathcal H}}\times{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$, defined by $$\label{Fitzpatrick function} F_A(x,v)=\sup_{(z,w)\in{\ensuremath{\operatorname{gra}}}A}\big({\langle{{w},{x}}\rangle}+{\langle{{v},{z}}\rangle}-{\langle{{w},{z}}\rangle}\big),\ \ \ \ \ \ (x,v)\in{\ensuremath{\mathcal H}}\times{\ensuremath{\mathcal H}}.$$ \[D:para rec def\] The monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is said to be *rectangular* [[@Simons2 Definition 31.5]]{} (also known as *$3^*$ monotone*) if for every $x\in{\ensuremath{\operatorname{dom}}}A$ and every $v\in{\ensuremath{\operatorname{ran}}}A$ we have $$\label{rectangular definition} \inf_{(z,w)\in{\ensuremath{\operatorname{gra}}}A} {\langle{{v-w},{x-z}}\rangle}>{\ensuremath{-\infty}},$$ equivalently, if $$\label{Fitzpatrick rectangular} {\ensuremath{\operatorname{dom}}}A\times{\ensuremath{\operatorname{ran}}}A\ \subseteq\ {\ensuremath{\operatorname{dom}}}F_A.$$ The mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is said to be *paramonotone* if whenever we have a pair of points $(x,v)$ and $(y,u)$ in ${\ensuremath{\operatorname{gra}}}A$ such that ${\langle{{x-y},{v-u}}\rangle}=0$, then $(x,u)$ and $(y,v)$ are also in ${\ensuremath{\operatorname{gra}}}A$. *[@bbw2007 Remark 4.11], [@bwy2012 Corollary 4.11]*\[f:pararecsame\] Let $A\in {\ensuremath{\mathbb R}}^{N\times N}$ be monotone and set $A_{+} = \tfrac{1}{2}A+\tfrac{1}{2}A^\intercal$. Then the following assertions are equivalent: 1. $A$ is paramonotone; 2. $A$ is rectangular; 3. ${\ensuremath{\operatorname{rank}}}A={\ensuremath{\operatorname{rank}}}A_{+}$; 4. ${\ensuremath{\operatorname{ran}}}A={\ensuremath{\operatorname{ran}}}A_{+}$. \[linear\]*[@Cross I.2.3 and I.4]* Let $A,B$ be linear relations. Then $A^{-1}$ and $A+B$ are linear relations. Basic properties of the resolvent average {#basic} ========================================= In this section we present several basic properties of the resolvent average. These will stand as the foundation of our entire discussion in the present paper and will be applied repeatedly. The inverse of the resolvent average ------------------------------------ We begin our discussion by recalling the following fact [@BGLW Theorem 5.1]: suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. Then $$\label{proximal conjugate} \big(p_\mu({\bf f},{\boldsymbol \lambda})\big)^*=p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda}).$$ Now for each $i\in I$ we set $A_i=\partial f_i$. Then $A_i^{-1}=\partial f_i^*$. Thus, recalling equation , we see that equation turns into $$\label{proximal conjugate sub} \big({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\big)^{-1}=\partial \big(p_\mu({\bf f},{\boldsymbol \lambda})\big)^*=\partial p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})={\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}},$$ that is, we have an inversion formula for the resolvent average in the case where we average subdifferential operators. We now aim at extending this result into our, more general, framework of the present paper. To this end, we will need the following general property of resolvents: Let $A:{\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$. Then $$\begin{aligned} (A+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}&=\big({\ensuremath{\operatorname{Id}}}-\mu(A^{-1}+\mu{\ensuremath{\operatorname{Id}}})^{-1}\big)\circ(\mu{\ensuremath{\operatorname{Id}}})\label{mu resolvent identity}\\ &=\mu\big({\ensuremath{\operatorname{Id}}}-(\mu^{-1}A^{-1}+{\ensuremath{\operatorname{Id}}})^{-1}\big).\label{mu resolvent identity 2}\end{aligned}$$ Consequently, if $B:{\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$ is a mapping such that $$\label{mu resolvent identity character} (A+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}=\big({\ensuremath{\operatorname{Id}}}-\mu(B+\mu{\ensuremath{\operatorname{Id}}})^{-1}\big)\circ(\mu{\ensuremath{\operatorname{Id}}}),$$ then $B=A^{-1}$. For any mapping $F:{\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$ we have $(\mu F)^{-1}=F^{-1}\circ(\mu^{-1}{\ensuremath{\operatorname{Id}}})$. Employing this fact, the fact that $A+\mu^{-1}{\ensuremath{\operatorname{Id}}}=\mu^{-1}(\mu A+{\ensuremath{\operatorname{Id}}})$ and the resolvent identity , we obtain the following chain of equalities: $$\begin{aligned} (A+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}&=J_{\mu A}\circ(\mu{\ensuremath{\operatorname{Id}}})=({\ensuremath{\operatorname{Id}}}-J_{(\mu A)^{-1}})\circ(\mu{\ensuremath{\operatorname{Id}}})=({\ensuremath{\operatorname{Id}}}-J_{A^{-1}\circ(\mu^{-1}{\ensuremath{\operatorname{Id}}})})\circ(\mu{\ensuremath{\operatorname{Id}}})\\ &=\Big({\ensuremath{\operatorname{Id}}}-\big(A^{-1}\circ(\mu^{-1}{\ensuremath{\operatorname{Id}}})+{\ensuremath{\operatorname{Id}}}\big)^{-1}\Big)\circ(\mu{\ensuremath{\operatorname{Id}}})\\ &=\Big({\ensuremath{\operatorname{Id}}}-\big(A^{-1}\circ(\mu^{-1}{\ensuremath{\operatorname{Id}}})+(\mu{\ensuremath{\operatorname{Id}}})\circ(\mu^{-1}{\ensuremath{\operatorname{Id}}})\big)^{-1}\Big)\circ(\mu{\ensuremath{\operatorname{Id}}})\\ &=\Big({\ensuremath{\operatorname{Id}}}-\big((A^{-1}+\mu{\ensuremath{\operatorname{Id}}})\circ(\mu^{-1}{\ensuremath{\operatorname{Id}}})\big)^{-1}\Big)\circ(\mu{\ensuremath{\operatorname{Id}}})=\big({\ensuremath{\operatorname{Id}}}-\mu(A^{-1}+\mu{\ensuremath{\operatorname{Id}}})^{-1}\big)\circ(\mu{\ensuremath{\operatorname{Id}}})\\ &=\mu{\ensuremath{\operatorname{Id}}}-\mu(\mu^{-1}A^{-1}+{\ensuremath{\operatorname{Id}}})^{-1}\circ(\mu^{-1}{\ensuremath{\operatorname{Id}}})\circ(\mu{\ensuremath{\operatorname{Id}}})=\mu\big({\ensuremath{\operatorname{Id}}}-(\mu^{-1}A^{-1}+{\ensuremath{\operatorname{Id}}})^{-1}\big).\end{aligned}$$ This completes the proof of and . Now, suppose that $B:{\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$ is a mapping which satisfies equation . Then, by employing equation to the right hand side of equation , we arrive at $(A+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}=(B^{-1}+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}$, which implies that $A+\mu^{-1}{\ensuremath{\operatorname{Id}}}=B^{-1}+\mu^{-1}{\ensuremath{\operatorname{Id}}}$. Since $\mu^{-1}{\ensuremath{\operatorname{Id}}}$ is single-valued, we conclude that $B^{-1}=A$ and complete the proof. \[mainresult\] Suppose that for each $i\in I$, $A_{i}:{\ensuremath{\mathcal H}}{\ensuremath{\rightrightarrows}}{\ensuremath{\mathcal H}}$ is a set-valued mapping. Then $$\label{niceselfdual} \big({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\big)^{-1}={\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}.$$ By employing the definition (see ) of ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$, ${\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}$ in ,  and by also employing equation  in  below, we obtain the following chain of equalities: $$\begin{aligned} {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}&=\bigg(\sum_{i\in I}\lambda_i(A_i+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}\bigg)^{-1}-\mu^{-1}{\ensuremath{\operatorname{Id}}}\label{first application of ResDef}\\ &=\bigg(\sum_{i\in I}\lambda_i\big({\ensuremath{\operatorname{Id}}}-\mu(A_i^{-1}+\mu{\ensuremath{\operatorname{Id}}})^{-1}\big)\circ(\mu{\ensuremath{\operatorname{Id}}})\bigg)^{-1}-\mu^{-1}{\ensuremath{\operatorname{Id}}}\label{application of mu resolvent identity}\\ &=\bigg(\mu{\ensuremath{\operatorname{Id}}}-\mu\sum_{i\in I}\lambda_i(A_i^{-1}+\mu{\ensuremath{\operatorname{Id}}})^{-1}\circ(\mu{\ensuremath{\operatorname{Id}}})\bigg)^{-1}-\mu^{-1}{\ensuremath{\operatorname{Id}}}\nonumber\\ &=\bigg(\mu{\ensuremath{\operatorname{Id}}}-\mu\Big({\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}+\mu{\ensuremath{\operatorname{Id}}}\Big)^{-1}\circ(\mu{\ensuremath{\operatorname{Id}}})\bigg)^{-1}-\mu^{-1}{\ensuremath{\operatorname{Id}}}\label{second application of ResDef}\end{aligned}$$ which, in turn, implies $$({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}+\mu^{-1}{\ensuremath{\operatorname{Id}}})^{-1}=\big({\ensuremath{\operatorname{Id}}}-\mu[{\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}+\mu{\ensuremath{\operatorname{Id}}}]^{-1}\big)\circ(\mu{\ensuremath{\operatorname{Id}}}).$$ Finally, letting $A={\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ and $B={\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}$, we may now apply characterization  in order to obtain $A^{-1}=B$ and completes the proof. We see that, indeed, Theorem \[mainresult\] extends the reach of formula . Consequently, since the subdifferential of a proper, lower semicontinuous and convex function determines its antiderivative uniquely up to an additive constant, we note that, in fact, Theorem \[mainresult\] recovers formula  up to an additive constant. Basic properties of ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ and common solutions to monotone inclusions --------------------------------------------------------------------------------------------------------------------------------- \[basic properties\] Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. 1. Let $x$ and $u$ be points in ${\ensuremath{\mathcal H}}$. Then $$\label{average of shift} \mathcal{R}_\mu((A_1-u,\dots,A_n-u),\boldsymbol\lambda)={\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}-u$$ and $$\label{average of shifted argument} \mathcal{R}_\mu\Big(\big(A_1\big((\cdot)-x\big),\dots,A_n\big((\cdot)-x\big)\big),\boldsymbol\lambda\Big)={\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\big((\cdot)-x)\big).$$ 2. Let $0<\alpha$. Then $$\label{scaled average} \mathcal{R}_\mu(\alpha\bold{A},\boldsymbol\lambda)=\alpha\mathcal{R}_{\alpha\mu}(\bold{A},\boldsymbol\lambda)\ \ \ \text{in particular,}\ \ \ \ {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=\mu^{-1}\mathcal{R}(\mu\bold{A},\boldsymbol\lambda).$$ 3. Let $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ be maximally monotone and suppose that for each $i\in I$ , $A_i=A$. Then $$\label{same mapping average} {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=A.$$ (i): For a mapping $B:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ we have $(B-u)^{-1}=B^{-1}((\cdot)+u)$. Thus, $$J_{\mu\mathcal{R}_\mu((A_1-u,\dots,A_n-u),\boldsymbol\lambda)}=\sum_{i\in I}\lambda_iJ_{\mu(A_i-u)}=\sum_{i\in I}\lambda_i J_{\mu A_i}((\cdot)+\mu u)=J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}((\cdot)+\mu u).$$ Consequently, follows. A similar argument also implies equation . (ii): Since $J_{\mu\mathcal{R}_\mu(\alpha\bold{A},\boldsymbol\lambda)}=\sum_{i\in I}\lambda_i J_{\mu\alpha A_i}=J_{\mu\alpha\mathcal{R}_{\mu\alpha}(\bold{A},\boldsymbol\lambda)} $, equation  follows. (iii): Since $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_i J_{\mu A_i}=J_{\mu A}$, we obtain ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=A$. Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex and we set $A_i=\partial f_i$. In this particular case formula  can be obtained by subdifferentiating the following formula [@BGLW Remark 4.2(iv)]: $$\label{scaled proximal average} p_\mu({\bf f},{\boldsymbol \lambda})=\mu^{-1}p({\mu\bf f},{\boldsymbol \lambda}).$$ A strong motivation for studying the resolvent average stems from the fact that it also captures common solutions to monotone inclusions: Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. Let $x$ and $u$ be points in ${\ensuremath{\mathcal H}}$. If $\bigcap_{i\in I}A_i(x)\neq\varnothing$, then $$\label{common solution} {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}(x)=\bigcap_{i\in I}A_i(x).$$ If $\bigcap_{i\in I}A_i^{-1}(u)\neq\varnothing$, then $$\label{common inverse solution} {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}^{-1}(u)=\bigcap_{i\in I}A_i^{-1}(u).$$ First we prove that if $\bigcap_{i\in I}A_i^{-1}(0)\neq\varnothing$ then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}^{-1}(0)=\bigcap_{i\in I}A_i^{-1}(0)$. Our assumption that $\bigcap_{i\in I}A_i^{-1}(0)\neq\varnothing$ means that $\bigcap_{i\in I}{\ensuremath{\operatorname{Fix}}}J_{A_i}\neq\varnothing$. Since for each $i\in I$, $J_{A_i}$ is nonexpansive, Corollary \[equality in convex combination of firmly nonexpansive\]\[fixed points of average of firmly nonexpansive mappings\] guarantees that $${\ensuremath{\operatorname{Fix}}}J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}={\ensuremath{\operatorname{Fix}}}\sum_{i\in I}\lambda_i J_{\mu A_i}=\bigcap_{i\in I}{\ensuremath{\operatorname{Fix}}}J_{\mu A_i}$$ which implies that $\big(\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\big)^{-1}(0)=\bigcap_{i\in I}(\mu A_i^{-1})(0)$ and, consequently, that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}^{-1}(0)=\bigcap_{i\in I}A_i^{-1}(0)$. Now, let $u\in{\ensuremath{\mathcal H}}$. Given a mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$, then $A^{-1}(u)=(A-u)^{-1}(0)$. Consequently, if $\bigcap_{i\in I}A_i^{-1}(u)\neq\varnothing$, then $\bigcap_{i\in I}(A_i-u)^{-1}(0)\neq\varnothing$. By employing equation  we obtain $$\begin{aligned} {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}^{-1}(u)&=({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}-u)^{-1}(0)=\mathcal{R}_\mu((A_1-u,\dots,A_n-u),\boldsymbol\lambda)^{-1}(0)\\ &=\bigcap_{i\in I}(A_i-u)^{-1}(0) =\bigcap_{i\in I}A_i^{-1}(u)\end{aligned}$$ which completes the proof of equation . Let $x\in{\ensuremath{\mathcal H}}$. If $\bigcap_{i\in I}A_i(x)\neq\varnothing$, then by employing Theorem \[mainresult\] we now obtain $${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}(x)=\big({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}^{-1}\big)^{-1}(x)=\big({\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}\big)^{-1}(x)=\bigcap_{i\in I}(A_i^{-1})^{-1}(x)=\bigcap_{i\in I}A_i(x)$$ which completes the proof of equation . (**convex feasibility problem**) Suppose that for each $i\in I$, $C_i$ is a nonempty, closed and convex subset of ${\ensuremath{\mathcal H}}$ and set $A_i=N_{C_i}$. If $\bigcap_{i\in I}C_i\neq\varnothing$, then $${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}^{-1}(0)=\bigcap_{i\in I}C_i.$$ Monotonicity, domain, range and the graph of the resolvent average ------------------------------------------------------------------ We continue our presentation of general properties of the resolvent average by focusing our attention on monotone operators. \[neededlater\] Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is a set-valued mapping. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is maximally monotone if and only if for each $i\in I$, $A_i$ is maximally monotone. In this case $$\label{resolvents graph} {\ensuremath{\operatorname{gra}}}\mu {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\subseteq \sum_{i\in I}\lambda_{i}{\ensuremath{\operatorname{gra}}}\mu A_{i},$$ and, consequently, $$\label{domain and range containment} {\ensuremath{\operatorname{ran}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\subseteq\sum_{i\in I}\lambda_{i}{\ensuremath{\operatorname{ran}}}A_{i}\ \ \ \ \text{and}\ \ \ \ {\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\subseteq\sum_{i\in I}\lambda_{i}{\ensuremath{\operatorname{dom}}}A_{i}.$$ By employing equation  we see that $$\label{montreal} {\ensuremath{\operatorname{dom}}}J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}= \bigcap_{i\in I}{\ensuremath{\operatorname{dom}}}J_{\mu A_{i}}.$$ As a consequence, we see that ${\ensuremath{\operatorname{ran}}}({\ensuremath{\operatorname{Id}}}+\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}})={\ensuremath{\mathcal H}}$ if and only if for each $i\in I$, ${\ensuremath{\operatorname{ran}}}({\ensuremath{\operatorname{Id}}}+\mu A_i)={\ensuremath{\mathcal H}}$. Recalling Minty’s Theorem (Fact \[MintyThm\]), we see that $\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is maximally monotone if and only if for each $i\in I$, $\mu A_i$ is maximally monotone, that is, ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is maximally monotone if and only if for each $i\in I$, $A_i$ is maximally monotone. Finally, applying Minty’s parametrization we obtain $$\begin{aligned} {\ensuremath{\operatorname{gra}}}\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}&=\Big\{(J_{\mu {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}(x),\ x-J_{\mu {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}x)\ \Big|\ x\in{\ensuremath{\operatorname{dom}}}J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}\Big\}\\ &= \bigg\{\Big(\sum_{i\in I}\lambda_i J_{\mu A_{i}}x,\ \sum_{i\in I}\lambda_i({\ensuremath{\operatorname{Id}}}-J_{\mu A_{i}})x\Big)\ \bigg|\ x \in \bigcap_{i\in I}{\ensuremath{\operatorname{dom}}}J_{\mu A_{i}}\bigg\}\\ &=\bigg\{\sum_{i\in I}\lambda_i\big( J_{\mu A_{i}}x,\ ({\ensuremath{\operatorname{Id}}}-J_{\mu A_{i}})x\big)\ \bigg|\ x \in \bigcap_{i\in I}{\ensuremath{\operatorname{dom}}}J_{\mu A_{i}}\bigg\}\ \subseteq\ \sum_{i\in I} \lambda_i {\ensuremath{\operatorname{gra}}}\mu A_i.\end{aligned}$$ which implies inclusion . As an example, we now discuss the case where we average a monotone mapping with its inverse. It was observed in [@BGLW Example 5.3] that when $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex, then $$\label{proximal average of function with its conjugate} p(f,f^*)=q.$$ Employing equation  we see that $$\mathcal{R}(\partial f,\partial f^*)=\partial p(f,f^*)=\partial q={\ensuremath{\operatorname{Id}}}.$$ We now extend the reach of this fact in order for it to hold in the framework of the resolvent average of monotone operators. To this end we first note the following fact: \[self dual is I\] Let $R:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ be a set-valued and monotone mapping such that $R=R^{-1}$. Then $ {\ensuremath{\operatorname{gra}}}R\subseteq {\ensuremath{\operatorname{gra}}}{\ensuremath{\operatorname{Id}}}$. Consequently, if $R$ is maximally monotone, then $R={\ensuremath{\operatorname{Id}}}$. Suppose that $(x,u)\in{\ensuremath{\operatorname{gra}}}R$, then we also have $(u,x)\in{\ensuremath{\operatorname{gra}}}R$. Because of the monotonicity of $R$ we now have $0\leq{\langle{{x-u},{u-x}}\rangle}=-\|x-u\|^2\leq 0$, that is, $x=u$. \[resolvent of mapping with inverse\] The mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows {\ensuremath{\mathcal H}}$ is maximally monotone if and only if $\mathcal{R}(A,A^{-1})={\ensuremath{\operatorname{Id}}}$. If $\mathcal{R}(A,A^{-1})={\ensuremath{\operatorname{Id}}}$, then since ${\ensuremath{\operatorname{Id}}}$ is maximally monotone, it is clear from Theorem \[neededlater\] that $A$ is maximally monotone. Conversely, suppose that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. Then Theorem \[mainresult\] guarantees that $\big(\mathcal{R}(A,A^{-1})\big)^{-1}=\mathcal{R}(A^{-1},A)=\mathcal{R}(A,A^{-1})$ while Theorem \[neededlater\] guarantees that $\mathcal{R}(A,A^{-1})$ is maximally monotone. Consequently, Proposition \[self dual is I\] implies that $\mathcal{R}(A,A^{-1})={\ensuremath{\operatorname{Id}}}$. Now we address domain and range properties of the resolvent average. A natural question is whether more precise Formulae than Formulae  can be attained. A precise formula for the domain of the proximal average of functions is [@BGLW Theorem 4.6]. It asserts that given proper, convex and lower semicontinuous functions $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}},\ i\in I$, then $${\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})=\sum_{i\in I}\lambda_i{\ensuremath{\operatorname{dom}}}f_i.$$ As we shall see shortly, such precision does not hold in general for the resolvent average. However, we now aim at obtaining *nearly* precise formulae for the domain and range of the resolvent average. To this end, we first recall that the *relative interior* of a subset $C$ of ${\ensuremath{\mathcal H}}$, which is denoted by ${\ensuremath{\operatorname{ri}}}C$, is the subset of ${\ensuremath{\mathcal H}}$ which is obtained by taking the interior of $C$ when considered a subset of its closed affine hull. We say that the two subsets $C$ and $D$ of ${\ensuremath{\mathcal H}}$ are *nearly equal* if $\overline{C}=\overline{D}$ and ${\ensuremath{\operatorname{ri}}}C={\ensuremath{\operatorname{ri}}}D$. In this case we write $C\simeq D$. In order to prove our near precise domain and range formulae we will employ the following fact which is an extension (to the case where we sum arbitrary finitely many rectangular mappings) of the classical and far reaching result of Brezis and Haraux [@BH] regarding the range of the sum of two rectangular mappings: [[@Pen Corollary 6]]{}\[Pennanen\] Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is monotone and rectangular. If $A=\sum_{i\in I}A_i$ is maximally monotone, then $$\label{BH near equality} {\ensuremath{\operatorname{ran}}}\sum_{i\in I}A_i\simeq\sum_{i\in I}{\ensuremath{\operatorname{ran}}}{A_i}.$$ \[domain and range of the resolvent average\] Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. Then $$\label{finite dimension domain and range of the resolvent average} {\ensuremath{\operatorname{ran}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\simeq\sum_{i\in I}\lambda_i{\ensuremath{\operatorname{ran}}}A_i,\ \ \ \ \ \ \ {\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\simeq\sum_{i\in I}\lambda_i{\ensuremath{\operatorname{dom}}}A_i.$$ We first recall that a firmly nonexpansive mapping $T:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is maximally monotone (see [@BC2011 Example 20.27]) and rectangular (see [@BC2011 Example 24.16]). Thus, we see that all of the mappings $J_{\mu A_i},\ i\in I$ as well as the firmly nonexpansive mapping $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_iJ_{\mu A_i}$ are maximally monotone and rectangular. As a consequence, we may now apply Fact \[Pennanen\] in order to obtain $${\ensuremath{\operatorname{ran}}}J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}\simeq\sum_{i\in I}{\ensuremath{\operatorname{ran}}}\lambda_i J_{\mu A_i}=\sum_{i\in I}\lambda_i{\ensuremath{\operatorname{ran}}}J_{\mu A_i}.$$ Since, given a maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$, we have ${\ensuremath{\operatorname{dom}}}A={\ensuremath{\operatorname{dom}}}\mu A={\ensuremath{\operatorname{ran}}}J_{\mu A}$ (see Minty’s parametrization ), we arrive at the domain formula in . We now combine the domain near equality with Theorem \[mainresult\] in order to obtain $${\ensuremath{\operatorname{ran}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}={\ensuremath{\operatorname{dom}}}({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}})^{-1}={\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}\simeq\sum_{i\in I}\lambda_i{\ensuremath{\operatorname{dom}}}A_i^{-1}=\sum_{i\in I}\lambda_i{\ensuremath{\operatorname{ran}}}A_i.$$ We see that by employing the resolvent average we avoid constraint qualifications we had while employing the arithmetic average, one of the most obvious of which is we can now average mappings the domains of which do not intersect. At this point we demonstrate why even in the finite-dimensional case the near equality in  cannot be replaced by an equality. To this end we consider the following example. [@BBBRW Example 5.4] discussed the function $f:{\ensuremath{\mathbb R}}^2\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$, defined by $$f(x,y)= \begin{cases} -\sqrt{xy}, & x\geq 0,y\geq 0; \\ +\infty, & \text{otherwise.} \end{cases}$$ In [@BBBRW] the function $f$ was presented as an example of a proper, lower semicontinuous and sublinear function which is not subdifferentiable at certain points of the boundary of its domain, where the boundary is $({\ensuremath{\mathbb R}}_+\times\{0\})\cup(\{0\}\times{\ensuremath{\mathbb R}}_+)$. In fact, it was observed in [@BBBRW] that the only point on the boundary of ${\ensuremath{\operatorname{dom}}}\partial f$ which belongs to ${\ensuremath{\operatorname{dom}}}\partial f$ is the origin. We now consider the function $g:{\ensuremath{\mathbb R}}^2\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ defined by $$g(x,y)=\max\{f(1-x,y),f(1+x,y)\}= \begin{cases} -\sqrt{(1-|x|)y}, & -1\leq x\leq 1 ,0\leq y; \\ +\infty, & \text{otherwise.} \end{cases}$$ We set $D={\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}g=\{(x,y)|\ -1<x<1,\ 0<y\}$. Then it follows that $g$ is lower semicontinuous, convex and ${\ensuremath{\operatorname{dom}}}\partial g=D\cup\{(-1,0),(1,0)\}$. Now, we set $n=2$, $A_1=A_2=\partial g,\ \mu=1,\ 0<\lambda<1,\ \lambda_1=\lambda$ and $\lambda_2=1-\lambda$, then ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=\partial g$ (see formula ) and hence $$\begin{aligned} {\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}&={\ensuremath{\operatorname{dom}}}\partial g=D\cup\{(-1,0),(1,0)\}\\ &\neq D\cup\{(-1,0),(1-2\lambda,0),(2\lambda-1,0),(1,0)\}=\lambda_1{\ensuremath{\operatorname{dom}}}A_1+\lambda_2{\ensuremath{\operatorname{dom}}}A_2.\end{aligned}$$ Letting $A_1=A_2=\partial g^*=(\partial g)^{-1}$ yields the same inequality with ranges instead of domains. Finally, we note that equality fails already in , that is, since $\partial g^*$ is the subdifferential of a proper, lower semicontinuous and convex function, it is rectangular (see [@BC2011 Example 24.9]), maximally monotone and we have $${\ensuremath{\operatorname{ran}}}(\partial g^*+\partial g^*)=2D\cup\{(-2,0),(2,0)\}\neq2D\cup\{(-2,0),(0,0),(2,0)\}={\ensuremath{\operatorname{ran}}}\partial g^*+{\ensuremath{\operatorname{ran}}}\partial g^*.$$ (For another example of this type, see [@BM Example 3.14]). The Fitzpatrick function of the resolvent average ------------------------------------------------- We relate the Fitzpatrick function of the resolvent average with the Fitzpatrick functions of the averaged mappings in the following result: Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. Then $$\label{resolvent average Fitzpatrick function inequality} F_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}\leq\sum_{i\in I}\lambda_i F_{\mu A_i}\ \ \ \text{in particular},\ \ \ F_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}\leq\sum_{i\in I}\lambda_i F_{ A_i}$$ and $$\label{resolvent average Fitzpatrick function domain} \sum_{i\in I}\lambda_i{\ensuremath{\operatorname{dom}}}F_{\mu A_i}\subseteq{\ensuremath{\operatorname{dom}}}F_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}\ \ \ \text{in particular},\ \ \ \sum_{i\in I}\lambda_i{\ensuremath{\operatorname{dom}}}F_{ A_i}\subseteq{\ensuremath{\operatorname{dom}}}F_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}.$$ For each $\in I$, let $x_i,u_i$ and $z$ be points in ${\ensuremath{\mathcal H}}$ and let $T_i=J_{\mu A_i}$. We set $(x,u)=\sum_{i\in I}\lambda_i(x_i,u_i)$ and $T=\sum_{i\in I}\lambda_i T_i=J_{\mu {\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}$. Then we employ equation  in order to obtain $$\begin{aligned} &{\langle{{z-Tz},{x}}\rangle}+{\langle{{u},{Tz}}\rangle}-{\langle{{z-Tz},{Tz}}\rangle} =\sum_{i\in I}\lambda_i\big({\langle{{z-T_i z},{x_i}}\rangle}+{\langle{{u_i},{T_iz}}\rangle}-{\langle{{z-T_iz},{T_iz}}\rangle}\big)\nonumber\\ &-\sum_{(i,j)\in I\times I}\frac{\lambda_i\lambda_j}{2}\big({\langle{{T_jz-T_iz},{x_i-x_j}}\rangle}+{\langle{{u_i-u_j},{T_iz-T_jz}}\rangle}-{\langle{{T_jz-T_iz},{T_iz-T_jz}}\rangle}\big)\nonumber\\ =&\sum_{i\in I}\lambda_i\big({\langle{{z-T_i z},{x_i}}\rangle}+{\langle{{u_i},{T_iz}}\rangle}-{\langle{{z-T_iz},{T_iz}}\rangle}\big)\nonumber\\ &-\sum_{(i,j)\in I\times I}\frac{\lambda_i\lambda_j}{2}\bigg(-\frac{1}{4}\|(u_i-u_j)-(x_i-x_j)\|^2+\Big\|\frac{(u_i-u_j)-(x_i-x_j)}{2}+(T_iz-T_jz)\Big\|^2\bigg)\nonumber\\ \leq&\sum_{i\in I}\lambda_i\big({\langle{{z-T_i z},{x_i}}\rangle}+{\langle{{u_i},{T_iz}}\rangle}-{\langle{{z-T_iz},{T_iz}}\rangle}\big)+\sum_{(i,j)\in I\times I}\frac{\lambda_i\lambda_j}{8}\|(u_i-u_j)-(x_i-x_j)\|^2.\label{Fitzpatrick inequality}\end{aligned}$$ Given a maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$, combining the definition of the Fitzpatrick function $F_A$ with Minty’s parametrization  implies that $$F_A (x,u)=\sup_{z\in{\ensuremath{\mathcal H}}}\big({\langle{{z-J_A z},{x}}\rangle}+{\langle{{u},{J_A z}}\rangle}-{\langle{{z-J_A z},{J_A z}}\rangle}\big).$$ Thus, by employing inequality  we obtain $$\label{Fitzpatrick function inequality} F_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}(x,u)\leq\sum_{i\in I}\lambda_i F_{\mu A_i}(x_i,u_i)+\sum_{(i,j)\in I\times I}\frac{\lambda_i\lambda_j}{8}\|(u_i-u_j)-(x_i-x_j)\|^2.$$ For each $i\in I$, letting $(x_i,u_i)=(x,u)$ in inequality  we arrive at inequality . For each $i\in I$, letting $(x_i,u_i)\in{\ensuremath{\operatorname{dom}}}F_{\mu A_i}$ in inequality  we see that $F_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}(x,u)<\infty$, that is, we obtain inclusions  and complete the proof. Dominant properties of the resolvent average {#dominant} ============================================ Domain and range properties --------------------------- The following domain and range properties of the resolvent average are immediate consequences of Theorem \[domain and range of the resolvent average\]: \[t:FDdom\] ***(nonempty interior of  the domain,  fullness of the domain and surjectivity are dominant)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. 1. \[t:NEdom\] If there exists $i_0\in I$ such that ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}A_{i_0}\neq\varnothing$, then ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\neq\varnothing$. 2. \[t:Fdom\] If there exists $i_0\in I$ such that ${\ensuremath{\operatorname{dom}}}A_{i_0}={\ensuremath{\mathcal H}}$, then ${\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}={\ensuremath{\mathcal H}}$. 3. \[t:surjdom\] If there exists $i_0\in I$ such that $A_{i_0}$ is surjective, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is surjective. \[t:FDdom functions\] ***(nonempty interior and fullness of domain are dominant w.r.t. $p_\mu$)*** Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. If there exists $i_0\in I$ such that ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}f_{i_0}\neq\varnothing$, then ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})\neq\varnothing$. If there exists $i_0\in I$ such that ${\ensuremath{\operatorname{dom}}}f_{i_0}={\ensuremath{\mathcal H}}$, then ${\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})={\ensuremath{\mathcal H}}$. For each $i\in I$, we set $A_i=\partial f_i$. Suppose that for some $i_0\in I$, ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}f_{i_0}\neq\varnothing$. Since ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}f_{i_0}\subseteq{\ensuremath{\operatorname{dom}}}\partial f_{i_0}$, we see that ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}A_{i_0}\neq\varnothing$ and it now follows from Theorem \[t:FDdom\]\[t:NEdom\] that ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\neq\varnothing$. Now equation  implies that $\varnothing\neq{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}={\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}\partial p_\mu({\bf f},{\boldsymbol \lambda})\subseteq{\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})$. A similar argument implies that if ${\ensuremath{\operatorname{dom}}}f_{i_0}={\ensuremath{\mathcal H}}$, then ${\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})={\ensuremath{\mathcal H}}$. We recall that the function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is said to be *coercive* if $\lim_{\|x\|\to\infty}f(x)=\infty$. The function $f$ is said to be *supercoercive* if $f/\|\cdot\|$ is coercive. As a consequence of Theorem \[t:FDdom\]\[t:surjdom\] in finite-dimensional spaces we obtain the following result: ***(supercoercivity is dominant w.r.t.$\ p_\mu$ in ${\ensuremath{\mathbb R}}^n$)*** [[@GHW Lemma 3.1(iii)]]{} Suppose that ${\ensuremath{\mathcal H}}$ is finite-dimensional and that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. If there exists $i_0\in I$ such that $f_{i_0}$ is supercoercive, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is supercoercive. For each $i\in I$, we set $A_i=\partial f_i$. We now recall that in finite-dimensional spaces, a proper, lower semicontinuous and convex function $f$ is supercoercive if and only if ${\ensuremath{\operatorname{dom}}}f^*={\ensuremath{\mathcal H}}$, which is equivalent to ${\ensuremath{\mathcal H}}={\ensuremath{\operatorname{dom}}}\partial f^*={\ensuremath{\operatorname{ran}}}\partial f$ (combine [@Rock Corollary 13.3.1] with [@Rock Corollary 14.2.2], or, alternatively, see [@BB1997 Proposition 2.16]). Thus, since ${\ensuremath{\operatorname{ran}}}A_{i_0}={\ensuremath{\mathcal H}}$, then Theorem \[t:FDdom\]\[t:surjdom\] together with equation guarantee that ${\ensuremath{\operatorname{ran}}}\partial p_\mu({\bf f},{\boldsymbol \lambda})={\ensuremath{\operatorname{ran}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}={\ensuremath{\mathcal H}}$. Consequently, we see that $p_\mu({\bf f},{\boldsymbol \lambda})$ is supercoercive. Single-valuedness ----------------- We shall say that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is at most single-valued if for every $x\in{\ensuremath{\mathcal H}}$, $Ax$ is either empty or a singleton. \[single valued\] Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. If there exists $i_0\in I$ such that $A_{i_0}$ is at most single-valued, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is at most single-valued. We recall that a maximally monotone mapping is at most single-valued if and only if its resolvent is injective (see [@bmw12 Theorem 2.1(iv)]). Thus, if $A_{i_0}$ is at most single-valued, then $\mu A_{i_0}$ is at most single-valued and $J_{\mu A_{i_0}}$ is injective. We now apply Corollary \[equality in convex combination of firmly nonexpansive\]\[firmly injective\] in order to conclude that $ J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_{i}J_{\mu A_{i}}$ is injective and, consequently, that $\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is at most single-valued and so is ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$. Recall that the proper, lower semicontinuous and convex function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is said to be *essentially smooth* if the interior of its domain is nonempty and if $\partial f$ is at most single-valued. This definition of essential smoothness coincides with the classical notions of the same name in finite-dimensional spaces (see the paragraph below preceding Corollary \[ess conv\]). In terms of essential smoothness of the proximal average, we recover the following result: *[@BGLW Corollary 7.7]*\[ess smooth\] Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. If there exists $i_0\in I$ such that $f_{i_0}$ is essentially smooth, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is essentially smooth. For each $i\in I$, we set $A_i=\partial f_i$. Suppose that for some $i_0\in I$, $f_{i_0}$ is essentially smooth, then $A_{i_0}=\partial f_{i_0}$ is at most single-valued and ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}A_{i_0}\neq\varnothing$. It follows follows from Corollary \[t:FDdom functions\] that ${\ensuremath{\operatorname{int}}}{\ensuremath{\operatorname{dom}}}p_\mu({\bf f},{\boldsymbol \lambda})\neq\varnothing$. Furthermore, Theorem \[single valued\] guarantees that $\partial p_\mu({\bf f},{\boldsymbol \lambda})={\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is at most single-valued. Consequently, $p_\mu({\bf f},{\boldsymbol \lambda})$ is essentially smooth. Strict monotonicity ------------------- Recall that $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is said to be *strictly monotone* if whenever $u\in Ax$ and $v\in Ay$ are such that $x\neq y$, then $0<{\langle{{u-v},{x-y}}\rangle}$. The resolvent perspective of this issue is addressed in the following proposition. \[averaged strictly firmly\] Suppose that for each $i\in I$, $T_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is firmly nonexpansive and set $T=\sum_{i\in I}\lambda_i T_i$. If there exits $i_0\in I$ such that $$\label{strictly firmly definition} T_{i_0}x\neq T_{i_0}y\ \ \ \ \ \Rightarrow\ \ \ \ \ \|T_{i_0}x-T_{i_0}y\|^2<{\langle{{x-y},{T_{i_0}x-T_{i_0}y}}\rangle},$$ then $T$ has property  as well. Suppose that $x$ and $y$ are points in ${\ensuremath{\mathcal H}}$ such that $\|Tx-Ty\|^2 = {\langle{{x-y},{Tx-Ty}}\rangle}$. Then Corollary \[equality in convex combination of firmly nonexpansive\] implies that $Tx-Ty=T_{i_0} x-T_{i_0} y$. In particular, we see that $$\|T_{i_0}x-T_{i_0}y\|^2=\|Tx-Ty\|^2 = {\langle{{x-y},{Tx-Ty}}\rangle}={\langle{{x-y},{T_{i_0}x-T_{i_0}y}}\rangle}.$$ Consequently, since $T_{i_0}$ has property , we see that $Tx-Ty=T_{i_0} x-T_{i_0} y=0$, as claimed. \[t:strmonodom\] ***(strict monotonicity is dominant)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. If there exists $i_0\in I$ such that $A_{i_0}$ is strictly monotone, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is strictly monotone. We recall that the maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is strictly monotone if and only if $J_A$ has property  (see [@bmw12 Theorem 2.1(vi)]). Thus, since $J_{\mu A_{i_0}}$ has property , then Proposition \[averaged strictly firmly\] guarantees that $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_{i}J_{\mu A_{i}}$ has property , which, in turn, implies that $\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is strictly monotone and, therefore, so is ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$. Recall that the proper, lower semicontinuous and convex function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is said to be *essentially strictly convex* if $f^*$ is essentially smooth; $f$ is said to be *Legendre* if $f$ is both, essentially smooth and essentially strictly convex. These definitions of essential strict convexity and Legendreness coincide with the classical notions of the same names in finite-dimensional spaces (see the next paragraph). Now suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. We suppose further that there exists $i_0\in I$ such that $f_{i_0}$ is essentially strictly convex. Then $f^*_{i_0}$ is essentially smooth. Consequently, Corollary \[ess smooth\] guarantees that $p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})$ is essentially smooth. Thus, according to formula , the function $p_\mu({\bf f},{\boldsymbol \lambda})=(p_\mu({\bf f},{\boldsymbol \lambda}))^{**}=(p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda}))^*$ is essentially convex. Consequently, we see that essential strict convexity is dominant w.r.t. the proximal average. This line of proof of this fact was carried out in [@BGLW]. Since in the present paper formula , up to an additive constant, was recovered by Theorem \[mainresult\], and since essential strict convexity is not affected by the addition of a constant to the function, we see that our discussion here does, indeed, recover the dominance of essential strict convexity w.r.t. the proximal average. Classically, when ${\ensuremath{\mathcal H}}$ is finite-dimensional, a different path that leads to the same conclusion is now available. Indeed, in this case, recall that the proper, lower semicontinuous and convex function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is said to be essentially smooth if the interior of its domain is nonempty, if $f$ is Gâteaux differentiable there and the norm of its Gâteaux gradient blows up as we tend from the interior to a point on the boundary of its domain. $f$ is said to be essentially strictly convex if $f$ is strictly convex on every convex subset of ${\ensuremath{\operatorname{dom}}}\partial f$, which is equivalent to $\partial f$ being strictly monotone (see [@RockWets Theorem 12.17]). Thus, given that $f_{i_0}$ is essentially strictly convex, then $\partial f_{i_0}$ is strictly monotone. Setting $A_i=\partial f_i$ for every $i\in I$, Theorem \[t:strmonodom\] guarantees that $\partial p_\mu({\bf f},{\boldsymbol \lambda})={\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is strictly monotone and, consequently, that $p_\mu({\bf f},{\boldsymbol \lambda})$ is essentially strictly convex, as asserted. Summing up both of these discussions, we have recovered the following result: \[ess conv\]***(essential strict convexity is dominant w.r.t. $p_\mu$)*** *[@BGLW Corollary 7.8]* Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. If there exists $i_0\in I$ such that $f_{i_0}$ is essentially strictly convex, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is essentially strictly convex. Combining Corollary \[ess smooth\] together with Corollary \[ess conv\], we recover the following result: *[@BGLW Corollary 7.9]*\[Leg\] Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. If there exists $i_1\in I$ such that $f_{i_1}$ is essentially smooth and there exists $i_2\in I$ such that $f_{i_2}$ is essentially strictly convex, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is Legendre. In particular, if there exists $i_0\in I$ such that $f_{i_0}$ is Legendre, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is Legendre. Uniform monotonicity -------------------- We say that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is monotone with modulus ${\ensuremath{\varphi}}:[0,\infty[\to[0,\infty]$ if for every two points $(x,u)$ and $(y,v)$ in ${\ensuremath{\operatorname{gra}}}A$, $${\ensuremath{\varphi}}\big(\|x-y\|\big)\leq{\langle{{u-v},{x-y}}\rangle}.$$ Clearly, if ${\ensuremath{\operatorname{gra}}}A\neq\varnothing$, then ${\ensuremath{\varphi}}(0)=0$. We recall that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is said to be *uniformly monotone* with modulus ${\ensuremath{\varphi}}$, ${\ensuremath{\varphi}}$-uniformly monotone for short, if $A$ is monotone with modulus ${\ensuremath{\varphi}}$ and ${\ensuremath{\varphi}}(t)=0 \Leftrightarrow t=0$. We say that the mapping $T:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}:[0,\infty[\to[0,\infty]$ if for every pair of points $x$ and $y$ in ${\ensuremath{\mathcal H}}$, $$\|Tx-Ty\|^2+{\ensuremath{\varphi}}\big(\|Tx-Ty\|\big)\leq{\langle{{Tx-Ty},{x-y}}\rangle}.$$ We also recall that the mapping $T:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is said to be *uniformly firmly nonexpansive* with modulus ${\ensuremath{\varphi}}$, ${\ensuremath{\varphi}}$-uniformly firmly nonexpansive for short, if $T$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}$ and ${\ensuremath{\varphi}}(t)=0 \Leftrightarrow t=0$. For the sake of convenience, we will identify a modulus ${\ensuremath{\varphi}}:[0,\infty[\to[0,\infty]$ with ${\ensuremath{\varphi}}(|\cdot|)$, its symmetric extension to ${\ensuremath{\mathbb R}}$. With this convention, when we say that ${\ensuremath{\varphi}}$ is increasing we mean that $0\leq t_1<t_2\ \Rightarrow\ {\ensuremath{\varphi}}(t_1)\leq{\ensuremath{\varphi}}(t_2)$. \[proximal average of positive functions which vanish at zero\] Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to[0,\infty]$ is proper, lower semicontinuous, convex and $f_i(0)=0$. Then $p_\mu({\bf f},{\boldsymbol \lambda}):{\ensuremath{\mathcal H}}\to[0,\infty]$ is proper, lower semicontinuous, convex and $p_\mu({\bf f},{\boldsymbol \lambda})(0)=0$. If there exists $i_0\in I$ such that $f_{i_0}(x)=0\Leftrightarrow x=0$, then $p_\mu({\bf f},{\boldsymbol \lambda})(x)=0\Leftrightarrow x=0$. Fact \[resolvents of subs and proximal reformulation\] guarantees that $p_\mu(\boldsymbol{f},\boldsymbol{\lambda})$ is lower semicontinuous, convex and that for every $x\in{\ensuremath{\mathcal H}}$, $$\label{applied proximal reformulation} p_\mu(\bold{f},\boldsymbol{\lambda})(x)=\inf_{\sum_{i\in I}\lambda_j y_j=x}\sum_{i\in I}\lambda_if_i(y_i)+\frac{1}{\mu}\bigg(\Big(\sum_{i\in I}\lambda_iq(y_i)\Big)-q(x)\bigg).$$ Since the bracketed term in  is greater or equal to zero and vanishes when for each $i\in I$, $y_i=x=0$, and since for each $i\in I$, $f_i$ is a function which is greater or equal to zero and which vanishes at zero, formula  implies that $p_\mu(\boldsymbol{f},\boldsymbol{\lambda})$ is greater or equal to zero and vanishes at zero. Finally, for each $i\in I$, we now set $A_i=\partial f_i$. Since for each $i\in I$, 0 is a minimizer of $f_i$, we see that $\bigcap_{i\in I}A_i^{-1}(0)\neq\varnothing$. If $f_{i_0}(x)=0\Leftrightarrow x=0$, then $\{0\}=A^{-1}_{i_0}(0)=\bigcap_{i\in I}A_i^{-1}(0)$. Consequently, equation  together with equation  imply that $\partial p_\mu(\boldsymbol{f},\boldsymbol{\lambda})^{-1}(0)=\mathcal{R}_\mu(\bold{A},\boldsymbol{\lambda})^{-1}(0)=\{0\}$. We conclude that 0 is the only minimizer of $p_\mu(\boldsymbol{f},\boldsymbol{\lambda})$ and, therefore, $p_\mu(\boldsymbol{f},\boldsymbol{\lambda})(x)=0\Leftrightarrow x=0$. \[uniform firmly dominance\] Suppose that for each $i\in I$, $T_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}_i$ which is lower semicontinuous and convex and set $T=\sum_{i\in I}\lambda_i T_i$. Then $T$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}=p_{\frac{1}{2}}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$ which is proper, lower semicontinuous and convex. In particular, if there exists $i_0\in I$ such that $T_{i_0}$ is ${\ensuremath{\varphi}}_{i_0}$-uniformly firmly nonexpansive, then $T$ is ${\ensuremath{\varphi}}$-uniformly firmly nonexpansive. Fact \[resolvents of subs and proximal reformulation\] guarantees that ${\ensuremath{\varphi}}=p_{\frac{1}{2}}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$ is lower semicontinuous, convex and that for every $t\in[0,\infty[$, $$\label{real line proximal average reformulation} {\ensuremath{\varphi}}(t)=p_{\frac{1}{2}}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})(t)=\inf_{\sum_{i\in I}\lambda_i t_i=t}\sum_{i\in I}\lambda_i{\ensuremath{\varphi}}_i(t_i)+2\bigg(\Big(\sum_{i\in I}\lambda_iq(t_i)\Big)-q(t)\bigg).$$ Proposition \[proximal average of positive functions which vanish at zero\] guarantees that ${\ensuremath{\varphi}}$ is greater or equal to zero and vanishes at zero. Furthermore, if ${\ensuremath{\varphi}}_{i_0}$ vanishes only at zero, then ${\ensuremath{\varphi}}$ vanishes only at zero. Since ${\ensuremath{\varphi}}$ is convex, we now see that it is increasing. Now, let $x$ and $y$ be points in ${\ensuremath{\mathcal H}}$. Then by employing formula  we obtain the following evaluation $$\begin{aligned} {\ensuremath{\varphi}}\big(\|Tx-Ty\|\big)&\leq{\ensuremath{\varphi}}\Big(\sum_{i\in I}\lambda_i\|T_ix-T_iy\|\Big)\nonumber\\ &\leq\sum_{i\in I}\lambda_i{\ensuremath{\varphi}}_i\big(\|T_ix-T_iy\|\big)+\sum_{i\in I}\lambda_i\|T_ix-T_iy\|^2-\Big(\sum_{i\in I}\lambda_i\|T_ix-T_iy\|\Big)^2\nonumber\\ &\leq\sum_{i\in I}\lambda_i{\ensuremath{\varphi}}_i\big(\|T_ix-T_iy\|\big)+\sum_{i\in I}\lambda_i\|T_ix-T_iy\|^2-\Big\|\sum_{i\in I}\lambda_i(T_ix-T_iy)\Big\|^2\end{aligned}$$ which implies that $$\begin{aligned} {\ensuremath{\varphi}}\big(\|Tx-Ty\|\big)+\|Tx-Ty\|^2&\leq\sum_{i\in I}\lambda_i\Big({\ensuremath{\varphi}}_i\big(\|T_ix-T_iy\|\big)+\|T_ix-T_iy\|^2\Big)\\ &\leq\sum_{i\in I}\lambda_i{\langle{{T_ix-T_iy},{x-y}}\rangle}={\langle{{Tx-Ty},{x-y}}\rangle}.\end{aligned}$$ Thus, we see that $T$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}$. As a consequence, we obtain the following result: \[uniform monotonicity with 1-coercive modulus is recessive\] ***(uniform monotonicity with a convex modulus is dominant)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone with modulus ${\ensuremath{\varphi}}_i$ which is lower semicontinuous and convex. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is monotone with modulus ${\ensuremath{\varphi}}=p_{\frac{\mu}{2}}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$ which is lower semicontinuous and convex. In particular, if there exists $i_0\in I$ such that $A_{i_0}$ is ${\ensuremath{\varphi}}_{i_0}$-uniformly monotone, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is ${\ensuremath{\varphi}}$-uniformly monotone. First we consider the case $\mu=1$. To this end we will employ the fact that the maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is monotone with modulus ${\ensuremath{\varphi}}$ if and only if $J_A$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}$. Indeed, employing Minty’s parametrization , we see that $A$ is uniformly monotone with modulus ${\ensuremath{\varphi}}$ if and only if for every $x$ and $y$ in ${\ensuremath{\mathcal H}}$ we have $ {\ensuremath{\varphi}}\big(\|J_Ax-J_Ay\|\big)\leq{\langle{{J_Ax-J_Ay},{(x-y)-(J_Ax-J_Ay)}}\rangle} $ which is precisely the firm nonexpansiveness of $J_A$ with modulus ${\ensuremath{\varphi}}$. Thus, we see that for each $i\in I$, $J_{A_i}$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}_i$. Consequently, Proposition \[uniform firmly dominance\] guarantees that $J_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_i J_{A_i}$ is firmly nonexpansive with modulus ${\ensuremath{\varphi}}=p_{\frac{1}{2}}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$, which, in turn, implies that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is monotone with modulus ${\ensuremath{\varphi}}$. For an arbitrary $0<\mu$, we employ formulae  and as follows: any mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is ${\ensuremath{\varphi}}$-monotone if and only if $\mu A$ is $\mu{\ensuremath{\varphi}}$-monotone. Thus, since we already have that $\mathcal{R}(\mu\bold{A},\boldsymbol{\lambda})$ is $p_{\frac{1}{2}}(\mu\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$-monotone, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=\mu^{-1}\mathcal{R}(\mu\bold{A},\boldsymbol{\lambda})$ is ${\ensuremath{\varphi}}$-monotone where ${\ensuremath{\varphi}}=\mu^{-1} p_{\frac{1}{2}}(\mu\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})=\frac{2}{\mu} p(\frac{\mu}{2}\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})=p_{\frac{\mu}{2}}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$. Finally, Proposition \[proximal average of positive functions which vanish at zero\] guarantees that if ${\ensuremath{\varphi}}_{i_0}$ vanishes only at zero, then ${\ensuremath{\varphi}}$ vanishes only at zero. We recall (see [@Zal Section 3.5]) that the proper function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is said to be *uniformly convex* if there exists a function ${\ensuremath{\varphi}}:[0,\infty[\to[0,\infty]$ with the property ${\ensuremath{\varphi}}(t)=0 \Leftrightarrow t=0$ such that for every two points $x$ and $y$ in ${\ensuremath{\mathcal H}}$ and every $\lambda\in\ ]0,1[$, $$\label{uniform convexity} f\big((1-\lambda)x+\lambda y\big)+\lambda(1-\lambda){\ensuremath{\varphi}}\big(\|x-y\|\big)\leq(1-\lambda)f(x)+\lambda f(y).$$ The largest possible function ${\ensuremath{\varphi}}$ satisfying  is called the *gauge of uniform convexity* of $f$ and is defined by $${\ensuremath{\varphi}}_f(t)=\inf\bigg\{\frac{(1-\lambda)f(x)+\lambda f(y)-f((1-\lambda)x+\lambda y)}{\lambda(1-\lambda)}\ \bigg|\ \lambda\in\ ]0,1[,\ x,y\in{\ensuremath{\operatorname{dom}}}f,\ \|x-y\|=t \bigg\}.$$ We also recall that the proper and convex function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is said to be *uniformly smooth* if there exists a function $\psi:[0,\infty[\to[0,\infty]$ with the property $\lim_{t\to 0}{\psi(t)}/{t}=0$ such that for every two points $x$ and $y$ in ${\ensuremath{\mathcal H}}$ and every $\lambda\in\ ]0,1[$ such that $(1-\lambda)x+\lambda y\in{\ensuremath{\operatorname{dom}}}f$, $$\label{uniform smoothness} f\big((1-\lambda)x+\lambda y\big)+\lambda(1-\lambda)\psi\big(\|x-y\|\big)\geq(1-\lambda)f(x)+\lambda f(y).$$ The smallest possible function $\psi$ satisfying  is called the *gauge of uniform smoothness* of $f$ and is defined by $$\psi_f(t)=\sup\bigg\{\frac{(1-\lambda)f(x)+\lambda f(y)-f((1-\lambda)x+\lambda y)}{\lambda(1-\lambda)}\ \bigg|\ \begin{array}{c} \lambda\in\ ]0,1[,\ x,y\in{\ensuremath{\mathcal H}},\ \|x-y\|=t, \\ \big((1-\lambda)x+\lambda y\big)\in{\ensuremath{\operatorname{dom}}}f \end{array}\ \bigg\}.$$ \[uniform convexity equivalent to uniform monotonicity\]*[@Zal Theorem 3.5.10, (i)$\Leftrightarrow$(v)]* Suppose that $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. Then $f$ is uniformly convex if and only if $\partial f$ is uniformly monotone, in which case $\partial f$ is ${\ensuremath{\varphi}}$-uniformly monotone with ${\ensuremath{\varphi}}=2{\ensuremath{\varphi}}_f^{**}$. Consequently, we arrive at the following results: \[uniform convexity is dominant or recessive\] ***(uniform convexity is dominant w.r.t. $p_\mu$)*** Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous, convex and there exists $i_0\in I$ such that $f_{i_0}$ is uniformly convex. Then $p_\mu({\bf f},{\boldsymbol \lambda})$ is uniformly convex. For each $i\in I$ we set $A_i=\partial f_i$. Since $f_{i_0}$ is uniformly convex, Fact \[uniform convexity equivalent to uniform monotonicity\] guarantees that there exists a lower semicontinuous and convex modulus ${\ensuremath{\varphi}}_{i_0}$ such that $A_{i_0}$ is ${\ensuremath{\varphi}}_{i_0}$-uniformly monotone. Consequently, Theorem \[uniform monotonicity with 1-coercive modulus is recessive\] together with equation  guarantee that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=\partial p_\mu({\bf f},{\boldsymbol \lambda})$ is uniformly monotone, which, in turn, implies that $p_\mu({\bf f},{\boldsymbol \lambda})$ is uniformly convex. \[gage conjugation\]*[@Zal Theorem 3.5.5]* Suppose that $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. Then *(i)* $f$ is uniformly convex if and only if $f^*$ is uniformly smooth and *(ii)* $f$ is uniformly smooth if and only if $f^*$ is uniformly convex. (Within our reflexive settings *(i)* and *(ii)* are equivalent.) \[uniform smoothness is dominant or recessive\] ***(uniform smoothness is dominant w.r.t. $p_\mu$)*** Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous, convex and there exists $i_0\in I$ such that $f_{i_0}$ is uniformly smooth. Then $p_\mu({\bf f},{\boldsymbol \lambda})$ is uniformly smooth. Fact \[gage conjugation\] guarantees that $f^*_{i_0}$ is uniformly convex. Consequently, by Theorem \[uniform convexity is dominant or recessive\], $p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})$ is uniformly convex. Finally, Applying Fact \[gage conjugation\] together with formula  implies that $p_\mu({\bf f},{\boldsymbol \lambda})=p_\mu({\bf f},{\boldsymbol \lambda})^{**}=p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})^*$ is uniformly smooth. A remark regarding the sharpness of our results is now in order. Under the generality of the hypotheses of Theorem \[uniform monotonicity with 1-coercive modulus is recessive\], even when given that for each $i\in I$, ${\ensuremath{\varphi}}_i$ is the largest possible modulus of monotonicity of $A_i$, it does not hold that ${\ensuremath{\varphi}}=p_{\frac{\mu}{2}}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$ is necessarily the largest possible modulus of monotonicity of ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$. In fact, the largest modulus of monotonicity of ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ cannot, in general, be expressed only in terms of $\mu,\boldsymbol{{\ensuremath{\varphi}}}$ and $\boldsymbol{\lambda}$ but depends also on $\bold{A}$. To this end, we consider the following example which illustrates a similar situation for functions and, consequently, for their subdifferential operators. We will illustrate this issue outside the class of subdifferential operators in Example \[strong sharpness counter outside subdeifferentials\] of the following subsection. \[uniform counter example\] Let $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ be a proper, lower semicontinuous and convex function such that ${\ensuremath{\varphi}}_f=0$ and $\psi_f=\iota_{\{0\}}$. Among a rich variety of choices, one can choose $f=\iota_C$ where $C$ is a closed half-space. For such a choice of $f$, one easily verifies that, indeed, by their definitions, ${\ensuremath{\varphi}}_f=0$ and $\psi_f=\iota_{\{0\}}$. Then it follows that $f$ is neither uniformly convex nor uniformly smooth. Consequently, Fact \[gage conjugation\] implies that also $f^*$ is neither uniformly convex nor uniformly smooth. However, employing formula , we see that $p(f,f^*)=q$ which is both, uniformly convex and uniformly smooth (in fact, as we recall in the following subsection, $q$ is strongly convex and strongly smooth). Thus, we see that ${\ensuremath{\varphi}}_{p_\mu(\bold{f},\boldsymbol{\lambda})}$ and $\psi_{p_\mu(\bold{f},\boldsymbol{\lambda})}$ cannot, in general, be expressed only in terms of $\mu,\boldsymbol{{\ensuremath{\varphi}}_f},\boldsymbol{\psi_f}$ and $\boldsymbol{\lambda}$ since, for example, in the case of $p(f,f)=f$ we have the same weights $\lambda_1=\lambda_2=\frac{1}{2}$, the same $\mu=1$ and the averaged functions have the same gages of uniform convexity and smoothness which are 0 and $\iota_{\{0\}}$, respectively, as in the case $p(f,f^*)=q$. However, ${\ensuremath{\varphi}}_{p(f,f)}=0$ and $\psi_{p(f,f)}=\iota_{\{0\}}$, that is, $p(f,f)$ is neither uniformly convex nor uniformly smooth. Before ending the current discussion a remark regarding the hypothesis of Theorem \[uniform monotonicity with 1-coercive modulus is recessive\] is in order. In the general framework of monotone operators, we are not aware of a study of finer properties of the modulus of uniform monotonicity. In particular, existence of a convex modulus (like in the case of uniformly monotone subdifferential operators for which there exists a modulus which is convex and which possesses also other attractive properties (see [@Zal Section 3.5]), which is crucial in the proof of Theorem \[uniform monotonicity with 1-coercive modulus is recessive\], is unavailable to us. At this point we relegate such a finer study for future research. Strong monotonicity and cocoercivity ------------------------------------ We say that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is *$\epsilon$-monotone*, where $\epsilon\geq 0$, if $A-\epsilon{\ensuremath{\operatorname{Id}}}$ is monotone, that is, if for any two points $(x,u)$ and $(y,v)$ in ${\ensuremath{\operatorname{gra}}}A$, $$\epsilon\|x-y\|^2\leq{\langle{{v-u},{x-y}}\rangle}.$$ In particular, a $0$-monotone mapping is simply a monotone mapping. We now recall that $A$ is said to be strongly monotone with constant $\epsilon$, $\epsilon$-strongly monotone for short, if it is $\epsilon$-monotone with $0<\epsilon$. Clearly, $\epsilon$-monotone implies $\epsilon'$-monotone for any $0\leq\epsilon'<\epsilon$. Letting ${\ensuremath{\varphi}}(t)=\epsilon t^2$, we see that $A$ is $\epsilon$-monotone if and only if $A$ is monotone with modulus ${\ensuremath{\varphi}}$. Thus, the subject matter of our current discussion is a particular case of our discussion in the proceeding subsection. However, in view of the importance of strong monotonicity, cocoercivity, strong convexity, strong smoothness and Lipschitzness of the gradient (all which will be defined shortly), we single out these subjects and treat them separately. Moreover, in the present discussion we add quantitative information in terms of explicit constants of the above properties. To this end, we will make use of the following notations and conventions. First we fix the following conventions in $[0,\infty]$: $0^{-1}=\infty,\ \infty^{-1}=0$ and $0\cdot\infty=0$. For $S\subseteq{\ensuremath{\mathcal H}}$ we fix $\infty\cdot S={\ensuremath{\mathcal H}}$ if $0\in S$ and $\infty\cdot S=\varnothing$ if $0\notin S$. We will apply these conventions in the case where ${\ensuremath{\mathcal H}}={\ensuremath{\mathbb R}}$ in order to calculate terms of the form $\alpha{\ensuremath{\operatorname{Id}}}$ and $\alpha q$ where $\alpha\in[0,\infty]$, however, these calculation hold in any Hilbert space ${\ensuremath{\mathcal H}}$. With these conventions at hand, it now follows that $\infty q=\iota_{\{0\}}$, $(\infty q)^*=0$ and $\partial(\infty q)=\partial\iota_{\{0\}}=N_{\{0\}}=\infty\cdot{\ensuremath{\operatorname{Id}}}=\infty\partial q$. Consequently, for every $\alpha\in[0,\infty]$ we deduce the following formulae: $(\alpha q)^*=\alpha^{-1}q$, $\partial\alpha q=\alpha{\ensuremath{\operatorname{Id}}}=\alpha\partial q$ and $\partial(\alpha q)^*=\alpha^{-1}{\ensuremath{\operatorname{Id}}}=(\alpha{\ensuremath{\operatorname{Id}}})^{-1}=(\partial\alpha q)^{-1}$. Suppose that for each $i\in I$, $0\leq\alpha_i\leq\infty$ and set $\boldsymbol{\alpha}=(\alpha_1\cdots,\alpha_n)$. Then we define $$\label{r average} r_\mu(\boldsymbol{\alpha},\boldsymbol{\lambda})=\big[\sum_{i\in I}\lambda_i(\alpha_i+\mu^{-1})^{-1}\big]^{-1}-\mu^{-1}\ \ \ \text{and}\ \ \ \ r(\boldsymbol{\alpha},\boldsymbol{\lambda})=r_1(\boldsymbol{\alpha},\boldsymbol{\lambda}).$$ We note that $0<r_\mu(\boldsymbol{\alpha},\boldsymbol{\lambda})$ if and only if there exists $i_0\in I$ such that $0<\alpha_{i_0}$ and $r_\mu(\boldsymbol{\alpha},\boldsymbol{\lambda})<\infty$ if and only if there exists $i_0\in I$ such that $\alpha_{i_0}<\infty$. For each $i\in I$ we set $f_i=\alpha_i q$ and $A_i=\alpha_i{\ensuremath{\operatorname{Id}}}=\partial f_i$. Then by combining our settings and calculations above with either a direct computations or with our formulae from Section 2 (namely, formulae , ,  and ), the following properties of $r_\mu$ follow: $$\begin{aligned} &{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=r_\mu(\boldsymbol{\alpha},\boldsymbol{\lambda}){\ensuremath{\operatorname{Id}}},\label{r and R}\\ &p_\mu(\bold{f},\boldsymbol{\lambda})=r_\mu(\boldsymbol{\alpha},\boldsymbol{\lambda})q,\label{r and p}\\ &r_\mu(\boldsymbol{\alpha},\boldsymbol{\lambda})^{-1}=r_{\mu^{-1}}(\boldsymbol{\alpha^{-1}},\boldsymbol{\lambda}),\ \ \ \ \ \text{that is,}\ \ \ \ r_{\mu^{-1}}(\boldsymbol{\alpha^{-1}},\boldsymbol{\lambda})^{-1}=r_\mu(\boldsymbol{\alpha},\boldsymbol{\lambda}),\label{r inverse}\\ &r_{\mu}(\boldsymbol{\alpha},\boldsymbol{\lambda})=\mu^{-1}r(\mu\boldsymbol{\alpha},\boldsymbol{\lambda}).\label{scaled r}\end{aligned}$$ For positive real numbers $\alpha_i$, formula  was obtained in [@BGLW formula (25)]. \[strong monotonicity dominance\]***(strong monotonicity is dominant)*** Suppose that for each $i\in I$, $\epsilon_i\geq0$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone and $\epsilon_i$-monotone. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is $\epsilon$-monotone where $\epsilon=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. In particular, if there exists $i_0\in I$, such that $A_{i_0}$ is $\epsilon_{i_0}$-strongly monotone, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is $\epsilon$-strongly monotone. For each $\in I$ we let ${\ensuremath{\varphi}}_i(t)=\epsilon_i t^2=2\epsilon_i(t^2/2)$. Then $A_i$ is monotone with modulus ${\ensuremath{\varphi}}_i$. Consequently, Theorem \[uniform monotonicity with 1-coercive modulus is recessive\] guarantees that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is monotone with modulus ${\ensuremath{\varphi}}=p_\frac{\mu}{2}(\boldsymbol{{\ensuremath{\varphi}}},\boldsymbol{\lambda})$. By employing formulae  and  we see that for every $t\geq0$, ${\ensuremath{\varphi}}(t)=r_\frac{\mu}{2}(2\boldsymbol{\epsilon},\boldsymbol{\lambda})(t^2/2)=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})t^2$ which means that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is $\epsilon$-monotone where $\epsilon=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. In particular, if there exists $i_0\in I$, such that $\epsilon_{i_0}>0$, then $r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})>0$. We recall that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is said to be $\epsilon$-cocoercive, where $\epsilon>0$, if $A^{-1}$ is $\epsilon$-strongly monotone, that is, if for every pair of points $(x,u)$ and $(y,v)$ in ${\ensuremath{\operatorname{gra}}}A$, $\epsilon\|u-v\|^2\leq{\langle{{u-v},{x-y}}\rangle}$. The following result is an immediate consequence of Theorem \[strong monotonicity dominance\]. \[cocoecivity dominance\]***(cocoerciveness is dominant)*** Suppose that for each $i\in I$, $\epsilon_i\geq 0$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone and $A_i^{-1}$ is $\epsilon_i$-monotone. Then $({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}})^{-1}$ is $\epsilon$-monotone where $\epsilon=r_{\mu^{-1}}(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. In particular, if there exists $i_0\in I$ such that $A_{i_0}$ is $\epsilon_{i_0}$-cocoercive, then ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is $\epsilon$-cocoercive. Since for each $i\in I$, $A_{i}^{-1}$ is $\epsilon_i$-monotone, then Theorem \[strong monotonicity dominance\] together with Theorem \[mainresult\] guarantee that $({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}})^{-1}={\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}$ is $\epsilon$-monotone. We say that the proper function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is $\epsilon$-convex, where $\epsilon\geq 0$, if $f$ is convex with modulus ${\ensuremath{\varphi}}(t)=\frac{\epsilon}{2}t^2$ (see ). In particular, a $0$-convex function is simply a convex function. We recall that $f$ is said to be *strongly convex* with constant $\epsilon$, $\epsilon$-strongly convex for short, if $f$ is $\epsilon$-convex and $\epsilon>0$. We say that the proper and convex function $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is $\epsilon$-smooth, where $\epsilon\geq 0$, if $f$ is smooth with modulus $\psi(t)=\frac{1}{2\epsilon}t^2$ (see ). In particular, a 0-smooth function is any proper and convex function. We now recall that $f$ is said to be *strongly smooth* with constant $\epsilon$, $\epsilon$-strongly smooth for short, if $f$ is $\epsilon$-smooth and $\epsilon>0$. \[Zal strong convexity iff\]*[@Zal Corollary 3.5.11 and Remark 3.5.3]* Suppose that $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and convex. Then the following assertions are equivalent: 1. $f$ is $\epsilon$-convex; 2. $\partial f$ is $\epsilon$-monotone; 3. $f^*$ is $\epsilon$-smooth. If $\epsilon>0$, assertions *(i), (ii)* and *(iii)* above are equivalent to the following assertion: 4. ${\ensuremath{\operatorname{dom}}}f^*={\ensuremath{\mathcal H}}$, $f^*$ is Fréchet differentiable on ${\ensuremath{\mathcal H}}$ and $\partial f^*=\nabla f^*$ is $\frac{1}{\epsilon}$-Lipschitz. As a consequence, we obtain the following results: \[strong convexity dominance\]***(strong convexity is dominant w.r.t. $p_\mu$)*** Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and $\epsilon_i$-convex. Then $p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-convex where $\epsilon=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. In particular, if there exists $i_0\in I$ such that $f_{i_0}$ is $\epsilon_{i_o}$-strongly convex, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-strongly convex. For each $i\in I$ we set $A_i=\partial f_i$. Then Fact \[Zal strong convexity iff\] guarantees that $A_i$ is $\epsilon_i$-monotone. Consequently, Theorem \[strong monotonicity dominance\] together with equation  imply that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=\partial p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-monotone which, in turn, implies that $p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-convex. In particular, if there exists $i_0\in I$ such that $f_{i_0}$ is $\epsilon_{i_0}$-strongly convex, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-strongly convex. \[strong smoothness dominance\]***(strong smoothness is dominant w.r.t. $p_\mu$)*** Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous and $\epsilon_i$-smooth. Then $p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-smooth where $\epsilon=r_{\mu^{-1}}(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. In particular, if there exists $i_0\in I$ such that $f_{i_0}$ is $\epsilon_{i_0}$-strongly smooth, then $p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-strongly smooth. Fact \[Zal strong convexity iff\] asserts that each $f_i^*$ is $\epsilon$-convex. Consequently, Theorem \[strong convexity dominance\] and formula  guaranty that $p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})=p_\mu({\bf f},{\boldsymbol \lambda})^*$ is $\epsilon$-convex which, in turn, imply that $p_\mu({\bf f},{\boldsymbol \lambda})=p_\mu({\bf f},{\boldsymbol \lambda})^{**}=p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})^*$ is $\epsilon$-smooth. \[Lip grad\]***(having a Lipschitz gradient is dominant w.r.t. $p_\mu$)*** Suppose that for each $i\in I$, $f_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ is proper, lower semicontinuous, convex and set $A_i=\partial f_i$. Suppose further that there exist $\varnothing\neq I_0\subseteq I$ such that for every $i\in I_0$, $f_{i}$ is Fréchet differentiable on ${\ensuremath{\mathcal H}}$, $\nabla f_{i}$ is $\epsilon_{i}$-Lipschitz and for every $i\notin I_0$ set $\epsilon_i=\infty$. Then $p_\mu({\bf f},{\boldsymbol \lambda})$ is Fréchet differentiable on ${\ensuremath{\mathcal H}}$ and ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=\nabla p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-Lipschitz where $\epsilon=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. Fact \[Zal strong convexity iff\] guarantees that for each $i\in I$, $f_i^*$ is $\frac{1}{\epsilon_i}$-convex. By applying Theorem \[strong convexity dominance\] we see that $p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})$ is $\frac{1}{\epsilon}$-convex where $\frac{1}{\epsilon}=r_{\mu^{-1}}(\boldsymbol{\epsilon}^{-1},\boldsymbol{\lambda})$. Furthermore, since $I_0\neq\varnothing$, we see that $0<\frac{1}{\epsilon}<\infty$ and, consequently, that $p_{\mu^{-1}}({\bf f^*},{\boldsymbol \lambda})$ is $\frac{1}{\epsilon}$-strongly convex. By applying Fact \[Zal strong convexity iff\] together with equation  and equation  we conclude that $p_\mu({\bf f},{\boldsymbol \lambda})=p_{\mu^{-1}}({\bf f}^*,{\boldsymbol \lambda})^*$ is Fréchet differentiable on ${\ensuremath{\mathcal H}}$ and that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=\partial p_\mu({\bf f},{\boldsymbol \lambda})=\nabla p_\mu({\bf f},{\boldsymbol \lambda})$ is $\epsilon$-Lipschitz where, by applying formula , $\epsilon=r_{\mu^{-1}}(\boldsymbol{\epsilon}^{-1},\boldsymbol{\lambda})^{-1}=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$. A remark regarding the sharpness of our results is now in order. Although in particular cases, our constants of monotonicity, Lipschitzness and cocoercivity of the resolvent average as well as the constants of convexity and smoothness of the proximal average are sharp, in general, they are not. Furthermore, such sharp constants cannot be determined only by $\mu$, the weight $\boldsymbol{\lambda}$ and the given constant $\boldsymbol{\epsilon}$ even if the latter is sharp, but depend also on the averaged objects as well. We now illustrate these situations by the following examples. \[strongly firmly example\] For each $i\in I$ let $0<\epsilon_i<\infty$, $f_i=\epsilon_i q$ and $A_i=\epsilon_i{\ensuremath{\operatorname{Id}}}=\nabla f_i$. Then for each $i\in I$, $A_i$ is $\epsilon_i$-strongly monotone which is equivalent to $f_i$ being $\epsilon$-strongly convex. As we have seen by formulae  and  we have ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda}){\ensuremath{\operatorname{Id}}}$ which is $r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$-strongly monotone and $p_\mu(\bold{f},\boldsymbol{\lambda})=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})q$ which is $r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$-strongly convex. Furthermore, for any $\epsilon'>r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$ we see that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}-\epsilon'{\ensuremath{\operatorname{Id}}}$ is not monotone, that is, ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is not $\epsilon'$-monotone, and that $p_\mu(\bold{f},\boldsymbol{\lambda})-\epsilon' q$ is not convex, that is, $p_\mu(\bold{f},\boldsymbol{\lambda})$ is not $\epsilon'$-convex. Thus, we conclude that $\epsilon=r_\mu(\boldsymbol{\epsilon},\boldsymbol{\lambda})$ is a sharp constant of strong monotonicity of the resolvent average and as a constant of strong convexity of the proximal average in this example. \[strong sharpness counter for subdifferentials\] We consider the settings in Example \[uniform counter example\]. We see that neither $f$ nor $f^*$ is strongly convex nor strongly smooth. Also, neither $\partial f$ nor $\partial f^*$ is strongly monotone or Lipschitz continuous. However, $p(f,f^*)=q$ is 1-strongly convex and 1-strongly smooth and $\mathcal{R}(\partial f,\partial f^*)={\ensuremath{\operatorname{Id}}}$ is 1-strongly monotone and 1-Lipschitz. Letting $\epsilon_1=\epsilon_2=0$ (which is the only possible constant of monotonicity of $\partial f$ and of $\partial f^*$, and the only possible constant of convexity and smoothness of $f$ and $f^*$), we see that $\epsilon=r(\boldsymbol{\epsilon},\boldsymbol{\lambda})=0$. Thus, we conclude that $\epsilon$ is not sharp in this case. On the other hand, $\epsilon=0$ will be the sharp constant of convexity when we consider $p(0,0)=0$ and a sharp constant of monotonicity and Lipschitzness of $\mathcal{R}(0,0)=0$ where the given $\mu$, $\boldsymbol{\lambda}$ and $\boldsymbol{\epsilon}$ are the same. Thus we conclude that the sharp constants of monotonicity, Lipschitzness and cocoercivity of the resolvent average as well as the sharp constants of convexity and smoothness of the proximal average cannot, in general, be determined only by the corresponding constants of the averaged objects, the parameter $\mu$ and the weight $\boldsymbol{\lambda}$. \[strong sharpness counter outside subdeifferentials\] Outside the class of subdifferential operators, we consider the following settings: let $\lambda_1=\lambda_2=\frac{1}{2}$, let $A_1$ be the counterclockwise rotation in the plane by the angle $0<\alpha\leq\frac{\pi}{2}$ and let $A_2=A_1^{-1}$ be the clockwise rotation by the angle $\alpha$. Then $A_1$ and $A_2$ are $\epsilon$-monotone where $\epsilon=\cos\alpha<1$ is sharp. In particular, for $\alpha=\frac{\pi}{2}$, $A_1$ and $A_2$ are not strongly monotone. However, by employing formula , we see that $\mathcal{R}(A_1,A_2)={\ensuremath{\operatorname{Id}}}$ which is 1-strongly monotone. On the other hand, $\mathcal{R}(A_1,A_1)=A_1$ is $\epsilon$-monotone where, again, $\epsilon=\cos\alpha$ is sharp. Thus, this is another demonstration to the fact that the sharp constant of monotonicity of the resolvent average does not depend only on the constants of monotonicity of the averaged mappings, the weight $\boldsymbol{\lambda}$ and the parameter $\mu$. We will analyze a variant of this example in order to discuss Lipschitzness outside the class of subdifferential operators in Example \[scaled rotations\]. Disjoint injectivity -------------------- We recall that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is said to be *disjointly injective* if for any two distinct point $x$ and $y$ in ${\ensuremath{\mathcal H}}$, $Ax\cap Ay=\varnothing$. \[t:disjinjdom\]***(disjoint injectivity is dominant)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. If there exists $i_0\in I$ such that $A_{i_0}$ is disjointly injective, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is disjointly injective. We recall that the maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is disjointly injective if and only if $J_A$ is strictly nonexpansive, that is, for any two distinct points $x$ and $y$ in ${\ensuremath{\mathcal H}}$, $\|J_A x-J_A y\|<\|x-y\|$ (see [@bmw12 Theorem 2.1(ix)]). We also note that $A$ is disjointly injective if and only if $\mu A$ is disjointly injective. Thus, since for every $i\in I$, $J_{\mu A_i}$ is nonexpansive and since $J_{\mu A_{i_0}}$ is strictly nonexpansive, then the convex combination $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_i J_{\mu A_i}$ is strictly nonexpansive, which, in turn, implies that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is disjointly injective. Let ${\ensuremath{\mathcal H}}$ be finite-dimensional and let $f:{\ensuremath{\mathcal H}}\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ be proper, lower semicontinuous and convex function. Then the disjoint injectivity of $\partial f$ is equivalent to the essential strict convexity of $f$ which is equivalent to the essential smoothness of $f^*$ (see [@Rock Theorem 26.3]). Thus, in the finite-dimensional case, once again, we recover Corollary \[ess smooth\], Corollary \[ess conv\] and Corollary \[Leg\], that is, essential smoothness, essential strict convexity and Legendreness are dominant properties w.r.t. the proximal average. Recessive properties of the resolvent average {#recessive} ============================================= We begin this section with the following example of a recessive property: **(being a constant mapping is recessive)** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is the constant mapping $x\mapsto z_i$. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is the constant mapping $x\mapsto\sum_{i\in I}\lambda_i z_i$. Indeed, the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is the constant mapping $x\mapsto z$ if and only if $J_A$ is the shift mapping $x\mapsto x-z$. Thus, we see that $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_i J_{\mu A_i}$ is the shift mapping $x\mapsto x-\mu\sum_{i\in I}\lambda_i z_i$ and, consequently, that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is the constant mapping $x\mapsto\sum_{i\in I}\lambda_i z_i$. However, if we let $A_1$ be any constant mapping, $A_2$ be any maximally monotone mapping which is not constant, $0<\lambda<1,\ \lambda_1=\lambda$ and $\lambda_2=1-\lambda$, then $J_{\mu A_1}$ is a shift, $J_{\mu A_2}$ is not a shift and $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\lambda_1 J_{\mu A_1}+(1-\lambda_1)J_{\mu A_2}$ is not a shift. Consequently, ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is not a constant mapping. Summing up, we see that being a constant mapping is recessive w.r.t. the resolvent average. Linearity and affinity ---------------------- We begin our discussion with the following example: \[linearcounter\] We set $f=\| \cdot\|$, $A_1 = \partial f$ and $A_2 = {\bf 0}$. Then $$J_{A_1} x = \begin{cases}\begin{matrix} \left(1-\frac{1}{\|x\|}\right)x, & \text{if } \|x\|>1; \\ 0, & \text{if } \|x\| \leq 1 \end{matrix} \end{cases}$$ and $J_{A_2} = {\ensuremath{\operatorname{Id}}}$ (see [@BC2011 Example 23.3 and Example 14.5]). We see that $J_{A_1}$ is not an affine relation and $J_{A_2}$ is linear. However, letting $0<\lambda<1,\ \lambda_1=\lambda$ and $\lambda_2=1-\lambda$, then $$J_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}x = \lambda J_{A_1}x + (1-\lambda) J_{A_2}x =\begin{cases}\begin{matrix} \left(1-\lambda \frac{1}{\|x\|}\right)x, & \text{if } \|x\|>1; \\ (1-\lambda)x, & \text{if } \|x\| \leq 1, \end{matrix} \end{cases}$$ which is not an affine relation. Thus, by employing Fact \[linear\], we conclude that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is not an affine relation. Example \[linearcounter\] demonstrates that linearity and affinity are not dominant properties w.r.t. the resolvent average. On the other hand, since the sums and inversions in the definition of ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ preserve linearity and affinity, we arrive at the following observation: \[linearrelation\] Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is a maximally monotone linear (resp. affine) relation. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is a maximally monotone linear (resp. affine) relation. Rectangularity and paramonotonicity ----------------------------------- We recall Definition \[D:para rec def\] of rectangular and paramonotone mappings and consider the following example: \[ParaRecDominant\] In ${\ensuremath{\mathbb R}}^2$, let $A_{1}=N_{{\ensuremath{\mathbb R}}\times \{0\}}.$ Then by Fact \[NCresolvent\], $J_{A_1}$ is the projection on ${\ensuremath{\mathbb R}}\times \{0\}$. Since $A_1$ is a subdifferential of a proper, lower semicontinuous and convex function, it is rectangular (see [@BC2011 Example 24.9]) and paramonotone (see [@BC2011 Example 22.3]). Let $A_{2}:{\ensuremath{\mathbb R}}^2\rightarrow{\ensuremath{\mathbb R}}^2$ be the counterclockwise rotation by $\pi/2$. Then by employing standard matrix representation we write $$J_{A_{1}}=P_{{\ensuremath{\mathbb R}}\times \{0\}}=\begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix},\ A_{2}=\begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix},\ A_{2+}=\frac{1}{2}(A_2+A_2^{\intercal})=0 \ \text{ and }\ \ J_{A_{2}}=\begin{pmatrix} \tfrac{1}{2} & \tfrac{1}{2}\\[+0.8mm] -\tfrac{1}{2} & \tfrac{1}{2} \end{pmatrix}.$$ Letting $\lambda_1=\lambda_2=\tfrac{1}{2}$, we obtain $$\mathcal{R}({\bf{A}}) =(\frac{1}{2} J_{A_{1}}+\frac{1}{2}J_{A_{2}})^{-1}-{\ensuremath{\operatorname{Id}}}= \begin{pmatrix} 0 & -1\\ 1 & 2 \end{pmatrix} \ \ \ \text{and}\ \ \ \mathcal{R}({\bf{A}})_+=\frac{1}{2} \big(\mathcal{R}(\bf{A})+\mathcal{R}(\bf{A})^{\intercal}\big)=\begin{pmatrix} 0 & 0\\ 0 & 2 \end{pmatrix}.$$ By employing Fact \[f:pararecsame\] we see that $A_2$ as well as ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ are neither rectangular nor paramonotone. In view of Example \[ParaRecDominant\] it is clear that rectangularity and paramonotonicity are not dominant properties w.r.t. the resolvent average. We now prove the recessive nature of these properties. We begin with rectangularity. Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is rectangular and maximally monotone. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is rectangular. The mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is rectangular if and only if $\mu A$ is rectangular as can be seen from . Thus, by employing  we see that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is rectangular as soon as ${\ensuremath{\operatorname{dom}}}\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\times{\ensuremath{\operatorname{ran}}}\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\subseteq{\ensuremath{\operatorname{dom}}}F_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}$. Indeed, since for each $i\in I$, ${\ensuremath{\operatorname{dom}}}\mu A_i\times{\ensuremath{\operatorname{ran}}}\mu A_i\subseteq {\ensuremath{\operatorname{dom}}}F_{\mu A_i}$, by employing inclusion  and then inclusion  we arrive at $$\begin{aligned} {\ensuremath{\operatorname{dom}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\times{\ensuremath{\operatorname{ran}}}{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}&\subseteq \sum_{i\in I}\lambda_i({\ensuremath{\operatorname{dom}}}\mu A_i\times{\ensuremath{\operatorname{ran}}}\mu A)\subseteq \sum_{i\in I}\lambda_i{\ensuremath{\operatorname{dom}}}F_{\mu A_i}\\ &\subseteq {\ensuremath{\operatorname{dom}}}F_{{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}},\end{aligned}$$ as asserted. In order to prove that paramonotonicity is recessive, we will make use of the following result: \[convex combination of resolvents of paramonotone\] Suppose that for each $i\in I,\ T_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is firmly nonexpansive and set $T=\sum_{i\in I}\lambda_i T_i$. Then: 1. If for each $i\in I$, given points $x$ and $y$ in ${\ensuremath{\mathcal H}}$, $$\label{resolvent of paramonotone} \|T_i x-T_i y\|^2={\langle{{x-y},{T_i x-T_iy}}\rangle}\ \ \ \ \Rightarrow\ \ \ \ \begin{cases} T_ix=T_i(T_ix+y-T_i y)\\ T_i y=T_i(T_i y+x-T_i x), \end{cases}$$ then $T$ also has property . 2. \[resolvent of paramonotone pluse injective\] If there exists $i_0\in I$ such that $T_{i_0}$ has property  and is injective, then $T$ has property  and is injective. \(i) Suppose that $x$ and $y$ are points in ${\ensuremath{\mathcal H}}$ such that $\|Tx-Ty\|^2={\langle{{x-y},{Tx-Ty}}\rangle}$. Then Corollary \[equality in convex combination of firmly nonexpansive\] guarantees that $T_i x-T_i y=Tx-Ty$ for every $i\in I$. Employing property  of the $T_i$’s, we obtain the set of equalities in . Consequently, we see that $$Tx=\sum_{i\in I}\lambda_i T_i x=\sum_{i\in I}\lambda_i T_i(T_ix+y-T_i y)=\sum_{i\in I}\lambda_i T_i(Tx+y-Ty)=T(Tx+y-Ty)$$ and, similarly, that $Ty=T(Ty+x-Tx)$. (ii) In the same manner as in the proof of (i), if $\|Tx-Ty\|^2={\langle{{x-y},{Tx-Ty}}\rangle}$, then $T_i x-T_i y=Tx-Ty=T_{i_0}x-T_{i_0}y$ for every $i\in I$. Since $T_{i_0}$ is injective and has property , then, in fact, $x-y=T_{i_0}x-T_{i_0}y$. Thus, we conclude that $x=Tx+y-Ty$, equivalently $y=Ty+x-Tx$, which confirms that $T$ has property . Finally, Proposition \[equality in convex combination of firmly nonexpansive\]\[firmly injective\] guarantees that $T$ is also injective. \[paramonotonicity is recessive\]***(paramonotonicity is recessive)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone and paramonotone. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is paramonotone. First we consider the case $\mu=1$. We recall that the maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is paramonotone if and only if $J_A$ has property  (see [@bmw12 Theorem 2.1(xv)]). Thus, since each $J_{A_i}$ has property , then Proposition \[convex combination of resolvents of paramonotone\](i) guarantees that $J_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_i J_{A_i}$ also has property , which, in turn, implies that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is paramonotone. For arbitrary $0<\mu$, we note that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is paramonotone if and only if $\mu A$ is paramonotone. Since for each $i\in I$, $\mu A_i$ is paramonotone, so is $\mathcal{R}\bold(\mu A,\boldsymbol{\lambda})$. Consequently, formula  implies that $\mu^{-1}\mathcal{R}(\bold\mu A,\boldsymbol{\lambda})={\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is paramonotone. $k$-cyclic monotonicity and cyclic monotonicity ----------------------------------------------- We recall that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is said to be *$k$-cyclically monotone*, where $k\in\{2,3,\dots\}$, if any $k$ pairs $(x_1,u_1),\dots,(x_k,u_k)\in{\ensuremath{\operatorname{gra}}}A$, letting $x_{k+1}=x_1$, satisfies $0\leq\sum_{i=1}^k{\langle{{u_i},{x_i-x_{i+1}}}\rangle}$. The mapping $A$ is said to be cyclically monotone if it is $k$-cyclically monotone for every $k\in\{2,3,\dots\}$. Rockafellar’s well known characterization from [@Rock; @cyc] asserts that proper, maximally monotone and cyclically monotone mappings are precisely the subdifferentials of proper, convex and lower semicontinuous functions. \[ex:cyclicmonocounter\] Let ${\ensuremath{\mathcal H}}= {\ensuremath{\mathbb R}}^2$. We set $n=2,\ {\boldsymbol \lambda}=(1/3,2/3)$, we let $A_1$ be the identity and we let $A_2$ be the mapping obtained by the counter-clockwise rotation by $\pi/2$. Then $A_1$ is cyclically monotone while $A_2$ is monotone but not $3$-cyclically monotone (see [@BBBRW Example 4.6] for further discussion of $k$-cyclic monotonicity of rotations by $\pi/k$). Then, by employing matrix representation, we obtain $${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}= \frac{1}{13}\begin{pmatrix}5 & -12\\12 & 5\end{pmatrix}.$$ We now let $x_1=x_4=0$, $x_2=e_1$ and $x_3=e_2$ where $(e_1,e_2)$ is the standard basis of ${\ensuremath{\mathbb R}}^2$. Then $$\sum_{i=1}^3{\langle{{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}x_i},{x_i-x_{i+1}}}\rangle}={\langle{{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}e_1},{e_1-e_2}}\rangle}+{\langle{{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}e_2},{e_2}}\rangle}=-2/13<0.$$ Thus, we see that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is not $3$-cyclically monotone. In view of Example \[ex:cyclicmonocounter\], the property of $k$-cyclic monotonicity is not dominant w.r.t. the resolvent average. In order to conclude the recessive nature of $k$-cyclic monotonicity and of cyclic monotonicity we recall the following: By employing cyclic monotonicity, the convexity of the set of proximal mappings was recovered in [@BBBRW Theorem 6.7]. In other words, if we suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone and cyclically monotone, then ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is maximally monotone and cyclically monotone. Since the proof of this result was actually carried out for every fixed $k$, it, in fact, holds also for $k$-cyclically monotone mappings. Summing up, we arrive at the following conclusion: ***($k$-cyclic monotonicity and cyclic monotonicity are recessive)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. If for every $i\in I$, $A_i$ is $k$-cyclically monotone, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is $k$-cyclically monotone. If for every $i\in I$, $A_i$ is cyclically monotone, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is cyclically monotone. The case where $\mu=1$ follows from our discussion above. For arbitrary $\mu>0$, we note that the mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is $k$-cyclically monotone (cyclically monotone) if and only if the mapping $\mu A$ is $k$-cyclically monotone (cyclically monotone). Thus, we see that $\mathcal{R}(\mu\bf{A},\boldsymbol{\lambda})$ is $k$-cyclically monotone (cyclically monotone), and, consequently, by employing formula , so is $\mu^{-1}\mathcal{R}(\mu\bf A,\boldsymbol{\lambda})={\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$. Weak sequential closedness of the graph --------------------------------------- The mapping $T:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is said to be *weakly sequentially continuous* if it maps a weakly convergent sequence to a weakly convergent sequence, that is, $x_n\rightharpoonup x\ \Rightarrow\ Tx_n\rightharpoonup Tx$. We recall that the maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ has a weakly sequentially closed graph if and only if $J_A$ is weakly sequentially continuous (see [@bmw12 Theorem 2.1(xxi)]). Within the settings of Example \[linearcounter\], we consider the case where ${\ensuremath{\mathcal H}}=\ell^2({\ensuremath{\mathbb N}})$ and $\{e_1,e_2,\dots\}$ is its standard basis. Then, for every $n\in{\ensuremath{\mathbb N}}$, we set $x_n=e_1+e_n$. Consequently, $x_n\rightharpoonup e_1$ and $J_{A_1}x_n=(1-1/\sqrt{2})x_n\rightharpoonup(1-1/\sqrt{2})e_1$ while $J_{A_1}e_1=0$. Thus, $J_{A_1}$ is not weakly sequentially continuous and $J_{A_2}$ is. Furthermore, $$\begin{aligned} J_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}x_n&=\lambda_1 J_{A_1}x_n+\lambda_2 J_{A_2}x_n=\lambda_1(1-1/\sqrt{2})x_n+\lambda_2 x_n\\ &\rightharpoonup \lambda_1(1-1/\sqrt{2})e_1+\lambda_2e_1=(1-\lambda_1/\sqrt{2})e_1.\end{aligned}$$ Since $J_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}e_1=\lambda_2e_1$, we see that $J_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}$ is not weakly sequentially continuous, which, in turn, implies that the graph of ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is not weakly sequentially closed, although, the graph of $A_2$ is. We see that the weak sequential closedness of the graph is not a dominant property w.r.t. the resolvent average. ***(weak sequential closedness of the graph is recessive)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone with a weakly sequentially closed graph. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ has a weakly sequentially closed graph. Since for each $i\in I$, $J_{\mu A_i}$ is weakly sequentially continuous, so is $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in }\lambda_i J_{\mu A_i}$. Consequently, ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ has a weakly sequentially closed graph. Displacement mappings --------------------- The mapping $D:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is said to be a *displacement mapping* if there exists a nonexpansive mapping $N:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ such that $D={\ensuremath{\operatorname{Id}}}-N$, in which case $D$ is maximally monotone. \[displacement not dominant\] Let $n=2,\ \lambda_1=\lambda_2=\frac{1}{2},\ 0<\alpha_1=\alpha,\ 0<\alpha_2=\beta,\ A_1=\alpha{\ensuremath{\operatorname{Id}}}$ and $A_2=\beta{\ensuremath{\operatorname{Id}}}$. Then by applying formula  we obtain $${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=r(\boldsymbol{\alpha},\boldsymbol{\lambda}){\ensuremath{\operatorname{Id}}}=\frac{\alpha+\beta+2\alpha\beta}{2+\alpha+\beta}{\ensuremath{\operatorname{Id}}}.$$ Now we let $\alpha=2$ and $\beta=5$. Then $A_1$ is a displacement mapping, $A_2$ is not a displacement mapping and ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=3{\ensuremath{\operatorname{Id}}}$ is not a displacement mapping. In view of Example \[displacement not dominant\], we see that being a displacement mapping is not a dominant property w.r.t. the resolvent average. In order to prove that being a displacement mapping is recessive we will make use of the following result: \[displacement characterization\] The maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is $\frac{1}{2}$-strongly monotone if and only if $A^{-1}$ is a displacement mapping. The assertion that the mapping $A$ is $\frac{1}{2}$-strongly monotone is equivalent to $A=\frac{1}{2}{\ensuremath{\operatorname{Id}}}+M_1$ for a maximally monotone mapping $M_1$, which, in turn, is equivalent to $A=\frac{1}{2}{\ensuremath{\operatorname{Id}}}+M_2\circ(\frac{1}{2}{\ensuremath{\operatorname{Id}}})$ for a maximally monotone mapping $M_2$ (let $M_2=M_1\circ(2{\ensuremath{\operatorname{Id}}}$)). By employing equation , this is equivalent to $A^{-1}=2[{\ensuremath{\operatorname{Id}}}-(M^{-1}_2+{\ensuremath{\operatorname{Id}}})^{-1}]$. Finally, this is equivalent to $A^{-1}=2({\ensuremath{\operatorname{Id}}}-F)$ for a firmly nonexpansive mapping $F$ (let $F=J_{M_2^{-1}}$), which, in turn, is equivalent to $A^{-1}=2({\ensuremath{\operatorname{Id}}}-({\ensuremath{\operatorname{Id}}}+N)/2)={\ensuremath{\operatorname{Id}}}-N$ for a nonexpansive mapping $N$ (let $N=2F-{\ensuremath{\operatorname{Id}}}$). \[monotone and nonexpansive char\] The maximally monotone mapping $N:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is nonexpansive if and only if $N=2J_B-{\ensuremath{\operatorname{Id}}}$ for a maximally monotone and nonexpansive mapping $B$. Since $\frac{1}{2}(N+{\ensuremath{\operatorname{Id}}})$ is $\frac{1}{2}$-strongly monotone, the assertion $(B+{\ensuremath{\operatorname{Id}}})^{-1}=J_B=\frac{1}{2}(N+{\ensuremath{\operatorname{Id}}})$ is equivalent to $B+{\ensuremath{\operatorname{Id}}}$ being a displacement mapping, that is ${\ensuremath{\operatorname{Id}}}+B={\ensuremath{\operatorname{Id}}}-N'$ for a nonexpansive mapping $N'$, that is, $B$ is a nonexpansive mapping. \[displacement recessiveness\] ***(being a displacement mapping is recessive)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is a displacement mapping. Then ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is a displacement mapping. By employing Proposition \[displacement characterization\] we see that for each $i\in I$, $A_i^{-1}$ is $(1/2)$-strongly monotone. Consequently, Theorem \[strong monotonicity dominance\] guarantees that $({\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}})^{-1}={\ensuremath{\mathcal{R}({\bf A}^{-1},{\boldsymbol \lambda})}}$ is $(1/2)$-strongly monotone, which, in turn, implies that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is a displacement mapping. Nonexpansive monotone operators ------------------------------- In Example \[displacement not dominant\], we let $\alpha=1$ and $\beta=5$. Then $A_1={\ensuremath{\operatorname{Id}}}$ is nonexpansive, $A_2=5{\ensuremath{\operatorname{Id}}}$ is not nonexpansive and ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=2{\ensuremath{\operatorname{Id}}}$ which is not nonexpansive. Thus, we see that nonexpansiveness is not a dominant property. \[nonexpansive average\] ***(nonexpansiveness is recessive)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is a nonexpansive and monotone mapping. Then ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is nonexpansive. Furthermore, for each $i\in I$, $A_i=2J_{B_i}-{\ensuremath{\operatorname{Id}}}$ where $B_i$ is maximally monotone, nonexpansive and ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=2J_B-{\ensuremath{\operatorname{Id}}}$ where $B$ is the maximally monotone and nonexpansive mapping given by $B=\sum_{i\in I}\lambda_iB_i$. Since the $A_i$’s have full domain and are continuous, they are maximally monotone. Employing Corollary \[monotone and nonexpansive char\], we see that the $B_i$’s are maximally monotone and nonexpansive (and have full domain). Furthermore, we have $$\begin{aligned} J_{{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}}&=\sum_{i\in I}\lambda_iJ_{A_i}=\sum_{i\in I}\lambda_i(2J_{B_i})^{-1}=\sum_{i\in I}\lambda_i(J_{B_i})^{-1}\circ(\tfrac{1}{2}{\ensuremath{\operatorname{Id}}})=\sum_{i\in I}\lambda_i(B_i+{\ensuremath{\operatorname{Id}}})\circ(\tfrac{1}{2}{\ensuremath{\operatorname{Id}}})\\ &=(B+{\ensuremath{\operatorname{Id}}})\circ(\tfrac{1}{2}{\ensuremath{\operatorname{Id}}})=(2J_B)^{-1}.\end{aligned}$$ Thus, we see that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=2J_B-{\ensuremath{\operatorname{Id}}}$, as asserted. Miscellaneous observations and remarks {#neither} ====================================== In our last section, we consider combinations of properties, indeterminate properties and other observations and remarks. Paramonotonicity combined with single-valudeness ------------------------------------------------ \[singlevalued paramonotone\] ***(paramonotonicity combined with single-valudeness is dominant)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is maximally monotone. If for some $i_0\in I$, $A_{i_0}$ is paramonotone and at most single-valued, then so is ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$. We recall that the maximally monotone mapping $A:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ is at most single-valued if and only if $J_A$ is injective (see [@bmw12 Theorem 2.1(iv)]). We also recall that $A$ is paramonotone if and only if $J_A$ has property . Thus, we see that $J_{\mu A_{i_0}}$ is injective and has property . Consequently, Proposition \[convex combination of resolvents of paramonotone\]\[resolvent of paramonotone pluse injective\] implies that $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}=\sum_{i\in I}\lambda_iJ_{\mu A_i}$ is injective and has property , which, in turn, implies that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is paramonotone and at most single-valued. Linear relations, bounded linear operators and linear operators on ${\ensuremath{\mathbb R}}^n$ ----------------------------------------------------------------------------------------------- Within the class of maximally monotone and linear relation on ${\ensuremath{\mathcal H}}$, which we will denote by $MLR({\ensuremath{\mathcal H}})$, lies the class of classical (single-valued) monotone and bounded linear operators which we will denote by $BML({\ensuremath{\mathcal H}})$. Within $BML({\ensuremath{\mathcal H}})$ we will denote by $BMLI({\ensuremath{\mathcal H}})$ the class of invertible operators, that is, bounded linear monotone surjective operators with a bounded inverse. We begin our discussion of these classes of operators with the following result: \[classical linearity dominance within\] ***(in $MLR({\ensuremath{\mathcal H}})$, being $BML({\ensuremath{\mathcal H}})$ and being $BMLI({\ensuremath{\mathcal H}})$ are dominant)*** Suppose that for each $i\in I$, $A\in MLR({\ensuremath{\mathcal H}})$ and there exists $i_0\in I$ such that $A_{i_0}\in BML({\ensuremath{\mathcal H}})$. Then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\in BML({\ensuremath{\mathcal H}})$. Furthermore: 1. If $A_{i_0}\in BMLI({\ensuremath{\mathcal H}})$, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}\in BMLI({\ensuremath{\mathcal H}})$; 2. If $A_{i_0}$ is paramonotone, then ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is paramonotone. Corollary \[linearrelation\] guarantees that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is a monotone linear relation. Since $A_{i_0}$ has a full domain, then Theorem \[t:FDdom\]\[t:Fdom\] guarantees that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ has full domain. Since $A_{i_0}$ is single-valued, then Theorem \[single valued\] guarantees that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is single-valued. Finally, since a monotone operator is locally bounded on the interior of its domain (see [@Rock; @bndd Theorem 1]) and since ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ has full domain, then it is everywhere locally bounded, which, in the case of single-valued linear operators, is equivalent to boundedness. If $A_{i_0}$ is a bounded linear operator which is also invertible, then $A_{i_0}^{-1}$ is bounded, which implies that ${\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}=({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}})^{-1}$ is bounded. Finally, if $A_{i_0}$ is paramonotone, then Theorem \[singlevalued paramonotone\] guarantees that ${\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}$ is paramonotone. Within the settings and hypothesis of Theorem \[classical linearity dominance within\], we note that when ${\ensuremath{\mathcal H}}={\ensuremath{\mathbb R}}^n$, paramonotonicity is interchangeable with rectangularity since, in this case, the two properties are equivalent (see Fact \[f:pararecsame\]). We now discuss the class of maximally monotone linear relations on ${\ensuremath{\mathbb R}}^n$, which we denote by $MLR(n)$, its subclass of monotone linear mappings (classical, single-valued), which we denote by $ML(n)$, and its subclass of monotone, linear and (classically) invertible operators which we denote by $MLI(n)$. We will identify operators in $ML(n)$ with their standard matrix representation. A subclass of $ML(n)$ is the class of positive semidefinite matrices and a subclass of $MLI(n)$ is the class of positive definite matrices. The resolvent average of positive semidefinite and definite matrices was studied in [@bmw-res]. We now recall that any positive semidefinite matrix $A$ has the property that $0\leq\det A$. This fact holds in the larger class $ML(n)$ as well: \[det\] Suppose that $A\in ML(n)$. Then for every $0\leq\lambda<1$, $0<\det(\lambda A+(1-\lambda){\ensuremath{\operatorname{Id}}})$. Consequently, $0\leq\det A$. Let $x\in{\ensuremath{\mathbb R}}^n,\ x\neq 0$. Since $0\leq{\langle{{x},{Ax}}\rangle}$, then $$0<\lambda{\langle{{x},{Ax}}\rangle}+(1-\lambda)\|x\|^2={\langle{{x},{(\lambda A+(1-\lambda){\ensuremath{\operatorname{Id}}})x}}\rangle}.$$ We conclude that $0\neq(\lambda A+(1-\lambda){\ensuremath{\operatorname{Id}}})x$ for every $x\in{\ensuremath{\mathbb R}}^n,\ x\neq0 $. We now define ${\ensuremath{\varphi}}:[0,1]\to{\ensuremath{\mathbb R}}$ by ${\ensuremath{\varphi}}(\lambda)=\det(\lambda A+(1-\lambda){\ensuremath{\operatorname{Id}}})$. It follows that ${\ensuremath{\varphi}}$ is continuous, ${\ensuremath{\varphi}}$ does not vanish on $[0,1[$ and ${\ensuremath{\varphi}}(0)=\det{\ensuremath{\operatorname{Id}}}=1$. Consequently, $0<{\ensuremath{\varphi}}(\lambda)$ for every $0\leq\lambda<1$, which, in turn, implies that $\det A={\ensuremath{\varphi}}(1)=\lim_{\lambda\to1^-}{\ensuremath{\varphi}}(\lambda)\geq 0$. We now focus our attention on $ML(n)$. To this end, we first recall the case where a mapping $A\in MLR(n)$ is, in fact, in $ML(n)$. Such a characterization is the combination of [@bwy2012 Fact 2.2] and [@bwy2012 Fact 2.3]: For $A\in MLR(n)$ the following assertions are equivalent: 1. $A\in ML(n)$; 2. $A$ is at most single-valued; 3. $A(0)$ is a singleton; 4. ${\ensuremath{\operatorname{dom}}}A={\ensuremath{\mathbb R}}^n$. In our current discussion it is crucial to distinguish between the notion of the inverse of a mapping in $MLR(n)$ (multivalued settings) and the classical notion of the inverse of a single-valued mapping in $MLI(n)$. In the case where $A\in MLI(n)$, we will abuse the notation and write $A^{-1}$ for its inverse, which can then be viewed as the same in both settings, the multivalued and the classical single-valued. In order for our results in the present paper to be relevant for studies within classical linear algebra settings in $ML(n)$, we now support our claim that: > When taking the resolvent average of operators in ML(n), all of the inversion operations involved in the averaging operation are classical inversions. Furthermore, for operators in MLI(n), all of the inverses in the formula $({\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}})^{-1}={\ensuremath{\mathcal{R}_{\mu^{-1}}({\bf A}^{-1},{\boldsymbol \lambda})}}$ are classical inverses. Indeed, the “Furthermore" part follows from the invertibility of the $A_i$’s when we recall Theorem \[classical linearity dominance within\]. Now, if $0<\mu$ and $A_i\in ML(n)$, then $\mu A_i+{\ensuremath{\operatorname{Id}}}$ is invertible (since $0<\|x\|^2\leq{\langle{{x},{(\mu A_i+{\ensuremath{\operatorname{Id}}})x}}\rangle}$ for every $0\neq x\in{\ensuremath{\mathbb R}}^n$). Thus, since $A_i$ is monotone, then $J_{\mu A_i}$ is now seen to be invertible and firmly nonexpansive, that is, for any $x\in{\ensuremath{\mathbb R}}^n,\ x\neq0$, we have $0<\|J_{\mu A_i}x\|^2\leq{\langle{{x},{J_{\mu A_i}x}}\rangle}$. As a consequence, $${\langle{{x},{J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}x}}\rangle}=\sum_{i\in I}\lambda_i{\langle{{x},{J_{\mu A_i}x}}\rangle}>0.$$ Since this is true for every $x\in{\ensuremath{\mathbb R}}^n,\ x\neq0$, we conclude that $J_{\mu{\ensuremath{\mathcal{R}_{\mu}({\bf A},{\boldsymbol \lambda})}}}$ is, indeed, invertible, as asserted. We now focus our attention further on $ML(n)\cap O(n,{\ensuremath{\mathbb R}})$. Here $O(n,{\ensuremath{\mathbb R}})$ is the group of orthogonal matrices in ${\ensuremath{\mathbb R}}^{n\times n}$; in particular, $A\in O(n{\ensuremath{\mathbb R}})\Rightarrow\det A\pm 1$. Since, as we saw in Proposition \[det\], the determinant is greater than zero in $ML(n)$, then, in fact, $ML(n)\cap O(n,{\ensuremath{\mathbb R}})=ML(n)\cap SO(n,{\ensuremath{\mathbb R}})$, where $SO(n)$ is the group of the special orthogonal matrices, that is, the matrices $A\in O(n)$ such that $\det A=1$. Thus, $ML(n)\cap O(n,{\ensuremath{\mathbb R}})=ML(n)\cap SO(n,{\ensuremath{\mathbb R}})$ can be viewed as the subset of rotations which consists of the rotations by an acute or right angles. The next result demonstrates that the resolvent average of such rotations is again such a rotation. This, of course, fails when taking the arithmetic average. \[rotations\] ***(being a rotation by an acute or right angle is recessive)*** Suppose that for each $i\in I$, $A_i\in ML(n)\cap O(n,{\ensuremath{\mathbb R}})=ML(n)\cap SO(n,{\ensuremath{\mathbb R}})$. Then ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}\in ML(n)\cap O(n,{\ensuremath{\mathbb R}})=ML(n)\cap SO(n,{\ensuremath{\mathbb R}})$. By employing the inversion formula  (which, in this case, was seen to be a classical inversion), we obtain $$\big({\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}\big)^{-1} = {\ensuremath{\mathcal{R}({\bf A}^{-1},{\boldsymbol \lambda})}}=\mathcal{R}({\bf A}^{\intercal},\lambda) ={\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}^{\intercal}.$$ That is, ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is orthogonal. The fact that $\det{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=1$ follows from the monotonicity of ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ (see Proposition \[det\]). When $n=2$, we consider the case where $A_1={\ensuremath{\operatorname{Id}}}$, $A_2$ is the counter clockwise rotation by $\frac{\pi}{2}$, $0<\lambda<1$, $\lambda_1=\lambda$ and $\lambda_2=1-\lambda$. Then $${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}= \frac{1}{\lambda^2-2\lambda+2}\begin{pmatrix}\lambda(2-\lambda) & -2(1-\lambda)\\2(1-\lambda) & \lambda(2-\lambda)\end{pmatrix}$$ is a counter-clockwise rotation matrix by an angle of a right triangle with sides $$a(\lambda)=\lambda(2-\lambda),\ \ b(\lambda)=2(1-\lambda)\ \ \ \ \text{and}\ \ \ \ c(\lambda)=\sqrt{a^2+b^2}=\lambda^2-2\lambda+2.$$ Now $\lambda\mapsto{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is a smooth and one-to-one curve, in particular, $\lambda\mapsto a(\lambda)/c(\lambda)$ is a bijection of $[0,1]$ to itself. (In fact, for any two monotone mappings $A_1:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ and $A_2:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$ such that $A_1\neq A_2$, $\lambda\mapsto{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is one-to-one). Consequently, up to rescaling, all possible right triangles can be recovered in this manner. In particular, considering all of the possible rational values of $\lambda$, our procedure recovers all possible *primitive Pythagorean triples* $(a,b,c)$ (other values of $\lambda$ should be considered as well in order to recover all possible Pythagorean triples). Indeed, letting $\lambda=\frac{p}{q}$ where $0<p<q$ are natural numbers, we obtain the matrix $${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}= \frac{1}{\lambda^2-2\lambda+2}\begin{pmatrix}\lambda(2-\lambda) & -2(1-\lambda)\\2(1-\lambda) & \lambda(2-\lambda)\end{pmatrix} =\frac{1}{p^2-2pq+2q^2}\begin{pmatrix} p(2q-p) & -2q(q-p)\\2q(q-p)& p(2q-p) \end{pmatrix}$$ which is a counter-clockwise rotation matrix by an angle of a right triangle with sides $$a=p(2q-p),\ \ b=2q(q-p)\ \ \ \ \text{and}\ \ \ \ \ c=p^2-2pq+2q^2.$$ Letting $k=q-p$ and $l=q$, we obtain the formula $$\label{Euclid's formula} a=l^2-k^2,\ \ b=2kl\ \ \ \ \ \text{and}\ \ \ \ c=k^2+l^2,$$ which is a well known formula for generating all of the primitive Pythagorean triples and is attributed to Euclid. For further, more accurate, relations between formula  and primitive Pythagorean triples as well as for an extensive historical overview see [@MathExp Section 4.2]. Nonexpansive monotone operators and Banach contractions ------------------------------------------------------- We continue our discussion of nonexpansive monotone operators. Within this class of mappings, being a Banach contraction is a dominant property: \[Banach contraction dominance within\] ***(within the class of nonexpansive mappings, being a Banach contraction is dominant)*** Suppose that for each $i\in I$, $A_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is nonexpansive and monotone. If there exists $i_0\in I$ such that $A_{i_0}$ is a Banach contraction, then ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is a Banach contraction. [@bmw12 Corollary 4.7 ] asserts that given a maximally monotone operator $B:{\ensuremath{\mathcal H}}\rightrightarrows{\ensuremath{\mathcal H}}$, letting $A=2J_B-{\ensuremath{\operatorname{Id}}}$, then $A$ is a Banach contraction if and only if $B$ and $B^{-1}$ are strongly monotone. We now employ the settings and the outcome of Theorem \[nonexpansive average\]: for every $i\in I$, $A_i:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is monotone and nonexpansive, $A_i=2J_{B_i}-{\ensuremath{\operatorname{Id}}}$ for a maximally monotone and nonexpansive mapping $B_i$ and ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=2J_B-{\ensuremath{\operatorname{Id}}}$ where $B:{\ensuremath{\mathcal H}}\to{\ensuremath{\mathcal H}}$ is the monotone and nonexpansive mapping $B=\sum_{i\in I}\lambda_iB_i$. We also see that $B_{i_0}$ is strongly monotone, say with $\epsilon_{i_0}$ being its constant of strong monotonicity. Now, let $x$ and $y$ be points in ${\ensuremath{\mathcal H}}$. Then $$\begin{aligned} {\langle{{Bx-By},{x-y}}\rangle}&=\sum_{i\in I}\lambda_i{\langle{{B_i x-B_i y},{x-y}}\rangle}\geq\lambda_{i_0}{\langle{{B_{i_0}x-B_{i_0}y},{x-y}}\rangle}\\ &\geq\lambda_{i_0}\epsilon_{i_0}\|x-y\|^2\geq\lambda_{i_0}\epsilon_{i_0}\|Bx-By\|^2.\end{aligned}$$ Thus, we see that both, $B$ and $B^{-1}$ are $\lambda_{i_0}\epsilon_{i_0}$-strongly monotone, which, in turn, implies that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=2J_B-{\ensuremath{\operatorname{Id}}}$ is a Banach contraction. Projections and normal cones ---------------------------- We will say that a property $(p)$ is ***indeterminate*** (with respect to the resolvent average) if $(p)$ is neither dominant nor recessive. \[201406121\] **(being a projection is indeterminate)** Let $A_1$ and $A_2$ be the projections in ${\ensuremath{\mathbb R}}^2$ onto ${\ensuremath{\mathbb R}}\times \{0\}$ and $\{0\} \times {\ensuremath{\mathbb R}}$, respectively. That is, $$A_1=\begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}\text{ and }A_2=\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix}.$$ Let $0<\lambda<1$, $\lambda_1=\lambda$ and $\lambda_2=1-\lambda$. Then $${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}= \begin{pmatrix} \frac{\lambda}{2-\lambda} & 0\\[6 pt] 0 & \frac{\lambda-1}{\lambda+1} \end{pmatrix},$$ which is not a projection since ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}^2\neq{\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$. \[201406122\] **(being a normal cone operator is indeterminate)** Let $f_1:{\ensuremath{\mathbb R}}^2\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ be the function $f_1=\iota_{{\ensuremath{\mathbb R}}\times\{0\}}$ and let $f_2:{\ensuremath{\mathbb R}}^2\to{\ensuremath{\,\left]-\infty,+\infty\right]}}$ be the function $f_1=\iota_{\{0\}\times{\ensuremath{\mathbb R}}}$. Let $A_1:{\ensuremath{\mathbb R}}^2\rightrightarrows{\ensuremath{\mathbb R}}^2$ be the normal cone operator $A_1=N_{{\ensuremath{\mathbb R}}\times\{0\}}=\partial f_1$ and let $A_2:{\ensuremath{\mathbb R}}^2\rightrightarrows{\ensuremath{\mathbb R}}^2$ be the normal cone operator $A_2=N_{\{0\}\times{\ensuremath{\mathbb R}}}=\partial f_2$. Let $0<\lambda<1$, $\lambda_1=\lambda$ and $\lambda_2=1-\lambda$, then $$J_{A_1}=\begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}\text{, }J_{A_2}=\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix} \ \text{\ and\ }\ {\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}= \begin{pmatrix}\frac{1-\lambda}{\lambda} & 0\\ 0 & \frac{\lambda}{1-\lambda}\end{pmatrix}.$$ Thus, we see that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}=\partial p({\bf f},{\boldsymbol \lambda}) =\nabla p({\bf f},{\boldsymbol \lambda})$ where $p({\bf f},{\boldsymbol \lambda}):{\ensuremath{\mathbb R}}^2\to{\ensuremath{\mathbb R}}$ is the parabola $p({\bf f},{\boldsymbol \lambda})(x,y)=\frac{1-\lambda}{2\lambda}x^2+\frac{\lambda}{2(1-\lambda)}y^2$. Since the antiderivative of ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is unique up to an additive constant and since $p({\bf f},{\boldsymbol \lambda})\neq\iota_C$ for any subset $C$ of ${\ensuremath{\mathbb R}}^2$, we see that ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is not of the form $\partial\iota_C$, that is, ${\ensuremath{\mathcal{R}({\bf A},{\boldsymbol \lambda})}}$ is not a normal cone operator. Lipschitz monotone operators {#Lipschitz} ---------------------------- The purpose of this last subsection is to point out that further study is still required in order to determine if Lipschitzness is a property which is dominant, recessive or indeterminate. Within certain classes of monotone operators we do, however, have several conclusions: (i) Within the class of monotone linear relations, Theorem \[classical linearity dominance within\](i) guarantees that Lipschitzness is dominant w.r.t. the resolvent average. However, no quantitative result is currently available with regard to the Lipschitz constant. (ii) Theorem \[nonexpansive average\] guarantees that 1-Lipschitzness is recessive in the case where $\mu=1$. However, no such result is available for other Lipschitz constants. Furthermore, within the class of nonexpansive monotone mappings, we saw that being a Banach contraction is dominant, as asserted by Theorem \[Banach contraction dominance within\], still, with no quantitative information regarding the Lipschitz constant. (iii) Within the class of subdifferential operators we do have dominance of Lipschitzness w.r.t. the resolvent average with an explicit Lipschitz constant, namely, Theorem \[Lip grad\]. However, as implied by Example \[strong sharpness counter for subdifferentials\], our explicit Lipschitz constant is not. Let $\alpha>0$. In the following example, within the class of monotone linear operators and outside the class of subdifferential operators, we take the resolvent average of two mappings with a common sharp Lipschitz constant $\alpha$ such that their resolvent average has a sharp Lipschitz constant $\alpha^2$. This is in stark contrast to the constant in Theorem \[Lip grad\] for subdifferential operators. \[scaled rotations\] Let $A_1=\begin{pmatrix} 0 & -\alpha\\ \alpha & 0 \end{pmatrix}=A_2^{\intercal} $. Then $$J_{A_1}=\frac{1}{\alpha^2+1}\begin{pmatrix} 1 & \alpha\\-\alpha & 1 \end{pmatrix}=J_{A_2}^{\intercal}\ \ \text{ and }\ \ \ \mathcal{R}(\bf{A})=\begin{pmatrix} \alpha^2 & 0\\ 0 & \alpha^2 \end{pmatrix}.$$ We see that, indeed, the sharp Lipschitz constant of $A_1$ and $A_2$ is $\alpha$ and the sharp Lipschitz constant of $\mathcal{R}(\bf{A})$ is $\alpha^2$. 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--- abstract: 'We can obtain one solution of the Hamiltonian constraint equation in the local sense. The form of the state is suggested from the up-to-down method in our previous work. The up-to-down method works for different way in treating the general metrics. In the mini-superspace approach there appears additional constraint in the 4-dimensional quantum gravity Hilbert space. However, in the general treatment of the metrics this method works as only solving technique.' author: - Shintaro Sawayama title: 'One Local Solution of the Wheeler-DeWitt equation' --- Introduction {#sec1} ============ There are many treatment of the quantum gravity, e.g. string theories [@AL] and mini-superspace approach [@Hart] or loop gravity [@Rov; @As; @Thi; @As2]. The theory of the quantum gravity has not yet been constructed. The theory of canonical quantum gravity is based on ADM decomposition. Form the ADM decomposition [@ADM]; we obtain constraint equations i.e. Hamiltonian and diffeomorphism constraint equations. To solve these constraint equations is the orthodox way of the canonical quantum gravity. The Hamiltonian constraint is the generator of the time translation and the diffeomorphism constraint are the generator of space translations [@Di2]. The theory of quantum gravity contains many unsolved problems which contains problem of the time and problem of the norm. However, most important problem is the difficulty of the constraint equations, i.e. Wheeler-DeWitt equation [@De]. Our motivation is simple and that is to find at least one local solution of the Wheeler-DeWitt equation. Our work is not motivated from the higher dimensional gravity. Although the Wheeler-DeWitt equation is difficult to solve, because it is the second order functional differential equation with diffeomorphism constraints, we had created a method to solve it that we call the up-to-down method [@Sa]. The introduced method contains many problems which is peculiar to the quantum gravity. And some long standing problems e.g. the problems of the norm or the problem of the time has not solved yet. In this paper we show one of the special local solutions of quantum gravity state for general spacetime metrics on the basis of the up-to-down method. The obtained state is a solution which satisfy Hamiltonian constraint. The main progress of the up-to-down method is the fact that we can find at least one local solution of the Hamiltonian constraint equation. In this paper, we reconstruct a technical method to solve the Wheeler-DeWitt equation, which we call up-to-down method. The up-to-down method consists of the following steps. First we add another dimension as an external time to the usual 4-dimensional metric and create an artificial functional space which has support of the spacetime metrics, and then we reduce this quantum state to the physical 4-dimensional state, we can simply solve the usual 3+1 Wheeler-DeWitt equation. The same method, however does not work for Klein-Gordon systems. The work of this method is different from the mini-superspace models, if we treat the Wheeler-DeWitt in the general sense. If we treat the mini-superspace models, the additional constraint appears. The simplification comes from the embedding of the 4-dimensional metric in the arbitrary 5-dimensional metric. The ideas of the up-to-down method come from the work of dynamical horizon [@AK; @Sa2] and the problem of the time. The problem of the time is inversely used such that if we add an additional dimension and treat it external time, then this dimension or all the components of additional dimension must be vanish. Although we use the up-to-down method as a derivation, the obtained state does not depend on the derivation. In section \[sec2\] we introduces the local quantum gravity and reconstruct the up-to-down method which is solving technique of the Wheeler-DeWitt equation. In section \[sec3\] we derive one local solution of the Hamiltonian constraint equation without fixing spacetime metrics. In section \[sec4\] we summarize the obtained result and comment on the problems of the quantum gravity. Local Quantum Gravity and Up-to-down Method {#sec2} =========================================== The local quantum gravity introduced in this section is considered to simply treat the Wheeler-DeWitt equation. The method of the local quantum gravity starts from decomposition of the Einstein-Hilbert action as, $$\begin{aligned} S=\int RdM=\sum_i \int R_i[g^{(i)}_{\mu\mu}]dS_i.\end{aligned}$$ Here $S_i$ is the subset of the hypersurface of $\Sigma$ with constant time. And $S_i$ is defined such that metric become diagonal by the local coordinate transformation. The Local quantum gravity starts from decomposition of $R_i$ as usual 3+1 sense. Then we obtained the Hamiltonian constraint and diffeomorphis constraint only in terms of the diagonal metric components. Although the local quantum gravity only uses diagonal components of the metrics, boundary condition appears and this condition is still not well defined. We introduce what we call the up-to-down method in self-contained way and in more strict way more than the previous paper. Some mistaken are corrected in this section. We should say some sentence is same as the previous paper. We start by introducing an additional dimension which is an external euclidean time with positive signature, and thus create an artificial functional space corresponding to this external time. We write such external dimension as $s$. We dare to start with artificial 5-dimensional action whose metric is created from the usual 4-dimensional metric components and arbitrary additional dimensional components as, $$\begin{aligned} S=\int _{M\times s}{}^{(5)}RdMds.\end{aligned}$$ Where ${}^{(5)}R$ is the 5-dimensional Ricci scalar. Although we stat from higher dimensional action, we do not motivated from higher dimensional gravity. Rewriting the action by a 4+1 slicing of the 5-dimensional spacetime with lapse functionals given by the $s$ direction, we obtain the 4+1 Hamiltonian constraint and the diffeomorphism constraints as, $$\begin{aligned} \hat{H}_S\equiv \hat{R}-\hat{K}^2+\hat{K}^{ab}\hat{K}_{ab} \\ \hat{H}_V^a\equiv \hat{\nabla} _b(\hat{K}^{ab}-\hat{K}\hat{g}^{ab}),\end{aligned}$$ where a hat means 4-dimensional, e.g. the $\hat{K}_{ab}$ is extrinsic curvature defined by $\hat{\nabla}_a s_b$ and $\hat{K}$ is its trace, while $\hat{R}$ is the 4-dimensional Ricci scalar, and $\hat{\nabla} _a$ is the 4-dimensional covariant derivative.\ \ [*Definition.* ]{} The artificial functional state is defined by $\hat{H}_S|\Psi^{5} (g)\rangle =\hat{H}_V^a|\Psi^{5} (g)\rangle =0$, where $g$ is the 4-dimenstional spacetime metrics $g_{\mu\nu}$ with ($\mu =0,\cdots ,3$). We write this functional space as ${\cal H}_5$.\ Here, the definition of the canonical momentum $P$ is different from the usual one. Note in fact that the above state in ${\cal H}_5$ is not the usual 5-dimensional quantum gravity state, because the 4+1 slicing is along the $s$ direction. It is not defined by $\partial {\cal L}/(\partial dg/dt)$ but by $\partial {\cal L}/(\partial dg/ds)$, where ${\cal L}$ is the 5-dimensional Lagrangian. Although whether this state is the Hilbert space or $l^2$ norm space is open question and this problem does not matter below, because what we would like to treat is the physical 4-dimensional quantum gravity state. In addition, we impose that 4-dimensional quantum gravity must be recovered from the above 5-dimensional action. The 3+1 Hamiltonian constraint and diffeomorphism constraint are, $$\begin{aligned} H_S\equiv {\cal R}+K^2-K^{ab}K_{ab} \\ H_V^a\equiv D_b(K^{ab}-Kq^{ab}).\end{aligned}$$ Here $K_{ab}$ is the usual extrinsic curvature defined by $D_at_b$ and $K$ is its trace, while ${\cal R}$ is the 3-dimensional Ricci scalar, and $D_a$ is the 3-dimensional covariant derivative. Then we can define a subset of the auxiliary Hilbert space on which the wave functional satisfies the usual 4-dimensional constraints. In order to relate the 4 and 5 dimensional spaces we should define projections.\ \ [*Definition.*]{} The subset of ${\cal H}_5$ in which the five dimensional quantum state satisfies the extra constraints $H_SP|\Psi ^5(g)\rangle=H_V^aP|\Psi ^5(g)\rangle =0$ is called ${\cal H}_{5lim}$, where $P$ is the projection defined by $$\begin{aligned} P:{\cal H}_5 \to L^2_4 \ \ \ \{ P|\Psi^5(g)\rangle=|\Psi^5(g_{0\mu}={\rm const})\rangle \} ,\end{aligned}$$ where $L^2_4$ is a functional space. And ${\cal H}_4$ is the usual four dimensional state with the restriction that $H_S|\Psi^4(q)\rangle=H_V^a|\Psi ^4(q)\rangle=0$. Here $q$ stands for the 3-dimensional metric $q_{ij}(i=1,\cdots ,3)$, and $P^{\dagger}$ is defined by $$\begin{aligned} P^{\dagger}:{\cal H}_{5lim} \to {\cal H}_4 .\end{aligned}$$ The above definition contains some extra definition such as $L^2_4,{\cal H}_{5lim}$. These functional spaces never appear in the following discussion and whether these space is norm space is open question and does not matter. We concern that some confusion occur by these symbols. However, ${\cal H}_4$ should be Hilbert space and there are problems of the norm. This problem can not have been solved yet, even if we use the up-to-down method. From now on we consider the recovery of the 4-dimensional vacuum quantum gravity from the 5-dimensional functional. We assume that the constraint, $$\begin{aligned} \hat{R}|\Psi^{5}(g)\rangle =0\end{aligned}$$ holes. Here $\hat{R}$ is the operator, corresponding to the usual 4-dimensional Ricci scalar. We use this constraint for the recovery of 4-dimensional quantum gravity in the sense of Dirac. And as we know in the previous paper physical meaning of this constraint is static restriction. Then artificial 5-dimensional Hamiltonian constraint becomes as $$\begin{aligned} \hat{H}_S=-\hat{K}^2+\hat{K}^{ab}\hat{K}_{ab}:=m\hat{H}_S\end{aligned}$$ We call this simplified 4+1 Hamiltonian constraint as modified Hamiltonian constraint. The simplification comes from the recovery of 4-dimensional quantum gravity i.e. $\hat{R}|\Psi ^5(g)\rangle =0$. Finally, the simplified Hamiltonian constraint in terms of the canonical representation becomes $$\begin{aligned} m\hat{H}_S =(-g_{ab}g_{cd}+g_{ac}g_{bd})\hat{P}^{ab}\hat{P}^{cd}.\end{aligned}$$ The magic constant factor $-1$ for the term $g_{ab}g_{cd}$ is a consequence of the choice of the dimensions for ${\cal H}_5,{\cal H}_4$. In the derivation of above formula we use the fact that the dimension of our universe is 4 with signature $(-,+,+,+)$. Here $\hat{P}^{ab}$ is the canonical momentum of the 4-dimensional metric $g_{ab}$, that is $\hat{P}^{ab}=\hat{K}^{ab}-g^{ab}\hat{K}$. And as we mentioned above, this canonical momentum is defined by the external time and not by the usual time. We now give a more detailed definition of the auxiliary 5-dimensional Hilbert space as follows:\ \ [*Definition.*]{} The subset ${\cal H}_{5(4)}\subset {\cal H}_5$ is defined by the constraints, $ \hat{R}|\Psi ^5(g)\rangle =0$, and we write its elements as $|\Psi ^{5(4)}(g)\rangle$. We also define a projection $P^*$ as $$\begin{aligned} P^* : {\cal H}_{5(4)} \to {\cal H}_{4(5)} \ \ \ \{ P^*|\Psi ^{5(4)}(g)\rangle =|\Psi ^{5(4)}(g_{0\mu}={\rm const})\rangle =: |\Psi ^{4(5)}(q)\rangle \} ,\end{aligned}$$ where ${\cal H}_{4(5)}$ is a subset of ${\cal H}_4$. There are reason why we choose such projection $P^*$. Although there are the problem of measure, this definition of the projection does not produce any additional constraint in ${\cal H}_4$ for the solution of $|\Psi^{5(4)}(g)\rangle$. So the modified Hamiltonian constraint and the 4+1 diffeomorphism constraint does not become additional constraint if they are projected by $P^*$.\ \ [*Definition.*]{} In ${\cal H}_{4(5)}$ there is a subset whose state satisfy $H_S|\Psi^{4(5)}(q)\rangle=H_V|\Psi ^{4(5)}(q)\rangle=0$. We write such Hilbert space as ${\cal H}_{4(5)phys}$ and its elements as $|\Psi_{phys}^{4(5)}(q)\rangle$. If there are relations $P^*|\Psi^{5(4)}(g)\rangle=|\Psi_{phys}^{4(5)}(q)\rangle$, we write such $|\Psi^{5(4)}(g)\rangle$ as $|\Psi^{5(4)}_{phys}(g)\rangle$ and we write such Hilbert space as ${\cal H}_{5(4)phys}$.\ \ We can summarize the procedure to solve the usual Wheeler-DeWitt equation.\ 1) Solve $m\hat{H}_SP^*|\Psi^4(g)\rangle =0$ and obtain $|\Psi ^{4(5)}(g)\rangle$.\ 2) Solve $H_S|\Psi ^{4(5)}(q)\rangle =H_V^a|\Psi^{4(5)}(q)\rangle =0$ to obtain $|\Psi ^{4(5)}_{phys}(q)\rangle$.\ \  We carry on step 1). At first we choose coordinate such that metric become diagonal because we treat local quantum gravity. Then modified Hamiltonian constraint at this limit becomes as, $$\begin{aligned} m\hat{H}_SP^*=\sum_{a\not= b}q_{ii}q_{jj}\frac{\delta}{\delta q_{ii}}\frac{\delta}{\delta q_{jj}}=0.\end{aligned}$$ We can easily find the solution of this equation and one of the states is, $$\begin{aligned} |\Psi^{4(5)}(q)\rangle =f_1[g_{11}]+f_2[g_{22}]+f_3[g_{33}].\end{aligned}$$ Here $f_{i}[g_{ii}]$ for $i=0,\cdots 3$ are functional of $g_{ii}$. Moreover, if we use a 3+1 diffeomorphism constraints, we can know that the $f_{i}$ are only the function of $g_{ii}$ at this point. However, we do not use this fact now. One solution of the Hamiltonian constraint {#sec3} ========================================== In this section we solve the local Hamiltonian constraint equation without fixing spacetime metrics, i.e. without treating mini-superspace models. Although this section motivated by the up-to-down method, what we derive in this section independently holeds. We use the fact that the metric becomes diagonal in the local coordinate. We start with assumption as, $$\begin{aligned} |\Psi^4(q)\rangle =f_1[q_{11}]+f_2[q_{22}]+f_3[q_{33}]\end{aligned}$$ with $$\begin{aligned} \hat{q}_{ii,k}f_j[q_{jj}]=\delta_{ij}q_{ii,k}f_j[q_{jj}] \ \ \ {\rm for} \ \ \ j\not= i.\end{aligned}$$ The parameter separated solution is the superposition of three local solutions. The assumption (16) means that the eigenvalue of the $q_{jj}$ of the state $f_i[q_{ii}]$ for $j\not=i$ is constant. The local Hamiltonian constraint equation can be written as, $$\begin{aligned} \sum_{ij}\frac{1}{2}\frac{\delta^2}{\delta \phi_i\delta \phi_j}+\sum_{i\not= j}(\hat{\phi}_{i,jj}+\hat{\phi}_{j,i}\hat{\phi}_{i,i})e^{\hat{\phi}_i}=0,\end{aligned}$$ where $q_{ii}=e^{\phi_i}$. The consistency of the additional constraint and the Hamiltonian constraint is clear from the next commutation relation. $$\begin{aligned} \bigg[ \sum_{k\not= l}\frac{\delta ^2}{\delta \phi_k\delta \phi_l}, \sum_{ij}\frac{1}{2}\frac{\delta^2}{\delta \phi_i\delta \phi_j} +\sum_{i\not= j}(\hat{\phi}_{i,jj}+\hat{\phi}_{j,i}\hat{\phi}_{i,i})e^{\hat{\phi}_i}\bigg] \sum_i f_i[\phi_i] =\bigg[ \sum_{k\not= l}\frac{\delta ^2}{\delta \phi_k\delta \phi_l},\sum_{i\not= j}\hat{\phi}_{j,i}\hat{\phi}_{i,i}e^{\hat{\phi}_i}\bigg]\sum_i f_i[\phi_i] \nonumber \\ =\sum_{k\not= l}\frac{\delta ^2}{\delta \phi_k\delta \phi_l}\sum_{i\not= j}\hat{\phi}_{j,i}\hat{\phi}_{i,i}e^{\hat{\phi}_i}\sum_i f_i[\phi_i ] .\end{aligned}$$ If the term $\hat{\phi}_{j,i}\hat{\phi}_{i,i}e^{\hat{\phi}_i}$ vanish, we can carry on simultaneous quantization. And this relation is same as the equation (16), so the assumption (16) is consistent. If we insert the equation (15), we obtain the following differential equation as, $$\begin{aligned} \frac{1}{2}\frac{\delta^2}{\delta \phi_i^2}f_i[\phi_i] +\sum_{j(\not=i)}\phi_{i,jj}e^{\phi_i}f_i[\phi_i]=0 \ \ \ {\rm for} \ \ \ i=1,2,3.\end{aligned}$$ Other components are dropped out because of the assumption (16). The important point is the fact that the above equation is algebraically closed for $\phi_i$. Because we ignore the operator ordering, we can rewrite the above equation as, $$\begin{aligned} \frac{\delta^2}{\delta a_i^2}+4\partial^j\partial_j\ln \hat{a_i} &=& \frac{\delta^2}{\delta a_i^2}+2i(\partial^j\partial_j\ln \hat{a_i})^{1/2}\frac{\delta}{\delta a_i} -2i(\partial ^j \partial _j \ln \hat{a_i})^{1/2}\frac{\delta}{\delta a_i} +4\partial^j \partial_j \ln \hat{a_i} \nonumber \\ &=&\frac{\delta^2}{\delta a_i^2}+2i(\partial^j\partial_j\ln \hat{a_i})^{1/2}\frac{\delta}{\delta a_i} -2i\frac{\delta}{\delta a_i}(\partial ^j \partial _j \ln \hat{a_i})^{1/2} +4\partial^j \partial_j \ln \hat{a_i} \nonumber \\ &=&\bigg( \frac{\delta}{\delta a_i}+2i(\partial^j\partial_j \ln \hat{a_i})^{1/2}\bigg) \bigg( \frac{\delta}{\delta a_i}-2i(\partial^j\partial_j\ln {\hat a_i})^{1/2}\bigg).\end{aligned}$$ Here, $a_i=q_{ii}^{1/2}$. We can find the solution of the above second order ordinal functional differential equation as, $$\begin{aligned} f_i[g_{ii}]=E_{1i}\exp (2i\int (\partial^j\partial_j\ln a_i)^{1/2}\delta a_i) +E_{2i}\exp (-2i\int (\partial^j\partial_j\ln a_i)^{1/2}\delta a_i) .\end{aligned}$$ Here $E_{1i}$ and $E_{2i}$ are constants. Then we can obtained the local Hamiltonian constraint as, $$\begin{aligned} |\Psi^4(q)\rangle =\sum_iE_i\cos (2\int (\partial^j\partial_j\ln a_i)^{1/2}\delta a_i).\end{aligned}$$ We can say local solution of the Hamiltonian constraint is the cosine wave. And above equation is the main progress of our work. Conclusion and discussions {#sec4} ========================== We can find at least one local solution of the Hamiltonian constraint. Before the up-to-down method we do not know any solution of this equation. The peculiar problem in the quantum gravity is still unsolved. Even if we use this method we can still not solve the problem of the norm and problem of the time [@Ha1; @Ha2]. We should find all state to solve the problem of norm. Otherwise we obtained one solution it may create other solutions. We hope such kinds of work succeed. Because our work succeeded the quantization of the inhomogeneous spacetime, we can enter this state symmetry for example spherical symmetry. Then we may obtain the state of the black holes [@K2]. Or we can enter symmetry of the homogeneous and isotropy, as loop gravity [@Bo]. Simple analysis is carried out by inserting symmetry in $q_{ii}$ such that $q_{11}=q_{22}=q_{33}=a$, where the $a$ is the function of only the time $t$. Then we can obtain a simple cosine wave. Such cosine wave does not occur if the cosmological constant is zero. We can not find such wave from the mini-superspace approach. 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--- abstract: 'I present some reminiscences, both personal and scientific, over a lifetime of admiration and friendship with one of the Grandmasters of our subject.' address: | Walter Burke Institute for Theoretical Physics,\ California Institute of Technology, Pasadena, CA 91125;\ Physics Department, Brandeis University, Waltham, MA 02454;\ [[email protected]]{} author: - Stanley Deser title: 'Julian Schwinger — Recollections from many decades' --- {#section .unnumbered} Dear students, friends, and admirers of Julian Schwinger, or all three. We are here to celebrate and commemorate a century since Julian’s birth. He only lived three quarters of that period, unfortunately dying far too young at 76, but left us a great legacy. Being in this Conference’s history section, I will try to discuss the life and work as I saw it, minus technicalities. I knew Julian for three-fifths of his life, a reasonable fraction. It began when I arrived as a graduate student in the fall of 1949. I didn’t know much physics, nor did I know who Julian was, but I was soon educated on the latter. In fact, I sat in on three of his quantum mechanics courses, all different. Like everybody else who wanted to do theory, I was convinced that Julian should be my mentor. He was willing to accept just about anybody, but he was chary with his time, as all his students know. Since he has saved my life so many times, I feel I should begin by giving some examples. The system at Harvard in my day, if you wanted to do theory, required you to take a qualifying exam, usually in something called Math and Mechanics, which covered various sins. One was supposed to bone up on that during one’s second academic year; a jury of one’s would-be advisor plus two other people was then convened. The day duly came and Julian arrived, flanked by Abe Klein and Bob Karplus, two up-and-coming assistant professors whose careers depended critically on sufficiently impressing Julian so they could get good positions elsewhere — one didn’t get promoted from within. They had discovered something in their latest calculations, some particularly uninteresting but technical stuff called dilogarithms, which are now, I suppose, taught in kindergarten but in those days unknown to anyone — certainly to me. They proceeded to show Julian how brilliant and clever they were, at my expense, so that after the first few words, I was totally excluded from everything, and after an hour and a half of this, they turned to me and pityingly asked me a question like what two plus two was, at which point I couldn’t even have answered one plus one. And so this terrible ordeal ended, I walked out, and two minutes later Julian came and said, “you realize you failed your qualifying exam,” and I said “yes,” and there was a little pause and he went on, “don’t worry about it.” I think this miracle (and miracle it was — no one else failed M&M) may have been due to my performance on an advanced electrodynamics course I had just taken with him. Then I started on my thesis. I think I probably saw Julian for a total — just on the upper limit — of about ten hours during those two years. One day, in the spring of my fourth year, I asked Julian, when would I could possibly think of finishing up. When he replied “right now if you want” — this was shortly before the strict Harvard deadline for submitting a thesis, I was not going to let this opportunity slide; somehow it all got done and typed on a Bible paper, only available in one place in the world, and bound in one particular way, and all the rest of it. Although the thesis was mediocre, I was handed my Ph.D. by James Bryant Conant, in his last year of a long tenure as president of the university. Rescue number two was a bit more indirect. In those days, Julian would simply phone Oppenheimer at the Institute for Advanced Study, tell him who his latest graduates were and Oppenheimer would take them, no applications or recommendations. Unfortunately, the year before mine, Julian’s choice at that point was a very strange guy, we’ll not name names, who was found in his first year at the Institute climbing the wall of some estate in Princeton, something frowned upon at such a rich community. The whole thing was handled very well, all airbrushed out. He disappeared, and I’m told became a successful psychoanalyst, but that could be apocryphal. In any case, Oppenheimer was taking no chances, so he told Julian that his two picks, Roger Newton and I, had better show up and pass a psychiatric exam. In those days Oppenheimer still had his clearance, so two FBI agents were guarding his files; I walked past with trepidation, but all Oppenheimer did was ask me what my thesis was about, the title of which I told him. He immediately told me (a) what was in the thesis and (b) why it was wrong. He was way off the mark, at least on point (a); he had no idea whatsoever, but that was in his style. At least I didn’t have any obvious tics. I was vetted also by the younger permanent people at the Institute, and Roger also passed with flying colors. We were installed at the Institute where I had my two years, and not so much contact with Julian. However he saved me because when I arrived at the Institute not too sure what to do, I was immediately pounced on by Murph Goldberger and Walther Thirring who were both visiting there. They said, “you must know all of Julian’s tricks, so let’s get moving and apply them to the following project.” Of course I didn’t know Julian’s tricks, but it in fact provided my first successful extra-Ph.D. experience and did use some of them after all. I should mention — going back a bit — that before you start on a thesis, you’re given a little test problem by your advisor. Julian gave me the little test problem, of which I had no idea whatsoever at all what to do, the reason was that this little problem was the beginning of his celebrated National Academy of Sciences series that to this very day is a standard tool. So when he showed me what he had done, I realized why I hadn’t a clue as to what to do. Well, that too he accepted. So I learned from that that one should do onto others and give would-be graduate students a certain amount of leeway, perhaps not as much as he gave me, but still. Then came my second postdoc stage. After the two years at the Institute, I went off for two years to the Niels Bohr Institute in Denmark, which was a difficult period for me. I only wrote one paper, which was furthermore wrong, although wrong in an interesting way. In any case, in those days especially, I hadn’t realized that, once you go into exile, you no longer exist in the United States, because you’re not in any loop. Fortunately, Julian came by that summer, visiting Denmark with Clarice, and he again saved my life by offering me one year as his assistant as an Instructor at Harvard, while I found my footing back home. That was truly critical, because being married having a baby, it was clear that I needed some sort of a job. He then also recommended me for my first faculty position at Brandeis. So, this was the support I got from Julian: his faith in me was truly beyond any requirement. His greatest confidence in me occurred much later. I was an invited visitor to UCLA, where Julian had moved, and used to stay in his house. Once I came it was during one of those oil embargos when you couldn’t get any gas for your car, especially in California. Julian lived in Bel Air, which had, and has still, for all I know, one and only one gas station, at some chi-chi little shopping center. It was going to open at 7AM until the gas ran out by 7:30, and Julian was of course in a terrible quandary because 7AM is too late for staying up and far too early for getting up. I was still on Eastern time, so 7AM suited me fine, but would he entrust his precious sports car? He agonized all evening and then finally handed me the keys, gave me a three-hour lecture on how to drive, and I’m sure had a very restless night. I arrived at 7AM, surrounded by all the neighborhood Bentleys and Rolls-Royces, chauffeurs waiting in line, but I did manage to snag sufficient gas for the next period and avoided having any dents in the car, which Julian inspected carefully. Our relations became more even with time. In particular, after the birth of supergravity in ’76, Julian asked me to come for a weekend tutorial for him and his entourage at UCLA. So there, on a Saturday at some ungodly early hour like 10AM, we started on a full Soviet-style two day session; Julian would say, “I don’t understand what this is,” and I replied “come on, Julian, you invented it all,” and reminded him of the Rarita–Schwinger equation, which he did indeed vaguely remember — they had actually a fairly ugly form for it — but they had found it. I suggested that in fact Julian should have discovered supergravity, as he had all the tools. He was of course one of the few people in the United States, back when he was student in the ’30s, who even knew general relativity. Just like he knew quantum mechanics. You all know the famous story of how when he was flunking out of City College, Lloyd Motz brought him to Rabi to try to get him a transfer to Columbia; he must have been in the teens. He was put on a bench while they argued about some quantum mechanics problem unsuccessfully, and then he spoke up, adding the one word that explained it all. Motz and Rabi finally realized he was in the room, and the rest is history. He was also an extremely cultured person, and though very shy, he was really quite up on a number of extra-physical things that one might not have expected. I realized that also during the year that I was his assistant in ’57–58, because I was then in the position that we graduate students used to envy. When we were students, with his assistants, instead of seeing us he would walk off to the restaurant of his choice, there was only one at Harvard Square those days that was even semi-edible, and leave us in the lurch. So this time it was my turn to leave the students in the lurch, and we would talk about all sorts of things, physics, and non; a great educational opportunity. The range of Julian’s discoveries and inventions, formal as well as directly physical, is, as we know, enormous. He was of course a great master of Green functions and everything related to gauge fields. He was one of the predecessors of the weak interaction theories that were then soon developed — Shelly Glashow was his student. Julian was extremely active not only during the great triumph of QED, quantum electrodynamics, but after that stayed very much in touch with developments. However, and this is my own theory, the Moses complex, that great men who are handed the truth from up above, are fated never to set foot in the promised land, and Julian is certainly an example. He felt at a certain point, especially in his last times at Harvard, that he was more or less sidetracked from what was then the main line, and that’s when he made his motto “If you can’t join ’em, beat ’em,” the origin of source theory, which engaged him for quite a long time, and which of course was a very interesting way to look at quantum field theory, although not really as productive as he might have hoped. He also had his engagements with cold fusion, and I think it was all part of a reaction — the Moses reaction, not being able quite to go to the next stage, such as it is, our standard model “promised land.” But he provided an enormous amount of impetus through his students. I remind you that Julian had something like 72 or so Ph.D.s to his credit. That is an amazing number, four of whom — if you count that way — were Nobel Prize winners, not bad. Roy Glauber, Shelly Glashow, Ben Mottelson and Walter Kohn, although he got it in Chemistry, was very much a Schwinger product. In fact, my first quantum mechanics course was taught by Walter, it was one of the many Schwinger QM variants. So, his influence both with the early Ph.D.s and of course later on with the cohort at UCLA, many of whom will be speaking and reminiscing here, should be very much counted as part of his contribution to our subject. But really — when you think about it, there’s almost nothing we do in theory that doesn’t somewhere bear Schwinger’s imprint. In fact, in the old days, there was a great form that you could fill out in order to publish a paper: it started with “According to Schwinger …,” then you would put in the equation of your choice, and then went on and said “… now using the Green function appropriate to this problem we discover that …,” and the paper was guaranteed to get into the Physical Review. That was not so far from true. But of course, Julian was much more than that. He was involved in early postwar nuclear physics; I took a full year’s nuclear physics course from him. This was really dirty nuclear physics and its phenomenology, effective range theory, scattering and bound states, again from an effective theory point of view, it was a very powerful tool. It was not quarks, but it was a way to understand, classify, and normalize low-energy nuclear physics, as it was then practiced, a forefront field. He had an enormous effect on classical electrodynamics. We all know that during the war his work on waveguides and propagation was not only very useful to the war effort, but really began a whole field of investigations in that time — and then of course, QED as I mentioned. I still remember his lecturing, his derivation, first of course of $\alpha/2\pi$, as he did it in class, and then of the Lamb shift, the really great achievements of QED. He did it in real time on the blackboard, sans notes. I have since taught the Lamb shift and I couldn’t do any better than Julian had done 20 years earlier. Sometimes, of course, like all great men he faltered. There was an early famous incident. Every Wednesday afternoon the joint theoretical seminar used to oscillate between Harvard and MIT. Julian started one, claiming to have completely solved the closed form of QED, which was clearly going a little too far. Finally, at some point Francis Low pointed out that in order to do this Julian had assumed that a four-point function was simply the product of two two-point functions, and of course if you assume that, you’re talking about a free theory. So that particular attempt did not work. I mention this to say that Julian, like all great men, was of course fallible, but fallible in a very trivial way, never on the real essentials. I think the only person who was infallible was Enrico Fermi; he was called the pope because of that. But in Julian’s frontier explorations, you had to take the risk of being wrong in order to get anywhere. He also was interested in general relativity later, so when we — ADM: Arnowitt, Misner, and I — especially Arnowitt and I at the end of my year at Harvard began our work on the theory, Julian got interested and wrote a couple of papers. He also tried to do something about supersymmetry which didn’t quite work out. There is actually very little in modern theory that he was not at least cognizant of if not actively pursuing, so that the brain never really stopped working. For example, he set Wally Gilbert, George Sudarshan and me on dispersion work during that Harvard year that is still useful. As I said, later on he became a little bit isolated from the mainstream community, but that’s not that he was unaware of what they were doing, he knew what was going on, and although he never made it into the standard model, he certainly laid its groundwork in many directions. The work that he did in electrodynamics evolved into the heroic calculations that his graduate students — Charlie Sommerfield, for example — utilized to do the two-loop magnetic anomalous moment corrections. That Julian stayed away from Feynman diagrams is a trivial difference which has wrongly been blown up. Whether you have a Green function or a line makes no difference; at the end of the day, you are doing the same integrals. I give these — among many other — examples to indicate the sheer scope of Julian’s inventions. In the early days, the Rarita–Schwinger equation, which was the first serious attempt at going beyond spin 1 — was completely virgin ground at that point, and of course the significance of going beyond spin 1 and its problems did not emerge until very much later, again very much before its time as often was with Julian and his work. During the war, he, like all the other physicists, concentrated on applications — all were either in Los Alamos or the Rad Lab, where Julian chose to go. It was not until after the war and he was lured to Harvard that he was able to really go full speed again in fundamental physics. The aura that Julian had around him at Harvard was really unparalleled and well-earned. It was a sad day for Harvard when he was lured to UCLA, although they did quite well with Steven Weinberg as a replacement, I should immediately add. UCLA was greatly enlivened by the presence of Julian and his group, and although I suspect that eternal sunshine and tennis had something to do with his move, he was really quite happy at UCLA. He died of a cancer with a guaranteed lethal outcome, but I think he enjoyed his life to the very end. Clarice was his ideal companion. He has left the memories we all know for his successors. He has made us all his ex-students. Even people who were not yet alive when he died benefited from the foundations that he laid to our field. I can think of no better exemplar in every respect. It was, for me, a great honor to be his student, as I’m sure will be echoed by everyone in this room. Thank you. Aknowledgments {#aknowledgments .unnumbered} ============== This work was supported by grants NSF PHY-1266107 and DOE\#desc0011632. It is a transcription of an invited video contribution to the Schwinger 100 Symposium, Singapore 2018.

--- abstract: | We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted $\ell^1$ space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted $\ell^1$ space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions, and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.\ *Keywords:* discrete fragmentation, positive semigroup, analytic semigroup, long-time behaviour, Sobolev towers\ *Mathematics Subject Classification (2010):* 47D06; 34G10, 80A30, 34D05 author: - 'Lyndsay Kerr, Wilson Lamb and Matthias Langer' title: | Discrete Fragmentation Systems in\ Weighted $\ell^1$ Spaces --- Introduction {#Introduction} ============ There are many diverse situations arising in nature and industrial processes where clusters of particles can merge together (coagulate) to produce larger clusters, and can break apart (fragment) to produce smaller clusters. Particular examples can be found in polymer science, [@aizenman1979convergence; @ziff1980kinetics; @ziffmcgrady1985kinetics], in the formation of aerosols, [@drake1972aerosol], and in the powder production industry, [@verdurmen2004simulation; @wells2018thesis]. It is often appropriate when modelling such processes to regard cluster size as a discrete variable, with a cluster of size $n$, an $n$-mer, composed of $n$ identical units (monomers). By scaling the mass, we can assume that each monomer has unit mass and so an $n$-mer has mass $n$. The aim is to use the mathematical model to obtain information on how clusters of different sizes evolve. In this paper we restrict our attention to the case when no coagulation occurs, and consequently the evolution of clusters can be described by a linear, infinite system of ordinary differential equations. With the number density of clusters of size $n$ (i.e. mass $n$) at time $t$ denoted by $u_n(t)$, this fragmentation system is given by $$\label{full frag system} \begin{split} u_n'(t)&=-a_nu_n(t)+\sum\limits_{j=n+1}^{\infty} a_jb_{n,j}u_j(t), \qquad t>0; \\ u_n(0)&=\mathring{u}_n, \qquad n=1,2,\ldots, \end{split}$$ where $a_n$ is the rate at which clusters of size $n$ are lost, $b_{n,j}$ is the rate at which clusters of size $n$ are produced when a larger cluster of size $j$ fragments and $\mathring{u}_n$ is the initial density of clusters of size $n$ at time $t=0$. Equation was first introduced in [@ziffmcgrady1985kinetics] to deal with the case of binary fragmentation, where it is assumed that each fragmentation event results in the creation of exactly two daughter clusters. As in [@banasiak2011irregular; @banasiakjoelshindin2019_onlinefirst; @mcbride2010strongly; @smith2012discrete], we consider the more general case, where each fragmentation event can result in the creation of two or more clusters. Since is an infinite system, it is convenient to express solutions as time-dependent sequences of the form $u(t) \coloneqq (u_n(t))_{n=1}^{\infty}$. Throughout this paper we need various assumptions on the fragmentation coefficients $a_n$ and $b_{n,j}$. We list these assumptions here and will refer to them in the sequel when required. \[A1.1\] ------------------------------------------------------------------------ For all $n \in \mathbb{N}$, $$\label{fragmentation rate assumption} a_n \ge 0.$$ For all $n,j \in \mathbb{N}$, $$\label{a_b_nonnegative} b_{n,j} \ge 0 \qquad\text{and}\qquad b_{n,j} = 0 \quad \text{when} \ n \ge j.$$ The total mass of daughter clusters resulting from the fragmentation of a $j$-mer is given by $\sum_{n=1}^{j-1} nb_{n,j}$. In most papers that have dealt with discrete fragmentation systems it is assumed that $$\label{local_mass_non_increasing} \sum\limits_{n=1}^{j-1} nb_{n,j} \le j \qquad\text{for all} \ j=2,3,\ldots,$$ i.e. there is no increase in mass at fragmentation events. If there is strict inequality in , then mass is lost by some other mechanism. However, for most of our results we do not assume that holds; this means that mass could even be gained at fragmentation events. We can specify the local mass loss or mass gain with real parameters $\lambda_j$, $j=2,3,\ldots$, such that $$\label{local mass conservation lambda} \sum\limits_{n=1}^{j-1} nb_{n,j} = (1-\lambda_j)j, \qquad j=2,3,\ldots.$$ In terms of the densities $u_n(t)$, the total mass of all clusters in the system at time $t$ is given by the first moment, $M_1(u(t))$, of $u(t)$, where $$\label{total mass} M_1\bigl(u(t)\bigr) \coloneqq \sum\limits_{n=1}^{\infty} nu_n(t).$$ A formal calculation establishes that if $u$ is a solution of , then $$\label{massode} \frac{{\mathrm{d}}}{{\mathrm{d}}t}M_1\bigl(u(t)\bigr) = - a_1u_1(t) - \sum_{j=2}^\infty j \lambda_j a_ju_j(t).$$ The expression in gives the rate at which mass may be lost from the system or gained, and also shows that, at least formally, the total mass is conserved when $a_1=0$ and $\lambda_j=0$ for all $j=2,3,\ldots$, i.e. when $$\label{mass_conserved} a_1 = 0 \qquad\text{and}\qquad \sum_{n=1}^{j-1} n b_{n,j} = j \quad \text{for all} \ j=2,3,\ldots.$$ Note that monomers cannot fragment to produce smaller clusters, and hence the case when $a_1 > 0$ is interpreted as a situation in which monomers are removed from the system. In this paper, the approach we use to investigate relies on the theory of semigroups of bounded linear operators, and entails formulating as an abstract Cauchy problem (ACP) in an appropriate Banach space. The existence and uniqueness of solutions to the ACP are established via the application of perturbation results for operator semigroups. Of particular relevance is the Kato–Voigt perturbation theorem for substochastic semigroups [@banasiak2001extension; @voigt1987onsubstochastic] that was first applied to in [@mcbride2010strongly], and subsequently in similar semigroup-based investigations into , such as [@banasiak2012global; @smith2012discrete]. We use a refined version of this theorem proved by Thieme and Voigt in [@thieme2006stochastic]. In previous studies, including [@mcbride2010strongly; @smith2012discrete], the ACP associated with the fragmentation system has been formulated in the space $$\label{X1space} X_{[1]} \coloneqq \biggl\{f=(f_n)_{n=1}^{\infty}: f_n \in \mathbb{R} \ \text{for all} \ n \in \mathbb{N} \ \text{and} \ \sum\limits_{n=1}^{\infty} n|f_n|<\infty\biggr\}.$$ Equipped with the norm $$\label{X1norm} \Vert f \Vert_{[1]} = \sum\limits_{n=1}^{\infty} n|f_n|, \qquad f \in X_{[1]},$$ $X_{[1]}$ is a Banach space, and $$\label{X_1functional} \Vert f \Vert_{[1]} = M_1(f)$$ if $f \in X_{[1]}$ is such that $f_n \ge 0$, $n \in \mathbb{N}$. This means that whenever $u:[0,\infty) \to X_{[1]}$ is a non-negative solution of the fragmentation system, the norm, $\Vert u(t)\Vert_{[1]}$, gives the total mass at time $t$. Other Banach spaces, with norms related to higher order moments, have also played a prominent role [@banasiak2012global; @banasiaklamb2012analytic], with $X_{[1]}$ being replaced by $X_{[p]}$, $p > 1$, where $$\label{Xpspace} X_{[p]} \coloneqq \biggl\{f=(f_n)_{n=1}^{\infty}: f_n \in \mathbb{R} \ \text{for all} \ n \in \mathbb{N} \ \text{and} \ \Vert f \Vert_{[p]} \coloneqq \sum\limits_{n=1}^{\infty} n^p|f_n|<\infty\biggr\}.$$ Rather than restricting our investigations to spaces of the type $X_{[p]}$, we choose to work within the framework of more general weighted $\ell^1$ spaces. As we shall demonstrate, this additional flexibility will enable us to establish desirable semigroup properties and results that may not always be possible in an $X_{[p]}$ setting. Therefore, we let $w=(w_n)_{n=1}^{\infty}$ be such that $w_n>0$ for all $n \in \mathbb{N}$, and define $$\label{weighted l^1 space} \ell_w^1 = \biggl\{f=(f_n)_{n=1}^{\infty}: f_n \in \mathbb{R} \ \text{for all} \ n \in \mathbb{N} \ \text{and} \ \sum\limits_{n=1}^{\infty} w_n|f_n|<\infty\biggr\}.$$ Equipped with the norm $$\label{weighted l^1 space norm} \Vert f \Vert_w=\sum\limits_{n=1}^{\infty} w_n|f_n|, \qquad f \in \ell_w^1,$$ $\ell_w^1$ is a Banach space, which we refer to as the weighted $\ell^1$ space with weight $w$. Motivated by the terms in , we introduce the formal expressions $$\begin{aligned} \mathcal{A}: (f_n)_{n=1}^{\infty} \mapsto (-a_nf_n)_{n=1}^{\infty} \qquad \text{and} \qquad \mathcal{B}: (f_n)_{n=1}^{\infty} \mapsto \Biggl(\sum\limits_{j=n+1}^{\infty} a_j b_{n,j}f_j\Biggr)_{n=1}^{\infty}.\end{aligned}$$ Operator realisations, $A^{(w)}$ and $B^{(w)}$, of $\mathcal{A}$ and $\mathcal{B}$ respectively, are defined in $\ell_w^1$ by $$\label{A^w equation} A^{(w)}f = \mathcal{A}f, \qquad \mathcal{D}(A^{(w)}) = \bigl\{f \in \ell_w^1: \mathcal{A}f \in \ell_w^1\bigr\}$$ and $$\label{B^w equation} B^{(w)}f = \mathcal{B}f, \qquad \mathcal{D}(B^{(w)}) = \bigl\{f \in \ell_w^1: \mathcal{B}f \in \ell_w^1\bigr\}.$$ Here, and in the sequel, $\mathcal{D}(T)$ denotes the domain of the designated operator $T$. Similarly, we shall represent the resolvent, $(\lambda I-T)^{-1}$, of $T$ by $R(\lambda,T)$. An ACP version of , posed in the space $\ell_w^1$, can be formulated as $$\label{weighted frag ACP} u'(t) = A^{(w)}u(t)+B^{(w)}u(t), \quad t>0; \qquad u(0)=\mathring{u}.$$ Note that this reformulation of imposes additional constraints on both the initial data and the sought solutions since we now require $\mathring{u} \in \ell_w^1$ and also that the solution $u(t) \in \mathcal{D}(A^{(w)}) \cap \mathcal{D}(B^{(w)})$ for all $t > 0$. Moreover, as the derivative on the left-hand side of is defined in terms of ${\lVert\,\cdot\,\rVert_{w}}$, it is customary to look for a solution $u \in C^1((0,\infty), \ell_w^1) \cap C([0,\infty), \ell_w^1)$. Such a solution is referred to as a classical solution of , and has the property that $\Vert u(t) - \mathring{u}\Vert_w \to 0$ as $t \to 0^+$. It turns out that often, instead of using the operator $A^{(w)}+B^{(w)}$ on the right-hand side of , one has to use its closure, which leads to the ACP $$\label{ACP_with_Gw} u'(t) = \overline{(A^{(w)}+B^{(w)})}u(t), \quad t>0; \qquad u(0)=\mathring{u}.$$ Yet another option for an operator on the right-hand side is the maximal operator, ${{G_{\textup{\textsf{max}}}}^{(w)}}$, which is defined by $$\label{definition_Gmax} {{G_{\textup{\textsf{max}}}}^{(w)}}f = \mathcal{A}f + \mathcal{B}f, \qquad \mathcal{D}({{G_{\textup{\textsf{max}}}}^{(w)}}) = \bigl\{f \in \ell_w^1: \mathcal{A}f + \mathcal{B}f \in \ell_w^1\bigr\}.$$ However, the domain of this operator is too large in general to ensure uniqueness of solutions; see Example \[example: random scission\] below, and also [@banasiak2002unique] where a continuous fragmentation equation is studied. There are a number of benefits to be gained by working in more general weighted $\ell^1$ spaces, least of which is the derivation of existence and uniqueness results for in $\ell_w^1$ that reduce to those established in earlier $X_{[p]}$-based investigations by choosing $w_n=n^p$. For example, in Theorem \[G=closure for frag\] we prove that $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$ is the generator of a substochastic $C_0$-semigroup. While this result has already been shown for the specific case $w_n=n^p$ for $p \ge 1$, see [@banasiak2012global; @mcbride2010strongly], Theorem \[G=closure for frag\] is formulated for more general weights, and is proved by means of an alternative and novel argument that is based on theory presented in [@thieme2006stochastic]. Our approach also leads to an additional invariance result, which can be used to establish the existence of solutions to the fragmentation system for a certain specified class of initial conditions. A further major advantage of working in the more general setting of $\ell_w^1$ is that it yields results on the analyticity of the related fragmentation semigroups, which do not necessarily hold in the restricted case of $w_n = n^p,\, p \geq 1$. In particular, in Theorem \[can always find analytic semigroup\] we prove that, for *any* fragmentation coefficients, we can *always* find a weight $w$ such that $A^{(w)}+B^{(w)}$ is the generator of an analytic, substochastic $C_0$-semigroup on $\ell_w^1$. In connection with this, it should be noted that there are no known general results that guarantee the analyticity of the fragmentation semigroup on the space $X_{[1]}$. Indeed, this provided the motivation for previous investigations into fragmentation ACPs posed in higher moment spaces, which led to a sufficient condition being found in [@banasiak2012global] for $A^{(w)}+B^{(w)}$ to generate an analytic semigroup on $X_{[p]}$ for some $p > 1$. However, simple examples are also given in [@banasiak2012global] of fragmentation coefficients where the semigroup is not analytic in $X_{[p]}$ for any $p\ge1$; see Example \[finding weights for binary fragmentation\]. The importance of establishing the analyticity of the semigroup associated with the fragmentation system is that analytic semigroups have extremely useful properties. For example, if $A^{(w)}+B^{(w)}$ generates an analytic semigroup on $\ell^1_w$, then it follows immediately that the ACP has a unique classical solution for any $\mathring{u} \in \ell_w^1$. In addition, when coagulation is introduced into the system, the analyticity of the semigroup generated by $A^{(w)}+B^{(w)}$ can be used to weaken the assumptions that are required on the cluster coagulation rates to obtain the existence and uniqueness of solutions to the corresponding coagulation–fragmentation system of equations. Such coagulation–fragmentation systems will be considered in a subsequent publication. Once the well-posedness of the fragmentation ACP has been satisfactorily dealt with, the next question to be addressed is that of the long-term behaviour of solutions. Results on the asymptotic behaviour of solutions to are given in [@banasiak2011irregular; @banasiaklamb2012discrete; @CadC94] for the specific case where the weight is $w_n=n^p$ for $p \ge 1$, $n \in \mathbb{N}$. In particular, for mass-conserving fragmentation processes, where holds, it is shown that the solution of converges to a state where there are only monomers present if and only if $a_n>0$ for all $n \geq 2$. In Section \[Asymptotic Behaviour of Solutions\] we continue to work with more general weights and, in the mass-loss case, show that the solution of decays to the zero state over time if and only if $a_n>0$ for all $n \in \mathbb{N}$. This mass-loss result can then be used to deduce that the solution, in the mass conserving case, converges to the monomer state if and only if $a_n>0$ for all $n \geq 2$, this result now holding in the general weighted space $\ell_w^1$. Regarding the rate at which solutions approach the steady state, the case where mass is conserved and $w_n=n^p$ for $p>1$ is examined in [@banasiaklamb2012discrete Section 4], and it is shown that solutions decay to the monomer state at an exponential rate, which, however, is not quantified. In Section \[Asymptotic Behaviour of Solutions\] we obtain results regarding the exponential rate of decay of solutions, both for the mass-conserving and mass-loss cases, by working in a space $\ell^1_w$ in which $A^{(w)}+B^{(w)}$ generates an analytic semigroup. The approach we use enables us to quantify the exponential decay rate. In [@smith2012discrete], the theory of Sobolev towers is used to investigate a specific example of that has been proposed as a model of random bond annihilation. Of particular note is the fact that the resulting analysis provides a rigorous explanation of an apparent non-uniqueness of solutions that emanate from a zero initial condition. We shall establish that an approach involving Sobolev towers can also be used to obtain results on for general fragmentation coefficients. By writing as an ACP in $\ell_w^1$, where $w$ is such that $A^{(w)}+B^{(w)}$ generates an analytic, substochastic $C_0$-semigroup on $\ell_w^1$, we are able to construct a Sobolev tower and then use this to prove the existence of unique, non-negative solutions of for a wider class of non-negative initial conditions than those in $\ell_w^1$; see Theorem \[semigroup from tower solves weighted ACP\]. The paper is structured as follows. In Section \[Preliminaries\] we provide some prerequisite results and definitions. Following this, we begin our examination of in Section \[The Generator of the Fragmentation Semigroup\], obtaining, in particular, the aforementioned Theorem \[G=closure for frag\], which is then used to draw conclusions on the existence and uniqueness of solutions to and , both in the space $X_{[1]}$ and in more general $\ell_w^1$ spaces. We consider the pointwise system in Section \[Pointwise Problem\] and show that for any $\mathring{u} \in \ell_w^1$, a solution of can be expressed in terms of the semigroup generated by $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$. We then use this result to show that $G^{(w)}$ is a restriction of the maximal operator ${{G_{\textup{\textsf{max}}}}^{(w)}}$. This is important in investigations into the full coagulation–fragmentation system as it allows the fragmentation terms to be completely described by the operator $G^{(w)}$. Results on the analyticity of the fragmentation semigroup are presented in Section \[Analyticity of the Fragmentation Semigroup\], and then applied both in Section \[Asymptotic Behaviour of Solutions\], where the asymptotic behaviour of solutions is investigated, and in Section \[Sobolev towers\], where the theory of Sobolev towers is applied to establish the well-posedness of for more general initial conditions. Preliminaries {#Preliminaries} ============= We begin by recalling some terminology. The following notions are well known and can be found in various sources, including [@banasiak2006perturbations; @batkai2017positive]. Let $X$ be a real vector lattice with norm ${\lVert\,\cdot\,\rVert}$. The positive cone, $X_+$, of $X$ is the set of non-negative elements in $X$ and, similarly, for a subspace $D$ of $X$, we denote the set of non-negative elements in $D$ by $D_+$. If $X$ is a vector lattice, then for each $f \in X$ the vectors $f_{\pm} \coloneqq \sup\{\pm f,0\}$ are well defined and satisfy $f_+,f_- \in X_+$ and $f=f_+-f_-$. A vector lattice, equipped with a lattice norm ${\lVert\,\cdot\,\rVert}$, is said to be a *Banach lattice* if $X$ is complete under ${\lVert\,\cdot\,\rVert}$. Moreover, if the lattice norm satisfies $$\Vert f+g \Vert=\Vert f \Vert + \Vert g \Vert$$ for all $f, g \in X_+$, then $X$ is an *AL-space*. It can be shown that, when $X$ is an AL-space, there exists a unique, bounded linear functional, $\phi$, that extends ${\lVert\,\cdot\,\rVert}$ from $X_+$ to $X$; see [@banasiak2006perturbations Theorems 2.64 and 2.65]. We now turn our attention to $C_0$-semigroups which are crucial to our investigation into the pure fragmentation system. The notions and results given here can be found in [@engel1999one]. First we note that if $(S(t))_{t \ge 0}$ is a $C_0$-semigroup on a Banach space $X$, then there exist $M \ge 1$ and $\omega \in \mathbb{R}$ such that $\Vert S(t) \Vert \le Me^{\omega t}$ for all $t \ge 0$, and the growth bound, $\omega_0$, of $(S(t))_{t \ge 0}$ is defined by $$\omega_0 \coloneqq \inf\bigl\{\omega \in \mathbb{R}: \text{there exists} \ M_{\omega} \ge 1 \ \text{such that} \ \Vert S(t) \Vert \leq M_{\omega}e^{\omega t} \ \text{for all} \ t \ge 0\bigr\}.$$ Analytic semigroups, see [@engel1999one Definition .4.5], are of particular importance in Section \[Analyticity of the Fragmentation Semigroup\]. Semigroups of this type have a number of useful properties that make them desirable to work with. For example, if $G$ is the generator of an analytic semigroup, $(S(t))_{t \ge 0}$, on a Banach space $X$, then $S(t)f \in \mathcal{D}(G^n)$ for all $t>0$, $n \in \mathbb{N}$ and $f \in X$, and $S(\cdot)$ is infinitely differentiable. When dealing with many physical problems, such as the fragmentation system, meaningful solutions must be non-negative, and this requirement has to be taken into account in any semigroup-based investigation. In connection with this, we say that a $C_0$-semigroup $(S(t))_{t \ge 0}$ on an ordered Banach space $X$, such as a Banach lattice, is *positive* if $S(t)f \ge 0$ for all $f \in X_+$; it is called *substochastic* (resp. *stochastic*) if, additionally, $\Vert S(t)f \Vert \le \Vert f \Vert$ (resp. $\Vert S(t)f \Vert=\Vert f \Vert$) for all $f \in X_+$. It follows that if $G$ generates a substochastic semigroup $(S(t))_{t \ge 0}$, then the associated ACP $$u'(t)= Gu(t), \;\; t>0; \qquad u(0)=\mathring{u},$$ has a unique, non-negative classical solution, given by $u(t) = S(t)\mathring{u}$, for any $\mathring{u} \in D(G)_+$. A result on substochastic semigroups and their generators that we shall exploit is due to Thieme and Voigt, [@thieme2006stochastic Theorem 2.7]. This result gives sufficient conditions under which the closure of the sum of two operators, such as $A^{(w)}+B^{(w)}$ in , generates a substochastic semigroup. The existence of an invariant subspace under the resulting semigroup is also established. As we demonstrate below in Proposition \[corollary of G=closure\], it is possible to adapt the Thieme–Voigt result to produce a modified version that is ideally suited for applying to the fragmentation system. We first provide some prerequisite results that are used in the proof of this proposition. \[uniqueness of extension generator\] Let $A$ be a closable operator in a Banach space $X$. If $G=\overline{A}$ is the generator of a $C_0$-semigroup on $X$, then no other extension of $A$ is the generator of a $C_0$-semigroup on $X$. Suppose that $G=\overline{A}$ and $H \supseteq A$ are generators of $C_0$-semigroups with growth bounds $\omega_1$ and $\omega_2$ respectively, and assume that $H \ne G$. Clearly, $H \supseteq G$ since $H$ is closed. Let $\lambda>\max\{\omega_1,\omega_2\}$. Then $\lambda\in\rho(G)\cap\rho(H)$ and hence $\lambda I-G: \mathcal{D}(G) \to X$ and $\lambda I-H: \mathcal{D}(H) \to X$ are both bijective. This is a contradiction since $\lambda I-H$ is a proper extension of $\lambda I-G$. The following lemma, which is a special case of [@banasiak2006perturbations Remark 6.6], will also be used. For the convenience of the reader we present a short proof. \[splitting D(G) into difference of two positives\] Let $G$ be the generator of a positive $C_0$-semigroup on a Banach lattice $X$. Then, for every $f \in \mathcal{D}(G)$, there exist $g$, $h \in \mathcal{D}(G)_+$ such that $f=g-h$. Let $f \in \mathcal{D}(G)$. Further, let $\omega_0$ be the growth bound of the semigroup generated by $G$, fix $\lambda>\omega_0$ and set $f_0 \coloneqq (\lambda I-G)f$. Since $X$ is a Banach lattice, we have $f_0 = f_+ - f_-$ with $f_+,f_- \in X_+$. Now let $g \coloneqq R(\lambda,G)f_+$ and $h \coloneqq R(\lambda,G)f_-$. The fact that $G$ generates a positive semigroup implies that $R(\lambda,G)$ is a positive operator, and therefore $g,h \in \mathcal{D}(G)_+$. Moreover, $$f = R(\lambda,G)f_0 = R(\lambda,G)(f_+-f_-) = R(\lambda,G)f_+-R(\lambda,G)f_- = g-h,$$ which proves the result. When the fragmentation coefficients satisfy Assumption \[A1.1\] and , then, as mentioned in the previous section, a formal calculation shows that the total mass is conserved. Consequently, if $u$ is a non-negative solution of the fragmentation system, and it is known that $u(t) \in X_{[1]}$ for $t \ge 0$, then we would expect $u$ to satisfy $$\Vert u(t)\Vert_{[1]} = \sum_{n=1}^\infty nu_n(t) = \sum_{n=1}^\infty n\mathring{u} = \Vert \mathring{u} \Vert_{[1]} \qquad \text{for all} \ t \ge 0.$$ Clearly this mass-conservation property will hold whenever the solution can be written in terms of a stochastic semigroup on $X_{[1]}$. To this end, the following proposition will prove useful. \[prop:stochastic\_semigroup\] Let $(S(t))_{t \ge 0}$ be a positive $C_0$-semigroup on an AL-space, $X$, with generator $G$, and let $\phi$ be the unique bounded linear extension of the norm ${\lVert\,\cdot\,\rVert}$ from $X_+$ to $X$. The semigroup $(S(t))_{t \ge 0}$ is stochastic if and only if $$\label{phi conserved} \phi\bigl(S(t)f\bigr) = \phi(f) \qquad \text{for all} \ f \in X.$$ If $\phi(Gf)=0$ for all $f \in \mathcal{D}(G)_+$, then holds and hence the semigroup $(S(t))_{t \ge 0}$ is stochastic. Let $G_0$ be an operator such that $G=\overline{G_0}$. If $\phi(G_0f)=0$ for all $f \in \mathcal{D}(G_0)_+$ and each $f \in \mathcal{D}(G_0)$ can be written as $f=g-h$, where $g, h \in \mathcal{D}(G_0)_+$, then holds and hence $(S(t))_{t \ge 0}$ is stochastic. () Assume that $(S(t))_{t \ge 0}$ is stochastic and let $f \in X$ and $t \ge 0$. Then $f=f_+-f_-$, where $f_+,f_- \in X_+$, and therefore $$\begin{aligned} \phi\bigl(S(t)f\bigr) &= \phi\bigl(S(t)f_+\bigr)-\phi\bigl(S(t)f_-\bigr) = \lVert S(t)f_+\rVert - \lVert S(t)f_-\rVert = \lVert f_+\rVert - \lVert f_-\rVert \\ &= \phi(f_+)-\phi(f_-) = \phi(f).\end{aligned}$$ Conversely, when holds, we have $\lVert S(t)f \rVert = \phi(S(t)f) = \phi(f) = \Vert f \Vert$ for $f \in X_+$ and $t \ge 0$. () Let $f \in \mathcal{D}(G)$. From Lemma \[splitting D(G) into difference of two positives\], there exist $g, h \in \mathcal{D}(G)_+$ such that $f=g-h$. Then $$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}t}\bigl(\phi(S(t)f)\bigr) &= \phi\biggl(\frac{{\mathrm{d}}}{{\mathrm{d}}t}\bigl(S(t)f\bigr)\biggr) = \phi\bigl(GS(t)f\bigr) \\ &= \phi\bigl(GS(t)g\bigr)-\phi\bigl(GS(t)h\bigr) = 0\end{aligned}$$ since $S(t)g$, $S(t)h \in \mathcal{D}(G)_+$. Thus $\phi(S(t)f)=\phi(f)$ for all $f \in \mathcal{D}(G)$, and hence also for all $f \in X$, since $\mathcal{D}(G)$ is dense in $X$. () Let $f \in \mathcal{D}(G_0)$. Then $f=g-h$ for some $g, h \in \mathcal{D}(G_0)_+$ by assumption, and $$\phi(G_0f) = \phi\bigl(G_0(g-h)\bigr) = \phi(G_0g)-\phi(G_0h) = 0.$$ Thus $\phi(G_0f)=0$ for all $f \in \mathcal{D}(G_0)$. Now let $f \in \mathcal{D}(G)$. Then there exist $f^{(n)} \in \mathcal{D}(G_0)$, $n \in \mathbb{N}$, such that $f^{(n)} \to f$ and $G_0f^{(n)} \to Gf$ as $n \to \infty$. Therefore $$\phi(Gf) = \phi\Bigl(\lim_{n \to \infty} G_0f^{(n)}\Bigr) = \lim_{n \to \infty} \phi(G_0f^{(n)})=0,$$ and the result follows from part (). We now use [@thieme2006stochastic Theorem 2.7] to obtain the following proposition, which will later be applied to the fragmentation problem. \[corollary of G=closure\] Let $(X,{\lVert\,\cdot\,\rVert})$ and $(Z,{\lVert\,\cdot\,\rVert_{Z}})$ be AL-spaces, such that $Z$ is dense in $X$, $(Z, {\lVert\,\cdot\,\rVert_{Z}})$ is continuously embedded in $(X, {\lVert\,\cdot\,\rVert})$. Also, let $\phi$ and $\phi_Z$ be the linear extensions of ${\lVert\,\cdot\,\rVert}$ from $X_+$ to $X$ and of ${\lVert\,\cdot\,\rVert_{Z}}$ from $Z_+$ to $Z$ respectively. Let $A: \mathcal{D}(A) \to X$, $B: \mathcal{D}(B) \to X$ be operators in $X$ such that $\mathcal{D}(A) \subseteq \mathcal{D}(B)$. Assume that the following conditions are satisfied. $-A$ is positive; $A$ generates a positive $C_0$-semigroup, $(T(t))_{t \geq 0}$, on $X$; the semigroup $(T(t))_{t \ge 0}$ leaves $Z$ invariant and its restriction to $Z$ is a (necessarily positive) $C_0$-semigroup on $(Z, {\lVert\,\cdot\,\rVert_{Z}})$, with generator $\widetilde{A}$ given by $$\widetilde{A}f = Af \qquad \text{for all} \ f \in \mathcal{D}(\widetilde{A}) = \bigl\{f \in \mathcal{D}(A) \cap Z: Af \in Z\bigr\};$$ $B|_{\mathcal{D}(A)}$ is a positive linear operator; $\phi((A+B)f) \le 0$ for all $f \in \mathcal{D}(A)_+$; $(A+B)f \in Z$ and $\phi_{Z}((A+B)f) \le 0$ for all $f \in \mathcal{D}(\widetilde{A})_+$; $\Vert Af \Vert \le \Vert f \Vert_Z$ for all $f \in \mathcal{D}(\widetilde{A})_+$. Then there exists a unique substochastic $C_0$-semigroup on $X$ which is generated by an extension, $G$, of $A+B$. The operator $G$ is the closure of $A+B$. Moreover, the semigroup $(S(t))_{t \ge 0}$ generated by $G$ leaves $Z$ invariant. If $\phi((A+B)f)=0$ for all $f \in \mathcal{D}(A)_+$, then $(S(t))_{t \ge 0}$ is stochastic. We first show that the conditions of [@thieme2006stochastic Theorem 2.7] hold. From (ii) and the fact that $(Z,{\lVert\,\cdot\,\rVert_{Z}})$ is an AL-space, it is clear that [@thieme2006stochastic Assumption 2.5] is satisfied. Also, from (f) and (g) we obtain that $$\phi_Z\bigl((A+B)f\bigr) \le 0 \le \Vert f \Vert_Z - \Vert Af \Vert$$ for all $f \in \mathcal{D}(\widetilde{A})_+$. Moreover, (f) and the definition of $\widetilde{A}$ imply that $Bf \in Z$ for all $f \in \mathcal{D}(\widetilde{A})_+$. Consequently, if we now take $f \in D(\widetilde{A})$ and use Lemma \[splitting D(G) into difference of two positives\] to express $Bf$ as $Bg-Bh$, where $g, h \in \mathcal{D}(\widetilde{A})_+$, then it follows easily that $B(\mathcal{D}(\widetilde{A})) \subseteq Z$. Thus, all the assumptions of [@thieme2006stochastic Theorem 2.7] are satisfied and therefore $G=\overline{A+B}$ is the generator of a substochastic semigroup $(S(t))_{t \ge 0}$, which leaves $Z$ invariant. That no other extension of $A+B$ can generate a $C_0$-semigroup on $X$ is an immediate consequence of Lemma \[uniqueness of extension generator\]. Finally, since $A$ generates a substochastic $C_0$-semigroup, it follows from Lemma \[splitting D(G) into difference of two positives\] that we can write any $f \in \mathcal{D}(A)=\mathcal{D}(A+B)$ as $f=g-h$, where $g, h \in \mathcal{D}(A)_+$. An application of Proposition \[prop:stochastic\_semigroup\]() then yields the stochasticity result. The fragmentation semigroup {#The Generator of the Fragmentation Semigroup} =========================== In this section, we begin our analysis of the fragmentation system by investigating the associated ACP , which we recall takes the form $$u'(t) = A^{(w)}u(t)+B^{(w)}u(t), \quad t>0; \qquad u(0)=\mathring{u},$$ where $A^{(w)}$ and $B^{(w)}$ are defined in $\ell_w^1$ by and respectively. A direct application of Proposition \[corollary of G=closure\] will establish that, under appropriate conditions on the weight $w$, $G^{(w)} =\overline{A^{(w)}+B^{(w)}}$ generates a substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$. As no other extension of $A^{(w)}+B^{(w)}$ generates a $C_0$-semigroup on $\ell_w^1$, we shall refer to $(S^{(w)}(t))_{t \ge 0}$ as *the* fragmentation semigroup on $\ell_w^1$. In the process of proving the existence of the fragmentation semigroup, we shall also obtain explicit subspaces of $\ell_w^1$ which are invariant under $(S^{(w)}(t))_{t \ge 0}$. First we note that $\ell_w^1$ is an AL-space, with positive cone $$(\ell_w^1)_+ = \bigl\{f = (f_n)_{n=1}^\infty \in \ell_w^1: f_n \ge 0 \ \text{for all} \ n \in \mathbb{N}\bigr\},$$ whenever $w=(w_n)_{n=1}^{\infty}$ is a positive sequence. Moreover, in this case the unique bounded linear functional, $\phi_w$, that extends ${\lVert\,\cdot\,\rVert_{w}}$ from $(\ell_w^1)_+$ to $\ell_w^1$ is given by $$\label{phi_w} \phi_w(f) = \sum\limits_{n=1}^{\infty} w_n f_n \qquad \text{for all} \ f \in \ell_w^1.$$ We recall also that if we take $w_n=n$ for all $n \in \mathbb{N}$, then $\ell_w^1 = X_{[1]}$ and ${\lVert\,\cdot\,\rVert_{w}} = {\lVert\,\cdot\,\rVert_{[1]}}$. For this specific case, we shall represent $\phi_w,\,A^{(w)}$ and $B^{(w)}$ by $M_1$, $A_1$ and $B_1$ respectively, and consequently the ACP on $X_{[1]}$ will be written as $$\label{ACP in X} u'(t) = A_1u(t)+B_1u(t), \quad t>0; \qquad u(0)=\mathring{u}.$$ From physical considerations, it is clear that the initial condition, $\mathring{u}$, in the ACP must necessarily be non-negative, and similarly, if $u:[0,\infty) \to \ell_w^1$ is the corresponding solution, then we require $u(t)$ to be non-negative for all $t \ge 0$. Moreover, if we assume to hold, or, equivalently, with $\lambda_j \in [0,1]$, we expect from that mass is either lost or conserved during fragmentation. From and the definition of the norm on $X_{[1]}$, this is equivalent to $$\label{mass loss/conservation in solution} \Vert u(t) \Vert_{[1]} \le \Vert \mathring{u} \Vert_{[1]} \qquad \text{for all} \ t \ge 0,$$ with equality being required in the mass-conserving case, provided that $w$ is such that $\ell_w^1 \subseteq X_{[1]}$. For convenience, we include the following elementary result which states that the operator $A^{(w)}$ generates a substochastic semigroup on $\ell_w^1$ for any non-negative weight $w$. \[A is a generator\] Let $\ell_w^1$ and ${\lVert\,\cdot\,\rVert_{w}}$ be defined by and , respectively, and let hold. Then the operator $A^{(w)}$, defined by , is the generator of a substochastic $C_0$-semigroup, $(T^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$, which is given, for $t \ge 0$, by the infinite diagonal matrix $\operatorname{diag}(v_1(t),v_2(t),\ldots)$, where $v_n(t) = e^{-a_nt}$ for all $n \in \mathbb{N}$. For the remainder of this section, the weight, $w$, will be required to satisfy the following assumption. \[assumption on weight for generation\] ------------------------------------------------------------------------ $w_n \ge n$ for all $n \in \mathbb{N}$. There exists $\kappa \in (0,1]$ such that $$\label{condition on weights} \sum\limits_{n=1}^{j-1} w_nb_{n,j} \leq \kappa w_j \qquad \text{for all} \ j=2,3,\ldots.$$ \[remark\_w\_n/n\_increasing\] Let $w$ be such that $\left(w_n/n\right)_{n=1}^{\infty}$ is increasing and let hold. Then $$\sum\limits_{n=1}^{j-1} w_n b_{n,j} = \sum\limits_{n=1}^{j-1} \frac{w_n}{n}nb_{n,j} \le \frac{w_j}{j} \sum\limits_{n=1}^{j-1} nb_{n,j} \le \frac{w_j}{j}j = w_j.$$ Hence is satisfied with $\kappa=1$. In particular, if holds, then Assumption \[assumption on weight for generation\] is automatically satisfied by any weight of the form $w_n=n^p$, $p \ge 1$. It is an immediate consequence of Assumption \[assumption on weight for generation\] that, for any $f \in \mathcal{D}(A^{(w)})_+$, we have $$\label{phiwBwf_le_kappa_phiwAw} \begin{aligned} \phi_w\bigl(B^{(w)}f\bigr) &= \sum\limits_{n=1}^{\infty} w_n\sum\limits_{j=n+1}^{\infty} a_jb_{n,j}f_j = \sum\limits_{j=2}^{\infty} \Biggl(\sum\limits_{n=1}^{j-1} w_n b_{n,j}\Biggr)a_jf_j \\ &\le \kappa\sum\limits_{j=1}^{\infty} w_ja_jf_j = -\kappa\phi_w\bigl(A^{(w)}f\bigr). \end{aligned}$$ Consequently, for all $f \in \mathcal{D}(A^{(w)})$, $$\label{B bounded by A} \begin{aligned} \Vert B^{(w)}f \Vert_w &= \sum_{n=1}^\infty w_n\bigg|\sum\limits_{j=n+1}^{\infty} a_jb_{n,j}f_j\bigg| \le \phi_w\bigl(B^{(w)}|f|\bigr) \\ &\le -\kappa\phi_w\bigl(A^{(w)}|f|\bigr) = \kappa\Vert A^{(w)}f \Vert_w, \end{aligned}$$ from which it follows that $$\label{domAw_Bw_incl} \mathcal{D}(A^{(w)}) \subseteq \mathcal{D}(B^{(w)}) \quad\text{and}\quad \mathcal{D}\bigl(A^{(w)}+B^{(w)}\bigr) = \mathcal{D}(A^{(w)}) \cap \mathcal{D}(B^{(w)}) =\mathcal{D}(A^{(w)}).$$ We now apply Proposition \[corollary of G=closure\] to the operators $A^{(w)}$ and $B^{(w)}$. This involves the construction of a suitable subspace of $\ell_w^1$, and to this end we require a sequence $(c_n)_{n=1}^\infty$ that satisfies $$\label{conditions for c_n} c_n \le c_{n+1} \qquad \text{and} \qquad a_n \le c_n \qquad \text{for all} \ n \in \mathbb{N}.$$ Note that such a sequence can always be found. For example, we can take $$\label{maximal choice of cn} c_n = \max\{a_1,\ldots,a_n\} \qquad \text{for} \ n=1,2,\ldots.$$ Let $C^{(w)}$ be the corresponding multiplication operator, defined by $$\label{C^w definition} [C^{(w)}f]_n = -c_n f_n, \;\; n \in \mathbb{N}, \qquad \mathcal{D}(C^{(w)}) = \biggl\{f \in \ell_w^1: \sum\limits_{n=1}^{\infty} w_nc_n|f_n|<\infty\biggr\},$$ and equip $\mathcal{D}(C^{(w)})$ with the graph norm $$\label{graph_norm_C^w} \Vert f \Vert_{C^{(w)}} = \Vert f \Vert_w + \Vert C^{(w)}f \Vert_w = \sum_{n=1}^\infty (w_n+w_nc_n)|f_n|, \qquad f \in \mathcal{D}(C^{(w)}).$$ Clearly, $(\mathcal{D}(C^{(w)}), {\lVert\,\cdot\,\rVert_{C^{(w)}}}) = (\ell_{\widetilde w}^1,{\lVert\,\cdot\,\rVert_{\widetilde w}})$ with weight $\widetilde{w} = (\widetilde{w}_n)_{n=1}^\infty$ where $$\label{def_tilde_w} \widetilde w_n = w_n + w_n c_n, \qquad n \in \mathbb{N},$$ and hence $(\ell_{\widetilde w}^1, {\lVert\,\cdot\,\rVert_{\widetilde{w}}})$ is an AL-space, and the unique linear extension of ${\lVert\,\cdot\,\rVert_{\widetilde{w}}}$ from $(\ell_{\widetilde w}^1)_+$ to $\ell_{\widetilde w}^1$ is given by $\phi_{\widetilde w}(f)=\sum\limits_{n=1}^\infty \widetilde{w}_n f_n$ for $f \in \ell_{\widetilde w}^1$. We note that the choice for $(c_n)_{n=1}^{\infty}$ is ‘maximal’ in the sense that if $(\hat{c}_n)_{n=1}^{\infty}$ is any other monotone increasing sequence that dominates $(a_n)_{n=1}^{\infty}$, and $\widehat{C}$ is defined analogously to , then $\mathcal{D}(\widehat{C}^{(w)}) \subseteq \mathcal{D}(C^{(w)})$. \[G=closure for frag\] Let Assumptions \[A1.1\] and \[assumption on weight for generation\] hold. Then $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$ is the generator of a substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$. Moreover, $(S^{(w)}(t))_{t \ge 0}$ leaves $\mathcal{D}(C^{(w)})=\ell_{\widetilde w}^1$ invariant, where $\mathcal{D}(C^{(w)})$ and $\widetilde w$ are defined in and , respectively, and $(c_n)_{n=1}^{\infty}$ satisfies . If, in addition, holds and $w_n=n$ for all $n \in \mathbb{N}$, then the semigroup, $(S_1(t))_{t \ge 0}$, generated by $G_1=\overline{A_1 +B_1}$ is stochastic on $X_{[1]}$. We show that the conditions (), () and (a)–(g) of Proposition \[corollary of G=closure\] are all satisfied when $A=A^{(w)}$, $B=B^{(w)}$ and the AL-spaces $(X, {\lVert\,\cdot\,\rVert})$ and $(Z,{\lVert\,\cdot\,\rVert_{Z}})$ are, respectively, $\ell_w^1$ and $(\mathcal{D}(C^{(w)}),{\lVert\,\cdot\,\rVert_{C^{(w)}}}) =(\ell_{\widetilde w}^1,{\lVert\,\cdot\,\rVert_{\widetilde w}})$. Clearly, $\ell_{\widetilde w}^1$ is dense in $\ell_w^1$ and continuously embedded since $w_n \le \widetilde w_n$, $n \in \mathbb{N}$. It follows that (i) and (ii) both hold. Condition (a) is obviously satisfied by $A^{(w)}$, and, for (b), we apply Lemma \[A is a generator\] to establish that $A^{(w)}$ generates a substochastic $C_0$-semigroup, $(T^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$. It is easy to see that the semigroup $(T^{(w)}(t))_{t \ge 0}$ leaves $\ell_{\widetilde w}^1$ invariant and the generator of the restriction to $\ell_{\widetilde w}^1$ is $A^{(\widetilde w)}$, the part of $A^{(w)}$ in $\ell_{\widetilde w}^1$; this shows (c). It is also clear that $B^{(w)}$ is positive. From we obtain that, for $f \in \mathcal{D}(A^{(w)})_+$, $$\label{phiwAwBwle0} \begin{aligned} \phi_w\bigl((A^{(w)}+B^{(w)})f\bigr) &= \phi_w(A^{(w)}f) + \phi_w(B^{(w)}f) \\ &\le \phi_w(A^{(w)}f) - \kappa\phi_w(A^{(w)}f) \le 0. \end{aligned}$$ Hence (d) and (e) hold. Since $w_n \ge n$, by Assumption \[assumption on weight for generation\](i), we have $\widetilde w_n = w_n+w_n c_n \ge n$, $n\in\mathbb{N}$. Moreover, the monotonicity of $(c_n)_{n=1}^\infty$ and Assumption \[assumption on weight for generation\](ii) imply that $$\sum_{n=1}^{j-1} \widetilde{w}_n b_{n,j} = \sum\limits_{n=1}^{j-1} (1+c_n)w_n b_{n,j} \le (1+c_j)\sum\limits_{n=1}^{j-1} w_n b_{n,j} \le \kappa (1+c_j)w_j = \kappa \widetilde{w}_j$$ for all $j \in \mathbb{N}$. This means that Assumption \[assumption on weight for generation\] also holds for the weight $\widetilde w$. Therefore we obtain from and that $\mathcal{D}(A^{(\widetilde w)}) \subseteq \mathcal{D}(B^{(\widetilde w)})$ and $\phi_{\widetilde w}((A^{(\widetilde w)}+B^{(\widetilde w)})f) \le 0$ for $f \in \mathcal{D}(A^{(\widetilde w)})_+$, and so (f) is also satisfied. That (g) holds follows from $$\Vert A^{(w)}f \Vert_w = \sum_{n=1}^\infty w_n a_n|f_n| \le \sum_{n=1}^\infty w_n c_n|f_n| \le \sum_{n=1}^\infty \widetilde{w}_n|f_n| = \|f\|_{\widetilde w}$$ for $f \in \mathcal{D}(\tilde{A}^{(w)})_+$. Thus, the conditions of Proposition \[corollary of G=closure\] are all satisfied and therefore $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$ is the generator of a substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$, which also leaves $\mathcal{D}(C^{(w)})=\ell_{\widetilde w}^1$ invariant. Finally, assume that is satisfied and $w_n=n$ for all $n \in \mathbb{N}$. Then equality holds in with $\kappa=1$ and hence also in , and so, from Proposition \[corollary of G=closure\], the semigroup generated in this case is stochastic. \[stochastic in X\] Consider the case where $w_n=n$ for all $n \in \mathbb{N}$, so that $\ell_w^1 = X_{[1]}$, and let Assumption \[A1.1\] and hold. Then, by Remark \[remark\_w\_n/n\_increasing\], is also satisfied, and therefore, from Theorem \[G=closure for frag\], the operator $G_1=\overline{A_1+B_1}$ is the generator of a substochastic $C_0$-semigroup, $(S_1(t))_{t \geq 0}$, on $X_{[1]}$. It follows that the ACP $$\label{GACP in X} u'(t)=G_1u(t), \quad t>0; \qquad u(0)=\mathring{u},$$ with $\mathring{u} \in \mathcal{D}(G_1)$, has a unique classical solution, given by $u(t)=S_1(t)\mathring{u}$ for all $t \ge 0$. Moreover, if $\mathring{u}\ge0$, then this solution is non-negative. Now suppose that $\mathring{u} \in \mathcal{D}(G_1)_+$ and, in addition, assume that holds. Then the semigroup $(S_1(t))_{t \ge 0}$ is stochastic on $X_{[1]}$ and so, from , $$M_1\bigl(u(t)\bigr) = \Vert u(t) \Vert_{[1]} = \Vert S_1(t)\mathring{u} \Vert_{[1]} = \Vert \mathring{u} \Vert_{[1]} = M_1(\mathring{u}) \qquad \text{for all} \ t \ge 0,$$ showing that $u(t)$ is a mass-conserving solution. With the help of Remark \[stochastic in X\] we obtain the following corollary. \[solution of GACP\] Let Assumptions \[A1.1\] and \[assumption on weight for generation\] hold and let $\mathring{u} \in \mathcal{D}(G^{(w)})$, where $G^{(w)} = \overline{A^{(w)}+B^{(w)}}$ as in Theorem \[G=closure for frag\]. Then the ACP $$\label{GACP} u'(t) = G^{(w)}u(t), \quad t>0; \qquad u(0) = \mathring{u}$$ has a unique classical solution, given by $u(t)=S^{(w)}(t)\mathring{u}$. This solution is non-negative if $\mathring{u} \in \mathcal{D}(G^{(w)})_+$. Moreover, if holds and $\mathring{u} \in \mathcal{D}(G^{(w)})_+$, then this solution is mass conserving. It follows immediately from Theorem \[G=closure for frag\] that $u(t)=S^{(w)}(t)\mathring{u}$ is the unique classical solution of for all $\mathring{u} \in \mathcal{D}(G^{(w)})$. Moreover, since $(S^{(w)}(t))_{t \ge 0}$ is substochastic, this solution is non-negative if $\mathring{u} \in \mathcal{D}(G^{(w)})_+$. Now assume that holds and $\mathring{u} \in \mathcal{D}(G^{(w)})_+$. Then $(S_1(t))_{t \ge 0}$ is a stochastic $C_0$-semigroup on $X_{[1]}$. Additionally, since $w_n \ge n$ for all $n \in \mathbb{N}$, $\ell_w^1$ is continuously embedded in $X_{[1]}$ and so, as $u(t)$ is differentiable in $\ell_w^1$, $u(t)$ is also differentiable in $X_{[1]}$ and the derivatives must coincide. Moreover, since $G^{(w)}$ is the part of $G_1$ in $\ell_w^1$, we have $u(t) \in \mathcal{D}(G_1)$. Therefore, $u(t)=S^{(w)}(t)\mathring{u}$ is also a solution of , and, by uniqueness of solutions, it follows that $S^{(w)}(t)\mathring{u} = S_1(t)\mathring{u}$ for $t \ge 0$. Remark \[stochastic in X\] then establishes that $u(t)=S^{(w)}(t)\mathring{u}$ is a mass-conserving solution. Note that even if $\mathring{u} \in \mathcal{D}(A^{(w)})$, the solution, $u(t)$, of need not belong to $\mathcal{D}(A^{(w)})$ for any $t > 0$. Hence the existence of a solution of is not guaranteed in general; one only has uniqueness of solutions. However, the next theorem shows that under the stronger assumption $\mathring{u} \in \mathcal{D}(C^{(w)})$ on the initial condition, the ACP is well posed. \[solution for IC in D(C)\] Let Assumptions \[A1.1\] and \[assumption on weight for generation\] hold. For $\mathring{u} \in \mathcal{D}(C^{(w)})$, the ACP has a unique classical solution given by $u(t)=S^{(w)}(t)\mathring{u}$, $t \ge 0$. If $\mathring{u} \in \mathcal{D}(C^{(w)})_+$, then this solution is non-negative. Moreover, if holds and $\mathring{u} \in \mathcal{D}(C^{(w)})_+$, then the solution is mass conserving. We know that $G^{(w)}$ and $A^{(w)}+B^{(w)}$ coincide on $\mathcal{D}(A^{(w)})$ and also that $u(t)=S^{(w)}(t)\mathring{u}$ is the unique solution of for $\mathring{u} \in \mathcal{D}(C^{(w)}) \subseteq \mathcal{D}(G^{(w)})$. Since $(S^{(w)}(t))_{t \ge 0}$ leaves $\mathcal{D}(C^{(w)})$ invariant, it follows that $S^{(w)}(t)\mathring{u} \in \mathcal{D}(C^{(w)}) \subseteq \mathcal{D}(A^{(w)})$. The result then follows from Corollary \[solution of GACP\]. The next proposition shows that if the sequence $(a_n)_{n=1}^{\infty}$ has a certain additional property, then a unique solution of exists for $\mathring{u} \in \mathcal{D}(A^{(w)})$. \[when does D(C)=D(A)\] Let $(a_n)_{n=1}^{\infty}$ be an unbounded sequence such that holds. Further, define the sequence $(c_n)_{n=1}^{\infty}$ by and let $w=(w_n)_{n=1}^\infty$ be such that $w_n>0$ for all $n \in \mathbb{N}$. Then $\mathcal{D}(C^{(w)})=\mathcal{D}(A^{(w)})$ if and only if $$\label{liminf condition for a and c} \liminf_{n \to \infty} \frac{a_n}{c_n}>0.$$ Note first that the unboundedness of $(a_n)_{n=1}^\infty$ implies that $c_n \to \infty$ as $n \to \infty$. Since $c_n \ge a_n$ for all $n \in \mathbb{N}$, we have $\mathcal{D}(C^{(w)}) \subseteq \mathcal{D}(A^{(w)})$. If holds, then there exist $\gamma > 0$, $N \in \mathbb{N}$ such that $a_n \ge \gamma c_n$ for all $n \geq N$. Let $f \in \mathcal{D}(A^{(w)})$. Then $$\begin{aligned} \Vert C^{(w)}f \Vert_w = \sum\limits_{n=1}^{\infty} w_nc_n|f_n| &\le \sum\limits_{n=1}^{N-1} w_n c_n|f_n| + \frac{1}{\gamma} \sum\limits_{n=N}^{\infty} w_n a_n|f_n| \\ &\le \sum\limits_{n=1}^{N-1} w_n c_n|f_n| + \frac{1}{\gamma} \Vert A^{(w)}f \Vert_w <\infty,\end{aligned}$$ and so $\mathcal{D}(A^{(w)})=\mathcal{D}(C^{(w)})$. Now suppose that $\liminf_{n \to \infty}(a_n/c_n) = 0$. Then there exists a subsequence, $\left(a_{n_k}/c_{n_k}\right)_{k=1}^{\infty}$, such that $$c_{n_k} \ne 0, \quad \frac{a_{n_k}}{c_{n_k}} \le \frac{1}{k} \quad\text{and}\quad \frac{1}{c_{n_k}} \le \frac{1}{k} \qquad \text{for all} \ k \in \mathbb{N}.$$ Let $f$ be such that $$f_j = \begin{cases} 1/(c_{n_k}w_{n_k}k) \qquad &\text{when} \ j=n_k, \\[0.5ex] 0 \qquad &\text{otherwise}. \end{cases}$$ Then $$\begin{aligned} & \sum\limits_{n=1}^{\infty}a_nw_n|f_n| = \sum\limits_{k=1}^{\infty} a_{n_k}w_{n_k}\frac{1}{c_{n_k}w_{n_k}k} \le \sum\limits_{k=1}^{\infty} \frac{1}{k^2} < \infty, \\[1ex] & \sum\limits_{n=1}^\infty w_n|f_n| = \sum_{k=1}^\infty w_{n_k}\frac{1}{c_{n_k}w_{n_k}k} \le \sum_{k=1}^\infty \frac{1}{k^2} < \infty \qquad\text{and}\qquad \sum\limits_{n=1}^{\infty}c_nw_n|f_n| = \sum\limits_{k=1}^{\infty} \frac{1}{k} = \infty.\end{aligned}$$ It follows that $f \in \mathcal{D}(A^{(w)})\backslash \mathcal{D}(C^{(w)})$, showing that $\mathcal{D}(C^{(w)})$ is a proper subset of $\mathcal{D}(A^{(w)})$. If $(a_n)_{n=1}^{\infty}$ is unbounded and eventually monotone increasing, then $(c_n)_{n=1}^{\infty}$, given by , satisfies . Note that, in $X_{[1]}$, the invariance of $\mathcal{D}(A^{(w)})$ under the fragmentation semigroup has already been established in [@mcbride2010strongly Theorem 3.2] for the case when $(a_n)_{n=1}^{\infty}$ is monotone increasing. We end this section by obtaining an infinite matrix representation of the fragmentation semigroup $(S^{(w)}(t))_{t \geq 0}$ on $\ell_w^1$, which is used in Section \[Asymptotic Behaviour of Solutions\]. Let Assumptions \[A1.1\] and \[assumption on weight for generation\] be satisfied so that $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$ is the generator of a substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$. For $n \in \mathbb{N}$, let $e_n \in \ell_w^1$ be given by $$\label{basis} \left(e_n\right)_k = \begin{cases} 1 \qquad &\text{if} \ n=k, \\[0.5ex] 0 \qquad &\text{otherwise}, \end{cases}$$ and let $(s_{m,n}(t))_{m, n \in \mathbb{N}}$, be the infinite matrix defined by $$s_{m,n}(t)=(S^{(w)}(t)e_n)_m \qquad \text{for all} \ m, n \in \mathbb{N}.$$ Note that, since $(S^{(w)}(t))_{t \ge 0}$ is positive, $s_{m,n}(t) \ge 0$ for all $m, n \in \mathbb{N}$. Now, each $f \in \ell_w^1$ can be expressed as $f=\sum\limits_{n=1}^{\infty} f_n e_n$, where the infinite series is convergent in $\ell_w^1$. Hence $$\bigl(S^{(w)}(t)f\bigr)_m = \Biggl(\sum\limits_{n=1}^{\infty} f_n S^{(w)}(t)e_n\Biggr)_{\!m} = \sum\limits_{n=1}^{\infty} f_n s_{m,n}(t) \qquad \text{for all} \ m \in \mathbb{N},$$ and therefore $(S^{(w)}(t))_{t \ge 0}$ can be represented by the matrix $(s_{m,n}(t))_{m, n \in \mathbb{N}}$. To determine $s_{m,n}(t)$ more explicitly, fix $n \in \mathbb{N}$ and let $(u_1(t),\ldots,u_n(t))$ be the unique solution of the $n$-dimensional system $$\begin{aligned} & u_m'(t) = -a_m u_m(t) + \sum_{j=m+1}^n a_j b_{m,j}u_j(t), \quad t>0; \qquad m = 1, 2, \ldots, n; \label{finite_system} \\[1ex] & u_n(0) = 1; \qquad u_m(0) = 0 \quad\text{for} \ m<n.\end{aligned}$$ It is straightforward to check that $u(t)=(u_1(t),\ldots,u_n(t),0,0,\ldots)$ solves with $\mathring{u}=e_n$. Since $u(t) \in \mathcal{D}(A^{(w)}) \subseteq \mathcal{D}(G^{(w)})$, the function $u$ coincides with the unique solution of , and hence $u(t)=S^{(w)}(t)e_n$, which yields $$\label{s_mn_u_m} s_{m,n}(t) = \begin{cases} u_m(t), \quad & m=1,2,\ldots,n, \\[0.5ex] 0, & m>n. \end{cases}$$ For $m=n$, the differential equation in reduces to $u_n'(t)=-a_n u_n(t)$, which implies that $s_{n,n}(t)=u_n(t)=e^{-a_nt}$. Since $n$ was arbitrary, it follows that, for all $t \ge 0$, $$\label{S/Sw matrix} S^{(w)}(t) = \left[\begin{array}{c|ccc} e^{-a_1t} & s_{1,2}(t) & s_{1,3}(t) & \cdots \\[1ex] \hline 0 & e^{-a_2t} & s_{2,3}(t) & \cdots \rule{0ex}{3ex} \\[1ex] 0 & 0 & e^{-a_3t} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{array}\right] = \begin{bmatrix} e^{-a_1t} & S^{(w)}_{(12)}(t) \\[2ex] \mathbf{0} & S^{(w)}_{(22)}(t) \end{bmatrix},$$ where $\mathbf{0}$ is an infinite column vector consisting entirely of zeros, $S^{(w)}_{(12)}(t)$ is a non-negative infinite row vector and $S^{(w)}_{(22)}(t)$ is an infinite-dimensional, non-negative, upper triangular matrix. We note that, in the particular case when $\ell_w^1 = X_{[1]}$ and mass is conserved, Banasiak obtains the infinite matrix representation for the semigroup $(S_1(t))_{t \ge 0}$ in [@banasiak2011irregular Equation (10) and Lemma 1]. In [@banasiak2011irregular], an explicit expression is also found for $s_{m,n}(t)$, $m<n$, but we omit this here since it is not required for the results that follow. As observed in [@banasiak2011irregular pp. 363], it follows from that, for all $N \in \mathbb{N}$, we have $S(t)f \in \operatorname{span}\{e_1, e_2, \ldots, e_N\}$ for all $f \in \operatorname{span}\{e_1, e_2, \ldots, e_N\}$. Also note that the functions $s_{m,n}$ are independent of the weight $w$, which implies that, whenever $\widehat w$ is another weight satisfying Assumption \[assumption on weight for generation\], $S^{(w)}(t)$ and $S^{(\widehat w)}(t)$ coincide on $\ell_w^1 \cap \ell_{\widehat w}^1$. The pointwise fragmentation problem and the fragmentation generator {#Pointwise Problem} =================================================================== We established in Theorem \[solution for IC in D(C)\] that if Assumptions \[A1.1\] and \[assumption on weight for generation\] are satisfied, then $u(t)=S^{(w)}(t)\mathring{u}$ is the unique, non-negative classical solution of the fragmentation ACP for all $\mathring{u} \in \mathcal{D}(C^{(w)})_+$. Moreover, when holds, then this solution is mass conserving. Clearly, $u(t)=S^{(w)}(t)\mathring{u}$ will also satisfy the fragmentation system in a pointwise manner when $\mathring{u} \in \mathcal{D}(C^{(w)})_+$. However, at this stage we do not know in what sense, if any, the semigroup $(S^{(w)}(t))_{t \geq 0}$ provides a non-negative solution for a general $\mathring{u} \in (\ell_w^1)_+$. In this section we show that a non-negative solution of the pointwise system can be determined for any given initial condition in $(\ell_w^1)_+$ by using the semigroup $(S^{(w)}(t))_{t \ge 0}$. As before, we require Assumptions \[A1.1\] and \[assumption on weight for generation\] to hold, and we define a sequence $(c_n)_{n=1}^{\infty}$ by , with the associated multiplication operator $C^{(w)}$ given by . Then $a_n \le c_n$ for all $n \in \mathbb{N}$ and it follows that $\mathcal{D}(C^{(w)}) \subseteq \mathcal{D}(A^{(w)})$. From Proposition \[G=closure for frag\], $\mathcal{D}(C^{(w)})$ is invariant under the substochastic semigroup $(S^{(w)}(t))_{t \geq 0}$ generated by $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$. Consequently, $u(t)=S^{(w)}(t)\mathring{u}$ is the unique, non-negative classical solution of for each $\mathring{u} \in \mathcal{D}(C^{(w)})_+$, and therefore $$\label{integrated pointwise system} u_n(t)-\mathring{u}_n=-a_n\int_0^t u_n(s)\,{\mathrm{d}}s + \int_0^t \sum_{j=n+1}^{\infty} a_j b_{n,j}u_j(s)\,{\mathrm{d}}s,$$ for $n=1,2,\ldots$. We use this integrated version of the pointwise fragmentation system to prove the following result. \[pointwise system satisfied for any IC\] Let Assumptions \[A1.1\] and \[assumption on weight for generation\] hold, and let $\mathring{u} \in \ell_w^1$. Then $u(t) = S^{(w)}(t)\mathring{u}$ satisfies the system for almost all $t \ge 0$. Moreover, if $\mathring{u} \ge 0$, then $u(t) \ge 0$ for $t \ge 0$. Let $\mathring{u} \in (\ell_w^1)_+$ and, for $N \in \mathbb{N}$, define the operator $P_N : \ell_w^1 \to \ell_w^1$ by $$P_Nf \coloneqq \sum_{n=1}^N f_n e_n = (f_1,f_2,\ldots,f_N,0,\ldots), \qquad f \in \ell_w^1.$$ Then $P_N\mathring{u} \in \mathcal{D}(C^{(w)})_+$ for all $N \in \mathbb{N}$, and so, on setting $u^{(N)}(t)=S^{(w)}(t)P_N\mathring{u}$, we have $$\label{truncated integral form} u^{(N)}_n(t) = P_N\mathring{u}_n - a_n\int_0^t u^{(N)}_n(s)\,{\mathrm{d}}s + \int_0^t \sum_{j=n+1}^{\infty} a_j b_{n,j}u^{(N)}_j(s)\,{\mathrm{d}}s,$$ for $n = 1,2,\ldots,N$. Clearly, $P_N\mathring{u} \to \mathring{u}$ in $\ell_w^1$ as $N \to \infty$, and so, by the continuity of $S^{(w)}(t)$, it follows that $u^{(N)}_n(t) \to u_n(t)$ as $N \to \infty$ for all $n \in \mathbb{N}$ and $t \ge 0$. Moreover, if $N_2 \ge N_1$ then $u^{(N_2)}(t)-u^{(N_1)}(t) \ge 0$ for all $t \ge 0$, since $(S^{(w)}(t))_{t \geq 0}$ is linear and positive. Similarly, $u(t)-u^{(N)}(t) \ge 0$ for all $N \in \mathbb{N}$ and $t \ge 0$. Hence $(u^{(N)}(t))_{N=1}^{\infty}$ is monotone increasing and bounded above by $u(t)$, and therefore, for each fixed $n \in \mathbb{N}$, $(u^{(N)}_n(t))_{N=1}^{\infty}$ is monotone increasing and bounded above by $u_n(t)$. On allowing $N \to \infty$ in , and using the monotone convergence theorem, we obtain $$u_n(t) = \mathring{u}_n - a_n\int_0^t u_n(s)\,{\mathrm{d}}s + \lim_{N \to \infty}\int_0^t \sum_{j=n+1}^{\infty} a_j b_{n,j}u^{(N)}_j(s)\,{\mathrm{d}}s.$$ From this, we deduce that $$\lim\limits_{N \to \infty} \int\limits_0^t \sum_{j=n+1}^{\infty} a_j b_{n,j}u^{(N)}_j(s)\,{\mathrm{d}}s$$ exists, and a further application of the monotone convergence theorem shows that $$\begin{aligned} \lim\limits_{N \to \infty} \int\limits_0^t \sum\limits_{j=n+1}^{\infty} a_j b_{n,j}u^{(N)}_j(s)\,{\mathrm{d}}s &= \int\limits_0^t \sum\limits_{j=n+1}^{\infty} a_jb_{n,j}u_j(s)\,{\mathrm{d}}s.\end{aligned}$$ Thus, for all $\mathring{u} \in (\ell_w^1)_+$, $$\label{integrated pointwise equation} (S^{(w)}(t)\mathring{u})_n = \mathring{u}_n + \int_0^t \biggl(-a_n(S^{(w)}(s)\mathring{u})_n + \sum_{j=n+1}^{\infty}a_jb_{n,j}(S^{(w)}(s)\mathring{u})_j \biggr)\,{\mathrm{d}}s.$$ It follows that $(S^{(w)}(t)\mathring{u})_n$ is absolutely continuous with respect to $t$ for each $n=1,2,\ldots$ and so $$\label{Final pointwise equation} \frac{{\mathrm{d}}}{{\mathrm{d}}t}\bigl(S^{(w)}(t)\mathring{u}\bigr)_n = -a_n\bigl(S^{(w)}(t)\mathring{u}\bigr)_n + \sum_{j=n+1}^{\infty} a_j b_{n,j}(S^{(w)}(t)\mathring{u})_j, \qquad n \in \mathbb{N},$$ for all $\mathring{u} \in (\ell_w^1)_+$ and almost every $t \ge 0$. When $\mathring{u}$ is a general, and therefore not necessarily non-negative, sequence in $\ell_w^1$, we can express $\mathring{u}=\mathring{u}_+-\mathring{u}_- \in \ell_w^1$. It then follows immediately from the first part of the proof that $u(t)=S^{(w)}(t)\mathring{u}$ also satisfies for almost all $ t \ge 0$. The last statement of the theorem follows immediately from the positivity of the semigroup $(S^{(w)}(t))_{t\ge0}$. Note that, in general, solutions of are not unique; see the discussion in Example \[example: random scission\] below. We now turn our attention to obtaining a simple representation of the generator $G^{(w)}$. Although we know that $G^{(w)}$ coincides with $A^{(w)}+B^{(w)}$ on $\mathcal{D}(A^{(w)})$, and also that $u(t)=S^{(w)}(t)\mathring{u}$ is the unique classical solution of for $\mathring{u} \in \mathcal{D}(G^{(w)})$, we have yet to ascertain an explicit expression that describes the action of $G^{(w)}$ on $\mathcal{D}(G^{(w)})$. This matter is resolved by the following theorem, which shows that $G^{(w)}$ is a restriction of the maximal operator, ${{G_{\textup{\textsf{max}}}}^{(w)}}$, defined in . In the specific case of $X_{[p]}$, the result has been obtained from [@banasiak2006perturbations Theorem 6.20], which uses extension techniques first introduced by Arlotti in [@arlotti1991], and which is applied in [@banasiak2012global Theorem 2.1]. We present an alternative proof, which avoids the use of such extensions. Let Assumptions \[A1.1\] and \[assumption on weight for generation\] hold. Then, for all $g \in \mathcal{D}(G^{(w)})$, we have $$\label{action of G} \bigl[G^{(w)}g\bigr]_n = -a_ng_n+\sum\limits_{j=n+1}^{\infty} a_j b_{n,j}g_j, \qquad n \in \mathbb{N}.$$ It follows from Lemma \[splitting D(G) into difference of two positives\] and its proof that, for every $g\in\mathcal{D}(G^{(w)})$, there exist $g_1,g_2\in\mathcal{D}(G^{(w)})_+$ such that $g=g_1-g_2$ and $f_j\coloneqq(I-G^{(w)})g_j \in (\ell_w^1)_+$ for $j=1,2$. This and the linearity of $G^{(w)}$ allow us to assume that $g\in\mathcal{D}(G^{(w)})_+$ such that $f\coloneqq(I-G^{(w)})g\in(\ell_w^1)_+$. Defining $u(t)=S^{(w)}(t)f$, we have from that $$\begin{aligned} & \bigl[R(1,G^{(w)})f\bigr]_n = \int\limits_0^{\infty} e^{-t}[S^{(w)}(t)f]_n\,{\mathrm{d}}t = \int\limits_0^{\infty} e^{-t}u_n(t)\,{\mathrm{d}}t \\ &= f_n -\int\limits_0^{\infty} \int\limits_0^t e^{-t}a_nu_n(s)\,{\mathrm{d}}s\,{\mathrm{d}}t + \int\limits_0^{\infty}\int\limits_0^t \sum\limits_{j=n+1}^{\infty} e^{-t}a_jb_{n,j}u_j(s)\,{\mathrm{d}}s\,{\mathrm{d}}t.\end{aligned}$$ By Tonelli’s theorem, we have $$\begin{aligned} \int\limits_0^{\infty} \int\limits_0^t e^{-t} a_nu_n(s)\,{\mathrm{d}}s\,{\mathrm{d}}t &= \int\limits_0^{\infty} \int\limits_s^{\infty} e^{-t} a_nu_n(s)\,{\mathrm{d}}t\,{\mathrm{d}}s \\ &= a_n \int\limits_0^{\infty} e^{-s}u_n(s)\,{\mathrm{d}}s =a_n\bigl[R(1,G^{(w)})f\bigr]_n.\end{aligned}$$ Using Tonelli’s theorem and the monotone convergence theorem we obtain $$\begin{aligned} \int\limits_0^{\infty} e^{-t} \int\limits_0^t \sum\limits_{j=n+1}^{\infty} a_j b_{n,j}u_j(s)\,{\mathrm{d}}s\,{\mathrm{d}}t = \sum\limits_{j=n+1}^{\infty} a_j b_{n,j}\bigl[R(1,G^{(w)})f\bigr]_j.\end{aligned}$$ Thus $$\begin{aligned} g_n &= \bigl[R(1,G^{(w)})f\bigr]_n = f_n-a_n\bigl[R(1,G^{(w)})f\bigr]_n + \sum\limits_{j=n+1}^{\infty} a_j b_{n,j}\bigl[R(1,G^{(w)})f\bigr]_j \\ &= \bigl[(I-G^{(w)})g\bigr]_n-a_n\bigl[R(1,G^{(w)})f\bigr]_n + \sum\limits_{j=n+1}^{\infty} a_j b_{n,j}\bigl[R(1,G^{(w)})f\bigr]_j \\ &= g_n - \bigl[G^{(w)}g\bigr]_n - a_ng_n + \sum_{j=n+1}^\infty a_j b_{n,j}g_j,\end{aligned}$$ and follows. We note that the formula is independent of the weight $w=(w_n)_{n=1}^{\infty}$. Being able to express the action of $G^{(w)}$ in this way is important when investigating the full coagulation–fragmentation system, as it enables the fragmentation terms to be described by means of an explicit formula for the operator $G^{(w)}$. We shall return to this in a subsequent paper. \[example: random scission\] Let us consider the system $$\label{system random scission} \begin{split} & u_n'(t) = -(n-1)u_n(t) + 2\sum_{j=n+1}^\infty u_j(t), \qquad t>0; \\ & u_n(0) = \mathring{u}_n, \qquad n=1,2,\ldots, \end{split}$$ which coincides with if one sets $$\label{anbnj random scission} a_n = n-1, \quad b_{n,j} = \frac{2}{j-1}\,, \qquad n,j \in \mathbb{N}, \quad j > n.$$ The system models random scission; see, e.g. [@ziffmcgrady1985kinetics equation (49)] and [@caiedwardshan1991 equation (10)]. It is easily seen that is satisfied, and hence, mass is conserved. The example is closely related to the example that is studied in [@smith2012discrete §3] and which models random bond annihilation. More precisely, if we denote the operators for the example from [@smith2012discrete] by $\widetilde{A}^{(w)}$, $\widetilde{B}^{(w)}$, $\widetilde{G}^{(w)}$ etc., then $$A^{(w)} = \widetilde{A}^{(w)}+I, \quad B^{(w)} = \widetilde{B}^{(w)}, \quad G^{(w)} = \widetilde{G}^{(w)}+I,$$ and hence $S^{(w)}(t) = e^t\widetilde{S}^{(w)}(t)$, $t\ge0$. For the particular case when $w_n=n$, $n \in \mathbb{N}$, we have similar relations for the operators $A_1$, $\widetilde{A}_1$ etc. It follows from [@smith2012discrete Lemma 3.6] that every $\lambda > 0$ is an eigenvalue of the maximal operator ${G_{1,\textup{\textsf{max}}}}$ (i.e. the operator ${{G_{\textup{\textsf{max}}}}^{(w)}}$ defined in for $w_n=n$) with eigenvector $g^{(\lambda)}=(g_n^{(\lambda)})_{n\in\mathbb{N}}$ where $$\label{eigenvector_Gmax} g_n^{(\lambda)} = \frac{1}{(\lambda+n-1)(\lambda+n)(\lambda+n+1)}\,, \qquad n \in \mathbb{N}.$$ The existence of positive eigenvalues of ${G_{1,\textup{\textsf{max}}}}$ implies that ${G_{1,\textup{\textsf{max}}}}$ is a proper extension of $G_1$. Note that the domain of $\widetilde G_1$ is determined explicitly in [@smith2012discrete Theorem 3.7], from which we obtain that $$\label{domain G1} \mathcal{D}(G_1) = \biggl\{f=(f_k)_{k\in\mathbb{N}} \in \mathcal{D}({G_{1,\textup{\textsf{max}}}}): \lim_{n\to\infty} \biggl(n^2\sum_{k=n+1}^\infty f_k\biggr) = 0\biggr\}.$$ Using the eigenvectors $g^{(\lambda)}$ from we can define the function $$u^{(\lambda)}(t) \coloneqq e^{\lambda t}g^{(\lambda)}, \qquad t \ge 0,$$ which is a solution of the ACP $$\label{ACP_Gonemax} u'(t) = {G_{1,\textup{\textsf{max}}}}u(t), \quad t>0; \qquad u(0) = \mathring{u}$$ with $\mathring{u}=g^{(\lambda)}$. On the other hand, since the semigroup $(S_1(t))_{t\ge0}$ is analytic by [@smith2012discrete Theorem 3.4], the function $u(t)=S_1(t)g^{(\lambda)}$, $t\ge0$, is also a solution of and is distinct from $u^{(\lambda)}$. This shows that, in general, one does not have uniqueness of solutions of the ACP, , corresponding to the maximal operator, ${G_{1,\textup{\textsf{max}}}}$, and hence, also solutions of are not unique. More generally, a specific characterisation of $\mathcal{D}(G^{(w)})$ is given by [@banasiak2006perturbations Theorem 6.20], but this does not lead to an explicit description, such as that obtained in Example \[example: random scission\]. Analyticity of the fragmentation semigroup {#Analyticity of the Fragmentation Semigroup} ========================================== In Section \[The Generator of the Fragmentation Semigroup\] we established that Assumptions \[A1.1\] and \[assumption on weight for generation\] are sufficient conditions for $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$ to be the generator of a substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$. This enabled us to obtain results on the existence and uniqueness of solutions to . We now investigate the analyticity of $(S^{(w)}(t))_{t \geq 0}$ and prove that, given *any* fragmentation coefficients, it is always possible to construct a weight, $w$, such that $A^{(w)}+B^{(w)}$ is the generator of an analytic, substochastic $C_0$-semigroup on $\ell_w^1$. This particular result, which is one of the main motivations for carrying out an analysis of the fragmentation system in general weighted $\ell^1$ spaces, requires a stronger assumption on the weight $w$. Note that when dealing with analytic semigroups, we use complex versions of the spaces $\ell_w^1$. \[assumption for analyticity\] ------------------------------------------------------------------------ $w_n \ge n$ for all $n \in \mathbb{N}$. There exists $\kappa \in (0,1)$ such that $$\label{inequ_for_analyticity} \sum\limits_{n=1}^{j-1} w_nb_{n,j} \leq \kappa w_j \qquad \text{for all} \ j=2,3,\ldots.$$ Note that Assumption \[assumption for analyticity\] is obtained from Assumption \[assumption on weight for generation\] by simply replacing $\kappa \in (0,1]$ with $\kappa \in (0,1)$. By removing the possibility of $\kappa = 1$, we can obtain the following improved version of Theorem \[G=closure for frag\]. \[thm for analyticity\] Let Assumptions \[A1.1\] and \[assumption for analyticity\] hold. Then the operator $G^{(w)}=A^{(w)}+B^{(w)}$ is the generator of an analytic, substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$. Let $(T^{(w)}(t))_{t\ge0}$ be as in Lemma \[A is a generator\]. For $\alpha > 0$ and $f \in \mathcal{D}(A^{(w)})_+$, we obtain from that $$\begin{aligned} \int\limits_0^{\alpha} \big\Vert B^{(w)}T^{(w)}(t)f \big\Vert\,{\mathrm{d}}t &\le \kappa \int\limits_0^{\alpha} \big\Vert A^{(w)}T^{(w)}(t)f \big\Vert_{w}\,{\mathrm{d}}t \\ &= \kappa\int\limits_0^{\alpha} \phi_w\bigl(-A^{(w)}T^{(w)}(t)f\bigr)\,{\mathrm{d}}t = \kappa \phi_w\biggl(-\int\limits_0^{\alpha} A^{(w)}T^{(w)}(t)f \, {\mathrm{d}}t \biggr) \\ &= \kappa \phi_w\biggl(-\int\limits_0^{\alpha} \frac{{\mathrm{d}}}{{\mathrm{d}}t}\bigl(T^{(w)}(t)f\bigr)\,{\mathrm{d}}t\biggr) = \kappa \phi_w\bigl(f-T^{(w)}(\alpha)f\bigr) \\ &= \kappa \Vert f \Vert_w-\kappa \Vert T^{(w)}(\alpha)f \Vert_w \le \kappa \Vert f \Vert_w.\end{aligned}$$ Since $\kappa<1$, it follows from [@thieme2006stochastic Theorem A.2] that $G^{(w)}=A^{(w)}+B^{(w)}$ is the generator of a positive $C_0$-semigroup. The proof of [@thieme2006stochastic Theorem A.2], establishes that this semigroup is substochastic since $\kappa<1$. Moreover, by Lemma \[A is a generator\], $A^{(w)}$ is also the generator of a substochastic $C_0$-semigroup, $(T^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$, and a routine calculation shows that $$\bigl\Vert R(\lambda,A^{(w)})f \bigr\Vert_w = \sum\limits_{n=1}^{\infty} w_n\frac{1}{|\lambda+a_n|}|f_n| \le \frac{1}{|{\operatorname{Im}}\lambda|}\Vert f \Vert_w, \qquad \lambda\in\mathbb{C}\setminus\mathbb{R} \ \text{with} \ {\operatorname{Re}}\lambda > 0,$$ for all $f\in\ell_w^1$. Therefore, by [@engel1999one Theorem .4.6], $(T^{(w)}(t))_{t \ge 0}$ is an analytic semigroup. Also, the positivity of $(S^{(w)}(t))_{t \ge 0}$ implies that $A^{(w)}+B^{(w)}$ is resolvent positive. Hence, by [@arendtrhandi1991perturbation Theorem 1.1], $(S^{(w)}(t))_{t \ge 0}$ is analytic. \[can always find weight remark\] ------------------------------------------------------------------------ Although Assumption \[assumption for analyticity\] is never satisfied when holds and $w_n=n$ for all $n \in \mathbb{N}$, this does not rule out the possibility of an analytic fragmentation semigroup on $X_{[1]}$ existing. Indeed, the semigroup $(S_1(t))_{t \ge 0}$ in Example \[example: random scission\] is analytic, which follows from [@smith2012discrete Theorem 3.4] as mentioned above. If there exists $\lambda_0>0$ such that holds with $\lambda_j \ge \lambda_0$ for all $j \ge 2$ (which corresponds to a ‘uniform’ mass loss case), then Assumption \[assumption for analyticity\] immediately holds with $w_n=n$ for all $n \in \mathbb{N}$, and $\kappa=1-\lambda_0$. The following lemma gives sufficient conditions under which Assumption \[assumption for analyticity\] holds. \[increasing weight condition for analyticity\] Let $w$ be such that $$\label{suff_cond_analyticity} 1 \le \frac{w_n}{n} \le \delta\frac{w_{n+1}}{n+1} \qquad \text{for all} \ n \in \mathbb{N},$$ where $\delta \in (0,1)$. Moreover, let hold. Then Assumption \[assumption for analyticity\] is satisfied with $\kappa=\delta$. Since $$\frac{w_n}{n} \le \delta^{j-n} \frac{w_j}{j} \le \delta\frac{w_j}{j} \qquad \text{for all} \ n=1,\ldots,j-1,$$ it follows that $$\sum\limits_{n=1}^{j-1} w_nb_{n,j} = \sum\limits_{n=1}^{j-1} \frac{w_n}{n}nb_{n,j} \le \delta \frac{w_j}{j} \sum\limits_{n=1}^{j-1} nb_{n,j} \le \delta w_j$$ for $j=2,3,\ldots$, where is used to obtain the last inequality. Since $\delta \in (0,1)$, the result follows immediately. This leads to the main result of this section. \[can always find analytic semigroup\] For any given fragmentation coefficients for which Assumption \[A1.1\] holds we can always find a weight, $w=(w_n)_{n=1}^{\infty}$, such that $A^{(w)}+B^{(w)}$ is the generator of an analytic, substochastic $C_0$-semigroup on $\ell_w^1$. If, in addition, holds, we can choose $w_n=r^n$ with arbitrary $r>2$ and $\kappa=2/r$ so that holds. For the first statement note that we can choose $w_n \ge n$ iteratively so that is satisfied. The claim then follows from Theorem \[thm for analyticity\]. Now assume that holds. Let $r>2$, $w_n=r^n$ for $n \in \mathbb{N}$, and $\delta=2/r$, which satisfies $\delta<1$. Then $w_n \ge n$ and $$\delta\frac{w_{n+1}}{n+1} = \frac{2}{r}\cdot\frac{r^{n+1}}{n+1} = \frac{2r^n}{n+1} \ge \frac{2r^n}{n+n} = \frac{r^n}{n} = \frac{w_n}{n}\,,$$ which shows that is satisfied. Hence Lemma \[increasing weight condition for analyticity\] implies that Assumption \[assumption for analyticity\] is fulfilled. As mentioned earlier, analytic semigroups have a number of desirable properties, and Theorem \[can always find analytic semigroup\] will play an important role when we investigate the full coagulation–fragmentation system in a subsequent paper. In particular, Theorem \[can always find analytic semigroup\] will enable us to relax the usual assumptions that are imposed on the coagulation rates in order to obtain the existence and uniqueness of solutions to the full coagulation–fragmentation system. It should be noted that a condition that is equivalent to Assumption \[assumption for analyticity\] has previously been used as a condition for analyticity in the mass-conserving case by Banasiak; see [@banasiak2012global Theorem 2.1]. However, the choice of weights in [@banasiak2012global] is restricted to $w_n=n^p$, $p>1$, and Assumption \[assumption for analyticity\] need not be satisfied for these weights for any $p>1$ as the following example shows. \[finding weights for binary fragmentation\] Consider the mass-conserving case where a cluster of mass $n$ breaks into two clusters, with respective masses $1$ and $n-1$. The corresponding fragmentation coefficients take the form $$\label{binary fragmentation coefficients} b_{1,2}=2; \quad b_{1,j}=b_{j-1,j}=1, \;\; j\ge3; \quad b_{n,j}=0, \;\; 2 \le n \le j-2.$$ For the choice $$a_0 = 0; \quad a_n = n, \quad n \ge 2; \qquad w_n=n^p, \quad n \in \mathbb{N}; \qquad p \ge 1,$$ it is proved in [@banasiak2011irregular Theorem 3] (for $p=1$) and [@banasiak2012global Theorem A.3] (for $p>1$) that the semigroup generated by $G^{(w)}$ is not analytic. On the other hand, Theorem \[can always find analytic semigroup\] guarantees the existence of exponentially growing weights $w_n$ such that $G^{(w)}=A^{(w)}+B^{(w)}$ generates an analytic semigroup. It is easy to show that for this particular example one can also choose powers of $2$, namely, $w_1=1$ and $w_n=2^n$ for $n \ge 2$, in which case $\kappa=5/8$. Asymptotic behaviour of solutions {#Asymptotic Behaviour of Solutions} ================================= There have been several earlier investigations into the long-term behaviour of solutions to the mass-conserving fragmentation system , when holds. In particular, the case of mass-conserving binary fragmentation is dealt with in [@CadC94], where it is shown that, under suitable assumptions, the unique solution emanating from $\mathring{u}$ must converge in the space $X_{[1]}$ to the expected steady-state solution $M_1(\mathring{u})e_1$, where $M_1(\mathring{u})$ and $e_1$ are given by and respectively. This was followed by [@banasiak2011irregular; @banasiaklamb2012discrete] where, once again, the expected long-term steady-state behaviour is established, but now for the mass-conserving multiple-fragmentation system. More specifically, in [@banasiak2011irregular], a semigroup-based approach is used to prove that, for any $\mathring{u} \in X_{[1]}$, $$\lim_{t \to \infty}\Vert S_1(t)\mathring{u}- M(\mathring{u})e_1\Vert_{[1]} = 0 \qquad\text{if and only if}\qquad a_n > 0 \;\; \text{for all} \ n = 2,3,\ldots.$$ That the corresponding result is also valid in the higher moment spaces $X_{[p]}$, $p > 1$, is established in [@banasiaklamb2012discrete], and, under additional assumptions on the fragmentation coefficients, it is shown in [@banasiaklamb2012discrete Theorem 4.3] that there exist constants $L>0$ and $\alpha>0$ such that the fragmentation semigroup $(S_p(t))_{t \ge 0}$ on $X_{[p]}$, $p>1$, satisfies $$\label{eq AEG} \Vert S_p(t)\mathring{u} - M_1(\mathring{u})e_1\Vert_{[p]} \le Le^{-\alpha t}\Vert \mathring{u} \Vert_{[p]},$$ for all $\mathring{u} \in X_{[p]}$. It follows from [@arino1992], that the fragmentation semigroup $(S_p(t))_{t \geq 0}$ has the asynchronous exponential growth (AEG) property (with $\lambda^*=0$ in [@arino1992 equation (3)], i.e. with trivial growth). The assumptions required in [@banasiaklamb2012discrete] to prove that holds in some $X_{[p]}$ space are somewhat technical and not straightforward to check. Moreover no information on the size of the constant $\alpha$, and hence the exponential rate of decay to the steady state, is provided. Our aim in this section is to address these issues. Working within the framework of more general weighted $\ell^1$ spaces, we study the long-term dynamics of solutions in both the mass-conserving and mass-loss cases. When mass is conserved, we establish simpler conditions under which the fragmentation semigroup $(S^{(w)}(t))_{t \geq 0}$ satisfies an inequality of the form on some space $\ell^1_w$, and also quantify $\alpha$. We begin by considering the general fragmentation system , where the coefficients $a_n$ and $b_{n,j}$ satisfy Assumption \[A1.1\], and recall that $G^{(w)}=\overline{A^{(w)}+B^{(w)}}$ is the generator of a substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$ whenever Assumption \[assumption on weight for generation\] holds. Furthermore, $(S^{(w)}(t))_{t \ge 0}$ is analytic, with generator $A^{(w)}+B^{(w)}$ when the more restrictive Assumption \[assumption for analyticity\] is satisfied. \[mass loss decay of solution\] Let Assumptions \[A1.1\] and \[assumption on weight for generation\] hold. Then $$\label{decay of solution to zero} \lim\limits_{t \to \infty} \Vert S^{(w)}(t)\mathring{u} \Vert_w = 0$$ for all $\mathring{u} \in \ell_w^1$ if and only if $a_n > 0$ for all $n \in \mathbb{N}$. If, additionally, we choose $w$ such that Assumption \[assumption for analyticity\] is satisfied, and set $a_0 \coloneqq \inf_{n \in \mathbb{N}} a_n$, then $$\label{semigroup exponential bound} \Vert S^{(w)}(t) \Vert \le e^{-(1-\kappa)a_0t},$$ and hence, if $a_0>0$ and $\alpha \in [0,(1-\kappa)a_0)$, we have $$\label{exponential decay to zero} \lim\limits_{t \to \infty} e^{\alpha t}\Vert S^{(w)}(t)\mathring{u} \Vert_w = 0 \qquad \text{for every} \ \mathring{u} \in \ell_w^1.$$ If $\alpha > a_0$, then does not hold. In particular, if $a_0=0$, then does not hold for any $\alpha>0$. () First assume that $a_n>0$ for all $n \in \mathbb{N}$. Let $\mathring{u} \in \ell_w^1$, and, as in Section \[Pointwise Problem\], let $P_N\mathring{u} = (\mathring{u}_1,\mathring{u}_2,\ldots, \mathring{u}_N,0, \ldots )$, $N \in \mathbb{N}$. For each fixed $n \in \mathbb{N}$, we know from that $(S^{(w)}(t)e_n)_m=s_{m,n}(t)=0$ for $m>n$. Furthermore, $(s_{1,n},s_{2,n},\ldots, s_{n,n})$, with the identification , is the unique solution of the $n$-dimensional system . Our assumption on the coefficients $a_n$ means that all eigenvalues of the matrix associated with are negative. It follows that $s_{m,n}(t) \to 0$ as $t \to \infty$ for $m=1,\ldots,n$, and therefore $$\lim\limits_{t \to \infty} \Vert S^{(w)}(t)e_n \Vert_w = \lim\limits_{t \to \infty} \sum\limits_{m=1}^{n} w_ms_{m,n}(t) = 0,$$ for all $n \in \mathbb{N}$. This in turn implies that $$\Vert S^{(w)}(t)P_N\mathring{u} \Vert_w \le \sum\limits_{n=1}^N |\mathring{u}_n| \Vert S^{(w)}(t)e_n \Vert_w \to 0 \qquad \text{as} \ t \to \infty,$$ for each $N \in \mathbb{N}$. Given any $\varepsilon >0$, we can always find $N \in \mathbb{N}$ and $t_0>0$ such that $$\Vert \mathring{u}-P_N\mathring{u} \Vert_w < \frac{\varepsilon}{2} \qquad\text{and}\qquad \Vert S^{(w)}(t)P_N\mathring{u} \Vert_w < \frac{\varepsilon}{2} \quad \text{for all} \ t \ge t_0.$$ Then $$\begin{aligned} \Vert S^{(w)}(t)\mathring{u} \Vert_w &\le \bigl\Vert S^{(w)}(t)\bigl(\mathring{u}-P_N\mathring{u}\bigr) \bigr\Vert_w + \Vert S^{(w)}(t)P_N\mathring{u} \Vert_w \\[0.5ex] &\le \Vert \mathring{u}-P_N\mathring{u} \Vert_w + \Vert S^{(w)}(t)P_N\mathring{u} \Vert_w < \varepsilon \hspace*{7ex} \text{for all} \ t \geq t_0,\end{aligned}$$ which establishes . On the other hand, suppose that $a_N=0$ for some $N \in \mathbb{N}$. Then we have that the unique solution of , with $\mathring{u}=e_N$, is $u(t)=S^{(w)}(t)e_N=(s_{m,N}(t))_{m=1}^{\infty}$. Since $s_{N,N}(t)=e^{-a_N t}=1$, it is clear that $u(t)\nrightarrow 0$ as $t \to \infty$. () Now let Assumption \[assumption for analyticity\] hold and let $\mathring{u} \in (\ell_w^1)_+$. From Theorem \[thm for analyticity\], $A^{(w)}+B^{(w)}$ generates an analytic, substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \ge 0}$, on $\ell_w^1$, and $u(t)=S^{(w)}(t)\mathring{u}$ is the unique, non-negative classical solution of . Let $ t > 0$. Using we obtain that $$\begin{aligned} \frac{{\mathrm{d}}}{{\mathrm{d}}t} \phi_w\bigl(u(t)\bigr) &= \phi_w\bigl(u'(t)\bigr) = \phi_w\bigl(A^{(w)}u(t)\bigr)+\phi_w\bigl(B^{(w)}u(t)\bigr) \\ &\le \phi_w\bigl(A^{(w)}u(t)\bigr)-\kappa\phi_w\bigl(A^{(w)}u(t)\bigr) \\ &= -(1-\kappa)\sum\limits_{n=1}^{\infty} w_na_nu_n(t)\\ &\le -(1-\kappa)a_0\phi_w\bigl(u(t)\bigr).\end{aligned}$$ Therefore, $$\phi_w\bigl(u(t)\bigr) \le \phi_w(\mathring{u})e^{-(1-\kappa)a_0t} \quad\text{and hence}\quad \Vert S^{(w)}(t)\mathring{u} \Vert_w \leq e^{-(1-\kappa)a_0t}\Vert \mathring{u} \Vert_w,$$ and then follows from the positivity of $(S^{(w)}(t))_{t \ge 0}$ and [@banasiak2006perturbations Proposition 2.67]. If $a_0>0$ and $\alpha \in [0,(1-\kappa)a_0)$, then holds. On the other hand if we choose $\alpha>a_0$, then there exists $N \in \mathbb{N}$ such that $a_N<\alpha$, in which case $(S^{(w)}(t)e_N)_N = e^{-a_Nt} > e^{-\alpha t}$ for $t>0$, and so $$e^{\alpha t}\Vert S^{(w)}(t)e_N \Vert_w \ge e^{\alpha t}w_N\bigl(S^{(w)}(t)e_N\bigr)_N > e^{\alpha t}w_Ne^{-\alpha t} = w_N.$$ Hence cannot hold for any $\alpha>a_0$. \[mass loss remark about equilibrium points\] When the assumptions of Theorem \[mass loss decay of solution\] are satisfied and $a_n>0$ for all $n \in \mathbb{N}$, then shows that the only equilibrium solution of is $u(t)\equiv 0$, and this equilibrium is a global attractor for the system. On the other hand, if $a_n=0$ for at least one $n \in \mathbb{N}$, then $u(t)\equiv 0$ is not a global attractor. We now examine the mass-conserving case and assume that holds. Note that, in this mass-conserving case, the fragmentation semigroup $(S_1(t))_{t \ge 0}$ is stochastic on the space $X_{[1]}$. Our aim is to establish an $\ell_w^1$ version of the results obtained in [@banasiak2011irregular; @banasiaklamb2012discrete; @CadC94]. To this end, we recall the matrix representation of $S^{(w)}(t)$ given by , and also define a sequence space $Y^{(w)}$, and its norm ${\lVert\,\cdot\,\rVert_{Y^{(w)}}}$, by $$Y^{(w)} = \bigl\{\tilde{f}=(f_n)_{n=2}^{\infty}: f=(f_n)_{n=1}^{\infty} \in \ell_w^1\bigr\} \qquad\text{and}\qquad \Vert f \Vert_{Y^{(w)}} = \sum\limits_{n=2}^{\infty} w_n|f_n|,$$ respectively. Clearly, $Y^{(w)}$ is a weighted $\ell^1$ space, and can be identified with $\ell_{\widehat{w}}^1$, where $\widehat{w}_n=w_{n+1}$ for $n \in \mathbb{N}$. Moreover, we define the embedding operator $J:Y^{(w)}\to\ell_w^1$ by $$Jf = (0,f_2,f_3,\ldots) \qquad \text{for all} \ f \in \ell_w^1.$$ \[convergence lemma\] Let $\alpha \ge 0$ and $f \in \ell_w^1$ be fixed, and define $\tilde{f} \coloneqq (f_n)_{n=2}^{\infty}$. If Assumptions \[A1.1\], \[assumption on weight for generation\] and hold, then $$\label{equivalence} \Vert S^{(w)}_{(22)}(t)\tilde{f} \Vert_{Y^{(w)}} \le \Vert S^{(w)}(t)f-M_1(f)e_1 \Vert_w \le (w_1+1)\Vert S^{(w)}_{(22)}(t)\tilde{f} \Vert_{Y^{(w)}}$$ for all $t \ge 0$. It follows from that $$\label{split_Stf} S^{(w)}(t)f = \bigl(f_1+S^{(w)}_{(12)}(t)\tilde{f}\bigr)e_1+JS_{(22)}^{(w)}(t)\tilde{f}.$$ From this we deduce that $$\label{link between semigroup norms} \Vert S^{(w)}(t)f-M_1(f)e_1 \Vert_w = w_1\Bigl\lvert f_1+S^{(w)}_{(12)}(t)\tilde{f}-M_1(f)\Bigr\rvert + \Vert S^{(w)}_{(22)}(t)\tilde{f} \Vert_{Y^{(w)}}$$ and so $$\Vert S^{(w)}(t)f-M_1(f)e_1 \Vert_w \ge \Vert S_{(22)}^{(w)}(t)\tilde{f} \Vert_{Y^{(w)}},$$ which is the first inequality in . On the other hand, from Proposition \[prop:stochastic\_semigroup\](i) and the stochasticity of $(S_1(t))_{t \ge 0}$ on $X_{[1]}$, we know that $M_1(S^{(w)}(t)f)=M_1(S_1(t)f)=M_1(f)$. Using we obtain $$\begin{aligned} \Big|f_1+S^{(w)}_{(12)}(t)\tilde{f}- M_1(f)\Big| &= \Big|M_1(f)- M_1\Bigl(\bigl(f_1+S_{(12)}^{(w)}(t)\tilde{f}\bigr)e_1\Bigr)\Big| \\ &= \Big|M_1\bigl(S^{(w)}(t)f\bigr) - M_1\Bigl(\bigl(f_1+S^{(w)}_{(12)}(t)\tilde{f}\bigr)e_1\Bigr)\Big| \\ &\le M_1\Bigl(\Big| S^{(w)}(t)f - \bigl(f_1+S^{(w)}_{(12)}(t)\tilde{f}\bigr)e_1\Big|\Bigr) \\ &\le \phi_w\Bigl(\Big|S^{(w)}(t)f-\bigl(f_1+S^{(w)}_{(12)}(t)\tilde{f}\bigr)e_1\Big|\Bigr) \\ &= \phi_w\Bigl(\Big|(JS_{(22)}^{(w)}(t)\tilde{f}\Big|\Bigr) \\ &= \bigl\Vert S^{(w)}_{(22)}(t)\tilde{f} \bigr\Vert_{Y^{(w)}}.\end{aligned}$$ The second inequality in then follows from . We are now in a position to prove the main theorem of this section. The first part confirms that $S^{(w)}(t)\mathring{u} \to M_1(\mathring{u})e_1$ in $\ell^1_w$ as $t \to \infty$, for all $\mathring{u} \in \ell_w^1$, provided that Assumption \[assumption on weight for generation\] holds and the fragmentation rates, $a_n$, are positive for all $n \geq 2$. In the second part, which deals with quantifying the rate of convergence to equilibrium, the fragmentation coefficients are assumed additionally to be bounded below by a positive constant, and Assumption \[assumption on weight for generation\] is strengthened to Assumption \[assumption for analyticity\]. In this case, the decay to zero of $\Vert S^{(w)}(t)\mathring{u} - M_1(\mathring{u})e_1\Vert_w$ is shown to occur at an exponential rate, defined explicitly in terms of the rate coefficients and the constant $\kappa \in (0,1)$ in Assumption \[assumption for analyticity\]. \[mass conservation decay theorem\] Let Assumptions \[A1.1\] and \[assumption on weight for generation\], and hold and let $M_1$ be as in . We have $$\label{decay to monomer state} \lim\limits_{t \to \infty} \Vert S^{(w)}(t)\mathring{u}-M_1(\mathring{u})e_1 \Vert_w = 0$$ for all $\mathring{u} \in \ell_w^1$ if and only if $a_n>0$ for all $n \ge 2$. Choose $w$ such that Assumption \[assumption for analyticity\] holds and let $\widehat{a}_0 \coloneqq \inf_{n \in \mathbb{N}: n \ge 2} a_n$. Then, for all $\mathring{u} \in \ell_w^1$, $$\label{exponential bound for S-phi} \Vert S^{(w)}(t)\mathring{u}-M_1(\mathring{u})e_1 \Vert_w \le (w_1+1)e^{-(1-\kappa)\widehat{a}_0t}\Vert \mathring{u} \Vert_w,$$ and so $$\label{exponential decay to monomer state} \lim\limits_{t \to \infty} e^{\alpha t} \Vert S^{(w)}(t)\mathring{u}-M_1(\mathring{u})e_1 \Vert_w = 0,$$ whenever $\widehat{a}_0>0$ and $\alpha \in [0,(1-\kappa)\widehat{a}_0)$. Equation does not hold for any $\alpha>\widehat{a}_0$. In particular, if $\widehat{a}_0=0$, then does not hold for any $\alpha>0$. Removing the equation for $u_1$ from leads to a reduced fragmentation system that can be formulated as an ACP in $Y^{(w)}=\ell_{\widehat{w}}^1$, where, as before, $\widehat{w}_n=w_{n+1}$ for all $n \in \mathbb{N}$. The fragmentation coefficients, $(\widehat{a}_n)_{n=1}^{\infty}$ and $(\widehat{b}_{n,j})_{n, j \in \mathbb{N}: n<j}$, associated with the reduced system are given by $\widehat{a}_n=a_{n+1}$ and $\widehat{b}_{n,j}=b_{n+1,j+1}$. Clearly, $\widehat{a}_n \ge 0$ and $\widehat{b}_{n,j} \ge 0$ for all $n,j \in \mathbb{N}$ and $\widehat{b}_{n,j} = 0$ if $n \ge j$, and $\widehat{w}_n=w_{n+1} \ge n+1 > n$ for all $n \in \mathbb{N}$. Moreover, for $j=2,3,\ldots$, $$\begin{aligned} \sum\limits_{n=1}^{j-1} \widehat{w}_n\widehat{b}_{n,j} &= \sum\limits_{n=1}^{j-1} w_{n+1}b_{n+1,j+1} = \sum\limits_{k=2}^{j} w_kb_{k,j+1} \le \sum\limits_{k=1}^{j} w_kb_{k,j+1} \\ &\le \kappa w_{j+1}=\kappa \widehat{w}_j.\end{aligned}$$ Hence Assumptions \[A1.1\] and \[assumption on weight for generation\] are satisfied by $\widehat{w}$, $\widehat{a}_n$ and $\widehat{b}_{n,j}$, and it follows from Theorem \[G=closure for frag\] and that associated with the reduced system is a substochastic $C_0$-semigroup on $Y^{(w)}$, which can be represented by the infinite matrix $$\begin{bmatrix} e^{-\widehat{a}_1t} & \widehat{s}_{1,2}(t) & \widehat{s}_{1,3}(t) & \cdots \\[1ex] 0 & e^{-\widehat{a}_2t} & \widehat{s}_{2,3}(t) & \cdots \\[1ex] 0 & 0 & e^{-\widehat{a}_3t} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix} =\begin{bmatrix} e^{-a_2t} & \widehat{s}_{1,2}(t) & \widehat{s}_{1,3}(t) & \cdots \\[1ex] 0 & e^{-a_3t} & \widehat{s}_{2,3}(t) & \cdots \\[1ex] 0 & 0 & e^{-a_4t} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix},$$ where, for all $n \in \mathbb{N}$, $m=1, \ldots, n-1$, $t \ge 0$, $\widehat{s}_{m,n}(t)$ is the unique solution of $$\begin{aligned} \widehat{s}_{m,n}'(t) &= -\widehat{a}_m\widehat{s}_{m,n}(t) + \sum\limits_{j=m+1}^n \widehat{a}_j\widehat{b}_{m,j}\widehat{s}_{j,n}(t) \\ &= -a_{m+1}\widehat{s}_{m,n}(t) + \sum\limits_{k=m+2}^{n+1} a_{k}b_{m+1,k}\widehat{s}_{k-1,n}(t).\end{aligned}$$ An inspection of , together with , shows that $\widehat{s}_{m,n}(t)=s_{m+1,n+1}(t)$ for all $n \in \mathbb{N}$, $m=1,\ldots, n-1$, $t \ge 0$, and therefore the substochastic semigroup on $Y^{(w)}$ is given by $(S^{(w)}_{(22)}(t))_{t \ge 0}$, where $(S^{(w)}_{(22)}(t))_{t \ge 0}$ is the infinite matrix that features in . () Let $\widehat{\mathring{u}}=(\mathring{u}_2,\mathring{u}_3,\ldots)$ for each $\mathring{u} \in \ell_w^1$. From Theorem \[mass loss decay of solution\], we deduce that $$\lim\limits_{t \to \infty} \bigl\Vert S^{(w)}_{(22)}(t)\widehat{\mathring{u}}\bigr\Vert_{Y^{(w)}} = \lim\limits_{t \to \infty} \bigl\Vert S^{(w)}_{(22)}(t)\widehat{\mathring{u}}\bigr\Vert_{\widehat{w}} = 0,$$ if and only if $a_n>0$ for all $n \ge 2$, and the result is then an immediate consequence of Lemma \[convergence lemma\]. () The calculations above show that, when Assumption \[assumption on weight for generation\] holds for $w$ and the coefficients $(b_{n,j})$, it is also satisfied by $\widehat{w}$ and $(\widehat{b}_{n,j})$ with exactly the same value of $\kappa$. Therefore, from Theorem \[mass loss decay of solution\], $$\Vert S^{(w)}_{(22)}(t) \Vert \leq e^{-(1-\kappa)\widehat{a}_0t},$$ and follows immediately from Lemma \[convergence lemma\]. Moreover, if $\widehat{a}_0>0$ and $\alpha \in [0,(1-\kappa)\widehat{a}_0)$, then we obtain . If $\alpha>\widehat{a}_0$, then, from Theorem \[mass loss decay of solution\], the result $$\lim\limits_{t \to \infty} e^{\alpha t} \bigl\Vert S^{(w)}_{(22)}(t)\widehat{\mathring{u}}\bigr\Vert_{Y^{(w)}} = \lim\limits_{t \to \infty} e^{\alpha t} \bigl\Vert S^{(w)}_{(22)}(t)\widehat{\mathring{u}}\bigr\Vert_{\widehat{w}} = 0,$$ does not hold for all $\mathring{u} \in \ell_w^1$. Hence, from Lemma \[convergence lemma\], does not hold if $\alpha>\widehat{a}_0$. When the assumptions of Theorem \[mass conservation decay theorem\] are satisfied, then it follows from that $\overline{u}_M = Me_1$ is an equilibrium solution of the mass-conserving fragmentation system for all $M \in \mathbb{R}$. In addition, the basin of attraction for $\overline{u}_M $ is given by $\{\mathring{u} \in \ell_w^1 : M_1(\mathring{u}) = M\}$ provided that the assumptions of Theorem \[mass conservation decay theorem\] hold and $a_n>0$ for all $n \ge 2$. On the other hand, if $a_N=0$ for some $N \ge 2$, then $Me_N$ is also an equilibrium solution for every $M \in \mathbb{R}$. Sobolev towers {#Sobolev towers} ============== In this section we use a Sobolev tower construction to obtain existence and uniqueness results relating to the pure fragmentation system for a larger class of initial conditions. Sobolev towers appear to have been first applied to the discrete fragmentation system in [@smith2012discrete], where the authors examine a specific example and use Sobolev towers to explain an apparent non-uniqueness of solutions. As we demonstrate below, the theory of Sobolev towers is applicable to more general fragmentation systems and, in the following, the only restrictions that are imposed are that the fragmentation coefficients satisfy Assumption \[A1.1\], and also that a weight, $w=(w_n)_{n=1}^{\infty}$, has been chosen so that Assumption \[assumption for analyticity\] holds. These restrictions imply that $G^{(w)}=A^{(w)}+B^{(w)}$ is the generator of an analytic, substochastic $C_0$-semigroup, $(S^{(w)}(t))_{t \geq 0}$, on $\ell_w^1$. Let $\omega_0$ be the growth bound of $(S^{(w)}(t))_{t \ge 0}$. Choosing $\mu >\omega_0$, we rescale $(S^{(w)}(t))_{t \ge 0}$ to obtain an analytic semigroup, $(\mathscr{S}^{(w)}(t))_{t \ge 0}=(e^{-\mu t}S^{(w)}(t))_{t \ge 0}$, with a strictly negative growth bound. The generator of $(\mathscr{S}^{(w)}(t))_{t \ge 0}$ is $\mathcal{G}^{(w)}=G^{(w)}-\mu I$. We set $X^{(w)}_0=\ell_w^1$, ${\lVert\,\cdot\,\rVert_{0}} \coloneqq {\lVert\,\cdot\,\rVert_{w}}$, $\mathscr{S}^{(w)}_0(t)=\mathscr{S}^{(w)}(t)$, $S^{(w)}_0(t)=S^{(w)}(t)$, and $\mathcal{G}^{(w)}_0=\mathcal{G}^{(w)}$. As described in [@engel1999one §.5(a)], $(\mathscr{S}^{(w)}(t))_{t \ge 0}$ can be used to construct a Sobolev tower, $(X_n^{(w)})_{n \in \mathbb{N}}$, via $$X_n^{(w)} \coloneqq \bigl(\mathcal{D}\bigl((\mathcal{G}^{(w)})^n\bigr), {\lVert\,\cdot\,\rVert_{n}}\bigr); \qquad \Vert f \Vert_n = \big\Vert (\mathcal{G}^{(w)})^nf \big\Vert_{w}, \;\; f \in \mathcal{D}\bigl((\mathcal{G}^{(w)})^n\bigr), \quad n \in \mathbb{N}.$$ For each $n \in \mathbb{N}$, $X_n^{(w)}$ is referred to as the Sobolev space of order $n$ associated with the semigroup $(\mathscr{S}^{(w)}(t))_{t \ge 0}$. We also define the operator $\mathcal{G}^{(w)}_n: X^{(w)}_n \supseteq \mathcal{D}(\mathcal{G}^{(w)}_n) \to X^{(w)}_n$ to be the restriction of $\mathcal{G}^{(w)}$ to $$\mathcal{D}(\mathcal{G}^{(w)}_n) = \bigl\{f \in X^{(w)}_n: \mathcal{G}^{(w)}f \in X^{(w)}_n\bigr\} = \mathcal{D}\bigl((\mathcal{G}^{(w)})^{n+1}\bigr) = X^{(w)}_{n+1},$$ for each $n \in \mathbb{N}$. Sobolev spaces of negative order, $-n$, $n \in \mathbb{N}$, are defined recursively by $$\label{Sobolev tower of negative order} X^{(w)}_{-n} = \bigl(X^{(w)}_{-n+1}, {\lVert\,\cdot\,\rVert_{-n}}\bigr) \widetilde{\rule{0ex}{1.5ex}\rule{1.5ex}{0ex}}; \qquad \Vert f \Vert_{-n} = \big\Vert (\mathcal{G}^{(w)}_{-n+1})^{-1}f \big\Vert_{-n+1}, \quad f \in X^{(w)}_{-n+1},$$ where $(X,{\lVert\,\cdot\,\rVert})\widetilde{\rule{0ex}{1.2ex}\rule{1.5ex}{0ex}}$ denotes the completion of the normed vector space $(X,{\lVert\,\cdot\,\rVert})$. Operators $\mathcal{G}^{(w)}_{-n}$ can then be obtained in a similar recursive manner for each $n \in \mathbb{N}$, with $\mathcal{G}^{(w)}_{-n}$ defined as the unique extension of $\mathcal{G}_{-n+1}^{(w)}$ from $\mathcal{D}(\mathcal{G}^{(w)}_{-n+1})=X_{-n+2}^{(w)}$ to $\mathcal{D}(\mathcal{G}^{(w)}_{-n})=X^{(w)}_{-n+1}$; see [@engel1999one §.5(a)]. From [@engel1999one §.5(a)], it follows that $\mathcal{G}^{(w)}_n$ is the generator of an analytic, substochastic $C_0$-semigroup, $(\mathscr{S}_n^{(w)}(t))_{t \ge 0}$, on $X^{(w)}_n$ for all $n \in \mathbb{Z}$, where $\mathscr{S}_{-n}^{(w)}(t)$ is the unique, continuous extension of $\mathscr{S}^{(w)}(t)$ from $X_0^{(w)}$ to $X^{(w)}_{-n}$ for each $t \ge 0$ and $n \in \mathbb{N}$. Since $\mathscr{S}^{(w)}(t)=e^{-\mu t}S^{(w)}(t)$, we also obtain the analytic, substochastic $C_0$-semigroup, $(S_{-n}^{(w)}(t))_{t \ge 0}$, defined on $X^{(w)}_{-n}$ by $S^{(w)}_{-n}(t) = e^{\mu t}\mathscr{S}_{-n}^{(w)}(t)$. More generally, it is known that $\mathscr{S}^{(w)}_n(t)$ is the unique, continuous extension of $\mathscr{S}^{(w)}_m(t)$ from $X^{(w)}_m$ to $X^{(w)}_n$ when $m,n\in \mathbb{Z}$ with $m \ge n$. The analyticity of $(\mathscr{S}_n^{(w)}(t))_{t \ge 0}$ on $X^{(w)}_n$, also enables us to prove the following key result. \[semigroup is in all higher levels of tower\] Let $\mathring{u} \in X^{(w)}_n$ for some fixed $n \in \mathbb{Z}$. Then $\mathscr{S}^{(w)}_n(t)\mathring{u} \in X^{(w)}_m$ for all $m \ge n$ and $t>0$. It is obvious that $\mathscr{S}^{(w)}_n(t)\mathring{u} \in X^{(w)}_n$ for all $t \geq 0$ and $\mathring{u} \in X^{(w)}_n$. Also, if $\mathscr{S}^{(w)}_n(t)\mathring{u} \in X^{(w)}_m$ for some $m \ge n$ and all $t > 0$, then, on choosing $t_0 \in (0,t)$, we have $$\mathscr{S}^{(w)}_n(t)\mathring{u} = \mathscr{S}^{(w)}_m(t-t_0)\mathscr{S}^{(w)}_n(t_0)\mathring{u} \in \mathcal{D}(\mathcal{G}^{(w)}_{m}) = X_{m+1}^{(w)},$$ where we have used the fact that $\mathscr{S}^{(w)}_n(t)$ and $\mathscr{S}^{(w)}_m(t)$ coincide on $X_{m}^{(w)}$ together with the analyticity of $\mathscr{S}^{(w)}_m(t)$. The result then follows by induction. We can now prove the following result regarding the solvability of . \[semigroup from tower solves weighted ACP\] Let Assumptions \[A1.1\] and \[assumption for analyticity\] hold. Further, let $n \in \mathbb{N}$. Then the ACP has a unique, non-negative solution $u \in C^1((0,\infty), \ell_w^1) \cap C([0,\infty),X^{(w)}_{-n})$ for all $\mathring{u} \in (X^{(w)}_{-n})_+$. This solution is given by $u(t)=S^{(w)}_{-n}(t)\mathring{u},\ t \ge 0$. Let $\mathring{u} \in (X^{(w)}_{-n})_+$ and let $u(t)=S^{(w)}_{-n}(t)\mathring{u} = e^{\mu t}v(t)$, $t \ge 0$, where $v(t) = \mathscr{S}^{(w)}_{-n}(t)\mathring{u}$. Then, $v \in C^1((0,\infty), X^{(w)}_{-n})\cap C([0,\infty), X^{(w)}_{-n})$ is the unique classical solution of $$\label{G(-n) ACP} v'(t) = \mathcal{G}^{(w)}_{-n}v(t), \quad t>0; \qquad v(0) = \mathring{u}.$$ Also, from Lemma \[semigroup is in all higher levels of tower\], $\mathscr{S}^{(w)}_{-n}(t)\mathring{u} \in X_1^{(w)}=\mathcal{D}(\mathcal{G}^{(w)})$ for all $t > 0$. Since $(\mathscr{S}^{(w)}_{-n}(t))_{t \geq 0}$ coincides with $(\mathscr{S}^{(w)}(t))_{t \geq 0}$ on $\mathcal{D}(\mathcal{G}^{(w)})$, it follows that $$\mathscr{S}^{(w)}_{-n}(t)\mathring{u} = \mathscr{S}^{(w)}(t-t_0)\mathscr{S}^{(w)}_{-n}(t_0)\mathring{u}, \qquad \text{where} \ t_0 \in (0,t).$$ Consequently, $$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\bigl(\mathscr{S}^{(w)}_{-n}(t)\mathring{u}\bigr) = \mathcal{G}^{(w)}\mathscr{S}^{(w)}(t-t_0)\mathscr{S}^{(w)}_{-n}(t_0)\mathring{u} = \mathcal{G}^{(w)}\mathscr{S}^{(w)}_{-n}(t)\mathring{u}, \qquad t>0,$$ where the derivative is with respect to the norm on $X^{(w)}_{0} = \ell_w^1$. This establishes that $u \in C^1((0,\infty), \ell_w^1) \cap C([0,\infty),X^{(w)}_{-n})$ and also that $u$ satisfies . The non-negativity of $u$ follows from the substochasticity of the semigroups. For uniqueness, we observe first that the construction of the Sobolev tower ensures that $X_0^{(w)}$ is continuously embedded in $X_{-n}^{(w)}$. Moreover, $\mathcal{G}^{(w)}$ is the restriction of $\mathcal{G}^{(w)}_{-n}$ to $X^{(w)}_1=\mathcal{D}(\mathcal{G}^{(w)})$. Consequently, if $u_1, u_2 \in C^1((0,\infty), \ell_w^1) \cap C([0,\infty),X^{(w)}_{-n})$ both satisfy , and we set $v_i(t) = e^{-\mu t}u_i(t)$, $i =1,2$, then the difference $v_1-v_2$ is the unique classical solution of with $\mathring{u} = 0$, and so $v_1= v_2$, from which it follows that $u_1=u_2$. Finally, we make the following remark on the solvability of . \[remark about general IC in X with Sobolev tower\] For fixed $n \in \mathbb{N}$, the previous theorem establishes that the ACP has a unique, non-negative solution $u \in C^1((0,\infty), \ell_w^1) \cap C([0,\infty),X^{(w)}_{-n})$, given by $u(t)=S^{(w)}_{-n}(t)\mathring{u}$, for all $\mathring{u} \in (X^{(w)}_{-n})_+$, provided that Assumptions \[A1.1\] and \[assumption for analyticity\] are satisfied. 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--- abstract: 'Quark-lepton compositeness is a well-known beyond the Standard Model (SM) scenario with heavy exotic particles like leptoquarks (LQs) and leptogluons (LGs) etc. These particles can couple to leptons and jets simultaneously. In this letter, we use the recent CMS scalar LQ search data in the $eejj$ and $eej$ channels to probe this scenario. We recast the data in terms of a color octet partner of the SM electron (or a first generation spin-1/2 LG) that couples to an electron and a gluon via a dimension five operator suppressed by the quark-lepton compositeness scale ($\Lm$). By combining different production processes of the color octet electron ($e_8$) at the LHC, we use the CMS 8TeV data to obtain a simultaneous bound on $\Lm$ and the mass of the $e_8$ ($M_{e_8}$). We also study the reach of the 13 TeV LHC to discover the $e_8$ and interpret the required luminosity in terms of $M_{e_8}$ and $\Lm$.' author: - Tanumoy Mandal - Subhadip Mitra - Satyajit Seth title: | [\ ]{}Probing Compositeness with the CMS $eejj$ & $eej$ Data --- Introduction {#sec:intro} ============ The idea of quark-lepton compositeness [@Pati:1974yy; @Terazawa:1976xx; @Neeman:1979wp; @Harari:1979gi; @Shupe:1979fv; @Terazawa:1979pj; @Harari:1980ez; @Fritzsch:1981zh] goes along with our intention to describe nature in terms of its most fundamental building blocks. As its name suggests, in the models with quark-lepton compositeness, the Standard Model (SM) fermions are not elementary but rather have finer substructures. Similarities between the SM lepton and quark sectors (like, both come with three flavors and behave similarly under the $SU(2)_{\rm L}\times U(1)_{\rm Y}$ gauge symmetry with the same weak coupling) can be explained if they are assumed to be different bound states of some fundamental constituents. These fundamental constituents, called preons by Pati and Salam [@Pati:1974yy], are charged under some new strong force which confines them below a certain scale $\Lm$, known as the compositeness scale. As we have hadrons in QCD, in this scenario one expects a host of new exited preonic-condensates. Some of these condensates would be quite exotic, as they would carry both $SU(3)_{\rm c}$ color charges and lepton numbers, like the bosonic leptoquarks (LQs or ${\ell_q}$’s) that transform as triplets under $SU(3)_{\rm c}$ [@Buchmuller:1986zs; @Hewett:1997ce; @Kramer:1997hh] or the leptogluons (LGs or ${\ell_8}$’s) that are color-octet fermions [@Harari:1985cr; @Baur:1985ud; @Nir:1985ah; @Rizzo:1985dn; @Rizzo:1985ud; @Streng:1986my] etc. Because of their color charges, if these exotic condensates have TeV-range masses, they would be produced copiously at the Large Hadron Collider (LHC) making it possible to probe this scenario experimentally. The LHC has already put some constraints on the masses of scalar LQs decaying to SM quarks and leptons [@Aad:2015caa; @Khachatryan:2015vaa; @Khachatryan:2015bsa; @Khachatryan:2015qda]. Of these, we look at the most recent search by CMS, for the first and second generations of scalar LQs in the $\ell\ell jj$ and the $\ell\n_\ell jj$ channels with 19.7 fb$^{-1}$ of integrated luminosity at the 8 TeV LHC [@Khachatryan:2015vaa]. With pair production, the 95% confidence level (CL) exclusion limit on the mass of the first (second) generation scalar LQ now stands at $M_{{\ell_q}} = 1005$ (1080) GeV assuming it always decays to an electron (a muon) and a jet. Note that unless specified otherwise, we do not distinguish between any particle and its anti-particle. Hence, an electron here could mean a positron as well. In the first generation search, mild excesses of events compared to the SM background were observed in both the $eejj$ and the $eej$ channels for $M_{{\ell_q}}\sim$ 650 GeV. Currently, these excesses have attracted considerable attention in the literature. CMS has also performed a dedicated search for the single productions of the first two generations of LQs in the $\ell\ell j$ channels [@Khachatryan:2015qda]. However, unlike the mostly QCD mediated pair production, the single productions depend strongly on an unknown coupling $\lm$, the ${\ell_q}$-$\ell$-$q$ coupling. Hence, the exclusion limits from this search are $\lm$ dependent. For the first generation, the exclusion limit goes from 895 GeV to 1730 GeV when $\lm$ goes from 0.4 to 1.0 and for the second generation the data exclude $M_{{\ell_q}}$ below 530 GeV for $\lm=1.0$. In this letter, we recast the CMS 8 TeV $eejj$ [@Khachatryan:2015vaa] and $eej$ [@Khachatryan:2015qda] data in terms of the first generation spin-1/2 LG carrying unit electric charge, [*i.e.*]{}, the color octet partner of the SM electron ($e_8$) to probe the composite quark-lepton scenarios and obtain the most stringent limits available on the $e_8$. This is possible because a LG can also decay to a lepton and a jet (gluon) just like a LQ. Hence, the pair production of $e_8$’s would have $eejj$ final states.[^1] Earlier, there have been other phenomenological studies on LGs [@Celikel:1998dj; @Sahin:2010dd; @Akay:2010sw; @Jelinski:2015epa; @Acar:2015wxp] and the CMS 7 TeV $eejj$ data [@Chatrchyan:2012vza] were used to infer bounds on $M_{e_8}$ [@Goncalves-Netto:2013nla; @Mandal:2012rx]. Considering the pair production, Ref. [@Goncalves-Netto:2013nla] put the mass exclusion limit at about 1.2-1.3 TeV. Similarly, an $e_8$ could be produced singly in association with an electron and give rise to an $eej$ final state. Interestingly, the single productions of LGs open up a way to probe the compositeness scale. This is because, at the leading order (LO), the ${\ell_8}$-$\ell$-$g$ interaction comes from an effective operator of dimension five that is suppressed by the compositeness scale $\Lm$ [@Agashe:2014kda; @Mandal:2012rx] (see the next section). This is unlike the LQ interactions, where the LO terms are of dimension four and hence, apparently insensitive to $\Lm$. In a recent paper [@Mandal:2015vfa], we pointed out that the single productions of LQs can also lead to the $eejj$ final state and similarly, events from the pair productions could also pass the signal selection criteria of the single production search in the $eej$ channel. Combining these production processes in the signal simulations can provide better limits in the $M_{{\ell_q}}$-$\lm$ plane from both the $eejj$ and the $eej$ channels. The same argument applies for LGs too. Hence, following Ref. [@Mandal:2015vfa], here we systematically combine both the pair and the single production processes of the $e_8$ while reinterpreting the CMS $eejj$ and $eej$ data and obtain exclusion limits in the $M_{e_8}$-$\Lm$ plane. This way, we obtain the mass exclusion limits as well as the limits on the compositeness scale from both the $eejj$ and the $eej$ data and compare them. Our presentation is organized as follows. In the next section we discuss the details of the signal we consider, in section \[sec:three\], we present the results of our recast analysis, in section \[sec:futpros\] we investigate the prospect of discovering the color octet electron at the 13 TeV LHC and then in section \[sec:last\] we conclude. Leptogluon (Combined) Signals {#sec:two} ============================= If we assume $M_{e_8}$ is smaller than $\Lm$ and there is no violation of lepton flavor, we can write a generic effective Lagrangian for the $e_8$ allowed by the SM gauge symmetry as [@Mandal:2012rx], = |e\_8\^a i\^(\_\^[ac]{} + g\_s f\^[abc]{}G\^b\_) e\_8\^c - M\_[e\_8]{}|[e]{}\_8\^a e\_8\^a + \_[int]{},\[eq:lag\] with [@Agashe:2014kda], \_[int]{} = G\^a\_ + + ….\[eq:intlag\] In the Lagrangian, we have displayed only those dimension five terms that are important for our study.[^2] Here, $G^a_{\mu\nu}$ is the gluon field strength tensor, and $\et_{L/R}$ are the chirality factors. Since, the electron chirality conservation implies $\et_L\et_R = 0$, we set $\et_L = 1$ and $\et_R=0$ in our analysis without any loss of generality. This dimension five interaction opens two decay modes for the color octet electron: $e_8\to eg$ and $e_8\to egg$. However, since the three body decay is more suppressed than the two body one, we simply set the total width of the $e_8$ as [@Mandal:2012rx; @Celikel:1998dj], . The production processes of the $e_8$ at the LHC (see Fig. \[fig:fyndiags\] for some representative Feynman diagrams) are discussed in much detail in Ref. [@Mandal:2012rx]. Instead, here we focus on some essential points. The main contribution to the $e_8$ pair production comes from the purely QCD mediated diagrams (see [*e.g.*]{} Fig. \[fig:feyn\_pair\_1\]). At the LO, there is an additional $t$-channel electron exchange diagram whose amplitude is proportional to $1/\Lm^2$ (Fig. \[fig:feyn\_pair\_e\_ex\]) but, for the ranges of $M_{e_8}$ and $\Lm$ we consider in this letter, its contribution is small compared to the model independent QCD mediated contribution. That is why the pair production process is practically insensitive to the compositeness scale. On the other hand, all the single production diagrams contain at least one $e_8$-$e$-$g$ or $e_8$-$e$-$g$-$g$ vertex ($\sim 1/\Lm$) coming from the interaction term of Eq. (see [*e.g.*]{} Figs. \[fig:feyn\_sin\_2bd\] & \[fig:feyn\_sin\_3bd\]). We simulate the pair and the single productions of $e_8$ at the 8 TeV LHC to estimate their contributions to the $eejj$ and the $eej$ channels by modeling Eqs.  and in <span style="font-variant:small-caps;">Feynrules</span> [@Alloul:2013bka]. We use the CTEQ6L1 Parton Distribution Functions (PDFs) [@Pumplin:2002vw] to generate events with <span style="font-variant:small-caps;">MadGraph5</span> [@Alwall:2014hca] and then shower them with <span style="font-variant:small-caps;">Pythia</span>6 [@Sjostrand:2006za]. We set the factorization and the renormalization scales, $\mu_{\rm F}$ $=\mu_{\rm R}$ $=M_{e_8}$. We use [Delphes 3.3.1]{} [@deFavereau:2013fsa] to simulate the CMS detector environment and implement the selection cuts. In [Delphes]{}, jets are clustered with [FastJet]{} [@Cacciari:2011ma] using the anti-$k_{\rm T}$ jet clustering algorithm [@Cacciari:2008gp] with the clustering parameter, $R=0.4$. Since, we generate the pair and the single productions separately, any possible interference between them has been ignored. However, this is justified as, for the parameters considered, the $e_8$ decay width is much smaller than its mass ([*i.e.*]{}, narrow width regime). We generate events for the inclusive single production for certain $\Lm=\Lm_{\rm o}$ by combining the following processes, \_[=\_[o]{}]{}\[eq:matching\_ee\] where the curved connections indicate a pair of electron and gluon coming from an on-shell $e_8$. However, a straightforward computation of cross section for the combined single and pair production processes would lead to some difficulties. Like, the jets that are not coming from a LG could be soft and lead to divergences. Ideally, to handle these divergences, one has to go beyond a tree level computation while combining the different single production processes as in Eq. . Moreover, such combination can lead to double counting of some diagrams while showering. Following Ref. [@Mandal:2015vfa], we avoid these difficulties by employing the matrix element-parton shower matching (ME$\oplus$PS) technique with the shower-$k_{\rm T}$ scheme [@Alwall:2007fs; @Alwall:2008qv] which effectively provides a consistent interpolation between the hard partons and the [Pythia]{} parton showers (PS). It relies on the [Pythia]{} PS for the soft jets and the parton level matrix elements for the hard jets and thereby, bypasses the double counting and the soft jets problems. The cross section for any other value of $\Lm=\Lm_{\rm n}$ (say) is obtained by simply multiplying the cross section for $\Lm_{\rm o}$ by $\Lm_{\rm o}^2/\Lm_{\rm n}^2$, since, as explained earlier, the $\Lm$ dependence of the inclusive single production cross section ($\sg^s$) can be written as, \_[s]{}(M\_[e\_8]{},) |\_s(M\_[e\_8]{}),\[eq:sigmabar\] if we ignore terms of $\mc O\left(1/\Lm^4\right)$ or higher. In Table \[tab:xsec\], we show $\sg_{\rm s}( M_{e_8},\Lm)$ for four difference choices of $M_{e_8}=$ 0.5, 1.0, 1.5 & 2.0 TeV and two different choices of $\Lm=$ 2.5 & 5 TeV. There, we also show the LO values of the pair production cross-section ($\sg_{\rm p}^{\rm LO}$) for the four masses. While combining the pair and the single productions, we use the next-to-leading (NLO) in QCD $K$-factors only for the pair production, available from Ref. [@Goncalves-Netto:2013nla] for masses up to 1.5 TeV. Beyond this, guided by the trend, we assume a constant $K_{\rm NLO}=$ 2.0. [^3] Note, however, no $K$-factor is available for the single productions. Hence, for a particular $\Lm$, we utilize the available information to the best possible manner and use the combined signal with the following cross section, && \_[p]{}(M\_[e\_8]{}) + |\_[s]{}(M\_[e\_8]{})\ && = K\_[NLO]{}(M\_[e\_8]{})\_[p]{}\^[LO]{}(M\_[e\_8]{}) + \_[s]{}(M\_[e\_8]{},).\[eq:comxsec\] [|&gt;m[0.24]{} |&gt;m[0.24]{} |&gt;m[0.24]{} c|]{} $M_{e_8}$ & $\sg_{\rm p}^{\rm LO}$ (fb) &\ (TeV) & ($\Lm\to\infty$) & $\Lm=$ 2.5 TeV & $\Lm=$ 5.0 TeV\ 0.5 & 3.85$\times 10^{3}$ & 4.85$\times 10^{3}$ & 1.22$\times 10^{3}$\ 1.0 & 1.77$\times 10^{1}$ & 2.66$\times 10^{1}$ & 6.60$\times 10^{0}$\ 1.5 &   2.36$\times 10^{-1}$ & 3.03$\times 10^{0}$ &   7.62$\times 10^{-1}$\ 2.0 &   3.22$\times 10^{-3}$ &   4.41$\times 10^{-1}$ &   1.09$\times 10^{-1}$\ Recast Analysis and New Limits {#sec:three} ============================== In Fig. \[fig:eejj\_data\], we show the recast mass exclusion plots obtained from the CMS $eejj$ [@Khachatryan:2015vaa] data for three different values of $\Lm$, namely, $\Lm\to\infty$ ([*i.e.*]{} the pair production only) in Fig. \[fig:eejj\_data\_a\], $\Lm=5$ TeV in Fig. \[fig:eejj\_data\_b\] and $\Lm=2.5$ TeV in Fig. \[fig:eejj\_data\_c\]. To obtain the expected and the observed 95% CL upper limits (ULs) for the recast plot, we rescale the corresponding limits from the CMS plot [@Khachatryan:2015vaa] by multiplying with a factor [@Mandal:2015vfa], R\^[eejj]{}\_[[\_q]{}e\_8]{}(M\_[e\_8]{},) = ,\[eq:efficiency\_rescale\_eejj\] where $\ep_{\rm p}^{({\ell_q}|eejj)}\lt(M_{e_8}\rt)$ is the efficiency (yield) of the final event selection cuts optimized for the pair production of the first generation scalar LQ of mass $M_{{\ell_q}}=M_{e_8}$ [@Khachatryan:2015vaa] and $\ep_{\rm p+s}^{(e_8|eejj)}\lt(M_{e_8},\Lm\rt)$ is the efficiency of the same set of cuts estimated for the combined (pair+single, combined as in Eq. ) productions of $e_8$’s. In other words, $\ep_{\rm p+s}^{(e_8|eejj)}\lt(M_{e_8},\Lm\rt)$ denotes the fraction of the combined signal events that survives the selection cuts optimized for $M_{{\ell_q}}=M_{e_8}$. Since, the CMS $eejj$ optimized cuts stop at $M_{{\ell_q}}=1.2$ TeV, we extrapolate beyond this mass by assuming identical selection cuts for $M_{{\ell_q}}\geq 1.2$ TeV. Because of the single productions, the lower limit of the allowed mass increases with decreasing $\Lm$. For example, from the pure QCD mediated pair production ($\Lm\to\infty$) the limit stands at about 1.56 TeV and it improves to about 1.66 (1.90) TeV for $\Lm=5~(2.5)$ TeV.[^4] Note that with increasing mass, the pair production becomes more phase space suppressed compared to the single productions and hence, beyond a certain mass, the single productions dominate over the pair production. The crossover point depends on $\Lm$, since all the single productions depend on it. With this in mind, we can now understand the behavior shown by the 95% CL UL lines in the high $M_{e_8}$ limit for finite $\Lm$’s. We expect the single productions to take over the pair production earlier when $\Lm=2.5$ TeV than when $\Lm=5$ TeV. This can be seen from Figs. \[fig:eejj\_data\_b\] & \[fig:eejj\_data\_c\]: the small raise in any UL line with increasing $M_{e_8}$ (that it is indeed coming from the single productions can be confirmed from its absence in the pair only plot) comes earlier for $\Lm=2.5$ TeV than $\Lm=5$ TeV. [^5] In Fig. \[fig:eej\_data\], the recast plots for $\Lm=2.5$ and 5 TeV obtained from the CMS $eej$ [@Khachatryan:2015vaa] data are shown. For Fig. \[fig:eej\_data\_a\], we have considered only the single productions in the signal to compare the mass exclusion limits for the two values of $\Lm$ while in Figs. \[fig:eej\_data\_b\] & \[fig:eej\_data\_c\], we consider the combined productions. Here, we rescale the CMS limits [@Khachatryan:2015vaa] by \^[eej]{}\_[[\_q]{}e\_8]{}(M\_[e\_8]{}) = \[eq:efficiency\_rescale\_eej\_s\] for the single only plot (Fig. \[fig:eej\_data\_a\]) and by R\^[eej]{}\_[[\_q]{}e\_8]{}(M\_[e\_8]{},) = \[eq:efficiency\_rescale\_eej\] for the other two (Figs. \[fig:eej\_data\_b\] & \[fig:eej\_data\_c\]). Here, $\ep_{\rm s}^{({\ell_q}|eej)}\lt(M_{e_8}\rt)$ is the efficiency of the final event selection cuts optimized for the single productions of the first generation scalar LQ of mass $M_{{\ell_q}}=M_{e_8}$ [@Khachatryan:2015vaa]. Notice that though the single productions of the LQ depend on the unknown ${\ell_q}$-$\ell$-$q$ coupling $\lm$, the efficiency $\ep_{\rm s}^{({\ell_q}|eej)}$, being a ratio of the number of events, does not depend on any overall factor in the cross section like $\lm$ [@Mandal:2015vfa]. For the same argument $\ep_{\rm s}^{(e_8|eej)}$, which is the cut efficiency for the inclusive single production of the $e_8$, does not depend on $\Lm$ even though $\ep_{\rm p+s}^{(e_8|eej)}$ does. If we compare Fig. \[fig:eej\_data\_a\] with Figs. \[fig:eej\_data\_b\] & \[fig:eej\_data\_c\], it is clear how the inclusion of the pair production in the signal for the $eej$ search improves the mass exclusion limits. For example, for $\Lm=5 (2.5)$ TeV the $eej$ data disfavor $M_{e_8}$ below 1.28 (1.84) TeV when only the single productions are considered. But the same limit goes up to about 1.62 (1.86) TeV when the pair production is also included. Obviously, the improvement is more prominent when the single productions are relatively smaller because of larger $\Lm$. In Fig. \[fig:lm-me8-limit\], we show the rescaled 95% CL exclusion limits in the $M_{e_8}$-$\Lm$ plane. The blue shaded regions are disfavored by the data. We show the exclusion contours obtained from the CMS $eejj$ data (Fig. \[fig:lm-me8-limit\_a\]) and the $eej$ data (Fig. \[fig:lm-me8-limit\_b\]). We compare these two in Fig. \[fig:lm-me8-limit\_c\]. The pair production dominates in the lower mass region and gives a limit on $M_{e_8}$ that is practically independent of $\Lm$. From Fig. \[fig:lm-me8-limit\_a\] or \[fig:lm-me8-limit\_c\], it is clear that irrespective of $\Lm$, the $eejj$ data disfavor the $e_8$ with mass below $\sim 1.55$ TeV. In the high mass region, the pair production becomes negligible and the inclusive single production puts a strong limit on $\Lm$. However, what is remarkable is that the $eejj$ data give almost identical limit as the $eej$ data in this regime. In other words, in the high mass limit, the contamination of single production in a search optimized for pair production is very significant.[^6] As explained in the introduction, the $\Lm$-dependent mass exclusion limits can also be translated as limits on $\Lm$. The overlapping limits in Fig. \[fig:lm-me8-limit\_c\] indicate that the lightest limit on $\Lm$ stands about $\Lm\approx 2~\textrm{TeV}\approx M_{e_8}$ within the domain of the effective theory. If $M_{e_8}$ lies between 1.64 TeV and 2 TeV, $\Lm$ must be higher. Future prospects {#sec:futpros} ================ So far our discussions were centered on reinterpreting the available data. Now, let us look at the prospect of a discovery of the $e_8$ at the LHC in its 13 TeV runs. In this section, we assume a future search in the $eej$ channel optimized for finding the $e_8$ and estimate the discovery reach using the combined production. We expect two high-$p_{\rm T}$ electrons and at least one high-$p_{\rm T}$ jet as the typical signature of the combined production of $e_8$ [@Mandal:2012rx]. Therefore, taking a cue from the existing CMS $eej$ search [@Khachatryan:2015vaa], we use the following selection cuts: 1. two oppositely charged electrons ($e^{\pm}$) with transverse momentum $p_{\rm T}^e >$ 45 GeV and pseudorapidity $|\eta_e| <$ 2.1 excluding 1.442 $< |\eta_e| <$1.56, 2. the hardest jet must have $p_{\rm T}^{j_1} >$ 125 GeV & $|\eta_{j_1}| < $ 2.4, 3. separation between any electron and the hardest jet in the $\eta$-$\phi$ plane, $\Delta R_{ej_{1}} > $ 0.3. To suppress the inclusive-$Z$ background, we apply a strong cut on 4. the invariant mass of the electron pair, $M_{e_1e_2} >$ 400 GeV. In addition, we also apply some cuts optimized for the different $(M_{e_8},\Lm)$ combinations, 5. the scalar sum of the $p_{\rm T}$ of the two electrons and the hardest jet, S\_[T]{}=p\_[T]{}\^[e\_1]{}+p\_[T]{}\^[e\_2]{}+p\_[T]{}\^[j\_1]{} &gt; S\_[T]{}\^[opt]{}(M\_[e\_8]{},),\[eq:cut\_st\] 6. the maximum of the two electron-jet invariant mass combinations, M\_[ej]{}\^[max]{}=[Max(M\_[e\_1j\_1]{},M\_[e\_2j\_1]{})]{}&gt;M\_[ej]{}\^[opt]{}(M\_[e\_8]{},).\[eq:cut\_mej\] The values of $S_{\rm T}^{\rm opt}$ and $M_{ej}^{\rm opt}$ for some benchmark parameters are shown in Table \[tab:cut\]. The strong cut on $M_{e_1e_2}$ suppresses the inclusive $Z$ ($+n$ jets) contribution which is the most dominant background. The other significant backgrounds are the inclusive top-pair production, the inclusive diboson ($ZZ$, $ZW$, $WW$) productions etc. [@Mandal:2012rx]. [|&gt;m[0.1]{}|&gt;m[0.135]{}&gt;m[0.135]{}|&gt;m[0.135]{}&gt;m[0.135]{}|&gt;m[0.135]{} c|]{} $M_{e_8}$ & & &\ & $S_{\rm T}^{\rm opt}$ & $M_{ej}^{\rm max}$ & $S_{\rm T}^{\rm opt}$ & $M_{ej}^{\rm max}$ & $S_{\rm T}^{\rm opt}$ & $M_{ej}^{\rm max}$\ 1.5 & 0.7 & 0.5 & 0.7 & 0.5 & 0.5 & 0.5\ 2.0 & 1.4 & 0.6& 1.2 & 1.0 & 1.1 & 0.8\ 2.5 & 1.8 & 1.8 & 1.7 & 1.7 & 1.5 & 1.2\ 3.0 & 1.8 & 1.8 & 1.8 & 1.8 & 1.8 & 1.8\ [|&gt;m[0.1]{}|&gt;m[0.135]{}&gt;m[0.135]{}|&gt;m[0.135]{}&gt;m[0.135]{}|&gt;m[0.135]{} c|]{} $M_{e_8}$ & & &\ (TeV) & Sig.& Backg.& Sig.& Backg. & Sig.& Backg.\ & (fb)&(fb)&(fb)&(fb)&(fb)&(fb)\ 1.5 & 9.385 & 2.764 & 10.253 & 2.764 & 13.035 & 3.786\ 2.0 & 0.569 & 0.222 &   0.771 & 0.282 &   1.435 & 0.517\ 2.5 & 0.039 & 0.025 &   0.086 & 0.034 &   0.263 & 0.105\ 3.0 & 0.003 & 0.025 &   0.018 & 0.025 &   0.065 & 0.025\ ![The estimated 100 & 300 fb$^{-1}$ contours of the discovery luminosity $\mc L_{\rm D}$ (Eq. ) for the $e_8$ (first generation spin-1/2 leptogluon) at the 13 TeV LHC. The shaded region corresponds to $M_{e_8}>\Lm$ where our effective theory approach (Eqs.  & ) is not reliable.[]{data-label="fig:LD"}](LM_Mlg) ![The $\et$ distribution of the second hardest electron can be used to distinguish spin-0 LQs from spin-1/2 LGs. Here we set $M_{{\ell_q}}=M_{e_8}=$ 2 TeV and $\lm=$ 0.3 (for the LQ signal) and $\Lm=$ 10 TeV (for the LG signal) at the 13 TeV LHC.[]{data-label="fig:rapdist"}](LGvsLQ_dist) To figure out the optimized values of the $S_{\rm T}$ and $M_{ej}$ cuts, [*i.e.*]{}, $S_{\rm T}^{\rm opt}$ and $M_{ej}^{\rm opt}$, we scan a square grid in the $S_{\rm T}$-$M_{ej}^{\rm max}$ plane defined between 0.5 TeV & 1.8 TeV in steps of 0.1 TeV in both directions. For every point in this grid, we compute the combined signal and the background events to find the combination for which the required luminosity for a discovery ($\mc{L}_{\rm D}$) minimizes. We define $\mc{L}_{\rm D}$ as, \_[D]{} = [Max]{}(\_5,\_[10]{}).\[eq:ld\] Here, $\mc{L}_5$ is the luminosity required to attain a $5\sg$ statistical significance for $\lt({\rm Sig.}/\sqrt{\rm Backg.}\rt)$ and $\mc{L}_{10}$ is the luminosity required to observe 10 signal events. In Table \[tab:cut\_eff\], we display the ‘after-cut’ cross sections of the combined signal and the dominant $Z+nj$ background (including the contributions from $ZV$) for the benchmark points of Table \[tab:cut\]. Though we show only the dominant background in the table, we include other sub-dominant contributions [@Mandal:2012rx] (like inclusive top-pair etc.) while estimating $\mc L_{\rm D}$. We show two $\mc{L}_{D}$ contours estimated for the 13 TeV LHC in Fig. \[fig:LD\]. To obtain this, we use constant $K_{\textrm{NLO}}=2$ for $M_{e_8}$ beyond 1.5 TeV like the recast analysis in section \[sec:three\]. With 300 fb$^{-1}$ of integrated luminosity, the mass reach goes from about 2.5 TeV to about 3.5 TeV as $\Lm$ decreases to about 3.5 TeV ($\Lm\approx M_{e_8}$) from very large values. Obviously, this increase in reach with decreasing $\Lm$ happens because of the single productions whose cross sections go like $1/\Lm^2$. Before closing this section we make a note. Even though we have reinterpreted the CMS LQ data in terms of the $e_8$, it is also possible to separate them at the LHC. Let us suppose, a significant excess is found in the $eej$ data in future. In Fig. \[fig:rapdist\], we show the $\et$ distribution of the second hardest electron (as an example), which can be used to distinguish a spin-0 LQ from a spin-1/2 LG. Obviously, there are other possibilities as well. However, we do not pursue this issue further in this letter. Discussions and Conclusions {#sec:last} =========================== The quark-lepton compositeness scenario is one of the well-known BSM scenarios which can accommodate LQs. In this letter, we have used the CMS first generation scalar LQ data in the $eejj$ and the $eej$ channels to probe this scenario. In these models, there exist other exotic composite particles that can also decay to lepton-jet final states. We have recast the CMS data in terms of such a particle, the color octet partner of the SM electron. An $e_8$ decays to an electron and a gluon via a dimension five interaction, suppressed by the compositeness scale. This opens up the possibility of probing the compositeness scale with the $eejj$ and the $eej$ data. In a recent paper [@Mandal:2015vfa], we argued that at the LHC, a search for the pair production of a colored particle (generally, model independent) can get ‘contaminated’ from the model dependent single productions and [*vice versa*]{}. There, we used the examples of the CMS LQ searches to demonstrate how the pair and the single productions can be combined systematically in the signal simulations. As a result, even a search for the pair production can give information on the model parameters that control the single productions. In this letter too, we have adopted the same strategy, [*i.e.*]{}, we have recast both the $eejj$ and the $eej$ data with signals that are combinations of the pair and the single productions for different values of $\Lm$, the compositeness scale that controls the single productions. Hence, the analysis in this letter stands as yet another demonstration of our arguments in Ref. [@Mandal:2015vfa]. From the combined signal, we extract the exclusion limits in the $M_{e_8}$-$\Lm$ plane. The limits obtained by our analysis are not very precise as they are obtained by simple rescaling instead of a full statistical analysis. However, one can conclude that the $eejj$ data disfavor $e_8$’s with mass below $\sim 1.5$ TeV for any value of $\Lm$.[^7] Beyond this mass range, the limit becomes a function of $\Lm$. As the mass increases, the single productions dominate the combined signals in both $eejj$ and $eej$ channels giving almost overlapping limits that can also be interpreted as the limits on $\Lm$. Data in both channels indicate that $\Lm\gtrsim 2$ TeV for 1.5 TeV $\lesssim M_{e_8}\lesssim$ 2 TeV. Beyond this mass range, where the exclusion limits enter in the region with $M_{e_8} > \Lm$, our effective theory approach becomes unreliable. We clearly mark this region in all the relevant plots. This is an inherent limitation present in any effective theory approach. It might also happen that, in nature, the $e_8$ is actually heavier than the compositeness scale. In that case, all our limits/predictions would not be reliable except in the parameter region dominated by the (QCD mediated) pair production. For example, let us suppose that, in nature, $\Lm$ is actually smaller than 1.5 TeV, the mass range disfavored by the pair production data. In that case, we will still be able to say that the $e_8$ can not exist below 1.5 TeV but we would not be able to conclude anything definitively about $\Lm$ from our analysis. Notice that there are other higher dimensional operators (like $\mc O_{ggee}$ or $\mc O_{qqee}$ for contact interactions) that, in principle, could also connect $\Lm$ with the $eejj/eej$ data irrespective of the values of $M_{e_8}$. However, two points go against them – the first, the signal selection criteria are not designed to favor them, and the second, these operators are of dimensions higher than five (so unless $\Lm$ is very small, in which case the whole effective theory approach might break down, these terms are expected to be highly suppressed). Hence, despite the inherent limitation, our approach gives the best available limits on $\Lm$ and $M_{e_8}$ from the CMS 8TeV $eejj$ and $eej$ data within the domain of validity of the effective theory (compare the limit on $M_{e_8}$ with the limit quoted in the Particle Data Book [@Agashe:2014kda], $M_{e_8}>$ 86 GeV from old Tevatron data [@Abe:1989es]). Finally, we note that one can also analyze the second generation $\m\m jj/\m\m j$ data in terms of color octet muon. However, it will be a very similar exercise and we do not expect that it will provide very different limits on $\Lm$ than what we have obtained. In case of the LQ, production of the second generation is reduced compared to the first generation because of the relative suppression of the second generation quark PDFs. However, since the LG productions at the LHC are mainly gluon mediated, they remain roughly the same for any generation. 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[^1]: In absence of any BSM decay, the only two body decay a LG can have is either ${\ell_8}\to\ell\ g$ or $\n_8\to\n_\ell\ g$ ($\n_8$, color octet partner of a neutrino) but not both. Hence, unlike LQs, the QCD mediated pair production of LGs can not have a $\ell\n_{\ell}jj$ final state. However, depending on the underlying model, a charged $\ell_8$ and a neutral $\n_8$ might couple simultaneously with a SM $W$ boson allowing a weak interaction mediated process, pp(W\_8\_8) j \_ j ,with the $\ell\n_{\ell} jj$ final state. [^2]: As pointed out in Ref. [@Mandal:2012rx], there are more dimension five operators allowed by the SM gauge symmetries and lepton number conservation like, i f\^[abc]{} |[e]{}\_8\^a G\_\^[b]{} \^ e\_8\^c + |[e]{}\_8\^a B\_ \^ e\_8\^a .However, these terms lead to $e_8e_8V$ or $e_8e_8VV$ vertices (may contain form factors) that would affect the production cross section. For simplicity, we assume the unknown coefficients associated with these terms are negligible.\[fn:footnote1\] [^3]: As it is clear from Table \[tab:xsec\], $\sg_{\rm p}$ is too small for $M_{e_8} \gtrsim$ 1.5. Hence, in practice, this assumption matters little, though we make it for consistency. [^4]: Production processes for LGs generally have enhanced color factors than LQ production processes (color octet LGs vs. color triplet LQs). As a result, from the same data one generally obtains higher mass exclusion limits for LGs than LQs for similar choice of parameters. [^5]: It is not very straight forward to understand the reason behind the raise itself intuitively. When these selection cuts [@Khachatryan:2015vaa] are held fixed, both the efficiencies start to increase with increasing $M_{e_8}$ till they saturate. However, since they evolve differently, there is a competition between the numerator and the denominator of Eq. . [^6]: Since the pair production search is insensitive to the spin of the particle being probed, kinematically it does not matter much whether the search is for LQs or LGs, at least in the narrow widths regime. [^7]: If additional sources to the LG pair production (like the higher dimensional operators in footnote \[fn:footnote1\] or the LO electroweak gauge mediated pair production etc.) are considered, this limit would receive corrections and could acquire some $\Lm$-dependence even. However, it is normal to expect these corrections to be smaller than the QCD mediated LO pair production.

--- abstract: | On 26 May 1999, one of the Sloan Digital Sky Survey (SDSS) fiber–fed spectrographs saw astronomical first light. This was followed by the first spectroscopic commissioning run during the dark period of June 1999. We present here the first hour of extra–galactic spectroscopy taken during these early commissioning stages: an observation of the Coma cluster of galaxies. Our data samples the Southern part of this cluster, out to a radius of 1.5 degrees ($1.8\,h^{-1}$ Mpc, approximately to the virial radius) and thus fully covers the NGC 4839 group. We outline in this paper the main characteristics of the SDSS spectroscopic systems and provide redshifts and spectral classifications for 196 Coma galaxies, of which 45 redshifts are new. For the 151 galaxies in common with the literature, we find excellent agreement between our redshift determinations and the published values, [*e.g.*]{}, for the largest homogeneous sample of galaxies in common (63 galaxies observed by Colless & Dunn 1996) we find a mean offset of 3 ${\rm km\,s^{-1}}$ and an RMS scatter of only 24 ${\rm km\,s^{-1}}$. As part of our analysis, we have investigated four different spectral classification algorithms: measurements of the spectral line strengths, a principal component decomposition, a wavelet analysis and the fitting of spectral synthesis models to the data. We find that these classification schemes are in broad agreement and can provide physical insight into the evolutionary histories of our cluster galaxies. We find that a significant fraction (25%) of our observed Coma galaxies show signs of recent star–formation activity and that the velocity dispersion of these active galaxies (emission–line and post–starburst galaxies) is 30% larger than the absorption–line galaxies. We also find no active galaxies within the central (projected) $200\,h^{-1}$ Kpc of the cluster. The spatial distribution of our Coma active galaxies is consistent with that found at higher redshift for the CNOC1 cluster survey. Beyond the core region, the fraction of bright active galaxies appears to rise slowly out to the virial radius and are randomly distributed within the cluster with no apparent correlation with the potential merger or post-merger of the NGC 4839 group. We briefly discuss possible origins of this recent galaxy star-formation. author: - 'Francisco J. Castander, Robert C. Nichol, Aronne Merrelli, Scott Burles, Adrian Pope, Andrew J. Connolly, Alan Uomoto, James E. Gunn, John E. Anderson, James Annis, Neta A. Bahcall, William N. Boroski, Jon Brinkmann, Larry Carey, James H. Crocker, István Csabai, Mamoru Doi, Joshua A. Frieman, Masataka Fukugita, Scott D. Friedman, Eric J. Hilton, Robert B. Hindsley, Željko Ivezić, Steve Kent, Donald Q. Lamb, R. French Leger, Daniel C. Long, Jon Loveday, Robert H. Lupton, Harvey MacGillivray, Avery Meiksin, Jeffrey A. Munn, Matt Newcomb, Sadanori Okamura, Russell Owen, Jeffrey R. Pier, Constance M. Rockosi, David J. Schlegel, Donald P. Schneider, Walter Seigmund, Stephen Smee, Yehuda Snir, Larry Starkman, Chris Stoughton, Gyula P. Szokoly, Christopher Stubbs, Mark SubbaRao, Alex Szalay, Aniruddha R. Thakar, Christy Tremonti, Patrick Waddell, Brian Yanny and Donald G. York' title: 'The First Hour of Extra–galactic Data of the Sloan Digital Sky Survey Spectroscopic Commissioning: The Coma Cluster.' --- Introduction ============ The Coma cluster is the richest cluster of galaxies in our local universe and has thus attracted considerable attention over the last century (see reviews by Biviano 1998 & West 1998). In the optical, for example, Goldwin, Metcalfe & Peach (1983; hereafter GMP83) have published an extensive photometric study of the cluster providing accurate positions, colors, magnitudes and ellipticities for 6724 bright galaxies over 2.63 square degrees centered on Coma. Several authors have explored the fainter dwarf galaxy population of Coma (see Bernstein et al. 1995; Kashikawa et al. 1998; Adami et al. 1998 & 2000). The dynamics of the cluster have also been well studied. Kent & Gunn (1982) assembled approximately 300 optical redshifts from the literature to determine the cluster mass distribution. This initial work was extended by Colless & Dunn (1996; CD96), who collected 556 redshifts (based on new and literature redshifts), and Geller, Diaferio & Kurtz (1999) who have extended the dynamical study of Coma to large radii (10 degrees) and larger numbers (1693 redshifts) thus measuring the density profile of Coma well beyond the virial radius of the cluster. Hughes (1989) measured the total mass of Coma using early X-ray observations of the cluster. More recently, ROSAT observations of Coma have provided unprecedented detail of the intracluster gas morphology (e.g., [@bri92]; [@whi93]). ASCA observations of the cluster have provided important information on the temperature structure of the X–ray emitting gas (Honda et al. 1996) which have been recently complemented by XMM-Newton observations (Briel et al 2000; Arnaud et al 2000; Neumann et al 2000) The Coma cluster has long been regarded as the archetypal relaxed massive cluster of galaxies. However, recent studies of the cluster have shown otherwise. The current view of Coma is that the cluster is the product of one recent, and one ongoing, cluster–group merger. A group centered on NGC 4839, in the southwest region of the cluster, is falling into (CD96), or has just passed through ([@bur94]), the main body of the cluster and may have triggered new star–formation in galaxies in Coma (Caldwell & Rose 1997). The velocity dispersion of this southwest group is approximately $\frac{1}{3}$ of that of the main cluster. Meanwhile, the core of Coma has two dominant galaxies, NGC 4874 and NGC 4889, which seem to be the relic central galaxies of previous groups that have merged into the current cluster. The X–ray and dynamical data reveal that both of these dominant galaxies do not appear to sit at the bottom of the cluster potential (CD96; [@whi93]). The lack of a cooling flow and the existence of an extended radio halo support this merging history of the cluster. The Coma cluster was thus an ideal first target for the Sloan Digital Sky Survey (SDSS; [@yor00]) spectroscopic commissioning program because of its location (the North Galactic Pole), the pre–existence of wide–field galaxy photometry (e.g. GMP83), and the high–density of known redshifts for comparison and testing. Moreover, there remains interesting scientific questions that can be addressed using the unique SDSS spectroscopic hardware [*e.g.*]{} the influence of cluster merger events on the star–formation rates of galaxies. The quality and quantity of SDSS spectral data will allow us to study such problems in great detail and in this paper, we start by outlining and comparing the different analysis techniques for classifying SDSS galaxy spectra as well as quantifying their star–formation rates. Robust, automated, spectral classification methods are necessary to help us understand the distribution and evolution of the galaxies physical properties that define and characterize their spectral energy distributions. The problem of galaxy spectra classification has been copiously treated in the astronomical literature. In general, methods are based on the measurement of spectral continuum and line features (e.g., the Lick/IDS system: Faber et al 1985; Burstein, Faber & González 1986) which are then used to classify and derive the galaxies physical properties. Stellar population synthesis models can, for example, be compared to these measurements, or the entire spectrum using template fitting, to provide a physical understanding of the galaxy properties. Recently, other techniques have been investigated. Amongst them, the principal component analysis has attracted a lot of interest (e.g, Connolly et al 1995; Bromley et al 1998; Folkes et al 1999; Ronen et al 1999). The technique is based on decomposing the spectra into a basis that highlights the galaxy differences. Such decomposition can therefore be used to classify the spectra. Different implementations change the way the basis is constructed or how the resulting coefficients of the decomposition are used to generate a classification method. Some authors, for instance, use artificial neural networks to build a classification scheme from the principal component coefficients (e.g., Folkes, Lahav & Maddox 1996). Wavelets also provide an orthogonal basis in which the spectra can be decomposed and therefore can in principle be used as a classification method. Along these lines, Pando & Fang (1996) and Theuns & Zaroubi (2000) have used wavelets to study quasar spectra. In this paper, we present the first hour of extra–galactic data taken by the SDSS spectroscopic system. Only one of the ten plates originally designed in the Coma region has been observed producing nearly 200 Coma redshifts and thus illustrating the capabilities of this new instrumentation. These spectra will allow us to investigate several classification schemes and test their applicability to the future SDSS dataset. The paper is structured as follows. In §2, we briefly highlight the main characteristics of the SDSS spectroscopic system. In §3, we describe the selection of galaxy targets in the Coma cluster region. The observations of the Coma cluster plates are presented in §4. In §5, we describe the data analysis. We discuss the redshift measurements in §6 and classify the galaxy members in §7. Finally, we discuss our data and present our conclusions in §8. The SDSS Spectroscopic System ============================= The SDSS will use a dedicated 2.5m telescope with a 3 degree field–of–view and two fiber–fed double spectrographs to measure the redshifts of approximately a million galaxies and one hundred thousand quasars over the next 5 years. The SDSS will also image the same area of the sky to an approximate depth of $\sim$23 mag in five optical bands ($u'$, $g'$, $r'$, $i'$ and $z'$; [@fuk96]; [@fan00]). For more details on the SDSS, the reader is referred to [@yor00] for a brief overview of the survey hardware, software and strategy$\footnote{In addition, the reader can refer to the online SDSS Project Book at {\tt http://www.astro.princeton.edu/PBOOK/welcome.html}}$. The full details of the SDSS spectroscopic system will be presented in Uomoto et al. (2001; in preparation) and Frieman et al. (2001; in preparation). For completeness, we provide here an outline of the SDSS spectroscopic system. The SDSS spectrographs are designed to cover the wavelength range of 3900 $\rightarrow$ 9100 [Å]{}. This is achieved using a dichroic beamsplitter centered at $\simeq6000$ [Å]{} to separate the incoming light onto a red and blue camera in both spectrographs. The SDSS spectrographs achieve a spectral resolution of 1800 across its entire spectral range. Each spectrograph accepts 320 fibers, each of which subtends a diameter of 3$^{''}$ on the sky. The fibers are plugged into aluminum plug–plates sitting in the focal plane of the SDSS telescope. Each plate contains 640 science holes (one for each science fiber), light–trap holes (drilled to avoid the reflection of light from bright stars off the plug–plate back into the telescope optics), guide star holes (11 in total which are fed to the guide camera) and quality assurance holes. During normal operations, plates scheduled for observation are manually plugged during the day. Each plug–plate is mounted in an individual cartridge that possesses a full set of 640 science fibers and 11 coherent fiber bundles that are plugged into the guide star holes and used to guide the telescope. Once plugged, the cartridges are automatically mapped with a video system that measures the plug–plate location of each fiber ([*i.e.*]{} sky coordinates) as they are successively illuminated by a laser diode at the slithead. After mapping the cartridges are stored ready for mounting on the SDSS telescope during the night. Target Selection {#target} ================ In normal survey operations, the SDSS will select targets for spectroscopic observation using the SDSS photometric survey. However, in the case of the Coma cluster, the SDSS imaging photometric camera had not yet observed this region of the sky and thus target selection was performed using an external catalog. We have used the SuperCOSMOS scans of two photographic plates (J and O), centered on the Coma cluster (see http://www-wfau.roe.ac.uk/sss/ for details) kindly provided to us by Harvey MacGillivray. The passband of the J plate is the combination of the Kodak IIIaJ emulsion and a GG395 filter (same as the UK Schmidt $b_j$ system; see Nichol, Collins & Lumsden 2000), while the passband of the O plate is the combination of the IIIaF emulsion and a OG590 filter. The raw object lists of these two SuperCOSMOS scans contain approximately 150,000 objects per plate and therefore must be trimmed for our purposes. We first merged the two photographic plates catalogs keeping only objects that matched to within 1.5 arcseconds. This merged catalog was separated into a list of stars and galaxies using the SuperCOSMOS source classifier. We produced two lists; one of stellar sources, which were objects classified as stars on both plates, and one of galaxies, which were objects classified as galaxies on at least one of the plates. We used these criteria since star-galaxy separation becomes increasingly difficult at the fainter magnitudes. By including sources which were classified as a galaxy in one plate, but as a star in the other, we hoped to recover some of the faint galaxy population that would have otherwise been missed. Unfortunately, we can expect higher stellar contamination (especially from blended stars) in our galaxy list. We photometrically calibrated the SuperCOSMOS data using the GMP83 catalog. We matched our galaxy list to the GMP83 data and then derived a transformation to convert the SuperCOSMOS instrumental magnitudes into the GMP83 magnitude system which was calibrated using $B$-band photoelectric magnitudes for their blue band and $B-R$ colors for their red band (see GMP83 for details). Our photometry is thus measured in the blue and red “photographic bands” (emulsion + filter described above) but transformed to the GMP83 system with a zero-point and color terms. Hereafter, we will refer to these magnitudes as, $b$ and $r$, following the notation in GMP83. Finally, a small astrometric correction was applied by correlating our star catalog with the ACT star reference catalog.[^1]. The resulting list of galaxies contained nearly 33,000 objects. This was further trimmed to 8187 targets by imposing a magnitude cut of 14 $\le b \le$ 20. From these targets, a total of 10 plates were designed, with two plates at each of the five plate centers shown in Figure \[fig1\]. Each pair of plates was allocated unique targets, with one plate containing the brighter galaxies and the other containing the fainter galaxies (see figure \[fig2\]). This was done for two reasons; first, to accommodate the minimum spacing restriction between SDSS fibers (55 arcseconds) and second, to allow for different exposure times for these plates. Finally, each plate was allocated a set of 10 bright stars ( $14 < b < 15$ ) for the guide camera fibers. Since these were selected from the same SuperCOSMOS data there was little concern about their relative astrometric reference frame. Observations {#observe} ============ One of the SDSS spectrographs saw first astronomical light on 26 May 1999 in bright time. After this milestone, the first spectroscopic commissioning run took place in the dark period of June 1999. At the time of the Coma SDSS spectroscopic observations, several parts of the nominal SDSS spectroscopic system were either not available or were being commissioned for the first time. For example, only one of the two SDSS spectrographs was mounted on the back of the telescope. Therefore, only 320 science fibers were available per plate instead of the 640 which is now the standard number. In addition, the plate–mapper, which automatically matches the plugged fibers to the target catalog, was not fully automated and we were forced to map the plate by hand. The telescope was not properly collimated at the time of the observations and the hardware needed to obtain spectroscopic calibrations was not installed. Finally, condensation on the front of the red camera dewar produced a series of bright “doughnuts” in the center of the red CCD images rendering most of the red spectrograph data in this commissioning run almost impossible to use. On 6 June, the SDSS observed plate 133 (the bright plate at the center of Coma). We took two 900 seconds exposures at mean airmasses of 1.53 and 1.66 without guiding. On 8 June, plate 133 was re–observed taking three 1200 second exposures. These were the first astronomical exposures in which the SDSS spectroscopic system guided on the sky. The seeing was between 1.2 and 1.5 arcseconds and the mean airmasses were 1.07, 1.10 and 1.15 for the three exposures. Being near the zenith, the atmospheric differential refraction effects were small. In addition to plate 133, we also observed plates 135 and 136 on 8 June 1999, which were designed using SDSS photometry in areas of the sky unrelated to the Coma cluster and are thus more representative of the field galaxy population. Plate 133 is the only Coma plate that has been observed during the SDSS spectroscopic commissioning phase. Figure \[fig3\] shows the distribution on the sky of the identified cluster members in this plate. Coma will be re–observed as part of the main SDSS spectroscopic and photometric survey in the coming years. These data will be far superior to the data discussed herein. Data Reduction {#datared} ============== In normal production mode the SDSS will reduce the spectroscopic data through a specifically designed pipeline (Frieman et al 2001; in preparation). However, due to the uniqueness of these observations and the non–standard observing set–up, the Coma observations were processed using a modified version of an earlier copy of the spectroscopic pipeline. The reduction stages are, nevertheless, standard in multi–fiber spectroscopy. The blue and red camera data were reduced separately given the aforementioned problem with the red camera. First, we subtracted the bias signal using the overscan columns at the edge of the CCD and co–added the individual 2–dimensional images, rejecting cosmic rays in the process. We then traced the fibers and optimally extracted the 1-dimensional spectra using the Horne (1986) algorithm. In the red this procedure had to be individually supervised and sometimes changed for troublesome fibers. No flat-fielding of the data was possible given the lack of uniformly illuminated exposures either for the whole CCD (2-dimensional flat) or through the fibers (1-dimensional flat). We wavelength calibrated the spectra using the sky emission lines. In the blue, we used the Hg, \[OI\] and NaD lines, while in the red, we used the numerous OH lines as well. We fitted a third order polynomial to the wavelength dispersion, and in the blue we obtained residuals of $<0.1$ [Å]{} for most fibers. For a few fibers, we did witness residuals of 0.2 [Å]{} (which is $\simeq15$ ${\rm km\,s^{-1}}$). However, these errors are computed at the position of the sky lines and are therefore, probably under–representative of the wavelength calibration error for the whole spectrum (see later). We obtained a similarly accurate wavelength solution in the red. On average, the blue spectra spanned the wavelength range 3770 to 6100 [Å]{} with a median dispersion of 1.14 [Å]{}/pix, while in the red, we obtained a wavelength coverage of 5770 to 9120 [Å]{} with a median dispersion of 1.64 [Å]{}/pix. These are close to the original design specification. To sky–subtract the spectra, we combined the dedicated sky fibers into one “super–sky” spectrum which was rebinned to the resolution of the individual spectra and scaled to match the flux of the \[OI\] $\lambda$5577 night sky line of each science fiber to compensate for any fiber throughput differences. Then each re-scaled “super–sky” was subtracted from the corresponding science fiber. As we did not possess calibration frames, and given the poorer quality of the red side, we did not attempt to merge the red and blue spectra. For the rest of the paper, except on the galaxy classification in Section \[classifications\] when we use H$\alpha$, we only use the blue camera data. To correct for the intrinsic response of the instrument in the blue, we have used bright, early-type stars observed as part of the plate 133. This was achieved by selecting all stars of a spectral type earlier (hotter) than K0 which also possessed a signal-to-noise ratio per pixel of greater than 100 at 5000 [Å]{}. In total, this criterion provided 15 stars, the earliest type being a A9-F0 star, the latest a K0. We divided these stars by the best fit stellar template in the Jacoby et al. (1984) stellar atlas. We smoothed, normalized, and combined the resulting spectra to obtain the response function of the instrument. We then divided all the spectra by this response function. We note this procedure does not provide an absolute spectrophotometric calibration; it only corrects for the response function of the instrument. Redshift Determinations ======================= We first determined the redshifts of our spectra via visual inspection and fitted Gaussians to all obvious absorption/emission lines. The main absorption lines used were CaII H and K $\lambda\lambda$3968, 3934, CaI $\lambda$4227, the Balmer series lines, MgI $\lambda\lambda\lambda$5167, 5173, 5184 and NaD $\lambda\lambda$5790, 5796. In emission we used \[OII\] $\lambda\lambda$3726, 3729, the Balmer series lines and \[OIII\] $\lambda\lambda$4959, 5007. Our visual inspections of the spectra revealed 196 galaxies in the Coma cluster, 49 field galaxies, 5 quasars, 47 stars and 12 spectra that could not be classified. The remaining 11 fibers were sky fibers. We also used the cross-correlation technique to obtain a more accurate galaxy redshift. We utilized the RVSAO package ([@KM98]) within the IRAF environment. Given that the majority of synthetic spectra have resolution coarser than our data, we decided to construct our own templates for the cross-correlation using the stellar objects observed on Plate 133; this ensures that we have the same resolution for both the template and object spectra. We chose to use as templates the fifteen stars discussed above (for correcting the response function of the instrument) as well as two additional later-type stars (K1 and K4) that also passed the aforementioned signal-to-noise criterion. We also added emission lines to these stellar templates creating a set of 34 templates [*i.e.*]{}, 17 absorption line stars and 17 absorption and emission stellar templates. All templates were shifted to their rest–frame. To determine the true error on our wavelength solutions and thus our redshifts, we first re–calculated the wavelength solution for each of our stellar templates using at least 9 lines, and normally 14, per spectrum. The availability of more lines, and the better sampling of the wavelength range, allowed us to improved the wavelength solution compared to that derived from just the sky lines alone. We then cross-correlated these re–calibrated stellar templates against higher–quality stellar spectra taken by the SDSS in later commissioning observations in Spring 2000. From these tests, we estimated that our dispersion solutions had an error of $\simeq30$ ${\rm km\,s^{-1}}$ which we added in quadrature to the error resulting from the cross-correlation technique. In Figure \[fig4\], we present the redshift distribution of the 245 galaxies for which we obtained a redshift on plate 133. The Coma cluster can be clearly seen as the broad peak around $z=0.0232$. Figure \[fig5\] is an expanded view of this region together with a 3-$\sigma$ clipped Gaussian fit to the data. Cluster membership is easily assigned for our galaxy sample since there is only one galaxy in the 3 to 4$\sigma$ region of the distribution. This galaxy would not normally be assigned to the cluster based on our data alone, however, CD96 measured a higher number of galaxy redshifts obtaining a larger velocity dispersion which would put this galaxy inside our 3-$\sigma$ cut. Therefore, for the rest of our analysis, we included this galaxy as a cluster member, bringing the total number of Coma member on plate 133 to 196. Table \[tbl1\] lists the main galaxy data on our Coma galaxies. Column 1 gives the extraction ID number (running from 1 to 320) while Column 2 is the GMP83 number when appropriate (three of our galaxies do not have a corresponding GMP83 detection). Columns 3 and 4 give the right ascension and declination of the fiber center (not necessarily the galaxy centroid). In column 5 we present the cross-correlation redshift corrected to the heliocentric reference frame. The redshift error, in Column 6, includes the error in the wavelength calibration combined in quadrature with the error resulting from the cross-correlation technique. The usual $R$ cross–correlation coefficient is listed in column 7. In most cases, we simply chose the cross–correlation redshift with the largest $R$ value out of the 34 possible template cross–correlation redshifts. There were a few cases where the best $R$ was in clear disagreement with our visual redshift measurements and in these cases we re-inspected the spectrum and quote, as our final redshift, the template with the highest $R$ coefficient consistent with our visual redshift. These rare cases were typically due to non-corrected cosmic rays or sky residuals that produced a confident cross-correlation with one of the emission–line templates. Finally, in Columns 8 and 9, we provide the magnitudes for the galaxy taken from the SuperCOSMOS catalog and calibrated to the GMP83 system. We have compared our redshift measurements to the data available in the literature. We have matched our catalog with the redshift compilation of CD96 as well as data from the NED$\footnote{NED is the NASA/IPAC Extragalactic Database, operated by the Jet Propulsion Laboratory, Caltech, under contract with NASA}$ database. We found 151 galaxies that match previous known redshifts. The remaining 45 galaxies, approximately a quarter of our Coma redshifts, represent new measurements. In Figure \[fig6\], we show the comparison between our redshift measurements and those in the literature. The solid histogram represents the 151 matches, where 148 come from the compilation of CD96 (including their own measurements and values in the literature), and the other three are taken from NED. The solid histogram is a Gaussian fit to the entire distribution. The fit is poor because of the extended tails. We can improve the fit considerably if we reject 3-$\sigma$ outliers (15 galaxies; see Figure \[fig6\]). We attribute these outliers to the heterogeneity of the data where different data sets from the literature have different calibration errors. Also, these discrepancies could be in part attributed to the internal dynamics of the observed galaxies because of the different fiber and/or slit placements. We can improve the fit even further by restricting ourselves to the largest homogeneous dataset from the literature [*i.e.*]{}, the 63 new redshifts measurements of CD96. We then find excellent agreement between our measurements (the dotted histogram and Gaussian fit) with a mean offset of 3 ${\rm km\,s^{-1}}$ and an RMS scatter of 24 ${\rm km\,s^{-1}}$. Overall, the agreement of our redshift determinations with the literature is remarkably good. The RMS scatter between datasets indicates that our estimate of the errors in the wavelength calibration is accurate. Spectral Classifications {#classifications} ======================== In this section, we consider the spectral classification of our Coma galaxies. We have investigated four different algorithms, all of which could be implemented for the main SDSS galaxy survey. The first algorithm is based on visual inspections of the spectra but quantified using measurements of the equivalent widths[^2] of the lines seen in the spectrum. The next two algorithms, Wavelets and Principal Component Analysis (PCA), were used to objectively classify the spectra using the visual inspections to define relevant thresholds in wavelet and eigenspace. The final classification scheme attempts to define a physical classification scheme based on synthetic models of galaxies. Line Strength Classifications {#lines} ----------------------------- We started by visually classifying the spectra into five classes which could then be compared to the other algorithms. We first divide the spectra into two broad classes: emission and absorption line galaxies. Absorption line spectra were then sub–divided into normal absorption line systems and objects with strong Balmer lines or post–starburst galaxies. The spectra showing emission lines were sub–divided into three categories depending on the strength of the emission lines. For our visual and line–strength classification scheme, we were able to use both the blue side of the spectrum and the presence of H$\alpha$ in the red end of the spectrum. Unfortunately the other classification algorithms could not use the red side of the spectrum due to the condensation problems discussed above. Since H$\alpha$ is a powerful indicator of star–formation, we believe that one of the classification schemes should utilize these data even if the others could not. Moreover, the other algorithms used our line strength classifications to quantify star–formation in Coma and therefore, it was justified to make the line strength classification as strong as possible by using all data available. Using our equivalent width measurements, we divided the spectra in the five following types: 1) Absorption line galaxies ([**AB**]{}) which have spectra without H$\alpha$ in emission and with the sum of the equivalent widths of H$\delta$, H$\gamma$ and H$\beta$ smaller than 15 [Å]{}. 2) Post–starburst galaxies ([**PS**]{}) which have no H$\alpha$ in emission, $W_o({\rm H}\alpha)>0$, and $W_o({\rm H}\delta+{\rm H}\gamma+{\rm H}\beta) > 15$ [Å]{}. 3) Absorption line dominated galaxies but with H$\alpha$ in emission and modest H$\beta$ emission ([**AB+EM**]{}): $W_o(H{\alpha})<0$ and $W_o(H{\beta})>-5$ [Å]{}. 4) Emission and absorption line galaxies ([**EM+AB**]{}) which we define using $W_o({\rm H}\alpha)<0$ and $-5<W_o({\rm H}\beta)<-15$ [Å]{}. 5) Emission line dominated spectra ([**EM**]{}) which have $W_o({\rm H}\alpha)<0$ and $W_o({\rm H}\beta)<-15$ [Å]{}. We present these classifications in Table \[tbl2\] where Column 1 is the same extraction number as given Table \[tbl1\], Column 2 is the classification based on the line strength criteria given above, Column 3 is the PCA classification and Column 4 is the wavelet classification (see below). The results of the synthetic model fitting and the parameters of the PCA classification, both of which are discussed below, are listed in Columns 5 through 10 and Columns 11 through 16, respectively. In Figure \[fig3\], we show the distribution on the sky of the 196 galaxies given in Tables \[tbl1\] and  \[tbl2\]. The half circle shape observed on the data is due to the availability of only one spectrograph. The presence of the NGC 4839 group is obvious in the southwest region of the cluster. The plotting symbols reflect the 5 different galaxy types defined above. Principal Component Analysis {#PCA} ---------------------------- To quantify the visual classification discussed above, we have implemented a Principal Component Analysis (PCA) of our Coma spectra (Mittaz et al 1990; Connolly et al 1995). The basis of this method consists of describing a multi–dimensional distribution of variables with a minimum number of dimensions. In our case, we are searching for the minimum number of eigenspectra needed to describe the whole spectral dataset. We can thereafter classify each spectrum as a function of these eigenspectra. We have used here a version of the official SDSS spectroscopic reduction software which already implements the PCA analysis of Connolly et al. (1995). ### Data Preparation As mentioned before, only the blue spectra were used for the PCA analysis. To prepare the spectra, we first interpolated over the \[OI\] $\lambda$5577 and NaD $\lambda\lambda$5890, 5896 doublet sky emission lines since many of the spectra contained significant residuals of these strong sky lines. Next, each spectrum was blueshifted to the galaxy’s restframe and rebinned to the wavelength range $3750 \rightarrow5900$ [Å]{} at a dispersion of 1.125 [Å]{} per pixel. Then, we subtract the continuum from the spectra using a fourth order polynomial. We have also carried out the PCA analysis without continuum subtraction, but we obtain better discriminating power to reproduce the line strength classification employing the continuum subtracted sample. Finally, we smoothed each spectrum with a Gaussian with a width set by the spectrograph resolution of $R=1800$ which produced a set of 673 pixel spectra, binned over a common wavelength range, which could then be used as input to the PCA algorithm. ### Eigenspectra Derivation Using these prepared spectra, we performed the PCA methodology presented in Connolly et al (1995) and Connolly & Szalay (1999). We present a very brief summary below, but for a more detailed description see Connolly et al (1995). We regard each spectrum as a vector, $f_{\lambda}$, where $\lambda$ is an index running on wavelength. We represent the whole spectra set by a matrix $f_{\lambda,i}$, where the index $i$ runs through the individual spectra. We apply a uniform normalization to each spectrum, so that the weighted sum of the squared flux is unity. Thus, our normalized spectra are computed as $$F_{\lambda,i} = \frac{f_{\lambda,i}} {\sqrt{\sum_{\lambda}^{} f_{\lambda,i}^2 \, W_{\lambda} }}$$ In all cases a uniform weight was applied, so that $W_{\lambda} = 1$ for all wavelengths $\lambda$. We want to construct an orthogonal basis for the space spanned by the spectra, which we call eigenspectra. We derive these eigenspectra by diagonalizing the correlation matrix $C_{ij}$, given by $$C_{ij} = \sum_{\lambda}^{} F_{\lambda,i} F_{\lambda,j} \, W_{\lambda}$$ By diagonalizing the correlation matrix, we find the matrix $R$ that produces the diagonal matrix $\Lambda$ whose components are the eigenvalues, $\gamma_i$, of the orthogonal eigenspectra, $e_{\lambda,i}$, $$\begin{aligned} R^{\dagger} C R & = & \Lambda \\ R^{\dagger}_{i,j} (\sum_{\lambda}^{} F_{\lambda,i} F_{\lambda,j}) R_{i,j} \ & = & \Lambda_{i,j} \nonumber \\ R^{\dagger}_{i,j} F_{\lambda,j} F_{\lambda,j} R_{i,j} & = & \ e_{\lambda,i} e_{\lambda,j}^{\dagger} \;=\; \gamma_i\, \delta_{i,j}\nonumber\\ \Rightarrow R^{\dagger}_{i,j} F_{\lambda,j} & = & e_{\lambda,i} \end{aligned}$$ In Figure \[fig7\], we show the distribution of the first 20 eigenvalues. The two curves are the percentage contribution of the N$^{th}$ eigenvalue (decreasing curve on a log scale) and the percentage contribution of all the eigenvalues less than or equal to N (the increasing curve on a linear scale). As one can see the relative contribution of each eigenspectrum falls off rapidly; it is possible to account for $\sim90\%$ of the variance in our sample using a reasonably small number of eigenspectra ($\le 10$). The remaining $\sim10\%$ of the variance is probably mostly noise in the data, but may also partly be due to complex, real variations in the galaxy spectra. Following the methodology of Connolly et al. (1995), we limit our classification to the first three components, as these eigenspectra contain virtually all the usable information, from a classification standpoint, as we will illustrate below. In our analysis, these three components account for 87% of the variance in our sample. By contrast, the first 2-3 components for eigensystems derived from synthetic spectra (e.g. Connolly & Szalay 1999, Ronen et al. 1999), generally account for 98-99% of the variance, while the preliminary eigensystem derived for the 2dF Galaxy Redshift Survey (Folkes et al 1999) contained about 66% of the total variance within the first three components. Our galaxy sample lies between these two extremes in terms of complexity and signal to noise, so this result is qualitatively consistent with these other works. These first three eigenspectra are displayed in Figure \[fig8\]. The first eigenspectrum represents the mean spectrum in our sample, so as expected it is a typical elliptical galaxy spectrum. It has spectral features typical of an old stellar population [*e.g.*]{}, the strong absorption lines of Ca II H & K, G band and magnesium. The second eigenspectrum contains strong Balmer absorption and emission lines, and strong \[OIII\] emission. This component will clearly correlate with current star formation. The third eigenspectrum again contains strong Balmer absorption, but now the strong H$\beta$ and O\[III\] lines are shown in absorption. The eigenspectra for eigenvalues beyond the top three components do not show major spectral features, so we believe these are mostly shaped by the noise in the data. Also, the eigencoefficients for these higher order components do not show any significant trends with our visual spectral types above. ### Galaxy Classification The eigensystem can be used to define eigencoefficients for each galaxy spectrum, which are simply the dot products of the galaxy spectrum with each eigenspectra. Since we are using uniform weights, the relationship is straightforward. For example, the $i^{th}$ eigencoefficient of the $j^{th}$ galaxy spectrum is simply: $$c_{i,j} = \sum_{\lambda}^{} e_{\lambda,i} f_{\lambda,j}$$ The resulting set of coefficients are normalized such that sum of their squares is one, that is, $\sum_{}^{} c_{i}^2 = 1$, where the summation index i runs through all eigencoefficients. In Figure \[fig9\], we show the distribution of the first, second and third eigencoefficients. Immediately one can see that the different spectral types cluster into specific areas of eigenspace. The relationship between the three eigencoefficients is best related by using a projection scheme as in Connolly et al. (1995), where three coefficients are converted into spherical coordinates. By assuming that all the useful information is contained within the first three components, we throw away the remaining coefficients and then treat the three top coefficients as a vector in a three dimensional space and convert them into a spherical coordinate system. In this view, the spectra can be described very simply by the two spherical angles $\theta$ and $\phi$ (see Figure \[fig10\]). The relative lengths of the vectors are not important for classification, since they only represent how well each spectrum is represented by the three components. Figure \[fig10\] shows the distribution of projected angles. The dashed lines show where we have made cuts based on the aforementioned classifications except that we have merged the [**EM+AB**]{} and [**EM**]{} types into one class. In figure \[fig11\], we show the composite spectra from each of our four remaining classes (these composites are simply averages of the spectra). The cuts we have made in Figure \[fig10\] are arbitrary and were chosen to reproduce the line strength classification scheme. The spectra have a continuum of possible shapes, which follow as a function of these parameters. Figure \[fig10\] shows that the angle $\theta$ correlates with the strength of the Balmer features in absorption and therefore can be used to differentiate post–starburst galaxies. The angle $\phi$ correlates with the emission line strength and the K–star spectral features thus being a good discriminator of the overall star formation activity. The dependence on the K–star spectral features can be seen in Figure \[fig11\] where we investigate a further division of the [**AB**]{} class into two sub–groups [**ABa**]{} and [**ABb**]{}, along a line parallel to the [**PS**]{} boundary. The [**ABb**]{} class have stronger Balmer absorption but weaker MgI and G band features than the [**ABa**]{} type. We present the PCA classifications in Column 3 of Table \[tbl2\]. For completeness, we also provide the coefficients of the PCA used in this paper: Columns 11 to 13 contain the top three eigencoefficients, Columns 14 & 15 are the $\theta$ & $\phi$ angle coefficients, and the square of the radius for each spectrum is given in Column 16. We do not include any information on the eigencoefficients in the case where the continuum is not subtracted as the results are qualitatively the same although quantitatively they differ. Wavelet Classification ---------------------- In addition to the PCA analysis on the Coma spectra, we have also investigated the use of Wavelets, since they provide a multi-resolution approach of classifying spectra that allows us to study – in a single analysis – both the small–scale emission/absorption features together with low-frequency continuum information. The goal of our wavelet analysis was to automate the line strength classification given above using as few a number of wavelet coefficients as possible thus providing an overall compression of information as well as producing an objective classification scheme that could be replicated elsewhere. For the analysis discussed herein, we have used the IDL Wavelet Workbench[^3]. We do not provide here a detailed explanation of the theory behind Wavelet since there now exists a large volume of literature on this subject (see, for example, Press et al. 1992). To prepare the data for our wavelet analysis, analogously to the PCA analysis, we first shifted all spectra to the galaxy’s rest–frame and define a common (rest–frame) wavelength range ($3890 \rightarrow5901$ [Å]{}) which only uses data from the blue side of the SDSS spectrograph. We then computed the average pixel value for the whole spectrum and subtracted this off each pixel thus re–normalizing the spectrum to have a mean of zero. This procedure removed the large discontinuities at the ends of the spectrum. The final step was to “zero–pad” the spectrum on either side of the data to create a pixel array with an integer power of two (2048 in this case). We then applied cosine–bell smoothing to the edges of the original spectrum to gradually taper the original data to zero thus avoid sharp edge–effects at the ends of the spectrum which would result in “ringing” or the Gibb’s phenomena. For our spectra, we smoothed 5% of the original data at either end of the spectrum. We note here that as in the PCA analysis above (Section \[PCA\]), we did not use the red side of the spectrum because of the problems discussed in Section \[observe\] and thus we have no information about H$\alpha$ in emission. However, for the wavelets, we did retain the shape of the continuum which is different from the PCA analysis. The multi-resolution nature of the wavelets does not require continuum-subtracted data. Nonetheless, we also performed the same wavelet analysis with continuum subtracted spectra obtaining different parameters values but a similar classification using the same strategy as described below. We then performed a wavelet transform on these spectra which returned a series of wavelet coefficients at ten different resolution levels; the number of resolution levels is set by the size of the padded data array since each level has a factor of two lower resolution than the previous layer. An example of such a wavelet transform for one of the SDSS Coma spectra is shown in Figure \[fig12\] along with the different resolution levels and the corresponding amplitude of the wavelet coefficients at those resolution levels. The first decision faced in the implementation of our wavelet analysis was the choice of the Mother Wavelet [*e.g.*]{}, a Daubechies, Symmlet or Coiflet wavelet function. Moreover, we also needed to choose the level of smoothness for these wavelets [*e.g.*]{}, Daubechies2 (which is just a Haar Wavelet), Daubechies4, Daubechies8 [*etc.*]{}, with the higher numbered wavelets being smoother versions of the Mother Wavelet. To aid in these decisions, we performed a series of empirical tests on the data to determine which wavelet provided the maximal compression of the information for the smallest number of coefficients used in the reconstruction of the data. In these tests, we re–constructed the spectra using the largest $N$ coefficients (in absolute value), regardless of which level they were taken from, as well as implementing the hard thresholding scheme [*i.e.*]{}, all coefficients below the $N$ largest–valued coefficients were set to zero in the re-construction. In Figure \[fig13\], we show the measured mean $\chi^2$ for all 196 Coma spectra as a function of $N$, the number of wavelet coefficients used in the reconstruction of the spectra. For our thresholding scheme, the Daubechies4 wavelet provided the most information compression, or largest fractional change in $\chi^2$, for a given $N$ coefficients. As expected, the $\chi^2$ decreases exponentially as a function of $N$, and for small values of $N$, the Daubechies4 clearly provides significant gain over the Symmlet wavelets. Therefore, we have used this Mother Wavelet (Daubechies4) in our analysis of these SDSS Coma spectra. This Mother Wavelet is rather sharp compared to the other wavelets which is an advantage for quantifying sharp discontinuities in these spectra [*e.g.*]{}, the $4000$ [Å]{} Balmer break discontinuity. We note here that we are simply used the relative change in $\chi^2$ to differentiate between the Mother Wavelets available and the absolute goodness–of–fit from the $\chi^2$ statistic is not being used because these spectral data are slightly correlated (due to the sky–subtraction and the resolution element being approximately 3 times the dispersion of these spectra). For reference however, we note that each spectrum had, on average, 2000 degrees–of–freedom and thus a reduced $\chi^2$ equal to one, which suggests a reasonable fit between the model and observed data, is achieved after adding $\simeq30$ wavelet coefficients (see Figure \[fig13\]). The fact we can explain these spectra with a relatively small number of coefficients agrees with our visual inspections and is reasonable given that we are studying a rather homogeneous subsample of galaxy spectra [*i.e.*]{} a majority of the spectra are dominated by older spectral populations. We caution the reader to not over–interpret the absolute $\chi^2$ results since we have not accounted for the correlations in the data. In Figure \[fig14\], we plot the amplitude of the second and third largest wavelet coefficients for all 196 Coma spectra presented in Table \[tbl1\]. We have ignored the largest wavelet coefficient since it simply measures the amplitude of the whole spectrum and thus contains no information about the general shape the spectrum (this is because a wavelet is defined to sum to zero). Figure \[fig14\] demonstrates that we can extract worthwhile information about these spectra from just two of wavelet coefficients; galaxies with recent star–formation, as indicated by the presence of emission lines or post–starburst characteristics (A star spectrum), preferentially possess large (negative) second and third wavelet coefficients, while absorption, non–star–forming, galaxies populate the remainder of the phase space. The explanation for Figure \[fig14\] is that these two wavelet coefficients are measuring the “color” of the continuum and are providing a similar discrimination as one may obtain from broadband multi–color photometry. This is exemplified by the fact that all the post–starburst galaxies in Figure \[fig14\] reside in the low–left arm since they all have “blue” spectra. In Figure \[fig15\], we show one of the post–starburst galaxies in Coma along with the re–constructed spectrum based on the three largest wavelet coefficients which illustrates that we have just measured the tilt of the spectrum. In this case, we have not exploited any high–frequency emission or absorption features, but have still managed to broadly segregate the galaxies based on their recent star–formation histories. To extend our analysis further, we have exploited the multi–resolution nature of wavelets by designing an algorithm to classify galaxies as emission line, post–starburst and absorption line galaxies using as few a number of wavelet coefficients as possible. Here, we have merged the different sub–divisions of the emission lines as discussed in Section \[lines\] and focus on the [**AB**]{}, [**PS**]{} and [**EM**]{} classes. To help define our algorithm, we co-added all the obvious post–starburst ([**PS**]{}) Coma galaxies (based on our visual classification scheme) and identified the strongest features in that co-added spectrum. In Figure \[fig16\], we show this summed spectrum and, as expected, the Hydrogen Balmer absorption series is strong because of the young A stars in the galaxy. Also visible are the Calcium H & K absorption features (with the $4000\AA$ break Balmer discontinuity) which is indicative of an older stellar population (K star spectrum). As discussed by Dressler & Gunn (1992) and Zabludoff et al. (1996) these post–starburst galaxies are also known as [E+A]{} or [K+A]{} galaxies [*i.e.*]{}, Elliptical plus A star or K plus A star spectra. The algorithm we have adopted is as follows. For each spectrum, we first scanned for the presence of any emission lines. This was empirically found to be equivalent to having one of the eight largest coefficients be a negative value located in the first five resolution levels [*e.g.*]{}, levels with more that 32 wavelet coefficients per level. We further restricted the search to look at wavelengths $>4000$ [Å]{} thus avoiding problems with the Balmer discontinuity. To identify post–starburst galaxies, we then re-scanned the first twelve largest coefficients for positive wavelet coefficients near to the wavelengths of H$\delta$, H$\gamma$ and H$7$ (see Figure \[fig12\] for an example of this algorithm). These features were used since they are the strongest of the Balmer absorption lines as shown in Figure \[fig16\]. We did not use H$\beta$ because of possible contamination from emission especially if H$\alpha$ is present about which we have no information in the current application of the method. If three such coefficients were detected, the galaxy was defined to be a post–starburst galaxy. We present these automated classifications in Column 4 of Table \[tbl2\]. Synthetic Models ---------------- As a final test, we have fit synthetic models of galaxy stellar populations to the Coma spectra. The advantage of this approach is that the spectra can be directly related to physical phenomena and parameters. The disadvantage is that the parameterization of the model, [*e.g.*]{}, initial mass function (IMF) or the star formation history of the stellar population studied, can differ substantially from the real star formation processes thus introducing large uncertainties based on the particular model used. Moreover, it is difficult to compare this scheme to the aforementioned three empirical classification schemes. Despite these caveats, we present this work here which will be primarily used in a forthcoming companion paper. Our method for comparing the stellar models with our Coma spectra is to minimize the following $\chi^2$ function, $$\chi^2(a_j) = \sum_{i=1}^n \left(\frac{(F_{i\;observed}^{\lambda} - F_{i\;model}^{\lambda}(a_j))} {\sigma_i(F_{i\;observed}^{\lambda})}\right)^2$$ where $a_j$ are the free parameters of the synthetic spectrum being fitted, $i$ is an index running through pixels, $F_{i\;observed}^{\lambda}$ and $F_{i\;model}^{\lambda}(a_j)$ represent each pixel of the spectral energy distributions observed and modelled respectively and $\sigma_i(F_{i\;observed}^{\lambda})$ the error of the observed spectrum in pixel $i$. In practice, the synthetic spectral energy distributions (SEDs) have a lower spectral resolution than our observed Coma spectra. Thus, we smooth both energy distributions to the same resolution; this means that the errors of the observed spectrum are therefore no longer uncorrelated. We also mask the spectral regions around strong sky emission lines to avoid the residual spikes coming from sky subtraction. We used two sets of synthetic spectral energy distributions. First, we employed the evolutionary synthesis model PEGASE ([@FRV97]) to generate the model spectra. We minimized the $\chi^2$ using three free parameters: an overall normalization, the age of the stellar population and the star formation prescription. We assumed solar metallicity and did not fit for the internal extinction. The star formation prescription was parameterized using two different laws: 1) we assumed that the star formation rate (SFR) depends on the galaxy gas fraction as $\Phi(t)=\nu\,f_g(t)$ and we minimized with respect to the astration rate $\nu$. In fact, we minimized this astration rate parameter using 8 discrete values. Following Fioc & Rocca-Volmerange, we used the Rana & Basu (1992) IMF and values of the astration parameter that reproduce the colors of the different spectral type galaxies observed locally. We also added a one Gigayear starburst model. 2) we assumed an exponential SFR law $\Phi(t)=\tau^{-1}\,exp(-t/\tau)$ and minimized with respect to the time scale $\tau$. In this case, we used a standard Salpeter IMF with lower and upper mass limits of 0.1 and 120 $M_{\odot}$, respectively. The age and star formation prescription of the best fitting model spectrum are presented in Table \[tbl2\]. For the first SFR prescription we provide the age (Column 5), the number of the best fitting model (Column 6) and the reduced $\chi^2$ (Column 7) of this best fit. The numbering of the models presented in Column 6 is the following: number 1 corresponds to a one Gigayear starburst model; numbers from 2 to 8 correspond to decreasing values of the astration rate parameter $\nu$, where large $\nu$ values resemble star formation histories typical of early-type galaxies and small $\nu$ values, typical of late-type galaxies. The degrees of freedom of the fit depend on the actual masking of each spectrum but for the majority of them, there are 2005-2020 d.o.f. We note here that our figure of merit, the $\chi^2$ we compute, does not strictly behave as a real $\chi^2$ distribution. It is certainly close to one, but not exactly given that our errors have not been determined with high accuracy and there has been some smoothing which we have neglected not taking into account the correlated nature of the errors. For the second SFR prescription, we present the age (Column 8), the time scale $\tau$ (Column 9) and the reduced $\chi^2$ (Column 10). Discussion and Conclusions ========================== SDSS Spectroscopic System ------------------------- We present in this paper the first extra–galactic spectroscopic data taken by the SDSS during the initial spectroscopic commissioning run in June 1999. Although the data was primarily obtained to commission all the SDSS spectroscopic subsystems (guider, plug–plates, spectrographs, control software), and are therefore far from the nominal quality expected for the SDSS when it is fully operational, these Coma spectra are of reasonable quality and provide a new insight into the galactic content of the Coma cluster, especially at large distances from the cluster center. This is a combination of the high throughput of the SDSS spectrographs, the large field of view of the spectroscopic observations ($3^{\circ}$ diameter), the extensive wavelength coverage and resolution, and the ability to obtain many hundreds of spectra simultaneously. This single plug–plate observation of Coma illustrates the overall efficiency of the SDSS spectroscopic system and demonstrates the potential for further cluster and field galaxy observations. As of October 2000, the entire SDSS spectroscopic system has been tested and commissioned. Both spectrographs are operational and are routinely collecting data. The target selection procedure, from determining target objects through to drilling plug-plates and shipping them to APO, has been exercised several times and the emphasis has now shifted to improving the overall efficiency of the observations. At the timing of writing, the SDSS has obtained $\simeq50,000$ spectra (approximately two thirds galaxies and one third quasars) and these data are being used, amongst other things, to define our integration time, completeness limits as well as debug the SDSS spectroscopic software. These data also represent one of the largest samples of galaxies and quasars presently in existence and are being used for several scientific programs. We have obtained 320 spectra (half the number of a nominal SDSS plug-plate) in the central region of the Coma cluster. The availability of one spectrograph only allowed us to observed the South half of plate 133 (see Figure \[fig3\]). Other engineering tasks and bad weather conditions prevented us from obtaining further Coma spectra and future data taken in this region will be acquired as part of the regular SDSS survey (both imaging and spectroscopy). Nevertheless, in one hour of observations with only half of the spectroscopic system commissioned, and without optimizing the target selection, we obtained 196 spectra of Coma galaxies. This number represents approximately $\frac{1}{3}$ of the number of redshifts presently available in the literature within the same 1.5 deg radius from the cluster center (as estimated from the NED database in August 2000). Amongst the 320 fibers available, 11 were placed on blank sky (to allow sky-subtraction from the rest of the fibers), while the rest of the science fibers yielded 196 Coma galaxies spectra, 49 field galaxies, 5 quasars (which were previously known and targeted on purpose), 47 stars and 12 unidentified spectra. Four of the unidentified spectra were due to “spill-over” from bright adjacent fibers$\footnote{ Fibers are only separated on the CCD camera by approximately 6 pixels. The FWHM of a fiber on the CCD is approximately 2 pixels. Therefore, the light from bright target fibers can {\it contaminate} adjacent fibers.}$. We note here that our target selection attempted to maximize the number of available galaxy targets by including all candidate galaxies [*i.e.*]{}, galaxies that were only classified as galaxies on one of the two photographic plates scanned by SuperCOSMOS. In hindsight, this produced considerable stellar contamination even at bright magnitude, but does ensure a higher level of completeness. Of the 196 Coma redshifts obtained, 45 represent new identifications. Because of the SDSS large, circular field–of–view, these new redshifts are located at large cluster radii in areas scarcely sampled by other spectroscopic surveys of Coma, [*i.e.*]{} the difference between a survey instrument and a more targeted observation focused on a particular science goal. Given this uniformity, and the quality of our data, we are thus able to obtain a better understanding of the spatial distribution of galaxy classifications within Coma. For the 151 redshifts found in common with the literature, we found a negligible mean offset between our redshift determinations and those already published. In particular, when we compared our redshifts to the largest homogeneous sample of redshift available in the literature, the 63 galaxies measured by CD96, we find only a 3 ${\rm km\,s^{-1}}$ mean offset with an RMS scatter of only 24 ${\rm km\,s^{-1}}$ (this is less than 2 SDSS spectrograph pixels). This is remarkable given the non–optimal observing conditions, the lack of proper calibration frames, the difference in the spectral resolutions of the two instruments used, as well as the non–standard wavelength calibration used. Moreover, we have looked for correlations between the observed difference between CD96 and our redshift as a function of CCD position, plate position and matching distance and find no significant dependences with these observational parameters. In summary, it is likely that we have over–estimated the error due to our wavelength calibration given the low RMS difference with CD96; the SDSS spectroscopic systems appear to be remarkably stable and linear over a one hour observation. Dynamics and Spectral Classifications ------------------------------------- In Figure \[fig5\], we show the redshift distribution of our 196 Coma members which is qualitatively similar to that seen for the 465 redshifts in CD96 (their Figure 5). This is not surprising since we have 148 galaxies in common with their analysis. However, we do sample out to larger radius than CD96 and the fraction of galaxies measured in the NGC 4839 group (in the SW portion of the cluster), with respect to the total, is larger in our sample than in CD96. A single Gaussian fit to our data gives a mean velocity of $6950\pm66\: {\rm km\,s}^{-1}$ and a velocity dispersion of $\sigma_v=916\pm50\: {\rm km\,s}^{-1}$, where we have corrected for the fact that we measure the velocity dispersion with a 3-$\sigma$ clip criterion. These values are in agreement with previous measurements and, as expected, are an average of the mean velocities, and mean velocity dispersions, of the main cluster core and the NGC 4839 group (see CD96). As already noted by CD96, a single Gaussian fit to the data is an over-simplification with a clear excess of galaxies, compared to the single Gaussian, between $\simeq 6950 \rightarrow 7600\: {\rm km\,s}^{-1}$ due to the NGC 4839 group. The central velocity of these excess galaxies is approximately coincident with the mean redshift quoted by CD96 for the NGC 4839 group, $7339\: {\rm km\,s}^{-1}$. We have also investigated the dynamical differences between the absorption line galaxies and the active (emission–line and post–starburst galaxies) galaxies. We find that the [**AB**]{} galaxies given by the line strength classification have a mean velocity of $6923\pm70\: {\rm km\,s}^{-1}$ and a velocity dispersion of $\sigma_{\rm v}=846\pm54\: {\rm km\,s}^{-1}$, while the rest of the galaxies have $<cz>=7027\pm160\: {\rm km\,s}^{-1}$ and $\sigma_{\rm v}=1160\pm134\: {\rm km\,s}^{-1}$. The velocity dispersion of the active galaxies is then 30% higher than that of the absorption line galaxies. This is similar to the difference found by CD96 for Coma and Carlberg et al (1997) when comparing the velocity dispersions measured from the red and blue galaxies in the CNOC1 cluster survey. We defer a more detailed examination of the dynamics of the whole Coma region until the future when we have both high quality photometric and spectroscopic data from the main SDSS survey in hand. In this paper we present several different spectral classifications for our 196 Coma galaxies. The homogeneity and quality of the data allowed us to experiment with these different classification schemes to understand their applicability to the main SDSS survey. The first classification scheme was based on the presence or absence of emission lines. In this case, the H$\alpha$ line served as the main classifier. For typical IMF and metallicities, H$\alpha$ cannot be detected in emission $\sim 25$ Myr after the end of active star formation. Therefore, this criterion efficiently separates galaxies with current star formation from those without. The strength of the H$\beta$ line in emission, and the strength of the H$\delta$, H$\gamma$ and H$\beta$ in absorption, are used as additional diagnostics to improve the classification (see Section \[lines\]). Using these specific lines, we were able to sub–divide the galaxies into 5 spectral types which we discuss below. Soon after the termination of a star formation episode, even if it only includes a few percent of the stellar population, the blue part of the optical spectrum of a galaxy is dominated by A stars which are characterized by the Balmer absorption features. If there is no further star formation activity the strength of these lines diminishes gradually with time. The Balmer absorption lines strength thus provides a rough estimate of the age of the stellar population, although the lines are also somewhat sensitive to metallicity. We use this criterion to separate absorption line systems into two sub–classes. Galaxies without recent star formation, based on the absence of H$\alpha$ in emission and the absence of strong Balmer absorption, were classed as [**AB**]{}. Galaxies in which their last episode of star formation ended recently, or post–starburst galaxies ([**PS**]{}), were characterized by their strong Balmer absorption lines as well as the absence of H$\alpha$ in emission. The threshold between [**AB**]{} and [**PS**]{} was set at an equivalent width larger than 15 [Å]{} for the sum of H$\delta$, H$\gamma$ and H$\beta$ equivalent widths. This corresponds to roughly 400-700 million years after the truncation of the active star–formation for solar metallicity. We note here that our criteria are based on equivalent widths; thus this threshold also depends upon the underlying stellar population (or the previous star formation history) [*i.e.*]{}, how many stars are contributing to the continuum. Emission line galaxies were divided into three sub–classes ([**AB+EM**]{}, [**EM+AB**]{} and [**EM**]{}) depending upon the strength of the H$\beta$ line. The $W_o$(H$\beta$) cuts are difficult to translate into star formation rates given the different continuum strengths and the fact that we have not attempted to correct the intrinsic extinction. Nevertheless, the equivalent width of H$\beta$ represents a measure of the strength of the star formation process with respect to the underlying stellar population. This classification scheme has the advantage of being simple and easy to measure in optical spectra with sufficient wavelength coverage, and is meaningful as it classifies spectra according to their star formation activity relative to the overall stellar population. Undoubtedly, the use of H$\alpha$, instead of H$\beta$, to subdivide the emission line spectra would have been preferred, as it is a better indicator of star formation and is less affected by extinction. However, the nature of our data made us settle for H$\beta$. We can use this classification as the test base to which compare our other schemes. Other classification schemes based on spectral features have also been used in the literature. In particular, the MORPHS group (Dressler et al 1999; Poggianti et al 1999) and the CNOC group (Balogh et al 1999) have recently put forward similar classification schemes based on the \[OII\] and H$\delta$ lines, in which they provide a more physical interpretation of their types based on evolutionary spectral synthesis models. They use the \[OII\] and H$\delta$ lines as indicators of current star formation and the time elapsed after the last episode of star formation, respectively. The large wavelength coverage of our spectra allow us to use more spectral features, H$\alpha$, H$\beta$, H$\gamma$ and H$\delta$, for the same purpose. Here, we have taken the phenomenological approach of providing a simple classification scheme based on observational quantities which are directly related to the star formation processes taking place within galaxies. We defer to a future paper a more detailed interpretation of these classes in terms of the evolutionary phases of galaxy stellar populations. There are 44 galaxies (22% of our sample) that show evidence of current star formation to some degree. There are five extra galaxies, classified as post starbursts, which ceased their star formation process in the last $\sim500$ Myr. If we include these five galaxies into the active galaxy population, then 25% of our galaxy sample show some sign of recent star formation activity. In Figure \[fig3\], we show the spatial distribution of all 196 Coma galaxies; the plotting symbol type reflects our line strength classification scheme discussed above and in Section \[lines\]. Clearly, there is no obvious correlation in the position of the different types of galaxies except that the active star–forming galaxies avoid the cluster center. In addition to the line strength classification scheme, we have explored two objective schemes based on PCA and wavelets. For our PCA analysis, Figure \[fig7\] shows that the first three eigenspectra for our dataset contains 87% of the variance while most of the remaining variance is likely due to the noise in our spectra. Each spectrum can therefore be expressed as a linear combination of these three eigenspectra which in turn can be presented in a spherical coordinate system with only two important parameters (see Figure \[fig10\]). We have defined regions in this two–dimensional space according to our line strength classification scheme. In Table \[tbl3\], we compare the line strength and PCA classifications to check whether the two–dimensional space defined by the eigencoefficient projection angles can be understand in term of the absorption and emission lines of the spectra. The agreement between both classifications is obviously good but nevertheless remarkable as there was no a priori guarantee that such a good separation should be achieved given that the PCA classification is based on the overall continuum subtracted SED while the line strength classification only on the Balmer lines. Figure \[fig10\] demonstrates that simple arbitrary cuts can efficiently separate galaxies according to their spectral line strengths. The only cut which does not clearly reproduce the line strength classification above is the division between [**AB**]{} and [**AE**]{} galaxies. In fact this is mostly due to fact that we have used H$\alpha$ to split these two classes in the line strength classification but have only used the blue part of the spectrum in the PCA analysis. Some of these galaxies show H$\alpha$ in emission while the rest of the Balmer series is seen in absorption. PCA thus classifies the galaxy as an [**AB**]{} galaxy while in the line strength classification it is an [**AB+EM**]{} galaxy based on the presence of faint H$\alpha$ emission. In addition, two galaxies classified as [**AB+EM**]{} by their line strengths are classified as [**PS**]{} by the PCA analysis (fibers 91 and 140). Both galaxies show strong broad Balmer absorption features which are filled with narrower emission. In these cases, it is difficult to separately measure the absorption and emission line components equivalent widths. We estimate that galaxy 91 has $W_o(H{\delta}+H{\gamma}+H{\beta}) \sim 10$ [Å]{} in absorption and galaxy 140 has $W_o(H{\delta}+H{\gamma}+H{\beta}) \sim 15$ [Å]{}. The latter would have been classified as a [**PS**]{} galaxy by the line strength criteria if it did not have the emission. These two galaxies have certainly experienced star formation activity that has been substantially decreased in the last Gigayear thus giving rise to the strong Balmer absorption lines. However, there is still some residual star formation activity going on and thus the weak emission on top of the absorption. We have used a version of the PCA based on the implementation presented in Connolly et al (1995) and Connolly and Szalay (1999). Our method uses the data themselves to generate the eigenbasis in which the observed spectra are projected. We then use another classification scheme, the line strength classification, to subdivide the eigencoefficent parameter space according to it. The line strength classification can be translated strikingly well into separated regions on the two-parameter space given by the angles $\theta$ and $\phi$. Other classification schemes employing PCA have also been used in the literature. For example, Bromley et al (1998) use a PCA to classify the Las Campanas Redshift Survey. Their implementation is very similar to ours. They use the data themselves to generate the eigenbasis as we do. They weigh their spectra by a smoothed version of the mean spectrum, while we continuum subtract each spectrum with a low order polynomial. Their sample is composed of field galaxies instead of galaxy cluster members and therefore their first eigenspectrum (their Figure 1) is more emission line dominated. Unlike us, they only use their first two eigencoefficients to classify their spectra, neglecting the third eigencoefficient that appears to be able to differentiate post-starburst galaxies. They subdivide their two eigenspectra parameter space into six “clans” or spectral types, without using any other external classification scheme for that subdivision. Their mean spectra for their clans show (their Figure 3), nevertheless, a good correlation with star formation. Post-starburst galaxies are, however, not represented. Folkes et al (1999) classify a preliminary sample of the 2dF galaxy survey using a PCA as well. Their sample is made up of field galaxies. Their PCA implementation is similar to us except that they subtract a mean spectrum from their spectra before applying the PCA. Their eigenspectra (their Figure 5) reflect both facts with their mean spectrum being similar to our first eigenspectrum (modulo our continuum subtraction), their first eigenspectrum being similar to our second and their second similar to our third. They use another external classification to subdivide their first two eigencoefficient parameter space to generate their classification. They argue that their third eigencoefficients do not add much additional information for their classification scheme. The ESO-Sculptor Survey has also been classified with a PCA method by Galaz & de Lapparent (1998). They employ the Connolly et al (1995) implementation, that is, the same method we use. Again, their eigenspectra are more emission line dominated than ours as they study a field galaxy population. They decompose their spectra into the same projection angles we do. However, they classify their spectra comparing them to the Kennicutt (1992) spectra projected onto the same eigenbasis generated by the data themselves. They find that their spectra can be interpreted as a one parameter family in their $\theta$-$\phi$ space. Their sample is of comparable size and signal-to-noise to ours but of coarser resolution. On the other hand, both Bromley et al (1998) and Folkes et al (1999) samples are larger and contain noisier spectra than ours. Their classification (as well as Galaz & de Lapparent’s) rely on the spectra being a one parameter family going along stellar activity strength. Our spectra, however, being a cluster sample, has more absorption line galaxies and more post-starburst galaxies that require another parameter to be differentiated from the star forming sequence. We argue that our classification requires two parameters: one, describing the stellar formation activity; the other, the strength of the Balmer absorption features. There have been other approaches to construct classification schemes using PCA. Connolly et al (1995), Connolly & Szalay (1999), Sodré & Cuevas (1997) and Ronen et al (1999) use spectral synthesis models (or galaxy spectra already classified) to construct their PCA eigenbases. They, then, interpret their classifications in terms of their input, either their already-classified spectra or the synthesis model parameters. Given that spectral synthesis models currently available do not have the spectral resolution of our observed spectra, we have chosen to use our own spectra to build our eigenbasis and apply the line strength classification to subdivide the eigencoefficient parameter space. Therefore, not losing the spectral resolution of our data. We have also used wavelets to define a simple classification for these spectra which uses both the absorption and emission lines in conjunction with the continuum shape. As with the PCA analysis, we only utilize the blue part of the spectrum and thus have no information on H$\alpha$. We also do not try and differentiate between the [**EM**]{}, [**AB+EM**]{} and [**EM+AB**]{} line strength types since, at present, we cannot measure the equivalent width of lines directly from the wavelet analysis (although in principle this could be achieved). In Table \[tbl4\], we compare the wavelet classification to the line strength classification. As above, there is good agreement between the two where the line strength sub–classes of [**AB+EM**]{} and [**EM+AB**]{} classifications are mostly shared between the [**EM**]{} and [**AB**]{} wavelet classes (19 [**AB+EM**]{} galaxies were placed in the wavelet [**AB**]{} class as expected since there is no information about H$\alpha$). The only surprises in Table \[tbl4\] are the four galaxies (fibers 84, 138, 201 & 244) classified as [**AB**]{} in wavelet space but [**EM+AB**]{} by the line strength criteria. A visual inspection of these spectra shows that the signal–to–noise of these data is, on average, low. However, all four have moderately weak emission lines that probably missed the top 8 wavelet coefficients because of their size. All four do have strong Calcium H & K indicative of an old stellar population. We also note that the wavelet algorithm has more [**PS**]{} galaxies than the line strength classification. This is primarily due to the hard threshold of $W_o(H{\delta}+H{\gamma}+H{\beta})>15$ [Å]{} imposed in the line strength classification. These extra wavelet [**PS**]{} galaxies (three classified as [**AB**]{} and 7 as [**AB+EM**]{} galaxies by the line strength method) do have Balmer absorption lines, but either they have H$\alpha$ in emission and/or the Balmer absorption is weak (although still stronger than galaxies dominated by old stellar populations). This apparent discrepancy in the classification schemes can be easily understood in terms of the thresholds used in assigning the [**PS**]{} classification, [*i.e.*]{} if the 15 [Å]{} threshold used in the line classification scheme had been lowered, then many of these galaxies would have also been classified as [**PS**]{} by the line strength scheme. The wavelet scheme has simply picked extra [*older*]{} post–starburst galaxies with weaker Balmer absorption lines. In Table \[tbl5\], we show the cross-comparison between the PCA and wavelet classifications. Again, the agreement is very good. The only exceptions are the same cases discussed above given that the PCA classification cuts were applied to reproduce the line strength classification. It is worth mentioning here that the extra wavelet [**PS**]{} classified galaxies discussed above reside in the “bridge of galaxies” that join the [**AB**]{} and [**PS**]{} regions shown in Figure \[fig10\]. This again explains why we find more wavelet classified [**PS**]{} galaxies compared to the other schemes since we are simply sampling more of the galaxy population between these two classes. We have also fitted evolutionary spectral synthesis models to our Coma spectra (only the blue side of the spectrum) to understand the composition and history of the stellar populations in these galaxies. We note that these fits are performed for a particular model, with a chosen star formation prescription, and should be interpreted as such. The star formation history of a galaxy is more complex that the simple parameterized models used (e.g., Abraham et al 1999) and the best fit gives only an indication of the properties of the combined stellar population and helps to compress the spectral information into a handful of meaningful parameters. We have chosen to fit the age of the stellar population and a parameter describing the star formation prescription using two families of models (see section 7.4 for further details). Both parameterizations give similar results (see table \[tbl2\]). It is nevertheless clear from table \[tbl2\] that in some cases the star formation prescription utilized is unable to fit properly the observed spectrum and a different prescription with combinations of different bursts is necessary. We have chosen, at this stage of simple comparison with other classification schemes, not to fit for the intrinsic absorption or the galaxy metallicity, assuming fixed solar metallicity and no intrinsic absorption. Indeed, there are known degeneracies between age–metallicity and age–intrinsic absorption when fitting synthesized SED to observed spectra (e.g., Worthey 1994; Thompson, Weymann & Storrie-Lombardi 2000). In general, a higher metallicity and the effect of neglecting internal extinction will be compensated by an older computed age. Figure \[fig17\] presents the age and time scale, $\tau$, of the best fit synthesized SED using an exponential SFR law. The symbols represent the different line strength classifications while the size of the symbol represented the observed magnitudes of the galaxies. Most of the Coma galaxies cluster along a strip in the age–$\tau$ space. The large dynamic range shown for the $\tau$ parameter gives the impression that galaxies of different line strength classes populate similar regions in this parameter space. However, the general trends of age-$\tau$ values with respect to the spectral line strength classification are as expected: [**AB**]{} galaxies populate regions of old ages ($\ge3-4$ Gyr) and short time–scales of star formation which produce old stellar population while the [**AB+EM**]{}, [**EM+AB**]{} and [**EM**]{} galaxies occupy regions next to the [**AB**]{} galaxies but gradually moving to younger ages and longer $\tau$ as the emission lines strength increases. Finally, the [**PS**]{} galaxies populate the region of short time scale star formation, which correspond to short and intensive star formation histories, and short ages after the strong formation episode. To help visualize these separate regions of the age-$\tau$ space, we show in Figure \[fig18\] the 1, 2 and 3$\sigma$ contours of the age-$\tau$ regions occupied by three representative galaxies drawn from our three main spectral types [*i.e.*]{}, an [**AB**]{} type galaxy (fiber 143), a [**PS**]{} type galaxy (fiber 95) and a [**EM+AB**]{} type galaxy (fiber 117). It is clear from this plot that the [**AB**]{} and [**EM+AB**]{} galaxy inhabit different regions of the age-$\tau$ space (at the 3$\sigma$ level) while there is no intersection for the [**AB**]{} and [**PS**]{} galaxies either. Only the [**EM+AB**]{} and [**PS**]{} galaxies share a similar parameter space (at the 3$\sigma$ level). Nevertheless, it is clear from Figures \[fig18\] and \[fig17\] that the fits of the exponential decaying star formation models to the observed spectra are degenerate along a line in the log $tau$–log age parameter space of slope $\sim$ 3/2. In summary, regardless of this degeneracy, our spectral classifications have a broad relationship to expected physical models of star–formation and therefore, they can be used to study star–formation and galaxy evolution with the Coma cluster. Galaxy Star–Formation in Coma ----------------------------- Our analysis of the Coma cluster has revealed that a substantial fraction (25%) of galaxies in Coma (out to the virial radius) exhibit recent star–forming activity. Such analyses are at the heart of studies of galaxy evolution in clusters and, in more general terms, the “Butcher–Oemler” effect (Butcher & Oemler 1984). Our large fraction of recent star–forming galaxies is more than initially expected for such a low redshift cluster but is in reasonable agreement with other spectroscopic studies of galaxy star–formation in clusters. For example, Caldwell & Rose (1997) found that $\sim15\%$ of galaxies in five nearby clusters showed signs of recent star–formation while the work of Ellingson et al. (2000) shows that $\sim20\%$ of galaxies within the virial radius ($\sim r_{200}$) of CNOC1 clusters ($0.18<z<0.5$) show signs of recent star–formation. However, we do not address here the redshift dependence of the BO effect since that will require a large, homogeneous sample of clusters spanning a significant range in redshift (See Andreon & Ettori 1999; Margoninier & Carvalho 2000). We will address this issue using the main SDSS sample. We explore here possible mechanisms for triggering star–formation in clusters of galaxies. Several authors (Burns et al. 1994; Caldwell & Rose 1997; Metevier, Romer & Ulmer 1999; Wang, Connolly & Brunner 1997) have proposed that shocks caused by cluster merger events can trigger star–formation in cluster galaxies. However, Figures \[fig3\] and \[fig19\] show that our Coma galaxies with recent star–formation ([**EM+AB**]{},[**AB+EM**]{}, [**EM**]{} and [**PS**]{}) are random distributed throughout the cluster with no apparent correlation with the NGC 4839 group. This implies that this merger (or post–merger) has had little effect on the recent star–formation rates of Coma contrary to the scenario outlined in Burns et al. (1994) and Caldwell & Rose (1997). This difference may be the combination of several effects: [*1)*]{} We have a complete, homogeneous, spectroscopically confirmed, sample of Coma galaxies out to large cluster radii [*i.e.*]{} we have targeted all bright galaxies out to the virial radius of Coma regardless of their location, color or morphological type. Caldwell & Rose (1997) were forced to selectively sample specific areas in Coma, because of the smaller field–of–view of their instrument, as well as preferentially targeting early–type galaxies; [*2)*]{} We have used different schemes, and thresholds, for defining post–starburst and star–forming galaxies. For example, if we lowered our threshold for the definition of a [**PS**]{} galaxy (to older and weaker lines) we may find a correlations with the NGC 4839 group. We note however that the extra wavelet [**PS**]{} galaxies (which are older [**PS**]{} galaxies) show no sign of excess clustering with the NGC 4839 group; [*3)*]{} The NGC 4839 is approximately 5–10% (CD96) of the mass of the core of Coma and thus the merger of Coma and this group may not be severe enough to trigger the levels of star–formation seen in more extreme head–on cluster–cluster and group–group mergers (Metevier et al. 1999); [*4)*]{} If the NGC 4839 group is still “spiralling” into Coma then we could simply be seeing the combination of ram–pressure induced star–formation as the field galaxies first infall into the cluster (see Dressler & Gunn 1983) as well as the expected recent star–formation in the NGC 4839 group before it fell in (see Zabludoff et al. 1996). This would also explain the large velocity dispersion measured for the emission–line galaxies (Carlberg et al. 1997). We must await further data to fully discriminate between these possible models or others. It is important to note that we only sample galaxies brighter than $b=18$ and therefore our observations only apply to the brightest galaxy members and not to the dwarf population (see Rakos et al. 2000). Moreover, our observations do not distinguish between different morphological types (see Doi et al. 1995a,b); this will be addressed using the main SDSS survey data. The characteristic radius of the Coma cluster is $r_{200}\sim 1.5$ $h^{-1}$ Mpc (Geller et al 1999), or 1.25 degrees, and thus, we sample the galaxy population out to the virial radius and slightly beyond. This provides a unique opportunity to study the large–scale star–formation histories of galaxies in this nearby cluster. In Figure \[fig20\], we present the fraction of passive (absorption) and active (emission plus post–starburst) galaxies as a function of projected distance from the center of Coma. The azimuthally averaged fraction of active galaxies ([**PS, AB+EM, EM+AB, EM**]{} types) appears to increase slightly with radius from 0.15 to 1.2 degrees$\footnote{At the distance of Coma, one degree on the sky corresponds to 1.2 $h^{-1}$ Mpc.}$, but the observed percentages are also consistent with a constant radial fraction of active galaxies in this region. This trend is different from that observed in the CNOC sample at higher redshift (Ellingson et al 2000), [i.e.]{}, they observe a steeper radial profile for the fraction of active galaxies in their high redshift clusters. However, the results are consistent given the errors and the different analyses performed which render a detailed comparison difficult. Moreover, the target selection used different criteria, although similar, which need to be taken into account to provide a fair comparison. We find no active galaxies in the central (projected) $\sim 200$ $h^{-1}$ Kpc of the Coma cluster. This result is similar to that observed by Ellington et al. (2000) who found a deficit of star–forming galaxies within 0.5$r_{200}$ of the cores of CNOC1 clusters (see also Rakos et al. 2000 who found a similar trend in intermediate redshift clusters). Ellingson et al. (2000) propose that the dense intracluster medium in the cores of clusters inhibits star–formation producing a deficit of star–formation compared to the outskirts of the cluster and the general field population. Our data certainly agrees with this scenario except we do see a reduction of active galaxies beyond $r_{200}$, although this reduction is not statistically significant. We defer a detailed discussion of Coma’s star–forming galaxies, and their relation to the X–ray gas, to a future paper. 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--- author: - | Christian Kleiber\ Universität Basel Achim Zeileis\ Universität Innsbruck bibliography: - 'rootograms.bib' title: Visualizing Count Data Regressions Using Rootograms --- Introduction {#sec:introduction} ============ The area of count data regression has experienced rapid growth over the last two decades. More often than not, the standard Poisson model from the generalized linear model (GLM) toolbox does not suffice in empirical work. Specifically, many data sets are plagued by some form of overdispersion, often resulting from unobserved heterogeneity that can potentially be handled by, e.g., models with additional shape parameters such as the negative binomial distribution or from an excess of zeros for which hurdle and zero-inflation models are available [@rootograms:Mullahy:1986; @rootograms:Lambert:1992]. While various diagnostic tests of dispersion are also available – see, e.g., [@rootograms:Cameron+Trivedi:1990] or [@rootograms:Dean:1992] for some popular tests and [@rootograms:Cameron+Trivedi:2013] for an overview – they typically only identify general issues with model fit and rarely provide clear indications regarding the source of the problems. Suitable graphical tools can guide the search for more appropriate specifications, thereby supplementing and enhancing more formal approaches. If count data regressions are visualized at all, this is currently mainly done in the form of barplots of observed and expected frequencies; see, e.g., Figures 3.1 and 6.4 in [@rootograms:Cameron+Trivedi:2013] for examples and also Figure \[fig:CrabSatellites-comparison\] below. In the present paper, we explore the use of rootograms for assessing the fit. Rootograms are associated with the work of John W. Tukey on exploratory data analysis (EDA) and statistical graphics, culminating in [@rootograms:Tukey:1977]. However, rootograms do not figure prominently there. Instead, early applications, all confined to continuous data, appear in selected contributions to collected volumes and conference proceedings [@rootograms:Tukey:1965; @rootograms:Tukey:1972], which were often not easily available prior to the publication of Tukey’s collected works in the 1980s. Nonetheless, the ideas pertaining to rootograms were known in some circles at an early stage [@rootograms:Healy:1968], and an early paper popularizing the concept is [@rootograms:Wainer:1974]. For further information on the history of statistical graphics we refer to [@rootograms:Friendly+Denis:2001]. The following section introduces a generalized version of the rootogram for regression models (as opposed to univariate distributions) and allowing for weights that can be applied to new data or (weighted) subsamples of a data set. This is useful for assessing in-sample fits as well as out-of-sample predictions and also for situations with survey weights or model-based weights. Several styles of the rootogram, namely standing, hanging, and suspended versions, are briefly described. We also provide some guidelines for interpretation using simulated data. Section \[sec:example\] provides an empirical example, presenting a case where a hurdle model adjusts for excess zeros and also for overdispersion, while the final section \[sec:disc\] discusses how rootograms could be included in routine applications of count data regressions. In supplementary materials, we present two further examples, one involving a finite mixture model requiring the rootogram version with model-based weights mentioned above, the other involving underdispersed data. All analyses are run in [@rootograms:R:2016], and we briefly describe an implementation of our tools in the package in an appendix. Rootograms {#sec:rootograms} ========== Given observations $y_i$ ($i = 1, \dots, n$) we want to assess the goodness of fit of some parametric model $F(\cdot; \alpha_i)$, with corresponding density or probability mass function $f(\cdot; \alpha_i)$. For classic rootograms [see e.g., @rootograms:Friendly:2000 Chapter 2] the parameter vector $\alpha_i$ is the same for all observations $i = 1, \dots, n$. Here, we allow it to be observation-specific, e.g., through dependence on some covariates $x_i$ – a leading case being the GLM with $\alpha_i = g(x_i^\top \beta)$ for some monotonic function $g(\cdot)$. In practice, these parameters are typically unknown and have to be estimated from data. Hence, in the following we assume that we have fitted parameters $\hat \alpha_i$ where estimation may have been carried out on the same observations $i = 1, \dots, n$ (i.e., corresponding to an in-sample assessment) or on a different data set (i.e., out-of-sample evaluation). The estimation procedure itself may be fully parametric or semiparametric etc. as long as it yields fitted parameters $\hat \alpha_i$ for all observations of interest. To judge the goodness of fit of a model with estimated parameters $\hat \alpha_i$ to observations $y_i$ ($i = 1, \dots, n$), a natural idea is to assess whether observed frequencies match expected frequencies from the model. In the case of discrete observations frequencies for the observations themselves could be considered while somewhat more generally frequencies for intervals of observations may be used. Tukey’s original work often considered goodness of fit to the normal distribution on the basis of binned observations, see, e.g., his example involving the heights of 218 volcanos [@rootograms:Tukey:1972]. In this paper, we focus on discrete distributions. For assessing the goodness of fit in regression models, practitioners routinely check some type of residuals, i.e., (weighted) deviations of the observations $y_i$ from the corresponding predicted means. However, this focuses on the first moment of the fitted distribution only while for count data, which are non-negative and typically skewed, further aspects of the distribution are also of interest. Relevant aspects include the amount of (over-)dispersion, skewness (or further aspects of shape), and whether there are excess zeros. Hence, it is natural to consider observed and expected values for a range of counts $0, 1, 2, \dots$ in order to assess the entire fitted distribution. Specifically, in the case of count data with possible outcomes $j = 0, 1, 2, \dots$, the observed and expected frequencies for each integer $j$ are given by $$\begin{aligned} \text{obs}_j & = & \sum_{i = 1}^n I(y_i = j) , \\ \text{exp}_j & = & \sum_{i = 1}^n f(j; \hat \alpha_i) ,\end{aligned}$$ where $I(\cdot)$ is an indicator variable. More generally, one can use a set of breaks $b_0, b_1, b_2, \dots$ that span (a suitable subset of) the support of $y$. Here, we additionally also allow for observation-specific weights $w_i$ ($i = 1, \dots, n$), the observed and expected frequencies are then given by $$\begin{aligned} \text{obs}_j & = & \sum_{i = 1}^n w_i \, I(y_i \in (b_{j}, b_{j + 1}]) , \\ \text{exp}_j & = & \sum_{i = 1}^n w_i \, \{ F(b_{j + 1}; \hat \alpha_i) - F(b_{j}; \hat \alpha_i) \} .\end{aligned}$$ The weights are needed for survey data and also for situations with model-based weights. For example, the latter may represent class membership in mixture models, a case that is relevant in one of our supplementary examples. Styles of Rootograms {#subsec:styles} -------------------- The rootogram compares observed and expected values graphically by plotting histogram-like rectangles or bars for the observed frequencies and a curve for the fitted frequencies, all on a square-root scale. The square roots rather than the untransformed observations are employed to approximately adjust for scale differences across the $j$ values or intervals. Otherwise, deviations would only be visible for $j$’s with large observed/expected frequencies. Different styles of rootograms have been suggested, see Figure \[fig:styles\]: - *Standing:* The standing rootogram simply shows rectangles/bars for $\sqrt{\text{obs}_j}$ and a curve for $\sqrt{\text{exp}_j}$. To assess deviations across the $j$’s, the expected curve needs to be followed as the deviations are not aligned. - *Hanging:* To align all deviations along the horizontal axis, the rectangles/bars are drawn from $\sqrt{\text{exp}_j}$ to $\sqrt{\text{exp}_j} - \sqrt{\text{obs}_j}$ so that they are “hanging” from the curve representing expected frequencies, $\sqrt{\text{exp}_j}$. - *Suspended:* To emphasize mainly the deviations (rather than the observed frequencies), a third alternative is to draw rectangles/bars for the differences between expected and observed frequencies, $\sqrt{\text{exp}_j} - \sqrt{\text{obs}_j}$ (some authors use $\sqrt{\text{obs}_j} - \sqrt{\text{exp}_j}$ instead). The basic version, the standing rootogram, is perhaps the least useful among the three: it simply plots rectangles/bars and a curve representing the model, but the fit is not easily assessed. The other versions both make use of a horizontal reference line, a detail often emphasized by Tukey [e.g., @rootograms:Tukey:1972]. Here, it highlights the discrepancies between observed and expected frequencies. In a sense, hanging rootograms emphasize the fitted values and suspended rootograms the corresponding residuals. We recommend the hanging version as the default as long as residuals are not of main concern, and hence employ hanging rootograms below. We also note that the suspended rootogram exists in several versions; in [@rootograms:Tukey:1972] it was turned upside down, i.e., with a curve resembling expected values below the bars resembling residuals. Here we follow [@rootograms:Friendly:2000]. ![\[fig:styles\] Styles of rootograms for two Poisson models fitted to 100 artificial observations from a Poisson (top row) and negative binomial (bottom row) distribution. Top row: The Poisson model fit ($\hat \mu = 3.34$) captures the true mean ($\mu = 3$) as well as the distributional form with small deviations only. Bottom row: The Poisson model fit ($\hat \mu = 3.32$) does not capture the underlying distribution well ($\mu = 3$, $\theta = 2$), leading to clear deviations in the rootogram.](rootograms-styles) Interpreting Rootograms {#subsec:interpreting} ----------------------- In analyses employing rootograms, one is often interested in detecting patterns such as runs of positive or negative deviations, which highlight aspects of the model fit that might require further attention. For example, Figure \[fig:styles\] presents rootograms for a Poisson model fitted to two simulated data sets from a Poisson (top row) and negative binomial (bottom row) distribution. Both underlying distributions have mean $\mu = 3$, the negative binomial has a shape parameter $\theta = 2$ (while the Poisson is formally a negative binomial with $\theta =\infty$). When fitting a Poisson model to the Poisson data, all three versions of the rootogram in the top row exhibit only small deviations: The standing version shows that the curve representing expected frequencies closely tracks the histogram representing observed frequencies, there are also no clear patterns in the hanging and suspended versions. All this indicates that the model fits well. In contrast, when fitting a Poisson model to the negative binomial data in the bottom row there are substantial departures of the model from the data: in the standing version, the curve representing expected frequencies does not track the observed frequencies, there are also discernible patterns in both the hanging and suspended variants. Specifically, for the latter the rootogram bars form a ‘wave-like’ pattern around the horizontal reference line: the data exhibit too many small counts, notably zeros, as well as too many large counts for a Poisson model to provide an adequate fit. In summary, the patterns encountered in the bottom row of Figure \[fig:styles\] reflect a substantial amount of overdispersion that is not captured by the fitted Poisson distribution. The patterns seen in the bottom panel of Figure \[fig:styles\] are theoretically supported by results presented by [@rootograms:Mullahy:1997], who shows that Poisson mixtures exhibit a larger number of zeros (compared with a Poisson null model) as well as more mass in the upper tails and less mass in the center of the distribution. Mullahy’s results rely on earlier work of [@rootograms:Shaked:1980], who shows that mixing will generally spread out a distribution (from the exponential family) towards its tails. The negative binomial distribution is a gamma mixture of the Poisson distribution, hence these arguments are directly relevant in the case at hand. While excess zeros are strictly implied by overdispersion [compare @rootograms:Mullahy:1997 Prop. 1], there also exist situations in practice where the number of zeros is so large that merely correcting for overdispersion via, e.g., a negative binomial model does not solve the problem. These tend to exhibit a spike at zero in graphical displays and are often best treated by fitting a two-part model. We shall encounter an example below. An Example from Ethology {#sec:example} ======================== In this section we present an empirical illustration revisiting a well-known data set from ethology, for which excess zeros and, more generally, overdispersion require treatment. We select models using information criteria, notably the BIC, and use rootograms for highlighting deficiencies of fitted models. [@rootograms:Brockmann:1996] investigates horseshoe crab mating. The crabs arrive on the beach in pairs to spawn. Furthermore, unattached males also come to the beach, crowd around the nesting couples and compete with attached males for fertilizations. These so-called satellite males form large groups around some couples while ignoring others. [@rootograms:Brockmann:1996] shows that the groupings are not driven by environmental factors but by properties of the nesting female crabs. Larger females that are in better condition attract more satellites. @rootograms:Agresti:2013 [Chapter 4.3] reanalyzes these data, modeling the number of satellites using count data regression techniques. The main explanatory variable is the female crab’s carapace width, but its color and spine condition are also considered in some analyses – with the ordered factors for color and spine condition often treated as numeric variables. In his analysis, [@rootograms:Agresti:2013] starts out from a Poisson model with the standard log link and then goes on to consider both Poisson and negative binomial models with both log and identity links. He finds that among these the negative binomial model fits best but also notes that further refinements might be possible, e.g., by allowing for zero inflation. ![\[fig:CrabSatellites-rootograms\] Hanging rootograms for crab satellite models (counts $0, \dots, 15$).](rootograms-CrabSatellites-rootograms) To illustrate how rootograms can help in judging the goodness of fit of various count regression models for this data, we extend the analysis of [@rootograms:Agresti:2013] in the following way: we consider both Poisson and negative binomial regressions (with log link) and hurdle versions of these (with a logit-type binary part) to allow for excess zeros. The carapace width and a numeric coding of the color variable are used as regressors in all (sub-)models. To compare the relative performances of the four models, we employ the Bayesian information criterion (BIC), yielding: Poisson (BIC = 931.0, df = 3), negative binomial (BIC = 769.5, df = 4), hurdle Poisson (BIC = 755.1, df = 6), and hurdle negative binomial (BIC = 736.8, df = 7). These results already suggest that the hurdle negative binomial model fits best. However, a look at the corresponding hanging rootograms (for counts $0, \dots, 15$) in Figure \[fig:CrabSatellites-rootograms\] provides much more insight into the pros and cons of the various models: - *Poisson:* The wave-like pattern in the rootogram bars in the top left panel shows that the counts 1, …, 4 are overfitted while 0 and most counts from 6 onwards are underfitted. This indicates a substantial amount of overdispersion in the data, the clear lack of fit for 0 could be an additional indication of excess zeros. - *Negative binomial:* The rootogram does no longer exhibit the wave-like pattern of the Poisson model, showing that the overdispersion is accounted for much better in this model. However, the underfitting of the count 0 and clear overfitting for counts 1 and 2 is typical for data with excess zeros. Note also that the fitted negative binomial model implies a decreasing probability mass function (with $\theta = 0.93$), which is not in line with the data structure. - *Hurdle Poisson:* The rootogram now shows a perfect fit for the count 0 (by design of the hurdle model). However, there is still overdispersion in the remaining positive counts that is again reflected by a wave-like pattern, note also the clear underfitting of the count 1. - *Hurdle negative binomial:* The rootogram shows that this model fits the data quite well. There are no clear patterns of departure anymore and the deviations between observed and predicted frequences are very small for most of the counts. To highlight that the conclusions above are drawn more easily based on the proposed rootograms than from more traditional visualizations, Figure \[fig:CrabSatellites-comparison\] provides different types of displays for the poorly fitting Poisson model (left column) and the well-fitting hurdle negative binomial (NB) model (right column). More traditional analyses include visualizations of some sort of residuals, for example using quantile-quantile (or Q-Q) plots or plots of residuals vs. fitted values, and also of observed vs. expected frequencies. All three versions are used in sources such as [@rootograms:Cameron+Trivedi:2013]. The rows of Figure \[fig:CrabSatellites-comparison\] show: 1. Quantile-quantile (or Q-Q) plots of randomized quantile residuals [@rootograms:Dunn+Smyth:1996] vs. the corresponding theoretical standard normal quantiles – along with a gray shaded area corresponding to the range from the 5% up to the 95% quantile of the randomized distribution. The curvature of the Poisson model clearly indicates overdispersion but the excess zeros are not directly visible. The hurdle NB model, on the other hand, appears to fit rather well. 2. Barplots of observed and expected frequencies. The excess zeros in the Poisson model are rather obvious while the overdispersion is somewhat obscured due to the tiny frequencies of the larger counts. Also, the deviations are not aligned and hence are more difficult to track than in the hanging rootogram. 3. Scatterplots of Pearson residuals vs. fitted values (means). Such displays are generally more difficult to interpret than in linear regression models due to the discrete and asymmetric response distribution. While it can be seen that the fit for the Poisson is not as good as for the hurdle NB model, the overall quality is harder to judge than in the previous displays. For the same reason, a scatter plot of observations vs. fitted means (not shown) would not be straightforward to interpret. Overall, rootograms clearly bring out several aspects that are not as easily seen in traditional displays. The barplots of observed and expected frequencies are closest in spirit to the rootogram, but suffer from an overemphasis of the tails (addressed by the square-root transformation in the rootogram) and the curved shape of the mass functions (addressed by the special alignment of observed vs. expected values and the horizontal reference line in the rootogram). ![\[fig:CrabSatellites-comparison\] Alternative graphical model checks for the crab satellites data. Rows: Q-Q plot based on randomized quantile residuals, barplot of observed and fitted frequencies, and scatterplot of Pearson residuals vs. fitted values (means). Columns: Poisson model and negative binomial hurdle model.](rootograms-CrabSatellites-plots) [lD[.]{}[.]{}[3]{}D[.]{}[.]{}[3]{}cD[.]{}[.]{}[3]{}D[.]{}[.]{}[3]{}]{} &&&\ &&&&&\ (Intercept)&0.43&-10.07&&1.47&-10.07\ &(0.94)&(2.81)&&(0.07)&(2.81)\ width&0.04&0.46&&&0.46\ &(0.03)&(0.10)&&&(0.10)\ color&0.01&-0.51&&&-0.51\ &(0.09)&(0.22)&&&(0.22)\ Log(theta)&1.53&&&1.50&\ &(0.35)&&&(0.35)&\ N&173&&&173&\ Log-likelihood&-350.4&&&-351.0&\ AIC&714.7&&&712.1&\ BIC&736.8&&&727.8&\ To further explore, the well-fitting hurdle NB model, its parameter estimates (and standard errors) are reported in the first two columns of Table \[tab:CrabSatellites\]. Interestingly, this reveals that the female crab’s carapace width and color both clearly affect the probability of having any satellites (binary zero hurdle part). Specifically, larger crabs are much more likely to have satellites. However, given that there is at least one satellite neither carapace width nor color are individually significant (zero-truncated count part). Omitting both variables improves the fit in terms of both AIC and BIC (hurdle NB, model 2, Table \[tab:CrabSatellites\]). The rootogram of the simplified model (see Figure \[fig:CrabSatellites-boot\]) is very similar to that of the full hurdle model. Additionally, identity (rather than log) links or a zero-inflation (rather than hurdle) specification could be employed but are omitted here for compactness. Both lead to qualitatively identical insights and similar patterns in the rootograms while neither leads to improvements over the negative binomial hurdle model. We conclude with a comparison of predicted effects for the mean function from several models. (Figure \[fig:CrabSatellites-effects\]), evaluated for increasing carapace width at the mean color (= 2.5 in the center of the scale 1, …, 4). This shows that, compared to the identity link model preferred by [@rootograms:Agresti:2013], the hurdle model leads to very similar predictions at average widths while avoiding negative predictions for small widths and at the same time increasing even more slowly for large widths. This complements the findings from the rootograms and underlines that the hurdle model fits the data rather well. ![\[fig:CrabSatellites-effects\] Predicted effect for the mean number of satellite at increasing carapace width and mean color.](rootograms-CrabSatellites-effects) Discussion and Concluding Remarks {#sec:disc} ================================= Several flavors of rootograms have been discussed as graphical diagnostic tools for visualizing complex regression models for count data. They combine exploratory data analysis with model-based inference by bringing out discrepancies between observed and fitted distributions. Unlike other model-based graphics that often focus on effects on the mean of the fitted distribution (e.g., effect displays), rootograms capture deviations across the support of the entire distribution and hence can help to diagnose misfit regarding scatter and/or shape. This is particularly relevant for count data models, which are often affected by problems such as overdispersion and/or excess zeros. To incorporate rootograms into the model-building workflow for count data regression models, their graphical information can be used to complement standard techniques such as information criteria (AIC, BIC, …). Using a range of basic models – as done in Figure \[fig:CrabSatellites-rootograms\] for the crab satellites data – rootograms can guide the practitioner in deciding whether overdispersion (e.g., Poisson vs. negative binomial models) and/or extra zeros (e.g., hurdle or zero-inflation vs. ‘one-part’ models) are relevant issues in the data at hand. The models upon which the rootograms are based should use a reasonable first selection of regressors (e.g., a standard specification from the literature or a model involving all potentially relevant variables). Some users may want to include a built-in calibration of uncertainty. To this end, Figure \[fig:CrabSatellites-boot\] provides the rootogram of the simplified model from Table \[tab:CrabSatellites\] along with (pointwise) 95% confidence intervals obtained via a parametric bootstrap based on 10,000 replications from the estimated model. Note that the resulting intervals are not substantially different from the “warning limits” of @rootograms:Tukey:1972 [p. 61], set at $\pm 1$, hence the latter would seem to be a useful practical device at a minimum cost. ![\[fig:CrabSatellites-boot\] Hanging rootogram for the second hurdle NB model. Pointwise confidence intervals are added based on a rule of thumb ($\pm 1$, blue dashed) or the 2.5% and 97.5% quantiles from a parametric bootstrap (10,000 replications from the estimated model, blue solid).](rootograms-CrabSatellites-boot) Two further examples are available via supplements: the first uses a large data set from health economics, available from the package supplementing [@rootograms:Kleiber+Zeileis:2008] under the name and exhibiting a different type of unobserved heterogeneity, the second involves the less frequent case of underdispersion and is available from the package [@rootograms:Zeileis+Kleiber:2016] under the name . See the appendix below for more information on the latter package. Computational Details {#sec:computational .unnumbered} ===================== Our results were obtained using  3.2.4 [@rootograms:R:2016] with the packages  0.1-5 [@rootograms:Zeileis+Kleiber:2016; @rootograms:Zeileis+Kleiber+Jackman:2008],  7.3-45 [@rootograms:Venables+Ripley:2002], and  2.3-13 [@rootograms:Leisch:2004; @rootograms:Gruen+Leisch:2008], and were identical on various platforms including PCs running Debian GNU/Linux (with a 3.2.0-1-amd64 kernel) and Mac OS X, version 10.10.5. Acknowledgments {#sec:acknowledgments .unnumbered} =============== The authors thank the editor, the associate editor, and several anonymous reviewers for many valuable suggestions on earlier versions of this paper. Implementation =============== For an overview of count data regression models in we refer to [@rootograms:Zeileis+Kleiber+Jackman:2008], where implementations of hurdle and zero-inflation models are described in some detail. The corresponding fitting functions have now been moved to the package, a new package that is currently under development by the authors of the present paper. First versions are already available from <http://R-Forge.R-project.org/projects/countreg/>. The current implementation of rootograms in provides a generic function along with several methods for different types of models/data. The methods all proceed in the same way: They first compute the observed and expected frequencies, $\text{obs}_j$ and $\text{exp}_j$ respectively (see Section \[sec:rootograms\]), and then call the default method that computes all required coordinates for drawing the rootograms. The latter has the following arguments: rootogram(object, fitted, breaks = NULL, style = c(“hanging”, “standing”, “suspended”), scale = c(“sqrt”, “raw”), plot = TRUE, width = NULL, xlab = NULL, ylab = NULL, main = NULL, ...) The arguments and need to provide the tables/vectors of observed and fitted frequencies. (The first argument is called rather than for consistency with the generic function that only takes one required argument and .) The need to be specified if a continuous distribution is employed while for a discrete distribution one may want to set the of the bars to leave small gaps between the bars (as in our examples). Additionally, one of three s can be specified: (default), , or . The object returned is then a ‘’ with all the coordinates needed for plotting, and this is also drawn directly by default () along with the specified graphical arguments (, , , ). By default, the base graphics method is used for drawing rootograms. In addition, there is also an method for drawing rootograms using the package [@rootograms:Wickham:2009]. Above we used methods for objects of classes ‘’ and ‘’. There are further methods available, currently for univariate distributions fitted via [to objects of class ‘’, @rootograms:Venables+Ripley:2002], zero-inflated models [objects of class ‘’, @rootograms:Zeileis+Kleiber+Jackman:2008], zero-truncated models (objects of class ‘’, as fitted by the function in ), generalized additive models [objects of class ‘’, @rootograms:Wood:2006], and for selected count distributions falling within the framework of generalized additive models for location, scale and shape [objects of class ‘’, @rootograms:Rigby+Stasinopoulos:2005; @rootograms:Stasinopoulos+Rigby:2007]. Supplementary Material: Demand for Medical Care {#sec:NMES1988} =============================================== Here we provide an additional example, taken from health economics. Its purpose is to present a larger data set with a much greater range of values for the response and further to show how fitted models resulting from modern tools such as finite mixture models can also be assessed via rootograms. The data are cross-sectional data originating from the US National Medical Expenditure Survey (NMES) conducted in 1987 and 1988. The NMES is based upon a representative, national probability sample of the civilian non-institutionalized population and of individuals admitted to long-term care facilities during 1987. The subsample used here comprises only individuals aged 66 and over, all of whom are covered by Medicare (a public insurance program providing substantial protection against health-care costs). For users, these data are conveniently available from the package supplementing [@rootograms:Kleiber+Zeileis:2008] under the name . They have been explored originally by [@rootograms:Deb+Trivedi:1997] using finite mixtures of count data regressions. [@rootograms:Zeileis+Kleiber+Jackman:2008] employ the data for illustration of hurdle and zero-inflation models while [@rootograms:Cameron+Trivedi:2013] reinvestigate finite mixtures. Here, we follow the latter approach but employ a slightly reduced set of regressors to facilitate interpretation while still obtaining reasonably good fits. ![\[fig:NMES1988-rootograms\] Hanging rootograms for NMES 1988 models.](rootograms-NMES1988-rootograms) Figure \[fig:NMES1988-rootograms\] displays the rootograms for a single negative binomial regression as well as for a finite mixture of two negative binomial regressions. For the latter, the mixture model (upper panel, right) as well as both components (lower panel) are given. The corresponding parameter estimates (standard errors in parentheses) as well as the sums of the posterior weights (denoted $N$) are reported in Table \[tab:NMES1988\]. [lD[.]{}[.]{}[3]{}cD[.]{}[.]{}[3]{}cD[.]{}[.]{}[3]{}]{} &&&&&\ (Intercept)&0.80&&-0.96&& 1.46\ &(0.06)&&(0.39)&&(0.15)\ health: poor/average&0.34&& 0.40&& 0.29\ &(0.05)&&(0.14)&&(0.06)\ health: excellent/average&-0.38&&-0.22&&-0.45\ &(0.06)&&(0.16)&&(0.10)\ chronic&0.19&& 0.27&& 0.16\ &(0.01)&&(0.04)&&(0.02)\ gender: male/female&-0.09&&-0.16&&-0.08\ &(0.03)&&(0.09)&&(0.05)\ school&0.03&& 0.07&& 0.01\ &(0.00)&&(0.02)&&(0.01)\ insurance: yes/no&0.35&& 1.70&&-0.10\ &(0.04)&&(0.32)&&(0.10)\ medicaid: yes/no&0.31&& 0.80&& 0.17\ &(0.06)&&(0.26)&&(0.08)\ Log(theta)& 0.2&&-0.4&& 0.9\ N&4406&&1744.9&&2661.1\ Log-likelihood&-12215.0&&-12149.8&&\ AIC&24448.0&& 24337.7&&\ BIC&24505.5&& 24459.1&&\ The single NB regression clearly misfits, especially for the low counts $0, 1, 2$, while the mixture NB provides an improved fit. It is possible to study the mixture model in more detail by decomposing observed and expected frequencies into the individual components and visualizing them separately. To this end, the observed and expected frequencies are computed as weighted sums using the posterior probabilities for each component. Figure \[fig:NMES1988-rootograms\] (bottom panels) highlights nicely that both components fit rather well. It also brings out the different means and variances in the two components. Specifically, the first component contains a fraction of $0.4 = 1744.9/4406$ of all observations and is characterized by a zero-modal rootogram. On average, the corresponding individuals have fewer physician office visits but at the same time a rather high variance. The parameter estimates are mostly larger (in absolute values) than in the second component, especially for the insurance and medicaid parameters. In contrast, the second component is characterized by a unimodal rootogram with comparatively lighter tails. On average, the corresponding individuals have more physician office visits but at the same time a smaller variance. The first group may, therefore, be seen as the group of occasional users, for which the number of visits likely depends on the severity of the issues, while the second group may be seen as the group of regular users, for which the number of visits often results from the presence of chronic conditions. Indeed, when splitting the patients into two clusters (with hard assignment to the clusters according to the highest posterior probability), it can be seen that the second cluster has a lower proportion of persons with excellent health status (10.1% vs. 7.1%), or without chronic diseases (34.4% vs. 19.8%), and a higher proportion of insured persons (64.7% vs. 81.7%). Moreover, further unobserved factors such as the type of diseases and medication might be captured by the two latent components. Supplementary Material: Takeover Bids {#sec:TakeoverBids} ===================================== Our final example uses data from finance. Its purpose is to present an application with underdispersion and fewer zeros than in the Poisson model. The data comprise a set of firms that were targets of takeover bids during the period 1978–1985. They were initially analyzed by [@rootograms:Jaggia+Thosar:1993] using a standard Poisson regression and are reanalyzed in @rootograms:Cameron+Trivedi:2013 [Chapter 5]. The response variable is the number of takeover bids (after the initial bid received by the target firm) and a number of regressor variables capturing the defensive actions of the target management and firm-specific characteristics as well as potential interventions by federal regulators are employed. A more detailed description of the variables is provided in @rootograms:Cameron+Trivedi:2013 [Table 5.2]. Coefficient estimates (and standard errors) of the Poisson regression model are reported in the first column of Table \[tab:TakeoverBids\]. [lD[.]{}[.]{}[3]{}cD[.]{}[.]{}[3]{}cD[.]{}[.]{}[3]{}]{} &&&&&\ (Intercept)&0.99&&1.14&&2.15\ &(0.53)&&(0.76)&&(3.47)\ legalrest: yes/no&0.26&&0.44&&0.97\ &(0.15)&&(0.21)&&(0.98)\ realrest: yes/no&-0.20&&-0.00&&-2.72\ &(0.19)&&(0.25)&&(1.00)\ finrest: yes/no&0.07&&0.27&&-1.47\ &(0.22)&&(0.27)&&(1.17)\ whiteknight: yes/no&0.48&&0.88&&1.19\ &(0.16)&&(0.28)&&(0.87)\ bidpremium&-0.68&&-1.35&&0.82\ &(0.38)&&(0.53)&&(2.48)\ insthold&-0.36&&-0.66&&-1.84\ &(0.42)&&(0.61)&&(2.41)\ regulation: yes/no&-0.03&&-0.06&&-1.14\ &(0.16)&&(0.22)&&(0.98)\ size&0.18&&0.24&&0.35\ &(0.06)&&(0.07)&&(1.02)\ size$^2$&-0.01&&-0.01&&0.01\ &(0.00)&&(0.00)&&(0.18)\ N&126&&126&&\ Log-likelihood&-184.9&&-159.5&&\ AIC&389.9&&359.0&&\ BIC&418.3&&415.7&&\ The number of bids ranges from $0, \dots, 10$ so that a Poisson model might capture the data well. However, only 7.1% of the observations are zeros which is fewer than expected under a Poisson model, leading to underdispersion in the model. This is also brought out clearly by the corresponding hanging rootogram in the top left panel of Figure \[fig:TakeoverBids-plots\]. One strategy to improve the model is to employ a hurdle Poisson regression model, see the second and third column of Table \[tab:TakeoverBids\]. This appropriately captures the fewer zeros and leads to a satisfactory fit in the rootogram (top right panel of Figure \[fig:TakeoverBids-plots\]). ![\[fig:TakeoverBids-plots\] Hanging rootograms (top) and Q-Q residuals plots (bottom) for the Poisson (left) and hurdle Poisson (right) models for the takeover bids data.](rootograms-TakeoverBids-plots) In comparison, the corresponding Q-Q residuals plots (based on randomized quantile residuals) also bring out the underdispersion in the Poisson model – by way of the curvature – and the satisfactory fit of the hurdle Poisson model. However, it is less obvious that the underdispersion is mainly due to the fewer zeros in the data.

--- address: 'Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA' author: - 'Juan C. Migliore' title: The Geometry of Hilbert Functions --- Introduction ============ The title of this paper, “The geometry of Hilbert functions," might better be suited for a multi-volume treatise than for a single short article. Indeed, a large part of the beauty of, and interest in, Hilbert functions derives from their ubiquity in all of commutative algebra and algebraic geometry, and the unexpected information that they can give, very much of it expressible in a geometric way. Most of this paper is devoted to describing just one small facet of this theory, which connects results of Davis (e.g. [@davis]) in the 1980’s, of Bigatti, Geramita and myself (cf. [@BGM]) in the 1990’s, and of Ahn and myself (cf. [@AM]) very recently. On the other hand, we have an alphabet soup of topics that play a role here: UPP, WLP, SLP, ACM at the very least. It is interesting to see the ways in which these properties interact, and we also try to illustrate some aspects of this. There are almost as many different notations for Hilbert functions as there are papers on the subject. We will use the following notation. If $I$ is a homogeneous ideal in a polynomial ring $R$, we write $$h_{R/I}(t) := \dim (R/I)_t.$$ If $I$ is a saturated ideal defining a subscheme $V$ of $\mathbb P^n$ then we also write this function as $h_V(t)$ or $h_{R/I_V}(t)$. So where is the geometry? Of course $\dim R_t = \binom{t+n}{n}$, so the information provided by the Hilbert function is equivalent to giving the dimension of the degree $t$ component of $I$. This dimension is one more than the dimension of the linear system of hypersurfaces of degree $t$ defined by $I_t$ (since this latter dimension is projective). What is the base locus of this linear system? Of course $V$ is contained in this base locus, but it may contain more. The results in this paper (e.g. Theorem \[BGM general results\], Theorem \[AM general\], Theorem \[BGM UPP results\] and Theorem \[AM UPP results\]) can be viewed as describing the dimension, irreducibility and reducedness of this base locus, based on information about the Hilbert function, and other basic properties, of $V$. We will see that under some situations, just knowing the dimension of this linear system in two consecutive degrees can force the base locus to contain a hypersurface, or anything smaller. (We will concentrate on the curve case.) An important starting point for us (and indeed for almost any discussion of Hilbert functions of standard graded algebras) is Macaulay’s theorem bounding the growth of the Hilbert function. Once we have this, we need Gotzmann’s results about what happens when Macaulay’s bound is achieved. These are both discussed in Section \[preliminary section\], as are several other results related to these. In Section \[WLP section\] we recall the notions of the Uniform Position Property (UPP) and the Weak Lefschetz Property (WLP) and some of their connections. Subsequent sections, especially Section \[UPP results\], continue the discussion of UPP. WLP, while often less visible, lurks in the background of many of the results and computations of this paper, and in fact is an important object of study. We include a short discussion of the behavior of WLP in families of points in Section \[WLP section\], including a new example (Example \[WLP in families\]) showing how, for fixed Hilbert function, WLP can hold in one component of the postulation Hilbert scheme and not hold in another. See also Theorem \[delta 2\]. The focus in this article is the situation where the first difference of the Hilbert function of a set of points, $Z$, in projective space $\mathbb P^n$ attains the same value in two consecutive degrees: $\Delta h_Z(d) = \Delta h_Z(d+1) = s$. Depending on the relation between $d$, $s$ and certain invariants of $Z$, we will get geometric consequences for the base locus. In Section \[set stage\] we describe these relations, setting the stage for the main results. These main results are given in Sections \[general results\] and \[UPP results\]. Here we see that under certain assumptions on $d$, the condition $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ guarantees that the base locus of the linear system $|I_d|$ is a curve of degree $s$. This comes from work in [@davis], [@BGM] and [@AM]. Other results follow as well. What is surprising here is that the central condition of [@AM], namely that $d > r_2(R/I_Z)$ (see Section \[preliminary section\] for the definition), is much weaker than the central assumption of the comparable results in \[Bigatti-Geramita-Migliore\], namely $d \geq s$, but the results are very similar. Section \[general results\] focuses on the general results, while Section \[UPP results\] turns to the question of what can be said about this base locus when the points have UPP. There are some differences in the results of \[Bigatti-Geramita- Migliore\] and [@AM] as a result of the differences in these assumptions. Section \[example section\] studies these, and gives examples to show that they are not accidental omissions. Some very surprising behavior is exhibited here. I am grateful to Irena Peeva for asking me to write this paper, which I enjoyed doing. In part it is a greatly expanded version of a talk that I gave in the Algebraic Geometry seminar at Queen’s University in the fall of 2004, and I am grateful to Mike Roth and to Greg Smith for their kind invitation. I would like to thank Jeaman Ahn, Chris Francisco, Hal Schenck and especially Tony Iarrobino for helpful comments. And of course I am most grateful to my co-authors Anna Bigatti and Tony Geramita ([@BGM]) and Jeaman Ahn ([@AM]) for their insights and for the enjoyable times that we spent in our collaboration. During the writing of this paper, and some of the work described here, I was sponsored by the National Security Agency (USA) under Grant Number MDA904-03-1-0071. Maximal growth of the Hilbert function {#preliminary section} ====================================== We first collect the notation that we will use throughout this paper. Let $k$ be a field of characteristic zero and let $R = k[x_1,\dots, x_n]$. Let $Z \subset \mathbb P^{n-1}$ be any closed subscheme with defining (saturated) ideal $I = I_Z$. - The [*Hilbert function of $Z$*]{} is the function $$h_Z(t) = \dim(R/I_Z)_t$$ We also may write $h_{R/I}(t)$ for this function. If $A$ is Artinian then we write $$h_A(t) = \dim A_t$$ for its Hilbert function. - We say that $Z$ is [*arithmetically Cohen-Macaulay*]{} (ACM) if the coordinate ring $R/I_Z$ is a Cohen-Macaulay ring. Note that if $Z$ is a zero-dimensional scheme then it is automatically ACM. If $F$ is a homogeneous polynomial, by abuse of notation we will also denote by $F$ the hypersurface of $\mathbb P^{n-1}$ defined by $F$. \[alpha definition\] For a homogeneous ideal $I$ we define $$\alpha = \min \{ t \ | \ I_t \neq 0 \},$$ i.e. $\alpha$ is the [*initial degree*]{} of $I$. If $A = R/I$ is a standard graded $k$-algebra, then there is a famous bound, due to Macaulay (cf. [@fsmacaulay]), that describes the maximum possible growth of the Hilbert function of $A$ from any degree to the next. To give this bound, we need a little preparation. \[ibinomexp\] The [*$i$-binomial expansion*]{} of the integer $c$ ($i, c >0$) is the unique expression $$c = \binom{m_i}{i} + \binom{m_{i-1}}{i-1} + \dots + \binom{m_j}{j},$$ where $m_i > m_{i-1} > \dots > m_j \geq j \geq 1$. Note that the assertion that this representation is unique is something that has to be checked! If $c \in {\mathbb Z}$ ($c>0$) has $i$-binomial expansion as in Definition \[ibinomexp\], then we set $$c^{\langle i \rangle} = \binom{m_i+1}{i+1} + \binom{m_{i-1}+1}{i} + \dots + \binom{m_j+1}{j+1}.$$ Note that this defines a collection of functions $^{\langle i \rangle}: \mathbb Z \rightarrow \mathbb Z$. For example, the 5-binomial expansion of 76 is $$76 = \binom{8}{5} + \binom{6}{4} + \binom{4}{3} + \binom{2}{2},$$ so $$76^{\langle 5 \rangle} = \binom{9}{6} + \binom{7}{5} + \binom{5}{4} + \binom{3}{3} = 111.$$ \[o-seq\] A sequence of non-negative integers $\{ c_i : i \geq 0 \}$ is called an [*O-sequence*]{} if $$c_0 = 1 \ \hbox{ and } \ c_{i+1} \leq c_i^{\langle i \rangle},$$ for all $i$. An O-sequence is said to have [*maximal growth from degree $i$ to degree $i+1$*]{} if $c_{i+1} = c_i^{\langle i \rangle}$. The importance of the binomial expansions described above becomes apparent from the following beautiful theorem of Macaulay: \[macaulay thm\] The following are equivalent: - $\{ c_i : i \geq 0 \}$ is an O-sequence; - $\{ c_i : i \geq 0 \}$ is the Hilbert function of a standard graded $k$-algebra. In other words, a sequence of non-negative integers is the Hilbert function of a standard graded $k$-algebra if and only if the growth from any degree to the next is bounded as above. Remember that the Hilbert function is eventually equal to the Hilbert polynomial, at which point $c_{i+1} = c_i^{\langle i \rangle}$, but until that point, anything is allowed as long as we maintain $c_{i+1} \leq c_i^{\langle i \rangle}$. One can ask when such a sequence is the Hilbert function of a [*reduced*]{} $k$-algebra. This was answered by Geramita, Maroscia and Roberts (cf. [@GMR]), by introducing the [*first difference*]{} of the Hilbert function. Given a sequence of non-negative integers $ \underline{c} = $$\{ c_i : i \geq 0 \}$, the [*first difference*]{} of this sequence is the sequence $\Delta \underline{c} := \{ b_i \}$ defined by $b_i = c_i - c_{i-1}$ for all $i$. (We make the convention that $c_{-1} = 0$, so $b_0 = c_0 = 1$.) We say that $\underline{c}$ is a [*differentiable O-sequence*]{} if $\Delta \underline{c}$ is again an O-sequence. By taking successive first differences, we inductively define the [*$k$-th difference*]{} $\Delta^k \underline{c}$. \[first diff\] An important fact to remember is that if $Z$ is a zero-dimensional scheme with Artinian reduction $A$ then $$h_A(t) = \Delta h_Z(t) \hbox{ for all $t$}.$$ It follows that ($\Delta h_Z(t) : t \geq 0)$ is a finite sequence of positive integers, called the [*$h$-vector*]{} of $Z$. Similarly, if $V$ is ACM of dimension $d$ with Artinian reduction $A$ then $h_A(t) = \Delta^{d+1} h_V (t) \hbox{ for all $t$}$. The following is the theorem of Geramita, Maroscia and Roberts mentioned above. It guarantees the existence of a reduced subscheme of $\mathbb P^{n-1}$ with given Hilbert function (remember that $R = k[x_1,\dots,x_n]$), under a simple hypothesis on the Hilbert function. \[GMR theorem\] Let $ \underline{c} = \{ c_i \}$ be a sequence of non-negative integers, with $c_1 = n$. Then $\underline{c}$ is the Hilbert function of a standard graded $k$-algebra $R/I$, with $I$ radical, if and only if $\underline{c}$ is a differentiable O-sequence. Note that if $I$ is any saturated ideal (reduced or otherwise), and if $L$ is a general linear form, then we have $[I:L] \cong I(-1)$ as graded modules. It follows that $L$ induces an injection $ \times L : ((R/I)(-1))_t \rightarrow (R/I)_t$, i.e.$R/I$ has depth at least 1. Hence the first difference of the Hilbert function of $R/I$ is the Hilbert function of $R/(I,L)$, again a standard graded $k$-algebra, so the Hilbert function of $R/I$ is a differentiable O-sequence. This shows that Theorem \[GMR theorem\] would also be true if “radical" were replaced by “saturated" (since if a radical ideal can be constructed for a given differentiable O-sequence, this ideal is also saturated). The real heart of Theorem \[GMR theorem\] is that any such sequence can be achieved by a radical ideal. We should stress that Theorem \[GMR theorem\] guarantees the existence of a reduced (even non-degenerate) subscheme of $\mathbb P^n$, but it does not guarantee that what we get will be equidimensional. It also does not say anything about higher differences. (Note that even if $I$ is saturated, this is not necessarily true of $(I,L)$.) We give two examples. 1. Let $I = (x_1,x_2)$ in $k[x_0,x_1,x_2,x_3]$, so $I$ defines a line, $\lambda$, in $\mathbb P^3$. Let $F$ be a homogeneous polynomial of degree 3, non-singular along $\lambda$. Let $J = (I^2,F)$. Then $J$ is the saturated ideal of a non-reduced subscheme of degree 2 and genus $-2$ (cf. [@dble]). Its Hilbert function is the sequence $1, 4, 7, 9, 11, \dots$ (with Hilbert polynomial $2t+3$). The smallest genus for a reduced equidimensional subscheme of degree 2 is $-1$ (two skew lines), so this Hilbert function does not exist among reduced, equidimensional curves in $\mathbb P^3$. However, adding points reduces the genus while not affecting the degree, and in fact this Hilbert function occurs for the union of two skew lines and one point, which is indeed reduced. 2. Consider the sequence $\underline{c} = 1, 4, 10, 17, 26, 35, \dots $ (with Hilbert polynomial $9t - 11 +1$, so it corresponds to a curve of degree 9 and arithmetic genus 11). Its first difference is $1, 3, 6, 7, 9, 9, \dots$, which is again an O-sequence, so $\underline{c}$ is a differentiable O-sequence. However, the second difference is $1, 2, 3, 1, 2$ which is not an O-sequence. Theorem \[GMR theorem\] guarantees the existence of a reduced curve with Hilbert function $\underline{c}$, and indeed it can be achieved by the union $C_1 \cup C_2 \cup P_1 \cup P_2$, where $C_1$ is a plane curve of degree 3, $C_2$ is a plane curve of degree 6 not in the plane of $C_1$, $C_1$ and $C_2$ meet in 3 points, $P_1$ is a generally chosen point in $\mathbb P^3$, and $P_2$ is a generally chosen point in the plane of $C_1$. One sees that finding the reduced subscheme of Theorem \[GMR theorem\] can be a tricky matter! If $\underline{h} = \{ c_i : i \geq 0 \}$ is an O-sequence, Theorem \[macaulay thm\] guarantees that $\underline{h}$ is the Hilbert function of a standard graded $k$-algebra, say $R/I$. The following striking result of Gotzmann says what happens if we have maximal growth (see Definition \[o-seq\]), i.e. if the equality $c_{d+1} = c_d^{\langle d \rangle}$ holds. Let $\bar I$ be the ideal generated by the components of $I$ of degree $\leq d$, so that $\bar I_i = I_i$ for all $i \leq d$. We also sometimes write $\bar I = \langle I_{\leq d} \rangle$. \[gotzmann persistence\] Let $\underline{h} = \{ c_i : i \geq 0 \}$ be the Hilbert function of $R/\bar I$, and suppose that $c_d = \binom{m_d}{d} + \binom{m_{d-1}}{d-1} + \dots + \binom{m_j}{j}$ is the $d$-binomial expansion of $c_d$. Assume that $c_{d+1} = c_d^{\langle d \rangle}$. Then for any $l \geq 1$ we have $$c_{d+l} = \binom{m_d +l}{d+l} + \binom{m_{d-1}+l}{d-1+l} + \dots + \binom{m_j +l}{j+l}.$$ In particular, the Hilbert polynomial $p_{R/\bar I}(x)$ agrees with the Hilbert function $h_{R/\bar I}(x)$ in all degrees $\geq d$. It can be written as $$p_{R/\bar I}(x) = \binom{x+m_d-d}{m_d-d} + \binom{x+m_{d-1}-d}{m_{d-1}-(d-1)} + \dots + \binom{x+m_j-d}{m_j-j},$$ where $m_d-d \geq m_{d-1}-(d-1) \leq \dots \leq m_j-j$. \[interpret max gr\] Theorem \[gotzmann persistence\] is also known as the [*Gotzmann Persistence Theorem*]{}. Since $p_{R/\bar I}(x)$ has degree $m_d-d$, this means that the Krull dimension $\dim R/\bar I = m_d-d+1$, and $\bar I$ defines a subscheme of dimension $m_d-d$. Its degree can also easily be computed by checking the leading coefficient. Gotzmann also shows that $\bar I$ has regularity $\leq d$ and agrees with its saturation in degrees $\geq d$. \[gotzmann references\] An excellent reference for the Gotzmann Persistence Theorem and related results is section 4.3 of [@bruns-herzog2] (note that this is the revised edition; this material does not appear in the original book). A [*much*]{} more detailed and comprehensive expository treatment than that given here can be found in [@iarrobino-kanev], Appendix C (written by A. Iarrobino and S. Kleiman). This includes a detailed description of the associated Hilbert scheme. Much of the work described in this paper revolves around the following idea. If $I$ is a saturated homogeneous ideal and $L$ is a general linear form, then reducing modulo $L$ gives a new ideal, $J$, in $S=R/(L)$. Suppose that $S/J$ has maximal growth from degree $d$ to degree $d+1$. This does not necessarily imply that $R/I$ has maximal growth at that spot (see also Remark \[delta max not imply I max\]). Then what information [*can*]{} we get about $I$ if we know that $R/J$ has maximal growth? As a first step we have the following result from [@BGM]. We have noted in Remark \[interpret max gr\] that maximal growth of $J$ implies something about the saturation and regularity of $J$, thanks to Gotzmann, but we now show that it also says something about the saturation and regularity of $\langle I_{\leq d} \rangle$. [(\[Bigatti-Geramita-Migliore\] Lemma 1.4, Proposition 1.6)]{} \[BGM saturation result\] Let $I \subset R$ be a saturated ideal. Let $L$ be a general linear form, and let $J = \frac{(I,L)}{(L)} \subset S = R/(L)$. Suppose that $S/J$ has maximal growth from degree $d$ to degree $d+1$. Let $\bar I = \langle I_{\leq d} \rangle$ as above. Let $\bar I^{sat}$ be the saturation of $\bar I$. Then $\bar I = \bar I^{sat}$, i.e. $\bar I$ is a saturated ideal. Furthermore, $\bar I$ is $d$-regular. \[comparison\] Again, Proposition \[BGM saturation result\] differs from the Gotzmann results (cf. Theorem \[gotzmann persistence\], Remark \[interpret max gr\]) in that $S/J$ is assumed to have maximal growth, but we conclude something about $I$. But even beyond this, there is another difference between Proposition \[BGM saturation result\] and the part of Gotzmann’s work that we have presented (referring to the sources mentioned in Remark \[gotzmann references\] for a more complete exposition); indeed, in Theorem \[gotzmann persistence\] and Remark \[interpret max gr\] we only conclude that the ideal in question agrees with its saturation in degree $d$ and beyond, while here we make a conclusion about the whole ideal. (The price is that we have to assume that $I$ itself is saturated to begin with, although this does not necessarily hold for $J$.) Also, the maximal growth assumption for $S/J$ is necessary. For instance, let $I$ be the saturated ideal of a set, $Z$, of sixteen general points in $\mathbb P^3$. Then $I$ has four generators in degree 3 and three generators in degree 4. The four generators in degree 3 do define $Z$ scheme-theoretically, but this is not enough. One can check (e.g. using [macaulay]{} [@macaulay]) that if $\bar I = \langle I_{\leq 3} \rangle$, then $\bar I^{sat} = I \neq \langle I_{\leq 3} \rangle = \bar I$. Furthermore, $\bar I$ is 5-regular and $I$ itself is 4-regular, and neither is 3-regular. And indeed, the Hilbert function of $S/J$ is $1,3,6,6,0$, so we do not have maximal growth of $S/J$ from degree 3 to degree 4. Despite this, the results in [@AM] show that statements along the lines of Proposition \[BGM saturation result\] can be obtained even weakening the maximal growth assumption. UPP and WLP {#WLP section} =========== A very important property of (many) reduced sets of points in projective space is the following: \[UPP definition\] A reduced set of points $Z \subset \mathbb P^{n-1}$ has the [*Uniform Position Property (UPP)*]{} if, for any $t \leq |Z|$, all subsets of $t$ points have the same Hilbert function, which necessarily is the truncated Hilbert function. The last comment follows from the fact that it was also shown in [@GMR] that given any reduced set $Z$ of, say, $d$ points with known Hilbert function, and truncating this function at any value $k <d$, there is a subset $X$ of $Z$ consisting of $k$ points whose Hilbert function is this truncated function. Hence if $Z$ has UPP, then [*all*]{} subsets have this truncated Hilbert function. While it is known that the general hyperplane section of an irreducible curve has UPP (cf. [@HE]), at least in characteristic zero, much remains open. An important open question is to determine all possible Hilbert functions of sets of points in projective space with UPP. It is known in $\mathbb P^2$ but it is open even in $\mathbb P^3$. See [@GM3] for a discussion of several of the many papers that have contributed to this question for $\mathbb P^2$. For any given Hilbert function, there may or may not be a set of points with UPP having that Hilbert function. One of the contributions of the papers discussed here is toward showing some conditions on the Hilbert function that prohibit the existence of points with UPP having that function. If there [*is*]{} a set of points with UPP having a given Hilbert function, somehow “most" sets of points with that Hilbert function have UPP: consider the postulation Hilbert scheme parameterizing sets of points with that Hilbert function; then in the component containing the given set of points, there is an open subset corresponding to points with UPP. (We make this more precise in the discussion about WLP below.) Another important property is the following. \[WLP definition\] An Artinian algebra $A$ has the [*Weak Lefschetz Property (WLP)*]{} if, for a general linear form $L$, the map $\times L : A_t \rightarrow A_{t+1}$ has maximal rank, for all $t$. We say that $A$ has the [*Strong Lefschetz Property (SLP)*]{} if for every $d$, and for a general form $F$ of degree $d$, the map $\times F : A_t \rightarrow A_{t+d}$ has maximal rank, for all $t$. If $Z$ is a set of points, we sometimes say that $Z$ has WLP or SLP if its general Artinian reduction does We have the following comments about WLP and SLP: 1. The statement that SLP holds for an ideal of any number of general forms is equivalent to the well-known Fröberg conjecture. See [@anick], [@MMR2], [@MMR3], [@MMRN]. 2. [*Every*]{} Artinian complete intersection in $k[x_1,x_2,x_3]$ has WLP (cf. [@HMNW]). It is also known that SLP (and hence WLP) holds for [*every*]{} ideal in $k[x_1,x_2]$ (cf. [@iarrobino], [@HMNW]), and it (and hence Fröberg’s conjecture) holds for an ideal of general forms in $k[x_1,x_2,x_3]$ (cf. [@anick]). We now begin a short digression about the behavior of WLP in families of reduced zero-dimensional schemes. Fix a Hilbert function, $H$, that corresponds to a zero-dimensional scheme, and consider the $h$-vector, $\underline{h} = (a_0,a_1,a_2,\dots,a_s)$, associated to that Hilbert function (i.e. its first difference $\Delta H$ – cf. Remark \[first diff\]). Let $d = \sum_i a_i$ be the degree of the zero-dimensional scheme. Consider the postulation Hilbert scheme, ${\mathcal H}_{\underline{h}} := \hbox{Hilb}^{H}(\mathbb P^n)$, parameterizing zero-dimensional schemes in $\mathbb P^n$ with Hilbert function $H$ (inside the punctual Hilbert scheme, $\hbox{Hilb}^d(\mathbb P^n)$, of all zero-dimensional schemes in $\mathbb P^n$ with the given degree). It is known that the closure, $\overline{ {\mathcal H}}_{\underline{h}}$, of ${\mathcal H}_{\underline{h}}$ may have several irreducible components – see for instance Richert (cf. [@richert], and use Hartshorne’s lifting procedure, cf. [@hartshorne], [@MN2], on the Artinian monomial ideals that Richert gives), Ragusa-Zappalà (cf. [@RZ]) or Kleppe (cf. [@kleppe], Remark 27). We will consider only those components of $\overline{ {\mathcal H}}_{\underline{h}}$ for which the general element is reduced. One can show that, like UPP, WLP is an open condition in the sense that in any component of $\overline{ {\mathcal H}}_{\underline{h}}$, an open subset (possibly empty) corresponds to zero-dimensional schemes with WLP. We will now give an example that answers (positively) the following question. Namely, does there exist a Hilbert function with $h$-vector $\underline{h}$, and components ${\mathcal H}_1$ and ${\mathcal H}_2$ of the corresponding $\overline{{\mathcal H}}_{\underline{h}}$, for which - $\underline{h} = (a_0,a_1,a_2,\dots,a_s)$ is unimodal, and in fact satisfies $$\label{unimodal plus} a_0 < a_1 < a_2 \dots < a_t \geq a_{t+1} \geq \dots \geq a_s.$$ for some $t$ (this is a technical necessity for WLP– cf., Remark 3.3); - the general element of ${\mathcal H}_1$ corresponds to a reduced, zero-dimensional scheme with WLP; - [*no*]{} element of ${\mathcal H}_2$ corresponds to a zero-dimensional scheme with WLP (i.e. the open subset referred to above is empty)? The following example answers this question. \[WLP in families\] We will give an example of ideals $I_1$ and $I_2$ of reduced zero-dimensional schemes in $\mathbb P^3$, both with $h$-vector $$\underline{h} = (1,3,6,9,11,11,11),$$ such that the Artinian reduction of $R/I_1$ has WLP and the Artinian reduction of $R/I_2$ does not. Furthermore, the Betti diagrams for $R/I_1$ and $R/I_2$ are (respectively) total: 1 14 24 11 total: 1 14 25 12 -------------------------------- -------------------------------- 0: 1 - - - 0: 1 - - - 1: - - - - 1: - - - - 2: - 1 - - 2: - 1 - - 3: - 1 - - 3: - 1 1 - 4: - 1 2 - 4: - 2 2 1 5: - - - - 5: - - - - 6: - 11 22 11 6: - 10 22 11 Clearly these Betti diagrams allow no minimal element in the sense of Richert (cf. [@richert]) or Ragusa-Zappalà (cf. [@RZ]). Furthermore, neither can be a specialization of the other, so they correspond to different components of $\overline{{\mathcal H}}_{\underline{h}}$. For $I_1$ we begin with a line in $\mathbb P^3$ and link, using a complete intersection of type $(3,4)$, to a smooth curve, $C$, of degree 11. Since a line is ACM, the same is true of $C$ by liaison. We let $Z_1$ be a set of 52 general points on $C$, and let $I_1$ be the homogeneous ideal of $Z_1$. It is easy to check that $R/I_1$ has the desired $h$-vector, using liaison computations (see for instance [@migbook]) to compute the Hilbert function of $C$ and then the fact that $Z_1$ is chosen generically on $C$ so that the Hilbert function is the truncation. The rows 0 to 5 in the Betti diagram come only from $C$, and then the last row is forced from the Hilbert function. Because the Hilbert function of $Z_1$ agrees with that of $C$ up to and including degree 6, the WLP for $Z_1$ follows from the Cohen-Macaulayness of $C$. Hence the general element of the corresponding component has WLP. For $I_2$ we start with the ring $R = k[x,y,z]$ (dropping subscripts on the variables for convenience). Consider the monomial ideal $J$ consisting of $(x^3, x^2y^2, x^2yz^2, z^5)$ together with all the monomials of degree 7. One can check on [macaulay]{} (cf. [@macaulay]) that $R/J$ is Artinian with Hilbert function $\underline{h}$ and with the Betti diagram above to the right. Using the lifting procedure for monomial ideals (cf. [@hartshorne], [@MN2]), we lift $J$ to the ideal $I_2$ of a reduced zero-dimensional scheme, $Z_2$. Now, it is clear from the Betti diagram that $R/J$ has a socle element in degree 4. This means that the map from $(R/J)_4$ to $(R/J)_5$ (both of which are 11-dimensional) induced by a general linear form necessarily has a kernel, and so it is neither injective nor surjective. Hence $Z_2$ does not have WLP. But from the Betti diagram, it is clear from semicontinuity that the general element (hence every element) of the component corresponding to $Z_2$ similarly fails to have WLP. Incidentally, the lex-segment ideal corresponding to this Hilbert function (and hence having maximal Betti numbers) has Betti diagram total: 1 18 31 14 -------------------------------- 0: 1 - - - 1: - - - - 2: - 1 - - 3: - 1 1 - 4: - 2 4 2 5: - 2 3 1 6: - 12 23 11 One can check that this is a specialization of both of the Betti diagrams above. It would be nice to find an example where $\underline{h}$ is actually the $h$-vector of a complete intersection. We remark that it is possible to have a reduced zero-dimensional scheme with the Hilbert function of a complete intersection, that does not have WLP. For example, for the $h$-vector $(1,3,4,4,3,1)$, which is that of a complete intersection of type $(2,2,4)$, one can take the union, $Z$, of a general set of points in the plane with $h$-vector $(1,2,3,4,3,1)$ (easily produced with liaison) and two general points in $\mathbb P^3$, to produce a set of points with the desired $h$-vector. One can check that the Artinian reduction of $R/I_Z$ has a socle element in degree 2, so the multiplication from the component in degree 2 to the component in degree 3 (both of which are 4-dimensional) induced by a general linear form has no chance to be injective or surjective, i.e. $Z$ does not have WLP. We know that a complete intersection in $\mathbb P^3$ has Artinian reduction with WLP (cf. [@HMNW]), but we do not know if this example is contained in the same component as the complete intersection or not. In the preceding example, note that $Z_1$ has UPP, while $Z_2$ is very far from having UPP, being the lifting of a monomial ideal. This motivates the following natural questions: 1. Does a set of points with UPP automatically have WLP? 2. Does the general hypersurface section of a smooth curve necessarily have WLP (we know that it does have UPP, at least in characteristic zero)? 3. Does the general hyperplane section of a smooth curve necessarily have WLP? Of course if the curve is in $\mathbb P^3$ then the hyperplane section is in $\mathbb P^2$, so the Artinian reduction is an ideal in $k[x_1,x_2]$, and WLP and SLP are automatic (as we mentioned above). So the first place that this question is interesting is for a smooth curve in $\mathbb P^4$. 4. Does the Artinian reduction of every reduced, arithmetically Gorenstein set of points have WLP? - It is not true that every Artinian Gorenstein ideal in codimension $\geq 5$ has WLP, because the $h$-vector can fail to satisfy condition (\[unimodal plus\]) on page (cf. for instance [@ber-iar], [@BL]). But the question is open for the Artinian reduction of arithmetically Gorenstein points in any codimension. (Part of what is missing is knowledge of what Artinian algebras lift to reduced sets of points.) - It is an open question whether all height 3 Artinian Gorenstein ideals have WLP. It is true for all height 3 complete intersections (cf. [@HMNW]). In [@AM] we were able to answer the first three of the above questions in the negative, essentially with one example (cf. \[Ahn- Migliore\], Examples 6.9 and 6.10). We omit the details here, but the basic idea is as follows. We consider a smooth arithmetically Buchsbaum curve whose deficiency module $$M(C) = \bigoplus_{t \in {\mathbb Z}} H^1(\mathbb P^3, {\mathcal I}_C (t))$$ is one-dimensional in degrees 3 and 4. Any linear form induces a homomorphism from $H^1(\mathbb P^3, {\mathcal I}_C(3))$ to $H^1(\mathbb P^3, {\mathcal I}_C(4))$. The condition “arithmetically Buchsbaum" means that this map is zero for all linear forms $L$. It is known that such a smooth curve exists (cf. [@BM2]). Now, we put a lot of points on such a curve, placed generically, and call the resulting set of points $Z$. Note that $I_Z$ agrees with $I_C$ in low degrees. We then play with the coordinate ring of $C$, the coordinate ring of $Z$, the cohomology of ${\mathcal I}_C$ and the cohomology of ${\mathcal I}_Z$, and in the end we show that the Artinian reduction of $R/I_Z$ does not have WLP. Tweaking this example somewhat (looking at the cone over $C$ and taking a general hypersurface and hyperplane section) answers the remaining questions. Some of the results given below, in particular those coming from the paper [@AM], are given in terms of [*reduction numbers*]{}, which we now define. Let $I \subset R$ be a homogeneous ideal. Let $m \geq \dim R/I$. The $m$-reduction number of $R/I$ is $$\begin{array}{rcl} r_m(R/I) & = & \min \{ k \ | \ x_{n-m}^{k+1} \in {\hbox{\rm Gin}}(I) \} \\ & = & \min \{ k \ | \ h_{R/(I+J)} \hbox{ vanishes in degree } k+1 \} \end{array}$$ where $J$ is an ideal generated by $m$ general linear forms. We will use the second line as the definition, and we include the first line only for completeness. (For the definition of, and results on, the generic initial ideal, ${\hbox{\rm Gin}}(I)$, see for instance [@green2].) \[334 example\] Let $Z$ be a complete intersection of type $(3,3,4)$ in $\mathbb P^{3}$ and let $I = I_Z$. Note $\dim R/I = 1$. Let $L_1, L_2, L_3$ be general linear forms. It is known that any Artinian reduction of $R/I$ has WLP (cf. [@HMNW]). We have $$\begin{array}{rclcll} h_{R/(I+(L_1))} & : & 1 \ 3 \ 6 \ 8 \ 8 \ 6 \ 3 \ 1 & \Rightarrow & r_1(R/I) = 7. \\ h_{R/(I+(L_1,L_2))} & : & 1 \ 2 \ 3 \ 2 & \Rightarrow & r_2(R/I) = 3 & \hbox{(by WLP)} \\ h_{R/(I+(L_1,L_2,L_3))} & : & 1 \ 1 \ 1 & \Rightarrow & r_3(R/I) = 2 & \hbox{(by WLP)} \end{array}$$ The first use of WLP came because we had a complete intersection; the second came because we were in a ring with two variables. So we see that having WLP is extremely helpful in computing the reduction numbers. Setting the stage {#set stage} ================= The main goal of this paper is to sketch a progression of results on Hilbert functions for sets of points, starting in $\mathbb P^2$ (mostly due to Davis, cf.  [@davis]), then moving to some of the generalizations of these results to higher projective space obtained by Bigatti, Geramita and myself (cf. [@BGM]), and finally recent extensions of some of these results obtained by Ahn and myself (cf. [@AM]). In Section \[general results\] we will focus on the general case, while in Section \[UPP results\] we will specialize to the case of UPP. For simplicity of exposition we will usually focus on reduced schemes, but many of these results extend to the non-reduced case. Let $Z$ be a zero-dimensional scheme. The central assumption in this paper will be the following: $$\label{main assumption} \Delta h_Z(d) = \Delta h_Z(d+1) = s, \hbox{ for some $d, s$.}$$ In both Section \[general results\] and Section \[UPP results\], we will be focusing on two situations: - $d \geq s$ (which was used in [@BGM]) - $d > r_2(R/I_Z)$ (which was used in [@AM]). We will see that (a) corresponds to a certain kind of maximal growth (see Remark \[s = s max gr\]), while (b) in general does not (see Remark \[connection betw hyp\]). This is the striking aspect of the results centered around (b) (see especially Theorem \[AM general\] and Theorem \[AM UPP results\]), that strong results can be obtained even without the power of the Gotzmann machinery behind us. In this section we make a series of remarks in preparation for the discussion in the coming sections. \[comment on 334\] In Example \[334 example\] we have (\[main assumption\]), but neither $d \geq s$ nor $d > r_2(R/I_Z)$ holds! (Neither do the conclusions that we will mention in the coming sections.) \[connection betw hyp\] What is the connection between (a) and (b) above? If $d \geq s$ then it can be shown that $d > r_2(R/I_Z)$ (see [@AM] Remark 4.4). But in general $d > r_2(R/I_Z)$ does [*not*]{} imply $d \geq s$, and indeed it can happen that $d > r_2(R/I_Z)$ but $d$ is much smaller than $s$. In Example \[334 example\], $s=8$ and $r_2(R/I_Z) = 3$. \[max gr in p2\] We now remark that in $\mathbb P^2$, condition (\[main assumption\]) essentially always corresponds to case (a) (there is only one, very special, exception). Assume that $Z \subset \mathbb P^{2}$ is a zero-dimensional scheme. For $d \leq \alpha-1$ we have $\Delta h_Z(d) = d+1$ and for $d \geq \alpha$ we have that $\Delta h_Z$ is non-increasing, so in particular $\Delta h_Z(d) \leq \alpha$. Of course there may not be any flat part at all, but the point is that the function cannot increase past degree $\alpha-1$. In any case we have $$[\Delta h_Z(d) = \Delta h_Z(d+1) = s] \Rightarrow \left \{ \begin{array}{ll} d = s-1, & \hbox{if $d = \alpha-1$ and $s=\alpha$} \\ d \geq s, & \hbox{if $d \geq \alpha$} \end{array} \right.$$ In the first case in the above calculation, we have that $\dim(I_Z)_\alpha = 1$. Notice that in this case $I_Z$ is zero in degree $d = \alpha-1$, but $(I_Z)_{d+1} = (I_Z)_\alpha$ defines a hypersurface (or equivalently a curve) in $\mathbb P^2$. \[s = s max gr\] Assume that $Z \subset \mathbb P^{n}$ is a zero-dimensional scheme. If $d \geq s$ then the condition $\Delta h_Z (d) = \Delta h_Z(d+1) = s$ implies that the growth of the Artinian reduction from degree $d$ to degree $d+1$ is maximal in the sense of Definition \[o-seq\]. Indeed, the $d$-binomial expansion of $s$ is $$s = \binom{d}{d} + \dots + \binom{d-s+1}{d-s+1}$$ so $$s^{\langle d \rangle} = \binom{d+1}{d+1} + \dots + \binom{d-s+2}{d-s+2}=s$$ as claimed. \[delta max not imply I max\] In practice, it usually is the case that $\Delta h_Z$ having maximal growth from degree $d$ to degree $d+1$ does [*not*]{} imply that $h_Z$ itself has maximal growth from degree $d$ to degree $d+1$. If it did, Gotzmann’s results (Theorem \[gotzmann persistence\] and Remark \[interpret max gr\]) would immediately apply to $h_Z$. The interest in the results described below is that similar powerful conclusions come from this maximal growth of the first difference (see Proposition \[BGM saturation result\] and Remark \[comparison\]). And as we will see, even more surprising is that similar results can be deduced at times even when the first difference does not have maximal growth. General results {#general results} =============== A by-now classical result of Davis (cf. [@davis]) is the following. Note that there is no uniformity assumption on $Z$. In the next section we will discuss the refinements that are possible when we assume UPP. \[davis thm\] Let $Z \subset \mathbb P^{2}$ be a zero-dimensional scheme. If $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ for $d \geq s$, then $(I_Z)_d$ and $(I_Z)_{d+1}$ both have a GCD, $F$, of degree $s$. If $Z$ is reduced then so is $F$. The polynomial $F$ defines a $\left \{ \begin{array}{l} \hbox{hypersurface} \\ \hbox{curve} \end{array} \right. $ in $\mathbb P^{2}$. If $Z_1 \subset Z$ is the subscheme of $Z$ lying on $F$ (defined by $[I_Z + (F)]^{sat}$) and $Z_2$ is the “residual” scheme defined by $[I_Z :F]$, then there are formulas relating the Hilbert functions of $I_Z, I_{Z_1}$ and $I_{Z_2}$. \[iarrobino comment\] It is worth mentioning that the Artinian version of Theorem \[davis thm\] was proved earlier, by A. Iarrobino (cf. [@iarrobino; @artin], page 56). The paper [@BGM] began with the observation that Davis’ result, Theorem \[davis thm\], was really about maximal growth of the function $\Delta h_Z(t)$ from degree $d$ to degree $d+1$, thanks to Remark \[max gr in p2\] and Remark \[s = s max gr\] above. These remarks show that in $\mathbb P^2$, $\Delta h_Z$ can have two different kinds of maximal growth: either $h_Z$ takes on the value of the polynomial ring (before the initial degree of the ideal), or else $\Delta h_Z$ takes the constant value $s \geq d$. This latter is the only interesting case. Notice that since $Z$ is a zero-dimensional scheme, eventually the Hilbert function is constant, so eventually $\Delta h_Z$ is zero. So the condition that $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ directly gives us information not so much about $I_Z$ as about $(I_Z)_{\leq d}$, the ideal generated by the components of degree $\leq d$. We have to deduce properties of $I_Z$ from this, via Proposition \[BGM saturation result\]. Gotzmann’s results tell us that in a general Artinian reduction of $R/I_Z$ (which has Hilbert function $\Delta h_Z$), the degree $d$ component of the ideal $J = \frac{(I_Z,L)}{(L)}$ agrees with the degree $d$ component of the saturated ideal of a subscheme of degree $s$ in the line defined by $L$ (cf. Remark \[interpret max gr\]). Since this corresponds to a hyperplane section by $L$, that means that the base locus of the linear system $|I_d|$ contains a curve of degree $s$, which is the GCD in question. The results about the subscheme of $Z$ lying on this GCD and the “residual" subscheme come from some algebraic arguments chasing exact sequences. As noted in the way we phrased Davis’ theorem, this GCD can be viewed as a curve, or as a hypersurface in $\mathbb P^2$. These correspond, respectively, to the interpretations (that happen to coincide in this case) that the maximal growth for $\Delta h_Z$ takes the constant value $s \leq d$, and it takes the largest possible growth short of being equal to the Hilbert function of the whole polynomial ring itself. In [@BGM] we extended this to $\mathbb P^{n-1}$, and we called these kinds of maximal growth [*growth like a curve*]{}, and [*growth like a hypersurface*]{}, respectively. We proved many results for $Z \subset \mathbb P^{n-1}$. In particular, we showed that viewing $F$ in Theorem \[davis thm\] as a curve, and viewing it as a hypersurface, both extend in separate directions when we move to higher projective spaces. In this paper we focus on the former. We now give a summary of the results in the “curve" direction found in [@BGM] that do not assume UPP. \[BGM general results\] Let $Z \subset \mathbb P^{n-1}$ be a [reduced]{} zero-dimensional scheme. Assume that $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ for some $d \geq s$. Then - $\langle (I_Z)_{\leq d} \rangle$ is the [**saturated**]{} ideal of a curve $V$ of degree $s$. $V$ is not necessarily unmixed, but it is [**[reduced]{}**]{} and [**$d$-regular**]{}. Let $C$ be the unmixed one-dimensional part of $V$. Let $Z_1$ be the subscheme of $Z$ lying on $C$ (defined by $[I_Z + I_C]^{sat}$) and $Z_2$ the residual scheme (defined by $[I_Z : I_C]$). Then - $\langle (I_{Z_1})_{\leq d} \rangle = I_C$ and $I_C$ is . (Part (a) was for $V$. This one is not surprising, since $V$ consists of $C$ plus some points.) - There are formulas relating the Hilbert functions. How might these results be improved? Recall that $\Delta h_Z(d) =$ $\Delta h_Z(d+1) = s$ for $d \geq s$ guarantees that $\Delta h_Z$ has maximal growth from degree $d$ to degree $d+1$. 1. Weaken the condition $\Delta h_Z (d) = \Delta h_Z(d+1) = s$, e.g. to the condition $\Delta h_Z(d) = \Delta h_Z(d+1) +1$ (possibly even maintaining the assumption $d \geq s$). It may be that at least “usually" something similar will hold. But there will be important differences. This is still an open direction, although in her thesis Susan Cooper (cf. [@cooper]), a student of Tony Geramita, is working on questions related to this. 2. Weaken the assumption $d \geq s$. This is the approach we take. We will assume only $d > r_2(R/I_Z)$. This is a beautiful idea, conceived by my co-author, Jeaman Ahn. It is very striking to see how much carries over to this case. \[AM general\] Let $Z \subset \mathbb P^{n-1}$ be a reduced zero- dimensional scheme. Assume that $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ for some $d > r_2(R/I_Z)$. Then - $\langle (I_Z)_{\leq d} \rangle$ is the ideal of a curve $V$ of degree $s$. $V$ is not necessarily unmixed or reduced, but it is $d$-regular. Let $C$ be the unmixed one-dimensional part of $V$. Let $Z_1$ be the subscheme of $Z$ lying on $C$ (defined by $[I_Z + I_C]^{sat}$) and $Z_2$ the residual scheme (defined by $[I_Z : I_C]$). Then - $\langle (I_{Z_1})_{\leq d} \rangle = I_C$ and $C$ is $d$-regular. - There are formulas relating the Hilbert functions. - [**If we also assume that**]{} $h^1({\mathcal I}_{C_{red}}(d-1)) = 0$ then $V$ is reduced and $C = C_{red}$ is $d$-regular. We stress that in (a), we do not necessarily obtain that $V$ is reduced. This is the first new twist. If $V$ is not reduced, then the top dimensional part, $C$, is not necessarily reduced. This curve $C$ is $d$-regular. If it is not reduced, however, it is supported on a reduced curve, $C_{red}$. Surprisingly, $C_{red}$ being $d$-regular does not follow from $C$ being $d$-regular, and there are examples where it is not true. But with the extra assumption, (d) delivers this conclusion. Example \[not decr type\] below illustrates what can happen. Theorem \[AM general\] combines different results of [@AM]. The proofs heavily use generic initial ideals and results of Green, Bayer, Stillman, Galligo, etc. We end this section with a result that incorporates WLP and also higher differences of the Hilbert function and higher reduction number. \[delta 2\] Let $Z$ be a zero-dimensional subscheme of $\mathbb P^{n-1}$, $n>3$, with . Suppose that $$\Delta^2 h_Z (d)=\Delta^2 h_Z (d+1)=s$$ for $r_2(R/I_Z)>d>r_3(R/I_{Z})$. Then $\langle (I_Z)_{\leq d}\rangle$ is a saturated ideal defining a two-dimensional subscheme of degree $s$ in $\mathbb P^{n-1}$, and it is $d$-regular. Results on Uniform Position {#UPP results} =========================== We now investigate the effects of assuming that our points have UPP. We begin again in $\mathbb P^2$. \[P2 UPP results\] Let $Z \subset \mathbb P^2$ be a reduced set of points with UPP. Then we have - [([@ger-mar], [@MR])]{} The component of $I_Z$ of least degree, $\alpha$, contains an irreducible form (hence the general such form is irreducible). In particular, this holds if there is only one such form, up to scalar multiple. This is in fact true not only in $\mathbb P^2$ but also in $\mathbb P^{n-1}$. - [([@harris])]{} $\Delta h_Z$ is of , i.e. $$\hbox{if } \Delta h_Z(d) > \Delta h_Z(d+1) \hbox{ then } \Delta h_Z(t) > \Delta h_Z(t+1)$$ for all $t \geq d$ as long as $\Delta h_Z(t) > 0$. - If $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ then $s = \alpha$ (see Remark \[max gr in p2\]), $(I_Z)_t = (F)_t$ for all $t \leq d+1$, and the points of $Z$ all lie on the irreducible curve defined by $F$. As before, this theorem was also extended in \[Bigatti-Geramita-Migliore\] to $\mathbb P^{n-1}$ for the case of “maximal growth like a hypersurface," but here we focus on extending it for “maximal growth like a curve." (See the discussion preceding Theorem \[BGM general results\].) In the following we repeat part of Theorem \[BGM general results\] and underline the new features. \[BGM UPP results\] Let $Z \subset \mathbb P^{n-1}$ be a [reduced]{} zero-dimensional scheme with UPP. Assume that $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ for some $d \geq s$. Then - $\langle (I_Z)_{\leq d} \rangle$ is the saturated ideal of a curve $V$ of degree $s$. $V$ is reduced, $d$-regular, and . - Since $V$ is unmixed, its top dimensional part $C$ is equal to $V$. Hence $Z \subset C$. - $\langle (I_{Z})_{\leq d} \rangle = I_C$ and $I_C$ is [$d$-regular]{}. The hardest part is irreducibility, and it strongly uses the assumption $d \geq s$. Still under the hypothesis of [@BGM] that $d \geq s$, the following extension of “decreasing type" was observed in \[Ahn-Migliore\]. This completes the picture of extending the $\mathbb P^2$ result to higher projective space under the assumption of “maximal growth like a curve." \[decr type in Pn\] If $Z \subset \mathbb P^{n-1}$ has UPP and $\Delta h_Z(d) = $ $\Delta h_Z(d+1) > \Delta h_Z(d+2)$ for some $d \geq s$, then $\Delta h_Z(t) > \Delta h_Z(t+1)$ for all $t \geq d+1$ as long as $\Delta h_Z(t) > 0$. We now turn to the situation of [@AM], where we only assume that $d > r_2(R/I_Z)$. It is somewhat surprising to see what we retain and what we lose. Compare this with Theorem \[AM general\] (part of which is repeated here). \[AM UPP results\] Let $Z \subset \mathbb P^{n-1}$ be a reduced zero- dimensional scheme with UPP. Assume that $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ for some $d > r_2(R/I_Z)$. Then - $\langle (I_Z)_{\leq d} \rangle$ is the saturated ideal of an curve, $V$, of degree $s$, and it is $d$-regular. $V$ is not necessarily reduced or irreducible. - Since $V$ is unmixed, its top dimensional part $C$ is equal to $V$. Hence $Z \subset C$. - $\langle (I_{Z})_{\leq d} \rangle = I_C$ and $I_C$ is [$d$-regular]{}. - [**If we also assume that**]{} $h^1({\mathcal I}_{C_{red}}(d-1)) = 0$ then $V$ is reduced and $C = C_{red}$ is $d$-regular. Why might we have hoped that “reduced and irreducible” would still hold in part (a) of Theorem \[AM UPP results\]? Recall the following. - We saw in Theorem \[P2 UPP results\] that if $Z \subset \mathbb P^{n-1}$ is a set of points in $\mathbb P^{n-1}$ with UPP, it is known that a general element of smallest degree $\alpha$ is reduced and irreducible. When this element is unique (up to scalar multiple), say $F$, we have $\langle (I_Z)_{\leq \alpha} \rangle$ is the saturated ideal of a reduced, irreducible hypersurface which is $\alpha$-regular. This is the base locus in degree $\alpha$, and just having UPP is enough to guarantee the irreducibility. - Recall from Theorem \[BGM UPP results\] that if $d \geq s$ then it [*is*]{} true that the base locus, $C$, is reduced and irreducible. In the next section we will give examples to show that in fact “reduced and irreducible" does not necessarily hold (as opposed to merely being a gap in the theorem). Examples {#example section} ======== These examples will serve to illustrate that some of our results above are apparently close to optimal. We omit some details, and refer the reader to [@AM]. We will see that 1. In contrast to Corollary \[decr type in Pn\], for a set of points in $\mathbb P^{n-1}$ with UPP satisfying $\Delta h_Z(d) = \Delta h_Z(d+1) > \Delta h_Z(d+2)$ for $d > r_2(R/I_Z)$ (instead of $d \geq s$), it is not necessarily true that $\Delta h_Z$ is strictly decreasing beyond this point. This is shown in Example \[not decr type\]. 2. In Theorem \[AM UPP results\], when we assume only $d > r_2(R/I_Z)$, it is in fact true (as claimed) that $V$ is not necessarily reduced or irreducible. This stands in contrast to Theorem \[BGM UPP results\], when we assumed $d \geq s$. This is shown in Example \[not decr type\]. 3. Upon realizing that in Theorem \[AM UPP results\] $V$ is not necessarily reduced and irreducible, one might hope that at least the points of $Z$ are restricted to one component of $V$ that [*is*]{} reduced and irreducible. Examples \[not irred\] and \[two sextics\] show that even this is not true. 4. Examples \[can’t extend\] and \[back to ex\] illustrate some of the difficulties of extending these results past the case $d > r_2(R/I_Z)$, showing that these results are optimal in a sense. We begin with an example of Chardin and D’Cruz. It settles an old question of independent interest. \[CD example\] Consider the family of complete intersection ideals $$I_{m,n} := (x^mt - y^mz, z^{n+2}-xt^{n+1}) \subset k[x,y,z,t].$$ Then for $m,n \geq 1$, ${\hbox{\rm reg}}(I_{m,n}) = m+n+2$ while ${\hbox{\rm reg}}(\sqrt{I_{m,n}}) =$ $ mn+2$. Hence the regularity of the radical may be much larger than the regularity if the ideal itself!! In particular, if we take $m=n=4$ then we obtain the following: $I_{4,4}$ is a complete intersection of type $(5,6)$. As such, it has degree 30, and has a Hilbert function whose first difference is ------------ --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- deg 0 1 2 3 4 5 6 7 8 9 10 11 12 $\Delta h$ 1 3 6 10 15 20 24 27 29 30 30 … ------------ --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- On the other hand, $\sqrt{I_{4,4}}$ can be computed on a computer program, e.g. (cf. [@macaulay]). It has degree 26 (hence $I_{4,4}$ is not reduced), and its Hilbert function has first difference deg 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ------------ --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h$ 1 3 6 10 15 20 24 27 29 29 29 29 28 28 28 27 deg 16 17 18 19 20 21 ------------ ---- ---- ---- ---- ---- ---- --- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h$ 27 27 26 26 26 26 … In [@CD], the authors were not so concerned with the geometry of this curve. We will modify this slightly, taking a more geometric approach, in Example \[not decr type\]. Example \[CD example\] shows that a curve $C \subset \mathbb P^3$ can have the property that ${\hbox{\rm reg}}(\sqrt{I_C})$ may be (much) larger than ${\hbox{\rm reg}}(I_C)$. However, it seems to still be an open question whether this can happen, for instance, for a smooth curve. Chardin and D’Cruz also study the surface case, where other interesting phenomena occur. For curves, see also Ravi [@ravi]. \[not decr type\] In this example we recall that $k$ has characteristic zero. We will be basing our example on the case $m = n = 4$ of the example of Chardin and D’Cruz. It is obtained by taking a geometric interpretation. Let $R = k[x,y,z,t]$. Let $I_\lambda = (z,t)$. Let $F \in (I_\lambda)_5$ be a homogeneous polynomial that is smooth along $\lambda$. Let $I' = I_\lambda^5 + (F)$. $I'$ is the saturated ideal of a non-ACM curve of degree 5 corresponding to the divisor $D := 5 \lambda$ on $F$. In particular, viewed as a subscheme of $\mathbb P^3$, $D$ has degree 5. Now, choose a general element $C$ in the linear system $|6H-D|$ on $F$. $C$ is smooth and irreducible. Furthermore, $C$ has degree 25 (by liaison) and Hilbert function with first difference deg 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ------------ --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h$ 1 3 6 10 15 20 24 27 29 29 29 29 28 28 28 27 deg 16 17 18 19 20 21 ------------ ---- ---- ---- ---- ---- ---- --- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h$ 27 27 26 26 26 25 … We now let $Z$ consist of sufficiently many points of $C$, chosen generally, so that $I_Z$ agrees with $I_C$ up to and including degree 21. $Z$ has UPP (since $C$ is smooth), and one checks that $r_2(R/I_Z) = r_2(R/I_C) = 8$. Taking $d=9$, we see that Theorem \[AM UPP results\] applies. We obtain that $\langle (I_Z)_{\leq 9} \rangle$ is the saturated ideal of an unmixed curve $V$ of degree 29 consisting of the union of $C$ and a subcurve of $D$ of degree 4 supported on $\lambda$, hence $V$ is neither irreducible nor reduced. Taking $d = 12$, $d = 15$, $d = 18$ slices away at the non-reduced part, and taking $d = 21$ gives just $C$. A computation on [macaulay]{} reveals that in fact the regularity of $I_C$ is 21. On the other hand, the curve $V$ of degree 29 has ideal $I_V$ with regularity 9. Actually, it is worth recording the Betti diagram of $I_C$ and $I_V$, because they display a surprising pattern: For $I_V$ we have total: 1 3 2 -------------------------- 0: 1 - - 1: - - - 2: - - - 3: - - - 4: - 1 - 5: - 1 - 6: - - - 7: - - - 8: - 1 2 while for $I_C$ we have total: 1 7 10 4 -------------------------------- 0: 1 - - - 1: - - - - 2: - - - - 3: - - - - 4: - 1 - - 5: - 1 - - 6: - - - - 7: - - - - 8: - 1 2 - 9: - - - - 10: - - - - 11: - 1 2 1 12: - - - - 13: - - - - 14: - 1 2 1 15: - - - - 16: - - - - 17: - 1 2 1 18: - - - - 19: - - - - 20: - 1 2 1 Notice that the diagram for $I_V$ is a subdiagram of the diagram for $I_C$, and that there is a striking simplicity to the diagram for $I_C$. Notice also that $V$ is ACM, while $C$ is not. Note that in the previous example, the points of $Z$ all lie on one reduced, irreducible curve ($C$). It is simply the case that $V$ contains another component, which happens to also be non-reduced. However, now we give another example to show that it is not necessarily true that all the points lie on one irreducible component of the base locus. \[not irred\] In Example \[not decr type\], instead of choosing “sufficiently many" points on $C$, instead choose $Z$ to consist of 192 general points on $C$ and one general point of $\lambda$. The first difference of the Hilbert function of $Z$ is deg 0 1 2 3 4 5 6 7 8 9 10 11 -------------- --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h_Z$ 1 3 6 10 15 20 24 27 29 29 29 0 Note that this is exactly the same as what we would have if we had taken 193 general points of $C$. Again, $r_2(R/I_Z) = 8$. The base locus of $(I_C)_{9}$ and $(I_C)_{10}$ is exactly the non-reduced and reducible curve of degree 29 mentioned in Example \[not decr type\], so the Hilbert function of $Z$ is the truncation of the Hilbert function given above. And by the general choice of the points, this will continue to be true regardless of which subsets we take. Hence $Z$ has UPP and satisfies $\Delta h_Z(d) = \Delta h_Z(d+1) = s$ for some $d > r_2(R/I_Z)$, but not all of the points of $Z$ lie on one reduced and irreducible component of the curve of degree $s$ obtained by our result (since one point lies on the non-reduced component). [\[Ahn-Migliore\] Example 5.9\]]{} \[two sextics\] We produced Example \[not irred\] by taking almost all of the points on $C$, and just one off of $C$ (in fact, it was on $\lambda$). In this example we show a surprising instance where half of the points of $V$ are on one irreducible component and the other half are on another irreducible component, but we still have UPP. We again omit some of the technical details. Let $Q$ be a smooth quadric surface in $\mathbb P^3$, and as usual by abuse of notation we use the same letter $Q$ to denote the quadratic form defining this surface. Let $C_1$ be a [*general*]{} curve on $Q$ of type $(1,15)$, and let $C_2$ be a [*general*]{} curve on $Q$ of type $(15,1)$. Hence both $C_1$ and $C_2$ are smooth rational curves of degree 16, and $C := C_1 \cup C_2$ is the complete intersection of $Q$ and a form of degree 16. Note that $C$ is arithmetically Cohen-Macaulay, but $C_1$ and $C_2$ are not. It is not difficult to compute the Hilbert functions of these curves. We record their first differences (of course there is no difference between behavior of $C_1$ and behavior of $C_2$; this is important in the argument given in [@AM]): degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ------------------ --- --- --- --- --- ---- ---- ---- ---- ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h_C$ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 $\Delta h_{C_i}$ 1 3 5 7 9 11 13 15 17 19 21 23 25 27 degree 14 15 16 17 18 ------------------ ---- ---- ---- ---- ---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- $\Delta h_C$ 29 31 32 32 32 $\Delta h_{C_i}$ 29 16 16 16 16 We now observe: 1. These first differences (hence the ideals themselves) agree through degree 14, and in fact the only generator before degree 15 is $Q$. 2. \[hf values\] By adding these values, we see that $h_C(18) = 352$ and $h_C(15) = 241$. 3. Since $C$ and $C_i$ are curves, these values represent the Hilbert functions of $I_C + (L)$ and $I_{C_i} +(L)$ for a general linear form $L$. 4. \[value of r2\] $r_2(R/I_C) = 16$ since $C$ is an arithmetically Cohen-Macaulay curve. Let $Z_1$ (respectively $Z_2$) be a general set of 176 points on $C_1$ (respectively $C_2$). So $Z := Z_1 \cup Z_2$ is a set of 352 points whose Hilbert function agrees with that of $C$ through degree 18. In particular, we have $\Delta h_Z(17) = \Delta h_Z(18) = 32 = \deg C$. Furthermore, $r_2(R/I_Z) = 16$ by our observation \[value of r2\]. above. Hence Theorem \[AM UPP results\] applies, and we indeed have that the component of $I_Z$ in degree 17 defines $C$. However, $C$ is not irreducible. In [@AM] Example 5.9, it is shown that $Z$ has the Uniform Position Property. Thus there is no chance of showing that all the points must lie on a unique irreducible component in Theorem \[AM UPP results\], under our hypothesis that $d > r_2(R/I_Z)$ (as was done in [@BGM] when $d \geq s$). To show UPP, it is enough to show that the union of any choice of $t_1$ points of $Z_1$ (i.e. $t_1$ general points of $C_1$ for $t_1 \leq 176$) and $t_2$ points of $Z_2$ (i.e. $t_2$ general points of $C_2$ for $t_2 \leq 176$) has the truncated Hilbert function. For example, if $t_1 = 150$ and $t_2 = 160$ then we have to show that $\Delta H(R/I_Z)$ has values $$1 \ \ 3 \ \ 5 \ \ 7 \ \ 9 \ \ 11 \ \ 13 \ \ 15 \ \ 17 \ \ 19 \ \ 21 \ \ 23 \ \ 25 \ \ 27 \ \ 29 \ \ 31 \ \ 32 \ \ 22 \ \ 0.$$ Notice that we know that some subset has this Hilbert function, by [@GMR]. We have to show that [*all*]{} subsets have this Hilbert function. See [@AM], Example 5.9, for the details. \[can’t extend\] We saw that the condition $d \geq s$ of \[Bigatti-Geramita-Migliore\] was improved in [@AM] to $d > r_2(R/I_Z)$. There was some loss in the strength of the result, but a surprising amount of it did go through. One might wonder if the condition $d > r_2(R/I_Z)$ can be further improved. But in fact, we saw already in Example \[334 example\] and Remark \[comment on 334\] that this is not the case. Similarly, we have the following examples: $$\begin{array}{l} \hbox{7 general points in $\mathbb P^3$ have $h$-vector 1 \ 3 \ 3} \\ \hbox{16 general points in $\mathbb P^3$ have $h$-vector 1 \ 3 \ 6 \ 6} \\ \hbox{30 general points in $\mathbb P^3$ have $h$-vector 1 \ 3 \ 6 \ 10 \ 10} \\ \hbox{etc.} \end{array}$$ In each case, $r_2(R/I_Z)$ has the “expected" value, say $d$. We have $\Delta h_Z (d) = \Delta h_Z(d+1) $ for $d = r_2(R/I_Z)$, but clearly the base locus of $(I_Z)_d$ is not one-dimensional. See Remark \[comparison\] as well. Similar examples can easily be found in higher projective spaces. A different sort of example can be found in Example \[334 example\]. \[back to ex\] We return to Example \[WLP in families\] to see how the theorems mentioned above apply to that example. Recall that we have two ideals, $I_1$ and $I_2$, both with $h$-vector $(1,3,6, 9, 11,11,11)$. Taking $s=11$, it is clear that the results from [@BGM] (Theorem \[BGM general results\] and Theorem \[BGM UPP results\]) do not apply because of the hypothesis $d \geq s$. As for the results of [@AM] (Theorem \[AM general\] and Theorem \[AM UPP results\]), one can check that $r_2(R/I_1) = 4$ while $r_2(R/I_2) = 5$. Hence both results from [@AM] apply to $I_1$, taking $d = 5$, but not to $I_2$. And indeed, we have seen that $\langle (I_1)_{\leq 5} \rangle$ is the saturated ideal of the curve $C$ described in that example. 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--- abstract: 'We generalize to topologically non-trivial gauge configurations the description of the Einstein–Yang–Mills system in terms of a noncommutative manifold, as was done previously by Chamseddine and Connes. Starting with an algebra bundle and a connection thereon, we obtain a spectral triple, a construction that can be related to the internal Kasparov product in unbounded KK-theory. In the case that the algebra bundle is an endomorphism bundle, we construct a $PSU(N)$-principal bundle for which it is an associated bundle. The so-called internal fluctuations of the spectral triple are parametrized by connections on this principal bundle and the spectral action gives the Yang–Mills action for these gauge fields, minimally coupled to gravity. Finally, we formulate a definition for a topological spectral action.' address: 'Institute for Mathematics, Astrophysics and Particle Physics, Faculty of Science, Radboud University Nijmegen, Heyendaalseweg 135, 6525AJ Nijmegen, The Netherlands' author: - 'Jord Boeijink and Walter D. van Suijlekom' title: 'The noncommutative geometry of Yang–Mills fields' --- Introduction ============ One of the main applications of noncommutative geometry to theoretical physics is in deriving the Yang–Mills action from purely geometrical data [@ChamseddineConnes]. In fact, the full Lagrangian of the Standard Model of high-energy physics – including the Higgs potential – can be derived by starting with a noncommutative Riemannian spin manifold [@CCM07]. It is interesting to confront this with the geometrical approach to Yang–Mills theory (*cf*. [@Ati79]), using the language of principal fiber bundles and connections thereon. It turns out that the noncommutative geometrical description of [@CC97] corresponds to topologically trivial $SU(N)$-principal bundles. It is the goal of this paper to generalize this to topologically non-trivial gauge configurations. As a matter of fact, we derive the Yang–Mills action for gauge fields defined on a non-trivial principal bundle from a noncommutative Riemannian spin manifold, that is, from a spectral triple. Since spectral triples – and more generally, (unbounded) KK-theory – form a natural setting for doing index theory, our construction has potential applications to [*e.g.*]{} the study of moduli spaces of instantons in noncommutative geometry. Our construction will naturally involve algebra bundles and connections thereon, for which – after some preliminaries – we will give their definition in Section \[sect:algebrabundles\]. There, we will also construct a spectral triple from this data. The above connection plays the same role as it does in the internal Kasparov product in KK-theory and we will explore this relation in some detail in Section \[sect:KK\]. In the case that the algebra bundle has typical fiber $M_N(\C)$ – [*i.e.*]{} it is an endomorphism bundle – it is possible to construct a $PSU(N)$-principal bundle, with the algebra bundle as an associated bundle. We will explore this case in Section \[sect:ym\]. The so-called internal fluctuations of the above spectral triple are parametrized by connections on this principal bundle. Finally, we show that the spectral action principle applied to the spectral triple gives the Yang–Mills action on a topologically non-trivial $PSU(N)$-principal bundle, minimally coupled to gravity. In the concluding section, we sketch the definition of a so-called topological spectral action. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank Simon Brain for a careful proofreading of the manuscript, as well as valuable suggestions and remarks. Preliminaries ============= Spectral triples and the spectral action principle -------------------------------------------------- Spectral triples, as they are introduced in [@Connes] are at the heart of noncommutative geometry. In fact, they generalize $spin^c$-structures to the noncommutative world. A *spectral triple* $(\mathcal{A}, \mathcal{H}, D$) is given by an involutive algebra $\mathcal{A}$ represented faithfully on the Hilbert space $\mathcal{H}$, together with a densely defined, self-adjoint operator $D$ on $\mathcal{H}$ with the following properties: - The resolvent operators $(D - \lambda)^{-1}$ are compact on $\mathcal{H}$ for all $\lambda \notin \mathbb{R}$, - For all $a \in \mathcal{A}$ the operator $[D,a]$ extends to a bounded operator defined on $\mathcal{H}$ . \[dfn:spectraltriple\] The triple is said to be *even* if there exists an operator $\Gamma$ on $\mathcal{H}$ with the properties $$\Gamma^*=\Gamma, \quad \Gamma^2=1, \quad \Gamma D + D \Gamma = 0, \quad \Gamma a - a \Gamma = 0.\label{eq:eventriple}$$ If such an operator does not exist, then the triple is said to be *odd*. \[ex:canonicaltriple\] The motivating example for the definition of a spectral triple is formed by the *canonical triple* $$({C^{\infty}(M)}, L^2(M,S), {{D\mkern-11.5mu/\,}})$$ associated to any compact Riemannian spin-manifold $M$.[^1] The Hilbert space $L^2(M,S)$ consists of square-integrable sections of the spinor bundle $S \to M$. The operator ${{D\mkern-11.5mu/\,}}$ is the Dirac operator on the spinor bundle. For even dimensional spin-manifolds there exists a grading $\gamma$ on $L^2(M,S)$. A spectral triple can have additional structure such as reality. A *real structure* on a spectral triple $({\mathcal}{A}, {\mathcal}{H},D)$ is an anti-unitary operator $J: {\mathcal}{H} \rightarrow {\mathcal}{H}$, with the property that $$J^2 = {\varepsilon}, \quad JD = {\varepsilon}' DJ, \quad and \quad J\gamma = {\varepsilon}'' \gamma J, \text{ (even} \text{ case}),$$ where the numbers ${\varepsilon}$,${\varepsilon}'$, ${\varepsilon}''$ are $\pm 1$. Moreover, there are the following relations between $J$ and elements of ${\mathcal}{A}$: $$[a,b^0] = 0, \qquad [[D,a], b^0] = 0 \text{ for all } a,b \in \mathcal{A}. \label{eq:jreq1}$$ where $b^0 = J b^* J^{-1} \text{ for all } b \in \mathcal{A}$. A spectral triple $(\mathcal{A},\mathcal{H},D)$ endowed with a real structure $J$ is called a *real spectral triple*. \[dfn:realstructure\] The signs ${\varepsilon}, {\varepsilon}'$ and ${\varepsilon}''$ determine the so-called KO-dimension (modulo 8) of the real spectral triple (see [@ConnesGravity] for more details). \[ex:canonical\] For a spin-manifold and a given spinor bundle $S$ there exists an operator $J_M$ – called charge conjugation – on $L^2(M,S)$ such that $$({C^{\infty}(M)}, L^2(M,S), {{D\mkern-11.5mu/\,}}, J_M) $$ is a real spectral triple. Here the $KO$-dimension is equal to the dimension of the spin-manifold $M$. For more details on the construction of $J_M$ the reader is referred to [*e.g.*]{} [@Varilly]. When the dimension $n$ is even, the inclusion of the grading operator $\gamma$ of Example \[ex:canonicaltriple\] to the datum $$({C^{\infty}(M)}, L^2(M,S), {{D\mkern-11.5mu/\,}}, J_M, \gamma) \label{eq:canonicaltripleJeven}$$ yields a real and even spectral triple. Note that the existence of a real structure $J$ turns ${\mathcal}{H}$ into a bimodule over ${\mathcal}{A}$. Indeed, condition (\[eq:jreq1\]) implies that the right action of ${\mathcal}{A}$ on ${\mathcal}{H}$ defined by $$\xi a := Ja^*J^* \xi, \quad (\xi \in {\mathcal}{H}, a \in {\mathcal}{A})$$ commutes with the left action of ${\mathcal}{A}$. ### Spectral triples and gauge theories In this subsection we show how noncommutative spectral triples naturally give rise to gauge theories, following [@ConnesGravity]. First of all, note that the most natural notion of equivalence of (unital) noncommutative ($C^*$-)algebras is Morita equivalence ([@Rieffel]). A unital algebra ${\mathcal}{A}$ is [*Morita equivalent*]{} to a unital algebra ${\mathcal}{B}$ if and only if there exists a ${\mathcal}{B}-{\mathcal}{A}$-module ${\mathcal}{E}$ which is finitely generated and projective as an ${\mathcal}{A}$-module such that ${\mathcal}{B} = \text{End}_{{\mathcal}{A}} {\mathcal}{E}$. Commutative algebras are Morita equivalent if and only if they are isomorphic, justifying this notion of equivalence for noncommutative algebras. If $({\mathcal}{A}, {\mathcal}{H}, D,J,\gamma)$ is a real and even spectral triple and ${\mathcal}{B}$ is a unital algebra Morita equivalent to ${\mathcal}{A}$, then there is natural way to construct a real and even spectral triple for the algebra ${\mathcal}{B}$. If this is done for the case ${\mathcal}{B} = {\mathcal}{A}$ (any algebra is Morita equivalent to itself) through the module ${\mathcal}{E} = {\mathcal}{A}$, the obtained spectral triple is of the form $$({\mathcal}{A}, {\mathcal}{H}, D_A := D + A + {\varepsilon}' JAJ^{-1}),$$ where $A = \sum_j a_j [D,b_j]$ for $a_j, b_j \in {\mathcal}{A} $ is a bounded self-adjoint operator on ${\mathcal}{H}$ (see [@ConnesMarcolli], Section 10.8 for more details). It is a straightforward verification that this is again a real and even spectral triple. Thus we get another spectral triple consisting of the same algebra and Hilbert space but with the operator $D$ fluctuated by an element $A$. Note that the terms $A$ and $JAJ^{-1}$ cancel each other if ${\mathcal}{A}$ is a commutative algebra (this follows by a small calculation using the compatibility conditions of $J$ with $D$ and the action of ${\mathcal}{A}$). The occurrence of fluctuations of the operator $D$ by Morita equivalences is therefore a purely noncommutative phenomenon. The element $A$ will be interpreted as the *gauge potential*. The *gauge group* of the triple $({\mathcal}{A}, {\mathcal}{H}, D)$ is the subgroup $\text{Inn}({\mathcal}{A})$ of $*$-automorphism of ${\mathcal}{A}$ consisting of all automorphisms of the form $a \mapsto uau^*$ where $u \in {\mathcal}{A}$ satisfies $uu^*=u^*u=1$ ([@ConnesMarcolli], Section 9.9). This inner automorphism group acts naturally on the constituents of a spectral triple as an intertwiner. The gauge potential transforms accordingly as $A \mapsto uAu^* + u[D,u^*]$. The action of the gauge group on the Hilbert space is given by $\psi \mapsto uJuJ^* \psi$, where $\psi \in {\mathcal}{H}$ and $u \in {\mathcal}{U}(A)$. ### Spectral action principle {#ssct:spectralaction} Associated to a spectral triple we have a gauge group, a gauge potential and gauge transformations and in this way a spectral triple forms the setting of a gauge theory. To obtain the dynamics of the theory, the spectral action principle [@ConnesGravity; @ChamseddineConnes] is used to calculate an action from the spectral triple. The action consists of two parts: the first part is a fermion part, which is defined by $$S_f[\psi, A] = \langle \psi, D_A \psi \rangle,$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on ${\mathcal}{H}$. Note that the fermionic action depends on the gauge potential $A$ in $D_A$ but that it is invariant under gauge transformations. The other part of the action is the bosonic action which is defined by $$\label{eq:spectralaction} S_b[A] =\operatorname{Tr}(f(D_A / \Lambda)),$$ where $\operatorname{Tr}$ denotes the trace in $\H$, $f$ is a suitable cut-off function with $\Lambda > 0$. Note that, just as for the fermionic action, the expression of $S_b$ is invariant under the transformations $D_A \mapsto uJuJ^* D_A Ju^*Ju^*$ for $u \in {\mathcal}{A}$ unitary. Einstein–Yang–Mills theories and spectral triples {#ssct:EYM} ------------------------------------------------- Chamseddine and Connes showed in [@ChamseddineConnes] that Yang–Mills gauge theory over a compact Riemannian spin-manifold $M$ can be obtained from a spectral triple built from the canonical triple associated to this manifold and a matrix algebra. In this subsection we will briefly review their results and we will relate it to the description of gauge theories in terms of principal fiber bundles. Let us first recall how such fiber bundles enter gauge theories. \[dfn:connection1form\] Let $G$ be a matrix Lie group and let $P$ be a principal $G$-bundle. A *connection* $\omega$ assigns to each local trivialization $\phi_U: \pi^{-1}(U) \rightarrow U \times G$ a $\mathfrak{g}$-valued one-form $\omega_u$ on $U$. If $\phi_V$ is another local trivialization and $g_{uv}: U\cap V \rightarrow G$ is the transition function from $(U,\phi_U)$ to $(V,\phi_V)$, then we require the following transformation rule for $\omega$: $$\omega_u = g_{uv}^{-1} dg_{uv} + g^{-1}_{uv} \omega_v g_{uv}. \label{eq:76}$$ More generally, that is in the case of arbitrary Lie groups, the gauge potential is defined as a global $\mathfrak{g}$-valued connection 1-form on $P$ satisfying some extra conditions. In the case of matrix Lie groups this definition coincides with Definition \[dfn:connection1form\] (see for instance [@Bleecker] for more details). The local one-forms $\omega_u$ are the gauge potentials one encounters in physics. \[dfn:gaugetheory\] A *gauge theory* with group $G$ over a manifold $M$ consists of a principal $G$-bundle together with a connection 1-form $\omega$ on $P$. The connection 1-form $\omega$ on $P$ is also called the *gauge potential*. If $G=(P)SU(N)$ then the gauge theory is called a $(P)SU(N)$-Yang–Mills theory. We now briefly summarize the results of [@ChamseddineConnes] that obtained Yang–Mills theory on a manifold $M$ from a well-chosen spectral triple. From now on the manifold $M$ is assumed to be a compact $4$-dimensional spin-manifold. Consider the following objects: $$\begin{gathered} \mathcal{A} = C^{\infty}(M) \otimes M_N(\mathbb{C}) , \qquad \mathcal{H} = L^2(M,S) \otimes M_N(\mathbb{C}), \qquad D = {{D\mkern-11.5mu/\,}}_M \otimes 1, \nonumber \\ J= J_M \otimes ( \cdot)^*, \qquad \gamma = \gamma_5 \otimes 1. \label{eq:conclusion0}\end{gathered}$$ The bundle $S$ is the spinor bundle whose fibers are isomorphic to $\mathbb{C}^4$ as in Example \[ex:canonicaltriple\] and the operator ${{D\mkern-11.5mu/\,}}$ is the Dirac operator on the bundle $S$. Observe that this triple forms a spectral triple, being the product of the canonical triple $({C^{\infty}(M)}, L^2(M,S), {{D\mkern-11.5mu/\,}})$ and the matrix algebra $M_N(\mathbb{C})$ that acts on itself by left multiplication. We will describe this product structure in more detail in Subsection \[ssct:kkproduct\]. The spectral triple in Equation is real and even using the fact that the canonical triple is real and even. Let us now determine the fluctuated Dirac operator $D_A = D + A + {\varepsilon}'JAJ^*$ for this spectral triple. The fact that ${\varepsilon}' =1$ in 4 dimensions implies that $$\begin{aligned} A + {\varepsilon}' JAJ^* = \gamma^{\mu} A_{\mu} + J \gamma^{\mu} A_{\mu} J^*. \label{eq:bpp}\end{aligned}$$ In even dimensions one has $$J_M \gamma^{\mu} J_M^* = -\gamma^{\mu},$$ and if we use that left-multiplication by $JA_{\mu}J^*$ is right multiplication by $A^*_{\mu}$, Equation ($\ref{eq:bpp}$) turns into $A + JAJ^* = \gamma^{\mu} \cdot \text{ad}(A_{\mu})$, since $A$ is self-adjoint. Thus the fluctuated Dirac operator is of the form: $$D_A = D + i \gamma^{\mu} \mathbb{A}_\mu $$ where $\bA_{\mu} = -i\operatorname{ad}A_{\mu}$. The self-adjointness of $A$ implies that $\bA_\mu$ is an anti-hermitian one-form. Since $A$ acts in the adjoint representation the $u(1)$-part drops out and we effectively have a $su(N)$-gauge potential. It was shown in [@ChamseddineConnes] that the spectral action applied to the above spectral triple describes the Einstein–Yang–Mills system. It contains the Einstein-Hilbert action and higher-order gravitational terms, as well as the Yang–Mills action for a global $su(N)$-valued 1-form $A_{\mu}$. This is in line with the interpretation of the fluctuation $A$ as a gauge potential. Comparing this with the definition of a $PSU(N)$-Yang–Mills theory as in Definition \[dfn:gaugetheory\], the fact that the gauge potential $A_{\mu}$ is globally an $su(N)$-valued 1-form means that this corresponds to a gauge theory with a trivial principal $PSU(N)$-bundle $P$. The goal of this paper is to generalize the spectral triple in such a way that it determines a topologically nontrivial $PSU(N)$-gauge theory. Algebra bundles and spectral triples {#sect:algebrabundles} ==================================== In this section we will generalize the above spectral triple to obtain a gauge theory on a non-trivial $PSU(N)$-bundle. The important observation here is that in the trivial case we started with the algebra ${C^{\infty}(M)} \otimes M_N(\mathbb{C})$ which is precisely the algebra of sections of a trivial $M_N(\mathbb{C})$-bundle over $M$. This suggests for the non-trivial case that the algebra in the spectral triple is given by $\Gamma(M,B)$, where $B$ is an arbitrary locally trivial algebra bundle where the fiber is the $*$-algebra $M_N(\mathbb{C})$. In fact, we will construct such a real and even spectral triple $({\mathcal}{A}, {\mathcal}{H}, D, J, \gamma)$ where the algebra ${\mathcal}{A}$ is isomorphic to $\Gamma(M,B)$. This allows for a derivation of Yang–Mills theory for a gauge connection on a non-trivial principal fiber bundle in the next section. Definition of algebra bundles {#ssct:definitionalgebrabundle} ----------------------------- In this paper we take the following definition of an algebra bundle. An *algebra bundle* $B$ is a vector bundle together with a vector bundle homomorphism $\mu: B \otimes B \rightarrow B$ such that for all $x \in M$: $$\mu(p_x \otimes(\mu(q_x \otimes r_x )) = \mu(\mu(p_x \otimes q_x) \otimes r_x), \quad \forall p_x,q_x, r_x \in B_x,$$ inducing an associative algebra structure on each of the fibers of $B$ by setting $p_x \cdot q_x = \mu(p_x \otimes q_x)$ for two section $p$ and $q$ evaluated at $x \in M$. If $B_1$ and $B_2$ are two algebra bundles, then a map $\phi: B_1 \rightarrow B_2$ is called an *algebra bundle morphism* if it is a vector bundle morphism such that the restriction $f_{|(B_1)_x}: (B_1)_x \rightarrow (B_2)_x$ is an homomorphism of algebras. An algebra bundle $B$ is called an [involutive]{} or [$\ast$-algebra bundle]{}, if there exists in addition an algebra bundle homomorphism $J: B \rightarrow {\overline}{B}^{op}$ such that $J^2=1$,[^2] giving each fiber the structure of an involutive algebra by setting $p^*_x = J(p_x)$. If $B_1, B_2$ are two $*$-algebra bundles, then an $*$-*algebra bundle homomorphism* is a vector bundle homomorphism $f: B_1 \rightarrow B_2$ such that the restriction $f_{|(B_1)_x}: (B_1)_x \rightarrow (B_2)_x$ is a $*$-algebra homomorphism for every base point $x \in M$. Let $AlgB(M)$ ($AlgB^*(M)$) denote the category whose objects are all (involutive) algebra bundles (over $M$), and where the morphisms are all (involutive) algebra bundle morphisms. We note here that we do not require that the algebra in each fiber is the same. However, the way we introduced the associative algebra structures on the fibers guarantees that the product of two smooth sections is again smooth. This turns $\Gamma(M,B)$ into an associative algebra. In general, the space of smooth sections of a vector bundle on $M$ is a module over $C^\infty(M)$. In the case of algebra bundles, this action is compatible with the multiplication in the fiber. Thus, if $B$ is an (involutive) algebra bundle, then $\Gamma(M,B)$ is a finitely generated (involutive) module algebra over $C^\infty(M)$. Recall that an $R$-module algebra is an $R$-module ${\mathcal}{A}$ with an associative multiplication ${\mathcal}{A} \times {\mathcal}{A} \mapsto {\mathcal}{A}: (a,b) \mapsto ab$ which is $R$-bilinear: $$r(ab)=(ra)b= a(rb) \quad \forall a,b \in {\mathcal}{A}, r \in R.$$ An $R$-module algebra is called involutive if there exists a map $*: {\mathcal}{A} \rightarrow {\mathcal}{A}$ such that $$\begin{gathered} (ab)^* = b^* a^* ; \qquad (a + b)^* = a^* + b^* ; \qquad (r a)^* = r^* a^* ; \qquad \qquad (r,s \in R, a,b \in {\mathcal}{A}).\end{gathered}$$ Recall the well-known Serre–Swan Theorem for (complex) vector bundles over compact manifolds [@Swan]. For every vector bundle $E$ over a compact manifold $M$ the space of sections $\Gamma(M,E)$ is a finitely generated projective ${C^{\infty}(M)}$-module. The association $\Gamma$ of the space of sections to the vector bundle $E$ establishes an equivalence of categories between the complex vector bundles over $M$ and the category of finitely generated projective ${C^{\infty}(M)}$-modules. We now extend this result to arrive at an equivalence between ($*$)-algebra bundles and finitely generated projective ${C^{\infty}(M)}$-module algebras (with involution). The idea is that the ${C^{\infty}(M)}$-linear multiplicative structure on ${\mathcal}{P}$, where ${\mathcal}{P}$ is now considered as the space of sections of some vector bundle $B$ (which is unique up to isomorphism), induces a product on the fibers $B_x$ such that $(s\cdot t)(x) = s(x) \cdot t(x)$. The next lemma is crucial for lifting the multiplication structure on $\Gamma(M,B)$ to the fibers of $B$. [@Nestruev Lemma 11.8b] \[lem:sectionzero\] Let $\pi: B \rightarrow M$ be a vector bundle. Suppose $s$ is a section with $s(x)=0$ for some $x \in M$. Then there exist functions $f_i$ with $f_i(x)=0$ and sections $s_i \in \Gamma(M,B)$ so that $s$ can be written as a finite sum $s = \sum_i f_i s_i$. Suppose that ${\mathcal}{P}$ is a finitely generated projective ${C^{\infty}(M)}$-module which is at the same time an (involutive) $C^{{\infty}}(M)$-module algebra. The Serre–Swan Theorem gives a vector bundle $B$ so that ${\mathcal}{P} \simeq \Gamma(M,B)$ as $C^\infty(M)$-modules. We will now step-by-step introduce an (involutive) algebra bundle structure on $B$. For $x \in M$, let $p,q \in B_x$ be given and suppose $s,t \in {\mathcal}{P}$ are such that $p = s(x)$ and $q = t(x)$. There exists a well-defined fiber multiplication $\mu(p \otimes q) : =st(x)$ turning $B$ into an algebra bundle. Consequently, we have $s t (y) = s(y) t(y)$ for all $y \in M$ and $s, t \in \Gamma(M,B)$. \[prp:dfnproduct\] We need to show that the definition of the fiber product is independent of the choice of sections $s, t$ with $s(x) = p$ and $t(x)=q$. Therefore, let $s', t'$ be two other sections of the bundle $B$ with $s'(x) = p$ and $t'(x) =q$. Then $s_0 = s' - s$ and $t_0 = t' - t$ are sections for which $s_0(x)=t_0(x)=0$. According to Lemma \[lem:sectionzero\] $s_0$ and $t_0$ can be written as $s_0 = \sum_i f_i s_i$, $t_0 = \sum_i g_i t_i$ where $f_i(x)=g_i(x)=0$ for every $i$. This gives $$\begin{aligned} s't' - st = (s' - s) t' + s(t' - t) = \sum f_i s_i t' + \sum_i g_i s t_i ,\end{aligned}$$ which evaluated at $x$ gives zero because of the module structure of $\Gamma(M,B)$. This argument shows that $s' t'(x) = s t(x)$ and the product is well-defined. Actually, the map $(s,t) \mapsto st$ is ${C^{\infty}(M)}$-bilinear so it can be considered as a ${C^{\infty}(M)}$-linear map from $\Gamma(M,B) \otimes_{{C^{\infty}(M)}} \Gamma(M,B)$ to $\Gamma(M,B)$. This corresponds to a vector bundle homomorphism $\mu: B \otimes B \rightarrow B$. If ${\mathcal}{P} \cong \Gamma(M,B)$ is unital with unit $1_P$, then we can fix a unit in the fiber $B_x$ by setting $1_{B_x}=1_{{\mathcal}{P}}(x)$. For given $p \in B_x$, let $s \in {\mathcal}{P}$ be such that $s(x)=p$. Define $p^* := J p := s^*(x)$. This is a well-defined involutive structure on the fiber $B_x$, turning $B$ into an involutive algebra bundle. \[prp:dfnstar\] We will use the same argument as before. Let $s'$ be another such section with $s'(x)=p$. Then with Lemma \[lem:sectionzero\] $s_0 = s - s'$ can be written as a sum $\sum_i f_i s_i$ where $s_i \in {\mathcal}{P}$, $f_i\in C^{\infty}(M)$ and $f_i(x)=0$ for all $i$. This gives $$s^*(x) - s'^*(x) = (s - s')^*(x) = \sum_i (f_i s_i)^*(x) = \sum_i f^*_i(x) s^*_i(x) = 0,$$ so that the star structure is well-defined. That this is indeed a star structure on the fiber $B_x$ compatible with the algebra structure of the fiber, follows immediately from the definition of a module $*$-algebra. The functor $\Gamma: \text{Vect}_M \rightarrow \text{FGP}_{{C^{\infty}(M)}} \text{-mod}$ can be restricted to a functor $\hat{\Gamma}$ from the category $AlgB(M)$ of algebra bundles to the category of finitely generated projective ${C^{\infty}(M)}$-algebras $\text{FGP}_{{C^{\infty}(M)}} \text{-alg-mod}$. A similar statement applies to involutive algebra bundles and involutive module algebra. It follows from Propositions \[prp:dfnproduct\] and \[prp:dfnstar\] that the restricted functor $\hat{\Gamma}$ is still essentially surjective. As a restriction of a faithful functor, $\hat \Gamma$ is of course also faithful. To show that $\Gamma$ is full, let $B_1, B_2$ be two ($*$-)algebra bundles and $F: \Gamma(M,B_1) \rightarrow \Gamma(M,B_2)$ be a ($*$-preserving) ${C^{\infty}(M)}$-algebra-homomorphism. We will prove that the bundle homomorphism $\phi$ defined by $$\phi(e) = F(s)(x), \quad (e \in E_1), \label{eq:full}$$ where $s \in \Gamma(M,B_1)$ satisfies $s(x) =e$, is a $*$-algebra bundle homomorphism which is mapped to $F$ by $\hat{\Gamma}$. Firstly, observe that the map $\phi$ is well-defined: let $s'$ be another section with $s'(x) = e$. Then $s - s' = \sum_i f_i s_i$, where the $s_i$ are in $\Gamma(M,B_1)$ and where the $f_i$ are smooth functions on $M$ vanishing at $x$. This implies that indeed $F(s - s')(x) = 0$. Secondly, $\phi$ is a $*$-algebra bundle homomorphism, since $$\begin{aligned} \phi(pq) &=& F(st)(x) = F(s)F(t)(x) = F(s)(x)\cdot F(t)(x) = \phi(p) \phi(q), \nonumber \\ \phi(p^*) &=& F(s^*)(x) = (F(s))^*(x) = (F(s)(x))^* = \phi(p)^*. \nonumber\end{aligned}$$ where $s,t \in \Gamma(M,E)$ are such that $p=s(x)$, $q=t(x)$. Finally, by construction $(\hat{\Gamma}(\phi)s)(x) = \phi(s(x)) = F(s)(x)$, so that $\hat{\Gamma}(\phi) = F$ as required. Hence, $\hat{\Gamma}$ is a full functor. If $B_1$, $B_2$ are unital ($*$-)algebra bundles and $\phi: B_1 \rightarrow B_2$ is a unital ($*$-)algebra bundle homomorphism, then $\hat{\Gamma}(\phi)$ is a unital ($*$-preserving) ${C^{\infty}(M)}$-algebra-homomorphism. Conversely, if $F: \Gamma(M,B_1) \rightarrow \Gamma(M,B_2)$ is a unital ($*$-preserving) ${C^{\infty}(M)}$-algebra-homomorphism, and $\phi: B_1 \rightarrow B_2$ is defined by (\[eq:full\]), then $$\phi(1_x) = F(1)(x) = 1_{x}.$$ We summarize the results in this subsection in the following theorem. \[thm:serreswan2\] Let $M$ be a compact manifold. The functor $\hat{\Gamma}$ furnishes an equivalence between the category of (unital) (involutive) algebra bundles over $M$ and the category of (unital) finitely generated projective (involutive) ${C^{\infty}(M)}$-algebras. Spectral triple obtained from an algebra bundle {#ssct:algebrabundletospectraltriple} ----------------------------------------------- In this subsection we construct a real and even spectral triple whose algebra is isomorphic to $\Gamma(M,B)$. Here $B$ is some locally trivial $*$-algebra bundle whose fibers are copies of a fixed (finite-dimensional) $*$-algebra $A$. Furthermore, we require that for each $x$ the fiber $B_x$ is endowed with a faithful tracial state $\tau_x$ so that for all $s \in \Gamma(M,B)$ the function $x \mapsto \tau_x{s(x)}$ is smooth. The corresponding Hilbert–Schmidt inner product on the fiber $B_x$ induced by $\tau_x$ is denoted by $\langle \cdot , \cdot \rangle_{B_x}$. Consequently, the ${C^{\infty}(M)}$-valued form $$( \cdot, \cdot )_B: \Gamma(M,B) \times \Gamma(M,B) \rightarrow {C^{\infty}(M)}, \quad (s,t)_B(x) = \langle s(x), t(x) \rangle_{B_x},$$ turns $\Gamma(M,B)$ into a pre-Hilbert ${C^{\infty}(M)}$-module. As in the previous sections, we assume that $M$ is a Riemannian spin manifold, on which $S \to M$ is a spinor bundle and ${{D\mkern-11.5mu/\,}}= c \circ \nabla^S$ a Dirac operator. Combining the inner product on spinors with the above hermitian structure naturally induces the following inner product on $\Gamma(M,B \otimes S)$: $$\langle \xi_1, \xi_2 \rangle_{\Gamma(M,B \otimes S)} := \int_M \langle \xi_1(x) , \xi_2(x) \rangle_{B_x \otimes S_x} \quad (\xi_1,\xi_2 \in \Gamma(M,B \otimes S)), \label{eq:innerproductbundle}$$ turning it into a pre-Hilbert space. The completion with respect to the norm induced by this inner product consists of all square-integrable sections of $B \otimes S$, and is denoted by $L^2(M,B \otimes S)$. Note that we can identify $\Gamma(M,B \otimes S ) \cong \Gamma(M,B) \otimes_{{C^{\infty}(M)}} \Gamma(M,S)$ as ${C^{\infty}(M)}$-modules. In what follows, we will use this isomorphism without further notice. The above inner product (\[eq:innerproductbundle\]) can be written as $$\langle s_1 \otimes \psi_1, s_2 \otimes \psi_2 \rangle_{\Gamma(M,B) \otimes_{{C^{\infty}(M)}} \Gamma(M,S)} = \langle \psi_1 , (s_1,s_2)_B \psi_2 \rangle_S,$$ where $(s_1, s_2)_B \in {C^{\infty}(M)}$ acts on $\Gamma(M,S)$ by point-wise multiplication. \[thm:spectraltriple1\] In the above notation, let $\nabla^B$ be a hermitian connection (with respect to the Hilbert–Schmidt inner product) on the algebra bundle $B$ and let $D_B = c \circ(\nabla^B \otimes 1 + 1 \otimes \nabla^S) $ be the twisted Dirac operator on $B \otimes S$. Then $$(\Gamma(M,B),L^2(M,B \otimes S), D_B)$$ is a spectral triple with summability equal to the dimension of $M$. First, it is obvious that fiber-wise multiplication of $a \in \Gamma(M,B)$ on $\Gamma(M,B \otimes S)$ extends to a bounded operator on $L^2(M,B \otimes S)$ since $$\|as \otimes \psi \|^2 = \int_M \langle \psi(x), \langle a(x)s(x), a(x)s(x) \rangle_{B_x} \psi(x) \rangle_{S_x} dx \leq \sup_{x \in M} \{|a(x)|^2\} \| s \otimes \psi \|^2.$$ Compactness of the resolvent and summability is clear from ellipticity of the twisted Dirac operator $D_B$, $M$ being a compact manifold. Moreover, the commutator $[D_B,a]$ is bounded for $a \in \Gamma(M,B)$ since $D_B$ is a first-order differential operator. More precisely, in local coordinates one computes $$[D_B,a] (s \otimes \psi) = \left(\partial_{\mu} a + [\omega^B_{\mu},a]\right) s \otimes \gamma^{\mu} \psi. $$ where $\nabla^B_\mu = \partial_\mu + \omega^B_\mu$, locally. This operator is clearly bounded on $L^2(M,B \otimes S)$, provided $a$ is differentiable and $\omega^B_\mu$ is a smooth connection one-form. Next, we would like to extend our construction to arrive at a real spectral triple. For this, we introduce an anti-linear operator on $L^2(M,B \otimes S)$ of the form $$J ( s \otimes \psi ) = s^* \otimes J_M \psi$$ with $J_M$ charge conjugation on $M$ (*cf*. Example \[ex:canonical\]). For this operator to be a real structure on our spectral triple $(\Gamma(B), L^2(B \otimes S), D_B)$, we need some extra conditions on the connection $\nabla_B$ on $B$. Let $B$ be a $*$-algebra bundle over a manifold $M$. A $*$*-algebra connection* $\nabla$ on $B$ is a connection on $B$ that satisfies $$\begin{aligned} \nabla(st) = s\nabla t + (\nabla s)t, \qquad (\nabla s)^* = \nabla s^*; \qquad (s,t \in \Gamma(M,B)).\end{aligned}$$ If $B$ is a hermitian $*$-algebra bundle and $\nabla$ is also a hermitian connection, then $\nabla$ is called a *hermitian* $*$-algebra connection. Before we proceed we need to know whether a hermitian $*$-algebra connection exists on any given locally trivial $*$-algebra bundle. A partition of unity argument easily shows how to construct hermitian $*$-algebra connections on arbitrary $*$-algebra bundles. Every locally trivial hermitian $*$-algebra bundle $B$ defined over a paracompact space $M$ admits a hermitian $*$-algebra connection. \[lma:existenceconnection\] Let $\{U_i\}$ be a locally finite open covering of $M$ such that $B$ is trivialized over $U_i$ for each $i$. Then on each $U_i$ there exists a hermitian $*$-algebra connection $\nabla_i$, for instance the trivial connection $d$ on $U_i$. Now, let $\{f_i\}$ be a partition of unity subordinate to the open covering $\{U_i\}$ (all $f_i$ are real-valued). Then the linear map $\nabla$ defined by $$(\nabla s)(x) = \sum_i f_i(x) (\nabla_i s)(x), \quad (x \in M),$$ is a hermitian $*$-algebra connection on $\Gamma(M,B)$. \[rmk:derivation1form\] The fact that locally, on some trivializing neighborhood, the exterior derivative $d$ is a hermitian $*$-algebra connection shows that on such a local patch every hermitian $*$-algebra connection is of the form $$d + \omega,$$ where $\omega$ is a real connection 1-form with values in the real Lie algebra of $*$-derivations of the fiber that are anti-hermitian with respect to the inner product on the fiber. For instance, when the fiber is the $*$-algebra $M_N(\mathbb{C})$ endowed with the Hilbert–Schmidt inner product, this Lie algebra is precisely $\text{ad}(\mathfrak{u}(N)) \cong \mathfrak{su}(N)$. \[thm:spectraltriple2\] Suppose in addition to the conditions of Theorem \[thm:spectraltriple1\] that $\nabla^B$ is hermitian $*$-algebra connection and set $\gamma_B = 1 \otimes \gamma_5$ as a self-adjoint operator on $L^2(M,B \otimes S)$. Then ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ is a real and even spectral triple with $KO$-dimension equal to the dimension of $M$. First of all, we check that $J$ is anti-unitary: $$\begin{aligned} \langle J( s\otimes \psi), J (t\otimes \eta) \rangle &= \langle J_M \psi, (s^*,t^*) J_M \eta \rangle = \langle J_M \psi, J_M {\overline}{(s^*,t^*)}\eta \rangle \nonumber \\ &= \langle {\overline}{(s^*,t^*)} \eta, \psi \rangle = \langle (s, t) \eta, \psi \rangle = \langle t \otimes \eta, s \otimes \psi \rangle, \nonumber\end{aligned}$$ where we have in the second step that $J_M f = \bar{f} J_M$ for every $f \in {C^{\infty}(M)}$, in the third step that $J_M$ is anti-unitary and in the fourth step that $(s,t) = (t^*,s^*)$ (by definition of the hermitian structure as a fiber-wise trace). Moreover, since $J_M^2=-1$ it follows that $J^2=-1$. We next establish $DJ =JD$ by a local calculation: $$\begin{aligned} (JD - DJ)(s \otimes \psi) &=&J (\nabla^B_{\mu}s \otimes i\gamma^{\mu}\psi + s \otimes {{D\mkern-11.5mu/\,}}\psi) - D_B(s^* \otimes J_M \psi) \nonumber \\ &=& (\nabla^B_{\mu} s)^* \otimes (-i)J_M \gamma^{\mu} \psi + s^* \otimes J_M {{D\mkern-11.5mu/\,}}\psi - \nabla^B_{\mu} s^* \otimes i\gamma^{\mu} J_M \psi -s^* \otimes {{D\mkern-11.5mu/\,}}J_M \psi \nonumber \\ &=& -i\left( (\nabla^B_{\mu} s)^* - \nabla^B_{\mu} s^* \right) \otimes J_M \gamma^{\mu} \psi = 0 \nonumber,\end{aligned}$$ since in four dimensions $\{J_M, \gamma^{\mu}\}=[{{D\mkern-11.5mu/\,}}, J_M]=0$, and the last step is established by the condition of a $*$-algebra connection, [*i.e.*]{} $(\nabla s)^* = \nabla s^*$ for all $s \in \Gamma(M,B)$. The commutant property follows easily: $$\begin{aligned} [a,b^0](s \otimes \psi) &=& aJb^*J^*(s \otimes \psi) - Jb^*J^*a(s \otimes \psi) = aJ(b^*s^* \otimes J_M^* \psi) - Jb^* (s^*a^* \otimes J_M^* \psi) \nonumber \\ &=& asb \otimes \psi - asb \otimes \psi = 0, \nonumber\end{aligned}$$ where $a,b \in \Gamma(M,B)$ and $s \otimes \psi \in \Gamma(M,B) \otimes_{{C^{\infty}(M)}} \Gamma(M,S)$. Since $[a,b^0]=0$ on $\Gamma(M,B) \otimes_{{C^{\infty}(M)}} \Gamma(M,S)\cong \Gamma(M,B \otimes S)$, it is zero on the entire Hilbert space $L^2(M,B\otimes S)$. It remains to check the order one condition for the Dirac operator. First note that $$[[D,a],b^0](s \otimes \psi) = i c([[\nabla, a], b^0]( s\otimes \psi)) \quad (a,b,s \in \Gamma(M,B)).$$ This is zero because $[[\nabla, a], b^0]( s\otimes \psi)$ is zero: $$\begin{aligned} ([\nabla,a]sb) \otimes \psi - Jb^*J^* ([\nabla, a] s \otimes \psi) &=& \nabla(asb) \otimes \psi - a \nabla(sb) \otimes \psi - Jb^* J^* \nabla(as) \otimes \psi + Jb^*J^* a (\nabla s) \otimes \psi \nonumber \\ &=& \nabla(asb) \otimes \psi - a \nabla(sb) \otimes \psi - \nabla(as)b \otimes \psi + a (\nabla s) b \otimes \psi \nonumber \\ &=& \left( (\nabla a) sb + a (\nabla s)b + as (\nabla b) - a (\nabla s)b \right.\nonumber \\ && \quad - \left. as (\nabla b) - (\nabla a)sb - a(\nabla s)b + a (\nabla s)b \right) \otimes \psi, \nonumber \\ &=& 0 \nonumber\end{aligned}$$ using the defining property for $\nabla^B$ to be a $*$-algebra connection. Thus, $J$ fulfils all of the necessary conditions of a real structure on $(\Gamma(M,B), L^2(M,B \otimes S), D_B)$. The conditions on $\gamma_B$ to be a grading operator for this spectral triple are easily checked. In the next section we show that the triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ gives a non-trivial Yang–Mills theory over the manifold $M$. The Serre–Swan Theorem \[thm:serreswan2\] plays an essential role in the proof. First, we explore the form of this spectral triple in the context of Kasparov’s KK-theory. Relation with the unbounded Kasparov internal product {#sect:KK} ----------------------------------------------------- \[ssct:kkproduct\] In this section we establish that the spectral triple of Theorem \[thm:spectraltriple1\] is an unbounded Kasparov product of two unbounded KK-cycles [@KasparovKK; @BaajJulg]. Let us briefly recall some elementary notions from (unbounded) KK-theory. Denote by ${\mathcal}{B}(E)$ the bounded endomorphisms of a right Hilbert $B$-module $E$ and by ${\mathcal}{K}(E)$ the compact endomorphisms. Let $A$ and $B$ be $\mathbb{Z}_2$-graded $C^*$-algebras. A Kasparov $A$-$B$-module consists of a triple $(E, \phi, F)$ where $E$ is a countably generated $\mathbb{Z}_2$-graded Hilbert-$B$-module, $\phi$ is a graded $*$-homomorphism $A \rightarrow {\mathcal}{B}(E)$ and $F$ is a bounded operator of degree $1$, such that $[F, \phi(a)]$, $(F^2 -1)\phi(a)$, and $(F - F^*)\phi(a)$ are in ${\mathcal}{K}(E)$. There are the natural notions of unitary and homotopy equivalence and under the direct sum the set of equivalence classes of Kasparov $A-B$-modules forms an abelian group which is denoted by $KK(A,B)$ [@Kasparov]. One of the key properties of $KK$-theory is the existence of the internal Kasparov product. \[dfn:boundedkasparovproduct\] Let $E_1$ be an $A$-$B$-module and $E_2$ a $B$-$C$-module, and define an $A$-$C$-module by $E:= E_1 \otimes_B E_2$. A Kasparov module $(E,\phi,F)$ is called a *Kasparov product* for $(E_1, \phi_1, F_1)$ and $(E_2, \phi_2, F_2)$ if - $(E, \phi_1 \otimes \text{Id}, F) \in KK(A,C)$; - for every $x \in E_1$ of homogeneous degree $\#x$, the operator $T_x: E_2 \rightarrow E$ defined by $T_x(e) = x\otimes e$ satisfies $$\begin{aligned} T_x \circ F_2 - (-1)^{\#x} F \circ T_x \in {\mathcal}{K}(E_2, E), \nonumber \\ F_2 \circ T_x^* - (-1)^{\#x} T^*_x \circ F \in {\mathcal}{K}(E,E_2);\nonumber\end{aligned}$$ - for all $a \in A$ the graded commutator $\phi(a)[F_1 \otimes \text{Id}, F]\phi(a^*) \geq 0 \text{ mod }{\mathcal}{K}(E)$. If $A$ is separable and $B$ is $\sigma$-unital, then there is a Kasparov-product for $(E_1, \phi_1, F_1)$ and $(E_2, \phi_2, F_2)$ and any of these products are homotopic.[^3] Therefore, the internal Kasparov product defines a bilinear map $\otimes_B: KK(A,B) \times KK(B,C) \rightarrow KK(A,C)$. The Kasparov internal product of Definition \[dfn:boundedkasparovproduct\] can be captured in terms of unbounded Kasparov-modules [@BaajJulg]. Let $A$ and $B$ be graded $C^*$-algebras. An *unbounded Kasparov module* is a triple $(E, \phi, D)$ where $E$ is a graded Hilbert-$B$-module, $\phi: A \rightarrow {\mathcal}{B}(E)$ a graded $*$-homomorphism, and $D$ a self-adjoint regular operator in $E$, homogeneous of degree $1$ such that $(1 + D^2)^{-1} \phi(a)$ extends to an element of ${\mathcal}{K}(E)$ for all $a \in A$, and the set of all $a \in A$ such that $[D,\phi(a)]$ extends to an element in ${\mathcal}{B}(E)$ is dense in $A$. The set of all unbounded Kasparov modules is denoted by $\Psi(A,B)$. The canonical spectral triple $({C^{\infty}(M)}, L^2(M,S), {{D\mkern-11.5mu/\,}}, \gamma^5)$ is an element in $\Psi({C^{\infty}(M)},\mathbb{C})$. Another example is given as follows: let $B$ be a locally trivial $*$-algebra bundle with a smoothly-varying faithful tracial state on the fibers. Then $(\Gamma(M,B), L^2(M,B), 0)$ is an element of $\Psi(\Gamma(M,B),{C^{\infty}(M)})$, and even in $KK(\Gamma(M,B),{C^{\infty}(M)})$. Note that the algebras $\Gamma(M,B)$ and ${C^{\infty}(M)}$ are trivially graded. If $(E,\phi,D) \in \Psi(A,B)$ then $(E, \phi, F) \in \mathbb{E}(A,B)$ where $F = D(1+D^2)^{-1}$. If $A$ is separable, the map $(E,\phi,D) \mapsto [(E, \phi, D(1+D^2)^{-1})]$ is a surjective map $\Psi(A,B) \rightarrow KK(A,B)$. Thus, classes in $KK(A,B)$ can be represented by unbounded cycles in $\Psi(A,B)$. The following theorem is due to Kucerovsky [@KucerovskyUnboundedKKmodules] and introduces a Kasparov product for unbounded $KK$-modules. This was further worked out by Mesland [@Mesland]. \[thm:unboundedkasparovproduct\] Suppose that $(E_1 \otimes_B E_2, \phi_1 \otimes_B \text{Id}, D) \in \Psi(A,C)$, $(E_1, \phi_1, D_1) \in \Psi(A,B)$ and $(E_2, \phi_2, D_2) \in \Psi(B,C)$ are such that 1. for all $x$ in some dense subset of $\phi_1(A)E_1$, the operator $$\left[ \left( \begin{array}{cc} D & 0 \\ 0 & D_2 \end{array} \right), \left( \begin{array}{cc} 0 & T_x \\ T_x^* & 0 \end{array} \right)\right]$$ is bounded on $\text{Dom }D \oplus \text{Dom }D_2$; 2. The resolvent of $D$ is compatible with $D_1$: that is, there is a dense submodule $W$ such that $D_1(i\mu + D)^{-1}(i\mu_1 + D_1)^{-1}$ is defined on $W$ for all $\mu, \mu_1 \in \mathbb{R} - \{0\}$; 3. There exists a $c \geq 0$ such that $\langle D_1 x, Dx \rangle + \langle Dx , D_1 x \rangle \geq c \langle x,x\rangle$, for all $x$ in the domain; where $x \in E_1$ is homogeneous and $T_x: E_2 \rightarrow E$ maps $e \mapsto x \otimes_B e$. Then $(E_1 \otimes_B E_2, \phi_1 \otimes_B \text{Id}, D)$ represents the Kasparov-product of $(E_1, \phi_1, D_1) \in \Psi(A,B)$ and $(E_2, \phi_2, D_2) \in \Psi(B,C)$. Using Theorem \[thm:unboundedkasparovproduct\] we show that the spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B)}$ of Theorem \[thm:spectraltriple1\] can be considered as a Kasparov product. Let $B$ be a locally trivial hermitian unital $*$-algebra bundle on a compact Riemannian spin manifold $M$ with fibers isomorphic to some complex $*$-algebra $A$. Let $\nabla^B$ be a hermitian connection on $B$ and $D_B$ the corresponding twisted Dirac operator. Then ${(\Gamma(M,B), L^2(M,B\otimes S),D_B)}$ is an unbounded Kasparov product of $(L^2(M,B), \lambda, 0) \in \Psi(\Gamma(M,B),{C^{\infty}(M)})$ and $(L^2(M,S), m, {{D\mkern-11.5mu/\,}}) \in \Psi({C^{\infty}(M)}, \mathbb{C})$, where $\lambda$ is the representation of $\Gamma(M,B)$ on $L^2(M,B)$ induced by left-multiplication and where $m$ denotes the representation of ${C^{\infty}(M)}$ on $L^2(M,S)$ by point-wise multiplication with elements in $\Gamma(M,S)$. Most of the assertions are straightforward to prove. To prove the last statement we will check the first condition of Theorem \[thm:unboundedkasparovproduct\] since the other two are trivial (because $D_1 = 0$). It suffices to check that $$\begin{aligned} D \circ T_a - T_a {{D\mkern-11.5mu/\,}}&\in {\mathcal}{B}(L^2(M,S), L^2(M, B \otimes S)), \\ {{D\mkern-11.5mu/\,}}T_a^* - T_a^* D &\in {\mathcal}{B}(L^2(M,B\otimes S), L^2(M,S)) ,\end{aligned}$$ for all $a \in \Gamma(M,B)$. For the first condition, we have for $\psi \in L^2(M,S)$ that $$(D \circ T_a - T_a D_2)(\psi) = D(a \otimes \psi) - a \otimes {{D\mkern-11.5mu/\,}}\psi = c(\nabla^B a) \otimes \psi $$ so that $D \circ T_a - T_a {{D\mkern-11.5mu/\,}}$ extends to a bounded operator. Now the second one: $$\begin{aligned} ({{D\mkern-11.5mu/\,}}T_a^* - T_a^* D) (s \otimes \psi) &= {{D\mkern-11.5mu/\,}}( \langle a, s \rangle \psi ) - \langle a, s \rangle {{D\mkern-11.5mu/\,}}\psi - \langle a, c(\nabla^B s) \rangle \psi \\ &= [{{D\mkern-11.5mu/\,}}, \langle a ,s \rangle]\psi - ( [{{D\mkern-11.5mu/\,}}, \langle a,s\rangle] - \langle c(\nabla^B a), s \rangle)\psi \\ &= \langle c(\nabla^B a), s \rangle \psi, \end{aligned}$$ which is again uniformly bounded. This completes the proof. Another proof of this fact follows by adopting the direct construction of the unbounded Kasparov products by Mesland [@Mesland]. Indeed, the spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B)}\in \Psi({C^{\infty}(M)}, \C)$ is by construction the internal product of $(\Gamma(M,B), L^2(M,B), 0) \in \Psi(\Gamma(M,B),{C^{\infty}(M)})$ and $({C^{\infty}(M)}, L^2(M,S), {{D\mkern-11.5mu/\,}}, \gamma^5) \in \Psi({C^{\infty}(M)},\C)$. Yang–Mills theory as a noncommutative manifold {#sect:ym} ============================================== The spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ that we obtained in Theorem \[thm:spectraltriple2\] will turn out to be the correct triple to describe a non-trivial $PSU(N)$-gauge theory on the manifold $M$ if the fibers of $B$ are taken to be isomorphic to the $*$-algebra $M_N(\mathbb{C})$. Not only does it describe a non-trivial $PSU(N)$-gauge theory, every $PSU(N)$-gauge theory on $M$ is described by such a triple. In this section we will prove these claims by first showing how a principal $PSU(N)$-bundle can be constructed from this spectral triple (in fact, the algebra $\Gamma(M,B)$ is already sufficient for this). As in the topologically trivial case [@ChamseddineConnes], the spectral action applied to this triple will give the Einstein–Yang–Mills action, but now the gauge potential can be interpreted as a connection 1-form on the $PSU(N)$-bundle $P$. In fact, the original algebra bundle $B$ will turn out to be an associated bundle of the principal bundle $P$. From now on, the fibers of $B$ are assumed to be $M_N(\mathbb{C})$. From the spectral triple to principal bundles --------------------------------------------- According to Theorem \[thm:serreswan2\] we are able to reconstruct the unital $*$-algebra bundle $B$ from $\Gamma(M,B)$. Note that in this theorem the ($*$-)algebra bundles are not required to be locally trivial as a ($*$-)algebra bundle (they *are* locally trivial as a *vector* bundle). For the rest of this section we assume that $B$ is a locally trivial $*$-algebra bundle with fiber $M_N(\mathbb{C})$. In order to construct a principal $PSU(N)$-bundle $P$ out of $B$, first of all note that since all $*$-automorphisms of $M_N(\mathbb{C})$ are obtained by conjugation with a unitary element $u \in M_N(\mathbb{C})$ the transition functions of the bundle $\Gamma(M,B)$ have their values in $\text{Ad }U(N) \cong U(N) / Z(U(N)) \cong PSU(N)$. Thus the bundle $B$ provides us with a open covering of $\{U_i\}$ and transition functions $\{g_{ij}\}$ with values in $PSU(N)$. Using the reconstruction theorem for principal bundles we can construct a principal $PSU(N)$-bundle. By construction, the bundle $B$ is an associated bundle to $P$. Furthermore, for the real and even spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ of Theorem \[thm:spectraltriple2\] the hermitian connection $\nabla^B$ on the bundle $B$ can locally be written as $\nabla^B = d + \omega$, where $\omega$ is a $\mathfrak{su}(N)$-valued 1-form, (*cf*. Eq. \[rmk:derivation1form\]). Moreover, the transformation rule for $\omega$ is $\omega_i = g^{-1}_{ij} dg_{ij} + g^{-1}_{ij} \omega_j g_{ij}$ with $g_{ij}$ the $PSU(N)$-valued transition function of $B$. Comparing this expression with the transformation property of a connection 1-form in Definition \[dfn:connection1form\] one concludes that the hermitian $*$-algebra connection $\nabla^B$ on $B$ induces a connection 1-form on the principal bundle $P$ constructed in the previous paragraph (and vice versa). Conversely, given a $PSU(N)$-gauge theory $(P, \omega)$ on some compact Riemannian spin-manifold, then we can construct the locally trivial hermitian $*$-algebra bundle $B:=P \otimes_{PSU(N)} M_N(\mathbb{C})$, where $PSU(N)$ acts on $M_N(\mathbb{C})$ in the usual way. Moreover, the connection $\omega$ on $P$ induces a hermitian $*$-algebra connection on $B$. By following the steps in the previous section it is not difficult to see that the gauge theory $(P_B, \omega_B)$ obtained from this spectral triple $(\Gamma(M,B), L^2(M,B\otimes S), c(\nabla^B \otimes 1 + 1 \otimes \nabla^S), J, \gamma)$ is isomorphic to $(P,\omega)$. This is in accordance with the approach to almost noncommutative manifolds taken in [@Branimir]. \[cor:reconstructiontheorempfb\] Let ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ be as before, with $B$ an endomorphism bundle. Then there exists a principal $PSU(N)$-bundle $P$ such that $B$ is an associated bundle of $P$, and a connection 1-form $\omega$ on $P$. Moreover, every $PSU(N)$-gauge theory on $M$ is determined by such a spectral triple. Spectral action --------------- In this section, we will calculate the spectral action for the real spectral triple of Theorem \[thm:spectraltriple2\]. We will show that the spectral action applied to the spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ produces the Einstein–Yang–Mills action for a 1-form $A$ that defines a connection 1-form on the $PSU(N)$-bundle $P$. If $B$ is a trivial algebra bundle, this reduces to the result of [@ChamseddineConnes]. In fact, much of their local computations can be adopted in this case as well, since locally, the bundle $B$ is trivial. Nevertheless, for completeness we include the computation in the case at hand. First of all, already in Remark \[rmk:derivation1form\] we noticed that locally, on some local trivialization $U_i$, $\nabla^B$ is expressed as $d + \omega_i$ where $\omega$ is an $su(N)$-valued 1-form that acts in the adjoint representation on $\Gamma(M,B)$. Therefore, according to Definition \[dfn:connection1form\] $\omega$ already induces a connection 1-form on $P$. To get the full gauge potential we need to take the fluctuation of the Dirac operator into account as well. Inner fluctuations of the Dirac operator are given by a perturbation term of the form $$A = \sum_j a_j[D,b_j], \qquad (a_j, b_j \in \Gamma(M,B)), \label{eq:nontrivinnerfluc}$$ with the additional condition that $\sum_j a_j[D,b_j]$ is a self-adjoint operator. Explicitly, we have $$A = \sum_j c \circ (a_j [\nabla,b_j] \otimes 1),$$ where $c: \Omega^1(M) \otimes_{{C^{\infty}(M)}} \Gamma(M, B \otimes S) \rightarrow \Gamma(M,B\otimes S)$ is given by $$c(\omega \otimes s \otimes \psi) = s \otimes c(\omega)\psi, \quad (\omega \in \Omega^1(M), s \otimes \psi \in \Gamma(M,B\otimes S)).$$ Also, $\sum_j a_j [\nabla, b_j]$ is an element of $\Gamma(T^*M \otimes B)$. Locally, on some trivializing neighborhood $U$, the expression in Eq. can be written as $$A= \gamma^{\mu} A_{\mu},$$ where $A_{\mu}$ are the components of the 1-form $\sum_j a_j [\nabla, b_j]$ with values in $\Gamma(M,B)$. Since $A$ is self-adjoint the 1-form $A_{\mu}$ can be considered as a real 1-form taking values in the hermitian elements $\Gamma(M,B)$. Similarly, the expression $A + JAJ^*$ is locally written as $$\gamma^{\mu} A_{\mu} - \gamma^{\mu} J A_{\mu} J^*,$$ since $\gamma^{\mu}$ anti-commutes with $J$ in 4 dimensions. Writing out the second term gives: $$(\gamma^{\mu} J A_{\mu} J^*)(s\otimes \psi) = s A_{\mu} \otimes \gamma^{\mu} \psi, \quad \forall s \otimes \psi \in \Gamma(M,B\otimes S).$$ so that on this local patch $A + JAJ^*$ can be written as $$\gamma^{\mu} \operatorname{ad}A_{\mu}.$$ Consequently, $A + JAJ^*$ eliminates the $i u(1)$-part of $A$, making it natural to impose the uni-modularity condition $$\text{Tr } A = 0.$$ Thus, $-i \operatorname{ad}A_{\mu}$ is a one-form on $M$ with values in $\Gamma(M,\operatorname{ad}P)$. We denote this 1-form by $- i \bA^{pert}$; it is defined on the whole of $M$. The local and global expression for $D + A + JAJ^*$ are given respectively by $$D_A = i\gamma^{\mu} (\nabla^B_{\mu} \otimes 1 + 1 \otimes \nabla^S_{\mu} - i \operatorname{ad}A_{\mu} \otimes 1) \\ $$ and $$D_A = ic \circ (1 \otimes \nabla^S + \nabla_B \otimes 1 + \bA^{pert}),$$ On some trivializing neighborhood $U_i$ ($i \in I$) the connection $\nabla^B$ can be expressed as $d + \bA_i^0$ for a unique $su(N)$-valued 1-form $\bA^0_i$ on $U_i$. Thus, on $U_i$ the fluctuated Dirac operator can be rewritten as $$D_A = i c \circ (d + 1 \otimes \omega^s + (\bA_i^0 + \bA^{pert}) \otimes 1).$$ We interpret $(\bA_i^0 + \bA^{pert})$ as the full gauge potential on $U_i$; it acts in the adjoint representation on the spinors. The natural action of $g \in \textup{Inn}(\Gamma(M,B) \simeq \Gamma(M,\operatorname{Ad}P)$ by conjugation on $D_A$ induces the familiar gauge transformation: $$\mathbb{A}^0 + \mathbb{A}^{pert} \mapsto (g^{-1}\mathbb{A}^0g + g^{-1}(dg)) + g^{-1}\mathbb{A}^{pert}g = g^{-1} (dg) + g ( \mathbb{A}^0 + \mathbb{A}^{pert})g^{-1},$$ where the first two terms are the transformation of $A^0$ under a change of local trivialization, and the last term is the transformation of $A^{pert}$. Since $P$ is an associated bundle of $B$ it follows from Definition $\ref{dfn:connection1form}$ that $\mathbb{A}^0 + \mathbb{A}^{pert}$ induces a $su(N)$-valued connection 1-form on the principal $PSU(N)$-bundle $P$ that acts on $\Gamma(M,B)$ in the adjoint representation. Let us summarize what we have obtained so far. Let ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$ and $P$ be as before, so that $P \times_{PSU(N)} M_N(\C) \simeq B$. Then 1. The group of inner automorphisms $\textup{Inn}(\Gamma(M,B)) \simeq \Gamma(M,\operatorname{Ad}P)$ where $\operatorname{Ad}P = P \times_{PSU(N)} SU(N)$. 2. The inner fluctuations of $D_B$ are parametrized by a section $\bA^{pert}$ of $\Gamma(M,\operatorname{ad}P)$ where $\operatorname{ad}P = P \times_{PSU(N)} su(N)$. Moreover, the action of $\textup{Inn}(\Gamma(M,B))$ on the inner fluctations $D_B + A + JAJ^{-1}$ by conjugation coincides with the adjoint action of $\Gamma(M,\operatorname{Ad}P)$ on $\Gamma(M,\operatorname{ad}P)$. Let us now proceed to compute the spectral action for these inner fluctuations. First, we recall some results on heat kernel expansions and Seeley–DeWitt coefficients, which will be useful later on; for more details we refer to [@Gilkey]. If $V$ is a vector bundle on a compact Riemannian manifold $(M, g)$ and if $Q:C^{\infty}(V)\to C^{\infty}(V)$ is a second-order elliptic differential operator of the form $$Q = - \big(g^{\mu\nu}\partial_{\mu}\partial_{\nu} + K^{\mu}\partial_{\mu} + L)\label{eq:elliptic}$$ with $K^{\mu}, L \in \Gamma(\textup{End}(V))$, then there exist a unique connection $\nabla$ and an endomorphism $E$ on $V$ such that $Q = \nabla\nabla^* - E$. In this situation we can make an asymptotic expansion (as $t \to 0$) of the trace of the operator $e^{-tQ}$ in powers of $t$: $$\operatorname{Tr}\,e^{-tQ} \sim \sum_{n \geq 0}t^{(n-m)/2}a_n(Q),\qquad a_n(Q) := \int_{M}a_n(x, Q)\sqrt{g}d^m x\label{eq:gilkey},$$ where $m$ is the dimension of $M$ and the coefficients $a_n(x, Q)$ are called the *Seeley–DeWitt coefficients*. It turns out [@Gilkey Theorem 4.8.16] that $a_n(x, Q) = 0$ for $n$ odd and that the first three even coefficients are given (modulo boundary terms) by $$\begin{aligned} a_0(x,Q) &= (4 \pi)^{-m/2} \text{Tr}(\text{Id}) \\ a_2(x,Q) &= (4 \pi)^{-m/2} \text{Tr}(-\frac{R}{6}\text{Id} + E) \\ a_4(x,Q) &= (4 \pi)^{-m/2} \frac{1}{360}\text{Tr}\left( 5R^2 - 2R^{\mu \nu} R_{\mu \nu} + 2R_{\mu \nu \rho \sigma} R^{\mu \nu \rho \sigma} - 60RE + 180E^2 + 30\Omega_{\mu \nu} \Omega^{\mu \nu} \right),\end{aligned}$$ where $\Omega_{\mu \nu}$ is the curvature of the connection $\nabla$. This can be used in the computation of the spectral action as follows. Assume that the inner fluctuations give rise to an operator $D_A$ for which $D_A^2$ is of the form on some vector bundle $V$ on a compact Riemannian manifold $M$. Then, on writing $f$ as a Laplace transform, we obtain $$\label{eq:laplace} f (D_A/\Lambda) = \int_{t>0} \tilde g(t) e^{-t D_A^2/\Lambda^2} ~ d t.$$ One calculates that in 4 dimensions the heat expansion (up to order $n=4$) of the spectral action (\[eq:spectralaction\]) is given by $$\begin{aligned} \text{Tr}\left(f(D/\Lambda)\right) &\sim f(0) \Lambda^0 a_4(D^2) + \sum_{n=0,2} \Lambda^{4-n} a_n(D^2) \frac{1}{\Gamma(\frac{4-n}{2})} \int_0^{{\infty}} k(v) v^{\frac{4-n}{2}-1} dv \\ &= f(0) \Lambda^0 a_4(D^2) + 2 f_2 \Lambda^2 a_2(D^2) + 2\Lambda^4 f_4 a_0(D^2). \label{eq:Diracexpansion}\end{aligned}$$ where the $f_k$ are moments of the function $f$: $$\begin{aligned} f_{k} := \int_{0}^{\infty} f(w)w^{k-1}dw; \qquad (k>0) \nonumber. \end{aligned}$$ \[lem:ED2\] For the spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$, the square of the fluctuated Dirac operator $D_A^2$ is locally of the form $-g_{\mu \nu} \partial_{\mu} \partial_{\nu} + K_{\mu} \partial_{\mu} + L$ and we have the following expressions for $\Omega_{\mu\nu}$ and $E$: $$\begin{aligned} E = -\frac{1}{4} R \otimes 1_{N^2} - \sum_{\mu < \nu} \gamma^{\mu} \gamma^{\nu} \otimes F_{\mu \nu} \nonumber \\ \Omega_{\mu \nu} = \frac{1}{4} R^{ab}_{\mu \nu} \gamma_{ab} \otimes 1_{n^2} + id_4 \otimes F_{\mu \nu} \nonumber,\end{aligned}$$ where $F_{\mu \nu}$ is the curvature of the connection $\nabla^B + \bA^{pert}$. This result allows us to compute the bosonic spectral action for the fluctuated Dirac operator $D_A$, essentially reducing the computation in terms of a local trivialization to the trivial case of [@ChamseddineConnes] with the following result. \[thm:actionnontriv\] For the spectral triple ${(\Gamma(M,B), L^2(M,B\otimes S),D_B,J,\gamma_B)}$, the spectral action equals the Yang–Mills action for $\nabla^B + \bA^{pert}$ minimally coupled to gravity: $$\operatorname{Tr}\left( f(D_A/\Lambda) \right) \sim \frac{f(0)}{24 \pi^2} \int_M \operatorname{Tr}F_{\mu\nu} F^{\mu\nu} \sqrt{g} d^4x + \frac{1}{(4\pi)^2} \int_M \mathcal{L}(g_{\mu \nu}) \sqrt{g} d^4x + \mathcal{O}(\Lambda^{-2}),$$ where $\mathcal{L}(g^{\mu \nu})$ is given by $$\mathcal{L}(g^{\mu \nu}) = 2N^2 \Lambda^4 f_4 + \frac{N^2}{6} \Lambda^2 f_2 R - \frac{N^2 f_0}{80} C_{\mu \nu \rho \sigma} C^{\mu \nu \rho \sigma},$$ ignoring topological and boundary terms. Here $C$ denotes the Weyl-tensor and $f_i$ are the $i$’th moments of the function $f$. Conclusions and outlook ======================= We have generalized the noncommutative description of the Einstein–Yang–Mills system by Chamseddine and Connes [@CC97] to the case where the principal bundle describing the gauge field is non-trivial. We have obtained a spectral triple from an algebra bundle and related its construction to the internal Kasparov product in unbounded KK-theory. If the typical fiber of the algebra bundle is $M_N(\C)$, we have showed that its internal fluctuations are parametrized by a $PSU(N)$-gauge field. In fact, we reconstructed a $PSU(N)$-principal bundle for which the algebra bundle is an associated bundle, and on which the gauge field defines a connection one-form. Finally, we have applied the spectral action principle to these inner fluctuations of the spectral triple and derived the Yang–Mills action for a $PSU(N)$-gauge field, minimally coupled to gravity. A natural question that arises in this topologically non-trivial context is how to incorporate, besides the Yang–Mills action, a topological action functional. Given an (even) spectral triple $(\A,\H,D,\gamma)$, we introduce – besides the spectral action – an invariant by $$\label{eq:topspectralaction} S_{\top} [A] = \operatorname{Tr}\left( \gamma f(D_A/\Lambda) \right).$$ We will call this the [*topological spectral action*]{}. It is clearly invariant under the action of the group of unitaries in the algebra $\A$, acting on $\gamma$ by conjugation. If we again write $f$ as a Laplace transform and use the McKean–Singer formula, $$\operatorname{Tr}e^{-t D_A^2} = \operatorname{Index}D_A,$$ then we can prove that asymptotically $$S_{\top} [A] \sim f(0) \operatorname{Index}D_A$$ In our case of interest, [*i.e.*]{} the setting of Theorem \[thm:actionnontriv\], we thus find with the Atiyah–Singer index theorem an extra contribution of the form $$S_{\top} [A] \sim \frac{f(0)}{(2 \pi i)^{n/2}} \int_M \hat{A}(M) \ch (B).$$ in terms of the $\hat A$ genus of $M$ and the Chern character of the algebra bundle $B$. [99]{} M. F. Atiyah. . Fermi Lectures. Scuola Normale, Pisa, 1979. S. Baaj and P. Julg. Théorie bivariante de Kasparov et opérateurs non bornés dans les $C^*$-modules hilbertiens. 296 (1983) 875–878. B. Blackadar. . Cambridge University Press, 1998. D. Bleecker. . Addison-Wesley Publishing Company, 1981. B. Ćaćić. Almost-commutative spectral triples and noncommutative-geometric field theory. Poster at Bonn International Graduate School in Mathematics, 2009. A. H. Chamseddine and A. Connes. The spectral action principle. 186 (1997) 731–750. A. Chamseddine and A. Connes. The spectral action principle. 186 (1997) 731–750. A. H. Chamseddine, A. Connes, and M. Marcolli. . 11 (2007) 991–1089. A. Connes. . Academic Press, London and San Diego, 1994. A. Connes. Gravity coupled with matter and the foundation of noncommutative geometry. 182 (1996). A. Connes and M. Marcolli. . AMS, Providence, 2008. P. Gilkey. . CRC Press, 1984. J. Gracia-Bondia, J. Várilly, and H. Figueroa. . Birkhauser Boston, 2001. G. Kasparov. The operator K-functor and extensions of $C^*$-algebras. 44 (1980) 571–636. G. Kasparov. Equivariant KK-theory and the Novikov conjecture. 91 (1988) 147–201. D. Kucerovsky. The KK-product of unbounded modules. 11 (1997) 17–34. B. Mesland. . PhD thesis, Universität Bonn, 2009. J. Nestruev. . Springer, 2003. M. Rieffel. Morita equivalence for ${C}^*$-algebras and ${W}^*$-algebras. 5 (1974) 51–96. R. Swan. Vector bundles and projective modules. 105 (1962) 264–277. [^1]: Here and in what follows we work in the category of smooth manifolds. [^2]: Here ${\overline}{B}^{op}$ is as a vector bundle conjugate to $B$ and has opposite multiplication in the fibers [^3]: Actually, they are even operator homotopic (*cf*. [@Kasparov] or [@Blackadar]).

--- abstract: 'We experimentally explore solutions to a model Hamiltonian dynamical system recently derived to study frequency cascades in the cubic defocusing nonlinear Schrödinger equation on the torus. Our results include a statistical analysis of the evolution of data with localized amplitudes and random phases, which supports the conjecture that energy cascades are a generic phenomenon. We also identify stationary solutions, periodic solutions in an associated problem and find experimental evidence of hyperbolic behavior. Many of our results rely upon reframing the dynamical system using a hydrodynamic formulation.' address: - 'Department of Mathematics, University of Toronto' - 'Department of Mathematics, University of North Carolina, Chapel Hill' - 'Department of Mathematics, Princeton University' - 'School of Mathematics, University of Minnesota' author: - 'James E. Colliander' - 'Jeremy L. Marzuola' - Tadahiro Oh - Gideon Simpson bibliography: - 'ToyModel.bib' title: Behavior of a Model Dynamical System with Applications to Weak Turbulence --- Introduction {#s:intro} ============ Recent investigations in [@CKSTT] reduced the study of the nonlinear Schrödinger equation (NLS), $$\begin{aligned} \label{e:dcnls} i u_t + \Delta u - |u|^2 u = 0, \ \ u(0,x) = u_0(x) \ \text{for} \ x \in \mathbb{T}^2,\end{aligned}$$ to the “Toy Model” dynamical system given by the equation $$\label{e:toy_model} -i\dt b_j (t) = -\abs{b_j(t)}^2 b_j(t) + 2 b_{j-1}^2 \overline{b_j}(t) + 2 b_{j+1}^2 \overline{b_j}(t)$$ for $j = 1,\ldots, N$, with boundary conditions $$\label{e:dirichletbc} b_0(t) = b_{N+1}(t) = 0.$$ The $b_j$’s approximate the energy of families of resonantly interacting frequencies to be described below. The main purpose of this paper is to study the evolution equation , both to gain additional insight into and for its own sake. In addition to showing how approximates , a key result of [@CKSTT] is the construction of a solution to which transfers mass from low index $j$ to high $j$. In the underlying NLS problem, this implies there exist arbitrarily large, but finite, energy cascades. Thus, [@CKSTT] showed that Hamiltonian dispersive equations posed on tori can have “weakly turbulent dynamics,” the phenomenon by which arbitrarily high index Sobolev norms can grow to be arbitrarily large in finite time. The question of energy cascades in infinite dimensional dynamical systems was considered by Bourgain [@B04], who asked if there was a solution to with an initial condition $u_0 \in H^s$, $s > 1$, such that $$\label{e1} \limsup_{t \to \infty} \|u(t)\|_{H^s} = \infty.$$ This corresponds to a weakly turbulent dynamic, as there is growth in high Sobolev norms, but no finite time singularity. Indeed, since is defocusing it has a bounded $H^1$ norm. One can view this behavior as an “infinite-time blowup.” Although the result in [@CKSTT] does not answer Bourgain’s question, it makes significant progress. The result says that given a threshold $K\gg1$ and $\delta >0$ there exists $u_0 \in H^s$ with $\| u_0\|_{H^s} \leq \delta$ and $T>0$ such that $\|u(T)\|_{H^s} \geq K$, where $u$ is the solution to the NLS with $u(0) = u_0$. This establishes $$\label{e2} \inf_{\delta > 0} \bigg\{ \limsup_{t \to \infty} \Big(\sup_{\|u_0\|_{H^s} \leq \delta }\|u(t)\|_{H^s}\Big) \bigg\} = \infty,$$ but not . This is one of the first rigorous result exhibiting the shift of energy from low to high frequencies for a nonlinear Hamiltonian PDE viewed as an infinite-dimensional Hamiltonian dynamical system, see also work by Kuksin [@Ku1]. The works Carles-Faou [@Carles:2012jv], Hani [@H11], and Guardia-Kaloshin [@GK12] have also recently treated . A particular achievement of these newer works is their careful construction of error estimates on the non-resonant terms. The dynamics in [@CKSTT] were not shown to be generic. Rather, the authors constructed a single solution with the desired properties. The stability of this solution to the flow is unknown. One purpose of this note is to explore this question of “genericity”, by investigating ensembles of data for , and finding that, on average, there is a transfer of energy from low to high indices. In addition to this statistical study, we seek out other interesting dynamics in . Notable behaviors we found include: - Compactly supported, time harmonic, structures; - Spatially and temporally periodic solutions subject to the adoption of periodic boundary conditions, $$\label{e:periodicbc} b_0(t) = b_N(t), \quad b_{N+1}(t) = b_1(t);$$ - Nonlinear hyperbolic behavior with both rarefactive waves and dispersive shock waves. Many of these solutions are obtained by going to the hydrodynamic formulation of the problem. Making the Madelung transformation, $$b_j(t) = \sqrt{\rho_j(t)}\exp(i\phi_j(t))$$ with $\rho_j \geq 0$ and $\phi_j \in \mathbb{R}$, we obtain evolution equations for $\rho_j$ and $\phi_j$: \[e:toy\_model\_hydro\] $$\begin{aligned} \dot\phi_j & = -\rho_j + 2 \rho_{j-1} \cos\bracket{2(\phi_{j-1}-\phi_j)} + 2 \rho_{j+1} \cos\bracket{2(\phi_{j+1}-\phi_j)} , \\ \dot\rho_j & = -4 \rho_j \rho_{j-1} \sin\bracket{2(\phi_{j-1}-\phi_j)} -4 \rho_j \rho_{j+1} \sin\bracket{2(\phi_{j+1}-\phi_j)}. \end{aligned}$$ From this perspective, it is clear that phase interactions play a key role in the dynamics. Properties of the Toy Model {#s:properties} =========================== In this section, we briefly review the connection between and , and review some important structural properties of . Relationship to NLS ------------------- First, we summarize the argument from [@CKSTT] which relates NLS to the Toy Model. This begins by studying NLS in Fourier space, $$u(t,x) = \sum_{n \in \mathbb{Z}^2} a_n (t) e^{i n \cdot x + |n|^2 t}.$$ After a choice of gauge eliminating certain trivial interactions, the Fourier amplitudes $\{a_n\}$ are seen to evolve according to $$\label{e:fnls} -i \partial_t a_n = -a_n |a_n|^2 + \sum_{ n_1, n_2, n_3 \in \Gamma (n)} a_{n_1} \bar{a}_{n_2} a_{n_3} e^{i \omega_4 t},$$ where $$\begin{gathered} \omega_4 = |n_1|^2 -|n_2|^2+|n_3|^2-|n|^2,\\ \Gamma (n) = \left\{ (n_1,n_2,n_3) \in (\mathbb{Z}^2)^3 | n_1 - n_2 + n_3 = n, \ n_1 \neq n, \ n_3 \neq n \right\}.\end{gathered}$$ For any $n$, the most significant contributions in the summation will be the elements of $\Gamma(n)$ belonging to the resonant set, $$\Gamma_{\rm res} (n) = \left\{ (n_1,n_2,n_3) \in \Gamma (n) \mid |n_1|^2 - |n_2|^2 + |n_3|^2 - |n|^2 = 0 \right\}.$$ Restricting to the resonant modes, we have $$\label{e:resfnls} -i \partial_t r_n = -r_n |r_n|^2 + \sum_{ n_1, n_2, n_3 \in \Gamma_{\rm res} (n)} r_{n_1} \bar{r}_{n_2} r_{n_3}.$$ A union of disjoint sets, $\Lambda_j$, of resonantly interacting frequencies is constructed, $$\boldsymbol{\Lambda} = \Lambda_1 \cup \Lambda_2 \cup \cdots \cup \Lambda_N,$$ where the mass from modes in generation $\Lambda_j$, $r_{n_1}$ and $r_{n_3}$, can mix to transfer mass to modes $r_{n}$ and $r_{n_2}$ in generation $\Lambda_{j+1}$, where again $n_1-n_2+n_3=n$. Subject to certain additional conditions, we will have that for all $t$ and $j$, $$r_n(t) = r_{n'}(t),\quad \forall \ n, n' \in \Lambda_j.$$ Once these sets have been constructed, a nontrivial step, the relationship between the toy model and is $$b_j(t) = r_n(t), \quad \forall \ n \in \Lambda_j.$$ Hence, $\abs{b_j(t)}^2$ is a measure of the spectral energy density of generation $\Lambda_j$. To show that NLS has an energy cascade, the authors used ideas inspired from studies of Arnold diffusion (see [@Arnold]) and explicit ODE manipulations to show that the Toy Model admits an instability mechanism transferring mass from a low index node to a high index node. By the construction of the initial data set $ \boldsymbol{\Lambda}$, such a mass transfer in the Toy Model yielded growth of high Sobolev norms of the solution to the resonant system , which in turn implied an energy cascade for NLS via a stationary phase argument. Structural Properties --------------------- We recall some of the results from Section 3 of [@CKSTT]. The toy model is a Hamiltonian dynamical system with Hamiltonian given by $$\label{e:hamiltonian} H[{\bf b} = (b_1, b_2,\ldots, b_N)] = \sum_{j=1}^N \left( \frac14 | b_j |^4 - \text{Re} ( \bar{b}_j^2 b_{j-1}^2) \right),$$ and symplectic structure, $$\label{e:symplectic} i \frac{db_j}{dt} = 2 \frac{\partial H[{\bf b}]}{\partial \bar b_j}, \quad j = 1,\ldots N.$$ This structure applies to both the original Dirichlet boundary conditions, , and the periodic boundary conditions, , studied below. The Toy Model, , admits many of the symmetries of , including phase invariance, scaling, time translation and time reversal. However, many of these symmetries are redundant, and the only known invariant, other than , is the mass quantity, $$\label{e:mass} M[{\bf b} ]= \sum_{j=1}^N |b_j|^2.$$ These invariants are useful in assessing the performance of our numerical schemes. A robust algorithm should preserve them within a controllable error. Since is a finite-dimensional Hamiltonian system, the behavior can studied statistically. By Liouville’s theorem, the Lebesgue measure $$\prod_{j = 1}^N d b_j = \prod_{j = 1}^N d\, \text{Re}\, b_j d\, \text{Im}\, b_j$$ on $\R^{2N}$ is invariant under the dynamics of (2). Moreover, in view of the mass conservation, the white noise $$\begin{aligned} d \mu_N & = Z_N^{-1} e^{-\frac{1}{2} \sum_{j = 1}^N |b_j|^2} \prod_{j = 1}^N d b_j \\ & = (2\pi)^{-N} \prod_{j = 1}^N e^{-\frac{1}{2} (\text{Re}\,b_j)^2 + (\text{Im}\,b_j)^2} d\, \text{Re}\, b_j d\, \text{Im}\, b_j \end{aligned}$$ is an invariant probability measure for . In particular, the Poincaré recurrence theorem (see, for example, p.106 of [@Z01]) ensures that almost every point $\bf b$ in the phase space is Poisson stable. That is to say, there exists $\{t_n\}_{n = 1}^\infty$ tending to $\infty$ (and another sequence tending to $-\infty$) such that $$\lim_{n\to \infty} {\bf b}(t_n) = {\bf b}$$ where ${\bf b}(t)$ is the solution to with ${\bf b}(0) = {\bf b}$. Here, “almost every” is with respect to both the white noise $\mu$ and the Lebesgue measure $\prod_{j = 1}^N d b_j$, since they are absolutely continuous with respect to one another. Of course, this is only an “almost every” statement, but it does says that the solution of the toy model constructed in [@CKSTT] is destined to return to the original configuration. Random Phase Interactions and Ensemble Dynamics =============================================== In this section, we present the results of running an ensemble of initial conditions. The statistics of the results indicate that there is generic movement of mass from low to high nodes. For our first ensemble, we took as the initial conditions, $$\label{e:ensemble_ic} b_j(\theta_j) = \begin{cases} \frac{\eps}{(N-1)} \exp \set{i \theta_j},& j \neq j_\star,\\ \sqrt{1 - \eps^2}\exp \set{i \theta_j}, & j = j_\star. \end{cases}$$ $\eps \in (0,1)$, $1< j_\star < N$ and $\theta_j$ are identical independently distributed random variables, $\theta_j \sim U(0,2\pi)$, the uniform distribution on $[0,2\pi)$. Thus, has mass one, with the majority of the mass concentrated at $j_\star$, and random phases on each node. To study the spreading of energy in this system, we introduce the Sobolev type norms, $h^s$, defined as $$\label{eqn:hsnorm} \norm{{\bf b}}_{h^s}^2 = \sum_{j=1}^N j^{2 s} \abs{b_j}^2.$$ We note that this norm will measure shift of mass to higher $j$ indices, but is difficult to connect directly to the corresponding $H^s$ norm of a solution to on the torus since that requires specifying the placement function from [@CKSTT]. Another candidate is $$\big(\sum_{j = 1}^N 2^{(s-1)j} |b_j|^2\big)^\frac{1}{2}, \label{e3}$$ which can be more closely connected to the construction in [@CKSTT]. However, since we only simulate a finite number of generations, if one grows, the other must too. Thus, we employ . We now proceed with our simulations for . Taking $\eps = .1$, $N=100$ and $j_\star = 10$, we simulated this initial condition until $t=10000$ with $10000$ realizations of the random phases. Generic slow growth in Sobolev norms appears in Figure \[fig:rand\]. These were computed using the explicit Runge-Kutta Prince-Dormand $(8,9)$ method, with a relative error tolerance of $10^{-12}$ and an absolute tolerance of $10^{-14}$, see [@GSL]. Over the entire ensemble, the maximum absolute and relative error in the invariants remained below $10^{-9}$. In Figure \[f:singlerealization\], we show the evolution of a particular realization to show how this growth in norms occurs. As the figure shows, there is a spreading of the mass away from the initial site of high mass. Additionally, there is local exchange between sites. Of course, as $N\to \infty$, the nodes in decouple. As an alternative, we consider $$\label{e:ensemble_weightedic} b_j(\theta_j) = \begin{cases} \frac{\eps}{j} \exp \set{i \theta_j},& j \neq j_\star,\\ \frac{\sqrt{1 - \eps^2}}{j}\exp \set{i \theta_j}, & j = j_\star. \end{cases}$$ The decay in $j$ ensures that as $N\to \infty$, the solution has finite mass. The results, plotted in Figure \[fig:randweighted\], are similar to the case. There is somewhat more growth in $h^1$, but less growth in the other $h^s$ norms. Particular Solutions ==================== In this section, we consider several particular solutions to , including localized solutions, periodic solutions and a “hyperbolic” solution. These were motivated by the hydrodynamic formulation of the problem, . Localized, Uniform Phase Solutions ---------------------------------- We first seek solutions which stay in phase for all time, $$\label{e:inphase} \phi_j(t) = \phi_{j+1}(t).$$ Such solutions are said to be [*phase locked*]{}. Assuming this holds, becomes $$\begin{aligned} \dot\phi_j & = -\rho_j + 2 \rho_{j-1} + 2 \rho_{j+1} , \\ \dot\rho_j & = 0. \end{aligned}$$ We now need a solution to $$-\rho_j + 2 \rho_{j-1} + 2 \rho_{j+1}=\omega \in \R, \quad\text{for $j=1,\ldots N$},$$ where $\omega$ is independent of $t$ and $\rho_0 = \rho_{N+1} = 0$. This corresponds to the linear system $$\label{e:upmat} \begin{pmatrix} -1 & 2 & 0 &\cdots & 0 \\ 2 & -1 & 2 & \cdots &0 \\ \vdots & \vdots & \ddots &\ddots & \vdots\\ 0 & 0 & \cdots & 2 & -1 \end{pmatrix} \boldsymbol{\rho} = \omega \boldsymbol{1}$$ Though the matrix is tri-diagonal, it is not diagonally dominant, so its solvability is not immediately clear. However, The matrix in has no kernel for any $N$. Letting $A_N$ denote this matrix, proving it has no kernel is equivalent to showing $\det A_N \neq 0$ for any $N$. Indeed, $$\label{e:matrecursion} \begin{split} \det A_{N} &= -1 \cdot \det A_{N-1} -2 \cdot \begin{vmatrix} 2 & 2 &0 & \cdots & 0\\ 0 & -1 &2 & \cdots & 0\\ 0 & 2 & -1 &\cdots & 0\\ \vdots & \ddots & \ddots & \ddots&\vdots\\ 0 & 0 & \cdots & 2 & -1 \end{vmatrix}\\ & = - \det A_{N-1} - 4 \det A_{N-2}, \end{split}$$ giving us a recursion relation for the determinant. By inspection, $$\det A_1= -1, \quad \det A_2 = -3.$$ By induction, recursion relation demands that for all $N$, the determinant of $A_N$ must be odd. Hence, it is never zero. A consequence of this is the following Corollary. Any nontrivial phase locked solution has $\omega \neq 0$. Moreover, given any real valued function $\omega(t) \neq 0$ for all $t \in \mathbb{R}$, there exists a nontrivial phase locked solution $\{ (\phi_j, \rho_j) \}_{j=1}^N$ to . For a given $N$, the linear system can always be solved, and a phase matched solution of exists. However, this will not always yield a solution of . As Figure \[f:compact\_solns1\] shows, at $N=5$, the solution is not strictly positive, and the Madelung transformation cannot be inverted. Despite the obstacle at $N=5$, we can again obtain a strictly positive solution at $N=8$ and higher, as Figure \[f:compact\_solns2\] shows. It is possible to ask whether or not, there exist positive solutions for specific, but arbitrarily large, values of $N$.[^1] Since the equations for the toy model, the hydrodynamic formulation and are autonomous in $j$, one can concatenate these localized solutions together to form new solutions. For example, one could place the $N=2$ solution at lattice sites 55 and 56, and the rest to zero, within with $N=100$. Thus, one can construct explicit solutions with isolated regions of support on arbitrarily large systems. Time Harmonic Periodic Solutions -------------------------------- Here, we consider the problem with periodic boundary conditions, . Assume for all $j$ the initial condition satisfies $$\begin{aligned} \rho_{j+1} (0) &= \rho_{j-1} (0),\\ \phi_{j+1} (0) & = \phi_{j-1} (0). \end{aligned}$$ Furthermore, assume one or both of the variables, $\boldsymbol{\rho}$ or $\boldsymbol{\phi}$, are not uniformly constant. This will result in time harmonic solutions. This can be observed by computing $$\begin{gathered} \begin{split} &\frac{d}{dt}\paren{\phi_{j+1} - \phi_{j-1}} \\ &\quad = - \paren{\rho_{j+1}-\rho_{j-1}} \\ &\quad\quad + 2 \rho_j \bracket{\cos(2(\phi_{j+1}-\phi_j))-\cos(2(\phi_{j}-\phi_{j-1}))}\\ &\quad \quad +2\rho_{j+2} \cos(2(\phi_{j+2}-\phi_{j+1}))\\ &\quad\quad -2\rho_{j-2} \cos(2(\phi_{j-1}-\phi_{j-2})) , \end{split}\\ \begin{split} \frac{d}{dt}\paren{\rho_{j+1} - \rho_{j-1}} &=4 \rho_j \rho_{j+1}\sin(2(\phi_{j+1}-\phi_j))\\ &\quad-4 \rho_{j+2} \rho_{j+1}\sin(2(\phi_{j+2}-\phi_{j+1}))\\ &\quad -4 \rho_{j-1} \rho_{j-2}\sin(2(\phi_{j-1}-\phi_{j-2}))\\ &\quad+ 4 \rho_{j} \rho_{j-1}\sin(2(\phi_{j}-\phi_{j-1})). \end{split} \end{gathered}$$ If, initially, $\phi_{j+1} = \phi_{j-1}$ and $\rho_{j+1} = \rho_{j-1}$ for all $j$, then they will remain so. We also have: - $\rho_j + \rho_{j-1}$ is constant in $j$ and $t$; - $\cos(2 (\phi_j-\phi_{j-1}))$ is constant in $j$, but may vary in $t$; - $\sin(2 (\phi_j-\phi_{j-1}))$ is constant in $j$, but may vary in $t$; - $\phi_j+\phi_{j-1}$ is constant in $j$, but may vary in $t$. These symmetries reduce the problem to four variables, $\rho_j$, $\rho_{j-1}$, $\phi_j$, and $\phi_{j-1}$: $$\begin{aligned} \dot{\phi}_j&=-\rho_j +4\rho_{j-1} \cos(2(\phi_j - \phi_{j-1})) , \\ \dot{\phi}_{j-1}&=-\rho_{j-1} +4\rho_{j} \cos(2(\phi_j - \phi_{j-1})) , \\ \dot{\rho}_j&=8 \rho_j \rho_{j-1} \sin(2(\phi_j - \phi_{j-1})) , \\ \dot{\rho}_{j-1}&=-8 \rho_j \rho_{j-1} \sin(2(\phi_j - \phi_{j-1})) . \end{aligned}$$ Defining $$\begin{aligned} \bar{\phi} &\equiv \phi_j + \phi_{j-1},\\ \Delta\phi&\equiv\phi_j - \phi_{j-1},\\ \bar{\rho}&\equiv\rho_j + \rho_{j-1},\\ \Delta{\rho}&\equiv\rho_j - \rho_{j-1}, \end{aligned}$$ we have: $$\begin{aligned} \frac{d}{dt}{\bar{\phi} }& =-{\bar\rho} + 4 \bar{\rho}\cos(2\Delta\phi), \\ \frac{d}{dt}{\Delta\phi}&=-{\Delta\rho} - 4 \Delta{\rho}\cos(2\Delta\phi),\\ \frac{d}{dt}{\bar{\rho} }& =0, \\ \frac{d}{dt}{\Delta\rho}&=16 \rho_j \rho_{j-1} \sin(2 \Delta\phi). \end{aligned}$$ Since $\bar\rho^2 - \Delta\rho^2=4 \rho_j \rho_{j-1}$ and $\bar \rho$ is invariant, we have a closed system of $2$ equations for $\Delta \phi$ and $\Delta \rho$. $$\begin{aligned} \frac{d}{dt}{\Delta\phi}&=-{\Delta\rho}\bracket{ 1+ 4 \cos(2\Delta\phi)}, \\ \frac{d}{dt}{\Delta\rho}&=4 \paren{\bar\rho^2 - \Delta\rho^2} \sin(2 \Delta\phi). \end{aligned}$$ The system is Hamiltonian with $$H = \frac{1}{2}\paren{1 + 4 \cos(2 \Delta \phi)}\paren{\bar\rho^2 - \Delta\rho^2}$$ and symplectic structure $$\frac{d}{dt} \Delta \phi= \frac{\partial H }{\partial \Delta \rho}, \quad \frac{d}{dt} \Delta \rho = -\frac{\partial H }{\partial \Delta \phi}.$$ Thus, we anticipate time harmonic motion. An example appears in Figure \[f:period2harmonic\], where the initial condition is \[e:per2ic\] $$\begin{gathered} \rho_1 = \rho_2 = 1, \\ \phi_1 = \frac{\pi}{4}, \quad \phi_2 = 0. \end{gathered}$$ Discrete Rarefaction Waves and Weak Turbulence ============================================== In this section, we explore the dynamics when the initial configuration is given by the out of phase initial condition $\phi_j = \phi_{j+1} - \frac{\pi}{4}$. If this phase relation were to somehow persist, the resulting equations hints at the discrete Burger’s formulation of the hydrodynamic equations $$\begin{aligned} \label{e:burgers} \left\{ \begin{array}{l} \dot\phi_j = 0, \\ \dot\rho_j = -4 \rho_j \rho_{j-1} + 4 \rho_j \rho_{j+1} = -8 \rho_j \left( \frac{ \rho_{j+1} - \rho_{j-1} }{ 2} \right). \end{array} \right. \end{aligned}$$ This has discrete rarefaction and shock wave dynamics. We call the discrete Burger’s equation since, were we to discretize $$\rho_t = -8 \rho \nabla \rho,$$ in space and take the gridpoint spacing paramter equal to one, we would recover the above equation. We note here that the rarefaction waves we observe have similar dynamics to those found in Fermi-Pasta-Ulam chains, see [@HerrmannRademacher]. However in , there are additional terms which prevent this phase relation from persisting. We show here how the solutions evolve, with the initial condition $$\label{e:shock_ic} b_j = \exp i \set{ (j-1)\pi/4}.$$ As a first example, we solve with the initial condition over $N=100$ lattice sites. The results appear in Figures \[f:shock1\] and \[f:shock1\_norms\]. As can be seen, the $h^s$ norms eventually cease to be monotonic. To see persistent growth in the norms, we can look at a system with $N=5000$ sites and for longer a time; see Figures \[f:shock2\] and \[f:shock2\_norms\]. ![Growth in the $h^s$ norms for the dynamics of Figure \[f:shock1\].[]{data-label="f:shock1_norms"}](shock_N100_tmax10_norms){width="6cm"} ![Growth in the $h^s$ norms for the dynamics of Figure \[f:shock2\].[]{data-label="f:shock2_norms"}](shock_N5000_tmax500_norms){width="6cm"} The simulation on $N=5000$ reveals that the rarefaction portion of the solution has more structure than is apparent in the case of $N=100$. As shown in Figure \[f:shock2\_zoom\], the rarefaction wave evolves with several different slopes. Unfortunately, as $N\to \infty$, will not correspond to a finite mass solution. Thus, we studied the weighted initial condition, $$\label{e:weightedshock_ic} b_j = \exp i \set{ (j-1)\pi/4}/j.$$ This, too, results in energy transfer, though it is not monotonic. Several frames from this simulation appear in Figure \[f:weightedshock\], and the growth of the norms can be seen in Figure \[f:weightedshock\_norms\]. The norm growth is quite pronounced; this may be due to an inability of mass to propagate backwards, against the weight $1/j$. In similar calculations with $$\label{e:generalizedweightedshock_ic} b_j = \exp i \set{ (j-1)\pi/4}/{(w(j))^\sigma}.$$ for $0<\sigma<1$, where $w(j) \to \infty$ as $j \to \infty$, we observe that some form of the rarefaction front propagates forward even with a tail in the higher nodes and that the structure of the rarefaction wave persists longer as $\sigma \to 0$. Since the rarefaction wave and the backward dispersive shock travel at finite speeds in the simulation, we observe motion to large $j$ on much longer time scales for large $N$. Here, the weights in allow us to study rarefaction waves in a setting with $h^1$ norms of order $1$ instead of order $N$. In addition, we observe that the rarefaction wave solution is robust even for initial data of the form , which has less back scattering thanks to the smaller jump at the right endpoint. As the rarefaction front enters the decaying tail, it does however begin to lose some mass at regular intervals, but continues to propagate weakly. All rarefaction wave solutions presented in this section result in norm growth when mapped back to solutions of . This is due to Proposition $2.1$ of [@CKSTT], which states that a mass shift to the higher nodes in the Toy Model results in growth of higher Sobolev norms of the corresponding solution to . This is fundamental to showing the importance of tracking the rarefaction wave front moving toward large $j$ in . However, a more detailed result relating to the frequency scales at each generation $\Lambda_j$ and a better categorization of families of resonant frequencies would be required to address this issue in its entirety and observe a rarefaction front in the resonant frequencies of the torus. It is unclear at this point how to directly translate solutions with frequency cascades in to computationally observable solutions with frequency cascades leading to $H^s$ norm growth for $s>1$ in . Here, however, we have demonstrated the robustness of solutions that move mass in from low to high $j$. ![Growth in the $h^s$ norms for the dynamics of Figure \[f:weightedshock\].[]{data-label="f:weightedshock_norms"}](weightedshock_N100_tmax10000_norms){width="5cm"} Continuum Limit & Compacton Type Solutions ========================================== As a final observation, let us introduce the parameter $0<h\ll 1$, such that $$B(x_j,t) = b_j(t),\quad x_j = h j.$$ Taylor expanding, $$- i\dt B = \bracket{3 B^2 + 4h^2 \paren{ (\dx B)^2 + B \partial_x^2 B} + \bigo(h^4)}\overline{B}.$$ Neglecting $\bigo(h^4)$ terms, $$\label{e:continuum} - i\dt B = 3 \abs{B}^2B + 4h^2 \overline{B}\dx\paren{ B \dx B}.$$ This retains the toy model scaling that if $B(x,t)$ is a solution, then so is $\lambda B(x, \lambda^2 t)$. It is also invariant to multiplication by an arbitrary phase. The equation is, formally, degenerately dispersive, and it can be viewed as an NLS analog of the compacton equations, [@Rosenau:1993uy; @Rosenau:1994ua; @Rosenau:2005tx; @Rosenau:2010wx]. One of the interesting features of the compacton equations is that they admit compactly supported nonlinear bound states, which we now seek for . We begin with the ansatz $B = e^{it} Q(x)$, $Q > 0$. Consequently, $Q$ solves $$Q = 3 Q^3 + 4 h^2 Q (Q Q')'$$ which can be expressed as $$\label{e:compacton} Q = 3 Q^3 + 2 h^2 Q (Q^2)''$$ Letting $U = Q^2$, $$2 h^2 U'' + 3 U -1 = 0$$ This can be integrated up once to $$h^2 U'^2 + {\frac{3}{2}U^2-U} = C$$ $h$ can easily be scaled out by changing the dependent variable, thus we set $h=1$. We always have a potential well in this equation, so there should be homoclinic orbits. If $C=0$, then $ 0 < U < 2/3$. The explicit solution is $$Q(x) = \sqrt{\frac{2}{3}}\sin\left(\frac{1}{2}\sqrt{\frac{3}{2}} x \right).$$ Putting $h$ in, $$Q_h(x) = \sqrt{\frac{2}{3}}\sin\left(h^{-1}\frac{1}{2}\sqrt{\frac{3}{2}} x \right).$$ Phase shifting the solution by $\pi/2$, we can alternatively have $$Q_h(x) = \sqrt{\frac{2}{3}}\cos\left(h^{-1}\frac{1}{2}\sqrt{\frac{3}{2}} x \right).$$ We can turn this into a compact structure if we now define $$\label{e:compacton_cos} Q_h^c(x) =\begin{cases}\sqrt{\frac{2}{3}}\cos\left(h^{-1}\frac{1}{2}\sqrt{\frac{3}{2}} x \right) & \abs{x} < h \pi \sqrt{\frac{2}{3}}\\ 0 & \abs{x} \geq h \pi \sqrt{\frac{2}{3}}. \end{cases}$$ This structure, with $h =1$, appears in Figure \[f:compacton\]. ![The compacton solution given by .[]{data-label="f:compacton"}](compacton1){width="8cm"} Note that this will satisfy the equation in the strong sense. If we go back to the sine formulation and look at $x=0$ with $h=1$, then $$Q^c(x) =\begin{cases}\sqrt{\frac{2}{3}}\sin\left(\frac{1}{2}\sqrt{\frac{3}{2}} x \right) & 0< x < \pi \sqrt{\frac{2}{3}} \\ 0 & \textrm{otherwise}. \end{cases}$$ Then, near $x=0$, $$\begin{aligned} Q^c(x) &\sim x H(x)\\ (Q^c(x))^2 &\sim x^2 H(x)\\ \frac{d^2}{dx^2} (Q^c(x))^2 & \sim H(x)\\ Q^c(x) \frac{d^2}{dx^2} (Q^c(x))^2 & \sim x H(x), \end{aligned}$$ where $H(x)$ is the Heaviside function. Hence, the most degenerate term in is continuous. [^1]: Indeed, using recursive linear algebra techniques, this question has been answered affirmatively thanks to an observation of Stefan Steinerberger about the recursive nature of the sequence of $N$ for which positive solutions exist and a clever argument from Sergei Ivanov through the Math Overflow Project at <http://mathoverflow.net/questions/106816>.

--- abstract: 'A new era of directly imaged extrasolar planets has produced a three-planet system [@2008M], where the masses of the planets have been estimated by untested cooling models. We point out that the nominal circular, face-on orbits of the planets lead to a dynamical instability in $\sim$$10^5$ yr, a factor of at least $100$ shorter than the estimated age of the star. Reduced planetary masses produce stability only for unreasonably small planets ($\lesssim 2$ $M_{\rm Jup}$). Relaxing the face-on assumption, but still requiring circular orbits while fitting the observed positions, makes the instability time even shorter. A promising solution is that the inner two planets have a 2:1 commensurability between their periods, and they avoid close encounters with each other through this resonance. That the inner resonance has lasted until now, in spite of the perturbations of the outer planet, leads to a limit $\lesssim 10 M_{\rm Jup}$ on the masses unless the outer two planets are *also* engaged in a 2:1 mean-motion resonance. In a double resonance, which is consistent with the current data, the system could survive until now even if the planets have masses of $\sim20$ $M_{\rm Jup}$. Apsidal alignment can further enhance the stability of a mean-motion resonant system. A completely different dynamical configuration, with large eccentricities and large mutual inclinations among the planets, is possible but finely tuned.' author: - 'Daniel C. Fabrycky and Ruth A. Murray-Clay' bibliography: - 'ms.bib' title: ' Stability of the directly imaged multiplanet system HR 8799: resonance and masses ' --- Introduction {#sec:intro} ============ The method of direct imaging for the discovery of extrasolar planets has yielded spectacular first results over the last several years [@2004C; @2008L; @2008M; @2008K; @2009Lagrange]. Direct imaging is a method for discovering planets located far from their host stars, an as-yet unexplored region of parameter space, and it promises new opportunities to characterize the planets using their own radiation. However, because the gravitational influence of directly-imaged planets is not measured and the astrometric orbital arcs obtained so far are short, determining the planetary masses and orbital architectures of these systems is challenging. In the newly-discovered planetary system HR 8799 ($=$ HD 218396), three planets have been imaged at projected separations of 24, 38, and 68 AU from their host star [@2008M]. The best current estimate of their masses is derived from the planetary luminosities, measured in the infrared. Because these planets are young and massive, they are still radiating prodigiously as they contract, cool, and become more gravitationally bound. The masses are estimated using untested models of this contraction and cooling process. One class of such models, the “hot-start” models, provides the largest luminosity possible at a certain mass and age, given assumptions about opacities in the planetary atmosphere. Hot-start models have initially extended envelopes and a large entropy per baryon; even hotter models converge to a common track after a few Myr [@2002B]. Therefore, for a given age and luminosity, these models should provide a lower limit on the mass. For HR 8799, the lower-limit masses are $5$-$11$, $7$-$13$, and $7$-$13$ $M_{\rm Jup}$ for planets b, c, and d, respectively, based on a rather uncertain stellar age of $30$-$160$ Myr[^1], which is presumably also roughly the age of the planets. The following simple calculation illustrates why a lower mass limit can be inferred from a planet’s contraction luminosity. For HR 8799, the planetary luminosities have been measured to be $L \simeq 10^{-5} L_\odot$, and radii of $R \simeq 1.2 R_{\rm Jup}$ were derived from the objects’ temperatures, measured by fitting photometry with a variety of synthetic spectral energy distributions [@2008M]. Because the objects are cooling, they were more luminous in the past, so they have radiated at least $L t_{\rm age} \gtrsim 4 \times 10^{43}$ erg. Their current binding energy, which supplied this luminosity, is $\simeq G M^2 R^{-1} \simeq 3 \times 10^{43}(M/M_{\rm Jup})^2$ erg, where the radius is roughly independent of the mass for Jupiter-mass objects. Consequently, $M > 1 M_{\rm Jup}$. Cooling models also take into account that $L$ diminishes with time, and thus arrive at a considerably larger mass. Whether this larger calculated mass is a robust lower limit depends on the accuracy of the model. Recently, [@2009D] measured the dynamical masses for a system of brown dwarfs (both of mass $\approx 57 M_{\rm Jup}$) and showed that cooling models overpredict the component masses by $\sim$25%. If energy is lost during the process of planet formation, then an even larger planet mass would be needed to generate the currently observed luminosity. For example, in the planetary core-accretion models of [@2007M], considerable luminosity is radiated in the accretion stream and shock, and that energy is not internalized by the planet. At the end of formation the planet has less gravitational potential energy to later supply its luminosity. The integrated luminosity since formation would not account for the planet’s current binding energy, so the mass needed to supply an observed luminosity at a given age may be much bigger. The HR 8799 system has survived an order of magnitude longer than the primordial gas disk, which, if typical of disks of A stars, lasted $\lesssim3$ Myr [@1993H; @2005H]. The system has therefore had time to dynamically evolve in the absence of gas. Though the planets orbiting HR 8799 are separated by tens of AU, the inferred minimum masses of the planets were large enough that their mutual gravitational interactions are important. For example, a planet with mass $M_p = 10$ $M_{\rm Jup}$ orbiting a star of mass $M_* = 1.5$ $M_\odot$ at semi-major axis $a = 40$ AU dominates gravitational dynamics within its Hill radius of size $R_H = a(M_p/3M_*)^{1/3} = 5$ AU. Because $R_H$ is a large fraction of the planetary separation, gravitational interactions among the planets can substantially modify the dynamical evolution of the system. In fact, the nominal orbits reported in the discovery paper [@2008M] are unstable. We integrated the Newtonian equations of motion of the proposed system using the Bulirsh-Stoer (BS) algorithm of the [*Mercury*]{} [@1999C] package (version 6.2), with an accuracy parameter of $10^{-12}$. The planets are assigned circular, face-on orbits, and we used the nominal masses for all four bodies: $7$, $10$, $10$ $M_{\rm Jup}$ for planets b, c, and d, respectively, and $1.5$ $M_\odot$ for the star[^2]. Figure \[fig:ae\] shows the results for the semi-major axis and maximum radial excursion of each planet as a function of time. A close encounter between planets c and d at $0.298$ Myr (i.e., they enter within one Hill radius of one another) leads to a brief interval of strong scattering which ejects planet b at $0.316$ Myr (i.e., it reaches $>500$ AU with positive energy, and is removed from the simulation). Planets c and d swap orbits and finish in a stable configuration, with no further semi-major axis evolution, but they exhibit a regular secular eccentricity cycle with a period of $1.5$ Myr. This evolution is not unique in its details since the orbital evolution is chaotic. However, qualitatively similar evolutions are common for simulated planetary systems constructed to match the discovery data: instability usually sets in well before the star’s age of $\gtrsim 30$ Myr. The goal of this paper is to determine orbits that are consistent with the astrometric data, the inferred planetary masses, *and* with dynamical stability over the system’s age. Neglecting stability considerations, there is a large amount of freedom in fitting orbits to the discovery data, because (1) the measured astrometric arcs cover only $\sim$2% of the middle orbit and $\sim$1% of the outer orbit, (2) the velocity of the inner planet is almost entirely unconstrained, and (3) the line-of-sight positions and velocities of the planets relative to the star are unknown. [*A priori*]{}, two classes of orbital architectures are possible—those in which the planets occupy roughly coplanar orbits and those with large mutual inclinations. Since planets form in disks, it is likely that they initially occupy nearly coplanar orbits, and systems that remain stable indefinitely are likely to stay roughly coplanar. Alternatively, the system may not be indefinitely stable. While old compared to the lifetime of the protoplanetary disk, the current age of the planetary system is probably less than one tenth the main-sequence lifetime of the star ($\sim$$1.5$ Gyr; @1967I). Without further analysis, it is thus possible that the planets are in the process of scattering off of one another, currently have large eccentricities and mutual inclinations, and will not be stable over the lifetime of the star. In fact, current models predict that planetary systems undergo periods of strong mutual excitation, perhaps generically leading to the ejection of planets (e.g., @lld98, @gls04, @2009SM, @2009VCF). [lcccc]{} b &\[$60.16(6), 31.50(6)$\] & $67.91(6)$& \[$1.49(0.14), -2.18(0.14)$\] & $ 2.64(0.14)$\ c &\[$-25.90(6), 27.76(6)$\] &$37.97(6)$& \[$2.15(0.14), 2.47(0.14)$\]& $ 3.27(0.14)$\ d &\[$-8.45(6), -22.93(6)$\] &$ 24.44(6)$& \[$-3.5(2.7), 0.0(2.7)$\] & — To explore these possibilities systematically, we take the following approach. We start with restrictive assumptions about the orbital architecture of the system, and we then progressively relax those assumptions. At each stage, we find parameters that maximize the stability, and we finally argue that a resonant configuration is most likely for the system to have survived to its current age. In §§\[sec:astrom\]–\[sec:planarei\], we assume that the orbits of the planets are close to coplanar. In §\[sec:astrom\] we discuss astrometric constraints on the orbits and show that, although the data are consistent with circular and coplanar orbits, the system orientations that fit the data best do not generate stable orbits. In §\[sec:masses\] we determine what planetary masses would be needed for circular, coplanar orbits to be stable and argue that they are too low given the observed luminosities. Having thus ruled out circular, coplanar solutions, we next allow the planetary eccentricities to vary. Since the inner planet’s eccentricity is unconstrained by the data, we first scan over non-circular orbits for the inner planet while keeping the outer two planets on circular orbits (§\[sec:stable\]). The suggestive results of this experiment led us to our preferred configuration for the planetary system: a mean motion resonance between the inner two planets (§\[sec:mmr\]). An initial exploration of the parameter space of possible resonant orbits shows that if the outer two planets are also in a mean motion resonance, the system could be stable even if the companion masses are twice as large as the nominal masses. In §\[sec:montecarlo\], we allow all the orbital parameters to vary via a Monte-Carlo method. We confirm that mean-motion resonance is the most likely reason the planetary system has survived. We also find a scattering-type configuration that is stable for $30$ Myr, but we argue that it is unlikely. In §\[sec:conclude\] we discuss our conclusions. Astrometric Constraints {#sec:astrom} ========================= In Figure \[fig:data\], we plot the sky-projected position and velocity vectors (Table \[tab:nom\]) of the three planets, at the epoch 2008 Aug. 12, as determined by least-squares fit to the astrometry in Table 1 of [@2008M]. The distance to the star is $39.4 \pm 1.1$ pc, based on the [*Hipparchos*]{} parallax [@2007VL]. We use this nominal distance to convert observed angular separations to AU. The 3% error thus introduced into the distances and velocities does not change our qualitative conclusions; in §\[sec:montecarlo\] we take this error into account. [cccccccccc]{} A & $\equiv 1.5$ & $\equiv 0^\circ$ & $\equiv 0^\circ$ & $12.63$ & $6$ & $0.049$ & $67.91$, $ 62.36^\circ$ & $37.97$, $ 316.99^\circ$ & $24.44$, $ 200.23^\circ$\ B & $1.44 $ & $\equiv 0^\circ$ & $\equiv 0^\circ$ & $12.19$ & $5$ & $0.032$ & $67.91$, $ 62.36^\circ$ & $37.97$, $ 316.99^\circ$ & $24.44$, $ 200.23^\circ$\ C & $\equiv 1.5$ & $21.3^\circ$ & $151.5^\circ$ & $9.07$ & $4$ & $0.059$ & $ 72.89$, $ 62.30^\circ$ & $38.15$, $ 315.97^\circ$ & $ 25.47$, $ 202.23^\circ$\ D & $1.86$ & $33.2^\circ$ & $145.9^\circ$ & $5.67$ & $4$ & $0.225$ & $ 81.04$, $ 61.30^\circ$ & $38.16$, $ 315.29^\circ$ & $ 27.69$, $ 204.92^\circ$\ E & $2.28$ & $41.4^\circ$ & $143.6^\circ$ & $2.77$ & $3$ & $0.429$ & $ 90.02$, $ 60.20^\circ$ & $38.16$, $ 314.81^\circ$ & $ 30.33$, $ 207.30^\circ$ The impression given by Figure \[fig:data\] is that we are seeing the planetary system face-on, with counter-clockwise, nearly circular orbits. This is what we call the “nominal model,” and we plot the implied orbits and velocity vectors for a $1.5$ $M_{\odot}$ star, also in Figure \[fig:data\]. If all of the orbits are truly face-on and circular, their sky-projected separation $s\equiv\sqrt{x_{\rm E}^2+x_{\rm N}^2}=a$, and their sky-projected velocity $v_p\equiv \sqrt{v_{\rm E}^2 + v_{\rm N}^2} = v_{\rm orb}$, the orbital velocity. Since all of the planets are bound mostly by the mass of the star, they should follow circular orbits at semi-major axis $a$ with velocities $v_{\rm orb} = 2 \pi$ AU yr$^{-1}$ $(M_\star/M_\odot)^{1/2} (a/\rm{AU})^{-1/2}$. For the outer two planets, $s$ and $v_p$ are measured with high precision (Table \[tab:nom\]), providing two independent measurements of the stellar mass. Given this nominal model, the stellar mass binding planet b is $M_{\star b}=1.60 \pm 0.17 M_\odot$ and the stellar mass binding planet c is $M_{\star c}=1.38 \pm 0.12 M_\odot$. These values bracket the value of $M_{\star}=1.47 \pm 0.30 M_\odot$ preferred by combining parallax, magnitude, and spectroscopic information [@1999GK] and are in reasonable agreement: $\Delta M_{\star} \equiv M_{\star b}-M_{\star c} = 0.22 \pm 0.21 M_{\odot}$. However, there is some tension in the observed velocities. For both planets b and c, the observed velocity vector is $\sim$$2 \sigma$ away from perpendicular to the separation vector (from the star to the planet). The instability reported in the introduction is, however, the main failing of the nominal model. To address this failing, we first search for another model in which the planets are still coplanar and circular, but the system plane is inclined by an angle $i$ to the plane of the sky, with an ascending node $\Omega$ measured East of North, and a to-be-determined consistent mass $M_\star$. The sky projection changes the magnitudes and directions of the velocity vectors and the inferred spacings of the planets, and taking it into account could lead us to infer a wider-spaced, more stable system. We focus only on circular and coplanar models in this section, saving more complicated direct fits to the data for §\[sec:montecarlo\]. The velocity field on the sky due to this model is: $$\left( \begin{array}{c} v_E \\ v_N \end{array} \right) = n(x_E, x_N) \left( \begin{array}{c} - \alpha \sin \Omega \cos i -\beta \cos \Omega (\cos i)^{-1} \\ \alpha \cos \Omega \cos i -\beta \sin \Omega (\cos i)^{-1} \end{array} \right),$$ where $$\left( \begin{array}{c} \alpha \\ \beta \end{array} \right)= \left( \begin{array}{cc} \cos \Omega & \sin \Omega \\ -\sin \Omega & \cos \Omega \end{array} \right) \left( \begin{array}{c} x_E \\ x_N \end{array} \right),$$ and $$n(x_E, x_N) = (G M_\star)^{1/2} (\alpha^2 + \beta^2 (\cos i)^{-2})^{-3/4}$$ is the mean motion as a function[^3] of position. We solve for the three parameters $i$, $\Omega$, and $M_\star$, assuming that the planets are on non-interacting Keplerian orbits, each of which only feels the mass of the central star. We calculate $\chi^2$ values using $v_E$ and $v_N$, and their associated measurement errors, for all three planets (Table \[tab:nom\]; 6 data points)—we neglect the errors on $x_E$ and $x_N$, which are too small to affect our results. Solutions are reported in Table \[tab:orient\]. In model A, which is the nominal model, we fix the parameters to their nominal values to serve as a baseline. In model B, we require face-on orbits, but let $M_\star$ float, the result being not far from the nominal stellar mass. In model C, we fix $M_\star=1.5 M_\odot$, but let the orientation float. The orbits depart from face-on by $\sim$$20^\circ$, and $\chi^2$ improves a little. In model D, we let all three parameters float, but respect the independently measured stellar mass by including $[(M_\star/M_{\odot}-1.5)/0.3]^2$ in $\chi^2$. In model E, all three parameters float with no such mass constraint. The orientation-dependence of $\chi^2$ is shown in Figure \[fig:xisqcontour\], and the mass-dependence is shown in Figure \[fig:mass\]. Interestingly, the best fits are for $M_\star$ much larger than the nominal value $1.5 \pm 0.3 M_{\odot}$. This is not surprising given the good agreement of circular orbits because we are introducing line-of-sight offsets and velocities, so a more massive star is needed to make such orbits circular. Figure \[fig:185\] shows how the velocity vectors of model D falls into the $1$-$\sigma$ error ellipse for each planet. However, the inner two orbits are closer spaced than the nominal model, and the instability is even more rapid: in an integration a close encounter occurred between c and d on their second conjunction (see Table \[tab:logI\] for initial conditions). This integration and all those in §§\[sec:astrom\]–\[sec:stable\] were performed using the HYBRID integrator of [*Mercury*]{} with a timestep of 100 days. Each integration was terminated when any two planets passed within 1 Hill radius of each other, one was ejected (distance to the star $>500$ AU with positive energy), or the system lasted $160$ Myr. Before the onset of close encounters, energy was conserved to 1 part in $\sim$$10^5$ and angular momentum was conserved to 1 part in $\sim$$10^{12}$. Though we used the HYBRID integrator, because the integrations were halted at the first close encounter, the integrator’s treatment of close encounters did not affect our results. In §\[sec:ejection\], we verify that after a close encounter, at least one planet would be quickly ejected. Similarly, we fit the best orientation for $M_\star$ values between $1.1$-$3.0 M_\odot$, spaced by $0.01 M_\odot$, and integrated those orbits. No three-planet systems generated in this way were stable for more than $1.5\times10^5$ yr. Therefore we find that more careful fits to the data, under the hypothesis of circular coplanar orbits, do not simply lead to a stable solution. Much Lower Planetary Masses? {#sec:masses} ============================== Before relaxing the assumption that the planets’ orbits are circular and coplanar, we ask how low the planets’ masses must be for the nominal orbits to be stable. Intuitively, if their masses are very small, the planets will not significantly perturb each other on the timescale of $30-160$ Myr. There is a well-developed framework for quantifying long-term stability in systems with only two planets. In three-body systems, conservation of total energy and angular momentum constrains the possible motions [@1982MB]. Applied to a system of a star and two planets, we may define Hill stability as a constraint that the planet that is initially closer to the star stays closer to the star for all time. When the criterion for Hill stability is satisfied, a close encounter between the planets is prohibited (although escape of the outer planet to infinity, or the collision of the inner planet with the star, is not forbidden). Qualitatively, stability requires that the planets be separated by more than a few mutual Hill radii: $R_{\rm H} \equiv \onehalf ({a_{\rm in}}+{a_{\rm out}}) \epsilon$, with $\epsilon \equiv [({M_{\rm in}}+ {M_{\rm out}}) / (3 M_\star)]^{1/3}$. Define $\Delta$ as the planets’ difference in semi-major axes in terms of $R_{\rm H}$. [@1993G] gave the Hill stability criterion as: $$\Delta > \Delta_{crit} \equiv 2 \sqrt{3} [ 1 + 3^{1/2} \epsilon - \left( \frac{11 {M_{\rm in}}+7 {M_{\rm out}}}{18 M_\star} \right) 3^{-2/3} \epsilon^{-2} + ...] . \label{eq:hillbound}$$ Evaluating these numbers using the nominal system with nominal masses $7$, $10$, $10$ $M_{\rm Jup}$, we have $\Delta_{cd}=2.68$ and $\Delta_{{\rm crit}, cd}=4.03$ for the inner two, and $\Delta_{bc}=3.69$ and $\Delta_{{\rm crit}, bc}=3.98$ for the outer two. Apparently *both* sub-systems fail to satisfy the Hill stability criterion. We performed numerical simulations to find just how small the planets would need to be to remain stable. We are helped by the long orbital periods and short system age (only $\sim$$10^6$ dynamical times), which allows suites of integrations to be rather inexpensive. First we surveyed the instability near the nominal orbits (“A”), as the search of §\[sec:astrom\] did not reveal any more stable starting points. Let us focus on the inner sub-system (c-d), as it is further from stability, and ask the question: by what factor must we multiply the nominal masses for stability over $30$ Myr? In Figure \[fig:massmult\] we plot the time to instability — when the first Hill-sphere entry occurs — versus this common mass scaling. Vertical lines represent two-planet systems consisting of planets c and d on circular, face-on orbits. We note that below $M_p = 0.33 M_{\rm nominal}$, where Hill’s stability criterion is satisfied, all of the two-planet systems last for $160$ Myr, when the integrations were stopped. @1993G found that if the planets initially have small eccentricity (radial excursions comparable or less than a Hill radius) and are not in resonance, then the timescale for instability drops rapidly after this boundary (eq. \[\[eq:hillbound\]\]) is crossed. However, instability does not appear on timescales relevant for the c-d subsystem until $M_p \simeq 0.5 M_{\rm nominal}$. In a separate suite of integrations (not plotted), we found that the nominal orbits and masses of the outer pair of planets can be stable for 160 Myr. We also plot the instability timescale of the three-planet system (dark gray region), with each of the three planetary masses scaled by a common factor. The masses must be lower than about $1/5$ of the nominal masses to remain stable $30$ Myr, the lower limit on the stellar age (depicted by the light gray stripe). When considering three or more planets, there are no sharp stability boundaries, but there are well-established empirical scaling relations between semi-major axis separation and instability timescale [@1996C; @2007Z]. Applying the scaling relation of @2008Chatterjee [appendix A, fit 1] to the HR 8799 system implies $\Delta \gtrsim 4.4$ if the system is to remain stable $\gtrsim 30$ Myr. Let us assume the instability between the inner two planets is dominant, so this limit applies to $\Delta_{cd}$; then the masses must be $\lesssim 1/4$ of the nominal masses, in good agreement with the non-resonant systems of Figure \[fig:massmult\]. We note, however, that none of the published scaling relations extend to planetary-to-stellar mass ratios of $5\times10^{-3}$, nor do they strictly apply if adjacent planets are unequally spaced in Hill radii ($\Delta_{cd} \neq \Delta_{bc}$), both of which are relevant for HR 8799. For masses $>1/4$ of the nominal masses, our results give a longer lifetime than the scaling of @2008Chatterjee, sometimes orders of magnitude longer. Regardless, we confirm that instability can occur even if the sub-systems are initially Hill stable. In circular, face-on orbits, the implied upper limits of masses — $1.5$, $2$, and $2$ $M_{\rm Jup}$ — are incompatible with any cooling model at ages greater than $30$ Myr, even extreme hot-start models. Non-circular inner orbit? {#sec:stable} =========================== In the previous section, we found that face-on, circular orbits, consistent with the astrometric constraints, could only be stable if the planetary masses were implausibly low. In this section, we choose the lowest planetary masses that are compatible with hot-start models, and we choose a non-circular orbit for planet d (its orbit is currently unconstrained by observations). We seek systems that remain stable until the lower limit on the stellar age of $30$ Myr. We first simulate the inner two planets, each of $7 M_{\rm Jup}$, in the absence of planet b. They are given coplanar orbits, with planet c on a circular orbit at $a_c=s_c =37.97$ AU. The initial longitudinal separation is given by the observed positions, assuming face-on orbits ($\lambda_c - \lambda_d \approx 117^\circ$). We scan over a grid of semi-major axes for the inner planet. For $a_d<s_d = 24.44$ AU, $e_d$ is chosen so that apastron is at $24.44$ AU, and for $a_d>24.44$ AU, $e_d$ is chosen so that periastron is at $24.44$ AU (see Table \[tab:logII\] for how the initial conditions are generated). These choices maximize the chance that the two-planet system will be stable, while matching the constraint of the currently-observed separations from the star. We plot the instability times in Figure \[fig:instabgridelm\] as vertical lines. We repeat this calculation with planet b present with its nominal orbital elements (see Table \[tab:logII\]) and with mass $5$ $M_{\rm Jup}$, and plot those instability timescales in Figure \[fig:instabgridelm\] as a gray region. We observe that a very narrow range of $a_d$ is compatible with both the observed astrometry of the planets and with dynamical stability. The presence of the third planet narrows this range still further. The center of this range corresponds with the 2:1 mean motion resonance between planets c and d. (The position is offset from the location $a_d = (1/2)^{2/3} a_c$ because the large mass ratios induce fast precession.) In Figure \[fig:protect\] we show how this resonance protects the planets from close encounters. We ran identical simulations with planetary masses of $7$, $10$, and $10~M_{\rm Jup}$, and found qualitatively similar results, except the most stable three-planet simulation lasted only $10$ Myr. In the next section we examine this resonant protection mechanism and find initial conditions that produce acceptably long survival times even for these and even higher masses. Mean motion resonance {#sec:mmr} ===================== Inspired by the fact that Figure \[fig:instabgridelm\] shows a region of greater stability in the vicinity of the 2:1 resonance between c and d, we search for a face-on system near the center of the resonance. We use the BS integrator throughout this section. We find a solution in the absence of planet b in which the resonance angle, $$\phi_d = 2\lambda_c - \lambda_d - \varpi_d, \label{eqn:phi}$$ librates with small amplitude around $0^\circ$. When evaluating resonant angles, we compute the orbital elements with astrocentric coordinates. Resonance requires that $a_d$ is low enough for planet d’s period to be commensurate with planet c’s, and the observations require that $e_d$ is high enough for planet d to reach its currently observed separation from the star. Currently, we observe $\lambda_c - \lambda_d \approx 117^\circ$, so $\phi_d\approx0^\circ$ implies $\lambda_d-\varpi_d\approx 126^\circ$. Thus we find small libration is compatible with planet d being closer to apoastron than to periastron at the current time, in which case the velocity should be smaller than that of a circular orbit at the same distance. Integration with these initial conditions for planets c and d, in the absence of b, indeed shows libration and long-term stability (at least 160 Myr), for initial $a_d$ and $e_d$ values that place the planets in resonance (e.g., the system labeled “two-planet resonance” in Table \[tab:logI\]). In such solutions, the resonance angle for planet c usually does not librate. The resonance involves only the eccentricity of planet d. When planet b is added, planets b and c excite each other’s eccentricities and cause the libration amplitude of $\phi_d$ to fluctuate. Sometimes these excited eccentricities cause an encounter between b and c; sometimes the loss of libration in $\phi_d$ allows an encounter between c and d. In Figure \[fig:unstabres\] we show an example of this instability, where planets b and c start in their nominal orbits, $a_d=23.32$ AU, $e_d=0.09$, $\phi_d = 0^\circ$, and all bodies have their nominal masses (see Table \[tab:logI\]). Panel (a) shows the range of motion of each orbit versus time, panel (b) shows the resonance angle versus time, and the bottom panels show brief segments ($3\times 10^4$ yr, at times labeled above panel b) of the motion of the resonance angle through phase space. Over such brief intervals, the libration amplitude holds rather steady, except at the very end of the integration. In this example, the instability causes an encounter between planets c and d at $35.6$ Myr. Compared to the non-resonant cases, this system showed considerable longevity: it lasts long enough to be a plausible model for the observed system. We have found a way to calm the strongest interactions, those that cause instability after a few thousand orbits: a resonance between planets c and d that protects them from close encounters. This resonance protects the system until the somewhat longer timescale interactions between b and c cause an instability. But those interactions can also be suppressed by postulating yet another resonance. We integrated the nominal masses with initial conditions as above except $a_d=23.42$ AU instead of $23.32$ AU (Table \[tab:logI\]). The resulting system showed resonance protection between planets b and c. The 2:1 resonance is active, which this is possible far from its nominal location because the pericenters are precessing on nearly orbital timescales. In Figure \[fig:boundres\] we show this system lasting for 160 Myr. In this example, the resonance angle $\phi_d$ is librating with small amplitude the whole time (panels b and c), and the resonance angle $\phi_{c,out} = 2\lambda_b-\lambda_c-\varpi_c$ spends more time near $0^\circ$ (panels d and e), indicating the system is protected by both resonances. Even after 160 Myr of evolution, we have verified that there are epochs at which this solution fits the astrometric data of Table \[tab:nom\]. We found print-outs for which a rotation in the plane of the sky matched the simulated to the observed positions within a fractional error of 1% (more print-outs would likely find a closer match), and then we calculated $\chi^2$ based on the velocities of Table \[tab:nom\]. The resulting $\chi^2=11.4$ was both acceptable and quite competitive with the models of §\[sec:astrom\]. The next step is to understand how these resonances protect the system as a function of planetary mass. For instance, Figure \[fig:instabgridelm\] shows four integrations in which the resonance allows planets of masses $M_b = 5$, and $M_c = M_d = 7$ $M_{\rm Jup}$ to be stable for $30$ Myr, which is consistent with the observed system. But can the system survive at the nominal masses with only one resonance? How high can the masses go, in the double resonance? In Figure \[fig:massfactor\], we plot the time to instability for a wide range of planetary mass scalings. We use initial conditions corresponding to the nominal face-on, circular orbits (non-resonant), the initial conditions for Figure \[fig:unstabres\] (singly resonant), and parameters chosen to maximize stability of the double resonance for massive planets. All are listed in Table \[tab:logI\]. Because the resonant locations shift with increasing planetary mass, the ideal orbital parameters for stable resonance depend on the masses. In a suite of integrations we slightly vary the initial conditions (see Table \[tab:logI\]) to sample the chaotic outcomes. We find that systems with the nominal masses rarely survive 30 Myr with a single resonance, but can easily survive at least 160 Myr with a double resonance. In fact, our integrations show that a doubly-resonant system can be stable for 160 Myr, even for planetary masses a factor of two larger than the nominal values. That is, if this doubly-resonant configuration is correct, the planets could even have the masses of brown dwarfs. We find it remarkable that a double 2:1 resonance can allow planetary masses an order of magnitude larger than the $\sim$$2 M_{\rm Jup}$ allowed in a stable, non-resonant system. In some of these integrations, we have found the three-body Laplace resonance, with angle $\phi_L = \lambda_d - 3 \lambda_c + 2 \lambda_b$, librating temporarily (see also §\[sec:mode\] below). Such solutions are also consistent with the astrometric data. The Laplace angle was first observed to librate in the satellites of Jupiter (e.g., @1999MD). Besides HR 8799, two other extrasolar planetary systems have been proposed to inhabit the Laplace resonance. Extra peaks in the periodogram of radial velocity residuals of the GJ 876 system [@2005R] and the HD 82943 system [@2008Beau] could correspond to planets in the Laplace resonance with the known planets. Although each of these three extrasolar systems taken separately is merely suggestive of 4:2:1 and Laplace resonances, taken together they are quite intriguing. They may point to a new area of research in multiplanet systems that has been explored rather little so far, both theoretically and observationally. Monte-Carlo search {#sec:montecarlo} ==================== In the integrations so far, we have systematically varied a few parameters, concluding that a mean motion resonance is a promising solution to stabilize the system. Now we seek alternatives by allowing all the other orbital parameters to vary. The objective is to survey what orbits are allowed when the age of the system and the planetary masses are presumed to be robust. To be conservative, we adopt the youngest system age ($30$ Myr), corresponding to the least massive planets $(5, 7, 7)$ $M_{\rm Jup}$, as in §\[sec:stable\]. We select all the other variables with a Monte Carlo method. We draw the stellar mass from a normal distribution with mean $1.47 M_\odot$ and standard deviation $0.30 M_\odot$ [@1999GK], and we draw the system distance from a normal distribution with mean $39.4$ pc and standard deviation $1.1$ pc [@2007VL]. The planetary sky-projected positions and velocities are drawn from normal distributions according to the observed parameters of Table \[tab:nom\]. Note that these parameters are derived from the discovery observations of [@2008M] only; in §\[sec:newdata\] we check which systems are consistent with the important precovery observation of planet b by [@2009M]. What remains is to draw $z$ and $v_z$ for each planet, which are not constrained by the observations; we make that choice in various ways in the following subsections. In this section, we follow some planets with very high eccentricities and in some cases integrate through close approaches between planets. We use the BS integrator as before, and we follow the integration until one planet is ejected or collides with the star. Over 30 Myr, energy is typically conserved to better than one part in $10^6$, and angular momentum is typically conserved to better than one part in $10^7$. From crossing orbits to ejection {#sec:ejection} ---------------------------------- First, we wish to verify that once planets’ orbits begin crossing, at least one of them is ejected in a timescale much shorter than the age of the system. To do so, note that the expression for orbital energy of a single planet around a star is a monotonically increasing function of $|z|$ or $|v_z|$. Therefore, selecting non-zero values for those parameters will lead to a planet that is less bound than in the case of a face-on orbit. If $v_z=0$, there is a maximum value of $|z|$, called $|z|_{\rm max}$, that permits a bound orbit. If $z$ is given, then there is a maximum value of $|v_z|$, called $|v_z|_{\rm max}$, that permits a bound orbit. We wish to find how long it takes for planets on crossing orbits that are *not* marginally bound to be ejected by each other, so we first selected $z$ from a distribution uniform in the interval $[-|z|_{\rm max}/3, |z|_{\rm max}/3]$, after which we selected $v_z$ from a distribution uniform in the interval $[-|v_z|_{\rm max}/3, |v_z|_{\rm max}/3]$. This choice of distribution has the advantage of being connected to the observables (actually, complementary to them), being easy to implement, and being tuned to answer the question. It has the disadvantage of not corresponding to a simple distribution in orbital element space; nevertheless, a very wide range of orbital elements are sampled. We ran 1530 systems generated in this way, integrating to an ejection of one component (defined as reaching an orbital distance $>500$ AU with positive energy). The median time to ejection was $0.22$ Myr, and the maximum time was $7.7$ Myr. These timescales are longer than the $\sim$$0.02$ Myr scattering phase of Figure \[fig:ae\] due to significant mutual inclinations. In any case, the scattering phase will not contribute significant longevity to the system, and we are justified in stopping integrations at the first close approach in other sections of this paper. [@2009VCF] have also reached this conclusion for the planets of HR 8799. Moderately eccentric, coplanar planets {#sec:mode} ---------------------------------------- Next, we extend the analysis of §\[sec:stable\] to the case in which all three planets have non-zero eccentricities. For simplicity, and acknowledging that the planets probably formed in a common, flattened disk, we first investigate coplanar systems. There are several steps to generating a ($z$, $v_z)$ pair for each planet: 1. [draw stellar mass, distance, and planetary sky-projected positions and velocities, as described above;]{} 2. [draw a vector uniformly from the unit sphere, which serves as the direction of all the planets’ angular momenta;]{} 3. [compute $z$ and $v_z$ for each planet, consistent with the already-chosen spatial variables;]{} 4. [ discard the system if $x$, a number drawn from a uniform distribution in $[0,1]$, is greater than $\mathcal{L}/\mathcal{L}_{\rm ML}$, where $$\mathcal{L} \equiv \prod_{j=b,c,d} e_j \exp [ - (e_j/\sigma)^2/2 ] \label{eq:ray}$$ and $$\mathcal{L}_{\rm ML} \equiv (\sigma \exp[-1/2] )^3$$ with $\sigma = 0.05$. ]{} If a system is discarded at steps 3-4, the process begins anew with step 1. Step 4 is a technique called rejection sampling, and its purpose is to impose a prior distribution on the selected orbital elements $e_b$, $e_c$, and $e_d$. We sought planetary orbits with low to moderate eccentricity, using the Rayleigh distribution (eq. \[\[eq:ray\]\]), as recommended by recent work on the generation of eccentricities by planetary perturbations [@2007Z; @2008JT]. We chose $\sigma$ consistent with the dynamically “inactive” population of [@2008JT]. We ran $16,581$ systems generated this way, until a close approach or $30$ Myr elapsed. The median time until close approach was 3,100 yr, and only $49$ survived $30$ Myr. Of the surviving systems, we checked for 2:1 resonances, two resonant arguments for the inner pair and two resonant arguments for the outer pair. We considered the resonance to be dynamically significant if $h \equiv e \cos \phi$ had a non-zero average value (as in Figure \[fig:boundres\], panel e): $|\langle h \rangle| > 2.5 \sqrt{\langle h^2 / n \rangle}$, where the averages were performed over $n$ ($\sim$$100$) printouts of astrocentric orbital elements. This criterion is considerably looser than traditional definitions of being “in a resonance,” either libration of a resonant angle or lying interior to a separatrix in phase space. Nevertheless, this criterion indicates (i) protection against close encounters in the sense of Figure \[fig:protect\] and (ii) enhanced coherency to energy and angular momentum transfers during conjunction, as conjunctions occur at preferential phases of the orbit. All $49$ survivors had at least one of the four angles fulfilling this criterion: for $26$ only the inner pair were engaged in the resonance, for $2$ only the outer pair were engaged in the resonance, for $21$ both resonances were active. We expect that most of these $49$ survivors will eventually be disrupted by the perturbations of planet b; in no case was the motion as periodic as in Figure \[fig:boundres\]. Even systems stable for 160 Myr may become unstable over the main-sequence lifetime of the star [@2009GM]. Although 21 integrations displayed the 4:2:1 double resonance, in only one case did the Laplace angle $\phi_L$ librate the entire time (about $180^\circ$), and that system shows the strongest inner and outer resonances of the entire set. It is unclear if or how libration of $\phi_L$ enhances stability, over and above each pair of 2:1 resonances. One surviving system showed only a very weak inner 2:1 mean motion resonance. We examined the first 1.6 Myr of this integration in detail, finding that $P_c/P_d$ fluctuated in the range $2.30-2.45$ and $P_b/P_c$ fluctuated in the range $2.7-3.0$. With thousands of print-outs we determined that $h$ associated with $\phi_d$ had a non-zero average of $+0.012$. However, its large range $-0.078$ to $+0.105$ suggests weaker protection by the 2:1 resonance than that enjoyed by the other stable systems. The outer two planets occupied the 3:1 mean-motion resonance associated with the angle $\phi_{3:1} = 3\lambda_b-\lambda_c-2\varpi_c$ at a similarly weak level. Most strikingly, all three planets of this system maintained apsidal alignment with one another throughout the integration: their relative apsidal angle $\Delta \varpi$ librated around $0^\circ$. Apsidal locking apparently provides additional protection against close approaches: an inner planet only comes to apocenter at the same spatial location where an outer planet is at apocenter, so the two bodies do not come too close together. This system[^4] is illustrated in Figure \[fig:seclock\], which gives a pictorial representation of the apsidal protection mechanism. We also searched all of the stable systems for a tendency towards apsidal alignment, as quantified by $|\langle \cos \Delta \varpi \rangle|$ being non-zero in the same way as $h$ above. We found that apsidal alignment was also common for systems with strong mean-motion resonance; in the set of 49 survivors it occurred 3 times between the inner planets only, 12 times between the outer planets only, and 10 times among all three planets. Arbitrarily eccentric, coplanar planets {#sec:planarei} ----------------------------------------- We repeated the procedure of §\[sec:mode\] without step 4, imposing no prior on the eccentricities. However, to ensure that close encounters were not already happening, we only accepted each system if $a_d (1+e_d) < 0.85 a_c (1-e_c)$ and $a_c (1+e_c) < 0.85 a_b (1-e_b)$. We integrated 25,280 systems, of which only 5 lasted 30 Myr, and the median time to a close approach was 37,000 yr. Of the 5 survivors, one had the 2:1 mean-motion resonance active and secular alignment between the inner two planets. In the other 4, all three planets tended towards alignment, and two of these had the 2:1 mean-motion resonance active between the outer two planets. In 3 cases, the inner planet had $e_d>0.95$ and a current position near apocenter, and the apsidal alignment between d and c was rather tight: $|\varpi_c - \varpi_d| \lesssim 45^\circ$. These systems may correspond to the non-linear secular resonance identified by [@2004MM]. We consider it unlikely that configurations with strong apsidal alignment but no mean-motion resonance correspond to the true system, as explained below (§\[sec:newdata\]). If all three planets orbit in the same plane, one may wonder whether the debris disk and the stellar equator share it as well. From the observed positions and velocities, we can use this Monte Carlo study to constrain the orientation of the planets. Coplanar systems that fit the observed positions and velocities with arbitrary eccentricities, but non-crossing orbits, have line-of-sight inclinations less than $45^\circ$. The subset of stable systems obey this same limit. Note that this is not substantially different from the limit for circular orbits, cf. Figure \[fig:xisqcontour\]. Constraints on the stellar spin orientation—an expected rotational velocity $v$ and a measured $v \sin i$—led [@2009R] to derive a stellar inclination of $13^\circ-30^\circ$, consistent with this limit. Circular, non-coplanar orbits {#sec:nonplanar} ------------------------------- Next, we investigate whether moderately non-coplanar orbits are substantially more stable, even in the absence of any resonance. If not, we expect the conclusion that a resonance is needed applies to any roughly coplanar systems, not just strictly coplanar ones. The following series of integrations is not intended to match the currently observed positions, but to be a parametric study of mutual inclination. Relative to the nominal case (A), but now with lower-limit masses, we varied the initial inclination and node of planets b and d (see initial conditions in Table \[tab:logI\]). Both planets b and d are given the same inclination (relative to c, which always starts at $i_c=0$). However, they are given different nodes, to sample the same inclination 36 times, so that the spread of chaotic outcomes are represented. The systems were integrated until a close approach or 160 Myr. In Figure \[fig:mutinc\] we plot the resulting times of instability. At low inclination the spread of these times is several orders of magnitude. Varying the initial orientation angles, which also control the initial longitude of each planet, causes some systems to have a close approach within a few tens of orbits and causes other systems to last $>1$ Myr due to the protection afforded by the 2:1 resonance between planets d and c. As the initial inclination increases, one might expect Hill-sphere encounters will be delayed, as the motion out of the plane exceeds a Hill radius at $i \simeq 6^\circ$ for these masses. Nevertheless, we observe that the median instability time modestly decreases until $40^\circ$, is constant between $50^\circ$ and $140^\circ$, and increases dramatically from $150^\circ$ to $180^\circ$, perfectly retrograde. The shorter instability timescale for substantially non-planar systems is likely due to the [@1962K] effect, which causes inclination to decrease and eccentricity to increase on a secular timescale of $\sim$$10^5$ years. We conclude that we can likely extend our conclusions from strictly coplanar systems to systems with large prograde inclinations. Systems with adjacent planets orbiting in the opposite sense can avoid instability; i.e., retrograde systems are inherently more stable than prograde systems, as has been shown in other contexts [@2003N; @2008G]. No mean-motion resonance or apsidal locking appears to be active in protecting retrograde systems from instability, but the brief timescales of conjunction may be responsible. We do not expect a retrograde configuration for planet d, nor is it consistent with the data (see §\[sec:newdata\]). Arbitrary orbits {#sec:arbitrary} ------------------ In this subsection, we remove all restrictions on individual eccentricity and orbital orientation, to have a completely data-driven sampling of orbits. We expect that not many systems generated this way are realistic, as planet d’s orbit is so poorly constrained. Nevertheless, after drawing the stellar and observable planetary parameters, we drew each $z$ uniformly from the interval $[-|z|_{\rm max}, |z|_{\rm max}]$ and then drew each $v_z$ uniformly from the interval $[-|v_z|_{\rm max}, |v_z|_{\rm max}]$. We rejected the resulting system if either: 1. [ any of the planets initially had positive energy (despite trying to avoid this case by construction of the osculating orbital elements); or]{} 2. [ planetary orbits crossed, i.e., $a_d (1+e_d) > a_c (1-e_c)$ or $a_c (1+e_c) > a_b (1-e_b)$.]{} We integrated 3010 systems until the first ejection (not stopping at close approaches), or until $30$ Myr elapsed. Of these, 13 systems survived the entire time. Three cases had inner apsidal alignment, and another two cases had outer apsidal alignment. A common characteristic is that planet d has a moderate-to-large eccentricity and comes to apocenter at its currently-observed position. Thus the period ratio $P_c / P_d$ can be quite large, and the system rather hierarchical. In fact, we found that using the full formulation of Hill stability [@1982MB], the Star-d-c sub-system (neglecting planet b) was usually Hill stable if planet d is prograde with respect to the others. With only small perturbations from planet b, the system remains stable for $30$ Myr, with the inner subsystem fulfilling Hill’s stability criterion for most of that time. In contrast, as mentioned above, no particular protection mechanism was apparent for the retrograde systems. The greater stability of retrograde orbits is not seen in Hill’s stability calculations, just as the greater stability of retrograde satellites is not reflected in the usual Jacobi constant in the circular restricted three-body problem [@1997H]. A common attribute is that in *all* 13 stable systems, planet b is in a very wide orbit, but it comes to pericenter at its currently-observed location. One way to quantify this is to note that, in all cases, the mean anomaly of $b$ at the present epoch is within $10^\circ$ of $0^\circ$, which would happen only $1/18$ of the time for randomly phased orbits. In such an orbit, it perturbs the inner two planets minimally, contributing to the system stability. The orbits of one of these 13 systems at the current epoch is displayed in Figure \[fig:noncoplanar\]. We regard these systems, with very large mutual inclinations and eccentricities, to be a possible, but not plausible, explanation of the observational data. The outer orbit being very near periastron seems finely tuned, because it does not spend much time there. It is as if it is swooping in from several hundred AU, just in time to have its picture taken by [@2008M]. We also note that orbits like those of Figure \[fig:noncoplanar\] are currently consistent with circular, coplanar orbits, but they do not fulfill this condition at most other orbital phases. Since the observed velocities are consistent with circular and coplanar, we can quantify the likelihood that the true system actually has very non-coplanar orbits, as follows. We produced outputs for the system displayed in Figure \[fig:noncoplanar\] at every $\sim3500$ years of a $10^8$ year integration, during which the semi-major axes and eccentricities showed no qualitative long-term changes. For each of the outputs, we found the $\chi^2$ of the best-fitting circular, coplanar model, minimizing over $(M_\star, i, \Omega)$ as in §\[sec:astrom\], with $M_\star$ in the range $[1,2]\times M_\odot$ with no penalty for being far from the nominal value. We assign the observed error bars on velocities to the sky-projected velocities in the simulation, then compare them to the best-fitting velocities that come from a circular, coplanar hypothesis. With the current snapshot (Figure \[fig:noncoplanar\]), the system fits a circular, coplanar model with $\chi^2=2.18$. In only 639 of the 28,996 snapshots was $\chi^2$ better than this value. So we conclude that the current phase of this particular non-coplanar system is fine-tuned to the $\sim$$2.2\%$ level. We thus prefer models that actually *are* close to circular and coplanar. Another consideration disfavors these solutions with large mutual inclination and large eccentricities. Fits to the infrared spectral energy distribution of HR 8799 suggest a population of colliding, dust-forming bodies with a semimajor axis of $\sim$$100$ AU, though this distance is still uncertain [@2006WA; @srs+09; @2009R]. It is unlikely that planet b’s orbit actually crosses this belt of bodies, which may constrain $e_b$ to less than a few tenths. This constraint will be observationally accessible in the near future. Comparison to New Data {#sec:newdata} ------------------------ The foregoing analysis was based solely on the astrometric measurements reported in the discovery paper. Between the original submission of this work and now, several new measurements were reported based on careful analysis of previously-collected images [@2009L; @2009F; @2009M]. In particular, the measurement by [@2009M] of positions of planet d over a one-year baseline determines its sky-velocity to be $$[v_E, v_N] = [-4.5\pm0.8, 0.7 \pm 0.8]\times10^{-3} {\rm AU\,day}^{-1}. \nonumber$$ These values come from a regression of position versus time, in combination with the [@2008M] data, as in Table \[tab:nom\]. For the purposes of this section, that value eliminates some of the previously possible dynamical configurations, as follows. In Figure \[fig:predict\] we plot the 1-$\sigma$ and 2-$\sigma$ contours of the measured velocity of planet b, and overlay the orbits from the preceding sections. We find that solutions with a retrograde planet d, and those for which planet d is at apastron of an very eccentric orbit, are now ruled out. However, the 2:1 resonant solutions are still quite consistent with the new data, including the weakly-resonant, apsidally-locked system (Fig \[fig:seclock\]) and some of the very non-coplanar (yet non-resonant) solutions (e.g., Fig \[fig:noncoplanar\]). Although these latter solutions fit the data, recall that we are seeing these systems at a special time, so we doubt they correspond to the true system. Discussion {#sec:conclude} ========== We have investigated the orbital stability of the newly-imaged planetary system HR 8799. The nominal orbital model and masses are not stable. In fact, no model with circular, coplanar orbits that also fits the astrometry well is stable, regardless of the inclination and orientation of the system on the sky. To overcome this problem by reducing the planetary masses, values $\lesssim 2$ $M_{\rm Jup}$ are required. This can happen if the cooling models under-predict the luminosity, though that is difficult to understand, as even hot-start models cannot produce the observed luminosity at such low masses (see §\[sec:intro\]). Such masses would be plausible if the system is considerably younger than expected, yet the star has reached the main sequence [@2008M]. Our favored solution is that a 2:1 resonance between the inner two planets preserves stability. Assuming that the inner pair of planets are in resonance, two qualitatively different configurations are possible: - [The outer two planets are not in resonance. ]{} This configuration remains stable in the perturbing presence of planet b only if the planetary masses are $\lesssim 10$ $M_{\rm Jup}$ (Fig. \[fig:massfactor\]). (It is also possible, given a less likely system orientation, that only the outer resonance is active. This configuration leads to a similar mass limit.) - [The outer pair of planets are also in 2:1 resonance. ]{} This solution fits all the current data for the system. At the nominal masses, the system can easily survive for the age of the star. In fact, the planetary masses could be up to $\sim1.9$ times bigger than their nominal values without violating stability constraints (Fig. \[fig:massfactor\]). This value is similar to the maximum planetary masses that can stably exist at the fixed point of the 2:1 resonance [@2003B fig. 8]. It will be very interesting to find a test of this hypothesis. The Laplace angle can also librate in this system, though this is not a requirement for stability. Strong apsidal alignment, while unnecessary for system stability, allows planetary survival in a weaker 2:1 mean-motion resonance. One final type of system architecture is, in principle, consistent with stability of the HR 8799 system. The planets may be hierarchically spaced, with large eccentricities and mutual inclinations, but inhabit phases of their orbits that look closely-packed at the moment. Such systems fit the current data but are finely tuned both in their orbital parameters and in the time at which we are viewing the system. It is possible that other stabilizing resonant configurations exist, but were missed because they occupy small regions of phase space. The 2:1 mean-motion resonance dominates our randomly-generated stable systems and would naturally yield the observations without fine tuning, so we consider it to be the most likely stabilizing mechanism. This study brings up several issues for future observations of HR 8799, and directly-imaged multiplanet systems in general, as follows. First, it serves as the first test of hot-start cooling models for exoplanets. They barely pass the test if only planets d and c are in resonance, and they comfortably pass the test if all three planets are in resonance. We hope more detailed dynamical studies of this system will sharpen this test as more data are collected. If the doubly-resonant configuration can somehow be verified, it would considerably weaken this test. We have not yet directly used dust observations as a dynamical constraint. The spectral energy distribution reveals a massive debris disk surrounding the planetary system, with an orbital radius of $\gtrsim 66$ AU [@2006WA]. Given that planet b is observed at $a_b \gtrsim 68$ AU, we expect that the inner edge of the debris disk must be $\gtrsim 90$ AU, and that future measurements and modeling will find that orbital radius to be plausible and even preferred. Such a model could in turn serve as a complementary test of planet b’s mass, in analogy to the test of the mass of Fomalhaut b [@2009C]. Perhaps other directly-imaged systems will fortuitously arrange for complementary tests. Second, we found evidence of a mean motion resonance at very large orbital separations, much farther than those found by the radial velocity technique, of which there are many (e.g., @2001Marcy [@2004Mayor]). Sometimes resonant identification is based on stability arguments in those systems (e.g., @2005C), as it is here. The most commonly invoked evolutionary mechanism for trapping planets into resonance with one another is convergent migration in the protoplanetary disk. The properties of migration in a disk with multiple massive planets deserve further investigation to determine the conditions under which convergent migration and resonance capture are possible at the locations of the HR 8799 planets. We verified that if planets d and c were initially placed in circular orbits exterior to the 2:1 resonance, they are stable to collisions or ejections for $\gtrsim 30$ Myr, so getting into the resonance without first becoming unstable is not a problem in this case. One difficulty with differential migration is that any additional migration, after the resonance is reached, efficiently increases the eccentricities. That this requires implausible fine-tuning in the absence of eccentricity damping by the gas disk has been discussed for the 2:1-resonant GJ876 system [@2002LP]. In a trial integration, we introduced a force to simulate outward migration of the inner planet, following [@2002LP], with a timescale $a/(da/dt) = 10^7$ yr and no eccentricity damping, starting from double-2:1 resonance at the nominal masses (fig. \[\[fig:boundres\]\], table \[tab:logI\]). The planetary eccentricities rapidly increased and the system began scattering after a semi-major axis change of $\sim15$%, illustrating its fragility. We expect calculations of migration into resonance will be very interesting for this system. We also showed how perturbations by a third planet tend to disrupt a mean-motion resonance (when only the inner sub-system is in resonance). This mechanism adds to a growing list of ways to disrupt resonances among planets, including turbulent fluctuations in a protoplanetary disk [@2008A], tidal dissipation [@2007T], and scattering of planetesimals [@2006M; @2007Morby]. Third, it may seem surprising that dynamical stability arguments are needed to correctly solve the orbits of the first directly-imaged multiplanet system. However, this situation is also common for multiplanet systems discovered by radial velocity. For instance, [@2005V] and [@2006Lee] have found that dynamical stability can constrain orbital parameters more tightly than radial-velocity data alone. Furthermore, planets that are discovered by direct imaging of their self-luminosity are biased to have high masses, making stability less assured. The bias of direct imaging towards large angular separation implies that very long orbital periods will be common for such discoveries, so we foresee many stability analyses predicated on only the sky-projected positions and velocity vectors of planets. We hope this paper proves to be a useful example of how to conduct such an analysis. We thank Matthew Holman, Eric Ford, Jack Lissauer, Guillermo Torres, Sean Andrews, Yanqin Wu, Andrew Shannon, and members of CITA and U of Toronto Astrophysics department for helpful discussions. We thank Scott Tremaine, Eugene Chiang, and the anonymous referee for suggesting that we expand the investigation to mutually inclined solutions. DF is grateful for funding through a Michelson Fellowship, which is supported by the National Aeronautics and Space Administration and administered by the Michelson Science Center. RM is grateful for an Institute for Theory and Computation Fellowship at Harvard University. [ccccccccc]{} §\[sec:astrom\], D &$1.86$ & $0.0067$ & $81.04$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $61.30^\circ$\ Fig. \[fig:185\] & & $0.0095$ & $38.16$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $315.29^\circ$\ & & $0.0095$ & $27.69$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $204.92^\circ$\ §\[sec:mmr\] &$1.5$ & $0.0095$ & $37.97$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $226.99^\circ$\ two-planet resonance & & $0.0095$ & $23.32$ & $0.09$ & $0^\circ$ & $344.0^\circ$ & $0^\circ$ & $126.00^\circ$\ §\[sec:mmr\] &$1.5$ & $0.0067$ & $67.91$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $332.36^\circ$\ Fig. \[fig:unstabres\] & & $0.0095$ & $37.97$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $226.99^\circ$\ & & $0.0095$ & $23.32$ & $0.09$ & $0^\circ$ & $344.0^\circ$ & $0^\circ$ & $126.00^\circ$\ §\[sec:mmr\] &$1.5$ & $0.0067$ & $67.91$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $332.36^\circ$\ Fig. \[fig:boundres\] & & $0.0095$ & $37.97$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $226.99^\circ$\ & & $0.0095$ & $23.42$ & $0.09$ & $0^\circ$ & $344.0^\circ$ & $0^\circ$ & $126.00^\circ$\ §\[sec:mmr\]\*\* & $1.5$ & $0.0067x$ & $67.91$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $332.36^\circ$\ Fig. \[fig:massfactor\] & & $0.0095x$ & $37.97$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $226.99^\circ$\ non-resonant & & $0.0095x$ & $24.44$ & $0.0$ & $0^\circ$ & $344.0^\circ$ & $0^\circ$ & $126.00^\circ$\ & & & $\pm10^{-4}$ & $\pm10^{-4}$ & $\pm0.01^\circ$ & $\pm0.01^\circ$ & $\pm0.01^\circ$ & $\pm0.01^\circ$\ §\[sec:mmr\]\*\* & $1.5$ & $0.0067x$ & $67.91$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $332.36^\circ$\ Fig. \[fig:massfactor\] & & $0.0095x$ & $37.97$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $226.99^\circ$\ single resonance & & $0.0095x$ & $23.32$ & $0.09$ & $0^\circ$ & $344.0^\circ$ & $0^\circ$ & $126.00^\circ$\ & & & $\pm10^{-4}$ & $\pm10^{-4}$ & $\pm0.01^\circ$ & $\pm0.01^\circ$ & $\pm0.01^\circ$ & $\pm0.01^\circ$\ §\[sec:mmr\]\*\* & $1.5$ & $0.0067x$ & $67.91$ & $0.002$ & $0^\circ$ & $180^\circ$ & $0^\circ$ & $180^\circ$\ Fig. \[fig:massfactor\] & & $0.0095x$ & $37.97$ & $0.005$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $0^\circ$\ double resonance & & $0.0095x$ & $23.52$ & $0.083$ & $0^\circ$ & $180^\circ$ & $0^\circ$ & $0^\circ$\ & & & $\pm10^{-4}$ & $\pm10^{-4}$ & $\pm0.01^\circ$ & $\pm0.01^\circ$ & $\pm0.01^\circ$ & $\pm0.01^\circ$\ §\[sec:nonplanar\]\*\*\* &$1.5$ & $0.0048$ & $67.91$ & $0.0$ & $i$ & $0^\circ$ & $\Omega_b$ & $62.36^\circ$\ Fig. \[fig:mutinc\] & & $0.0067$ & $37.97$ & $0.0$ & $0^\circ$ & $0^\circ$ & $0^\circ$ & $316.99^\circ$\ & & $0.0067$ & $24.44$ & $0.0$ & $i$ & $0^\circ$ & $\Omega_d$ & $200.23^\circ$\ [ccccccccc]{} §\[sec:intro\], nominal & $1.5$ & $0.0067$ & $60.16$&$31.50$&$0.0$&$1.186$&$-2.265$&$0.0$\ §\[sec:astrom\], A & & $0.0095$ & $-25.90$ & $27.76$ & $0.0$ & $2.500$ & $2.332$ & $0.0$\ Fig. \[fig:ae\],\[fig:data\] & & $0.0095$ & $-8.45$ & $-22.93$ & $0.0$ & $-3.998$ & $1.473$ & $0.0$\ §\[sec:masses\] & $1.5$ & $0.0067x$ & $60.16$&$31.50$&$0.0$&$1.186$&$-2.265$&$0.0$\ Fig. \[fig:massmult\]\* & & $0.0095x$ & $-25.90$ & $27.76$ & $0.0$ & $2.496$ & $2.329$ & $0.0$\ & & $0.0095x$ & $-8.45$ & $-22.93$ & $0.0$ & $-3.996$ & $1.473$ & $0.0$\ §\[sec:stable\]\*\* & $1.5$ & $0.00435$ & $60.16$&$31.50$&$0.0$&$1.186$&$-2.265$&$0.0$\ Fig. \[fig:instabgridelm\],\[fig:protect\] & & $0.0062$ & $-25.90$ & $27.76$ & $0.0$ & $2.496$ & $2.329$ & $0.0$\ & & $0.0062$ & $-8.45$ & $-22.93$ & $0.0$ & $-3.996 \gamma$ & $1.473 \gamma$ & $0.0$\ §\[sec:mode\]& $1.60905$ & $0.0048$ & $60.18$ & $31.47$ & $27.21$ & $1.464$ & $-2.244$ & $0.4295$\ Fig. \[fig:seclock\] & & $0.0067$ & $-25.94$ & $27.89$ & $-8.527$ & $ 2.398$ & $2.460$ & $1.178$\ & & $0.0067$ &$-8.601$ & $-22.90$ & $-5.311$ & $-3.577$ & $1.891$ & $-1.327$\ §\[sec:arbitrary\] & $1.54796$ & $0.0048$ & $57.54$ & $30.22$ & $9.765$ & $1.293$ & $-2.034$ & $0.2204$\ Fig. \[fig:noncoplanar\] & & $0.0067$ & $-24.74$ & $26.61$ & $-23.15$ & $2.228$ & $2.327$ & $1.334$\ & & $0.0067$ & $-8.084$ & $-22.00$ & $-9.609$ & $-3.490$ & $1.518$ & $0.2634$ [^1]: This estimate is given by [@2008M] based on four age indicators: Galactic space motion, main-sequence fitting, stellar pulsations, and the massive debris disk. The first is circumstantial but consistent with the quoted ages; the others suggest an age $\lesssim 100$ Myr. That the star has reached the main sequence suggests that it is at least several $10$s of Myr old. [^2]: For the specific initial conditions of this and other integrations herein, see Tables \[tab:logI\] and \[tab:logII\]. [^3]: Here we neglect the few-percent contribution of the planetary mass [^4]: Initial conditions are given in Table \[tab:logII\].

--- abstract: 'We show that higher-order nonlinear indices ($n_4$, $n_6$, $n_8$, $n_{10}$) provide the main defocusing contribution to self-channeling of ultrashort laser pulses in air and Argon at 800 nm, in contrast with the previously accepted mechanism of filamentation where plasma was considered as the dominant defocusing process. Their consideration allows to reproduce experimentally observed intensities and plasma densities in self-guided filaments.' author: - 'P. Béjot$^{1,2}$' - 'J. Kasparian$^{1}$' - 'S. Henin$^1$' - 'V. Loriot$^{2}$' - 'T. Vieillard$^{2}$' - 'E. Hertz$^{2}$' - 'O. Faucher$^{2}$' - 'B. Lavorel$^{2}$' - 'J.-P. Wolf$^{1}$' title: 'Higher-order Kerr terms allow ionization-free filamentation in gases' --- The filamentation of ultrashort laser pulses in gases [@BraunKLDSM95] attracted a lot of interest in the last years because of its physical interest as well as its potential applications [@KasparianW08; @BergeSNKW07; @CouaironM07; @ChinHLLTABKKS05]. Filaments are self-channeled structures propagating over many Rayleigh lengths without diffraction. They are generally considered to stem from a dynamic balance between Kerr focusing and defocusing by the plasma generated at the non-linear focus. Numerical simulations based on this balance report a core intensity of several 10$^{13}$ W/cm$^{2}$ and typical electron densities of several 10$^{16}$ cm$^{-3}$ [@BergeSNKW07; @CouaironM07]. Consequently, plasma ionization is generally admitted as necessary for an ultrashort pulse to experience self-channeling in gases. But the plasma density provided by this description of filamentation appears overestimated as compared with experimental measurements. As reviewed in [@KasparianSC00], such measurements are dispersed over several orders of magnitude, especially due to different focusing conditions and divergent assumptions about the core diameter of the filaments, but the electron density in a filament generated by a slightly focused beam is more likely to amount to $10^{14} - 10^{15}$ cm$^{-3}$ [@KasparianSC00]. This value, as well as the discrepancy by more than one order of magnitude with numerical simulations, was recently confirmed [@ThebergeLSBC06]. The observation of so-called plasma-free filamentation [@MechainCADFPTMS04; @DubietisGTT04], as well as the consideration that a balance between the instantaneous Kerr term and the time-integrated plasma contribution implies strongly asymmetric pulse shapes [@StibenzZS06], periodically led to challenge the role of plasma in laser filamentation. However, up to now, no other process seriously challenged plasma as the main defocusing process balancing the Kerr self-focusing. Nurhuda et al. proposed that the saturation of the nonlinear susceptibility $\chi^{(3)}$ should be taken into account [@NurhudaSM08]. Such saturation can be described as negative higher-order Kerr terms. The nonlinear index of air induced by high-power femtosecond laser pulses can be written as $\Delta\text{n}_\text{Kerr}=n_2I+n_4I^2+n_6I^3+n_8I^4+ ...$ , where $I$ is the incident intensity and the $n_{2*j}$ coefficients are related to $\chi^\text{(2*j+1)}$ susceptibilities. This nonlinear index is generally truncated after its first term, $n_2$ [@KasparianW08; @BergeSNKW07; @CouaironM07; @ChinHLLTABKKS05], mostly because of the lack of data about the values of the subsequent terms. Numerical works have investigated the influence of the quintic nonlinear response on the propagation dynamics in gases, although without knowledge of its value [@AkozbekSBC01; @Couairon03; @VincotteB04; @FibichI04; @Centurion05]. They showed that $n_4$ is negative, *i.e.* the $\chi^{(5)}$ susceptibility is a defocusing term. It tends to stabilize the propagation of ultrashort laser pulses in air and to decrease both the electron density and the maximal on-axis intensity. Consequently, the losses due to multiphoton absorption (MPA), which lead to the end of the filamentation, are reduced and pulse self-channeling is sustained over longer distances. However, plasma generation still appeared as necessary for filament stabilization. Moreover, the value of $n_4$ was set arbitrarily, which limits the conclusiveness of these studies. Finally, the lack of data prevented any evaluation of a possible effect of the further-order nonlinear refractive indices. However, the higher-order Kerr indices have recently been measured in N$_2$, O$_2$ and Ar by Loriot *et al.* [@Loriot09]. The reader is referred to this work for a detailed description of this experimental determination. In this Letter, we investigate their influence on numerical simulations of laser filamentation. We show that their values are sufficient to provide the dominant contribution to the defocusing terms of self-channeling. Their implementation in numerical simulations yields the experimentally observed plasma density. As a consequence, contrary to previously held beliefs, a plasma is not required for the observation of filamentation. Rather, plasma generation can be considered as a by-product of the self-guiding of laser filaments. We implemented these nonlinear coefficients into a numerical model describing the propagation of ultrashort high power pulses [@BejotBBW07]. We consider a linearly polarized incident electric field at $\lambda_0$=$800$ nm with cylindrical symmetry around the propagation axis $z$. The scalar envelope $\varepsilon(r,t,z)$, assumed to vary slowly in time and along $z$, evolves according to the propagation equation: $$\begin{aligned} \begin{aligned} \label{Equation3} &\partial_z\varepsilon =\frac{i}{2k_0}\triangle_{\bot}\varepsilon-i\frac{k''}{2}\partial_t^2\varepsilon+i\frac{k_0}{n_0}\left(\sum_{j=1}^{4}{n_{2*j}|\varepsilon|^{2*j}}\right)\varepsilon \\ &-i\frac{k_0}{2n_0^2\rho_c}\rho\varepsilon-\frac{\varepsilon}{2}\sum_{l=\mathrm{O}_2,\mathrm{N}_2}{\left(\sigma_l\rho+\frac{W_l(|\varepsilon|^2)U_l}{|\varepsilon|^2}(\rho_{at_l}-\rho)\right)} \end{aligned}\end{aligned}$$ where $k_0$=${2\pi n_0}/{\lambda_0}$ and $\omega_0={2\pi c}/{\lambda_0}$ are the wavenumber and the angular frequency of the carrier wave respectively, $n_0$ is the linear refractive index at $\lambda_0$, $k''=\frac{\partial^2 k}{\partial\omega^2}|_{\omega_0}$ is the second order dispersion coefficient, $\rho_{at}$ the neutral atoms density, $\rho$ the electron density, $\rho_c=\epsilon_0 m \omega_0^2/e^2$ is the critical electron density, $m$ being the electron mass and $e$ its charge. $W_l(|\varepsilon|^2)$ and $\sigma_l$ are the photoionization probability and the inverse Bremsstrahlung cross-section of species $l$ respectively (with ionization potential $U_l$), and $t$ refers to the retarded time in the reference frame of the pulse. The right-hand terms of Eq.(\[Equation3\]) account for spatial diffraction, second order group-velocity dispersion (GVD), instantaneous nonlinear effects (*i.e.* the nonlinear refractive index of air, up to the $n_8$ term), plasma defocusing, inverse Bremsstrahlung and multiphoton absorption respectively. As compared with previously published data [@Loriot09], we used values of the higher-order refractive indices (Table \[tab1\]) incorporating the correction for the coherent artifact [@Oudar82], *i.e.* adequately substracting its electronic contribution at play in the original measurement of Ref. [@Loriot09]. This correction results in dividing each $n_{2*j}$ term by $j+1$. Owing to the short pulse duration ($30\ fs$) used in the simulations, the delayed orientational response is disregarded. The propagation dynamics of the electric field is coupled with the density of the electrons originating from the ionization of both O$_2$ and N$_2$: $\rho=\rho_{\mathrm{O}_2}+\rho_{\mathrm{N}_2}$. This density is governed by the muti-species generalized Keldysh-PPT (Perelomov, Popov, Terent’ev) formulation [@KasparianSC00; @BergeSNKW07]. --------- ----------------------- ----------------------- ----------------------- ----------------------- $n_2\ (10^\text{-19}$ $n_4\ (10^\text{-33}$ $n_6\ (10^\text{-46}$ $n_8\ (10^\text{-59}$ Species $cm^2/W)$ $cm^4/W^2)$ $cm^6/W^3)$ $cm^8/W^4)$ N$_2$ 1.1$\pm0.2$ -0.5$\pm0.27$ 1.4$\pm0.15$ -0.44$\pm0.04$ O$_2$ 1.6$\pm0.35$ -5.2$\pm0.5$ 4.8$\pm0.5$ -2.1$\pm0.14$ Air 1.2$\pm0.23$ -1.5$\pm0.3$ 2.1$\pm0.2$ -0.8$\pm0.06$ --------- ----------------------- ----------------------- ----------------------- ----------------------- : Coefficients of the nonlinear refractive index expansion of N$_2$ and O$_2$ at 1 bar pressure, and interpolation to air, as used in the present work [@Loriot09][]{data-label="tab1"} We used this model to simulate the propagation of an ultrashort pulse typical of laboratory-scale experiments: $1$ mJ energy, $30$ fs FWHM pulse duration without initial chirp (hence, about 3.9 critical powers $P_{\mbox{cr}}$), an initial waist of $\sigma_{\mbox{r}} = 4$ mm, a focal length $f=1$ m and a pressure of $1$ bar. Figures  \[intensite\_plasma\] and \[intensite\_z\_r\] compare the numerical results of the full model implementing Kerr terms up to $n_8$ and of the classical model, where the Kerr term is truncated to $n_2$. Both models lead to self-guided filaments. The full model yields a lower maximum intensity ($31.6$ TW/cm$^{2}$ *vs.* $78$ TW/cm$^{2}$), although these values lie within the range of published experimental data in comparable conditions [@BergeSNKW07; @CouaironM07; @ChinHLLTABKKS05; @KasparianW08]. On the other hand, the full model predicts an electron density 40 times below the classical one ($1.1\cdot10^{15}$ cm$^{-3}$ *vs.* $4.2\cdot10^{16}$ cm$^{-3}$). While the latter value is comparable with the output of other numerical works [@BergeSNKW07; @CouaironM07; @ChinHLLTABKKS05; @KasparianW08], the full model agrees with the available experimental measurements of the electron density [@KasparianSC00; @ThebergeLSBC06]. Note that, with the considered parameters, the full model yields a more strict intensity clamping than the classical one [@BeckerAVOBC01]. It predicts an intensity constant within $20\%$ over $15$ cm (*vs* 9.5 cm in the case of the classical model), a length well comparable to experimental data reported to date in air for mJ-pulses [@BergeSNKW07; @CouaironM07; @HosseiniYLC04; @MechainOFCPM06]. This stricter clamping can be explained by the lower electron density, which results in weaker multiphoton losses, allowing a slower decay of the filament intensity and ionization. The full model also yields a narrower output spectrum (Figure \[Spectre\_2m\]), which better fits experimental data in air [@BergeSNKW07; @CouaironM07]. It should therefore be considered as the reference model for numerical simulations of filamentation. Note that the almost symmetric shape of the spectrum is due to the neglection of self-steepening ![(a) On-axis intensity and (b) Plasma density as a function of the propagation distance for the classical model (considering only $n_2$ term of the Kerr index and the plasma defocusing), the full model, as well as the full model without plasma.[]{data-label="intensite_plasma"}](Figure_1.eps){width="12cm"} ![Fluence distribution in air as a function of the propagation distance for the full model (a) and the classical model including $n_2$, ionization and GVD only (b). The white lines display the quadratic radius as a function of the propagation distance.[]{data-label="intensite_z_r"}](Figure_2.eps){width="12cm"} On the other hand, neglecting the ionization in the full model (see Fig. \[intensite\_plasma\](a)) almost does not affect the simulation output. This shows that, in contrast to the classical understanding of filamentation in gases, the self-guiding process and plasma generation are almost decoupled. Instead, the negative higher-order nonlinear indices $n_4$ and $n_8$ constitute the dominant regularization terms leading to filamentation in air at atmospheric pressure. This limited influence of the ionization on the filamentation dynamics when higher-order non-linear indices are adequately considered sheds a new light on the possibility of ionization-free filamentation [@MechainCADFPTMS04], which appears as a natural possibility in the context of the full model. Still, the dominant contribution of higher-order Kerr terms does not prevent ionization (Fig. \[intensite\_plasma\]b), which may contribute *e.g.* to the conical emission. ![Spectrum after 2 m propagation in air at atmospheric pressure.[]{data-label="Spectre_2m"}](Figure_3.eps){width="12cm"} ----------------------- ----------------------- ----------------------- ----------------------- -------------------------- $n_2\ (10^\text{-19}$ $n_4\ (10^\text{-33}$ $n_6\ (10^\text{-45}$ $n_8\ (10^\text{-59}$ $n_{10}\ (10^\text{-74}$ $cm^2/W)$ $cm^4/W^2)$ $cm^6/W^3)$ $cm^8/W^4)$ $cm^{10}/W^5)$ 1.0$\pm0.09$ -0.37$\pm1$ 0.4$\pm0.05$ -1.7$\pm0.1$ 8.8$\pm0.5$ ----------------------- ----------------------- ----------------------- ----------------------- -------------------------- : Coefficients of the nonlinear refractive index expansion of Ar at 1 bar pressure, as used in the present work [@Loriot09][]{data-label="tab2"} We checked that the above conclusions are not restricted to a particular set of values of the non-linear refractive indices. Indeed, qualitatively comparable results have been obtained when varying the indices by several tens of percent, comparable with the experimental uncertainties on the non-linear indices. Furthermore, to compare the above molecular results with an atomic gas where no molecular orientation occurs, we performed simulations for Argon, where the ionization potential is close to that of the air molecules [@LoriotHLF08], thus behaving in a similar manner as far as ionization is concerned. As in the case of air, we refined the corresponding indices to take the coherent artifact into account. The resulting values are summarized in Table \[tab2\]. Like in air, the full model yields lower filament intensity ($28.5$ TW/cm$^{2}$ *vs.* $80.9$ TW/cm$^{2}$) and electron density ($5.2\cdot10^{13}$ cm$^{-3}$ *vs.* $4.1\cdot10^{16}$ cm$^{-3}$) than the classical model (Figure \[intensite\_plasma\_Ar\]). Also, the evolution of the fluence profile as a function of propagation distance (Figure \[intensite\_z\_r\_Ar\]) is quite similar in both models. ![(a) On-axis intensity and (b) Plasma density as a function of the propagation distance for the classical model (considering only $n_2$ term of the Kerr index and the plasma defocusing) and the full model, in Argon under 1 bar pressure.[]{data-label="intensite_plasma_Ar"}](Figure_4.eps){width="12cm"} ![Fluence distribution in 1 bar Argon as a function of the propagation distance for the full model (a), and the classical model including $n_2$, ionization and GVD only (b).[]{data-label="intensite_z_r_Ar"}](Figure_5.eps){width="12cm"} The space-time dynamics shows more differences between the full and the classical models (Figure \[intensite\_z\_t\_Ar\]). In both cases, the pulse splits into two sub-pulses around 1.05 m propagation, but the full model predicts an almost symmetrical temporal profiles pattern all along propagation, while the classical model yields a largely asymmetric one. This behavior illustrates the different temporal dynamics of higher-order Kerr terms as compared with the plasma generation. The former is an instantaneous phenomenon depending only on the intensity. In contrast, the plasma generated during the pulse accumulates, resulting in an ever growing contribution. As a consequence, the leading edge of the pulse propagates in a low plasma density while the trailing edge is more defocused by the much higher electron concentration it encounters. Moreover, the lower losses due to the lower plasma density in the full model allows a slight refocusing cycle around 1.15 m, which is not predicted by the classical model. The results of the full model stay unaffected when the plasma is not taken into account (*e.g* the peak intensity only increases by 0.6 %), which confirms that the filamentation process, including the pulse splitting is indeed driven by the higher order Kerr terms when they are considered. ![Space-time dynamics of filamentation in 1 bar Argon for the full model (a), and the classical model including $n_2$, ionization and GVD only (b). Both models yield pulse splitting around 1.05 m propagation distance, but the full model where filamentation is driven by the instantaneous Kerr effect results in a more symmetrical temporal dynamics.[]{data-label="intensite_z_t_Ar"}](Figure_6.eps){width="12cm"} These differences in time-space dynamics illustrate the interest of implementing all orders of the Kerr effect in numerical simulations of filamentation in gases. Since successive terms $n_{2*j}I^j$ of the Kerr index are of alternate signs and have comparable values at an intensity of about $30-35$ TW/cm$^{2}$ [@Loriot09], the inclusion of all terms up to $n_8$ in air (resp. $n_{10}$ in Argon) is necessary to adequately simulate the propagation of filamenting ultrashort pulses. The observation that ionization, as well as GVD, almost do not affect the results of the full model provides an opportunity to speed up the numerical simulations. Neglecting the ionization typically cuts the computation time by a factor of 3 with little impact on the result in the conditions shown above. A parametric study would be necessary to determine the conditions, and especially the wavelengths and materials where such approximation is legitimate. Such study shall compare the intensities yielding a dynamic balance of the Kerr terms on one side, and between Kerr and plasma contributions on the other side. In air, where these intensities amount to $31.6$ TW/cm$^2$ and $\sim78$ TW/cm$^2$, respectively, the lower intensity for pure Kerr balance ensures the domination of the latter process. Depending on the respective values of the higher-order non-linear indexes and ionization rates, the respective balance intensities may switch, leading to the domination of the Kerr-plasma balance. In conclusion, we have shown that the recently measured higher-order nonlinear indices of air (up to $n_8$) or Argon (up to $n_{10}$) dominate both the focusing and defocusing terms implied in the self-guiding of ultrashort laser pulses in these gases. As a consequence, contrary to previously held beliefs, a plasma is not required to generate filamentation in gases, and its generation is quite decoupled from the self-guiding process. Instead, filamentation is, at least in the considered conditions, governed by higher-order nonlinear indices. 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--- abstract: 'The BabyAI platform is designed to measure the sample efficiency of training an agent to follow grounded-language instructions. BabyAI 1.0 presents baseline results of an agent trained by deep imitation or reinforcement learning. BabyAI 1.1 improves the agent’s architecture in three minor ways. This increases reinforcement learning sample efficiency by up to $3 \times$ and improves imitation learning performance on the hardest level from $77 \%$ to $90.4 \%$. We hope that these improvements increase the computational efficiency of BabyAI experiments and help users design better agents.' author: - 'David Yu-Tung Hui' - 'Maxime Chevalier-Boisvert' - Dzmitry Bahdanau - Yoshua Bengio bibliography: - 'references.bib' title: 'BabyAI 1.1' --- Introduction ============ The BabyAI platform [^1] is an environment designed to evaluate how well an agent follows grounded-language instructions. The quality of an agent is measured with two metrics: its success rate at following instructions and the number of episodes or demonstrations required to train it. BabyAI 1.0, [@babyai_iclr19], presents results of a baseline agent trained by reinforcement and imitation learning (RL and IL) methods. In this technical report, we present three modifications that significantly improved the baseline results. Two modifications are to the network’s architecture and the third to the representation of the visual input. The network is modified by removing maxpooling at lower levels in the visual encoder and adding residual connections around FiLM layers [@perez_film:_2017]. The visual representation is modified to use learned embeddings in a Bag-of-Words fashion [@mikolov2013efficient]. Proposed Architectural and Representational Modifications {#sec:arch} ========================================================= This section describes the network architecture and BabyAI 1.0 visual representation before detailing the two architectural modifications and two alternate visual representations. The BabyAI platform has nineteen levels which can be categorised into two types: small and big [@babyai_iclr19]. Small levels are single-room but big levels are usually $3 \times 3$ rooms. The BabyAI 1.0 baseline agent has two architectures used on the small and big levels. These architectures have the same structure and are illustrated by Figure \[fig:models\].a. The architecture takes two inputs, a visual input and a linguistic instruction. We use FiLM to combine the outputs of a convolutional ‘visual encoder’ with a GRU [@cho_learning_2014] embedding of the instruction. We refer the reader to Appendix \[section:arch\] for more details concerning the distinction between ‘big’ and ‘small’. Figures \[fig:models\].b and \[fig:models\].c respectively present two architectural modifications: removing pooling in the visual encoder and adding residual connections around the image convolution and the FiLM layers. To ensure that the shape of the visual encoder is consistent after pooling is removed, we change filter size from $2\times2$ to $3\times3$. We expect these changes to improve sample efficiency because they enable more information to be transmitted to higher layers. At every timestep, the agent receives visual information about a $7 \times 7$ grid of tiles which are immediately in the direction it is facing. BabyAI 1.0 represents a tile by a triple-integer value. The first integer describes the type of object in the tile and the second integer the object’s color. The third integer is only used if the object is a door, and describes whether it is open, closed or locked. BabyAI 1.0 represents a visual input by concatenating all tile representations together. This results in a tensor of size $7 \times 7 \times 3$. A gridworld tile (and thus the visual input) can also be represented in two other ways: as a “bag of words” or by an RGB image. In the Bag-of-Words (“BOW”) approach a set of symbols that describes the tile is embedded in a trainable lookup table. This approach is commonly used in gridwords such as [@leike2017ai], [@rajendran2015attend] and [@SchraderSokoban2018]. Because a tile in BabyAI can be represented by three integers, we use three look-up tables and use each integer as a key. A tile is then represented by the mean of the three looked-up feature vectors. As with the BabyAI 1.0 visual representation, the BOW representation is formed from combining all tile representations into a 3D tensor. As we set the dimensionality of a feature vector to 128, the dimensionality of the BOW visual representation is $7 \times 7 \times 128$. This is depicted in Figure \[fig:models\].d. The contents of a tile can also be represented by a 3-channel Red-Green-Blue (RGB) image. As we choose the size of an image to be $8$ pixels, a grid is thus represented by an image stored in a $8 \times 8 \times 3$ tensor. The entire $7 \times 7$ visual input can be represented in a 3-channel RGB image with dimensionality $56 \times 56 \times 3$. An architecture using this visual representation is illustrated in Figure \[fig:models\]. Architectures names are structured in two parts. The first part is either “original”, “bow” or “pixels”, indicating which visual representation is used. The second part is optional and describes whether “\_endpool” (because the only source of pooling is at the end) or “\_res” (adding residual connections) are present in the architecture. Experiments =========== To determine the best architecture and visual representation, we follow BabyAI 1.0 and experiment on the six easiest BabyAI levels. These six levels consist of five single-room levels and one multi-room level. Then, we present IL performance benchmarks on all levels. Finding the Best Architecture ----------------------------- We measure RL sample and computational efficiency and IL performance with varying number of demonstrations. RL experiments were structured in two stages. The first set of experiments investigated architectural modifications. Results in Table \[tbl:arch\] showed that removing pooling in the visual encoder had a significant improvement on sample efficiency, but adding residual connections effected both increases and decreases. Nevertheless, we adopted residual connections for further experiments because the sample efficiency increase for PutNextLocal greatly outweighted the total decrease in GoToLocal and GoTo. The second set of experiments investigated visual representations. Results in Table \[tbl:visual\] still do not show a wide variation in sample efficiency. Because training from pixels was hard on the two most difficult levels (GoTo, PutNextLocal), we halved the learning rate ($\alpha$ in Adam [@kingma_adam:_2015] from $1 \times 10^{-4}$ to $5 \times 10^{-5}$) and reran the second set of experiments. The resulting statistics in Table \[tbl:visual2\] do not show much variation between the three visual representations. We now consider the computational efficiency of training each of these five architectures. Training from pixels has a slower throughput than the other visual representations (Table \[tbl:fps\]). Because of this and no clear advantage in RL sample efficiency (Tables \[tbl:visual\], \[tbl:visual2\]), we drop further experiments on pixels. Now, we investigate whether changing the visual representation to BOW and two architectural modifications improve IL performance. @babyai_iclr19 measures sample efficiency using an interpolated function fitted with a Gaussian Process (GP) [@rasmussen_gaussian_2005]. In our experiments we found that an infeasibly large number of training runs would be required in order to obtain a sufficiently confident sample efficiency estimate from the GP. Instead, we follow Table 6 in [@zolna2020combating], who evaluate IL by observing its success rate trained with varying number of demonstrations. @zolna2020combating use $1/64$^th^, $1/8$^th^ and all of 1 million demonstrations. We use 5, 10, 50, 100 and 500 thousand demonstrations, which correspond to $1/200$^th^, $1/100$^th^, $1/20$^th^, $1/10$^th^ and $1/2$^th^ of the total 1 million demonstrations. IL results in Table \[tbl:il\] show that training from BOW is advantageous to the original BabyAI 1.0 visual representation. Interestingly, we find that for hard levels with a few number of demos, the architectural modifications are not beneficial for training. This is offset by changing the visual representation to BOW. Benchmarking the Best Modifications ----------------------------------- Having constructed the BabyAI 1.1 agent, we benchmark its performance over all nineteen BabyAI levels. Table \[tbl:baseline\] shows that modifications found by the previous section yielded improvements in performance over all levels. Four more levels (Unlock, Putnext, Synth and SynthLoc) were solved, and success rate increased by $13.7 \%$ in the hardest level (BossLevel) from $77 \%$ to $90.4 \%$. Conclusion ========== As BabyAI was intended to be a lightweight experimental platform, BabyAI 1.0 used a specific hand-crafted representation rather than a more realistic pixel-based representation. We have shown that training from other visual representations (BOW and pixels) is feasible, and is sometimes more sample efficient (Table \[tbl:il\]). However, learning from pixels took longer to compute (Table \[tbl:fps\]) and was more sensitive to hyperparameters. Besides, using pixel representations for the tiles still does not bridge the reality gap between the gridworld and the 3D real world, for almost all challenging aspects of visual perception, such as e.g. occlusion, illumination, different viewpoints are still not modelled. For this reason we keep the BabyAI input representation symbolic, though we switch to the more standard BOW approach for encoding the symbolic input. This report introduces BabyAI 1.1, the latest version of the BabyAI platform. This modifies the previous version of BabyAI with minor changes to the baseline agent, but major improvements to baseline statistics. We hope that this change encourages researchers to (re-)use similar architectures within novel agents, so that research into grounded language learning may be conducted in more computationally efficient ways. Acknowledgements {#acknowledgements .unnumbered} ---------------- This research was mostly performed at Mila with funding by the Government of Quebec and CIFAR, and enabled by Compute Canada ([www.computecanada.ca](www.computecanada.ca)). -- ------------------------------------------------------------------------------------------------------------------- accept null hypothesis that there is *no significant difference* reject null hypothesis at 1 % significance due to *significant increase in sample efficiency* (number is smaller) reject null hypothesis at 1 % significance due to *significant decrease in sample efficiency* (number is bigger) -- ------------------------------------------------------------------------------------------------------------------- : RL sample efficiency for different visual representations with *learning rate = $5 \times 10^{-5}$*. See the last paragraph of Section \[sec:arch\] for the explanation of architecture names.[]{data-label="tbl:visual2"} ----------------- ---------------- ------------------- ------------------------ Level (lr)[2-4]{} original original\_endpool original\_endpool\_res GoToRedBallGrey 21 $\pm$ 5 21 $\pm$ 6 21 $\pm$ 5 GoToRedBall 273 $\pm$ 27 200 $\pm$ 16 179 $\pm$ 17 GoToLocal 1311 $\pm$ 251 381 $\pm$ 30 437 $\pm$ 45 PickupLoc 1797 $\pm$ 290 743 $\pm$ 132 710 $\pm$ 166 PutNextLocal 2984 $\pm$ 172 2169 $\pm$ 739 1009 $\pm$ 128 GoTo 1601 $\pm$ 463 454 $\pm$ 69 813 $\pm$ 278 ----------------- ---------------- ------------------- ------------------------ : RL sample efficiency for different visual representations with *learning rate = $5 \times 10^{-5}$*. See the last paragraph of Section \[sec:arch\] for the explanation of architecture names.[]{data-label="tbl:visual2"} ----------------- ------------------------ ------------------- ---------------------- Level (lr)[2-4]{} original\_endpool\_res bow\_endpool\_res pixels\_endpool\_res GoToRedBallGrey 21 $\pm$ 5 24 $\pm$ 2 34 $\pm$ 4 GoToRedBall 179 $\pm$ 17 177 $\pm$ 2 172 $\pm$ 2 GoToLocal 437 $\pm$ 45 611 $\pm$ 760 242 $\pm$ 15 PickupLoc 710 $\pm$ 166 982 $\pm$ 266 1082 $\pm$ 385 PutNextLocal 1092 $\pm$ 143 876 $\pm$ 104 Not Trainable GoTo 813 $\pm$ 278 817 $\pm$ 502 Not Trainable ----------------- ------------------------ ------------------- ---------------------- : RL sample efficiency for different visual representations with *learning rate = $5 \times 10^{-5}$*. See the last paragraph of Section \[sec:arch\] for the explanation of architecture names.[]{data-label="tbl:visual2"} ----------------- ------------------------ ------------------- ---------------------- Level (lr)[2-4]{} original\_endpool\_res bow\_endpool\_res pixels\_endpool\_res GoToRedBallGrey 35 $\pm$ 5 30 $\pm$ 4 44 $\pm$ 11 GoToRedBall 263 $\pm$ 22 164 $\pm$ 3 155 $\pm$ 5 GoToLocal 606 $\pm$ 81 449 $\pm$ 176 336 $\pm$ 28 PickupLoc 1732 $\pm$ 579 1461 $\pm$ 422 1308 $\pm$ 421 PutNextLocal 1277 $\pm$ 252 876 $\pm$ 104 1301 $\pm$ 320 GoTo 984 $\pm$ 484 803 $\pm$ 525 845 $\pm$ 329 ----------------- ------------------------ ------------------- ---------------------- : RL sample efficiency for different visual representations with *learning rate = $5 \times 10^{-5}$*. See the last paragraph of Section \[sec:arch\] for the explanation of architecture names.[]{data-label="tbl:visual2"} Architecture original original\_endpool original\_endpool\_res bow\_endpool\_res pixels\_endpool\_res -------------- ---------------- ------------------- ------------------------ ------------------- ---------------------- RL (FPS) 1139 $\pm$ 128 927 $\pm$ 72 907 $\pm$ 69 855 $\pm$ 58 540 $\pm$ 67 : Frames Per Second (mean $\pm$ std) of RL training with different architectures, averaged across the six easiest BabyAI levels. Inter-level differences are negligible.[]{data-label="tbl:fps"} -- ------------------------------------------------------------------------------------------------------------- accept null hypothesis that there is *no significant difference* reject null hypothesis at 1 % significance due to *significant increase in performance* (number is bigger) reject null hypothesis at 1 % significance due to *significant decrease in performance* (number is smaller) -- ------------------------------------------------------------------------------------------------------------- : A cell is shaded depending on whether its values are statistically significant from those of the cell *to its left*. Statistical significance is computed using a two-tailed T-test with inequal variance. ----------------- ----------------------------- ------------------------ ------------------------ ------------------------ Level Number of Demos (thousands) (lr)[3-5]{} original original\_endpool\_res bow\_endpool\_res GoToRedBallGrey 5 [**99.5 $\pm$ 0.1**]{} [**99.5 $\pm$ 0.1**]{} 10 [**99.7**]{} [**99.8 $\pm$ 0.1**]{} 50 [**100**]{} [**100**]{} 100 [**100**]{} 500 [**100**]{} [**100**]{} [**100**]{} GoToRedBall 5 89.6 $\pm$ 0.3 91.3 $\pm$ 0.6 10 93.1 $\pm$ 0.8 95.6 $\pm$ 0.7 50 [**99.2 $\pm$ 0.2**]{} 100 [**99.7**]{} [**100**]{} 500 [**99.9**]{} [**100**]{} GoToLocal 5 72.5 $\pm$ 1.0 71.6 $\pm$ 1.4 84.2 $\pm$ 2.0 10 79.9 $\pm$ 1.2 79.7 $\pm$ 1.8 94.2 $\pm$ 0.8 50 95.3 $\pm$ 0.5 100 [**97.8 $\pm$ 0.3**]{} [**99.9**]{} 500 [**99.6 $\pm$ 0.1**]{} [**100**]{} PutNextLocal 5 22.3 $\pm$ 1.7 12.0 $\pm$ 1.8 12.5 $\pm$ 1.2 10 39.1 $\pm$ 3.5 16.2 $\pm$ 2.3 24.9 $\pm$ 3.2 50 80.8 $\pm$ 1.4 90.6 $\pm$ 3.5 88.6 $\pm$ 11.0 100 93.9 $\pm$ 0.4 [**99.5 $\pm$ 0.5**]{} 500 [**99.3 $\pm$ 0.2**]{} PickupLoc 5 53.0 $\pm$ 1.3 35.9 $\pm$ 1.5 60.3 $\pm$ 1.8 10 65.3 $\pm$ 1.5 53.7 $\pm$ 1.2 74.9 $\pm$ 3.9 50 90.8 $\pm$ 1.5 96.2 $\pm$ 0.5 97.0 $\pm$ 0.3 100 96.4 $\pm$ 0.5 [**98.6 $\pm$ 0.3**]{} 500 [**99.5 $\pm$ 0.2**]{} [**99.8 $\pm$ 0.1**]{} GoTo 10 70.4 $\pm$ 1.1 76.3 $\pm$ 5.0 96.1 $\pm$ 0.4 100 94.9 $\pm$ 0.3 [**99.4**]{} ----------------- ----------------------------- ------------------------ ------------------------ ------------------------ : A cell is shaded depending on whether its values are statistically significant from those of the cell *to its left*. Statistical significance is computed using a two-tailed T-test with inequal variance. ----------------- -------------- ------------------------------ ----------------- Level Demo Length (Mean $\pm$ Std) (lr)[2-3]{} BabyAI 1.0 BabyAI 1.1 GoToObj [**100**]{} [**100**]{} 5.18 $\pm$ 2.38 GoToRedBallGrey [**100**]{} [**100**]{} 5.81 $\pm$ 3.29 GoToRedBall [**100**]{} [**100**]{} 5.38 $\pm$ 3.13 GoToLocal [**99.8**]{} [**100**]{} 5.04 $\pm$ 2.76 PutNextLocal [**99.2**]{} [**100**]{} 12.4 $\pm$ 4.54 PickupLoc [**99.4**]{} [**100**]{} 6.13 $\pm$ 2.97 GoToObjMaze [**99.9**]{} [**100**]{} 70.8 $\pm$ 48.9 GoTo [**99.4**]{} [**100**]{} 56.8 $\pm$ 46.7 Pickup [**99**]{} [**100**]{} 57.8 $\pm$ 46.7 UnblockPickup [**99**]{} [**100**]{} 57.2 $\pm$ 50 Open [**100**]{} [**100**]{} 31.5 $\pm$ 30.5 Unlock 98.4 [**100**]{} 81.6 $\pm$ 61.1 PutNext 98.8 [**99.6**]{} 89.9 $\pm$ 49.6 Synth 97.3 [**100**]{} 50.4 $\pm$ 49.3 SynthLoc 97.9 [**100**]{} 47.9 $\pm$ 47.9 GoToSeq 95.4 96.7 72.7 $\pm$ 52.2 SynthSeq 87.7 93.9 81.8 $\pm$ 61.3 GoToImpUnlock 87.2 84.0 110 $\pm$ 81.9 BossLevel 77 90.4 84.3 $\pm$ 64.5 ----------------- -------------- ------------------------------ ----------------- : Comparision of baseline IL results for all BabyAI levels. Experiments with a success rate $\geqslant 99 \%$ are successful and are **bolded**. On all levels, the ‘big’ configuration was trained on 1 million demonstrations until the loss has converged. As running these experiments are computationally expensive, we present results on 1 seed. Success rate is calculated as with 512 trials once the loss has converged.[]{data-label="tbl:baseline"} Agent Architecture {#section:arch} ================== At every timestep, an agent receives a visual input and linguistic command of variable length which compels the agent to execute an action. The BabyAI baseline agent is implemented by a deep neural network which processes the visual input and linguistic command, producing an action. The visual input and linguistic instruction are respectively encoded by a Gated Recurrent Unit (GRU) and convolutional network. These two encodings are then combined by two batch-normalised FiLM layers. A Long-Short-Term-Memory cell (LSTM) [@hochreiter_long_1997] integrates the output of FiLM across timesteps. Finally, the integrated output is passed to policy and value heads. The agent can be trained by RL or IL methods in conjunction with BackPropagation Through Time (BPTT) [@werbos_bptt]. The ‘small’ configuration uses a unidirectional GRU and LSTM of dimensionality 128 for memory. The ‘big’ configuration uses a 128-dimensional bidirectional GRU with attention [@bahdanau_neural_2015] and the memory LSTM with dimensionality 2048. Reinforcement Learning Experiments {#section:rl_exps} ================================== Sample efficiency is defined by the number of RL training episodes needed to train an agent to $\geqslant 99 \%$ success rate. @babyai_iclr19 defines success as whether an agent can follow an instruction within $n_{max}$ steps, a figure pre-defined for each level. We use Advantage-Actor Critic (A2C) [@wu_a2c] with Proximal Policy Optimisation (PPO) [@schulman_proximal_2017] and Generalised Advantage Estimation (GAE) [@schulman_high-dimensional_2015]. Data for A2C is collected in batches of 64 rollouts of length 40. These were used in 4 epochs of PPO. $\lambda=0.99$ was used in GAE. If an agent completes a task after $n$ steps, it is rewarded with $1 - 0.9 n / n_{max}$. Otherwise, no reward is given. The returns were discounted by $\gamma=0.99$. Results in Table \[tbl:arch\] and Table \[tbl:visual\] was optimised by Adam with the hyperparameters $\alpha=10^{-4}$, $\beta_1=0.9$, $\beta_2=0.999$ and $\epsilon=10^{-5}$. Results in Table \[tbl:visual2\] used $\alpha=5 \times 10^{-5}$. Imitation Learning Experiments {#section:il_exps} ============================== Different hyperparameters were used for training ‘small’ and ‘big’ models. The small model was trained with a batch size of 256 and an epoch consisting of 25600 demos. Backpropagation Through Time (BPTT) was truncated at 20 steps. The large model had a batch size of 128 and an epoch of 102400 demonstrations. BPTT was truncated at 80 steps. In addition, the model was trained with an entropy regulariser, which had a coefficient of $0.01$. These were optimised by Adam with $\alpha=10^{-4}$ for small architectures, $\alpha=5 \times 10^{-5}$ for large architectures and $\beta_1=0.9$, $\beta_2=0.999$ and $\epsilon=10^{-5}$. [^1]: (<https://github.com/mila-iqia/babyai>

--- abstract: 'A search for RR Lyrae stars has been conducted in the publicly available data of the Northern Sky Variability Survey ([*NSVS*]{}). Candidates have been selected by the statistical properties of their variation; the standard deviation, skewness and kurtosis with appropriate limits determined from a sample 314 known RRab and RRc stars listed in the GCVS. From the period analysis and light curve shape of over 3000 candidates 785 RR Lyrae have been identified of which 188 are previously unknown. The light curves were examined for the Blazhko effect and several new stars showing this were found. Six double-mode RR Lyrae stars were also found of which two are new discoveries. Some previously known variables have been reclassified as RR Lyrae stars and similarly some RR Lyrae stars have been found to be other types of variable, or not variable at all.' author: - | Patrick Wils$^{1}$, Christopher Lloyd$^{2}$, Klaus Bernhard$^{3,4}$\ $^{1}$Vereniging voor Sterrenkunde, Belgium, email: [email protected]\ $^{2}$Space Science & Technology Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon. OX11 0QX, UK, email: [email protected]\ $^{3}$A-4030 Linz, Austria, email: [email protected]\ $^{4}$Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV), Munsterdamm 90, D-12169 Berlin, Germany date: 'Accepted 2006 February 23. Received 2006 February 21; in original form 2006 January 16' title: A Catalogue of RR Lyrae Stars from the Northern Sky Variability Survey --- \[firstpage\] stars: variables: other - stars: Population II Introduction ============ Statistical studies of the relative numbers of the different classes of RR Lyrae variable stars and especially the incidence rate of multi-periodicity may give indications of the metallicity of different stellar systems and of their evolution [see e.g. @mos]. Exhaustive studies have been done in the Magellanic Clouds by the [*MACHO*]{} collaboration [@macho and 2003] and the [*OGLE*]{} survey [@oglelmc]. Other studies have searched for RR Lyrae stars in the Galaxy, such as [*QUEST*]{} [@quest] and also [*OGLE*]{} [@collinge]. Most of the stars found in these studies are faint, and limited to a small region of sky (the galactic bulge and the equator for [*OGLE*]{} and [*QUEST*]{} respectively). The Robotic Optical Transient Search Experiment [[*ROTSE-1*]{}, @rotse] found a fairly large number of previously unknown bright (magnitude $< 15$) RR stars in a part of the sky. This paper sets out to extend this search for field galactic RR Lyrae stars to the whole northern sky in the [*ROTSE-1*]{} data, made publicly available via the Internet [Northern Sky Variability Survey - [*NSVS*]{}, @skydot]. Methodology =========== With only photometric data, and no spectral information, the type of a short period variable can only be determined once the (phased) light curve is known, and hence only once the period is known. However, determining the period from sparse data is a computationally demanding process. Therefore it was decided to limit the number of objects by statistical parameters involving much less computation: standard deviation, skewness, kurtosis and the mean square of successive differences of the magnitude data. By looking at the values for these statistics for the RR Lyrae stars listed in the General Catalogue of Variable Stars [GCVS, @gcvs and its online edition], limiting conditions were then derived. The aim was to set these limits as strict as possible, so that not too many objects needed to be checked, but still to include the majority of RR Lyrae stars. Those GCVS stars that did not follow the criteria, can then provide an estimate for the completeness of the survey. Data ---- The [*ROTSE-1*]{} was an unfiltered CCD survey of the sky from the north pole to declination $\sim -38$, reaching magnitude $\sim 15$ with varying levels of completeness. The survey lasted nominally for one year but depending on the circumstances coverage of individual objects may be significantly less than this. Objects typically have 100 to 500 measurements with a median photometric accuracy of 0.02 mag for 10th magnitude stars and a positional accuracy of 2“. The spatial resolution of 14” compromises the photometry in crowded fields, typically with $|b| < 20$, but also at higher galactic latitudes for stars with companions within $\sim 45"$. The data are publicly available from the Sky Database for Objects in Time-Domain (SkyDOT) web site [@skydot] and it is possible to select data with respect to 8 extraction flags and 7 photometric correction flags. The default selection for good data sets all but one, [PATCH]{}, of the photometric correction flags and only one, [SATURATED]{}, of the extraction flags. However, experience of working with the data has shown that observations with the extraction flag [APINCOMPL]{} set are often completely out of range and should be rejected. On the other hand data with the photometric correction flag [RADECFLIP]{} set are often indistinguishable from the other data. So the data have been selected with the [SATURATED]{} and [APINCOMPL]{} flags set and the [PATCH]{} and [RADECFLIP]{} flags unset. It was also decided that only stars with 100 good [*NSVS*]{} observations or more were to be considered, in order to get good statistics and reliable period determinations, and to possibly detect multiperiodicity (double-mode pulsation or a Blazhko effect). With fewer data points, statistics may be influenced substantially by erroneous observations, such as those introduced by e.g. a close companion. Also, it is then not always possible to derive the correct period: in view of the sampling frequency, and the rather short total time span of the available data (less than a year) alias frequencies will be more important. RR Lyrae stars are fairly blue stars (spectral types A and F). The [*NSVS*]{} survey however observed only in one colour (unfiltered CCD), so colour information has to be retrieved from another source, such as [*2MASS*]{} [@2mass]. The [*NSVS*]{} positions are not very accurate however, and matching them to [*2MASS*]{} coordinates may be troublesome in crowded fields. In view of this and of reddening aspects, it was decided not to use colour information as a filter. Control group: GCVS stars ------------------------- The GCVS stars that were to be considered for the control group, had to be well-known and have an accurate position. Therefore only the GCVS types RRab or RRc were taken (no RR, RR:, RRab: or RRc: stars, i.e. the GCVS classification should be precise enough to give the exact subtype). Because of the limited number of RRd stars in the GCVS (the RR(B) class), these were not taken into account either. For practical purposes, as far as their statistical parameters are concerned, these double-mode stars can be considered to be RRc stars. The known RR Lyrae stars in the GCVS were further limited to the constellations And to Ori for which precise positions had been determined by the GCVS team at the time this study started. Many stars in other constellations did not have accurate enough coordinates, which could lead to misidentifications with the [*NSVS*]{} stars. It is however still possible that a faint RR Lyrae star unobservable by the [*ROTSE*]{} camera, lies close to a brighter (constant) companion, leading to a false identification. Because of this, the success rates for finding an RR star, may be underestimated. On the other hand, especially at fainter magnitudes, some stars which should have been detectable, will not have been registered, thus overestimating the success rate. With the above restrictions imposed, 582 [*NSVS*]{} objects were identifed as GCVS RR Lyrae stars by their [HTM]{} identification [see @skydot]. [*NSVS*]{} synonyms, the same object observed in overlapping [*NSVS*]{} fields, have been counted separately here. This will be done also for the remainder of this section, as it will not change the statistics very much. The further restriction that there needed to be at least 100 good data points limited the sample to 314 objects (273 RRab and 41 RRc), or 54% of the total number of stars identified. Compare this to the overall 42% of objects with at least 100 good points (8393519 out of a total of 19995106 [*NSVS*]{} objects). 60% of the GCVS stars that are on average brighter than magnitude 14, have more than 100 observations, and 68% if only objects North of the equator are counted. Skewness -------- RRab stars have a typical light curve, spending more time near minimum than near maximum, while RRc stars have more symmetric light curves. It distinguishes them from eclipsing binaries, by far the most common type of variable star found in the [*NSVS*]{} database. As a result, the distribution of magnitudes of an RRab star shows a negative skewness, while those of an RRc star shows a skewness near zero (note that a perfect sine curve has a skewness equal to 0). The distribution of the skewness values for the selected GCVS stars is given in Fig. \[skew\]. ![Distribution of the skewness values of the [*NSVS*]{} data for RRab and RRc stars from the GCVS.[]{data-label="skew"}](RRNSVSfig1.eps){width="84mm"} The maximum value for the skewness statistic was set to 0.5, which selects 303 out of 314 stars (96%). Most of the GCVS objects with higher skewness values did not satisfy the standard deviation criterion either. Therefore the chosen restriction seems to be an appropriate one. The objects not selected include HM Aql (probably a constant star, see below) and UU Cam (often referred to as an EW type variable) and a few objects near the magnitude limit resulting in a disproportionally high number of bright observations, and hence a larger skewness. One star had two very faint data points, outside its normal range, most probably data errors, also influencing skewness to rise abnormally. Some of the objects selected with positive skewness are eclipsing variables of the W UMa type (EW). Especially for skewness values $> 0.5$, most of the variables are EW stars or other eclipsing binaries. Some of these have been reported by [@ecl]. In a number of cases it is impossible to distinguish between EW and RRc on the basis of the light curve alone. These stars have not been withheld. Mean square of successive differences ------------------------------------- The mean square of successive differences ([MSSD]{}) gives an indication for the timescale of the variation of a star as compared to the sampling timescale. The statistic considered further in this paper is in fact the rescaled value $$\theta = 1-{\sc MSSD}/2\sigma^2,$$ with $\sigma$ the standard deviation. For a purely random distribution of the data $\theta = 0$ [@cuypers], and for a star that only brightens or fades during the total time interval, $\theta$ will approach 1. RR Lyrae stars are rapidly changing stars (their periods range between about 0.2 and 1 day), compared to the frequency of observation by [*ROTSE*]{} (up to 5 points per night). Therefore a raw light curve will show almost random variation. The criterion $\theta < 0.65$ was chosen. 307 (98%) of the selected GCVS stars satisfy this criterion. Because of the [*ROTSE*]{} observing regime, in some cases it is possible that a star with a period close to an integer fraction of a day, will show a raw light curve that resembles one of a much longer period star. This extreme aliasing is the case for RU Boo (period 0.4927d), V1949 Cyg (0.4989d), AW Lyr (0.4975d), BT Leo (0.4997d) and FY Aqr. For the latter, a period has not been given in the GCVS, but from [*ASAS3*]{} [@asasI] and [*NSVS*]{} data a period of 1.0229 days may be derived; see also [@dom04] for another period determination of this star. Kurtosis -------- Kurtosis is a measure of the “peakedness” of a distribution and it was found to be useful here as a discriminator against stars with extreme values. A maximum value of 4.5 was set for the kurtosis value of the magnitudes (a perfect sine has kurtosis = 1.5). It excludes stars which have some unusually faint or bright data points. This criterion effectively removes stars with bad data points making the standard deviation erroneously large. It may however hide true variation in some particular cases, as some stars with highly deviating points which are really variable are excluded as well. This did not affect the majority of the RR Lyrae stars as the criterion selects 301 (96%) of the GCVS stars, All of the objects that failed this test also violated other criteria as well. Overall with all criteria so far applied (without limits on the standard deviation) 287 (255 RRab and 32 RRc) out of 314 (91%) are left. Standard deviation ------------------ Being the most important parameter for the recognition of a variable star, a cut is difficult to define for the standard deviation, as it strongly depends on the average magnitude of the stars, especially for fainter objects. From a Fourier fit to the observations, “theoretical” standard deviations (i.e. without observational errors) for the selected RR Lyrae stars were found to be always larger than 0.1 mag, unless the star was in a close pair which could not be resolved by the instrument. Most RRab stars have “theoretical” standard deviations between 0.15 and 0.30 mag, RRc variables between 0.10 and 0.20 mag (note that in the case of a perfect sine curve, the standard deviation $\sigma = \Delta m/2\sqrt2$, with $\Delta m$ the total amplitude from minimum to maximum). A survey for Cepheids in Milky Way fields of the [*NSVS*]{} [@cep] has shown that the standard deviation needed to be at least about twice that of the average standard deviation at the star’s magnitude, to be certain the star is a genuine variable star, and not one with unusually high observational scatter. Therefore this restriction was chosen. Without imposing it, the number of stars to be checked grows exponentially with decreasing standard deviation. True variables exist which have a lower standard deviation, as shown by some of the GCVS stars, but these cases are rather rare and/or hard to confirm, as true variation gets masked by observational scatter. In addition, inaccuracies on the other statistics calculated, increase as well. The cutoff for the standard deviation $\sigma$ is graphically illustrated in Fig. \[stdev\]. It plots $\sigma$ for the 314 GCVS RR Lyrae stars against their average magnitude. The thin full lines represent the average $\sigma$ (lower line) and the chosen lower limit $2\sigma$ (upper line) at the given magnitude for all stars with more than 100 data points in the [*NSVS*]{} database. 0.1% of the stars have a value of $\sigma$ above the upper dashed line, and 1% above the lower dashed line. Almost all the RR Lyrae stars belong to the latter group. It is probably worth looking at the brightest stars with low standard deviation. HM Aql [@harwood] looks constant in [*NSVS*]{} data as well as in [*ASAS3*]{} data [@asasV]. Also V1510 Cyg and HU Cam appear to be constant in [*NSVS*]{} data. There might be an identification problem for these stars or a bright close companion. ![[*NSVS*]{} standard deviation compared to magnitude for the RRab (open circles) and RRc stars (filled circles) selected from the GCVS. The lower and upper dashed lines represent respectively the 99 and 99.9 percentiles of the [*NSVS*]{} standard deviations $\sigma$ as a function of magnitude. The bottom thin full line gives the average standard deviation for a given magnitude ($1 \sigma$ level). The upper thin full line is the $2 \sigma$ level and it represents the chosen cutoff limit. The total number of stars with at least 100 data points, is given per 0.1 mag bin by the thick line (right axis).[]{data-label="stdev"}](RRNSVSfig2.eps){width="84mm"} Detection probability --------------------- Only 205 RRab and 16 RRc stars out of the total of 314 GCVS objects are left after applying the criterion on the standard deviation. So the overall detection possibility for RRab stars is 75%, for RRc stars it is only 40%. However, for stars on average brighter than mag 13, the success rate is 91 and 83% respectively. For stars brighter than magnitude 14, these rates are reduced to 83 and 57%. If these numbers are then applied to the requirement that the stars needed to be observed at least 100 times, it follows that 56% of the RRab stars on average brighter than magnitude 14 in the northern hemisphere are detectable with the assumed criteria and nearly 40% of the RRc stars. These detection probabilities will be compared with the results from the [*ASAS3*]{} survey. Because of the different detection possibilities, it is hard to accurately determine the ratio between the number of RRab and RRc stars. Results ======= More than 3000 objects satisfy all the criteria. These are not all RR Lyrae stars however, some are not even genuine variables. The final selection was based on the period and the shape of the phased light curve. Periods were determined using the PDM technique [@pdm]. For RRab stars the default setting $N_b=5$ gives only approximate values, because the rise to maximum is so steep both minimum and maximum observations may appear in the same bin, with an excessively large value for the PDM statistic as a consequence. Therefore $N_b=10$ was used. The RR Lyrae subtype was determined by inspection of the phased light curve: stars for which the ascending branch takes less than 30% of the period were considered to be of type RRab (in most cases it is much less), others of type RRc. Stars with a symmetric light curve, which could also be contact binaries, were rejected from the final catalogue. All in all 785 RR Lyrae stars were detected, for which the following details are given in the main catalogue (Table \[maintable\]; the complete table is available electronically only): [*NSVS*]{} number, coordinates in degrees, magnitude range (brightest and faintest observed magnitude), type, period in days, epoch of maximum light (HJD - 2450000) and a cross identification if one exists. References for some of the abbreviations used in the cross identifications are given below the table. [*NSVS*]{} synonyms have not been included in this table. 714 of these stars are of type RRab (of which 469 are brighter than magnitude 14 on average), 65 of type RRc (all brighter than magnitude 14; not surprisingly since the lower limit for the standard deviation at magnitude 14 is larger than the average standard deviation for RRc stars) and 6 are of type RRd. Of the 785 stars, 188 are previously unknown. Many others have only been suspected of variability before or had been wrongly classified. For others, it is the first time a period has been given. The catalogue contains 342 RR Lyrae stars already known in the GCVS and 23 stars with an incorrect type in the GCVS. However it does not contain 178 GCVS RR Lyrae stars with 50 or more good data points that could be confirmed to be RR Lyrae stars from the [*NSVS*]{} data, but that did not satisfy all of the selection criteria given above. For completeness these variables are given in Table \[missed\] (available in full electronically only), with the same layout as Table \[maintable\]. Blazhko stars ------------- Some RR Lyrae stars show a cyclic modulation of the amplitude and shape of their light curves known as the Blazhko effect [@blazhko]. The effect manifests itself as one or two additional frequencies in the periodogram, very close to the main frequency. The total timespan of available [*NSVS*]{} data is only about 9 months, which is short to accurately determine Blazhko periods. However, it can provide reasonable indications, as a number of the known Blazhko stars were easily picked up, and gave Blazhko periods in good agreement with the values from the literature. All stars from Tables \[maintable\] and \[missed\] were checked for a Blazhko effect. Objects for which no effect was found include SW And, RS Boo (known Blazhko period $>500$ days, so it is not surprising its effect was not noticed), SW Boo, XZ and DM Cyg, XZ Dra, RU Psc, and RR Lyr itself (in the latter case too few data points are available). It has been impossible to detect a Blazhko effect in stars fainter than about magnitude 13.5. The results are summarized in Table \[blazhko\]. Stars for which at least one linear combination of the main frequency and the additional frequency have been found, can be positively identified as Blazhko stars. Stars for which no such combination has been detected are indicated with a “:” after the Blazhko period, and need further confirmation. The stars which are known to be Blazhko stars in the GCVS are marked with an asterisk after their name. Except for RZ Lyr and AR Ser, the Blazhko periods given are not much different from those derived here. For RZ Lyr the Blazhko period given in the GCVS is 59 days. For AR Ser, a variable Blazhko period between 80 and 120 days has been given. [@lee1] give a Blazhko period of 122 days for FM Per and 57.5 days for DR And [@lee2]. [@lee0] already noted the Blazhko effect in V421 Her. Finally, the Blazhko effect of OV And has been studied in more detail by [@ovand], but the data were inconclusive. Double mode stars ----------------- Galactic field double-mode RR Lyrae stars (RRd) are very rare with less than twenty currently known. These are low-amplitude variables very similar to RRc stars but showing two periods, the fundamental $P_0$ and first overtone, $P_1$, with a ratio $P_1/P_0 \approx 0.744$. In most cases, the first overtone mode has the highest amplitude. Six RRd stars have been detected. The double-mode nature of V372 Ser [@enrique] was already known before. The true nature of GSC 3047-0176 [@koppelman], GSC 3059-0636 [@smith] and GSC 4868-0831 [@rrdasas] was recently found in other studies of the [*NSVS*]{} or [*ASAS3*]{} data. The two remainig RRd stars are identified here for the first time. Table \[rrd\] gives the fundamental period $P_0$ and the first overtone period $P_1$, as well as the period ratio for the six double mode stars. Misclassified GCVS stars {#misclass} ------------------------ Ten stars classified as RR Lyrae variables in the GCVS, turned out to be of another type. V1180 Aql is a close double [@hoffmeister], one star of the pair is a Mira variable, not an RR. AU and V556 Cas, V811 Oph, V421 Per and BQ Pup are Cepheids and V1823 Cyg and IT Her are probably EW-type stars. V1069 Sgr may be constant or its position may be in error. Also BU UMa is most likely a constant star (P. Van Cauteren, private communication). 23 genuine RR Lyrae stars (included in Table \[maintable\]) are also misclassified in the GCVS. These include DP Aqr, AL and TZ Cap, V939 Cyg, SX and FM Del, AQ and KO Dra, V534 Her and V1429 Oph. Others like V344 Ser and KQ UMa have been identified earlier also by [@khruslov]. Period corrections ------------------ Some stars in the GCVS are correctly classified but have a wrong period, mostly an alias of the correct period. These stars include GT Aqr [@diethelm], X CMi, RW Equ, DG Hya, V418 Her and V784 Oph. Estimates of the number of RR Lyrae stars ----------------------------------------- From the analysis and the detection possibilities discussed above, it is found that there should be about 650 RRab stars in the northern hemisphere brighter than magnitude 14 (of which 365 have been detected in this study) and about 140 RRc stars (56 have been detected). One could state that about four out of five galactic field RR Lyrae stars are of type RRab. However, because of selection effects which favour RRab stars, the exact fraction of the different subtypes are uncertain. Only about 2% of the RR Lyrae stars are double-mode stars. But because of the low number of RRd stars, this frequency is even more uncertain. These estimates can be compared to those obtained from a comparison of the current catalogue with the results of the All Sky Automated Survey [[*ASAS3*]{}, @asasV]. That survey is based in the southern sky and detected variables south of declination $\sim +28$. In the overlapping region from declination $-15$ to $+15$, [*ASAS3*]{} detected 323 objects unambiguously identified as RRab stars on average brighter than magnitude 14 (assuming that the average magnitude of an RRab star equals the maximum magnitude plus 0.65 times the full amplitude), and 73 RRab stars fainter than magnitude 14. Of these respectively 150 and 28 were detected in the [*NSVS*]{} data, giving detection probabilities of 46 and 38%. These probabilities do not change when only the region from declination $-10$ to $+10$ is considered. However if the declination zone between $0$ and $+15$ is considered only, the detection probability for RRab stars brighter than magnitude 14 rises to 53% (89 out of 168 stars were identified), very close to the 56% found from the GCVS sample. It is clear that the efficiency of the search deteriorates fast for negative declinations, probably for a large part due to the requirement of at least 100 data points. The estimate for the RRc stars differs more. [*ASAS3*]{} identified 83 stars unambiguously as RRc stars in the considered overlapping region (all brighter than magnitude 14), of which only 10 were found in the [*NSVS*]{} data. This results in a detection probability of only 12% (14% if the region from declination $-10$ to $+10$ is considered, and 19% for stars in the declination zone between $0$ and $+15$). The large difference with the estimate from the GCVS sample can probably be explained by the small number of RRc stars found overall, so that the chosen samples are not really representative for the total population. Again, it can be concluded that the estimate for the number of RRc stars is not very reliable, and may in reality be considerably higher. Figures for the southern sky can be obtained from the reverse comparison. Of the 124 RRab stars brighter than magnitude 14 between declination $-15$ to $+15$ found in this study, 114 were identified as well by [*ASAS3*]{} (including four that have an ambiguous variability type, but not including five stars with a different type in [*ASAS3*]{}). This results in a detection probability of 92% for [*ASAS3*]{}. As the survey unambiguously identified 817 objects south of the equator and brighter than magnitude 14 as RRab stars (1015 if ambiguous types are considered as well), there will be about 890 in total (or 1100 when all the ambiguously typed stars are taken into account). It may be concluded that there is an over-abundance of RRab stars in the southern sky compared to the northern sky. For the RRc stars the figures are as follows: of the 16 RRc stars found in this study in the overlapping region between declinations $-15$ and $+15$, 12 or 75% were identified by [*ASAS3*]{} as RR Lyrae stars as well. With 259 stars brighter than magnitude 14 unambiguously identified as RRc stars, there may be about 345 in total south of the equator. This may be considered as a lower limit as there are 1047 possible RRc stars given by [*ASAS3*]{}. However a large part of those are probably not genuine RRc stars. Conclusion ========== This paper has enlarged the known number of galactic field RR Lyrae stars in the northern sky. The number of known stars showing multiperiodicity (Blazhko effect and double-mode pulsation) has been enlarged as well. The number of RR Lyrae stars brighter than magnitude 14 has been estimated, showing that there is an over-abundance of RRab stars in the southern hemisphere. Acknowledgments {#acknowledgments .unnumbered} =============== The authors thank John Greaves, Sebastián Otero, Gisela Maintz and Christoph Kaser for helpful suggestions. Doug Welch is acknowledged for granting access to the copy of the [*NSVS*]{} database at McMaster University. This publication makes use of the data from the Northern Sky Variability Survey created jointly by the Los Alamos National Laboratory and University of Michigan. The [*NSVS*]{} was funded by the US Department of Energy, the National Aeronautics and Space Administration and the National Science Foundation. 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Pojmanski G., Maciejewski G., 2004, Acta Astron., 54, 153 Pojmanski G., Maciejewski G., 2005, Acta Astron., 55, 97 Pojmanski G., Pilecki B., Szczygiel D., 2005, Acta Astron., 55, 275 Soszynski I., Udalski A., Szymanski M., Kubiak M., Pietrzynski G., Wozniak P., Zebrun K., Szewczyk O., Wyrzykowski L., 2003, Acta Astron., 53, 93 Stellingwerf R.F., 1978, ApJ, 224, 953 Vivas A.K. et al, 2004, AJ, 127, 1158 Welsh B.Y., Wheatley J.M., Heafield K., Seibert M., Browne S.E., Salim S., Rich R.M., Barlow T.A., Bianchi L., Byun Y.-I., et al., 2005, ApJ, 633, 447 Wils P., Greaves J., 2004, IBVS, 5512 Wils P., Otero S.A., 2005, IBVS, 5593 Wozniak P.R., Vestrand W.T., Akerlof C.W., Balsano R., Bloch J., Casperson D., Fletcher S., Gisler G., Kehoe R., Kinemuchi K., Lee B.C., Marshall S., McGowan K.E., McKay T.A., Rykoff E.S., Smith D.A., Szymanski J., Wren J., 2004, AJ, 127, 2436 ---------- --------- ----------- ------------- ------ --------- ---------- ------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 6313547 0.0164 +35.3631 12.71-13.88 RRab 0.70669 1465.50 GM And 14654207 0.9480 -11.4764 12.58-13.62 RRab 0.74084 1438.52 ASAS000348-1128.6 17443023 1.5857 -35.2884 13.43-14.12 RRab 0.36595 1487.71 NSV 29 1616266 3.8114 +60.3403 10.38-10.77 RRc 0.54647 1505.61 V363 Cas 9088466 4.4523 + 9.8886 11.94-12.56 RRab 0.70119 1462.67 ASAS001748+0953.3 3656843 5.1850 +40.8276 10.90-11.26 RRab 0.47060 1403.69 OV And 9091171 5.8449 +13.7610 13.22-14.14 RRab 0.53151 1497.69 ASAS002323+1345.6 6302330 5.9297 +29.4009 9.55-10.29 RRab 0.44223 1416.32 SW And 3703390 6.2954 +51.5924 13.30-14.04 RRab 0.48325 1540.62 3707286 6.9956 +49.1627 12.22-12.62 RRc 0.39202 1541.62 6381354 8.2156 +26.3040 14.30-15.37 RRab 0.51203 1532.63 3735016 8.2257 +47.1466 12.73-13.39 RRab 0.56741 1443.72 14669480 8.4908 -13.5221 12.98-14.63 RRab 0.43731 1422.20 NSV 201 14685164 10.1027 -21.5763 13.42-14.06 RRab 0.56485 1491.73 NSV 252 258744 10.1815 +76.9803 12.98-14.03 RRab 0.45746 1505.61 1704599 10.5062 +54.2509 11.14-11.47 RRc 0.33440 1413.66 9129904 10.9981 +22.1527 14.19-15.34 RRab 0.52516 1403.89 11967992 11.3361 - 4.3236 13.34-13.91 RRab 0.56684 1488.83 9130769 11.4220 +18.9520 13.14-13.97 RRab 0.55024 1522.69 ASAS004541+1857.1 9149730 11.9844 +11.7048 11.81-12.54 RRab 0.45577 1508.80 11955058 12.2681 + 2.4212 13.38-14.86 RRab 0.52649 1497.71 ST Cet 6392776 12.3952 +27.0221 12.82-13.93 RRab 0.55456 1442.63 ZZ And 11955315 12.3961 + 2.3817 13.69-15.06 RRab 0.54260 1416.23 SU Cet 1637092 12.9709 +65.1806 10.28-10.52 RRc 0.42333 1459.61 219255 13.6495 +74.5280 10.52-10.73 RRab 0.64968 1401.76 9154383 14.3927 + 7.7813 12.45-12.98 RRab 0.73444 1427.86 ASAS005734+0746.9 9175633 15.4768 +15.9058 14.01-14.97 RRab 0.56204 1482.60 11990964 15.6369 + 5.3952 12.62-13.64 RRab 0.67368 1512.72 SY Psc 6429832 16.2945 +34.2184 12.03-13.15 RRab 0.56307 1453.66 DR And 6406215 17.1789 +24.9359 12.79-13.28 RRc 0.30215 1483.59 6407234 17.5893 +28.5500 13.60-14.33 RRab 0.38213 1491.90 1731450 17.7636 +57.3450 11.90-12.91 RRab 0.41157 1354.23 HU Cas 11996471 18.4184 + 2.1608 13.31-14.38 RRab 0.45169 1484.83 ASAS011341+0209.6 6410298 18.6080 +24.4148 10.28-10.70 RRc 0.39026 1486.63 RU Psc 3814115 18.9508 +39.7512 13.62-14.99 RRab 0.64810 1542.20 NX And 272580 19.3054 +74.8554 12.11-13.11 RRab 0.51410 1425.73 V568 Cas 3817373 19.3642 +38.9505 10.52-11.37 RRab 0.72281 1462.50 XX And 14698706 19.6058 -17.4162 13.23-14.02 RRab 0.62095 1462.71 ASAS011825-1724.9 9186813 20.2182 +21.7288 11.04-11.44 RRc 0.28782 1512.69 SS Psc 17498574 21.2494 -34.6500 13.57-14.58 RRab 0.42636 1478.67 NSV 501 233619 21.6062 +73.2219 12.67-13.47 RRab 0.43612 1414.73 6446901 21.6812 +34.0686 13.07-14.14 RRab 0.39877 1488.80 XY And 14718412 22.2010 -11.4534 12.68-13.96 RRab 0.51675 1495.74 ASAS012848-1127.2 12022798 22.5695 - 2.7111 14.32-15.35 RRab 0.61088 1535.64 12007999 23.0339 + 1.3416 9.60-10.39 RRab 0.55310 1481.66 RR Cet 9199238 23.2316 +20.7805 14.11-15.10 RRab 0.48833 1415.89 1751271 23.8758 +55.7153 13.27-14.51 RRab 0.44367 1515.81 PS Cas 12010602 24.4232 + 7.0556 13.23-14.03 RRab 0.54794 1382.93 ASAS013742+0703.4 6456202 24.5268 +33.0098 14.23-15.13 RRab 0.65268 1514.85 NSV 573 12011841 25.0748 + 1.6654 14.10-15.10 RRab 0.48770 1450.69 17532230 25.6099 -30.4603 12.45-12.85 RRab 0.37720 1488.71 ASAS014226-3027.6 6471887 26.3959 +31.3801 13.07-14.50 RRab 0.46711 1478.65 UX Tri 14728161 28.2544 - 8.0724 14.08-15.54 RRab 0.49150 1460.23 NSV 646 3962806 28.7327 +42.2158 12.79-13.26 RRab 0.50699 1415.88 3851192 28.7850 +43.7656 12.00-12.92 RRab 0.48472 1408.85 CI And 6480340 28.8795 +33.7696 12.10-12.97 RRab 0.44721 1474.61 U Tri 3852316 29.0346 +43.2915 12.65-13.17 RRab 0.77890 1456.78 12034783 29.4679 - 5.5342 12.20-12.57 RRc 0.30148 1460.70 ASAS015752-0532.1 9250068 30.8306 +22.0892 13.72-14.51 RRab 0.50921 1409.93 12056041 31.8808 + 5.6849 12.66-13.48 RRc 0.22232 1443.92 ASAS020731+0541.1 ---------- --------- ----------- ------------- ------ --------- ---------- ------------------- : RR Lyrae objects identified in the NSVS database[]{data-label="maintable"} ---------- --------- ----------- ------------- ------ --------- ---------- ------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 9253268 32.2680 +21.1914 13.83-14.97 RRab 0.47157 1467.68 TU Ari 3939460 33.6697 +49.8887 12.48-13.01 RRab 0.55289 1425.75 14762124 33.8121 -10.8001 10.70-11.36 RRab 0.62348 1521.73 RV Cet 9240848 34.9677 + 9.2793 12.77-13.39 RRab 0.46273 1454.83 ASAS021952+0916.8 14768131 37.1354 - 8.3582 11.51-12.42 RRab 0.51066 1490.19 RZ Cet 3994861 37.6315 +40.8429 13.17-14.28 RRab 0.60687 1558.66 DU And 12073506 38.9843 + 2.7745 13.54-14.26 RRab 0.64540 1403.92 ASAS023556+0246.5 14778728 41.6213 -13.9510 13.85-14.61 RRab 0.56122 1542.65 NSV 15578 12096933 42.4827 - 1.4199 12.38-13.21 RRab 0.54804 1462.70 BK Eri 6587240 42.6422 +35.4056 13.93-15.48 RRab 0.53515 1494.13 V447 Per 6624544 45.8146 +27.2782 11.64-12.04 RRc 0.32831 1593.75 ASAS030316+2716.7 1945920 47.2697 +53.1932 12.02-13.03 RRab 0.60708 1607.16 TU Per 17591360 47.7510 -35.3456 13.23-13.98 RRab 0.60822 1497.74 NSV 1068 12107391 47.9906 - 3.8085 13.54-14.36 RRab 0.45581 1478.66 NSV 15648 14818781 48.4403 -14.2134 12.61-13.68 RRab 0.64877 1566.68 UZ Eri 9338384 50.7730 +18.5791 12.68-13.66 RRab 0.45534 1437.77 ASAS032306+1834.7 4160840 51.7137 +40.2162 12.79-14.11 RRab 0.54609 1404.40 V375 Per 12140780 54.1744 + 5.3616 12.19-13.26 RRab 0.36994 1555.07 SS Tau 14853469 55.7787 -19.4400 13.34-14.63 RRab 0.61503 1493.94 ASAS034307-1926.5 14837671 56.3560 - 8.7927 13.59-15.09 RRab 0.49739 1599.10 NSV 1284 9362709 57.0823 +20.3355 13.68-14.89 RRab 0.56845 1623.65 AI Tau 17623705 57.4619 -23.9304 12.26-12.74 RRab 0.49751 1536.74 BI Eri 9364556 57.6972 +20.4694 13.65-15.10 RRab 0.48134 1565.81 CV Tau 6705218 58.7552 +32.5950 12.49-14.00 RRab 0.39874 1426.85 V378 Per 9390460 59.4783 + 9.1401 13.96-14.83 RRab 0.52674 1630.65 433158 59.5218 +81.2351 12.19-13.00 RRab 0.37898 1572.59 EZ Cep 6746535 60.0563 +28.9798 13.69-14.48 RRab 0.57148 1531.68 4241456 60.8624 +47.9978 12.02-13.24 RRab 0.48928 1553.85 FM Per 2094283 61.6628 +55.4999 11.31-12.18 RRab 0.36866 1522.82 AH Cam 2058479 62.4178 +62.4532 11.18-11.74 RRab 0.57203 1449.21 LP Cam 14889929 64.0241 -14.5670 13.59-15.16 RRab 0.48199 1487.86 AC Eri 4275988 64.3218 +47.3999 10.24-10.94 RRab 0.42555 1508.14 AR Per 4255930 64.9284 +50.1242 13.42-14.08 RRab 0.83723 1421.80 NU Per 14925916 67.0182 -17.1758 13.24-14.70 RRab 0.65722 1486.86 AL Eri 12248878 68.4789 - 0.4253 12.50-13.55 RRab 0.48770 1490.72 BN Eri 9446612 68.6782 +21.7763 12.40-13.43 RRab 0.39062 1515.08 BR Tau 14908250 69.5010 - 8.7559 13.11-14.44 RRab 0.48158 1582.72 BP Eri 12252067 69.5146 - 1.9958 12.65-14.67 RRab 0.57956 1496.90 BE Eri 9480936 70.0421 +11.7213 12.64-13.10 RRc 0.33120 1536.73 ASAS044010+1143.3 14941697 73.4070 -19.4361 11.30-12.00 RRab 0.57002 1526.75 BB Eri 14943428 74.0753 -21.2172 10.33-11.35 RRab 0.58150 1492.74 U Lep 12240344 74.1310 + 5.5928 14.04-15.55 RRab 0.53490 1494.94 GO Ori 6899770 77.2601 +28.6812 12.74-13.94 RRab 0.53939 1573.19 NU Aur 6860144 78.0179 +33.9631 11.50-12.25 RRab 0.45610 1491.69 BH Aur 14990008 81.0611 -14.1011 12.12-13.06 RRab 0.56002 1514.46 NSV 1965 15004447 83.0522 -13.0915 13.44-14.42 RRab 0.53640 1540.66 ASAS053212-1305.6 2225114 83.3597 +62.4876 12.79-13.81 RRab 0.66999 1467.77 TY Cam 12387435 84.9381 - 3.4218 13.78-14.82 RRab 0.45427 1514.91 TZ Ori 2230726 84.9910 +64.8547 13.98-15.00 RRab 0.52946 1601.79 15044339 85.6261 -16.3863 11.93-12.70 RRab 0.53895 1536.67 ASAS054230-1622.9 15016350 86.4864 -14.6919 12.99-14.35 RRab 0.59938 1582.74 ASAS054557-1441.5 15050684 87.1573 -19.4935 13.09-14.16 RRab 0.55868 1631.22 ASAS054838-1929.5 15021360 87.9033 -14.5363 12.59-13.65 RRab 0.51896 1509.32 ASAS055137-1432.2 12397825 88.1605 - 5.8591 12.10-12.46 RRc 0.37440 1553.73 ASAS055239-0551.6 12370363 88.6547 + 4.9031 12.39-13.17 RRab 0.47927 1571.86 NSV 2724 17765420 89.2109 -27.6672 12.38-13.25 RRab 0.46864 1553.76 NSV 2740 2284759 92.6941 +52.7789 13.52-14.38 RRab 0.56447 1593.61 MisV 1268 15119319 93.2337 -14.6690 11.90-12.34 RRc 0.32007 1542.66 ASAS061256-1440.1 15122083 93.7820 -14.9186 12.20-13.15 RRab 0.66682 1490.53 NSV 2888 653554 93.8262 +67.3353 13.03-13.90 RRab 0.51762 1579.85 ---------- --------- ----------- ------------- ------ --------- ---------- ------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- -------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 9753568 95.6834 +18.5305 12.27-12.71 RRab 0.28671 1540.80 ASAS062244+1831.9 2336392 98.4985 +67.0255 12.47-13.73 RRab 0.48042 1541.73 RZ Cam 15206917 98.6774 -15.7776 13.19-14.54 RRab 0.40858 1486.87 ASAS063442-1546.7 15172858 101.3309 - 8.8388 12.77-13.64 RRab 0.41343 1563.76 DV Mon 4694846 107.8960 +40.7769 11.47-12.73 RRab 0.39166 1593.62 TZ Aur 9934090 108.0236 +17.3801 13.69-14.37 RRab 0.72400 1504.71 NSV 3449 4661442 108.0785 +47.3256 13.10-14.05 RRab 0.46850 1633.67 4700161 109.1990 +43.3050 12.86-13.29 RRab 0.61773 1277.68 12673234 109.3303 + 1.7278 11.31-12.11 RRab 0.47633 1559.20 AA CMi 7267813 109.8559 +22.9663 12.76-13.28 RRab 0.52311 1612.68 ASAS071926+2258.0 12678416 110.1606 + 6.6822 12.05-12.49 RRab 0.61591 1599.83 7231540 110.3899 +30.8831 11.04-12.05 RRab 0.39730 1558.71 RR Gem 12680189 110.4360 + 2.3569 12.50-13.45 RRab 0.57141 1515.95 X CMi 12680352 110.4608 + 4.2443 12.70-13.70 RRab 0.62540 1629.28 RV CMi 7235547 111.2184 +35.5244 12.17-13.04 RRab 0.57932 1532.63 10004229 112.0671 + 9.8235 14.04-14.85 RRab 0.58979 1540.80 ASAS072816+0949.5 683310 114.6598 +69.3466 13.30-14.50 RRab 0.80633 1287.65 12728157 117.2378 + 5.6366 11.86-12.55 RRab 0.55052 1603.83 AL CMi 12778073 117.5894 - 1.2418 12.38-13.22 RRab 0.51328 1593.77 NSV 3757 12730938 117.7436 + 6.9928 14.19-15.21 RRab 0.46320 1557.81 ASAS075058+0659.6 9995854 118.4305 +19.2734 11.34-12.44 RRab 0.50116 1600.84 SZ Gem 687170 119.7440 +72.7877 11.71-12.10 RRc 0.26705 1633.63 UY Cam 4760771 119.7991 +45.3268 13.22-13.84 RRab 0.50010 1542.63 691329 120.0165 +67.3796 13.76-15.28 RRab 0.57254 1609.18 763728 120.2170 +70.3175 14.37-15.52 RRab 0.45031 1541.70 4762875 121.0896 +48.3467 13.11-13.77 RRab 0.57534 1567.83 10083413 121.2231 +19.7532 12.71-13.19 RRab 0.60338 1602.70 ASAS080454+1945.2 2431224 122.0091 +53.6584 12.34-12.97 RRab 0.57546 1602.79 12843003 123.1325 + 2.8348 11.83-12.93 RRab 0.50153 1578.87 DD Hya 10094864 124.1497 +18.3496 13.66-14.56 RRab 0.66093 1572.85 ASAS081636+1821.1 771106 124.7231 +73.0966 13.45-15.22 RRab 0.46689 1593.75 UZ UMa 12853552 125.2163 + 6.4739 13.85-14.77 RRab 0.55045 1554.73 NSV 4017 95599 125.6806 +86.0812 12.04-13.16 RRab 0.65148 1456.74 AQ Cep 4807769 125.9209 +37.4696 11.58-12.01 RRc 0.49489 1603.76 DQ Lyn 2480497 126.1037 +65.7181 10.90-11.34 RRab 0.59910 1558.65 NSV 4034 7394811 126.4263 +25.7189 12.31-13.29 RRab 0.61755 1572.68 AS Cnc 768200 127.1301 +67.4978 12.60-13.18 RRab 0.66107 1515.80 7370759 127.1426 +32.0676 14.16-15.16 RRab 0.53040 1623.64 12900309 127.6247 - 2.7110 12.75-13.93 RRab 0.50866 1536.74 ASAS083030-0242.6 10141483 128.2299 +13.1912 11.10-11.92 RRab 0.56346 1514.89 TT Cnc 12872212 128.2375 + 2.9845 13.35-13.99 RRab 0.56168 1517.91 ASAS083257+0259.0 15549182 128.7716 - 8.8395 11.52-12.16 RRab 0.68550 1275.76 ET Hya 4792324 129.3573 +49.2689 12.26-12.69 RRc 0.31127 1511.79 7404884 130.0100 +27.7248 12.22-13.10 RRab 0.50200 1603.79 ASAS084002+2743.5 10121973 130.1974 +15.4139 14.07-15.00 RRab 0.60102 1555.74 12880090 130.2471 + 2.6233 13.05-14.20 RRab 0.50596 1288.25 GL Hya 15603190 130.8930 -17.4533 13.35-14.61 RRab 0.57401 1526.93 ASAS084335-1727.2 2446419 131.3066 +56.6089 11.10-11.50 RRab 0.54281 1613.18 EX UMa 12919472 131.9456 - 3.6501 10.78-11.26 RRd 0.42078 1554.73 GSC 4868-0831 $^1$ 2448910 132.8109 +56.3553 11.46-11.80 RRc 0.34528 1600.77 777167 133.0631 +70.4399 13.45-14.35 RRab 0.51869 1565.84 12892357 133.7231 + 6.4368 12.33-12.90 RRab 0.63647 1581.71 GO Hya 2487311 133.9123 +60.4929 13.74-14.55 RRab 0.47414 1504.78 2488168 134.0067 +62.2457 13.81-14.74 RRab 0.50510 1623.76 12929279 134.5273 - 5.4380 11.92-12.65 RRab 0.75422 1582.61 DG Hya 15657370 135.0616 - 9.7787 11.77-13.02 RRab 0.48897 1526.93 DH Hya 12967520 137.5868 + 5.3480 13.65-14.90 RRab 0.57678 1604.81 CY Hya 12968541 137.9065 + 4.0418 12.86-13.42 RRc 0.34666 1581.71 ASAS091138+0402.5 12969975 138.3450 + 4.5440 12.06-12.54 RRc 0.20084 1615.74 ASAS091323+0432.6 15671446 138.4533 - 9.3211 10.89-11.78 RRab 0.53731 1550.94 SZ Hya ---------- ---------- ----------- ------------- ------ --------- ---------- -------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- --------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 2496194 139.3336 +68.6355 14.18-15.34 RRab 0.48599 1275.81 KQ UMa 7472263 139.7749 +29.0656 11.40-12.48 RRab 0.54720 1274.73 RW Cnc 7454553 139.9645 +33.8733 13.50-14.24 RRab 0.52174 1577.67 15679507 140.5880 -13.6473 13.07-14.10 RRab 0.53218 1502.92 IV Hya 4838955 141.1273 +42.3047 12.79-13.51 RRab 0.56211 1606.66 15681946 141.2336 -13.2002 13.27-14.46 RRab 0.61129 1558.78 ASAS092456-1312.0 10203058 141.3207 +18.6321 13.15-13.95 RRab 0.59370 1601.86 ASAS092517+1837.9 7458705 141.9222 +36.9732 13.08-14.26 RRab 0.52614 1598.20 VY LMi 12985724 142.2805 + 1.4568 11.90-12.24 RRab 0.63441 1578.71 ASAS092907+0127.4 4887740 142.3199 +39.6698 14.59-15.77 RRab 0.46430 1619.81 10228105 142.6088 + 7.2057 12.45-13.21 RRab 0.60272 1558.93 WW Leo 12990485 144.1241 + 4.1113 11.96-13.11 RRab 0.52383 1598.64 UU Hya 15698098 144.2404 -13.6201 13.76-15.15 RRab 0.63779 1604.84 2543921 144.8000 +61.1737 12.97-13.67 RRab 0.63103 1540.77 13015287 145.3948 - 5.8856 14.48-15.51 RRab 0.56406 1598.77 10235939 145.7650 +10.3170 12.82-13.95 RRab 0.67389 1607.72 DL Leo 13016369 145.7705 - 4.9725 13.58-14.53 RRab 0.58782 1278.75 7484924 145.8701 +29.4541 12.11-12.68 RRab 0.59072 1518.87 13017409 146.1153 - 3.6967 13.16-14.00 RRab 0.64980 1549.74 ASAS094428-0341.8 13018296 146.4197 - 6.7333 10.97-11.39 RRc 0.35025 1604.81 RU Sex 2545935 146.7356 +63.6954 11.04-11.38 RRc: 0.29852 1549.65 4878717 146.8860 +51.4567 12.59-13.03 RRc 0.32306 1600.80 10239478 147.2552 +11.7336 14.03-15.08 RRab 0.48675 1577.69 4879576 147.4812 +51.7366 12.58-13.20 RRab 0.57205 1540.78 4879610 147.7964 +47.1534 13.92-14.82 RRab 0.51615 1279.68 GSC 3433-1003 $^2$ 13000978 148.3684 + 2.0575 10.21-10.62 RRc 0.32471 1607.73 T Sex 10241978 148.4073 + 7.9798 13.10-14.50 RRab 0.47225 1634.19 SU Leo 13001999 148.8165 + 3.7590 13.36-14.39 RRab 0.53092 1590.87 13002451 149.0316 + 4.8526 12.32-13.44 RRab 0.48738 1530.88 V Sex 15742386 149.0441 -17.0030 13.41-14.53 RRab 0.55012 1554.47 ASAS095611-1700.1 2549611 149.4537 +60.7419 12.92-13.72 RRab 0.61895 1574.61 7512213 151.1516 +31.8806 13.09-13.89 RRab 0.55725 1277.85 7496253 152.0579 +26.3756 13.38-14.42 RRab 0.46202 1330.75 NSV 4745 10252385 153.3326 +11.1025 13.49-14.70 RRab 0.52865 1598.22 DM Leo 15781328 153.3505 -13.1382 10.67-11.45 RRab 0.53770 1603.86 WZ Hya 13057921 153.6427 + 6.5567 14.22-15.21 RRab 0.57933 1627.86 7517104 153.9644 +32.8592 12.21-13.42 RRab 0.52440 1615.85 Y LMi 7540719 156.3565 +28.7855 11.46-12.72 RRab 0.54391 1307.72 V LMi 4911199 156.6541 +43.5755 13.58-14.34 RRab 0.46716 1601.66 4930110 157.8820 +50.2499 13.57-14.41 RRab 0.64634 1278.69 15820267 158.3370 -20.4544 13.20-14.26 RRab 0.61792 1280.76 ASAS103321-2027.3 10284155 158.4708 +19.2597 12.29-12.68 RRab 0.50412 1578.85 ASAS103353+1915.6 13071196 158.4788 + 1.8254 14.15-15.10 RRab 0.48470 1629.67 2589438 158.9091 +58.2636 13.18-14.20 RRab 0.52424 1324.69 13075952 160.7608 + 6.5797 14.12-14.99 RRab 0.51591 1628.66 15827761 161.0284 -18.8771 13.09-14.11 RRab 0.59776 1627.89 ASAS104407-1852.6 13094757 161.4817 - 7.3602 13.21-14.28 RRab 0.50354 1599.84 ASAS104556-0721.6 13095984 162.0022 - 5.5091 13.11-13.78 RRab 0.60463 1526.91 ASAS104800-0530.5 2595353 164.5309 +56.1192 11.32-11.83 RRab 0.62730 1390.69 KT UMa 15863239 165.8752 -10.9873 12.35-12.92 RRc 0.37696 1532.29 ASAS110330-1059.2 10329395 166.3979 +15.6363 13.13-13.73 RRab 0.59214 1497.89 ASAS110536+1538.2 4966885 168.2751 +40.3605 11.69-12.02 RRc 0.31801 1573.81 GSC 3010-01290 $^3$ 7576787 169.4581 +33.6678 13.34-14.06 RRab 0.57667 1507.91 2600797 169.8653 +58.3145 14.34-15.35 RRab 0.51907 1325.72 NSV 5171 13112988 170.0811 - 2.3099 13.56-14.79 RRab 0.49542 1628.80 ASAS112019-0218.5 10337184 170.7924 +15.6983 12.50-13.18 RRab 0.59941 1612.72 BU Leo 7564333 170.9930 +26.6148 11.73-12.43 RRab 0.65339 1325.81 RX Leo 15876052 171.2555 -10.2687 13.86-15.20 RRab 0.64036 1599.42 10338360 171.5509 +17.6609 12.06-13.08 RRab 0.62671 1558.42 AE Leo 10356202 171.9972 +13.3641 12.18-12.56 RRc: 0.34567 1620.87 ASAS112759+1321.8 ---------- ---------- ----------- ------------- ------ --------- ---------- --------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- --------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 10356449 172.2964 + 9.5362 13.46-15.01 RRab 0.54505 1288.89 AS Leo 15901811 172.4048 -15.7028 13.33-14.18 RRab 0.66435 1572.91 ASAS112937-1542.2 10357384 172.7233 +13.3245 13.05-13.98 RRab 0.33241 1573.86 ASAS113053+1319.4 13138797 173.0785 + 3.2874 13.94-14.79 RRab 0.51509 1582.87 ASAS113219+0317.2 10358222 173.2656 +12.1538 12.13-12.89 RRab 0.72683 1332.71 AX Leo 4974265 173.9029 +38.7660 11.83-12.27 RRc 0.26796 1633.68 GSC 3013-01988 $^3$ 859250 174.0348 +81.2897 9.33- 9.65 RRab 0.62149 1628.65 CN Cam 877852 174.4837 +67.3301 9.62-10.52 RRab 0.66029 1598.61 SU Dra 10346622 174.5086 +16.5433 11.45-12.27 RRc 0.36285 1311.75 BX Leo 4990595 174.6116 +45.9362 12.94-13.82 RRab 0.53483 1556.41 AX UMa 10360494 174.6362 +10.5615 11.17-12.24 RRab 0.47799 1558.00 ST Leo 10360816 174.8093 +10.3273 12.05-13.17 RRab 0.59865 1601.44 AA Leo 10360976 174.9303 + 9.1358 13.14-13.74 RRab 0.65054 1358.70 ASAS113943+0908.2 10347357 174.9413 +15.7945 13.33-14.66 RRab 0.50236 1601.33 BD Leo 15889153 175.2784 -10.6185 12.50-13.48 RRab 0.62234 1605.42 ASAS114107-1037.1 13144199 175.8841 + 2.6987 12.82-13.40 RRab 0.59497 1597.84 ASAS114332+0241.9 10363811 176.4398 +11.8689 13.29-14.48 RRab 0.67931 1342.42 GP Leo 15893671 177.2343 -10.4412 11.38-11.93 RRab 0.73277 1550.94 X Crt 5001653 177.2585 +49.1933 13.37-14.61 RRab 0.55906 1606.83 BB UMa 5003300 177.8919 +52.9730 13.81-15.69 RRab 0.52145 1356.77 CD UMa 10352409 177.9047 +21.8783 11.67-12.31 RRab 0.54719 1612.46 BO Leo 7607943 178.2927 +23.2812 13.36-14.85 RRab 0.46224 1526.32 BP Leo 2610847 178.4968 +55.2430 13.40-14.12 RRab 0.59312 1630.63 10354012 178.9564 +19.3437 14.33-15.71 RRab 0.54375 1530.88 BS Leo 15897530 179.0211 - 9.3444 13.21-14.41 RRab 0.54384 1627.88 ASAS115605-0920.7 5003784 179.2783 +48.4069 13.16-14.37 RRab 0.68114 1342.40 BD UMa 4981888 179.3512 +38.4265 13.85-14.57 RRab 0.50190 1335.72 Tmz V773 13186818 180.0663 + 3.7190 13.73-14.83 RRab 0.82598 1526.91 5005431 180.9356 +46.6715 12.99-14.37 RRab 0.50101 1568.32 BF UMa 7614130 182.5659 +27.4315 13.00-14.37 RRab 0.46915 1603.79 V Com 5007980 182.8108 +47.8289 10.92-11.32 RRab 0.59957 1274.72 AB UMa 2644855 183.1269 +65.5068 13.01-13.52 RRab 0.34905 1557.73 7630016 183.4453 +30.9853 13.52-14.55 RRab 0.46185 1554.35 TU Com 887047 184.4439 +69.5105 10.33-11.19 RRab 0.56968 1630.78 SW Dra 2661529 185.0400 +56.9143 13.11-14.65 RRab 0.64487 1357.39 UU UMa 13197687 185.7201 + 1.2974 12.41-13.37 RRab 0.54298 1612.89 2632654 186.2689 +66.6444 13.68-14.57 RRab 0.54113 1354.69 7640672 187.2812 +34.6473 12.28-13.40 RRab 0.55861 1607.86 RR CVn 5013761 187.4033 +47.8215 12.10-12.98 RRab 0.56368 1341.72 BN CVn 5028198 187.6156 +40.5089 11.91-12.79 RRab 0.69781 1549.88 UZ CVn 15985120 187.8411 -19.0011 13.43-14.66 RRab 0.55295 1280.77 ASAS123122-1900.0 7654632 188.1900 +27.0293 11.32-12.41 RRab 0.58660 1311.74 S Com 5015284 188.7253 +53.6338 14.15-15.08 RRab 0.64447 1331.68 NSV 5746 909273 188.9797 +82.6981 13.20-14.71 RRab 0.44156 1354.69 BU Cam 15989121 189.5435 -15.0000 11.73-12.10 RRc 0.32904 1280.76 Y Crv 13207964 189.9855 + 6.0794 13.17-14.31 RRab 0.68155 1559.60 DH Vir 7657416 190.0167 +27.5038 11.44-11.81 RRc 0.29273 1608.85 U Com 5047910 190.9676 +47.6417 12.93-13.58 RRab 0.56257 1280.84 15972929 191.0680 -12.0517 12.49-13.34 RRab 0.68115 1286.79 ASAS124416-1203.1 893894 191.8309 +69.0982 13.65-14.35 RRab 0.57523 1616.78 15975025 192.0188 - 8.3465 12.26-13.13 RRab 0.51046 1616.25 ASAS124805-0820.8 18728647 192.1005 -33.6593 10.46-11.03 RRab 0.55148 1597.85 V746 Cen 13228663 192.2790 - 1.0725 14.03-15.44 RRab 0.47066 1620.75 BW Vir 5034191 192.4388 +43.7736 11.62-12.55 RRab 0.65395 1620.83 Z CVn 15977771 193.1913 -10.2603 11.59-12.57 RRab 0.55340 1620.93 AS Vir 13230978 193.3859 - 2.1413 13.87-14.80 RRab 0.54391 1600.00 QUEST-230 7649369 193.5899 +32.2426 13.07-14.59 RRab 0.51345 1631.15 TY CVn 15979690 194.0445 -11.2953 12.55-13.23 RRab 0.59716 1631.41 ASAS125611-1117.7 10427234 194.5185 + 9.8453 13.72-14.67 RRab 0.54166 1554.91 ASAS125804+0950.7 13233333 194.6136 - 3.6020 13.99-15.18 RRab 0.90617 1332.76 ---------- ---------- ----------- ------------- ------ --------- ---------- --------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 7651881 195.3717 +32.0869 13.77-15.15 RRab 0.55187 1618.39 TZ CVn 7666039 195.5569 +24.2388 13.17-14.42 RRab 0.66141 1318.70 BF Com 960209 195.8796 +71.1122 10.40-10.73 RRc 0.33297 1280.64 10447559 197.0760 +18.5406 13.45-14.96 RRab 0.54666 1616.26 Z Com 2690430 197.1737 +65.6668 12.39-12.80 RRc 0.31299 1390.73 10433141 197.8103 +11.1578 13.66-14.48 RRab 0.53627 1307.75 ASAS131114+1109.5 10451725 199.4639 +20.7808 11.26-11.95 RRab 0.59893 1360.33 ST Com 2676459 199.8443 +58.7839 12.82-13.35 RRab 0.74198 1284.67 10452666 199.9771 +19.8991 13.14-13.84 RRab 0.60101 0000.00 ASAS131954+1954.0 10437331 200.0481 + 9.1878 11.73-12.38 RRab 0.65692 1607.72 AV Vir 10453785 200.4946 +16.7049 12.57-13.70 RRab 0.58413 1278.75 BH Com 13264396 200.5881 + 5.8865 11.73-12.83 RRab 0.56454 1309.32 BC Vir 16018300 201.1297 -15.8915 13.58-14.70 RRab 0.81903 1330.71 NSV 6223 13266152 201.4303 + 6.0554 13.71-14.60 RRab 0.60391 1531.92 OQ Vir 2678878 201.5560 +56.2568 10.97-11.45 RRc 0.30712 1276.79 SX UMa 13287613 202.0993 - 5.2859 13.06-14.26 RRab 0.65155 1577.50 WW Vir 13288117 202.3435 - 5.8833 11.50-12.41 RRab 0.57642 1332.76 ASAS132922-0553.0 7678033 203.0557 +26.4252 14.33-15.90 RRab 0.51323 1617.85 BQ Com 2723669 203.3252 +53.9872 10.58-11.49 RRab 0.46806 1615.81 RV UMa 13291861 204.0441 - 3.2366 13.74-14.54 RRab 0.54193 1613.98 10464170 204.2280 +22.5133 13.26-14.72 RRab 0.55430 1354.35 AH Boo 2698338 205.0732 +65.0133 14.38-15.71 RRab 0.46337 1318.67 2713175 205.1119 +67.9345 12.76-14.14 RRab 0.50381 1295.83 IS Dra 7703550 206.2626 +32.6546 11.29-12.26 RRab 0.56742 1632.30 RZ CVn 13298164 206.7268 - 3.1265 14.28-15.49 RRab 0.50725 1334.79 16058306 206.8721 -12.2112 13.46-14.97 RRab 0.54211 1278.78 5089959 207.0666 +39.9010 11.53-12.62 RRab 0.47851 1332.74 SS CVn 5090119 207.1779 +41.3851 12.52-13.28 RRab 0.54002 1307.24 RX CVn 7706107 207.6842 +37.6187 12.62-13.81 RRab 0.54969 1326.27 SZ CVn 13282687 207.9201 + 6.4311 11.09-11.81 RRab 0.47111 1603.84 BB Vir 13301497 208.1859 - 2.3082 13.86-14.75 RRab 0.57549 1304.91 ASAS135245-0218.5 16061651 208.2150 -13.8435 13.74-14.51 RRab 0.54183 1616.91 NSV 6478 10471171 208.3040 +17.2126 13.48-15.43 RRab 0.61430 1629.84 AY Boo 2731334 208.8511 +60.1460 13.30-14.01 RRab 0.58593 1341.71 7723299 209.3914 +29.8576 11.45-11.88 RRc 0.32906 1334.75 ST CVn 10488921 209.6931 +12.9519 10.84-11.56 RRab 0.65089 1617.52 UY Boo 13304749 209.7097 - 2.4063 13.71-14.91 RRab 0.44063 1320.78 ASAS135850-0224.4 7709214 209.8884 +31.6511 11.72-12.69 RRab 0.57327 1287.72 RU CVn 7724594 210.2063 +28.3315 13.89-15.41 RRab 0.63158 1357.36 UV CVn 7725370 210.7420 +25.5366 14.02-15.56 RRab 0.60091 1325.79 BH Boo 13328982 211.0290 + 2.4041 13.90-14.69 RRab 0.56413 1330.76 ASAS140407+0224.3 10491546 211.0342 +12.1767 13.15-14.23 RRab 0.45484 1601.00 ASAS140408+1210.5 10492035 211.2929 +14.0803 13.47-14.12 RRab 0.53627 1359.81 7726596 211.5071 +24.5705 12.75-13.89 RRab 0.55339 1360.34 CS Boo 7711864 211.6167 +37.8282 10.41-11.10 RRab 0.55177 1277.87 W CVn 5113607 212.2733 +47.8245 13.09-13.97 RRab 0.61579 1426.67 13311477 212.5283 - 5.5333 13.34-14.42 RRab 0.55810 1604.97 ASAS141007-0532.0 5115340 213.4983 +47.4450 13.47-14.54 RRab 0.45879 1359.80 FT Boo 7730377 213.8561 +28.2822 13.78-14.59 RRab 0.53524 1597.83 ROTSE1\_J141525.52+281656.0 10514612 213.9438 +16.9346 14.20-15.27 RRab 0.57031 1359.81 10514625 214.0130 +20.0602 12.38-13.53 RRab 0.60908 1364.80 CM Boo 16100816 214.1075 -17.0918 12.57-13.11 RRab 0.56537 1330.77 NSV 6606 5099771 214.1517 +42.3597 11.03-11.59 RRc 0.31256 1308.71 TV Boo 16128088 214.3777 -10.3014 13.79-15.15 RRab 0.56917 1319.23 975559 214.4808 +71.6857 12.95-13.75 RRab 0.65252 1330.67 GALEX J141755.4+714107.6 7732274 214.9131 +25.7898 14.26-15.42 RRab 0.59165 1615.70 ROTSE1\_J141939.17+254723.2 5126042 215.0474 +49.8658 12.96-13.52 RRab 0.51100 1288.85 2737870 215.5643 +60.4848 12.97-13.77 RRab 0.60781 1321.70 13340626 215.7734 + 1.9001 13.28-14.01 RRc 0.44692 1304.92 ASAS142306+0154.1 10519227 216.3725 +20.9629 12.92-13.65 RRab 0.51212 1364.74 ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 13343222 216.8725 + 3.7795 12.90-14.09 RRab 0.63387 1607.89 AE Vir 7752385 216.8958 +36.0457 12.09-13.16 RRab 0.51354 1616.22 SW Boo 13322485 216.9128 - 0.9014 11.23-12.32 RRab 0.41082 1341.19 ST Vir 13343644 217.0412 + 6.5455 11.38-12.35 RRab 0.48372 1311.78 AF Vir 5129241 217.6285 +50.3393 13.17-13.85 RRab 0.57067 1567.83 10526447 217.7533 +20.4249 13.71-14.48 RRab 0.61712 1631.76 7755407 218.3883 +31.7545 10.09-11.11 RRab 0.37736 1318.68 RS Boo 5144531 218.5261 +39.1087 13.07-13.95 RRab 0.58143 1612.46 SV Boo 16144061 218.6268 - 8.3090 13.35-14.75 RRab 0.47436 1296.39 7740195 219.2874 +25.7461 13.41-14.57 RRab 0.46675 1287.87 ROTSE1\_J143708.98+254445.7 13352013 219.3077 + 2.7726 14.26-15.47 RRab 0.60405 1356.72 7762964 219.4739 +34.9901 13.24-13.97 RRab 0.56435 1330.75 ROTSE1\_J143753.84+345924.8 10547761 219.5908 +14.4156 13.00-13.67 RRab 0.55297 1295.89 ASAS143822+1424.9 2742937 219.6875 +53.7827 14.25-15.22 RRab 0.52238 1390.73 7741096 219.7487 +25.1448 13.12-14.46 RRab 0.59107 1326.29 VX Boo 2743022 219.7583 +53.9622 14.44-15.42 RRab 0.51600 1286.67 13373557 219.8639 - 3.4603 12.40-12.98 RRab 0.60158 1310.92 ASAS143927-0327.6 13373761 219.9491 - 4.1372 12.80-13.39 RRab 0.68280 1306.77 979937 219.9645 +74.7504 11.96-12.47 RRab 0.58509 1612.62 10531096 220.1359 +17.5993 12.94-13.69 RRab 0.56341 1573.87 ASAS144033+1736.0 16175190 220.4302 -20.8409 13.66-14.69 RRab 0.64605 1628.81 NSV 20158 7780955 220.5558 +28.2057 12.24-13.39 RRab 0.52283 1573.36 SZ Boo 5133136 220.6740 +50.1066 12.60-13.46 RRab 0.61179 1615.83 5148448 220.7015 +42.5252 13.92-14.79 RRab 0.49586 1353.78 5148765 221.0883 +40.6522 13.80-14.88 RRab 0.33334 1580.84 Tmz V787 13377176 221.2136 - 4.6786 13.18-14.40 RRab 0.61273 1357.39 CY Vir 5149127 221.2749 +41.0289 11.02-11.92 RRab 0.53226 1560.50 TW Boo 10534826 221.8969 +16.8455 10.81-11.22 RRc 0.31490 1608.87 AE Boo 7783856 222.4371 +24.4862 12.99-13.99 RRab 0.66172 1603.80 VY Boo 16155644 222.5887 - 9.0974 12.98-13.77 RRab 0.64395 1391.28 ASAS145021-0905.9 7785360 222.9066 +29.3575 11.83-12.43 RRab 0.58139 1323.88 XX Boo 2745644 222.9359 +60.0689 11.33-12.26 RRab 0.58857 1295.84 BT Dra 5152466 223.1675 +45.6616 12.28-13.18 RRab 0.50108 1287.69 13383049 223.3877 - 2.1144 13.86-14.71 RRab 0.44151 1608.89 QUEST-374 5152353 223.4165 +40.5286 12.65-13.18 RRd 0.35299 1312.71 GSC 3047-0176 $^4$ 10556950 223.5202 +15.6298 12.05-12.54 RRc 0.28198 1332.76 CQ Boo 13362900 223.6671 + 2.6951 12.67-13.64 RRab 0.53030 1307.30 AR Vir 13362912 223.6737 + 2.7391 13.53-14.84 RRab 0.66271 1331.37 DV Vir 13384149 223.8125 - 6.2927 13.77-15.51 RRab 0.48317 1553.95 UX Lib 13363373 223.8639 + 6.2480 14.04-15.16 RRab 0.47199 1608.89 7787461 224.2281 +25.5835 12.81-14.08 RRab 0.47927 1312.73 WW Boo 13386557 224.7849 - 2.7073 13.65-14.70 RRab 0.45558 1320.78 ASAS145908-0242.5 10582966 225.0263 +21.7953 14.05-15.10 RRab 0.52934 1312.90 NSV 6881 16224996 225.8831 - 9.7423 13.05-14.03 RRab 0.51640 1335.80 KK Lib 13414163 226.0419 + 4.3618 14.23-15.17 RRab 0.53028 1390.70 5140510 226.1143 +51.8678 13.64-14.76 RRab 0.47949 1334.72 13415462 226.5510 + 2.9165 12.74-13.79 RRab 0.49584 1612.90 10586519 226.6916 +21.4382 14.45-15.61 RRab 0.47456 1324.90 NSV 6940 5157470 226.7562 +41.3814 14.22-15.22 RRab 0.50577 1443.62 10564776 226.8914 +10.0464 14.58-15.69 RRab 0.35734 1620.88 7776015 227.0300 +33.0103 14.01-14.92 RRab 0.51100 1332.75 ROTSE1\_J150807.19+330036.4 13416891 227.0723 + 3.0617 13.99-14.85 RRab 0.58343 1601.89 16230737 227.6577 - 9.5036 13.45-14.46 RRab 0.56272 1332.71 NSV 20262 7777448 227.8057 +34.4377 13.38-14.20 RRab 0.55604 1339.91 ROTSE1\_J151113.37+342615.5 13419554 227.9948 + 6.0389 13.30-14.24 RRab 0.46510 1364.75 V344 Ser 10567740 228.0915 +11.9092 13.43-14.01 RRab 0.74834 1318.69 ASAS151222+1154.6 10568870 228.5037 + 9.9807 11.21-11.68 RRc 0.34078 1630.80 AP Ser 10591457 228.7544 +19.4432 12.17-13.51 RRab 0.43456 1374.20 BH Ser 16234523 228.7754 -10.4748 13.44-14.91 RRab 0.51411 1318.76 10591671 228.8320 +18.6566 11.97-12.66 RRab 0.43778 1318.77 DF Ser ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 7818286 229.2717 +35.1157 11.92-13.10 RRab 0.45694 1286.72 UU Boo 13399252 229.3958 - 1.0889 11.18-11.88 RRd 0.35072 1604.00 V372 Ser 5181225 229.5150 +46.7040 11.14-11.69 RRab 0.45866 1275.69 DG Boo 13399707 229.5561 - 1.3191 13.68-14.46 RRab 0.60488 1617.97 FASTT 701 16237482 229.5912 - 8.4619 11.61-12.67 RRab 0.26963 1614.00 TV Lib 13425078 229.8271 + 7.8869 12.65-13.48 RRab 0.54764 1337.79 ASAS151919+0753.1 16239734 230.3295 -13.2127 13.30-14.51 RRab 0.63475 1318.72 ASAS152119-1312.7 16240333 230.4710 -12.1128 12.65-13.74 RRab 0.44182 1305.21 ASAS152153-1206.8 7799921 230.5889 +26.8767 11.56-12.52 RRab 0.58460 1602.43 TV CrB 7801632 231.4762 +27.8715 14.34-15.32 RRab 0.70131 1343.75 ROTSE1\_J152554.26+275217.7 16213133 231.5045 -15.5465 12.42-13.10 RRab 0.57943 1311.79 NSV 7072 13405704 231.6146 - 5.4222 13.92-14.73 RRab 0.57185 1274.93 13430779 231.7433 + 5.6377 13.71-14.76 RRab 0.50948 1364.75 ASAS152658+0538.3 7823145 231.8096 +34.2479 12.92-13.80 RRab 0.59522 1573.84 ROTSE1\_J152714.27+341452.5 5184769 231.9250 +49.8045 13.67-14.59 RRab 0.55776 1356.70 2790003 232.1507 +64.7126 14.02-14.88 RRab 0.60489 1327.73 13436086 232.1512 + 3.0898 11.90-13.20 RRab 0.52680 1630.33 CS Ser 13436119 232.1633 + 5.0141 13.49-14.44 RRab 0.53530 1361.72 ASAS152839+0500.9 7824714 232.6637 +35.7847 10.75-11.62 RRab 0.62229 1362.72 ST Boo 13437791 232.7579 + 1.6837 10.12-10.69 RRab 0.71410 1391.40 VY Ser 991712 233.0034 +71.4529 12.87-13.53 RRab 0.56962 1620.79 10628157 233.2777 +11.4154 13.54-14.37 RRab 0.61723 1318.69 ASAS153307+1124.9 13439617 233.3782 + 2.7768 11.88-12.46 RRab 0.57547 1589.88 AR Ser 7826675 233.5240 +33.8306 14.08-14.97 RRab 0.51679 1353.84 ROTSE1\_J153405.66+334950.9 13467815 234.3870 - 1.7311 12.65-13.19 RRab 0.51758 1275.91 ASAS153733-0143.9 10631629 234.5332 +12.0281 13.95-14.84 RRab 0.50604 1325.79 13444031 234.7996 + 3.9239 13.23-14.10 RRab 0.46348 1359.76 ASAS153912+0355.4 7839519 235.7937 +36.8948 12.80-13.50 RRab 0.51212 1287.72 ROTSE1\_J154310.50+365341.3 5222076 236.6077 +44.3125 12.71-13.22 RRd 0.49402 1336.88 GSC 3059-0636 $^5$ 7861270 236.8569 +29.6621 13.16-14.21 RRab 0.44867 1391.12 SZ CrB 7841955 237.2507 +35.2665 13.48-14.48 RRab 0.47943 1325.73 ROTSE1\_J154900.16+351559.2 13453021 237.6797 + 2.4642 12.93-13.64 RRab 0.64716 1304.77 Tmz V57 13479854 238.0461 - 0.9393 12.87-13.97 RRab 0.62597 1334.79 ASAS155211-0056.4 10619565 238.1035 +19.6087 14.13-15.18 RRab 0.49782 1348.84 10642060 238.1222 +10.1427 12.89-13.64 RRab 0.51155 1286.74 ASAS155229+1008.6 10620030 238.3155 +17.5698 13.79-14.74 RRab 0.65297 1448.63 10642885 238.3795 +12.9611 10.75-11.61 RRab 0.52208 1287.90 AN Ser 10643584 238.6194 +15.3539 13.48-14.16 RRab 0.50240 1325.79 ASAS155428+1521.4 13456881 238.9184 + 7.9892 11.26-12.11 RRab 0.74649 1312.52 AT Ser 5206588 239.0027 +49.7991 12.76-13.22 RRc 0.37831 1328.74 7866620 239.3826 +28.6334 12.19-12.86 RRab 0.66540 1389.74 NSV 7366 10647104 239.9001 +12.7737 12.98-13.57 RRab 0.57133 1389.75 ASAS155936+1246.3 7847696 240.0147 +34.9724 13.70-15.20 RRab 0.51909 1340.91 VX CrB 5207933 240.1339 +46.9240 10.92-11.99 RRab 0.46996 1615.83 AR Her 1048201 240.4528 +77.9673 13.24-14.84 RRab 0.49043 1630.25 RX UMi 10678189 240.4696 +22.3784 13.70-14.54 RRab 0.56396 1307.75 10678349 240.5157 +17.4807 11.04-11.53 RRc 0.23078 1343.77 LS Her 10679775 241.0818 +18.1424 14.15-15.17 RRab 0.47607 1361.77 5230227 241.4498 +39.5569 13.67-14.40 RRab 0.54081 1325.72 7850909 241.5482 +33.3711 13.81-15.37 RRab 0.46293 1612.37 VY CrB 10652319 241.6202 +15.3683 12.52-13.84 RRab 0.59716 1353.85 AW Ser 7872649 242.0172 +24.9886 13.16-14.58 RRab 0.47574 1331.17 V677 Her 10653751 242.0292 +11.3748 13.92-14.81 RRab 0.55322 1275.89 2844577 242.0884 +62.4982 13.80-14.72 RRab 0.53451 1318.73 7853244 242.4923 +36.9979 13.90-14.95 RRab 0.60742 1448.68 ROTSE1\_J160958.14+365952.3 13494100 242.5298 - 3.1310 13.72-15.23 RRab 0.53885 1318.72 CF Ser 10687094 243.5969 +17.9431 13.56-15.00 RRab 0.51101 1627.86 V686 Her 10661257 244.2524 +10.2920 13.12-13.81 RRab 0.54791 1274.91 ASAS161701+1017.5 5216042 244.5045 +51.1980 13.43-14.04 RRab 0.60236 1630.79 5216390 244.7434 +49.9089 14.02-15.26 RRab 0.45949 1481.59 ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 5237136 244.8092 +39.5020 13.30-14.75 RRab 0.48014 1451.12 WX CrB 7879638 244.8578 +29.7133 11.48-12.02 RRc 0.33162 1278.72 RV CrB 10664602 245.1852 + 9.7402 14.37-15.47 RRab 0.47708 1356.77 2824910 245.2730 +58.4503 12.53-13.32 RRc 0.32106 1414.68 VZ Dra 2869677 246.4965 +65.2371 14.05-15.06 RRab 0.62220 1328.73 13544287 247.0452 + 3.0701 13.45-14.48 RRab 0.59701 1312.77 ASAS162811+0304.3 10705897 247.2824 +19.1061 13.67-15.11 RRab 0.51887 1415.24 V650 Her 7886196 247.3113 +24.9939 13.76-14.76 RRab 0.44032 1318.74 ROTSE1\_J162914.72+245937.7 13551180 247.3744 + 4.4879 13.48-14.17 RRab 0.63941 1321.78 ASAS162930+0429.3 10706127 247.3796 +18.4957 14.04-15.82 RRab 0.53609 1360.31 V698 Her 7914362 247.5774 +34.4605 13.48-14.52 RRab 0.61372 1274.75 GT Her 10706906 247.6697 +18.3665 10.30-11.44 RRab 0.45537 1277.90 VX Her 13552432 247.7080 + 4.8697 14.00-15.13 RRab 0.43916 1359.76 5270816 247.8985 +48.7267 11.96-12.43 RRab 0.60983 1483.59 2830506 248.8814 +57.8394 11.28-12.32 RRab 0.44282 1375.71 RW Dra 13559365 249.4054 + 6.8036 13.83-15.44 RRab 0.63782 1296.38 ASAS163737+0648.2 7919184 249.5747 +37.8018 12.32-13.41 RRab 0.52430 1456.66 GY Her 7919987 249.6305 +33.0417 13.27-14.77 RRab 0.48306 1324.75 GZ Her 7893196 249.7054 +24.8627 13.87-15.26 RRab 0.59810 1475.15 V545 Her 5248400 249.9094 +41.1122 12.56-13.59 RRab 0.63035 1359.80 AF Her 5249020 250.1368 +40.6182 12.34-13.62 RRab 0.64946 1329.33 AG Her 5250335 250.6600 +39.8773 13.19-14.21 RRab 0.65353 1453.30 V448 Her 5249837 250.8700 +45.3928 13.18-14.08 RRab 0.62112 1308.71 10746565 250.9622 + 9.8904 14.31-15.53 RRab 0.44191 1306.90 ROTSE1\_J164350.93+095325.7 13567202 251.1842 + 3.1588 12.26-13.27 RRab 0.63519 1403.76 ASAS164444+0309.6 13571614 252.1766 + 6.1300 13.43-14.19 RRab 0.46252 1389.75 NSV 7978 10751420 252.1784 + 9.9397 12.54-12.97 RRab 0.78762 1604.95 ROTSE1\_J164842.78+095623.2 13572505 252.3723 + 4.8794 13.37-14.03 RRab 0.72649 1325.74 ASAS164929+0452.8 10723064 252.5586 +19.0616 14.09-15.89 RRab 0.61347 1422.34 V616 Her 7937813 252.6591 +35.4512 12.34-13.34 RRab 0.62384 1608.86 CW Her 16438535 252.8542 -20.3517 13.14-13.59 RRab 0.66745 1332.71 NSV 7996 13574816 252.8749 + 6.3743 13.01-14.09 RRab 0.77875 1627.86 V2509 Oph 2899416 253.7489 +54.7087 13.93-14.88 RRab 0.62979 1328.73 10758203 253.9046 +12.4326 13.43-14.12 RRab 0.52892 1358.71 ROTSE1\_J165537.07+122557.9 7940699 254.0147 +32.2674 14.09-15.72 RRab 0.51590 1415.25 HN Her 2858831 254.2580 +66.5866 14.01-14.92 RRab 0.56172 1415.82 7970686 254.3368 +30.3579 12.96-14.14 RRab 0.47274 1325.79 HO Her 13658631 255.0320 + 6.6896 13.75-14.69 RRab 0.62075 1359.76 NSV 8101 1115590 255.0977 +80.6107 12.77-13.53 RRab 0.69112 1279.64 2903305 255.6830 +55.9284 14.14-15.04 RRab 0.57202 1467.77 10767118 256.1358 +14.4427 12.33-13.19 RRab 0.35513 1288.90 NSV 8170 13664588 256.2904 + 4.2275 13.77-14.77 RRab 0.41080 1330.76 ASAS170510+0413.6 10806471 256.4160 +21.5167 12.78-13.91 RRab 0.61325 1389.75 V365 Her 7948150 256.4537 +33.5881 14.20-15.76 RRab 0.54503 1353.85 V646 Her 10769205 256.6713 +15.6754 13.31-14.53 RRab 0.53940 1414.73 NSV 8208 13667153 256.8133 + 5.2516 13.45-14.33 RRab 0.36509 1348.86 V1429 Oph 10809333 257.1295 +18.5213 13.10-13.58 RRd 0.35999 1478.60 V458 Her 1117770 257.5905 +78.2495 13.55-14.36 RRab 0.54440 1457.61 10773701 257.7051 +12.8805 12.93-14.05 RRab 0.51307 1402.75 V461 Her 7953371 258.1896 +32.4275 13.70-14.65 RRab 0.50620 1364.80 ROTSE1\_J171245.49+322538.8 7954068 258.2668 +35.9786 11.04-12.35 RRab 0.44033 1348.83 VZ Her 1088957 258.3326 +69.1317 12.75-13.30 RRd 0.40320 1332.73 10814487 258.4166 +20.9801 14.26-15.79 RRab 0.51063 1443.16 V468 Her 5332615 258.7112 +41.8561 13.65-15.38 RRab 0.51245 1357.24 V759 Her 5307610 258.8338 +52.9113 14.23-15.33 RRab 0.47717 1415.82 ROTSE1\_J171520.08+525441.2 10781556 259.4986 +11.0742 12.00-12.84 RRab 0.55718 1356.74 V452 Oph 5308842 259.5913 +51.2925 14.47-15.51 RRab 0.60625 1483.65 10782845 259.7505 + 9.5073 12.40-13.59 RRab 0.63825 1414.70 V865 Oph 10783891 260.0929 +14.5113 11.86-12.89 RRab 0.59153 1362.71 DL Her 7990147 260.2133 +26.5377 12.66-13.59 RRab 0.52972 1333.28 V392 Her ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 10824662 260.6612 +17.8844 12.21-13.68 RRab 0.43605 1600.42 V394 Her 10824626 260.7845 +22.6587 13.24-14.59 RRab 0.55620 1325.77 V397 Her 13688631 260.8057 + 1.8644 12.20-12.56 RRab 0.73777 1352.79 ASAS172313+0151.8 13644721 260.9004 - 1.1914 12.14-12.88 RRab 0.66793 1403.76 ASAS172336-0111.5 7962128 260.9231 +37.3258 13.10-13.96 RRab 0.45402 1478.65 8038923 262.1327 +38.3771 14.34-15.30 RRab 0.45834 1415.83 ROTSE1\_J172831.79+382236.9 7998287 262.2804 +27.8242 12.85-13.50 RRab 0.46883 1356.71 ROTSE1\_J172907.35+274928.6 10885457 262.5485 +14.3762 12.35-13.44 RRab 0.37854 1338.78 V552 Her 5344664 263.0036 +45.0288 14.02-15.72 RRab 0.47737 1354.70 V495 Her 5344210 263.0232 +39.7586 13.33-14.64 RRab 0.55677 1414.74 V421 Her 10835507 263.0521 +18.2359 12.44-13.08 RRab 0.60411 1359.83 V418 Her 10887397 263.1248 + 9.3929 12.89-13.89 RRab 0.58569 1476.16 V773 Oph 13710973 263.3536 + 2.9931 13.42-14.86 RRab 0.52143 1359.76 V777 Oph 13758492 263.4974 - 1.0812 11.71-13.10 RRab 0.45035 1425.67 ST Oph 10848453 263.6831 +17.9547 13.43-14.78 RRab 0.63071 1370.97 V424 Her 13759699 263.7355 - 0.3856 13.65-14.76 RRab 0.51408 1443.64 ASAS173457-0023.2 13713760 263.8546 + 7.7557 12.13-13.31 RRab 0.60336 1310.79 V784 Oph 10891108 263.8860 + 9.5654 13.43-14.61 RRab 0.44958 1605.40 V785 Oph 10853322 264.6236 +19.8029 14.00-14.98 RRab 0.46492 1359.81 5348956 264.6561 +37.8991 10.72-11.02 RRc 0.27269 1421.80 ROTSE1\_J173837.40+375357.1 10856198 265.1378 +22.8175 13.80-15.06 RRab 0.51453 1480.07 V434 Her 13721373 265.1609 + 1.6071 12.24-13.07 RRab 0.59825 1311.37 NSV 9504 1181281 265.3613 +71.9995 13.96-14.87 RRab 0.57614 1482.57 8011052 265.4533 +25.1567 12.99-13.98 RRc 0.32220 1356.71 LW Her 2956689 265.5514 +63.5671 13.36-14.01 RRc 0.35762 1276.79 8013685 266.0578 +25.2482 13.07-15.12 RRab 0.46174 1401.25 FS Her 10904025 266.1810 +15.0321 13.65-14.67 RRab 0.46261 1318.69 NSV 9631 10905293 266.4963 +10.5037 12.53-13.95 RRab 0.53765 1322.76 V822 Oph 5355947 266.7764 +38.5513 13.21-13.89 RRab 0.26956 1474.59 NSV 9697 10864468 266.8813 +16.8308 13.95-15.06 RRab 0.55437 1359.76 NSV 9696 13735067 267.4951 + 7.1735 13.47-14.08 RRc 0.30934 1358.77 NSV 9748 10869886 267.9291 +17.1407 12.95-13.65 RRab 0.43571 1356.72 ASAS175143+1708.4 2927472 268.0055 +53.9370 12.84-13.47 RRab 0.54557 1414.68 ROTSE1\_J175201.56+535615.1 8023423 268.2041 +24.7539 12.52-12.98 RRc 0.29139 1325.73 ROTSE1\_J175248.98+244513.7 5398346 268.5535 +51.0229 12.17-12.54 RRab 0.54587 1611.89 ROTSE1\_J175412.89+510123.5 8023770 268.6309 +30.4098 10.96-12.06 RRab 0.39958 1353.81 TW Her 8025632 268.7933 +26.6050 12.51-13.41 RRab 0.42573 1345.91 EP Her 10919329 268.9865 +12.0340 13.95-15.01 RRab 0.56349 1353.80 NSV 9872 5401723 269.9339 +51.8839 12.50-13.57 RRab 0.55560 1322.72 AV Dra 1127822 269.9416 +77.6960 13.51-14.28 RRab 0.54619 1318.66 5403829 270.2376 +47.6392 12.74-13.19 RRc 0.28025 1611.89 ROTSE1\_J180057.01+473820.5 10981416 270.3383 +22.8440 12.64-13.24 RRab 0.58357 1455.72 ASAS180121+2250.6 2930719 270.4686 +60.1119 12.22-12.69 RRab 0.57927 1318.73 IBVS5700-28 $^6$ 2936819 271.7967 +53.2608 13.22-13.97 RRab 0.63431 1603.92 ROTSE1\_J180711.26+531539.5 8094208 272.3402 +36.7545 12.00-12.41 RRc 0.27026 1330.70 ROTSE1\_J180921.64+364516.2 10994615 272.7031 +17.2083 12.74-13.60 RRab 0.52233 1460.68 ASAS181048+1712.5 5377086 273.2430 +42.0625 12.63-13.75 RRab 0.44216 1443.10 V442 Her 1132203 273.5794 +76.6856 11.46-12.15 RRab 0.71957 1338.69 BC Dra 5378312 273.6560 +42.5100 14.31-15.40 RRab 0.51477 1442.62 8101030 273.9553 +35.4637 14.13-15.59 RRab 0.49522 1299.75 V335 Lyr 1132043 274.4687 +77.2975 12.05-13.10 RRab 0.58902 1296.21 BD Dra 10955592 274.8896 +10.9028 11.35-11.95 RRab 0.43589 1328.76 V408 Oph 13822241 275.4979 - 5.6495 12.47-13.20 RRab 0.35606 1323.13 NSV 10697 8108615 275.6589 +32.9591 11.57-12.44 RRab 0.57710 1427.42 IO Lyr 11015122 276.1591 +19.8725 12.90-13.47 RRab 0.59982 1604.96 V534 Her 2947743 276.7795 +55.4896 12.12-13.25 RRab 0.60266 1416.42 AE Dra 11037662 277.3025 +21.0728 11.37-11.81 RRab 0.59114 1390.69 ASAS182913+2104.3 3037033 278.3514 +58.7065 9.19- 9.47 RRc 0.29872 1356.76 HI Dra 8226518 279.2883 +33.3942 13.76-15.13 RRab 0.50895 1479.61 CG Lyr 8173179 279.2951 +26.1553 12.08-12.68 RRc 0.35860 1453.67 EX Lyr ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 3038588 279.4839 +56.8290 13.62-14.54 RRab 0.59435 1475.61 3040364 279.9083 +58.1000 13.95-14.87 RRab 0.52746 1389.73 5494746 280.0971 +41.0403 13.06-14.12 RRab 0.54555 1603.96 LX Lyr 8178485 280.3172 +28.7233 11.23-11.61 RRab 0.41141 1360.13 CN Lyr 8234128 280.9082 +32.7979 11.20-12.35 RRab 0.51131 1600.99 RZ Lyr 11061929 281.2414 +23.0543 12.61-13.85 RRab 0.53722 1307.44 V347 Her 8238513 281.9195 +35.9916 10.60-11.03 RRab 0.52527 1283.78 EZ Lyr 8192312 282.8218 +28.7963 12.50-13.74 RRab 0.61678 1442.69 CX Lyr 8195470 283.2941 +27.0783 13.49-14.64 RRab 0.37260 1319.04 DD Lyr 5508571 283.3585 +43.1552 13.80-15.18 RRab 0.47378 1471.07 V355 Lyr 5568044 285.2001 +50.0917 12.53-13.55 RRab 0.68718 1275.56 AW Dra 5568340 285.2447 +48.7450 11.83-12.26 RRab 0.61324 1415.82 ROTSE1\_J190058.77+484441.5 3057786 286.8226 +55.3707 13.93-14.93 RRab 0.59924 1450.67 5525581 287.1147 +38.8118 12.27-12.97 RRab 0.68165 1322.72 NR Lyr 3022100 287.4273 +64.8596 9.94-10.84 RRab 0.47651 1439.71 XZ Dra 5526813 287.5916 +42.4595 12.34-13.28 RRab 0.52743 1421.80 FN Lyr 3117499 289.5369 +65.5886 12.44-13.27 RRc 0.32963 1360.01 3117504 289.5858 +66.4134 10.80-11.87 RRab 0.59203 1340.88 BK Dra 5587648 289.9911 +46.8890 13.37-13.99 RRab 0.54860 1576.78 NSV 11924 1213492 291.2899 +75.5495 12.62-13.27 RRc 0.30031 1325.81 8360537 291.9842 +24.3479 10.34-11.35 RRab 0.59416 1415.89 BN Vul 3078093 293.1227 +56.3883 9.33-10.35 RRab 0.46669 1327.74 XZ Cyg 1251330 293.3314 +80.9295 12.30-13.62 RRab 0.58894 1336.86 WY Dra 3127879 293.7645 +62.3296 13.89-14.90 RRab 0.47122 1324.70 KO Dra 3083006 294.6013 +56.5401 12.01-12.40 RRc 0.38756 1353.78 V939 Cyg 3087038 296.5217 +59.5739 12.64-13.56 RRab 0.53487 1599.92 CY Dra 1225600 297.4739 +68.1720 13.85-14.88 RRab 0.40882 1475.60 NSV 12492 5692678 297.6740 +39.4803 12.47-12.98 RRab 0.55013 1512.66 NS Cyg 1223703 297.7914 +71.4493 13.72-14.50 RRab 0.50715 1452.67 8394280 297.9980 +27.2765 12.74-13.45 RRab 0.46730 1427.85 EW Vul 5654620 300.1124 +48.9938 12.83-14.23 RRab 0.35999 1478.94 V759 Cyg 1227429 300.1575 +71.0594 13.45-14.05 RRab 0.44772 1442.67 1225869 300.7087 +73.4630 12.82-13.81 RRab 0.55025 1321.84 AQ Dra 14183136 301.1749 - 1.2494 11.55-12.25 RRc 0.27557 1354.95 ASAS200442-0114.8 11350502 301.6528 + 9.4980 11.92-12.29 RRc 0.32965 1491.71 ASAS200637+0929.9 14189160 302.2019 - 1.0742 13.06-14.57 RRab 0.63026 1355.24 V788 Aql 5666135 302.7898 +51.6160 13.10-13.98 RRab 0.56556 1475.63 V1369 Cyg 8487853 303.1729 +32.2124 10.57-10.81 RRc 0.37110 1361.70 11379205 306.2582 +11.2102 11.93-13.14 RRab 0.44279 1421.77 CK Del 14215653 306.9074 - 1.6664 11.80-12.22 RRab 0.72470 1448.64 17130622 307.1700 -12.4772 12.46-13.42 RRab 0.66762 1420.76 SX Cap 11507357 307.4849 + 9.5943 13.78-14.60 RRab 0.59103 1311.92 NSV 13112 11510376 307.8098 +15.0750 12.66-13.47 RRab 0.52010 1338.29 ZZ Del 17134774 308.1015 -11.4212 13.57-15.00 RRab 0.68503 1452.72 1259373 308.1395 +82.2560 11.77-12.10 RRc 0.32827 1421.74 14285465 308.3655 + 4.6527 14.06-15.09 RRab 0.47645 1463.70 11462023 308.4309 +16.2720 12.02-12.38 RRab 0.79688 1449.78 FM Del 17176197 309.3811 -15.6228 13.18-13.83 RRab 0.40499 1426.75 AL Cap 14335375 309.5627 - 2.8938 11.41-12.42 RRab 0.36181 1410.17 AA Aql 17142018 309.6263 -12.0179 13.52-14.98 RRab 0.40101 1449.79 RZ Cap 17142150 309.7109 - 9.7656 13.34-14.68 RRab 0.59881 1421.42 8572437 309.7988 +25.6566 13.11-14.31 RRab 0.43702 1509.63 BC Vul 14336635 309.8226 - 5.5081 13.66-14.55 RRab 0.53388 1408.91 14294563 309.9124 + 3.8741 13.69-14.61 RRab 0.50077 1356.73 17147355 310.8157 - 9.1581 13.17-14.41 RRab 0.51231 1390.72 GSC 5756-373 $^7$ 11527987 310.8701 +14.5720 13.13-14.20 RRab 0.47620 1486.12 DS Del 14342579 311.0149 - 3.3852 13.92-14.86 RRab 0.59102 1408.76 3200728 311.1382 +63.0393 12.77-13.33 RRab 0.45749 1453.78 11533493 311.8681 +12.4641 9.83-10.46 RRab 0.47263 1390.76 DX Del 17154709 312.3918 -14.5786 13.45-14.41 RRab 0.54097 1450.70 TZ Cap ---------- ---------- ----------- ------------- ------ --------- ---------- ----------------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 8588795 312.4413 +24.2144 11.83-12.25 RRc 0.33609 1324.89 ASAS204946+2412.8 14311711 312.9558 + 2.3913 13.94-15.40 RRab 0.44490 1450.63 DP Aqr 11491144 313.0251 +21.8229 13.38-13.95 RRc 0.31332 1337.78 ASAS205206+2149.3 11491797 313.1282 +22.4361 12.23-13.08 RRab 0.43406 1496.13 FK Vul 14352725 313.2229 - 0.6331 13.10-13.81 RRab 0.33361 1421.77 ASAS205254-0038.0 3306145 313.7241 +66.4448 12.02-12.89 RRab 0.46789 1322.87 FP Cep 14316194 313.8468 + 2.9601 13.94-15.50 RRab 0.59549 1356.73 EL Del 8651515 314.1176 +30.4278 10.97-11.67 RRab 0.56074 1496.22 UY Cyg 11547113 314.3702 +10.0418 13.89-15.15 RRab 0.58262 1443.74 EM Del 14357981 314.4509 - 5.6849 12.28-13.41 RRab 0.40639 1461.11 BT Aqr 11599377 315.9427 +19.9652 13.02-13.77 RRab 0.61354 1325.80 SX Del 8609914 316.3444 +28.2969 13.68-14.77 RRab 0.47609 1456.66 NSV 13519 11559602 316.5425 +10.3862 12.90-13.66 RRab 0.55717 1402.75 RW Equ 11561025 316.8096 + 9.8916 13.92-14.82 RRab 0.54696 1453.84 14371461 317.5533 - 1.7215 11.50-12.69 RRab 0.46337 1359.77 CP Aqr 5890743 317.7904 +39.5571 13.02-14.33 RRab 0.52975 1329.24 V1664 Cyg 14411402 318.6801 + 4.4937 12.69-13.45 RRab 0.44463 1484.69 RT Equ 14415962 319.6645 + 6.2049 11.33-11.71 RRc 0.29152 1415.90 11622872 320.1423 +18.6219 11.88-12.35 RRab 0.56234 1408.75 ASAS212034+1837.2 8702063 320.2984 +32.1911 11.31-12.23 RRab 0.41985 1448.63 DM Cyg 8752554 320.7471 +23.8901 13.51-15.06 RRab 0.55545 1454.34 BL Peg 8706201 321.0736 +32.0573 12.64-13.22 RRab 0.70912 1415.89 11631563 321.7645 +18.5996 12.61-13.74 RRab 0.54721 1415.91 AO Peg 14433600 322.4773 + 1.0059 13.30-13.93 RRab 0.92732 1421.82 ASAS212955+0100.3 11647105 322.5734 +18.7343 13.05-13.66 RRab 0.85285 1452.64 ASAS213017+1843.9 14439051 323.9002 + 3.5767 14.19-15.23 RRab 0.57609 1364.81 14439564 324.0351 + 3.2304 11.45-12.44 RRab 0.53566 1421.78 SX Aqr 8770586 324.0988 +26.3025 11.70-12.20 RRab 0.55678 1404.28 BT Peg 8724692 324.4381 +34.6204 13.17-13.85 RRab 0.60418 1362.70 17288024 324.7581 -21.2129 12.25-12.74 RRab 0.66207 1488.70 ASAS213902-2112.7 14471347 325.0531 - 1.3806 13.85-14.73 RRab 0.48838 1455.74 FASTT 1542 11659896 325.1713 +21.0802 13.63-15.17 RRab 0.57108 1359.29 CD Peg 11696549 325.3026 +11.3554 14.32-15.53 RRab 0.56641 1505.72 8776607 325.3201 +24.7734 10.98-11.76 RRab 0.46711 1514.69 CG Peg 19875041 325.5267 -25.4755 11.68-12.02 RRc 0.31687 1378.82 NSV 13844 11698204 325.7765 + 8.0584 14.05-14.87 RRab 0.65565 1503.70 8779392 325.8450 +26.8655 13.60-14.88 RRab 0.63211 1454.43 CN Peg 14447742 326.2217 + 4.3621 13.20-14.06 RRab 0.58697 1473.70 8782731 326.5493 +26.9403 13.20-14.51 RRab 0.57197 1414.77 CQ Peg 1363671 326.5950 +69.1856 12.74-14.24 RRab 0.41659 1475.61 EL Cep 8783506 326.7193 +24.7212 12.89-14.01 RRab 0.56451 1353.80 CS Peg 11668679 327.0538 +22.2433 13.17-14.39 RRab 0.56288 1444.30 CV Peg 1409068 327.2606 +83.0560 13.70-14.42 RRab 0.45571 1414.73 11670320 327.4470 +21.1441 12.23-13.39 RRab 0.64813 1460.49 CY Peg 14480356 327.9223 - 3.1424 13.01-13.68 RRab 0.48667 1504.72 ASAS215141-0308.5 11672832 328.0118 +22.5750 10.23-11.13 RRab 0.39037 1426.88 AV Peg 11677350 329.1108 +15.5777 12.78-13.40 RRab 0.56041 1475.69 ASAS215626+1534.7 14512124 330.6547 + 3.7032 13.00-13.66 RRab 0.54971 1402.77 ASAS220237+0342.3 11747454 330.7662 +16.7384 13.36-14.18 RRab 0.62111 1442.69 ASAS220304+1644.4 14512963 330.9157 + 3.9518 14.10-15.06 RRab 0.61326 1466.70 1277310 331.1397 +80.1874 13.09-14.16 RRab 0.48390 1403.83 NSV 14038 11753752 332.3296 +19.7325 13.30-14.20 RRab 0.76960 1537.64 6080428 332.6396 +37.2198 12.50-13.67 RRab 0.51247 1426.26 PW Lac 14495866 333.2550 - 1.7286 12.81-13.50 RRab 0.58824 1370.79 FX Aqr 11757689 333.2661 +18.4512 11.49-12.54 RRab 0.48835 1375.93 VV Peg 8817269 333.4019 +26.7785 12.47-13.45 RRab 0.36493 1413.72 ASAS221336+2646.8 17331061 333.6274 -10.9296 12.41-13.41 RRab 0.55184 1452.72 YZ Aqr 19930778 333.9781 -25.3784 11.58-12.23 RRab 0.54707 1403.80 ASAS221556-2522.6 1376651 334.3503 +70.8947 12.81-13.64 RRab 0.43092 1459.80 NSV 14111 8824058 334.5928 +22.1364 13.38-14.02 RRab 0.56948 1448.68 ASAS221822+2208.2 ---------- ---------- ----------- ------------- ------ --------- ---------- ------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 6094125 335.2934 +39.7200 12.13-13.30 RRab 0.62000 1438.36 CQ Lac 11734397 335.5116 + 8.8227 14.35-15.31 RRab 0.60895 1421.76 14529130 335.9223 + 4.1039 13.61-14.40 RRab 0.51799 1475.64 14504106 336.2604 - 4.9495 13.64-15.09 RRab 0.52778 1414.75 FH Aqr 8895403 336.3465 +34.8858 13.02-14.23 RRab 0.50344 1361.78 GY Peg 11770210 336.3669 +20.1519 13.14-13.81 RRab 0.64271 1421.76 11780560 336.8397 +16.8046 12.22-13.50 RRab 0.49679 1448.69 AE Peg 14557029 337.0722 - 0.9885 13.90-14.73 RRab 0.54289 1459.87 GT Aqr 19941416 337.5287 -28.1706 12.54-13.21 RRab 0.51261 1449.79 NSV 14176 17344991 337.7187 - 7.7112 13.03-14.21 RRab 0.52612 1511.64 GW Aqr 17346485 338.2630 -11.2805 13.60-14.83 RRab 0.60515 1506.15 8903283 338.3266 +30.1683 11.92-12.72 RRab 0.53870 1425.30 ES Peg 14560815 338.7393 - 3.7089 13.29-14.64 RRab 0.54723 1533.13 GX Aqr 17348072 339.0161 -10.0151 12.60-13.71 RRab 0.60882 1348.89 AA Aqr 14561490 339.0302 - 1.6320 13.34-14.60 RRab 0.85383 1538.49 GY Aqr 6111508 339.1869 +37.4114 12.79-13.39 RRab 0.46117 1486.63 17368467 339.7450 -19.8072 13.13-14.41 RRab 0.52569 1487.70 BH Aqr 8951390 340.1763 +26.3837 12.78-14.15 RRab 0.48988 1374.24 ET Peg 14564730 340.3814 - 6.4777 12.03-13.05 RRab 0.57451 1454.71 HH Aqr 1428559 340.8642 +74.3721 13.65-14.45 RRab 0.51366 1520.62 47457 341.0807 +83.9500 11.81-12.90 RRab 0.52609 1378.69 DX Cep 19963108 341.1731 -31.9689 13.34-13.93 RRab 0.47200 1467.72 NSV 14301 8956175 341.4715 +24.1433 12.36-13.73 RRab 0.49595 1489.65 BF Peg 17354267 341.5993 -12.9157 12.83-13.50 RRab 0.51097 1421.77 ASAS224624-1254.8 11798682 341.9148 +15.2954 13.89-14.75 RRab 0.52564 1448.69 14550660 342.5646 + 4.4785 13.48-14.09 RRab 0.35772 1475.64 ASAS225015+0428.6 11803134 343.2544 +15.7877 10.39-10.92 RRab 0.64187 1487.67 BH Peg 11828257 343.3463 + 8.7683 13.14-14.15 RRab 0.49319 1514.71 ASAS225323+0846.1 17358580 343.5346 -12.3606 11.94-12.86 RRab 0.69405 1475.70 BO Aqr 14553058 343.5686 + 3.0800 13.80-14.82 RRab 0.60048 1450.69 11804971 343.8301 +17.7495 14.06-15.07 RRab 0.54723 1475.63 14554494 344.2004 + 5.3688 13.21-14.15 RRab 0.53628 1505.64 ASAS225649+0522.1 19972018 347.3769 -35.7883 12.44-13.00 RRab 0.59344 1475.64 ASAS230931-3547.3 11840238 347.8033 +13.8484 13.15-14.63 RRab 0.60099 1476.29 IX Peg 11864350 347.9984 +19.7417 12.59-13.09 RRab 0.69042 1537.64 ASAS231159+1944.5 6155448 348.3022 +36.9012 13.29-14.83 RRab 0.52880 1494.67 GV And 14602495 348.9862 + 3.0496 14.42-15.61 RRab 0.59359 1375.79 17403992 349.2786 -12.6380 13.89-15.39 RRab 0.56532 1403.95 139245 349.4676 +75.7292 13.10-13.69 RRab 0.68034 1507.88 11869182 349.5273 +15.8824 12.76-13.38 RRab 0.60225 1456.84 ASAS231807+1553.0 11870672 350.0292 +16.0685 11.68-12.73 RRab 0.60740 1537.24 DZ Peg 11847482 350.4621 +12.7892 13.81-15.39 RRab 0.55515 1444.69 11872086 350.5334 +17.5284 13.11-14.47 RRab 0.54575 1466.39 IY Peg 17407758 350.8917 - 8.0124 13.56-14.93 RRab 0.51409 1422.25 17407837 351.0546 -12.6330 12.73-13.87 RRab 0.51895 1457.73 HQ Aqr 6231459 353.0025 +35.1973 11.69-12.53 RRab 0.63038 1577.27 DM And 17393366 353.0752 -17.3979 12.59-13.28 RRab 0.65524 1475.70 NSV 14603 9063965 353.7455 +13.7350 13.07-13.81 RRab 0.69425 1536.63 3611792 353.7776 +41.1038 12.72-14.13 RRab 0.42160 1505.20 BK And 14616219 353.9484 -10.9902 13.62-15.09 RRab 0.55269 1515.73 HR Aqr 11902532 354.1554 - 2.2119 12.66-13.14 RRab 0.64497 1470.68 ASAS233637-0212.7 14617415 354.6370 - 9.3187 11.12-12.07 RRab 0.48188 1461.19 BR Aqr 9047406 355.4950 +18.2166 12.55-13.11 RRab 0.52960 1450.68 ASAS234159+1813.0 6270560 355.5680 +24.9162 12.01-12.52 RRc 0.30647 1496.67 VZ Peg 3619356 355.6285 +43.0144 10.61-11.04 RRab 0.61693 1474.38 AT And 1534647 356.1724 +52.3481 11.99-12.50 RRab 0.64946 1510.23 IU Cas 9049886 356.4618 +15.9886 13.91-14.91 RRab 0.62221 1535.72 9050492 356.6716 +17.6340 13.11-14.01 RRab 0.58908 1494.69 ASAS234641+1738.1 9052467 357.4046 +20.4290 14.21-15.31 RRab 0.58141 1353.85 9052681 357.5211 +17.8956 13.99-15.02 RRab 0.60372 1415.89 NSV 14723 ---------- ---------- ----------- ------------- ------ --------- ---------- ------------------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ------------------- NSVS ID NSVS Mag. Type Epoch Cross ID 6248008 357.6928 +33.3512 12.87-13.73 RRab 0.53393 1402.87 14624322 358.5058 - 7.6773 12.46-13.12 RRab 0.52890 1401.88 NSV 14744 11895079 358.5340 + 0.9635 10.76-11.14 RRc 0.30623 1463.69 NSV 14745 9075933 359.0035 +10.8884 11.93-12.60 RRab 0.56249 1382.75 ASAS235601+1053.3 3633664 359.6761 +41.4886 12.94-14.15 RRab 0.60298 1521.65 DY And ---------- ---------- ----------- ------------- ------ --------- ---------- ------------------- : continued [rp[200mm]{}]{}\ $^1$ & [@rrdasas]\ $^2$ & [@nichoejv]\ $^3$ & [@pz]\ $^4$ & [@koppelman]\ $^5$ & [@smith]\ $^6$ & [@nicholson]\ $^7$ & [@bernasconi]\ ASAS & [@asasV]\ FASTT & [@fastt]\ GALEX & [@galex]\ MisV & http://www.aerith.net/misao\ ROTSE & [@rotse]\ QUEST & [@quest]\ ---------- ---------- ----------- ------------- ------ --------- ---------- ---------- NSVS ID NSVS Mag. Type Epoch Cross ID 14641662 1.0211 -16.9979 12.03-12.54 RRab 0.60589 1456.70 UU Cet 14682480 8.4095 -15.4874 11.27-11.94 RRab 0.57347 1514.73 RX Cet 17478336 13.9943 -26.3831 13.27-14.28 RRab 0.52074 1402.95 UV Scl 3763277 15.7183 +45.1897 14.28-15.66 RRab 0.48525 1492.84 IY And 3860939 19.6106 +50.6715 13.03-13.61 RRc 0.37355 1456.63 V830 Cas 17510932 20.7232 -26.2927 13.25-14.51 RRab 0.45151 1536.65 RW Scl 6452465 23.2962 +32.5946 14.22-15.64 RRab 0.70559 1336.90 TV Tri 17530684 23.8484 -35.1285 11.94-12.75 RRab 0.63706 1486.71 VX Scl 17532959 26.2484 -30.0577 11.48-11.91 RRc 0.37738 1463.88 SV Scl 17528524 31.9667 -26.8660 9.86-10.90 RRab 0.49550 1455.92 SS For 9254456 32.7862 +20.4500 13.28-14.12 RRab 0.57992 1414.76 TV Ari 9243493 36.2171 + 8.4007 14.39-15.77 RRab 0.69251 1483.60 BP Cet 17570708 42.5412 -26.2645 12.66-13.43 RRab 0.37049 1488.87 Z For 17572476 43.6260 -28.4005 12.41-13.51 RRab 0.60311 1496.75 TX For 9310412 47.1286 +10.4457 9.31-10.09 RRab 0.65119 1597.70 X Ari 17578705 47.8051 -26.4832 11.76-12.73 RRab 0.59705 1453.89 RX For 4149951 48.6468 +43.2470 13.79-15.76 RRab 0.54103 1560.67 V433 Per 4157182 50.6321 +42.0993 13.34-14.47 RRab 0.53980 1504.78 V460 Per 14910118 70.1662 - 9.1896 13.28-14.38 RRab 0.66038 1536.74 BG Eri 17719289 76.9460 -33.8651 12.32-13.33 RRab 0.48711 1498.93 SU Col 15215875 100.2258 -19.3971 13.89-14.96 RRab 0.58614 1594.64 EN CMa 679069 111.8654 +72.7035 9.57- 9.99 RRab 0.62839 1606.77 EW Cam 4717532 115.1903 +39.3141 14.17-15.20 RRab 0.24094 1603.76 WZ Lyn 15369473 116.1950 -13.0989 11.29-11.87 RRab 0.73426 1597.71 HK Pup 4720405 116.2761 +43.1115 11.79-12.72 RRab 0.48186 1601.81 TW Lyn 7315103 121.6063 +23.2506 11.90-12.78 RRab 0.36737 1606.83 SS Cnc 716865 121.8966 +76.4150 14.20-15.74 RRab 0.20060 1536.70 HU Cam 10125458 131.3420 +15.2749 12.90-13.93 RRab 0.52446 1558.72 CQ Cnc 7411608 132.8328 +25.5563 13.10-14.24 RRab 0.51017 1615.69 SX Cnc 7412671 133.2403 +23.7984 12.14-12.96 RRab 0.54581 1623.64 EZ Cnc 10134374 134.5419 +15.8019 12.82-14.07 RRab 0.54317 1577.69 AN Cnc 4828497 135.7823 +44.5855 9.83-10.41 RRab 0.59850 1603.93 TT Lyn 10175327 139.4111 +12.6518 11.69-12.32 RRab 0.54852 1631.75 AQ Cnc 2500172 143.7256 +65.4483 14.40-15.67 RRab 0.48536 1631.68 DE UMa 7496220 151.9311 +23.9917 10.37-11.47 RRab 0.45234 1612.69 RR Leo 10258228 155.9672 + 9.7568 13.39-15.07 RRab 0.51515 1597.83 RV Leo 4958650 162.2346 +42.6710 14.02-16.00 RRab 0.46221 1325.78 AG UMa 4959081 162.5788 +42.5689 12.68-13.34 RRab 0.63569 1349.72 BK UMa 4966144 167.6012 +42.8148 13.61-14.91 RRab 0.57702 1354.69 AP UMa 13108143 167.8412 - 5.8921 11.84-12.89 RRab 0.67288 1608.73 TV Leo 10332183 168.0193 +18.5014 12.97-14.03 RRab 0.49969 1601.00 BT Leo 4967969 169.0956 +41.2337 13.27-14.19 RRc 0.39964 1598.75 BN UMa 13132177 170.8437 + 6.6348 12.39-12.96 RRab 0.57207 1601.88 AN Leo 15854955 171.6235 -17.9144 11.39-12.27 RRab 0.41147 1278.77 W Crt 7585322 172.4522 +30.0674 9.67-10.46 RRab 0.55685 1629.65 TU UMa 13155114 173.4760 - 0.0338 10.78-11.65 RRab 0.62640 1517.92 SS Leo 10349008 175.8061 +21.1358 14.76-15.98 RRab 0.60045 1557.80 BI Leo 10350274 176.6358 +16.2312 12.20-12.74 RRab 0.73879 1557.80 CF Leo 7606568 177.2952 +28.0071 14.70-15.95 RRab 0.59774 1311.88 EF Leo 18621888 177.4848 -35.6476 13.30-14.70 RRab 0.41643 1630.82 DQ Hya 18624468 178.5012 -31.2608 12.83-13.73 RRab 0.56797 1573.91 DT Hya 10354196 179.0594 +21.2585 13.47-14.51 RRab 0.36168 1629.80 CM Leo 10388636 181.8916 +20.1027 14.39-15.88 RRab 0.64606 1616.86 WZ Com 10389967 182.7059 +20.2942 14.02-15.42 RRab 0.73668 1353.71 YY Com 18669749 183.0693 -22.9093 13.34-14.33 RRab 0.35763 1600.89 SY Crv 7630300 183.7107 +33.1017 14.35-15.48 RRab 0.69408 1627.84 CK Com 10395082 185.5892 +16.2926 13.40-14.29 RRab 0.59795 1607.71 TY Com 10395583 185.8594 +16.0834 13.56-14.30 RRab 0.71836 1582.86 CR Com 7621236 187.0831 +24.9551 14.28-15.36 RRab 0.75790 1353.70 CY Com 10397808 187.2322 +16.7505 14.29-15.25 RRc 0.31014 1580.88 AH Com ---------- ---------- ----------- ------------- ------ --------- ---------- ---------- : Confirmed RR Lyrae stars from the GCVS excluded by the selection criteria[]{data-label="missed"} ---------- ---------- ----------- ------------- ------ --------- ---------- ----------- NSVS ID NSVS Mag. Type Epoch Cross ID 18691063 188.9169 -26.6618 12.78-13.57 RRab 0.57738 1318.70 EM Hya 7643255 188.9835 +37.2072 12.37-13.18 RRab 0.66815 1288.71 SV CVn 10404312 189.0420 +22.3952 14.08-14.94 RRab 0.64553 1310.90 AO Com 7656842 189.4881 +29.9685 14.07-15.41 RRab 0.47242 1311.73 FV Com 10405541 189.8281 +22.0545 14.11-15.30 RRab 0.59366 1631.75 AP Com 7644920 190.2293 +37.0855 12.44-13.69 RRab 0.44166 1307.73 SW CVn 10408059 191.4180 +18.2132 13.87-15.98 RRab 0.34449 1567.79 AT Com 7647755 192.5028 +31.1399 13.91-15.33 RRab 0.53657 1558.89 TX Com 7663103 193.7313 +23.2571 14.24-15.20 RRab 0.54165 1630.65 BD Com 7665219 194.9679 +30.2426 13.97-15.22 RRab 0.53270 1310.73 UW Com 10446173 196.2832 +23.2786 12.12-13.17 RRab 0.46891 1283.79 RY Com 10451446 199.2228 +17.1083 13.61-14.83 RRab 0.56519 1286.89 RT Com 13247042 200.9107 - 4.3612 13.79-15.10 RRab 0.68409 1287.75 ZZ Vir 10456372 202.0413 +20.2261 13.73-15.11 RRab 0.45226 1356.72 BK Com 7678638 203.6287 +29.3042 14.55-15.88 RRab 0.52373 1318.74 WW CVn 7679165 203.7309 +26.4499 14.29-15.85 RRab 0.57358 1364.74 BT Com 7695165 203.9410 +31.8613 14.61-15.99 RRab 0.72926 1321.88 WX CVn 7719920 207.4268 +29.0945 13.51-14.49 RRc 0.32811 1598.76 XZ CVn 7721875 208.6414 +27.5261 13.57-14.46 RRab 0.54854 1414.69 BD Boo 2730126 209.1790 +52.9028 14.36-15.75 RRab 0.58656 1553.85 DQ UMa 16119799 211.4042 - 7.2480 12.47-13.53 RRab 0.55188 1280.77 AD Vir 10534195 221.5012 +23.3126 13.33-15.07 RRab 0.49276 1370.76 RU Boo 7786875 223.7970 +28.0422 14.15-15.52 RRab 0.62210 1328.75 DF Boo 16238330 229.7946 - 6.9753 13.06-14.08 RRab 0.51414 1338.80 TX Lib 5182357 230.5902 +52.4788 12.94-13.42 RRc 0.29619 1356.76 CU Boo 16244467 231.6604 -11.5411 13.34-14.18 RRab 0.58170 1335.80 BM Lib 16253000 232.4847 -12.8839 13.14-14.12 RRc 0.38110 1364.76 KN Lib 16283173 232.5322 -20.7175 13.01-14.17 RRab 0.49308 1320.80 BR Lib 16290714 234.6066 -16.3606 13.92-15.33 RRab 0.65303 1606.91 CM Lib 16292602 234.9454 -21.0137 12.72-13.86 RRab 0.47814 1287.93 VV Lib 16299448 236.6865 -16.6028 13.83-14.78 RRab 0.33063 1617.99 DF Lib 16267797 236.7840 -12.4134 13.59-14.67 RRab 0.64175 1288.93 DH Lib 16300075 236.8419 -16.1075 13.91-14.65 RRab 0.67832 1275.92 DI Lib 16300388 236.8881 -18.3146 13.70-15.04 RRab 0.48669 1332.71 DK Lib 13488696 240.9241 + 0.5999 11.07-11.93 RRab 0.48755 1390.75 AV Ser 7871804 241.5910 +28.1177 12.60-13.59 RRab 0.92924 1606.85 UY CrB 16355596 242.1513 -12.1078 13.90-15.30 RRab 0.52936 1318.76 V783 Sco 13501020 244.4379 - 5.0474 13.75-14.63 RRab 0.59085 1389.75 V688 Oph 13504164 245.2522 - 4.2659 13.13-14.06 RRab 0.57106 1307.77 V1021 Oph 13504802 245.4169 - 1.5241 13.50-14.40 RRab 0.59099 1600.01 EN Ser 13504921 245.4456 - 4.0974 12.29-13.40 RRab 0.62394 1414.71 V1023 Oph 16372426 246.1724 - 6.5410 10.63-11.22 RRab 0.39716 1359.77 V445 Oph 13540829 246.2069 + 8.0703 12.81-13.75 RRab 0.64510 1318.70 V1013 Her 16372448 246.2963 -10.5235 11.57-12.53 RRab 0.44901 1335.80 V413 Oph 7911816 246.5109 +31.7949 14.19-15.80 RRab 0.55870 1325.73 GR Her 13586443 247.5126 - 0.9989 14.00-15.23 RRab 0.55571 1287.91 V714 Oph 13588639 248.0947 - 1.5887 14.09-14.96 RRab 0.61898 1325.80 V722 Oph 7888073 248.1430 +30.3420 13.97-15.44 RRab 0.51723 1414.75 V596 Her 16389121 249.1000 - 8.2700 14.30-15.66 RRab 0.52874 1340.81 V731 Oph 7891261 249.1260 +26.5370 13.81-14.76 RRab 0.75062 1330.75 V599 Her 7919042 249.2419 +31.7355 14.60-15.99 RRab 0.51815 1452.63 GW Her 7918765 249.2806 +34.9485 14.28-15.60 RRab 0.54962 1275.86 GX Her 7922831 250.9580 +37.4633 14.32-15.50 RRab 0.51756 1353.84 HI Her 7923973 251.2888 +36.3140 14.60-15.96 RRab 0.63909 1390.74 HK Her 7901008 252.1392 +23.5834 14.39-15.40 RRab 0.63741 1390.68 V711 Her 7927408 252.4163 +34.9720 14.78-15.95 RRab 0.54281 1482.59 HL Her 5259123 254.3946 +41.5291 13.15-13.86 RRab 0.68237 1353.78 V863 Her 5300615 256.3005 +47.9874 13.32-14.65 RRab 0.64115 1364.79 V715 Her 7948267 256.5882 +31.8884 13.83-15.28 RRab 0.54664 1358.70 V619 Her 5300936 256.7977 +45.8238 13.76-15.19 RRab 0.71430 1450.62 V716 Her ---------- ---------- ----------- ------------- ------ --------- ---------- ----------- : continued ---------- ---------- ----------- ------------- ------ --------- ---------- ----------- NSVS ID NSVS Mag. Type Epoch Cross ID 5329963 257.7005 +41.6200 14.30-15.46 RRab 0.52792 1495.66 V721 Her 5331289 257.8341 +44.9010 14.07-14.98 RRab 0.62099 1390.68 V794 Her 10815361 258.6470 +22.3959 14.10-15.26 RRab 0.67960 1325.74 V379 Her 5334360 259.4222 +41.3285 14.11-15.72 RRab 0.57093 1328.74 V727 Her 13682138 259.6576 + 8.2280 13.27-14.19 RRab 0.50969 1373.78 V864 Oph 7995290 261.6600 +26.9379 12.99-13.82 RRab 0.80619 1300.89 V486 Her 10846112 263.0557 +21.3888 14.18-15.79 RRab 0.71793 1299.75 V494 Her 13714797 264.0379 + 8.1653 13.92-15.33 RRab 0.54693 1312.92 V788 Oph 8008136 264.8599 +29.2925 14.01-15.82 RRab 0.48305 1421.81 EG Her 8010287 265.2337 +24.0480 13.14-13.83 RRab 0.63393 1362.70 V514 Her 8012178 265.8278 +28.2539 14.30-15.74 RRab 0.62887 1336.91 LX Her 13733129 267.1761 + 7.0953 13.17-14.13 RRab 0.63806 1306.78 V2210 Oph 10910335 267.3721 +12.2314 13.70-15.00 RRab 0.56922 1382.92 V829 Oph 8062879 268.3597 +31.7131 14.37-15.42 RRab 0.62467 1397.73 V523 Her 8071159 270.7539 +36.4794 13.75-14.60 RRab 0.58324 1404.72 OW Her 5378465 273.1863 +38.7990 14.30-15.50 RRab 0.62388 1474.59 QQ Her 5380594 274.0383 +40.7120 14.79-15.98 RRab 0.56746 1296.74 HW Lyr 8148286 274.6596 +28.4195 13.86-15.13 RRab 0.40380 1481.65 V576 Her 5484203 277.6858 +38.3985 13.43-14.44 RRab 0.60141 1461.80 KN Lyr 11096176 278.3474 +13.3719 13.46-14.31 RRab 0.48995 1456.68 V633 Her 8179855 280.5579 +28.3785 13.19-14.50 RRab 0.49748 1615.86 AW Lyr 8187901 282.1015 +30.5718 13.72-14.87 RRab 0.51740 1414.75 CT Lyr 5462833 282.4888 +50.5869 13.39-14.70 RRab 0.52666 1328.74 DT Dra 8249293 283.9339 +33.5673 14.00-15.07 RRab 0.61270 1485.64 BH Lyr 8213965 286.4686 +26.5574 13.07-13.77 RRab 0.44686 1359.75 ZZ Lyr 8291277 288.4421 +33.4922 12.49-12.98 RRab 0.51582 1408.87 WW Lyr 8293252 288.7248 +34.7667 13.00-13.88 RRab 0.70298 1439.82 EN Lyr 5543125 291.3664 +42.7855 7.67- 8.46 RRab 0.56689 1448.67 RR Lyr 5595928 292.5520 +50.8060 12.55-14.01 RRab 0.49896 1308.72 V1949 Cyg 11327183 298.1224 +12.1705 13.16-14.27 RRab 0.48085 1489.66 OZ Aql 11336237 299.5947 + 8.2429 12.99-14.06 RRab 0.26805 1450.69 V1070 Aql 8469354 303.0252 +34.6433 11.72-12.25 RRab 0.73421 1328.75 V1823 Cyg 11415440 303.3187 +17.5105 13.27-14.03 RRab 0.50474 1448.69 FI Sge 11451943 306.7503 +19.2081 12.31-12.80 RRab 0.61540 1453.84 FF Del 14284108 308.1314 + 0.5852 10.65-11.71 RRab 0.57812 1448.64 V341 Aql 14329706 308.4255 - 5.6472 13.12-14.39 RRab 0.38910 1382.79 CH Aql 11468866 309.4474 +18.9241 13.66-14.60 RRab 0.58007 1325.80 EO Del 17144390 310.1009 -13.0663 13.45-14.84 RRab 0.49423 1364.76 WZ Aqr 14312324 313.0795 + 7.1463 12.82-14.81 RRab 0.56671 1415.90 LX Del 17191429 313.5147 -17.4642 13.65-14.50 RRab 0.62568 1356.79 UX Cap 11544059 313.7589 +13.8015 14.25-15.45 RRab 0.53523 1335.78 EH Del 17206972 318.1859 -17.4081 14.22-15.58 RRab 0.46562 1448.70 WY Cap 14412076 318.8245 + 0.0762 10.89-11.97 RRab 0.45937 1452.64 SW Aqr 17212681 319.8852 -15.1171 11.34-11.79 RRc 0.27347 1483.61 YZ Cap 8777826 325.5937 +22.1973 13.92-15.10 RRab 0.71856 1465.82 CI Peg 17291543 326.1727 -17.2097 13.30-14.51 RRab 0.66138 1437.81 UV Cap 17329438 333.0125 - 8.7628 13.65-14.68 RRab 0.63686 1483.67 FV Aqr 14522663 333.8568 + 6.8228 9.63-10.03 RRc 0.25552 1478.61 DH Peg 14498391 334.1456 - 3.8156 12.14-12.93 RRab 1.02268 1400.84 FY Aqr 14499328 334.4718 - 0.0929 13.94-15.04 RRab 0.65569 1348.87 GG Aqr 6044959 334.7435 +49.7400 12.97-14.07 RRab 0.63014 1353.84 XZ Lac 17339901 336.9534 - 7.4837 12.28-13.37 RRab 0.46966 1403.79 BN Aqr 17367595 339.3586 -18.3720 14.32-15.93 RRab 0.50594 1494.74 BG Aqr 8981440 348.4161 +27.9947 14.08-15.25 RRab 0.56901 1414.70 AL Peg 19991511 349.8217 -24.2162 11.06-11.70 RRab 0.63391 1426.76 DN Aqr 1521546 351.5644 +57.3984 13.73-15.00 RRab 0.57077 1505.61 V845 Cas 17438805 358.6738 -33.4813 12.49-13.69 RRab 0.61673 1449.79 TW Scl 17425254 358.8430 -26.3020 12.37-13.08 RRab 0.72742 1409.83 TX Scl ---------- ---------- ----------- ------------- ------ --------- ---------- ----------- : continued --------------- ---------- ---------- ---------- ---------------- ------ Name NSVS ID Blazhko period OV And 3656843 5.1850 +40.8276 0.47060  27: GSC 0607-0591 9149730 11.9844 +11.7048 0.45570  55: DR And 6429832 16.2945 +34.2184 0.56307  57: UX Tri 6471887 26.3959 +31.3801 0.46690  45  RV Cet 14762124 33.8121 -10.8001 0.62340 117  FM Per 4241456 60.8624 +47.9978 0.48927  20  AH Cam \* 2094283 61.6628 +55.4999 0.36873  11  NSV 2724 12370363 88.6547 + 4.9031 0.47908  28  NSV 4034 2480497 126.1037 +65.7181 0.59909  65: TT Cnc \* 10141483 128.2299 +13.1912 0.56344  88  GSC 1948-1733 7404884 130.0100 +27.7248 0.50202  42: GSC 4378-1934 777167 133.0631 +70.4399 0.51871  46: GSC 0275-0090 13144199 175.8841 + 2.6987 0.59500  59: Z CVn \* 5034191 192.4388 +43.7736 0.65387  21: GSC 1454-0093 10452666 199.9771 +19.8991 0.60100  74  RV UMa \* 2723669 203.3252 +53.9872 0.46808  93: SS CVn 5089959 207.0666 +39.9010 0.47854  97  TV Boo \* 5099771 214.1517 +42.3597 0.31256  10  GSC 0318-0905 13340626 215.7734 + 1.9001 0.44693  48  AR Ser \* 13439617 233.3782 + 2.7768 0.57540  63: AR Her \* 5207933 240.1339 +46.9240 0.47000  32  LS Her 10678349 240.5157 +17.4807 0.23081  13  RW Dra \* 2830506 248.8814 +57.8394 0.44295  41  NSV 8170 10767118 256.1358 +14.4427 0.55102  39  V365 Her \* 10806471 256.4160 +21.5167 0.61303  40: DL Her \* 10783891 260.0929 +14.5113 0.59164  34  V421 Her 5344210 263.0232 +39.7586 0.55675  56: AV Dra 5401723 269.9339 +51.8839 0.55556  96  BD Dra 1132043 274.4687 +77.2975 0.58901  24  GSC 1581-1784 11037662 277.3025 +21.0728 0.59112  23  RZ Lyr \* 8234128 280.9082 +32.7979 0.51126  26: NR Lyr 5525581 287.1147 +38.8118 0.68201  27: GSC 1667-1182 11622872 320.1423 +18.6219 0.56232  84  AE Peg 11780560 336.8397 +16.8046 0.49675  23: --------------- ---------- ---------- ---------- ---------------- ------ : RR Lyrae stars showing the Blazhko effect[]{data-label="blazhko"} --------------- ---------- ---------- ---------- --------- -------------- -------- Name NSVS ID Period ratio GSC 4868-0831 12919472 131.9456 - 3.6501 0.56392 0.42081 0.7462 GSC 3047-0176 5152353 223.4165 +40.5286 0.47462 0.35298 0.7437 V372 Ser 13399252 229.3958 - 1.0889 0.47135 0.35072 0.7441 GSC 3059-0636 5222076 236.6077 +44.3125 0.49404 0.36691 0.7427 V458 Her 10809333 257.1295 +18.5213 0.48374 0.35998 0.7442 GSC 4421-1234 1088957 258.3326 +69.1317 0.54080 0.40320 0.7456 --------------- ---------- ---------- ---------- --------- -------------- -------- : Double mode RR Lyrae stars[]{data-label="rrd"} \[lastpage\]

--- abstract: 'Before the launch of the Rossi X-ray Timing Explorer ([*RXTE*]{}) it was recognized that neutron star accretion disks could extend inward to very near the neutron star surface, and thus be governed by millisecond timescales. Previous missions lacked the sensitivity to detect them. The kilohertz quasi-periodic oscillations (QPO) that [*RXTE*]{} discovered are often, but not always, evident in the X-ray flux. In 8 years [*RXTE*]{} has found kilohertz signals in about a fourth of 100 low-mass X-ray binaries (LMXB) containing neutron stars. The observed power spectra have simple dominant features, the two kilohertz oscillations, a low frequency oscillation, and band-limited white noise. They vary systematically with changes in other source properties and offer the possibility of comparison with model predictions. New information from the millisecond pulsars resolves some questions about the relations of the QPO and the spin. Coherence, energy spectrum and time lag measurements have indicated systematic behaviors, which should constrain mechanisms.' author: - Jean Swank bibliography: - 'swankj.bib' title: 'Quasi-Periodic Oscillations from Low-mass X-Ray Binaries with Neutron Stars' --- [ address=[Goddard Space Flight Center, Greenbelt, MD 20740]{} ]{} A Brief History of LMXB QPO =========================== Soon after the discoveries of Sco X–1 and Cyg X–2, it was realized that accretion onto a neutron star in a binary was a likely source of the X-ray emission. But while clear pulsations were seen in the flux from Hex X–1, these sources exhibited no periodic signal. The possibility was raised that accretion over a long lifetime had spun up the neutron star to frequencies higher than could have been measured in the early observations [@alpar82]. Successive missions strove to increase their sensitivity to higher frequencies. [*EXOSAT*]{} and [*Ginga*]{} pushed the frontier to about 200 Hz. The world before [*RXTE*]{} is below 200 Hz. [*EXOSAT*]{} discovered timing signals, but quasi-periodic signals rather than the coherent clock of the neutron star. QPO were found in many X-ray sources in the galactic bulge. The frequencies varied in the 1-50 Hz range. [*EXOSAT*]{} proportional counter data provided spectral information at the same time. @HasvdK89 showed that the spectral variations fell in two categories, denoted “Z” and “Atoll” and that the QPO frequencies depended on the source’s position in a plot of hard versus soft “colors” or energy ratios. The widths and amplitudes varied also in systematic ways. @Wijnands01 compiled the [*RXTE*]{} version of a figure summarizing the properties of both Z and Atoll sources. At first the frequency appeared to be positively correlated with the X-ray luminosity and a simple explanation was attractive, the magnetic beat frequency model [@alpar85]. The accretion rate through the disc, should be stopped eventually, by a magnetosphere due to the neutron star, but closer to the neutron star because the magnetic field was much weaker. The Kepler period of gas in the disk would beat with the spin frequency of the neutron star to cause brightness oscillations. Changing the accretion rate would change the magnetospheric radius, the Kepler frequency at the boundary, and thus the beat frequency. It implied spin rates of 50-350 Hz in several cases [@GL92]. However the model was not a satisfactory fit to the data from several sources and there was evidence that the luminosity was not a good measure of the accretion rate. The character of bursts and their recurrence rate changed in 4U 1636–53 as it moved through the “Atoll” pattern [@vdK90], while the luminosity did not increase smoothly. In Cyg X–2 [@Has90; @Vrtilek90] and Sco X–1 [@Vrtilek91] UV emission decreased as the X-ray flux increased, while the magnetospheric beat frequency model implied it should increase [@Has90]. Nevertheless coherent oscillations were sought [@Vaughan94] and upper limits of less than 0.5 % were achieved for frequencies below 200 Hz. The idea that the magnetic fields of the neutrons stars are $10^{2} - 10^{4}$ lower than the $10^{12}$ G of “classical” pulsars was advanced to explain the failure to detect strongly channeled accretion flow that should show up as pulsations and the higher frequencies. Kluzniak and Wagoner saw that accretion disks around low magnetic field neutron stars could be very interesting if the equation of state of nuclear matter meant that neutron stars were inside the innermost stable orbit of orbiting material. The accretion disk could extend down to the innermost stable orbit and be truncated there rather than at the magnetosphere. A signal might even indicate the Kepler frequency of the inner most stable orbit [@Kluz85; @Kluz90]. These papers foresaw that signals bearing the imprint of General Relativity could come from these sources. In 1996, [*RXTE*]{} began observing and the first observations of the Atoll source 4U 1728–34 by @Stroh96 and the Z source Sco X-1 by @vdK96 showed signals with frequencies in the range that orbits close to neutron stars would have. Figure 1 shows several of the important aspects of the kilohertz QPO discovered in the flux from 21 LMXB. As the count rates vary, the QPO center frequencies vary significantly compared to the widths of the features. The features in the Atoll and the Z source are very similar. The phenomena and the physical models that have been explored during the 8 yr since the discovery have been described in several review articles [See @vdK00; @Wijnands01]. Looking back at why [*RXTE*]{} could detect the signals while previous missions did not, the increase in sensitivity came from several factors. The number of sigmas of the detection of a QPO feature can be expressed as $$n_{\sigma} = (1/2) S^{2}/(S+B)(rms/S)^{2} \surd (T/\Delta \nu)$$ Here, S is the source count rate, B the background rate, rms the root mean square variance in S, T the duration of the observation and $\Delta \nu$ the width of the QPO feature. This scales as the detector area. The PCA has observed with a maximum of 6250 cm$^2$, compared to [*EXOSAT*]{}’s 1600 cm$^2$, but [*Ginga*]{} had 4000 cm$^2$, and did not detect kiloHertz oscillations because of insufficient time resolution. Other factors - background, noise, dead time, low duty cycle of observations - have influenced the sensitivity to these phenomena. So far, [*RXTE*]{} has been the only instrument to detect them. Observed Characteristics of the Two KHz QPO =========================================== Frequency Range --------------- Low-mass X-ray binaries have a wide range in X-ray luminosity, from apparently exceeding the Eddington limit for a neutron star of 1.4 M to 0.5 % of it. Yet for sources at both extremes the frequencies observed for the upper of the two kilohertz oscillations range from approximately 300 Hz to 1100 Hz. This was apparent early in the exploration of QPOs and remains true now [@SSZ98; @vdK00]. (The highest frequency, although only 2.6 $\sigma$, is 1330 Hz for 4U 0614+09 [@vdK00; @Straaten00]). @Zhang97 deduced from this that the frequency must depend only on properties of the neutron star, independent of the mass accretion rate. It could either be the radius of the neutron star or the radius of the inner-most stable orbit (ISCO). It seemed more likely to be the ISCO than the radius, in that surface behavior would be more likely to depend on the accretion rate. @Kaaret97 also pointed out that if the ISCO was responsible for the peak frequency, it was a test of General Relativity. As the number of sources accumulated, @Ford00 exhibited that this independence of the frequency range on the luminosity continued to hold, using fits to the simultaneous spectral data to determine more accurate luminosities. Figure 2 shows that there is a slight trend for the lower luminosity bursters to exhibit highest frequencies a little higher than those of the Z sources at high luminosity. Interestingly, none of the bright Atoll sources (e.g. GX 3+1, GX 9+9, GX 9+1, GX 13+1) have shown oscillations. They fill in the luminosity range between the brightest of the bursters, 4U 1820–30 and the Z sources. [![Frequencies of upper kilohertz oscillations for LMXB of the range of luminosities. These values are for a subset of all the observations. The highest frequency points for 4U 1636-53 are of disputed significance.[]{data-label="fig:2"}](swankj_f2.ps "fig:")]{} We now believe we know the rotation period P for 16 of the LMXB which either have coherent oscillations or have oscillations during bursts, For some of these we have a good estimate of the distance and therefore the X-ray luminosity, L$_X$. The accretion rate onto the neutron star could be through the disk or from a corona, so that for the accretion rate in the disk, $dM/dt \leq L_X/(GM/R)$. If we suppose, as did @White97, that adiabatic evolution keeps the neutron star rotation period approximately in equilibrium with an average L$_X$, this is related to P and $\mu = B R^3$. Table 1 gives values of B and the corotation radius R for 4 burst sources and also for 5 Z sources (for M=1.4 M$_{\odot}$ and R = 10 km). We don’t have any burst oscillations for a Z source. But, the spin period is believed to be within 15 % of either the difference between the two kilohertz frequencies, or twice it. The values of $\Delta \nu$ are relatively high, so that twice it would make these rates the highest. The values for B and the corotation radius R$_{co}$ in Table 1 assume a spin rate $\nu_s \approx \Delta \nu$. (If $\nu_s$ is twice that, B would be 2.25 times lower, and R$_{co}$ 1.6 times lower.) In this simple treatment the magnetospheric radius is $\propto L_{X}^{-2/7}$. Radiation drag and magnetospheric reaction are neglected and approximate values used, so that the estimates can only be expected to hold to about a factor of 2. The resulting values for B imply that for the Atoll bursters B is a few times $10^8$ Gauss, and for the Z sources a few times $10^9$ Gauss. For both cases R$_{co}$ is about 2-3 times R$_{ISCO}$ = 6 G M/c$^2$ =12.5 (M/1.4M$_{\odot}$) km, neglecting relatively small relativistic corrections. A picture in which the disk extends close to the ISCO and approaches it according to some scaling of the luminosity appears quantitatively justified. ------------ ----- ----- ---- ----- 4U 1728–34 363 1.0 3 2.6 4U 1636–53 581 3.2 3 1.9 KS 1731–26 524 6.0 5 2.0 4U 1702–43 330 1.2 4 2.7 Sco X–1 307 40 25 1.2 GX 5–1 298 37 25 2.9 GX 17+2 294 22 19 3.0 Cyg X–2 346 29 17 2.7 GX 340+0 339 30 19 2.7 ------------ ----- ----- ---- ----- : LMXB parameters assuming Spin Equilibrium[]{data-label="tab:a"} Size of the Emission Region --------------------------- Models that have been advanced to explain the oscillations do not agree on the mechanism for producing them. It is not in the scope of this paper to delve into them. But if the disk extends inside a radius 2-3 times the ISCO, the inner rim of the disk as well as the surface of the neutron star could be a source of X-rays. A statistically demanding measurements is that of time delays of photons of different energy bands. This has been done for data in which a QPO is very strong, in particular the lower of the two frequencies, for 4U 1636–53, 4U 1608–52, 4U 1828–34, and 4U 1702–429. The result was initially surprising. In the case of the black hole Cygnus X-1, in the low hard state hard photons are delayed behind soft photons and it has been understood in terms of a coronal model, in which a corona of high temperature electrons is cooled by Compton scattering low energy photons originating in the disk. In order to explain some parts of the energy spectra of the low mass X-ray binaries, Comptonization off a small corona has been discussed. Some of the same elements undoubtedly should be in both Cyg X-1 and the low mass X-ray binaries with neutron stars. The sign of the delay for the QPO in several sources is the opposite. This suggests the intrinsic properties of the emission regions are more likely to cause the time delay than is Comptonization. Figure 3 shows the delay as a function of energy for 4U 1636–536 and 4U 1608–522 [@Kaaret99]. Converting the time delay to a light travel time, the contributions can be no more than 20 km apart. This would be consistent with emission from the neutron star surface and from the inner edge, or from different parts of the disk. Such measurements would benefit from a higher throughput instrument. [![ Time delays of soft behind hard photons.[]{data-label="fig:3"}](swankj_f3.ps "fig:")]{} Saturation of the Frequency --------------------------- Several of the LMXB go through strong quasi-regular long time-scale modulations. For 4U 1820–30 and 4U 1705–44 the time scale is about half a year and the modulations are factors of 5 and 10, respectively. For the transients 4U 1608–52 and Aql X–1 the amplitude is higher and the time scale longer (1-2 yr). For 4U 1636–54 and 4U 1728–34 the amplitude is a factor of 2 and the time scale 30 days. Figure 4 (top) shows some [*RXTE*]{} ASM data for 4U 1820–30 [@Bloser00]. For 4U 1820–30 the QPO frequencies increase as the source goes from the “Island” to “Lower Banana” part of the color-color diagram (CCD) shown in Figure 4 (middle). In the “Upper Banana” part of the diagram, QPO cannot be detected. The parameter S$_a$ tracks position in the CCD and the increase of QPO frequency. For 4U 1820–30, the $\geq$ 2 keV flux and luminosity approximately track S$_a$ as well, while in other sources, the situation is more complicated. But in 4U 1820–30, as shown in Figure 4 (bottom), the frequency of both of the kilohertz oscillations increases with S$_a$ to a certain point and while S$_a$ increases further, the QPO frequency has saturated. Because in this case S$_a$ is correlated with luminosity, the source behaves as if the accretion rate increases with the luminosity and pushes the inner disk inwards, with corresponding increases in the Keplerian frequency at the inner edge of the disk. Once it reaches the ISCO the disk could not exist inside of it, so no higher frequencies would be possible. It should be possible to continue to have higher flux moving through the disk. @Zhang98 first found that the frequency saturates. It was confirmed by @Kaaret2Zhang99 in a different data set. @Bloser00 has changed the plot abscissa from luminosity to S$_a$. Unification of Atoll Source Parallel Tracks ------------------------------------------- One of the confusing aspects of the kilohertz oscillations has been that frequency was correlated with luminosity locally in time, but not over long time scales [@Mendez99; @Zhang2Zhang98]. As shown in Figure 5, multiple X-ray luminosities have the same frequency. In fact these tracks map onto a color- color diagram with an S$_a$ defined such that the frequency is a monotonic function of S$_a$. The frequency and the spectrum are determined by S$_a$, while the luminosity is not. It is supposed that S$_a$ represents a relevant accretion rate. While the X-ray luminosity is apparently not the independent variable determining the frequency and other properties of the QPO, the system is simple enough that there is one controlling independent variable. S$_a$ is unsatisfactory in not yet being understood in terms of physical quantities. But it does determine the properties. @Mendez01 have shown that the QPO of 3 Atoll bursters have almost identical energy spectra and runs of rms amplitude as a function of the centroid frequency. They have examined examples of multiple tracks and shown that for QPO on different tracks, when the frequency is the same, the rms amplitude is nearly the same. It is not the case that an extra source of luminosity varies that does not participate in the oscillation. A suggested simple picture was that there were two different flows, one through the disk, one a radial or coronal flow. This model is clearly ruled out. A phenomenological model that generates multiple tracks was constructed by @vdK01. He assumed there are two independent time scales, one on the order of hours on which frequency and luminosity is correlated, and one on the other of days or weeks, which changes the scale of the luminosity. Thus the frequency is correlated, not to the luminosity or an accretion rate directly, but to a form of a fractional accretion rate. The whole surface of an accreting disk can contribute to evaporation into a radial flow, while fluctuations in the disk must diffuse through the disk. A quantitative physical model has not been worked out, but some general predictions are made which could be borne out by persistent observations. Méndez is exploring the characteristics of transitions, which should be sequential in that model. The spectrum determines the place in the color-color diagram. The frequency is determined by that as well. These both could depend on the radius of a ring where oscillations occur. Different luminosities may correspond to different rates of flow through this region. Van der Klis hypothesizes that a range of local equilibria of radiation pressure, ram pressure and magnetic pressure are possible which allow the spectrum and oscillation forming conditions to be the same. Long term modulations have in several sources (Her X–1, LMC X–4, SMC X–1, and SS433) appeared to be due to shadowing by a tilted and precessing accretion disk. Energy independent obscuration might be able to explain aspects of the multiple tracks, but the time scales for the long term changes are generally very irregular, if not chaotic. The simulations carried out by @vdK01, which show promise of representing important aspects of the behavior, assume a random walk in the disk accretion rate. Further, some long-term variations are the same as motion in the Atoll diagram. [![Properties of the kilohertz QPO in the Z source GX 17+2, (left) $\nu 1$ and (right) $\nu 2$. S$_Z$ is defined along the color-color diagram, such as to have the value 1.0 at the Horizontal Branch to Normal Branch inflection and the value 2.0 at the Normal Branch to Flaring Branch transition.[]{data-label="fig:6"}](swankj_f6.ps "fig:")]{} Z Sources --------- While there have been suggestions of how Atoll and Z sources may be related, the color-color diagrams and the ranges of luminosity differ. The Z sources have their own family characteristics. Ever since observations of Cyg X–2 showed changes in X-ray luminosity anticorrelated with UV flux [@Has90; @Vrtilek90], we have had to live with the idea that the accretion rate appears to increase along the Z, although the observed X-ray luminosity increases along the “Horizontal Branch”, decreases along the “Normal Branch” and increases again in the “Flaring Branch”. In Figure 6 @Homan02 shows typical dependences of the two kilohertz QPO properties on the S$_Z$ parameter as it moves along the “Z” for GX 17+2. The shape of the color versus intensity diagram that inspired the name “Z” depends on the instrumental definition of the color. As in the case of the Atoll sources, some variable determines the spectrum and the frequencies, a parameter with which the X-ray luminosity is locally correlated or anticorrelated. Summary Parameters ------------------ In general the QPO have higher amplitude at higher energies (at least to about 20 keV) both in the case of the Atoll and the Z sources. The amplitudes are higher for the Atoll sources than for the Z sources. Although not invariably, there is a tendency for the lower of the two kilohertz oscillations to be stronger and narrower than the upper one. Table 2 shows the range of properties. -------------------- ----------- ----------- rms (E$\leq$6 keV) 15 % 5 % rms (E $>$ 6 keV) 40 % 13 % $\Delta \nu 1$ 5-100 Hz 70-200 Hz $\Delta \nu 2$ 10-200 HZ 60-300 Hz -------------------- ----------- ----------- : Amplitudes and Widths of the Two Kilohertz QPO.[]{data-label="tab:b"} Relation of the Difference Frequency to the Neutron Star Spin ------------------------------------------------------------- The [*RXTE*]{} observation of SAX J1808.4-3658 in October 2002 confirmed the evidence of the BeppoSax observation [@in'tZand01] that bursts from this source can exhibit pulsations. [*RXTE*]{} measured the frequency and amplitude of the burst oscillations accurately. In addition two kilohertz QPO were detected during a part of the outburst. It was not known whether these QPO would be possible under the flow conditions that allow the coherent pulsations to be seen. We now know that they can coexist under some conditions. @Wijnands03 found that two QPO appeared near the peak of the outburst. As the outburst luminosity declined a single, broad, QPO persisted and drifted lower in frequency, moving through the pulse frequency with little apparent affect, as shown in Figure 7. When two QPO were present, the difference was consistent with being half the spin frequency. SAX J1808.4-3658 thus has shown that the spin frequency is the frequency seen in the bursts, while the difference can be about half that. A strong upper limit was established on power at the 401 Hz sub-harmonic. [![SAX J1808.4-3658 power spectra[]{data-label="fig:7"}](swankj_f7_c.ps "fig:")]{} To first approximation for a given source in which two kilohertz oscillations have been seen, the difference is approximately constant. The data are accurate enough that to the next approximation the differences can be seen to vary, in particular as functions of one of the two frequencies. While the differences are as low as 1% and seldom larger than 15 %, they are significant and systematic. In the case of the relativistic precession model, the difference is specified in terms the central mass and spin, and the radius, as is the azimuthal frequency. It is possible to find fits “in the ball park” of what is required, although the fits are not statistically acceptable. However it is not at all clear whether any physical mechanism can give rise to x-ray emission modulated with the frequencies characteristic of eccentric particle orbits. The Atoll source difference frequencies are plotted in Figure 8 against the lower kilohertz frequency, along with the observed values of Sco X-1. We do not know at this point, whether the fact that the difference frequencies bear such a close relation to the neutron star spins means that they are physically related, or whether the closeness can be coincidental. The Sonic Point Beat Frequency Model appeared to imply that the difference frequency should be the spin. That is now known to be incorrect, although explanations are being explored. Perhaps there is a resonance which drives the spin to stabilize at a multiple of a specific $\Delta \nu$. [![$\Delta \nu$ for Atoll Sources, with comparison to Sco X–1. The dashed line is 1/2 the spin frequency for 4U 1636–536. The dot-dash line is 1/2 the spin frequency for 4U 1608–522. The dotted line is the spin frequency of 4U 1728–34.[]{data-label="fig:8"}](swankj_f8.ps "fig:")]{} Low Frequency Oscillations ========================== Low frequency QPO (below about 50 Hz) are present in various forms in LMXBs with neutron stars and in systems with black holes. They exhibit a variety of strengths and coherence ($\nu/\Delta \nu$). Some characteristics, notably band-limited white noise with a break frequency, and a QPO near the break frequency and closely correlated with it, appear in Atoll sources in the Island state, in Z sources on the Horizontal Branch, and in black hole sources in “low hard” and “intermediate” states. In some cases this QPO is strong and a first harmonic is observed. Sometimes a weak sub-harmonic appears (e.g.[@JonkerGX5; @JonkerGX340]). The strongest low frequency QPO is notable for being correlated with the kilohertz QPO. In particular, it sometimes is proportional to the square of the upper kilohertz frequency [@vdK00]. This correlation in the case of GX 17+2 has been shown to reverse at the highest frequencies [@Homan02]. [*RXTE*]{} observations have shown that the Normal Branch oscillations of Z sources can also move in frequency with the kilohertz oscillations [@vdK00; @JonkerGX5]. It has been pointed out that with certain identifications of features in the power spectra of black hole candidates, the correlations appear to be the same for neutron stars and black holes. It has recently been claimed that even the the white dwarf SS Cygni exhibits oscillations that fall on the same line. The similarity of correlations clearly seems to imply that their explanations are related. At first this may seem to imply that neutron star characteristics, surface and spin, can not be playing a role. If it holds for a white dwarf system, it could imply that the relations come from disk characteristics that are not related to General Relativity. It is likely that phenomena occur in a disk because of properties that do not depend on the nature of the compact object, but whose exact values do depend on how strong gravity is and what the boundary conditions are. A careful evaluation is needed of how exact and general are the correlations. Conclusions =========== In the complex of QPO observed in data from the LMXB containing neutron stars [*RXTE*]{} has found very clearly signatures which are generally characteristic of the sources. The signatures are related to those seen in black hole sources and perhaps to phenomena seen in a few white dwarf systems. But the neutron star systems have exhibited definitive characteristics that allow very quantitative study of the dependences on parameters. Timing is a tool for obtaining spatial resolution and understanding of dynamics. Even without final detailed interpretation, the QPO are clearly entirely consistent with the neutron stars being inside their ISCOs and with the accretion disks penetrating so close to the neutron star as to be affected by General Relativity. The set of signatures is very simple compared to, say, atomic spectra, but intricate enough that the correct explanation of the details will be interesting and important. I thank the many [*RXTE*]{} users whose results I discuss here, the [*RXTE*]{} Science Operations Center members for the roles they play in making the observations successful, and NASA for its support of all aspects of [*RXTE*]{}.

--- abstract: 'In this research, we apply ensembles of Fourier encoded spectra to capture and mine recurring concepts in a data stream environment. Previous research showed that compact versions of Decision Trees can be obtained by applying the Discrete Fourier Transform to accurately capture recurrent concepts in a data stream. However, in highly volatile environments where new concepts emerge often, the approach of encoding each concept in a separate spectrum is no longer viable due to memory overload and thus in this research we present an ensemble approach that addresses this problem. Our empirical results on real world data and synthetic data exhibiting varying degrees of recurrence reveal that the ensemble approach outperforms the single spectrum approach in terms of classification accuracy, memory and execution time.' author: - 'Sripirakas Sakthithasan^\*^ , Russel Pears^\*^, Albert Bifet^\#^ and Bernhard Pfahringer^\#^' title: Use of Ensembles of Fourier Spectra in Capturing Recurrent Concepts in Data Streams --- Introduction ============ In many real world applications, patterns or concepts recur over time. Machine learning applications that model, capture and recognize concept re-occurrence gain significant efficiency and accuracy advantages over systems that simply re-learn concepts each time they re-occur. When such applications include safety and time critical requirements, the need for concept re-use to support decision making becomes even more compelling. Auto-pilot systems sense environmental changes and take appropriate action (classifiers, in the supervised machine learning context) to avoid disasters and to fly smoothly. As environmental conditions change, appropriate actions must be taken in the shortest possible time in the interest of safety. Thus for example, a situation that involves the occurrence of a sudden low pressure area coupled with high winds (a concept that would be captured by a classifier) would require appropriate action to keep the aircraft on a steady trajectory. A machine learning system that is coupled to a flight simulator can learn such concepts in the form of classifiers and store them in a repository for timely re-use when the aircraft is on live flying missions. In live flying mode the autopilot system can quickly re-use the stored classifiers when such situations re-occur. Additionally, in live flying mode, new potentially hazardous situations not experienced in simulator mode can also be learned and stored as classifiers in the repository for future use. In a real world setting, there is an abundance of applications that exhibit such recurring behavior such as stock and sales applications where timely decision making results in improved productivity. Our research setting is a data stream environment where we seek to capture concepts as they occur, store them in highly compressed form in a repository and to re-use such concepts for classification when the need arises in the future. A number of challenges need to be overcome. Firstly, a compression scheme that captures concepts using minimal storage is required as in a high volatile high dimensional environment. Memory overhead will be a prime concern as the number of concepts will grow continuously in time given the unbounded nature of data streams. Secondly, in real-world environments, concepts rarely, if ever, occur in exactly their original form and so a mechanism is needed to recognize partial re-occurrence of concepts. Thirdly, the concept encoding scheme needs to be efficient in order to support high speed data stream environments. In order to meet the above challenges, we extend the work proposed in [@sak:mrc] in a number of ways. In [@sak:mrc] concepts were initially captured using decision trees and the Discrete Fourier Transform (DFT) was applied to encode them into spectra yielding compressed versions of the original decision trees. Firstly, instead of encoding each concept using its own Fourier spectrum, we use an ensemble approach to aggregate individual spectra into a single unified spectrum. This has two advantages, the first of which is reduction of memory overhead. Memory is further reduced as Fourier coefficients that are common between different spectra can be combined into a single coefficient, thus eliminating redundancy. The second advantage arises from the use of an ensemble: new concepts that manifest as a combination of previously occurring concepts already present in the ensemble have a higher likelihood of being recognized, resulting in better accuracy and stability over large segments of the data stream. Secondly, we devise an efficient scheme for spectral energy thresholding that directly controls the degree of compression that can be obtained in encoding concepts in the repository. Thirdly, we optimize the DFT encoding process by removing the need for computing a potentially expensive inner product operation on vectors. Related Research {#sec:relatedresearch} ================ While a vast literature on concept drift detection exists [@pea:dci], only a small body of work exists so far on exploitation of recurrent concepts. The methods that exist fall into two broad categories. Firstly, methods that store past concepts as models and then use a meta-learning mechanism to find the best match when a concept drift is triggered [@joa:trc], [@gom:trc]. Secondly, methods that store past concepts as an ensemble of classifiers. The method proposed in this research belongs to the second category where ensembles remember past concepts. An algorithm called REDDLA is presented in [@pli:mrc]. This algorithm is designed to handle recurring concepts with unlabeled data instances. One of the key issues is that explicit domain is required on the concept recurrence interval. The other issue is high memory overhead. Lazarescu in [@laz:aml] proposed an evidence forgetting mechanism based on a multiple window approach and a prediction module to adapt classifiers based on estimating future rate of change. Whenever the difference between observed and estimated rates of change are above a threshold, a classifier that best represents the current concept is stored in a repository. Experimentation on the STAGGER data set showed that the proposed approach outperformed the FLORA method on classification accuracy with re-emergence of previous concepts in the stream. Ramamurthy and Bhatnagar [@ram:trc] use an ensemble approach based on a set of classifiers in a global set G. An ensemble of classifiers is built dynamically from a collection of classifiers in G, if none of the existing individual classifiers are able to meet a minimum accuracy threshold based on a user defined acceptance factor. Whenever the ensemble accuracy falls below the accuracy threshold, G is updated with a new classifier trained on the current chunk of data. Another ensemble based approach by Katakis et al. is proposed in [@kat:aeo]. A mapping function is applied on data stream instances to form conceptual vectors which are then grouped together into a set of clusters. A classifier is incrementally built on each cluster and an ensemble is formed based on the set of classifiers. Experimentation on the Usenet data set showed that the ensemble approach produced better accuracy than a simple incremental version of the Naive Bayes classifier. Gomes et al. [@gom:trc] used a two layer approach with the first layer consisting of a set of classifiers trained on the current concept, while the second layer contains classifiers created from past concepts. A concept drift detector flags when a warning state is triggered and incoming data instances are buffered to prepare a new classifier. If the number of instances in the warning window is below a threshold, the classifier in layer 1 is used instead of re-using classifiers in layer 2. One major issue with this method is validity of the assumption that explicit contextual information is available in the data stream. Gama and Kosina also proposed a two layered system in [@joa:trc] which is designed for delayed labelling, similar in some respects to the Gomes et al. [@gom:trc] approach. In their approach, Gama and Kosina pair a base classifier in the first layer with a referee in the second layer. Referees learn regions of feature space which its corresponding base classifier predicts accurately and is thus able to express a level of confidence on its base classifier with respect to a newly generated concept. The base classifier which receives the highest confidence score is selected, provided that it is above a user defined hit ratio parameter; if not, a new classifier is learnt. Just-in-Time classifiers is the solution proposed by Allipi et al. [@ali:jit] to deal with recurrent concepts. Concept change detection is carried out on the classification accuracy as well as by observing the distribution of input instances. The drawback is that this model is designed for abrupt drifts and is weak at handling gradual changes. Recently, Sakthithasan and Pears in [@sak:mrc] used the Discrete Fourier Transform (DFT) to encode decision trees into a highly compressed form for future use. They showed that DFT encoding is very effective in improving classification accuracy, memory usage and processing time in general. It maintains a pool of Fourier spectra and a decision tree forest in parallel. The decision tree forest dominates the model, when none of the existing Fourier spectra matches the current concept, otherwise classification is done by the best performing Fourier spectrum. Application of the Discrete Fourier Transform on Decision Trees {#sec:dftapplication} =============================================================== The Discrete Fourier Transform (DFT) has a vast area of application in diverse domains such as time series analysis, signal processing, image processing and so on. It turns out as Park [@par:kdf] and Kargupta [@hhil:afs] show, that the DFT is very effective in terms of classification when applied on a decision tree model. Kargupta et al. [@hhil:afs], working in the domain of distributed data mining, showed that the Fourier spectrum fully captures a decision tree in algebraic form, meaning that the Fourier representation preserves the same classification power as the original decision tree. Transforming Decision Tree into Fourier Spectrum ------------------------------------------------ A decision tree can be represented in compact algebraic form by applying the DFT to paths of the tree. Each Fourier coefficient $\omega_{j}$ is given by: $$\\ \small \label{coeffciientequation} \omega_j=\frac{1}{2^d}{\sum_x{f(x)}\psi^{\overline\lambda}_j(x)};$$ $\psi^{\overline\lambda}_j(x)=\prod_m{\exp^{\frac{2{\pi}i}{\lambda_m}{x_m}{j_m}}}$ where j and x are strings of length $d$; $x_m$ and $j_m$ represent the $m^{th}$ attribute value in $j$ and $x$respectively; $f(x)$ is the classification outcome of path vector x and $\psi^{\overline\lambda}_j(x)$ is the Fourier basis function. ![Decision Tree with 3 binary features[]{data-label="fig:tree"}](Fig1.png){width="50.00000%"} Figure 1 shows a simple example with 3 binary valued features $x_1$, $x_2$ and $x_3$, out of which only $x_1$ and $x_3$ are actually used in the classification. With the wild card operator \* in place we can use equations (1) and (2) to calculate non zero coefficients. Thus for example we can compute: $$\begin{aligned} \omega_{000}&=\frac{4}{8}f(**0)\psi_{000}(**0)+\frac{2}{8}f(0*1)\psi_{000}(0*1)\\ &+\frac{2}{8}f(1*1)\psi_{000}f(1*1)=\frac{3}{4}\\ \omega_{001}&=\frac{4}{8}f(**0)\psi_{001}(**0)+\frac{2}{8}f(0*1)\psi_{001}(0*1)\\ &+\frac{2}{8}f(1*1)\psi_{001}f(1*1)=\frac{1}{4} %c_{100}&=\frac{4}{8}f(**0)\phi_{100}(**0)+\frac{2}{8}f(0*1)\phi_{100}(0*1)+\frac{2}{8}f(1*1)\phi_{100}f(1*1)=-\frac{1}{4}\\ %c_{101}&=\frac{4}{8}f(**0)\phi_{101}(**0)+\frac{2}{8}f(0*1)\phi_{101}(0*1)+\frac{2}{8}f(1*1)\phi_{101}f(1*1)=\frac{1}{4}\\ %c_{111}&=\frac{4}{8}f(**0)\phi_{111}(**0)+\frac{2}{8}f(0*1)\phi_{111}(0*1)+\frac{2}{8}f(1*1)\phi_{111}f(1*1)=\frac{0}{4}\end{aligned}$$ and so on. Kargupta et al in [@hhil:afs] showed that the Fourier spectrum of a given decision tree can be approximated by computing only a small number of [*low order coefficients*]{}, thus reducing storage overhead. With a suitable thresholding scheme in place, the Fourier spectrum consisting of the set of low order coefficients is thus an ideal mechanism for capturing past concepts. Furthermore, classification of unlabeled data instances can be done directly in the Fourier domain as it is well known that the inverse of the DFT defined in expression \[coeffciientequation\] can be used to recover the classification value, thus avoiding the need for expensive reconstruction of a decision tree from its Fourier spectrum. The inverse Fourier Transform is given by $$\label{inversefouriertransform} \small f(x)=\sum_j{\omega_j\overline\psi^{\overline\lambda}_j(x)} % \vspace{-5pt}$$ where $\overline\psi^{\overline\lambda}_j(x)$ is the complex conjugate of $\psi^{\overline\lambda}_j(x)$. ‘ An instance can be transformed into binary vector through the symbolic mapping between the actual attribute value and mapped value ( either 0 or 1 in binary case). It can then be classified using the inverse function in equation \[inversefouriertransform\]. Suppose the instance is 010, the classification value $f(010)$ can be calculated as follows: $$\begin{aligned} f(010)=&\frac{1}{2^d}(-1)^{000.010} \omega_{000} + \frac{1}{2^d}(-1)^{001.010}\omega_{001} \nonumber \\ &+ \frac{1}{2^d}(-1)^{010.010}\omega_{010} + \frac{1}{2^d}(-1)^{011.010}\omega_{011} \nonumber \\ &+ \frac{1}{2^d}(-1)^{100.010}\omega_{100}+ \frac{1}{2^d}(-1)^{101.010}\omega_{101} \nonumber \\ &+\frac{1}{2^d}(-1)^{110.010}\omega_{110} + \frac{1}{2^d}(-1)^{111.010} \omega_{111} = 1\end{aligned}$$ Exploitation of the Fourier Transform for Recurrent Concept Capture {#sec:exploitationofdft} =================================================================== We first present the basic algorithm used in Section \[subsec:ep\] and then go on to discuss an optimization that we used for energy thresholding in Section \[subsec:dftoptimization\]. We use CBDT [@hoe:acb] as the base classifier which maintains a forest of Hoeffding Trees [@ped:mhs] CBDT is dynamic in the sense that it can adapt to changing concepts at drift detection points. ![An Architecture for Recurrent Concept Capture[]{data-label="fig:eparchitecture"}](EPStructure){width="\textwidth"} As shown in Figure \[fig:eparchitecture\], the memory is divided into two segments: the forest of Hoeffding trees; and a pool of Fourier Spectra. The forest learns and undergoes structural modification on a continuous basis. The pool maintains a collection of Fourier Spectra encoded from Hoeffding Trees, each of which had the best classification accuracy across the forest at a particular concept drift point. Each Hoeffding Tree and Fourier Spectrum is equipped with an instance of a drift detector. In this research, we use the SeqDrift2 drift detector [@pea:dci] as the default option. In [@sak:mrc], each Fourier spectrum is represented individually as a Fourier Concept Tree (FCT). In this work, we aggregate spectra and maintain a pool of ensemble spectra known as Ensemble Pool (EP). The aggregation process is carried out in two different ways. Algorithm $EP_a$ aggregates with reference to similarity based on accuracy whereas $EP$ aggregates based on structural similarity. We describe the $EP$ process in Algorithm [*EP*]{} and discuss how FCT can be generated from it as a special case. In practice, any incremental decision tree approach that uses a forest of trees can be used in place of CBDT base classifier. EP Algorithm {#subsec:ep} ------------ \[h\] [EP]{}[ ]{} Plant a Hoeffding tree rooted on each attribute found in the data stream\ **C** is set to a randomly selected Hoeffding tree model from forest\ Initialise an empty pool\ Read an instance *I* from the data stream\ \ Apply all classifiers in forest and pool to classify *I*\ Append *0* to the embedded drift detector’s window for each classifier if classification is correct, else *1*\ \ Identify best performing Fourier Spectrum **F** in pool\ \ Apply DFT on model **C** to produce Fourier Spectrum **F\*** using energy threshold [*$E_T$*]{}\ \ Call [*Aggregation*]{}\ Identify best classifier **C** across forest and pool\ GoTo 4 \[alg:ep\] [Aggregation]{}[ ]{}\ \ $d(E)=d(E)+ |c(\textbf{F*},i)-c(E,i)|$\ $E*=\underset{E}{\operatorname{arg\,min}}(distance(E))$\ \ \[alg:aggregation\] In step 1, a Hoeffding Tree rooted on each attribute is created. In step 2, a tree is randomly chosen as the best performing classifier **C**. Next, an empty pool is created in step 3. Each incoming instance is routed to all trees in the forest and pool until a concept drift signal is triggered by the drift detector instance attached to the best classifier **C** (steps 4 to 8). At the first concept drift point, the best performing tree **C** (in terms of drift detector’s estimate of accuracy) is transformed into a Fourier Spectrum **F\*** after energy thresholding [@sak:mrc]. In this method, the assumption is that the best tree that has the highest accuracy helps locate conceot changes precisely than other trees because it is the tree that captures concepts at a greater detail than others, thus the highest accuracy. Thereafter, **F\*** is stored in the repository for reuse whenever the concept recurs. The spectra stored in the repository are fixed in nature as the intention is to capture past concepts. A new best performing classifier is then identified as shown in step 15. At each subsequent drift point, if the best classifier is from the pool then that classifier is applied to classify data instances until a new best classifier emerges at a subsequent drift point. Otherwise, if the best classifier is from forest, two tests are made prior to applying the DFT to reduce redundancy in the pool. Firstly, we check whether the difference in accuracy between the best Hoeffding tree in forest (C) and the best performing Fourier Spectrum (F) in the pool (from step 10) is greater than a user defined tie threshold $\tau$ (step 11). If this test succeeds, the DFT is applied to C to produce (F\*) (step 12). Furthermore, a second test is made to ensure that its Fourier representation (F\*) is not already in the pool (step 13). If this test is also passed, algorithm [*Aggregation*]{} is called to integrate F\* into a selected existing Fourier Spectrum (E\*) or plant (F\*) as a separate Fourier spectrum in the pool (step 14). Algorithm [*Aggregation*]{} searches for the spectrum (E\*) that has the greatest structural similarity to the currently generated spectrum (F\*) (step 3). Step 3 evaluates the degree of disagreement (d) between the classification decisions (c) for F\* and E on data instance i. Degree of disagreement between (F\*) and each of the existing ensemble (E) in pool can easily be updated incrementally in Algorithm [*EP*]{} using a single counter variable at each ensemble E. This removes the steps from 2 to 4 in Algorithm [*Aggregation*]{}. As an alternative to aggregating [*structurally similar spectra*]{}, we used accuracy as the measure that defines similarity. Similarity based on accuracy leads to aggregating similar performing Fourier Spectra together. Thus, we test the hypothesis, *aggregation of two spectra based on structural similarity produces better performing trees than the one based on accuracy*. As stated earlier, FCT omits the call to Algorithm [*Aggregation*]{} and inserts (F\*) as it is, and is thus a special case of EP. Optimising the Energy Thresholding Process {#subsec:dftoptimization} ------------------------------------------ Sakthihasan et al. in [@sak:mrc] showed that classification accuracy is sensitive to spectral energy, which is given by the total of the sum of squares of the coefficients[@hhil:afs]); the higher the energy the greater is the classification accuracy in general. Thresholding on spectral energy is thus an effective method of obtaining a compact spectrum while retaining the classification power inherent in the decision tree counterpart. A solution described in [@sak:mrc] was to iterate through each order of the spectrum and compute ratio of energy at orders $i-1$ to that of $i$ respectively. Thresholding can then be implemented at order O when the ratio is less than some small tolerance value, say $0.01$. The drawback of this simple solution is that it does not guarantee that the cumulative energy up to order O contains a proportion ($\epsilon$) of the total energy. Fortunately, a solution exists for this problem. Theorem 1 proves that E(T) (total energy of Fourier Spectrum) equals to $\omega_{\overline 0}$ (The $0^{th}$ coefficient). Thus, total energy can be computed efficiently, without having to enumerate all the single coefficients.\ \ **Theorem 1** The total spectral energy $E =\sum_j{\omega_{{j}}^2} = \omega_{\overline 0}$, where $\omega_{\overline 0}$ denotes the coefficient with order 0, which is easily computed as its Fourier basis function is unity.\ \ **Proof:** For case 1 we prove the result when exactly one such combination exists and discuss the extension to the case when more than one combination is present. Without loss of generality we illustrate the proof when the wild card characters occur at the beginning of vector x; if they occur in any other position then a simple reordering operation can be used without affecting the validity of the proof.\ \ Suppose that the cardinality of the attributes after reordering are $\lambda_i$ where $i \in [0,d-1]$, where d is the dimensionality of the dataset. $$\begin{aligned} \sum_{x \in S}\psi_j(x)&=\exp \Bigl(\frac{2{\pi}i{j_0}0}{\lambda_0} \Bigr) \times \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr) \nonumber \\ &+\exp\Bigl(\frac{2{\pi}i{j_0}1}{\lambda_0}\Bigr) \times \exp\Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1}\Bigr) \times \cdots \times \exp\Bigl(\frac{2{\pi}i{j_{d-1}}{x_{d-1}}}{\lambda_{d-1}}\Bigr) \nonumber \\ &\vdots \nonumber \\ &+\exp\Bigl(\frac{2{\pi}i{j_0}(\lambda_0-1)}{\lambda_0}\Bigr) \times \exp\Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1}\Bigr) \times \cdots \times \exp\Bigl(\frac{2{\pi}i{j_{d-1}}{x_{d-1}}}{\lambda_{d-1}}\Bigr) \nonumber \\ &= \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\exp \Bigl(\frac{2{\pi}i{j_0}k}{\lambda_0} \Bigr)\Bigr) \nonumber \\ %&= \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\Bigl(\sum_k{ \psi_0 \psi_{j_0}}\Bigr) \mbox { as $$} \nonumber \\ &= \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\psi_o\psi_k\Bigr) \mbox { as $\psi_0=1$} \nonumber \\ &=0 \mbox{ due to orthogonality of the Fourier basis functions} \nonumber\end{aligned}$$ When n wild cards are present, the extension is straightforward and results in: $$\begin{aligned} \sum_{x \in S}\psi_j(x)&= \exp \Bigl(\frac{2{\pi}i{j_{n}}{x_{n}}}{\lambda_n} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr) \prod_{l=0}^{n-1}\sum_{k=0}^{\lambda_l-1}\Bigl(\exp \Bigl(\frac{2{\pi}i{j_l}k}{\lambda_l} \Bigr)\Bigr) \\ \sum_{x \in S}\psi_j(x)&= \exp \Bigl(\frac{2{\pi}i{j_{n}}{x_{n}}}{\lambda_n} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr) \prod_{l=0}^{n-1}\sum_{k=0}^{\lambda_l-1}\Bigl(\psi_0\psi_k\Bigr) \nonumber \\ %\Bigl(1+\sum_{k=m}^{d-1}\Bigl(\exp \frac{2{\pi}i{j_0}k}{\lambda_0} \Bigr)\Bigr) \nonumber \\ &=0 \mbox{ due to orthogonality of Fourier basis functions } \nonumber\end{aligned}$$ This optimization significantly increases processing speed, especially in high dimensional data stream environments.\ \ The next optimization is applied to optimize the Fourier Basis function calculation in equation \[coeffciientequation\] especially when wildcard characters (denoting absence of a feature) are present in a path vector $x$ of a Hoeffding Tree. Optimizing the Computation of the Fourier Basis Function -------------------------------------------------------- The computation of a Fourier basis function for a given partition $j$ in a generic $n-ary$ ( $n\ge2$) domain is given by: $$\begin{aligned} \sum_{x \in S}\psi_j(x)&=\sum_{x \in S}\prod_m{\exp^{\Bigl(\frac{2{\pi}i{j_m}{x_m}}{{\lambda_m}}\Bigr)}} \end{aligned}$$ Thus we can see from (4.3) that the computation of $ \sum_{x \in S}\psi(j)$ over a set of schema S requires the computation of an expensive inner product operation between the $x$ and $j$. However, it is possible to optimize this inner product computation as defined in Theorem 2.\ \ **Theorem 2** The computation of $ \sum_{x \in S}\psi_j(x)$ can be optimized as follows:\ \ Case 1: If there exists at least one $(p,*)$ combination with $p \in j$, $p\ne 0$ and $*$ a wild card character defining a set of schema $S$, then $ \sum_{x \in S}\psi_j(x)=0$.\ \ Case 2: else if there exists $n$ combinations of $(0,*)$ pairs in the $j$ and $x$ vectors respectively, then $$\sum_{x \in S}\psi_j(x)= {\lambda}\prod_{k=n}^{\lambda_k-1}{\exp^{\Bigl(\frac{2{\pi}i{j_k}{x_k}}{{\lambda_k}}\Bigr)}} \mbox{ where $\lambda=\prod_{l=0}^{n-1}{\lambda_l}$}$$ **Proof:** For case 1 we prove the result when exactly one such combination exists and discuss the extension to the case when more than one combination is present. Without loss of generality we illustrate the proof when the wild card characters occur at the beginning of vector x; if they occur in any other position then a simple reordering operation can be used without affecting the validity of the proof.\ \ Suppose that the cardinality of the attributes after reordering are $\lambda_i$ where $i \in [0,d-1]$, where d is the dimensionality of the dataset. $$\begin{aligned} \sum_{x \in S}\psi_j(x)&=\exp \Bigl(\frac{2{\pi}i{j_0}0}{\lambda_0} \Bigr) \times \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr) \nonumber \\ &+\exp\Bigl(\frac{2{\pi}i{j_0}1}{\lambda_0}\Bigr) \times \exp\Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1}\Bigr) \times \cdots \times \exp\Bigl(\frac{2{\pi}i{j_{d-1}}{x_{d-1}}}{\lambda_{d-1}}\Bigr) \nonumber \\ &\vdots \nonumber \\ &+\exp\Bigl(\frac{2{\pi}i{j_0}(\lambda_0-1)}{\lambda_0}\Bigr) \times \exp\Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1}\Bigr) \times \cdots \times \exp\Bigl(\frac{2{\pi}i{j_{d-1}}{x_{d-1}}}{\lambda_{d-1}}\Bigr) \nonumber \\ &= \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\exp \Bigl(\frac{2{\pi}i{j_0}k}{\lambda_0} \Bigr)\Bigr) \nonumber \\ %&= \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\Bigl(\sum_k{ \psi_0 \psi_{j_0}}\Bigr) \mbox { as $$} \nonumber \\ &= \exp \Bigl(\frac{2{\pi}i{j_1}{x_1}}{\lambda_1} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr)\sum_{k=0}^{\lambda_0-1}\Bigl(\psi_o\psi_k\Bigr) \mbox { as $\psi_0=1$} \nonumber \\ &=0 \mbox{ due to orthogonality of the Fourier basis functions} \nonumber\end{aligned}$$ When n wild cards are present, the extension is straightforward and results in: $$\begin{aligned} \sum_{x \in S}\psi_j(x)&= \exp \Bigl(\frac{2{\pi}i{j_{n}}{x_{n}}}{\lambda_n} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr) \prod_{l=0}^{n-1}\sum_{k=0}^{\lambda_l-1}\Bigl(\exp \Bigl(\frac{2{\pi}i{j_l}k}{\lambda_l} \Bigr)\Bigr) \\ \sum_{x \in S}\psi_j(x)&= \exp \Bigl(\frac{2{\pi}i{j_{n}}{x_{n}}}{\lambda_n} \Bigr) \times \cdots \times \exp \Bigl(\frac{2{\pi}i{j_{d-1}}x_{d-1}} {\lambda_{d-1}}\Bigr) \prod_{l=0}^{n-1}\sum_{k=0}^{\lambda_l-1}\Bigl(\psi_0\psi_k\Bigr) \nonumber \\ %\Bigl(1+\sum_{k=m}^{d-1}\Bigl(\exp \frac{2{\pi}i{j_0}k}{\lambda_0} \Bigr)\Bigr) \nonumber \\ &=0 \mbox{ due to orthogonality of Fourier basis functions } \nonumber\end{aligned}$$ The value of Case 1 is that a simple scan of the $j$ and $x$ vectors will save a total of $d$ multiplications and $d-1$ additions.\ \ We now turn our attention to Case 2. Since $\prod_l{\lambda_l}$ is a constant for all possible values of $j$ and $y$, the value of Case 2 is that a scan of the two vectors will avoid the overhead of $n$ multiplications and $n-1$ additions. Even with these optimizations, coefficient calculation may be expensive in a large dimensional data set. In the next section we present a strategy to further optimize the derivation of the spectrum. Localized Approach to Ensemble Learning in the Fourier Domain {#sec:agg} ------------------------------------------------------------- In order to realize the full benefits of ensemble learning in the Fourier domain, we aggregate individual spectra $s_i(x)$ that represent different concepts which manifest at different points in the stream. $$\begin{aligned} \label{eqn:agg1} s_c(x)&=\sum_i{A_i\sum_i{s_i(x)}} \nonumber \\ &=\sum_i{A_i\sum_{j \in P_i}{\omega_j}^{(i)}\overline\psi_{{{j}}(x)}} \end{aligned}$$ where $s_c(x)$ denotes the ensemble spectrum produced from the individual spectra $s_i(x)$ produced at different points $i$ in the stream; $A_i$ is the classification accuracy of its corresponding spectrum and $P_i$ is the set of partitions for non zero coefficients in spectrum $s_i$. Park in [@par:kdf] used ensemble learning with Fourier spectra in a setting different to ours. They considered a distributed system with each node $i$ producing its own spectrum $s_i(x)$ and aggregation taking place at a central node. In our setting of a data stream environment, we do not have all spectra in advance but we can still use the same principle due to the distributive nature of the linear weighted sum expressed by $(\ref{eqn:agg1})$. Hence, we use: $$\label{eqn:agg2} s_c^{(i+1)}(x)=s_c^{(i)}(x)+A_{i+1}{s_{i+1}(x)}$$ where $s_c^{(i+1)}(x)$, $s_c^{(i)}$ represent the ensemble spectra at concept drift points $i+1$ and $i$ respectively in the stream and $s_{i+1}(x)$ is the spectrum produced at drift point $i+1$ with accuracy $A_{i+1}$. We use expression $(\ref{eqn:agg2})$ for implementing ensemble learning but with one essential difference. A direct application of $(\ref{eqn:agg2})$ using the entire (global) set of attributes $G$ comprising the data set would be inefficient. As there are an exponential number of coefficients with respect to the number of attributes, this could cause a bottleneck in high dimensional environments. One practical solution is to populate the spectrum using only attributes present in a given tree. The major advantage of this approach is smaller computational overhead as the Fourier transform effort is directly proportional to the size of the attribute set used. Then this initial spectrum can be extended to a full length spectrum containing the attributes that are absent in the given tree, using a simple transformation scheme. We define an attribute set of a Decision Tree as that subset of attributes which define splits in the tree. Suppose that we are integrating spectra from trees $D_1$ and $D_2$, having attribute sets $L$ and $M$ respectively. We apply the DFT on $D_1$ to obtain $S_1$ using *only the attributes in its attribute set $L$ and not all attributes in $G$*. Similarly we generate $S_2$ from $D_2$ using only the attributes defined in $M$. Now, in order to integrate $S_1$ with $S_2$, we need to account for differences in the attribute sets $L$ and $M$. To do this, we take $S_1$ and expand the spectrum by incorporating attributes in the set $M \setminus L$. The expansion is defined by a single operation: For each schema instance in the spectrum (say $S_1$) expand the spectrum by adding $0$ to all attribute index positions in set $M \setminus L$. The [*coefficient value after expansion will remain it the same as the classification $f$*]{} value for all of these added index positions remains unchanged. We are now in a position to integrate two spectra produced from their own localized set of attributes. Essentially, this means that we now have a more efficient method of implementing ensemble learning using expression $(\ref{eqn:agg2})$. The next section presents the empirical outcomes of the proposed models with the above mentioned optimizations. Experimental Study {#sec:experimentalstudy} ================== The main focus of the study is to assess the effectiveness of the ensemble EP approach vis-a-vis FCT in respect of classification accuracy, memory consumption, processing speed, tolerance to noise. We also assessed the sensitivity of EP’s accuracy on two significant factors, pool size and impact of drift detector. All experimentation was done with the following parameter values:\ \ *Tree Forest:* Max Node Count=5000, Max Number of Fourier spectra=10, Tie Threshold $\tau$=0.01\ *SeqDrift2/ADWIN [@bit:lft]:* drift significance value=0.01 Datasets Used for the Experimental Study {#sec:datasets} ---------------------------------------- ### Synthetic Data We experimented with the Rotating Hyperplane data generator that is commonly used in drift detection and recurrent concept mining. The dataset was generated within the MOA data stream tool [@bit:moa]. We injected concept recurrence into the stream at known points so that we could evaluate the capabilities of FCT and EP to recognize and exploit such recurrences. For this dataset 10 different concepts were generated, each of which spanned 5,000 instances and each occurred a total of 3 times at different points in the stream. In order to challenge the concept recognition process, we added 10% noise by inverting the class labels of 10% of randomly selected instances. ### Real World Data [ *Spam Data Set:* ]{} The Spam dataset was used in its original form [^1] which encapsulates an evolution of Spam messages. There are 9,324 instances and 499 informative attributes. [*Electricity Data Set:* ]{} NSW Electricity dataset is also used in its original form [^2]. There are two classes [*Up*]{} and [*Down*]{} that indicate the change of price with respect to the moving average of the prices in last 24 hours. [*Flight Data Set:*]{} This dataset is generated through the use of NASA’s FLTz flight simulator which was designed to simulate flight conditions experienced with commercial flights. It consists of a set of 20 separate files, each containing data about a single flight with four scenarios: take off, climb, cruise and landing. Data is recorded every second and a data instance is produced. The “Velocity” feature is chosen as the class feature as it needs to be adjusted in order to maintain aircraft stability during various maneuvers such as take off and landing. Velocity was discretized into binary outcomes “UP” or “DOWN” depending on the directional change of the moving average in a window of size 10 data instances. Comparative Study: Ensemble versus Single Spectrum Approach ----------------------------------------------------------- Previous research on the use of Fourier spectrum revealed accuracy and memory advantages over meta learning approaches such as the one employed by Gama and Kosina in storing past concepts in a repository [@sak:mrc]. For details of the advantages of the Fourier approach and experimentation with it the reader is referred to [@sak:mrc]. Our focus here is a comparative study of the Ensemble approach versus the single spectrum approach. With this in mind we designed three types of experiments. ### Accuracy Accuracy is a critical performance measure in many practical applications. Due to the dynamic nature of data streams classification accuracy on the current concept was taken as the performance measure. ![Accuracy Profiles[]{data-label="accuracycurves"}](accuracygraphs.eps){width="\textwidth"} Figure \[accuracycurves\] presents Accuracy values of all algorithms at 10 equal-sized sub-divisions of the stream. We also present overall mean and standard deviations of accuracy taken across the entire stream for each dataset. Fig \[accuracycurves\] shows that the individual accuracies across segments and overall accuracy across the entire stream are consistent with each other. MetaCT, which uses a referee based strategy was found to be the worst performing algorithm on all datasets. In contrast EP outperforms the other algorithms in general, followed by $EP_a$ and FCT. These results show clearly that DFT based methods are superior in a dynamic data stream environment. FCT does not exploit aggregation of Fourier Spectra and is hence challenged in a memory constrained environment where the number of models stored for reuse is limited. Figure \[accuracycurves\] depicts the performance in such an environment where memory is severely limited. This introduces a large burden on FCT to re-learn concepts after change. EP is more resilient at small pool sizes as any given concept that recurs can be approximated by a linear combination of spectra embedded in the ensemble, just as a waveform of arbitrary shape can be approximated by a large enough sum of sine functions in signal processing. Examining model usage statistics, EP was 3.7 times higher in model re-use on the Flight dataset. The corresponding value was 2.6 for $EP_a$ on the same dataset. This provides empirical support for the claim that an aggregation-based model such as EP has a significant advantage in reducing the degree of relearning. For Rotating Hyperplane with known recurrence points the advantage of EP over its counterparts is very explicit. We display the stream segment for the third round of concept occurrences, spanning the 10 concepts. Each of the 10 intervals represent the second recurrence of a concept and the Figure shows that EP outperforms FCT on 8/10 concepts; $EP_a$ and MetaCT on 7/10 concepts; and CBDT on all 10 concepts. The next key aspect in a memory constrained environment is memory consumption which is assessed in the following section. ### Memory Memory consumption is influenced by the degree of generalizability of a given algorithm. A greater degree of generalizability promotes higher re-use and reduces the number of spectra that need to be stored in the repository to achieve a given level of classification accuracy. In this context it will be interesting to compare the consumption of EP with that of FCT as they have contrasting model re-use characteristics. MetaCT(SeqDrift2) and CBDT were excluded from memory comparison due to their relatively poor performance in the previous experiment. \[h\] Dataset ----------------- ------ -------- -------------- -- FCT $EP_a$ EP Flight 32.1 20.2 [**18.1**]{} Electricity 31.6 16.1 [**14.1**]{} Rot. Hyperplane 48.4 38.6 [**27.9**]{} Spam 17.3 17.2 [**16.4**]{} : Memory Usage with Pool size set to 10 \[tab:memorycomparison\] Table \[tab:memorycomparison\] presents the average memory consumption of the pool over the entirety of each dataset. As mentioned in Section \[sec:dftapplication\], each of the above algorithms in Table \[tab:memorycomparison\] has two components: a forest and a repository pool. Memory consumed by forest is not a distinguishing factor as there was a very marginal difference between the algorithms and thus the focus was on the repository pool. Without exception, EP consumed the least memory compared to the other algorithms. This was expected as EP structurally examines instance vectors (i.e. corresponding to classification paths in Hoeffding tree) and aggregates similar vectors together. On the other hand in $EP_a$, structural similarity is not guaranteed and two structurally very different Spectra producing similar accuracy could be chosen as the candidates to be aggregated, thus resulting in larger spectra. Table \[tab:memorycomparison\] provides evidence to support this premise as the memory consumed by $EP_a$ is higher than that of EP but lower than FCT. On average over all datasets, EP achieved a 41% reduction in memory consumption in relation to FCT; the corresponding figure for Electricity was 55%. This represents a significant benefit of applying aggregation in Fourier space. ### Processing Speed DFT application is a potential performance bottleneck when compared to classification, especially in high dimensional data streams. Processing speed is dependent on a variety of factors: maintaining and classifying relatively larger number of Fourier Spectra in FCT compared to EP and $EP_a$, aggregation in EP and $EP_a$ that generalize models thus reducing re-learning and the need for DFT application, and finally the computational overheads of aggregation. Therefore, this section assesses the trade off between single and aggregated Fourier approaches in terms of processing speed. \[h\] Dataset FCT $EP_a$ EP --------------------- ------------- -------- ------------ Flight 797.2 731.2 **836.9** Electricity **11600.3** 9002.5 11402.5 Rotating Hyperplane 5647.8 5413.8 **5804.5** Spam **4.2** 3.9 **4.2** : Processing Speed in instances per second \[tab:processingspeed\] Table \[tab:processingspeed\] shows that EP is the fastest most of the time. $EP_a$, even though it has the potential to be faster due to its simple aggregation strategy, suffers from inappropriate aggregations that introduce instability, thus triggering more drift points than its EP counterpart. EP, on the other hand, efficiently does structural similarity comparison by incrementally updating simple counters that remembers the number of disagreements in classification between the current winner tree and every Fourier Spectra in pool. On the other hand, although EP through its aggregation strategy requires more computational effort than $EP_a$, that effort is compensated for by its stability, which triggers fewer false drift alarms than either $EP_a$ or FCT. Therefore, this experiment demonstrates that an expensive operation such as aggregation if applied appropriately will yield a direct processing speed advantage over a period of time. ### Effects of Noise Algorithms that work well in noise-free environments will fail on noisy environments if they lack the ability to generalize to new data by removing minor variations which often correspond to noise. DFT application, as mentioned earlier, extracts significant coefficients by ignoring minor coefficients that may capture noise inherent in data. It was shown in [@sak:mrc] that DFT application provides robustness in a noisy environment as opposed to a non-DFT based approach such as MetaCT. Therefore, this experiment is aimed at testing whether aggregation has an added advantage over a non-aggregation based method such as FCT. ![The impact of noise on accuracy[]{data-label="noiseaccuracy"}](noiseaccuracy){width="\textwidth"} Figure \[noiseaccuracy\] shows percentage accuracy decrease for noise levels 20% and 30% on FCT and EP relative to accuracy on the original Flight dataset. It is clear that the decrease in accuracy is higher at the 30% noise level. What is interesting is the higher tolerance of EP to noise compared to FCT. In 8/10 intervals, for 20% noise, EP is found to be having a lesser decrease than its counterpart. Similarly at the 30% noise level, the fraction is 4/10, with the two being tied in performance in two other intervals. Again, as with the other metrics that we tracked, the superior performance of EP can be explained in terms of its power to generalize making it more robust to the effects of noise [^3]. Next we examine the sensitivity of EP on key parameters that significantly affect performance. Due to the superiority of EP over the other algorithms, the study was confined to this algorithm. Please refer [@sak:mrc] for sensitivity analysis on FCT’s parameters. Sensitivity Analysis -------------------- EP(SeqDrift2) has two key parameters of its own: pool size and choice of drift detector. ### Pool Size In this experiment we contrasted classification accuracy at two different ends of the pool size scale, namely 1 and 10. In the context of the Flight dataset which has four concepts, a pool size of 1 represents an extremely limiting memory environment and the size of 10 represents a situation where memory is plentiful. ![The impact of pool size on flight dataset[]{data-label="poolsizeaccuracy"}](accuracypoolsize){width="\textwidth"} Figure \[poolsizeaccuracy\] shows accuracy values over 10 intervals. Interestingly, EP, with pool size1, has the highest accuracy in 8/10 intervals. There is a 7.6% and 7.2% gain in accuracy compared to FCT over pool sizes 1 and 10 respectively. This is a significant outcome of this research. Even in an extreme memory challenged environment, EP achieves its best accuracy over a setting with a much higher memory capacity. The implication is that ensemble accuracy increases with greater diversity and resonates with the research conducted by [@gas:dte]. This illustrates the strength of aggregation applied in the EP algorithm. As more memory becomes available at pool size 10, FCT’s accuracy converges to that of its counterpart, as expected. At the higher memory setting FCT can accommodate more spectra in its pool that are tailored to specific concepts. ### Impact of Drift Detector A drift detector that incorrectly triggers change points leads to partial learning of a concept and under developed classifiers being stored in the pool. This introduces fluctuations in accuracy, which in turn trigger change detections, causing even more fluctuations and so on. This is a cyclic problem. On the other hand, if a drift detector fails to detect changes, classifiers are not updated in a timely fashion, thus leading to poor performance. This situation may arise if a drift detector has significantly high detection delay in signaling changes. The ADWIN and SeqDrift2 drift detectors, as shown in [@pea:dci] have contrasting properties. SeqDrift2 has a lower false positive rate than ADWIN while having similar sensitivity to ADWIN. Therefore, the comparative study is largely governed by false positive detections. ![The impact of drift detector on EP with pool size 10[]{data-label="driftdetectoraccuracy"}](driftdetectoraccuracygraph){width="\textwidth"} Figure \[driftdetectoraccuracy\] reveals that SeqDrift2 helped EP to reduce the frequency of sudden accuracy drops seen with ADWIN, due to the latter signaling false changes in concepts. In the segment shown in Figure \[driftdetectoraccuracy\], there is a 5% gain in accuracy by using SeqDrift2 and it is 3.4% over the entire data set. Conclusions and Future Work {#sec:conclusion} =========================== In this research we proposed a novel approach for capturing and exploiting recurring concepts in data streams. We optimized the derivation of the Fourier spectrum by employing two mechanisms: one for energy thresholding and the other for speeding up computation of the Fourier basis functions. This research revealed that the ensemble approach outperformed the single spectrum approach and is thus the method of choice in high speed dynamic environments that generate large amounts of concepts over the progression of the stream. In such environments FCT would be challenged in terms of memory capacity and would be forced to flush portions of its repository sooner that EP, thus losing its ability to exploit concept recurrences and in turn leading to a loss of accuracy. However, as shown in the experimentation care needs to be taken on how spectra are combined: a naive approach of simply combining similarly performing spectra in terms of accuracy can be worse than maintaining single spectra. We showed that the structural similarity scheme outperformed the other two approaches on a broad set of criteria including accuracy, robustness to noise and over-fitting, memory consumption and processing speed. In terms of future work there are two promising directions. 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Machine Learning, 97:3, pp 259–293, 2014. Peipei Li, Xindong Wu, and Xuegang Hu, “Mining recurring concept drifts with limited labeled streaming data,” ACM Trans. Intell. Syst. Technol.,vol. 3, no. 2, pp. 29:1-29:32, Feb. 2012 Ramamurthy, S. & Bhatnagar, R. . In 6th International Conference on Machine Learning Applications, pages 404–409, Dec 2007. Sripirakas, S. & Pears, R. . In DaWaK’14, vol(8646) of [*Lecture Notes in Computer Science*]{}, pp 439–451. Springer International Publishing, 2014. [^1]: from [*http://www.liaad.up.pt/kdus/products/datasets-for-concept-drift*]{} [^2]: from [*http://moa.cms.waikato.ac.nz/datasets/*]{} [^3]: The other 3 datasets that we experimented with displayed similar trends to that of the Flight dataset and were thus not included in interests of space constraints

--- author: - | O. L. Creevey, T. S. Metcalfe, M. Schultheis, D. Salabert,\ M. Bazot, F. Thévenin, S. Mathur, H. Xu, R. A. García bibliography: - 'seismic.bib' date: 'Received ; accepted' title: 'Characterizing solar-type stars from full-length *Kepler* data sets using the Asteroseismic Modeling Portal' --- Introduction\[sec1\] ==================== ---------- -------------------- --------------------- ------------------- ------------------- ------------------ ------ -- -- -- -- -- -- -- -- -- -- -- -- -- KIC ID [$T_{\rm eff}$]{}  [\[M/H\]]{} $K_s$ $A_{K_S}$ P$_{\rm ROT}$ Ref. (K) (dex) (mag) (mag) (days) 1435467 6326 $\pm$ 77 $+$0.01 $\pm$ 0.10 7.718 $\pm$ 0.009 0.011 $\pm$ 0.004 6.68 $\pm$ 0.89 1,A 2837475 6614 $\pm$ 77 $+$0.01 $\pm$ 0.10 7.464 $\pm$ 0.023 0.008 $\pm$ 0.002 3.68 $\pm$ 0.36 1,A 3427720 6045 $\pm$ 77 $-$0.06 $\pm$ 0.10 7.826 $\pm$ 0.009 0.020 $\pm$ 0.019 13.94 $\pm$ 2.15 1,B 3656476 5668 $\pm$ 77 $+$0.25 $\pm$ 0.10 8.008 $\pm$ 0.014 0.022 $\pm$ 0.050 31.67 $\pm$ 3.53 1,A 3735871 6107 $\pm$ 77 $-$0.04 $\pm$ 0.10 8.477 $\pm$ 0.016 0.018 $\pm$ 0.027 11.53 $\pm$ 1.24 1,A 4914923 5805 $\pm$ 77 $+$0.08 $\pm$ 0.10 7.935 $\pm$ 0.017 0.017 $\pm$ 0.029 20.49 $\pm$ 2.82 1,A 5184732 5846 $\pm$ 77 $+$0.36 $\pm$ 0.10 6.821 $\pm$ 0.005 0.012 $\pm$ 0.007 19.79 $\pm$ 2.43 1,A 5950854 5853 $\pm$ 77 $-$0.23 $\pm$ 0.10 9.547 $\pm$ 0.017 0.002 $\pm$ 0.004 1 6106415 6037 $\pm$ 77 $-$0.04 $\pm$ 0.10 5.829 $\pm$ 0.017 0.003 $\pm$ 0.020 1 6116048 6033 $\pm$ 77 $-$0.23 $\pm$ 0.10 7.121 $\pm$ 0.009 0.013 $\pm$ 0.020 17.26 $\pm$ 1.96 1,A 6225718 6313 $\pm$ 76 $-$0.07 $\pm$ 0.10 6.283 $\pm$ 0.011 0.003 $\pm$ 0.001 1 6603624 5674 $\pm$ 77 $+$0.28 $\pm$ 0.10 7.566 $\pm$ 0.019 0.008 $\pm$ 0.008 1 6933899 5832 $\pm$ 77 $-$0.01 $\pm$ 0.10 8.171 $\pm$ 0.015 0.023 $\pm$ 0.017 1 7103006 6344 $\pm$ 77 $+$0.02 $\pm$ 0.10 7.702 $\pm$ 0.015 0.007 $\pm$ 0.010 4.62 $\pm$ 0.48 1,A 7106245 6068 $\pm$ 102 $-$0.99 $\pm$ 0.19 9.419 $\pm$ 0.006 0.015 $\pm$ 0.029 4 7206837 6305 $\pm$ 77 $+$0.10 $\pm$ 0.10 8.575 $\pm$ 0.011 0.004 $\pm$ 0.005 4.04 $\pm$ 0.28 1,A 7296438 5775 $\pm$ 77 $+$0.19 $\pm$ 0.10 8.645 $\pm$ 0.009 0.012 $\pm$ 0.018 25.16 $\pm$ 2.78 1,A 7510397 6171 $\pm$ 77 $-$0.21 $\pm$ 0.10 6.544 $\pm$ 0.009 0.018 $\pm$ 0.010 1 7680114 5811 $\pm$ 77 $+$0.05 $\pm$ 0.10 8.673 $\pm$ 0.006 0.011 $\pm$ 0.013 26.31 $\pm$ 1.86 1,A 7771282 6248 $\pm$ 77 $-$0.02 $\pm$ 0.10 9.532 $\pm$ 0.010 0.005 $\pm$ 0.001 11.88 $\pm$ 0.91 1,A 7871531 5501 $\pm$ 77 $-$0.26 $\pm$ 0.10 7.516 $\pm$ 0.017 0.023 $\pm$ 0.021 33.72 $\pm$ 2.60 1,A 7940546 6235 $\pm$ 77 $-$0.20 $\pm$ 0.10 6.174 $\pm$ 0.011 0.023 $\pm$ 0.009 11.36 $\pm$ 0.95 1,A 7970740 5309 $\pm$ 77 $-$0.54 $\pm$ 0.10 6.085 $\pm$ 0.011 0.003 $\pm$ 0.013 17.97 $\pm$ 3.09 1,A 8006161 5488 $\pm$ 77 $+$0.34 $\pm$ 0.10 5.670 $\pm$ 0.015 0.009 $\pm$ 0.006 29.79 $\pm$ 3.09 1,A 8150065 6173 $\pm$ 101 $-$0.13 $\pm$ 0.15 9.457 $\pm$ 0.014 0.010 $\pm$ 0.013 4 8179536 6343 $\pm$ 77 $-$0.03 $\pm$ 0.10 8.278 $\pm$ 0.009 0.005 $\pm$ 0.016 24.55 $\pm$ 1.61 1,A 8379927 6067 $\pm$ 120 $-$0.10 $\pm$ 0.15 5.624 $\pm$ 0.011 0.004 $\pm$ 0.012 16.99 $\pm$ 1.35 2,A 8394589 6143 $\pm$ 77 $-$0.29 $\pm$ 0.10 8.226 $\pm$ 0.016 0.013 $\pm$ 0.010 1 8424992 5719 $\pm$ 77 $-$0.12 $\pm$ 0.10 8.843 $\pm$ 0.011 0.016 $\pm$ 0.018 1 8694723 6246 $\pm$ 77 $-$0.42 $\pm$ 0.10 7.663 $\pm$ 0.007 0.003 $\pm$ 0.001 1 8760414 5873 $\pm$ 77 $-$0.92 $\pm$ 0.10 8.173 $\pm$ 0.009 0.016 $\pm$ 0.012 1 8938364 5677 $\pm$ 77 $-$0.13 $\pm$ 0.10 8.636 $\pm$ 0.016 0.003 $\pm$ 0.009 1 9025370 5270 $\pm$ 180 $-$0.12 $\pm$ 0.18 7.372 $\pm$ 0.025 0.041 $\pm$ 0.030 3 9098294 5852 $\pm$ 77 $-$0.18 $\pm$ 0.10 8.364 $\pm$ 0.009 0.011 $\pm$ 0.021 19.79 $\pm$ 1.33 1,A 9139151 6302 $\pm$ 77 $+$0.10 $\pm$ 0.10 7.952 $\pm$ 0.014 0.002 $\pm$ 0.011 10.96 $\pm$ 2.22 1,B 9139163 6400 $\pm$ 84 $+$0.15 $\pm$ 0.09 7.231 $\pm$ 0.007 0.013 $\pm$ 0.007 6 9206432 6538 $\pm$ 77 $+$0.16 $\pm$ 0.10 8.067 $\pm$ 0.013 0.032 $\pm$ 0.037 8.80 $\pm$ 1.06 1,A 9353712 6278 $\pm$ 77 $-$0.05 $\pm$ 0.10 9.607 $\pm$ 0.011 0.011 $\pm$ 0.010 11.30 $\pm$ 1.12 1,A 9410862 6047 $\pm$ 77 $-$0.31 $\pm$ 0.10 9.375 $\pm$ 0.013 0.011 $\pm$ 0.001 22.77 $\pm$ 2.37 1,A 9414417 6253 $\pm$ 75 $-$0.13 $\pm$ 0.10 8.407 $\pm$ 0.009 0.010 $\pm$ 0.010 10.68 $\pm$ 0.66 7,A 9955598 5457 $\pm$ 77 $+$0.05 $\pm$ 0.10 7.768 $\pm$ 0.017 0.002 $\pm$ 0.001 34.20 $\pm$ 5.64 1,A 9965715 5860 $\pm$ 180 $-$0.44 $\pm$ 0.18 7.873 $\pm$ 0.012 0.005 $\pm$ 0.005 3 10079226 5949 $\pm$ 77 $+$0.11 $\pm$ 0.10 8.714 $\pm$ 0.012 0.015 $\pm$ 0.025 14.81 $\pm$ 1.23 1,A 10454113 6177 $\pm$ 77 $-$0.07 $\pm$ 0.10 7.291 $\pm$ 9.995 0.042 $\pm$ 0.019 14.61 $\pm$ 1.09 1,A 10516096 5964 $\pm$ 77 $-$0.11 $\pm$ 0.10 8.129 $\pm$ 0.015 0.000 $\pm$ 0.012 1 10644253 6045 $\pm$ 77 $+$0.06 $\pm$ 0.10 7.874 $\pm$ 0.021 0.008 $\pm$ 0.015 10.91 $\pm$ 0.87 1,A 10730618 6150 $\pm$ 180 $-$0.11 $\pm$ 0.18 7.874 $\pm$ 0.021 0.008 $\pm$ 0.015 3 10963065 6140 $\pm$ 77 $-$0.19 $\pm$ 0.10 7.486 $\pm$ 0.011 0.003 $\pm$ 0.016 12.58 $\pm$ 1.70 1,A 11081729 6548 $\pm$ 82 $+$0.11 $\pm$ 0.10 7.973 $\pm$ 0.011 0.005 $\pm$ 0.001 2.74 $\pm$ 0.31 1,A 11253226 6642 $\pm$ 77 $-$0.08 $\pm$ 0.10 7.459 $\pm$ 0.007 0.017 $\pm$ 0.013 3.64 $\pm$ 0.37 1,A 11772920 5180 $\pm$ 180 $-$0.09 $\pm$ 0.18 7.981 $\pm$ 0.014 0.008 $\pm$ 0.005 3 12009504 6179 $\pm$ 77 $-$0.08 $\pm$ 0.10 8.069 $\pm$ 0.019 0.005 $\pm$ 0.034 9.39 $\pm$ 0.68 1,A 12069127 6276 $\pm$ 77 $+$0.08 $\pm$ 0.10 9.494 $\pm$ 0.012 0.016 $\pm$ 0.005 0.92 $\pm$ 0.05 1,A 12069424 5825 $\pm$ 50 $+$0.10 $\pm$ 0.03 4.426 $\pm$ 0.009 0.005 $\pm$ 0.006 23.80 $\pm$ 1.80 5,B 12069449 5750 $\pm$ 50 $+$0.05 $\pm$ 0.02 4.651 $\pm$ 0.005 0.005 $\pm$ 0.006 23.20 $\pm$ 6.00 5,B 12258514 5964 $\pm$ 77 $+$-0.00 $\pm$ 0.10 6.758 $\pm$ 0.011 0.021 $\pm$ 0.021 15.00 $\pm$ 1.84 1,A 12317678 6580 $\pm$ 77 $-$0.28 $\pm$ 0.10 7.631 $\pm$ 0.009 0.027 $\pm$ 0.021 1 ---------- -------------------- --------------------- ------------------- ------------------- ------------------ ------ -- -- -- -- -- -- -- -- -- -- -- -- -- Spectroscopic references: $^1$[@Buchhave2015], $^2$[@Ramirez2009], $^3$[@Pinsonneault2012], $^4$[@Huber2013], $^5$[@Chaplin2014], $^6$[@Pinsonneault2014],$^7$[@Casagrande2014]\ Rotation period references: $^{\rm A}$[@garcia2014], $^{\rm B}$[@cellier2016] Solar-like oscillations are stochastically excited and intrinsically damped by turbulent motions in the near-surface layers of stars with substantial outer convection zones. The sound waves produced by these motions travel through the interior of the star, and those with resonant frequencies drive global oscillations that modulate the integrated brightness of the star by a few parts per million and change the [surface]{} radial velocity by several meters per second. The characteristic timescale of these variations is determined by the sound travel time across the stellar diameter, which is around 5 minutes for a star like the Sun. With sufficient precision, more than a dozen consecutive overtones can be detected for each set of oscillation modes with radial, dipole, quadrupole, and sometimes even octupole geometry [(i.e., for $l=0, 1, 2,$ and 3, respectively, where $l$ is the angular degree)]{}. The technique of asteroseismology uses these oscillation frequencies [combined]{} with other observational constraints to measure the stellar radius, mass, age, and other properties of the stellar interior [for a recent review, see @ChaplinMiglio2013]. The [*Kepler*]{} space telescope yielded unprecedented data for the study of solar-like oscillations in other stars. Ground-based radial velocity data had previously allowed the detection of solar-like oscillations in some of the brightest stars in the sky [e.g., @Brown1991; @Kjeldsen1995; @Bedding2001; @Bouchy2002; @Carrier2003], but [intensive]{} multi-site campaigns were required to measure and identify the frequencies unambiguously [e.g., @Arentoft2008]. The [*Convection Rotation and planetary Transits*]{} satellite (CoRoT, @Baglin2006) achieved the [photometric]{} precision necessary to detect solar-like oscillations in main-sequence stars [e.g., @Michel2008], and it obtained continuous photometry for up to five months. NASA’s [*Kepler*]{} mission [@Borucki2010] extended these initial successes to a larger sample of solar-type stars, with observations eventually spanning up to several years [@Chaplin2010]. Precise photometry from [*Kepler*]{} led to the detection of solar-like oscillations in nearly 600 main-sequence and subgiant stars [@Chaplin2014], including the measurement of individual frequencies in more than 150 targets [@Appourchaux2012; @Davies2016; @Lund2016]. Asteroseismic modeling has become more sophisticated over time, with better methods gradually developing alongside the [extended]{} observations and [improved]{} data analysis techniques. Initial efforts attempted to reproduce the observed large and small frequency separations with models that simultaneously matched constraints from spectroscopy [e.g., @JCD1995; @Thevenin2002; @Fernandes2003; @Thoul2003]. As individual oscillation frequencies became available, modelers started to match the observations in échelle diagrams that [highlighted variations around the average frequency separations]{} [e.g., @DiMauro2003; @Guenther2004; @Eggenberger2004]. This approach continued until the frequency precision from longer space-based observations became sufficient to reveal systematic errors in the models that are known as [****]{}, which arise from incomplete modeling of the near-surface layers where the mixing-length treatment of convection is [approximate]{}. [@Kjeldsen2008] proposed an empirical correction for [the]{} surface effects based on the discrepancy for [the]{} standard solar model, and applied it to ground-based observations of several stars with different masses and evolutionary states. The correction was subsequently implemented using stars observed by [CoRoT]{} and [*Kepler*]{} [@Kallinger2010; @Metcalfe2010]. During the [*Kepler*]{} mission, asteroseismic modeling methods were adapted as longer data sets became available. The first year of short-cadence data [[sampled at]{} 58.85 s, @Gilliland2010] was devoted to an asteroseismic survey of 2000 solar-type stars observed for one month each. The survey initially yielded [frequencies for]{} 22 stars, [allowing]{} detailed modeling [@Mathur2012], and hundreds of targets were flagged for extended observations during the remainder of the mission. Longer data sets improved the signal-to-noise ratio (S/N) [of the power spectrum]{} for stars with [previously]{} marginal detections, and yielded additional oscillation frequencies for the best targets in the sample. The first coordinated analysis of nine-month data sets yielded individual frequencies in 61 stars [@Appourchaux2012], though many were subgiants with complex patterns of dipole mixed-modes. The larger set of radial orders observed in each star began to reveal the limitations of the empirical correction for surface effects [@Metcalfe2014]. This situation [motivated the implementation]{} of a Bayesian method that marginalized over the unknown systematic error [for each frequency]{} [@Gruberbauer2012], as well as a method for fitting ratios of frequency separations that are insensitive to surface effects [@Roxburgh2003; @Bazot2013; @SilvaAguirre2013]. It also inspired the development of a more physically motivated correction [based on an analysis of frequency shifts induced by the solar magnetic cycle]{} [[@Gough1990; @Ball2014; @schmittbasu2015]]{}. The [*Kepler*]{} telescope completed its primary mission in 2013, but the large samples of multi-year observations posed an enormous data analysis challenge that has only recently been surmounted [@benomar2014a; @benomar2014b; @Davies2015; @Davies2016; @Lund2016]. The first modeling of these full-length data sets appeared in [@SilvaAguirre2015] and [@Metcalfe2015]. In this paper we apply the latest version of the Asteroseismic Modeling Portal [hereafter AMP, see @Metcalfe2009] to [oscillation frequencies derived from]{} the full-length [*Kepler*]{} [observations]{} for 57 stars, [as determined by [@Lund2016]]{}. The new fitting method relies on ratios of frequency separations rather than the individual frequencies, so that we can use the modeling results to investigate the empirical amplitude and character of [the]{} surface effects within the sample. We describe the sources of our adopted observational constraints in Section \[sec2\]. We outline updates to the AMP input physics and fitting methods in Section \[sec3\], including an overview of how the optimal stellar properties and their uncertainties are determined. In Section \[sec4\] we present the modeling results, and in Section \[sec:sec5\] we use them to establish the limitations of the [@Kjeldsen2008] correction for surface effects. Finally, after summarizing in Section \[sec6\], we discuss our expectations for asteroseismic modeling of future observations from the Transiting Exoplanet Survey Satellite, [TESS, @tess-ricker2015] and PLanetary Transits and Oscillations of stars [PLATO, @plato-rauer2014] missions. Observational constraints\[sec2\] ================================= To constrain the properties of each star in our sample, we adopted the solar-like oscillation frequencies determined by [@Lund2016] from a uniform analysis of the full-length [*Kepler*]{} data sets. [For each target]{}, the power spectrum of the time-series photometry shows the oscillations embedded in several background components attributed to granulation, faculae, and shot noise. The [power spectral distribution of]{} individual [modes]{} were modeled as Lorentzian functions, and the background components were optimized simultaneously in a Bayesian manner using the procedure described in [@Lund2014]. For the targets presented here, this analysis resulted in sets of oscillation modes spanning 7 to 20 radial orders. In most cases, the identified frequencies included only $l=0$, 1, and 2 modes, but [for]{} 14 stars, [the mode-fitting procedure]{} also [identified]{} limited sets of $l=3$ modes spanning 2 to 6 radial orders. Complete tables of the identified frequencies for each star are published in [@Lund2016]. To complement the [oscillation frequencies]{}, we also adopted spectroscopic constraints on the effective temperature, $T_{\rm eff}$, and metallicity, \[M/H\], for each star. For 46 of the targets in our sample, we used the uniform spectroscopic analysis of [@Buchhave2015]. In this case, the values and uncertainties on $T_{\rm eff}$ and \[M/H\] were determined using the Stellar Parameters Classification (SPC) method described in detail by [@Buchhave2012; @Buchhave2014]. For the other 11 stars in our sample, [which were not included in [@Buchhave2015]]{}, we adopted constraints from a variety of sources, including [@Ramirez2009; @Pinsonneault2012; @Huber2013; @Chaplin2014; @Pinsonneault2014], and from the SAGA survey [@Casagrande2014]. The [57 stars in our sample]{} span a range of $T_{\rm eff}$ from 5180 to 6642 K and \[M/H\] from $-$0.99 to 0.36 dex. These atmospheric constraints are listed in Table \[tab1\] along with [the K-band magnitude from 2MASS, $K_s$ [@2MASS], the derived interstellar absorption, $A_{Ks}$ (see Section \[sec:asteroseismicdistances\]),]{} and rotational periods from @garcia2014 and @cellier2016. Although independent determinations of the radius and luminosity are available for a few of the stars in our sample, we excluded these constraints from the modeling so that we could use them to assess the accuracy of our results (see Section \[sec4\]). AMP$_1$ AMP$_2$ AMP$_3$ AMP$_4$ $\langle P \rangle$ $\sigma$ ----------------------- --------- --------- --------- --------- --------------------- ---------- $R$ ([R$_{\odot}$]{}) 1.002 1.003 1.003 1.010 1.001 0.005 $M$ ([M$_{\odot}$]{}) 1.01 1.01 1.01 1.03 1.001 0.02 Age (Gyr) 4.59 4.38 4.41 4.69 4.38 0.22 [$Z_i$]{} 0.019 0.021 0.020 0.024 0.017 0.002 [$Y_i$]{} 0.266 0.281 0.278 0.282 0.265 0.023 $\alpha$ 2.16 2.24 2.24 2.30 2.12 0.12 $L$ ([L$_{\odot}$]{}) 0.96 0.99 0.99 1.00 0.97 0.03 $\log g$ (dex) 4.441 4.439 4.439 4.442 4.438 0.003 $\chi^2$ 1.047 0.968 0.995 1.058 : Reference solar parameters from AMP using the updated method and physics \[tab:solar-reference\] ![\[fig:hrdiag\]HR diagram showing the position of the sample of stars used for this work. Evolutionary tracks for solar-metallicity models with 1.0, 1.2, and 1.4 [M$_{\odot}$]{} stellar masses are shown.](hrdiag){width="48.00000%"} Asteroseismic modeling\[sec3\] ============================== [Based on]{} the observational constraints described in Section \[sec2\], [we determined]{} the properties of each star in our sample using the latest version of AMP. [The method]{} [relies on]{} a parallel genetic algorithm [hereafter GA, see @Metcalfe2003] to optimize the match between the [properties of a stellar]{} model and a given set of observations. The asteroseismic models [are]{} generated by the Aarhus stellar evolution and adiabatic pulsation codes [@JCD08a; @JCD08b]. The search procedure generates thousands of models that can be used to evaluate the stellar properties and their uncertainties. Unlike the usual grid-modeling approach, the GA preferentially samples combinations of model parameters that provide a better than average match to the observations. This approach allows us not only to identify the globally optimal solution, but also to include the effects of parameter correlations and non-uniqueness into reliable uncertainties. Below we outline recent updates to the input physics and [model-fitting]{} methods, and we describe improvements to our statistical analysis of the results. Updated physics and methods --------------------------- The AMP code has been in development since 2004. Details about previous versions are outlined in [@Metcalfe2015]. For this paper we use version 1.3, which includes input physics that are mostly unchanged from version 1.2 [@Metcalfe2014]. It uses the [2005 release of the]{} OPAL equation of state [@Rogers2002], with opacities from OPAL [@Iglesias1996] supplemented by [@Ferguson2005] at low temperatures. Nuclear reaction rates come from the NACRE collaboration [@Angulo1999]. The prescription of [@Michaud1993] for diffusion and settling is applied to helium, but not to heavier elements because some models are numerically instable. Convection is described using the mixing-length treatment of [@BohmVitense1958] with no overshoot. There have been several minor updates to the model physics for version 1.3 of the AMP code. First, it incorporates the revised $^{14}N+p$ reaction from NACRE [@Angulo2005], which is particularly important for more evolved stars. Second, it uses the solar mixture of [@GS1998] instead of [@GN1993]. This requires different opacity tables and a slight modification to the calculation of metallicity \[$\log(Z_\odot/X_\odot) = -1.64$ instead of $-$1.61\]. Finally, following the suggestion of [@SilvaAguirre2015], diffusion and settling is only applied to models with $M < 1.2\ M_\odot$, to avoid potential biases that are due to the short diffusion timescales in the envelopes of more massive stars. {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} {width="45.00000%"} [The frequency separation ratios $r_{01}$ and $r_{02}$ were defined by [@Roxburgh2003] as]{} $$r_{01}(n) = \frac{\nu_{n-1,0} - 4\nu_{n-1,1} + 6\nu_{n,0} - 4\nu_{n,1} + \nu_{n+1,0}}{8(\nu_{n,1} - \nu_{n-1,1})} \label{eqn:r01}$$ and $$r_{02}(n) = \frac{\nu_{n,0} - \nu_{n-1,2}}{\nu_{n,1} - \nu_{n-1,1}},$$ [where $\nu$ is the mode frequency, $n$ is the radial order, and $l$ is the angular degree.]{} These ratios were first included as observational constraints in AMP 1.2. Version 1.3 uses these ratios exclusively, omitting the individual oscillation frequencies to avoid potential biases from the empirical correction for surface effects. AMP 1.3 also [calculates]{} the full covariance matrix of $r_{01}$, which is necessary to properly account for correlations induced by the five-point smoothing that is [implicit in Eq. (\[eqn:r01\])]{}. [For each stellar model, AMP 1.2 defined the quality of the match to observations using a combination of metrics from four different sets of constraints. For AMP 1.3, we]{} combine all observational constraints into a single $\chi^2$ metric $$\chi^2 = (x - x_M)^T C^{-1} (x - x_M), \label{eqn:chisq}$$ where $C$ is the covariance matrix of the observational constraints $x$, and $x_M$ are the corresponding observables from the model. For the results presented here, $x$ includes only the ratios $r_{01}$ and $r_{02}$ [augmented by]{} the atmospheric constraints [$T_{\rm eff}$]{} and [\[M/H\]]{}. [$C$ is assumed to be diagonal for all observables except $r_{01}$. Like all previous versions of AMP, the individual frequencies are]{} used to calculate the average large separation of the radial modes $\Delta\nu_0$, allowing us to optimize the [stellar]{} age [along each model sequence]{} and then match the lowest observed radial mode frequency [see @Metcalfe2009]. Statistical analysis -------------------- Versions [1.0 and 1.1]{} of the AMP code performed a local analysis near the optimal model to determine the uncertainties on each parameter [@Metcalfe2009; @Mathur2012]. This approach failed to capture the uncertainties due to parameter correlations and non-uniqueness of the solution, so that it typically [produced]{} implausibly small error bars, although formally correct. To [derive]{} more realistic uncertainties [in version 1.2]{}, [@Metcalfe2014] began using the thousands of models sampled by the GA [during the optimization procedure. As the GA approaches the optimal model, each parameter is densely sampled with a uniform spacing in stellar mass ($M$), initial metallicity ($Z_i$), initial helium mass fraction ($Y_i$), and mixing-length ($\alpha$)]{}. Each sampled model is assigned a likelihood $$\mathcal{L}=\exp\left( \frac{-\chi^2}{2} \right), \label{eqn:likelihooddefn}$$ where $\chi^2$ is calculated from Eq. (\[eqn:chisq\]). By assuming flat priors on each of the model parameters, we then construct posterior probability functions (PPF) for each of the stellar properties to obtain more reliable estimates of the values and uncertainties from the dense ensemble of models sampled by the GA. We adopt the median value of the PPF as the best estimate for the parameter value, $\langle P \rangle$. We use the 68% credible interval [of the PPF]{} to define the associated uncertainty, $\sigma$. Sample PPFs for the radius, mass, and age of KIC 12069424 are shown in Fig. \[fig:examplelikelihood\]. Combining the best estimates for each of the stellar properties generally will not produce the best stellar model. For many purposes it is useful to identify a [**]{}, an individual stellar model that is representative of the PPF. The optimal model identified by AMP, $P_{\rm AMP}$, is used as the reference model, but it can sometimes fall near the edge of one or more of the distributions. A comparison of the masses and ages estimated from $\langle P \rangle$ and $P_{\rm AMP}$ yields differences much smaller than 1$\sigma$ for most cases. Validation with solar data -------------------------- To validate our new approach, we used AMP 1.3 to [match a set of solar oscillation frequencies comparable to the [*Kepler*]{} observations of 16 Cyg A and B [@Metcalfe2015]. The frequencies were derived from observations obtained with]{} the Variability of solar IRradiance and Gravity Oscillations (VIRGO) instrument [@virgo] [using 2.5 years of data [@Davies2015]]{}. The best models identified by the four independent runs of the GA are listed in Table \[tab:solar-reference\] under the headings AMP$_N$ along with their individual $\chi^2$ values[^1]. The model with the lowest value of $\chi^2$ is the optimal solution identified by AMP, and this is adopted as the reference model. The remaining models reveal intrinsic parameter correlations, in particular between the mass and initial composition. The final two columns of Table \[tab:solar-reference\] show the values of $\langle P \rangle$ and $\sigma$ derived from the PPFs, showing excellent agreement with the known solar properties: [$R, M, L \equiv 1$, age$=4.60\pm0.04$ Gyr [@Houdek2011].]{} ![Comparison of measured radii (top), luminosities (middle), and parallaxes (lower) with those deduced from the asteroseismic parameters. The interferometric radii are denoted by the red circles in the top panel, and the green triangle is the value from @masana06[]{data-label="fig:lumrad"}](lumrad "fig:"){width="48.00000%"} ![Comparison of measured radii (top), luminosities (middle), and parallaxes (lower) with those deduced from the asteroseismic parameters. The interferometric radii are denoted by the red circles in the top panel, and the green triangle is the value from @masana06[]{data-label="fig:lumrad"}](comparedist "fig:"){width="48.00000%"} Results\[sec4\] =============== ---------- ------------------- ------------------- ------- -------- ------- ---------- ----------- ------- ------------------ ------------------ ----------------------- -- -- -- -- -- -- -- -- KIC ID $R$ $M$ Age $Z_i$ $Y_i$ $\alpha$ $X_c/X_i$ $a_0$ $\chi^2_{N,r01}$ $\chi^2_{N,r02}$ $\chi^2_{N,\rm spec}$ ([R$_{\odot}$]{}) ([M$_{\odot}$]{}) (Gyr) Sun 1.003 1.01 4.38 0.0210 0.281 2.24 0.50 -2.54 1.03 0.78 0.71 1435467 1.704 1.41 1.87 0.0231 0.284 1.84 0.43 -3.95 2.68 1.64 1.49 2837475 1.613 1.41 1.70 0.0168 0.247 1.70 0.53 -4.48 1.29 2.07 0.32 3427720 1.125 1.13 2.17 0.0168 0.259 2.10 0.64 -2.41 1.10 1.26 0.15 3656476 1.326 1.10 8.48 0.0231 0.248 2.30 0.00 -2.22 2.35 0.68 1.57 3735871 1.089 1.08 1.57 0.0157 0.292 2.02 0.71 -3.64 1.47 0.67 0.05 4914923 1.326 1.01 7.15 0.0121 0.260 1.68 0.02 -4.51 0.56 1.50 3.35 5184732 1.365 1.27 4.70 0.0340 0.242 1.92 0.27 -4.43 6.98 2.32 0.85 5950854 1.257 1.01 9.01 0.0147 0.249 2.16 0.00 -1.27 0.60 4.61 1.30 6106415 1.213 1.06 4.43 0.0184 0.295 2.04 0.18 -3.48 0.93 2.81 0.54 6116048 1.239 1.06 5.84 0.0114 0.242 2.16 0.11 -3.27 3.27 2.48 0.44 6225718 1.194 1.06 2.30 0.0117 0.286 2.02 0.49 -5.99 3.47 0.97 0.64 6603624 1.159 1.03 8.64 0.0455 0.313 2.12 0.01 -2.34 3.42 135.14 5.90 6933899 1.535 1.03 6.58 0.0152 0.296 1.76 0.00 -4.38 1.45 1.25 0.21 7103006 1.957 1.56 1.94 0.0224 0.239 1.66 0.36 -7.28 1.15 0.69 1.33 7106245 1.120 0.97 6.05 0.0070 0.242 1.98 0.22 -4.02 2.96 0.73 4.41 7206837 1.579 1.41 1.72 0.0255 0.249 1.52 0.60 -4.61 1.48 1.43 1.52 7296438 1.371 1.10 5.93 0.0309 0.315 2.04 0.02 -2.76 0.74 0.53 0.47 7510397 1.828 1.30 3.58 0.0129 0.248 1.84 0.08 -2.37 0.75 2.23 0.55 7680114 1.395 1.07 7.04 0.0197 0.277 2.02 0.00 -3.00 1.63 0.74 0.00 7771282 1.645 1.30 3.13 0.0168 0.257 1.78 0.19 -4.03 2.10 0.75 0.33 7871531 0.859 0.80 9.32 0.0125 0.296 2.02 0.34 -4.15 1.06 0.65 1.25 7940546 1.917 1.39 2.58 0.0152 0.259 1.74 0.07 -6.26 2.47 0.82 1.45 7970740 0.779 0.78 10.59 0.0094 0.244 2.36 0.45 -2.55 4.93 5.09 3.34 8006161 0.954 1.06 4.34 0.0485 0.288 2.66 0.61 -0.63 2.33 1.21 1.26 8150065 1.394 1.20 3.33 0.0162 0.252 1.62 0.21 -3.97 2.03 2.30 0.66 8179536 1.353 1.26 2.03 0.0157 0.249 1.88 0.50 -3.89 1.51 0.62 0.01 8379927 1.105 1.08 1.65 0.0162 0.287 1.82 0.71 -4.98 1.87 1.63 0.33 8394589 1.169 1.06 3.82 0.0094 0.247 1.98 0.37 -3.14 0.71 0.70 0.01 8424992 1.056 0.94 9.62 0.0162 0.264 2.30 0.14 -1.38 0.70 0.30 0.22 8694723 1.493 1.04 4.22 0.0085 0.309 2.36 0.00 -2.23 0.70 1.46 3.18 8760414 1.028 0.82 12.09 0.0042 0.239 2.14 0.07 -2.42 0.52 1.69 4.43 8938364 1.361 1.00 11.00 0.0217 0.272 2.14 0.00 -2.09 1.44 3.52 3.26 9025370 1.000 0.97 5.50 0.0184 0.253 1.60 0.54 -6.01 1.45 3.78 0.27 9098294 1.151 0.99 8.22 0.0129 0.245 2.14 0.11 -3.13 1.93 0.96 0.23 9139151 1.167 1.20 1.84 0.0203 0.265 2.48 0.63 -1.58 1.66 1.26 0.17 9139163 1.582 1.49 1.26 0.0330 0.245 1.64 0.71 -9.60 0.95 1.89 4.25 9206432 1.499 1.37 1.32 0.0247 0.285 1.82 0.65 -2.37 1.68 1.10 0.72 9353712 2.183 1.56 2.17 0.0203 0.249 1.76 0.08 -1.89 2.57 0.73 1.16 9410862 1.159 0.99 6.15 0.0091 0.247 1.90 0.20 -3.11 1.28 0.75 0.74 9414417 1.896 1.40 2.67 0.0147 0.244 1.70 0.11 -5.41 1.01 0.78 0.39 9955598 0.876 0.87 6.38 0.0203 0.308 2.16 0.48 -2.71 1.15 2.13 0.13 9965715 1.224 0.99 3.00 0.0080 0.310 1.58 0.33 -5.57 0.78 0.65 1.76 10079226 1.135 1.09 2.35 0.0203 0.291 1.84 0.61 -4.10 1.39 0.73 0.12 10454113 1.282 1.27 2.03 0.0217 0.244 2.02 0.58 -0.79 2.07 4.38 1.79 10516096 1.407 1.08 6.44 0.0168 0.270 2.04 0.00 -2.81 1.29 1.14 0.65 10644253 1.073 1.04 1.14 0.0162 0.319 1.78 0.78 -4.91 0.78 0.62 0.31 10730618 1.729 1.33 2.55 0.0147 0.253 1.34 0.30 -2.14 2.04 3.36 0.14 10963065 1.210 1.04 4.28 0.0114 0.277 2.04 0.22 -3.53 1.41 0.98 0.00 11081729 1.393 1.25 1.88 0.0143 0.271 1.86 0.51 -5.62 6.03 5.17 1.56 11253226 1.635 1.53 1.06 0.0224 0.248 1.90 0.69 -4.76 2.76 1.83 2.00 11772920 0.839 0.81 11.11 0.0143 0.254 1.82 0.43 -3.90 2.28 0.35 0.33 12009504 1.379 1.13 3.44 0.0157 0.294 1.96 0.26 -4.67 0.81 0.88 0.10 12069127 2.262 1.58 1.89 0.0203 0.262 1.64 0.12 -4.46 3.00 0.79 0.02 12069424 1.223 1.07 7.35 0.0179 0.241 2.12 0.09 -4.41 3.78 1.02 1.39 12069449 1.105 1.01 6.88 0.0217 0.278 2.14 0.22 -2.90 4.93 0.94 0.69 12258514 1.601 1.25 6.11 0.0247 0.229 1.64 0.00 -4.04 2.45 0.92 9.89 12317678 1.749 1.27 2.18 0.0107 0.302 1.74 0.13 -5.26 1.22 1.09 0.65 ---------- ------------------- ------------------- ------- -------- ------- ---------- ----------- ------- ------------------ ------------------ ----------------------- -- -- -- -- -- -- -- -- Notes: The parameters are radius, mass, age, initial metallicity $Z_i$ and helium $Y_i$ mass fraction, mixing-length parameter $\alpha$, ratio of current central hydrogen to initial hydrogen mass fraction, $X_c/X_i$, the $a_0$ parameter in Eq. \[eqn:kjeldsen\], and the normalized $\chi^2$ values for the $r_{01}$, $r_{02}$ and spectroscopic data. [The sample of stars analyzed in this work span the main-sequence and early subgiant phase, as illustrated by their position in the Hertzsprung-Russell diagram (Fig. \[fig:hrdiag\]). They cover a range in mass of about 0.6 [M$_{\odot}$]{}, with about half of the sample being within 10% of the solar value. For a representative set of four stars, Fig. \[fig:fitdata0\] compares the measured frequency separation ratios (crosses) with the corresponding values from the reference models (red filled dots). Here it can be seen that the agreement with the seismic observations is in general excellent, but some of the models do not necessarily reproduce features of the observed data. One example is KIC 10454113, which is shown in the lower right panel. It displays an oscillation as a function of frequency that the models fail to reproduce. These discrepancies are indeed noted in the normalized [$\chi^2$]{} value, $\chi^2_N = \chi^2/N = 3.2$, where $N$ is the number of frequency ratios. For KIC 8006161, shown in the top left panel, the fit is of higher quality with $\chi^2_N = 1.8$.]{} The parameters of the reference models that are used to compare with the observations are listed in Table \[tab:referencemodels\] along with the individual $\chi^2_N$ values for $r_{01}$, $r_{02}$, and combined [$T_{\rm eff}$]{} and [\[M/H\]]{}. [For the Sun and each star in our sample, we derived a best estimate and uncertainty for the stellar radius, mass, age, metallicity, luminosity, and surface gravity using the method described in Sec. \[sec3\] (see Table \[tab:properties\_derived\]). Using the rotation periods given in Table \[tab1\] and the derived radius, we also computed their rotational velocities.]{} [Since the AMP 1.3 method uses only one set of physics in the stellar modeling]{}, the derived uncertainties do not include possible systematic errors arising from errors in the model physics, such as the equation of state, heavy element settling, and convective overshoot. However, the uncertainties include sources of errors arising from free parameters that are often fixed in [the stellar codes used in other methods]{}, for example, the mixing-length parameter $\alpha$, the initial chemical composition $(X_i, Y_i, Z_i),$ or a chemical enrichment law. The [uncertainty on these]{} parameters contributes substantially to the error budget, and in some cases more so, for example, changing the equation of state or the opacities. The effect of such changes in the physics has been studied in detail for HD52265 by [@lebreton2014]. A similar detailed [analysis]{} for each star in the sample we studied is beyond the scope of this paper. [We refer to @silvaaguirre2016, who also analyzed data from @Lund2016 using seven distinct modeling methods and codes.]{} The accuracy, namely the bias and not [the]{} precision, of our results can be [ascertained]{} by an analysis of the solar observations. As [stated]{} above, we [derived a best-matched model with values for the mass of]{} a 1 [M$_{\odot}$]{} model and a radius of 1 [R$_{\odot}$]{}, and an age that, within the derived uncertainty, matches the solar value. A second accuracy test, at least for the age, can be established based on the independently derived ages for the binary system 16 Cyg A and B (also known as KIC 12069449 and KIC 12069424). [The ages that we derive agree to within 1$\sigma$.]{} ---------- ------------------- ------------------- ------------------ ------------------- ---------------- ------------------- ------------------ ------------------ ------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- KIC ID $R$ $M$ Age $L$ $T_{\rm eff}$ $\log g$ \[M/H\] $\pi$ $v$ ([R$_{\odot}$]{}) ([M$_{\odot}$]{}) (Gyr) ([L$_{\odot}$]{}) (K) (dex) (dex) (mas) (km s$^{-1}$) Sun 1.001 $\pm$ 0.005 1.001 $\pm$ 0.019 4.38 $\pm$ 0.22 0.97 $\pm$ 0.03 5732 $\pm$ 43 4.438 $\pm$ 0.003 0.07 $\pm$ 0.04 1435467 1.728 $\pm$ 0.027 1.466 $\pm$ 0.060 1.97 $\pm$ 0.17 4.29 $\pm$ 0.25 6299 $\pm$ 75 4.128 $\pm$ 0.004 0.09 $\pm$ 0.09 6.99 $\pm$ 0.24 13.09 $\pm$ 1.76 2837475 1.629 $\pm$ 0.027 1.460 $\pm$ 0.062 1.49 $\pm$ 0.22 4.54 $\pm$ 0.26 6600 $\pm$ 71 4.174 $\pm$ 0.007 0.05 $\pm$ 0.07 8.18 $\pm$ 0.29 22.40 $\pm$ 2.22 3427720 1.089 $\pm$ 0.009 1.034 $\pm$ 0.015 2.37 $\pm$ 0.23 1.37 $\pm$ 0.08 5989 $\pm$ 71 4.378 $\pm$ 0.003 -0.05 $\pm$ 0.09 11.04 $\pm$ 0.40 3.95 $\pm$ 0.61 3656476 1.322 $\pm$ 0.007 1.101 $\pm$ 0.025 8.88 $\pm$ 0.41 1.63 $\pm$ 0.06 5690 $\pm$ 53 4.235 $\pm$ 0.004 0.17 $\pm$ 0.07 8.49 $\pm$ 0.30 2.11 $\pm$ 0.24 3735871 1.080 $\pm$ 0.012 1.068 $\pm$ 0.035 1.55 $\pm$ 0.18 1.45 $\pm$ 0.09 6092 $\pm$ 75 4.395 $\pm$ 0.005 -0.05 $\pm$ 0.04 8.05 $\pm$ 0.31 4.74 $\pm$ 0.51 4914923 1.339 $\pm$ 0.015 1.039 $\pm$ 0.028 7.04 $\pm$ 0.50 1.79 $\pm$ 0.12 5769 $\pm$ 86 4.198 $\pm$ 0.004 -0.06 $\pm$ 0.09 8.64 $\pm$ 0.35 3.31 $\pm$ 0.46 5184732 1.354 $\pm$ 0.028 1.247 $\pm$ 0.071 4.32 $\pm$ 0.85 1.79 $\pm$ 0.15 5752 $\pm$ 101 4.268 $\pm$ 0.009 0.31 $\pm$ 0.06 14.53 $\pm$ 0.67 3.46 $\pm$ 0.43 5950854 1.254 $\pm$ 0.012 1.005 $\pm$ 0.035 9.25 $\pm$ 0.68 1.58 $\pm$ 0.11 5780 $\pm$ 74 4.245 $\pm$ 0.006 -0.11 $\pm$ 0.06 4.41 $\pm$ 0.18 6106415 1.205 $\pm$ 0.009 1.039 $\pm$ 0.021 4.55 $\pm$ 0.28 1.61 $\pm$ 0.09 5927 $\pm$ 63 4.294 $\pm$ 0.003 -0.00 $\pm$ 0.04 25.35 $\pm$ 0.87 6116048 1.233 $\pm$ 0.011 1.048 $\pm$ 0.028 6.08 $\pm$ 0.40 1.77 $\pm$ 0.13 5993 $\pm$ 73 4.276 $\pm$ 0.003 -0.20 $\pm$ 0.08 13.31 $\pm$ 0.57 3.61 $\pm$ 0.41 6225718 1.234 $\pm$ 0.018 1.169 $\pm$ 0.039 2.23 $\pm$ 0.20 2.08 $\pm$ 0.11 6252 $\pm$ 63 4.321 $\pm$ 0.005 -0.09 $\pm$ 0.06 19.32 $\pm$ 0.60 6603624 1.164 $\pm$ 0.024 1.058 $\pm$ 0.075 8.66 $\pm$ 0.68 1.23 $\pm$ 0.11 5644 $\pm$ 91 4.326 $\pm$ 0.008 0.24 $\pm$ 0.05 11.89 $\pm$ 0.59 6933899 1.597 $\pm$ 0.008 1.155 $\pm$ 0.011 7.22 $\pm$ 0.53 2.63 $\pm$ 0.06 5815 $\pm$ 47 4.093 $\pm$ 0.002 0.11 $\pm$ 0.03 6.48 $\pm$ 0.15 7103006 1.958 $\pm$ 0.025 1.568 $\pm$ 0.051 1.69 $\pm$ 0.12 5.58 $\pm$ 0.36 6332 $\pm$ 89 4.048 $\pm$ 0.006 0.09 $\pm$ 0.10 6.19 $\pm$ 0.23 21.44 $\pm$ 2.25 7106245 1.125 $\pm$ 0.009 0.989 $\pm$ 0.023 6.05 $\pm$ 0.39 1.56 $\pm$ 0.09 6078 $\pm$ 74 4.327 $\pm$ 0.003 -0.44 $\pm$ 0.11 4.98 $\pm$ 0.20 7206837 1.556 $\pm$ 0.018 1.377 $\pm$ 0.039 1.55 $\pm$ 0.50 3.37 $\pm$ 0.15 6269 $\pm$ 87 4.191 $\pm$ 0.008 0.07 $\pm$ 0.15 5.28 $\pm$ 0.15 19.49 $\pm$ 1.37 7296438 1.370 $\pm$ 0.009 1.099 $\pm$ 0.022 6.37 $\pm$ 0.60 1.85 $\pm$ 0.08 5754 $\pm$ 55 4.205 $\pm$ 0.003 0.21 $\pm$ 0.07 6.09 $\pm$ 0.18 2.76 $\pm$ 0.30 7510397 1.823 $\pm$ 0.018 1.309 $\pm$ 0.037 3.51 $\pm$ 0.24 4.19 $\pm$ 0.20 6119 $\pm$ 69 4.031 $\pm$ 0.004 -0.14 $\pm$ 0.06 11.75 $\pm$ 0.36 7680114 1.402 $\pm$ 0.014 1.092 $\pm$ 0.030 6.89 $\pm$ 0.46 2.07 $\pm$ 0.09 5833 $\pm$ 47 4.181 $\pm$ 0.004 0.08 $\pm$ 0.07 5.73 $\pm$ 0.17 2.70 $\pm$ 0.19 7771282 1.629 $\pm$ 0.016 1.268 $\pm$ 0.040 2.78 $\pm$ 0.47 3.61 $\pm$ 0.18 6223 $\pm$ 73 4.118 $\pm$ 0.004 -0.03 $\pm$ 0.07 3.24 $\pm$ 0.10 6.94 $\pm$ 0.54 7871531 0.871 $\pm$ 0.008 0.834 $\pm$ 0.021 8.84 $\pm$ 0.46 0.60 $\pm$ 0.05 5482 $\pm$ 69 4.478 $\pm$ 0.006 -0.16 $\pm$ 0.04 16.81 $\pm$ 0.81 1.31 $\pm$ 0.10 7940546 1.974 $\pm$ 0.045 1.511 $\pm$ 0.087 2.42 $\pm$ 0.17 5.69 $\pm$ 0.35 6330 $\pm$ 43 4.023 $\pm$ 0.005 0.00 $\pm$ 0.06 12.16 $\pm$ 0.44 8.79 $\pm$ 0.76 7970740 0.776 $\pm$ 0.007 0.768 $\pm$ 0.019 10.53 $\pm$ 0.43 0.42 $\pm$ 0.04 5282 $\pm$ 93 4.546 $\pm$ 0.003 -0.37 $\pm$ 0.09 36.83 $\pm$ 1.71 2.19 $\pm$ 0.38 8006161 0.930 $\pm$ 0.009 1.000 $\pm$ 0.030 4.57 $\pm$ 0.36 0.64 $\pm$ 0.03 5351 $\pm$ 49 4.498 $\pm$ 0.003 0.41 $\pm$ 0.04 37.89 $\pm$ 1.18 1.58 $\pm$ 0.16 8150065 1.402 $\pm$ 0.018 1.222 $\pm$ 0.040 3.15 $\pm$ 0.49 2.52 $\pm$ 0.19 6138 $\pm$ 105 4.230 $\pm$ 0.005 -0.04 $\pm$ 0.15 3.94 $\pm$ 0.18 8179536 1.350 $\pm$ 0.013 1.249 $\pm$ 0.031 1.88 $\pm$ 0.25 2.63 $\pm$ 0.11 6318 $\pm$ 59 4.274 $\pm$ 0.005 -0.04 $\pm$ 0.07 6.91 $\pm$ 0.20 2.78 $\pm$ 0.18 8379927 1.102 $\pm$ 0.012 1.073 $\pm$ 0.033 1.64 $\pm$ 0.12 1.39 $\pm$ 0.10 5971 $\pm$ 91 4.382 $\pm$ 0.005 -0.04 $\pm$ 0.05 30.15 $\pm$ 1.40 3.28 $\pm$ 0.26 8394589 1.155 $\pm$ 0.009 1.024 $\pm$ 0.030 3.82 $\pm$ 0.25 1.68 $\pm$ 0.09 6103 $\pm$ 61 4.324 $\pm$ 0.003 -0.28 $\pm$ 0.07 8.47 $\pm$ 0.28 8424992 1.048 $\pm$ 0.005 0.930 $\pm$ 0.016 9.79 $\pm$ 0.76 0.99 $\pm$ 0.04 5634 $\pm$ 57 4.362 $\pm$ 0.002 -0.12 $\pm$ 0.06 7.52 $\pm$ 0.23 8694723 1.463 $\pm$ 0.023 1.004 $\pm$ 0.036 4.85 $\pm$ 0.22 3.15 $\pm$ 0.18 6347 $\pm$ 67 4.107 $\pm$ 0.004 -0.38 $\pm$ 0.08 8.18 $\pm$ 0.28 8760414 1.027 $\pm$ 0.004 0.814 $\pm$ 0.011 11.88 $\pm$ 0.34 1.15 $\pm$ 0.06 5915 $\pm$ 54 4.329 $\pm$ 0.002 -0.66 $\pm$ 0.07 9.83 $\pm$ 0.32 8938364 1.362 $\pm$ 0.007 1.015 $\pm$ 0.023 10.85 $\pm$ 1.22 1.65 $\pm$ 0.15 5604 $\pm$ 115 4.174 $\pm$ 0.004 0.06 $\pm$ 0.06 6.27 $\pm$ 0.31 9025370 0.997 $\pm$ 0.017 0.969 $\pm$ 0.036 5.53 $\pm$ 0.43 0.71 $\pm$ 0.11 5296 $\pm$ 157 4.424 $\pm$ 0.006 0.01 $\pm$ 0.09 15.66 $\pm$ 1.44 9098294 1.150 $\pm$ 0.003 0.979 $\pm$ 0.017 8.23 $\pm$ 0.53 1.34 $\pm$ 0.05 5795 $\pm$ 53 4.312 $\pm$ 0.002 -0.17 $\pm$ 0.07 8.30 $\pm$ 0.23 2.94 $\pm$ 0.20 9139151 1.137 $\pm$ 0.027 1.129 $\pm$ 0.091 1.94 $\pm$ 0.31 1.81 $\pm$ 0.11 6270 $\pm$ 63 4.375 $\pm$ 0.008 0.05 $\pm$ 0.10 9.57 $\pm$ 0.34 5.25 $\pm$ 1.07 9139163 1.569 $\pm$ 0.027 1.480 $\pm$ 0.085 1.23 $\pm$ 0.15 3.51 $\pm$ 0.24 6318 $\pm$ 105 4.213 $\pm$ 0.004 0.11 $\pm$ 0.00 9.85 $\pm$ 0.39 9206432 1.460 $\pm$ 0.015 1.301 $\pm$ 0.048 1.48 $\pm$ 0.31 3.47 $\pm$ 0.18 6508 $\pm$ 75 4.219 $\pm$ 0.009 0.06 $\pm$ 0.07 7.03 $\pm$ 0.26 8.39 $\pm$ 1.01 9353712 2.240 $\pm$ 0.061 1.681 $\pm$ 0.125 1.91 $\pm$ 0.14 7.27 $\pm$ 1.02 6343 $\pm$ 119 3.965 $\pm$ 0.008 0.12 $\pm$ 0.08 2.21 $\pm$ 0.16 10.03 $\pm$ 1.03 9410862 1.149 $\pm$ 0.009 0.969 $\pm$ 0.017 5.78 $\pm$ 0.82 1.56 $\pm$ 0.08 6017 $\pm$ 69 4.304 $\pm$ 0.003 -0.34 $\pm$ 0.08 5.05 $\pm$ 0.16 2.55 $\pm$ 0.27 9414417 1.891 $\pm$ 0.015 1.401 $\pm$ 0.028 2.53 $\pm$ 0.17 4.98 $\pm$ 0.22 6260 $\pm$ 67 4.028 $\pm$ 0.004 -0.07 $\pm$ 0.12 4.65 $\pm$ 0.13 8.96 $\pm$ 0.56 9955598 0.881 $\pm$ 0.008 0.885 $\pm$ 0.023 6.47 $\pm$ 0.45 0.58 $\pm$ 0.03 5400 $\pm$ 57 4.494 $\pm$ 0.003 0.06 $\pm$ 0.04 14.98 $\pm$ 0.53 1.30 $\pm$ 0.22 9965715 1.234 $\pm$ 0.015 1.005 $\pm$ 0.033 3.29 $\pm$ 0.33 1.85 $\pm$ 0.15 6058 $\pm$ 113 4.258 $\pm$ 0.004 -0.27 $\pm$ 0.11 8.81 $\pm$ 0.51 10079226 1.129 $\pm$ 0.016 1.082 $\pm$ 0.048 2.75 $\pm$ 0.42 1.41 $\pm$ 0.10 5915 $\pm$ 89 4.364 $\pm$ 0.005 0.07 $\pm$ 0.06 7.05 $\pm$ 0.29 3.86 $\pm$ 0.33 10454113 1.272 $\pm$ 0.006 1.260 $\pm$ 0.016 2.06 $\pm$ 0.16 2.07 $\pm$ 0.08 6134 $\pm$ 61 4.325 $\pm$ 0.003 0.04 $\pm$ 0.04 11.94 $\pm$ 0.63 4.41 $\pm$ 0.33 10516096 1.398 $\pm$ 0.008 1.065 $\pm$ 0.012 6.59 $\pm$ 0.37 2.11 $\pm$ 0.08 5872 $\pm$ 43 4.173 $\pm$ 0.003 -0.06 $\pm$ 0.06 7.53 $\pm$ 0.21 10644253 1.090 $\pm$ 0.027 1.091 $\pm$ 0.097 0.94 $\pm$ 0.26 1.45 $\pm$ 0.09 6033 $\pm$ 67 4.399 $\pm$ 0.007 0.01 $\pm$ 0.10 10.45 $\pm$ 0.39 5.05 $\pm$ 0.42 10730618 1.763 $\pm$ 0.040 1.411 $\pm$ 0.097 1.81 $\pm$ 0.41 4.04 $\pm$ 0.56 6156 $\pm$ 181 4.095 $\pm$ 0.011 0.05 $\pm$ 0.18 3.35 $\pm$ 0.27 10963065 1.204 $\pm$ 0.007 1.023 $\pm$ 0.024 4.33 $\pm$ 0.30 1.80 $\pm$ 0.08 6097 $\pm$ 53 4.288 $\pm$ 0.003 -0.24 $\pm$ 0.06 11.46 $\pm$ 0.34 4.84 $\pm$ 0.65 11081729 1.423 $\pm$ 0.009 1.257 $\pm$ 0.045 2.22 $\pm$ 0.10 3.29 $\pm$ 0.07 6474 $\pm$ 43 4.215 $\pm$ 0.026 0.07 $\pm$ 0.03 7.48 $\pm$ 0.17 26.28 $\pm$ 2.98 11253226 1.606 $\pm$ 0.015 1.486 $\pm$ 0.030 0.97 $\pm$ 0.21 4.80 $\pm$ 0.20 6696 $\pm$ 79 4.197 $\pm$ 0.007 0.10 $\pm$ 0.05 8.07 $\pm$ 0.23 22.32 $\pm$ 2.28 11772920 0.845 $\pm$ 0.009 0.830 $\pm$ 0.028 10.79 $\pm$ 0.96 0.42 $\pm$ 0.06 5084 $\pm$ 159 4.502 $\pm$ 0.004 -0.06 $\pm$ 0.09 14.82 $\pm$ 1.24 12009504 1.382 $\pm$ 0.022 1.137 $\pm$ 0.063 3.44 $\pm$ 0.44 2.46 $\pm$ 0.25 6140 $\pm$ 133 4.213 $\pm$ 0.006 -0.04 $\pm$ 0.05 7.51 $\pm$ 0.42 7.44 $\pm$ 0.55 12069127 2.283 $\pm$ 0.033 1.621 $\pm$ 0.084 1.79 $\pm$ 0.14 7.26 $\pm$ 0.42 6267 $\pm$ 79 3.926 $\pm$ 0.010 0.15 $\pm$ 0.08 2.35 $\pm$ 0.08 125.54 $\pm$ 7.07 12069424 1.223 $\pm$ 0.005 1.072 $\pm$ 0.013 7.36 $\pm$ 0.31 1.52 $\pm$ 0.05 5785 $\pm$ 39 4.294 $\pm$ 0.001 -0.04 $\pm$ 0.05 47.44 $\pm$ 1.00 2.60 $\pm$ 0.20 12069449 1.113 $\pm$ 0.016 1.038 $\pm$ 0.047 7.05 $\pm$ 0.63 1.21 $\pm$ 0.11 5732 $\pm$ 83 4.361 $\pm$ 0.007 0.15 $\pm$ 0.08 46.77 $\pm$ 2.10 2.43 $\pm$ 0.63 12258514 1.593 $\pm$ 0.016 1.251 $\pm$ 0.016 5.50 $\pm$ 0.40 2.63 $\pm$ 0.12 5808 $\pm$ 61 4.129 $\pm$ 0.002 0.10 $\pm$ 0.09 12.79 $\pm$ 0.40 5.37 $\pm$ 0.66 12317678 1.788 $\pm$ 0.014 1.373 $\pm$ 0.030 2.30 $\pm$ 0.20 5.49 $\pm$ 0.28 6587 $\pm$ 97 4.064 $\pm$ 0.005 -0.26 $\pm$ 0.09 6.89 $\pm$ 0.23 ---------- ------------------- ------------------- ------------------ ------------------- ---------------- ------------------- ------------------ ------------------ ------------------- -- -- -- -- -- -- -- -- -- -- -- -- -- Notes: The mean model parameters are radius, mass, age, luminosity, effective temperature, surface gravity, metallicity, parallax, and rotational velocity. The latter two are derived using data from this table and Table \[tab1\]. Accuracy of radii and luminosities\[sec:comparison\] ---------------------------------------------------- To test the accuracy of the derived radii and luminosities, we have compiled measured values of these properties for nine stars (Table \[tab:lumradobs\]). [These stars have reliable Hipparcos parallaxes and are not members of close binary systems.]{} Only three of the radii of the subsample of stars have been measured interferometrically [@huber12; @white13]. [The angular diameters from @masana06 and @huber14 were derived from broadband photometry and from literature atmospheric properties and stellar evolution models, respectively.]{} @met1216cyg and @Metcalfe2014 derived the luminosities using [extinction estimates from @ammons2006 and the bolometric corrections from @flower1996 ([-@flower1996], see @torres2010).]{} ---------- ------------------- ----------------------- ------------------ -- -- -- KIC ID $L$ $R$ $\pi$ ([L$_{\odot}$]{}) ([R$_{\odot}$]{}) (mas) 8006161 0.61 $\pm$ 0.02 0.950$^1$ $\pm$ 0.020 37.47 $\pm$ 0.49 9139151 1.63 $\pm$ 0.40 1.160$^3$ $\pm$ 0.020 9.46 $\pm$ 1.15 9139163 3.88 $\pm$ 0.69 1.570$^3$ $\pm$ 0.030 9.49 $\pm$ 0.83 9206432 4.95 $\pm$ 1.48 1.520$^3$ $\pm$ 0.030 5.85 $\pm$ 0.87 10454113 2.60 $\pm$ 0.36 1.240$^3$ $\pm$ 0.020 9.95 $\pm$ 0.67 11253226 4.22 $\pm$ 0.61 1.576$^4$ $\pm$ 0.143 8.52 $\pm$ 0.60 12069424 1.56 $\pm$ 0.05 1.220$^1$ $\pm$ 0.020 47.44 $\pm$ 0.27 12069449 1.27 $\pm$ 0.04 1.120$^1$ $\pm$ 0.020 47.14 $\pm$ 0.27 12258514 2.84 $\pm$ 0.25 1.590$^3$ $\pm$ 0.040 12.32 $\pm$ 0.51 ---------- ------------------- ----------------------- ------------------ -- -- -- : Luminosities, radii, and parallaxes from independent sources[]{data-label="tab:lumradobs"} The luminosities are from @met1216cyg [@Metcalfe2014]. The references to the radii are $^1$@huber12 $^2$@white13 $^3$@huber14 $^4$@masana06. The parallaxes are from @hipparcos07. A comparison of these independent measures of stellar radii and luminosities with those derived using our asteroseismic methodology is shown in the top two panels of Fig. \[fig:lumrad\]. This comparison, using measurement differences relative to their uncertainty as listed in the literature, shows no systematic biases or trends for this subsample of nine stars. The mean relative difference is –0.40 with a root mean square (rms) around the mean of 0.59 for the interferometrically measured radii (references 1 and 2, red filled circles) and –0.28 $\pm$ 1.03 for the radii derived using photometry and isochrones (references 3 and 4). For the luminosity the mean relative difference is –0.35 with an rms around the mean of 1.1. Asteroseismic parallaxes \[sec:asteroseismicdistances\] ------------------------------------------------------- We used the luminosity $L$ that was derived from the asteroseismic analysis to compute the stellar distance as a parallax. Using the modeled surface gravity and the observed [$T_{\rm eff}$]{} and [\[M/H\]]{}, we derived the amount of interstellar absorption between the top of the Earth’s atmosphere and the star, $A_{Ks}$, using the isochrone method described in @schultheis2014. Here, the subscript $Ks$ refers to the 2MASS $K_s$ filter [@2MASS]. With the same observed [$T_{\rm eff}$]{} , we computed the corresponding bolometric correction $BC_{Ks}$ for this band, using $BC_{Ks} = 4.51465 -0.000524461 T_{\rm eff}$ [@marigo2008] where the solar bolometric magnitude is 4.72 mag. The $K_s$-band magnitude and $A_{Ks}$ are listed in Table \[tab1\]. The distance, $d$, or parallax, $\pi$, is then computed directly from $L$, $K_s$, $BC_{Ks}$ , and $A_{Ks}$. [The parallaxes and uncertainties of the stars in our sample are listed in Table \[tab:properties\_derived\]. They were derived using Monte Carlo simulations, described as follows. We perturbed each of the input data measures $L$, $A_{Ks}$, $K_s$, and $BC_K$, using noise sampled from a Gaussian distribution with zero mean and standard deviation equivalent to their errors to calculate a parallax. By repeating the perturbations 10,000 times, we obtained a distribution of parallaxes, which is modeled by a Gaussian function. The mean and standard deviation are adopted as the parallax value and its uncertainty. In most cases, the derived parallax error is dominated by the luminosity error.]{} A comparison between the derived parallaxes and existing literature values (@hipparcos07, Table \[tab:lumradobs\]) again validates our results, as shown in [the lower panel of]{} Fig. \[fig:lumrad\], where no significant trend can be seen. In particular, we note that for the binary KIC 12069424 and KIC 12069449 (16 Cyg A&B), we obtain almost identical parallaxes of [47.4 mas and 46.8 mas,]{} equivalent to a difference of 0.3 pc at a distance of 21.2 pc. [This result provides further evidence of the accuracy of our derived properties.]{} Trends in stellar properties \[sec:stellartrends\] -------------------------------------------------- Performing a homogenous analysis on a [relatively large sample allows us to check for trends in some stellar parameters and compare them to trends derived or established by other methods. We performed this check for two parameters: the mixing-length parameter and the stellar age.]{} ### Mixing-length parameter versus [$T_{\rm eff}$]{} and [$\log g$]{} The mixing-length parameter $\alpha$ is usually calibrated for a solar model and then applied to all models for a set range of masses and metallicities. However, several authors have shown that this approach is not correct, for instance, @yildiz2006 [@bonaca2012; @creevey2012A]. The values of $\alpha$ resulting from a GA [analysis]{} offer an optimal approach to effectively test and subsequently constrain this parameter, since [by design]{} the GA only restricts $\alpha$ to be between 1.0 and 3.0, [a range large enough to encompass all plausible values]{}. The color-coded distribution of $\alpha$ with [$\log g$]{} and [$T_{\rm eff}$]{} in the top panel of Fig. \[fig:poster\_teffalpha\], using the results derived from our sample of 57 stars and the Sun. It is evident from this figure that for a given value of [$\log g$]{}, the value of $\alpha$ has an upper limit. This upper limit can be represented by the equation $\alpha < 1.65 \log g - 4.75$, and this is denoted by the dashed line in the figure. A regression analysis considering the model values of [$\log g$]{}, $\log$[$T_{\rm eff}$]{} and \[M/H\] yields $$\begin{aligned} \indent \alpha &=& 5.972778 + 0.636997 \log g \nonumber \\ && - 1.799968 \log T_{\rm eff} + 0.040094 [{\rm M}/{\rm H}], \nonumber \label{eqn:alpha_regression} \\\end{aligned}$$ with a mean and rms of the residual to the fit of –0.01 $\pm$ 0.15 for the 58 stars. The residuals of this fit scaled by the uncertainties in $\alpha$ are shown in the lower panel of Fig. \[fig:poster\_teffalpha\] as a function of [$T_{\rm eff}$]{}. No trend with this parameter can be seen. This equation yields a value of $\alpha = 2.03$ for the known solar properties, within 1$\sigma$ of its mean value (2.12). ![[*Top:*]{} Distribution of the [$\log g$]{} and $\alpha$ for the full sample. The color coding shows in red [$T_{\rm eff}$]{} $<$ 5600 K, in yellow 5600 K $<$ [$T_{\rm eff}$]{} $<$ 6000 K, in green 6000 K $<$ [$T_{\rm eff}$]{} $<$ 6300 K, and blue [$T_{\rm eff}$]{} $>$ 6300 K. [*Bottom:*]{} Residuals of the regression analysis scaled by the uncertainties in $\alpha$ as a function of [$T_{\rm eff}$]{}. \[fig:poster\_teffalpha\]](poster_teffalpha "fig:"){width="48.00000%"} ![[*Top:*]{} Distribution of the [$\log g$]{} and $\alpha$ for the full sample. The color coding shows in red [$T_{\rm eff}$]{} $<$ 5600 K, in yellow 5600 K $<$ [$T_{\rm eff}$]{} $<$ 6000 K, in green 6000 K $<$ [$T_{\rm eff}$]{} $<$ 6300 K, and blue [$T_{\rm eff}$]{} $>$ 6300 K. [*Bottom:*]{} Residuals of the regression analysis scaled by the uncertainties in $\alpha$ as a function of [$T_{\rm eff}$]{}. \[fig:poster\_teffalpha\]](residual-alpha "fig:"){width="48.00000%"} [These results agree in part with those derived by @magic2015, who used a full 3D radiative hydrodynamic simulation for modeling convective envelopes.]{} These authors found that $\alpha$ increases with [$\log g$]{} and decreases with [$T_{\rm eff}$]{}, [which is qualitatively]{} in agreement with our results. [The size of the variation that they inferred]{}, however, is smaller than the values we find. In our sample, $\alpha$ varies between 1.7 and 2.4, while for the same range in [$\log g$]{}, [$T_{\rm eff}$]{}, @magic2015 see variations in $\alpha$ from 1.9 to 2.3. We note that the range of metallicity [in our sample is much smaller than the range in their work. This could be the reason of the weak and opposite dependence on $\alpha$ that we find]{}. ### Age and $\langle r_{02} \rangle$ ![Age determination as a function of the mean value of $r02$. \[fig:res\_Age\_r02\]](res_Age_r02){width="48.00000%"} The $r_{02}$ frequency ratios [contain what is known as [**]{} and]{} these are effective at probing the gradients near the core of the star [@Roxburgh2003]. As the core is most sensitive to nuclear processing, $r_{02}$ are a diagnostic of the evolutionary state of the star. Using theoretical models, @lebMon2009 showed a relationship between the mean value of $r_{02}$ and the [stellar]{} age. This relationship was recently used by @appourchaux2015 to estimate the age of the binary KIC 7510397 (HIP 93511). Figure \[fig:res\_Age\_r02\] shows the distribution [of the mean of the $r_{02}$ ratios, that is, $\langle r_{02} \rangle$,]{} versus the derived ages for the sample of stars studied here. A linear fit to these data leads to the following estimate of the stellar age, $\tau$ in Gyr, based on $ \langle r_{02} \rangle$ $$\tau = 17.910 - 193.918 \langle r_{02} \rangle, \label{eqn:r02}$$ [This is, of course, only valid for the range covered by our sample. The range of radial orders used for calculating $\langle r_{02} \rangle$ has [almost no impact on this result (an effect lower than a 1%)]{}. We note that when inserting the value of $\langle r_{02} \rangle = 0.068$ for the Sun, Eq. \[eqn:r02\] yields an age of 4.7 Gyr, in excellent agreement with the Sun’s age as determined by other means.]{} Characterizing surface effects\[sec:sec5\] ========================================== It is known that a direct comparison of observed frequencies with model frequencies derived from 1D stellar structure models reveals a systematic discrepancy [that increases with the mode frequency]{}; this is commonly referred to as (@rosenthal1997, see Section \[sec1\]). This discrepancy arises because a 1D stellar atmosphere does not represent the [actual structural and thermal properties of the stellar atmosphere in the layers close to the surface and because non-adiabatic effects that are present immediately below the surface are not included when computing resonant frequencies using an adiabatic code.]{} Some recent works [have attempted]{} to produce more realistic stellar atmospheres by replacing the outer layers of a 1D stellar envelope by an averaged 3D surface simulation and by including the effects of turbulent pressure in the equation of hydrostatic support and opacity changes from the temperature fluctuations, and by also considering non-adiabatic effects [@Trampedach2014; @Trampedach2016; @Houdek2016]. [This reduced the approximately $-$15 $\mu$Hz discrepancy to around +2 $\mu$Hz near 4,000 $\mu$Hz when including both structural and modal effects. ]{} While progress is being made, we are still not in a position to apply these calculations for a large sample of stars. To sidestep this problem, several authors [have suggested]{} the use of combination frequencies that are insensitive to this systematic offset in frequency, see for example, @Roxburgh2003, hence the exclusive use of $r_{01}$ and $r_{02}$ in the AMP 1.3 method. However, [since individual frequencies contain more information than ratios of frequency separations,]{} some authors have derived simple prescriptions to mitigate [the surface effects]{}. One such parametrization is that of @Kjeldsen2008, who suggested a simple correction to the 1D model frequencies $\delta\nu_{n,l}$ of the form [of a power law,]{} $$\delta\nu_{n,l} = a_0 \left ( \frac{\nu^{\rm obs}_{n,l}}{\nu_{\rm max}} \right )^{b} \label{eqn:kjeldsen} ,$$ where $b = 4.82$ is a fixed value, [calibrated by a solar model]{}, [ $\nu_{\rm max}$ is the frequency corresponding to the highest amplitude mode, see [@Lund2016]]{}, $a_0$ is [computed from the differences between the]{} observed and model frequencies [@Metcalfe2009; @Metcalfe2014], $$a_0 = \frac{\langle \nu_{n,0}^{\rm obs} \rangle - \langle \nu_{n,0}^{\rm mod} \rangle}{N^{-1}_{0} \sum_{i=1}^{N_{0}} [\nu^{\rm obs}_i / \nu_{\rm max} ]^b} .$$ [$\text{Here, }\nu^{\rm obs}_{n,l}$ and $\nu^{\rm mod}_{n,l}$ are the observed and model frequency of radial order $n$ and degree $l$, respectively, and $N_{0}$ is the number of $l=0$ frequencies.]{} In the absence of perfect 3D simulations, the interest in using such a surface correction becomes evident when we consider [not only]{} that the individual frequencies contain a higher information content, but more importantly, [that]{} the $r_{01}$ and $r_{02}$ frequency ratios are only useful if the precision on these derived quantities is high enough. [A precision like this on the ratio requires not only having a high precision on the individual frequencies, but enough radial order modes to constrain the stellar modeling.]{} This is not necessarily the case for some stars, where, for example, ground-based campaigns are limited in time-domain coverage, such as the case of $\nu$ Ind [@nuind], or even for space-based missions such as the TESS mission, where [only one month of continuous data will be available for stars at certain galactic latitudes]{}. [Similary, limited precision will also be achieved for the stars observed in the PLATO [**]{} phase, since the observation window will only be two to three months each.]{} The AMP 1.3 method exclusively uses the $r_{01}$ and $r_{02}$ frequency ratios, and our results [are therefore expected to be]{} insensitive to [surface effects]{}. [Hence]{}, using the resulting models and the observed frequencies, we can explore the nature of the surface term for a large sample of stars, and in particular, we can test to which extent the @Kjeldsen2008 prescription is useful. Surface [effects]{} as observed in the Sun at low degrees --------------------------------------------------------- [The magnitude of the surface [effects]{} on the frequency discrepancy for the Sun is on the order of 10-15 $\mu$Hz around 4000 $\mu$Hz for the low degrees ($l=0,1,2,\text{and }3$)]{}. Our analysis using the solar data reveals a similar offset. In the top panel of Fig. \[fig:best100\_surfaceterm\] we show the solar surface term by comparing the input frequencies with those of the models. The term of the reference model is shown by the thick line with filled black dots, and in gray we show those for 100 of the best solar models, with the mean of these 100 shown as the thick dashed line. At [$\nu_{\rm max}$]{}, the value of $a_0 = -2.5 \mu$Hz for the reference model, and for 100 of the representative models it spans –2.3 to –4.6 $\mu$Hz. ![[*Top:*]{} S[urface term]{} for the reference solar model (connected black dots) for the $l=0$ frequencies as a function of observed frequency scaled by [$\nu_{\rm max}$]{}. The surface term for a sample of 100 of the best models is also shown for the Sun (gray), with the mean value highlighted by the dashed line. The blue connected squares show the empirical surface correction $\delta\nu_{n,l}$ (Eq. \[eqn:kjeldsen\]) based on the reference model. [*Lower:*]{} The differences between the observed and corrected model frequencies as a function of scaled frequency, with the solar observational errors overplotted in blue. The shaded gray areas represent the mean and standard deviation of $q$ for the same 100 models shown in the top panel. The dotted vertical lines delimit the region used to calculate the quality metric $Q$. []{data-label="fig:best100_surfaceterm"}](best100_surface9999999){width="48.00000%"} When we apply Eq. \[eqn:kjeldsen\] to the reference solar model, we calculate a correction $\delta\nu_{n,l}$ that successfully mitigates the surface [effects]{}. This is clearly shown in the top panel of Fig. \[fig:best100\_surfaceterm\] , where the surface [term]{} for the reference model (black connected dots) is traced by the scaled surface correction $\delta\nu$ (blue connected squares) for the $l=0$ modes alone. By applying the proposed corrections $\delta\nu_{n,l}$ to the observed frequencies, we can then make a quantitive comparison between the model and the data. This agreement is shown in the lower panel for $l=0,$ and we denote it as $q_{n,l} = \nu^{\rm obs}_{n,l} - \nu^{\rm mod}_{n,l} + \delta\nu_{n,l}$. To quantify the agreement between the corrected model frequencies and the observed ones, we define the metric $Q$ as the median of the [absolute value]{} of the residuals, $$ \\ Q = \mathrm{median}\left | {q_{n,l}} \right |, \label{eqn:chit}$$ for all observed $n$ and $l$ defined in the region of $0.7 \le \nu_{n,l}^{\rm obs} / \nu_{\rm max} \le 1.3$. This region is delimited in the lower panel by the vertical dotted lines. We note that we purposely exclude any reference to an observational error in the definition of $q$, as the surface correction results from an error in the models and is not related to the precision of the frequency data. In the ideal case and in the absence of errors in the data, $Q \rightarrow 0~\mu$Hz, which means that the model is perfect. The value of $Q$ is 0.38 $\mu$Hz for the reference solar model, and the mean value for the 100 solar models shown in Fig. \[fig:best100\_surfaceterm\] is 0.51 $\mu$Hz. From this figure and the low value of the quality metric, it is expected that the @Kjeldsen2008 empirical surface correction $\delta\nu_{n,l}$ (Eq. \[eqn:kjeldsen\]) is useful for mitigating [the surface effects]{} for this solar model. [Surface effects]{} for other stars ----------------------------------- Is the simplified surface correction useful in other stars? And if so, to what extent? These are the questions that we aim to answer by inspecting the reference models (Table \[tab:referencemodels\]) of the best-fit stars within our sample. We define a subset of stars by selecting those with $\chi^2_N \leq 3.0$[^2] for both $r_{01}$ and $r_{02}$. This [selection results in a subset of]{} 44 stars. The differences between the observed frequencies and the frequencies of the reference models for this subset are shown in Fig. \[fig:star\_surfacecorr\], [and we assume]{} that these differences are dominated by [the surface effects.]{} For the stars represented by the continuous lines it can be noted that the [remaining discrepancies are]{} quite similar in [magnitude]{} and shape for the [less evolved stars]{}. [For]{} the more evolved stars [([$\log g$]{} $>$ 4.2, indicated by dashed lines), the remaining discrepancies are larger and [of a different nature]{}, and cannot readily be modeled by a simple power law.]{} ![Surface terms for the stars in our subsample defined by the criteria of $\chi_{N}^2 (r_{01},r_{02}) \le 3$. For clarity, the more evolved stars are shown by the dashed lines.[]{data-label="fig:star_surfacecorr"}](Star_surfacecorr){width="48.00000%"} For each of the stars, [a value of $a_0$ is derived directly from the comparison of model and observed frequencies (see Table \[tab:properties\_derived\]),]{} and Eq. \[eqn:kjeldsen\] is used to calculate the surface correction $\delta\nu_{n,l}$ to apply to the model frequencies. We then calculate the metric $Q$ for each star in the subsample, [and these values]{} are shown as a function of $a_0$ in Fig. \[fig:dif\_cut\]. We see very clearly that as the difference between the observed and model frequency at [$\nu_{\rm max}$]{} increases (i.e., $a_0$ becomes more negative), $Q$ also increases, indicating that the @Kjeldsen2008 correction becomes less [adequate]{} to mitigate the surface [effects]{}. It seems then quite likely that there is a value of $Q$ (and $a_0$) that defines a limit where the surface correction is useful. By inspecting the residuals between observed and corrected model frequencies for this subset of stars, we found that when $Q \lesssim 1.0~\mu$Hz, we obtained a very good match to the observed frequencies when the surface correction was included. These stars also have values of $a_0$ that are typically lower than [$-$6.0 $\mu$Hz]{}, as shown in Fig. \[fig:dif\_cut\], just like the solar case. For [an illustration]{}, we present some échelle diagrams in Fig. \[fig:someechelles\] with different values of $Q$ to [show]{} the validity of this criterion. [A visual inspection of the residuals and the échelle diagrams for this subsample of stars led to the same conclusion.]{} ![Metric $Q$ versus $a_0$ for the stars in our subsample. [The dashed lines highlight the approximate limitation in $Q$ and $a_0$ where the surface correction enables a useful comparison between the observed and corrected model frequencies]{}.[]{data-label="fig:dif_cut"}](dif_cut){width="48.00000%"} When we rely on the criteria of $Q \lesssim 1$ $\mu$Hz, we can trace the ranges of the stellar parameters where the surface correction mitigates the surface [effects]{}. This is [presented]{} in Fig. \[fig:whichparameterswork0\], [which shows]{} the distribution of observed and inferred stellar properties of stars from this subsample (open circles) along with the stars that satisfy the criterion of $Q \lesssim 1.0~\mu$Hz (filled dark blue circles) and $Q \lesssim 1.2~\mu$Hz (filled light blue circles). We also delimit the regions (dashed lines) where we infer that the correction is no longer useful. More concretely, we find that the limit of the solar-like regime in terms of observed properties is approximately at $\log g = 4.2$, [$T_{\rm eff}$]{} $= 6250$ K, [$\langle \Delta \nu \rangle$]{} $= 70 \mu$Hz and [$\nu_{\rm max}$]{} $= 1600 \mu$Hz. In terms of physical properties of the star, the limit is around $R = 1.6$ [R$_{\odot}$]{}, $M = 1.35$ [M$_{\odot}$]{}, and $L = 3.0$ [L$_{\odot}$]{}, with no evidence that the absolute age (not evolution state) or the metallicity playing any role. In Table \[tab:appliedsurface\] we summarize these [limiting regions, but adopt a slightly]{} more conservative limit. The limit in [$T_{\rm eff}$]{} can probably be attributed to the Kraft break (e.g., @Kraft1967), where at around $6250$ K, these hotter stars rotate much faster as a result of a lack of a deep convective envelope, in which magnetic braking could slow the star down. The depth of the convective region is shown as a function of [$T_{\rm eff}$]{} in Fig. \[fig:teffdcz\], and stars with regions larger than approximately 0.2 stellar radii satisfy this criterion. This limit is also compatible with the proposed mass limit of approximately 1.3 [M$_{\odot}$]{} where a transition in envelope convection takes place. The negative slope of the surface correction at [$\nu_{\rm max}$]{} is also found to increase with increasing mass (becoming flatter), again indicating a change in convective zone properties and [$T_{\rm eff}$]{}. The limit in [$\log g$]{} points toward a transition from the main-sequence to the subgiant phase where the convective envelope begins to deepen. These limits are imposed by the physical structure of the star itself, but no quantitative measure of $a_0$ can be deduced from the observed and/or inferred stellar properties at this stage, except for a slight linear dependence of $a_0$ with [$\langle \Delta \nu \rangle$]{}, [$\nu_{\rm max}$]{}, or [$\log g$]{} with a rather large scatter. {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} Property ------------------------------------------------ ------- ------ -- -- -- -- -- [$\log g$]{} (cgs) $\ge$ 4.2 [$T_{\rm eff}$]{} (K) $\le$ 6200 [$\langle \Delta \nu \rangle$]{} ([$\mu$Hz]{}) $\ge$ 80 [$\nu_{\rm max}$]{} ([$\mu$Hz]{}) $\ge$ 1700 $a_0$ ([$\mu$Hz]{}) $\le$ -6 $R$ ([R$_{\odot}$]{}) $\le$ 1.5 $M$ ([M$_{\odot}$]{}) $\le$ 1.3 $L$ ([L$_{\odot}$]{}) $\le$ 2.5 : Stellar property regimes where the @Kjeldsen2008 surface correction is useful. \[tab:appliedsurface\] Summary\[sec6\] =============== The high-quality and long-term photometric time series provided by [*Kepler*]{} has enabled an unprecedented precision on asteroseismic data of stars like the Sun. Thanks to the very high precision, we could [use]{} the frequency separation ratios along with spectroscopic temperatures and metallicities to infer stellar properties of the Sun and 57 [*Kepler*]{} stars, comprising solar analogs, active stars, components of binaries, and planetary hosts, with a precision of the same quality when using the individual frequencies. Median uncertainties on radius and mass are 1% and 3%, while uncertainties on the age compared to the estimated main-sequence lifetime are typically 7% or 11% compared to the absolute age. These realistic uncertainties [account for unbiased determinations of mixing-length parameter and initial chemical composition]{}. Along with the physical stellar properties, we also derived the interstellar absorption and distances to each star, and where the rotation period was available, we derived the rotational velocity. For nine stars [our derivation of radii, luminosities, and distances are in very good agreement with independently measured values.]{} [Our inferred ages are]{} validated for the Sun and by comparing the ages of the individual components of the binary system 16 Cyg A and B. From an analysis of our derived properties for the full sample we investigated the [dependence]{} of the mixing-length parameter with stellar properties and found it to correlate with [$\log g$]{} and [$T_{\rm eff}$]{} , just as proposed by @magic2015 from 3D RHD simulations of convective envelopes. We also derived a linear expression relating the mean value of the $r_{02}$ frequency separation ratios directly to the age of the star, which yields an age of 4.7 Gyr for the Sun. By selecting a subsample of the stars using a $\chi^2_N$ [threshold]{}, we investigated the usefulness of the @Kjeldsen2008 empirical [correction for the surface effects]{} across a broad range of stellar parameters, and we found that it is useful, [but only]{} in certain regimes, [as also suggested by the theoretical study of @schmittbasu2015]{}. This is of particular interest for stars with much shorter time series, where the precision on the individual frequencies or the number of radial orders is not high enough to constrain the stellar modeling. In particular, this will be the case for the forthcoming NASA TESS mission, where some stars with ecliptic latitude $|b| \lesssim 60^{\circ}$ will be observed continuously for only 27 days, along with the [**]{} phase of the future PLATO mission (launch 2024). Perspectives\[sec7\] ==================== In this work we used [$T_{\rm eff}$]{} and [\[M/H\]]{} as the only complementary data to the asteroseismic data. However, within a year from now, we will have a homogenous set of microarcsecond precision parallaxes that will give access to the intrinsic luminosity of the star. [This quantity is sensitive to the interior stellar composition.]{} While today we have very high precision radii along with other properties, degeneracies in model parameters, such as the mass and initial helium abundance (e.g., @Metcalfe2009 [@lebreton2014]) limit the full exploitation of asteroseismic data for testing stellar interior models and improving precision on model parameters. The forthcoming Gaia data in Release 2 promise to overcome this obstacle and thus provide even higher precision radii and ages, along with constraints on interior and initial chemical composition, and thus pushing stellar models to their limit. We highlight the importance of the precise characterisation of exoplanetary systems using asteroseismic data. In this work, we determined the radius and age of three planetary hosts (KIC 9414417, KIC 9955598, and KIC 10963065). Combining our data with those of @batalha2013 constrains the planetary and orbital parameters. We illustrate this in Fig. \[fig:planetages\], where we depict the separation of the planet and host as a function of stellar age (including the Earth). The sizes of the symbols [are indicative of]{} the planetary radius, and the equilibrium temperature decreases with distance from the host. The diversity of planetary systems can be easily noted, and such an analysis of a larger sample of planetary candidates will yield important constraints on the formation and evolution of planetary systems. The future TESS and PLATO missions targeting bright stars with asteroseismic characterization promise to be a goldmine for not only exoplanetary physics, but with access to microarcsecond parallaxes and homogenous multiband photometry, also for stellar and Galactic physics. ![Fractional depth of the convection zone as a function of [$T_{\rm eff}$]{} for our selected subsample of stars. The color-coding is the same as Fig. \[fig:whichparameterswork0\].[]{data-label="fig:teffdcz"}](interesting_teffdcz){width="48.00000%"} ![Age of planet and separation from host. Symbol sizes represent planetary radius, and equilibrium temperature decreases with distance from the host. Age and radius are taken from this work, while other parameters are taken from [@batalha2013]. The Earth is shown at 1 AU. \[fig:planetages\]](planetages){width="48.00000%"} This work is based on data collected by the Kepler mission. Funding for the Kepler mission is provided by the NASA Science Mission directorate. This collaboration was partially supported by funding from the Laboratoire Lagrange 2015 BQR. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23. This work was supported in part by NASA grants NNX13AE91G and NNX16AB97G. Computational time at the Texas Advanced Computing Center was provided through XSEDE allocation TG-AST090107. DS and RAG acknowledge the financial support from the CNES GOLF grants. DS acknowledges the Observatoire de la Côte d’Azur for support during his stays. Some of these computations have been done on the ’Mesocentre SIGAMM’ machine, hosted by the Observatoire de la Cote d’Azur. The authors wish to thank Sylvain Korzennik for his very careful reading of the paper and valuable suggestions for improving the presentation and the scientific arguments. Supplementary material\[sec:app0\] ================================== {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} {width="48.00000%"} -------------- ------------------- ------------------- ------- ------------------- ---------- --------- -- -- -- KIC ID $R$ $M$ Age $L$ $\log g$ \[M/H\] ([R$_{\odot}$]{}) ([M$_{\odot}$]{}) (Gyr) ([L$_{\odot}$]{}) (dex) (dex) no diffusion 9139151 1.132 1.11 1.96 1.80 4.375 -0.01 12009504 1.366 1.10 3.38 2.39 4.210 -0.01 6225718 1.227 1.15 2.29 2.09 4.320 -0.10 diffusion 1225814 1.595 1.26 5.04 2.81 4.129 0.05 5184732 1.356 1.25 4.68 1.82 4.269 0.25 8150065 1.397 1.21 3.12 2.54 4.228 -0.05 8179536 1.348 1.25 1.93 2.64 4.274 -0.05 7771282 1.631 1.26 3.34 3.65 4.116 -0.03 10454113 1.250 1.20 1.98 2.04 4.320 -0.04 -------------- ------------------- ------------------- ------- ------------------- ---------- --------- -- -- -- [^1]: [^2]: The limit of 3.0 is rather arbitrary and was chosen as a compromise between having an adequate sample size and the best match to the data. Using a threshold of 2.0 or 4.0 does not change the results significantly.

--- author: - Maximilien Pindao - Daniel Schaerer - 'Rosa M. González Delgado' - Grażyna Stasińska date: 'Received 20 june 2002/ Accepted 12 august 2002' title: 'VLT observations of metal-rich extra galactic HII regions. I. Massive star populations and the upper end of the IMF [^1] ' --- Introduction {#s_intro} ============ Wolf-Rayet stars (WR) are the descendants of the most massive stars. Although they live during a short time (Maeder & Conti 1994) these stars have been detected in young stellar systems, such as extragalactic HII regions (Kunth & Schild 1986) and the so-called WR galaxies (Conti 1991, Schaerer  1999b). They are recognized by the presence of broad stellar emission lines at optical wavelengths, mainly at 4680 Å (known as the blue WR bump) and at 5808 Å  (red WR bump). The blue bump is a blend of N [v]{} $\lambda\lambda$4604,4620, N [iii]{} $\lambda\lambda$4634,4641, C [iii/iv]{} $\lambda\lambda$4650,4658 and [He [ii]{} $\lambda$4686]{} lines, that are produced in WR stars of the nitrogen (WN) and carbon (WC) sequences. In contrast, the red bump is formed only by [C [iv]{} $\lambda$5808]{} and it is mainly produced by WC stars. The detection of these features in the integrated spectrum of a stellar system provides a powerful tool to date the onset of the burst, and it constitutes the best direct measure of the upper end of the initial mass function (IMF). Thus, if WR features are found in the spectra of star forming systems, stars more massive than $M_{\rm WR}$, where $M_{\rm WR} \sim$ 25  for solar metallicity, must be formed in the burst. The IMF is one of the fundamental ingredients for studies of stellar populations, which has an important bearing on many astrophysical studies ranging from cosmology to the understanding of the local Universe. In particular the value of the IMF slope and the upper mass cut-off () strongly influences the mechanical, radiative, and chemical feedback from massive stars to the ISM such as the UV light, the ionizing radiation field, and the production of heavy elements. A picture of a universal IMF has emerged from numerous works performed in the last few years (e.g. Gilmore & Howell 1998 and references therein). Indeed, these studies derive a slope of the IMF close to the Salpeter value for a mass range between 5 and 60 . This result seems to hold for a variety of objects and metallicities from very metal poor up to the solar metallicity, with the possible exception of a steeper field IMF (Massey  1995, Tremonti  2002). However, the IMF in high metallicity (12+log (O/H) $\ga$ (O/H)$_\odot \approx$ 8.92) systems is much less well constrained. Different indirect methods to derive the slope and  give contradictory results. The detection of strong wind resonance UV lines in the integrated spectrum of high metallicity nuclear starbursts clearly indicate the formation of massive stars (Leitherer 1998; Schaerer 2000; González Delgado 2001). In contrast, the analysis of the nebular optical and infrared lines of IR-luminous galaxies and high metallicity [H [ii]{}]{} regions indicates a softness of the ionizing radiation field that has beeninterpreted as due to the lack of stars more massive than $\sim$ 30  (Goldader  1997; Bresolin  1999; Thornley  2000; Coziol  2001). However, the interpretation of these indirect probes relies strongly on a combination of models for stellar atmospheres and interiors, evolutionary synthesis, and photoionisation, each with several potential shortcomings/difficulties (cf. García-Vargas 1996, Schaerer 2000, Stasińska 2002). For example, recently González Delgado  (2002) have shown that the above conclusion could be an artifact of the failure of WR stellar atmospheres models to correctly predict the ionizing radiation field of high metallicity starbursts (see also Castellanos 2001, Castellanos  2002b). A more direct investigation of the stellar content of metal-rich nuclear starbursts has been performed by Schaerer  (2000, hereafter SGIT00), using the detection of WR features to constrain . They found that the observational data are compatible with a Salpeter IMF extending to masses  $\ga$ 40 . Most recently, a similar conclusion has been obtained by Bresolin & Kennicutt (2002, hereafter BK02) from observations of high-metallicity HII regions in M83, NGC 3351 and NGC 6384. Here, we present a direct attempt to determine  based on the detection of WR features in metal-rich [H [ii]{}]{} regions of a sample of spiral galaxies. To obtain statistically significant conclusions about  and the slope of the IMF, a large sample of [H [ii]{}]{} regions needs to be observed. For coeval star formation with a Salpeter IMF and =120  at metallicities above solar, $\sim$ 60 to 80 % (depending on the evolutionary scenario and age of the region) of the [H [ii]{}]{} regions are expected to exhibit WR signatures (Meynet 1995; Schaerer & Vacca 1998, hereafter SV98). Thus, to find $\ga$ 40 regions with WR stars (our initial aim) a sample of at least 5-7 galaxies with $\ga$ 10 [H [ii]{}]{} regions per galaxy needs to be observed. Spectra of high S/N (at least 30) in the continuum are also required to obtain an accurate measure of the WR features. For this propose, we have selected the nearby spiral galaxies NGC 3351, NGC 3521, NGC 4254, NGC 4303 and NGC4321, which have have sufficient number of disk [H [ii]{}]{} regions of high-metallicity, as known from earlier studies. Our observations have indeed allowed to find a large number of metal-rich WR [H [ii]{}]{} regions. The analysis of their massive star content is the main aim of the present paper. Quite independently of the detailed modeling undertaken below, our sample combined with additional WR regions from Bresolin & Kennicutt (2002) allow us to derive a fairly robust [*lower limit*]{} on the upper mass cut-off of the IMF in these metal-rich environments (see Sect. \[s\_imf\]). The structure of the paper is as follows: The sample selection, observations and data reduction are described in Sect. \[s\_obs\]. The properties of the [H [ii]{}]{} regions are derived in Sect. \[s\_props\]. Section \[s\_wroh\] discusses the trends of the WR populations with metallicity. Detailed comparisons of the observed WR features with the evolutionary synthesis models are presented in Sect. \[s\_models\]. More model independent constraints on  are derived in Sect. \[s\_imf\]. Our main results and conclusions are summarised in Sect. \[s\_conclude\]. Sample selection, observations and reduction {#s_obs} ============================================ ---------- ----------------------- ------------------ ------------------ ----------------- ---------- -- -- -- -- Galaxy NED type and activity $\alpha$ (J2000) $\delta$ (J2000) $v_r$ distance \[km s$^{-1}$\] \[Mpc\] NGC 3351 SB(r)b, HII Sbrst 10h43m57.8s +11d42m14s 778 10.0 NGC 3521 SAB(rs)bc, LINER 11h05m48.6s -00d02m09s 805 7.2 NGC 4254 SA(s)c 12h18m49.5s +14d24m59s 2407 16. NGC 4303 SAB(rs)bc, HII Sy2 12h21m54.9s +04d28m25s 1566 16. NGC 4321 SAB(s)bc, LINER HII 12h22m54.9s +15d49m21s 1571 15.21 ---------- ----------------------- ------------------ ------------------ ----------------- ---------- -- -- -- -- Selection of the HII regions ---------------------------- Our target galaxies (see Table \[tab\_sample\]) are selected among nearby spiral galaxies where a sufficient number of disk [H [ii]{}]{} regions of high metallicity are known from the previous studies of Shields et al.(1991), Oey & Kennicutt (1993), and Zaritsky et al. (1994). Inspection of spectra from the two latter studies kindly made available to us showed that the vast majority of their spectra are not deep enough to allow the detection of WR or other stellar signatures in the continuum. Metallicities [12 + ([O/H]{}) $12 + \log({\rm O/H})$]{} of all known regions were estimated from the published  and  intensities using the standard $R_{23}$ “strong line” method and various empirical calibrations. For the FORS1 multi-object spectroscopic observations described below [H [ii]{}]{} regions with metallicities above solar ($\log R_{23} \la$ 0.6) were given first priority. Secondary criteria taken into account in the choice of the known [H [ii]{}]{} regions were a large [H$\beta$]{} equivalent width, and bright continuum flux at $\sim$ 4650 Å as determined from inspection of the spectra. This procedure lead to a first selection of 4 to 7 [H [ii]{}]{} regions per galaxy. Other regions with lower metallicities and/or lower [H$\beta$]{} equivalent widths were retained as secondary targets. Up to 19 slitlets per exposure can be used for spectroscopy with FORS1. Our primary targets were first positioned using the R-band images (see below) and the remaining slitlets were filled whenever possible with secondary targets. If a slitlet was left without any of our selected regions, we attempted to target other [H [ii]{}]{} regions selected from the [H$\alpha$]{} images of Hodge & Kennicutt (1983). For each galaxy a nuclear spectrum, to be reported upon later, was also obtained. Observations ------------ R band imaging was obtained with FORS1/VLT in april 2000, and was used to determine the positions of our targeted regions with sufficient accuracy. Subtracting a local average emission from the host galaxy the R band magnitudes of our target [H [ii]{}]{} regions were determined; typical magnitudes of $m_R \sim$ 19–21 are found. The spectroscopic observations of our sample of [H [ii]{}]{} regions were carried out with FORS1/VLT in the second 2001 trimester. Table \[tab\_log\] gives informations about the exact dates and meteorological conditions during the observations. -------- ------ --------- -------------------------------------------------------------- -- -- galaxy date weather seeing \["\] & exp. time blue \[s\]& exp. time red \[s\]\ NGC 3351 & 19.04.2001 & photometric & 0.8-1.0 & 1700 & 1700\ NGC 3521 & 25.04.2001 & clear & 1.6-2.0 & 1800 & 1800\ NGC 4254 & 23.05.2001 & clear & 1.1-1.4 & 900 & 900\ NGC 4303 & 23.05.2001 & clear & 0.8-1.1 & 750 & 750\ NGC 4321 & 19.06.2001 & photometric & 1.3-1.5 & 1050 & 1050\ -------- ------ --------- -------------------------------------------------------------- -- -- \[tab\_log\] The spectral range from 3600 Å to 1 $\mu$m was covered with a “blue” spectrum from 3600 to 6500 Å with grism 300V+10, and a “red” spectrum from 6000 to 10000 Å  with grism 300I+11. The use of a 1 slit width allowed to get medium spectral resolution of around 6 Å in the blue and 12 Åin the red. Due to the limited slit size, a fraction of the total nebular emission of the regions may be lost. This effect is acounted for in our interpretation of the data (Sect. \[s\_models\]). Unless WR stars follow systematically a different spatial distribution than other stars responsible for the continuum emission, a possible loss of continuum light does not alter our analysis. Exposure times for each galaxy (see Table \[tab\_log\]) were adapted to obtain in the continuum S/N $\sim$ 30 in the blue, (needed for a precise measure of the WR bump) and $\sim$ 10 in the red (needed to measure the  lines). Spectrophotometric standard stars data were also acquired. Data reduction and analysis --------------------------- Reduction was carried out using the IRAF and MIDAS packages. The first steps consisted in the usual bias subtraction, flatfield division, and 2D wavelength calibration. Flux calibration was done using a standard atmospheric extinction curve and spectrophotometric standard stars. Given that the spectrophotometric standards were not always obtained during the night of the observations, we estimate an absolute flux accuracy of $\sim$ 10 %. In addition, due to the optimisation for a maximum multiplex, the observations were not taken at parallactic angle, leading to a slight mismatch between the blue and red spectra. A quantitative analysis of the effetcs of differential refraction has not been undertaken here. As the main diagnostics used in the present paper lie in a limited wavelength range, and the observations have been taken at small airmass, this should represent a negligible source of uncertainty. For each [H [ii]{}]{} region, a background including sky emission and underlying emission from the galaxy was extracted from the slitlet sub-image. This procedure was non-trivial as this background spectrum had in most cases to be determined near the edges of the sub-image, where the wavelength calibration may slightly deviate from the one of the [H [ii]{}]{} region. Special care has been taken for the red spectra, since the sky emission was often several times brighter than the [H [ii]{}]{}region emission. We thus re-calibrated the background emission spectrum according to the [H [ii]{}]{} region by comparing the position (and sometimes the intensity) of the sky emission lines. This time-consuming operation gave very satisfying results and useable spectra up to 1 $\mu$m for almost all [H [ii]{}]{}regions. The final 1D spectra were generally extracted with a 4  wide aperture. Line intensities and equivalent width were obtained by visually placing a continuum on both sides of the line and then integrating all over this range. Errors were estimated by moving the continuum upwards by half the value of the noise near the line position and re-computing the intensity and equivalent width. Where possible the following nebular emission lines were measured: [\[O [ii]{}\]]{} $\lambda$3727, the H Balmer line series including [H$\alpha$]{} to H9, [He [i]{} $\lambda$4471]{}, [\[O [iii]{}\]]{} $\lambda$4959,5007, [\[N [ii]{}\]]{} $\lambda$5201, [He [i]{}]{} $\lambda$5876, [\[O [i]{}\]]{}  $\lambda$6300, [\[N [ii]{}\]]{} $\lambda$6548,6584, [He [i]{}]{} $\lambda$6678, , [He [i]{}]{} $\lambda$7065, \[Ar [iii]{}\] $\lambda$7136, [\[O [ii]{}\]]{} $\lambda$7325, and . If present, broad emission lines at $\lambda \sim$ 4680 Å  (referred to subsequently as the (blue) WR bump), [C [iii]{} $\lambda$5696]{}, and [C [iv]{} $\lambda$5808]{}indicative of Wolf-Rayet (WR) stars were also measured. The spectra were also inspected for the presence of stellar absorption lines like the Ca [ii]{} triplet, the CH G band at $\sim$ 4300 Å, Mg lines at $\sim$ 5200 Å, or TiO bands. The spectra were deredened using the Whitford  (1958) extinction law as parametrised by Izotov  (1994) assuming an underlying absorption of $W({\ifmmode {\rm H}\beta \else H$\beta$\fi})=$ 2 Å and an intrinsinc Balmer decrement ratio of $I({\ifmmode {\rm H}\alpha \else H$\alpha$\fi})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})=2.86$. All detailed results including finding charts, line measurements, and a detailed analysis of the nebular properties will be published in a forthcoming paper. ---------- -------------- ------------------------------- -------------------------------- ----------------- ---------------------------------- ----------------------------------- -- -- -- Galaxy \# blue bump \# [C [iv]{} $\lambda$5808]{} \# [C [iii]{} $\lambda$5696]{} cand. blue bump cand. [C [iv]{} $\lambda$5808]{} cand. [C [iii]{} $\lambda$5696]{} NGC 3351 2 4 2 NGC 3521 4 2 1 6 1 1 NGC 4254 9 8 1 1 1 NGC 4303 9 4 3 3 2 NGC 4321 3 3 5 1 1 total 27 14 8 15 6 10 ---------- -------------- ------------------------------- -------------------------------- ----------------- ---------------------------------- ----------------------------------- -- -- -- Properties of the HII region sample {#s_props} =================================== The properties of our galaxy sample as given by the NED database and the adopted distances, are summarised in Table \[tab\_sample\]. For NGC 3351 and the Virgo cluster member NGC 4321 we adopt the Cepheid distances from Freedman  (2001). The other two Virgo galaxies (NGC 4254, NGC 4303) are member of the same subgroup as NGC 4321 (Boselli, private communication). We therefore adopt an identical, approximate distance of 16 Mpc. The distance of NGC 3521 is taken from Tully’s (1998) Nearby Galaxy Catalog. A total of 121 spectra were extracted from the 95 slitlets. Nebular emission lines were detected in 88 spectra; 85 correspond to extra-nuclear regions. Metallicities ------------- The metallicity O/H of the [H [ii]{}]{} regions has been estimated using the following empirical calibrations: the calibrations of Kobulnicky  (1999, hereafter KKP) using /  and (+)/[H$\beta$]{} (=$R_{23}$) based on the photoionisation model grid of McGaugh (1994), the similar $P$-method of Pilyugin (1991), and the older $R_{23}$ calibrations of Edmunds & Pagel (1984) and Zaritsky  (1994). The O/H abundances obtained from these methods are compared in Fig. \[fig\_oh\_compare\]. Unsurprisingly rather large differences are obtained. As well known, at abundances ${\ifmmode 12 + \log({\rm O/H}) \else$12 + \log({\rm O/H})$\fi}\la$ 8.5–8.6 the various $R_{23}$ methods yield similar results, while the differences increase towards higher metallicities (see e.g. comparison in Pilyugin 2001). Systematically lower values are found from the $P$-method of Pilyugin (2001). Although calibrated only for regions with ${\ifmmode 12 + \log({\rm O/H}) \else$12 + \log({\rm O/H})$\fi}\la 8.6$, this could indicate a systematic overestimate of the absolute metallicities using the other methods. To ease comparisons with the recent study of BK02 of metal-rich [H [ii]{}]{} regions we subsequently adopt the KKP calibration by default except otherwise stated. The metallicity distribution of our entire sample is shown in Fig. \[fig\_oh\_histogram\]. The mean metallicity is $<{\ifmmode 12 + \log({\rm O/H}) \else$12 + \log({\rm O/H})$\fi}>= 8.88 \pm 0.22$ (8.57 $\pm$ 0.24) using the KKP (Pilyugin’s $P$) calibration. The vertical dashed line indicates the solar value ([12 + ([O/H]{}) $12 + \log({\rm O/H})$]{}=8.92) adopted in McGaugh’s calculations used for the calibration of KKP. ### Regions with direct $T_e$ determinations The transauroral \[O [ii]{}\] $\lambda$7325 line has been detected in 11 [H [ii]{}]{} regions allowing thus a direct determination of the electron temperature from \[O [ii]{}\] $\lambda$7325/. Other potential electron temperature indicators, e.g. \[S [iii]{}\] $\lambda$6312, \[N [ii]{}\] $\lambda$5755, are too weak or could not be measured due to the limited spectral reolution. Electron densities are determined from . $T_e($O [ii]{}$)$ and the resulting ionic abundance ratios of O$^{++}$/H$^+$ and O$^+$/H$^+$ were derived using this temperature for both ions (the atomic data are those listed in Stasińska & Leitherer (1996). As shown in Fig. \[fig\_oh\_toii\] the resulting O/H abundances (assuming O/H $=$ O$^{++}$/H$^+$+O$^+$/H$^+$) are on average found to be lower than those derived from the KKP calibration, the largest metallicity being closer to solar. However, the O/H derived here are lower limits, due to the strong temperature gradients expected at high metallicities (see Stasińska 2002). A deeper discussion of the abundances in our objects taking into account the observational constraints from the entire emission line spectrum is deferred to a forthcoming publication. For the purpose of the present paper, it is sufficient to note that the bulk of our [H [ii]{}]{} region sample with low values of $R_{23}$ have metallicities close to and above solar. [H$\alpha$]{} luminosities, WR and O star populations ----------------------------------------------------- The histogram of the [H$\alpha$]{} luminosity of the [H [ii]{}]{} regions, as measured from our spectra, is shown in the upper panel of Fig. \[fig\_plot\_both\]. As seen from this figure, $\sim$ 75 % of the [H [ii]{}]{} regions correspond to giant extra-galactic [H [ii]{}]{} regions characterised by $L({\ifmmode {\rm H}\alpha \else H$\alpha$\fi}) \ga 10^{38}$ erg s$^{-1}$ (Kennicutt 1984, 1991), while the remainder are less luminous objects similar to normal Galactic [H [ii]{}]{} regions. The corresponding number of equivalent O7V stars [^2], $N_{\rm O7V}$ plotted in the lower panel of Fig. \[fig\_plot\_both\], ranges from $\sim$ 0.15 O7V stars (i.e. presumably corresponding to $\sim$ 1 late O or early B stars) to $\sim$ 400 O7V stars for the brightest region. Our search for WR features in metal-rich [H [ii]{}]{} regios proved quite successful yielding with 27 WR detections a sample of unprecedented size (cf. Castellanos 2001, Bresolin & Kennicutt 2002). The number of regions where different WR features were detected (hereafter called “WR regions”) at various levels of confidence are listed in Table \[tab\_wr\]. The certain WR detections (defined as $\ge 2 \sigma$ detections) are listed in columns 2-4; “candidate” WR regions with emission line detections 1.1 $\le \sigma < $ 2 are given in cols.5-7. Visual inspection of the spectra yield essentially the same detection of the “certain” WR regions. To illustrate the quality of our data sample spectra of a secure WR region and a candidate region are shown in Fig. \[fig\_spectra\]. As also clear from Table \[tab\_wr\], a large fraction of the WR regions shows signatures of WR stars of both WN and WC types as anticipated from theoretical expectations (Meynet 1995, SV98) and earlier studies of WR galaxies (Schaerer  1997, 1999ab, Guseva  2000). In our sample $\sim$ 50 % of WR regions show WC signatures; predictions from the Meynet (1995) and SV98 models yield $\sim$ 30–77 % at metallicities $Z \sim$ 0.008 – 0.040. At least 1/3 of the WR regions harbour WC stars of late subtypes (WCL), characterised by their strong [C [iii]{} $\lambda$5696]{} emission[^3]. The [C [iii]{} $\lambda$5696]{}/[C [iv]{} $\lambda$5808]{} ratio indicates subtypes WC7 or WC8 assuming that the contribution of WN stars to [C [iv]{} $\lambda$5808]{} is negligible; if this were not the case the mean spectral type could be of later subtype. So far relatively few WR “galaxies” (true starbursts or extra-galactic giant [H [ii]{}]{} regions) with WCL stars are known (cf. Schaerer  1999b). However, as late WC types are expected to occur preferentially in metal-rich environments (Smith & Maeder 1991, Maeder 1991, Philipps & Conti 1992) the high detection rate of [C [iii]{} $\lambda$5696]{} is not surprising. The [H$\alpha$]{} luminosity distribution of the WR regions is shown in Fig. \[fig\_plot\_both\] (upper panel, dashed line). Clearly, WR stars are only detected in the brightest regions. This is [*not*]{} due to the flux limit of our observations as can easily be seen by comparison of the smallest WRbump fluxes ($F({\rm WR})_{2 \sigma} \sim 4. 10^{-16}$ erg s$^{-1}$ cm$^{-2}$) with the detection limit of the faintest emission lines ($F_{\rm lim}({\ifmmode {\rm H}\beta \else H$\beta$\fi}) \sim 10^{-16}$ erg s$^{-1}$ cm$^{-2}$). In fact our observations are essentially deep enough to allow in all galaxies [^4] the detection of the blue WR bump of just $\sim$ 2–3 WNL stars, assuming the average 4650-4686 Å bump luminosity of a WN7 stars of $10^{36.5}$ erg s$^{-1}$ (cf. Smith 1991, Schaerer & Vacca 1998). The number of WNL stars derived in this way is plotted in the lower panel of Fig. \[fig\_plot\_both\] for regions with certain WR detections (filled squares) and “candidate” WR regions (open circles). As our detection limit allows for the detection of few ($\sim$ 2–3) average WNL stars, the subsample of the brightest regions with $F({\ifmmode {\rm H}\beta \else H$\beta$\fi}) \ga 5. 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ could therefore represent a fairly complete sample of [H [ii]{}]{} regions “massive”/bright enough to allow a meaningful comparison between WR detections and non-detections. However, a possible bias against regions with small [W([H]{}) $W({\rm H}\beta)$]{} may exist (Sect. \[s\_obs\]). In this subsample containing a total of 47 regions we find 20 objects without WR signatures, or a fraction of $\sim$ 57 % regions with WR signatures. Such a high fraction of WR detections compares fairly well with the predictions of 60–80 % by Meynet (1995) and Schaerer & Vacca (1998) using the high mass loss stellar evolution tracks at metallicities $1/2.5 \la Z/\zsun \la 2$ for bursts with a standard Salpeter IMF and an upper mass cut-off =120 . Given the fact that very young regions (ages 0 to $\sim$ 1.5–2 Myr) with large expected [H$\beta$]{} equivalent widths are notoriously absent (in the present sample and other samples of [H [ii]{}]{} regions and galaxies) it is, however, not clear how significant this finding is. Trends of WR populations with metallicity {#s_wroh} ========================================= Behaviour of the “WR bump” -------------------------- Figure \[fig\_bump\_oh\] shows the WR bump intensities and equivalent widths as a function of metallicity for our metal-rich [H [ii]{}]{} regions (large filled triangles) and the 11 WR regions in spiral galaxies recently reported by Bresolin & Kennicutt (2002, large filled squares), together with data compiled by Schaerer (1999, small crosses) and Schaerer  (2000). Our new measurements at high O/H are found to fill in the range from the previously observed maximum intensities/equivalent widths down to lower values. Physically the maxima of [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{} and [W([WR]{}) $W({\rm WR})$]{} are expected to reflect the maximum WR/O star ratio achieved in bursts. No lower limit is expected; if present in a given sample, such a lower limit presumably reflects the detection limit of the WR features. The increase of the upper envelope of [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{} with metallicity is known since the work of Arnault  (1989) and has been reviewed by Schaerer (1999). With few exceptions, max([W([WR]{}) $W({\rm WR})$]{}) also seems to show an increase with O/H as shown here for the first time. The increase of max([I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{}) is naturally interpreted as due to the increase of stellar wind mass loss with metallicity leading to lower minimum mass limit for the formation of WR stars, $M_{\rm WR}$, thereby favouring the presence of WR stars at high metallicity (cf. Maeder  1981, Arnault  1989, Maeder 1991). Other effects, e.g. a lowering of the [H$\beta$]{} flux due to a) increasing amounts of dust absorbing ionising radiation or b) lower average stellar temperatures at high O/H due to modified stellar evolution, could also play a role (cf. Schaerer 1999), but are likely secondary. The maxima of the predicted WRbump intensities and equivalent widths computed with the code of SV98  with a “standard” Salpeter IMF for instantaneous bursts (solid line), and extended bursts of duration $\Delta t$ = 2 Myr (dotted), and 4 Myr (long dashed) are overplotted on Fig. \[fig\_bump\_oh\]. As already shown earlier (cf. Schaerer 1996, 1999, Mas-Hesse & Kunth 1999, Guseva  2000) the range of observations at subsolar metallicities ([12 + ([O/H]{}) $12 + \log({\rm O/H})$]{} $\la$ 8.6) is fairly well reproduced by the models, when accounting for the various uncertainties (e.g. missing [H$\beta$]{} flux in slit observations, some objects with small numbers of WR stars, some poor spectra; cf. discussion in Guseva ). The new sample of metal-rich objects plotted here shows WRbump strengths smaller than the maxima predicted by the “standard” models. The possible reasons for this behaviour are discussed in Sect. \[s\_models\] where detailed model comparisons are undertaken. WC/WN ratio ----------- We have estimated the relative number ratio of WC and WN stars, shown in Fig.\[fig\_wcwn\_oh\], in several ways. First the number of WN stars, $N({\rm WNL})$ assuming late WN subtypes dominate, is derived from the luminosity of the blue WR bump, as described above. The number of WC stars, $N({\rm WC})$, is estimated from the [C [iv]{} $\lambda$5808]{} or [C [iii]{} $\lambda$5696]{} luminosity where measured, again assuming that WN stars do not contribute to these lines. As the observed average luminosity of WC stars in these lines varies strongly with subtype (see SV98), the estimated $N({\rm WC})$ depends on the assumption of the dominant WC subtype. As the observations (see above, Guseva  2000, Schaerer  1999a) indicate that early types ($\sim$ WC4) dominate at low metallicity, while WC7-8 dominate at high [12 + ([O/H]{}) $12 + \log({\rm O/H})$]{}, we assume these mean WC subtypes for the sample of Guseva  (2000). For our high metallicity sample, the estimated $N({\rm WC})/N({\rm WNL})$ ratios is estimated adopting different assumptions on the WC subtype and using [C [iv]{} $\lambda$5808]{} or [C [iii]{} $\lambda$5696]{}(see Fig. \[fig\_wcwn\_oh\]). The resulting estimates show a fairly clear trend of an increasing upper envelope for $N({\rm WC})/N({\rm WNL})$ with metallicity. Furthermore, and in contrast with the limited sample of Guseva  (2000), we now find at the high metallicity end a number of objects with $N({\rm WC})/N({\rm WNL}) \ga$ 0.5–1. and a WC/WN number ratio larger than the observed trend in Local Group galaxies by Massey & Johnson (1998), indicated by the dash-dotted line in Fig. \[fig\_wcwn\_oh\]. Indeed, while the regions observed by these authors are thought to correspond to averages large enough to represent the equilibrium $N({\rm WC})/N({\rm WNL})$ value at constant star formation, larger (and obviously also smaller) values should be found in regions with fairly short bursts. A more quantitative interpretation of the observed WC to WN ratio appears difficult for the following reasons. First the uncertainties in the estimated $N({\rm WC})/N({\rm WNL})$ are quite large (cf. above); second, detailed evolutionary synthesis model predictions of $N({\rm WC})/N({\rm WNL})$ depend quite strongly on the adopted interpolation techniques (cf. SV98, comparison between results from SV98 models and [*Starburst99*]{} (Leitherer  1999), also Massey 2002); third, other comparisons with synthesis models reveal potential difficulties (cf. below). In any case the SV98 models predict the maximum WC/WN number ratios indicated in Fig. \[fig\_wcwn\_oh\] by the solid line for instantaneous bursts, and burst durations of $\Delta t=$ 2 Myr (dotted) and 4 Myr (dashed) respectively. Detailed comparison of WR populations with synthesis models {#s_models} =========================================================== Procedure {#s_procedure} --------- To interpret quantitatively the observational data we use evolutionary synthesis models and proceed essentially as in SGIT00. The following main observational constraints are used: 1. [*[H$\beta$]{} and [H$\alpha$]{} equivalent widths.*]{} The former is used as a primary age indicator; once [W([H]{}) $W({\rm H}\beta)$]{} is reproduced $W({\ifmmode {\rm H}\alpha \else H$\alpha$\fi})$ may serve as an independent consistency test for the predicted spectral energy distribution (SED) in the red (cf. SGIT00). 2. [*Nebular line intensities.*]{} $F$([H$\alpha$]{})/$F$([H$\beta$]{}) determines the extinction of the gas. The use of other line intensities requires detailed photoionization modeling which is beyond the scope of this paper. 3. [*Intensities and equivalent widths of the main WR features.*]{} The blue bump and [C [iv]{} $\lambda$5808]{} (red bump) serve as main constraints on the WR population. To avoid uncertainties in deblending individual contributions of the blue bump we prefer to use measurements for the entire bump. In contrast to the spectra of metallicity objects our spectra show no evident contamination from nebular lines (e.g. \[Fe [iii]{}\] $\lambda$ 4658, nebular [He [ii]{}]{}). To potentially disentangle between various effects (underlying “non-ionizing” population, loss of photons, differential extinction between gas and stars) it is important to use both equivalent widths and relative [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{} intensities (cf. Schaerer  1999a). For the model comparisons we use calculations based on the evolutionary synthesis code of SV98, which in particular includes the most recent calibration of WR line luminosities used to synthesize the WR features, up-to-date stellar tracks, [*CoStar*]{} stellar atmospheres for O stars, pure H-He models for WR stars and Kurucz models for cooler stars (see SV98 for a full description). Except for the improved O star atmospheres used by SV98 the [*Starburst99*]{} synthesis models (Leitherer  1999) use the same basic input physics. New generation stellar atmosphere models for O and WR stars including a full treatment of non-LTE line blanketing and stellar winds have just now become available for the use in synthesis models (Smith  2002). However, as the quantities of interest here depend only on the total number of Lyman continuum photons which is not altered, the use of these more sophisticated atmosphere models does not affect our results. It is important to stress that in all cases the high-mass loss stellar tracks of Meynet  (1994) are used. It is thought that this adjustment of mass-loss, treated like a free parameter, will become ultimately obsolete when a proper treatment of the various effects of stellar rotation is made in the stellar evolution models. First results tend to indicate that this may indeed be the case (Meynet 1999). The Meynet  (1994) tracks are chosen as they reproduce a large number of properties of individual WR stars and WR populations (including especially relative WR/O ratios for a standard Salpeter IMF) in Local Group galaxies (Maeder & Meynet 1994). The use of other tracks (e.g. the “normal” mass loss tracks) which are known to disagree with these basic constraints on WR and O star populations, would imply a strong inconsistency with the Local Group data. The basic model parameters we consider are: 1. [*Metallicity.*]{} Stellar tracks covering metallicities $Z=$, 0.008, 0.02 (solar), and 0.04. 2. [*IMF slope and upper mass cut-off ().*]{} We adopt a Salpeter IMF (slope $\alpha$=2.35), and =120  as our standard model. 3. [*Star formation history (SFH).*]{} Models for instantaneous bursts (coeval population), extended burst durations (constant SF during period [t $\Delta t$]{}; in this case age=0 is defined at the onset of SF, i.e. corresponds to that of the oldest stars present), and constant SF are considered. 4. [*Fraction of ionizing Lyman continuum photons ($f_\gamma$).*]{} $f_\gamma$ indicates the fraction of ionizing photons absorbed by the gas. Our standard value is $f_\gamma=1$. Values $f_\gamma < 1$ are used to simulate various effects (e.g. dust absorption, photon leakage outside regions, etc.) leading to a reduction of photons available for photoionization. Unless stated otherwise our models are calculated assuming an IMF fully sampled over the entire mass range (as in SV98). For some cases we have also done model calculations based on a Monte Carlo sampling of the IMF, in order to quantify the effects of statistical fluctuations due to the finite number of massive stars. We have verified our calculations by comparison with the Monte Carlo models and analytical results of Cerviño  (2000, 2002). Results {#s_result} ------- A comparison of the observed equivalent widths and relative intensity of the WR bump with standard model predictions at different metallicities is presented in Fig. \[fig\_std\]. The following points can be seen from this Figure: - The observed trends of decreasing (increasing) equivalent width (line intensities) of the WR features with decreasing [H$\beta$]{} equivalent width agree with the instantaneous burst model predictions over the same [W([H]{}) $W({\rm H}\beta)$]{} range. Though shorter, part of the initial phase of increasing (decreasing) [W([WR]{}) $W({\rm WR})$]{} ([I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{}) corresponding to the onset of the WR rich phase does not seem to be detected. - Most importantly, [*essentially all the observed WR features are weaker than the predictions of our standard models for instantaneous bursts at solar or higher metallicity.*]{} Very similar observational trends are also found in the metal-rich [H [ii]{}]{} region samle of BK02 and the metal-rich starbursts of Schaerer  (2000). This is in stark contrast with observations at lower metallicity where generally an good agreement is found with short burst models (e.g.  Schaerer  1999a, Guseva  2000). - The above result is found independently from the observed WR equivalent widths and relative line intensities WRbump/[H$\beta$]{}. This indicates that the discrepancy between models and observations is not related to the possible presence of an underlying older stellar population, which would dilute (reduce) equivalent widths but not alter the relative line intensities. The following possibilities (one or a combination thereof) could be invoked to explain the discrepancy between our observations and models: 1. [*The metallicities of our HII regions are overestimated.*]{} Indeed the observations could be reconciled with burst models with a “standard” IMF for metallicities $Z \sim$ (1/2.5–1) , as shown in the left panel of Fig. \[fig\_std\] (short dashed line). However, despite the uncertainties in the O/H determinations (cf. Sect.  \[s\_props\]) such low average metallicities seem very implausible. 2. [*Extended bursts.*]{} Such a scenario has been invoked by SGIT00 for the sample of metal-rich starbursts based on the finding of red supergiant features in their spectra and the fact that these distant objects are mostly nuclear starbursts observed through apertures corresponding to relatively large spatial scales. In this case all observed properties could quite well be fitted with “standard” solar metallicity models for burst durations $\Delta t \sim$ 4–10 Myr. However, in view of the different nature (disk [H [ii]{}]{} regions) of the present sample, indications of relatively short formation time scales of [H [ii]{}]{} regions (e.g. Massey  1995), and the lack of direct signatures of older/red supergiant populations (cf. below) it seems quite unjustified to appeal to extended burst to solve the observed discrepancy. 3. [*A modified IMF (upper mass cut-off and/or slope).*]{} In a plot like Fig. \[fig\_std\], a Salpeter IMF with a lower upper mass cut-off simply implies that the curve plotted here (for =120) is joined at lower [W([H]{}) $W({\rm H}\beta)$]{} as the WR stars from the most massive stars are absent. This is illustrated for the cases of  $=$ 30 and 60  by the shaded domains in Fig. \[fig\_alpha\]. The shape of the predicted WR equivalent width or line intensity remains, however, unchanged. Therefore the observed discrepancy cannot be resolved with an IMF of Salpeter slope and a lower value of  (see also Sect. \[s\_imf\]). Models with steeper, variable IMF slopes ($2.35 < \alpha \la 3.3$) and $\sim$ 60–120  could reproduce most of the objects, with the exception of the lowest [W([H]{}) $W({\rm H}\beta)$]{} objects (see Fig. \[fig\_alpha\]). As the least metal-poor objects in our sample are probably of similar nature as young clusters or [H [ii]{}]{} regions in our Galaxy whose stellar content has been studied in detail, we may presume that their IMF (slope and ) should be similar. Since none of the Galactic regions have shown convincing evidence of a strong deviation of the IMF slope from the Salpeter value (see Massey 1998 and references therein), we think that such a steeper slope is an unlikely explanation. 4. [*Incorrect stellar evolution models and/or “calibration data”:*]{} Although the adopted tracks (Meynet  1994) compare fairly well with various observations, several failures of the non-rotating stellar models are also known (see e.g. Maeder 1999). However, the used tracks have essentially been calibrated/adjusted to fit the observed WR/O ratio in various regions of our Galaxy and Local Group objects which are though to be at equilibrium, i.e. showing relative populations corresponding to constant star formation (see compilation in Maeder & Meynet 1994). The relative WR/O star ratio is the one most directly related to our (time resolved) observables. As this calibration yields a fairly good agreement over a large metallicity range ($1/10 \la Z/\zsun\ \la 2$) there seems little room for changes in the tracks which could reduce the predicted WRbump by the required factor of $\sim$ 2 without violating the WR/O constraints in the Local Group. One could argue that the calibration data, the observed WR/O number ratio at solar metallicity and above could be incorrect due to possible incompleteness or biases in the stellar counts (see e.g. related discussions in Massey & Johnson 1998). However, to reconcile our WR observations in [H [ii]{}]{} regions with the corresponding counts for our Galaxy and M31 would require a downward revision of the relative WR/O ratio by up to a factor of 2, which seems highly unlikely. 5. [*Uncertainties in synthesis of the WR bump:*]{} Presently the calculation of observables related to WR stars is simply done in the following way in evolutionary synthesis models. The different emission line strengths are computed by multiplying the predicted number of WR stars (grouped in different types and/or subtypes) with their average line luminosity as derived from observations of a sample of WR stars (see SV98). Interestingly the intrinsic line luminosity of the strongest line of the WR bump, [He [ii]{} $\lambda$4686]{}, shows a rather large scatter, namely $L_{\rm 4686}=(1.6 \pm 1.5) \times 10^{36}$ erg s$^{-1}$ in the Galactic and LMC WNL calibration sample of SV98 with a possible increase of $L_{\rm 4686}$ with the stellar bolometric luminosity $L$ (see Fig. 1 of SV98). Such a luminosity dependence of $L_{\rm 4686}$ with $L$ could in fact (partly or fully) explain the observed discrepancy as we will now show. Splitting the WNL calibration in two domains with luminosities above/below $\log L/\lsun = 6$, SV98 found average line luminosities $L_{\rm 4686} = 5.6 \times 10^{35}$ ($\log L < 6$) and $L_{\rm 4686} = 3.1 \times 10^{36}$ ($\log L > 6$). Replacing in the synthesis models the overall average for WNL stars by these quantities leads to an important reduction of [W([WR]{}) $W({\rm WR})$]{} in solar metallicity bursts with [W([H]{}) $W({\rm H}\beta)$]{} $\la$ 60–70 Å, as shown in Fig. \[fig\_wnl\_mod\][^5]. At larger [H$\beta$]{} equivalent widths (corresponding to ages $\la$ 4–5 Myr for $Z=$0.02) the WRbump predictions are less modified, since [*a)*]{} WC stars contribute more importantly to the bump and [*b)*]{} only the youngest bursts with very high [W([H]{}) $W({\rm H}\beta)$]{} are dominated by very luminous WNL stars showing thus larger [W([WR]{}) $W({\rm WR})$]{}. The last option (5) seems the most likely explanation to explain the surprisingly low WR equivalent widths and intensities in our sample of metal-rich [H [ii]{}]{} regions. Implications on earlier studies of WR galaxies are briefly discussed in Sect. \[s\_discuss\]. In contrast, the following hypothesis or effects altering observed equivalent widths and/or relative line intensities [**cannot**]{} be the cause of the discrepancy: - [*Underlying (old) populations diluting the [W([WR]{}) $W({\rm WR})$]{} measurements.*]{} The line intensities are unaffected by underlying populations and show the same discrepancy as [W([WR]{}) $W({\rm WR})$]{} (cf. above). Furthermore, inspection of our spectra show little or no indication of signatures of older stellar populations. - [*Differential extinction between gas and stars*]{} as frequently observed in [H [ii]{}]{} galaxies and starbursts (cf. Fanelli  1988, Calzetti 1997, Mas-Hesse & Kunth 1999, SGIT00). If present such an effect alters [W([H]{}) $W({\rm H}\beta)$]{} and [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{} but not [W([WR]{}) $W({\rm WR})$]{}. To bring the observations to agreement with standard models would require that the gas suffered a [*lower*]{} extinction than the stars (implying thus lower corrected [W([H]{}) $W({\rm H}\beta)$]{} and larger [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{}), opposite to what is found empirically. - [*Leakage of ionising photons outside the observed regions, dust absorption of ionising photons*]{}, or other mechanisms reducing the fraction $f_\gamma$ of Lyman continuum photons used for photoionisation. Correcting the observations for such an effect would increase the observed [W([H]{}) $W({\rm H}\beta)$]{} and decrease [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{}, which does not reconcile observations and theory (see Fig. \[fig\_std\]). - [*Effects due to varying seeing conditions and limited slit widths*]{} could lead to a loss of nebular emission in the aperture or a fraction of the stellar light. The former was just discussed (“leakage”). If the WR distribution were systematically more extended than the other stars responsible of the continuum, this effect could lead to reduced WR bump equivalent widths. No such trend between [W([WR]{}) $W({\rm WR})$]{} and the seeing conditions is found. - [*Stochastical fluctuations of the IMF*]{} due to small number statistics of the massive stars, as invoked by Bresolin & Kennicutt (2002). Although indeed expected to introduce some scatter (see Cerviño  2000, 2002) this effect cannot explain the discrepancy for several reasons. First, the sample discussed here is sufficiently large and shows clear observational trends with relatively small scatter. In addition, no systematic trend of [W([WR]{}) $W({\rm WR})$]{} (and [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{}) with absolute scale, such as measured by the continuum luminosity or [H$\alpha$]{} luminosity, is observed in our sample as shown in Fig. \[fig\_abs\_scales\]. Second, Monte Carlo simulations we have undertaken for cluster sizes corresponding roughly to the observed average continuum luminosity of $<\log L_{\rm cont}> \sim 36.8 \pm 0.4$ [^6] predict a typical relative scatter of $\sigma(W({\rm WRbump}))/{\ifmmode W({\rm WR}) \else $W({\rm WR})$\fi}\sim$ 25 % (cf. Cerviño  2002) – too small to account for the discrepancy – and no significant bias towards lower values as illustrated in Fig. \[fig\_mc\_wrbump\]. - The use of different stellar tracks, such as e.g. the Geneva tracks with standard mass loss tracks adopted by Bresolin & Kennicutt (2002) for the Monte Carlo models, which [*do not*]{} reproduce basic constraints of massive stars and stellar populations in Local Group objects cannot be a solution as they would lead to important inconsistencies (cf.  discussion in Sect. \[s\_procedure\]). Discussion {#s_discuss} ---------- In Section \[s\_result\] we have argued that, compared to the normal prescription used in our SV98 synthesis models, a different prescription should preferrably be adopted to predict more accurately the [He [ii]{} $\lambda$4686]{}  emission from WN stars. As several earlier studies including ours (e.g. Schaerer 1996, 1999, Schaerer  1999a, Guseva  2000, SGIT00) are based on the use of the simple average [He [ii]{} $\lambda$4686]{} line luminosity of SV98 for WNL stars, it is important to assess if or to what extent the use of a luminosity dependent prescription would affect the results from previous studies. To verify this we have recomputed several sets of models for sub-solar metallicities. The maxima of the WRbump intensity and [W([WR]{}) $W({\rm WR})$]{} (cf. Fig. \[fig\_bump\_oh\]) are only slightly modified (increased at [12 + ([O/H]{}) $12 + \log({\rm O/H})$]{} $\la$ 8.5, and decreased above) and lead to a somewhat smaller increase with O/H, improving the agreement with the observations. For metallicities $Z \la$ 1/2  the predicted WR bump is found to be larger at all ages (as the bulk of WN stars are of high luminosity), whereas for higher metallicities both larger/smaller WRbump strengths are predicted depending on the burst age ([W([H]{}) $W({\rm H}\beta)$]{}), as for the cases shown in Fig. \[fig\_wnl\_mod\]. These changes improve the comparison with observations at low $Z$ (see e.g.Fig. 7 of Guseva  2000). No clear statement can be made for intermediate metallicities. A better understanding of the dependence of the WR emission lines on the stellar parameters appears necessary to improve the accuracy of the predictions of WR features in evolutionary synthesis models. The impact of newly available stellar evolution models including the effects of rotation on interior mixing and mass loss on massive star populations remains also to be explored. Constraining the upper end of the IMF {#s_imf} ===================================== As shown above, and in contrast to previous studies considering WR populations in objects with sub-solar metallicities, the quantitative modeling of the WR features in metal-rich [H [ii]{}]{} regions is not straightforward. The reliability of constraints on the IMF obtained from modeling of the WR features appears thus unclear at present and other constraints are therefore desirable. The maximum observed equivalent width of hydrogen recombination lines (e.g. [W([H]{}) $W({\rm H}\beta)$]{}) of a large sample of star forming regions provide in principal a constraint on the upper mass cut-off of the IMF (e.g Kennicutt 1983, Leitherer  1999). In practice, however, as well known but not understood to date, there is a notable absence of regions with [W([H]{}) $W({\rm H}\beta)$]{} as large as predicted for very young bursts with ages between $\sim$ 0 to 1.5–2 Myr. As the onset of the WR phase is expected after $\sim$ 2–3 Myr quite independently of the exact adopted stellar tracks (e.g. different mass loss scenarios) this sets a new clock, and therefore the [H$\beta$]{} equivalent width of the youngest observed WR region also contains information on the value of . In Fig. \[fig\_imf\] we show the dependence of the predicted [W([H]{}) $W({\rm H}\beta)$]{} at the beginning of the WR phase (i.e. the maximum of [W([H]{}) $W({\rm H}\beta)$]{} during this phase) on the upper mass cut-off for different IMF slopes in instantaneous bursts. The maximum [W([H]{}) $W({\rm H}\beta)$]{} depends little on metallicity (see the dotted lines) and on the choice of stellar tracks (not shown here). Overplotted are the observed [W([H]{}) $W({\rm H}\beta)$]{} in our WR region sample (triangles) and the sample of BK02 (squares) drawn at arbitrary . The observed max([W([H]{}) $W({\rm H}\beta)$]{}) ($\log {\ifmmode W({\rm H}\beta) \else $W({\rm H}\beta)$\fi}\ \sim$ 2.2–2.4) indicates an upper mass cut-off of $\sim$ 80–90  for a Salpeter IMF or $\mup \ga$ 120  for a steeper IMF with $\alpha=$3.3. From all the above considerations (Sect. \[s\_models\]) flatter slopes seem excluded. If the bulk of the regions were forming stars in extended bursts the deduced value of  has to be lower; for the example illustrated here (burst duration $\Delta t=$ 4 Myr) this would correspond to $\mup \sim$ 60 for the Salpeter IMF. It is important to note that the value of  derived in this way represents a [*lower limit*]{}. This is the case since all observational effects known to affect potentially the [H$\beta$]{} equivalent width (loss of photons in slit or leakage, dust inside [H [ii]{}]{} regions, differential extinction, underlying population) can only reduce the observed [W([H]{}) $W({\rm H}\beta)$]{}. The observed [W([H]{}) $W({\rm H}\beta)$]{} represent therefore lower limits when compared to evolutionary synthesis models. We thus conclude that the available [W([H]{}) $W({\rm H}\beta)$]{} measurements in metal-rich [H [ii]{}]{} regions with WR stars yield a [*lower limit of $\mup \ga$ 60–90* ]{} for the upper mass-cut off of the IMF. Larger values of  are not excluded. This result is also compatible with our favoured models presented in Sect.\[s\_models\] (see Fig. \[fig\_wnl\_mod\]). Our new estimate of , based only on a sample of WR regions, provides a more stringent limit than previous studies (SGIT00, BK02). Conclusions {#s_conclude} =========== We have obtained high quality FORS1/VLT optical spectra of 85 disk [H [ii]{}]{} regions in the nearby spiral galaxies NGC 3351, NGC 3521, NGC 4254, NGC 4303, and NGC 4321. This sample, consisting in particular of a good fraction of objects with oxygen abundances presumably above solar (as estimated from $R_{23}$ using the calibration reported by Kobulnicky  1999), provides an unprecedented opportunity to study stellar populations, nebular properties and ISM abundances in [H [ii]{}]{} regions at the high metallicity end. In this first paper we have presented the observational findings on spectral signatures from massive stars, and compared these with evolutionary synthesis models with the main aim of constraining the upper part of the IMF. The average metallicity of our [H [ii]{}]{} region sample is [12 + ([O/H]{}) $12 + \log({\rm O/H})$]{} $\sim 8.9 \pm 0.2$ using the calibration of Kobulnicky  (1999). For 12 regions we are able to determine the electron temperature from the transauroral [\[O [ii]{}\]]{} $\lambda$7325 line, yielding lower limits on O/H (Sect. \[s\_props\]). For 6 regions we have been able to confirm a high metallicity ([12 + ([O/H]{}) $12 + \log({\rm O/H})$]{} $\ga$ 8.8–8.9). Detailed photoionisation modeling will be undertaken in the future to improve our abundance determinations and to include the full sample of [H [ii]{}]{} regions. The spectra of a large number (27) of regions show clear signatures of the presence of Wolf-Rayet (WR) stars as indicated by broad emission in the blue WR bump ($\sim$ 4680 Å). Including previous studies (Castellanos 2001, Bresolin & Kennicutt 2002, Castellanos  2002b) our observations now nearly quadrupel the number of metal-rich [H [ii]{}]{} regions where WR stars are known. Approximately half (14) of the WR regions also show broad [C [iv]{} $\lambda$5808]{} emission attributed to WR stars of the WC subtype. The simultaneous detection of [C [iii]{} $\lambda$5696]{} emission in 8 of them allows us to determine an average late WC subtype ($\sim$ WC7-WC8) compatible with expectations for high metallicities (Sect. \[s\_props\]). Combined with existing observations of WR regions and WR galaxies at sub-solar our data confirm the continuation of previously known trends of increasing WRbump/[H$\beta$]{} intensity with metallicity, establish also such a trend for [W([WR]{}) $W({\rm WR})$]{}, and allow us to estimate the trend of the WC/WN ratio with [12 + ([O/H]{}) $12 + \log({\rm O/H})$]{} in extra-galactic [H [ii]{}]{} regions (Sect. \[s\_props\]) The observed strength of the blue WR bump (relative line intensities and equivalent widths) shows quite clear trends with [W([H]{}) $W({\rm H}\beta)$]{}. Both [W([WR]{}) $W({\rm WR})$]{} and [I([WR]{})/I([H$\beta$]{}) $I({\rm WR})/I({\ifmmode {\rm H}\beta \else H$\beta$\fi})$]{} are found to be smaller than “standard” predictions from state-of-the-art evolutionary synthesis models (Schaerer & Vacca 1998) at corresponding metallicities. Various possibilities (including deviations of the IMF from a Salpeter slope and a “normal” high upper mass cut-off) which could explain this discrepancy have been discussed. The most likely solution is found with an improved prescription to predict the line emission from WN stars in synthesis models (Sect. \[s\_models\]). Using this new prescription the observed WR features are found to be broadly consistent with short bursts and a “standard” Salpeter IMF extending to high masses, as indicated by earlier studies at sub-solar metallicities. Independently of the difficulties encountered to model the WR features in detail, the availability of a fairly large sample of metal-rich WR regions allows us to improve existing estimates (Schaerer  2000, Bresolin & Kennicutt 2002) of the upper mass cut-off of the IMF. Independently of the exact tracks and metallicity we derive a [**lower limit for  of 60–90** ]{} in the case of a Salpeter slope, and larger values for steeper IMF slopes, from the observed maximum [H$\beta$]{} equivalent width of the WR regions. This constitutes a lower limit on  as all observational effects known to affect potentially the [H$\beta$]{} equivalent width (loss of photons in slit or leakage, dust inside [H [ii]{}]{} regions, differential extinction, underlying population) can only reduce the observed [W([H]{}) $W({\rm H}\beta)$]{}. From our probe of the massive star content we therefore conclude that there is at present no direct evidence for systematic variations of the upper mass cut-off of the IMF in metal-rich environments, in contrast to some claims based on indirect nebular diagnostics (e.g. Goldader  1997, Bresolin  1999, Coziol  2001). What the origin of this “universality” of the IMF is, remains an open question. 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Hidayat, (Dordrecht: Kluwer), p. 601 Smith, L.F., Maeder, A., 1991, , 241, 77 Smith, L.J., Norris, R.P.F., Crowther, P.A., 2002, , in press Stasińska, G., 2002, in “Cosmochemistry: the melting pot of elements”, XII Canary Islands Winter School of Astrophysics, in press Stasińska, G., Leitherer, C., 1996, , 107, 661 Thornley, M.D., Forster Schreiber, N.M., Lutz, D., Genzel, R., Spoon, H. W. W., Kunze, D., & Sternberg, A. 2000, , 539, 641 Tremonti, C., Calzetti, D., Leitherer, C., Heckman, T., 2001, , 555, 322 Tully, R.B., 1998, Nearby Galaxies Catalog, Cambridge: Cambridge Univ. Press Vacca, W. D., Garmany, C.D., Shull, M.J., 1996, , 460, 914 Whitford, A.E., 1958, , 63, 201 Zaritsky, D., Kennicutt, R.C., Huchra, J.P., 1994, , 420, 87 [^1]: Based on observations ESO/VLT service observations (65.N-0308 and 67.B-0197) [^2]: We compute $N_{\rm O7V} = Q_0 / 10^{49.12} = L({\ifmmode {\rm H}\alpha \else H$\alpha$\fi}) / (1.36\, 10^{-12} \times 10^{49.12}$ taking the Lyman continuum flux $Q_0$ of an O7V star from Vacca  (1996) and assuming case B recombination for $T_e=$ 10000 K and $n_e =$ 100 cm$^{-3}$. [^3]: It can be seen that [C [iv]{} $\lambda$5808]{} is not formally detected in all regions with [C [iii]{} $\lambda$5696]{} detections. However, in most of these cases (2 exceptions), [C [iv]{} $\lambda$5808]{} is weak, but likely present. [^4]: The few remaining “candidate” WR regions at high $L({\ifmmode {\rm H}\alpha \else H$\alpha$\fi})$ are all located in NGC 4321, have formally a WR detection at $1.4 \la \sigma \la 2$, and appear as rather clear WR detections at visual inspection. The observations of NGC 4321 may be of less good quality due to limited seeing conditions. [^5]: At $Z=$0.04 one has a reduction of [W([WR]{}) $W({\rm WR})$]{} both in the early burst phase (ages $\la$ 3.5 Myr) at $\log {\ifmmode W({\rm H}\beta) \else $W({\rm H}\beta)$\fi}\ \ga$ 1.8 Åand after $\ga$ 6 Myr ($\log {\ifmmode W({\rm H}\beta) \else $W({\rm H}\beta)$\fi}\ \la$ 1.35), while in between the models predict that WC stars dominate the bump (cf. SV98, Figs. 2 and 11). [^6]: The model $L_{\rm cont}$ at young ages ($\la$ 1 Myr) is scaled to the observed value, implying a stellar mass of $\sim 5. (3.3) \times 10^4$  for a Salpeter IMF between 0.8 (2.) and 120 . Accounting for somewhat older ages would yield a slightly larger mass.

--- author: - 'E. Moraux' - 'C. Clarke' title: Kinematics of stars and brown dwarfs at birth --- Introduction ============ So far, most studies of the star formation process have dealt with the formation of stellar systems but with the recent discovery of brown dwarfs in 1995 (Nakajima et al. 1995; Rebolo et al. 1995) new perspectives have opened regarding the formation of condensed objects in molecular clouds. Today more than a hundred brown dwarfs (BDs) in various environments are known. However, their mode of formation is still controversial and the theoretical framework describing the stellar and substellar formation process(es) is not completely satisfactory. Regions of high density ($n(H_{2})\sim 10^{7}$ cm$^{-3}$) in molecular clouds are needed to form proto-[brown dwarfs ]{}but then for their mass to remain substellar their reservoir of gas has to be small or the accretion not very efficient. Two main competing scenarios have been proposed so far to account for the formation of substellar objects. One assumes that brown dwarfs form like solar-mass stars by gravitational collapse of small, dense molecular cloud core and subsequent accretion. The supporting argument is that in the opacity limited regime the Jeans mass can be as low as a few Jupiter masses (Low & Lynden-Bell 1976). The alternative view assumes that brown dwarfs are ejected “stellar embryos” as proposed by Reipurth & Clarke (2001). In this scenario, molecular cloud cores fragment to form unstable protostellar multiple systems which decay dynamically. The lowest mass fragments are ejected from their birth place and deprived of surrounding gas to accrete remain substellar objects. The brown dwarf properties predicted by these two different formation scenarios may in principle be quite different. In the former case, both stars and brown dwarfs form predominantly as single or binary ($N=2$) systems; in this case there are no obvious reasons why properties such as the binary fraction or kinematics should depend on mass. In the latter scenario, by contrast, the dominant formation mechanism (again for both stars and brown dwarfs) is in small $N$ ($>2$) clusters, and the gravitational interplay that precedes the break up of the system into stable entities implies a potentially strong mass dependence for resulting properties like the binary fraction and kinematics. In particular it has been suggested (Reipurth and Clarke 2001) that low mass objects (e.g. brown dwarfs) ejected from such clusters would have a higher velocity dispersion than higher mass objects. Reipurth and Clarke’s initial suggestion - that brown dwarfs in star forming regions may have a detectably higher velocity distribution from stars - has [*not*]{} been borne out by radial velocity studies (Joergens & Guenther 2001). Meanwhile, successive simulations have modified the predictions of small $N$ clusters models. Delgado et al. (2003) and Sterzik and Durisen (2003) have emphasised that, in their simulations, the main difference in velocity dispersion is between single stars and binaries, and that brown dwarfs attain rather larger velocities - with respect to their parent cores - because they are more likely to be ejected as single objects. This dependence of ejection speed on binarity may readily be understood, since one binary is typically formed in each cluster in these simulations: this binary is able to eject the remaining stars from the cluster by sling-shot gravitational encounters, whilst itself remaining close to the center of mass of the natal cluster. In the turbulent fragmentation calculations of Bate, Bonnell and Bromm (2003) and Delgado, Clarke and Bate (2004), by contrast, more than one binary is formed per cluster and so binaries are able to eject each other from the natal cluster. Consequently, in these simulations, the kinematics of the resulting objects do not depend strongly on either mass [*or*]{} binarity. Evidently, the relative kinematics of stars and brown dwarfs and of single stars and binaries can shed some light on the conditions in star forming cores and could ultimately answer the question of whether stars (and brown dwarfs) are formed as isolated single and binary systems, as small $N$ aggregates containing typically one binary or as aggregates containing more than one binary. \[Note that this question is not easy to answer by direct observations, since the timescale for the break up of putative small clusters implies that this process occurs in the deeply embedded phase. However, high resolution imaging of the driving sources of Herbig Haro objects by Reipurth (2000) suggests that the multiplicity of stars in deeply embedded regions is indeed high\]. Direct observations of the kinematics of young stars and brown dwarfs is unlikely to be fruitful however. The differences in velocity dispersion predicted by theoretical models are small (of the order of a km/s). When one bears in mind that these velocities are measured with respect to star forming cores, which are in themselves in relative motion at $\sim1$ km/s, it is unsurprising that the study of Joergens and Guenther (2001) - involving small numbers of objects, with velocity resolution of $\sim0.2$ km/s and rather small dynamic range in mass - did not detect any differences. In this paper, we propose another approach that could potentially detect any mass dependence of the kinematics of stars (and brown dwarfs) at birth. Here we examine the statistical consequences of such an effect on the spatial distribution of stars and brown dwarfs in clusters. This approach has the advantage that one can work with large samples of stars and brown dwarfs, whose positions and masses are known with high accuracy. On the other hand, we cannot predict how initial variations in velocity dispersion affect the spatial distributions of stars and brown dwarfs in a cluster at a given age without further, N-body, modeling. This is partly because two body relaxation leads to mass segregation in older clusters, even in the absence of a mass dependent initial velocity dispersion. Our purpose in this paper, therefore, is to use ‘toy’ N-body models (in which brown dwarfs are introduced with a velocity dispersion that is a variable multiple of the stellar velocity dispersion) in order to establish under what circumstances could one detect a higher velocity dispersion at birth for low mass objects. We stress that this toy models for the kinematics is not supposed to correspond to the outcome of any particular numerical star formation model but is designed to provide a ready parameterization of the problem. We also underline that in no models are any sudden discontinuities in kinematic properties expected at the hydrogen burning mass limit. We use the Pleiades as the testbed for our calculations. This is because the brown dwarf population of the Pleiades has been the subject of intensive scrutiny in recent years (Moraux et al. 2003, Dobbie et al. 2002, Pinfield et al. 2000, Zapatero-Osorio et al. 1999, Bouvier et al. 1998) so that the present day mass function in this cluster is reasonably well constrained. We shall proceed by first placing an upper limit on the initial velocity dispersion of brown dwarfs in the Pleiades, based on the broad similarity between the normalization of stars to brown dwarfs in the Pleiades and in the field, which limits the number of brown dwarfs that can have left the cluster to date. We shall then explore whether the radial distribution of brown dwarfs in the cluster can place meaningful limits on their initial velocity distribution. Numerical simulations ===================== We performed numerical simulations of the dynamical evolution of a Pleiades-like cluster using the code [Nbody2]{} (Aarseth 2001) on a Sun workstation. This code is an algorithm for direct integration of N-body problem based on the neighbour scheme of Ahmad & Cohen (1973) and employs a softened potential $\phi$ of the form $$\phi = -\frac{m}{(r^{2}+\epsilon^{2})^{1/2}}$$ to reduce the effects of close encounters. The model of cluster we used is defined as follows. At time $t=0$ the stellar density $n(r,t)$ conforms to Plummer’s model $$n(r,0) = \frac{3}{4\pi r_{0}^{3}} N \left[ 1 + \left( \frac{r}{r_{0}} \right)^{2} \right]^{-5/2}$$ where $N=1900$ is the number of cluster members and $r_{0}=2.2$ pc is a scale factor determining the dimensions of the cluster. It is related to the half-mass radius $r_{h}$ by $r_{h}\simeq1.3\, r_{0}=2.86$ pc (Aarseth & Fall 1980). This leads to an overall initial central density $n(0,0)=42.6$ objects/pc$^{3}$. Initially, the stellar population is assumed to be in virial equilibrium with a velocity distribution everywhere isotropic (cf. Aarseth, Hénon & Wielen 1974 for a practical scheme for the generation of the initial positions and velocities). Note that the true initial conditions of a cluster are not very well known and are likely to be very complex. Isothermal models are sometimes preferred to describe open cluster density initial states, however Plummer models are also used and have already been proved to reproduce reasonably well the Pleiades cluster (Kroupa et al. 2001). In section \[radii\] we compare our results to observational data and find a reasonable agreement. The system is assumed to be isolated. No external potential is included but any object which reaches the cluster tidal radius would in reality be stripped off by the galactic tide and lost by the cluster. For simplicity and in order to focus on how the initial kinematics affects the the spatial distribution of the cluster population, our model does not include gas. We assume that the gas has already left the cluster when we start the simulations and thus the original cluster may have expanded and lost a large fraction of its primordial members because of the change of the gravitational potential (Adams 2000, Boily & Kroupa 2003a, 2003b). This explains in particular why our initial system is not as concentrated as e.g. the cluster models used by Kroupa et al. (2001) to reproduce the Pleaides. We shall discuss how the presence of gas would affect our results later on. The smoothing length employed for the gravitational interactions on small scales is $\epsilon=5\times10^{-4}\, r_{0}\sim200$ A.U.. To justify the choice of this value, one can estimate the rate of encounters per star closer than $\epsilon$ by $$f = 4\sqrt{\pi}n \left( \sigma_{V}\epsilon^{2} + \frac{Gm\epsilon}{\sigma_{V}} \right)$$ (e.g. Binney & Tremaine 1987) where $n$ is the local stellar density, $\sigma_{V}$ is the velocity dispersion and $m$ is the stellar mass, assumed to be the same for all stars. Assuming $n$ does not change with time, we find the probability for a star located at $r=0$ to have encounters closer than $\epsilon$ is $\sim9$% in 120 Myr – wich is about the age of the [Pleiades ]{}(Stauffer et al. 1998, Martín et al. 1998). Since the stellar density decreases with time, this value is an upper limit indicating $\epsilon\sim200$ A.U. is appropriate for our simulations. The mass of the $N=1900$ objects are distributed over a three power-law mass function $$\xi(m) = \frac{dn}{dm} \propto m^{-\alpha_{p}}\,,\,\, p\in[1..3] \label{imf}$$ with $$\begin{aligned} \alpha_{1} &=& 0.6 \,\,{\textrm{ for }} m_{1,inf}=0.01\le m \le m_{1,sup}=0.3 {M_{\odot}}, \\ \alpha_{2} &=& 1.3 \,\,{\textrm{ for }} m_{2,inf}=0.3\le m \le m_{2,sup}=1.0 {M_{\odot}}, \\ \alpha_{3} &=& 2.3 \,\,{\textrm{ for }} m_{3,inf}=1.0\,\le m \le m_{3,sup}=10.0 {M_{\odot}},\end{aligned}$$ which corresponds to the [Pleiades ]{}mass function determined by Moraux et al. (2003). This MF estimate has not been corrected for binarity which means that this mass function corresponds to the [*system*]{} mass function. Likewise, our simulations do not include any treatment of binarity. Using a softened potential inhibits binary formation and each object issued from this mass distribution is considered as a single object. This choice can be justified by results obtained by de la Fuente Marcos & de la Fuente Marcos (2000) who performed simulations to investigate the dynamical evolution of substellar population in cluster with Aarseth’s [Nbody5]{} code (Aarseth 1985). Some of their models include a population of hard binaries, which are the most important binary stars in term of dynamics, but the results are all very similar (see their fig.2). This suggests that close two body encounters are not too important and that a softened potential $\phi$ with an adequate $\epsilon$ can be used. In practice, the mass of each member $i$ is readily obtained by $$m_{i} = m_{p,sup}^{-(\alpha_{p}-1)} - (i-1) g_{N_{p}}$$ with $$g_{N_{p}} = (m_{p,sup}^{-(\alpha_{p}-1)} - m_{p,inf}^{-(\alpha_{p}-1)})) / (N_{p} - 1)$$ where $N_{p}$ is the number of objects having a mass between $m_{p,inf}$ and $m_{p,sup}$. A convenient representation for this distribution is the mass-generating function $$m(X) = \left\{ \begin{array}{ll} \frac{0.3}{[1-1.35(X-0.55)]^{-2.5}} & \textrm{if } 0<X\le 0.55,\\ \frac{1}{[1-1.45(X-0.85)]^{3.33}} & \textrm{if } 0.55<X\le 0.85,\\ \frac{0.24}{[1-0.99X]^{0.77}} & \textrm{if } 0.85<X\le 1,\\ \end{array} \right.$$ where $X$ is distributed uniformly in \[0,1\]. The intervals here are the same as the mass intervals in Eq. (\[imf\]). This function has been computed in the way described in Kroupa et al. (1991). In such a model, [brown dwarfs ]{}($0.01\le m\le 0.08{M_{\odot}}$) constitute 25% by number but only less than 3% by mass. Initially these proportions are the same everywhere in the cluster, i.e. there is no mass segregation, so that stellar and substellar objects have the same radial distributions. The initial density of stars and brown dwarfs in the center are $n_{\star}(0,0)=32$ and $n_{\rm{BD}}=10.6$ objects/pc$^{3}$ respectively. In our toy model we assume that substellar objects represent a peculiar population having its own velocity dispersion $\sigma_{V_{\rm{BD}}}$. We arbitrarily choose $$\sigma_{V_{\rm{BD}}}=k\times\sigma_{V_{\star}}$$ initially where $k\in [1.0-3.0]$ and $\sigma_{V_{\star}}$ is the stellar velocity dispersion. Then we let the cluster evolve dynamically under the effect of gravitational interactions between members over 12 crossing times. Since $t_{cr}\simeq 10$ Myr for the Pleiades (Pinfield et al. 1998), this corresponds to about the age of the cluster, i.e. 120 Myr. Thus, we can follow the evolution of the star and [brown dwarf ]{}population within the cluster from their birth to the age of the Pleiades depending on their initial kinematics. Note that we only present the results for one set of initial conditions but we also performed other simulations using different numbers of objects $N$, which means in particular using various seed numbers to initialize the cluster model. The results are similar to those described in the following sections. Results and discussion {#results} ====================== Characteristic radii {#radii} -------------------- Over the timescale of the simulations the half mass radius does not change much through the simulations. It goes up to $\sim3.2-3.4$ pc after 12 crossing times which is consistent with observational results obtained for the Pleiades ($r_{h}=3.6$ pc; Pinfield et al. 1998). This result does not depend on the initial substellar velocity dispersion which is indeed expected considering the fact that [brown dwarf ]{}represent only 3% of the cluster mass. A nominal core radius $r_{c}$ is calculated in [Nbody2]{} using the definition of the density radius given by Casertano & Hut (1985) slightly modified in order to obtain a convergent result using a smaller sample ($n\simeq N/2$). It is determined by the rms expression (Aarseth 2001) $$r_{c}= \left( \frac{\sum_{i=1}^{n} |\mathbf{r}_{i} - \mathbf{r}_{d}|^{2} \, \rho_{i}^{2}}{\sum \rho_{i}^{2}} \right)^{1/2},$$ where $\mathbf{r}_{i}$ is the three-dimensional position vector of the $i$th star and $\mathbf{r}_{d}$ denotes the coordinates of the density centre. The density estimator $\rho_{i}=3\,M_{5}/(4\pi r_{6}^{3})$ is defined with respect to the sixth nearest particle $r_{6}$ and takes into account the total mass ($M_{5}$) of the five nearest neighbours. \[See Casertano & Hut (1985) for definitions of density centre and density estimator.\] In our simulations, $r_{c}=1.3$ pc at $t=0$ and $r_{c}\sim0.8$ pc at $t=12\,t_{cr}\simeq 120$ Myr. However, to be compared to the observational core radius $R_{c}$ these values have to be divided by $\sim0.8$ (Heggie & Aarseth 1992) and we obtain $R_{c}$ between 1.6 and 1 pc. For the Pleiades, Raboud & Mermilliod (1998) found $R_{c}=0.6\degr=1.3$ pc for a cluster distance of 125 pc which is reasonably consistent with our model. We also compare the projected radial density profiles obtained at $t\simeq 120$ Myr to the observed Pleiades profiles for $0.1\le m \le 0.3{M_{\odot}}$ (see Fig. \[density\]) and we find a reasonable agreement. ![The projected radial density profiles for $0.1\le m\le3{M_{\odot}}$. The different thin curves correspond to different values of $\sigma_{V_{\rm{BD}}}=k\times\sigma_{V_{\star}}$ with $k\in [1.0-3.0]$, the symbols are the same than in Fig. \[cumul120\]. The thick curve correspond to the best-fitting King profile of observational data for the Pleiades from Pinfield et al. (1998, their fig. 2.)[]{data-label="density"}](1439fig1.ps){width="45.00000%"} Dynamical evolution of the substellar population ------------------------------------------------ Figure \[cumul120\] illustrates the effect of the initial velocity dispersion of [brown dwarfs ]{}on their spatial distribution at the age of the [Pleiades ]{}cluster ($\sim 120$ Myr). The vertical dashed line corresponds to the tidal radius $r_{t}$ of the [Pleiades ]{}($\sim 13$ pc, Pinfield et al. 1998). All the objects located at larger radii are in reality lost by the cluster. ![Cumulative radial distribution of [brown dwarfs ]{}after $\sim 120$ Myr calculated by our [Nbody2]{} numerical simulations of the dynamical evolution of a Pleiades-like cluster. The different curves correspond to the different values of the substellar initial velocity dispersion, $\sigma_{V_{\rm{BD}}} = k \times \sigma_{V_{\star}}$ with $k\in [1.0-3.0]$ ($k$ increases from top to bottom). The vertical dashed line represents the extend of the Pleiades cluster.[]{data-label="cumul120"}](1439fig2.ps){width="45.00000%"} If the stellar and substellar initial velocity dispersion are similar ($k=1.0$), then as many [brown dwarfs ]{}as low mass stars have been lost after 120 Myr (about 10%). This result is perfectly consistent with the simulations performed by de la Fuente Marcos & de la Fuente Marcos (2000), which indicates in particular that our choice of the softening parameter $\epsilon$ is adequate. Like the stars, these brown dwarfs have diffused as a result of successive two body interactions to beyond the notional tidal radius. However, as soon as $k\ge2.0$ then 40% or more of the initial [brown dwarf ]{}population have left the cluster. If it does happen, this loss of [brown dwarfs ]{}occurs quite quickly, in a couple of crossing times, i.e. a few tens of Myr. We note from Fig. \[cumul20\] that the depletion rates obtained after only 20 Myr are already important and of the same order as those obtained after 120 Myr. This means in particular that our results do not depend much on the age of the cluster and are not affected by the errors in open cluster age determinations, typically in the range of 30 to 50%. This can be explained by the fact that if an object has a velocity larger than the cluster escape velocity $v_{esc}$ it will be lost after a few crossing times $t_{cr}$. The depletion rates found at 20 Myr correspond indeed to about the fraction of [brown dwarfs ]{}initially unbound in our Plummer distribution. For our Pleiades-like model, we have $t_{cr}\simeq10$ Myr and $v_{esc}=\sqrt{2}v_{vir}=1.6$ km/s, with the virialised velocity $v_{vir}=1.15$ km/s. Thus, if [brown dwarfs ]{}form following the ejection scenario proposed by Reipurth & Clarke (2001) with a resulting average velocity larger than the cluster escape velocity or with a velocity dispersion larger than a few km/s, then a large number of substellar objects will be lost relatively quickly. ![Cumulative radial distribution of [brown dwarfs ]{}after $\sim$ 20 Myr for different values of initial substellar velocity dispersions $\sigma_{V_{\rm{BD}}}=k\times\sigma_{V_{\star}}, k\in [1.0-3.0]$.[]{data-label="cumul20"}](1439fig3.ps){width="45.00000%"} The fact that basically all the initially unbound objects are lost by the cluster is quite straightforward and constitutes a robust result that does not depend much on the initial conditions, like the IMF or the initial spatial distribution. Evidently the actual fractions of stars and brown dwarfs that would be bound at the start of our simulations would partly depend on the prior dynamical history of the cluster when it contained significant quantities of gas, an issue that is beyond the scope of the present paper. We note however that changes in the cluster potential due to gas loss would not affect the kinematic properties of the stars and brown dwarfs [*differentially*]{} and in this study we focus on the possible observable consequences of stars and brown dwarfs having different kinematical properties at birth. However, an important observational constraint is that the [Pleiades ]{}mass function is found to be similar to those of several star forming regions (where the gas is still present), such as the Trapezium (Muench et al. 2002) or IC 348 (e.g. Luhman et al. 2003), and of the field (cf. Moraux et al. 2003). This indicates that it is still representative of its initial mass function and that the dynamical effects did not affect the shape of the IMF in 120 Myr. This means in particular that there was [*no*]{} significant preferential escape of [brown dwarfs ]{}in the Pleiades. Therefore, the substellar initial velocity dispersion must be less than twice the stellar velocity dispersion and cannot exceed a few km/s. Radial distribution ------------------- As discussed above, the vast majority of objects has a typical velocity of a few km/s and remains bound to Pleiades-like clusters after $\sim100$ Myr. However, if [brown dwarfs ]{}still have a larger dispersion velocity than that of stars, even if only slighty larger, we may hope to find a signature of their ejection by looking at their radial distribution. Figure \[cumul\_cut120\] shows the effect of the initial velocity dispersion on the [brown dwarf ]{}radial distribution for a Pleiades-like cluster after $12 t_{cr} \sim 120$ Myr. The number of objects inside a sphere having a radius $R=13$ pc has been normalised to one. The radius has been chosen to correspond to the Pleiades tidal radius so that the shown radial distributions represent what an observer would obtain if he studied the substellar population in the cluster. ![Cumulative radial distribution of [brown dwarfs ]{}inside the cluster after $12 t_{cr}\sim 120$ Myr for several initial substellar velocity dispersions $\sigma_{V_{\rm{BD}}}=k\times\sigma_{V_{\star}}, k\in [1.0-3.0]$.[]{data-label="cumul_cut120"}](1439fig4.ps){width="45.00000%"} Strikingly, we find that at the age of the Pleiades, the spatial distribution of brown dwarfs in the cluster provides [*no*]{} information about the velocity dispersion of brown dwarfs at birth. This is because those brown dwarfs whose velocity exceeds the escape velocity of the cluster will have already left the cluster (on a crossing timescale $\sim 10$ Myr) and those that remain have a velocity distribution (and hence spatial distribution) that is very similar to that of the low mass stars. In order to find possible evidence of a population of brown dwarfs with high velocities at birth, it is instead necessary to examine clusters that are only about a crossing time old. This expectation is borne out by Figures \[cumul\_cut10\] and \[bd\_star10\] which compare the normalised distributions of stars and brown dwarfs within a Pleiades-like cluster at an age of $10$ Myr for various ratios of $\sigma_{V_{\rm{BD}}}$ to $\sigma_{V_{\star}}$. Evidently, at this age, two-body relaxation is ineffective in producing a more diffuse brown dwarf distribution (as evidenced by the fact that the brown dwarf and stellar distributions are very similar at this age if $\sigma_{V_{\rm{BD}}} = \sigma_{V_{\star}}$). Thus any differences in the spatial distribution of stars and brown dwarfs at such a young age would be indicative of different velocity distributions at birth. ![Cumulative radial distribution of [brown dwarfs ]{}inside the cluster at $t=1 t_{cr}\sim 10$ Myr for several initial velocity dispersions $\sigma_{V_{\rm{BD}}}=k\times\sigma_{V_{\star}}, k \in [1.0-3.0]$.[]{data-label="cumul_cut10"}](1439fig5.ps){width="45.00000%"} Figure \[cumul\_cut10\] shows the effect of the initial velocity dispersion on the [brown dwarf ]{}radial distribution for a Pleiades-like cluster after $1 t_{cr} \sim 10$ Myr. We note that as soon as the [brown dwarf ]{}velocity dispersion is larger than that of stars the substellar radial distributions are different from the one obtained for $\sigma_{V_{\rm{BD}}}=\sigma_{V_{\star}}$. Figure \[bd\_star10\] compares the spatial distribution of low mass stars to that of brown dwarfs for two different values of $\sigma_{V_{\rm{BD}}}$ after 10 Myr. When the stellar and substellar velocity dispersions are similar then the radial distributions are also the same, whereas this is not the case for $\sigma_{V_{\rm{BD}}}=2\times \sigma_{V_{\star}}$. In that case about 90% of the low mass stars ($0.08\le m\le 0.5{M_{\odot}}$) are located at less than 5 pc from the cluster center compared with only 65% of the [brown dwarfs ]{}at the same location. We have to reach a radius of $\sim 9$ pc to include 90% of the substellar population. ![Radial distribution of low mass stars ($0.08-0.5{M_{\odot}}$; solid line) and brown dwarfs (dashed lines) inside the cluster after $1t_{cr}\sim 10$ Myr. The long-dashed line corresponds to $\sigma_{V_{\rm{BD}}}=\sigma_{V_{\star}}$ and the short-dashed line to $\sigma_{V_{\rm{BD}}}=2\times\sigma_{V_{\star}}$.[]{data-label="bd_star10"}](1439fig6.ps){width=".45\textwidth"} In the presence of gas, the results would be similar. One would expect gas expulsion due to e.g. photoionisation or stellar winds to occur within a few Myr, i.e. on a timescale less than the crossing timescale of the cluster, and for the stars and brown dwarfs to respond to such gas loss on about a crossing timescale. If brown dwarfs have a larger velocity dispersion than stars, we therefore expect to find a signature of this initial kinematics in studying the spatial distribution of stars and brown dwarfs in clusters that are about a crossing timescale old. Conclusion ========== We have shown that the observed similarity between the brown dwarf to star ratio in the Pleiades, in star forming regions and in the field implies that the velocity dispersion of brown dwarfs at birth cannot exceed $2\times$ the stellar velocity dispersion in the Pleiades. This imposes an upper limit on the brown dwarf velocity dispersion at birth of a few km/s. Thus either brown dwarfs are not dynamically ejected from their natal cores or else the ejection velocities with respect to their natal cores is low. Such a velocity dispersion limit would rule out the common incidence of very hard encounters (i.e. at $<$ 4 A.U., Delgado et al. 2004) in the natal cores, but would be consistent with the results of current hydrodynamical simulations (Delgado et al. 2003, Bate et al. 2003). We stress that the velocities quoted here are limits on the 3D velocity dispersion for a Maxwellian distribution and so do not necessarily rule out a tail of objects with significantly higher velocities (such, for example, as the (not substellar) object PV Ceph, for which a velocity of $20$ km/s has recently been claimed (Goodman & Arce 2003). We have shown that the radial density profile of brown dwarfs and stars in the Pleiades provides [*no*]{} information about the velocity dispersion of brown dwarfs at birth. This is because any brown dwarfs with velocities greater than the escape velocity of the cluster would have long ago escaped the cluster and thus the residual brown dwarf population has a velocity distribution - and hence spatial distribution - that is very similar to the stars. We therefore conclude that if we seek [*positive*]{} evidence for a modestly higher velocity distribution of brown dwarfs at birth, we need to look at clusters that are significantly different from the Pleiades. There are two possibilities here. Firstly, the upper limits we have derived could still imply a significant preferential loss of brown dwarfs from loosely bound regions. For example, Preibisch et al. (2003) suggested that this could be the cause of the small fraction of [brown dwarfs ]{}observed in IC348 for which the escape velocity is only $\sim 0.8$ km/s (Herbig 1998). Kroupa & Bouvier (2003) found also that the ejection mechanism could explain a part of the deficit of substellar objects in the Taurus association. Secondly, one might seek evidence of high velocity brown dwarfs at birth by instead examining clusters that are about a crossing timescale old. In this case, the brown dwarfs would be expected to have a more extended distribution than the stars, which could [*not*]{}, at such a young age, be confused with the signature of two body relaxation. Even if the brown dwarf velocity dispersion exceeds that of stars by only 50%, the spatial distributions of the two populations are quite different at this age (contrast dashed lines in Fig. \[cumul\_cut10\] with solid line in Fig. \[bd\_star10\]) Thus brown dwarf surveys in clusters of various ages and degrees of richness will afford further opportunities to constrain the velocities of brown dwarfs at birth. We thank S. Aarseth for allowing us access to his N-body codes. 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--- abstract: | We propose to describe higher spins as invariant subspaces of the Casimir operators of the Poincaré Group, $P^{2}$, and the squared Pauli-Lubanski operator, $W^{2}$, in a properly chosen representation, $\psi (\mathbf{\mathbf{p}})$ (in momentum space), of the Homogeneous Lorentz Group. The resulting equation of motion for any field with $s\neq0$ is then just a specific combination of the respective covariant projectors. We couple minimally electromagnetism to this equation and show that the corresponding wave fronts of the classical solutions propagate causally. Furthermore, for $(s,0)\oplus(0,s)$ representations, the formalism predicts the correct gyromagnetic factor, $g_{s}=\frac{1}{s}$. The advocated method allows to describe any higher spin without auxiliary conditions and by one covariant matrix equation alone. This master equation is only quadratic in the momenta and its dimensionality is that of $\psi(\mathbf{\mathbf{p}})$. We prove that the suggested master equation avoids the Velo-Zwanziger problem of superluminal propagation of higher spin waves and points toward a consistent description of higher spin quantum fields. author: - Mauro Napsuciale - Mariana Kirchbach title: 'Avoiding superluminal propagation of higher spin waves via projectors onto $W^{2}$ invariant subspaces. ' --- Introduction. ============= The field theoretical description of interacting particles with spin $>1$ is a long standing problem. The interaction of a spin $\frac{3}{2}$ Rarita-Schwinger (RS) field minimally coupled to an external electromagnetic field was shown to be inconsistent more than forty years ago [@sudarshan1]. Later on, Velo and Zwanziger observed superluminal propagation of the RS wave front in the presence of a minimally coupled electromagnetic field [@VZ1] and studied also the conditions under which the Proca field interacting with an external electromagnetic field propagates causally [@VZ2]. After these works many authors have addressed above problem from different perspectives and for different interactions [@todos] and the general feeling seems to be that it is not possible to construct a consistent quantum theory for massive particles with $s >$ 1. At several decades of distance in looking afresh onto the equations of motion can lead to different understanding of this fundamental problem. Weinberg emphasizes in his textbook on quantum field theory [@Weinberg:mt] that the equation of motion satisfied by the Dirac field is nothing but the record about the way how one puts together the two irreducible representations, (1/2,0), and (0,1/2), of the proper orthochronous Lorentz group to form a field that transforms invariantly under parity. In a wider understanding, this means that the equations of motion satisfied by a field are just a consequence of the properties of the representations of the Homogeneous Lorentz Group (HLG) chosen by us to accommodate the field and the discrete symmetries we require to be realized in this space. Closely related arguments can be found, among others in [@WKT], [@ryder], [@MK97], and [@prinind]. More recently, Refs. [@MK03; @Gaby] studied covariant projectors onto invariant subspaces of the squared Pauli-Lubanski operator in the representation space of the four-vector–spinor and showed that the associated equations are free from the Velo-Zwanziger problem. The corresponding projectors for the $(s,0)\oplus(0,s)$ representation space were studied in [@MC] where it was shown that under minimal coupling a particle in this representation has the correct value for the spin gyromagnetic factor, $g_{s}=\frac{1}{s}$, thus proving Belinfante’s conjecture [@belinfante] from 1953. In this work we explore the projectors onto the invariant subspaces of the Poincaré Casimir operators, the squared four-momentum and thesquared Pauli-Lubanski operator, for any $s$, and study propagation of the corresponding wave fronts along the lines of Refs. [@VZ1; @VZ2]. The paper is organized as follows. In the next Section we recall in brief current description of higher spins and its relation to the Poincaré group. In Section III we suggest to describe higher spins as invariant subspaces of the Poincaré Casimirs. In Section IV we show that particles within this framework propagate causally in the presence of an electromagnetic field, thus avoiding the classical Velo-Zwanziger problem. The paper closes with a brief Summary. Current description of fields and its relation to Poincaré group representations. ================================================================================= The primary classification of elementary systems is usually done by identifying them (up to form factors) with the irreducible representations (irreps) of the Poincaré group ($PG$). If so, then one necessarily has to consider particles as invariant spaces of the Casimir operators of this group– the squared four-momentum $P^{2}$, on the one side, and the squared Pauli-Lubanski operator $W^{2}$, on the other side and label them by their respective eigenvalues, $p^{2}$, and $-p^{2}s(s+1)$, as $|p^{2},s(s+1)>$. Further quantum numbers can be associated with the Casimir invariants of the underlying Homogeneous Lorentz Group (HLG), $SO(1,3)$, and are approached by the reduction chain $PG\supset SO(1,3)$. For finite dimensional representations, the Casimir invariants of $SO(1,3)$ are frequently expressed in terms of two $SU(2)$ Casimirs, in turn denoted by $\mbox{\bf S}_{L}^{2}$, and $\mbox{\bf S}_{R}^{2}$, of $SU(2)_{L}\otimes SU(2)_{R}$, a group that is locally isomorphic to $SL(2,C)$, the universal covering of HLG. The two additional quantum labels gained in this manner are the well known left– and right handed “angular momenta”, $s_{L}$, and $s_{R}$, respectively. Therefore, a covariant state labeling can be introduced as: $|p^{2},s(s+1);s_{L},s_{R}>$, with $s=|s_{L}-s_{R}|,...,s_{L}+s_{R}$. In so doing one encounters essentially two types of finite dimensional HLG representations. 1. The first ones contain just one $W^{2}$ invariant subspace, and correspond to the case when one of the $s_{L}$, $s_{R}$ labels vanishes (i.e. either $(s_{L},0)$, or $(0,s_{R})$), and $s_{R}=s_{L}$. In such a case, $s_{L/R}(s_{L/R}+1)=s(s+1)$, equals the $\left( -\frac{1}{m^{2}}W^{2}\right) $ eigenvalue in the space under consideration (see Eq. (\[W2\_rest\]) below) and $W^{2}$ – and $\mbox{\bf S}_{L/R}^{2}$ invariant spaces coincide. Irreps of the above type are suggestive of replacing $W^{2}$– by $SU(2)$ spin labels. As long as the basic fields in physics are precisely of the above type (the Dirac field is $(1/2,0)\oplus(0,1/2)$, the electromagnetic field strength tensor is $(1,0)\oplus(0,1)$, and scalars are just $(0,0)$) identifying Poincaré labels with $SU(2)$ spins works out without any harm. 2. The second ones are HLG irreps containing several $W^{2}$ invariant subspaces. In this case, both $s_{L}$, and $s_{R}$ are non-vanishing, and the irreps are of the type $\left( s_{L},s_{R}\right) $ with $s_{L}\not =0$, and $s_{R}\not =0$. Examples are the vector–, and tensor gauge fields, $(1/2,1/2)$, and $(1,1)$, respectively. In the rest frame, $W^{2}=-\frac {1}{m^{2}}\mbox{ \bf S}^{2}$ hence $W^{2}$ and $\mbox{ \bf S}^{2}$ invariant sub-spaces coincide. However, beyond rest frame, in flight, $W^{2}$ and $\mbox{\bf S}^{2}$ invariant sub-spaces are no longer identical, a situation caused by the property of the boost to mix up SU(2) spins differing by one unit. Often, Lorentz representations that contain as building blocks irreps of the second type, appear attractive for the description of higher spins, the classical examples being the totally symmetric $K$ rank Lorentz tensors with Dirac spinor components, generically denoted by $\psi_{\mu_{1}...\mu_{K}}$. They are exploited for the description of fields that have been labeled in the rest frame by the highest spin $J=K+1/2$. The separation between Lorentz and spinor indices inherent to such tensors makes them especially appealing for the construction of covariant fermion-boson vertices. However, one has to face the problem how to pick up the favored degrees of freedom and exclude interference with the unwanted ones. It seems inevitable to return back to the Poincaré invariants, if one wishes to distinguish all the degrees of freedom contained in $\psi_{\mu_{1}...\mu_{K}}$ in a covariant and transitionally invariant fashion. Yet, for one reason or the other, this is not the path tenaciously pursued by the theory. Rather, one still prefers to stay within the elaborated scheme of substituting $W^{2}$ by $SU(2)$ labels, but, yes, modify the latter scheme to account for the new situation in introducing constraints, considered as appropriate. To be specific, in order to select out of $\psi_{\mu}$ (a field belonging to $[(1/2,0)\oplus(0,1/2)]\otimes(1/2,1/2)$) the $W^{2}$ invariant subspace that relates to spin $3/2$ at rest, one requires $$\begin{aligned} (i\partial^{\mu}\gamma_{\mu}-m)\psi_{\mu} & =0\,,\nonumber\\ \partial^{\mu}\psi_{\mu} & =0\,,\nonumber\\ \gamma^{\mu}\psi_{\mu} & =0\,. \label{RS_set}$$ Exploiting constraints (some times termed to as auxiliary, or, supplementary, conditions) in place of $W^{2}$ quantum numbers brings the advantage to remain within the framework of equations linear in the momenta, and to work with four-dimensional Dirac spinors. However, these advantages reveal themselves as deceptive at the moment one has to face grave worries about compatibility of constraints and dynamics. Recall, that the constraints change upon gauging and one has to make sure that the modification is preserved in time by the equation of motion and the latter does not violate causality. Notice that covariance alone is indeed a necessary but not a sufficient condition for special relativity. For example, space-like intervals are doubtlessly covariant objects, but they are unacceptable for the description of *free* physical fields as they prescribe the particle to violate causality during propagation. Precisely a flaw of that very type was revealed by Velo and Zwanziger in Ref. [@VZ1] regarding the $\gamma^{\mu}\psi_{\mu }=0$ constraint onto the four–vector spinor. Velo and Zwanziger showed that above constraint triggers acausal propagation of Rarita-Schwinger particles crossing an electromagnetic field. In the present article we shall avoid above inconsistencies in developing a different view on form and content of wave equations for higher spins. Namely, we take the position that the equation of motion for whatever free particle has to be (i) a function of $P^{2}$ and $W^{2}$, the Casimir invariants of the Poincaré group, (ii) operates immediate, i.e. without any supplementary constraints, on the HLG representation chosen to embed the field as one of its covariant sectors[^1]. Within this context, there are two primordial equations of motion to be satisfied by any field. One of them searches for $P^{2}$ invariant subspaces. It is nothing more but the Klein-Gordon equation. The other one secures in addition invariance under pseudo–rotations and pins-down $W^{2}$ invariant subspaces by means of appropriately constructed covariant projectors. It is that very latter type of equations on which we focus attention here. For the sake of self-sufficiency of the presentation, the subsequent Section opens with a brief review of the basics of space-time symmetries. Covariant wave equations for higher spins from $W^{2}$ invariant subspaces. ============================================================================ Basics of Space-Time Transformations. ------------------------------------- A general Poincaré transformation in space time can be written in the factorized form $$g(b,\Lambda)=T(b)\ \Lambda\, ,$$ where $T(b)=g(b,E)$ (E denotes the unit matrix) is a translation and  $\Lambda=g(0,\Lambda)$ is a proper Lorentz transformation. In the standard convention, the generators of the translation group in 1+3 time-space dimensions, $\mathcal{T}$ $_{1,3}$ , are $P_{\mu}$ in $T(b)$, which are commuting, $$\qquad\left[ P_{\mu},P_{\nu}\right] =0.$$ The HLG transformation in coordinate space, $$x_{\mu}^{\prime} =\Lambda_{\mu}^{\quad\nu}\quad x_{\nu},\quad\Lambda_{\mu }^{\quad\nu} =\exp\left[ -\frac{i}{2}\theta^{\mu\nu}L_{\mu\nu}\right] \, ,\quad L_{\mu\nu}=X_{\mu}P_{\nu}-X_{\nu}P_{\mu}\, ,$$ induces the following transformation for a field $\psi(x)$, $$\psi^{\prime}(x)=\exp\left[ -\frac{i}{2}\theta^{\mu\nu}M_{\mu\nu}\right] \psi(\Lambda^{-1}x).$$ Here, $\theta^{\mu\nu}$are continuous parameters, while the $n\times n$ matrices $M_{\mu\nu}$ represent a totally antisymmetric 2nd rank Lorentz tensor. They are the generators of the homogeneous Lorentz group in the representation space of interest, and satisfy the commutation relations of the associated algebra :$$\left[ M_{\mu\nu},M_{\alpha\beta}\right] =-i(g_{\mu\alpha}M_{\nu\beta }-g_{\mu\beta}M_{\nu\alpha}+g_{\nu\beta}M_{\mu\alpha}-g_{\nu\alpha}M_{\mu \beta}). \label{M_comm}$$ Their commutators with the generators of the translation group read $$\left[ P_{\mu},M_{\alpha\beta}\right] =i(g_{\mu\alpha}P_{\beta}-g_{\mu\beta }P_{\alpha}), \label{M_P_comm}$$ where $g_{\mu\nu}=diag(1,-1,-1,-1)$ is the metric tensor. The $M_{\mu\nu}$ generators consist of $$M_{\mu\nu}=L_{\mu\nu}+S_{\mu\nu},\qquad\left[ L_{\mu\nu},S_{\mu\nu}\right] =0,$$ where $L_{\mu\nu}$, and $S_{\mu\nu}$ in turn generate rotations in external coordinate– and internal representation spaces. The generators of boosts ($\mathcal{K}_{x},\mathcal{K}_{y},\mathcal{K}_{z}$) and rotations ($J_{x},J_{y},J_{z}$) are related to $M_{\mu\nu}$ via $$\mathcal{K}_{i}=M_{0i}\ ,\qquad J_{i}=\frac{1}{2}\epsilon_{ijk}M_{jk}\ ,$$ respectively. Pauli Lubanski Vector and Associated Casimir Invariant. ------------------------------------------------------- The Pauli–Lubanski (PL) vector is now defined as $$W_{\mu}=-\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}M^{\nu\alpha}P^{\beta}, \label{paulu}$$ where $\epsilon_{0123}=1$. This operator can be shown to satisfy the commutators $$\lbrack W_{\alpha},M_{\mu\nu}]=i(g_{\alpha\mu}W_{\nu}-g_{\alpha\nu}W_{\mu }),\qquad\lbrack W_{\alpha},P_{\mu}]=0, \label{conmrelpl}$$ i.e. it transforms as a four-vector under Lorentz transformations. The remarkable point is that the external coordinate part of $M_{\mu\nu}$, namely the “orbital” momentum  $L_{\mu\nu}$, does not contribute to $W_{\mu}$ due to the anti-symmetric Levi-Civita tensor. As a result, $W_{\mu}$ restricts to $$W_{\mu}=-{\frac{1}{2}}\epsilon_{\mu\nu\rho\tau}S^{\nu\rho}P^{\tau}\,, \label{PauLu1}$$ and its squared (in covariant form) is calculated to be $$W^{2}=-{\frac{1}{2}}S_{\mu\nu}S^{\mu\nu}P^{2}+G^{2}\,,\quad G_{\mu}:=S_{\mu \nu}P^{\nu}\,. \label{paulu2}$$ The operators $S_{\mu\nu}$ act exclusively in the internal spin space and commute like $$\left[ S_{\mu\nu},S_{\alpha\beta}\right] =-i(g_{\mu\alpha}S_{\nu\beta }-g_{\mu\beta}S_{\nu\alpha}+g_{\nu\beta}S_{\mu\alpha}-g_{\nu\alpha}S_{\mu \beta})\,. \label{S_comm}$$ As long as Eq. (\[S\_comm\]) has same form as Eq. (\[M\_comm\]), one may view $S_{\mu\nu}$ as generators of Lorentz transformations in the intrinsic space. However, in contrast to Eq. (\[M\_P\_comm\]), $S_{\mu\nu}$ *commute* with the operators of translations $$\quad\left[ P_{\alpha},S_{\mu\nu}\right] =0\,. \label{Poinc_contr}$$ In effect, one does not find precisely Poincaré transformations in the internal space but rather a contracted form of them. Hereafter we will refer to the group generated by $S_{\mu\nu}$ as the Internal Homogeneous Lorentz Group ($\mathcal{I}$HLG) to distinguish it from the HLG spanned by $M_{\mu\nu}$. In summary, one can write down generators of boosts and rotations in the internal space as $$K_{i}=S_{0i},\qquad\ S_{i}=\frac{1}{2}\epsilon_{ijk}S_{jk}\,.$$ The internal HLG has by itself two Casimir invariants, in turn given by $C_{1}={\frac{1}{4}}S_{\mu\nu}S^{\mu\nu}$, and $C_{2}=$ $S_{\mu\nu}\widetilde{S}^{\mu\nu}$, with $\widetilde{S}_{\mu\nu}=\epsilon_{\mu\nu\rho \tau}S^{\rho\tau}$. In terms of $\mbox{\bf K}$, and $\mbox{\bf S}$ one finds $$C_{1}=\frac{1}{2}(\mathbf{S}^{2}-\mathbf{K}^{2})\,,\quad C_{2}=i\mathbf{S\cdot K}\,. \label{casimirs}$$ The latter equation allows to cast $W^{2}$ into the form $$W^{2}=-2C_{1}P^{2}+G^{2}\ . \label{paso_1}$$ For irreps of the type $(s,0)\oplus(0,s)$ where $K_{i}=\mp iS_{i}$, one finds the insightful relation [@Gaby] $$G^{2}=-W^{2}\,. \label{P:_s0_0s}$$ Insertion of Eq. (\[P:\_s0\_0s\]) into Eq. (\[paso\_1\]) amounts to $$W^{2}=-\mathbf{S}^{2}P^{2}\,. \label{W2_rest}$$ The latter relation explains the privileged position of $(s,0)\oplus(0,s)$ states to carry unique SU(2) spin both at rest (where $W^{2}$ any way reduces to $-\mathbf{\mathbf{S}}^{2}\,m^{2}$ in accord with Eq. (\[W2\_rest\])) and in flight. However, for all the other types of Lorentz representations, $W^{2}\not =G^{2}$ and the $\left( -\frac{1}{m^{2}}W^{2}\right) $ labels for particles in flight do not have the interpretation of ordinary $SU(2)$ spin. In the following we label $W^{2}$ invariant sub-spaces by $s$ but in general without any reference to SU(2) spin. Covariant projectors onto $W^{2}$ invariant subspaces. ------------------------------------------------------ To begin with we recall that the interpretation of elementary systems as Poincaré group irreducible representations requires any field to transform invariantly under the action of  both $P^{2}$ and $W^{2}$. In the following we work with massive fields. The former invariance leads to the Klein-Gordon equation for any arbitrary field $$\left( P^{2}-m^{2}\right) \psi(\mathbf{\mathbf{p}})=0\,. \label{kl-gr}$$ Invariance under the action of $W^{2}$ results into the new condition$$\Pi^{s}(\mathbf{\mathbf{p}})\psi(\mathbf{\mathbf{p}})=\psi(\mathbf{\mathbf{p}})\,, \label{dummy}$$ where $\Pi^{s}(\mathbf{\mathbf{p}})$ stands for an appropriately constructed covariant projector onto the $\left( -p^{2}s(s+1)\right) $ invariant subspace of $W^{2}$ in $\psi(\mathbf{\mathbf{p}})$. To be specific, for the case of the four-vector spinor, such projectors have been presented in Ref. [@MK03]. In general, equations of the type (\[dummy\]) are equivalent to $$\left[ W^{2}+P^{2}s(s+1)\right] \ \psi(\mathbf{\mathbf{p}})=0. \label{master}$$ Next, it is necessary to account for the mass shell condition in Eq. (\[kl-gr\]). For this purpose, we sum up Eqs. (\[master\]) and (\[kl-gr\]) to obtain$$\left[ \frac{1}{s}W^{2}+sP^{2}+m^{2}\right] \ \psi(\mathbf{\mathbf{p}})=0, \label{master1}$$ and cast the latter equation into the explicitly covariant form$$\left[ t_{\mu\nu}P^{\mu}P^{\nu}-m^{2}\right] \psi(\mathbf{\mathbf{p}})=0\,. \label{ma-eq}$$ Here $t_{\mu\nu}$ stands for$$t_{\mu\nu}=\frac{1}{s}(2C_{1}g_{\mu\nu}-S_{\alpha\nu}S^{\alpha}\,_{\mu })\ -s\ g_{\mu\nu}\,,$$ $C_{1}$ denotes the first Casimir in Eq.(\[casimirs\]) and $S^{\beta\rho}$ are the $\mathcal{I}HLG$ generators in the particular representation chosen for $\psi(\mathbf{\mathbf{p}})$. Their construction as solutions of the algebra of the Lorentz group for the representation space under consideration is straightforward [@ryder; @prinind; @MK03]. Using now the gauge principle for electromagnetism in this equation we obtain$$\left[ \left( \frac{1}{s}(2C_{1}g_{\mu\nu}-S_{\alpha\nu}S^{\alpha}\,_{\mu })-s\ g_{\mu\nu}\right) \ \pi^{\mu}\ \pi^{\nu}-m^{2}\right] \psi (\mathbf{\mathbf{p}})=0, \label{eom}$$ with $\pi^{\mu}=P^{\mu}+eA^{\mu}$, and $e$ denoting the charge of the field $\psi(\mathbf{\mathbf{p}})$. Notice that Eq. (\[eom\]) is a covariant matrix equation that operates in the vector space of the dimensionality of $\psi(\mathbf{\mathbf{p}})$. For example, when $\psi(\mathbf{\mathbf{p}})$ stands for the four-vector– spinor, $W^{2}$ is represented by a $16\times16$ matrix.  For the sake of illustration, we here bring the Lagrangian density  for the lowest Rarita-Schwinger representation. It reads $$\mathcal{L}(x)=\overline{\psi}(x)\ t_{\mu\nu}\ \pi^{\mu}\ \pi^{\nu}\ \psi(x)-m^{2}\overline{\psi}(x)\psi(x),\label{Lagr_RS}$$ where $\overline{\psi}(x)=\psi^{\dagger}(x)(\gamma^{0}\otimes g)$ where $g$ is the matrix of the metric tensor. The definition of $\overline{\psi}(x)$ has to be performed for each representation individually. When applied to the Dirac representation $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ , Eq. (\[eom\]) also has the great advantage to yield the correct value of the gyromagnetic factor, $g_{s}=2$. This is not fortuitous but reflects the general property of our master equation (\[eom\]) to predict the correct value for the gyromagnetic ratio as $g_{s}=\frac{1}{s}$ for fields in $(s,0)\oplus(0,s)$ [@MC]. Had we used instead Eq. (\[master\]) alone, we would have found the problematic case of $g_{s}=\frac{1}{s(s+1)}$. With respect to $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$, Eq.(\[master1\]) is nothing more but the Klein-Gordon equation for each field component. This is due to the fact that the squared Pauli-Lubanski vector for all $(s,0)\oplus (0,s)$ fields is just $-s(s+1)P^{2}\mathbf{1}_{(2s+1)\times(2s+1)}$. The $W^{2}$ Casimir invariant identifies only the spin content and remains indifferent to the discrete $C$, $P$, or, $T$ properties of the representation of interest. Recall that one has different options to stick together, say, $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ in depending on whether one wants $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ to diagonalize the parity–, $\gamma^{0}\mathcal{R}$ or, the charge conjugation–, $i\gamma_{2}K$, operator. For parity eigenstates one ends up with the standard Dirac equation, while for $C-$ parity states one finds again the Dirac equation but with a Majorana mass term, respectively. Notice however that, under gauging, this equation gives the right magnetic properties for $(s,0)\oplus(0,s)$ fields . This means that solutions to Dirac equation are solutions to Eq.(\[eom\]) although the converse is not necessarily true since our equation specifies just the value of the spin. For product representation spaces of the type $\psi_{\mu_{1}\mu_{2}...\mu_{K}}$, the most interesting representation space for applications in hadron physics, the situation is different provided, one is tracking the highest spin. As long as the highest spins are non-degenerate, there is no confusion with parity doubling, as would be the case for the lower spins. For these reasons, Eq. (\[eom\]) has its major merits with respect to the highest spins in the representations. Next we study wave front propagation of particles described by means of Eq. (\[eom\]) along the line of Refs. [@VZ1; @VZ2]. Avoiding superluminal propagation of higher spin waves. ======================================================= Wave propagation is associated with a hyperbolic system of partial differential equations [@CH]. For such a class of differential equations the initial value problem can be posed on a class of surfaces ( “space like” surfaces with respect to the equation of motion). The equations possess solutions with wave fronts traveling along rays at finite velocities. At any point on the surface, the rays form a cone that is entirely determined by the coefficients of the highest derivatives in the equation of motion [@CH]. The wave front can be characterized by $n_{\text{ }}^{\mu}=(n^{0},\mathbf{n)}$, the vectors normal to the characteristic surface. The system of equations is hyperbolic if $n^{0}$ is real for any $\mathbf{n}$. To find the normal vectors it is sufficient to first replace in the highest derivatives of the equation of motion $P_{\mu}$ by $n_{\mu\text{ }}$ and then calculate the determinant $D(n)$ (so called “characteristic determinant” [@VZ2]) of the matrix given by the corresponding coefficients. Wave front propagation of the Klein-Gordon, Dirac and Rarita-Schwinger equations. --------------------------------------------------------------------------------- In cases when the coupling to external fields is carried by the lower derivatives in the equation of motion, such as, say, the Klein–Gordon equation, the ray cones for interacting and free fields coincide. Indeed, in the latter case and under minimal coupling one finds $$\left[ \pi^{\mu}\pi_{\mu}-m^{2}\right] \psi(\mathbf{p})=\left[ P^{\mu }P_{\mu}+e(P^{\mu}A_{\mu}+A^{\mu}P_{\mu})+e^{2}A^{\mu}A_{\mu}-m^{2}\right] \psi(\mathbf{p})=0. \label{KL_Go}$$ The vanishing of the characteristic determinant in this case yields$$D(n)=\mbox{Det}(n^{2})=n^{2}=0\,, \label{DET_KLG}$$ which has real $n^{0}$ for any $\mathbf{n}$. Same is true for Dirac particles, though not as obvious. As is well known, a Dirac particle coupled minimally to the electromagnetic field is described by $$\left[ \gamma^{\mu}(P_{\mu}+eA_{\mu})-m\right] \psi(\mathbf{\mathbf{p}})=0\,. \label{DIR_EQ}$$ Now, the resulting characteristic determinant is found to be the squared of Eq. (\[DET\_KLG\]) $$D(n)=\mbox{Det}(\gamma^{\mu}n_{\mu})=(n^{2})^{2}\,. \label{DET_DIR}$$ The vanishing of this determinant results once again into a ray cone that coincides with the light cone. The wave front propagation of the solution of the Rarita-Schwinger set of equations was studied in great detail in Ref. [@VZ1]. To understand the essence of the latter work recall that Eqs. (\[RS\_set\]) or the analogous equation in the interacting case, can be derived from a Lagrangian, a method suggested by Fierz and Pauli [@FP]. Within the latter framework not all the Euler-Lagrange equations appear as genuine equations of motion, meaning that some of them may not involve time derivatives, a property that qualifies them only as constraints onto the fields. Precisely this is the case for the Rarita-Schwinger framework discussed in Section II. As a consequence, any surface in space-time is a characteristic surface [@CH]. The Rarita-Schwinger system of coupled equations turns to be equivalent to a system of hyperbolic equations supplemented by constraints that are conserved in time. In this case, the wave fronts of the constrained system are no longer given by the characteristic determinant of the Euler-Lagrange equations. Rather, it is necessary to find the genuine equation of motion, i.e. the one which (i) contains all the higher order derivatives needed for the complete characterization of the system, (ii) preserves the constraints in time. Finding such an equation in general introduces, in addition to the derivatives already present in the system of coupled equations, also new ones which as a rule spoil causal propagation, a result due to [@VZ1; @VZ2]. Wave front propagation of $W^{2}$ invariant subspaces. ------------------------------------------------------ In the present work we suggested an alternative formalism to the Rarita-Schwinger framework. Our proposal was to pin down the degrees of freedom of interest by means of Eq. (\[eom\]). This equation was build upon the covariant projector onto the $W^{2}$ invariant vector spaces in the representation under consideration, and did not invoke any supplementary conditions. In this concern, it is worth to remark that the formalism does not deal with the whole representation space but only with one of its $W^{2}$ invariant sub-spaces. Below we prove that equations of the latter type do not suffer the Velo-Zwanziger problem upon gauging. Firstly, we have to check that for all the degrees of freedom of $\psi(\mathbf{p})$, the second order time-derivatives enter Eq. (\[eom\]) with non-vanishing coefficients. This can be done in full generality in momentum space where $$t_{00}=\frac{1}{s}(2C_{1}g_{00}-S_{\alpha0}S^{\alpha}\,_{0})-sg_{00}=\mathbf{1}. \label{t00}$$ Therefore, for all $W^{2}$ invariant subspaces, the time derivative of each field component in Eq. (\[eom\]) does not vanish. This equation will be hyperbolic if the solutions $n^{0}$ to $D(n)=0$ are real for any $\mathbf{n}$. In this case we must solve $$\mbox{Det}\left[ -\frac{1}{s}W^{2}(n)-s\ n^{2}\right] =0. \label{Determinant}$$ In order to demonstrate that (\[eom\]) is a hyperbolic equation in the HLG representation space chosen for $\psi(\mathbf{p})$ we here exploit decomposition of the latter into invariant subspaces of $W^{2}$. The most transparent representation of $W^{2}$ is obtained in the basis of $\mathbf{p}$-dependent $W^{2}$ eigenstates where $W^{2}$ is block diagonal and equal to $$W^{2}(P)=-P^{2}\mbox{Diag}\left[ s_{1}(s_{1}+1)\mathbf{1}_{s_{1}},\ s_{2}(s_{2}+1)\mathbf{1}_{s_{2}},...s_{N}(s_{N}+1)\mathbf{1}_{s_{N}}\right] \,. \label{sis}$$ Here $\left\{ s_{1},s_{2}...s_{N}\right\} $ label the different eigensubspaces of $W^{2}$ (one of them being $s$) in the representation of interest, while $\mathbf{1}_{s_{j}}$ denotes the unit matrix of dimensionality $(2s_{j}+1)\times(2s_{j}+1)$. Notice that the dimensionality, $(d)$, of the representation space $\psi(\mathbf{p})$ relates to the $W^{2}$ quantum numbers via $d=\sum_{i}m_{i}(2s_{i}+1)$, where $m_{i}$ is the multiplicity of $s_{i}$. The latter accounts for possible degeneracies of the $W^{2}$ invariant subspaces in $\psi(\mathbf{p})$ with respect to further symmetries such like, say, one of the discrete space–time symmetries. The determinant (\[Determinant\]) is calculated as$$\mbox{Det}\left[ -\frac{1}{s}W^{2}(n)-s\ n^{2}\right] ={\displaystyle\prod\limits_{k=0}^{N}} \left( n^{2}(\frac{s_{k}(s_{k}+1)}{s}-s)\right) ^{2s_{k}+1}. \label{Det2}$$ As long as for the integer/half-integer $s$ under consideration, there are no positive integers and half-integers $s_{k}$ satisfying$$\frac{s_{k}(s_{k}+1)}{s}-s=0,$$ the roots of the characteristic determinant are $n^{2}=0$. Thus the solutions have $n^{0}$ real for any $\mathbf{n}$, and Eq.(\[eom\]) is a set of hyperbolic equations for the $\psi(\mathbf{p})$ components. The characteristic surfaces are same for free and interacting particles, and the ray cone coincides with the light cone. In other words, the wave front propagation of $W^{2}$ invariant subspaces is free from the Velo–Zwanziger problem. Conclusions and perspectives. ============================= In the present article we advocate the idea to consider higher spins as invariant subspaces of the Casimir operators of the Poincaré group, the squared four-momentum and the squared Pauli-Lubanski vector, in a properly chosen representation of the HLG, $\psi(\mathbf{p})$. In executing the idea we demonstrated that any higher spin is described in terms of one covariant matrix equation that (i) is determined exclusively by the HLG generators in $\psi(\mathbf{p})$, (ii) is of the dimensionality of $\psi(\mathbf{p})$, (iii) is always of second order in the momenta. 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--- abstract: 'Carrying out clinical diagnosis of retinal vascular degeneration using Fluorescein Angiography (FA) is a time consuming process and can pose significant adverse effects on the patient. Angiography requires insertion of a dye that may cause severe adverse effects and can even be fatal. Currently, there are no non-invasive systems capable of generating Fluorescein Angiography images. However, retinal fundus photography is a non-invasive imaging technique that can be completed in a few seconds. In order to eliminate the need for FA, we propose a conditional generative adversarial network (GAN) to translate fundus images to FA images. The proposed GAN consists of a novel residual block capable of generating high quality FA images. These images are important tools in the differential diagnosis of retinal diseases without the need for invasive procedure with possible side effects. Our experiments show that the proposed architecture outperforms other state-of-the-art generative networks. Furthermore, our proposed model achieves better qualitative results indistinguishable from real angiograms.' bibliography: - 'egbib.bib' title: 'Fundus2Angio: A Novel Conditional GAN Architecture for Generating Fluorescein Angiography Images from Retinal Fundus Photography' --- Introduction {#sec:intro} ============ For a long time Fluorescein Angiography (FA) combined with Retinal Funduscopy have been used for diagnosing retinal vascular and pigment epithelial-choroidal diseases [@mary2016retinal]. The process requires the injection of a fluorescent dye which appears in the optic vein within 8-12 seconds depending on the age and cardiovascular structure of the eye and stays up to 10 minutes [@mandava2004fluorescein]. Although generally considered safe, there have been reports of mild to severe complications due to allergic reactions to the dye [@kwiterovich1991frequency; @brockow2014hypersensitivity; @torres20091]. Frequent side effects can range from nausea, vomiting, anaphylaxis, heart attack, to anaphylactic shock and death [@lira2007adverse; @kwan2006fluorescein; @lieberman2005diagnosis; @el1996anaphylactic; @fineschi1999fatal]. In addition, leakage of fluorescein in intravaneous area is common. However, the concentration of fluorescein solutions don’t have any direct impact on adverse effects mentioned above.[@yannuzzi1986fluorescein]. Given the complications and the risks associated with this procedure, a non-invasive, affordable, and computationally effective procedure is quite imperative. The only current alternatives to flourecein angigraphy (FA) is carried out by Optical Coherence Tomography and basic image processing technique. These systems are generally quite expensive. Without a computationally effective and financially viable mechanism to generate reliable and reproducible flourecein angiograms, the only alternative is to utilize retina funduscopy for differential diagnosis. Although automated systems consisting of image processing and machine learning algorithms have been proposed for diagnosing underlying conditions and diseases from fundus images [@gurudath2014machine; @fu2018disc; @poplin2018prediction; @lira2007adverse], there has not been an effective effort to generate FA images from retina photographs. In this paper, we propose a novel conditional Generative Adversarial Network (GAN) called Fundus2Angio, capable of synthesizing fluorescein angiograms from retinal fundus images. The procedure is completely automated and does not require any human intervention. We use both qualitative and quantitative metrics for testing the proposed architecture. We compare the proposed architecture with other state-of-the-art conditional GANs [@wang2018high; @isola2017image; @zhu2017unpaired]. Our model outperforms these networks in terms of quantitative measurement. For qualitative results, expert ophthalmologists were asked to distinguish fake angiograms from a random set of balanced real and fake angiograms over two trials. Results show that the angiograms generated by the proposed network are quite indistinguishable from real FA images. Literature Review ================= Generative adversarial networks have revolutionized many image manipulation tasks such as image editing [@zhu2016generative; @dekel2018sparse], image styling [@chen2018sketchygan; @sangkloy2017scribbler], and image style transfer [@zhu2017unpaired; @wang2018high; @xian2018texturegan]. Multi-resolution architectures are common practice in computer vision, while coupled architectures have the capability to combine fine and coarse information from images [@burt1983laplacian; @brown2003recognising]. Recently, techniques on Conditional [@huang2017stacked; @denton2015deep] and Unconditional GANs [@chen2017photographic; @zhang2017stackgan] have explored the idea of combined-resolutions within the architecture for domain specific tasks. Inspired by this, we propose an architecture that extract features at different scales. Some approaches also used multi-scale discriminators for style-transfer [@wang2018high; @karras2017progressive; @zhang2018densely]. However, they only attached discriminators with generator that deals with fine features while ignoring discriminators for coarse generator completely. In order to learn useful features at coarsest scale, separate multi-scale discriminators are necessary. Our proposed architecture employs this for both coarse and fine generators. For high quality image synthesis, a pyramid network with multiple pairs of discriminators and generators has also been proposed, termed SinGAN [@shaham2019singan]. Though it produces high quality synthesized images, the model works only on unpaired images. To add to this problem, each generator’s input is the synthesized output produced by the previous generator. As a result, it can’t be employed for pair-wise image training that satisfies a condition. To alleviate from this problem, a connection needs to be established that can propagate feature from coarse to fine generator. In this paper, we propose such an architecture that has a feature appending mechanism between the coarse and fine generators, making it a two level pyramid network with multi-scale discriminators as illustrated in Fig. \[fig1\]. ![Proposed Generative Adversarial Network[]{data-label="fig1"}](Fig1.png){width="9cm"} The Proposed Methodology ======================== This paper proposes a new conditional generative adversarial network (GAN) comprising of a novel residual block for producing realistic FA from retinal fundus images. First, we introduce the residual block in section \[subsec:residualblock\]. We then delve into the proposed conditional GAN encompassing of fine and coarse generators and four multi-scale discriminators in sections \[subsec:generators\] and \[subsec:discriminators\]. Lastly, in section \[subsec:objective\], we discuss the objective function and loss weight distributions for each of the architectures that form the proposed architecture. Novel Residual Block {#subsec:residualblock} -------------------- Recently, residual blocks have become the norm for implementing many image classification, detection and segmentation architectures [@he2016deep; @he2016identity]. Generative architectures have employed these blocks in interesting applications ranging from image-to-image translation to super-resolution [@johnson2016perceptual; @wang2018high; @ledig2017photo]. In its atomic form, a residual unit consists of two consecutive convolution layers. The output of the second layers is added to the input, allowing for deeper networks. Computationally, regular convolution layers are expensive compared to a newer convolution variant, called separable convolution [@chollet2017xception]. Separable convolution performs a depth-wise convolution followed by a point-wise convolution. This, in turn helps to extract and retain depth and spatial information through the network. It has been shown that interspersing convolutional layers allows for more efficient and accurate networks [@opticnet19]. We incorporate this idea to design a novel residual block to retain both depth and spatial information, decrease computational complexity and ensure efficient memory usage, as shown in Table. \[table1\]. Residual Block Equation Activation No. of Parameters$^{1}$ ---------------- -------------------------------------------------------------------------- ------------------------------ ------------------------- Original $\big[R_{i} \circledast F_{Conv} \circledast F_{Conv} \big] + R_{i}$ ReLU (Pre) [@he2016identity] 18,688 Proposed $\big[R_{i} \circledast F_{Conv} \circledast F_{SepConv} \big] + R_{i}$ Leaky-ReLU (Post) 10,784 : Comparison between Original and Proposed Residual Block[]{data-label="table1"} \ $^1$ $F_{Conv}$ and $F_{SepConv}$ has kernel size $K=3$, stride $S=1$, padding $P=0$ and No. of channel $C=32$. ![Proposed Residual Block[]{data-label="fig2"}](Fig2.png){width="9cm"} As illustrated in Fig. \[fig2\], we replace the last convolution operation with a separable convolution. We also use Batch-normalization [@ioffe2015batch] and Leaky-ReLU as post activation mechanism after both convolution and separable Convolution layers. For better results, we incorporate reflection padding as opposed to zero-padding before each convolution operation. The entire operation can be formulated as shown in Eq. \[eq1\]: $$\begin{split} R_{i+1} &= \big[R_{i} \circledast F_{Conv} \circledast F_{SepConv} \big] + R_{i} \\ &= F(R_{i}) + R_{i} \label{eq1} \end{split}$$ Here, $\circledast$ refers to convolution operation while $F_{conv}$ and $F_{SepConv}$ signify the back-to-back convolution and separable convolution operations. By exploiting convolution and separable convolution layer with Leaky-ReLU, we ensure that two distinct feature maps (spatial & depth information) can be combined to generate fine fluorescein angiograms. ![Generator and Discriminator Architectures[]{data-label="fig3"}](Fig3.png){width="10cm" height="6cm"} Coarse and Fine Generators {#subsec:generators} -------------------------- Using a coarse-to-fine generator for both conditional and unconditional GANs results in very high quality image generation, as observed in recent architectures, such as pix2pixHD [@wang2018high] and SinGan [@shaham2019singan]. Inspired by this idea, we use two generators ($G_{fine}$ and $G_{coarse}$) in the proposed network, as illustrated in Fig. \[fig3\]. The generator $G_{fine}$ synthesizes fine angiograms from fundus images by learning local information, including retinal venules, arterioles, hemorrhages, exudates and microaneurysms. On the other hand, the generator $G_{coarse}$ tries to extract and preserve global information, such as the structures of the macula, optic disc, color, contrast and brightness, while producing coarse angiograms. The generator $G_{fine}$ takes input images of size $512\times 512$ and produces output images with the same resolution. Similarly, the generator $G_{coarse}$ network takes an image with half the size ($256\times 256$) and outputs an image of the same size as the input. In addition, the $G_{coarse}$ outputs a feature vector of the size $256\times 256 \times 64$ that is eventually added with one of the intermediate layers of $G_{fine}$. These hybrid generators are quite powerful for sharing local and global information between multiple architectures as seen in [@johnson2016perceptual; @shaham2019singan; @wang2018high]. Both generators use convolution layers for downsampling and transposed convolution layers for upsampling. It should be noted that $G_{coarse}$ is downsampled twice ($\times 2$) before being upsampled twice again with transposed convolution. In both the generators, the proposed residual blocks are used after the last downsampling operation and before the first upsampling operations as illustrated in Fig. \[fig3\]. On the other hand, in $G_{fine}$, downsampling takes place once with necessary convolution layer, followed by adding the feature vector, repetition of residual blocks and then upsampling to get fine angiography image. All convolution and transposed convolution operation are followed by Batch-Normalization [@ioffe2015batch] and Leaky-ReLU activations. To train these generators, we start with $G_{coarse}$ by batch-training it on random samples once and then we train the $G_{fine}$ once with a new set of random samples. During this time, the discriminator’s weights are frozen, so that they are not trainable. Lastly, we jointly fine-tune all the discriminator and generators together to train the GAN. Multi-scale PatchGAN as Discriminator {#subsec:discriminators} ------------------------------------- For synthesizing fluorescein angiography images, GAN discriminators need to adapt to coarse and fine generated images for distinguishing between real and fake images. To alleviate this problem, we either need a deeper architecture or, a kernel with wider receptive field. Both these solutions result in over fitting and increase the number of parameters. Additionally, a large amount of processing power will be required for computing all the parameters. To address this issue, we exploit the idea of using two Markovian discriminators, first introduced in a technique called PatchGAN [@li2016precomputed]. This technique takes input from different scales as previously seen in [@wang2018high; @shaham2019singan]. We use four discriminators that have a similar network structure but operate at different image scales. Particularly, we downsample the real and generated angiograms by a factor of $2$ using the Lanczos sampling [@duchon1979lanczos] to create an image pyramid of three scales (original and $2\times$downsampled and $4\times$downsampled). We group the four discriminators into two, $D_{fine}=[D1_{fine},D2_{fine}]$ and $D_{coarse}=[D1_{coarse},D2_{coarse}]$ as seen in Fig. \[fig1\]. The discriminators are then trained to distinguish between real and generated angiography images at the three distinct resolutions respectively. The outputs of the PatchGAN for $D_{fine}$ are $64\times64$ and $32\times32$ and for $D_{coarse}$ are $32\times32$ and $16\times16$. With the given discriminators, the loss function can be formulated as given in Eq. \[eq2\]. It’s a multi-task problem of maximizing the loss of the discriminators while minimizing the loss of the generators. $$\min \limits_{G_{fine},G_{coarse}} \max \limits_{D_{fine},D_{coarse}} \mathcal{L}_{cGAN}(G_{fine},G_{coarse}, D_{fine},D_{coarse}) \label{eq2}$$ Despite discriminators having similar network structure, the one that learns feature at a lower resolution has the wider receptive field. It tries to extract and retain more global features such as macula, optic disc, color and brightness etc to generate better coarse images. In contrast, the discriminator that learns feature at original resolution dictates the generator to produce fine features such as retinal veins and arteries, exudates etc. By doing this we combine feature information of global and local scale while training the generators independently with their paired multi-scale discriminators. Weighted Object Function and Adversarial Loss {#subsec:objective} --------------------------------------------- We use LSGAN [@mao2017least] for calculating the loss and training our conditional GAN. The objective function for our conditional GAN is given in Eq. \[eq3\]. $$\mathcal{L}_{cGAN}(G,D) = \mathbb{E}_{x,y} \big[\ (D(x,y) -1)^2 \big]\ + \mathbb{E}_{x} \big[\ (D(x,G(x)+1))^2 \big]\ \label{eq3}$$ where the discriminators are first trained on the real fundus, $x$ and real angiography image, $y$ and then trained on the the real fundus, $x$ and fake angiography image, $G(x)$. We start with training the discriminators $D_{fine}$ and $D_{coarse}$ for couple of iterations on random batches of images. Next, we train the $G_{coarse}$ while keeping the weights of the discriminators frozen. Following that, we train the the $G_{fine}$ on a batch of random samples in a similar fashion. We use Mean-Squared-Error (MSE) for calculating the individual loss of the generators as shown in Eq. \[eq4\]. $$\mathcal{L}_{L2}(G) = \mathbb{E}_{x,y} \Vert G(x) - y \Vert^2 \label{eq4}$$ where, $\mathcal{L}_{L2}$ is the reconstruction loss for a real angiogram, $y$, given a generated angiogram, $G(x)$. We use this loss for both $G_{fine}$ and $G_{coarse}$ so that the model can generate high quality angiograms of different scales. Previous techniques have also exploited this idea of combining basic GAN objective with a MSE loss [@pathak2016context]. From Eq. \[eq3\] and \[eq4\] we can formulate our final objective function as given in Eq. \[eq5\]. $$\min \limits_{G_{fine},G_{coarse}} \max \limits_{D_{fine},D_{coarse}} \mathcal{L}_{cGAN}(G_{fine},G_{coarse}, D_{fine},D_{coarse}) + \lambda\big[\ \mathcal{L}_{L2}(G_{fine}) + \mathcal{L}_{L2}(G_{coarse})\big]\ \label{eq5}$$ Here, $\lambda$ dictates either to prioritize the discriminators or the generators. For our architecture, more weight is given to the reconstruction loss of the generators and thus we pick a large $\lambda$ value. Experiments =========== In the following section, different experimentation and evaluation is provided for our proposed architecture. First we elaborate about the data preparation and pre-prossessing scheme in Sec. \[subsec:dataset\]. We then define our hyper-parameter settings in Sec. \[subsec:hyper\]. Following that, different architectures are compared based on some quantitative and qualitative evaluation metrics in Sec. \[subsec:quant\]. Lastly, and Sec. \[subsec:qual\], Dataset {#subsec:dataset} ------- For training, we use the funuds and angiography data-set provided by Hajeb et al. [@hajeb2012diabetic]. The data-set consists of 30 pairs of diabetic retinopathy and 29 pairs normal of angiography and fundus images from 59 patients. Because, not all of the pairs are perfectly aligned, we select 17 pairs for our experiment based on alignment. The images are either perfectly aligned or nearly aligned. The resolution for fundus and angiograms are as follows $576\times720$. Fundus photographs are in RGB format, whereas angiograms are in Gray-scale format. Due to shortage of data, we take 50 random crops of size $512\times512$ from each images for training our model. So, the total number of training sample is 850 ($17\times50$). Hyper-parameter tuning {#subsec:hyper} ---------------------- LSGAN [@mao2017least] was found to be effective for generating desired synthetic images for our tasks. We picked $ \lambda =10$ (Eq. \[eq5\]). For optimizer, we used Adam [@kingma2014adam], with learning rate $\alpha=0.0002$, $\beta_1=0.5$ and $\beta_2=0.999$. We train with mini-batches with batch size, $b=4$ for 100 epochs. It took approximately 10 hours to train our model on an NVIDIA RTX2070 GPU. ![Angiogram generated from transformed Fundus images[]{data-label="fig4"}](Fig4.png){width="9cm"} Qualitative Evaluation {#subsec:quant} ---------------------- For evaluating the performance of the network, we took 14 images and cropped 4 sections from each quadrant of the image with a size of $512\times512$. We conducted two sets of experiments to evaluate both the network’s robustness to global changes to the imaging modes and its ability to adapt to structural changes to the vascular patterns and structure of the eye. We used GNU Image Manipulation Program (GIMP) [@gimp2019gimp] for transforming and distorting images. In the first set of experiments, three transformations were applied to the images: 1) blurring to represent out of focus funduscopy or fundus photography in the presence of severe cataracts, 2) sharpening to represent pupil dilation, and 3) noise to represent interference during photography. Good robustness is represented by the generated angiograms similarity to the real FA image since these transformation do not affect the vascular structure of the retina. A side by side comparison of different architecture’s prediction is shown in Fig. \[fig4\]. As it can be observed from the image, the proposed architecture produces images very similar to the ground-truth (GT) under these global changes applied to the fundus image. In the case of **blurred** fundus images, our model is less affected compared to other architectures, as seen in the second row of Fig. \[fig4\]– structure of smaller veins are preserved better compared to Pix2Pix and Pix2PixHD. In the case of **sharpened** images, the angiogram produced by Pix2Pix and Pix2PixHD show vein-like structures introduced in the back, which are not present in our prediction. These are seen in the third row of Fig. \[fig4\]. In the case of **noisy** images, as seen in the last row of Fig. \[fig4\] our prediction is still unaffected with this pixel level alteration. However, both Pix2Pix and Pix2PixHD fails to generate thin and small vessel structures by failing to extract low level features. ![Angiogram generated from distorted Fundus images with biological markers[]{data-label="fig5"}](Fig5.png){width="9cm"} In the second set of experiments we modified the vascular pattern of the retina and the fundus images. These structural changes are represented by two different types of distortions: 1) pinch, representing the flattening of the retina resulting in the pulled/pushed retinal structure, and 2) whirl, representing retina distortions caused by increased intra-ocular pressure (IOP). Good adaptation to structural changes in the retina is achieved if the generated angiograms are similar to the angiograms with changed vascular structure. The effects of Pinch and Whirl on predicted angiogram is illustrated in Fig. \[fig5\]. **Pinch** represents the globe flattening condition, manifesting vascular changes on the retina as a result of distortions of retinal subspace. This experiment shows the adaptability and reproduciblity of the proposed network to uncover the changes in vascular structure. From the first row in Fig. \[fig5\] it is evident that our model can effectively locates the retinal vessels compared to other proposed techniques. **Whirl** represented changes in the IOP or vitreous changes in the eye that may result in twists in the vascular structure. Similar to pinch, the network’s ability to adapt to this structural change can be measured if the generated FA image is similar to the real angiogram showing the changed vascular structure. As seen in the last row of Fig. \[fig5\] our network encodes the feature information vessel structures, and is much less affected this kind of distortion. The other architectures failed to generate micro vessel structure as it can be seen in Fig. \[fig5\]. Quantitative Evaluations {#subsec:qual} ------------------------ For quantitative evaluation, we also performed two experiments. In the first experiment we use the Fréchet inception distance (FID) [@heusel2017gans] that has been used to evaluate similar style-transfer GANs [@brock2018large; @karras2017progressive; @karras2019style]. We computed the FID scores for different architectures on the generated FA image and original angiogram, and those generated from the changed fundus images by the five global and structural changes –i.e., blurring, sharpening, noise, pinch, and whirl. The results are reported in Table. \[table2\]. It should be noted that, lower FID score means better results. Architecture Orig. Noise Blur Sharp Whirl Pinch --------------------------- ---------- --------------------------- -------------------------- -------------------------- -------------------------- -------------------------- **Ours** **30.3** **41.5** (11.2$\uparrow$) **32.3** (2.0$\uparrow$) **34.3** (4.0$\uparrow$) **38.2** (7.9$\uparrow$) **33.1** (2.8$\uparrow$) Pix2PixHD [@wang2018high] 42.8 53.0 (10.2$\uparrow$) 43.7 (1.1$\uparrow$) 47.5 (4.7$\uparrow$) 45.9 (3.1$\uparrow$) 39.2 (3.6$\downarrow$) Pix2Pix [@isola2017image] 48.6 46.8 (1.8 $\downarrow$) 50.8 (2.2$\uparrow$) 47.1 (1.5$\downarrow$) 43.0 (5.6$\downarrow$) 43.7 (4.9$\downarrow$) : Fréchet inception distance (FID) for different architectures[]{data-label="table2"} From Table. \[table2\], using the original fundus image, the FID of our network angiogrm is 30.3, while other techniques are at least 10 points worse, Pix2PixHD (42.8) and Pix2Pix (48.6). For the case of noisy images, the FID for Pix2Pix dropped slightly but increased for both Pix2PixHD and our technique. Notice that the FID for our technique is still better than both Pix2Pix and Pix2PixHD. For all other changes, the FID score of our technique increased slightly but still outperformed Pix2Pix and Pix2PixHD in both robustness and adaptation to the structural changes. ------ --------- ----------- -------- ------- ----------- Correct Incorrect Missed Found Confusion Fake 15% 85% Real 80% 20% ------ --------- ----------- -------- ------- ----------- : Results of Qualitative with Undisclosed Portion of Fake/Real Experiment[]{data-label="table3"} In the next experiment we evaluate the quality of the generated angiograms by asking experts (e.g. ophthalmologists) to identify fake angiograms among a collection of 40 balanced (50%, 50%) and randomly mixed angiograms. For this experiment, the experts were not told how many of the images are real and how many are fake. The non-disclosed ratio of fake and real images was a significant design choice for this experiment, as it will allow us to evaluate three metrics: 1) incorrectly labeled fake images representing how real the generated images look, 2) correctly labeled real images representing how accurate the experts recognized angiogram salient features, and 3) the confusion metric representing how effective the overall performance of our proposed method was in confusing the expert in the overall experiment. The results are shown in Table \[table3\]. As it can be seen from Table \[table3\], experts assigned 85% of the fake angiogams as real. This result shows that experts had difficulty in identifying fake images, while they easily identified real angiograms with 80% accuracy. Overall, the experts misclassified 53% of all images. This resulted in a confusion factor of 52.5%. This is significant, as the confusion factor of 50% is the best achievable result. Conclusion ========== In this paper, we introduced Fundus2Angio, a novel conditional generative architecture that capable of generating angiograms from retinal fundus images. We further demonstrated its robustness, adaptability, and reproducibility by synthesizing high quality angiograms from transformed and distorted fundus images. Additionally, we illustrated how changes in biological markers do not affect the adaptability and reproducibility of synthesizing angiograms by using our technique. This ensures that the proposed architecture effectively preserves known biological markers (e.g. vascular patterns and structures). As a result, the proposed network can be effectively utilized to produce accurate FA images for the same patient from his or her fundus images over time. This allows for a better control on patient’s disease progression monitoring or to help uncover newly developed diseases or conditions. One future direction to this work is to improve upon this work to incorporate retinal vessel segmentation and exudate localization. The proceedings of BMVC are published only in electronic form, but it is still assumed that\cite{} readers of the papers may wish to print the paper. This document illustrates the required paper format, which is designed to read well either printed with two pages per sheet (“2- up”), or on screen. Note that printing with one page per sheet will produce a “large print” version, which in many cases is not what is desired. To approximate the old BMVC format, print at one page per sheet, but do not choose the option to “scale to fit paper”. LATEX users should use this template in order to prepare their paper. Users of other packages should emulate the style and layout of this example. Note that best results will be achieved using pdflatex, which is available in most modern distributions. 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It must [*not*]{} include annotations, figures, or any other paraphernalia intended to subvert the paper length requirement. ----- ----- ----- (a) (b) (c) ----- ----- ----- Citations --------- When citing a multi-author paper, you may save space by using “[*et alia*]{}”, shortened to “” (not “[*et. al.*]{}” as “[*et*]{}” is a complete word.) The provided `\etal` macro is a useful [*aide memoire*]{} in this regard. However, use it only when there are three or more authors. Thus, the following is correct: “ Frobnication has been trendy lately. It was introduced by Alpher [@Alpher02], and subsequently developed by Alpher and Fotheringham-Smythe [@Alpher03], and Alpher  [@Alpher04].” This is incorrect: “... subsequently developed by Alpher  [@Alpher03] ...” because reference [@Alpher03] has just two authors. If you use the `\etal` macro, then you need not worry about double periods when used at the end of a sentence as in Alpher . We use [natbib]{}, so citations in random order are nicely sorted: [@Alpher03; @Alpher02; @Authors06b; @Authors06]. However, we don’t use the compress option, as we want each reference to have its own hyperlink and popup window. Footnotes --------- Please use footnotes[^1] sparingly. Indeed, try to avoid footnotes altogether and include necessary peripheral observations in the text (within parentheses, if you prefer, as in this sentence). If you wish to use a footnote, place it at the bottom of the column on the page on which it is referenced. Use Times 8-point type, single-spaced. Method Frobnability -------- ------------------------ Theirs Frumpy Yours Frobbly Ours Makes one’s heart Frob : Results. Ours is better. Mathematics ----------- Please number all of your sections and displayed equations. It is important for readers to be able to refer to any particular equation. Just because you didn’t refer to it in the text doesn’t mean some future reader might not need to refer to it. It is cumbersome to have to use circumlocutions like “the equation second from the top of page 3 column 1”. (Note that the ruler will not be present in the final copy, so is not an alternative to equation numbers). All authors will benefit from reading Mermin’s description [@Mermin89] of how to write mathematics. References ---------- List and number all bibliographical references in 9-point Times, single-spaced, at the end of your paper. When referenced in the text, enclose the citation number in square brackets, for example [@Authors06]. Where appropriate, include the name(s) of editors of referenced books. Color ----- Color is valuable, and will be visible to readers of the electronic copy. However ensure that, when printed on a monochrome printer, no important information is lost by the conversion to grayscale. [^1]: This is what a footnote looks like. It often distracts the reader from the main flow of the argument.

--- abstract: 'We investigate angular correlations in multi-jet final states at high-energy colliders and discuss their sensitivity to initial-state showering effects, including QCD coherence and corrections to collinear ordering [@url].' author: - | F. Hautmann$^1$ and H. Jung$^2$\ 1 - Oxford University, Theoretical Physics Department\ Oxford OX1 3NP, UK\ 2 - Deutsches Elektronen Synchrotron\ Hamburg D-22603, Germany\ title: ' Dijet azimuthal distributions and initial-state parton showers' --- 0.5 cm 0.8 cm [*Presented at the Workshop DIS08, University College London, April 2008*]{} 0.5 cm Events with multiple hadronic jets are central to many aspects of the LHC physics program and their analysis will require realistic Monte Carlo simulations. See e.g. [@alwalletal]. In a multi-jet event the correlation in the azimuthal angle $\Delta \phi$, defined to be between the two hardest jets, provides a useful measurement, sensitive to how well QCD multiple-radiation effects are described, and has been used to tune shower Monte Carlo event generators [@albrow]. The Tevatron $\Delta \phi$ measurements [@d02005] admit a reasonable description by Monte Carlo, see   and   results in Fig. \[Fig:d0az\] [@d02005]. In particular the data are sensitive to initial-state showering parameters and have been used for re-tuning of these parameters in  [@albrow]. On the other hand, the [HERA]{} $\Delta \phi$ measurements [@h1deltaphi; @zeus1931] are not well described by the standard   and   Monte Carlo showers in most of the data kinematic range (see below). At the LHC, measurements of $\Delta \phi$ distributions in multi-jet events may become accessible relatively early. Such complex hadronic final states at LHC energies are potentially sensitive to corrections to the collinear ordering implemented in standard parton showers [@ictppaper]. In particular, for jets of given $E_T$ the partonic momentum fraction $x$ is reduced as the energy increases, and angular correlations probe coherence effects in the spacelike branching [@hj_angjet], associated with non-collinear radiation at $ x \ll 1$ and not included in   or . Monte Carlo generators designed to take these effects into account are based (see e.g. [@jepp06; @hann04] and early studies in [@mw]) on transverse-momentum dependent parton distributions and matrix elements, defined via high-energy factorization [@hef]. General formulations for these distributions in initial-state showers are studied in [@collinszu]. Ref. [@hj_angjet] investigates the effects of corrections to collinear-ordered showers on correlations in multi-jet final states, using the precise ep measurements [@zeus1931] that have recently become available. These measurements are characterized by large phase space available for jet production and by small $x$ kinematics, potentially relevant for extrapolation of initial-state showering effects to the LHC. In Fig. \[Fig:azz\] we report results [@hj_angjet] for the azimuthal $\Delta \phi$ distribution in two-jet and three-jet cross sections. In Fig. \[Fig: 3\] we give results for the $\Sigma p_t$ and $\Delta p_t$ distributions [@zeus1931; @hj_angjet] measuring the transverse-momentum imbalance between the leading jets. ![\[Fig:d0az\] Dijet azimuthal correlations measured by D0 along with the   and   results [@d02005].](d0az.eps){width="0.35\columnwidth"} These results show that the shape of the distributions is different for  and for the k$_\perp$-shower Monte Carlo  [@jung02], with the largest differences occurring at small $\Delta \phi$ and small $\Delta p_t$, where the two highest $E_T$ jets are far from back to back and one has effectively three hard, well-separated jets. Ref. [@hj_angjet] also analyzes the angular distribution of the third jet and finds significant contributions from regions where the transverse momenta in the initial state shower are not ordered. The description of the measurement by the k$_\perp$-shower is good, whereas the collinear-based   shower is not sufficient to describe it. ![\[Fig:azz\] Azimuthal correlations [@hj_angjet] by the k$_\perp$-shower  and by  compared with the ep data [@zeus1931]: (left) two-jet cross section; (right) three-jet cross section.](az_2jet.eps "fig:"){width="0.45\columnwidth"} ![\[Fig:azz\] Azimuthal correlations [@hj_angjet] by the k$_\perp$-shower  and by  compared with the ep data [@zeus1931]: (left) two-jet cross section; (right) three-jet cross section.](az_3jet.eps "fig:"){width="0.45\columnwidth"} The physical picture underlying the k$_\perp$-shower method involves both transverse momentum dependent pdfs and matrix elements [@ictppaper]. The angular and momentum correlations of Figs. \[Fig:azz\],\[Fig: 3\] are found [@hj_angjet; @hjradcor] to be sensitive in particular to the large-k$_\perp$ tail in the hard matrix elements [@hef]. More detailed studies of these off-shell contributions are currently underway, including comparisons with results of next-to-leading order (NLO) event generators, see single-jet and di-jet distributions in Fig. \[Fig: 4\]. Here we see in particular that the dijet $p_t$ spectrum at high $p_t$ is close for the NLO calculation and the k$_\perp$-shower (at low $p_t$ we see the effect of the Sudakov form factor in the shower). Ref. [@hj_angjet] illustrates that the collinear approximation to the matrix element does not describe the shape of the angular distribution at small $\Delta \phi$. We note that the inclusion of the perturbatively computed high-k$_\perp$ correction distinguishes the calculation [@hj_angjet] of multi-jet cross sections from other shower approaches (see e.g. [@hoeche]) that include transverse momentum dependence in the pdfs but not in the matrix elements. ![\[Fig: 3\] Transverse momentum correlations [@hj_angjet] by the k$_\perp$-shower  and by   compared with the 3-jet data [@zeus1931]. The variables $\Sigma p_t$ (left) and $\Delta p_t$ (right) are as defined in [@zeus1931; @hj_angjet].](zeus-ptsum-3jet_may5.eps "fig:"){width="0.45\columnwidth"} ![\[Fig: 3\] Transverse momentum correlations [@hj_angjet] by the k$_\perp$-shower  and by   compared with the 3-jet data [@zeus1931]. The variables $\Sigma p_t$ (left) and $\Delta p_t$ (right) are as defined in [@zeus1931; @hj_angjet].](zeus-deltapt-3jet_1.eps "fig:"){width="0.45\columnwidth"} It is worth emphasizing that the coherence effects in the angular distributions computed above are associated with multi-gluon radiation terms to the initial-state shower that become non-negligible at high energy (small $x$) and small $\Delta \phi$. These can be incorporated using the formulation at fixed transverse momentum. Near the back-to-back region of large $\Delta \phi$ [@delenda], corrections due to mixed Coulomb/radiative terms may also become important and affect the basic picture: see recent studies in [@0708pap]. See also [@manch] for a related discussion of Coulomb contributions. More general issues on unintegrated pdfs in parton showers are discussed in [@ictppaper; @collinszu; @endp]. Applications to semi-inclusive processes and spin asymmetries are reviewed in [@murgiarev]. ![\[Fig: 4\] Comparison of the k$_\perp$-shower   with the NLO di-jet calculation : (left) single-jet distributions; (right) di-jet distributions.](jjbar-etjets.eps "fig:"){width="0.45\columnwidth"} ![\[Fig: 4\] Comparison of the k$_\perp$-shower   with the NLO di-jet calculation : (left) single-jet distributions; (right) di-jet distributions.](jjbar-ptsum.eps "fig:"){width="0.45\columnwidth"} Besides jet final states, the corrections to collinear-ordered showers discussed in this article will also be relevant to heavy particle production [@hann04; @hef; @hgs], including phenomenological studies of small-$x$ broadening in W and Z $p_\perp$ distributions [@cpyuan1], kinematical relations of DIS event shapes with Drell-Yan production [@dasgqt], heavy flavor production. First results on top-antitop pair production at the LHC may be found in the first paper of reference [@ictppaper]. 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--- author: - | Balázs Hidasi [^1]\ Gravity R&D Inc.\ Budapest, Hungary\ `[email protected]` Alexandros Karatzoglou\ Telefonica Research\ Barcelona, Spain\ `[email protected]` Linas Baltrunas [^2]\ Netflix\ Los Gatos, CA, USA\ `[email protected]` Domonkos Tikk\ Gravity R&D Inc.\ Budapest, Hungary\ `[email protected]` bibliography: - 'citations.bib' title: | Session-based Recommendations with\ Recurrent Neural Networks --- ### Acknowledgments {#acknowledgments .unnumbered} The work leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under CrowdRec Grant Agreement n$^\circ$ 610594. [^1]: The author spent 3 months at Telefonica Research during the research of this topic. [^2]: This work was done while the author was a member of the Telefonica Research group in Barcelona, Spain

--- author: - 'Chao Yeh Chen and Kristen Grauman [^1]' bibliography: - 'strings.bib' - 'ref.bib' title: 'Efficient Activity Detection in Untrimmed Video with Max-Subgraph Search' --- Acknowledgment {#acknowledgment .unnumbered} ============== We thank the anonymous reviewers for their feedback, and Sudheendra Vijayanarasimhan for helpful discussions. This research is supported in part by ONR PECASE N00014-15-1-2291. [^1]:

--- abstract: 'We introduce a generalization of the Temperley–Lieb algebra. This generalization is defined by adding certain relations to the algebra of braids and ties. A specialization of this last algebra corresponds to one small Ramified Partition algebra, this fact is the motivation for the name of our generalization.' address: 'Departamento De Matemáticas, Universidad de Valparaíso, Gran Bretaña 1091, Valparaíso, Chile.' author: - Jesús Juyumaya date: 'April 1, 2013' title: 'A Partition Temperley–Lieb Algebra ' --- [^1] Introduction {#introduction .unnumbered} ============ The Temperley–Lieb algebra appears originally in Statistical Mechanics as well as in Knot theory, quantum groups and subfactors of von Neumann algebras. This algebra was discovered by Temperley and Lieb by building transfer matrices[@tl]. Further, this algebra was rediscovered by V. Jones[@jo83] who used it in the construction of his polynomial invariant for knots known as the Jones polynomial[@jo]. From a purely algebraic point of view, the Temperley–Lieb algebra is a quotient of the Iwahori–Hecke algebra by the two–sided ideal generated by the Steinberg elements $h_{ij}$ associated to $h_i$’s, where $\vert i-j \vert =1$ and $h_i$’s denote the usual generators of the Iwahori–Hecke algebra, view p. 35[@gohajo]. In other words, the Temperley–Lieb algebra can be defined by the usual presentation of the Iwahori–Hecke algebra but by adding the relations $h_{ij}=0$, for all $\vert i-j \vert =1$. Using this point of view, there are several generalizations of the Temperley–Lieb algebra, e.g. see [@fan; @gojula]. This paper proposes a generalization of the Temperley–Lieb algebra by adding relations of Steinberg types to the [*algebra of braid and ties*]{}[@aj; @ry]. The algebra of braid and ties ${\mathcal E}_n(u)$, where $u$ is a parameter and $n$ denotes a positive integer, can be regarded as a generalization of the Hecke algebra and recently E. O. Banjo proved that ${\mathcal E}_n(1)$ is isomorphic to a small ramified partition algebra, see Theorem 4.2[@ba]. The possible connexion of the ${\mathcal E}_n(u)$ and the Partition algebras [@joPA; @mar1] was speculated first by S. Ryom–Hansen[@ry]. The algebra ${\mathcal E}_n(u)$ is defined by two sets of generators and relations. One set of generators $T_1,\ldots , T_{n-1}$ reflects the braid generators of the Yokonuma–Hecke algebra[@yo; @th; @chda] of type $A$ and the other set of generators $E_1, \ldots ,E_{n-1}$ reflects the behavior of the monoid $\mathsf{P}_n$ associated to the set partitions of $\{1, \dots , n\}$. Thus, ${\mathcal E}_n(u)$ also can be thought as a $u$–deformation of an amalgam among the symmetric group on $n$ symbols and $\mathsf{P}_n$. In short, in this paper we define and study the [*Partition Temperley–Lieb algebra*]{}, denoted ${\rm PTL}_n(u)$, which is defined by adding to the presentation of ${\mathcal E}_n(u)$ mentioned above the following relations $$E_iE_jT_{ij}=0 \quad \text{for all}\quad \vert i-j\vert=1$$ where $T_{ij}$ is the Steinberg element associated to the $T_i$’s. This work is organized as follows. In Section 1 we fix notations and we recall the definition of the Jimbo representation. In Section 2 we recall the definition of the algebra ${\mathcal E}_n(u)$, we have included also some results from [@ry] which are used in the paper. In Section 3 we construct a non–faithful tensor representation of the algebra ${\mathcal E}_n(u)$ which is used in Section 4 for the definition of our Partition Temperley–Lieb algebra ${\rm PTL}_n(u)$. The Section 5 shows two presentations of the ${\rm PTL}_n(u)$. By using one of these presentations we constructed a span linear set of ${\rm PTL}_n(u)$ which is conjectured that is a basis for the Partition Temperley–Lieb algebra. Finally, based on a conjecture that the algebra ${\mathcal E}_n(u)$ supports a Markov trace, we prove in Section 7 under which condition this trace could pass to ${\rm PTL}_n(u)$. Preliminaries ============= Along the paper algebra means unital associative algebra, with unity $1$, over the field of rational function $K:={\Bbb C}(\sqrt{u})$ in the variable $\sqrt{u}$. Consequently, we put $u = (\sqrt{u})^2$. Let $ {\rm H}_n = {\rm H}_n(u)$ be the Iwahori–Hecke algebra of type $A$, that is, the algebra presented by generators $1, h_1, \ldots , h_{n-1}$ subject to braid relations among the $h_i$’s and the quadratic relation $h_i^2 = u + (u-1)h_i$, for all $i$. We shall recall the Jimbo representation of the Hecke algebra. Set $V$ the $K$–vector space with basis $\{v_1, v_2\}$. Denotes by ${\bf J}$ the endomorphism of $V\otimes V$ defined through the mapping $$\begin{array}{ccl} {\bf J}(v_i\otimes v_j) & = & -v_i \otimes v_j \qquad \text{for } \quad i=j\\ {\bf J}(v_1\otimes v_2) & = & (u-1)\,v_1\otimes v_2 + \sqrt{u}\, v_2\otimes v_1 \\ {\bf J} (v_2\otimes v_1) & = & \sqrt{u}\, v_1\otimes v_2. \end{array}$$ The Jimbo representation of ${\rm H}_n$ in $V^{\otimes n}$ is defined by mapping $h_i\mapsto {\bf J}_i$, where ${\bf J}_i$ acts as the identity, with exception of the factors $i$ and $i+1$, where acts by ${\bf J}$. \[kerJ\] The kernel of the Jimbo representation is the two–sided ideal generated by $h_{ij}$, where $\vert i-j\vert =1$ and $$h_{ij}:= 1 + h_i + h_j + h_ih_j + h_jh_i + h_ih_jh_i.$$ It is well known that the Temperley–Lieb algebra can be defined as the quotient of the Iwahori–Hecke algebra by the Kernel of Jimbo representation. Thus, the Temperley–Lieb algebra can be defined by adding extra non–redundant relations to the above presentations of the Hecke algebra. More precisely, we have the following definition. The Temperley–Lieb algebra ${\rm TL}_n = {\rm TL}_n(u)$ is the algebra generated by $1, h_1, \ldots , h_{n-1}$ subject to the following relations: $$\begin{aligned} h_i^2 & = & u + (u-1)h_i \qquad \text{ for all $i$}\label{tl1}\\ h_ih_j & = & h_j h_i \qquad \text{ for $\vert i - j\vert >1$}\label{tl2}\\ h_ih_jh_i & = & h_jh_i h_j \qquad \text{ for $\vert i - j\vert =1$}\label{tl3}\\ h_{ij} & = & 0\qquad \text{ for $\vert i - j\vert =1$}.\label{tl4}\end{aligned}$$ It is well known that the dimension of ${\rm TL}_n$ is the $n$th Catalan number $C_n: = \frac{1}{n+ 1}$ $2n\choose{n}$ [@jo83] and that ${\rm TL}_n$ has a presentation (reduced) with idempotents generators. Indeed, making $$f_i := \frac{1}{1+u}(1+h_i)$$ we have the following proposition. \[pretl\] ${\rm TL}_n$ can be presented by generators $1, f_1, \ldots ,f_{n-1}$ satisfying the following relations $$\begin{aligned} f_i^2 & = & f_i \qquad \text{ for all $i$}\label{pretl1}\\ f_if_j & = & f_j f_i \qquad \text{ for $\vert i - j\vert >1$}\label{pretl2}\\ f_if_jf_i & = & \frac{u}{(1+u)^ 2} f_i\qquad \text{ for $\vert i - j\vert =1$}. \label{pretl3}\end{aligned}$$ By virtue Proposition \[kerJ\], the Jimbo representation of the Iwahori–Hecke algebra defines a representation of the Temperley–Lieb algebra. In terms of the generators $f_i$’s, this representation, called also the Jimbo representation, acts on $V^{\otimes n}$ by mapping $f_i\mapsto {\bf F}_i$. The endomorphism ${\bf F}_i$ acts as the identity, with exception of the factors $i$ and $i+1$, where acts by ${\bf F}\in {\rm End}(V^{\otimes 2})$, $$\begin{array}{ccl} {\bf F} (v_i\otimes v_j) & = & 0 \qquad \text{for } \quad i=j\\ {\bf F} (v_1\otimes v_2) & = & (u + 1)^{-1}(u\, v_1\otimes v_2 + \sqrt{u}\, v_2\otimes v_1) \\ {\bf F} (v_2\otimes v_1) & = & (u + 1)^{-1} (\sqrt{u}\,v_1\otimes v_2 + v_2\otimes v_1 ). \end{array}$$ The algebra of braids and ties ============================== Let $\bf n$ be the poset $\{1, \dots , n\}$. A partition of $\bf n$ is a collection of pairwise disjoint subposets (called parts) whose union is equal to $\bf n$. We shall denote $\mathsf{P}_n$ the set formed by the partitions of $\bf n$. The cardinal $b_n$ of $\mathsf{P}_n$ is known as the $n$th Bell number. Let $I\in\mathsf{P}_n$, an arc $i\frown j $ of $I$ is an ordered pair $(i,j)\in \{1, \dots , n\}\times \{1, \dots , n\}$ such that 1. $i<j$ 2. $i$ and $j$ are in the same part of $I$ 3. if $k$ is in the same part as $i$ and $i<k\leq j$, then $k=j$ In other words the arcs are pairs of adjacent elements in each part of $I$. Therefore the elements of $ \mathsf{P}_n$ can be represented by a graph with vertices $\{1, \dots , n\}$ and whose edge connecting the vertices $i$ and $j$ if and only if $i\frown j $ is an arc of $I$. For example, for $n=3$ we have $$\{\{1,2\}, \{3\}\} \qquad\text{is represented by} \qquad \begin{picture}(50,25) %\put(0,5){\line(1,0){25}} \put(-2,2){$\bullet$} \put(23,2){$\bullet$} \put(-1, -5){\tiny{1}} \put(24, -5){\tiny{2}} \qbezier(0,5)(12,25)(25,5) % \put(-2,2){$\bullet$} \put(48,2){$\bullet$} \put(49, -5){\tiny{3}} \end{picture}$$ and so on. The set $ \mathsf{P}_n$ can be regarded naturally as a poset, where the partial order $\preceq$, is defined by: $ I= (I_1, \dots I_k) \preceq J=(J_1, \dots J_l)$ if and only if each $J_i$ is a union of certain $I_i$’s. By using $\preceq$ we give to $ \mathsf{P}_n$ a structure of commutative monoid by defining the product $I\ast J$, of $I$ with $J$, as the minimum element of the poset $ \mathsf{P}_n$ containing $I$ and $J$. Clearly the unity is $\{\{1\}, \{2\}, \dots , \{n\}\}\}$ which is represented by $\begin{picture}(80,9) \put(-2,2){$\bullet$} \put(23,2){$\bullet$} %\put(48,2){$\bullet$} \put(73, 2){$\bullet$} \put(-1, -5){\tiny{$1$}} \put(24, -5){\tiny{$2$}} %\put(44, -5){\tiny{$i+1$}} \put(74, -5){\tiny{$n$}} %\qbezier(25,5)(37,25)(50,5) %\put(7,3){\dots} \put(40,3){\dots} \end{picture}$. The monoid $ \mathsf{P}_n$ is generated by the unity and the elements: $$\begin{picture}(80,25) \put(-2,2){$\bullet$} \put(23,2){$\bullet$} \put(48,2){$\bullet$} \put(73, 2){$\bullet$} \put(-1, -5){\tiny{$1$}} \put(24, -5){\tiny{$i$}} \put(44, -5){\tiny{$i+1$}} \put(74, -5){\tiny{$n$}} \qbezier(25,5)(37,25)(50,5) \put(7,3){\dots} \put(56,3){\dots} \end{picture} \qquad \text{for all $1\leq i\leq n$}$$ The Hasse diagram for $ \mathsf{P}_3$ is: $$\begin{picture}(200,110) %\put(90,100){d} \put(88,97){$\bullet$} \put(113,97){$\bullet$} \put(138,97){$\bullet$} \put(89, 90){\tiny{1}} \put(114, 90){\tiny{2}} \put(139, 90){\tiny{3}} \qbezier(115, 100)(127,120)(140,100) \qbezier(90, 100)(102,120)(115,100) %%% %\put(0,100){\line(1,0){25}} \put(-2,47){$\bullet$} \put(23,47){$\bullet$} \put(48,47){$\bullet$} \put(-1,40){\tiny{1}} \put(24, 40){\tiny{2}} \put(49, 40){\tiny{3}} \qbezier(0, 50)(12,70)(25,50) %%% %\put(90,50){d} \put(88,47){$\bullet$} \put(113,47){$\bullet$} \put(138,47){$\bullet$} \put(89, 40){\tiny{1}} \put(114, 40){\tiny{2}} \put(139, 40){\tiny{3}} \qbezier(115, 50)(127,70)(140,50) %%% %\put(180,50){d} \put(178,47){$\bullet$} \put(203,47){$\bullet$} \put(228,47){$\bullet$} \put(179, 40){\tiny{1}} \put(204, 40){\tiny{2}} \put(229, 40){\tiny{3}} \qbezier(180, 50)(205,70)(230,50) %%% %\put(90,0){d} \put(88,-3){$\bullet$} \put(113,-3){$\bullet$} \put(138,-3){$\bullet$} \put(89, -10){\tiny{1}} \put(114, -10){\tiny{2}} \put(139, -10){\tiny{3}} %\qbezier(115, 0)(127,20)(140,0) %\qbezier(90, 0)(102,20)(115,0) %%% \put(117,13){\line(0,1){18}} \put(117,63){\line(0,1){18}} \put(80,13){\line(-1,1){18}} \put(150,13){\line(1,1){18}} \put(60,63){\line(1,1){18}} \put(170,63){\line(-1,1){18}} \end{picture}$$ And we have, for example: $$\begin{picture}(221,25) %\put(0,5){\line(1,0){25}} \put(-2,2){$\bullet$} \put(23,2){$\bullet$} \put(48,2){$\bullet$} \put(-1, -5){\tiny{1}} \put(24, -5){\tiny{2}} \put(49, -5){\tiny{3}} \qbezier(0,5)(12,25)(25,5) %% \put(63, 5){$\ast$} %% \put(78,2){$\bullet$} \put(104,2){$\bullet$} \put(129,2){$\bullet$} \put(79, -5){\tiny{1}} \put(104, -5){\tiny{2}} \put(129, -5){\tiny{3}} \qbezier(105,5)(118,25)(133,5) %% \put(155, 5){$=$} %% \put(178,2){$\bullet$} \put(204,2){$\bullet$} \put(229,2){$\bullet$} \put(179, -5){\tiny{1}} \put(204, -5){\tiny{2}} \put(229, -5){\tiny{3}} \qbezier(180,5)(193,25)(208,5) \qbezier(208,5)(221,25)(232,5) \end{picture}$$ As usual we denote $S_n$ the symmetric group on symbols and we denote $s_i$ the transposition $(i,\, i+1)$. For $I = \{ I_1, \dots , I_m\}\in \mathsf{P}_n$ and $w\in S_n$ we define $wI = \{wI_1,\ldots , wI_m\}$, where $wI_i$ is the subposet of $\bf n$ obtained by applying $w$ to the elements of $I_i$. We denote ${\mathcal E}_n={\mathcal E}_n(u)$ the algebra generated by $1, T_1, \ldots, T_{n-1}, E_1, \ldots , E_{n-1}$ satisfying the following relations: $$\begin{aligned} T_iT_j & = & T_jT_i \qquad \text{for $\vert i-j \vert >1$}\label{E1}\\ T_iT_jT_i & = & T_jT_iT_i \qquad \text{ for $\vert i - j\vert =1$}\label{E2} \\ T_i^2 & = & 1 + (u-1) E_{i} \left( 1+ T_i\right)\qquad \text{ for all $i$}\label{E3}\\ E_iE_j & = & E_j E_i \qquad \text{ for all $i,j$}\label{E4}\\ E_i^2 & = & E_i \qquad \text{ for all $i$}\label{E5} \\ E_iT_j & = & T_j E_i \qquad \text{ for $\vert i - j\vert >1$}\label{E6}\\ E_iT_i & = & T_i E_i \qquad \text{ for all $i$}\label{E7}\\ E_iE_jT_i & = & T_iE_iE_j \quad = \quad E_jT_iE_j\qquad \text{ for $\vert i - j\vert =1$}\label{E8}\\ E_iT_jT_i & = & T_jT_i E_j \qquad \text{ for $\vert i - j\vert =1$}\label{E9}.\end{aligned}$$ If $w= s_{i_1}\cdots s_{i_k}\in S_n$ is reduced form for $w$, we write $T_w := T_{i_1}\cdots T_{i_k}$ (this is a possible debt to a well known result of H. Matsumoto). For $i<j$, we define $E_{ij}$ as $$E_{ij} = \left\{\begin{array}{ll} E_i & \text{for} \quad j = i +1\\ T_i \cdots T_{j-2}E_{j-1}T_{j-2}^{-1}\cdots T_{i}^{-1}& \text{otherwise} \end{array}\right.$$ For any $J=\{i_1, i_2, \ldots , i_k\}$ subposet of $\bf n$ we define $E_J=1$ if $k=1$ and $$E_J := E_{i_1i_2}E_{i_2i_3}\cdots E_{i_{k-1}i_k} \quad \text{for}\quad k>1$$ Note that $E_{\{i,j\}} = E_{ij}$. Also we note that in Lemma 4[@ry] it is proved that $E_J$ can be computed as $$E_{J} = \prod_{j\in J,\, j\not= i_0}E_{i_0j}\qquad (i_0:= \text{min}\{i\,;\,i\in J\})$$ For $I = \{I_1, \ldots , I_m\}\in \mathsf{P}_n$, we define $E_I$ as $$E_I = \prod_{k}E_{I_k}$$ The Corollary 2[@ry] implies the following proposition. The mapping $E_i\mapsto %\{\{1,2\}, \{3\}\} \qquad\text{is represented by} \qquad \begin{picture}(80,25) \put(-2,2){$\bullet$} \put(23,2){$\bullet$} \put(48,2){$\bullet$} \put(73, 2){$\bullet$} \put(-1, -5){\tiny{$1$}} \put(24, -5){\tiny{$i$}} \put(44, -5){\tiny{$i+1$}} \put(74, -5){\tiny{$n$}} \qbezier(25,5)(37,25)(50,5) \put(7,3){\dots} \put(56,3){\dots} \end{picture} $ defines a monoid isomorphism between the monoid generated by $1, E_1, \dots , E_{n-1}$ and $\mathsf{P}_n$. For $I \in \mathsf{P}_n$ and $w\in S_n$, we have $$T_w E_I T_w^{-1} = E_{wI}.$$ \[basEn\] The set ${\Bbb S}_n:=\{E_I T_w \,; \,w\in S_n,\, I\in \mathsf{P}_n\}$ is a linear basis of ${\mathcal E}_n$. Hence the dimension of ${\mathcal E}_n$ is $b_nn!$. A tensorial representation for ${\mathcal E}_n$ =============================================== In this section we will define a tensorial representation for ${\mathcal E}_n$. This representation is nothing more than a variation of that constructed by S. Ryom–Hansen in Section 3[@ry]. We note that, contrary to the representation constructed by Ryom–Hansen, our variation is a non–faithful representation. This fact is the key point in order to define the Partition Temperley–Lieb algebra as a quotient of ${\mathcal E}_n$. Let $V$ be the $K$–vector space with basis $\{v_i^r \,;\, 1\leq i,r\leq n\}$, we define the endomorphisms ${\bf E}$ and $\bf{T}$ of $V^{\otimes 2}$ through the following mapping, $${\bf E}(v_{i}^{r}\otimes v_{j}^{s}) := \left\{\begin{array}{lr}0 & \qquad \text{for } \quad r\not= s\\ v_{i}^{r}\otimes v_{j}^{s} & \qquad \text{for } \quad r=s \end{array}\right.$$ $${\bf T}(v_{i}^{r}\otimes v_{j}^{s}) := \left\{\begin{array}{ll} -v_{j}^{s}\otimes v_{i}^{r} & \qquad \text{for } \quad r\not=s \\ -v_{i}^{r}\otimes v_{j}^{s} & \qquad \text{for } \quad r=s, \, i = j\\ (u-1)\,v_i^r\otimes v_j^s + \sqrt{u}\, v_j^s\otimes v_i^r & \qquad \text{for } \quad r=s, \, i < j \\ \sqrt{u}\, v_j^s\otimes v_i^r & \qquad \text{for } \quad r=s, \, i>j \end{array}\right.$$ Define now, ${\bf E}_i$ (respectively ${\bf T}_i$) as the endomorphism of $V^{\otimes n}$ that acts as the identity with exception on the factors $i$ and $i+1$ where acts by ${\bf E}$ (respectively ${\bf T}$). \[JimboEn\] The mapping $E_i \mapsto {\bf E}_i$, $T_i \mapsto {\bf T}_i$ defines a representation $\mathcal{J}_n$ of $\mathcal{E}_n$ in $V^{\otimes n}$. The proof uses the same strategy as Theorem 1[@ry]. We only need to check that the operators $ {\bf E}_i$ and ${\bf T}_i$ satisfy the respective relations (\[E1\])–(\[E9\]). The relations (\[E1\]), (\[E4\])–(\[E7\]) clearly hold. To check relation (\[E3\]) it is enough to take $n=2$. Evaluating the relation in $ v_i^r\otimes v_j^s$ with $r=s$, the relation becomes the Hecke quadratic relation. In the case $r\not= s$, the operator ${\bf E}(1 + {\bf T})$ acts as zero and ${\bf T}^2$ as the identity, hence the relation holds. To check the remaining of the relations, without loss of generality, we can suppose $n=3$. Also we observe that it is enough to check the relations in question on the basis elements $x= v_{i}^{r}\otimes v_{j}^{s}\otimes v_{k}^{t}$. By simplicity we shall introduce the following notation: whenever we have two repetitions in the upper indices in the basis elements, we omit the two repeated upper indices and we replace the remaining indices by a prime, e.g. $v_{i}^{r}\otimes v_{j}^{s}\otimes v_{k}^{r}$ is written simply as $v_{i}\otimes v_{j}^{\prime}\otimes v_{k}$. Then when we have two repetitions in the upper indices we shall distinguish three forms of elements: $$\label{3form} v_{i}^{\prime}\otimes v_{j}\otimes v_{k} \qquad v_{i}\otimes v_{j}^{\prime}\otimes v_{k} \qquad v_{i}\otimes v_{j}\otimes v_{k}^{\prime}$$ Further, in these elements we can suppose that the lower indices are $1$ or $2$ since ${\bf T}$ acts according the order in the pair formed by lower indices. Now, the action of ${\bf T}$ on primed and unprimed elements is, up to sign, a transposition, so we can suppose that the lower index of the primed factor is always $1$. Therefore, the elements in the form as (\[3form\]) can be reduced to consider the following cases: $$\label{12form} \begin{array}{lll} v_{1}^{\prime}\otimes v_{1}\otimes v_{1}\qquad & v_{1}\otimes v_{1}^{\prime}\otimes v_{1}\qquad & v_{1}\otimes v_{1}\otimes v_{1}^{\prime}\\ v_{1}^{\prime}\otimes v_{1}\otimes v_{2}\qquad & v_{1}\otimes v_{1}^{\prime}\otimes v_{2} \qquad& v_{1}\otimes v_{2}\otimes v_{1}^{\prime}\\ v_{1}^{\prime}\otimes v_{2}\otimes v_{1} \qquad & v_{2}\otimes v_{1}^{\prime}\otimes v_{1} \qquad & v_{2}\otimes v_{1}\otimes v_{1}^{\prime}\\ v_{1}^{\prime}\otimes v_{2}\otimes v_{2} \qquad & v_{2}\otimes v_{1}^{\prime}\otimes v_{2}\qquad & v_{2}\otimes v_{2}\otimes v_{1}^{\prime} \end{array}$$ The checking of (\[E8\]) and (\[E9\]) are similar and routine. Thus we shall check only the first relation of (\[E8\]). If all upper indices in $x$ are distinct the operator ${\bf E}_i{\bf E}_j$ acts as zero and as the identity if all upper indices are equals. Hence ${\bf E}_1{\bf E}_2{\bf T}_1$ and ${\bf T}_1{\bf E}_1{\bf E}_2$ coincide on such $x$’s. Now it is easy to check the relation whenever $x$ is an element of (\[12form\]) whose unprimed factor has equal lower indices. The checking on the other elements of (\[12form\]) results from a direct computation, e.g., for $x= v_{1}\otimes v_{2}\otimes v_{1}^{\prime}$ we have $${\bf E}_1{\bf E}_2{\bf T}_1 (x)=(u-1){\bf E}_1{\bf E}_2(x) + \sqrt{u}\,{\bf E}_1{\bf E}_2( v_{2}\otimes v_{1}\otimes v_{1}^{\prime} )= 0 = {\bf T}_1{\bf E}_1{\bf E}_2(x)$$ Finally we will check the relation (\[E2\]). If in the basis elements the upper indices are all equal we are in the situation of Jimbo representation ${\bold J}$. If all upper indices are different the action becomes, up to sign, in the permutation action on the factors of the basis elements. Therefore, it only remains to check that (\[E2\]) is true when one evaluates on the elements of (\[12form\]). Now, it is easy to see that the evaluation of both sides of (\[E2\]) on the elements of (\[12form\]) whose unprimed factors are equal is $-\sigma_{13}$, where $\sigma_{13}$ permutes the the first with the third factor in the tensor product. The check of (\[E2\]) on the remaining elements of (\[12form\]) is all similar for all. We shall do, as a representative case, the case $x= v_{1}^{\prime}\otimes v_{1}\otimes v_{2}$: $$\begin{aligned} {\bf T}_2{\bf T}_1{\bf T}_2(x) & = & (u-1)\,{\bf T}_2{\bf T}_1(v_{1}^{\prime}\otimes v_{1}\otimes v_{2}) + \sqrt{u}\,{\bf T}_2{\bf T}_1(v_{1}^{\prime}\otimes v_{2}\otimes v_{1} )\\ & = & -(u-1)\,{\bf T}_2(v_{1}\otimes v_{1}^{\prime}\otimes v_{2}) - \sqrt{u}\,{\bf T}_2(v_{2}\otimes v_{1}^{\prime}\otimes v_{1} )\\ & = & (u-1)\,(v_{1}\otimes v_{2}\otimes v_{1}^{\prime}) + \sqrt{u}\,( v_{2}\otimes v_{1}\otimes v_{1}^{\prime} )\\ & = & {\bf T}_1(v_{1}\otimes v_{2}\otimes v_{1}^{\prime}) \\ & = & -{\bf T}_1{\bf T}_2(v_{1}\otimes v_{1}^{\prime}\otimes v_{2}) = {\bf T}_1{\bf T}_2{\bf T}_1(x).\end{aligned}$$ The [PTL]{} algebra =================== We want to define a generalization of Temperley–Lieb algebra by using the algebra $\mathcal{E}_n$, we shall call this generalization the Partition Temperley–Lieb algebra which is denoted ${\rm PTL}_n$. A first natural attempt of definition ${\rm PTL}_n$ is as the algebra that results by adding to defining relations of $\mathcal{E}_n$ the relations $T_{ij}= 0$, where $T_{ij}$ are the Steinberg elements $T_{ij}$’s associated to the $T_i$’s, $$T_{ij} := 1 + T_i + T_j +T_iT_j + T_j T_i +T_iT_jT_j\quad \text{where }\qquad \vert i-j \vert =1$$ As in the classical case we want that the Jimbo representation $\mathcal{J}$ of $\mathcal{E}_n$ passes to ${\rm PTL}_n$, hence the $T_{ij}$’s must be killed by $\mathcal{J}$. But unfortunately this does not happen. In fact, for $n=3$ and by taking $x= v_1\otimes v_2 \otimes v_1^{\prime}$, we have $${\bf T}_1 x = (u-1)v_1\otimes v_2 \otimes v_1^{\prime} + \sqrt{u}\,v_2 \otimes v_1 \otimes v_1^{\prime}\qquad {\bf T}_2x = - v_1\otimes v_1^{\prime}\otimes v_2$$ $${\bf T}_2{\bf T}_1 x = -(u-1)v_1\otimes v_1^{\prime} \otimes v_2 - \sqrt{u}\,v_2 \otimes v_1^{\prime} \otimes v_1\qquad {\bf T}_1{\bf T}_2x = v_1^{\prime}\otimes v_1\otimes v_2$$ $${\bf T}_1{\bf T}_2{\bf T}_1 x = (u-1)v_1^{\prime}\otimes v_1\otimes v_2 + \sqrt{u}\,v_1^{\prime} \otimes v_2 \otimes v_1$$ Then $$\begin{aligned} (\mathcal{J} T_{12} )x & = & u\, v_1\otimes v_2 \otimes v_1^{\prime} - u\, v_1\otimes v_1^{\prime} \otimes v_2 + \sqrt{u}\,v_2 \otimes v_1 \otimes v_1^{\prime} \\ & & - \sqrt{u}\,v_2 \otimes v_1^{\prime} \otimes v_1 + u\,v_1^{\prime}\otimes v_1\otimes v_2 + \sqrt{u}\,v_1^{\prime} \otimes v_2 \otimes v_1\end{aligned}$$ Therefore $\mathcal{J}$ does not kill $T_{12}$. Having in mind the above discussion we make the following definition. The Partition Temperley–Lieb algebra ${\rm PTL}_n={\rm PTL}_n(u)$ is defined by adding to the defining presentation of ${\mathcal E}_n$ the relations: $$\label{rptl} E_iE_jT_{i,j} = 0 \quad \text{for all}\quad \vert i -j \vert =1.$$ Clearly, from (\[E8\]) we have that $E_iE_jT_{i,j} = 0$ is equivalent to $T_{i,j} E_iE_j= 0$. Notice that by taking $E_i=1$ the algebra ${\rm PTL}_n$ coincides with the classical Temperley–Lieb algebra. Also, we note that the defining relations of ${\rm PTL}_n$ hold when $T_i$ is replaced by the generators $h_i$ of the Temperley–Lieb algebra and $E_i$ is replaced by 1, thus the mapping $E_i\mapsto 1$ and $T_i\mapsto h_i$ defines an algebra homomorphism from ${\rm PTL}_n$ onto ${\rm TL}_n$. \[JimboPTLn\] The Jimbo representation $\mathcal{J}_n$ of $\mathcal{E}_n$ factors through the algebra ${\rm PTL}_n$. Without loss of generality we can suppose that $n=3$. Thus, we must prove that $\mathcal{J}_3 (E_1E_2T_{12})=0$. Now, keeping the notations used during the proof of Theorem \[JimboEn\], to prove the theorem it is enough to see that $\mathcal{J}_3 (E_1E_2T_{12})$ kill the basis elements $x= v_{i}^{r}\otimes v_{j}^{s}\otimes v_{k}^{t}$. If all upper indices in $x$ are equal, $\mathcal{J}_3$ is the Jimbo representation of the Hecke algebra, so $\mathcal{J}_3(T_{12})$ kill $x$; hence $\mathcal{J}_3 (E_1E_2T_{12})$ kill $x$ too. If the upper indices of $x$ are not all equal, we have that $x$ is killed by $\bf{E}_1$ or $\bf{E}_2$, hence $\mathcal{J}_3 (E_1E_2T_{12})(x)=0$. We are going to prove now that the set of relations (\[rptl\]) can be reduced to only one. To do this we need to introduce the following element $\Gamma$, $$\Gamma := T_1T_2\cdots T_{n-1}$$ \[gam\] For all $1\leq i, j\leq n-1$ we have: 1. $T_i=\Gamma^{i-1}T_1\Gamma^{-(i-1)}$ 2. $T_{i, i+1}=\Gamma^{i-1}T_{1,2}\Gamma^{-(i-1)}$ 3. $E_{i} = \Gamma^{i-1}E_1\Gamma^{-(i-1)}$ 4. $T_{i+1}\Gamma^{i-1} = \Gamma^{i-1}T_2 $ 5. $E_{\{i, i+2\}} = \Gamma^{i-1}E_{\{1,3\}}\Gamma^{-(i-1)}$ The statement (1) results from an inductive argument on $i$ and the braid relations of $T_i$’s. The statement (2) is a result applying (1). The proof of statement (3) is analogous to the proof of (1), that is: an argument inductive on $i$ and using the relation (\[E6\]). The statement (4) is clear, since (1). Finally, we have: $$\begin{aligned} \Gamma^{i-1}E_{\{1,3\}}\Gamma^{-(i-1)} & =& \Gamma^{i-1}T_2 E_1 T_2^{-1}\Gamma^{-(i-1)}\\ & =& \Gamma^{i-1}T_2 (\Gamma^{-(i-1)}E_i\Gamma^{(i-1)}) T_2^{-1}\Gamma^{-(i-1)}\\ &=& T_{i+1} E_iT_{i+1}^{-1}\end{aligned}$$ Thus, the statement (5) is proved. The relation $E_1E_2T_{1,2} = 0$ implies the relations $E_iE_jT_{i,j} = 0$, for all $\vert i-j\vert =1$. We can suppose $j=i+1$, since $T_{ij}=T_{ji}$ and $E_i$ and $E_j$ commute. From the statements (1) and (3)Lemma \[gam\], we have: $$\begin{aligned} E_i E_{i+1} T_{i, i+1} & = & (\Gamma^{i-1}E_1\Gamma^{-(i-1)})(\Gamma^{i}E_1\Gamma^{-i}) (\Gamma^{i-1}T_{1,2}\Gamma^{-(i-1)}) \\ & = & \Gamma^{i-1}E_1 \Gamma E_1\Gamma^{-1} T_{1,2}\Gamma^{-(i-1)}) = \Gamma^{i-1}E_1E_2 T_{1,2}\Gamma^{-(i-1)})\end{aligned}$$ Hence the proof follows. \[PTLquo\] The Partition Temperley–Lieb algebra ${\rm PTL}_n$ can be regarded as the quotient of ${\mathcal E}_n$ by the two–sided ideal generated by $E_1E_2T_{12}$. Others presentations for ${\rm PTL}_n$ ====================================== In order to have more comfortable notations we shall introduce the following element $\delta$, $$\delta := \frac{1-u}{1+u}\in K$$ Having in mind the definition of the idempotents generators $f_i$ of the Temperley–Lieb algebra, it is natural to consider the following definition. $$F_i := \frac{1}{u+1}(1 + T_i)\qquad (1\leq i \leq n-1)$$ It is obvious that $F_i$ commute with $E_i$ (and $T_i$) and that they form a set of generators for the algebra ${\rm PTL}_n$, but notice that the $F_i$’s are not idempotents. In fact, from (\[E3\]) we have $$F_i^2 = \frac{1}{(u+1)^2}(1 + 2T_i + 1 + (u-1)E_i + (u-1)E_iT_i)$$ then $$F_i^2 = (1+\delta)F_i - \delta E_iF_i$$ We have the following proposition \[preptlF\] ${\rm PTL}_n$ can be presented by the generators $1, E_1, \ldots , E_{n-1}, F_1, \ldots , F_{n-1}$ subject to the relations (\[E4\]), (\[E5\]) together with the following relations $$\begin{aligned} F_i^2 & = & (1+\delta)F_i - \delta E_iF_i \label{ptlF4}\\ F_i F_j & = & F_j F_i \quad \text{ for all $\vert i-j\vert >1$} \label{ptlF1}\\ F_i E_j & = & E_j F_i \quad \text{ for all $\vert i-j\vert >1$} \label{ptlF2}\\ E_i F_i & = & F_i E_i \label{ptlF3}\end{aligned}$$ and for all $\vert i - j\vert =1$: $$\begin{aligned} E_iE_jF_i & = & \quad F_i E_iE_j \quad = \quad E_jF_iE_j + \frac{1}{u+1} (E_iE_j - E_j)\label{ptlF5}\\ E_iF_jF_i & = & F_jF_iE_j + \frac{1}{u+1}\left[(E_i - E_j)F_j + F_i(E_i-E_j)\right]- \frac{1}{(u+1)^2}(E_i-E_j) \label{ptlF6}\\ F_iF_jF_i & = & \frac{1}{(u+1)^2}\left(F_i - (1-u)E_iF_i\right)\label{ptlF7}\end{aligned}$$ It is easy to check that (\[E1\]) (respectively (\[E6\])) is equivalent to (\[ptlF1\]) (respectively (\[ptlF2\])), so having in mind the previous discussion to the theorem, it only remains to prove that the relations (\[ptlF5\])–(\[ptlF7\]) hold and that relations (\[E8\]), (\[E9\]), (\[rptl\]) and (\[E2\]) can be deduced from the relations of the theorem. We have that $T_i= (u+1)F_i -1$. Now replacing this expression of $T_i$ in (\[E8\]) (respectively (\[E9\])) it is a routine to check that (\[E8\]) becomes (\[ptlF5\]) (respectively (\[ptlF6\])). We have to check that relation (\[rptl\]) is equivalent to relation (\[ptlF7\]). We have $$T_iT_j = ((u+1)F_i -1)((u+1)F_j -1) = (u+1)^2 F_iF_j - (u+1)F_i - (u+1)F_j +1$$ then $$\begin{aligned} T_iT_jT_i & = & (u+1)^3 F_iF_jF_i - (u+1)^2F_i^2 - (u+1)^2F_jF_i + (u+1)F_i\\ & & -(u+1)^2 F_iF_j + (u+1)F_i + (u+1)F_j -1\end{aligned}$$ Therefore, by using (\[ptlF4\]), we deduce $$\begin{aligned} T_iT_jT_i & = & (u+1)^3 F_iF_jF_i + (1-u^2)E_iF_i \\ & & - (u+1)^2F_jF_i - (u+1)^2F_iF_j + (u+1)F_j- 1 \end{aligned}$$ Now, substituting each summand of $T_{ij}$ by its expression in term of $F_i$’s one obtains $$T_{ij} = (u+1)^3 F_iF_jF_i + (1-u^2)E_iF_i - (u+1)F_i$$ Hence (\[rptl\]) is equivalent (\[ptlF7\]). Finally notice that (\[ptlF7\]) implies (\[E2\]), since the above expression of $T_iT_jT_i$ in terms of $F_i$’s tells us that (\[E2\]) is equivalent to $$(u+1)^2 F_iF_jF_i + (1-u)E_iF_i + F_j = (u+1)^2 F_jF_iF_j + (1-u)E_jF_j + F_i$$ Thus the proof is concluded. In this subsection we shall show a presentation of ${\rm PTL}_n$ by idempotent generators. For $1\leq i <j\leq n-1$, we define $$L_i := \frac{1}{1+u}\left( T_i + 1\right) \left(\alpha +(1-\alpha)E_i\right)\qquad \text{where}\quad \alpha := \frac{1+u}{2}$$ notice that $$\label{adL} L_i =\frac{1}{2}\left(T_i + \delta T_iE_i + \delta E_i + 1 \right) = \frac{1}{2}(1+T_i)(1+ \delta E_i)$$ Also we have $$\label{L->F} L_i = \frac{u+1}{2}F_i + \frac{1-u}{2}E_iF_i$$ It is clear that $L_i$ commute with $E_i$, $T_i$ and $F_i$. We have the following useful lemma. \[ide\] For all $i$ we have: 1. $L_i^2 = L_i$ 2. $(1+u)E_iL_i = E_i(1+ T_i)$ 3. $T_i = 2L_i +(u-1)E_iL_i -1$ 4. $E_iL_i = E_iF_i $ 5. $F_i = (1+\delta)L_i -\delta E_iL_i $. We have: $$L_i^2 = 4^{-1} (1+T_i)^2 (1+\delta E_i)^2 = 4^{-1}(2(1+ T_i) +(u-1)E_i(1+T_i))(1 +(2\delta+ \delta^2)E_i)$$ then $$\begin{aligned} L_i^2 & = & 4^{-1}(1+ T_i)(2 +(u-1)E_i)(1 +(2\delta+ \delta^2)E_i)\\ & = & 4^{-1}(1+ T_i)(2+ (2(2\delta+ \delta^2) + (u-1) + (u-1)(2\delta + \delta^2))E_i)\\ & = & 4^{-1}(1+ T_i)(2 + (2\delta + \delta^2)(1+u) + u-1 )E_i) \\ & = & 4^{-1}(1+ T_i)(2 + 2\delta E_i) = L_i.\end{aligned}$$ The second assertion follows by multiplying the formula of $L_i$ by $E_i$. To prove the third assertion, we bring first $E_iT_i$ from the second assertion and then we substitute this expression of $E_iT_i$ in (\[adL\]), thus the third assertion follows. The fourth assertion results by multiplying (\[L-&gt;F\]) by $E_i$. The fifth assertion result directly from (4) and (\[L-&gt;F\]). \[preptl\] ${\rm PTL}_n$ can be presented by the generators $1, E_1, \ldots , E_{n-1}, L_1, \ldots , L_{n-1}$ subject to the relations (\[E4\]), (\[E5\]) together with the following relations $$\begin{aligned} L_i^2 & = & L_i \label{ptl1}\\ L_i L_j & = & L_j L_i \quad \text{ for all $\vert i-j\vert >1$} \label{ptl2}\\ L_iE_j & = & E_j L_i \quad \text{ for all $\vert i-j\vert >1$} \label{ptl3}\\ L_i E_i & = & E_iL_i \label{ptl4}\end{aligned}$$ and for all $\vert i - j\vert =1$: $$\begin{aligned} & & E_iE_jL_i \, = \, L_i E_iE_j \, = \, E_jL_iE_j +2^{-1}(E_iE_j -E_j)\label{ptl5}\\ & & 4L_iL_jE_i + 2 E_j(L_j + L_i) + E_i \, = \, 4E_jL_iL_j + 2(L_i + L_j)E_i + E_j \label{ptl6}\end{aligned}$$ $$\label{ptl7} \begin{array}{c} 8L_iL_jL_i + 4(u-1) \left[L_iE_jL_jL_i +E_iL_iL_jL_i+ L_iL_jE_iL_i\right] \\ + (u-1)^2(u+5)E_iE_j L_iL_jL_i \, = \, 2L_i + 3(u-1)E_iL_i + (u-1)^2E_iE_jL_i %\label{ptl7} \end{array}$$ We will use the presentation of Theorem \[preptlF\]. From (5)Lemma \[ide\], follows that ${\rm PTL}_n$ is generated by $1$, $E_i$’s and $L_i$’s. Checking that (\[ptlF4\])–(\[ptlF6\]) are equivalent, respectively, to (\[ptl1\])–(\[ptl6\]) is a straight forward and just a routine, so we leave the computation to the reader. Thus, to finish the proof it only remains to check that (\[ptl7\])is equivalent to (\[ptlF7\]). We have $$\begin{aligned} F_iF_j & = & ((1+\delta )L_i -\delta E_iL_i)((1+\delta )L_j -\delta E_jL_j) \\ & =& (1+ \delta)^2L_iL_j -\delta (1+\delta)L_iE_jL_j - \delta (1+\delta)E_iL_iL_j +\delta^2 E_iL_iE_jL_j)\end{aligned}$$ Hence $$\begin{aligned} F_iF_jF_i & = & (1+\delta)^3 L_iL_jL_i -\delta (1+\delta)^2 \left[ L_iE_jL_jL_i + E_iL_iL_jL_i + L_iL_jE_iL_i\right] \\ & & + \delta^2(1+\delta)E_iL_iE_jL_jL_i + \delta^2(1+\delta)L_iE_jL_jE_iL_i + \delta^2(1+\delta)E_iL_iL_jE_iL_i \\ & & - \delta^3 E_iL_iE_jL_jE_iL_i\end{aligned}$$ Using now (\[E4\]), (\[E5\]), (\[ptl4\]) and (\[ptl5\]) we get $$\begin{aligned} F_iF_jF_i & = & (1+\delta)^3 L_iL_jL_i -\delta (1+\delta)^2 \left[ L_iE_jL_jL_i + E_iL_iL_jL_i + L_iL_jE_iL_i\right] \\ & & (2\delta^2(1+\delta)- \delta^3)E_iE_jL_iL_jL_i + \delta^2(1+\delta)E_iL_iL_jE_iL_i\end{aligned}$$ Now applying on the last term the relation (\[ptl4\]) and using later (\[ptl5\]), we get $E_iL_iL_jE_iL_i = L_i(E_iL_jE_i)L_i$, so $$\begin{aligned} E_iL_iL_jE_iL_i & = & L_i \left[ E_iE_jL_j -\frac{1}{2}(E_iE_j - E_i) \right]L_i \\ & = & E_iE_jL_iL_jL_i - \frac{1}{2}E_iE_jL_i^2 +\frac{1}{2}L_iE_iL_i \quad (\text{by using (\ref{ptl5})})\\ & = & E_iE_jL_iL_jL_i - \frac{1}{2}E_iE_jL_i +\frac{1}{2}E_iL_i \quad (\text{by using (\ref{ptl1}) and (\ref{ptl4})})\end{aligned}$$ Then $$\begin{aligned} F_iF_jF_i & = & (1+\delta)^3 L_iL_jL_i -\delta (1+\delta)^2 \left[L_iE_jL_jL_i + E_iL_iL_jL_i+L_iL_jE_iL_i\right] \\ & & + (2\delta^3 + 3\delta^ 2)E_iE_jL_iL_jL_i - \delta^2(1+\delta)\left[ \frac{1}{2}E_iE_jL_i -\frac{1}{2}E_iL_i\right]\end{aligned}$$ On the other side, from (4)Lemma \[ide\], we have $$F_i+ (u-1)E_iF_i = (1+\delta )L_i - \delta E_iL_i + (u-1)E_iL_i = (1+\delta )L_i - (u+2) \delta E_iL_i$$ Therefore, the relation (\[ptl7\]) is equivalent to $$\begin{aligned} & & (1+\delta)^3 L_iL_jL_i -\delta (1+\delta)^2 \left[L_iE_jL_jL_i +E_iL_iL_jL_i+ L_iL_jE_iL_i\right] + (2\delta^3 + 3\delta^ 2)E_iE_jL_iL_jL_i \\ & = & \frac{1}{(u+1)^2}\left[(1+\delta )L_i - (u+2) \delta E_iL_i \right] + \delta^2(1+\delta)\left[ \frac{1}{2}E_iE_jL_i -\frac{1}{2}E_iL_i\right]\end{aligned}$$ which is reduced, after multiplication by $(u+1)^2$, to $$\begin{aligned} & & \frac{8}{(u+1)} L_iL_jL_i -4\delta \left[L_iE_jL_jL_i +E_iL_iL_jL_i+ L_iL_jE_iL_i\right] + (1-u)^2(2\delta +3)E_iE_jL_iL_jL_i \\ & = & (1+\delta )L_i - (u+2) \delta E_iL_i + (1-u)^2(1+\delta)\left[ \frac{1}{2}E_iE_jL_i -\frac{1}{2}E_iL_i\right]\end{aligned}$$ or equivalently $$\begin{aligned} & & \frac{8}{(u+1)} L_iL_jL_i -4\delta \left[L_iE_jL_jL_i +E_iL_iL_jL_i+ L_iL_jE_iL_i\right] + (1-u)^2(2\delta +3)E_iE_jL_iL_jL_i \\ & = & (1+\delta )L_i - 3\delta E_iL_i + (1-u)\delta E_iE_jL_i \end{aligned}$$ Multiplying this last equation by $u+1$ we obtain (\[ptl7\]). By taking $E_i=1$ the elements $L_i$’s become $f_i$’s and the Theorem \[preptl\] and Theorem \[preptlF\] become Theorem \[pretl\]. A linear basis for ${\rm PTL}_n$ ================================ By using essentially Theorems \[preptlF\], \[JimboPTLn\] we shall construct a linear basis of ${\rm PTL}_n$. Further we use also the following lemmas. \[FE=EF\] For all $i,j$ such that $\vert i-j \vert=1$, we have: 1. $F_iE_{j} = T_iE_jT_i^{-1} F_i + \frac{1}{u+1}\left(E_j - T_iE_jT_i^{-1}\right)$ 2. $E_jF_i = F_i T_iE_jT_i^{-1} + \frac{1}{u+1}\left(E_j - T_iE_jT_i^{-1}\right)$ It is enough to expand $F_i$ in both side of the equality. \[aiju\] Any word in $1, F_1,\dots , F_{n-1}$, $E_1,\dots , E_{n-1}$ can be expressed as a $K$–linear combination of words in $E_i$’s and $F_i$’s having at most one $F_{n-1}$, $E_{n-1}$, or $F_{n-1}E_{n-1}$. It is a consequence of Proposition 1[@aj] and the fact that $F_i$ is a linear expression of $1$ and $T_i$. A word in $F_1, \dots , F_{n-1}$ is called $F$–reduced (or simply reduced) if and only if has the form $$\label{Fred} (F_{i_1}\cdots F_{j_1})(F_{i_2}\cdots F_{j_2})\cdots (F_{i_k}\cdots F_{j_k})$$ where $0\leq k\leq n-1$ and $$\begin{aligned} 1\leq i_1 < i_2<\cdots <i_k\leq n-1\\ 1\leq j_1 < j_2<\cdots <j_k\leq n-1\\ i_1\geq i_2, i_2\geq j_2 ,\ldots ,i_k\geq j_k\end{aligned}$$ Any word in $1, E_1, \dots , E_{n-1}, F_1, \dots , F_{n-1}$ may be written as $K$–linear combination of words in the form $E_IF$, where $I\in \mathsf{P}_n$ and $F$ is $F$–reduced. We have adapted the proof of Proposition 2.8[@gohajo]. We will use induction on $n$. The assertion is clearly valid for $n=2$. We assume now that the proposition is valid for $n$. Let $W$ a word in $1, E_1, \dots , E_n, F_1, \dots , F_n$. By using Lemma \[FE=EF\] we can move the $E_i$’s appearing in $W$ to the front position, obtaining in this way that $W$ is a linear combination of words in the form $E_IW^{\prime}$, where $W^{\prime}$ is a word in $1, F_1, \ldots , F_n$. Thus, to prove the proposition it is enough to show that $W^{\prime}$ is a linear combination of words in the form desired. Now, if $W^{\prime}$ does not contain $F_n$ then we are done. If $W^{\prime }$ contains $F_n$, according to Lemma \[aiju\], we have that $W^{\prime}$ is a linear combination of words in the form $$W_1R_nW_2$$ where $R_n = E_n$, $F_n$ or $E_nF_n$ and $W_i$ are words in $1, E_1, \dots , E_{n-1}, F_1, \dots , F_{n-1}$. If $R_n=E_n$, according to Lemma \[FE=EF\], we can move $R_n$ to the front position and then using the induction hypothesis we are done. Suppose $R_n = F_n$, we note that by induction hypothesis $W_2$ is a linear combination of words in the form $$E_JV(F_nF_{n-1}\cdots F_{j_k})$$ where now $J\in\mathsf{P}_{n-1}$, $V$ is a word reduced in $1, F_1, \dots , F_{n-2}$ (notice that $F_nF_{n-1}\cdots F_{j_k}$ could be empty). Hence $W^{\prime}$ is a linear combination of words of the form $$W_1F_n E_JV(F_nF_{n-1}\cdots F_{j_k})$$ Now, $F_n E_J= E_{s_nJ}F_n $, so using (\[ptlF4\]) and (\[ptlF2\]) follows that $W^{\prime}$ can be written as a linear combination $ (1 + \delta )N_1 - \delta N_2 $ with $N_1:= E_{J^{\prime}}V^{\prime}(F_nF_{n-1}\dots F_{j_k})$ and $N_2:= E_{J^{\prime}}V^{\prime}(E_nF_nF_{n-1}\dots F_{j_k})$, where $J^{\prime}\in\mathsf{P}_n$ and $V^{\prime}$ is a word in $1, F_1, \dots F_{n-1}$. Again we note that in $N_2$, $E_n$ can move to the front position, so $N_2$ is in fact in the form of $N_1$. Therefore, $W^{\prime}$ is a linear combination of words in the form $$E_{J^{\prime}}V^{\prime}(F_nF_{n-1}\dots F_{j_k})$$ where $J^{\prime}\in\mathsf{P}_n$ and $V^{\prime}$ is a word in $1, F_1, \dots F_{n-1}$. Applying the induction hypothesis, on $V^{\prime}$, we deduce that $W^{\prime}$ is a linear combination of words in the form $ E_I F $, where $I\in\mathsf{P}_n$ and $F$ has the form $$F= (F_{i_1}\cdots F_{j_1})(F_{i_2}\cdots F_{j_2})\cdots (F_{i_k}\cdots F_{j_k})$$ with $i$’s increasing and $i_l\geq j_l$, for all $1\leq l\leq k $. Thus it remains to prove that in $F$’s the $j$’s can be taken increasing. Suppose $j_1\geq j_2$, so $$F = (F_{i_1}\cdots F_{j_1 + 1})(F_{i_2}\cdots (F_{j_1}F_{j_1 +1}F_{j_1})\cdots F_{j_2})\cdots (F_{i_k}\cdots F_{j_k})$$ Then, by using (\[ptlF7\]), we have $F = (u + 1)^{-2}F_1 - (u+1)^{-1}\delta F_2$, where $$F_1 := (F_{i_1}\cdots F_{j_1 + 1})(F_{i_2}\cdots F_{j_1}\cdots F_{j_2})\cdots (F_{i_k}\cdots F_{j_k})$$ and $$F_2:=(F_{i_1}\cdots F_{j_1 + 1})(F_{i_2}\cdots (E_{j_1}F_{j_1})\cdots F_{j_2})\cdots F_{i_k}\cdots F_{j_k})$$ Clearly (applying Lemma \[FE=EF\]), $E_{j_1}$ in $F_2$ can be moved to the front position. Therefore, by using an inductive argument we deduce that $F$ can be expressed as a linear combination in the desired form. Hence $W^{\prime}$ can be written in the desired form. Thus, the proof is concluded. The set formed by the elements $E_IF$, where $I\in \mathsf{P}_n$ and $F$ is reduced , is a linear basis of ${\rm PTL}_n$. Hence the dimension of ${\rm PTL}_n$ is $b_nC_n$. Markov trace ============ For $d$ a positive integer we denote ${\rm Y}_{d,n}={\rm Y}_{d,n}(u)$ the Yokonuma–Hecke algebra, i.e. the algebra presented by braid generators $g_1, \ldots , g_{n-1}$ together with the framing generators $t_1, \ldots , t_{n}$ which satisfies the following defining relations: braids relation (of type $A$) among the $g_i$’s, $t_it_j=t_jt_i$, $g_it_j = t_{s_i(j)}g_i$ and $$g_i^2 = 1 + (u-1) e_i (1 + g_i)$$ where $e_i$ is defined as $$e_i :=\frac{1}{d}\sum_{s=1}^{d}t_i^st_{i+1}^{-s}$$ We have a natural algebra morphism $\psi : {\mathcal E}_n\mapsto {\rm Y}_{d,n}$ defined through the mapping $T_i\mapsto g_i$ and $E_i \mapsto e_i$. According to Lemma 2.1[@jula3] the defining relations of ${\mathcal E}_n$ are satisfied by changing $T_i$ by $g_i$ and $E_i$ by $e_i$. Hence the proof follows. \[trace\] Let $z$, $x_1, \ldots , x_{d-1}$ be in ${\Bbb C}$. There exists a unique family of linear map $ \{{\rm tr}_n\}_n $ on inductive limit associated to the family $\left\{{\rm Y}_{d,n}\right\}_n$ with values in ${\Bbb C}$ satisfying the rules: $$\begin{array}{rcll} {\rm tr}_n(ab) & = & {\rm tr}_n(ba) & \\ {\rm tr}_n(1) & = & 1 & \\ {\rm tr}_{n+1}(ag_n) & = & z\,{\rm tr}_n(a) & \qquad {\text for}\quad a \in {\rm Y}_{d,n} \\ {\rm tr}_{n+1}(at_{n+1}^m) & = & x_m{\rm tr}_n(a) & \qquad {\text for} \quad a \in {\rm Y}_{d,n}, \, 1\leq m\leq d-1. \end{array}$$ It is natural to consider the composition ${\rm tr}_n\circ \psi$ which defines a Markov trace on ${\mathcal E}_n$. This supports the following conjecture. \[conaj\]\[Aicardi, Juyumaya\] The algebra ${\mathcal E}_n$ supports a Markov trace. I.e. for all $n\in {\Bbb N}$ we have a unique linear map $\rho_n :{\mathcal E}_n\longrightarrow K( A, B)$ such that for all $x, y\in {\mathcal E}_n$, we have: 1. $\rho_n(1) = 1$ 2. $\rho_n(xy) = \rho_n(yx)$ 3. $\rho_{n+1}(xT_n) =\rho_{n+1}(xE_nT_n)=A \rho_n(x)$ 4. $\rho_{n+1}(xE_n) =B \rho_n(x)$ where $A$ and $B$ are parameters. According to the rule (3)Conjecture\[conaj\] of $\rho$ we have, $\rho(E_1T_1T_2T_1)= A\rho(E_1T_1^2)$. Now, $E_1T_1^2 = E_1 (1 + (u-1)E_1(1 + T_1))= uE_1 + (u-1) E_1T_1$. So $$\rho(E_1T_1T_2T_1) = A(uB + (u-1)A) = u AB + (u-1)A^2.$$ Assuming that Conjecture \[conaj\] is true, we are going to study when the Markov trace $\rho_n$ passes to ${\rm PTL}_n$. According to Corollary \[PTLquo\], studying the factorization of $\rho_n$ to ${\rm PTL}_n$ is reduced to studying the values of $\rho_n$ on the two–sided ideal generated by $E_iE_jT_{12}$. For this study we need the following lemmas. \[TwT12\] 1. $T_1T_{12} = \left[ 1 + (u-1)E_1\right] T_{12}$ 2. $T_2T_{12} = \left[ 1 + (u-1)E_2\right] T_{12}$ 3. $T_1T_2T_{12} = \left[ 1 + (u-1)E_1 + (u-1)E_{1,3} + (u-1)^2 E_1E_2\right]T_{12}$ 4. $T_2T_1T_{12} =\left[ 1 + (u-1)E_2 + (u-1)E_{1,3} + (u-1)^2 E_1E_2\right]T_{12}$ 5. $T_1T_2T_1 T_{12}=\left[ 1 + (u-1)(E_1 + E_2 + E_{1,3}) + (u-1)^2(u+2)E_1E_2\right]T_{12}$ The proof of the statements results by expanding the left side and then using the defining relations of ${\mathcal E}_n$. As example we shall check the first statement: $$\begin{aligned} T_1 T_{12} & = & T_1 + T_1^2 + T_1T_2 + T_1^2T_2 + T_1T_2T_1 + T_1^2 T_2T_1\\ & = & T_1 + 1 + (u-1)E_1 +(u-1)E_1T_1 +T_1T_2+ T_2 \\ &\, & + (u-1)E_1T_2 +(u-1)E_1T_1T_2 + T_1T_2T_1 + T_2T_1\\ &\, & + (u-1)E_1T_2T_1 + (u-1)E_1T_1T_2T_1 = T_{12} + (u-1)E_1T_{12}.\end{aligned}$$ \[rhoEI\] 1. $\rho_3(T_{12}) = (u+1)A^2+3A +(u-1)AB + 1$ 2. $\rho_3(E_{\{1,2,3\}}T_{12}) = (u+1)A^2+(u+2)AB + B^2$ 3. $\rho_3(E_IT_{12}) = (u+1)A^2 +( u+1) AB + A + B$, for all $I\in \mathsf{P}(3)$ of cardinal $2$. The proof is only a routine of computations. We shall prove, as an example, the third claim. Suppose $I=\{\{1,2\},\{3\}\}$, hence $E_I = E_1$. Then, by linearity and using the example above we have $$\rho_3(E_IT_{12}) = B + A + AB + A^2 + A^2 + u AB + (u-1)A^2$$ Hence we have proved the claim. The Markov trace ${\rho}_n:{\mathcal E}_n \longrightarrow K( A, B)$ passes to ${\rm PTL}_n$ if only if $A= -B$ or $A=- B/(1+u)$. From Corollary \[PTLquo\] we have that ${\rho}_n$ pass to ${\rm PTL}_n$ if only if ${\rho}_n(xE_1E_2T_{12}y) = 0$, for all $x,y\in {\mathcal E}_n$. Now, by linearity and trace properties of ${\rho}_n$ follows that it is enough to study the conditions to have ${\rho}_n(xE_1E_2T_{12}) = 0$, for all $x$ in a linear basis of ${\mathcal E}_n$. We consider now the basis ${\Bbb S}_n$ of ${\mathcal E}_n$, see Theorem \[basEn\]. Using the rules that define ${\rho}_n$ we deduce that the computation of ${\rho}_n(xE_1E_2T_{12})$, for $x\in {\Bbb S}_n$, results in a $K(A, B)$–linear combination of $\rho_3 (zE_1E_2T_{12})$ with $z\in {\Bbb S}_3$. Now, $z$ is of the form $E_IT_w$, with $w\in S_3$ and $I\in \mathsf{P}({\bf 3})$; since $T_w$ commutes with $E_1E_2$ having in mind the Lemma \[TwT12\] and the fact that $E_1E_2$ is the maxim element of $\mathsf{P}({\bf 3}) $, we obtain that $z E_1E_2 T_{12}$ is a $K$–scalar multiple of $E_1E_2T_{12}$. Therefore, ${\rho}_n(xE_1E_2T_{12}y) = 0$, for all $x,y\in {\mathcal E}_n$ is equivalent to have $\rho_3( E_1E_2T_{12}) =0$. Now, from (2)Lemma \[rhoEI\], we have $\rho(E_1E_2T_{12})= 0$ is equivalent to $(u+1)A^2 + (u+2)AB + B^2=0$, then $A= -B$ or $A=- B/(1+u)$. [25]{} F. Aicardi, J. Juyumaya, [*An algebra involving braids and ties*]{}, ictp preprint IC2000179, http://streaming.ictp.trieste.it/preprints/P/00/179.pdf. E. O. Banjo, [*The generic representation theory of the Juyumaya algebra of braids and ties*]{}. Algebr. Represent. Theor. DOI 10.1007/s10468-012-9361-3 M. Chlouveraki, L. P. D’Andecy, [ *Representation theory of the Yokonuma–Hecke algebra*]{}, arXiv:1302.6225v1. C. K. Fan, [*A Hecke algebra quotient and some combinatorial applications*]{}, J. Algebr. Comb. [**5**]{}, (1996), 175–189. F. M. Goodman, P. de la Harpe, V.F.R. Jones, [*Coxeter Graphs and Towers of Algebras*]{}, Springer–Verlag (1989). D. Goundaroulis, J. Juyumaya, S. Lambropoulou, [*Yokonuma–Temperley–Lieb algebras.*]{} arXiv: 1012.1557. M. Jimbo, [*A $q$–analogue of $U(gl(N+1))$, Hecke algebra, and the Yang–Baxter equation*]{}, Lett. in Math. Phys. [**11**]{} (1986) 247–252. V.F.R. Jones, [*Index for subfactors*]{}, Invent. Math. [**72**]{} (1983), 1–25. V.F.R. Jones, [*Hecke algebra representations of braid groups and link polynomials*]{}, Ann. Math. [**126**]{} (1987), 335–388. V.F.R. Jones, [*The Potts models and the symmetric group*]{}, in Subfactors: Proceedings of the Taniguchi Symposium on Operators Algebras, Kyuzeso, 1993, 259–267. World Scientific, River Edge (1994). J. Juyumaya, [*Markov trace on the Yokonuma–Hecke algebra*]{}, J. Knot Theory Ramif. [**13**]{} (2004), 25–39. J. Juyumaya, S. Lambropoulou, [*An invariant for singular knots*]{}, J. Knot Theory Ramif. , [**18**]{} (2009) 825–840. P.P. Martin,[*Temperley–Lieb algebras for non–planar statistical mechanics–the partition algebra construction*]{}, J. Knot Theory Ramif. [**3**]{} (1994), 51–82. S. Ryom–Hansen, [*On the Representation Theory of an Algebra of Braids and Ties*]{}, J. Algebr. Comb. [**33**]{} (2011), 57–79. H.N.V. Temperley, E. H. Lieb, [*Relations between the percolation and couloring  problem and other graph–theoretical problem associated with regular planar lattice: some exact results for the percolations problems*]{}, Proc. Roy. Soc. London Ser. A 322 (1971), 251–280. N. Thiem, [*Unipotent Hecke algebras of $GL_n({\Bbb F}_q)$*]{}, J. of Algebra, [**284**]{} (2005) 559–577. T. Yokonuma, [*Sur la structure des anneaux de Hecke d’un groupe de Chevalle y fini*]{}, C.R. Acad. Sc. Paris, 264 (1967) 344–347. [^1]: This research has been supported in part by DIUV Grant Nº1-2011. Also, co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALES: Reinforcement of the interdisciplinary and/or inter-institutional research and innovation.

--- abstract: 'An explicit CMC Schwarzschildean line element is derived near the critical point of the foliation, the lapse is shown to decay exponentially, and the coefficient in the exponent is calculated.' address: - '$^{*}$ ESI, A-1090 Wien, Boltzmangasse 9, Austria.' - ' Institute of Physics, Jagiellonian University, 30-059 Cracow, Reymonta 4, Poland.' - '$^{+}$ Physics Department, University College, Cork, Ireland.' author: - 'Edward Malec$^{*,**}$ and Niall Ó Murchadha$^{*,+}$' title: '**Constant mean curvature slices in the extended Schwarzschild solution and collapse of the lapse. Part II**' --- \#1[[$\backslash$\#1]{}]{} Introduction ------------ This is a sequel of our previous work on the constant mean curvature (CMC) slices of the extended Schwarzschild geometry. Here we get CMC foliations by solving Einstein equations in a particular gauge. A crucial role is played by a condition (Eq. (\[5\]) below) imposed on the lapse. While this method is completely equivalent to the other more geometric approach (see [@Niall]), it seems to be more straightforward and technically simpler. We focus on the concise derivation of the explicit CMC foliation near the critical point of the CMC foliation. The final result is identical to the result derived in [@Niall]. The constant mean curvature foliations have been recently investigated numerically in the simulation of a single spherically-symmetric black hole [@CMC2]. We hope that our analytic results appear helpful in the verification of the numerical schemes. CMC slicing of the Schwarzschild spacetime ------------------------------------------ The notation is the same as in the preceding paper [@Niall]. We define $$(pR)^2=4\left[ 1 -{2m \over R} + \left( {KR\over 3}- {C\over R^2} \right)^2\right] , \label{1}$$ $$\gamma(R,t) =1+ 8\partial_tC\int_{R}^{\infty }dr{1\over r^5p^3}. \label{2}$$ and $$N =\gamma {pR\over 2}. \label{3}$$ Here $m$ is the mass, $K$ (the trace of the extrinsic curvature) is a constant and C is a time-dependent parameter which measures the transverse part of the extrinsic curvature. The Schwarzschild line element, expressed in terms of coordinates adapted to the constant mean curvature foliation, is given by [@Iriondo] $$\begin{aligned} ds^2&=&-dt^2\Biggl( N^2 -\gamma^2\Bigl( {KR\over 3}- {C\over R^2}\Bigr)^2\Biggr) +4N {{C\over R^3}-{K\over 3}\over p^2R}dtdR+ {4\over (pR)^2}dR^2+ R^2d\Omega^2. \label{4}\end{aligned}$$ The hypersurfaces of constant time are CMC slices, asymptotic to the CMC slices of Minkowskian geometry. Elliptic slicing condition --------------------------- A minimal surface is a locus of points defined by the condition $p =0$. Choose a CMC Cauchy hypersurface $\Sigma_C$ of the extended Schwarzschild manifold corresponding to a parameter $C$ and let $R_0$ be an areal radius corresponding to a simple zero of $p^2$; that is $p^2(R_0)=0$ but $\partial_{R}p^2|_{R_0}\ne 0$. Futhermore, assume that $${\partial_rN \over \sqrt{a}}|_{R_0}=0 \label{5}$$ at $R_0$. The condition (\[5\]) yields $$\partial_tC={1\over 8I(R_0)}. \label{6}$$ Here $$I(R_0)\equiv \int_{R_0}{dr\over pr}{6{C^2\over r^4}+{K^2r^2\over 3}\over \Biggl( 2m+{2KC\over 3}+{2K^2r^2\over 9} -{4C^2\over r^3} \Biggr)^2}. \label{7}$$ The value of the lapse function $N$ at the minimal surface, that is at the areal radius $R_0$, can be shown to be equal (using Eqs. (\[1\] – \[3\])) to $$N = {dC\over dt} {1\over m+{KC\over 3} +{K^2R^3_0\over 9}-2{C^2\over R^3_0}}. \label{8}$$ The lapse $N $ is strictly positive at the minimal surface corresponding to a simple zero $R_0$. Eqs. (\[1\] – \[3\]) imply that $N (R)> N (R_0)$ if $R>R_0$ and therefore the lapse exists on all of $\Sigma_C$. Equation (\[6\]) dictates the rate of change of the parameter $C$. It is clear that one can uniquely construct a foliation of a part of the extended Schwarzschild geometry by imposing the condition (\[5\]) at minimal surfaces on all slices to the future of a given one. The leaves of the resulting foliation connect two null infinities of the extended Schwarzschild spacetime. This gives us a curve $R_0(t)$ of zeroes of the mean curvature $p$. It is evident, just by inspecting the explicit solution presented above, that the line running along the locations of minimal surfaces $R_0(t)$ can be arranged to be smooth. It can be chosen to coincide with the ‘vertical’ $t = 0$ axis in standard Schwarzschild coordinates. This construction breaks down when $R_0$ ceases to be a simple zero of $p^2$, since expressions appearing in Eqs. (\[6\]) and (\[8\]) become unboundedly large. The goal of this paper is to show the asymptotic behaviour of the lapse at the critical minimal surface. The evolution of C near critical point -------------------------------------- Let $C_*$ and $R_*$ be degenerate, that is such that the zero of $p^2$ ceases to be simple. In this case both $p$ and its derivative $\partial_Rp$ vanish; that means that $$\begin{aligned} && 1- {2m\over R_*} -{2KC_*\over 3R_*}+{K^2R^2_*\over 9}+{C^2\over R^4_*}=0, \nonumber\\ && 2m +{2KC_*\over 3} + {2K^2R^3_*\over 9} -{4C^2_*\over R^3_*}=0. \label{9}\end{aligned}$$ One can easily show, if $C_*$ and $R_*$ are critical, then the sign of $$\beta \equiv -2C_*+{2\over 3}KR^3_* \label{10}$$ is the same as the sign of $-C_*$. There exists critical values of $C_*$ that are positive ($C_*^+$) or negative ($C_{*-}$). For definiteness we shall consider only the case when $C(t=0) > C_{*-}$, therefore the only limiting case we consider is that with $C\rightarrow C_*^+$. (That choice corresponds to a foliation formed by leaves connecting two null infinities which moves forward in time - see a discussion in Sec IV in [@Niall]). For simplicity we will drop the $^+$ suffix and $C_*$ will mean a positive critical parameter. From the dynamical equation (\[6\]) follows that $C$ can only increase. Next, let us introduce the notation that $$\begin{aligned} &&\epsilon\equiv C_*-C \nonumber\\ && R_0\equiv R_* +\delta . \label{11}\end{aligned}$$ where both $\delta $ and $\epsilon $ are positive and small. The equation $p(R_0)=0$ yields a nonlinear algebraic equation whose truncation gives $$\delta^2 A +\epsilon \beta =0. \label{12}$$ Here $A \equiv 2R^2_*+K^2R^4_*$. Eq. (\[12\]) is in fact the Lyapunov - Schmidt reduced equation constructed according to the standard rules [@Trenogin]. Therefore in the vicinity of the critical point we have $$\delta =\sqrt{-\beta \epsilon \over A}. \label{13}$$ The function $p$ can be expressed in a form $${pr\over 2} =\sqrt{1-{R_0\over r}}\Biggl[ {\kappa \delta \over R_0}+ {K^2\over 9}(rR_0+r^2-2R_0^2 )-{C^2\over R^4_0} ({R_0\over r} + {R_0^2\over r^2} +{R_0^3\over r^3}-3)\Biggr]^{1/2}. \label{p}$$ The insertion of (\[11\]), (\[13\]) and (\[p\]) into the equation (\[7\]) and the change of the integration variable to $y=\sqrt{1-{R_0\over r}}$, yield after a simple but tedious algebra $$I(R_0)\approx \sqrt{R_*}\int_0^1dy{F_1\over \sqrt{ \kappa \delta +y^2F_2}( \kappa \delta +y^2F_3)^2}. \label{14}$$ Here the functions $F_1, F_2$ and $F_3$ are given by $$\begin{aligned} F_1(y)&=&{K^2R_*^3\over 3(1-y^2)^4}+{6C^2_*\over R^3_*}(1-y^2)^2, \nonumber \\ F_2(y)&=& {K^2R^3_*(3-2y^2)\over 9(1-y^2)^2}+{C^2_*\over R_*^3} \Bigl( 6-4y^2+y^4\Bigr) , \nonumber \\ F_3(y)&=& (3-3y^2+y^4)\Bigl( {2K^2R^3_*\over 9(1-y^2)^3} +{4C^2_*\over R^3_*}\Bigr) , \label{15}\end{aligned}$$ while $\kappa $ reads $$\kappa = {2K^2R^2_*\over 3}+{12C^2_*\over R^4_*}. \label{16}$$ Limiting behaviour of the foliation. ------------------------------------ The asymtotic behaviour of $C$ will be dominated by the $1/\delta^2$ part of $I(R_0)$. As will be shown later, $C$ tends exponentially to $C_*$; the attenuation factor in the exponent depends only on the leading term of $I(R_0)$. It is useful to define $z=y /\sqrt{\kappa \delta }$. Then one obtains $I(R_0)={\sqrt{R_*}\over \kappa^2\delta^2}\times I_d$, where $$I_d\equiv \int_0^{1/\sqrt{\kappa \delta }}dz{F_1(\sqrt{\kappa \delta}z)\over \sqrt{ 1 +{z^2\over R_*}F_2(\sqrt{\kappa \delta }z) }( 1 +{z^2\over R_*}F_3(\sqrt{\kappa \delta }z))^2}. \label{17}$$ One can split the integral $\int_0^{1/\sqrt{\kappa \delta }}$ into two parts: $\int_0^{1/\sqrt{\kappa \delta }}= \int_0^{1/\sqrt{10^4\kappa \delta }}+ \int_{1/\sqrt{10^4\kappa \delta }}^{1/\sqrt{\kappa \delta }}$. It is easy to check that the contribution coming from the second integral goes to zero as $\delta $ approaches zero. Therefore $F_1\approx R_*\kappa /2$, $F_2\approx R_*\kappa /2$ and $F_3\approx R_*\kappa $. Thus the first integral (and also the integral $I_d$) is well approximated by $$\begin{aligned} I&=&{\kappa R_*\over 2} \int_0^{\infty } dz {1\over \sqrt{1+ {R_*\kappa \over 2}z^2}(1+R_*\kappa z^2)^2}= \nonumber \\ && {\sqrt{\kappa R_*}\over 2} \int_0^{\infty } dz {1\over \sqrt{1+ {z^2\over 2}}(1+z^2)^2}. \label{18}\end{aligned}$$ The integral $I_z=\int_0^{\infty } dz {1\over \sqrt{1+ {z^2\over 2}}(1+z^2 )^2}$ can be explicitly evaluated and gives $\sqrt{2}/2$. In summary, near the critical point we have $$\begin{aligned} I(R_0) &=&{\sqrt{R_*}\over \kappa^2\delta^2}I \approx \nonumber \\ && {\sqrt{2}\over 4\epsilon } {R_*A \over \kappa^{3/2}|\beta |} \label{19}\end{aligned}$$ The insertion of (\[11\]) and (\[19\]) into (\[6\]) yields $$\partial_t\epsilon = - \Gamma \epsilon , \label{20}$$ where $$\Gamma = {|\beta |\kappa^{3/2} \over 2\sqrt{2} A R_*}. \label{21}$$ Eq. (\[20\]) immediately implies that $\epsilon $ approaches 0 exponentially as $$\epsilon (t) =\epsilon_0e^{-t\Gamma }, \label{23}$$ where $\epsilon_0 $ is an initial value of the parameter. Taking into account relations (\[11\]) and (\[13\]), one can conclude that the parameter $C$ and the minimal radius $R_0$ tend exponentially to their critical values $C_*$ and $R_*$, respectively. Finally, collecting the above information and putting it into Eq. (\[8\]), we obtain the asymptotic behaviour of the lapse function near the critical point $$N = N_0 e^{-t\Gamma /2}. \label{24}$$ This is exactly the same result as that obtained in [@Niall]; in order to show equivalence, use expressions for the extrinsic curvatures (valid in the upper quadrant of the extended Schwarzschild geometry), which imply $K=(2mR_*-3m)/\sqrt{2mR^3_*-R^4_*}$ and $C_*=(3mR^3_*-R^4_*)/ \sqrt{2mR^3_*-R^4_*}$. In the case of maximal slicing ($K =0$) the decay constant $\Gamma /2$ is equal to $4/(3\sqrt{6})$, in agreement with the analytic derivation of [@MAX] and close to the numerical result of [@CMC1a] . The asymptotic behaviour of $\gamma $ and $p$ in a region close to the line $R_0$ is given by $$\begin{aligned} \gamma &=& \gamma_0 {e^{-{t\Gamma \over 4}}\over \sqrt{1-{R_0\over r}}}\nonumber \\ {pr\over 2}&=&p_0 \sqrt{1-{R_0\over r}} e^{-t\Gamma \over 4} \label{25}\end{aligned}$$ Here $\gamma_0$ and $p_0$ are initial values of $\gamma \sqrt{1-{R_*\over r}}$ and $pr/(2\sqrt{1-{R_*\over r}})$, respectively. The four constants ($\epsilon_0, N_0, \gamma_0$ and $p_0$) can be expressed in terms of one free parameter (say, $\epsilon_0$) and $A, \beta, \kappa $. Equations (\[3\]), (\[14\]), (\[23\]) - (\[25\]) and $C=C_*-\epsilon_0e^{-t\Gamma }$ suffice to construct the metric (\[4\]) near the line $R_0(t)$ of minimal surfaces. Acknowledgments. This work has been suported in part by the KBN grant 2 PO3B 006 23. E. Malec and N. Ó Murchadha, [*Constant mean curvature slices in the extended Schwarzschild solution and collapse of the lapse. Part I*]{}. A. P. Gentle, D. E. Holz, A. Kheyfets, P. Laguna, W. A. Miller and D. M. Shoemaker, [*Phys. Rev.* ]{} [**D**]{}(2001)064024. M. Iriondo, E. Malec and N. Ó Murchadha, gr-qc/9503030; notice that our $C$ is equal to $C_1/2$ therein. The published version, Phys. Rev. [**D54**]{}, 4792(1996), contains i) a misprint in equation (52), which should read $\partial_t \Bigl( R^3(K_r^r-K/3)\Bigr) =4\partial_tC_1$); ii) instead of $C_2(R,t)$ should be $C_2(t)$. M. M. Vajnberg and W. A. Trenogin, Teorija vetvlenija reshenij nelinejnykh uravnenij, Nauka, Moscow 1969 (there exists also an English edition). R. Beig and N. Ó Murchadha, [*Phys. Rev.*]{} [**D57**]{}, 4728(1998). L. Smarr and J. York, [*Phys. Rev.*]{} [**D17**]{}, 2529(1978).

--- abstract: 'A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for Itô processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; and some Itô-type expressions are given here. Furthermore, there is some analysis of these results for their applications to problems using Rosenblatt noise.' address: - 'University of Kansas, Department of Mathematics, 1460 Jayhawk Blvd., Lawrence, 660 45, Kansas, USA' - 'Charles University, Faculty of Mathematics and Physics, Sokolovská 83, Prague 8, 186 75, Czech Republic' author: - Petr Čoupek - 'Tyrone E. Duncan' - 'Bozenna Pasik-Duncan' title: A Stochastic Calculus for Rosenblatt Processes --- Rosenblatt process ,stochastic calculus ,Itô formula ,Skorokhod integral ,forward integral 60H05 ,60H07 ,60G22 *This paper is dedicated to the memory of Larry Shepp.* Introduction ============ Self-similar stochastic processes, that are processes whose distributions are invariant under suitable scalings, can be used as mathematical models of various physical phenomena. These processes have been used for modeling in hydrology, biophysics, geophysics, telecommunication, turbulence, cognition, and finance. Typically, these self-similar processes exhibit long-range dependence, that is, their autocorrelations decay slower than exponentially. Some bibliographical guides are given by Taqqu [@Taqqu86]; and Willinger, Taqqu, and Erramili [@WilTaqErr96]; that provide applications of self-similar stochastic processes and many references. The family of fractional Brownian motions is among the most studied self-similar stochastic processes. Fractional Brownian motion indexed by the Hurst parameter $0<H<1$, that is denoted by $B^H$ here, is a centered Gaussian stochastic process whose covariance function is given by $$\mathbb{E} B^H_sB_t^H = \frac{1}{2}\left(|s|^{2H}+|t|^{2H}-|s-t|^{2H}\right), \quad s,t\in\mathbb{R}.$$ There are at least two reasons why fractional Brownian motions are of interest. First, these processes are self-similar, have stationary increments, and exhibit long-range dependence for $\sfrac{1}{2}<H<1$. These properties make them very attractive for practical modeling and applications. The second reason is the fact that they are Gaussian processes which makes some mathematical models using fractional noise feasible for analysis. In fact, stochastic calculus for fractional Brownian motions is fairly developed, e.g. [@AlosMazNua01; @AlosNua03; @BiaHuOksZha08; @DecrUstu99; @DunJakDun06; @DunHuDun00]. However, non-Gaussian data with fractal features have also been observed empirically, e.g. [@Dom15] where control error in single-input-single-output (SISO) loops is analyzed. Domański has shown from data of some physical systems that the Gaussian assumption is not always appropriate. In such cases, it does not seem reasonable to use a Gaussian process such as a fractional Brownian motion as a model for these physical phenomena and the use of a Rosenblatt process can provide a useful alternative. A Rosenblatt process with the Hurst parameter $\sfrac{1}{2}<H<1$, denoted here as $R^H$, can arise as a non-Gaussian limit of suitably normalized sums of long-range dependent random variables in a non-central limit theorem, see e.g. [@DobMaj79; @Ros61; @Taqq79]. This process admits a version with Hölder continuous sample paths (up to order $H$), has stationary increments, and is $H$-self-similar with long-range dependence. Moreover, its covariance function is the same as that of the fractional Brownian motion $B^H$. However, unlike the family of fractional Brownian motions, the family of Rosenblatt processes is not Gaussian. A detailed history, construction, and many properties of Rosenblatt processes are given in the survey article of Taqqu [@Taqqu11]. Some stochastic analysis of Rosenblatt pocesses is given by Tudor in [@Tud08] and some properties of Rosenblatt processes are given in [@AbrPip06; @Alb98; @Pip04]. Furthemore, stochastic (partial) differential equations with additive Rosenblatt noise have also been studied, e.g. [@BonTud11; @Cou18; @CouMas17; @CouMasOnd18]. However, despite the considerable attention that Rosenblatt processes have received, there is only a limited development of a stochastic calculus and especially Itô-type formulas for these processes. In the pioneering work of Tudor [@Tud08], a representation of Rosenblatt processes on a finite time interval is given and used to construct both Wiener-type and stochastic integrals in which Rosenblatt processes appear as the integrators. Furthemore, an Itô-type formula for functionals of a Rosenblatt process is given for some general conditions. However, these conditions seem to be difficult to verify in specific cases and in fact in [@Tud08], the conditions are only verified for the square and cube of a Rosenblatt process. In [@Arr15], a stochastic calculus with respect to Rosenblatt processes is developed by means of white-noise theory [@HidaKuoPottStr94]. In this framework, an Itô-type formula for functionals of a Rosenblatt process is proved. However, this formula is given as an infinite series that involves derivatives of all orders and white-noise integrals with respect to stochastic processes obtained from the Rosenblatt process. An important contribution is made by Arras in [@Arr16] that improves the results in [@Tud08] and provides an Itô-type formula for functionals of Rosenblatt processes by means of Malliavin calculus on the white-noise probability space which allows to use techniques from white-noise distribution theory. This Itô-type formula is valid for infinitely differentiable functionals with at most polynomial growth. In the approach used here, some methods of [@Arr16] are used without relying on the white-noise setting. Not only functionals of Rosenblatt processes but functionals of stochastic integrals with respect to them are considered. A main results of this paper is an Itô-type formula for $\mathscr{C}^3$ functionals with at most polynomial growth of the stochastic processes with second-order fractional differential of the form $$\label{eq:x_t_intro} x_t = x_0+ \int_0^t\vartheta_s\,\mathrm{d}{s} + 2c_{H}^{B,R}\int_0^t\varphi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} + \int_0^t\psi_s\delta R_s^H$$ that have Hölder continuous sample paths of an order greater than $\sfrac{1}{2}$. Here, $c_H^{B,R}$ is a suitable normalizing constant. The integrals are defined using (multiple) Skorokhod integrals with respect to a Wiener process and suitable (fractional) transfer operators similar to [@Tud08]. This formula generalizes the results of [@Tud08] and [@Arr16]. There are two noteworthy properties of the obtained Itô-type formula: - The formula shows that the form of the process $x$ is preserved under compositions with $\mathscr{C}^3$ functions. - There is a term that involves the third derivative. Both of these properties result from the second-order nature of Rosenblatt processes; that is, from the fact that Rosenblatt processes are defined as second-order Wiener-Itô integrals. As suggested in [@Arr16], it seems plausible that the method used to obtain the general Itô-type formula can be employed to obtain analogous formulas for Hermite processes of any order $k$, see [@Tud13 Definition 3.1], and it is conjectured to expect the appearance of stochastic integrals with respect to related Hermite processes up to order $k$ (such as $B^{\frac{H}{2}+\frac{1}{2}}$ is related to $R^H$) as well as derivatives up to order $k+1$ in such formulas. Further discussion of this phenomenon can be found on [@Arr15 p. 548] and in [@Tud08 Remark 8]. The method used to obtain the Itô-type formula has already been used in the literature, see e.g. [@BiaOks08] for the case of fractional Brownian motions and [@Tud08; @Arr16] for the case of Rosenblatt processes. It can be briefly outlined as follows: 1. \[step:intro\_1\] Initially, two types of integrals with respect to fractional Brownian motions and Rosenblatt processes are defined: a pathwise forward integral defined by regularization of the integrator, see [@RusVal93], and a Skorokhod-type integral defined by means of Malliavin calculus. These definitions are given in , and in and , respectively. Moreover, the relationship between these two integrals is established. The forward integral with respect to the fractional Brownian motion equals the Skorokhod-type integral plus an additional correction term, see . In the case of the Rosenblatt process with Hurst parameter $\sfrac{1}{2}<H<1$, there appear two correction terms and one of these terms is identified as a Skorokhod-type integral with respect to the fractional Brownian motion of Hurst parameter $\sfrac{H}{2}+\sfrac{1}{2}$, see . 2. The Itô formula itself is then proved as follows: Starting with a process that has the Skorokhod-type differential , write the integrals as forward integrals by adding the appropriate correction terms using the relationship from \[step:intro\_1\] The important feature of forward integrals is that, unlike the Skorokhod-type integrals, they commute with random variables. Use this property in the approximation procedure that leads to the Itô formula and that is based on Taylor’s formula. The resulting forward integrals are then rewritten as Skorokhod-type integrals using results from \[step:intro\_1\] Finally, convergence of the approximation is verified. This paper is organized as follows. In , the general setting is introduced and several notions from Malliavin calculus (Malliavin derivative, Sobolev-Watanabe spaces, and the Skorokhod integral) are reviewed. Furthermore, the definitions of fractional Brownian motions and Rosenblatt processes are also given here. In , some methods that are used throughout the paper but which may also be of independent interest are collected. Initially, the forward integral is recalled which is followed by the definitions of the Skorokhod-type integrals with respect to fractional Brownian motions and Rosenblatt processes. Subsequently, some mapping properties of the Skorokhod-type integrals together with their relationship with the corresponding forward integrals are described. This section is concluded with several technical lemmas useful for computations (product and chain rules for fractional stochastic derivatives, the duality relationship between fractional stochastic derivatives and Skorokhod-type integrals, and a Fubini-type result). The main results are collected in . A general Itô-type formula for functionals of stochastic processes with second-order fractional differential is given in . The assumptions for which the formula is satisfied are kept broad not to limit the possible applications. However, some cases where the assumptions can be simplified in special cases are described. More precisely, in and its corollaries, an Itô formula for functionals of the stochastic integral with respect to a Rosenblatt process which is valid under assumptions formulated in terms of the integrand rather then in terms of the integral is given. This section also contains four technical lemmas which can be useful for computations and three short examples which show how the formulas can be used. The paper is concluded with where the second moment of stochastic integral with respect to a Rosenblatt process is computed and where an estimate for its higher absolute moments is given. Preliminaries {#sec:prelim} ============= Some notation is briefly described now. The expression $A\lesssim B$ means that that there exists a finite positive constant $c$ such that $A\leq c B$. The constant is independent of the variables which appear in the expressions $A$ and $B$ and it can change from line to line. Similarly, $A\eqsim B$ means that there exists a finite positive constant $c$ such that $A=c B$. These symbols are used when the exact value of a constant is not important to simplify the exposition. The symbol $\otimes$ denotes the tensor product of two Hilbert spaces or their elements. Some notions from Malliavin calculus ------------------------------------ Let $W=(W_t,t\in\mathbb{R})$ be the two-sided Wiener process defined on a complete probability space $(\Omega,\mathscr{F},\mathbb{P})$ and assume that the $\sigma$-algebra $\mathscr{F}$ is generated by $W$. Several notions of Malliavin calculus that are needed in this paper are now recalled, see e.g. [@NouPec12; @Nua06] for a complete exposition of the topic. Let $I(\xi)$ be the first-order Wiener-Itô integral and let $\mathcal{P}$ be the algebra of polynomial random variables generated by $\{I(\xi), \xi\in L^2(\mathbb{R})\}$, i.e. $\mathcal{P}$ is the set of random variables of the form $$\label{eq:X_polynomial} P = p(I(\xi_1), I(\xi_2), \ldots, I(\xi_n))$$ where $p$ is a polynomial, $n\in\mathbb{N}$, and $\xi_i\in L^2(\mathbb{R})$ for every $i=1,2,\ldots, n$. For $k\in\mathbb{N}$ and $P\in\mathcal{P}$ of the form , define the *$k$-th Malliavin derivative of $P$ at point* $x=(x_1,x_2, \ldots,x_k)\in\mathbb{R}^k$ by $$D^k_xP\overset{\textnormal{Def.}}{=} \sum_{i_1=1}^n\sum_{i_2=1}^n\cdots \sum_{i_k=1}^n\partial_{i_1}\partial_{i_2}\,\cdots\,\partial_{i_k} p(I(\xi_1), I(\xi_2), \ldots ,I(\xi_n))\xi_{i_1}(x_1)\xi_{i_2}(x_2)\cdot\ldots\cdot \xi_{i_k}(x_k).$$ It follows that for every $k\in\mathbb{N}$ and every $p\geq 1$, the operator $D^k$ is closable from $\mathcal{P}\subset L^p(\Omega)$ to $L^p(\Omega;L^2(\mathbb{R}^k))$, see e.g. [@NouPec12 Proposition 2.3.4], and its domain is the *Sobolev-Watanabe space* $\mathbb{D}^{k,p}$ defined as the closure of $\mathcal{P}$ with respect to the norm $$\|P\|_{\mathbb{D}^{k,p}}\overset{\textnormal{Def.}}{=} \left(\mathbb{E}|P|^p + \mathbb{E}\|D P\|_{L^2(\mathbb{R})}^p + \cdots + \mathbb{E}\|D^kP\|_{L^2(\mathbb{R}^k)}^p\right)^\frac{1}{p}.$$ The operator $D^k$ admits an adjoint denoted by $\delta^k$. Define the set $\mathrm{Dom}(\delta^k)$ as the set of $u\in L^2(\Omega;L^2(\mathbb{R}^k))$ for which there exists a constant $c>0$ such that $$\left|\left\langle D^k P;u\right\rangle_{L^2(\Omega;L^2(\mathbb{R}^k))}\right|\leq c\|P\|_{L^2(\Omega)}$$ is satisfied for every $P\in\mathcal{P}$. For $u\in\mathrm{Dom}(\delta^k)$, the symbol $\delta^k(u)$ denotes the unique element of $L^2(\Omega)$ such that $$\label{eq:duality_formula} \langle P,\delta^k(u)\rangle_{L^2(\Omega)} = \langle u;D^kP\rangle_{L^2(\Omega;L^2(\mathbb{R}^k))}$$ holds for every $P\in\mathcal{P}$, see e.g. [@NouPec12 Definition 2.5.1 and Definition 2.5.2]. The existence of $\delta^k(u)$ is ensured by the Riesz representation theorem. The operator $\delta^k:\mathrm{Dom}(\delta^k)\rightarrow L^2(\Omega)$ is called the *(multiple) Skorokhod integral (with respect to $W$)*. Fractional Brownian motions and Rosenblatt processes ---------------------------------------------------- The definitions of a fractional Brownian motion and a Rosenblatt process are now recalled. Denote by $(u)_+ = \max\{u,0\}$ the positive part of $u$. \[def:fBm\] Let $H\in(\sfrac{1}{2},1)$. The *fractional Brownian motion $B^H=(B^H_t)_{t\in\mathbb{R}}$ of the Hurst parameter $H$* is defined by $$B_t^H\overset{\textnormal{Def.}}{=} C_H^B\int_{\mathbb{R}}\left(\int_0^t (u-y)_+^{H-\frac{3}{2}}\,\mathrm{d}{u}\right)\,\mathrm{d}{W}_y, \quad t\in\mathbb{R},$$ where $C_H^B$ is a normalizing constant such that $\mathbb{E}(B_1^H)^2=1$ and the stochastic integral is the Wiener-Itô integral of order one. \[def:Rosenblatt\_process\] Let $H\in(\sfrac{1}{2},1)$. The *Rosenblatt process $R^H=(R^H_t)_{t\in\mathbb{R}}$ of the Hurst parameter $H$* is defined by $$R_t^H\overset{\textnormal{Def.}}{=} C^R_H\int_{\mathbb{R}^2}\left(\int_0^t(u-y_1)_+^{\frac{H}{2}-1}(u-y_2)_+^{\frac{H}{2}-1}\,\mathrm{d}{u}\right)\,\mathrm{d}{W}_{y_1}\,\mathrm{d}{W}_{y_2}, \quad t\in\mathbb{R},$$ where $C^R_H$ is a normalizing constant such that $\mathbb{E}(R_1^H)^2=1$ and the double stochastic integral is the Wiener-Itô multiple integral of order two. \[rem:constants\] The normalizing constants $C_H^B$ and $C_H^R$ are given by $$C_H^B = \sqrt{\frac{H(2H-1)}{\mathrm{B}\left(2-2H,H-\frac{1}{2}\right)}},\quad\quad C_H^R = \frac{\sqrt{2H(2H-1)}}{2\mathrm{B}\left(1-H,\frac{H}{2}\right)}$$ where $\mathrm{B}$ is the Beta function. It will be also convenient to denote $$c_H^B \overset{\textnormal{Def.}}{=} C_H^B\,\Gamma\left(H-\frac{1}{2}\right), \quad\quad c_H^R \overset{\textnormal{Def.}}{=} C_H^R\,\Gamma\left(\frac{H}{2}\right)^2,$$ and $$c_H^{B,R}\overset{\textnormal{Def.}}{=}\frac{c_H^R}{c_{\frac{H}{2}+\frac{1}{2}}^B} = \sqrt{\frac{(2H-1)}{(H+1)}\frac{\Gamma\left(1-\frac{H}{2}\right)\Gamma\left(\frac{H}{2}\right)}{\Gamma(1-H)}}$$ where $\Gamma$ is the Gamma function. Stochastic integration {#sec:stoch_int} ====================== In this section, stochastic integration with respect to fractional Brownian motions and Rosenblatt processes is reviewed and formulas relating the respective forward and Skorokhod integrals are given. Forward-type integrals ---------------------- One possible approach to stochastic integration with respect to these processes is via regularization of the integrand in the sense of Russo and Vallois [@RusVal93]. This is a natural approach due to the fact that both the processes $B^H$ and $R^H$ have versions with Hölder continuous sample paths of every order smaller than $H$ (recall that $H$ is assumed to be larger than $\sfrac{1}{2}$ in this paper). The definition of the forward integral follows. \[def:forward\_integral\] Let $M\subseteq\mathbb{R}$ be an interval. An integrable stochastic process $g=(g_s)_{s\in M}$ is said to be *forward integrable* on $M$ with respect to a continuous process $h=(h_s)_{s\in\mathbb{R}}$ if the limit in probability $$\operatorname*{\mathbb{P}-lim}_{\varepsilon\downarrow 0}\int_M g_s\frac{h_{s+\varepsilon}-h_s}{\varepsilon}\,\mathrm{d}{s}$$ exists. The limit is called the *forward integral of $g$ with respect to $h$ on $M$* and it is denoted by $\int_Mg_s\,\mathrm{d}^{-}h_s$. Skorokhod-type integrals ------------------------ On the other hand, stochastic calculus with respect to fractional Brownian motions and Rosenblatt processes can be approached from the perspective of Malliavin calculus. For fractional Brownian motions, such stochastic calculus is now well-developed, e.g. [@AlosMazNua01; @AlosNua03; @DecrUstu99; @DunJakDun06]. Here, the definition of the Skorokhod-type integral is recalled and some properties that are useful for the subsequent analysis are reviewed. Note, however, that in the above references, the finite time interval representation of fractional Brownian motions from [@DecrUstu99 Corollary 3.1] is used to define the stochastic integral whereas here, its infinite time interval representation from is used as in [@Arr15; @Arr16; @CheNua05; @Cou18]. Consider the first-order fractional integrals defined for $f\in L^p(\mathbb{R})$ by $$I_+^\alpha (f)(x) = \frac{1}{\Gamma(\alpha)}\int_{-\infty}^xf(u)(x-u)^{\alpha-1}\,\mathrm{d}{u}$$ and $$I_{-}^\alpha(f)(x) = \frac{1}{\Gamma(\alpha)}\int_{x}^\infty f(u)(u-x)^{\alpha-1}\,\mathrm{d}{u}$$ provided that $\alpha\in (0,1)$ and $1\leq p<\sfrac{1}{\alpha}$, cf. [@SamKilMar93 formulas (5.2) and (5.3)]. If the function $f$ takes values in a Banach space, then the integrals above are interpreted as Bochner integrals. The Skorokhod integral with respect to the fractional Brownian motion $B^H$ is defined via the transfer operator $I_{-}^{H-\frac{1}{2}}$ as follows. \[def:Skor\_int\_B\] Let $H\in (\sfrac{1}{2},1)$ and let $M\subseteq\mathbb{R}$ be an interval. Define $$\varLambda_{B^H}(M) \overset{\textnormal{Def.}}{=} \left\{g:\mathbb{R}\rightarrow L^2(\Omega)\mbox{ such that } I_{-}^{H-\frac{1}{2}}(\textbf{1}_Mg)\in \mathrm{Dom}(\delta)\right\}.$$ A stochastic process $g:\mathbb{R}\rightarrow L^2(\Omega)$ is said to be *Skorokhod integrable with respect to the fractional Brownian motion $B^H$ on $M$* if $g\in\varLambda_{B^H}(M)$. For such integrands, the *Skorokhod integral* is defined by $$\int_Mg_s\delta B_s^H \overset{\textnormal{Def.}}{=} c_H^B \left(\delta\circ I_-^{H-\frac{1}{2}}\right)\left(\textbf{1}_Mg\right).$$ The following proposition provides a mapping property of the Skorokhod integral with respect to the fractional Brownian motion with the Hurst parameter $H\in (\sfrac{1}{2},1)$. In particular, it ensures that stochastic processes from the space $L^\frac{1}{H}(M;\mathbb{D}^{1,2})$ are Skorokhod integrable with respect to the fractional Brownian motion and the stochastic integral is a square-integrable random variable. \[prop:boundedness\_of\_int\_B\] Let $H\in(\sfrac{1}{2},1)$ and $M\subseteq\mathbb{R}$ be an interval. The linear operator $\int_M(\cdots)\delta B^H$ is bounded from $L^\frac{1}{H}(M;\mathbb{D}^{k,p})$ to $\mathbb{D}^{k-1,p}$ for every integer $k\geq 1$ and every $p$ such that $1\leq pH<\infty$. The proof is similar to the proof of the following result in . The main difference between the two is that [@SamKilMar93 Theorem 5.3] is used instead of to obtain the estimate $$\mathbb{E}\left\|I_{-}^{H-\frac{1}{2}}(\bm{1}_MD^lg)\right\|_{L^2(\mathbb{R})\otimes L^2(\mathbb{R}^l)}^p\lesssim \left(\int_M\left(\mathbb{E}\left\|D^lg_s\right\|_{L^2(\mathbb{R}^l)}\right)^\frac{1}{pH}\,\mathrm{d}{s}\right)^{pH}.$$ For non-fractional Sobolev-Watanabe spaces, improves [@Arr16 Proposition 16]. There it is shown that the integral $\int_M(\cdots)\delta B^H$ is bounded from $L^2(M;\mathbb{D}^{s,2})$ to $\mathbb{D}^{s-1,2}$ for every real $s\geq 1$ in the white-noise setting, that is, when the probability space is the space of tempered distributions, e.g. [@HidaKuoPottStr94]. is used in the above form for the proofs of the Itô-type formulas in . The relationship between the forward and the Skorokhod integral with respect to a fractional Brownian motion is given in the next proposition. For the comparison of these integrals, the first-order fractional stochastic derivative $\nabla^{\alpha}$ is defined, following [@Arr16], by $$\nabla^\alpha\overset{\textnormal{Def.}}{=} I_{+}^\alpha\circ D.$$ It can be shown that if $\alpha\in (0,\sfrac{1}{2})$, then the operator $\nabla^\alpha$ extends to a bounded linear operator from the Sobolev-Watanabe space $\mathbb{D}^{k,p}$ to the space $L^\frac{2}{1-2\alpha}(\mathbb{R};\mathbb{D}^{k-1,2})$ for $k\geq 1$ and $p\geq 2$ (and consequently to $L^\frac{2}{1-2\alpha}(M;\mathbb{D}^{k-1,2})$ for a bounded interval $M\subset\mathbb{R}$), cf. [@Arr16 Proposition 14]. \[ex:nabla\_W\] As an example, an explicit expression for $\nabla^\frac{H}{2}W$ is given where $W$ is the underlying Wiener process and $H\in(\sfrac{1}{2},1)$ is a fixed constant. For every $s\geq 0$, $W_s=I(\bm{1}_{[0,s]})$ and thus $$\begin{aligned} \nabla^\frac{H}{2}W_s(u) & = \frac{1}{\Gamma\left(\frac{H}{2}\right)}\int_{-\infty}^u (u-x)^{\frac{H}{2}-1}D_x W_s\,\mathrm{d}{x} \\ & = \frac{1}{\Gamma\left(\frac{H}{2}\right)}\int_{-\infty}^u (u-x)^{\frac{H}{2}-1}\bm{1}_{[0,s]}(x)\,\mathrm{d}{x} \\ & = \frac{2}{H\Gamma\left(\frac{H}{2}\right)}\left[(u-s)_+^{\frac{H}{2}}-u_+^\frac{H}{2}\right] \end{aligned}$$ is satisfied for $s\geq 0$ and $u\in\mathbb{R}$. The relationship between the forward and the Skorokhod integral with respect to a fractional Brownian motion is given now. \[prop:relationship\_for\_Skor\_B\] Let $H\in (\sfrac{1}{2},1)$ and let $g=(g_s)_{s\in [0,T]}$ be a stochastic process such that the following conditions are satisfied: 1. The following integrability is satisfied $$\int_0^T\|g_s\|_{\mathbb{D}^{1,2}}^\frac{1}{H}\,\mathrm{d}{s} <\infty.$$ 2. \[ass:(2)dB\] 1. For almost every $s\in [0,T]$, the following limit is satisfied $$\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (s,s+\varepsilon)}\|\nabla^{H-\frac{1}{2}}g_s(u)-\nabla^{H-\frac{1}{2}}g_s(s)\|_{L^2(\Omega)}=0.$$ 2. There exists a non-negative function $p\in L^1(0,T)$ such that, for almost every $s\in [0,T]$, the inequality $$\|(\nabla^{H-\frac{1}{2}}g_s)(u)\|_{L^2(\Omega)}\leq p(s)$$ is satisfied for almost every $u\in [s,T]$. Then $g$ is forward integrable with respect to $B^H$ on $[0,T]$ and the following equality is satisfied in $L^2(\Omega)$: $$\label{eq:relationship_for_Skor_B} \int_0^Tg_s\,\mathrm{d}^{-}B_s^H = \int_0^Tg_s\delta B_s^H + c_H^B\int_0^T(\nabla^{H-\frac{1}{2}}g_s)(s)\,\mathrm{d}{s}.$$ The proof follows the proof of [@Arr16 Theorem 1]. In particular, instead of the $\varepsilon$-approximation of the forward integral on the left-hand side of , its $S$-transform is used which allows to employ the duality to determine a new representation of the integral. Then, convergence is verified and the limits are identified as the two terms on the right-hand side of . The $S$-transform is given in e.g. [@Ben03a section 2.2]. An expression related to formula is given in [@BiaOks08 Theorem 3.7] though proved by a different method. The proof here follows the method used in [@Arr16] where the equality is verified only in the case $g_s=F(B_s^H)$, cf. [@Arr16 Theorem 1]. However, the white-noise setting of [@Arr16] is not necessary and it is possible to use a generic probability space as in [@Ben03a]. The Skorokhod integral with respect to Rosenblatt processes is reviewed now. The definition is similar to [@Tud08 Definition 1] where the finite time interval representation of Rosenblatt processes from [@Tud08 Proposition 1] is used. To simplify computations, the infinite time interval representation given in is used here. The Skorokhod integral with respect to a Rosenblatt process is defined via a transfer operator similar to the case of a fractional Brownian motion. Consider the second-order fractional integral given by $$(I_{+,+}^{\alpha_1,\alpha_2}f)(x_1,x_2) \overset{\textnormal{Def.}}{=} \frac{1}{\Gamma(\alpha_1)\Gamma(\alpha_2)}\int_{-\infty}^{x_1}\int_{-\infty}^{x_2}f(u,v)(x_1-u)^{\alpha_1-1}(x_2-v)^{\alpha_2-1}\,\mathrm{d}{u}\,\mathrm{d}{v}$$ for sufficiently nice $f:\mathbb{R}^2\rightarrow\mathbb{R}$ and $\alpha_i\in (0,1)$, $i=1,2$, as in [@SamKilMar93 formula (24.20)]; and define also $$(I_{-,\mathrm{tr}}^{\alpha_1,\alpha_2}f)(x_1,x_2) \overset{\textnormal{Def.}}{=} \frac{1}{\Gamma(\alpha_1)\Gamma(\alpha_2)}\int_{x_1\vee x_2}^\infty f(u)(u-x_1)^{\alpha_1-1}(u-x_2)^{\alpha_2-1}\,\mathrm{d}{u}$$ for sufficiently nice $f:\mathbb{R}\rightarrow \mathbb{R}$. If the function $f$ takes values in a Banach space, then the integrals above are interpreted as Bochner integrals. The operator $I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}$ plays the role of the transfer operator in the following definition of the Skorokhod integral with respect to a Rosenblatt process. \[def:Skor\_int\_R\] Let $H\in (\sfrac{1}{2},1)$ and let $M\subseteq\mathbb{R}$ be an interval. Define $$\varLambda_{R^H}(M) \overset{\textnormal{Def.}}{=} \left\{g:\mathbb{R}\rightarrow L^2(\Omega)\mbox{ such that } I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\textbf{1}_Mg)\in \mathrm{Dom}(\delta^2)\right\}.$$ A stochastic process $g:\mathbb{R}\rightarrow L^2(\Omega)$ is said to be *Skorokhod integrable with respect to the Rosenblatt process $R^H$ on $M$* if $g\in\varLambda_{R^H}(M)$. For such integrands, the *Skorokhod integral* is defined by $$\label{eq:Skor_int_def_Rosenblatt} \int_{M}g_s\delta R_s^H \overset{\textnormal{Def.}}{=} c_H^R\,\,\left(\delta^2\circ I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}\right)(\textbf{1}_Mg).$$ To investigate the mapping properties of the Skorokhod integral with respect to the Rosenblatt process, a technical lemma that is given below is used. Recall the equality $$\label{eq:integral_beta} \int_{-\infty}^{u\wedge v} (u-x)^{\alpha -1}(v-x)^{\alpha-1}\,\mathrm{d}{x} = \mathrm{B}(\alpha, 1-2\alpha)|u-v|^{2\alpha-1}, \quad u\neq v,$$ where $\mathrm{B}$ is the Beta function, that is satisfied for $0<\alpha<\sfrac{1}{2}$, see e.g [@Tud13 Lemma 3.1]. \[lem:I\_tr\_bounded\] Let $H\in (\sfrac{1}{2},1)$. The linear operator $I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}$ is bounded from $L^\frac{1}{H}(\mathbb{R})$ to $L^2(\mathbb{R}^2)$. Let $g\in L^\frac{1}{H}(\mathbb{R})$. It follows that $$\begin{aligned} \|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}g\|_{L^2(\mathbb{R}^2)}^2 & \eqsim \int_{\mathbb{R}^2}\left(\int_{x\vee y}^\infty g(u)(u-x)^{\frac{H}{2}-1}(u-y)^{\frac{H}{2}-1}\,\mathrm{d}{u}\right)^2\,\mathrm{d}{x}\,\mathrm{d}{y}\\ & \eqsim\int_{\mathbb{R}^2}g(u)g(v)\left(\int_{-\infty}^{u\wedge v}(u-x)^{\frac{H}{2}-1}(v-x)^{\frac{H}{2}-1}\,\mathrm{d}{x}\right)^2\,\mathrm{d}{u}\,\mathrm{d}{v}\\ & \eqsim\int_{\mathbb{R}^2}g(u)g(v)|u-v|^{2H-2}\,\mathrm{d}{u}\,\mathrm{d}{v}\\ &\eqsim\int_{\mathbb{R}}g(v)\int_{-\infty}^vg(u)(u-v)^{2H-2}\,\mathrm{d}{u}\,\mathrm{d}{v}\\ &\lesssim\left(\int_{\mathbb{R}}|g(u)|^\frac{1}{H}\,\mathrm{d}{u}\right)^{H}\left(\int_{\mathbb{R}}\left|\int_{-\infty}^vg(u)(u-v)^{2H-2}\,\mathrm{d}{u}\right|^\frac{1}{1-H}\,\mathrm{d}{v}\right)^{1-H}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:lem:I_tr_bounded:split_int} \end{aligned}$$ where equation and Hölder’s inequality are used. The rightmost expression in is (modulo a constant) the $L^\frac{1}{1-H}(\mathbb{R})$-norm of $I_+^{2H-1}g$. The use of [@SamKilMar93 Theorem 5.3] gives the estimate $$\|I_{+}^{2H-1}g\|_{L^\frac{1}{1-H}(\mathbb{R})}\lesssim \|g\|_{L^\frac{1}{H}(\mathbb{R})}.$$ The following proposition provides a mapping property of the Skorokhod integral with respect to a Rosenblatt process. It ensures that stochastic processes from the space $L^\frac{1}{H}(M;\mathbb{D}^{2,2})$ are Skorokhod integrable with respect to the Rosenblatt process $R^H$ and the stochastic integral is a square-integrable random variable. Integrability is also treated in [@Tud08 Lemma 1 and Corollary 3]. \[prop:boundedness\_of\_int\_R\] Let $H\in(\sfrac{1}{2},1)$ and $M\subseteq\mathbb{R}$ be an interval. The linear operator $\int_M(\cdots)\delta R^H$ is bounded from $L^{\frac{1}{H}}(M;\mathbb{D}^{k,p})$ to $\mathbb{D}^{k-2,p}$ for every integer $k\geq 2$ and every $p$ such that $1\leq pH<\infty$. Let $g$ be a $\mathcal{P}$-valued step function. By [@NouPec12 Theorem 2.5.5], it follows that $$\begin{aligned} \left\|\int_Mg_s\delta R_s^H\right\|_{\mathbb{D}^{k-2,p}}^p & \eqsim \left\|\delta^2\left(I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_{M}g)\right)\right\|_{\mathbb{D}^{k-2,p}}^p \\ & \lesssim \left\|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_{M}g)\right\|_{\mathbb{D}^{k,p}(L^2(\mathbb{R}^2))}^p \\ & = \mathbb{E}\left\|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_{M}g)\right\|_{L^2(\mathbb{R}^2)}^p+ \sum_{l=1}^k\mathbb{E}\left\|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_MD^lg)\right\|_{L^2(\mathbb{R}^2)\otimes L^2(\mathbb{R}^l)}^p{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:prop:intR:Dnorm_integral} \end{aligned}$$ Using and Minkowski’s inequality [@HarLittlePol34 Theorem 202] twice successively, it follows that $$\begin{aligned} \mathbb{E}\left\|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_MD^lg)\right\|_{L^2(\mathbb{R}^2)\otimes L^2(\mathbb{R}^l)}^p & = \mathbb{E}\left(\int_{\mathbb{R}^l}\left\|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_{M}D^l_xg)\right\|^2_{L^2(\mathbb{R}^2)}\,\mathrm{d}{x}\right)^\frac{p}{2}\\ & \lesssim \mathbb{E}\left(\int_{\mathbb{R}^l}\left(\int_M\left|D_x^lg_s\right|^\frac{1}{H}\,\mathrm{d}{s}\right)^{2H}\,\mathrm{d}{x}\right)^\frac{p}{2}\\ & \lesssim \mathbb{E}\left(\int_M\left(\int_{\mathbb{R}^l}\left|D_x^lg_s\right|^2\,\mathrm{d}{x}\right)^\frac{1}{2H}\,\mathrm{d}{s}\right)^{pH}\\ & \lesssim \left(\int_M\left(\mathbb{E}\left(\int_{\mathbb{R}^l}\left|D_x^lg_s\right|^2\,\mathrm{d}{x}\right)^\frac{p}{2}\right)^\frac{1}{pH}\,\mathrm{d}{s}\right)^{pH}\\ & \eqsim \left(\int_M\left(\mathbb{E}\left\|D^lg_s\right\|_{L^2(\mathbb{R}^l)}^p\right)^\frac{1}{pH}\,\mathrm{d}{s}\right)^{pH}. \end{aligned}$$ Hence, by using Meyer’s inequality [@Wat84 Theorem 1.8] and the embedding $\mathbb{D}^{k,p}\hookrightarrow \mathbb{D}^{l,p}$, it follows that $$\label{eq:prop:intR:estimate_on_Dl} \mathbb{E}\left\|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_MD^lg)\right\|_{L^2(\mathbb{R}^2)\otimes L^2(\mathbb{R}^l)}^p \lesssim \left(\int_M \|g_s\|_{\mathbb{D}^{l,p}}^\frac{1}{H}\,\mathrm{d}{s}\right)^{pH}\lesssim \left(\int_M \|g_s\|_{\mathbb{D}^{k,p}}^\frac{1}{H}\,\mathrm{d}{s}\right)^{pH}$$ is satisfied for every integer $l\leq k$. Consequently, it follows that $$\left\|\int_Mg_s\delta R_s^H\right\|_{\mathbb{D}^{k-2,p}} \lesssim \|g\|_{L^\frac{1}{H}(M;\mathbb{D}^{k,p})}$$ is satisfied by estimating the terms in by . The claim for general $g\in L^\frac{1}{H}(M;\mathbb{D}^{k,p})$ is proved by a standard limit argument. The relationship between the forward and the Skorokhod integral with respect to a Rosenblatt process is given in the next proposition. For this relationship, define the second-order fractional stochastic derivative $\nabla^{\alpha,\alpha}$, following [@Arr16], by $$\nabla^{\alpha,\alpha} \overset{\textnormal{Def.}}{=} I_{+,+}^{\alpha,\alpha}\circ D^2.$$ It can be shown that if $\alpha\in (0,\sfrac{1}{2})$, the operator $\nabla^{\alpha,\alpha}$ extends to a bounded linear operator from the Sobolev-Watanabe space $\mathbb{D}^{k,p}$ to the space $L^\frac{2}{1-2\alpha}(\mathbb{R}^2;\mathbb{D}^{k-2,2})$ for $k\geq 2$ and $p\geq 2$ (and, consequently, to $L^\frac{2}{1-2\alpha}(M^2;\mathbb{D}^{k-2,2})$ for a bounded interval $M\subset\mathbb{R}$), cf. [@Arr16 Proposition 16]. The relationship between the forward and the Skorokhod integral with respect to a Rosenblatt process follows. \[prop:relationship\_for\_Skor\_R\] Let $H\in (\sfrac{1}{2},1)$ and let $g=(g_s)_{s\in [0,T]}$ be a stochastic process such that the following conditions are satisfied: 1. It holds that $$\int_0^T\|g_s\|_{\mathbb{D}^{2,2}}^\frac{1}{H}\,\mathrm{d}{s} <\infty.$$ 2. \[ass:(2)dR\] 1. For almost every $s\in [0,T]$, it follows that $$\begin{aligned} & \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (s,s+\varepsilon)}\|(\nabla^\frac{H}{2}g_s)(u)-(\nabla^\frac{H}{2}g_s)(s)\|_{\mathbb{D}^{1,2}}=0,\\ & \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (s,s+\varepsilon)}\|(\nabla^{\frac{H}{2},\frac{H}{2}}g_s)(u,u)-(\nabla^{\frac{H}{2},\frac{H}{2}}g_s)(s,s)\|_{L^2(\Omega)}=0. \end{aligned}$$ 2. There exist functions $p_1\in L^\frac{2}{1+H}(0,T)$ and $p_2\in L^1(0,T)$ such that, for almost every $s\in [0,T]$, the estimates $$\begin{aligned} & \|(\nabla^\frac{H}{2}g_s)(u)\|_{\mathbb{D}^{1,2}}\leq p_1(s),\\ & \|(\nabla^{\frac{H}{2},\frac{H}{2}}g_s)(u,u)\|_{L^2(\Omega)}\leq p_2(s) \end{aligned}$$ are satisfied for almost every $u\in[s,T]$. Then $g$ is forward integrable with respect to $R^H$ on $[0,T]$ and the following equality is satisfied in $L^2(\Omega)$: $$\label{eq:relationship_for_Skor_R} \int_0^Tg_s\,\mathrm{d}^-R_s^H = \int_0^Tg_s\delta R_s^H + 2c_H^{B,R}\int_0^T (\nabla^\frac{H}{2}g_s)(s)\delta{B}_s^{\frac{H}{2}+\frac{1}{2}} + c_H^R\int_0^T(\nabla^{\frac{H}{2},\frac{H}{2}}g_s)(s,s)\,\mathrm{d}{s}.$$ The proof follows similarly to the proof of [@Arr16 Theorem 3] where equality is only proved in the case $g_s=F(R_s^H)$. Equality should be compared to the equality given in [@Tud08 Theorem 2] where two correction terms arise as a result of the integration by parts formula for the double Skorokhod integral $\delta^2$, see [@Tud08 formula (34)] or [@NuaZak87]. These two terms have a similar structure as the ones in expression and they are given in terms of two limits, called the *trace of order 1* and *trace of order 2*. The result in [@Tud08 Theorem 2] is then proved under the assumption that these limits exist. In , sufficient conditions for this convergence are given, and, moreover, the term that corresponds to the trace of order 1 is identified as the Skorokhod integral with respect to the fractional Brownian motion of the Hurst parameter $\sfrac{H}{2} + \sfrac{1}{2}$. Additional lemmas ----------------- This section contains several results that are useful in the rest of the paper and that can also be useful for applications. Initially, a product rule for the fractional stochastic derivatives $\nabla^\alpha$ and $\nabla^{\alpha,\alpha}$ is given. \[lem:product\_rule\] Let $0<\alpha<\sfrac{1}{2}$. 1. \[lem:prod\_H\_1\] If $G_1,G_2\in\mathbb{D}^{1,4}$, then $G_1G_2\in\mathbb{D}^{1,2}$ and $$\nabla^{\alpha}(G_1G_2) = (\nabla^\alpha G_1)G_2 + G_1(\nabla^\alpha G_2).$$ 2. \[lem:prod\_H\_2\] If $G_1,G_2\in\mathbb{D}^{2,4}$, then $G_1G_2\in\mathbb{D}^{2,2}$ and $$\nabla^{\alpha,\alpha}(G_1G_2) = (\nabla^{\alpha,\alpha}G_1)G_2 + 2(\nabla^\alpha G_1)\tilde{\otimes}(\nabla^\alpha G_2) + G_1(\nabla^{\alpha,\alpha}G_2)$$ where $\tilde{\otimes}$ denotes the symmetrization of the tensor product $\otimes$. The proof follows directly from [@NouPec12 Exercise 2.3.10]. The following chain rule for the Malliavin derivative implies a chain rule for the fractional stochastic derivatives. Furthermore, its proof gives estimates that are used in the proof of . \[lem:chain\_rule\] Let $p>1$. 1. \[lem:chain\_1\] Let $f\in\mathscr{C}^1(\mathbb{R})$ be such that $$\label{eq:f'_poly_growth} |f'(x)|\leq c(1+|x|^\beta), \quad x\in\mathbb{R},$$ is satisfied for some $c\geq 0$ and $\beta\geq 0$. If $G\in\mathbb{D}^{1,p(\beta+1)}$, then $f(G)\in\mathbb{D}^{1,p}$ and $$D f(G) = f'(G)DG.$$ 2. \[lem:chain\_2\] Let $f\in\mathscr{C}^2(\mathbb{R})$ be such that $$\label{eq:f''_poly_growth} |f''(x)|\leq c(1+|x|^\beta), \quad x\in\mathbb{R},$$ is satisfied for some $c\geq 0$ and $\beta\geq 0$. If $G\in\mathbb{D}^{2,p(\beta+2)}$, then $f(G)\in\mathbb{D}^{2,p}$ and $$\label{eq:chain_2} D^2f(G) = f''(G)D G\otimes D G + f'(G)D^2 G.$$ The proof of \[lem:chain\_1\] is given in [@NuaNua18] and \[lem:chain\_2\] is proved similarly. Initially, equality is verified for $f$ smooth and $G\in\mathcal{P}$. The result is then extended for general $f$ and $G$ by approximation using the estimates $$\label{eq:estimates_on_norms_of_f} \|f(G)\|_{\mathbb{D}^{1,p}} \lesssim(1+\|G\|_{\mathbb{D}^{1,p(\beta+2)}}^{\beta+2})\quad\mbox{and}\quad \|f(G)\|_{\mathbb{D}^{2,p}} \lesssim (1+\|G\|_{\mathbb{D}^{2,p(\beta+2)}}^{\beta+2}),$$ which follow by Hölder’s inequality because there exist constants $c', c''\geq 0$ such that $$|f'(x)|\leq c'(1+|x|^{\beta+1})\quad\mbox{and}\quad |f(x)|\leq c''(1+|x|^{\beta+2}).$$ \[cor:chain\_rule\_H\] Let $0<\alpha<\sfrac{1}{2}$ and $p>1$. 1. \[cor:chain\_1\_H\] Let $f$ be as in \[lem:chain\_1\] of . If $G\in\mathbb{D}^{1,p(\beta+2)}$, then $$\nabla^{\alpha}f(G) = f'(G)(\nabla^\alpha G).$$ 2. \[cor:chain\_2\_H\] Let $f$ be as in \[lem:chain\_2\] of . If $G\in\mathbb{D}^{2,p(\beta+2)}$, then $$\nabla^{\alpha, \alpha}f(G) = f''(G)(\nabla^\alpha G)\otimes(\nabla^\alpha G) + f'(G)(\nabla^{\alpha,\alpha}G).$$ The proof of \[cor:chain\_1\_H\] follows directly from \[lem:chain\_1\] of . The proof of \[cor:chain\_2\_H\] follows from \[lem:chain\_2\] of using the fact that for $f(u,v)=f_1(u) f_2(v)$, the double integral $I_{+,+}^{\alpha,\alpha}(f)(u,v)$ equals $ I_+^\alpha(f_1)(u) I_+^\alpha (f_2)(v)$. The following lemma relates the Skorokhod integrals with respect to the fractional Brownian motion $B^H$ and the Rosenblatt process $R^H$ to the fractional stochastic derivatives $\nabla^{H-\frac{1}{2}}$ and $\nabla^{\frac{H}{2},\frac{H}{2}}$, respectively. Let $H\in (\sfrac{1}{2},1)$ and $M\subset\mathbb{R}$ be an interval. \[lem:adjoint\_property\] 1. \[lem:adjoint\_property\_B\] If $g\in L^\frac{1}{H}(M;\mathbb{D}^{1,2})$ and $G\in\mathbb{D}^{1,2}$, then the following equality is satisfied: $$\mathbb{E}\left[ G\int_Mg_u\delta B_u^H\right]= c_H^{B}\int_M\mathbb{E}\left[(\nabla^{H-\frac{1}{2}}G)(u)g_u\right]\,\mathrm{d}{u}.$$ 2. \[lem:adjoint\_property\_R\] If $g\in L^\frac{1}{H}(M;\mathbb{D}^{2,2})$ and $G\in\mathbb{D}^{2,2}$, then the following equality is satisfied: $$\mathbb{E}\left[ G\int_Mg_u\delta R_u^H\right]= c_H^R\int_M\mathbb{E}\left[ (\nabla^{\frac{H}{2},\frac{H}{2}}G)(u,u)g_u\right]\,\mathrm{d}{u}.$$ The adjoint property in \[lem:adjoint\_property\_B\] follows directly from the duality formula and the integration by parts formula for fractional integrals $I_{-}^\alpha$ and $I_{+}^\alpha$, see [@SamKilMar93 formula (5.16) on p. 96]. [@Arr16 Proposition 18] provides the proof. To prove \[lem:adjoint\_property\_R\], use the duality formula and interchange the order of the integrals. Thus $$\begin{aligned} \mathbb{E}\left[ G\int_Mg_s\delta R_s^H\right] & = c_H^R\left\langle G;\delta ^2\left(I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_Mg)\right)\right\rangle_{L^2(\Omega)}\\ & = c_H^R\left\langle D^2G;I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_Mg)\right\rangle_{L^2(\mathbb{R}^2;L^2(\Omega))}\\ & = \frac{c_H^R}{\Gamma\left(\frac{H}{2}\right)^2}\int_{\mathbb{R}^2}\mathbb{E}\left[D^2_{x,y}G\int_{x\vee y}^\infty\bm{1}_M(u)g_u(u-x)^{\frac{H}{2}-1}(u-y)^{\frac{H}{2}-1}\,\mathrm{d}{u}\right]\,\mathrm{d}{x}\,\mathrm{d}{y}\\ & = \frac{c_H^R}{\Gamma\left(\frac{H}{2}\right)^2}\int_M \mathbb{E} \left[g_u\int_{-\infty}^u\int_{-\infty}^uD_{x,y}^2G(u-x)^{\frac{H}{2}-1}(u-y)^{\frac{H}{2}-1}\,\mathrm{d}{x}\,\mathrm{d}{y}\right]\,\mathrm{d}{u}\\ & = c_H^R\int_M\mathbb{E}\left[ (\nabla^{\frac{H}{2},\frac{H}{2}}G)(u,u)g_u\right]\,\mathrm{d}{u}. \end{aligned}$$ The use of Fubini’s theorem follows by the estimate $$\begin{aligned} \frac{c_H^R}{\Gamma\left(\frac{H}{2}\right)^2}\int_{\mathbb{R}^2}\mathbb{E}\left[|D^2_{x,y}G|\int_{x\vee y}^\infty\bm{1}_M(u)|g_u|(u-x)^{\frac{H}{2}-1}(u-y)^{\frac{H}{2}-1}\,\mathrm{d}{u}\right]\,\mathrm{d}{x}\,\mathrm{d}{y} & \lesssim \\ & \hspace{-4cm} \lesssim \|D^2G\|_{L^2(\mathbb{R}^2;L^2(\Omega))}\|I_{-,\mathrm{tr}}^{\frac{H}{2},\frac{H}{2}}(\bm{1}_Mg)\|_{L^2(\mathbb{R}^2;L^2(\Omega))}\\ & \hspace{-4cm} \lesssim\|G\|_{\mathbb{D}^{2,2}}\|g\|_{L^\frac{1}{H}(M;L^2(\Omega))} \end{aligned}$$ which follows by Hölder’s inequality and [@NouPec12 Proposition 2.5.5] with . This section ends with a Fubini theorem for the Skorokhod integral with respect to Rosenblatt processes. \[lem:fubini\] Let $H\in (\sfrac{1}{2},1)$ and let $(E,\mu)$ be a measurable space equipped with a finite positive measure $\mu$. Let further $g:[0,T]\times E\rightarrow \mathbb{D}^{2,2}$ be a random field such that $$\int_E\left(\int_0^T\|g(s,x)\|_{\mathbb{D}^{2,2}}^\frac{1}{H}\,\mathrm{d}{s}\right)^{2H}\mu(\mathrm{d}{x}) <\infty.$$ Then the equality $$\int_E\left(\int_0^T g(s,x)\delta R_s^H\right)\mu(\mathrm{d}{x}) = \int_0^T\left(\int_Eg(s,x)\mu(\mathrm{d}{x})\right)\delta R_s^H$$ is satisfied almost surely. By [@NuaZak87 Proposition 2.6], the Fubini-type theorem for the Skorokhod integral $\delta$ [@NuaLeo98 Lemma 2.10] can be used iteratively. A general Itô formula {#sec:Ito_formulas} ===================== In this section, a generalized Itô-type formula for functionals of processes with second-order fractional differentials is given. Moreover, several results useful for its applications are given here as well. Let $H\in (\sfrac{1}{2},1)$ and $T>0$ be fixed for the remainder of the paper. \[prop:Ito\_formula\] Let $f\in\mathscr{C}^2([0,T]\times\mathbb{R})$ be such that for every $t\in [0,T]$, the function $f(t,\cdot)$ belongs to $\mathscr{C}^3(\mathbb{R})$ and its third derivative has at most polynomial growth, i.e. there exist constants $C_t\geq 0$ and $\alpha\geq 0$ such that $$\left|\frac{\partial^3 f}{\partial x^3}(t,x)\right| \leq C_t(1+|x|^\alpha), \quad x\in\mathbb{R}.$$ Let $x_0\in\mathbb{R}$ and $(\vartheta_t)_{t\in[0,T]}$, $(\varphi_t)_{t\in [0,T]}$, and $(\psi_t)_{t\in [0,T]}$ be stochastic processes satisfying the following: 1. \[ass:gen\_1\] $(\vartheta_t)$ belongs to the space $L^1(0,T;\mathbb{D}^{2,2(\alpha+2)})$. 2. $(\varphi_t)$ belongs to the space $L^\frac{2}{1+H}(0,T;\mathbb{D}^{3,2(\alpha+2)})$ and satisfies the following conditions: 1. For almost every $u\in[0,T]$, the following equality is satisfied $$\lim_{\varepsilon\downarrow0}\operatorname*{ess\,sup}_{v\in (u,u+\varepsilon)}\|(\nabla^\frac{H}{2}\varphi_u)(v) - (\nabla^\frac{H}{2}\varphi_u)(u)\|_{L^4(\Omega)} = 0.$$ 2. There exists a function $p_1\in L^1(0,T)$ such that, for almost every $u\in [0,T]$, the estimate $$\|\nabla^\frac{H}{2}\varphi_u(v)\|_{L^4(\Omega)}\leq p_1(u)$$ is satisfied for almost every $v\in [u,T]$. 3. \[ass:gen\_3\] $(\psi_t)$ belongs to the space $L^\frac{1}{H}(0,T;\mathbb{D}^{4,4(\alpha+2)})$ and satisfies the following conditions: 1. For almost every $u\in [0,T]$, it holds that $$\begin{aligned} & \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{v\in (u,u+\varepsilon)}\|(\nabla^\frac{H}{2}\psi_u)(v)-(\nabla^\frac{H}{2}\psi_u)(u)\|_{\mathbb{D}^{1,8}}=0,\\ & \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{v\in (u,u+\varepsilon)}\|(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(v,u) -(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(u,u)\|_{L^4(\Omega)} = 0,\\ & \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{v\in (u,u+\varepsilon)}\|(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(v,v) -(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(u,u)\|_{L^4(\Omega)} = 0. \end{aligned}$$ 2. There exist functions $p_2\in L^{\frac{2}{1+2H}}(0,T)$, $p_3\in L^1(0,T)$, and $p_4\in L^1(0,T)$ such that, for almost every $u\in [0,T]$, the inequalities $$\begin{aligned} & \|(\nabla^\frac{H}{2}\psi_u)(v)\|_{\mathbb{D}^{1,8}}\leq p_2(u),\\ & \|(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(u,v)\|_{L^4(\Omega)} \leq p_3(u),\\ & \|(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(v,v)\|_{L^4(\Omega)} \leq p_4(u), \end{aligned}$$ are satisfied for almost every $v\in [u,T]$. Define the stochastic process $(x_t)_{t\in [0,T]}$ by $$\label{eq:fractional_differential} x_t \overset{\mathrm{Def.}}{=} x_0+\int_0^t\vartheta_s\,\mathrm{d}{s} + 2c_H^{B,R}\int_0^t\varphi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}}+\int_0^t\psi_s\delta{R}^H_s$$ and assume that it is Hölder continuous of an order greater than $\sfrac{1}{2}$. Furthermore assume the following: 1. \[ass:gen\_4\] The following finiteness conditions are satisfied $$\operatorname*{ess\,sup}_{s\in [0,T]} \left\|\frac{\partial f}{\partial x}(s,x_s)\right\|_{\mathbb{D}^{2,4}} <\infty, \quad \operatorname*{ess\,sup}_{s\in [0,T]} \left\|\frac{\partial^2 f}{\partial x^2}(s,x_s)\right\|_{\mathbb{D}^{1,8}} <\infty, \quad \operatorname*{ess\,sup}_{s\in [0,T]} \left\|\frac{\partial^3 f}{\partial x^3}(s,x_s)\right\|_{L^8(\Omega)} <\infty.$$ 2. \[ass:gen\_5\] For almost every $v\in [0,T]$, the following equalities are satisfied $$\begin{aligned} & \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)} \|(\nabla^{\frac{H}{2}}X_u)(v) - (\nabla^\frac{H}{2}X_v)(v)\|_{\mathbb{D}^{1,8}}=0,\\ &\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)} \|(\nabla^{\frac{H}{2}, \frac{H}{2}}X_u)(v,v) - (\nabla^\frac{H}{2}X_v)(v,v)\|_{L^8(\Omega)}=0.\\ \end{aligned}$$ 3. \[ass:gen\_6\] For almost every $s\in [0,T]$ and almost every $v\in [0,T]$, the following equalities are satisfied $$\begin{aligned} & \lim_{\varepsilon\downarrow 0} \operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)}\|(\nabla^\frac{H}{2}X_s)(u) - (\nabla^\frac{H}{2}X_s)(v)\|_{\mathbb{D}^{1,8}} = 0,\\ & \lim_{\varepsilon\downarrow 0} \operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)}\|(\nabla^{\frac{H}{2},\frac{H}{2}}X_s)(u,u) - (\nabla^{\frac{H}{2},\frac{H}{2}}X_s)(v,v)\|_{L^{8}(\Omega)} = 0. \end{aligned}$$ 4. \[ass:gen\_7\] The following finiteness conditions are satisfied $$\operatorname*{ess\,sup}_{s,u\in [0,T]} \|(\nabla^\frac{H}{2}X_s)(u)\|_{\mathbb{D}^{1,8}}<\infty \quad \mbox{and} \quad \operatorname*{ess\,sup}_{s,u\in [0,T]}\|(\nabla^{\frac{H}{2},\frac{H}{2}}X_s(u,u)\|_{L^8(\Omega)}<\infty.$$ Then for the process $(y_t)_{t\in [0,T]}$ defined by $y_t=f(t,x_t)$ the equality $$\label{eq:Ito_formula_full} y_t = y_0 + \int_0^t\tilde{\vartheta}_s\,\mathrm{d}{s} + 2c_H^{B,R}\int_0^t\tilde{\varphi}_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} + \int_0^t\tilde{\psi}_s\delta R^H_s$$ is satisfied with $$\begin{aligned} \tilde{\vartheta}_s & = \frac{\partial f}{\partial s}(s,x_s) + \frac{\partial f}{\partial x}(s,x_s)\vartheta_s \\ & \hspace{2cm} +\,\, 2c_H^R\frac{\partial^2 f}{\partial x^2}(s,x_s)(\nabla^\frac{H}{2}x_s)(s)\varphi_s\\ & \hspace{4cm} + \,\, c_H^R \frac{\partial^2f}{\partial x^2}(s,x_s) (\nabla^{\frac{H}{2},\frac{H}{2}}x_s)(s,s)\psi_s \\ & \hspace{6cm} + \,\, c_H^R\frac{\partial^3f}{\partial x^3}(s,x_s) [(\nabla^\frac{H}{2}x_s)(s)]^2\psi_s,\\ \tilde{\varphi}_s & = \frac{\partial f}{\partial x}(s,x_s)\varphi_s + \frac{\partial^2 f}{\partial x^2}(s,x_s)(\nabla^\frac{H}{2}x_s)(s)\psi_s,\\ \tilde{\psi}_s & = \frac{\partial f}{\partial x}(s,x_s)\psi_s. \end{aligned}$$ for every $t\in [0,T]$ almost surely. Let $t\in[0,T]$. Initially, the proof of the Itô formula for the forward integral in [@RusVal95 Theorem 1.2] is used. Since the process $(x_t)$ has continuous sample paths, the equality $$f(t,x_t)-f(0,x_0) = \lim_{\varepsilon\downarrow 0} \frac{1}{\varepsilon}\int_0^t\left[f(s+\varepsilon,x_{s+\varepsilon}) - f(s,x_s)\right]\,\mathrm{d}{s} \overset{\mathrm{Def.}}{=} \lim_{\varepsilon\downarrow 0} A_{t,\varepsilon}$$ is satisfied almost surely. Using Taylor’s formula to expand the integrand in $A_{t,\varepsilon}$ yields, for $\varepsilon>0$, that $$\begin{aligned} A_{t,\varepsilon} & = \int_0^t\frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}{s}+ \frac{1}{\varepsilon}\int_0^t\frac{\partial f}{\partial x}(s,x_s)(x_{s+\varepsilon}-x_s)\,\mathrm{d}{s}\\ & \hspace{1cm} + \varepsilon \int_0^tR_1(s,\varepsilon)\,\mathrm{d}{s} + \int_0^tR_2(s,\varepsilon)(x_{s+\varepsilon}-x_s)\,\mathrm{d}{s} + \frac{1}{\varepsilon}\int_0^tR_3(s,\varepsilon)(x_{s+\varepsilon}-x_s)^2\,\mathrm{d}{s}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:Ito_4} \end{aligned}$$ is satisfied almost surely. Here, the processes $R_1,R_2,R_3$ are remainders in the integral form. All the three terms containing these remainders tend to zero as $\varepsilon \downarrow 0$; the second by continuity of $x$, the third by the fact that sample paths of $x$ are Hölder continuous of an order greater than $\sfrac{1}{2}$. Therefore focus on the second term of $A_{t,\varepsilon}$ that tends to the forward integral. An idea from [@BiaOks08] is used. By and , the Skorokhod-type integrals in $x$ can be written as forward integrals by adding the appropriate correction terms: $$\begin{aligned} x_{s+\varepsilon} -x_s & =\int_{s}^{s+\varepsilon}\psi_u\,\mathrm{d}^{-}R_u^H + 2c_{H}^{B,R}\int_s^{s+\varepsilon} \left[\varphi_u-(\nabla^\frac{H}{2}\psi_u)(u)\right]\,\mathrm{d}^{-}{B}_u^{\frac{H}{2}+\frac{1}{2}}\\ & \hspace{1.5cm} + \int_s^{s+\varepsilon}\left[\vartheta_u - 2c_H^R(\nabla^\frac{H}{2}\varphi_u)(u)+c_H^R(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(u,u)\right]\,\mathrm{d}{u}. \end{aligned}$$ Forward integrals commute with random variables so that the equality $$\begin{aligned} \frac{\partial f}{\partial x}(s,x_s)(x_{s+\varepsilon}-x_s) & = \int_s^{s+\varepsilon} \frac{\partial f}{\partial x}(s,x_s)\psi_u\,\mathrm{d}^{-}R_u^H \\ & \hspace{1cm} + 2c_H^{B,R}\int_s^{s+\varepsilon}\frac{\partial f}{\partial x}(s,x_s)\left[\varphi_u-(\nabla^\frac{H}{2}\psi_u)(u)\right]\,\mathrm{d}^{-}{B}_u^{\frac{H}{2}+\frac{1}{2}}\\ & \hspace{1cm} + \int_s^{s+\varepsilon} \frac{\partial f}{\partial x}(s,x_s)\left[\vartheta_u - 2c_H^R(\nabla^\frac{H}{2}\varphi_u)(u)+c_H^R(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(u,u)\right]\,\mathrm{d}{u} \end{aligned}$$ is satisfied almost surely. Denote $$B_{s,\varepsilon} \overset{\mathrm{Def.}}{=} \frac{\partial f}{\partial x}(s,x_s)(x_{s+\varepsilon}-x_s).$$ By and , the forward integrals above can be written as Skorokhod-type integrals and thus $$\begin{aligned} B_{s,\varepsilon} & = \int_s^{s+\varepsilon} \frac{\partial f}{\partial x}(s,x_s)\psi_u\delta R_u^H \\ & \hspace{1cm} + 2c_H^{B,R}\int_s^{s+\varepsilon}\left[\frac{\partial f}{\partial x}(s,x_s)\left(\varphi_u-(\nabla^\frac{H}{2}\psi_u)(u)\right)+\nabla^\frac{H}{2}\left(\frac{\partial f}{\partial x}(s,x_s)\psi_u\right)(u)\right]\delta{B}_u^{\frac{H}{2}+\frac{1}{2}}\\ & \hspace{1cm} + \int_s^{s+\varepsilon}\bigg[\frac{\partial f}{\partial x}(s,x_s)\vartheta_u+ 2c_H^R\nabla^\frac{H}{2}\left(\frac{\partial f}{\partial x}(s,x_s)\varphi_u\right)(u)-2c_H^R\frac{\partial f}{\partial x}(s,x_s)(\nabla^\frac{H}{2}\varphi_u)(u) \\ & \hspace{3cm} + c_H^R\frac{\partial f}{\partial x}(s,x_s)\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u(u,u) + c_H^R\nabla^{\frac{H}{2},\frac{H}{2}}\left(\frac{\partial f}{\partial x}(s,x_s)\psi_u\right)(u,u)\\ &\hspace{3cm} -2c_H^R\nabla^\frac{H}{2}\left(\frac{\partial f}{\partial x}(s,x_s)\nabla^\frac{H}{2}\psi_u(u)\right)(u)\bigg]\,\mathrm{d}{u}.{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:Ito_5} \end{aligned}$$ The terms in the above expression are now simplified. Using the product and chain rules from and , it follows that $$\nabla^\frac{H}{2}\left(\frac{\partial f}{\partial x}(s,x_s)\psi_u\right)(u) = \frac{\partial^2f}{\partial x^2}(s,x_s)(\nabla^\frac{H}{2} x_s)(u)\psi_u + \frac{\partial f}{\partial x}(s,x_s)(\nabla^\frac{H}{2} \psi_u)(u).$$ Similarly, it follows that $$\begin{aligned} \nabla^\frac{H}{2}\left(\frac{\partial f}{\partial x}(s,x_s)\nabla^\frac{H}{2}\psi_u(u)\right)(u) = \frac{\partial^2f}{\partial x}(s,x_s)\nabla^\frac{H}{2}x_s(u)\nabla^\frac{H}{2}\psi_u(u) + \frac{\partial f}{\partial x}(s,x_s)\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u(u,u) \end{aligned}$$ and, for the second-order term, that $$\begin{aligned} \nabla^{\frac{H}{2},\frac{H}{2}}\left(\frac{\partial f}{\partial x}(s,x_s)\psi_u\right)(u,u) & = \frac{\partial^3f}{\partial x^3}(s,x_s)\left[(\nabla^\frac{H}{2}x_s)(u)\right]^2\psi_u + \\ & \hspace{1cm} + \frac{\partial^2f}{\partial x^2}(s,x_s) \left[2(\nabla^\frac{H}{2}\psi_u)(u)(\nabla^\frac{H}{2}x_s)(u) + \psi_u(\nabla^{\frac{H}{2},\frac{H}{2}}x_s)(u,u) \right]\\ & \hspace{1cm} + \frac{\partial f}{\partial x}(s,x_s)(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_u)(u,u). \end{aligned}$$ Substituting these formulas in yields that $$\begin{aligned} B_{s,\varepsilon} & = \int_s^{s+\varepsilon}\frac{\partial f}{\partial x}(s,x_s)\psi_u\delta R_u^H \\ & \hspace{1cm} + 2c_H^{B,R}\int_s^{s+\varepsilon}\left[\frac{\partial f}{\partial x}(s,x_s)\varphi_u+ \frac{\partial^2f}{\partial x^2}(s,x_s)(\nabla^\frac{H}{2} x_s)(u)\psi_u\right]\delta{B}_u^{\frac{H}{2}+\frac{1}{2}}\\ & \hspace{1cm} + \int_s^{s+\varepsilon}\bigg[\frac{\partial f}{\partial x}(s,x_s)\vartheta_u + 2c_H^R\frac{\partial^2f}{\partial x^2}(s,x_s)(\nabla^\frac{H}{2}x_s)(u)\varphi_u \\ & \hspace{3cm} + c_H^R\frac{\partial^2 f}{\partial x^2}(s,x_s)(\nabla^{\frac{H}{2},\frac{H}{2}}x_s)(u,u)\psi_u +c_H^R\frac{\partial^3f}{\partial x^3}(s,x_s)[(\nabla^\frac{H}{2}x_s)(u)]^2\psi_u\bigg]\,\mathrm{d}{u} \end{aligned}$$ is satisfied almost surely. In order to finish the proof, it suffices to verify the convergence $$\label{eq:Ito_3} \frac{1}{\varepsilon}\int_0^tB_{s,\varepsilon}\,\mathrm{d}{s} \quad \overset{L^2(\Omega)}{\underset{\varepsilon\downarrow 0}{\longrightarrow }}\quad \int_0^t \left[\tilde{\vartheta}_s-\frac{\partial f}{\partial s}(s,x_s)\right]\,\mathrm{d}{s} + 2c_H^{B,R}\int_0^t\tilde{\varphi}_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} + \int_0^t \tilde{\psi}_s\delta R_s^H$$ where the processes $(\tilde{\vartheta})$, $(\tilde{\varphi})$, and $(\tilde{\psi})$ are given in . Only the following convergence is proved: $$\label{eq:Ito_2} \frac{1}{\varepsilon}\int_0^t\int_s^{s+\varepsilon}\frac{\partial f}{\partial x}(s,x_s)\psi_u\delta R_u^H\,\mathrm{d}{s}\quad \overset{L^2(\Omega)}{\underset{\varepsilon\downarrow 0}{\longrightarrow }}\quad\int_0^t \frac{\partial f}{\partial x}(s,x_s)\psi_s\delta R_s^H.$$ The convergence of the other terms can be shown in a similar manner. By using the Fubini-type theorem in , it follows that $$\frac{1}{\varepsilon}\int_0^t\int_s^{s+\varepsilon}\frac{\partial f}{\partial x}(s,x_s)\psi_u\delta R_u^H\,\mathrm{d}{s} = \frac{1}{\varepsilon}\int_0^{t+\varepsilon}\left(\int_{(u-\varepsilon)\vee 0}^{u\wedge t}\frac{\partial f}{\partial x}(s,x_s)\psi_u\,\mathrm{d}{s}\right)\delta R_u^H$$ is satisfied almost surely. By using and [@Nua06 Proposition 1.5.6], it follows that $$\begin{aligned} \left\|\frac{1}{\varepsilon}\int_\varepsilon^{t}\int_{u-\varepsilon}^u \frac{\partial f}{\partial x}(s,x_s)\psi_u\,\mathrm{d}{s}\delta R_u^H - \int_\varepsilon^t \frac{\partial f}{\partial x}(u,x_u)\psi_u\delta R_u^H\right\|_{L^2(\Omega)}^{H} & \lesssim\\ & \hspace{-4cm} \lesssim \int_\varepsilon^t\left\|\psi_u\left[\frac{1}{\varepsilon}\int_{u-\varepsilon}^u\frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}{s} - \frac{\partial f}{\partial x}(u,x_u)\right]\right\|_{\mathbb{D}^{2,2}}^\frac{1}{H}\,\mathrm{d}{u} \\ &\hspace{-4cm} \lesssim \int_\varepsilon^t\|\psi_u\|_{\mathbb{D}^{2,4}}^\frac{1}{H}\left\|\frac{1}{\varepsilon}\int_{u-\varepsilon}^u\frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}{s} - \frac{\partial f}{\partial x}(u,x_u)\right\|_{\mathbb{D}^{2,4}}^\frac{1}{H}\,\mathrm{d}{u}. \end{aligned}$$ The convergence $$\frac{1}{\varepsilon}\int_{u-\varepsilon}^u\frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}{s} \quad\overset{\mathbb{D}^{2,4}}{\underset{\varepsilon\downarrow 0}{\longrightarrow}} \quad \frac{\partial f}{\partial x}(u,x_u)$$ is satisfied for almost every $u\in [0,t]$ by the Lebesgue differentiation theorem because $$\int_0^t\left\|\frac{\partial f}{\partial x}(s,x_s)\right\|_{\mathbb{D}^{2,4}}\,\mathrm{d}{s}\leq t\,\operatorname*{ess\,sup}_{u\in (0,t)}\left\|\frac{\partial f}{\partial x}(s,x_s)\right\|_{\mathbb{D}^{2,4}} <\infty.$$ Moreover, the estimate $$\|\psi_u\|_{\mathbb{D}^{2,4}}^\frac{1}{H}\left\|\frac{1}{\varepsilon}\int_{u-\varepsilon}^u\left[\frac{\partial f}{\partial x}(s,x_s) - \frac{\partial f}{\partial x}(u,x_u)\right]\,\mathrm{d}{s}\right\|_{\mathbb{D}^{2,4}}^\frac{1}{H}\lesssim \|\psi_u\|_{\mathbb{D}^{2,4}}^\frac{1}{H}\left(\operatorname*{ess\,sup}_{s\in (0,t)}\left\|\frac{\partial f}{\partial x}(s,x_s)\right\|_{\mathbb{D}^{2,4}}\right)^\frac{1}{H}$$ is satisfied for sufficiently small $\varepsilon>0$, and because the right-hand side is integrable (with respect to $\,\mathrm{d}{u}$) on the interval $(0,t)$, Lebesgue’s dominated convergence theorem yields the desired convergence . The significance of is two-fold. Firstly, it describes the general structure of the Itô-type formula that should be expected when a Rosenblatt integrator is involved. Secondly, it gives a general method of proof that could be employed in concrete situations. Moreover, it is expected that the equality will be satisfied even under a different set of assumptions that are more suitable in specific cases. Some remarks of the assumptions used here follow. 1. The assumption of Hölder continuity of $x$ is rather natural since it is known that the Skorokhod-type integrals retain Hölder continuity of the integrator (under suitable conditions on the integrand), see [@AlosNua03 Theorem 5] for fractional integrators and [@Tud08 Proposition 4] for Rosenblatt integrators. 2. Polynomial growth of $f$ as well as the corresponding integrability of the processes $(\vartheta)$, $(\varphi)$, and $(\psi)$ is assumed for the purposes of the chain rule for Malliavin derivative in . However, for a specific problem, a more suitable version of the chain rule can be used, see e.g. [@Nua06 Proposition 1.2.3 or Proposition 1.2.4]. This would lead to a different set of assumptions. 3. It is not assumed that any of the processes $(\vartheta)$, $(\varphi)$, or $(\psi)$ are adapted. 4. Assumptions \[ass:gen\_4\] - \[ass:gen\_7\] are not independent of assumptions \[ass:gen\_1\] - \[ass:gen\_3\] and they can pose even stronger conditions on the processes $(\vartheta)$, $(\varphi)$, and $(\psi)$. However, in many cases, it can be convenient to verify conditions that are formulated in terms of the process $x$. In , it is shown how to obtain conditions solely in terms of the integrand. Note that the convergence of the terms with the remainders $R_1, R_2$, and $R_3$ in the equality shows that the forward integral $$\int_0^t\frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}^{-}x_s$$ exists and that $$\label{eq:Ito_forward} f(t,x_t)-f(0,x_0) = \int_0^t\frac{\partial f}{\partial s}(s,x_s)\,\mathrm{d}{s} + \int_0^t\frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}^{-}x_s$$ is satisfied almost surely (this is, in fact, the statement of [@RusVal95 Theorem 1.2] for processes with zero quadratic variation). Therefore, if the equality $$\begin{aligned} \int_0^t\frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}^{-}x_s & = \int_0^t \frac{\partial f}{\partial x}(s,x_s)\,\mathrm{d}^{-}R_s^H + 2c_H^{B,R}\int_0^t \frac{\partial f}{\partial x}(s,x_s)[\varphi_s-(\nabla^\frac{H}{2}\psi_s)(s)]\,\mathrm{d}^{-}B_s^{\frac{H}{2}+\frac{1}{2}} \\ & \hspace{1cm} + \int_0^t \frac{\partial f}{\partial x}(s,x_s)\bigg[\vartheta_s - 2c_H^R(\nabla^\frac{H}{2}\varphi_s)(s) + c_H^R (\nabla^{\frac{H}{2},\frac{H}{2}}\psi_s)(s,s)\bigg]\,\mathrm{d}{s}, {\addtocounter{equation}{1}\tag{\theequation}}\label{eq:forward_transition} \end{aligned}$$ is satisfied almost surely, the proof of could be as follows: Rewrite the Skorokhod integrals in as forward integrals by means of and , use the Itô formula for processes with forward differential and equality , and rewrite the resulting forward integrals back in their Skorokhod form. To be able to employ this strategy, it would be desirable to find general conditions under which the equality $$\label{eq:interchange_formula} \int_0^T g_s^{(1)}\,\mathrm{d}^{-}H_s = \int_0^T g_s^{(1)}g_s^{(2)}\,\mathrm{d}^{-}h_s$$ is satisfied almost surely for some continuous process $(g_t^{(1)})_{t\in[0,T]}$ and the process $(H_t)_{t\in [0,T]}$ that is given by $$H_t = \int_0^t g_s^{(2)}\,\mathrm{d}^{-}h_s$$ with some integrable process $(g_t^{(2)})_{t\in [0,T]}$ and continuous process $(h_t)_{t\in [0,T]}$. This problem is also the subject of [@Zahle99 Remark 5.9]. Formally, there are the following equalities $$\begin{aligned} \int_0^t g_s^{(1)}\,\mathrm{d}^{-}H_s & = \operatorname*{\mathbb{P}-lim}_{\varepsilon\downarrow 0} \frac{1}{\varepsilon}\int_0^t g_s^{(1)}\int_s^{s+\varepsilon}g_u^{(2)}\,\mathrm{d}^{-}H_u\,\mathrm{d}{s}\\ & =\operatorname*{\mathbb{P}-lim}_{\varepsilon\downarrow 0}\frac{1}{\varepsilon}\int_0^tg_s^{(1)}\left(\operatorname*{\mathbb{P}-lim}_{\delta\downarrow 0}\frac{1}{\delta }\int_s^{s+\varepsilon}g_u^{(2)}(h_{u+\delta}-h_u)\,\mathrm{d}{u}\right)\,\mathrm{d}{s}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:interchange_1}\\ & = \operatorname*{\mathbb{P}-lim}_{\varepsilon\downarrow 0}\operatorname*{\mathbb{P}-lim}_{\delta\downarrow 0}\frac{1}{\varepsilon}\int_0^tg_s^{(1)}\left(\frac{1}{\delta }\int_s^{s+\varepsilon}g_u^{(2)}(h_{u+\delta}-h_u)\,\mathrm{d}{u}\right)\,\mathrm{d}{s}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:interchange_2}\\ & = \operatorname*{\mathbb{P}-lim}_{\delta\downarrow 0}\operatorname*{\mathbb{P}-lim}_{\varepsilon\downarrow 0}\frac{1}{\delta}\int_0^tg_s^{(1)}\left(\frac{1}{\varepsilon }\int_s^{s+\varepsilon}g_u^{(2)}(h_{u+\delta}-h_u)\,\mathrm{d}{u}\right)\,\mathrm{d}{s}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:interchange_3}\\ & = \operatorname*{\mathbb{P}-lim}_{\delta\downarrow 0}\frac{1}{\delta}\int_0^tg_s^{(1)}\left(\operatorname*{\mathbb{P}-lim}_{\varepsilon\downarrow 0}\frac{1}{\varepsilon }\int_s^{s+\varepsilon}g_u^{(2)}(h_{u+\delta}-h_u)\,\mathrm{d}{u}\right)\,\mathrm{d}{s}\\ & = \operatorname*{\mathbb{P}-lim}_{\delta\downarrow 0}\frac{1}{\delta}\int_0^tg_s^{(1)}g_s^{(2)}(h_{s+\varepsilon}+h_{s})\,\mathrm{d}{s}\\ & = \int_0^tg_s^{(1)}g_s^{(2)}\,\mathrm{d}^{-}h_s. \end{aligned}$$ The above formal computation would be correct provided that the probability limit can be interchanged with the outer integral in and , and that the interchange of the probability limits in is also possible. Some criteria for the interchange of probability limits and the integrals can be found using Markov’s inequality and a Moore-Osgood-type argument could allow to interchange the two probability limits. In the case of , however, the validity of is shown at the level of $\varepsilon$-approximations via the $L^2(\Omega)$-convergence . is compared to the Itô formula for regular fractional Brownian motions in [@DunHuDun00 Theorem 4.5]. In this remark, the same symbols as in are used to complete the analogy, however note that the objects can be different. Thereom 4.5 of [@DunHuDun00] states that if $(x_t)$ is the stochastic process defined by $$x_t = x_0 + \int_0^t\vartheta_s\,\mathrm{d}{s} + \int_0^t\varphi_s\delta B^{H}_s,$$ where $H\in (\sfrac{1}{2},1)$ and $(\vartheta)$ and $(\varphi)$ are stochastic process that satisfy suitable integrability assumptions, then the equality $$\label{eq:Ito_fbm} f(t,x_t) = f(0,x_0) + \int_0^t\left[\frac{\partial f}{\partial s}(s,x_s) + \frac{\partial f}{\partial x}(s,x_s)\vartheta_s + \frac{\partial f}{\partial x}(s,x_s)\varphi_s D^\phi_sx_s\right]\,\mathrm{d}{s} + \int_0^t\frac{\partial f}{\partial x}(s,x_s)\varphi_s\delta B_s^{H}.$$ is satisfied almost surely. In formula , the operator $D^\phi$ is the operator $\nabla^{H-\frac{1}{2}}$ (up to a constant). Therefore, the structure of a process with a first-order fractional differential is preserved under compositions with $\mathscr{C}^2$ functions; that is, for $y_t=f(t,x_t)$ it follows that $$y_t=y_0+ \int_0^t \tilde{\vartheta}_s\,\mathrm{d}{s} + \int_0^t \tilde{\varphi}_s\delta B_s^{H}.$$ In the case where $(x_t)$ is defined by $$x_t = x_0 + \int_0^t\vartheta_s\,\mathrm{d}{s} + 2c_H^{B,R}\int_0^t \varphi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} + \int_0^t\psi_s\delta R_s^H,$$ the situation is analogous to the case of fractional Brownian motions because the following formula for $y_t=f(t,x_t)$ is obtained: $$y_t = y_0 + \int_0^t\tilde{\vartheta}_s\,\mathrm{d}{s} + 2c_H^{B,R}\int_0^t \tilde{\varphi}_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} + \int_0^t\tilde{\psi}_s\delta R_s^H.$$ On the other hand, a notable difference between formula and formula is the appearance of a term that involves the third derivative $\frac{\partial^3f}{\partial x^3}(s,x_s)$ and a term that involves the second-order fractional stochastic derivative $\nabla^{\frac{H}{2},\frac{H}{2}}x_s$. Both of these terms arise as a result of the chain rule used to compute the second-order fractional stochastic derivative $\nabla^{\frac{H}{2},\frac{H}{2}}(\frac{\partial f}{\partial x}(s,x_s))$. Thus, the appearance of these new terms is a direct consequence of the second-order nature of Rosenblatt processes. These phenomena are also discussed in [@Arr15 p. 548] and in [@Tud08 Remark 8]. For applications of , it is necessary to compute explicit formulas for $\nabla^\frac{H}{2}x_t$ and $\nabla^{\frac{H}{2},\frac{H}{2}}x_t$ where $x_t$ is given by $$x_t = x_0 + \int_0^t\vartheta_s\,\mathrm{d}{s} + 2c_H^{B,R}\int_0^t\varphi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} + \int_0^t\psi_s\delta R_s^H.$$ This requires being able to compute both the first and second-order fractional stochastic derivative of the Skorokhod integral with respect to both the fractional Brownian motion and the Rosenblatt process. Explicit formulas are given in the following four lemmas. These lemmas are proved by, possibly iterative, use of [@Nua06 Proposition 1.3.8] and formula . The first two lemmas give expressions for the first and second-order fractional stochastic derivatives of the Skorokhod integral with respect to a fractional Brownian motion. \[prop:1der1int\] Let $g\in L^\frac{2}{1+H}(0,T;\mathbb{D}^{2,2})$. Then the following equality is satisfied for almost every $x\in\mathbb{R}$: $$\nabla^\frac{H}{2}\left(\int_0^Tg_s\delta B_s^{\frac{H}{2}+\frac{1}{2}}\right)(x) = \int_0^T(\nabla^\frac{H}{2}g_s)(x)\delta B_s^{\frac{H}{2}+\frac{1}{2}} + c_{\frac{H}{2}+\frac{1}{2}}^B\frac{\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2}\int_0^Tg_s|s-x|^{H-1}\,\mathrm{d}{s}.$$ The above should be compared to [@DunHuDun00 Theorem 4.2]. Note, in particular, that the constant appearing in front of the second integral is different. \[prop:2der\_1int\] Let $g\in L^\frac{2}{1+H}(0,T;\mathbb{D}^{3,2})$. Then the following equality is satisfied for almost every $x,y\in\mathbb{R}$: $$\begin{aligned} \nabla^{\frac{H}{2},\frac{H}{2}}\left(\int_0^Tg_s\delta B_s^{\frac{H}{2}+\frac{1}{2}}\right)(x,y) & = \int_0^T(\nabla^{\frac{H}{2},\frac{H}{2}}g_s)(x,y)\delta B_s^{\frac{H}{2}+\frac{1}{2}}\\ & \hspace{-3cm} + c_{\frac{H}{2}+\frac{1}{2}}^B\frac{\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2} \left(\int_0^T(\nabla^\frac{H}{2}g_s)(x)|s-y|^{H-1}\,\mathrm{d}{s} + \int_0^T(\nabla^\frac{H}{2}g_s)(y)|s-x|^{H-1}\,\mathrm{d}{s}\right). \end{aligned}$$ The following two propositions can be used to compute the first and second-order fractional stochastic derivative of the Skorokhod integral with respect to a Rosenblatt process. \[prop:1der\_2int\] Let $g\in L^\frac{1}{H}(0,T;\mathbb{D}^{3,2})$. Then the following equality is satisfied for almost every $x\in\mathbb{R}$: $$\begin{aligned} \nabla^{\frac{H}{2}}\left(\int_0^Tg_s\delta R_s^H\right)(x) & = \int_0^T(\nabla^\frac{H}{2}g_s)(x)\delta R_s^H + 2c_{H}^{B,R}\frac{\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2}\int_0^Tg_s|s-x|^{H-1}\delta B_s^{\frac{H}{2}+\frac{1}{2}}.\end{aligned}$$ Let $g\in L^\frac{1}{H}(0,T;\mathbb{D}^{4,2})$. Then the following equality is satisfied for almost every $x,y\in\mathbb{R}$: \[prop:2der\_2int\] $$\begin{aligned} \nabla^{\frac{H}{2},\frac{H}{2}}\left(\int_0^Tg_s\delta R_s^H\right)(x,y) & = \int_0^T(\nabla^{\frac{H}{2},\frac{H}{2}}g_s)(x,y)\delta R_s^H\\ & \hspace{-3.1cm} + 2c_H^{B,R}\frac{\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2}\left(\int_0^T(\nabla^\frac{H}{2}g_s)(x)|s-y|^{H-1}\delta{B}_s^{\frac{H}{2}+\frac{1}{2}} + \int_0^T(\nabla^\frac{H}{2}g_s)(y)|s-x|^{H-1}\delta{B}_s^{\frac{H}{2}+\frac{1}{2}}\right) \\ & \hspace{-3.1cm} + 2c_H^R\frac{\mathrm{B}\left(\frac{H}{2},1-H\right)^2}{\Gamma\left(\frac{H}{2}\right)^4}\int_0^Tg_s|s-x|^{H-1}|s-y|^{H-1}\,\mathrm{d}{s}.\end{aligned}$$ Some special cases of are given now. Since the Itô formula for functionals of Skorokhod integrals with respect to the fractional Brownian motion is well-known, e.g. [@DunHuDun00], functionals of the Skorokhod integral with respect to a Rosenblatt process are considered. Moreover, in this case, it is possible to formulate sufficient conditions for the Itô formula in terms of the integrand rather than in terms of the integral. \[prop:Ito\_for\_integral\_only\] Let $f$ be a function in $\mathscr{C}^3(\mathbb{R})$ such that $$\label{eq:poly_growth} |f'''(x)|\leq c(1+|x|^\alpha), \quad x\in\mathbb{R},$$ is satisfied for some $c\geq 0$ and $\alpha\geq 0$. Let $\psi=(\psi_s)_{s\in [0,T]}$ be a stochastic process such that the following three assumptions are satisfied: 1. \[ass:psi\_1\] The proces $\psi$ is in $L^\infty(0,T;\mathbb{D}^{4,p})$ for some $$p>\max\left\{\frac{2}{(2H-1)}, 8(\alpha+1)\right\}.$$ 2. \[ass:psi\_4\] For almost every $r,v\in [0,T]$, the following equalities are satisfied $$\begin{aligned} & \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)}\|(\nabla^\frac{H}{2}\psi_r)(u)-(\nabla^\frac{H}{2}\psi_r)(v)\|_{\mathbb{D}^{3,8}} = 0,\\ &\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)} \|(\nabla^{\frac{H}{2}, \frac{H}{2}}\psi_r)(u,u)-(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(v,v)\|_{\mathbb{D}^{2,8}}=0, \\ &\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)} \|(\nabla^{\frac{H}{2}, \frac{H}{2}}\psi_r)(r,u)-(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(r,v)\|_{\mathbb{D}^{2,8}}=0. \end{aligned}$$ 3. \[ass:psi\_2\] There exist functions $p_1\in L^\frac{6}{1+H}(0,T)$, $p_2\in L^\frac{1}{H}(0,T)$, and $p_3\in L^\frac{1}{H}(0,T)$ such that for almost every $r\in [0,T]$, the estimates $$\begin{aligned} & \|(\nabla^\frac{H}{2}\psi_r)(v)\|_{\mathbb{D}^{3,8}}\leq p_1(r)\\ & \|(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(v,v)\|_{\mathbb{D}^{2,8}}\leq p_2(r)\\ & \|(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(r,v)\|_{\mathbb{D}^{2,8}}\leq p_3(r) \end{aligned}$$ are satisfied for almost every $v\in [0,T]$. Define the process $(Z_t)_{t\in [0,T]}$ by $$\label{eq:Z} Z_t \overset{\mathrm{Def.}}{=}\int_0^t\psi_r\delta R_r^H.$$ Then the following equality is satisfied for every $t\in [0,T]$ almost surely: $$\begin{aligned} f(Z_t) - f(0) & = \int_0^t f'(Z_s)\psi_s\delta R_s^H \\ & \hspace{1cm} + 2c_H^{B,R}\int_0^tf''(Z_s)(\nabla^\frac{H}{2}Z_s)(s)\psi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} \\ & \hspace{2cm} + c_H^R\int_0^t \left(f''(Z_s)(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + f'''(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\psi_s\,\mathrm{d}{s}.{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:Ito_for_integral} \end{aligned}$$ The conditions of are verified. Condition \[ass:gen\_3\] of is clearly satisfied. It is therefore necessary to verify that the process $Z$ has a version with Hölder continuous sample paths of order greater than $\sfrac{1}{2}$ (that is considered in the sequel) and that conditions \[ass:gen\_4\] - \[ass:gen\_7\] of are satisfied. *Claim 1:* The process $Z$ has a version with Hölder continuous sample paths of an order greater than $\sfrac{1}{2}$. *Proof of Claim 1:* By using , the embedding $\mathbb{D}^{4,p}\hookrightarrow \mathbb{D}^{2,p}$, and assumption \[ass:psi\_1\] successively, it follows that $$\mathbb{E}|Z_t-Z_s|^p=\mathbb{E}\left|\int_s^t \psi_r\delta R_r^H\right|^p \lesssim \left(\int_s^t \|\psi_r\|_{\mathbb{D}^{2,p}}^\frac{1}{H}\,\mathrm{d}{r}\right)^{pH} \leq \|\psi\|_{L^\infty(0,T;\mathbb{D}^{4,p})}^p (t-s)^{pH}$$ is satisfied for $0\leq s<t\leq T$. This implies, by the Kolmogorov-Chentsov criterion, that the process $(Z_t)_{t\in [0,T]}$ has a version with Hölder continuous sample paths of every order less than $H-\sfrac{1}{p}$. Clearly, $H-\sfrac{1}{p} > \sfrac{1}{2}$, see also [@Tud08 Proposition 4]. *Claim 2:* The process $Z$ satisfies condition \[ass:gen\_4\] of . *Proof of Claim 2:* By using the estimate , and the estimate $$\begin{aligned} \operatorname*{ess\,sup}_{s\in[0,T]}\|f'(Z_s)\|_{\mathbb{D}^{2,4}} & \lesssim 1 + \operatorname*{ess\,sup}_{s\in [0,T]}\|Z_s\|_{\mathbb{D}^{2,4(\alpha+2)}}^{\alpha+2}\\ & \lesssim 1 + \operatorname*{ess\,sup}_{s\in [0,T]}\left(\int_0^s\|\psi_r\|_{\mathbb{D}^{4,4(\alpha+2)}}^\frac{1}{H}\,\mathrm{d}{r}\right)^{H(\alpha+2)}\\ & \lesssim T^{H(\alpha+2)}\|\psi\|_{L^\infty(0,T;\mathbb{D}^{4,4(\alpha+2)})}^{\alpha+2}. \end{aligned}$$ is obtained. The last expression is finite by the embedding $\mathbb{D}^{4,p}\hookrightarrow \mathbb{D}^{4(\alpha+2)}$ and assumption \[ass:psi\_1\]. Similarly, it follows that $$\operatorname*{ess\,sup}_{s\in[0,T]}\|f''(Z_s)\|_{\mathbb{D}^{1,8}} \lesssim 1 + \operatorname*{ess\,sup}_{s\in [0,T]}\|Z_s\|^{\alpha+1}_{\mathbb{D}^{1,8(\alpha +1)}} \lesssim 1 + \left(\int_0^T\|\psi_r\|^\frac{1}{H}_{\mathbb{D}^{1,8(\alpha+1)}}\,\mathrm{d}{r}\right)^{H(\alpha+1)} <\infty$$ and also $\operatorname*{ess\,sup}_{s\in[0,T]} \|f'''(Z_s)\|_{L^8(\Omega)} <\infty$. *Claim 3:* The process $Z$ satisfies condition \[ass:gen\_5\] of . *Proof of Claim 3:* By , the inequality $$\|(\nabla^\frac{H}{2}Z_u)(v)-(\nabla^\frac{H}{2}Z_v)(v)\|_{\mathbb{D}^{1,8}} \lesssim \left\|\int_u^v (\nabla^\frac{H}{2}\psi_r)(v)\delta R_r^H\right\|_{\mathbb{D}^{1,8}} + \left\|\int_u^v\psi_r|v-r|^{H-1}\delta{B}_r^{\frac{H}{2}+\frac{1}{2}}\right\|_{\mathbb{D}^{1,8}}$$ is satisfied for almost every $u,v\in [0,T]$ such that $0<u<v<T$. By using and assumption \[ass:psi\_2\], it follows for the norm of the first integral that $$\begin{aligned} \lim_{\varepsilon\downarrow 0} \operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)}\left\|\int_u^v (\nabla^\frac{H}{2}\psi_r)(v)\delta R_r^H\right\|_{\mathbb{D}^{1,8}} & \lesssim\lim_{\varepsilon\downarrow 0} \operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)} \left(\int_u^v\|(\nabla^\frac{H}{2}\psi_r)(v)\|_{\mathbb{D}^{3,8}}^\frac{1}{H}\,\mathrm{d}{r}\right)^H \\ & \leq \lim_{\varepsilon\downarrow 0} \operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)} \left(\int_{u}^vp_1(r)^\frac{1}{H}\,\mathrm{d}{r}\right)^H \\ & = \lim_{\varepsilon\downarrow 0} \left(\int_{v-\varepsilon}^vp_1(r)^\frac{1}{H}\,\mathrm{d}{r}\right)^H \end{aligned}$$ for almost every $v\in [0,T]$. The last limit is zero since $p_1$ belongs to the space $L^\frac{6}{H+1}(0,T)$ by \[ass:psi\_2\]. Similarly for the norm of the second integral, it follows that $$\begin{aligned} \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)} \left\|\int_u^v\psi_r|v-r|^{H-1}\delta{B}_r^{\frac{H}{2}+\frac{1}{2}}\right\|_{\mathbb{D}^{1,8}} & \lesssim \lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)} \left(\int_u^v\|\psi_r\|_{\mathbb{D}^{2,8}}^\frac{2}{H+1}(v-r)^\frac{2(H-1)}{H+1}\,\mathrm{d}{r}\right)^\frac{H+1}{2}\\ & \lesssim \|\psi\|_{L^\infty(0,T;\mathbb{D}^{2,8})}\lim_{\varepsilon\downarrow 0}\left(\int_{v-\varepsilon}^v(v-r)^\frac{2(H-1)}{H+1}\,\mathrm{d}{r}\right)^\frac{H+1}{2}\end{aligned}$$ is satisfied for almost every $v\in [0,T]$ by using . The last expression is clearly zero since the norm is finite by the embedding $\mathbb{D}^{4,p}\hookrightarrow \mathbb{D}^{2,8}$ and assumption \[ass:psi\_1\]. Similarly, by using and and successively, the inequality $$\begin{aligned} \|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_u)(v,v)-(\nabla^{\frac{H}{2},\frac{H}{2}}Z_v)(v,v)\|_{L^8(\Omega)} & \lesssim \left(\int_u^v\|(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(v,v)\|_{\mathbb{D}^{2,8}}^\frac{1}{H}\,\mathrm{d}{r}\right)^H \\ & \hspace{1cm} + \left(\int_u^v\|(\nabla^\frac{H}{2}\psi_r)(v)\|_{\mathbb{D}^{1,8}}^\frac{2}{H+1}|v-r|^\frac{2(H-1)}{H+1}\,\mathrm{d}{r}\right)^\frac{H+1}{2} \\ & \hspace{1cm} + \int_u^v\|\psi_r\|_{L^8(\Omega)}|v-r|^{2H-2}\,\mathrm{d}{r}\\ & \leq \left(\int_u^vp_2(r)^\frac{1}{H}\,\mathrm{d}{r}\right)^H \\ & \hspace{1cm} + \left(\int_u^vp_1(r)^\frac{6}{H+1}\,\mathrm{d}{r}\right)^\frac{H+1}{6}\left(\int_u^v|v-r|^\frac{3(H-1)}{H+1}\,\mathrm{d}{r}\right)^\frac{H+1}{3(H-1)} \\ & \hspace{1cm} + \|\psi\|_{L^\infty(0,T;L^8(\Omega))}\int_u^v|v-r|^{2H-2}\,\mathrm{d}{r}\end{aligned}$$ is satisfied for almost every $u,v\in [0,T]$ such that $v>u$. The second inequality is obtained by using assumption \[ass:psi\_2\] and Hölder’s inequality. By the embedding $\mathbb{D}^{4,p}\hookrightarrow L^8(\Omega)$ and assumptions \[ass:psi\_1\] and \[ass:psi\_2\], it follows from the last inequality that $$\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v-\varepsilon,v)}\|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_u)(v,v)-(\nabla^{\frac{H}{2},\frac{H}{2}}Z_v)(v,v)\|_{L^8(\Omega)}=0$$ is satisfied for almost every $v\in [0,T]$. *Claim 4:* The process $Z$ satisfies condition \[ass:gen\_6\] of . *Proof of Claim 4:* By using , it follows that the estimate $$\begin{aligned} \|(\nabla^\frac{H}{2}Z_s)(u)-(\nabla^\frac{H}{2}Z_s)(v)\|_{\mathbb{D}^{1,8}} & \lesssim \left\|\int_0^s\left[(\nabla^\frac{H}{2}\psi_r)(u)-(\nabla^\frac{H}{2}\psi_r)(v)\right]\delta{R}_r^H\right\|_{\mathbb{D}^{1,8}} \\ & \hspace{1cm} + \left\|\int_0^s\psi_r\left[|u-r|^{H-1}-|v-r|^{H-1}\right]\delta B_r^{\frac{H}{2}+\frac{1}{2}}\right\|_{\mathbb{D}^{1,8}}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:claim_4_1} \end{aligned}$$ is satisfied for every $s\in [0,T]$ and almost every $u,v\in [0,T]$ such that $u>v$. By using , the following estimate for the norm of the first stochastic integral in is obtained: $$\left\|\int_0^s\left[(\nabla^\frac{H}{2}\psi_r)(u)-(\nabla^\frac{H}{2}\psi_r)(v)\right]\delta{R}_r^H\right\|_{\mathbb{D}^{1,8}} \lesssim \left(\int_0^s\|(\nabla^\frac{H}{2}\psi_r)(u)-(\nabla^\frac{H}{2}\psi_r)(v)\|_{\mathbb{D}^{3,8}}^\frac{1}{H}\,\mathrm{d}{r}\right)^H.$$ The integrand in the last expression tends to zero as $u\downarrow v$ by assumption \[ass:psi\_4\] and it is dominated by $p_1$ by assumption \[ass:psi\_2\]. Consequently, it follows that $$\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)} \left\|\int_0^s\left[(\nabla^\frac{H}{2}\psi_r)(u)-(\nabla^\frac{H}{2}\psi_r)(v)\right]\delta{R}_r^H\right\|_{\mathbb{D}^{1,8}} = 0$$ is satisfied for every $s\in [0,T]$ by the Lebesgue’s dominated convergence theorem. Similarly, by , the following estimate for the norm of the second integral in is obtained: $$\left\|\int_0^s\psi_r\left[|u-r|^{H-1}-|v-r|^{H-1}\right]\delta B_r^{\frac{H}{2}+\frac{1}{2}}\right\|_{\mathbb{D}^{1,8}} \lesssim \left(\int_0^s\|\psi_r\|_{\mathbb{D}^{2,8}}^\frac{2}{H+1}\left||u-r|^{H-1}-|v-r|^{H-1}\right|^{\frac{2}{H+1}}\,\mathrm{d}{r}\right)^\frac{H+1}{2}.$$ This estimate yields, by using the embedding $\mathbb{D}^{4,p}\hookrightarrow \mathbb{D}^{2,8}$ and assumption , that $$\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)} \left\|\int_0^s\psi_r\left[|u-r|^{H-1}-|v-r|^{H-1}\right]\delta B_r^{\frac{H}{2}+\frac{1}{2}}\right\|_{\mathbb{D}^{1,8}} =0$$ is satisfied for every $s\in [0,T]$. By similar arguments, it can also be shown that $$\lim_{\varepsilon\downarrow 0}\operatorname*{ess\,sup}_{u\in (v,v+\varepsilon)} \|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(u,u)-(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(v,v)\|_{L^8(\Omega)} = 0.$$ *Claim 5:* The process $Z$ satisfies condition \[ass:gen\_7\] of . *Proof of Claim 5:* As above, by using and , the estimate $$\begin{aligned} \operatorname*{ess\,sup}_{s,u\in [0,T]}\|(\nabla^\frac{H}{2}Z_s)(u)\|_{\mathbb{D}^{1,8}} & \lesssim \operatorname*{ess\,sup}_{s,t\in [0,T]}\left[ \left(\int_0^s\|(\nabla^{\frac{H}{2}}\psi_r)(u)\|_{\mathbb{D}^{3,8}}^\frac{1}{H}\,\mathrm{d}{r}\right)^H + \left(\int_0^s\|\psi_r\|_{\mathbb{D}^{2,8}}^\frac{2}{H+1}|u-r|^{\frac{2(H-1)}{H+1}}\,\mathrm{d}{r}\right)^\frac{H+1}{2}\right]\\ & \leq \|p_1\|_{L^\frac{1}{H}(0,T)} + \|\psi\|_{L^\infty(0,T;\mathbb{D}^{2,8})}\operatorname*{ess\,sup}_{u\in [0,T]}\left(\int_0^s|u-r|^\frac{2(H-1)}{H+1}\,\mathrm{d}{r}\right)^\frac{H+1}{2}. \end{aligned}$$ is obtained. The essential supremum in the expression above is clearly finite and the norms are finite by assumptions \[ass:psi\_1\] and \[ass:psi\_2\]. The finiteness of $$\operatorname*{ess\,sup}_{s,u\in [0,T]} \|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(u,u)\|_{L^8(\Omega)}$$ can be shown similarly. It is now shown that the Itô-type formula is satisfied for any function $f$ in $\mathscr{C}^3(\mathbb{R})$ with bounded third derivative and the process $(Z_W(t))_{t\in [0,T]}$ given by $$Z_W(t)\overset{\mathrm{Def.}}{=}\int_0^tW_s\delta R_s^H$$ where $W$ is the underlying Wiener process. It is necessary to show that the Wiener process $W$ satisfies the conditions on $\psi$ in with $\alpha=0$. For every $r\in [0,T]$, $W_s = I(\bm{1}_{[0,s]})$. Thus $$\|W_r\|_{\mathbb{D}^{4,p}} = \left(\mathbb{E}|W_s|^p+\|\bm{1}_{[0,s]}\|_{L^2(\mathbb{R})}^p\right)^\frac{1}{p} \lesssim \sqrt{s}$$ is satisfied for every $p\geq 2$ by the fact that the process $W$ has equivalent moments. This implies that \[ass:psi\_1\] is satisfied. From , it further follows that $$\nabla^{\frac{H}{2}}W_s(u) \eqsim \left[u_+^\frac{H}{2}-(u-s)_+^\frac{H}{2}\right]$$ is satisfied for every $u\in\mathbb{R}$ and $\nabla^{\frac{H}{2},\frac{H}{2}}W_s(u,v)=0$ for every $u,v\in\mathbb{R}$. Therefore, \[ass:psi\_4\] and \[ass:psi\_2\] are satisfied as well. In this example, it is shown that the Itô-type formula is satisfied for any function $f$ in $\mathscr{C}^3(\mathbb{R})$ with bounded third derivative and the process $(Z_R(t))_{t\in[0,T]}$ given by $$Z_R(t) \overset{\mathrm{Def.}}{=} \int_0^tR_s^H\delta R_s^H.$$ Recall from that for every $s\in[0,T]$, the Rosenblatt process is given by $R_s^H = C_H^RI_2(h_s^H)$ where $$h_s^H(x,y) = \int_0^s(u-x)_+^{\frac{H}{2}-1}(u-y)_+^{\frac{H}{2}-1}\,\mathrm{d}{u}.$$ Since the Rosenblatt process is defined as the second-order multiple Wiener-Itô integral, all of its moments can be estimated by its second moment, see [@NouPec12 Corollary 2.8.14]. By using this property and formula , it follows that $$\|R_s^H\|_{\mathbb{D}^{4,p}} \lesssim \|h^H_s\|_{L^2(\mathbb{R}^2)} \eqsim s^{H}$$ is satisfied for every $p\geq 2$. This implies that \[ass:psi\_1\] is satisfied with $\alpha=0$. Using and , it follows that $$\begin{aligned} & \nabla^{\frac{H}{2}}R_s^H(u) \eqsim \int_0^s|u-r|^{H-1}\delta B_{r}^{\frac{H}{2}+\frac{1}{2}}\\ & \nabla^{\frac{H}{2},\frac{H}{2}}R_s^H(u,v) \eqsim \int_0^s|u-r|^{H-1}|v-r|^{H-1}\,\mathrm{d}{r} \end{aligned}$$ from which, appealing to , it follows that both conditions \[ass:psi\_4\] and \[ass:psi\_2\] are satisfied as well. The following corollary of provides an Itô-type formula for functionals of Wiener integrals with respect to the Rosenblatt process, i.e. when the integrand is deterministic. \[cor:Ito\_for\_Wiener\_integral\] Let $f\in\mathscr{C}^3(\mathbb{R})$ be such that its third derivative has at most polynomial growth and let $(\psi_t)_{t\in [0,T]}$ be a bounded deterministic function. Let $(Z_t)_{t\in [0,T]}$ be the integral process defined by . Then the formula $$\begin{aligned} f(Z_t) & = f(0) + \int_0^tf'(Z_s)\psi_s\delta R_s^H \\ &\hspace{5mm} + H(2H-1)\int_0^tf''(Z_s)\psi_s\int_0^s\psi_r(s-r)^{2H-2}\,\mathrm{d}{r}\,\mathrm{d}{s}\\ &\hspace{1.5cm} + c_1(H)\int_0^tf''(Z_s)\psi_s\left(\int_0^s\psi_r(s-r)^{H-1}\delta B_r^{\frac{H}{2}+\frac{1}{2}}\right)\delta B_s^{\frac{H}{2}+\frac{1}{2}}\\ & \hspace{5mm} + (\sqrt{2H(2H-1)})^3\int_0^tf'''(Z_s)\psi_s\int_0^s\psi_u(s-u)^{H-1}\int_0^u\psi_v(s-v)^{H-1}(u-v)^{H-1}\,\mathrm{d}{v}\,\mathrm{d}{u}\,\mathrm{d}{s}\\ & \hspace{1.5cm} + c_2(H) \int_0^t f'''(Z_s)\psi_s\left(\int_0^s\psi_u(s-u)^{H-1}\left(\int_0^u\psi_v(s-v)^{H-1}\delta B_v^{\frac{H}{2}+\frac{1}{2}}\right)\delta B_u^{\frac{H}{2}+\frac{1}{2}}\right)\,\mathrm{d}{s} \end{aligned}$$ is satisfied almost surely for every $t\in [0,T]$ with the constants $c_1(H)$ and $c_2(H)$ given by $$\label{eq:constants_c_12(H)} c_1(H) \overset{\textnormal{Def.}}{=} \frac{4(2H-1)}{H+1}, \qquad c_2(H)\overset{\textnormal{Def.}}{=} \frac{8(2H-1)}{H+1}\sqrt{\frac{H(2H-1)}{2}}.$$ By , it follows that $$\begin{aligned} f(Z_t) & =f(0) + \int_0^t f'(Z_s)\psi_s\delta R_s^H \\ & \hspace{1cm} + 2c_H^{B,R}\int_0^tf''(Z_s)(\nabla^\frac{H}{2}Z_s)(s)\psi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} \\ & \hspace{2cm} + c_H^R\int_0^t \left(f''(Z_s)(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + f'''(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\psi_s\,\mathrm{d}{s}.{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:not_refined_formula} \end{aligned}$$ For the verification of this corollary, it is necessary to compute $\nabla^\frac{H}{2}Z_s(s)$, $[\nabla^\frac{H}{2}Z_s(s)]^2$, and $\nabla^{\frac{H}{2},\frac{H}{2}}Z_s(s,s)$. By , it follows that $$\nabla^{\frac{H}{2}}Z_s(s) = \frac{2c_H^{B,R}\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2}\int_0^s\psi_r(s-r)^{H-1}\delta B_r^{\frac{H}{2}+\frac{1}{2}}$$ is satisfied for almost every $s\in[0,T]$. Since $\nabla^\frac{H}{2}Z_s(s)$ is a Wiener integral with respect to $B^{\frac{H}{2}+\frac{1}{2}}$, its square can be computed by using the chain rule for functionals of Wiener integrals with respect to the fractional Brownian motion, see [@DunHuDun00 Corollary 4.4]. In particular, it follows that the equality $$\begin{aligned} [\nabla^{\frac{H}{2}}Z_s(s)]^2 & = \frac{8(c_H^{B,R})^2\mathrm{B}\left(\frac{H}{2},1-H\right)^2}{\Gamma\left(\frac{H}{2}\right)^4}\int_0^s\psi_u(s-u)^{H-1}\left(\int_0^u\psi_v(s-v)^{H-1}\delta B_v^{\frac{H}{2}+\frac{1}{2}}\right)\delta B_u^{\frac{H}{2}+\frac{1}{2}}\\ &\hspace{1mm} + \frac{4H(H+1)(c_H^{B,R})^2\mathrm{B}\left(\frac{H}{2},1-H\right)^2}{\Gamma\left(\frac{H}{2}\right)^4}\int_0^s\psi_u(s-u)^{H-1}\int_0^u\psi_v(s-v)^{H-1}(u-v)^{H-1}\,\mathrm{d}{v}\,\mathrm{d}{u} \end{aligned}$$ is satisfied for almost every $s\in [0,T]$. Note that, alternatively, can be used to compute the square. Finally, by , it follows that $$\nabla^{\frac{H}{2},\frac{H}{2}}Z_s(s,s) = \frac{2c_H^R\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2}\int_0^s\psi_r(s-r)^{2H-2}\,\mathrm{d}{r}$$ is satisfied for almost every $s\in [0,T]$. The claim of the corollary is thus verified by substituting the three expressions above in and simplifying the constants (see for their values). For the sake of completeness, it is noted that from , the Itô formula for functionals of the Rosenblatt process itself can be obtained. This formula is given in [@Arr16 Theorem 3] and proved in the white-noise setting for the case when $f$ is an infinitely differentiable function whose derivatives have at most polynomial growth. Here, it is only required that $f$ be $\mathscr{C}^3$. \[prop:Ito\_R\] Let $f\in\mathscr{C}^3(\mathbb{R})$ be such that its third derivative has at most polynomial growth. Then the equality $$\begin{aligned} f(R_t^H) & = f(0) + \int_0^tf'(R_s^H)\delta R_s^H \\ &\hspace{1cm} + H\int_0^tf''(R_s^H)s^{2H-1} \,\mathrm{d}{s} + \\ &\hspace{2cm} + c_1(H)\int_0^tf''(R_s^H)\left(\int_0^s(s-u)^{H-1}\delta B_u^{\frac{H}{2}+\frac{1}{2}}\right)\delta B_s^{\frac{H}{2}+\frac{1}{2}}\\ &\hspace{1cm} +\frac{H}{2}\kappa_3(R_1^H)\int_0^tf'''(R_s^H)s^{3H-1}\,\mathrm{d}{s} \\ &\hspace{2cm} +c_2(H)\int_0^tf'''(R_s^H)\left(\int_0^s(s-u)^{H-1}\left(\int_0^u(s-v)^{H-1}\delta B_v^{\frac{H}{2}+\frac{1}{2}}\right)\delta B_u^{\frac{H}{2}+\frac{1}{2}}\right)\,\mathrm{d}{s} \end{aligned}$$ is satisfied for $t\in [0,T]$ almost surely with the constants $c_1(H)$ and $c_2(H)$ given by and with $\kappa_3(R_1^H)$ being the third cumulant of $R_1^H$ given by $$\kappa_3(R_1^H) = \frac{4\sqrt{2H(2H-1)^3}}{3H-1}\mathrm{B}(H,H).$$ where $\mathrm{B}$ is the Beta function. Set $\psi\equiv 1$ in . The cumulant $\kappa_3(R_1^H)$ is computed in [@TaqVei13 formula (12)]. As an example, the square of $R^H$ is computed. The following equality is obtained: $$(R_t^{H})^2 = 2\int_0^tR_s^H\delta R_s^H + \frac{8(2H-1)}{H+1} \int_0^t\left(\int_0^s(s-u)^{H-1}\delta B_{u}^{\frac{H}{2}+\frac{1}{2}}\right)\delta B_{s}^{\frac{H}{2}+\frac{1}{2}} + t^{2H}.$$ The formula has the same structure as given in [@Tud08 Theorem 4] for the case when $R^H$ is defined by its finite time interval representation from [@Tud08 Proposition 1]. See also [@Arr15 Theorem 3.12] where the square is computed in the white-noise setting. It is interesting to note that the above formula can be written as $$\begin{aligned} (R_t^H)^2 & = 2\int_0^t\left(\int_0^s\delta R_u^H\right)\delta R_s^H \\ & \hspace{1cm} + \frac{8(2H-1)}{H+1}\int_0^t\left(\int_0^s(s-u)^{H-1}\delta B_u^{\frac{H}{2}+\frac{H}{2}}\right)\delta B_s^{\frac{H}{2}+\frac{1}{2}}\\ & \hspace{2cm} + 2H(2H-1)\int_0^t\left(\int_0^s(s-u)^{2H-2}\,\mathrm{d}{u}\right)\,\mathrm{d}{s}. \end{aligned}$$ An application {#sec:corollaries} ============== The Itô-type formula from can be used to compute the second moment of the stochastic integral with respect to a Rosenblatt process. \[prop:L2norm\_of\_int\] Let $\psi$ be a stochastic process that satisfies \[ass:psi\_1\] - \[ass:psi\_2\] and $(Z_t)_{t\in[0,T]}$ be defined by . Then we have that $$\begin{aligned} \mathbb{E} \left(Z_t\right)^2 & = H(2H-1)\int_0^t\int_0^t \mathbb{E}[\psi_r\psi_s]|s-r|^{2H-2}\,\mathrm{d}{r}\,\mathrm{d}{s} \\ & \hspace{1cm} + 2H(2H-1)c_3(H) \int_0^t\int_0^t\mathbb{E}\left[\nabla^{\frac{H}{2}}\psi_r(s) \nabla^\frac{H}{2}\psi_s(r)\right]|s-r|^{H-1}\,\mathrm{d}{r}\,\mathrm{d}{s} \\& \hspace{1cm} + \frac{1}{2}H(2H-1)c_3(H)^2\int_0^t\int_0^t\mathbb{E}\left[\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r(s,s) \nabla^{\frac{H}{2},\frac{H}{2}}\psi_s(r,r)\right]\,\mathrm{d}{r}\,\mathrm{d}{s}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:second_moment} \end{aligned}$$ is satisfied with the constant $c_3(H)$ given by $$c_3(H)\overset{\textnormal{Def.}}{=}\frac{\Gamma\left(\frac{H}{2}\right)\Gamma\left(1-\frac{H}{2}\right)}{\Gamma\left(1-H\right)}.$$ Using with $f(x) = x^2$, it follows that $$\mathbb{E}Z_t^2 = 2c_H^R\mathbb{E}\int_0^t\left[(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s)\psi_s\right]\,\mathrm{d}{s}$$ because the stochastic integrals have zero expectation. Using , it follows that $$\begin{aligned} \mathbb{E}\left[ (\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s)\psi_s\right] & = \mathbb{E}\left[\psi_s\left(\int_0^s(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(s)\delta R_r^H\right)\right] \\ & \hspace{1cm} + \frac{4c_H^{B,R}\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2} \mathbb{E}\left[\psi_s\left(\int_0^s(\nabla^\frac{H}{2}\psi_r)(s)(s-r)^{H-1}\delta B_r^{\frac{H}{2}+\frac{1}{2}}\right)\right] \\ & \hspace{1cm} + \frac{2c_H^R\mathrm{B}\left(\frac{H}{2},1-H\right)^2}{\Gamma\left(\frac{H}{2}\right)^4} \int_0^s \mathbb{E}[ \psi_r\psi_s](s-r)^{2H-2}\,\mathrm{d}{r}. \end{aligned}$$ Using \[lem:adjoint\_property\_R\] of yields $$\mathbb{E}\left[\psi_s\left(\int_0^s(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(s,s)\delta R_r^H\right)\right] = c_H^R\int_0^s \mathbb{E}\left[(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_s)(r,r)(\nabla^{\frac{H}{2},\frac{H}{2}}\psi_r)(s,s)\right]\,\mathrm{d}{r}$$ and by \[lem:adjoint\_property\_B\] of the same lemma, it follows that $$\mathbb{E}\left[\psi_s\left(\int_0^s(\nabla^\frac{H}{2}\psi_r)(s)(s-r)^{H-1}\delta B_r^{\frac{H}{2}+\frac{1}{2}}\right)\right] = c_{\frac{H}{2}+\frac{1}{2}}^B\int_0^s \mathbb{E}\left[(\nabla^\frac{H}{2}\psi_s)(r) (\nabla^\frac{H}{2}\psi_r)(s)\right](s-r)^{H-1}\,\mathrm{d}{r}.$$ is satisfied for almost every $s\in [0,t]$. A few remarks are made now. 1. Formula holds under weaker assumptions: it is sufficient if the integrand $\psi$ belongs to the space $L^\frac{1}{H}(0,T;\mathbb{D}^{4,2})$. This follows because the duality formula from can be used instead of the Itô formula in which case, the assumptions \[ass:psi\_1\] - \[ass:psi\_2\] are not needed. 2. Formula has the same structure as the one given in [@Arr15 Theorem 5.8] for the white-noise-type integral with respect to Rosenblatt processes. However, in contrast to the proof of [@Arr15 Theorem 5.8], the proof of is relatively straightforward. 3. If $\psi$ is deterministic, then the well-known expression for the second moment of the Wiener integral with respect to the Rosenblatt process, e.g. [@Tud08 p. 236], is obtained: $$\mathbb{E} Z_t^2 = H(2H-1)\int_0^t\int_0^t\psi_r\psi_s|s-r|^{2H-2}\,\mathrm{d}{r}\,\mathrm{d}{s}.$$ An estimate for higher absolute moments of the stochastic integral with respect to the Rosenblatt process is now given. Let $q\geq 3$ and let $\psi$ be a stochastic process that satisfies \[ass:psi\_1\] - \[ass:psi\_2\] with $\alpha=q-2$. Let $(Z_t)_{t\in[0,T]}$ be the stochastic process defined by . Then the estimate $$\|Z_t\|_{L^q(\Omega)}^3 \leq 3(q-1)c_H^R \int_0^t \left\|\psi_s\left(|Z_s|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + \mathrm{sgn}(Z_s)(q-2)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\right\|_{L^\frac{q}{3}(\Omega)}\,\mathrm{d}{s}.$$ is satisfied for every $t\in [0,T]$. Initially, assume that $q>3$. Using with $f(x)=|x|^q$ ($f$ is $\mathscr{C}^3$ since $q>3$), it follows that $$\begin{aligned} |Z_t|^q & = \int_0^tq|Z_s|^{q-1}\mathrm{sgn}(Z_s)\psi_s\delta R_s^H \\ & \hspace{10mm} + 2c_H^{B,R}\int_0^tq(q-1)|Z_s|^{q-2}\nabla^\frac{H}{2}Z_s(s)\psi_s\delta B_s^{\frac{H}{2}+\frac{1}{2}} \\ & \hspace{10mm} + c_H^R\int_0^t\psi_s\left(q(q-1)|Z_s|^{q-2}(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + \right. \\ & \hspace{4cm} \left.\phantom{\nabla^\frac{H}{2}} + q(q-1)(q-2)|Z_s|^{q-3}\mathrm{sgn}(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\,\mathrm{d}{s}{\addtocounter{equation}{1}\tag{\theequation}}\label{eq:power} \end{aligned}$$ where $\mathrm{sgn}$ denotes the sign function. Taking the expectation of both sides of , it follows that $$\mathbb{E}|Z_t|^q = q(q-1)c_H^R\int_0^t\mathbb{E} \left[|Z_s|^{q-3}\psi_s\left(|Z_s|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + (q-2)\mathrm{sgn}(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\right]\,\mathrm{d}{s}$$ because the stochastic integrals have zero expectation. Thus, Hölder’s inequality yields $$\begin{aligned} \mathbb{E} |Z_t|^q & \leq q(q-1)c_H^R\int_0^t (\mathbb{E} |Z_s|^q)^\frac{q-3}{q} \cdot \\ &\hspace{2cm} \cdot \left\|\psi_s\left(|Z_s|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s)+(q-2)\mathrm{sgn}(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\right\|_{L^\frac{q}{3}(\Omega)}\,\mathrm{d}{s}. \end{aligned}$$ The desired inequality is proved by using Bihari’s inequality, see [@BecBel61 Theorem 3, p. 135], which gives $$\mathbb{E} |Z_t|^q \leq \left(\frac{3}{q}q(q-1)c_H^R\int_0^t\left\|\psi_s\left(|Z_s|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s)+(q-2)\mathrm{sgn}(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\right\|_{L^\frac{q}{3}(\Omega)}\,\mathrm{d}{s}\right)^\frac{q}{3}.$$ For the case $q=3$, cannot be used directly, since the function $f(x)=|x|^3$ does not belong to $\mathscr{C}^3(\mathbb{R})$. Instead, for $\varepsilon>0$, consider the function $$f_\varepsilon(x) \overset{\textnormal{Def.}}{=} (x^2+\varepsilon^2)^\frac{3}{2}, \quad x\in\mathbb{R}.$$ The function $f_\varepsilon$ is a smooth approximation of $f(x)=|x|^3$ with bounded third derivative. Hence, by it follows that $\mathbb{E} f_\varepsilon(Z_t)$ satisfies the formula $$\label{eq:approximate_eq} \mathbb{E} f_\varepsilon(Z_t) = \varepsilon^3 + c_H^R\int_0^t \mathbb{E}\left[\psi_s\left(f''_\varepsilon(Z_s)(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + f'''_\varepsilon(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\right]\,\mathrm{d}{s}$$ similarly as in the case $q>3$. Since $$\lim_{\varepsilon\downarrow 0}f''_\varepsilon(x)=6|x|\quad\mbox{and}\quad\lim_{\varepsilon\downarrow 0}f'''_\varepsilon(x) = 6\mathrm{sgn}(x),$$ taking the limit $\varepsilon\downarrow 0$ in equality and using Lebesgue’s dominated convergence theorem to interchange the limit and the integrals yields $$\mathbb{E} |Z_t|^3 = 6c_H^R \int_0^t \mathbb{E}\left[\psi_s\left(|Z_s|(\nabla^{\frac{H}{2},\frac{H}{2}}Z_s)(s,s) + \mathrm{sgn}(Z_s)[(\nabla^\frac{H}{2}Z_s)(s)]^2\right)\right]\,\mathrm{d}{s}$$ which concludes the proof. A concluding remark =================== Stochastic processes with second-order fractional stochastic differential arise naturally in the Itô-type formula for functionals of (stochastic integral with respect to) Rosenblatt processes. Such formulas will in general contain stochastic integrals with respect to a fractional Brownian motion as well as derivatives up to the third order. It seems that the method given here can be also used to obtain Itô-type formulas for higher-order processes, e.g. Hermite processes, and it appears that such formulas will contain stochastic integrals with respect to related fractional processes of lower Hermite order as well as derivatives up to the Hermite order of the considered process increased by one. Another topic which should be further studied is non-linear stochastic differential equations with Rosenblatt noise. The general Itô-type formula given here should provide a convenient tool for their analysis. However, important questions of existence of the solutions need to be answered and further properties of the solutions and of fractional stochastic derivatives of the solutions need to be investigated before the formula can be applied. 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--- abstract: 'Several models of dark matter suggest the existence of hidden sectors consisting of $SU(3)_C \times SU(2)_L \times U(1)_Y$ singlet fields. The interaction between the ordinary and hidden sectors could be transmitted by new Abelian $U''(1)$ gauge bosons $A''$ (dark or hidden photons) mixing with ordinary photons. If such $A''$’s have masses below the $\pi^0$ meson mass, they would be produced through $\gamma - A''$ mixing in the $\pi^0\to \gamma \gamma$ decays and be observed via decays $A'' \to {e^+e^-}$. Using bounds from the SINDRUM experiment at the Paul Scherrer Institute that searched for an excess of ${e^+e^-}$ pairs in $\pi^- p$ interactions at rest, the area excluding the $\gamma - A''$ mixing $\epsilon \gtrsim 10^{-3}$ for the $A''$ mass region $ 25 \lesssim M_{A''} \lesssim 120$ MeV is derived.' author: - 'S.N. Gninenko' title: 'Constraints on dark photons from $\pi^0$ decays' --- The origin of dark matter is still a great puzzle in particle physics and cosmology. Several models dealing with this problem suggest the existence of ‘hidden’ sectors consisting of $SU(3)_C \times SU(2)_L \times U(1)_Y$ singlet fields. These sectors do not interact with our world directly and couple to it by gravity. It is also possible that there exist new very-weak forces between the ordinary and dark worlds transmitted by new Abelian $U'(1)$ gauge bosons $A'$ (dark or hidden photons for short) mixing with our photons [@hop], as discussed first by Okun in his model of paraphotons [@okun]. In a class of recent interesting models the $\gamma-A'$ mixing strength may be large enough to be experimentally tested. This makes searches for $A'$’s very attractive; for a recent review see [@jr] and references therein. It should be noted, that many models of physics beyond the Standard Model (SM) such as GUTs [@1], superstring models [@2] (see also Ref.[@khlop]), supersymmetric [@3], and models including the fifth force [@carl] also predict an extra U$^{'}$(1) factor and the corresponding new gauge $X$ boson. The $X$’s could interact directly with quarks and/or leptons. If the $X$ mass is below the pion mass, the $X$ could be effectively searched for in the decays $P\to \gamma X$, where $P = \pi^{0},\eta$, or $\eta^{\prime}$. This is due to the fact, that the decay rate of $P\to \gamma~+~$ $\it any~new~particles~with~spin~0~or~\frac{1}{2}$ is proved to be negligibly small [@di]. Hence, an observation of these decay modes could unambiguously signal the discovery of a new spin-1 boson, in contrast with other searches for new light particles in rare K, $\pi$ or $\mu$ decays [@di; @md; @gkx2]. The allowed $\gamma - A'$ interaction is given by the kinetic mixing [@okun; @jr; @holdom; @foot1] $$L_{int}= -\frac{1}{2}\epsilon F_{\mu\nu}A'^{\mu\nu} \label{mixing}$$ where $F^{\mu\nu}$, $A'^{\mu\nu}$ are the ordinary and the dark photon fields, respectively, and $\epsilon$ is their mixing strength. In some recent dark matter models the dark photon could be massless; see, e.g. Refs.[@Cline:2012is; @Cline:2012ei]. If the $A'$ has a mass, the kinetic mixing of Eq.(\[mixing\]) can be diagonalized resulting in a nondiagonal mass term and $\gamma - A'$ mixing. Hence, any $\gamma$-source could produce a kinematically allowed massive $A'$ boson according to the appropriate mixings. Then, if the mass difference is small, ordinary photons may oscillate into dark photons-similarly to neutrino oscillations- or, if the mass difference is large, dark photons could decay, e.g. into ${e^+e^-}$ pairs. Experimental constaints on dark photons in the meV-keV mass range can be derived from searches for the fifth force [@okun; @c1; @c2], from experiments based on the photon regeneration technique [@phreg; @bober; @sik; @rs; @vanb], and from astrophysical considerations [@seva1; @seva2]. For example, the results of experiments searching for solar axions [@cast1; @cast2] can be used to set limits on the ${{\gamma }}- {{A'}}$ mixing in the keV part of the solar spectrum of dark photons [@jr1; @jr2; @gr; @st]. Stringent bounds on the low mass $A'$s could be obtained from astrophysical considerations [@blin]-[@david]. There are plans to test the existence of sub-eV dark photons at new facilities, such as, for example, SHIPS [@ships] and IAXO [@igor]. The $A'$’s with the masses in the sub-GeV range, see e.g. [@bpr; @rw; @will], can be searched for through their $A'\to {e^+e^-}$ decays in beam-dump experiments [@jdb; @e137; @brun; @e141; @e774; @apex], or in particle decays [@bes; @kloe; @babar; @mami]. Recently, stringent bounds on the mixing $\epsilon$ have been obtained from searches for decay modes $\pi^0,\eta,\eta' \to \gamma A'(X)$, $A'(X)\to {e^+e^-}$ with existing data of neutrino experiments [@sngpi0; @sngeta]. These limits are valid for the relatively long-lived $A'$s with a mixing strength in the range $10^{-4}\lesssim \epsilon \lesssim 10^{-7}$. The goal of this note is to show that new bounds on the decay $\pi^0 \to \gamma A'$ of neutral pions into a photon and a short-lived $A'$ followed by the rapid decay $A'\to {e^+e^-}$ due to the relatively large $\gamma-A'$ mixing can be obtained from the results of sensitive searches for an excess of single isolated ${e^+e^-}$ pairs from decays of the weakly interacting neutral boson $X$ by the SINDRUM Collaboration at the Paul Scherrer Institute (PSI, Switzerland) [@sindrum]. The SINDRUM experiment- specifically designed to search for rare particle decays in the SINDRUM magnetic spectrometer- was performed by using the $\pi^- p $ interactions at rest as the source of $\pi^0$’s. The $\pi^0$’s were produced in the charge exchange reaction $\pi^- p \to \pi^0 n $ of 95 MeV/c $\pi^-$’s stopped in a small liquid hydrogen target in the center of the SINDRUM magnetic spectrometer. The magnetic field was 0.33 T, resulting in a transverse-momentum threshold of roughly 17 MeV/c for particles reaching the scintillator hodoscope surrounding the target. The trigger required an ${e^+e^-}$ pair with an opening angle in the plane perpendicular to the beam axis of at least 35$^o$; this corresponds to a lower threshold in the invariant mass of 25 MeV/c [@sindrum]. A total of 98 400 ${\pi^0 \to \gamma {e^+e^-}}$ decays were observed. The signature of the $X\to {e^+e^-}$ decay would be seen as a peak in the continuous ${e^+e^-}$ invariant mass distribution. ![ The 90 % C.L. area (shaded) in the $\bigl(M_{X}; Br(\pi^0\to \gamma X, X\to {e^+e^-})\bigr)$ plane excluded by the SINDRUM experiment (from Ref.[@sindrum]).[]{data-label="limit"}](sindrum.eps){width="50.00000%"} No such peak events were found and upper limits on the branching ratio $Br(\pi^0\to \gamma X, X\to {e^+e^-})=\frac{\Gamma(\pi^0\to \gamma X, X\to {e^+e^-})}{\Gamma(\pi^0\to \gamma \gamma)}$ in the range $\simeq 10^{-6}-10^{-5}$ have been placed for the $X$-mass region $25 \lesssim M_X \lesssim 120$ MeV. The corresponding 90% C.L. exclusion area in the $\bigl(M_{X}; Br(\pi^0\to \gamma X, X\to {e^+e^-})\bigr)$ plane is shown in Fig.\[limit\]. The limits were obtained assuming the $X$ lifetimes to be in the range $$10^{-23} \lesssim \tau_{X} \lesssim 10^{-11} ~{\rm s}. \label{lifetime}$$ For lower values of $\tau_X$ in Eq.(\[lifetime\]) the ${e^+e^-}$ mass peak would be smeared out beyond recognition; for larger values most $X$’s would decay outside the target region and thus the detector would not be triggered [@sindrum]. If the $A'$ exists and is a short-lived particle, it would decay in the SINDRUM target and be observed in the detector via the ${A' \rightarrow e^+ e^-}$ decay similar to the decays of $X$’s. The occurrence of ${A' \rightarrow e^+ e^-}$ decays would appear as an excess of ${e^+ e^-}$ pairs in the SINDRUM spectrometer above those expected from standard decays of $\pi^0$ produced in $\pi^- p$ interactions. As the final states of the decays $\pi^0\to \gamma X, X\to {e^+e^-}$ and ${\pi^0 \to \gamma A'}, {A' \rightarrow e^+ e^-}$ are identical, the results of the searches for the former can be used to constrain the latter for the same ${e^+e^-}$ invariant mass regions. ![ Exclusion region in the ($M_{A'}; \epsilon$) plane obtained in the present work from the results of the SINDRUM experiment [@sindrum]. Shown are areas excluded from the muon (g-2) considerations, by the results of the electron beam-dump experiments E137 [@jdb; @e137], E141 [@e141], E774 [@e774], the searches in APEX [@apex], KLOE[@kloe], BaBar[@babar], and MAMI [@mami], and from the data of the neutrino experiments NOMAD [@sngpi0] and CHARM [@sngeta]. Expected sensitivities of the planned APEX (full run) and DarkLight experiments are also shown for comparison. For a review of all experiments, which intend to probe a similar parameter space, see Ref.[@hif] and references therein.[]{data-label="plot"}](plot.eps){width="50.00000%"} For a given number $N_{\pi^0}$ of $\pi^0$’s produced in the target the expected number of ${A' \rightarrow e^+ e^-}$ (or $X\to {e^+e^-}$) decays occuring within the fiducial volume of the SINDRUM detector is given by $$\begin{aligned} N_{{A' \rightarrow e^+ e^-}}(M_{A'}) = \int f\Bigl[1-{\rm exp}\Bigl(-\frac{r M_{A'}}{P\tau_{A'}}\Bigr)\Bigr]\zeta A dr d\Omega \nonumber \\ =N_{\pi^0}Br({\pi^0 \to \gamma A'}) Br(A' \to{e^+e^-})\zeta A~~ \label{nev}\end{aligned}$$ where $M_{A'},~ P, ~f ,~r, ~\tau_{A'}$ are the $A'$ mass, momentum, flux, the distance between the $A'$ decay vertex and the target, and the lifetime at rest, respectively and $\zeta$ and $A$ are the ${e^+ e^-}$ pair reconstruction efficiency and the acceptance of the SINDRUM spectrometer, respectively [@sindrum]. Here it is assumed that the $A'$ is a short-lived particle with $\frac{r M_{A'}}{P\tau_{A'}} \gg 1 $ for $r$ values larger than the effective size of the target, in accordance with Eq.(\[lifetime\]). Taking Eq.(\[nev\]) into account and using the relation $ N_{{A' \rightarrow e^+ e^-}}(M_{A'}) < N_{{e^+e^-}}^{90\%}(M_{A'}) $, where $N_{{e^+e^-}}^{90\%}(M_{A'})$ is the 90% C.L. upper limit for the number of signal events from the decays of the $A'$ with a given mass $M_{A'}$, results in the $90\%$ C.L. exclusion area in the ($M_{A'};Br({\pi^0 \to \gamma A'}, A' \to{e^+e^-}) $) plane obtained by the SINDRUM experiment and shown in Fig.\[limit\]. The upper limit $N_{{e^+e^-}}^{90\%}$ as a function of $M_{A'}$ was obtained from the fit of the measured ${e^+e^-}$ mass distribution in the vicinity of each selected value of $M_{A'}$, to a sum of the signal peak from the $A'\to {e^+e^-}$ decays and a flat background distribution. The obtained results can be used to impose bounds on the $\gamma-A'$ mixing strength as a function of the dark photon mass. For $A'$ masses smaller than the mass $M_{\pi^0}$ of the $\pi^0$ meson, the branching fraction of the decay $\pi^0 \to \gamma A'$ is given by [@bpr]: $$Br(\pi^0 \to \gamma A') = 2\epsilon^2 Br(\pi^0 \to \gamma \gamma) \Bigl( 1- \frac{M_{A'}^2}{M_{\pi^0}^2}\Bigr)^3. \label{br}$$ Assuming that the dominant $A'$-decay is into a ${e^+e^-}$ pair, the corresponding decay rate is given by: $$\Gamma ({A' \rightarrow e^+ e^-}) = \frac{\alpha}{3} \epsilon^2 M_{A'} \sqrt{1-\frac{4m_e^2}{M_{A'}^2}} \Bigl( 1+ \frac{2m_e^2}{M_{A'}^2}\Bigr) \label{rate}$$ Taking into account Eq.(\[br\]), one can determine the $90\%$ C.L. exclusion area in the ($M_{A'}; \epsilon $) plane from the results of the SINDRUM experiment. This area is shown in Fig. \[plot\], together with regions excluded by the results of the electron beam-dump experiments E137, E141, E774 [@jdb; @e137; @e141; @e774], by recent measurements from APEX [@apex], KLOE [@kloe], BaBar [@babar], and MAMI [@mami], and from the data of the neutrino experiments NOMAD [@sngpi0] and CHARM [@sngeta]. For a recent, more detailed review of existing and planned limits, see Refs. [@hif; @sarah1; @sarah2]. The shape of the exclusion contour from the SINDRUM experiment corresponding to the $A'$ masses $M_{A'} \gtrsim$ 100 MeV is defined mainly by the phase-space factor in Eq.(\[br\]). The $A'$ lifetime values calculated by using Eq.(\[rate\]) for the mass range $25 \lesssim M_X \lesssim 120$ MeV are found to be within the allowed range of Eq.(\[lifetime\]). 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--- abstract: 'A new method is presented for the construction of a natural continuous wavelet transform on the sphere. It incorporates the analysis and synthesis with the same wavelet and the definition of translations and dilations on the sphere through the spherical harmonic coefficients. We construct a couple of wavelets as an extension of the flat [*Mexican Hat Wavelet*]{} to the sphere and we apply them to the detection of sources on the sphere. We remark that no projections are used with this methodology.' address: | $^1$ Instituto de Fí[sica]{} de Cantabria (CSIC-UC), 39005, Santander, Spain\ email: [email protected]\ $^2$ Departamento de Fí[sica]{} Moderna, Universidad de Cantabria, 39005, Santander, Spain\ $^3$ Departamento de Matemáticas, Universidad de Oviedo, 33007, Oviedo, Spain title: | Wavelets on the sphere.\ Application to the detection problem --- Introduction ============ Multiscaling analysis techniques dealing with the analysis/synthesis of nD-images defined on intervals of $R^n$ have been applied in many fields of physics in the last 15 years. For instance, in the case $n = 1$ one has electronics and audio signals, in the case $n = 2$ one has optical or infrared images whereas for $n = 3$ one deals with fluid dynamics or the large-scale structure of the universe as 3D-images. However, there are data given on other manifolds like the circle $S_1$ (e. g. scanning along circles the microwave sky) and the sphere $S_2$ (e. g. geophysics). In this paper, we are interested in data distributed on the sphere. Trivially, for the study of local properties (e. g. detection of objects) one can project on the tangent plane at any point on the sphere to make this type of analysis but when global properties are taken into account the curvature of the sphere can not be neglected. A first approach to deal with these global properties is to make some global projection of all the points of the sphere. The stereographic projection has been recently used dealing with the continuous wavelet transform. In this case, to get the wavelet coefficient at any point on the sphere, one projects from the opposite point to the local tangent plane. \[1\] have made a connection to group theory. The translations and dilations in the wavelet have their definition on the plane. Clearly, such a projection does not take into account the topological structure of the sphere. Some applications to cosmology, in particular the study of anisotropies of the cosmic microwave background radiation have been done by some authors (\[3\],\[7\],\[10\]) using the projection of the [*Mexican hat wavelet*]{}. A drawback of such projection is the obvious deformation of the pixels and wavelets near the projection pole. We remark that the synthesis can be done in terms of another biorthogonal wavelet \[11\]. Another approach uses some analyzing wavelet functions that are defined in terms of spherical harmonics \[5\] with a definition of the dilation operator and conditions on the wavelets in such a way to get a synthesis formula. The drawback of such methodology is: the dilations do not satisfy the appropriate flat limit in general. Also some examples of wavelet functions are poorly localized (e. g. Abel-Poison wavelets). A different approach assumes from the beginning discrete wavelets incorporating tensor product approaches in polar coordinates, then the two poles are singular points regarding approximation/stability properties (\[4\],\[6\]). Another approach is adapted to arbitrary point systems or triangulations on the spheres, then there is no efficient tool as fast wavelet algorithms. In the approach by \[8\] basis are defined on a quasi-uniform icosahedral triangulation on the sphere allowing for a fast algorithm. However, biorthogonal wavelets are needed and a lifting scheme for the multiresolution is applied avoiding the concepts of translations and dilations and also it is not clear whether the construction leads to a stable $L_2 (S_2)$ basis. Haar-type wavelets have been developed using different pixel combinations (\[2\],\[7\],\[9\]). The first case uses the lifting scheme weighting for the area of the pixels whereas in the other two cases an equal area pixelization is used but the Haar-type transform is only applied on regions of the sphere covering only $\frac{1}{12}$ of the total area. Clearly, with any pixelization the symmetry on the sphere is lost. In this paper we will consider a continuous approach, we will introduce a methodology that incorporates the analysis and synthesis of any function defined on the sphere $S_2$ using the same circularly-symmetric wavelet and also we will introduce the generalization of the translations/dilations. In this sense we follow Freeden’s approach working with spherical harmonics. Examples will be given that have the appropriate flat limit. Finally, the application to the detection of a spot is given, studying the concentration of the wavelet coefficients. Properties of the wavelet ========================= We will consider a circularly-symmetric filter defined on the sphere $ S_2$ $$\label{eq:cc} \Psi(\vec{n}\cdot \vec{\gamma}; R),$$ where $\vec{n}$ is a fixed direction. $\vec{\gamma}$ is another fixed, but arbitrary direction, therefore $\vec{n}\cdot \vec{\gamma}$ will represent a rotation on the sphere with respect to the direction $\vec{n}$ defined by the angle $\theta$ ($\cos(\theta)\equiv \vec{n}\cdot \vec{\gamma})$. $R>0$ will represent a dilation, which will be defined later on through the spherical harmonics. We assume the following properties of the filter: \(i) the analysis of any function $f(\vec{n})$ will be done with the wavelets $\Psi(\vec{n}\cdot \vec{\gamma}; R)$, \(ii) the synthesis of any function $f(\vec{n})$ will be done with the wavelets coefficients and the wavelets $\Psi(\vec{n}\cdot \vec{\gamma}; R)$, \(iii) it will incorporate the definition of translation and dilation on the sphere. We remark that no assumption about compensation of the filter (i. e. $\int d\Omega (\vec{n})\,\Psi(\vec{n}\cdot \vec{\gamma}; R) = 0$) and projection from $R^2$ to $S_2$ is imposed. Analysis with the filter $\Psi$ =============================== We define the wavelet coefficients associated to the translation $\vec{\gamma}$ and dilation $R$ for the function $f(\vec{n})$ defined on $S_2$ $$\label{eq:cd} w(R, \vec{\gamma} ) = \int d\Omega (\vec{n})\,f(\vec{n}) \Psi(\vec{n}\cdot \vec{\gamma}; R).$$ Let us assume the standard decomposition of $f(\vec{n})$ in spherical harmonics $Y_{lm}(\vec{n})$ $$\label{eq:ce} f(\vec{n}) = \sum_{lm} f_{lm}Y_{lm}(\vec{n}), \ \ \ f_{lm} = \int d\Omega (\vec{n})\,f(\vec{n})Y_{lm}^*(\vec{n}).$$ By introducing Eq.(\[eq:ce\]) into Eq.(\[eq:cd\]) and taking into account that $Y_{lm}(\vec{n})$ is an orthonormal base of $S_2$, we obtain $$\label{eq:cf} w(R, \vec{\gamma} ) = \sum_{lm}(\frac{4\pi}{2l+1})f_{lm}\Psi_l(R)Y_{lm}(\vec{\gamma}),$$ where the Legendre coefficients associated to the circularly-symmetric filter $\Psi$ are given by $$\begin{aligned} \Psi(\vec{n}\cdot \vec{\gamma}; R) & = & \sum_l \Psi_l(R)P_l(\vec{n}\cdot \vec{\gamma}),\nonumber \\ \label{eq:cg} \Psi_l(R) & = & (l+\frac{1}{2})\int_{-1}^1dy\,P_l(y)\Psi(y; R).\end{aligned}$$ Synthesis with the filter $\Psi$ ================================ Now, let us show that in order to have a reconstruction equation, i. e. $f(\vec{n})$ as a functional integral of the wavelet coefficients and the wavelet base $\Psi$ one can impose the condition $$\label{eq:ch} \Psi_l(R) \equiv (\frac{2l+1}{4\pi})\psi (lR),$$ i. e. $\Psi_l(R)$ depends on the product $lR$ and $\psi(l)$ satisfies the admissibility condition $$\label{eq:cl} C_{\psi} \equiv \int_0^{\infty} \frac{dl}{l}\psi^2(l) < \infty,$$ where $l$ runs in the interval $[0, \infty)$. We remark that the analogous condition to have a reconstruction on the plane by substituting $l \rightarrow q$, $q$ being the wave number in Fourier space. Therefore, the filter $\Psi$ -given by Eq. (\[eq:cg\])- can be rewritten as $$\label{eq:ci} \Psi(\vec{n}\cdot \vec{\gamma}; R) = \sum_l \Psi_l(R)P_l(\vec{n}\cdot \vec{\gamma}) = \sum_{lm}\psi (lR)Y_{lm}^*(\vec{n})Y_{lm}(\vec{\gamma}).$$ Firstly, we remark that the previous equation defines a dilation on the sphere in terms of dilation of the number $l$ and a translation on the sphere in terms of a rotation through the spherical harmonics $Y_{lm}(\vec{\gamma})$. We think that such a definition is the most natural on the sphere and generalizes the one associated to dilations and translations in the plane $R^2$ via Fourier space. Secondly, we can write the following equation $$\begin{aligned} \int \frac{dR}{R}\int d\Omega (\vec{\gamma})\,w(R, \vec{\gamma)}) \Psi(\vec{n}\cdot \vec{\gamma}; R) = \nonumber \\ \sum_{lm}f_{lm}Y_{lm}(\vec{n})[\int \frac{dR}{R}{(\frac{4\pi}{2l+1})}^2\Psi_l^2(R)], \label{eq:cj}\end{aligned}$$ where we have taken the harmonic expansions for $w(R, \vec{\gamma})$ and $\Psi(\vec{n}\cdot \vec{\gamma}; R)$. If one wants to have this equation proportional to $\sum_{lm}f_{lm}Y_{lm}(\vec{n}) = f(\vec{n})$, i.e. to be able to reconstruct $f(\vec{n})$, then it is obvious that necessarily $$\label{eq:ck} \int \frac{dR}{R}{(\frac{4\pi}{2l+1})}^2\Psi_l^2(R) = C_{\psi},$$ where $C_{\psi} \neq 0$ must be a constant. A particular solution to the previous equation is given by Eq. \[eq:ch\] and the admissibility condition. In this case, the synthesis equation can be written as $$\label{eq:cm} f(\vec{n}) = \frac{1}{C_{\psi}}\int \frac{dR}{R}\int d\Omega (\vec{\gamma}) \,w(R, \vec{\gamma})\,\Psi(\vec{n}\cdot \vec{\gamma}; R).$$ and the Equation (\[eq:cf\]) can be rewritten as $$\label{eq:cn} w(R, \vec{\gamma} ) = \sum_{lm}f_{lm}\psi (lR)Y_{lm}(\vec{\gamma}).$$ These equations are the analysis/synthesis counterparts on $S_2$ of the corresponding ones on $R^2$. Properties of the filter $\Psi$ =============================== Let us focus on some of the properties of the filter: [**$\Psi$ is a [*compensated*]{} filter**]{} Taking into account Eq. (\[eq:cl\]), $C_{\psi} < \infty$ implies that $\psi (l)\rightarrow l^{\epsilon},\epsilon >0$ as $l\rightarrow 0$, i. e. $\psi (l=0) = 0$. Now, taking into account Eq. (\[eq:ci\]), one obtains $$\label{eq:co} \int d\Omega (\vec{n})\,\Psi(\vec{n}\cdot \vec{\gamma}; R) = \psi (0) = 0,$$ the filter is compensated (hereinafter wavelet). [**Energy of the wavelet**]{} Taking into account Eq. (\[eq:ci\]) and the orthonormal property of the spherical harmonics, one obtains $$\label{eq:cp} \int d\Omega (\vec{n})\,\Psi^2(\vec{n}\cdot \vec{\gamma}; R) = \sum_l \frac{2l+1}{4\pi}\psi^2(lR).$$ [**Energy of the function $f(\vec{n})$**]{} Taking into account the standard properties of the spherical harmonics $$\label{eq:cq} \parallel f \parallel ^2 \equiv \int d\Omega (\vec{n})\,f^{2}(\vec{n}) = \sum_{lm}f^2_{lm},\ \ \ f^2_{lm}\equiv f_{lm}f_{lm}^*.$$ We can also prove the following equivalence $$\label{eq:cr} \parallel f \parallel ^2 = \frac{1}{C_{\psi}}\int \frac{dR}{R}\int d\Omega (\vec{n})\,w^2(R, \vec{n}).$$ An example ========== As an example of the previous ideas we will consider the generalization to the sphere of the [*Mexican Hat wavelet*]{} (MHW). We focus on the MHW because it is a widely used tool in Astronomy, well suited for the detection of pointlike objects such as extragalactic sources (\[3\],\[7\],\[10\]), but the same ideas can be applied to other wavelet families as well. Two natural generalizations of the mother Mexican Hat wavelet on the plane are possible $$\label{eq:cs} \psi_1(l)\propto l^2e^{-\frac{1}{2}l^2},\ \ \ \psi_2(l)\propto l(l+1)e^{-\frac{1}{2}l(l+1)},$$ where we have taken a unit width to define the mother wavelet. For $l \gg 1$ both functions approach the MHW on the plane. We have represented the harmonic coefficients $\psi_1(lR), \psi_2 (lR)$ for different values of $R=0.2\times 2^j, j=-2,-1,0,1,2$, as well as the profile of the wavelets on real space for the same cases, in Figure \[fig:fig1\]. The differences between wavelets $\psi_1$ and $\psi_2$ is shown in Figure \[fig:fig2\]. Next, in order to study the concentration in wavelet space we will consider a spot with spherical symmetry placed on the north pole, i. e. it is defined by a function $f(\theta)$. In this case, we get for the wavelet coefficients $$\begin{aligned} w(R, \theta ) & = & \sum_lw_l(R)\,P_l(\cos \theta ), \ \ \ w_l(R)\equiv f_l\,\psi (lR), \nonumber \\ f_l & \equiv & (l+\frac{1}{2}) \int_0^{\pi}d\theta\,\sin \theta P_l(\cos \theta )f(\theta ). \label{eq:ct}\end{aligned}$$ Let us now consider a very simple spot defined by a top hat $f(\theta ) = 1_{[0, \theta_o]}$. A simple way to test how the wavelets allow us to concentrate the information in a few number of coefficients is to measure somehow the width of the curve $w(R, \theta )$ of wavelet coefficients for the spot. An intuitive way to do this is to define the “energy” as the integral under the squared curve $w^2(R,\theta)$ and to see which is the radius $\theta_e$ that contains a given fraction of the total energy. The smaller $\theta_e$ is, the more concentrated the coefficients are. We have performed numerical simulations with a simple toy model to see which of the two generalizations of the MHW, $\psi_1$ or $\psi_2$, concentrates more the coefficients. We have placed a top hat spot of size $\theta_0=0.2$ rad in the North Pole and we have filtered it with the wavelets $\psi_1$ and $\psi_2$ using different scales ranging from $R=0.1$ to $R=1.6$ rad. The results are shown in the upper panel of Figure \[fig:fig3\]. We have ploted the radius $\theta_e(R)$ such that $68\%$ of the energy is inside the circle of radius $\theta_e(R)$ as a function of the dilation scale $R$. As can be seen, the wavelet $\psi_1$ concentrates more the coefficients, that is, produces smaller values of $\theta_e$. Now we repeat the same process but using a Gaussian spot instead of a top hat. The Gaussian spot is given by $f(\theta ) = \mathrm{exp}(-\theta^2/2 \theta_0^2)$. This case is interesting since most of the detectors that operate in microwave Astronomy experiments have approximately Gaussian response. The lower panel of Figure \[fig:fig3\] show the results, that are very similar to the top hat case. Again, $\psi_1$ concentrates more the coefficients. We have tested the resuts for different levels of concentration and we have found that up to $75\%$ the wavelet $\psi_1$ concentrates more the coefficients. Above $75\%$ this is still true for large values of the dilation $R$, while for small $R$ $\psi_2$ concentrates more the coefficients. This is due to the fact that $\psi_1$ produces more small oscilations in the tail of the curve $w(R, \theta )$ for small $R$. Conclusions =========== We have developed a constructive wavelet approach on the sphere without any projection from the plane. It is a continuous transform that allows the analysis and synthesis of any function defined on the sphere and incorporates the concepts of translation and dilation as generalizations of the elementary ones defined on the plane. It is a compensated filter that conserves the energy of any function. We have considered some natural generalizations of the plane Mexican hat wavelet and we have applied them to the detection of a big spot. The conclusion is that one of the wavelets ($\psi_1$) concentrates more the information than the other one. Acknowledgements ================ We acknowledge partial financial support from the Spanish Ministry of Education (MEC) under project ESP2004–07067–C03–01. MLC acknowledge a FPI fellowship of the Spanish Ministry of Education and Science (MEC). DH acknowledges the Spanish MEC for a “Juan de la Cierva” postdoctoral fellowship. 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--- author: - 'M.-C. ARNAUD [^1] [^2] [^3]' title: '[ Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures ]{}' --- Keywords: Minimizing orbits and measures, Lyapunov exponents, weak KAM theory, Green bundles, regularity of solutions to Hamilton-Jacobi equations. [**Résumé**]{} Dans cet article, on étudie les mesures minimisantes de Hamitoniens de Tonelli. Plus précisément, on explique quelles relations existent entre les fibrés de Green et différentes notions comme : 1. les exposants de Lyapunov des mesures minimisantes; 2. les solutions KAM faibles. On en déduit par exemple que si tous les exposants de Lyapunov d’une mesure minimisante $\mu$ sont nuls, alors le support de cette mesure est $C^1$-régulier en $\mu$-presque tout point. Mots clefs: Orbites et mesures minimisantes, exposants de Lyapunov, théorie KAM faible, fibrés de Green, régularité des solutions de l’équation de Hamilton-Jacobi. MSC: 37J50, 35D40, 37C40, 34D08, 35D65 Introduction ============ In this article, $M$ is a closed $n$-dimensional manifold and $\pi~: T^*M\rightarrow M$ its cotangent bundle. We consider a Tonelli Hamiltonian $H~: T^*M\rightarrow {\mathbb {R}}$, i.e. a $C^2$ function that is strictly $C^2$-convex and superlinear in the fiber. The Hamiltonian flow associated with such a function is denoted by $(\varphi_t)_{t\in{\mathbb {R}}}$ or $(\varphi_t^H)_{t\in{\mathbb {R}}}$. To such a Hamiltonian, there corresponds a Lagrangian function $L~: TM\rightarrow {\mathbb {R}}$ that has the same regularity as $H$ and is also superlinear and strictly convex in the fiber. The corresponding Euler-Lagrange flow is denoted by $(f_t)_{t\in{\mathbb {R}}}$. For such a Hamiltonian system, it is usual to study its “minimizing objects”; more precisely, a piece of orbit $(\varphi_t(q,p))_{t\in [a,b]}=(q_t, p_t)_{t\in[a,b]}$ is minimizing if the arc $(q_t)_{t\in[a, b]}$ minimizes the action functional $A_L$ defined by $A_L(\gamma )=\int_a^bL(\gamma (t), \dot\gamma (t))dt$ among the $C^2$-arcs joining $q_a$ to $q_b$. More generally, if $I$ is an interval and $(\varphi_t)_{t\in I}=(q_t, p_t)_{t\in I}$ is an orbit piece, we say that it is minimizing if for every segment $[a,b]\subset I$, its restriction to $[a,b]$ is minimizing. Then we call the set of points of $T^*M$ whose (complete) orbit is minimizing the [*Mañé set*]{}. We denote it by ${\mathcal {N}}^*(H)$ and its projection, the [*projected Mañé set*]{}, is denoted by: ${\mathcal {N}}(H)=\pi ({\mathcal {N}}^*(H))$. The Mañé set is non empty, compact and invariant by the Hamiltonian flow (see [@Fa1]). The first proof of the non-emptiness of the Mañé set is due to J. Mather: he proved in the 90’s in [@mather1] the existence of minimizing measures. We are interested in invariant subsets of the Mañé set, i.e. subsets that are the union of some minimizing orbits. More precisely, we would like to know if we can say something about the regularity of such subsets (we will be more precise very soon. It’s a kind of differentiability) and particularly if there is a link between the dynamic of the flow restricted to such a set and the regularity of the set. The oldest result in this direction concerns the time-dependent case : considering a symplectic twist map of the annulus $T^*{\mathbb {S}}$, G. Birkhoff proved in the 1920’s that any essential invariant curve is the graph of a Lipschitz map (see [@Bir1] or [@He1]). It is easy to prove that such a curve is action minimizing. In the case of higher dimensions, M. Herman proved in [@He2] that any $C^0$-Lagrangian graph of $T^*{\mathbb {T}}^n$ that is invariant by a symplectic twist map is, in fact, the graph of a Lipschitz map. A related result in the autonomous case is that any $C^1$-Hamilton-Jacobi solution of a Tonelli Hamiltonian is, in fact, $C^{1,1}$ (see [@Fa2]). As Rademacher’s theorem says to us that any Lipschitz function is differentiable Lebesgue almost everywhere, these results are a kind of regularity result. In [@Arna2], we did, in fact, improve these results of regularity in the autonomous case, proving that if a $C^0$-Lagrangian graph is invariant by a Tonelli flow, and if one of the two following hypotheses is satisfied: 1. $\dim M=2$ and all the singularities of $H$ are non degenerate; 2. the dynamic of the restriction of the flow to the invariant graph is Lipschitz conjugate to a translations’ flow; then the invariant graph is, in fact, $C^1$ almost everywhere (this is stronger than just differentiable). Let us point out that any of the two previous hypotheses implies that the dynamic of the restricted flow to the graph is soft on a certain sense (our arguments are not very precise, but we only want to give a certain intuition of the forthcoming result); indeed, when $\dim M=2$, if we reduce the dynamic modulo the vector field, we obtain a 1-dimension dynamic, and it is known at least in the differentiable case that the Lyapunov exponents of a dynamic on the circle are zero. The same is true for any dynamic that is Lipschitz conjugate to a translation. We gave a similar results for the invariant curves of the twist maps of the annulus in [@Arna1], proving that Birkhoff’s result can be improved: any essential invariant curve of a symplectic twist map of the annulus $T^*{\mathbb {S}}$ is the graph of a Lipschitz map that is $C^1$ Lebesgue almost everywhere. Hence, it seems reasonable to try to find a relationship between the Lyapunov exponents of any minimizing measure and the regularity of its support, where an invariant measure is [*minimizing*]{} if its support is in the Mañé set. For a twist map of the annulus $T^*{\mathbb {S}}$, we studied the ergodic minimizing measures in [@Arna3] and proved that the $C^1$-regularity (we will be more precise very soon) of its support is equivalent to the fact that the Lyapunov exponents are zero. Hence, in a certain way, in this case, “$C^1$-irregularity” is equivalent to non-vanishing Lyapunov exponents. The question that we ask now ourselves is the following: what can we say for higher dimensions? Is the irregularity (in a sense we will soon specify) of the support of a minimizing ergodic measure equivalent to non-vanishing exponents? A first and obvious answer is: no. Indeed, let us consider the following example: $(\psi_t)$ is an Anosov flow defined on the cotangent bundle $T^*{\mathcal{S}}$ of a closed surface ${\mathcal{S}}$. Let ${\mathcal {N}}=T^*_1{\mathcal{S}}$ be its unitary cotangent bundle, which is a 3-manifold invariant by $(\psi_t)$. Then a method due to Mañé (see [@Man1]) allows us to define a Tonelli Hamiltonian $H$ on $T^*{\mathcal {N}}$ such that the restriction of its flow $(\varphi_t)$ to the zero section ${\mathcal {N}}$ is $(\psi_t)$: the Lagrangian $L$ associated with $H$ is defined by: $L(q,v)=\frac{1}{2}\| \dot\psi (q)-v\|^2$ where $\| .\|$ is any Riemannian metric on ${\mathcal {N}}$. In this case, the zero section is very regular (even $C^\infty$), but the Lyapunov exponents of every invariant measure whose support is contained in ${\mathcal {N}}$ are non zero (except two, the one corresponding to the flow direction and the one corresponding to the energy direction). Hence, it may happen that some exponents are non zero and the support of the measure is very regular…\ In fact, the other implication is true: we will see that the nullity of the Lyapunov exponents implies the regularity of the support of the considered measure.\ Let us now explain in a detailed way in which kind of regularity we are interested: Let $A$ be a subset of a manifold $M$ and let $a$ belong to $A$. The contingent cone to $A$ at $a$ is the set of the tangent vectors $v\in T_aM$ such that there exist a sequence $(a_n)$ of elements of $A$ and a sequence $(t_n)$ of positive real numbers such that (we write everything in a chart, but this is independent of the chosen chart): $$\lim_{n\rightarrow \infty} \frac{1}{t_n}(a_n-a)=v.$$ We denote it by: ${\mathcal {C}}_aA$. This notion of contingent cone is due to Bouligand (see [@Bou]). The contingent cone is never empty (it always contains the null vector), and it is equal to the null vector if, and only if, $a$ is an isolated point of $A$. We will see later that the sets in which we are interested are contained in some (weak) Lagrangian manifolds. Our definitions of $1$-regularity and $C^1$-regularity seems very natural for such sets: Let $A$ be a subset of a symplectic manifold $M$ and let $a$ belong to $A$. We say that $A$ is $1$-regular at $a$ if the contingent cone to $A$ at $a$ is contained in a Lagrangian subspace of $T_aM$.\ We say that $A$ is $C^1$ regular at $a$ if there exists a Lagrangian subspace ${\mathcal {L}}$ of $T_aM$ such that: for every sequence $(a_n, v_n\in {\mathcal {C}}_{a_n}A)$ such that $\displaystyle{\lim_{n\rightarrow \infty}a_n=a}$ and the sequence $(v_n)$ converges to an element $v$ of $T_aM$, then $v\in {\mathcal {L}}$. Let us notice that this notion of $C^1$-regularity is slightly different from the ones given in [@Arna1], [@Arna2] and [@Arna3]: the notions given in these former articles are a little stronger. This notion of $C^1$-regularity is stronger than the notion of $1$-regularity, which is nothing else but the notion of differentiability for the $C^0$- Lagrangian graphs (see [@Arna2] for a definition of $C^0$-Lagrangian graphs).\ The measures that we study are the minimizing ones, that is the ones that are invariant and whose supports are contained in the Mañé set. Then we prove: \[th1\] Let $H~: T^*M\rightarrow {\mathbb {R}}$ be a Tonelli Hamiltonian and let $\mu$ be an ergodic minimizing probability measure all of whose Lyapunov are zero. Then, at $\mu$-almost every point of the support ${\rm supp}(\mu)$ of $\mu$, the set ${\rm supp}(\mu)$ is $C^1$-regular. Hence: 1. we succeed in proving that a kind of “soft dynamic” implies some $C^1$-regularity; 2. we know that we can have simultaneously a strong dynamic (for example hyperbolic) and a $C^\infty$-regularity. In fact, we obtain more precise results than this theorem; for example, an interesting question is: what happens if there are simultaneously some zero and non-zero exponents?\ To explain what happens, we need to introduce some other notions. Let us begin by recalling what the Green bundles are. These Lagrangian bundles were introduced by L. Green in 1958 in [@Gr] for geodesic flows to prove some rigidity results. For the existence and the construction of these bundles, the reader is referred to [@Arna2], [@C-I] or [@It1]. We recall: Here, $V(x)=\ker D\pi (x)$ designates the linear vertical.\ Let $(\varphi_t(q,p))_{t\in ]-\infty, 0]}$ be a minimizing negative orbit; then the positive Green bundle $G_+$ is defined along this orbit by: $\displaystyle{G_+(x)=\lim_{t\rightarrow +\infty} D\varphi_t.V(\varphi_{-t}x)}$.\ Let $(\varphi_t(q,p))_{t\in [0, +\infty[}$ be a minimizing positive orbit; then the negative Green bundle $G_-$ is defined along this orbit by: $\displaystyle{G_-(x)=\lim_{t\rightarrow +\infty} D\varphi_{-t}.V(\varphi_{t}x)}$. Hence, at every point of the Mañé set, the two Green bundles are defined. Let us recall that the two Green bundles are Lagrangian, invariant under the linearized flow $D\varphi_t$, transverse to the vertical, that they depend semi-continuously on the considered point (see [@Arna2] for the definition of semi-continuity of Lagrangian subspaces transverse to the vertical), that $G_-\leq G_+$ (see [@Arna2] for the definition of the order between two planes transverse to the vertical; in coordinates, this corresponds to the usual order on the set of symmetric matrices whose Lagrangian subspaces are the graphs.). Hence, if $\mu$ is an ergodic minimizing probability measure, the integer $\dim(G_-(x)\cap G_+(x))$ is constant $\mu$ almost everywhere. We obtain a result linking the dimension of the intersection of the two Green bundles to the number of non zero Lyapunov exponents: \[thdimlyap\] Let $H~: T^*M\rightarrow {\mathbb {R}}$ be a Tonelli Hamiltonian and let $\mu$ be an ergodic minimizing probability measure. Then the two following assertions are equivalent: 1. at $\mu$ almost every point, $\dim (G_-(x)\cap G_+(x))=p$; 2. $\mu$ has exactly $2p$ zero Lyapunov exponents, $n-p$ positive ones and $n-p$ negative ones. Let us mention some former related results: 1. in [@C-I], the authors prove that the transversality of the two Green bundles along an energy level implies that the restriction of the flow to this level is Anosov; they use some ideas about quasi-Anosov dynamics due to R. Mañé that are contained in [@Man2]; in [@Eb1], P. Eberlein gives the same statement for the geodesic flows; 2. we proved in [@Arna3] that any quasi-hyperbolic symplectic cocycle above a compact set is hyperbolic; we can apply this result to any minimizing compact invariant subset $K$ contained in an energy level ${\mathcal {E}}$ without singularity: considering the restricted/reduced dynamical system to the energy level ${\mathcal {E}}$ modulo the vector-field (see [@Arna2] p 899 for the construction), we deduce that the transversality of the Green bundles in the energy level above $K$ is equivalent to the partial hyperbolicity of the linearized flow along $K$ with a center bundle’s dimension equal to 2; 3. concerning the non-uniform case (i.e. the case of minimizing measures), the only known result was a formula giving the entropy due to A. Freire & R. Mañé (see [@F-M]). Roughly speaking, by integrating some functional along one of the two Green bundles, they compute the sum of the positive Lyapunov exponents. This formula was generalized in [@C-I] to any Tonelli Hamiltonian. But this formula doesn’t say to us how many non-zero Lyapunov exponents exist: it only gives the sum of the positive Lyapunov exponents. Let us mention too that G. Knieper gives a nicer formula in his (non-published) thesis. To prove theorem \[th1\], we recall in section \[sec3\] some points of the recent weak KAM theory developped by A. Fathi in [@Fa1]. In this section too, we give some statements concerning the relationships between weak KAM solutions and the Green bundles. We don’t give them in the introduction because we would need all the notions that will be defined in section \[sec3\], but the interested reader can go to section \[sec3\]. Roughly speaking, the theorem asserts that along the support of the minimizing measures, the contingent cones to the weak KAM pseudographs is not far from some cone delimited by the two Green bundles.\ Theorem \[thdimlyap\] is proved in section \[sec2\]. The statement concerning the relationships between the weak KAM solutions and the Green bundles are contained in section \[sec3\] and the proofs are in section \[sec4\]. Green bundles and Lyapunov exponents {#sec2} ==================================== In this section, we prove theorem \[thdimlyap\]. We consider an ergodic minimizing measure $\mu$ that is not the Dirac measure at a critical point and we denote the integer such that we have $\mu$ almost everywhere: $\dim G_-\cap G_+=p$ by $p$. Let us recall the dynamical criterion that is proved in [@Arna2]: \[dyncrit\] Let $(x_t)$ be a minimizing and relatively compact orbit. Let $v\in T_{x_0}(T^*M)$. Then:\ – if $v\notin G_-(x_0)$, then $\displaystyle{\lim_{t\rightarrow +\infty}\| D\pi\circ D\varphi_t.v\|=+\infty}$;\ –if $v\notin G_+(x_0)$, then $\displaystyle{\lim_{t\rightarrow +\infty}\| D\pi\circ D\varphi_{-t}.v\|=+\infty}$. and some direct consequences of this criterion: 1\) We deduce from the dynamical criterion that the Hamiltonian vector-field $X_H$ belongs to the two Green bundles. This implies that $p\geq 1$. Because these two Green bundles are Lagrangian, this implies that $G_+$ and $G_-$ are tangent to the Hamiltonian levels $\{ H=c\}$.\ 2) Moreover, we deduce also that if there is an Oseledet splitting (this will be precisely defined very soon) $T(T^*M)=E^s\oplus E^c\oplus E^u$ above a minimizing compact set $K$, then $E^s\subset G_-$ and $E^u\subset G_+$. Because the flow is symplectic, $E^u$ and $E^s$ are isotropic and orthogonal to $E^c$ for the symplectic form (see [@bochi-viana1]). Moreover, $E^{s\bot}=E^s\oplus E^c$ (where $\bot$ designates the orthogonal subspace for the symplectic form) and $E^{u\bot}=E^u\oplus E^c$; we deduce that: $G_-(x)=G_-(x)^\bot\subset E^{s\bot}=E^s\oplus E^c$ and similarly that $G_+(x)\subset E^u(x)\oplus E^c(x)$. Hence, finally: $$E^s(x)\subset G^-(x)\subset E^s(x)\oplus E^c(x)\ {\rm and}\ E^u(x)\subset G^+(x)\subset E^u(x)\oplus E^c(x)$$ and then: $G_-(x)\cap G_+(x)\subset E^c(x)$. Hence, $G_-\cap G_+$ being an isotropic subspace of the symplectic subspace $E^c$, we obtain: $\dim E^c\geq 2\dim (G_-\cap G_+)$. The dimension of the intersection of the two Green bundles gives a lower bound to the number of zero Lyapunov exponents. Theorem \[thdimlyap\] says to us that this inequality is, in fact, an equality. Let us notice that when $p=n$, we directly have the conclusion of the theorem because $\dim E^c\geq 2\dim M$ implies that $\dim E^c=2n$.\ We have the same results for a hyperbolic or partially hyperbolic dynamic. Let us notice that in the hyperbolic case, $G_-$ (resp. $G_+$) is nothing else but the stable (resp. unstable) bundle $E^s$ (resp. $E^u$)\ 3) Let us consider the case of a K.A.M. torus that is a graph (when $M={\mathbb {T}}^n$): the dynamic on this torus is $C^1$ conjugated to a flow of irrational translations on the torus ${\mathbb {T}}^n$; M. Herman proved in [@He2] that such a torus is Lagrangian, and it is well-known that any invariant Lagrangian graph is locally minimizing. Then the orbit of every vector tangent to the K.A.M. torus is bounded, and belongs to $G_-\cap G_+$. In this case, the two Green bundles are equal to the tangent space to the invariant torus. Let us introduce some notations: Oseledet’s theorem implies that there exist an invariant subset $N$ of $T^*M$ with full $\mu$-measure, some real numbers $0<\lambda _1 <\lambda_2 <\dots < \lambda_{q }$ and a (measurable) splitting with constant dimensions above $N$: $$T_x(T^*M)=E^s_1(x)\oplus E^s_2(x)\oplus\dots\oplus E^s_{q }(x)\oplus E^c(x)\oplus E^u_1(x)\oplus E^u_2(x)\oplus \dots \oplus E^u_{q }(x)$$ such that: 1. for every $v\in E^s_j(x)\backslash\{ 0\}$; $\displaystyle{\lim_{t\rightarrow\pm \infty}\frac{1}{t}\log\left( \| D\varphi_t(x)v\| \right)=-\lambda_j}$; 2. for every $v\in E^c(x)\backslash\{ 0\}$; $\displaystyle{\lim_{t\rightarrow\pm \infty}\frac{1}{t}\log\left( \| D\varphi_t(x)v\| \right)=0}$; 3. for every $v\in E^u_j(x)\backslash\{ 0\}$; $\displaystyle{\lim_{t\rightarrow\pm \infty}\frac{1}{t}\log\left( \| D\varphi_t(x)v\| \right)=+\lambda_j}$.\ We may ask, too, that: $\forall x\in N, \dim(G_-(x)\cap G_+(x))=p$. Let us recall that the stable bundle $E^s(x)=E^s_1(x)\oplus E^s_2(x)\oplus\dots\oplus E^s_{q }(x)$ and the unstable one $E^u(x)=E^u_1(x)\oplus E^u_2(x)\oplus\dots\oplus E^u_{q }(x)$ are isotropic (for the symplectic form) and that $E^c(x)$ is a symplectic subspace of $T_x(T^*M)$ that is orthogonal (for $\omega$) to $E^s(x)\oplus E^u(x)$. Moreover, we have: $\dim E^s_i=\dim E^u_i$. Reduction of the problem ------------------------ As in the statement of theorem \[thdimlyap\], we assume that $\mu$ is a minimizing ergodic measure whose support is not reduced to a point and that $p\in [1, n]$ is so that at $\mu$-almost every point $x$, the intersection of the Green bundles $G_+(x)$ and $G_-(x)$ is $p$-dimensional. We deduce from the previous remark that for every $x\in N$: $G_+(x)\cap G_-(x)\subset E^c(x)$ and $E^s(x)\oplus E^u(x)= \left( E^c(x)\right)^\bot\subset G_+(x)^\bot+ G_-(x)^\bot=G_-(x)+ G_+(x)$. We introduce the two notations: $E(x)=G_-(x)+G_+(x)$ and $R(x)= G_-(x)\cap G_+(x)$. We denote the reduced space: $F(x)=E(x)/R(x)$ by $F(x)$ and we denote the canonical projection $p~: E\rightarrow F$ by $p$. As $G_-$ and $G_+$ are invariant by the linearized flow $D\varphi_t$, we may define a reduced cocycle $M_t~: F\rightarrow F$. But $(M_t)$ is not continuous, because $G_-$ and $G_+$ don’t vary continuously.\ Moreover, we introduce the notation: ${\mathcal {V}}(x)=V(x)\cap E(x)$ is the trace of the linearized vertical on $E(x)$ and $v(x)=p({\mathcal {V}}(x))$ is the projection of ${\mathcal {V}}(x)$ on $F(x)$. We introduce a notation for the images of the reduced vertical $v(x)$ by $M_t$: $g_t(\varphi_tx)=M_tv(x)$. The subspace $E(x)$ of $T_x(T^*M)$ is co-isotropic with $E(x)^{\bot}=R(x)$. Hence $F(x)$ is nothing else than the symplectic space that is obtained by symplectic reduction of $E(x)$. We denote its symplectic form by $\Omega$. Hence we have: $\forall (v, w)\in E(x)^2, \Omega (p(v), p(w))=\omega (v,w)$. Moreover, $(M_t)$ is a symplectic cocycle.\ We can notice, too, that $\dim E(x)=\dim (G_-(x)+G_+(x))=\dim G_-(x)+\dim G_+(x)-\dim (G_-(x)\cap G_+(x))=2n-p$ and deduce that $\dim F(x)=\dim E(x)-\dim (G_-(x)\cap G_+(x))=2(n-p)$. If $L$ is any Lagrangian subspace of $T_x(T^*M)$, we denote $(L\cap E(x))+R(x)$ by $\tilde{L}$ and $p(\tilde{L})$ by $l$. \[bernard1\] If $L\subset T_x(T^*M)$ is Lagrangian, then $\tilde{L}$ is also Lagrangian and $l=p(\tilde L)=p(L\cap E(x))$ is a Lagrangian subspace of $F(x)$. Moreover, $p^{-1}(l)=\tilde{L}$ . In particular, $v(x)$ is a Lagrangian subspace of $F(x)$ and $p^{-1}(v(x))={\mathcal {V}}(x)+R(x)$. We just have to prove that $\tilde L$ is Lagrangian, the other assertions being easy consequences of this fact.\ We begin by proving that $\tilde L$ is isotropic. If $u, u'\in L\cap E(x)$ and $v, v'\in R(x)$, then $\omega (u+v, u'+v')=0$ because $L$ is Lagrangian and then $\omega (u, u')=0$ and because $R(x)\subset E(x)^\bot$.\ Let us determine $\dim \tilde{L}$. Let $L'$ be such that: $L=(E(x)\cap L)\oplus L'$. Then the dimension of $L\cap R(x)=(L+E(x))^\bot$ is: $2n-(\dim L+E(x))=2n-(2n-p+\dim L')=p-\dim L'$. We deduce: $\dim \tilde L=\dim (L\cap E(x))+\dim R(x)-\dim (L\cap R(x))=\dim (L\cap E(x))+p-(p-\dim L')=\dim (L\cap E(x))+\dim L'=\dim L$. \[L2\] The subspace $v(x)$ is a Lagrangian subspace of $F(x)$. Moreover, for every $t\not=0$, $g_t(\varphi_tx)=M_tv(x)$ is transverse to $v(\varphi_t(x))$ The first sentence is contained in lemma \[bernard1\].\ Let us consider $t\not=0$ and let us assume that $ M_tv(x) \cap v(\varphi_tx)\not=\{ 0\}$. We may assume that $t>0$ (or we replace $x$ by $\varphi_t(x)$ and $t$ by $-t$). Then there exists $v\in {\mathcal {V}}(x)\backslash \{ 0\}$ such that$D\varphi_t(x)v\in {\mathcal {V}}(\varphi_t x)+(G_-(\varphi_tx)\cap G_+(\varphi_tx))$. Let us write $D\varphi_t(x)v=w+g$ with $w\in {\mathcal {V}}(\varphi_tx)$ and $g\in R(\varphi_tx)$. We know that the orbit has no conjugate vector (because the measure is minimizing); hence $g\not=0$. Moreover, we proved in [@Arna2] that $D\varphi_t V(x)$ is strictly above $G_-(\varphi_tx)$, i.e. that: $$\forall h\in G_-(\varphi_tx), \forall k\in V(\varphi_tx), h+k\in D\varphi_t V(x)\backslash \{ 0\}\Rightarrow \omega (h, h+k)> 0.$$ We deduce that: $\omega (g, w+g)>0$. This contradicts: $D\varphi_t(x)v\in E(\varphi_tx)=\left(G_+(\varphi_tx)\cap G_-(\varphi_tx)\right)^\bot\subset ({\mathbb {R}}g)^\bot$. As in [@Arna2], we ask ourselves what the order between the different Lagrangian subspaces $g_t(x)=M_tv(\varphi_{-t}x)$ is. Let us recall how we define this order: Let $g_1$ and $g_2$ be two subspaces of $F(x)$ that are transverse to the (reduced) vertical $v(x)$. Let $f(x)=F(x)/v(x)$ be the reduced space and $P(x)~: F(x)\rightarrow f(x)$ the canonical projection. Then to every $w\in f(x)$, we can associate a unique $\ell_1(w)\in g_1$ (resp. $\ell_2(w)\in g_2$) such that: $P( \ell_1(w))= w$ (resp. $P(\ell_2(w))= w$). We then define the altitude of $g_2$ above $g_1$, which is a quadratic form defined on $f(x)$, by: $q(g_1,g_2)(w)=\Omega (\ell_1(w), \ell_2(w))$.\ We say that $g_2$ is above (resp. strictly above) $g_1$ when $q(g_1, g_2)$ is positive semi-definite (resp. positive definite). We write $g_1\leq g_2$ (resp. $g_1<g_2$). \[bernard2\] Let $L_1$, $L_2$ be two Lagrangian subspaces of $T_x(T^*M)$ transverse to $V(x)$ such that at least one of them is contained in $E(x)$. Then, if $L_1<L_2$ (resp. $L_1\leq L_2$), we have: $l_1$ and $l_2$ are transverse to $v(x)$ and $l_1 < l_2$ (resp. $l_1\leq l_2$). We deduce that $p(G_-)<p(G_+)$. We assume that $L_2\subset E(x)$ and that $L_1<L_2$. Let $v_1\in L_1\cap E(x)$ be a non-zero vector of $L_1\cap E(x)$. As $L_1$ and $L_2$ are transverse to $V(x)$, there exists a unique $v_2\in L_2$ such that $v_2-v_1\in V(x)$. Moreover, as $v_1, v_2\in E(x)$, we have $v_2-v_1\in {\mathcal {V}}(x)$ and $p(v_2)-p(v_1)\in v(x)$. Hence: $$\Omega (p(v_1), p(v_2))=\omega (v_1, v_2)>0.$$ This means exactly that $l_1<l_2$.\ To deduce the assertion for $\leq$, we can use a limit.\ As $G_-\leq G_+$, we deduce that $p(G_-)\leq p(G_+)$. Because of the definition of $E(x)$, $R(x)$ and $F(x)$, $p(G_-)$ and $p(G_+)$ are transverse and then $p(G_-)<p(G_+)$. If $\mu$ is a minimizing measure, for every $x\in{\rm supp}\mu$, for all $0<t<s$, we have: $$g_{-t}(x)<g_{-s}(x)<g_s(x)<g_t(x).$$ The map $(t\in {\mathbb {R}}^*\rightarrow g_t(x))$ is continuous; moreover, we know by lemma \[L2\] that if $t\not=s$, then $g_t(x)$ is transverse to $g_s(x)$. Hence, the index of $q(g_s(x), g_t(x))$ is constant for $(s, t)\in{\mathcal {E}}$ where ${\mathcal {E}}$ is one of the sets: $\{ (s,t); 0<s<t\}$; $\{ (s,t); s<0<t\}$, $\{ (s,t); s<t<0\}$. Hence, we only have to determine this index for one point $(s,t)$ of each of these three sets.\ We prove the result only for the first set, the other inequalities being very similar. Let us fix $s>0$ and introduce the notation $G_s(x)=D\varphi_sV(\varphi_{-s}x)$. Then $\tilde G_s(x)$ is a Lagrangian subspace of $E(x)$ that is transverse to the vertical because $\tilde G_s(x)\cap V(x)=\tilde G_s(x)\cap {\mathcal {V}}(x)=(\tilde G_s(x)\cap \tilde V(x))\cap {\mathcal {V}}(x)=p^{-1}(g_s(x)\cap v(x))\cap {\mathcal {V}}(x)=R(x)\cap {\mathcal {V}}(x)=\{ 0\}$. We assume that $t>0$ is very small and we work in a chart, with symplectic coordinates defined in [@Arna2] (p 897) such that the “horizontal” subspace of $T_x(T^*M)$ is $G_-(x)$. A vector of $ G_t(x)=D\varphi_t(\varphi_{-t}x)V(\varphi_{-t}x)$ is $(h, S^+_t(x)h)$ and it is proved in [@Arna2] (p 894) that $S^+_{t}(x)\sim \frac{1}{t}D$ where $D$ is a fixed positive definite matrix. Hence, for $t>0$ small enough, we have $\tilde G_s<G_t$. We deduce from lemma \[bernard2\] that $g_s=p(\tilde G_s)<p(G_t)=g_t$.\ As in [@Arna2], when $t$ tends to $\pm \infty$, we find two $M_t$-invariant Lagrangian sub-bundle of $F(x)$ that are: $\displaystyle{g_-(x)=\lim_{t\rightarrow -\infty} g_t(x)}$ and $\displaystyle{g_+(x)=\lim_{t\rightarrow +\infty} g_t(x)}$; they are transverse to $v(x)$ and satisfy: $g_-(x)\leq g_+(x)$. We call them the reduced Green bundles. Then we have: $\forall t>0, g_{-t}(x)<g_-(x)\leq g_+(x)<g_t(x)$. If we use the notations $\tilde G_\pm(x)=p^{-1}(g_\pm(x))$, then $\tilde G_\pm$ are transverse to the vertical because $\tilde G_\pm(x) \cap V(x)=\tilde G_\pm (x)\cap {\mathcal {V}}(x)=(\tilde G_\pm\cap \tilde V(x))\cap {\mathcal {V}}(x)=p^{-1}(g_\pm (x)\cap v(x))\cap {\mathcal {V}}(x)=R(x)\cap {\mathcal {V}}(x)=\{ 0\}$. Moreover, $\tilde G_-(x)\leq \tilde G_+(x)$ and the two bundles $\tilde G_-$, $\tilde G_+$, are invariant by the linearized flow $(D\varphi_t)$. Theorem 3.11 of [@Arna2] asserts that any invariant Lagrangian bundle that is transverse to the vertical is between the two Green bundles. We deduce that $G_-(x)\leq \tilde G_-(x)\leq \tilde G_+(x)\leq G_+(x)$. We can then use lemma \[bernard2\] and we obtain: $p(G_-(x))\leq g_-(x)\leq g_+(x)\leq p(G_+(x))$. \[derlem\] We have: $\forall x\in{\rm supp}\mu, g_-(x)=p(G_-(x))<p(G_+(x))= g_+(x) $. Because of the last remark, we just have to prove that on ${\rm supp} \mu$: $ g_- \leq p(G_- )<p(G_+ )\leq g_+ $. Because of lemma \[bernard2\], we just have to prove that $ g_- \leq p(G_- )$ and $p(G_+ )\leq g_+ $. But $p(G_\pm)$ is a lagrangian subspace of $F(x)$ whose orbit is transverse to the vertical. We can use a similar statement to proposition 3.11 of [@Arna2] to deduce the inequalities. Hence we have proved that $\tilde G_\pm=G_\pm$, the notation $\tilde G_\pm$ will disappear from tnow on. Reduced Green bundles and Lyapunov exponents -------------------------------------------- We have to be careful because the bundles that we consider are not continuous and, as this is noted in [@Arna2], we don’t use a continuous change of coordinates, but just a bounded one when we say that $G_-$ or $G_+$ is the horizontal subspace (the matrix $P$ that is necessary to change the coordinates is uniformly bounded, as $P^{-1}$).\ We choose at every point $x\in N$ some (linear) symplectic coordinates $(Q,P)$ of $F(x)$ such that $v(x)$ has for equation: $Q=0$ and $g_+(x)$ has for equation $P=0$. We will be more precise on this choice later. Then the matrix of $M_t(x)$ in these coordinates is a symplectic matrix: $M_t(x)=\begin{pmatrix} a_t(x)&b_t(x)\\ 0&d_t(x)\\ \end{pmatrix}$. As $M_t(x)v(x)=g_t(\varphi_tx)$ is a Lagrangian subspace of $E(\varphi_tx)$ that is transverse to the vertical, then $\det b_t(x)\not=0$ and there exists a symmetric matrix $s_t^+(\varphi_tx)$ whose graph is $g_t(\varphi_tx)$, i.e: $d_t(x)=s_t^+(\varphi_t(x))b_t(x)$. Moreover, the family $(s^+_t(x))_{t>0}$ being decreasing and tending to zero (because by hypothesis the horizontal is $g_+$), the symmetric matrix $s^+_t(\varphi_tx)$ is positive definite. Moreover, the matrix $M_t(x)$ being symplectic, we have: $$\left(M_t(x)\right)^{-1}=\begin{pmatrix} {}^td_t(x)&-{}^tb_t(x)\\ 0&{}^ta_t(x)\\ \end{pmatrix}$$ and by definition of $g_{-t}(x)$, if it is the graph of the matrix $s^-_t(x)$ (that is negative definite), then: ${}^ta_t(x)=-s_t^-(x){}^tb_t(x)$ and finally: $$M_t(x)=\begin{pmatrix} -b_t(x)s_t^-(x)& b_t(x)\\ 0& s_t^+(\varphi_tx)b_t(x)\\ \end{pmatrix}$$ Let us be now more precise in the way we choose our coordinates; we may associate an almost complex structure $J$ and then a Riemannian metric $(.,.)_x$ defined by: $(v,u)_x=\omega (x)(v, Ju)$ with the symplectic form $\omega$ of $T^*M$; from now on, we work with this fixed Riemannian metric of $T^*M$. We choose on $ G_+(x)=p^{-1}(g_+(x))$ an orthonormal basis whose last vectors are in $R(x)$ and complete it in a symplectic base whose last vectors are in $V(x)$. We denote the associated coordinates of $T_x(T^*M)$ by $(q_1, \dots, q_n, p_1, \dots, p_n)$. These (linear) coordinates don’t depend in a continuous way on the point $x$ (because $G_+$ doesn’t), but in a bounded way. Then $G_-(x)=p^{-1}(g_-(x))$ is the graph of a symmetric matrix whose kernel is $R(x)$ and then on $G_-(x),$ we have: $p_{n-p+1}=\dots =p_n=0$. An element of $E(x)$ has coordinates such that $p_{n-p+1}=\dots =p_n=0$, and an element of $F(x)=E(x)/R(x)$ may be identified with an element with coordinates $(q_1, \dots , q_{n-p}, 0, \dots , 0, p_1, \dots , p_{n-p}, 0, \dots , 0)$. We then use on $F(x)$ the norm $\displaystyle{\sum_{i=1}^{n-p}(q_i^2+p_i^2)}$, which is the norm for the Riemannian metric of the considered element of $F(x)$. Then this norm depends in a measurable way on $x$. Let us now notice the following fact: $\mu$ being ergodic for the flow $(\varphi_t)$, there exists a dense $G_\delta$ subset $A$ of ${\mathbb {R}}$ such that, for every $t\in A$, the diffeomorphism $\varphi_t$ is ergodic. As it is simpler for us to work with a diffeomorphism instead of a flow, we fix such a $t\in A$. We assume that $t=1$ (if not we replace $H$ by $\frac{1}{t}H$).\ \[LJ\] For every $\varepsilon >0$, there exists a measurable subset $J_\varepsilon$ of $N$ such that: 1. $\mu (J_\varepsilon)\geq 1-\varepsilon$; 2. on $J_\varepsilon$, $(s_n^+)$ and $(s_n^-)$ converge uniformly ; 3. there exists two constants $\beta=\beta(\varepsilon)>\alpha=\alpha(\varepsilon)>0$ such that: $\forall x\in J_\varepsilon,\beta{\bf 1}\geq -s_-(x)\geq \alpha {\bf 1}$ where $g_-$ is the graph of $s_-$. This is a consequence of Egorov theorem and of the fact that on $N$, $g_+$ and $g_-$ are transverse and then $-s_-$ is positive definite. We deduce: \[LCVU\] Let $J_\varepsilon$ be as in the previous lemma. On the set $\{ (n,x)\in{\mathbb {N}}\times J_\varepsilon, \varphi_n(x)\in J_\varepsilon\}$, the sequence of conorms $(m(b_n(x))$ converge uniformly to $+\infty$, where $m(b_n)=\| b_n^{-1}\| ^{-1}$. Let $n, x$ be as in the lemma.\ The matrix $M_n(x)=\begin{pmatrix} -b_n(x)s_n^-(x)& b_n(x)\\ 0& s_n^+(\varphi_nx)b_n(x)\\ \end{pmatrix} $ being symplectic, we have:\ $-s_n^-(x){}^tb_n(x)s_n^+(\varphi_nx)b_n(x)={\bf 1}$ and thus $-b_n(x)s_n^-(x){}^tb_n(x)s_n^+(\varphi_nx)={\bf 1}$ and:\ $b_n(x)s_n^-(x){}^tb_n(x)=-\left(s_n^+(\varphi_nx)\right)^{-1}$.\ We know that on $J_\varepsilon$, $(s_n^+)$ converges uniformly to zero. Hence, for every $\delta>0$, there exists $N=N(\delta) $ such that: $n\geq N\Rightarrow \| s_n^+(\varphi_nx)\|\leq \delta$. Moreover, we know that $\| s_n^-(x)\|\leq \beta$. Hence, if we choose $\delta'=\frac{\delta^2}{\beta}$, for every $n\geq N=N(\delta')$ and $x\in J_\varepsilon$ such that $\varphi_nx\in J_\varepsilon$, we obtain: $$\forall v\in{\mathbb {R}}^p,\beta \| {}^tb_n(x)v\|^2= {}^tv b_n(x)(\beta{\bf 1}){}^tb_n(x)v\geq - {}^tv b_n(x)s_n^-(x){}^tb_n(x)v={}^tv\left(s_n^+(\varphi_nx)\right)^{-1}v$$ and we have: ${}^tv\left(s_n^+(\varphi_nx)\right)^{-1}v\geq \frac{\beta}{\delta^2}\| v\|^2$ because $s_n^+(\varphi_nx)$ is a positive definite matrix that is less than $\frac{\delta^2}{\beta}{\bf 1}$. We finally obtain: $\| {}^tb_n(x)v\|\geq \frac{1}{\delta}\| v\|$ and then the result that we wanted. From now we fix a small constant $\varepsilon>0$, associate a set $J_\varepsilon$ with $\varepsilon$ via lemma \[LJ\] and two constants $0<\alpha<\beta$; then there exists $N\geq 0$ such that $$\forall x\in J_\varepsilon, \forall n\geq N, \varphi_n(x)\in J_\varepsilon\Rightarrow m(b_n(x))\geq \frac{2}{\alpha}.$$ Let $J_\varepsilon$ be as in lemma \[LJ\]. For $\mu$-almost point $x$ in $J_\varepsilon$, there exists a sequence of integers $(j_n)=(j_n(x))$ tending to $+\infty$ such that: $$\forall n\in {\mathbb {N}}, m(b_{j_n}(x)s_{j_n}(x))\geq \left( 2^\frac{1-\varepsilon}{2N}\right)^{j_n}.$$ As $\mu$ is ergodic for $\varphi_1$, we deduce from Birkhoff ergodic theorem that for almost every point $x\in J_\varepsilon$, we have: $$\lim_{\ell\rightarrow +\infty}\frac{1}{\ell}\sharp \{ 0 \leq k\leq \ell-1; \varphi_k(x)\in J_\varepsilon\}=\mu (J_\varepsilon)\geq 1-\varepsilon.$$ We introduce the notation: $N(\ell)=\sharp \{ 0 \leq k\leq \ell-1; \varphi_k(x)\in J_\varepsilon\}$.\ For such an $x$ and every $\ell\in{\mathbb {N}}$, we find a number $n(\ell)$ of integers: $$0=k_1\leq k_1+N\leq k_2\leq k_2+N\leq k_3\leq k_3+N\leq \dots \leq k_{n(\ell)}\leq \ell$$ such that $\varphi_{k_i}(x)\in J_\varepsilon$ and $n(\ell)\geq [\frac{N(\ell)}{N}]\geq \frac{N(\ell)}{N}-1$. In particular, we have: $\frac{n(\ell)}{\ell}\geq\frac{1}{N}(\frac{N(\ell)}{\ell}-\frac{N}{\ell})$, the right term converging to $\frac{\mu (J_\varepsilon)}{N}\geq \frac{1-\varepsilon}{N}$ when $\ell$ tends to $+\infty$. Hence, for $\ell$ large enough, we find: $n(\ell)\geq 1+ \ell \frac{1-\varepsilon}{2N}$.\ As $\varphi_{k_i}(x)\in J_\varepsilon$ and $k_{i+1}-k_i\geq N$, we have: $m(b_{k_{i+1}-k_i}(\varphi_{k_i}(x)))\geq \frac{2}{\alpha}$. Moreover, we have: $m(s_{k_{i+1}-k_i}^-(\varphi_{k_i}x))\geq \alpha$; hence: $$m(b_{k_{i+1}-k_i}(\varphi_{k_i}x)s_{k_{i+1}-k_i}^-(\varphi_{k_i}x))\geq 2.$$ But the matrix $-b_{k_{n(\ell)}}(x)s^-_{k(n(\ell))}(x)$ is the product of $n(\ell)-1$ such matrix. Hence: $$m(b_{k_{n(\ell)}}(x)s^-_{k(n(\ell))}(x))\geq 2^{n(\ell)-1}\geq 2^{\ell\frac{1-\varepsilon}{2N}}\geq \left( 2^\frac{1-\varepsilon}{2N}\right)^{k_{n(\ell)}}.$$ Let us now come back to the whole tangent space $T_x(T^*M)$ with a slight change in the coordinates that we use. We defined the symplectic coordinates $(q_1, \dots , q_n, p_1, \dots , q_n)$ and now we use the non symplectic ones:\ $(Q_1, \dots, Q_n,P_1, \dots , P_n)=(q_{n-p+1}, \dots, q_n, q_1, \dots, q_{n-p}, p_1, \dots , p_n)$. Then: 1. $(Q_1, \dots , Q_p)$ are coordinates in $R(x)$; 2. $(Q_1, \dots , Q_n)$ are coordinates in $G_+(x)$; 3. $(Q_1, \dots , Q_n, P_{1}, \dots , P_{n-p})$ are coordinates of $E(x)=G_+(x)+G_-(x)$. We write then the matrix of $D\varphi_t(x)$ in these coordinates $(Q_1, \dots , Q_n, P_1, \dots s , P_n)$ (which are not symplectic): $$\begin{pmatrix} A^1_t(x)&A^2_t(x)&A^3_t(x)&A^4_t(x)\\ 0&b_t(x)s_t^-(x)&b_t(x)&A^5_t(x)\\ 0&0& s_t^+(\varphi_tx)b_t(x)&A^6_t(x)\\ 0&0&0&A^9_t(x)\\ \end{pmatrix}$$ where the blocks correspond to the decomposition $T_x(T^*M)=E_1(x)\oplus E_2(x)\oplus E_3(x)\oplus E_4(x)$ with $\dim E_1(x)=\dim E_4(x)=p$ and $\dim E_2(x)=\dim E_3(x)=n-p$.\ We have noticed that $E_1(x)=E(x)\subset E^c(x)$ and that $G_+(x)=E_1(x)\oplus E_2(x)$.\ If $x\in J_\varepsilon$, we have found a sequence $(j_n)$ of integers tending to $+\infty$ so that: $$\forall n\in {\mathbb {N}}, m(b_{j_n}(x)s^-_{j_n}(x))\geq \left( 2^\frac{1-\varepsilon}{2N}\right)^{j_n}.$$ We deduce: $$\forall v\in E_2(x)\backslash \{ 0\}, \frac{1}{j_n}\log\left( \| b_{j_n}(x)s^-_{j_n}(x)v\|\right)\geq \frac{1-\varepsilon}{2N}\log 2 +\frac{\|v\|}{j_n};$$ and because $E_1(x)\subset E^c(x)$: $$\forall v\in G_+(x)\backslash E_1(x), \liminf_{n\rightarrow \infty}\frac{1}{n}\log \| D\varphi_n(x)v\|\geq \frac{1-\varepsilon}{2N}\log 2.$$ Hence there are at least $n-p$ Lyapunov exponents bigger than $ \frac{1-\varepsilon}{2N}\log 2$ and then bigger than $0$ for the linearized flow. Because this flow is symplectic, we deduce that it has at least $n-p$ negative Lyapunov exponents (see [@bochi-viana1]). As we noticed that the linearized flow has at least $2p$ zero Lyapunov exponents, we deduce that $\mu$ has exactly $n-p$ positive Lyapunov exponents, exactly $n-p$ negative Lyapunov exponents and exactly $2p$ zero Lyapunov exponents.\ This finishes the proof of theorem \[thdimlyap\]. Let us notice that we proved too that for $x\in N$ (i.e. generic in the Oseledet’s sense), we have: $E^u(x)\subset G_+(x)$, and then $G_+(x)=E^u(x)\oplus R(x) $ Weak K.A.M. solutions and Green bundles {#sec3} ======================================= In this section, we recall the weak KAM theory and give a relationship between some tangent cones to the pseudographs of the weak KAM solutions and the Green bundles. These results imply theorem \[th1\]. The proofs are given in section \[sec4\]. Weak KAM theory --------------- We don’t give any proof in this section, but all the results that we give are proved in [@Fa1] or [@Be1]. If $t>0$, the function $A_t~: M\times M\rightarrow {\mathbb {R}}$ is defined by: $$A_t(q_0, q_1)=\inf_\gamma \int_0^tL(\gamma (s), \dot\gamma (s))ds=\min_\gamma \int_0^tL(\gamma (s), \dot\gamma (s))ds$$ where the infimum is taken on the set of $C^2$ curves $\gamma~: [0, t]\rightarrow M$ such that $\gamma (0)=q_0$ and $\gamma (t)=q_1$. 1. A function $v~: V\rightarrow {\mathbb {R}}$ defined on a subset $V$ of ${\mathbb {R}}^d$ is [*$K$-semi-concave*]{} if for every $x\in V$, there exists a linear form $p_x$ defined on ${\mathbb {R}}^d$ so that: $$\forall y\in V, v(y)\leq v(x)+p_x(y-x)+K\| y-x\|^2.$$ Then we say that $p_x$ is a [*K-super-differential*]{} of $v$ at $x$. 2. Let us fix a finite atlas ${\cal A}$ of the manifold $M$; a function $u~: M\rightarrow {\mathbb {R}}$ is [*$K$-semi-concave*]{} if for every chart $(U, \phi)$ belonging to ${\cal A}$, $u\circ\phi^{-1}$ is $K$-semi-concave. Then a [*$K$-super-differential*]{} of $u$ at $q$ is $p_x\circ D\phi(q)$ where $p_x$ is a $K$-super-differential of $u\circ \phi^{-1}$ at $x=\phi(q)$. A semi-concave function is always Lipschitz and then differentiable almost everywhere and for such a function, we define its pseudograph: a [*pseudograph*]{} is the graph ${\cal G}(du)$ of $du$, where $u~: M\rightarrow {\mathbb {R}}$ is a semi-concave function.\ A function $u~: M\rightarrow {\mathbb {R}}$ is $K$-semi-convex if $-u$ is $K$-semi-concave. We have a notion of sub-differential and the anti-pseudograph of a semi-convex function $u$ is ${\mathcal {G}}(du)$. It is proved in [@Be1] that $A_t$ is semi-concave and that for every minimizing curve $\gamma~: [0, t]\rightarrow M$ between $q_0$ and $q_1$, $(-\frac{\partial L}{\partial v}(\gamma (0), \dot\gamma (0)), \frac{\partial L}{\partial v}(\gamma (t), \dot\gamma (t)))$ is a super-differential of $A_t$ at $(q_0, q_1)$. It is proved, too, that $A_t(.,q_1)$ is differentiable at $q_0$ if, and only if, $A_t(q_0,.)$ is differentiable at $q_1$ if, and only if, there exists a unique minimizing curve $\gamma~: [0, t]\rightarrow M$ joining $q_0$ to $q_1$.\ We denote the two Lax-Oleinik semi-groups associated with $L$ by $(T_t)_{t>0}$ and $(\breve T_t)_{t>0}$; for $u\in C^0(M, {\mathbb {R}})$ , they are defined by: $$T_tu(q)=\min_{q'\in M} (u(q')+A_t(q',q))\ {\rm and}\ \breve T_tu(q)=\max_{q'\in M} (u(q')-A_{t}(q, q'))$$ A function $u~: M\rightarrow {\mathbb {R}}$ is a negative (resp. positive) weak KAM solution if there exists $c\in{\mathbb {R}}$ such that: $\forall t>0, T_tu=u-ct$ (resp. $\forall t>0, \breve T_t u=u+ct$). Then there exist at least one positive and one negative weak K.A.M. solutions (see [@Fa1] or [@Be1]). The constant $c$ is unique and is called Mañé’s critical value. If $u_-$ is a negative weak KAM solution and $u_+$ a positive one, then $u_-$ is semi-concave and $u_+$ is semi-convex. Let us introduce the Mather set: The Mather set, denoted by ${\mathcal {M}}^*(H)$, is the union of the supports of the minimizing measures. The projected Mather set is ${\mathcal {M}}(H)=\pi({\mathcal {M}}^*(H))$. J. Mather proved that ${\mathcal {M}}^*(H)$ is compact, non-empty and that it is a Lipschitz graph above a compact part of the zero-section of $T^*M$. A. Fathi proved in [@Fa1] that if $u_-$ is a negative weak KAM solution, there exists a unique positive weak KAM solution $u_+$ such that $u_{-|{\mathcal {M}}(H)}=u_{+|{\mathcal {M}}(H)}$. Such a pair $(u_-, u_+)$ is called a pair of conjugate weak KAM solutions. For such a pair, we have: 1. $\forall q\in {\mathcal {M}}(H), u_-(q)=u_+(q)$; let us denote the set of equality: ${\mathcal {I}}(u_-, u_+)=\{ q; u_-(q)=u_+(q)\}$ by ${\mathcal {I}}(u_-,u_+)$; then ${\mathcal {M}}(H)\subset {\mathcal {I}}(u_-, u_+)$; 2. $u_-$ and $u_+$ are differentiable at every point $q\in {\mathcal {I}}(u_-, u_+)$; for such a $q$ we have $(q, du_-(q))\in {\mathcal {N}}^*(H)$; when $q\in{\mathcal {M}}(H)$ and $(q, p)\in {\mathcal {M}}^*(H)$ is its lift to ${\mathcal {M}}^*(H)$, then $ du_-(q)= du_+(q)=p$; 3. $u_+\leq u_-$. Moreover, it is proved in [@Be1] that if $q$ is a point of differentiability of $T_tu$ (resp. $\breve T_tu$), then the minimum (resp. maximum) in the definition of $T_tu(q)$ (resp. $\breve T_tu$) is attained at a unique $q'$ and there is a unique curve $\gamma~: [0, t]\rightarrow M$ minimizing between $q'$ and $q$ (resp. $q$ and $q'$); in this case: $\frac{\partial L}{\partial v}(q, \dot\gamma (t))=dT_tu(q)$ (resp. $\frac{\partial L}{\partial v}(q, \dot\gamma (0))=d\breve T_tu(q)$). Comparison between the weak KAM solutions and the Green bundles ---------------------------------------------------------------- If $(u_-, u_+)$ is a pair of conjugate weak KAM solutions, if $q\in {\mathcal {I}}(u_-, u_+)$, we have seen that $(q, du_-(q))=(q, du_+(q))\in{\mathcal {N}}^*(H)$. Hence, the two Green subspaces $G_-(q, du_-(q))$ and $G_+(q, du_+(q))$ exist. Let us introduce two other Lagrangian subspaces: If the orbit of $x$ is minimizing, if $G_-(x)$ is the graph of the symmetric matrix $s_-(x)$ and $G_+(x)$ the graph of the symmetric matrix $s_+(x)$, we denote the graph of $\tilde s_-(x)= 2s_-(x)-s_+(x)$ (resp. $\tilde s_+(x)= 2s_+(x)-s_-(x)$) by $\tilde G_-(x)$ (resp. $\tilde G_+(x)$).\ If $\Delta s(x)= s_+(x)- s_-(x)$, then $\Delta s (x)$ is positive semi-definite and we have: $\tilde s_-=s_- -\Delta s$ and $\tilde s_+=s_++\Delta s$.\ Moreover, if $s$ is a positive semi-definite matrix, we will denote by $p_s$ the orthogonal projection on its image ${\mathrm{Im}}(s)$ and by $\Lambda (s)$ is greatest eigenvalue: $\Lambda (s)=\| s\|$. Let us notice that $G_-(x)=G_+(x)$ if, and only if, $\tilde G_-(x)=G_-(x)=G_+(x)=\tilde G_+(x)$. Moreover, we always have: $\tilde G_-(x)\leq G_-(x)\leq G_+(x)\leq \tilde G_+(x)$. The bundle $\tilde G_-$ is lower semi-continuous and the bundle $\tilde G_+ $ is upper semi-continuous, and they are continuous at the points where $G_-=G_+$. Let us recall that if $x\in A\subset T^*M$, ${\mathcal {C}}_xA$ designates the contingent cone to $A$ at $x$, that was defined in the introduction. \[greenkam\] Let $(u_-, u_+)$ be a pair of conjugate weak KAM solutions and let $q$ belong to ${\mathcal {I}}(u_-, u_+)$. Then we have: $\forall (X, Y)\in {\mathcal {C}}_{(q, du_-(q))}{\mathcal {G}}(du_-), $ $$\| Y-\tilde s_-(q, du_-(q))X\|\leq 2\sqrt{\|\Delta s(q, du_-(q))\|} .\sqrt{\Delta s(q, du_-(q))(X,X)}$$ $$\leq 2\Lambda (\Delta s(q, du_-(q))). \| p_{\Delta s(q, du_-(q))}(X)\|$$ and: $\forall (X, Y)\in {\mathcal {C}}_{(q, du_+(q))}{\mathcal {G}}(du_+), $ $$\| Y-\tilde s_+(q, du_+(q))X\|\leq 2\sqrt{\|\Delta s(q, du_+(q))\|} .\sqrt{\Delta s(q, du_+(q))(X,X)}$$ $$\leq 2\Lambda (\Delta s(q, du_+(q))). \| p_{\Delta s(q, du_+(q))}(X)\|$$ We postpone the proof of this theorem to section \[sec4\].\ As ${\mathcal {M}}^*(H)\subset {\mathcal {G}}(du_-)\cap {\mathcal {G}}(du_+)$, we deduce: \[cormes\] If $x$ is an element of ${\mathcal {M}}^*(H)$, then we have: $\forall (X, Y)\in {\mathcal {C}}_x{\mathcal {M}}^*(H)$, $$\max \{ \| Y-\tilde s_-(x)X\|, \| Y-\tilde s_+(x)X\|\}\leq 2\sqrt{\|\Delta s(x)\|} .\sqrt{\Delta s (x) (X,X)}\leq 2\Lambda (\Delta s(x)). \| p_{\Delta s(x)}(X)\|$$ Now, we use theorem \[thdimlyap\]: if $\mu$ is an ergodic minimizing measure whose Lyapunov exponents are zero, then we have $\mu$-almost everywhere: $G_-=G_+$ i.e. $\Delta s=0$. We deduce from corollary \[cormes\] that ${\mathcal {C}}_x({\rm supp} \mu) \subset G_-(x)=G_+(x)$ at $\mu$ almost every point. This implies that ${\rm supp}\mu$ is $1$-regular at $x$, and even that it is $C^1$-regular at $x$. Indeed, if $(x_n)$ is a sequence of points of ${\rm supp}(\mu)$ that converges to $x$ and $v_n=(X_n, Y_n)\in {\mathcal {C}}_{x_n}({\rm supp}\mu)$ converges to $v=(X,Y)$, we have for every $n$: $$\| Y_n-\tilde s_-(x_n)X_n\|\leq 2\sqrt{\Delta s(x_n)}\sqrt{\Delta s(x_n)(X_n, X_n)}.$$ As $G_-(x)=G_+(x)$, $\tilde s_-$ and $\Delta s$ are continuous at $x$. We deduce that $\| Y-s_-(x)Y\|=0$ and then $(X, Y)\in G_-(x)$. We have then proved: If $\mu$ is an ergodic minimizing measure all of whose Lyapunov exponents are zero, then, ${\rm supp}\mu$ is $C^1$ regular at $\mu$-almost every point. This is exactly theorem \[th1\]. Proof of the results of section \[sec3\] {#sec4} ======================================== In this section, we use the images of the physical verticals to obtain a control of the weak KAM solutions. More precisely, we can choose a graph in the image of a vertical, the graph of $da$ for a certain function $a$, and prove a certain inequality between $a$ and the considered weak KAM solution $u$. Then we deduce an inequality along some subset of the Mañé set between the “second derivatives” of $a$ and $u$. This gives a relationship between the Green bundles and the Bouligand’s contingent cones to the pseudograph of any weak KAM solution along some subset of the Mañé set . Selection of some graphs in the images of the verticals ------------------------------------------------------- 1. If $q\in M$, we denote the (physical) vertical $\pi^{-1}(\{ q\} )$ by ${\mathcal {V}}(q)\subset T^*M$.\ 2. If $t>0$, the function $A_t~: M\times M\rightarrow {\mathbb {R}}$ is defined by: $$A_t(q_0, q_1)=\inf_\gamma \int_0^tL(\gamma (s), \dot\gamma (s))ds=\min_\gamma \int_0^tL(\gamma (s), \dot\gamma (s))ds$$ where the infimum is taken on the set of $C^2$ curves $\gamma~: [0, t]\rightarrow M$ such that $\gamma (0)=q_0$ and $\gamma (t)=q_1$. 3. if $u~: M\rightarrow {\mathbb {R}}$ is a Lipschitz function, then by Rademacher’s theorem, it is differentiable (Lebesgue) almost everywhere and the graph of its derivative is denoted by: $${\mathcal {G}}(du)= \{ (q, du(q)); u\ {\rm is} \ {\rm differentiable}\ {\rm at}\ q\} .$$ Tonelli’s theorem asserts that for every $t\not=0$, $\pi\circ\varphi_t({\mathcal {V}}(q))=M$ (i.e. for every $q'\in M$ there exists a solution to the Euler-Lagrange equations $\gamma$ such that $\gamma (0)=q$ and $\gamma (t)=q'$); but in general $\varphi_t({\mathcal {V}}(q))$ is not a graph. To select a graph in $\varphi_t({\mathcal {V}}(q))$, we prove: Let $H~: T^*M\rightarrow {\mathbb {R}}$ be a Tonelli Hamiltonian and $L~: TM\rightarrow {\mathbb {R}}$ be the associated Lagrangian. Then for every $t>0$ and every $q\in M$, the function $v_q^t=A_t(q,.)$ and $v_q^{-t}= A_t(.,q)$ are semi-concave, and satisfy: $${\mathcal {G}}(dv_q^t) \subset \varphi_t({\mathcal {V}}(q))\ {\rm and}\ {\mathcal {G}}(-dv_q^{-t})\subset \varphi_{-t}({\mathcal {V}}(q)).$$ Because $A_t$ is semi-concave, the two functions $v^t_q $ and $v^{-t}_q$ are semi-concave and then Lipschitz. By Rademacher’s theorem they are differentiable almost everywhere.\ Moreover, if $q_0$ is a point where $v^t_q$ is differentiable, then $v^t_q$ has exactly one super-differential at this point, there is only one minimizing arc $\gamma$ joining $(0, q)$ to $(t, q_0)$, and we have: 1. $dv_q^t(q_0)=\frac{\partial L}{\partial v}(\gamma (t), \dot\gamma (t))$; 2. $(\gamma(0), \frac{\partial L}{\partial v}(\gamma (0), \dot\gamma (0)))=(q, \frac{\partial L}{\partial v}(\gamma (0), \dot\gamma (0)))\in {\mathcal {V}}(q)$; 3. $\varphi_t \left( q, \frac{\partial L}{\partial v}(\gamma (0), \dot\gamma (0))\right)=(\gamma (t), \frac{\partial L}{\partial v}(\gamma (t), \dot\gamma (t))) =(q_0, dv_q^t(q_0))$. Then we have proved that: $\varphi_t({\mathcal {V}}(q))\supset {\mathcal {G}}(dv_q^t)$. Hence, we have selected a pseudograph in the image $\varphi_t({\mathcal {V}}(q))$ of the vertical.\ In a very similar way, we may see that the anti-pseudograph of the semi-convex function $-v_q^{-t}$ is a subset of $\varphi_{-t}({\mathcal {V}}(q))$: $ {\mathcal {G}}(-dv_q^{-t})\subset \varphi_{-t}({\mathcal {V}}(q))$. Local smoothness of some of these graphs ----------------------------------------- For every $x\in T^*M$, we denote the [*linear*]{} vertical at $x$ by $V(x)$: $V(x)=\ker D\pi (x)=T_x{\mathcal {V}}(\pi(x))\subset T_x(T^*M)$.\ The images of the linear vertical are denoted by: $G_t(x)=D\varphi_tV(\varphi_{-t}x)$. We recall that an orbit piece $(\varphi_t(x))_{t\in[a, b]}$ has no conjugate vectors if: $$\forall s\not=t\in[a,b], G_{t-s}(\varphi_tx)\cap V(\varphi_tx)=D\varphi_{t-s}(V(\varphi_s(x))\cap V(\varphi_t(x))=\{ 0\}.$$ Let us now fix a minimizing arc $\gamma~: [-t, 0]\rightarrow M$ such that: 1. there is only one minimizing arc between $(-t, \gamma(-t))$ and $(0, \gamma(0))$ (then it is $\gamma$); 2. the orbit piece $\left( \gamma(\tau), \frac{\partial L}{\partial v}(\gamma (\tau),\dot\gamma (\tau))\right)_{\tau\in[-t, 0]}$ has no conjugate vectors. Let us notice that when $(q, p)\in{\mathcal {N}}^*(H)$, then any piece of the curve $(t\rightarrow \pi\circ\varphi_t(q,p))$ satisfies the previous hypotheses.\ We define a function $a_t^+~: M\rightarrow {\mathbb {R}}$ by: $a_t^+(q)=v_{\gamma(-t)}^t(q)=A_t(\gamma(-t), q)$ (this function depends on $\gamma$).\ In a similar way, we can consider $x_0=(q_0, p_0)$ such that the orbit $(\varphi_s(x_0))_{s\in[0, t]}$ has no conjugate points and so that there is only one minimizing arc $\gamma~: [0, t]\rightarrow M$ joining $q_0$ to $q_t$. We define a function $ a_t^- ~: M\rightarrow {\mathbb {R}}$ by: $ a_t^-=-v^{-t}_{q_{t}}(q)=-A_t(q, q_t)$. Let $\gamma~: [-t, 0]\rightarrow M$ (resp. $\gamma~: [0, t]\rightarrow M$) be a minimizing arc such that: 1. $\gamma$ is the only minimizing arc joining its two ends; 2. the orbit piece $(\gamma, \frac{\partial L}{\partial v}(\gamma, \dot\gamma))$ has no conjugate vectors. Then there exists a neighborhood $V_0$ of $q_0=\gamma (0)$ in $M$ such that $a^+_{t|V_0}$ (resp. $a_{t|V_0}^-$) is as regular as $H$ is (then at least $C^2$). We have seen that: ${\mathcal {G}}(da_t^+)\subset \varphi_t(V(q_{-t}))$. Let us now prove that $a_t^+$ is smooth near $q_0$. We use now the so-called “a priori compactness lemma” (see [@Fa1]) that says to us that there exists a constant $K_t=K>0$ such that the velocities $(\dot\gamma (s))_{s\in[0, t]}$ of any minimizing arc between any points $q\in M$ and $q'\in M$ are bounded by $K$; hence if we denote the set of the minimizing arcs that are parametrized by $[0, t]$ by ${\mathcal {K}}$, ${\mathcal {K}}$ is a compact set for the $C^1$ topology because it is the image by the projection $\pi$ of a closed set of bounded orbits. Let us denote the set of $\gamma\in {\mathcal {K}}$ such that $\gamma (0)=q_{-t}$ by ${\mathcal {K}}_0$; then ${\mathcal {K}}_0$ is compact. Let us introduce another notation: ${\mathcal {K}}(q)=\{ \gamma\in {\mathcal {K}}_0; \gamma (t)=q\}$. Then ${\mathcal {K}}(q_0)=\{\gamma_0\}$ and hence, because ${\mathcal {K}}_0$ is closed, for $q$ close enough to $q_0$, all the elements of ${\mathcal {K}}(q)$ are $C^1$ close to $\gamma_0$. Moreover, $\varphi_t({\mathcal {V}}(q_{-t}))$ is a sub-manifold of $M$ that contains $(q_0, \frac{\partial L}{\partial v}(q_0, \dot\gamma_0(0)))=(q_0, p_0)$. Its tangent space at $(q_0, p_0)$ is $G_t(q_0, p_0)$, which is transverse to the vertical because $(q_s,p_s)_{s\in [-t, 0]}$ has no conjugate vectors. Hence, the manifold $\varphi_t({\mathcal {V}}(q_{-t}))$ is, in a neighborhood $U_0$ of $(q_0, p_0)$, the graph of a $C^1$ section of $T^*M$ defined on a neighborhood $V_0$ of $q_0$ in $M$. Moreover, because this sub-manifold is Lagrangian (indeed, ${\mathcal {V}}(q_{-t})$ is Lagrangian and $\varphi_t$ is symplectic), it is the graph of $du_0$ where $u_0~: V_0\rightarrow {\mathbb {R}}$ is a $C^2$ function. Now, if $q$ is close enough to $q_0$, we know that all the elements $\gamma$ of ${\mathcal {K}}(q)$ are $C^1$ close to $\gamma_0$, and then that $(q, \frac{\partial L}{\partial v}(\gamma (t), \dot\gamma (t)))$ belongs to the neighborhood $U_0$ of $(q_0, p_0)= (q_0, \frac{\partial L}{\partial v}(\gamma_0 (t), \dot\gamma_0 (t)))$ and to $\varphi_t({\mathcal {V}}(q_{-t}))$. Because $\varphi_t({\mathcal {V}}(q_{-t}))\cap U_0$ is a graph, this element is unique: ${\mathcal {K}}(q)$ has only one element and $a_t^+$ is differentiable at $q$, with $da_t^+(q)= \frac{\partial L}{\partial v}(\gamma (t), \dot\gamma (t)))=du_0(q) $ . We deduce that near $q_0$, on the set of differentiability of $a_t^+$, $da_t^+$ is equal to $du_0$; because $a_t^+$ and $u_0$ are Lipschitz on $V_0$ and their differentials are equal almost everywhere, we deduce that on $V_0$, $a_t^+-u_0$ is constant. Hence, on a neighborhood $V_0$ of $q_0$, $a_t^+$ is $C^2$. In a similar way, using the fact that $ {\mathcal {G}}(da_t^-)\subset \varphi_{-t}(V(q_t))$, we obtain that $a_t^-$ is $C^2$ near $q_0$. If $x_0=(q_0, p_0)$ is a point of the Mañé set, $(q_t,p_t)_{t\in{\mathbb {R}}}=(\varphi_t(q_0, p_0))_{t\in{\mathbb {R}}}$ has no conjugate vectors and for every $t<\tau$, there is only one minimizing arc $\gamma~:[t, \tau]\rightarrow M$ joining $q_t$ to $q_\tau$, hence for every $t>0$ the two functions $a_{x_0, t}^+$ and $a_{x_0, t}^-$ are smooth near $q_0$ (of course the neighborhood of $q_0$ where they are smooth depends on $t$). Comparison between the weak K.A.M. solutions and the maps $a_t^+$ and $a_t^-$ ----------------------------------------------------------------------------- We assume that $u_-$ is a negative weak K.A.M. solution and that $u_+$ is a positive weak K.A.M. solution. Let $q_0\in M$ be a point of differentiability of $u_-$ (resp. $u_+$) and $a_t^+$ (resp. $a_t^-$) be the function built in the previous subsection for the arc $\gamma=(\pi\circ \varphi_s(q_0, du_-(q_0)))_{s\in [-t, 0]}$ (resp. $\gamma=(\pi\circ \varphi_s(q_0, du_+(q_0)))_{s\in [0, t]}$). Then, in a chart: $u_-(q)-u_-(q_0)-du_-(q_0)(q-q_0)\leq a_t^+(q)-a_t^+(q_0)-da_t^+(q_0)(q-q_0)$ (resp. $a_t^-(q)-a_t^-(q_0)-da_t^-(q_0)(q-q_0)\leq u_+(q)-u_+(q_0)-du_+(q_0)(q-q_0)$). Let us consider $q_0$ in $M$ that is a point of differentiability of a weak K.A.M. solution $u_-$ and let us denote the point above $q_0$ on the pseudograph ${\mathcal {G}}(du_-)$ of $u_-$ by $x_0$: $x_0=(q_0, du_-(q_0))$. Then, for every $t>0$, because $T_tu_-=u_--ct$ is differentiable at $q_0$, there is only one point $q\in M$ such that $u_-(q_0)=T_tu_-(q_0)+ct=u(q)+A_t(q, q_0)+ct$ and only one minimizing arc $\gamma~: [-t, 0]\rightarrow M$ joining $q$ to $q_0$. We introduce the notation: $x_t=(q_t, p_t)=\varphi_t(x_0)$. Then: $T_tu_-(q_0)=u_-(q_{-t})+A(q_{-t}, q_0) $; moreover: $T_tu_-(q)\leq u_-(q_{-t})+A(q_{-t}, q) =T_tu_-(q_0)+A(q_{-t}, q)-A(q_{-t}, q_0)$. Finally: $u_-(q)-u_-(q_0)\leq a_t^+(q)-a_t^+(q_0)$. Because these two maps $a_t^+$ and $u_-$ are differentiable at $q_0$, they have the same differential at this point and we obtain (in chart): $u_-(q)-u_-(q_0)-du_-(q_0)(q-q_0)\leq a_t^+(q)-a_t^+(q_0)-da_t^+(q_0)(q-q_0)$. Using the same argument for $u_+$, we obtain: if $q_0$ is a point of differentiability of $u_+$:\ $a_t^-(q)-a_t^-(q_0)-da_t^-(q_0)(q-q_0)\leq u_+(q)-u_+(q_0)-du_+(q_0)(q-q_0)$. Now we would like to use these inequalities at different points $q_0$; we have to be careful, because $a_t^+$ and $a_t^-$ depend on the point $q_0$ we choose. That is why we change now our notation, replacing $a_t^+$ by $a_{q_0, t}^+$ if the considered point is $(q_0, du_-(q_0))$ and $a_t^-$ by $a_{q_0, t}^-$ if the considered point is $(q_0, du_+(q_0))$. We assume that $u_-$ is a negative weak K.A.M. solution and that $u_+$ is a positive weak K.A.M. solution. Let $y\in{\mathcal {I}}(u_-, u_+)$ be a point, $(x_n)$ be a sequence of points of $M$ converging to $y$, and $(t_n)$ be a sequence of positive real numbers so that the two limits (written in charts) $\displaystyle{\lim_{n\rightarrow \infty} \frac{x_n-y}{t_n}=X}$ and $\displaystyle{Y=\lim_{n\rightarrow \infty} \frac{du_-(x_n)-du_-(y)}{t_n} }$ (resp. $\displaystyle{ \lim_{n\rightarrow \infty} \frac{du_+(x_n)-du_+(y)}{t_n} }$) exist. Then we have: $$\forall k\in{\mathbb {R}}^n, Y.k \leq \frac{1}{2}(d^2a_{y,t}^+(y)(k,k)+d^2a_{y,t}^+(y)(X,X)-d^2a_{y,t}^-(y)(X-k,X-k))$$ (resp: $$\forall k\in{\mathbb {R}}^n, \frac{1}{2}(d^2a_{y,t}^-(y)(k,k)+d^2a_{y,t}^-(y)(X,X)-d^2a_{y,t}^+(y)(k-X,k-X))\leq Y.k)$$ We work in a chart, and we have, if $ y\in {\mathcal {I}}(u_-, u_+)$ and $x$ is a point of differentiability of $u_-$: 1. $u_-(x+h)-u_-(x)-du_-(x)h\leq a_{x,t}^+(x+h)-a_{x,t}^+(x)-da_{x,t}^+(x)h$; 2. $u_-(x)-u_-(y)-du_-(y)(x-y)\leq a_{y,t}^+(x)-a_{y,t}^+(y)-da_{y,t}^+(y)(x-y)$; 3. $a^-_{y,t}(x+h)-a^-_{y,t}(y)-da_{y,t}^-(y)(x+h-y)\leq u_+(x+h)-u_+(y)-du_+(y)(x+h-y)$. Hence, by adding these three inequalities and using that $u_-(y)=u_+(y)$, $du_-(y)=du_+(y)$ and $u_+\leq u_-$:\ $(du_-(y) -du_-(x))h \leq a_{x,t}^+(x+h)-a_{x,t}^+(x)-da_{x,t}^+(x)h+a_{y,t}^+(x)-a_{y,t}^+(y)-da_{y,t}^+(y)(x-y) -a^-_{y,t}(x+h)+a^-_{y,t}(y)+da_{y,t}^-(y)(x+h-y)$.\ We now need to precise the regularity of the maps: $x\rightarrow da^-_{x,t}$ and $x\rightarrow da^+_{x,t}$. To do that, we prove a lemma. We fix a finite atlas of $M$ to write that $u_-$ is $K$-semi-concave and that $u_+$ is $K$-semi-convex. The proof is very similar to the one given by A. Fathi in [@Fa1] to prove that the Aubry set is a Lipschitz graph. \[Aubrylipschitz\]There exists a constant $K>0$ such that, for every $y\in{\mathcal {I}}(u_-, u_+)$ and every $x\in M$ where $u_-$ (resp. $u_+$) is differentiable, then $\| du_-(y)-du_-(x)\|\leq K\| y-x\|$ (resp. $\| du_+(y)-du_+(x)\|\leq K\| y-x\|$ ). In particular, $du_-$ and $du_+$ are continuous at every point of ${\mathcal {I}}(u_-, u_+)$. Because $u_+\leq u_-$, $u_-$ is semi-concave and $u_+$ is semi-convex, then $u_-$ is $K$ semi-convex at every point of ${\mathcal {I}}(u_-, u_+)$; hence: 1. $u_-(x+h)-u_-(x)-du_-(x)h\leq K\| h\|^2$; 2. $u_-(x)-u_-(y)-du_-(y)(x-y)\leq K\|x-y\|^2$; 3. $-K\| x+h-y\|^2 \leq u_-(x+h)-u_-(y)-du_-(y)(x+h-y)$. Adding these three inequalities, we obtain:\ $(du_-(y)-du_-(x))h\leq K\| h\|^2+K\|x-y\|^2+K\| x+h-y\|^2$.\ We choose $h$ such that $\| h\|=\| x-y\|$: $(du_-(y)-du_-(x))\frac{h}{\| h\|}\leq 6K\| x-y\|$ and then: $\| du_-(x)-du_-(y)\|\leq 6K\| x-y\|$. We have found a constant for $y$ close to $x$, this is enough to conclude because ${\mathcal {I}}(u_-, u_+)$ is compact and $du_-$ is bounded on $M$. Let us now fix $y\in{\mathcal {I}}(u_-, u_+)$. For $x$ close to $y$ that is a point of differentiability of $u_-$, we have: 1. $a_{x,t}^+(z)=A_t(\pi\circ\varphi_{-t}(x, du_-(x)), z)$; 2. $\{ (z, da_{x,t}^+(z))\}={\mathcal {V}}(z)\cap \varphi_t({\mathcal {V}}_{\rm loc}(\pi\circ\varphi_{-t}(x, du_-(x))))$; 3. ${\rm graph} (d^2a_{x,t}^+(z)) $ $= $ $T_{(z, dg_{x,t}^+(z))}D\varphi_t({\mathcal {V}}(\pi\circ\varphi_{-t}(x, du_-(x))))=G_t(x, du_-(x))$ and then the previous intersection is transverse. These three quantities depend on $x$ and $z$; because $du_-$ is continuous at $y$, we have: for every $\varepsilon>0$, there exists $\delta>0$ such that, if $\| x-y\|<\delta$ and $z$ is in the chart near $y$: $\|d^2a^+_{x,t}(z)-d^2a_{y,t}^+(z)\|\leq \varepsilon$.\ Moreover, by Taylor-Lagrange inequality, we have:\ $\displaystyle{ \|a_{x,t}^+(x+h)-a_{x,t}^+(x)-da_{x,t}^+(x)h-\frac{1}{2}d^2a_{x,t}^+(x)(h,h)\|\leq\max_{z\in [x, x+h]}\| d^2a_{x,t}^+(z)-d^2a^+_{x,t}(x)\|\|h\|^2.}$ Hence, if $x$ is close enough to $y$ and $h$ small enough :\ $\displaystyle{ \|a_{x,t}^+(x+h)-a_{x,t}^+(x)-da_{x,t}^+(x)h-\frac{1}{2}d^2a_{y,t}^+(x)(h,h)\|\leq \varepsilon \|h\|^2.} $\ We have of course a similar result for $a_{x,t}^-$ and $x$ any differentiability point of $u_+$.\ Let us now consider a sequence $(x_n)$ of points of differentiability of $u_-$ that converges to $y$ so that: $\forall n, x_n\not=y$, a vector $k$ with fixed norm $\| k\|=\lambda>0$ and $(h_n)=(t_nk)$ where $(t_n)$ is a sequence of positive numbers tending to $0$. We have proved that: $(du_-(y) -du_-(x_n))h_n \leq a_{x_n,t}^+(x_n+h_n)-a_{x_n,t}^+(x_n)-da_{x_n,t}^+(x_n)h_n+a_{y,t}^+(x_n)-a_{y,t}^+(y)-da_{y,t}(y)(x_n-y) -a^-_{y,t}(x_n+h_n)+a^-_{y,t}(y)+da_{y,t}^-(y)(x_n+h_n-y)$.\ We assume that $\displaystyle{\lim_{n\rightarrow \infty} \frac{ x_n-y}{t_n}=X}$ and $\displaystyle{Y=\lim_{n\rightarrow \infty} \frac{du_-(x_n)-du_-(y)}{t_n} }$. We divide by $t_n^2$ the previous inequality and take the limit when $n$ tends to $+\infty$ and we obtain: $$-Y.k \leq \frac{1}{2}(d^2a_{y,t}^+(y)(k,k)+d^2a_{y,t}^+(y)(X,X)-d^2a_{y,t}^-(y)(X+k,X+k))$$ changing $k$ into $-k$, this gives the wanted result. In a similar way we obtain for $u_+$: $$\forall k\in{\mathbb {R}}^n, \frac{1}{2}(d^2a_{y,t}^-(y)(k,k)+d^2a_{y,t}^-(y)(X,X)-d^2a_{y,t}^+(y)(k-X,k-X))\leq Y.k$$ Links between the Green bundles and the weak K.A.M. solutions ------------------------------------------------------------- Near every point $q\in M$, we choose some coordinates $(q_1, \dots , q_n)$ of $M$ and associate to them their dual coordinates $(p_1, \dots , p_n)$ such that $(q_1, \dots, q_n, p_1, \dots , p_n)$ are symplectic coordinates on $T^*M$. Then we can associate to these coordinates their infinitesimal coordinates $(Q_1, \dots, Q_n, P_1, \dots , P_n)$.\ Then any Lagrangian subspace $G$ of $T_x(T^*M)$ that is transverse to the vertical is the graph of a linear map whose matrix $s$ in the coordinates $(Q_1, \dots, Q_n, P_1, \dots , P_n)$ is symmetric. We can then associate to $G$ the unique quadratic form $Q$ whose matrix (as a quadratic form) in coordinates $(Q_1, \dots , Q_n)$ is $s$. For example, if $q\in M$ is a point of differentiability of $u_-$ (resp. $u_+$ then the Green bundle $G_+(q, du_-(q))$ (resp. $G_-(q, du_+(q))$) is well defined and transverse to the vertical. We denote by $Q_-$ (resp. $Q_+$) its associated quadratic form and by $s_-$ (resp. $s_+$) its matrix. Let us recall that if $x\in A\subset T^*M$, ${\mathcal {C}}_xA$ designates the contingent cone to $A$ at $x$, that was defined in the introduction. We assume that $(u_-, u_+) $ is a pair of conjugate weak K.A.M. solutions. Let $y\in{\mathcal {I}}(u_-, u_+)$ be a point and $(X,Y)\in {\mathcal {C}}_{(y, du_-(y))}{\mathcal {G}}(du_-)$. Then we have: $$\forall k\in{\mathbb {R}}^n, Y.k \leq \frac{1}{2}(Q_+(k,k)+Q_+(X,X)-Q_-(X-k,X-k))$$ and if $(X,Y)\in {\mathcal {C}}_{(y, du_+)}{\mathcal {G}}(du_+)$: $$\forall k\in{\mathbb {R}}^n, \frac{1}{2}(Q_-(k,k)+Q_-(X,X)-Q_+(X-k,X-k))\leq Y.k .$$ We know that $\displaystyle{G_+(q,p)=\lim_{t\rightarrow +\infty}G_t(q,p)}$ (resp. $\displaystyle{G_-(q,p)=\lim_{t\rightarrow -\infty}G_t(q,p)}$). Hence, if $q$ is a point of differentiability of $u_-$, we have: $\displaystyle{Q_+(q, du_-(q))=\lim_{t\rightarrow +\infty} d^2g_{q,t}^+(q)}$ and if $q$ is a point of differentiability of $u_+$: $\displaystyle{Q_-(q, du_+(q))=\lim_{t\rightarrow +\infty} d^2g_{q,t}^-(q)}$. If we use the inequalities that we proved in the previous section, we obtain: $$\forall k\in{\mathbb {R}}^n, Y.k \leq \frac{1}{2}(Q_+(X,X)+Q_+(k,k)-Q_-(X-k,X-k)).$$ Let us now look for the contingent cone to the pseudograph ${\mathcal {G}}(du_-)$ at $(y, du_-(y))\in {\mathcal {I}}(u_-, u_+)$. Working in a chart, we assume that $(X, Y)\in {\mathcal {C}}_{y, du_-(y))}{\mathcal {G}}(du_-)$ is not the null vector. Hence, there exists a sequence $(t_n)$ of positive numbers that converges to $0^+$ and a sequence $(x_n)$ of points of differentiability of $u_-$ that converges to $y$ so that: $$(X, Y)=\lim_{n\rightarrow \infty} \frac{1}{t_n}(x_n-y, du_-(x_n)-du_-(y)).$$ This corresponds exactly to the limit that we computed in the previous subsection. Hence, we proved: *If $y\in {\mathcal {I}}(u_-, u_+)$, if $(X, Y)$ is a vector of the contingent cone to ${\mathcal {G}}(du_-)$ at $(y, du_-(y))$, then:* $$\forall k\in {\mathbb {R}}^n, Y.k\leq \frac{1}{2}(Q_+(k,k)+Q_+(X,X)-Q_-(X-k, X-k)).$$ In a similar way, we obtain: *If $y\in {\mathcal {I}}(u_-, u_+)$, if $(X, Y)$ is a vector of the contingent cone to ${\mathcal {G}}(du_+)$ at $(y, du_+(y))$, then:* $$\forall k\in {\mathbb {R}}^n, \frac{1}{2}(Q_-(k,k)+Q_-(X,X)-Q_+(X-k, X-k)\leq Y.k .$$ Proof of theorem \[greenkam\] ----------------------------- Let $(u_-, u_+)$ be a pair of conjugate weak KAM solutions and let $q$ belong to ${\mathcal {I}}(u_-, u_+)$. We want to prove that: $\forall (X, Y)\in {\mathcal {C}}_{(q, du_-(q))}{\mathcal {G}}(du_-), $ $$\| Y-\tilde s_-(q, du_-(q))X\|\leq 2\sqrt{\|\Delta s(q, du_-(q))\|} .\sqrt{\Delta s(q, du_-(q))(X,X)}$$ $$\leq 2\Lambda (\Delta s(q, du_-(q))). \| p_{\Delta s(q, du_-(q))}(X)\|$$ We denote the quadratic form associated with $G_-$ (resp. $G_+$) by $Q_-$ (resp. $Q_+$). Then the quadratic form associated with $\tilde G_-$ (resp. $\tilde G_+$) is $\tilde Q_-=2Q_--Q_+$ (resp. $\tilde Q_+=2Q_+-Q_-$). Let $(X, Y)\in{\mathcal {C}}_{(q, du_-(q))}{\mathcal {G}}(du_-)$ be a vector of the contingent cone. We have proved that: $$\forall k\in {\mathbb {R}}^n, Y.k\leq \frac{1}{2}(Q_+(k,k)+Q_+(X,X)-Q_-(X-k, X-k)).$$ Then we write: $Y={}^tQ_+X+\Delta Y$ and $\Delta Q=Q_+-Q_-$. The previous inequality can be rewritten as follows: $$\forall k\in {\mathbb {R}}^n, \Delta Y.k\leq \frac{1}{2}\Delta Q(X-k, X-k)\quad (*).$$ We have the following splitting: ${\mathbb {R}}^n=\ker {}^t \Delta Q\oplus {\mathrm{Im}}{}^t\Delta Q$ and $\Delta Y=Y_1+Y_2$ with $Y_1\in \ker {}^t\Delta Q$ and $Y_2\in {\mathrm{Im}}{}^t\Delta Q$. We deduce from $(*)$: $$\forall k\in\ker{}^t\Delta Q, Y_1. k\leq \frac{1}{2}\Delta Q(X, X).$$ This implies: $Y_1=\vec 0$. Hence $\Delta Y=Y_2\in {\mathrm{Im}}{}^t\Delta Q$ and there exists a unique $y\in -2X+{\mathrm{Im}}{}^t\Delta Q$ such that $\Delta Y={}^t \Delta Q y$. Then $(*)$ becomes: $$\forall k\in{\mathbb {R}}^n, \Delta Q(y, k)\leq \frac{1}{2} \Delta Q(X-k, X-k)$$ i.e: $$\forall k\in{\mathbb {R}}^n, \Delta Q(X+\frac{y}{2}, X+\frac{y}{2})-\Delta Q(X,X)\leq \Delta Q(X-k+\frac{y}{2}, X-k+\frac{y}{2})$$ As $G_-\leq G_+$, the quadratic form $\Delta Q$ is positive semi-definite. Hence the previous inequality is equivalent to: $$\Delta Q(2X+y, 2X+y)\leq 4 \Delta Q(X,X).$$ Let us write $y=-2X+\Delta y$. We have $\Delta y\in {\mathrm{Im}}{}^t\Delta Q$. Then $Y= {}^t\tilde Q_-X+{}^t\Delta Q\Delta y$ and $\Delta Q(\Delta y, \Delta y)\leq 4 \Delta Q(X,X)$.\ Then we can write: $\Delta y=2\sqrt{Q(X,X)} u$ with $\Delta Q(u, u)\leq 1$ and: $$\| {}^t\Delta Q\Delta y\|^2=4\Delta Q(X, X)({}^t\Delta Q u).({}^t\Delta Q u)$$ with: $$({}^t\Delta Q u).({}^t\Delta Q u)\leq \sup \frac{{}^t\Delta Q v .{}^t\Delta Q v }{\Delta Q(v,v)}= \| \Delta Q\|.$$ We then obtain: $$\| Y-{}^t\tilde Q_-X\| \leq 2\sqrt{\| \Delta Q\|}.\sqrt{\Delta Q(X, X)}.$$ If we denote by $\Lambda(\Delta Q)$ the biggest eigenvalue of $\Delta Q$ and by $p_{\Delta Q}$ the orthogonal projection on the image of ${}^t\Delta Q$, we deduce: $$\| Y-{}^t\tilde Q_-X\| \leq 2\Lambda (\Delta Q) \| p_{\Delta Q}(X)\|Ê.$$ [cc]{} M.-C. Arnaud, [*Fibrés de Green et régularité des graphes $C^0$-Lagrangiens invariants par un flot de Tonelli*]{}, Ann. Henri Poincaré [**9**]{} (2008), no. 5, 881–926. M.-C. Arnaud, [*Three results on the regularity of thecurves that are invariant by an exact symplectic twist map*]{}, Publ. Math. Inst. Hautes Études Sci. No. [**109**]{} , 1–17 (2009). M.-C. Arnaud, [*The link between the shape of the Aubry-Mather sets and their Lyapunov exponents*]{}, preprint arXiv:0902.3266 P. Bernard. *The dynamics of pseudographs in convex Hamiltonian systems.* J. Amer. Math. Soc. 21 (2008), no. 3, 615–669. G. D. Birkhoff, *Surface transformations and their dynamical application*, [Acta Math.]{} [**43**]{} (1920) 1-119. J. Bochi & M. Viana, *Lyapunov exponents: how frequently are dynamical systems hyperbolic?* Modern dynamical systems and applications, 271–297, Cambridge Univ. Press, Cambridge, 2004. G. Bouligand. Introduction à la géométrie infinitésimale directe (1932) Librairie Vuibert, Paris. G. Contreras & R. Iturriaga, *Convex Hamiltonians without conjugate points*, [ Ergodic Theory Dynam. Systems]{} [**19**]{} (1999), no. 4, 901–952. P. Eberlein, *When is a geodesic flow of Anosov type?* I,II. J. Differential Geometry 8 (1973), 437–463; ibid. 8 (1973), 565–577 A. Fathi, [*Weak KAM theorems in Lagrangian dynamics*]{}, book in preparation. A . Fathi, [*Regularity of $C\sp 1$ solutions of the Hamilton-Jacobi equation*]{}. Ann. Fac. Sci. Toulouse Math. (6) [**12** ]{} no. 4, 479–516 (2003). A. Freire & R. Mañé. *On the entropy of the geodesic flow in manifolds without conjugate points.* Invent. Math. 69 (1982), no. 3, 375–392. L. W. Green, *A theorem of E. Hopf* Michigan Math. J. [**5**]{} 31–34 (1958). ÊM. Herman, [*Sur les courbes invariantes par les difféomorphismes de l’anneau*]{}, Vol. 1, Asterisque [**103-104**]{} (1983). M. Herman, [*In' egalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques.* ]{}, vol. I, Inst. Hautes Études Sci. Publ. Math. No. [**70**]{}, 47–101 (1989) R. Iturriaga, [*A geometric proof of the existence of the Green bundles*]{}. Proc. Amer. Math. Soc. [**130**]{} , no. 8, 2311–2312 (2002). R. Mañé, [*Global Variational Methods in Conservative Dynamics*]{}, 18 Coloquio Brasileiro de Matematica, IMPA, 1991. R. Mañé. *Quasi-Anosov diffeomorphisms and hyperbolic manifolds.* Trans. Amer. Math. Soc. 229 (1977), 351–370. J. N. Mather. *Action minimizing invariant measures for positive definite Lagrangian systems* Math. Z. 207 (1991), no. 2, 169–207. [^1]: ANR Project BLANC07-3\_187245, Hamilton-Jacobi and Weak KAM Theory [^2]: ANR DynNonHyp [^3]: Université d’Avignon et des Pays de Vaucluse, Laboratoire d’Analyse non lin' eaire et G' eom' etrie (EA 2151), F-84 018Avignon, France. e-mail: [email protected]

--- author: - 'Christian Kanzow$^{\dagger}$' - 'Daniel Steck [^1]' bibliography: - 'VI\_ALMinf.bib' date: 'March 27, 2018' title: ' On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces [^2] ' --- **.** This paper deals with a general form of variational problems in Banach spaces which encompasses variational inequalities as well as minimization problems. We prove a characterization of local error bounds for the distance to the (primal-dual) solution set and give a sufficient condition for such an error bound to hold. In the second part of the paper, we consider an algorithm of augmented Lagrangian type for the solution of such variational problems. We give some global convergence properties of the method and then use the error bound theory to provide estimates for the rate of convergence and to deduce boundedness of the sequence of penalty parameters. Finally, numerical results for optimal control, Nash equilibrium problems, and elliptic parameter estimation problems are presented. **Keywords.** Variational problem, variational inequality, error bound, augmented Lagrangian method, local convergence, global convergence, Nash equilibrium problem. **AMS subject classifications.** 49K, 49M, 65K, 90C. Introduction {#Sec:Intro} ============ This paper deals with the following variational problem: $$\label{Eq:VI} \text{Find }x\in M\text{ such that}\quad{\mleft\langle F(x),v \mright\rangle}\ge 0 \quad\forall v\in{\mathcal{T}_{M}(x)},$$ where $M\subseteq X$ is a nonempty closed set, $X$ a real Banach space, and $F:X\to X^*$ a given mapping. The set ${\mathcal{T}_{M}(x)}$ denotes the (Bouligand) tangent cone [@Bonnans2000] to $M$ at $x$. If $M$ is additionally convex, then is equivalent to $$\label{Eq:VI_Convex} \text{Find }x\in M\text{ such that}\quad{\mleft\langle F(x),y-x \mright\rangle}\ge 0 \quad\forall y\in M,$$ which is often regarded as the standard form of a variational inequality (VI). Throughout this paper, we will use the terms “variational inequality” and “variational problem” interchangeably, and often refer to as a VI. Note that, in the absence of convexity, is the canonical formulation of variational problems; in particular, this form encompasses first-order necessary conditions for nonlinear optimization problems of the type $$\label{Eq:Opt} \min\ f(x) \quad\text{s.t.}\quad x\in M$$ by choosing $F:=f'$. Throughout this paper, we assume that $M$ is given in the form $$\label{Eq:M} M=\{ x\in X: g(x)\in K \},$$ where $g:X\to H$ is a given mapping, $H$ a real Hilbert space, and $K\subseteq H$ a nonempty closed convex set (not necessarily a cone). We make no blanket convexity assumptions on $g$ (although some of our results do pertain to the convex case). Hence, the set $M$ is nonconvex in general, and is the natural framework for our setting. Variational inequalities are a well-known and popular class in both finite and infinite-dimensional optimization since they unify various problem types such as constrained minimization and equilibrium-type problems, in particular Nash and (certain) generalized Nash equilibrium problems [@Facchinei2007; @Facchinei2010; @Fischer2014; @Hintermueller2015; @Kanzow2017a]. This opens up a broad spectrum of applications including optimal control, parameter estimation, differential games, and problems in mechanics or shape optimization. Many further applications are given in [@Baiocchi1984; @Glowinski2015; @Glowinski1981; @Kinderlehrer2000]. As a result, VIs have gained considerable attention in the literature and a variety of algorithms have been developed for their solution, e.g. [@Facchinei2003; @Fortin1983; @Glowinski2008; @Ulbrich2011]. On the other hand, the augmented Lagrangian method (ALM, also called multiplier-penalty method or simply multiplier method) is one of the classical methods for nonlinear optimization, see [@Conn1991; @Hestenes1969; @Powell1969; @Rockafellar1973; @Rockafellar1974] and the textbooks [@Bertsekas1982; @Nocedal2006]. In recent years, ALMs have seen a certain resurgence [@Andreani2007; @Andreani2008; @Birgin2012; @Birgin2010; @Birgin2014] in the form of modified methods which use a slightly different update of the Lagrange multiplier and turn out to have very strong global convergence properties [@Birgin2014]. A comparison of the classical and modified ALMs is given in [@Kanzow2017]. We also note that ALMs have been generalized to VIs in finite dimensions [@Andreani2008] and to infinite-dimensional optimization problems in certain restricted settings [@Hintermueller2006; @Ito1990a; @Ito1990b; @Ito2000; @Ito2008; @Kanzow2016; @Wierzbicki1977]. However, most of these papers either consider rather specific problem settings [@Hintermueller2006; @Ito1990a; @Ito1990b; @Ito2000; @Ito2008] or deal with global convergence properties only [@Kanzow2016]. The main purpose of the present paper is to analyze the local convergence properties of ALMs for variational inequalities in the general (possibly infinite-dimensional) setting . To accomplish this, we will need certain elements of perturbation and error bound theory for generalized equations and KKT systems, some of which are refinements of the corresponding results in finite dimensions [@Ding2017; @Dontchev1998; @Fischer2002; @Izmailov2012a]. Using these, we will prove that, given a KKT point which admits a primal-dual error bound, the ALM converges locally to this point with a rate of convergence that is essentially $1/\rho_k$ (where $\rho_k$ is the penalty parameter), and that $\{\rho_k\}$ remains bounded if updated suitably. Sufficient conditions for the primal-dual error bound include a suitable second-order sufficient condition (SOSC) together with a strict version of the Robinson constraint qualification (see Section \[Sec:Prelims\]). These assumptions are akin to those used in [@Birgin2012] for ALMs in finite-dimensional nonlinear programming (NLP), where the authors obtain results similar to ours. Interestingly, however, it turns out that these results (for standard NLP) can be established under SOSC only [@Fernandez2012] by using the specific structure of the constraints. In particular, when transferred to our notation, the set $K$ arising from NLP is polyhedral and this yields, roughly speaking, the dual part of the error bound without any constraint qualification [@Fernandez2012; @Izmailov2012a]. However, apart from the NLP setting, polyhedrality is a rare property which is usually violated, e.g. in optimal control or semidefinite programming. As a result, SOSC alone does not yield a primal-dual error bound, see the example in Section \[Sec:ErrorBounds\]. We solve this issue by using SOSC together with a suitable constraint qualification. The paper is organized as follows. We start with some preliminary material in Section \[Sec:Prelims\] and give some results on primal-dual error bounds in Section \[Sec:ErrorBounds\]. Section \[Sec:Method\] contains a precise statement of our algorithm and we continue with some global convergence results in Section \[Sec:GlobalConv\]. In Section \[Sec:LocalConv\], we prove the main results of this paper, i.e. local convergence of the ALM under the error bound hypothesis. We then give some numerical results in Section \[Sec:Applic\] and final remarks in Section \[Sec:Final\]. **Notation:** Throughout the paper, $X$ is always a real Banach space, $H$ a real Hilbert space, and their duals are denoted by $X^*$ and $H^*$, the latter of which we usually identify with $H$. Fréchet-derivatives are denoted by a prime $'$ or by $D_x$ if the variable is emphasized, and we use the abbreviation lsc for lower semicontinuity. Strong and weak convergence are denoted by $\to$ and ${\rightharpoonup}$, respectively. Duality pairings are written as ${\mleft\langle \cdot,\cdot \mright\rangle}$, scalar products as ${\mleft( \cdot,\cdot \mright)}$, and norms are denoted by $\|\cdot\|$ with an appropriate subscript to emphasize the corresponding space (e.g. $\|\cdot\|_X$). If $S$ is a nonempty subset of some normed space, we write $d_S={\operatorname{dist}(\cdot,S)}$ for the distance to $S$. Additionally, if $S\subseteq H$ is closed and convex, we write $P_S$ for the projection onto $S$. Preliminaries {#Sec:Prelims} ============= This section is dedicated to establishing some preliminary results as well as fixing the setting we will consider later. Recall that the set $M$ is given by the formula with a nonempty closed convex set $K\subseteq H$. Cones and Convexity ------------------- If $S$ is a nonempty closed subset of some space $Z$, then $S^{\circ}:=\{ \psi\in Z^*: {\mleft\langle \psi,s \mright\rangle}\le 0~\forall s\in S \}$ denotes the *polar cone* of $S$. If $Z$ is a Hilbert space, we of course treat $S^{\circ}$ as a subset of $Z$. Moreover, if $x\in S$ is a given point, we denote by $${\mathcal{T}_{S}(x)}:= \bigl\{ d\in Z : \exists x^k\to x,\, t_k\downarrow 0 \text{ such that } x^k\in S \text{ and } (x^k-x)/t_k \to d \bigr\}$$ the *tangent cone* of $S$ at $x$. If $S$ is additionally convex, we also define the *normal cone* $${\mathcal{N}_{S}(x)}:= \mleft\{ \psi\in Z^*: {\mleft\langle \psi,y-x \mright\rangle}\le 0~\forall y\in S \mright\} ={\mathcal{T}_{S}(x)}^{\circ}.$$ If $x\notin S$, we define ${\mathcal{T}_{S}(x)}$ and ${\mathcal{N}_{S}(x)}$ to be empty. Note that, if $S$ is a convex set, then ${\mathcal{T}_{S}(x)}$ and ${\mathcal{N}_{S}(x)}$ are closed convex cones for all $x\in S$. Recall that the constraint system of the VI is given by $g(x)\in K$ with $K\subseteq H$ a nonempty closed convex set. A natural question is what the appropriate notion of convexity is in this general setting. In particular, we would like to give sufficient conditions for the convexity of the feasible set $M$. To this end, consider the recession cone $$\label{Eq:RecessionCone} K_{\infty}:=\{ y\in H: y+K\subseteq K \}.$$ It is well-known that $K_{\infty}$ is a nonempty closed convex cone [@Bauschke2011; @Bonnans2000]. If $K$ itself is a cone, then $K_{\infty}=K$. We associate with $K$ (and $K_{\infty}$) the (partial) order relation $$\label{Eq:OrderK} y \le_K z :\Longleftrightarrow z-y\in K_{\infty}.$$ Note that we use the notation $\le_K$ for the sake of convenience, even though the order is actually induced by the cone $K_{\infty}$. We also note that $K_{\infty}$ may not be pointed (that is, $K_{\infty}\cap (-K_{\infty})$ may contain a nonzero element) and, hence, the relation $\le_K$ does not necessarily satisfy the antisymmetry property $$a\le_K b \wedge b\le_K a \implies a=b.$$ In the terminology of order theory, this makes $\le_K$ a so-called preorder. We will simply call it an order relation due to the descriptiveness of the term. Note also that, throughout this paper, the symbol $\le$ without any index is always the standard ordering in ${\mathbb{R}}$. The order relation allows us to extend various familiar concepts from finite-dimensional optimization to our setting. For instance, we say that $g$ is convex if $$g(\alpha x+(1-\alpha)y) \le_K \alpha g(x) + (1-\alpha) g(y)$$ holds for all $x,y\in X$ and $\alpha\in[0,1]$. Other notions which involve an order such as increasing, decreasing or concave functions are also defined in a straightforward way. For example, the distance function $d_K:H\to{\mathbb{R}}$ is decreasing since $z\ge_K y$ implies $z=y+k$, $k\in K_{\infty}$, and $$d_K(z)=d_K(y+k)\le \|y+k-(P_K(y)+k)\| = \|y-P_K(y)\| = d_K(y),$$ where the inequality uses the fact that $P_K(y)+k\in K$ by definition of $K_{\infty}$. Some other results pertaining to convexity, concavity, etc. are given in the following lemma. Note that, in the context of our constraint set with $g(x)\in K$, it is more natural to consider concavity of $g$ with respect to the ordering as opposed to convexity. \[Lem:GeneralizedConvexity\] Assume that $g:X\to H$ is concave. If $m:H\to{\mathbb{R}}$ is convex and decreasing, then $m\circ g$ is convex. In particular: 1. The function $d_K\circ g:X\to{\mathbb{R}}$ is convex. 2. If $\lambda\in K_{\infty}^{\circ}$, then $x\mapsto{\mleft( \lambda,g(x) \mright)}$ is convex. 3. The set $M=\{x\in X: g(x)\in K\}$ is convex. Let $x,y\in X$ and $x_{\alpha}=\alpha x+(1-\alpha) y$, $\alpha\in[0,1]$. Then $g(x_{\alpha})\ge_K \alpha g(x)+(1-\alpha) g(y)$ by the concavity of $g$. Applying $m$ on both sides yields $$m(g(x_{\alpha}))\le m(\alpha g(x)+(1-\alpha)g(y))\le \alpha m(g(x))+(1-\alpha)m(g(y)),$$ where we used the monotonicity and the convexity of $m$. Hence, $m\circ g$ is convex. Assertion (a) now follows because $d_K$ is decreasing (see above) and convex [@Bauschke2011 Cor. 12.12]. Similarly, for (b), the function $y\mapsto{\mleft( \lambda,y \mright)}$ with $\lambda\in K_{\infty}^{\circ}$ is obviously a convex function, and it is decreasing because ${\mleft( \lambda,k \mright)}\le 0$ for all $k\in K_{\infty}$. Finally, for (c), note that $$M=\{ x\in X: g(x)\in K \}=\{ x\in X: d_K(g(x))\le 0 \}.$$ Hence, $M$ is a lower level set of the convex function $d_K\circ g$ and therefore a convex set. Note that the extreme case $K_{\infty}=\{0\}$ can occur, e.g. if $K$ is bounded. In this case, monotonicity becomes trivial and convexity and concavity reduce to linearity. It is possible to characterize $K_{\infty}^{\circ}$ by means of the so-called barrier cone to $K$, see [@Bauschke2011]. Here, we will only need the following observation. \[Lem:RecessionConePolar\] If $y\in H$, then $y-P_K(y)\in K_{\infty}^{\circ}$. Let $k\in K$ be fixed and let $z\in K_{\infty}$, $\alpha\ge 0$. Then $\alpha z\in K_{\infty}$ and therefore $\alpha z+k\in K$. A standard projection inequality yields ${\mleft( y-P_K(y),\alpha z+k-P_K(y) \mright)}\le 0$. But this clearly cannot hold for all $\alpha$ if ${\mleft( y-P_K(y),z \mright)}>0$. Hence, ${\mleft( y-P_K(y),z \mright)}\le 0$. The KKT Conditions ------------------ We now turn to the variational inequality and discuss its KKT conditions. Starting with this section, we assume that the mapping $F$ is continuously differentiable and that $g$ is twice continuously differentiable. Consider now the Lagrange function $$\label{Eq:L} {\mathcal{L}}: X\times H\to X^*, \quad {\mathcal{L}}(x,\lambda):=F(x)+g'(x)^* \lambda.$$ Note that, if the VI originates from a minimization problem, then ${\mathcal{L}}$ is actually the derivative of the conventional Lagrange function. The following are the standard first-order conditions which we will use throughout this paper. \[Dfn:KKT\] A tuple $(\bar{x},\bar{\lambda})\in X\times H$ is a *KKT point* of , if $$\label{Eq:KKT} {\mathcal{L}}(\bar{x},\bar{\lambda})=0 \quad\text{and}\quad \bar{\lambda}\in{\mathcal{N}_{K}(g(\bar{x}))}.$$ We call $\bar{x}\in X$ a stationary point if $(\bar{x},\bar{\lambda})$ is a KKT point for some $\bar{\lambda}\in H$, and denote by $\mathcal{M}(\bar{x})$ the corresponding set of multipliers. Note that $\bar{\lambda}\in{\mathcal{N}_{K}(g(\bar{x}))}$ implies $g(\bar{x})\in K$, since otherwise the normal cone would be empty. Moreover, we remark that, if $K$ is a cone, then $\bar{\lambda}\in{\mathcal{N}_{K}(g(\bar{x}))}$ is equivalent to the complementarity conditions $g(\bar{x})\in K$, $\bar{\lambda}\in K^{\circ}$, and ${\mleft( \bar{\lambda},g(\bar{x}) \mright)}=0$, see [@Bonnans2000 Ex. 2.62]. The relationship between the VI and its KKT conditions is given as follows: if $\bar{x}$ solves the VI and a suitable constraint qualification holds in $\bar{x}$, then there exists a multiplier $\bar{\lambda}$ such that $(\bar{x},\bar{\lambda})$ is a KKT point [@Bonnans2000 Remark 5.8]. On the other hand, it is easy to see that the KKT conditions are always sufficient for the VI , even if $M$ is nonconvex. This result is contained in the following theorem and crucially depends on the fact that the VI uses the tangent cone $\mathcal{T}_M$ and not $M$ itself. \[Thm:SufficiencyKKT\] If $(\bar{x},\bar{\lambda})$ is a KKT point of the VI, then $\bar{x}$ is a solution of the VI. Let $(\bar{x},\bar{\lambda})$ be a KKT point and $d\in{\mathcal{T}_{M}(\bar{x})}$. Then $d=\lim_{k\to\infty}(x^k-\bar{x})/t_k$ with $\{x^k\}\subseteq M$, $x^k\to\bar{x}$, and $t_k\downarrow 0$. Hence, $${\mleft\langle F(\bar{x}),d \mright\rangle}={\mleft\langle -g'(\bar{x})^* \bar{\lambda}, \lim_{k\to\infty} \frac{x^k-\bar{x}}{t_k} \mright\rangle}= -\lim_{k\to\infty} \frac{1}{t_k} {\mleft( \bar{\lambda},g'(\bar{x})(x^k-\bar{x}) \mright)}.$$ But $g'(\bar{x})(x^k-\bar{x})=g(x^k)-g(\bar{x})+o(t_k)$ and therefore $${\mleft\langle F(\bar{x}),d \mright\rangle}=-\lim_{k\to\infty}\frac{1}{t_k} {\mleft( \bar{\lambda},g(x^k)-g(\bar{x}) \mright)}\ge 0,$$ where we used $\bar{\lambda}\in {\mathcal{N}_{K}(g(\bar{x}))}$ and $g(x^k)\in K$ for all $k$. For a given KKT point $(\bar{x},\bar{\lambda})$ and $\eta\ge 0$, we define the extended critical cone $$C_{\eta}(\bar{x}):= \bigl\{ d\in X: {\mleft\langle F(\bar{x}),d \mright\rangle}\le \eta\|d\|_X,~ g'(\bar{x})d\in{\mathcal{T}_{K}(g(\bar{x}))} \bigr\}.$$ The following is the second-order condition which we will use throughout this paper. \[Dfn:SOSC\] Let $(\bar{x},\bar{\lambda})$ be a KKT point of the VI. We say that the *second-order sufficient condition (SOSC)* holds in $(\bar{x},\bar{\lambda})$ if there are $\eta,c>0$ such that $${\mleft\langle D_x {\mathcal{L}}(\bar{x},\bar{\lambda})d,d \mright\rangle}\ge c\|d\|_X^2 \quad\text{for all }d\in C_{\eta}(\bar{x}).$$ Note that we use the terminology “second-order sufficient condition” mainly for the sake of consistency with a similar condition from nonlinear optimization, e.g. [@Bonnans2000 Def. 3.60]. For variational problems such as , there is actually no need for sufficiency conditions to complement the KKT system because the latter always implies that $\bar{x}$ is a solution of the VI (see Theorem \[Thm:SufficiencyKKT\]). Let us also note that Definition \[Dfn:SOSC\] is slightly different from the second-order sufficient condition for nonlinear optimization [@Bonnans2000 Def. 3.60] because our extended critical cone is slightly smaller. However, under the Robinson constraint qualification (see below and [@Bonnans2000 Def. 2.86]), the corresponding second-order conditions coincide [@Bonnans2000 Remark 3.68]. Moreover, and more importantly, our subsequent analysis will be based on [@Bonnans2000 Thm. 5.9] which directly uses the “smaller” critical cone together with the following condition. \[Dfn:SRC\] Let $(\bar{x},\bar{\lambda})$ be a KKT point of the VI, and $K_0:=\bigl\{ y\in K: {\mleft( \bar{\lambda},y-g(\bar{x}) \mright)}=0 \bigr\}$. We say that the *strict Robinson condition (SRC)* holds in $(\bar{x},\bar{\lambda})$ if $$\label{Eq:SRC} 0\in \operatorname{int}\bigl(g(\bar{x})+g'(\bar{x})X-K_0\bigr). $$ Note that the standard Robinson constraint qualification arises if we replace $K_0$ in by the larger set $K$. Hence, SRC is stronger than the Robinson constraint qualification and the equivalent regularity condition of Zowe and Kurcyusz [@Zowe1979]. On the other hand, SRC implies the uniqueness of $\bar{\lambda}$ and is weaker than the surjectivity of $g'(\bar{x})$, which is a typical regularity assumption for infinite-dimensional problems. It should be noted that the definition of SRC presupposes the existence of $\bar{\lambda}$ and therefore depends not only on the constraints but also on the function $F$. Hence, we refrain from calling a constraint qualification (in contrast to [@Bonnans2000], where SRC is called the *strict constraint qualification*). A similar condition which is occasionally used in the finite-dimensional literature is the strict Mangasarian-Fromovitz condition [@Birgin2012; @Floudas2001; @Kyparisis1985]. This condition turns out to be a special case of SRC [@Bonnans2000 Remark 4.49] and is also not a constraint qualification [@Wachsmuth2013]. Error Bounds for the Variational Problem {#Sec:ErrorBounds} ======================================== Recall that the KKT conditions of the VI are given by $${\mathcal{L}}(\bar{x},\bar{\lambda})=0 \quad\text{and}\quad \bar{\lambda}\in{\mathcal{N}_{K}(g(\bar{x}))},$$ where $(\bar{x},\bar{\lambda})\in X\times H$. The last condition is well-known [@Bauschke2011 Prop. 6.46] to be equivalent to $g(\bar{x})=P_K(g(\bar{x})+\bar{\lambda})$. This suggests defining the residual mapping $$\label{Eq:KKTResidual} \sigma(x,\lambda):=\|{\mathcal{L}}(x,\lambda)\|_{X^*} +\|g(x)-P_K(g(x)+\lambda)\|_H.$$ Clearly, the KKT conditions of the VI are equivalent to $\sigma(\bar{x},\bar{\lambda})=0$. We will use this relationship to construct suitable error bounds for the primal-dual variables. In order to establish the error bound we are looking for, we first need a characterization of local error bounds in terms of a local upper Lipschitz property (or calmness) of the KKT system. This result has appeared in various forms in the literature [@Ding2017; @Fischer2002; @Izmailov2012a], albeit mostly in a finite-dimensional setting. In our notation, it involves certain perturbations of the KKT system with a parameter pair $p=(\alpha,\beta)\in X^*\times H$. Without loss of generality, we equip this product space with the norm $\|(\alpha,\beta)\|_{X^*\times H}:=\|\alpha\|_{X^*}+\|\beta\|_H$. Recall also that $\mathcal{M}(\bar{x})$ denotes the set of Lagrange multipliers corresponding to $\bar{x}$. \[Thm:ErrorBoundEquivalence\] Let $(\bar{x},\bar{\lambda})\in X\times H$ be a KKT point of the VI. Then the following assertions are equivalent: 1. There are a neighborhood $U$ of $\bar{x}$ and $c>0$ such that, for all $p=(\alpha,\beta)\in X^*\times H$ close to $(0,0)$, any solution $(x_p,\lambda_p)\in U\times H$ of the perturbed KKT system $$\label{Eq:PerturbedKKT} {\mathcal{L}}(x,\lambda)=\alpha, \quad \lambda\in {\mathcal{N}_{K}(g(x)-\beta)}$$ satisfies the estimate $\|x_p-\bar{x}\|_X+{\operatorname{dist}(\lambda_p,\mathcal{M}(\bar{x}))} \le c\|p\|_{X^*\times H}$. 2. There are a neighborhood $U$ of $\bar{x}$ and $c>0$ such that, for all $(x,\lambda)\in U\times H$ with $\sigma(x,\lambda)$ sufficiently small, $$\|x-\bar{x}\|_X+{\operatorname{dist}(\lambda,\mathcal{M}(\bar{x}))} \le c \sigma(x,\lambda).$$ (b)$\Rightarrow$(a): Let $p=(\alpha,\beta)\in X^*\times H$. It is an easy consequence of [@Bauschke2011 Cor. 4.10] that the mapping $y\mapsto y-P_K(y+\lambda_p)$ is nonexpansive. Hence, we obtain the inequality $$\|g(x_p)-P_K(g(x_p)+\lambda_p)\|_H \le \|\beta\|_H + \|g(x_p)-\beta-P_K(g(x_p) -\beta+\lambda_p)\|_H.$$ Since $\lambda_p\in{\mathcal{N}_{K}(g(x_p)-\beta)}$, the last term is equal to zero [@Bauschke2011 Prop. 6.46] and we obtain $\sigma(x_p,\lambda_p)\le\|\alpha\|_{X^*}+\|\beta\|_H=\|p\|_{X^*\times H}$. Choosing $p=(\alpha,\beta)$ sufficiently close to $0$, we see that $\sigma(x_p,\lambda_p)$ becomes arbitrarily small. Hence, we can apply (b) and obtain $$\|x_p-\bar{x}\|_X+{\operatorname{dist}(\lambda_p,\mathcal{M}(\bar{x}))} \le c\sigma(x_p,\lambda_p)\le c\|p\|_{X^*\times H}.$$ (a)$\Rightarrow$(b): Shrinking $U$ if necessary, we may assume that $\|g'(x)^*\|_{\mathcal{L}(H,X^*)}\le c_1$ for all $x\in U$ with some constant $c_1\ge 0$. Let $(x,\lambda)\in U\times H$, set $\delta:=\sigma(x,\lambda)$, and define $$\hat{g}:=P_K(g(x)+\lambda),\quad \hat{\lambda}:=g(x)+\lambda-\hat{g}.$$ Now, let $\alpha:={\mathcal{L}}(x,\hat{\lambda})$ and $\beta:=g(x)-\hat{g}$. Then $\hat{\lambda}\in{\mathcal{N}_{K}(\hat{g})}$ and, hence, $(x,\hat{\lambda})$ solves the perturbed KKT system corresponding to $\sigma:=(\alpha,\beta)$. Moreover, we have $\|\beta\|_H=\|\hat{g}-g(x)\|_H=\|g(x)-P_K(g(x)+\lambda)\|_H\le\delta$ and $\|\hat{\lambda}-\lambda\|_H=\|\beta\|_H\le\delta$. This implies $$\|\sigma\|_{X^*\times H}=\|{\mathcal{L}}(x,\hat{\lambda})\|_{X^*}+\|\beta\|_H\le \|{\mathcal{L}}(x,\lambda)\|_{X^*}+(c_1+1)\|\beta\|_H \le (c_1+2)\delta.$$ Hence, if $\delta=\sigma(x,\lambda)$ is small enough, then $\sigma$ becomes arbitrarily close to $0$. We can therefore apply (a) to $(x,\hat{\lambda})$ and obtain $$\|x-\bar{x}\|_X+{\operatorname{dist}(\hat{\lambda},\mathcal{M}(\bar{x}))} \le c\|\sigma\|_{X^*\times H}\le c(c_1+2)\delta.$$ But $\|\hat{\lambda}-\lambda\|_H\le\delta$ and, hence, ${\operatorname{dist}(\hat{\lambda},\mathcal{M}(\bar{x}))}\ge {\operatorname{dist}(\lambda,\mathcal{M}(\bar{x}))}-\delta$ by the nonexpansiveness of the distance function. This finally yields $$\|x-\bar{x}\|_X+{\operatorname{dist}(\lambda,\mathcal{M}(\bar{x}))}\le \bigl[c(c_1+2)+1\bigr]\delta,$$ and the proof is complete. Let us stress that the distance estimate provided by the above theorem holds if $x$ is close to $\bar{x}$; in particular, no assumption on the proximity of $\lambda$ to $\mathcal{M}(\bar{x})$ is necessary. We also remark that (a) does not make any assertion about the existence of solutions to the perturbed KKT conditions . These may have solutions for some but not all $\sigma$. Theorem \[Thm:ErrorBoundEquivalence\] is our main tool for establishing local error bounds for the distance of $(x,\lambda)$ to the primal-dual solution set in terms of the residual mapping $\sigma$. To verify such an error bound, we only need to prove property (a) of the theorem. The following result does precisely that and is based on the perturbation theory from [@Bonnans2000]. \[Thm:ErrorBound\] Assume that $(\bar{x},\bar{\lambda})$ is a KKT point which satisfies SOSC and the strict Robinson condition. Then $\mathcal{M}(\bar{x})=\{\bar{\lambda}\}$ and there is a $c>0$ such that, for all $(x,\lambda)\in X\times H$ with $x$ sufficiently close to $\bar{x}$ and $\sigma(x,\lambda)$ sufficiently small, $$\label{Eq:ErrorBound} \|x-\bar{x}\|_X+\|\lambda-\bar{\lambda}\|_H \le c \sigma(x,\lambda).$$ The uniqueness of $\bar{\lambda}$ follows as in [@Bonnans2000 Prop. 4.47], see also the discussion in Section 5.1.2 of that reference. For the error bound result, we essentially need to apply [@Bonnans2000 Thm. 5.9] and Theorem \[Thm:ErrorBoundEquivalence\]. Since some technical details need to be considered, we give a formal proof here. To this end, assume that the error bound in question does not hold. Then property (a) from Theorem \[Thm:ErrorBoundEquivalence\] does not hold either; hence, there are sequences $x^k\to\bar{x}$, $\{\lambda^k\}\subseteq H$ and $\{\sigma^k\}\subseteq X^*\times H$ with $\sigma^k=(\alpha^k,\beta^k)\to 0$ such that, for all $k$, $(x^k,\lambda^k)$ satisfies the perturbed KKT conditions corresponding to $\sigma^k$, and $$\label{Eq:CorErrorBound1} \|x^k-\bar{x}\|_X+\|\lambda^k-\bar{\lambda}\|_H \ge k \|\sigma^k\|_{X^*\times H}.$$ Now, let $F(x,\sigma):=F(x)-\alpha$ and $G(x,\sigma):=g(x)-\beta$ for $\sigma=(\alpha,\beta)\in X^*\times H$. Then $(x^k,\lambda^k)$ satisfies $$F(x^k,\sigma^k)+D_x G(x^k,\sigma^k)^* \lambda^k=0, \quad \lambda^k\in {\mathcal{N}_{K}(G(x^k,\sigma^k))}$$ for all $k$. Applying [@Bonnans2000 Thm. 5.9] yields a contradiction to . The function $\sigma$ is locally Lipschitz-continuous with respect to $(x,\lambda)$, and globally so with respect to $\lambda$. Hence, we can extend the one-sided error bound to $$\label{Eq:DoubleErrorBound} c_1 \sigma(x,\lambda) \le \|x-\bar{x}\|_X+\|\lambda-\bar{\lambda}\|_H \le c_2 \sigma(x,\lambda)$$ for suitable constants $c_1,c_2>0$ and all $(x,\lambda)\in X\times H$ with $x$ near $\bar{x}$. For certain problem classes, it is possible to establish error bounds under weaker assumptions than those given above. The most important example in this direction is if the set $K$ is (generalized) polyhedral, e.g. in nonlinear programming. Roughly speaking, one can use Hoffman’s lemma [@Bonnans2000 Thm. 2.200] to get the “dual part” of the error bound for free, while the primal part again follows from SOSC. As a result, one obtains a primal-dual error bound under SOSC alone (with the restriction that the multiplier is not necessarily unique). Unsurprisingly, this result does not extend to the non-polyhedral case, which shows that additional assumptions such as SRC are inevitable. Let $X:=H:=\ell^2({\mathbb{R}})$ be the space of square-summable real sequences. Consider the optimization problem , with $f(x):=\|x\|_X^2/2$, $g(x):=(x_i/i)_{i=1}^{\infty}$, and $K$ the nonnegative cone in $X$. It is easy to see that $(\bar{x},\bar{\lambda}):=(0,0)$ is the unique KKT point of this problem, and that SOSC holds. Now, let $x^k:=e^k/k$ and $\lambda^k:=-e^k$, where $\{e^k\}$ is the sequence of unit vectors. Then $$\sigma(x^k,\lambda^k)=\|{\mathcal{L}}(x^k,\lambda^k)\|_{X^*}+\|g(x^k)-P_K(g(x^k)+\lambda^k)\|_H =k^{-2}$$ for all $k$. Moreover, $x^k\to \bar{x}$, but $\lambda^k\not\to \bar{\lambda}$. Hence, a local error bound does not hold. (In particular, SRC cannot hold, even though the Lagrange multiplier is actually unique.) A slightly different example is obtained by setting $\hat{x}^k:=e^k/k^2$ and $\hat{\lambda}^k:=-e^k/k$. In this case, $(\hat{x}^k,\hat{\lambda}^k)\to (\bar{x},\bar{\lambda})$, but an easy calculation shows that $$\sigma(\hat{x}^k,\hat{\lambda}^k)=k^{-3} \quad\text{and}\quad \|\hat{x}^k-\bar{x}\|_X+\|\hat{\lambda}^k-\bar{\lambda}\|_H=k^{-2}+k^{-1}.$$ In particular, the error bound is violated even if the multiplier is close to $\bar{\lambda}$. We close this section by noting that the error bound in Theorem \[Thm:ErrorBound\] necessarily implies that the Lagrange multiplier $\bar{\lambda}$ is unique. It is natural to ask whether sufficient conditions can be established which guarantee the error bound property with a nonunique multiplier (as in the statement of Theorem \[Thm:ErrorBoundEquivalence\]). However, it turns out that the resulting conditions are often of technical nature, see [@Bonnans2000 Thm. 4.51], and not easily verified for common problem classes. Therefore, and since the case covered by Theorem \[Thm:ErrorBound\] suffices for our applications, we restrict ourselves to the situation where $\bar{\lambda}$ is unique. The Augmented Lagrangian Method {#Sec:Method} =============================== We now present the augmented Lagrangian method for the variational inequality . The main approach is to penalize the function $g$ and therefore reduce the VI to a sequence of (unconstrained) nonlinear equations. Consider the augmented Lagrangian $$\label{Eq:AL} {\mathcal{L}}_{\rho}:X\times H\to X^*, \quad {\mathcal{L}}_{\rho}(x,\lambda):= F(x)+\rho g'(x)^* \mleft[ g(x)+\frac{\lambda}{\rho}-P_K\mleft( g(x)+\frac{\lambda}{\rho}\mright) \mright].$$ Note that, if $K$ is a cone, then we can simplify the above formula to ${\mathcal{L}}_{\rho}(x,\lambda)=F(x)+g'(x)^* P_{K^{\circ}}(\lambda+\rho g(x))$ by using Moreau’s decomposition [@Bauschke2011; @Moreau1962]. For the construction of our algorithm, we will need a means of controlling the penalty parameters. To this end, we define the utility function $$\label{Eq:V} V(x,\lambda,\rho):=\|{\mathcal{L}}_{\rho}(x,\lambda)\|_{X^*}+ \left\|g(x)-P_K\mleft(g(x)+\frac{\lambda}{\rho}\mright)\right\|_H.$$ This function requires some elaboration. The first term in measures the precision with which the subproblem was solved in the current iteration. The second term is a composite measure of feasibility and complementarity; it arises from an inherent slack variable transformation which is often used to define the augmented Lagrangian for inequality or cone constraints. As a result, the function $V$ measures optimality, feasibility and complementarity at the current iterate. \[Alg:ALM\] - Let $(x^0,\lambda^0)\in X\times H$, $B\subseteq H$ bounded, $\rho_0>0$, $\gamma>1$, $\tau\in(0,1)$, and set $k:=0$. - If $(x^k,\lambda^k)$ satisfies a suitable termination criterion: STOP. - Choose ${w}^k\in B$ and compute an inexact zero (see below) $x^{k+1}$ of ${\mathcal{L}}_{\rho_k}(\cdot,{w}^k)$. - Update the vector of multipliers to $$\label{Eq:MultUpdate} \lambda^{k+1}:=\rho_k \mleft[ g(x^{k+1})+\frac{{w}^k}{\rho_k} -P_K\mleft(g(x^{k+1})+\frac{{w}^k}{\rho_k}\mright) \mright].$$ - If $k=0$ or $$\label{Eq:RhoTest} V(x^{k+1},{w}^k,\rho_k)\le \tau V(x^k,{w}^{k-1},\rho_{k-1})$$ holds, set $\rho_{k+1}:=\rho_k$; otherwise, set $\rho_{k+1}:=\gamma\rho_{k}$. - Set $k\leftarrow k+1$ and go to [(S.1)]{.nodecor}. Let us make some simple observations. First, regardless of the primal iterates $\{x^k\}$, the multipliers $\{\lambda^k\}$ always lie in the polar cone $K_{\infty}^{\circ}$ by Lemma \[Lem:RecessionConePolar\]. Moreover, if $K$ is a cone, then the Moreau decomposition [@Moreau1962] implies that $\lambda^{k+1}=P_{K^{\circ}}({w}^k+\rho_k g(x^{k+1}))$. Secondly, we note that Algorithm \[Alg:ALM\] uses a safeguarded multiplier sequence $\{{w}^k\}$ in certain places where classical augmented Lagrangian methods use the sequence $\{\lambda^k\}$. This bounding scheme goes back to [@Andreani2007; @Pang2005] and is crucial to establishing strong global convergence results for the method [@Andreani2007; @Birgin2010; @Birgin2014; @Kanzow2016]. In practice, one usually tries to keep ${w}^k$ as “close” as possible to $\lambda^k$, e.g. by defining ${w}^k:=P_B(\lambda^k)$, where $B$ (the bounded set from the algorithm) is chosen suitably to allow cheap projections. The third observation is that if the sequence of penalty parameters $\{\rho_k\}$ remains bounded, then yields $V(x^{k+1},{w}^k,\rho_k)\to 0$. In this case, the definition of $V$ implies that both the residual $\|{\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)\|_{X^*}$ of the subproblems and the composite feasibility-complementarity measure converge to zero. Hence, from a theoretical point of view, the case of bounded $\{\rho_k\}$ is the “good” case. In Section \[Sec:LocalConv\], we will actually prove the boundedness of $\{\rho_k\}$ under certain assumptions, and this result crucially depends on the fact that the function $V$ involves both terms from . For the remainder of this paper, we make the following assumption. \[Asm:Subproblems\] There is a null sequence $\{\varepsilon_k\}\subseteq [0,\infty)$ such that $$\|{\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)\|_{X^*}\le \varepsilon_{k+1} \quad\text{for all }k.$$ This assumption is fairly natural and basically asserts that $x^{k+1}$ is an approximate zero point of ${\mathcal{L}}_{\rho_k}(\cdot,{w}^k)$, and that the degree of inexactness vanishes as $k\to\infty$. Global Convergence {#Sec:GlobalConv} ================== In this section, we discuss the global convergence properties of Algorithm \[Alg:ALM\]. Some general results in this direction were obtained in [@Kanzow2017a; @Kanzow2016] for optimization and generalized Nash equilibrium problems by assuming that the sequence $\{x^k\}$ has a limit point which satisfies a suitable constraint qualification. Here, we pursue a slightly different approach. Since the constraints occurring in VIs are often convex, we can use this convexity to directly show that (weak) limit points are solutions of the VI. This idea has the advantage that we do not need any constraint qualification (in return, we do not get much information on the sequence $\{\lambda^k\}$). Recall that we have already assumed $F$ to be continuously differentiable and $g$ twice continuously differentiable (for this section, one degree less would actually be sufficient). We now make the following additional assumptions. \[Asm:GeneralConv\] We assume that $g$ is concave with respect to $K_{\infty}$ (see Section \[Sec:Prelims\]) and that ${\mleft\langle F(x),x-y \mright\rangle}$ is weakly sequentially lsc with respect to $x$ for all $y\in X$. The first of the above conditions ensures the convexity of the set $M$, see Lemma \[Lem:GeneralizedConvexity\]. The second assumption implies, roughly speaking, that weak limit points of a sequence of “approximate solutions” of the VI are exact solutions. Note that this condition has also been used in certain existence results for VIs [@Isac1992]. \[Lem:Feasibility\] Let Assumptions \[Asm:Subproblems\], \[Asm:GeneralConv\] hold, and let $\bar{x}$ be a weak limit point of $\{x^k\}$. Then $\bar{x}$ is a minimizer of the convex function $d_K\circ g$. In particular, if the feasible set $M$ is nonempty, then $\bar{x}$ is feasible. Note that the function $d_K\circ g$ is convex by Lemma \[Lem:GeneralizedConvexity\] and continuous, hence weakly sequentially lower semicontinuous [@Bauschke2011 Thm. 9.1]. If $\{\rho_k\}$ remains bounded, then the penalty updating scheme implies $$d_K(g(x^{k+1})) \le \mleft\| g(x^{k+1})-P_K \mleft( g(x^{k+1})+\frac{{w}^k}{\rho_k} \mright) \mright\|_H \le V(x^{k+1},{w}^k,\rho_k)\to 0$$ and therefore $d_K(g(\bar{x}))=0$. We now assume that $\rho_k\to\infty$ and define the auxiliary functions $h_k(x)=d_K^2(g(x)+{w}^k/\rho_k)$. Note that $h_k$ is continuously differentiable [@Bauschke2011 Cor. 12.30]. Let $x^{k+1}{\rightharpoonup}_{\mathcal{K}}\bar{x}$ for some index set $\mathcal{K}\subseteq {\mathbb{N}}$ and assume that there is a point $y\in X$ with $d_K(g(y))<d_K(g(\bar{x}))$. The weak sequential lower semicontinuity of $d_K\circ g$ and the boundedness of $\{{w}^k\}$ imply that $$\liminf_{k\in\mathcal{K}} h_k(x^{k+1}) =\liminf_{k\in\mathcal{K}} d_K^2 \bigl(g(x^{k+1})+{w}^k/\rho_k\bigr) \ge d_K^2(g(\bar{x}))$$ and $h_k(y)\to d_K^2(g(y))$. Hence, there is a constant $c_1>0$ such that $h_k(x^{k+1})-h_k(y)\ge c_1$ for all $k\in\mathcal{K}$ sufficiently large. Since $h_k$ is convex by Lemma \[Lem:GeneralizedConvexity\], it follows that $$\label{Eq:LemFeasibility1} {\mleft\langle h_k'(x^{k+1}),y-x^{k+1} \mright\rangle}\le h_k(y)-h_k(x^{k+1})\le -c_1$$ for all $k\in\mathcal{K}$ sufficiently large. Now, let $\{\varepsilon_k\}$ be the sequence from Assumption \[Asm:Subproblems\]. Using [@Bauschke2011 Cor. 12.30] for the derivative of $h_k$, we obtain $$\begin{aligned} -\varepsilon_{k+1} \|y-x^{k+1}\|_X & \le {\mleft\langle {\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k),y-x^{k+1} \mright\rangle} \\ & = {\mleft\langle F(x^{k+1}),y-x^{k+1} \mright\rangle}+\frac{\rho_k}{2}{\mleft\langle h_k'(x^{k+1}),y-x^{k+1} \mright\rangle}. \end{aligned}$$ By Assumption \[Asm:GeneralConv\], the function ${\mleft\langle F(x),x-y \mright\rangle}$ is weakly sequentially lsc with respect to $x$. Hence, there is a constant $c_2\in{\mathbb{R}}$ such that ${\mleft\langle F(x^{k+1}),y-x^{k+1} \mright\rangle}\le c_2$ for all $k\in\mathcal{K}$. This together with implies $$-\varepsilon_{k+1} \|y-x^{k+1}\|_X\le c_2-\frac{\rho_k c_1}{2}\to -\infty.$$ Since $\{x^{k+1}\}_{\mathcal{K}}$ is bounded and $\varepsilon_k\to 0$, this is a contradiction. Note that Lemma \[Lem:Feasibility\] guarantees that every weak limit point $\bar{x}$ automatically minimizes the constraint violation even if the feasible set $M$ is empty. We now prove the optimality of limit points. To this end, we first need a technical lemma which essentially asserts some sort of “approximate normality” of $\lambda^{k+1}$ with respect to $K$ and $g(x^{k+1})$, the latter not necessarily being an element of $K$. Note that the result does not require any assumptions but directly follows from the definition of $\lambda^{k+1}$ as well as the updating scheme . \[Lem:Limsup\] We have $\limsup_{k\to\infty}{\mleft( \lambda^{k+1},y-g(x^{k+1}) \mright)}\le 0$ for all $y\in K$. Let $y\in K$ and define the sequence $s^{k+1}:=P_K(g(x^{k+1})+{w}^k/\rho_k)$. Then $s^{k+1}\in K$ and it follows from [@Bauschke2011 Prop. 6.46] that $\lambda^{k+1}\in{\mathcal{N}_{K}(s^{k+1})}$. Moreover, we have $$\label{Eq:LemLimsup1} g(x^{k+1})=\frac{\lambda^{k+1}-{w}^k}{\rho_k}+s^{k+1}.$$ This yields $$\begin{aligned} {\mleft( \lambda^{k+1},y-g(x^{k+1}) \mright)} & = {\mleft( \lambda^{k+1},y-\frac{1}{\rho_k}(\lambda^{k+1}-{w}^k)-s^{k+1} \mright)} \notag \\ & \le \frac{1}{\rho_k} \Bigl[{\mleft( \lambda^{k+1},{w}^k \mright)}-\|\lambda^{k+1}\|_H^2\Bigr] \label{Eq:LemLimsup2}, \end{aligned}$$ where we used $\lambda^{k+1}\in{\mathcal{N}_{K}(s^{k+1})}$ for the last inequality. Now, if $\{\rho_k\}$ is bounded, then and imply $\|\lambda^{k+1}-{w}^k\|_H/\rho_k\to 0$ and therefore $\|\lambda^{k+1}-{w}^k\|_H\to 0$. This yields the boundedness of $\{\lambda^{k+1}\}$ in $H$ as well as ${\mleft( \lambda^{k+1},{w}^k \mright)}-\|\lambda^{k+1}\|_H^2={\mleft( \lambda^{k+1},{w}^k-\lambda^{k+1} \mright)}\to 0$. Hence, the desired results follows from . We now assume that $\rho_k\to\infty$. Note that is a quadratic function in $\lambda$. A simple calculation therefore shows that $${\mleft( \lambda^{k+1},y-g(x^{k+1}) \mright)}\le \frac{1}{4\rho_k}\|{w}^k\|_H^2.$$ The boundedness of $\{{w}^k\}$ now implies $\limsup_{k\to\infty} {\mleft( \lambda^{k+1},y-g(x^{k+1}) \mright)}\le 0$. The above result can be stated more concisely if $K$ is a cone. By inserting $0\in K$ into the inequality, it is easy to see that it is equivalent to $\liminf_{k\to\infty} {\mleft( \lambda^{k+1},g(x^{k+1}) \mright)}\ge 0$. We now turn to the main global convergence result. \[Thm:Optimality\] Let Assumptions \[Asm:Subproblems\], \[Asm:GeneralConv\] hold, and let $\bar{x}$ be a weak limit point of $\{x^k\}$. If the feasible set $M$ is nonempty, then $\bar{x}$ is feasible and solves the VI. Let $x^{k+1}{\rightharpoonup}_{\mathcal{K}}\bar{x}$ for some $\mathcal{K}\subseteq{\mathbb{N}}$. The feasibility claim follows from Lemma \[Lem:Feasibility\]. For the optimality, let $y\in M$ be any feasible point. Then ${\mleft\langle {\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k),y-x^{k+1} \mright\rangle}\ge -\varepsilon_{k+1} \|y-x^{k+1}\|_X$ by Assumption \[Asm:Subproblems\] and, since ${\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)={\mathcal{L}}(x^{k+1},\lambda^{k+1})$, we get $$\begin{aligned} -\varepsilon_{k+1} \|y-x^{k+1}\|_X & \le {\mleft\langle F(x^{k+1})+g'(x^{k+1})^* \lambda^{k+1},y-x^{k+1} \mright\rangle} \\ & \le {\mleft\langle F(x^{k+1}),y-x^{k+1} \mright\rangle}+{\mleft( \lambda^{k+1},g(y)-g(x^{k+1}) \mright)}, \end{aligned}$$ where we used the fact that $x\mapsto {\mleft( \lambda^{k+1},g(x) \mright)}$ is convex by Lemma \[Lem:GeneralizedConvexity\] (recall that $\lambda^{k+1}\in K_{\infty}^{\circ}$). Using $\varepsilon_k\to 0$ and Lemma \[Lem:Limsup\], we now obtain $\liminf_{k\in\mathcal{K}}{\mleft\langle F(x^{k+1}),y-x^{k+1} \mright\rangle}\ge 0$. Since ${\mleft\langle F(x),x-y \mright\rangle}$ is weakly sequentially lsc, this implies ${\mleft\langle F(\bar{x}),y-\bar{x} \mright\rangle}\ge 0$. Local Convergence {#Sec:LocalConv} ================= We will now consider the local convergence characteristics of Algorithm \[Alg:ALM\]. A key ingredient is the error bound property from Section \[Sec:ErrorBounds\] which allows us to estimate the distance from $(x^k,\lambda^k)$ to $(\bar{x},\bar{\lambda})$ by using the function $\sigma$ from . \[Lem:ConvSOSC\] Let Assumption \[Asm:Subproblems\] hold and let $(\bar{x},\bar{\lambda})$ be a KKT point satisfying the error bound . Then there is an $r>0$ such that, if $x^k\in B_r(\bar{x})$ for all $k$ and $d_K(g(x^k))\to 0$, then $(x^k,\lambda^k)\to (\bar{x},\bar{\lambda})$. By Assumption \[Asm:Subproblems\], we have ${\mathcal{L}}(x^{k+1},\lambda^{k+1})={\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)\to 0$. Hence, in view of the error bound property, it suffices to show that $g(x^{k+1})-P_K(g(x^{k+1})+\lambda^{k+1})\to 0$. To this end, define the sequence $s^{k+1}:=P_K(g(x^{k+1})+{w}^k/\rho_k)$. Then $s^{k+1}\in K$ and, as noted before, $\lambda^{k+1}\in{\mathcal{N}_{K}(s^{k+1})}$ for all $k$. We now use the fact that $y\mapsto y-P_K(y+\lambda^{k+1})$ is nonexpansive, which is an easy consequence of [@Bauschke2011 Cor. 4.10]. Therefore, the inverse triangle inequality yields $$\label{Eq:LemConvSOSC1} \begin{aligned} \lefteqn{\|g(x^{k+1})-P_K(g(x^{k+1})+\lambda^{k+1})\|_H} & \\ & \le \|g(x^{k+1})-s^{k+1}\|_H+\|s^{k+1}-P_K(s^{k+1}+\lambda^{k+1})\|_H. \end{aligned}$$ The last term is equal to zero since $\lambda^{k+1}\in{\mathcal{N}_{K}(s^{k+1})}$, cf. [@Bauschke2011 Cor. 6.46]. Hence, to complete the proof, we only need to show that $\|s^{k+1}-g(x^{k+1})\|_H\to 0$. If $\{\rho_k\}$ is bounded, then this readily follows from the penalty updating scheme . On the other hand, if $\rho_k\to\infty$, then $$\|s^{k+1}-g(x^{k+1})\|_H \le \|s^{k+1}-P_K(g(x^{k+1}))\|_H + d_K(g(x^{k+1}))\to 0,$$ where we used the nonexpansiveness of the projection operator. The above lemma gives us some information about the behavior of zeros of the augmented Lagrangian in a neighborhood of $\bar{x}$. Note that the assumption $d_K(g(x^{k+1}))\to 0$ asserts that the iterates become (asymptotically) feasible and is often satisfied in practice, see also Lemma \[Lem:Feasibility\]. For the remaining analysis, we now make the following assumption. \[Asm:LocalConv\] We assume that $(\bar{x},\bar{\lambda})$ is a KKT point of the VI which satisfies the local error bound . Moreover, the sequence $\{(x^k,\lambda^k)\}$ from Algorithm \[Alg:ALM\] converges strongly to $(\bar{x},\bar{\lambda})$, and we have ${w}^k=\lambda^k$ for all $k$ sufficiently large. One of the above assumptions which might require some elaboration is ${w}^k=\lambda^k$ for all $k$. The boundedness of $\{{w}^k\}$ is key to establishing global convergence of the algorithm, see Section \[Sec:GlobalConv\]. Since $\lambda^k\to\bar{\lambda}$ in our setting, we do not need to force boundedness of $\{{w}^k\}$ and can simply set ${w}^k:=\lambda^k$ for all $k$. (In the context of Algorithm \[Alg:ALM\], we formally need to choose the bounded set $B$ sufficiently large to allow this.) We will now prove convergence rates for the primal-dual sequence $\{(x^k,\lambda^k)\}$. Since the distance of $(x^k,\lambda^k)$ to $(\bar{x},\bar{\lambda})$ admits both upper and lower estimates relative to the residual terms $\sigma_k:=\sigma(x^k,\lambda^k)$ by , we will largely base our analysis on the sequence $\{\sigma_k\}$, and the results on the primal-dual sequence $\{(x^k,\lambda^k)\}$ will follow directly. \[Lem:SigmaConv\] Let Assumptions \[Asm:Subproblems\], \[Asm:LocalConv\] hold, and let $\sigma_k:=\sigma(x^k,\lambda^k)$. Then there is a constant $c_1>0$ such that $$\mleft(1-\frac{c_1}{\rho_k}\mright) \sigma_{k+1} \le \varepsilon_{k+1} + \frac{c_1}{\rho_k}\sigma_k$$ for all $k\in{\mathbb{N}}$ sufficiently large. Observe that ${\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)={\mathcal{L}}(x^{k+1},\lambda^{k+1})$ for all $k$. By Assumption \[Asm:Subproblems\] and the definition of $\sigma_k$, we therefore have $$\label{Eq:LemSigmaConv1} \sigma_{k+1}\le \varepsilon_{k+1}+\|g(x^{k+1})-P_K(g(x^{k+1})+\lambda^{k+1})\|_H.$$ Now, let $k\in{\mathbb{N}}$ be large enough so that ${w}^k=\lambda^k$. Consider again the sequence $s^{k+1}:=P_K(g(x^{k+1})+\lambda^k/\rho_k)$. Using , we see that $$\label{Eq:LemSigmaConv2} \|g(x^{k+1})-P_K(g(x^{k+1})+\lambda^{k+1})\|_H \le \|g(x^{k+1})-s^{k+1}\|_H = \frac{\|\lambda^{k+1}-\lambda^k\|_H}{\rho_k}.$$ Inserting this into and using the triangle inequality yields $$\sigma_{k+1}\le \varepsilon_{k+1}+\frac{1}{\rho_k} \bigl(\|\lambda^{k+1}-\bar{\lambda}\|_H+\|\lambda^k-\bar{\lambda}\|_H\bigr).$$ Now, by Assumption \[Asm:LocalConv\] and since $x^k\to\bar{x}$, there is a $c_1>0$ such that $\|\lambda^k-\bar{\lambda}\|_H\le c_1 \sigma_k$ for all $k\in{\mathbb{N}}$ sufficiently large. Hence, $$\sigma_{k+1}\le \varepsilon_{k+1}+\frac{c_1}{\rho_k}\sigma_{k+1} +\frac{c_1}{\rho_k}\sigma_k,$$ again for $k\in{\mathbb{N}}$ sufficiently large. Reordering gives the desired result. With the above lemma, it is easy to deduce convergence rates for the primal-dual sequence $\{(x^k,\lambda^k)\}$. \[Thm:ConvRate\] Let Assumptions \[Asm:Subproblems\], \[Asm:LocalConv\] hold, and let $\varepsilon_{k+1}=o(\sigma_k)$. Then: 1. For every $q\in(0,1)$, there is a $\bar{\rho}_q>0$ such that, if $\rho_k\ge \bar{\rho}_q$ for sufficiently large $k$, then $(x^k,\lambda^k)\to (\bar{x},\bar{\lambda})$ Q-linearly with rate $q$. 2. The sequence of penalty parameters $\{\rho_k\}$ remains bounded. Let $k\in{\mathbb{N}}$ be sufficiently large so that ${w}^k=\lambda^k$. By Lemma \[Lem:SigmaConv\], if $\rho_k$ is large enough so that $1-c_1/\rho_k>0$, then $$\label{Eq:ThmConvRate1} \frac{\sigma_{k+1}}{\sigma_k}\le \frac{c_1}{\rho_k-c_1}+o(1).$$ Using and the local Lipschitz-continuity of $\sigma$ (e.g. equation ), it is easy to derive (a). For (b), let us again consider the sequence $s^{k+1}=P_K(g(x^{k+1})+\lambda^k/\rho_k)$, and define $V_{k+1}:=V(x^{k+1},{w}^k,\rho_k)=\|{\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)\|_{X^*}+\|g(x^{k+1})-s^{k+1}\|_H$. To prove boundedness of $\{\rho_k\}$, we need to show that $V_{k+1}\le\tau V_k$ for sufficiently large $k$. Using and ${\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)={\mathcal{L}}(x^{k+1},\lambda^{k+1})$, we obtain $$V_{k+1}\ge \|{\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)\|_{X^*} + \|g(x^{k+1})-P_K(g(x^{k+1})+\lambda^{k+1})\|_H=\sigma_{k+1}$$ for all $k\in{\mathbb{N}}$ and, from and Assumption \[Asm:Subproblems\], $$\begin{aligned} V_{k+1} =\|{\mathcal{L}}_{\rho_k}(x^{k+1},{w}^k)\|_{X^*}+ \frac{\|\lambda^{k+1}-\lambda^k\|_H}{\rho_k} & \le \varepsilon_{k+1} + \frac{\|\lambda^{k+1}-\bar{\lambda}\|_H+ \|\lambda^k-\bar{\lambda}\|_H}{\rho_k} \\ & \le \varepsilon_{k+1} + \frac{c}{\rho_k}(\sigma_{k+1}+\sigma_k) \end{aligned}$$ for all $k\in{\mathbb{N}}$ sufficiently large, where $c$ is the constant from (recall that $x^k\to\bar{x}$). Putting these inequalities together yields $$\frac{V_{k+1}}{V_k}\le \frac{\varepsilon_{k+1}}{\sigma_k}+\frac{c}{\rho_k} \frac{\sigma_{k+1}+\sigma_k}{\sigma_k}=\frac{\varepsilon_{k+1}}{\sigma_k} +\frac{c}{\rho_k}\mleft( 1+\frac{\sigma_{k+1}}{\sigma_k} \mright).$$ If we now assume that $\rho_k\to\infty$, then it is easy to deduce from and $\varepsilon_{k+1}=o(\sigma_k)$ that $V_{k+1}/V_k\to 0$. Hence, $V_{k+1}/V_k\le\tau$ for all $k$ sufficiently large, which contradicts the assumption that $\rho_k\to\infty$. The assumption $\varepsilon_{k+1}=o(\sigma_k)$ in the above theorem says that, roughly speaking, the degree of inexactness should be small enough to not affect the rate of convergence. Note that we are comparing $\varepsilon_{k+1}$ to the optimality measure $\sigma_k$ of the previous iterates $(x^k,\lambda^k)$. Hence, it is easy to ensure this condition in practice, for instance, by always computing the next iterate $x^{k+1}$ with a precision $\varepsilon_{k+1}\le z_k \sigma_k$ for some fixed null sequence $z_k$. Let us also note that one can easily adapt the proof of Theorem \[Thm:ConvRate\](a) to conclude that $(x^k,\lambda^k)\to (\bar{x},\bar{\lambda})$ Q-superlinearly if $\rho_k\to\infty$. However, the resulting assertion would be redundant because part (b) of the theorem actually implies the boundedness of $\{\rho_k\}$. On the other hand, the proof of (b) uses the specific penalty updating scheme with the function $V$ from , whereas the proof of (a) does not depend on the penalty updating rule at all. If we replace $V$ by the function $$\tilde{V}(x,\lambda,\rho):=\left\|g(x)-P_K\mleft(g(x)+ \frac{\lambda}{\rho}\mright)\right\|_H$$ (which is just the second term from the definition of $V$), it is rather easy to see that the assertions of Lemmas \[Lem:ConvSOSC\], \[Lem:SigmaConv\] and Theorem \[Thm:ConvRate\](a) remain true. Additionally, we obtain superlinear convergence if $\rho_k\to\infty$, but we do not get boundedness of $\{\rho_k\}$. Let us close this section by mentioning two special cases for which different or stronger rate of convergence results can be obtained. The first case is that of convex optimization. In this case, the augmented Lagrangian algorithm is essentially equivalent to a proximal-point method (applied to the dual problem), and this duality can be used to establish certain rate of convergence results, see [@Dong2015; @Guler1991; @Kanzow2017b; @Rockafellar1976]. The second special case, which was already mentioned in the introduction, is that of nonlinear programming-type (NLP) constraints. Here, it is possible to prove local linear convergence under SOSC only [@Fernandez2012]. Constraint qualifications are not needed since the set $K$ is polyhedral, see the discussion in the introduction and in [@Bonnans2000 Section 4.4]. However, the techniques used in [@Fernandez2012] rely heavily on finite-dimensional arguments and the specific structure of NLP constraints, and thus cannot readily be adapted to our setting. Applications {#Sec:Applic} ============ This section describes some applications of our method. Recall that our variational setting encompasses constrained optimization problems . This opens up a broad spectrum of applications, including, as mentioned before, standard nonlinear programming (NLP). However, there already is a plethora of literature on this topic, in particular the recent paper [@Fernandez2012]. Moreover, the discussion in Section \[Sec:ErrorBounds\] indicates that NLP is actually a very confined special case which does not allow us to demonstrate the full generality of our approach. In particular, NLPs are inherently finite-dimensional and the corresponding set $K$ is polyhedral, which is very restrictive. As a result, we focus on problems in function space settings where the constraint set is almost never polyhedral. This section contains two examples in this direction: we begin with a simple linear-quadratic optimal control problem and then continue with multiobjective optimal control in a Nash equilibrium framework. For both examples, we first present the general problem setting and then explain why the regularity properties from Assumption \[Asm:LocalConv\] are satisfied. To verify our theoretical results in practice, we follow a standard approach by which we discretize the respective problems and then analyze the behavior of the algorithm for increasingly fine levels of discretization. As we shall see, the assertions of the previous section can be verified in both examples, and independently of the dimension $n$, which indicates that our results are valid. An Optimal Control Problem {#Sec:ApplicOptCont} -------------------------- Let $\Omega\subseteq{\mathbb{R}}^d$, $d\in\{2,3\}$, be a bounded domain. The example presented in this section consists of minimizing $$J(y,u):=\frac{1}{2}\|y-y_d\|_{L^2(\Omega)}^2+\frac{\alpha}{2}\|u\|_{L^2(\Omega)}^2$$ subject to $y\in H_0^1(\Omega)\cap C(\bar{\Omega})$ and $u\in L^2(\Omega)$ satisfying the partial differential equation (PDE) and pointwise control constraints $$-\Delta y=u+f \quad\text{and}\quad u_a\le u\le u_b.$$ Here, $y_d,u_a,u_b\in L^2(\Omega)$ are problem-specific and $\alpha>0$ is a regularization parameter. It is well-known that, for every right-hand side $w\in L^2(\Omega)$, the Poisson equation $-\Delta y=w$ admits a uniquely determined weak solution $y=S w\in H_0^1(\Omega)\cap C(\bar{\Omega})$, and the resulting operator $S:L^2(\Omega)\to H_0^1(\Omega)\cap C(\bar{\Omega})$ is linear and compact [@Troeltzsch2010 Thm. 4.17]. Writing $y_u:=S(u+f)$, we can now restate the objective function as $$\bar{J}(u):=J(y_u,u)=\frac{1}{2}\|y_u-y_d\|_{L^2(\Omega)}^2 +\frac{\alpha}{2}\|u\|_{L^2(\Omega)}^2.$$ This function together with the control constraints $u_a\le u\le u_b$ is typically called the *reduced formulation* of the optimal control problem and directly fits into our variational framework by setting $X:=H:=L^2(\Omega)$, $F(u):=\bar{J}'(u)$, and $$g(u):=u, \quad K:=\{ u\in X: u_a\le u\le u_b \}.$$ Since $\bar{J}$ is strongly convex and $g$ is just the identity mapping on $X=H$, it is easy to show that the above problem admits a unique primal-dual solution, and that both SOSC and SRC hold. Hence, by Theorem \[Thm:ErrorBound\], the KKT system is upper Lipschitz stable and the control problem admits a local error bound. We now present a numerical example which is constructed in such a way that the optimal solution is known analytically. Let $\Omega:=(0,1)^2$ be the unit square and define $\alpha:=1$, $u_a:= -0.5$, $u_b:= 0.5$. Consider the functions $$\bar{y}(x):=\sin (\pi x_1)\sin (\pi x_2), \quad \bar{p}(x):=\sin (2\pi x_1)\sin (2\pi x_2),$$ and set $y_d:=\bar{y}+\Delta \bar{p}$. Now, using $\bar{u}:=P_{[u_a,u_b]}(-\bar{p}/\alpha)$ and $f:=-\Delta \bar{y}-\bar{u}$, it is easy to see that $\bar{u}$ is a solution to the problem. Moreover, $\bar{y}$ is the corresponding state, $\bar{p}$ the so-called adjoint state [@Troeltzsch2010], and the Lagrange multiplier is given by $\bar{\lambda}:=-\bar{p}-\alpha \bar{u}$. For the numerical testing, we discretized the problem by means of a uniform grid with $n\in{\mathbb{N}}$ interior points per row or column (i.e., $n^2$ points in total) and approximated the Laplace operator by a standard five-point finite difference scheme. It is easy to argue that the resulting discretized versions of $\bar{J}$ and $g$ again satisfy the (now finite-dimensional) SOSC and SRC assumptions (since $\bar{J}$ is strongly convex and $g$ is the identity mapping). Hence, we can expect locally fast convergence of the augmented Lagrangian method, both from a continuous and a discrete point of view. The implementation of the algorithm was done in MATLAB^^ and uses the parameters $$(u^0,\lambda^0):=(0,0), \quad B:=[-10^6,10^6]^{n^2}, \quad \rho_0:=1, \quad \gamma:=10, \quad \tau:=0.5,$$ together with the formula ${w}^k:=P_B(\lambda^k)$ for the safeguarded multipliers (see the discussion in Section \[Sec:Method\]). Moreover, we use the termination criteria $\sigma(x,\lambda)\le 10^{-8}$ and $\|{\mathcal{L}}_{\rho_k}(x,{w}^k)\|\le 10^{-10}$ for the outer and inner iterations, respectively, where the norm is the discrete $L^2$-norm. The subproblems are nonlinear equations which we solve with a standard semismooth Newton method. It should be noted that, while the discrete Laplacian is a sparse matrix, the solution operator $S$ which occurs in the function $F$ is nearly dense. To circumvent this issue, we use a sparse Cholesky factorization of the negative Laplacian to obtain an “implicit” form of $S$ and solve the Newton equations with the MATLAB^^ conjugate gradient method `pcg`. ----- ---------- ------------ ------------------------- ---------- ------------ ------------------------- ---------- ------------ ------------------------- $k$ $\rho_k$ $\sigma_k$ $\operatorname{dist}_k$ $\rho_k$ $\sigma_k$ $\operatorname{dist}_k$ $\rho_k$ $\sigma_k$ $\operatorname{dist}_k$ 0 1 5.08e-01 5.43e-01 1 5.02e-01 5.37e-01 1 5.01e-01 5.35e-01 1 1 8.58e-02 1.71e-01 1 8.47e-02 1.69e-01 1 8.44e-02 1.69e-01 2 1 4.29e-02 8.55e-02 1 4.23e-02 8.46e-02 1 4.22e-02 8.44e-02 3 10 2.15e-02 4.26e-02 10 2.12e-02 4.23e-02 10 2.11e-02 4.22e-02 4 10 1.95e-03 3.57e-03 10 1.92e-03 3.83e-03 10 1.92e-03 3.84e-03 5 10 1.77e-04 4.44e-04 10 1.75e-04 3.29e-04 10 1.74e-04 3.48e-04 6 10 1.61e-05 5.08e-04 10 1.59e-05 2.85e-05 10 1.59e-05 3.04e-05 7 10 1.47e-06 5.21e-04 10 1.45e-06 3.18e-05 10 1.44e-06 2.08e-06 8 10 1.33e-07 5.22e-04 10 1.31e-07 3.29e-05 10 1.31e-07 1.96e-06 9 10 1.21e-08 5.22e-04 10 1.20e-08 3.30e-05 10 1.19e-08 2.06e-06 10 10 1.10e-09 5.22e-04 10 1.09e-09 3.30e-05 10 1.08e-09 2.07e-06 ----- ---------- ------------ ------------------------- ---------- ------------ ------------------------- ---------- ------------ ------------------------- : Numerical results for the optimal control problem.[]{data-label="Tab:OptCont"} Table \[Tab:OptCont\] lists some numerical results for different values of $n$, where each line contains the penalty parameter $\rho_k$, the optimality measure $\sigma_k$ and the distance $\operatorname{dist}_k$ of $(u^k,\lambda^k)$ to $(\bar{u},\bar{\lambda})$. The results suggest that the algorithm works very well for this problem; in particular, the number of required iterations remains constant as $n$ increases. Moreover, we also observe that the rate of convergence appears to be proportional to $1/\rho_k$, as suggested by the theory. It should be noted, however, that the distances $\operatorname{dist}_k$ stop decreasing after a certain point because of the inexactness induced by the discretization; in particular, if we discretize the (known) optimal solution pair $(\bar{u},\bar{\lambda})$, we do not obtain an *exact* solution of the discretized problem. This phenomenon is also evidenced by the fact that the “limit” value of $\operatorname{dist}_k$ decreases as $n$ increases. We close this section with an important remark on the analytical representation of the feasible set. This observation is crucial and was in fact one of our main motivations to consider constraint sets $K$ which are not necessarily cones. \[Rem:BoxConstraints\] It is important that we define the constraint system with $g$ and $K$ as above. Indeed, the alternative formulation of the box constraints as $\hat{g}(u)\in \hat{K}$ with $$\hat{g}(u):=(u-u_a, u_b-u), \quad \hat{K}:=\{(v,w)\in L^2(\Omega)^2: v,w\ge 0 \},$$ may seem advantageous at first glance (since $\hat{K}$ is a closed convex cone, whereas $K$ is not). However, in this formulation, the strict Robinson condition is not satisfied. In fact, the function $\hat{g}$ does not even satisfy the standard Robinson constraint qualification (RCQ) [@Bonnans2000] or the equivalent regularity condition of Zowe and Kurcyusz [@Zowe1979]. We refer the reader to [@Troeltzsch2010] for a formal proof; an alternative way to verify this irregularity is to note that if RCQ holds, then it remains stable under small perturbations of the constraint function [@Bonnans2000]. However, even if $u_a$ and $u_b$ are “well separated”, it is fairly easy to construct small perturbations (in the sense of $L^2$) which make the lower and upper bounds coincide on some set of positive measure. If this happens, then the set of Lagrange multipliers corresponding to a local minimum is unbounded, and RCQ is violated. Optimal Control in a Nash Equilibrium Framework ----------------------------------------------- We now present a generalization of the optimal control problem from the previous section by considering it in a multi-player framework [@Borzi2013; @Dreves2016; @Kanzow2017a]. The result is a Nash equilibrium problem (NEP) of two players with control variables $u_1,u_2\in L^2(\Omega)$ and a state variable $y\in H_0^1(\Omega)\cap C(\bar{\Omega})$, where $\Omega\subseteq{\mathbb{R}}^d$, $d\in\{2,3\}$, is again a bounded domain. Similarly to before, each player attempts to minimize the objective function $$J_i(y,u_i):=\frac{1}{2}\|y-y_d^i\|_{L^2(\Omega)}^2+ \frac{\alpha_i}{2}\|u_i\|_{L^2(\Omega)}^2$$ with respect to $u_i$, subject to the partial differential equation $-\Delta y=u_1+u_2+f$ and the pointwise control constraints $a_i \le u_i \le b_i$ with $a_i,b_i\in L^2(\Omega)$. The remaining problem parameters satisfy $\alpha_i>0$ and $y_d^i\in L^2(\Omega)$ for all $i$. As in Section \[Sec:ApplicOptCont\], we can use the compact linear solution operator $S:L^2(\Omega)\to H_0^1(\Omega)\cap C(\bar{\Omega})$ and the resulting control-to-state mapping $y_u:=S(u_1+u_2+f)$ to transform the objective functions to $$\bar{J}_i(u):=J_i(y_u,u_i) =\frac{1}{2}\|y_u-y_d^i\|_{L^2(\Omega)}^2 +\frac{\alpha_i}{2}\|u_i\|_{L^2(\Omega)}^2,$$ where $u:=(u_1,u_2)$. To establish the connection with our variational problem , , we only need to make some definitions and use the well-known correspondence between NEPs and VIs [@Facchinei2007; @Kanzow2017a]. Define $X:=H:=L^2(\Omega)^2$, $F(u):=\bigl(D_{u_1} \bar{J}_1(u),D_{u_2} \bar{J}_2(u)\bigr)$, and $$g(u_1,u_2):=(u_1,u_2), \quad K:=\{(u_1,u_2)\in X: a_i\le u_i\le b_i \}.$$ Then it is easy to see that the NEP is equivalent to the VI (and , since the feasible set is convex). The existence of a solution of the NEP (and of the VI) can be shown as in [@Borzi2013]; moreover, since $g$ is the identity operator on $X=H$, SRC holds and the problem admits a unique Lagrange multiplier. Finally, an easy calculation shows that $$F'(u)= \begin{pmatrix} S^* S+\alpha_1 I & S^* S \\ S^* S & S^* S+\alpha_2 I \end{pmatrix},$$ where $I$ is the identity operator on $L^2(\Omega)$, see [@Kanzow2017a]. It follows that $F$ is strongly monotone and, since $g$ is linear, the problem automatically satisfies SOSC and therefore admits a local error bound by Theorem \[Thm:ErrorBound\]. Moreover, it is easy to see that the same holds for the discretized problems presented below. We now present some numerical results for the example from [@Borzi2013]. The setting is again constructed in such a way that the optimal solution is known. In fact, the construction is very similar to the one from the previous section: let $\Omega:=(0,1)^2$ be the unit square and define $\alpha_i:=1$, $a_i:=-0.5$, and $b_i:=0.5$ for all $i$. Consider the functions $$\bar{y}(x) :=\sin (\pi x_1)\sin (\pi x_2), \quad \begin{aligned} \bar{p}_1(x) & :=-\sin (2\pi x_1)\sin (2\pi x_2), \\ \bar{p}_2(x) & :=-\sin (3\pi x_1)\sin (3\pi x_2), \end{aligned}$$ as well as $y_d^i:=\bar{y}+\Delta \bar{p}_i$, $\bar{u}_i:=P_{[a_i,b_i]}(-\bar{p}_i/\alpha_i)$ for all $i$, and finally $f:=-\Delta \bar{y}-\bar{u}_1-\bar{u}_2$. Then it is easy to see that $\bar{u}$ is a Nash equilibrium. The corresponding state is given by $\bar{y}$, the variables $\bar{p}_i$ are the adjoint states of the players, and the Lagrange multiplier is given by $\bar{\lambda}:=(-\bar{p}_1-\alpha_1\bar{u}_1,-\bar{p}_2-\alpha_2\bar{u}_2)$. ----- ---------- ------------ ------------------------- ---------- ------------ ------------------------- ---------- ------------ ------------------------- $k$ $\rho_k$ $\sigma_k$ $\operatorname{dist}_k$ $\rho_k$ $\sigma_k$ $\operatorname{dist}_k$ $\rho_k$ $\sigma_k$ $\operatorname{dist}_k$ 0 1 5.08e-01 5.43e-01 1 5.02e-01 5.37e-01 1 5.01e-01 5.35e-01 1 1 8.59e-02 1.71e-01 1 8.47e-02 1.69e-01 1 8.44e-02 1.69e-01 2 1 4.30e-02 8.54e-02 1 4.23e-02 8.46e-02 1 4.22e-02 8.44e-02 3 10 2.15e-02 4.24e-02 10 2.12e-02 4.23e-02 10 2.11e-02 4.22e-02 4 10 1.95e-03 3.41e-03 10 1.92e-03 3.81e-03 10 1.92e-03 3.83e-03 5 10 1.78e-04 8.13e-04 10 1.75e-04 3.17e-04 10 1.74e-04 3.47e-04 6 10 1.61e-05 8.95e-04 10 1.59e-05 5.08e-05 10 1.59e-05 2.96e-05 7 10 1.47e-06 9.08e-04 10 1.45e-06 5.63e-05 10 1.44e-06 3.23e-06 8 10 1.33e-07 9.09e-04 10 1.31e-07 5.74e-05 10 1.31e-07 3.50e-06 9 10 1.21e-08 9.09e-04 10 1.20e-08 5.75e-05 10 1.19e-08 3.59e-06 10 10 1.10e-09 9.09e-04 10 1.09e-09 5.75e-05 10 1.08e-09 3.60e-06 ----- ---------- ------------ ------------------------- ---------- ------------ ------------------------- ---------- ------------ ------------------------- : Numerical results for the optimal control Nash equilibrium problem.[]{data-label="Tab:OptContNEP"} The implementation of the augmented Lagrangian method for the above problem is similar to that of the previous section. More precisely, we use the same set of parameters, the same termination criteria, and the same method for the solution of the subproblems. The corresponding numerical results are given in Table \[Tab:OptContNEP\], where each line contains the values of the penalty parameter $\rho_k$, the optimality measure $\sigma_k$, and the distance $\operatorname{dist}_k$ of $(u^k,\lambda^k)$ to $(\bar{u},\bar{\lambda})$. We observe good consistency of the results with our established theory; in particular, the rate of convergence is roughly proportional to $1/\rho_k$. We also highlight once again that the distances $\operatorname{dist}_k$ do not converge to zero because of the inexactness induced by the discretization. We close this section by noting that, as explained in Remark \[Rem:BoxConstraints\] for the standard (single-objective) optimal control problem, it is very important that we define $g$ and $K$ precisely as we did in order to ensure the fulfillment of the strict Robinson condition. Parameter Estimation in Elliptic Systems {#Sec:ApplicPara} ---------------------------------------- This example is based on the theory in [@Ito1990b; @Ito1991]. For the sake of simplicity, we restrict ourselves to the one-dimensional case. Let $\Omega\subseteq {\mathbb{R}}$ be a bounded interval and consider the elliptic differential equation $$\label{Eq:ParaEstPDE} -\nabla (q \nabla u )=f, \quad u\in H_0^1(\Omega),$$ where $q\in H^1(\Omega)$ and $f\in H^{-1}(\Omega)$. The parameter estimation problem now consists of the minimization of the tracking-type functional $$\label{Eq:ParaEstOpt} J(q,u):= \frac{1}{2}\|u-z\|_{H_0^1(\Omega)}^2+\frac{\beta}{2}\|q\|_{H^1(\Omega)}^2$$ subject to and $q\ge \alpha$, where $z\in H_0^1(\Omega)$ and $\alpha,\beta>0$. To formulate this problem in our variational framework, let $X:=H:=H^1(\Omega)\times H_0^1(\Omega)$, $F:=( D_q J, D_u J )$, and $$g(q,u):= \begin{pmatrix} q-\alpha \\ -\Delta^{-1}\bigl( \nabla (q \nabla u)+f \bigr) \end{pmatrix} , \quad K:=H_+^1(\Omega) \times \{0\},$$ where $H_+^1(\Omega)$ is the nonnegative cone in $H^1(\Omega)$. Note that the second component of $g$ is essentially the differential equation , but premultiplied with $-\Delta^{-1}$ to map the result back into $H_0^1(\Omega)$. ![Computed solutions $q$ of the parameter estimation problem for $n=256$ (left) and $n=1024$ (right).[]{data-label="Fig:ParaEst"}](img/ParameterEstimation1Df256.png "fig:") ![Computed solutions $q$ of the parameter estimation problem for $n=256$ (left) and $n=1024$ (right).[]{data-label="Fig:ParaEst"}](img/ParameterEstimation1Df1024.png "fig:") The existence of solutions to can be shown by eliminating $u$ in and using the coercivity of $J$, see [@Ito1990b]. Let $(\bar{q},\bar{u})$ be a solution of the problem. Then $$g'(\bar{q},\bar{u})= \begin{pmatrix} \operatorname{id}_{H^1} & 0 \\ T_{\bar{u}} & T_{\bar{q}} \end{pmatrix},$$ where $T_{\bar{u}}(q):=-\Delta^{-1}( \nabla (q\nabla \bar{u}) )$ and $T_{\bar{q}}(u):=-\Delta^{-1}(\nabla (\bar{q}\nabla u))$. Observe now that $T_{\bar{q}}:H_0^1(\Omega)\to H_0^1(\Omega)$ is surjective. This follows from the fact that $\Delta:H_0^1(\Omega)\to H^{-1}(\Omega)$ is an isomorphism and that $u\mapsto \nabla (\bar{q}\nabla u)$ is surjective onto $H^{-1}(\Omega)$ by the Lax–Milgram theorem (since $\bar{q}\ge \alpha>0$), see [@Troeltzsch2010]. It therefore follows that the whole operator $g'(\bar{q},\bar{u})$ is surjective, and thus the strict Robinson condition is satisfied in $(\bar{q},\bar{u})$. Let us furthermore assume that the second-order sufficient condition holds in $(\bar{q},\bar{u})$. The precise verification of this condition would require the knowledge of the solution, but the second-order condition is very plausible since the objective in is strongly convex (by virtue of the $H^1$-regularization term). Under the present assumptions, the problem admits the local error bound from Theorem \[Thm:ErrorBound\]. The corresponding residual mapping $\sigma:X\times H\to {\mathbb{R}}$ takes on the form $$\sigma(q,u,\mu,\lambda):=\|F(q,u)+g'(q,u)^* (\mu,\lambda)\|_{X^*}+ \|g(q,u)-P_K (g(q,u)+(\mu,\lambda)) \|_H,$$ where $(\mu,\lambda)\in H=H^1(\Omega)\times H_0^1(\Omega)$ is the pair of Lagrange multipliers. The vector $\mu$ corresponds to the lower bound constraint $q\ge \alpha$ (the first component of $g$), whereas $\lambda$ belongs to the partial differential equation (the second component of $g$). ----- ---------- ------------ ---------- ------------ ---------- ------------ ---------- ------------ $k$ $\rho_k$ $\sigma_k$ $\rho_k$ $\sigma_k$ $\rho_k$ $\sigma_k$ $\rho_k$ $\sigma_k$ 0 1 2.54e+04 1 2.54e+04 1 2.05e+05 1 2.05e+05 1 1 4.64e-01 1 1.21e-01 1 4.44e-01 1 6.59e-02 2 1 7.48e-02 1 5.01e-02 1 5.83e-02 1 2.52e-02 3 1 2.75e-02 1 2.50e-02 1 1.94e-02 1 1.17e-02 4 1 1.12e-02 10 1.30e-02 1 7.54e-03 1 5.64e-03 5 1 4.62e-03 10 1.30e-03 1 3.00e-03 1 2.73e-03 6 1 1.93e-03 10 1.30e-04 1 1.21e-03 1 1.33e-03 7 1 8.16e-04 10 1.31e-05 1 4.97e-04 1 6.46e-04 8 1 3.46e-04 1 4.80e-03 1 3.14e-04 9 1 1.48e-04 1 8.67e-05 1 1.53e-04 10 1 6.34e-05 1 7.48e-05 ----- ---------- ------------ ---------- ------------ ---------- ------------ ---------- ------------ : Iteration histories for the parameter estimation problem.[]{data-label="Tab:ParaEst"} We now present some numerical results. For practical purposes, we slightly alter the penalization scheme from Algorithm \[Alg:ALM\] in the sense that we augment the equality constraint only and leave the inequality constraint $q\ge \alpha$ unchanged. This has the benefit that we avoid the computation of projections and distance functions involving $H_+^1(\Omega)$. The resulting modifications to Algorithm \[Alg:ALM\] are fairly straightforward (see, for instance, [@Birgin2012; @Birgin2014; @Kanzow2017a]). Indeed, the augmented subproblems are now (constrained) variational inequalities over the set $\{q\in H^1(\Omega): q\ge \alpha \}$. Moreover, in the updating scheme of the penalty parameter, we have to take into account the multiplier corresponding to the lower inequality constraint, which has to be recovered from the solution process of the corresponding constrained subproblem. The example we present is [@Ito1991 Ex. 6]. The domain $\Omega:=(0,1)$ is discretized by means of $n\in{\mathbb{N}}$ points, including boundary points, and the derivative operators are approximated by forward differences. The problem is constructed by setting $$q_0(x):=1+x, \quad z(x):=u_0(x):=\sin (\pi x), \quad f(x):=(1+x)\pi^2 \sin (\pi x)-\pi \cos (\pi x),$$ so that $-\nabla (q_0 \nabla u_0)=f$. Since $z=u_0$, an exact solution of for $\beta=0$ is simply given by $(q_0,u_0)$. For $\beta>0$, which is the preferable case from a numerical perspective, the solutions are different in general. The implementation of the algorithm was done in MATLAB^^ and uses the parameters $$(q^0,u^0,\mu^0,\lambda^0):=(1,0,0,0), \quad \alpha:=0.1, \quad \rho_0:=1, \quad \gamma:=10, \quad \tau:=0.5,$$ together with ${w}^k:=P_B(\lambda^k)$ and $B$ the closed ball with radius $10^6$ around zero in $H_0^1(\Omega)$. The termination criteria for the outer and inner iterations are $\sigma(q,u,\mu,\lambda)\le 10^{-4}$ and $\|{\mathcal{L}}_{\rho_k}(q,u,{w}^k)+\mu^k\|_{X^*}\le 10^{-6}$, respectively, where $\mu^k$ is the Lagrange multiplier corresponding to the constraint $q\ge \alpha$. Finally, the augmented subproblems were solved by the `fmincon` routine which takes into account the lower box constraint. Table \[Tab:ParaEst\] contains the corresponding iteration numbers for different values of $n$ and $\beta$. We again observe linear convergence of the optimality measures $\sigma_k$, and the sequences of penalty parameters remain bounded. The only exception is the eighth iteration for $n=1024$ and $\beta=1$, which may be due to the subproblem routine `fmincon` failing to find a sufficiently exact minimizer. Finally, Figure \[Fig:ParaEst\] compares the computed solutions $q$ for different $n$ and $\beta$ to the exact solution $q_0$ for $\beta=0$. Final Remarks {#Sec:Final} ============= We have presented a method of augmented Lagrangian type for the solution of variational problems in Banach spaces. In particular, we have shown global and local convergence of the algorithm under suitable assumptions. The assumptions needed for the local convergence results include, in particular, a local error bound for the distance of a pair $(x,\lambda)$ to a KKT point $(\bar{x},\bar{\lambda})$. This property has played a central role in our analysis and is a consequence of the second-order sufficient condition together with a strict version of the Robinson constraint qualification. The above results suggest that error bounds are the natural framework for the local convergence analysis of augmented Lagrangian methods. We therefore hope that the results in this paper will find applications in other areas of optimization. In particular, an interesting idea would be to specialize some of the assumptions and results for problem classes such as optimal control or semidefinite programming. Another aspect which could lead to further developments is the concept of *partial penalization* which arises when additional constraints are present in the problem formulation which are not penalized, see [@Andreani2007; @Birgin2012; @Birgin2014] and the example in Section \[Sec:ApplicPara\]. [^1]: University of Würzburg, Institute of Mathematics, Campus Hubland Nord, Emil-Fischer-Str. 30, 97074 Würzburg, Germany; {kanzow,daniel.steck}@mathematik.uni-wuerzburg.de [^2]: This research was supported by the German Research Foundation (DFG) within the priority program “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962) under grant number KA 1296/24-1.

--- abstract: 'We describe the far from equilibrium non-local transport in a diffusive superconducting wire with a Zeeman splitting, taking into account the different spin relaxation mechanisms. We demonstrate that due to the Zeeman splitting an injection of a current in a superconducting wire creates a spin accumulation that can only relax via thermalization. In addition the Zeeman splitting also causes a suppression of the spin-orbit and spin-flip scattering rates. These two effects lead to long-range spin and charge accumulations detectable in the non-local signal. Our model explains the main qualitative features of recent experimental results in terms of realistic parameters and predicts a strong dependence of the non-local signal on the orbital depairing effect from an induced magnetic field.' author: - 'M. Silaev' - 'P. Virtanen' - 'F.S. Bergeret' - 'T.T. Heikkilä' title: 'Long-range spin and charge accumulation in mesoscopic superconductors with Zeeman splitting' --- Hybrid ferromagnetic/superconducting (FS) structures reveal a rich physics originating from the interplay between magnetism and superconductivity [@BuzdinRMP; @Bergeret2001]. While most of the research activity has been focused on the study and detection of proximity induced triplet superconducting correlations in an equilibrium situation [@Bergeret2001; @triplet], more recent experiments addressed the problem of spin and charge accumulation in superconducting wires[@Fukuma2011; @HanleSuper; @Poli; @SpinInjectionNb; @Aprili2013; @Beckman2012; @Beckman2014]. Figure \[Fig:Sketch\] shows a typical experimental setup, in which a spin accumulation is generated by a spin-polarized current injected from a ferromagnetic electrode. This spin accumulation observed in the experiments can be quite large. Two puzzling findings motivate this Letter: First, in superconductors with a strong Zeeman splitting, the induced spin accumulation has been detected at distances from the injector much larger than the spin-relaxation length in the normal state [@Aprili2013; @Beckman2012; @Beckman2014]. Second, the non-local conductance $g_{nl}$ depends drastically on the origin of the Zeeman splitting. Such a splitting can be caused either by an applied (strong) external magnetic field [@Aprili2013; @Beckman2012] or by the proximity of a ferromagnetic insulator [@Beckman2014proximity]. In this Letter, we develop a microscopic model based on the well-established Keldysh kinetic equations for superconductors extended to spin-dependent phenomena, and solve this puzzle. In particular we show that: (i) The observed long-range spin accumulation can be understood as a thermoelectric effect for Bogoliubov quasiparticles. The heating of a superconducting wire, originated for example from an injected current, produces a spin accumulation which can be detected as an electric signal by a spin-filter detector. The spin accumulation created in such a way can relax only due to the thermalization of injected quasiparticles and therefore the spin relaxation length is determined by inelastic electron-phonon and electron-electron scattering that can well exceed the usual spin diffusion length. (ii) Besides generating a large thermoelectric effect the Zeeman splitting also suppresses the spin-flip and spin-orbital scattering which are the main sources of charge imbalance relaxation in superconductors at low temperatures [@PairBreakingChImb]. Hence the different behaviors observed for the non-local conductance $g_{nl}$ as a function of the injection voltage $V_{inj}$, depend on the value of the orbital depairing parameter $\alpha_{orb}$ defined below. For large enough values of $\alpha_{orb}$, at large applied fields the contribution from the charge imbalance to the non-local conductance is suppressed and the $g_{nl}(V_{inj})$ dependence is almost antisymmetric with respect to $V_{inj}$ [@Aprili2013; @Beckman2012; @Beckman2014]. In contrast, if the Zeeman splitting is caused by the proximity of a ferromagnetic insulator [@Beckman2014proximity], $\alpha_{orb}$ is small and the charge imbalance contribution to $g_{nl}$ becomes important. In this case, we predict a qualitative change of the non-local conductance as function of the injected current, that can be experimentally proven. We consider the nonlocal spin valve shown in Fig. \[Fig:Sketch\]. A spin-polarized current is injected in the superconducting wire from a ferromagnetic electrode with polarization ${\bm P_{inj}}$, pointing in the direction of the magnetization. The detector is also a ferromagnet with a polarization vector ${\bm P_{det}}$ and located at a distance $L_{det}$ from the injector. Both the injector and the detector are coupled to the wire via tunnel contacts. A magnetic field ${\bm B}$ is applied in $z$ direction. ![\[Fig:Sketch\] (Color online) Schematic view of the setup for nonlocal conductance measurements. Here we assume that the polarizations of the magnetic contacts are collinear to the magnetic field, ${\bm P_{inj}}\parallel {\bm P_{det}}\parallel {\bm B}$. ](Sketch5.eps){width="1.0\linewidth"} When ${\bm P_{inj}}\parallel {\bm B}\parallel {\bm P_{det}}$ (for the non-collinear case, see Ref. [@silaevup14]), the tunnelling current at the detector is given by $$\label{Eq:ZeroCurrentYGen} R_{det}I_{det}= \mu + P_{det} \mu_z$$ where $R_{det}$ is the detector interface resistance in the normal state, $\mu$ is the charge imbalance and $\mu_z$ the spin imbalance. Here we assume that the detector current is measured at zero bias $V_{det}=0$. The nonlocal differential conductance measured in the experiment is $ g_{nl}= d I_{det}/d V_{inj} $. The charge imbalance $\mu$ and spin accumulation $\mu_z$ can be expressed in terms of the Keldysh quasiclassical Green function (GF) as $ \mu = \int_{0}^\infty {\rm Tr}(g^K) d\varepsilon/16 $ and $ \mu_{z} = \int_{0}^\infty {\rm Tr}[\tau_3 \sigma_3 (g^K-g^K_{eq})] d\varepsilon/16 $. Here $\tau_3$ ($\sigma_3$) is the third Pauli matrix in Nambu (spin) space, $g^K$ is the (4$\times$4 matrix) Keldysh component of the quasiclassical GF matrix $\check{g} = \left(% \begin{array}{cc} g^R & g^K \\ 0 & g^A \\ \end{array}% \right)$, and $g^{R(A)}$ is the retarded (advanced) GF. We denote $g^K_{eq}=g^K$ at the equilibrium state. The matrix GF satisfies the normalization condition $\check g^2=1$ that allows writing the Keldysh component as $ g^K= g^R \hat f - \hat f g^A$, where $\hat f$ is the distribution function with a general spin structure $ \hat f= f_L +f_T\tau_3 + f_{T3} \sigma_3+ f_{L3} \tau_3\sigma_3$[@SupplMat]. With the help of the above notations we obtain the expressions for the charge and spin imbalance in the superconductor (here and below, $\hbar=k_B=1$) $$\begin{aligned} \label{Eq:ChPot0} % \nonumber to remove numbering (before each equation) \mu = \frac{1}{2}\int_{0}^{\infty} d\varepsilon ( N_+ f_T+ N_- f_{L3}) \\\label{Eq:ChPotZ} \mu_z = \frac{1}{2}\int_{0}^{\infty} d\varepsilon [ N_+ f_{T3}+ N_-(f_{L}-n_0)], \end{aligned}$$ where $N_+$ is the total density of states (DOS), $N_-$ is the DOS difference between the spin subbands, and $n_0(\varepsilon) = \tanh(\varepsilon/2T)$. According to Eq. (\[Eq:ChPotZ\]) there are two contributions to the spin signal. One is generated from the longitudinal component $f_L$. This contribution is only finite in the presence of a Zeeman splitting of the DOS ($N_-\neq 0$). The second contribution is described by the first term in the integrand of Eq. (\[Eq:ChPotZ\]) and it is finite even in the absence of an exchange field. While this latter contribution has been analyzed in Ref. [@Beckman2012], we show below that in several cases it is the longitudinal contribution that dominates the spin signal due to its long-range character. In order to obtain the kinetic equations in a diffusive spin-polarized superconductor we start from the general Usadel equation [@Bergeret2001] $$\label{Eq:Usadel1} D\nabla\cdot(\check{g}\nabla\check{g})+ [\check\Lambda - \check\Sigma_{so} - \check\Sigma_{sf} - \check\Sigma_{orb}, \check{g}] =0.$$ Here $D$ is the diffusion constant, $\check\Lambda = i\varepsilon \tau_3-i({\bm{ h\cdot S})}\tau_3 - \check{\Delta}$, $\varepsilon$ is the energy, $ \check{\Delta}=\Delta\tau_1$ the spatially homogeneous order parameter in the wire, ${\bm h }$ is the Zeeman field, and ${\bm S}= (\sigma_1,\sigma_2,\sigma_3)$ the vector of Pauli matrices in spin space. The last three terms in Eq. , $\check\Sigma_{so} = \tau_{so}^{-1} ({\bm {S}}\cdot\check{g} {\bm {S}})$, $\check\Sigma_{sf} = \tau_{sf}^{-1} ({\bm {S}}\cdot\tau_3\check{g} \tau_3 {\bm {S}})$ and $\check\Sigma_{orb} = \tau_{orb}^{-1} \tau_3\check{g} \tau_3$ describe spin and charge imbalance relaxation due to the spin-orbit scattering, exchange interaction with magnetic impurities and orbital magnetic depairing, characterized by the relaxation times $\tau_{so}$, $\tau_{sf}$ and $\tau_{orb}$, respectively. The orbital depairing rate can be written in the form $\tau^{-1}_{orb} = T_c \alpha_{orb} (h/T_c)^2$ where $\alpha_{orb}$ is the dimensionless parameter measuring the relative strength of orbital and paramagnetic effects and $T_c$ is the critical temperature of the superconductor for $h=0$. If the Zeeman field is provided by an external magnetic field [@Beckman2012; @Aprili2013] ${\bm h}= \mu_B {\bm B}$ where $\mu_B$ is the Bohr magneton then $\alpha_{orb}= T_c\Delta/(\mu^2_B B^2_c) $, where $B_c = \sqrt{12} \phi_0/ (\pi\xi W)$ is the critical field of a thin superconducting film of width $W$, $\xi = \sqrt{D/\Delta}$ is the superconducting coherence length and $\phi_0=h/(2e)$ is the magnetic flux quantum [@SchmidtDepairing]. Assuming $\Delta=240$ $\mu$eV and diffusion constant $D=40$ cm$^2$/s [@Beckman2012; @Beckman2014], we obtain $\alpha_{orb} \approx 210 (W/\xi)^2$ where $\xi \approx 100$ nm. This estimation yields $\alpha_{orb}=1.33$ for the film width $W=8$ nm. The Zeeman field can also be induced by an exchange field in a FS proximity system [@Beckman2014proximity]. In this case $\tau_{orb}$ is not directly related to the Zeeman field and we describe this with $\alpha_{orb}=0$ in the numerical results below. We assume that the transparencies of the detector and injector interfaces are small, so that up to leading order the retarded and advanced GF are the bulk ones determined by the nonlinear equation $ [\Lambda^R-\Sigma^R_{sf}-\Sigma^R_{so}-\Sigma^R_{orb}, g^R]=0$. In the presence of an exchange field ${\bm h} = h{\bm z}$, the spectral functions read $ g^R = g_{01}\tau_1+ g_{31} \sigma_3\tau_1 + g_{03} \tau_3 + g_{33} \sigma_3\tau_3$ and $ g^A=-\tau_3 g^{R\dag}\tau_3$. While the terms diagonal in Nambu space ($\tau_3$) correspond to the normal GFs, the $g_{01},g_{31}$ describe the singlet and zero-spin triplet anomalous components [@Bergeret2001]. From these GFs we get $N_+={\rm Re} g_{03}$, $N_-={\rm Re} g_{33}$ in Eqs. (\[Eq:ChPot0\],\[Eq:ChPotZ\]). From Eq. (\[Eq:Usadel1\]) we obtain two decoupled sets of kinetic equations complemented by boundary conditions (BC) at the spin-polarized injector interface $z=0$. We use the BC of Ref. [@bergeret12] that generalizes the Kupriyanov-Lukichev [@KL] one to the case of spin-dependent barrier transmission. We start analyzing the set of equations that couple the components $f_T$ and $f_{L3}$. These determine the charge ${\bm {j_c}}$ and spin-heat ${\bm {j_{se}}}$ currents according to: $$\begin{aligned} {\bm {j_c}}= {\mathcal{D}}_{T}\nabla f_{T}+{\mathcal{D}}_{L3}\nabla f_{L3} \\ {\bm {j_{se}}}={\mathcal{D}}_{T}\nabla f_{L3}+{\mathcal{D}}_{L3}\nabla f_{T}, \end{aligned}$$ where the diffusion coefficients are $$\begin{aligned} {\mathcal{D}}_{T} &=& D\left( 1+|g_{01}|^2 + |g_{03}|^2 + |g_{31}|^2 + |g_{33}|^2 \right)\\ {\mathcal{D}}_{L3} &=& 2D{\rm Re} \left(g_{03}g_{33}^* + g_{01}g_{31}^* \right) . \end{aligned}$$ These currents satisfy a pair of coupled diffusion equations $$\begin{aligned} \label{Eq:fTKinC} {\bm {\nabla\cdot j_c}} &=& R_{T} f_{T} + R_{L3} f_{L3} \\ \label{Eq:fTKinCS} {\bm {\nabla\cdot j_{se}}} &=& (R_{T} + S_{L3}) f_{L3} + R_{L3} f_{T}, \end{aligned}$$ supplemented by the BC at the injector electrode $x=0$: $$\begin{aligned} \label{Eq:fTBCC}\nonumber {\bm {j_c}} = \frac{\kappa_I}{D} \{ N_+(f_T-n_-) + N_- [ P_I (n_0 - n_+) +f_{L3} ] \} \\ \label{Eq:fTBCS}\nonumber {\bm {j_{se}}} = \frac{\kappa_I}{D} \{ N_-(f_T-n_-) + N_+ [ P_I (n_0 - n_+) +f_{L3} ] \}, \end{aligned}$$ where $n_{\pm} = [ n_0 (\varepsilon + V_{inj}) \pm n_0 (\varepsilon - V_{inj}) ]/2$. The coefficients in Eqs.  (\[Eq:fTKinC\],\[Eq:fTKinCS\]) are given by $ R_{T}=4\Delta{\rm Re} g_{01}$, $R_{L3}=4\Delta{\rm Re} g_{31}$, and $$S_{L3} = \frac{16}{\tau_\Sigma} \{({\rm Re}g_{03})^2 -({\rm Re}g_{33})^2 + \beta\left[ ({\rm Re}g_{31})^2 -({\rm Re}g_{01})^2\right] \}.$$ Here $\tau^{-1}_\Sigma = \tau^{-1}_{so} +\tau^{-1}_{sf}$ and the parameter $\beta= (\tau_{so}-\tau_{sf})/(\tau_{so}+\tau_{sf})$ characterizes the relative strength of spin-orbit and spin-flip scattering. For example, in [Al]{} wires used in the spin-transport experiments, the typical spin relaxation time is $\tau_\Sigma \approx 800$ ps $\approx 40/T_{c}$ where $T_{c} \approx 1.6$ K. For the spin accumulation experiments [@Beckman2012; @Beckman2014; @Beckman2014proximity] the value of $\beta$ can be inferred from the magnetic depairing parameter $\zeta =8 \cdot10^{-4}$ in the absence of a magnetic field. It is proportional to the spin-flip scattering rate $\zeta = 3(1+\beta)/(2\tau_\Sigma T_{c}) $ which yields $\beta\approx -0.98$. We use $\beta=-0.9$ to obtain the qualitative effects. The solution of the system (\[Eq:fTKinC\],\[Eq:fTKinCS\]) is given by the superposition of two exponentially decaying functions $e^{-k_{T1,2}x}$ with amplitudes determined by the BC. The energy dependencies of $k_{T1,2}$ are shown in Fig. \[Fig:Scales\]a,b for the cases of strong and weak orbital depairing. The charge and spin-heat imbalance relaxation is non-zero above the gap $k_{T1,2}\neq 0$ due to the magnetic pair breaking effects [@ShcmidSchoen1975; @PairBreakingChImb]. The other set of equations are for the components $f_L,f_{T3}$ that determine the energy and pure spin currents $$\begin{aligned} {\bm {j_e}}= {\mathcal{D}}_{L}\nabla f_{L}+{\mathcal{D}}_{T3}\nabla f_{T3} \\ {\bm {j_{s}}}={\mathcal{D}}_{L}\nabla f_{T3}+{\mathcal{D}}_{T3}\nabla f_{L} \end{aligned}$$ satisfying the diffusion equations $$\begin{aligned} \label{Eq:fLKinE} % \nonumber to remove numbering (before each equation) {\bm {\nabla\cdot j_e}} &=& 0\\ \label{Eq:fLKinS} {\bm {\nabla\cdot j_s}} &=& S_{T3} f_{T3}, \end{aligned}$$ where the diffusion coefficients are $$\begin{aligned} {\mathcal{D}}_{L} &=& D\left(1-|g_{01}|^2 + |g_{03}|^2 - |g_{31}|^2 + |g_{33}|^2 \right) \\ {\mathcal{D}}_{T3} &=& 2D{\rm Re} \left(g_{03}g_{33}^* - g_{01}g_{31}^*\right) \end{aligned}$$ and $$S_{T3} = \frac{16}{\tau_\Sigma} \{ ({\rm Re}g_{03})^2 - ({\rm Re}g_{33})^2 + \beta \left[ ({\rm Im}g_{01})^2 - ({\rm Im}g_{31})^2 \right] \} .$$ The boundary conditions at $x=0$ are $$\begin{aligned} \label{Eq:fLBCE} %\nonumber {\bm {j_e}} = \frac{\kappa_I}{D} [ N_+(f_L-n_+) + N_- (f_{T3} -P_I n_- ) ] \\ \label{Eq:fLBCS} %\nonumber {\bm {j_s}} = \frac{\kappa_I}{D} [ N_-(f_L-n_+) + N_+ (f_{T3} -P_I n_- ) ], %%%%%%%%%%%%%%% \end{aligned}$$ where $n_\pm=n_\pm(V_{inj})$. The solution of the system (\[Eq:fLKinE\],\[Eq:fLKinS\]) is given by a superposition of two qualitatively different terms $$\label{Eq:SolFL} \left( f_L\atop f_{T3} \right) = A\left( 1 \atop - {\mathcal{D}}_L/{\mathcal{D}}_{T3} \right) e^{-k_Lx}+ \left( \alpha (x-L) \atop 0 \right),$$ where the amplitudes $(\alpha, A)$ can be found from the BC (\[Eq:fLBCE\],\[Eq:fLBCS\]). The first term in (\[Eq:SolFL\]) describes a decay of the (spectral) spin imbalance with a characteristic length scale $ k_L= \sqrt{S_{T3}{\mathcal{D}}_{L}/ ( {\mathcal{D}}_{L}^2-{\mathcal{D}}_{T3}^2)}$. The second term in (\[Eq:SolFL\]) describes the rise of quasiparticle temperature generated by the applied voltage $V_{inj}$. This decays only via inelastic scattering disregarded in the above equations, but discussed in more detail below.  (a,b) Energy dependence of the inverse length scales $k_{T1,2}$ and $k_L$ for: (a) $\alpha_{orb}=1.33$ and $h=0.3 T_c$; (b) $\alpha_{orb}=0$, and $h=0.8 T_c$ (here $\lambda_{sf} = \sqrt{D\tau_\Sigma/8}$ is the normal state spin relaxation length). Panels (c) and (d) show the nonlocal conductance as a function of the injecting voltage, $g_{nl}(V_{inj})$ for: (c) $\alpha_{orb}=1.33$ and $h=0;\; 0.1;\;0.3;\; 0.4 T_c $. (d) $\alpha_{orb}=0$ and $h=\;0.8T_c$. Red solid line shows the total signal, blue dashed and green dash-dotted lines show the spin and charge imbalance contributions respectively. The parameters common to all panels are $T=0.05\; T_{c}$, $\beta=-0.9$, $\tau_\Sigma T_c =40$, $P_{inj}=-P_{det}=0.1$, $\kappa_I\xi=0.02$, the inelastic relaxation length $L=20\lambda_{sf}$ and $L_{det}=5\lambda_{sf}$. ](Fig2.eps){width="1.0\linewidth"} We now calculate the non-local conductance from Eq. (\[Eq:ZeroCurrentYGen\]) and the solutions of kinetic equations (\[Eq:fTKinC\],\[Eq:fTKinCS\],\[Eq:fLKinE\],\[Eq:fLKinS\]). First, we assume a strong orbital depairing $\alpha_{orb}=1.33$. Figure (\[Fig:Scales\])c shows the non-local conductance and describes several features observed in recent experiments [@Aprili2013; @Beckman2012; @Beckman2014] that we discuss below. In the absence of a Zeeman field, $N_-=0$, and therefore only the first terms in the r.h.s of Eqs. (\[Eq:ChPot0\]-\[Eq:ChPotZ\]) contributes to $g_{nl}$. For $h=0$ the contribution stemming from the spin accumulation is finite only if $P_{inj}\neq0$, which is the condition to obtain a finite $f_{T3}$. However, this function decays over the short spin diffusion length and therefore is negligibly small at the distances $L_{det}= 3\lambda_{sf}$ from the injector. Thus, the detected signal in this case is mostly determined by the charge imbalance, Eq. (\[Eq:ChPot0\]). This explains the symmetry with respect to the injecting voltage: $g_{nl} (V_{inj})= g_{nl} (- V_{inj})$. The charge imbalance contribution to $g_{nl}$ grows monotonically when $|V_{inj}|>\Delta_g$. This behavior is determined by the increase of the charge relaxation scale at large energies $k^{-1}_{T2}\rightarrow\infty$ shown in Fig. \[Fig:Scales\]a. In the presence of a magnetic field, on the one hand, the charge relaxation is strongly enhanced due to the orbital depairing effect. This explains a strong suppression of the charge imbalance background signal by increased $h$ in Fig. \[Fig:Scales\]c. On the other hand the spin imbalance contribution stemming from the second term in the r.h.s of Eq. (\[Eq:ChPotZ\]) is large. As shown above, this term describes the heat injection in the presence of a finite Zeeman field $h$ and has a long-range behavior. This contribution leads to the large peaks in $g_{nl} (V_{inj})$ shown in Fig. \[Fig:Scales\]c. In contrast to the linear thermoelectric effect [@ThermoelectriEschrig; @ThermoelectriOsaeta], which is exponentially small for temperatures well below the energy gap $T\ll\Delta$, a non-linear heating produced by quasiparticles injected at voltages exceeding the energy gap explains the large electric signal observed in the experiments [@Aprili2013; @Beckman2012; @Beckman2014]. Notice that the peaks do not have exactly the same form so that $g_{nl}(V_{inj})\neq -g_{nl}(-V_{inj})$. The small deviation from the antisymmetric case is due to the small but finite injector polarization $P_{inj}$, as well as to the presence of an admixture of the charge imbalance signal. Next let us consider the case of no orbital depairing $\alpha_{orb}=0$. This may correspond to the case of a Zeeman field caused by the proximity of a ferromagnetic insulator [@Beckman2014proximity]. Figure \[Fig:Scales\]d shows a clearly different behavior for $g_{nl} (V_{inj})$ with respect to the large $\alpha_{orb}$ case, and can be observed in the experiments [@Beckman2014proximity]. Now the asymmetry of the $g_{nl} (V_{inj})$ curve is much more pronounced and the peaks are much broader. This occurs due to a significant admixture of the long-range charge imbalance contribution (blue dashed curves) with the spin imbalance one (green dash-dotted curve). While the latter is almost perfectly antisymmetric the former produces symmetric peaks of $g_{nl}(V_{inj})$ at voltages within the interval $\Delta-h <V_{inj} <\Delta+h$. These peaks appear due to the strong suppression of charge relaxation by the Zeeman splitting at the energy interval $\Delta-h <\varepsilon <\Delta+h$, in accordance to Fig. \[Fig:Scales\]b. Our results give a qualitative explanation of experiments with large magnetic fields \[Fig. \[Fig:Scales\]c\]. Including an additional constant orbital depairing which can originate from the stray fields of ferromagnetic contacts [@Aprili2013] we are able to obtain accurate fits of the experimental curves shown in Fig.3a of Ref. [@Beckman2012] using realistic parameters. Notice that the decay of the component $f_L$, responsible for the long-range spin imbalance, is only limited by inelastic relaxation, which have not been taken into account in our kinetic equations. The observed relaxation length $\lambda\sim 1$ $\mu$m likely cannot be explained by electron-phonon scattering, which already in the normal state leads to a much larger value $\lambda_{ph} = \sqrt{\tau_{ph} D}\approx 20$ ${\rm \mu m} $[@Beckman2012; @Beckman2014]. Electron-electron scattering on the other hand can redistribute the total energy in the electron system and damp nonequilibrium components of the signal. In order to obtain the observed relaxation length $\lambda_{ee}\sim 1$ $\mu$m one should assume that the e-e scattering time is $\tau_{ee} \sim 10^{-10} s$ which is significantly less than the known value in bulk dirty Al [@KlapwijkRelTime] but can be achieved in low-dimensional samples [@KlapwijkRelTime2D]. The e-e thermalization process as well as nonuniversal properties of the heat transport in real experimental setups could explain the suppression of the spin imbalance relaxation by the Zeeman field [@Beckman2012; @Beckman2014]. To conclude, we have developed a theoretical framework to study the transport properties of superconductors with a Zeeman splitting. We have demonstrated that the splitting field leads to a strong suppression of the relaxation of charge and spin imbalances created by the injected current. In particular, the long-range spin accumulation observed in recent experiments is shown to be a manifestation of a non-linear thermoelectric effect and it is only limited by the inelastic relaxation length which can be larger than the spin relaxation time in normal metals by several orders of magnitude. Our model gives a qualitative explanation for a wide range of experiments on SF nonlocal spin valves, and predicts a strong dependence of the non-local conductance on orbital depairing, characterized by $\alpha_{\rm orb}$. Besides explaining the properties of superconductor-ferromagnet structures, our theory may be straightforwardly extended for the general description of thermoelectric effects in far from equilibrium situations in terms of the well-established theory of non-equilibrium GFs. We thank Detlef Beckmann for discussions. The work of T.T.H was supported by the Academy of Finland Center of Excellence program and the European Research Council (Grant No. 240362-Heattronics). P.V. acknowledges the Academy of Finland for financial support. The work of F.S.B. was supported by the Spanish Ministry of Economy and Competitiveness under Project No. FIS2011-28851-C02- 02 and the Basque Government under UPV/EHU Project No. IT-756-13. [99]{} A.I. Buzdin, Rev. Mod. Phys. [**77**]{}, 935 (2005). F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. [**77**]{}, 1321 (2005). See for example R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao, and A. Gupta, Nature [**439**]{}, 825 (2006); J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Science [**329**]{}, 59 (2010); T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O. Birge, Phys. Rev. Lett. [**104**]{}, 137002 (2010); M. Anwar, F. Czeschka, M. Hesselberth, M. Porcu, and J. Aarts, Phys. Rev. B [**82**]{}, 100501 (R) (2010). Y. Fukuma [*et al*]{}, Nat. Mat. [**10**]{}, 527 (2011). H. Yang, S.-H. Yang, S. Takahashi, S. Maekawa, and S. S. P. Parkin, Nature Mat. [**9**]{}, 586 (2010). 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Supplementary Material ====================== We express the charge density, spin density, energy density and spin energy density in terms of the Nambu-Keldysh Green’s function and show that in the presence of a spin-splitting field their parametrization differs from the usual case. We wish to characterize the state of the electron system in the superconductor via different matrix elements of the Nambu-Keldysh Green’s function. We choose the Nambu vector to be of the form $$\Psi = \begin{pmatrix} \psi_\uparrow({\mathbf r},t) & \psi_\downarrow({\mathbf r},t) & -\psi_\downarrow^\dagger({\mathbf r},t) & \psi_\uparrow^\dagger({\mathbf r},t)\end{pmatrix},$$ where $\psi_\sigma^{(\dagger)}({\mathbf r},t)$ annihilates (creates) an electron of spin $\sigma$ in position ${\mathbf r}$ at time $t$. The Keldysh Green’s function is written in terms of the Nambu vector as $$G^K(1,1') = -i \tau_3 \langle [\Psi(1),\Psi^\dagger(1')]\rangle,$$ where the argument $1$ refers to position and time ${\mathbf r}_1$, $t_1$. Up to a state independent factor, the charge density can then be written as $$\rho(1)=-ie {\rm Tr}[G^K(1,1)]/4 = -ie \int \frac{d {\mathbf p}}{(2\pi)^3} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} {\rm Tr}[G^K(\epsilon,{\mathbf p},{\mathbf r},t)]/4,$$ where the latter form is expressed in the Wigner representation. This is thus the particle density averaged over spin. The spin density in direction $\hat u_j \in \{\hat u_x,\hat u_y, \hat u_z\}$ is obtained from $$s_j(1)=-i {\rm Tr}[\tau_3 \sigma_j G^K]/4 = -i \int \frac{d {\mathbf p}}{(2\pi)^3} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} {\rm Tr}[\tau_3 \sigma_j G^K(\epsilon,{\mathbf p},{\mathbf r},t)]/4,$$ where $\tau_j (\sigma_j)$ is the $j$th Pauli spin matrix in Nambu (spin) space. The multiplication with $\tau_3$ takes care of the chosen order of spins in the Nambu vector. In order to characterize the non-equilibrium spin accumulation we introduce the difference between the total spin density and the one at the equilibrium state $$s_{ej}(1)= -i \int \frac{d {\mathbf p}}{(2\pi)^3} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} {\rm Tr} \{ \tau_3 \sigma_j [ G^K(\epsilon,{\mathbf p},{\mathbf r},t) - G^K_{eq}(\epsilon,{\mathbf p},{\mathbf r},t) ] \}/4.$$ As shown in the main text of the paper the spin accumulation ${\bf s}_{ej}$ determines the tunnelling current at the spin-polarized detector electrode in the non-local measurement scheme. The spin-averaged energy density involves a multiplication by the Nambu matrix $\tau_3$ to take care of the fact that whereas particles and holes contribute to the charge an opposite amount, they contribute an equal amount to the excitation energy. We thus can write in the Wigner representation the (internal) energy density $$\epsilon({\mathbf r},t)= -ie \int \frac{d {\mathbf p}}{(2\pi)^3} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \epsilon {\rm Tr}[\tau_3 G^K(\epsilon,{\mathbf p},{\mathbf r},t)]/4.$$ In the limit of a large bandwidth this becomes very large, so it is convenient to describe only the excess energy density compared to some equilibrium value, $$\epsilon_e({\mathbf r},t)= -ie \int \frac{d {\mathbf p}}{(2\pi)^3} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \epsilon {\rm Tr} \{ \tau_3 [G^K(\epsilon,{\mathbf p},{\mathbf r},t)-G^K_{\rm eq}(\epsilon,{\mathbf p},{\mathbf r},t) ] \}/4.$$ Similarly to the charge density, $\epsilon_e$ is a spin-averaged quantity. We can thus also define the energy density difference in the two spin ensembles by $$\epsilon_j({\mathbf r},t)= -ie \int \frac{d {\mathbf p}}{(2\pi)^3} \int_{-\infty}^\infty \frac{d\epsilon}{2\pi} \{\epsilon {\rm Tr}[\sigma_j G^K(\epsilon,{\mathbf p},{\mathbf r},t)]\}/4.$$ In the absence of a ferromagnetic transition, the contribution to this quantity comes from the small energy region related to voltage or temperature, and therefore there is no need to remove the equilibrium value. We now define the quasiclassical Keldysh Green’s function in the diffusive limit (and a spherical Fermi surface) via $$g^K(1)=\frac{i}{\pi} \int \frac{d \hat p}{4\pi}\int d\xi G^K(\epsilon,\xi,\hat{p},{\mathbf r},t),$$ where ${\mathbf p}=p \hat p$ and $\xi=p^2/(2m)-\epsilon_F$, where $\epsilon_F$ is the Fermi energy. Employing charge neutrality (charge density vanishes on length scales that are large compared to the usually microscopic screening length) and including the non-quasiclassical corrections in the standard way [@RammerBook], the electrostatic potential (instead of the charge density that vanishes) can be written in terms of the quasiclassical Green’s function as $$\mu(\vec{r},t)=-\frac{1}{16} \int_{-\infty}^\infty d\epsilon {\rm Tr}[g^K(\epsilon,\vec{r},t)].$$ In the absence of a ferromagnetic transition, the other expressions can be then straightforwardly written in terms of the quasiclassical Green’s function by using $$\int \frac{d{\mathbf p}}{(2\pi)^3} = \int d\xi N(\xi) \int \frac{d\hat p}{4\pi} \approx N_0 \int d\xi \int \frac{d\hat p}{4\pi},$$ where $N(\xi)$ is the density of states in the normal state, and we assume $N(\xi) \approx N_0$ for excitation energies close to the Fermi surface. We hence get $$\begin{aligned} s_{ej}(1)&=-N_0 \int_{-\infty}^\infty d\epsilon {\rm Tr} \{\tau_3 \sigma_j [ g^K(\epsilon,{\mathbf r},t) - g^K_{eq}(\epsilon,{\mathbf r},t) ]\}/8.\\ \epsilon_e(1)&=-N_0 \int_{-\infty}^\infty d\epsilon \epsilon {\rm Tr}\{\tau_3 [g^K(\epsilon,{\mathbf r},t)-g_{\rm eq}^K(\epsilon,\mathbf{r},t)]\}/8.\\ \epsilon_j(1)&=-N_0 \int_{-\infty}^\infty d\epsilon {\rm Tr}[\sigma_j g^K(\epsilon,{\mathbf r},t)]/8. \end{aligned}$$ The full quasiclassical Keldysh Green’s function $$\check g = \begin{pmatrix} g^R & g^K\\0 & g^A\end{pmatrix}$$ satisfies the normalization condition $\check g^2=1$. This allows parameterizing $g^K=g^R f - f g^A$, where in the spin-dependent case the distribution function $f$ is parameterized by eight functions, $$f=f_L+f_T \tau_3 + \sum_j (f_{Tj} \sigma_j + f_{Lj} \sigma_j \tau_j).$$ Here the L-labelled functions denote the (spin) energy degrees of freedom and are antisymmetric in energy with respect to the Fermi level $\varepsilon=0$ in the superconductor. The T-labelled functions are symmetric in energy and describe the charge/spin imbalance. In the main text, we only concentrate on the case of collinear spin configurations, and thereby it is enough to choose one of the spin directions, say $j=3$. In this case the above expressions for the local potential, nonequilibrium spin density, energy density and spin energy density can be written as $$\begin{aligned} \mu(\vec{r},t)&=-\frac{1}{2} \int_{0}^\infty d\epsilon (N_+ f_T+N_- f_{L3})\\ s_{e3}(\vec{r},t)&=-N_0 \int_0^\infty d\epsilon [N_+ f_{T3}+N_- (f_L-n_0)]\\ \epsilon_e(\vec{r},t)&=-N_0 \int_0^\infty d\epsilon \epsilon [N_+(f_{L}-n_0)+N_- f_{T3}]\\ \epsilon_3(\vec{r},t)&=-N_0 \int_0^\infty d\epsilon \epsilon (N_+ f_{L3}+N_- f_{T}),\end{aligned}$$ where $n_0$ is the equilibrium distribution function describing $G^K_{\rm eq}$, $\epsilon$ has been redefined with respect to the chemical potential of the superconductor, and we have used the fact that the integrands are symmetric with respect to $\epsilon=0$. These expressions are also used to define the charge and spin imbalances in Eqs. (2-3) of the main text. When entering the current (Eq. (1) of the main text), the prefactors are absorbed in the definition of the normal-state interface resistance. Note that in the absence of the exchange field, the density of states (per spin) is electron-hole symmetric. In that case the particle/spin/energy densities could be written directly in terms of the individual distribution functions, instead of their combinations, whose presence reveals the strong thermoelectric effect.

--- abstract: 'We prove that NIP valued fields of positive characteristic are henselian. Furthermore, we partially generalize the known results on dp-minimal fields to dp-finite fields. We prove a dichotomy: if $K$ is a sufficiently saturated dp-finite expansion of a field, then either $K$ has finite Morley rank or $K$ has a non-trivial ${\operatorname{Aut}}(K/A)$-invariant valuation ring for a small set $A$. In the positive characteristic case, we can even demand that the valuation ring is henselian. Using this, we classify the positive characteristic dp-finite pure fields.' author: - Will Johnson bibliography: - 'mybib.bib' title: 'Dp-finite fields I: infinitesimals and positive characteristic' --- Introduction ============ The two main conjectures for NIP fields are - The *henselianity conjecture*: any NIP valued field $(K,\mathcal{O})$ is henselian. - The *Shelah conjecture*: any NIP field $K$ is algebraically closed, real closed, finite, or admits a non-trivial henselian valuation. By generalizing the arguments used for dp-minimal fields (for example in Chapter 9 of [@myself]), we prove the henselianity conjecture in positive characteristic, and the Shelah conjecture for positive characteristic dp-finite fields. This yields the positive-characteristic part of the expected classification of dp-finite fields. We also make partial progress on dp-finite fields of characteristic zero. Let $(K,+,\cdot,\ldots)$ be a sufficiently saturated dp-finite field, possibly with extra structure. Then either - $K$ has finite Morley rank, or - There is an ${\operatorname{Aut}}(K/A)$-invariant non-trivial valuation ring on $K$ for some small set $A$. Unfortunately, we can only prove henselianity of this valuation ring in positive characteristic. Following the approach used for dp-minimal fields, there are three main steps to the proof: 1. Construct a type-definable group of infinitesimals. 2. Construct a valuation ring from the infinitesimals. 3. Prove henselianity. We discuss each of these steps, explaining the difficulties that arise when generalizing from rank 1 to rank $n$. Constructing the infinitesimals ------------------------------- Mimicking the case of dp-minimal fields, we would like to define the group $I_M$ of $M$-infinitesimals as $$\bigcap_{X \text{ ``big'' and $M$-definable}} \{\delta \in K ~|~ X \cap (X + \delta) \text{ is ``big''}\}$$ for some notion of “big.” In the dp-minimal case, “big” was “infinite.” By analyzing the proof for dp-minimal fields, one can enumerate a list of desiderata for bigness: 1. Non-big sets should form an ideal. 2. Bigness should be preserved by affine transformations. 3. \[definability-condition\] Bigness should vary definably in families. 4. The universe $K$ should be big. 5. \[mininf-condition\] If $X, Y$ are big, the set $\{\delta ~|~ X \cap (Y + \delta) \text{ is big}\}$ should be big. 6. Bigness should be coherent on externally definable sets, to the extent that: - If $X$ is $M$-definable and big for a small model $M \preceq K$, if $Y$ is $K$-definable, and if $X(M) \subseteq Y$, then $Y$ is big. - If $X$ is big and $X \subseteq Y_1 \cup \cdots \cup Y_n$ for externally definable sets $Y_1, \ldots, Y_n$, then there is a big definable subset $X' \subseteq X$ such that $X' \subseteq Y_i$ for some $i$. The intuitive guess is that for a field of dp-rank $n$, “big” should mean “rank $n$.” But there is no obvious proof of the definability condition (\[definability-condition\]), as noted in §3.2 of Sinclair’s thesis [@sinclair]. An alternative, silly guess is that “big” should mean “infinite.” This fails to work in some of the simplest examples, such as $({\mathcal{C}},+,\cdot,{\mathbb{R}})$. However, the silly guess *nearly* works; the only requirement that can fail is (\[mininf-condition\]). The key insight that led to the present paper was the realization that in rank 2, any failure of (\[mininf-condition\]) for the silly option (“big”=“infinite”) fixes (\[definability-condition\]) for the intuitive option (“big”=“rank 2”). Indeed, if $X, Y$ are infinite sets but $X \cap (Y + \delta)$ is finite for almost all $\delta$, then - By counting ranks, $X, Y$ must be dp-minimal. - The map $X \times Y \to X - Y$ is almost finite-to-one. - By a theorem of Pierre Simon [@surprise], “rank 2” is definable on $X \times Y$.[^1] This ensures that “rank 2” is definable on $X - Y$. - A definable set $D \subseteq K$ has rank 2 if and only if some translate of $D$ has rank 2 intersection with $X - Y$. So, in the rank-2 setting, one can first try “big”=“infinite,” and if that fails, take “big”=“rank 2.” A prototype of this idea appears in Peter Sinclair’s thesis [@sinclair]. In his §3.3, he observes that the machinery of infinitesimals goes through when “rank $n$”=“infinite,” and conjectures that this always holds in the pure field reduct. By extending this line of thinking to higher ranks, we obtain a notion of *heavy* sets satisfying the desired properties.[^2] See §\[sec:heavy-light\] for details; the technique is reminiscent of Zilber indecomposability in groups of finite Morley rank. Once heavy and light sets are defined, the construction of infinitesimals is carried out in §\[sec:infinitesimals\] via a direct generalization of the argument for dp-minimal fields. For certain non-triviality properties, we need to assume that $K$ does not have finite Morley rank. The relevant dichotomy is proven in §\[sec:likeTT\]; it is closely related to Sinclair’s Large Sets Property (Definition 3.0.3 in [@sinclair]), but with heaviness replacing full dp-rank. For ranks greater than 2, we need to slightly upgrade Simon’s results in [@surprise]. We do this in §\[sec:broad-narrow\]. Say that an infinite definable set $Q$ is *quasi-minimal* if ${\operatorname{dp-rk}}(D) \in \{0, {\operatorname{dp-rk}}(Q)\}$ for every definable subset $D \subseteq Q$. The main result is the following: Let $M$ be an NIP structure eliminating $\exists^\infty$. Let $Q_1, \ldots, Q_n$ be quasi-minimal sets, and $m = {\operatorname{dp-rk}}(Q_1 \times \cdots \times Q_n)$. Then “rank $m$” is definable in families of definable subsets of $Q_1 \times \cdots \times Q_n$. This is a variant of Corollary 3.12 in [@surprise]. Note that dp-minimal sets are quasi-minimal, and quasi-minimal sets are guaranteed to exist in dp-finite structures. Getting a valuation ------------------- Say that a subring $R \subseteq K$ is a *good Bezout domain* if $R$ is a Bezout domain with finitely many maximal ideals, and ${\operatorname{Frac}}(R) = K$. This implies that $R$ is a finite intersection of valuation rings on $K$. Ideally, we could prove the following The $M$-infinitesimals $I_M$ are an ideal in a good Bezout domain $R \subseteq K$. Assuming the Conjecture, one can tweak $R$ and arrange for $I_M$ to be the Jacobson radical of $R$. This probably implies that - The ring $R$ is $\vee$-definable, and so are the associated valuation rings. - The canonical topology is a field topology, and has a definable basis of opens. This would put us in a setting where we could generalize the infinitesimal-based henselianity proofs from the dp-minimal case, modulo some technical difficulties in characteristic zero. The strategy would be to prove that $R$ is the intersection of just one valuation ring; this rules out the possibility of $K$ carrying two definable valuations, leading easily to proofs of the henselianity and Shelah conjectures.[^3] As it is, the Conjecture is still unknown, even in positive characteristic. Nevertheless, we *can* prove that a non-trivial ${\operatorname{Aut}}(K/A)$-invariant good Bezout domain $R$ exists for some small set $A$. Unfortunately, the connection between $R$ and $I_M$ is too weak for the henselianity arguments, and we end up using a different approach, detailed in the next section. How does one obtain a good Bezout domain? It is here that we diverge most drastically from the dp-minimal case. Say that a type-definable group $G$ is *00-connected* if $G = G^{00}$. The group $I_M$ of $M$-infinitesimals is 00-connected because it is the minimal $M$-invariant heavy subgroup of $(K,+)$. In the dp-minimal case (rank 1), the 00-connected subgroups of $(K,+)$ are totally ordered by inclusion. Once one has *any* non-trivial 00-connected proper subgroup $J < (K,+)$, the set $\{x \in K ~|~ x \cdot J \subseteq J\}$ is trivially a valuation ring. In higher rank, the lattice $\mathcal{P}$ of 00-connected subgroups is no longer totally ordered. Nevertheless, dp-rank bounds the complexity of the lattice. Specifically, it induces a rank function ${\operatorname{rk}}_{dp}(A/B)$ for $A \ge B \in \mathcal{P}$ probably[^4] satisfying the following axioms: - ${\operatorname{rk}}_{dp}(A/B)$ is a nonnegative integer, equal to zero iff $A = B$. - If $A \ge B \ge C$, then ${\operatorname{rk}}_{dp}(A/C) \le {\operatorname{rk}}_{dp}(A/B) + {\operatorname{rk}}_{dp}(B/C)$. - If $A \ge A' \ge B$, then ${\operatorname{rk}}_{dp}(A/B) \ge {\operatorname{rk}}_{dp}(A'/B)$. - If $A \ge B' \ge B$, then ${\operatorname{rk}}_{dp}(A/B) \ge {\operatorname{rk}}_{dp}(A/B')$. - For any $A, B$ $$\begin{aligned} {\operatorname{rk}}_{dp}(A \vee B/B) & = {\operatorname{rk}}_{dp}(A/A \wedge B) \\ {\operatorname{rk}}_{dp}(A \vee B/A \wedge B) &= {\operatorname{rk}}_{dp}(A/A \wedge B) + {\operatorname{rk}}_{dp}(B/A \wedge B). \end{aligned}$$ This rank constrains the lattice substantially. For example, if $n = {\operatorname{dp-rk}}(K)$ there can be no “$(n+1)$-dimensional cubes” in $\mathcal{P}$, i.e., no injective homomorphisms of unbounded lattices from ${\mathcal{P}ow}([n])$ to $\mathcal{P}$. Somehow, the rank needs to be leveraged to recover a valuation. First we need some lattice-theoretic way to recover valuations. Recall that in a lower-bounded modular lattice $(P,\wedge,\vee,\bot)$, there is a modular pregeometry on the set of atoms. More generally, if $x$ is an element in an unbounded modular lattice $(P,\wedge,\vee)$, there is a modular pregeometry on the set of “relative atoms” over $x$, i.e., elements $y \ge x$ such that the closed interval $[x,y]$ has size two. If $(K,\mathcal{O},\mathfrak{m})$ is a model of ACVF and $P_n$ is the lattice of definable additive subgroups of $K^n$, then the pregeometry of relative atoms over $\mathfrak{m}^n \in P_n$ is projective $(n-1)$-space over the residue field $k$, because the relative atoms all lie in the interval $[\mathfrak{m}^n,\mathcal{O}^n]$ whose quotient $\mathcal{O}^n/\mathfrak{m}^n$ is isomorphic to $k^n$. Furthermore, the specialization map on Grassmannians $${\operatorname{Gr}}(\ell, K^n) \to {\operatorname{Gr}}(\ell, k^n)$$ is essentially the map sending $V$ to the set of relative atoms below $V + \mathfrak{m}^n$. For dp-finite fields, we do something formally similar: we consider the lattice $\mathcal{P}_n$ of 00-connected type-definable subgroups of $K^n$, we consider the pregeometry $\mathcal{G}_n$ of relative atoms over $(I_M)^n$, and we extract a map $f_n$ from the lattice of $K$-linear subspaces of $K^n$ to the lattice of closed sets in $\mathcal{G}_n$. Morally, this leads to a system of maps $$\label{n-ell-eq} f_{n, \ell} : {\operatorname{Gr}}(\ell, K^n) \to {\operatorname{Gr}}(r \cdot \ell, k^n)$$ where $r = {\operatorname{dp-rk}}(K)$ and $k$ is some skew field. The maps $f_{n, \ell}$ satisfy some compatibility across $n$ and $\ell$. Call this sort of configuration an *$r$-fold specialization*. The hope was that $r$-fold specializations would be classified by valuations (or good Bezout domains). Unfortunately, this turns out to be far from true. All we can say is that there is a way of “mutating” $r$-fold specializations such that, in the limit, they give rise to Bezout domains. See Remark \[r-fold-specs\] and Section \[sec:mutate\] for further details.[^5] The above picture is complicated by three technical issues. First and most glaringly, the modular lattices we are considering don’t really have enough atoms. For example, there are usually no minimal non-zero type-definable subgroups of $K$. So instead of using atoms, we use equivalence classes of quasi-atoms. In a lower-bounded modular lattice $(P,\wedge,\vee,\bot)$, say that an element $x > \bot$ is a *quasi-atom* if the interval $(\bot,x]$ is a sublattice, and two quasi-atoms $x, y$ are *equivalent* if $x \wedge y > \bot$. There is always a modular geometry on equivalence classes of quasi-atoms. In the presence of a subadditive rank such as ${\operatorname{rk}}_{dp}(-/-)$, there are always “enough” quasi-atoms, and the modular geometry has bounded rank. We verify these facts in §\[sec:geometry\], in case they are not yet known. Second of all, equation (\[n-ell-eq\]) says that an $\ell$-dimensional subspace $V \subseteq K^n$ must map to a closed set in $\mathcal{G}_n$ of rank exactly $r \cdot \ell$, where $r = {\operatorname{dp-rk}}(K)$. This only works if the pregeometry of relative atoms over $I_M$ has the same rank as the dp-rank of the field. This turns out to be false in general, necessitating two modifications: - Rather than using dp-rank, we use the minimum subadditive rank on the lattice. We call this rank “reduced rank,” and verify its properties in §\[sec:reduced-rank\]. In equation (\[n-ell-eq\]), $r$ should be the reduced rank rather than the dp-rank. - Rather than using $I_M$, we use some other group $J$ which maximizes the rank of the associated geometry. We call such $J$ *special*, and consider their properties in §\[sec:special\]. With these changes, equation (\[n-ell-eq\]) can be recovered; see Lemma \[special-lemma-1\]. Lastly, there is an issue with the lattice meet operation. Note that the lattice operations on the lattice of 00-connected type-definable subgroups are $$\begin{aligned} A \vee B & := A + B \\ A \wedge B & := (A \cap B)^{00}.\end{aligned}$$ The 00 in the definition of $A \wedge B$ is a major annoyance[^6]; it would be nicer if $A \wedge B$ were simply $A \cap B$. The following trick clears up this headache: There is a small submodel $M_0 \preceq K$ such that $J = J^{00}$ for every type-definable $M_0$-linear subspace $J \le K$. Consequently, if $\mathcal{P}' \subseteq \mathcal{P}$ is the sublattice of 00-connected $M_0$-linear subspaces of $K$, then the lattice operations on $\mathcal{P}'$ are given by $$\begin{aligned} A \vee B & = A + B \\ A \wedge B & = A \cap B. \end{aligned}$$ This is a corollary of the following uniform bounding principle for dp-finite abelian groups: If $H$ is a type-definable subgroup of a dp-finite abelian group $G$, then $|H/H^{00}|$ is bounded by a cardinal $\kappa(G)$ depending only on $G$. We prove both facts in §\[sec:bounds\]. Henselianity ------------ There is comparatively little to say about henselianity. In the dp-minimal case, henselianity of definable valuations was first proven by Jahnke, Simon, and Walsberg [@JSW]. Independently around the same time, the author proved henselianity using the machinery of infinitesimals. The outline of the proof is 1. Assume non-henselianity 2. Pass to a finite extension and get multiple incomparable valuations. 3. Pass to a coarsening and get multiple independent valuations. 4. Consider the multiplicative homomorphism $f(x) = x^2$ in characteristic $\ne 2$ or the additive homomorphism $f(x) = x^2 - x$. 5. Use strong approximation to produce an element $x$ such that $x \notin I'_M$ but $f(x) \in I'_M$. 6. Argue that $f(I_M')$ is strictly smaller than $I_M'$, contradicting the minimality of the group of infinitesimals among $M$-definable infinite type-definable groups. In the last two steps, $I_M'$ denotes the multiplicative infinitesimals $1 + I_M$ or the additive infinitesimals $I_M$ as appropriate. The “strong approximation” step requires the infinitesimals to be an intersection of valuation ideals. As noted in the previous section, this property is unknown for higher rank, so we are not able to prove much. But then a miracle occurs. In the additive Artin-Schreier case, the only real properties of $I_M$ being used are that $I_M$ is the Jacobson radical of a good Bezout domain, and $I_M = I_M^{00}$. If $\mathcal{O}_1$ and $\mathcal{O}_2$ are two incomparable valuation rings on an NIP field, then $R = \mathcal{O}_1 \cap \mathcal{O}_2$ is a good Bezout domain whose Jacobson radical is easily seen to be 00-connected, provided that the residue fields are infinite. This yields an extremely short proof of the henselianity conjecture for NIP fields in positive characteristic. See §\[sec:henselianity\], specifically Lemma \[hensel-key\]. This argument does not directly apply to the $A$-invariant valuation rings constructed using modular lattices. Nevertheless, a variant can be made to work leveraging the infinitesimals, utilizing the existence of $G^{000}$ for type-definable $G$. See §\[sec:further-hensel\] for details. Putting everything together, we find that if $K$ is a sufficiently saturated dp-finite field of positive characteristic, then either $K$ has finite Morley rank, or $K$ admits a non-trivial henselian valuation. This in turn yields a proof of the Shelah conjecture for dp-finite positive characteristic fields, as well as the expected classification. See §\[sec:the-end\]. Outline ------- In §\[sec:henselianity\] we prove the henselianity conjecture in positive characteristic. In §\[sec:broad-narrow\] we generalize Pierre Simon’s results [@surprise] about definability of dp-rank in dp-minimal theories eliminating $\exists^\infty$. In §\[sec:heavy-light\], we apply this to define a notion of “heavy” sets which take the place of infinite sets in the construction of the group of infinitesimals in §\[sec:infinitesimals\]. Section \[sec:likeTT\] verifies the additional property of heavy sets that holds when the field is not of finite Morley rank. Section \[sec:further-hensel\] applies the infinitesimals to the problem of proving henselianity of $A$-invariant valuation rings in positive characteristic. In §\[sec:bounds\] we prove the technical fact that $|H/H^{00}|$ is uniformly bounded as $H$ ranges over type-definable subgroups of a dp-finite abelian group. Section \[sec:modular-lats\] is a collection of abstract facts about modular lattices, probably already known to experts. Specifically, we verify that “independence” and “cubes” make sense (§\[sec:independence\]-\[sec:cubes\]), that there is a minimum subadditive rank (§\[sec:reduced-rank\]), and that there is a modular pregeometry on quasi-atoms (§\[sec:geometry\]). Then, in §\[sec:valuations\], we apply the abstract theory to construct a valuation ring. Finally, in §\[sec:the-end\] we verify the Shelah conjecture for positive characteristic dp-finite fields, and enumerate the consequences. Henselianity in positive characteristic {#sec:henselianity} ======================================= In this section, we prove that definable NIP valuations in positive characteristic must be henselian. We also consider the situation of ${\operatorname{Aut}}({\mathbb{M}}/A)$-invariant valuation rings. \[incomparables\] Let $\mathcal{O}_1$ and $\mathcal{O}_2$ be two incomparable valuation rings on a field $K$. For $i = 1, 2$ let $\mathfrak{m}_i$ be the maximal ideal of $\mathcal{O}_i$. Consider the sets $$\begin{aligned} R &= \mathcal{O}_1 \cap \mathcal{O}_2 \\ I_1 &= \mathfrak{m}_1 \cap \mathcal{O}_2 \\ I_2 &= \mathcal{O}_1 \cap \mathfrak{m}_2 \\ J &= \mathfrak{m}_1 \cap \mathfrak{m}_2. \end{aligned}$$ Then $R$ is a Bezout domain with exactly two maximal ideals $I_1, I_2$ and Jacobson radical $J$. The quotient $R/I_i$ is isomorphic to the residue field $\mathcal{O}_i/\mathfrak{m}_i$. Moreover, $$(a + \mathfrak{m}_1) \cap (b + \mathfrak{m}_2) \ne \emptyset$$ for any $a \in \mathcal{O}_1$ and $b \in \mathcal{O}_2$. (This is mostly or entirely well-known.) Let ${\operatorname{val}}_i : K^\times \to \Gamma_i$ denote the valuation associated to $\mathcal{O}_i$. Note that $x|y$ in $R$ if and only if ${\operatorname{val}}_1(x) \le {\operatorname{val}}_1(y)$ and ${\operatorname{val}}_2(x) \le {\operatorname{val}}_2(y)$. \[3-bezout\] For any $x, y \in R$, the ideal $(x,y)$ is generated by $x$ or $y$ or $x - y$. The following cases are exhaustive: - If ${\operatorname{val}}_1(x) \le {\operatorname{val}}_1(y)$ and ${\operatorname{val}}_2(x) \le {\operatorname{val}}_2(y)$ then $(x,y)$ is generated by $x$. - If ${\operatorname{val}}_1(x) \ge {\operatorname{val}}_1(y)$ and ${\operatorname{val}}_2(x) \ge {\operatorname{val}}_2(y)$, then $(x,y)$ is generated by $y$. - If ${\operatorname{val}}_1(x) < {\operatorname{val}}_1(y)$ and ${\operatorname{val}}_2(x) > {\operatorname{val}}_2(y)$, then $$\begin{aligned} {\operatorname{val}}_1(x - y ) = &{\operatorname{val}}_1(x) < {\operatorname{val}}_1(y) \\ {\operatorname{val}}_2(x - y) = &{\operatorname{val}}_2(y) < {\operatorname{val}}_2(x) \end{aligned}$$ so $(x,y)$ is generated by $x- y$. - If ${\operatorname{val}}_1(x) > {\operatorname{val}}_1(y)$ and ${\operatorname{val}}_2(x) < {\operatorname{val}}_2(y)$, then similarly $(x,y)$ is generated by $x - y$. By the claim, $R$ is a Bezout domain. \[u-claim\] There is an element $u \in \mathfrak{m}_1 \cap (1 + \mathfrak{m}_2)$. By incomparability, we can find $a \in \mathcal{O}_1 \setminus \mathcal{O}_2$ and $b \in \mathcal{O}_2 \setminus \mathcal{O}_1$. Then ${\operatorname{val}}_1(a) \ge 0 > {\operatorname{val}}_1(b)$ and ${\operatorname{val}}_2(b) \ge 0 > {\operatorname{val}}_2(a)$, so $a/b \in \mathfrak{m}_1$ and $b/a \in \mathfrak{m}_2$. Let $u = a/(a + b)$. Then $$\begin{aligned} u & = \frac{a/b}{a/b + 1} \in \mathfrak{m}_1 \\ 1 - u & = \frac{b/a}{b/a + 1} \in \mathfrak{m}_2 \qedhere \end{aligned}$$ If $u$ is as in Claim \[u-claim\], then $u \in I_1$ but $u \notin I_2$, so $I_1 \ne I_2$. \[m-i-surj\] For any $a \in \mathcal{O}_1$, $(a + \mathfrak{m}_1) \cap \mathcal{O}_2 \ne \emptyset$. If $a \in \mathfrak{m}_1$ then $0$ is in the intersection. If $a \in \mathcal{O}_2$ then $a$ is in the intersection. So we may assume $a \in \mathcal{O}_1^\times \setminus \mathcal{O}_2$. Then for $u$ as in Claim \[u-claim\], $$(a^{-1} + u)^{-1} \in (a + \mathfrak{m}_1) \cap \mathcal{O}_2.$$ Indeed, $a^{-1}$ and $a^{-1} + u$ have the same residue class modulo $\mathfrak{m}_1$, so their inverses have the same residue class. And $$a \notin \mathcal{O}_2 \implies a^{-1} \in \mathfrak{m}_2 \implies a^{-1} + u \in 1 + \mathfrak{m}_2 \implies (a^{-1} + u)^{-1} \in 1 + \mathfrak{m}_2 \subseteq \mathcal{O}_2. \qedhere$$ Claim \[m-i-surj\] says that the natural inclusion $$R/I_1 = (\mathcal{O}_1 \cap \mathcal{O}_2)/(\mathfrak{m}_1 \cap \mathcal{O}_2) \hookrightarrow \mathcal{O}_1/\mathfrak{m}_1$$ is onto, hence an isomorphism. Therefore $R/I_1$ is a field and $I_1$ is a maximal ideal. Similarly $I_2$ is a maximal ideal and $R/I_2 \cong \mathcal{O}_2/\mathfrak{m}_2$. There cannot exist a third maximal ideal $I_3$. Otherwise use the Chinese remainder theorem to find $x \in (1 + I_1) \cap (1 + I_2) \cap I_3$ and $y \in I_1 \cap (1 + I_2) \cap (1 + I_3)$. Then $$\begin{aligned} (x) \subseteq & I_3 \not \supseteq (x,y) \\ (y) \subseteq & I_1 \not \supseteq (x,y) \\ (x - y) \subseteq & I_2 \not \supseteq (x,y) \end{aligned}$$ contradicting Claim \[3-bezout\]. Consider the composition $$R/(I_1 \cap I_2) \twoheadrightarrow (R/I_1) \times (R/I_2) \stackrel{\sim}{\to} (\mathcal{O}_1/\mathfrak{m}_1) \times (\mathcal{O}_2/\mathfrak{m}_2).$$ The first map is surjective by the Chinese remainder theorem (as $I_1$ and $I_2$ are distinct maximal ideals). The second map was shown to be an isomorphism earlier. Therefore the composition is surjective, which exactly means that for any $a \in \mathcal{O}_1$ and $b \in \mathcal{O}_2$, the intersection $(a + \mathfrak{m}_1) \cap (b + \mathfrak{m}_2)$ is non-empty. \[hensel-key\] Let $K$ be a field and $\mathcal{O}_i$ be a valuation ring on $K$ for $i = 1,2$. If the structure $(K,\mathcal{O}_1,\mathcal{O}_2)$ is NIP and $K$ has characteristic $p > 0$, then $\mathcal{O}_1$ and $\mathcal{O}_2$ are comparable. Otherwise, let $\mathfrak{m}_1, \mathfrak{m}_2, R, I_1, I_2, J$ be as in Lemma \[incomparables\]. The incomparability of the $\mathcal{O}_i$ implies that $K$ is infinite, hence Artin-Schreier closed (by the Kaplan-Scanlon-Wagner theorem [@NIPfields]). Consider the homomorphism $$f : (J,+) \to ({\mathbb{Z}}/p,+)$$ defined as follows: given $x \in J = \mathfrak{m}_1 \cap \mathfrak{m}_2$, take an Artin-Schreier root $y \in K$ such that $y^p - y = x$. Then $y \in (a + \mathfrak{m}_1) \cap (b + \mathfrak{m}_2)$ for unique $a, b \in {\mathbb{Z}}/p$. Define $f(x) = a - b$. This is independent of the choice of the root $y$, and the homomorphism $f$ is definable. By Lemma \[incomparables\] there is some $y$ such that $y \in (1 + \mathfrak{m}_1) \cap \mathfrak{m}_2$; then $f(y^p - y) = 1$. Therefore $f$ is onto and $\ker f$ is a definable subgroup of $J$ of index $p$. Recall that $J^{00}$ exists in NIP theories. It follows that $J^{00} \subsetneq J$. Now for any $a \in R$, we have $$a \cdot J^{00} = (a \cdot J)^{00} \subseteq J^{00},$$ and so $J^{00}$ is an ideal in $R$. Choose $b \in J \setminus J^{00}$ and let $I = \{x \in R | xb \in J^{00}\}$. Then $I$ is a proper ideal of $R$, so $I \le I_i$ for $i = 1$ or $i = 2$. The maps $$\begin{aligned} (R \cdot b)/(R \cdot b \cap J^{00}) &\cong R/I \twoheadrightarrow R/I_i \cong \mathcal{O}_i/\mathfrak{m}_i \\ (R \cdot b)/(R \cdot b \cap J^{00}) &\cong (R \cdot b + J^{00})/J^{00} \hookrightarrow J/J^{00} \end{aligned}$$ together show that $|J/J^{00}| \ge |\mathcal{O}_i/\mathfrak{m}_i|$. But the fact that $K$ is Artin-Schreier closed forces the residue field $\mathcal{O}_i/\mathfrak{m}_i$ to be Artin-Schreier closed, hence infinite, hence unbounded in elementary extensions. This contradicts the definition of $J^{00}$. Let $(K,\mathcal{O})$ be a valued field and $L/K$ be a finite normal extension. Every extension of $\mathcal{O}$ to $L$ is definable (identifying $L$ with $K^d$ for $d = [L : K]$). This essentially follows from Beth implicit definability. Naming parameters, we may assume that $\mathcal{O}$ is 0-definable and $L$ is 0-interpretable. Recall that ${\operatorname{Aut}}(L/K)$ acts transitively on the set of extensions (essentially because valued fields can be amalgamated, i.e., ACVF has quantifier elimination). So there are only finitely many extensions to $L$, and it suffices to show that at least one extension is definable. For any formula $\phi(x;y)$, the condition “$\phi(L;a)$ is a valuation ring on $L$ extending $\mathcal{O}$” is expressed by a formula $\psi(a)$, so we may replace the original $(K,\mathcal{O})$ with an elementary extension. Fix some $\mathcal{O}_L$ on $L$ extending $K$, and pass to an elementary extension if necessary to ensure that - $M^+ := (L,K,\mathcal{O}_L,\mathcal{O})$ is $\aleph_0$-saturated. - $M^- := (L,K,\mathcal{O})$ is $\aleph_0$-homogeneous and $\aleph_0$-saturated. Because there are only finitely many extensions of $\mathcal{O}$ to $\mathcal{O}_L$ we can find $a_1, \ldots, a_n, b_1, \ldots, b_m \in L$ such that $\mathcal{O}_L$ is the unique extension of $\mathcal{O}$ to $L$ containing $A = \{a_1,\ldots,a_n\}$ and disjoint from $B = \{b_1, \ldots, b_m\}$. Now if $\sigma \in {\operatorname{Aut}}(M^-/AB)$, then $\sigma$ preserves $\mathcal{O}$ setwise, and therefore moves $\mathcal{O}_L$ to *some* extension $\mathcal{O}_L'$ of $\mathcal{O}$. But $\sigma$ fixes $A$ and $B$ so $\mathcal{O}_L'$ must still contain $A$ and be disjoint from $B$. Thus $\mathcal{O}_L' = \mathcal{O}_L$, so ${\operatorname{Aut}}(M^-/AB)$ fixes $\mathcal{O}_L$ setwise. By $\aleph_0$-homogeneity, we see that $${\operatorname{tp}}_{M^-}(c/AB) = {\operatorname{tp}}_{M^-}(c'/AB) \implies (c \in \mathcal{O}_L \iff c \in \mathcal{O}_L')$$ for $c, c' \in L$. Then by considering the map of type spaces $$S_{M^+}^L(AB) \to S_{M^-}^L(AB)$$ the clopen set in $S_{M^+}^L(AB)$ corresponding to $\mathcal{O}_L$ must be the preimage of some subset (necessarily clopen) in $S_{M^-}^L(AB)$, implying that $\mathcal{O}_L$ is $AB$-definable in $M^-$. Here we are using $\aleph_0$-saturation to ensure that every $M^+$ and $M^-$ type is realized in $L$. \[henselianity-conjecture\] Let $(K,\mathcal{O})$ be an NIP valued field of positive characteristic. Then $(K,\mathcal{O})$ is henselian. Otherwise there is some finite normal extension $L/K$ such that $\mathcal{O}$ has multiple extensions to $L$. If $\mathcal{O}_1, \mathcal{O}_2$ are two distinct extensions, then $(L,\mathcal{O}_1,\mathcal{O}_2)$ is an NIP 2-valued field. By Lemma \[hensel-key\], $\mathcal{O}_1$ and $\mathcal{O}_2$ must be comparable. But ${\operatorname{Aut}}(L/K)$ acts transitively on the set of extensions, and a finite group cannot act transitively on a poset unless all elements are incomparable. The same proof should work when $\mathcal{O}$ is $\vee$-definable, or equivalently, $\mathfrak{m}$ is type-definable. We will also need a variant of the above results for invariant valuation rings. Here and in what follows, $A$-invariant means ${\operatorname{Aut}}({\mathbb{M}}/A)$-invariant, and we will often use “invariant” to mean “$A$-invariant for some small $A \subseteq {\mathbb{M}}$”. \[span-a-valuation\] A valuation ring $\mathcal{O} \subseteq K$ is *spanned* by a set $S$ if - Every element of $S$ has positive valuation. - For every $x \in K$ of positive valuation, there is $y \in S$ such that ${\operatorname{val}}(y) \le {\operatorname{val}}(x)$. In other words, ${\operatorname{val}}(S)$ is downwards-cofinal in the interval $(0,+\infty) \subseteq \Gamma$. Equivalently, $S$ is a set of generators for the maximal ideal of $\mathcal{O}$. Recall that $G^{000}$ exists for type-definable abelian $G$ in NIP theories, by a theorem of Shelah ([@shelah-g000], Theorem 1.12). \[invariant-hensel-key\] Let $(K,\ldots)$ be a monster NIP field of positive characteristic, and let $\mathcal{O}_1, \mathcal{O}_2$ be two invariant valuation rings. Suppose that $\mathcal{O}_1$ and $\mathcal{O}_2$ are both spanned by some type-definable subgroup $(I,+) \le (K,+)$ with $I = I^{000}$. Then $\mathcal{O}_1$ and $\mathcal{O}_2$ are comparable. Assume not. Take a small set $A$ such that $\mathcal{O}_1, \mathcal{O}_2$ are $A$-invariant and $I$ is type-definable over $A$. Let $\mathfrak{m}_1, \mathfrak{m}_2, R, I_1, I_2, J$ be as in Lemma \[incomparables\]. Define $f : (J,+) \to ({\mathbb{Z}}/p,+)$ as in the proof of Lemma \[hensel-key\], and let $J'$ be the kernel of $f$. As in the proof of Lemma \[hensel-key\] $f$ is a surjective homomorphism, and so $J'$ is an index p subgroup of $J$. Moreover, $J'$ is $A$-invariant (though not necessarily definable). We claim that $R \cdot I = J$. Indeed, if $x \in J$, then ${\operatorname{val}}_1(x) > 0$ and ${\operatorname{val}}_2(x) > 0$, so we may find $y, z \in I$ such that ${\operatorname{val}}_1(y) \le {\operatorname{val}}_1(x)$ and ${\operatorname{val}}_2(z) \le {\operatorname{val}}_2(x)$. As $R$ is a Bezout domain, there is $w \in R$ such that $(w) = (y,z)$. Then $$\begin{aligned} w|y &\implies {\operatorname{val}}_1(w) \le {\operatorname{val}}_1(y) \implies {\operatorname{val}}_1(w) \le {\operatorname{val}}_1(x) \\ w|z &\implies {\operatorname{val}}_2(w) \le {\operatorname{val}}_2(z) \implies {\operatorname{val}}_2(w) \le {\operatorname{val}}_2(x) \end{aligned}$$ so $w|x$. This in turn implies that $(x) \subseteq (w) = (y,z) \subseteq R \cdot I$. Thus $J \subseteq R \cdot I$, and the converse holds because $I \subseteq \mathfrak{m}_i$ for $i = 1, 2$. For any $r \in R$, the group homomorphism $$I \stackrel{x \mapsto r \cdot x}{\longrightarrow} r \cdot I \hookrightarrow J \twoheadrightarrow J/J' \cong {\mathbb{Z}}/p$$ is $rA$-invariant. The kernel of this map is $rA$-invariant of finite index in $I$, so the kernel must be all of $I$ because $I = I^{000}$. Therefore $r \cdot I \subseteq J'$ for any $r \in R$. It follows that $R \cdot I \subseteq J'$, contradicting the fact that $R \cdot I = J \supsetneq J'$. \[invariant-henselian\] Let $(K,\ldots)$ be a monster NIP field of positive characteristic, and $\mathcal{O}$ be an invariant valuation ring, spanned by some type-definable group $I \le (K,+)$ with $I = I^{000}$. Then $\mathcal{O}$ is henselian. Let $\Gamma$ be the value group of $\mathcal{O}$. Then $\Gamma$ is $p$-divisible (because $K$ is Artin-Schreier closed or finite). For any positive $\gamma \in \Gamma$ and positive integer $n$, we can find $\varepsilon \in I$ such that $$\label{division-cofinal} {\operatorname{val}}(\varepsilon) \le \frac{\gamma}{p^n} < \frac{\gamma}{n},$$ because $\gamma/p^n \in \Gamma_{> 0}$. If henselianity fails, there is a finite normal extension $L/K$ such that $\mathcal{O}$ has at least two (and at most $[L : K]$) extensions $\mathcal{O}_1$ and $\mathcal{O}_2$ to $L$. Let $A$ be a set such that $\mathcal{O}$ and $I$ are $A$-invariant and $L$ is interpretable over $A$. Any $\sigma \in {\operatorname{Aut}}(K/A)$ fixes $\mathcal{O}$ setwise, hence permutes the finitely many extensions of $\mathcal{O}$ to $L$. After adding finitely many parameters to $A$, we may assume that ${\operatorname{Aut}}(K/A)$ fixes $\mathcal{O}_1$ and $\mathcal{O}_2$. Thus $\mathcal{O}_1$ and $\mathcal{O}_2$ are $A$-invariant. If $\Gamma, \Gamma_1, \Gamma_2$ are the value groups of $\mathcal{O}, \mathcal{O}_1, \mathcal{O}_2$ respectively, then $|\Gamma_i/\Gamma| \le [L : K] < \aleph_0$ for $i = 1, 2$. Thus $\Gamma_i \otimes {\mathbb{Q}}= \Gamma \otimes {\mathbb{Q}}$. By (\[division-cofinal\]) it follows that $\mathcal{O}_1$ and $\mathcal{O}_2$ are both spanned by $I$. By Lemma \[invariant-hensel-key\] the valuation rings $\mathcal{O}_1$ and $\mathcal{O}_2$ must be comparable, which is absurd. Broad and narrow sets {#sec:broad-narrow} ===================== Almost everything in this section is a straightforward generalization of the second half of [@surprise]. The general setting ------------------- Let $T$ be a theory, not assumed to eliminate imaginaries. Work in a monster model ${\mathbb{M}}$. Let $X_1, \ldots, X_n$ be definable sets and $Y \subseteq X_1 \times \cdots \times X_n$ be type-definable. Then $Y$ is *broad* if there exist $a_{i,j} \in X_i$ for $1 \le i \le n$ and $j \in {\mathbb{N}}$ such that the following conditions hold: - For fixed $i$, the $a_{i,j}$ are pairwise distinct. - For any function $\eta : [n] \to {\mathbb{N}}$, $$(a_{1,\eta(1)},\ldots,a_{n,\eta(n)}) \in Y.$$ Otherwise, we say that $Y$ is *narrow*. \[n-is-1\] If $n = 1$, then a type-definable set $Y \subseteq X_1$ is broad if and only if it is infinite. \[n-form\] By compactness and saturation, $Y$ is broad if and only if the following holds: for every $m \in {\mathbb{N}}$ there exist $a_{i,j} \in X_i$ for $1 \le i \le n$ and $1 \le j \le m$ such that - For fixed $i$, the $a_{i,j}$ are pairwise distinct. - For any function $\eta : [n] \to [m]$, $$(a_{1,\eta(1)},\ldots,a_{n,\eta(n)}) \in Y.$$ An equivalent condition is that for every $m \in {\mathbb{N}}$ there exist subsets $S_1, \ldots, S_n$ with $S_i \subseteq X_i$, $|S_i| = m$, and $S_1 \times \cdots \times S_n \subseteq Y$. \[type-definability\] Fix a product of definable sets $\prod_{i = 1}^n X_i$. 1. A partial type $\Sigma(\vec{x})$ on $\prod_{i = 1}^n X_i$ is broad if and only if every finite subtype is broad. \[continuity\] 2. If $\{D_b\}_{b \in Y}$ is a definable family of definable subsets of $\prod_{i = 1}^n X_i$, then the set of $b$ such that $D_b$ is broad is type-definable. \[true-type-definability\] 3. Let $A$ be some small set of parameters and $D$ be a definable set. The set of tuples $(a_1, \ldots, a_n, b) \in X_1 \times \cdots \times X_n \times D$ such that ${\operatorname{tp}}(\vec{a}/bA)$ is broad is type-definable. \[pairs\] <!-- --> 1. This follows immediately by compactness. 2. The statement that $D_b$ is broad is equivalent to the small conjunction $$\bigwedge_{m \in {\mathbb{N}}} \exists a_{1,1}, \ldots, a_{n,m} \left( \bigwedge_{i \in [n], 1 \le j < j' \le m} a_{i,j} \ne a_{i,j'} \wedge \bigwedge_{\eta : [n] \to [m]} (a_{1,\eta(1)},\ldots,a_{n,\eta(m)}) \in D_b \right).$$ 3. The statement that ${\operatorname{tp}}(\vec{a}/bA)$ is broad is equivalent to the type-definable condition $$\begin{aligned} \bigwedge_{m \in {\mathbb{N}}} \exists a_{1,1}, \ldots, a_{n,m} \Biggl( & \bigwedge_{i \in [n], 1 \le j < j' \le m} a_{i,j} \ne a_{i,j'} \\ & \wedge \bigwedge_{\eta : [n] \to [m]} (a_{1,\eta(1)},\ldots,a_{n,\eta(m)}) \equiv_{bA} (a_1,\ldots,a_n) \Biggr). \end{aligned}$$ Note that $\vec{a} \equiv_{A\vec{c}} \vec{b}$ is a type-definable condition on $(\vec{a},\vec{b},\vec{c})$ for a fixed small set $A$, and that in a monster model the type-definable conditions are closed under quantification and small conjunctions. Let $A$ be a small set of parameters. Suppose $X_1, \ldots, X_n$ are $A$-definable sets and $Y \subseteq X_1 \times \cdots \times X_n$ is type-definable over $A$. Then $Y$ is broad if and only if there exists a mutually $A$-indiscernible array $\langle a_{ij} \rangle_{i \in [n], j \in {\mathbb{N}}}$ such that for fixed $i$ the $a_{ij}$ are pairwise distinct elements of $X_i$, and such that for any $\eta : [n] \to {\mathbb{N}}$, the tuple $(a_{1,\eta(1)},a_{2,\eta(2)},\ldots,a_{n,\eta(n)})$ is an element of $Y$. In other words, the witnesses of broadness can be chosen to be mutually $A$-indiscernible. This follows from the fact that we can extract mutually indiscernible arrays. \[ideal\] Let $X_1, \ldots, X_n$ be infinite definable sets, and let $Y, Z$ be type-definable subsets. 1. If $Y \subseteq Z$ and $Y$ is broad, then $Z$ is broad. 2. \[ideal-unions\] If $Y \cup Z$ is broad, then $Y$ is broad or $Z$ is broad. Equivalently, narrow sets form an ideal: 1. If $Y \subseteq Z$ and $Z$ is narrow, then $Y$ is narrow. 2. If $Y$ and $Z$ are narrow, then $Y \cup Z$ is narrow. If $Y$ is broad then $Z$ is broad because the witnesses of broadness of $Y$ show broadness of $Z$. If $Y \cup Z$ is broad, we can take a small set $A$ over which the $X_i, Y, Z$ are defined and then find a mutually $A$-indiscernible array $\langle a_{i,j} \rangle_{i \in [n], j \in {\mathbb{N}}}$ such that the $i$th row is a sequence of distinct elements, and such that for every $\eta : [n] \to {\mathbb{N}}$, $$(a_{1,\eta(1)},\ldots,a_{n,\eta(n)}) \in Y \cup Z.$$ Choose some fixed $\eta_0 : [n] \to {\mathbb{N}}$. Without loss of generality (interchanging $Y$ and $Z$), $$(a_{1,\eta_0(1)},\ldots,a_{n,\eta_0(n)}) \in Y.$$ By mutual $A$-indiscernibility, it follows that $$(a_{1,\eta(1)},\ldots,a_{n,\eta(n)}) \in Y$$ for all $\eta$, and so $Y$ is broad. \[completion\] Let $A$ be a small set of parameters, let $X_1,\ldots,X_n$ be $A$-definable, and let $Y \subseteq X_1 \times \cdots \times X_n$ be type-definable over $A$. Then $Y$ is broad if and only if ${\operatorname{tp}}(\vec{a}/A)$ is broad for some $\vec{a} \in Y$. If ${\operatorname{tp}}(\vec{a}/A)$ is broad then $Y$ is broad as it contains (the set of realizations of) ${\operatorname{tp}}(\vec{a}/A)$. Conversely, suppose $Y$ is broad. Choose a mutually $A$-indiscernible array $\{\alpha_{i,j}\}_{i \in [n], j \in {\mathbb{N}}}$ witnessing broadness. Let $\vec{a} = (\alpha_{1,1},\alpha_{2,1},\ldots,\alpha_{n,1})$. By mutual indiscernibility, for any $\eta : [n] \to {\mathbb{N}}$ the $n$-tuple $(\alpha_{1,\eta(1)},\ldots,\alpha_{n,\eta(n)})$ realizes ${\operatorname{tp}}(\vec{a}/A)$, so ${\operatorname{tp}}(\vec{a}/A)$ is broad. The trick in the following proof is taken from Theorem 3.10 in [@surprise]. \[simon-lemma\] Assume NIP. Let $X_1, \ldots, X_n$ be infinite definable sets, and let $Y \subseteq X_1 \times \cdots \times X_n$ be *definable*, not just type-definable. Assume $Y$ is broad and $n \ge 2$. 1. \[sl-1\] There exists some $b \in X_n$ such that the slice $$\{(a_1,\ldots,a_{n-1}) \in X_1 \times \cdots \times X_{n-1} ~|~ (a_1,\ldots,a_{n-1},b) \in Y\}$$ is broad as a subset of $X_1 \times \cdots \times X_{n-1}$. 2. \[sl-2\] There exists a broad definable subset $D_{< n} \subseteq X_1 \times \cdots \times X_{n-1}$ and an infinite definable subset $D_n \subseteq X_n$ such that $$(D_{< n} \times D_n) \setminus Y$$ is a “hyperplane,” in the sense that for every $b \in D_n$, the definable set $$\{(a_1,\ldots,a_{n-1}) \in D_{< n} ~|~ (a_1,\ldots,a_{n-1},b) \notin Y\}$$ is narrow in $X_1 \times \cdots \times X_{n-1}$. Choose a small set $A$ over which the $X_i$ and $Y$ are definable. Choose a mutually indiscernible array $\{a_{i,j}\}_{i \in [n], j \in {\mathbb{N}}}$ witnessing broadness of $Y$. Let $b_j = a_{n,j}$ be the elements of the bottom row. Note that the top $n-1$ rows $\{a_{i,j}\}_{i \in [n-1], j \in {\mathbb{N}}}$ form a mutually indiscernbile array over $A\vec{b}$. In particular, if $\vec{e}$ denotes the first column $$\vec{e} = (e_1,\ldots,e_{n-1}) := (a_{1,1},a_{2,1},\ldots,a_{n-1,1}),$$ then ${\operatorname{tp}}(\vec{e}/A\vec{b})$ is broad (within $X_1 \times \cdots \times X_{n-1}$). Note that the $n$-tuple $\vec{e}b_j$ lies in $Y$ for any $j \in {\mathbb{N}}$. In particular, $\vec{e}$ lies in the slice over $b_1$, so the slice over $b_1$ is broad, proving the first point. For the second point, consider sequences $c_1, c_2, \ldots, c_m$ satisfying the following constraints: 1. The sequence $I = b_1 c_1 b_2 c_2 \cdots b_m c_m b_{m+1} b_{m+2} b_{m+3} \cdots$ is $A$-indiscernible. 2. The type ${\operatorname{tp}}(\vec{e}/AI)$ is broad. 3. The tuple $\vec{e}c_j$ is *not* in $Y$ for $j = 1, \ldots, m$. There is at least one such sequence, namely, the empty sequence. Because of NIP, the first and third conditions imply some absolute bound on $m$, so we can find such a sequence with $m$ maximal. Fix such a sequence $$I = b_1 c_1 b_2 c_2 \cdots b_m c_m b_{m+1} b_{m+2} b_{m+3} \cdots$$ and choose $c_{m+1}$ such that $b_1 c_1 \cdots b_m c_m b_{m+1} c_{m+1} b_{m+2} b_{m+3} \cdots$ is $A$-indiscernible. If $\vec{\alpha} \in X_1 \times \cdots \times X_{n-1}$ and $\beta \in X_n$ satisfy $$\vec{\alpha} \equiv_{AI} \vec{e},~ \beta \equiv_{AI} c_{m+1},\textrm{ and } (\vec{\alpha},\beta) \notin Y$$ then ${\operatorname{tp}}(\vec{\alpha}/AI\beta)$ is narrow. Applying an automorphism over $AI$, we may assume $\vec{\alpha} = \vec{e}$. If ${\operatorname{tp}}(\vec{e}/AI\beta)$ is broad, then 1. The sequence $I' = b_1 c_1 b_2 c_2 \cdots b_m c_m b_{m+1} \beta b_{m+2} b_{m+3} \cdots$ is $A$-indiscernible, because $\beta \equiv_{AI} c_{m+1}$. 2. The type ${\operatorname{tp}}(\vec{e}/AI')$ is broad. 3. The tuple $\vec{e}\beta$ is not in $Y$, and neither are $\vec{e}c_j$ for $j = 1,\ldots,m$. This contradicts the maximality of $m$. By Remark \[type-definability\].\[pairs\] and compactness, there must be formulas $\varphi(\vec{x}) \in {\operatorname{tp}}(\vec{e}/AI)$ and $\psi(y) \in {\operatorname{tp}}(c_{m+1}/AI)$ such that $$\varphi(\vec{\alpha}) \wedge \psi(\beta) \wedge \left((\vec{\alpha},\beta) \notin Y\right)$$ implies narrowness of ${\operatorname{tp}}(\vec{\alpha}/AI\beta)$. Let $D_{<n}$ be the subset of $X_1 \times \cdots \times X_{n-1}$ cut out by $\varphi(\vec{x})$, and let $D_n$ be the subset of $X_n$ cut out by $\psi(y)$. Because the sequence $$b_1 c_1 b_2 c_2 \cdots b_m c_m b_{m+1} c_{m+1} b_{m+2} b_{m+3} b_{m+4} \cdots$$ is $A$-indiscernible and non-constant (as the $b_i$ are distinct), it follows that no term is in the algebraic closure of the other terms. In particular $$c_{m+1} \notin {\operatorname{acl}}(Ab_1c_1 \ldots b_mc_m b_{m+1} b_{m+2} \cdots) = {\operatorname{acl}}(AI).$$ Therefore $D_n$ is infinite. And $D_{<n}$ is a broad subset of $X_1 \times \cdots \times X_{n-1}$ because ${\operatorname{tp}}(\vec{e}/AI)$ is broad (by choice of $c_1,\ldots,c_m$). It remains to show that if $\beta \in D_n$, then the set $$\{ \vec{\alpha} \in D_{< n} ~|~ (\vec{\alpha},\beta) \notin Y\}$$ is narrow as a subset of $X_1 \times \cdots \times X_{n-1}$. Indeed, the set in question is $AI\beta$ definable, and if $\vec{\alpha}$ belongs to the set then ${\operatorname{tp}}(\vec{\alpha}/AI\beta)$ is narrow by choice of $\varphi(\vec{x})$ and $\psi(y)$. By Proposition \[completion\] the set in question is narrow. \[main-characterization\] Assume NIP. Let $X_1, \ldots, X_n$ be definable and $Y$ be a definable subset of $X_1 \times \cdots \times X_n$. Then $Y$ is broad if and only if there exist infinite subsets $D_i \subseteq X_i$ such that $(D_1 \times \cdots \times D_n) \setminus Y$ is a “hyperplane,” in the sense that for every $b \in D_n$ the set $$\{(a_1,\ldots,a_{n-1}) \in D_1 \times \cdots \times D_{n-1} ~|~ (a_1,\ldots,a_{n-1},b) \notin Y\}$$ is narrow as a subset of $X_1 \times \cdots \times X_{n-1}$. For the “if” direction, first note that $D_1 \times \cdots \times D_n$ is broad: for each $i \in [n]$ we can choose $a_{i,1}, a_{i,2}, \ldots$ to be an arbitrary infinite sequence of distinct elements in $D_i$; the array $\{a_{i,j}\}_{i \in [n], j \in {\mathbb{N}}}$ then witnesses broadness of $D_1 \times \cdots \times D_n$. Consider the sets $$\begin{aligned} Y' &:= (D_1 \times \cdots \times D_n) \cap Y \\ H &:= (D_1 \times \cdots \times D_n) \setminus Y. \end{aligned}$$ Then $D_1 \times \cdots \times D_n$ is the (disjoint) union of $Y'$ and $H$, so at least one of $Y'$ and $H$ must be broad. If $H$ were broad, by Lemma \[simon-lemma\].\[sl-1\] it could not be a hyperplane. Therefore $H$ is narrow and $Y'$ is broad, and so $Y$ is broad, proving the “if” direction. We prove the “only if” direction by induction on $n$. For the base case $n = 1$, we can take $D_1 = Y$ by Remark \[n-is-1\]. Next suppose $n > 1$. By Lemma \[simon-lemma\].\[sl-2\] there exist definable sets $D_{<n} \subseteq X_1 \times \cdots \times X_{n-1}$ and $D_n \subseteq X_n$ with $D_{< n}$ broad, $D_n$ infinite, and $(D_{<n} \times D_n) \setminus Y$ a hyperplane. By induction there exist infinite definable sets $D_i \subseteq X_i$ for $1 \le i < n$ such that the set $H' = (D_1 \times \cdots \times D_{n-1}) \setminus D_{<n}$ is a hyperplane. By Lemma \[simon-lemma\].\[sl-1\] the set $H'$ is narrow. For any $b \in D_n$, the two sets $$\begin{aligned} & \{\vec{a} \in D_1 \times \cdots \times D_{n-1} ~|~ \vec{a} \notin D_{< n} \} = H' \\ & \{\vec{a} \in D_{< n} ~|~ (\vec{a},b) \notin Y\} \end{aligned}$$ are both narrow (the latter because $(D_{< n} \times D_n) \setminus Y$ is a hyperplane). The union of these two sets contains $$\{\vec{a} \in D_1 \times \cdots \times D_{n-1} ~|~ (\vec{a},b) \notin Y\}$$ which must therefore be narrow. Therefore $(D_1 \times \cdots \times D_n) \setminus Y$ is a hyperplane, completing the proof. Assume that $T$ is NIP and eliminates $\exists^\infty$. Then “broadness is definable in families” on the product $X_1 \times \cdots \times X_n$. In other words, if $\{Y_b\}_{b \in Z}$ is a definable family of definable subsets of $X_1 \times \cdots \times X_n$, then the set $\{b \in Z ~|~ Y_b \textrm{ is broad}\}$ is definable. In fact, we only need $T$ to eliminate $\exists^\infty$ on the sets $X_1,\ldots,X_n$. We proceed by induction on $n$. The base case $n = 1$ is equivalent to elimination of $\exists^\infty$ by Remark \[n-is-1\]. Suppose $n > 1$ and $\{Y_b\}_{b \in Z}$ is a definable family of definable subsets of $X_1 \times \cdots \times X_n$. Let $A$ be a set of parameters over which everything is defined. The set $\{b \in Z ~|~ Y_b \textrm{ is broad}\}$ is type-definable by Remark \[type-definability\].\[true-type-definability\]. It remains to show that the set is also $\vee$-definable: if $Y_{b_0}$ is broad, then there is some $A$-definable neighborhood $N$ of $b_0$ such that $Y_b$ is broad for $b \in N$. Indeed, by Theorem \[main-characterization\] there exist formulas $\phi_i(x,z)$ and elements $c_1,\ldots,c_n$ such that $$\bigwedge_{i = 1}^n \exists^\infty x \in X_i : \phi_i(x,c_i)$$ and $$\begin{aligned} \forall y \in X_n : \phi_n(x,c_n) \implies \neg \exists^{broad} & (x_1,\ldots,x_{n-1}) \in X_1 \times \cdots \times X_{n-1} : \\ & \left((x_1,\ldots,x_{n-1},y) \notin Y_{b_0}\right) \wedge \bigwedge_{i = 1}^{n-1} \phi_i(x_i,c_i) \end{aligned}$$ where $\exists^{broad} \vec{x} : P(\vec{x})$ means that the set of $\vec{x}$ such that $P(\vec{x})$ holds is broad. (This quantifier is eliminated, by induction.) We take $N$ to be the $A$-definable set of $b$ such that $$\begin{aligned} \exists c_1, \ldots, c_n : & \left( \bigwedge_{i = 1}^n \exists^\infty x \in X_i : \phi_i(x,c_i) \right) \\ &\wedge \Biggl( \forall y \in X_n : \phi_n(x,c_n) \implies \neg \exists^{broad} (x_1,\ldots,x_{n-1}) \in X_1 \times \cdots \times X_{n-1} : \\ & \left((x_1,\ldots,x_{n-1},y) \notin Y_b\right) \wedge \bigwedge_{i = 1}^{n-1} \phi_i(x_i,c_i)\Biggr). \end{aligned}$$ If $b \in N$, then the sets $\phi_i({\mathbb{M}};c_i)$ show that $Y_b$ is broad by Theorem \[main-characterization\]. This proves $\vee$-definability of the set of $b$ such that $Y_b$ is broad, completing the inductive step and the proof. \[n-bound\] Assume $T$ is NIP and eliminates $\exists^\infty$. Let $X_1, \ldots, X_n$ be definable sets and $\{D_b\}_{b \in Y}$ be a definable family of subsets of $X_1 \times \cdots \times X_n$. Then there is some constant $m$ depending on the family such that for any $b \in Y$, the set $D_b$ is broad if and only if there exist $\{a_{i,j}\}_{i \in [n],~j \in [m]}$ such that - For fixed $i$, the $a_{i,j}$ are pairwise distinct elements of $X_i$. - For any $\eta : [n] \to [m]$, the tuple $(a_{1,\eta(1)},\ldots,a_{n,\eta(n)})$ belongs to $D_b$. Equivalently, there is an $m$ such that for every $b$, $D_b$ is broad if and only if there exist finite subsets $S_i \subseteq X_i$ of cardinality $m$ such that $S_1 \times \cdots \times S_n \subseteq D_b$. This follows by compactness and Remark \[n-form\], once we know that “broad” is a definable condition. Externally definable sets {#sec:extern1} ------------------------- It is a theorem of Chernikov and Simon that in NIP theories eliminating $\exists^\infty$, any infinite externally definable set contains an infinite internally definable set (Corollary 1.12 in [@edsdp]). As a consequence, if an infinite (internally) definable set $Y$ is covered by finitely many externally definable sets $D_1, \ldots, D_\ell$, then one of the $D_i$ contains an infinite internally definable set. The following lemma is an analogue for broad sets. \[chernikov-1\] Assume $T$ is NIP and eliminates $\exists^\infty$. Let $M \preceq {\mathbb{M}}$ be a small model. Let $X_1, \ldots, X_n$ be $M$-definable infinite sets and $Y \subseteq X_1 \times \cdots \times X_n$ be an $M$-definable broad set. Let $D_1, \ldots, D_\ell$ be ${\mathbb{M}}$-definable subsets of $X_1 \times \cdots \times X_n$ such that $$Y(M) \subseteq \bigcup_{k = 1}^\ell D_k.$$ Then there exists some $k$ and some $M$-definable broad set $Y' \subseteq Y$ such that $Y'(M) \subseteq D_k$. For every $m \in {\mathbb{N}}$ we can find $\{a_{i,j}\}_{i \in [n], j \in [m]}$ satisfying the following conditions: - $a_{i,j} \in X_i$ - $a_{i,j} \ne a_{i,j'}$ for $j \ne j'$. - For any $\eta : [n] \to [m]$ the tuple $(a_{1,\eta(1)},\ldots,a_{n,\eta(n)})$ lies in $Y$. Because $X_i$ and $Y$ are $M$-definable, we can even choose the $a_{i,j}$ to lie in $M$. By the Ramsey-theoretic statement underlying Proposition \[ideal\].\[ideal-unions\], there is some fixed $k \in [\ell]$ such that for every $m \in {\mathbb{N}}$ we can find $\{a_{i,j}\}_{i \in [n], j \in [m]}$ satisfying the following conditions: - $a_{i,j} \in X_i(M)$ - $a_{i,j} \ne a_{i,j'}$ for $j \ne j'$. - For any $\eta : [n] \to [m]$ the tuple $(a_{1,\eta(1)},\ldots,a_{n,\eta(n)})$ lies in $Y \cap D_k$. By honest definitions ([@NIPguide], Remark 3.14), the externally definable set $Y(M) \cap D_k$ can be approximated by internally definable sets in the following sense: there is an $M$-definable family $\{F_b\}$ such that for every finite subset $S \subseteq Y(M) \cap D_k$ there is a $b \in {\operatorname{dcl}}(M)$ such that $$S \subseteq F_b(M) \subseteq Y(M) \cap D_k.$$ Take $m$ as in Corollary \[n-bound\] for the family $\{F_b\}$. Take $a_{i,j}$ for $i \in [n]$ and $j \in [m]$ such that - $a_{i,j} \in X_i(M)$ - $a_{i,j} \ne a_{i,j'}$ for $j \ne j'$. - For any $\eta : [n] \to [m]$ the tuple $(a_{1,\eta(1)},\ldots,a_{n,\eta(n)})$ lies in $Y \cap D_k$ (hence in $Y(M) \cap D_k$). Let $S$ be the finite subset of tuples of the form $(a_{1,\eta(1)},\ldots,a_{n,\eta(n)})$. Take $b \in {\operatorname{dcl}}(M)$ such that $S \subseteq F_b(M) \subseteq Y(M) \cap D_k$. The $a_{i,j}$ show that $F_b$ is broad, by choice of $m$. Let $Y' = F_b$. The fact that $Y'(M) \subseteq Y(M)$ implies that $Y' \subseteq Y$, as $Y$ and $Y'$ are both $M$-definable and $M \preceq {\mathbb{M}}$. \[something-similar-1\] Let $M \preceq {\mathbb{M}}$ be a small model and $X_1, \ldots, X_n$ be infinite $M$-definable sets. Suppose that $Y \subseteq X_1 \times \cdots \times X_n$ is broad and $M$-definable, and that $W \subseteq X_1 \times \cdots \times X_n$ is ${\mathbb{M}}$-definable. If $Y(M) \subseteq W$, then $W$ is broad. For every $m$, we can find $\{a_{i,j}\}_{i \in [n], j \in [m]}$ satisfying the following conditions: 1. $a_{i,j} \in X_i$. 2. $a_{i,j} \ne a_{i,j'}$ for $j \ne j'$. 3. For any $\eta : [n] \to [m]$ the tuple $(a_{1,\eta(1)},\ldots,a_{n,\eta(n)})$ lies in $Y$. The requirements on the $a_{i,j}$ are $M$-definable, and $[n] \times [m]$ is finite. Therefore, for any $m$ we can choose the $a_{i,j}$ to lie in $M$. Having done so, $$(a_{1,\eta(1)},\ldots,a_{n,\eta(n)}) \in Y(M) \subseteq W$$ for any $\eta$. The existence of the $a_{i,j}$ for all $m$ imply that $W$ is broad. The finite rank setting ----------------------- We now turn to proving Theorem \[finite-rank-case\], which relates broadness and narrowness to dp-rank, under certain assumptions. \[narrow-confusion\] Let $X_1, \ldots, X_n$ be $A$-definable sets, and let $(a_1,\ldots,a_n)$ be a tuple in $X_1 \times \cdots \times X_n$. Suppose that there exists a sequence $b_1, b_2, \ldots \in X_n$ of pairwise distinct elements such that $$(a_1,\ldots,a_{n-1},b_i) \equiv_A (a_1,\ldots,a_{n-1},a_n)$$ for every $i$, and such that ${\operatorname{tp}}(a_1,\ldots,a_{n-1}/A\vec{b})$ is broad. Then ${\operatorname{tp}}(a_1,\ldots,a_n/A)$ is broad. Let $\{e_{i,j}\}_{i \in [n-1], j \in {\mathbb{N}}}$ witness broadness of ${\operatorname{tp}}(a_1,\ldots,a_{n-1}/A\vec{b})$. For any $\eta : [n-1] \to {\mathbb{N}}$, $$(e_{1,\eta(1)},\ldots,e_{n-1,\eta(n-1)}) \equiv_{A \vec{b}} (a_1,\ldots,a_{n-1}).$$ Thus, for any $\eta : [n-1] \to {\mathbb{N}}$ and any $j$, $$\begin{aligned} (e_{1,\eta(1)},\ldots,e_{n-1,\eta(n-1)},b_j) & \equiv_A (a_1,\ldots,a_{n-1},b_j) \\ & \equiv_A (a_1,\ldots,a_{n-1},a_n). \end{aligned}$$ If we set $e_{n,j} := b_j$, then the $e_{i,j}$ show that ${\operatorname{tp}}(\vec{a}/A)$ is broad. \[faux-independence\] Let $X_1, X_2, \ldots, X_n$ be $A$-definable sets of finite dp-ranks $r_1, \ldots, r_n$. Let $(a_1,a_2,\ldots,a_n)$ be a tuple in $X_1 \times \cdots \times X_n$. Then $${\operatorname{dp-rk}}(a_1,\ldots,a_n/A) \le r_1 + \cdots + r_n$$ by subadditivity of dp-rank, and if equality holds then $${\operatorname{dp-rk}}(a_i/Aa_1a_2 \cdots a_{i-1} a_{i+1} \cdots a_n) = r_i$$ for every $i$. Indeed, otherwise $${\operatorname{dp-rk}}(a_i/Aa_1a_2 \cdots a_{i-1} a_{i+1} \cdots a_n) < r_i$$ and so $$\begin{aligned} r_1 + \cdots + r_n & \le {\operatorname{dp-rk}}(a_i/Aa_1\cdots a_{i-1}a_{i+1}\cdots a_n) + {\operatorname{dp-rk}}(a_1,\ldots,a_{i-1}, a_{i+1}, \ldots, a_n/A) \\ & < r_i + (r_1 + \cdots + r_{i-1} + r_{i+1} + \cdots + r_n) \end{aligned}$$ which is absurd. The technique for the next proof comes from Proposition 3.4 in [@surprise]. \[mass-confusion\] Let $X, Y$ be infinite $A$-definable sets of finite dp-rank $n$ and $m$, respectively. Let $(a,b)$ be a tuple in $X \times Y$ with ${\operatorname{dp-rk}}(a,b/A) = n + m$. Then there exist pairwise distinct $b_1, b_2, \ldots$ such that $ab_i \equiv_A ab$ and such that ${\operatorname{dp-rk}}(a/Ab_1 \ldots, b_\ell) = n$ for every $\ell$. Take an ict-pattern of depth $n + m$ in ${\operatorname{tp}}(ab/A)$, extract a mutually $A$-indiscernible ict-pattern, and extend the pattern to have columns indexed by ${\mathbb{N}}\times {\mathbb{N}}$ (with lexicographic order). This yields formula $\phi_i(x,y;z)$ for $1 \le i \le n + m$ and elements $c_{i,j,k}$ for $1 \le i \le n + m$ and $j, k \in {\mathbb{N}}$ such that $c_{i;j,k}$ is a mutually $A$-indiscernible array, and such that $$\bigwedge_{i = 1}^{n + m} \left( \phi_i(x,y;c_{i;0,0}) \wedge \bigwedge_{(j,k) \ne (0,0)} \neg \phi_i(x,y;c_{i;j,k}) \right)$$ is consistent with ${\operatorname{tp}}(a,b/A)$. Moving the $c$’s by an automorphism over $A$, we may assume that $(a,b)$ realizes this type, so that $\phi_i(a,b;c_{i;j,k})$ holds if and only if $(j,k) = (0,0)$. By the proof of subadditivity of dp-rank, there exist $m$ rows which form a mutually $Aa$-indiscernible array. Without loss of generality, these are the rows $i = n + 1, \ldots, n + m$. For $j \in {\mathbb{N}}$ let $d_j$ be an enumeration of $\{ c_{i;j,k} \}_{n < i \le n+m,~ k \in {\mathbb{N}}}$. The $d_j$ form an indiscernible sequence over $Aa$, so we may choose $b_j$ such that $$b_jd_j \equiv_{Aa} bd_0$$ and such that $b_0 = b$. In particular, $ab_j \equiv_A ab$ for any $j$. For any $\ell > 0$ the tuple $(a,b_0,b_1,\ldots,b_{\ell - 1})$ has dp-rank $n + \ell m$ over $A$. The dp-rank is at most $n + \ell m$ by subadditivity of dp-rank (as $b_j \equiv_A b \implies b_j \in Y$). It thus suffices to exhibit $(n + \ell m)$-many mutually $A$-indiscernible sequences, none of which are indiscernible over $A a b_0 b_1 \ldots b_{\ell - 1}$: - For $1 \le i \le n$, the sequence $$c_{i,0,0}, c_{i,0,1}, c_{i,0,2}, \ldots,$$ which fails to be indiscernible over $ab_0 = ab$ on account of the fact that $$\phi_i(a,b,c_{i,0,k}) \iff k = 0.$$ - For $n < i \le n + m$ and for $0 \le j < \ell$, the sequence $$c_{i,j,0}, c_{i,j,1}, c_{i,j,2}, \ldots$$ which fails to be indiscernible over $ab_j$ on account of the fact that $$\phi_i(a,b_j,c_{i,j,k}) \iff \phi_i(a,b,c_{i,0,k}) \iff k = 0.$$ These sequences are indeed mutually $A$-indiscernible, because we can split the mutually indiscernible array $c_{i;j,k}$ into a mutually indiscernible array $c_{i,j;k}$. Now by Remark \[faux-independence\], it follows that $${\operatorname{dp-rk}}(a/Ab_0 b_1 \cdots b_{\ell - 1}) = n$$ for each $\ell$. Moreover, $${\operatorname{dp-rk}}(b_\ell/Aa b_0 b_1 \cdots b_{\ell - 1}) = m > 0,$$ implying that $b_\ell \notin {\operatorname{acl}}(A a b_0 \cdots b_{\ell - 1})$. In particular, $b_\ell \ne b_j$ for $j < \ell$, and the $b_j$ are pairwise distinct. \[halfway-there\] For $i = 1, \ldots, n$ let $X_i$ be a definable set of finite dp-rank $r_i > 0$. Let $Y \subseteq X_1 \times \cdots \times X_n$ be type-definable. If ${\operatorname{dp-rk}}(Y) = r_1 + \cdots + r_n$ then $Y$ is broad. We proceed by induction on $n$. Assume ${\operatorname{dp-rk}}(Y) = r_1 + \cdots + r_n$. For the base case $n = 1$, we see that ${\operatorname{dp-rk}}(Y) = r_1 > 0$, so $Y$ is infinite. By Remark \[n-is-1\], $Y$ is broad. Suppose $n > 1$. Let $A$ be a small set of parameters such that the $X_i$ are definable over $A$ and $Y$ is type-definable over $A$. Take $(a_1,\ldots,a_n) \in Y$ such that ${\operatorname{dp-rk}}(a_1,\ldots,a_n/A) = r_1 + \cdots + r_n$. By Lemma \[mass-confusion\] applied to $(a_1,\ldots,a_{n-1};a_n)$, there exist pairwise distinct $b_1, b_2, \ldots$ such that $$(a_1,\ldots,a_{n-1},a_n) \equiv_A (a_1,\ldots,a_{n-1},b_j)$$ for every $j$ and such that $${\operatorname{dp-rk}}(a_1,\ldots,a_{n-1}/Ab_1\ldots b_j) = r_1 + \cdots + r_{n-1}$$ for every $j$. By induction, ${\operatorname{tp}}(a_1,\ldots,a_{n-1}/Ab_1 \ldots b_j)$ is broad for every $j$. By Remark \[type-definability\].\[continuity\], it follows that ${\operatorname{tp}}(a_1,\ldots,a_{n-1}/Ab_1 b_2 \cdots)$ is broad. By Lemma \[narrow-confusion\], it follows that ${\operatorname{tp}}(a_1,\ldots,a_n/A)$ is broad, and so $Y$ is broad. A definable set $D$ is *quasi-minimal* if $D$ has finite dp-rank $n > 0$, and every definable subset $D' \subseteq D$ has dp-rank $0$ or $n$. Equivalently, every infinite definable subset of $D$ has the same (finite) dp-rank as $D$. If $D$ is a definable set of dp-rank 1, then $D$ is quasi-minimal. \[they-exist\] If $D$ is an infinite definable set of finite dp-rank, then $D$ contains a quasi-minimal definable subset. \[finite-rank-case\] Assume NIP. Let $X_1, \ldots, X_n$ be quasi-minimal definable sets of rank $r_1, \ldots, r_n$, and let $Y \subseteq X_1 \times \cdots \times X_n$ be definable. Then $Y$ is broad if and only if ${\operatorname{dp-rk}}(Y) = r_1 + \cdots + r_n$. Note that the $r_i > 0$ by definition of quasi-minimal. The “if” direction then follows by Proposition \[halfway-there\]. We prove the “only if” direction by induction on $n$. For $n = 1$, $Y$ is broad if and only if $Y$ is infinite (by Remark \[n-is-1\]) if and only if ${\operatorname{dp-rk}}(Y) = r_1$ (by definition of quasi-minimal). Assume $n > 1$. Let $Y$ be broad. By Theorem \[main-characterization\], there are infinite definable subsets $D_1 \times \cdots \times D_n$ such that for every $b \in D_n$, the set $$H_b := \{(a_1,\ldots,a_{n-1}) \in D_1 \times \cdots \times D_{n-1} ~|~ (a_1,\ldots,a_{n-1},b) \notin Y\}$$ is narrow (as a subset of $X_1 \times \cdots \times X_{n-1}$). Note that ${\operatorname{dp-rk}}(D_i) = r_i$ by quasi-minimality. Let $$\begin{aligned} H &= (D_1 \times \cdots \times D_n) \setminus Y = \coprod_{b \in D_n} H_b \\ Y' &= (D_1 \times \cdots \times D_n) \cap Y. \end{aligned}$$ By induction, ${\operatorname{dp-rk}}(H_b) < r_1 + \cdots + r_{n-1}$ for every $b \in D_n$. By subadditivity of the dp-rank, it follows that $${\operatorname{dp-rk}}(H) < r_1 + \cdots + r_{n-1} + {\operatorname{dp-rk}}(D_n) = r_1 + \cdots + r_n.$$ Now $D_1 \times \cdots \times D_n = H \cup Y'$, so $$r_1 + \cdots + r_n = {\operatorname{dp-rk}}(H \cup Y') = \max({\operatorname{dp-rk}}(H),{\operatorname{dp-rk}}(Y')).$$ Given the low rank of $H$, this forces $Y'$ to have rank $r_1 + \cdots + r_n$. Then $$r_1 + \cdots + r_n = {\operatorname{dp-rk}}(Y') \le {\operatorname{dp-rk}}(Y) \le {\operatorname{dp-rk}}(X_1 \times \cdots \times X_n) = r_1 + \cdots + r_n$$ completing the inductive step. Combining Theorem \[finite-rank-case\] with Corollary \[n-bound\] yields the following: \[finite-rank-final\] Assume $T$ is NIP and eliminates $\exists^\infty$, and let $X_1, \ldots, X_n$ be quasi-minimal sets of finite dp-rank. Let $r = {\operatorname{dp-rk}}(X_1 \times \cdots \times X_n)$. Given a definable family $\{D_b\}_{b \in Y}$ of subsets of $X_1 \times \cdots \times X_n$, the set of $b$ such that ${\operatorname{dp-rk}}(D_b) = r$ is definable. In fact, there is some $m$ depending on the family such that ${\operatorname{dp-rk}}(D_b) = r$ if and only if there exist finite subsets $S_i \subseteq X_i$ of cardinality $m$ such that $S_1 \times \cdots \times S_n \subseteq D_b$. Heavy and light sets {#sec:heavy-light} ==================== In this section, we assume that ${\mathbb{M}}$ is a dp-finite field, possibly with additional structure. The goal is to define a notion of “heavy” and “light” sets, and prove Theorem \[heavy-light\], which verifies that heaviness satisfies the properties needed for the construction of infinitesimals. The heavy sets turn out to be exactly the definable sets of full rank, but this will not be proven until a later paper ([@prdf2], Theorem 5.9.2). By the assumption of finite dp-rank, $T$ is NIP. Moreover, $T$ eliminates $\exists^\infty$ by a theorem of Dolich and Goodrick ([@dolich-goodrick-strong-oags], Corollary 2.2). Therefore, the results of the previous section apply. Coordinate configurations ------------------------- A *coordinate configuration* is a tuple $(X_1,\ldots,X_n,P)$ where $X_i \subseteq {\mathbb{M}}$ are quasi-minimal definable sets, where $P \subseteq X_1 \times \cdots \times X_n$ is a broad definable set, and where the map $$\begin{aligned} P & \to {\mathbb{M}}\\ (x_1,\ldots,x_n) & \mapsto x_1 + \cdots + x_n \end{aligned}$$ has finite fibers. The *target* of the coordinate configuration is the image of this map. The *rank* of the coordinate configuration is the sum $\sum_{i = 1}^n {\operatorname{dp-rk}}(X_i)$. If $(X_1,\ldots,X_n,P)$ is a coordinate configuration with target $Y$ and rank $r$, then $${\operatorname{dp-rk}}(Y) = {\operatorname{dp-rk}}(P) = {\operatorname{dp-rk}}(X_1 \times \cdots \times X_n) = r$$ by Theorem \[finite-rank-case\] and the subadditivity of dp-rank. \[basic-properties\] Let $(X_1,\ldots,X_n,P)$ be a coordinate configuration with target $Y$. 1. \[bp-1\] Full dp-rank is definable on $Y$: if $\{D_b\}$ is a definable family of subsets of $Y$, the set of $b$ such that ${\operatorname{dp-rk}}(D_b) = {\operatorname{dp-rk}}(Y)$ is definable. \[full-rank-def\] 2. If $Y'$ is a definable subset of $Y$ and ${\operatorname{dp-rk}}(Y') = {\operatorname{dp-rk}}(Y)$, then $Y'$ is the target of some coordinate configuration. Let $r$ be the rank of the coordinate configuration, or equivalently, $r = {\operatorname{dp-rk}}(Y)$. Let $\pi : P \to Y$ be the map $\pi(x_1,\ldots,x_n) = \sum_{i = 1}^n x_i$. Because $\pi$ is surjective with finite fibers, a subset $D_b \subseteq Y$ has rank $r$ if and only if $\pi^{-1}(D_b)$ has rank $r$. Therefore, the definability of rank $r$ on $Y$ follows from the definability of rank $r$ on $X_1 \times \cdots \times X_n$, which is Corollary \[finite-rank-final\]. If $Y'$ is a full-rank subset of $Y$, then $\pi^{-1}(Y)$ is a broad subset of $X_1 \times \cdots \times X_n$, and so $(X_1,\ldots,X_n,\pi^{-1}(Y))$ is a coordinate configuration with target $Y'$. \[upgrades\] Let $X_1, \ldots, X_n$ be quasi-minimal and $A$-definable. Let $(a_1,\ldots,a_n)$ be an element of $X_1 \times \cdots \times X_n$ such that ${\operatorname{tp}}(a_1,\ldots,a_n/A)$ is broad. Let $s = a_1 + \cdots + a_n$. If $\vec{a} \in {\operatorname{acl}}(sA)$, then there is a broad set $P \subseteq X_1 \times \cdots \times X_n$ containing $\vec{a}$ such that $(X_1,X_2,\ldots,X_n,P)$ is a coordinate configuration with target containing $s$. We can find some formula $\phi(x_1,\ldots,x_n,y)$ (with suppressed parameters from $A$) such that $\phi(a_1,\ldots,a_n,s)$ holds, and $\phi({\mathbb{M}},s)$ is finite of size $k$. Strengthening $\phi$, we may assume that $\phi({\mathbb{M}},s')$ has size at most $k$ for any $s'$. We may also assume that $\phi(x_1,\ldots,x_n,y) \implies y = x_1 + \cdots + x_n$. Let $P$ be the definable set $$P = \{(x_1,\ldots,x_n) \in X_1 \times \cdots \times X_n ~|~ \phi(x_1,\ldots,x_n,x_1 + \cdots + x_n)\}.$$ Then $P$ is $A$-definable, and broad by virtue of containing $\vec{a}$. The map $$\begin{aligned} P &\to {\mathbb{M}}\\ (x_1,\ldots,x_n) & \mapsto x_1 + \cdots + x_n \end{aligned}$$ has fibers of size at most $k$. Therefore $(X_1,\ldots,X_n,P)$ is a coordinate configuration. \[coheir-magic-1\] Let $X_1,\ldots,X_n$ be quasi-minimal $A$-definable sets. For each $i$ let $p_i$ be a global $A$-invariant type in $X_i$. Assume $p_i$ is not realized (i.e., not an algebraic type). Then $(p_1 \otimes \cdots \otimes p_n) | A$ is broad. Because $p_i$ is not algebraic, any realization of $p_i^{\otimes \omega}$ is an indiscernible sequence of pairwise distinct elements. Let $\{a_{i,j}\}_{i \in [n], j \in {\mathbb{N}}}$ be a realization of $p_1^{\otimes \omega} \otimes \cdots \otimes p_n^{\otimes \omega}$ over $A$. Then for every $\eta : [n] \to {\mathbb{N}}$, the tuple $$(a_{1,\eta(1)},\ldots,a_{n,\eta(n)})$$ is a realization of $p_1 \otimes \cdots \otimes p_n$ over $A$, and the rows in this array are sequences of distinct elements. \[coheir-magic-2\] Let $(X_1,\ldots,X_n,P)$ be a coordinate configuration. There exists a small set $A$ and non-algebraic global $A$-invariant types $p_i$ on $X_i$ such that if $(a_1,\ldots,a_n) \models p_1 \otimes \cdots \otimes p_n|A$ then $(a_1,\ldots,a_n) \in P$. By Lemma \[simon-lemma\].\[sl-1\] and Theorem \[main-characterization\], we can find infinite definable sets $D_1, D_2, \ldots, D_n$ such that $(D_1 \times \cdots \times D_n) \setminus P$ is a hyperplane, hence narrow. Choose a model $M$ defining the $X_i$’s, $D_i$’s, and $P$, and let $p_i$ be any $M$-invariant non-algebraic global type in $D_i$, such as a coheir of a non-algebraic type in $D_i$ over $M$. If $(a_1,\ldots,a_n) \models \bigotimes_{i = 1}^n p_i|M$, then ${\operatorname{tp}}(a_1,\ldots,a_n/M)$ is broad, hence cannot live in the hyperplane. But $a_i \in D_i$, so $\vec{a} \in P$. Critical coordinate configurations ---------------------------------- In this section, we carry out something similar to the Zilber indecomposability technique in groups of finite Morley rank. We consider ways to expand coordinate configurations to higher rank, and deduce consequences of being a maximal rank coordinate configuration. The *critical rank* $\rho$ is the maximum rank of any coordinate configuration. A coordinate configuration is *critical* if its rank is the critical rank $\rho$. A set $Y$ is a *critical set* if $Y$ is the target of some critical coordinate configuration. \[subs\] If $Y$ is a critical set, then ${\operatorname{dp-rk}}(Y) = \rho$. If $Y' \subseteq Y$ is definable of rank $\rho$, then $Y'$ is critical. Indeed, $Y'$ is the target of some coordinate configuration by Proposition \[basic-properties\].\[bp-1\]. This coordinate configuration has rank $\rho$ and is thus critical. \[translate\] If $Y$ is a critical set and $\alpha \in {\mathbb{M}}$, the translate $\alpha + Y$ is critical. If $(X_1,\ldots,X_n,P)$ is a coordinate configuration with target $Y$, then $(X_1 + \alpha, X_2, \ldots, X_n, P')$ is a coordinate configuration with target $Y'$, where $$P' := \{(a_1 + \alpha, a_2, a_3, \ldots, a_n) ~|~ (a_1,\ldots,a_n) \in P\}.$$ Because ${\operatorname{dp-rk}}(Y) = {\operatorname{dp-rk}}(Y + \alpha)$, $Y + \alpha$ is critical. \[expander\] Let $(X_1,\ldots,X_n,P)$ be a critical coordinate configuration and $Q$ be a quasi-minimal set. Let $A$ be a small set of parameters over which the $X_i,P,$ and $Q$ are defined. Let $(a_1,\ldots,a_n,b)$ be a tuple in $X_1 \times \cdots \times X_n \times Q$ such that ${\operatorname{tp}}(a_1,\ldots,a_n,b/A)$ is broad and $(a_1,\ldots,a_n) \in P$. Let $s = a_1 + \cdots + a_n + b$. Then $b \notin {\operatorname{acl}}(sA)$. Suppose otherwise. Let $s' = a_1 + \cdots + a_n = s - b$. If $b \in {\operatorname{acl}}(sA)$, then $s' \in {\operatorname{acl}}(sA)$. Because $\vec{a} \in P$, we have $\vec{a} \in {\operatorname{acl}}(s'A)$ by definition of coordinate configuration. It follows that $(\vec{a},b) \in {\operatorname{acl}}(sA)$. By Lemma \[upgrades\], there is a coordinate configuration $(X_1,\ldots,X_n,Q,P')$. This coordinate configuration has higher rank than $(X_1,\ldots,X_n,P)$, contradicting criticality. \[shuffle-chaos\] Let $A$ be a small set. Suppose $b \notin {\operatorname{acl}}(A)$, and suppose $a \models p|Ab$ for some global $A$-invariant type $p$. Then there is a small model $M$ containing $A$ and a global $M$-invariant *non-constant* type $r$ such that $(a,b) \models p \otimes r | M$, i.e., $b \models r | M$ and $a \models p|Mb$. Let $b_1, b_2, \ldots$ be an $A$-indiscernible sequence of pairwise distinct realizations of ${\operatorname{tp}}(b/A)$. Let $M_0$ be a small model containing $A$. Let $b_1', b_2', \ldots$ be an $M_0$-indiscernible sequence extracted from $b_1, b_2, \ldots$. Thus each $b_i'$ realizes ${\operatorname{tp}}(b_i/A)$ and the $b'_i$ are pairwise distinct. Let $\sigma \in {\operatorname{Aut}}({\mathbb{M}}/A)$ move $b'_1$ to $b$, and let $M_1 = \sigma(M_0)$. Then $\sigma(b'_1), \sigma(b'_2), \ldots$ is an $M_1$-indiscernible sequence whose first entry is $b$ and whose entries are pairwise distinct. Therefore $M_1$ contains $A$ but not $b$. Let $r_0$ be a global coheir of $tp(b/M_1)$; note that $r_0$ is non-constant as $b \notin {\operatorname{acl}}(M_1)$. Let $a'$ realize $p|M_1b$; thus $(a',b)$ realizes $p \otimes r_0 | M_1$. Now $a'$ and $a$ both realize $p|Ab$, because $A \subseteq M_1$. Therefore, there is an automorphism $\tau \in {\operatorname{Aut}}({\mathbb{M}}/Ab)$ such that $\tau(a') = a$. Let $M = \tau(M_1)$ and $r = \tau(r_0)$. Then $(a,b) \models p \otimes r|M$ because $(a',b) \models p \otimes r_0|M_1$. \[key-argument\] Let $Y$ be a critical set, let $Q$ be quasi-minimal, and let $t \ge 1$ be an integer. There exist pairwise distinct $q_1, q_2, \ldots, q_t \in Q$ such that $${\operatorname{dp-rk}}\left(\bigcap_{i = 1}^t (Y + q_i)\right) = {\operatorname{dp-rk}}(Y).$$ In particular, by Remarks \[subs\] and \[translate\], the intersection $\bigcap_{i = 1}^t (Y + q_i)$ is a critical set. Suppose otherwise. Then for any distinct $q_1, \ldots, q_t$ in $Q$, the intersection $\bigcap_{i = 1}^t (Y + q_i)$ has rank at most $\rho - 1$. Choose a critical coordinate configuration $(X_1,\ldots,X_n,P)$ with target $Y$. Note that $\sum_i {\operatorname{dp-rk}}(X_i) = \rho$. By Lemma \[coheir-magic-2\], we may find a small set $A$ and global $A$-invariant non-algebraic types $p_i$ on $X_i$ and $p_0$ on $Q$ such that - The sets $X_i, P, Q$ are $A$-definable. - The type $p_1 \otimes \cdots \otimes p_n$ lives on $P$. For any $k$, let $\Omega_k$ be the set of $(q_0,a_{1,1},\ldots,a_{1,n},\ldots,a_{k,1},\ldots,a_{k,n}) \in Q \times (X_1 \times \cdots \times X_n)^k$ such that - For each $i \in [k]$, the tuple $(a_{i,1},\ldots,a_{i,n})$ lies on $P$. - There are infinitely many $q \in Q$ such that $$\bigwedge_{i = 1}^k \left( q_0 + \sum_{j = 1}^n a_{i,j} \in Y + q \right).$$ Then $\Omega_k$ is narrow as a subset of $Q \times (X_1 \times \cdots \times X_n)^k$ for sufficiently large $k$. Let $h = {\operatorname{dp-rk}}(Q) > 0$ and take $k$ so large that $th + k(\rho - 1) < h + k\rho$. The set $\Omega_k$ is definable (because $\exists^\infty$ is eliminated). If $\Omega_k$ were broad, it would have dp-rank $h + k\rho$ by Theorem \[finite-rank-case\]. Therefore $\Omega_k$ would contain a specific tuple $(q_0,a_{1,1},\ldots,a_{k,n})$ of dp-rank $h + k\rho$ over $A$. Let $s_i = a_{i,1} + \cdots + a_{i,n}$ for $i \in [k]$. Because $(a_{i,1},\ldots,a_{i,n}) \in P$, we have $s_i \in Y$. Moreover, by definition of coordinate configuration, $$(a_{i,1},\ldots,a_{i,n}) \in {\operatorname{acl}}(s_iA).$$ By definition of $\Omega_k$ there are infinitely many $q$ such that $$\{q_0 + s_1, q_0 + s_2, \ldots, q_0 + s_k\} \subseteq Y + q.$$ Choose $q_1, \ldots, q_{t-1}$ distinct such $q$, not equal to $q_0$. Then $$q_0 + s_i \in (Y + q_0) \cap (Y + q_1) \cap \cdots \cap (Y + q_{t-1})$$ for every $i$. Therefore $${\operatorname{dp-rk}}(a_{i,1},\ldots,a_{i,n}/Aq_0q_1 \cdots q_{t-1}) = {\operatorname{dp-rk}}(s_i/Aq_0q_1 \cdots q_{t-1}) \le {\operatorname{dp-rk}}\left(\bigcap_{i = 0}^{t-1}(Y + q_i)\right) < \rho.$$ By subadditivity of dp-rank, $${\operatorname{dp-rk}}(a_{1,1},\ldots,a_{k,n},q_0,q_1,\ldots,q_{t-1}/A) \le k(\rho - 1) + th < k\rho + h$$ contradicting the fact that $${\operatorname{dp-rk}}(a_{1,1},\ldots,a_{i,n},q_0/A) = k\rho + h. \qedhere$$ Fix $k$ as in the Claim. Choose $$(a_{k,1},\ldots,a_{k,n},\ldots,a_{1,1},\ldots,a_{1,n},q_0)$$ realizing $(p_1 \otimes \cdots \otimes p_n)^{\otimes k} \otimes p_0$ over $A$. Let $s_i = a_{i,1} + \cdots + a_{i,n}$. By choice of $p_1 \otimes \cdots \otimes p_n$, each $(a_{i,1},\ldots,a_{i,n})$ lives on $P$, and therefore $s_i \in Y$. By Lemma \[coheir-magic-1\], the type of $(\vec{a},q_0)$ over $A$ is broad in $(X_1 \times \cdots \times X_n)^k \times Q$. Therefore the tuple $(q_0,\vec{a})$ cannot lie in the narrow set $\Omega_k$. Consequently, there are only finitely many $q \in Q$ such that $$\{s_1 + q_0, \ldots, s_k + q_0\} \subseteq Y + q.$$ As $s_i \in Y$, one such $q$ is $q_0$. Therefore, $$q_0 \in {\operatorname{acl}}(A,s_1 + q_0,\ldots, s_k + q_0).$$ Choose $j$ minimal such that $$q_0 \in {\operatorname{acl}}(A,s_1 + q_0, \ldots, s_j + q_0).$$ Note that $j \ge 1$ because ${\operatorname{tp}}(q_0/A)$ is the non-algebraic type $p_0$. Let $A' = A \cup \{s_1 + q_0, \ldots, s_{j-1} + q_0\}$. By choice of $j$, $q_0 \notin {\operatorname{acl}}(A')$. Note that $(a_{j,1},\ldots,a_{j,n})$ realizes the $A$-invariant type $p_1 \otimes \cdots \otimes p_n$ over $A'q_0 \subseteq {\operatorname{dcl}}(q_0,a_{1,1},\ldots,a_{j-1,n}A)$. Applying Lemma \[shuffle-chaos\], we find a small model $M \supseteq A'$ and a non-constant global $M$-invariant type $r$ such that $$(a_{j,1},\ldots,a_{j,n},q_0) \models p_1 \otimes \cdots \otimes p_n \otimes r|M.$$ The type $r$ extends ${\operatorname{tp}}(q_0/M)$ and therefore lives in the $M$-definable set $Q$. By Lemma \[coheir-magic-1\], it follows that $${\operatorname{tp}}(a_{j,1},\ldots,a_{j,n},q_0/M)$$ is broad, as a type in $X_1 \times \cdots \times X_n \times Q$. This contradicts Lemma \[expander\]: the $X_i, P, Q$ are all $M$-definable, the type of $(a_{j,1},\ldots,a_{j,n},q_0)$ over $M$ is broad, the tuple $(a_{j,1},\ldots,a_{j,n})$ lives on $P$, and for $$s = a_{j,1} + \cdots + a_{j,n} + q_0 = s_j + q_0,$$ we have $$q_0 \in {\operatorname{acl}}(A',s_j + q_0) \subseteq {\operatorname{acl}}(sM). \qedhere$$ \[stacking\] Let $Y$ be a critical set and $Q_1, \ldots, Q_n$ be quasi-minimal. Then for every $m$ there exist $\{q_{i,j}\}_{i \in [n], j \in [m]}$ such that - For fixed $i \in [n]$, the sequence $q_{i,1},\ldots,q_{i,m}$ consists of $m$ distinct elements of $Q_i$. - The intersection $$\bigcap_{\eta : [n] \to [m]} \left(Y + \sum_{i = 1}^n q_{i,\eta(i)}\right)$$ is critical. We prove this by induction on $n$; the base case $n = 1$ is Lemma \[key-argument\]. If $n > 1$, by induction there are $q_{1,1},\ldots,q_{n-1,m}$ in the correct places, such that $$Y' = \bigcap_{\eta : [n-1] \to [m]} \left(Y + \sum_{i = 1}^n q_{i,\eta(i)}\right)$$ is critical. By Lemma \[key-argument\] there are $q_{n,1},\ldots,q_{n,m} \in Q_n$, pairwise distinct, such that $$Y'' = (Y' + q_{n,1}) \cap \cdots \cap (Y' + q_{n,m})$$ is critical. But $x \in Y''$ if and only if for every $j \in [m]$ and every $\eta : [n-1] \to [m]$ the element $x - q_{n,j} - q_{n-1,\eta(n-1)} - \cdots - q_{1,\eta(1)}$ lies in $Y$. Thus $$Y'' = \bigcap_{\eta : [n] \to [m]} \left(Y + \sum_{i = 1}^n q_{i,\eta(i)} \right). \qedhere$$ \[stacking-2\] Let $Y$ be a critical set and $Q_1,\ldots,Q_n$ be quasi-minimal. There exists a $\delta \in {\mathbb{M}}$ such that $$\{ (x_1,\ldots,x_n) \in Q_1 \times \cdots \times Q_n ~|~ x_1 + \cdots + x_n \in Y + \delta\}$$ is broad, as a subset of $Q_1 \times \cdots \times Q_n$. For any $m \ge 1$, we can find $\{q_{i,j}\}_{i \in [n], j \in [m]}$ such that - $q_{i,j} \in Q_i$ - $q_{i,j} \ne q_{i,j'}$ for $j \ne j'$. - The intersection $$\bigcap_{\eta : [n] \to [m]} \left(Y - \sum_{i = 1}^n q_{i,\eta(i)} \right)$$ is non-empty. Indeed, this follows from Proposition \[stacking\] applied to the quasi-minimal sets $-Q_i$. If $-\delta$ is an element of the intersection, then for any $\eta : [n] \to [m]$ we have $$-\delta \in Y - \sum_{i = 1}^n q_{i,\eta(i)}$$ or equivalently, $$\sum_{i = 1}^n q_{i, \eta(i)} \in Y + \delta.$$ Thus, for every $m$ we can find $\{q_{i,j}\}_{i \in [n], j \in [m]}$ and $\delta \in {\mathbb{M}}$ such that - $q_{i,j} \in Q_i$. - $q_{i,j} \ne q_{i,j'}$ for $j \ne j'$. - For any $\eta : [n] \to [m]$ the sum $\sum_{i = 1}^n q_{i,\eta(i)}$ lies in $Y + \delta$. By compactness, we can find $\{q_{i,j}\}_{i \in [n],~j \in {\mathbb{N}}}$ and $\delta$ satisfying these same properties. This means that $$\{(q_1,\ldots,q_n) \in Q_1 \times \cdots \times Q_n ~|~ q_1 + \cdots + q_n \in Y + \delta\}$$ is broad as a subset of $Q_1 \times \cdots \times Q_n$. Heavy sets ---------- We are nearly ready to define heavy and light sets. Let $Y$ be a critical set. Say that a definable set $X \subseteq {\mathbb{M}}$ is *$Y$-heavy* if there is $\delta \in {\mathbb{M}}$ such that ${\operatorname{dp-rk}}(Y \cap (X + \delta)) = {\operatorname{dp-rk}}(Y)$. \[def-in-fams\] Let $Y$ be a critical set and $\{D_b\}$ be a definable family of subsets of ${\mathbb{M}}$. Let $S$ be the set of $b$ such that $D_b$ is $Y$-heavy. Then the set $S$ is definable. Indeed, this follows almost immediately from Proposition \[basic-properties\].\[full-rank-def\]. Let $Y, Y'$ be two critical sets and $X \subseteq {\mathbb{M}}$ be definable. Then $X$ is $Y$-heavy if and only if $X$ is $Y'$-heavy. Suppose $X$ is $Y$-heavy. Let $X' := Y \cap (X + \delta_0)$ have full rank in $Y$. Then $X'$ is critical by Remark \[subs\]. We claim that $X'$ is $Y'$-heavy. Let $(A_1,\ldots,A_n,P)$ be a coordinate configuration for $Y'$. The set $P$ is broad in $A_1 \times \cdots \times A_n$, so there exist infinite definable subsets $Q_i \subseteq A_i$ such that $(Q_1 \times \cdots \times Q_n) \setminus P$ is a hyperplane, and in particular is narrow. By Corollary \[stacking-2\], there is a $\delta_1 \in {\mathbb{M}}$ such that $$\{(q_1,\ldots,q_n) \in Q_1 \times \cdots \times Q_n ~|~ q_1 + \cdots + q_n \in X' + \delta_1\}$$ is broad as a subset of $Q_1 \times \cdots \times Q_n$, hence broad as a subset of $A_1 \times \cdots \times A_n$. Because $(Q_1 \times \cdots \times Q_n) \setminus P$ is narrow, it follows that $$\{(q_1,\ldots,q_n) \in P ~|~ q_1 + \cdots + q_n \in X' + \delta_1\}$$ is broad as a subset of $A_1 \times \cdots \times A_n$. Because the map $(q_1,\ldots,q_n) \mapsto q_1 + \cdots + q_n$ has finite fibers on $P$, it follows that $$\{q_1 + \cdots + q_n ~|~ \vec{q} \in P \text{ and } q_1 + \cdots + q_n \in X' + \delta_1\} = Y' \cap (X' + \delta_1)$$ has full rank. Therefore, $X'$ is $Y'$-heavy. Now $X' \subseteq (X + \delta_0)$, and so $Y' \cap (X + \delta_0 + \delta_1)$ has full rank in $Y'$, implying that $X$ is $Y'$-heavy. The converse follows by symmetry. A definable subset $X \subseteq {\mathbb{M}}$ is *heavy* if it is $Y$-heavy for any/every critical set $Y \subseteq {\mathbb{M}}$, and *light* otherwise. \[heavy-light\] Let $X, Y$ be definable subsets of ${\mathbb{M}}$. 1. (Assuming ${\mathbb{M}}$ is infinite) If $X$ is finite, then $X$ is light. 2. \[light-union\] If $X, Y$ are light, then $X \cup Y$ is light. 3. If $Y$ is light and $X \subseteq Y$, then $X$ is light. 4. If $\{D_b\}$ is a definable family of subsets of ${\mathbb{M}}$, then $$\{b ~|~ D_b \text{ is light}\}$$ is definable. 5. ${\mathbb{M}}$ is heavy. 6. If $X$ is heavy, then $\alpha \cdot X$ is heavy for $\alpha \in {\mathbb{M}}^\times$. 7. \[light-translate\] If $X$ is heavy, then $\alpha + X$ is heavy for $\alpha \in {\mathbb{M}}$. 8. If $X$ and $Y$ are heavy, the set $$X {-_\infty}Y := \{\delta \in {\mathbb{M}}~|~ X \cap (Y + \delta) \text{ is heavy}\}$$ is heavy. <!-- --> 1. Indeed, if $X$ is heavy then some translate of $X$ contains a critical set, and so $X$ has rank at least $\rho$. The critical rank $\rho$ cannot be 0 because there is at least one quasi-minimal set $Q_i$ (by Remark \[they-exist\]), and $(Q_i,Q_i)$ is a coordinate configuration with target $Q_i$. Therefore heavy sets are infinite. 2. We show the contrapositive: if $X \cup Y$ is heavy, then $X$ is heavy or $Y$ is heavy. Fix some critical set $Z$. The fact that $X \cup Y$ is $Z$-heavy implies that there is a $\delta$ such that $Z \cap ((X \cup Y) + \delta)$ has full rank. But $$Z \cap ((X \cup Y) + \delta) = (Z \cap (X + \delta)) \cup (Z \cap (Y + \delta))$$ so at least one of $Z \cap (X + \delta)$ and $Z \cap (Y + \delta)$ has full rank $\rho$. 3. If $X \subseteq Y$ and $X$ is $Z$-heavy because $Z \cap (X + \delta)$ has full rank $\rho$, then certainly $Z \cap (Y + \delta)$ has full rank $\rho$. 4. This is Remark \[def-in-fams\]. 5. Critical sets exist, and if $Y$ is a critical set, then ${\operatorname{dp-rk}}(Y \cap {\mathbb{M}}) = {\operatorname{dp-rk}}(Y)$. 6. Heaviness was defined without any reference to multiplication, so it is preserved by definable automorphisms of the additive group $({\mathbb{M}},+)$. 7. Heaviness is translation-invariant by definition. 8. Shrinking $X$ and $Y$ we may assume that $X$ and $Y$ are translates of critical sets–or merely critical sets by Remark \[translate\]. Let $(A_1,\ldots,A_n,P)$ be a coordinate configuration for $Y$. By Theorem \[main-characterization\] we can find infinite definable subsets $Q_i \subseteq A_i$ such that $(Q_1 \times \cdots \times Q_n) \setminus P$ is narrow. By Corollary \[stacking-2\], there is a $\delta \in {\mathbb{M}}$ such that the set $$\{(q_1,\ldots,q_n,q'_1,\ldots,q'_n) \in (Q_1 \times \cdots \times Q_n)^2 ~|~ q_1 + \cdots + q_n + q'_1 + \cdots + q'_n \in X + \delta\}$$ is broad as a subset of $Q_1 \times \cdots \times Q_n \times Q_1 \times \cdots \times Q_n$. In particular, we can find $q_{i,j}$ and $q'_{i,j}$ for $i \in [n]$ and $j \in {\mathbb{N}}$ such that - $q_{i,j}$ and $q'_{i,j}$ are in $Q_i$. - $q_{i,j} \ne q_{i,j'}$ and $q'_{i,j} \ne q'_{i,j'}$ for $j \ne j'$. - For any $\eta : [n] \to {\mathbb{N}}$ and $\eta' : [n] \to {\mathbb{N}}$, we have $$\sum_{i = 1}^n (q_{i,\eta(i)} + q'_{i,\eta'(i)}) \in X + \delta.$$ Let $s_\eta$ denote the sum $\sum_{i = 1}^n q_{i,\eta(i)}$. Note that for any $\eta$, the $q'_{i,j}$ show that the set $$\{(q'_1,\ldots,q'_n) \in Q_1 \times \cdots \times Q_n ~|~ s_\eta + q'_1 + \cdots + q'_n \in X + \delta\}$$ is broad as a subset of $Q_1 \times \cdots \times Q_n$. It follows that $$\{(q'_1,\ldots,q'_n) \in P ~|~ s_\eta + q'_1 + \cdots + q'_n \in X + \delta\}$$ has full rank $\rho$, and so $(Y + s_\eta) \cap (X + \delta)$ has full rank $\rho$ for any $\eta$. The translate $X \cap (Y + s_\eta - \delta)$ has full rank and is trivially $X$-heavy. Therefore $s_\eta - \delta \in X {-_\infty}Y$. The fact that this holds for all $s_\eta$ implies that $$\{(q_1,\ldots,q_n) \in Q_1 \times \cdots \times Q_n ~|~ q_1 + \cdots + q_n + \delta \in X {-_\infty}Y\}$$ is broad as a subset of $Q_1 \times \cdots \times Q_n$. Again, this implies that $(Y + \delta) \cap (X {-_\infty}Y)$ has full rank $\rho$, and so $X {-_\infty}Y$ is heavy. Externally definable sets {#externally-definable-sets} ------------------------- We prove the analogues of §\[sec:extern1\] for heavy and light sets. \[chernikov-2\] Let $M \preceq {\mathbb{M}}$ be a small model defining a critical coordinate configuration. Let $Z$ be an $M$-definable heavy set. Let $D_1, \ldots, D_m$ be ${\mathbb{M}}$-definable sets such that $$Z(M) \subseteq D_1 \cup \cdots \cup D_m.$$ Then there is a $j \in [m]$ and a $M$-definable heavy set $Z' \subseteq Z$ such that $Z'(M) \subseteq D_j$. Let $(X_1,\ldots,X_n,P)$ be an $M$-definable critical coordinate configuration, with target $Y$. Because $Z$ is $Y$-heavy, there is some $\delta$ such that $Y \cap (Z + \delta)$ has full rank in $Y$. Full rank in $Y$ is a definable condition, clearly $M$-invariant, so we may take $\delta \in M$. Replacing $Z$ and the $D_i$’s with $Z + \delta$ and $D_i + \delta$, we may assume $\delta = 0$. The fact that $Y \cap Z$ has full rank implies that the set $$R := \{(x_1,\ldots,x_n) \in P ~|~ x_1 + \cdots + x_n \in Z\}$$ is broad. Furthermore, $R(M)$ is a union of the externally-definable sets $$\{(x_1,\ldots,x_n) \in P(M) ~|~ x_1 + \cdots + x_n \in Z \cap D_j.\}$$ By Lemma \[chernikov-1\], there is an $M$-definable $R' \subseteq R$ and a $j \in [m]$ such that $R'$ is broad and $$(x_1,\ldots,x_n) \in R'(M) \implies x_1 + \cdots + x_n \in D_j.$$ Let $Z'$ be the $M$-definable set $$\{x_1 + \cdots + x_n ~|~ (x_1,\ldots,x_n) \in R' \}.$$ Because $R' \subseteq R \subseteq P$, the set $Z' \subseteq Y \cap Z$. The set of $(x_1,\ldots,x_n) \in P$ such that $x_1 + \cdots + x_n \in Z'$ contains the broad set $R'$, and is therefore broad. It follows that $Z'$ has full rank in $Y$, and so $Z'$ is $Y$-heavy. Lastly, we claim that $$Z'(M) \subseteq D_j.$$ Indeed, suppose $y \in Z'(M)$. Then there exists an $n$-tuple $(x_1, \ldots, x_n) \in R'$ such that $y = x_1 + \cdots + x_n$. The fact that $M \preceq {\mathbb{M}}$ implies that we can take $\vec{x} \in {\operatorname{dcl}}(M)$. Then by choice of $R'$, it follows that $$y = x_1 + \cdots + x_n \in D_j.$$ Therefore $Z'(M) \subseteq D_j$. \[something-similar-2\] Let $M$ be a small model defining a critical coordinate configuration. Let $Z$ be an $M$-definable heavy set, and let $W$ be an ${\mathbb{M}}$-definable set. If $Z(M) \subseteq W$, then $W$ is heavy. Take an $M$-definable coordinate configuration $(X_1,\ldots,X_n,P)$ with target $Y$. As in the proof of Lemma \[chernikov-2\] we may assume that $Z \cap Y$ has full rank in $Y$, and so the set $$\{(x_1,\ldots,x_n) \in P ~|~ x_1 + \cdots + x_n \in Z\}$$ is broad in $X_1 \times \cdots \times X_n$. This set is $M$-definable, and its $M$-definable points lie in the set $$\{(x_1,\ldots,x_n) \in P ~|~ x_1 + \cdots + x_n \in W\},$$ which is therefore broad by Lemma \[something-similar-1\]. This implies that $W \cap Y$ has full rank, and so $W$ is ($Y$-)heavy. The edge case: finite Morley rank {#sec:likeTT} ================================= If our end goal is to construct non-trivial valuation rings on dp-finite fields, we had better quarantine the stable fields at some point. In other words, we need to find some positive consequence of the hypothesis “not stable.” We prove the relevant dichotomy in this section. \[morley-rank\] Let $X_1, \ldots, X_n$ be infinite definable sets. If the number of broad global types in $X_1 \times \cdots \times X_n$ is finite, then every $X_i$ has Morley rank 1. Suppose, say, $X_1$ has Morley rank greater than 1. Then we can find a sequence $D_1, D_2, D_3, \ldots$ of length $\omega$ where each $D_i$ is an infinite definable subset of $X_1$ and the $D_i$ are pairwise disjoint. Let $P_i = D_1 \times X_2 \times \cdots \times X_n$. Each $P_i$ is broad and the $P_i$ are pairwise disjoint. By Proposition \[completion\], each $P_i$ is inhabited by a broad type over ${\mathbb{M}}$, so there are at least $\aleph_0$ broad global types. If there are a finite number of broad global types, it follows that every $X_i$ has Morley rank 1. \[tt-dichotomy\] At least one of the following holds: 1. The field ${\mathbb{M}}$ has finite Morley rank. 2. There are two disjoint heavy sets. Let $(X_1,\ldots,X_n,P)$ be a critical coordinate configuration. We may find infinite definable subsets $Q_i \subseteq X_i$ such that for $$\begin{aligned} H &:= (Q_1 \times \cdots \times Q_n) \setminus P \\ P' &:= (Q_1 \times \cdots \times Q_n) \cap P \end{aligned}$$ the set $H$ is narrow in $Q_1 \times \cdots \times Q_n$, and therefore $P'$ is broad. Then $(Q_1,\ldots,Q_n,P')$ is a coordinate configuration of the same rank as $(X_1,\ldots,X_n,P)$. In particular $(Q_1,\ldots,Q_n,P')$ is critical and its target $Y$ is a critical set. First suppose that at least one $Q_i$ has Morley rank greater than 1 (possibly $\infty$). By Lemma \[morley-rank\], there are infinitely many broad global types in $Q_1 \times \cdots \times Q_n$. Let $p_1, p_2, p_3, \ldots$ be pairwise distinct broad global types. Because $H$ is narrow, each $p_i$ must live in $P$. Let $m$ bound the fiber size of the map $\Sigma : P' \to Y$. There is a pushforward map $\Sigma_*$ from the space of ${\mathbb{M}}$-types in $P'$ to the space of ${\mathbb{M}}$-types in $Y$, and this map has fibers of size no greater than $m$. Consequently, we can find $i, j$ such that $\Sigma_* p_i \ne \Sigma_* p_j$. Without loss of generality $\Sigma_* p_1 \ne \Sigma_* p_2$. We can then write $Y$ as a disjoint union $Y = Z_1 \cup Z_2$ where $\Sigma_* p_i$ lives on $Z_i$. For $i = 1, 2$, the definable set $$\{(x_1,\ldots,x_n) \in P' ~|~ x_1 + \cdots + x_n \in Z_i\}$$ is broad as a subset of $Q_1 \times \cdots \times Q_n$ because it contains the broad type $p_i$. Therefore this set has dp-rank $\rho$ and so ${\operatorname{dp-rk}}(Z_i) = \rho = {\operatorname{dp-rk}}(Y)$. So the two sets $Z_1, Z_2$ are full-rank subsets of $Y$, and are therefore $Y$-heavy. We have constructed two disjoint heavy sets. Next suppose that every $Q_i$ has Morley rank 1. Let $$W = Q_1 + \cdots + Q_n.$$ The set $W$ contains the critical set $Y$, so $W$ is heavy. If $W - W = {\mathbb{M}}$, then there is a surjective map $$\begin{aligned} Q_1 \times \cdots \times Q_n \times Q_1 \times \cdots \times Q_n &\to {\mathbb{M}}\\ (x_1,\ldots,x_n,x'_1,\ldots,x'_n) &\mapsto x_1 + \cdots + x_n - x'_1 - \cdots - x'_n. \end{aligned}$$ The set $(Q_1 \times \cdots \times Q_n)^2$ has finite Morley rank,[^7], and so ${\mathbb{M}}$ must have finite Morley rank. Otherwise, take $x \notin W - W$. Then the sets $W$ and $W + x$ are disjoint, and both are heavy. Theorem \[tt-dichotomy\] is closely related to Sinclair’s *Large Sets Property* (LSP). A field ${\mathbb{M}}$ of dp-rank $n$ has the LSP if there do not exist two disjoint subsets of ${\mathbb{M}}$ of dp-rank $n$ ([@sinclair], Definition 3.0.3). Sinclair conjectures that fields with LSP are stable, hence algebraically closed. Theorem \[tt-dichotomy\] says something similar, with “heavy” in place of dp-rank $n$. Infinitesimals {#sec:infinitesimals} ============== In this section, we carry out the construction of infinitesimals. The arguments are copied directly from the dp-minimal setting ([@myself], Chapter 9). Basic neighborhoods ------------------- If $X, Y$ are definable sets, $X {-_\infty}Y$ denotes the set $$\{ \delta \in {\mathbb{M}}~|~ X \cap (Y + \delta) \text{ is heavy}\}.$$ \[mininf-props\] Let $X, Y$ be definable sets. 1. $X {-_\infty}Y$ is definable. 2. $X {-_\infty}Y \subseteq X - Y$, where $$X - Y = \{x - y ~|~ x \in X, ~ y \in Y\}.$$ 3. If $X$ or $Y$ is light, then $X {-_\infty}Y$ is empty. 4. If $X$ and $Y$ are heavy, then $X {-_\infty}Y$ is heavy. 5. For any $\delta_1, \delta_2 \in {\mathbb{M}}$, $$(X + \delta_1) {-_\infty}(Y + \delta_2) = (X {-_\infty}Y) + (\delta_1 - \delta_2).$$ 6. For any $\alpha \in {\mathbb{M}}^\times$, $$(\alpha \cdot X) {-_\infty}(\alpha \cdot Y) = \alpha \cdot (X {-_\infty}Y).$$ 7. If $X' \subseteq X$ and $Y' \subseteq Y$ then $$X' {-_\infty}Y' \subseteq X {-_\infty}Y.$$ Indeed, all of these properties follow immediately from Theorem \[heavy-light\]. A *basic neighborhood* is a set of the form $X {-_\infty}X$, where $X$ is heavy. If $M$ is a small model, then every $M$-definable basic neighborhood is of the form $X {-_\infty}X$ with $X$ heavy and $M$-definable. Let $U = X {-_\infty}X$ be a basic neighborhood which is $M$-definable. Write $X$ as $\phi({\mathbb{M}};b)$. Because heaviness is definable in families (by Theorem \[heavy-light\]), the set of $b'$ such that $\phi({\mathbb{M}};b') {-_\infty}\phi({\mathbb{M}};b') = U$ is definable. This set is $M$-invariant, and non-empty because of $b$, so it intersects $M$. Therefore, we can find $M$-definable $X' = \phi({\mathbb{M}};b')$ such that $U = X' {-_\infty}X'$. \[basic-nbhds\]   1. \[heavy-nbhd\] If $U$ is a basic neighborhood, then $U$ is heavy. 2. \[zero-in\] If $U$ is a basic neighborhood, then $0 \in U$. 3. \[scaling\] If $U$ is a basic neighborhood and $\alpha \in {\mathbb{M}}^\times$, then $\alpha \cdot U$ is a basic neighborhood. 4. \[filtered-isect\] If $U_1, U_2$ are basic neighborhoods, there is a basic neighborhood $U_3 \subseteq U_1 \cap U_2$. If $U_1, U_2$ are $M$-definable, we can choose $U_3$ to be $M$-definable. 5. \[non-degen\] If ${\mathbb{M}}$ is not of finite Morley rank, then for every $\alpha \ne 0$ there is a basic neighborhood $U \not\ni \alpha$. If $\alpha$ is $M$-definable, we can choose $U$ to be $M$-definable. <!-- --> 1. This is part of Remark \[mininf-props\]. 2. If $X$ is heavy, then $0 \in X {-_\infty}X$ because $X \cap (X + 0)$ is the heavy set $X$. 3. If $U = X {-_\infty}X$, then $\alpha \cdot U = (\alpha \cdot X) {-_\infty}(\alpha \cdot X)$ by Remark \[mininf-props\], and $\alpha \cdot X$ is heavy by Theorem \[heavy-light\]. 4. Let $U_i = X_i {-_\infty}X_i$. By Theorem \[heavy-light\] the set $X_1 {-_\infty}X_2$ is heavy, hence non-empty. Take $\delta \in X_1 {-_\infty}X_2$. Then $X_3 := X_1 \cap (X_2 + \delta)$ is heavy. By Remark \[mininf-props\] $$\begin{aligned} X_3 {-_\infty}X_3 &\subseteq X_1 {-_\infty}X_1 = U_1 \\ X_3 {-_\infty}X_3 & \subseteq (X_2 + \delta) {-_\infty}(X_2 + \delta) = X_2 {-_\infty}X_2 = U_2. \end{aligned}$$ If $U_1$ and $U_2$ are $M$-definable and $X_3 = \phi({\mathbb{M}};b_0)$, the set of $b$ such that $$\emptyset \ne \phi({\mathbb{M}};b) {-_\infty}\phi({\mathbb{M}};b) \subseteq U_1 \cap U_2$$ is $M$-definable and non-empty. Therefore we can move $X_3$ to be $M$-definable. 5. If ${\mathbb{M}}$ is not of finite morley rank, we can find two disjoint heavy sets $X, Y$. By Theorem \[heavy-light\], $X {-_\infty}Y \ne \emptyset$. Take $\delta \in X {-_\infty}Y$. Then the sets $$\begin{aligned} X' & := X \cap (Y + \delta) \\ Y' & := (X - \delta) \cap Y \end{aligned}$$ are heavy—$X'$ is heavy by choice of $\delta$ and $Y'$ is a translate of $X'$. Also, $$Y' \cap (Y' + \delta) = Y' \cap X' \subseteq Y \cap X = \emptyset,$$ so $\delta \notin Y' {-_\infty}Y'$. Because $Y'$ is heavy, there is a basic neighborhood $U_0 = Y' {-_\infty}Y'$ such that $\delta \notin U_0$. By part 2, $\delta \ne 0$. If $\alpha$ is a non-zero element of ${\mathbb{M}}$, then we can find $\beta \in {\mathbb{M}}^\times$ such that $\alpha = \beta \cdot \delta$. Then $\alpha = \beta \cdot \delta$ is not in $U = \beta \cdot U_0$, which is a basic neighborhood by part 3. Finally, suppose that $\alpha$ is a non-zero element of a small model $M$. We know that there is some heavy set $\phi({\mathbb{M}};b)$ such that $$\alpha \notin \phi({\mathbb{M}};b) {-_\infty}\phi({\mathbb{M}};b).$$ As before, the set of $b$ with this property is $M$-definable, hence we can pull $b$ into $M$, producing an $M$-definable basic neighborhood that avoids $\alpha$. The set of infinitesimals ------------------------- Let $M$ be a small model. An element $\varepsilon \in {\mathbb{M}}$ is an *$M$-infinitesimal* if $\varepsilon$ lies in every $M$-definable basic neighborhood. Equivalently, for every $M$-definable heavy set $X$, the intersection $X \cap (X + \varepsilon)$ is heavy. We let $I_M$ denote the set of $M$-infinitesimals. Note that $I_M$ is type-definable over $M$, as it is the small intersection $$I_M = \bigcap_{X \text{ heavy and $M$-definable}} X {-_\infty}X.$$ Moreover, this intersection is *directed*, by Proposition \[basic-nbhds\]. \[change-of-M\] If $M' \succeq M$, then $I_{M'} \subseteq I_M$ because there are more neighborhoods in the intersection. \[basic-infs\] Let $M$ be a small model. 1. \[non-trivial\] If $X$ is a definable set containing $I_M$, then $X$ is heavy. In particular, if ${\mathbb{M}}$ is infinite, then $I_M \ne 0$. 2. $0 \in I_M$. 3. \[infinitesimal-times-standard\] $I_M \cdot M \subseteq I_M$. 4. If ${\mathbb{M}}$ is not of finite Morley rank, then $I_M \cap M = \{0\}$. <!-- --> 1. The fact that $X \supseteq I_M$ implies by compactness that there exist $M$-definable basic neighborhoods $U_1, \ldots, U_n$ such that $$X \supseteq U_1 \cap \cdots \cap U_n.$$ However, the fact that the intersection is directed (i.e., Proposition \[basic-nbhds\].\[filtered-isect\]) implies that we can take $n = 1$. Then $X \supseteq U_1$ and $U_1$ is heavy (by Proposition \[basic-nbhds\].\[heavy-nbhd\]), so $X$ is heavy. 2. Every basic neighborhood contains $0$ by Proposition \[basic-nbhds\].\[zero-in\]. 3. Suppose $a \in M$. We claim that $a \cdot I_M \subseteq I_M$. If $a = 0$ this is the previous point. Otherwise, the two collections $$\begin{aligned} \{U & ~|~ U \text{ an $M$-definable basic neighborhood}\} \\ \{a \cdot U &~|~ U \text{ an $M$-definable basic neighborhood}\} \end{aligned}$$ are equal by Proposition \[basic-nbhds\].\[scaling\], so they have the same intersection. But the intersection of the first is $I_M$ and the intersection of the second is $a \cdot I_M$. Thus $a \cdot I_M = I_M$. 4. This follows directly from Proposition \[basic-nbhds\].\[non-degen\]. The group of infinitesimals --------------------------- Let $M$ be a small model and $X \subseteq {\mathbb{M}}$ be $M$-definable. An element $\delta \in {\mathbb{M}}$ is said to *$M$-displace* the set $X$ if $$x \in X \cap x \in M \implies x + \delta \notin X.$$ \[heirs\] Let $M \preceq M'$ be an inclusion of small models, and $\varepsilon, \varepsilon'$ be elements of ${\mathbb{M}}$. Suppose that ${\operatorname{tp}}(\varepsilon'/M')$ is an heir of ${\operatorname{tp}}(\varepsilon/M)$. 1. If $\varepsilon$ is an $M$-infinitesimal, then $\varepsilon'$ is an $M'$-infinitesimal. 2. If $X \subseteq {\mathbb{M}}$ is $M$-definable and $M$-displaced by $\varepsilon$, then $X$ is $M'$-displaced by $\varepsilon'$. The assumptions imply that $\varepsilon' \equiv_M \varepsilon$, and the statements about $\varepsilon$ are $M$-invariant, so we may assume $\varepsilon' = \varepsilon$. 1. Suppose $\varepsilon$ fails to be $M'$-infinitesimal. Then there is a tuple $b \in {\operatorname{dcl}}(M')$ such that $\phi({\mathbb{M}};b)$ is heavy and $\varepsilon \notin \phi({\mathbb{M}};b) {-_\infty}\phi({\mathbb{M}};b)$. These conditions on $b$ are $M\varepsilon$-definable. Because ${\operatorname{tp}}(b/M\varepsilon)$ is finitely satisfiable in $M$, we can find such a $b$ in $M$. Then $\varepsilon$ fails to be $M$-infinitesimal because it lies outside $\phi({\mathbb{M}};b) {-_\infty}\phi({\mathbb{M}};b)$. 2. Suppose $\varepsilon$ fails to $M'$-displace $X$. Then there exists $b \in X(M')$ such that $b + \varepsilon \in X = X({\mathbb{M}})$. These conditions on $b$ are $M\varepsilon$-definable, so we can find such a $b$ in $M$. Then $\varepsilon$ fails to $M$-displace $X$. \[inf-heart\] Let $M$ be a small model defining a critical coordinate configuration. Let $\varepsilon$ be an $M$-infinitesimal, and $X \subseteq {\mathbb{M}}$ be an $M$-definable set that is $M$-displaced by $\varepsilon$. Then $X$ is light. Build a sequence $\varepsilon_0, \varepsilon_1, \ldots$ and $M = M_0 \preceq M_1 \preceq M_2 \preceq \cdots$ so that for each $i$, - $M_{i+1} \ni \varepsilon_i$ - ${\operatorname{tp}}(\varepsilon_i/M_i)$ is an heir of ${\operatorname{tp}}(\varepsilon/M)$. By Lemma \[heirs\], - $\varepsilon_i$ is $M_i$-infinitesimal. - The set $X$ is $M_i$-displaced by $\varepsilon_i$. For $\alpha$ a string in $\{0,1\}^{< \omega}$, define $X_\alpha$ recursively as follows: - $X_{\{\}} = X$. - If $\alpha$ has length $n$, then $X_{\alpha 0} = \{x \in X_\alpha ~|~ x + \varepsilon_n \notin X\}$. - If $\alpha$ has length $n$, then $X_{\alpha 1} = \{x \in X_\alpha ~|~ x + \varepsilon_n \in X\}$. For example $$\begin{aligned} X_0 = \{x \in X ~|~ &x + \varepsilon_0 \notin X\} \\ X_1 = \{x \in X ~|~ &x + \varepsilon_0 \in X\} \\ X_{011} = \{x \in X ~|~ &x + \varepsilon_0 \notin X,\\ &x + \varepsilon_1 \in X,\\&x + \varepsilon_2 \in X\}. \end{aligned}$$ Note that some of the $X_\alpha$ must be empty, by NIP. Note also that if $\alpha$ has length $n$, then $X_\alpha$ is $M_n$-definable. If $X_\alpha$ is heavy, then $X_{\alpha 1}$ is heavy. Let $\alpha$ have length $n$. Then $X_\alpha$ is heavy and $M_n$-definable, and $\varepsilon_n$ is $M_n$-infinitesimal. Consequently $X_\alpha \cap (X_\alpha - \varepsilon_n)$ is heavy. But $$X_{\alpha 1} = X_\alpha \cap (X - \varepsilon_n) \supseteq X_\alpha \cap (X_\alpha - \varepsilon_n),$$ and so $X_{\alpha 1}$ must be heavy. If $X_\alpha$ is heavy, then $X_{\alpha 0}$ is heavy. Let $\alpha$ have length $n$. The set $X_\alpha$ is $M_n$-definable. Note that $$x \in X_\alpha(M_n) \implies x \in X(M_n) \implies x + \varepsilon_n \notin X(M_n)$$ because $X$ is $M_n$-displaced by $\varepsilon_n$. Therefore $$X_\alpha(M_n) \subseteq X_{\alpha 0}.$$ By Lemma \[something-similar-2\] it follows that $X_{\alpha 0}$ is heavy. If $X = X_{\{\}}$ is heavy, then the two claims imply that every $X_\alpha$ is heavy, hence non-empty, for every $\alpha$. This contradicts NIP. \[inf-subtr\] Let $M$ be a model defining a critical coordinate configuration. Let $\varepsilon_1, \varepsilon_2$ be two $M$-infinitesimals. Then $\varepsilon_1 - \varepsilon_2$ is an $M$-infinitesimal. Let $X$ be an $M$-definable heavy set; we will show that $X \cap (X + \varepsilon_1 - \varepsilon_2)$ is heavy. Note that $X$ is covered by the union of the following three sets: $$\begin{aligned} D_0 &:= \{x \in X ~|~ x + \varepsilon_1 \in X,~ x + \varepsilon_2 \in X\} \\ D_1 & := \{x \in {\mathbb{M}}~|~ x + \varepsilon_1 \notin X \} \\ D_2 & := \{x \in {\mathbb{M}}~|~ x + \varepsilon_2 \notin X \}. \end{aligned}$$ By Lemma \[chernikov-2\], there is a $j \in \{0,1,2\}$ and an $M$-definable heavy set $X' \subseteq X$ such that $X'(M) \subseteq D_j$. If $j > 0$, then $$x \in X'(M) \implies x \in D_j \implies x + \varepsilon_j \notin X \implies x + \varepsilon_j \notin X'.$$ In other words, $X'$ is $M$-displaced by $\varepsilon_j$. But $X'$ is heavy and $\varepsilon_j$ is an $M$-infinitesimal, so this would contradict Lemma \[inf-heart\]. Therefore $j = 0$. The fact that $X'(M) \subseteq D_0$ implies that $D_0$ is heavy, by Lemma \[something-similar-2\]. By definition of $D_0$, $$\begin{aligned} D_0 + \varepsilon_1 &\subseteq X \\ D_0 + \varepsilon_2 &\subseteq X \\ D_0 + \varepsilon_1 &\subseteq X + \varepsilon_1 - \varepsilon_2 \\ D_0 + \varepsilon_1 & \subseteq X \cap (X + \varepsilon_1 - \varepsilon_2). \end{aligned}$$ Heaviness of $D_0$ then implies heaviness of $X \cap (X + \varepsilon_1 - \varepsilon_2)$. \[pull-down\] Let $U$ be an $M$-definable basic neighborhood. Then there is an $M$-definable basic neighborhood $V$ such that $V - V \subseteq U$. Let $M' \succeq M$ be a larger model that defines a critical coordinate configuration. By Lemma \[inf-subtr\] and Remark \[change-of-M\], $$I_{M'} - I_{M'} \subseteq I_{M'} \subseteq I_M \subseteq U.$$ The $M'$-definable partial type asserting that $$\begin{aligned} x & \in I_{M'} \\ y & \in I_{M'} \\ x - y & \notin U \end{aligned}$$ is inconsistent. Because $I_{M'}$ is the directed intersection of $M'$-definable basic neighborhoods, there is an $M'$-definable basic neighborhood $V'$ such that $$x \in V' \wedge y \in V' \implies x - y \in U,$$ i.e., $V' - V' \subseteq U$. In particular, there is a formula $\phi(x;z)$ and $c \in {\operatorname{dcl}}(M')$ such that $\phi({\mathbb{M}};c)$ is heavy and $$\label{c-cond} (\phi({\mathbb{M}};c) {-_\infty}\phi({\mathbb{M}};c)) - (\phi({\mathbb{M}};c) {-_\infty}\phi({\mathbb{M}};c)) \subseteq U.$$ But heaviness of $\phi({\mathbb{M}};c)$ and condition (\[c-cond\]) are both $M$-definable constraints on $c$, so we may find $c_0 \in {\operatorname{dcl}}(M)$ satisfying the same conditions. Then $$V := \phi({\mathbb{M}};c_0) {-_\infty}\phi({\mathbb{M}};c_0)$$ is an $M$-definable basic neighborhood such that $V - V \subseteq U$. \[inf-add\] If $M$ is any model, the set $I_M$ of $M$-infinitesimals is a subgroup of the additive group $({\mathbb{M}},+)$. The fact that $0 \in I_M$ follows by \[basic-infs\], and the fact that $I_M - I_M \subseteq I_M$ is a consequence of Lemma \[pull-down\]. Consequences ------------ We mention a few corollaries of Theorem \[inf-add\]. \[exists-topology\] If $M$ is a field of finite dp-rank but not finite Morley rank, then there is a (unique) Hausdorff non-discrete group topology on $(M,+)$ such that 1. The basic neighborhoods $X {-_\infty}X$ form a basis of neighborhoods of 0. 2. For any $\alpha \in M$, the map $x \mapsto x \cdot \alpha$ is continuous. Indeed, this follows formally from the fact that basic neighborhoods are filtered with intersection $I_M$, and $$\begin{aligned} I_M - I_M & \subseteq I_M \\ 0 & \in I_M \\ \{0\} & \ne I_M \\ I_M \cap M & = \{0\} \\ M \cdot I_M & \subseteq I_M. \end{aligned}$$ On the other hand, we have not yet shown that there is a definable basis of opens, or that multiplication is continuous in general (which would require $I_M \cdot I_M \subseteq I_M$). Later (Corollary \[ring-topology\]), we will see that multiplication is continuous. \[minimal-heavy-subgroup\] Let $J$ be a subgroup of $({\mathbb{M}},+)$, type-definable over a small model $M$. Suppose that every $M$-definable set containing $J$ is heavy. Then $J \supseteq I_M$. Let $D$ be any $M$-definable set containing $J$. We claim $D$ contains $I_M$. Because $J - J \subseteq J \subseteq D$ there is, by compactness, an $M$-definable set $D'$ such that $D' \supseteq J$ and $D' - D' \subseteq D$. By assumption, $D'$ is heavy. Then $D' {-_\infty}D'$ is an $M$-definable basic neighborhood, and so $$I_M \subseteq D' {-_\infty}D' \subseteq D' - D' \subseteq D.$$ As this holds for all $M$-definable $D \supseteq J$, we see that $I_M \subseteq J$. Because we are in an NIP setting, $G^{00}$ and $G^{000}$ exist for any type-definable group $G$, by [@shelah-g000], Theorem 1.12 in the abelian case and [@gismatullin-g000] in general. \[triply-connected\] For any small model $M$, $I_M = I_M^{00} = I_M^{000}$. Let $M' \supseteq M$ be a model containing a representative from each coset of $I_M^{000}$ in $I_M$. Take $a \in I_M$ such that $a$ does not lie in any $M'$-definable light set; this is possible by Theorem \[heavy-light\].\[light-union\] and Proposition \[basic-nbhds\].\[heavy-nbhd\]. If $\delta$ is the $M'$-definable coset representative for $a + I_M^{000}$, then $a' := a - \delta$ lies in $I_M^{000}$, but not in any $M'$-definable light set (by Theorem \[heavy-light\].\[light-translate\]). Let $Y$ be the set of realizations of ${\operatorname{tp}}(a'/M)$. By $M$-invariance of $I_M^{000}$ we have $Y \subseteq I_M^{000}$. On the other hand, every $M$-definable neighborhood $D \supseteq Y$ is heavy, so $I_M \subseteq D {-_\infty}D \subseteq D - D$. As $Y$ is the directed intersection of $M$-definable $D \supseteq Y$, it follows that $I_M \subseteq Y - Y \subseteq I_M^{000}$. Further comments on henselianity {#sec:further-hensel} ================================ Now that we have a type-definable group of infinitesimals, we can reduce the Shelah conjecture for dp-finite fields of positive characteristic to the construction of invariant valuation rings. \[full-heavy\] Let $X$ be a definable set of full dp-rank. Then $X$ is heavy. Let $n$ be ${\operatorname{dp-rk}}({\mathbb{M}})$. Let $(Q_1,\ldots,Q_m,P)$ be a critical coordinate configuration of rank $\rho$. Shrinking $Q_i$, we may assume that $(Q_1 \times \cdots \times Q_m) \setminus P$ is narrow. Let $M$ be a small model over which $X, Q_1, \ldots, Q_m, P$ are defined. Take a tuple $$(a,b_1,\ldots,b_m) \in X \times Q_1 \times \cdots \times Q_m$$ such that $${\operatorname{dp-rk}}(a,b_1,\ldots,b_m/M) = {\operatorname{dp-rk}}(X) + {\operatorname{dp-rk}}(Q_1) + \cdots + {\operatorname{dp-rk}}(Q_m) = n + \rho.$$ By Remark \[faux-independence\], ${\operatorname{dp-rk}}(\vec{b}/M) = \rho$. Let $\delta = a - (b_1 + \cdots + b_m)$. Then $$n + \rho = {\operatorname{dp-rk}}(a, \vec{b}/M) = {\operatorname{dp-rk}}(\delta,\vec{b}/M) \le {\operatorname{dp-rk}}(\vec{b}/\delta M) + {\operatorname{dp-rk}}(\delta/M).$$ As ${\operatorname{dp-rk}}(\delta/M) \le {\operatorname{dp-rk}}({\mathbb{M}}) = n$, it follows that $$\rho \le {\operatorname{dp-rk}}(\vec{b}/\delta M) \le {\operatorname{dp-rk}}(\vec{b}/M) = \rho.$$ By Proposition \[halfway-there\], ${\operatorname{tp}}(\vec{b}/\delta M)$ is broad. It follows that the set $$\{(b_1,\ldots,b_m) \in Q_1 \times \cdots \times Q_m ~|~ b_1 + \cdots + b_m + \delta \in X\}$$ is broad. As we arranged $(Q_1 \times \cdots \times Q_m) \setminus P$ to be narrow, the set $$\{(b_1,\ldots,b_m) \in P ~|~ b_1 + \cdots + b_m + \delta \in X\}$$ is also broad, and therefore has dp-rank $\rho$. If $Y$ is the target of $(Q_1,\ldots,Q_n,P)$, then $$\{y \in Y ~|~ y + \delta \in X\}$$ has dp-rank $\rho$, as the map $P \to Y$ has finite fibers. Therefore $X$ is $Y$-heavy. In fact, the converse to Lemma \[full-heavy\] is true. See Theorem 5.9.2 in [@prdf2]. \[primality-trick\] Let $M$ be a small model and $\mathcal{O}$ be a non-trivial $M$-invariant valuation ring. Then the maximal ideal $\mathfrak{m}$ contains $I_M$, and the subideal of $\mathcal{O}$ generated by $I_M$ is prime. Let $n = {\operatorname{dp-rk}}({\mathbb{M}})$. Take an element $a$ of positive valuation. Take an element $b \in {\mathbb{M}}$ with ${\operatorname{dp-rk}}(b/aM) = n$. Replacing $b$ with $a/b$ if necessary, we may assume $b \in \mathfrak{m}$. Let $Y$ be the type-definable set of realizations of ${\operatorname{tp}}(b/M)$. Note ${\operatorname{dp-rk}}(Y) \ge {\operatorname{dp-rk}}(b/aM) = n$, so ${\operatorname{dp-rk}}(Y) = n$. Also note that $Y \subseteq \mathfrak{m}$ because $\mathfrak{m}$ is $M$-invariant. Every $M$-definable set $D \supseteq Y$ is heavy, by Lemma \[full-heavy\]. Therefore, $I_M \subseteq D - D$. As $D$ is arbitrary, $$I_M \subseteq Y - Y \subseteq \mathfrak{m} - \mathfrak{m} \subseteq \mathfrak{m}.$$ Now let $J$ be the ideal of $\mathcal{O}$ generated by $I_M$. Evidently $J \le \mathfrak{m}$ and $J$ is $M$-invariant. Suppose $J$ is not prime. By the characterization of ideals in valuation rings in terms of cuts in the value group, non-primality of $J$ implies that the ideal $J' := J \cdot J$ is a strictly smaller non-zero ideal in $\mathcal{O}$, and in particular $J' \not \supseteq I_M$ (by choice of $J$). On the other hand, $J'$ is $M$-invariant. Because $J'$ is non-zero, there exists $c \in {\mathbb{M}}^\times$ such that $c \cdot \mathfrak{m} \subseteq J'$. Let $d$ be an element of $Y$ such that ${\operatorname{dp-rk}}(d/cM) = n$. Then ${\operatorname{dp-rk}}(d \cdot c / cM) = {\operatorname{dp-rk}}(d / cM) = n$, so ${\operatorname{dp-rk}}(d \cdot c / M) = n$. Let $Z$ be the set of realizations of ${\operatorname{tp}}(d \cdot c / M)$. As $d \cdot c \in c \cdot Y \subseteq c \cdot \mathfrak{m} \subseteq J'$, the usual argument shows that $I_M \subseteq Z - Z \subseteq J' - J' = J'$, a contradiction. Therefore $J$ is prime. \[upgrade-to-span\] Let $M$ be a small model and $\mathcal{O}$ be a non-trivial $M$-invariant valuation ring. Then there is an $M$-invariant coarsening $\mathcal{O}' \supseteq \mathcal{O}$ such that $I_M$ spans $\mathcal{O}'$, in the sense of Definition \[span-a-valuation\]. It is a general fact that if $\mathfrak{p}$ is a prime ideal in a valuation ring $\mathcal{O}$, then $\mathfrak{p}$ is the maximal ideal of some (unique) coarsening $\mathcal{O}' \supseteq \mathcal{O}$. In our specific case, $$\begin{aligned} I_M & \subseteq \mathfrak{p} = \mathcal{O} \cdot I_M \\ \implies I_M & \subseteq \mathfrak{p} \subseteq \mathcal{O}' \cdot I_M \subseteq \mathcal{O}' \cdot \mathfrak{p} = \mathfrak{p} \\ \implies I_M & \subseteq \mathfrak{p} = \mathcal{O}' \cdot I_M \end{aligned}$$ so $I_M$ spans $\mathcal{O}'$. \[invariant-valuations-are-enough\] If ${\mathbb{M}}$ has characteristic $p > 0$ and if at least one non-trivial invariant valuation ring exists, then ${\mathbb{M}}$ admits a non-trivial henselian valuation. By Proposition \[upgrade-to-span\] there is a small model $M$ and an $M$-invariant valuation ring $\mathcal{O}$ spanned by $I_M$. The fact that $I_M \ne 0$ prevents $\mathcal{O}$ from being the trivial valuation ring. Moreover, $I_M = I_M^{000}$ by Corollary \[triply-connected\]. By Proposition \[invariant-henselian\] it follows that $\mathcal{O}$ is henselian. Bounds on connected components {#sec:bounds} ============================== In this section, $(G,+,\ldots)$ is a monster-model abelian group, possibly with additional structure, of finite dp-rank $n$. \[baldwin-saxl-variant\] Let $G_0, \ldots, G_n$ be type-definable subgroups of $G$. There is some $0 \le k \le n$ such that $$\left( \bigcap_{i = 0}^n G_i \right)^{00} = \left( \bigcap_{i \ne k} G_i \right)^{00}.$$ This follows by the proof of Proposition 4.5.2 in [@CKS]. \[up-bound\] Let $H$ be a type-definable subgroup of $G$. There is a cardinal $\kappa$ depending only on $H$ and $G$ such that if $H < H' < G$ for some type-definable subgroup $H'$, and if $H'/H$ is bounded, then $H'/H$ has size at most $\kappa$. This $\kappa$ continues to work in arbitrary elementary extensions. Naming parameters, we may assume that $H$ (but not $H')$ is type-definable over $\emptyset$. By Morley-Erdős-Rado there is some cardinal $\kappa$ with the following property: for any sequence $\{a_\alpha\}_{\alpha < \kappa}$ of elements of $G$, there is some $\emptyset$-indiscernible sequence $\{b_i\}_{i \in {\mathbb{N}}}$ such that for any $i_1 < \cdots < i_n$ there is $\alpha_1 < \cdots < \alpha_n$ such that $$a_{\alpha_1} \cdots a_{\alpha_n} \equiv_\emptyset b_{i_1} \cdots b_{i_n}.$$ Let $H'$ be a subgroup of $G$, containing $H$, type-definable over some small set $A$. Suppose that $|H'/H| \ge \kappa$. We claim that $H'/H$ is unbounded. Suppose for the sake of contradiction that $|H'/H| < \lambda$ in all elementary extensions. Take a sequence $\{a_\alpha\}_{\alpha < \kappa}$ of elements of $H'$ lying in pairwise distinct cosets of $H$. Let $\{b_i\}_{i \in {\mathbb{N}}}$ be an $\emptyset$-indiscernible sequence extracted from the $a_\alpha$ by Morley-Erdős-Rado. Because the $a_\alpha$ live in pairwise distinct cosets of $H$ and $H$ is $\emptyset$-definable, the $b_i$ live in pairwise distinct cosets of $H$. By indiscernibility, there is a 0-definable set $D \supseteq H$ such that $b_i - b_j \notin D$ for $i \ne j$. Consider the $\ast$-type over $A$ in variables $\{x_\alpha\}_{\alpha < \lambda}$ asserting that 1. $x_\alpha \in H'$ for every $\alpha < \lambda$ 2. If $\alpha_1 < \cdots < \alpha_n$, then $$x_{\alpha_1} \cdots x_{\alpha_n} \equiv_\emptyset b_1 \cdots b_n.$$ This type is consistent. Indeed, if $\Sigma_{\alpha_1,\ldots,\alpha_n}(\vec{x})$ is the sub-type asserting that $$\begin{aligned} x_{\alpha_1}, \ldots, x_{\alpha_n} &\in H' \\ x_{\alpha_1} \cdots x_{\alpha_n} &\equiv_\emptyset b_1 \cdots b_n \end{aligned}$$ then $\Sigma_{\alpha_1,\ldots,\alpha_n}(\vec{x})$ is satisfied by $(a_{\beta_1},\ldots,a_{\beta_n})$ for some well chosen $\beta_i$, by virtue of how the $b_i$ were extracted. Moreover, the full type is a filtered union of $\Sigma_{\vec{\alpha}}(\vec{x})$’s, so it is consistent. Let $\{c_\alpha\}_{\alpha < \lambda}$ be a set of realizations. Then every $c_\alpha$ lies in $H'$, but $$c_{\alpha} - c_{\alpha'} \notin D \supseteq H$$ for $\alpha \ne \alpha'$. Therefore, the $c_\alpha$ lie in pairwise distinct cosets of $H$, and $|H'/H| \ge \lambda$, a contradiction. \[extract-of-mer\] For any cardinal $\kappa$ there is a cardinal $\tau(\kappa)$ with the following property: given any family $\{H_\alpha\}_{\alpha < \tau(\kappa)}$ of type-definable subgroups of $G$, there exist subsets $S_1, S_2 \subseteq \tau(\kappa)$ such that $S_1$ is finite, $|S_2| = \kappa$, and $$\left( \bigcap_{\alpha \in S_1} H_\alpha \right)^{00} \subseteq \bigcap_{\alpha \in S_2} H_\alpha.$$ Without loss of generality $\kappa \ge \aleph_0$. By the Erdős-Rado theorem (or something weaker), we can choose $\tau(\kappa)$ such that any coloring of the $n+1$-element subsets of $\tau(\kappa)$ with $n+1$ colors contains a homogeneous subset of cardinality $\kappa^+$. Now suppose we are given $H_\alpha$ for $\alpha < \tau(\kappa)$. Given $\alpha_1 < \cdots < \alpha_{n+1}$, color the set $\{\alpha_1, \ldots, \alpha_{n+1}\}$ with the smallest $k \in \{1, \ldots, n+1\}$ such that $$\left( \bigcap_{i = 1}^{n+1} H_{\alpha_i} \right)^{00} = \left( \bigcap_{i = 1}^{k-1} H_{\alpha_i} \cap \bigcap_{i = k + 1}^{n+1} H_{\alpha_i} \right)^{00}.$$ This is possible by Fact \[baldwin-saxl-variant\]. Passing to a homogeneous subset and re-indexing, we get $\{H_\alpha\}_{\alpha < \kappa^+}$ such that every $(n+1)$-element set has color $k$ for some fixed $k$. In particular, for any $\alpha_1 < \cdots < \alpha_{n+1} < \kappa^+$, we have $$H_{\alpha_k} \supseteq \left( \bigcap_{i = 1}^{n+1} H_{\alpha_i} \right)^{00} = \left( \bigcap_{i = 1}^{k - 1} H_{\alpha_i} \cap \bigcap_{i = k + 1}^{n+1} H_{\alpha_i} \right)^{00}.$$ Thus, for any $\beta_1 < \beta_2 < \cdots < \beta_{2n+1} < \kappa^+$ we have $$H_{\beta_{n+1}} \supseteq \left( \bigcap_{i = n - k + 2}^n H_{\beta_i} \cap \bigcap_{i = n + 2}^{2n - k + 2} H_{\beta_i} \right)^{00} \supseteq \left ( \bigcap_{i = 1}^n H_{\beta_i} \cap \bigcap_{i = n+ 2}^{2n + 1} H_{\beta_i}\right)^{00}$$ by taking $$(\alpha_1, \ldots, \alpha_{n+1}) = (\beta_{n-k+2},\ldots,\beta_{2n-k+2}).$$ Then, for any $\beta \in [n+1,\kappa]$, $$H_\beta \supseteq \left( H_1 \cap \cdots \cap H_n \cap H_{\kappa + 1} \cap H_{\kappa + n}\right)^{00},$$ so we may take $S_1 = \{1, \ldots, n, \kappa + 1, \ldots, \kappa + n\}$ and $S_2 = [n+1,\kappa]$. \[bounding-theorem\] There is a cardinal $\kappa$, depending only on the ambient group $G$, such that for any type-definable subgroup $H < G$, the index of $H^{00}$ in $H$ is less than $\kappa$. This $\kappa$ continues to work in arbitrary elementary extensions. Say that a subgroup $K \subseteq G$ is *$\omega$-definable* if it is type-definable over a countable set. Note that if $K$ is $\omega$-definable, so is $K^{00}$. Moreover, if $K_1, K_2$ are $\omega$-definable, then so are $K_1 \cap K_2$ and $K_1 + K_2$. Also note that if $H$ is any type-definable group, then $H$ is a small filtered intersection of $\omega$-definable groups. Up to automorphism, there are only a bounded number of $\omega$-definable subgroups of $G$, so by Lemma \[up-bound\] there is some cardinal $\kappa_0$ with the following property: if $K$ is an $\omega$-definable group and if $K'$ is a bigger type-definable group, then either $|K'/K| < \kappa_0$ or $|K'/K|$ is unbounded. \[diamond-slide-claim\] If $H$ is a type-definable group and $K$ is an $\omega$-definable group containing $H^{00}$, then $|H/(H \cap K)| < \kappa_0$. Note that $$H^{00} \subseteq H \cap K \subseteq H,$$ so $H/(H \cap K)$ is bounded. On the other hand, $H/(H \cap K)$ is isomorphic to $(H + K)/K$, which must then have cardinality less than $\kappa_0$. Let $\kappa_1 = \tau((2^{\kappa_0})^+)$ where $\tau(-)$ is as in Lemma \[extract-of-mer\]. If $H$ is a type-definable subgroup of $G$, then there are fewer than $\kappa_1$ subgroups of the form $H \cap K$ where $K$ is $\omega$-definable and $K \supseteq H^{00}$. Otherwise, choose $\{K_\alpha\}_{\alpha \in \kappa_1}$ such that $K_\alpha$ is $\omega$-definable, $K_\alpha \supseteq H^{00}$, and $$H \cap K_\alpha \ne H \cap K_{\alpha'}$$ for $\alpha < \alpha' < \kappa_1$. By Lemma \[extract-of-mer\], there are subsets $S_1, S_2 \subseteq \kappa_1$ such that $|S_1| < \aleph_0$, $|S_2| = (2^{\kappa_0})^+$, and $$\left( \bigcap_{\alpha \in S_1} K_\alpha \right)^{00} \subseteq \bigcap_{\alpha \in S_2} K_\alpha.$$ Let $J$ be the left-hand side. Then $J$ is an $\omega$-definable group containing $H^{00}$, so $|H/(H \cap J)| < \kappa_0$ by Claim \[diamond-slide-claim\]. Now for any $\alpha \in S_2$, $$J \subseteq K_\alpha \implies H \cap J \subseteq H \cap K_\alpha \subseteq H.$$ There are at most $2^{|H/(H \cap J)|} \le 2^{\kappa_0}$ groups between $H \cap J$ and $J$, so there are at most $2^{\kappa_0}$ possibilities for $H \cap K_\alpha$, contradicting the fact that $|S_2| > 2^{\kappa_0}$ and the $H \cap K_\alpha$ are pairwise distinct for distinct $\alpha$. Now given the claim, we see that the index of $H^{00}$ in $H$ can be at most $\kappa_0^{\kappa_1}$. Indeed, let $\mathcal{S}$ be the collection of $\omega$-definable groups $K$ such that $K \supseteq H^{00}$, and let $\mathcal{S}'$ be a subcollection containing a representative $K$ for every possibility of $H \cap K$. By the second claim, $|\mathcal{S}'| < \kappa_1$. Every type-definable group is an intersection of $\omega$-definable groups, so $$H^{00} = \bigcap_{K \in \mathcal{S}} K = \bigcap_{K \in \mathcal{S}} (H \cap K) = \bigcap_{K \in \mathcal{S}'} (H \cap K).$$ Then there is an injective map $$H/H^{00} \hookrightarrow \prod_{K \in \mathcal{S}'} H/(H \cap K),$$ and the right hand size has cardinality at most $\kappa_0^{\kappa_1}$. But $\kappa_0^{\kappa_1}$ is independent of $H$. \[big-enough-model\] Let ${\mathbb{M}}$ be a field of finite dp-rank. There is a cardinal $\kappa$ with the following property: if $M \preceq {\mathbb{M}}$ is any small model of cardinality at least $\kappa$, and if $J$ is a type-definable $M$-linear subspace of ${\mathbb{M}}$, then $J = J^{00}$. More generally, if $J$ is a type-definable $M$-linear subspace of ${\mathbb{M}}^k$, then $J = J^{00}$. Note that we are not assuming $J$ is type-definable over $M$. Take $\kappa$ as in the Theorem, $M$ a small model of size at least $\kappa$, and $J$ a type-definable $M$-linear subspace of ${\mathbb{M}}$. For any $\alpha \in {\mathbb{M}}^\times$, we have $(\alpha \cdot J)^{00} = \alpha \cdot J^{00}$. Restricting to $\alpha \in M^\times$, we see that $\alpha \cdot J^{00} = J^{00}$. In other words, $J^{00}$ is an $M$-linear subspace itself. The quotient $J/J^{00}$ naturally has the structure of a vector space over $M$. If it is non-trivial, it has cardinality at least $\kappa$, contradicting the choice of $\kappa$. Therefore, $J/J^{00}$ is the trivial vector space, and $J^{00} = J$. For the “more generally” claim, apply Theorem \[bounding-theorem\] to the groups ${\mathbb{M}}^k$ and take the supremum of the resulting $\kappa$. Generalities on modular lattices {#sec:modular-lats} ================================ Recall that a a lattice is *modular* if the identity $$(x \vee a) \wedge b = (x \wedge b) \vee a$$ holds whenever $a \le b$. Modularity is equivalent to the statement that for any $a, b$, the interval $[a \wedge b, a]$ is isomorphic as a poset to $[b, a \vee b]$ via the maps $$\begin{aligned} [a \wedge b, a] & \to [b, a \vee b] \\ x & \mapsto x \vee b\end{aligned}$$ and $$\begin{aligned} [b, a \vee b] & \mapsto [a \wedge b, a] \\ x & \mapsto x \wedge a.\end{aligned}$$ Independence {#sec:independence} ------------ Let $(P,<)$ be a modular lattice with bottom element $\bot$. \[def-of-ind\] A finite sequence $a_1, \ldots, a_n$ of elements of $P$ is *independent* if $a_k \wedge \bigvee_{i = 1}^{k-1} a_i = \bot$ for $2 \le k \le n$. Note that if $a$ and $b$ are independent (i.e., $a \wedge b = \bot$), then there is an isomorphism between the poset $[\bot, a]$ and $[b, a \vee b]$ as above. \[3-perm\] If the sequence $\{a, b, c\}$ is independent, then the sequence $\{a, c, b\}$ is independent. By assumption, $$\begin{aligned} b \wedge a &= \bot \\ c \wedge (a \vee b) &= \bot. \end{aligned}$$ First note that $$c \wedge a \le c \wedge (a \vee b) = \bot,$$ so $\{a,c\}$ is certainly independent. Let $x = b \wedge (a \vee c)$. We must show that $x = \bot$. Otherwise, $$\bot < x \le b.$$ Applying the isomorphism $[\bot, b] \cong [b, a \vee b]$, $$a < x \vee a \le a \vee b.$$ Applying the isomorphism $[\bot, a \vee b] \cong [c, a \vee b \vee c]$, $$a \vee c < x \vee a \vee c \le a \vee b \vee c.$$ But $x \vee (a \vee c) = a \vee c$, as $x \le (a \vee c)$. \[perm-invar\] Independence is permutation invariant: if $\{a_1,\ldots,a_n\}$ is independent and $\pi$ is a permutation of $[n]$, then $\{a_{\pi(1)},\ldots,a_{\pi(n)}\}$ is independent. It suffices to consider the case where $\pi$ is the transposition of $k$ and $k+1$. Let $b = a_1 \vee \cdots \vee a_{k-1}$. Then we know $$\begin{aligned} a_k \wedge b &= \bot \\ a_{k+1} \wedge (b \vee a_k) &= \bot \end{aligned}$$ and we must show $$\begin{aligned} a_{k+1} \wedge b & \stackrel{?}{=} \bot \\ a_k \wedge (b \vee a_{k+1}) &\stackrel{?}{=} \bot. \end{aligned}$$ This is exactly Lemma \[3-perm\]. \[1-collapse\] Let $c_1, \ldots, c_n$ and $a_1, \ldots, a_m$ be two sequences. Set $a_0 = c_1 \vee \cdots \vee c_n$. Then the following are equivalent: 1. The sequence $c_1, \ldots, c_n$ is independent and the sequence $a_0, a_1, \ldots, a_m$ is independent. 2. The sequence $c_1, \ldots, c_n, a_1, \ldots, a_m$ is independent. Trivial after unrolling the definition. \[1-grouping\] Let $a_1, \ldots, a_n$ be an independent sequence and $S$ be a subset of $[n]$. Let $b = \bigvee_{i \in S} a_i$ (understood as $\bot$ when $S = \emptyset$). Then $\{b\} \cup \{a_i ~|~ i \notin S\}$ is an independent sequence. Modulo Proposition \[perm-invar\], this is the (2) $\implies$ (1) direction of Proposition \[1-collapse\]. \[grouping\] Let $a_1, \ldots, a_n$ be an independent sequence. Let $S_1, \ldots, S_m$ be pairwise disjoint subsets of $[n]$. For $j \in [m]$ let $b_j = \bigvee_{i \in S_j} a_i$, or $\bot$ if $S_j$ is empty. Then $b_1, \ldots, b_m$ is an independent sequence. Iterate Remark \[1-grouping\]. \[3-odd\] If $a, b, c$ is an independent sequence, then $(a \vee c) \wedge (b \vee c) = c$. As $c \le b \vee c$, $$(a \vee c) \wedge (b \vee c) = (a \wedge (b \vee c)) \vee c = \bot \vee c = c,$$ where the first equality is by modularity, and the second equality is by independence of $\langle b,c,a \rangle$. \[proto-cube\] Let $a_1, \ldots, a_n$ be an independent sequence. For $S \subseteq [n]$, let $a_S = \bigvee_{i \in S} a_i$. Then the following facts hold: 1. $a_{S \cup S'} = a_S \vee a_{S'}$. 2. $a_\emptyset = \bot$. 3. $a_{S \cap S'} = a_S \wedge a_{S'}$. In other words, $S \mapsto a_S$ is a homomorphism of lower-bounded lattices from ${\mathcal{P}ow}([n])$ to $P$. The first two points are clear by definition. For the third point, let $$\begin{aligned} S_1 &= S \setminus S' \\ S_2 &= S' \setminus S \\ S_3 &= S \cap S'. \end{aligned}$$ By Lemma \[grouping\], the sequence $a_{S_1}, a_{S_2}, a_{S_3}$ is independent. By Lemma \[3-odd\], $$a_{S \cap S'} = a_{S_3} \stackrel{!}{=} (a_{S_1} \vee a_{S_3}) \wedge (a_{S_2} \vee a_{S_3}) = a_{S_1 \cup S_3} \wedge a_{S_2 \cup S_3} = a_S \wedge a_{S'}. \qedhere$$ \[proto-sharp-cube\] In Proposition \[proto-cube\], suppose in addition that $a_i > \bot$ for $1 \le i \le n$. Then the map $S \mapsto a_S$ is an isomorphism onto its image: $$S \subseteq S' \iff a_S \le a_{S'}.$$ The $\implies$ direction follows formally from the fact that $S \mapsto a_S$ is a lattice morphism. Conversely, suppose $S \not\subseteq S'$ but $a_S \le a_{S'}$. Take $i \in S \setminus S'$. Then $a_i \le a_S \le a_{S'}$, so $$a_i = a_i \wedge a_{S'} = a_{S' \cap \{i\}} = a_\emptyset = \bot,$$ contradicting the assumption. \[down-smashing\] Let $(P,\le)$ and $(P',\le)$ be two bounded-below modular lattices, and $f : P \to P'$ be a map satisfying the following conditions: $$\begin{aligned} f(x \wedge y) &= f(x) \wedge f(y) \\ f(\bot_P) &= \bot_{P'}. \end{aligned}$$ If $a_1, \ldots, a_n$ is an independent sequence in $P$ then $f(a_1), \ldots, f(a_n)$ is an independent sequence in $P'$. Note that $$x \le y \iff x = x \wedge y \implies f(x) = f(x) \wedge f(y) \iff f(x) \le f(y),$$ so $f$ is weakly order-preserving. In particular, for any $x, y$ $$\begin{aligned} f(x) & \le f(x \vee y) \\ f(y) & \le f(x \vee y) \\ f(x) \vee f(y) & \le f(x \vee y). \end{aligned}$$ Therefore for $2 \le k \le n$ we have $$f(a_k) \wedge \bigvee_{i < k} f(a_i) \le f(a_k) \wedge f\left( \bigvee_{i < k} a_i \right) = f\left(a_k \wedge \bigvee_{i < k} a_i \right) = f(\bot_P) = \bot_{P'}. \qedhere$$ Cubes and relative independence {#sec:cubes} ------------------------------- We continue to work in a modular lattice $(P,\le)$, but drop the assumption that a bottom element exists. Let $b$ be an element of $P$. 1. A sequence $a_1, \ldots, a_n$ of elements of $P$ is *independent over $b$* if it is an independent sequence in the bounded-below modular lattice $\{x \in P ~|~ x \ge b\}$. Equivalently, $a_1, \ldots, a_n$ is independent over $b$ if $a_i \ge b$ for each $i$, and $$a_k \wedge \bigvee_{i < k} a_i = b$$ for $2 \le k \le n$. 2. A sequence $a_1, \ldots, a_n$ of elements of $P$ is *co-independent under $b$* if $a_i \le b$ for each $i$, and $$a_k \vee \bigwedge_{i < k} a_i = b$$ for $2 \le k \le n$. \[relative-smashing\] Let $b$ and $c$ be elements of $P$. 1. If $a_1, \ldots, a_n$ is an independent sequence over $b$, then $a_1 \wedge c, a_2 \wedge c, \ldots, a_n \wedge c$ is an independent sequence over $b \wedge c$. 2. If $a_1, \ldots, a_n$ is a co-independent sequence under $b$, then $a_1 \vee c, a_2 \vee c, \ldots, a_n \vee c$ is a co-independent sequence under $b \vee c$. Part 1 follows from Lemma \[down-smashing\] applied to the function $$\begin{aligned} f : \{x \in P ~|~ x \ge b\} & \to \{x \in P ~|~ x \ge b \wedge c\} \\ x & \mapsto x \wedge c, \end{aligned}$$ and part 2 follows by duality. In what follows, we require “lattice homomorphisms” to preserve $\vee$ and $\wedge$, but not necessarily $\top$ and $\bot$ when they exist. An *$n$-cube* in $P$ is a family $\{a_S\}_{S \subseteq [n]}$ of elements of $P$ such that $S \mapsto a_S$ is a lattice homomorphism from ${\mathcal{P}ow}([n])$ to $P$. A *strict $n$-cube* is an $n$-cube such that this homomorphism is injective. The *top* and *bottom* of an $n$-cube are the elements $a_\emptyset$ and $a_{[n]}$, respectively. \[lattice-hom\] Lattice homomorphisms are weakly order-preserving, and injective lattice homomorphisms are strictly order-preserving. \[cubes-and-independence\] Let $b$ be an element of $P$. 1. If $\{a_S\}_{S \subseteq [n]}$ is an $n$-cube with bottom $b$, then $a_1, \ldots, a_n$ is an independent sequence over $b$. 2. This establishes a bijection from the collection of $n$-cubes with bottom $b$ to the collection of independent sequences over $b$ of length $n$. 3. If $a_1, \ldots, a_n$ is an independent sequence over $b$, the corresponding $n$-cube is strict if and only if every $a_i$ is strictly greater than $b$. <!-- --> 1. By definition of $n$-cube and Remark \[lattice-hom\], $$\begin{aligned} a_i & \ge a_\emptyset = b \\ a_i \wedge \bigvee_{j < i} a_j & = a_i \wedge a_{[i-1]} = a_{i \cap [i-1]} = a_\emptyset = b. \end{aligned}$$ 2. Injectivity: if $\{a_S\}_{S \subseteq [n]}$ and $\{a'_S\}_{S \subseteq [n]}$ are two $n$-cubes such that $a_S = a'_S$ when $|S| \le 1$, then $$a_S = \bigvee_{i \in S} a_i = \bigvee_{i \in S} a'_i = a'_S$$ when $|S| > 1$. Surjectivity: let $a_1, \ldots, a_n$ be a sequence independent over $b$. By Proposition \[proto-cube\] applied to the sublattice $\{x \in P ~|~ x \ge b\}$, there is an $n$-cube with bottom $b$. 3. Let $\{a_S\}_{S \subseteq [n]}$ be the corresponding $n$-cube. If $a_i > b$ for $i \in [n]$, then the cube is strict by Proposition \[proto-sharp-cube\]. Conversely, if the cube is strict, then $a_i > a_\emptyset = b$. Dually, \[cubes-and-coindependence\] Let $b$ be an element of $P$. 1. Let $\{a_S\}_{S \subseteq [n]}$ be an $n$-cube with top $b$. For $i \in [n]$ let $c_i = a_{[n] \setminus \{i\}}$. Then $c_1, \ldots, c_n$ is a co-independent sequence under $b$. 2. This establishes a bijection from the collection of $n$-cubes with top $b$ to the collection of co-independent sequences under $b$ of length $n$. 3. If $c_1, \ldots, c_n$ is a co-independent sequence under $b$, the corresponding $n$-cube is strict if and only if every $c_i$ is strictly less than $b$. Subadditive ranks {#sec:reduced-rank} ----------------- The *reduced rank* of a modular lattice $(P,\le)$ is the supremum of $n \in {\mathbb{Z}}$ such that a strict $n$-cube exists in $P$, or $\infty$ if there is no supremum. If $a \ge b$ are two elements of $P$, the *reduced rank* ${\operatorname{rk}_0}(a/b)$ is the reduced rank of the sublattice $[b,a] := \{x \in P ~|~ b \le x \le a\}$. \[subs-and-products\]   1. Let $P_1$ and $P_2$ be two modular lattices. If the reduced rank of $P_1$ is at least $n$ and the reduced rank of $P_2$ is at least $m$, then the reduced rank of $P_1 \times P_2$ is at least $n + m$. 2. \[subs2\] If $P_1$ is a modular lattice and $P_2$ is a sublattice, the reduced rank of $P_1$ is at least the reduced rank of $P_2$. 3. If $a, b$ are two elements of a modular lattice $(P,\le)$, then there is an injective lattice homomorphism $$\begin{aligned} [a \wedge b, a] \times [a \wedge b, b] &\to [a \wedge b, a \vee b] \\ (x,y) & \mapsto x \vee y. \end{aligned}$$ 4. \[product-goal\] If $a, b$ are two elements of a modular lattice $(P,\le)$ such that ${\operatorname{rk}_0}(a/a \wedge b) \ge n$ and ${\operatorname{rk}_0}(b/ a \wedge b) \ge m$, then ${\operatorname{rk}_0}(a \vee b/a \wedge b) \ge n + m$. <!-- --> 1. Let $\{a_S\}_{S \subseteq[n]}$ and $\{b_S\}_{S \subseteq [m]}$ be a strict $n$-cube in $P_1$ and a strict $m$-cube in $P_2$. Then $$(S,T) \mapsto (a_S,b_T)$$ is an injective lattice homomorphism ${\mathcal{P}ow}([n]) \times {\mathcal{P}ow}([m]) \to P_1 \times P_2$. But ${\mathcal{P}ow}([n]) \times {\mathcal{P}ow}([m])$ is isomorphic to ${\mathcal{P}ow}([n+m])$. 2. Any $n$-cube in $P_2$ is already an $n$-cube in $P_1$. 3. The map is clearly a homomorphism of $\vee$-semilattices. If $(x, y) \in [a \wedge b, a] \times [a \wedge b, b]$, then $x \vee y = (x \vee b) \wedge (y \vee a)$. By modularity, $$(b \vee x) \wedge a = (b \wedge a) \vee x.$$ Simplifying, we see that $a \wedge (x \vee b) = x$. Because $y \le b$, we have $x \vee y \le x \vee b$. By modularity, $$(a \vee (x \vee y)) \wedge (x \vee b) = (a \wedge (x \vee b)) \vee (x \vee y).$$ Simplifying, we see that $$(y \vee a) \wedge (x \vee b) = x \vee x \vee y = x \vee y. \qedhere$$ In light of the claim, the map $(x,y) \mapsto x \vee y$ is the composition $$[a \wedge b, a] \times [a \wedge b, b] \stackrel{\sim}{\to} [b, a \vee b] \times [a, a \vee b] \to [a \wedge b, a \vee b]$$ where the first map is the product of the poset isomorphisms $$\begin{aligned} [a \wedge b, a] & \stackrel{\sim}{\to} [b, a \vee b] \\ [a \wedge b, b] & \stackrel{\sim}{\to} [a, a \vee b] \end{aligned}$$ and the second map is the $\wedge$-semilattice homomorphism $(x',y') \mapsto x' \wedge y'$. As a composition of a poset isomorphism and a $\wedge$-semilattice homomorphism, the original map in question is a $\wedge$-semilattice homomorphism. Finally, for injectivity, note that the composition of $(x,y) \mapsto x \vee y$ with $z \mapsto (z \vee b, z \vee a)$ is $$(x,y) \mapsto (x \vee b, y \vee a)$$ which is the aforementioned isomorphism $$[a \wedge b, a] \times [a \wedge b, b] \stackrel{\sim}{\to} [b, a \vee b] \times [a, a \vee b].$$ Therefore $(x,y) \mapsto x \vee y$ is injective on $[a \wedge b, a] \times [a \wedge b, b]$. 4. This follows by combining the previous three points. Work in a modular lattice $(P,\le)$. \[subadditive-lemma\] Let $x \le y \le z$ be three elements of $P$. If there is a strict $n$-cube $\{a_S\}_{S \subseteq [n]}$ in $[x,z]$, then we can write $n = m + \ell$ and find a strict $m$-cube in $[x,y]$ and a strict $\ell$-cube in $[y,z]$. Passing to the sublattice $[x,z]$, we may assume that $\bot, \top$ exist and equal $x, z$ respectively. \[one-or-the-other\] For any strict inequality $b < c$ in $P$, at least *one* of the following strict inequalities holds: $$\begin{aligned} b \wedge y &< c \wedge y \\ b \vee y &< c \vee y. \end{aligned}$$ Otherwise, $b \wedge y = c \wedge y$ and $b \vee y = c \vee y$. Then $$c = (y \vee c) \wedge c = (y \vee b) \wedge c = (y \wedge c) \vee b = (y \wedge b) \vee b = b$$ where the middle equality is the modular law. Take $S_0 \subseteq [n]$ maximal such that $a_{S_0} \wedge y = a_\emptyset \wedge y$. Applying a permutation, we may assume $S_0 = [\ell]$ for some $\ell \le n$. Let $m = n - \ell$. Note that there are injective lattice homomorphisms $$\begin{aligned} {\mathcal{P}ow}([\ell]) &\to P \\ S &\mapsto a_S \\ {\mathcal{P}ow}([n] \setminus [\ell]) &\to P \\ S &\mapsto a_{S \cup [\ell]} \end{aligned}$$ obtained by restricting the original lattice homomorphism $S \mapsto a_S$. By Proposition \[cubes-and-coindependence\] applied to the first of these two cubes, there is a co-independent sequence $b_1, \ldots, b_\ell$ under $a_{[\ell]}$ given by $$b_i = a_{[\ell] \setminus \{i\}},$$ and $b_i < a_{[\ell]}$ for $i \in [\ell]$. Similarly, by Proposition \[cubes-and-independence\] applied to the second of these two cubes, there is an independent sequence $c_1, \ldots, c_m$ over $a_{[\ell]}$ given by $$c_i = a_{[\ell] \cup \{\ell + i\}},$$ and $c_i > a_{[\ell]}$ for $i \in [m]$. Note that $$a_\emptyset \wedge y \le b_i \wedge y \le a_{[\ell]} \wedge y \le c_j \wedge y$$ for $i \in [\ell]$ and $j \in [m]$. By choice of $S_0$, we in fact have $$a_\emptyset \wedge y = b_i \wedge y = a_{[\ell]} \wedge y < c_j \wedge y.$$ Then $b_i \wedge y = a_{[\ell]} \wedge y$ and $b_i < a_{[\ell]}$ imply $b_i \vee y < a_{[\ell]} \vee y$ by Claim \[one-or-the-other\]. Thus $$\begin{aligned} b_i \vee y & < a_{[\ell]} \vee y \\ a_{[\ell]} \wedge y & < c_j \wedge y \end{aligned}$$ for all $i \in [\ell]$ and $j \in [m]$. By Lemma \[relative-smashing\], the sequence $\{b_i \vee y\}$ is co-independent under $\{a_{[\ell]} \vee y\}$, yielding a strict $\ell$-cube in $[y,\top]$. Similarly, the sequence $\{c_j \wedge y\}$ is independent over $\{a_{[\ell]} \wedge y\}$, yielding a strict $m$-cube in $[\bot,y]$. \[subadditive-corollary\] Let $a \ge b \ge c$ be three elements of a modular lattice $(P,\le)$. If ${\operatorname{rk}_0}(a/b) < \infty$ and ${\operatorname{rk}_0}(b/c) < \infty$, then $${\operatorname{rk}_0}(a/c) \le {\operatorname{rk}_0}(a/b) + {\operatorname{rk}_0}(b/c) < \infty.$$ \[def-of-subadd-rank\] A *subadditive rank* on a modular lattice is a function assigning a non-negative integer ${\operatorname{rk}}(a/b)$ to every pair $a \ge b$, satisfying the following axioms 1. \[not-in-weak\] ${\operatorname{rk}}(a/b) = 0$ if and only if $a = b$. 2. If $a \ge b \ge c$, then $$\begin{aligned} {\operatorname{rk}}(a/c) & \le {\operatorname{rk}}(a/b) + {\operatorname{rk}}(b/c) \\ {\operatorname{rk}}(a/c) & \ge {\operatorname{rk}}(a/b) \\ {\operatorname{rk}}(a/c) & \ge {\operatorname{rk}}(b/c). \end{aligned}$$ 3. If $a, b$ are arbitrary, then $$\begin{aligned} {\operatorname{rk}}(a/a \wedge b) &= {\operatorname{rk}}(a \vee b/b) \\ {\operatorname{rk}}(b/a \wedge b) &= {\operatorname{rk}}(a \vee b/a) \\ {\operatorname{rk}}(a \vee b / a \wedge b) & = {\operatorname{rk}}(a/a \wedge b) + {\operatorname{rk}}(b/a \wedge b). \end{aligned}$$ A *weak subadditive rank* is a function satisfying the axioms other than (\[not-in-weak\]). \[abelian-category-motivation\] We could also define the notion of a subadditive rank on an abelian category. This should be a function ${\operatorname{rk}}(-)$ from objects to nonnegative integers satisfying the axioms: 1. ${\operatorname{rk}}(A) = 0$ if and only if $A = 0$. 2. If $f : A \to B$ is epic (resp. monic) then ${\operatorname{rk}}(A) \ge {\operatorname{rk}}(B)$ (resp. ${\operatorname{rk}}(A) \le {\operatorname{rk}}(B)$). 3. In a short exactly sequence $$0 \to A \to B \to C \to 0,$$ we have ${\operatorname{rk}}(B) \le {\operatorname{rk}}(A) + {\operatorname{rk}}(C)$ with equality if the sequence is split. A weak subadditive rank would satisfy all the axioms except the first. With these definitions, a weak or strong subadditive rank on an abelian category $\mathcal{C}$ should induce a weak or strong subadditive rank on the subobject lattice of any $A \in \mathcal{C}$.[^8] \[subadditive-rk-in-cubes\] If ${\operatorname{rk}}(-/-)$ is a subadditive rank and $a_1, a_2, \ldots, a_n$ is relatively independent over $b$, then $${\operatorname{rk}}(a_1 \vee \cdots \vee a_n/b) = \sum_{i = 1}^n {\operatorname{rk}}(a_i/b),$$ by induction on $n$. \[dp-rank-example\] Let $K$ be a $\kappa$-saturated field of dp-rank $n < \infty$. Let $P$ be the lattice of subgroups of $(K,+)$ that are type-definable over sets of size less than $\kappa$. There is probably a weak subadditive rank on $P$ such that ${\operatorname{rk}}_{dp}(A/B)$ is the dp-rank of the hyper-definable set $A/B$. To verify this, we would merely need to double-check that subadditivity of dp-rank continues to hold on hyperimaginaries. Although this example motivates the notion of “subadditive rank,” we will never specifically need it. 1. Let $P$ be a modular lattice with a weak subadditive rank. Define $x \approx y$ if ${\operatorname{rk}}(x/x \wedge y) = {\operatorname{rk}}(y/x \wedge y) = 0$. Then $\approx$ is probably a lattice congruence, so we can form the quotient lattice $P/\approx$. The weak rank on $P$ should induce a (non-weak) subadditive rank on $P/\approx$. 2. In the example of Remark \[dp-rank-example\], $G \approx H$ if and only if $G^{00} = H^{00}$, and $P/\approx$ is presumably isomorphic to the subposet $$P' := \{G \in P ~|~ G = G^{00}\},$$ which is a lattice under the operations $$\begin{aligned} G \vee H & := G + H \\ G \wedge H & := (G \cap H)^{00}. \end{aligned}$$ The fact that dp-rank is weak and $P'$ is not a sublattice of $P$ is an annoyance that motivates §\[sec:bounds\]. By taking a large enough small model $K_0 \preceq K$ and restricting to the sublattice $$P'' := \{G \in P ~|~ G \text{ is a $K_0$-linear subspace of } K\}$$ the equation $G = G^{00}$ is automatic, and dp-rank presumably induces a (non-weak) subadditive rank on $P''$. \[redrk-subadd\] Let $(P,\le)$ be a modular lattice. 1. If the reduced rank ${\operatorname{rk}_0}(a/b)$ is finite for all pairs $a \ge b$, then ${\operatorname{rk}_0}$ is a (non-weak) subadditive rank. 2. If there is a subadditive rank ${\operatorname{rk}}$ on $P$, then ${\operatorname{rk}_0}(a/b) \le {\operatorname{rk}}(a/b) < \infty$ for all $a \ge b$. <!-- --> 1. Assume that ${\operatorname{rk}_0}(a/b)$ is finite for all pairs $a \ge b$. We verify the definition of subadditive rank. - Note that a strict 1-cube is just a pair $(x,y)$ with $x < y$. Therefore, ${\operatorname{rk}_0}(a/b) \ge 1$ if and only if there exist $x, y \in [b,a]$ such that $x < y$, if and only if $a > b$. Conversely, ${\operatorname{rk}_0}(a/b) = 0$ if and only if $a = b$. - Suppose $a \ge b \ge c$. The inequalities $$\begin{aligned} {\operatorname{rk}_0}(a/c) & \ge {\operatorname{rk}_0}(a/b) \\ {\operatorname{rk}_0}(a/c) & \ge {\operatorname{rk}_0}(b/c) \end{aligned}$$ follow by Lemma \[subs-and-products\].\[subs2\] applied to the inclusions $[c,b] \subseteq [c,a]$ and $[b,a] \subseteq [c,a]$. The requirement that ${\operatorname{rk}_0}(a/c) \le {\operatorname{rk}_0}(a/b) + {\operatorname{rk}_0}(b/c)$ is Corollary \[subadditive-corollary\]. - Let $a, b$ be arbitrary. By modularity, we have lattice isomorphisms $$\begin{aligned} [a \wedge b, a] & \cong [b, a \vee b] \\ [a \wedge b, b] & \cong [a, a \vee b], \end{aligned}$$ so certainly $$\begin{aligned} {\operatorname{rk}_0}(a/a \wedge b) &= {\operatorname{rk}_0}(a \vee b/b) \\ {\operatorname{rk}_0}(b/a \wedge b) &= {\operatorname{rk}_0}(a \vee b/a). \end{aligned}$$ By subadditivity, it remains to show that $$\begin{aligned} {\operatorname{rk}_0}(a \vee b / a \wedge b) \ge {\operatorname{rk}_0}(a / a \wedge b) + {\operatorname{rk}_0}(b / a \wedge b). \end{aligned}$$ This is Lemma \[subs-and-products\].\[product-goal\]. 2. Suppose ${\operatorname{rk}_0}(a/b) \ge n$; we will show ${\operatorname{rk}}(a/b) \ge n$. By definition of reduced rank, there is a strict $n$-cube $\{c_S\}_{S \subseteq [n]}$ in the interval $[b, a]$. By definition of strict $n$-cube, we have $$\begin{aligned} c_i \wedge c_{[i-1]} &= c_\emptyset \\ c_i \vee c_{[i-1]} &= c_{[i]} \\ c_i &> c_\emptyset \end{aligned}$$ for $0 < i \le n$. By the axioms of subadditive rank, $${\operatorname{rk}}(c_{[i]}/c_\emptyset) = {\operatorname{rk}}(c_{[i-1]}/c_\emptyset) + {\operatorname{rk}}(c_i/c_\emptyset) > {\operatorname{rk}}(c_{[i-1]}/c_\emptyset).$$ By induction on $i$, we see that ${\operatorname{rk}}(c_{[i]}/c_\emptyset) \ge i$. In particular, $${\operatorname{rk}}(a/b) \ge {\operatorname{rk}}(c_{[n]}/c_\emptyset) \ge n. \qedhere$$ A modular lattice $(P,\le)$ admits a subadditive rank if and only if ${\operatorname{rk}_0}(a/b) < \infty$ for all pairs $a \ge b$, in which case ${\operatorname{rk}_0}(a/b)$ is the unique minimum subadditive rank. \[expected-properties\] Let $K$ be a $\kappa$-saturated field of dp-rank $n$, and $P$ be the lattice of subgroups of $(K,+)$ that are type-definable over sets of size less than $\kappa$. 1. $P$ need not have finite reduced rank. For example, if $K$ has positive characteristic, there are arbitrarily big cubes made of *finite* subgroups of $(K,+)$. 2. Let $P'$ be the subposet of $G \in P$ such that $G = G^{00}$. Then $P'$ should be a modular lattice with lattice operations given by $G \wedge H = (G \cap H)^{00}$ and $G \vee H = G + H$. The map $G \mapsto G^{00}$ should be a surjective lattice homomorphism from $P$ to $P'$. The lattice $P'$ should have reduced rank at most $n$ by Fact \[baldwin-saxl-variant\]. Moreover, the same facts should hold when $K$ is a field of finite *burden* $n$. However, $G^{00}$ can fail to exist in this case, and we can only realize $P'$ as the quotient of $P$ modulo the 00-commensurability equivalence relation where $G \approx H$ iff $G/(G \cap H)$ and $H/(G \cap H)$ are bounded.[^9] In particular, we get a weak subadditive rank on the lattice $P$ and a subadditive rank on the lattice $P'$ without needing the conjectural subadditivity of burden. We will not directly use Remark \[expected-properties\]. Instead, we will take the following variant approach which avoids all $G^{00}$ issues: \[reduced-rank-vs-dp-rk\] Let ${\mathbb{M}}$ be a field, possibly with additional structure. Assume ${\mathbb{M}}$ is a monster model, and ${\operatorname{dp-rk}}({\mathbb{M}}) = n < \infty$. Let $M_0$ be a small submodel as in Corollary \[big-enough-model\]. Let $P$ be the modular lattice of $M_0$-linear subspaces of ${\mathbb{M}}$, type-definable over small[^10] parameter sets. Then $P$ has reduced rank at most $n$. Otherwise, there is an $(n+1)$-cube $\{H_S\}_{S \subseteq [n+1]}$ of type-definable subgroups which happen to be $M_0$-linear subspaces of ${\mathbb{M}}$. For $i = 1, \ldots, n+1$, let $G_i = H_{[n+1] \setminus \{i\}}$. Note that $$\begin{aligned} \bigcap_{i = 1}^{n+1} G_i &= H_\emptyset \\ \bigcap_{i \ne k} G_i &= H_k, \end{aligned}$$ for any $k \in [n+1]$, by definition of $(n+1)$-cube. Because the cube is strict, we have $H_\emptyset \subsetneq H_k$, so $$\bigcap_{i = 1}^{n+1} G_i \subsetneq \bigcap_{i \ne k} G_i$$ for each $k$. Both sides are $M_0$-linear type-definable subspaces. By choice of $M_0$ (i.e., by Corollary \[big-enough-model\]), for every $k$ we have $$\left( \bigcap_{i = 1}^{n+1} G_i\right)^{00} = \bigcap_{i = 1}^{n+1} G_i \subsetneq \bigcap_{i \ne k} G_i = \left( \bigcap_{i \ne k} G_i \right)^{00}$$ contradicting Fact \[baldwin-saxl-variant\]. The modular pregeometry on quasi-atoms {#sec:geometry} -------------------------------------- Work in a modular lattice $(P,\le)$ with bottom element $\bot$. An element $a > \bot$ is a *quasi-atom* if the set $$\{x \in P ~|~ \bot < x \le a\}$$ is downwards directed. Equivalently, if $\bot < x \le a$ and $\bot < y \le a$, then $\bot < x \wedge y$. \[quasi-atoms-basics\]   1. If $a$ is a quasi-atom and $\bot < a' < a$ then $a'$ is a quasi-atom. 2. Every atom is a quasi-atom. 3. \[sym-trans\] Non-independence is an equivalence relation on quasi-atoms. Two quasi-atoms $a, a'$ are equivalent if $a \wedge a' > \bot$. This equivalence relation is the transitive symmetric closure of $\ge$ on the set set of quasi-atoms. 4. \[enough-quasiats\] Suppose a subadditive rank exists. For every $a > \bot$, there is a quasi-atom $a' \le a$. <!-- --> 1. If $\bot < x, y$ and $x,y \le a'$ then $x,y \le a$ so $x \wedge y > \bot$. 2. If $a$ is an atom, then $\{x \in P ~|~ \bot < x \le a\} = \{a\}$ which is trivially directed. 3. By definition, two arbitrary elements $a, b$ are non-independent exactly if $a \wedge b > \bot$. Now restrict to the case of quasi-atoms. The relation $a \wedge a' > \bot$ is clearly symmetric. It is reflexive because $a > \bot$ is part of the definition of quasi-atom. For transitivity, suppose that $a, a', a''$ are quasi-atoms, and $a \wedge a' > \bot < a' \wedge a''$. Then $$a \wedge a'' \ge (a \wedge a') \wedge (a'' \wedge a') > \bot$$ where the strict inequality holds because $a'$ is a quasi-atom and $$\begin{aligned} \bot &< a \wedge a' \le a' \\ \bot &< a'' \wedge a' \le a'. \end{aligned}$$ Thus transitivity holds. This equivalence relation contains the restriction of $\ge$ to quasi-atoms, because if $a$ and $a'$ are quasi-atoms such that $a \ge a'$, then $a \wedge a' = a' > \bot$. Finally, it is contained in the transitive symmetric closure of $\ge$ because if $a \wedge a' > \bot$, then $a'' := a \wedge a'$ is a quasi-atom by part 1, and $a \ge a'' \le a$. 4. Among the elements of the set $\{x \in P ~|~ \bot < x \le a\}$, choose an element $a'$ such that ${\operatorname{rk}}(a'/\bot)$ is minimal. We claim that $a'$ is a quasi-atom. Otherwise, there exist $x, y \le a'$ such that $x, y > \bot = x \wedge y$. Then $${\operatorname{rk}}(a'/\bot) \ge {\operatorname{rk}}(x \vee y / \bot) = {\operatorname{rk}}(x/\bot) + {\operatorname{rk}}(y/\bot).$$ As $y > \bot$, it follows that ${\operatorname{rk}}(y/\bot) > 0$ and so ${\operatorname{rk}}(x/\bot) < {\operatorname{rk}}(a'/\bot)$, contradicting the choice of $a'$. In particular, if $K$ is a field of finite dp-rank and $P$ is the lattice of type-definable subgroups $G \le (K,+)$ such that $G = G^{00}$, then every non-zero element $G \in P$ has a quasi-atomic subgroup $G' < G$. The same fact might hold in strongly dependent fields, even if a finite subadditive rank is lacking. Otherwise, one can take a counterexample $G$ and split off an infinite independent sequence $H_1, H_2, \ldots$ of subgroups of $G$. This sequence might violate strong dependence. Say that two quasi-atoms $a, a'$ are *equivalent* if $a \wedge a' > \bot$, as in Proposition \[quasi-atoms-basics\].\[sym-trans\]. \[proto-respect-equivalence\] Let $a, a'$ be equivalent quasi-atoms, and $b$ be an arbitrary element. Then $\{a,b\}$ is independent if and only if $\{a',b\}$ is independent. By Proposition \[quasi-atoms-basics\].\[sym-trans\], we may assume $a \le a'$. Then $$a' \wedge b = \bot \implies a \wedge b = \bot$$ trivially. Conversely, suppose $a \wedge b = \bot$ but $a' \wedge b > \bot$. Note that $$\{a, a' \wedge b\} \subseteq \{x \in P ~|~ \bot < x \le a'\}.$$ As $a'$ is a quasi-atom, we have $$\bot = a \wedge b = (a \wedge a') \wedge b = a \wedge (a' \wedge b) \stackrel{!}{>} \bot. \qedhere$$ \[respect-equivalence\] Let $a_1, \ldots, a_n$ be an independent sequence. Suppose that $a_k$ is a quasi-atom for some $k$. Let $a'_k$ be an equivalent quasi-atom. Then $a_1, \ldots, a_{k-1}, a'_k, a_{k+1}, \ldots, a_n$ is an independent sequence. By Proposition \[perm-invar\] we may assume $k = n$. Let $b = a_1 \vee \cdots \vee a_{n-1}$. Then $\{b, a_n\}$ is independent and it suffices to show that $\{b, a'_n\}$ is independent. This is Lemma \[proto-respect-equivalence\]. In the next few lemmas, we adopt the following notation. The set of quasi-atoms will be denoted by $Q$. If $x \in P$, then $V(x)$ will denote the set of $a \in Q$ such that $a \wedge x > \bot$. Thus $Q \setminus V(x)$ is the set of quasi-atoms independent from $x$. \[2-ary-v\] $V(x) \cap V(y) = V(x \wedge y)$ for any $x, y \in P$. Let $a$ be a quasi-atom. Note that $$(a \wedge x) \wedge (a \wedge y) > \bot \iff \left( (a \wedge x > \bot) \text{ and } (a \wedge y > \bot)\right);$$ the left-to-right implication holds generally and the right-to-left implication holds because $a$ is a quasi-atom. The left hand side is equivalent to $a \in V(x \wedge y)$ and the right hand side is equivalent to $a \in V(x) \cap V(y)$. \[main-lemma-for-pregeometry\] Let $S$ be a finite subset of $Q$. 1. There is a closure operation on $S$ whose closed sets are exactly the sets of the form $V(x) \cap S$. 2. \[tfae\] Let $a_1, \ldots, a_n$ be a sequence of elements of $S$, independent in the sense of Definition \[def-of-ind\]. The following are equivalent for $b \in S$: 1. \[option-a\] $b$ lies in the closure of $\{a_1,\ldots,a_n\}$. 2. \[option-b\] $b \in V(a_1 \vee \cdots \vee a_n)$. 3. \[option-c\] The sequence $a_1, \ldots, a_n, b$ is *not* independent. 3. If $a_1, \ldots, a_n$ is a sequence in $S$, there is an subsequence $b_1, \ldots, b_m$ having the same closure, and independent in the sense of Definition \[def-of-ind\]. 4. The closure operation on $S$ satisfies exchange, i.e., it is a pregeometry. 5. A sequence $a_1, \ldots, a_n$ in $S$ is independent with respect to the pregeometry if and only if it is independent in the sense of Definition \[def-of-ind\]. <!-- --> 1. Let $\mathcal{C}$ be the collection of sets of the form $V(x) \cap S$ for $x \in P$. Then $\mathcal{C}$ is closed under 2-ary intersections by Lemma \[2-ary-v\]. It is closed under 0-ary intersections because if $\{a_1,\ldots,a_n\}$ is an enumeration of $S$, then every $a_i$ lies in $V(a_1 \vee \cdots \vee a_n)$. Therefore, $\mathcal{C}$ is the set of closed sets with respect to some closure operation on $S$. 2. The set $V(a_1 \vee \cdots \vee a_n) \cap S$ is closed and contains each $a_i$, so it contains the closure of $\{a_1,\ldots,a_n\}$. This shows that (\[option-a\])$\implies$(\[option-b\]). Since $a_1, \ldots, a_n$ is independent, the sequence $a_1, \ldots, a_n, b$ is non-independent if and only if $b \wedge (a_1 \vee \cdots \vee a_n) > \bot$, if and only if $b \in V(a_1 \vee \cdots \vee a_n)$. Thus (\[option-b\])$\iff$(\[option-c\]). Finally, suppose (\[option-c\]) holds. Let $x$ be an element of $P$ such that $V(x) \cap S$ equals the closure of $\{a_1,\ldots,a_n\}$. Let $a_i' = a_i \wedge x$. By definition of $V(x)$ these elements are non-zero, and $a_i'$ is a quasi-atom equivalent to $a_i$ by Proposition \[quasi-atoms-basics\]. By Lemma \[respect-equivalence\], the sequence $a_1', a_2', \ldots, a_n', b$ is *not* independent, but $a_1', a_2', \ldots, a_n'$ *is* independent. Therefore, $$\bot < b \wedge (a_1' \vee \cdots \vee a_n') \le b \wedge x$$ because $a_i' \le x$. Then $b \in V(x) \cap S$, so (\[option-a\]) holds. 3. Let $b_1, \ldots, b_m$ be the subsequence consisting of those $a_i$ which do not lie in the closure of the preceding $a_i$’s. This subsequence has the same closure as the original sequence. The sequence $\{b_1,\ldots,b_j\}$ is independent for $j \le m$, by induction on $j$, using the equivalence (\[option-a\])$\iff$(\[option-c\]) in the previous point. 4. Let $a_1, \ldots, a_n, b, c$ be elements of $S$. Suppose that $b$ is *not* in the closure of $\{a_1, \ldots, a_n\}$, and $c$ is *not* in the closure of $\{a_1,\ldots,a_n,b\}$. We must show that $b$ is *not* in the closure of $\{a_1,\ldots,a_n,c\}$. By the previous point, we may assume that the $a_i$’s are independent. In this case, the equivalence (\[option-a\])$\iff$(\[option-c\]) in (\[tfae\]) immediately implies that the sequence $\{a_1,\ldots,a_n,b\}$ is independent, and then that $\{a_1,\ldots,a_n,b,c\}$ is independent. By permutation invariance, it follows that $\{a_1,\ldots,a_n,c\}$ and $\{a_1,\ldots,a_n,c,b\}$ are independent. By the equivalence, this means that $b$ is not in the closure of $\{a_1,\ldots,a_n,c\}$. 5. We proceed by induction on $n$. For $n = 1$, note that $V(\bot) \cap S = \emptyset$ and so every sequence of length 1 is independent with respect to the pregeometry (and vacuously independent with respect to the lattice). Suppose $n > 1$. Let $a_1, \ldots, a_{n-1}, a_n$ be a sequence. We may assume that $a_1, \ldots, a_{n-1}, a_n$ is independent with respect to the pregeometry or the lattice. Either way, $a_1, \ldots, a_{n-1}$ is independent with respect to the pregeometry or the lattice, hence with respect to the lattice or the pregeometry (by induction). Then $a_1, \ldots, a_n$ is independent with respect to the pregeometry if and only if $a_n$ is not in the closure of $a_1, \ldots, a_{n-1}$. By (\[tfae\])’s equivalence, this is the same as $a_1, \ldots, a_n$ being independent with respect to the lattice. \[the-pregeometry-exists\] Let $Q$ be the set of quasi-atoms in $(P,\le)$. 1. There is a finitary pregeometry on the set $Q$ of quasi-atoms, characterized by the fact that a finite set $\{a_1,\ldots,a_n\}$ of quasi-atoms is independent with respect to the pregeometry if and only if it is independent in the lattice-theoretic sense of Definition \[def-of-ind\]. 2. If $x$ is any element of $P$, then the set $V(x) = \{a \in Q ~|~ a \wedge x > \bot\}$ is a closed subset of $Q$. 3. Two quasi-atoms $a, b$ are parallel (i.e., have the same closure) if and only if they are equivalent in the sense of Proposition \[quasi-atoms-basics\]. Consequently, there is an induced *geometry* on the equivalence classes of quasi-atoms. To specify a finitary closure operation on an infinite set $Q$, it suffices to give a closure operation on each finite subset $S \subseteq Q$, subject to the compatibility requirements that when $S' \subseteq S$, the closed sets on $S'$ are exactly the sets of the form $V \cap S'$ for $V$ a closed set on $S$. We clearly have this compatibility. Furthermore, the induced finitary closure operation on $Q$ satisfies exchange if and only if the closure operation on each finite $S \subseteq Q$ satisfies exchange. When this holds, a finite set $I$ is independent with respect to the pregeometry on $Q$ if and only if it is independent with respect to the pregeometry on $S$, for any/every finite $S$ containing $I$. Moreover, a set $V \subseteq Q$ is closed if and only if $V \cap S$ is a closed subset of $S$ for every finite $S \subseteq Q$. Therefore, the sets $V(x)$ are certainly closed. For the final point, it follows on general pregeometry grounds that two non-degenerate elements $a, b$ are parallel if and only if $\{a,b\}$ is independent. Because independence in the pregeometry agrees with lattice-theoretic independence, we see that $a, b$ are parallel if and only if they are equivalent in the sense of Proposition \[quasi-atoms-basics\]. \[3-fold-lemma\] Let $x, a, b$ be three elements of $P$, such that the sets $\{x, a\}, \{x, b\}, \{a,b\}$ are independent, but $\{x, a, b\}$ is not independent. Suppose that $a$ is a quasi-atom. Then there is a quasi-atom $w \le x$ such that $\{w, a, b\}$ is not independent (but $\{w, a\}, \{w, b\}, \{a, b\}$ are independent). Let $w = (a \vee b) \wedge x$. As $\{a, b\}$ is independent but $\{a, b, x\}$ is not, we must have $w > \bot$. As $w \le x$ it follows that $\{w, b\}$ and $\{w, a\}$ are independent. From the independence of $\{b, w\}$, there is an isomorphism of posets $$(\bot,w] \cong (b, b \vee w].$$ Now $b \vee w \le a \vee b$ because $w \le a \vee b$ (by definition) and $b \le a \vee b$. Therefore $(b, b \vee w] \subseteq [b, a \vee b]$. By the independence of $a$ and $b$, there is a poset isomorphism $$\begin{aligned} [b, a \vee b] & \stackrel{\sim}{\to} [\bot, a] \\ x & \mapsto x \wedge a. \end{aligned}$$ This isomorphism induces an isomorphism of the subposets $$(b, b \vee w] \cong (\bot, a \wedge (b \vee w)].$$ In particular, there is a chain of poset isomorphisms $$(\bot,w] \cong (b, b \vee w] \cong (\bot, a \wedge (b \vee w)].$$ The left hand side is non-empty, because $w > \bot$. Therefore, the right hand side is non-empty, and so $a' := a \wedge (b \vee w)$ is greater than $\bot$. On the other hand, $a'$ is less than or equal to the quasi-atom $a$, so $a'$ is a quasi-atom. Therefore, the right hand side is a downwards directed poset. The same must hold for the left hand side, implying that $w$ is a quasi-atom. Note that $w \wedge (a \vee b) = w > \bot$, so the sequence $\{a, b, w\}$ is not independent. \[prop:is-modular\] The pregeometry of Corollary \[the-pregeometry-exists\] is modular. \[the-previous-claim\] Let $a, b, c_1, \ldots, c_n$ be elements of $Q$, with $n > 0$. Suppose that $a \in {\operatorname{cl}}\{b, c_1, \ldots, c_n\}$. Then there is $c' \in {\operatorname{cl}}\{c_1, \ldots, c_n\}$ such that $a \in {\operatorname{cl}}\{b, c'\}$. We may assume that the $c_i$ are independent, passing to an independent subsequence otherwise[^11]. We may assume that $a \notin {\operatorname{cl}}\{b\}$, or else take $c' = c_1$. We may assume that $a \notin {\operatorname{cl}}\{c_1,\ldots,c_n\}$, or else take $c' = a$. Then $b \notin {\operatorname{cl}}\{c_1,\ldots,c_n\}$. It follows that the sequences - $\{a, b\}$ - $\{c_1,\ldots,c_n,a\}$ - $\{c_1,\ldots,c_n,b\}$ are independent, but the sequence $\{c_1, \ldots, c_n, a, b\}$ is *not* independent. Let $x = c_1 \vee \cdots \vee c_n$. Then $$\begin{aligned} a \wedge b &= \bot \\ a \wedge x &= \bot \\ b \wedge x &= \bot \end{aligned}$$ but $\{a, b, x\}$ is *not* independent. By Lemma \[3-fold-lemma\], there is a quasi-atom $w \le x$ such that $\{a, w\}, \{b, w\}, \{a, b\}$ are independent but $\{a, b, w\}$ is not. Now $w \le x = c_1 \vee \cdots \vee c_n$, so the sequence $\{c_1, \ldots, c_n, w\}$ is not independent. Therefore $w \in {\operatorname{cl}}\{c_1,\ldots,c_n\}$. The fact that $\{b, w\}$ is independent but $\{a, b, w\}$ is not implies that $a \in {\operatorname{cl}}\{b, w\}$. Take $c' = w$. Given the claim, modularity follows on abstract grounds. First, one upgrades the claim to the following: If $X, Y$ are two non-empty closed subsets of $Q$ and $a \in {\operatorname{cl}}(X \cup Y)$, then there is $x \in X$ and $y \in Y$ such that $a \in {\operatorname{cl}}\{x,y\}$. Because closure is finitary, there exist $b_1, \ldots, b_n \in X$ and $c_1, \ldots, c_m \in Y$ such that $a \in {\operatorname{cl}}\{b_1,\ldots,b_n,c_1,\ldots,c_m\}$. By repeated applications of the previous claim one can find - $p_1 \in {\operatorname{cl}}\{b_2, \ldots, b_n, c_1, \ldots, c_m\}$ such that $a \in {\operatorname{cl}}\{b_1,p_1\}$. - $p_2 \in {\operatorname{cl}}\{b_3, \ldots, b_n, c_1, \ldots, c_m\}$ such that $p_1 \in {\operatorname{cl}}\{b_2,p_2\}$. - … - $p_k \in {\operatorname{cl}}\{b_{k+1},\ldots,b_n,c_1,\ldots,c_m\}$ such that $p_{k-1} \in {\operatorname{cl}}\{b_k,p_k\}$. - … - $p_n \in {\operatorname{cl}}\{c_1,\ldots,c_m\}$ such that $p_{n-1} \in {\operatorname{cl}}\{b_n,p_n\}$. Then ${\operatorname{cl}}\{b_1, b_2, \ldots, b_n, p_n\}$ contains $p_{n-1}$, hence also $p_{n-2}, p_{n-3}, \ldots, p_2, p_1,$ and $a$. Thus $a \in {\operatorname{cl}}\{b_1,b_2,\ldots,b_n,p_n\}$. By one final application of Claim \[the-previous-claim\], there is $q \in {\operatorname{cl}}\{b_1,b_2,\ldots,b_n\}$ such that $a \in {\operatorname{cl}}\{p_n,q\}$. Take $x = q$ and $y = p_n$. Finally, let $V_1, V_2, V_3$ be closed subsets of $Q$ with $V_1 \subseteq V_2$. Write $+$ for the lattice join on the lattice of closed sets. We must show one direction of the modular equation: $$(V_1 + V_3) \cap V_2 \subseteq V_1 + (V_3 \cap V_2)$$ as the $\supseteq$ direction holds in any lattice. If $V_1$ or $V_3$ is empty, the equality is clear, so we may assume both are non-empty. Take $x \in (V_1 + V_3) \cap V_2$. By the previous claim, there are $(y,z) \in V_1 \times V_3$ such that $x \in {\operatorname{cl}}\{y,z\}$. If $x \in {\operatorname{cl}}\{y\}$, then $x \in V_1$ so certainly $x \in V_1 + (V_3 \cap V_2)$. Otherwise, $x \in {\operatorname{cl}}\{y,z\} \setminus {\operatorname{cl}}\{y\}$, so $z \in {\operatorname{cl}}\{x,y\}$. Now $x \in V_2$ by choice of $x$, and $y \in V_1 \subseteq V_2$. Therefore, $z \in {\operatorname{cl}}\{x,y\} \subseteq V_2$. The element $z$ is also in $V_3$, so $z \in (V_3 \cap V_2)$. Thus $x \in {\operatorname{cl}}\{y,z\}$ where $y \in V_1$ and $z \in V_3 \cap V_2$. It follows that $x \in V_1 + (V_3 \cap V_2)$, completing the proof of modularity. In a later paper ([@prdf3]), we will see that equivalence classes of quasi-atoms in $M$ correspond to atoms in the category ${\operatorname{Pro}}M$ obtained by formally adding filtered infima to $M$. The category ${\operatorname{Pro}}M$ is itself a modular lattice, and the modular pregeometry constructed in Corollary \[the-pregeometry-exists\] and Proposition \[prop:is-modular\] comes from the usual modular geometry on atoms. See §7.1-7.2 of [@prdf3] for details. Let $P$ be the modular lattice of type-definable subgroups of $(K,+)$ in a dp-finite field $K$. Or let $P$ be an interval inside that lattice. In these cases, the modular geometry on quasi-atoms in $P$ should consist of finitely many pieces, each of which is canonically a projective space over a division ring. In particular, - Exotic non-Desarguesian projective planes shouldn’t appear. - Even if a component has rank 1, it should still carry the structure of ${\mathbb{P}}^1(D)$ for some division ring $D$. These nice properties should hold because of the fact that we can embed $P$ into a larger lattice $P_n$ of type-definable subgroups of $K^n$. The existence of the family of $P_n$’s for $n \ge 1$ should ensure that all the connected components of the modular geometry can be embedded into connected modular geometries of arbitrarily high rank, forcing Desargues’ theorem to hold. Similar statements should hold in the abstract setting of Remark \[abelian-category-motivation\].[^12] Miscellaneous facts ------------------- \[covering-lemma-home\] Let $(P,\le)$ be a modular lattice with top and bottom elements $\top$ and $\bot$. Suppose that $P$ admits a subadditive rank. 1. The rank of the modular pregeometry on quasi-atoms is at most the reduced rank ${\operatorname{rk}_0}(\top/\bot)$. In particular, it is finite. 2. Every closed set is of the form $V(x)$ for some $x$. 3. Let $x$ be any element of $P$, and $y_1, \ldots, y_n$ be a basis for $V(x)$. Let $z_1, \ldots, z_m$ be a sequence of quasi-atoms. Then $\{x, z_1, \ldots, z_m\}$ is independent if and only if $$\{y_1, \ldots, y_n, z_1, \ldots, z_m\}$$ is independent. 4. \[covering-lemma\] If $x$ is any element of $P$, there is a pregeometry basis of the form $y_1, \ldots, y_n, z_1, \ldots, z_m$ where - Each $y_i \le x$. - The set $\{y_1,\ldots,y_n\}$ is a basis for $V(x)$. - The set $\{x, z_1, \ldots, z_m\}$ is independent. <!-- --> 1. If the pregeometry rank is at least $n$, then there is a sequence of independent quasi-atoms $a_1, \ldots, a_n$. Each $a_i$ is greater than $\bot$, so this determines a strict $n$-cube by Proposition \[cubes-and-independence\]. 2. Let $V$ be a closed set. Because the pregeometry rank is finite, we can find a basis $a_1, \ldots, a_n$ for $V$. Note that $\{a_1,\ldots,a_n\}$ is an independent set. Let $x = a_1 \vee \cdots \vee a_n$. If $b$ is any element of $Q$, then the following statements are equivalent - $b$ is in $V$. - $b$ is in the closure of $\{a_1,\ldots,a_n\}$. - The set $\{a_1,\ldots,a_n,b\}$ is *not* independent. - $b \wedge x > \bot$. - $b \in V(x)$. Therefore $V = V(x)$. 3. Let $y'_i = y_i \wedge x$. By definition of $V(x)$, each $y'_i > \bot$, and so $y'_i$ is a quasi-atom equivalent to $y_i$. By Lemma \[respect-equivalence\] we may replace $y_i$ with $y'_i$ and assume that each $y_i \le x$. Let $x' = y_1 \vee \cdots \vee y_n \le x$. If $\{x, z_1, \ldots, z_m\}$ is independent, then so is $\{x', z_1, \ldots, z_m\}$. As the $y_i$ are independent, it follows by Proposition \[1-collapse\] that $\{y_1, \ldots, y_n, z_1, \ldots, z_m\}$ is independent. Conversely, suppose that $\{y_1,\ldots,y_n,z_1,\ldots,z_m\}$ is independent, but $\{x,z_1,\ldots,z_m\}$ is not. Choose $m$ minimal such that this occurs. Let $b = z_1 \vee \cdots \vee z_{m-1}$. Then $(x \vee b) \wedge z_m > \bot$, but $$\begin{aligned} x \wedge b &= \bot \\ b \wedge z_m &= \bot \\ x \wedge z_m &= \bot \end{aligned}$$ respectively: by choice of $m$; because $\vec{z}$ is independent; and because $z_m$ is independent from $y_1,\ldots,y_n$ hence not in ${\operatorname{cl}}\{y_1,\ldots,y_n\} = V(x)$. By Lemma \[3-fold-lemma\] there is a quasi-atom $w \le x$ such that $\{w, b, z_m\}$ is not independent. But $w \in V(x)$, so $w \in {\operatorname{cl}}\{y_1,\ldots,y_n\}$. The fact that $\{y_1,\ldots,y_n,z_1,\ldots,z_{m-1},z_m\}$ is independent then implies on pregeometry-theoretic grounds that $\{w,z_1,\ldots,z_{m-1},z_m\}$ is independent, which implies on lattice-theoretic grounds that $\{w, b, z_m\}$ is independent, a contradiction. 4. Take $\{y_1, \ldots, y_n\}$ a basis for $V(x)$ and extend it to a basis $\{y_1,\ldots,y_n,z_1,\ldots,z_m\}$ for the entire pregeometry. As in the proof of the previous point, we may replace $y_i$ with $y_i \wedge x$, and thus assume that $y_i \le x$. By the previous point, independence of $\{y_1,\ldots,y_n,z_1,\ldots,z_m\}$ implies independence of $\{x,z_1,\ldots,z_m\}$. \[bound-on-independence\] Let $(P,\le)$ be a modular lattice with bottom element $\bot$ and reduced rank $n < \infty$. The pregeometry on quasi-atoms has rank at most $n$. If $a_1, \ldots, a_m$ is a basis, then the following facts hold: 1. Any independent sequence $\{b_1, \ldots, b_\ell\}$ of non-$\bot$ elements has size $\ell \le m$. 2. If $\ell = m$ then each $b_i$ is a quasi-atom. <!-- --> 1. For each $b_i$, we may find a quasi-atom $b'_i \le b_i$, by Proposition \[quasi-atoms-basics\].\[enough-quasiats\]. The sequence $b'_1, \ldots, b'_\ell$ is pregeometry-independent, so $\ell \le m$. 2. Suppose $\ell = m$. If, say, $b_\ell$ is not a quasi-atom, we can find $x, y \in (\bot,b_\ell]$ such that $x \wedge y = \bot$. Replacing $b_\ell$ with $x \vee y \le b_\ell$, and then applying Proposition \[1-collapse\], we see that $b_1, \ldots, b_{\ell - 1}, x, y$ is an independent sequence contradicting part 1. If $(P,\le)$ is a modular lattice and $a \ge b$ are elements, we define ${\operatorname{rk}_\bot}(a/b)$ to be the supremum of $n$ such that there is a strict $n$-cube in $[b,a]$ with bottom $b$. Equivalently (by Proposition \[cubes-and-independence\]), ${\operatorname{rk}_\bot}(a/b)$ is the supremum over $n$ such that there exist $c_1, \ldots, c_n \in (b,a]$ relatively independent over $b$. The rank ${\operatorname{rk}_\bot}(-)$ is not a subadditive rank in the sense of Definition \[def-of-subadd-rank\]. \[botrks\] If $a \ge b$, then 1. \[botrk-vs-redrk\] ${\operatorname{rk}_\bot}(a/b) \le {\operatorname{rk}_0}(a/b)$ 2. If ${\operatorname{rk}_0}(a/b) < \infty$, there exists $b' \in [b,a]$ such that ${\operatorname{rk}_\bot}(a/b') = {\operatorname{rk}_0}(a/b')$. Indeed, take a strict $n$-cube in $[b,a]$ for maximal $n$ and let $b'$ be the bottom of the cube. 3. If ${\operatorname{rk}_0}(a/b) < \infty$, then ${\operatorname{rk}_\bot}(a/b)$ is the rank of the pregeometry of quasi-atoms in $[b,a]$, by Remark \[bound-on-independence\]. \[botrk-superadd\] For any $a, b$ $${\operatorname{rk}_\bot}(a \vee b / a \wedge b) \ge {\operatorname{rk}_\bot}(a/a \wedge b) + {\operatorname{rk}_\bot}(b/a \wedge b).$$ Also, ${\operatorname{rk}_\bot}(a \vee b/a) = {\operatorname{rk}_\bot}(b/a \wedge b)$. If $c_1, \ldots, c_n$ is a strict independent sequence in $[a \wedge b, a]$ and $d_1, \ldots, d_m$ is a strict independent sequence in $[a \wedge b, b]$, then $c_1, \ldots, c_n, d_1, \ldots, d_m$ is a strict independent sequence in $[a \wedge b, a \vee b]$ by a couple applications of Proposition \[1-collapse\]. The “also” clause is unrelated and follows immediately from the isomorphism of lattices between $[a, a \vee b]$ and $[a \wedge b, b]$. Invariant valuation rings {#sec:valuations} ========================= Let $({\mathbb{M}},+,\cdot,\ldots)$ be a monster-model finite dp-rank expansion of a field. Assume that ${\mathbb{M}}$ is not of finite Morley rank. Fix a small model $M_0$ large enough for Corollary \[big-enough-model\] to apply. Thus, for any type-definable $M_0$-linear subspace $J \le {\mathbb{M}}^n$, we have $J = J^{00}$. Let $\mathcal{P}_n$ be the poset of type-definable $M_0$-linear subspaces of ${\mathbb{M}}^n$, let $\mathcal{P} = \mathcal{P}_1$, and let $\mathcal{P}^+$ be the poset of non-zero elements of $\mathcal{P}$. We collect the basic facts about these posets in the following proposition: \[those-posets\]   1. For each $n$, $\mathcal{P}_n$ is a bounded lattice. 2. \[p-nontriviality\] For any small model $M \supseteq M_0$, the group $I_M$ is an element of $\mathcal{P}$. In particular, $\mathcal{P}$ contains an element other than $\bot = 0$ and $\top = {\mathbb{M}}$. 3. If $J \in \mathcal{P}$ is non-zero, every definable set $D$ containing $J$ is heavy. 4. \[infinitesimals-below\] If $J \in \mathcal{P}^+$ is $M$-definable for some small model $M \supseteq M_0$, then $J \supseteq I_M$. 5. If $J \in \mathcal{P}_n$, then $J = J^{00}$. 6. \[nonzero-intersect\] $\mathcal{P}^+$ is a sublattice of $\mathcal{P}$, i.e., it is closed under intersection. Thus, $\mathcal{P}^+$ is a bounded-above lattice. 7. \[p-plus-vs-p\] $\mathcal{P}$ has reduced rank $r$ for some $0 < r \le {\operatorname{dp-rk}}({\mathbb{M}})$. The reduced rank of $\mathcal{P}^+$ is also $r$, and the reduced rank of $\mathcal{P}^n$ is $rn$. <!-- --> 1. Clear—the lattice operations are given by $$\begin{aligned} G \vee H &= G + H \\ G \wedge H &= G \cap H \\ \bot &= 0 \\ \top &= {\mathbb{M}}^n. \end{aligned}$$ 2. By Remark \[basic-infs\] and Theorem \[inf-add\], $I_M$ is a non-zero $M_0$-linear subspace of ${\mathbb{M}}$, distinct from $0$ and ${\mathbb{M}}$. 3. Replacing $J$ with $a \cdot J$ for some $a \in {\mathbb{M}}^\times$, we may assume $1 \in J$. Then $M_0 = M_0 \cdot 1 \subseteq J$ by $M_0$-linearity. Let $D$ be any definable set containing $J$. By Lemma \[something-similar-2\] with $Z = {\mathbb{M}}$ and $W = D$, the set $D$ is heavy. 4. Corollary \[minimal-heavy-subgroup\]. 5. By choice of $M_0$. 6. Let $J_1, J_2$ be two non-zero elements of $\mathcal{P}$. Let $M$ be a small model containing $M_0$, over which both $J_1$ and $J_2$ are type-definable. Then $J_1 \cap J_2 \ge I_M > \bot$. Therefore $\mathcal{P}^+$ is closed under intersection. 7. Proposition \[reduced-rank-vs-dp-rk\] gives the bound $r \le n$. Then $$0 < {\operatorname{rk}_0}(\mathcal{P}^+) \le {\operatorname{rk}_0}(\mathcal{P}) = r \le n,$$ where the left inequality is sharp because $\mathcal{P}$ has at least three elements by part \[p-nontriviality\]. If $r > {\operatorname{rk}_0}(\mathcal{P}^+)$, there is a strict $r$-cube in $\mathcal{P}$ which does not lie in $\mathcal{P}^+$. The bottom of this cube must be $\bot$, the only element of $\mathcal{P} \setminus \mathcal{P}^+$. Then $r \le {\operatorname{rk}_\bot}(\mathcal{P})$. However, part \[nonzero-intersect\] says ${\operatorname{rk}_\bot}(\mathcal{P}) \le 1$, so $r \le 1 \le {\operatorname{rk}_0}(\mathcal{P}^+)$, a contradiction. Therefore ${\operatorname{rk}_0}(\mathcal{P}^+) = r = {\operatorname{rk}_0}(\mathcal{P})$. Finally, in $\mathcal{P}_n$, if we let $J_i = 0^{\oplus (i - 1)} \oplus {\mathbb{M}}\oplus 0^{\oplus (n - i)}$ for $i = 1, \ldots, n$, then the sequence $J_1, \ldots, J_n$ is independent and $J_1 \vee \cdots \vee J_n = {\mathbb{M}}^n$. Thus $${\operatorname{rk}_0}({\mathbb{M}}^n/0) = \sum_{i = 1}^n {\operatorname{rk}_0}(J_i/0)$$ by Remark \[subadditive-rk-in-cubes\]. However, ${\operatorname{rk}_0}(J_i/0) = r$ because of the isomorphism of lattices $$\begin{aligned} \mathcal{P} &\to [0,J_i] \\ X &\mapsto 0^{\oplus(i-1)} \oplus X \oplus 0^{\oplus(n-i)}. \qedhere \end{aligned}$$ In what follows, we will let $r$ be ${\operatorname{rk}_0}({\mathbb{M}}/0)$. If $r = 1$, then $\mathcal{P}$ is totally ordered and we can reuse the arguments for dp-minimal fields to immediately see that $I_M$ is a valuation ideal. Usually we are not so lucky. Special groups {#sec:special} -------------- An element $J \in \mathcal{P}_n$ is *special* if ${\operatorname{rk}_\bot}({\mathbb{M}}^n/J) = {\operatorname{rk}_0}({\mathbb{M}}^n/J) = rn$. Equivalently, $J \in \mathcal{P}_n$ is special if ${\operatorname{rk}_\bot}({\mathbb{M}}^n/J) \ge rn$. This follows by Proposition \[those-posets\].\[p-plus-vs-p\] and Remark \[botrks\].\[botrk-vs-redrk\]. \[special-proposition\]   1. There is at least one non-zero special $J \in \mathcal{P} = \mathcal{P}_1$. 2. \[guards\] Let $J \in \mathcal{P}$ be special. Let $A_1, \ldots, A_r$ be a basis of quasi-atoms over $J$. Let $G \in \mathcal{P}$ be arbitrary. If $G \cap A_i \not\subseteq J$ for each $i$, then $G \supseteq J$. 3. \[guard-application\] If $J \in \mathcal{P}$ is special and nonzero and type-definable over a small model $M \supseteq M_0$, then $$I_M \cdot J \subseteq I_M \subseteq J$$ 4. \[oplus\] If $I \in \mathcal{P}_n$ and $J \in \mathcal{P}_m$ are special, then $I \oplus J \in \mathcal{P}_{n+m}$ is special. 5. \[special-scale\] If $I \in \mathcal{P}_n$ is special and $\alpha \in {\mathbb{M}}^\times$, then $\alpha \cdot I$ is special. <!-- --> 1. By Proposition \[those-posets\].\[p-plus-vs-p\] the reduced rank of $\mathcal{P}^+$ is exactly $r$, so we can find a strict $r$-cube in $\mathcal{P}^+$. The base of such a cube is a non-zero special element of $\mathcal{P}$. 2. For $i = 1, \ldots, r$ take $a_i \in (G \cap A_i) \setminus J$. Then for each $i$, we have $$\begin{aligned} & a_i \in G \cap A_i \subseteq (G + J) \cap A_i \supseteq J \cap A_i = J \\ & a_i \notin J \\ (\implies) & (G + J) \cap A_i \supsetneq J. \end{aligned}$$ Consequently, $$A_1 \cap (G + J), A_2 \cap (G + J), \ldots, A_r \cap (G + J)$$ is a strict independent sequence over $J$. It follows that $${\operatorname{rk}_0}(G/(G \cap J)) = {\operatorname{rk}_0}((G+J)/J) \ge {\operatorname{rk}_\bot}((G+J)/J) \ge r.$$ On the other hand $${\operatorname{rk}_0}(G/(G \cap J)) + {\operatorname{rk}_0}(J/(G \cap J)) = {\operatorname{rk}_0}((G+J)/(G \cap J)) \le r,$$ and so ${\operatorname{rk}_0}(J/(G \cap J)) = 0$. This forces $J = G \cap J$, so $J \subseteq G$. 3. The inclusion $I_M \subseteq J$ is Proposition \[those-posets\].\[infinitesimals-below\]. Let $A_1, \ldots, A_r$ be a basis of quasi-atoms over $J$ as in part \[guards\]. For each $A_i$ choose an element $a_i \in A_i\setminus J$. Let $M'$ be a small model containing $M$ and the $a_i$’s. We first claim that $I_{M'} \cdot J \subseteq I_{M'}$. Let $\varepsilon$ be a non-zero element of $I_{M'}$. As $I_{M'}$ is closed under multiplication by $(M')^\times$, we have $M' \subseteq \varepsilon^{-1} \cdot I_{M'}$. In particular, $a_i \in \varepsilon^{-1} \cdot I_{M'}$ for each $i$. Then $$(\varepsilon^{-1} \cdot I_{M'}) \cap A_i \not \subseteq J$$ for each $i$, so by part \[guards\] we have $$\varepsilon^{-1} \cdot I_{M'} \supseteq J.$$ In other words, $\varepsilon \cdot J \subseteq I_{M'}$. As $\varepsilon$ was an arbitrary non-zero element of $I_{M'}$, it follows that $I_{M'} \cdot J \subseteq I_{M'}$. Now suppose that $D$ is an $M$-definable basic neighborhood. Then $D$ is an $M'$-definable basic neighborhood. By the above and compactness, there is an $M'$-definable basic neighborhood $X {-_\infty}X$ and a definable set $D_2 \supseteq J$ such that $(X {-_\infty}X) \cdot D_2 \subseteq D$. Furthermore, $D_2$ can be taken to be $M$-definable, because $J$ is a directed intersection of $M$-definable sets. Having done this, we can then pull the parameters defining $X$ into $M$, and assume that $X$ is $M$-definable. (This uses the fact that heaviness is definable in families). Then we have an $M$-definable basic neighborhood $X {-_\infty}X$ and an $M$-definable set $D_2 \supseteq J$ such that $(X {-_\infty}X) \cdot D_2 \subseteq D$. As $D$ was arbitrary, it follows that $$I_M \cdot J \subseteq I_M.$$ 4. The interval $[I,{\mathbb{M}}^n]$ in $\mathcal{P}_n$ is isomorphic to $[I \oplus J, {\mathbb{M}}^n \oplus J]$ in $\mathcal{P}_{n+m}$, so $$\begin{aligned} rn = {\operatorname{rk}_\bot}({\mathbb{M}}^n/I) &= {\operatorname{rk}_\bot}(({\mathbb{M}}^n \oplus J)/(I \oplus J)) \\ rm = {\operatorname{rk}_\bot}({\mathbb{M}}^m/J) &= {\operatorname{rk}_\bot}((I \oplus {\mathbb{M}}^m)/(I \oplus J)), \end{aligned}$$ where the second line is true for similar reasons. By Lemma \[botrk-superadd\], $${\operatorname{rk}_\bot}(({\mathbb{M}}^n \oplus {\mathbb{M}}^m)/(I \oplus J)) \ge rn + rm.$$ On the other hand $${\operatorname{rk}_\bot}(({\mathbb{M}}^n \oplus {\mathbb{M}}^m)/(I \oplus J)) \le {\operatorname{rk}_0}(({\mathbb{M}}^n \oplus {\mathbb{M}}^m)/(I \oplus J)) \le r(n+m)$$ so equality holds and $I \oplus J$ is special. 5. For any $\alpha \in {\mathbb{M}}^\times$, the map $X \mapsto \alpha \cdot X$ is an automorphism of $\mathcal{P}_n$. \[ring-topology\] For any model $M$, $I_M \cdot I_M \subseteq I_M$. The topology in Remark \[exists-topology\] is a ring topology, not just a group topology. First suppose $M \supseteq M_0$. Let $J$ be a non-zero special element of $\mathcal{P}_1$. Then $I_M \subseteq J$, and so $$I_M \cdot I_M \subseteq I_M \cdot J \subseteq I_M$$ by Proposition \[special-proposition\].\[guard-application\]. Then we can shrink $M$ using the technique of the proof of Proposition \[special-proposition\].\[guard-application\]. With much more work, one can show that the canonical topology is a field topology. See Corollary 5.15 in [@prdf2]. Say that $J \in \mathcal{P}_1$ is *bounded* if $J \le J'$ for some special $J'$. Based on the argument in Proposition \[special-proposition\].\[guards\]-\[guard-application\], it seems that $J$ is bounded if and only if $\alpha \cdot J \subseteq I_M$ for some $\alpha \in {\mathbb{M}}^\times$ and some small model $M$. Bounded elements should form a sublattice of $\mathcal{P}_1$.[^13] Let $I \in \mathcal{P}_n$ be special and $D \in \mathcal{P}_n$ be arbitrary. Then $D$ *dominates* $I$ if $D \ge I$ and ${\operatorname{rk}_\bot}(D/I) = nr$. \[characterization-of-domination\] Let $J \in \mathcal{P}_n$ be special, let $A_1, \ldots, A_{nr}$ be a basis of quasi-atoms in $[J,{\mathbb{M}}^n]$, and $D$ be arbitrary. Then $D$ dominates $J$ if and only if $D \cap A_i \supsetneq J$ for each $i$. In particular, this condition doesn’t depend on the choice of the basis $\{A_1,\ldots,A_{nr}\}$. Suppose $D$ dominates $J$. Let $B_1, \ldots, B_{nr}$ be a basis of quasi-atoms in $[J,D]$. The $B_i$ are independent quasi-atoms in the larger interval $[J,{\mathbb{M}}^n]$, so $\{B_1,\ldots,B_{nr}\}$ is another basis of quasi-atoms in $[J,{\mathbb{M}}^n]$. Therefore, for every $i$ the sequence $\{B_1,\ldots,B_{nr},A_i\}$ is *not* independent over $J$. Consequently $$D \cap A_i \supseteq (B_1 + \cdots + B_{nr}) \cap A_i \supsetneq J.$$ Conversely, suppose $D \cap A_i \supsetneq J$ for each $i$. Then certainly $D \supseteq J$, and it remains to show ${\operatorname{rk}_\bot}(D/J) \ge nr$. Let $A'_i := D \cap A_i$. Then each $A'_i$ is a quasi-atom over $J$, equivalent to $A_i$, and so $\{A'_1,\ldots,A'_{nr}\}$ is another basis of quasi-atoms over $J$. As each $A'_i$ lies in $[J,D]$, it follows that ${\operatorname{rk}_\bot}(D/J) \ge nr$. \[special-lemma-1\] Let $I \in \mathcal{P}_n$ be special, and $V$ be a $k$-dimensional ${\mathbb{M}}$-linear subspace of ${\mathbb{M}}^n$. Then $$\begin{aligned} {\operatorname{rk}_\bot}((V + I)/I) &= {\operatorname{rk}_0}((V + I)/I) = kr \\ {\operatorname{rk}_\bot}(V/(V \cap I)) &= {\operatorname{rk}_0}(V/(V \cap I)) = kr \\ {\operatorname{rk}_\bot}({\mathbb{M}}^n/(V + I)) &= {\operatorname{rk}_0}({\mathbb{M}}^n/(V + I)) = (n-k)r. \end{aligned}$$ Moreover, there exist $A_1, \ldots, A_{kr}, B_1, \ldots, B_{(n-k)r} \in \mathcal{P}_n$ such that the following conditions hold: 1. The set $\{A_1, \ldots, A_{kr}, B_1, \ldots, B_{(n-k)r}\}$ is a basis of quasi-atoms in $[I, {\mathbb{M}}^n]$. 2. Let $\tilde{A}_i = A_i \cap V$. Then $\{\tilde{A}_1, \ldots, \tilde{A}_{kr}\}$ is a basis of quasi-atoms in $[V \cap I, V]$. 3. Let $\tilde{B}_i = B_i + V$. Then $\{\tilde{B}_1, \ldots, \tilde{B}_{(n-k)r}\}$ is a basis of quasi-atoms in $[V + I, {\mathbb{M}}^n]$. Given a $D \in \mathcal{P}_n$ dominating $I$, we may choose the $A_i$ and $B_i$ to lie in $[I,D]$. Let $W$ be a complementary $(n-k)$-dimensional ${\mathbb{M}}$-linear subspace, so that $V + W = {\mathbb{M}}^n$. Let $V' = V + I$ and $W' = W+I$. Then $$\begin{aligned} nr = {\operatorname{rk}_0}({\mathbb{M}}^n/I) & \le {\operatorname{rk}_0}({\mathbb{M}}^n/V') + {\operatorname{rk}_0}(V'/I) \\ & = {\operatorname{rk}_0}((V'+W')/V') + {\operatorname{rk}_0}(V'/I) \\ & = {\operatorname{rk}_0}(W'/(W' \cap V')) + {\operatorname{rk}_0}(V'/I) \\ & \le {\operatorname{rk}_0}(W'/I) + {\operatorname{rk}_0}(V'/I) \\ & = {\operatorname{rk}_0}(W/(W \cap I)) + {\operatorname{rk}_0}(V/(V \cap I)) \\ & \le {\operatorname{rk}_0}(W/0) + {\operatorname{rk}_0}(V/0). \end{aligned}$$ Now any ${\mathbb{M}}$-linear isomorphism $\phi : {\mathbb{M}}^k \stackrel{\sim}{\to} V$ induces an isomorphism of posets from $\mathcal{P}_k$ to $[0,V] \subseteq \mathcal{P}_n$, so $$\begin{aligned} {\operatorname{rk}_0}(V/0) &= {\operatorname{rk}_0}(\mathcal{P}_k) = kr \\ {\operatorname{rk}_0}(W/0) &= {\operatorname{rk}_0}(\mathcal{P}_{n-k}) = (n-k)r, \end{aligned}$$ where the second line follows similarly. Therefore the inequalities above are all equalities, and $$\begin{aligned} {\operatorname{rk}_0}((V+I)/I) &= {\operatorname{rk}_0}(V/(V \cap I)) = kr \\ {\operatorname{rk}_0}({\mathbb{M}}^n/(V+I)) &= {\operatorname{rk}_0}({\mathbb{M}}^n/V') = (n-k)r. \end{aligned}$$ By Proposition \[covering-lemma-home\].\[covering-lemma\], there is a basis $\{A_1, \ldots, A_m, B_1, \ldots, B_{nr-m}\}$ of independent quasi-atoms in $[I, {\mathbb{M}}^n]$ such that - Each $A_i \subseteq V+I$. - The sequence $(V+I), B_1, \ldots, B_{nr-m}$ is independent over $I$. Given a bound $D$ dominating $I$, we may replace each $A_i$ with $A_i \cap D$ and $B_i$ with $B_i \cap D$, and assume henceforth that $A_i, B_i \subseteq D$. By Remark \[subadditive-rk-in-cubes\], $$\begin{aligned} nr = {\operatorname{rk}_0}({\mathbb{M}}^n/I) & \ge {\operatorname{rk}_0}((V+I)/I) + {\operatorname{rk}_0}(B_1/I) + \cdots + {\operatorname{rk}_0}(B_{nr-m}/I) \\ & = kr + {\operatorname{rk}_0}(B_1/I) + \cdots + {\operatorname{rk}_0}(B_{nr-m}/I). \end{aligned}$$ Each $B_i$ is strictly greater than $I$, so $$nr \ge kr + nr - m,$$ and thus $m \ge kr$. On the other hand, the set $\{A_1, \ldots, A_m\}$ is a set of independent quasi-atoms in $[I, V + I]$, so $$m \le {\operatorname{rk}_\bot}((V+I)/I) \le {\operatorname{rk}_0}((V+I)/I) = kr.$$ Thus equality holds, $m = kr$, and the set $\{A_1, \ldots, A_m\}$ is a basis of quasi-atoms in $[I, V + I]$. Applying the isomorphism $$\begin{aligned} [I, V + I] & \stackrel{\sim}{\to} [V \cap I, V] \\ X & \mapsto X \cap V \end{aligned}$$ shows that the $\tilde{A}_i$ form a basis of quasi-atoms in $[V \cap I, V]$. Next, let $Q = B_1 \vee \cdots \vee B_{(n-k)r}$. (Note that $nr - m = (n-k)r$.) The fact that $(V+I), B_1, \ldots, B_{(n-k)r}$ is independent over $I$ implies that $(V+I) \cap Q = I$. Therefore, there is an isomorphism $$\begin{aligned} [I,Q] &\stackrel{\sim}{\to} [V+I,V+I+Q] \\ X & \mapsto X + (V + I) = X + V. \end{aligned}$$ The elements $\{B_1, \ldots, B_{(n-k)r}\}$ are independent quasi-atoms in $[I,Q]$, and therefore the $\tilde{B}_i$ are a set of independent quasi-atoms in $[V+I,V+I+Q]$ or even in $[V+I,{\mathbb{M}}^n]$. It follows that $$(n-k)r \le {\operatorname{rk}_\bot}({\mathbb{M}}^n/(V+I)) \le {\operatorname{rk}_0}({\mathbb{M}}^n/(V+I)) = (n-k)r,$$ so equality holds and the $\tilde{B}_i$ are a basis of quasi-atoms in $[V+I,{\mathbb{M}}^n]$. \[r-fold-specs\] Fix a special element $I$ of $\mathcal{P}_1$. For every $n$, $I^n$ is a special element of $\mathcal{P}_n$. Let $\mathcal{G}_n$ be the modular geometry associated to the pregeometry of quasi-atoms over $I^n$ in $\mathcal{P}_n$. By utilizing the embeddings between the $\mathcal{G}_n$, one should be able to prove the following: there is an object $T$ in a semi-simple abelian category $\mathcal{C}$ such that $T$ has length $r$, and for every $n$ the geometry $\mathcal{G}_n$ is the geometry of atoms in the lattice of subobjects of $T^n$. The category $\mathcal{C}$ should be enriched over $M_0$-vector spaces. For every $n$ there is a map $f_n$ from the lattice of ${\mathbb{M}}$-linear subspaces of ${\mathbb{M}}^n$ to the lattice of subobjects of $T^n$, defined in such a way that $f_n(V)$ corresponds to the closed set cut out by $V + I^n$ in the pregeometry of quasi-atoms in $[I^n,{\mathbb{M}}^n]$. These maps $f_n$ should have the following properties: - $V \le W$ should imply $f_n(V) \le f_n(W)$. - The length of $f_n(V)$ should be $r$ times the dimension of $V$, essentially by Lemma \[special-lemma-1\]. In particular, $f_1(0) = 0$ and $f_1({\mathbb{M}}) = T$. - For each $n$, the map $f_n$ should be $GL_n(M_0)$-equivariant. - $f_n(V) \oplus f_m(W)$ should equal $f_{n+m}(V \oplus W)$. When $r = 1$, this should secretly amount to a valuation on ${\mathbb{M}}$. Indeed, $T$ is then a simple object so we may assume that $\mathcal{C}$ is the category of finite-dimensional $D$-modules for some skew field $D$ extending $M_0$, and $T$ is just $D^1$. Then the above configuration should necessarily come from a map ${\mathbb{M}}\to k \to D$ where ${\mathbb{M}}\to k$ is a valuation/place, and $k \to D$ is an inclusion. The hope was that similar configurations for $r > 1$ should come from multi-valued fields, but several things go wrong. 1. The idea is to recover a valuation on ${\mathbb{M}}$ where the residue of $a$ is $\alpha \in End(T)$ if $$f_2(\{(x,ax) ~|~ x \in {\mathbb{M}}\}) = \{(x, \alpha x) ~|~ x \in T\}.$$ This line of thinking is what underlies the ring $R_J$ and ideal $I_J$ in Proposition \[rings-and-ideals\] below–$R_J$ is the set of $a$ such that the left hand side is the graph of any endomorphism of $T$ and $I_J$ is the set of $a$ such that the left hand side is the graph of $0$. 2. Ideally, we would get $R_J$ equal to a Bezout domain with finitely many maximal ideals. This amounts to the requirement that for any $\alpha \in {\mathbb{M}}$ and any distinct $q_0, \ldots, q_n \in M_0$, at least one of $1/(\alpha - q_i)$ lies in $R_J$. 3. However, this fails when the system of maps looks like $$\left(V \le ({\mathbb{Q}}(\sqrt{2}))^n\right) \mapsto (V + V^\dag, V \cap V^\dag),$$ where the right hand side should be thought of as an element of $FinVec({\mathbb{Q}}) \times FinVec({\mathbb{Q}})$, and $V^\dag$ is the image of $V$ under the non-trivial element of ${\operatorname{Aut}}({\mathbb{Q}}(\sqrt{2})/{\mathbb{Q}})$. 4. There is a certain way to mutate $f$ that improves the situation, namely $$f'_n(V) := f_{2n}(\{(\vec{x},a \cdot \vec{x}) ~|~ \vec{x} \in V\})$$ for well-chosen $a$, e.g., $a = \sqrt{2}$. This mutation is what underlies Lemma \[twists\] below. 5. Originally, I had hoped that finitely many mutations would reduce to the “good” Bezout case, but there are genuine type-definable $J$ in dp-finite fields for which this fails. For example, if $\Gamma$ is the group ${\mathbb{Q}}(1/3)$ and $K$ is ${\mathcal{C}}((\Gamma))$, we can make $K$ have dp-rank 2 by adding a predicate for the subfield ${\mathcal{C}}((2 \cdot \Gamma))$. Any Hahn series $x \in K$ can be decomposed as $x_{odd} + x_{even}$ where $x_{odd}$ has the terms with odd exponents and $x_{even}$ has the terms with even exponents. The decomposition is definable. Taking $J$ to be the set of $x$ such that ${\operatorname{val}}(x_{even}) > 0$ and ${\operatorname{val}}(x_{odd}) \ge 1$ (or something similar), one arrives at a situation where $R_J$ and $I_J$ cannot be fixed via any finite sequence of mutations. 6. Nevertheless, one can still recover a good Bezout domain in the limit, which is Theorem \[bezout-theorem\]. Unfortunately, the resulting Bezout domain is not closely tied to the infinitesimals—not enough to run the henselianity machine from the dp-minimal case. For details on the above construction, see [@prdf3]. We now return to actual proofs. \[special-lemma-2\] Let $I, J \in \mathcal{P}_n$ be special. Then $I + J$ and $I \cap J$ are special. Furthermore, there exists - a basis of quasi-atoms $\hat{A}_1, \ldots, \hat{A}_n$ in $[I \cap J, {\mathbb{M}}^n]$, - a basis of quasi-atoms $\hat{B}_1, \ldots, \hat{B}_n$ in $[I + J, {\mathbb{M}}^n]$, and - a basis of quasi-atoms $A_1, \ldots, A_n, B_1, \ldots, B_n$ in $[I \oplus J, {\mathbb{M}}^{2n}]$ related as follows: $$\begin{aligned} \hat{A}_i & = \{ \vec{x} \in {\mathbb{M}}^n ~|~ (\vec{x},\vec{x}) \in A_i\} \\ \hat{B}_i & = \{ \vec{x} - \vec{y} ~|~ (\vec{x},\vec{y}) \in B_i\}. \end{aligned}$$ Given $D \in \mathcal{P}_{2n}$ dominating $I \oplus J$, we may choose the $A_i$ and $B_i$ to lie in $[I \oplus J, D]$. For any $J \in \mathcal{P}_n$, define $$\begin{aligned} \Delta(C) &= \{(\vec{x},\vec{x}) ~|~ \vec{x} \in C\} \in \mathcal{P}_{2n} \\ \nabla(C) &= \{(\vec{x}, \vec{x} + \vec{y}) ~|~ \vec{x} \in {\mathbb{M}}^n, ~ \vec{y} \in C\} \in \mathcal{P}_{2n}. \end{aligned}$$ Let $V = \Delta({\mathbb{M}}^n) = \nabla(0)$. The maps $\Delta(-), \nabla(-)$ yield isomorphisms $$\begin{aligned} \Delta : \mathcal{P}_n &\stackrel{\sim}{\to} [0,V] \subseteq \mathcal{P}_{2n} \\ \nabla : \mathcal{P}_n &\stackrel{\sim}{\to} [V,{\mathbb{M}}^{2n}] \subseteq \mathcal{P}_{2n}. \end{aligned}$$ Indeed, the inverses are given by $$\begin{aligned} \Delta^{-1} : [0, V] & \stackrel{\sim}{\to} [0, {\mathbb{M}}^n] \\ C & \mapsto \{\vec{x} ~|~ (\vec{x},\vec{x}) \in C\} \\ \nabla^{-1} : [V, {\mathbb{M}}^{2n}] & \stackrel{\sim}{\to} [0, {\mathbb{M}}^n] \\ C & \mapsto \{\vec{x} - \vec{y} ~|~ (\vec{x},\vec{y}) \in C\}. \end{aligned}$$ Note that $\Delta^{-1}(V \cap (I \oplus J)) = I \cap J$ and $\nabla^{-1}(V + (I \oplus J)) = I + J$. Therefore, $\Delta^{-1}$ and $\nabla^{-1}$ restrict to isomorphisms $$\begin{aligned} \Delta^{-1} : [V \cap (I \oplus J), V] & \stackrel{\sim}{\to} [I \cap J, {\mathbb{M}}^n] \\ \nabla^{-1} : [V + (I \oplus J), {\mathbb{M}}^{2n}] & \stackrel{\sim}{\to} [I + J, {\mathbb{M}}^n]. \end{aligned}$$ It follows that $$\begin{aligned} {\operatorname{rk}_\bot}({\mathbb{M}}^n/(I \cap J)) &= {\operatorname{rk}_\bot}(V/(V \cap (I \oplus J))) = {\operatorname{rk}_\bot}((V + (I \oplus J))/V) \\ {\operatorname{rk}_\bot}({\mathbb{M}}^n/(I + J)) &= {\operatorname{rk}_\bot}({\mathbb{M}}^{2n}/(V + (I \oplus J))/V). \end{aligned}$$ Now $I \oplus J$ is special in $\mathcal{P}_{2n}$ by Proposition \[special-proposition\].\[oplus\], and $V$ is an $n$-dimensional ${\mathbb{M}}$-linear subspace of ${\mathbb{M}}^{2n}$, so by Lemma \[special-lemma-1\], $$\begin{aligned} {\operatorname{rk}_\bot}({\mathbb{M}}^n/(I \cap J)) &= {\operatorname{rk}_\bot}((V + (I \oplus J))/V) = rn \\ {\operatorname{rk}_\bot}({\mathbb{M}}^n/(I + J)) &= {\operatorname{rk}_\bot}({\mathbb{M}}^{2n}/(V + (I \oplus J))) = rn. \end{aligned}$$ Therefore $I \cap J$ and $I + J$ are special. Furthermore, by Lemma \[special-lemma-1\] there exists a basis $\{A_1,\ldots,A_{rn},B_1,\ldots,B_{rn}\}$ of quasi-atoms over $I \oplus J$ such that - The elements $\tilde{A}_i := A_i \cap V$ form a basis of quasi-atoms in $[V \cap (I \oplus J), V]$. - The elements $\tilde{B}_i := B_i + V$ form a basis of quasi-atoms in $[V + (I \oplus J), {\mathbb{M}}^{2n}]$. (Additionally, the $A_i$ and $B_i$ can be chosen below any given $D$ dominating $I \oplus J$.) Applying $\Delta^{-1}$ and $\nabla^{-1}$ we see that the elements $$\begin{aligned} \hat{A}_i = \Delta^{-1}(A_i \cap V) & = \{\vec{x} ~|~ (\vec{x},\vec{x}) \in A_i \cap V\} \\ & = \{\vec{x} ~|~ (\vec{x},\vec{x}) \in A_i\} \\ \hat{B}_i = \nabla^{-1}(A_i + V) & = \{\vec{x} - \vec{y} ~|~ (\vec{x},\vec{y}) \in B_i + V\} \\ & = \{\vec{x} - \vec{y} ~|~ (\vec{x},\vec{y}) \in B_i\} \end{aligned}$$ form bases of quasi-atoms for $[I \cap J, {\mathbb{M}}^n]$ and $[I + J, {\mathbb{M}}^n]$, respectively. By Lemma \[special-lemma-2\] special elements of $\mathcal{P}_n$ form a sublattice. Can this be proven directly (lattice theoretically) within $\mathcal{P}_n$ without using the larger lattice $\mathcal{P}_{2n}$? The associated rings and ideals ------------------------------- Let $J \in \mathcal{P}_n$ be special, and $a \in {\mathbb{M}}^\times$. Say that $a$ *contracts* $J$ if $a = 0$ or $J$ dominates $a \cdot J$ (i.e., ${\operatorname{rk}_\bot}(J/a \cdot J) = nr$). Note that when $a \ne 0$, $J$ dominates $a \cdot J$ if and only if $a^{-1} \cdot J$ dominates $J$. \[contraction\]   1. \[c-1\] Let $A_1, \ldots, A_{nr}$ be a basis of quasi-atoms over $J \in \mathcal{P}_n$ and $a$ be an element of ${\mathbb{M}}$. If $a \cdot A_i \subseteq J$ for all $i$, then $a$ contracts $J$. Conversely, suppose $a$ contracts $J$. Then there exists $A'_i \in (J,A_i]$ such that $A'_1, \ldots, A'_{nr}$ is a basis of quasi-atoms over $J$ and $a \cdot A'_i \subseteq J$ for each $i$. 2. \[c-2\] If $a$ contracts $J$ and $b \in {\mathbb{M}}^\times$, then $a$ contracts $b \cdot J$. 3. \[c-3\] If $a, b$ contract $J$ then $a + b$ contracts $J$. 4. \[c-4\] If $a$ contracts $J$ and $b \cdot J \subseteq J$, then $a \cdot b$ contracts $J$. 5. If $a$ contracts both $I \in \mathcal{P}_n$ and $J \in \mathcal{P}_m$, then $a$ contracts $I \oplus J \in \mathcal{P}_{n+m}$. 6. \[c-6\] If $a$ contracts $I, J \in \mathcal{P}_n$, then $a$ contracts $I \cap J$ and $I + J$. <!-- --> 1. First suppose $a \cdot A_i \subseteq J$. If $a = 0$ then $a$ contracts $J$ by definition, so suppose $a \ne 0$. Then $$(a^{-1} \cdot J) \cap A_i = A_i \supsetneq J$$ for each $i$, so by Lemma \[characterization-of-domination\] the group $a^{-1} \cdot J$ dominates $J$, or equivalently, $J$ dominates $a \cdot J$. Thus $a$ contracts $J$. Conversely, suppose that $a$ contracts $J$. If $a = 0$ then $a \cdot A_i \subseteq J$ so we may take $A'_i = A_i$. Otherwise, note that $a^{-1} \cdot J$ dominates $J$, so by Lemma \[characterization-of-domination\], $$A'_i := (a^{-1} \cdot J) \cap A_i \supsetneq J.$$ Then $A'_i$ is a quasi-atom over $J$, equivalent to $A_i$, so $\{A'_1, \ldots, A'_{nr}\}$ is another basis of quasi-atoms over $J$. Furthermore $A'_i \subseteq a^{-1} \cdot J$, so $a \cdot A'_i \subseteq J$. 2. Multiplication by $b$ induces an automorphism of $\mathcal{P}_n$ sending the interval $[a \cdot J, J]$ to $[a \cdot (b \cdot J), b \cdot J]$, so ${\operatorname{rk}_\bot}(J/a \cdot J) = {\operatorname{rk}_\bot}(b \cdot J / (ab) \cdot J)$. 3. Take a basis of quasi-atoms $A_1, \ldots, A_{rn}$ over $J$. By part 1, we may shrink the $A_i$ and assume that $a \cdot A_i \subseteq J$. Shrinking again, we may assume $b \cdot A_i \subseteq J$. Then $$(a + b) \cdot A_i \subseteq a \cdot A_i + b \cdot A_i \subseteq J + J = J$$ so by part 1, $a+b$ contracts $J$. 4. Suppose $a$ contracts $J$ and $b \cdot J \subseteq J$. Then $${\operatorname{rk}_\bot}(J/a \cdot b \cdot J) \ge {\operatorname{rk}_\bot}(b \cdot J/ a \cdot b \cdot J) = {\operatorname{rk}_\bot}(J/a \cdot J) = nr.$$ 5. Let $A_1, \ldots, A_{rn}$ be a basis of quasi-atoms in $[I,{\mathbb{M}}^n]$, and $B_1, \ldots, B_{rm}$ be a basis of quasi-atoms in $[J,{\mathbb{M}}^m]$. Shrinking the $A_i$ and $B_i$, we may assume $a \cdot A_i \subseteq I$ and $a \cdot B_i \subseteq J$. Note that the sequence $$A_1 \oplus J, A_2 \oplus J, \ldots, A_{rn} \oplus J, I \oplus B_1, I \oplus B_2, \ldots, I \oplus B_{rm}$$ is a basis of quasi-atoms in $[I \oplus J, {\mathbb{M}}^{n+m}]$. Multiplication by $a$ collapses each of these quasi-atoms into $I \oplus J$ (using the fact that $a \cdot I \subseteq I$ and $a \cdot J \subseteq J$). Therefore $a$ contracts $I \oplus J$. 6. We may assume $a \ne 0$. By the previous point, $a^{-1} \cdot (I \oplus J)$ dominates $I \oplus J$. By Lemma \[special-lemma-2\], $I + J$ and $I \cap J$ are special. Moreover, there is a basis of quasi-atoms $A_1, \ldots, A_n, B_1, \ldots, B_n$ in $[I \oplus J, {\mathbb{M}}^{2n}]$ such that for $$\begin{aligned} \hat{A}_i &= \{ \vec{x} \in {\mathbb{M}}^n ~|~ (\vec{x},\vec{x}) \in A_i\} \\ \hat{B}_i &= \{ \vec{x} - \vec{y} ~|~ (\vec{x},\vec{y}) \in B_i\} \end{aligned}$$ the set $\{\hat{A}_1, \ldots, \hat{A}_n\}$ is a basis of quasi-atoms over $I \cap J$ and the set $\{\hat{B}_1, \ldots, \hat{B}_n\}$ is a basis of quasi-atoms over $I + J$. Furthermore Lemma \[special-lemma-2\] ensures that the $A_i$ and $B_i$ can be chosen in $[I \oplus J, a^{-1} \cdot (I \oplus J)]$. Thus $a \cdot A_i \subseteq I \oplus J$ and $a \cdot B_i \subseteq I \oplus J$. Then $$\vec{x} \in \hat{A}_i \iff (\vec{x},\vec{x}) \in A_i \implies (a \cdot \vec{x}, a \cdot \vec{x}) \in I \oplus J \iff a \cdot \vec{x} \in I \cap J,$$ so $a \cdot \hat{A}_i \subseteq I \cap J$. As the $\hat{A}_i$ form a basis of quasi-atoms over $I \cap J$, it follows that $a$ contracts $I \cap J$. Similarly, $$(\vec{x},\vec{y}) \in B_i \implies (a \cdot \vec{x}, a \cdot \vec{y}) \in I \oplus J \implies a \cdot (\vec{x} - \vec{y}) \in I + J$$ so $a \cdot \hat{B}_i \subseteq I + J$. Thus $a$ contracts $I + J$. \[rings-and-ideals\] For any special $J \in \mathcal{P} = \mathcal{P}_1$, let $R_J$ be the set of $a \in {\mathbb{M}}$ such that $a \cdot J \subseteq J$, and let $I_J$ be the set of $a \in {\mathbb{M}}$ that contract $J$. 1. $R_J$ is a subring of ${\mathbb{M}}$, containing $M_0$. 2. $I_J$ is an ideal in $R_J$. 3. \[r-scale\] If $b \in {\mathbb{M}}^\times$ then $R_J = R_{b \cdot J}$ and $I_J = I_{b \cdot J}$. 4. If $J$ is type-definable over $M \supseteq M_0$, then $R_J$ and $I_J$ are $M$-invariant. 5. \[ij-im\] If $J$ is non-zero and type-definable over $M \supseteq M_0$ then $I_M \subseteq I_J$. 6. \[cap-not-vee\] If $J_1$ and $J_2$ are special, then $$\begin{aligned} R_{J_1} \cap R_{J_2} & \subseteq R_{J_1 \cap J_2} \\ I_{J_1} \cap I_{J_2} & \subseteq I_{J_1 \cap J_2} \end{aligned}$$ 7. \[jacobson\] $(1 + I_J) \subseteq R_J^\times$. Consequently, $I_J$ lies inside the Jacobson radical of $R_J$. <!-- --> 1. Straightforward. 2. The set $I_J$ is a subset of $R_J$. The fact that $I_J \lhd R_J$ is exactly Lemma \[contraction\].\[c-3\]-\[c-4\]. 3. For $I_J$ this is Lemma \[contraction\].\[c-2\]. For $R_J$ this is clear: $$a \cdot J \subseteq J \implies (ab) \cdot J \subseteq b \cdot J.$$ 4. The definitions are ${\operatorname{Aut}}({\mathbb{M}}/M)$-invariant. 5. Let $A_1, \ldots, A_r$ be a basis of quasi-atoms over $J$. For each $i$ let $a_i$ be an element of $A_i\setminus J$. Let $M'$ be a small model containing $M$ and the $a_i$’s. Any $\varepsilon \in I_{M'}$ contracts $J$. We may assume $\varepsilon \ne 0$. Let $D = \varepsilon^{-1} \cdot I_{M'}$. By Remark \[basic-infs\].\[infinitesimal-times-standard\], $a_i \cdot \varepsilon \in I_{M'}$, or equivalently $a_i \in D$. Thus $$D \cap A_i \not \subseteq J$$ By Proposition \[special-proposition\].\[guards\], $D \supseteq J$. Then $$D \cap A_i \supsetneq J$$ so by Lemma \[characterization-of-domination\], $D$ dominates $J$. By Proposition \[those-posets\].\[infinitesimals-below\], $\varepsilon^{-1} \cdot J \supseteq \varepsilon^{-1} \cdot I_{M'} = D$. Thus $\varepsilon^{-1} \cdot J$ dominates $J$. Let $\varepsilon$ be a realization of the partial type over $M'$ asserting that $\varepsilon \in I_{M'}$ and $\varepsilon \notin X$ for any light $M'$-definable set $X$. This type is consistent because $M'$-definable basic neighborhoods are heavy (Proposition \[basic-nbhds\].\[heavy-nbhd\]) and no heavy set is contained in a finite union of light sets (Theorem \[heavy-light\]). Then $\varepsilon \in I_{M'} \subseteq I_J$. As $I_J$ is $M$-invariant, every realization of ${\operatorname{tp}}(\varepsilon/M)$ is in $I_J$. Let $Y$ be the $M$-definable set of realizations of ${\operatorname{tp}}(\varepsilon/M)$. For any $M$-definable $X \supseteq Y$ we have $$I_M \subseteq X {-_\infty}X \subseteq X - X.$$ Therefore $I_M \subseteq Y- Y$. But $Y - Y \subseteq I_J - I_J = I_J$. 6. If $a \in R_{J_1}$ and $a \in R_{J_2}$, then $$a \cdot (J_1 \cap J_2) = (a \cdot J_1) \cap (a \cdot J_2) \subseteq J_1 \cap J_2$$ so $a \in R_{J_1 \cap J_2}$. The inclusion $I_{J_1} \cap I_{J_2} \subseteq I_{J_1 \cap J_2}$ is Lemma \[contraction\].\[c-6\]. 7. First note that $1$ does not contract $J$. Indeed, ${\operatorname{rk}_\bot}(J/J) = 0 \ne r$. Thus $1 \notin I_J$. As $I_J$ is an ideal, it follows that $-1 \notin I_J$. If $\varepsilon \in I_J$ then $\varepsilon/(1 + \varepsilon) \in I_J$. We may assume $\varepsilon \ne 0$. Using Lemma \[contraction\].\[c-1\] choose a basis $\{A_1,\ldots,A_r\}$ of quasi-atoms over $J$ such that $\varepsilon \cdot A_i \subseteq J$. For each $i$ choose $a_i \in A_i \setminus J$. Then $\varepsilon \cdot a_i \in J$, so $(1 + \varepsilon) \cdot a_i \in A_i \setminus J$. Let $\beta = (1 + \varepsilon)/\varepsilon$. Then $$\begin{aligned} \beta \cdot (\varepsilon \cdot a_i) & \in \beta \cdot J \\ (\beta \cdot \varepsilon) \cdot a_i = (1 + \varepsilon) \cdot a_i & \in A_i \setminus J. \end{aligned}$$ In particular $$(\beta \cdot J) \cap A_i \not \subseteq J,$$ for every $i$, so $\beta \cdot J \supseteq J$ by Proposition \[special-proposition\].\[guards\]. Then $(\beta \cdot J) \cap A_i \supsetneq J$ for every $i$, so $\beta \cdot J$ dominates $J$ by Lemma \[characterization-of-domination\]. This means that $\beta^{-1} = \varepsilon/(1 + \varepsilon)$ lies in $I_J$. Now if $\varepsilon \in I_J$, then $$\frac{1}{1 + \varepsilon} = 1 - \frac{\varepsilon}{1 + \varepsilon} \in 1 + I_J \subseteq R_J. \qedhere$$ Proposition \[rings-and-ideals\].\[cap-not-vee\] also holds for $R_{J_1 + J_2}$ and $I_{J_1 + J_2}$. In Proposition \[rings-and-ideals\].\[ij-im\], not only is $I_M$ a subset of $I_J$, it is a sub*ideal* in the ring $R_J$. One can probably prove this by first increasing $M$ to contain a non-zero element $j_0$ of $J$. Then for any $\varepsilon \in I_M$ and $a \in R_J$, we have $$\varepsilon \cdot a \cdot j_0 \in I_M \cdot R_J \cdot J \subseteq I_M \cdot J \subseteq I_M,$$ so $\varepsilon \cdot a \in j_0^{-1} I_M = I_M$. Thus $I_M \cdot R_J \subseteq I_M$. Then one can probably shrink $M$ back to the original model by the usual methods. Now, the important thing is that if $R_J$ were already a good Bezout domain, then there would be some valuations ${\operatorname{val}}_i : {\mathbb{M}}\to \Gamma_i$ and cuts $\Xi_i$ in $\Gamma_i$ such that $$I_M = \{x \in {\mathbb{M}}~|~ {\operatorname{val}}_1(x) > \Xi_1 \wedge {\operatorname{val}}_2(x) > \Xi_2 \wedge \cdots\}.$$ By replacing $\Xi_i$ with $2 \Xi_i$ or $\Xi_i/2$ we could obtain alternate $M$-invariant subgroups smaller than $I_M$, contradicting the analogue of Corollary \[minimal-heavy-subgroup\] for invariant (not type-definable) subgroups. Unless, of course, the $\Xi_i$ lie on the edges of convex subgroups. Then, by coarsening, one could essentially arrive at a situation where $I_M$ is the Jacobson radical of an $M$-invariant good Bezout domain. This would allow some of the henselianity arguments used in the dp-minimal case to directly generalize, though there are some additional complications in characteristic 0.[^14] Mutation and the limiting ring {#sec:mutate} ------------------------------ The next two lemmas provide a way to “mutate” a special group $J$ and obtain a better special group $J'$ for which $R_{J'}$ is closer than $R_J$ to being a good Bezout domain. \[twists\] Let $J \in \mathcal{P}$ be special and non-zero. Let $a_1, \ldots, a_n$ be elements of ${\mathbb{M}}^\times$. Let $J' = J \cap a_1 \cdot J \cap a_2 \cdot J \cap \cdots \cap a_n \cdot J$. Then $J'$ is special and non-zero, $R_J \subseteq R_{J'}$, and $I_J \subseteq I_{J'}$. By Proposition \[special-proposition\].\[special-scale\], each $a_i \cdot J$ is special, so the intersection $J'$ is special by Lemma \[special-lemma-2\]. It is nonzero by Proposition \[those-posets\].\[nonzero-intersect\]. By Proposition \[rings-and-ideals\].\[r-scale\] we have $R_J = R_{a_i \cdot J}$ and $I_J = I_{a_i \cdot J}$ for each $i$. Then the inclusions $R_J \subseteq R_{J'}$ and $I_J \subseteq I_{J'}$ follow by an interated application of Proposition \[rings-and-ideals\].\[cap-not-vee\]. Recall that $r$ is the reduced rank of $\mathcal{P}$. \[vandermonde\] Let $J \in \mathcal{P}$ be special and non-zero. Let $\alpha \in {\mathbb{M}}^\times$ be arbitrary. Let $J' = J \cap (\alpha \cdot J) \cap \cdots \cap (\alpha^{r-1} \cdot J)$. Let $q_0, q_1, \ldots, q_r$ be $r+1$ distinct elements of $M_0$. Then there is at least one $i$ such that $\alpha \ne q_i$ and $$\frac{1}{\alpha - q_i} \in R_{J'}.$$ For each $0 \le i \le r$ let $$\begin{aligned} \alpha_i & := \alpha - q_i \\ G_i & := \{x \in {\mathbb{M}}~|~ \alpha_ix \in J \wedge \alpha_i^2 x \in J\wedge \cdots \wedge \alpha_i^rx \in J\} \\ H_i & := J \cap G_i = \{x \in {\mathbb{M}}~|~ x \in J \wedge \alpha_ix \in J \wedge \cdots \wedge \alpha_i^rx \in J\}. \end{aligned}$$ Also let $$\begin{aligned} H = \{x \in {\mathbb{M}}~|~ x \in J \wedge \alpha x \in J \wedge \cdots \wedge \alpha^r x \in J\}. \end{aligned}$$ \[claim-hi\] $H_i = H$ for any $i$. Note $\alpha = \alpha_i + q_i$. If $x \in H_i$ then $$\alpha^n x = (\alpha_i + q_i)^n x = \sum_{k = 0}^n \binom{n}{k} \alpha_i^k q_i^{n-k} x \in J$$ for $0 \le n \le r$, because $\alpha_i^k x \in J$, $q_i^{n-k} \in M_0$, and $J$ is an $M_0$-vector space. Thus $H_i \subseteq H$; the reverse inclusion follows by symmetry. Because the $q_i$ are distinct, the $(r+1) \times (r+1)$ Vandermonde matrix built from the $q_i$ is invertible. Let $f : {\mathbb{M}}^{r+1} \to {\mathbb{M}}^{r+1}$ be the ${\mathbb{M}}$-linear map sending $(1,q_i,\ldots,q_i^r)$ to the $i$th basis vector. Let $g : {\mathbb{M}}\to {\mathbb{M}}^{r+1}$ be the map $$g(x) = (x, \alpha x, \ldots, \alpha^r x).$$ \[vandermonde-key\] The composition $${\mathbb{M}}\stackrel{g}{\to} {\mathbb{M}}^{r+1} \stackrel{f}{\to} {\mathbb{M}}^{r+1} \twoheadrightarrow ({\mathbb{M}}/J)^{r+1}$$ has kernel $H$, and maps $G_i$ into $0^i \oplus ({\mathbb{M}}/J) \oplus 0^{r-i}$. The invertible matrix defining $f$ has coefficients in $M_0$, and $J$ is closed under multiplication by $M_0$, so $f$ maps $J^{r+1}$ isomorphically to $J^{r+1}$. Therefore, $$f(g(x)) \in J^{r+1} \iff g(x) \in J^{r+1} \iff x \in H,$$ where the second $\iff$ is the definition of $H$. Now suppose $x \in G_i$. Then $g(x) - (x, q_ix, \ldots, q_i^r x) \in J^{r+1}$. Indeed, for any $0 \le n \le r$ we have $$\alpha^n x = (\alpha_i + q_i)^n x = q_i^n x + \sum_{k = 1}^n \binom{n}{k} q_i^{n-k} (\alpha_i^k x),$$ and the sum is an element of $J$ by definition of $G_i$. As $f$ preserves $J^{r+1}$, it follows that $$f(g(x)) \equiv f(x, q_ix, \ldots, q_i^r x) = x \cdot e_i \pmod{ J^{r+1}},$$ where $e_i$ is the $i$th basis vector. \[ult\] If $(x_0, x_1, \ldots, x_r) \in G_0 \times \cdots \times G_r$ has $x_0 + \cdots + x_r \in H$, then each $x_i \in H$. For $0 \le i \le r$ let $p_i : {\mathbb{M}}^{r+1} \to {\mathbb{M}}/J$ be the composition of the $i$th projection and the quotient map ${\mathbb{M}}\to {\mathbb{M}}/J$. Claim \[vandermonde-key\] implies that $$\begin{aligned} x \in H & \implies p_i(f(g(x))) = 0 \\ x \in G_j &\implies p_i(f(g(x))) = 0 \qquad \text{if } i \ne j. \end{aligned}$$ Thus $$0 = p_i(f(g(x_0 + \cdots + x_r))) = p_i(f(g(x_i))).$$ As $p_j(f(g(x_i))) = 0$ for $j \ne i$, it follows that $p_j(f(g(x_i))) = 0$ for all $j$. In other words, $f(g(x_i)) \in J^{r+1}$. By Claim \[vandermonde-key\], $x_i \in H$. Now Claim \[ult\] implies that the map $$\begin{aligned} (G_0/H) \times \cdots \times (G_r/H) & \to {\mathbb{M}}/H \\ (x_0,\ldots,x_r) & \mapsto x_0 + \cdots + x_r \end{aligned}$$ is injective. The image is $D/H$ for some type-definable $D \in \mathcal{P}$, namely $D = G_0 + \cdots + G_r$. Then the interval $[H^{r+1},G_0 \oplus \cdots \oplus G_r]$ in $\mathcal{P}_{r+1}$ is isomorphic to the interval $[H,D]$ in $\mathcal{P}_1$. Thus $$r \ge {\operatorname{rk}_0}(D/H) = {\operatorname{rk}_0}(G_0/H) + \cdots + {\operatorname{rk}_0}(G_r/H).$$ Therefore $G_i = H = H_i$ for at least one $i$. By definition of $G_i$ and $H_i$, this means that $$\alpha_i x \in J \wedge \cdots \wedge \alpha_i^r x \in J \implies x \in J \label{callback}$$ for any $x \in {\mathbb{M}}$. As $J \ne 0$, this implies $\alpha_i \ne 0$. Then (\[callback\]) can be rephrased as $$\alpha_i^{-1} \cdot J \cap \cdots \cap \alpha_i^{-r} \cdot J \subseteq J. \label{callback2}$$ Define $$\begin{aligned} J'' & := J \cap \alpha_i^{-1} J \cap \cdots \cap \alpha_i^{-(r-1)} J \\ & = J \cap \alpha^{-1} J \cap \cdots \cap \alpha^{-(r-1)} J, \end{aligned}$$ where the second equality follows by the proof of Claim \[claim-hi\]. By (\[callback2\]), $$\alpha_i^{-1} \cdot J'' = \alpha_i^{-1} J \cap \cdots \cap \alpha_i^{-r} \stackrel{!}{\subseteq} J \cap \alpha_i^{-1} J \cap \cdots \cap \alpha_i^{-(r-1)} J = J''.$$ Therefore $\alpha_i^{-1} \in R_{J''}$. But $$J' = J \cap \cdots \cap \alpha^{r-1} J = \alpha^{r-1} \cdot (J \cap \cdots \cap \alpha^{-(r-1)} J) = \alpha^{r-1} J''.$$ Thus, by Proposition \[rings-and-ideals\].\[r-scale\] $$\alpha_i^{-1} \in R_{J''} = R_{J'}. \qedhere$$ \[bezout-theorem\] Let $J \in \mathcal{P}_1$ be special, non-zero, and type-definable over $M \supseteq M_0$. Then there is an $M$-invariant ring $R^\infty_J$ and ideal $I^\infty_J \lhd R^\infty_J$ satisfying the following properties: - $R^\infty_J$ and $I^\infty_J$ are $M$-invariant. - $(1 + I^\infty_J) \subseteq (R^\infty_J)^\times$, so $I^\infty_J$ is a subideal of the Jacobson radical of $R^\infty_J$. - The $M$-infinitesimals $I_M$ are a subgroup of $I^\infty_J$ (and therefore of the Jacobson radical). - $M_0 \subseteq R^\infty_J$. - $R^\infty_J$ is a Bezout domain with at most $r$ maximal ideals. - The field of fractions of $R^\infty_J$ is ${\mathbb{M}}$. Let $P$ be the set of finite $S \subseteq {\mathbb{M}}^\times$ such that $1 \in S$. Then $P$ is a commutative monoid with respect to the product $S \cdot S' = \{x \cdot y ~|~ x \in S, y \in S'\}$. For any $S \in P$ and $G \in \mathcal{P}_1$, define $$G^S := \bigcap_{s \in S} s \cdot G.$$ Note that $(G^S)^{S'} = G^{S \cdot S'}$. If $G$ is special and non-zero then by Lemma \[twists\] $G^S$ is special and non-zero, and there are inclusions $R_G \subseteq R_{G^S}$ and $I_G \subseteq I_{G^S}$. Define sets $$\begin{aligned} R_J^\infty & := \bigcup_{S \in P} R_{J^S} \\ I_J^\infty & := \bigcup_{S \in P} I_{J^S}. \end{aligned}$$ These sets are clearly $M$-invariant. Moreover, the unions are directed: given any $S$ and $S'$ we have $$\begin{aligned} R_{J^S} \cup R_{J^{S'}} &\subseteq R_{J^{S \cdot S'}} \\ I_{J^S} \cup I_{J^{S'}} &\subseteq I_{J^{S \cdot S'}}. \end{aligned}$$ Therefore $R^\infty_J$ is a ring and $I^\infty_J$ is an ideal. The fact that $(1 + I^\infty_J) \subseteq (R^\infty_J)^\times$ also follows (using Proposition \[rings-and-ideals\].\[jacobson\]). Taking $S = \{1\}$, we see that $I_J \subseteq I^\infty_J$. Proposition \[rings-and-ideals\].\[ij-im\] says $I_M \subseteq I_J$, so $I_M \subseteq I^\infty_J$ as desired. Similarly, $M_0 \subseteq R_J \subseteq R^\infty_J$. \[key-to-bezout\] If $q_0, q_1, \ldots, q_r$ are distinct elements of $M_0$ and $\alpha \in {\mathbb{M}}^\times$, then at least one of $1/(\alpha - q_i)$ is in $R^\infty_J$. By Lemma \[vandermonde\], at least one of $1/(\alpha - q_i)$ lies in $R_{J^S}$ for $S = \{1, \alpha, \ldots, \alpha^{r-1}\}$. It follows formally that $R^\infty_J$ is a Bezout domain with no more than $r$ maximal ideals. Let $a, b$ be two elements of $R^\infty_J$. We claim that the ideal $(a,b)$ is principal. This is clear if $a = 0$ or $b = 0$. Otherwise, let $\alpha = a/b$. As $M_0$ is infinite, Claim \[key-to-bezout\] implies that $$\frac{b}{a - q b} = \frac{1}{\frac{a}{b} - q} \in R^\infty_J$$ for some $q \in M_0$. Then the principal ideal $(a - qb) \lhd R^\infty_J$ contains $b$, hence $qb$ and thus $a$. Therefore $(a - qb) = (a, b)$. Next, we show that $R^\infty_J$ has at most $r$ maximal ideals. Suppose for the sake of contradiction that there were distinct maximal ideals $\mathfrak{m}_0, \ldots, \mathfrak{m}_r$ in $R^\infty_J$. As $R^\infty_J$ is an $M_0$-algebra, each quotient $R^\infty_J/\mathfrak{m}_i$ is a field extending $M_0$. Take distinct $q_0, \ldots, q_r \in M_0$, and find an element $x \in R^\infty_J$ such that $x \equiv q_i \pmod{\mathfrak{m}_i}$ for each $i$, by the Chinese remainder theorem. Then $x - q_i \in \mathfrak{m}_i \subseteq R^\infty_J \setminus (R^\infty_J)^\times$ for each $i$. So $1/(x - q_i)$ does not lie in $R^\infty_J$ for any $0 \le i \le r$, contrary to Claim \[key-to-bezout\]. Lastly, note that if $x$ is any element of ${\mathbb{M}}^\times$, then $1/(x - q) \in R^\infty_J$ for some $q \in M_0$, $q \ne x$. As $q \in M_0 \subseteq R^\infty_J$, the field of fractions of $R^\infty_J$ contains $x$. So the field of fractions must be all of ${\mathbb{M}}$. From Bezout domains to valuation rings -------------------------------------- We double-check some basic facts about Bezout domains. \[bezout-basics\] Let $R$ be a Bezout domain. 1. For each maximal ideal $\mathfrak{m}$, the localization $R_{\mathfrak{m}}$ is a valuation ring on the field of fractions of $R$. 2. $R$ is the intersection of the valuation rings $R_{\mathfrak{m}}$. <!-- --> 1. Given non-zero $a, b \in R$, we must show that $a/b$ or $b/a$ lies in $R_{\mathfrak{m}}$. If $c$ is such that $(c) = (a,b)$, then we may replace $a$ and $b$ with $a/c$ and $b/c$, and assume that $(a,b) = (1) = R$. Both of $a$ and $b$ cannot lie in $\mathfrak{m}$, and so either $a/b$ or $b/a$ is in $R_{\mathfrak{m}}$. 2. Given $a \in \bigcap_{\mathfrak{m}} R_{\mathfrak{m}}$, we will show $a \in R$. Let $I = \{x \in R ~|~ ax \in R\}$. This is an ideal in $R$. If $I$ is proper, then $I \le \mathfrak{m}$ for some $\mathfrak{m}$. As $a \in R_{\mathfrak{m}}$ we can write $a = b/s$ where $b \in R$ and $s \in R \setminus \mathfrak{m}$. Then $s \in R$ and $as = b \in R$, so $s \in I \le \mathfrak{m}$, a contradiction. Therefore $I$ is an improper ideal, so $1 \in I$, and $a = a \cdot 1 \in R$. \[any-valuation-at-all\] Let ${\mathbb{M}}$ be a sufficiently saturated dp-finite field, possibly with extra structure. Suppose ${\mathbb{M}}$ is not of finite Morley rank. Then there is a small set $A \subseteq {\mathbb{M}}$ and a non-trivial $A$-invariant valuation ring. Take $M_0$ as usual in this section. By Proposition \[special-proposition\] there is a non-zero special $J \in \mathcal{P}_1$. The group $J$ is type-definable over some small $M \supseteq M_0$. Let $R$ be the $R^\infty_J$ of Theorem \[bezout-theorem\]. Then $R$ is an $M$-invariant Bezout domain with at most $r$ maximal ideals, the Jacobson radical of $R$ is non-zero (because it contains $I_M$), and ${\operatorname{Frac}}(R) = {\mathbb{M}}$. Let $\mathfrak{m}_1, \ldots, \mathfrak{m}_k$ enumerate the maximal ideals of $R$. Let $\mathcal{O}_i$ be the localization $R_{\mathfrak{m}_i}$. By Remark \[bezout-basics\], each $\mathcal{O}_i$ is a valuation ring on ${\mathbb{M}}$, and $$R = \mathcal{O}_1 \cap \cdots \cap \mathcal{O}_k.$$ At least one $\mathcal{O}_i$ is non-trivial; otherwise $R = {\mathbb{M}}$ and has Jacobson radical 0.[^15] Without loss of generality $\mathcal{O}_1$ is non-trivial. By the Chinese remainder theorem, choose $a \in R$ such that $a \equiv 1 \pmod{ \mathfrak{m}_1}$ and $a \equiv 0 \pmod{\mathfrak{m}_i}$ for $i \ne 1$. We claim that $\mathcal{O}_1$ is ${\operatorname{Aut}}({\mathbb{M}}/aM)$-invariant. If $\sigma \in {\operatorname{Aut}}({\mathbb{M}}/aM)$, then $\sigma \in {\operatorname{Aut}}({\mathbb{M}}/M)$ so $\sigma$ preserves $R$ setwise. It therefore permutes the finite set of maximal ideals. As $\mathfrak{m}_1$ is the unique maximal ideal not containing $a$, it must be preserved (setwise). Therefore $\sigma$ preserves the localization $\mathcal{O}_1$ setwise. Stable fields do not admit non-trivial invariant valuation rings ([@prdf2], Lemma 2.1). Consequently, Theorem \[any-valuation-at-all\] can be used to give an extremely roundabout proof of Halevi and Palacín’s theorem that stable dp-finite fields have finite Morley rank ([@Palacin], Proposition 7.2). Shelah conjecture and classification {#sec:the-end} ==================================== \[proto-shelah\] Let $K$ be a sufficiently saturated dp-finite field of positive characteristic. Then one of the following holds: - $K$ has finite Morley rank (and is therefore finite or algebraically closed). - $K$ admits a non-trivial henselian valuation. This is Theorem \[invariant-valuations-are-enough\] and \[any-valuation-at-all\]. \[o-infty\] Let $K$ be a sufficiently saturated dp-finite field of positive characteristic. Assume $K$ is infinite. Let $\mathcal{O}_\infty$ be the intersection of all $K$-definable valuation rings on $K$. Then $\mathcal{O}_\infty$ is a henselian valuation ring on $K$ whose residue field is algebraically closed. The proof for dp-minimal fields (Theorem 9.5.7 in [@myself]) goes through without changes, using Lemma \[hensel-key\], Theorem \[henselianity-conjecture\], and Proposition \[proto-shelah\]. Additionally, we must rule out the possibility that the residue field is real closed or finite. The first cannot happen because we are in positive characteristic. The second cannot happen because $K$ is Artin-Schreier closed, a property which transfers to the residue field. \[cor-infty\] Let $K$ be a sufficiently saturated infinite dp-finite field of positive characteristic. If every definable valuation on $K$ is trivial, then $K$ is algebraically closed. \[shelah-conjecture\] Let $K$ be a dp-finite field of positive characteristic. Then one of the following holds: - $K$ is finite. - $K$ is algebraically closed. - $K$ admits a non-trivial definable henselian valuation. Suppose $K$ is neither finite nor algebraically closed. Let $K' \succeq K$ be a sufficiently saturated elementary extension. Then $K'$ is neither finite nor algebraically closed. By Corollary \[cor-infty\] there is a non-trivial definable valuation $\mathcal{O} = \phi(K',a)$ on $K'$. The statement that $\phi(x;a)$ cuts out a valuation ring is expressed by a 0-definable condition on $a$, so we can take $a \in {\operatorname{dcl}}(K)$. Then $\phi(K,a)$ is a non-trivial valuation ring on $K$, henselian by Theorem \[henselianity-conjecture\]. So the Shelah conjecture holds for dp-finite fields of positive characteristic. By Proposition 3.9, Remark 3.10, and Theorem 3.11 in [@halevi-hasson-jahnke], this implies the following classification of dp-finite fields of positive characteristic: up to elementary equivalence, they are exactly the Hahn series fields $\mathbb{F}_p((\Gamma))$ where $\Gamma$ is a dp-finite $p$-divisible group. Dp-finite ordered abelian groups have been algebraically characterized and are the same thing as strongly dependent ordered abelian groups ([@sd-groups-dolich-goodrick], [@sd-groups-farre], [@sd-groups-halevi-hasson]). The author would like to thank - Meng Chen, for hosting the author at Fudan University, where this research was carried out. - Jan Dobrowolski, who provided some helpful references. - John Goodrick, for suggesting that the ideal of finite sets could be replaced with other ideals in the construction of infinitesimals. - Franziska Jahnke, for convincing the author that the proof in §\[sec:henselianity\] was valid. - Shichang Song, who invited the author to give a talk on dp-minimal fields. The ideas for §\[sec:broad-narrow\]-\[sec:heavy-light\] arose while preparing that talk. - The UCLA model theorists, who read parts of this paper and pointed out typos. [This material is based upon work supported by the National Science Foundation under Award No. DMS-1803120. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.]{} [^1]: He proves this under the assumption that the entire theory is dp-minimal, but the proof generalizes to products of definable sets of dp-rank 1. [^2]: In later work, it has been shown that the heavy sets are exactly the sets of full rank ([@prdf2], Theorem 5.9.2). However, the proof relies on the construction of the infinitesimals, and only works in hindsight. [^3]: The details of this strategy have been verified in [@prdf2], which proves the classification of dp-finite fields contingent on the above Conjecture. The “technical difficulties in characteristic zero” involve showing that $1 + I_M$ is exactly the group of “multiplicative infinitesimals.” This is [@prdf2], Proposition 5.12. [^4]: One must verify that the basic properties of dp-rank, including subadditivity, go through for hyperimaginaries. [^5]: Luckily, we can avoid the machinery of $r$-fold specializations in the present paper. We defer a proper treatment of $r$-fold specializations until a later paper, [@prdf3]. [^6]: Specifically, it would derail the proof of Proposition \[special-proposition\].\[guard-application\], among other things. [^7]: See Proposition 1.5.16 in [@PillayGST] or the proof of Corollary 2.14 in [@stab-groups]. [^8]: These ideas are developed further in [@prdf3], Appendix E. [^9]: These claims have now been verified in [@prdf3], §6.4. [^10]: Here and in what follows, we will be a bit sloppy with what exactly “small” means, but this can probably be fixed. [^11]: This preserves $n > 0$ because no quasi-atom is in the closure of the empty set, as sequences of length 1 are always independent [^12]: All these claims have now been verified in [@prdf3] §2.2 and §7.4. [^13]: These ideas have been developed in §8 of [@prdf2]. [^14]: The details are worked out in [@prdf2]. [^15]: Tracing through the proof, here is what explicitly happens. If $\varepsilon \in I_M$ then $-1/\varepsilon$ cannot be in $R^\infty_J$, or else $\varepsilon \in I_M \subseteq I_J \subseteq I_{J^S} \lhd R_{J^S}$ and $-1/\varepsilon \in R_{J^S}$ for large enough $S$, so that $-1 \in I_{J^S}$, contradicting Proposition \[rings-and-ideals\].\[jacobson\].

--- abstract: 'Results obtained with the HADES dielectron spectrometer at GSI are discussed, with emphasis on dilepton production in elementary reactions.' address: - 'Institut de Physique Nucléaire, CNRS/IN2P3-Université Paris Sud, F-91406 Orsay Cedex, France' - | [$^1$Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali del Sud, 95125 Catania, Italy,\ $^{2}$LIP-Laboratório de Instrumentação e Física Experimental de Partículas , 3004-516 Coimbra, Portugal\ $^3$Smoluchowski Institute of Physics, Jagiellonian University of Cracow, 30-059 Kraków, Poland,\ $^4$GSI Helmholtzzentrum für Schwerionenforschung, 64291 Darmstadt, Germany,\ $^5$Institut für Strahlenphysik, Forschungszentrum Dresden-Rossendorf, 01314 Dresden, Germany,\ $^6$Joint Institute of Nuclear Research, 141980 Dubna, Russia,\ $^7$Institut für Kernphysik, Johann Wolfgang Goethe-Universität, 60438  Frankfurt, Germany,\ $^8$II.Physikalisches Institut, Justus Liebig Universität Giessen, 35392 Giessen, Germany,\ $^9$Istituto Nazionale di Fisica Nucleare, Sezione di Milano, 20133 Milano, Italy,\ $^10$Institute for Nuclear Research, Russian Academy of Science, 117312 Moscow, Russia,\ $^{11}$Physik Department E12, Technische Universität München, 85748 München, Germany,\ $^{12}$Department of Physics, University of Cyprus, 1678 Nicosia, Cyprus,\ $^{13}$Institut de Physique Nucléaire , CNRS/IN2P3 - Université Paris Sud, F-91406 Orsay Cedex, France,\ $^{14}$Nuclear Physics Institute, Academy of Sciences of Czech Republic, 25068 Rez, Czech Republic,\ $^{15}$Dep. de Física de Partículas, Univ. de Santiago de Compostela, 15706 Santiago de Compostela, Spain,\ $^{16}$Instituto de Física Corpuscular, Universidad de Valencia-CSIC, 46971 Valencia, Spain,\ $^a$Also at Dipartimento di Fisica e Astronomia, Università di Catania, 95125 Catania, Italy,\ $^b$Also at ISEC Coimbra,  Coimbra, Portugal,\ $^c$Also at Technische Universität Dresden, 01062 Dresden, Germany,\ $^d$Also at Dipartimento di Fisica, Università di Milano, 20133 Milano, Italy,\ $^e$Also at Panstwowa Wyzsza Szkola Zawodowa , 33-300 Nowy Sacz, Poland.]{} author: - 'B. Ramstein' - '[G. Agakichiev$^{\,8}$, C. Agodi$^{\,1}$, A. Balanda$^{\,3,e}$, G. Bellia$^{\,1,a}$, D. Belver$^{\,15}$, A. Belyaev$^{\,6}$, A. Blanco$^{\,2}$, M. Böhmer$^{\,11}$, J. L. Boyard$^{\,13}$, P. Braun-Munzinger$^{\,4}$, P. Cabanelas$^{\,15}$, E. Castro$^{\,15}$, T. Christ$^{\,11}$, M. Destefanis$^{\,8}$, J. Díaz$^{\,16}$, F. Dohrmann$^{\,5}$, A. Dybczak$^{\,3}$, L. Fabbietti$^{\,11}$, O. Fateev$^{\,6}$, P. Finocchiaro$^{\,1}$, P. Fonte$^{\,2,b}$, J. Friese$^{\,11}$, I. Fröhlich$^{\,7}$, T. Galatyuk$^{4}$, J. A. Garzón$^{\,15}$, R. Gernhäuser$^{\,11}$, A. Gil$^{\,16}$, C. Gilardi$^{\,8}$, M. Golubeva$^{\,10}$, D. González-Díaz$^{\,4}$, E. Grosse$^{\,5,c}$, F. Guber$^{\,10}$, M. Heilmann$^{\,7}$, T. Hennino$^{\,13}$, R. Holzmann$^{\,4}$, A. Ierusalimov$^{\,6}$, I. Iori$^{\,9,d}$, A. Ivashkin$^{\,10}$, M. Jurkovic$^{\,11}$, B. Kämpfer$^{\,5}$, K. Kanaki$^{\,5}$, T. Karavicheva$^{\,10}$, D. Kirschner$^{\,8}$, I. Koenig$^{\,4}$, W. Koenig$^{\,4}$, B. W. Kolb$^{\,4}$, R. Kotte$^{\,5}$, A. Kozuch$^{\,3,e}$, A. Krása$^{\,14}$, F. Křížek$^{\,14}$, R. Krücken$^{\,11}$, W. Kühn$^{\,8}$, A. Kugler$^{\,14}$, A. Kurepin$^{\,10}$, J. Lamas-Valverde$^{\,15}$, S. Lang$^{\,4}$, J. S. Lange$^{\,8}$, K. Lapidus$^{\,10}$, L. Lopes$^{\,2}$, M. Lorenz$^{\,7}$, T. Liu$^{\,13}$, L. Maier$^{\,11}$, A. Mangiarotti$^{\,2}$, J. Marín$^{\,15}$, J. Markert$^{\,7}$, V. Metag$^{\,8}$, B. Michalska$^{\,3}$, J. Michel$^{\,7}$, D. Mishra$^{\,8}$ E. Morinière$^{\,13}$, J. Mousa$^{\,12}$, C. Müntz$^{\,7}$, L. Naumann$^{\,5}$, R. Novotny$^{\,8}$, J. Otwinowski$^{\,3}$, Y. C. Pachmayer$^{\,7}$, M. Palka$^{\,4}$, Y. Parpottas$^{\,12}$, V. Pechenov$^{\,8}$, O. Pechenova$^{\,8}$, T. Pérez Cavalcanti$^{\,8}$, J. Pietraszko$^{\,4}$, W. Przygoda$^{\,3,e}$, A. Reshetin$^{\,10}$, A. Rustamov$^{\,4}$, A. Sadovsky$^{\,10}$, P. Salabura$^{\,3}$, A. Schma$^h{\,11}$, R. Simon$^{\,4}$, Yu.G. Sobolev$^{\,14}$, S. Spataro$^{\,8}$, B. Spruck$^{\,8}$, H. Ströbele$^{\,7}$, J. Stroth$^{\,7,4}$, C. Sturm$^{\,7}$, M. Sudol$^{\,13}$, A. Tarantola$^{\,7}$, K. Teilab$^{\,7}$, P. Tlustý$^{\,14}$, M. Traxler$^{\,4}$, R. Trebacz$^{\,3}$, H. Tsertos$^{\,12}$, I. Veretenkin$^{\,10}$, V. Wagner$^{\,14}$, M. Weber$^{\,11}$, M. Wisniowski$^{\,3}$, J. Wüstenfeld$^{\,5}$, S. Yurevich$^{\,4}$, Y.V. Zanevsky$^{\,6}$, P. Zhou$^{\,4}$, P. Zumbruch$^{\,5}$]{}' title: 'Study of elementary reactions with the HADES dielectron spectrometer [^1]' --- Introduction ============ The main objective of the High-Acceptance di-Electron Spectrometer at GSI is the study of in-medium modifications of $\rho$ and $\omega$ vector mesons in hot and/or dense baryonic matter. Despite the challenging instrumental requirements, the dilepton probe provides the most direct information on the hadronic matter. Being complementary to the ones performed at higher energy facilities (SPS,RHIC) or looking for effects at normal density with photon or proton beams (JLab, KEK), the HADES experiments explore the 1-2 AGeV energy domain, where moderate temperatures (T$ <$ 100 MeV) and baryonic densities up to 3 times the normal nuclear matter density can be achieved, with expected sizeable modifications of $\rho$ and $\omega$ meson spectral functions. In contrast to reactions at ultrarelativistic energies, the multiplicity of produced pions per participant remains quite small, of the order of 10$\%$. This presents major advantages, since the main source of combinatorial background is the conversion of photons from $\pi^0 \rightarrow \gamma\gamma$ or $\pi^0 \rightarrow \gamma e^+e^-$. Another specificity of the SIS-18 energy regime is the important role played by baryonic resonances. Due to the very long life-time (15 fm/c) of the dense hadronic matter phase, the resonance can propagate and regenerate and the modification of its spectral function inside the baryonic medium is therefore an important issue for transport model calculations. The $\Delta(1232)$ resonance, which is responsible for a dominant part of the pion production, is the most copiously produced, but as the incident energy increases, higher lying resonances will play an increasing role. While all of them contribute to pion production, the N(1535) for example is important for the $\eta$ production and the N(1520), $\Delta(1620)$ and others for the $\rho$ production. Through the direct dilepton decay ($\rho / \omega \rightarrow $ [e$^+$e$^-$]{}) or Dalitz decay ($\pi^0 / \eta \rightarrow \gamma$[e$^+$e$^-$]{} or $\omega \rightarrow \pi^0 $[e$^+$e$^-$]{}) modes of these mesons, the baryonic resonances therefore play a crucial role in dilepton emission. They are also expected to contribute directly to dilepton emission via their own Dalitz decay modes. For example, the $\Delta(1232)$ should present a Dalitz decay ($\Delta \rightarrow N e^+e^-$) branching ratio of 4.2 10$^{-5}$, according to QED calculations. As it has never been measured up to now, the experimental study of this decay mode is an experimental challenge. In addition, the [$\Delta$]{} Dalitz decay process is in principle sensitive to the electromagnetic structure of the N-$\Delta$ transition and the kinematics is suited to test the Vector Dominance Models. On the other hand, another important dilepton source in this energy range is the nucleon-nucleon Bremsstrahlung NN$\rightarrow$NN[e$^+$e$^-$]{}, which adds coherently to the [$\Delta$]{} Dalitz decay. Section 2 will show how the first results of HADES in heavy-ion experiments motivate the study of elementary pp and pd reactions. In sect. 3, the [$\Delta$]{} Dalitz decay and NN virtual bremsstrahlung are discussed. Section 4 is devoted to the description of the experimental set-up and results from inclusive pp and quasi-free pn reactions. Exclusive measurements in pp collisions and perspectives of pion beam experiments are presented in sect. 5 and 6, respectively. First results from heavy-ion experiments ======================================== The first results from the HADES collaboration for the $^{12}$C+$^{12}$C reaction at 1 and 2 AGeV [@Agakishiev06_CC2GeV; @Agakishiev07_CC1GeV] have marked an important turning point. In these reactions, the dilepton production shows an excess in the intermediate mass range 0.15-0.6 GeV/c$^2$ with respect to the long-lived source contribution, which is mainly due to the $\eta$ Dalitz decay and is well constrained by experimental measurements. Such a dilepton excess had been already observed, more than 10 years ago, by the DiLepton Spectrometer (DLS) experiment at Berkeley [@Porter97] in the $^{12}$C+$^{12}$C reaction at 1 AGeV and remained unexplained over years, the situation being known as the “DLS Puzzle”. Taking into account the much smaller DLS acceptance, a direct comparison of the two data sets was performed, showing very good agreement [@Agakishiev07_CC1GeV]. This confirmation of the DLS controversial results triggered new transport model calculations [@Bratko08; @Thomere07Santini08Schmidt09Barz09] which are now able to reproduce the dilepton spectra measured in the $^{12}$C+$^{12}$C reactions by DLS and HADES. In particular, the new HSD results are using the recent bremsstrahlung calculation from [@Kaptari06], which is a factor 2-4 higher than other calculations. According to the authors of [@Bratko08], this provided the solution to the “DLS puzzle”. However, this bremstrahlung prediction is contradicted by other approaches [@Shyam03]. In addition, other transport model calculations reproduce quite well the excess with different relative contributions of bremstrahlung and [$\Delta$]{} Dalitz decay processes. More selective experimental constraints on these specific dilepton sources seem therefore necessary to achieve a satisfactory explanation of the intermediate mass dilepton production in the C+C system. This is even more important for heavier systems, like Ar+KCl, recently investigated by HADES [@Krizek09] and where medium effects are looked for, since a reliable reference for “vacuum” dilepton production above $\eta$ contribution is needed. This motivates the study of the p+p and quasi-free n+p reactions with HADES experiments at 1.25 GeV, i.e. below the $\eta$ production threshold. Another unavoidable requirement for interpretation of dilepton spectra in terms of medium effects, is a careful description of the vector meson production, which implies the knowledge of inclusive meson production cross-section. The first results [@Krizek09] from the Ar+KCl reaction indeed show that this input might need to be readjusted in transport models. To measure it, the dilepton spectra in the p+p reaction at 3.5 GeV have been analyzed. The results are still too preliminary, therefore, the focus will be in the following on the analysis of p+p and n+p reactions at 1.25 GeV. Results obtained in exclusive analysis at 2.2 GeV will also be presented in sect. 6. [$\Delta$]{} Dalitz decay and NN Bremsstrahlung =============================================== Different theoretical approaches -------------------------------- The description of these processes has to combine the electromagnetic vertex, including the electromagnetic structure of the involved hadrons, parametrized by form factors and the nucleon-nucleon interaction. The Soft Photon Approximation (SPA) [@Gale87] offers a possible way to take both aspects into account, with a factorization of the photon emission probability and of the strong interaction process. It was found to be in reasonable agreement, at least for the pn case, where this process is the most important, with more complete calculations [@Shyam03]. As a consequence, SPA is still widely used in transport model calculations, the Dalitz decay dilepton yield being calculated independantly and added incoherently. Although the differential decay width of the [$\Delta$]{} Dalitz decay process derives in principle unambiguously from the QED vertex, different expressions can be found in the litterature, as stressed in [@Krivoruchenko01]. We checked ourselves these calculations and could confirm the expressions of [@Krivoruchenko01] and [@Zetenyi02]. Consistent spectra are provided by the other descriptions at the [$\Delta$]{} resonance mass pole. However, taking as an example a mass of 1.480 GeV/c$^2$, corresponding to the kinematical limit for a pp reaction at 1.25 GeV, variations of the differential Dalitz decay width of 30$\%$ for M$_{ee}$ close to zero to 65$\%$ at M$_{ee}$=0.5 GeV/c$^2$ are observed[@Wolf90]. A reliable description of these processes can only be accessed in full quantum mechanical and gauge invariant calculations. Two One Boson Exchange (OBE) calculations [@Kaptari06; @Shyam09], which fulfill these requirements were performed recently and the yields were found a factor about 2-3 higher in [@Kaptari06] than in [@Shyam09] for both pp and pn reactions, the second calculation being much closer to the SPA predictions. An experimental check of these predictions is therefore needed to clarify the situation. As seen on fig.\[fig|kaptari\] for an incident energy of 1.25 GeV, the [$\Delta$]{} graphs are widely dominant in the case of the pp reaction, except above 400 MeV/c$^2$, while, for the pn reaction, the nucleon graphs are much more important. These qualitative features are common to both models, and show that, by measuring dilepton spectra in pp and np reactions, a selective sensitivity to these different graphs can be obtained. Electromagnetic form-factors ---------------------------- Further important elements are the electromagnetic form factors, which are of two types, namely the elastic nucleon form factor and the N-[$\Delta$]{} transition form factors. In both cases, the electromagnetic vertex is time-like, since the four-momentum transfer squared q$^2$, which is equal to the squared dilepton mass, is a positive quantity. The influence of elastic nucleon form factors taken in Vector Dominance Models (VDM) has been studied in [@Kaptari06]. We will here discuss in more details the case of the N-[$\Delta$]{} transition form factors. While, for negative four-momentum transfer squared (space-like region), the three N-[$\Delta$]{} transition form factors (G$_E$, G$_M$ and G$_C$ as electric, magnetic and Coulomb form factors respectively) have been measured in pion- or photo-production experiments in a quite wide range of q$^{2}$, the time-like region is unexplored. Here, the q$^2$ dependence can therefore only be given by models, constrained by the fact that the form factors have to be analytical functions of q$^{2}$, and should reproduce the available space-like data. Due to the small q$^2$ values probed by the dilepton production in our reactions, the major requirement is that the values of the form factors at q$^2$=0 should be in agreement with the values from pion photoproduction experiments (photon point) and correlatively the radiative decay width ($\Gamma(\Delta \rightarrow \gamma N)=0.66$ MeV $\pm$ 0.06 MeV) should be well reproduced, as in [@Krivoruchenko01; @Zetenyi02]. However, the kinematical region probed by the [$\Delta$]{} Dalitz decay (q$^2<$ 0.3 (GeV/c)$^2$ for an incident energy of 1.25 GeV) is of high interest to check the Vector Dominance. In such a model, the electromagnetic baryon form factors present structures in the vicinity of the $\rho$ meson mass, which might be probed by the [$\Delta$]{} Dalitz decay process. Analysis tool for p+p and quasi-free n+p reactions {#sec|simul} -------------------------------------------------- The new developments of our event generator PLUTO [@Froehlich07; @Dohrmann09] were exploited in order to build efficient tools for the interpretation of our data. Two different approaches were followed: The first one is based on the observation that, at an energy of 1.25 GeV/nucleon, pions are mostly produced through intermediate [$\Delta$]{} resonances. In analogy with the description of the [$\pi^0$]{} and [$\Delta$]{} Dalitz decay in transport model calculations, it provides a description of the following channels: $$\begin{aligned} \Delta^+\rightarrow p\pi^0 \rightarrow p\gamma e^+ e^- \ ; \qquad \Delta^+\rightarrow pe^+e^-\\ \Delta^0\rightarrow n\pi^0 \rightarrow n\gamma e^+ e^- \ ; \qquad \Delta^0\rightarrow n e^+e^- \end{aligned}$$ The cross sections of all the [$\pi^0$]{} and related [$\Delta$]{}$^+$ and [$\Delta$]{}$^0$ channels are taken from the resonance model [@Teis97], which describes the existing data [@Teis97; @Bistricky81], including as we will see in sect.\[exclusive\], the new measurements by HADES in the hadronic channels pp $\rightarrow pp\pi^{0}$. The details of the [$\Delta$]{} production and decay are given in [@Froehlich07; @Dohrmann09]. For the [$\Delta$]{} Dalitz decay differential width (d$\Gamma$/dM), the expression from [@Krivoruchenko01] was adopted, as explained above and two options for the N-[$\Delta$]{} transition form factors are provided: either a constant magnetic form factor (G$_M$=3, in agreement with the photon-point measurements), or the two-component quark-model (fig.\[fig|iachello\]) [@Wan0508], which is mainly driven by the Vector Dominance in our energy range. The second approach aims at a direct comparison with the OBE predictions. Hence, the differential cross sections (d$\sigma$/dM) provided by the models [@Kaptari06; @Shyam09] have been parameterized, an isotropic virtual photon emission was further assumed and corrections due to Final State Interaction of the two outgoing nucleons were included. To simulate the quasi-free n+p reaction, the available energy in the center of mass was smeared to include the neutron momentum distribution in the deuteron using the Paris potential and the energy dependence of the cross-sections was taken into account. When the center of mass energy of the pn system exceeds the $\eta$ threshold, its production is also taken into account, with cross sections taken from existing data [@Moskal08]. The generated events are then filtered by the detector acceptance in order to compare to the experimental data. Experimental set-up ------------------- The HADES (High Acceptance Dielectron Spectrometer) detector (fig. \[fig|set-up\]) consists in 6 identical sectors covering the full azimuthal range and polar angles between 18$^{\circ}$ and 85$^{\circ}$, hence providing a lepton pair acceptance of the order of 0.35. A detailed description can be found in [@Agakishiev09_techn], thus only the main features are given here. Momentum measurement derives from the particle trajectory reconstruction using four Mini-Drift Chambers (two before and two after the magnetic field zone) providing a position resolution of about 140 $\mu$m per cell and a measured dilepton invariant mass resolution of about 2.4$\%$ at the $\omega$ meson mass. A hadron-blind Ring Imaging CHerenkov detector (RICH), made by a C$_4$F$_{10}$ gas radiator and CsI photocathodes placed around the target region is used for electron [b]{}[6.cm]{} identification, together with Time Of Flight (TOF/TOFINO) and an electromagnetic pre-shower detector (Pre-Shower). Particle identification is also provided using the correlation between time-of-flight and momentum for charged pions and protons and using in addition the width of the time signal in the MDC’s for charged kaons. Time of flight measurements in a Forward Wall scintillator hodoscope (FW) located 7 m downstream the target was used in d+p reactions. It allowed indeed for the detection of forward emitted particles with the characteristics of spectator protons in order to select quasi-free n+p reactions. The first level trigger selects events within a defined charged particle multiplicity range, while the second level trigger corresponds to electron candidates defined by RICH and Pre-Shower/TOF information. In the case of the d+p experiment, the first level trigger also requires a coincidence with at least one particle in the FW. A 5 cm long liquid-hydrogen target (1$\%$ interaction probability) and proton and deuteron beams with intensities up to 10$^7$ particles/s were used. Data analysis ------------- e$^+$e$^-$ pairs are selected using different criteria to check the track and ring qualities, as well as the identification of the electron and positron. The combinatorial background, which arises from double conversion of [$\pi^0$]{} decay photons, conversion of the photon emitted in the [$\pi^0$]{} Dalitz decay, or multi-pion decays, was obtained as the arithmetic mean of like-sign [e$^+$e$^+$]{} and [e$^-$e$^-$]{} pairs and was subtracted from the measured [e$^+$e$^-$]{} sample. The correlated pairs from photon conversion are also removed, using a lower limit of 9$^{\circ}$ on the opening angle of the pair. Detection and efficiency corrections, based on GEANT simulations, are also applied, and the final spectra are normalized using the elastic (or quasi-elastic) pp scattering measured simultaneously by HADES. The overall normalization error is estimated to be 9$\%$, the systematic error to about 20$\%$, with a possible smooth invariant mass dependence. In the case of the d+p experiment, a condition on the momentum (1.6 GeV/c $<$ p$_{FW} <$ 2.6 GeV/c) and on the angle ($0.3^{\circ}<\theta_{FW}<6^{\circ}$) of the particle detected in the FW is added. Results and comparison to models --------------------------------- Figure \[fig|pppn\] shows the dilepton mass spectra measured in the pp and quasi-free np reactions [@Agakishiev09_elem] compared to the simulations as described in sect. \[sec|simul\]. For both reactions, there is a good agreement between the dilepton yield measured at low invariant masses and the simulation of the [$\pi^0$]{} Dalitz decay, which confirms the normalization and analysis procedures. In the case of the pp reaction, the region of invariant masses larger than 140 MeV/c$^2$ is also well described by the simulation of the [$\Delta$]{} Dalitz decay. An even better agreement is obtained when the two-component quark model is used instead of the constant magnetic form factor(G$_M$=3), which illustrates the sensitivity of these data to the electromagnetic structure of the N-[$\Delta$]{} transition. However, the description of [$\Delta$]{} Dalitz decay in this resonance model is to crude to extract direct information on the time-like N-[$\Delta$]{} transition form factor. A more accurate description is expected from the OBE models, since they take into account all graphs involving intermediate [$\Delta$]{} or nucleons. In these models, constant form factors are used, but defined using different covariants than the usual magnetic, electric and coulomb form factors. This induces a different q$^2$ dependence of the differential width. This effect has again an influence at the high invariant mass end of the spectrum. The predictions of [@Shyam09] (shown as dashed line) are indeed in pretty good agreement with the data. The other OBE model [@Kaptari06] (full line) overestimates the data. The shape of the spectrum changes dramatically when going from p+p to n+p interactions. In the mass region between 0.15 and 0.35 GeV/c$^2$, the yield is about a factor 9 higher in the case of the n+p reaction, while only a factor 2 is expected for the [$\Delta$]{} Dalitz decay contribution due to the isospin factors. The resonance model simulation widely underestimates the measured dilepton yield. The $\eta$ Dalitz decay contribution is rather small and the inclusion of the N-[$\Delta$]{} transition form factor model does not help either. Nevertheless, this simulation is missing the nucleon-nucleon bremsstrahlung contribution which is expected to be in the case of the pn system much larger than in the case of pp. The comparison to the OBE exchange models is thus more relevant, but no satisfactory agreement is achieved, even with the model of [@Shyam09], despite its good behaviour in the case of the pp data. To select more strictly quasi-free reactions, a smaller angular selection ($0.3^{\circ} < \theta_{FW} < 2 ^{\circ}$) has been applied, with no change in the shape of the invariant mass distribution. No clarification was provided either by the transverse momentum and rapidity spectra, which present very similar shapes as compared to the p+p reaction. These results are also in agreement with the DLS spectra measured in pp and pd reactions [@Wilson98], with lower statistics and precision. In the case of the pd reaction, only indirect confirmation could be obtained through the comparison of the same models, while in the case of the pp reaction at 1.04 GeV and 1.27 GeV, the direct comparison was possible [@Galatyuk09], showing a very good agreement. The interpretation of the pn dilepton spectra is still the subject of theoretical investigations, related for example, to possible $\rho$ or $\omega$ meson off-shell production by higher-lying resonances. These dilepton spectra measured in pp and quasi-free np experiments are used to build a reference spectrum defined by $0.5/\sigma^{NN}_{\pi^0}(d\sigma^{pp}_{ee}/dM_{ee}+ d\sigma^{pn}_{ee}/M_{ee})$, where $\sigma^{NN}_{\pi^0}$ is the mean inclusive [$\pi^0$]{} cross-section in a nucleon-nucleon collision. After subtraction of the $\eta$ contributions and normalization to the [$\pi^0$]{} multiplicities, the dilepton spectra measured in the C+C systems at 1 and 2 AGeV [@Agakishiev09_elem; @Galatyuk09; @Stroth09] are compatible with this reference spectrum [@Stroth09], which hints to the fact that the excess dilepton yield measured in C+C systems is due to some additionnal dilepton source already present in the np system. Exclusive channels in elementary reactions {#exclusive} ========================================== Dedicated exclusive channels can be isolated, by exploiting the capability of HADES to measure charged hadrons. For example, the $\pi^0$ and $\eta$ Dalitz decays could be studied in pp$\rightarrow $pp[e$^+$e$^-$]{}$\gamma$ reactions at 2.2 GeV, where the four charged particles are detected, the photon is reconstructed using missing four-momentum and the meson [b]{}[7.5 cm]{} is identified by the 2 proton missing mass. The helicity angle $\alpha$ is then defined as the angle between the momentum vectors of the lepton in the virtual photon ($\gamma^{\star}$) frame and of the $\gamma^{\star}$ in the decay meson rest frame. After acceptance corrections, these angular distributions are in agreement with a 1+ cos$^2\alpha$ distribution (fig.\[fig|helicities\]). This is a very nice experimental check of the trend which is expected from QED, considering that, in the decay of these pseudoscalar mesons, only transverse photons can be produced. In the pp reaction at 1.25 GeV, the on-going analysis of the pp $\rightarrow$ pp[e$^+$e$^-$]{} channel is expected to bring more detailed information on the [$\Delta$]{} Dalitz decay process and pp Bremsstrahlung, like p[e$^+$e$^-$]{} invariant masses, or lepton helicity angular distribution. Hadronic channels are also intensively studied, since they provide analysis checks, but also new physics results. The detection of both protons from p+p elastic scattering allows for tracking efficiency and momentum resolution measurements. Moreover, the exclusive production of unstable particles, which present a known leptonic or Dalitz decay branching ratio, can be studied both in pp$\rightarrow$ pp[e$^+$e$^-$]{}X channels, and in purely hadronic channels, which is suited for a cross-check of the analysis efficiencies, while producing new measurements of the production cross-sections. This possibility has been exploited in the case of $pp\rightarrow pp\pi^0$ reaction at 1.25 and 2.2 GeV and $pp\rightarrow pp\eta$ reactions at 2.2 GeV. While the $\eta$ case is still investigated, the exclusive $\pi^0$ production cross-section determined both in hadronic and Dalitz decay channel is found in very good agreement with existing data. Moreover, the exclusive pp$\rightarrow$pn$\pi^+$ and pp$\rightarrow $pp$\pi^0$ measurements provide very detailed checks of the resonance model used for the analysis of the dilepton spectra [@WisniaLiu09]. As shown on fig.\[fig|invariantmass\], the yields and invariant mass distributions are in good agreement with the simulations using an event generator based on the resonance model (see sec.\[sec|simul\]). The dominant contribution comes from the [$\Delta$]{} resonance excitation, but N(1440) and N(1520) play also a significant role at 2.2 GeV. At 1.25 GeV, the relation $\sigma$(pp$\rightarrow$ pn$\pi^+$) = 5 $\sigma$(pp$\rightarrow$ pp$\pi^0$) is fulfilled, as expected from the isospin factors in the different [$\Delta$]{} decay channels. The angular distributions are also carefully studied, since they carry detailed information on the mechanisms of [$\Delta$]{} resonance excitation beyond the one-pion exchange. Perspectives from pion induced reactions ======================================== The dilepton spectroscopy in $\pi$ induced experiments on nuclei is proposed in order to study medium effects on $\rho$ and $\omega$ mesons, with the advantages, with respect to heavy-ion induced reactions, of higher expected effects on $\omega$ meson and reduced combinatorial background. This would also complement the on-going studies of $\rho / \omega$ production at normal nuclear density in the p+Nb system. Due to the well-known interaction and the possibility of exclusive channel measurements, the reactions on nucleon constitute a unique tool to study $\omega$ and $\rho$ production, with a special interest of subthreshold production via the coupling to baryonic resonances. In particular, in [@Lutz03Titov01], a spectacular destructive $\rho$/$\omega$ interference is predicted below the $\omega$ threshold. As these couplings are related to the electromagnetic structure of the resonances, these measurements present a fundamental interest. Strangeness production measurements with pion beam induced reactions is also possible. This includes $\Lambda(1405)$ production in $\pi$-p reactions at an incident momentum around 1.7 GeV/c, in medium modifications and K$^-$ absorption in $\pi^-$-p and $\pi^-$-A reactions at 1.7 GeV/c, and K$^0_S$ production in $\pi^-$+p, $\pi^-$+C and $\pi^-$+Pb at lower energies. From the technical point of view, some developments are still needed to check the feasibility of these experiments. An intensity of 10$^6$ particles/s is needed, which is in principle accessible and fast and thin position sensitive beam detectors are under study to fully reconstruct the trajectory and momentum of incident pions. Conclusion ========== Recent HADES results have been discussed, with emphasis on the elementary reactions, which allow to build a reference for the heavy-ion measurements, and help to clarify the controversial problem of contribution of [$\Delta$]{} Dalitz decay and Bremsstrahlung processes. The interpretation of the inclusive pn spectra remains however challenging. More selective information on the [$\Delta$]{} Dalitz decay and bremsstrahlung processes is expected from the analysis of the exclusive pp$\rightarrow$pp[e$^+$e$^-$]{} reaction. HADES is currently being upgraded in order to handle more efficiently the higher multiplicities related to the heavier systems, like Ag+Ag and Au+Au. Pion beam experiments also offer interesting perspectives to clarify the role of higher lying resonances. Although the specificity of HADES is the dilepton spectroscopy with a large angular acceptance and good precision, the variety of all these measurements demonstrates the power of HADES as a multipurpose detector. [10]{} G. Agakichiev [*et al.*]{} (HADES coll.), Phys. Rev. Lett. [**98**]{}, 052302 (2007). G. Agakichiev [*et al.*]{} (HADES coll.), Phys. Lett. [**B663**]{}, 43 (2008). R. Porter [*et al.*]{} (DLS coll.), Phys. Rev. Lett. [**79**]{}, 1229 (1997). E. Bratkovskaya and W. Cassing, Nucl. Phys. [**A807**]{}, 214 (2008), [**]{}. M. Thomere, C. Hartnack, G. Wolf, and J. Aichelin, Phys. Rev. [**C75**]{}, 064902 (2007). E. Santini [*et al.*]{}, Phys. Rev. [**C78**]{}, 034910 (2008). K. Schmidt [*et al.*]{}, Phys. Rev. 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--- abstract: 'We explore a few-body mixture of two bosonic species confined in quasi-one-dimensional parabolic traps of different length scales. The ground state phase diagrams in the three-dimensional parameter space spanned by the harmonic length scale ratio, inter-species coupling strength and particle number ratio are investigated. As a first case study we use the mean-field ansatz (MF) to perform a detailed analysis of the separation mechanism. It allows us to derive a simple and intuitive rule predicting which of the immiscible phases is energetically more favorable at the miscible-immiscible phase boundary. We estimate the critical coupling strength for the miscible-immiscible transition and perform a comparison to correlated many-body results obtained by means of the Multi-Layer Multi-Configuration Time Dependent Hartree method for bosonic mixtures (ML-X). At a critical ratio of the trap frequencies, determined solely by the particle number ratio, the deviations between MF and ML-X are very pronounced and can be attributed to a high degree of entanglement between the components. As a result, we evidence the breakdown of the effective one-body picture. Additionally, when many-body correlations play a substantial role, the one-body density is in general not sufficient for deciding upon the phase at hand which we demonstrate exemplarily.' author: - Maxim Pyzh - Peter Schmelcher title: 'Phase separation of a Bose-Bose mixture: impact of the trap and particle number imbalance' --- Introduction {#sec:intro} ============ Binary mixtures of ultra-cold gases have been extensively studied over the past years. They represent a unique platform for the investigation of complex interacting many-body quantum systems in a well controlled environment. In particular, it is experimentally possible to shape the geometry of the trap [@TailoredTraps2000], to reduce the dimensionality of the relevant motion [@1Dgases2008; @1Dgases2011], to tune the inter-particle interactions [@Feshbach2010; @CIR1998; @CIR2000; @CIR2003; @CIR2010] and prepare samples of only a few atoms [@fewbody2012; @fewbody2019]. Numerous experiments have been conducted with different hyperfine states [@Myatt1997; @Hall1998; @Ketterle1999; @Inguscio2000; @Aspect2001; @Hall2007; @Hirano2010; @Becker2008; @Engels2011; @Oberthaler2015; @Hirano2016collision; @Hirano2016quench; @dropletsCabrera2018; @dropletsInguscio2018], different elements [@Inguscio2002; @Weidemuller2002; @Inguscio2008; @Ospelkaus2008; @Cornish2011; @Nagerl2011; @Grimm2013; @Nagerl2014; @Cornish2014; @Arlt2015; @Wang2015a; @Proukakis2018; @Wang2015b; @Minardi2010] or different isotopes [@Papp2008; @Takahashi2011] to reveal how the interplay between two condensates impacts their stationary properties and non-equilibrium dynamics. Highlights of these explorations include among others the phase separation between the components and symmetry-breaking phenomena [@Hall1998; @Papp2008; @Hirano2010; @Wang2015b; @Proukakis2018], the observation of Efimov physics [@Minardi2010] and creation of deeply bound dipolar molecules [@Ospelkaus2008; @Nagerl2014; @Cornish2014], as well as dark-bright solitary waves [@Becker2008; @Engels2011] and quantum droplets [@dropletsCabrera2018; @dropletsInguscio2018]. One of the key properties, which makes the multi-component systems attractive and their physics very rich, is the miscibility, which has significant implications for sympathetic cooling [@Aspect2001; @Weidemuller2002], coarse graining dynamics [@coarseGraining2004; @coarseGraining2008; @coarseGraining2010; @coarseGraining2014] and vortex formation [@vortex2003; @vortex2011] to name a few. In the very early theoretical investigations a very rich phase space for the ground state of the Bose-Bose mixture has been identified. These investigations [@TFAShenoy1996; @cGPEBigelow1998; @phasesChui2003; @phasesOhberg1999; @phasesTrippenbach2000] are based on the one-body densities obtained from solving the underlying mean-field equations, commonly known as Gross-Pitaevskii equations. In case of a weak inter-component coupling one finds a miscible phase with a high spatial overlap between the components. For a sufficiently large repulsive coupling there are three types of segragated phases with a rather small overlap. Two of them are core-shell phases with one component being symmetrically surrounded by the other component, whereas the third is an asymmetrical phase, where the rotational or parity symmetry of the underlying trapping potential is broken. Neglecting the kinetic energy (Thomas-Fermi approximation) a simple separation criterion for the miscible-immiscible transition has been derived [@separationRuleTimmermans1998; @separationRuleChui1998; @separationRuleEsry1997]. It depends solely on the intra-species and inter-species interactions strengths, which are easily adjustable by Feshbach or confinement induced resonances [@Feshbach2010; @CIR1998; @CIR2000; @CIR2003; @CIR2010]. However, it has been shown that this separation criterion, while valid in homogeneous systems, should be applied with care in inhomogeneous geometries. There, system parameters such as trap frequency, particle numbers or mass ratio, have also an impact on the miscible-immiscible phase boundary [@brokenRuleTrapKevrekidis2009; @brokenRuleTrapHu2012; @brokenRuleTrapProukakis2016; @brokenRuleMassBoronat2018; @brokenRuleImbalanceZhang2020]. From the intuitive point of view the trap pressure favors miscibility, since it costs energy to extend in space. Thus, it requires stronger inter-component repulsion for the species to separate. However, there are still open questions regarding the impact of different length scales, the characterization of boundaries between the immiscible phases and what type of separation will occur once the critical coupling is reached. Another relevant topic affecting the critical coupling strength for a transition as well as the resulting type of phase are the inter-species correlations, which generate entanglement between the components and lead to bunching of particles of the same species. Although a mean-field treatment is often justified in experimental setups, a very thorough numerical analysis of 1D few-body systems has revealed that an asymmetric immiscible phase is one of the two possible configurations of an entangled many-body state, the other one being the mirror image. The one body densities of this so-called composite fermionization phase [@phasesZollner2008; @phasesHao2008; @phasesPolls2014; @phasesZinner2015; @phasesPyzh2018] preserve parity symmetry of the underlying trapping potential and have a high spatial overlap, which is uncharacteristic for an immiscible phase. Nevertheless, the components are indeed separated, which is encoded in the inter-species two-body density matrix. In experiments, the single-shots do not represent one-body densities but are projections on one of the two mutually exclusive configurations. An averaging procedure would reveal a parity preserving density, unless the Hamiltonian itself violates that symmetry, such as not coinciding trap centers of the one-body potentials. Apart from composite fermionization, there is a whole class of so called spin-chain phases with an even higher degree of entanglement [@spinchain2014; @spinchain2015; @spinchain2016]. When all interactions in the system become nearly-resonant, many states become quasi-degenerate and particles, being bosons, gain fermionic features like the Pauli exclusion principle. Considering the above, our work addresses three different points. First, we characterize the phase diagram in a three-dimensional parameter space spanned by the ratio of the harmonic trap lengths, the inter-species coupling strength and the particle number ratio. We switch off intra-component interactions to reduce the complexity and gain a better understanding of the separation process. A very rich phase diagram is revealed admitting two tri-critical points, where three phases may coexist. Second, within the framework of a mean-field approximation, we perform a detailed analysis of the separation mechanism. Equipped with this knowledge we derive a selection rule for phase separation processes and a simple algorithm to estimate the miscible-immiscible phase boundary. Finally, we investigate the deviations of the mean-field picture to a many-body approach. For this we use the Multi-Layer Multi-Configuration Time-Dependent Hartree method for bosonic mixtures [@MLX2013; @MLX2013b; @MLX2017]. We find that in the vicinity of the high-entanglement regime the phase diagram is indeed greatly affected. The symmetry-broken phase is replaced by the composite fermionization, while the onset of symmetry-breaking is linked to the degree of entanglement reaching a certain threshold. Furthermore, the location of this beyond mean-field regime strongly depends on the harmonic length scale ratio and the particle number ratio. We also find that the one-body density is in general not sufficient to distinguish between a core-shell phase and the composite fermionization. This work is organized as follows. In Sec. \[sec:general\_setup\] we introduce our physical setup and in Sec. \[sec:methodology\] our computational approach. Sec. \[sec:few\_body\] is dedicated to a detailed study of a few-body mixture. Sec. \[sec:few\_body\_MF\] provides intuitive insights in the framework of the mean-field approximation, while Sec. \[sec:few\_body\_MLX\] focuses on correlation and entanglement effects using Multi-Layer Multi-Configuration Time-Dependent Hartree method for bosonic mixtures. The few-to-many-body transition is subject of Sec. \[sec:many\_body\]. Finally, we summarize our findings in Sec. \[sec:conclusions\]. General Framework {#sec:general_setup} ================= Our system consists of a particle-imbalanced mixture of two distinguishable bosonic components, denoted by $\sigma \in \{M,I\}$, with $N_M$ particles in the majority component and $N_I$ impurities. All particles are assumed to be of equal mass $m$ and the intra-component interactions are assumed to be zero or negligibly small. The latter can be achieved by means of magnetic Feshbach or confinement induced resonances [@CIR1998; @CIR2000; @CIR2003; @CIR2010; @Feshbach2010]. The majority species interacts with the impurities via s-wave contact interaction of coupling strength $g_{_{MI}}$. The species are confined in separate quasi-1D harmonic traps of different length scales $a_{\sigma}=\sqrt{\hbar/m \omega_{\sigma}}$ with trap frequency $\omega_{\sigma}$ and coinciding trap centers. This can be realized by using two different hyperfine states of $^{87}\rm{Rb}$ and species-dependent optical potentials [@speciestrap2007; @speciestrap2019]. We remark that many of the present results can be translated to the mass-imbalanced case. By choosing $a_{M}$ and $\hbar \omega_{M}$ as length and energy scales we arrive at the rescaled Hamiltonian: $$\begin{aligned} \label{eq:hamiltonian} H && = H_M + H_I + H_{MI} \\ && = \sum_{i=1}^{N_M} \left( -\frac{1}{2} \frac{\partial^2}{\partial x_i^2} + \frac{1}{2} x_i^2 \right) + \sum_{i=1}^{N_I} \left( -\frac{1}{2} \frac{\partial^2}{\partial y_i^2} + \frac{1}{2} \eta^2 y_i^2 \right) + \nonumber \\ && \quad + g_{_{MI}} \sum_{i=1}^{N_M} \sum_{j=1}^{N_I} \delta(x_i-y_j), \nonumber\end{aligned}$$ where $x_i$ labels the spatial coordinate of the $i$-th majority particle, $y_i$ of the $i$-th impurity particle and $\eta=\omega_I/\omega_M$ denotes the trap frequency ratio. In the present work we focus on the ground state characterization and consider both attractive and repulsive interactions ranging from weak to intermediate couplings $g_{_{MI}}\in[-2,2]$ with the impurity being localized or delocalized w.r.t. the majority species, i.e.  $a_I/a_M=\sqrt{1/\eta} \in [0.5, 1.5]$. We also study the impact of the particle number ratio $N_I/N_M$ on the system’s properties. Computational Approach {#sec:methodology} ====================== To find the ground state of our binary mixture we employ imaginary time propagation by means of the Multi-Layer Multi-Configurational Time-Dependent Hartree Method for atomic mixtures (ML-MCTDHX). For reasons of brevity we call it ML-X from now on. This multi-configurational wave function based method for efficiently solving the time-dependent Schroedinger equation was first developed for distinguishable degrees of freedom [@MCTDH2000] and ML-X is an extension to indistinguishable particles such as bosons or fermions and mixtures thereof [@MLX2013; @MLX2013b; @MLX2017]. ML-X is an ab-initio method, whose power lies in expanding the wave-function in time-dependent basis functions. Let us demonstrate the underlying ansatz for the system at hand: $$\ket{\Psi(t)} =\sum_{i=1}^{S}\sqrt{\lambda_i(t)} \ket{\Psi_i^M(t)} \otimes \ket{\Psi_i^I(t)}, \label{eq:wfn_ansatz_species_layer}$$ $$\ket{\Psi_i^{\sigma}(t)} = \sum_{\vec{n}^{\sigma}|N_\sigma} C_{i, \vec{n}^{\sigma}}(t) \ket{\vec{n}^{\sigma}(t)}. \label{eq:wfn_ansatz_particle_layer}$$ The time-dependent many-body wave-function $\ket{\Psi(t)}$ has two layers: the so-called species layer and the particle layer . In the first step we separate majority and impurity species and assign them to $S \in \mathbb{N}$ corresponding species wave-functions $\ket{\Psi_i^{\sigma}(t)}$. The time-dependent coefficients $\lambda_i(t)$ are normalized $\sum_{i=1}^{S} \lambda_i(t) = 1$ and describe the degree of entanglement between the components. In case $\exists \ i\in\{1,\dots,S\} : \lambda_i(t) \approx 1$ the components are said to be disentangled. In the second step each species wave-function $\ket{\Psi_i^{\sigma}(t)}$, which depends on $N_{\sigma}$ indistinguishable coordinates, is expanded in terms of species-dependent symmetrized product states, also known as permanents or number states, $\ket{\vec{n}^{\sigma}}=\ket{n_1^{\sigma},\ldots, n_{s_{\sigma}}^{\sigma}}$ admitting $s_{\sigma}\in \mathbb{N}$ normalized single particle functions (SPF) $\ket{\varphi_{j}^{\sigma}(t)}$. The sum is over all possible configurations $\vec{n}^{\sigma}|N_\sigma$ fulfilling the constraint $\sum_{i=1}^{s_{\sigma}} n_i^{\sigma} = N_{\sigma}$. The time dependence of number states is meant implicitly through the time-dependence of the underlying SPFs. Finally, each SPF is represented on a primitive one-dimensional time-independent grid [@DVR1985]. When one applies the Dirac-Frenkel variational principle [@DiracFrenkel2000] to the above ansatz, one obtains coupled equations of motion for the expansion coefficients $\lambda_i(t)$, $C_{i, \vec{n}^{\sigma}}(t)$ and the SPFs $\ket{\varphi_{j}^{\sigma}(t)}$. This procedure allows to considerably reduce the size of the basis set as compared to choosing time-independent SPFs constituiting the number states on the particle layer . We note that $S=1 \land s_{\sigma}=1$ is equivalent to solving coupled Gross-Pitaevskii equations. We will show parameter regions, where the mean-field description is valid and regions where it fails as a result of increasing interspecies correlations. These generate entanglement between the components and decrease the degree of condensation of the non-interacting majority atoms. In the following we will often refer to the one-body density $\rho_1^{\sigma}(z)$ of species $\sigma$, two-body density matrix $\rho_2^{\sigma}(z,z')$ of species $\sigma$ and inter-species two-body density matrix $\rho_2^{MI}(x,y)$ of the many-body density operator $\hat{\rho} = \ket{\Psi}\bra{\Psi}$ defined as: $$\begin{aligned} \rho_1^{\sigma}(z) &&= \bra{z} \tr_{_{N_{\sigma} \setminus 1}} \{\tr_{_{N_{\bar{\sigma}}}} \{{\hat{\rho}}\}\} \ket{z} \label{eq:rho1}\\ \rho_2^{\sigma}(z,z') &&= \bra{z,z'} \tr_{_{N_{\sigma} \setminus 2}} \{\tr_{_{N_{\bar{\sigma}}}} \{\hat{\rho}\}\} \ket{z,z'} \label{eq:rho2}\\ \rho_2^{MI}(x,y) &&= \bra{x,y} \tr_{_{N_M \setminus 1}}\{\tr_{_{N_I \setminus 1}} \{\hat{\rho}\}\} \ket{x,y},\label{eq:rho2inter}\end{aligned}$$ where $N_{\sigma} \setminus n$ stands for integrating out $N_{\sigma}-n$ coordinates of component $\sigma$ and $\bar{\sigma} \neq \sigma$. Phase separation: few body mixture {#sec:few_body} ================================== We start our analysis with a few-body system consisting of $N_M = 5$ majority particles with $N_I\in\{1,2\}$ impurities. First, we uncover the mechanism responsible for the phase separation on the mean-field level and subsequently we perform a comparison to the correlated many-body treatment by means of ML-X. Mean-field approach: Basic mechanism of phase separation {#sec:few_body_MF} -------------------------------------------------------- For the mean field description we choose a single species orbital $S=1$, yielding a non-entangled state $\ket{\Psi(t)} = \ket{\Psi^M(t)} \otimes \ket{\Psi^I(t)}$ on the species layer. On the particle layer a single SPF $s_{\sigma}=1$ is used for each component, meaning that particles of the same species are forced to condense into the same single-particle state $\varphi^{\sigma}(z,t)$ and $\ket{N_{\sigma}}$ is the only possible number state on the particle level. Thus, our ansatz is $\ket{\Psi(t)} = \ket{N_{M}(t)} \otimes \ket{N_{I}(t)}$ and only $\varphi^{\sigma}(z,t)$ are time-dependent. As a result of imaginary time propagation we end up with the ground state orbitals $\varphi_{_{MF}}^{\sigma}(z)$. The interpretation of the mean-field treatment is that each species feels in addition to its own external potential an averaged one-body potential induced by the other component. To obtain the effective mean-field Hamiltonian $H_{\sigma}^{MF}$ of species $\sigma$ we need to integrate out the other component $\bar{\sigma}$. For convenience we also subtract the energy offset $c_{\bar{\sigma}}=\braket{N_{\bar{\sigma}} | H_{\bar{\sigma}} | N_{\bar{\sigma}}}$ caused by the one-body energy of component $\bar{\sigma}$: $$\begin{aligned} \label{eq:MF_hamiltonian} \nonumber H_{\sigma}^{MF} && = \braket{N_{\bar{\sigma}}|H|N_{\bar{\sigma}}} - c_{\bar{\sigma}} = H_{\sigma} + N_{\bar{\sigma}} g_{_{MI}} \sum_{i=1}^{N_{\sigma}} \rho_{_{MF}}^{\bar{\sigma}}(z_i) \\ \nonumber && = \sum_{i=1}^{N_{\sigma}} \left(-\frac{1}{2} \frac{\partial^2}{\partial z_i^2} + V_{\sigma} (z_i) + V_{\sigma}^{ind} (z_i)\right) \\ && = \sum_{i=1}^{N_{\sigma}} \left(-\frac{1}{2} \frac{\partial^2}{\partial z_i^2} + V_{\sigma}^{eff} (z_i)\right),\end{aligned}$$ where $\rho_{_{MF}}^{\sigma}(z) = |\varphi_{_{MF}}^{\sigma}(z)|^2$ is the one-body density of species $\sigma$ normalized as $\int dz \ \rho_{_{MF}}^{\sigma}(z) = 1$, $V_{\sigma}^{ind} (z)=N_{\bar{\sigma}} g_{_{MI}} \rho_{_{MF}}^{\bar{\sigma}}(z)$ the induced one-body potential, $V_{\sigma}^{eff} (z)=V_{\sigma} (z)+V_{\sigma}^{ind} (z)$ the effective one-body potential and $\bar{\sigma}\neq\sigma$. To systematically distinguish between different phases we define the following two functions, applicable also in the more general case of a many-body treatment in Sec. \[sec:few\_body\_MLX\]: $$\label{quali1} \Delta_{\sigma} = \frac{\rho^{\sigma}_{1} (z=0)}{\underset{z}{\rm{max}} \ \rho^{\sigma}_1 (z)}$$ $$\label{quali2} d = |\int_{-\infty}^{\infty} dz \ z \rho_{1}^{M}(z) - \int_{-\infty}^{\infty} dz \ z \rho_{1}^{I}(z)|,$$ with the one-body density $\rho_1^{\sigma}(z)$ of component $\sigma$ . Eq.  compares the one-body density $\rho_1^{\sigma}(z)$ at the trap center with its maximum value, while eq.  checks for parity asymmetry, as we will argue below. The above equations are motivated from the literature on binary mixtures and we provide a brief summary on the discovered ground state phases and some of their properties, which will be relevant in the following discussions. For weak couplings there is a miscible phase $M$ with a high spatial overlap of the one-body densities $\rho_1^{\sigma}(z)$. As a result both components exhibit a Gaussian profile ($\Delta_{\sigma}=1$) and occupy the center of their trap ($d=0$). The state is disentangled and both species are condensed. For negative couplings, i.e. attractive interactions, the phase remains miscible and the widths of the Gaussian densities shrink with decreasing coupling strength. For stronger positive couplings three different phase separation scenarios are possible. In case the majority species occupies the trap center ($\Delta_{M}=1$), pushing the impurities outside in a way that the impurity density forms a shell around the majority density with two parity-symmetric humps ($\Delta_{I}<1$ and $d=0$), we have a core-shell $IMI$ phase. When the impurities remain at the trap center instead ($\Delta_{I}=1$) with the majority species forming a shell ($\Delta_{M}<1$ and $d=0$), we have a core-shell $MIM$ phase. Finally, when both species develop two parity-symmetric humps with a local minimum at the trap center ($\Delta_{\sigma}<1$ and $d=0$) we have a composite fermionization phase $CF$. On the level of one-body densities $CF$ appears to be miscible owing to the high spatial overlap between the components. However, the deviations to the miscible phase become evident upon investigating the two-body density matrices and . Namely, two particles of the same component can be found either on the left or the right side w.r.t. trap center, while two particles of different components are always on opposite sides. While the core-shell phases $IMI$ and $MIM$ do not rely on entanglement between the components, $CF$ is always an entangled many-body state made out of two major species orbitals $S=2$ and two major SPFs $s_{\sigma}=2$ on the particle layer. Thus, $CF$ cannot be obtained within the mean-field approximation. In fact, we observe that once the entanglement of the true many-body state, characterized by the von Neumann entropy (see Sec. \[sec:few\_body\_MLX\]), reaches a certain threshold, a collapse to a phase with broken parity symmetry ($d>0$) will take place in the mean-field picture. We abbreviate this phase with $SB$ from now on. The origin of $SB$ is the onset of a quasi-degeneracy between the ground state and the first excited state of the many-body spectrum, which becomes an exact degeneracy in the limit of $g_{_{MI}} \to \infty$. Once this limit is reached, any superposition of those two states is also an eigenstate of . Since they are of different parity symmetry $P$ and $[H,P]=0$, it is possible to choose the superposition to be parity symmetric or to break the parity symmetry of . It was suggested [@phasesZollner2008] that the corresponding many-body wave-function may be written in terms of number states as $\ket{\Psi} =c_1 \ket{N_M}_L \ket{0_M}_R \otimes \ket{0_I}_L \ket{N_I}_R + c_2 \ket{0_M}_L \ket{N_M}_R \otimes \ket{N_I}_L \ket{0_I}_R$ with two parity-broken SPFs $\varphi^{\sigma}_j(z)$ featuring an asymmetric Gaussian shape with a maximum on the j=L(eft) or j=R(ight) side w.r.t. the trap center. Within the mean-field approximation the eigenenergy of the first excited state coincides with the ground state energy already for a finite coupling $g_{_{MI}}$. Since mean-field does not incorporate entanglement, the state collapses either to $\ket{N_M}_L \otimes \ket{N_I}_R$ or to $\ket{N_M}_R \otimes \ket{N_I}_L$, resulting in a phase with broken parity symmetry. With this we end our overview over different phases and showcase a compact summary of the phases: $$\label{eq:classification} \begin{cases} \text{M} & \mbox{:} \ d = 0 \ \land \Delta_{_{M}} = 1 \ \land \Delta_{_{I}} = 1 \\ \text{IMI} & \mbox{:} \ d = 0 \ \land \Delta_{_{M}} = 1 \ \land \Delta_{_{I}} < 1 \\ \text{MIM} & \mbox{:} \ d = 0 \ \land \Delta_{_{M}} < 1 \ \land \Delta_{_{I}} = 1 \\ \text{CF} & \mbox{:} \ d = 0 \ \land \Delta_{_{M}} < 1 \ \land \Delta_{_{I}} < 1 \\ \text{SB} &\mbox{:} \ d > 0 \ \land \Delta_{_{M}} < 1 \ \land \Delta_{_{I}} < 1 \end{cases}$$ In Fig. \[fig:MF\_few\_body.eps\] we depict the ground state phases within the mean-field approximation for $N_B=5$ with a) $N_I=1$ and b) $N_I=2$ impurities as a function of the inter-component coupling strength $g_{_{MI}}$ and the impurity localization $a_{I}/a_{M}$. As expected $CF$ is not among the phases in Fig. \[fig:MF\_few\_body.eps\]. The transition region on the $a_{I}/a_{M}$ axis, where core-shell $MIM$ is replaced by core-shell $IMI$, can be tuned by variation of the particle number ratio such that for $N_I=N_M$ it lies at $a_{I}/a_{M}=1$ (not shown), while for increasing particle imbalance $N_I/N_M<1$ it is shifted towards a lower $a_{I}/a_{M}$ ratio. This is also the point, where the coupling strength $g_{_{MI}}$, required for the realization of the $SB$ phase, is the smallest. We will see later in Sec. \[sec:few\_body\_MLX\] that the species entropy has here its global maximum. Note that each phase diagram features critical points, where three different phases can coexist (green circles). {width="0.75\linewidth"} Now that we have identified the phases, we are going to shed some light on the mechanism behind the phase separation taking place for different specific coupling strength $g_{_{MI}}$ for a fixed trap ratio $\eta$. In particular, we will provide a simple formula, which determines which of the core-shell structures is energetically more favorable. Additionally, we provide an estimate on the miscible-immiscible transition region and on the $SB$ phase boundary. Let us make two horizontal cuts across the phase diagram of Fig. \[fig:MF\_few\_body.eps\]b) at $a_I/a_M=0.5$ and at $a_I/a_M=1.1$. In Fig. \[fig:dw\_picture.eps\] we take a closer look at the variation of the one-body densities $\rho_{_{MF}}^{\sigma}$ being part the effective one-body potential $V_{\sigma}^{eff}$ when increasing the coupling strength $g_{_{MI}}$. First, let us focus on columns 1 and 2, corresponding to $a_I/a_M=0.5$. For very weak coupling (first row), every atom to a good approximation populates the energetically lowest harmonic oscillator orbital of the respective potential $V_{\sigma}$. The induced potential $V_{\sigma}^{ind}$ gains an amplitude linearly with $g_{_{MI}}$ and with the density profiles being Gaussians of different widths we observe the appearance of a small barrier in $V_M^{eff}$ at $g_{_{MI}}=0.2$ (Fig. \[fig:dw\_picture.eps\] a2). This barrier grows with $g_{_{MI}}$ and at $g_{_{MI}}=0.4$ (Fig. \[fig:dw\_picture.eps\] a3) it becomes comparable to the ground state energy of the effective potential, while the one-body density $\rho_{_{MF}}^M$ turns flat at the trap origin. Once the ground state energy drops below the barrier height, two density humps appear and core-shell $MIM$ is established (Fig. \[fig:dw\_picture.eps\] a4). Meanwhile, the effective potential of the impurity $V_I^{eff}$ does not show significant deviations from the harmonic case (second column). Especially, the induced part $V_I^{ind}$, being initially also a Gaussian, is not capable to produce a barrier at the trap center. Similar statements can be made for columns 3 and 4, corresponding to $a_I/a_M=1.1$. The only difference is that $V_I^{eff}$ develops a barrier instead, whereas $V_M^{eff}$ shows only a slight variation, which finally leads to the core-shell $IMI$ phase. {width="1.0\linewidth"} Motivated by the above observation we define an alternative phase classification from an energetical point of view: $$\label{eq:classification_energy} \begin{cases} \text{M} & \mbox{:} \ E_{_{0,\sigma}}^{MF}-V_{\sigma}^{eff}(0) > 0 \\ \text{IMI} & \mbox{:} \ E_{_{0,M}}^{MF}-V_{M}^{eff}(0) > 0 \ \land \ E_{_{0,I}}^{MF}-V_{I}^{eff}(0) < 0 \\ \text{MIM} & \mbox{:} \ E_{_{0,M}}^{MF}-V_{M}^{eff}(0) < 0 \ \land \ E_{_{0,I}}^{MF}-V_{I}^{eff}(0) > 0 \\ \text{SB} &\mbox{:} \ d > 0 \end{cases}$$ where $E_{_{0,\sigma}}^{MF}$ is the ground state energy of . As long as the ground state energy of the effective species Hamiltonian exceeds the effective potential height at the trap center, the species remains at the trap center. Phase diagrams produced this way match exactly the ones shown in Fig. \[fig:MF\_few\_body.eps\]. The interpretation is now as follows. For a very weak coupling both the majority and the impurity reside in the ground state of the harmonic oscillator. Once the induced potential $V_{\sigma}^{ind}$ of species $\sigma$ becomes large enough to produce a barrier in $V_{\sigma}^{eff}$, the corresponding density $\rho_{_{MF}}^{\sigma}$ will start to expand. By growing in width it will prevent the other component $\bar{\sigma}$ from developing a barrier of its own. When the height of the potential barrier becomes of the same magnitude as the lowest energy of the corresponding effective potential, the species $\sigma$ splits into two fragments. Then it starts squeezing the other component $\bar{\sigma}$ by increasing the effective trap frequency of the renormalized harmonic oscillator $V_{\bar{\sigma}}^{eff}$. The barrier in $V_{\sigma}^{eff}$ appears once the following condition is fulfilled: $$\label{eq:barrier_criterion} \exists \ x_0 \neq 0 : \frac{d}{dx} V_{\sigma}^{eff} \Bigr\rvert_{x_0}= 0.$$ Assuming one-body densities to be unperturbed harmonic oscillator ground states, we obtain the following effective potentials: $$\begin{aligned} \label{eq:barrier_criterion_M_pot} && V_{M}^{eff}(z) \approx \frac{1}{2}z^2 + g_{_{MI}} N_I \sqrt{\frac{\eta}{\pi}} e^{-\eta z^2} \\ \label{eq:barrier_criterion_I_pot} && V_{I}^{eff}(z) \approx \frac{1}{2}\eta^2 z^2 + g_{_{MI}} N_M \sqrt{\frac{1}{\pi}} e^{-z^2},\end{aligned}$$ and the corresponding barrier conditions: $$\begin{aligned} \label{eq:barrier_criterion_M} && \frac{\sqrt{\pi}}{2 N_I \sqrt{\eta^3}} \hat{=} g_{_{MI}}^M < g_{_{MI}} \\ \label{eq:barrier_criterion_I} && \frac{\sqrt{\pi} \eta^2}{2 N_M} \hat{=} g_{_{MI}}^I< g_{_{MI}}.\end{aligned}$$ For given particle numbers $N_M$, $N_I$ and trap ratio $\eta$ either condition or condition will be fulfilled first upon increasing the coupling $g_{_{MI}}$ and thus either the majority or the impurity will form a shell. We remark that the above criterion for barrier formation is inversely proportional to the particle number of the other component, while the dependence on the trap ratio $\eta$ for the majority differs substantially from the one for the impurity. Furthermore, for a fixed particle number ratio there is a critical trap ratio $\eta_c$, for which and can be fulfilled simultaneously. $$\sqrt{1/\eta_c}=\sqrt[7]{N_I/N_M}$$ Around this critical region we expect that none of the components will occupy the trap center. We summarize our findings in a simple formula, which determines the type of phase separation at the miscible-immiscible phase boundary: $$\label{eq:critical_trap} \begin{cases} \text{core shell MIM} & \mbox{:} \ \eta \gg \eta_c \\ \text{core shell IMI} & \mbox{:} \ \eta \ll \eta_c \\ \text{CF or SB} & \mbox{:} \ \eta \approx \eta_c \\ \end{cases}$$ For particle number ratios discussed in this section, the critical region lies at $a_I/a_M \approx 0.8$ (Fig. \[fig:MF\_few\_body.eps\]a) and at $a_I/a_M \approx 0.9$ (Fig. \[fig:MF\_few\_body.eps\]b). Next, we want to find an estimate for the miscible-immiscible phase boundary $g^c_{_{MI}}$. To this end we combine the energetical separation criterion in with approximate effective potentials from and . Specifically, for a given particle number ratio $N_I/N_M$ we determine the critical trap ratio $\eta_c$. Then depending on the choice of $\eta$ we solve numerically for the ground state energy of a single particle inside the effective potential or . Finally, we compare this energy to the potential height at the trap center: $$\label{eq:critical_g} \begin{cases} a_I/a_M < \sqrt[7]{N_I/N_M} & \mbox{:} \ H_M^{eff}= -\frac{1}{2} \frac{\partial^2}{\partial x^2} + V_{M}^{eff}(x) \begin{cases} E^{eff}_{_{0,M}} > g_{_{MI}} N_I \sqrt{\frac{\eta}{\pi}} & \Rightarrow \ \text{M} \\ E^{eff}_{_{0,M}} < g_{_{MI}} N_I \sqrt{\frac{\eta}{\pi}} & \Rightarrow \ \text{MIM} \\ \end{cases} \\ \\ a_I/a_M > \sqrt[7]{N_I/N_M} & \mbox{:} \ H_I^{eff}=-\frac{1}{2} \frac{\partial^2}{\partial y^2} + V_{I}^{eff}(y) \begin{cases} E^{eff}_{_{0,I}} > g_{_{MI}} N_B \sqrt{\frac{1}{\pi}} & \Rightarrow \ \text{M} \\ E^{eff}_{_{0,I}} < g_{_{MI}} N_B \sqrt{\frac{1}{\pi}} & \Rightarrow \ \text{IMI} \\ \end{cases} \\ \end{cases}$$ The results are plotted as blue solid curves in Fig. \[fig:MF\_few\_body.eps\]. We recognize that it performs quite well except for $\eta \approx \eta_c$, where it underestimates $g^c_{_{MI}}$. We can also get a rough estimate on the $SB$ phase boundary $g_{_{MI}}^{^{SB}}$ by using the following Gaussian ansatz: $$\varphi^{\sigma}(z)= \sqrt[4]{\frac{\beta_{\sigma}}{\pi}} e^{-\frac{\beta_{\sigma}(z-z_\sigma)^2}{2}},$$ with the width $\beta_{\sigma}$ and the displacement $z_\sigma$ of component $\sigma$ being variational parameters. We evaluate the expectation value of and minimize the energy w.r.t. the above variation parameters. By looking further at the special case when the relative position $|z_M-z_I|$ becomes zero one arrives after some algebraic transformations at: $$\label{eq:SP_boundary} g_{_{MI}}^{^{SB}} N_I = \frac{\sqrt{\pi}}{2 \eta} \sqrt[4]{\frac{\gamma}{1+\gamma \eta^2}} (1+\eta^2 \sqrt{\gamma})^{\frac{3}{2}},$$ with particle number ratio $\gamma=N_I/N_M$. We remark that this equation reduces to eq. (8) from [@brokenRuleImbalanceZhang2020] for $\eta=1$. Although this equation describes well the qualitative behavior of the $SB$ phase boundary, quantitatively it scales badly when the trap ratio $\eta$ deviates from $\eta_c$ (blue dotted line in Fig. \[fig:MF\_few\_body.eps\]). There are two possible reasons for this. First, our ansatz incorporates only $M$ and $SB$ phases, while ignoring the core-shell phases. Thus, as one draws away from $\eta_c$ the core-shell parameter region, which lies in-between $M$ and $SB$, grows in size making the estimate inefficient. The other reason is that the mean-field solution $\varphi^{\sigma}_{_{MF}}$ of the $SB$ phase is rather an asymmetric Gaussian. Finally, we discuss the limiting cases. When $\eta \rightarrow \infty$ ($a_{I}/a_{M} \rightarrow 0$) the impurity becomes highly localized at $z=0$. It will not be affected by the majority atoms. Meanwhile the majority species will be subject to an additional delta-potential at $z=0$ with potential strength $g_{_{MI}} N_I$. This analytically solvable one-body problem results in a Weber differential equation. Upon increasing the delta-potential pre-factor $g_{_{MI}} N_I$ the initially unperturbed Gaussian solution develops a cusp at the trap center, whose depth tends to zero as the pre-factor goes to infinity. When $\eta \rightarrow 0$ ($a_{I}/a_{M} \rightarrow \infty$), we can change our perspective by rescaling the Hamiltonian in impurity harmonic units and argue in a similar way as above. In the following section we compare to the results obtained for the corresponding correlated many-body approach of ML-X. ML-X: modifications of the phase diagram due to correlations and entanglement {#sec:few_body_MLX} ----------------------------------------------------------------------------- For the total wave-function in we use $S=8$ species orbitals and $s_{\sigma}=8$ SPFs for each component. We perform again an imaginary time-propagation of an initially chosen wave-function and obtain the ground state of . In Fig. \[fig:MLX\_few\_body.eps\] we show the resulting ground state phases based on the selection rules for $N_B=5$ and a) $N_I=1$ or b) $N_I=2$. We remark that the alternative selection scheme defined in does not apply here and below we provide an explanation why it fails. The first eye-catching feature is that the $SB$ phase has completely disappeared, as expected, since it is an artifact of the mean-field treatment. Additionally, we observe the presence of composite fermionization $CF$ for the case of two impurities in Fig. \[fig:MLX\_few\_body.eps\] b). Overall, the transition between the miscible phase and separated phases takes places at a different coupling strength $g_{_{MI}}^c$ for a fixed trap ratio $\eta$. {width="0.75\linewidth"} In order to better understand why the phase diagram is altered this way, we investigate in Fig. \[fig:entanglement.eps\] the von-Neumann entropy $S_{vN}$ on the species layer (first row) as well as the von-Neumann entropy of the majority species $S_{vN}^{M}$ (second row) and the impurity species $S_{vN}^{I}$ (third row). $S_{vN}$ characterizes the degree of entanglement between the components (entanglement entropy) while $S_{vN}^{\sigma}$ reflects the degree of species fragmentation (fragmentation entropy). The definitions are as follows: $$\begin{aligned} \label{eq:species_entropy} S_{vN} && = - \sum_{i=1}^{S} \lambda_i \ln{\lambda_i} \\ \label{eq:particle_entropy} S_{vN}^{\sigma} && = - \sum_{i=1}^{s_{\sigma}} m_i \ln{m_i} \; \; \text{with} \ \hat{\rho}_1^{\sigma} =\sum_{i=1}^{s_{\sigma}} m_i \ket{m_i}\bra{m_i}\end{aligned}$$ where $\lambda_i$ are expansion coefficients from , $m_i$ natural populations satisfying $\sum_{i=1}^{s_{\sigma}}m_i=1$ and $\ket{m_i}$ natural orbitals of the spectrally decomposed one-body density operator $\hat{\rho}_1^{\sigma}$. The entanglement entropy is bounded by the equal distribution of orbitals $S_{vN} \leq \ln(S)$, whereas for two dominantly occupied orbitals we expect $S_{vN} \leq \ln(2)\approx0.7$. If $S_{vN}=0$, then there is no entanglement between the species and the wave-function is a simple product state on the species layer. Similarly, fragmentation entropy $S_{vN}^{\sigma}=0$ means that all particles occupy the same SPF and the species is thus condensed. For parameter values where this is fulfilled a mean-field treatment is well justified. However, in Fig. \[fig:entanglement.eps\] we recognize that for stronger couplings $g_{_{MI}}$ this is not the case. Particularly, in the vicinity of the critical region $a_I/a_M \approx \sqrt[7]{N_I/N_M}$ at positive $g_{_{MI}}$, identified in the previous section as highly competitive, the entanglement entropy $S_{vN}$ is very pronounced (Fig. \[fig:entanglement.eps\] first row). The fragmentation entropy of the majority species $S_{vN}^M$ is comparatively weaker and slightly shifted towards a smaller length scale ratio $a_I/a_M$ at positive $g_{_{MI}}$ (Fig. \[fig:entanglement.eps\] second row). The fragmentation entropy of the impurity species $S_{vN}^I$ for $N_I=1$ (not shown) coincides with the entanglement entropy $S_{vN}$ (Fig. \[fig:entanglement.eps\]a), while for $N_I=2$ there are substantial differences (see Fig. \[fig:entanglement.eps\]e). Namely, the impurity shows a higher degree of fragmentation when it is less confined compared to the majority species and vice versa. In contrast to positive couplings $g_{_{MI}}$, for negative couplings the entanglement and species fragmentation build up with a much slower rate. Finally, we emphasize that phase separation like core-shell $MIM$ or $IMI$ are not necessarily related to a high degree of entanglement or species depletion, whereas $CF$ is located in the parameter region, where $S_{vN}$ takes the highest values. Another striking observation is that the onset of the $SB$ phase from Fig. \[fig:MF\_few\_body.eps\] is related to the entanglement entropy reaching some threshold value around $S_{vN}\approx0.5$ at positive couplings $g_{_{MI}}$ (compare to Fig. \[fig:entanglement.eps\] first row). {width="0.75\linewidth"} Now that we have identified the parameter space where deviations from mean-field are to be expected, we want to gain a deeper insight into how the effective picture is affected as a result of increasing correlations. For this purpose we define an effective single-body Hamiltonian of species $\sigma$ similar to the one in , except that we use the exact many-body densities $\rho_1^{\sigma}$ instead of the mean-field densities $\rho_{_{MF}}^{\sigma}$: $$\begin{aligned} \label{eq:projection_hamiltonian} \nonumber H_{\sigma}^{eff} && = H_{\sigma} + N_{\bar{\sigma}} g_{_{MI}} \sum_{i=1}^{N_{\sigma}} \rho_1^{\bar{\sigma}}(z_i) \qquad \text{with} \ \bar{\sigma}\neq\sigma\\ &&= \sum_{i=1}^{N_{\sigma}} \left(-\frac{1}{2} \frac{\partial^2}{\partial z_i^2} + V_{\sigma}^{eff} (z_i)\right)\end{aligned}$$ Next, we diagonalize and use the obtained eigenfunctions $\tilde{\varphi}_i^{\sigma}$ as SPFs for number states $\ket{\vec{n}^{M}} \otimes \ket{\vec{n}^{I}}$ on which we project our many-body ground state $\ket{\Psi}$. The reader should distinguish the latter SPFs $\tilde{\varphi}_i^{\sigma}$ from the numerical SPFs $\varphi_i^{\sigma}$ obtained by improved relaxation which define the permanents contained in our ML-X total wave-function. Thus, we decompose our ground state in terms of disentangled product states made out of single permanents. We anticipate that $\ket{N_M}\ket{N_I}$ represents dominant contribution to $\ket{\Psi}$, which should be the case whenever a mean-field approach is valid. From the previous analysis we observed that the entanglement entropy values were mostly $S_{vN}^{\sigma} \leq 0.7$, which suggests two relevant SPFs. Indeed, our many-body state consists of two major orbitals and two major SPFs. Furthermore, taking parity symmetry into account and considering at most two-particle excitations, we conclude that number states $\ket{N_M-1,1}\ket{N_I-1,1}$, $\ket{N_M-2,2}\ket{N_I}$ and $\ket{N_M}\ket{N_I-2,2}$ may become of relevance too at stronger couplings. We remark that the one-body density operator of number state $\ket{N_{\sigma}-n_2^{\sigma},n_2^{\sigma}}$ with $n_2^{\sigma}$ particles in the odd orbital $\tilde{\varphi}_2^{\sigma}$ will be a mixed state of one even and one odd orbital, eventually featuring two humps in the corresponding one-body density. Thus, depending on the occupation amplitude of such states, they may either accelerate or slow down the development of humps in $\rho_1^{\sigma}(z)$ thereby quantitatively shifting the critical coupling $g^c_{_{MI}}$, at which the mixed phase transforms into one of the species-separated phases. In Fig. \[fig:number\_states.eps\] we show the projection on number state $\ket{N_M}\ket{N_I}$ (first row) and a sum over projections on the above mentioned permanents (second row) for $N_B=5$ and $N_I=1$ (first column) or $N_I=2$ (second column). For negative couplings the state $\ket{N_M}\ket{N_I}$ provides a major contribution and the effective picture holds. Let us focus in the following on positive couplings. In Fig. \[fig:number\_states.eps\]a ($N_I=1$), we observe that the state $\ket{N_M}\ket{N_I}$ has indeed a major contribution at coupling strength below $1.0$. Once inter-species correlations build up with increasing coupling strength, the state $\ket{N_M-1,1}\ket{N_I-1,1}$ grows in importance, which corresponds to a simultaneous single-particle excitation within each component. This is mostly pronounced around $\eta_c$. Double excitations within the majority species $\ket{N_M-2,2}\ket{N_I}$ are of minor amplitude and rather of relevance for a localized impurity $a_I/a_M \ll 1$. All in all, the low-lying excitations of the effective potentials provide a good description (Fig. \[fig:number\_states.eps\]b). In Fig. \[fig:number\_states.eps\]c ($N_I=2$), we observe that the state $\ket{N_M}\ket{N_I}$ loses its contribution very quickly as one goes deeper into the regime of strong entanglement. Although we are able to get a better understanding for weak entanglement by including two-particle excitations mentioned above, our effective picture clearly breaks down for strong entanglement. There we may account only for as much as $\approx 50\%$ of the ground state, even though the one-body density in incorporates beyond mean-field corrections. {width="0.75\linewidth"} Let us take a closer look at this regime, where the single-particle picture tends to break down. We show in Fig. \[fig:eff\_picture\_breakdown\_densities.eps\] the one-body densities for $N_B=5$ majority particles and $N_I=2$ impurities in the strong entanglement region at $g_{_{MI}}=2$. The first row corresponds to the $CF$ phase at $a_I/a_M=0.8$. Here we recognize immediately, why the effective picture fails. The origin of the two humps in the one-body density is counter-intuitive considering that they are at the position of local maxima of the effective potential. One would rather expect a density profile with three peaks at the positions of the potential minima. The second row ($a_I/a_M=0.9$) seems at first glance to be an $IMI$ phase. The majority is at the core, while the impurity forms a shell. Upon a more detailed investigation we notice that the majority species is broader than it should be inside the squeezed “harmonic” trap. The humps of the impurity also do not coincide with the positions of the minima of the respective effective potential. As a matter of fact this phase is a latent $CF$ phase, which becomes clear when we analyse the corresponding two-body density matrices in Fig. \[fig:eff\_picture\_breakdown\_dmats.eps\]. The intra-species two-body density matrices (Fig. \[fig:eff\_picture\_breakdown\_dmats.eps\]a-b) indicate that particles of the same component avoid the trap center and form a cluster either on the right or the left side w.r.t. trap center. Moreover, the inter-species density matrix (Fig. \[fig:eff\_picture\_breakdown\_dmats.eps\]c) tells us that the two different clusters of majority and impurity will always be found on opposite sides of the trap with a rather small spatial overlap between them. It allows to diminish the impact of the repulsive energy on the total energy at the cost of paying potential energy. These are clear signatures of the $CF$, which are blurred in the reduced one-body density. We note that the parameter space where ML-X predicts an $IMI$ phase, whereas $MF$ produces $SB$ phase, we have in fact a latent $CF$, hidden behind a one-body quantity. {width="0.8\linewidth"} {width="0.9\linewidth"} Above, we have mentioned that in the literature the $CF$ phase was suggested to be a superposition of two parity-broken mean-field states $\ket{\Psi} =c_1 \ket{N_M}_L \ket{0_M}_R \otimes \ket{0_I}_L \ket{N_I}_R + c_2 \ket{0_M}_L \ket{N_M}_R \otimes \ket{N_I}_L \ket{0_I}_R$ as a result of the degeneracy onset. Indeed, ML-X has two prominent orbitals on the species layer and two major SPFs on the particle layer. Nevertheless, the other occupied species orbitals and SPFs provide a minor contribution, as we have evidenced in Fig. \[fig:entanglement.eps\] second column, where the entropies take values beyond $\ln(2)$. To provide an illustrative example we displace the trap centers in by a small amount to energetically separate the two symmetry broken configurations. For parameter values for which the $CF$ phase is observed, we perform again the improved relaxation to the find ground state of the system in order to check whether it is indeed a MF state. It turns out that the majority species and the impurity species are still fragmented states though the degree of depletion is much less compared to the parity-symmetric ground state. The species entropy $S_{vN}$ is greatly reduced, but still appreciable. The impact of correlations is also visible in Fig. \[fig:SB.eps\]. The ground state of the effective potential is different from the one-body density of the many-body ML-X wave-function. This is caused by induced attractive interactions mediated by the inter-component coupling, a beyond-mean field effect [@induced2018]. To conclude our discussion about the high-entaglement regime, we state that the mean-field approach, being an effective one-body model, fails to explain a one-body quantity such as reduced one-body density. Nevertheless, it manages to characterize quite well one of the two possible configurations of the entangled many-body state. The latter is not just a simple superposition of two mean-field states describing two different parity-broken configurations. A thourough analysis showed that on the many-body level the $SB$ phase is in fact slightly entangled, while each species is partially fragmented. We also evidenced that $CF$ completely dominates the highly correlated regime and made a link of its appearance to the onset of $SB$ on the mean-field level. Sometimes $CF$ is even camouflaged behind core-shell $IMI$ or $MIM$ densities, indicating that the one-body density is not enough to distinguish between them. {width="0.8\linewidth"} Phase separation: Impact of particle numbers {#sec:many_body} ============================================ When increasing the number of majority atoms $N_M$, while keeping $N_I$ fixed, one might expect two properties based on an intuition for few-body systems. First, the location of the strong entanglement regime will be shifted towards lower values of $a_I/a_M\approx\sqrt{1/\eta_c}=\sqrt[7]{N_I/N_M}$. Thus, the $IMI$ phase will cover the most part of our parameter space for positive $g_{_{MI}}$. Second, at a fixed $\eta$ the critical coupling $g_{_{MI}}^c$ for the miscible-immiscible transition will decrease, because according to the majority species will be able to induce a barrier for the impurity species already for a much weaker coupling. The induced barrier of the majority on the other hand will not be affected according to . Indeed, this is what we observe in the phase diagrams depicted in Fig. \[fig:many\_body.eps\]. In the mean-field (first column) the location of the $SB$ phase relocates from $\sqrt{1/\eta_c}\approx0.79$ ($N_M=5$ Fig. \[fig:MF\_few\_body.eps\]a) to $\sqrt{1/\eta_c}\approx0.72$ ($N_M=10$ Fig. \[fig:many\_body.eps\]a), then to $\sqrt{1/\eta_c}\approx0.65$ ($N_M=20$ Fig. \[fig:many\_body.eps\]b) and finally moves outside our parameter space $\sqrt{1/\eta_c}\approx0.37$ ($N_M=1000$ Fig. \[fig:many\_body.eps\]c). The blue curve, which estimates the miscible-immiscible transition according to is in good agreement (except for the critical region $\eta_c$) with the mean field phase boundary. We also recognize that for a fixed trap ratio $\eta$, the critical coupling strength $g_{_{MI}}^c$ decreases with increasing $N_M$ and at $N_M=1000$ a very small $g_{_{MI}}^c<0.05$ is sufficient to cause phase separation, which is below our resolution. We have also performed the corresponding ML-X calculations (second column) with $S=s_{\sigma}=6$ (first row), $S=s_{\sigma}=4$ (second row) and $S=s_{\sigma}=2$ (third row) orbitals. We remark that the latter case might not be converged to the exact solution, which is beyond numerical capabilities to verify. Still it provides valuable beyond mean-field corrections. The deviations to the mean-field, still clearly visible at $N_M=10$, are most pronounced near $\eta_c$. They become less as the particle-imbalance is increased until finally at $N_M=1000$ the phase diagrams almost coincide except for a small $SB$ region. This is mainly attributed to the fact that the strong entanglement regime, where deviations are to be expected, moves outside our parameter space ($a_I/a_M<0.5$). Furthermore, the deviations may still be there, but on a finer coupling scale $g_{_{MI}}<0.05$ according to and . {width="0.6\linewidth"} Conclusions {#sec:conclusions} =========== In this work we have investigated the phase-separation of a quasi-1D inhomogeneous Bose-Bose mixture in a three dimensional parameter space spanned by the inter-component coupling $g_{_{MI}}$, harmonic length scale ratio $a_I/a_M=\sqrt{1/\eta}$ and the particle number ratio $N_I/N_M$, when the intra-component couplings $g_{\sigma}$ are switched off. Although we have concentrated on the case of equal masses, our results may be easily extended to the more general case of unequal masses. We expect some quantitative changes, but the qualitative picture and the line of argumentation will remain unchanged. The commonly used separation criterion $g_{_{MI}}>\sqrt{g_{_M} g_{_I}}$, which is valid for homogeneous mixtures, would predict a miscible-immiscible transition for any finite coupling $g_{_{MI}}>0$. However, this separation rule does not apply here, since we have harmonic traps of different length scales. We have analyzed the mechanism, which leads to phase separation, by using an effective mean-field picture. Within this description each species is subject to an additional induced potential caused by the other component. This potential has initially a Gaussian shape and grows linearly with the coupling strength $g_{_{MI}}$. However, it does not immediately trigger a barrier at the center of the harmonic trap. In fact, the species, which first manages to induce a barrier for the other component upon increasing the coupling $g_{_{MI}}$, will stay at the center of its parabolic trap. Meanwhile the other species will split up, once the ground state energy of the effective potential drops below the barrier height. Thus, we end up with either a core-shell $IMI$ or a core-shell $MIM$ phase, except for a highly competitive region, where the barrier conditions can be met simultaneously for both components. We have derived a simple rule to predict the type of phase separation, developed a straightforward algorithm to identify the miscible-immiscible phase boundary $g_{_{MI}}^c$ and gave a rough estimate on the phase boundary between the segregated phases $g_{_{MI}}^{^{SB}}$. As a next step, we compared mean-field (MF) results to the numerically exact many-body calculations based on Multi-Layer Multi-Configurational Time-Dependent Hartree Method for atomic mixtures (ML-X). It turns out that MF agrees well with ML-X far away from the critical region $\sqrt{1/\eta_c}=\sqrt[7]{N_I/N_M}$. At $\eta_c$ there are considerable quantitative deviations and sometimes the two methods do not even agree on the type of phase separation. This is caused by the growing inter-particle correlations, which generate entanglement between the components and increase the degree of species fragmentation. We have seen that symmetry-broken phase ($SB$) is replaced by composite fermionization ($CF$), which is an entangled parity symmetric ground state. Furthermore, we have linked the onset of $SB$ to the fact that the entanglement entropy reaches a certain threshold and saw a clear breakdown of the effective single-particle picture in the strong entanglement region in terms of a corresponding number state analysis. This led to the discovery of a latent $CF$ phase in the $IMI$ region. The latent $CF$ phase has the characteristic one-body density of the $IMI$ phase, but a thorough analysis of the two-body densities reveals typical $CF$ features. We have argued that at a finite coupling $g_{_{MI}}$ the $CF$ is not a simple superposition of two $SB$ states given by mean-field. We have studied the impact of particle number variations, which confirmed our intuition that $\eta_c$ and thus the location of the strong entanglement regime can be manipulated as a function of the particle number ratio. Furthermore, for a fixed particle number ratio the critical coupling $g_{_{MI}}^c$ of the miscible-immiscible transition can be tuned to lower values by increasing the number of particles while keeping the particle number ratio fixed. Finally, we remark that an intriguing next step would be to perform a similar study of phase-separation at finite intra-component coupling $g_{\sigma}$. The broadening or shrinking of the density profiles, depending on the sign and strength of $g_{\sigma}$, will definitely modify the barrier conditions and . Another interesting but challenging direction would be the non-equilibrium dynamics by quenching the trap ratio across the phase boundaries. M. P. acknowledges fruitful discussions with K. Keiler and M. Roentgen. M. P. gratefully acknowledges a scholarship of the Studienstiftung des deutschen Volkes. 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--- abstract: '[ We define a class of orthosymplectic $osp(m;j|2n;\omega)$ and unitary $sl(m;j|n;\epsilon)$ superalgebras which may be obtained from $osp(m|2n)$ and $sl(m|n)$ by contractions and analytic continuations in a similar way as the special linear, orthogonal and the symplectic Cayley-Klein algebras are obtained from the corresponding classical ones. Casimir operators of Cayley-Klein superalgebras are obtained from the corresponding operators of the basic superalgebras. Contractions of $sl(2|1)$ and $osp(3|2)$ are regarded as an examples. ]{}' author: - | N. A. GROMOV, I. V. KOSTYAKOV, V. V. KURATOV\ Department of Mathematics,\ Syktyvkar Branch of IMM UrD RAS,\ Chernova st., 3a, Syktyvkar, 167982, Russia\ E-mail: [email protected] title: ON CONTRACTIONS OF CLASSICAL BASIC SUPERALGEBRAS --- §[ + ]{} Introduction ============ Since their discovery [@1], [@2], [@3] in 1971 the supersymmetry is used in different physical theories such as Kaluza–Klein supergravity [@W-86], supersymmetric field theories of the Wess–Zumino type [@K-75], massless higher-spin field theories [@Vas-90]. Recently the secret theory [@B-96] (or S-theory) that includes superstring theory and its super p-brane and D-brane [@BIK] generalizations was discussed. All these theories are build algebraically with the help of some superalgebra in their base. In this work we wish to present a wide class of Cayley–Klein (CK) superalgebras which may be used for constructions of different sypersymmetric models. For an ordinary Lie groups (or algebras) the title CK was initially used for the short name of the set of a motion groups of a spaces of constant curvature. It is well known that there are $3^n$ n-dimensional spaces of constant curvature and their motion groups may be obtained from the orthogonal group $SO(n+1)$ with the help of contractions and analytical continuations [@NG]. Later the notion CK was extend to the case of unitary and symplectic groups (algebras) [@JMP]. The typical (and attractive) property of CK groups is that all of them are depend on the same number of independent parameters as the corresponding simple classical group. On the level of Lie algebras this means that all CK algebras of the same type have the equal dimensions. A basic superalgebras include a simple algebras as an even subalgebras, so it looks quite natural to introduce a new class of superalgebras with CK algebras as an even subalgebras. A superalgebra as an algebraic structure contain (as compared with Lie algebra) a new additional operation, namely, $Z_2$-grading. So under contraction of superalgebra this $Z_2$-grading must be conserved. To our knowledge contraction of orthosymplectic superalgebra to the superkinematics was first regarded in [@Rem]. The detailed investigation a class of contraction of $osp(1|2)$ and $osp(1|4)$ to the kinematical Poincar$\acute{e}$ and Galilei superalgebras was made in [@Val-99]. Contraction of unitary superalgebra $Gsu(2)=sl(2|1)$ as well as their representations was described in [@Pat]. Later the notion of contraction was generalized [@MP] to the case of Lie algebra with an arbitrary finite grading group and is known as graded contractions. Nevertheless the particular case of the simplest $Z_2$-grading deserve an independent interest. The preliminary results was reported in [@GKK]. The paper is organized as follows. In section 2 the orthogonal, symplectic and special linear CK groups and algebras are briefly remind. Section 3 is devoted to the orthosymplectic CK superalgebras. CK unitary superalgebras are regarded in section 4. Casimir operators of the CK unitary and orthosymplectic superalgebras are described in section 5. Orthogonal, symplectic and special linear Cayley-Klein algebras ================================================================ Special linear $sl(m),$ orthogonal $so(m)$ and symplectic $sp(2n)$ algebras are even subalgebras of classical basic superalgebras. On the other hand all of them may be contracted and analytically continued to the set of CK algebras. Lie groups and algebras are in close relations. CK group $SO(m;j)$ is defined as the set of transformations of vector space ${\bf R}_m(j),$ which preserve the quadratic form $x^2(j)=x^t(j)x(j) =x_1^2+\sum_{k=2}^{m}(1,k)^2x^2_k, $ where $ (i,k)=\prod^{\max(i,k)-1}_{p=\min(i,k)}j_p, \, (i,i)=1, $ each parameter $j_k=1,\iota_k,i,$ where $\iota_k$ are nilpotent $ \iota^2_k=0,$ commutative $\iota_k\iota_p=\iota_p\iota_k \neq 0$ generators of Pimenov algebra ${\bf}P(\iota).$ Cartesian components of vector $x(j)\in {\bf R}_m(j)$ are $x^t(j)=(x_1,j_1x_2, \ldots ,(1,m)x_m)^t, $ as it is easily follows from $x^2(j).$ For $m\times m$ matrix $g(j) \in SO(m;j)$ the transformation $g(j): {\bf R}_m(j) \rightarrow {\bf R}_m(j)$ means that the vector $x'(j)=g(j)x(j)$ has exactly the same distribution of parameters $j$ among its components as $x(j).$ This requirement give an opportunity to obtain the distribution of parameters $j$ among elements of matrix $g(j),$ i.e. to build the fundamental representation of CK group $SO(m;j)$ starting from the quadratic form. It is remarkable that the same distribution of the parameters $j$ is hold for CK Lie algebra $so(m;j),$ namely $A_{ik}=(i,k)a_{ik},$ for $A \in so(m;j).$ The set of transformations $ L(j): {\bf R}_m(j)\to {\bf R}_m(j) $ with the property $ \det L(j)=1 $ form CK special linear group $ SL(m;j)$ and the corresponding CK algebras $ sl(m;j)$ are given by the $m \times m $ matricies $l(j),$ tr $ l(j)=0.$ Let us stress that in Cartesian basis all matricies from $ SL(m;j), SO(m;j), sl(m;j), so(m;j) $ have identical distribution of parameters $j$ among its elements, i.e. they are of the same type as the matricies with elements from Pimenov algebra $P(j).$ CK symplectic group $Sp(2n;\omega)$ is defined as the set of transformations of ${\bf R}_n(\omega) \times {\bf R}_n(\omega),$ which preserve the bilinear form $S(\omega)=S_1+ \sum_{k=2}^{n}[1,k]^2S_k,$ where $S_k(y,z)=y_kz_{n+k}-y_{n+k}z_k, \, [i,k]=\prod^{\max(i,k)-1}_{p=\min(i,k)} \omega_k, \, [i,i]=1, \, \omega_k=1,\xi_k,i, \, \xi^2_k=0, \, \xi_k\xi_p=\xi_p\xi_k.$ The distribution of parameters $\omega_k$ among matrix elements of the fundamental representation $M(\omega)=\left( \begin{array}{cc} H(\omega) & E(\omega) \cr F(\omega) & -H^t(\omega) \end{array} \right)$ of the CK symplectic algebra $sp(2n;\omega)$ may be obtained as for orthogonal CK algebras and is as follows: $B_{ik}=[i,k]b_{ik},$ where $B=H(\omega),E(\omega),F(\omega).$ Orthosymplectic superalgebras $osp(m;j|2n;\omega)$ =================================================== Let $e_{IJ} \in M_{m+2n}$ satisfying $(e_{IJ})_{KL}=\delta_{IK}\delta_{JL}$ are elementary matrices. One defines the following graded matrix [@Fra] $$G=\left ( \begin{array}{c|c} I_m & 0 \cr \hline 0 & 0 \quad I_n \cr & -I_n \quad 0 \end{array} \right ) \label{0}$$ where $I_m,I_n$ are identity matrices. Let $i,j,\ldots=1,\ldots ,m, \, \bar i,\bar j=m+1, \ldots, m+2n.$ The generators of the orthosymplectic superalgebra $osp(m|2n)$ are given by $$E_{ij}=-E_{ji}=\sum_{k}(G_{ik}e_{kj}-G_{jk}e_{ki}),\;\; E_{\bar i\bar j}=E_{\bar j\bar i}=\sum_{\bar k} (G_{\bar i\bar k}e_{\bar k\bar j}+ G_{\bar j\bar k}e_{\bar k\bar i}),\;\;$$ $$E_{i\bar j}=E_{\bar ji}=\sum_{k}G_{ik}e_{k\bar j}+ \sum_{\bar k}G_{\bar j\bar k}e_{\bar ki}, \label{1}$$ where the even (bosonic) $E_{ij}$ generate the $so(m)$ part, the even (bosonic) $E_{\bar i\bar j}$ generate the $sp(2n)$ part and the rest $E_{i\bar j}$ are the odd (fermionic) generators of superalgebra. They satisfy the following (super) commutation relations $$[E_{ij},E_{kl}]=G_{jk}E_{il}+G_{il}E_{jk}-G_{ik}E_{jl}-G_{jl}E_{ik},\;\;$$ $$[E_{\bar i\bar j},E_{\bar k\bar l}]=-G_{\bar j\bar k}E_{\bar i\bar l}- G_{\bar i\bar l}E_{\bar j\bar k}-G_{\bar j\bar l}E_{\bar i\bar k}- G_{\bar i\bar k}E_{\bar j\bar l},$$ $$[E_{ij},E_{k\bar l}]=G_{jk}E_{i\bar l}- G_{ik}E_{j\bar l},\;\; [E_{i\bar j},E_{\bar k\bar l}]=-G_{\bar j\bar k}E_{i\bar l}- G_{\bar j\bar l}E_{i\bar k},$$ $$[E_{ij},E_{\bar k\bar l}]=0, \quad \{E_{i\bar j},E_{k\bar l}\}= G_{ik}E_{\bar j\bar l}- G_{\bar j\bar l}E_{ik}. \label{2}$$ In the matrix form $ osp(m|2n)=\{M \in M_{m+2n}|M^{st}G+GM=0\}. $ If the matrix $M$ has the following form: $ %\begin{equation} M=\sum_{i,j}a_{ij}E_{ij} + \sum_{\bar i,\bar j}b_{\bar i\bar j} E_{\bar i\bar j} + \sum_{i\bar j}\mu_{i\bar j}E_{i\bar j}, %\label{3} %\end{equation} $ with $a_{ij}, b_{\bar i \bar j}\in $ [**R**]{} or [**C**]{} and $\mu_{i\bar j}$ as the odd nilpotent elements of Grassmann algebra: $\mu^2_{i\bar j}=0,\, \mu_{i\bar j}\mu_{i'\bar j'}=-\mu_{i'\bar j'}\mu_{i\bar j},$ then the corresponding supergroup $Osp(m|2n)$ is obtained by the exponential map $ {\cal M}=\exp M $ and act on (super)vector space by matrix multiplication ${\cal X}'={\cal M}{\cal X},$ where ${\cal X}^t=(x|\theta)^t,$ $x$ is a $n$–dimentsional even vector and $\theta$ is a $2m$–dimensional odd vector with odd Grassmann elements. The form $inv=\sum^m_{i=1}x^2_i+2\sum^n_{k=1} \theta_{+k}\theta_{-k}=x^2+2\theta^2$ is invariant under this action of orthosymplectic supergroup. We shall define CK orthosymplectic superalgebras starting with the invariant form $$inv(j;\omega) = u^2\sum^m_{k=1}(1,k)^2x^2_k + 2v^2\sum^{m+n}_{\bar k=m+1}[1,\hat {\bar k}]^2 \theta_{\hat {\bar k}}\theta_{-\hat {\bar k}}\equiv u^2x^2(j) + 2v^2\theta^2(\omega), \label{4}$$ $\hat {\bar k}=\bar k-m,$ when $ \bar k=m+1,\ldots ,m+n $ and $\hat {\bar k}=\bar k-m-n,$ when $ \bar k=m+n+1, \ldots ,m+2n,$ which is the natural unification of CK orthogonal and symplectic forms. The distributions of contraction parameters $j,\omega $ among matrix elements of the fundamental representation of $osp(m;j|2n;\omega)$ and the transformations of the generators (\[1\]) are obtained in a standart CK manner and are as follows: $$E_{ik}=(i,k)E^*_{ik}, \;\; E_{\bar i\bar k}=[\hat {\bar i},\hat {\bar k}]E^*_{\bar i\bar k},\;\; E_{i\bar k}=u(1,i)v[1,\hat {\bar k}]E^*_{i\bar k}, \label{5}$$ where $E^*$ are generators (\[1\]) of the starting superalgebra $osp(m|2n).$ The transformed generators are subject of the (super) commutation relations: $$[E_{ij},E_{kl}] = (i,j)(k,l) \left ( {{G_{jk}E_{il}} \over {(i,l)}} + {{G_{il}E_{jk}} \over {(j,k)}} - {{G_{ik}E_{jl}} \over {(j,l)}} - {{G_{jl}E_{ik}} \over {(i,k)}} \right ),$$ $$[E_{\bar{i}\bar{j}},E_{\bar{k}\bar{l}}] = -[\hat{\bar{i}},\hat{\bar{j}}][\hat{\bar{k}},\hat{\bar{l}}] \left ( {{G_{\bar{j}\bar{k}}E_{\bar{i}\bar{l}}} \over {[\hat{\bar{i}},\hat{\bar{l}}]}} + {{G_{\bar{i}\bar{l}}E_{\bar{j}\bar{k}}} \over {[\hat{\bar{j}},\hat{\bar{k}}]}} + {{G_{\bar{i}\bar{k}}E_{\bar{j}\bar{l}}} \over {[\hat{\bar{j}},\hat{\bar{l}}]}} + {{G_{\bar{j}\bar{l}}E_{\bar{i}\bar{k}}} \over {[\hat{\bar{i}},\hat{\bar{k}}]}} \right ),$$ $$[E_{ij},E_{\bar{k}\bar{l}}] = 0, \quad [E_{ij},E_{k\bar{l}}] = (i,j)(1,k) \left ( {{G_{jk}E_{i\bar{l}}} \over {(1,i)}} - {{G_{ik}E_{j\bar{l}}} \over {(1,j)}} \right ),$$ $$[E_{i\bar{j}},E_{\bar{k}\bar{l}}] = -[1,\hat{\bar{j}}][\hat{\bar{k}},\hat{\bar{l}}] \left ( {{G_{\bar{j}\bar{k}}E_{i\bar{l}}} \over {[1,\hat{\bar{l}}]}} + {{G_{\bar{j}\bar{l}}E_{i\bar{k}}} \over {[1,\hat{\bar{k}}]}} \right ),$$ $$\{E_{i\bar{j}},E_{k\bar{l}}\} =u^2v^2 (1,i)[1,\hat{\bar{j}}](1,k)[1,\hat{\bar{l}}] \left ( {{G_{ik}E_{\bar{j}\bar{l}}} \over {[\hat{\bar{j}},\hat{\bar{l}}]}} - {{G_{\bar{j}\bar{l}}E_{ik}} \over {(i,k)}} \right ). \label{6}$$ For $u=\iota$ or $v=\iota, \iota^2=0$ superalgebra $osp(m|2n)$ is contracted to inhomogeneous superalgebra, which is semidirect sum $ \{E_{i\bar{j}}\} \S (so(m) \bigoplus sp(2n)),$ with all anticommutators of the odd generators equal to zero $\{E_{i\bar{j}},E_{k\bar{p}} \} = 0.$ Example: CK contractions of $osp(3|2)$ -------------------------------------- This superalgebra has $so(3)$ as even subalgebra therefore their contractions to the kinematical $(1+1)$ Poincare, Newton and Galilei superalgebras may be fulfilled according to general CK scheme of the first section. But unlike of two odd generators of $osp(1|2)$ the superalgebra $osp(3|2)$ has six odd generators. In the basis $X_{ik}=E_{ki}, \, k,i=1,2,3, \, F=\displaystyle{\frac{1}{2}}E_{44}, \, E=-\displaystyle{\frac{1}{2}}E_{55}, \, H=-E_{45}, \, Q_k=E_{k4}, \, Q_{-k}=E_{k5}$ the generators are affected by the contraction coefficients $j_1,j_2$ in the following way $$X_{ik}\to (i,k)X_{ik}, \quad Q_{\pm k}\to (1,k)Q_{\pm k} \label{12}$$ and $H,F,E $ are remained unchanged. Then superalgebra $osp(3;j|2)$ is given by $$[X_{12},X_{13}]=j_1^2X_{23}, \quad [X_{13},X_{23}]=j_2^2X_{12}, \quad [X_{23},X_{12}]=X_{13},$$ $$[H,E]=2E, \quad [H,F]=-2F, \quad [E,F]=H,$$ $$[X_{ik},Q_{\pm i}]=Q_{\pm k}, \quad [X_{ik},Q_{\pm k}]=-(i,k)^2Q^2_{\pm i}, \;\; i<k,$$ $$[H,Q_{\pm k}]=\mp Q_{\pm k}, \quad [E,Q_k]=-Q_{-k}, \quad [F,Q_{-k}]=-Q_k,$$ $$\{Q_k,Q_k\}=(1,k)^2F, \quad \{Q_{-k},Q_{-k}\}=-(1,k)^2E,$$ $$\{Q_k,Q_{-k}\}=-(1,k)^2H, \quad \{Q_{\pm i},Q_{\mp k}\}=\pm (1,k)^2X_{ik}. \label{12-}$$ The non-minimal Poincare superalgebra is obtained for $j_1=\iota_1, \, j_2=i $ and has the structure of the semidirect sum $T \S (\{X_{23}\}\oplus osp(1|2)),$ where abelian $T=\{X_{12},X_{13},Q_{\pm 2},Q_{\pm 3}\}$ and $osp(1|2)=\{H,E,F,Q_{\pm 1}\}.$ The Newton superalgebra $osp(3;\iota_2|2)=T_2 \S osp(2|2),$ where $T_2=\{X_{13},X_{23},Q_{\pm 3}\} $ and $osp(2|2) $ is generated by $X_{12},H,E,F,Q_{\pm 1},Q_{\pm 2}.$ Finally the non-minimal Galilei superalgebra may be presented as semidirect sums $osp(3;\iota_1,\iota_2|2)=(T \S\{X_{23}\})\S osp(1|2)= T \S (\{X_{23}\}\oplus osp(1|2)).$ Unitary superalgebras $sl(m;j|n;\epsilon)$ ========================================== The superalgebras $sl(m|n)$ can be generated as a matrix superalgebras by taking matrices of the form [@Fra] $$M= \left( \begin{array}{cc} X_{mm} & T_{mn} \cr T_{nm} & X_{nn} \cr \end{array} \right )$$ where $X_{mm}$ and $X_{nn}$ are $gl(m)$ and $gl(n)$ matrices, $T_{mn}$ and $T_{nm}$ are $m \times n$ and $n \times m$ matrices respectively, with the supertrace condition $${\rm str}(M)={\rm tr}(X_{mm}) -{\rm tr}(X_{nn})=0. \label{str}$$ This matrix superalgebra is the set of transformations of the superspace with $m$ even coordinates $x_1,\ldots,x_m$ and $n$ odd ones $\theta_1,\ldots,\theta_n$. A basis of superalgebra $sl(m|n)$ can be constructed as follows. Define the $(m+n)^2-1$ generators $$\begin{aligned} E_{ij}=e_{ij}-{1 \over m-n} \delta_{ij}\left (\sum_{k=1}^{m}e_{kk}+ \sum_{\bar{k}=m+1}^{m+n}e_{\bar{k}\bar{k}}\right), & \qquad &E_{i\bar{j}}= e_{i\bar{j}}, \cr E_{\bar{i}\bar{j}}=e_{\bar{i}\bar{j}}+{1 \over m-n} \delta_{\bar{i}\bar{j}}\left(\sum_{k=1}^{m} e_{kk}+ \sum_{\bar{k}=m+1}^{m+n} e_{\bar{k}\bar{k}}\right), & \qquad & E_{\bar{i}j}= e_{\bar{i}j},\end{aligned}$$ where the indices $i,j,\ldots$ run from $1$ to $m$ and $\bar{i},\bar{j},\ldots$ from $m+1$ to $m+n$. The generators of $sl(m|n)$ in the Cartan-Weyl basis are given by $$\begin{aligned} H_i &=& E_{ij}-E_{i+1, j+1}, \quad \ 1 \leq i \leq m-1, \nonumber \\ H_{\bar{i}} &=& E_{\bar{i}\bar{i}}-E_{\bar{i}+1, \bar{i}+1}, \quad \ \ m+1 \leq \bar{i} \leq m+n-1, \nonumber \\ H_m &=& E_{mm}+E_{m+1, m+1}, \nonumber \\ E_{ij} & & {\rm for} \ \ sl(m), \qquad E_{\bar{i}\bar{j}} \qquad {\rm for} \quad sl(n), \nonumber \\ E_{i\bar{j}} & & {\rm and} \qquad E_{\bar{i}j} \qquad {\rm for \ the \ odd \ part } \label{gn}\end{aligned}$$ and their commutation relations looked as $$\begin{aligned} [H_I,H_J] & = & 0, \cr [H_K,E_{IJ}] & = & \delta_{IK}E_{KJ}-\delta_{I,K+1}E_{K+1,J}- \delta_{KJ}E_{IK}+\delta_{K+1,J}E_{I,K+1}, \; (K\neq m), \cr [H_m,E_{IJ}]&=&\delta_{Im}E_{mJ}-\delta_{I,m+1}E_{m+1,J}- \delta_{mJ}E_{Im}+\delta_{m+1,J}E_{I,m+1}, \cr [E_{IJ},E_{KL}] & = & \delta_{JK}E_{IL}-\delta_{IL}E_{KJ} \ \qquad {\rm for}\ E_{IJ} \ {\rm and} \ E_{KL} \ {\rm even}, \cr [E_{IJ},E_{KL}] & = &\delta_{JK}E_{IL}-\delta_{IL}E_{KJ} \ \qquad {\rm for}\ E_{IJ} \ {\rm even} \ {\rm and} \ E_{KL} \ {\rm odd}, \cr \{E_{IJ},E_{KL}\} & = & \delta_{JK}E_{IL}+\delta_{IL}E_{KJ} \ \qquad {\rm for}\ E_{IJ} \ {\rm and} \ E_{KL} \ {\rm odd}. \label{st}\end{aligned}$$ CK special linear (or unitary) superalgebras $sl(m;j|n;\epsilon) $ are consistent with the transformations of (super) vectors $${\cal X}^t(j,\epsilon)=(x_1,j_1x_2,\ldots ,(1,m)x_m|\nu(x_{m+1}, \epsilon_1x_{m+2},\ldots ,[1,n]x_{m+n}))^t, \label{sv}$$ where the odd components are denote as $ x_{m+1}=\theta_1,\ldots, x_{m+n}=\theta_{n} $ and $ \hat{\bar i}={\bar i}-m, \, \hat{\bar k}={\bar k}-m=1,\ldots ,n,\, [\hat{\bar i},\hat{\bar k}]= \prod^{\max(\hat{\bar i},\hat{\bar k})-1}_{l=\min(\hat{\bar i},\hat{\bar k})} \epsilon_l, \, \epsilon_l=1,\xi_l,i, \, \xi^2_l=0, \, \xi_l\xi_p=\xi_p\xi_l \neq 0. $ The components of ${\cal X}(j;\epsilon) $ are choosen in such a way that the contraction parameters $\epsilon_l $ of the odd components were independent of the contraction parameters $j_l $ of the even ones. The transformations of the standart generators (\[gn\]) (marked with star) of the special linear superalgebra $ sl(m|n) $ to the generators of $sl(m;j|n,\epsilon)$ are given by $$H_I=H^*_I, \; E_{ij}=(i,j)E^*_{ij}, \; E_{\bar i\bar j}=[\hat{\bar i},\hat{\bar j}]E^*_{\bar i\bar j}, \; i \neq j, \; \bar i \neq \bar j,$$ $$E_{i\bar j}=\nu(1,i)[1,\hat{\bar j}]E^*_{i\bar j}, \quad E_{\bar ij}=\nu(1,j)[1,\hat{\bar i}]E^*_{\bar ij}. \label{5.1+}$$ Nonzero commutators and anticommutators are easily obtained from the corresponding commutation relations (\[st\]) of the initial superalgebra $ sl(m|n) $ in the form $$[H_K,E_{IJ}] = \delta_{IK}E_{KJ}-\delta_{I,K+1}E_{K+1,J}- \delta_{KJ}E_{IK}+\delta_{K+1,J}E_{I,K+1},$$ $$[E_{ij},E_{jl}]=\left\{ \begin{array}{ll} E_{il}, &i<j<l,\; l<j<i,\; l\neq i, \\ (l,j)^2E_{il}, & i<l<j \ {\rm or} \ j<l<i, \\ (i,j)^2E_{il}, & l<i<j \ {\rm or} \ j<i<i, %%\\ \end{array} \right.$$ $$[E_{ij},E_{kj}]=\left\{ \begin{array}{ll} -E_{kj}, &k<i<j,\; j<i<k,\; k\neq j, \\ -(i,j)^2E_{kj}, & i<j<k {\ \rm or \ } k<j<i, \\ -(i,k)^2E_{kj}, & i<k<j {\ \rm or \ } j<k<i, \end{array} \right.$$ $$[E_{ij},E_{ji}]=(i,j)^2(E_{ii}-E_{jj}),$$ $$[E_{\bar{i}\bar{j}},E_{\bar{j}\bar{l}}]=\left\{ \begin{array}{ll} E_{\bar{i}\bar{l}}, & \bar{i} < \bar{j} < \bar{l},\; \bar{l} < \bar{j} < \bar{i},\; \bar{l}\neq \bar{i}, \cr [\hat{\bar{l}},\hat{\bar{j}}]^2 E_{\bar{i}\bar{l}}, & \bar{i}< \bar{l}< \bar{j} {\ \rm or \ } \bar{j} < \bar{l} < \bar{i}, \cr [\hat{\bar{i}},\hat{\bar{j}}]^2 E_{\bar{i}\bar{l}}, & \bar{l}< \bar{i}< \bar{j} {\ \rm or \ } \bar{j}< \bar{i}< \bar{k}, \end{array} \right.$$ $$[E_{\bar{i}\bar{j}},E_{\bar{k}\bar{j}}]=\left\{ \begin{array}{ll} -E_{\bar{k}\bar{j}}, & \bar{k} < \bar{i} < \bar{j},\; \bar{j} < \bar{i} < \bar{k},\; \bar{k}\neq \bar{j}, \\ -[\hat{\bar{i}},\hat{\bar{j}}]^2 E_{\bar{k}\bar{j}}, & \bar{i}< \bar{j}< \bar{k} {\ \rm or \ } \bar{k}< \bar{j}< \bar{i},\\ -[\hat{\bar{i}},\hat{\bar{k}}]^2 E_{\bar{k}\bar{j}}, & \bar{i}< \bar{k}< \bar{j} {\ \rm or \ } \bar{j}< \bar{k}< \bar{i}, \end{array} \right.$$ $$[E_{\bar{i}\bar{j}},E_{\bar{j}\bar{i}}]=[\hat{\bar{i}},\hat{\bar{j}}]^2 (E_{\bar{i}\bar{i}}-E_{\bar{j}\bar{j}}),$$ $$[E_{ij},E_{j\bar l}]=\left\{ \begin{array}{ll} (i,j)^2E_{i\bar l}, & i<j, \\ E_{i\bar l}, & i>j, \end{array} \right. \quad [E_{ij},E_{\bar ki}]=\left\{ \begin{array}{ll} -E_{\bar kj}, & i<j, \\ -(j,i)^2E_{\bar kj}, & i>j, \end{array} \right. ,$$ $$[E_{\bar i\bar j},E_{k\bar i}]=\left\{ \begin{array}{ll} -E_{k\bar j}, & \bar i<\bar j, \\ -[\hat{\bar j},\hat{\bar j}]^2 E_{k\bar j}, & \bar i>\bar j, \end{array} \right. \quad [E_{\bar i\bar j},E_{\bar jl}]=\left\{ \begin{array}{ll} [\hat{\bar i},\hat{\bar j}]^2 E_{\bar il}, & \bar i<\bar j, \\ E_{\bar il}, & \bar i>\bar j, \end{array} \right. ,$$ $$\left\{ E_{i\bar j},E_{\bar jl}\right\}= \left\{ \begin{array}{ll} \nu^2[1,\hat{\bar j}]^2(1,i)^2E_{il}, & i<l, \\ \nu^2[1,\hat{\bar j}]^2(1,l)^2E_{il}, & i>l, \end{array} \right.$$ $$\left\{ E_{i\bar j},E_{\bar ki}\right\}= \left\{ \begin{array}{ll} \nu^2(1,i)^2[1,\hat{\bar j}]^2E_{\bar k\bar j}, & \bar j<\bar k, \\ \nu^2(1,i)^2[1,\hat{\bar k}]^2E_{\bar k\bar j}, & \bar j>\bar k, \end{array} \right.$$ $$\left\{ E_{i\bar j},E_{\bar ji}\right\}= \nu^2(1,i)^2[1,\hat{\bar j}]^2(E_{ii}+E_{\bar j\bar j}). \label{scr}$$ For $\nu=\iota$ superalgebra $sl(m|n)$ is contracted to inhomogeneous superalgebra, which is semidirect sum $ \{E_{i\bar{j}}, E_{\bar{i}j} \} \S (sl(m) \bigoplus sl(n)),$ with all anticommutators of the odd generators equal to zero. Example: CK contractions of $sl(2|1)$ ------------------------------------- The generators of superalgebra $sl(2;j_1;\nu|1)$ is given by [@Fra] $$H=\left ( \begin{array}{cc|c} {1 \over 2} & 0 & 0 \cr 0 & -{1 \over 2} & 0 \cr \hline 0 & 0 & 0 \cr \end{array} \right ), \quad Z=\left ( \begin{array}{cc|c} {1 \over 2} & 0 & 0 \cr 0 & {1 \over 2} & 0 \cr \hline 0 & 0 & 1 \cr \end{array} \right ), \;$$ $$E_{12}=E^+=\left ( \begin{array}{cc|c} 0 & j_1 & 0 \cr 0 & 0 & 0 \cr \hline 0 & 0 & 0 \cr \end{array} \right ), \quad E_{21}=E^-=\left ( \begin{array}{cc|c} 0 & 0 & 0 \cr j_1 & 0 & 0 \cr \hline 0 & 0 & 0 \cr \end{array} \right ),$$ $$E_{13}=\bar{F}^+=\left ( \begin{array}{cc|c} 0 & 0 & \nu \cr 0 & 0 & 0 \cr \hline 0 & 0 & 0 \cr \end{array} \right ), \; E_{31}=F^-=\left ( \begin{array}{cc|c} 0 & 0 & 0 \cr 0 & 0 & 0 \cr \hline \nu & 0 & 0 \cr \end{array} \right ), \;$$ $$E_{32}=F^+=\left ( \begin{array}{cc|c} 0 & 0 & 0 \cr 0 & 0 & 0 \cr \hline 0 & \nu j_1 & 0 \cr \end{array} \right ), \; E_{23}=\bar{F}^-=\left ( \begin{array}{cc|c} 0 & 0 & 0 \cr 0 & 0 & \nu j_1 \cr \hline 0 & 0 & 0 \cr \end{array} \right ) \label{gen}$$ and acts on the superspace $(x_1,j_1x_2|\nu \theta_1).$ The commutation relations are represented as $$[H,E^{\pm}]=\pm E^{\pm} ,\; [E^+,E^-]=2j_1^2 H, \; [Z,H]=[Z,E^{\pm}]= [E^{\pm},\bar{F}^{\pm}]= [E^{\pm},F^{\pm}]=0 ,\;$$ $$[H,\bar{F}^{\pm}]=\pm {1 \over 2} \bar{F}^{\pm} , \; [H,{F}^{\pm}]=\pm{1 \over 2} {F}^{\pm},\; [Z,F^{\pm}]={1 \over 2} F^{\pm}, \; [Z,\bar{F}^{\pm}]=-{1 \over 2} \bar{F}^{\pm},$$ $$[E^{+},F^{-}]=-F^{+}, \; [E^{-},F^{+}]=-j_1^2F^{-}, \; [E^{+},\bar{F}^{-}]=j_1^2\bar{F}^{+} , \; [E^{-},\bar{F}^{+}]=\bar{F}^{-} ,$$ $$\{F^{+},\bar{F}^{-}\}= \nu^2 j_1^2 (Z - H), \quad \{F^{-},\bar{F}^{+}\}=\nu^2 (Z + H),$$ $$\{\bar{F}^{+},F^{+}\}=\nu^2E^{+}, \; \{\bar{F}^{-},F^{-}\}=\nu^2E^{-}, \; \{\bar{F}^{+},\bar{F}^{-}\}= \{{F}^{+},{F}^{-}\}=0. \label{screl}$$ For $\nu=\iota$ we obtain the semidirect sum of the abelian odd subalgebra with the direct sum of the even subalgebras, namely, $sl(2;j_1;\iota|1)=\{F^{\pm},\bar{F}^{\pm}\} \S (u(1)\oplus sl(2)).$ Two-dimensional contraction $\nu=\iota, j_1=\iota_1$ give in result similar semidirect sum $ sl(2;\iota_1;\iota|1)=\{F^{\pm},\bar{F}^{\pm}\} \S (u(1)\oplus sl(2;\iota_1))$ but with the subalgebra $ sl(2;\iota_1)=\{H,E^{\pm}\}$ instead of $sl(2).$ Under contraction $j_1=\iota_1$ we have the semidirect sum $sl(2;\iota_1;\nu|1)= \{E^{\pm}, F^+, \bar{F}^- \} \S \{H, Z, F^-, \bar{F}^+ \} $ of the subsuperalgebras each of them generated both even and odd generators. Casimir operators ================= The study of Casimir operators plays a grate role in the representation theory of simple Lie algebras since their eigenvalues characterize a representations. In the case of Lie superalgebras their eigenvalues completely characterize a typical representation while they are identically vanishing on an atypical representation. An element $C$ of universal enveloping superalgebra $U(A)$ commuting with all elements of $U(A)$ is called a Casimir operator of superalgebra $A.$ The algebra of the Casimir operators of $A$ is the $Z_2$-center of $U(A).$ Casimir operators of the basic Lie superalgebras can be constructed as follows [@Fra], [@LS], [@ACF]. Let $A=sl(m|n)$ with $m\neq n$ or $osp(m|n)$ be a basic Lie superalgebra. Let $\{E_{IJ}\}$ be a matrix basis of generators of $A$ where $I,J=1,\ldots,m+n$ with ${\rm deg}I=0$ for $I=1,\ldots,m$ and ${\rm deg}I=1$ for $I=m+1,\ldots,m+n.$ Then defining $(\bar{E})_{IK}=(-1)^{{\rm deg}K}E_{IK},$ a standard sequence of Casimir operators is given by $$C_p=str(\bar{E}^p)=\sum_{I=1}^{m+n} (-1)^{{\rm deg}I}(\bar{E}^p)_{II}=$$ $$=\sum_{I,I_1,\ldots,I_{p-1}=1}^{m+n}E_{II_1}(-1)^{{\rm deg}I_1}\ldots E_{I_kI_{k+1}}(-1)^{{\rm deg}I_{k+1}}\ldots E_{I_{p-1}I}. \label{6.1}$$ In the case of $sl(m|n)$ with $m \neq n$ one finds for example $ C_1 =0$ and $$C_2= \sum_{i,j=1}^{m}E_{ij}E_{ji}- \sum_{\bar{k},\bar{l}=m+1}^{m+n}E_{\bar{k}\bar{l}}E_{\bar{l}\bar{k}} +\sum_{i=1}^{m}\sum_{\bar{k}=m+1}^{m+n}(E_{\bar{k}i}E_{i\bar{k}}- E_{i\bar{k}}E_{\bar{k}i})-{m-n \over mn} Y^2. \label{6.2}$$ The diagonal elements of matrix $\bar{E}$ are taken in the form $ (\bar{E})_{ii}=E_{ii}+{{1}\over {m}}Y, \; (\bar{E})_{\bar{k}\bar{k}}=-E_{\bar{k}\bar{k}}+{{1}\over {n}}Y$ and two conditions on generators: $\sum_{i=1}^{m}E_{ii}=0,\; \sum_{\bar{k}=m+1}^{m+n}E_{\bar{k}\bar{k}}=0$ are taken into consideration. In the case of $osp(m|n)$ one finds $ C_1 =0$ and $$C_2= \sum_{i,j=1}^{m}E_{ij}E_{ji}- \sum_{\bar{k},\bar{l}=m+1}^{m+n}E_{\bar{k}\bar{l}}E_{\bar{l}\bar{k}} +\sum_{i=1}^{m}\sum_{\bar{k}=m+1}^{m+n}(E_{\bar{k}i}E_{i\bar{k}}- E_{i\bar{k}}E_{\bar{k}i}). \label{6.3}$$ One has to stress that unlike the algebraic case, the center of $ U(A)$ for the classical Lie superalgebras is in general not finitely generated. For only Lie superalgebra $osp(1|2n)$ the center of its universal enveloping superalgebra is generated by $n$ Casimir operators of degree $2,4, \ldots ,2n.$ To obtain Casimir operators of superalgebra $sl(m;j|n;\epsilon)$ we shall proceed in the standart manner. First we get the matrix $\bar{E}(j;\epsilon).$ For this we put in matrix $\bar{E}$ the new generators of $sl(m;j|n;\epsilon)$ instead of the old ones of $sl(m;n)$ according to (\[5.1+\]) and denote the obtained matrix as $\bar{E}(\rightarrow).$ In general its elements are undefined for nilpotent values of parameters $j,\epsilon,\nu.$ So it is necessary to multiply $\bar{E}(\rightarrow)$ on minimal multiplier which eliminate all undefined expressions in matrix elements, namely, $\nu(1,m)[1,n].$ Finally we have $$\bar{E}(j;\epsilon)= \nu(1,m)[1,n]\bar{E}(\rightarrow) \label{6.4}$$ with matrix elements $(k\neq p, \; \bar{k} \neq \bar{p})$ $$(\bar{E}(j;\epsilon))_{kk} = \nu(1,m)[1,n](E_{kk}+{{1}\over{m}}Y), \quad (\bar{E}(j;\epsilon))_{\bar{k} \bar{k}}= \nu(1,m)[1,n](-E_{\bar{k}\bar{k}}+{{1}\over{n}}Y),$$ $$(\bar{E}(j;\epsilon))_{kp} = \nu(1,k)(p,m)[1,n]E_{kp}, \quad (\bar{E}(j;\epsilon))_{\bar{k} \bar{p}}= \nu(1,m)[1,\hat{\bar{k}}] [\hat{\bar{p}},n]E_{\bar{k}\bar{p}},$$ $$(\bar{E}(j;\epsilon))_{i \bar{k}}= -(i,m)[\hat{\bar{k}},n]E_{i\bar{k}},\quad (\bar{E}(j;\epsilon))_{\bar{i}k}= (k,m)[\hat{\bar{i}},n]E_{\bar{i}k}. \label{6.5}$$ Maximal multiplier $\nu(1,m)[1,n]$ have the diagonal elements and minimal unit multiplier have the matrix elements $ (\bar{E}(j;\epsilon))_{m,m+n} = E_{m,m+n}, \; (\bar{E}(j;\epsilon))_{m+n,m} = E_{m+n,m}. $ The sequence of Casimir operators of $sl(m;j|n;\epsilon)$ is given by $$C_p(j;\epsilon)={\rm str}\bar{E}^p(j;\epsilon)= \nu^p(1,m)^p[1,n]^p {\rm str} (\bar{E}(\rightarrow))^p. \label{6.6}$$ Indeed, let $X^{\star}$ be an arbitrary generator of $sl(m|n)$. Under computing $[C_p,X^{\star}]=0$ we get identical terms but with different signs (plus and minus) so their sum is equal to zero. Under transformation of this commutator to the corresponding commutator of $sl(m;j|n;\epsilon)$ identical terms are multiplied on identical multipliers therefore their sum remains equal to zero, i.e. $[C_p(j;\epsilon),X]=0.$ Let us illustrate the above expressions on the simple example of $sl(2;j_1|1)$ superalgebra. The generators are transformed as follows $$E_{11}=E_{11}^{\star},\; Y= Y^{\star}, \; E_{12}=j_1E_{12}^{\star},\; E_{21}=j_1E_{21}^{\star},\; E_{13}=\nu E_{13}^{\star},\;$$ $$E_{31}=\nu E_{31}^{\star},\; E_{23}=\nu j_1E_{23}^{\star}, \; E_{32}=\nu j_1E_{32}^{\star}. \label{6.7}$$ and matrix $\bar{E}(j_1)$ according to (\[6.4\]),(\[6.5\]) is given by $$\bar{E}(j_1)=\nu j_1\bar{E}(\rightarrow)= \nu j_1 \left( \begin{array}{cc|c} E_{11}+{1 \over 2} Y & {1 \over j_1}E_{12} &-{1\over \nu}E_{13} \cr {1 \over j_1}E_{21} & -E_{22}+{1 \over 2}Y &-{1\over \nu j_1} E_{23} \cr \hline {1 \over \nu}E_{31} & {1\over \nu j_1} E_{32} & Y \cr \end{array} \right)=$$ $$= \left( \begin{array}{cc|c} \nu j_1(E_{11}+{1 \over 2}Y) & \nu E_{12} &-j_1E_{23} \cr \nu E_{21} & \nu j_1(-E_{22}+{1 \over 2}Y) &-E_{23} \cr \hline j_1E_{31} & E_{32} & \nu j_1 Y \cr \end{array} \right). \label{6.8}$$ The first order Casimir operator disappear $C_1(j_1)={\rm str}\bar{E}(j_1)=0.$ The second order Casimir operator is as follows $$C_2(j_1)={\rm str} (\bar{E}(j_1))^2= %\left(\bar{E}^2(j)\right)_{11}+ %\left(\bar{E}^2(j)\right)_{22}- %\left(\bar{E}^2(j)\right)_{33}= \nu^2 j_1^2\left(2E^2_{11}-{1\over 2} Y^2 \right)+ \nu^2 \left(E_{12}E_{21}+E_{21}E_{12} \right)+$$ $$+j_1^2 \left(E_{31}E_{13}+E_{13}E_{31} \right)+ E_{32}E_{23}-E_{23}E_{32}.$$ In the case of superalgebras $osp(M|N)$ the multiplier in (\[6.5\]) is equal to $\nu (1,M)[1,{N \over 2}]$ and all formulas for matrix $\bar{E}(j;\epsilon)$ and matrix elements $\left(\bar{E}(j;\epsilon)\right)_{kp}$ appear as for the $sl(m;j|n;\epsilon)$ with substitution $m=M$ and $n={N \over 2}$. Let us consider the $osp(1|2;\nu)$ superalgebra as an example. Their generators are transformed as $$E_{12}=\nu E_{12}^{\star},\; %E_{21}=\nu E_{21}^{\star},\; E_{13}=\nu E_{13}^{\star},\; %E_{31}=\nu E_{31}^{\star},\; E_{23}= E_{23}^{\star}, \; E_{32}= E_{32}^{\star}, \; E_{22}= E_{22}^{\star}, %\label{6.10}$$ and matrix $\bar{E}(\nu)$ is given by $$\bar{E}(\nu)= %\nu \bar{E}(\rightarrow)= -\nu \left( \begin{array}{c|cc} 0 & {1 \over \nu }E_{12} & {1\over \nu }E_{13} \cr \hline {1 \over \nu }E_{13} & E_{22} & E_{23} \cr -{1 \over \nu }E_{12} & E_{32} & -E_{22} \cr \end{array} \right) =- \left( \begin{array}{c|cc} 0 & E_{12} & E_{13} \cr \hline E_{13} & \nu E_{22} & \nu E_{23} \cr - E_{12}& \nu E_{32} & -\nu E_{22} \cr \end{array} \right). \label{6.11}$$ The first order Casimir operator is equal to zero $C_1(\nu)={\rm str}\bar{E}(\nu)=0$ and the second order Casimir operator is represented as $$C_2(\nu)= \nu^2 E^2_{22}+ \left(E_{12}E_{13}-E_{13}E_{12} \right)- {1 \over 2}\nu^2 \left(E_{32}E_{23}+E_{23}E_{32}\right).$$ Conclusion ========== Using classical CK Lie algebras of different type we have built basic CK superalgebras. Unlike standard procedure [@IW] of zero tending parameter contractions in this work are described with the help of nilpotent valued parameters. Such approach gives an opportunity to obtain the distribution of contraction parameters among superalgebra generators starting from quadratic form and hence to build CK superalgebras by means of pure algebraic tools without limiting procedure. Contracted superalgebras are connected with transformations of superspaces with nilpotent cartesian coordinates and represent a wide class of different semidirect sums for different possible contractions. An infinite sequences of Casimir elements of CK superalgebras have been obtained by a suitable transformations of the standard expressions of the corresponding operators of the basic superalgebras. It is our hope that CK superalgebras will be relevant for construction of supersymmetric physical models. 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Unified Approach*]{} (Syktyvkar: Komi SC) (in Russian) Gromov N A and Man’ko V I 1990 [*J.Math.Phys.*]{} [**31**]{} 1054, 1060 Rembielinski J, Tybor W 1984 [*Acta Physica Polonica*]{} [**B15**]{} 611 Hussin V, Negro J and del Olmo M A 1999 [*J.Phys.A: Math.Gen.*]{} [**32**]{} 5097 Patra M K and Tripathy K C 1989 [*Lett.Math.Phys*]{} [**17**]{} 1 Moody R V and Patera J 1991 [*J.Phys.A: Math.Gen.*]{}[**24**]{} 2227 Gromov N A, Kostyakov I V and Kuratov V V [*Preprint*]{} arXiv:hep-th/0110257 Frappat L, Sciarrino A and Sorba P 1996 [*Dictionary on Lie Superalgebras*]{} (hep-th/9607161, ENSLAPP-AL-600/96 and DSF-T-30/96) Leites D and Sergeev A 2002 [*Preprint*]{} arXiv:math.RT/0202180 v1 Arnaudon D, Chryssomalakos C and Frappat L 1995 [*Preprint*]{} arXiv:q-alg/9503021 v2, ENSLAPP-A-505/95 In[ö]{}n[ü]{} E and Wigner E P 1953 [*Proc.Nat.Acad.Sci. USA*]{} [**39**]{} 510

--- abstract: | The purpose of this paper is to calculate explicitly the volumes of Siegel sets which are coarse fundamental domains for the action of ${\mathrm{SL} _n (\mathbb{Z})}$ in $\mathrm{SL} _n (\mathbb{R})$, so that we can compare these volumes with those of the fundamental domains of ${\mathrm{SL} _n (\mathbb{Z})}$ in $\mathrm{SL} _n (\mathbb{R})$, which are also computed here, for any $n\geq 2$. An important feature of this computation is that it requires keeping track of normalization constants of the Haar measures. We conclude that the ratio between volumes of fundamental domains and volumes of Siegel sets grows super-exponentially fast as $n$ goes to infinity. As a corollary, we obtained that this ratio gives a super-exponencial lower bound, depending only on $ n $, for the number of intersecting Siegel sets. We were also able to give an upper bound for this number, by applying some results on the heights of intersecting elements in $ {\mathrm{SL} _n (\mathbb{Z})}$.\ **Keywords:** Arithmetic Groups, Siegel Sets, Coarse Fundamental Domains, Volumes. author: - Gisele Teixeira Paula title: 'Comparison of Volumes of Siegel Sets and Fundamental Domains for $\mathrm{SL}_n (\mathbb{Z})$ ' --- [Correspondence to be sent to: e-mail: [email protected]]{} \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] Introduction {#intro} ============ Siegel sets were first introduced in the study of quadratic forms by Siegel [@siegel2] in 1939, with some results following from previous works of Hermite and Korkine-Zolotarreff. In a fundamental paper [@borelharish], Borel and Harish-Chandra have generalised this notion and used Siegel domains to prove finiteness of covolumes of non-cocompact arithmetic subgroups. The simple structure of Siegel sets, compared to those of the actual fundamental domains makes them appealing for applications. For example, in his recent paper [@young], R. Young exploited their properties to obtain new results in geometric group theory. Still very little is known about the geometry of Siegel sets in general. In his book [@morris], Morris describes algebraically examples of Siegel sets not only for $\mathrm{SL} _n (\mathbb{R})$, with $n\geq 2$ , but also in the case of any semisimple Lie group G with a given Iwasawa decomposition. In this paper we recall one of the main properties of Siegel sets – the finiteness of their volumes. We evaluate these volumes explicitely in the basic case of Siegel sets for ${\mathrm{SL} _n (\mathbb{Z})}$ in $\mathrm{SL} _n (\mathbb{R})$ for any $n\geq 2$. We then compare these volumes with the actual covolumes of ${\mathrm{SL} _n (\mathbb{Z})}$. To this end, we have to deal with an essential difficulty related to the normalization of the Haar measure. For calculating the volumes of Siegel sets, the main difficulty is to find a nice way to describe the region of integration, which we solve with an appropriate change of coordinates. Most of the volume computations that followed Siegel’s original approach were not careful about the normalization constants, just noting that they are computable and could be calculated from the proof. In Section \[domfund\], we follow Garret’s notes on Siegel’s method [@garret] to compute the volumes of the quotients ${\mathrm{SL} _n (\mathbb{Z})}\backslash \mathrm{SL} _n (\mathbb{R})$ for $n\geq 3$ using induction and the volume of $\mathrm{SL}_2({\mathbb{Z}}) \backslash\mathrm{SL}_2({\mathbb{R}})$, that is computed in [@garret]. Our main goal here is to keep a careful track of the normalization constants. The main tools we use are the Poisson Summation formula, the Iwasawa decomposition of $G$ and the choice of a good Haar measure normalization on each group. At the end of the section we discuss the relation between the normalization of the measure we used and the canonical normalization that comes from the metric associated to the Killing form on $\mathfrak{sl}_n({\mathbb{R}})$. By comparing the volumes of Siegel sets and the volumes of fundamental domains of ${\mathrm{SL} _n (\mathbb{Z})}$, we conclude that somewhat surprinsingly the ratio between them grows super-exponentially fast with $n$. As an application of the computations presented here, in Section \[morr\] we show that given a Siegel set $\Sigma$ of ${\mathrm{SL} _n (\mathbb{Z})}$, we have an explicit lower bound for the number of elements $\gamma \in {\mathrm{SL} _n (\mathbb{Z})}$ such that $\gamma \Sigma$ intersects $\Sigma$. This bound is given by the ratio between $\mathrm{vol}(\Sigma)$ and $\mathrm{vol}(\mathrm{SL}_n({\mathbb{Z}}) \backslash\mathrm{SL}_n({\mathbb{R}}))$ – see Corollary \[corol1\]. We also give a proof that this result is consistent with a recent work of M. Orr [@martinorr], which generalizes a previous result of P. Habegger and J. Pila [@habegger] on the height of such elements $\gamma$, motivated by the study of Shimura varieties and their unlikely intersections. More precisely, Orr’s result gives, as a corollary, an upper bound for the number of intersecting Siegel sets while our work provides a lower bound for this number (see Corollary \[final\]). It would be interesting to compute the volumes of Siegel sets in other cases, for example for the action of well known Bianchi groups $\Gamma_d = \mathrm{SL}_2(\mathcal{O}_d)$ on the hyperbolic three-dimensional space $\mathbb{H}^3$. In this case we should have to deal with another difficulty when describing Siegel sets, because of the fact that as $d$ grows the quotients $\Gamma_d \backslash \mathbb{H}^3$ have a growing number of cusps. It would be worth doing these computations in the future, and then comparing them to the results obtained in this paper. The Iwasawa decomposition of . {#iwasawa} ============================== Let $n\geq 2$, $G=\mathrm{SL}_n(\mathbb{R})$ and $\Gamma = \mathrm{SL}_n(\mathbb{Z})$. Consider the action of $\Gamma$ by left translations on $G$ and let $$K = \mathrm{SO}_n;$$ $$A =\left\{\mbox{diag}(a_1,\ldots ,a_n); \displaystyle{ \prod_{i=1}^n{a_i} = 1} ; a_i > 0, \mbox{ for any } i=1,\ldots, n\right\};$$ $$N =\left\{(n_{ij})_{i,j} \in G ; n_{ii}=1 \mbox{ and } n_{ij}= 0 \mbox{ for } i>j\right\}.$$ The product map $$\Phi: K\times A \times N \longrightarrow G$$ $$(k,a,n)\mapsto kan$$ is a homeomorphism. We can construct an inverse map for $\Phi$ by using the Gram-Schmidt orthonormalization process. Take $g\in G$ and let $x_1, \ldots ,x_n$ be its columns. Then define inductively $y_1, \ldots ,y_n$ by $$y_1 = \frac{x_1}{\left\|x_1\right\|};$$ $$y_i = \frac{\widetilde{y}_i}{\left\|\widetilde{y}_i\right\|}, \mbox{ where } \widetilde{y}_i = x_i - \displaystyle{\sum_{l=1}^{i-1}{\left\langle x_i,y_l\right\rangle}y_l}; \mbox{ for } i = 2, \ldots, n.$$ Let $e_1, \ldots ,e_n$ be the standard orthonormal basis of $\mathbb{R}^n$. Then there exists an unique $k\in {\mathrm{SO} _n}$ such that $k(y_i) = e_i$, $\mbox{ for any } i = 1, \ldots n$. Therefore $$k(\widetilde{y}_i) = k(\left\|\widetilde{y}_i\right\| y_i) = \left\|\widetilde{y}_i\right\| k(y_i) = \left\|\widetilde{y}_i\right\|e_i,\mbox{ for any } i=1, \ldots , n.$$ So there is a diagonal matrix $a = \mbox{diag}(\left\|\widetilde{y}_1\right\|, \ldots, \left\|\widetilde{y}_n\right\|)$, such that $$k(\widetilde{y}_i) =a(e_i), \mbox{ for any } i=1, \ldots , n.$$ Also, it is easy to see that $y_i \in \left\langle x_1,\ldots , x_i\right\rangle, \mbox{ for any } i=1, \ldots , n$. Thus we have: $$g^{-1}\widetilde{y}_i= g^{-1}(x_i - \displaystyle{\sum_{l=1}^{i-1}{\left\langle x_i,y_l\right\rangle}y_l}) \in g^{-1}x_i + g^{-1}\left\langle x_1, \ldots , x_{i-1}\right\rangle$$ $$\Rightarrow g^{-1}\widetilde{y}_i \in e_i + \left\langle e_1, \ldots , e_{i-1}\right\rangle.$$ From this, we conclude that there exists $u \in N $ such that $g^{-1}\widetilde{y}_i= u e_i$, for every $i= 1, \ldots, n$. Therefore, $$u^{-1}g^{-1}\widetilde{y}_i= e_i = a^{-1}k(\widetilde{y}_i), \mbox{ for any } i \Rightarrow u^{-1}g^{-1} = a^{-1}k \Rightarrow g=k^{-1}a u^{-1}.$$ It is easy to see now that $\mathrm{det}(a) =1$, so $a \in A$ and thus we can define a continuous inverse map $g\in G \mapsto (k^{-1}, a, u^{-1} ) \in K\times A \times N$. The previous lemma gives us the Iwasawa decomposition $G=KAN$ of $\mathrm{SL} _n (\mathbb{R})$. Note that $K\cap A = K\cap N = A \cap N = \{I\}$ and that for this Iwasawa decomposition, $ AN = NA $ and $ K(AN) = (AN)K $ (see [@morris], page 148). Haar measure on . {#haar} ================= Given a locally compact Hausdorff topological group $G$, a left invariant Haar measure on $G$ is, by definition, a regular Borel measure $\mu$ on $G$ such that for all $g \in G$ and all Borel sets $E \subset G$ we have $\mu (gE) = \mu (E)$. It is well known that every connected Lie group admits such a Haar measure. Moreover, it is unique up to scalar multiples. We can define analogously right-invariant Haar measures. See [@venka] for more results about Haar measures on Lie groups. Since $G = \mathrm{SL} _n (\mathbb{R})$ is unimodular, i.e. the left and right invariant Haar measures coincide, and $dg$ is invariant under left translation by elements of $K$ and under right translation by elements of $AN$, we get that the Haar measure of $G$ in $k, v, a$ coordinates is given by the product measure $dg=dk\, du\, da$, where $da$, $du$ and $dk$ are the Haar measures on $A$, $N$ and $K$, repectively. This means that for every compactly supported and continuous function $f$ on $G$, we have $$\int_G{f(g) dg} = \int_K \int_{A} \int_{N} {f(kau) du \, da \, dk}.$$ It can be proved by induction on $n$ that the Haar measure on $N$ is given by $du= \displaystyle \prod_{i<j}{du_{ij}}$. It is usually convenient to change the order of integration on the variables $a$ and $u$, and to this end we can change the coordinates from $u$ to $v=aua^{-1}$. Then $v$ is also an upper triangular unipotent matrix of the form $$v = Id + \displaystyle \sum_{i<j\leq n} \frac{a_i}{a_j}u_{ij}E_{ij}.$$ It is easily seen that $dv= \displaystyle \prod_{i<j}dv_{ij} = \displaystyle \prod_{i<j}\frac{a_i}{a_j}du_{ij}$. This gives us $$\int_G{f(g) dg} = \int_K \int_{A} \int_{N} {f(kva) dv \, da \, dk}= \int_K \int_{N} \int_{A} {f(kau)\displaystyle \prod_{i<j}\frac{a_i}{a_j} da \, du \, dk}.$$ Also for convenience, we change coordinates from $ au $ to $ k^{-1}auk $ in the last integral. This has Jacobian equal to 1 (for each $ k\in K $), so we get: $$\int_G{f(g) dg} = \int_{N} \int_{A}\int_K {f(auk)\displaystyle \prod_{i<j}\frac{a_i}{a_j} dk \, da \, du}.$$ In this work, we will consider the Haar measure in $K$ to be the following: it is easy to see that the isotropy group of $e_n =(0, \ldots, 0, 1)$ by the action of ${\mathrm{SO} _n}$ in ${\mathbb{S}}^{n-1}$ is isomorphic to $\mathrm{SO}_{n-1}$. Then ${\mathbb{S}}^{n-1} \cong \mathrm{SO}_{n-1} \backslash {\mathrm{SO} _n}$. We have that the natural map $\pi : {\mathrm{SO} _n}\rightarrow {\mathbb{S}}^{n-1}$ is a Riemannian submersion if we rescale it by a factor of $\frac{1}{\sqrt{2}}$. Thus $$\mathrm{vol}({\mathrm{SO} _n}) = 2^{\frac{1}{2}(n-1)} \mathrm{vol}(\mathbb{S}^{n-1})\cdot \mathrm{vol}(\mathrm{SO}_{n-1}).$$ By using induction and the fact that $\mathrm{vol}({\mathbb{S}}^{n-1}) = \frac{2\pi ^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}$, we obtain $$\mathrm{vol}({\mathrm{SO} _n}) = 2^{\frac{1}{4}n(n-1)} \mathrm{vol}(\mathbb{S}^{n-1})\cdot \mathrm{vol}(\mathbb{S}^{n-2}) \ldots \mathrm{vol}(\mathbb{S}^{1}) = 2^{(n-1)(\frac{n}{4}+1)} \prod^n_{i=2}{\frac{\pi^\frac{i}{2}}{\Gamma (\frac{i}{2})}}.$$ It remains to define a Haar measure on $A$. We claim that $da = \displaystyle \prod _{i=1}^{n-1} {\frac{da_i}{a_i}}$ is such a measure. Indeed, let $\phi: A \rightarrow {\mathbb{R}}^{n-1}$ be the map $$a = \left( \begin{array}{ccccc} a_{1} & 0 & \ldots & 0 & 0 \\ 0 & a_{2} & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & a_{n-1} & 0\\ 0 & 0 & \cdots & 0 & \displaystyle \prod _{i=1}^{n-1}{a_i^{-1}}\\ \end{array} \right) \mapsto (t_1 , \ldots , t_{n-1}) = (\log a_1, \ldots ,\log a_{n-1}).$$ As $\phi$ is a group isomorphism and Haar measure is preserved by isomorphisms we get that $da = \displaystyle \prod _{i=1}^{n-1} dt_i = \displaystyle \prod _{i=1}^{n-1} {\frac{da_i}{a_i}}$ is a Haar measure on $A$. Siegel Sets for . {#Siegelsets} ================= Let $\Gamma$ be some group acting properly discontinuously on a topological space $X$. We call $\mathcal{F} \subset X$ a coarse fundamental domain for $\Gamma$ if: - $\Gamma \mathcal{F} = X$; - $\left\{\gamma \in \Gamma ; \gamma \mathcal{F} \cap \mathcal{F} \neq \emptyset \right\}$ is finite. A Siegel set in $\mathrm{SL} _n (\mathbb{R})$ is a set ${\Sigma _{t,\lambda}}$ of the form $$\Sigma _{t,\lambda} = A_tN_\lambda K,$$ where $t,\lambda$ are positive real numbers, $$A_t = \left\{a \in A ; \frac{a_i}{a_{i+1}} \leq t , \mbox{ for any } i =1, \ldots , n\right\}$$ and $$N_\lambda = \left\{u \in N | \left|u_{ij}\right|\leq \lambda , \mbox{ for any } i, j =1, \ldots , n\right\}.$$ For certain parameters $t, \lambda$ the Siegel sets ${\Sigma _{t,\lambda}}$ are coarse fundamental domains for ${\mathrm{SL} _n (\mathbb{Z})}$. Another important property is that they have finite volume. Siegel sets can be also defined in a more general way for lattices in other semisimple Lie groups, as it can be seen in Chapter 19 of Morris [@morris]. In many cases, a finite union of copies of Siegel sets glue together to form coarse fundamental domains for general lattices. In this section we compute the volumes of the Siegel sets in $\mathrm{SL} _n (\mathbb{R})$. We will use the Haar measure on $G$ given in Section \[haar\] in $v,a,k$ coordinates. $$\mathrm{vol}({\Sigma _{t,\lambda}}) = \frac{1}{2} \mathrm{vol}(\mathrm{SO}_n) (2\lambda)^{\frac{n(n-1)}{2}}\frac{t^{\frac{n(n^2-1)}{6}}}{((n-1)!)^2} . \label{eq1}$$ $$\mathrm{vol}({\Sigma _{t,\lambda}}) = \int_{\left|u_{ij}\right| \leq \lambda }\int_{\frac{a_i}{a_{i+1}} \leq t} \int_{K} \prod_{i<j}{\frac{a_i}{a_j}} dk \prod_{i=1}^{n-1} {\frac{da_i}{a_i}} \prod_{1\leq i <j \leq n}{du_{ij}}.$$ $$= \mathrm{vol}(K) (2\lambda)^{\frac{n(n-1)}{2}} \int_{\frac{a_i}{a_{i+1}} \leq t}{\prod_{i<j}{\frac{a_i}{a_j}}\prod_{i=1}^{n-1}{\frac{da_i}{a_i}}}.$$ To compute the integral over $a_1, \ldots, a_n$ (with the condition $\displaystyle \prod_{i=1}^{n}{a_i} = 1$), we change variables from $a_1, \ldots, a_n$ to the variables $$b_i = \frac{a_i}{a_{i+1}}, \mbox{ for any } i = 1, \ldots , n-1.$$ By elementary computation, we get $$\displaystyle \prod_{i<j}{\frac{a_i}{a_j}} = \prod_{i=1}^{n-1}{b_i ^{i(n-i)}}.$$ Moreover, as $a_i = b_i a_{i+1}$, the Jacobian of the change of coordinates from $a_i$ to $b_i$ is $\frac{1}{2a_1}$. The integral then becomes $$\int_{b_i \leq t} {\displaystyle \prod_{i=1}^{n-1}{b_i ^{i(n-i)}} \frac{1}{b_1 a_2b_2 a_3 \ldots b_{n-1} a_n} \frac{1}{2a_1}\prod_{i\leq n-1}{db_i}}$$ $$=\frac{1}{2} \int_{b_i \leq t} {\prod_{i=1}^{n-1}{b_i ^{[i(n-i)-1]}} \prod_{i\leq n-1}{db_i}} = \frac{1}{2} \prod_{i=1}^{n-1}{\frac{t^{ni-i^2}}{(ni - i^2)}} = \frac{1}{2} \frac{t^{\frac{n(n^2-1)}{6}}}{((n-1)!)^2}.$$ Thus we get to $$\mathrm{vol}({\Sigma _{t,\lambda}}) = \frac{1}{2} \mathrm{vol}(\mathrm{SO}_n) (2\lambda)^{\frac{n(n-1)}{2}}\frac{t^{\frac{n(n^2-1)}{6}}}{((n-1)!)^2} . \label{eq1}$$ Borel proves in [@borel] the following theorem: For $t\geq \frac{2}{\sqrt{3}}$ and $\lambda \geq \frac{1}{2}$, one has $\Sigma _{t,\lambda} \Gamma = G$. Moreover, $ \Sigma _{t,\lambda} $ is a coarse fundamental domain for $ \Gamma $ in $ G $. The quotient $ {\Gamma\backslash G}$ has finite volume, which satisfies $$\mathrm{vol}({\Gamma\backslash G}) \prec e^{cn^3}, \mbox{ as } n \rightarrow \infty,$$ for some positive constant $ c $. It is clear that $\mathrm{vol}({\Gamma\backslash G}) < \infty$, since $\Sigma _{t,\lambda}$ has finite volume and it contains a fundamental domain for ${\mathrm{SL} _n (\mathbb{Z})}$ if $t\geq \frac{2}{\sqrt{3}}$ and $\lambda \geq \frac{1}{2}$. Thus $\mathrm{vol}({\Gamma\backslash G}) \leq \mathrm{vol}(\Sigma_{t,\lambda})$ for these values of $ t $ and $ \lambda $. By taking $\lambda=\frac{1}{2}$ and $t= \frac{2}{\sqrt{3}}$ in formula , we get that $$\mathrm{vol}(\Sigma_{{\scriptscriptstyle{\frac{2}{\sqrt{3}}, \frac{1}{2}}}}) = 2^{(n-1)(\frac{n}{4}+1)-1} \Big( \prod ^n_{i=2}{\frac{\pi^\frac{i}{2}}{\Gamma (\frac{i}{2})}} \Big) \frac{(\frac{2}{\sqrt{3}})^{\frac{n(n^2-1)}{6}}}{((n-1)!)^2}$$ $$=\frac{ 2^{\frac{2n^3 +3n^2+ 7n -24}{12}} \pi^{\frac{n^2+n-2}{2}}}{3^{\frac{n(n^2-1)}{12}} ((n-1)!)^2 \displaystyle \prod^n_{i=2}{\Gamma (\frac{i}{2})}}$$ Using Stirling’s formula, this volume is easily seen to grow assymptotically like $e^{cn^3}$, for some positive constant $c$ and this finishes the proof. On the other hand, as we will see in the next section, $\mathrm{vol}({\mathrm{SL} _n (\mathbb{Z})}\backslash \mathrm{SL} _n (\mathbb{R}))$ computed with respect to the same normalization of the Haar measure goes to zero as $n$ grows. Volume of . {#domfund} =========== It is a well-known fact that $\mathrm{vol}({\mathrm{SL} _n (\mathbb{Z})}\backslash \mathrm{SL} _n (\mathbb{R}))$ is finite. Our goal is to calculate it, with respect to the same normalization of the Haar measure used in the previous section. The whole computation follows the original approach of Siegel [@siegel], but we have to be careful with the normalization constants. We use Poisson summation, induction and the previously known fact that $\mathrm{vol}(\mathrm{SL}_2({\mathbb{Z}}) \backslash \mathrm{SL}_2({\mathbb{R}})) = \sqrt{2}\zeta (2)$, which can be proved in a similar way (see [@garret], being careful with respect to the different normalization of $\mathrm{vol}(\mathrm{SO} _2)$ we are considering). We will state first the Poisson Summation Formula, which will play a fundamental role in the computations, and for which the reader can refer to [@psf]. Given a lattice $\Lambda$ in ${\mathbb{R}}^n$, we define $\left|\Lambda \right|$ to be the covolume of $\Lambda$, i.e. the volume of ${\mathbb{R}}^n/ \Lambda$ and the dual lattice of $\Lambda$ by $$\Lambda^* = \left\{y \in {\mathbb{R}}^n; \left\langle x,y\right\rangle \in {\mathbb{Z}}\mbox{ for any } x \in \Lambda \right\}.$$ \[psf\] Given any lattice $\Lambda$ in ${\mathbb{R}}^n$, a vector $w \in {\mathbb{R}}^n$ and an adimissible function $f:{\mathbb{R}}^n \rightarrow {\mathbb{R}}$ in $\mathcal{L}^1$, we have $$\displaystyle \sum_{x \in \Lambda}{f(x+w)} = \frac{1}{\left|\Lambda\right|}\sum_{t \in \Lambda^*}{e^{-2\pi i \left\langle w,t\right\rangle}\hat{f}(t)},$$ Here, $\hat{f}(t) = \int_{{\mathbb{R}}^n}{f(x) e^{2\pi i\left\langle x,t\right\rangle} dx}$ is the Fourrier transform of $f$ and admissibilty of $f$ means that there exist constants $\epsilon, \delta > 0$ such that $\left|f(x)\right|$ and $\left|\hat{f}(x)\right|$ are bounded above by $\epsilon(1+\left|x\right|)^{-n-\delta}$. Let then $f \in \mathcal{L}^1$ be an admissible function on ${\mathbb{R}}^n$. We can ask $ f $ to be a $ C^{\infty} $ function with compact support. We then define $F:G\mapsto{\mathbb{R}}$ by $$F(g) = \displaystyle \sum_{v \in {\mathbb{Z}}^n}{f(vg)}.$$ Here we are considering the multiplication of line-vectors $v \in {\mathbb{R}}^n$ by elements of $ G $ by the right. Clearly, $F$ is left $\Gamma$-invariant, as ${\mathbb{Z}}^n \Gamma = {\mathbb{Z}}^n$ under the action of ${\mathrm{SL} _n (\mathbb{Z})}$ on ${\mathbb{R}}^n$ by right multiplication of line vectors by the inverse elements of $ {\mathrm{SL} _n (\mathbb{Z})}$. Consider $\int_{{\Gamma\backslash G}}{F(g)dg}$. We will use this integral to calculate $\mathrm{vol}({\Gamma\backslash G})$. Let $$Q = \mathrm{stab}_G(e) = \left\{ \left( \begin{array}{cc} h & v \\ 0 & 1 \\ \end{array} \right); h \in \mathrm{SL}_{n-1}({\mathbb{R}}), v \in {\mathbb{R}}^{n-1}\right\},$$ where $e=(0,\ldots, 0,1) \in {\mathbb{R}}^{n}$, and write $Q_{{\mathbb{Z}}} = Q \cap \Gamma $. Using linear algebra over ${\mathbb{Z}}$, note that $${\mathbb{Z}}^{n} - \{0\} = \displaystyle\bigcup_{\ell >0} \displaystyle\bigcup_{\gamma \in {Q_{\mathbb{Z}} \backslash \Gamma}} \ell e \gamma,$$ where $\ell$ runs over positive integers. Then we can write $$\displaystyle \int_{{\Gamma\backslash G}}{F(g)dg} = \int_{{\Gamma\backslash G}}{f(0)dg} + {\int_{{\Gamma\backslash G}}{\sum_{\ell >0}\sum_{\gamma \in {Q_{\mathbb{Z}} \backslash \Gamma}}f(\ell e \gamma g)dg}}$$ $$= \mathrm{vol}({\Gamma\backslash G})f(0) + \sum_{\ell >0} \int_{Q_{{\mathbb{Z}}} \backslash G}{f(\ell eg)dg}.$$ For the second equality note that a fundamental domain for $Q_{{\mathbb{Z}}} $ in $G$ is the union of images of a fundamental domain for $\Gamma$ in $G$ by representatives of classes in ${Q_{\mathbb{Z}} \backslash \Gamma}$. In addition, the Schwartz condition on $f$ ensures that the integral over $Q_{{\mathbb{Z}}} \backslash G$ is finite. Indeed, in his article [@siegel45], Siegel proves that the function $ F(g) $ is integrable over a fundamental domain for $ {\Gamma\backslash G}$. On pages 344-345 of \[loc. cit.\] we can see that the integrals over $ \mathbb{Q}_{\mathbb{Z}}\backslash G$ are also convergent (for any fixed $ l\in \mathbb{N} $). We observe that although he uses different decomposition of $ G $ and normalization of Haar measures, this does not change the finiteness of the integrals. Write $$P = \left\{ \left( \begin{array}{cc} h & * \\ 0 & \frac{1}{\mbox{det}(h)} \\ \end{array} \right); h \in \mathrm{GL}_{n-1}({\mathbb{R}}) , \mbox{det}(h) > 0 \mbox{ and } * \in {\mathbb{R}}^{n-1} \right\};$$ $$N' = \left\{ \left( \begin{array}{cc} I_{n-1} & v \\ 0 & 1 \\ \end{array} \right); v \in {\mathbb{R}}^{n-1} \right\}, N'_{{\mathbb{Z}}} = N' \cap \Gamma.$$ $$M = \left\{ \left( \begin{array}{cc} h & 0 \\ 0 & 1 \\ \end{array} \right); h \in \mathrm{SL}_{n-1}( {\mathbb{R}}) \right\}, M_{{\mathbb{Z}}} = M \cap \Gamma;$$ $$A' = \left\{ \left( \begin{array}{cc} t^{{\scriptscriptstyle\frac{1}{n-1}}}I_{n-1} & 0 \\ 0 & t^{-1} \\ \end{array} \right); t > 0 \right\};$$ Note that $P = N'MA' \supset NA$, $Q= N'M$ and $G = N'MA'K$. However this time we have that $N'MA'$ intersects $K$ non-trivially, i.e. this is not an Iwasawa decomposition. The product $N'MA'K$ projects on $G$ with fiber $\mathrm{SO} (n-1)$. Therefore we get the following For every left $G$-invariant function $\Phi$, we have $$\displaystyle \int_{{Q_{\mathbb{Z}} \backslash G}}{\Phi (g)dg} = \frac{1}{\mathrm{vol}(\mathrm{SO}_{n-1})} \int_{Q_{{\mathbb{Z}}} \backslash ( N'MA'K)}{\Phi (n'ma'k)dn'\, dm\,da' \, dk}$$ where $dg$ is the Haar measure in $ G $ coming from its Iwasawa decomposition (as in Section \[Siegelsets\]). Here $dn'$, $dm$, $da'$ and $dk$ are the left Haar measures on $N'$, $M$, $A'$ and $K$, respectively. We see that $ dn' = \displaystyle \prod_{i=1}^{n-1}v_i $ and that $ M $ is isomorphic to $ \mathrm{SL}_{n-1}({\mathbb{R}}) $ and thus $ dm $ will appear as the measure of this group. This allows us to use induction in the calculations. On the other hand, $ A' $ is isomorphic to $ {\mathbb{R}}_{>0} $ via the isomorphism $$\left( \begin{array}{cc} t^{{\scriptscriptstyle\frac{1}{n-1}}}I_{n-1} & 0 \\ 0 & t^{-1} \\ \end{array} \right) \in A' \mapsto t \in {\mathbb{R}}_{>0}.$$ Thus we have $ da'= \frac{dt}{t} $ where $dt$ is the usual measure in $ {\mathbb{R}}$. Again it will be convenient to change the order of integration, by letting the variable $a' \in A'$ to be the last one. This will give us $d(a' q a'^{-1}) = t^n dq$, for $q = n'm \in N'M$. Indeed, for $ a' = \left( \begin{array}{cc} t^{{\scriptscriptstyle\frac{1}{n-1}}}I_{n-1} & 0 \\ 0 & t^{-1} \\ \end{array} \right) \in A'$ and $ q = \left( \begin{array}{cc} h & v \\ 0 & 1 \\ \end{array} \right) \in Q $, we have $ a'q (a')^{-1} = \left( \begin{array}{cc} h & t^{\frac{n}{n-1}}v \\ 0 & 1 \\ \end{array} \right)$, and thus the $ M $-contribuction to the measure doesn’t change, but the $ N' $-contribution is multiplied by $ (t^{\frac{n}{n-1}})^{n-1} = t^n $ and we get to $d(a' q a'^{-1}) = t^n dq$ as stated. Then, if we require $f$ to be $K$-invariant, the integral $\int_{{\Gamma\backslash G}} {F(g)dg}$ becomes equal to $$\mathrm{vol}({\Gamma\backslash G})f(0) + \frac{1}{\mathrm{vol}(\mathrm{SO}_{n-1})}\sum_{\ell >0} \int_{Q_{{\mathbb{Z}}} \backslash ( N'MA' K)}{f(\ell en'ma'k) dn' \, dm\, da' \, dk}$$ $$= \mathrm{vol}({\Gamma\backslash G})f(0) + \frac{\mathrm{vol}(Q_{{\mathbb{Z}}} \backslash K )}{\mathrm{vol}(\mathrm{SO}_{n-1})}\sum_{\ell >0} \int_{Q_{{\mathbb{Z}}} \backslash ( N'M)}\int_{A'} {f(\ell en'ma')\displaystyle t^n da' \, dn' \, dm }.$$ We have $K \cap Q_{{\mathbb{Z}}} = \mathrm{SO}_{n-1}({\mathbb{Z}})$. Noting that $${\mathbb{S}}^{n-1} \cong \mathrm{SO}_{n-1} \backslash \mathrm{SO}_n \cong \frac{\mathrm{SO}_{n-1}({\mathbb{Z}}) \backslash \mathrm{SO}_n}{\mathrm{SO}_{n-1}({\mathbb{Z}}) \backslash \mathrm{SO}_{n-1}},$$ we get $$\mathrm{vol}({\mathbb{S}}^{n-1}) = \mathrm{vol}(\mathrm{SO}_{n-1} \backslash \mathrm{SO}_n) = \frac{\mathrm{vol}(\mathrm{SO}_{n-1}({\mathbb{Z}}) \backslash \mathrm{SO}_n)}{\mathrm{vol}(\mathrm{SO}_{n-1}({\mathbb{Z}}) \backslash \mathrm{SO}_{n-1})}.$$ As $\mathrm{SO}_{n}({\mathbb{Z}})$ acts properly and freely in $\mathrm{SO}_n$, for any $n \in \mathbb{N}$ we have that $$\mathrm{SO}_n \longrightarrow \mathrm{SO}_{n}({\mathbb{Z}}) \backslash \mathrm{SO}_n$$ is a finite covering with $\# \mathrm{SO}_{n}({\mathbb{Z}})$ sheets, which gives us $$\mathrm{vol}(\mathrm{SO}_n) = \#(\mathrm{SO}_{n}({\mathbb{Z}})) \mathrm{vol}(\mathrm{SO}_{n}({\mathbb{Z}}) \backslash \mathrm{SO}_n).$$ Altogether, we obtain: $$\mathrm{vol}(Q_{{\mathbb{Z}}} \backslash K ) = \mathrm{vol}(\mathrm{SO}_{n-1}({\mathbb{Z}}) \backslash \mathrm{SO}_n) = \frac{\mathrm{vol}({\mathbb{S}}^{n-1}) \mathrm{vol}(\mathrm{SO}_{n-1})}{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))}.$$ As the integrand is invariant under $N'M$ ($en'm = e$, for any $n' \in N'$ and $m \in M$) and the volume of $N'_{{\mathbb{Z}}} \backslash N'$ is $1$, this implies $$\mathrm{vol}({\Gamma\backslash G})f(0) + \frac{\mathrm{vol}(Q_{{\mathbb{Z}}} \backslash K )}{\mathrm{vol}(\mathrm{SO}_{n-1})}\sum_{\ell >0} \int_{Q_{{\mathbb{Z}}} \backslash ( N'M)}\int_{A'} {f(\ell en'ma')\displaystyle t^n da'\, dn' \, dm}$$ $$=\mathrm{vol}({\Gamma\backslash G})f(0) + \frac{\mathrm{vol}(Q_{{\mathbb{Z}}} \backslash K )}{\mathrm{vol}(\mathrm{SO}_{n-1})} \mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}( {\mathbb{R}})) \sum_{\ell >0} \int_{A'} \! \! \! \! \! {f(\ell ea')t^n da'}$$ $$= \mathrm{vol}({\Gamma\backslash G})f(0) + \frac{\mathrm{vol}({\mathbb{S}}^{n-1})}{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))} \mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}( {\mathbb{R}})) \sum_{\ell >0} \int_{A'} f(\ell ea')t^n da'$$ By replacing $a' \in A'$ by $t \in {\mathbb{R}}_{>0}$ and using the description of $ da' $, we get to $$\mathrm{vol}({\Gamma\backslash G})f(0) + \frac{\mathrm{vol}({\mathbb{S}}^{n-1}) }{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))} \mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}( {\mathbb{R}})) \sum_{\ell >0} \int_{0}^{\infty} f(\ell et)t^n \frac{dt}{t}.$$ By replacing $t$ by $\frac{t}{\ell}$, we obtain $$\mathrm{vol}({\Gamma\backslash G})f(0) + \frac{\mathrm{vol}({\mathbb{S}}^{n-1}) }{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))} \mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}( {\mathbb{R}})) \sum_{\ell >0}\frac{1}{\ell ^n} \int_{0}^{\infty} f(et)t^n \frac{dt}{t}.$$ By using polar coordinates in ${\mathbb{R}}^n = \{ (v, t), v \in \mathbb{S}^{n-1}, t \in {\mathbb{R}}_{>0}\}$, we get $$\mathrm{vol}({\mathbb{S}}^{n-1})\int_{0}^{\infty} f(et)t^n \frac{dt}{t} = \int_{{\mathbb{S}}^{n-1}}\int_{0}^{\infty} f(v,t)t^{n-1} dtdv = \int_{{\mathbb{R}}^{n}} f(x)dx = \hat{f}(0).$$ Thus what we get until now is the following The initial integral becomes $$\displaystyle \int_{{\Gamma\backslash G}}{F(g)dg} = \mathrm{vol}({\Gamma\backslash G})f(0) + \frac{\mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}( {\mathbb{R}}))}{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))} \zeta(n) \hat{f}(0),$$ where $\zeta(n) = \displaystyle \sum_{l\in{\mathbb{Z}}}{\frac{1}{l^n}}$ is the Riemman zeta function. The previous result allows us to compute explicitely the value of $ \mathrm{vol}({\Gamma\backslash G}) $: $$\mathrm{vol}({\Gamma\backslash G}) = \frac{ \mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}({\mathbb{R}}))}{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))} \zeta(n) = \sqrt{2} \displaystyle \prod_{i=2}^{n}\zeta(i) \prod_{i=1}^{n-1}{\frac{1}{\# (\mathrm{SO}_{i}( {\mathbb{Z}}))}} .$$ For every $g \in G$, we are going to apply the Poisson summation formula to the lattice $\Lambda = \left\{vg;v \in {\mathbb{Z}}^n \right\}$ in ${\mathbb{R}}^n$, the vector $w=0$ and the initial function $f$. Note that $\Lambda^* = \left\{vg^*; v \in {\mathbb{Z}}^n \right\}$, where $g^* = ^{\top}\! \! g^{-1}$. Then we get $$F(g) = \displaystyle \sum_{v \in {\mathbb{Z}}^n}{f(vg)} = \displaystyle \sum_{v \in {\mathbb{Z}}^n}{\hat{f}(vg^*)} = \hat{F}(g^*), \mbox{ for any } g \in G.$$ The automorphism $g \mapsto g^*$ preserves the measure on $G$ and stabilizes $\Gamma$, so we can do an analogous computation with the roles of $f$ and $\hat{f}$ reversed. Since $\hat{\hat{f}}(0) = f(0)$ and $\int_{\Gamma \backslash G}{F(g)dg} = \int_{\Gamma \backslash G}{\hat{F}(g)dg}$, we obtain $$\mathrm{vol}({\Gamma\backslash G})f(0) + \frac{ \mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}( {\mathbb{R}}))}{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))} \zeta(n) \hat{f}(0) = \int_{{\Gamma\backslash G}}\!\!\!{F(g)dg}$$ $$=\int_{\Gamma \backslash G}\!\!\!{\hat{F}(g)dg} = \mathrm{vol}({\Gamma\backslash G})\hat{f}(0) + \frac{\mathrm{vol}(\mathrm{SL}_{n-1}( {\mathbb{Z}}) \backslash \mathrm{SL}_{n-1}( {\mathbb{R}}))}{\# (\mathrm{SO}_{n-1}({\mathbb{Z}}))} \zeta(n) f(0).$$ By asking additionally that $f$ is such that $f(0) \neq \hat{f}(0)$ and using indution on $n$, we get to the desired result. We observe that for every $i\in \mathbb{N}$, $\# (\mathrm{SO}_{i}( {\mathbb{Z}})) = 2^{i-1}i!$. Indeed, the group $\mathrm{SO}_{i}( {\mathbb{Z}})$ consists of monomial matrices whose nonzero entries are equal to $\pm 1$ and which have determinant equal to $1$. The first condition gives us $2^i i!$ matrices. Now if we look at the surjective group homomorphism $$\mathrm{det}: B=\{\mbox{monomial matrices with nonzero entries} \in \{\pm 1\}\} \rightarrow \{\pm 1 \},$$ we get $B/Ker(\mathrm{det}) \cong \{\pm 1 \}$, which implies $$\# (\mathrm{SO}_{i}( {\mathbb{Z}})) = \#(Ker (\mathrm{det})) = \frac{\#(B)}{2} = \frac{2^{i}i!}{2} = 2^{i-1}i! .$$ Thus we have proved the following The explicit volume of $ {\Gamma\backslash G}$, by considering the Haar measures described in Section \[haar\] is given by $$\mathrm{vol}({\mathrm{SL} _n (\mathbb{Z})}\backslash \mathrm{SL} _n (\mathbb{R})) =\sqrt{2} \displaystyle \prod_{i=2}^{n}\zeta(i) \displaystyle \prod_{i=1}^{n-1}{\frac{1}{2^{i-1}i!}} = \frac{\displaystyle \prod_{i=2}^{n}\zeta(i)}{2^{{\scriptscriptstyle\frac{n^2-3n+1}{2}}} \displaystyle \prod_{i=2}^{n} i!} . \label{eq2}$$ It is not difficult to see that this function goes to zero like $e^{-c'n^2}$ as $n$ grows, where $c'$ is a positive constant. It has a completely different behaviour from the volume growth of Siegel Sets described by formula . What we can conclude directly from all this is that although the geometry of a Siegel set is simpler than that of the actual fundamental domain for a lattice, their volumes can differ dramatically as $n$ grows. Thus we should be careful if we want to replace fundamental domains of any lattice by simpler structures such as Siegel sets, due to the possibility that some of their relevant geometric features, e.g. volume, may have different behavior to that of fundamental domains. As a consequence of Sections \[Siegelsets\] and \[domfund\], we obtain: The ratio between volumes of the minimal Siegel sets $\Sigma = \Sigma_{{\scriptscriptstyle{\frac{1}{2}, \frac{2}{\sqrt{3}}}}}$ for $ {\mathrm{SL} _n (\mathbb{Z})}$ and the actual fundamental domains for these groups in $ {\mathrm{SL} _n (\mathbb{R})}$ is given by $$C(n) = \frac{\mathrm{vol}(\Sigma)}{\mathrm{vol}({\Gamma\backslash G})} = \frac{2^{{\scriptscriptstyle\frac{2n^3+ 9n^2+25n-30}{12}}}\pi^{{\scriptscriptstyle\frac{n^2+n-2}{4}}} \displaystyle \prod_{i=1}^{n-1}{i!}}{3^{{\scriptscriptstyle\frac{n^3-n}{12}}}((n-1)!)^2 \displaystyle \prod_{i=2}^{n}{\Gamma(\frac{i}{2})} \displaystyle \prod_{i=2}^n{\zeta(i)}}.$$ Moreover, $ C(n) \sim e^{\tilde{c}n^3} $ for some constant $\tilde{c}$ that does not depend on $n$. A natural question arising here is the following: “How is our normalization of the Haar measure related to the canonical normalization defined by using the Killing form on $\mathfrak{sl}_n({\mathbb{R}})$?” To answer to this question we can compare our formula with a result of Harder [@harder], who computed the volume of ${\mathrm{SL} _n (\mathbb{Z})}\backslash X$, where $X$ is the symmetric space $\mathrm{SL} _n (\mathbb{R})/{\mathrm{SO} _n}$. In order to do this comparison, note that by equation we have $$\mathrm{vol}({\mathrm{SL} _n (\mathbb{Z})}\backslash X) =\frac{\mathrm{vol}({\Gamma\backslash G})}{\mathrm{vol}({\mathrm{SO} _n})} = \frac{\sqrt{2} \displaystyle \prod_{i=1}^{n-1}{\frac{1}{2^{i-1}i!}}\displaystyle \prod_{i=2}^{n}{\zeta(i)}}{2^{(n-1)(\frac{n}{4}+1)} \displaystyle \prod^n_{i=2}{\frac{\pi^\frac{i}{2}}{\Gamma (\frac{i}{2})}}}. \label{eq3}$$ By Harder’s formula, we obtain that this volume in the canonical normalization is given by $$\mathrm{vol}_1 ({\mathrm{SL} _n (\mathbb{Z})}\backslash X) = \frac{\displaystyle \prod_{i=1}^{n-1}{i!}\displaystyle \prod_{i=2}^{n}{\zeta(i)}}{(2\pi)^{{\scriptscriptstyle\frac{n(n+3)}{2}}} 2^{\tau}n!}, \label{eq4}$$ where $\tau = n$ if $n$ is odd and $\tau = n-1$ if $n$ is even. We see that these volumes differ by a factor given by $$C_1(n) = \frac{\mathrm{vol}_1 ({\mathrm{SL} _n (\mathbb{Z})}\backslash X)}{\mathrm{vol}({\mathrm{SL} _n (\mathbb{Z})}\backslash X)}= \frac{2^{{\scriptscriptstyle\frac{n^2-5n-2}{4} - \tau }} \Bigl(\displaystyle \prod_{i=1}^{n-1}{i!}\Bigl)^2 }{ n!\pi^{{\scriptscriptstyle\frac{n^2+5n+2}{4}}} \displaystyle \prod^n_{i=2}{\Gamma \Bigl(\frac{i}{2}\Bigl)}},$$ where $\tau = n$ if $n$ is odd and $\tau = n-1$ if $n$ is even. We note that again by using Stirling’s formulas, we obtain that $ C_1(n) $ grows assymptotically with $ n $ like $ e^{\kappa n^2} $, for some positive constant $ \kappa $. The same renormalization can be applied to in order to obtain the volumes of Siegel sets in the symmetric spaces with respect to the standard normalization of the measure. Bounding the number of intersecting domains {#morr} =========================================== Another relevant consequence of this work is the following corollary: \[corol1\] Let $N$ be the cardinality of the set $\mathcal{I} :=\left\{\gamma \in \Gamma ; \gamma \Sigma \cap \Sigma \neq \emptyset\right\}$, where $\Sigma = \Sigma_{{\scriptscriptstyle{\frac{1}{2}, \frac{2}{\sqrt{3}}}}}$ . Then $N\geq C(n) = \frac{\mathrm{vol}(\Sigma)}{\mathrm{vol}({\Gamma\backslash G})}$. As $\Sigma$ is a Siegel set, it must contain a fundamental domain $\mathcal{F}$ for $\Gamma$. We affirm that $\Sigma \subset \underset{{\scriptscriptstyle \gamma \in \mathcal{I}}}{\bigcup} \gamma\mathcal{F}$. Indeed, given $x \in \Sigma$, if $x\in \mathcal{F}$, there is nothing to prove. If $x \notin \mathcal{F}$, as the images of $\mathcal{F}$ tesselate $\mathrm{SL} _n (\mathbb{R})$ we must have $x \in \gamma \mathcal{F}$, for some $Id \neq \gamma \in \Gamma$. As $\gamma \mathcal{F} \subset \gamma \Sigma$, we obtain $x \in \gamma \Sigma \cap \Sigma$, and thus $\gamma \in \mathcal{I}$. Therefore the inclusion above is true. From this we obtain $N \mathrm{vol}({\Gamma\backslash G}) = N \mathrm{vol}(\mathcal{F}) \geq \mathrm{vol}(\Sigma)$ and thus $N \geq C(n)$, as stated. In his recent work [@martinorr], Martin Orr shows in a more general setting that given a reductive algebraic group $G$ defined over $\mathbb{Q}$, a general Siegel set $\Sigma \subset G({\mathbb{R}})$ for some arithmetic subgroup $\Gamma \subset G(\mathbb{Q})$, and $\theta \in G(\mathbb{Q})$, there exists an upper bound for the height of elements $\gamma\in \Gamma$ such that $\theta \Sigma \cap \gamma \Sigma \neq \emptyset$. The height of an element is defined by: $$H(\gamma) = \displaystyle \max_{1\leq i,j\leq n} H(\gamma_{ij}),$$ where given a rational number $a/b$, $H(a/b)$ is defined as the maximum of the absolute values of $a$ and $b$. Orr shows that, given any element $\gamma$ of the set $$\Sigma_{N,D} := \Sigma \Sigma^{-1}\cap \left\{\gamma \in G(\mathbb{Q}), \mathrm{det} \gamma \leq N\mbox{ and the denominators of } \gamma \mbox{ are } \leq D\right\},$$ there exists some constant $C_1$, depending on the group $G$, on the Siegel set $\Sigma$ and on the way the group $G$ is embedded in some $GL_n({\mathbb{R}})$, such that $$H(\gamma) \leq C_1N^nD^{n^2},$$ where $N = \left|\mathrm{det} \gamma\right|$ and $D$ is the maximum of the denominators of entries of $\gamma$. Note that for $\Gamma = {\mathrm{SL} _n (\mathbb{Z})}$, the set $\mathcal{I}$ defined above is contained in $\Sigma_{N,D}$. In this section we are going to compare this result with ours, i.e., to see what happens in the case when $G= \mathrm{SL} _n (\mathbb{R})$ and $\Gamma = {\mathrm{SL} _n (\mathbb{Z})}$. Note that in this case, for any $\gamma \in \Gamma$, we have $N= \left|\mathrm{det} \gamma\right| = 1$ and also $D = 1$ because the entries of $\gamma$ are all integers. Thus Orr’s result gives us, for this case, $$H(\gamma)\leq C_1(n).$$ By the definition, the height of an element $\gamma \in {\mathrm{SL} _n (\mathbb{Z})}$ is equal to $\left|\gamma\right|_{max}$. Therefore, his result turns to $$\left|\gamma \right|_{max} \leq C_1(n), \mbox{ for any } \gamma \in \Sigma \Sigma^{-1}.$$ By Example $1.6$ on page $5$ of [@sarnack], the set $\left\{\gamma \in {\mathrm{SL} _n (\mathbb{Z})};\left\|\gamma \right\| \leq C_1(n) \right\}$ has cardinality of assymptotic order $c_nC_1(n)^{(n^2-n)}$, with $c_n \rightarrow 0$ as $n\rightarrow\infty$. Thus if we assume that $n$ is sufficiently large, we can suppose that $c_n < \epsilon$ for some $\epsilon>0$ fixed. Therefore, we have $$\left|\left\{\gamma \in {\mathrm{SL} _n (\mathbb{Z})};\left\|\gamma \right\| \leq C_1(n) \right\}\right| \prec C_1(n)^{(n^2-n)},$$ where the notation $f(n) \prec g(n)$ used above means that there exists a positive constant $C$ such that for sufficiently big $n$, we have $f(n)\leq Cg(n)$. Note that the result in [@sarnack] is proved for the Euclidean norm $\left\|.\right\|$ in $M_{n\times n}$ and we know that $\left\|\gamma\right\| \leq n\left|\gamma\right|_{max}$. Thus $$\left|\left\{\gamma \in {\mathrm{SL} _n (\mathbb{Z})};\left|\gamma\right|_{max} \leq C_1(n) \right\}\right| \prec (n C_1(n))^{(n^2-n)}.$$ We are going to show that $$C_1(n)\leq e^{\frac{n^2-n}{2} ln(n)}.$$ From this we obtain that $\left|\mathcal{I}\right| \prec e^{\frac{n^4}{2}ln(n)}$. Hence we have: \[final\] For $\Gamma = {\mathrm{SL} _n (\mathbb{Z})}$ in $\mathrm{SL} _n (\mathbb{R})$ and $\mathcal{I}$ defined above, there exist constants $c_1, c_2 >0$ such that $$e^{c_1 n^3} \leq \left|\mathcal{I}\right| \leq e^{c_2 n^4 ln(n)}.$$ In order to obtain the second inequality we adapt the proofs in [@martinorr] for the $\mathrm{SL} _n (\mathbb{R})$ case, with the difference that we give explicit values for the constants. Let $\gamma \in {\mathcal{I}}$. From this element, we can define: - A partition of ${\left\{1,\ldots,n\right\}}$ (with respect to $\gamma$) is a list of disjoint subintervals of ${\left\{1,\ldots,n\right\}}$, which we call components, whose union is all of ${\left\{1,\ldots,n\right\}}$ and such that: - $\gamma $ is block upper triangular with respect to the chosen partition; - $\gamma $ is not block upper triangular with respect to any other finer partition of ${\left\{1,\ldots,n\right\}}$; - A leading entry of $\gamma$ is a pair $(i,j) \in {\left\{1,\ldots,n\right\}}^2$ such that $\gamma_{ij}$ is the leftmost non-zero entry of the $i$-th row of $\gamma$. For a concrete description of what are the possible partitions in the $\mathrm{GL}_3$ case see Section 3.2 of [@martinorr]. We will make use of the following lemma whose proof can be found in [@martinorr]: \[lema1\] If $i, j$ are in the same component, then there exists a sequence of indices $i_1, \ldots, i_s$ such that $i_1 =i, i_s = j$ and $$(*) \mbox{ For every } p \leq s-1, \mbox{ either } i_p \leq i_{p+1} \mbox{ or } (i_p, i_p+1) \mbox{ is a leading entry.}$$ In the proof of the following lemmas for the $\mathrm{GL}_n$ case, Martin Orr uses the notation $A\ll B$ meaning that there exists a constant $C$, depending on $n$, such that $\left|A\right| \leq C\left|B\right|$. Our point here is to compute such constants so that we can make explicit the value of $C_1(n)$. \[lema2\] If $(i,j)$ is a leading entry of $\gamma$, then $\alpha_j \leq \sqrt{n} \beta_i $. For any $\gamma \in \Sigma \Sigma^{-1}$, we can write $\gamma= \nu \beta \kappa \alpha^{-1}\mu^{-1}$, with $\kappa \in {\mathrm{SO} _n}$, $\nu, \mu \in N_{\frac{1}{2}}$ and $\alpha, \beta \in A_{\frac{2}{\sqrt{3}}}$. This gives us the equation $\gamma \mu \alpha = \nu \beta \kappa$. We will compare the lengths of the $i$-th rows on each side of this equation. As $\kappa \in {\mathrm{SO} _n}$, multiplying by $\kappa$ on the right does not change the length of each row. If we expand out lengths we obtain $$\displaystyle \sum_{p=1}^{n}{\Big(\sum_{q=1}^{n}{\gamma_{iq} \mu_{qp}}\Big)\alpha_{p}^2} = \sum_{p=1}^{n}{\nu_{ip}^2\beta_p^{2}}.$$ As $\nu$ is upper triangular, the non-zero terms on the right hand side of the last equation must have $p \geq i$. By the definition of $A_t$, for all $p \geq i$ we have $$\beta_p \leq \frac{1}{t^{(p-i)}}\beta_i \leq \beta_i,$$ where in the second inequality we used that $t=\frac{2}{\sqrt{3}}$ and $p\geq i $ imply $ \frac{1}{t^{(p-i)}} \leq 1$. Since $\nu \in N_{\frac{1}{2}}$, $\left|\nu_{ip}\right| \leq 1$ for any $i,p$. Alltogether, $$\sum_{p=1}^{n}{\nu_{ip}^2\beta_p^{2}} \leq \left|\sum_{p=1}^{n}{\nu_{ip}^2\beta_p^{2}}\right| \leq \sum_{p\geq i}^{n}{\beta_p^{2}} \leq (n-i)\beta_i^2 \leq n \beta_i^2.$$ On the other hand, by looking at the left hand side of the equation, we obtain: $$\Big(\sum_{q=1}^{n}{\gamma_{iq} \mu_{qj}}\Big)\alpha_{j}^2 \leq \displaystyle \sum_{p=1}^{n}{\Big(\sum_{q=1}^{n}{\gamma_{iq} \mu_{qp}}\Big)\alpha_{p}^2}.$$ As $(i,j)$ is a leading entry, we can only have $\gamma_{iq} \neq 0$ if $q \geq j$. But as $\mu$ is upper triangular, $\mu_{qj} \neq 0$ implies $q\leq j$. Thus the only non-zero term in the first sum is the one for $q=j$ and then we get $$\Big(\sum_{q=1}^{n}{\gamma_{iq} \mu_{qj}}\Big)\alpha_{j}^2 = \gamma_{ij}^2\mu_{jj}^2\alpha_j^2 = \gamma_{ij}^2 \alpha_j^2.$$ Note that $\gamma_{ij}\neq 0$ and that as $\gamma$ has integer entries, we must have $\left|\gamma_{ij}\right| \geq 1$, which implies $\gamma_{ij}^2\geq 1$. Altogether, we obtain $$\alpha_j^2 \leq \alpha_j^2\gamma_{ij}^2 \leq n\beta_i^2 \Rightarrow \alpha_j \leq \sqrt{n}\beta_i,$$ from what we conclude the proof. \[lema3\] For all $k\in {\left\{1,\ldots,n\right\}}$, $\alpha_k \leq \sqrt{n} \beta_k $. We affirm that there must exist a leading entry $(i,j)$ such that $j\leq k \leq i$. To prove this notice that as $\gamma$ is invertible, there must exist $i\geq k$ such that the $i$-th row of $\gamma$ contains a non-zero entry in the $k$-th column or to its left (otherwise the leftmost k columns of $\gamma$ would have rank less than k). Choose j so that $\gamma_{ij}$ is the leading entry of $\gamma$ in the $i$-th line and it will satisfy $j\leq k$ as claimed. By Lemma \[lema2\] and by the definition of $A_t$ we obtain $$\alpha_k \leq \frac{1}{t^{(k-j)}}\alpha_j \leq \sqrt{n} \beta_i \leq \sqrt{n}\frac{1}{t^{(i-k)}}\beta_k \leq \sqrt{n} \beta_k.$$ \[lema4\] For all $j\in {\left\{1,\ldots,n\right\}}$, $\beta_j \leq (\sqrt{n})^{n-1} \alpha_j $. As $\alpha$ and $\beta$ are diagonal with positive real entries, we have (by using Lemma \[lema3\] in the inequality) $$\beta_j \mathrm{det}(\alpha) = \beta_j \displaystyle \prod_{k=1}^n{\alpha_k} \leq \beta_j \alpha_j (\sqrt{n})^{n-1}\prod_{k\neq j}{\beta_k} = (\sqrt{n})^{n-1} \alpha_j \mathrm{det}(\beta).$$ But as $\mathrm{det}(\beta) = \mathrm{det}(\alpha)= 1$, $$\beta_j \leq (\sqrt{n})^{n-1} \alpha_j$$ and the lemma is proved. \[lema5\] If $i$ and $j$ are in the same component, $\beta_j \leq (\sqrt{n})^{n^2-n}\alpha_{i}$. We can apply Lemma \[lema1\] to obtain a sequence $i_1=i, \ldots, i_s=j$ such that for any $p\in \left\{1,\ldots,s\right\},$ we have either $i_p\leq i_{p+1}$ or $(i_p,i_{p+1})$ is a leading entry. We take this subsequence as the smallest possible. If $i_p\leq i_{p+1}$ then as $\alpha \in A_t$ and $(\sqrt{n})^{n}\geq 1$, we get $$\frac{\alpha_{i_p}}{\alpha_{i_{p+1}}}\geq t^{i_{p+1}-i_p}\geq 1 \Rightarrow \alpha_{i_{p+1}}\leq \alpha_{i_{p}} \leq (\sqrt{n})^{n}\alpha_{i_{p}}.$$ On the other hand if $(i_p,i_{p+1})$ is a leading entry then by Lemmas \[lema2\] and \[lema4\] we have $$\alpha_{i_{p+1}}\leq \sqrt{n}\beta_{i_p} \leq (\sqrt{n})^{n}\alpha_{i_p}.$$ If we apply the last inequality successively we get to $$\alpha_j = \alpha_{i_s} \leq (\sqrt{n})^{n(s-1)}\alpha_{i}.$$ Now we just apply Lemma \[lema4\] and notice that $s\leq n$ to obtain $$\beta_j\leq (\sqrt{n}))^{n-1}\alpha_{j}\leq (\sqrt{n})^{n^2-1}\alpha_{i}.$$ We write $$Q = \left\{g \in G; g \mbox{ is block upper triangular according to the components of }\gamma \right\};$$ $$L = \left\{g \in G; g \mbox{ is block diagonal according to the components of }\gamma \right\}.$$ We affirm that $\kappa\in L$. Indeed, as the matrices $\gamma, \mu, \alpha, \beta$ and $\nu$ are in Q by the construction, we also have $\kappa \in Q$. On the other hand, if a matrix is block upper triangular and is also orthogonal, then it is block diagonal. Thus $\kappa \in L$. \[lema6\] If $i, j \in {\left\{1,\ldots,n\right\}}$, then $\left|\gamma_{ij} \right| \leq C_1(n) = n^{\frac{n^2-n}{2}}$. Write $\gamma = \nu\beta\kappa\alpha^{-1}\mu^{-1}$. Because $\alpha, \beta$ are diagonal, the $pq$-th entry of $\beta\kappa\alpha^{-1}$ is $\beta_p\kappa_{pq}\alpha_q^{-1}$. If $p$ and $q$ are not in the same component, as $\kappa \in L$, we get that $\kappa_{pq} = 0$. On the other hand, if they are in the same component, then by Lemma \[lema5\] $$\beta_p\kappa_{pq}\alpha_q^{-1} \leq \kappa_{pq} (\sqrt{n})^{n^2-1}.$$ By the definition of ${\mathrm{SO} _n}$, $\left|\kappa\right|_{max}\leq 1$ for every $\kappa\in {\mathrm{SO} _n}$. Therefore $$\beta_p\kappa_{pq}\alpha_q^{-1} \leq (\sqrt{n})^{n^2-1}.$$ As we have $\mu, \nu \in N_{\frac{1}{2}}$, we have $\left|\mu\right|_{\infty}, \left|\nu\right|_{\infty} \leq 1.$ Altogether, we obtain $$\left|\gamma_{ij} \right| \leq (\sqrt{n})^{n^2-1}.$$ Therefore we conclude the proof that $H(\gamma) \leq C_1(n)$, where $$C_1(n) = (\sqrt{n})^{n^2-1} = e^{\frac{n^2-1}{2} ln(n)}$$ and this finishes the proof of Corollary \[final\]. Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thank Professor Mikhail Belolipetsky for several suggestions on the development of this paper and also on the text. I also thank Paul Garret and Martin Orr for their very helpful works and for always answering my emails with good suggestions, and Cayo Dória for helping me to understand better some topics. Finally, I also thank the refferee for carefully reading the paper and for giving suggestions that improved the presentation of the results. Borel, A.: Introduction aux Groupes Arithmètiques. *Hermann*, Paris (1969). Borel, A. and Harish-Chandra.: Arithmetic subgroups of algebraic groups. *Ann. of Math*. 75, 485–535 (1962). Duke, W.; Rudinick, Z. and Sarnak, P.: Density of Integer Points on Affine Homogeneous Varieties. *Duke Math. J.*, Vol.71, No.1, 143-179 (1993). Garret, P.: Volume of ${\mathrm{SL} _n (\mathbb{Z})}\backslash \mathrm{SL} _n (\mathbb{R})$ and $Sp_n({\mathbb{Z}})\backslash Sp_n({\mathbb{R}})$. Paul Garrett’s homepage http://www-users.math.umn.edu/$\sim$garrett/m/v/volumes.pdf (2014). Accessed 07 November 2017. Habegger, P. and Pila, J.: Some unlikely intersections beyond André–Oort. *Compos. Math.* 148, 1-27 (2012). Harder, G.: A Gauss-Bonnet formula for discrete arithmetically defined groups. *Ann. Sci. Ec. Norm. Supér.* $4^e$ série, tome 4, n° 3, 409-455 (1971). Morris, D. W.: Introduction to Arithmetic Groups. *Deductive Press*, United States (2015). Orr, M.: Height bounds and the Siegel property. Preprint, arXiv:1609.01315v3 (2016). Siegel, C.L.: Einführung in die Theorie der Modulfunktionen n-ten Grades. *Math. Ann* 116, 617–657 (1939). Siegel, C.L.: A Mean Value Theorem in Geometry of Numbers. *Annals of Mathematics* Vol 45, No 2 (1945). Siegel, C.L.: Lectures on the Geometry of Numbers. *Springer-Verlag*, Berlin (1989). Stein, E. M. and Weiss, G. L.: Introduction to Fourier Analysis on Euclidean Spaces. *Princeton Math. Ser.* 32, *Princeton Univ. Press*, Princeton, NJ (1971). Venkataramana, T.N.: Lattices in Lie Groups. In Workshop on Geometric Group Theory, India (2010). Young, R.: The Dehn function of $\mathrm{SL} _n (\mathbb{Z})$. *Ann. of Math*. 177, 969–1027 (2013).

--- abstract: 'We analyze the spatial and velocity distributions of confirmed members in five massive clusters of galaxies at intermediate redshift ($0.5 < z < 0.9$) to investigate the physical processes driving galaxy evolution. Based on spectral classifications derived from broad- and narrow-band photometry, we define four distinct galaxy populations representing different evolutionary stages: red sequence (RS) galaxies, blue cloud (BC) galaxies, green valley (GV) galaxies, and luminous compact blue galaxies (LCBGs). For each galaxy class, we derive the projected spatial and velocity distribution and characterize the degree of subclustering. We find that RS, BC, and GV galaxies in these clusters have similar velocity distributions, but that BC and GV galaxies tend to avoid the core of the two $z\approx0.55$ clusters. GV galaxies exhibit subclustering properties similar to RS galaxies, but their radial velocity distribution is significantly platykurtic compared to the RS galaxies. The absence of GV galaxies in the cluster cores may explain their somewhat prolonged star-formation history. The LCBGs appear to have recently fallen into the cluster based on their larger velocity dispersion, absence from the cores of the clusters, and different radial velocity distribution than the RS galaxies. Both LCBG and BC galaxies show a high degree of subclustering on the smallest scales, leading us to conclude that star formation is likely triggered by galaxy-galaxy interactions during infall into the cluster.' author: - 'Steven M. Crawford' - 'Gregory D. Wirth' - 'Matthew A. Bershady' title: Spatial and Kinematic Distributions of Transition Populations in Intermediate Redshift Galaxy Clusters --- We thank the referee for the careful reading of our manuscript and the constructive criticism that improved our paper. S.M.C. acknowledges the South African Astronomical Observatory and the National Research Foundation of South Africa for support during this project. M.A.B. acknowledges suppport from NSF grant AST-1009471. This work made use of IRAF, a software package distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. [*Facilities:*]{}

--- abstract: 'We have developed a formalism to study non-adiabatic, non-radial oscillations of non-rotating compact stars in the frequency domain, including the effects of thermal diffusion in the framework of general relativistic perturbation theory. When a general equation of state depending on temperature is used, the perturbations of the fluid result in heat flux which is coupled with the spacetime geometry through the Einstein field equations. Our results show that the frequency of the first pressure ($p$) and gravity ($g$) oscillation modes is significantly affected by thermal diffusion, while that of the fundamental ($f$) mode is basically unaltered due to the global nature of that oscillation. The damping time of the oscillations is generally much smaller than in the adiabatic case (more than two orders of magnitude for the $p-$ and $g-$modes) reflecting the effect of thermal dissipation. Both the isothermal and adiabatic limits are recovered in our treatment and we study in more detail the intermediate regime. Our formalism finds its natural astrophysical application in the study of the oscillation properties of newly born neutron stars, neutron stars with a deconfined quark core phase, or strange stars which are all promising sources of gravitational waves with frequencies in the band of the first generation and advanced ground-based interferometric detectors.' address: - | $^1$ Dipartimento di Fisica “G.Marconi", Universit\` a di Roma “La Sapienza"\ and Sezione INFN ROMA1, piazzale Aldo Moro 2, I-00185 Roma, Italy - | $^2$ Departament de Física Aplicada, Universitat d’Alacant,\ Apartat de correus 99, 03080 Alacant, Spain - '$^3$ Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA' author: - 'L. Gualtieri$^1$, J.A. Pons $^2$, and G. Miniutti $^3$' title: 'Non-adiabatic oscillations of compact stars in general relativity' --- ł ø Introduction ============ The theory of stellar oscillations has been a fundamental tool in the study of stellar interiors and stellar properties during decades. For Newtonian stars the theory is well established and observationally tested. In some areas such as helioseismology, very high precision measurements of the normal oscillation frequencies allow for a detailed understanding of the properties of the solar interior (see [@ChDa] for a review). It is usually assumed that stellar pulsations are adiabatic because the thermal relaxation timescale in stellar interiors is orders of magnitude larger than the pulsation periods. However, in some particular situations, energy transfer in the external regions of stars is fast enough to affect the pulsation properties, and some work has been devoted to study, for example, non-adiabatic oscillations of white dwarfs [@LB93] or the coupling between the different non-linear modes in non–adiabatic situations [@vH94]. Concerning relativistic stars, the study of pulsation properties of neutron stars or compact objects such as strange stars or hybrid stars has become more popular in the last decade (see e.g. the reviews in Ref. [@rev1; @rev2; @rev3]), mainly due to the expectations of detecting the gravitational emission from pulsating nearby compact stars with the current or next generation ground-based interferometric detectors (LIGO, VIRGO, GEO600, TAMA). The theory of non-adiabatic relativistic stellar pulsations is not as well developed as its Newtonian cousin, due the higher complexity of the formalisms, but also due to the fact that most attention has been paid to the study of old neutron stars, in which the thermal conductivity is too small to have visible non-adiabatic effects. Nevertheless, there might be situations in which this is not entirely true. For example, in a newly born neutron star the thermal structure is determined by neutrino diffusion [@BL86; @Pon99], instead of electron conduction as in old neutron stars [@BHY01], and the effects of non-adiabaticity are likely to be relevant in the outer layers. Another interesting possibility is the existence of deconfined quark matter in neutron star cores, or in the form of strange stars. There seems to be common agreement in that, if strange matter exists, it has to be in a color superconducting phase. It has been recently pointed out [@SE02] that the thermal conductivity of such exotic matter is many orders of magnitude larger than in normal neutron star matter, and for sufficiently high temperatures the timescale for thermal relaxation can be as short as $10^{-4}$ s., comparable with typical oscillation periods of compact objects. If this or other exotic scenarios (hyperons, kaon condensates) happen to be true, our common belief that compact star pulsations are adiabatic must be modified, and some care must be taken to understand how non-adiabatic effects change the oscillation properties and therefore the predicted gravitational wave signals of pulsating compact stars. With this motivation, we have derived a formalism that includes the effects of heat transfer and chemical diffusion (important in proto-neutron stars, where lepton diffusion is the driving force of thermodynamical changes) in a relativistic analysis of stellar perturbations. Some work in this line has been done in the past for radial oscillations [@MMIH]. We have considered the case of non-radial oscillations, since this is the case of interest for gravitational wave emission, by extending in a simple way the formalism of Lindblom and Detweiler [@LD; @DL], and complementing the system of equations in the frequency domain with the additional equations for thermal or chemical diffusion. In the present study we focused on the effects of heat transfer and chemical diffusion, neglecting the rotation of the star. The paper is structured as follows. In Sec. II we derive the equations of non-adiabatic stellar perturbations in general relativity. In Sec. III we describe the equation of state we have used and we define the thermodynamical quantities we need in our derivation. The additional equations (transport of energy) that close the system are discussed in section IV. Section V is devoted to the numerical implementation of the complete set of equations. In Sec. VI we discuss in detail the results of the numerical integrations and in Sec. VII we draw the main conclusions and comment on possible future extensions of this work. Derivation of the equations =========================== The stress–energy tensor of a non–perfect fluid[^1], including heat flux but without viscosity, has the general form [@MTW] T\_ = (+p)u\_u\_+pg\_ +u\_q\_+u\_q\_ where $\rho$ is the energy density, $p$ is the pressure, $u^{\alpha}$ is the matter four–velocity, and $q^{\alpha}$ is the heat flux which satisfies $u_{\alpha}q^{\alpha}=0\,.$ In addition, we will also consider the conservation equation for the baryon number density $n$ (equation of continuity) \[cont\] (nu\^)\_[;]{}=0. Background configuration ------------------------ Hereafter, we will neglect the heat flux in the background configuration. This is justified if the background is assumed to be in thermal equilibrium. Even in the case that thermal or chemical gradients are present, the assumption of stationary background is valid if the timescale for global thermodynamical changes is much larger than the timescale of variation of the perturbations. For example, in newly born neutron stars, structural changes happen on timescales of the order of 0.1-1 second, while we are interested in oscillations of periods of the order of milliseconds. Therefore, labeling background quantities by the superscript $\z$ , in the following we assume that q\_=0 such that, our background is a perfect fluid, spherically symmetric star, and is described by the well known TOV equations: g\_&=&[diag]{}(-e\^[(r)]{},e\^[ł(r)]{},r\^2\_[ab]{})\ u\^[(0)]{}&=&(e\^[-/2]{},0,0,0)\[defu0\]\ T\_&=&(+p)u\_u\_+pg\_\ ’&=&-e\^[ł]{}(M-4r\^3)\ ’&=&e\^[ł]{}(M+4pr\^3)\ p\^[(0)]{}&=&-(+p)\_[,r]{} where we denote with a prime $\pa/\pa r$, and we have defined \_[ab]{}(1,\^2) (here and in the following, greek indexes $\mu=0,\dots,3$ run on spacetime, latin indexes $i=1,\dots,3$ run on the spatial subspace, and latin indexes $a=\theta,\phi$ run on the sphere). Perturbations ------------- The equations of stellar perturbations in the perfect fluid case have been derived by many authors in different formalisms [@altri; @altri2]. In this paper, we follow the notation of Lindblom & Detweiler [@LD; @DL], (LD hereafter) and we will try to take the parallelism between their and our equations as far as possible. The perturbed stress–energy tensor has the form T\_&=&(+p)u\_u\_+ (+p)(u\_u\_+u\_u\_)\ &&+p g\_+pg\_ +(q\_u\_+u\_q\_).\[pertT\] Following the conventions in [@LD; @DL], we expand in spherical harmonics the metric perturbations with polar symmetry (we will not discuss perturbations with axial symmetry, which are not related with fluid oscillations in non–rotating stars) as g\_=-r\^le\^[øt]{}( [c|c]{} [cc]{} H\_[0lm]{}e\^ Y\^[lm]{} & ørH\_[1lm]{}Y\^[lm]{}\ ørH\_[1lm]{}Y\^[lm]{} & H\_[0lm]{}e\^Y\^[lm]{}\ & 0\ 0 & r\^2\_[ab]{}K\_[lm]{}Y\^[lm]{} ).\[metricexpansion\] The Lagrangian displacements are expanded as \^r&=&W\_[lm]{}Y\^[lm]{}r\^le\^[øt]{}\ \^a&=&-V\_[lm]{}\^[ab]{}Y\^[lm]{}\_[,b]{}r\^le\^[øt]{}, and they are related to the four–velocity perturbation by u\^i=u\^[(0)0]{}\^i\_[,t]{}=øe\^[-/2]{}\^i,    (i=1,2,3) so that its expansion in harmonics is u\^=(-e\^[-/2]{}H\_[0lm]{}Y\^[lm]{}, W\_[lm]{}Y\^[lm]{},-V\_[lm]{} \^[ab]{}Y\_[,b]{}\^[lm]{})r\^le\^[t]{} . We also expand the perturbed heat flux in harmonics as follows q\_=e\^[-/2]{} (0,Q\_[1lm]{}(r)r\^[l-1]{}Y\^[lm]{}(,) ,Q\_[2lm]{}(r)r\^lY\^[lm]{}\_[,a]{}(,))e\^[t]{}, ($\kappa$ thermal conductivity) and the Lagrangian perturbations of the pressure and the energy density as p&&p+p\^[(0)]{}\_[,]{}\^=p\_[lm]{}Y\^[lm]{}r\^le\^[t]{}\ &&+\^[(0)]{}\_[,]{}\^=\_[lm]{}Y\^[lm]{}r\^l e\^[t]{} (the same holds for the Lagrangian perturbations of the other thermodynamical quantities). Finally, following and generalizing LD, we will use a set of variables ($X_{lm},E_{lm}, \Sigma_{lm}$) related to the Lagrangian perturbations of the pressure, energy density, and entropy, defined as follows: X\_[lm]{}&=&-e\^[/2]{}r\^[-l]{}p\_[lm]{}\ E\_[lm]{}&=&-e\^[/2]{}r\^[-l]{}\_[lm]{}\ \_[lm]{}&=&-e\^[/2]{}r\^[-l]{}s\_[lm]{}.\[XESY\] At this point, we already can see that the presence of the new dissipative terms affects the resulting equations in two main ways. First, we have a number of additional terms in the perturbed energy–stress tensor, related all of them to the heat flux. Second, the thermodynamical relation between the pressure and the energy density is now more complex because the equation of state (EOS) is not only a function of the energy density, but also of the entropy, say, $p=p(\rho,s)$. For clarity, we will discuss the above points separately. ### Taking into account the heat flux in the stress–energy tensor. The new terms appearing in the perturbed equations can be greatly simplified by working in a suitable reference frame. Introducing the four–velocity of an observer in the frame in which the [*energy*]{} (as opposed to the [*matter*]{}) is at rest u\_u\_+,\[defhatu\] then, the stress–energy tensor (\[pertT\]) has formally the expression of the perfect fluid case: T\_=(+p)u\_u\_+pg\_+pg\_+ (u\_u\_+u\_u\_).\[standardtmunu\] Accordingly, we can redefine the original variables of Lindblom & Detweiler in the following way: W\_[lm]{}&&W\_[lm]{} +e\^[-ł/2]{}\[Wh\]\ V\_[lm]{}&&V\_[lm]{} -\[Vh\]\ X\_[lm]{}&&X\_[lm]{} -e\^[/2-ł]{}\[Xh\]\ E\_[lm]{}&&E\_[lm]{} -e\^[/2-ł]{}, \[Eh\] and now the harmonic expansions of $\d u_{\mu}$ and of the thermodynamical quantities are also formally identical to those in the perfect fluid case: u\^&=&(-e\^[-/2]{}H\_[0lm]{}Y\^[lm]{}, W\_[lm]{}Y\^[lm]{}, -V\_[lm]{}\^[ab]{}Y\_[,b]{}\^[lm]{})r\^le\^[t]{}\ &&\ X\_[lm]{}&=& -e\^[/2]{}(r\^[-l]{}p\_[lm]{} +W\_[lm]{}p\^[(0)]{})\ E\_[lm]{}&=& -e\^[/2]{}(r\^[-l]{}\_[lm]{} +W\_[lm]{}\^[(0)]{}) . Thus, we can follow the formal derivation of the perturbation equations in the adiabatic case, but using variables in the energy frame, the [*hatted*]{} variables; in this way we take easily into account the extra terms in the stress–energy tensor. The final system of equations (see section IV) will be analogous to the original system in the formalism of LD, if the perturbation of the energy $\hat E$ is left explicitly in the equations. ### Relation between $\Delta\rho$ and $\Delta p$ In the adiabatic case, the system of equations is closed by using the EOS to express the energy perturbation in terms of the pressure perturbation, [*i.e.*]{} $ \D\rho=c_s^{-2}\D p $ where $c_s^{2}$ is the adiabatic speed of sound c\_s\^[2]{}()\_[s]{}. This relation is used in the derivation of the LD equations and by many other authors in order to eliminate the $\D\rho$ from the equations, so that in the final system only the pressure perturbation appears. For general EOSs (of the form, i.e., $\rho=\rho(p,s)$), the Lagrangian perturbation of the energy density can be expressed in terms of the Lagrangian perturbations of the other variables by using the thermodynamical relation: d=c\_s\^[-2]{}dp+nT\_1ds,\[relalpha\] where $T$ is the matter temperature, $n$ is the particle density (baryon density for our purposes in neutron stars) and \_1&&()\_[p]{}. Therefore, the Lagrangian perturbation of the energy density is given by =c\_s\^[-2]{}p+nT\_1s. \[Deltarho2\] In the following we will omit the $\z$ super-indexes that label background quantities ($n, T, \rho, p$) for clarity. More details about the equation of state and thermodynamics are given in the next section. The perfect–fluid limit is recovered when the element of fluid does not exchange heat with its surroundings, i.e., $\Delta s=0$. In the general case, $\Delta s$ does not vanish, so we have to take into account all terms in equation (\[Deltarho2\]), which couples the entropy perturbation to the other perturbations. Therefore, instead of simply substituting $E_{lm}=c_s^{-2} X_{lm}$, one must now use E\_[lm]{}=c\_s\^[-2]{}X\_[lm]{}+n T \_1\_[lm]{} .\[exprhE\] But this introduces a new variable, the perturbation of the entropy, so that we need additional information. ### Using the energy and baryon conservation equations The new variable which is now included in our system of equations can be rearranged by using the first law of thermodynamics and the continuity equation, which was not necessary in the perfect fluid case. The time component of the divergence of the stress–energy tensor, i.e. the energy density conservation equation, up to first order in the perturbations, reads u\_T\^\_[;]{}= -e\^[-/2]{}-(+p)u\^\_[;]{} -(q\^\_[ ;]{}+q\^u\^ u\_[;]{})=0. In order to simplify this expression we can use the equation of continuity (\[cont\]) (nu\^)\_[;]{}=u\^n\_[, ]{}+nu\^\_[;]{}=e\^[-/2]{}n +nu\^\_[;]{}=0 and the first principle of thermodynamics in the form = n+nT s, to obtain n Ts=-[e\^[/2]{}]{} (q\^\_[ ;]{}+q\^u\^u\_[;]{}) \[peqSY1\] that[^2], in terms of $\hat\Sigma_{lm}$, becomes && n T \_[lm]{}= .\[eqSY1\] Equation (\[eqSY1\]) gives the entropy perturbation in terms of the perturbed energy flux $Q_1$. In order to close the system, we need to supplement our system with the equations that describe how the heat flux depends on the local gradients of the perturbations: the transport equations, which are given by the relativistic theory of dissipative thermodynamics. A simple two–parameter Equation of State. ========================================= In this section we describe the EOS that we have used and we define all the thermodynamical quantities that appear in the final form of the equations. We begin by considering a general equation of state depending on the baryon density $n$ and the entropy per baryon $s$: p=p(n,s) ;   =(n,s) . Let us define the specific internal energy $e$ (energy per particle, excluding the rest mass) and the speed of sound $c_s$: e &=& - m\_B\ c\_s\^2 &=& ()\_s ($m_B$ baryon mass). Next, let us assume that the EOS can be written as follows p(n,e) = (-1) n e \[perfect0\] with $\gamma$ constant. Using thermodynamical identities, one can show that $\left(\frac{\partial p}{\partial n}\right)_s=\frac{\gamma p}{n}$, so that (\[perfect0\]) is equivalent to p(n,s)=K(s)(m\_B n)\^\[perfect\] where $K(s)$ is an arbitrary function of integration that only depends on the specific entropy. If one chooses $K(s)=K_0 \exp{(\gamma-1)s}$, this leads to the perfect gas law $p=n T$ (in units with the Boltzmann constant $k_B=1$). The form (\[perfect\]) allows to include finite temperature effects with a simple equation of state, just giving an explicit dependence of $K(s)$ with the entropy. From the first law of thermodynamics d= dn + n T ds ,\[Ilaw\] the temperature can be found according to T=()\_n =()\_n =.\[defT\] The rest of thermodynamical quantities we need, in terms of $K$, are the following: &=&()\^+\ c\_s\^2&=&\^[-1]{}\ \_1&=&()\_p =1-\ C\_p&=&T()\_p= \^[-1]{} where $C_p$ is the specific heat at constant pressure. In order to study the effects of heat transport in the non–adiabatic case, we want to give a simple form of EOS which depends non trivially on the entropy profile and such that, consistently with the assumption that the background flux is negligible, it is isothermal in the relativistic sense ($T e^{\nu/2}$=const.)[^3]. This can be done as follows. By using the first law of thermodynamics (\[Ilaw\]) it is easy to prove that = + (1-\_1) .\[dToverT\] If we look for relativistic isothermal profiles, this implies that = = where the last equality comes from the TOV equation (hydrostatic equilibrium). Therefore, we have = \_1 = - ( 1 - ) and, substituting the explicit expression of the speed of sound, = - . Defining $x= \ln{p^{1-1/\gamma}}$, we finally obtain = - where $f(s)= K(s)^{1/\gamma}$. In principle this differential equation is not separable, but by numerical integration we can find the entropy profile $s(p)$ that leads to isothermality in the relativistic sense, for any given function $f(s)$. In particular, we have chosen f(s) = C + e\^[s/a]{} that gives a constant specific heat $C_p=a$. Here, $C$ and $\beta$ are arbitrary constants fixed to reproduce similar masses, radii, and temperatures as for newly born neutron stars. In particular, the temperature of the model that we analyze in the results section varies between (5-8) $\times 10^{11} K$, and $C_p=0.5$. Dissipative relativistic fluids and heat transfer. ================================================== The theory of standard irreversible thermodynamics was first extended from Newtonian to relativistic by Eckart in 1940. This theory shares with its Newtonian counterpart the problem that perturbations propagate at infinite speeds, which results in unstable equilibrium states. We must notice, however, that the non–causal Newtonian theory has been used for years in the study of non–relativistic stars with remarkable success, and with observational confirmation. Therefore, there is no reason to believe that the standard theory cannot be used as a first order approach to the problem and can give reasonable estimates of the main effects, even for relativistic stars. In general relativity, causality can be restored within the framework of an extended theory developed by Israel and Stewart [@IS]. This extended theory is also known as causal thermodynamics or second order thermodynamics because of the appearance of second order terms of the dissipative variables in the entropy. The problems associated to non–causal heat transfer become more relevant when the mean free path becomes larger than the spatial scale of the problem (in our case, the radius of the star), that is, for large thermal conductivities or diffusivity. The extended theory automatically incorporates transient phenomena on the timescale of the mean free path that cure the inconsistency. In this paper, we are interested in understanding qualitatively the effects of dissipative terms on the oscillation properties of stars, rather than developing a fully consistent theory, which is a more ambitious and complex task. Nevertheless, we start with the extended theory of dissipative fluids, and later on we discuss which second order terms are neglected and their relative importance. We begin with the equation that describes the variation of the entropy nT u\^ s\_[,]{}=-q\^\_[ ;]{}-q\^u\^u\_[;]{}\[ek1\] , and the definition of the heat flux, including the relaxation term that restores causality; this is of covariant Maxwell–Cattaneo form (a truncated version of Israel–Stewart equations): \_1 h\_\^ u\^ q\_[;]{} + q\_ =-(h\_\^ T\_[,]{}+Tu\^u\_[;]{})\[ek2\] where $h_{\alpha}^{\beta}$ is the projector onto the spatial subspace, $h_{\alpha}^{\beta}=\delta_{\alpha}^{\beta}+u_{\alpha}u^{\beta}$, $\kappa$ is the thermal conductivity, and $\tau_1$ is the relaxation timescale in which the system restores thermal equilibrium. Let us consider equations (\[ek1\]), (\[ek2\]) on a perturbed static spherical background. The perturbation of equation (\[ek1\]) gives the equation previously derived (\[eqSY1\]). The perturbed version of equation (\[ek2\]) must satisfy $q_{\alpha}u^{\alpha}=0$ and, since we are imposing a zero background flux ($q\z_{j}=0,~j=1,2,3$), we have $\d q_0=0$ and (1 + e\^[-/2]{} \_1) q\_[j]{}&=&-(T\_[,j]{}+T u\^[(0)]{}u\_[j;]{}+T(u\^u\_[j;]{})) .\[pert\] Expanding in spherical harmonics and after some manipulations we find (1 + e\^[-/2]{} \_1) q\_[j]{} &=&-e\^[-/2]{}\_[,j]{}-T \^2e\^[-]{}v\_[j]{}\ &=&-e\^[-/2]{}\_[,j]{}-T \^2e\^[-]{}v\_[j]{} \[ekpert\] where v\_[j]{}(Y\^[lm]{},V\_[lm]{}Y\^[lm]{}\_[,a]{})r\^le\^[t]{}. \[defvj\] Applying the thermodynamical relation (\[dToverT\]), which implies = + (1-\_1) , on Eq. (\[ekpert\]), changing to the energy frame (perturbations in terms of hatted variables), and applying the Einstein equation X\_[lm]{}-\^2(+p)e\^[-/2]{}V\_[lm]{}-e\^[(-)/2]{} (+p)W\_[lm]{}-(+p)e\^[/2]{}H\_[0lm]{}=0,\[einsteineqnew\] we obtain a first algebraic relation between the entropy perturbation, $Q_2$, and $\hat X$, (1 + e\^[-/2]{} \_1) Q\_[2lm]{}=T.\[sigmainv0\] The exact evaluation of the relaxation time $\tau_1$ involves complicated collision integrals and depends on the microscopical details of the interaction between particles. Notice though, that it is usually estimated as $\approx \kappa T/ \rho$ (see i.e. [@IS]). Therefore we do know that this second order correction is of the same order as the last term on the right hand side of Eq. (\[sigmainv0\]). Indeed, if one defines the relaxation timescale as $\tau_1 = \kappa T / (\rho+p)$, these two terms cancel each other. In the following, we will neglect all second order terms, and Equation \[sigmainv0\] becomes Q\_[2lm]{}=T .\[sigmainv\] We have checked that the numerical results presented in section VI are not altered by including the second order corrections, being the relative difference between the two solutions less than $1\%$. Let us come back to Eq. (\[ekpert\]). After neglecting the second order term, the quantity $e^{\nu/2}\d q_{j}/\kappa+T \o^2e^{-\nu/2}v_{j}$ is a gradient, thus the harmonic components of the perturbed heat flux satisfy Q\_[2lm]{}’&=&-Q\_[2lm]{}+Q\_[2lm]{}\ &&-T\^2e\^[-/2]{}.\[eqQ20\] Next, we can make use of the following equation, that is derived directly from the LD equations [@LD; @DL] (\^2e\^[-]{}V\_[lm]{})’ = \^2e\^[-]{}(rH\_[1lm]{}-W\_[lm]{}- V\_[lm]{}) - (X\_[lm]{}\^ -E\_[lm]{}p\^) to simplify the right hand side of Eq. (\[eqQ20\]) Q\_[2lm]{}’&=&-Q\_[2lm]{}+Q\_[1lm]{} +(X\_[lm]{}’-E\_[lm]{}p’)  ,\[Q2parz\] and, finally, introducing the thermodynamical derivatives defined in the previous section ($\alpha_1, C_p$) and keeping only first order terms we obtain Q\_[2lm]{}’=-Q\_[2lm]{}+Q\_[1lm]{} +nT\_1 Q\_[2lm]{}. Numerical implementation ======================== The complete set of equations ----------------------------- We can now write the full set of equations. For simplicity, in the following we will omit the sub-indexes ${lm}$, keeping in mind that one has a complete set of equations for each multipole in the expansion. Our variables are H\_[0]{},H\_[1]{},K,W,V,X, E,,Q\_[1]{},Q\_[2]{}. Four of these variables $H_{0},\,\hat V,\,\hat E,\,\hat\Sigma,$ are given in terms of the others by the following algebraic relations: H\_[0]{}&=&{ 8r\^3e\^[-/2]{}[X]{} - r\^3 e\^[-ł]{} H\_[1]{} .\ && . + K}\ &&\[exH0\]\ &&\ \^2 e\^[-]{} V&=& -- H\_[0]{} \[exV\]\ &&\ E&=&c\_s\^[-2]{}X+n T\_1\[exE\]\ &&\ &=&C\_p. \[exSigma\] The remaining variables $ H_{1},\,K,\,{\hat W},\,{\hat X},\,Q_{1},\,Q_{2},\,$ satisfy the following differential equations: H\_[1]{}’&=&- H\_[1]{}\ &&+e\^[ł]{}\[eqH1\]\ &&\ K’&=&H\_[0]{}+H\_[1]{} -K -8(+p)\[eqK\]\ &&\ [W]{}’&=&-+re\^[ł/2]{} \[eqW\]\ &&\ [X]{}’&=&-+(+p)e\^[/2]{}{( -’)H\_[0]{}.\ &&+H\_[1]{} +(’-)K -’[V]{}\ &&. -}\[eqX\]\ &&\ Q\_2’&=&-Q\_2+Q\_1 -nT\_1p’Q\_2 \[eqq2\]\ &&\ Q\_1’&=&nT -(++ -)Q\_1+e\^Q\_2.\[eqq1\] It must be noticed that the first four equations are formally identical to those of Lindblom & Detweiler [@LD; @DL], but in terms of the new variables defined by Eq. (\[Wh\])–(\[Eh\]). Boundary conditions ------------------- We seek for solutions of the perturbation equations which are regular at the origin. Assuming that all variables near the center of the star have the form $x(r)=x(0)+O(r^2)$, and expanding equations (\[exH0\])–(\[eqq1\]) we find that the following relations must be satisfied at the origin: H\_[0]{}(0)&=&K(0)\[H00\]\ [V]{}(0)&=&-(0)\[V0\]\ [X]{}(0)&=&(\_0+p\_0)e\^[\_0/2]{} {(0)+K(0)}\ &&\[X0\]\ H\_[1]{}(0)&=&\[H10\]\ Q\_[2]{}(0)&=&Q\_[1]{}(0).\[q20\] The values of $\hat E_{lm}$ and $\hat\Sigma_{lm}$ at the origin are given by (\[exE\]), (\[exSigma\]) evaluated at $r=0$. Therefore, our equations admit three independent solutions, which are reduced to one by imposing appropriate boundary conditions at the surface of the star. The first is the vanishing of the Lagrangian pressure perturbation X(R\_s)=0;\[Xs0\] and the second is the vanishing of the radial flux Q\_1(R\_s)=0. Once we know the only independent solution at the surface of the star, the Zerilli function and its derivative can be computed in terms of the $H_1$ and $K$, as usual (see for example [@IP] and [@koj] for the expression of the Zerilli function in terms of $H_0$, $K$; the expression in terms of $H_1$ follows by applying Eq.(\[exH0\]) ): Z&=&r\^l(K-e\^H\_[1]{})\ Z’&=&r\^l(K .\ &&.+e\^[2]{} H\_[1]{}).\ Notice that the metric variables $H_0, H_1, K$ had not been redefined by the presence of heat flux. Finally, we integrate the Zerilli equation outside the star using the continued fraction method (see e.g. [@Sotani; @gdisc]), to obtain the amplitude of the ingoing and outgoing parts of the gravitational wave. The quasi-normal modes will correspond to the solutions for which the wave is purely outgoing. Results ======= In the previous sections we have ignored the effects of chemical diffusion. Our equations can then be applied to a physical situation in which heat conduction is relevant, while chemical diffusion is not. This could be the case of a young, hot, neutron star about 30 seconds after birth, once the proto-neutron star has lost its lepton content, and the temperature gradient is not far from isothermal. It is also the situation that one expects to find in a cold neutron star that undergoes a phase transition to deconfined quark matter in a color superconductor state. In this latter scenario, pairing between quarks inhibits the presence of leptons [@AR02]. For reference, in Appendix \[nm\] we derive the equations including the terms related to chemical diffusion, leaving for future work their application in a realistic scenario. In this paper, we will focus only on the effects of finite thermal conductivity, from which we can understand most of the qualitative differences between the adiabatic and non–adiabatic cases. Hereafter we consider a profile for the thermal conductivity such that $\kappa/n$ is constant throughout the star. The ratio $\kappa/n$ represents, roughly, a sort of mean free path, to be compared with the characteristic scale of the system, i.e., the radius of the star. In the diffusion limit $\kappa/n \ll R$, perturbations can be considered basically adiabatic, while when $\kappa/n$ becomes of the order or larger than the radius of the star, thermal equilibrium is achieved in a timescale shorter than the dynamical characteristic timescale. We begin by studying the effects of finite thermal conductivity on the $p-$modes. In Fig. \[figp\], we show, in the complex plane, the variation of the complex frequency as the ratio $\kappa/n$ varies from the adiabatic limit to the isothermal limit. The oscillation frequency (real part) is represented in the horizontal axis while the vertical axis corresponds to the inverse damping time (imaginary part). From the results, we can see that the real part of the frequency of the first acoustic mode is shifted to lower values in about 200 Hz, as the mean free path becomes larger, an the isothermal limit is approached. The damping time in both, the adiabatic and isothermal limit is very similar (not visible in the scale of the figure), being in both cases of about 1 s. However, in the semi-transparent regime, thermal dissipation is so effective that the damping time is reduced by about 3 orders of magnitude (to 2-3 ms). These results can be easily understood by looking at a simple toy model that describes how acoustic modes are affected by thermal diffusion. We describe this simple Newtonian problem in Appendix \[toy\]. It makes more manifest the relevant physics without the complications of the full set of equations in General Relativity. By analogy with this toy model, one can understand that the oscillation frequency in the isothermal limit is smaller than that of the adiabatic case, depending on the ratio between the adiabatic speed of sound $c_s$ and the isothermal speed of sound $c_T$, defined as c\_T\^2()\_T = ( + )\^[-1]{}. For the EOS we used, this difference is about a 5% in average, which is consistent with the shift in the $p-$mode frequency. Several comments about the adiabatic and the isothermal limits are in order. In both cases, the heat flux tends to zero as we approach the limits, but for different reasons, that can be understood by looking at Eq. (\[sigmainv\]). In the diffusion limit (adiabatic perturbations) $\kappa \rightarrow 0$, but in the isothermal limit, the flux vanishes because the term within brackets (i.e. the perturbation of the temperature, including relativistic corrections) vanishes. Therefore, in both limits Eqs. (\[eqq2\])–(\[eqq1\]) are decoupled from the rest of the system. The difference between both limits appears only in Eq. (\[eqW\]). Explicitly, in the adiabatic limit Eq. (\[eqW\]) reads: ’=-+re\^[ł/2]{} while in the isothermal limit it becomes ’=-+re\^[ł/2]{} where = H\_0 + . In other words, one just needs to substitute the adiabatic speed of sound by the isothermal speed of sound (consistently with what happens in Newtonian perturbation theory) but now there appears an additional term (proportional to the Lagrangian perturbation of $\nu/2$) because of the fact that the concept of isothermality in general relativity involves the red-shifted temperature, $T e^{\nu/2}$. The damping time when dissipation is most effective, i.e. (as we can see from Fig.\[figp\]) when $\kappa/n \approx 1$ km, can be also estimated by \_[diss]{}= = n C\_p ,  \[ediss\] and taking typical values at the interior of the star, such as $C_p=1$, $R=10$ km, we get that for $\kappa/n=1$ km, the dissipative timescale is $\tau_{diss}=0.001$ s, in agreement with the results shown in Fig. \[figp\]. These results show explicitly and quantitatively that thermal dissipation affects the damping of the non-radial pulsations, competing with the other main dissipation mechanism, i.e., GW emission. The damping times we compute take for the first time into account both mechanisms in a self–consistent manner, by including the appropriate physics in the equations. In the semitransparent regime, when diffusion is more effective, we must expect short-lived, strongly damped, GW signals, more similar to a GW burst than a proper $\sim$ kHz oscillation lasting for several hundred oscillations. Notice, however, that if the physical conditions are such that the thermal relaxation timescale becomes shorter than the oscillation period, thermal damping does not affect much the damping time, being the only remarkable difference a shift in the oscillation frequency. Let us now focus on the fundamental mode. The $f-$mode is a global oscillation mode that depends essentially on the average density of the star, rather than on the velocity at which acoustic waves propagate. In fact, it is well known that an incompressible fluid (infinite speed of sound) does not have $p-$modes but the frequency of the $f-$mode is similar to that of a realistic star with the same average density. For these reasons, one expects similar results to those of the $p-$mode but with smaller effects on the shifts of frequencies. In Fig. \[figf\] we show our results for the $f-$mode, when this is clearly visible. Now, the shift in frequency is only of a few Hz, and the effect on the damping time is also smaller, although in the semitransparent regime, it can be as much as a factor 2.5 smaller than that of the adiabatic case (0.25 s). Analogously to what happened for the $p-$mode, once we enter in the isothermal regime, thermal damping is less effective and the damping time becomes quite similar to the adiabatic value. One must keep in mind that these results depend on the particular details of the EOS, or more precisely, on the temperature and specific heat. In cases with higher temperatures and lower specific heat, the frequency of the fundamental mode could be quite different from the adiabatic case. It remains to be analyzed what are the quantitative differences for realistic models at different stages of neutron star evolution, which is beyond the scope of this paper. Finally, let us discuss the $g-$modes. In Fig. \[figg\] we show the results for the first gravity mode, which in the adiabatic case has a frequency of 364 Hz and a damping time of thousands of seconds. The existence of these modes is related to the presence of thermal (or chemical) gradients in the star, therefore it is not surprising that as we increase the thermal conductivity and, consequently, heat interchange between displaced fluid elements is more effective, the $g-$mode frequencies decrease and the damping time increases. Contrary to what happens with the acoustic and fundamental modes, in the limit of infinite thermal conductivity the $g-$modes degenerate at zero frequency. It is interesting to note that the damping time does not decrease indefinitely as the mean free path increases, being bounded by the minimum dissipative timescale that is, again, close to the estimate of Eq. \[ediss\] (of the order of ms), in our scenario. The last important difference with the other modes is that for values of $\kappa/n$ as low as $10^{-3}$km, although the change in frequency is small, the damping time is sensibly shorter (0.01 s). For the $f-$ or $p-$modes it was needed to have higher values of $\kappa/n$ to obtain significant differences with respect to the adiabatic damping time. Final remarks ============= In this paper we have developed the formalism to study non-radial oscillations of relativistic stars in the frequency domain giving one step forward from the perfect fluid case by including the effects of thermal diffusion in a fully relativistic formalism. This allows us to understand one of the non–adiabatic processes (we did not consider viscosity) that may be relevant in the study of the oscillation properties of newly born neutron stars, strange stars, or neutron stars with deconfined quark matter in the core (hybrid stars). When a general equation of state that depends on temperature is used, the perturbations of the fluid result in perturbations of temperature (or chemical composition) and, consequently, in heat flux (or chemical diffusion), that is coupled with the geometry through the Einstein field equations. We have analyzed separately the $p-$modes, the fundamental mode and the $g-$modes, each one being affected in a different way by thermal diffusion. As expected from a simplified model discussed in Appendix \[toy\], the $p-$mode frequency is shifted to lower values in a factor roughly proportional to the ratio between the [*isothermal*]{} and [*adiabatic*]{} speeds of sound $c_T/c_s$, thus depending very much on the EOS employed and the values of the local temperature. The frequency of the $f-$mode, however, is barely affected because it is a global oscillation mode that does not depend much on the local value of the speed of sound. Both $f-$ and $p-$modes are more efficiently damped in the semitransparent regime, when the mean free path of the particles responsible of the heat transfer is smaller, but close to the typical length scale of the system. The reason is that once the timescale to reach thermal equilibrium becomes shorter than the oscillation period, the fluid oscillates keeping the temperature constant, and further increasing the thermal conductivity does not change this situation. In this limit, there is no additional thermal damping at first order (as shown in Appendix \[toy\]), so that the damping times in the adiabatic and isothermal limits are very similar. The response of the $g-$modes to heat transfer is quite different. Since these modes exist because of the presence of thermal gradients, it is naturally found that, as heat transfer becomes more effective, and the temperature perturbations are smeared out, the frequency is shifted to lower values and the damping time becomes shorter. In the isothermal limit the $g-$modes have degenerated to zero frequency. In this first approach to the problem, we have only presented results about the effects of heat transfer. It must be remarked that in newly born neutron stars the effects of lepton diffusion are also important, and in Appendix \[nm\] we discuss the relevant equations to study that case. The coupling between thermal and chemical diffusion is one of the complications that arise if one wants to study the realistic case of proto–neutron stars a few seconds after birth, but there is a second one. In this early stage the background gradients of temperature and chemical potential are important, and in some cases, or for some of the modes, the overall evolution timescale may become similar to the characteristic oscillation periods or damping times. Furthermore, proto–neutron stars are expected to rotate fastly, while in the present work we focused on thermal effects neglecting rotation. We defer for future work a more rigorous study of the realistic scenario, that must consider the non–trivial issue of coupling between rotational and thermal effects. This must probably be done in the time domain instead of the frequency domain, or including second order terms. Our results can also be directly applied to another interesting case: hot strange stars in which quark matter is in a color superconductor state. As discussed in the paper, this appears to be the natural state of deconfined quark matter, and recent calculations show that the thermal conductivities are many orders of magnitude larger than those of standard neutron star matter. In this latter scenario, if the strange star is born after the mini–collapse of an old, evolved neutron star that has been accreting matter, the initial lepton content is small, and the effects of thermal diffusion are the dominant non–adiabatic correction. We are indebted to J.A. Miralles and V. Ferrari for many useful comments and suggestions. We also thank L. Rezzolla and M. Bruni for useful discussions. This work has been supported by the Spanish MCyT grant AYA 2001-3490-C02-02, and the [*Acción Integrada Hispano–Italiana*]{} HI2003-0284. J.A.P. is supported by a [*Ramón y Cajal*]{} contract from the Spanish MCyT. GM thanks the PPARC for support. A toy model for non-adiabatic oscillations. {#toy} =========================================== Consider a 1-dimensional problem consisting of a box at constant density, and constant pressure, and let us study the normal acoustic modes of the fluid. The linearized continuity and momentum equations in the adiabatic case are + v\_1 = 0 ,     + p\_1 = 0 . Integrating in time the continuity equation gives ${\rho_1} + \rho \nabla \xi = 0$, with $\xi =\delta r$, and substituting ${\rho_1}$ in the momentum equation leads to = - p\_1 = - \_1 = [c\_s\^2]{} \^2 \[mom1\] which is a simple wave equation that, when we consider perturbations of the form $exp{(i \sigma t - i k x)}$ gives the dispersion equation: - \^2 + k\^2 [c\_s\^2]{} = 0. Its solution consists of oscillatory modes with frequencies $\pm k c_s$. Consider now the non–adiabatic case. The perturbation of the pressure is p\_1 = c\_s\^2 \_1 + s\_1 with $\beta= \frac{1}{\rho}(dp/ds)_\rho$, and equation (\[mom1\]) becomes: = [c\_s\^2]{} \^2 - s\_1 . \[mom2\] We have to introduce the additional equation of conservation of energy n T = - F where $T$ is the temperature, $n=\rho/m$ is the particle density, and F = - T . Considering first order perturbations of the previous equations we have n T = \^2 T\_1,   that can be used to replace $\frac{d s_1}{dt}$ after taking the time derivative of Eq. (\[mom2\]), to obtain = [c\_s\^2]{} \^2 - (\^2 T\_1). \[mom3\] Next, we only need to find an expression for the last term in Eq. (\[mom3\]). The perturbation of the temperature can be written in terms of $\rho_1$ and $s_1$ as follows T\_1 = ()\_s c\_s\^2 \_1 + s\_1 , where $c_v=T\left(\frac{\partial s}{\partial T}\right)_\rho$ is the specific heat at constant volume. Thus , T\_1 = ()\_s c\_s\^2 \_1 + s\_1 = - ()\_s c\_s\^2 \^2 - ( - [c\_s\^2]{} \^2 ) . Inserting this in Eq. (\[mom3\]) one obtains = [c\_s\^2]{} \^2 + ()\_s \^4 + \^2 . \[mom4\] Considering again perturbations of the form $exp{(i \sigma t - i k x)}$ we can derive the dispersion equation: - \^3 + [c\_s\^2]{} k\^2 + i k\^2 = 0. \[disp1\] The adiabatic limit is recovered when $\kappa /n c_v \ll 1/k$ while in the limit $\kappa / n c_v \gg 1/k$, the dispersion equation becomes \^2 - k\^2 c\_T\^2 = 0 , which simply states that sound waves propagate now at $c_T$ instead of $c_s$. It must be remarked that in the isothermal limit, the solution is always real (no damping), and $\sigma$ acquires an imaginary part only in the intermediate regime. Notice also that dissipation is most effective for modes at short wavelengths (large $k$). In Fig. \[fappendix\] we show the solutions of the dispersion equation (\[disp1\]) as a function of the parameter $D =\kappa k/nc_v$. The oscillation frequency (normalized to $k c_s$) is represented by the solid line, and the inverse damping time by the dashed line. The qualitative behaviour is very similar to the results discussed in the text. The ratio of the the limiting frequencies depends on the ratio $c_T/c_s$. For this example, we have taken $c_T/c_s=0.8$. A final remark is in order. The dispersion relation can be written as \^3 + a\_2 \^2 + a\_1 \_1 + a\_0 = 0 a\_2 = D   ,    a\_1 = c\_s\^2   ,    a\_0 = D c\_T\^2 where $\gamma=i \sigma/ k$. The condition that at least one of the roots has a positive real part (unstable) is equivalent to one of the following inequalities a\_[2]{} &lt; 0, a\_[0]{} &lt; 0, a\_[1]{} a\_[2]{} &lt; a\_[0]{} . Since all coefficients are positive defined, only the third condition could apply. It is simply c\_s\^2 &lt; c\_T\^2 which is generally not the case, for reasonable realistic EOSs. Introducing neutrino diffusion in our equations {#nm} =============================================== In order to study non-adiabatic perturbations in proto-neutron stars, we need to generalize our equations to the case when both, energy flux and particle number flux due to neutrino diffusion are present. In this appendix we derive the equations in a similar way as it was done in the main body of the paper for the case with only heat transfer. Thermodynamical variables and relations {#thermorel} --------------------------------------- We choose as independent thermodynamical variables the pressure $p$, the entropy per baryon $s$, and the lepton fraction $Y_L=n_L/n$, with $n$ being baryon number density and $n_L$ the lepton number density. Given these three thermodynamical quantities, the EOS gives the rest of variables. In particular, the EOS provides n=n(p,s,Y\_L)     =(p,s,Y\_L). The reason for taking $Y_L$ as the extensive variable describing the chemical composition is the following. The work associated with a variation of chemical composition at constant entropy is \_i\_idY\_i. Here, $Y_i=n_i/n$, and $\mu_i,n_i$ are the chemical potentials and number densities of the specie $i$, respectively. In our case, the process that changes the chemical composition is the capture of electrons by protons and its reciprocal (inverse $\beta$ decay) e+p n+, so that chemical equilibrium results in $\mu_{\nu}=\mu_e+\mu_p-\mu_n$. Since only neutrinos can diffuse throughout the star, we also have $dY_e=dY_p=-dY_n$, while $dY_{\nu}$ is independent from the others. Consequently, \_i\_idY\_i=(\_e+\_p-\_n)dY\_e+\_dY\_= \_(dY\_e+dY\_)=\_dY\_L. Notice that in this system the variable conjugate to $Y_L$ is $\mu_{\nu}$. According to this result, the first principle of thermodynamics (see for example [@MTW]) takes the form d=dn+nTds+n\_dY\_L.\[first\] Analogously, since we use $(p,s,Y_L)$ as independent variables, we can write, in place of (\[relalpha\]), d=c\_s\^[-2]{}dp+nT\_1 ds+n\_\_2dY\_L\[drho\] where the sound speed $c_s$ is defined by c\_s\^[2]{}()\_[s,Y\_L]{}. and we have defined the following thermodynamical derivatives: \_1()\_[p,Y\_L]{} ,      \_2 ()\_[p,s]{}. We also define \_3()\_[p,s]{}= ()\_[p,Y\_L]{} where the last equality can be easily proved from Maxwell relations. Let us now consider the heat function $\d Q$ Q= Tds +\_i\_idY\_i = Tds+\_dY\_L. Then, we define ()\_[p,Y\_L]{}&=&T()\_[p,Y\_L]{} C\_p\ ()\_[p,s]{}&=&\_ ()\_[p,s]{} \_p, where $C_p$ is the specific heat at constant pressure and constant number fraction, $\Lambda_p$ is a coefficient describing the heat associated to a change in composition at constant entropy and pressure. With all the above definitions, and using the first law of thermodynamics, the expressions of $dT$ and $d\mu_{\nu}$ in terms of $s,p,Y_L$ can be written as dT&=&ds+(1-\_1)dp+\_3 dY\_L\[dT\]\ d\_&=&dY\_L +(1-\_2)dp+\_3 ds.\[dmu\] Equations (\[drho\]), (\[dT\]), and (\[dmu\]) allow us to express the perturbations of density, temperature and chemical potentials in terms of the independent variables. Similar relations apply for the gradients of the background quantities. For example, from (\[drho\]) &=&c\_s\^[-2]{}p+nT\_1s+n\_\_2 Y\_L\[deltarho\]\ \^[(0)]{}&=&c\_s\^[-2]{}p\^[(0)]{}+nT\_1s\^[(0)]{} +n\_\_2Y\_L\^[(0)]{}\[rhoprime\] and the same applies for $\Delta T,\,\Delta\mu_{\nu},\, T^{(0)\prime},\, \mu_{\nu}^{(0)\prime}$. Dissipative thermodynamics with energy and number transport ----------------------------------------------------------- Next, we need to generalize Eckart dissipative thermodynamics to the case when also chemical diffusion, (lepton diffusion in this case) is present. Therefore, we need to consider the lepton conservation equation (n Y\_eu\^+ n Y\_(u\^+n\^))\_[;]{}=0, where $n^{\alpha}$ is the neutrino four–velocity with respect to the frame comoving with matter (see [@Thorne], [@IS]), with $n^{\mu}u_{\mu}=0$ and $n^{\mu}n_{\mu}=1$. Defining the neutrino flux as $f^{\alpha}=n Y_{\nu}n^{\alpha}$, that satisfies $u_{\alpha}f^{\alpha}=0\,$, and using the continuity equation (\[cont\]) we obtain the equation of lepton conservation $n u^{\alpha} (Y_L)_{,\alpha}=-f^{\beta}_{\,\,;\beta}$. This latter equation is particularized for lepton diffusion, which is relevant for nascent neutron stars, but it is a general conservation equation for massless particles. Therefore, the set of equations we have is the following: - Continuity equation: (nu\^)\_[;]{}=0. - Energy conservation equation (stress-energy tensor projected onto $u$): u\^\_[,]{}+u\^\_[ ;]{}(+p)+q\^\_[ ;]{} +q\^u\^u\_[;]{}=0. - First law of thermodynamics: d=dn+nTds+n\_ dY\_L.\[newfirst\] - Lepton conservation: nu\^Y\_[L,]{}=-f\^\_[ ;]{}.\[Ydot\] Together they give: nTu\^s\_[,]{}=-q\^\_[ ;]{}+\_ f\^\_[ ;]{} -q\^u\^u\_[;]{} .\[sdot\] Let us now define the entropy flux as in [@Israel], S\^=snu\^+-\_. Notice that when degenerate neutrinos dominate the transport of energy and particles, $q_\alpha=\mu_{\nu} f_\alpha$ and there is no entropy flux. By taking the divergence of the entropy flux, we have TS\^\_[ ;]{}= -(D\_T+Tu\^u\_[;]{}) -f\^TD\_ , where we have defined $\eta\equiv\frac{\mu_{\nu}}{T}$. The second law of thermodynamics, $S^{\alpha}_{~;\alpha}\ge 0$, must be always satisfied. The simplest assumption for the form of the energy and lepton fluxes that satisfies $S^{\alpha}_{~;\alpha}\ge 0$ is q\_=-\_E(D\_T+Tu\^u\_[;]{}) ,      f\_=-\_ND\_\[tqf\] with $\kappa_E$, $\kappa_N$ positive defined coefficients, which are, respectively, the thermal conductivity and the diffusivity governing the transport of particles. In the static, radially symmetric case, our equations coincide with a particular case of those of [@Pon99]. Perturbations on a static, radially symmetric, non stratified background ------------------------------------------------------------------------ Let us consider a static, non–stratified background space–time, in which the lepton and energy fluxes satisfy f\_&=&0       q\_=0  . Consistently, we will have $(Te^{\nu/2})'=0,\,\eta'=0,\,(\mu_{\nu} e^{\nu/2})'=0$. Next, we expand the fluxes in spherical harmonics as follows: q\_i&=&\_Ee\^[-/2]{}(Q\_[1lm]{}r\^[l-1]{}Y\^[lm]{},Q\_[2lm]{}r\^lY\^[lm]{}\_[,a]{}) e\^[t]{}\ f\_i&=&\_N (F\_[1lm]{}r\^[l-1]{}Y\^[lm]{},F\_[2lm]{}r\^lY\^[lm]{}\_[,a]{})e\^[t]{}  , \[hexpansions\] as well as the Lagrangian perturbation of $Y_L$ Y\_L&=&-e\^[-/2]{}r\^l[Y]{}\_[lm]{}Y\^[lm]{}e\^[t]{}. Analogously to the procedure in subsection 2.1, we define \_[lm]{}=[Y]{}\_[lm]{}-e\^[/2-]{} . By expanding the transport equations (\[tqf\]), we obtain q\_i&=&-e\^[-/2]{}T\_E\_[,i]{} -\_ET\^2e\^[-]{}v\_i\ f\_i&=&-\_N\_i,\[transporteqns\] where $v_j$ has been defined in (\[defvj\]). At this point, we can derive the equations for the perturbations $\hat E,Q_1,Q_2,F_1,F_2,\hat\Sigma,\hat{\cal Y}$. The differences with respect to the case with only thermal diffusion are the following: - Expression for $\hat E$. From (\[drho\]) we have E\_[lm]{}=c\_s\^[-2]{}X\_[lm]{}+nT\_1\_[lm]{}+ n\_\_2\_[lm]{}.\[EqEnumber\] - Equations for $Q_1$, $F_1$. Perturbing the conservation equations (\[sdot\]), (\[Ydot\]) we have nTs+n\_Y\_L&=&- (q\^\_[ ;]{}+q\^u\^u\_[;]{} )\[consen\]\ nY\_L&=&-f\^\_[ ;]{} \[consY\] that, after expanding in harmonics, give n T \_[lm]{}+n\_\_[lm]{}&=& \_E ,\[eqSY2\]\ n\_\_[lm]{}&=&\_N\^2.\[expdoty\] - Expressions for $\hat\Sigma$, $\hat{\cal Y}$. Comparing the $i=\theta,\phi$ components of (\[hexpansions\]) and (\[transporteqns\]) we obtain e\^[-/2]{}+\^2e\^[-]{}V\_[lm]{} &=& -(+)\ e\^[-/2]{}F\_[2lm]{}&=& -. Then, using the thermodynamical relations derived in Section \[thermorel\], the Einstein equation (\[einsteineqnew\]), and the fact that the background is static, that leads to -p’+Y\_L’&=&0\ -p’+s’ &=&0,\[derivatives\] we find \_[lm]{}+\_[lm]{}&=&C\_pA\_E\[EqSigmanumber\]\ \_[lm]{}+\_[lm]{}&=& \_pA\_N\[EqYnumber\] where A\_E=X\_[lm]{}+ ,      A\_N=X\_[lm]{}+. The solution of this system is \_[lm]{}&=&\ \_[lm]{}&=& . - Equations for $Q_2,F_2$. The equations for $Q_2$, $F_2$ follow from the fact that $e^{\nu/2}\delta q_j/\kappa_E+T\omega^2e^{-\nu/2}v_j$ and $\delta f_j/\kappa_N$ are gradients, which can be used to relate the radial and angular components of (\[hexpansions\]), F\_[2lm]{}’&=&-F\_[2lm]{}+F\_[1lm]{}.\[EqF2number\]\ Q\_[2lm]{}’&=&-Q\_[2lm]{}+Q\_[1lm]{} +(X\_[lm]{}’-E\_[lm]{}p’). \[EqQ2n\] The last term in (\[EqQ2n\]) can be expanded in terms of the rest of variables. Using (\[rhoprime\]) and (\[EqEnumber\]), and after some algebra, one finds Q\_[2lm]{}’&=&-Q\_[2lm]{}+Q\_[1lm]{}\ &&+{nT\_1C\_p .\ &&.+n\_\_2\_p }.\[EqQ2number\] Equations (\[EqEnumber\]), (\[eqSY2\]), (\[expdoty\]), (\[EqSigmanumber\]), (\[EqYnumber\]), (\[EqF2number\]), (\[EqQ2number\]), together with equations (\[exH0\]) to (\[eqX\]), form a closed system that allows to compute the non-radial perturbations of a star with energy and lepton transport. 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Misner C W, Thorne K S, Wheeler J A, [*Gravitation*]{}, W H Freeman & C, New York (1973). Thorne K S, Campolattaro A, Astrophys. J. [**149**]{} 591, (1967). Chandrasekhar S, Ferrari V, Proc. R. Soc. London [**A432**]{} 247, (1990). Israel W, Stewart J M, Ann. Phys. [**118**]{} 341, (1979). Ipser J R, Price R H, Phys. Rev. D[**43**]{}, 1768 (1990). Kojima Y, Phys. Rev. D[**46**]{}, 4289 (1992). Sotani H, Tominaga K, Maeda K, Phys. Rev. D[**65**]{}, 024010 (2001). Miniutti G, Pons J A, Berti E, Gualtieri L, Ferrari V, Mon. Not. R. Astron. Soc., [**338**]{} 389 (2003). Alford M, and Reddy S, Phys. Rev. D [**67**]{} 074024 (2003). Thorne K S, Mon. Not. R. Astron. Soc., [**194**]{} 439, (1981). Israel W, Ann. Phys. [**100**]{} 310, (1976). [^1]: Throughout the paper, we use units in which the Newton gravitational constant, the speed of light, and the Boltzmann constant are equal to one, $G=c=k_B=1$. [^2]: Notice that when $\delta q^{\alpha}=0$, equation (\[peqSY1\]) gives $\Delta s=0$: the adiabaticity of the perturbations directly follows from the form of the perfect fluid stress–energy tensor. [^3]: We stress that, as we said in the previous section, the equations we derive do apply also for background profiles not perfectly isothermal, as long as the background evolution timescale is longer than the dynamical timescale.

--- abstract: 'Stationary wave functions at the transition between plateaus of the integer quantum Hall effect are known to exhibit multi-fractal statistics. Here we explore this critical behavior for the case of scattering states of the Chalker-Coddington network model with point contacts. We argue that moments formed from the wave amplitudes of critical scattering states decay as pure powers of the distance between the points of contact and observation. These moments in the continuum limit are proposed to be correlation functions of primary fields of an underlying conformal field theory. We check this proposal numerically by finite-size scaling. We also verify the CFT prediction for a 3-point function involving two primary fields.' author: - | R. Bondesan, D. Wieczorek, M.R. Zirnbauer\ *Institut für Theoretische Physik, Universität zu Köln, Zülpicher Stra[ß]{}e 77, 50937 Köln, Germany* title: Pure scaling operators at the integer quantum Hall plateau transition --- #### Introduction. A revealing monitor of quantum critical behavior driven by disorder is multi-fractal wave-function statistics. In this vein, theory and experiment have focused on the multi-fractality at Anderson localization transitions between different topological phases of disordered electrons in two dimensions, the prime example being the transition between plateaus of the Hall conductance in the integer quantum Hall (IQH) effect [@Evers2008]. There has long been a consensus that it should be possible to describe the IQH transition by a conformal-invariant effective field theory. Yet, in spite of many efforts [@Bhaseen2000; @Ikhlef2011; @Bettelheim2012] it remains an unsolved problem to identify that conformal field theory (CFT) description. To make progress with the search for it, one needs to find the conformal fields and determine their scaling dimensions. A step in this direction was taken in [@Janssen1999; @Klesse2001], where the moments of the point-contact conductance were introduced and studied as correlation functions. Alas, these are coherent sums of conformal field correlators and therefore do not give direct access to individual conformal fields in pure form; see [@Obuse2013] for a recent discussion. The purpose of this Letter is to put forth a large (and so far unrecognized) class of multi-fractal observables that correspond directly to correlators of CFT primary fields. Our results are motivated by a recent $\sigma$-model based classification of scaling fields at Anderson transitions [@Gruzberg2011; @Gruzberg2013]. The new feature here is that we focus on the scattering states of an *open* system, while the previous work concerned moments of the local density of states for *closed* systems. For concreteness and simplicity, we work with the Chalker-Coddington (CC) network model. The CC model is known to be related by a duality transformation to a statistical mechanical system of vertex-model type [@Gruzberg1997; @Zirnbauer1997]. The main advance of our work is to construct lattice approximations for pure scaling fields on both sides of the duality – as scattering observables of the CC model and, equivalently, as operators of the vertex model. Both representations serve a purpose. Based on the latter, we argue that our lattice operators indeed are discretizations of pure scaling fields, while the former makes it possible to compute their conformal dimensions numerically by finite-size scaling. #### CC model and scattering states. We begin with a quick review of the CC model [@Chalker1988]. This is a network model for the quantum dynamics of an electron moving in two dimensions under the influence of a strong magnetic field and a random electric potential. Formulated on a square lattice, the model is built from elementary plaquettes with a definite sense of circulation that alternates between neighboring plaquettes. The links of the network are directed accordingly, so that each site has two incoming and two outgoing links. The electron wave function lives on the links and evolves in discrete time as $\vert \psi(t+1) \rangle = {U}\vert \psi(t) \rangle$ by a unitary operator ${U}= {U_\text{s}}\, {U_\text{r}}$. The factor ${U_\text{r}}$ is a diagonal matrix modeling the propagation along the links; it assigns to each link a random, independent and uniformly distributed ${\mbox{U}}(1)$ phase. The factor ${U_\text{s}}$ is non-random and consists of $2 \times 2$ matrices that describe the transfer from incoming to outgoing links at each site. When the probabilities for transfer to the left/right are equal, the model is critical and falls into the universality class of the IQH transition [@Chalker1988]. While it is of some interest to study the spectral properties and stationary wave function statistics of the closed network, here we turn to an open network. One major advantage of the open setting is that it allows one to formulate and study CFT correlators right at the critical point. (In contrast, Green’s functions of the closed system are defined by introducing a regularization which places the system slightly off criticality.) The network is opened up by severing a subset of links $C = \{ {\bm{c}}_1, \dots, {\bm{c}}_n\}$, which we call point contacts. Each cut makes for one network-incoming and one network-outgoing link where electric current is injected resp. drained by connecting the network to charge reservoirs. The dynamics in the presence of the point contacts is [@Janssen1999] $$\begin{aligned} \left|\psi(t+1)\right> &= {U}\Big( Q \left|\psi(t) \right> + \sum\nolimits_{l=1}^n \left|{\bm{c}}_l \right> a_l \Big) ,\end{aligned}$$ where the projector $Q = 1 - \sum_{l=1}^n | {\bm{c}}_l \rangle \langle {\bm{c}}_l|$ implements the draining action at the outgoing open ends, and $a_l$ is the amplitude of the flux per time step fed into the incoming end at ${\bm{c}}_l$. We then consider stationary states of this open-network dynamics. Without loss we take the quasi-energy to be zero, as the statistical properties of the network model are independent of it. We refer to the solutions of the stationarity condition $\vert \psi(t+1) \rangle \equiv \vert \psi(t) \rangle$ as *scattering states*. For a system with $n$ point contacts, a basis of scattering states is furnished by $$\begin{aligned} \label{eq:scatt_state} \vert \psi_k \rangle \equiv {U}(1 - Q {U})^{-1} \vert {\bm{c}}_k \rangle \quad (k = 1, \ldots n) .\end{aligned}$$ Note that $|\!| Q {U}|\!| < 1$, which ensures that the inverse exists as a convergent power series $(1 - Q {U})^{-1} = \sum_{t = 0}^\infty (Q {U})^t$. #### Main results and numerics. The first result to be announced is a statement about two-point functions, allowing one to measure the scaling dimensions of primary fields. Consider a set of links $R = \{ {\bm{r}}_1,\dots, {\bm{r}}_n\}$ for the purpose of (non-invasive) observation, and define for $i, j, m = 1,\dots,n$: $$\begin{aligned} \label{eq:An} A_m &= \operatorname{Det}K^{(m)} , \quad K_{ij}= \sum_{k=1}^n \psi_k({\bm{r}}_i)\, \overline{\psi_k({\bm{r}}_j)} ,\end{aligned}$$ where $K^{(m)}$ denotes the upper-left $m\times m$ sub-matrix of $K$. These observables are the open-network counterparts of those considered in [@Gruzberg2013]. Suppose now that coarse graining of the lattice takes the contact and observation regions ($C$ and $R$) to single points, i.e. ${\bm{r}}_i \to {\bm{r}}$ and ${\bm{c}}_i \to {\bm{c}}$ for all $i$, while ${\bm{r}}$ and ${\bm{c}}$ remain distinct. Denoting disorder averages by ${\mathbb{E}}\{ \ldots \}$ and CFT correlators as $\langle \ldots \rangle$, we then claim that [@footnote1] $$\label{eq:2pt} \begin{split} &{\mathbb{E}}\left\{ \left(A_1^{q_1-q_2} A_2^{q_2-q_3} \cdots A_n^{q_n}\right)(R,C) \right\} \\ &= a^{2\Delta_{q_1 \ldots\, q_n}} \big\langle \varphi_{q_1 \ldots\, q_n}({\bm{r}}) \, \Phi({\bm{c}}) \big\rangle \,, \end{split}$$ where $q_1, \ldots, q_n$ are complex numbers, $\varphi_{q_1\dots\, q_n}$ is a CFT primary field with scaling dimension $\Delta_{q_1\dots\, q_n}$, the operator $\Phi({\bm{c}})$ represents the contacts, and $a$ is the non-universal scale parameter of the network. Even though $\Phi({\bm{c}})$ is not a pure scaling field, it here contributes a definite scaling dimension $\Delta_{q_1\dots\, q_n}$ due to the orthogonality principle for two-point functions. Thus for an infinite planar network we predict that the observable in (\[eq:2pt\]) depends on the distance between the contact and observation regions as a pure power $\vert {\bm{r}}- {\bm{c}}\vert^{-2\Delta_{q_1 \dots\, q_n}}$. For the special choice of $q_2 = \dots = q_n = 0$ this prediction reduces to ${\mathbb{E}}\left\{ |\psi_{{\bm{c}}} ({\bm{r}})|^{2q_1} \right\} \propto \vert {\bm{r}}- {\bm{c}}\vert^{-2\Delta_{q_1 0\, \dots\, 0}}$, which strongly suggests that $\Delta_{q_1 0\, \dots \, 0}$ coincides with the multi-fractality spectrum of the local density of states [@Evers2008]. The analytical arguments leading to (\[eq:2pt\]) are sketched below. Next we support our proposal by computing numerically some of the observables above. We consider cylindrical networks of length $L = 400$ (with reflecting boundary conditions) and eight different circumferences $W\in\{ 19, 22, \ldots, 40\}$. We use Eq.  to compute the scattering states for an ensemble of $10^6$ disorder realizations. To illustrate the result (\[eq:2pt\]), we focus on the example of ${\mathbb{E}}\{ A_n^q (R,C) \}$ for $n$ contact and $n$ observation links. Assuming (\[eq:2pt\]) and using the CFT prediction for the correlator of (spinless) primary fields on an infinite cylinder of width $W$ (see, e.g., [@DiFrancesco1997]), we have: $$\begin{aligned} \label{eq:An-scaling} &{\mathbb{E}}\{ A_n^q(R,C) \} = \alpha_{q,n}\, \zeta_{{\bm{r}}{\bm{c}}}^{-2\Delta_{q,n}} , \\ \label{eq:zeta} &\zeta_{{\bm{r}}{\bm{r}}'} = \left| \frac{W}{\pi} \sinh \frac{\pi}{W} (\tau - \tau^\prime + \mathrm{i}\sigma - \mathrm{i}\sigma^\prime) \right| ,\end{aligned}$$ where $\Delta_{q,n} \equiv \Delta_{q \dots q}$, and the form factor $\alpha_{q,n}$ is due to the contact operator $\Phi({\bm{c}})$. The variables $\sigma$ and $\tau$ are the angular and longitudinal cylindrical coordinates of ${\bm{r}}$. Detailed numerical investigations were performed of the correlator ${\mathbb{E}}\{ A_n^q (R,C) \}$ for $n = 1, 2, 3$. For $n > 1$ we place the contacts on equivalent links of $n$ plaquettes next to each other; we checked for $n = 2$ that our results do not change significantly when this choice is modified. Data for the numerically most demanding case of $n = 3$ are shown in Fig. \[fig:A3\] for $q = 0.5$ as an example. We see an excellent agreement over the whole range of distances $|{\bm{r}}- {\bm{c}}|$ between the numerical data (circles) and the functional behavior (solid line) predicted by . In order to extract exponents and make an estimate of the statistical errors, we use the following procedure. Given $q$ and $n$, we fit the data for ${\mathbb{E}}\{ A_n^q (R,C) \}$ by the prediction for different $W$. This fit yields eight “raw” exponents $\Delta_{q,n}(W)$ from which we calculate the mean value and the standard deviation. For $n = 3$ and $q = 0.5$, the inset of Fig.\[fig:A3\] shows the data collapse by the optimal value of $\Delta_{0.5,3}$ thus obtained. We interpret the excellent quality of these fits as strong evidence that the $A_n^q$ indeed give rise to CFT primary fields in the continuum limit. ![Fits of ${\mathbb{E}}\{A_3^{0.5}\}$ by the CFT prediction for $W \in \{ 19, 22, \ldots, 40\}$ (bottom to top) as a function of $\tau = |{\bm{r}}- {\bm{c}}|$. Inset: Collapse of rescaled curves $\ln {\mathbb{E}}\{A_3^{0.5}\}+2\Delta_{0.5,3} \ln W$ vs. $\ln x/W$ using $\Delta_{0.5,3} = 2.15$.[]{data-label="fig:A3"}](A3.pdf){width="8cm"} Having established the existence of these fields, we now turn to a systematic analysis of the scaling dimensions $\Delta_{q,n}$. These are constrained by $\Delta_{q,n} = \Delta_{n-q,n}$ due to an argument [@Gruzberg2013] using Weyl group invariance. The simplest ansatz compatible with that constraint would be $\Delta_{q,n} \propto C_2(q,n)$, with $C_2 = n q (n - q)$ the quadratic Casimir eigenvalue of the symmetry group at hand, but it is known from [@Evers2008b; @Obuse2008] that this so-called “parabolic approximation” needs improvement by including in $\Delta_{q,1}$ the square of $C_2$ with a small coefficient. Focusing on $n = 1$, we find that $\Delta_{q,1}$ is in fact described reasonably well by the parameter set of [@Evers2008b; @Obuse2008]. The situation changes, however, when we take into account our results for $n = 2, 3$, as shown in Fig. \[fig:alln\]. ![Scaling exponents $\Delta_{q,n}$ for ${\mathbb{E}}\{ A_n^q \}$. The solid curves are plots of $\Delta_{q,n}=0.28 \, C_2 - 0.0011\, C_2^2 + 0.0014\, C_4$. Inset: zoom in for $n=1$.[]{data-label="fig:alln"}](alln.pdf){width="8cm"} To get a good fit of all data $n = 1, 2, 3$ simultaneously, we find it necessary to include in $\Delta_{q,n}$ the Casimir eigenvalue of degree four [@Scheunert]. Evaluated on $A_n^q$ this is $$C_4(q,n) = - n (q (n - q))^2 + n (n^2 - 1/2) q (n-q) .$$ We leave it for future work to decide whether this is a real effect or might have another explanation, e.g. by the presence of irrelevant operators perturbing the CC model away from the CFT fixed point [@Obuse2013]. To strengthen our claim that the operators in (\[eq:2pt\]) behave as CFT primary fields, we present a second result, this time for a three-point function. Here we sacrifice generality for simplicity and open the network at just a single contact link ${\bm{c}}_0$ to study the scalar-type observable $A_1({\bm{r}}) = \vert \psi_0 ({\bm{r}}) \vert^2$ at two observation links ${\bm{r}}_1$ and ${\bm{r}}_2$. Our assertion is that, after coarse graining, $$\label{eq:3pt-scalar} {\mathbb{E}}\left\{ |\psi_0({\bm{r}}_1)|^{2q_1} |\psi_0({\bm{r}}_2)|^{2q_2} \right\} \propto \left\langle \varphi_{q_1}({\bm{r}}_1) \varphi_{q_2}({\bm{r}}_2) \Phi({\bm{c}}_0) \right\rangle$$ depends on ${\bm{r}}_1, {\bm{r}}_2, {\bm{c}}_0$ as a CFT three-point function. Because of the special nature of the operators $\varphi_{q_1}$, $\varphi_{q_2}$ as “highest-weight vectors” (see below), the contact operator $\Phi({\bm{c}}_0)$ still contributes a definite scaling dimension $\Delta_{q_0}$, where $q_0=q_1 + q_2$. The CFT prediction for three-point functions [@DiFrancesco1997] then gives $$\label{eq:3pt-scaling} \begin{split} &{\mathbb{E}}\left\{|\psi_0({\bm{r}}_1)|^{2 q_1} |\psi_0({\bm{r}}_2)|^{2 q_2} \right\} \propto \\ & \times\zeta_{{\bm{r}}_1{\bm{r}}_2}^{-\Delta_{q_1}-\Delta_{q_2}+\Delta_{q_0}} \zeta_{{\bm{r}}_2{\bm{c}}_0}^{-\Delta_{q_2}-\Delta_{q_0}+\Delta_{q_1}} \zeta_{{\bm{c}}_0 {\bm{r}}_1}^{-\Delta_{q_0}-\Delta_{q_1}+\Delta_{q_2}}\, , \end{split}$$ where $\zeta_{{\bm{r}}{\bm{r}}'}$ was defined in Eq. . In our numerical test we take ${\bm{r}}_1$ and ${\bm{r}}_2$ to have the same $\tau$-coordinates and ${\bm{r}}_1$ to share the $\sigma$-coordinate of the point contact ${\bm{c}}_0$, while ${\bm{r}}_2$ moves along the circumference of the cylinder. Fig. \[fig:3pt\] shows the angular dependence observed for $W = 50$ together with the prediction at $q_1 = q_2 = 1/4$. The exponents $\Delta_{1/4,1}$ and $\Delta_{1/2,1}$ used are those extracted from the study of the two-point function. ![Comparison between the CFT prediction and the angular dependence of the 3-point function computed for $W = 50$ and $\tau_1 = \tau_2 = 10, 20, \ldots, 60$ (top to bottom).[]{data-label="fig:3pt"}](3pt.pdf){width="8cm"} #### Analytical argument. Finally, we sketch the reasoning that leads to Eqs. (\[eq:2pt\]), (\[eq:3pt-scalar\]); details and generalizations will be discussed in [@Longpaper]. We use a variant of the Efetov-Wegner method to pass from the CC model to a supersymmetric vertex model (see e.g.[@Gruzberg1997; @Zirnbauer1997]) as follows. For each link ${\bm{r}}$ of the network we introduce $n$ replicas of charged $(\pm)$ canonical bosons and fermions: $b_{\pm,k} ({\bm{r}})$, $f_{\pm,k} ({\bm{r}})$ ($k = 1, \ldots, n$), acting on a Fock space with vacuum $\vert 0_{{\bm{r}}} \rangle$. The time-evolution operator $U$ of the closed network is then replaced by its second quantization $\rho(U)$ acting on the tensor product of all these Fock spaces. This factors as $\rho(U \equiv \mathrm{e}^X) = \rho_+( \mathrm{e}^X) \rho_-(\mathrm{e}^X)$ where, assuming the summation convention, $$\begin{split} \rho_+ (\mathrm{e}^X) &= \mathrm{e}^{b^\dagger_{+,k} ({\bm{r}}) X_{{\bm{r}}{\bm{r}}'} b_{+,k} ({\bm{r}}') + f^\dagger_{+,k} ({\bm{r}}) X_{{\bm{r}}{\bm{r}}'} f_{+,k} ({\bm{r}}')}, \\ \rho_- (\mathrm{e}^X) &= \mathrm{e}^{- b_{-,k} ({\bm{r}}) X_{{\bm{r}}{\bm{r}}'} b_{-,k}^\dagger({\bm{r}}') + f_{-,k}({\bm{r}}) X_{{\bm{r}}{\bm{r}}'} f_{-,k}^\dagger ({\bm{r}}')} . \end{split}$$ It is important that second quantization preserves operator products. In particular, $\rho(U_{\rm s} U_{\rm r}) = \rho(U_{\rm s}) \rho(U_{\rm r})$. Statistical averages in this Fock representation are defined by $\langle A \rangle_\mathcal{F} := \operatorname{STr}\rho(U) A$, where $\operatorname{STr}$ is the supertrace over the total Fock space. For simplicity, we now specialize to $n = 1$ and return to $n \geq 1$ below. Given $Q_\varepsilon = Q + \varepsilon \vert {\bm{c}}\rangle \langle {\bm{c}}\vert$, let $$\pi_0({\bm{c}}) = \vert 0_{{\bm{c}}} \rangle \langle 0_{{\bm{c}}} \vert = \lim_{\varepsilon \to 0+} \rho_+(Q_\varepsilon) \rho_-(Q_\varepsilon^{-1})$$ be the projector on the vacuum state at ${\bm{c}}$. The second-quantized formalism is connected to observables of the first-quantized network model by the basic identities $$\label{eq:BasicID} \begin{split} \langle b_+^\dagger ({\bm{r}}) b_+({\bm{r}}) \, \pi_0({\bm{c}}) \rangle_\mathcal{F} &= \langle {\bm{r}}\vert Q U (1 - Q U)^{-1} \vert {\bm{r}}\rangle , \\ \langle b_-({\bm{r}}) b_-^\dagger({\bm{r}}) \, \pi_0({\bm{c}}) \rangle_\mathcal{F} &= \langle {\bm{r}}\vert (1 - U^{-1} Q)^{-1} \vert {\bm{r}}\rangle . \end{split}$$ To exploit these, we introduce the following key objects for an observation link ${\bm{r}}$: $$\begin{aligned} &{\mathcal{Z}}_q({\bm{r}},{\bm{c}}) = \langle (B^\dagger B)^q ({\bm{r}}) \, \pi_0({\bm{c}}) \rangle_\mathcal{F} , \label{eq:13} \\ &B^\dagger = b_+^\dagger - \mathrm{e}^{\mathrm{i} \alpha} b_- \,, \quad B = b_+ - \mathrm{e}^{-\mathrm{i} \alpha} b_-^\dagger \,,\end{aligned}$$ where $\mathrm{e}^{\mathrm{i} \alpha}$ is any (fixed) unitary number. Note the vanishing commutator $[ B , B^\dagger ] = 0$. Note also that neither $B$ nor $B^\dagger$ annihilates any state in Fock space. Therefore the operator $B^\dagger B$ is strictly positive and $(B^\dagger B)^q$ makes good sense for any $q \in \mathbb{C}$. Moreover, Wick’s theorem holds in our noninteracting-particle situation before disorder averaging. Thus for $q$ a positive integer we have $$\label{eq:15} {\mathcal{Z}}_q({\bm{r}},{\bm{c}}) = q! \, {\mathcal{Z}}_1({\bm{r}},{\bm{c}})^q .$$ This extends to complex $q$ by analytic continuation. The basic correlator ${\mathcal{Z}}_1({\bm{r}},{\bm{c}})$ is expressed in terms of the scattering state $\vert \psi_{{\bm{c}}} \rangle$ by the following computation; it is based on the identities (\[eq:BasicID\]) and in the last step uses that scattering states with incoming-wave and outgoing-wave boundary conditions are unitarily related to each other: $$\begin{aligned} &{\mathcal{Z}}_1({\bm{r}},{\bm{c}}) = \langle (b_+^\dagger b_+^{\vphantom{\dagger}} + b_-^{\vphantom{\dagger}} b_-^\dagger)({\bm{r}}) \, \pi_0({\bm{c}}) \rangle_\mathcal{F} \cr &= \langle {\bm{r}}\vert (1 - U^{-1} Q)^{-1} U^{-1} (1 - Q ) U (1 - Q U)^{-1} \vert {\bm{r}}\rangle \cr &= \left\vert \langle {\bm{r}}\vert (1 - U^{-1} Q)^{-1} U^{-1} \vert {\bm{c}}\rangle \right\vert^2 = \vert \psi_{{\bm{c}}}({\bm{r}}) \vert^2 \label{eq:16} .\end{aligned}$$ Next we take the disorder average. This is straightforward in the Fock representation since $\rho(U)=\rho(U_\mathrm{s}) \rho (U_\mathrm{r})$ and averaging over the random phases in $\rho (U_\mathrm{r})$ simply kills all states with non-zero charge. For any charge-conserving operator $A$ we thus obtain $${\mathbb{E}}\left\{ \langle A \rangle_\mathcal{F} \right\} = {\mathbb{E}}\left\{ \operatorname{STr}A \rho(U_\mathrm{s}) \rho(U_\mathrm{r}) \right\} = \operatorname{STr}^\prime A \rho(U_\mathrm{s}) ,$$ where $\operatorname{STr}^\prime$ is $\operatorname{STr}$ restricted to the zero-charge sector. In this way we arrive at what is called a *vertex model*. We denote vertex-model averages by $\langle A \rangle_\mathcal{V} \equiv \operatorname{STr}^\prime A \rho(U_\mathrm{s})$. The operators $B^\dagger B$ and $\pi_0$ conserve charge, so by taking the disorder average of and using (\[eq:15\],\[eq:16\]) we get $${\mathbb{E}}\left\{ \vert \psi_{{\bm{c}}}({\bm{r}}) \vert^{2q} \right\} = q!^{-1} \langle (B^\dagger B)^q ({\bm{r}}) \, \pi_0({\bm{c}}) \rangle_\mathcal{V} .$$ This is an exact result. Although our focus has been on $n = 1$, the general case $n \geq 1$ can be handled in a similar way. The outcome is an exact relation expressing network-model averages as vertex-model averages: $$\begin{aligned} \label{eq:19} \begin{split} &{\mathbb{E}}\left\{ ( A_1^{q_1-q_2} A_2^{q_2-q_3} \cdots A_n^{q_n}) (R,C) \right\} \\ &= f(q) \langle (D_1^{q_1-q_2} D_2^{q_2-q_3} \cdots D_n^{q_n}) (R) \, \pi_0 (C) \rangle_\mathcal{V} , \end{split}\\ &D_m(R) = \operatorname{Det}\left( \sum\nolimits_{k=1}^m B_k^\dagger({\bm{r}}_i) B_k({\bm{r}}_j) \right)_{i,j=1,\ldots,m} .\end{aligned}$$ Here $f(q)$ is a combinatorial factor. The determinants $D_m(R)$ are well-defined because the matrix elements $\sum B_k^\dagger({\bm{r}}_i) B_k({\bm{r}}_j)$ all commute. The analytic continuation to complex powers of $D_m$ is well-defined because $D_m > 0$. We will now argue that the expression in becomes a pure scaling function in the continuum limit. The key ingredient here is symmetry: the statistical average $\langle \ldots \rangle_\mathcal{V}$ is invariant under the global action of a group with Lie superalgebra $\mathfrak{g} \equiv \mathfrak{gl} (2n|2n)$ generated by all charge-conserving bilinears in $b_{\pm,k}$, $f_{\pm,k}$, and their adjoints. What is most remarkable about the operators $D_m$ is their property of being *highest-weight vectors* for $\mathfrak{g}$. By this we mean that there exists a maximal abelian subalgebra $\mathfrak{h} \subset \mathfrak{g}$ such that the $D_m$ are (i) eigenoperators w.r.t. the commutator action by all generators from $\mathfrak{h}$ and (ii) are annihilated by all the raising operators, i.e. the operators from $\mathfrak{g}$ which are positive root vectors for $\mathfrak{h}$. Since the operation of taking the commutator satisfies the Leibniz rule, these properties carry over to products of powers of $D_m$. Thus $\varphi_{q_1 \ldots q_n}^\mathrm{lat}(R) \equiv D_1^{q_1 - q_2}(R) D_2^{q_2 - q_3}(R) \cdots D_n^{q_n}(R)$ is a highest-weight vector for $\mathfrak{g}$. The operators $\varphi_{q_1 \ldots q_n}^\mathrm{lat}(R)$ for different $q = (q_1, \ldots, q_n)$ (modulo Weyl transformations) lie in inequivalent representations of $\mathfrak{g}$. Therefore, by a Schur lemma argument they cannot be mixed by the transfer matrix of the $\mathfrak{g}$-invariant vertex model. Thus we expect them to become pure scaling fields of the renormalization flow in the continuum limit. (This has to taken with a grain of salt since our CFT has a logarithmic sector [@Saleur2007].) Finally, our methods generalize to any multi-point function of the scaling operators given here. In particular, one can derive Eq.  by the reasoning above. To arrive at with $\Delta_{q_0} = \Delta_{q_1 + q_2}$ one uses that the product of two highest-weight vectors with weights $q_1$ and $q_2$ is another highest-weight vector with weight $q_0 = q_1 + q_2$. #### Summary and outlook. We have presented new methods and relations by which to make a systematic study of the IQH plateau transition. For the first time in the context of Anderson transitions, we have identified and studied operators whose correlation functions decay as pure powers at criticality and thus are candidates for CFT primary fields. Although we applied our techniques to the specific case of the CC model, they are rather general and can be used for other situations as well. Future applications will include the spin quantum Hall transition [@Gruzberg1999] and the study of boundary criticality [@Bondesan2012]. #### Acknowledgment. We thank R. Klesse and A.W.W. Ludwig for useful discussions and the former also for help with the numerical simulations. Financial support by DFG grant ZI-513/1-2 is gratefully acknowledged. [40]{} F. Evers, A.D. Mirlin, Rev. Mod. Phys. [**80**]{}, 1355 (2008). M.J. Bhaseen, I.I. Kogan, O.A. Soloviev, N. Taniguchi, A.M. Tsvelik, Nucl. Phys. B [**580**]{}, 688 (2000). Y. Ikhlef, P. Fendley, J. Cardy, Phys. Rev. B [**84**]{}, 144201 (2011). E. Bettelheim, I.A. Gruzberg, A.W.W. Ludwig, Phys. Rev. B [**86**]{}, 165324 (2012). M. Janssen, M. Metzler, M.R. Zirnbauer, Phys. Rev. B [**59**]{}, 15836 (1999). R. Klesse, M.R. Zirnbauer, Phys. Rev. Lett. [ **86**]{}, 2094 (2001). H. Obuse, S. Bera, A.W.W. Ludwig, I.A. Gruzberg, F. Evers, Europhys. Lett. [**104**]{}, 27014 (2013). I.A. Gruzberg, A.W.W. Ludwig, A.D. Mirlin, M.R. Zirnbauer, Phys. Rev. Lett. [**107**]{}, 086403 (2011). I.A. Gruzberg, A.D. Mirlin, M.R. Zirnbauer, Phys. Rev. B [**87**]{}, 125144 (2013). M.R. Zirnbauer, J. Math. Phys. [**38**]{}, 2007 (1997). I.A. Gruzberg, N. Read, S. Sachdev, Phys. Rev. B [**55**]{}, 10593 (1997). J.T. Chalker, P.D. Coddington, J. Phys. C: Solid State Physics [**21**]{}, 2665 (1988). Eqs. (\[eq:2pt\],\[eq:3pt-scalar\]) are meant to hold on large enough scales, where the contribution of irrelevant operators [@Obuse2013] is negligible. P. Di Francesco, P. Mathieu, D. Senechal, *Conformal Field Theory* (Springer, Berlin, 1997). F. Evers, A. Mildenberger, A.D. Mirlin, Phys. Rev. Lett. [**101**]{}, 116803 (2008). H. Obuse, A.R. Subramaniam, A. Furusaki, I.A. Gruzberg, A.W.W. Ludwig, Phys. Rev. Lett. [**101**]{}, 116802 (2008). M. Scheunert, J. Math. Phys. [**24**]{}, 2681 (1983). 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--- abstract: | We present a finite difference method to compute the principal eigenvalue and the corresponding eigenfunction for a large class of second order elliptic operators including notably linear operators in nondivergence form and fully nonlinear operators.\ The principal eigenvalue is computed by solving a finite-dimensional nonlinear min-max optimization problem. We prove the convergence of the method and we discuss its implementation. Some examples where the exact solution is explicitly known show the effectiveness of the method. author: - 'Isabeau Birindelli[^1]' - 'Fabio Camilli[^2]' - 'Italo Capuzzo Dolcetta[^3]' date: 'version: ' title: On the approximation of the principal eigenvalue for a class of nonlinear elliptic operators --- [**MSC 2000**]{}: : 35J60, 35P30, 65M06. [**Keywords**]{}: : Principal eigenvalue, nonlinear elliptic operators, finite difference schemes, convergence. Introduction {#intro} ============ Consider the elliptic self-adjoint operator $$\label{Lself} Lu(x)={{\partial}}_i\left(a_{ij}(x){{\partial}}_{j}u(x)\right),$$ where $a_{ij}=a_{ji}$ are smooth functions in $ \Omega$, a smooth bounded open subset of ${{\mathbb R}}^n$, satisfying $a_{ij}\xi_i \xi_j\ge \alpha|\xi|^2$ for some $\alpha>0$. It is well-known that the minimum value $\lambda_1$ in the Rayleigh-Ritz variational formula $$\lambda_1= \inf_{{\varphi}\in H^1_0(\Omega), {\varphi}\not\equiv 0} \frac{- \int_{\Omega} {\varphi}(x)\, L{\varphi}(x) \,\,dx\, }{ \|{\varphi}\|^2_{L^2(\Omega)}}=\inf_{{\varphi}\in H^1_0(\Omega), {\varphi}\not\equiv 0} \frac{\int_{\Omega} a_{ij}(x){{\partial}}_{j} {\varphi}(x){{\partial}}_i{\varphi}(x) \,\,dx\, }{ \|{\varphi}\|^2_{L^2(\Omega)}}$$ is attained at some function $w_1$ satisfying $$\left\{ \begin{array}{ll} Lw_1(x)+\lambda_1 w_1(x)=0 \qquad& x\in {\Omega}, \\ w_1(x)=0 & x\in \partial{\Omega}. \end{array} \right.$$ The number $\lambda_1$ is usually referred to as the principal eigenvalue of $L$ in $\Omega$ and $w_1$ is the corresponding principal eigenfunction. For operators of the form and also more general linear operator in divergence form there is a vast literature on computational methods for the principal eigenvalue, see for example [@BO], [@B], [@H], [@W]. General non-divergence type elliptic operators, namely $$\label{Lnonself} Lu(x)=a_{ij}(x){{\partial}}_{ij}u(x)+b_i(x){{\partial}}_i u(x)+c(x)u$$ are not self-adjoint and the spectral theory is then much more involved: in particular, the Rayleigh-Ritz variational formula is not available anymore. In the seminal paper [@DV2] by M.D. Donsker and S.R.S. Varadhan, a min-max formula for the principal eigenvalue of a class of elliptic operators $L$ including (\[Lnonself\]) was proved, namely $$\label{PE2intro} {\lambda}_{1}=-\inf_{{\varphi}\in C^2({\Omega}), {\varphi}>0}\;\sup_{x\in{\Omega}}\frac{L{\varphi}(x)}{{\varphi}(x)}.$$ In that papers other representation formulas for ${\lambda}_{1}$ were also proposed in terms of large deviations and of the average long run time behavior of the positive semigroup generated by $L$. A further crucial step in that direction is the paper [@BNV] by H. Berestycki, L. Nirenberg and S.R.S. Varadhan, where the validity of formula (\[PE2intro\]) is proved under mild smoothness assumptions ($\Omega$ a bounded open set and $a_{ij}\in C^0(\Omega)$, $b_i$, $c\in L^\infty(\Omega)$). Moreover it is proved that is equivalent to $${\lambda}_1:=\sup\{{\lambda}\in{{\mathbb R}}:\, \exists\, {\varphi}>0 \;\text{ such that}\;L{\varphi}+{\lambda}{\varphi}\le 0\quad\text{in ${\Omega}$}\}.\,$$ Following this path of ideas, notions of principal eigenvalue for fully nonlinear uniformly elliptic operators of the form $$F[u]= F(x, u(x), Du(x), D^2 u(x))$$ have been introduced and analyzed in [@A], [@BCDPR], [@BD], [@BEQ], [@IY], [@L]. A by now established definition of principal eigenvalue is given by $$\label{PEC} {\lambda}_1:=\sup\{{\lambda}\in{{\mathbb R}}:\, \exists\, {\varphi}>0 \;\text{ such that}\;F[{\varphi}]+{\lambda}{\varphi}\le 0\quad\text{in ${\Omega}$}\}\,$$ where the inequality in is intended in viscosity sense. It is possible to prove under appropriate assumptions, see -, that there exists a viscosity solution $w_1$ of $$\label{PE} \left\{ \begin{array}{ll} F[w_1]+ {\lambda}_1 w_1(x)=0 \qquad& x\in {\Omega}, \\ w_1(x)=0 & x\in \partial{\Omega}. \end{array} \right.$$ Moreover the characterization still holds in this nonlinear setting. As it is well-known, the principal eigenvalue plays a key role in several respects, both in the existence theory and in the qualitative analysis of elliptic partial differential equations as well in applications to large deviations [@A], [@DV2], bifurcation issues [@L], ergodic and long run average cost problems in stochastic control [@BEN]. For linear non self-adjoint operators and, a fortiori, for nonlinear ones the principal eigenvalue can be explicitly computed only in very special cases, see e.g. [@BL; @Pu], hence the importance to devise numerical algorithms for the problem. But, apart some specific case (see [@BEM] for the $p$-Laplace operator), approximation schemes and computational methods are not available in the literature, at least at our present knowledge. The aim of this paper is to define a numerical scheme for the principal eigenvalue of nonlinear uniformly elliptic operators via a finite difference approximation of formula . More precisely, denoting by ${{\mathbb Z}}^n_h=h{{\mathbb Z}}^n$ the orthogonal lattice in ${{\mathbb R}}^n$ where $h>0$ is a discretization parameter, we consider a discrete operator $F_h$ acting on functions defined on a discrete subset ${\Omega}_h\subset{{\mathbb Z}}^n_h$ of $\Omega$ and the corresponding approximated version of , namely $$\label{PE3intro} {\lambda}_{1,h}=-\inf_{{\varphi}>0}\sup_{x\in{\Omega}_h}\frac{F_h[{\varphi}](x)}{{\varphi}(x)}.$$ As for the approximating operators $F_h$, we consider a specific class of finite difference schemes introduced in [@KT0], [@KT1] since they satisfy some useful properties for the convergence analysis. We prove that if $F$ is uniformly elliptic and satisfies in addition some quite natural further conditions, then it is possible to define a finite difference scheme $F_h$ such that the discrete principal eigenvalues ${\lambda}_{1,h}$ and the associated discrete eigenfunctions $w_{1,h }$ converge uniformly in ${\Omega}$, as the mesh step $h$ is sent to $0$, respectively to the principal eigenvalue ${\lambda}_1$ and to the corresponding eigenfunction $w_1$ for the original problem (\[PE\]). It is worth pointing out that the proof of our main convergence result, Theorem \[main\], cannot rely on standard stability results for fully nonlinear partial differential equations, see [@BS], since the limit problem does not satisfy a comparison principle (see Remark \[convergence\] for details). We mention that our approach is partially inspired by the paper [@GO] where a similar approximation scheme is proposed for the computation of effective Hamiltonians occurring in the homogenization of Hamilton-Jacobi equations which can be characterized by a formula somewhat similar to . In Section \[sect2\] we introduce the main assumptions and we investigate some issues related to the Maximum Principle for discrete operators. In Section \[sect3\] we study the approximation method for a class of finite difference schemes and we prove the convergence of the scheme. In Section \[sect4\] we show that under some additional structural assumptions on $F_h$ the inf-sup problem can be transformed into a convex optimization problem on the nodes of the grid and we discuss its implementation. A few tests which show the efficiency of our method on some simple examples are reported in Section \[sect4\] as well. The Maximum Principle for discrete operators {#sect2} ============================================ We start by fixing some notations and the assumptions on the operator $F$. Set $\Gamma={\Omega}\times{{\mathbb R}}\times{{\mathbb R}}^n\times S^n$ , where $S^n$ denotes the linear space of real, symmetric $n\times n$ matrices. The function $F(x,z,p,r)$ is assumed to be continuous on $\Gamma$ and locally uniformly Lipschitz continuous with respect to $z,p,r$ for each fixed $x\in {\Omega}$. We will also suppose that the partial derivatives $F_r$, $F_p$, $F_z$ satisfy the following structure conditions: $$\label{Hyp_gen} 0<a I \le F_r\le A I, \quad |F_p|\le \mu_1, \quad -\mu_0\le F_z\le 0.$$ for some constants $a$, $A$, $\mu_0$, $\mu_1$. A further condition is the positive homogeneity of degree $1$, that is $$\label{Hyp_om} F(x,tz,tp,tr)=tF(x,z,p,r)\qquad \forall t\geq 0.$$ The principal eigenvalue of problem is defined by $${\lambda}_1=\sup\{{\lambda}:\, \exists\, {\varphi}>0 \; { \rm such \,that} \;F[{\varphi}]+{\lambda}{\varphi}\le 0\ \mbox{in}\ {\Omega}\},$$ where the differential inequality $F[{\varphi}]+{\lambda}{\varphi}\le 0$ is meant in the viscosity sense. Under assumptions -, there exists a viscosity solution of and the characterization of ${\lambda}_1$ holds (see [@BD], [@BEQ]). \[negativeigenv\] It is possible to define $${\lambda}_1^-=\sup\{{\lambda}:\, \exists\, {\varphi}<0 \; { \rm such \,that} \;F[{\varphi}]+{\lambda}{\varphi}\ge 0\ \mbox{in}\ {\Omega}\}.$$ When $F$ is not odd in its dependence on the Hessian, then in general ${\lambda}_1\neq{\lambda}_1^-$. Of course it is possible to see ${\lambda}_1^-$ as ${\lambda}_1$ of some other operator. Hence we will only consider in this paper ${\lambda}_1$. For example, for the extremal Pucci operators ${\mathcal M}^+_{a,A}(D^2u):=\sup_{aI\leq B\leq AI}tr(AD^2u)$ and ${\mathcal M}^-_{a,A}(D^2u):=\inf_{aI\leq B\leq AI}tr(AD^2u)$, since ${\mathcal M}^+_{a,A}(-M)=-{\mathcal M}^-_{a,A}(M)$, the following holds $${\lambda}_1^-({\mathcal M}^+_{a,A})={\lambda}_1({\mathcal M}^-_{a,A}).$$ \[cpositive\] The assumption $ F_z\le 0$, i.e. the monotonicity of the differential operator in the zero-order term, could be removed. Indeed $\bar F:=F-c_0 z$, with $c_0$ large, satisfies this assumption, moreover $\bar F$ and $F$ have the same principal eigenfunction and the eigenvalues differ by $c_0$. We now describe the discrete setting that we shall consider. Given $h>0$, let ${{\mathbb Z}}^n_h=h{{\mathbb Z}}^n$ denote the orthogonal lattice in ${{\mathbb R}}^n$. Let $F_h$ be a discrete operator acting on functions defined in ${\Omega}_h\subset{{\mathbb Z}}^n_h$. We shall consider an approximation of (which can be seen also as an eigenvalue problem for the discrete operator $F_h$). We look for a number ${\lambda}$ and a positive function $w$ such that $$\label{PEd} \left\{ \begin{array}{ll} F_h(x,w (x),[w]_x)+{\lambda}w(x) =0 \qquad &x\in {\Omega}_h,\\ w(x)=0 & x\in \partial{\Omega}_h, \end{array} \right.$$ where - $h>0$ is the discretization parameter ($h$ is meant to tend to $0$), - $x\in {\Omega}_h$ is the point where is approximated, - $w$ is a real valued mesh function in ${{\mathbb Z}}^n_h$ meant to approximate the viscosity solution of , - $[\cdot]_x$ represents the stencil of the scheme, i.e. the points in ${\Omega}_h \backslash \{x\} $ where the value of $u$ is computed for writing the scheme at the point $x$ (we assume that $[w]_x$ is independent of $w(y)$ for $|x-y|>Mh$ for some fixed $M\in{{\mathbb N}}$). We denote by ${{\cal C}}_h$ the space of the mesh functions defined on $\overline {\Omega}_h$ and we introduce some basic assumptions for the scheme $F_h$ (see [@KT0], [@KT1]). - The operator $F_h$ is of positive type, i.e. for all $x\in{\Omega}_h$, $z,\tau\in{{\mathbb R}}$, $u,\eta\in {{\cal C}}_h$ satisfying $0\le \eta(y)\le \tau$ for each $y\in{\Omega}_h$, then $$F_h(x,z,[u+\eta]_x)\ge F_h(x,z,[u]_x)\ge F_h(x,z+\tau, [u+\eta]_x)$$ - The operator $F_h$ is positively homogeneous, i.e. for all $x\in{\Omega}_h$, $z\in{{\mathbb R}}$, $u\in {{\cal C}}_h$ and $t\geq 0$, then $$F_h(x,tz,[tu]_x)= t F_h(x,z, [u]_x).$$ - The family of operators $\{F_h, 0<h\le h_0\}$, where $h_0$ is a positive constant, is consistent with the operator $F$ on the domain ${\Omega}\subset{{\mathbb R}}^n$, i.e. for each $u\in C^2({\Omega})$ $$\sup_{{\Omega}_h} \left|F(x,u(x), Du(x),D^2u(x))-F_h(x,u(x),[u]_x)\right|\to 0\quad\text{as $h\to 0$,}$$ uniformly on compact subset of ${\Omega}$. We study below some properties related to the maximum principle and a comparison result for the operator $F_h$. Let us start by the following definitions: A function $u\in {{\cal C}}_h$ is a subsolution (respectively $v\in {{\cal C}}_h$ is a supersolution) of $$\label{PEd2} F_h(x,u (x),[u]_x) =f(x) \qquad x\in{\Omega}_h$$ if $$\begin{aligned} &F_h(x,u (x),[u]_x) \ge f(x),\qquad x\in{\Omega}_h\\ & \left(\text{respectively, } F_h(x,v (x),[v]_x) \le f(x),\qquad x\in{\Omega}_h\right). \end{aligned}$$ \[def\_MP\] The Maximum Principle holds for the operator $F_h$ in ${\Omega}_h$ if $$\label{MP} \left\{\begin{array}{ll} F_h(x,u(x),[u]_x)\ge 0\qquad &\text{in ${\Omega}_h$,}\\ u\le 0&\text{on $\partial {\Omega}_h$,} \end{array} \right.$$ implies $u\le 0$ in ${\Omega}_h$. \[prop\_MP\] Assume that $F_h$ is of positive type and positive homogeneous and satisfies either $$\label{C5_1} \begin{split} \text{for all $z\in{{\mathbb R}}$, $u,\eta\in {{\cal C}}_h$ satisfying $0\le \eta(y)$ and $max_{y\in [\cdot]_x}\eta(y)>0$,}\\ \text{then }\quad F_h(x,z,[u+\eta]_x)>F_h(x,z,[u]_x)\qquad \end{split}$$ or $$\label{C5_2} \begin{split} \text{for all $z,\tau\in{{\mathbb R}}$, $u,\eta\in {{\cal C}}_h$ satisfying $0\le \eta(y)\le \tau$ for each $y$, then}\\ F_h(x,z,[u]_x)\ge F_h(x,z+\tau, [u+\eta]_x)+c_0\tau\qquad \end{split}$$ for some positive constants $c_0$. Then the Maximum Principle holds for the operator $F_h$ in ${\Omega}_h$ . Assume by contradiction that $u$ satisfies and $M:=\max_{\overline {\Omega}_h} u>0$. Let $\bar x\in{\Omega}_h$ be such that $u(\bar x)=M$. Since $u\le0$ on $\partial{\Omega}_h$, it is not restrictive to assume that there exists $y\in {\Omega}_h$ such that $u(y)<u(\bar x)=M$. Hence $$\begin{aligned} 0&\le F_h(\bar x,u(\bar x), [u]_{\bar x})\le F_h(\bar x,u(\bar x)-M, [u-M]_{\bar x}) \\ & < F_h(\bar x,0,[0]_{\bar x})= 0,\end{aligned}$$ a contradiction. A similar proof can be done with the assumption . The assumptions and correspond to the uniform ellipticity and, respectively, to the strict monotonicity of the operator $F$ with respect to the zero-order term. The following proposition shows that, as it is known in the continuous case (see for example [@BNV; @BD]), the validity of the Maximum Principle for subsolutions of the operator $F_h$ is equivalent to the positivity of the principal eigenvalue for $F_h$. \[prop\_MP2\] Assume that the scheme $F_h$ is of positive type and that it is positively homogeneous. Suppose that for ${\lambda}\in{{\mathbb R}}$, there exists a nonnegative grid function ${\varphi}$ with ${\varphi}>0$ in ${\Omega}_h$ such that $F_h[{\varphi}]+{\lambda}{\varphi}\le 0$. If, for $\tau<{\lambda}$, the function $u$ satisfies $$\left\{\begin{array}{ll} F_h(x,u(x),[u]_x)+\tau u\ge 0\qquad &\text{in ${\Omega}_h$}\\ u\le 0&\text{on $\partial {\Omega}_h$,} \end{array} \right.$$ then $u\le 0$ in $\Omega_h$, i.e. $F_{h,\tau}[\cdot]=F_h[\cdot]+\tau \cdot$ satisfies the Maximum Principle. Suppose by contradiction that $\max_{\overline {\Omega}_h}\{u\}>0$. Let ${\varphi}$ as in the statement and set $L({{\gamma}})=\max_{ {\Omega}_h}\{u-{{\gamma}}{\varphi}\}$ (note that the maximum is taken only with respect to the internal points). Then $L: [0,\infty)\to {{\mathbb R}}$ is continuous, decreasing, $L(0)>0$ and $L({{\gamma}})\to -\infty$ for ${{\gamma}}\to+\infty$. Hence there exists ${{\gamma}}'>0$ such that $L({{\gamma}}')=0$. Moreover, since $u-{{\gamma}}'{\varphi}\le 0$ on $\partial{\Omega}$, we also have $\max_{\overline {\Omega}_h}\{u-{{\gamma}}'{\varphi}\}=0$. Let $0<{{\gamma}}< {{\gamma}}'$ be such that $$\label{mp2} \frac{{{\gamma}}}{{{\gamma}}'}{\lambda}>\tau$$ and set $\psi={{\gamma}}{\varphi}$. Then $F_h[\psi]+{\lambda}\psi\le0$ and $M=\max_{\overline {\Omega}_h}\{u-\psi\}=(u-\psi)(\bar x)>0$ for some $\bar x\in {\Omega}_h$. Hence $\psi(\bar x)+M=u(\bar x)$ and $\psi(x)+M\ge u(x)$. Since $F_h$ is of positive type, it follows that $$\begin{aligned} F_h(x,\psi(\bar x),[\psi]_{\bar x})&\ge F_h((x,\psi(\bar x)+M,[\psi+M]_{\bar x})=F_h(x,u(\bar x),[\psi+M]_{\bar x})\\ &\ge F_h(x,u(\bar x),[u]_{\bar x}).\end{aligned}$$ Then $$\begin{aligned} \tau u(\bar x)\ge -F_h[u](\bar x)\ge -F_h[\psi](\bar x)\ge {\lambda}\psi(\bar x) ={\lambda}{{\gamma}}{\varphi}(\bar x)\ge{\lambda}\frac{{{\gamma}}}{{{\gamma}}'}u(\bar x)\end{aligned}$$ and therefore a contradiction to . The following result gives a comparison principle for . \[prop\_comp\] Assume that $F_h$ is of positive type and it satisfies either or . Let $u$ and $v$ be a subsolution and respectively a supersolution of such that $u\le v$ on ${{\partial}}{\Omega}_h$. Then $u\le v$ in $\overline{\Omega}_h$. Suppose by contradiction that $M:=\max_{\overline {\Omega}_h} \{u-v\}>0$ and let $\bar x\in{\Omega}_h$ be such that $u(\bar x)-v(\bar x)=M$. Hence $v+M\ge u$ in ${\Omega}_h$ and it is not restrictive to assume that $\max_{y_\in [\cdot]_{\bar x}}(v+M-u)>0$. It follows that $$\begin{aligned} f(\bar x)\le F_h(\bar x, u(\bar x), [u]_{\bar x})&=F_h(\bar x, v(\bar x)+M, [u]_{\bar x}) < F_h(\bar x, v(\bar x)+M, [v+M]_{\bar x})\\ &\le F_h(\bar x, v(\bar x), [v]_{\bar x})\le f(\bar x)\end{aligned}$$ and therefore a contradiction. A similar proof can be carried on under assumption . Approximation of the principal eigenvalue {#sect3} ========================================= In this section we consider a specific class of finite difference schemes introduced in [@KT1]. These schemes satisfy certain pointwise estimates which are the discrete analogues of those valid for a general class of fully nonlinear, uniformly elliptic equations.\ We assume that for all $x\in{{\mathbb Z}}^n_h$, the stencil $[\cdot]_x$ of the scheme is given by $x+h Y$ where $Y=\{y_1,\dots,y_k\}\subset {{\mathbb Z}}^n$ is a finite set containing all the vectors of the canonical basis of ${{\mathbb R}}^n$. Then we consider a discrete operator $F_h$ in given by a finite difference scheme written in the form $$\label{finitediff} F_h[u]={{{\cal F}}} (x, u, {{\delta}}_h u,{{\delta}}^2_h u),$$ where ${{\cal F}}:{{\mathbb R}}^n\times{{\mathbb R}}\times{{\mathbb R}}^Y\times{{\mathbb R}}^Y\to {{\mathbb R}}$ and for $y\in Y$, $u\in {{\cal C}}_h$ $$\begin{aligned} \delta_{h,y}^\pm u(x)&=\pm\frac{u(x\pm h y)-u(x)}{h|y|},\\ \delta_{h,y} u(x)&=\frac{1}{2}\{ \delta_{h,y}^+ u(x)+ \delta_{h,y}^- u(x)\}=\frac{u(x+hy)-u(x-hy)}{2h|y|},\\ \delta^2_{h,y} u(x)&=\delta_{h,y}^+\delta_{h,y}^- u(x)=\frac{u(x+hy)+u(x-hy)-2u(x) }{h^2|y|^2},\\ {{\delta}}_h u&=\{{{\delta}}_{h,y} u:\, y\in Y\}, \quad {{\delta}}^2_{h} u=\{{{\delta}}^2_{h,y} u:\, y\in Y\}.\end{aligned}$$ Set $\tilde\Gamma:={{\mathbb R}}^n\times{{\mathbb R}}\times{{\mathbb R}}^Y\times{{\mathbb R}}^Y$ and denote by $(x,z,q,s)$ the generic points in $\tilde \Gamma$. The operator $F_h$ given by is of positive type if $$\begin{aligned} & \frac{\partial{{\cal F}}}{\partial s_y}-\frac{|hy|}{2}\left|\frac{\partial{{\cal F}}}{\partial q_y}\right|\ge 0\quad \forall y\in Y,\label{P1_scheme}\\[4pt] &\frac{\partial{{\cal F}}}{\partial z}\le 0, \label{P2_scheme}\end{aligned}$$ and positively homogeneous if $${{\cal F}}(x,tz,tq,ts)=t {{\cal F}}(x,z,q,s) \qquad \forall t\geq 0.$$ Moreover if $F$ in satisfies the assumptions , then it is always possible to find a scheme of type which is consistent with $F$ and which, besides -, satisfies for all $y\in Y$, the bounds $$\begin{aligned} \frac{\partial{{\cal F}}}{\partial s_y}-\frac{|hy|}{2}\left|\frac{\partial{{\cal F}}}{\partial q_y}\right|\ge {{\alpha}}_0,\qquad \frac{\partial{{\cal F}}}{\partial s_y}\le a_0, \qquad \left|\frac{\partial{{\cal F}}}{\partial q_y}\right|\le b_0 \label{P3_scheme}\end{aligned}$$ where ${{\alpha}}_0$, $a_0$, $b_0$ are constants depending on $a$, $A$, $\mu_0$, $\mu_1$ in (see [@KT0], [@KT1]). Note that in particular implies .\ We recall some important properties of the previous scheme (for the proof we refer to [@KT1]) \[prop\_wellposed\] Assume - and let $f$, $g$ be two given mesh functions. Then for every $h>0$ sufficiently small there exists a unique solution $u_h:{\Omega}_h\to{{\mathbb R}}$ to the Dirichlet problem $$\label{Dird} \left\{ \begin{array}{ll} F_h(x,u(x),[u]_x)=f\quad& x\in {\Omega}_h, \\ u=g & x\in {{\partial}}{\Omega}_h. \end{array} \right.$$ \[Prop\_ABP\] Assume - and let $u_h$ be a subsolution of . Then $$\label{ABP} \max_{{\Omega}_h} u_h\le \max_{{{\partial}}{\Omega}_h} g+ \frac{C}{{{\alpha}}_0}\left\{\sum_{x\in{\Omega}_h}h^n|f(x)|^{n}\right\}^{\frac{1}{n}},$$ where the constant $C$ is independent of $h$. Moreover if $u_h$ is a solution of , then for any $x,y\in{\Omega}_h$ $$\label{Holder} |u_h(x)-u_h(y)|\le C \frac{|x-y|^{{\delta}}}{R}\left(\max_{B_R^h} u_h+\frac{R}{{{\alpha}}_0}\left\{\sum_{x\in{\Omega}_h}h^n|f(x)|^{n}\right\}^{\frac{1}{n}}\right),$$ where $R=\min\{\mathrm{dist}(x,{{\partial}}{\Omega}_h),\mathrm{dist}(x,{{\partial}}{\Omega}_h)\}$, $B_R^h=B(0,R)\cap{\Omega}_h$, ${{\delta}}$ and $C$ are positive constants independent of $h$. We give an example of a scheme of the form . Consider the Hamilton-Jacobi-Bellman operator $$F(x,u,D u(x), D^2u(x))=\sup_{{{\alpha}}\in A}\inf_{{{\beta}}\in B} L^{{{\alpha}}{{\beta}}}u(x)$$ where $$\label{op_L} L^{{{\alpha}}{{\beta}}}u(x)= a_{ij}^{{{\alpha}}{{\beta}}}(x)D_{ij}u+b_i^{{{\alpha}}{{\beta}}}(x)D_iu(x)+c^{{{\alpha}}{{\beta}}}(x)u(x).$$ It is always possible to rewrite the operator $L^{{{\alpha}}{{\beta}}}$ in in the following form (see [@KT1]) $$\overline L^{{{\alpha}}{{\beta}}} u(x)= \bar a_{k}^{{{\alpha}}{{\beta}}}(x)D^2_{y_k}u+\bar b_k^{{{\alpha}}{{\beta}}}(x)D_{y_k}u(x)+\bar c^{{{\alpha}}{{\beta}}}(x)u(x)$$ where $D_{y_k}u =\langle Du,y_k\rangle$ and $Y=\{y_1,\dots,y_k\}\subset {{\mathbb Z}}^n$ is a finite set containing all the vectors of the canonical basis in ${{\mathbb R}}^n$. Moreover the coefficients $\bar a_{k}^{{{\alpha}}{{\beta}}}$, $\bar b_k^{{{\alpha}}{{\beta}}}$ and $\bar c^{{{\alpha}}{{\beta}}}$ satisfy the same properties of $ a_{ij}^{{{\alpha}}{{\beta}}}$, $ b_{ij}^{{{\alpha}}{{\beta}}}$ and $c^{{{\alpha}}{{\beta}}}$. Then we consider $$\label{HJBd} F_h[u](x):=\sup_{{{\alpha}}\in A}\inf_{{{\beta}}\in B} L_h^{{{\alpha}}{{\beta}}}u(x)$$ where $$\label{op_Ld} L_h^{{{\alpha}}{{\beta}}}u(x)= \bar a_{k}^{{{\alpha}}{{\beta}}}(x) \delta^2_{h,y_k} u(x)+\bar b_k^{{{\alpha}}{{\beta}}}(x) \delta_{h,y_k}u(x)+\bar c^{{{\alpha}}{{\beta}}}(x)u(x).$$ For $x\in{{\mathbb R}}$ with $Y=\{1\}$ the previous scheme reads as $$\begin{split} \sup_{{{\alpha}}\in A}\inf_{{{\beta}}\in B}\Big\{& a^{{{\alpha}}{{\beta}}}(x)\frac{u(x+h)+u(x-h)-2u(x)}{h^2}+ b^{{{\alpha}}{{\beta}}}(x)\frac{u(x+h)-u(x-h)}{2h}+\\ &+ c^{{{\alpha}}{{\beta}}}(x)u(x)\Big\}=0. \end{split}$$ The linear case --------------- In this part we assume that the operator $F$ in is linear, i.e. $F[u]=Lu$ with $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu(x)+c(x)u(x)$$ and we consider a scheme defined as in –, obviously without the dependence on ${{\alpha}}$, ${{\beta}}$. \[KR\] Under the assumption the eigenvalue problem has a simple eigenvalue ${\lambda}_{1,h}\in{{\mathbb R}}$ which corresponds to a positive eigenfunction. The other eigenvalues correspond to sign changing eigenfunctions. Choose $\xi>0$ large enough so that $c(x)-\xi<0$ and set $$L_{h,\xi}(x,t,[ u]_x)=L_h (x,t,[u]_x)-\xi t.$$ Let $K$ be the positive cone of the nonnegative grid functions in ${{\cal C}}_h$. For a given grid function $f$, by Proposition \[prop\_wellposed\] and Proposition \[prop\_comp\] there exists a unique solution $u\in {{\cal C}}_h$ to $$\left\{ \begin{array}{ll} L_{h,\xi}(y,u(y),[u]_y)+f=0\qquad &\text{in ${\Omega}_h$,}\\ u= 0&\text{on $\partial {\Omega}_h$.} \end{array} \right.$$ Since ${{\cal C}}_h$ is a finite dimensional space it follows that $T:{{\cal C}}_h\to {{\cal C}}_h$ defined by $Tf=u$ is a compact linear operator. Moreover, if $f\ge 0$, then by Proposition \[prop\_MP\] $u\ge 0$ and if $f\in K\setminus \{0\}$, $u=Tf>0$.\ Therefore, by the Krein-Rutman theorem [@KR], $r(T)$ the spectral radius of $T$ is a simple real eigenvalue $r(T)>0$ with a positive eigenfunction $u$ such that $Tu=r(T)u$. Hence for ${\lambda}_{1,h}=r(T)^{-1}-\xi$, $w_1=Tu$ satisfies $$\left\{\begin{array}{ll} L_h (x,w_1(x),[w_1]_x) +{\lambda}_{1,h}w_1=0\qquad &\text{in ${\Omega}_h$,}\\ w_1= 0&\text{on $\partial {\Omega}_h$}. \end{array} \right.$$ The following characterization of ${\lambda}_{1,h}$ is a simple consequence of Proposition \[prop\_MP2\]. We have $$\label{PEd_char1} {\lambda}_{1,h}=\sup\left\{{\lambda}:\,\text {$\exists$ ${\varphi}> 0$ s.t. $L_h[{\varphi}]+{\lambda}{\varphi}\le 0\ \mbox{in}\ {\Omega}$}\right\},$$ or, equivalently, $$\label{PEd_char2} {\lambda}_{1,h}=-\inf_{{\varphi}>0}\sup_{x\in{\Omega}_h}\left\{\frac{L_h[{\varphi}](x)}{{\varphi}(x)}\right\}.$$ Denote by $\bar {\lambda}$ the right hand side of . Clearly ${\lambda}_{1,h}\le \bar{\lambda}$. If ${\lambda}_{1,h}< \bar{\lambda}$ then there exist $\mu\in ({\lambda}_{1,h}, \bar{\lambda})$ and ${\varphi}>0$ such that $L_h[{\varphi}]+\mu {\varphi}\le 0$. A contradiction follows immediately by Proposition \[prop\_MP2\] since the eigenfunction corresponding to ${\lambda}_{1,h}$ is positive. Hence we have .\ Let ${\varphi}>0$ such that $L_h[{\varphi}](x)+{\lambda}{\varphi}(x)\le 0$ for $x\in{\Omega}_h$. Hence $${\lambda}\le \inf_{{\Omega}_h}\left\{-\frac{L_h[{\varphi}]}{{\varphi}}\right\}= -\sup_{{\Omega}_h}\left\{\frac{L_h[{\varphi}]}{{\varphi}}\right\}.$$ Consequently $${\lambda}_{1,h}=\sup_{{\varphi}>0}\left(-\sup_{x\in{\Omega}_h}\left\{\frac{L_h[{\varphi}](x)}{{\varphi}(x)}\right\}\right)=-\inf_{{\varphi}>0}\sup_{x\in{\Omega}_h}\left\{\frac{L_h[{\varphi}](x)}{{\varphi}(x)}\right\}.$$ We give next an upper bound for ${\lambda}_{1,h}$ (compare with the corresponding estimate for ${\lambda}_1$ in [@BNV], Lemma 1.1). Let $n=1$ and assume that $B_R=\{|x|<R\}$ lies in ${\Omega}$ with $R\le 1$. Then $${\lambda}_{1,h}({\Omega}_h)\le \frac{C}{R^2}$$ Given the linear operator $$Lu=a(x)u''+b(x)u'(x)+c(x)u(x),$$ let ${{\gamma}}_0$, ${{\Gamma}}_0$, $b$ be positive constants such that ${{\gamma}}_0\le a (x)\le {{\Gamma}}_0$ and $|b(x)|,\, |c(x)|\le b$ in ${\Omega}$. Let $r=R/2$ and assume for simplicity that $r=Nh$ for some $N\in {{\mathbb N}}$. Set $B_r=\{|x|<r\}$ and consider the grid function $${{\sigma}}_i=(r^2-|ih|^2)^2 \qquad i=-N+1,\dots\,N-1$$ Then for $i=-N+1,\dots, N-1$ we have $$\begin{aligned} &\frac{{{\sigma}}_{i+1}-{{\sigma}}_{i-1}}{2h}=-4hi(r^2-|ih|^2)+4h^3i\\ &\frac{{{\sigma}}_{i+1}+{{\sigma}}_{i-1}-2{{\sigma}}_i}{h^2}=- 4(r^2-|ih|^2)+2h^2(2i^2+1).\end{aligned}$$ Denote by $a_i$, $b_i$ and $c_i$ the coefficients of the linear operator computed at the point $x=ih$. Since $h^2/(r^2-|ih|^2)\le 1$ it follows that $$\label{st_auto1} \begin{split} -\frac{ L_h[{{\sigma}}](ih)}{4{{\sigma}}_i}&\le \frac{ a_i}{(r^2-|ih|^2)}-\frac{a_i |hi|^2}{(r^2-|ih|^2)^2} +|b_i|\frac{2r}{(r^2-|ih|^2)}+\frac{c_i}{4}\\ & \le \frac{ {{\Gamma}}_0+br}{(r^2-|ih|^2)}- \frac{{{\gamma}}_0|hi|^2}{(r^2-|ih|^2)^2}+\frac{b}{4}. \end{split}$$ If $$|ih|^2({{\gamma}}_0+{{\Gamma}}_0+br)>r^2({{\Gamma}}_0+br),$$ then the second term in dominates the first one and therefore $$\label{st_auto2} -\frac{ L_h[{{\sigma}}](ih)}{4{{\sigma}}_i}\le \frac{b}{4}.$$ In the remaining part of $B_r$, $$\label{st_auto3} -\frac{ L_h[{{\sigma}}](ih)}{4{{\sigma}}_i}\le \frac{ {{\Gamma}}_0+br}{(r^2-|ih|^2)}+\frac{b}{4}\le \frac{b}{4}+\frac{1}{{{\gamma}}_0r^2}({{\Gamma}}_0+br)({{\gamma}}_0+{{\Gamma}}_0+br).$$ By and , we get $$\sup_{B_r}\left(-\frac{L_h[{{\sigma}}](ih)}{{{\sigma}}_i}\right)\le\frac{C}{R^2}\quad\text{for $i=-N+1,\dots, N-1$.}$$ To conclude the proof, we show that if for some positive function ${\varphi}$ and ${\lambda}\in{{\mathbb R}}$, $L_h[{\varphi}]+{\lambda}{\varphi}\le 0$, then ${\lambda}\le \sup_{B_r}(-L_h[{{\sigma}}]/{{\sigma}})$. For this purpose, assume that $ {\lambda}> \sup_{B_r}(-L_h[{{\sigma}}]/{{\sigma}}):=\tau$; then $ L_h[{{\sigma}}]+\tau {{\sigma}}\ge 0$ in $B_r$ and ${{\sigma}}=0$ on $\partial B_r$, while $L_h[{\varphi}]+\tau {\varphi}\le 0$. Hence by Proposition \[prop\_MP2\], it follows ${{\sigma}}\le 0$ in $B_r$, a contradiction, and therefore ${\lambda}\le \tau$. The nonlinear case ------------------ We consider now a general discrete operator $F_h$ given by and we study the corresponding eigenvalue problem . In analogy with formula , we define $$\label{PEd1} {\lambda}_{1,h}=\sup\left\{{\lambda}:\,\text {$\exists$ ${\varphi}> 0$ \;{\rm such that} $F_h[{\varphi}]+{\lambda}{\varphi}\le 0$}\right\}$$ We prove for each $h$ the existence of a pair $({\lambda}_{1,h}, w_{1,h})$ satisfying with $w_{1,h}>0$ in ${\Omega}_h$. \[PEd:prop\_exlower\] Assume that $F_h$ satisfies , $f\le 0$ and ${\lambda}< {\lambda}_{1,h}$. Then there exists a nonnegative solution to $$\label{PEd3} \left\{ \begin{array}{ll} F_h(x,u (x),[u]_x)+ {\lambda}u(x) =f(x) \qquad &x\in {\Omega}_h,\\ u(x)=0 & x\in \partial{\Omega}_h. \end{array} \right.$$ We can assume ${\lambda}\ge 0$, since for ${\lambda}<0$, $F_h[u]+{\lambda}u$ satisfies and therefore by Propositions \[prop\_wellposed\] and there exists a unique solution to problem .\ Let us define by induction a sequence $u_n$ by setting $u_1\equiv 0$ and, for $n\ge1$ we consider the equation: $$\label{PEd4} \left\{ \begin{array}{ll} F_h(x,u_{n+1} (x),[u_{n+1}]_x) =f(x)-{\lambda}u_{n},\qquad &x\in {\Omega}_h,\\ u_{n+1}(x)=0 & x\in \partial{\Omega}_h. \end{array} \right.$$\ For any $n\in N$ there exists a non negative solution $u_{n+1}$ to . For $n=1$, existence follows by Proposition \[prop\_wellposed\]. Moreover since $u_1\equiv 0$ is a subsolution to , by Proposition \[prop\_comp\] we get $u_2\ge 0$. The existence of a non negative solution at the $(n+1)$-step is proved in a similar way; moreover the solution is non negative since $f-{\lambda}u_{n}\le 0$.\ We claim now that, for any $n\geq 1$, $u_n\leq u_{n+1}$. For $n=1$ the claim is trivially true since $u_2\ge 0$. Assume then by induction that $u_n\ge u_{n-1} $. Since $f(x)-{\lambda}u_{n}\le f(x)-{\lambda}u_{n-1}$ it follows that $u_n$ is a subsolution of . By Proposition \[prop\_comp\], we get that $u_n\le u_{n+1}$.\ Let us show now that the sequence $u_n$ is bounded. Assume by contradiction that it is false and set $\overline u_n=u_n/|u_n|_\infty$. Then, by positive homogeneity, $\overline u_n$ is a solution of $$F_h(x,\overline u_{n+1} (x),[\overline u_{n+1}]_x) =\frac{f(x)}{|u_{n+1}|_\infty}-{\lambda}\frac{u_{n}}{|u_{n+1}|_\infty},\qquad x\in {\Omega}_h.$$ Since the sequence $\overline u_n$ is bounded, then up to a subsequence it converges to a function $\overline u $, while $u_{n}/|u_{n+1}|_\infty$ converges to $k \overline u$ where $k=\lim_{n\to \infty} |u_{n}|_\infty/|u_{n+1}|_\infty\le 1$. Hence $\overline u \ge 0$, $|\overline u |_\infty=1$, $\overline u =0$ on ${{\partial}}{\Omega}_h$ and $$F_h(x,\overline u (x),[\overline u ]_x) +k{\lambda}\overline u =0,\qquad x\in {\Omega}_h.$$ Since $0\le k{\lambda}\le {\lambda}$ and using the fact that for ${\lambda}<{\lambda}_{1,h}$ there exists by definition ${\varphi}>0$ such that $F_h[{\varphi}]+{\lambda}{\varphi}\ge 0$ in ${\Omega}_h$, we get a contradiction to Proposition \[prop\_MP2\]. Hence the sequence $u_n$ is bounded, and being in addition monotone, it converges pointwise to a function $u$ which solves . The next result shows that ${\lambda}_{1,h}$ is indeed an eigenvalue for the approximated operator $F_h$. \[PEd:prop\_ex\] Assume that $F_h$ satisfies . Then there exists $w_{1,h}>0$ in ${\Omega}_h$ satisfying $$\label{PEd8} \left\{ \begin{array}{ll} F_h(x,w_{1,h}(x),[w_{1,h}]_x)+ {\lambda}_{1,h}w_{1,h}(x) =0 \qquad &x\in {\Omega}_h,\\ w_{1,h}=0 & x\in \partial{\Omega}_h. \end{array} \right.$$ Moreover that the characterization is still valid for the nonlinear operator $F_h$. Let ${\lambda}_n$ be an increasing sequence converging to ${\lambda}_{1,h}$. By Proposition \[PEd:prop\_exlower\] there exists a positive solution $u_n$ of $$\left\{ \begin{array}{ll} F_h(x,u_{n} (x),[u_{n}]_x)+{\lambda}_n u_{n} =-1,\qquad &x\in {\Omega}_h,\\ u_{n}(x)=0 & x\in \partial{\Omega}_h. \end{array} \right.$$ We claim that $u_n$ is not bounded. Assume by contradiction that $u_n$ is bounded so that, up to a subsequence, $u_n$ converges to a function $u>0$ which solves $$\left\{ \begin{array}{ll} F_h(x,u (x),[u]_x)+{\lambda}_{1,h} u =-1,\qquad &x\in {\Omega}_h,\\ u(x)=0 & x\in \partial{\Omega}_h. \end{array} \right.$$ Then, for ${\varepsilon}>0$ small enough, $u$ satisfies $$F_h(x,u (x),[u]_x)+({\lambda}_{1,h}+{\varepsilon}) u =-1+{\varepsilon}u\le 0$$ which gives a contradiction to the definition . Hence $|u_n|_\infty\to \infty$.\ Define now $w_n=u_n/|u_n|_\infty$ that solves $$\left\{ \begin{array}{ll} F_h(x,w_{n} (x),[w_{n}]_x)+{\lambda}_n w_{n} =-\frac{1}{|u_n|_\infty},\qquad &x\in {\Omega}_h,\\ w_{n}(x)=0 & x\in \partial{\Omega}_h. \end{array} \right.$$ Then, up to a subsequence, $w_n$ converges to a bounded function $w_{1,h}$ which has norm 1 and which satisfies , so that $w_{1,h}>0$.\ It is immediate that is still valid for $F_h$. \[convergence\] There is a huge literature about the approximation of viscosity solutions of first and second order PDEs. In this framework a well established technique to prove the convergence of a numerical scheme is the Barles-Souganidis’method [@BS]: besides some natural properties of the scheme (stability, consistency, monotonicity), a key ingredient for this technique is a *strong comparison result* for the continuous problem which allows to show that a subsolution is always lower than or equal to a supersolution. The comparison principle implies in particular that there is at most one viscosity solution of the problem. But it is immediate that cannot satisfy a comparison principle since $w\equiv 0$ and the principal eigenfunction $w_1$ are two distinct solutions of the problem, hence the convergence proof cannot rely on the Barles-Souganidis’method and it needs a different argument. We now discuss the convergence of the discrete principal eigenvalue ${\lambda}_{1,h}$ to the continuous one defined by . We recall the definition of weak limits in viscosity sense (see [@BS]) $$\begin{aligned} { {\limsup\limits_{h\to 0}}^*}u_h(x):=\lim_{h\to 0^+}\sup\{u_{{\delta}}(y):\,|x-y|\le h,\, {{\delta}}\le h\},\\ { {\liminf\limits_{h\to 0}}\phantom{\hskip -1pt}_*}u_h(x):=\lim_{h\to 0^+}\inf\{u_{{\delta}}(y):\,|x-y|\le h,\, {{\delta}}\le h\}.\end{aligned}$$ \[main\] Assume -, - and that $F_h$ is consistent with $F$. Let $({\lambda}_{1,h}, w_{1,h})$ be the sequence of the discrete eigenvalues and of the corresponding eigenfunctions, solutions of . Then ${\lambda}_{1,h}\to {\lambda}_{1}$ and $w_{1,h}\to w_{1}$ uniformly in $\overline {\Omega}$ as $h\to 0$, where ${\lambda}_1$ and $w_1$ are respectively the principal eigenvalue and a corresponding eigenfunction associated to $F$. \ By the positive homogeneity of the scheme, it is not restrictive to assume that $\max_{{\Omega}_h}\{w_{1,h}\}=1$, hence the sequence $w_{1,h}$ is bounded. We first prove that $$\label{conv1} \liminf_{h\to 0}{\lambda}_{1,h}\ge {\lambda}_1.$$ Assume by contradiction that $\liminf_{h\to 0}{\lambda}_{1,h}=\tau$ for some some $\tau<{\lambda}_1$. Consider a subsequence, still denoted by ${\lambda}_{1,h}$, such that $\lim_{h\to 0} {\lambda}_{1,h}=\tau$. Set $\overline w= { {\limsup\limits_{h\to 0}}^*}w_{1,h}$. By standard stability results in viscosity solution theory, see [@BS], $\overline w$ satisfies in viscosity sense $$\label{conv1a} F[\overline w]+\tau \overline w\ge 0\qquad \text{in ${\Omega}_h$,}$$ and $$\label{conv1b} \max_{{\Omega}} \overline w = 1.$$ Let $\eta>0 $ be such that for $h$ sufficiently small, ${\lambda}_{1,h}\le \tau+\eta$. Hence $$\begin{aligned} F_h[w_{1,h}]=-{\lambda}_{1,h}w_{1,h}\ge -\tau-\eta, \qquad x\in {\Omega}_h.\end{aligned}$$ Set $f=-\tau-\eta$, $g\equiv 0$ and let $u_h$ be the corresponding solution of , while $u$ is the solution of $$\left\{ \begin{array}{ll} F (x,u(x),Du, D^2u)=-\tau-\eta\quad& x\in {\Omega}, \\ u=0 & x\in {{\partial}}{\Omega}. \end{array} \right.$$ Then by Proposition \[prop\_comp\] and the consistency of the scheme for $h$ sufficiently small $$\label{conv1d} 0\le w_{1,h}\le u_h\le u+o(1)\qquad \text{in $\overline{\Omega}_h$}$$ and therefore $$\label{conv1c} \overline w=0\quad \text{on ${{\partial}}{\Omega}$}.$$ By , and we get a contradiction to the maximum principle for the operator $F$ (see [@BD], [@BEQ]) and therefore .\ We now prove that $$\label{conv2} \limsup_{h\to 0}{\lambda}_{1,h}\le {\lambda}_1.$$ Assume by contradiction that there exists $\eta >0$ such that $$\bar {\lambda}:=\limsup_{h\to 0}{\lambda}_{1,h}\ge {\lambda}_1+\eta.$$ We consider a subsequence, still denoted by ${\lambda}_{1,h}$, such that $\lim_{h\to 0}{\lambda}_{1,h}=\bar {\lambda}$ and we set $\underline w= { {\liminf\limits_{h\to 0}}\phantom{\hskip -1pt}_*}w_{1,h}$. By standard stability results $\underline w$ satisfies $0\le \underline w\le 1$ and $$\label{conv3} \left\{ \begin{array}{ll} F[\underline w ]+({\lambda}_1+\eta) \underline w\le 0\qquad &\mbox{ in } {\Omega},\\ \underline w= 0&\mbox{ on }\ \partial {\Omega}, \end{array} \right.$$ in viscosity sense. Let $x_h\in{\Omega}_h$ be a sequence such that $x_h\to x_0\in\overline{\Omega}$ and $w_{1,h}(x_h)=1$ for all $h>0$. By , $x_0\in{\Omega}$. We claim that $$\label{conv6} \underline w(x_0)>0.$$ Assume by contradiction that $\underline w(x_0)=0$, hence there exists a sequence $y_h\to x_0$ such that $\lim_{h\to 0} w_{1,h}(y_h)=0$. By with $u_h=w_{1,h}$ and $f=-{\lambda}_{1,h}w_{1,h}$ we get $$\begin{aligned} |w_{1,h}(x_h)-w_{1,h}(y_h)|&\le C \frac{|x_h-y_h|^{{\delta}}}{R}\left(\max_{B_R} w_{1,h}+\frac{R}{{{\alpha}}_0}\left\{\sum_{x\in{\Omega}_h}h^n|{\lambda}_{1,h}w_{1,h}|^{n}\right\}^{\frac{1}{n}}\right)\\ &\le C \frac{|x_h-y_h|^{{\delta}}}{R}\left(1+\frac{R}{{{\alpha}}_0}|{\lambda}_{1,h}|\right).\end{aligned}$$ Since $\lim_{h\to 0}(w_{1,h}(x_h)-w_{1,h}(y_h))=1$ we get a contradiction for $h$ sufficiently small and therefore .\ We are in a position to apply the maximum principle for the continuous problem (see [@BD]), and so we obtain that $\underline w>0$. But and the positivity of $\underline w$ give a contradiction to the definition of ${\lambda}_1$. By and we get $\lim_{h\to 0}{\lambda}_{1,h}={\lambda}_1$.\ By and a local boundary estimate for $w_{1,h}$, see [@KT0 Thm. 5.1] and [@KT1 Thm.3.], we get the equi-continuity of the family $\{w_{1,h}\}$ and therefore the uniform convergence, up to a subsequence, of $w_{1,h}$ to $w_1$ with $\|w_{1}\|_\infty=1$. The simplicity of the eigenfunction associated to the principal eigenvalue ${\lambda}_1$ gives the uniform convergence of all the sequence $w_{1,h}$ to $w_1$. An algorithm for computing the principal eigenvalue {#sect4} =================================================== In this section we discuss an algorithm for the computation of the principal eigenvalue based on the inf-sup formula . In fact we show that this formula results in a finite dimensional nonlinear optimization problem. Discretization in one dimension. -------------------------------- We first present the scheme in one dimension. Note that since the eigenfunction corresponding to the principal eigenvalue vanishes on the boundary of ${\Omega}_h$ and it is strictly positive inside, then the minimization in can be restricted to the internal points. By the formula and the homogeneity of ${{\cal F}}$, we have $$\frac{F_h[u](x_i)}{u(x_i)}={{{\cal F}}}\left(x_i, 1, \frac{u(x_i+h)-u(x_i-h)}{2hu(x_i)},\frac{u(x_i+h)+u(x_i+h)}{h^2u(x_i)}-\frac{2}{h^2}\right).$$ We identify the function $u(x)$ with the values $U_i$, $i=0,\dots, N_h+1$, at the points of the grid (with $U_0=U_{N_h+1}=0$). Assume that ${{\cal F}}(x,z,q,s)$ is linear or more generally convex in $(q,s)$. Then the functions $G:{{\mathbb R}}^{N_h}\to{{\mathbb R}}^{N_h}$, defined by $$G_i(x,U_1, \dots, U_{N_h})={{{\cal F}}}\left(x_i, 1, \frac{U_{i+1}-U_{i-1}}{2hU_i},\frac{U_{i+1}+U_{i-1}}{h^2U_i}-\frac{2}{h^2}\right).$$ for $i=1,\dots,{N_h}$, is either linear or respectively convex in $U_{i+1}$, $U_{i-1}$. Moreover, since $U_i>0$, $G$ is also convex in $U_i$. Taking the maximum of the functions $G_i$ over the internal nodes of the grid gives a convex function ${{\cal G}}:{{\mathbb R}}^{N_h}\to{{\mathbb R}}$ defined by $$\label{minmax_2} {{\cal G}}(U_1,\dots, U_{N_h}) =\max_{i=1,\dots, {N_h}}G_i(x_i, U_1, \dots, U_{N_h})$$ Hence the computation of ${\lambda}_{1,h}$ is equivalent to the minimization of the convex function ${{\cal G}}$ of ${N_h}$ variables: this problem can be solved by means of standard algorithms in convex optimization. Note also that the minimum is unique and the map is sparse, in the sense that the value of ${{\cal G}}$ at $U_i$ depends only on the values at $U_{i-1}$ and $U_{i+1}$.\ In general, if ${{\cal F}}(x,z,q,s)$ is not convex, the computation of the principal eigenvalue is equivalent to the solution of a min-max problem in ${{\mathbb R}}^{N_h}$.\ To solve min-max problem we use the routine `fminmax` available in the Optimization Toolbox of MATLAB. This routine is implemented on a laptop and therefore the number of variables is modest. A better implementation of the minimization procedure which takes advantage of the sparse structure of the problem would allow to solve larger problems. **Example 1.** To validate the algorithm we begin by studying the eigenvalue problem: $$\left\{ \begin{array}{ll} w^{\prime\prime}+{\lambda}w=0\quad& x\in (0,1), \\ w(x)=0 & x=0,\,1. \end{array} \right.$$ In this case the eigenvalue and the corresponding eigenfunction are given by $${\lambda}_1= \pi^2, \qquad w_1(x)= \sin(\pi x).$$ Note that since the eigenfunctions are defined up to multiplicative constant, we normalize the value by taking $\|w_1\|_\infty=\|w_{1,h}\|_\infty=1$ (the constraint for $w_{1,h}$ is included in the routine `fminmax`). Given a discretization step $h$ and the corresponding grid points $x_i=ih$, $i=0,\dots,N_h+1$, the minimization problem is $${\lambda}_{1,h}=-\min_{U\in{{\mathbb R}}^{N_h}}\left[\max_{i=1,\dots,N_h}\frac{U_{i+1}+U_{i-1}-2U_{i}}{h^2 U_i}\right]$$ (with $U_0=U_{N_h+1}=0$). In Table 4.1, we compare the exact solution with the approximate one obtained by the scheme . We report the approximation error for ${\lambda}_1$ and $w_1$ (in $L^\infty$-norm and $L^2$-norm) and the order of convergence for ${\lambda}_1$. We can observe an order of convergence close to $2$ for ${\lambda}_1$ and therefore equivalent to one obtained by discretization of the Rayleigh quotient via finite elements (see [@B]). $h$ $Err({\lambda}_1)$ $Order({\lambda}_1)$ $Err_\infty(w_1)$ $Err_2(w_1)$ ------------------------ ------------------------ ---------------------- ------------------------- ------------------------ $1.00 \cdot 10^{-1} $ $8.0908 \cdot 10^{-2}$ $3.3662\cdot 10^{-11}$ $5.7732\cdot 10^{-11}$ $5.00 \cdot 10^{-2} $ $2.0277 \cdot 10^{-2}$ 1.9964 $1.4786\cdot 10^{-10} $ $3.8119\cdot 10^{-10}$ $2.50 \cdot 10^{-2} $ $5.0723 \cdot 10^{-3}$ 1.9991 $6.6613\cdot 10^{-16}$ $1.8731\cdot 10^{-15}$ $1.25 \cdot 10^{-2} $ $1.2683\cdot 10^{- 3}$ 1.9998 $1.5543\cdot 10^{-15}$ $6.2524\cdot 10^{-15}$ $6.25 \cdot 10^{-3} $ $3.1708\cdot 10^{-4}$ 1.9999 $1.2212 \cdot 10^{-15}$ $7.1576\cdot 10^{-15}$ : Space step (first column), eigenvalue error (second column), convergence order (third column), eigenfunction error in $L^\infty$ (fourth column), eigenfunction error in $L^2$ (last column) \[tab1\] **Example 2.** In this example we consider the eigenvalue problem for a linear equation with a discontinuous coefficient $$\label{eqdisc} \left\{ \begin{array}{ll} a(x) w^{\prime\prime}+{\lambda}w=0\quad& x\in (0,\pi), \\ w(x)=0 & x\in \{0,\,\pi\}, \end{array} \right.$$ where $$a(x)= \left\{\begin{array}{lc} 1 & \mbox{for }\ x\in [0,\frac{\pi}{2k}),\\ 2 & \mbox{for }\ x\in [\frac{\pi}{2k},\pi], \end{array} \right.$$ and $k:=\frac{2+\sqrt{2}}{2\sqrt{2}}>1$. The principal eigenvalue ${\lambda}_1$ associated to problem is given by $k^2=\frac{3+2\sqrt{2}}{4}$. Let $$w(x)=\left\{\begin{array}{lc} \sin(kx) & \mbox{for }\ x\in [0,\frac{\pi}{2k}),\\ b\sin(\frac{kx}{\sqrt 2}+c) & \mbox{for }\ x\in [\frac{\pi}{2k},\pi]. \end{array} \right.$$ We choose $b$ and $c$ such that $w(0)=w(\pi)=0$ and $w$ continuous in $\frac{\pi}{2k}$. Imposing these conditions we get $$\frac{k\pi}{\sqrt 2}+c=\pi,\quad \mbox{and}\quad b\sin(\frac{\pi}{2\sqrt 2}+c)=1,$$ i.e. $$c=\pi(1-\frac{k}{\sqrt 2})=\pi(\frac{2-\sqrt{2}}{4}).$$ Furthermore, using that $\frac{\pi}{2\sqrt{2}}+c=\frac{\pi}{2}$, we get $$\lim_{x\rightarrow \frac{\pi}{2k}^-}w^\prime(x)=k\cos(\frac{\pi}{2})=0,\ \lim_{x\rightarrow \frac{\pi}{2k}^+}w^\prime(x)= bk\cos(\frac{\pi}{2\sqrt{2}}+c)=0,$$ hence $w\in C^1([0,\pi])$.\ On the other hand, since $w$ is not $C^2$ in $\frac{\pi}{2k}$, we show that it satisfies in the sense of viscosity solutions. For any $(p,q)\in J^{2,+}w(\frac{\pi}{2k})$, we get $p=0$ and $q\geq-\frac{k^2}{2}$. This implies that for both $a=1$ and $a=2$: $$aq\geq -k^2w(\frac{\pi}{2k}),$$ so $w$ is a subsolution. For any $(p,q)\in J^{2,-}w(\frac{\pi}{2k})$, we get $p=0$ and $q\leq-k^2$. This implies that for both $a=1$ and $a=2$: $$aq\leq -k^2w(\frac{\pi}{2k}),$$ so $w$ is a supersolution. In Table 4.2, we compare the exact solution with the approximate one obtained by means of the scheme $${\lambda}_{1,h}=-\min_{U\in{{\mathbb R}}^{N_h}}\left[\max_{i=1,\dots,N_h}a(ih)\frac{U_{i+1}+U_{i-1}-2U_{i}}{h^2 U_i}\right]$$ (with $U_0=U_{N^h+1}=0$). The rates are not very good, but the problem is out of our setting since $F$ is discontinuous and the error is very sensible to the chosen grid. In Figure 4.1, we report the graph of the exact and approximate eigenfunctions for $h=0.1$. $h$ $Err({\lambda}_1)$ $Order({\lambda}_1)$ $Err_\infty(w_1)$ $Err_2(w_1)$ ------------- -------------------- ---------------------- ------------------- -------------- $ 0.1571 $ $0.1197$ $0.0213$ $0.0563$ $ 0.0785 $ $ 0.0476$ $1.3303$ $0.0090$ $0.0383$ $ 0.0393 $ $0.0347$ $0.4576$ $0.0065$ $0.0391$ $ 0.0196 $ $ 0.0157$ $1.1417$ $0.0030$ $0.0264$ $ 0.0098 $ $0.0061$ $1.3596$ $0.0012$ $0.0149$ : Space step (first column), eigenvalue error (second column), eigenfunction error in $L^\infty$ (fourth column), eigenfunction error in $L^2$ (last column) \[tab2\] \[fig1\] **Example 3.** The Fucîk spectrum of $\Delta$ is the set of pairs $(\mu, {{\alpha}}\mu) \in{{\mathbb R}}^2$ for which the equation $$-\Delta u=\mu u^+-{{\alpha}}\mu u^-$$ has a non-zero solution, where $u^+(x) = \max\{u(x), 0\}$ and $u^-(x) = \max\{-u(x), 0\}$. For fixed${{\alpha}}>0$ the previous problem is equivalent to $$\begin{aligned} \min\{\Delta u,\frac 1 {{\alpha}}\Delta u\}+\mu u=0\quad &\text{if ${{\alpha}}\ge 1$},\\ \max\{\Delta u,\frac 1 {{\alpha}}\Delta u\}+\mu u=0\quad &\text{if ${{\alpha}}\le1$}.\end{aligned}$$ For details see [@BEQ]. Hence the Fucîk spectrum can be seen as the spectrum of a nonlinear operator involving the maximum or minimum of two linear operators. To find the corresponding principal eigenvalue we apply the scheme . In Table 4.3, we report the corresponding approximation error for ${\lambda}_1$ in the case ${{\alpha}}=1/2$ and $\Omega=[0,\pi]$ (by the convexity of the solution the eigenvalue for the continuous problem coincides with the one of the second derivative operator in $[0,\pi]$ i.e. ${\lambda}_1=1$). \[tab3\] $h$ $Err({\lambda}_1)$ $Order({\lambda}_1)$ ------------------------ -------------------- ---------------------- $1.00 \cdot 10^{-1} $ $0.0809$ $5.00 \cdot 10^{-2} $ $0.0203$ 1.9964 $2.50 \cdot 10^{-2} $ $0.0051$ 1.9991 $1.25 \cdot 10^{-2} $ $0.0013$ 1.9998 $6.25 \cdot 10^{-3} $ $0.0003$ 2.0000 : Space step (first column), eigenvalue error (second column), convergence order (third column) for the Fucîk spectrum with ${{\alpha}}=1/2$ **Example 4.** Consider the eigenvalue problem for the $p$-Laplace operator $$\mathrm{div}(|Dw_1|^{p-2}Dw_1)+\lambda_1 |w_1|^{p-2}w_1=0.$$ This example does not fit exactly in the framework of this paper since the operator is not uniformly elliptic. However, the following formula $${\lambda}_{p,h}:=-\inf_{{\varphi}>0}\sup_{y\in{\Omega}_h}\left\{\frac{F_{h,p}[{\varphi}](y)}{{\varphi}(y)^{p-1}}\right\}$$ where $F_{h,p}$ is a finite-difference approximations of $F_p$ produces a good approximation of the principal eigenvalue of the $p$-Laplace operator in the interval $(a,b)$ whose exact value is given by $$\sqrt[p]{{\lambda}_p} =\frac{2\pi\sqrt[p]{p-1}}{(b-a)p\sin(\frac{\pi}{p})}.$$ In Table 4.4 we report the approximation error and the corresponding order of convergence for the principal eigenvalue of the $p$-Laplace operator for $p=4$ (in this case ${\lambda}_4\approx 73.0568 $).\ \[tab4\] $h$ $Err({\lambda}_4)$ $Order({\lambda}_4)$ ------------------------ -------------------- ---------------------- $1.00 \cdot 10^{-1} $ $2.6770$ $5.00 \cdot 10^{-2} $ $0.6210$ 2.1079 $2.50 \cdot 10^{-2} $ $0.1457$ 2.0912 $1.25 \cdot 10^{-2} $ $0.0347$ 2.0724 $6.25 \cdot 10^{-3} $ $0.0083$ 2.0581 : Space step(first column), eigenvalue error (second column), convergence order (third column) for the $p$-Laplace operator with $p=4$ It is also known (see [@JPM]) that if ${\Omega}$ is a ball, the eigenfunction $w_p$ corresponding to the eigenvalue ${\lambda}_p$ converges for $p\to \infty$ to $d(x,\partial{\Omega})$. In Figure 4.2, we draw approximations of $w_p$ computed by the scheme for various values of $p$ and we observe the convergence of these functions to $d(x,\{0,1\})$ for $p$ increasing, as expected by the theory. \[fig2\] Discretization in higher dimension. ----------------------------------- We now consider the eigenvalue problem in ${{\mathbb R}}^N$. Arguing as in the 1-dimensional case we write $$\label{minmax_3} \small{\begin{split} \frac{F_h[u](x)}{u(x)} = {{{\cal F}}}\Big(x, 1, &\left\{\frac{u(x+hy)-u(x-hy)}{2h|y|u(x)}\right\}_{y\in Y}, \left\{\frac{u(x+hy)+u(x-hy)}{h^2|y|^2u(x)}-\frac{2}{h^2}\right\}_{y\in Y}\Big) \end{split}}$$ for $i=1,\dots,N_h$ and $F_h$ defined as in , $Y$ the stencil and ${N_h}$ the cardinality of ${\Omega}_h$. Hence if the function ${{\cal F}}(x,z,\{q_y\}_{y\in Y},\{s_y\}_{y\in Y})$ is linear or more generally convex in the variables $q_y$ and $s_y$, $y\in Y$, then the computation of the principal eigenvalue ${\lambda}_{1,h}$ is equivalent to the minimization with respect to the vector $U\in{{\mathbb R}}^{N_h}$ of the convex function ${{\cal G}}:{{\mathbb R}}^{N_h}\to{{\mathbb R}}$ obtained by taking the maximum with respect to $x\in {\Omega}_h$ in . Therefore this problem can be solved by means of some standard algorithms in convex optimization. **Example 5.** Consider the problem $$\left\{ \begin{array}{ll} \Delta w+{\lambda}w=0\quad& x\in (0,1)^2, \\ w(x)=0 & x\in \partial ((0,1)^2). \end{array} \right.$$ The eigenvalue and the corresponding eigenfunction are given by $${\lambda}_1=2 \pi^2, \qquad w(x_1,x_2)= \sin(\pi x_1)\sin(\pi x_2)$$ (the eigenfunctions are normalized by taking $\|w\|_\infty=\|w_{h}\|_\infty=1$). We use a standard five-point formula for the discretization of the Laplacian. In Table 4.5, we compare the exact solution with the approximate one obtained by the scheme . We report the approximation error for ${\lambda}_1$ and $w_1$ (in $L^\infty$-norm and $L^2$-norm) and the order of convergence for ${\lambda}_1$. We can observe an order of convergence close to $2$ for ${\lambda}_1$ and therefore equivalent to one obtained by discretization of the Rayleigh quotient via finite elements, see [@B]. \[tab5\] $h$ $Err({\lambda}_1)$ $Order({\lambda}_1)$ $Err_\infty(w)$ $Err_2(w)$ ------------------------ -------------------- ---------------------- ----------------- ------------ $2.00 \cdot 10^{-1} $ $0.4469 $ $ 0.0801$ $0.2256$ $1.00 \cdot 10^{-1} $ $ 0.1338$ $1.7397$ $ 0.0203$ $0.1137$ $5.00 \cdot 10^{-2} $ $0.0368 $ $1.8629$ $ 0.0056$ $0.0590$ $2.50 \cdot 10^{-2} $ $0.0097$ $1.9297$ $ 0.0015$ $0.0301$ : Space step (first column), eigenvalue error (second column), convergence order (third column), eigenfunction error in $L^\infty$ (fourth column), eigenfunction error in $L^2$ (last column) **Example 6.** We consider the eigenvalue problem for the Ornstein-Uhlenbeck operator $$\Delta w-x\cdot Dw+{\lambda}w=0, \qquad x\in (-1,1)^2$$ with homogeneous boundary conditions. The eigenvalue and the corresponding eigenfunction are given by $${\lambda}_1=4, \qquad w(x_1,x_2)=(1-x_1^2)(1-x_2^2),$$ with the eigenfunctions normalized by taking $\|w\|_\infty=\|w_{1,h}\|_\infty=1$. The Laplacian is discretized by a five-point formula. In Table 4.6, we report the approximation error for ${\lambda}_1$ and the corresponding order of convergence. \[tab6\] $h$ $Err({\lambda}_1)$ $Order({\lambda}_1)$ ------------------------ -------------------- ---------------------- -- -- $4.00 \cdot 10^{-1} $ $0.1524$ $2.00 \cdot 10^{-1} $ $0.0392$ $1.9592$ $1.00 \cdot 10^{-1} $ $0.0103$ $1.9250$ $5.00 \cdot 10^{-2} $ $0.0027$ $1.9580$ : Space step (first column), eigenvalue error (second column), convergence order (third column) [99]{} S.N. Armstrong, The Dirichlet problem for the Bellman equation at resonance. J. Differential Equations 247 (2009), 931-955. I. Babuska, J. E. Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems. 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Weinberger,*Variational methods for eigenvalue approximation.* Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, no. 15, SIAM, Philadelphia, 1974. [^1]: Dip. di Matematica, “Sapienza" Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy [[email protected]]{}. [^2]: Dip. di Scienze di Base e Applicate per l’Ingegneria, “Sapienza" Universit[à]{} di Roma, via Scarpa 16, 00161 Roma, Italy, ([e-mail:[email protected]]{}) [^3]: Dip. di Matematica, “Sapienza" Università di Roma, P.le Aldo Moro 2, 00185 Roma, Italy [[email protected]]{}.

--- abstract: 'We address the relation between star formation and AGN activity in a sample of 231 nearby ($0.0002<z<0.0358$) early type galaxies by carrying out a multi-wavelength study using archival observations in the UV, IR and radio. Our results indicate that early type galaxies in the current epoch are rarely powerful AGNs, with $P<10^{22}\,WHz^{-1}$ for a majority of the galaxies. Only massive galaxies are capable of hosting powerful radio sources while less massive galaxies are hosts to lower radio power sources. Evidence of ongoing star formation is seen in approximately 7% of the sample. The SFR of these galaxies is less than 0.1 $M_{\odot}yr^{-1}$. They also tend to be radio faint ($P<10^{22}\,WHz^{-1}$). There is a nearly equal fraction of star forming galaxies in radio faint ($P<10^{22}\,WHz^{-1}$) and radio bright galaxies ($P\geq10^{22}\,WHz^{-1}$) suggesting that both star formation and radio mode feedback are constrained to be very low in our sample. We notice that our galaxy sample and the Brightest Cluster Galaxies (BCGs) follow similar trends in radio power versus SFR. This may be produced if both radio power and SFR are related to stellar mass.' author: - 'Sravani Vaddi, Christopher P. O’Dea, Stefi A. Baum, Samantha Whitmore, Rabeea Ahmed, Katherine Pierce , Sara Leary' title: 'Constraints on Feedback in the local Universe: The relation between star formation and AGN activity in early type galaxies' --- Introduction ============ It is now well known that supermassive black holes (SBH) are present in the centers of massive galaxies [@kormendy1995] and share interesting correlations with the host galaxy properties such as the velocity dispersion [@ferrarese2000; @gebhardt2000], bulge mass [@haring2004], bulge luminosity [@kormendy1995; @magorrian1998] and galaxy light concentration [@graham2001]. These empirical correlations suggest that the growth of the central SBH and the host galaxy are fundamentally interlinked. AGN feedback may be responsible for the correlations observed [@silkrees1998; @king2003; @fabian2012], although it has also been argued that the origin of the observed relations is entirely non-causal and is a natural consequence of merger driven galaxy growth [@peng2007; @jahnke2011; @graham2013]. The energy released from the central SBH is several orders more than the binding energy of massive galaxies [@fabian2012]. This energy has the potential to expel gas from the galaxy (radiative-mode feedback) or deposit energy into the surroundings and thus heat up the inter galactic medium (mechanical feedback). These two modes may operate at different redshifts and accretion rates and ensure to regulate the growth of the black hole and the galaxy [review of @mcnamara2007; @fabian2012; @churazov2005].\ Various theoretical models that invoke AGN feedback in galaxy evolution are also able to successfully reproduce the observed galaxy luminosity function [@silkrees1998; @king2003; @granato2004; @dimatteo2005; @springel2005; @croton2006; @hopkins2008]. This theoretical picture has been supported by numerous observations. The strongest evidence comes from the brightest cluster galaxies (BCG) of cool core clusters, whose powerful radio jets have swept out cavities in the intracluster medium (ICM)[@rosner1989; @allen2001; @mcnamara2007]. While in some individual galaxies, energy transportation into the ISM via AGN driven outflows are observed to remove gas from the central regions of the galaxy [@crenshaw2003; @nesvadba2007; @alexander2010; @morganti2013]. All these show the negative effect of AGN feedback by removing/heating up the gas and eventually suppressing the star formation and regulating the galaxy growth. However, several other theoretical studies [@begelman1989; @rees1989; @silk2005; @santini2012; @silk2013] have reported an increased star formation rate in AGN especially at high redshifts via induced pressure by jets/winds. All this evidence thus far has been obtained mostly from studies of large groups, clusters and galaxies at higher redshifts. At low redshifts, the majority of the luminous AGNs reside in early type galaxies [@mclure1999; @bahcall1997]. But how common AGN feedback is in the local universe is not yet well understood. To explore this, we focused our attention on a carefully selected sample of nearby early type galaxies and studied them at multiple wavelengths. We describe the sample selection in Section \[sample\]. In Section \[data\], we discuss the data, steps carried out to retrieve the magnitude in the UV and in the IR, flux from radio images and extinction correction. We present our results in Section \[results\]. In Section \[summary\] we discuss the implications of the results. In the Appendix, we describe the photometry technique in more detail. The sample {#sample} ========== This study is focused on a sample of early type (ellipticals and S0) galaxies that are present at low redshift. The sample was selected from the *Two Micron All Sky Survey* ([*2MASS*]{}; [@jarrett2003]) that have an apparent $K_{s}$ band (2.2 $\mu$m) magnitude of 13.5 and brighter and whose positions correlate with the Chandra archive of ACIS-I and ACIS-S observations (C. Jones, private communication). A total of 231 galaxies were identified. The Chandra selection criteria was used to create a sample which would allow a study of the nature of AGN activity in early type galaxies. X-ray emission is detected for approximately 80% of the galaxies. The X-ray luminosities of the nuclei range from $10^{38}$ to $10^{41}$ erg $s^{-1}$. The Eddington ratios are measured to be small $\sim$ $10^{-5}$ to $10^{-9}$ suggesting that these galaxies are low-luminosity AGNs [@jones2013]. In this paper, we present a parallel study on the star formation in the sample. In a second paper in this series, we will study the relation between the X-ray properties and the star formation. The data are homogeneous and since the sample is not selected based on specific properties in radio or UV, the data can be considered to be unbiased regarding their star formation and radio source properties. Our large dataset at low redshift allows us to study the interplay between star formation and AGN activity in typical galaxies in the current epoch.\ Figure  \[fig:redshift\_hist\] shows the redshift distribution of the sample. All the galaxies are nearby galaxies. The redshift range of the galaxies in the sample is $0.0002<z<0.0358$ with a median of $z=0.006$, of which, 63% are at a redshift of less than 0.01. Adopting a Hubble constant of $71 \,km\, s^{-1}Mpc^{-1}$ [@jarosik2011], $z=0.01$ corresponds to a distance of 42 Mpc and 1 correspond to a scale of 210 pc.\ ![Figure shows the histogram of redshift for the sample indicating that most of the galaxies are at low redshifts. Redshift of 0.01 corresponds to a distance of 42 Mpc.[]{data-label="fig:redshift_hist"}](fig1.pdf){width="50.00000%"} The Data {#data} ======== We examine observations in multiple wavelengths for this study namely, radio, IR and UV. Infrared observations were collected from *Wide-field Infrared Survey Explorer* [*WISE*]{}[@wright2010] and [*2MASS*]{}[@jarrett2003]. We use the [$K_s$]{}band to trace the stellar mass distribution; [*WISE*]{}and [*GALEX*]{}data to study star formation and radio data at 1.4 GHz to study AGN properties. Basic and observational properties for a subset of the sample are given in Table  \[tab:sample\]. A complete list of the sample properties in machine readable format can be obtained in the online version.\ IR data ------- The Two Micron All Sky Survey ([*2MASS*]{}) was conducted in the near-infrared $J(1.25 \mu m)$, $H(1.65 \mu m)$ and $K_{s}(2.16 \mu m)$ wavebands using two 1.3 m diameter telescopes with a resolution of $\sim$ 2-3. The detectors are sensitive to point sources brighter than 1 mJy at the 10$\sigma$ level. The astrometric accuracy is on order of 100 mas. The camera contains three NICMOS 256$\times$256 HgCdTe arrays. The *WISE* mission observed the entire sky at four infrared wavebands - $W1$ at 3.4 $\mu$m, $W2$ at 4.6 $\mu$m, $W3$ at 12 $\mu$m, and $W4$ at 22 $\mu$m with an angular resolution of 6.1, 6.4, 6.5and 12.0respectively. The field of view (FOV) is 47. The short wavelength detectors are HgCdTe arrays whereas long wavelength detectors are SiAs BIB arrays. The arrays are 1024$\times$1024 pixels in size. WISE has 5$\sigma$ point source sensitivity higher than 0.08, 0.11, 1 and 6 mJy at 3.4, 4.6,12 and 22 $\mu m$ wavelengths respectively.\ Rather than use the existing cataloged values, we chose to perform photometry on [*2MASS*]{}[$K_s$]{}band images and [*WISE*]{}band images for the following reasons: 1. Underestimation of the WISE flux: The extended source photometry for WISE is based on the [*2MASS*]{}aperture. Since WISE is most sensitive to galaxies in W1 and W2 bands, the extended emission in these bands is much larger than the [*2MASS*]{}aperture. The [*2MASS*]{}aperture is too small by 10-20% for resolved sources, thus resulting in an underestimate of the total flux by about 30-40%  [@cutri2012]. 2. Contamination of the flux from nearby sources: In galaxies that have close companions, such as stars or other galaxies, there is an over estimation of the total flux since no masking was employed to remove the flux from unwanted sources. Surface photometry of our galaxy sample was performed using the ELLIPSE task in IRAF. The task reads an input image and initial guess isophotal parameters and then returns the fitted isophote parameters and several other geometrical parameters. A program has been written in Pyraf to automate the photometry for the entire sample. The program runs the ELLIPSE task in two stages. In the first stage, the RA and DEC positions of [*2MASS*]{}as the initial central values were used and allowed for a non-linear increase of the semi-major axis step size. In images where the target galaxy has close companions (e.g. saturated stars or or other galaxies), masks were created for those regions and the ELLIPSE task was allowed to flag the pixels in the mask. The region masks were created by inspecting each galaxy by eye and is not done by the program itself. These regions are then given as input to the program that calls the MSKREGIONS task. This task creates pixel masks which in turn are used by the ELLIPSE task.\ In the first stage, the ellipse parameters are allowed to change freely. From the output of the first stage, we extract the ellipse parameters (center, PA and ellipticity) of the isophote at which the intensity is 3$\sigma$ above the mean of the sky. This gives the set of ellipse parameters that best describes the outer isophotes of the galaxy. The mean and the standard deviation of the sky background is estimated by calculating the mode of an annulus region around the galaxy using the task FITSKY. The task also allows for k-sigma clipping to reject any deviant pixels.\ The program uses the new set of parameters obtained in the first stage as the initial guess for the second stage of the ELLIPSE run while keeping the center, PA and the ellipticity fixed. The task outputs a list of isophotes that have the same center, PA and ellipticity at different semi-major axes. Ideally, total brightness of a galaxy is obtained by integrating the light from the entire galaxy. Since galaxies do not have well defined edges and additionally our observations are limited by the sensitivity of the telescope, we integrate the light out to an isophote of specified brightness or to a specified radius. For our analysis, the isophote whose intensity is one standard deviation above the mean of the sky background is considered the best aperture for that galaxy, encloseing all of the visible galaxy light. Flux enclosed by this aperture is considered to be the $\lq$total flux’ of the galaxy. This process is repeated for all the galaxies in each band ($K_s,$ W1, W2, W3 and W4).\ A sample galaxy is shown in Figure  \[fig:modelfit\]. The top panel shows the ellipse fit to a sample galaxy NGC 4476. The middle panel shows the 2D smooth model image obtained from the isophotal analysis. A residual image is obtained by subtracting the model from the galaxy image. This is shown in the right panel. The residual is nearly smooth. Also shown is a plot of the mean isophotal intensity with respect to the semi-major axis. The mean intensity is normalized with the peak intensity. The solid red line marks the semi-major axis at which the intensity drops to one $\sigma$ above the mean of the sky. The ellipticity and the position angle as a function of the semi-major axis are also shown.\ Shown in the bottom panel of Figure  \[fig:modelfit\] is the ellipse fit to a sample galaxy in which masking of close companion has been done. The ellipticity and the position angle are quite robust. {width="70.00000%"} Radio data ---------- Radio data for our sample are drawn from several resources. It is obtained from NRAO VLA Sky Survey (NVSS; [@condon1998])[^1], Faint Images of the Radio Sky at Twenty-Centimeters (FIRST; [@becker1995])[^2], Sydney University Molonglo Sky Survey (SUMSS; [@mauch2003])[^3]. We retrieved total flux density for the galaxies in the sample.\ NVSS is a radio continuum survey conducted using Very Large Array (VLA) at 1.4 GHz and covers the entire sky north of $-40^{\circ}$ declination. The array is used in D (with baseline of 1 km) and DnC (a hybrid configuration in which antennas in the east and the west arms are maintained in D configuration while the northern arm remain in C configuration with a baseline of 3.6 km) configuration and has angular resolution of 45. The catalog has 3$\sigma$ detection limit of $S\sim1.35$ mJy with typical RMS of 0.45 mJy/beam. FIRST is the survey conducted over 10,000 square degrees of the North and South Galactic Caps. The array is used in B-configuration with frequency centered at 1365 and 1435 MHz which gives a resolution of 5. The 5$\sigma$ sensitivity of the survey is $\sim$1 mJy. The typical rms is 0.2 mJy. If NVSS images are unavailable, FIRST images were used. We estimated the total flux density using the TVWIN+IMSTAT tasks in AIPS.\ For sources that lie in the southern hemisphere of the sky, data from SUMSS is used. SUMSS is a survey carried out at 843 MHz with the Molonglo Observatory Synthesis Telescope (MOST). The survey covers 8000 square degrees from $-30$ degrees declination southwards. The RMS noise level is $\sim$1 mJy/beam with a detection limit of 5 mJy. The SUMSS resolution of 43$\arcsec$ matches with NVSS resolution and thus, SUMSS and NVSS together provide a complete survey of the radio sky. To match the 843 MHz flux density to 1.4 GHz flux density, we use a simple linear extrapolation using the relation $S = S_{o}\nu^{-\alpha}$ with a power law index of 0.83 [@mauch2003]. Hence we use the following relation to estimate the flux density at 1.4 GHz using the flux density at 843 MHz, $$S_{1.4GHz} = S_{843MHz}\left(\frac{1400 MHz}{843 MHz}\right)^{-0.83}$$ The total radio power is calculated using the following equation $$P_{1.4GHz} = S_{1.4GHz} 4 \pi D_L^2\;,$$ where $D_L$ is the luminosity distance calculated using Equation   in the Appendix.\ For sources that are not detected in any of the surveys, we searched in the VLA image archive. We also searched the literature for the total flux density measurements. If the radio flux density could not be determined using the above mentioned resources, the catalog detection limit at the source position is taken as an upper limit in the detection. UV data ------- Far-UV (FUV) and Near-UV (NUV) magnitudes are collected from Galaxy Evolution Survey (GALEX; [@martin2005]). GALEX is a wide-field UV imaging survey performed in two UV bands: FUV ($\lambda_{eff} = 1539\AA, \Delta\lambda=1344-1786\AA$) and NUV ($\lambda_{eff} = 2316\AA, \Delta\lambda=1771-2831\AA$) with angular resolution of 4.2and 5.3respectively. The data is retrieved from GALEX GR6 release[^4]. The GALEX pipeline photometry estimates the magnitudes using the Kron-type SExtractor MAG\_AUTO aperture. It uses an elliptical aperture with the characteristic radius of the ellipse given by the first moment of the source brightness distribution. The limiting magnitude for FUV and NUV is 24.7 and 25.5 respectively.\ Galactic Extinction Correction ------------------------------ FUV, NUV and [$K_s$]{}band magnitudes are corrected for galactic extinction using the corrections from [@wyder2005]. For FUV, NUV and K bands, the ratio of $\frac{A(\lambda)}{E(B-V)}$ is 8.376, 8.741 and 0.347 where $A_{\lambda}$, the extinction at wavelength $\lambda$, is the difference between the observed magnitude and the actual magnitude of the source. Values of color excess $E(B-V)$ are retrieved from the [*GALEX*]{}GR6 catalog which uses the Schlegel maps for the reddening. For galaxies whose $E(B-V)$ are unavailable in the GR6 catalog, we obtained it from NED (which also estimates reddening using Schlegel maps). Because the extinction for the infrared WISE bands is minimal, the correction for these bands have been ignored.\ \ Results ======= Properties of star formation in the sample galaxies {#Radio-FUV-K} --------------------------------------------------- A color magnitude plot such as the one shown in Figure  \[fig:fuv-k\_mk\] can be used to distinguish between actively star-forming and more passive galaxies. The galaxies are color coded according to the strength of the radio power. Those with high radio power are shown in red and low radio power galaxies are shown in blue. We do not see a systematic change in the [$[FUV-K_s]$]{}color with absolute [$K_s$]{}magnitude. An [$[FUV-K_s]$]{}color of 8.8 mag is defined as the transition point between non-star-forming and star-forming galaxies [@gilde2007]. Consistent with the color properties of typical early type galaxies, most of the galaxies in our sample lie in a band of [$[FUV-K_s]$]{}$\sim$ 9-11 [@gilde2007]. A majority of the galaxies are UV weak, and only $\sim$7% of the galaxies in our sample have [$[FUV-K_s]$]{}bluer than 8.8 mag suggesting signs of recent star formation or bright accretion disks. Furthermore, these FUV bright galaxies are not powerful in the radio ($P_{1.4GHz} < 10^{22}\,WHz^{-1}$) and are also less luminous in the [$K_s$]{}band and thus less massive. ![**[$[FUV-K_s]$]{}versus absolute [$K_s$]{}magnitude**. The green dashed line at 8.8 mag defines the separation between star forming and non-star forming as per [@gilde2007]. The galaxies are color coded to represent the strength of the radio power. Galaxies that are redder than 8.8 mag do not show signs of star formation; these consist of $\sim$92% of the sample. Galaxies that have [$[FUV-K_s]$]{}bluer than 8.8 mag show indications of significant star formation; they are less luminous (thus less massive) and are also weak in the radio. The more massive galaxies tend to be FUV faint but are more luminous in the radio. Note here that [$K_s$]{}magnitudes are in the Vega system and FUV are in the AB system.[]{data-label="fig:fuv-k_mk"}](fig3.pdf){width="50.00000%"} We have examined the 13 FUV bright galaxies for evidence of star formation. These fall into three categories: - Galaxies with on-going star formation: The two bluest galaxies (NGC 3413, NGC 1705) are known to be undergoing a strong star burst (evidence from SDSS strong H$\alpha$ emission and [@annibali2003] respectively). The radio power at 1.4 GHz for these galaxies is less than $10^{20}\,WHz^{-1}$ indicating that the FUV emission is dominated via star formation and not by AGN. Also, NGC 855 shows CO emission [@nakanishi2007], NGC 3928 has a starburst nucleus [@balzano1983] and IC5267 has a large number of star formation sites[@caldwell1991], indicating ongoing star formation activity. NGC 7252 is a merger remnant[@chien2010] that has old and new star forming population residing in the nuclear regions of the galaxy. - AGN contribution: NGC 5252 and NGC 5283 are AGNs with Seyfert type Sy1.9 and Sy2, respectively. They show slight excess in the FUV light. Similarly, NGC 4457 hosts a bright UV nucleus which is attributed to the central AGN [@flohic2006]. - Unknown FUV origin: In the rest of the galaxies, NGC 3955, NGC 4344, NGC 4627, and UGC 3097 do not have any strong evidence of ongoing star formation or AGN activity. Thus far, the origin of the excess UV emission in these galaxies is unclear. Similar to the UV, the mid-infrared (MIR) emission is also a good indicator of star formation in a galaxy. A color-color diagram in [$[12\mu m-22\mu m]$]{}vs [$[FUV-K_s]$]{}for the galaxy sample is shown in Figure  \[fig:w3-w4\_fuv-k\]. Only 160 galaxies have the photometry for 12 $\mu$m, 22 $\mu$m, FUV and [$K_s$]{}band. The green dashed vertical line at [$[FUV-K_s]$]{}= 8.8 mag separates star forming and non-star forming galaxies. We draw the horizontal line at [$[12\mu m-22\mu m]$]{}= 2.0 mag to emphasize the concentration of galaxies centered at (10, 0.5). The galaxies are color coded with ellipticals in pink circles and lenticular galaxies in blue plus symbols. The four quadrants are named I, II, III and IV for convenience. The galaxies in quadrant II and III (\[[$[FUV-K_s]$]{}$<$ 8.8 mag) are bright in the FUV and show signs of a young stellar population. The galaxies in quadrant I are bright in the IR but faint in the FUV, indicating star formation that is obscured by dust. Galaxies in IV quadrant are redder in [$[FUV-K_s]$]{}and are not undergoing substantial star formation. The majority of the galaxies that show star formation (i.e. in quadrants I, II and III) are lenticular galaxies.\ ![**Plot of [*WISE*]{}[$[12\mu m-22\mu m]$]{}color versus [$[FUV-K_s]$]{}color**. The vertical green line is as defined in Figure  \[fig:fuv-k\_mk\]. The horizontal line defines the redder galaxies. Galaxies to the left of the vertical line (quadrant II and III) are star forming, those in the top right quadrant (quadrant I) are dust obscured star forming galaxies. Non-star forming galaxies tend to occupy the bottom right quadrant (IV). Elliptical galaxies are colored in red and lenticular galaxies in blue. $\sim$7% of the galaxies show indication of ongoing and obscured star formation.[]{data-label="fig:w3-w4_fuv-k"}](fig4.pdf){width="50.00000%"} Using the IR and FUV colors, we examine the star forming properties of our sample. We notice a small fraction of star forming galaxies ($\sim$7%) that are identified based on the FUV excess in the [$[FUV-K_s]$]{}and IR excess in the [$[12\mu m-22\mu m]$]{}. These star forming galaxies are also less massive and weaker in radio power than that of the galaxies without excess [$[FUV-K_s]$]{}.\ \ SFR estimation using FUV ------------------------ We now proceed to estimate the star formation rate (SFR) for our galaxy sample. The most frequently used SFR indicators are UV continuum, recombination lines (primarily H$\alpha$, but H$\beta$, P$\alpha$, P$\beta$ have been used) , forbidden lines (\[OII\]$\lambda$3727), mid to far IR dust emission and radio continuum emission at 1.4 GHz [@kennicutt1998]. The calibration of SFR for these different star formation tracers are prone to systematic uncertainties from uncertainties in IMF, dust content and distribution and metallicity. However, the scaling relations offer a convenient method to compare the SFR properties in a large galaxy sample. To estimate the SFR in our galaxy sample, we use the calibration in [@salim2007] which was derived to suit the GALEX wavebands. This relation is valid in the $\lq$constant star formation approximation’ where the SFR is assumed to remain constant over the life time of the UV emitting population ($<10^8$ year). It also assumes a Salpeter IMF with mass limits from 0.1 to 100$M_{\odot}$.\ {width="90.00000%"} The FUV emission can be from young stars as well as the evolved stellar population and from the accretion disks of AGNs. To account for the FUV luminosity ($L_{FUV}$) from the young stars alone, we use the following technique to remove the contribution from the evolved stellar population: We chose galaxies that have [$[FUV-K_s]$]{}above the median [$[FUV-K_s]$]{}and treat them as non-star forming galaxies (which is a fair assumption to make, since star forming galaxies are defined to occupy the region below [$[FUV-K_s]$]{}$<$8.8 mag). We then compare the [$K_s$]{}and the FUV luminosity. A fit to the $L_{FUV}$ vs $L_{K_s}$ gives an indication of the amount of FUV emission from the evolved stars. We obtained the following relation $$\label{eq:fuv_evolved} \log L_{FUV,evolved*} = a\,\log L_{K _s}+ b,$$ with a =1.0072 and b = $-3.63$. This fit is used to estimate the FUV contribution from the evolved stellar population for the rest of the galaxies and subtract it from the observed FUV luminosity. This gives the FUV luminosity that is preferentially due to young stars ($L_{FUV,young*}$), which is then used to estimate the SFR. For galaxies that have not been detected in the FUV, we use the [*GALEX*]{}detection limit magnitude of 24.7. FUV emission can also be contaminated by the AGN accretion disk especially from unobscured Sy1 galaxies. There are five Sy1 in our sample and these have been removed in the SFR estimation. Also, these Sy1 have [$[FUV-K_s]$]{}$<8.8$ mag.\ We estimate the SFR using the following relation from [@salim2007], $$\label{eq:sfr} SFR (M_{\odot}yr^{-1}) = 1.08 \times 10^{-28} L_{FUV} \quad (ergs^{-1}Hz^{-1}).$$ The FUV luminosity, $L_{FUV}$ has been corrected for Galactic extinction alone (ignoring internal extinction due to dust). Thus our SFR estimates can be considered as lower limits. From here on, we label $SFR_{FUV,young*}$ as just $SFR_{FUV}$. All the derived quantities (stellar mass, SFR, radio power and the absolute [$K_s$]{}band magnitude) are available online in machine readable format. Table  \[tab:derivedprop\] lists these quantities for a subset of the sample.\ In Figure  \[fig:sfr\], the left panel shows the plot of the SFR obtained using $L_{FUV,young*}$ against [$K_s$]{}band luminosity. The SFR for our galaxy sample with FUV detections is less than 0.4 $M_{\odot}yr^{-1}$. The green diamonds are the galaxies whose SFR is estimated using the total FUV luminosity, i.e., before subtracting the UV light expected from evolved stars. Galaxies that are marked with blue cross are the star forming galaxies identified with [$[FUV-K_s]$]{}$<8.8$ mag. There is an overlap of green diamonds and the blue crosses suggesting that there is no significant change in the SFR before and after subtracting the evolved stellar contribution. All of the detected FUV emission is probably from young stellar population. The red circles are the galaxies with [$[FUV-K_s]$]{}$\geq8.8$ and their SFR are less than the $SFR_{FUV,total}$ indicating a significant contribution to FUV from evolved stellar population. The black dots show the SFR estimated from FUV using equation . The down arrows are the galaxies with FUV upper limits. The right panel shows the distribution of $SFR_{FUV}$ for our sample. The distribution is asymmetric and left skewed. Most of the galaxies have SFR within 0.1-1 $M_{\odot}$/yr with a median at 0.4 $M_{\odot}$/yr and tails off at lower star formation rates. The median SFR for ellipticals is higher at 0.5 whereas for the lenticulars, the median SFR is 0.2. This probably is due to the fact that SFR correlates with the stellar mass [@brinchmann2004], and the average (and the median) stellar mass for the ellipticals in our sample is greater than for the lenticulars.\ ![**Histogram of the specific SFR**. sSFR for ellipticals and lenticular galaxies is shown in red and blue lines respectively. The total sSFR is shown with the black line. The upper limits in the FUV are not considered in the making of the histogram.[]{data-label="fig:ssfr"}](fig6.pdf){width="49.00000%"} Figure  \[fig:ssfr\] shows the normalized distribution of the SFR per unit stellar mass, known as the specific SFR ($sSFR_{FUV}$). It is color coded in red for ellipticals and in blue for lenticulars. The median sSFR for ellipticals and lenticular is roughly equal ($1.1\times 10^{-13}$ and $1.3 \times 10^{-13}$ respectively). We perform a simple KS (Kolmogorov-Smirnov) statistical two-sample test to verify the claim (null hypothesis) that the distribution of the two population is the same. The test gives a p-value of 0.05 and a KS statistic of 0.2. The p-value gives us the probability of the strength of evidence against or in favor of the null hypothesis and the KS statistic tells the maximum distance between the cumulative distribution function of the two samples. In this test, the small p-value suggests that there is only 5% probability that the distribution of ellipticals and lenticulars appear to be same. This suggests that there is significant difference between the distribution of the two populations. Relation between radio power and host galaxy properties ------------------------------------------------------- ### Radio power versus stellar mass {#Radio-Kband} Galaxy luminosities in the [$K_s$]{}-band are 5 to 10 times less sensitive to dust than in the optical band, which allows them to be used as excellent tracers of stellar luminosity and thus stellar mass [@bell2001]. Assuming constant mass to light ratio, which is a good approximation to make especially in the [$K_s$]{}-band, the [$K_s$]{}-band luminosity gives an estimate of the stellar mass of the galaxy [e.g., @bell2003].\ Shown in Figure  \[fig:radio\_k\] is a plot of 1.4 GHz radio power and absolute [$K_s$]{}band magnitude. The radio power ranges between $10^{17}\,WHz^{-1}$ to $10^{25}\,WHz^{-1}$ and $M_{K_s}$ range from $-18$ to $-27$. Galaxies that are fainter than 21 mag have radio power less than $10^{21}\, WHz^{-1}$. Star forming galaxies are known to produce up to $10^{22}\,WHz^{-1}$ radio power at 5 GHz [@wrobel1991] which corresponds to $\sim$ $10^{21}\, WHz^{-1}$ at 1.4 GHz . The weakest radio source detected is NGC 855, and it has a radio power of $3.86\times10^{19}\, WHz^{-1}$ and $M_{K_s} =-19.4$. This is a blue star forming dwarf elliptical galaxy [@nakanishi2007; @walsh1990].\ ![**Plot of total radio power (1.4 GHz) versus absolute k magnitude**. Out of 231 sources, only 195 galaxies have [$K_s$]{}band magnitudes. Sources with upper limits to the radio power are indicated with $\downarrow$. About 56% of the sources have radio flux measurements. The dashed line shows the median radio power binned by the absolute [$K_s$]{}magnitude. The median is calculated considering both the detected and undetected sources. The plots shows that the upper envelope of radio power is a steep function of the total mass of the galaxy. This indicates that massive galaxies are capable of hosting powerful radio sources.[]{data-label="fig:radio_k"}](fig7.pdf){width="50.00000%"} Figure \[fig:radio\_k\] shows the relationship between the radio power and the stellar luminosity (mass). Our results are consistent with previous investigations [@hummel1983; @heckman1983; @feretti1984; @sadler1987; @calvani1989; @brown2011]. But, a relation such as this between two luminosities need to be addressed carefully. Since there is a tight correlation between luminosity and distance, any property that is related to distance can appear as a luminosity dependent relation (Malmquist Bias). Before we claim to see any correlation between radio power and stellar luminosity, it is important to correct for this bias. Following [@singal2014], we examine whether the observed relation is due to Malmquist bias or is an intrinsic property of the sample. We show in Figure \[fig:bias\] the normalized cumulative distribution of $M_{K_s}$ at different radio power bins and at different distance bins. We divided the sample into equal distance bins, except that we combined the last two bins into one bin due to the small number of sources. In each distance bin, the sources are separated into two bins of radio power defined by the median radio power of the sample in that distance bin. Each graph in the plot is the normalized cumulative distribution of $M_{K_s}$. We notice that the median value of $M_{K_s}$, indicated by the dashed line, is higher for the higher radio power bin except in the third distance bin. This suggests that the relation that we see in the Figure \[fig:radio\_k\], i.e., luminous galaxies have higher radio power, is most likely intrinsic to the sample.\ {width="110.00000%"} Compared to previous studies, our sample extends the investigation of radio power and galaxy absolute magnitude to fainter galaxies. There is a broad distribution of radio power at a fixed [$K_s$]{}band absolute magnitude. However, the two quantities show a strong correlation with a correlation coefficient of 0.75, and the probability of it arising by chance is $10^{-37}$. There is an upper envelope of radio power that is a steep function of absolute [$K_s$]{}magnitude. The median radio power (shown in the dotted blue line) also increases monotonically as a function of the galaxy brightness. These results suggest that the maximum radio power from the galaxy is dependent on the mass of the galaxy. Less massive galaxies appear to be capable of hosting only low radio power sources, while more massive galaxies are capable of hosting more powerful radio sources [@kauffmann2003; @best2005; @best2007]. There is an apparent change in the slope of the median radio power around [$K_s$]{}magnitude of $-24$, which suggests that there may be two distinct processes that are responsible for the radio emission in a galaxy. In the fainter galaxies, radio power can be attributed to star formation, i.e., a young stellar population going supernovae, whereas for massive galaxies, the radio power may be dominated by an AGN (see section \[radio\_mir\]).\ This type of relation between galaxy mass and radio power is also found to exist in high redshift quasars [@browne1987; @carballo1998; @serjeant1998; @willott1998; @sanchez2003] and in radio galaxies [@yates1986; @vanvelzen2014]. Galaxies have to be massive enough to be powerful radio sources. The fact that such a correlation exists for AGN and non-AGN population, from faint to bright galaxies is interesting. Since the black hole mass scales with the bulge mass [e.g., @haring2004], the correlation also suggests that the black hole mass closely relates to the maximum radio power [@franceschini1998; @laor2000; @liu2006]. The plot also shows that there is a broad dispersion in radio power at a given absolute magnitude even for the most massive galaxies in our sample. This is consistent with the hypothesis that high black hole mass is necessary but not sufficient to produce a powerful radio source. Other physical parameters such as the spin of the black hole, accretion efficiency and other large-scale environmental effects may be responsible for the broad dispersion in radio power [e.g., @baum1995; @meier1999; @wold2007]. ### Radio power and nuclear activity {#sec:radio_wise} The origin of [*WISE*]{}mid-IR emission is associated with a combination of continuum emission from dust, atomic and molecular emission lines and features associated with PAHs that are heated by young stars and AGN, as well as Gyr old evolved stellar population [@jarrett2013]. Shown in Figure \[fig:radio\_w1-w2\] is a plot of 1.4 GHz radio power against [$[3.4\mu m-4.6\mu m]$]{}infrared color. The [$[3.4\mu m-4.6\mu m]$]{}color is sensitive to warm/hot dust and thus to optical/UV nuclear activity. An excess in this color identifies galaxies in which hot dust surrounding the AGNs produces a strong mid-IR continuum that dominates the host galaxy emission [@stern2012]. This enhanced mid-IR continuum may be associated with the dusty torus heated by the radiation from an accretion disk. About 83% of the galaxies in our sample lie in a narrow color range (between $-0.3$ and 0.1 mag), and only a few galaxies show excess IR emission. The fact that most of the galaxies do not show a color excess indicates that the galaxies in our sample are not associated with bright accretion disks, and if they are AGN, they are accreting in a radiatively inefficient process[e.g. ADAF @narayan1994].\ ![**Plot of radio power at 1.4 GHz versus WISE [$[3.4\mu m-4.6\mu m]$]{}color**. Galaxies marked with down arrow have upper limit in the radio. Most of the galaxies do not show a color excess. This indicates that the galaxies in the sample are not associated with bright accretion disks.[]{data-label="fig:radio_w1-w2"}](fig9.pdf){width="50.00000%"} ### Radio power and SFR relation The supply of cold gas to fuel star formation and AGN activity can include galaxy mergers/interactions, the cooling of gas in the ISM or surrounding hot halo, and the mass loss from stars [e.g., @heckman2014]. The relation between radio power and SFR can provide some insight into the source of the fueling (Figure  \[fig:radio\_sfr\]).\ We see a clear correlation (spearman correlation coefficient of 0.45 at a significance level of $10^{-6}$) although with a bit of scatter, between radio power and the SFR (Figure \[fig:radio\_sfr\]). Figure  \[fig:radio\_k\] shows a weak correlation between radio power and galaxy mass, while Figure  \[fig:sfr\_mass\] shows a correlation between SFR and galaxy mass [see also @brinchmann2004]. This suggests that the weak correlation between radio power and SFR may be due to a correlation of both radio power and SFR with galaxy mass. Such a correlation with galaxy mass would be consistent with an origin of the gas supply which fuels the AGN and star formation associated with the host galaxy itself rather than a predominantly external origin (e.g., major mergers). In this case the gas supply might be due to stellar mass loss [e.g., @faber1976; @knapp1992; @voit2011] or perhaps cooling from the ISM or halo [@binney1981; @forman1985; @canizares1987; @voit2015].\ The figure also shows a broad dispersion of several orders of magnitude between the radio power and SFR (especially between 0.01 and 0.1 $M_{\odot}yr^{-1}$). A similar large dispersion is observed in the radio power vs stellar mass relation (Figure \[fig:radio\_k\]), which indicates that, producing a radio source is a complicated process that depends not only on the gas supply but also on the gas transport mechanism, black hole spin and black hole mass, accretion rate and the external environment [e.g., @baum1995; @meier1999; @wold2007]. These will naturally add dispersion to the relation between radio power and the SFR. In addition irregular fuel supply [@tadhunter2011; @kaviraj2014] and variability in the AGN $\lq$on’ phase [@hickox2014] will further weaken the correlation.\ Determining the connection between AGN feedback and star formation from this relation is not straightforward. Although we notice that galaxies that have low SFR do not have high radio power, it is not clear whether AGN is responsible for the low SFRs by suppressing the star formation via feedback. If AGN feedback is ongoing and has suppressed the star formation even though the radio power is low, this would imply that radio power is not a good indicator of AGN feedback. There can be other possibilities for the observed low radio power at low SFRs. Absence of major mergers (as suggested above) can leave a galaxy with less cold gas, that is insufficient for high SFR and power nuclear accretion. ![**Relation between radio power and the estimated rate of star formation.**Upper limits in the radio are shown with a down arrow, FVU with a left arrow and upper limits in both FUV and radio are shown with an oplus symbol. Radio detections are shown in pink. The observed weak correlation between the radio power and SFR is likely due to the correction of both radio power and SFR with galaxy mass (Figure  \[fig:radio\_k\] and Figure  \[fig:sfr\_mass\] respectively).[]{data-label="fig:radio_sfr"}](fig10.pdf){width="50.00000%"} ![**Plot of SFR against stellar mass of our sample.** []{data-label="fig:sfr_mass"}](fig11.pdf){width="50.00000%"} ### Radio-MIR flux correlation {#radio_mir}  Radio emission observed in a galaxy is non-thermal from relativistic electrons accelerated either by the AGN or by the supernovae [@dejong1985; @condon1992]. One way to identify the origin of the radio emission is to compare the radio flux with the far-IR flux. A correlation between radio and infrared emission suggests that sources that are responsible for IR emission, are also responsible for emission in the radio. Young stars with $M\sim8M_{\odot}$ and above emit most of their energy in the UV which is then absorbed and re-radiated in the IR by the dust. At the end of their life, these massive stars explode as supernovae which accelerate the electrons to relativistic speeds resulting in radio emission due to synchrotron. Thus, the relationship between radio and IR emission can trace star formation activity [@dejong1985; @helou1985].\ Figure  \[fig:radio\_w3flux\] shows the relationship between 1.4 GHz radio flux and [*WISE*]{}mid-IR apparent magnitudes. At both 12 and 22 [$\mu$m]{}, we find a correlation similar to the radio-FIR (see also @appleton2004). We compute the radio-MIR regression coefficients using Kaplan-Meier method so as to consider the upper limits in the radio during the fit. We used iraf task *buckleyjames* for this purpose and obtained the following relation:\ $$\begin{aligned} \log S_{1.4GHz} &= -0.33(\pm0.04) m_{12\mu m} + 2.87 \\ \log S_{1.4GHz} &= -0.30(\pm0.04) m_{22\mu m} + 1.96, \end{aligned}$$ where $S_{1.4GHz}$ is the radio flux at 1.4 GHz and $m_{12\mu m}$, $m_{22\mu m}$ are the apparent magnitudes at 12 and 22$\mu$m waveband respectively. The correlation coefficient for both the relations is 0.83 and the dispersion in the regression is $\sim$ 0.35 dex. In terms of the MIR flux, the slope of the radio-12 [$\mu$m]{}and radio-22 [$\mu$m]{}relation is $\sim$ 0.8 and 0.75 respectively which are comparable to the slope obtained in the previous studies of radio-MIR [@gruppioni2003] and radio-FIR relation (which is $\sim$ 0.9). Unlike the tight radio-FIR correlation, the radio-MIR relation has a high dispersion [@appleton2004], possibly because the $12\mu m$ emission is not solely due to the dust heated by young stars, but can arise from the PAHs heated by young/evolved stars/AGN or from the dust shells of AGB population.\ Galaxies shown in pink triangles in Figure  \[fig:radio\_w3flux\] are the galaxies with $P_{1.4GHz}\geq 10^{22} \, WHz^{-1}$. Although this radio power cut was chosen arbitrary (but with the knowledge that star forming galaxies have $P_{1.4GHz}\sim 10^{21} \, WHz^{-1}$ [@wrobel1991]), these galaxies show a noticeable departure from the normal radio-MIR and follow a different relation with a slope of 0.4. Despite the huge scatter, we notice a trend in the radio-MIR relation. These galaxies show an excess radio emission relative to its MIR flux, which can be attributed to AGN origin. Thus, these galaxies are potential candidates for being an AGN. We define $10^{22}\,WHz^{-1}$ as the threshold radio power and for the rest of the sections, we define galaxies above this threshold as radio bright’ and below the threshold as radio faint’ galaxies.\ The rest of the galaxies are identified by pale blue dots. These are fainter than 5 mag in 12[$\mu$m]{}and 4 mag in 22[$\mu$m]{}. Although with a large scatter, these galaxies appear to follow the radio-12[$\mu$m]{}relation. On the other hand, in the radio-22[$\mu$m]{}plot, these galaxies do not seem to follow the radio-22[$\mu$m]{}relation for star forming galaxies. Also, these fall in the region that is mid-way between the star forming galaxies and the radio bright galaxies. About 28% of the radio faint galaxies are star forming galaxies and fall on the radio-MIR correlation.\ ### Specific SFR and stellar mass Specific star formation rate (sSFR), which is the SFR normalized by the stellar mass, traces the star formation efficiency. The sSFR indicates fractional galaxy growth due to star formation. Figure  \[fig:ssfr\_k\] shows the sSFR with respect to the absolute [$K_s$]{}magnitude. Radio bright galaxies with $P_{1.4GHz}\geq10^{22}\,WHz^{-1}$ are indicated with pink triangles. The mean and one sigma deviation above mean for the sSFR is shown in solid and dashed lines respectively. The mean sSFR for the radio bright and radio faint galaxies is nearly the same. Most of the galaxies have a low sSFR of $10^{-13}\,yr^{-1}$. The least square regression for the two quantities gives a flat slope slope of $10^{-12}$. This flat relation suggests that the sSFR is not a strong function of galaxy stellar mass.\ There are a few outlier galaxies (above the dashed lines) that show increased sSFR which is observed in both galaxy populations i.e. radio bright and radio faint galaxies. The standard deviation in the sSFR is noticeably different in the two populations. But the proportion of high to low sSFR (i.e. above and below the dashed line) in radio bright galaxy is nearly the same as that in radio faint galaxies. A statistical test using z-score proportionality is calculated for these two populations on the null hypothesis that the two population proportions are the same. The test statistic gives a p-value of 0.4 indicating that the null hypothesis cannot be rejected. What this tells us is that, the fraction of star forming galaxies in high radio power galaxies is similar to the fraction of galaxies forming stars in low radio power galaxies. In addition, the KS two-sample test gives a high p-value of 0.9 indicating that the two samples are drawn from the same distribution further supporting the idea that the distribution of the sSFR in both the samples is nearly the same. Thus, we find that star formation efficiency is small and independent of radio power in our sample. This suggests that our sample galaxies are not experiencing significant growth or significant AGN feedback.\ \ ![**Specific SFR vs absolute [$K_s$]{}band magnitude**. Pink triangles indicate galaxies that have an excess radio power ($P_{1.4GHz} \geq 10^{22}WHz^{-1}$) and the blue circles indicate low radio power ($P_{1.4GHz} \le 10^{22}WHz^{-1}$) galaxies. The average sSFR is indicated with solid line and the deviation from the mean is indicated with dashed line. The average sSFR for both the groups is small and almost equal. The results indicate that galaxies are not experiencing significant growth or significant AGN feedback.[]{data-label="fig:ssfr_k"}](fig13.pdf){width="50.00000%"} Comparision to galaxy clusters ------------------------------ Observational evidence exists for AGN feedback in action in the Brightest Cluster Galaxies (BCG) in the form of X-ray cavities [@mcnamara2007; @fabian2012] and shocks [@mcnamara2005; @fabian2006; @forman2007] in the ICM . Here we compare our sample with the Brightest Cluster Galaxies (BCG) in cool cores. We selected the BCGs from [@odea2008] (hereafter Odea08) and used their 1.4GHz radio data and IR derived SFR. The Odea08 sample consists of BCGs that are located in the cores of X-ray luminous clusters that have optical line emission, thus preferentially selecting BCGs in cool cores. We also considered BCGs from [@rafferty2006] (hereafter R06) sample. The R06 sample consists of BCGs that show evidence of X-ray cavities indicating AGN feedback. The R06 data set provides X-ray cavity power and mass cooling rate which are related to AGN jet power and the rate of star formation respectively. The cavity power scales to jet radio power according to the following relation (Equation 1 of [@cavagnolo2010] ) : $$\log P_{cav} = 0.75(\pm 0.14) \log P_{1.4} + 1.91(\pm 0.18),$$ and the average mass cooling rate is $\sim$ 4 times the SFR (section 4.3 of [@rafferty2006]) which is within the range of 3-10 suggested by [@odea2008].\ In Figure  \[fig:radio\_sfr\_bcg\] we plot the radio power as a function of SFR extending the relation to the BCGs. Our galaxy sample follows broadly along the R06 line thus extending the relation from weak radio power galaxies to more radio powerful galaxies in clusters . It covers about eight orders of magnitude in both radio power and in SFR. However, there is a scatter about this relation which spans roughly four orders of magnitude. A Spearman test for the combined sample indicates a strong correlation with a correlation coefficient of 0.78 and the probability of it arising by chance is $\sim 10^{-32}$. Thus, the plot shows a general trend between SFR and the radio power across several orders of magnitude. The relation in the plot suggests that the fuel supply for the triggering of star formation and the AGN has a common origin. We suggest that the same factors that introduce scatter to the radio power vs stellar mass relation, contribute to the scatter here as well, which include gas supply mechanisms, the amount of gas available for fueling AGN and star formation, and the relative differences in life cycles of the star formation and AGN activity. Although these factors contribute to the scatter, the relative contribution of these factors remains uncertain.\ The average radio power of the BCGs is $\sim 10^{24}\,WHz^{-1}$. For an effective mechanical feedback via heating, the radio source should have $P_{1.4GHz}\sim10^{24}-10^{25}\,WHz^{-1}$ [@best2006]. The majority of the galaxies in our sample have radio powers below this value. These results are consistent with weak (or negligible) AGN feedback in our sample. ![**Radio power at 1.4GHz versus SFR - Comparing our sample with the BCGs**. The markers in pink and black are the galaxies in this study where detections are indicated in pink and upper limits with black arrow. The blue triangles and green stars are the BCGs from [@rafferty2006] and [@odea2008]. The solid blue line is the best fit line ($\log P_{1.4GHz} = 1.08\log SFR+24.0$)to the BCGs in R06 sample. The mean radio power of the BCGs is $\sim10^{24}\, WHz^{-1}$. The spread in the correlation suggests various possibilities such as different sources of gas supply, black hole spin, accretion rate and the time delay between the triggering of star formation and AGN activity.[]{data-label="fig:radio_sfr_bcg"}](fig14.pdf){width="50.00000%"} ### Comparision of sSFR To estimate the total stellar mass for the Odea08 sample, we used the flux measurements at IRAC bands 3.6 [$\mu$m]{}and 4.5 [$\mu$m]{}. The redshift range of this sample is between 0.017 to 0.25. The redshifted light at the these central wavelengths fall within the IRAC bandwidth. A simple and yet robust conversion between the stellar mass and infrared flux is given by [@eskew2012]: $$\begin{aligned} M_* = 10^{5.65} F_{3.6}^{2.85}F_{4.5}^{-1.85}\left(\frac{D}{0.05}\right)^2 \end{aligned}$$ where $M_*$ is in $M_{\odot}$, D is in Mpc, $F_{3.6}$ and $F_{4.5}$ are in Jy. We used the luminosity distance estimates to these BCGs from NED with cosmological parameters for $H_o$=71 $km s^{-1}Mpc^{-1}$, $\omega_{matter}$=0.27 and $\omega_{vaccum}$=0.73. The $F_{3.6}$ and F$_{4.5}$ estimates are taken from [@quillen2008].\ The average stellar mass we obtained for Odea08 sample is $2.3\times10^{11}\,M_{\odot}$, which is comparable to the average stellar mass of BCG at low redshifts [@liu2012; @fraser2014]. We plot the radio power against the sSFR in Figure \[fig:bcg\_radio\_ssfr\]. The average sSFR for the BCG is $3.6\times10^{-11}\, yr^{-1}$ which is roughly two orders of magnitude higher relative to the local early type galaxies. We see that both the star formation efficiency and the radio power are higher in the BCGs than in our sample. This suggests that although feedback is likely present in the BCGs [e.g., @fabian2012], it is not sufficient to completely suppress star formation [e.g., @odea2008; @tremblay2012b; @tremblay2015]. ![**Plot of radio power versus sSFR - Comparing our sample with Odea08 sample**. The description of the legend is same as that of Figure \[fig:radio\_sfr\_bcg\]. []{data-label="fig:bcg_radio_ssfr"}](fig15.pdf){width="50.00000%"} Summary ======= We collected multiple wavelength data (radio, IR and UV ) for a sample of 231 early type galaxies at $z<0.04$ and analyzed the properties of star formation and radio mode feedback. The main results are as follows. Properties of Star Formation ---------------------------- The SFR in our sample tend to be low ($< 1M_{\odot}yr^{-1}$) and only 7% of the galaxies show obvious signs of ongoing star formation via the [$[FUV-K_s]$]{}and [$[12\mu m-22\mu m]$]{}colors. In addition, the sSFR is very small. The star forming galaxies trace a radio-MIR correlation similar to that seen in other samples of star forming galaxies. These results indicate that galaxy building in early type galaxies has essentially ceased at the present epoch. AGN Radio Properties --------------------- The Radio-MIR relation shows that galaxies with $P\geq10^{22}\,WHz^{-1}$ have radio power in excess of that expected to be produced by the estimated star formation rates and thus are potential candidates for being radio AGN. Only $\sim 20\%$ of the galaxies in our sample have $P\geq10^{22}\,WHz^{-1}$. Only a few of the high radio power galaxies show excess 4.6 [$\mu$m]{}flux, an indication of hot dust heated by an accretion disk. This indicates that the majority of the radio AGN are accreting gas in a radiatively inefficient manner [@ho2009].\ There is an upper envelope of radio power that is a function of galaxy stellar mass (and thus BH mass) suggesting that the maximum radio power scales with galaxy (BH) mass. The large scatter in the relation between radio power and galaxy stellar mass suggests that high black hole mass is necessary but not sufficient for producing a radio loud AGN. This is consistent with additional parameters (such as BH spin, accretion rate) playing an important role in determining radio power. Relation between Radio and Star Formation Properties ---------------------------------------------------- The sSFR is roughly independent of radio power in our sample, suggesting that radio mode feedback is not having a significant effect on star formation efficiency in these galaxies. Alternately, radio power may not be a good proxy for radio mode feedback or the feedback is episodic.\ The correlation between radio power and SFR is weak, and if real may be due to a correlation of both radio power and SFR with galaxy stellar mass. This would suggest that the host galaxy is the source for the fuel (e.g., stellar mass loss for lower mass galaxies and cooling from the ISM/halo for more massive galaxies) for star formation and AGN activity in these galaxies.\ Two samples of cool core BCGs lie on the same relation for radio power and SFR as our sample over a range of eight orders of magnitude. Although both star formation and radio mode feedback are constrained to be very low in our sample, the BCG samples exhibit both at high levels. The relatively low radio power in our sample compared to the average radio power of the BCGs (i.e. $\sim10^{24}\, WHz^{-1}$) suggest that there may be a threshold in the radio power that is needed for the feedback from the AGN to affect the star formation in the host galaxy. Acknowledgements ================ This work was supported in part by the Radcliffe Institute for Advanced Study at Harvard University. We thank Dr. Christine Jones for constructing the sample and sharing it with us. We thank Dr. Bill Forman for his invaluable comments and suggestions which helped us to improve the paper. We are grateful to the facilities of Harvard-Smithsonian Center for Astrophysics (CfA), Cambridge where the project took its initial shape. This work made use of the archival data from [*GALEX*]{}, [*2MASS*]{}, [*WISE*]{}, NRAO and SUMSS and supplemented with information from Hyperleda and NASA/IPAC Extragalactic Database (NED). The National Radio Astronomy Observatory (NRAO) is operated by Associated Universities, Inc., under cooperative agreement with the National Science Foundation. We thank the NSF funded REU (Research Experience for Undergraduate) program in the Chester F. Carlson Center for Imaging Science at RIT. We also wish to acknowledge helpful conversations with colleagues. Note: Column description: (1) Name of the galaxy; (2) RA in J2000; (3) DEC in J2000; (4) Redshift; (5) Radial Velocity [@makarov2014]; (6) Morphological type (de Voucouleur’s scale) [@makarov2014]; (7) Limit on the radio flux at 1.4 GHz; (8) Radio flux at 1.4 GHz; (9) - (12) Photometric estimates in [*WISE*]{}bands; (13) Photometric estimate in [$K_s$]{}band; (14) GALEX FUV; (15) GALEX NUV; (16) Galactic reddening from [*GALEX*]{} ------------ -------------------- ------------- --------- ------------ ----------- Name SFR $M_{*}$ l\_P1.4 P1.4 $M_{K_s}$ $M_{\odot}yr^{-1}$ $M_{\odot}$ $WHz^{-1}$ mag 7ZW700 1.64E-02 1.17E+11 $<$ 1.96E+21 -24.40 ESO137006 1.54E-03 4.84E+11 $<$ 9.00E+20 -25.95 ESO269 8.69E-04 6.52E+10 — 2.22E+21 -23.77 ESO3060170 1.77E-03 4.00E+11 — 3.32E+22 -25.74 ESO351030 9.41E-05 5.97E+08 $<$ 3.62E+17 -18.68 ESO428 1.53E-04 3.76E+10 — 5.30E+21 -23.17 ESO443G024 2.91E-02 2.92E+11 — 8.14E+23 -25.40 ESO495G021 — 7.39E+09 — 1.39E+21 -21.41 ESO552G020 5.10E-02 5.56E+11 $<$ 2.67E+21 -26.10 IC1262 4.35E-02 1.96E+11 — 1.49E+23 -24.97 IC1459 2.15E-02 2.23E+11 — 9.25E+22 -25.10 IC1633 2.02E-01 8.29E+11 — 1.88E+21 -26.53 IC1729 2.65E-03 1.03E+10 $<$ 6.84E+19 -21.77 IC4296 9.09E-03 5.75E+11 — 2.59E+21 -26.14 IC5267 1.25E-01 9.77E+10 $<$ 3.96E+20 -24.21 IC5358 — 1.59E+11 — 4.62E+22 -24.74 NGC1023 — 7.78E+10 $<$ 1.26E+19 — NGC1052 4.69E-03 1.03E+11 — 5.02E+22 -24.27 NGC1132 4.91E-02 4.14E+11 — 5.86E+21 -25.78 NGC1199 — 1.18E+11 $<$ 2.18E+20 — NGC1265 8.21E-02 8.21E+11 — 6.70E+24 -26.52 NGC1316 — 2.71E+11 — 8.54E+24 — NGC1332 2.00E-02 1.37E+11 — 2.40E+20 -24.58 NGC1340 2.82E-03 5.28E+10 $<$ 4.24E+19 -23.54 NGC1381 2.40E-03 5.19E+10 $<$ 9.03E+19 -23.52 NGC1386 8.22E-03 1.76E+10 — 6.78E+20 -22.35 NGC1387 1.69E-02 6.65E+10 — 1.43E+20 -23.79 NGC1389 2.04E-04 1.04E+10 $<$ 2.57E+19 -21.78 NGC1395 2.09E-02 1.96E+11 — 7.14E+19 -24.96 NGC1399 — 1.76E+11 — 2.91E+22 — NGC1404 1.72E-02 2.41E+11 — 3.31E+20 -25.19 NGC1407 3.48E-02 2.79E+11 — 6.31E+21 -25.35 NGC1426 1.10E-03 2.79E+10 $<$ 6.32E+19 -22.85 NGC1427 4.95E-03 3.89E+10 $<$ 5.84E+19 -23.21 NGC1439 9.30E-04 4.09E+10 $<$ 8.43E+19 -23.26 NGC1482 1.95E-02 5.25E+10 — 1.85E+22 -23.54 NGC1521 2.37E-02 1.63E+11 — 1.68E+21 -24.77 NGC1549 — 1.48E+11 $<$ 1.37E+20 — NGC1550 5.48E-02 1.55E+11 — 6.75E+21 -24.71 NGC1553 2.88E-03 1.53E+11 — 1.38E+20 -24.70 NGC1587 8.43E-03 1.83E+11 — 3.95E+22 -24.89 NGC1600 7.29E-02 6.44E+11 — 3.06E+22 -26.26 NGC1638 2.95E-02 5.88E+10 $<$ 3.28E+20 -23.66 NGC1700 — 2.89E+11 $<$ 4.59E+20 -25.39 NGC1705 1.40E-01 6.41E+08 $<$ 3.49E+19 -18.75 NGC1800 4.89E-02 1.34E+09 $<$ 1.94E+19 -19.55 NGC205 5.00E-03 1.05E+10 $<$ 1.76E+18 -21.79 NGC2110 1.10E-03 1.14E+11 — 3.58E+22 -24.38 NGC221 1.63E-03 1.48E+10 $<$ 1.21E+18 -22.16 NGC2305 1.14E-02 1.64E+11 $<$ 1.13E+21 -24.77 NGC2314 3.65E-03 1.47E+11 — 7.82E+21 -24.66 NGC2329 4.11E-02 2.09E+11 — 5.51E+23 -25.03 NGC2340 2.81E-02 4.01E+11 — 3.94E+20 -25.74 NGC2434 1.55E-02 4.59E+10 $<$ 1.86E+20 -23.39 NGC2563 2.37E-02 1.79E+11 $<$ 4.24E+20 -24.87 NGC2768 1.25E-02 1.17E+11 — 6.36E+20 -24.40 NGC2778 8.63E-04 2.50E+10 $<$ 8.88E+19 -22.73 NGC2787 — 1.30E+10 — 1.19E+20 — NGC2832 5.00E-02 4.32E+11 — 5.34E+21 -25.83 ------------ -------------------- ------------- --------- ------------ ----------- : Derived properties of a subset of the sample.[]{data-label="tab:derivedprop"} Note: Column description: (1) Name of the galaxy; (2) Star Formation Rate; (3) Stellar Mass; (4) Limit on the radio flux at 1.4 GHz; (5) Radio power at 1.4 GHz; (6) Absolute [$K_s$]{}band magnitude Calculations ============ From the apparent magnitude, absolute magnitude is calculated using the following distance modulus relation: $$M_{k} = m_{k}-5logD_{L} + 5 \;,$$ where, $m_{k}$ is the apparent magnitude in [$K_s$]{}band, distance $D_{L}$ is the luminosity distance calculated using the relation $$D_L = v/H_o\;, \label{eq:distance}$$ where $v$, the radial velocity is obtained from hyperleda [^5], and $H_o$ is the Hubble constant.\ Redshift is calculated using the radial velocity with the relation $z=v/c$.\ Flux densities for $K_{s}$ and *WISE* bands in the units of $W/cm^{2}/\mu m$ are calculated using the zero-magnitude fluxes given in Table  \[tab:zeromagflux\]. [Band]{} [$F_{\lambda}-0\;mag\;(W/cm^{2}/\mu m$)]{} [$F_{\nu}-0\;mag\;(Jy)$]{} ------------- -------------------------------------------- ---------------------------- [$K_{s}$]{} [4.283E-14]{} [666.7]{} [W1]{} [8.1787E-15]{} [309.540]{} [W2]{} [2.4150E-15]{} [171.787]{} [W3]{} [6.5151E-17]{} [31.674]{} [W4]{} [5.0901E-18]{} [8.363]{} : [*2MASS*]{}[@cohen2003] and *WISE* [@jarrett2011] fluxes for zero-magnitude.[]{data-label="tab:zeromagflux"} Photometry comparisions ======================= Comparision with the 2MASS -------------------------- A comparison of our [$K_s$]{}band measurements with that of the 2MASS XSC catalog has been done. We compare our results with the isophotal measurements that are set to 20 mag [$\mathrm{arcsec^{-2}}$ ]{}surface brightness isophote at [$K_s$]{}using elliptical apertures (identified with k\_m\_k20fe in the catalog). This corresponds to roughly 1$\sigma$ of the typical background noise in the [$K_s$]{}images.\ The 2MASS ellipse fitting pipeline is described in [@jarrett2000]. The basic step in the fitting method is to first isolate an approximate 3$\sigma$ isophote which at [$K_s$]{}band magnitude is at 18.55. This step is called isovector operationand is done by analyzing the radial profiles at different position angles and determining the pixels that correspond to the 3$\sigma$isophote. The center of the isophote corresponds to the pixel with peak intensity. A best-fit ellipse to the pixel distribution is obtained by minimizing the ratio of the standard deviation to the mean of the radial distribution at the 3$\sigma$ isophote. Using the axis ratio and position angle of the 3$\sigma$ isophote, an ellipse fit to [$K_s$]{}band at 20 mag [$\mathrm{arcsec^{-2}}$ ]{}is obtained by varying the semi-major axis such that the mean surface brightness along the ellipse is 20 mag [$\mathrm{arcsec^{-2}}$ ]{}. The integrated flux within this ellipse after background subtraction is the 2MASS isophotal magnitude.\ The Figure  \[fig:k\] shows the comparison with the 2MASS. Our magnitude measurements match closely with that of 2MASS catalog. Galaxies that do not have 2MASS magnitudes are shown with green stars. Out of 231 galaxies in the sample, estimates for 47 galaxies were excluded, either because they are close to the edge of the image or due to the presence of a very close companion which could not be masked. ![Comparison between 2MASS [$K_s$]{}magnitude from the catalog ($m_{K_{2MASS}}$ ) and our measurements ($m_{K_{1\sigma aper}}$). Solid blue line shows the one-to-one relation between the x and y. The yellow dashed line is the linear fit with a slope of 0.97 and an intercept of 0.23. Our magnitude measurements match with that of the 2MASS magnitude. The green stars are the galaxies whose 2MASS magnitudes are not present. The outlier galaxy is IC5358 whose 2AMSS mag is brighter than our measurement. This is because of the presence of a companion galaxy which was not excluded in the catalog estimates.[]{data-label="fig:k"}](fig16.pdf){width="50.00000%"} Comparison with WISE -------------------- The WISE extended source photometry pipeline uses the aperture that is based on the elliptical shape reported by the 2MASS XSC. Due to the larger beam size of WISE, the aperture is scaled accordingly. They sum the pixel fluxes within this aperture and subtract it with the background to obtain the elliptical aperture photometry measurement which is indicated by w?gmag in their catalog. Since W1 is the most sensitive wavelength for all galaxies where the emission is from the evolved stellar population, the 2MASS [$K_s$]{}aperture is typically 3 to 4 times smaller than the 1$\sigma$ aperture for W1. This underestimates the integrated flux by about 30-40%. Hence, we performed photometry on the WISE images to estimate the galaxy magnitude within an elliptical aperture fit to 1$\sigma$ isophote.\ Shown in Figure  \[fig:w1\] is the comparison with WISE W1 magnitude. The WISE W1 catalog magnitudes are faint compared to our measurements on an average by 0.31 mag. There are few outliers whose WISE magnitudes are brighter to our measurements. NGC4467 at mag of  11 is brighter by 0.1 mag. This is because the WISE aperture is bigger than the size of the galaxy because of which some of the light from the adjacent galaxy is also included. In case of NGC4782, NGC5353 NGC821, the emission from the nearby source increased the WISE W1 flux. These are consistently brighter in W2 band as well.\ ![Comparison between WISE W1 magnitude with our measurements. Notation used in this figure is same as that used in Figure  \[fig:k\]. The WISE catalog measurements give galaxy magnitudes that are faint compared to the magnitudes that we measured. The uncertainty in magnitude is smaller than the point size. The mean difference between WISE catalog and our measurements is $\sim$0.34 mag with a range of percentage difference between $\sim$5% to 70%. The slope of the fit is 1.05 with an intercept of $-0.06$[]{data-label="fig:w1"}](fig17.pdf){width="50.00000%"} Figures  \[fig:w2\] through  \[fig:w4\] show the comparison of W2, W3 and W4 bands. The WISE W2 magnitudes also are fainter than our measured values. In the case of W3 band, the extended emission from the galaxy is contained within the WISE apertures. Hence we see that the one-to-one correspondence matches with the line fit for W3 magnitude.\ ![Comparison between WISE W2 magnitude with our measurements. The WISE magnitudes are fainter by  0.32 mag. the slope of the fit is 1.04 with an intercept of $-0.07$. []{data-label="fig:w2"}](fig18.pdf){width="50.00000%"} We modified our program when we measure W3 and W4 magnitudes. We do not go through the two stage process in these cases and ignore the process of obtaining the 3$\sigma$ aperture and fixing the ELLIPSE parameters. This is because, the flux from the galaxy is very faint and is almost always about 3$\sigma$ or less.\ ![Comparison between W3 WISE magnitude with our measurements. It can be noticed that the WISE catalog measurements give galaxy magnitudes that are close to the magnitudes that we measured. The mean difference is $\sim$0.095.[]{data-label="fig:w3"}](fig19.pdf){width="50.00000%"} Figure  \[fig:w4\] shows the W4 magnitude comparison. Majority of the galaxies have very faint emission at this wavelength. Since the apertures used in WISE measurements are larger, the magnitude measurements indicate them as bright galaxies due to contamination from faint foreground stars that are not subtracted in the images. In our method, we use an aperture at 1$\sigma$ which gives an aperture that is just right to measure the galaxy flux without adding noise.\ ![Comparison between W4 WISE magnitude with our measurements. There is huge scatter in the measurements for faint galaxies and the catalog estimates small magnitudes for the faint galaxies. The slope of the fit is 0.92 with an intercept of 0.39.[]{data-label="fig:w4"}](fig20.pdf){width="50.00000%"} We show the difference in the aperture between our method and that from WISE in W1 mag in Figure  \[fig:smadiff\]. About 95% of the galaxies have small WISE apertures. ![Difference between the aperture sizes obtained using our method and that used in the WISE catalog.[]{data-label="fig:smadiff"}](fig21.pdf){width="50.00000%"} Error Analysis -------------- The uncertainty in the flux measurement include contribution from possion noise and error in the sky background estimation. 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--- abstract: 'We report results of Raman scattering experiments on twin-free with the main focus placed on understanding the influence of electronic and spin degrees of freedom on the lattice dynamics. In particular, we scrutinize the [$E_{g}$]{}modes and the As [$A_{1g}$]{}mode. Each of the two [$E_{g}$]{}phonons in the tetragonal phase is observed to split into a [$B_{2g}$]{}and a [$B_{3g}$]{}mode upon entering the orthorhombic stripe-magnetic phase. The splitting amounts to approximately 10cm$^{-1}$ and less than 5cm$^{-1}$ for the low- and the high-energy [$E_{g}$]{}mode, respectively. The detailed study of the fully symmetric As mode using parallel incident and outgoing photon polarizations along either the antiferromagnetic or the ferromagnetic Fe-Fe direction reveals an anisotropic variation of the spectral weight with the energy of the exciting laser indicating a polarization-dependent resonance effect. Along with the experiments we present results from density functional theory calculations of the phonon eigenvectors, the dielectric function, and the Raman tensor elements. The comparison of theory and experiment indicates that (i) orbital-selective electronic correlations are crucial to understand the lattice dynamics and (ii) all phonon anomalies originate predominantly from the magnetic ordering and the corresponding reconstruction of the electronic bands at all energies.' author: - 'A. Baum' - Ying Li - 'M. Tomić' - 'N. Lazarević' - 'D. Jost' - 'F. Löffler' - 'B. Muschler' - 'T. Böhm' - 'J.-H. Chu' - 'I. R. Fisher' - 'R. Valentí' - 'I.I. Mazin' - 'R. Hackl' title: | Interplay of lattice, electronic and spin degrees of freedom in detwinned BaFe2As2:\ a Raman scattering study --- Introduction ============ One of the most debated issues in Fe-based superconductors is the interplay of spin, orbital and lattice degrees of freedom at the onset of magnetism, nematicity and superconductivity. [@Sefat:2011; @Wang:2015; @Gallais:2016a; @YiM:2017; @Bohmer:2018] Actually, phonons may play a decisive role for probing subtle changes of the electronic and magnetic properties. For instance, soon after the discovery of Fe-based superconductors the magnetic moment was predicted to couple to the As position. [@Yildirim:2009a] Zbiri *et al.* found a modulation of the electronic density of states at the Fermi energy $E_\mathrm{F}$ by the two [$E_{g}$]{}and the [$A_{1g}$]{}modes. [@Zbiri:2009i] Various anomalies were observed experimentally using neutron, Raman and optical spectroscopy, [@Chauviere:2009; @Chauviere:2011; @Rahlenbeck:2009; @Kumar:2010; @Mittal:2009; @Gnezdilov:2013; @Gnezdilov:2011; @Akrap:2009] but are not fully understood yet. One particular effect is the observation of substantial Raman scattering intensity of the As phonon below the magneto-structural transition in crossed polarizations with the electric fields oriented along the axes of the pseudo-tetragonal 2Fe unit cell [@Chauviere:2011] \[For the definition of the axes see Fig. \[fig:orth\_phase\](a)\]. García-Martínez *et al.* argued that magnetism sufficiently modifies the low-energy electronic structure to explain this anomalous intensity.[@Garcia:2013] Recent experiments seem to support this view [@Wu:2017dec] upon comparing spectra obtained with parallel and crossed polarizations in twinned samples with the incident field oriented along the $a$ and the scattered field either along the $a$ or $b$ axis, respectively. Yet, to which extent the phonons are affected by correlations and magnetic-ordering-induced changes in the electronic structure at energies in the range of the photon energies is still unclear. In this work we address this issue both experimentally and theoretically and investigate how magnetism and the combination of moderately correlated Fe $d$ states and uncorrelated As $p$ states affect such complex spectroscopic properties as, for instance, resonant Raman scattering. In particular, we try to clarify whether the observed anomalous intensity of the As mode is a low- or a high-energy phenomenon and aim at identifying the driving force behind the ordering instabilities. In our study we find that very good agreement between experimental observations and density functional theory (DFT) calculations can be achieved in both the paramagnetic and the antiferromagnetic state of if two physically motivated modifications are being made to the standard DFT electronic bands. On the one hand, we need to account for the fact that the high-temperature tetragonal phase is paramagnetically disordered, and cannot be simulated by calculations with suppressed local magnetism.[@Mazin:2008a] Besides, it appears necessary not only to introduce an antiferromagnetic order in the calculations, but also to account for strong correlations. The latter is achieved by separating the energy bands into two regions, a high-energy region with predominantly As states and a low-energy region with predominantly Fe states. The Fe states are then appropriately renormalized. With these two assumptions we can reproduce (i) the positions of the Raman active phonons and their splitting and evolution in the (mechanically detwinned) orthorhombic antiferromagnetic state and (ii) Raman intensities, including the $\tilde{a}-\tilde{b}$ anisotropy as well as the complex resonant evolution with the laser light frequency. This agreement gives an experimental justification to the proposed computational procedure and convincingly substantiates the physical concepts it was derived from, namely the pivotal role of local moments in the lattice dynamics of Fe-based superconductors, and the importance of band renormalizations for $d$-electrons. ![(Color online) FeAs layer of and detwinning clamp. (a) The As-atoms (grey) in the center and at the edges are below and, respectively, above the Fe plane (red). For this reason, the 2Fe unit cell with the axes $a$ and $b$ (green) is determined by the As atoms. In the orthorhombic phase the Fe-Fe distances become inequivalent with the distortion strongly exaggerated here. The magnetic unit cell is twice as large as the 2Fe unit cell and has the axes $\tilde{a}$ and $\tilde{b}$. (b) Schematic sketch and (c) photograph of the detwinning clamp. The sample (4) is glued on the copper plate (1) which is in good thermal contact with the sample holder (3). Upon tightening the screws (5) the force exerted by the copper-beryllium cantilever (2) can be adjusted. (d) Schematic representation of the geometry of our Raman scattering experiment. All incoming light polarizations which are not parallel to $y$ have finite projections on the $c$ axis (red arrow).[]{data-label="fig:orth_phase"}](./Baum_fig1.pdf){width="8.5cm"} Methods ======= Samples {#sec:samples} ------- The crystal was prepared using a self-flux technique. Details of the crystal growth and characterization are described elsewhere.[@Chu:2009] is a parent compound of double-layer iron-based superconductors and orders in a stripe-like spin-density-wave (SDW) below ${\ensuremath{T_\mathrm{SDW}}\xspace}\approx \mathrm{135\,K}$. Superconductivity can be obtained by substituting any of the ions or by pressure.[@Kimber:2009i] In ($0<x\lesssim 0.06$) the SDW is preceded by a structural phase transition from a tetragonal ($I4/mmm$) to an orthorhombic ($Fmmm$) lattice at ${\ensuremath{T_\mathrm{s}}\xspace}> {\ensuremath{T_\mathrm{SDW}}\xspace}$.[@Chu:2009] It remains a matter of debate as to whether or not [$T_\mathrm{SDW}$]{}and [$T_\mathrm{s}$]{}coincide in .[@Chu:2009; @Kim:2011b] Fig. \[fig:orth\_phase\](a) shows the relation of the various axes. The axes of the tetragonal crystal ($T > T_{\mathrm{s}}$, green lines) are denoted $a$ and $b$ with $a = b$. The axes of the magnetically ordered structure (4Fe per unit cell, black lines), $\tilde{a}$ and $\tilde{b}$, differ by approximately 0.7% below [$T_\mathrm{SDW}$]{}[@Rotter:2008] and the Fe-Fe distance along the $\tilde{b}$ axis becomes shorter than along the $\tilde{a}$ axis as sketched in Figure \[fig:orth\_phase\](a). As a result, the angle between $a$ and $b$ differs from 90$^{\circ}$ by approximately 0.4$^{\circ}$. Below [$T_\mathrm{SDW}$]{}the spins order ferromagnetically along $\tilde{b}$ and antiferromagnetically along $\tilde{a}$. Due to the small difference between $\tilde{a}$ and $\tilde{b}$ the crystals are twinned below [$T_\mathrm{s}$]{}, and the orthogonal $\tilde{a}$ and $\tilde{b}$ axes change roles at twin boundaries running along the directions of the tetragonal $a$ and $b$ axes. The orthorhombic distortion makes the proper definition of the axes important as has been shown for twin-free crystals by longitudinal and optical transport as well as by ARPES.[@Chu:2010; @Ying:2011; @Dusza:2011; @Dusza:2012; @Nakajima:2011; @Yi:2011] In order to obtain a single-domain orthorhombic crystal we constructed a sample holder for applying uniaxial pressure parallel to the Fe-Fe direction. Detwinning clamp {#sec:clamp} ---------------- The detwinning clamp is similar to that used by Chu *et al.*[@Chu:2010] Fig. \[fig:orth\_phase\](b) and (c) show, respectively, a schematic drawing and a photograph of the clamp. The sample is attached to a thermally sinked copper block (1) with GE varnish, which remains sufficiently elastic at low temperatures and maintains good thermal contact between the holder (3) and the sample (4). The stress is applied using a copper-beryllium cantilever (2) which presses the sample against the body of the clamp. Upon tightening the screws (5) the force on the sample can be adjusted. In our experiment, the pressure is applied along the Fe-Fe bonds. The $c$ axis of the sample is perpendicular to the force and parallel to the optical axis. The uniaxial pressure can be estimated from the rate of change of the tetragonal-to-orthorhombic phase transition at $T_{\mathrm{s}}$. Using the experimentally derived rate of per [@Liang:2011; @Blomberg:2012] we find approximately for our experiment to be sufficient to detwin the sample. Light scattering {#sec:ls} ---------------- The experiment was performed with a standard light scattering setup. We used two ion lasers (Ar$^{+}$ Coherent Innova 304C and Kr$^{+}$ Coherent Innova 400) and two diode pumped solid state lasers (Coherent Genesis MX SLM, Laser Quantum Ignis) providing a total of 14 lines ranging from to , corresponding to incident energies $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}$ between 3.1 and 1.8eV. Due to this wide range the raw data have to be corrected. The quantity of interest is the response function $R\chi^{\prime\prime}(\Omega)$ where $\Omega={\ensuremath{\omega_\mathrm{I}}\xspace}-{\ensuremath{\omega_\mathrm{S}}\xspace}$ is the Raman shift, [$\omega_\mathrm{S}$]{}is the energy of the scattered photons and $R$ is an experimental constant. Details of the calibration are described in Appendix \[asec:dataevaluation\]. Application of the Raman selection rules requires well-defined polarizations for the exciting and scattered photons. The polarizations are given in Porto notation with the first and the second symbol indicating the directions of the incoming and scattered photons’ electric fields and , respectively. We use $xyz$ for the laboratory system \[see Fig. \[fig:orth\_phase\](d)\]. The $xz$ plane is vertical and defines the plane of incidence, $yz$ is horizontal, $xy$ is the sample surface, and the $z$ axis is parallel to the optical axis and to the crystallographic $c$ axis. For the sample orientation used here (see Fig. \[fig:orth\_phase\]) the Fe-Fe bonds are parallel to $x$ and $y$, specifically $\tilde{a} = (1,0,0) \parallel x$ and $\tilde{b} = (0,1,0) \parallel y$. Since the orthorhombicity below [$T_\mathrm{s}$]{}is small the angle between $a$ and $\tilde{a}$ deviates only by 0.2$^{\circ}$ from 45$^{\circ}$. It is therefore an excellent approximation to use $a \parallel x^{\prime}=1/\sqrt{2}(x+y) \equiv 1/\sqrt{2}(1,1,0)$ and $b \parallel y^{\prime}=1/\sqrt{2}(y-x) \equiv 1/\sqrt{2}(1,\bar{1},0)$. As the angle of incidence of the exciting photons is as large as 66$^{\circ}$ in our setup \[see Fig. \[fig:orth\_phase\](d)\] the orientations of parallel and perpendicular to the $xz$ plane are inequivalent. In particular, has a projection on the $c$ axis for ${\mbox{{\bf e}$_\mathrm{I}$}\xspace}\parallel xz$. This effect was used before[@Chauviere:2009] and allows one to project out the [$E_{g}$]{}phonons in the $x^{\ast}x$ and $x^{\ast}y$ configurations, where $x^{\ast} \parallel (x+\alpha z)$ inside the crystal \[see Fig. \[fig:orth\_phase\](d)\]. For the index of refraction is $n^{\prime} = 2.2+2.1i$ at 514nm resulting in $\alpha \approx 0.4$ for an angle of incidence of 66$^{\circ}$. The corresponding intensity contribution is then 0.16. As a consequence, $x^{\ast}x$ and $yy$ are inequivalent whereas ${\mbox{{\bf e}$_\mathrm{I}$}\xspace}= x^{\prime\ast} \parallel (x^{\prime} + \alpha\,z/\sqrt{2})$ and ${\mbox{{\bf e}$_\mathrm{I}$}\xspace}= y^{\prime\ast} \parallel (y^{\prime} + \alpha\,z/\sqrt{2})$ are equivalent for having the same projection on the $c$ direction. Upon comparing $x^{\ast}y$ and $yx$ the leakage of the $c$-axis polarized contributions to the electronic continuum can be tested. In the case here, they are below the experimental sensitivity. The effect of the finite angle of acceptance of the collection optics ($\pm 15^{\circ}$ corresponding to a solid angle $\tilde{\Omega}$ of 0.21sr) on the projections of the scattered photons can be neglected. ![(Color online) Raman-active phonons in with the symmetry assignments in the tetragonal crystallographic unit cell $abc$.[]{data-label="fig:mode_tetra"}](./Baum_fig2.pdf){width="8.5cm"} Theoretical Calculations {#sec:theory} ------------------------ The phonon eigenvectors $Q^{(\nu)}$ (displacement patterns of the vibrating atoms in branch $\nu$) and the energies of all Raman-active phonons of in the tetragonal ($I4/mmm$) and the orthorhombic ($Fmmm$) phases were obtained from *ab initio* DFT calculations within the Perdew-Burke-Ernzerhof parameterization [@Perdew:1996] of the generalized gradient approximation. The phonon frequencies were calculated by diagonalizing the dynamical matrices using the *phonopy* package. [@Togo:2008; @Togo:2015] The dynamical matrices were constructed from force constants determined via the finite displacement method in 2 $\times$ 2 $\times$ 1 supercells. [@Parlinski:1997] As a basis for the calculations we used the projector augmented wave approximation, [@Bloechl:1994] as implemented in the Vienna package (VASP). [@Kresse:1993; @Kresse:1996; @Kresse:1996b] The Brillouin zone for one unit cell was sampled with a $10 \times 10 \times 10$ **k** point mesh, and the plane wave cutoff was set at 520eV. For the tetragonal phase, we used a Néel-type magnetic order to relax the structure and to obtain the experimental lattice parameters. [^1] For the orthorhombic phase, we used the stripe-like magnetic order shown in Fig. \[fig:orth\_phase\](a). In addition, we studied the resonant phonon-photon interaction by exploring the dielectric tensor $\hat{\varepsilon}$. The latter was determined using the Optics code package [@Ambrosch-Draxl:2006] implemented in WIEN2k (Ref. ) with the full-potential linearized augmented plane-wave (LAPW) basis. The Perdew-Burke-Ernzerhof generalized gradient approximation [@Perdew:1996] was employed as the exchange correlation functional and the basis-size controlling parameter RK$_{\mathrm{max}}$ was set to 8.5. A mesh of 400 **k** points in the first Brillouin zone for the self-consistency cycle was used. The density of states (DOS) and dielectric tensors were computed using a $10\times10\times10$ **k** mesh. For the dielectric tensor a Lorentzian broadening of 0.1eV was introduced. The (generally complex) Raman tensor ${\alpha}_{jk}^{(\nu)}({\ensuremath{\omega_\mathrm{I}}\xspace}) = \alpha^{(\nu)\prime}_{jk}({\ensuremath{\omega_\mathrm{I}}\xspace}) + i\alpha^{(\nu)\prime\prime}_{jk}({\ensuremath{\omega_\mathrm{I}}\xspace})$ is determined by the derivative of the dielectric tensor elements ${\varepsilon}_{jk}({\ensuremath{\omega_\mathrm{I}}\xspace})=\varepsilon^\prime_{jk}({\ensuremath{\omega_\mathrm{I}}\xspace}) + i\varepsilon^{\prime\prime}_{jk}({\ensuremath{\omega_\mathrm{I}}\xspace})$ with respect to the normal coordinate of the respective phonon, $Q^{(\nu)}$. Since we are interested only in the resonance behavior of the As phonon, we are only concerned with the derivative with respect to $Q^{\mathrm{(As)}}$, $$\label{eq:de_dQ} {\alpha}_{ll}^{\mathrm{(As)}}({\ensuremath{\omega_\mathrm{I}}\xspace}) = \frac{\partial\varepsilon^\prime_{ll}({\ensuremath{\omega_\mathrm{I}}\xspace})}{\partial Q^{\mathrm{(As)}}} + i\frac{\partial\varepsilon^{\prime\prime}_{ll}({\ensuremath{\omega_\mathrm{I}}\xspace})}{\partial Q^{\mathrm{(As)}}}.$$ Results and Discussion {#sec:results} ====================== -------------- ------------- -------- --------------------------------- -------------------------------------- ------------ -------- Exp. (140K) Theory Exp. (60K) Theory [$A_{1g}$]{} 180 168 $\xrightarrow{\makebox[5mm]{}}$ [$A_{g}$]{} 180 172 [$B_{1g}$]{} 215 218 $\xrightarrow{\makebox[5mm]{}}$ [$B_{1g}$]{} 215 221 ${\ensuremath{B_{2g}}\xspace}^{(1)}$ 125 110 ${\ensuremath{B_{3g}}\xspace}^{(1)}$ 135 133 ${\ensuremath{B_{2g}}\xspace}^{(2)}$ 270 272 ${\ensuremath{B_{3g}}\xspace}^{(2)}$ 273 287 -------------- ------------- -------- --------------------------------- -------------------------------------- ------------ -------- : Raman-active phonons in . The experimental and theoretically determined energies are given in cm$^{-1}$. In addition, the symmetry correlations between the tetragonal ($I4/mmm$) and orthorhombic ($Fmmm$) structures are shown.[]{data-label="table:phonon"} Lattice dynamics {#sec:phononcalc} ---------------- The energies and symmetries as obtained from lattice dynamical calculations for tetragonal and orthorhombic are compiled in Table \[table:phonon\]. The four modes in tetragonal $I4/mmm$ symmetry obey [$A_{1g}$]{}+ [$B_{1g}$]{}+ 2[$E_{g}$]{}selection rules. The eigenvectors are depicted in Fig. \[fig:mode\_tetra\]. In the orthorhombic $Fmmm$ phase, the two [$E_{g}$]{}modes are expected to split into [$B_{2g}$]{}and [$B_{3g}$]{}modes. Thus, there are six non-degenerate modes in the orthorhombic phase, [$A_{g}$]{}+ [$B_{1g}$]{}+ 2[$B_{2g}$]{}+ 2[$B_{3g}$]{}. Table \[table:phonon\] shows the symmetry relations between the tetragonal and orthorhombic phonons. Since the [$A_{g}$]{}and [$B_{1g}$]{}eigenvectors remain unchanged upon entering the orthorhombic phase, only those of the [$B_{2g}$]{}and [$B_{3g}$]{}phonons are shown in Fig. \[fig:mode\_orthor\]. For the [$B_{2g}$]{}and [$B_{3g}$]{}phonons the As and Fe atoms move perpendicular to the $c$ axis and perpendicular to each other. The calculated phonon vibrations agree with previous results for , [@Zbiri:2009i] however our energies differ slightly from those reported by Zbiri *et al.* [@Zbiri:2009i] In particular, we find a splitting between the ${\ensuremath{B_{2g}}\xspace}^{(1)}$ and ${\ensuremath{B_{3g}}\xspace}^{(1)}$ phonons. - ![(Color online) [$B_{2g}$]{}and [$B_{3g}$]{}phonon modes in with the symmetry assignments in the orthorhombic crystallographic unit cell $\tilde{a}\tilde{b}c$. []{data-label="fig:mode_orthor"}](./Baum_fig3 "fig:"){width="8.5cm"} Eg phonons {#sec:Egphonon} ---------- ![(Color online) Phonons in detwinned . The spectra at 60K (red and blue) are displayed with the experimental intensity. The spectrum at 140K (black) is downshifted by 1.4countss$^{-1}$mW$^{-1}$ for clarity. Each of the two tetragonal [$E_{g}$]{}phonons (vertical dashed lines) splits into two lines below [$T_\mathrm{s}$]{}. The ${\ensuremath{B_{2g}}\xspace}^{(1)}$ and ${\ensuremath{B_{3g}}\xspace}^{(1)}$ lines appear at distinct positions for polarizations of the scattered light parallel (blue) and perpendicular (red) to the applied pressure as indicated in the insets. The ${\ensuremath{B_{2g}}\xspace}^{(2)}$ and ${\ensuremath{B_{3g}}\xspace}^{(3)}$ phonons are shifted only slightly upwards with respect to the ${\ensuremath{E_{g}}\xspace}^{(2)}$ mode. Violet and orange arrows indicate the polarizations of the incident and scattered photons, respectively. The black triangles indicate the direction of the applied pressure. The shorter $\tilde{b}$ axis is parallel to the stress.[]{data-label="fig:Eg_Phonon_Split"}](./Baum_fig4){width="8.5cm"} Table \[table:phonon\] displays the experimental phonon energies as measured above and below the magneto-structural transition along with the theoretical values. The ${\ensuremath{E_{g}}\xspace}^{(1)}$ phonon found at 130cm$^{-1}$ above [$T_\mathrm{s}$]{}splits into two well separated lines as predicted (Table \[table:phonon\]) and shown in Fig. \[fig:Eg\_Phonon\_Split\]. The splitting of the ${\ensuremath{E_{g}}\xspace}^{(2)}$ mode at 268cm$^{-1}$ is small, and the ${\ensuremath{B_{2g}}\xspace}^{(2)}$ and ${\ensuremath{B_{3g}}\xspace}^{(2)}$ modes are shifted to higher energies by 2[$\mathrm{cm}^{-1}$]{}, and 5cm$^{-1}$, respectively. The theoretical and experimental phonon energies are in agreement to within 14% for both crystal symmetries. The splitting between the [$B_{2g}$]{}and [$B_{3g}$]{}modes is overestimated in the calculations. Previous experiments were performed on twinned crystals, [@Chauviere:2009; @RenX:2015; @ZhangWL:2016] and the [$B_{2g}$]{}and [$B_{3g}$]{}modes were observed next to each other in a single spectrum. An equivalent result can be obtained in de-twinned samples by using $x^{\prime \ast}y^{\prime}$, $x^{\prime \ast}x^{\prime}$ or $RR$ polarizations where the $x$ and $y$ axes are simultaneously projected (along with the $z$ axis). In neither case the symmetry of the [$B_{2g}$]{}and [$B_{3g}$]{}phonons can be pinned down. Only in a de-twinned sample where the $xz$ and $yz$ configurations are projected separately the [$B_{2g}$]{}and [$B_{3g}$]{}modes can be accessed independently. Uniaxial pressure along the Fe-Fe direction, as shown by the black arrows in the insets of Fig. \[fig:Eg\_Phonon\_Split\], determines the orientation of the shorter $\tilde{b}$ axis. This configuration enables us to observe the ${\ensuremath{B_{2g}}\xspace}^{(1)}$ mode at 125cm$^{-1}$ and the ${\ensuremath{B_{3g}}\xspace}^{(1)}$ mode at 135cm$^{-1}$ in $x^{\ast}x$ and, respectively, $x^{\ast}y$ polarization configurations thus augmenting earlier work. With the shorter axis determined by the direction of the stress (insets of Fig. \[fig:Eg\_Phonon\_Split\]) the assignment of the [$B_{2g}$]{}and [$B_{3g}$]{}modes is unambiguous. Since the $x^{\ast}x$ spectrum (red) comprises $\tilde{a}\tilde{a}$ and $c\tilde{a}$ polarizations both the [$A_{g}$]{}and the [$B_{2g}$]{}phonons appear. The $x^{\ast}y$ spectrum (blue) includes the [$B_{1g}$]{}($\tilde{a}\tilde{b}$) and [$B_{3g}$]{}($c\tilde{b}$) symmetries. The calculated splitting between the [$B_{2g}$]{}and [$B_{3g}$]{}modes is smaller for the ${\ensuremath{E_{g}}\xspace}^{(2)}$ than for the ${\ensuremath{E_{g}}\xspace}^{(1)}$ mode, qualitatively agreeing with the experiment. However, in the calculations this difference is entirely due to the different reduced masses for these modes since the ${\ensuremath{E_{g}}\xspace}^{(1)}$ and ${\ensuremath{E_{g}}\xspace}^{(2)}$ phonon are dominated by As and Fe motions, respectively. In the experiment the splitting for the ${\ensuremath{E_{g}}\xspace}^{(2)}$ mode is close to the spectral resolution, indicating an additional reduction of the splitting below that obtained in the calculation. The source of this additional reduction is unclear at the moment. As phonon intensity {#sec:Agphonon} ------------------- ![(Color online) Raman spectra of twinned at temperatures as indicated. (a) In parallel $RR$ polarization configuration (see inset) the As phonon appears at all temperatures. (b) For crossed light polarizations ($ab$) the As phonon is present only below the magneto-structural transition at ${\ensuremath{T_\mathrm{s}}\xspace}= 135$K as reported before. [@Chauviere:2009] Asterisks mark the [$E_{g}$]{}modes discussed in Sec. \[sec:Egphonon\].[]{data-label="fig:Ag_Phonon_in_B1g"}](./Baum_fig5.pdf){width="8.5cm"} Fig. \[fig:Ag\_Phonon\_in\_B1g\] shows low-energy spectra of twinned for (a) $RR$ and (b) $ab$ polarization configurations at 310 (orange), 150 (green), and 60K (blue). The As phonon at 180[$\mathrm{cm}^{-1}$]{}is the strongest line in the $RR$ spectra at all temperatures as expected and gains intensity upon cooling. In $ab$ polarizations there is no contribution from the As mode above [$T_\mathrm{s}$]{}. Below [$T_\mathrm{s}$]{}(blue spectrum) the As phonon assumes a similar intensity as in the $RR$ polarization as reported earlier. [@Chauviere:2009; @Wu:2017dec] Due to a finite projection of the incident light polarizations onto the $c$ axis \[see Fig. \[fig:orth\_phase\](d)\] in both $RR$ and $ab$ configurations the [$E_{g}$]{}phonons appear in all spectra (asterisks). The electronic background has been extensively discussed in previous works [@Choi:2008; @Chauviere:2010; @Sugai:2012; @Kretzschmar:2016; @Thorsmolle:2016] and is not a subject of the study here. In order to understand the appearance of the As line in the crossed $ab$ polarizations it is sufficient to consider the in-plane components of the [$A_{g}$]{}Raman tensor, $$\label{eq:tensor-Ag} \hat{\alpha}^{(Ag)} = \left( \begin{array}{cc} {\alpha}_{11} & 0 \\ 0 & {\alpha}_{22}\end{array} \right).$$ The response of this phonon for the polarization configuration (,) is given by $\chi^{\prime\prime \mathrm{(As)}}_{IS} \propto \left| \mathbf{e}_{\mathrm{S}}^{\ast} \cdot \hat{\alpha}^{(Ag)} \cdot {\mbox{{\bf e}$_\mathrm{I}$}\xspace}\right|^2$ (where $^\ast$ means conjugate transposed). In the tetragonal ([$A_{1g}$]{}) case the two elements are equal, ${\alpha}_{11} = {\alpha}_{22}$, and the phonon appears only for ${\mbox{{\bf e}$_\mathrm{S}$}\xspace}\! \parallel\! {\mbox{{\bf e}$_\mathrm{I}$}\xspace}$. In the orthorhombic phase the tensor elements are different, and one can expect the phonon to appear for ${\mbox{{\bf e}$_\mathrm{S}$}\xspace}\perp {\mbox{{\bf e}$_\mathrm{I}$}\xspace}$ since the intensity then depends on the difference between $\alpha_{11}$ and $\alpha_{22}$. In detwinned samples $\alpha_{11}$ and $\alpha_{22}$ can be accessed independently by using parallel polarizations for the incident and scattered light oriented along either the $\tilde{a}$ or the $\tilde{b}$ axis. In addition, putative imaginary parts of $\alpha_{ii}$ may be detected by analyzing more than two polarization combinations as discussed in Appendix \[asec:complex\_tensor\]. Spectra for $\tilde{a}\tilde{a}$ and $\tilde{b}\tilde{b}$ configurations are shown in Figure \[Afig:Ag\_all\] of Appendix \[asec:rawdata\]. ![(Color online) Spectral weight $A_{IS}^{\mathrm{(As)}} ({\ensuremath{\omega_\mathrm{I}}\xspace})$ of the As phonon as a function of excitation energy and polarization. The top axis shows the corresponding wavelength of the exciting photons. (a) Experimental data. The intensity for parallel light polarizations along the ferromagnetic axis ($\tilde{b}\tilde{b}$, blue squares) is virtually constant for $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}< 2.7$eV and increases rapidly for $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}> 2.7$eV. For light polarizations along the antiferromagnetic axis ($\tilde{a}\tilde{a}$, red squares), the phonon intensity increases monotonically over the entire range studied. The solid lines are Lorentzian functions whose extrapolations beyond the measured energy interval are shown as dashed lines. (b) Theoretical prediction of $A^{\mathrm{(As)}}_{\tilde{a}\tilde{a}}$ (red) and $A^{\mathrm{(As)}}_{\tilde{b}\tilde{b}}$ (blue). The curves qualitatively reproduce the experimental data shown in panel (a).[]{data-label="fig:Ag_resonance_energy"}](./Baum_fig6){width="8.5cm"} We proceed now with the analysis of the phonon spectral weight $A_{IS}^{\mathrm{(As)}}({\ensuremath{\omega_\mathrm{I}}\xspace})$ as a function of the incident photon excitation energy ($\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}$) and polarization. Fig. \[fig:Ag\_resonance\_energy\](a) shows $A_{IS}^{\mathrm{(As)}}({\ensuremath{\omega_\mathrm{I}}\xspace})$ as derived by fitting the peak with a Voigt function, after subtracting a linear background. Measurements were repeated several times in order to check the reproducibility. The variation of the spectral weight between different measurements can be taken as an estimate of the experimental error. For light polarizations parallel to the antiferromagnetic $\tilde{a}$ axis, $A^{\mathrm{(As)}}_{\tilde{a}\tilde{a}}({\ensuremath{\omega_\mathrm{I}}\xspace})$ (red squares) increases continuously with increasing $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}$ whereas $A^{\mathrm{(As)}}_{\tilde{b}\tilde{b}}$ (blue squares) stays virtually constant for incident photons in the red and green spectral range, $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}< 2.7 \,\mathrm{eV}$, and increases rapidly for $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}> 2.7 \,\mathrm{eV}$. For all wavelengths the spectral weight is higher for the $\tilde{a}\tilde{a}$ than for the $\tilde{b}\tilde{b}$ configuration. The variations of $A^{\mathrm{(As)}}_{\tilde{a}\tilde{a}}({\ensuremath{\omega_\mathrm{I}}\xspace})$ and $A^{\mathrm{(As)}}_{\tilde{b}\tilde{b}}({\ensuremath{\omega_\mathrm{I}}\xspace})$ display a typical resonance behavior, [@Cardona1982_ResonancePhenomena] which is expected when the intermediate state of the Raman scattering process is an eigenstate of the electronic system. Then in second order perturbation theory the intensity diverges as $|\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}-E_0|^{-2}$ where $E_0$ is the energy difference between an occupied and an unoccupied electronic Bloch state. In real systems having a finite electronic lifetime a Lorentzian profile is expected. We therefore approximated $A_{IS}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ with Lorentzians centered at $E_{0,IS}$ as shown by solid lines in Fig. \[fig:Ag\_resonance\_energy\](a). From these model functions we determine $E_{0,\tilde{a}\tilde{a}} = 3.1\,\mathrm{eV}$ and $E_{0,\tilde{b}\tilde{b}} = 3.3\,\mathrm{eV}$. In order to compare the experimental observations with theoretical calculations of the phonon spectral weights the band structure needs to be renormalized so as to account for correlation effects (for details see Appendix \[asec:theory\]). Specifically, we differentiate three regions: (i) the unoccupied Fe $3d$ bands near the Fermi energy that we renormalize via a rescaling factor, (ii) the occupied bands below -2.7eV of predominantly As $4p$ character that remain unchanged and (iii) the occupied bands between -2.7eV and the Fermi level derived from hybridized Fe $3d$ and As $4p$ orbitals. Due to this hybridization, the renormalization of the latter bands cannot be performed by simple rescaling. One can anticipate that the optical absorption would set in at energies below 1.8eV, smaller than our minimal laser energy, if the occupied Fe bands would have been renormalized prior to hybridization with the As bands. Due to the small density of states of the As bands in the range from -2.7eV to $E_\mathrm{F}$ their contribution to the dielectric function would be small. With this in mind, we simply excluded all occupied bands in this range from the calculations. The effect of these bands, although small, could be accounted for using the DMFT method, which, however, is beyond the scope of our present work. We then determine the dielectric tensor and the Raman tensor (section \[sec:theory\]) on the basis of this renormalized band structure for the $(\pi,0)$ ordered state. Since our resonances lie in the range $\hbar {\ensuremath{\omega_\mathrm{I}}\xspace}>2.7$eV our calculations can capture the intensities and the $\tilde{a}-\tilde{b}$ anisotropy in this range of energies rather well as can be seen in Fig. \[fig:Ag\_resonance\_energy\](b). With this analysis, we interpret our joint experimental and theoretical results as evidence that resonance effects are the main source of the anomalous intensity of the As phonon in crossed polarizations. The main experimental argument is based on the anisotropic variation of the phonon intensities with $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}$ in $\tilde{a}\tilde{a}$ and $\tilde{b}\tilde{b}$ polarization configurations, while the theoretical derivation of the tensor elements demonstrates the importance of magnetic order and correlation effects for the band reconstruction. As proposed previously,[@Garcia:2013] magnetism appears to be the origin of the anisotropy. However, the intensity anisotropy cannot be explained without taking into account the high-energy electronic states. Finally, we briefly looked into the effect of doping on the anomaly and found further support for its magnetic origin. In the transition temperature [$T_\mathrm{SDW}$]{}is several degrees below [$T_\mathrm{s}$]{}for finite $x$, and one observes that the anomaly of the As phonon does not commence at [$T_\mathrm{s}$]{}, but rather at the magnetic transition. For $x=0.025$ the phonon assumes intensity in crossed polarizations only below [$T_\mathrm{SDW}$]{}(see supplementary information of Ref. ). For $x=0.051$ the anomaly starts to appear at [$T_\mathrm{s}$]{}, as displayed in Fig. \[Afig:BFCA\] in the Appendix, but the spectral weight does not show an order-parameter-like temperature dependence. The increase is nearly linear and saturates below [$T_\mathrm{SDW}$]{}at a value which is smaller by a factor of approximately 7 than that in $RR$ polarization projecting [$A_{g}$]{}/[$A_{1g}$]{}symmetry. In FeSe, with a structural transition at ${\ensuremath{T_\mathrm{s}}\xspace}=89.1$K but no long range magnetism, [@Baek:2014] the anomalous intensity can also be observed below [$T_\mathrm{s}$]{}but the intensity relative to that in the [$A_{g}$]{}projection is only 1%, as shown in Fig. \[Afig:FeSe\]. Similar to Ba(Fe$_{0.949}$Co$_{0.051}$)$_2$As$_2$, the spectral weight increases approximately linearly but does not saturate, presumably because FeSe does not develop long-ranged magnetic order. Conclusion {#sec:conclusion} ========== We studied the two [$E_{g}$]{}phonons and the fully symmetric As vibration in twin-free by Raman scattering. The tetragonal [$E_{g}$]{}phonon at 130[$\mathrm{cm}^{-1}$]{}(${\ensuremath{E_{g}}\xspace}^{(1)}$) splits into two modes in the orthorhombic phase. The detwinning allows us to identify the modes at 125[$\mathrm{cm}^{-1}$]{}and 135[$\mathrm{cm}^{-1}$]{}as [$B_{2g}$]{}and [$B_{3g}$]{}phonons, respectively. DFT calculations predict the symmetries correctly and show that the splitting occurs because of the stripe magnetic order. The As [$A_{g}$]{}phonon was studied for various laser lines in the range 1.8 to 3.1eV. In the ordered phase the spectral weight of the phonon resonates for an excitation energy of ($3.2 \pm 0.1$)eV. The resonance energy is almost the same for the light polarized along the ferro- or antiferromagnetic directions $\tilde{b}$ and $\tilde{a}$ \[for the definition of the axes see Fig. \[fig:orth\_phase\](a)\], whereas the variation of the spectral weight with the energy of the incident photon is rather different for the $\tilde{b}\tilde{b}$ and $\tilde{a}\tilde{a}$ configurations. We find that our DFT calculations reproduce the anisotropy and the resonance very well for energies above 2.7eV if we include both the effects of the magnetism and of the strong correlations in the Fe $3d$ orbitals, responsible for the band renormalization. For energies below 2.7eV our approximation is only semi-quantitative, because the occupied Fe $3d$ bands are strongly hybridized with the As bands and cannot be renormalized by simple rescaling. As in the case of the [$E_{g}$]{}phonons, all effects are strongly linked to magnetism. However, in the case of the As phonon the inclusion of electronic states at high energies is essential because of the resonance behavior. Therefore, low-energy physics with magnetism-induced anisotropic electron-phonon coupling [@Garcia:2013] is probably insufficient for explaining the anomalous intensity in crossed polarizations. Acknowledgement {#acknowledgement .unnumbered} =============== We gratefully acknowledge discussions with L. Degiorgi and thank him for providing us with raw and analyzed IR data of . The work was supported by the German Research Foundation (DFG) via the Priority Program SPP1458, the Transregional Collaborative Research Centers TRR80, TRR49 and by the Serbian Ministry of Education, Science and Technological Development under Project III45018. We acknowledge support by the DAAD through the bilateral project between Serbia and Germany (grant numbers 56267076 and 57142964). The collaboration with Stanford University was supported by the Bavaria California Technology Center BaCaTeC (grant-no. A5\[2012-2\]). Work in the SIMES at Stanford University and SLAC was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Contract No. DE-AC02-76SF00515. Y.L. and R.V. acknowledge the allotment of computer time at the Centre for Scientific Computing (CSC) in Frankfurt. I.I.M. was supported by ONR through the NRL basic research program and by the Alexander von Humboldt foundation. \[sec:appendix\] Calibration of the sensitivity {#asec:dataevaluation} ============================== Scattering experiments performed over a wide energy range necessitate an appropriate correction of the data. The quantity of interest is the response function $R\chi^{\prime\prime}_{IS}(\Omega)$ where $\Omega={\ensuremath{\omega_\mathrm{I}}\xspace}-{\ensuremath{\omega_\mathrm{S}}\xspace}$ is the Raman shift, and [$\omega_\mathrm{S}$]{}is the energy of the scattered photons. $R$ includes all experimental constants and units in a way that $R\chi^{\prime\prime}_{IS}(\Omega)$ is as close as possible to the count rate $\dot{N}^\ast_{I,S}$, measured for a given laser power $P_\mathrm{I}=I_\mathrm{I}\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}$ absorbed by the sample. $I_\mathrm{I}$ is the number of incoming photons per unit time and $I,S$ refer to both photon energies and polarizations. With $A_\mathrm{f}$ the (nearly) energy-independent area of the laser focus the cross section is given by [@Muschler:2010a] $$\label{eq:cts} \frac{\dot{N}^\ast_{IS}(\Delta{\ensuremath{\omega_\mathrm{S}}\xspace},\Delta\tilde{\Omega})}{P_I} \hbar{\ensuremath{\omega_\mathrm{I}}\xspace}A_\mathrm{f} = R^\ast r({\ensuremath{\omega_\mathrm{S}}\xspace})\frac{d^2\sigma}{d{\ensuremath{\omega_\mathrm{S}}\xspace}d\tilde\Omega} \Delta{\ensuremath{\omega_\mathrm{S}}\xspace}\Delta\tilde{\Omega}.$$ $R^\ast$ and $r({\ensuremath{\omega_\mathrm{S}}\xspace})$ are a constant and the relative sensitivity, respectively. $r({\ensuremath{\omega_\mathrm{S}}\xspace})$ is assumed to be dimensionless and includes energy-dependent factors such as surface losses, penetration depth, and the monochromatic efficiency of the setup. $\Delta{\ensuremath{\omega_\mathrm{S}}\xspace}$ and $\Delta\tilde{\Omega}$ are the bandwidth and the solid angle of acceptance, respectively, and depend both on ${\ensuremath{\omega_\mathrm{S}}\xspace}$. $r({\ensuremath{\omega_\mathrm{S}}\xspace}) \Delta {\ensuremath{\omega_\mathrm{S}}\xspace}\Delta \tilde{\Omega}$ is determined by calibration and used for correcting the raw data. The resulting rate $\dot{N}_{IS}$ is close to $\dot{N}^\ast_{IS}$ in the range $\Omega \le 1,000$cm$^{-1}$ but increasingly different for larger energy transfers mainly for the strong variation of $\Delta{\ensuremath{\omega_\mathrm{S}}\xspace}$. Applying the fluctuation-dissipation theorem, one obtains $$\begin{aligned} \frac{\dot{N}_{IS}}{P_I}\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}A_\mathrm{f} &=& R^\prime\frac{d^2\sigma}{d{\ensuremath{\omega_\mathrm{S}}\xspace}d\tilde\Omega}\nonumber\\ \label{eq:fd} &=& R^\prime\frac{\hbar}{\pi} r_0^2\frac{{\ensuremath{\omega_\mathrm{S}}\xspace}}{{\ensuremath{\omega_\mathrm{I}}\xspace}}\{1+n(\Omega,T)\} \chi^{\prime\prime}(\Omega),\end{aligned}$$ where $R^\prime$ is another constant, which is proportional to $\Delta {\ensuremath{\omega_\mathrm{S}}\xspace}(\omega_0) \Delta \tilde{\Omega}(\omega_0)$, $n(\Omega,T) = [\exp(\frac{\hbar\Omega}{k_BT}) - 1]^{-1}$ is the thermal Bose factor and $r_0$ is the classical electron radius. Finally, after collecting all energy-independent factors in $R$ we obtain $$\label{eq:Rchi} R\chi^{\prime\prime}_{IS}(\Omega) = \frac{\dot{N}_{IS}}{P_\mathrm{I}} \frac{{\ensuremath{\omega_\mathrm{I}}\xspace}^2}{\omega_0{\ensuremath{\omega_\mathrm{S}}\xspace}} \left\{1-\exp\left(-\frac{\hbar\Omega}{k_BT}\right)\right\}.$$ Here, $\omega_0 = 20,000\,\mathrm{cm}^{-1}$ is inserted for convenience to get a correction close to unity. Therefore, the spectra shown reflect the measured number of photon counts per second and mW absorbed power as closely as possible, thus approximately obeying counting statistics as intended. Since the spectra are taken with constant slit width the spectral resolution depends on energy, and narrow structures such as phonons may change their shapes but the spectral weight is energy independent. Ag spectra {#asec:rawdata} ========== Fig. \[Afig:Ag\_all\] shows the complete set of the [$A_{g}$]{}spectra we measured for detwinned . All spectra were corrected as described in Appendix \[asec:dataevaluation\]. For all spectra the same constant width of $550\,\mu$m of the intermediate slit of the spectrometer was used. This results in an energy-dependent resolution varying between approximately 12[$\mathrm{cm}^{-1}$]{}at 24,630[$\mathrm{cm}^{-1}$]{}(3.05eV or 406nm) and 3[$\mathrm{cm}^{-1}$]{}at 14,793[$\mathrm{cm}^{-1}$]{}(1.83eV or 676nm). Accordingly, the width of the peak changes as a function of the excitation wavelength and does not reflect the intrinsic line width of the phonon, in particular not for blue photons. The intensity of the peak monotonically increases towards short wavelengths for the $\tilde{a}\tilde{a}$ spectra (solid lines). For light polarized parallel to the ferromagnetic axis ($\tilde{b}\tilde{b}$, dashed lines) the intensity is low for $\lambda_\mathrm{I} > 450\,\mathrm{nm}$, but strongly increases for $\lambda_\mathrm{I} < 450\,\mathrm{nm}$. The underlying electronic continuum, which is not a subject of this paper, also changes in intensity as a function of the excitation wavelength. From the spectra the spectral weight $A_{IS}^{\rm (As)} ({\ensuremath{\omega_\mathrm{I}}\xspace})$ of the phonon can be derived by fitting a Voigt function to the phonon peak after subtracting a linear background. The width of the Gaussian part of the Voigt function is given by the known resolution of the spectrometer while that of the Lorentzian part reflects the line width of the phonon. ![(Color online) [$A_{g}$]{}spectra of detwinned at various laser wavelenghts $\lambda_\mathrm{I}$. We used laser lines between 406 and 676nm and parallel polarizations of incoming and outgoing photons along the antiferromagnetic ($\tilde{a}\tilde{a}$, solid lines) and the ferromagnetic ($\tilde{b}\tilde{b}$, dashed lines) direction. []{data-label="Afig:Ag_all"}](./Baum_figA1.pdf){width="8.5cm"} Spectral weight for aa and ab polarizations {#asec:complex_tensor} =========================================== . The intensity for crossed ($ab$, purple diamonds) and for $aa$ polarizations (orange dots) is comparable to the intensity found for $\tilde{b}\tilde{b}$ polarization. The purple dashed line is the intensity for $ab$ polarization calculated from the fitted resonance profiles (solid lines) assuming a Raman tensor with real elements. The orange dashed line shows the same calculation for $aa$ polarization. []{data-label="afig:Ag_resonance_all_pol"}](./Baum_figA2){width="8.5cm"} For clarity, Fig. \[fig:Ag\_resonance\_energy\](a) displays only part of the data we collected. We also measured spectra in $aa$ and $ab$ configurations (cf. Figs. \[fig:orth\_phase\](a) and \[afig:Ag\_resonance\_all\_pol\] for the definitions) and find them instructive for various reasons. The $aa$ and $ab$ data (i) can be compared directly with results presented recently [@Wu:2017dec] and (ii) indicate that the Raman tensor has large imaginary parts. Fig. \[afig:Ag\_resonance\_all\_pol\] shows the spectral weights of the As phonon mode for $aa$ and $ab$ polarizations, $A_{aa}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ (orange circles) and $A_{ab}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ (open purple diamonds), respectively, for selected wavelengths together with the data and model functions from Fig. \[fig:Ag\_resonance\_energy\](a) of the main text. Given the experimental error the respective intensities for $aa$ and $ab$ polarizations are rather similar and are also comparable to $A^{\mathrm(As)}_{\tilde{b}\tilde{b}}({\ensuremath{\omega_\mathrm{I}}\xspace})$ (blue squares) in the range $1.9 < \hbar{\ensuremath{\omega_\mathrm{I}}\xspace}\le 2.7$eV. For $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}=3.05$eV (406nm) $A^{\mathrm(As)}_{ab}({\ensuremath{\omega_\mathrm{I}}\xspace})$ is very small, for the yellow-green spectral range $A^{\mathrm(As)}_{ab}({\ensuremath{\omega_\mathrm{I}}\xspace})$ may be even larger than $A^{\mathrm(As)}_{aa}({\ensuremath{\omega_\mathrm{I}}\xspace})$ in qualitative agreement with Ref. . The elements of a real Raman tensor $\hat{\alpha}^{(\rm Ag)}$ (Eq.  of the main text) can be derived directly from the experimental data as ${\alpha}_{11} = \sqrt{A^{\mathrm(As)}_{\tilde{a}\tilde{a}}}$ and ${\alpha}_{22} = \sqrt{A^{\mathrm(As)}_{\tilde{b}\tilde{b}}}$. Then, the phonon’s spectral weight expected for all other polarizations can be calculated right away, and $A^{\mathrm(As)}_{{a}{a}}$ is just the average of $A^{\mathrm(As)}_{\tilde{a}\tilde{a}}$ and $A^{\mathrm(As)}_{\tilde{b}\tilde{b}}$ (dashed orange line in Fig. \[afig:Ag\_resonance\_all\_pol\]). Obviously, there is no agreement with the experimental values for $A^{\mathrm(As)}_{aa}$ (orange circles). $A^{\mathrm(As)}_{ab}$ can be determined in a similar fashion. In Fig. \[afig:Ag\_resonance\_all\_pol\] we show the expected spectral weight as purple dashed line. The dependence on [$\omega_\mathrm{I}$]{}is again derived from the model functions describing the resonance (full red and blue lines). Also for $A^{\mathrm(As)}_{ab}$ the mismatch between experiment (open purple diamonds) and expectation (purple dashed line) is statistically significant, and one has to conclude that the assumption of real tensor elements in the orthorhombic phase is not valid. This effect is not particularly surprising in an absorbing material and was in fact discussed earlier for the cuprates. [@Strach:1998; @Ambrosch:2002] For the Fe-based systems, the possibility of complex Raman tensor elements for the As phonon was not considered yet. Our experimental observations show that the complex nature of $\hat{\alpha}^{\rm (Ag)}$ is crucially important and that the imaginary parts of ${\alpha}_{11}$ and ${\alpha}_{22}$ must have opposite sign to explain the observed enhancement of $A_{ab}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ and the suppression of $A_{aa}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ with respect to the values expected for real tensor elements (dashed orange and purple lines in Fig. \[afig:Ag\_resonance\_all\_pol\]). In summary, the results for $A^{\mathrm(As)}_{aa}({\ensuremath{\omega_\mathrm{I}}\xspace})$ and $A^{\mathrm(As)}_{ab}({\ensuremath{\omega_\mathrm{I}}\xspace})$ support our interpretation that absorption processes are important for the proper interpretation of the Raman data. Currently, we cannot imagine anything else but resonance effects due to interband transitions as the source. Band structure and PDOS {#asec:theory} ======================= ![(Color online) DFT band structure. Bands predominantly from Fe states are shown in brown, bands predominantly from As states in black. The shaded region from -2.7eV to $E_\mathrm{F}$ contains bands of mixed character and is blacked out for the calculation of the dielectric tensor. Only transitions between the bands within the turquois frames are included. []{data-label="Afig:bandstructure"}](./Baum_figA3){width="8.5cm"} The DFT band structure is shown in Fig. \[Afig:bandstructure\]. Bands above $E_\mathrm{F}$ stem predominantly from Fe $3d$ orbitals (brown) while for $E < -2.7\,\mathrm{eV}$ As $4p$ orbitals prevail (black). For a suitable comparison to the experiment these Fe bands are renormalized by a factor between 2 and 3 (Refs. ) while no renormalization is needed for the As bands. The bands between -2.7eV and $E_\mathrm{F}$ are of mixed Fe/As character and are left out when calculating the dielectric tensor as is illustrated by the grey shade in Fig. \[Afig:bandstructure\]. Only transitions between the ranges \[-5.5eV,-2.7eV\] and \[0,2.6eV\], highlighted by turqouis rectangles, are taken into account. Thus for photon energies below 2.7eV the absorption in our calculations originates predominantly from the Drude response whereas for $\hbar{\ensuremath{\omega_\mathrm{I}}\xspace}>2.7$eV the results become increasingly realistic since they include interband absorption. In either case we use a phenomenological damping of 0.1eV. We determine the dielectric tensor and the Raman tensor as described in section \[sec:theory\] on the basis of this renormalized band structure. While the $\tilde{a}-\tilde{b}$ anisotropy is qualitatively reproduced for all energies [$\omega_\mathrm{I}$]{}as shown in Fig. \[fig:Ag\_resonance\_energy\](b) of the main text, the two other experimental quantities, $A_{ab}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ (purple) and $A_{aa}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ (orange) shown in Fig. \[afig:Ag\_resonance\_all\_pol\] here, are not captured properly simply because the imaginary parts of the theoretically determined tensor elements ${\alpha}^{\mathrm{(As)}\prime\prime}_{ii}$ become very small below 2.7eV. In order to describe $A_{aa}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ and $A_{ab}^\mathrm{(As)}({\ensuremath{\omega_\mathrm{I}}\xspace})$ absorption processes which lead to imaginary parts of the Raman tensor are necessary. Upon phenomenologically introducing imaginary parts of $\hat{\alpha}$ for low energies, which cut off at 2.7eV where the correct absorption takes over, full agreement can be achieved. However, a solution on a microscopic basis becomes possible only by using LDA$+$DMFT schemes which are beyond the scope of the present work. Ag phonon in crossed polarizations in BaFeCoAs and FeSe {#asec:Agnematic} ======================================================= ![(Color online) As phonon in crossed polarizations for at $x=0.051$. (a) Raw data for temperatures as indicated. The phonon position is shown as vertical dash-dotted line. The spectra are shifted vertically for clarity. (b) Temperature dependence of the spectral weight. The spectral weight in [$A_{1g}$]{}symmetry (orange circles) was multiplied by 0.15. [$T_\mathrm{s}$]{}and [$T_\mathrm{SDW}$]{}are indicated as vertical dashed lines. []{data-label="Afig:BFCA"}](./Baum_figA4){width="8.5cm"} Similarly as in , the [$A_{g}$]{}phonon can be observed for crossed polarizations. Fig. \[Afig:BFCA\](a) shows Raman spectra in $ab$ polarization of with $x=5.1$% having ${\ensuremath{T_\mathrm{s}}\xspace}= 60.9$K and ${\ensuremath{T_\mathrm{SDW}}\xspace}= 50.0$K. The As mode appears below [$T_\mathrm{s}$]{}and gains strength upon cooling. Fig. \[Afig:BFCA\](b) shows the corresponding spectral weight as a function of temperature. In the nematic phase ${\ensuremath{T_\mathrm{SDW}}\xspace}< T < {\ensuremath{T_\mathrm{s}}\xspace}$ the phonon spectral weight increases almost linearly upon cooling and saturates in the magnetic phase for $T < {\ensuremath{T_\mathrm{SDW}}\xspace}$ at approximately 15% of that in the fully symmetric channel ([$A_{g}$]{}/[$A_{1g}$]{}). ![(Color online) Se phonon in crossed polarization for FeSe. (a) Raw data for temperatures as indicated. The phonon position is shown as dash-dotted line. The spectra are shifted vertically for clarity. (b) Temperature dependence of the spectral weight. The spectral weight in [$A_{1g}$]{}symmetry (orange circles) was multiplied by 0.01. [$T_\mathrm{s}$]{}is indicated as vertical dotted line. []{data-label="Afig:FeSe"}](./Baum_figA5){width="8.5cm"} In FeSe the Se phonon appears also in the $ab$ spectra as shown in Fig. \[Afig:FeSe\](a) when the temperature is lowered below the structural phase transition at ${\ensuremath{T_\mathrm{s}}\xspace}\approx 90$K. Upon cooling \[Fig. \[Afig:FeSe\](b)\] the spectral weight of the phonon increases almost linearly for crossed polarizations ($ab$, black squares), but stays virtually constant across the phase transition for parallel light polarizations \[$RR$, orange circles in Fig. \[Afig:FeSe\](b)\]. As opposed to no saturation of the spectral weight in $ab$ polarizations is found, likely because FeSe shows no long range magnetic order down to lowest temperatures. [@Baek:2014] Only about 1% of the spectral weight of the [$A_{1g}$]{}spectra ($RR$) is found in crossed polarizations here, in contrast to and , where the spectral weight of the phonon is larger (Figs. \[fig:Ag\_Phonon\_in\_B1g\] and \[Afig:BFCA\]). [57]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1557/mrs.2011.175) [****,  ()](\doibase 10.1038/nphys3456) [****, ()](\doibase http://dx.doi.org/10.1016/j.crhy.2015.10.001) [****,  ()](\doibase 10.1038/s41535-017-0059-y) [****,  ()](\doibase 10.1088/1361-648X/aa9caa) [****,  ()](\doibase 10.1103/PhysRevLett.102.037003) [****,  ()](\doibase 10.1103/PhysRevB.79.064511) [****,  ()](\doibase 10.1103/PhysRevB.80.094504) [****,  ()](\doibase 10.1103/PhysRevB.84.104508) [****,  ()](\doibase 10.1103/PhysRevB.80.064509) [****,  ()](\doibase 10.1016/j.ssc.2010.01.033) [****,  ()](\doibase 10.1103/PhysRevLett.102.217001) [****,  ()](\doibase 10.1103/PhysRevB.87.144508) [****,  ()](\doibase 10.1103/PhysRevB.83.245127) [****,  ()](\doibase 10.1103/PhysRevB.80.180502) [****,  ()](\doibase 10.1103/PhysRevB.88.165106) @noop [“,” ]{} (),  [****,  ()](\doibase 10.1103/PhysRevB.78.085104) [****,  ()](\doibase 10.1103/PhysRevB.79.014506) [****,  ()](\doibase 10.1038/nmat2443) [****,  ()](\doibase 10.1103/PhysRevB.83.134522) [****,  ()](\doibase 10.1103/PhysRevLett.101.107006) [****,  ()](\doibase 10.1126/science.1190482) [****,  ()](\doibase 10.1103/PhysRevLett.107.067001) [****,  ()](\doibase 10.1209/0295-5075/93/37002) [****,  ()](\doibase 10.1088/1367-2630/14/2/023020) [****, ()](\doibase 10.1073/pnas.1100102108) [****,  ()](\doibase 10.1073/pnas.1015572108) [****,  ()](\doibase 10.1016/j.jpcs.2010.10.080) [****,  ()](\doibase 10.1103/PhysRevB.85.144509) [****,  ()](\doibase 10.1103/PhysRevLett.77.3865) [****,  ()](\doibase 10.1103/PhysRevB.78.134106) [****,  ()](\doibase 10.1016/j.scriptamat.2015.07.021) [****,  ()](\doibase 10.1103/PhysRevLett.78.4063) [****,  ()](\doibase 10.1103/PhysRevB.50.17953) [****,  ()](\doibase 10.1103/PhysRevB.47.558) [****,  ()](\doibase 10.1103/PhysRevB.54.11169) [****,  ()](\doibase http://dx.doi.org/10.1016/0927-0256(96)00008-0) [****,  ()](\doibase 10.1016/j.cpc.2006.03.005) [“,” ](http://susi.theochem.tuwien.ac.at/) () [****,  ()](\doibase 10.1103/PhysRevLett.115.197002) [****,  ()](\doibase 10.1103/PhysRevB.94.014513) [****,  ()](\doibase 10.1103/PhysRevB.78.212503) [****,  ()](\doibase 10.1103/PhysRevB.82.180521) [****,  ()](\doibase 10.1143/JPSJ.81.024718) [****,  ()](\doibase 10.1038/NPHYS3634) [****,  ()](\doibase 10.1103/PhysRevB.93.054515) “,”  (, ) Chap. , pp. ,  ed. [****,  ()](\doibase 10.1038/nmat4138) [****,  ()](\doibase 10.1140/epjst/e2010-01302-4) [****,  ()](\doibase 10.1103/PhysRevB.57.1292) [****,  ()](\doibase 10.1103/PhysRevB.65.064501) [****,  ()](\doibase 10.1103/PhysRevB.80.092501) [****,  ()](\doibase 10.1103/PhysRevB.84.245112) [****,  ()](\doibase 10.1103/PhysRevB.85.094505) [****,  ()](\doibase 10.1103/PhysRevB.92.195128) [^1]: The local correlations in the tetragonal phase are of the stripe type; however, we had to use a pattern that does not break the symmetry, and it is known [@Mazin:2008a] that the difference in the calculated elastic properties calculated within different magnetic orders is much smaller than between magnetic and nonmagnetic calculations

--- abstract: 'In this study, we consider an empirical Bayes method for Boltzmann machines and propose an algorithm for it. The empirical Bayes method allows estimation of the values of the hyperparameters of the Boltzmann machine by maximizing a specific likelihood function referred to as the empirical Bayes likelihood function in this study. However, the maximization is computationally hard because the empirical Bayes likelihood function involves intractable integrations of the partition function. The proposed algorithm avoids this computational problem by using the replica method and the Plefka expansion. Our method does not require any iterative procedures and is quite simple and fast, though it introduces a bias to the estimate, which exhibits an unnatural behavior with respect to the size of the dataset. This peculiar behavior is supposed to be due to the approximate treatment by the Plefka expansion. A possible extension to overcome this behavior is also discussed.' author: - Muneki Yasuda - Tomoyuki Obuchi bibliography: - 'citation.bib' title: Empirical Bayes Method for Boltzmann Machines --- Introduction {#sec:intro} ============ *Boltzmann machine learning* (BML) [@Ackley_etal1985] has been actively studied in the field of machine learning and also in statistical mechanics. In statistical mechanics, the problem of BML is sometimes referred to as the *inverse Ising problem*, because a Boltzmann machine is the same as an Ising model, and BML can be regarded as an inverse problem for the Ising model. The framework of the *usual* BML is as follows. Given a set of observed data points (e.g., spin snapshots), we estimate appropriate values of the parameters, the external field and couplings, of our Boltzmann machine through maximum likelihood (ML) estimation (cf. Sec. \[sec:BM\]). Because BML involves intractable multiple summations (i.e., evaluation of the partition function), many approximations for it were proposed from the viewpoint of statistical mechanics [@Roudi2009]: for example, methods based on mean-field approximations (such as the Plefka expansion [@Plefka1982] and the cluster variation method [@CVM-review2005])  and methods based on other approximations [@MPF2011; @SMCI2015]. In this study, we focus on another type of learning problem. We consider prior distributions of parameters of the Boltzmann machine and assume that the prior distributions are governed by some hyperparameters. The introduction of the prior distributions is strongly connected with the regularized ML estimation (cf. Sec. \[sec:BM\]). As mentioned above, the aim of the *usual* BML is to optimize the values of the parameters of the Boltzmann machine by using a set of observed data points. Meanwhile, the aim of the problem investigated in this study is the estimation of appropriate values of the hyperparameters from the dataset without estimating specific values of the parameters. One way to allow us to accomplish this from the Bayesian point of view is the *empirical Bayes method* (or also called type-II ML estimation or evidence approximation) [@MacKay1992; @Bishop2006] (cf. Sec. \[sec:Framework\_EB\]). The schemes of the *usual* BML and of our problem are illustrated in Fig. \[fig:Scheme\_of\_EBM\]. ![Illustration of scheme of empirical Bayes method considered in this study.[]{data-label="fig:Scheme_of_EBM"}](framework_EBM.eps){height="3.4cm"} However, the evaluation of the likelihood function in the empirical Bayes method is again intractable, because it involves intractable multiple integrations of the partition function. In this study, we analyze the empirical Bayes method for fully-connected Boltzmann machines, using statistical mechanical techniques based on the replica method [@ParisiBook1987; @Nishimori2001] and the Plefka expansion to derive an algorithm for it. We consider two types of cases of the prior distribution of $\bm{J}$: the cases of Gaussian and Laplace priors. The rest of this paper is organized as follows. The formulations of the *usual* BML and the empirical Bayes method are presented in Sec. \[sec:BM&EB\]. In Sec. \[sec:StatisticalMechanicalAnalysis\], we describe our statistical mechanical analysis for the empirical Bayes method. The proposed inference algorithm obtained from our analysis is shown in Sec. \[sec:algorithm\] with its pseudocode. In Sec. \[sec:experiment\], we examine our proposed method through numerical experiments. Finally, the summary and some discussions are presented in Sec. \[sec:summary\]. Boltzmann Machine and Empirical Bayes Method {#sec:BM&EB} ============================================ Boltzmann machine and prior distributions {#sec:BM} ----------------------------------------- Consider a fully-connected Boltzmann machine with $n$ Ising variables $\bm{S}:= \{S_i \in \{-1,+1\} \mid i = 1,2,\ldots, n\}$ [@Ackley_etal1985]: $$\begin{aligned} P(\bm{S} \mid h,\bm{J}):=\frac{1}{Z(h,\bm{J})}\exp\Big(h \sum_{i=1}^n S_i + \sum_{i<j}J_{ij}S_iS_j\Big), \label{eqn:BoltzmannMachine}\end{aligned}$$ where $\sum_{i<j}$ is the sum over all the distinct pairs of variables; i.e., $\sum_{i<j} = \sum_{i=1}^n\sum_{j = i+1}^n$. $Z(h,\bm{J})$ is the partition function defined by $$\begin{aligned} Z(h,\bm{J}):= \sum_{\bm{S}}\exp\Big(h \sum_{i=1}^n S_i + \sum_{i<j}J_{ij}S_iS_j\Big),\end{aligned}$$ where $\sum_{\bm{S}}$ is the sum over all the possible configurations of $\bm{S}$; i.e., $\sum_{\bm{S}} := \prod_{i=1}^n \sum_{S_i = \pm 1}$. The parameters, $h \in (-\infty, +\infty)$ and $\bm{J} := \{J_{ij} \in (-\infty, +\infty) \mid i<j\}$, denote the external field and couplings, respectively. Given $N$ observed data points, $\mcal{D}:=\{\mbf{S}^{(\mu)} \in \{-1,+1\}^n \mid \mu = 1,2,\ldots, N\}$, we define the log-likelihood function: $$\begin{aligned} L_{\mrm{ML}}(h,\bm{J}):=\frac{1}{n N}\sum_{\mu = 1}^N \ln P(\mbf{S}^{(\mu)} \mid h,\bm{J}). \label{eqn:log-likelihood}\end{aligned}$$ Maximizing the log-likelihood function with respect to $h$ and $\bm{J}$ (i.e., the ML estimation) just corresponds to the BML (or the inverse Ising problem), i.e., $$\begin{aligned} \{\hat{h}_{\mrm{ML}},\hat{\bm{J}}_{\mrm{ML}}\} = \argmax_{h, \bm{J}}L_{\mrm{ML}}(h,\bm{J}). \label{eqn;InverseIsing}\end{aligned}$$ Now, we introduce prior distributions for the parameters $h$ and $\bm{J}$ as $P_{\mrm{prior}}(h\mid H)$ and $$\begin{aligned} P_{\mrm{prior}}(\bm{J} \mid \gamma)&:= \prod_{i<j} P_{\mrm{prior}}(J_{ij} \mid \gamma), \label{eqn:prior_J}\end{aligned}$$ respectively. $H$ and $\gamma$ are the hyperparameters of these prior distributions. One of the most important motivations for introducing the prior distributions is for a Bayesian interpretation of the regularized ML estimation [@Bishop2006]. Given the observed dataset $\mcal{D}$, by using the prior distributions, the posterior distribution of $h$ and $\bm{J}$ is expressed as $$\begin{aligned} &P_{\mrm{post}}(h,\bm{J} \mid \mcal{D}, H, \gamma) \nn &= \frac{P(\mcal{D} \mid h, \bm{J})P_{\mrm{prior}}(h\mid H)P_{\mrm{prior}}(\bm{J} \mid \gamma)}{P(\mcal{D} \mid H, \gamma)}, \label{eqn;posterior_H&J}\end{aligned}$$ where $$\begin{aligned} P(\mcal{D} \mid h, \bm{J}):= \prod_{\mu = 1}^N P(\mbf{S}^{(\mu)} \mid h,\bm{J}).\end{aligned}$$ The distribution in the denominator in Eq. (\[eqn;posterior\_H&J\]), $P(\mcal{D} \mid H, \gamma)$, is sometimes referred to as the evidence. By using the posterior distribution, the maximum a posteriori (MAP) estimation of the parameters is obtained as $$\begin{aligned} \{\hat{h}_{\mrm{MAP}},\hat{\bm{J}}_{\mrm{MAP}}\} = \argmax_{h, \bm{J}}L_{\mrm{MAP}}(h,\bm{J}), \label{eqn;MAP}\end{aligned}$$ where $$\begin{aligned} &L_{\mrm{MAP}}(h,\bm{J}):= \frac{1}{nN}\ln P_{\mrm{post}}(h,\bm{J} \mid \mcal{D}, H, \gamma)\nn &=L_{\mrm{ML}}(h,\bm{J}) + \frac{1}{nN} R_0(h) + \frac{1}{nN} R_1(\bm{J}) + \mrm{constant}.\end{aligned}$$ The MAP estimation in Eq. (\[eqn;MAP\]) corresponds to the regularized ML estimation, in which $R_0(h):=\ln P_{\mrm{prior}}(h\mid H)$ and $R_1(\bm{J}):=\ln P_{\mrm{prior}}(\bm{J} \mid \gamma)$ work as a penalty. For example, (i) when the prior distribution of $\bm{J}$ is the Gaussian prior, $$\begin{aligned} P_{\mrm{prior}}(J_{ij} \mid \gamma)= \sqrt{\frac{n}{2 \pi \gamma}} \exp\Big(-\frac{n J_{ij}^2}{2 \gamma}\Big),\quad \gamma > 0, \label{eqn:GaussPrior}\end{aligned}$$ $R_1(\bm{J})$ corresponds to the $L_2$ regularization term, and $\gamma$ corresponds to its coefficient; (ii) when the prior distribution of $\bm{J}$ is the Laplace prior, $$\begin{aligned} P_{\mrm{prior}}(J_{ij} \mid \gamma)= \sqrt{\frac{n}{2 \gamma}} \exp\Big(-\sqrt{\frac{2n}{\gamma}} |J_{ij}|\Big),\quad \gamma > 0 \label{eqn:LaplacePrior}\end{aligned}$$ $R_1(\bm{J})$ corresponds to the $L_1$ regularization term, and $\gamma$ again corresponds to its coefficient. The variances of these prior distributions are identical, $\mrm{Var}[J_{ij}]=\gamma /n$. In this study, as a simple test case, we use these two prior distributions for $\bm{J}$ and $$\begin{aligned} P_{\mrm{prior}}(h\mid H) = \delta(h - H), \label{eqn:prior_H}\end{aligned}$$ where $\delta(x)$ is the Dirac delta function, for $h$. Framework of the empirical Bayes method {#sec:Framework_EB} --------------------------------------- Using the empirical Bayes method, we can infer the values of the hyperparameters, $H$ and $\gamma$, from the observed dataset $\mcal{D}$. We define a marginal log-likelihood function as $$\begin{aligned} L_{\mrm{EB}}(H,\gamma)&:=\frac{1}{nN} \ln \big[ P(\mcal{D} \mid h, \bm{J})\big]_{h,\bm{J}}, \label{eqn:EmpiricalBayesLikelihood}\end{aligned}$$ where $[\cdots]_{h,\bm{J}}$ is the average over the prior distributions; i.e., $$\begin{aligned} [\cdots]_{h,\bm{J}}:= \int d\bm{J}\int d h (\cdots) P_{\mrm{prior}}(h\mid H)P_{\mrm{prior}}(\bm{J} \mid \gamma).\end{aligned}$$ We refer to the marginal log-likelihood function as the *empirical Bayes likelihood function* in this study. From the perspective of the empirical Bayes method, the optimal values of the hyperparameters, $\hat{H}$ and $\hat{\gamma}$, are obtained by maximizing of the empirical Bayes likelihood function; i.e., $$\begin{aligned} \{\hat{H},\hat{\gamma}\} = \argmax_{H, \gamma} L_{\mrm{EB}}(H,\gamma). \label{eqn:Maximizing_EBL}\end{aligned}$$ It is noteworthy that $[P(\mcal{D} \mid h, \bm{J})]_{h,\bm{J}}$ in Eq. (\[eqn:EmpiricalBayesLikelihood\]) is identified as the evidence appearing in Eq. (\[eqn;posterior\_H&J\]). The marginal log-likelihood function can be rewritten as $$\begin{aligned} L_{\mrm{EB}}(H,\gamma)=\frac{1}{nN}\ln \Big[\exp\big(n N L_{\mrm{ML}}(h,\bm{J})\big)\Big]_{h, \bm{J}} \label{eqn:EmpiricalBayesLikelihood_ML-representation}\end{aligned}$$ Consider the case $N\gg n$. In this case, by using the saddle point evaluation, Eq. (\[eqn:EmpiricalBayesLikelihood\_ML-representation\]) is reduced to $$\begin{aligned} L_{\mrm{EB}}(H,\gamma)&\approx \frac{1}{n N} \ln P_{\mrm{prior}}(\hat{h}_{\mrm{ML}}\mid H) \nn \aleq + \frac{1}{nN} \ln P_{\mrm{prior}}(\hat{\bm{J}}_{\mrm{ML}} \mid \gamma)+\mrm{constant}.\end{aligned}$$ In this case, the empirical Bayes’ estimates $\{\hat{H},\hat{\gamma}\}$ thus converge to the maximum likelihood estimates of the hyperparameters in the prior distributions in which the maximum likelihood estimates of the parameters $\{\hat{h}_{\mrm{ML}},\hat{\bm{J}}_{\mrm{ML}}\}$ (i.e., the solution to the BML) are inserted. This indicates that the parameter estimations can be conducted independently of the hyperparameter estimation. In this study, we do not concern ourselves with this trivial case. Statistical Mechanical Analysis {#sec:StatisticalMechanicalAnalysis} =============================== The empirical Bayes likelihood function in Eq. (\[eqn:EmpiricalBayesLikelihood\]) involves intractable multiple integrations. In this section, we evaluate the empirical Bayes likelihood function using a statistical mechanical analysis. We consider the two types of the prior distribution of $\bm{J}$: one is the Gaussian prior in Eq. (\[eqn:GaussPrior\]), and the other is the Laplace prior in Eq. (\[eqn:LaplacePrior\]). First, we evaluate the empirical Bayes likelihood function on the basis of the Gaussian prior in Secs. \[sec:ReplicaMethod\]–\[sec:algorithm\], after which we describe the evaluation based on the Laplace prior in Sec. \[sec:Analysis\_LaplacePrior\]. Replica method {#sec:ReplicaMethod} -------------- The empirical Bayes likelihood function in Eq. (\[eqn:EmpiricalBayesLikelihood\]) can be represented as $$\begin{aligned} L_{\mrm{EB}}(H,\gamma)=\frac{1}{nN}\ln \lim_{x \to -1} \Psi_x(H,\gamma), \label{eqn:EmpiricalBayesLikelihood_ReplicaMethod}\end{aligned}$$ where $$\begin{aligned} &\Psi_x(H,\gamma)\nn &:=\Big[ Z(h,\bm{J})^{x N} \exp N\Big(h \sum_{i=1}^n d_i + \sum_{i<j}J_{ij} d_{ij}\Big) \Big]_{h, \bm{J}}, \label{eqn:def_Psi_x}\end{aligned}$$ and $$\begin{aligned} d_i := \frac{1}{N} \sum_{\mu = 1}^N \mrm{S}_i^{(\mu)},\quad d_{ij} := \frac{1}{N} \sum_{\mu = 1}^N \mrm{S}_i^{(\mu)}\mrm{S}_j^{(\mu)}\end{aligned}$$ are the sample averages of the observed data points. We assume that $\tau_x := x N$ is a natural number, and therefore Eq. (\[eqn:def\_Psi\_x\]) can be expressed as $$\begin{aligned} \Psi_x(H,\gamma) &=\Big[ \sum_{\mcal{S}_x}\exp \Big\{h \sum_{i=1}^n\Big( \sum_{a = 1}^{\tau_x}S_i^{\{a\}}+ N d_i\Big) \nn \aleq + \sum_{i<j}J_{ij}\Big(\sum_{a = 1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}} + N d_{ij}\Big)\Big\} \Big]_{h, \bm{J}}, \label{eqn:Psi_x_naturalnumberAssumption}\end{aligned}$$ where $a ,b \in \{1,2,\ldots, \tau_x\}$ are replica indices, and $S_i^{\{a\}}$ is the Ising variable on site $i$ in the $a$th replica. $\mcal{S}_x:= \{S_i^{\{a\}} \mid i = 1,2,\ldots, n;\, a = 1,2,\ldots, \tau_x\}$ is the set of all the Ising variables in the replicated system, and $\sum_{\mcal{S}_x}$ is the sum over all the possible configurations of $\mcal{S}_x$; i.e., $\sum_{\mcal{S}_x} := \prod_{i=1}^n\prod_{a=1}^{\tau_x} \sum_{S_i^{\{a\}} = \pm 1}$. We evaluate $\Psi_x(H,\gamma)$ under the assumption that $\tau_x$ us a natural number, after which we take the limit of $x \to -1$ of the evaluation result to obtain the empirical Bayes likelihood function (this is the so-called *replica trick*). By employing the Gaussian prior in Eq. (\[eqn:GaussPrior\]), Eq. (\[eqn:Psi\_x\_naturalnumberAssumption\]) becomes $$\begin{aligned} \Psi_x^{\mrm{Gauss}}(H,\gamma) &=\exp\Big\{ n N H M + \frac{\gamma (n-1) N^2}{4}\Big(C_2 + \frac{x}{N}\Big)\nn \aleq - F_x(H, \gamma)\Big\}, \label{eqn:Psi_x_Gauss}\end{aligned}$$ where $$\begin{aligned} M:= \frac{1}{n}\sum_{i=1}^n d_i,\quad C_k:= \frac{2}{n(n-1)}\sum_{i<j}d_{ij}^k, \label{eqn:def_M&Ck}\end{aligned}$$ and $$\begin{aligned} F_x(H, \gamma):=-\ln \sum_{\mcal{S}_x}\exp\big(-E_x(\mcal{S}_x;H,\gamma)\big) \label{eqn:ReplicatedFreeEnergy}\end{aligned}$$ is the replicated (Helmholtz) free energy [@RCVM2010; @YKT2012; @RCVM2013; @Yasuda2015]; here, $$\begin{aligned} &E_x(\mcal{S}_x;H,\gamma)\nn &:=-H \sum_{i=1}^n\sum_{a=1}^{\tau_x} S_i^{\{a\}} - \frac{\gamma N}{n}\sum_{i<j}d_{ij} \sum_{a = 1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}} \nn \aldef - \frac{\gamma}{n}\sum_{i<j}\sum_{a < b}S_i^{\{a\}}S_j^{\{a\}}S_i^{\{b\}}S_j^{\{b\}} \label{eqn:ReplicatedHamiltonian}\end{aligned}$$ is the Hamiltonian of the replicated system, where $\sum_{a<b}$ is the sum over all the distinct pairs of replicas; i.e., $\sum_{a<b} = \sum_{a=1}^{\tau_x}\sum_{b = a+1}^{\tau_x}$. Plefka expansion {#sec:PlefkaExpansion} ---------------- Because the replicated free energy in Eq. (\[eqn:ReplicatedFreeEnergy\]) includes intractable multiple summations, an approximation is needed to proceed with our evaluation. In this section, we approximate the replicated free energy using the Plefka expansion [@Plefka1982]. In brief, the Plefka expansion is the perturbative expansion in a Gibbs free energy that is a dual form of a corresponding Helmholtz free energy. The Gibbs free energy is obtained as $$\begin{aligned} G_x(m,H,\gamma) &=- n \tau_x H m + \extr_{\lambda}\Big\{\lambda n \tau_x m \nn \aleq -\ln \sum_{\mcal{S}_x}\exp\big( - E_x(\mcal{S}_x;\lambda,\gamma)\big)\Big\}. \label{eqn:GibbsFreeEnergy}\end{aligned}$$ The derivation of this Gibbs free energy is described in Appendix \[sec:app:GibbsFreeEnergy\]. It is noteworthy that this type of expression of the Gibbs free energy implies the replica-symmetric (RS) assumption. To take the replica-symmetry breaking (RSB) into account, explicit treatments of overlaps between different replicas are needed [@YKT2012]. By expanding $G_x(m,H,\gamma)$ around $\gamma = 0$, we obtain $$\begin{aligned} \frac{G_x(m,H,\gamma)}{nN} &=-x H m+ x e(m) + \phi_x^{(1)}(m) \gamma \nn \aleq + \phi_x^{(2)}(m)\gamma^2 + O(\gamma^3), \label{eqn:PlefkaExpansion}\end{aligned}$$ where $e(m)$ is the negative mean-field entropy defined by $$\begin{aligned} e(m):=\frac{1+m}{2} \ln \frac{1+m}{2} + \frac{1-m}{2} \ln \frac{1-m}{2}, \label{eqn:MeanFieldEntropy}\end{aligned}$$ and the coefficients, $\phi_x^{(1)}(m)$ and $\phi_x^{(2)}(m)$, are expressed as Eqs. (\[eqn:PlefkaExpansion\_1st\]) and (\[eqn:PlefkaExpansion\_2nd\]), respectively. The detailed derivation of these coefficients is presented in Appendix \[sec:app:PlefkaExpansion\]. From Eqs. (\[eqn:EmpiricalBayesLikelihood\_ReplicaMethod\]), (\[eqn:Psi\_x\_Gauss\]), (\[eqn:PlefkaExpansion\]), and (\[eqn:relation\_F&G\]), we obtain the empirical Bayes likelihood function as $$\begin{aligned} L_{\mrm{EB}}(H,\gamma)&\approx HM -\extr_{m}\Big[ Hm - e(m)+ \Phi(m)\gamma\nn \aleq +\phi_{-1}^{(2)}(m)\gamma^2\Big]. \label{eqn:EmpiricalBayesLikelihood_Gauss_Result}\end{aligned}$$ where $$\begin{aligned} \Phi(m):=\phi_{-1}^{(1)}(m) - \frac{(n-1)N}{4n}\Big(C_2 - \frac{1}{N}\Big).\end{aligned}$$ From Eqs. (\[eqn:PlefkaExpansion\_1st\]) and (\[eqn:PlefkaExpansion\_2nd\]), $\Phi(m)$ and $\phi_{-1}^{(2)}(m)$ are $$\begin{aligned} \Phi(m) &= \frac{(n-1)NC_1}{2n}m^2- \frac{(n-1)N}{4n}\Big\{C_2 \nn \aleq + \frac{N+1}{N}\Big(m^4 - \frac{1}{N+1}\Big)\Big\} \label{eqn:Phi(m)}\end{aligned}$$ and $$\begin{aligned} \phi_{-1}^{(2)}(m)&=\frac{(n-1)^2 N^2 \Omega}{2n^2}m^2(1 - m^2) +\frac{(n-1) N^2 C_2}{4n^2}(1-m^2)^2 - \frac{(n-1)N(N+1)C_1}{2n^2}m^2(1-m^2)^2\nn \aleq -\frac{(n-1) (N+1)}{4n^2}\big(n - N - 3\big)m^4(1-m^2)^2 -\frac{(n-1)(N+1)}{8n^2}(1 - m^4)^2, \label{eqn:PlefkaExpansion_2nd_x=-1}\end{aligned}$$ respectively. The coefficient $\Omega$ appearing in the above equation is defined by $$\begin{aligned} \Omega:=\frac{1}{n}\sum_{i=1}^n \omega_i^2, \label{eqn:def_Omega}\end{aligned}$$ where $$\begin{aligned} \omega_i &:=\frac{1}{n-1}\sum_{j\in \partial(i)}d_{ij} - C_1; \label{eqn:def_omega_i}\end{aligned}$$ here, $\partial(i):= \{1,2,\ldots,n\} \setminus \{i\}$. Inference algorithm {#sec:algorithm} ------------------- As mentioned in Sec. \[sec:Framework\_EB\], the empirical Bayes inference is achieved by maximizing $L_{\mrm{EB}}(H,\gamma)$ with respect to $H$ and $\gamma$ (cf. Eq. (\[eqn:Maximizing\_EBL\])). From the extremum condition of Eq. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\]) with respect to $H$, we obtain $$\begin{aligned} \hat{m} = M, \label{eqn:Extr_H}\end{aligned}$$ where $\hat{m}$ is the value of $m$ that satisfies the extremum condition in Eq. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\]). From the extremum condition of Eq. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\]) with respect to $m$ and Eq. (\[eqn:Extr\_H\]), we obtain $$\begin{aligned} \hat{H} =\tanh^{-1}M -\Big(\frac{\partial \phi_{-1}^{(1)}(m)}{\partial m}\gamma +\frac{\partial \phi_{-1}^{(2)}(m)}{\partial m}\gamma^2\Big) \Big|_{m = M}. \label{eqn:determine_H-hat}\end{aligned}$$ From Eqs. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\]) and (\[eqn:Extr\_H\]), the optimal value of $\gamma$ is obtained by $$\begin{aligned} \hat{\gamma}&=\argmax_{\gamma}\big[-\Phi(M)\gamma -\phi_{-1}^{(2)}(M)\gamma^2\big]. \label{eqn:determine_gamma-hat}\end{aligned}$$ From Eq. (\[eqn:determine\_gamma-hat\]), $\hat{\gamma}$ is immediately obtained as follows: (i) when $\phi_{-1}^{(2)}(M) > 0$ and $\Phi(M) \geq 0$ or when $\phi_{-1}^{(2)}(M) = 0$ and $\Phi(M) > 0$, $\hat{\gamma} = 0$, (ii) when $\phi_{-1}^{(2)}(M) > 0$ and $\Phi(M) < 0$, $\hat{\gamma} = - \Phi(M) / (2 \phi_{-1}^{(2)}(M))$, and (iii) $\hat{\gamma} \to \infty$ elsewhere. Here, we ignore the case $\phi_{-1}^{(2)}(M) = \Phi(M) = 0$, because it hardly occurs in realistic settings. By using Eqs. (\[eqn:determine\_H-hat\]) and (\[eqn:determine\_gamma-hat\]), we can obtain the solution to the empirical Bayes inference without any iterative processes. The pseudocode of the proposed procedure is shown in Algorithm \[alg:EmpiricalBayes\]. **Input** Observed data set: $\mcal{D}:=\{\mbf{S}^{(\mu)} \in \{-1,+1\}^n \mid \mu = 1,2,\ldots, N\}$. Compute $M$, $\Omega$, $C_1$, and $C_2$ using the data set according to Eqs. (\[eqn:def\_M&Ck\]) and (\[eqn:def\_Omega\]). Determine $\hat{\gamma}$ using Eq. (\[eqn:determine\_gamma-hat\]): $$\begin{aligned} \hat{\gamma}= \begin{cases} 0 & \mbox{case (i)}\\ - \Phi(M) / (2 \phi_{-1}^{(2)}(M)) & \mbox{case (ii)}\\ \infty & \mrm{elsewhere} \end{cases} ,\end{aligned}$$ where case (i): $\phi_{-1}^{(2)}(M) > 0, \>\Phi(M) \geq 0$ or $\phi_{-1}^{(2)}(M) = 0, \> \Phi(M) > 0$ and case (ii): $\phi_{-1}^{(2)}(M) > 0, \> \Phi(M) < 0$. Using $\hat{\gamma}$, determine $\hat{H}$ using Eq. (\[eqn:determine\_H-hat\]). **Output** $\hat{\gamma}$ and $\hat{H}$. In the proposed method, the value of $\hat{H}$ does not affect the determination of $\hat{\gamma}$. Many mean-field-based methods for BML (e.g., listed in Sec. \[sec:intro\]) have similar procedures, in which $\hat{\bm{J}}_{\mrm{ML}}$ are determined separately from $\hat{h}_{\mrm{ML}}$. This is seen as one of the common properties of the mean-field-based methods for BML including the current empirical Bayes problem. Evaluation based on Laplace prior {#sec:Analysis_LaplacePrior} --------------------------------- The above evaluation was for the Gaussian prior in Eq. (\[eqn:GaussPrior\]). Here, we explain the evaluation for the Laplace prior in Eq. (\[eqn:LaplacePrior\]). By employing the Laplace prior in Eq. (\[eqn:LaplacePrior\]), Eq. (\[eqn:Psi\_x\_naturalnumberAssumption\]) becomes $$\begin{aligned} &\Psi_x^{\mrm{Laplace}}(H,\gamma)\nn &=\xi^{n(n-1)}e^{nNHM}\sum_{\mcal{S}_x}\exp\Big[ H\sum_{i=1}^n\sum_{a = 1}^{\tau_x}S_i^{\{a\}} \nn \aleq - \sum_{i<j}\ln \Big\{ \xi^2 -\Big(\sum_{a=1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}}+ N d_{ij}\Big)^2\Big\}\Big], \label{eqn:Psi_x_Laplace}\end{aligned}$$ where $\xi:=\sqrt{2n/\gamma}$. Here, we assume $$\begin{aligned} \xi > \max_{i<j}\Big(\sum_{a=1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}}+ N d_{ij}\Big). \label{eqn:assumption_Laplace}\end{aligned}$$ By using the perturbative approximation, $$\begin{aligned} &\ln \Big\{ \xi^2 -\Big(\sum_{a=1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}}+ N d_{ij}\Big)^2\Big\}\nn &= \ln \xi^2 + \ln \Big\{1 -\xi^{-2}\Big(\sum_{a=1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}}+ N d_{ij}\Big)^2\Big\}\nn &\approx \ln \xi^2 -\xi^{-2}\Big(\sum_{a=1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}}+ N d_{ij}\Big)^2,\end{aligned}$$ we obtain the approximation of Eq. (\[eqn:Psi\_x\_Laplace\]) as $$\begin{aligned} \Psi_x^{\mrm{Laplace}}(H,\gamma) &\approx e^{nNHM}\sum_{\mcal{S}_x}\exp\Big[ H\sum_{i=1}^n\sum_{a = 1}^{\tau_x}S_i^{\{a\}} \nn \aleq + \xi^2\sum_{i<j}\Big(\sum_{a=1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}}+ N d_{ij}\Big)^2\Big], %\label{eqn:Psi_x_Laplace_approx}\end{aligned}$$ The right-hand side of this equation coincides with $\Psi_x^{\mrm{Gauss}}(H,\gamma)$ in Eq. (\[eqn:Psi\_x\_Gauss\]). This means that the empirical Bayes inference based on the Laplace prior in Eq. (\[eqn:LaplacePrior\]) is (approximately) equivalent to that based on the Gaussian prior in Eq. (\[eqn:GaussPrior\]) (i.e., $\Psi_x^{\mrm{Laplace}}(H,\gamma) \approx \Psi_x^{\mrm{Gauss}}(H,\gamma)$) when the assumption of Eq. (\[eqn:assumption\_Laplace\]) is justified. Thus, we can also use the algorithm presented in Sec. \[sec:algorithm\] for the case of the Laplace prior. Numerical Experiments {#sec:experiment} ===================== In this section, we describe the results of our numerical experiments. In these experiments, the observed dataset $\mcal{D}$ are generated from the generative Boltzmann machine, which has the same form as Eq. (\[eqn:BoltzmannMachine\]), by using annealed importance sampling (AIS) [@Neal2001]. In AIS, we controlled the annealing schedule using a series of inverse temperature $0 = \beta_0 < \beta_1 < \cdots < \beta_T = 1$, where $\beta_{t + 1} = \beta_t + 0.03$. The parameters of the generative Boltzmann machine are drawn from the prior distributions in Eqs. (\[eqn:prior\_J\]) and (\[eqn:prior\_H\]). That is, we consider the model-matched case (i.e., the generative and learning models are identical). In the following, we use the notations $\alpha := N / n$ and $J := \sqrt{\gamma}$. The standard deviations of the Gaussian prior in Eq. (\[eqn:GaussPrior\]) and of the Laplace prior in Eq. (\[eqn:LaplacePrior\]) are then $J / \sqrt{n}$. We express the hyperparameters for the generative Boltzmann machine by $H_{\mrm{true}}$ and $J_{\mrm{true}}$. Gaussian prior case ------------------- Here, we consider the case in which the prior distribution of $\bm{J}$ is the Gaussian prior in Eq. (\[eqn:GaussPrior\]). In this case, the Boltzmann machine corresponds to the Sherrington-Kirkpatrick (SK) model [@SK1975], and therefore it shows the spin-glass transition at $J = 1$ when $h = 0$ (i.e., when $H = 0$). First, we consider the case $H_{\mrm{true}} = 0$. We show the scatter plots for the estimation of $\hat{J}$ for various $J_{\mrm{true}}$ when $H_{\mrm{true}} = 0$ and $\alpha = 0.4$ in Fig. \[fig:J\_Gauss\_alpha0.4\]. ![Scatter plots of $J_{\mrm{true}}$ (horizontal axis) versus $\hat{J}$ (vertical axis) when $H_{\mrm{true}} = 0$ and $\alpha = 0.4$: (a) $n = 300$ and (b) $n = 500$. Plots are the average values over 300 experiments.[]{data-label="fig:J_Gauss_alpha0.4"}](J_Gauss_alpha0.4.eps){height="4.4cm"} The detailed values of the plots for some $J_{\mrm{true}}$ values are shown in Tab. \[tab:J\_Gauss\_alpha0.4\]. -- ----------- ------------------ ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- 0 0.2 0.4 0.6 0.8 1 1.2 $n = 300$ $0.048 \pm 0.06$ $0.20 \pm 0.04$ $0.41 \pm 0.02$ $0.62 \pm 0.02$ $0.82 \pm 0.02$ $0.96 \pm 0.02$ $1.03 \pm 0.02$ $n = 500$ $0.038 \pm 0.05$ $0.20 \pm 0.03$ $0.40 \pm 0.01$ $0.62 \pm 0.01$ $0.82 \pm 0.01$ $0.96 \pm 0.01$ $1.03 \pm 0.01$ -- ----------- ------------------ ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- \[tab:J\_Gauss\_alpha0.4\] When $J_{\mrm{true}} < 1$, our estimates of $\hat{J}$ are in good agreement with $J_{\mrm{true}}$. This implies that the validity of our perturbative approximation is lost in the spin-glass phase, as is often the case with many mean-field approximations. Fig. \[fig:J\_Gauss\] shows the scatter plots for various $\alpha$. ![Scatter plots of $J_{\mrm{true}}$ (horizontal axis) versus $\hat{J}$ (vertical axis) for various $\alpha = N / n$ when $H_{\mrm{true}} = 0$: (a) $n = 300$ and (b) $n = 500$. Plots are the average values over 300 experiments.[]{data-label="fig:J_Gauss"}](J_Gauss.eps){height="4.4cm"} A smaller $\alpha$ causes $\hat{J}$ to be overestimated and a larger $\alpha$ causes it to be underestimated. At least in our experiments, the optimal value of $\alpha$ seems to be $\alpha_{\mrm{opt}} \approx 0.4$ when $H_{\mrm{true}} = 0$. Our method can estimate $\hat{H}$ together with $\hat{J}$. The results for the estimation of $\hat{H}$ when $H_{\mrm{true}} = 0$ and $\alpha = 0.4$ are shown in Fig. \[fig:H\_Gauss\_alpha0.4\]. ![Results of estimation of $\hat{H}$ against $J_{\mrm{true}}$ when $H_{\mrm{true}} = 0$ and $\alpha = 0.4$: (a) the mean absolute error and (b) standard deviation. Plots are the average values over 300 experiments.[]{data-label="fig:H_Gauss_alpha0.4"}](H_Gauss_alpha0.4.eps){height="4.4cm"} Figs. \[fig:H\_Gauss\_alpha0.4\](a) and (b) show the average of $|H_{\mrm{true}} - \hat{H}|$ (i.e., the mean absolute error (MAE)) and the standard deviation of $\hat{H}$ over 300 experiments, respectively. The MAE and standard deviation increase in the region $J_{\mrm{true}} > 1$. Next, we consider the cases $H_{\mrm{true}} > 0$. The scatter plots for the estimation of $\hat{J}$ for various $J_{\mrm{true}}$ values when $H_{\mrm{true}} = 0.2$ and $H_{\mrm{true}} =0.4$ are shown in Fig. \[fig:J\_Gauss\_H0.2&0.4\]. ![Scatter plots of $J_{\mrm{true}}$ (horizontal axis) versus $\hat{J}$ (vertical axis) for various $\alpha = N / n$ for $n = 300$ and $500$: (a) $H_{\mrm{true}} = 0.2$ and (b) $H_{\mrm{true}} = 0.4$. Plots are the average values over 300 experiments. The notation in the legend means $(n, N)$.[]{data-label="fig:J_Gauss_H0.2&0.4"}](J_Gauss_H0.2_and_0.4.eps){height="4.4cm"} The appropriate values of $\alpha$ when $H_{\mrm{true}} = 0.2$ and $H_{\mrm{true}} = 0.4$ “approximately” seem to be $\alpha_{\mrm{opt}} \approx 30/n$ and $\alpha_{\mrm{opt}} \approx 5/n$, respectively. The detailed values of these plots for some $J_{\mrm{true}}$ values are shown in Tabs. \[tab:J\_Gauss\_H0.2\_N30\] and \[tab:J\_Gauss\_H0.4\_N5\]. The results for the estimation of $\hat{H}$ when $H_{\mrm{true}} = 0.2$ and $\alpha = 30/n$ and when $H_{\mrm{true}} = 0.4$ and $\alpha = 5/n$ are shown in Figs. \[fig:H\_Gauss\_H0.2\] and \[fig:H\_Gauss\_H0.4\], respectively. ![Results of estimation of $\hat{H}$ against $J_{\mrm{true}}$ when $H_{\mrm{true}} = 0.2$ and $\alpha = 30/ n$: (a) the mean absolute error and (b) standard deviation. Plots are the average values over 300 experiments.[]{data-label="fig:H_Gauss_H0.2"}](H_Gauss_H0.2.eps){height="4.4cm"} ![Results of estimation of $\hat{H}$ against $J_{\mrm{true}}$ when $H_{\mrm{true}} = 0.4$ and $\alpha = 5 / n$: (a) the mean absolute error and (b) standard deviation. Plots are the average values over 300 experiments.[]{data-label="fig:H_Gauss_H0.4"}](H_Gauss_H0.4.eps){height="4.4cm"} The increases in the MAE and standard deviations occur earlier than for the case in Fig. \[fig:H\_Gauss\_alpha0.4\]. -- ----------- ------------------ ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- 0 0.2 0.4 0.6 0.8 1 1.2 $n = 300$ $0.083 \pm 0.10$ $0.17 \pm 0.12$ $0.38 \pm 0.07$ $0.58 \pm 0.05$ $0.79 \pm 0.06$ $1.05 \pm 0.12$ $1.35 \pm 0.16$ $n = 500$ $0.075 \pm 0.09$ $0.16 \pm 0.11$ $0.38 \pm 0.06$ $0.57 \pm 0.04$ $0.78 \pm 0.06$ $1.05 \pm 0.10$ $1.39 \pm 0.16$ -- ----------- ------------------ ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- \[tab:J\_Gauss\_H0.2\_N30\] -- ----------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- 0 0.2 0.4 0.6 0.8 1 1.2 $n = 300$ $0.15 \pm 0.17$ $0.17 \pm 0.17$ $0.33 \pm 0.19$ $0.53 \pm 0.14$ $0.75 \pm 0.12$ $0.95 \pm 0.14$ $1.22 \pm 0.20$ $n = 500$ $0.12 \pm 0.15$ $0.17 \pm 0.17$ $0.33 \pm 0.17$ $0.55 \pm 0.12$ $0.76 \pm 0.10$ $0.98 \pm 0.11$ $1.20 \pm 0.16$ -- ----------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- \[tab:J\_Gauss\_H0.4\_N5\] One of the largest qualitative differences between the cases $H_{\mrm{true}} = 0$ and $H_{\mrm{true}} > 0$ is the scale of $\alpha$. In the case $H_{\mrm{true}} = 0$, the optimal $\alpha$ was scaled by $O(1)$ with respect to $n$ (i.e., $N = O(n)$). Meanwhile, in the case $H_{\mrm{true}} > 0$, the optimal $\alpha$ is scaled by $O(1/n)$ with respect to $n$ (i.e., $N = O(1)$). This change of scale can be understood from a scale evaluation for the terms in the empirical Bayes likelihood function in Eq. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\]). The detailed reasoning is given in Appendix \[sec:app:OrdarEvaluation\]. Laplace prior case ------------------ Here, we consider the case in which the prior distribution of $\bm{J}$ is the Laplace prior in Eq. (\[eqn:LaplacePrior\]). The scatter plots for the estimation of $\hat{J}$ for various $J_{\mrm{true}}$ values when $H_{\mrm{true}} = 0$ are shown in Fig. \[fig:J\_Laplace\]. ![Scatter plots of $J_{\mrm{true}}$ (horizontal axis) versus $\hat{J}$ (vertical axis) for various $\alpha = N/n$, when $H_{\mrm{true}} = 0$, in the case of the Laplace prior: (a) $n = 300$ and (b) $n = 500$. Plots are the average values over 300 experiments.[]{data-label="fig:J_Laplace"}](J_Laplace.eps){height="4.4cm"} The plots shown in Fig. \[fig:J\_Laplace\] almost completely overlap with those in Fig. \[fig:J\_Gauss\]. Furthermore, all the numerical results in the case $H_{\mrm{true}} > 0$ also almost completely overlap with the corresponding results obtained in the above Gaussian prior case, and therefore we do not show those results. Summary and Discussions {#sec:summary} ======================= In this study, we proposed a hyperparameters inference algorithm by analyzing the empirical Bayes likelihood function in Eq. (\[eqn:EmpiricalBayesLikelihood\]) using the replica method and the Plefka expansion. The validity of our method was examined in numerical experiments for the Gaussian and Laplace priors, which demonstrated the existence of an appropriate scale in the size of the dataset that can accurately recover the values of the hyperparameters. However, some problems remain. The first one is the scale of $N$. In our experiments, we found that an appropriate $N$ is scaled by $O(n)$ when $H_{\mrm{true}} = 0$ or by $O(1)$ when $H_{\mrm{true}} \neq 0$. However, such scales seem to be unnatural, because they should not appear in the original framework of the empirical Bayes method. As discussed in Sec. \[sec:Framework\_EB\], when $N\gg n$, maximizing the empirical Bayes likelihood function is reduced to the maximum likelihood estimation of the prior distributions for the solution to BML. This must lead to the correct $\hat{\gamma}$ and $\hat{H}$, because the solution to BML is perfect when $N \to \infty$. Therefore, such unnatural scales appear due to our approximation, which is also supported by a scale analysis given in Appendix \[sec:app:OrdarEvaluation\]. An improvement of the approximation (e.g., by evaluating the leading terms in the Plefka expansion or using some other approximations) might reduce these unnatural behaviors. The second problem is the optimal setting $\alpha = N/n$. Empirically, we found that $\alpha_{\mrm{opt}} \approx 0.4$ when $H_{\mrm{true}} = 0$ and that it decreases as $H_{\mrm{true}}$ increases (e.g., $\alpha_{\mrm{opt}} \approx 30/n$ when $H_{\mrm{true}} = 0.2$ and $\alpha_{\mrm{opt}} \approx 5/n$ when $H_{\mrm{true}} = 0.4$). As can be seen in the results of our experiments, the solution to our method is robust for the choice of $\alpha$ when $J_{\mrm{true}}$ is small ($J_{\mrm{true}} < J_c$) and is sensitive to it when $J_{\mrm{true}}$ is large ($J_{\mrm{true}} > J_c$), where $J_c \approx 0.4$. The estimation of $\alpha_{\mrm{opt}}$ is very important for our method, and it will make our method more practical. This problem would be strongly related to the first problem. The third problem is the degradation of the estimation accuracy in the spin-glass phase. In our experiments, the estimation accuracies of $\hat{\gamma}$ and $\hat{H}$ were obviously degraded in the spin-glass phase. This means that our Plefka expansion based on the RS assumption loses its validity in the spin-glass phase. In Ref. [@YKT2012], a Plefka expansion for the one-step RSB was proposed. Employing this expansion instead of the current expansion could reduce the degradation in the spin-glass phase. These three problems should be addressed in our future studies. In this study, we used fully-connected Boltzmann machines whose variables are all visible. We are also interested in an extension of our method to other types of Boltzmann machines such as Boltzmann machines having specific structures or hidden variables. Furthermore, we considered the model-matched case (i.e., the case in which the generative mode and learning model are the same model) in the current study, but model-mismatched cases are more practical and important. Gibbs Free Energy {#sec:app:GibbsFreeEnergy} ================= In this appendix, we derive the Gibbs free energy for the replicated (Helmholtz) free energy in Eq. (\[eqn:ReplicatedFreeEnergy\]). The replicated free energy is obtained by minimizing the variational free energy, defined by $$\begin{aligned} f[Q]:=\sum_{\mcal{S}_x}E_x(\mcal{S};H,\gamma)Q(\mcal{S}_x) + \sum_{\mcal{S}_x}Q(\mcal{S}_x)\ln Q(\mcal{S}_x), \label{eqn:VariationalFreeEnergy}\end{aligned}$$ under the normalization constraint, i.e., $\sum_{\mcal{S}_x}Q(\mcal{S}_x) = 1$, where $Q(\mcal{S}_x)$ is a test distribution over $\mcal{S}_x$, and $E_x(\mcal{S}_x;H,\gamma)$ is the Hamiltonian for the replicated system defined in Eq. (\[eqn:ReplicatedHamiltonian\]). The Gibbs free energy is obtained by adding new constraints to the minimization of $f[Q]$. Here, we add the relation $(n\tau_x)^{-1}\sum_{i=1}^n\sum_{a = 1}^{\tau_x}\sum_{\mcal{S}_x}S_i^{\{a\}}Q(\mcal{S}_x) = m$ as the constraint. By using Lagrange multipliers, the Gibbs free energy is obtained as $$\begin{aligned} G_x(m,H,\gamma)&:=\extr_{Q,\lambda, r} \Big\{ f[Q] - r \Big(\sum_{\mcal{S}_x}Q(\mcal{S}_x) - 1\Big)\nn &-\lambda \Big(\sum_{i=1}^n\sum_{a = 1}^{\tau_x}\sum_{\mcal{S}_x}S_i^{\{a\}}Q(\mcal{S}_x) - n \tau_x m\Big)\Big\}, \label{eqn:def_GibbsFreeEnergy}\end{aligned}$$ where “$\extr$” denotes the extremum with respect to the assigned parameters. By performing the extremum operation with respect to $Q(\mcal{S})$ and $r$ in Eq. (\[eqn:def\_GibbsFreeEnergy\]), we obtain $$\begin{aligned} &G_x(m,H,\gamma) \nn &= \extr_{\lambda}\Big\{\lambda n \tau_x m -\ln \sum_{\mcal{S}_x}\exp\big( - E_x(\mcal{S}_x;H+\lambda,\gamma)\big)\Big\}. \label{eqn:def_GibbsFreeEnergy_trans}\end{aligned}$$ The replicated free energy in Eq. (\[eqn:ReplicatedFreeEnergy\]) coincides with the extremum of this Gibbs free energy with respect to $m$; i.e., $$\begin{aligned} F_x(H, \gamma) = \extr_{m}G_x(m,H,\gamma). \label{eqn:relation_F&G}\end{aligned}$$ By performing the shift $H + \lambda \to \lambda$ in Eq. (\[eqn:def\_GibbsFreeEnergy\_trans\]), we obtain Eq. (\[eqn:GibbsFreeEnergy\]). Derivation of Coefficients of Plefka Expansion {#sec:app:PlefkaExpansion} ============================================== The Plefka expansion considered in this study can be obtained by expanding the Gibbs free energy in Eq. (\[eqn:GibbsFreeEnergy\]) around $\gamma = 0$. When $\gamma = 0$, we have $$\begin{aligned} G_x(m,H,0) &=- n \tau_x H m+ n \tau_x\extr_{\lambda}\big(\lambda m - \ln 2\cosh \lambda\big) \nn &=- n \tau_x H m + n \tau_x e(m), \label{eqn:PlefkaExpansion_0th}\end{aligned}$$ where $e(m)$ is defined in Eq. (\[eqn:MeanFieldEntropy\]). For the derivations of the coefficients $\phi_x^{(1)}(m)$ and $\phi_x^{(2)}(m)$, we decompose $E_x(\mcal{S}_x;H,\lambda)$ in Eq. (\[eqn:GibbsFreeEnergy\]) into two parts: $$\begin{aligned} E_x(\mcal{S}_x;\lambda,\gamma)=-\lambda \sum_{i=1}^n\sum_{a=1}^{\tau_x} S_i^{\{a\}} + \gamma E_x^{\mrm{int}}(\mcal{S}_x),\end{aligned}$$ where $$\begin{aligned} E_x^{\mrm{int}}(\mcal{S}_x)&:=- \frac{N}{n}\sum_{i<j}d_{ij} \sum_{a = 1}^{\tau_x}S_i^{\{a\}}S_j^{\{a\}} \nn \aldef - \frac{1}{n}\sum_{i<j}\sum_{a < b}S_i^{\{a\}}S_j^{\{a\}}S_i^{\{b\}}S_j^{\{b\}}.\end{aligned}$$ Coefficient $\phi_x^{(1)}(m)$ is defined by $$\begin{aligned} \phi_x^{(1)}(m):=\frac{1}{nN}\frac{\partial G_x(m,H,\gamma)}{\partial \gamma} \Big|_{\gamma = 0}.\end{aligned}$$ The derivative leads to $$\begin{aligned} \frac{\partial G_x(m,H,\gamma)}{\partial \gamma}&=\Ave{E_x^{\mrm{int}}(\mcal{S}_x)}_{\gamma}, \label{eqn:1st-derivative_Gx}\end{aligned}$$ where $\ave{\cdots}_{\gamma}$ denotes the average for the distribution $$\begin{aligned} P(\mcal{S}_x \mid \gamma, m) \propto \exp\big( - E_x(\mcal{S}_x;\lambda^*,\gamma)\big),\end{aligned}$$ where $\lambda^*$ is the value of $\lambda$ that satisfies the extremum condition in Eq. (\[eqn:GibbsFreeEnergy\]) and which is the function relating $\gamma$ and $m$; i.e., $\lambda^* = \lambda^*(\gamma,m)$. From the extremum condition for $\lambda$ in Eq. (\[eqn:GibbsFreeEnergy\]), we obtain the equation $$\begin{aligned} m = \frac{1}{n\tau_x}\sum_{i=1}^n\sum_{a = 1}^{\tau_x}\ave{S_i^{\{a\}}}_{\gamma}, \label{eqn:Extr_lambda}\end{aligned}$$ which holds for any $\gamma$. In the derivation of Eq. (\[eqn:1st-derivative\_Gx\]), we used Eq. (\[eqn:Extr\_lambda\]). When $\gamma = 0$, Eq. (\[eqn:Extr\_lambda\]) reduces to $m = \tanh \lambda^*$. This means that $\ave{S_i^{\{a\}}}_0 = m$ for any $i$ and $a$. Therefore, we obtain $$\begin{aligned} \phi_x^{(1)}(m) &=- \frac{x(n-1) N C_1}{2n} m^2 - \frac{(n-1) K_x}{2n N}m^4, \label{eqn:PlefkaExpansion_1st}\end{aligned}$$ where $K_x := \tau_x(\tau_x - 1) / 2$. In the derivation of Eq. (\[eqn:PlefkaExpansion\_1st\]), we used the relation $\ave{S_i^{\{a\}}S_j^{\{b\}}}_0 = \ave{S_i^{\{a\}}}_0\ave{S_j^{\{b\}}}_0$ if $i\not= j$ or $a \not=b$. The coefficient $\phi_x^{(2)}(m)$ is defined by $$\begin{aligned} \phi_x^{(2)}(m):=\frac{1}{2nN}\frac{\partial^2 G_x(m,H,\gamma)}{\partial \gamma^2} \Big|_{\gamma = 0}.\end{aligned}$$ From Eq. (\[eqn:1st-derivative\_Gx\]), the second derivative is $$\begin{aligned} \frac{\partial^2 G_x(m,H,\gamma;\mcal{D})}{\partial \gamma^2} &=\frac{\partial}{\partial \gamma}\AVE{E_x^{\mrm{int}}(\mcal{S}_x)}_{\gamma}\nn &=\AVE{E_x^{\mrm{int}}(\mcal{S}_x)U_x(\gamma)}_{\gamma}, \label{eqn:2nd-derivative_Gx}\end{aligned}$$ where $$\begin{aligned} U_x(\gamma)&:= \AVE{\frac{\partial E_x(\mcal{S}_x;\lambda^*,\gamma)}{\partial \gamma}}_{\gamma} -\frac{\partial E_x(\mcal{S}_x;\lambda^*,\gamma)}{\partial \gamma} %\label{eqn:GeorgesOperator}\end{aligned}$$ is Georges’s operator, proposed in Ref. [@Georges1991]. To simplify the notation, we omit the explicit description of the dependency of the operator on $\mcal{S}_x$ and $m$. By using this operator, the derivative of $\ave{A}_{\gamma}$ with respect to $\gamma$ is obtained as $$\begin{aligned} \frac{\partial \ave{A}_{\gamma}}{\partial \gamma} = \AVE{\frac{\partial A}{\partial \gamma}}_{\gamma} + \AVE{A U_x(\gamma)}_{\gamma}. \end{aligned}$$ This immediately leads to $\ave{S_i^{\{a\}} U_x(\gamma)}_{\gamma} = 0$, because $\partial \ave{S_i^{\{a\}}}_{\gamma} /\partial \gamma = \partial m / \partial \gamma = 0$. Therefore, $$\begin{aligned} \AVE{U_x(\gamma)^2}_{\gamma} &=-\AVE{U_x(\gamma)\frac{\partial E_x(\mcal{S}_x,\lambda^*,\gamma)}{\partial \gamma}}_{\gamma}\nn &=-\AVE{E_x^{\mrm{int}}(\mcal{S}_x)U_x(\gamma)}_{\gamma} \label{eqn:Ux^2}\end{aligned}$$ is obtained, where we have used $\ave{U_x(\gamma)}_{\gamma} = 0$. From Eqs. (\[eqn:2nd-derivative\_Gx\]) and (\[eqn:Ux\^2\]), we have $$\begin{aligned} \frac{\partial^2 G_x(m,H,\gamma)}{\partial \gamma^2} =-\AVE{U_x(\gamma)^2}_{\gamma}. \label{eqn:2nd-derivative_Gx_OperatorRepresentation}\end{aligned}$$ Because $$\begin{aligned} \frac{\partial \lambda^*}{\partial \gamma} \Big|_{\gamma = 0} &= \frac{1}{n \tau_x}\frac{\partial }{\partial \gamma}\frac{\partial G_x(m,H,\gamma)}{\partial m}\Big|_{\gamma = 0} =\frac{N}{\tau_x}\frac{\partial \phi_x^{(1)}(m)}{\partial m},\end{aligned}$$ when $\gamma = 0$, we obtain $$\begin{aligned} &U_x(0)=\frac{(n-1)N}{n}\sum_{i=1}^n \omega_i m\sum_{a = 1}^{\tau_x}\big(S_i^{\{a\}} - m\big)\nn &+\frac{N}{n}\sum_{i<j}\Big(d_{ij} + \frac{\tau_x - 1}{N}m^2\Big)\sum_{a = 1}^{\tau_x}\big(S_i^{\{a\}} - m\big)\big(S_j^{\{a\}} - m\big)\nn &+\frac{1}{n}\sum_{i<j}\sum_{a<b}\big(S_i^{\{a\}}S_j^{\{a\}} - m^2\big)\big(S_i^{\{b\}}S_j^{\{b\}} - m^2\big), \label{eqn:U0}\end{aligned}$$ where $\omega_i$ is defined in Eq. (\[eqn:def\_omega\_i\]). By using Eqs. (\[eqn:2nd-derivative\_Gx\_OperatorRepresentation\]) and (\[eqn:U0\]), we obtain $$\begin{aligned} \phi_x^{(2)}(m)&=-\frac{(n-1)^2 \tau_x N \Omega}{2n^2}m^2(1 - m^2) -\frac{(n-1)\tau_x N C_2}{4n^2}(1-m^2)^2 - \frac{(n-1)K_x C_1}{n^2}m^2(1-m^2)^2\nn \aleq -\frac{(n-1)K_x}{2n^2 N}\big(n + \tau_x-3\big)m^4(1-m^2)^2 -\frac{(n-1)K_x}{4n^2 N}(1 - m^4)^2, \label{eqn:PlefkaExpansion_2nd}\end{aligned}$$ where $\Omega$ is defined in Eq. (\[eqn:def\_Omega\]). Evaluation of Orders of Each Term in the Empirical Bayes Likelihood {#sec:app:OrdarEvaluation} =================================================================== Here, we evaluate the orders of each term in Eq. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\]) with $m = M$, with respect to $n\gg 1$, that is, the orders of each term in $$\begin{aligned} L_{\mrm{EB}}(H,\gamma)&\approx e(M)- \Phi(M)\gamma -\phi_{-1}^{(2)}(M)\gamma^2. \label{eqn:EmpiricalBayesLikelihood_Gauss_Result_M}\end{aligned}$$ In the following, we assume that $N = O\big(n^{\rho} \big)$ ($\rho \geq 0$) and that $\{\mrm{S}_i^{(\mu)}\}$ are i.i.d. samples from a certain distribution. First, we consider the case $H_{\mrm{true}} = 0$ in which the distribution of $\{\mrm{S}_i^{(\mu)}\}$ is unbiased. In this case, we obtain $M = O\big(n^{-(1+\rho)/2} \big)$, $C_1 = O\big(n^{-1-\rho/2}\big)$, and $$\begin{aligned} C_2 &= \frac{1}{N} + \frac{1}{n(n-1) N^2}\sum_{\mu < \nu} \sum_{i < j}\mrm{S}_i^{(\mu)}\mrm{S}_j^{(\mu)}\mrm{S}_i^{(\nu)}\mrm{S}_j^{(\nu)}\nn &=O\big(n^{-\rho} \big).\end{aligned}$$ Similarly, we obtain $$\begin{aligned} \Omega&=\frac{1}{n(n-1)^2 N^2}\sum_{i=1}^n\sum_{\mu,\nu = 1}^N \sum_{j,k \in \partial(i)}\mrm{S}_i^{(\mu)}\mrm{S}_j^{(\mu)}\mrm{S}_i^{(\nu)}\mrm{S}_k^{(\nu)} - C_1^2\nn &=\frac{1}{(n-1)N} + \frac{1}{n(n-1)^2 N^2}\sum_{i=1}^n\sum_{\mu=1}^N \sum_{j\neq k \in \partial(i)}\mrm{S}_j^{(\mu)}\mrm{S}_k^{(\mu)}\nn &+ \frac{1}{n(n-1)^2 N^2}\sum_{i=1}^n\sum_{\mu\neq \nu} \sum_{j,k \in \partial(i)}\mrm{S}_i^{(\mu)}\mrm{S}_j^{(\mu)}\mrm{S}_i^{(\nu)}\mrm{S}_k^{(\nu)}- C_1^2\nn &=O\big(n^{-1-\rho} \big),\end{aligned}$$ because $C_1^2 = O\big( n^{-2-\rho}\big)$. Using the above results and Eqs. (\[eqn:MeanFieldEntropy\]), (\[eqn:Phi(m)\]), and (\[eqn:PlefkaExpansion\_2nd\_x=-1\]), we obtain $e(M) = O(1)$, $\Phi(M) = O(1)$, and $\phi_{-1}^{(2)}(M) = O\big(n^{\rho-1}\big)$, respectively. Therefore, when $\rho = 1$, the orders of all the terms in Eq. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\_M\]) are just $O(1)$ with respect to $n$. Next, we consider the case $H_{\mrm{true}} \neq 0$ in which the distribution of $\{\mrm{S}_i^{(\mu)}\}$ is biased. In this case, $M$, $C_1$, and $C_2$ are $O(1)$, and furthermore, $\Omega$ is $O(1)$ because $\omega_i = O(1)$. This leads to $e(M) = O(1)$, $\Phi(M) = O\big(n^{\rho}\big)$, and $\phi_{-1}^{(2)}(M) = O\big(n^{2\rho}\big)$. Therefore, when $\rho = 0$, the orders of all the terms in Eq. (\[eqn:EmpiricalBayesLikelihood\_Gauss\_Result\_M\]) are just $O(1)$ with respect to $n$. This consideration and the experiments in Sec. \[sec:experiment\] imply that our method based on the Plefka expansion can be validated when all the terms in the empirical Bayes likelihood are $O(1)$. The introduction of the external field changes the condition to satisfy this criterion, leading to the appropriate scaling of $\alpha$. This statement is consistent with the numerical observation that a stable result is obtained even for different $n$’s as long as the appropriate scale in $\alpha$ is maintained, as shown in Sec. \[sec:experiment\]. Acknowledgment {#acknowledgment .unnumbered} -------------- This work was partially supported by JSPS KAKENHI (Grant Numbers: 15H03699, 18K11459, 18H03303, 25120013, and 17H00764), JST CREST (Grant Number: JPMJCR1402), and the COI Program from the JST (Grant Number JPMJCE1312). TO is also supported by a Grant for Basic Science Research Projects from the Sumitomo Foundation.

--- abstract: 'Magnetic resonance imaging (MRI) has great potential to improve prostate cancer diagnosis. It can spare men with a normal exam from undergoing invasive biopsy while making biopsies more accurate in men with lesions suspicious for cancer. Yet, the subtle differences between cancer and confounding conditions, render the interpretation of MRI challenging. The tissue collected from patients that undergo pre-surgical MRI and radical prostatectomy provides a unique opportunity to correlate histopathology images of the entire prostate with MRI in order to accurately map the extent of prostate cancer onto MRI. Such mapping will help improve existing MRI interpretation schemes, e.g. PIRADS, and will facilitate the development of quantitative image analysis methods to assess the imaging characteristics of prostate cancer on MRI. Here, we introduce the RAPSODI (diology athology patial pen-Source multi-imensional ntegration) framework for the registration of radiology and pathology images. RAPSODI relies on a three-step procedure that first reconstructs in three dimensions (3D) the resected tissue using the serial whole-mount histopathology slices, then registers corresponding histopathology and MRI slices, and finally maps the cancer outlines from the histopathology slices onto MRI. We tested RAPSODI in a phantom study where we simulated various conditions, e.g., tissue specimen rotation upon mounting on glass slides, tissue shrinkage during fixation, or imperfect slice-to-slice correspondences between histopathology and MRI images. Our experiments showed that RAPSODI can reliably correct for rotations within $\pm15^{\circ}$ and shrinkage up to 10%. We also evaluated RAPSODI in 89 patients from two institutions that underwent radical prostatectomy, yielding 543 histopathology slices that were registered to corresponding T2 weighted MRI slices. We found a Dice similarity coefficient of 0.98$ \pm $0.01 for the prostate, prostate boundary Hausdorff distance of 1.71$ \pm $0.48 mm, a urethra deviation of 2.91$ \pm $1.25 mm, and a landmark deviation of 2.88$ \pm $0.70 mm between registered histopathology images and MRI. Our robust framework successfully mapped the extent of disease from histopathology slices onto MRI and created ground truth labels for characterizing prostate cancer on MRI. Our open-source RAPSODI platform is available as a 3D Slicer plugin or as a stand-alone program and can be downloaded from <https://github.com/pimed/Slicer-RadPathFusion>.' author: - 'Mirabela Rusu[^1]' - 'Christian A. Kunder' - 'Nikola C. Teslovich' - 'Jeffrey B. Wang' - 'Rewa R. Sood' - Wei Shao - 'Leo C. Chen' - Robert West - Richard Fan - Pejman Ghanouni - 'James D. Brooks' - 'Geoffrey A. Sonn' bibliography: - 'draft.bib' title: 'Registration of pre-surgical MRI and whole-mount histopathology images in prostate cancer patients with radical prostatectomy via RAPSODI' --- Keywords: radiology-pathology registration $|$ prostate cancer $|$ whole-mount histopathology $|$ radical prostatectomy $|$ magnetic resonance imaging Introduction ============ Despite advances in diagnosis and treatment, prostate cancer remains the second leading cause of cancer death in American men [@siegel_cancer_2019]. Overdiagnosis of low-grade cancers that do not require treatment and the underdiagnosis of aggressive cancers are still a concern [@futterer_can_2015], even after the changes in the recommendation of prostate biopsy for elevated Prostate Specific Antigen (PSA). Magnetic Resonance Imaging (MRI) can help address all of these problems [@ahmed_diagnostic_2017]. When MRI is normal, up to 50% of men can safely avoid prostate biopsy, thereby reducing overdiagnosis of low-grade cancer and infectious complications of biopsy. However, this is only true when MRI is interpreted by world-leading experts [@van_der_leest_head--head_2019]. In practice, lack of widespread expertise and alarming levels of inter-reader variation greatly reduce the potential of MRI to revolutionize prostate cancer diagnosis [@sonn_prostate_2017]. Both false negatives and false positives, even when using the recommended PIRADS reporting system [@weinreb_pi-rads_2016], are very common and the vast majority of men who undergo MRI still undergo biopsy. Finally, MRI has yet to supplant biopsy which is still required to confirm the presence and aggressiveness of prostate cancer [@barentsz_synopsis_2016]. In men diagnosed with prostate cancer on biopsy, radical prostatectomy remains the most common treatment [@cooperberg_trends_2015]. The resected prostate provides a unique opportunity to correlate pre-surgical MRI with digitized histopathology images and map the exact extent of cancer from histopathology images onto MRI. Developing a large dataset of prostatectomy cases where cancer and Gleason grade is accurately mapped on MRI has two potentially transformative applications. First is helping to improve existing MRI interpretation schemes that are still affected by many false positive and false negative findings. Second, it may facilitate the development of machine learning methods to identify prostate cancer on MRI by accurately labeling of cancer for model training and validation. [|m[0.8in]{}|m[0.5in]{}|m[1.70in]{}|m[1.2in]{}|m[0.6in]{}|m[0.8in]{}|]{} Publication & Subject \# & Approach & Additional Input & Dice Coef. & Landmark Error (mm)\ Park 2008 [@park_registration_2008] & 2& 3D reconstruction + affine and TPS registration & block face picture, ex vivo MRI & NA & 3-3.74\ Chappelow 2011[@chappelow_elastic_2011] & 25 & Feature Based Mutual Information + BSpline & - & NA & NA\ Ward 2012 [@ward_prostate:_2012] & 13 & 2D Affine + TPS Registration & Strand-shaped fiducials, Ex vivo MRI & NA & 1.1\ Kalavagunta 2014 [@kalavagunta_registration_2015] & 35 & Local affine registration & Internal landmarks, 3D Printed Molds & 0.99 & 1.54$\pm$0.64\ Reynolds 2015 [@reynolds_development_2015] &6& 2D TPS registration + deformable registration & Control Points, ex vivo MRI, sectioning box & 0.93 & 3.3\ Li 2017 [@li_co-registration_2017] & 19 & Multi-Scale Representation + deformable registration & - & 0.96$\pm$0.01 &2.96$\pm$0.76\ Losnegard 2018 [@losnegard_intensity-based_2018] & 12 & 3D histopathology reconstruction, 3D affine and deformable registration & - & 0.94 & 5.4\ Wu 2019 [@wu_system_2019] & 17 & 2D Rigid, TPS Registration (automatic landmarks) & ex vivo MRI, 3D printed molds & 0.87$\pm$0.04&2.0$\pm$0.5\ Rusu 2019 [@rusu_framework_2019] & 15 & 3D histopathology reconstruction, 2D Affine+Deformable & 3D printed Molds &0.94$\pm0.02$& 1.11$\pm$0.34\ Although numerous approaches for the radiology-pathology registration in the prostate have been introduced (see section “Prior Work“), these approaches have not been widely adopted and have not been carefully tested by scientists outside the developer teams. Recent publications using histopathology images as reference to improve MRI and automatically detect cancer [@penzias_identifying_2018; @hurrell_optimized_2018; @sumathipala_prostate_2018; @cao_joint_2019; @reynolds_voxel-wise_2019] still use manual approaches to align the histopathology to MRI images, which are known to be labor-intensive and subjective. The reduced adoption of previous methods is due to the challenges associated with managing and registering the histopathology and Magnetic Resonance (MR) images, the lack of open source release of existing methods, and the time constraints associated with running these methods. Specifically, the registration of histopathology images and prostate MRI has the following challenges. Histologic processing of the resected tissue causes artifacts, e.g., deformations, shrinkage, and tissue ripping. Some of these artifacts (e.g., deformation and shrinking) can be corrected through registration, while others (e.g. tissue ripping) are challenging to correct and may result in discarding slices when such artifacts are major. Furthermore, our method and many others [@kalavagunta_registration_2015; @reynolds_development_2015; @wu_system_2019] assume slice-to-slice correspondence between histopathology and MRI images, which can be improved through the use of customized 3D printed molds based on pre-operative MRI [@turkbey_multiparametric_2011]. However, this approach requires a change in clinical protocol that is not present in the vast majority of institutions performing radical prostatectomy. Finally, the acquired data is different between the histopathology images and MRI. Histopathology images provide a discontinuous serial stack of 4$\mu m$ high-resolution colored images with a pixel size of 0.0005 mm separated by roughly 4 mm spaces, while MRI has a typical resolution of 0.4$\times$0.4$\times$4 mm$^3$. Here, we introduce the RAPSODI (diology athology patial pen-Source multi-imensional ntegration) framework for the registration of histopathology slices and pre-operative MRI. RAPSODI includes a dictionary-based data management system, a memory-efficient registration methodology and a Graphical User Interface Plugin to 3D Slicer [@fedorov_3d_2012]. Our registration approach relies on the 3D reconstruction of the histopathology specimen to create a digital representation of the tissue before gross sectioning. Next, RAPSODI registers corresponding histopathology and MRI slices. Finally, the optimized transforms are applied to the cancer regions outlined on the histopathology images to project those labels onto the pre-operative MRI. We evaluated our methodology using a digital phantom study where we simulated various conditions resulting from the histologic preparation of the excised tissue, e.g., rotation of the tissue when mounting on the glass slide or shrinking of the tissue. Moreover, we tested RAPSODI in 89 prostate cancer patients that underwent radical prostatectomy from two institutions, ours and a public cohort [@madabhushi_fused_2016]. RAPSODI is open-source and can be downloaded from <https://github.com/pimed/Slicer-RadPathFusion>, while the phantom data is available at <https://github.com/pimed/rad-path-phantom>. Prior Work and Our Contribution ------------------------------- Although numerous automated approaches for the registration of radiology and histopathology images have been developed, manual approaches are still employed, even in recent publications [@penzias_identifying_2018; @hurrell_optimized_2018; @sumathipala_prostate_2018; @cao_joint_2019; @reynolds_voxel-wise_2019]. Some manual or semi-automatic approaches utilize landmark-based registration approaches, either alone [@penzias_identifying_2018; @hurrell_optimized_2018; @reynolds_voxel-wise_2019] or in combination with automated registration steps [@reynolds_development_2015; @hurrell_optimized_2018]. These approaches are labor-intensive and require the human operator to possess expertise in both MRI and histopathology, and necessitate identification of corresponding landmarks on both modalities. Other such approaches [@turkbey_multiparametric_2011; @sumathipala_prostate_2018] employ cognitive alignment in which a radiologist with the help of a pathologist directly outlines the cancer region on MRI considering the histopathology images as reference. Such methods are tedious to apply and may be prone to underestimating the dimensions of the lesion [@priester_magnetic_2017] while MRI invisible lesions are hard if not impossible to outline and thereby they are often omitted from follow-up analysis. A few approaches use interactive image transformations [@costa_improved_2017], in which a user indicate scaling, rotations and translations to be applied to the images. Such approaches are also tedious to utilize and require extensive knowledge in both radiology and pathology of the prostate. The automated registration of histopathology images with pre-surgical prostate MRI has been performed in proof-of-concept studies, which usually only include a small number of subjects, often &lt; 20 ( \[tab:prior\_work\]). Most approaches assume a slice-to-slice correspondence between the histopathology images and T2 weighted (T2w) MRI slices. Some partial correspondence commonly results from the gross sectioning of the prostate in histologic preparation which is done perpendicular to the urethra. Yet, more advanced methods have been introduced to enforce such correspondences. For example, three dimensional (3D) printed patient-specific molds [@turkbey_multiparametric_2011] have been used [@kalavagunta_registration_2015; @reynolds_development_2015; @wu_system_2019] to help preserve the correspondences during tissue sectioning. Some studies additionally included blockface picture [@park_registration_2008], ex vivo MRI [@reynolds_development_2015; @wu_system_2019; @park_registration_2008; @ward_prostate:_2012] or external fiducials [@ward_prostate:_2012] to help improve the accuracy of the registration. Yet, these approaches required modifications of the clinical protocols usually resulting in only a small number of subjects to be recruited for such research studies. Once correspondences between the histopathology images and T2w MRI are identified, their registration can still be challenging, partially due to the artifacts induced by the tissue preparation. Textural features [@chappelow_elastic_2011; @li_co-registration_2017] have been proposed, yet they may be cumbersome to use due to the high-dimensional scoring function optimization and the choice of textural features. Other approaches rely solely on image intensity to drive the deformable alignment [@losnegard_intensity-based_2018; @rusu_framework_2019], but require accurate affine alignment prior to the deformable registration. Previous work in the lung [@rusu_framework_2015; @rusu_co-registration_2017], breast [@rusu_rad-path_2019] or prostate [@losnegard_intensity-based_2018; @rusu_framework_2019], has relied on approaches that reconstruct the sequential histopathology slices and created a 3D volume representing the histopathology specimen prior to sectioning, which facilitates the spatial registration with the 3D volumetric MRI and alleviates the need for slice correspondences. However, these methods are prone to overfitting the histopathology reconstruction due to the large number of degrees of freedom and may suffer from partial volume effects due to the missing data associated with thick MRI slices and the histopathology slice spacing. Our approach makes the following contributions: 1) Our registration methodology combines a 3D reconstruction of the histopathology specimen with 2D affine and deformable registration of corresponding histopathology and MRI slices and was optimized for an accurate alignment, 2) Our approach was tested in a digital phantom where the ground truth is known as well as in the largest cohort considered to date in a radiology-pathology registration study, and 3) To the best of our knowledge, we are the first to release the source code for the registration of histopathology and radiology images in the prostate, [which is essential to test the reproducibility and robustness of the approach while allowing the wide adaption.]{} Methods {#methods .unnumbered} ======= Notations {#notations .unnumbered} --------- Let $\mathcal{M}$ be the T2 weighted (T2w) MRI image, $\mathcal{M}:\mathcal{R}^3 \rightarrow \mathcal{R}$ and has a matrix size $K \times L \times M$, where $K$, $L$ and $M$ represent the width, height and number of slices of the axial T2w MRI. Let $\mathcal{H}$ be the stack of histopathology slices, $\mathcal{H}:\mathcal{R}^3\rightarrow\mathcal{R}^3$, obtained by stacking 2D histopathology slices, $\mathcal{H}_i:\mathcal{R}^2\rightarrow\mathcal{R}^3$. The histopathology images are colored images, with Red, Green and Blue channels. The volume $\mathcal{H}$ has $W\times H\times D$ voxels with three components corresponding to the Red, Green and Blue channels, where $D \in [3,9]$ represents the number of slices, while $W$ and $H$ are the width and height of the histopathology images. The index $i$, is used to indicate either an axial slice within the MRI volume or an image in the histopathology stack. $\mathcal{M}^{Pr}$ and $\mathcal{H}^{Pr}$ represent the prostate segmentation on MRI and histopathology images, respectively. [|m[1.0in]{}|m[1.10in]{}|m[1.00in]{}|m[1.10in]{}|m[1.00in]{}|]{} & &\ Variable & MRI & Pathology & MRI & Pathology\ Manufacturer: Coil type & GE: Surface & - & Siemens: Endorectal & -\ Sequence/Data Type & T2w & whole-mount & T2w & pseudo-whole mount\ Acquisition & TR: \[3.9, 6.3\]; & H&E & TR: \[3.7, 7.0\]; &H&E\ Characteristics & TE: \[122, 130\] & & TE: \[107\] &\ Number of Patients/Slices & 74/1994 & 74/478 & 16/430 & 16/65\ Matrix Size & $K,L\in[256,512]$ $M\in[24,43]$ & W,H $\in$ \[1663,7556\] & $K,L = 320$, $M \in [21,31]$ & W,H $\in$ \[2368,6324\]\ Pixel Spacing (mm) & $\in$ \[0.27,0.94\] & $\in$ {0.0081,0.0162} & $\in$ \[0.41,0.43\] & 0.0072\*\ Distance Between Slices (mm) & $\in$ \[3,5.2\] & Same as MRI via 3D printed mold & 4 & Free hand\ Annotations & Prostate, Anatomic Landmarks & Prostate, Anatomic Landmarks, Cancer & Prostate, Cancer, Urethra & Prostate, Urethra, Cancer\ Data Description {#data-description .unnumbered} ---------------- Our IRB approved study includes $N_1=73$ subjects from Stanford Hospital (Cohort C1) and $N_2=16$ patients from the “Prostate Fused MRI Pathology” collection, The Cancer Imaging Archive [@madabhushi_fused_2016] (Cohort C2) ( \[tab:data\]). A subset of 15 subjects from cohort C1 was previously utilized in [@rusu_framework_2019]. The subjects in C1 had an MRI acquired between 2016-2019 prior to the radical prostatectomy, and the excised prostate was submitted for histologic preparation to generate whole-mount sections. [ The subjects in C2 (description available at <https://wiki.cancerimagingarchive.net/display/Public/Prostate+Fused-MRI-Pathology>), underwent radical prostatectomy and had the prostate sectioned in quadrants before being submitted for histology processing.]{} *MRI:* The MRI exams for the patients in cohort C1 were acquired using 3 Tesla scanners (MR750, GE Healthcare, Waukesha, WI) with an external 32-channel body array coil without an endorectal coil. The imaging protocol included T2 weighted MRI (T2w), diffusion weighted imaging (DWI) and derived Apparent Diffusion Coefficient (ADC), and dynamic contrast-enhanced imaging sequences. For the patients in Cohort C2 were acquired on a 3T scanner (Siemens) using an endorectal coil. The public repository provides T2w MRI and dynamic contrast-enhanced MRI for these patients. In this study, we utilized the Axial T2w MRI, which is acquired using a 2D Spin Echo protocol ( \[tab:data\]) *Histopathology:* The cohort C1 patients, following resection, the prostate was fixed in formalin, sectioned using a patient-specific 3D printed mold built based on the pre-surgical MRI to maintain the correspondence of histopathology slices and T2w images, and embedded in paraffin. This process is now part of our clinical standard of care. The histopathology for the patients in cohort C2, was cut without the use of 3D printed molds, but seeking gross sectioning perpendicular to the urethra. Mounting of the 5$\mu m$ thick tissue on the glass slide can result in a rotation of the histopathology slice as well as mounting with either aligned or misaligned left-right orientation. To account for this variability, an expert indicated the gross rotation angle and whether the slice requires left-right flipping. The whole-mount slices in C1 and quadrants sections in C2 were stained using Hematoxylin & Eosin (H&E) and were digitized at 20x magnification (pixel size 0.5 $\mu m$). Pseudo-whole mounts were generated for the images in C2, by stitching adjacent quadrants as described in [@singanamalli_identifying_2016]. *Labels:* Our expert radiologist (PG) outlined the prostate on MRI, $\mathcal{M}^{Pr}$, while our expert pathologist (CK) outlined the prostate, $\mathcal{H}^{Pr}$, and the cancer on the high-resolution scanned images of the histopathology specimen. Two hundred fifty-seven matching anatomic landmarks were picked on both histopathology and radiology images for a subset of 12 subjects, targeting 3 landmarks for each corresponding pair of histopathology and MRI slices. Examples of anatomic landmarks include benign prostate hyperplasia nodules, ejaculatory ducts, predominant glands, etc. The urethra was outlined on 22 studies in cohort C1 and the 16 studies in cohort C2. Outlining the urethra on MRI is relatively straightforward at the apex of the prostate, yet it becomes challenging towards the base. Thereby, often urethra annotations are available on the MRI from the mid-gland to apex, but lacking between the mid-gland and the prostate base. Slice correspondences between the MRI and histopathology were identified by an expert urologist and a radiology-pathology registration expert and validated by a multi-disciplinary team of radiologists, pathologists and urologists. Radiology - Pathology Registration {#radiology---pathology-registration .unnumbered} ---------------------------------- Our approach is summarized in  \[fig:flowchart\] and described below: ![Summary of our approach. First, we align the serial histopathology slices relative to each other to reconstruct the 3D histopathology volume. Second, we register the histopathology slices relative to the T2w MRI using rigid, affine and deformable transforms. Finally, we map the extent of cancer from the histopathology images onto the radiology images.[]{data-label="fig:flowchart"}](figures/flowchart.pdf){width="\linewidth"} 1. **Pre-Processing:** We applied the prostate masks, $\mathcal{M}^{Pr}$ and $\mathcal{H}^{Pr}$ onto $\mathcal{M}$ and $\mathcal{H}$, respectively, to exclude the structures outside the prostate from image registration. The gross rotation angles or left-right flipping was applied as well. 2. **3D Histopathology Reconstruction:** We registered $\mathcal{H}_i$ relative each other, to ensure their 3D consistency within the 3D reconstruction, $\mathcal{H}$. We selected the middle slice $\mathcal{H}_i, i = \frac{M}{2}$ as the first fixed image and registered $\mathcal{H}_{i-1}$ to $\mathcal{H}_i$, $\mathcal{H}_{i-2}$ with $\mathcal{H}_{i-1}$, etc, as well as $\mathcal{H}_{i+1}$ to $\mathcal{H}_{i}$, $\mathcal{H}_{i+2}$ with $\mathcal{H}_{i+1}$, etc. With the exception of the middle slice, all histopathology images will have a corresponding rigid transform following the registration with the adjacent slice $\mathcal{T}^H_{Rig}$. 3. **2D Registration:** We registered $\mathcal{M}_i$ with $\mathcal{H}_i$, for $\forall i \in [1, D]$, by optimizing rigid $\mathcal{T}^M_{Rig}$, affine $\mathcal{T}^M_{Aff}$ and deformable $\mathcal{T}^M_{Def}$ transforms using gradient-based approaches. The rigid and affine registrations only use the prostate masks during the optimization and applied Sum of Square Differences as scoring function. The deformable registration used Free-Form Deformations [@rueckert_nonrigid_1999] to optimize the Mattes Mutual Information computed based on the image intensities. [We used a multi-resolution pyramid with three layers (with shrinking factors 16, 8, and 4 respectively, and a smoothing sigma of 4, 2, and 1 respectively). The affine transform optimization was done using a gradient descent optimizer with a learning rate of 0.01 and 250-500 iterations per resolution layer, while, the deformable registration employed a LBFGSB optimizer with 10-50 iterations per resolution layer.]{} 4. **Mapping Cancer onto MRI:** A composite transform of $\mathcal{T}^H_{Rig}$, $\mathcal{T}^M_{Rig}$, $\mathcal{T}^M_{Aff}$ and $\mathcal{T}^M_{Def}$ is applied to deform the histopathology image as well as the cancer label and anatomic landmarks into the coordinates of the T2w MRI. Our approach was developed using the Insight Toolkit (ITK) [@johnson_itk_2013]and its Simple ITK API in python. The approach is available as a 3D Slicer python plugin [@fedorov_3d_2012] ( \[fig:slicer\]) or as a stand-alone application to be run in batch mode. RAPSODI can be downloaded from <https://github.com/pimed/Slicer-RadPathFusion>. We measured the performance of the approach on an Intel i7-8700 CPU, 3.70GHz, 64GB Memory Computer. {width="\linewidth"} Digital Phantom for Radiology-Pathology Registration {#digital-phantom-for-radiology-pathology-registration .unnumbered} ---------------------------------------------------- [0.33]{} [0.33]{} [0.30]{} [0.31]{} [0.31]{} [0.31]{} We created a digital phantom to assess the quality of the alignment during the development of RAPSODI and to evaluate its performance when ground truth exists. The phantom is used to simulate artifacts known to affect the histopathology sample. We constructed the phantom by first outlining different prostatic regions, peripheral zone, cancer, and urethra in a 3D T2w MRI ( \[fig:Phantom\]a-c). Then, we synthesized the phantom T2 MRI by filling the segmented regions with the average intensities from the input T2 image ( \[fig:Phantom\]d). Moreover, we created the pathology phantom based on the histopathology images already registered to the T2w MRI (data not shown), by averaging their color intensities within the segmented regions ( \[fig:Phantom\]e-f). Our simulations included Gaussian noise on both the MRI and histopathology phantom slices. Using the T2w and pathology phantom, we tested three conditions: 1) the influence of the rotation angle when mounting the tissue slice on the glass slide, 2) the influence of shrinkage caused by fixation of the tissue during histology processing, and 3) the influence of imperfect slice correspondences between the MRI and histopathology slices, e.g.,  \[fig:Phantom\]d-e have a perfect correspondence, while  \[fig:Phantom\]d-f are 2mm apart from each other. To evaluate RAPSODI, we used one or multiple conditions and evaluated different quantitative metrics. Ten experiments were run for each condition to assess the mean and variance in performance of RAPSODI. When a random rotation of $r$ was assigned to the histopathology phantom, it resulted in applying a random angle ranging between $-r$ and $r$ to each slide and running 10 experiments with different noise and random angle conditions. When rotations were applied alone, no translation or scaling were applied. When a shrinkage factor $s$ is applied, all histopathology slices are shrunk by $s$ relative to their original appearance. Moreover, along with shrinking the images, we also apply a random translation of as much as 5% relative to the entire image in either x or y directions. Thereby, the experiments that include rotation and shrinkage also include random translation, and when combined with the imperfect slice correspondences represent the closest condition to the real data. Quantitative Evaluation {#quantitative-evaluation .unnumbered} ----------------------- The accuracy of the radiology-histopathology registration was evaluated using the Dice similarity coefficient, which assesses the overlap of the prostate outlined on T2w MRI and the outline of the prostate from the histopathology reconstruction: $$Dice (\mathcal{H}, \mathcal{M}) = \frac{1}{D}\sum^D_{i=1} \frac{2 \times |\mathcal{H}^{Pr}_i \cap \mathcal{M}^{Pr}_i|}{|\mathcal{H}^{Pr}_i| + |\mathcal{M}^{Pr}_i|} \label{eq:Dice}$$ where $D$ is the number of slices in the histopathology specimen, $\mathcal{H}^{Pr}_i$ represents the slice $i$ in the prostate segmentation on histopathology, while $\mathcal{M}^{Pr}_i$ represents the slice $i$ in the prostate segmentation on MRI. Additionally, we evaluated the Hausdorff distance between the prostate boundary, to asses how far the boundary is after performing the registration: $$\begin{split} Hausdorff^{Pr} (\mathcal{H}, \mathcal{M}) = max\{sup_{h \in \mathcal{H}^{Pr}} ~ inf_{m \in \mathcal{M}^{Pr}}d(\mathcal{H}, \mathcal{M}),\\sup_{m \in \mathcal{M}^{Pr}} inf_{h \in \mathcal{H}^{Pr}} d(\mathcal{H}, \mathcal{M}) \} \end{split} \label{eq:HDice}$$ where $sup$ represents the supremum operator and $inf$ represents the infimum operators. Moreover, we evaluated the landmark distance: $$Dist (\mathcal{L}^H,\mathcal{L}^M) = \frac{1}{X}\sum^X_{j=1}|\mathcal{L}^H_j,\mathcal{L}^M_j|_2 \label{eq:Dist}$$ where $|.|_2$ represents the Euclidean distance of the center of mass of the j landmark $\mathcal{L}^H_j$ on histopathology and center of mass of the j landmark $\mathcal{L}^M_j$ on MRI, while X represents the number of landmarks. Similarly, we computed the urethra distances, using per-slice correspondences, and slices where the urethra was visible on both MRI and histopathology slices. [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} Results {#results .unnumbered} ======= Phantom Study {#phantom-study .unnumbered} ------------- The phantom study is used to assess the average performance and variability of RAPSODI under conditions known to affect the tissue during the histopathology preparation. We ran our registration approach for 480 different conditions, to estimate the trends of the evaluation metrics as well as to their variations.  \[fig:PhantomResults\] summarizes our results in which we tested the effect of the rotation of histopathology slices while mounting on glass slide (range: 0-40$^\circ$), and the effect of shrinkage (range: 0-30%) when perfect slice correspondences exist between the histopathology and the MRI images in the phantom. Our approach is able to perfectly recover rotation angles ranging between 0-20$^\circ$ or shrinkage of 0-10% when applied alone ( \[fig:PhantomResults\]), indicated by the perfect $\tilde{}$ 1 dice coefficient and the sub-pixel error. When combined, either using 20% shrinkage and random rotation ( \[fig:PhantomResults\]b,f,j) or using 20$^\circ$ rotation and shrinkage ( \[fig:PhantomResults\]d,h,l), sub-pixel accuracy was observed for angles ranging between 0-15$^\circ$ or shrinkage of 0-5%. Beyond these conditions, RAPSODI is still able to recover induced rotation and shrinkage, yet with some misalignment as the initial starting conditions are far from the correct solution. Moreover, the limitations of the registration may be observed when perfect correspondences are lacking between the histopathology and MRI slices ( \[fig:PhantomResults3\]). Not surprising, the landmark and prostate border deviation are as large as $\sim$ 4 pixels (1.6 mm), as these features are not perfectly matching. Yet we can observe the relative stability of the approach for rotations ranging 0-30$^\circ$ and all tested shrinkage factors ranging between 0-30%, as the induced rotation and shrinkage are properly recovered. [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} Qualitative Results {#qualitative-results .unnumbered} ------------------- [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} [0.31]{} We applied RAPSODI to register the histopathology slices and T2w MRI in our radical prostatectomy cohorts of 89 patients.  \[fig:ResultsQualitativeC1\] shows the qualitative results for a subject in cohort C1 that had a Dice Coefficient of 0.98 and a Hausdorff distance of 1.75 mm of the prostate border ($\sim$ 4 pixels).  \[fig:ResultsQualitative2\] shows the same slice as  \[fig:ResultsQualitativeC1\] Raw 2, with the histopathology slice shown with progressive transparency from right-left ( \[fig:ResultsQualitative2\]a) and left to right ( \[fig:ResultsQualitative2\]b) to emphasize the alignment of the two modalities. The qualitative and quantitative evaluation suggest that a good alignment was obtained for this subject. The accurate registration allowed us to map the extent of two cancer foci with different Gleason groups ( \[fig:ResultsQualitative2\], blue - Gleason group 2; red - Gleason Group 3). Although the higher grade cancer is visible on MRI, its MRI visible borders are smaller than the histopathology projected lesion, confirming previous work showing that MRI underestimates actual tumor size [@priester_magnetic_2017]. The fusion enabled the mapping of the Gleason Group 2 cancer, which is not clearly visible on MRI, and would have been otherwise difficult to outline on MRI. [0.48]{} [0.48]{} [  \[fig:ResultsQualitativeC2\] shows a subject in cohort C2 for which the alignment of the prostate achieved a Dice coefficient of 0.98 and a Hausdorff distance of 2.50 mm on the prostate boundary. As for the results in  \[fig:ResultsQualitativeC1\]-\[fig:ResultsQualitative2\], the results for this subject are average and not outline. The five histopathology images ( \[fig:ResultsQualitativeC2\] Column 1) were registered with the MRI ( \[fig:ResultsQualitativeC2\] Column 2), and the cancer outline (red) was mapped onto MRI ( \[fig:ResultsQualitativeC2\] Columns 3-4). The public dataset includes the cancer annotation (blue) for this subject, which was obtained using a landmark-based registration [@singanamalli_identifying_2016]. The cancer annotations obtained via RAPSODI overlaps well with the labels provided by the dataset authors, with a dice overlap of 0.58 and a Hausdorff Distance of 2.71 mm. The relatively low overlap indicated by the dice coefficient may be accounted by the relatively small size of the tumor, and the misalignment of the regions in the apex slice ( \[fig:ResultsQualitativeC2\] Row 1).]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} [0.24]{} Quantitative Results {#quantitative-results .unnumbered} -------------------- An improvement in the alignment of the histopathology images and the T2w MRI can be observed across the different steps of our framework ( \[fig:ResultsQuantitative\]). Statistically significant differences in Dice coefficients and Hausdorff distances were found between the input and the results of the registration performed using RAPSODI(Mann-Whitney test is statistically significant for $\alpha<0.05$). [ These statistically significant differences were observed both between the input and the affine registration in step 2 as well as between the affine registration and the deformable registration in step 2, suggesting that both affine and deformable registrations are required to facilitate an accurate alignment.]{} The urethra ( \[fig:ResultsQuantitative\]c,f) and the landmark deviations failed to show a clear trend, but we observe a moderate decrease of landmark deviation between the input, 2.94$\pm$0.54 mm, and after applying RAPSODI, 2.88$\pm$0.7 mm, however, these differences were not statistically significant.. [ The comparison of results from cohorts C1 and C2 indicated that RAPSODI produces consistent results, with Dice coefficients on the prostate border of 0.98, and Hausdorff distances averaging 1.36-1.78 mm ( \[tab:ResultsQuantitative\]). The subjects in cohort C2 have MRIs acquired using an endorectal coil which causes larger deformations of the prostate. Thereby, the input data and affine registration results show worse alignment in cohort C2 compared to cohort C1. However, similar metrics are evaluated after the deformable registration in RAPSODI, suggesting that our approach generalizes even for larger deformations, as those induced by an endorectal coil.]{} [|m[0.50in]{}|m[0.7in]{}|m[1.2in]{}|m[1.0in]{}|m[1.2in]{}|m[0.8in]{}|]{} Cohort & Dice Prostate & Haussdorff Distance (mm) & Urethra Deviation (mm) & Landmarks Deviation (mm) & Dice Cancer\ C1 & 0.98$\pm$0.01 & 1.79$\pm$0.45 & 2.74$\pm$0.82 & 2.88$\pm$0.70 & -\ C2 & 0.98$\pm$0.01 & 1.37$\pm$0.47 & 3.14$\pm$1.71& - &0.53$\pm$0.18\ All & 0.98$\pm$0.01 & 1.71$\pm$0.48 & 2.91$\pm$1.25 & 2.88$\pm$0.70 & 0.53$\pm$0.18\ Additional evaluation was possible in cohort C2, since the authors of the dataset [@madabhushi_fused_2016] have provided the mapped cancer obtained via landmark-based registration [@singanamalli_identifying_2016]. Thereby, we compared the mapped cancer from RAPSODI with those provided by the dataset authors, and we observed a dice coefficient of $0.53\pm 0.18$ and deviation computed on the center of mass of $2.71\pm 1.31$ mm. The relatively reduced alignment of the cancer labels may be attributed to the general misalignment error, which is within 3 mm inside the prostate and 2 mm on the prostate border. This misalignment can have a significant effect on the value of the overlap evaluated via Dice coefficient for regions of small size, such as the cancer. Due to the use of stitched histopathology images, and of endorectal coil MRI, larger deformations needed to be recovered when aligning the histopathology images to MRI in the patients in cohort C2. The pseudo-whole mounts can have stitching artifacts that are absent in the whole-mount histopathology images. For example, the stitched pseudo-whole mount images are elongated in the anterior-posterior direction, e.g. slice C1234 of patient aaa0054. The affine parameters, i.e. scales, of the registration were relaxed, in order to enable the recovery of such large anisotropic stretching. Such modifications were only required for processing two patients, aaa0054 and aaa0072. [0.33]{} {width="\linewidth"} [0.33]{} {width="\linewidth"} [0.33]{} {width="\linewidth"} [0.33]{} {width="\linewidth"} [0.33]{} {width="\linewidth"} [0.33]{} {width="\linewidth"} In order to identify the optimal set of steps in our registration, we tested multiple combinations of processing steps (data not shown) for the subjects in cohort C1, e.g., skipping step 1 (histopathology reconstruction), or applying step 2 without performing the deformable registration. We found that the RAPSODI approach (Histopathology reconstruction followed by 2D affine and deformable transforms) achieves the highest accuracy, showing the highest Dice coefficients and lowest Hausdorff distances and landmark deviations compared to approaches where we skipped step 1 or the deformable registration in step 2. Discussion {#discussion .unnumbered} ========== Here, we introduced the RAPSODI platform that enables the registration of histopathology and MR images in the prostate. [RAPSODI first reconstructs the histopathology volume using pair-wise registration starting from the mid-gland slices towards the apex and base slices, respectively, followed by a slice-to-slice alignment between the corresponding histopathology and T2w images. The reconstruction ensures the consistent stacking of the histopathology slices relative to each other, independent of the MRI, which results in a better initialization of the histopathology slices in the registration with the MRI images.]{} We first evaluated RAPSODI in a digital phantom and showed that our framework can recover the rotation angles of the histopathology slices resulting from the slide mounting on glass slides when these angles are within 0-15$^\circ$ from the correct solution and with tissue shrinkage up to 10%. Correcting for large rotation angles can be achieved prior to applying RAPSODI either by using automated approaches, e.g., by aligning the major axis of the data [@rusu_framework_2019], or via manual approaches where the user indicates an angle, as was done in our study. The tissue shrinks during fixation with a factor that is outside our control. The affine transform helps identify the shrinkage factor, yet the accuracy of the registration declines as the initial conditions are further away from the optimal solution. Registration errors are most apparent at the prostate apex, where the prostate size, shape and textures are reduced. RAPSODI successfully registered histopathology images with corresponding T2w MR images in the 89 subjects (543 slices) achieving a prostate boundary error within 2 mm and an interior error within 3 mm. Through the use of the prostate segmentation during the registration, we emphasize the importance of the prostate border resulting in a better alignment compared to the interior landmarks. Moreover, picking the landmarks used for evaluation can be challenging as we sought to capture 3+ landmarks/slice, and the resolution of the MRI is relatively reduced due to the surface coil acquisition. We acknowledge the following limitations for our approach. Although the registration approach is fully automated and does not require patient-specific parameterization for general cases without unusual artifacts, similar to existing approaches, some manual interventions are needed to either segment the prostate on both MRI and histopathology images, to identify slice correspondences between the histopathology and T2w MRI or to correct the gross rotation of the histopathology slices. Unlike other approaches, RAPSODI does not require landmark selection, but only uses them to evaluate the accuracy of registration. The registration assumes that a slice-to-slice correspondence exists between the histopathology and MR images. While this is improved by using 3D printed molds, slice misalignment is possible due to the shrinking of the prostate during fixation and shifting in the mold during slicing. Such misalignment is occasionally observed at the base and apex of the prostate. The digital phantom allowed us to study the effect of such misalignment and showed that a perfect alignment cannot be obtained in this situation (we observed a $\sim$4 pixels error) yet the induced shrinkage and rotations are well recovered. The registration runtime for our approach is 6-8 minutes which is limiting for a Graphical User Interface execution, yet it is acceptable when running the approach in batch mode. We investigated a computationally efficient method that skips Step 1 - reconstructing the histopathology, and runs the registration at lower resolution. The fast approach run in 3.1 minutes and achieved a Dice coefficient of 0.97$\pm$0.01, a Hausdorff distance of 2.23$\pm$0.66 mm, urethra deviation 2.81$\pm$0.73mm and landmark deviation of 2.92$\pm$0.7 mm in a 66 patient subset from Cohort C1. Although the results are slightly less accurate, the fast approach is 2-3 times faster, and is more suited to running in a Graphical User Interface. Although our study only includes 89 patients from two institutions, to date it represents the largest study of this magnitude with data from multiple institutions, with MRI acquired either using surface or endorectal coils, and the only study to evaluate the approach in a digital histopathology-MRI prostate phantom. Compared to previous approaches outlined in  \[tab:prior\_work\], our quantitative results place us close to the method by Kalavagunta et. al. [@kalavagunta_registration_2015] in terms of Dice similarity coefficient. The latter approach relies on heavily annotated datasets that include the border of the transitional zone and the peripheral zone as well as other landmarks. Such landmarks are used to drive the registration at the interior of the prostate resulting in better landmark alignment, yet the approach is labor-intensive and requires careful examination of the data to identify matching landmarks in the pathology images and MRI, which is not trivial. [ RAPSODI aims at registering the histopathology and MRI images with the sole goal of mapping the extent and grade of cancer from histopathology images onto T2 weighted MRI, thus creating careful and objective spatial labels on pre-operative MRI. Such mapping may help develop advanced image analysis tools to reliably predict prostate cancer and its aggressiveness on MRI, help improve current MRI interpretation schemes as well as help validate novel MRI protocols or other imaging techniques. Better imaging accompanied by better interpretation schemes can have great impact in reducing overdiagnosis of low-grade cancers, the underdiagnosis of aggressive cancers, and infectious complications of biopsy.]{} Conclusion {#conclusion .unnumbered} ========== Our radiology-pathology registration framework, RAPSODI, allowed the alignment of histopathology slices and pre-surgical MRI, enabling the accurate mapping of the labels from histopathology onto MRI. The reconstruction of the 3D histopathology specimen followed by 2D registration of corresponding histopathology and T2w MRI slices ensured a robust alignment that provides accurate prostate cancer labels for MRI. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the Department of Radiology at Stanford University, for their support for this work. [^1]: To whom correspondence should be addressed. E-mail: [email protected]

--- abstract: | We have discovered three globular clusters beyond the Holmberg radius in Hubble Space Telescope Advanced Camera for Surveys images of the gas-rich dark matter dominated blue compact dwarf galaxy NGC2915. The clusters, all of which start to resolve into stars, have $M_{V606} = -8.9$ to –9.8 mag, significantly brighter than the peak of the luminosity function of Milky Way globular clusters. Their colors suggest a metallicity $[{\rm Fe/H}] \approx -1.9$ dex, typical of metal-poor Galactic globular clusters. The specific frequency of clusters is at a minimum normal, compared to spiral galaxies. However, since only a small portion of the system has been surveyed it is more likely that the luminosity and mass normalized cluster content is higher, like that seen in elliptical galaxies and galaxy clusters. This suggests that NGC2915 resembles a key phase in the early hierarchical assembly of galaxies - the epoch when much of the old stellar population has formed, but little of the stellar disk. Depending on the subsequent interaction history, such systems could go on to build-up larger elliptical galaxies, evolve into normal spirals, or in rare circumstances remain suspended in their development to become systems like NGC2915. author: - 'Gerhardt R. Meurer, J.P. Blakeslee, M. Sirianni, H.C. Ford, G.D. Illingworth, N. Benítez, M. Clampin, F. Menanteau, H.D. Tran, R.A. Kimble, G.F. Hartig, D.R. Ardila, F. Bartko, R.J. Bouwens, T.J. Broadhurst, R.A. Brown, C.J. Burrows, E.S. Cheng, N.J.G. Cross, P.D. Feldman, D.A. Golimowski, C. Gronwall, L. Infante, J.E. Krist, M.P. Lesser, A.R. Martel, G.K. Miley, M. Postman, P. Rosati, W.B. Sparks, Z.I. Tsvetanov, R.L. White, & W. Zheng' nocite: '[@s92; @cgcfp00]' title: 'Discovery of Globular Clusters in the Proto-Spiral NGC2915: Implications for Hierarchical Galaxy Evolution' --- Introduction {#s:intro} ============ All galaxies with massive old stellar populations are thought to contain globular clusters (GCs). They are particularly noticeable where the old stellar population is dominant such as in elliptical (E), dwarf elliptical (dE), and central Dominant (cD), as well as spiral galaxies with prominent bulges. Dwarf spheroidal galaxies generally do not contain GCs, probably because they have insufficient mass to make their formation likely. The exceptions are the most massive dwarf spheroidals Fornax [@h61] and Sagitarius [@igi94] which each have at least 4 GCs. Disk galaxies dominated by population I stars contain fewer GCs per unit luminosity, presumably because of star formation in the disk after the formation of the population II component. Galaxies with a high  ratio have yet to form much of their baryonic mass into stars. They typically are blue and not considered likely hosts for populous GC systems. NGC2915 is an extreme gas rich galaxy having ${\mbox{${\cal M}_{\rm HI}/L_B$}}= 1.7\, {\mbox{${\cal M}_\odot$}}/L_{B,\odot}$ [@mcbf96 hereafter MCBF96]. Its regularly rotating  disk extends to over 5 times beyond the readily detectable optical emission providing an excellent dynamical tracer for the mass distribution; not coincidentally NGC2915 has one of the highest known mass-to-light ratios in a single galaxy (MCBF96). Furthermore, while its optical morphology is that of a blue compact dwarf [BCD; @mmc94 hereafter MMC94], its  disk clearly shows spiral arms which are not apparent in the optical. In this Letter, we report the discovery of three luminous GCs found in Hubble Space Telescope Advanced Camera for Surveys [ACS; @ford_acs02] images of NGC2915 that were obtained in order to look for a stellar heating source for the  disk. That issue will be discussed in a separate article (Meurer [[*et al.*]{}]{} 2003; in preparation, hereafter Meu03). Data and analysis {#s:data} ================= ACS Wide Field Camera (WFC) images were obtained of a field centered at 09$^{\rm h}$ 25$^{\rm m}$ 36$\fs$48, –76$^\circ$ 35$'$ 52$\farcs$4 (J2000). The images cover projected radii of 45 to 257 , whereas the Holmberg radius $R_{Ho} = 114''$. We obtained 2, 2, 4 images for a total exposure of 2480$s$, 2600$s$, 5220$s$ in the filters F475W (), F606W (), and F814W (), respectively. The basic processing of the images was done using the [*CALACS*]{} pipeline [@hack99]. We used the program [[*Apsis*]{}]{} [@bambm02] to align and combine the images encorporating geometric correction and rejection of cosmic rays and hot pixels. Here we present photometry in the natural system of the filters, with zeropoints selected so that Vega would have a magnitude of 0.0 in all bands. In order to compare our observations with previous work, we convert the previous work to this system, as needed, using the calibrations of Sirianni et al (2003, in preparation). The most important correction is to the  photometry, since the F606W filter straddles the wavelength of traditional $V$ and $R$ filters. Results {#s:res} ======= Table \[t:prop\] presents adopted global properties for NGC2915. The foreground extinction, , is from the @sfd98 extinction maps. It is significantly larger than ${\mbox{$E(B-V)$}}= 0.15 \pm 0.05$ estimated by MMC94, but consistent with the position of the field star Red Giant Branch (RGB; Meu03). Extinction corrected photometry employing the @ccm89 extinction curve is denoted with a “0” subscript. The distance, $D$ was derived from the field star RGB tip (Meu03). It is consistent with but improves on previous estimates $D = 5.3 \pm 1.3$ Mpc (MMC94) and $D = 3.8 \pm 0.5$ Mpc [@k03]. The remaining quantities in Table \[t:prop\] were derived from MMC94 and MCBF96 after correcting to the new  and $D$. As shown in Fig. \[f:finders\], the three sources are clearly GCs whose brightest stars are resolved. Table \[t:clust\] compiles the properties of the clusters. The photometric quantities were measured using a circular aperture having a radius of $r = 3''$, with the local sky subtracted using an annulus having radii of 5 and 7.5. The cluster size  is the circular aperture radius encompassing half the  light as measured from curves of growth. Compared to Galactic GCs, these clusters are large and luminous, but not abnormally so. Only 16% of the clusters in the @h96 database[^1] have  luminosities brighter than G3; only three clusters are more luminous than G1. The clusters’  ranges from about 5 to 9 pc, placing them in the upper quartile of Galactic GCs which have  ranging from 0.3 to 24.7 pc [@h96]. The clusters are noticeably elongated with ellipticity $\epsilon \equiv 1 - b/a$ similar to the canonical flattened Galactic GCs M22 and $\omega$ Cen ($\epsilon = 0.14$, 0.17, respectively; Harris 1996). The combination of high luminosity and appreciable flattening is also seen in the cluster M31-G1 [@pvdb84; @msjdbr01]. The  and  colors of the clusters are compared to Milky Way GCs [@h96] in Fig. \[f:2cd\]. Their colors are virtually identical implying similar metallicities, assuming they are old and nearly coeval. We derive their metallicity by fitting the metallicty - color relationship from the Harris database after converting the colors to . We employed an unweighted least squares fit with an iterative $2.5\sigma$ rejection resulting in ${\mbox{[Fe/H]}}= -5.37 + 5.36{\mbox{$(V_{606} - I_{814})_0$}}$ with a dispersion of 0.29 dex. The metallicity for the three clusters is then ${\mbox{[Fe/H]}}= -1.9 \pm 0.4$ dex, consistent with [*low*]{} metallicity Galactic GCs. Our stellar population analysis, in progress (Meu03), indicates that the stars at the outskirts of the clusters have very similar  versus  color-magnitude diagrams, dominated by a narrow and blue RGB. This is also consistent with low , if the clusters are old. Discussion {#s:disc} ========== Cluster formation efficiency ---------------------------- Because of the blue core and gas rich nature of NGC2915 we had not expected to find GCs in our images. However, in retrospect this discovery should not have been surprising. GCs have previously been discovered around morphologically similar systems; @obr98 find a considerable population of at least 65 old GCs about the Blue Compact (albeit not dwarf) galaxy ESO 338-IG04. In addition, NGC2915 is not a dwarf system in terms of mass, and optical imaging shows that it to be dominated by an older stellar population for $R > 0.5$ Kpc (MMC94). These facts suggest that old GCs might have been expected. Are the number of clusters found anomalous? To address this we estimate the total number of clusters in the system. Unfortunately, our images sample only a small fraction of the galaxy. If the GCs are distributed spherically out to $R_{\rm HI} = 10'$ (where the  distribution ends; MCBF96), then we have only surveyed 3.5% of the available area. Here we consider two cases for the possible distribution of old clusters. Case (1) assumes that we are lucky and have managed to observe all the clusters in NGC2915. While this is unlikely, it provides a strict lower limit to the cluster formation efficiency. The more likely case (2) is that the globular cluster system is like that of more luminous systems - having a spherically symmetric power-law radial distribution in number per unit area - $N(R) \propto\ R^\alpha$ [@h91]. We assumes this extends between $R = 0.5$ Kpc and $R_{\rm HI}$, where the minimum $R$ insures that the predictions are finite and was chosen to correspond to the size of the central star forming population (MMC94). For $-2.5 < \alpha < -0.5$ we estimate that the total number of clusters in the system is 7 to 12 times higher than found on our images. For case (2) we adopt a correction factor of 9. Hence there are at least 3 GCs in the system (case 1), with a more likely number being $\sim 27$ (case 2). In principle, we should correct the total GC estimate for the finite luminosity sampling of the images. However, SExtractor [@ba96] catalogs of our images show that we can detect extended objects down to ${\mbox{$V_{606}$}}\approx 27$ which corresponds to an $M_{V606} \sim -2$ for sources in NGC2915. This is well below the peak of the GC luminosity function $M_{V} \sim -7.5$ (Secker 1992; revised to agree with the [*Hipparcos*]{} RR Lyrae zero point, e.g. Carretta [[*et al.*]{}]{}2000). Hence we apply no luminosity sampling correction. The fact that we only have found GCs much brighter than this peak is somewhat puzzling. We will consider three measures of cluster formation efficiency. First, the specific frequency, ${\mbox{$S_N$}}= N_T\, 10^{0.4(M_V + 15)}$, is the $V$ luminosity normalized cluster content [@hvdb81].  is typically around 1 for spiral galaxies and increases towards earlier galaxy types, with E galaxies having $S_N \sim 4$. In the center of galaxy clusters  ranges from 2.5 to 12.5 [@btm97; @jpb99]. We consider two measures of the mass normalized contribution:  the number of clusters per $10^9$  in dynamical mass (Blakeslee [[*et al.*]{}]{} 1997), and , the fractional baryonic (gas and stars) mass in GCs [@McL99]. As done by @McL99 we assume each cluster has an average mass $\langle {\cal M}_{\rm cl}\rangle = 2.4 \times 10^5\, {\mbox{${\cal M}_\odot$}}$ when calculating . Blakeslee [[*et al.*]{}]{} (1997) find an average ${\mbox{$\eta_{\rm GC}$}}= 0.7$ with a scatter of 30%, while @McL99 computes an average ${\mbox{$\epsilon_{\rm cl}$}}= 0.0026$ with a 20% uncertainty. Table \[t:gcs\] tabulates ,  and  for NGC2915 under the above two cases. Relevant quantities used for these calculations are given in Table \[t:prop\]. Table \[t:gcs\] also lists literature values and uncertainties of the efficiencies for “normal” systems. The uncertainty of our estimates are large: for a Poissonian distribution yielding a count of 3, the 95% confidence limits on the cluster formation efficiencies are 0.3 and 2.3 times the estimated value. We find that at a minimum (case 1)  is close to normal for a spiral galaxy while while  and  are low compared to normal globular cluster systems. In the more likely case (2),  is very high compared to normal gas rich systems.  and  are also somewhat high compared to literature values, but in reasonable agreement considering the Poissonian uncertainty of our measurements. While we can not rule out the possibility that NGC2915 is like a normal spiral galaxy in terms of , it is more likely that it has a high luminosity normalized cluster content, whereas the mass normalized content is closer to normal. A missing link in galaxy evolution? ----------------------------------- It is interesting to interpret these results within the framework of hierarchical evolution through comparison with other systems. @jpb99 has discussed how the  of the young Milky Way must have been fairly high, similar to the values for cluster E galaxies, after the formation of the Galactic halo but before stellar disk formation. The increase in luminosity from later star formation in the disk would cause a decrease in the Galactic  to its present low value, while leaving the number of GCs per unit mass unchanged. In NGC2915 we have an example of a present-day galaxy with an old stellar component, including GCs, but a spiral disk that is still mainly in the form of gas. In this sense, NGC2915, like the centers of rich galaxy clusters, has an elevated value of  compared to typical spirals because of a lower efficiency for converting gas into stars. Clusters typically have 2–5 times as much mass in gas as in stars [@arbvv92], similar to what we see in NGC2915. The difference, however, is that the gas in clusters now resides in the hot intracluster medium, while the gas in NGC2915 still retains the potential to be converted into stars. To a large extent, the fate of the gaseous disk (and the future evolution of  in NGC2915) must depend on the surrounding environment. Simulations show that when a disk galaxy enters the environment of a rich galaxy cluster, much of the gas in its disk and halo will be removed by the combination of tidal and ram pressure stripping [@amb99; @bcs02; @bcdg01; @g03]. For example, if a system similar to the Galaxy were to fall into a rich cluster it would have its gas disk truncated down to a radius of a few kpc within a few tens of Myr by ram pressure stripping alone (Abadi [[*et al.*]{}]{} 1999). Indeed, models of present day evolution of galaxy clusters invoke mildly truncated star formation in field galaxies accreting onto the cluster as the cause of the Butcher-Oemler effect [@bnm00; @kb01]. It is likely that in their [*early evolution*]{} clusters were assembled from building blocks similar to NGC2915 which had their ISM stripped from them by these mechanisms. The stripped gas was then virialized to become the cluster X-ray halo. In this scenario galaxy clusters have high  because they formed out of subclumps which had already efficiently formed star clusters but which never had the chance to form disks. In a less hostile environment, a building block similar to NGC2915 could go on to form a normal spiral galaxy. If the  disk of NGC2915 were to be perturbed enough to efficiently form stars in a fairly quiescent fashion this would result in an additional $7 \times 10^8 {\mbox{${\cal M}_\odot$}}$ of stars forming with no additional GCs. Assuming a $M/L_V \sim 1\, {\mbox{${\cal M}_\odot$}}/L_{V,\odot}$ for the additional stars then the system would evolve towards $S_N \sim 2$, a fairly normal value for spiral galaxies, while  and  remain fixed at their normal values. The  in NGC2915 is then anomalously high because the formation of its disk has not proceeded, presumably due to a lack of external perturbations (MCBF96). If so, then we may expect other galaxies with high  ratios and extended  disks to also have a significant GC population, especially if they have a strong old population. While this scenario seems compelling, we caution that it is possible that NGC2915 only superficially resembles the building block we describe. We have not proven that the number of clusters is anomalously high. Furthermore, we can not yet rule out the possibility that the clusters are of higher metallicity and younger than Galactic GCs. This is shown in figure \[f:2cd\] where we overplot @bc03 population models on the two color diagram illustrating the strong age-metallicity degeneracy for the filters used here. If the clusters are not old, they may represent the remnants of a starburst occurring as recently as a few Gyr ago. We are undertaking additional observations (imaging and spectroscopic) to get a more accurate census of star clusters in the NGC2915 system and determine their nature. The results of these studies should determine whether NGC2915 is just a gas rich galaxy with a peculiar star formation history, or whether it truly presents us with a rare local view of how galaxies looked in the epoch of cluster assembly. ACS was developed under NASA contract NAS 5-32864, and this research has been supported by NASA grant NAG5-7697. We thank the technical and administrative support staff of the ACS science team for their committed work on the ACS project. Abadi, M.G., Moore, B., & Bower, R.G. 1999, , 308, 947 Arnaud, M., Rothenflug, R., Boulade, O., Vigroux, L., & Vangioni-Flam. E. 1992, , 254, 49 Balogh, M.L., Navarro, J.F. & Moore, S.L. 2000, , 540, 113 Bekki, K., Couch, W.J., Drinkwater, M.J., & Gregg, M.D. 2001, , 557, L39 Bekki, K., Couch, W.J., & Shioya, Y. 2002, , 577, 651 Bertin, E. & Arnouts, S. 1996, , 117, 393 Blakeslee, J.P. 1999, , 118, 1506 Blakeslee, J.P., Tonry, J.L., & Metzger, M.R. 1997, , 114, 482 Blakeslee, J.P., Anderson, K.R., Meurer, G.R., Benitez, N., & Magee, D. 2002, ASP Conf. Ser. 295: ADASS XII, 257 Bruzual, G., & Charlot, S. 2003, , in press (astro-ph/0309134) Cardelli, J.A., Clayton, G.C., & Mathis, J.S. 1989, , 345, 245 Carretta, E., Gratton, R.G., Clementini, G., & Fusi Pecci, F. 2000, , 533, 215 Ford, H.C., [[*et al.*]{}]{} 2002, Proc. SPIE, 4854, 81 Gnedin, O.Y. 2003, , 589, 752 Hack, W.J. 1999, CALACS Operation and Implementation, ISR ACS-99-03 (Baltimore: STScI) Harris, W.E. & van den Bergh, 1981, , 86, 1627 Harris, W.E. 1991, , 29, 543 Harris, W.E. 1996, , 112, 1487 Hodge, P.W. 1961, , 66, 83 Ibata, R., Gilmore, G., & Irwin, M. 1994, , 370, 194 Kodama, T. & Bower, R.G. 2001, , 321, 18 Karachentsev, [[*et al.*]{}]{} 2003, , 398, 479 McLaughlin, D.E. 1999, , 117, 2398 Meurer, G.R., Mackie, G., & Carignan, C. 1994, , 107, 2021 (MMC94) Meurer, G.R., Carignan, C., Beaulieu, S., & Freeman, K.C. 1996, , 111, 1551 (MCBF96) Meylan, G., Sarajedeni, A., Jablonka, P., Djorgovski, S.G., Bridges, T., & Rich, R.M. 2001, , 122, 830 Östlin, G., Bergvall, N., & Rönnback, J. 1998, , 335, 85 Pritchet, C.J., & van den Bergh, S. 1984, , 96, 804 Secker, J. 1992, , 104, 1472 Schlegel, D.J., Finkbeiner, D.P., & Davis, M. 1998, , 500, 525 [lccl]{}   & $0.28 \pm 0.04$ & mag & foreground extinction\ $D$ & $4.1 \pm 0.3$ & Mpc & Distance\ ${\cal M}_g$ & $7.4 \times 10^8$ & & ISM mass\ ${\cal M}_\star$ & $3.2 \times 10^8$ & & Mass in stars\ ${\cal M}_T$ & $2.1 \times 10^{10}$ & & Total dynamical mass\ $M_V$ & –16.42 & mag & Absolute mag $V$ band\ $L_B$ & $3.4 \times 10^8$ & $L_{B,\odot}$ & $B$ band luminosity\   & 62 & solar & Mass to light ratio\ [l l l c c c c c c c]{} G1 & 09 25 56.273 & -76 35 14.53 & 3.0 & -9.82 & 0.62 & 0.66 & 8.8 & 0.11\ G2 & 09 25 27.445 & -76 36 31.55 & 3.3 & -9.04 & 0.57 & 0.65 & 4.6 & 0.16\ G3 & 09 25 41.954 & -76 36 33.70 & 2.4 & -8.92 & 0.59 & 0.64 & 5.9 & 0.13\ [l c c c ]{} & 0.81 & 7.3 & 0–12.5\ & 0.14 & 1.3 & $0.7 \pm 0.2$\ & 0.00067 & 0.0061 & $0.0026 \pm 0.0005$\ [^1]: http://physwww.physics.mcmaster.ca/$\sim$harris/mwgc.dat

--- author: - 'V. Charmandaris' - 'O. Laurent' - 'E. Le Floc’h' - 'I. F. Mirabel' - 'M. Sauvage' - 'S. Madden' - 'P. Gallais' - 'L. Vigroux' - 'C. J. Cesarsky' date: 'Received 1 March 2002 / Accepted 30 May 2002' title: ' Mid-infrared observations of the ultraluminous galaxies , , and [^1]' --- Introduction ============ It is currently widely accepted that the majority of the most luminous galaxies (L$_{bol}>10^{11}$L$_{\sun}$) in the local universe (z $<0.3$) are luminous in the infrared, and include the ultraluminous infrared galaxies (ULIRGs, L$_{\rm IR}>10^{12}$L$_{\sun}$) which emit the bulk of their energy at infrared wavelengths [@Houck1984; @Soifer1989; @Sanders1996 and references therein]. In those systems most of the infrared emission seems to originate from their dusty nuclear regions. Even though one of the principal heating mechanisms for the lowest luminosity ($\lesssim 10^{11}$ L$_{\sun}$) infrared galaxies is the stellar radiation field of young massive stars, it is still unclear if the star formation is also the dominant heating source for ULIRGs or whether one needs to invoke an active galactic nucleus (AGN) and its strong radiation field as the central engine responsible for the heating of the dust [see @Joseph99; @Sanders99]. The presence of large quantities of molecular gas has long been detected in the central regions of most ULIRGs [e.g. @Sanders1985; @Sanders1991] leading to high extinction of both their UV and optical radiation. As a result, since it appears that most galaxies do harbor a super-massive, though often quiescent, black hole [@Richstone1998], one would expect to find in their galactic nucleus observational evidence for a mixture of AGN [@Sanders1988] and/or strong compact starburst regions [@Condon1991] fueled by the high concentration of molecular gas [@Bryant1999]. Observations in the mid-infrared (MIR), which are less affected by absorption than shorter wavelengths [A$_{15\,\mu m}$ $\sim$A$_{V}$/70, @Mathis1990], thus provide a powerful probe of galactic central regions [@Soifer2000; @Soifer2001]. As we discussed in @Laurent2000, the integrated MIR emission in active galaxies is produced mainly by the interstellar dust which is heated directly by the ionization field from young stars or an AGN. This is in contrast to late type galaxies where the MIR (5–20$\mu$m) energy budget is dominated by the reprocessed emission of star forming regions in their disk and accounts for $\sim$15% of their luminosity [@Dale2001; @Helou2001; @Roussel2001]. However, the main difficulty in assessing the importance of the underlying physics in galactic nuclei, where the spatial resolution is typically poor, is in separating the contribution of star forming regions and the active nucleus from the integrated MIR emission. The development, application, and general utility of MIR diagnostics in nuclei of galaxies has already been demonstrated by @Roche1991 and more recently by @Genzel1998 [@Laurent2000], as well as by @Dudley1999 [@Imanishi2000]. This was mainly accomplished with the advent of ISOCAM and SWS on board ISO, with high spatial and spectral resolution, as well as improved sensitivity in the 3 to $\sim$40$ \mu$m wavelength range, thus allowing us to study the nature of the heating sources in ULIRGs. More specifically it has been shown by @Lutz1998 [@Laurent1999b; @Laurent2000; @Tran2001] that a nearby galaxy hosting a dominant AGN is clearly different in the MIR from a starburst or a late type spiral. The most striking difference is that the rather featureless MIR spectrum in AGN lacks the emission bands at 6.2, 7.7, 8.6, 11.3 and 12.7$\mu$m, which are seen in late type galaxies and are attributed to Polycyclic Aromatic Hydrocarbons (PAHs) – also often called Unidentified Infrared Bands (UIBs). One may consider that this is simply due to the fact that its elevated MIR continuum of the AGN overwhelms any UIB feature emission [@Pier1992; @Barvainis1987]. It seems inevitable that as the AGN heats its dusty torus at T$\sim$1000K and the dust grains approach sublimation temperatures, the more fragile molecules responsibly for the UIB emission could be partly destroyed by a photo-thermo-dissociation mechanism [@Leger1989]. Obviously this picture is more complicated in distant galaxies since due to limited spatial resolution the contribution of the star forming regions surrounding an AGN would progressively enter into the beam and dilute any AGN MIR signature [see @Laurent1999b]. When sufficient spatial resolution is available to directly view the active nucleus, as is often the case in Seyfert 1 galaxies, the non-thermal emission from the AGN will dominate the spectrum. Consequently, the spectrum can then be fitted by a power law and has a “bump” in the 4–5$\mu$m range. A 5–11$\mu$m study of a large sample of Seyfert galaxies with ISO by @Clavel2000 confirmed this picture, concluding that Seyfert 2 galaxies have weaker MIR continuum. However, a detailed analysis of the MIR spectra and images of the prototypical Seyfert 2 galaxy NGC1068 by @LeFloch2001 showed that if sufficient spatial resolution is available and the AGN is extremely strong, even in the case of a Seyfert 2 one can isolate the emission of the central engine from the star forming regions which surround it. In that case the MIR spectrum of the Seyfert 2 would also be a power law with the addition of a weak PAH emission. Despite this progress, several questions concerning the extent and spectral characteristics of the MIR emission in active nuclei, as well as the correlation between MIR and optical activity have not been fully examined. Could broad band MIR photometry be used to probe the physical characteristics of AGNs? In the present paper we try to address some of these issues by studying the MIR spectral energy distribution (SED) of three ultraluminous IRAS galaxies. Each IRAS source, the properties of which are presented in Table \[info\], consists of a merging pair of galaxies with different levels of nuclear activity. The targets were specifically selected as MIR bright and harboring an optically classified AGN. In section 2, we describe the observations and in section 3 we present the details of our study and analysis of the data for each system. A discussion followed by concluding remarks is presented in section 4. Throughout this paper we assume a Hubble constant H$_{0}$=75 kms$^{-1}$Mpc$^{-1}$ and q$_0$=1/2. Observations and data reduction =============================== Our MIR observations were obtained using ISOCAM, a 32$\times$32 pixel array [@CesarskyC1996] on board the ISO satellite [@Kessler1996]. Each system was observed with broad band filters ranging from 5 to 18$\mu$m in a 2$\times$2 raster with 6 pixel offsets and a lens producing a pixel field of view (PFOV) of 1.5, resulting in a final image of 57$\times$57. This enabled us to obtain images with a spatial resolution of 3 (at 6$\mu$m) to 4.5 (15$\mu$m) limited by the pixel size at 6$\mu$m and by the full width at half maximum (FWHM) of the point spread function (PSF) at 15$\mu$m. We note the ISOCAM filters by their name and central wavelength. The wavelength range in $\mu m$ covered by each filter was: LW2 (5.0 – 8.5), LW3 (12.0 – 18.0), LW4 (5.5 – 6.5), LW6 (7.0 – 8.5), LW7 (8.5 – 10.7), LW8 (10.7 – 12.0), LW9 (14.0 – 16.0). At subsequent sections in this paper we will refer to the measured flux densities using the various filters as f$_{x\,\mu m}$ where *x* is the central wavelength of each filter in microns. Spectrophotometric observations were also obtained with the circular variable filter (CVF) for IRAS 23128-5919, the brightest of our sources. The CVF covers a spectral range from 5 to 16.5$\mu$m with a 1.5 PFOV and a spectral resolution of  50. Each integration step was composed of 12 images with 5.04 second integration time and during the CVF scan the wavelength step varied between 0.05 and 1.11$\mu$m. Details on the observing parameters are summarized in Table \[param\]. The data were analyzed with the CAM Interactive Analysis software (CIA[^2]). A dark model taking into account the observing time parameters was subtracted. Cosmic ray contamination was removed by applying a wavelet transform method [@Starck1997]. Corrections of detector memory effects were done applying the Fouks-Schubert’s method [@Coulais2000]. The flat field correction was performed using the library of calibration data. Finally, individual exposures were combined using shift techniques in order to correct the effect of jittering due to the satellite motions (amplitude $\sim$1). A deconvolution using multiscale resolution techniques [@Starck1999] was subsequently applied to estimate the physical size of the quasi-point like sources responsible for the infrared emission in our data (see section \[sec:results\]). The details of the analysis of the ISOPHOT-S data of the three galaxies, which we also include in this paper for reasons of comparison, are published by @Rigopoulou1999. Based on three different observations of IRAS 19254-7245 taken with identical LW filters but with different roll angle, integration times per exposure (2s and 5s) and PFOVs, as well as on similar analysis of other ISOCAM-CVF and ISPHOT-S observations, we estimate that the uncertainty of our photometry measurements is $\sim$20$\%$ (see Table \[phot\]). ------------ ---------- ----------- -------- ------------------- ------------- -------------------- ------------------- ------- -------------- -------------- Target RA DEC z D$_{\rm L}$ log(L$_{\rm FIR}$) log(L$_{\rm IR}$) IRAS Name J2000.0 J2000.0 12$\mu$m 25$\mu$m 60$\mu$m 100$\mu$m (Mpc) (L$_{\sun}$) (L$_{\sun}$) 19254-7245 193121.6 -723920.8 0.0617 0.22 1.24 5.48 5.79 250 11.68 12.01 23128-5919 231546.9 -590314.2 0.0446 0.24 1.59 10.80 10.99 180 11.69 11.96 14348-1447 143738.2 -150023.9 0.0823 $\!\!\!\!\!<$0.14 0.49 6.87 7.07 335 12.05 12.27 ------------ ---------- ----------- -------- ------------------- ------------- -------------------- ------------------- ------- -------------- -------------- **Table note:** The far-infrared and infrared luminosities are calculated using L$_{\rm FIR}$=3.94$\times$10$^5\times$D(Mpc)$^2$(2.58$% \times$*f*$_{60}$+*f*$_{100}$) and L$_{\rm IR}$=5.62$\times$10$^5\times$D(Mpc)$^2$(13.48$\times$*f*$_{12}$+5.16$\times$ *f*$_{25}$+2.58$\times$*f*$_{60}$+*f*$_{100}$) respectively, where the luminosity distance is defined as D$_{\rm L}= \frac{c}{H_0q_0^2}(zq_0+(q_0-1)(\sqrt{(1+2q_0z)}-1)$ [see @Sanders1996]. --------------------- ---------------- ------------ ---------- --------- ------------ ------------ ------------ ---------- ------- -- Target ISOCAM Filter: LW2 LW3 LW4 LW6 LW7 LW8 LW9 CVF Filter Center: 6.75$\mu$m 15$\mu$m 6$\mu$m 7.75$\mu$m 9.62$\mu$m 11.4$\mu$m 15$\mu$m – IRAS 19254-7245$^1$ 15.3 15.3 15.4 15.2 15.4 15.3 15.4 – IRAS 19254-7245$^2$ 7.1 7.0 11.3 – 8.3 – – – IRAS 19254-7245$^3$ 3.4 3.6 – – – – – – IRAS 23128-5919$^4$ 7.2 7.0 11.5 – 8.5 – – – IRAS 23128-5919$^5$ – – – – – – – 148.7 IRAS 14348-1447$^6$ 8.6 8.4 – – – – – – --------------------- ---------------- ------------ ---------- --------- ------------ ------------ ------------ ---------- ------- -- **Table note:** The numbers following each galaxy denote the total on-source exposure time (in minutes) for each filter used, and two galaxies were observed more than once under different configurations, the details of which are: (1) IRAS19254-7245 observed in proposal CAMACTI2 (PI I.F. Mirabel), 7 LW filters, integration time per frame Tint=5s, pfov=1.5“. (2) IRAS19254-7245 observed in proposal CAMACTIV (PI I.F. Mirabel), 4 LW filters, Tint=2s, pfov=1.5”. (3) IRAS19254-7245 observed in proposal SAM12N\_2 proposal (PI L. Spinoglio), 2 LW filters, Tint=2s, pfov=3“. (4) IRAS23128-5919 observed in proposal CAMACTIV (PI I.F. Mirabel), 4 LW filters, Tint=2s, pfov=1.5”. (5) IRAS23128-5919 observed in proposal CAMACTI2 (PI I.F. Mirabel), CVF, Tint=5s, pfov=1.5“. (6) IRAS14348-1447 observed in proposal CAMACTIV (PI I.F. Mirabel), 2 LW filters, Tint=2s, pfov=1.5”. Results ======= Background and General Properties {#sec:results} --------------------------------- The sensitivity and spatial resolution capabilities of ISOCAM enable us to obtain deep maps of the MIR emission of each galaxy. Since the interacting members of the IRAS galaxies are very close and are point-like objects with one member typically dominating the MIR emission, photometry measurements were treated with extra care. Our approach was to fit the MIR PSF of the brightest component and to subtract its contribution from the location of the neighboring, fainter galaxy. We then performed aperture photometry on the fainter component, using an aperture $\sim4.5''\times4.5''$. In spite of the difference in their peak intensities, the relative positions of the nuclei were very well known from deep near-IR imaging [@Duc1997b]. Final aperture correction was applied to the flux of each galaxy to account for the overall extension of the PSF. Our measurements are presented in Table \[phot\]. We also include the equivalent broad-band filter fluxes estimated from the ISOPHOT-S spectra, which are found in good agreement with our data within the photometric uncertainties. Since the galaxies were observed several times under different ISOCAM configurations, more than one value is often quoted for the same filter. This was done in order to display the internal consistency of the different measurements and their median value should be considered as the nominal flux density of each galaxy. ISOCAM has detected nearly $\sim$100% of the 12$\mu$m IRAS flux (see Table \[info\]) of these galaxies. Moreover, as it can be seen from the images of the galaxies presented later in this section, no extended extra-nuclear emission, has been detected in any of the galaxies in the MIR. In all cases, the bulk of the flux coming from these objects originates from a region less than 3–4.5 in diameter (which corresponds to the FWHM of PSFs) associated with the nuclei of the interacting galaxies. [lrrrrrrrcc]{} Target & & & & & & & & &\ IRAS & & & & & & & & LW2 & LW4\ 19254S$^{1}$ & 106.9$\pm$10.7 & 284.0$\pm$28.4 & 90.0$\pm$9.1 & 150.1$\pm$15.0 & 91.2$\pm$9.1 & 107.5$\pm$10.8 & 337.5$\pm$33.8 & 2.7$\pm$0.4 & 1.2$\pm$0.2\ 19254S$^{2}$ & 103.6$\pm$11.0 & 278.9$\pm$29.1 & 87.3$\pm$11.2 & & 97.1$\pm$10.5 & & & 2.7$\pm$0.4 & 1.2$\pm$0.2\ \ 19254N$^{1}$ & 4.8$\pm$0.5 & 5.9$\pm$0.7 & 1.9$\pm$0.4 & 8.3$\pm$0.9 & 5.1$\pm$0.6 & 5.9$\pm$0.7 & 5.4$\pm$1.0 & 1.2$\pm$0.2 & 2.5$\pm$0.6\ 19254N$^{2}$ & 3.1$\pm$1.0 & 7.5$\pm$2.4 & 1.5$\pm$2.6 & & 4.5$\pm$1.7 & & & 2.4$\pm$1.1 & 2.1$\pm$3.6\ \ 19254$^{1}$ & 111.7$\pm$11.2 & 289.9$\pm$29.0 & 91.9$\pm$9.2 & 158.4$\pm$15.9 & 96.3$\pm$9.6 & 113.4$\pm$11.3 & 342.9$\pm$34.3 & 2.6$\pm$0.4 & 1.2$\pm$0.2\ 19254$^{2}$ & 106.7$\pm$11.0 & 286.4$\pm$29.2 & 88.8$\pm$11.5 & & 97.1$\pm$10.6 & & & 2.7$\pm$0.4 & 1.2$\pm$0.2\ 19254$^{3}$ & 114.8$\pm$12.5 & 290.2$\pm$11.5 & & & & & & 2.5$\pm$0.4 &\ 19254$^{\dag}$ & 113.0$\pm$2.8 & & 85.6$\pm$3.7 & 135.7$\pm$4.1 & 110.4$\pm$4.6 & 116.9$\pm$13.9 & & & 1.3$\pm$0.1\ 23128S$^{4}$ & 70.8$\pm$7.1 & 228.3$\pm$22.9 & 48.5$\pm$5.0 & & 67.3$\pm$6.8 & & & 3.2$\pm$0.5 & 1.5$\pm$0.2\ 23128S$^{5}$ & 77.5$\pm$1.6 & 262.3$\pm$4.3 & 50.6$\pm$2.3 & 116.0$\pm$2.6 & 79.9$\pm$2.0 & 106.2$\pm$3.6 & 277.1$\pm$6.6 & 3.4$\pm$0.1 & 1.5$\pm$0.1\ \ 23128N$^{4}$ & 38.6$\pm$3.9 & 88.5$\pm$9.1 & 19.8$\pm$2.2 & & 26.7$\pm$2.8 & & & 2.3$\pm$0.3 & 2.0$\pm$0.3\ 23128N$^{5}$ & 34.8$\pm$1.2 & 90.4$\pm$2.1 & 19.9$\pm$2.0 & 53.4$\pm$1.7 & 34.7$\pm$1.5 & 51.3$\pm$2.4 & 90.9$\pm$3.2 & 2.6$\pm$0.1 & 1.7$\pm$0.2\ \ 23128$^{4}$ & 109.4$\pm$11.0 & 316.8$\pm$31.7 & 68.3$\pm$7.0 & & 94.0$\pm$9.5 & & & 2.9$\pm$0.4 & 1.6$\pm$0.2\ 23128$^{5}$ & 112.3$\pm$2.6 & 352.7$\pm$7.2 & 70.5$\pm$3.4 & 169.4$\pm$4.4 & 114.5$\pm$3.1 & 157.5$\pm$6.1 & 368.0$\pm$10.7 & 3.1$\pm$0.1 & 1.6$\pm$0.1\ 23128$^{\dag}$ & 123.6$\pm$3.2 & & 70.0$\pm$3.6 & 163.0$\pm$4.9 & 134.5$\pm$4.8 & 140.4$\pm$14.2 & & & 1.8$\pm$0.1\ 14348S$^{6}$ & 21.8$\pm$4.6 & 73.9$\pm$9.0 & & & & & & 3.4$\pm$0.8 &\ \ 14348N$^{6}$ & 11.3$\pm$3.6 & 22.9$\pm$5.0 & & & & & & 2.0$\pm$0.8 &\ \ 14348$^{6}$ & 33.1$\pm$5.8 & 96.8$\pm$10.3 & & & & & & 2.9$\pm$0.6 &\ 14348$^{\dag}$ & 37.9$\pm$1.6 & & 16.9$\pm$2.0 & 52.6$\pm$2.4 & 34.4$\pm$3.3 & 14.1$\pm$10.7 & & & 2.2$\pm$0.3\ **Table note:** For each interacting system, we have measured the integrated flux of the individual galaxies resolved by ISOCAM and marked the southern and the northern galaxies with (S) and (N) respectively. We used the same notations as in Table 2 for identifying the different sets of ISOCAM observations, labeled (1) through (6). As all three galaxies were also observed with ISOPHOT-S and one with the CVF, we also provide the equivalent broad-band filter flux estimates (using the known filter band-passes and transmission curves) based on those spectra marked with a $\dag$ for ISOPHOT-S and a (5) for the CVF. The errors given for each measurement are statistical derived by adding the 1$\sigma$ rms map to the systematic error of 10% commonly associated with the transient correction. Absolute flux uncertainties are estimated to be $\pm$ 20$\%$. For the cases where we present multiple measurements for a target their median value should be considered as its nominal flux density. As it has been discussed in several papers describing ISO observations [i.e. @Laurent2000 and references therein] the MIR emission of spiral galaxies observed by ISOCAM originates from a number of physical processes, with two dust heating mechanisms typically prevailing. One is the thermal emission produced by thermally-fluctuating, small grains ($\sim$10nm) heated by the interstellar radiation field, observed between 12$\mu$m and 18$\mu$m in areas of strong radiation environments and is often sampled by the LW3 filter. The second is due to the UIBs, which originated from complex 2-dimensional aromatic molecules having C=C and C–H bonds and can be seen at 6.2, 7.7, 8.6, 11.3 and 12.7$\mu$m in the ISOCAM wavelength range. The emission in these bands can be observed either with the CVF or using a sequence of narrow-band filters. An absorption feature due to silicates is often observed at 9.7$\mu$m and can be measured using the LW7 (8.5-10.7$\mu$m) filter. Finally, two forbidden emission lines due to \[NeII\] at 12.8$\mu$m and \[NeIII\] at 15.5$\mu$m can be detected in the CVF mode. A contribution to the MIR spectrum by a third component, the Rayleigh-Jeans tail of an old stellar population, is generally negligible in late type galaxies where the hot dust emission dominates. This MIR emission directly arising from stellar photosphere is detected in early type galaxies [@Madden1997]. Analysis of a wealth of ISOCAM data has shown that the flux ratio of the broad band filters centered at 15$\mu$m and 6.75$\mu$m (LW3/LW2 or f$_{15\mu m}$/f$_{6.7\mu m}$) provides a diagnostic of the dominant global MIR emission characteristic of regions, the diffuse interstellar medium or photo-dissociation regions [@Verstraete1996; @CesarskyD1996b; @Dale2001; @Roussel2001]. It has been shown that while quiescent star forming regions typically have f$_{15\mu m}$/f$_{6.7\mu m}$$\sim$1, in active sites of massive star formation this ratio increases due to the increasing contribution of the continuum emission in the 15$\mu$m bandpass [@Sauvage1996; @Mirabel1998; @Vigroux1999; @Dale2001]. However, one should note that the use of this indicator alone is not sufficient to distinguish between the MIR spectrum due to star formation or an AGN, since in AGNs the hot dust continuum arising from the torus also has f$_{15\mu m}$/f$_{6.7\mu m}$ $>$1. Such a degeneracy may be resolved using the flux ratio of the 6.75$\mu$m LW2 filter (sampling the 6.2 and 7.7$\mu$m UIBs) to the narrower LW4 filter which is centered at 6.0$\mu$m only contains the 6.2$\mu$m UIB. As the continuum variation between these two filters is negligible, the f$_{6.7\mu m}$/f$_{6\mu m}$ (LW2/LW4) ratio estimates the intensity of UIBs relative to the underlying continuum [see Fig.5 of @Laurent2000]. The closer f$_{6.7\mu m}$/f$_{6\mu m}$ is to 1, the stronger the continuum is. Since AGNs have weaker UIBs than starbursts, @Laurent2000 proposed to use the combination of the f$_{15\mu m}$/f$_{6.7\mu m}$ and f$_{6.7\mu m}$/f$_{6\mu m}$ colors to differentiate between the two mechanisms contributing to the MIR emission. Clearly there is a redshift dependence of this diagnostic due to the K-correction of the SEDs, but since the redshifts of our targets are small, these indicators can be applied [@Laurent1999a]. Using a large sample of galaxies in the Virgo cluster @Boselli1997 studied the properties of their MIR emission, normalized to the mass of these galaxies. This was done by examining the ratios of the f$_{6.7\mu m}$ (LW2) and f$_{15\mu m}$ (LW3) flux densities to the K band light, which scales with stellar mass of the galaxy, and it was found that the typical f$_{15\mu m}$/K ratio for a late type spiral ranges between 1 and 10. In Table \[k\_ha\], we present those ratios for our sample and we find that even though their active nuclei must contribute some non-thermal emission in the K band the ratios are considerably larger. This can be attributed to a combination of increased thermal dust emission along with a wavelength dependent absorption, which, in highly obscured sources, may decrease their K band flux. Such an example is Arp220 which displays a ratios f$_{15\mu m}$/K$\sim $30 [@Charmandaris2002]. Two more ratios of the LW3 and LW2 over the H$\alpha$ line flux density are also included in Table \[k\_ha\] for reasons of completeness. It has been established that in normal spirals, both filters mostly trace the MIR flux arising from the reprocessing of ionising radiation which is observed in the optical via the H$\alpha$ line [@Sauvage1996; @Roussel2001; @Dale2001]. Since in more active galaxies, the H$\alpha$ emission is strongly affected by the absorption, these ratios could be used to quantify the level of absorption[^3] even though one should be cautious in their quantitative interpretation since the ratios may saturate toward extreme starbursts [@Roussel2001]. We present the LW2/H$\alpha$ mainly for comparison, as the most interesting indicator is clearly the one involving the LW3 filter which directly traces the continuum of hot dust emission emitted by the small grains. [lcccc]{}Target & & & &\ & H$\alpha$ & K & H$\alpha$ & K\ IRAS 19254-7245(S)& 91.3 & 12.4 & 224.5 & 30.4\ IRAS 19254-7245(N)& 78.4 & 1.2 & 85.4 & 1.3\ \ IRAS 23128-5919(S)& 63.5 & 9.9 & 283.8 & 44.3\ IRAS 23128-5919(N)& 31.2 & 8.2 & 85.2 & 22.5\ \ IRAS 14348-1447(S)& 125.7 & 8.1 & 332.9 & 21.5\ IRAS 14348-1447(N)& 161.2 & 6.3 & 263.3 & 10.3\ Finally, in Table \[global\] we also present the MIR luminosities of both the LW2 and LW3 filters for each galaxy of our sample. One can clearly see that despite the activity in these systems, the MIR spectrum contains only a small fraction ($<$5%) of their energy which is mostly emitted at longer wavelengths in the far-infrared (FIR). This is in sharp contrast from what is seen in normal late type galaxies where $\sim$15% of the luminosity is emitted between 5–20$\mu$m [@Dale2001]. In the same table we include the L$_{\rm IR}$(L$_{\sun}$)/M$_{\rm H_2} $(M$_{\sun}$) ratio which traces the efficiency of molecular gas consumption, via either star formation or AGN activity, as well as the production of high energy photons which in-turn are reprocessed into infrared via dust absorption and/or scattering. As expected the reported values for our sample are typical of ultraluminous galaxies while ormal spiral galaxies such as the Milky Way have a ratio of 1–10L$_{\sun}$M$_{\sun}$$^{-1}$, while starbursts such as M82 display higher $\sim$100L$_{\sun}$M$_{\sun}$$^{-1}$ values [see @Sanders1986; @Wild1992]. [lccccc]{}Target & L$_{\rm LW2}$ & L$_{\rm LW3}$ & & &\ IRAS name & (10$^9$L$_{\sun}$) &(10$^9$L$_{\sun}$)& L$_{\rm IR}$ & L$_{\rm IR}$ & M$_{\rm H_2}$\ 19254-7245(S) & 53.8 & 42.9 & – & – & –\ 19254-7245(N) & 1.1 & 0.9 & – & – & –\ 19254-7245 & 54.9 & 43.8 & 0.05 & 0.04 & 34.1\ \ 23128-5919(S) & 17.7 & 19.2 & – & – & –\ 23128-5919(N) & 9.6 & 7.5 & – & – & –\ 23128-5919 & 27.3 & 26.7 & 0.03 & 0.03 & 70.2\ \ 14348-1447(S) & 18.9 & 21.6 & – & – & –\ 14348-1447(N) & 9.8 & 6.7 & – & – & –\ 14348-1447 & 28.7 & 28.3 & 0.02 & 0.02 & 31.0\ Let us now review the MIR properties of each system in detail. IRAS19254-7245 -------------- The ultraluminous infrared galaxy IRAS19254-7245, also known as the “Superantennae” is the result of a collision between two gas-rich spiral galaxies separated by 10 kpc (8.5) in projection and displays extremely long tidal tails extending to 350kpc [@Mirabel1990]. Only the MIR emission originating from the nuclear regions of the galaxies is detected in our images (Figure \[ir19254\_im\]), and there is no evidence for emission extending toward the direction of the tails. Even the northern nucleus is marginally above the sensitivity limit $\sim$1mJy at 3$\sigma$ (see Table \[phot\]). Using optical spectroscopy, the southern galaxy has been classified as a Seyfert 2 with an observed FWHM of $\sim$1700kms$^{-1}$ in both permitted and forbidden lines [@Mirabel1990; @Duc1997a]. The presence of an active nucleus is further suggested by the IRAS criteria for selecting Seyferts, since the ratio of its 25$\mu$m to the 60$\mu$m IRAS flux density is greater than 0.2 [see @deGrijp1985], while its optical and near-infrared colors indicate a strong contribution from a non-thermal component, likely originating from an AGN, as well as emission from very hot dust ($\sim$1000K) [@Vanzi2002]. Evidence of massive star formation is also seen in the nuclear regions as emission line splitting which has been attributed to a biconical outflow [@Colina1991]. The kinetic energy necessary for this to occur can only be produced by supernova explosions or stellar winds further suggesting high star formation rates [150M$_{\sun}$yr$^{-1}$, @Colina1991]. Ground-based MIR observations at 10$\mu$m show that more than 80$\%$ of the total flux originates from the Seyfert 2 (the southern galaxy). The spectrum of the northern galaxy has much weaker emission lines in H$\alpha$ and \[NII\], typical of a starburst or LINER [@Colina1991]. More recently HST imaging provided new evidence that a double nucleus may be present in both the north and southern components of the Superantennae [@Borne1999], suggesting a multiple merger origin of the system. Based on the photometry of Table \[phot\], we present in Figures \[ir19254s\_spec\] and \[ir19254n\_spec\] the MIR spectral energy distribution for each galaxy, while the integrated MIR SED of the whole system is shown in Figure \[ir19254s\_phot\]. In the latter we also compare our data with the spectrum obtained with ISOPHOT-S, the beam of which spatially covered the full emission of IRAS19254-7245. The extreme difference in the MIR intensities between the southern and northern members are apparant as well as the constrasts in their spectral shape. More than 95 $\%$ of the MIR emission of IRAS19254-7245 originates from the southern Seyfert 2 galaxy which displays a peculiar spectrum with a dominant thermal emission at 15$\mu$m (f$_{15\mu m}$/f$_{6.7\mu m}$ $\sim$2.7) and weak UIBs (f$_{6.7\mu m}$/f$_{6\mu m}$$\sim$1.2). This strong continuum relative to the UIB emission can be the consequence of a high radiation field density mainly produced in ionized regions close to young stars [@Mirabel1998] or AGN [@Laurent2000]. On the contrary, the northern galaxy has strong UIBs (f$_{6.7\mu m}$/f$_{6\mu m}$$\sim$2.5) and faint thermal emission at 15$\mu$m (f$_{15\mu m}$/f$_{6.7\mu m}$$\sim$1.2), which is typical of MIR emission from normal spiral galaxies with cool IRAS colors [@Dale2001; @Roussel2001]. Comparison of in its broad-band SED with the template SED of a quiescent star forming region within the disk of M82 [@Laurent2000] illustrates that they are in a fairly good agreement (Figure \[ir19254n\_spec\]). The absolute luminosities presented in Table \[global\] shows that the MIR emission originating from the southern Seyfert 2 galaxy is by far the strongest in our sample although the most luminous FIR source is IRAS14348-1447 (see Table \[info\]). One may also note that the flux density near 5$\mu$m does not reach zero level but is $\sim$100mJy, suggesting the presence of a hot dust component, which as discussed in the previous section is a clear sign of a hot dusty torus of an AGN [@Laurent2000]. Similarly, one can draw the same conclusion by observing the combination of the f$_{15\mu m}$/f$_{6.7\mu m}$ and f$_{6.7\mu m}$/f$_{6\mu m}$ flux ratios. In IRAS19254-7245S, the low f$_{6.7\mu m}$/f$_{6\mu m}$ indicates weak UIB emission while f$_{15\mu m}$/f$_{6.7\mu m}$$\sim$2.7, a value somewhat lower than other well studied starburst galaxies such as Arp220 [f$_{15\mu m}$/f$_{6.7\mu m}$$\sim$3.9, @Charmandaris1999b] or the extremely strong starburst region in the Cartwheel [f$_{15\mu m}$/f$_{6.7\mu m}$$\sim$5.2, @Charmandaris1999a]. This effect can be understood since the hot continuum produced by an AGN at short MIR wavelengths would cause the flux in the 6–10$\mu$m range to increase more relative to the increase observed between 12–16$\mu$m and as result it would be added the UIB emission sampled by the LW2 filter. Could the large difference in the MIR brightness between the north and south component in IRAS19254-7245 be related to the additional contribution of the AGN? Studies of the dynamical evolution of this system suggest that the starburst time scale is much shorter than the dynamical age of the merger [@Mihos1998]. Even though we can not quantify accurately the fraction of MIR luminosity due to the AGN activity, it appears that the southern component of IRAS19254-7245 has reached an AGN dominant phase, however short this may be, after an initial phase of strong starburst activity (see @Laurent2000 and @Genzel1998 for details on the MIR AGN/starburst fraction of this and other galaxies). The MIR properties of the northern nucleus are similar to a normal spiral galaxy which indicates that even if a starburst did occur in it at some point, it has by now subsided and the star formation is progressing in a more quiescent rate. Finally, we note that the southern galaxy exhibits higher f$_{15\mu m}$/H$\alpha$ ($\sim$225) compared to that inthe north ($\sim$85). We interpret this effect as a consequence of higher dust concentration and stronger absorption in the southern nucleus since near AGNs high column densitis of molecular gas are typically observed. The southern galaxy also has a higher f$_{15\mu m}$/K ratio than that in the north, which has an f$_{15\mu m}$/K ratio of a normal spiral galaxy,consistent with its overall MIR spectral features (Table \[k\_ha\]). IRAS23128-5919 -------------- This system consists of two merging galaxies in a rather late stage of their interaction, the nuclei of which are separated by a projected distance of 4 kpc (5) (Figure \[ir23128\_im\]). Two tidal tails 40kpc stretch in opposite directions [@Bergvall1985; @Mihos1998]. Based on optical studies, the northern galaxy is classified as a starburst, while it is unclear whether the southern one is a Seyfert, a starburst or a LINER [@Duc1997a]. Optical spectroscopy of the southern nucleus shows a relatively high ionization state having emission lines with wings of $\sim$ 1500kms$^{-1}$ larger in the blue and extending $\sim$5kpc out from the nucleus. These emission lines, as well as other Wolf-Rayet features observed, could be caused by supernova winds and turbulent motions associated with the merger [@Johansson1988]. The northern galaxy on the other hand, has narrower emission lines and weaker starburst activity. In Figure \[ir23128\_spec\], we present the CVF spectra of each galaxy along with our flux measurements using the four broad-band filters. The integrated MIR SED of the whole system is displayed in Figure \[ir23128\_phot\], as well as the ISOPHOT-S spectrum which is in good agreement with our data. As in the case of IRAS19254-7245, no MIR emission is seen to be associated with the tidal tails down to our sensitivity limits (see Figure \[ir23128\_im\]). We find that approximately 75$\%$ of the MIR flux in IRAS23128-5919 originates from the southern galaxy. The spectrum reveals that the thermal continuum (12–16$\mu$m) is higher in the southern galaxy than that of the north, making the southern galaxy the dominant origin of the MIR emission. Since the SED of both components displays a rising spectrum with prominent UIBs and a weak continuum at 5–6$\mu$m, we conclude that the MIR emission in this system is mostly powered by massive star formation. The same conclusion can be reached using the broad-band filter flux ratios for the two galaxies. In the northern more quiescent galaxy of the pair, the MIR activity indicator f$_{15\mu m}$/f$_{6.7\mu m}$ (LW3/LW2) is 2.6, lower than the value of the southern galaxy ($\sim$3.3), while its ratio of f$_{6.7\mu m}$/f$_{6\mu m}$ is $\sim$2.0, higher than that of the southern galaxy which has an f$_{6.7\mu m}$/f$_{6\mu m}$$\sim$1.5. Following similar reasoning as for the southern component of the Superantennae, these results can be interpreted as an increase in the density of regions of the southern component, relative to the density of the photo-dissociation regions. Further comparisons of the properties of this galaxy to IRAS19254-7245 (see Table \[global\]) show that its ratio of L$_{\rm LW3}$/L$_{\rm IR}$ $\sim$0.03 is smaller despite is high L$_{\rm IR}$(L$_{\sun}$)/M$_{\rm H_2}$(M$_{\sun}$) of $\sim$70. This indicates that even though IRAS23128-5919 is more efficient in consuming the molecular gas, its radiation field is not sufficient to heat the large amount of dust at similarly high temperatures as does the AGN in the Superantennae. The data presented in Table \[k\_ha\] also indicate that the southern galaxy of the pair emits more MIR flux relatively to its stellar emission (f$_{15\mu m}$/K $\sim$284) and is apparently more obscured by dust (f$_{15\mu m}$/H$\alpha \sim $85). In conclusion, the more luminous galaxy is clearly undergoing a stronger star formation phase than its northern companion. The global MIR characteristics of this system are in agreement with the assertion that a starburst is the dominant heating mechanism for the dust and no evidence of an AGN contributing to the ISOCAM wavelength range are present. IRAS14348-1447 -------------- IRAS14348-1447 is the most distant object in the IRAS Bright Galaxy Sample with a redshift of 0.08 [@Soifer1987]. This system, shown in Figure \[ir14348\_im\], consists of two galaxies separated by a projected distance of 6kpc (4) with a tail extending to more than 10kpc away from the northern nucleus [@Melnick1990]. Strong H$_2$ emission, mainly triggered by shocks in molecular clouds, has been detected [@Geballe1988; @Nakajima1991]. The presence of large quantities of shocked molecular hydrogen is consistent with the detection of 6$\times10^{10}$ M[$_{\sun}$]{} of molecular gas in this system which makes it the most H$_{2}$-rich in the ultraluminous galaxy sample [@Sanders1991]. The large quantities of cold dust, inferred using the usual gas to dust conversion, lead us to believe that the reddening seen in both galaxies is a consequence of strong absorption and not due to an intrinsically old stellar population [@Carico1990a]. Based on near-infrared spectroscopic observations in Pa$\alpha$ and H$_{2}$ lines, the nucleus of the southern galaxy has been classified as a Seyfert 1.5 and the northern one as a Seyfert 2 [@Nakajima1991], while their optical line features are similar to those of LINERs [@Veilleux1995] or Seyfert 2 galaxies [@Sanders1988]. Due to its relatively weak MIR emission this source was only observed with the two ISOCAM broad band filters LW2 and LW3 (Table \[param\]). As in the other galaxies in this sample, MIR emission is detected only from the circumnuclear regions. We estimate that $\sim$75$\%$ of the MIR flux seen in both filters originates from the southern galaxy, which is also the more active one in the optical. Interestingly, this roughly scales with the fraction of the CO emission from the two components [@Evans2000]. The southern galaxy exhibits the higher hot dust component traced by 15$\mu$m (LW3) relative to the UIB emission at 7$\mu$m (LW2). Using the LW3/LW2 ratio to trace the MIR activity in this system we find that f$_{15\mu m}$/f$_{6.7\mu m}$$\sim$3.4 in the southern galaxy and f$_{15\mu m}$/f$_{6.7\mu m}$ $\sim$2.0 in the northern one. Since we only have one MIR color, we can not comment on the MIR contribution the AGN. Nevertheless, the low integrated L$_{\rm LW3}$/L$_{\rm IR}$ of IRAS14348-1447 ($\sim$0.02)would be consistent with a negligible AGN contribution in the MIR (Table \[global\]) while the high dust obscuration suggested by the increased f$_{15\mu m}$/H$\alpha$$\sim$333 is consistent with its large molecular gas content [@Mirabel1990]. Evidence that the starburst activity is the main heating mechanism can also be seen in Figure \[ir14348\_phot\] using the MIR spectrum of the whole system obtained with ISOPHOT. This spectrum reveals strong UIBs (f$_{6.7\mu m}$/f$_{6\mu m}$$\sim$2.2, see Table \[phot\]) likely caused by a starburst with only a weak contamination by an AGN [to the 25$\%$ level, see @Genzel1998; @Lutz1998]. Discussion and concluding remarks {#sec:discuss} ================================= A wealth of observational data available has shown that ULIRGs have high concentrations of gas and dust in their nuclei, sufficient to account for most of their observed infrared luminosity [see @Sanders1996 for a review]. Whether the energy source of ULIRGs is a dust enshrouded AGN or a starburst still remains an open issue. However, recent indirect evidence is beginning to favour the existence of bright extremely red point-like sources in the nuclear regions of ULIRGs. More specifically near-infrared observations of luminous infrared galaxies have shown that their flux at 2.2$\mu$m is more concentrated towards the center than at 1.3$\mu$m [@Carico1990b; @Scoville2000]. Furthermore, recent high resolution MIR observations using Keck of a sample of ULIRGs reveal compact sub-arcsecond sources (with linear scales of $\sim$100–300 pc) which contain 30% to 100% of the observed MIR energy of these galaxies [@Soifer2000]. This contrasts with the LIRGs ($10^{11}$L$_{\sun}$ $\leq$ L$_{\rm IR} \leq 10^{12}$L$_{\sun}$), in which the infrared energy seems to be generated over somewhat larger scales [$\sim$100pc–1kpc, @Soifer2001] and sometimes can be found in extra-nuclear regions associated with the physical interaction of merging pairs of galaxies. Furthermore, there are galaxies such as VV114 where it has even been found that a substantial fraction of the MIR flux originates from an extended component of hot dust emission spread over several kpc scales [@Soifer2001; @LeFloch2002]. ULIRGs are thus not simply a scaled-up version of LIRGs and require further dynamical compression of the molecular gas responsible for the IR luminosity within very compact regions. A plausible mechanism would be one where the shocks and tidal forces of the interaction first lead to star formation over galactic scales, leading to IR luminosities up to a few 10$^{11}$L$_{\sun}$. Subsequently, gravitational instabilities and the formation of a bar, strip the gas of its angular momentum, funneling large quantities towards the nuclear regions of galaxies, which can feed circumnuclear starbursts or AGNs and trigger the ultraluminous phase in the infrared [@Combes2001]. Even though the above scenario is appealing, given the high extinction in the nuclei of ULIRGs, the limited atmospheric transmission in the MIR windows, and the limited sensitivity of ground-based instruments, questions related to the direct probing of the nuclear activity such as “does all MIR emission from those systems originate from the nuclei?” and if not “what are the spectral properties of any extended component?” still remain unanswered. This is where the superb sensitivity of space instruments, such as ISO, is essential. We have found that in the ULIRGs studied here *more than $\sim $95% of the MIR emission seen by IRAS is confined within a few arcsecs of their central region*. Obviously the relatively large pixel size of the ISOCAM detector places limitations in interpreting these findings. However, deconvolution tests of the central point source in each galaxy suggest that the corresponding nuclei are resolved and the physical diameter of the emitting region is contained within 1 to 2kpc. Moreover, with the exception of the Superantennae where the MIR spectrum is dominated by the emission arising from the AGN of the southern galaxy, the bulk of the IR luminosity of IRAS23128-5919 and IRAS14348-1447 is powered by massive star formation. The fact that starbursts can dominate the MIR emission in galaxies with IR luminosities as high $\sim$10$^{12}$L$_{\sun}$ had already been demonstrated in other ISOCAM-CVF [@Tran2001] and ISO-SWS [@Genzel1998] observations of ULIRGs, and is supported by our results. Given that an active nucleus appears to be always present in the most energetic objects of the local Universe [@Lutz1998], our MIR data favor a luminosity threshold for the transition between starburst- and AGN-dominated galaxies which is higher than the IR luminosity of the galaxies in our sample. This is in agreement with the results of @Tran2001 who proposed that this transition takes place at L$_{\rm IR}\sim$10$^{12.5}$L[$_{\sun}$]{} and also found individual starbursts up to 10$^{12.65}$L[$_{\sun}$]{}. Our data also indicate is that such starbursts can be confined to the very central nuclear regions which may have important consequences in the probing how the instabilities fuel the inner regions of galaxies [e.g. @Combes2001], as well as determining the nature of high redshift dusty sources [e.g. @Ivison2000]. Another striking feature revealed in our observations is that in all three cases one galaxy seems to dominate the MIR energy output of the system by more than 75%. Could this be a record of the initial distribution of the amount of molecular gas available in each merging progenitor or could this suggest that in the later stages of interaction, the gas finally merges towards *one* component? If the latter were true one would expect that a sufficiently large quantity of gas could trigger and fuel both circumnuclear star forming activity and AGN-type activity at the core of a single object. This is evident in the southern galaxy of IRAS19254-7245 which harbors an active nucleus as well as numerous massive star forming regions. As we mentioned in the introduction though the presence of a Seyfert nucleus is correlated with a MIR flux increase relative to the FIR luminosity of the entire galaxy, which is what one can actually derive from our observations when we compare the Superantennae with IRAS14348-1447. IRAS14348-1447 has indeed a much higher total IR luminosity despite its MIR flux being lower than that of the southern source of IRAS19254-7245. Furthermore, using the f$_{15\mu m}$/H$\alpha$ and f$_{15\mu m}$/K ratios as probes of dust absorption and hot dust emission normalized to the mass of the galaxy, we find that in each interacting system it is always the most active galaxy of the system that exhibits the higher ratios. In each system, the most luminous galaxy contains a larger amount of molecular gas leading to the triggering/feeding of the starburst activity and/or an active nucleus. Finally, we wish to stress once more that because of the limited spatial resolution in studying such distant sources, the diagnostics we have used in this paper address only the integrated MIR emission of each galaxy. Our difficulties to identify whether an active nucleus is solely responsible for the increase in the MIR luminosity relative to the FIR emission will not be resolved unless we can either clearly map the extent of the emitting region or obtain MIR spectra using very narrow slits. 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--- abstract: 'The calculation of optimal structures in reaction-diffusion models is of great importance in many physicochemical systems. We propose here a simple method to monitor the number of interphases for long times by using a boundary flux condition as a control. We consider as an illustration a 1-D Allen-Cahn equation with Neumann boundary conditions. Numerical examples are given and perspectives for the application of this approach to electrochemical systems are discussed.' author: - 'J.-P. Chehab[^1], A. A. Franco[^2], Y. Mammeri[^3]' date: title: 'A simple mathematical approach to optimize the structure of reaction-diffusion physicochemical systems' --- Introduction ============ The dynamics of a large diversity of physicochemcal systems can be mathematically modeled as reaction-diffusion systems in which it is described how the composition of multiple chemical species distributed in space change under the influence of competitive chemical reactions between the species (giving origin to a new species) and the diffusion which causes the species to spread out in the space. It is well known that depending on the relative importance of the kinetics and the diffusion these systems can provide a large diversity of behaviors, including the formation of complex structures and patterns see [@Sachs]. Such a structure formation occurs for example during the solid phase formation and evolution in intercalation and conversion reactions in rechargeable lithium batteries [@Franco1; @Franco2], during the self-organisation of materials occuring with the fabrication process of composite electrodes for electrochemical devices applications [@Malek1], during the microstructural evolution of composite elecrodes upon their degradation [@Malek2] and in other competitive chemically reactive systems like in the Belousov-Zhabotinsky reaction [@Sirimungkala].\ Designing appropriate controllers of these reaction-diffusion systems can reveal of great relevance within a reverse engineering approach for example towards the optimization of discharge-charge of lithium batteries (by for example enhancing the formation of solid phases during discharge more reversible upon charge) and the optimization of the structure of the fabricated electrodes as function of the fabrication parameters (e.g. temperature dynamics, reactant flow, etc.).\ In this paper, we consider the one-dimensional Allen-Cahn equation $$\begin{aligned} {\frac{\textstyle \partial u} {\textstyle \partial t}} -{\frac{\textstyle \partial^2 u} {\textstyle \partial x^2}} +{\frac{\textstyle 1} {\textstyle \epsilon^2}}f(u)=0 & x\in]0,1[, t >0,\\ u_x(0,t)=\alpha(t), u_x(1,t)=0& \forall t >0,\\ u(x,0)=u_0(x) & x\in ]0,1[.\end{aligned}$$\ This reaction-diffusion equation describes the process of phase separation in many situations. It was originally introduced in [@AllenCahn] by Allen and Cahn to model the motion of anti-phase boundaries in crystalline solids. In equation (1), $u$ represents the concentration of one of the possible phases, $\epsilon$ represents the interfacial width, supposed to be small as compared to the characteristic length of the laboratory scale. The homogenous Neumann boundary condition (when $\alpha(t)=0$) traduces that there is no loss of mass across the boundary walls. However, the Allen-Cahn equation is invoqued in a large number of complicated moving interface problems in materials science through a phase-field approach, therefore a large litterature in mathematical analysis and in numerical analysis is devoted to the study of the mathematical properties of this equation and of its simulation (see [@MPierre; @JShen] and the references therein).\ In equation (1), $f(u)$ represents the potential energy and $\alpha(t)$ represents the control flux at one of the boundaries; $f(u)$ is assumed having stable roots $\rho_i$, $i=1,\cdots, r$ such that $f(\rho_i)=0$ and $f'(\rho_i)>0$. It is observed in many cases that when $\epsilon <<1$ and as $t$ goes to $+ \infty$ , the solutions tend to steady states ${\bar u}$ which consist in (almost) piecewise constant functions whose the different values are equal to the stable roots of $f$ which represent the different phase stripes. Hence ${\bar u}$ exhibits large gradient near $\rho_i$, as illustrated in Figure (\[fig1\]). ![Steady state for $\epsilon=0.004$ (left) and for $\epsilon=0.001$ (right).[]{data-label="fig1"}](solution1.png "fig:"){height="7.0cm" width="8.5cm"} ![Steady state for $\epsilon=0.004$ (left) and for $\epsilon=0.001$ (right).[]{data-label="fig1"}](solution2.png "fig:"){height="7.0cm" width="8.5cm"} \ \ An important issue in the conception of rechargeable lithium and post-lithium batteries, is the design of active materials providing upon the battery discharge a number of interphases as low as possible. The morphological simplicity of such discharged materials is expected to enhance the rechargeability of these type batteries and thus to increase their efficiency [@Franco2]. In this paper we propose a first numerical strategy to calculate the boundary flux function $\alpha(t)$ on a given time interval $[0,T]$, with $T$ large enough, in such a way the number of interphase of the steady state ${\bar u}$ is minimized. To this end, we consider as control function $\alpha(t)$, $\epsilon$ being constant. For the sake of simplicity, we first restrict ourselves to the case $f(u)=u(u^2-1)$ which possesses 3 roots: $u=\pm 1$ which are stable and $u=0$ which is unstable.\ The article is organized as follows: first, in Section 2, we present first the global numerical strategy by deriving the estimation of the number of interphases, which will be the merit function to minimize. Then, we present the finite differences discretization of the system in space and we describe the numerical solver, that includes the optimization process as well as the full discretized problem to be solved at each iteration. In Section 3, we present some numerical results demonstrating the numerical controllability of the problem: we calculate optimal$\alpha$ for different values of $u_0$, $T$ and $\epsilon$. Finally, in Section 4 we conclude and indicate further perspectives of development of our work. Numerical strategy ================== Estimation of the number of interphases --------------------------------------- We consider the finite differences discretization in space of the Allen-Cahn equation which leads to a differential system. The grid points $x_i$, $i=1,\cdots, N$ are regularly spaced for simplicity, $h$ is the corresponding stepsize. We assume that $h$ is small enough in order the discrete solution captures the strong gradients near the interphases. The steady solution ${\bar u}$ is considered to be almost piecewise constant, so its approximations at grid points ${\bar u}_i$, $i=1,\cdots, N$ take the values $\pm 1$. Hence $${\bar u}_{i+1}-{\bar u}_i =\left\{ \begin{array}{c} 0 \\ 2 \\ -2 \\ \end{array} \right.$$ Therefore, the number of interphases is $$\begin{aligned} \label{formula_changes} N({\bar u})&={\frac{\textstyle 1} {\textstyle 2}}\displaystyle{\sum_{i=0}^{N}\mid {\bar u}_{i+1}-{\bar u}_i \mid}.\end{aligned}$$ This quantity can be related to the $L^1$-norm of $u'$, indeed $$\begin{aligned} N({\bar u})={\frac{\textstyle 1} {\textstyle 2}}\displaystyle{\sum_{i=0}^{N}\mid {\frac{\textstyle {\bar u}_{i+1}-{\bar u}_i} {\textstyle h}} \mid h}\simeq {\frac{\textstyle 1} {\textstyle 2}}\displaystyle{\int_0^1 \mid {\bar u}'(x)\mid dx}.\end{aligned}$$ In Figure (\[fig1\]) (left), we count 10 changes, the result given by formula (\[formula\_changes\]) is 9.9968 and in Figure (\[fig1\]) (right) 48 changes are counted while (\[formula\_changes\]) estimation is 47.7475.\ We remark that an interesting numerical issue could be to plug an adaptive grid strategy since the steady solution needs only few points to be represented. Selection of given phases ------------------------- Our approach applies when more than 2 interphases are present. Indeed, consider for simplicity the case of $m$ stable phases. To obtain the number of interphases, it is sufficient to split the final signal profile into 4 parts, each of them reprensenting the state of one phase stirp (see figure below in the case of $m=4$). Once done, we can apply formula (\[formula\_changes\]) separately. This procedure allows using a weighted merit function $$\begin{aligned} F(u)=\displaystyle{\sum_{i=1}^m\omega_iN_i(u)},\end{aligned}$$ where $N_i(u)$ is the number of connex components for phase $i$ and $\omega_i \ge 0$ the associated weight: a large value of $\omega_i$ enforces the optimal state to provide small number of phases of type $i$. So it is possible to select a given profile. It has to be pointed out that $\displaystyle{\sum_{i=1}^mN_i(u)}\neq N(u)$, the total number of interphases, in fact $\displaystyle{\sum_{i=1}^mN_i(u)}\ge N(u)$, however the simplified summation formula allows selecting easily given phase stripes (Figure (\[fig3\])). ![The original signal[]{data-label="fig2"}](multi_phase_orig_signal.png){height="7.0cm" width="9cm"} ![Multiphase decomposition of a signal, superposition of the various phases is represented at the center[]{data-label="fig3"}](multi_u1.png "fig:"){height="7.0cm" width="8cm"} 0.2cm ![Multiphase decomposition of a signal, superposition of the various phases is represented at the center[]{data-label="fig3"}](multi_um1.png "fig:"){height="7.0cm" width="8cm"} ![Multiphase decomposition of a signal, superposition of the various phases is represented at the center[]{data-label="fig3"}](multi_supper.png "fig:"){height="7.0cm" width="8cm"}\ ![Multiphase decomposition of a signal, superposition of the various phases is represented at the center[]{data-label="fig3"}](multi_u12.png "fig:"){height="7.0cm" width="8cm"} 0.2cm ![Multiphase decomposition of a signal, superposition of the various phases is represented at the center[]{data-label="fig3"}](multi_um12.png "fig:"){height="7.0cm" width="8cm"} When the different phases $U_I, i=1,\cdots, m$ are known [*a priori*]{}, the merit function can be defined more precisely as $$\begin{aligned} \label{weighted} F(u)=\displaystyle{\sum_{i=1}^m\omega_i\parallel u-U_i\parallel^2}.\end{aligned}$$ For instance, with $m=2$ and $f(u)=u(u^2-1)$, the two stable phases are $U_{1,2}=\pm 1$ and the merit function associated to $U_1=+1$ is $$F(u)=\parallel u-1\parallel^2,$$ which can be considered directly in the continuous case with, e.g., $L^2$ norm, giving rise to the merit function $F(u)=\displaystyle{\int_0^1 (u(x)-1)^2dx}$. Global scheme ------------- We denote by $u_i(t)$ the approximation of $u(x_i,t)$ generated by the semi discrete scheme $$\begin{aligned} \label{eq1} \mbox{for } i=1,\cdots, N \ {\frac{\textstyle du_i(t)} {\textstyle dt}} +{\frac{\textstyle 2u_i(t)-u_{i-1}(t)-u_{i+1}(t)} {\textstyle h^2}}+{\frac{\textstyle 1} {\textstyle \epsilon^2}}u_i(t)\left(u_i^2(t)-1\right)=0& t>0,\\ \label{eq2} u_0(t)=u_1(t)-h\alpha(t) & t >0,\\ \label{eq3} u_{N+1}(t)=u_{N}(t) & t >0,\\ \label{eq4} u_i(0)=v_i. & \end{aligned}$$ in which we have implemented Neumann boundary condition to calculate the values $u_{N+1}(t)$ and $u_{0}(t)$. As a time marching scheme, we will use a semi implicit one, in order to have a good stability: it is important since, as we will see hereafter, the calculation of optimal $\alpha(t)$ requires a great number of numerical solutions of this system and a not too small time step $\Delta t$ must be used. We fix a value for the final time $T$, $T$ being large enough to obtain a steady state: in practice the solution converges toward equilibrium relatively fastly for small values of $\epsilon$, which is the case here. The time interval $[0,T]$ is subdivided into $M$ subintervals of length $\Delta t$, $[t_k,t_{k+1}]$, $k=0,\cdots M-1$, with $t_k=k\Delta t$ and $t_M=T$. We note $u^M$ the numerical approximation to ${\bar u}$. A first idea is to compute $\alpha(t)$ as a piecewise constant function in time, say $$\begin{aligned} \alpha(t)=\displaystyle{\sum_{k=0}^{M-1} \alpha_k\chi_{[t_k,t_{k+1}]}}.\end{aligned}$$\ We remark here that calculating $\alpha(t)$ as a piecewise constant function, is a simple and stable approach: other techniques allowing orthogonal polynomial (such as Fourier or Laguerre) could be used but the strong decreasing of the Fourier coefficients make the problem ill-conditioned. Also, an heuristic method can be used for accelerating the numerical convergence: once computed $\alpha^M(t)$ for a given $M$ and $\Delta t$, one can repeat the computation for $2M$ time steps (with $\Delta t/2$), in order to get a more precise result $\alpha^{2M}(t)$, starting from a mid-point interpolated value.\ The problem we want to solve is then expressed as\ $$\mbox{find $(\alpha_0, \cdots,\alpha_{M-1})$ such as minimizing $N(u^M)$}. \label{opt_pb}$$\ We can now describe the global approach under the following algorithm Initialization: Start from an initial guess $\alpha^{(0)}(t)=\displaystyle{\sum_{k=0}^{M-1} \alpha^{(0)}_k\chi_{[t_k,t_{k+1}]}}$ Compute $u^M$ by time integration of (\[eq1\])-(\[eq4\]) with $$\alpha^{(m)}(t)=\displaystyle{\sum_{k=0}^{M-1} \alpha^{(m)}_k\chi_{[t_k,t_{k+1}]}}$$ Compute $N(u^M)$. Update $(\alpha^{(m+1)}_0, \cdots,\alpha^{(m+1)}_{M-1})$ from $\alpha^{(m)}$ by a derivative free optimization process (nonlinear search) Also, we will have to establish numerical convergence by varying $M$, $\Delta t$. Practical solution to the optimization problem ---------------------------------------------- ### Full discretization scheme of the equations We subdivide $[0,T]$ into $M$ subintervals of length $\Delta t$ and note $u^k_i$ the approximation of $u_i(t_k)$ at time $t_k=k \Delta t$. Let $A$ be the discretization matrix of the negative seconde derivative en $x$ with homogeneous Neumann boundary conditions and $U^k=(u_0^k,\cdots,u^k_{n+1})^T$. We consider the following linearized implicit Euler scheme which reads, after usual simplifications as $$\begin{aligned} \label{Scheme1} U^{k+1}+\Delta t A U^{k+1}+{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}(U^k)^2U^{k+1}=U^k+{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}U^k +{\frac{\textstyle \Delta t} {\textstyle h}}F^k,\end{aligned}$$ where $F^k=(-\alpha(t_k),0\cdots,0)^T$. We have the Assume $\alpha=0$. If $\Delta t < \epsilon^2$ and if $-1\le U^0\le 1$ then the sequence $U^k$ defined by the scheme (\[Scheme1\]) satisfies $-1\le U^k\le 1, \forall k$. If $\alpha^k=0$, we set $\overline{U^k}=U^k-1$ and $\underline{U^k}=U^k+1$. We’ll show by induction that $\overline{U^k}\le 0$ and $\underline{U^k}\ge 0 \forall k \in \N$. Let us fix $k$ and assume that $-1\le U^k\le 1$ says $\overline{U^k}\le 0$ and $\underline{U^k}\ge 0$. We have, after simplifications $$\left(1+{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}} (\overline{U^k}+1)^2\right)\overline{U^{k+1}} +\Delta t A \overline{U^{k+1}}=\left(1-{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}\right)\overline{U^k} -{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}(\overline{U^k})^2.$$ If $\overline{U^k}\le 0$, then $\left(1+{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}} (1+\overline{U^k}^2+\overline{U^k}) \right) \ge 1-{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}>0$, so the matrix $$M^k=diag\left(1+{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}} (1+\overline{U^k}) ^2\right) +\Delta t A$$ has the discrete maximum property ($M_k$ is a M-matrix) and since $\overline{U^k}\le 0$ and ${\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}<1$, we have $\left(1-{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}\right)\overline{U^k} -{\frac{\textstyle \Delta t} {\textstyle \epsilon^2}}(\overline{U^k})^2 \le 0$ so that $$\overline{U^{k+1}}\le 0.$$ We proceed in a similar way for $\underline{U^k}$. As, we will see hereafter in the numerical simulations, this is observed for moderate values of $\alpha$.\ It is important to underline that it is crucial that the numerical scheme heritates of the intrinsic properties of the equation (maximum principle, asymptotic behavior) for calculating a numerical control: if not, non-physical control can be determined[@Ignat1]. ### Choice of the optimization method As pointed out before, the principal key of this approach is the choice of the optimization method to compute $\alpha(t)$. This is not an easy task since the merit function is not differentiable in the $L^1$ case; generally the gradient is not available as in the $L^2$ case. Hence, gradient methods cannot be used. We have then to address to derivative free optimization algorithms [@ConnScheinbergVicente].\ In order to illutrate the numerical cacutation of optimal $\alpha$, we use Matlab build-in house solver, such as [x = fminsearch(fun,x0)]{} or [x = fminunc(fun,x0)]{} where [fun]{} is the objective function which in our case is built numerically by solving the Allen–Cahn equation up to $t=T$ with $\alpha$ as an input. Numerical Results ================= The minimization procedure used is [fminunc]{} available in from the optimization toolbox of Matlab computational software, [@Matlab] . We fix $\epsilon$ and we start from $\alpha^0(t)=0, \forall t \in [0,T]$. We hereafter display results for different values of $\epsilon$ and $T$. The merit function is $N(u^M)$. Regular fixed initial data - numerical convergence -------------------------------------------------- Starting from a regular initial data allows to observe numerical convergence of the optimal control function $\alpha(t)$ as the number of discretisation points $N$ increases, and as, for a fixed $T$, the time step $\Delta t $ decreases. We can see, in Figures (\[fig4\])-(\[fig5\])-(\[fig6\])-(\[fig7\]) the good coherence of the results for fixed values of $\epsilon=0.01$ and $T$ when varying $\Delta t$ and the number $N$ of grid points. In all the cases, the global procedure allows minimizing the number of interphases or stripes. ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig4"}](alpha_127_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig4"}](merit_127_01_50.png "fig:"){height="7.0cm" width="8cm"}\ ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig4"}](init_sol_127_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig4"}](sol_127_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.005, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig5"}](alpha_127_01_100.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.005, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig5"}](merit_127_01_100.png "fig:"){height="7.0cm" width="8cm"}\ ![$\epsilon=0.01, \ \Delta t = 0.005, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig5"}](init_sol_127_01_100.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.005, \ T=0.5, u_0=\cos(20\pi x), \ N=127$[]{data-label="fig5"}](sol_127_01_100.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=255$[]{data-label="fig6"}](alpha_255_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=255$[]{data-label="fig6"}](merit_255_01_50.png "fig:"){height="7.0cm" width="8cm"}\ ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=255$[]{data-label="fig6"}](init_sol_255_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=255$[]{data-label="fig6"}](sol_255_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=511$[]{data-label="fig7"}](alpha_511_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=511$[]{data-label="fig7"}](merit_511_01_50.png "fig:"){height="7.0cm" width="8cm"}\ ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=511$[]{data-label="fig7"}](init_sol_511_01_50.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.5, u_0=\cos(20\pi x), \ N=511$[]{data-label="fig7"}](sol_511_01_50.png "fig:"){height="7.0cm" width="8cm"} Random initial data ------------------- ### Minimization of the number of interphases Here, in Figures (\[fig8\]) and (\[fig9\]) the discrete components of $u_0$ are randomly calculated following a uniform law on $[-1,1]$ and we start from $\alpha^0(t)=0, \forall t \in [0,T]$. As we see, here again, in all cases, the global procedure, illustrating the effective numerical controllability by the boundaries and the robustness of the approach, since the data are very oscillating. ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.8, u_0=rand, \ N=127$[]{data-label="fig8"}](rand_alpha_127_01_80.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.8, u_0=rand, \ N=127$[]{data-label="fig8"}](rand_merit_127_01_80.png "fig:"){height="7.0cm" width="8cm"}\ ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.8, u_0=rand, \ N=127$[]{data-label="fig8"}](init_rand_sol_127_01_80.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.01, \ \Delta t = 0.01, \ T=0.8, u_0=rand, \ N=127$[]{data-label="fig8"}](rand_sol_127_01_80.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.01, u_0=rand, \ N=255$[]{data-label="fig9"}](rand_alpha_255_005_100.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.01, u_0=rand, \ N=255$[]{data-label="fig9"}](rand_merit_255_005_100.png "fig:"){height="7.0cm" width="8cm"}\ ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.01, u_0=rand, \ N=255$[]{data-label="fig9"}](init_rand_sol_255_005_100.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.01, u_0=rand, \ N=255$[]{data-label="fig9"}](rand_sol_255_005_100.png "fig:"){height="7.0cm" width="8cm"} ### Selection of a given phase Finally, we give hereafter a numerical illustration of the optimization process when considering a weighted merit function as in (\[weighted\]): we adopt here $F(u)=10 \parallel u-1\parallel +\parallel u+1\parallel $. As we see in Figure (\[fig10\]), the merit function favors the formation of the phase $U=1$, the initial datum is, as above, randomly generated on $[-1,1]$ by an uniform law. ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.02, u_0=rand, \ N=255$[]{data-label="fig10"}](opt_alpha_U1.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.02, u_0=rand, \ N=255$[]{data-label="fig10"}](merit_iter_U1.png "fig:"){height="7.0cm" width="8cm"}\ ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.02, u_0=rand, \ N=255$[]{data-label="fig10"}](sol_alpha_0_U1.png "fig:"){height="7.0cm" width="8cm"} ![$\epsilon=0.005, \ \Delta t = 0.0001, \ T=0.02, u_0=rand, \ N=255$[]{data-label="fig10"}](sol_alpha_opt_U1.png "fig:"){height="7.0cm" width="8cm"} Concluding remarks and features =============================== In this paper, we have presented a simple approach to calculate numerically a boundary control that allows obtaining an optimal steady-sate configuration, i.e., with a minimal number of interphases. We have also demonstrated that we can also favor the formation of a given phase by following the same procedure. The results we obtained are encouraging and show the numerical faisability of the proposed method.. Of course, we have here considered first a relatively simple case, namely the one dimensional case before extending the approach to 2D or 3D models which correspond to more realistic situations found, for example, in electrochemistry. Furthermore, the monitoring of the number of interphases by $\epsilon$ (problem 2) is an important feature that we will study in a near future. Finally the integration of such optimization algorithms in an in-house multiscale simulator of electrochemical power generators will be also considered [@FrancoLiberT]. [99]{} S. M. Allen and J. W. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater., (27) (1979), pp 1085–-1095. A. R. Conn, K. Scheinberg, and L. N. 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[**** ]{}\ Stanisław Saganowski^1,\*^, Piotr Bródka^1^, Michał Koziarski^2^, Przemysław Kazienko^1^\ **1** Department of Computational Intelligence, Faculty of Computer Science and Management, Wrocław University of Science and Technology, Wrocław, Poland\ **2** Department of Electronics, Faculty of Computer Science, Electronics and Telecommunications, AGH University of Science and Technology, Kraków, Poland\ \* [email protected] Abstract {#abstract .unnumbered} ======== In the world, in which acceptance and the identification with social communities are highly desired, the ability to predict the evolution of groups over time appears to be a vital but very complex research problem. Therefore, we propose a new, adaptable, generic, and multistage method for Group Evolution Prediction (GEP) in complex networks, that facilitates reasoning about the future states of the recently discovered groups. The precise GEP modularity enabled us to carry out extensive and versatile empirical studies on many real-world complex / social networks to analyze the impact of numerous setups and parameters like time window type and size, group detection method, evolution chain length, prediction models, etc. Additionally, many new predictive features reflecting the group state at a given time have been identified and tested. Some other research problems like enriching learning evolution chains with external data have been analyzed as well. Introduction {#introduction .unnumbered} ============ Network science is a very interdisciplinary domain focusing on understanding the relational nature of various real-world phenomena using for that purpose diverse network models. Commonly, networks consist of smaller, more integrated structures called groups, communities, or clusters. In practice, both the groups and whole networks evolve and change their profiles over time. Hence, their analysis demands advanced computational methods to understand and predict their future behavior. For that reason, group evolution prediction is an essential component of computational network science. One of the domains explored by network science are biological networks[@zickenrott2017prediction; @barabasi2011network; @wu2008network; @goh2007human]. Viruses are as old as life on earth. At the same time, they are very young, as they constantly mutate to change their lethal attributes. Influenza, unlike other viruses which are rather stable, evolves much more rapidly[@Influenza_rate1:2017; @Influenza_rate2:1986] and kills up to one million people worldwide every year[@Influenza_kills:2009]. We can try to protect ourselves using vaccines. However, the rate of mutation is too rapid to provide an effective cure. What is more, the development of a new drug requires a huge amount of money and lasts from a few to a dozen or so years. Despite these difficulties, new drugs are introduced to the market every year. For example, antagonist drugs (also called blockers) are designed to bind to specific receptors to block the disease’s ability to attach to these particular receptors, thereby immunizing the body to the disease. Unfortunately, diseases react to drugs and eventually mutate, creating a variety that will bind to other receptors. Therefore, we need methods that will be able to track the evolution of the disease, and based on the history of its mutations, will be able to predict the most likely future mutations. To track diseases mutations, we can focus on the group of receptors that it binds to, and observe how such group evolves. Based on the history of changes in the lifetime of this group, we can try to predict what will be the next change. Predicting the direction of the mutation could significantly reduce the amount of time and money needed to study the disease. With such knowledge, we would be able to start preparing the drug in advance and bring it to the market much faster and cheaper. Another area that widely applies network science, especially its branch called social network analysis (SNA), is marketing, in particular advertising[@husnain2017impact; @antoniadis2016social; @guo2016effects; @barhemmati2015effects]. Let us imagine that a start-up company invented a new generation of diapers – *Smart Diapers*, which are extra soft, super absorbing, and additionally, can communicate with parents’ smartphones to notify when their change time comes. The company invested very much in their development, therefore, it has a limited budget to advertise the product. The owners decided to introduce the product to discussion groups on the Facebook platform where parents from different countries/cities create and join independent groups to talk about and comment on new products for babies, share general advice about raising children, sell used clothes, etc. Convincing members (parents) of such relevant, targeted groups to use and buy the new diaper product would be much more effective and cheaper than advertising the broader community using expensive TV commercials. Additionally, the word-of-mouth recommendation is commonly believed to be the most powerful marketing tool[@Kozinets:2010]. However, the vital question rises here: which Facebook groups the company should invest in its limited resources, i.e., time and money? In the newly created relatively small groups that might be very active and are expanding fast, or in the larger groups that might be not very active in the nearest future? Which of these groups will be still running or growing in a few weeks/months/years and which one will disappear? That is why the knowledge about the history, current state, and future evolution of groups is crucial at decision making on where to allocate the resources. In 2007, Palla et al. [@palla2007quantifying] have defined the problem of group evolution identification. In the following years, dozens of solutions to this problem have been proposed. One of them was the highly cited GED method [@brodka2011tracking]. Existing surveys describe as many as 12 [@saganowski2017community] or even over 60 methods [@rossetti2018community]. All of them are focused on defining possible events in the community life, hence, tracking the historical changes. This, in turn, has led to emerging a new problem – predicting future changes that will occur in the community lifetime. Some of the first methods concerning prediction of some aspects (e.g., determining lifespan) of the group evolution were: (1) Goldberg et al. [@goldberg2011tracking] – they focused on predicting the lifespan of evolution for a group; (2) Qin et al. [@qin2011evolution] – analyzed dynamic patterns to predict the future behavior of dynamic networks; and (3) Kairam et al. [@kairam2012life] – they investigated the possibility of prediction whether a community will grow and survive in the long term. Note that the methods for tracking group evolution can be also utilized to other similar prediction problems, like link prediction[@group_evolution_link_prediction:2017], churn prediction[@group_evolution_churn_prediction:2010], as well as to understand evolution of software (Unix operating system networks) [@evolution_linux:2017] or dynamics of social groups forming at coffee breaks[@evolution_coffee_breakes:2014]. In 2012, we proposed a new concept, in which the historical group changes were utilized to classify the next event in the group’s lifetime[@Brodka:2012]. In this first trial, we have used only event type and size of the group to describe its state at a given time. Over the next year, we have investigated the concept and adopted it to two methods for tracking group evolution – the GED[@brodka2013ged] method and the SGCI method[@Gliwa:2012]. This resulted in the first method for group evolution prediction[@Gliwa:2013]. It was the predecessor of the GEP (Group Evolution Prediction) method described in this paper. Since then, a few more methods have been proposed. At the end of 2013, İlhan et al. presented their research with several new measures describing the state of the community and a new method for tracking group evolution[@Ilhan:2013]. In 2014, Takafolli et al. applied the binary approach to classifying the next change that group will undergo[@Takaffoli:2014]. They used 33 measures to describe the state of the community. We have presented new results in 2015, where, apart from new measures, the influence of the length of the history used in the classification was examined[@Saganowski:2015]. Later the same year, Diakidis et al. adapted the GED method to conduct their research with 10 measures as predictive features[@Diakidis:2015]. In 2016, İlhan et al. presented new results and proposed a method to select measures, which should be the most useful as predictive features for a given data set[@Ilhan:2016]. More recently, Pavlopoulou et al. used 19 measures already validated in other works and studied whether employing the temporal features on top of the structural ones improves prediction, as well as what is the impact of using a different number of historical community states on the prediction quality[@Pavlopoulou:2017predicting]. Unfortunately, all of the methods proposed to this day have some drawbacks (see the Comparison with other methods section) and have been designed to solve a particular problem, hence, their application area is rather narrow. Therefore, in this paper, a new generic and comprehensive method to predict the future behavior of the groups, based on their historical structural changes as well as experienced events, is proposed, evaluated and discussed. Some of the contributions of this work are: decomposing the group evolution prediction problem, proposing and extensively evaluating the modular method that can be applied to any dynamic network data, proposing new predictive features, performing the features’ ranking, proposing a new concept of data set enriching, initial evaluation of the transfer learning technique, an example and discussion on the concept drift problem in group evolution prediction, reviewing all proposed methods in the field. Methods {#methods .unnumbered} ======= Decomposition of the group evolution prediction problem {#decomposition-of-the-group-evolution-prediction-problem .unnumbered} -------------------------------------------------------- The crucial matter in developing the modular method predicting group evolution, called *GEP*, was the identification and separation of the components of the entire group evolution prediction problem. The appropriate problem decomposition and information flow between particular components (dependencies) are depicted in Eq \[eq:decomposition\] and Fig. \[fig:GEP\_method\]. $$\label{eq:decomposition} IS\xrightarrow[S_1]{TWT}TW\xrightarrow[S_2]{NT}TSN\xrightarrow[S_3]{CDM}G\xrightarrow[S_4]{CETM}EC\xrightarrow[S_5]{FE}PF\xrightarrow[S_6]{classification(CH)}Q$$ The data from the input stream $IS$ is divided into time windows $TW$ using the time window type definition $TWT$. For each time window $TW$, a complex/social network is created using the network type definition $NT$, resulting in the temporal complex/social network $TSN$. Within each time window $TW$ in $TSN$, some groups $G$ are identified using a community detection method $CDM$. Next, similar and consecutive groups are matched using a community evolution tracking method $CETM$, as well as the transition is labeled with an event type out of the set of possible changes $CH$. The matched groups are combined into evolution chains $EC$ that may consist of many successive changes. For each community state in $EC$, the feature extraction process $FE$ is applied in order to obtain a set of predictive features $PF$ describing the community state at a given time. Using features $PF$ in the form of a vector representing each evolution chain $EC$, classification of possible changes $CH$ is performed. The classification task (stage $S_6$) is to learn and finally label the next change(s) in community lifetime. The output of the classification process is a set of classification quality (performance) measures $Q$, for example, F-measure, accuracy, precision, or recall. The identified components were converted into six stages $S_1$-$S_6$ of the GEP method, Fig. \[fig:GEP\_method\]. GEP method {#gep-method .unnumbered} ---------- The GEP framework consists of six main stages (Fig. \[fig:GEP\_method\]): (1) time window definition, (2) complex network extraction for the defined periods, (3) community detection in periods, (4) group evolution tracking, (5) evolution chain identification for communities together with feature extraction and computation for each chain and (6) classification, containing classification model learning and testing. Each of them can be implemented by means of different methods and approaches depending on research need and prerequisites, e.g., complexity level. The formal definition of the GEP method is as follows: The GEP method is defined as an octuple $<IS, S_1, S_2, S_3, S_4, S_5, S_6, Q>$, where: $IS$ is an input stream of activities, e.g., phone calls, linking two actors (network nodes) $x, y$ at time $t_i$; $S_1$ is a set of considered time windows of the given type $TWT$; $S_2$ is a set of considered approaches to temporal complex / social network $TSN$ creation from $IS$ using time window definitions from $S_1$; $S_3$ is a set of considered approaches to community detection methods $CDM$ for each time window in $TSN$ from $S_2$; $S_4$ is a set of considered approaches to tracking community evolution methods $CETM$ for communities from $S_3$; $S_5$ is a set of considered approaches to feature extraction for evolution chains from $S_4$; $S_6$ is a set of considered approaches to classification, including learning, training, validating, undersampling, oversampling, and feature selection techniques; $Q$ is a set of considered classification quality measures, for example, F-measure, accuracy, precision, recall, estimated based on the classification results from $S_6$. The methods enumerated especially in $S_1$, $S_3$, $S_4$, $S_6$ also include the space / set of their parameters. The output of one stage $S_i$ is the input for the next stage $S_{i+1}$, e.g., communities detected in $S_3$ are used to discover their evolution in $S_4$. All these stages, together with parameters of the methods used, are more in-depth described in . They also require an appropriate definition of data structures to facilitate hassle-free implementation. ![**The concept of the GEP method.** **Stage 1:** Data set is divided into time windows. **Stage 2:** A complex network for each time window is created. **Stage 3:** Groups are extracted within each time window using any community detection method. **Stage 4:** The evolution of communities is tracked with any group evolution tracking method, and the evolution chains are created. **Stage 5:** Features describing the previous group profile such as size, density, cohesion, etc. are calculated to capture community state at a given time. **Stage 6:** Supervised machine learning approach is applied to learn and predict the forthcoming event in the group’s lifetime.[]{data-label="fig:GEP_method"}](GEP){width="\linewidth"} CPM method {#cpm-method .unnumbered} ---------- The Clique Percolation Method (CPM) proposed by Palla et al.[@Palla:2005CPM] is the most widely used algorithm for extracting overlapping communities. The CPM method works locally, and its primary idea assumes that the internal edges of a group have a tendency to form cliques as a result of high density between them. Oppositely, the edges connecting different communities are unlikely to form cliques. A complete graph with $k$ members is called k-clique. Two k-cliques are treated as adjoining if a number of shared members is $k$–1. Lastly, a k-clique community is the graph achieved by the union of all adjoining k-cliques. Such an assumption is made to represent the fact that it is a crucial feature of a group that its nodes can be attained through densely joint subsets of nodes. Infomap method {#infomap-method .unnumbered} -------------- The Infomap method proposed by Rosvall and Bergstrom[@Rosvall:2008infomap] uses the information-theoretic approach to cluster nodes within a network. It focuses on information diffusion across the graph and compression of the information flow description obtained from a random walker, which is chosen as a mean of information diffusion. Infomap changes the problem of finding the best cluster structure into finding the partition with the minimum description length of an infinite random walk. It follows the intuitive idea that if the community structure is present, the random walker will spend more time inside the community because of its higher edges density. It means that the transition to another cluster will be less likely. GED method {#ged-method .unnumbered} ---------- The Group Evolution Discovery (GED) method[@brodka2013ged] is one of the best methods for tracking community evolution[@he2017comparative]. It uses inclusion measure to match similar communities from neighboring time windows. This measure takes into account both the quantity and quality of the group members. The quantity is reflected by the first part of the inclusion measure, i.e., what portion of the members from group $G_1$ also belongs to group $G_2$. The quality is expressed by the second part of the inclusion measure, namely, what contribution of important members from group $G_1$ is in $G_2$. It provides a balance between the groups that contain many of the less important members and groups with only few but key members. The inclusion measure and the group size determine the type of community change. The authors defined seven possible event types: forming, dissolving, continuing, growing, shrinking, merging, and splitting. The method can work with any community detection method and with any group similarity measure, thus, providing great flexibility. İlhan et al. method {#ilhan-et-al.-method .unnumbered} ------------------- The İlhan et al. method[@Ilhan:2016] works with the disjoint type of communities and utilizes the function by Hopcroft et al.[@Hopcroft:2004] to calculate the similarity between two communities. The event types that can occur in the community lifetime and also the classes being classified are: survive, growth, shrink, merge, split, and dissolve. The measures used as predictive features are divided into two categories: structural and temporal community measures. In total, nine features per timeframe are used, i.e., number of nodes and edges, intra and inter measure of community edges, betweenness, degree, conductance, aging, and activeness. If one calculates four network measures beforehand (average path length, betweenness, clustering coefficient, embeddedness), the method can also identify features that should be the most prominent for a given network profile. Results {#results .unnumbered} ======= Suitable decomposing the problem of group evolution prediction (see the Methods section and Fig. \[fig:GEP\_method\]) was crucial in solving the problem. It allowed to analyze distinct phases of the process and to propose multiple solutions for each phase. The GEP method was extensively analyzed on fifteen real-world data sets (see for their profiles), for which more than 1,000 different temporal networks were created, and in total, more than 5,000,000 individual classification tasks were performed. However, to keep the article clear and concise, only selected results are presented for each stage. Stage 1: Time windows creation {#stage-1-time-windows-creation .unnumbered} ------------------------------ At first, the data is divided into time windows. Three main approaches can be considered in this context: (1) equal length periods – the events and relations are segmented based on their timestamp; (2) the same number of relations in each time window; (3) the arbitrary division, based on the data context. Additionally, the type and size of time windows have to be decided, which may be a challenging task. There are three most common types of time windows: disjoint, overlapping, and increasing. A proper choice of the time window type and size has a direct impact on the following GEP stages, especially on the number of evolution chains discovered by the tracking method (Stage 4). If relations between individuals in a data set have a tendency to change rapidly, then disjoint time windows would be a poor choice since there may not be too many relations lasting between two consecutive time windows. As a result, the tracking method will not provide any events (Stage 4), so there will be no input to a classifier resulting in no event to predict (Stage 6). The too large size of the time window, in turn, might lose some information about community changes that occurred in the meantime. So far, there is no formula which determines the right type and size of the time window, but a few guidelines can be provided based on our extensive experiments: - If the network is sparse or changes rapidly, the overlapping time window should be used. Usually, the offset equal to 30% of the time window size is enough to obtain a reasonable number of events between the consecutive time windows; - The time window type and size should be adjusted to the context of the given data set, e.g., the co-authorship network, referring to researchers who often publish only once a year, should evolve smoothly with the 1-year disjoint time windows; - If the persistent groups are the goal of analyses, the increasing time window should be utilized, as it provides mostly the continuing and growing events; - If relations between individual nodes are recurrent and the network is rather dense, one may try using disjoint time windows to lower the computational cost; - It is acceptable and even preferable to repeat the selection of the time window type and size several times to see which approach yields the best results. The most common choice in our studies was the overlapping time windows with the offset between 30% - 50% of their size. Stage 2: Formation of networks {#stage-2-formation-of-networks .unnumbered} ------------------------------ The parameters that can be adjusted at the creation of networks for each time window is the set of edge attributes, in particular, their weights and direction. The weighted/unweighted, as well as directed/undirected profile of the network, did not yield a significant impact on computational complexity nor classification accuracy. Some community detection methods, however, may be incompatible with the networks of particular characteristics or may ignore some attributes, e.g., weights. The CPM[@Palla:2005CPM] and Infomap[@Rosvall:2008infomap] methods, used in the experimental studies, are capable of handling the most important network attributes. Stage 3: Community detection {#stage-3-community-detection .unnumbered} ---------------------------- Some community detection methods can produce both disjoint and overlapping communities, but there are only a few methods for tracking the evolution (Stage 4) that can deal with the overlapping groups. Overall, the methods extracting disjoint communities perform faster than the ones providing overlapping groups. In some extreme cases, when the network is very large, the CPM method is unable to extract groups due to its enormous memory requirements. It is hard to compare two types of the grouping methods in terms of their impact on the classification accuracy, as each type of clustering delivers a different set of communities resulting in a different distribution of evolution events. Besides, the profile of the groups may be diverse, e.g., networks grouped with the CPM method tend to have a single giant component with many small overlapping groups alongside. This method also inclines to leave out nodes that do not belong to any clique, thus, excluding them from further consideration. If the network is sparse, a major fraction of the network may be omitted. In the most extreme case, the CPM method neglected even as many as 97% of network nodes, what resulted in a deficient number of communities and evolutions (Fig. \[fig:Chain CPMvsInfomap GEPvsIlhan\]A), and eventually in very low classification accuracy, Fig. \[fig:Chain CPMvsInfomap GEPvsIlhan\]B. At the same time, the Infomap method performed very well, identifying a large number of communities. Furthermore, the overlapping groups are likely to generate more merging and splitting events in Stage 4, since there are plenty of similar and overlapping communities in the consecutive time windows. On the other hand, the Infomap method tends to produce many communities having only 2 or 3 nodes. In general, while considering which type of grouping method to use the data context should be a crucial factor. [-2.25in]{}[0in]{} ![**(A) CPM vs. Infomap.** The number of events tracked with the GED method for groups obtained with two different community detection methods applied to the Digg data set. The CPM method leaves out even 97% of nodes that do not belong to any clique, hence the small number of groups and events. **(B) CPM vs. Infomap.** The F-measure values achieved for the events presented in Fig. \[fig:Chain CPMvsInfomap GEPvsIlhan\]A. The results reflect the distribution of events. **(C) Chain length.** The F-measure values for different lengths of the evolution chains for the Facebook data set. For most of the events, the F-measure value was increasing with the increase of the chain length up to 6 or even 7 states (the continuing and growing events). Beyond that point, the number of evolution chains of the particular types dropped below 50 which was insufficient to train the classifier properly; **(D) GEP vs. Ilhan et al.** The F-measure values for the 9-state evolution chains obtained from the Slashdot data set with the different set of predictive features: only from the GEP method (GEP) - see , from the İlhan et al. method, combined from both GEP and İlhan et al. methods (All features), and from the GEP method, but only for the last 3 states out of all 9 states (GEP\*). The GEP\* and “All features” scenarios achieved slightly better overall scores.[]{data-label="fig:Chain CPMvsInfomap GEPvsIlhan"}](mix "fig:"){width="\linewidth"} Stage 4: Stepwise evolution tracking and chain identification {#stage-4-stepwise-evolution-tracking-and-chain-identification .unnumbered} ------------------------------------------------------------- Regardless of the method, tracking the evolution of community is a computationally demanding task. The method has to iterate over all time windows and compare all the communities in order to detect similar ones. Although the methods for tracking group evolution can be very distinct, especially while defining the possible event types, our earlier study showed that the selection of the method has no significant impact on classification accuracy[@Saganowski:2015]. In this evaluation, we use the GED method [@brodka2013ged] since, in the last evaluation of existing community evolution tracking method, it was selected as the one giving the most satisfying results[@he2017comparative]. The parameters of the selected method might influence the classification results, e.g., the alpha and beta parameters of the GED method have a direct impact on the number of evolution events discovered – the lower the threshold, the more events obtained (see for details). In the experimental studies, the most common value for the alpha and beta parameters was 50%. If the network is dense and relations are recurrent, the alpha and beta might be even increased to 70%. On the other hand, when the method provides a small number of the evolution events, the alpha and beta should be reduced to, e.g., 30%. Apart from the selection of the evolution tracking method, the length of the evolution chain has to be decided. The longer the evolution chain, the more predictive features for the classifier in Stage 6, hence, the higher computational complexity. Nevertheless, the results presented in Fig. \[fig:Chain CPMvsInfomap GEPvsIlhan\]C revealed that it is worth dedicating some more time and resources to extract longer chains since it can boost classification accuracy. The overall score achieved with the evolution chains containing six community states was 32% higher than the results achieved with shorter 2-state chains. In case of limited time or resources, the chains with the length of 2-3 states should be reasonably good. Stage 5: Feature extraction {#stage-5-feature-extraction .unnumbered} --------------------------- In order to predict the future evolution of the group, we need to describe its recent and historical states by means of predictive features. Based on these features and previous evolutionary changes used to learn the model, we are able to forecast the next changes. The crucial features that are at our disposal are structural network measures computed for the previous group states. Calculation of all measures may be a very demanding task since they need to be evaluated for every community state in the evolution chain. Additionally, some measures, e.g., betweenness centrality, require finding all shortest paths for each pair of nodes in the community or network. The experiments revealed, Fig. \[fig:Chain CPMvsInfomap GEPvsIlhan\]D, Fig. \[fig:Features selection\] that the set of predictive features has a significant impact on classification accuracy, as they are used to build the classification model, see also , Feature Selection section. Therefore, it is highly recommended to compute as many predictive features as possible to deliver to the classifier a wide variety of descriptions to choose from. To significantly enhance the already existing approaches, many new predictive features are proposed in this paper (see , Predictive Features section). We have clustered structural features into three general types: (1) *microscopic* – calculated for individual nodes, e.g., node degree, (2) *mesoscopic* – quantifying single groups, e.g., group size - no. of nodes, and (3) *macroscopic* – describing the whole network, e.g., network density. Mesoscopic features also include normalized group measures like the group size divided by the network size. Besides, node-based (microscopic) measures can be aggregated (usually averaged) at either *local* (group) or *global* (network) level resulting in *microscopic local* or *microscopic global* features, respectively. All computed features were thoroughly evaluated in terms of usefulness for the classifier and rankings of the most prominent features were built, see , Feature Selection section, especially Tab. 5-9. For the evolution chains of a variable length, different rankings were obtained. For the shortest 1-state evolution chains, only macroscopic (network) features were helpful, which may result from the fact that communities with a short history are considered unstable and vulnerable to the environment they are a part of. For the evolution chains with the increasing time windows, the features describing the local structure, especially the centrality- and distance-based measures, were more informative for the classifier, as the changes between the consecutive increasing time windows were delicate and occurred at the microscopic rather than macroscopic level. The neighborhood-based features were among the most valuable features for the longest 8- and 9-state chains, which lead to believe that for the long-lasting communities, the relations with their surroundings are a better predictor of the forthcoming change than, e.g., the macroscopic features. In general, the variations of the eigenvector-, eccentricity-, and closeness-based features were present in most of the selective rankings, which suggests that centrality- and distance-based measures obtained on the node level are the most prominent ones. Hence, in case of limited computational capacity, these features should be respected before any other. However, out of all features considered by the classifiers, the Backward Feature Elimination selected only up to 34% of them as prominent, i.e., used by the classifier to make a decision, Fig. \[fig:Features selection\]A. Additionally, it turned out that usually over 90% of the selected prominent features were obtained from the last three community states, Fig. \[fig:Features selection\]A1. For example, when the evolution chain length was 8, and the next change was classified, all the prominent features were from the 8th, 7th, and 6th group profiles. It means that the most recent history of the community has the most significant impact on its next change. This is an extremely useful conclusion if one has limited computational capabilities and cannot calculate community profiles for all states or does not possess data about older history. The number of features has a direct impact on the duration of the entire learning process, Fig. \[fig:Features selection\]C. [-2.25in]{}[0in]{} ![**(A) Feature selection.** Important features selection obtained by the Backward Feature Elimination for the DBLP data set. The total number of features increases with every state by 91, e.g., the 3-state evolution chain has 91\*3=273 features in total, out of which 34% were selected as prominent. **(A1)** Features selected only from those related to the last 3 time windows. **(B) Feature ranking.** The most frequently selected features for the 1-state evolution chains. All kinds of information are important to achieve a satisfactory prediction; microscopic features are focused on nodes, mesoscopic on groups, and macroscopic on entire network parameters. The ranking obtained by analyzing eight data sets and repeating feature selection 1000 times. **(C) Computational efficiency.** The time required to train a single Random Forest classifier in relation to the number of descriptive features used as the input data. The results obtained for the IrvineMessages data set.[]{data-label="fig:Features selection"}](features "fig:"){width="\linewidth"} Stage 6: Prediction {#stage-6-prediction .unnumbered} ------------------- In the last stage, the machine learning techniques, such as oversampling, undersampling, feature selection, and first of all, model training and adjustment are applied to achieve the highest possible prediction quality. The common problem with the training data is an imbalanced distribution of output classes, Fig. \[fig:Chain CPMvsInfomap GEPvsIlhan\]A. In extreme cases, when one class greatly dominates over the other ones, a trained model tends to assign the dominant class to most observations. Then, the solution is to apply additional preprocessing techniques like oversampling and undersampling to generate additional observations or to filter out predominant ones, thus providing a distribution closer to flat. Another common problem is overfitting the classifier by providing too many features or observations. In order to prevent from such case, feature elimination technique may be applied, which unfortunately is very expensive in terms of computational complexity. Additionally, the proper classifier should be selected, and its parameters need to be accordingly adjusted. In the experimental study, fifteen different classifiers were compared in terms of the classification accuracy, Fig. \[fig:Ranking of classifiers\]. The tree-based classifiers and meta-classifiers (equipped with decision trees) performed best. Many classifiers could not efficiently handle imbalanced data, so the undersampling and oversampling techniques were applied, resulting in notably better prediction quality, Fig. \[fig:Ranking of classifiers\]B. On the balanced data set, a classifier focuses on the predictive features computed for the community states instead of focusing on the event distribution. The Friedman statistical test [@Friedman:1937] with the Shaffer post-hoc multiple comparisons [@Shaffer:1986] was performed to obtain rankings of classifiers on the imbalanced and balanced data sets (cf. , Tab. 10). In both cases, the Bagging classifier (with the REPTree classifier) was the winner, and the Random Forest classifier was ranked second. What is essential, the p-values confirmed that the results were statistically significant. Furthermore, classifiers often have their parameters to tune them accordingly, which can substantially affect the classification accuracy, cf. for detailed discussion. For example, the logarithmic correlations were observed between the number of bagging iterations for the Bagging classifier and the average F-measure value, as well as between F-measure and the number of generated trees by the Random Forest classifier. The results prove that the process of adjusting the classifier parameters should always be performed, as long as the computational time and resources are available. [-2.25in]{}[0in]{} ![**The rankings of classifiers.** The heat-maps of the F-measure results for the 1-state evolution chains obtained from the Twitter data set. Classifiers are ordered by the overall score. The Bagging classifier and the SimpleCart classifier achieved the highest overall scores but failed to predict the growing and the merging events. Therefore, the tree-based classifiers are the best choice as all the events are successfully classified and the overall score is insignificantly lower.[]{data-label="fig:Ranking of classifiers"}](classifiers_ranking "fig:"){width="\linewidth"} Comparison with other methods {#comparison-with-other-methods .unnumbered} ----------------------------- The GEP method was compared to other approaches. The existing methods for group evolution prediction were additionally analyzed, and many of their drawbacks have been identified. The most severe were: a narrow application area, methodological issues (e.g., inappropriate computation of the conditional probability), insufficient validation of the methods (e.g., a single sampling into two folds instead of the 10-fold cross-validation), superficial descriptions of the methods and conducted experiments (often insufficient to repeat and validate the experiments), and lack or unreliable comparisons with other methods. Despite GEP is so flexible and has so many options, it is competitive with other approaches, designed to deal with a specific problem or data set. For example, a special version of the GEP method, in which only features from the last three states (out of all 8 or 9 states) were used as an input for the classifier, performed noticeably better than the method by İlhan et al.[@Ilhan:2016], Fig. \[fig:Chain CPMvsInfomap GEPvsIlhan\]D. After all, it needs to be emphasized that none of the existing methods is as adjustable and versatile as the GEP method. Discussion {#discussion .unnumbered} ========== Across its six stages, the GEP method utilizes various approaches, methods, and techniques, which can be adjusted with respect to a given data set and a particular study purpose. These approaches, methods, and techniques are considered as the GEP method parameters. To provide a concise summary of their impact on overall computational complexity, and first of all on the final classification accuracy, the crucial parameters were listed in Tab. \[tab:parametersInfluence\] and discussed throughout the article. [-2.25in]{}[0in]{} Parameter group Parameter Parameter value Impact on computational complexity Impact on classification accuracy ----------------- ----------------------------- ------------------------------------------------ ------------------------------------ ----------------------------------- window division timestamp / relations count / arbitrary none low window size time unit or number of relations medium low window type disjoint / overlapping / increasing medium medium network type edge attributes directed / undirected, weighted / unweighted low low group type disjoint / overlapping medium low grouping method a method high medium tracking method a method medium low GED alpha and beta (10%, 100%\] none low GED social position measure a measure medium low classifier used a classifier medium medium machine learning techniques undersampling, oversampling, feature selection high high evolution chain length number of community states medium medium predictive features a set of features high high : [**The GEP method parameters and their impact on computational complexity and classification accuracy.**]{} \[tab:parametersInfluence\] Many different classifiers were evaluated on various data sets. The tree-based classifiers and meta-classifiers (equipped with decision trees) performed best. Many classifiers could not handle imbalanced data sets, so the undersampling and oversampling techniques were applied. Balancing data sets notably improved the results confirming the usefulness of the undersampling and oversampling methods. The experimental studies showed that adjusting the classifier parameters can significantly improve classification accuracy. The logarithmic correlations were observed between the number of bagging iterations in Bagging classifier and the average F-measure value, as well as between the number of generated trees by the RandomForest classifier and the average F-measure value. The confidence factor parameter of the J48 classifier was found also correlated with the average F-measure value. The maximum improvement in average F-measure value achieved by adjusting the classifier parameter was 17%, and it was obtained by increasing the number of generated trees by the RandomForest classifier. The results prove that the process of adjusting the classifier parameters should always be performed, as long as the computational time and resources are not limited. The GEP method enables us to consider different new scenarios, which are hardly available without this generative framework like transfer learning, class balancing by adding external data, or decreasing the concept drift effect. The transfer learning technique was adapted to the problem of group evolution prediction for the first time in this field. Its main idea is to learn the classification model on one data set and test it on another one. Such an attempt was quite successful, and the preliminary results were satisfactory. The key to success is finding a data set with a likewise profile. Moreover, in some cases, learning the transferred model on the balanced data set can boost the classification quality for the data set to which the model is adapted. The initial experiments also suggest that the underlying similarity of two data sets (e.g., the same habits of actors or ideally the same set of actors) can help to create a model that if transferred can outperform the primary model built for a given data set. Very promising results, although at an early stage, were achieved at enriching the learning phase of the classification model with additional evolution chains from a different data set. By partially balancing the original training set with extra evolution chains from another external data set, it was possible to improve the model and thus produce better results for minority classes, without affecting the outcome for the dominating classes, Fig. \[fig:applications\]A. This phenomenon is especially important because the existing techniques of balancing a data set always affect the classification results of the dominating classes. [-2.25in]{}[0in]{} ![**Application of the GEP method.** **(A) Enriching the classification model** by partially balancing the original training set (Twitter) with extra evolution chains taken from another full external data set (MIT) or with chains from only selected event types, i.e. growing, merging and splitting (MIT\*); chains with these classes were the worst classified events for the original Twitter data – they had the lowest F-measure values. The results for these selectively enriched event types were significantly improved without worsening classification for other classes (green vs. blue bars). Data enriching was performed only for learning, not for testing. **(B) Concept drift.** Classification quality for the Facebook data from one longer period $T_1-T_{50}$ (the red bar); alternatively, the data was split into five smaller periods and separate classification models were built to catch concept drift phenomena between periods (blue bars). Independent models learned for smaller periods are better adapted to the changing environments.[]{data-label="fig:applications"}](enrich_model_concept_drift "fig:"){width="\linewidth"} Another way to enhance the classification model, initially considered, is an appropriate selection of the observation time span to reduce the effect of non-stationarity of data – a.k.a concept drift. Our preliminary research shows that for a network spanning over a long period or changing rapidly, updating the classification model every once in a while might improve the results, as the model reflects the current characteristics of the network in the better and more up to date way, Fig. \[fig:applications\]B. Nevertheless, in order to rebuild the model every now and then, the number of observations (evolution chains) extracted from such shorter time span must be high enough. The GEP framework can be applied to any dynamic network data, i.e., to any complex network changing over time. In this paper, we have explored popular social network data, see Table 2 in the Supporting information section. However, the entire GEP method, its stages and component solutions may be used for diverse complex networks [@Barzel2013; @BOCCALETTI2006] like evolving clusters of web pages [@Dezso2006], co-citation and bibliographic coupling networks extracted from citations between scientific papers [@kessler1963; @small1973], biological and medical networks [@Bode2007; @Barabasi2011], linguistic networks linking word meanings - WordNets [@Bartusiak2019], multimedia networks [@Indyk2013] and many more. Conclusion {#conclusion .unnumbered} ========== The main subject studied in this paper is group evolution prediction in social/complex networks. Its primary goal is to foresee a change like shrinking, growing, splitting, merging, or dissolving that the recently existing community will experience in the nearest future. To be able to perform any prediction, the most common approach is to process a temporal complex network $TSN$ extracted from the stream of user activity traces. Communities and their changes are identified and predicted within such $TSN$. However, the existing methods are often limited to operate on a particular data set or to solve a specific problem, which makes them useful only in a particular and narrow domain. Therefore, a new generic method called Group Evolution Prediction (GEP) has been proposed in this paper. The GEP method has a modular structure, which makes it very flexible and allows us to successfully apply it to any data set and under any specific requirements. The method consists of several stages; each of them involves a suitable selection of methods, algorithms, and attributes – the GEP method parameters. The evaluation process of the GEP method included: (1) analysis of numerous parameters (time window type and size, community detection method, evolution chain length, classifier used, set of features, and more), (2) comparative analysis against other existing methods, (3) adaptation of the transfer learning concept to group evolution prediction, (4) enriching the classification model with evolution chains from a different data set, and (5) enhancing the classification model with a more appropriate training set. Regarding the time window types and sizes, the main finding is that for rapidly changing or sparse social networks a shorter overlapping time windows (in relation to the context of the data) are a better choice than longer or disjoint periods. On the contrary, if relations between individuals are recurrent and the network is rather dense, one may try disjoint time windows to obtain more concise results and to lower the computational cost. If long-lasting, persistent communities are the goal, then the increasing type of time window is the best choice as it generates a high number of the continuing, growing, and shrinking events. Two most commonly used community detection approaches were analyzed: the CPM method detecting the overlapping communities, and the Infomap method identifying the disjoint communities. It turned out that the CPM method was not a proper choice for sparse networks, as it left out nodes that did not belong to any clique. However, if a network is not so sparse, then generating overlapping communities may be a better choice, especially if the context of the data suggests overlapping communities. For example, when the nodes tend to belong to more than one community at a given time. The Infomap method, however, performs better if computational complexity is an essential factor, and computational time is limited. The results yield that evolution chains with more community states (longer chains) provide better classification results. However, there seems to be a threshold of the number of states, which make the evolution chains too short, resulting in a lack of possibility of improving the accuracy level. Even over 70% of the most prominent features were obtained from the last three community states. It means that the most recent history of the community has the highest impact on its next change. This is an extremely useful conclusion if one has limited computational capabilities and cannot calculate community profiles for all states. Additionally, many new predictive features are proposed in this paper. In particular, some aggregations of node measures were used to compute the local and global microscopic features. Network structural measures were adopted as macroscopic features, and ratios of community measures to network measures were utilized as mesoscopic features. In general, the variations of the eigenvector-, eccentricity-, and closeness-based features were present in most of the selective rankings, which suggests that centrality- and distance-based measures obtained on the node level are the most valuable features. The GEP method flexibility enabled us to investigate some other interesting scenarios, i.e., (1) adapting the transfer learning technique to the group evolution prediction problem, (2) enriching the classification model with evolution chains from a different data set, (3) appropriate selection of the observation time span to reduce the concept drift effect. All of them appeared to be quite successful. Even though the GEP method is a flexible, generic framework, it is competitive with other approaches often dedicated to a specific problem or data set. Supporting information {#supporting-information .unnumbered} ====================== #### S1 File. {#SI_File .unnumbered} [**Supporting information file.**]{} Contains additional results and discussion. Authors Contributions {#authors-contributions .unnumbered} ===================== [**Conceptualization:**]{} SS, PB, PK. SS. SS, MK. PK. SS, MK. SS, PB, MK, PK. SS. SS, MK. PB, PK. SS, PB, MK, PK. SS, PB, PK. Acknowledgements {#acknowledgements .unnumbered} ================ This work was partially supported by The Polish National Science Centre, the projects no. 2016/21/D/ST6/02408 and 2016/21/B/ST6/01463; by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 691152 (RENOIR); and by the Polish Ministry of Science and Higher Education fund for supporting internationally co-financed projects in 2016-2019, agreement no. 3628/H2020/2016/2. This research was supported in part by PLGrid Infrastructure. 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--- abstract: 'We give a closed form of the discrete-time evolution of a recombination transformation in population genetics. This decomposition allows to define a Markov chain in a natural way. We describe the geometric decay rate to the limit distribution, and the quasi-stationary behavior when conditioned to the event that the chain does not hit the limit distribution.' author: - 'Servet Mart[í]{}nez' title: '*A probabilistic analysis of a discrete-time evolution in recombination*' --- [**Keywords: $\,$**]{} Markov chain; Population genetics; Recombination; geometric decay rate; quasi-stationary distributions. [**AMS Subject Classification:**]{} 60J10; 92D10. Introduction {#sec0} ============== Here we study the evolution of the following transformation $\Xi$ acting on the set of probability measures $\mu$ on a product measurable space $\prod_{i\in I}A_i$, $$\Xi[\mu]=\sum_{J\subseteq I} \rho_J \, \mu_J \otimes \mu_{J^c}.$$ Here $\rho=(\rho_J: J\subseteq I)$ is a probability vector, $\mu_J$ and $\mu_{J^c}$ are the marginals of $\mu$ on $\prod_{i\in J}A_i$ and $\prod_{i\in J^c}A_i$ respectively, and $\otimes$ means that these marginals are combined in an independent way. The analysis of $\Xi$ should give an insight in the study of the genetic composition of population under recombination. Genetic information is encoded in terms of sequences of symbols indexed by a finite set of sites. In the process of recombination the children sequences are derived from two parents, a subset of sites is encoded with the maternal symbols and the complementary set is encoded with the paternal symbols. The above equation expresses that these sets $(J,J^c)$ constitute a probabilistic object distributed according to $\rho$. A relevant feature is that recombination produces decorrelation between sites and this is expressed by the fact that the sequence distribution on these sets are grouped independently into $\Xi[\mu]$. The evolution $(\Xi^n[\mu])$ has been mainly studied in the context of single cross-overs, that is where $I=\{1,..,K\}$ and the pairs of sets $(J,J^c)$ are of the form $J=\{i: i<j\}$, $J^c=\{i: i\ge j\}$. This evolution was introduced by H. Geiringer [@ge], and firstly solved in the continuous-time case by E. Baake and M. Baake [@bb], where it is also supplied an important corpus of ideas and techniques to study the discrete-time evolution. In relation to the discrete-time evolution we refer to [@bvw]: [*’...the corresponding discrete-time dynamics, which is prevalent in the biological literature, is more difficult: its solution has, so far, required nontrivial transformations and recursions that have not been solved in closed form (Benett 1954; Dawson 2000, 2002; von Wangenheim et al. 2010).’*]{} These last works are cited in our list of references as [@be]; [@daw1], [@daw2]; [@vwbb]. Richer discussions on the interpretation of the above equation in a broader perspective of recombination in population genetics, are given in the introductory sections of references [@bb], [@bvw], [@bbs] and [@uvw]. When studying single cross-over recombination, one the main objectives in [@uvw] and [@bvw] is to express the iterated $\Xi^n[\mu]$ in a simple form. The main tools in these works are Möbius inversion formulae, similarly to the continuous case, and commutation relations between $\Xi$ and recombination operators. Some of the main results of these works are the one step recursive relation stated in Theorem 1 in [@bvw], Proposition 3.3 in [@uvw] stating that if one starts from a distribution $\mu$ then $\Xi^n[\mu]$ converges to the Bernoulli distribution having the marginals of $\mu$, and the relation to ancestry trees and Markov chains, summarized in Theorem 3 in [@bvw]. In our work we present two main results, these are Theorems \[theo0\] and \[theo1\]. In Theorem \[theo0\] we write $\Xi^n[\mu]$ as a weighted decomposition of $\otimes_{\ell \in \delta} \mu_\ell$, where $\mu_\ell$ is the marginal $\mu$ on the set $\ell$, and $\{\ell \in \delta\}$ are the atoms of some partition $\delta$ of $I$, and we give exactly the weights of this decomposition. This follows from a simple backward development of $\Xi^n[\mu]$ done in Lemma \[lemmafd\]. When looking in detail the formulae stated in Theorem \[theo0\] one realizes that they define a natural Markov chain $(Y_n)$ on the set of partitions of $I$, having the remarkable property that when it starts from the coarsest partition $\{I\}$, then the probability that $\{Y_n=\delta\}$ is equal to the sum of weights of all trees participating in the backward development of $\Xi^n[\mu]$ whose set of leaves is $\{\ell \in \delta\}$. These results are Lemmas \[lemma4\] and \[lemma5\]. In Theorem \[theo1\] we use this Markov chain to describe the geometric coefficient of convergence to the limit distribution $\otimes_{\ell \in {{\cal D}}^\rho} \mu_\ell$, where ${{\cal D}}^\rho$ is the partition generated by the sets $\{J: \rho_J>0\}$. In the single cross-over case the atoms of this partition are the singletons, so the limit probability measure is the Bernoulli distribution. A key result is formula (\[50e\]) that characterizes the geometric decay behavior. In this Theorem we also study in detail the limiting conditional behavior of the chain when conditioned to the fact that it has not hit the limit distribution. Besides giving the limiting conditional distribution, we state a ratio limit of the probabilities of not hitting the limit distribution. We emphasize that these last results are not a consequence of any known result in the theory of quasi-stationary distributions because the Markov chain $(Y_n)$ is not irreducible on the class of non-absorbing states, so we are not able to use the Perron-Frobenius theory. All these results require entirely new computations. Quasi-stationary distributions have been studied mostly in relation to population extinction, see for instance Section 2.6 in [@les], and [@pp; @cms] for a wide ranging bibliography on the subject. In our context the absorbing state is not the void population as happens in extinction, and a main interest of the quasi-limiting behavior is in the process that never hit the limit distribution which is given in Corollary \[cor1\]. In Section \[sec1\] we fix notation on partitions, atoms, and dyadic partitions. In Section \[sec2\] we define supply some technical lemmas on the transformation $\Xi$. Thus, in Lemma \[lemma3\] we get the marginal $\Xi[\mu]_K$ for $K$ a union of atoms, in terms of some iterated coefficients $\rho^K_M$ derived from $\rho$ which constitute key quantities along all our study. In Section \[sec3\] we introduce the dyadic family of trees depending on the support of $\rho$ participating in the tree decomposition of $\Xi^n[\mu]$. Finally, in Section \[sec4\] we introduce the Markov chain on partitions and state our main results on the quasi-limiting behavior. Let us discuss briefly the relations of our results with respect to previous literature mainly with respect to [@uvw] and [@bvw], which have been an important inspiration for our work. In these references a Markov chain on partitions was introduced for single cross-over recombination, by following the ancestry of the genetic material of a selected individual from a population and using some limits arguments. As a consequence of this rather complicated construction, a key relation between the Markov chain and the coefficients of the iterated $\Xi^n[\mu]$ is stated in Theorem 3 in [@bvw], which must be the same relation we state in Lemma \[lemma5\]. We note, that each backward step in ancestry involves a probabilistic object because the dyadic partition $(J,J^c)$ is randomly distributed. But, our approach differs with the one used in [@bvw] at some substantial points: we get a closed form of $\Xi^n[\mu]$ by using a simple backward decomposition and this decomposition suggests the definition of the Markov chain $(Y_n)$ in a very natural way. Our techniques are totally different to those used in [@uvw] and [@bvw]. Also, our result apply to all kind of dyadic partitions $(J,J^c)$ that can have a complex combinatorics and not only for the ones arising in the single cross-over case. Finally, the study of the quasi-stationary behavior of this chain is, to our knowledge, firstly studied in this monograph. We point out that even if our results are stated for a product of finite spaces, they can be stated for general product of measurable spaces as pointed out in Remark \[rem2a\]. Recently, in [@bbs], the continuous-time evolution was studied in a framework of general partitions other than dyadic partitions. The extension of our results to the analogous framework but for discrete-time, deserves a different study. It is worth mentioning, that in Section \[sec2\] and in the final comment of this work, we point out that all our results remain true when $\otimes$ is a commutative and associative operation, it has an identity element and is also stable under restriction. It could be explored the existence of good candidates for operations $\otimes$ other than the product of probability measures, that would be meaningful in population genetics. Partitions {#sec1} ============ In this section we fix some notation on partitions. Some emphasis is put in defining a partition from a family of sets, with a special care in defining the atoms, and we make the difference between dyadic and strictly dyadic partitions, the last ones having exactly two atoms. Let $I$ be a finite set and ${\mathbb{S}}(I)=\{L: L\subseteq I\}$ be the class of its subsets ($\subseteq$ means inclusion and $\subset$ strict inclusion). For any class of sets ${{\cal Z}}\subseteq {\mathbb{S}}(I)$ we put ${{\cal Z}}^{(\emptyset)}={{\cal Z}}\setminus \{\emptyset\}$, ${{\cal Z}}^{(I)}={{\cal Z}}\setminus \{I\}$ and ${{\cal Z}}^{(\emptyset,I)}={{\cal Z}}\setminus \{\emptyset, I\}$. So, when ${{\cal Z}}$ does not contain the empty set we have ${{\cal Z}}^{(I)}={{\cal Z}}^{(\emptyset,I)}$. A partition ${{\cal D}}$ of $I$ is a collection of nonempty sets (so ${{\cal D}}\subseteq {\mathbb{S}}(I)^{(\emptyset)})$, pairwise disjoint and covering $I$. We note ${{\cal D}}=\{L: L\in {{\cal D}}\}$ and any of the sets $L$ is called an atom of ${{\cal D}}$. We note by ${\mathbb{D}}(I)$ the family of partitions of $I$. For ${{\cal D}}, {{\cal D}}'\in {\mathbb{D}}(I)$, ${{\cal D}}'$ is said to be finer than ${{\cal D}}$ or ${{\cal D}}$ is coarser than ${{\cal D}}'$, if every atom of ${{\cal D}}'$ is contained in an atom of ${{\cal D}}$. In this case every atom in ${{\cal D}}$ is union of atoms of ${{\cal D}}'$. The finer partition is the class of singletons ${{\cal D}}_{I,si}=\{\{i\}: i\in I\}$, and the coarsest one is $\{I\}$. Let ${{\cal J}}\subseteq {\mathbb{S}}(I)^{(\emptyset)}$ be a nonempty family of nonempty sets which satisfies, $J\in {{\cal J}}, J\neq I \, \Rightarrow \, J^c\in {{\cal J}}$. Then, it defines a partition ${{\cal D}}({{\cal J}})\in {\mathbb{D}}(I)$ as follows. Let ${{\cal Y}}_1({{\cal J}})={{\cal J}}$ and define by recursion the following family of classes of nonempty sets, $$\forall\, n\ge 1: \quad {{\cal Y}}_{n+1}({{\cal J}})=\{K\cap J: K\in {{\cal Y}}_n({{\cal J}}), J\in {{\cal J}}, K\cap J\neq \emptyset\}.$$ Since $J\cap J=J$, we have ${{\cal Y}}_n({{\cal J}})\subseteq {{\cal Y}}_{n+1}({{\cal J}})$ for all $n\ge 1$. Also it stabilizes in a finite number of steps, that is there exists $n_0\ge 1$ such that ${{\cal Y}}_{n_0+k}({{\cal J}})={{\cal Y}}_{n_0}({{\cal J}})$ for all $k\ge 0$. Let $$\label{e7} {{\cal Y}}({{\cal J}})=\bigcup_{n\ge 1} {{\cal Y}}_n({{\cal J}}).$$ We define the atoms of the partition ${{\cal D}}({{\cal J}})$ by: $$\label{e8} L\in {{\cal D}}({{\cal J}})\Leftrightarrow \big[ \, L\in {{\cal Y}}({{\cal J}}) \hbox{ and } \forall J\in {{\cal J}}: \, J\cap L =L \vee J\cap L =\emptyset \, \big].$$ It is clear that the atoms $L\in {{\cal D}}({{\cal J}})$ are disjoint, on the other hand they cover $I$, because in the contrary the set $I\setminus (\bigcup_{L\in {{\cal D}}_I({{\cal J}})} L)$ would have a nonempty intersection with some $J\in {{\cal J}}$ and so it would contain an atom of the form (\[e8\]) leading to a contradiction. Then, $I=\bigcup_{L\in {{\cal D}}_I({{\cal J}})} L$. It is also straightforward to show that $$\label{e3} \forall K\in {{\cal Y}}({{\cal J}}): \quad K=\!\!\!\bigcup_{L\in {{\cal D}}({{\cal J}}): L\subseteq K}\!\!\!L.$$ It is useful to introduce [*dyadic*]{} partitions. The set of dyadic partitions on $I$ is noted by ${\mathbb{D}}_{1,2}(I)$ and it is given by $${\mathbb{D}}_{1,2}(I)=\{I\}\cup \{\{J,J^c\}: J\in {\mathbb{S}}^{\emptyset,I}(I)\}.$$ The ${\;}_{1,2}$ subscript is because a partition $\delta\in {\mathbb{D}}_{1,2}(I)$ can have one or two atoms. It has one atom only in the case $\delta=\{I\}$, in all other cases it contains two atoms. We will make the distinction with respect to the family of [*strictly dyadic*]{} partitions, which is the class of partitions having exactly two atoms, $${\mathbb{D}}_{2}(I)=\{\{J,J^c\}: J\in {\mathbb{S}}^{\emptyset, I}(I)\}.$$ From now on, we fix $I$ and call it the set of sites. In the notation of the variables we will often delete the dependence on $I$, thus we write ${\mathbb{S}}={\mathbb{S}}(I)$, ${\mathbb{D}}={\mathbb{D}}(I)$, ${\mathbb{D}}_{1,2}={\mathbb{D}}_{1,2}(I)$, ${\mathbb{D}}_2={\mathbb{D}}_2(I)$ and so on. But we keep the dependence of these quantities on a set $J$ when it is not necessarily $I$, in this case we write ${\mathbb{S}}(J)$, ${\mathbb{D}}(J)$, ${\mathbb{D}}_{1,2}(J)$, ${\mathbb{D}}_2(J)$ and so on. The recombination transformation {#sec2} ================================ In this section we define the action of $\Xi$ on the set of probability of measures of a product measurable space. For simplicity we assume the product space is finite, in fact all our results remain true for a product of general measurable spaces. But, the finiteness of the set of sites $I$ is crucial. We will supply some elementary properties of $\Xi$, a main one being the description of the marginal of the transformed probability measure, this is done in Lemma \[lemma3\]. This description is written in terms of some coefficients whose main properties are summarized in Lemma \[lemma2\]. We devote some time to state exactly the properties of $\otimes$ that will used in this work. Let $A_i$ be a finite set for $i\in I$, called the alphabet on site $i$. Let $\prod_{i\in I}A_i$ be the product space. In order that our statements are for non-trivial, we will assume that the sets $I$ and $A_i$ for $i=1,..,n$, contain at least two elements. We note by $x$ an element of $\prod_{i\in I}A_i$, so $x=(x_i\in A_i: i\in I)$. Denote by ${\cal P}_I$ the set of probability measures on $\prod_{i\in I}A_i$. Any $\mu\in {\cal P}_I$ is determined by the values $(\mu(x): x\in \prod_{i\in I}A_i)$. Let $J\in {\mathbb{S}}$. We note $x_J=(x_i: i\in J)$ and make the identification $x=(x_J,x_{J^c})$. We denote by ${\cal P}_J$ the set of probability measures on $\prod_{i\in J}A_i$. The marginal $\mu_J\in {\cal P}_J$ of $\mu\in {\cal P}_I$ on $J$ is, $$\label{eab2} \forall x_J\in \prod_{i\in J}A_i: \quad \mu_J(x_J) :=\mu(\{y\in \prod_{i\in I}A_i: y_i=x_i\})= \sum_{x_{J^c}\in \prod_{i\in J^c}A_i}\!\!\!\mu(x).$$ For $J=I$ we have $\mu_I=\mu$, and for $J=\emptyset$ we have $\mu_\emptyset(x_\emptyset)=1$. We put $\mu_\emptyset\equiv 1$ to get consistency in all the relations where it will appear. If $K\subseteq J$ then the marginal $\mu_{K}$ can be defined from $\mu_J$, that is it satisfies $$\label{eab3} \mu_K(x_K)=\sum_{x_{J\setminus K}\in \prod_{i\in J\setminus K}A_i}\mu_J(x_J).$$ We take $\otimes$ to be the product measure: for all $\mu^J\in {\cal P}_J$, $\mu^K\in {\cal P}_K$, $$\forall x_{J\cup K}\in \prod_{i\in J\cup K} A_i: \quad \mu^J\otimes \mu^K(x_{J\cup K})=\mu^J(x_J)\mu^K(x_K).$$ Let us explicit the properties we will use from $\otimes$. First, the operation $\otimes$ is defined in the domains $$\label{eab0} \forall J,K\in {\mathbb{S}}, J\cap K=\emptyset; \quad \otimes: {\cal P}_J\times {\cal P}_K\to {\cal P}_{J\cup K}.$$ The operation $\otimes$ is commutative and associative, that is for $J,K,M\in {\mathbb{S}}$ pairwise disjoint, $\mu^J\in {\cal P}_J$, $\mu^K\in {\cal P}_K$, $\mu^M\in {\cal P}_M$, it is satisfied $$\label{eab4} \mu^J\otimes \mu^K=\mu^K\otimes \mu^J \hbox{ and } (\mu^J\otimes \mu^K)\otimes\mu^M=\mu^J\otimes (\mu^K\otimes\mu^M).$$ Moreover, $\mu^\emptyset\equiv 1$ is an identity element for $\otimes$, and $\otimes$ satisfies the following stability property under restriction. \[lemma0\] For all $J, K, M\in {\mathbb{S}}$ with $J\cap K=\emptyset$ and $M\subseteq J\cup K$, $$\label{eab} (\mu_J\otimes \mu_{K})_M=\mu_{J\cap M}\otimes \mu_{K\cap M}.$$ This is a consequence of definition (\[eab2\]) and property (\[eab3\]). In fact $$\begin{aligned} (\mu_J\otimes \mu_{K})_M(x_{(J\cup K)\cap M}) &=& \sum_{x_{(J\cup K)\setminus M}} (\mu_J \otimes \mu_K)(x_{J\cup K})\\ &=&\sum_{(x_{J\setminus M}, x_{K\setminus M})} \mu_J(x_J) \cdot \mu_K(x_K)\\ &=&\left(\sum_{x_{J\setminus M}}\mu_J(x_J)\right) \left(\sum_{x_{K\setminus M}}\mu_K(x_K)\right)\\ &=&\mu_{J\cap M}(x_{J\cap M})\cdot \mu_{K\cap M}(x_{K\cap M})\\ &=& (\mu_{J\cap M}\otimes \mu_{K\cap M}) (x_{(J\cap M)\cup (K\cap M)}).\end{aligned}$$ The above properties (\[eab0\]), (\[eab4\]), (\[eab\]), and $\mu^\emptyset$ an identity, are all we need from $\otimes$ to get the results of this work. Note that commutation and associativity imply that $\bigotimes_{L\in {{\cal D}}}\mu_L\in {\cal P}_I$ is well-defined for a partition ${{\cal D}}$ of $I$. For $\otimes$ the product measure we have $$\forall x\in \prod_{i\in I}A_i: \quad \bigotimes_{L\in {{\cal D}}}\mu_L(x)=\prod_{L\in {{\cal D}}}\mu_L(x_L).$$ If ${{\cal D}}={{\cal D}}^{si}=\{\{i\}: i\in I\}$, $\bigotimes_{i\in I} \mu_{\{i\}}$ is called Bernoulli and $(\mu^{\{i\}}: i\in I)$ are the one-site marginals. From now on, we fix $\rho=(\rho_J: J\in {\mathbb{S}})$ a probability vector, so $\rho_J\ge 0$ for $J\in {\mathbb{S}}$ and $\sum_{J\in {\mathbb{S}}}\rho_J=1$, and that also satisfies $\rho_\emptyset=0$. \[def1\] Define the following transformation $\Xi: {\cal P}_I\to {\cal P}_I$, $$\label{e4} \Xi[\mu]= \sum_{J\in {\mathbb{S}}} \rho_J \, \mu_J\otimes \mu_{J^c}=\sum_{J\in {\mathbb{S}}^{(\emptyset)}} \rho_J \, \mu_J\otimes \mu_{J^c}.$$ $\Box$ Since $\mu_\emptyset\equiv 1$, then $$\label{e4x} \Xi[\mu]= \rho_I \, \mu + \sum_{J\in {\mathbb{S}}^{\emptyset,I}} \rho_J \, \mu_J\otimes \mu_{J^c}.$$ \[rem0ax\] We have $\rho_\emptyset=0$, but we can have $\rho_I>0$. On the other hand, since $\mu_J\otimes \mu_{J^c}=\mu_{J^c}\otimes \mu_J$, we can assume when it is needed that $\rho_J=\rho_{J^c}$ for $J\in {\mathbb{S}}^{(\emptyset,I)}$. Observe that formula (\[e4\]) can be written in terms of dyadic and strictly dyadic partitions as, $$\Xi[\mu]=\sum_{{{\cal D}}\in {\mathbb{D}}_{1,2}} (\sum_{K\in {{\cal D}}}\rho_K)\bigotimes_{K\in {{\cal D}}} \mu_K =\rho_I \mu+ \sum_{{{\cal D}}\in {\mathbb{D}}_{2}} (\sum_{K\in {{\cal D}}}\rho_K)\bigotimes_{K\in {{\cal D}}} \mu_K.$$ Let $${{\cal J}}_\rho=\{J\in {\mathbb{S}}^{(\emptyset)}: \rho_J>0\}$$ be the support of $\rho$, so ${{\cal J}}_\rho^{(I)}=\{J\in {\mathbb{S}}^{(\emptyset,I)}: \rho_J>0\}$ is the class of nonempty subsets strictly contained in $I$ which are in the support of $\rho$. Denote by $$D^\rho:={{\cal D}}({{\cal J}}_\rho)$$ the partition generated by the class of sets ${{\cal J}}_\rho$, whose atoms satisfy (\[e8\]) with ${{\cal J}}={{\cal J}}_\rho$. \[lemma1\] $(i)$ Let $M$ be contained in an atom in ${{\cal D}}^\rho$, then $\Xi$ preserves the marginal on $M$, that is $$\label{e6} [M\subseteq L, L\in {{\cal D}}^\rho]\, \Rightarrow \, \Xi[\mu]_M=\mu_M.$$ $(ii)$ Let ${{\cal D}}$ be a partition finer than ${{\cal D}}^\rho$ (so ${{\cal D}}={{\cal D}}^\rho$ is allowed), and $\mu^{L}\in {\cal P}_L$ for $L\in {{\cal D}}$. Then, $\mu=\bigotimes_{L\in {{\cal D}}} \mu^{L}$ is a fixed point for $\Xi$, that is $\, \Xi[\mu]=\mu$. $(i)$ From (\[eab\]) we have $(\mu_J\otimes \mu_{J^c})_M=\mu_{J\cap M}\otimes \mu_{J^c\cap M}$. From (\[e8\]) and since $M\subseteq L\in {{\cal D}}^\rho$, we get that every $J\in {{\cal J}}_\rho$ satisfies: $J^c\subseteq M^c$ or $J\subseteq M^c$. Then $(\mu_J\otimes \mu_{J^c})_M=\mu_M$. $(ii)$ For every $L\in {{\cal D}}$ we have $\mu_{L}=\mu^{L}$ for all $L\in {{\cal D}}$. From (\[e3\]) we get, $$\forall J\in {{\cal J}}_\rho:\quad \mu_J=\bigotimes_{L\in {{\cal D}}: L\subseteq J}\mu^{L}.$$ Since the sets in the families $\{L\in {{\cal D}}: L\subseteq J\}$ and $\{L\in {{\cal D}}: L\subseteq J^c\}$, are disjoint and their union is $I$, we get $\mu_J\otimes \mu_{J^c}=\bigotimes_{L\in {{\cal D}}}\mu^{L}=\mu$. Hence $\Xi[\mu]=\mu$. As consequence of Lemma \[lemma1\] $(ii)$ we get that the one-site marginals $\mu_{\{i\}}$ are preserved by $\Xi$, and the Bernoulli probability measures are fixed points of $\Xi$. When $\rho_I=1$, so $\Xi[\mu]=\mu$ and $\Xi$ is the identity transformation. Then, in the sequel we assume $$\rho_I<1 \hbox{ or equivalently } {{\cal J}}_\rho^{(I)}\neq \emptyset.$$ We recall the notation in (\[e7\]), ${{\cal Y}}({{\cal J}}_\rho)$ is the class of all nonempty intersections of sets in ${{\cal J}}_\rho$. For $K\in {{\cal Y}}({{\cal J}}_\rho)$ we define, $$K\cap {{\cal J}}_\rho=\{K\cap J: J\in {{\cal J}}\}.$$ We have $K\in K\cap {{\cal J}}_\rho$ and $K\cap {{\cal J}}_\rho \subseteq {{\cal Y}}({{\cal J}}_\rho)\cup \{\emptyset\}$. \[def2\] For all $K\in {{\cal Y}}({{\cal J}}_\rho)$, $M\in (K\cap {{\cal J}}_\rho)^{(\emptyset)}$ we define $$\label{e10} \rho^K_M=\sum_{J\in {{\cal J}}_\rho: J\cap K=M} \!\!\!\rho_J \hbox{ if } M\neq K \hbox{ and } \rho^K_K=\!\! \sum_{J\in {{\cal J}}_\rho: J\cap K=K\vee J\cap K=\emptyset} \!\!\!\! \rho_M.$$ $\Box$ By definition the above quantities are positive: $\rho^K_M>0$ and $\rho^K_K>0$. For all $K\in {{\cal Y}}({{\cal J}}_\rho)$ we have $$\label{e11} \sum_{M\in (K\cap {{\cal J}}_\rho)^{(\emptyset)}} \rho^K_{M}= \sum_{M\in K\cap {{\cal J}}_\rho} \; \sum_{J\in {{\cal J}}_\rho: J\cap K=M}\!\!\!\!\rho_J= \sum_{J\in {{\cal J}}_\rho}\rho_J=1.$$ Then, $\rho^K_\bullet=(\rho^K_{M}: M\in (K\cap {{\cal J}}_\rho)^{(\emptyset)})$ is a probability vector. In particular $\rho^J_J\ge \rho_J+\rho_{J^c}>0$ for all $J\in {{\cal J}}_\rho^{(I)}$ and when $I\in {{\cal J}}_\rho$ then, $\rho^I_I=\rho_I$ and $\rho^I_K=\rho_K \hbox{ for } K\in {{\cal J}}_\rho^{(I)}$. \[lemma2\] $(i)$ We have: $$\label{e12} \forall K\in {{\cal Y}}({{\cal J}}_\rho), M\in (K\cap {{\cal J}}_\rho)^{(\emptyset)}, M\neq K: \;\; \rho^K_K<\rho^M_M.$$ When we assume $\rho_J=\rho_{J^c}$ for $J\in {{\cal J}}_\rho^{(I)}$, we get $$\label{e12'} \forall K\in {{\cal Y}}({{\cal J}}_\rho), M\in (J\cap {{\cal J}}_\rho)^{(\emptyset)}, M\neq K: \;\; \rho^K_M=\rho^K_{K\setminus M}.$$ $(ii)$ The atoms of ${{\cal D}}^\rho$ are characterized by the following relation: $$\label{e13} \forall L\in {{\cal Y}}({{\cal J}}_\rho): \;\; L\in {{\cal D}}^\rho \Leftrightarrow \rho^L_L=1.$$ $(i)$ Let us show (\[e12\]). For $J\in {{\cal J}}_\rho$ we have: $$\big[ J\cap K=M\Rightarrow J\cap M=M \big] \hbox{ and } \big[ J\cap K=\emptyset\Rightarrow J\cap M=\emptyset\big].$$ Hence, from definition (\[e10\]) we get $\rho^K_K\le \rho^M_M$. For showing the strict inequality we use that there exists some $J\in {{\cal J}}_\rho^{(I)}$ such that $J\cap K=M$. Note that $J^c\cap K\neq \emptyset$ but $J^c\cap M=\emptyset$, then $\rho^K_K<\rho^M_M$, so (\[e12\]) is proven. The relation (\[e12’\]) follows from $$J\in {{\cal J}}_\rho: \;\; J\cap K=M \Rightarrow J^c\cap K=K\setminus M,$$ and $\rho_J=\rho_{J^c}$ for $J\in {{\cal J}}_\rho^{(I)}$. In fact, both relations imply $$\sum_{J\in {{\cal J}}_\rho^{(I)}: J\cap K=M} \rho_J= \sum_{J^c\in {{\cal J}}_\rho^{(I)}: J^c\cap K=K\setminus M} \rho_{J^c} \;.$$ $(ii)$ Let us show the equivalence (\[e13\]). The implication ($\Rightarrow$) is a direct consequence of $L\cap J=L$ or $L\cap J=\emptyset$ for $J\in {{\cal J}}_\rho$. The converse relation ($\Leftarrow$) is deduced from the fact that $\rho^L_L=1$ happens if and only if $L\cap J=L$ or $L\cap J=\emptyset$ for $J\in {{\cal J}}_\rho$, but since $L\in {{\cal Y}}({{\cal J}}_\rho)$, from (\[e8\]) we get that $L$ is necessarily an atom of ${{\cal D}}^\rho$. \[lemma3\] Let $K\in {{\cal Y}}({{\cal J}}_\rho)$. Then, the marginal $\Xi[\mu]_K$ of $\Xi[\mu]$ on $K$, satisfies $$\Xi[\mu]_K=\rho^K_K \, \mu+ \sum_{M\in (K\cap {{\cal J}}_\rho)^{(\emptyset,K)}} \!\! \rho^K_M \; \mu_M\otimes \mu_{K\setminus M}.$$ Since $J\cap K=M$ implies $J^c\cap K=K\setminus M$, from (\[eab\]) we obtain, $$\begin{aligned} \Xi[\mu]_K&=&\sum_{J\in {\mathbb{S}}^{\emptyset}} \rho_J (\mu_J\otimes \mu_{J^c})_K = \sum_{J\in {\mathbb{S}}^{\emptyset}} \rho_J \, \mu_{J\cap K}\otimes \mu_{J^c\cap K}\\ &=& \left(\sum_{J\in {\mathbb{S}}^{\emptyset}: J\cap K=K\vee J\cap K=\emptyset} \!\!\!\!\!\!\! \rho_J\right) \! \mu_K+ \sum_{M\in (K\cap {{\cal J}}_\rho)^{(\emptyset,K)}} \! \left(\sum_{J\in {\mathbb{S}}^{\emptyset}: J\cap K=M} \!\!\rho_J\right) \mu_M\otimes \mu_{K\setminus M}\\ &=& \rho^K_K \,\mu_K+ \sum_{M\in (K\cap {{\cal J}}_\rho)^{(\emptyset,K)}} \!\!\!\rho^K_M \, \mu_M\otimes \mu_{K\setminus M}.\end{aligned}$$ Let us define the following kernel $f^K_{{\cal D}}$ between sets $K\in {{\cal Y}}({{\cal J}}_\rho)$ and dyadic partitions ${{\cal D}}\in {\mathbb{D}}_{1,2}(K)$. We set $$\label{e15} f^K(\{K\})=\rho^K_K \hbox{ and } f^K(\{M, K\setminus M\})=\rho^K_M+\rho^K_{K\setminus M} \hbox{ if } M\in (K\cap {{\cal J}}_\rho)^{(\emptyset, K)}.$$ Then, equality (\[e11\]) can be written in terms of dyadic partitions, $$\begin{aligned} \nonumber \sum_{{{\cal D}}\in {\mathbb{D}}_{1,2}(K)}\!\!\!\! f^K_{{\cal D}}&=&f^K(\{K\})+ \sum_{\{M, K\setminus M\}\in {\mathbb{D}}_2(K)}\!\!\!\! f^K(\{M, K\!\setminus \!M\})\\ \label{e16} &=&\rho^K_K+ \!\! \sum_{\{M, K\setminus M\}\in {\mathbb{D}}_2(K)} \!\!\!(\rho^K_M+\rho^K_{K\setminus M})=1.\end{aligned}$$ In the following sections we will present our main results. We note that there will be cases in which these results will be trivial, for instance when ${{\cal J}}_\rho=\{I\}$ or ${{\cal J}}_\rho=\{J,J^c\}$ for some $J\in {\mathbb{S}}^{(\emptyset,I)}$, but they will be not listed in detail. We will assume that the sets $I$, $A_i$, $i\in I$, and ${{\cal J}}_\rho$, are sufficiently big in order that the statements of our results make sense and are not trivial. The recursive equation in terms of trees {#sec3} ======================================== In this Section we supply the first of our main results, the decomposition of $\Xi^n[\mu]$ in terms of product marginal measures, where the marginals are the atoms of some partitions. This is done in Theorem \[theo0\]. It requires to introduce some dyadic trees because the atoms of the partitions are exactly the set of leaves of some dyadic trees. To expand $\Xi^n[\mu]$ for $n\ge 1$, we require to introduce further notation. Let us describe a class of rooted dyadic trees whose nodes are sets, in fact they are elements of ${{\cal Y}}({{\cal J}}_\rho)$. The dyadic property means that each parent node has one or two children: if it has one children the set associated to the children is the same as the one of the parent, and when it has two children the set of the parent is partitioned into two disjoint nonempty sets by some set in ${{\cal J}}_\rho^{(I)}$, and these are the sets associated to the children. The set $I$ will be the root of all these trees. Let us be more precise in notation and concepts. We note by ${{\cal T}}={{\cal T}}({{\cal J}}_\rho)$ the family of dyadic trees rooted by $I$, which depends on ${{\cal J}}_\rho$, and that we will construct in an inductive way. The recursion will depend on the length $|T|$ of a tree $T\in {{\cal T}}$, so the classes ${{\cal T}}_n=\{T\in {{\cal T}}: \, |T|=n\}$ will be defined in a recursive way for $n\ge 0$. A tree $T\in {{\cal T}}$ is defined as the set of its branches. A branch $b\in T$ is a tuple of elements in ${{\cal Y}}({{\cal J}}_\rho)$ and its last component is called its leaf and noted $\ell(b)$. The set of leaves of the tree $T$ is $$\partial(T)=\{\ell(b): b\in T\},$$ and a leaf of $T$ is simply noted $\ell\in \partial(T)$. As a consequence of our construction of ${{\cal T}}$, all the branches $b$ of a tree $T\in {{\cal T}}$ will have the same length, so $|b|=|T|$. Any of these branches is written $b=(b_0,..,b_{|T|})$, and so $b_{|T|}=\ell(b)$. Below, the algorithm of construction of ${{\cal T}}$ is given as a recursive definition of $({{\cal T}}_n: n\ge 0)$. For $n=0$, the class ${{\cal T}}_0$ is a singleton formed by the unique tree $T=\{I\}$. So, it that has a unique branch $b=(I)$ with leaf $\ell(b)=I$. The length of $T$ is by definition $|T|=0$, so $|b|=0$ and $b_0=I$. Assume we have constructed the set of trees ${{\cal T}}_n$. We will construct ${{\cal T}}_{n+1}$ by using the following algorithm: Take $T\in {{\cal T}}_n$. It generates a family of trees in ${{\cal T}}_{n+1}$, where each one of these trees is the result of adding either one or two nodes, to each leaf of $T$. So, any of the choices made for the leaves $\ell \in \partial T$, defines a tree $T'\in {{\cal T}}_{n+1}$. To be precise let $b=(b_0,..,b_n)$ be a branch in $T$, then: - If we are not in the case $(n=0,\rho_I=0)$, $b$ can generate the branch $b'=(b_0,..,b_n,b_n)$, so with $b_{n+1}=b_n$; - If $\ell(b)$ is not an atom in ${{\cal D}}^\rho$, $b$ can generate two branches $b'$ and $b''$. This is done by partitioning the set $\ell(b)$ into a pair of nonempty sets $\{\ell(b)\cap J, \ell(b)\cap J^c\}$ with some $J\in {{\cal J}}_\rho^{(I)}$. The two branches generated by $b$ are respectively $b'=(b, \ell(b)\cap J)$ and $b''=(b, \ell(b)\cap J^c)$, so these branches share all the nodes with $b$ except that we have added to them and extra node, these are their leaves $\ell(b)\cap J$ and $\ell(b)\cap J^c$ respectively. We have specified the possible choices on the branches of $T$, but as it can be easily checked we could also performed it in terms of $\partial(T)$. As said, once a choice is made for the whole set of branches $\{b\in T\}$, or equivalently for all the leaves $\{\ell\in \partial(T)\}$, a tree $T'$ of length $|T'|=|T|+1$ is defined from $T$, or equivalently a partition $\partial(T')$ is defined from $\partial(T)$. When this happens we put $$\label{e17} T\rightarrow T' \hbox{ or equivalently } \partial(T)\rightarrow \partial(T'),$$ which defines a relation in ${{\cal T}}$ or equivalently in $\partial({{\cal T}})=\{\partial(T): T\in {{\cal T}}\}$. Thus, a tree $T\in {{\cal T}}_1$ can have the following shapes: either it has one branch $(I,I)$ in which case $I$ is the unique leaf (this can happen only when $\rho_I>0$); or it can have two branches $\{(I,J), (I,J^c)\}$ for some $J\in {{\cal J}}_\rho^{(I)}$, and so with leaves $J$ and $J^c$ respectively. The family of rooted trees constructed as above but with root $K$ instead of $I$, is noted by ${{\cal T}}^K$. So, ${{\cal T}}_n^K$ refers to the class of the trees in ${{\cal T}}^K$ of length $n$. With this notation the recursive step to construct ${{\cal T}}_{n+1}$ from ${{\cal T}}_{n}$, can be summarized by saying that a tree $T\in {{\cal T}}_n$ generates a family of trees $T'\in {{\cal T}}_{n+1}$, each $T'$ is the result of a choice of a family of trees $({T'}^\ell\in {{\cal T}}_1^\ell: \ell\in \partial(T))$, being ${T'}^\ell$ attached to $\ell$. In the next result, $\mu_\ell$ refers to the marginal probability measure $\mu$ on the set $\ell$. \[lemmafd\] For every $\mu\in {\cal P}_I$, for all $n\ge 1$ and all $j\le n$, the following relation is satisfied, $$\begin{aligned} \label{e18} \Xi^n[\mu]&=&\sum_{T\in {{\cal T}}_j} \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right) \bigotimes_{\ell\in \partial(T)} (\Xi^{n-j}[\mu])_\ell\\ \label{e22} &=&\sum_{T\in {{\cal T}}_j} \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right) \left(\bigotimes_{\ell\in \partial(T)\setminus {{\cal D}}^\rho} \!\!\! \Xi^{n-j}[\mu]_\ell\right)\! \otimes \! \left(\bigotimes_{\ell\in \partial(T)\cap D^\rho} \!\!\!\! \mu_\ell\right).\end{aligned}$$ (We note $\prod_{r=1}^{|b|}=1$ when $|b|=0$.) First of all, by using Lemma \[lemma1\] $(i)$ the expression (\[e18\]) becomes (\[e22\]). So, we only need to show (\[e18\]). This is done by recurrence on $n\ge 1$. Let $n=1$. The development made for the family of trees ${{\cal T}}_1$, implies that the relation (\[e4\]) can be written as $$\Xi[\mu]=\sum_{T\in {{\cal T}}_1} \left(\sum_{b\in T} \rho^{b_0}_{b_1}\right) \bigotimes_{\ell\in \partial(T)}\mu_\ell.$$ Then, (\[e18\]) is satisfied for $n=1$. Now, assume we have shown (\[e18\]) for some $n-1$, let show it for $n$. First take $j<n$. By recurrence hypothesis, we can apply formula (\[e18\]) to $n-1$, $j$ and $\Xi[\mu]$. Hence, $$\begin{aligned} \Xi^n[\mu]&=&\Xi^{n-1}(\Xi[\mu])\\ &=&\sum_{T\in {{\cal T}}_j} \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right) \bigotimes_{\ell\in \partial(T)} (\Xi^{n-1-j}[\Xi[\mu]])_{\ell}\\ &=&\sum_{T\in {{\cal T}}_j} \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right) \bigotimes_{\ell\in \partial(T)} (\Xi^{n-j}[\mu])_\ell.\end{aligned}$$ Then the formula (\[e18\]) holds for $n$, $j$ and $\mu$. Now take $j=n$. By recurrence hypothesis and by using Lemma \[lemma3\] we get $$\begin{aligned} \nonumber \Xi^n[\mu]&=&\Xi^{n-1}[\Xi[\mu]]\\ \nonumber &=& \sum_{T\in {{\cal T}}_{n-1}} \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right) \bigotimes_{\ell\in \partial(T)} \Xi[\mu]_\ell\\ &=& \label{e21} \sum_{T\in {{\cal T}}_{n-1}} \!\!\! \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\!\right)\! \bigotimes_{\ell\in \partial(T)} \! \left(\sum_{{T'}^\ell\in {{\cal T}}_1^\ell}\!\! \left(\sum_{b'\in {T'}^\ell} \!\!\!\rho^{b'_0}_{b'_1}\right)\!\! \bigotimes_{\ell'\in \partial({T'}^\ell)} \!\!\!\!\mu_{\ell'}\right)\\ \label{e21x} &=& \sum_{T^*\in {{\cal T}}_n} \left(\sum_{b\in T^*} \prod_{r=1}^{|b^*|}\rho^{b^*_{r-1}}_{b^*_r}\right) \bigotimes_{\ell^*\in \partial(T^*)} \!\!\!\! \mu_{\ell^*}. \end{aligned}$$ In (\[e21\]) we set $b_{|b|}=b'_0$. On the other hand, in (\[e21x\]) we used, $$\bigotimes_{\ell^*\in \partial(T^*)} \!\!\mu_{\ell^*}= \bigotimes_{\ell\in \partial(T)}\left(\bigotimes_{\ell'\in \partial({T'}^\ell)} \!\!\mu_{\ell'}\right),$$ and $$\prod_{r=1}^{|b^*|}\rho^{b^*_{r-1}}_{b^*_r}= \left(\prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right) \rho^{b'_0}_{b'_1} \hbox{ for } b^*_s=b_s \hbox{ for } s\le |b|, \hbox{ and } b^*_{|b|+1}=b'_1,$$ for the tree $T^*$ formed by adding ${T'}^\ell\in {{\cal T}}^\ell$ to each leaf $\ell\in \partial(T)$. Hence, the result is shown. We will note by $\partial({{\cal T}}_n)=\{\partial(T): T\in {{\cal T}}_n\}$. \[theo0\] For every probability measure $\mu\in {\cal P}_I$ and all $n\ge 1$, we get the following decomposition $$\label{eqwr1} \Xi^n[\mu]=\sum_{\delta\in \partial({{\cal T}}_n)} q^n_\delta \; \bigotimes_{\ell\in \delta} \mu_\ell,$$ where the vector $q^n=(q^n_\delta: \delta\in \partial({{\cal T}}_n)$ is given by, $$\label{eqwr2} q^n_\delta=\sum_{T\in {{\cal T}}_n: \partial(T)=\delta} \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right),$$ and it is a probability vector, so it satisfies $$\label{eqwr3} \sum_{\delta\in \partial({{\cal T}}_n)} q^n_\delta=1.$$ By taking $j=n$ in (\[e18\]) we get, $$\label{e19} \Xi^n[\mu]=\sum_{T\in {{\cal T}}_n} \left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right) \bigotimes_{\ell\in \partial(T)} \mu_\ell.$$ So, by using definition (\[eqwr2\]), the equality (\[eqwr1\]) is shown. Since (\[e19\]) expresses that the probability measure $\Xi^n[\mu]$ is a positive linear combination of the set of probability measures $(\bigotimes_{\ell\in \partial(T)} \mu_\ell: T\in {{\cal T}}_n)$, we deduce it is necessarily a convex linear combination, that is $$\sum_{T\in {{\cal T}}_n}\left(\sum_{b\in T} \prod_{r=1}^{|b|}\rho^{b_{r-1}}_{b_r}\right)=1.$$ But this is exactly (\[eqwr3\]). The result is shown. Then, in the expansion (\[eqwr1\]) $\Xi^n[\mu]$ has a weight $q^n_\delta$ of being the product probability measure $\otimes_{\ell\in \delta} \mu_\ell$. \[remirs\] In the following section we will use some properties of the relation $\rightarrow$ on $\partial({{\cal T}})$ defined in (\[e17\]). We have that $\rightarrow$ is an order relation and $\delta\rightarrow \delta'$ implies that $\delta'$ is finer than $\delta$ (finer includes equal). Also, for all $\delta\in \partial({{\cal T}})$, $\delta\neq \{I\}$, there exists a path $\delta_1=\{I\}\rightarrow...\rightarrow \delta_k=\delta$ from $\{I\}$ to $\delta$, and $\{I\}\rightarrow \{I\}$ only when $\rho_I>0$. On the ordered space $(\partial({{\cal T}}), \rightarrow)$ we can say that $\delta'$ is a successor of $\delta$ when $\delta\to \delta'$ in a consistent way because $(\delta_1\rightarrow...\rightarrow \delta_k, \, \delta_1\neq \delta_k)$ implies $\delta_k\not\rightarrow \delta_1$. But the ordered space $(\partial({{\cal T}}),\rightarrow)$ is in general not a tree. For instance if the elements $I, J_1, J_2$ are three different elements of ${{\cal J}}_\rho$, and the intersections $J_1\cap J_2$, $J_1^c\cap J_2$, $J_1^c\cap J_2$ and $J_1^c\cap J_2^c$ are nonempty, then $$\{I\}\to \{J_1, J_1^c\}\to \{J_1\cap J_2, J_1^c\cap J_2, J_2^c\}\to \{J_1\cap J_2, J_1^c\cap J_2, J_1\cap J_2^c, J_1^c\cap J_2^c\}$$ and $$\{I\}\to \{J_2, J_2^c\}\to \{J_1\cap J_2, J_1\cap J_2^c, J_1^c\}\to \{J_1\cap J_2, J_1\cap J_2^c, J_1^c\cap J_2, J_1^c\cap J_2^c\}$$ are two different paths from $\{I\}$ to $\{J_1\cap J_2, J_1^c\cap J_2, J_1\cap J_2^c, J_1^c\cap J_2^c\}$, having in common only the initial and final points. So, $\{J_1\cap J_2, J_1^c\cap J_2, J_1\cap J_2^c, J_1^c\cap J_2^c\}$ has at least two predecessors. Markov chain, geometric convergence and quasi-stationarity {#sec4} ========================================================== In this section we supply our main results. Firstly, the definition of a natural Markov chain associated to $(\Xi^n: n\ge 0)$ is done in Lemmas \[lemma4\] and \[lemma5\]. The main results are the description of this chain found in Theorem \[theo1\]. Since the orbit $(\Xi^n[\mu])$, converges to the product of the marginal probability measures on the atoms of the partition ${{\cal D}}^\rho$, we study geometric convergence to the limit probability measure. We give the geometric decay rate, and we study the ratio limit and the quasi-stationary behavior of the chain. This last study responds to the following question: if the chain has not arrived to the limit probability measure after a long time, which is its distribution? Finally, in Corollary \[cor1\] we supply the Markov chain that never hit the limit distribution. The relations (\[eqwr1\]), (\[eqwr2\]) and (\[eqwr3\]) of Theorem \[theo0\] will be at the basis of the construction of a Markov chain $Y=(Y_n: n\ge 0)$ taking values on $\partial({{\cal T}})$ and having the following remarkable property: if it starts from $Y_0=\{I\}$, then at time $n$, the event $\{Y_n=\delta\}$ has probability $q^n_\delta$. In this purpose we define the following transition matrix $P=(P_{\delta, \delta'}: \delta, \delta'\in \partial({{\cal T}}))$. First we put $P_{\delta, \delta'}=0$ when $\delta\not\rightarrow \delta'$. To define the transition probability $P_{\delta, \delta'}$ when $\delta\rightarrow \delta'$ it is useful to introduce the following notation: for each leaf $\ell\in \delta$ we denote by $\{\ell_1,\ell_2\}$ its corresponding dyadic partition in $\delta'$. We can either have $\{\ell_1,\ell_2\}=\{\ell\}$ that is $\ell_1=\ell_2=\ell$ which means $\ell\in \delta\cap \delta'$; or $\{\ell_1,\ell_2\}\in {\mathbb{D}}_2(\ell)$ is an strictly dyadic partition of $\ell$ and in this case $\ell\in \delta\setminus \delta'$. We define $$\label{e23} \forall \delta, \delta'\!\in \!\partial({{\cal T}}), \delta\rightarrow \delta':\; P_{\delta, \delta'}\!=\! \prod_{\ell\in \delta} f^\ell(\{\ell_1, \ell_2\}) \!=\!\left(\prod_{\ell\in \delta\cap \delta'}\! \rho^\ell_\ell\right) \left(\prod_{\ell\in \delta\setminus \delta'}\!\! (\rho^\ell_{\ell_1}\!+\!\rho^\ell_{\ell_2})\right).$$ In particular $$\label{e24} \forall \delta\in \partial({{\cal T}}):\;\, P_{\delta, \delta}=\prod_{\ell\in \delta} \rho^\ell_\ell.$$ \[lemma4\] $P$ is an stochastic transition matrix, that is $$\forall \delta\in \partial({{\cal T}}): \quad \sum_{\delta'\in \partial({{\cal T}}): \delta\to \delta'} \!\!\!P_{\delta, \delta'}=1.$$ We will use the following decomposition: $\delta=(\delta\cap {{\cal D}}^\rho) \cup (\delta\setminus {{\cal D}}^\rho)$, so the atoms of $\delta$ are partitioned according to the fact that if they belong or not to ${{\cal D}}^\rho$. We recall that ${\mathbb{D}}_2(\ell)$ excludes the partition $\{\ell\}$. For $U\subseteq \delta\setminus {{\cal D}}^\rho$ denote $${\cal D}(U,2)=\{((K_1^\ell, K_2^\ell): \ell\in U)\in \prod_{\ell\in U} {\mathbb{D}}_2(\ell): \forall \ell\in U, \exists J\in {{\cal J}}_\rho, K_1^\ell=\ell\cap J, K_2^\ell=\ell\cap J^c\}.$$ We have $$\begin{aligned} \sum_{\delta': \delta\rightarrow \delta'} \!\! P_{\delta, \delta'}&=& \!\!\left(\prod_{\ell\in \delta\cap {{\cal D}}^\rho} \!\!\! \rho^\ell_\ell\right)\! \times \! \left( \sum_{U\subseteq \delta\setminus {{\cal D}}^\rho} \!\!\left(\prod_{\ell\in U^c} \rho^\ell_\ell\right) \left(\sum_{((K_1^\ell, K_2^\ell): \ell\in U)\in {\cal D}(U,2)} \, \prod_{\ell\in U} (\rho^\ell_{K_1^\ell}\!+\!\rho^\ell_{K_2^\ell}) \right) \! \right)\\ &=&\!\sum_{U\subseteq \delta\setminus {{\cal D}}^\rho} \!\! \left(\prod_{\ell\in U} \rho^\ell_\ell\right) \left(\sum_{((K_1^\ell, K_2^\ell): \ell\in U)\in {\cal D}(U,2)} \; \prod_{\ell\in U}(\rho^\ell_{K_1^\ell}+ \rho^\ell_{K_2^\ell})\right).\end{aligned}$$ This last equality uses $\rho^\ell_\ell=1$ when $\ell\in {{\cal D}}^\rho$, see (\[e13\]) in Lemma \[lemma2\] $(ii)$. By using notation $f^\ell(\gamma_\ell)$ introduced in (\[e15\]) we have, $$\begin{aligned} \sum_{\delta': \delta\rightarrow \delta'} \!P_{\delta, \delta'}&=& \sum_{(\gamma^\ell: \ell\in \delta\setminus{{\cal D}}^\rho)\in \prod\limits_{\ell\in \delta\setminus{{\cal D}}^\rho}{\mathbb{D}}_{1,2}(\ell)} \;\, \prod_{\ell\in \delta\setminus{{\cal D}}^\rho}f^\ell(\gamma_\ell)\\ &=& \prod\limits_{\ell\in \delta\setminus{{\cal D}}^\rho} \left(\sum_{\gamma_\ell\in {\mathbb{D}}_{1,2}(\ell)} f^\ell(\gamma_\ell)\right)=1. \end{aligned}$$ In this last equality we use (\[e16\]). \[remirr\] From the positive properties of coefficients $\rho^K_M$ we get that $P_{\delta, \delta'}>0$ if and only if $\delta\rightarrow \delta'$. Since there exists a path $\delta_1=\{I\}\rightarrow...\rightarrow \delta_k=\delta$ for all $\delta\in \partial({{\cal T}})$, $\delta\neq \{I\}$, this path has positive probability. Let $Y=(Y_n: n\ge 0)$ be the Markov chain taking values in $\partial({{\cal T}})$ defined by the transition stochastic matrix $P$. Let $(\Omega, {\cal F})$ be the measurable space with $\Omega=\partial({{\cal T}})^{\mathbb{N}}$ and ${\mathbb{F}}$ the product $\sigma-$field. Let $({\mathbb{P}}_\delta: \delta\in \partial({{\cal T}}))$ be the family of probability Markov measures on $(\Omega, {\cal F})$, all of them with transition matrix $P$, and ${\mathbb{P}}_\delta$ starting from $\delta$. We will simply note ${\mathbb{P}}:={\mathbb{P}}_{\{I\}}$, because most of the time the chain will assume to start from $Y_0=\{I\}$, and this will be clear from the context or the notation. The mean expected values associated to ${\mathbb{P}}_\delta$ and ${\mathbb{P}}$ are noted by ${\mathbb{E}}_\delta$ and ${\mathbb{E}}$, respectively. The Markov chain $(Y_n: n\ge 0)$ can be also constructed from a probability space $({\widetilde{\Omega}}, {\widetilde{\cal F}},{\bf P})$ containing an independent family of random variables $\left(\delta^K_n: K\in {{\cal Y}}({{\cal J}}_\rho), n\ge 1 \right)$, where $\delta^K_n$ takes values in ${\mathbb{D}}_{1,2}(K)$ and $${\bf P}(\delta^K_n=\delta)=f^K_\delta,$$ where $f^K_\delta$ was defined in (\[e15\]). Thus, the random variables $(\delta^K_n: n\ge 1)$ are independent and identically distributed with law $f^K_\bullet$. It is easily checked that the random sequence given by $$Y_0=\delta, \;\; Y_n=(\delta^K_n: K\in Y_{n-1}) \;\, \forall n\ge 1,$$ defines a Markov chain $(Y_n)$ starting from $\delta$, and transition probability given by (\[e23\]). Let us show that the Markov chain $(Y_n)$ fulfills the first claim of this section: after $n-$steps of time the probability of the event $\{Y_n=\delta\}$ is the weight of all the trees of length $j$ whose set of leaves is $\delta$. \[lemma5\] For every $n\ge0$ and $\delta\in \partial({{\cal T}})$ it holds ${\mathbb{P}}(Y_n=\delta)=q^n_\delta$. We will use a recurrence argument. For $n=0$ the property holds because $Y_0=\{I\}$ and the class of trees of length $0$ is the singleton ${{\cal T}}_0=\{\{I\}\}$. Assume the property holds up to $n$ let us show it for $n+1$. We have $$\begin{aligned} {\mathbb{P}}(Y_{n+1}=\delta')&=& \sum_{\delta\in \partial({{\cal T}}): \delta \rightarrow \delta'} {\mathbb{P}}(Y_n=\delta) P_{\delta,\delta'}\\ &=&\sum_{\delta\in \partial({{\cal T}}): \delta \rightarrow \delta'} q^n_\delta \, \left(\prod_{\ell\in \delta\cap \delta'} \rho^\ell_\ell\right) \left(\prod_{\ell\in \delta\setminus \delta'}\!\! (\rho^\ell_{\ell_1}\!+\!\rho^\ell_{\ell_2})\right).\end{aligned}$$ Now we use the step (\[e21\]) of the proof of Lemma \[lemmafd\], which allows to get ${\mathbb{P}}(Y_{n+1}=\delta')=q^{n+1}_{\delta'}$. The result is proven. The partition ${{\cal D}}^\rho$ is an absorbing state for the chain $(Y_n)$ because $P_{{{\cal D}}^\rho,{{\cal D}}^\rho}=\prod_{L\in {{\cal D}}^\rho}\rho^L_L=1$, and so $Y_n={{\cal D}}^\rho$ implies $Y_{n+k}={{\cal D}}^\rho$ for all $k\ge 0$. Let us define the hitting times, $$\forall B\subseteq \partial({{\cal T}}): \quad \zeta_B=\inf\{n\ge 0: Y_n\in B\}.$$ For singletons we simply put, $$\forall \delta\in \partial({{\cal T}}): \quad \zeta_\delta= \zeta_{\{\delta\}}.$$ For $\delta=\{I\}$ we have ${\mathbb{P}}(\zeta_{\{I\}}=0)=1$. The random time for attaining ${{\cal D}}^\rho$, $$\zeta=\zeta_{{{\cal D}}^\rho}=\inf\{n\ge 0: Y_n={{\cal D}}^\rho\},$$ is an absorbing time because $Y_{\zeta+n}={{\cal D}}^\rho$ for all $n\ge 0$. Since $Y_n(\omega)\in \partial({{\cal T}})$ we can define the random probability: $$\forall \omega\in \Omega: \quad \Xi^n[\mu](\omega)=\bigotimes_{K\in Y_n(\omega)} \mu_K.$$ From above discussion and Lemma \[lemma1\] $(ii)$ we find, $$\forall n\ge 0: \;\; \Xi^{\zeta(\omega)+n}[\mu](\omega)=\bigotimes_{L\in {{\cal D}}^\rho} \mu_L.$$ \[eqtodas\] Note that $$\label{eq27} \{\Xi^n[\mu]\neq \otimes_{L\in {{\cal D}}^\rho}\}\subseteq \{\zeta\!>\!n\} \hbox{ and so } {\mathbb{P}}\left(\Xi^n[\mu]\neq \otimes_{L\in {{\cal D}}^\rho} \mu_L \, \right)\le {\mathbb{P}}(\zeta\!>\! n).$$ It can be checked that when the spaces $I$, $A_i$, $i=1,..,n$, have sufficiently many points we have the equivalence $$\big\{\forall \mu\in {\cal P}_I: \; \Xi^n[\mu]\neq \otimes_{L\in {{\cal D}}^\rho}\big\}=\{\zeta>n\}.$$ For some particular ${\widetilde{\mu}}\in {\cal P}_I$ the inequality (\[eq27\]) can be strict. For instance, if ${\widetilde{\mu}}=\otimes_{L\in {{\cal D}}^\rho} {\widetilde{\mu}}_L$ then $\Xi^n[{\widetilde{\mu}}](\omega)={\widetilde{\mu}}$ for all $n\ge0$, but ${\mathbb{P}}(\zeta>0)=1$ in the nontrivial case ${{\cal D}}^\rho\neq \{I\}$. In the next result we show that the random measure $\Xi^n[\mu](\omega)$ converges geometrically to a product measure with the marginals of $\mu$ at the atoms of ${{\cal D}}^\rho$. This is controlled with the geometric decay rate of ${\mathbb{P}}(\zeta> n)$. Also we give the quasi-limiting distribution which results from conditioning to the event $\{\zeta> n\}$ for $n\to \infty$. We will supply the notions of quasi-limiting distribution (and further of quasi-stationary distributions) in the context of the Markov chain $(Y_n)$. The definition and study of these concepts in the context of finite Markov chains which are irreducible on the non-absorbing states are found in the pioneer work [@ds] and the continuous time case can be seen in Chapter $3$ of monograph [@cms]. There is a large body of literature on quasi stationary distributions, in particular for extinction in population dynamics and we recommend addressing to [@pp] for an exhaustive list of references. We emphasize that $(Y_n)$ is not irreducible on $\partial({{\cal T}})\setminus \{{{\cal D}}^\rho\}$, because when $(Y_n)$ exits from some state it does never return to it. In fact, $\delta_1\rightarrow \delta_2\rightarrow ... \rightarrow \delta_k$ and $\delta_k\neq \delta_1$ implies $\delta_k\not\rightarrow \delta_1$ (see Remark \[remirs\]). Therefore, we cannot apply Perron-Frobenius theory which is in the theoretical basis of the main results of quasi-stationary distributions on finite Markov chains. So, we need to develop new elements to describe the quasi-limiting behavior and in particular the geometric decay rate. In this purpose we introduce a class of distinguished partitions in $\partial({{\cal T}})$. Any $K\in {{\cal Y}}({{\cal J}}_\rho)$ defines the partition $${{\cal D}}^{\rho,K}=\{L\in {{\cal D}}^\rho: L\cap J=\emptyset\}\cup \{K\}.$$ So, the partition ${{\cal D}}^{\rho,K}$ has the same atoms as ${{\cal D}}^\rho$ when they do not intersect $K$, and all the other atoms collapse into the unique atom $K\in {{\cal D}}^{\rho,K}$. For $a\in [0,1]$ define the following classes of sets and partitions, $$\label{e50d} {{\cal E}}(a)=\{K\in {{\cal Y}}({{\cal J}}_\rho): \rho^K_K=a\} \hbox{ and } \partial({{\cal T}})^{{{\cal E}}(a)}=\{{{\cal D}}^{\rho,K}: K\in {{\cal E}}(a)\}.$$ Note that ${{\cal E}}(a)$ and so $\partial({{\cal T}})^{{{\cal E}}(a)}$ can be empty. When $a=1$ we have ${{\cal E}}(1)={{\cal D}}^\rho$ and $\partial({{\cal T}})^{{{\cal E}}(1)}=\{{{\cal D}}^\rho\}$. When $\rho_I>0$ we have ${{\cal E}}(\rho_I)=\{I\}$ and $\partial({{\cal T}})^{{{\cal E}}(\rho_I)}=\{\{I\}\}$. \[theo1\] Assume $\rho_I<1$. Then, $$\label{e29} {\mathbb{P}}(\zeta<\infty)=1.$$ Define $$\eta=\max\{\rho^K_K: K\in {{\cal Y}}({{\cal J}}_\rho), K\not\in {{\cal D}}^\rho\}.$$ Then $\eta\in (0,1)$. Let $$\label{e31} {{\cal E}}={{\cal E}}(\eta),\quad \partial({{\cal T}})^{{\cal E}}=\partial({{\cal T}})^{{{\cal E}}(\eta)},\quad \zeta^{{{\cal E}}}=\zeta_{\partial({{\cal T}})^{{\cal E}}}.$$ Then $0<{\mathbb{P}}(\zeta^{{{\cal E}}}<\infty)<1$ and the geometric rate of decay of ${\mathbb{P}}(\zeta>n)$ satisfies, $$\label{50e} \lim\limits_{n\to \infty} \eta^{-n} {\mathbb{P}}(\zeta\!>\!n)= \lim\limits_{n\to \infty} \eta^{-n} {\mathbb{P}}(\zeta\!>\!n, Y_n\!\in \!\partial({{\cal T}})^{{\cal E}}) ={\mathbb{E}}\left(\eta^{-\zeta^{{\cal E}}}, \, \zeta^{{\cal E}}\!<\!\infty \right) \!\in \! (0,\infty).$$ The quasi-limiting distribution on $\partial({{\cal T}})\setminus \{{{\cal D}}^\rho\}$ is given by, $$\begin{aligned} \nonumber \forall \delta\in \partial({{\cal T}})^{{\cal E}}:&{}& \lim\limits_{n\to \infty} {\mathbb{P}}(Y_n=\delta\,| \, \zeta>n)= \frac{ {\mathbb{E}}\left(\eta^{-\zeta_\delta}, \, \zeta_{\delta}<\infty \right)} {{\mathbb{E}}\left(\eta^{-\zeta^{{{\cal E}}}}, \, \zeta^{{{\cal E}}}<\infty\right)},\\ \label{e32} \forall \delta\in \partial({{\cal T}})\setminus \partial({{\cal T}})^{{\cal E}}:&{}& \lim\limits_{n\to \infty} {\mathbb{P}}(Y_n=\delta\,| \, \zeta>n)=0.\end{aligned}$$ Furthermore, we have the following ratio limit relation for $\delta\in \partial({{\cal T}})\setminus \{{{\cal D}}^\rho\}$, $$\label{50b} \lim\limits_{n\to \infty} \frac{{\mathbb{P}}_\delta(\zeta>n)}{{\mathbb{P}}(\zeta>n)} =\frac{{\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)} {{\mathbb{E}}(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)}.$$ Both ratios vanish only when ${\mathbb{P}}_\delta(\zeta^{{\cal E}}<\infty)=0$. Finally, the vector $$\label{rev1} \varphi=(\varphi_\delta: \delta\in \partial({{\cal T}})\setminus \{{{\cal D}}^\rho\}) \hbox{ with } \varphi_\delta={\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty),$$ is a right eigenvector of the restriction of $P$ to $\partial({{\cal T}})\setminus \{{{\cal D}}^\rho\}$, and it has eigenvalue $\eta$. [*Proof*]{}. It is obvious that $\eta>0$ and from (\[e13\]) in Lemma \[lemma2\] $(ii)$ we have $\eta<1$. Then ${{\cal E}}\cap {{\cal D}}^\rho=\emptyset$ because $K\in {{\cal E}}$ and $L\in {{\cal D}}^\rho$ imply $\rho^K_K=\eta<1=\rho^L_L$. Note that if $\delta=D^{\rho,K}$ then $P_{\delta,\delta}=\rho^K_K$. Hence $$\label{e33} \forall \delta\in \partial({{\cal T}})^{{\cal E}}: \quad P_{\delta,\delta}=\eta.$$ We claim that $$\max\{P_{\delta,\delta}: \delta\in \partial({{\cal T}}), \delta\neq {{\cal D}}^\rho\}= \eta.$$ This follows from (\[e33\]) for partitions having at most one atom that is not in ${{\cal D}}^\rho$, and if $\delta'\in \partial({{\cal T}})$ has at least two different atoms $K,K'$ that are not elements of ${{\cal D}}^\rho$, from (\[e24\]) we get $P_{\delta',\delta'}\le \eta^2$. Let $$\label{eqlast} \beta_0=\max\{P_{\delta,\delta}: \delta\in \partial({{\cal T}}), \delta\neq {{\cal D}}^\rho, \delta\!\not\in \!\partial({{\cal T}})^{{\cal E}}\}.$$ We have $$\label{e33ax} \beta_0\le \max\{\beta,\eta^2\}<\eta \, \hbox{ where } \beta=\sup\{\rho^K_K: K\in {{\cal Y}}({{\cal J}}_\rho), \rho^K_K< \eta\}.$$ In fact, when $K\not\in {{\cal E}}\cup\{D^\rho\}$ we have $P_{{{\cal D}}^{\rho,K}, {{\cal D}}^{\rho,K}}=\rho^K_K\le \beta$ and if a partition has at least two different atoms $K,K'$ that are not in ${{\cal D}}^\rho$, then $P_{\delta',\delta'}\le \eta^2$. Then, (\[e33ax\]) is shown. Let us show (\[e29\]). We use that when $(Y_n)$ exits from some state it does never return to it and inequality $P_{\delta,\delta}<1$ for $\delta\neq {{\cal D}}^\rho$. In fact, they allow us to prove that the Markov chain $(Y_n)$ visits every state $\delta\neq {{\cal D}}^\rho$ only a finite number of times ${\mathbb{P}}-$a.s., $$\forall \delta\in \partial({{\cal T}}), \delta\neq {{\cal D}}^\rho: \quad {\mathbb{P}}(\#\{n: Y_n=\delta\}<\infty)=1.$$ Then, by using that ${{\cal D}}^\rho$ is an absorbing state, we obtain (\[e29\]), $${\mathbb{P}}(\exists n: Y_n={{\cal D}}^\rho)={\mathbb{P}}(\zeta<\infty)=1.$$ The existence of paths from $\{I\}$ to $\partial({{\cal T}})^{{\cal E}}$ with positive probability gives ${\mathbb{P}}(\zeta^{{\cal E}}<\infty)>0$. On the other hand there exists $\delta'\in \partial({{\cal T}})$ with $\delta'\rightarrow {{\cal D}}^\rho$ and $\#\{J\in \delta': J\not\in {{\cal D}}^\rho\}>1$. The existence of some path from $\{I\}$ to $\delta'$ with positive probability now gives ${\mathbb{P}}(\zeta^{{\cal E}}<\infty)<1$. We have shown ${\mathbb{P}}(\zeta^{{\cal E}}<\infty)\in (0,1)$. Let us now turn to the proof of relations (\[50e\]), (\[e32\]) and (\[50b\]). We have $$\label{eq41} \forall \, \delta\in \partial({{\cal T}})^{{\cal E}}, \, j\ge 0:\quad \delta\rightarrow \delta' \Leftrightarrow \, \big[\, \delta'=\delta \vee \delta'={{\cal D}}^\rho\big].$$ Then, the definition of ${{\cal E}}$ and $\partial({{\cal T}})^{{\cal E}}$ in (\[e31\]) and the fact that ${{\cal D}}^\rho$ is absorbing, allow us to get $$\forall n\ge 0,\, \delta\in \partial({{\cal T}})^{{\cal E}}:\quad {\mathbb{P}}_\delta(Y_n=\delta)=\eta^n.$$ We have $$\label{e34} {\mathbb{P}}(\zeta>n)={\mathbb{P}}(\zeta>n, Y_n\not\in \partial({{\cal T}})^{{\cal E}})+ {\mathbb{P}}(\zeta>n, Y_n\in \partial({{\cal T}})^{{\cal E}}).$$ Since $P_{\delta, \delta'}>0$ when $\delta\rightarrow \delta'$ and there exists paths of positive probability from $\{I\}$ to $\delta\in \partial({{\cal T}})$, $\delta\neq \{I\}$ (see Remark \[remirr\]), we obtain the existence of $k_0\ge 1$ such that $$\forall \, K\in {{\cal E}}:\quad {\mathbb{P}}(\zeta_{{{\cal D}}^{\rho,K}} \le k_0)>0.$$ So, $$\alpha({{\cal E}}):= \min\{{\mathbb{P}}(\zeta^{{{\cal D}}^{\rho,K}} \le k_0): K\in {{\cal E}}\}>0.$$ Then, from the Markov property we get, $$\begin{aligned} \nonumber {\mathbb{P}}(\zeta>n)&\ge& \sum_{j=1}^{k_0}{\mathbb{P}}(\zeta^{{{\cal D}}^{\rho,K}}=j, \zeta>n)\\ \nonumber &\ge & \sum_{j=1}^{k_0}{\mathbb{P}}(\zeta^{{{\cal D}}^{\rho,K}}=j) {\mathbb{P}}_{D^{\rho,K}}(\zeta>n-j)\\ \nonumber &\ge & \sum_{j=1}^{k_0}{\mathbb{P}}(\zeta^{{{\cal D}}^{\rho,K}}=j) {\mathbb{P}}_{D^{\rho,K}}(Y_{n-j}=D^{\rho,K})\\ \nonumber &\ge & \sum_{j=1}^{k_0}{\mathbb{P}}(\zeta^{{{\cal D}}^{\rho,K}}=j) \eta^{n-j}\\ \label{e35} &\ge & \alpha({{\cal E}}) \eta^n.\end{aligned}$$ To analyze the first term at the right hand side of equality (\[e34\]) it will useful to first prove the following result, which uses the quantity $\beta_0$ defined (\[eqlast\]) which satisfies $\beta_0<\eta<1$, see (\[e33ax\]). \[lemma6\] We have, $$\label{e36} \forall\, \theta\!>\!0 \, \exists C'\!=\!C'(\theta):\quad {\mathbb{P}}(\forall j\!\le \! n: \; Y_j\not\in (\partial({{\cal T}})^{{\cal E}}\cup \{{{\cal D}}^\rho\}) \le C'(\beta_0\!+\!\theta)^n.$$ [*Proof of Lemma \[lemma6\]*]{}. Let $U=\partial({{\cal T}})\setminus (\partial({{\cal T}})^{{\cal E}}\cup \{{{\cal D}}^\rho\})$. Put $\delta_1=\{I\}$. For every $s\ge 1$ denote by $${\cal C}(U,s)=\{(\delta_1,..,\delta_s)\in U^s: \forall r\le s-1,\; \delta_r\to \delta_{r+1}\hbox{ and } \delta_r\neq \delta_{r+1}\}.$$ (So, $P_{\delta_r,\delta_{r+1}}>0$ for all $r=1,..,s-1$, see Remark \[remirr\]). We have $$\begin{aligned} &{}& {\mathbb{P}}(\forall j\le n: \; Y_j\in U)\\ &{}& =\sum_{s\ge 1}\; \sum_{(\delta_1,..,\delta_s)\in {\cal C}(U,s)} \; \prod_{r=1}^{s-1} P_{\delta_r,\delta_{r+1}} \; \left( \sum_{k_1,..,k_s\ge 0: \sum_{r=1}^s k_r=n-s} P^{k_r}_{\delta_r,\delta_r}\right). \end{aligned}$$ When $(\delta_1,..,\delta_s)\in {\cal C}(U,s)$ we have that every $\delta_k$ with $k\le s$ satisfies $P_{\delta_k,\delta_k}\le \beta_0$. On the other hand, $$\#\{(k_1,..,k_s): \forall r\le s, \; k_r\ge 0; \; \sum_{r=1}^s k_r=n\!-\!s\}=\binom{n\!-\!1}{s}.$$ Then, $${\mathbb{P}}(\forall j\le n: \; Y_j\in U)\le \sum_{s\ge 1} \binom{n\!-\!1}{s} \beta_0^{n-s} \left(\sum_{(\delta_1,..,\delta_s)\in {\cal C}(U,s)} \; \prod_{r=0}^{s-1}P_{\delta_r,\delta_{r+1}} \right).$$ We claim that there exists a constant $k^*$ such that ${\cal C}(U,s)\neq \emptyset$ implies $s\le k^*$. Let us show it. Fix an atom $L\in {{\cal D}}^\rho$. Let $(K_n: n\ge 1)$ be a sequence of sets constructed in an inductive way and satisfying the following properties: $K_1=I$; $L\subseteq K_n$ for all $n$; $K_{n+1}=K_n\cap J_n$ for some $J_n\in {{\cal J}}_\rho$ and $K_{n+1}\subset K_n$ for all $n$. Then, after a number $n_0$ of steps bounded by $\#I-\#L-1$ one necessarily has $K_{n_0}=L$ and the construction is stopped. Now, define $k^*=\sum_{ L\in {{\cal D}}^\rho}(\#I-\#L-1)$. A consequence of the above argument is that the existence of some $(\delta_1,..,\delta_s)\in {\cal C}(U,s)$ implies $s\le k^*$. So, $$C_1=\sum_{s\ge 1} \sum_{(\delta_1,..,\delta_s)\in {\cal C}(U,s)} \; \prod_{r=0}^{s-1}P_{\delta_r,\delta_{r+1}}<\infty.$$ On the other hand, for $\theta'\in (0,1)$ we have $$C_2(\theta')=\max_{s\le k^*}\; \sup_{n\ge 1} \binom{n-1}{s} (1-\theta')^{n-k^*}<\infty.$$ Then $${\mathbb{P}}(\forall j\le n: \; Y_j\in U)\le C_1\cdot C_2(\theta') \beta_0^{n-k^*}/ (1-\theta')^{n-k^*}.$$ So by taking $\theta'\in (0,1)$ such that $\beta_0/ (1-\theta')<\beta_0+\theta$ we get that the constant $$C'=(\beta_0+\theta)^{-k^*} C_1\cdot C_2(\theta')$$ makes the job in (\[e36\]). $\Box$ [*Continuation with the proof of Theorem \[theo1\]*]{}. In (\[e36\]) we will always take $\theta>0$ such that $\beta_0+\theta<\eta$. Hence, from (\[e35\]) and (\[e36\]) we find, $$\label{50a} {\mathbb{P}}(Y_n\not\in \partial({{\cal T}})^{{\cal E}}\, | \, \zeta>n)\le C'' \left((\beta_0+\theta)/\eta\right)^n\to 0 \hbox{ as } n\to \infty,$$ with $C''=C'/\alpha({{\cal E}})$. Therefore, $$\label{50f} \lim\limits_{n\to \infty}{\mathbb{P}}(Y_n\in \partial({{\cal T}})^{{\cal E}}\, | \, \zeta>n)=1.$$ Let us examine the second term at the right hand side of equality (\[e34\]). For every $K\in {{\cal E}}$ we have $$\begin{aligned} {\mathbb{P}}(\zeta>n, Y_n=D^{\rho,K})&=&\sum_{j=1}^n {\mathbb{P}}(\zeta>n, \zeta_{D^{\rho,K}}=j)\\ &=&\sum_{j=1}^n{\mathbb{P}}(\zeta_{D^{\rho,K}}=j) {\mathbb{P}}_{D^{\rho,K}}(\zeta>n-j)\\ &=&\sum_{j=1}^n{\mathbb{P}}(\zeta_{D^{\rho,K}}=j)\eta^{n-j}\\ &=&\eta^n \left(\sum_{j=1}^n\eta^{-j}{\mathbb{P}}(\zeta_{D^{\rho,K}}=j)\right).\end{aligned}$$ Since $$\begin{aligned} {\mathbb{P}}(\zeta_{D^{\rho,K}}=j)&\le& {\mathbb{P}}(\zeta^{{\cal E}}=j)\\ &\le&{\mathbb{P}}(\forall n\le j-1: \; Y_n\not\in (\partial({{\cal T}})^{{\cal E}})\cup \{{{\cal D}}^\rho\}) \le C'(\beta_0+\theta)^{j-1},\end{aligned}$$ and $\beta_0+\epsilon<\eta$, we get $$\sum_{j=1}^\infty \eta^{-j}{\mathbb{P}}(\zeta_{D^{\rho,K}}=j)<\infty.$$ Hence, $$\begin{aligned} \label{e40y} \forall K\in {{\cal E}}:\; \lim\limits_{n\to \infty}\eta^{-n}{\mathbb{P}}(\zeta>n, Y_n=D^{\rho,K})&=& \sum_{j=1}^\infty\eta^{-j}{\mathbb{P}}(\zeta_{D^{\rho,K}}=j)\\ \nonumber &=& {\mathbb{E}}\left(\eta^{-\zeta_{D^{\rho,K}}}, \zeta_{D^{\rho,K}}<\infty \right)<\infty.\end{aligned}$$ We have $$\zeta_{D^{\rho,K}}<\infty \, \Rightarrow \, \big[\, \forall K'\in {{\cal E}}\setminus \{K\}: \; \zeta_{D^{\rho,K}}=\infty \hbox{ and } \zeta^{{\cal E}}=\zeta_{D^{\rho,K}} \, \big].$$ Then, $$\{\zeta^{{\cal E}}=j\}=\bigcup_{K\in {{\cal E}}} \{\zeta_{D^{\rho,K}}=j\}$$ and the union is disjoint. Hence, $$\eta^{-\zeta^{{\cal E}}} {\bf 1}_{\zeta^{{\cal E}}<\infty}= \sum_{K\in {{\cal E}}}\eta^{-\zeta_{D^{\rho,K}}} {\bf 1}_{\zeta_{D^{\rho,K}}<\infty}.$$ Then, $${\mathbb{E}}\left(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty\right)= \sum_{K\in {{\cal E}}} {\mathbb{E}}\left(\eta^{-\zeta_{D^{\rho,K}}}, \zeta_{D^{\rho,K}} <\infty\right)<\infty.$$ Hence, from (\[e40y\]), we deduce $$\label{e40yx} \lim\limits_{n\to \infty}\eta^{-n}{\mathbb{P}}(\zeta>n, Y_n\in \partial({{\cal T}})^{{\cal E}})= {\mathbb{E}}\left(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty\right).$$ Then, relations (\[50a\]), (\[e40y\]) and (\[e40yx\]), give (\[e32\]). Now, relation (\[50e\]) is a consequence of relations (\[50f\]) and (\[e40yx\]) because they imply $$\begin{aligned} \lim\limits_{n\to \infty} \eta^{-n} {\mathbb{P}}(\zeta>n)&=& \lim\limits_{n\to \infty} \eta^{-n} {\mathbb{P}}(\zeta>n, Y_n\in \partial({{\cal T}})^{{\cal E}})\\ &=&{\mathbb{E}}(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)\in (0,\infty).\end{aligned}$$ Let us show (\[50b\]). First, assume $\delta$ is such that ${\mathbb{P}}_\delta(\zeta^{{\cal E}}<\infty)>0$. Since there is a path with positive probability from $\delta$ to some nonempty subset of $\partial({{\cal T}})^{{\cal E}}$, a similar proof as the one showing (\[50e\]) gives that $$\lim\limits_{n\to \infty} \eta^{-n} {\mathbb{P}}_\delta(\zeta>n)= {\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}},\zeta^{{\cal E}}<\infty)\in (0,\infty),$$ and so the relation (\[50b\]) is satisfied. Now, let ${\mathbb{P}}_\delta(\zeta^{{\cal E}}<\infty)=0$. Then, ${\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)=0$ and in (\[50b\]) we have ${{\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)}/ {{\mathbb{E}}(\eta^{-\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)}=0$. We claim that in this case we also have $\lim\limits_{n\to \infty} {\mathbb{P}}_\delta(\zeta>n)/{\mathbb{P}}(\zeta>n)=0$. In fact, ${\mathbb{P}}_\delta(\zeta^{{\cal E}}<\infty)=0$ implies $$\begin{aligned} (\beta_0+\theta)^{-n}{\mathbb{P}}_\delta(\zeta>n)&=& (\beta_0+\theta)^{-n}{\mathbb{P}}_\delta(\zeta>n, \zeta^{{\cal E}}>n)\\ &=&(\beta_0+\theta)^{-n} {\mathbb{P}}(\forall j\le n: Y_j\not\in (\partial(T)^{{\cal E}}\cup \{{{\cal D}}^\rho\})<\infty. \end{aligned}$$ Since $\lim\limits_{n\to \infty}\eta^{-n}{\mathbb{P}}(\zeta>n)>0$ and $\beta_0+\theta<\eta$, the claim follows and (\[50b\]) is shown. Now, let $P^*$ be the restriction of $P$ to $\partial({{\cal T}})^*$. The last statement we must show is that the vector $\varphi$ defined in (\[rev1\]) is a right eigenvector of $P^*$ with eigenvalue $\eta$. First take $\delta\in \partial({{\cal T}})^{{\cal E}}$. We have ${\mathbb{P}}_\delta(\zeta^{{\cal E}}=0)=1$ and so ${\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}},\zeta^{{\cal E}}<\infty)=1$. Since $P_{\delta,\delta'}>0$ and $\delta'\neq D^\rho$ imply $\delta'=\delta$, from $P_{\delta,\delta}=\eta$ we get $$(P^* \varphi)_\delta= \sum_{\delta':\delta'\neq D^\rho, \delta\to \delta'} P_{\delta,\delta'} {\mathbb{E}}_{\delta'}(\eta^{-\zeta^{{\cal E}}},\zeta^{{\cal E}}<\infty)=\eta =\eta\, \varphi_\delta.$$ Now let $\delta$ be such that ${\mathbb{P}}_\delta(\zeta^{{\cal E}}<\infty)=0$, so $\varphi_\delta=0$. Then $P_{\delta,\delta'}>0$ implies ${\mathbb{P}}_{\delta'}(\zeta^{{\cal E}}<\infty)=0$ and so $(P^* \varphi)_\delta=0=\eta\, \varphi_\delta$. Now take $\delta\not\in \partial({{\cal T}})^{{\cal E}}$ with ${\mathbb{P}}_\delta(\zeta^{{\cal E}}<\infty)>0$. Then, from the Markov property we get, $$\begin{aligned} \varphi_\delta&=& {\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}},\zeta^{{\cal E}}<\infty)= \sum_{\delta':\delta'\neq D^\rho, \delta\to \delta'} {\mathbb{E}}_\delta(\eta^{-\zeta^{{\cal E}}},\zeta^{{\cal E}}<\infty, Y_1=\delta')\\ &=&\sum_{\delta':\delta'\neq D^\rho, \delta\to \delta'} P_{\delta,\delta'} \; \eta^{-1}\, {\mathbb{E}}_{\delta'}(\eta^{-\zeta^{{\cal E}}},\zeta^{{\cal E}}<\infty) =\eta^{-1}\, (P^* \varphi)_\delta.\end{aligned}$$ Hence, the result is shown. This finishes the proof of the theorem. $\Box$ We will get two results from Theorem \[theo1\]. In the first one we supply the $Q-$process, which is the Markov chain that avoids some forbidden region, in our case the singleton $\{\otimes_{L\in {{\cal D}}^\rho}\mu_L\}$. In the second one we give a class of quasi-stationary distributions, that must be compared with the irreducible case where there is a unique one. The $Q-$process in branching process was introduced in Section I.D.14 in [@an]. In [@cms] it can be found the construction of the $Q-$process for Markov chains and dynamical systems. In the sequel it is convenient to denote by $\partial({{\cal T}})^*=\partial({{\cal T}})\setminus \{D^\rho\}$ and by $P^*$ the restriction of $P$ to $\partial({{\cal T}})^*$. \[cor1\] The following limit exists $$\lim\limits_{n\to \infty}{\mathbb{P}}(Y_i=\delta_i, i=1,..,j \, | \, \zeta>n)$$ for all $\delta_i\in \partial({{\cal T}})\setminus \{D^\rho\}$, $i=1,..,k$, and it vanishes if some $\delta_i$ satisfies ${\mathbb{P}}_{\delta_i}(\zeta^{{\cal E}}<\infty)=0$. Denote $$\partial(\zeta^{{\cal E}})=\{\delta\in \partial({{\cal T}})^*: {\mathbb{P}}_{\delta}(\zeta^{{\cal E}}<\infty)>0\}.$$ Then, the matrix $Q=\left(Q_{\delta,\delta'}: \delta,\delta'\in \partial(\zeta^{{\cal E}})\right)$ given by $$Q_{\delta,\delta'}=\eta^{-1}\, P_{\delta,\delta'}\frac{{\mathbb{E}}_{\delta'}(\eta^{\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)}{{\mathbb{E}}_\delta(\eta^{\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)},$$ is an stochastic matrix on $\partial(\zeta^{{\cal E}})$, and it is satisfied $$\forall \delta_i\in \partial(\zeta^{{\cal E}}), i=0,..,j: \quad \lim\limits_{n\to \infty}{\mathbb{P}}_{\delta_0}(Y_i=\delta_i, i=1,..,j \, | \, \zeta>n)= \prod_{i=0}^{j-1} Q_{\delta_i,\delta_{i+1}}.$$ That is, $Q$ is the transition matrix of the Markov chain that never hits $\otimes_{L\in {{\cal D}}^\rho}\mu_L$. Let us prove that $Q$ is an stochastic matrix. Let $\varphi$ be the right eigenvector of $P^*$ with eigenvalue $\eta^{-1}$ given in (\[rev1\]). We have that $\varphi_\delta$ vanishes when ${\mathbb{P}}_{\delta}(\zeta^{{\cal E}}<\infty)=0$. Let $\delta\in \partial(\zeta^{{\cal E}})$. We will use that $P_{\delta,\delta'}=0$ if $\delta\not\rightarrow \delta'$ and that $${\mathbb{P}}_{\delta'}(\zeta^{{\cal E}}<\infty)=0 \hbox{ implies } \frac{\varphi_{\delta'}}{\varphi_\delta}=\frac{{\mathbb{E}}_{\delta'}(\eta^{\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)}{{\mathbb{E}}_\delta(\eta^{\zeta^{{\cal E}}}, \zeta^{{\cal E}}<\infty)}=0.$$ Hence $$\sum_{\delta'\in \partial(\zeta^{{\cal E}})}Q_{\delta,\delta'}= \eta^{-1}\left(\sum_{\delta'\in \partial(\zeta^{{\cal E}})} P_{\delta,\delta'}\frac{\varphi_{\delta'}}{\varphi_{\delta}}\right) =\eta^{-1} \left( \sum_{\delta'\in \partial({{\cal T}})^*} P_{\delta,\delta'}\frac{\varphi_{\delta'}}{\varphi_{\delta}}\right)=1.$$ The last equality because $\varphi$ is a right eigenvector with eigenvalue $\eta$. Now, from the Markov property we obtain for $n>j$, $${\mathbb{P}}(Y_i=\delta_i, i=1,..,j \, | \, \zeta>n)= {\mathbb{P}}(Y_i=\delta_i, i=1,..,j)\frac{{\mathbb{P}}_{\delta_j}(\zeta>n-j)}{{\mathbb{P}}(\zeta>n)}.$$ Now we use the ratio limit result (\[50b\]). This limit vanishes if ${\mathbb{P}}_{\delta_j}(\zeta^{{\cal E}}<\infty)=0$. Also it vanishes when ${\mathbb{P}}_{\delta_i}(\zeta^{{\cal E}}<\infty)=0$ for some $i<j$ because $P_{\delta_i,\delta_{i+1}}>0$ implies ${\mathbb{P}}_{\delta_{i+1}}(\zeta^{{\cal E}}<\infty)=0$. Finally, let $\delta_i\in \partial(\zeta^{{\cal E}})$ for $i=0,..,j$. We have $$\begin{aligned} \nonumber &{}&\lim\limits_{n\to \infty}{\mathbb{P}}_{\delta_0} (Y_i=\delta_i, i=1,..,j \, | \, \zeta>n)\\ \nonumber &{}& = \lim\limits_{n\to \infty} {\mathbb{P}}_{\delta_0}(Y_i=\delta_i, i=1,..,j) \frac{{\mathbb{P}}_{\delta_j}(\zeta>n-j)} {{\mathbb{P}}_{\delta_0}(\zeta>n)}\\ \label{rats} &{}&={\mathbb{P}}_{\delta_0}(Y_i=\delta_i, i=1,..,j) \frac{\varphi_{\delta_j}}{\varphi_{\delta_0}}\eta^{-j}\\ \nonumber &{}& =\prod_{l=0}^{j-1} \left(\eta^{-1}P_{\delta_l, \delta_{l+1}}\, \frac{\varphi_{\delta_{l+1}}}{\varphi_{\delta_l}}\right).\end{aligned}$$ In (\[rats\]) we used $\lim\limits_{n\to \infty}{\mathbb{P}}(\zeta>n-j)/{\mathbb{P}}(\zeta>n)=\eta^{-j}$, which is a consequence of (\[50e\]). Then the result follows. Let $\nu=(\nu_\delta: \delta\in \partial({{\cal T}})^*)$ be a probability measure on $\partial({{\cal T}})^*$. If necessary, $\nu$ will be identified with its extension on $\partial({{\cal T}})$ with $\nu_{D^\rho}=0$. We say that $\nu$ is supported by some subset ${\widetilde{\partial}}\subseteq \partial({{\cal T}})^*$ if $\nu({\widetilde{\partial}})=1$. We denote by $\nu'$ the row vector associated to $\nu$. \[cor2\] Every probability measure $\nu$ on $\partial({{\cal T}})^*$ supported on $\partial({{\cal T}})^{{\cal E}}$ satisfies $\nu'P^*=\eta \, \nu'$ and it is a quasi-stationary distribution, that is it satisfies $$\label{e43} \forall n\ge 1, \, \forall \delta\in \partial({{\cal T}})^{{\cal E}}: \quad {\mathbb{P}}_\nu(Y_n=\delta \, | \, \zeta>n)=\nu_\delta.$$ Moreover, if for some $a\le \eta$ we have ${{\cal E}}(a)=\{K\in {{\cal Y}}({{\cal J}}_\rho): \rho^K_K=a\}\neq \emptyset$, then any probability measure ${\widetilde{\nu}}$ supported on $\partial({{\cal T}})^{{{\cal E}}(a)}=\{{{\cal D}}^{\rho,K}: K\in {{\cal E}}(a)\}$ satisfies ${\widetilde{\nu}}'P^*=a\, {\widetilde{\nu}}'$ and it is a quasi-stationary distribution, $$\label{e44} \forall n\ge 1, \, \forall \delta\in \partial({{\cal T}})^{{{\cal E}}(a)}: \quad {\mathbb{P}}_{{\widetilde{\nu}}}(Y_n=\delta \, | \, \zeta>n)={\widetilde{\nu}}_\delta.$$ With the above notation and by using (\[eq41\]) we get, $$(\nu' P^*)_\delta= P_{\delta,\delta} \, \nu_\delta=\eta \, \nu_\delta,$$ so $\nu' P^*=\eta \nu'$. By iteration we find $\nu' P^{*n}=\eta^n \, \nu'$. Note that this is equivalent to $$(\nu' P^{*n})_\delta={\mathbb{P}}_{\nu}(Y_n=\delta)={\mathbb{P}}_{\nu}(\forall j\le n \; Y_j=\delta) =\eta^n \, \nu'_\delta.$$ Now $${\mathbb{P}}_\nu(\zeta>n)= \sum_{\delta\in \partial({{\cal T}})^{{\cal E}}} (\nu' P^{*n})_\delta=\eta^n \left(\sum_{\delta\in \partial({{\cal T}})^{{\cal E}}} \nu_\delta\right)=\eta^n.$$ Hence, relation (\[e43\]) is proven. The proof of (\[e44\]) is completely similar. This is analogous for positive eigenvectors. Let $\widetilde{\partial}\subseteq \partial({{\cal T}})^{{\cal E}}$ be a nonempty set, then the characteristic function ${\bf 1}_{\widetilde{\partial}}$ is a right eigenvector of $P^*$ with eigenvalue $\eta$. Also, if $\widetilde{\partial}\subseteq \partial({{\cal T}})^{{{\cal E}}(a)}$ is a nonempty subset for some $a\le \eta$, then ${\bf 1}_{\widetilde{\partial}}$ is a right eigenvector of $P^*$ with eigenvalue $a$. We notice that an analogous of the $Q-$process construction can be written on the class of states $\delta$’s that verify ${\mathbb{P}}_\delta(\zeta^{{\cal E}}<\infty))=0$ and ${\mathbb{P}}_\delta(\zeta_{{\widetilde{\partial}}}<\infty)>0$, being ${\widetilde{\partial}}=\{\delta\in \partial({{\cal T}})^*: P_{\delta,\delta}=\beta_0\}$. \[rem2a\] The results we have obtained can be easily extended to general probability spaces. In fact, let $((X_i,{{\cal B}}_i): i\in I)$ be a finite collection of measurable spaces and $(\prod_{i\in I}X_i,\otimes_{i\in I} {{\cal B}}_i,\mu)$ be a probability space. For $J\in {\mathbb{S}}$, let ${{\cal B}}_J=\otimes_{i\in J} {{\cal B}}_i$ be the product $\sigma$-field on $\prod_{i\in J}X_i$ and $\mu_J$ be the marginal on $(\prod_{i\in J}X_j, {{\cal B}}_J)$, $$\mu_J(V)=\mu\left(V\times \,\prod_{i\in I\setminus J}X_j\right) \hbox{ for } V\in {{\cal B}}_J.$$ Then, introduce the partitions on $I$ as in Sections \[sec1\] and \[sec2\] and as done in Definition \[def1\], for a probability vector $\rho=(\rho_J: J\in {\mathbb{S}}^{(\emptyset)})$ define $\Xi[\mu]=\rho_I\, \mu+ \sum_{J\in {\mathbb{S}}} \rho_J\, \mu_J\otimes \mu_{J^c}$. Then, all the results of this paper, in particular Theorem \[theo0\] and Theorem \[theo1\], can be written in this setting. The unique thing one must take care is to replace the expression $\sum\limits_{x_{K}\in \prod_{i\in K} A_i} \mu_K(x_K) g(x_k,x_{K^c})$ by $\int\limits_{\prod_{i\in K} X_i} g(x_k,x_{K^c}) d\mu_K$ when it is required, for instance in Lemma \[lemma3\]. \[rem3a\] The atoms of the partition ${{\cal D}}^\rho$ can always be assumed to be singletons, that is ${{\cal D}}^\rho={{\cal D}}_{si}$. We have not done it because on one hand, there is no substantial gain in notation, and on the other hand, this can be made only a posteriori because the input is the vector $\rho=(\rho_J: J\in {\mathbb{S}}^{(\emptyset)})$ and the atoms are obtained once computing the set of nonempty intersections (\[e7\]) from the support ${{\cal J}}_\rho$, the atoms are the minimal elements, so they satisfy (\[e8\]). [**Final comment**]{}. As already said, all we have done does not require the operation $\otimes$ to be the product between probability measures. As it can be checked, the results can be extended to any operation $\otimes$ defined in the domains (\[eab0\]) that satisfies commutativity, associativity (\[eab4\]), stability under restriction (\[eab\]) and $\mu_\emptyset$ is the identity element. In particular, commutativity and associativity imply that for any partition the probability measure $\otimes_{K\in \delta} \mu_K$ is well-defined. Obviously, the partition $\otimes_{L\in {{\cal D}}^\rho} \mu_L$ could have a meaning different from a product measure of the marginals on the atoms, but it continue to play the same central role in all our constructions and results. [**Acknowledgments**]{}. We thank support from the CMM Basal CONICYT Project PB-03. We acknowledge discussions with Dr. Thierry Huillet from CNRS and the hospitality of the Laboratoire de Physique Théorique et Modélisation at the Université de Cergy-Pontoise, where this work was started. [99]{} K. Athreya, P. Ney. Branching processes, 287 p. 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On quasi-stationary distribution in absorbing discrete-time finite Markov chains. Journal of Applied Probability, [**2**]{}, p. 88-100 (1965). K.J. Dawson. The decay of linkage disequilibria under random union of gametes. How to calculate Bennett’s principal components. Theor. Popul. Biol. [**58**]{}, p. 1-20 (2000). K.J. Dawson. The evolution of a population under recombination: How to linearise the dynamics. Linear Algebra Appl. [**348**]{}, p. 115-137 (2002). H. Geiringer. On the probability theory of linkage in Mendelian heredity. Ann. Math. Stat. [**15**]{}, p. 25-57 (1944). R. Lande, S. Engen, B. Saether. Stochastic population dynamics in ecology and conservation. Oxford Series in Ecology and Evolution, Oxford University Press, New York (2003). P. Pollett. Research Communications www.maths.uq.edu.au/ pkp/publist.html. U. von Wangenheim. Single-crossover recombination and ancestral recombination trees. PhD. Thesis, U. Bielelfeld (2011). U. von Wangenheim, E. Baake, M. Baake. Single-crossover recombination in discrete time. J. Math. Biol. [**60**]{}, No. 5, 727-760 (2010). SERVET MARTÍNEZ [*Departamento Ingenier[í]{}a Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile.*]{} e-mail: [email protected] \[lastpage\]

--- abstract: 'We Define moments of partitions of integers, and show that they appear in higher order derivatives of certain combinations of functions.' author: - Shaul Zemel title: Moments of Partitions and Derivatives of Higher Order --- Introduction and Statement of the Main Result {#introduction-and-statement-of-the-main-result .unnumbered} ============================================= Changes of coordinates grew, through the history of mathematics, from a powerful computational tool to the underlying object behind the modern definition of many objects in various branches of mathematics, like differentiable manifolds or Riemann surfaces. With the change of coordinates, all the objects that depend on these coordinates change their form, and one would like to investigate their behavior. For functions of one variable, like holomorphic functions on Riemann surfaces, this is very easy, but one may ask what happens to the derivatives of functions under this operation. The answer is described by the well-known formula of Faà di Bruno for the derivative of any order of a composite function. For the history of this formula, as well as a discussion of the relevant references, see [@[J]]. For phrasing Faà di Bruno’s formula, we recall that a partition $\lambda$ of some integer $n$, denoted by $\lambda \vdash n$, is defined to be a finite sequence of positive integers, say $a_{l}$ with $1 \leq l \leq L$, written in decreasing order, whose sum is $n$. The number $L$ is called the *length* of $\lambda$ and is denoted by $\ell(\lambda)$, and given a partition $\lambda$, the number $n$ for which $\lambda \vdash n$ is denoted by $|\lambda|$. Another method for representing partitions, which will be more useful for our purposes, is by the *multiplicities* $m_{i}$ with $i\geq1$, which are defined by $m_{i}=\big|\;\{1 \leq l \leq L|a_{l}=i\}\big|$, with $m_{i}\geq0$ for every $i\geq1$ and such that only finitely many multiplicities are non-zero. In this case we have $|\lambda|=\sum_{i\geq1}im_{i}$ and $\ell(\lambda)=\sum_{i\geq1}m_{i}$. Note that the empty partition, in which all the multiplicities $m_{i}$ vanish, is allowed. It is considered to be partition of 0, with length 0. Therefore when some partition $\lambda$ is known from the context, the numbers $m_{i}$ will denote the associated multiplicities, and in case several partitions are involved we may write $m_{i}(\lambda)$ for clarification. Assume that $f$ is a function of $z$ and the variable $z$ is a function of another variable $t$, say $z=\varphi(t)$, and we wish to differentiate the resulting function of $t$ successively. The formula of Faà di Bruno is the answer to this question, which we can write explicitly as $$(f\circ\varphi)^{(n)}(t)=\frac{d^{n}}{dt^{n}}\big((f(\varphi(t)\big)=\sum_{\lambda \vdash n}\frac{n!}{\prod_{i=1}^{n}(i!)^{m_{i}}m_{i}!}f^{(\ell(\lambda))}\big(\varphi(t)\big)\prod_{i=1}^{n}\big(\varphi^{(i)}(t)\big)^{m_{i}}. \label{FaadiBruno}$$ We remark that gathering these formulae for all $n$ together, and noticing that $\lambda$ appears in the derivative of order $|\lambda|$, yields a structure of a Hopf algebra on the polynomial ring of infinitely many variables, graded appropriately—see, e.g., [@[FGV]]. Equation can be viewed as describing the behavior of derivatives of functions on 1-dimensional objects (like Riemann surfaces, when the variables are locally taken from $\mathbb{C}$) under changing the coordinate. However, functions are not the only type of forms that can be defined on 1-dimensional objects, and the next forms to consider are differentials, and more generally $q$-differentials. These are defined such that their coordinate changes also involve the $q$th power of the derivative of the coordinate change, namely if a $q$-differential is expressed in a coordinate neighborhood as $f(z)$ times the formal symbol $(dz)^{q}$, then when we change the coordinate via $z=\varphi(t)$ the description in the coordinate $t$ is $f\big(\varphi(t)\big)\varphi'(t)^{q}$ times $(dt)^{q}$ (see, e.g., Section III.4.12 of [@[FK]]). While simply differentiating such expressions may seem a bit unnatural, this operation does appear, for example, in the proof of Proposition III.5.10 of [@[FK]], which states that if $d$ is the dimension of the space of $q$-differentials on a Riemann surface $X$ then the Wronskian of this space is an $m$-differential, where $m=\frac{d}{2}(d+2q-1)$. While the proof of the latter statement takes only the “essential terms” of this derivative, where no combinatorial calculations have to be carried out, it does leave open the question about the formula for the $n$th derivative of such a transformation rule, and whether some interesting combinatorial phenomena hide in it. The dependence on $q$ as a number becomes formal, and the expression that we investigate in this manner is the $n$th derivative of an expression like $f\big(\varphi(t)\big)g\big(\varphi'(t)\big)$, or just $(f\circ\varphi)\cdot(g\circ\varphi')$ when we omit the variable $t$. In fact, $g$ needs not be composed with the first derivative of $\varphi$, but can rather be composed with the derivative $\varphi^{(s)}$ of any order $s\geq0$. The question that we tackle in this paper is therefore finding an explicit formula for the $n$th derivative of the expression $(f\circ\varphi)\cdot\big(g\circ\varphi^{(s)}\big)$, in terms of the derivatives of $f$, $g$, and $\varphi$. The fact that the formula, which is given in Equation below, involves partitions, is, of course, no big surprise. But in addition to the combinatorial coefficients appearing in Faà di Bruno’s formula from Equation , the resulting coefficient involves some numbers that we call *moments* of partitions. More precisely, given an integer $k\geq1$ and a partition $\lambda$, with the summands $a_{l}$, $1 \leq l\leq\ell(\lambda)$ and the multiplicities $m_{i}$, we define its *$k$th moment* to be $$p_{k}(\lambda)=\sum_{l=1}^{\ell(\lambda)}a_{l}^{k}=\sum_{i\geq1}i^{k}m_{i}.$$ In particular the first moment of $\lambda$ is just $|\lambda|$ by definition. The notation $p_{k}$ comes from the theory of symmetric functions, as this moment is the value attained by the $k$th power sum function on the numbers $a_{l}$, $1 \leq l\leq\ell(\lambda)$. However, there are several natural bases for the ring of symmetric functions, and in particular one can take the basis arising from the *elementary symmetric functions* $\{e_{r}\}_{r=0}^{\infty}$, which appear, e.g., in the expressions for the coefficients of a polynomial in terms of its roots. We shall therefore denote by $e_{r}(\lambda)$ the *$r$th elementary moment* of $\lambda$, which is obtained by substituting the $a_{l}$s into the $r$th elementary symmetric function $e_{r}$. Note that every symmetric function with index 0 is the constant 1, so that the 0th moment of every partition is 1 (even though the formula for $p_{k}$ above would give $\ell(\lambda)$ when $k=0$). An interesting feature of the resulting formula is that for expressing the coefficient associated with $\lambda$, we first have to modify $\lambda$ in two different directions, and take the elementary moments of this modification. More explicitly, given an integer $s\geq0$ and a partition $\lambda$, we shall denote by $\lambda^{>s}$ the *$s$th truncation* of $\lambda$, which is obtained by eliminating any number $a_{l}$ which satisfies $a_{l} \leq s$, or equivalently by setting each $m_{i}$ with $i \leq s$ to 0 and leaving the multiplicities $m_{i}$ with $i>s$ at their value $m_{i}(\lambda)$. Note that this operation may transform some non-trivial partitions into the trivial one, all the moments of positive indices of which vanish by definition. In addition, for every partition $\mu$ and integer $s\geq0$ we denote by $(\mu)_{s}$ the partition obtained by replacing each number $a_{l}$ by its *Pochhammer symbol* $(a_{l})_{s}=\prod_{\upsilon=0}^{s-1}(a_{l}-\upsilon)=\frac{a_{l}!}{(a_{l}-s)!}$ (the latter equality holding also when $0 \leq a_{l}<s$, since then the numerator is finite and the denominator is infinite, but we shall use it for $\mu=\lambda^{>s}$ where no such indices appear). Using this notation, our main result states that the $n$th derivative of $(f\circ\varphi)\cdot\big(g\circ\varphi^{(s)}\big)$ is given by $$\sum_{r=0}^{n}\sum_{\substack{\lambda \vdash n+rs \\ \ell(\lambda^{>s}) \geq r}}\frac{n!e_{r}\big((\lambda^{>s})_{s}\big)}{\prod_{i=1}^{n}(i!)^{m_{i}}m_{i}!}\big(f^{(\ell(\lambda)-r)}\circ\varphi\big)\big(g^{(r)}\circ\varphi^{(s)}\big)\prod_{i=1}^{n}\big(\varphi^{(i)}\big)^{m_{i}}, \label{finres}$$ where $m_{i}=m_{i}(\lambda)$ are the multiplicities associated with the partition $\lambda$. An immediate corollary is that the coefficients from Equation are integers, a fact that is much less obvious than the integrality of the coefficients from Equation , which have combinatorial interpretations (these are also the coefficients appearing in the summands with $r=0$ in Equation ). In addition, we deduce a combinatorial identity involving these moments of partitions, by comparing the expression from Equation with the one arising from combining Leibnitz’s Rule with Equation for evaluating the $n$th derivative in question. The rest of the paper is divided into 3 sections. Section \[FdBSec\] presents a (well-known) proof of the formula of Faà di Bruno’s from Equation , the ideas of which will be later used for proving the main result. Section \[Propofer\] establishes some properties of the elementary symmetric functions that we shall need, and the Section \[Main\] proves Equation and deduces some consequences. A Proof of Faà di Bruno’s Formula \[FdBSec\] ============================================ We will prove our main result by induction on $n$, like one of the many proofs of Faà di Bruno’s formula, which we shall give here in Proposition \[FdBprop\]. The reason we include this proof is for introducing the tools that we shall use for the main result below. First we introduce a notation that will help us avoid undefined terms. Recall the Kronecker’s $\delta$-symbol $\delta_{x,y}$, which is defined to be 1 when $x=y$ and 0 otherwise. Following our previous paper [@[Z]], we shall use the complementary symbol $\overline{\delta}_{x,y}=1-\delta_{x,y}$, which equals 0 when $x=y$ and 1 otherwise. In addition, we make the following definition of partitions. Let an integer $j\geq1$ and a partition $\lambda$ be given, and assume that $m_{j}(\lambda)\geq1$. We denote by $\lambda-\varepsilon_{j}$ the partition obtained by omitting one of the instances of $j$ (i.e., by deleting one of the numbers $a_{l}$ which equals $j$, and re-indexing). We also write $\lambda_{j}$ for the partition obtained by subtracting 1 from one of the numbers $a_{l}$ that equal $j$, and then deleting trivial terms and again re-indexing in decreasing order. \[lambdajdef\] The partitions from Definition \[lambdajdef\] have the parameters given in the following simple lemma. For the partition $\lambda-\varepsilon_{j}$ we have $$|\lambda-\varepsilon_{j}|=|\lambda|-j,\ \ell(\lambda-\varepsilon_{j})=\ell(\lambda)-1,\mathrm{\ and\ }m_{i}(\lambda-\varepsilon_{j})=m_{i}-\delta_{i,j}\mathrm{\ for\ }i\geq1.$$ On the other hand, for $\lambda_{j}$ we get $$|\lambda_{j}|=|\lambda|-1,\ \ell(\lambda_{j})=\ell(\lambda)-\delta_{j,1},\mathrm{\ and\ }m_{i}(\lambda_{j})=\begin{cases}m_{i}(\lambda)-1 & i=j \\ m_{i}(\lambda)+1 & i=j-1\geq1 \\ m_{i}(\lambda) & \mathrm{otherwise}. \end{cases}$$ \[lambdaj\] Note that when $j=1$ we have $\lambda-\varepsilon_{1}=\lambda_{1}$ in Definition \[lambdajdef\], and the two lines from Lemma \[lambdaj\] coincide. In addition, we shall make the convention, which is appropriate by our definition, that $m_{0}(\lambda)=0$ for every partition $\lambda$ (this is why we wrote $i=j-1\geq1$ in the second case in Lemma \[lambdaj\]—we do not want $m_{0}(\lambda_{1})$ to be 1). This will be convenient for many statements below. We begin with establishing the combinatorial identity behind one of the proofs of Faà di Bruno’s formula. While this proof is well-known, it contains the ideas that will be used later for proving the main result as well. We denote the combinatorial coefficient from Equation by $C_{\lambda,r}^{(s)}$, so that the one from Equation is $C_{\lambda,0}^{(s)}$, regardless of the value of $s$. If $\lambda$ is any partition such that $|\lambda|>0$, with multiplicities $\{m_{i}\}_{i\geq1}$, then the coefficient $C_{\lambda,0}^{(s)}$ can be written as $\sum_{j\geq1}(m_{j-1}+1)\overline{\delta}_{m_{j},0}C_{\lambda_{j},0}^{(s)}$. \[recr0\] Lemma \[lambdaj\] shows that for every $j\geq1$ with $m_{j}\geq1$ the denominator of $C_{\lambda_{j},0}^{(s)}$ is the same as that of $C_{\lambda,0}^{(s)}$, except that $j!$ now appears only to the power $m_{j}-1$ and we have $(m_{j}-1)!$ instead of $m_{j}!$, and if $j\geq2$ then $(j-1)!$ comes with the power $m_{j-1}+1$ and the denominator also contains $(m_{j-1}+1)!$. The multiplier $m_{j-1}+1$ is trivial when $j=1$ and cancels the latter factorial to $m_{j-1}!$ otherwise, and after we multiply the numerator and denominator by $jm_{j}$, we also get the required denominator $m_{j}!$, and the powers of $j!$, as well as of $(j-1)!=\frac{j!}{j}$ when $j\geq2$, become the correct ones as well. This shows that after multiplying by the denominator of the left hand side, the right hand side becomes $$\sum_{j\geq1}|\lambda_{j}|!jm_{j}\overline{\delta}_{m_{j},0}=(|\lambda|-1)!\sum_{j\geq1}jm_{j}=(|\lambda|-1)!\cdot|\lambda|=|\lambda|!,$$ where the expression on the left is obtained via Lemma \[lambdaj\] again, the first equality is based on the fact that $\overline{\delta}_{m_{j},0}$ vanishes only when the multiplier $m_{j}$ vanishes and can therefore be ignored, and we then use the definition of $|\lambda|$. As this is the numerator on the left hand side as well, this proves the lemma. The details of the proof of Lemma \[recr0\] will be useful for the proof of the main result. As the latter proof will work inductively, we provide the full proof of Equation , since we shall use these arguments as well. The formula from Equation is valid for every $n$, i.e., for any $n\geq0$ we have $$(f\circ\varphi)^{(n)}=\sum_{\lambda \vdash n}C_{\lambda,0}^{(s)}\big(f^{(\ell(\lambda))}\circ\varphi\big)\prod_{i=1}^{n}\big(\varphi^{(i)}\big)^{m_{i}}.$$ \[FdBprop\] We argue by induction on $n$, where the case with $n=0$, consisting only of the empty partition with $C_{\lambda}=1$, is a tautology. Assuming that $n>0$ and that the result holds for $n-1$, we have to differentiate the result for $n-1$ with respect to $t$ and show that it gives the asserted expression. Now, for any $\mu \vdash n-1$ we can first differentiate $f^{(\ell(\mu))}\circ\varphi$ and get $f^{(\ell(\mu)+1)}\circ\varphi$ times $\varphi'$, which (up to the coefficient) corresponds to the partition of $n$ obtained by adding another number $a_{\ell(\mu)+1}=1$ to $\mu$, and we write $j=1$ in this case. We can also differentiate one of the other multipliers, which we write as $\varphi^{(j-1)}$ for some $j\geq2$, and render it $\varphi^{(j)}$, yielding (again up to the coefficient) a term like the one associated with the partition of $n$ in which one of the $a_{l}$s of $\mu$ which equal $j-1$ was increased to $j$ (there are $m_{j-1}(\mu)$ such numbers, and indeed this operation comes with multiplicity $m_{j-1}(\mu)$ because of the power to which $\varphi^{(j-1)}$ appears in the expression from the induction hypothesis). In any case the resulting partition $\lambda$ satisfies $m_{j}(\lambda)\geq1$, and the contributing partition $\mu$ is $\lambda_{j}$ from Definition \[lambdajdef\] (we write $\lambda_{1}$ also for $\lambda-\varepsilon_{1}$ in the case with $j=1$). Hence we indeed obtain a sum over $\lambda \vdash n$ of the required expressions, and we need to verify the coefficients. But given such $\lambda$, we get contributions exactly from those $\lambda_{j}$ for which $m_{j}(\lambda)\geq1$, and Lemma \[lambdaj\] shows that the contribution to the coefficient is $C_{\lambda_{j},0}^{(s)}$ times $m_{j-1}(\lambda)+1$ (also for $j=1$, where the latter multiplier is indeed 1). Therefore the resulting coefficient is the one from the right hand side of Lemma \[recr0\], which is the desired one by this lemma. This proves the proposition. Some Properties of the Functions $e_{r}$ \[Propofer\] ===================================================== The coefficients $C_{\lambda,r}^{(s)}$ do depend on $s$, and also involve the non-trivial elementary symmetric functions $\{e_{r}\}_{r\geq1}$ (the function $e_{0}$ appearing in $C_{\lambda,0}^{(s)}$ is just 1). We now present some of their properties that we shall need. Most of the material can be found in many places in the literature, e.g., Section 2 in Chapter 1 of [@[M]], but we include it here for completeness and since it is short and simple. We first consider what happens to the elementary symmetric function $e_{r}$, of the (finitely many) numbers $b_{l}$ with $1 \leq l \leq L$, say, when we subtract some number $c$ from one particular number $b_{l}$. This includes, as a special case, the formula for omitting $b_{l}$, where we simply take $c=b_{l}$. For any $r\geq0$, write $e_{r}$ for $e_{r}(b_{1},\ldots,b_{L})$ for some numbers $b_{l}$, $1 \leq l \leq L$. Then replacing $b_{l}$ for one index $l$ by $b_{l}-c$ sends $e_{r}$ to the expression $e_{r}-c\sum_{k=1}^{r}(-b_{l})^{k-1}e_{r-k}$, which for $c=b_{l}$ becomes $\sum_{k=0}^{r}(-b_{l})^{k}e_{r-k}$. \[ertrans\] The formula expressing the coefficients of a polynomial using its roots transforms, by a simple operation, to the equality $$\sum_{r=0}^{L}e_{r}X^{r}=\prod_{l=1}^{L}(1+b_{l}X). \label{prodfore}$$ Our operation replaces $b_{l}$ by $b_{l}-c$, so that we need the coefficients of the power series in $X$ obtained by multiplying by $\frac{1+(b_{l}-c)X}{1+b_{l}X}$. Since after expanding the denominator geometrically this multiplier becomes $1-c\sum_{k=1}^{\infty}(-b_{l})^{k-1}X^{k}$, multiplying by the left hand side gives the series with the asserted coefficients. The case with $c=b_{l}$ is now immediate. This proves the lemma. Another well-known identity that we shall need is the following one. With $e_{r}$ as in Lemma \[ertrans\], and with $p_{k}$ defined to be $p_{k}(b_{1},\ldots,b_{L})$ as well, the sum $\sum_{k=1}^{r}(-1)^{k-1}p_{k}e_{r-k}$ equals $re_{r}$ for every $r\geq1$. \[epiden\] Take the logarithm of Equation , and substitute the series for $\log(1-z)$ in the right hand side. The right hand side then becomes $\sum_{k=1}^{\infty}(-1)^{k-1}\frac{p_{k}}{k}X^{k}$, and after differentiating we get $\sum_{k=1}^{\infty}(-1)^{k-1}p_{k}X^{k-1}$. But differentiating the logarithm of the left hand side gives the quotient between $\sum_{r\geq1}re_{r}X^{r-1}$ and $\sum_{r\geq0}e_{r}X^{r}$, and after we multiply by the denominator, the result follows by comparing the coefficient of $X^{r-1}$ on both sides. This proves the lemma. We remark that the result of Lemma \[epiden\] is trivially true also for $r=0$, but we shall need it only for $r\geq1$. We can now establish the extension of Lemma \[recr0\] to $r\geq1$. For every $s\geq0$, $r\geq0$ and partition $\lambda$, we have the equality $$C_{\lambda,r}^{(s)}=\sum_{j\geq1}(m_{j-1}+1)\overline{\delta}_{m_{j},0}C_{\lambda_{j},r}^{(s)}+\overline{\delta}_{r,0}\overline{\delta}_{m_{s+1},0}C_{\lambda-\varepsilon_{s+1},r-1}^{(s)}.$$ \[recrpos\] The case $r=0$ is simply the equality from Lemma \[recr0\] (because $\overline{\delta}_{r,0}$ vanishes), so that we may assume $r\geq1$. The same argument from the proof of that lemma, and the fact that the numerators in all the terms on the right hand side involve $(n-1)!$ (by Lemma \[lambdaj\]), reduce us to proving the equality $$\sum_{j\geq1}jm_{j}e_{r}\big((\lambda_{j}^{>s})_{s}\big)+(s+1)!m_{s+1}e_{r-1}\big((\lambda^{>s}-\varepsilon_{s+1})_{s}\big)=ne_{r}\big((\lambda^{>s})_{s}\big) \label{sumnum}$$ (because the denominator associated with $\lambda-\varepsilon_{s+1}$ misses one power of $(s+1)!$ and one coefficient of $m_{s+1}$ to become equal to that of $\lambda$). Now, it is easy to verify that the partition $\lambda_{j}^{>s}$ is the same as $\lambda^{>s}$ when for $j \leq s$, it coincides with $\lambda^{>s}-\varepsilon_{s+1}$ for $j=s+1$, and it equals $(\lambda^{>s})_{j}$ when $j>s+1$. It follows that $(\lambda_{j}^{>s})_{s}$ is $(\lambda^{>s})_{s}$ in the first case, $(\lambda^{>s})_{s}-\varepsilon_{(s+1)_{s}}$ in the second one, and it obtained from $(\lambda^{>s})_{s}$ by replacing one instance of $(j)_{s}$ by $(j-1)_{s}$. We may therefore evaluate the summands with $j>s$ on the left hand side of Equation via Lemma \[ertrans\]. For $j=s+1$ we obtain $$(s+1)m_{s+1}\big[e_{r}\big((\lambda^{>s})_{s}\big)-(s+1)_{s}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big)\big],$$ and as the remaining sum becomes $$(s+1)!m_{s+1}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big)$$ after a summation index change, where $(s+1)!=(s+1)_{s}=(s+1)(s)_{s-1}$, the missing factor of $s+1$ in the latter equation in comparison with the preceding one shows that they combine to $$(s+1)m_{s+1}\big[e_{r}\big((\lambda^{>s})_{s}\big)-s(s)_{s-1}\sum_{k=1}^{r}\big(-(s+1)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big)\big]$$ (this is also valid when $s=0$, where the sum over $k$ is indeed multiplied by a vanishing coefficient). On the other hand, when $j>s+1$ we observe that $(j)_{s}-(j-1)_{s}=s(j-1)_{s-1}$ (once again this equality holds also when $s=0$, in the form of $1-1=0$), so that we get $$jm_{j}\big[e_{r}\big((\lambda^{>s})_{s}\big)-s(j-1)_{s-1}\sum_{k=1}^{r}\big(-(j)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big)\big],$$ of which the latter equality is the case with $j=s+1$. Altogether the left hand side of Equation equals $$\sum_{j\geq1}jm_{j}e_{r}\big((\lambda^{>s})_{s}\big)-\sum_{j>s}sjm_{j}(j-1)_{s-1}\sum_{k=1}^{r}\big(-(j)_{s}\big)^{k-1}e_{r-k}\big((\lambda^{>s})_{s}\big). \label{LHS}$$ But recalling that $j(j-1)_{s-1}=(j)_{s}$ for every such $j$, we deduce that for each $1 \leq k \leq r$, the summand $e_{r-k}\big((\lambda^{>s})_{s}\big)$ (which is independent of $j$) is multiplied by $-s\sum_{j>s}(-1)^{k-1}(j)_{s}^{k}$, which equals $-s(-1)^{k}p_{k}\big((\lambda^{>s})_{s}$ by definition. It thus follows from Lemma \[epiden\] that the second expression in Equation is just $-sre_{r-k}\big((\lambda^{>s})_{s}\big)$. Since the first term there is $|\lambda|e_{r-k}\big((\lambda^{>s})_{s}\big)$, and we know that $|\lambda|=n-rs$, we indeed establish Equation . This proves the proposition. The Main Result and Some Consequences \[Main\] ============================================== We can now prove our main result. For every three functions with derivatives of high enough order, and for every integer $s\geq0$, The $n$th derivative of $(f\circ\varphi)\cdot\big(g\circ\varphi^{(s)}\big)$ is $$\sum_{r\geq0}\sum_{\lambda \vdash n+rs}\frac{n!e_{r}\big((\lambda^{>s})_{s}\big)}{\prod_{i=1}^{n}(i!)^{m_{i}}m_{i}!}\big(f^{(\ell(\lambda)-r)}\circ\varphi\big)\big(g^{(r)}\circ\varphi^{(s)}\big)\prod_{i=1}^{n}\big(\varphi^{(i)}\big)^{m_{i}}.$$ Moreover, this expression coincides with the one from Equation , in which we pose the restrictions $r \leq n$ and $\ell(\lambda^{>s}) \geq r$, and is therefore finite. \[main\] We first prove that the two restrictions in Equation are redundant. Indeed, the elementary symmetric function $e_{r}$ is known to vanish when we substitute less than $r$ distinct parameters, and as $\ell\big((\lambda^{>s})_{s}\big)=\ell(\lambda^{>s})$, adding the restriction that this length is at least $r$ does not affect the resulting sum. Now, if $\lambda \vdash n+rs$ and $\ell(\lambda^{>s}) \geq r$ then $\lambda$ contains at least $r$ summands, all of which are at least $s+1$. We therefore obtain the inequality $n+rs=|\lambda| \geq r(s+1)$, which implies that such partitions exist only when $r \leq n$ as desired. The finiteness of the set of possible indices $r$ and of the number of partitions of $n+rs$ for every $0 \leq r \leq n$ thus yield the finiteness of our formula. For establishing the formula itself we follow the proof of Proposition \[FdBprop\] and argue by induction on $n$, where the case with $n=0$ is now clearly trivial (we only have $r=0$ and $\lambda\vdash0$). Assume now that $n>0$ and that our formula is true for $n-1$, and differentiate with respect to $t$ again. Given $0 \leq k \leq n-1$ and $\mu \vdash n-1+ks$, we first obtain contributions from differentiating $f^{(\ell(\mu)-k)}\circ\varphi$, yielding a summand associated with $r=k$ and with a partition $\lambda \vdash n+ks$, with $m_{1}(\lambda)\geq1$ and such that $\lambda_{1}=\lambda-\varepsilon_{1}=\mu$ via Definition \[lambdajdef\]. The differentiation of one of the multipliers $\varphi^{(j-1)}$ with $j\geq2$ will again produce a summand corresponding to $r=k$ and to a partition $\lambda \vdash n+ks$ for which $m_{j}(\lambda)\geq1$ and $\lambda_{j}=\mu$ as in Definition \[lambdajdef\], with an extra multiplier of $m_{j-1}(\mu)$. But here we can also differentiate the multiplier $g^{(k)}\circ\varphi^{(s)}$, whose derivative is $\big(g^{(k+1)}\circ\varphi^{(s)}\big)\cdot\varphi^{(s+1)}$, and the resulting summand is based on $r=k+1$ and on $\lambda \vdash n+(k+1)s$, where here $m_{s+1}(\lambda)\geq1$ and $\mu$ is $\lambda-\varepsilon_{s+1}$. Therefore we indeed obtain the asserted sum over $r$ and $\lambda$, and it remain to compare the coefficients. We take some $r\geq0$ and $\lambda \vdash n+rs$, and using Lemma \[lambdaj\] we deduce, from the induction hypothesis, that we have a contribution of $m_{j-1}(\lambda)+1$ times $C_{\lambda_{j},r}^{(s)}$ for every $j\geq1$ such that $m_{j}(\lambda)\geq1$, and when $r\geq1$ and $m_{s+1}(\lambda)\geq1$ we also obtain a contribution of $C_{\lambda-\varepsilon_{s+1},r-1}^{(s)}$, with no extra coefficient. In other words, the total coefficient multiplying the summand associated with $r$ and $\lambda$ is the one appearing in the right hand side of Proposition \[recrpos\], which therefore equals $C_{\lambda,r}^{(s)}$ by this proposition. This proves the theorem. We remark that the restriction $r \leq n$ in Theorem \[main\] and Equation corresponds to the fact that in such an $n$th derivative we cannot differentiate $g$ to an order exceeding $n$. We recall that the coefficients $C_{\lambda,0}^{(s)}$ from Equation and Proposition are integers. Indeed, each such coefficient $C_{\lambda,0}^{(s)}$ has a combinatorial meaning, where it counts the number of ways to put $n=|\lambda|$ numbered balls in boxes whose sizes are determined by $\lambda$, where boxes of the same size are identical. The first consequence that we draw from Theorem \[main\] is the integrality of the other coefficients, which is much less trivial in first sight. For every $r\geq0$, $s\geq0$, $n\geq0$, and partition $\lambda \vdash n+rs$, the rational number $C_{\lambda,r}^{(s)}$ from Equation and Theorem \[main\] is an integer. \[integer\] Theorem \[main\] shows that for $n=0$ this coefficient is 1 when $r=0$ and 0 otherwise, and Proposition \[recrpos\] evaluates each such coefficient as a combination of previous ones with integral coefficients. The assertion thus follows by induction on $n$ as in the proof of Theorem \[main\]. This proves the corollary. We remark that Corollary \[integer\] does not follow from the case $r=0$ by the obvious integrality of $e_{r}\big((\lambda^{>s})_{s}\big)$, because for $\lambda \vdash n+rs$ the coefficient $C_{\lambda,0}^{(s)}$ has $(n+rs)!$ in the numerator, while for $C_{\lambda,r}^{(s)}$ it is just $n!$. Note that the case $s=1$ in Corollary \[integer\] involves the moments of the partition $\lambda$ itself (up to the truncation to $\lambda^{>1}$), because the operation of taking $a$ to $(a)_{1}$ is trivial. In particular we obtain the integrality of $n!e_{r}(\lambda^{>1})\big/\prod_{j}j!^{m_{j}}m_{j}!$ for every partition $\lambda \vdash n+r$. Recall that Theorem \[main\] evaluates the $n$th derivative of a product of compositions, which we can also evaluate using Leibnitz’s Rule and the original formula of Faà di Bruno. We now use this fact for obtaining an identity, for describing which we recall the following definition. Let $\mu$ and $\nu$ be two partitions. Then $\mu\cup\nu$ is the partition obtained by taking all the summands in $\mu$ and all those of $\nu$, combining them together, and ordering the resulting sequence in decreasing order. \[union\] It is clear from Definition \[union\] that $$|\mu\cup\nu|=|\mu|+|\nu|,\ \ell(\mu\cup\nu)=\ell(\mu)+\ell(\nu),\mathrm{\ and\ }m_{i}(\mu\cup\nu)=m_{i}(\mu)+m_{i}(\nu)\mathrm{\ for\ }i\geq1. \label{unsum}$$ Equation now shows that given two partitions $\lambda$ and $\mu$, we can write $\lambda$ as $\mu\cup\nu$ for some partition $\nu$ if and only if $m_{i}(\lambda) \geq m_{i}(\mu)$ for every $i\geq1$, and then the partition $\nu$ for which $\lambda=\mu\cup\nu$ is uniquely determined. The identity that we deduce is the following one. For any $s\geq0$ and $r\geq0$ we have the equality $$\sum_{\substack{\ell(\mu)=r \\ m_{i}(\mu) \leq m_{i}(\lambda)\ \forall i\geq1 \\ m_{i}(\mu)=0\ \forall i \leq s}}\prod_{i>s}\binom{m_{i}(\lambda)}{m_{i}(\mu)} \cdot e_{r}\big((\mu)_{s}\big)=e_{r}\big((\lambda^{>s})_{s}\big).$$ \[combiden\] First of all, every partition $\mu$ in this set must satisfy $|\mu| \geq r(s+1)$, and since $|\lambda|\geq|\mu|$ this set can be non-empty only if $|\lambda|=n+rs$ for some $n \geq r$. The right hand side can be presented, via Theorem \[main\], as the coefficient of $\big(f^{(\ell(\lambda)-r)}\circ\varphi\big)\big(g^{(r)}\circ\varphi^{(s)}\big)\prod_{i=1}^{n}\big(\varphi^{(i)}\big)^{m_{i}}$ in the expansion of $\frac{d^{n}}{dt^{n}}(f\circ\varphi)\cdot\big(g\circ\varphi^{(s)}\big)$, with $n=|\lambda|-rs$, multiplied by $\prod_{i=1}^{n}(i!)^{m_{i}}m_{i}!\big/(|\lambda|-rs)!$. We therefore expand this derivative via Leibnitz’s Rule and Faà di Bruno’s formula from Equation and Proposition \[FdBprop\], and compare with the corresponding coefficient in the resulting expression. Now, Leibnitz’ Rule and the fact that $g$ is composed with $\varphi^{(s)}$ rather than $\varphi$ show that $\frac{d^{n}}{dt^{n}}(f\circ\varphi)\cdot\big(g\circ\varphi^{(s)}\big)$ equals $$\sum_{k=0}^{n}\!\binom{n}{k}\!\sum_{\nu \vdash k}C_{\nu,0}^{(s)}\big(f^{(\ell(\nu))}\circ\varphi\big)\!\prod_{i\geq1}\!\big(\varphi^{(i)}\big)^{m_{i}(\nu)}\!\!\!\sum_{\rho \vdash n-k}\!\!\!C_{\rho,0}^{(s)}\big(g^{(\ell(\rho))}\circ\varphi^{(s)}\big)\!\prod_{j\geq1}\big(\varphi^{(j+s)}\big)^{m_{j}(\rho)}\!,$$ where the denominators of $\binom{n}{k}$ cancel with the numerators of $C_{\nu,0}^{(s)}$ and $C_{\rho,0}^{(s)}$. We replace each partition $\rho$ of $n-k$ by the partition $\mu \vdash n-k+s\ell(\rho)$ obtained from $\rho$ by adding $s$ to each of the summands $a_{l}$ of which $\rho$ consists, with $\ell(\mu)=\ell(\rho)$, so that $m_{i}(\mu)$ is $m_{i-s}(\rho)$ when $i>s$ and just 0 if $i \leq s$. Then the product on the right hand side is $\prod_{i>s}\big(\varphi^{(i)}\big)^{m_{i}(\mu)}$, or equivalently $\prod_{i\geq1}\big(\varphi^{(i)}\big)^{m_{i}(\mu)}$, and Equation allows us to combine the latter expression with the other product over $i$ to give $\prod_{i\geq1}\big(\varphi^{(i)}\big)^{m_{i}(\lambda)}$, where $\lambda=\mu\cup\nu$ is a partition of $n+s\ell(\mu)$. The coefficient now includes $n!$ in the numerator and $\prod_{i\geq1}(i!)^{m_{i}(\nu)}m_{i}(\nu)!\prod_{j\geq1}(j!)^{m_{j+s}(\mu)}m_{j+s}(\mu)!$ in the denominator, where the latter multiplier can be written as $\prod_{i>s}\big((i-s)!\big)^{m_{i}(\mu)}m_{i}(\mu)!$ for $i=j+s$, and the similar multipliers with $i \leq s$ can be trivially added because $m_{i}(\mu)=0$ for such $i$. We therefore separate the resulting sum according to $r=\ell(\rho)=\ell(\mu)$, and then $\lambda \vdash n+rs$, regardless of the value of $k$, and we have seen that $r \leq n$. In addition, the coefficient does not involve $k$, and given $r$ and $\lambda$, the conditions on $\mu$ are precisely those in the left hand side of the asserted sum, and then $\nu$ is the unique partition such that $\lambda=\mu\cup\nu$, and it determines $k=|\nu|$. We therefore have to gather the coefficients, divide by $(|\lambda|-rs)!=n!$ (which cancels it), and multiply by $\prod_{i\geq1}(i!)^{m_{i}(\lambda)}m_{i}(\lambda)!$. Expressing each $m_{i}(\nu)$ via Equation , and recalling that $\frac{i!}{(i-s)!}$ is $(i)_{s}$ for any $i$ and $m_{i}(\mu)\geq0$ only for $i>s$, we obtain the product over $i>s$ of the binomial coefficient and of $(i)_{s}^{m_{i}(\mu)}$. But since $\ell(\mu)=r$, the $r$th elementary moment $e_{r}\big((\mu)_{s}\big)$ of $(\mu)_{s}$ is just the product over $(a_{l})_{s}$ for the numbers $a_{l}$, $1 \leq l \leq r$ appearing in $\mu$, which is indeed the product over $i>s$ of $(i)_{s}^{m_{i}(\mu)}$ by our condition on $\mu$. This proves the proposition. Note that the last two conditions on $\mu$ in Proposition \[combiden\] can be written as $m_{i}(\mu) \leq m_{i}(\lambda^{>s})$ (on the other hand, the condition $m_{i}(\mu) \leq m_{i}(\lambda)$ for $i>s$ may be omitted because of the binomial coefficients). In addition, the only effect of replacing $\lambda$ by $\lambda^{>s}$ in the partitions with the Pochhammer symbol of order $s$ is omitting the entries that are equal to $s$ (indeed, for $i<s$ we have $(i)_{s}=0$, and the parameters with $i>s$ remain the same). The assertion of Proposition \[combiden\] can therefore be rephrased to the statement that if $m_{s}(\lambda)=0$ then $e_{r}\big((\lambda)_{s}\big)$ is the sum over all the partitions $\mu$ of length $r$ such that $m_{i}(\mu) \leq m_{i}(\lambda)$ of the expression on the left hand side there. One may therefore ask whether this equality also holds without the assumption that $m_{s}(\lambda)=0$, but our proof establishes it only under this assumption. As an example of Proposition \[combiden\], we consider the case where $\ell(\lambda^{>s})=r$. Then the only possible partition $\mu$ in Proposition \[combiden\] is $\lambda^{>s}$, and since the binomial coefficients equal 1, the result of that proposition is trivially true in this case. Note that the case where $\lambda^{\geq s}$ (defined similarly) has length $r$ is, by a similar argument, an indication that the equality, presented in the form from the previous paragraph, may hold also without the assumption that $m_{s}(\lambda)=0$. We conclude by remarking about the case with $s=0$. In this case the partition $\lambda^{>0}$ is just $\lambda$, and as the expression $(a)_{0}$ is 1 for every $a\geq1$, we deduce that $e_{r}\big((\lambda^{>0})_{0}\big)=e_{r}(\lambda_{0})$ is just the binomial coefficient $\binom{\ell(\lambda)}{r}$ (which once again vanishes unless $\ell(\lambda^{>0})=\ell(\lambda) \geq r$). Since in this case all the partitions in Theorem \[main\] are of $n$, regardless of the value of $r$, we can invert the order of summation, and we indeed get for every $\lambda$ the coefficient $C_{\lambda,0}^{(0)}$ and the product $\prod_{i=1}^{n}\big(\varphi^{(i)}(t)\big)^{m_{i}}$ from Equation . Since the inner sum over $r$ is just $\sum_{r=0}^{n}\binom{\ell(\lambda)}{r}\big(f^{(\ell(\lambda)-r)}\circ\varphi\big)\big(g^{(r)}\circ\varphi\big)$, where the sum is essentially up to $\ell(\lambda)$ because of the binomial coefficient, Theorem \[main\] is in correspondence with the formula obtained from differentiating $(f\circ\varphi)\cdot\big(g\circ\varphi\big)$ as $(fg)\circ\varphi$ via Equation and then Leibnitz’ Rule. As for Proposition \[combiden\] in this case, the right hand side was seen to be $\binom{\ell(\lambda)}{r}$, the $e_{r}$-multipliers on the left hand side all equal 1 because $\ell(\mu)=r$, and we get the formula $$\binom{\ell(\lambda)}{r}=\sum_{\substack{\ell(\mu)=r \\ m_{i}(\mu) \leq m_{i}(\lambda)\ \forall i\geq1}}\prod_{i\geq1}\binom{m_{i}(\lambda)}{m_{i}(\mu)},$$ in which we may also omit the second condition on $\mu$ by the presence of the binomial coefficients. While this formula seems non-trivial algebraically, it has a straightforward combinatorial interpretation. Indeed, one may view the partition $\lambda$ as a marking of $\ell(\lambda)$ balls, where the number of balls that are marked by $i$ is $m_{i}(\lambda)$, and we ask in how many ways can one choose $r$ of these balls. Now, every such choice will correspond to a partition $\mu$ with $\ell(\mu)=r$ (and $m_{i}(\mu) \leq m_{i}(\lambda)$ for every $i\geq1$) according to the markings, and given such a partition $\mu$, the number of options to choose $m_{i}(\mu)$ balls out of the $m_{i}(\lambda)$ ones that are marked with $i$ is $\binom{m_{i}(\lambda)}{m_{i}(\mu)}$. The answer to our question, which is known to be $\binom{\ell(\lambda)}{r}$, is thus obtained by multiplying over $i$ for each $\mu$, and then summing over $\mu$ as desired. Therefore Proposition \[combiden\] may be viewed as a generalization of this combinatorial identity. Farkas, H. M., Kra, I., <span style="font-variant:small-caps;">Riemann Surfaces</span> (second edition), Graduate Texts in Mathematics 71, Springer–Verlag, 354pp (1992). Figueroa, H., Garcia–Bondía, J. M., Várilly, J. C., <span style="font-variant:small-caps;">Faà di Bruno Hopf Algebras</span>, pre-print. arXiv link: https://arxiv.org/abs/math/0508337. Johnson, W. P., <span style="font-variant:small-caps;">The Curious History of Faà di Bruno’s Formula</span>, Am. Math. Monthly, vol 109 no. 3, 217–234 (2002). MacDonald, I. G., <span style="font-variant:small-caps;">Symmetric Functions and Hall Polynomials</span> (second edition), Oxford Mathematical Monographs, Oxford Science Publications, x+475pp (1995). Zemel, S., <span style="font-variant:small-caps;">The Combinatorics of Higher Derivatives of Implicit Functions</span>, Monatsh. Math., vol 188 issue 4, 765–784 (2019). <span style="font-variant:small-caps;">Einstein Institute of Mathematics, the Hebrew University of Jerusalem, Edmund Safra Campus, Jerusalem 91904, Israel</span> E-mail address: [email protected]

--- abstract: 'In this short note, we show that homogeneous Ricci solitons are algebraic. As an application, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.' author: - Michael Jablonski title: Homogeneous Ricci solitons are algebraic --- [^1] Introduction ============ A Riemannian manifold $(M,g)$ is said to be a Ricci soliton if it satisfies the equation $$\label{eqn: ricci soliton} ric_g = cg + L_Xg$$ for some $c\in\mathbb R$ and some smooth vector field $X\in \mathfrak X(M)$. Such metrics are of interest as they correspond to self-similar solutions of the Ricci flow $$\frac{\partial}{\partial t} g = -2ric_g$$ That is, $g$ is the initial value of a solution to the Ricci flow of the form $g_t = c(t) \varphi_t^*g$, where $c(t)\in \mathbb R$ and $\varphi_t \in \mathfrak{Diffeo}(M)$. In this way, Ricci solitons are geometric fixed points of the flow and so are special metrics. Homogeneous Ricci solitons arise naturally as limits under the Ricci flow [@Lott:DimReductionAndLongTimeBehaviorOfRicciFlow; @Lauret:RicciFlowForSimplyConnectedNilmanifolds] and, independently, hold a distinguished place apart from other homogeneous metrics. For example, nilmanifolds cannot admit Einstein metrics, but do often admit Ricci solitons [@Jensen:TheScalarCurvatureOfLeftInvariantRiemannianMetrics; @Jablo:ModuliOfEinsteinAndNoneinstein], Ricci solitons on nilmanifolds are precisely the minima of a natural geometric functional [@LauretNilsoliton], and Ricci solitons are metrics of maximal symmetry on certain solvmanifolds [@Jablo:ConceringExistenceOfEinstein]. One natural kind of example arises as follows. Consider a homogeneous space $G/K$ where $K$ is closed and connected. For every derivation $D\in Der(\mathfrak g)$ such that $D:\mathfrak k \to \mathfrak k$, we have a well-defined map $D_{\mathfrak g/\mathfrak k} : \mathfrak g/\mathfrak k \to \mathfrak g / \mathfrak k$. Denote such derivations of $\mathfrak g$ by $Der(\mathfrak g/\mathfrak k)$. A homogeneous Ricci soliton $(G/K,g)$ is called *$G$-semi-algebraic* if the $(1,1)$ Ricci tensor is of the form $$\label{eqn: definition of semi-algebraic soliton} Ric = cId + \frac{1}{2}( D_{\mathfrak g/\mathfrak k} + D_{\mathfrak g/\mathfrak k} {}^t)$$ on $\mathfrak g/\mathfrak k \simeq T_eG/K$, for some $c\in \mathbb R$ and some $D\in Der(\mathfrak g/\mathfrak k)$. This definition is motivated by the idea of taking our family of diffeomorphisms $\{\varphi_t \}$ above to come from automorphisms of the group $G$ which leave $K$ invariant, see [@Jablo:HomogeneousRicciSolitons] or [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture] for more details. If our semi-algebraic Ricci soliton satisfies the seemingly stronger condition that $D_{\mathfrak g/\mathfrak k}$ is symmetric, then it is called a *$G$-algebraic Ricci soliton*. Up to this point, all known examples of semi-algebraic Ricci solitons were in fact algebraic and isometric to solvmanifolds. (This follows from [@Jablo:HomogeneousRicciSolitons] together with [@LauretLafuente:OnhomogeneousRiccisolitons].) Further, it was known that every homogeneous Ricci soliton must be semi-algebraic relative to its full isometry group [@Jablo:HomogeneousRicciSolitons]. We now present our main result. \[thm: main theorem\] Every $G$-semi-algebraic Ricci soliton is necessarily $G$-algebraic. Let $(M,g)$ be a homogeneous Ricci soliton. There exists a transitive group $G$, of isometries, such that $M=G/K$ is a $G$-algebraic Ricci soliton. The theorem above resolves questions raised by Lafuente-Lauret [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture] and He-Petersen-Wylie [@HePetersenWylie:WarpProdEinsteinMetricsOnHomogAndHomogRicciSolitons]. In these works, it was shown that one can always extend a simply-connected, algebraic soliton to an Einstein metric on a larger homogeneous space. There the goal was to relate the classical Alekseevskii conjecture on Einstein metrics to a more general version for Ricci solitons. More precisely, they showed that (among simply-connected manifolds) the Alekseevkii conjecture for Einstein metrics is equivalent to the (apriori) more general conjecture in the case of algebraic Ricci solitons. We state these conjectures for completeness. > **Alekseevskii Conjecture:** Every homogeneous Einstein metric with negative scalar curvature is isometric to a simply-connected solvmanifold. > **Generalized Alekseevskii Conjecture:** Every expanding homogeneous Ricci soliton is isometric to a simply-connected solvmanifold. Until now, it was not clear if these conjectures were equivalent. Applying [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture] or [@HePetersenWylie:WarpProdEinsteinMetricsOnHomogAndHomogRicciSolitons] in the simply-connected case together with [@Jablo:StronglySolvable] and the results here, we now know the following. The generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture. It is important to note that the Alekseevskii conjecture stated above is a more modern, geometric version than that given in [@Besse:EinsteinMflds]. The version given in [@Besse:EinsteinMflds] has the weaker, topological conclusion that a non-compact, homogeneous, Einstein space is only diffeomorphic to $\mathbb R^n$. It is still an open question as to whether the classical version stated in [@Besse:EinsteinMflds] is equivalent to the stronger version we pose above. *Acknowledgments:* It is our pleasure to thank Ramiro Lafuente for providing useful comments on a draft of this manuscript. Ricci solitons by type ====================== The analysis of (homogeneous) Ricci solitons varies depending on which of the following categories the metric falls into. A Ricci soliton is called *shrinking, steady, or expanding* (respectively) if the cosmological constant $c$ appearing in Eqn. \[eqn: ricci soliton\] satisfies $c>0$, $c=0$, or $c<0$ (respectively). Shrinking solitons {#shrinking-solitons .unnumbered} ------------------ The simplest example of a non-Einstein, homogeneous, shrinker is obtained by considering a compact homogeneous Einstein space $M'$ (which necessarily has positive scalar curvature) and taking a product with $\mathbb R^n$, i.e. $M=M'\times \mathbb R^n$. Here the vector field $X\in\mathfrak{X}(M)$ appearing in Eqn. \[eqn: ricci soliton\] generates a family of diffeomorphisms which simply dilate the $\mathbb R^n$ factor. Examples of this type are called trivial Ricci solitons and a result of Petersen-Wylie [@Petersen-Wylie:OnGradientRicciSolitonsWithSymmetry] says that every homogeneous shrinking Ricci soliton is finitely covered by a trivial one. Observe that such spaces are algebraic Ricci solitons. Steady solitons {#steady-solitons .unnumbered} --------------- A homogeneous steady soliton is necessarily flat. This well-known fact is proved as follows. Along the Ricci flow of any homogeneous manifold, the scalar curvature $sc$ evolves by the ODE $$\frac{d}{d t}sc = 2 |Ric|^2$$ As the scalar curvature of a steady soliton does not change along the flow, we see that the homogeneous, steady solitons are Ricci flat and so flat by [@AlekseevskiiKimelfeld:StructureOfHomogRiemSpacesWithZeroRicciCurv]. Such spaces are trivially algebraic Ricci solitons. Expanding solitons {#expanding-solitons .unnumbered} ------------------ Every homogeneous, expanding Ricci soliton is necessarily non-compact, non-gradient and all known examples of such spaces are isometric to solvable Lie groups with left-invariant metrics. While there is no characterization in this case as nice as the previous two cases, new structural results have recently appeared in [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture]. The results obtained there are essential in our proof and we briefly recall those which we need. We first observe that it suffices to prove the theorem for simply-connected manifolds. Now consider a simply-connected, expanding, semi-algebraic Ricci soliton on $G/K$. As $G/K$ is endowed with a $G$-invariant metric, $Ad(K)$ is contained in a compact subgroup of $Aut(G)$ and so we have a decomposition $\mathfrak g = \mathfrak p \oplus \mathfrak k$, where $\mathfrak p$ is an $Ad(K)$-complement to $\mathfrak k$. We fix the point $p = eK \in M = G/K$ and naturally identify $\mathfrak p$ with $T_pM$ as follows $$X\in\mathfrak p \quad \leftrightarrow \quad {{\left. \frac{d}{ds} \right|_{s=0}}}exp(sX)\cdot p = {{\left. \frac{d}{ds} \right|_{s=0}}}exp(sX)K.$$ Although there is more than one choice of $\mathfrak p$ that one can make, we apply the work [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture] in the sequel and so we choose, as they do, to have $B(\mathfrak k,\mathfrak p)=0$, where $B$ is the Killing form of $\mathfrak g$. As $G/K$ admits an expanding Ricci soliton, we know from [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture] that the group $G$ decomposes as $N\rtimes U$ where $N$ is the nilradical and $U$ is a reductive subgroup which contains the stabilizer $K$. Thus the underlying manifold of $M$ may be considered as $N \times U/K$ and we naturally identify the point $p=eK\in G/K$ with $(e,eK)\in N \times U/K$. The subalgebra $\mathfrak u$ contains a subspace $\mathfrak h$ which is complementary to $\mathfrak k$, and so we have $T_pM \simeq \mathfrak p = \mathfrak n \oplus \mathfrak h$. Furthermore, $\mathfrak n$ and $\mathfrak h$ are orthogonal subspaces of $T_pM$. For more details, see [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture]. Denote the restriction of our metric $g$ to $\mathfrak p \simeq T_eG/K$ by ${\langle \cdot,\cdot \rangle}$. Denote by $H\in\mathfrak p$ the ‘mean curvature vector’ of $G/K$ defined by $${\langle H,X \rangle} = tr\, (ad\, X) \quad \mbox{ for all } X\in\mathfrak p$$ Observe that $H\in\mathfrak h$. It is a useful fact that the subspace $\mathfrak h$ of $\mathfrak u$ is $(ad~H)$-stable [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture Prop. 4.1]. If $D$ is the soliton derivation appearing Eqn. \[eqn: definition of semi-algebraic soliton\], then we have $$D = -ad~H + D_1$$ where $D_1$ is the derivation which vanishes on $\mathfrak u$ and restricts to the nilsoliton derivation on $\mathfrak n$. In [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture Prop. 4.14], several conditions are given for when a semi-algebraic Ricci soliton is actually algebraic. One of those conditions is $$\label{eqn: s(ad H)=0} S(ad~H|_\mathfrak h) = 0$$ where $S(A) =\frac{1}{2}(A+A^t)$. This is the technical result that we will prove, from which the theorem follows. The proof of theorem \[thm: main theorem\] ========================================== The soliton inner product ${\langle \cdot, \cdot \rangle}$ on $T_pM$ above gives rise to a natural inner product on the endomorphisms of $T_pM$ given by ${\langle A,B \rangle} = tr(AB^t)$, where $B^t$ denotes the metric adjoint of $B$ relative to ${\langle \cdot,\cdot \rangle}$. Using the above inner product on endomorphisms we have $${\langle (0,ad~H|_\mathfrak h), Ric \rangle} = 0$$ where $(0,ad~H|_\mathfrak h)$ is the map on $T_pM$ defined as $0$ on $\mathfrak n$ and $ad~H|_\mathfrak h$ on $\mathfrak h$. As has been observed by R. Lafuente [@Lafuente:OnHomogeneousWarpedProductEinsteinMetrics], our proof of the lemma holds more generally. In fact, one simply needs the group to satisfy $G = U\ltimes N$ with $N$ nilpotent, $U$ reductive, and $K<U$, the metric to satisfy $N \perp U/K$ at $eK$, and the element $H$ may be replaced by any $Y\in\mathfrak u$ satisfying $[Y,\mathfrak k]\subset \mathfrak k$. Before proving the lemma, we use it to verify that Eqn. \[eqn: s(ad H)=0\] holds. Consider the mean curvature vector $H\in \mathfrak u$. As $\mathfrak u$ is reductive, $ad~H|_\mathfrak u$ is traceless. Furthermore, since $ad~H$ vanishes on the stabilizer $\mathfrak k$ (see Eqn. 26 of [@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture]) and $\mathfrak u = \mathfrak k \oplus \mathfrak h$, we see that $tr~ad~H|_\mathfrak h = 0$. Together with the above lemma we have $$\begin{aligned} 0 &=& {\langle (0,ad~H|_\mathfrak h), Ric \rangle}\\ &=& {\langle (0,ad~H|_\mathfrak h), cId -S(ad~H) + D_1 \rangle}\\ &=& {\langle ad~H|_\mathfrak h , cId|_\mathfrak h - S(ad~H|_\mathfrak h) \rangle}\\ &=& c\ tr(ad~H|_\mathfrak h) - tr~S (ad~H|_\mathfrak h)^2\\ &=& 0 - tr~S(ad~H|_\mathfrak h)^2 \end{aligned}$$ Thus $S(ad~H|_\mathfrak h) = 0$, as claimed. We now prove the lemma by considering a certain deformation of the metric $g$ on $M$. As $ad~H$ vanishes on $\mathfrak k$ and $K$ is connected, the family of automorphisms $\Phi_t = C_{exp(tH)} \in Aut(U)$ is the identity on $K$ and hence gives rise to well-defined diffeomorphisms $\phi_t$ on $U/K$ given by $$\phi_t(uK) = \Phi_t(u)K \quad \mbox{ for } u\in U$$ Note that $(\Phi_t)_* =Ad(exp(tH)) = e^{t\ ad~H} \in Aut(\mathfrak u)$. On the manifold $M = N\times U/K$, we consider the family of diffeomorphisms given by $$\varphi_t =(id, \phi_t) \quad \mbox{ on } N\times U/K$$ The deformations of $g$ of interest are $g_t = \varphi_t {}^* g$. As $\varphi_t$ fixes the point $p:= eK = (e,eK)\in M = N\times U/K$, and scalar curvature is an invariant, we have $${{\left. \frac{d}{dt} \right|_{t=0}}}sc(\varphi_t {}^* g) _p = 0$$ We use this in the following general equation which holds for any family of metrics $\{g_t\}$ with variation $h = \frac{\partial}{\partial t} g_t$ (see [@ChowKnopf Lemma 3.7]) $$\label{eqn: scalar curv variation} \frac{\partial}{\partial t} sc = - \Delta \overline H + div(div~h) -{\langle h,ric \rangle}$$ where in local coordinates we have $$\label{eqn: delta H} \Delta \overline{H} = g^{ij}g^{kl} \nabla_i \nabla_j h_{kl}$$ and $$\label{eqn: div div h} div(div~h) = g^{ij}g^{kl} \nabla_i \nabla_k h_{jl}$$ Observe, at the point $p :=eK = (e,eK)$ of $M$ we have $\frac{\partial}{\partial t}|_{t=0} (\varphi_t)_* = (0,ad~H|_\mathfrak h)$ and so the lemma follows from Eqn. \[eqn: scalar curv variation\] (evaluated at $p$) upon showing the terms $\Delta \overline{H}$ and $div(div\ h)$ vanish. Recall that, in local coordinates, we define the metric inverse $g^{ij}$ as the function satisfying $\delta _i^l = g^{ij}g_{jl}$. By choosing a frame which is $g$-orthonormal at every point, one would have that both $g_{ij}$ and $g^{ij}$ are the identity. We make such a choice below. To ease computational burden, we build a frame which is $g$-orthonormal at every point and exploits the property that our metric $g$ is $G$-invariant. We start with an orthonormal basis of $T_pM$. As $T_pM = \mathfrak n \oplus \mathfrak h$, we may choose a basis $\{e_i\}$ which is the union of an orthonormal basis of $\mathfrak n$ together with an orthonormal basis of $\mathfrak h$. Next, we extend the basis $\{e_i\}$ to a local frame nearby to $p\in M$. To do this, we first consider a slice $\mathfrak S$ of the right $K$ action on $G$ through $e\in G$. That is, we have a submanifold $\mathfrak S$ of $G$ containing $e$ such that $\dim \mathfrak S = \dim G/K$ and the map $$s\mapsto sK \quad s\in \mathfrak S$$ is a diffeomorphism of a neighborhood of $e\in \mathfrak S$ to a neighborhood of $eK \in G/K$. Now, for $q\in M$ nearby to $p$, there exists $s\in \mathfrak S$ such that $q=s\cdot p$ and we define $$e_i(q) = s_* e_i,$$ where $s_*$ denotes the differential of the translation $s: p \mapsto q$. We note that the frame is well-defined as our choice of $s\in\mathfrak S$ is unique, since $\mathfrak S$ is a slice. Furthermore, our frame is $g$-orthonormal as $g$ is $G$-invariant. Using the above choice of frame nearby to $p\in M$, we now study Eqns. \[eqn: delta H\] and \[eqn: div div h\]. We begin by computing the variation $h$ of $g_t = \varphi_t {}^* g$ in terms of $\{e_i\}$. For a point $q\in M$ near $p$, $$\label{eqn: general variation} h_{ij}(q) = {{\left. \frac{\partial}{\partial t} \right|_{t=0}}}(g_t)_{ij}(q) = {{\left. \frac{\partial}{\partial t} \right|_{t=0}}}(g_t) (e_i(q),e_j(q)) = {{\left. \frac{\partial}{\partial t} \right|_{t=0}}}g( (\varphi_t)_* e_i(q), (\varphi_t)_* e_j(q) )$$ Next we compute $(\varphi_t)_* v_q$ for a vector $v_q \in T_qM$. As $G=NU$, there exist $n\in N$ and $u\in U$ such that $s\in\mathfrak S$ may be written as $s=nu$ and $q = (nu)\cdot p$. Furthermore, there exists $X\in \mathfrak p = \mathfrak n \oplus \mathfrak h$ such that $v_q = (nu)_* {{\left. \frac{d}{ds} \right|_{s=0}}}exp( s X) \cdot p$. To understand Eqn. \[eqn: general variation\], we analyze separately the cases when $X$ is an element of $\mathfrak n$ or of $\mathfrak h$. For $X\in\mathfrak n$, we have $$\begin{aligned} (\varphi_t)_* v_q &=& (\varphi_t)_* (nu)_* X \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}\varphi_t( nu \ exp(sX)\cdot p ) \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}\varphi_t( n \ u \ exp(sX) \ u^{-1}\ u \cdot p ) \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}\varphi_t( n \ u \ exp(sX) \ u^{-1}, \ uK ) \label{eqn: X in n} \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}( n \ exp(sAd_uX), \Phi_t( u) K ) \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}( n \Phi_t(u) \ \Phi_t(u)^{-1} \ exp(sAd_uX) \Phi_t( u) K ) \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}( n \Phi_t(u) \ exp(sAd_{\Phi_t(u)^{-1}} Ad_u X) K ) \nonumber \\ &=& (n\Phi_t(u))_* \ Ad_{\Phi_t(u)^{-1} u} X \nonumber \end{aligned}$$ Here we have used that $N$ is normal in $G$. Note also that $Ad_{\Phi_t(u)^{-1} u} X \in \mathfrak n$. In the case when $X\in \mathfrak h \subset \mathfrak u$, we have $$\begin{aligned} (\varphi_t)_* v_q &=& (\varphi_t)_* ( nu )_* X \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}\varphi_t( nu \ exp(sX)\cdot p ) \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}\varphi_t( n \ u \ exp(sX) \ K ) \label{eqn: X in h}\\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}( n \ \Phi_t(u \ exp(sX) )\ K ) \nonumber \\ &=& {{\left. \frac{d}{ds} \right|_{s=0}}}( n \ \Phi_t(u) \ exp(s (\Phi_t)_*X) )\ K ) \nonumber \\ &=& (n\Phi_t(u))_* (\Phi_t)_*X \nonumber \end{aligned}$$ Observe that since $ad~H$ preserves $\mathfrak h$ ([@LauretLafuente:StructureOfHomogeneousRicciSolitonsAndTheAlekseevskiiConjecture] Eqn. 32), $(\Phi_t)_*X \in \mathfrak h$ and so the the last line is consistent with our identification of $\mathfrak p = \mathfrak n \oplus \mathfrak h$ with $T_pM$. From Eqns. \[eqn: general variation\], \[eqn: X in n\], and \[eqn: X in h\] we see that 1. If $e_i \in \mathfrak n$ and $e_j \in \mathfrak h$, then $g_{ij}(q) = 0$. 2. If $e_i \in \mathfrak n$ and $e_j \in \mathfrak h$, then $h_{ij}(q) = 0$. 3. If $e_i,e_j \in \mathfrak h$, then $h_{ij}(q)$ does not depend on $n$ and $u$, and so is constant in $q$. 4. If $e_i,e_j \in \mathfrak n$, then $h_{ij}(q)$ does not depend on $n$, but does depend on $u$. Using these observations, we see that the only possible non-zero terms of $$div(div~h) = g^{ij}g^{kl} \nabla_i \nabla_k h_{jl}$$ are when $e_j,e_l \in \mathfrak n$ and $e_i, e_k \in \mathfrak h$. However, $(g_{\alpha \beta}) = Id$ implies $(g^{\alpha \beta})=Id$ and so $g^{kl}=0$. This yields $$div(div~h) = 0$$ Next we study $\Delta \overline H = g^{ij}g^{kl} \nabla_i \nabla_j h_{kl} $. As above, the only possible non-zero terms occur when $e_k,e_l\in\mathfrak n$ and $e_i, e_j\in\mathfrak h$. Further, as our frame is orthonormal, we have $$\Delta \overline H (q) = g^{ii} (q) g^{kk} (q) ( \nabla_i \nabla_i h_{kk} )(q) = \sum_i \left( \nabla_i \nabla_i \sum_k h_{kk} \right) (q)$$ where the first sum is over the frame from $\mathfrak h$ and the second is over the frame from $\mathfrak n$. From Eqns. \[eqn: general variation\] and \[eqn: X in n\] we have $$\begin{aligned} h_{kk}(q) &=& {{\left. \frac{\partial}{\partial t} \right|_{t=0}}}g( (\varphi_t)_* e_k(q), (\varphi_t)_* e_k(q) ) \\ &=& {{\left. \frac{\partial}{\partial t} \right|_{t=0}}}{\langle Ad_{\Phi_t(u)^{-1} u} ( e_k) , Ad_{\Phi_t(u)^{-1} u} ( e_k) \rangle} \\ &=& 2 \ {\langle e_k, ({{\left. \frac{d}{dt} \right|_{t=0}}}Ad_{\Phi_t(u)^{-1} u} ) ( e_k) \rangle} \\ &=& 2 \ {\langle e_k, ad~M ( e_k) \rangle} \end{aligned}$$ where $M = {{\left. \frac{d}{dt} \right|_{t=0}}}\Phi_t(u)^{-1} u $. To see that this last line makes sense, observe that $\Phi_t(u)^{-1} u$ is a curve in $U$ with $\Phi_0(u)^{-1}u =e$ and thus ${{\left. \frac{d}{dt} \right|_{t=0}}}\Phi_t(u)^{-1} u \in \mathfrak u$. Although $M$ is a function of $u$, we suppress this detail as it does not impact the rest of our proof. We claim that $ad~M|_\mathfrak n$ is traceless. To see this, we use that fact that $U$ being reductive and connected implies $U = [U,U] Z(U)$, where $Z(U)$ is the center of $U$. Thus, we may write $u=u_1u_2$ where $u_1\in [U,U]$ and $u_2\in Z(U)$. As $u_2$ is central and $\Phi_t$ is an inner automorphism, $\Phi_t(u_2)=u_2$ and $$\Phi_t(u)^{-1}u = \Phi_t(u_1)^{-1} u_1 \in [U,U]$$ This gives $ad~M \in ad~[\mathfrak u,\mathfrak u]$ from which our claim immediately follows. 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Helv. 85 (2010) 485–534 **P Petersen**, **W Wylie**, [*On gradient [R]{}icci solitons with symmetry*](http://dx.doi.org/10.1090/S0002-9939-09-09723-8), Proc. Amer. Math. Soc. 137 (2009) 2085–2092 [^1]: This work was supported in part by NSF grant DMS-1105647.

**A direct method of solution for the Fokas-Lenells derivative** **nonlinear Schrödinger equation: II. Dark soliton solutions** Yoshimasa Matsuno[^1] *Division of Applied Mathematical Science,* *Graduate School of Science and Engineering* *Yamaguchi University, Ube, Yamaguchi 755-8611, Japan* In a previous study (Matsuno Y [ *J. Phys. A: Math. Theor.*]{} [**45**]{} (2012) 23202), we have developed a systematic method for obtaining the bright soliton solutions of the Fokas-Lenells derivative nonlinear Schrödinger equation (FL equation shortly) under vanishing boundary condition. In this paper, we apply the method to the FL equation with nonvanishing boundary condition. In particular, we deal with a more sophisticated problem on the dark soliton solutions with a plane wave boundary condition. We first derive the novel system of bilinear equations which is reduced from the FL equation through a dependent variable transformation and then construct the general dark $N$-soliton solution of the system, where $N$ is an arbitrary positive integer. In the process, a trilinear equation derived from the system of bilinear equations plays an important role. As a byproduct, this equation gives the dark $N$-soliton solution of the derivative nonlinear Schrödinger equation on the background of a plane wave. We then investigate the properties of the one-soliton solutions in detail, showing that both the dark and bright solitons appear on the nonzero background which reduce to algebraic solitons in specific limits. Last, we perform the asymptotic analysis of the two- and $N$-soliton solutions for large time and clarify their structure and dynamics. [*PACS:*]{} 05.45.Yv; 42.81.Dp; 02.30.Jr [*Keywords:*]{} derivative nonlinear Schrödinger equation; dark soliton; direct method of solution The Fokas-Lenells derivative nonlinear Schrödinger (NLS) equation (FL equation shortly) is a completely integrable nonlinear partial differential equation (PDE) which has been derived as an integrable generalization of the NLS equation using bi-Hamiltonian methods \[1\]. In the context of nonlinear optics, the FL equation models the propagation of nonlinear light pulses in monomode optical fibers when certain higher-order nonlinear effects are taken into account \[2\]. We employ the following equation which can be derived from its original version by a simple change of variables combined with a gauge transformation \[2\]: $$u_{xt}=u-2{\rm i}|u|^2u_x. \eqno(1.1)$$ Here, $u=u(x,t)$ is a complex-valued function of $x$ and $t$, and subscripts $x$ and $t$ appended to $u$ denote partial differentiations. The complete integrability of the FL equation has been demonstrated by means of the inverse scattering transform (IST) method \[3\]. Especially, a Lax pair and a few conservation laws associated with it have been obtained explicitly using the bi-Hamiltonian structure and the multisoliton solutions have been derived by applying the dressing method \[4\]. Another remarkable feature of the FL equation is that it is the first negative flow of the integrable hierarchy of the derivative NLS equation \[2, 5\]. In a previous study \[6\] which is referred to as I hereafter, the two different expressions of the bright $N$-soliton solution of the FL equation have been obtained by a direct method which does not recourse to the IST and their properties have been explored in detail. Here, we construct the dark $N$-soliton solution of the FL equation on the background of a plane wave. Explicitly, we consider the boundary condition $$u \rightarrow \rho\,{\rm exp}\left\{{\rm i}\left(\kappa x-\omega t+\phi^{(\pm)}\right)\right\}, \quad x \rightarrow \pm\infty, \eqno(1.2)$$ where $\rho(>0)$ and $\kappa$ are real constants representing the amplitude and wavenumber, respectively, $\phi^{(\pm)}$ are real phase constants and the angular frequency $\omega=\omega(\kappa)$ obeys the dispersion relation $\omega=1/\kappa+2\rho^2.$ Note that the plane wave given in (1.2) is an exact solution of the FL equation. As will be discussed later, the possible values of $\kappa$ must be restricted to assure the existence of the soliton solutions. A similar problem to that posed in this paper has been studied recently and an explicit formula for the dark $N$-soliton solution have been presented by an ingenious application of the Bäcklund transformation between solutions of the FL equation and the Ablowitz-Ladik hierarchy \[7\]. Nevertheless, the detailed analysis of the soliton solutions has not been undertaken as yet. An exact method of solution employed here which is sometimes called the direct method \[8\] or the bilinear transformation method \[9\] is a powerful tool for analyzing soliton equations and differs from the method used in \[7\]. Once the equation under consideration is transformed to a system of bilinear equations, the standard technique in the bilinear formalism is applied to obtain soliton solutions. A novel feature of the bilinearization of the FL equation is that one of the bilinear equations can be replaced by a [*trilinear*]{} equation, as already demonstrated in I. The same situation happens in the current dark soliton problem. However, the resulting trilinear equation will be used essentially in the process of performing the proof of the dark $N$-soliton solution. This paper is organized as follows. In section 2, we bilinearize the FL equation under the boundary condition (1.2). We then show that one of the resulting bilinear equations can be replaced by a trilinear equation. In section 3, we present the dark $N$-soliton solution of the bilinear equations. It has a simple structure expressed in terms of certain determinants. Subsequently, we perform the proof of the dark $N$-soliton solution using an elementary theory of determinants in which Jacobi’s identity will play a central role. As already noted, the proof of the trilinear equation turns out to be a core in the analysis. In accordance with the relation between the FL equation and the derivative NLS equation at the level of the Lax representation, we also demonstrate that the dark $N$-soliton solution obtained here yields the dark $N$-soliton solution of the derivative NLS equation by replacing simply the time dependence of the solution. As in the case of the defocusing NLS equation subjected to nonvanishing boundary conditions, it is necessary for the existence of dark solitons that the asymptotic state given by (1.2) must be stable. Hence, we perform the linear stability analysis of the plane wave solution (1.2) and provide a criterion for the stability. In section 4, we first investigate the properties of the one-soliton solution in detail. We find that depending on the sign of $\kappa$ and that of the real part of the complex amplitude parameter, the solution can be classified into two types, i.e., the dark and bright solitons. The latter soliton may be termed “anti-dark soliton” since the background field is nonzero. However, we use a term “bright soliton” throughout the paper. We demonstrate that regardless the sign of $\kappa$, the bright soliton has a limiting profile of algebraic type (or an algebraic bright soliton) whereas an algebraic dark soliton appears only if $\kappa<0$. We then analyze the asymptotic behavior of the two-soliton solution and derive the explicit formulas for the phase shift in terms of the amplitude parameters of solitons. In particular, we address the interaction between a dark soliton and a bright soliton as well as that of two dark solitons. Last, the similar asymptotic analysis to that of the two-soliton solution is performed for the general dark $N$-soliton solution. Section 5 is devoted to concluding remarks. In this section, we develop a direct method of solution for constructing dark soliton solutions of the FL equation (1.1) under the boundary condition (1.2). In particular, we show that it can be transformed to a system of bilinear equations by introducing the same type of the dependent variable transformation as that employed in I for the bilinearization of the FL equation under vanishing boundary condition. We also demonstrate that this system yields a trilinear equation which will play a crucial role in our analysis. The bilinearization of the FL equation (1.1) is established by the following proposition: [**Proposition 2.1.**]{} [*By means of the dependent variable transformation $$u=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega t)}\,{g\over f}, \eqno(2.1)$$ with $\omega=1/\kappa+2\rho^2$, equation (1.1) can be decoupled into the following system of bilinear equations for the tau functions $f$ and $g$ $$D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)=0, \eqno(2.2)$$ $$D_xD_tf\cdot f^*-{\rm i}\rho^2D_xg\cdot g^*+{\rm i}\rho^2D_xf\cdot f^*+2\kappa\rho^2(gg^*-ff^*)=0, \eqno(2.3)$$ $$D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm i}\omega D_xg\cdot f=0. \eqno(2.4)$$ Here, $f=f(x, t)$ and $g=g(x, t)$ are complex-valued functions of $x$ and $t$, and the asterisk appended to $f$ and $g$ denotes complex conjugate and the bilinear operators $D_x$ and $D_t$ are defined by $$D_x^mD_t^nf\cdot g=\left({\partial\over\partial x}-{\partial\over\partial x^\prime}\right)^m \left({\partial\over\partial t}-{\partial\over\partial t^\prime}\right)^n f(x, t)g(x^\prime,t^\prime)\Big|_{ x^\prime=x,\,t^\prime=t}, \eqno(2.5)$$ where $m$ and $n$ are nonnegative integers.*]{} [**Proof.**]{} Substituting (2.1) into (1.1) and rewriting the resultant equation in terms of the bilinear operators, equation (1.1) can be rewritten as $${1\over f^2}(D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm i}\omega D_xg\cdot f)$$ $$-{g\over f^3f^*}\bigl\{f^*D_xD_tf\cdot f-2\kappa\rho^2f^2f^*-2{\rm i}\rho^2g^*(g_xf-gf_x+{\rm i}\kappa fg)\bigr\}=0. \eqno(2.6)$$ Inserting the identity $$f^*D_xD_tf\cdot f=fD_xD_tf\cdot f^*-2f_xD_tf\cdot f^*+f(D_tf\cdot f^*)_x, \eqno(2.7)$$ which can be verified by direct calculation, into the second term on the left-hand side of (2.6), one modifies it in the form $${1\over f^2}(D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm i}\omega D_xg\cdot f)$$ $$-{g\over f^3f^*}\Bigl[f\bigl\{D_xD_tf\cdot f^*-{\rm i}\rho^2D_xg\cdot g^*+{\rm i}\rho^2D_xf\cdot f^*+2\kappa\rho^2(gg^*-ff^*)\bigr\}$$ $$-2f_x\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}+f\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}_x\Bigr]=0. \eqno(2.8)$$ By virtue of equations (2.2)-(2.4), the left-hand side of (2.8) vanishes identically. $\Box$ It follows from (2.1) and (2.2) that $$|u|^2=\rho^2+{\rm i}\,{\partial \over \partial t}\,{\rm ln}\,{f^*\over f}. \eqno(2.9)$$ The above formula gives the modulus of $u$ in terms of the tau function $f$. [**Proposition 2.2.**]{} [*The [*trilinear*]{} equation for $f$ and $g$ $$f^*\left\{g_{xt}f-(f_x-{\rm i}\kappa f)g_t-{\rm i}\left({1\over \kappa}+\rho^2\right)(g_xf-gf_x)\right\} =f_t^*(g_xf-gf_x+{\rm i}\kappa fg), \eqno(2.10)$$ is a consequence of the bilinear equations (2.2)-(2.4).*]{} [**Proof.**]{} By direct calculation, one can show the following trilinear identity among the tau functions $f$ and $g$: $$f^*\left\{g_{xt}f-(f_x-{\rm i}\kappa f)g_t-{\rm i}\left({1\over \kappa}+\rho^2\right)(g_xf-gf_x)\right\} -f_t^*(g_xf-gf_x+{\rm i}\kappa fg)$$ $$=f^*(D_xD_tg\cdot f+{\rm i}\kappa D_tg\cdot f-{\rm i}\omega D_xg\cdot f)$$ $$-{g\over 2}\Bigl[\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}_x+(D_xD_tf\cdot f^*-{\rm i}\rho^2D_xg\cdot g^*+{\rm i}\rho^2D_xf\cdot f^*-2{\rm i}\kappa D_tf\cdot f^*)\Bigr]$$ $$+g_x\bigl\{D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*)\bigr\}. \eqno(2.11)$$ Replacing a term $2{\rm i}\kappa D_tf\cdot f^*$ on the right-hand side of (2.11) by (2.2), the right-hand side becomes zero by (2.2)-(2.4). This yields (2.10). $\Box$ In view of proposition 2.2, the proof of the dark $N$-soliton solution is completed if one can prove any three equations among the three bilinear equations (2.2)-(2.4) and a trilinear (2.10). We will see later in section 3 that the proof of (2.4) is not easy to perform and hence we prove (2.10) instead. In this section, we show that the tau functions $f$ and $g$ representing the dark $N$-soliton solution admit the compact determinantal expressions. This statement is proved by an elementary calculation using the basic formulas for determinants. We first prove that the proposed dark $N$-soliton solution solves the bilinear equations (2.2) and (2.3) and then the trilinear equation (2.10) in place of (2.4). The implication of the equation (2.10) will be discussed in conjunction with the dark $N$-soliton solution of the derivative NLS equation. Last, we perform the linear stability analysis of the plane wave solution (1.2) and provide a criterion for the stability. [*3.1. Dark $N$-soliton solution*]{} The main result in this paper is given by the following theorem: [**Theorem 3.1.**]{} [*The dark $N$-soliton solution of the system of bilinear equations (2.2)-(2.4) is expressed by the following determinants $$f=|D|, \eqno(3.1a)$$ $$g=\begin{vmatrix} D & {\bf z}^T\\ {1\over \rho^2}{\bf z}_t^* & 1\end{vmatrix}=|D|+{1\over \rho^2}\begin{vmatrix} D & {\bf z}^T\\ {\bf z}_t^* & 0\end{vmatrix}. \eqno(3.1b)$$ Here, $D$ is an $N\times N$ matrix and ${\bf z}$ and ${\bf z}_t$ are $N$-component row vectors defined below and the symbol $T$ denotes the transpose: $$D=(d_{jk})_{1\leq j,k\leq N}, \quad d_{jk}=\delta_{jk}+{\kappa-{\rm i}p_j\over p_j+p_k^*}\,z_jz_k^*, \quad z_j={\rm exp}\left(p_jx+{\kappa\rho^2\over p_j}t+{1\over p_j+{\rm i}\kappa}\,\tau+\zeta_{j0}\right), \eqno(3.2a)$$ $${\bf z}=(z_1, z_2, ..., z_N), \quad {\bf z}_t=\left({\kappa\rho^2z_1\over p_1}, {\kappa\rho^2z_2\over p_2}, ..., {\kappa\rho^2z_N\over p_N}\right), \eqno(3.2b)$$ where $p_j$ are complex parameters satisfying the constraints $$(p_j+{\rm i}\kappa)(p_j^*-{\rm i}\kappa)={1+\kappa\rho^2\over \kappa\rho^2}p_jp_j^*,\quad j=1, 2, ..., N, \eqno(3.2c)$$ $\zeta_{j0}\ (j=1, 2, ..., N)$ are arbitrary complex parameters, $\delta_{jk}$ is kronecker’s delta and $\tau$ is an auxiliary variable.*]{} The dark $N$-soliton solution is parameterized by $2N$ complex parameters $p_j$ and $\zeta_{j0}\ (j=1, 2, ..., N)$. The parameters $p_j$ determine the amplitude and velocity of the solitons whereas the parameters $\zeta_{j0}$ determine the phase of the solitons. As opposed to the bright soliton case explored in I, however, the real and imaginary parts of $p_j$ are not independent because of the constraints (3.2c). Actually, it may be parameterized either by the velocity of the $j$th soliton or by a single angular variable, as will see in section 4. An auxiliary variable $\tau$ introduced in (3.2a) will be used conveniently in performing the proof of (2.10). It can be set to zero after all the calculations have been completed. [**Remark 3.1.**]{} The tau function $g$ given by (3.1b) is represented by the determinant of an $(N+1)\times (N+1)$ matrix. It can be rewritten by the determinant of an $N\times N$ matrix. To show this, we multiply the $(N+1)$th column of $g$ by $z_{k,t}^*/\rho^2$ and subtract it from the $k$th column for $k=1, 2, ..., N$ to obtain $$g=\left|\left(\delta_{jk}-{\kappa+{\rm i}p_k^*\over p_j+p_k^*}\,{p_j\over p_k^*}\,z_jz_k^*\right)_{1\leq j,k\leq N}\right|. \eqno(3.3)$$ Although the tau function from (3.1b) is used in the proof of the dark $N$-soliton solution, an equivalent form (3.3) will be employed in section 4 to analyze the structure of the solution. [**Remark 3.2.**]{} The complex parameters $p_j$ subjected to the constraints (3.2c) exist only if the condition $\kappa(1+\kappa\rho^2)>0$ is satisfied, as confirmed easily by putting $p_j=a_j+{\rm i}b_j$ with real $a_j$ and $b_j$. We will show in section 3.6 that this condition is closely related to the stability of the plane wave solution of the FL equation. [*3.2. Notation and basic formulas for determinants*]{} Before entering into the proof of the dark $N$-soliton solution, we first define the matrices associated with the dark $N$-soliton solution and then provide some basic formulas for determinants. Although these formulas have been used extensively in I, we reproduce them for convenience. The following bordered matrices appear frequently in our analysis: $$D({\bf a}; {\bf b})=\begin{pmatrix} D & {\bf b}^T\\ {\bf a} & 0\end{pmatrix},\quad D({\bf a},{\bf b};{\bf c},{\bf d})=\begin{pmatrix} D &{\bf c}^T & {\bf d}^T \\ {\bf a} & 0 &0\\ {\bf b} & 0& 0 \end{pmatrix}, \eqno(3.4)$$ where ${\bf a}, {\bf b}, {\bf c}$ and [**d**]{} are $N$ component row vectors. Let $D_{jk}$ be the cofactor of the element $d_{jk}$. The following formulas are well known in the theory of determinants \[10\]: $${\partial\over\partial x}|D|=\sum_{j,k=1}^N{\partial d_{jk}\over\partial x}D_{jk}, \eqno(3.5)$$ $$\begin{vmatrix} D & {\bf a}^T\\ {\bf b} & z\end{vmatrix}=|D|z-\sum_{j,k=1}^ND_{jk}a_jb_k, \eqno(3.6)$$ $$|D({\bf a}, {\bf b}; {\bf c}, {\bf d})||D|= |D({\bf a}; {\bf c})||D({\bf b}; {\bf d})|-|D({\bf a}; {\bf d})||D({\bf b}; {\bf c})|. \eqno(3.7)$$ The formula (3.5) is the differentiation rule of the determinant and (3.6) is the expansion formula for a bordered determinant with respect to the last row and last column. The formula (3.7) is Jacobi’s identity. The proof of lemmas described below is based on the above three formulas as well as a few fundamental properties of determinants. [*3.3. Differentiation rules and related formulas*]{} In terms of the notation (3.4), the tau functions $f$ and $g$ can be written as $$f=|D|, \eqno(3.8a)$$ $$\quad g=|D|+{1\over\rho^2}|D({\bf z}_t^*;{\bf z})|. \eqno(3.8b)$$ The differentiation rules of the tau functions with respect to $t$ and $x$ are given by the following formulas: [**Lemma 3.1.**]{} $$f_t={\rm i}|D({\bf z}_t^*;{\bf z})| - {1\over \rho^2}|D({\bf z}_t^*;{\bf z}_t)|, \eqno(3.9)$$ $$f_x=-\kappa |D({\bf z}^*;{\bf z})|+{\rm i}|D({\bf z}^*;{\bf z}_x)|, \eqno(3.10)$$ $$f_{xt}={\rm i}\kappa\rho^2|D({\bf z}^*;{\bf z})|-\kappa|D({\bf z}_t^*;{\bf z})|-\kappa|D({\bf z}^*;{\bf z}_t)| +{\rm i}|D({\bf z}_t^*;{\bf z}_x)|$$ $$-|D({\bf z}^*,{\bf z}_t^*;{\bf z}_x,{\bf z})|+{\kappa\over \rho^2}|D({\bf z}^*,{\bf z}_t^*;{\bf z},{\bf z}_t)| -{\rm i\over \rho^2}|D({\bf z}^*,{\bf z}_t^*;{\bf z}_x,{\bf z}_t)|, \eqno(3.11)$$ $$g_t={\rm i}|D({\bf z}_t^*;{\bf z})|+{1\over \rho^2}|D({\bf z}_{tt}^*;{\bf z})|, \eqno(3.12)$$ $$g_x={\rm i}|D({\bf z}_t^*;{\bf z})|+{1\over \rho^2}|D({\bf z}_{t}^*;{\bf z}_x)| +{\rm i\over \rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z},{\bf z}_x)|. \eqno(3.13)$$ [**Proof.**]{} We prove (3.9). Applying formula (3.5) to $f$ given by (3.1) with (3.2a), one obtains $$\begin{aligned} f_t &=\kappa\rho^2\sum_{j,k=1}^ND_{jk}{\kappa-{\rm i}p_j\over p_jp_k^*}z_jz_k^* \notag \\ &=-{\rm i}\sum_{j,k=1}^ND_{jk}z_{j}z_{k,t}^* +{1\over \rho^2}\sum_{j,k=1}^ND_{jk}z_{j,t}z_{k,t}^*, \notag\end{aligned}$$ where in passing to the second line, use has been made of the relation $z_{j,t}=(\kappa\rho^2/p_j)z_j$. Referring to formula (3.6) with $z=0$ and taking into account the notation (3.4), the above expression is equal to the right-hand side of (3.9). Formulas (3.10)-(3.13) can be proved in the same way if one uses (3.5), (3.6) and the relation ${\bf z}_{xt}=\kappa\rho^2{\bf z}$ as well as some basic properties of determinants. $\Box$ The complex conjugate expressions of the tau functions $f$ and $g$ and their derivatives are expressed as follows: [**Lemma 3.2.**]{} $$f^*=|D|-{\rm i}|D({\bf z}^*;{\bf z})|, \eqno(3.14)$$ $$f_t^*=-{\rm i}|D({\bf z}^*;{\bf z}_t)|- {1\over \rho^2}|D({\bf z}_t^*;{\bf z}_t)|+{\rm i\over \rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z}_t,{\bf z})|, \eqno(3.15)$$ $$g^*=|D|-{\rm i}|D({\bf z}^*;{\bf z})|+{1\over \rho^2}|D({\bf z}^*;{\bf z}_t)|. \eqno(3.16)$$ [**Proof.**]{} It follows from (3.2a) that $d_{jk}^*=d_{kj}+{\rm i}z_j^*z_k$ or in the matrix form, $D^*= D^T+{\rm i}(z_jz_k^*)_{1\leq j,k\leq N}^T$. Since $|D^T|=|D|$, one has $$f^*=|D+{\rm i}(z_jz_k^*)_{1\leq j,k\leq N}|=\begin{vmatrix} D & {\bf z}^T\\ -{\rm i}{\bf z}^* & 1\end{vmatrix}.$$ Applying formula (3.6) to the right-hand side, formula (3.14) follows immediately. Formulas (3.15) and (3.16) can be derived in the same way. $\Box$ [*3.4. Proof of the dark $N$-soliton solution*]{} [*3.4.1. Proof of (2.2)*]{} Let $$P_1=D_tf\cdot f^*-{\rm i}\rho^2(gg^*-ff^*). \eqno(3.17)$$ Substituting (3.8), (3.9) and (3.14)-(3.16) into (3.17), most terms are canceled, leaving the following three terms $$P_1={{\rm i}\over \rho^2}\Bigl\{-|D||D({\bf z}_t^*,{\bf z}^*;{\bf z}_t,{\bf z})|+|D({\bf z}^*;{\bf z})||D({\bf z}_t^*;{\bf z}_t)| -|D({\bf z}_t^*;{\bf z})||D({\bf z}^*;{\bf z}_t)|\Bigr\}.$$ This expression becomes zero by Jacobi’s identity. $\Box$ [*3.4.2. Proof of (2.3)*]{} Instead of proving (2.3) directly, we differentiate (2.2) by $x$ and add the resultant expression to (2.3) and then prove the equation $P_2=0$, where $$P_2=f_{xt}f^*-f_xf_t^*-{\rm i}\rho^2(g_xg^*-f_xf^*)+\kappa\rho^2(gg^*-ff^*). \eqno(3.18)$$ Substituting (3.8)-(3.11), (3.13) and (3.14)-(3.16) into (3.18) and rearranging, $P_2$ reduces to $$P_2={{\kappa}\over \rho^2}\Bigl\{|D||D({\bf z}^*,{\bf z}_t^*;{\bf z},{\bf z}_t)|-|D({\bf z}^*;{\bf z})||D({\bf z}_t^*;{\bf z}_t)| +|D({\bf z}^*;{\bf z}_t)||D({\bf z}_t^*;{\bf z})|\Bigr\}$$ $$+{{\rm i}\over \rho^2}\Bigl\{-|D||D({\bf z}^*,{\bf z}_t^*;{\bf z}_x,{\bf z}_t)|+|D({\bf z}^*;{\bf z}_x)||D({\bf z}_t^*;{\bf z}_t)| -|D({\bf z}^*;{\bf z}_t)||D({\bf z}_t^*;{\bf z}_x)|\Bigr\}$$ $$\!+{1\over \rho^2}\Bigl\{\!-|D({\bf z}^*;{\bf z})||D({\bf z}^*,{\bf z}_t^*;{\bf z}_x,{\bf z}_t)|+|D({\bf z}^*;{\bf z}_x)||D({\bf z}_t^*,{\bf z}^*;{\bf z}_t,{\bf z})| -|D({\bf z}^*;{\bf z}_t)||D({\bf z}^*,{\bf z}_t^*;{\bf z},{\bf z}_x)|\Bigr\}. \eqno(3.19)$$ The first and second terms on the right-hand side of (3.19) vanish by virtue of Jacobi’s identity. To show that the third term becomes zero as well, we consider the determinantal identity $$\begin{vmatrix}|D({\bf z}^*;{\bf z})| & |D({\bf z}^*;{\bf z})|& |D({\bf z}_t^*;{\bf z})|\\ |D({\bf z}^*;{\bf z}_x)| & |D({\bf z}^*;{\bf z}_x)|& |D({\bf z}_t^*;{\bf z}_x)| \\ |D({\bf z}^*;{\bf z}_t)| & |D({\bf z}^*;{\bf z}_t)|& |D({\bf z}_t^*;{\bf z}_t)| \end{vmatrix}=0.$$ It is obvious that this determinant is zero since the first two columns coincide. The above assertion follows immediately by expanding the determinant with respect to the first column and using Jacobi’s identity. Consequently, $P_2=0$. $\Box$ Before proceeding to the proof of (2.10), we emphasis that the constraints (3.2c) have not been used in the process of the proof of (2.2) and (2.3). On the other hand, we find that the proof of (2.4) depends crucially on the constraints. This is an obstacle which has never been encountered in performing the proof of the bright $N$-soliton solution (see I). In conclusion, a direct proof of (2.4) still remains open and hence we shall prove the trilinear equation (2.10) instead. It turns out, however that its proof is found to be unfeasible. As we shall now demonstrate, introduction of an auxiliary variable $\tau$ in the exponential function (3.2) would resolve this difficulty. [*3.4.3. Proof of (2.10)*]{} We first prepare the two lemmas to prove (2.10). The lemma 3.3 below gives a very simple relation between the partial derivatives $f_t$ and $f_\tau$. It is to be noted that the constraints (3.2c) are used only for the proof of this lemma. [**Lemma 3.3.**]{} $$f_t=(1+\kappa\rho^2)f_\tau, \eqno(3.20a)$$ $$g_t=(1+\kappa\rho^2)g_\tau. \eqno(3.20b)$$ [**Proof.**]{} Extracting the factor $z_j$ from the $j$th row and the factor $z_k^*$ from the $k$th column of the determinant $|D|$, respectively for $j, k=1, 2, ...,N$, one can rewrite the tau function $f$ into the form $$f=\prod_{j=1}^N{\rm e}^{\zeta_j}\left|\left({\rm e}^{-\zeta_j}\delta_{jk}+{\kappa-{\rm i}p_j\over p_j+p_k^*}\right)_{1\leq j,k\leq N}\right|,$$ where $$\zeta_j=(p_j+p_j^*)x+\kappa\rho^2\left({1\over p_j}+{1\over p_j^*}\right)t+{p_j+p_j^*\over (p_j+{\rm i}\kappa)(p_j^*-{\rm i}\kappa)}\,\tau+\zeta_{j0}+\zeta_{j0}^*,$$ showing that $f$ can be regarded as a function of $\zeta_j\ (j=1, 2, .., N)$. Thus, differentiation of $f$ with respect to $t$ gives $$f_t=\sum_{j=1}^N{\partial f\over\partial\zeta_j}{\partial\zeta_j\over\partial t}=\kappa\rho^2\sum_{j=1}^N\left({1\over p_j}+{1\over p_j^*}\right){\partial f\over\partial\zeta_j}.$$ Similarly, one has $$f_\tau=\sum_{j=1}^N{p_j+p_j^*\over (p_j+{\rm i}\kappa)(p_j^*-{\rm i}\kappa)}{\partial f\over\partial\zeta_j}.$$ The constraints (3.2c) are introduced into the above expression to give $$f_\tau={\kappa\rho^2\over 1+\kappa\rho^2}\sum_{j=1}^N\left({1\over p_j}+{1\over p_j^*}\right){\partial f\over\partial\zeta_j}={1\over 1+\kappa\rho^2\,}f_t.$$ This completes the proof of (3.20a). Repeating the similar procedure, one can show that the relation (3.20b) holds as well. $\Box$ The lemma 3.4 gives the differentiation rules of $f$ and $g$ with respect to $\tau$: [**Lemma 3.4.**]{} $$f_\tau={\rm i}|D({\bf z}_\tau^*;{\bf z})|, \eqno(3.21)$$ $$f_{\tau}^*=-{\rm i}|D({\bf z}^*;{\bf z}_\tau)|, \eqno(3.22)$$ $$g_\tau={{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z})|+{1\over \rho^2}|D({\bf z}_t^*;{\bf z}_\tau)|, \eqno(3.23)$$ $$g_{x\tau}={\rm i}|D({\bf z}^*;{\bf z})|+{1\over \rho^2}|D({\bf z}_t^*;{\bf z})|+\kappa |D({\bf z}^*;{\bf z}_\tau)|+{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)| -{{\rm i \kappa}\over \rho^2}|D({\bf z}_t^*;{\bf z}_\tau)|$$ $$-{1\over \kappa\rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z},{\bf z}_x)|-{\kappa\over \rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z}_\tau,{\bf z})| +{{\rm i}\over \rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z}_\tau,{\bf z}_x)|. \eqno(3.24)$$ [**Proof.**]{} If one notes the relations $${\bf z}_{t\tau}=-{{\rm i}\over \kappa}{\bf z}_t+{\rm i}\rho^2{\bf z}_{\tau},\qquad {\bf z}_{x\tau}={\bf z}-{\rm i}\kappa{\bf z}_{\tau},$$ which follows from the definition (3.2a) of $z_j$, the proof can be done straightforwardly along with the same procedure as that used in the proof of lemma 3.1 and lemma 3.2. $\Box$ With lemmas (3.2) and (3.3) at hand, we are now ready for starting the proof of (2.10). [**Proof of (2.10).**]{} If one replaces the $t$ derivative by the $\tau$ derivative in accordance with (3.20), the trilinear equation (2.10) can be rewritten in the form $$f^*P_3=f_\tau^*P_3^\prime, \eqno(3.25a)$$ with the bilinear forms $P_3$ and $P_3^\prime$ defined respectively by $$P_3=g_{x\tau}f-(f_x-{\rm i}\kappa f)g_\tau-{\rm i\over \kappa}(g_xf-gf_x), \eqno(3.25b)$$ $$P_3^\prime=g_xf-gf_x+{\rm i}\kappa fg. \eqno(3.25c)$$ The trilinear equation (3.25) is proved as follows. Substituting (3.8), (3.10), (3.13), (3.23) and (3.24) into (3.25b) and applying Jacobi’s identity to terms multiplied by $|D|$, $P_3$ is simplified considerably. After some elementary calculations, one finds that $$P_3=\kappa |D({\bf z}^*;{\bf z}_\tau)|\Bigl\{|D|+{1\over\rho^2}|D({\bf z}_t^*;{\bf z})|-{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)|\Bigr\}. \eqno(3.26a)$$ Performing the similar calculation for $P_3^\prime$, one obtains $$P_3^\prime={\rm i}\kappa\Bigl\{|D|-|D({\bf z}^*;{\bf z})|\Bigr\}\Bigl\{|D|+{1\over\rho^2}|D({\bf z}_t^*;{\bf z})|-{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)|\Bigr\}. \eqno(3.26b)$$ Taking into account the formulas (3.14) and (3.22), the expressions (3.26a) and (3.26b) yield (3.25). The trilinear equation (3.25) coupled with lemma 3.3 now completes the proof of the trilinear equation (2.10). $\Box$ [*3.5. Dark $N$-soliton solution of the derivative NLS equation*]{} In accordance with the fact that the FL equation is the first negative flow of the Lax hierarchy of the derivative NLS equation, the spatial part of the Lax pair associated with the former equation coincides with that of the latter equation with an identification $q=u_x\ [2, 5]$. This observation enables us to obtain the dark $N$-soliton solution of the derivative NLS equation $${\rm i}q_t+q_{xx}+2{\rm i}(|q|^2q)_x=0,\quad q=q(x,t), \eqno(3.27)$$ under the boundary condition $$q\rightarrow \rho\,{\rm exp}\left\{{\rm i}\left(\kappa x-\omega^\prime t+\psi^{(\pm)}\right)\right\}, \quad x\rightarrow \pm\infty, \eqno(3.28)$$ where $\omega^\prime=\kappa^2+2\kappa\rho^2$ and $\psi^{(\pm)}$ are real phase constants. In particular, we establish the following proposition: [**Proposition 3.1.**]{} [*The dark $N$-soliton solution of the derivative NLS equation (3.27) subjected to the boundary condition (3.28) is given in terms of the tau functions $f$ and $h$ by $$q=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega^\prime t)}\,{hf^*\over f^2}, \eqno(3.29a)$$ with $$f=|D|, \quad h=|H|. \eqno(3.29b)$$ Here, $D$ and $H$ are $N\times N$ matrices defined respectively by $$D=(d_{jk})_{1\leq j,k\leq N}, \quad d_{jk}=\delta_{jk}+{\kappa-{\rm i}p_j\over p_j+p_k^*}\,z_jz_k^*, \quad z_j={\rm exp}\left[p_jx\!+\!\{{\rm i}p_j^2-2(\kappa+\rho^2)p_j\}t+\zeta_{j0}\right], \eqno(3.30a)$$ $$H=(h_{jk})_{1\leq j,k\leq N}, \quad h_{jk}=\delta_{jk}-{\kappa-{\rm i}p_j\over p_j+p_k^*}\,{p_j\over p_k^*}\,z_jz_k^*, \eqno(3.30b)$$ where $p_j$ are complex parameters satisfying the constraints $$p_jp_j^*=\rho^2\{\kappa-{\rm i}(p_j-p_j^*)\},\quad j=1, 2, ..., N, \eqno(3.30c)$$ and $\zeta_{j0}\ (j=1, 2, ..., N)$ are arbitrary complex parameters.*]{} [**Proof.**]{} The correspondence between $q$ and $u_x$ mentioned above implies that the relation $$q=u_x={\partial\over\partial x}\left(\rho\,{\rm e}^{{\rm i}\kappa x}\,{g\over f}\right)=\rho\,{\rm e}^{{\rm i}\kappa x}{1\over f^2}(g_xf-gf_x+{\rm i}\kappa fg),$$ holds at $t=0$. On the other hand, the expression in the parentheses on the right-hand side is just $P_3^\prime$ defined by (3.25c) and hence it is equal to (3.26b). This fact and (3.14) lead, after applying the formula (3.6), to $$\begin{aligned} g_xf-gf_x+{\rm i}\kappa fg &={\rm i}\kappa\Bigl\{|D|-|D({\bf z}^*;{\bf z})|\Bigr\}\Bigl\{|D|+{1\over\rho^2}|D({\bf z}_t^*;{\bf z})|-{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)|\Bigr\} \notag \\ &={\rm i}\kappa f^*\begin{vmatrix} D & \left(z_j-{{\rm i}p_j\over\kappa}z_j\right)_{1\leq j\leq N}^T \\ \left({\kappa\over p_k^*}z_k^*\right)_{1\leq k\leq N} & 1\end{vmatrix}. \notag\end{aligned}$$ Multiplying the $(N+1)$th column of the determinant by $\kappa z_k^*/p_k^*$ and subtracting it from the $k$th column for $k=1, 2, ..., N$, one finds that the above expression becomes ${\rm i}\kappa f^*h$. Consequently, $$q={\rm i}\kappa \rho\,{\rm e}^{{\rm i}\kappa x}\,{f^*h\over f^2}\Bigg|_{t=0}.$$ If one replaces $q$ by ${\rm i}q$ and $\rho$ by $\rho/\kappa$, respectively and introduces the time dependence appropriately, one arrives at (3.29) with (3.30). The constraints (3.30c) follow from (3.2c) by the above replacement of $\rho$. The complex parameters $p_j$ subjected to the constraints (3.30c) exist only if the condition $\kappa+\rho^2>0$ is satisfied. $\Box$ It is instructive to perform the bilinearization of the derivative NLS equation under the boundary condition (3.28). This provides an alternative way to construct the dark $N$-soliton solution given by proposition 3.1, as we shall see now. To this end, following the procedure used in \[11, 12\], we introduce the gauge transformation $$q=v\,{\rm exp}\left[{\rm i}\int_{-\infty}^x(\rho^2-|v|^2)dx\right], \eqno(3.31a)$$ as well as the dependent variable transformation for $v$ $$v=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega^\prime t)}\,{h\over f}. \eqno(3.31b)$$ Then, equation (3.27) can be decoupled to the system of bilinear equations for $f$ and $h$ $$D_xf\cdot f^*-{\rm i}\rho^2(hh^*-ff^*)=0, \eqno(3.32)$$ $$D_x^2f\cdot f^*-{\rm i}\rho^2D_xh\cdot h^* +\rho^2(2\kappa+\rho^2)(hh^*-ff^*)=0, \eqno(3.33)$$ $${\rm i} D_th\cdot f+2{\rm i}(\kappa+\rho^2)D_xh\cdot f+D_x^2h\cdot f=0. \eqno(3.34)$$ In view of (3.32), the modulus of $v$ is given in terms of the tau function $f$ by $$|v|^2=\rho^2+{\rm i}\,{\partial \over \partial x}\,{\rm ln}\,{f^*\over f}, \eqno(3.35)$$ which, combined with (3.31), yields the formula (3.29). Note from (3.31a) that $|q|^2=|v|^2$. It may be checked by direct computation that the tau functions $f$ and $h$ from (3.29b) with (3.30) satisfy the above bilinear equations. It is important to realize that we can take the limit $\kappa\rightarrow 0$ for the solution (3.29) since the dispersion relation is not singular at $\kappa=0$. This gives the $N$-soliton solution of the derivative NLS equation on a constant background which has been studied extensively using various exact methods of solution such as the IST \[13-16\], Bäcklund transformation \[17, 18\] and Hirota’s direct method \[19\]. On the other hand, for the dark $N$-soliton solution given by (2.1), this limiting procedure is not relevant because of the singular nature of the dispersion relation. Last, we shall briefly describe the properties of the one-soliton solution for the purpose of comparison with those of the one-soliton solution of the FL equation. Introducing the new real parameters $a_1$ and $b_1$ by $p_1=a_1+{\rm i}b_1$, the square of the modulus of the one-soliton solution from (3.29) and (3.30) with $N=1$ can be written in the form $$|q_1|^2=\rho^2-{2a_1^2\,{\rm sgn}\, a_1\over \sqrt{a_1^2+(\kappa+b_1)^2}}\,{1\over \cosh\,2(\theta_1+\delta_1)+{(\kappa+b_1){\rm sgn}\, a_1\over \sqrt{a_1^2+(\kappa+b_1)^2}}}. \eqno(3.36a)$$ with $$\theta_1=a_1(x+c_1t)+\theta_{10},\qquad c_1=2(b_1+\kappa+\rho^2),\qquad {\rm e}^{4\delta_1}={a_1^2+(\kappa+b_1)^2\over 4a_1^2}, \eqno(3.36b)$$ where ${\rm sgn}\, a_1$ denotes the sign of $a_1$, i.e., $a_1=1$ for $a_1>0$ and $a_1=-1$ for $a_1<0$, and $\theta_{10}$ is a real constant. The constraint (3.30c) then becomes $$a_1^2+b_1^2=\rho^2(2b_1+\kappa). \eqno(3.37)$$ Using (3.36b) and (3.37), the parameters $a_1$ and $b_1$ are expressed in terms of the velocity $c_1$ of the soliton as $$a_1^2={1\over 4}\left(c_{\rm max}-c_1\right)\left(c_1-c_{\rm min}\right),\qquad b_1={c_1\over 2}-\kappa-\rho^2, \qquad c_{\rm min}<c_1<c_{\rm max}, \eqno(3.38a)$$ where $$c_{\rm max}=2(\kappa+2\rho^2)+2\rho\sqrt{\kappa+\rho^2},\qquad c_{\rm min}=2(\kappa+2\rho^2)-2\rho\sqrt{\kappa+\rho^2}. \eqno(3.38b)$$ One must impose the condition $\kappa+\rho^2> 0$ to assure the existence of soliton solutions. Recall that this condition coincides with a criterion for the stability of the plane wave (3.28) \[20\]. We see from (3.36) that if $a_1>0$, then $|q_1|$ takes the form of a dark soliton whereas if $a_1<0$, it becomes a bright soliton on a constant background $u=\rho$. Let $A_d$ and $A_b$ be the amplitudes of the dark and bright solitons, respectively with respective to the background. The amplitude-velocity relations follow from (3.36) and (3.38). They read $$A_d=\rho-\left|\sqrt{c_1-\kappa-2\rho^2}-\sqrt{\kappa+\rho^2}\right|, \eqno(3.39a)$$ $$A_b=\sqrt{c_1-\kappa-2\rho^2}+\sqrt{\kappa+\rho^2}-\rho. \eqno(3.39b)$$ The detailed analysis for the case $\kappa>0$ has been undertaken in \[21\]. To sum up, the solution has been shown to exhibit the spiky modulation of the amplitude and phase. It also has been demonstrated that the bright soliton reduces to an algebraic soliton for both limits $c_1\rightarrow c_{\rm max}$ and $c_1\rightarrow c_{\rm min}$ whereas the algebraic dark soliton never exists. In the case $\kappa<0$ which has not been treated in \[21\], however, a careful inspection of (3.36) and (3.38) reveals that the algebraic bright and dark solitons are produced in the limit $c_1\rightarrow c_{\rm max}$ and $c_1\rightarrow c_{\rm min}$, respectively. The latter new feature is pointed out here for the first time. [**Remark 3.3.**]{} Using the result obtained in proposition 3.1, we can construct the dark $N$-soliton solution of the modified NLS equation $${\rm i}q_t+q_{xx}+\mu |q|^2q+{\rm i}\gamma(|q|^2q)_x=0,\quad q=q(x,t), \eqno(3.40)$$ under the boundary condition $$q\rightarrow \rho\,{\rm exp}\left\{{\rm i}\left(\kappa x-\omega^{\prime\prime} t+\psi^{(\pm)}\right)\right\}, \quad x\rightarrow \pm\infty, \eqno(3.41)$$ where $\omega^{\prime\prime}=\kappa^2-\mu\rho^2+\gamma\kappa\rho^2$ and $\mu$ and $\gamma$ are real constants. To show this, we apply the gauge transformation $$q={\rm exp}\left[{\mu\over\gamma}\tilde x+\left({\mu\over\gamma}\right)^2\tilde t\right]\tilde q, \quad x=\tilde x+{2\mu\over\gamma}\,\tilde t, \quad t=\tilde t, \eqno(3.42)$$ to equation (3.40) and see that it can be recast to the derivative NLS equation ${\rm i}\tilde q_{\tilde t}+\tilde q_{\tilde x\tilde x}+\gamma (|\tilde q|^2\tilde q)_{\tilde x}=0$, which coincides with equation (3.27) with the identification $\tilde q=q,\ \tilde x=x,\ \tilde t=t$ and $\gamma=2$. The dark $N$-soliton solution of the equation (3.40) then takes the form $$q= \rho\,{\rm e}^{{\rm i}(\kappa x-\omega^{\prime\prime} t)}\,{{f^\prime}^*h^\prime\over {f^\prime}^2}, \eqno(3.43a)$$ where the tau functions $f^\prime$ and $h^\prime$ are given respectively by $$f^\prime=\left|\left(\delta_{jk}+{\kappa-{\mu\over\gamma}-{\rm i}p_j\over p_j+p_k^*}\,z_jz_k^*\right)_{1\leq j,k\leq N}\right|,\eqno(3.43b)$$ $$h^\prime=\left|\left(\delta_{jk}-{\kappa-{\mu\over\gamma}-{\rm i}p_j\over p_j+p_k^*}{p_j\over p_k^*}\,z_jz_k^*\right)_{1\leq j,k\leq N}\right|,\eqno(3.43c)$$ with $$z_j={\rm exp}\left[p_jx\!+\!\{{\rm i}p_j^2-(2\kappa+\gamma\rho^2)p_j\}t+\zeta_{j0}\right]. \eqno(3.43d)$$ The constraints for $p_j$ become $$p_jp_j^*={\gamma\rho^2\over 2}\left\{\kappa-{\mu\over\gamma}-{\rm i}(p_j-p_j^*)\right\},\quad j=1, 2, ..., N. \eqno(3.44)$$ The complex parameters $p_j$ exist only if the condition $\gamma\left(\kappa-{\mu\over\gamma}+{\gamma\rho^2\over 2}\right)>0$ is satisfied. The following two special cases are worth remarking. The case $\mu=0$ and $\gamma=2$ reduces to the result given by proposition 3.1. On the other hand, in the limit $\gamma\rightarrow 0$ while $\mu$ being fixed, we first replace $z_j$ by $\sqrt \gamma z_j$ for $j=1, 2, ..., N$ and then take the limit, producing the dark $N$-soliton solution of the NLS equation. Note, in this limit, that the constraints (3.44) reduce to $p_jp_j^*=-\mu\rho^2/2$ and hence the dark soliton solutions exist only if the condition $\mu<0$ is satisfied. [*3.6. Stability of the plane wave*]{} We have considered the dark solitons on the background of a plane wave $\rho\,{\rm e}^{{\rm i}(\kappa x-\omega t)}$ with $\omega =1/\kappa +2\rho^2$. It is important to see whether the background field is stable or not against perturbations. If unstable, then dark solitons would not exist, as will be demonstrated in the next section. To this end, we perform the linear stability analysis of the plane wave. Following the standard procedure, we seek a solution of the form $$u=(\rho+\Delta\rho)\,{\rm e}^{{\rm i}(\kappa x-\omega t+\Delta\phi)}, \eqno(3.45)$$ where $\Delta\rho=\Delta\rho(x,t)$ and $\Delta\phi=\Delta\phi(x,t)$ are small perturbations. Substituting (3.45) into the FL equation (1.1) and linearizing about the plane wave, we obtain the system of linear PDEs for $\Delta\rho$ and $\Delta\phi$ $$\Delta\rho_{xt}+\rho(\omega-2\rho^2)\Delta\phi_x-\kappa\rho\Delta\phi_t-4\kappa\rho^2\Delta\rho=0, \eqno(3.46a)$$ $$\rho\Delta\phi_{xt}-(\omega-2\rho^2)\Delta\rho_x+\kappa\Delta\rho_t=0. \eqno(3.46b)$$ Assume the perturbations of the form ${\rm e}^{{\rm i}(\lambda x-\nu t)}$ with $\lambda$ real and $\nu$ possibly complex and substitute them into (3.46) to obtain a homogeneous linear system for $\Delta\rho$ and $\Delta\phi$ $$(\lambda\nu-4\kappa\rho^2)\Delta\rho+{\rm i}\{\rho\lambda(\omega-2\rho^2)+\kappa\rho\nu\}\Delta\phi=0, \eqno(3.47a)$$ $$-{\rm i}\{(\omega-2\rho^2)\lambda+\kappa\nu\}\Delta\rho+\rho\lambda\nu\Delta\phi=0. \eqno(3.47b)$$ The nontrivial solution exists if $\nu$ satisfies the quadratic equation $$(\lambda^2-\kappa^2)\nu^2-2(2\kappa\rho^2+1)\lambda\nu-{\lambda^2\over\kappa^2}=0. \eqno(3.48)$$ Solving this equation, we obtain $$\nu={\lambda\over \lambda^2-\kappa^2}\left[2\kappa\rho^2+1 \pm {1\over\kappa}\sqrt{\lambda^2+4\kappa^3(\kappa\rho^2+1)\rho^2}\right]. \eqno(3.49)$$ Thus, if the condition $$\kappa(\kappa\rho^2+1)>0, \eqno(3.50)$$ is satisfied, then $\nu$ becomes real for all values of real $\lambda$, implying that the plane wave is neutrally stable. It is evident that this condition always holds for $\kappa>0$. For negative $\kappa$, on the other hand, we put $\kappa=-K$ with $K>0$ and see that the stability criterion turns out to be as $K\rho^2>1$. Last, we remark that a similar stability analysis has been performed recently in conjunction with a plane wave solution of the original version of the FL equation \[22, 23\]. [**4. Properties of the soliton solutions**]{} In this section, we detail the properties of the soliton solutions. To this end, we first parametrize the complex parameters $p_j$ and $\zeta_{j0}$ by the real quantities $a_j, b_j, \theta_{j0}$ and $\chi_{j0}$ as $$p_j=a_j+{\rm i}b_j, \qquad \zeta_{j0}=\theta_{j0}+{\rm i}\chi_{j0}, \qquad j=1, 2, ..., N,\eqno(4.1)$$ and introduce the new independent variables $\theta_j$ and $\chi_j$ according to the relations $$\quad \theta_j=a_j(x+c_jt)+\theta_{j0}, \qquad c_j={\kappa\rho^2\over a_j^2+b_j^2}, \qquad j=1, 2, ..., N. \eqno(4.2a)$$ $$\chi_j=b_j(x-c_jt)+\chi_{j0}, \qquad j=1, 2, ..., N. \eqno(4.2b)$$ In terms of these variables, the variables $z_j$ defined by (3.2a) are put into the form $$z_j={\rm e}^{\theta_j+{\rm i}\chi_j}, \qquad j=1, 2, ..., N, \eqno(4.2c)$$ after setting $\tau=0$. Substituting (4.1) into (3.2c), the constraints for $p_j$ can be rewritten as a quadratic equation for $b_j$ $$b_j^2-2\kappa^2\rho^2b_j+a_j^2-\kappa^3\rho^2=0, \qquad j=1, 2, ..., N.\eqno(4.3)$$ The solution to this equation is found to be as follows: $$b_j=(\kappa\rho)^2\pm\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)-a_j^2},\qquad j=1, 2, ..., N.\eqno(4.4)$$ We can see from the above expression that the real $b_j\ (j=1, 2, ..., N)$ exist only when the condition $\kappa^3\rho^2(1+\kappa\rho^2)>0$ is satisfied. This coincides with the criterion (3.50) for the stability of the plane wave, as discussed in section 3.6. Throughout the analysis, we assume this condition to assure the existence of soliton solutions. It is to be noted from (4.2) and (4.3) that the parameters $a_j$ and $b_j$ are expressed in terms of $c_j$ as $$a_j^2={\kappa^2\over 4c_j^2}\left(c_{\rm max}-c_j\right)\left(c_j-c_{\rm min}\right), \qquad b_j={1\over 2\kappa c_j}(1-\kappa^2 c_j), \qquad c_{\rm min}<c_j<c_{\rm max}, \eqno(4.5a)$$ where $$c_{\rm max}={1\over \kappa^2}\left\{1+2\kappa\rho^2+2\sqrt{\kappa\rho^2(1+\kappa\rho^2)}\right\}, \qquad c_{\rm min}={1\over \kappa^2}\left\{1+2\kappa\rho^2-2\sqrt{\kappa\rho^2(1+\kappa\rho^2)}\right\}. \eqno(4.5b)$$ The relations (4.5) correspond to (3.38) for those of the one-soliton solution of the derivative NLS equation. Thus, the dark $N$-soliton solution is characterized by the $N$ velocities $c_j\,(j=1, 2, ..., N)$ and the $2N$ real phase constants $\theta_{j0}$ and $\chi_{j0}\, (j=1, 2, ..., N)$, the total number of which is $3N$. Another parameterization of the solution is possible if one introduces the angular variables $\gamma_j$ by $$a_j=\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\, \sin\,\gamma_j, \eqno(4.6a)$$ $$b_j=(\kappa\rho)^2+\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\, \cos\,\gamma_j, \quad 0< \gamma_j <2\pi, \quad \gamma_j\not=\pi, \quad j=1, 2, ..., N. \eqno(4.6b)$$ In terms of $\gamma_j$, $p_j$ from (4.1) can be written in the form $$p_j={\rm i}\left\{(\kappa\rho)^2+\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\,{\rm e}^{-{\rm i}\gamma_j}\right\}, \quad j=1, 2, ..., N, \eqno(4.7)$$ and the velocity $c_j$ of the $j$th soliton given in (4.2a) is expressed as $$c_j={1\over \kappa^2\{1+4\kappa\rho^2(1+\kappa\rho^2)\,\sin^2\gamma_j\}}\left\{1+2\kappa\rho^2-2\,{\rm sgn}\,\kappa\,\sqrt{\kappa\rho^2(1+\kappa\rho^2)}\,\cos\,\gamma_j\right\}. \eqno(4.8)$$ It follows from the above parametric representation that $p_j$ lies on the circle of radius $\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}$ centered at ${\rm i}(\kappa\rho)^2$ in the complex plane. Let us first describe the properties of the one- and two-soliton solutions and then address the general $N$-soliton solution. [*4.1. One-soliton solution*]{} The tau functions $f=f_1$ and $g=g_1$ for the one-soliton solution follows from (3.1)-(3.3) with $N=1$. They read $$f_1=1+{\kappa-{\rm i}p_1\over p_1+p_1^*}\,z_1z_1^*,\qquad g_1=1-{\kappa+{\rm i}p_1^*\over p_1+p_1^*}{p_1\over p_1^*}\,z_1z_1^*. \eqno(4.9)$$ The one-soliton solution $u_1$ follows from (2.1) with (4.9), yielding $$u_1=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega t)}\,{1-{\kappa+b_1+{\rm i}a_1\over 2a_1}\,{a_1+{\rm i}b_1\over a_1-{\rm i}b_1}\,{\rm e}^{2\theta_1}\over 1+{\kappa+b_1-{\rm i}a_1\over 2a_1}\,{\rm e}^{2\theta_1}}. \eqno(4.10)$$ The above expression can be put into the form $$u_1=|u_1|\,{\rm e}^{{\rm i}(\kappa x-\omega t)}{\rm exp}\left\{{\rm i}\left(\phi+\phi^{(+)}\right)\right\}, \eqno(4.11)$$ where the square of the modulus of $u_1$ is represented by $$|u_1|^2=\rho^2-{2a_1^2c\,{\rm sgn}(\kappa a_1)\over \sqrt{a_1^2+(\kappa+b_1)^2}}\,{1\over \cosh\,2(\theta_1+\delta_1)+{(\kappa+b_1)\,{\rm sgn}\,a_1\over \sqrt{a_1^2+(\kappa+b_1)^2}}}, \qquad c=|c_1|, \eqno(4.12a)$$ with $$\theta_1=a_1(x+c_1t)+\theta_{10},\qquad c_1={\kappa\rho^2\over a_1^2+b_1^2},\qquad {\rm e}^{4\delta_1}={a_1^2+(\kappa+b_1)^2\over 4a_1^2}, \eqno(4.12b)$$ and the tangent of the phase $\phi$ and $\phi^{(+)}$ being given respectively by $$\tan\, \phi={\{a_1^2+b_1(\kappa+b_1)\}\,\cosh\,2(\theta_1+\delta_1)+b_1\,{\rm sgn}\, a_1\sqrt{a_1^2+(\kappa+b_1)^2}\over \kappa a_1\,\sinh\,2(\theta_1+\delta_1)}, \eqno(4.13a)$$ $$\tan\,\phi^{(+)}={a_1^2+b_1(\kappa+b_1)\over \kappa a_1}. \eqno(4.13b)$$ It can be confirmed by direct substitution that (4.11) indeed satisfies the FL equation. The one-soliton solution (4.10) is a one-parameter family of solutions. The parameterization in terms of $a_1$ will be employed in classifying the soliton solutions. The parameters $c_1$ and $b_1$ are then expressed by $a_1$. See (4.2a) and (4.4) whereas the parameters $\rho$ and $\kappa$ are fixed by the boundary condition (1.2). The relation (4.5) will be used conveniently when considering the generation of algebraic solitons in the limit $|a_1|\rightarrow 0$. The form of $|u_1|$ from (4.12) reveals that If $\kappa a_1>0$, then $|u_1|$ takes the form of a dark soliton whereas if $\kappa a_1<0$, it becomes a bright soliton on a constant background $u=\rho$. Note from (4.12) that the width of the soliton may be defined by $(2|a_1|)^{-1}$. The net change of the phase caused by the effect of nonlinear modulation is given by (4.13). Roughly speaking, the phase $\phi$ behaves like a step function as a function of $\theta_1$. Specifically, a rapid change of the phase occurs in the vicinity of the center position of the soliton ($\theta_1=-\delta_1$), yielding a phase difference $\pi$ (or $-\pi$). As a result, the phase of $u_1$ changes by a quantity $2\phi^{(+)}$ as $\theta_1$ varies from $-\infty$ to $+\infty$, where $\phi^{(+)}$ is given by (4.13b). Let us classify the one-soliton solutions in accordance with the sign of $\kappa$. We consider the two cases, i.e., case 1 ($\kappa>0, a_1\lessgtr 0$) and case 2 ($\kappa<0, a_1\lessgtr 0$) separately. For each sign of $\kappa$, both dark and bright solitons arise, as we shall show now. [*4.1.1. Case 1: $\kappa>0$* ]{} In this case, the velocity $c_1$ of the soliton is positive, as evidenced from (4.12b). Let $A_d$ and $A_b$ be the amplitudes of the dark and bright solitons, respectively with respective to the background. We then find from (4.5) and (4.12) that $$\begin{aligned} A_d &= \rho-\sqrt{\rho^2-2c_1\left\{\sqrt{a_1^2+(\kappa+b_1)^2}-(\kappa+b_1)\right\}} \notag \\ &= \rho-{1\over\sqrt{\kappa}}\left|\kappa\sqrt{c}-\sqrt{1+\kappa\rho^2}\right|, \qquad a_1>0, \quad c_1=c>0,\tag{4.14}\end{aligned}$$ $$\begin{aligned} A_b &=\sqrt{\rho^2+2c_1\left\{\sqrt{a_1^2+(\kappa+b_1)^2}+(\kappa+b_1)\right\}}-\rho \notag \\ &={1\over\sqrt{\kappa}}\left(\kappa\sqrt{c}+\sqrt{1+\kappa\rho^2}\right)-\rho, \qquad a_1<0, \quad c_1=c>0,\tag{4.15}\end{aligned}$$ where $c\equiv |c_1|$ lies in the interval $c_{\rm min}<c<c_{\rm max}$ with $c_{\rm max}$ and $c_{\rm min}$ being given by (4.5b). Note from (4.5a) that $\kappa+b_1=(1+\kappa^2c_1)/(2\kappa c_1)>0$ for $\kappa>0$ and $c_1>0$. This estimate will be used to judge the existence of algebraic solitons in the limit of infinite width. {width="10cm"} [**Figure 1.**]{} Amplitude-velocity relation for the dark soliton $A_d$ (solid line) and bright soliton $A_b$ (broken line) for $\rho=1$ and $\kappa=2$. Figure 1 plots the dependence of the amplitudes $A=A_d$ and $A=A_b$ on the velocity $c=|c_1|$ for $\rho=1$ and $\kappa=2$. {width="10cm"} [**Figure 2.**]{} Profile of the amplitude of the dark soliton $U=|u_1|$ at $t=0$. a: $c=c_0=0.75$, b: $c=0.33$, c: $c=0.098$. The profile a is a black soliton. [*(i) Dark soliton: $a_1>0$*]{} As seen from figure 1, the amplitude $A_d$ of the dark soliton becomes an increasing function of the velocity $c$ in the interval $c_{\rm min}< c\leq c_0$ and a decreasing function in the interval $c_0<c< c_{\rm max}$, where $c_{\rm max}\, (\gamma_1=\pi)$ and $c_{\rm min}\,(\gamma_1=0)$ are given by (4.5b) and a critical velocity $c_0$ and the corresponding angle $\gamma_0$ by $$c_0={1+\kappa\rho^2\over \kappa^2}, \qquad {\rm at}\quad \gamma_1=\gamma_0=\cos^{-1}\left[-{(\kappa\rho^2)^{1\over 2}(3+2\kappa\rho^2)\over 2(1+\kappa\rho^2)^{3\over 2}}\right], \qquad (0<\gamma_0<\pi). \eqno(4.16)$$ In the present numerical example ($\rho=1, \kappa=2$), $c_{\rm min}=0.025, c_0=0.75, c_{\rm max}=2.47$. The above observation shows that in the interval $c_0<c<c_{\rm max}$, a small dark soliton propagates faster than a large dark soliton. A similar behavior has also been found in I for the bright soliton solutions of the FL equation with zero background. Figure 2 depicts the profile of $U=|u_1|$ at $t=0$ for three different values of $c$, i.e., a: $c=c_0=0.75 (\gamma_1=\gamma_0=0.90\pi)$, b: $c=0.33 (\gamma_1=5\pi/6)$, c: $c=0.098 (\gamma_1=2\pi/3)$ with the parameters $\rho=1, \kappa=2, \theta_{10}=-\delta_1$ and $\chi_{10}=0$. When $c=c_0$, the amplitude of the dark soliton attains the maximum value $A_d=\rho$. See figure 2 a. It then turns out that the intensity of the soliton center falls to zero. Such a soliton is well-known in the field of nonlinear optics. It is sometimes called a [*black*]{} soliton. For this specific value of $c$, one finds from (4.5), (4.12) and (4.13) that $$a_1={\kappa^{3\over 2}\rho(4+3\kappa\rho^2)^{1\over 2}\over 2(1+\kappa\rho^2)}, \quad b_1=-{(\kappa\rho)^2\over 2(1+\kappa\rho^2)}, \eqno(4.17a)$$ $${\rm e}^{4\delta_1}={\kappa^2\over 4a_1^2}, \qquad \tan\,\phi=-{b_1\over a_1}\, \tanh(\theta_1+\delta_1),\qquad \tan\,\phi^{(+)}=-{b_1\over a_1}. \eqno(4.17b)$$ The profile of $|u_1|^2$ from (4.12) then becomes $$|u_1|^2=\rho^2\left[1-{4+3\kappa\rho^2\over 2(1+\kappa\rho^2)}{1\over \cosh\,2(\theta_1+\delta_1)+{2+\kappa\rho^2\over 2(1+\kappa\rho^2)}}\right]. \eqno(4.18)$$ As confirmed easily from the above expression, the minimum value of $|u_1|$ is zero at $\theta_1=-\delta_1$. The algebraic dark soliton may be produced from (4.12) by taking the limit $a_1\rightarrow +0$. However, as already noticed, the value of $\kappa+b_1$ is positive so that $|u_1|$ tends simply to a constant value $\rho$. Hence, this limiting procedure is irrelevant for the dark soliton solution under consideration, indicating that the algebraic dark soliton does not exist for $\kappa>0$ and $a_1>0$. {width="10cm"} [**Figure 3.**]{} Profile of a black soliton $u_{\rm R}={\rm Re}\,u_1$ at $t=1$. Figure 3 shows the profile of $u_{\rm R}={\rm Re}[u_1]$ at $t=1$ for the black soliton. The broken line indicates $\pm |u_1|$ (see figure 2 a). One can see that the dark soliton exhibits phase modulations near the center position of the soliton. This peculiar feature is in striking contrast to the bright soliton solution of the NLS equation for which no phase modulation occurs. A similar behavior has been observed for both dark and bright soliton solutions of the derivative NLS equation with the background of a plane wave \[21, 24\]. [*(ii) Bright soliton: $a_1<0$*]{} Figure 4 depicts the profile of the bright soliton $U=|u_1|$ at $t=0$ for three different values of $c$, i.e., a: $c=2.47 (\gamma_1=1.001\pi)$, b: $c=0.73 (\gamma_1=1.1\pi)$, c: $c= 0.025 (\gamma_1=1.999\pi)$ with $\rho=1$ and $\kappa=2$. The feature of the bright soliton differs substantially from that of the dark soliton. To be specific, the amplitude of the bright soliton always becomes an increasing function of the velocity (see figure 1). It takes the maximum value at $c=c_{\rm max} (\gamma_1\rightarrow\pi+0, a_1\rightarrow -0)$ and the minimum value at $c=c_{\rm min} (\gamma_1\rightarrow 2\pi-0, a_1\rightarrow -0)$. At these limiting values of the velocity, the algebraic soliton is produced from the soliton of hyperbolic type. Indeed, if we put $\theta_{10}=a_1x_0-\delta_1$ in (4.10) and (4.12) with $x_0$ being a real constant and then take the limit $a_1\rightarrow -0$, we find $$u_1=\rho\,{\rm e}^{{\rm i}(\kappa x-\omega t)}\,{x+ct+x_0-{\rm i}\,{2\kappa+b_1\over 2b_1(\kappa+b_1)} \over x+ct+x_0-{\rm i}\,{1\over 2(\kappa+b_1)}}, \eqno(4.19a)$$ $$|u_1|^2=\rho^2+{2\kappa c^2\over 1+\kappa^2c}\,{1\over (x+ct+x_0)^2+\left({\kappa c\over 1+\kappa^2c}\right)^2},\eqno(4.19b)$$ where $b_1=(1-\kappa^2c)/2\kappa c$ by (4.5a) and $c=c_{\rm max}$ or $c_{\rm min}$. Note from (4.12b) that $b_1^2=\kappa\rho^2/c$ when $a_1\rightarrow -0$. One can see that the algebraic soliton has no free parameters except a phase constant $x_0$ since the velocity $c$ is determined by $\rho$ and $\kappa$ which are fixed by the boundary condition. To derive (4.19a) from (4.10), we use the following expansion formulas for small $a_1$: $${\rm e}^{2\theta_1} ={2|a_1|\over \sqrt{a_1^2+(\kappa+b_1)^2}}\,{\rm e}^{2a_1(x+ct+x_0)} \sim {2|a_1|\over |\kappa+b_1|}\Big\{1+2a_1(x+ct+x_0)+O(a_1^2)\Big\}, \eqno(4.20a)$$ $${\kappa+b_1-{\rm i}a_1\over 2a_1}\,{\rm e}^{2\theta_1} \sim {\rm sgn}\, a_1\, {\rm sgn}(\kappa+b_1)\left[1+a_1\left\{2(x+ct+x_0)-{\rm i}\,{1\over \kappa+b_1}\right\}+O(a_1^2)\right], \eqno(4.20b)$$ $${\kappa+b_1+{\rm i}a_1\over 2a_1}{a_1+{\rm i}b_1\over a_1-{\rm i}b_1}\,{\rm e}^{2\theta_1}$$ $$\sim -{\rm sgn}\, a_1\, {\rm sgn}(\kappa+b_1)\left[1+a_1\left\{2(x+ct+x_0)-{\rm i}\,{2\kappa+b_1\over b_1(\kappa+b_1)}\right\}+O(a_1^2)\right]. \eqno(4.20c)$$ {width="10cm"} [**Figure 4.**]{} Profile of the amplitude of the bright soliton $U=|u_1|$ at $t=0$. a: $c=2.47$, b: $c=0.73$, c: $c=0.025$. The profiles a and c are algebraic solitons. Because of the inequalities $a_1<0$ and $\kappa+b_1>0$ in the current problem, one finds that the condition ${\rm sgn}\, a_1\, {\rm sgn}(\kappa+b_1)=-1$ is satisfied, which yields (4.19a) by taking the limit $a_1\rightarrow -0$ for (4.10). Actually, under the above condition, the leading-order terms of the denominator and numerator of (4.10) turn out to be of order $a_1$. Consequently, the expression (4.10) has a limiting form (4.19a) in the zero limit of $a_1$. On the other hand, the expression (4.19b) follows either directly from (4.19a) or from (4.12) by performing the similar limiting procedure. {width="10cm"} [**Figure 5.**]{} Profile of an algebraic bright soliton $u_{\rm R}={\rm Re}\,u_1$ at $t=1$. A representative profile of the algebraic bright soliton $U=|u_1|$ at $t=0$ and the corresponding profile of $u_{\rm R}={\rm Re}\,u_1$ at $t=1$ are shown in figure 4 a and figure 5, respectively. The novel feature of the bright soliton mentioned above deserves a few comments. First, the amplitude of the bright soliton tends to a finite value when its width tends to infinity, as opposed to the behavior of the dark soliton discussed just before for which the amplitude becomes zero in this limit. Second, the FL equation has an infinite number of conservation laws \[3\]. Among them, we evaluate the conserved quantity $I=\int_{-\infty}^\infty(|u_x|^2-\kappa^2\rho^2)dx$ for the one-soliton solution (4.10). This quantity may be termed the energy of the soliton in accordance with the correspondence between the solution $u$ of the FL equation and the solution $q$ of the derivative NLS equation. Using the relation $(|u_x|^2)_t=(|u|^2)_x$ which follows directly from the FL equation, we obtain $$I=-4\,{\rm sgn}\, a_1\,\tan^{-1}\left[{1\over |a_1|}\left\{\sqrt{a_1^2+(\kappa+b_1)^2}-(\kappa+b_1){\rm sgn}\, a_1\right\}\right].$$ We find from this expression that in the limit of infinite width $|a_1|\rightarrow 0$, $I$ becomes zero for the dark soliton $(a_1>0)$ and tends to a finite value $2\pi$ for the bright soliton $(a_1<0)$. See also an analogous calculation for the bright soliton solution of the derivative NLS equation with zero background \[25\]. [*4.1.2. Case 2: $\kappa<0$*]{} For negative $\kappa$, the expressions of the amplitude for the dark and bright solitons are given respectively by $$\begin{aligned} A_d &= \rho-\sqrt{\rho^2+2c_1\left\{\sqrt{a_1^2+(\kappa+b_1)^2}+(\kappa+b_1)\right\}} \notag \\ &= \rho-{1\over\sqrt{K}}\left|K\sqrt{c}-\sqrt{K\rho^2-1}\right|, \qquad a_1<0, \quad c_1=-c<0,\tag{4.21}\end{aligned}$$ $$\begin{aligned} A_b &=\sqrt{\rho^2-2c_1\left\{\sqrt{a_1^2+(\kappa+b_1)^2}-(\kappa+b_1)\right\}}-\rho \notag \\ &={1\over\sqrt{K}}\left(K\sqrt{c}+\sqrt{K\rho^2-1}\right)-\rho, \qquad a_1>0, \quad c_1=-c<0,\tag{4.22}\end{aligned}$$ where $K=-\kappa$ is a positive wavenumber and the velocity $c$ lies in the interval $c_{\rm min}^\prime<c<c_{\rm max}^\prime$ with $$c_{\rm max}^\prime={1\over K^2}\left\{2K\rho^2\!-\!1\!+2\sqrt{K\rho^2(K\rho^2\!-\!1)}\right\},\ c_{\rm min}^\prime={1\over K^2}\left\{2K\rho^2\!-\!1\!-\!2\sqrt{K\rho^2(K\rho^2\!-\!1)}\right\}. \eqno(4.23)$$ Recall that the condition $K\rho^2-1>0$ must be imposed to assure the existence of the soliton solutions. {width="10cm"} [**Figure 6.**]{} Amplitude-velocity relation for the dark soliton $A_d$ (solid line) and bright soliton $A_b$ (broken line) for $\rho=1$ and $\kappa=-2$. Figure 6 plots the dependence of the amplitudes $A=A_d$ and $A=A_b$ on the velocity $c=|c_1|$ for $\rho=1$ and $\kappa= -2$. When compared with figure 1 for $\kappa>0$, there appear several different features for $\kappa<0$. In particular, the algebraic [*dark*]{} soliton would arise in the limit $c\rightarrow c_{\rm min}^\prime$ since in this limit, the amplitude $A_d$ tends to a finite value. In addition, the algebraic bright soliton exists only in the limit $c\rightarrow c_{\rm max}^\prime$. We now proceed to the detailed description of the soliton solutions. {width="10cm"} [**Figure 7.**]{} Profile of the amplitude of the dark soliton $U=|u_1|$ at $t=0$. a: $c=c_0=0.25$, b: $c=0.16$, c: $c=0.043$. The profile a is a black soliton and the profile c is an algebraic soliton. [*(i) Dark soliton: $a_1<0$*]{} It follows from (4.5) with $\kappa=-K, c_1=-c$ that $\kappa+b_1=1/2Kc-K/2$. Since $c_{\rm min}^\prime<c<c_{\rm max}^\prime$ by (4.23), the possible value of $\kappa+b_1$ is restricted by the inequality $$K\left[K\rho^2-1-\sqrt{K\rho^2(K\rho^2-1)}\right]<\kappa+b_1<K\left[K\rho^2-1+\sqrt{K\rho^2(K\rho^2-1)}\right]. \eqno(4.24)$$ One can see that the upper limit of $\kappa+b_1$ is attained when $c=c_{\rm min}^\prime$ and its limiting value is positive by the condition $K\rho^2>1$ whereas the lower limit is attained when $c=c_{\rm max}^\prime$ and is negative. In view of this fact, the algebraic dark soliton would be produced in the limit $c\rightarrow c_{\rm min}^\prime$ for which ${\rm sgn}(\kappa+b_1)>0$. Actually, taking the limit $a_1\rightarrow -0$ for the solutions (4.10) and (4.12) and using the expansion formulas (4.20), we find that the hyperbolic soliton reduces to the limiting form $$u_1=\rho\,{\rm e}^{{\rm i}(-Kx-\omega t)}\,{x-ct+x_0-{\rm i}\,{-2K+b_1\over 2b_1(-K+b_1)} \over x-ct+x_0-{\rm i}\,{1\over 2(-K+b_1)}}, \eqno(4.25a)$$ $$|u_1|^2=\rho^2-{2K c^2\over 1-K^2c}\,{1\over (x-ct+x_0)^2+\left({K c\over 1-K^2c}\right)^2},\eqno(4.25b)$$ where $b_1=(1+K^2c)/2K c$ and $c=c_{\rm min}^\prime$. Since $1-Kc^\prime_{\rm min}>0$ by virtue of the condition $K\rho^2>1$, the expression (4.25b) actually represents an algebraic dark soliton. The black soliton appears when the velocity $c$ takes a specific value $c=c_0^\prime$, where $$c_0^\prime=(K\rho^2-1)/K^2 \qquad {\rm at}\quad \gamma_1=\gamma_0^\prime=\cos^{-1}\left[{(K\rho^2)^{1\over 2}(3-2K\rho^2)\over 2(K\rho^2-1)^{3\over 2}}\right], \qquad (\pi<\gamma_0^\prime<2\pi). \eqno(4.26)$$ Its profile is represented by $$|u_1|^2=\rho^2\left[1-{3K\rho^2-4\over 2(K\rho^2-1)}{1\over \cosh\,2(\theta_1+\delta_1)+{K\rho^2-2\over 2(K\rho^2-1)}}\right]. \eqno(4.27)$$ It is important to notice that the inequality $c_{\rm min}^\prime<c_0^\prime<c_{\rm max}^\prime$ requires the condition $K\rho^2>4/3$ for the wavenumber $K$. It then turns out that expression (4.27) takes the form of a black soliton. {width="10cm"} [**Figure 8.**]{} Profile of an algebraic dark soliton $u_{\rm R}={\rm Re}\,u_1$ at $t=1$. Figure 7 depicts the profile of $U=|u_1|$ at $t=0$ for three different values of $c$, i.e., a: $c=c_0^\prime=0.25 (\gamma_1=\gamma_0^\prime=5\pi/4)$, b: $c=0.16 (\gamma_1=4\pi/3)$, c: $c=0.043 (\gamma_1=2\pi)$ with the parameters $\rho=1, \kappa=-2, \theta_{10}=-\delta_1$ and $\chi_{10}=0$. In this example, $c_{\rm min}^\prime=0.043, c_0^\prime=0.25$ and $c_{\rm max}^\prime=1.46$ (see figure 6). An algebraic soliton appears at the lower limit of the velocity, i.e., $c=c_{\rm min}^\prime$ whereas a black soliton arises at $c=c_0^\prime$. Figure 8 shows the profile of $u_R={\rm Re}\ u_1$ at $t=1$ for an algebraic dark soliton. [*(ii) Bright soliton: $a_1>0$*]{} {width="10cm"} [**Figure 9.**]{} Profile of the amplitude of the bright soliton $U=|u_1|$ at $t=0$. a: $c=1.46$, b: $c=0.81$, c: $c=0.19$. The profiles a is an algebraic soliton. {width="10cm"} [**Figure 10.**]{} Profile of an algebraic bright soliton $u_R={\rm Re\, u_1}$ at $t=1$. The crucial difference between the case 1 and the case 2 for the bright solitons is observed if one compares figure 6 with figure 1. Notably, the bright soliton with $\kappa<0$ reduces to an algebraic soliton only at the upper limit of the velocity $c=c_{\rm max}^\prime$ whereas the bright soliton with $\kappa>0$ has two critical velocities $c_{\rm max}$ and $c_{\rm min}$ for which algebraic solitons are produced. Figure 9 depicts the profile of $U=|u_1|$ at $t=0$ for three different values of $c$, i.e., a: $c=1.46 (\gamma_1=0.998\pi)$, b: $c=0.73 (\gamma_1=0.9\pi)$, c: $c= 0.025 (\gamma_1=0.7\pi)$ with $\rho=1$ and $\kappa=-2$. Figure 10 shows the profile $u_{\rm R}={\rm Re}\,u_1$ of an algebraic bright soliton at $t=1$ which corresponds to the profile a in figure 9. [*4.1.3. Note on algebraic solitons* ]{} We have seen that the algebraic solitons arise from the hyperbolic solitons when certain conditions are satisfied. Here, we summarize the result. The algebraic bright solitons are produced when the conditions ${\rm sgn}\,a_1{\rm sgn}(\kappa+b_1)=-1$ and ${\rm sgn}(\kappa a_1)<0$ are satisfied simultaneously whereas the corresponding conditions for the dark algebraic solitons are given by ${\rm sgn}\,a_1{\rm sgn}(\kappa+b_1)=-1$ and ${\rm sgn}(\kappa a_1)>0$. Thus, for $\kappa>0$, the conditions ${\rm sgn}(\kappa+b_1)=1$ and ${\rm sgn}(\kappa+b_1)=-1$ are responsible for the generation of the algebraic bright and dark solitons, respectively. Since $\kappa+b_1>0$ in this case, only the bright algebraic soliton exists. See figure 1. For $\kappa<0$, on the other hand, the above conditions turn out to be ${\rm sgn}(\kappa+b_1)=-1$ and ${\rm sgn}(\kappa+b_1)=1$, respectively. Under this setting, the limiting value of $\kappa+b_1$ becomes negative for the bright soliton and positive for the dark soliton, respectively, implying the existence of both types of algebraic solitons. See figure 6. In conclusion, we emphasize that the criterion for the existence of solitons (which depends crucially on the sign of $\kappa$) plays an important role in our analysis. [*4.2. Two-soliton solution*]{} As clarified by the analysis of the ono-soliton solutions, both dark and bright solitons exist in our system. Therefore, the two-soliton solutions can be classified into three types, i.e., dark-dark solitons, dark-bright solitons and bright-bright solitons. Here, we focus our attention on the dark-dark solitons. Especially, we investigate the asymptotic behavior of the solution for large time. The two-soliton solution describing the interaction between a dark soliton and a bright soliton will be briefly discussed. For both cases, we assume that $\kappa>0$. [*4.2.1. Dark-dark solitons*]{} The tau functions $f_2$ and $g_2$ representing the dark two-soliton solution are given by (3.1)-(3.3) with $N=2$ subjected to the conditions $\kappa>0, a_1>0, a_2>0$. They read $$f_2=1+{\kappa-{\rm i}p_1\over p_1+p_1^*}\,z_1z_1^*+{\kappa-{\rm i}p_2\over p_2+p_2^*}\,z_2z_2^* +{(\kappa-{\rm i}p_1)(\kappa-{\rm i}p_2)(p_1-p_2)(p_1^*-p_2^*)\over (p_1+p_1^*)(p_1+p_2^*)(p_2+p_1^*)(p_2+p_2^*)}\,z_1z_2z_1^*z_2^*, \eqno(4.28a)$$ $$g_2\!=\!1\!-\!{\kappa+{\rm i}p_1^*\over p_1+p_1^*}{p_1\over p_1^*}\,z_1z_1^*\!-\!{\kappa+{\rm i}p_2^*\over p_2+p_2^*}{p_2\over p_2^*}\,z_2z_2^* \!+\!{(\kappa+{\rm i}p_1^*)(\kappa+{\rm i}p_2^*)(p_1-p_2)(p_1^*-p_2^*)\over (p_1+p_1^*)(p_1+p_2^*)(p_2+p_1^*)(p_2+p_2^*)}{p_1p_2\over p_1^*p_2^*}\,z_1z_2z_1^*z_2^*. \eqno(4.28b)$$ To investigate the interaction process of two solitons, we first order the magnitude of the velocity of each soliton in the $(x, t)$ coordinate system as $c_1>c_2>0$. Invoking the definition (4.2a) of the velocity of the solitons, this can be established by imposing the condition $|p_1|<|p_2|$ on the amplitude parameters. Now, we take the limit $t\rightarrow -\infty$ with $\theta_1$ being fixed. Since in this limit $|z_1|=$finite and $|z_2|\rightarrow \infty$, the leading-order asymptotics of $f_2$ and $g_2$ are found to be as $$f_2 \sim {\kappa-{\rm i}p_2\over p_2+p_2^*}\,z_2z_2^*\left\{1+{(\kappa-{\rm i}p_1)(p_1-p_2)(p_1^*-p_2^*)\over (p_1+p_1^*)(p_1+p_2^*)(p_2+p_1^*))}\,z_1z_1^*\right\}, \eqno(4.29a)$$ $$g_2 \sim -{\kappa+{\rm i}p_2^*\over p_2+p_2^*}{p_2\over p_2^*}\,z_2z_2^*\left\{1-{(\kappa+{\rm i}p_1^*)(p_1-p_2)(p_1^*-p_2^*)\over (p_1+p_1^*)(p_1+p_2^*)(p_2+p_1^*)}{p_1\over p_1^*}\,z_1z_1^*\right\}. \eqno(4.29b)$$ The asymptotic form of the two-dark soliton solution follows from (2.1) upon substituting (4.29) into it, giving rise to $$u_2 \sim \rho\,{\rm exp}\left\{{\rm i}\left(\kappa x-\omega t+\phi_1^{(-)}\right)\right\}{1-{\kappa+{\rm i}p_1^*\over p_1+p_1^*}{p_1\over p_1^*}\,z_1^\prime {z_1^\prime}^* \over 1 +{\kappa-{\rm i}p_1\over p_1+p_1^*}\,z_1^\prime {z_1^\prime}^*}, \eqno(4.30a)$$ where $$z_1^\prime=z_1\,{\rm exp}\left[-{\rm ln}\left({p_1+p_2^*\over p_1-p_2}\right)\right], \eqno(4.30b)$$ $$\phi_1^{(-)}={\rm arg}\left({\kappa+{\rm i}p_2^*\over \kappa-{\rm i}p_2}{p_2\over p_2^*}\right)+\pi. \eqno(4.30c)$$ Let $u_1(\theta_1)$ be the dark one-soliton solution (4.10). Then, the asymptotic form of $u_2$ can be written in terms of $u_1$ as $$u_2 \sim {\rm exp}\left({\rm i}\phi_1^{(-)}\right)u_1(\theta_1+\Delta \theta_1^{(-)}), \quad \Delta\theta_1^{(-)}=-{\rm ln}\left|{p_1+p_2^*\over p_1-p_2}\right|. \eqno(4.31)$$ Next, we take the limit $t\rightarrow +\infty$ with $\theta_1$ being fixed. In this limit, $|z_1|=$finite and $|z_2|\rightarrow 0$. Therefore, the tau functions $f_2$ and $g_2$ and the two-soliton solution $u_2$ behave like $$f_2\sim 1+{\kappa-{\rm i}p_1\over p_1+p_1^*}\,z_1z_1^*,\qquad g_2\sim 1-{\kappa+{\rm i}p_1^*\over p_1+p_1^*}{p_1\over p_1^*}\,z_1z_1^*, \eqno(4.32)$$ $$u_2 \sim \rho\,{\rm e}^{{\rm i}(\kappa x-\omega t)}\,{1-{\kappa+{\rm i}p_1\over p_1+p_1^*}{p_1\over p_1^*}\,z_1z_1^* \over 1+{\kappa-{\rm i}p_1\over p_1+p_1^*}\,z_1z_1^*}. \eqno(4.33)$$ It follows from (4.33) that $$u_2 \sim u_1(\theta_1+\Delta \theta_1^{(+)}), \quad \Delta\theta_1^{(+)}=0. \eqno(4.34)$$ The trajectory of the center position $x=x_c(t)$ of the $j$th soliton is described by the equation $\theta_j+\Delta\theta_j^{(\pm)}=0$, or $x_c=-c_jt-(\theta_{j0}+\Delta\theta_j^{(\pm)})/a_j$. Since the soliton propagates to the left, the phase shift $\Delta x_j$ of the $j$th soliton can be defined by the relation $$\Delta x_j=x_c(-\infty)-x_c(+\infty)={1\over a_j}\left(\Delta\theta_j^{(+)}-\Delta\theta_j^{(-)}\right), \quad j=1, 2. \eqno(4.35)$$ We see from (4.31) and (4.34) that the fast soliton suffers a phase shift $$\Delta x_1={1\over a_1}{\rm ln}\left|{p_1+p_2^*\over p_1-p_2}\right|. \eqno(4.36)$$ In terms of the angular variable $\gamma_1$ and $\gamma_2$ defined by (4.6) and (4.7), this expression can be rewritten in the form $$\Delta x_1={1\over a_1}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_1+\gamma_2\right)\over \sin{1\over 2}\left(\gamma_1-\gamma_2\right)}\right|, \quad a_1=\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\,\sin\,\gamma_1,\qquad 0<\gamma_1<\pi. \eqno(4.37)$$ We can perform the similar asymptotic analysis while keeping $\theta_2$ fixed. Hence, we quote only the final results. As $t\rightarrow -\infty$, the expressions corresponding to (4.29) and (4.31) read respectively $$f_2\sim 1+{\kappa-{\rm i}p_2\over p_2+p_2^*}\,z_2z_2^*,\qquad g_2\sim 1-{\kappa+{\rm i}p_2^*\over p_2+p_2^*}{p_2\over p_2^*}\,z_2z_2^*, \eqno(4.38)$$ $$u_2 \sim u_1(\theta_2+\Delta \theta_2^{(-)}), \quad \Delta\theta_2^{(-)}=0. \eqno(4.39)$$ As $t\rightarrow \infty$, on the other hand, they take the form $$f_2 \sim {\kappa-{\rm i}p_1\over p_1+p_1^*}\,z_1z_1^*\left\{1+{(\kappa-{\rm i}p_2)(p_1-p_2)(p_1^*-p_2^*)\over (p_2+p_2^*)(p_1+p_2^*)(p_2+p_1^*))}\,z_2z_2^*\right\}, \eqno(4.40a)$$ $$g_2 \sim -{\kappa+{\rm i}p_1\over p_1+p_1^*}{p_1\over p_1^*}\,z_1z_1^*\left\{1-{(\kappa+{\rm i}p_2^*)(p_1-p_2)(p_1^*-p_2^*)\over (p_2+p_2^*)(p_1+p_2^*)(p_2+p_1^*))}{p_2\over p_2^*}\,z_2z_2^*\right\}, \eqno(4.40b)$$ $$u_2 \sim {\rm exp}\left({\rm i}\phi_2^{(+)}\right)u_1(\theta_2+\Delta \theta_2^{(+)}),\eqno(4.41a)$$ $$\Delta\theta_2^{(+)}=-{\rm ln}\left|{p_2+p_1^*\over p_2-p_1}\right|, \quad \phi_2^{(+)}={\rm arg}\left({\kappa+{\rm i}p_1^*\over \kappa-{\rm i}p_1}{p_1\over p_1^*}\right)+\pi. \eqno(4.41b)$$ {width="10cm"} [**Figure 11.**]{} The interaction of two dark solitons. The phase shift of the slow soliton follows from (4.35), (4.39) and (4.41), resulting in $$\Delta x_2=-{1\over a_2}{\rm ln}\left|{p_2+p_1^*\over p_2-p_1}\right|, \eqno(4.42)$$ or equivalently in terms of the angular variables $\gamma_1$ and $\gamma_2$, it reads $$\Delta x_2=-{1\over a_2}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_2+\gamma_1\right)\over \sin{1\over 2}\left(\gamma_2-\gamma_1\right)}\right|, \quad a_2=\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\sin\,\gamma_2, \qquad 0<\gamma_2<\pi. \eqno(4.43)$$ An inspection of the formulas (4.36) and (4.42) reveals that $\Delta x_1>0$ and $\Delta x_2<0$ under the setting $a_1>0,\ a_2>0$. Figure 11 shows the intercaction of two dark solitons with the parameters $\rho=1, \kappa=2, c_1=0.75(\gamma_1=0.90\pi), c_2=0.24(\gamma_2=0.80\pi)$ and $\zeta_{10}=\zeta_{20}=0$ so that from (4.14), $A_{d1}=1.0$ and $A_{d2}=0.47$. It can be seen from figure 1 that the amplitude of each dark soliton is an increasing function of the velocity for the present choice of the parameters. Note, in this example, that the large soliton is a black soliton since its asymptotic amplitude is $A_{d1}=\rho=1$. The phase shifts evaluated from the formulas (4.37) and (4.43) are given by $\Delta x_1=0.70$ and $\Delta x_2=-0.36$, respectively. Figure 11 shows clearly a typical interaction process of solitons, i.e., as time goes, the large soliton gets close to the small soliton and overtakes it and after the collision, both solitons eventually separate each other without changing their profiles. The net effect of the collision is only the phase shift. [*4.2.2. Dark-bright solitons*]{} The two-soliton solution consisting of a dark soliton and a bright soliton is obtained by choosing the parameters such as $\kappa>0, a_1>0$ and $a_2<0$, for example. The asymptotic analysis can be performed as well for this solution and hence the detail will be omitted. {width="10cm"} [**Figure 12.**]{} The interaction between a dark soliton and a bright soliton. Figure 12 depicts the interaction between a dark soliton and a bright soliton with the parameters $\rho=1, \kappa=2, c_1=0.75(\gamma_1=0.90\pi), c_2=0.24(\gamma_2=1.2\pi)$ and $\zeta_{10}=\zeta_{20}=0$, showing that the dark soliton propagates faster than the bright soliton. The asymptotic amplitudes of the dark and bright solitons are given respectively by $A_{d1}=1.0$ and $A_{b2}=0.92$ and hence the former is a black soliton. The figure clearly shows the solitonic behavior of the solution. The dark soliton suffers a positive phase shift whereas the bright soliton suffers a negative phase shift. The formulas $\Delta x_1$ for the dark soliton and $\Delta x_2$ for the bright soliton for the phase shifts are given respectively by $$\Delta x_1=-{1\over a_1}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_1+\gamma_2\right)\over \sin{1\over 2}\left(\gamma_1-\gamma_2\right)}\right|, \quad a_1=\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\,\sin\,\gamma_1,\qquad 0<\gamma_1<\pi, \eqno(4.44a)$$ $$\Delta x_2=-{1\over a_2}\,{\rm ln}\left|{\sin{1\over 2}\left(\gamma_2+\gamma_1\right)\over \sin{1\over 2}\left(\gamma_2-\gamma_1\right)}\right|, \quad a_2=\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\sin\,\gamma_2, \qquad \pi<\gamma_2<2\pi. \eqno(4.44b)$$ As in the case of the dark-dark solitons, one can see that $\Delta x_1>0$ and $\Delta x_2<0$. In the present example, $\Delta x_1= 0.70$ and $\Delta x_2=-0.36$. [*4.3. Dark $N$-soliton solution*]{} The preceding analysis reveals that the asymptotic form of the $N$-soliton solution will be represented by a superposition of $n$ dark solitons and $N-n$ bright solitons where $n$ is an arbitrary nonnegative integer in the interval $0\leq n\leq N$. The derivation of the large time asymptotic for the general $N$-soliton solution can be done following the similar procedure to that used for the two-soliton case. Hence, we outline the result. We address the dark soliton solutions satisfying the conditions $\kappa>$ and $a_j>0\ (j=1, 2, ..., N)$. The analysis for the bright soliton solutions as well as an arbitrary combination of dark and bright solitons can be carried out in exactly the same way. To begin with, we order the magnitude of the velocity of each soliton as $c_1>c_2> ... >c_N>0$. We take the limit $t \rightarrow -\infty$ with $\theta_n$ being finite. Since in this limit, $|z_j|\rightarrow 0$ for $j<n$ and $|z_j|\rightarrow \infty$ for $n<j$, we find that the leading-order asymptotic of the tau function $f=f_N$ from (3.1) with (3.2) can be written in the form $$f_N\sim \left|(c_{jk})_{n+1\leq j,k\leq N}\right|\prod_{j=n+1}^N(z_jz_j^*)\left(1+{\left|(c_{jk})_{n\leq j,k\leq N}\right|\over \left|(c_{jk})_{n+1\leq j,k\leq N}\right|}\,z_nz_n^*\right). \eqno (4.45a)$$ Here, $(c_{jk})$ is a matrix of Cauchy type given by $$c_{jk}={\kappa-{\rm i}p_j\over p_j+p_k^*}, \quad 1\leq j,\ k\leq N. \eqno(4.45b)$$ Referring to the well-known Cauchy’s formula, the determinant of the matrix $(c_{jk})$ is evaluated as $$\left|(c_{jk})_{m\leq j,k\leq n}\right|=\prod_{j=m}^n(\kappa-{\rm i}p_j){\prod_{m\leq j<k\leq n}(p_j-p_k)(p_j^*-p_k^*)\over \prod_{m\leq j,k\leq n}(p_j+p_k^*)},\quad 1\leq m<n\leq N. \eqno(4.45c)$$ If we use (4.45c), we have $${\left|(c_{jk})_{n\leq j,k\leq N}\right|\over \left|(c_{jk})_{n+1\leq j,k\leq N}\right|}={\kappa-{\rm i}p_n\over p_n+p_n^*}\,{\rm exp}\left[-\sum_{j=n+1}^N{\rm ln}\left({p_n+p_j^*\over p_n-p_j}\right) -\sum_{j=n+1}^N{\rm ln}\left({p_n^*+p_j\over p_n^*-p_j^*}\right)\right]. \eqno(4.46)$$ Substitution of (4.46) into (4.45) now gives $$f_N\sim \left|(c_{jk})_{n+1\leq j,k\leq N}\right|\prod_{j=n+1}^N(z_jz_j^*)\left(1+{\kappa-{\rm i}p_n\over p_n+p_n^*}\,z_n^\prime {z_n^\prime}^*\right), \eqno(4.47a)$$ where $$z_n^\prime=z_n\,{\rm exp}\left[-\sum_{j=n+1}^N{\rm ln}\left({p_n+p_j^*\over p_n-p_j}\right)\right]. \eqno(4.47b)$$ The leading-order asymptotic of $g_N$ in the limit of $t\rightarrow -\infty$ can be derived in the same way. It takes the form $$g_N\sim \left|(c_{jk}^\prime)_{n+1\leq j,k\leq N}\right|\prod_{j=n+1}^N(z_jz_j^*)\left(1-{\kappa+{\rm i}p_n^*\over p_n+p_n^*}\,{p_n\over p_n^*}\,z_n^\prime {z_n^\prime}^*\right), \eqno(4.48a)$$ where $$c_{jk}^\prime=-{\kappa-{\rm i}p_j\over p_j+p_k^*}\,{p_j\over p_k^*}, \quad 1\leq j, k\leq N. \eqno(4.48b)$$ The asymptotic form of the dark $N$-soliton solution follows from (2.1), (4.47) and (4.48). It reads $$u_N\sim \rho\,{\rm exp}\left\{{\rm i} \left(\kappa x-\omega t+\phi_n^{(-)}\right)\right\}{1-{\kappa+{\rm i}p_n^*\over p_n+p_n^*}\,{p_n\over p_n^*}\,z_n^\prime {z_n^\prime}^*\over 1+{\kappa-{\rm i}p_n\over p_n+p_n^*}\,z_n^\prime {z_n^\prime}^*}, \eqno(4.49a)$$ with $$\phi_n^{(-)}={\rm arg}\left[\prod_{j=n+1}^N\left({\kappa+{\rm i}p_j^*\over \kappa-{\rm i}p_j}\,{p_j\over p_j^*}\right)\right]+(N-n)\pi. \eqno(4.49b)$$ This expression can be rewritten in terms of the one-soliton solution as $$u_N\sim {\rm exp}\left({\rm i}\phi_n^{(-)}\right)u_1(\theta_n+\Delta\theta_n^{(-)}), \eqno(4.50a)$$ with $$\Delta\theta_n^{(-)}=-\sum_{j=n+1}^N{\rm ln}\left|{p_n+p_j^*\over p_n-p_j}\right|. \eqno(4.50b)$$ By a similar asymptotic analysis, we can derive the asymptotic form of $u_N$ in the limit of $t\rightarrow +\infty$. We find that $$u_N\sim {\rm exp}\left({\rm i}\phi_n^{(+)}\right)u_1(\theta_n+\Delta\theta_n^{(+)}), \eqno(4.51a)$$ with $$\Delta\theta_n^{(+)}=-\sum_{j=1}^{n-1}{\rm ln}\left|{p_n+p_j^*\over p_n-p_j}\right|, \eqno(4.51b)$$ $$\phi_n^{(+)}={\rm arg}\left[\prod_{j=1}^{n-1}\left({\kappa+{\rm i}p_j^*\over \kappa-{\rm i}p_j}\,{p_j\over p_j^*}\right)\right]+(n-1)\pi. \eqno(4.51c)$$ We see from (4.50) and (4.51) that in the rest frame of reference, the asymptotic form of the dark $N$-soliton solution can be represented by a superposition of $N$ independent dark one-soliton solutions, the only difference being the phase shifts of each soliton caused by the collisions. It follows from (4.50b) and (4.51b) that the formula for the total phase shift of the $n$th soliton is given by $$\Delta x_n={1\over a_n}\left(\sum_{j=n+1}^N{\rm ln}\left|{p_n+p_j^*\over p_n-p_j}\right|-\sum_{j=1}^{n-1}{\rm ln}\left|{p_n+p_j^*\over p_n-p_j}\right|\right), \quad n=1, 2, ..., N. \eqno(4.52)$$ As in the two-soliton case, we can rewrite the above formula in terms of the variables $\gamma_j$ defined by (4.6) and (4.7). Explicitly, $$\Delta x_n={1\over a_n}\left(\sum_{j=n+1}^N{\rm ln}\left|{\sin{1\over 2}(\gamma_n+\gamma_j)\over \sin{1\over 2}(\gamma_n-\gamma_j)}\right| -\sum_{j=1}^{n-1}{\rm ln}\left|{\sin{1\over 2}(\gamma_n+\gamma_j)\over \sin{1\over 2}(\gamma_n-\gamma_j)}\right|\right),$$ $$a_n=\sqrt{\kappa^3\rho^2(1+\kappa\rho^2)}\sin\,\gamma_n, \qquad 0<\gamma_n<\pi,\qquad n=1, 2, ..., N. \eqno(4.53)$$ The formulas (4.52) and (4.53) reduce to (4.36), (4.37), (4.42) and (4.43) for the special case of $N=2$. They clearly show that each soliton has pairwise interactions with other solitons, i.e., there are no many-particle collisions among solitons. This feature is common to that of the bright $N$-soliton solution considered in I. In this paper, the system of bilinear equations reduced from the FL equation has been derived and used to construct the dark $N$-soliton solution. The corresponding $N$-soliton solution derived in \[7\] using the Bäcklund transformation follows from our solution (2.1) with (3.1) and (3.2) if one introduces the angular variables $\gamma_j$ according to the relations (4.7). We have found that unlike the bright soliton solutions obtained in I, the complex amplitude parameters $p_j$ are subjected to the constraints (3.2c) which have prevented the proof of the solution. To overcome this difficulty, we have employed a trilinear equation in place of one of the bilinear equations, in addition to an auxiliary variable $\tau$ in (3.2c). As a byproduct, this trilinear equation has led for the first time to a simple formula for the dark $N$-soliton solution of the derivative NLS equation on the background of a plane wave. Note that the dark soliton solutions on a constant background \[13-19\] stem simply from the above-mentioned solution in the zero limit of the wavenumber $\kappa$. However, this limiting procedure is found to be unable to perform for the dark $N$-soliton solution of the FL equation due to the singularity of the dispersion relation. We have seen that the soliton solutions presented here exhibit several new features. Specifically, both the dark and bright solitons exist depending on the sign of the wavenumber $\kappa$ and that of the real part of the complex amplitude parameter. Of particular interest is the existence of an algebraic dark soliton which appears only in the case of negative $\kappa$. 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--- abstract: '[ With the Keck Interferometer, we have studied at 2 um the innermost regions of several nearby, young, dust depleted “transitional” disks. Our observations target five of the six clearest cases of transitional disks in the Taurus/Auriga star-forming region (DM Tau, GM Aur, LkCa 15, UX Tau A, and RY Tau) to explore the possibility that the depletion of optically thick dust from the inner disks is caused by stellar companions rather than the more typical planet-formation hypothesis. At the 99.7% confidence level, the observed visibilities exclude binaries with flux ratios of at least 0.05 and separations ranging from 2.5 to 30 mas (0.35 - 4 AU) over $\gtrsim\,$94% of the area covered by our measurements. All targets but DM Tau show near-infrared excess in their SED higher than our companion flux ratio detection limits. While a companion has previously been detected in the candidate transitional disk system , we can exclude similar mass companions as the typical origin for the clearing of inner dust in transitional disks and of the near-infrared excess emission. Unlike CoKu Tau/4, all our targets show some evidence of accretion. We find that all but one of the targets are clearly spatially resolved, and UX Tau A is marginally resolved. Our data is consistent with hot material on small scales (0.1 AU) inside of and separated from the cooler outer disk, consistent with the recent SED modeling. These observations support the notion that some transitional disks have radial gaps in their optically thick material, which could be an indication for planet formation in the habitable zone ($\sim$ a few AU) of a protoplanetary disk. ]{}' author: - 'Jorg-Uwe Pott, Marshall D. Perrin, Elise Furlan, Andrea M. Ghez, Tom M. Herbst, Stanimir Metchev' title: Ruling out Stellar Companions and Resolving the Innermost Regions of Transitional Disks with the Keck Interferometer --- Introduction ============ Circumstellar disks are a natural outcome of the star-formation process: when a molecular cloud core collapses, it gives rise to a central star surrounded by a rotating circumstellar disk, which transports material towards the star. Over time, the disk material dissipates through [ processes such as]{} accretion onto the central star, [ disk winds]{} and the formation of planets. At an age of $\sim5~{\rm Myr}$, about 90% of disks have already dispersed, and within $10~{\rm Myr}$ of their formation, almost all pre-main-sequence stars are diskless [e.g. @2006ApJ...638..897S]. While it is now believed that such disks commonly give rise to planetary systems, the details of this process remain unclear. Theory predicts that disks evolve from the inside out: dust grain growth is expected to occur faster in the inner disk than in the outer disk , higher densities favor planet formation in the inner disk [@2002Icar..156..291B], and photoevaporation by the central star will cause the inner disk to dissipate first [@2001MNRAS.328..485C]. [ Possible observational support for inside out disk evolution has been found in a small number of so-called transitional disks.]{} These systems show a strong mid-infrared excess ($\gtrsim\,8\,\mu {\rm m}$) revealing the presence of dust [ but significantly reduced or no shorter wavelength infrared excess compared to typical classical T Tauri disks, indicating a depletion of optically thick inner dust out to a radius of a few AU]{} [@1990AJ.....99.1187S; @1997ApJ...489L.173M; @2001AJ....121.1003S; @2005ApJ...621..461D; @2005ApJ...630L.185C; @2006ApJ...636..932M; @2006ApJS..165..568F; @2006AJ....131.1574L; @2006ApJ...643.1003M; @2007ApJ...664L.111E_disk; @2007ApJ...664L.107B]. Therefore, these disks might be in the process of dispersing and this has often been assumed to be due to the influence of newly formed planets [e.g. @2004ApJ...612L.137Q]. Discussed explanations of the transitional disk phenomenon reveal two important features which can be tested directly by high angular resolution imaging observations. \(1) The depletion of dust inside of the outer, mid-infrared disk, could be caused by a close (AU-scale) binary system inside of the disk. [ Binary companions can perturb a circumstellar disk and create inner holes with diameters comparable to the binary separation [@1994ApJ...421..651A see also the discussion for DI Tau in Meyer et al. 1997].]{} To call such a circumbinary disk [*transitional*]{}, would be misleading, since circumbinary disks can be dynamically stable and longer-lasting than the short ($<$ Myr) time-scales derived from the small (few percent) fractional abundance of transitional disks around pre-main-sequence stars in nearby, a few Myr young, star-forming regions [@2006ApJS..165..568F; @2008AJ....135..966F; @2009ApJ...703.1964F note that the fractional abundancies of transitional disks might be as high as a few tens of percent, depending on the exact definition of transitional disks, in particular if and what type of [ residual]{} inner disk emission is [ permitted]{}]. [ Also, a close companion affects the SED interpretation of apparent transitional disk systems.]{} Unresolved infrared companions can create additional near-infrared and, if embedded, mid-infrared flux, that appears comparable to the infrared excess radiation seen in transitional disk systems, which typically is interpreted as disk emission [e.g. @2003ApJ...592..288D]. [ Indeed, recent diffraction limited NIR imaging with the Keck II telescope of the candidate transitional disk system indicates that its inner hole (10 AU radius) is actually caused by a newly discovered binary companion of $\sim\,8\,{\rm AU}$ separation, removing the need to invoke other processes like planet formation as the disk clearing mechanism in transitional disk systems [@2008ApJ...678L..59I].]{} While the census of very close companions [ of T Tauri stars (TTS) in star-forming regions]{} is far from complete, the companion star fraction in young, nearby star-forming regions is about 50% in the 15-1800 AU separation range and $\sim\,20\%$ at separations less than 10 AU . The companion star fraction typically decreases towards smaller separations (less than a few AU), but suggest that YSOs in Ophiuchus have a companion fraction of at least 10% at the 0.8-4 AU separation scale. [ These observational constraints suggest that there would be enough binaries to populate a large fraction of transitional disks, although the short-period binary frequency appears to vary between different sites of star formation . ]{} [ (2) While some transitional disks may be completely cleared of material in the inner region, the planet formation hypothesis suggests that disk clearing may often result in gaps between inner and outer dusty regions . Other possible disk clearing mechanisms such as photoevaporation would produce strictly inside-out clearing [@2000prpl.conf..401H; @2001MNRAS.328..485C], so evidence for gaps in disks (in contrast to totally cleared holes) tends to support the planet formation hypothesis. Many transitional disks show some infrared excess emission inside of the outer optically thick disk, which itself dominates at wavelengths longer than $\sim\,8~\mu$m. It has been shown for a few systems that this near-infrared excess can be explained by a small amount of emitting dust close to the star at $\sim\,0.1\,$AU-scales, leaving a gap between this innermost dust, and the outer mid-infrared disk [e.g. @2008ApJ...682L.125E_LkCa]. Therefore, the near-infrared excess in transitional disk systems might origin from such small size scales, if not emitted by a so far unresolved companion (case 1).]{} To [ directly]{} assess (1) the presence of close binary companions within transitional disks and [ (2) the emission size scale of the near-infrared excess over the stellar continuum]{}, we used the Keck Interferometer (KI) in $V^2$ mode[^1] to observe 5 transitional disks in the nearby ($\sim$ 140 pc) Taurus-Auriga young star-forming [ region]{}. The nominal interferometric resolution of $\sim\,2.7$ mas and the field of view of $\sim\,50\,{\rm mas}$, offered in the $V^2$ mode, is well suited to resolve any companion stars from about 0.5 to 5 AU [ distance]{} from the target primary stars. This angular resolution is a significant improvement over the resolution available with speckle or aperture mask interferometry and adaptive optics at 8-10 m class telescopes ($\gtrsim$ 25 mas). This article is organized as follows: [ detailed target properties ]{}are reported in Sect. \[sec:2\]. Observations and data reduction are give in Sect. \[sec:3\]. The results are discussed in Sect. \[sec:5\], and the conclusions of our experiment are given in Sect. \[sec:6\] \[sec:2\]Target selection and properties ======================================== [lccccc|c]{} & DM Tau$^a$ & GM Aur$^a$ & LkCa 15$^b$ & UX Tau A$^b$ & RY Tau$^c$ & CoKu Tau/4$^f$\ $M_* \,({\rm M_{\odot}})$ & 0.65 & 1.2 & 1.1 & 1.5 & 2.0 & 0.5\ $R_* \,({\rm R_{\odot}})$ & 1.2 & 1.5 & 1.7 & 2.0 & 3.6 & 1.9$\,^{h}$\ $T_* \,({\rm K)}$ & 3720& 4730 & 4350 &4900& 5945$\,^{d}$ & 3720\ $L_* \,({\rm L_{\odot}})$ & 0.25 &1.03 &0.96 &2.18 & 12.8 & 0.61$\,^{h}$\ $\dot{M} \, ({\rm M_{\odot}\,yr^{-1}})$& 2.0$\cdot 10^{-9}$ & 1.0$\cdot 10^{-8}$& 2.4$\cdot 10^{-9}$ & 9.6$\cdot 10^{-9}$ & 2.5$\cdot 10^{-7}$ & –\ Spectral type & M1 & K5 & K5 & K2 & G1$\,^{d}$ & M1.5\ $A_{\rm V}$ & 0.5 & 1.2 & 1.2 & 1.3 & 2.1 & 3.0 $^g$\ $K_{\rm s}$ mag. $^e$ & 9.5 & 8.3 & 8.2 & 7.5 & 5.4 & 8.3 $^g$\ Inclination (deg) & 40 & 55 & 42 & 60 & 25 & $50\,-\,75\,^{i}$\ $R_{\rm hole}\,^l\,{\rm (AU)}$ & 3 & 24 & 46 & 56 & $^{k}$ & 10\ $K_{\rm inner\,dust}/K_{\rm total }$ & $\lesssim\,0.05$ & 0.12 & 0.23 & 0.32 & 0.73 &$\lesssim\,0.05$\ $R_{\rm inner\,dust}\,^l\,{\rm (AU)}$ & – &$<\,5$& $0.13$ & $0.17$ & $0.25$ & –\ [ Transitional disks are defined as systems that are significantly depleted of optically thick dust on scales of a few AU compared to the majority of similarly aged stars with circumstellar disks.]{} However, the variety of observing constraints [ and their interpretation as well as their possible dependence on age and environment makes it difficult to describe a [*typical*]{} transitional disk]{}. Some disk data favor an inner [*hole*]{}, which refers to a true depletion of the inner optically thick dust [@2007ApJ...664L.111E_disk]. A disk [*gap*]{} is present, when optically thick dust close to the stellar photosphere, as traced by 2 $\mu$m excess emission, is separated from the outer, cooler, optically thick dust disk, which is detected at mid-to-far-infrared and ${\rm (sub-)mm}$ wavelengths [@2008ApJ...682L.125E_LkCa]. In general, it is assumed that dust (traced by the IR continuum) and gas (traced by emission lines) are well mixed in primordial disks. In individual systems however, gas has been found in disk regions devoid of optically thick dust [e.g. @2008ApJ...675L.109B; @2009ApJ...699..330S]. Thus, an observed depletion of optically thick dust does not need to correlate with a similarly complete depletion of disk material. Our sample consists of five low-mass pre-main sequence stars in the Taurus-Auriga star forming region showing mid-infrared dust excess characteristic of transitional disks: , , , , and . These objects comprise the best-studied disks to date (i) with strong evidence from observations or SED-modeling that inner disk holes or gaps are present [@2005ApJ...621..461D; @2005ApJ...630L.185C; @2007ApJ...670L.135E_LkUX see Table \[tab:1\]] and (ii) for which no close binary companions are known (i.e., all but CoKu Tau/4). Table \[tab:1\] lists the stellar and disk properties of our targets and associated references. The respective values of CoKu Tau/4 are given as well for comparison. DM Tau is the only object in our sample with no detectable excess emission below 8 $\mu$m, but it is still accreting, traced by hydrogen emission lines. This suggests that at least gas must exist in the innermost disk, feeding the accretion. GM Aur has a hole of 24 AU, which is partially filled with optically thin dust. Both LkCa 15 and UX Tau A seem to have gaps between optically thick inner and outer disk regions; in addition, LkCa 15 also has some optically thin dust in the gap. RY Tau could be a transitional disk with a gap, in part due to its somewhat similar SED shape to LkCa 15 [@2009ApJ...703.1964F]. On the other hand, RY Tau has an earlier spectral type than the rest of the sample, and therefore its disk structure could be different. Our data probe binary separations of 2.5-30 mas (0.35-4 AU). This matches the inner region, which cannot be resolved by single telescope imaging of 8-10 m class telescopes, and which lies inside the colder optically thick dust disks, responsible for the MIR-excess of transitional disks (see $R_{\rm hole}$ in Table \[tab:1\]). [ In particular, this region covers the range where a stellar companion might reside and be responsible for inner disk truncation or NIR excess flux.]{} A circumbinary disk is typically truncated at an inner radius of about 2-3 times the semi-major axis of the binary. @1994ApJ...421..651A investigated the gravitational impact of a central binary on a geometrically thin, non-self gravitating circumbinary disk, and found that the semi-major axis of the central binary is about half the inner disk edge-radius. The detailed relation between binary separation and disk hole size depends on the binary’s eccentricity. [ Our interferometric approach]{} is sensitive to companions with orbital periods ranging from several months to several years, a regime that could also be probed by radial velocity surveys. However, such surveys would take years to complete, as opposed to a single epoch of KI observations. Earth’s rotation of the KI baseline allows us to obtain visibilities at about 5 independent [*spatial frequencies*]{} (or [*u,v*]{}-points) per object per night, [ which suffices to exclude a large number of binary parameters.]{} For some of our targets, we are able to combine our data with previous interferometric measurements; RY Tau: @2005ApJ...622..440A_PTI, LkCa 15, GM Aur: @2005ApJ...635.1173A_KI. [ The earlier measurements of these objects’ visibility amplitudes indicate that they are spatially resolved and were modeled as inner disk structure in each case. However, those single visibility measurements of LkCa 15 and GM Aur are insufficient to distinguish such disk structure from a companion star. This ambiguity is easily resolved through the visibility amplitude measurements at multiple spatial frequencies presented here.]{} Observations and data reduction {#sec:3} =============================== [ccccccc]{} Target & date (UT) & H.A.$\,^{a}$ & $u,v$$\,^{b}$ & proj. B$\,^{b}$ & calibrators & $V^2$\ && $[$hr$]$ &$[{\rm m}]$ & $[$m,deg(EofN)$]$ & \[from Tab \[tab:3\]\]& \[calib.\]\ DM Tau & Mar.17,2008 & 3.3 & (16.7, 77.1) & (78.9, 12.2) & 1,2 & 0.91\ & Mar.17,2008 & 3.4 & (15.3, 77.3) & (78.8, 11.2) & 1,2 & 0.91\ & Dec.15,2008 & -1.2 & (56.3, 62.0) & (83.7, 42.2) & 6,7 & 0.90\ & Dec.15,2008 & 1.1 & (43.2, 71.7) & (83.7, 31.1) & 6,7 & 0.90\ & Dec.15,2008 & 2.2 & (30.9, 75.1) & (81.2, 22.4) & 6,7 & 0.88\ & Dec.15,2008 & 3.1 & (19.3, 76.9) & (79.3, 14.1) & 6,7 & 0.90\ & Dec.15,2008 & 3.5 & (13.0, 77.5) & (78.6, 9.5) & 6,7 & 0.89\ GM Aur & Mar.17,2008 & 2.6 & (25.7, 80.2) & (84.2, 17.8) & 1,2 & 0.95\ & Dec.15,2008 & -0.2 & (52.7, 65.0) & (83.7, 39.0) & 6,7 & 0.93\ & Dec.15,2008 & 1.7 & (36.9, 76.4) & (84.8, 25.8) & 6,7 & 0.95\ & Dec.15,2008 & 3.4 & (14.2, 82.4) & (83.6, 9.8) & 6,7 & 0.96\ LkCa 15 & Mar.17,2008 & 3.7 & (10.9, 79.8) & (80.5, 7.8) & 1,2 & 0.93\ & Dec.15,2008 & -1.0 & (55.9, 62.2) & (83.6, 41.9) & 6,7 & 0.91\ & Dec.15,2008 & 1.5 & (39.0, 74.2) & (83.8, 27.7) & 6,7 & 0.91\ & Dec.15,2008 & 2.6 & (26.3, 77.7) & (82.0, 18.7) & 6,7 & 0.91\ & Dec.15,2008 & 3.3 & (16.3, 79.3) & (81.0, 11.6) & 6,7 & 0.91\ UX Tau A & Dec.15,2008 & -1.3 & (56.4, 61.7) & (83.6, 42.4) & 3,4,5 & 0.96\ & Dec.15,2008 & 0.9 & (45.3, 70.8) & (84.1, 32.6) & 3,4,5 & 0.97\ & Dec.15,2008 & 1.8 & (35.7, 74.0) & (82.2, 25.8) & 3,4,5 & 0.97\ & Dec.15,2008 & 3.0 & (20.3, 76.8) & (79.4, 14.8) & 3,4,5 & 0.96\ RY Tau$\,^{d}$ & Mar.17,2008 & 2.6 & (25.6, 79.8) & (83.8, 17.8) & 1,2$\,^{c}$ & 0.30\ & Dec.15,2008, & 1.5 & (39.4, 75.1) & (84.8, 27.7) & 3,4,5 & 0.27\ & Dec.15,2008 & 2.7 & (24.4, 80.1) & (83.7, 16.9) & 3,4,5 & 0.28\ [clccc]{} \# &Calibrator & $V/H/K$ & Spec. Type$\,^{(a)}$ & Ang. diameter (mas)$\,^{(b)}$\ 1 & HD283798$^{(c)}$ & 9.5/8.1/8.0 & G2V & $0.11\,\pm\,0.01$\ 2 & HD283886 & 9.9/8.6/8.4 & G2V & $0.08\,\pm\,0.01$\ 3 & HD21379 & 6.3/6.3/6.3 & A0V & $0.12\,\pm\,0.01$\ 4 & HD254236 & 8.8/6.5/6.3 & K2III & $0.34\,\pm\,0.13$\ 5 & HD41076 & 6.1/6.1/6.1 & A0V & $0.13\,\pm\,0.01$\ 6 & HD250388 & 10.7/8.9/8.8 & K0 & $0.09\,\pm\,0.01$\ 7 & HD283934 & 10.6/9.0/9.0 & G5V & $0.07\,\pm\,0.01$\ The Keck Interferometer (KI) atop Mauna Kea combines the light of the two 10-meter Keck telescopes and has a baseline of 85 meters, oriented 38$^\circ$ east of north [@2004SPIE.5491..454C; @2004SPIE.5491.1678W]. We used the KI in the $V^2$ continuum mode. All data shown here are from the white-light channel of the beam combiner, which illuminates one pixel with the full $K'$-band (2-2.4 $\mu$m) to maximize sensitivity. The observations were conducted on the nights of Mar. 17, and Dec. 15, 2008 (UT). Details of the observations appear in Table \[tab:4\]. KI data are provided to the observer in a semi-raw state. The technical calibration, such as detector bias corrections, and some averaging has been applied by a pipeline reduction. The result are raw fringe contrast (visibility) measurements, which still need to be calibrated for the so-called system visibility (the visibility transfer function), and the ratio correction, which corrects for systematic flux biases between the telescopes. Both the system visibility and the flux ratios proved to be very stable over the two nights, indicating a reliable data calibration. Due to these stable conditions and good seeing, we did not follow up each target with a calibrator immediately, but rather alternated two targets with one visibility calibrator measurement. We used several different calibrator sources throughout the night to enable cross calibration, and to match the targets in brightness within $\pm~1\,{\rm mag}$ to minimize the impact of the known flux-dependence of the KI system visibility. [ Flux bias calibration of the visibilities was applied, which has been shown to push the systematic $V^2$ calibration errors from $\sim 0.05$ down to $\lesssim \, 0.03$ [^2] ]{}. The calibrators were selected using either the getCal planning tool [^3] or browsing [SIMBAD]{} directly within a radius of $\sim15^{\circ}$ around the science targets. We only used stars from the Hipparcos and Tycho catalogs to assure high coordinate precision. To ensure that all calibrator stars are unresolved with respect to the projected baseline, their photospheric diameters were estimated by fitting black-body SEDs to published photometry using NExScI’s fbol routine (Table \[tab:3\]). The fitted black-body model was compared in particular to the photometry at wavelengths longer than 2 ${\rm \mu m}$ to check for dust excess that would be indicative of extended structure. Our data reduction followed the standard procedures developed and suggested by the NExScI team, and we made ample use of their wb/nbCalib-software suite and the respective documentation. The individual transfer functions, derived from each calibrator measurement, are defined as the ratio of the measured raw squared visibility to the respective ideal uniform disk squared visibility [^4]. Then, a time and sky-location dependent average system transfer function is calculated, which picks for each target the calibrators closer than 15$^\circ$ and observed within two hours before or after the target measurement. The calibrators closer in time and space get a higher weight in the averaging process. We use the default time-weighting options. This averaging approach potentially minimizes the effect of a single bad calibration measurement on the data calibration, in contrast to using only the two calibrators immediately taken before and after the science measurement. This calculation of an average transfer function, based on all calibrator measurements, is particularly suitable for nights with stable observing conditions. Finally the raw data are divided by the transfer function to calculate the calibrated visibilities. We use the standard deviation of the raw measurements as a first estimate of the uncertainties of each data point. The resulting statistical visibility uncertainties of the individual measurements are $0.005-0.01$ in most cases, smaller than the canonical value of 0.03 (see the NExScI KI support websites), which includes margin for systematics such as slightly different observing conditions and Strehl between the calibrator and science measurements. The calibrated visibility measurements appear in Table \[tab:4\]. Although the observing conditions were very good and stable throughout the night, and the calibration uncertainty seems to be slightly better, we assume the standard $\delta\,V^2\,=\,0.03$ for the analysis in this paper. To get the highest precision, it is recommended to bracket a science object with two calibrator measurements at similar flux levels which would give a slightly better time-sampling of the system visibility than ours. However, since our goal was to detect visibility changes of 0.2 over the observed range in hour angles, our conclusions do not depend on the precise uncertainty adopted. Higher observing efficiency (i.e. time sampling) of the targets is more critical for our project than highest calibration precision, to sample as large a range of binary parameters as possible. Still, the stable conditions and good visibility precision helped us to achieve the sensitivity to detect circumstellar material on scales as small as 1 milliarcsecond. Fig. \[fig:1\] shows the [*u,v*]{}-plane coverage of . This is typical for the whole sample of targets. Results {#sec:5} ======= The calibrated visibilities appear in Fig. \[fig:2\], plotted as a function of the hour angle and the projected baseline length. Since we show broad band visibilities, observed at a fixed central wavelength of about 2.2 $\mu$m, the projected baseline length scales the interferometric resolution ($\lambda/2B$) and the spatial frequencies ($B/ \lambda$) directly for all data shown. We added previously published KI data points for GM Aur and LkCa 15. For , we also included in our analysis visibility measurements from a previous observation with the Palomar Testbed Interferometer [PTI, @2005ApJ...622..440A_PTI]. Due to the different baselines of the PTI measurements (see footnote ($d$) in Table \[tab:4\]), in particular different position angles at similar baseline lengths, we do not show those data together with our KI data in Fig. \[fig:2\]. They probe different spatial frequencies at the hour angles and baseline-lengths shown in the figure. The data in Fig. \[fig:2\] are shown as the measured squared visibility amplitudes. [ The photospheric stellar diameters of the observed program stars ($<~0.3~{\rm mas}$, Table \[tab:1\]) cannot be resolved by the KI. However, we measure mean calibrated visibilities ranging from $0.28\pm0.03$ for RY Tau to $0.96\pm0.03$ for UX Tau A.]{} Such visibilities below unity rule out that our targets are compact at 2 $\mu$m (apart for the marginally resolved UX Tau A). Thus, we can test our visibility data for each target against two contrary scenarios: 1. A binary companion is present, responsible both for clearing out the optically thick dust from the inner region of the disk, and for producing some or all of the observed near infrared excesses. In the following section, we demonstrate that a large range of binary parameters in the range of dynamical interaction between the binary and the inner disk edge can be excluded on basis of our KI data. 2. As an alternative to the binary scenario, we evaluate a simple model of a disk gap, simulating the inner dust disk by a face-on circumstellar ring which contributes to the $K$-band flux. The diameter estimates of such a pragmatic model approach give order-of-magnitude constraints on the location and extension of the observed emission. As seen in Fig. \[fig:2\], we mostly sample a range of similar projected baseline lengths (75 – 85 m). Therefore, we cannot firmly conclude from the visibility data alone whether we resolve or over-resolve the dust emission. While resolved emission would result in decreasing visibilities with longer baselines, an over-resolved emission would show constant visibilities below unity. More information, for instance about the $K$-band excess from the SED models, is needed to interpret the data best. Here, we aim at understanding at an order of magnitude level the physical location of the extended emission. Thus, details like scattered light contributions have not been considered in our analysis [@2008ApJ...673L..63P]. Constraints on the presence of Binary Companions {#sec:51} ------------------------------------------------ Our overall approach is to test binary models against the observed visibilities, in order to estimate the parts of the parameter space that are incompatible with the data. Due to the high maximum elevation of the Taurus-Auriga star forming region at Mauna Kea, the single baseline of the KI delivers sufficient [*u,v*]{}-coverage to probe a large range of binary parameters. We calculated theoretical visibilities for all position angles ($PA$) from $0\,-\,180^\circ$ (the visibility sensitivity is point-symmetric), star-star separations $\rho$ between 2.5 and 30 mas, and flux ratios ($FR$) between 1 and 0.05 times the brightness of the primary. For each model the reduced $\chi^2$ deviation to the data was calculated. Because the number of observed data points differ for each source (and thus the number of degrees of freedom for our model also varies) we calculate individually for each source the $\chi^2$ level that corresponds to a formal 99.7 % confidence level for the relevant number of degrees of freedom, based on the cumulative $\chi^2$-distribution of random measurements. The 99.7 % confidence limit is arbitrarily chosen to match the 3 $\sigma$ confidence level of a normal probability density function. Models whose (reduced) $\chi^2$ exceeds this threshold have a probability of less than 0.3% to be consistent with the data and are rejected [^5]. The thresholds are reported in Table \[tab:2\]. We chose 360 and 120 linearly spaced steps for PA and $\rho$, respectively. $FR$ is sampled in 80 logarithmic steps. Thus, we examine about 3.5 million possible binary configurations per star. [ These steps are small enough to adequately sample the parameter space and find a number of solutions if a binary would have been observed. This is demonstrated by analysing mock datasets. Fig. \[fig:80\] shows artificial visibility data at the $u,v$-coverage of the DM Tau observation. The data points are distributed around the theoretical visibility curve (solid line) using a Gaussian noise with a FWHM of 0.03, matching our measurement precision. This exercise shows several properties of our analysis. In case of flux ratios close to one the sensitiviy is very high, and only a few tens of solutions are found. The sparse $u,v$-coverage results in artificial solutions which are evenly separated by multiples of the KI-resolution ($\sim$ 2.7 mas), and along the position angle of the KI-baseline. Each of the colored patches in the central panels of the figure show where a companion cannot be rejected at the 99.7 % confidence limit. The solution histogram in the right panels shows how the number of solutions depend on the confidence level of the rejection criterium. The $PA$ of the mock binary datasets is with 120 $^\circ$ chosen to be orthogonal to the KI baseline. The KI observations are most sensitive to these $PA$ due to the rotation of the projected baseline, thus this $PA$ is best suited to test the sampling of the binary parameter search grid. A decent number of a few tens of solutions is found. For binary $PA$ more aligned to the KI baseline, the number of solutions would increase.]{} [ The combined PTI and KI dataset of RY Tau, the target with the largest NIR-excess flux, excludes the complete binary parameter space probed. The other four visibility datasets allow for a few solutions, which are visualized in Fig. \[fig:81\]&b. We argue in the following that these solutions are most likely artefacts of the sparse $u,v$-coverage of the observations, missing data from additional baselines as in the case of RY Tau.]{} [ In the left panels of the figure, we show all allowed solutions down to $FR\,=\,0.08$. The field of view (FoV) of our binary analysis is the 2.5 - 30 mas radius annulur patch of sky around each central star, marked in the plots]{}. The slight extension of the inner boundary of the FoV, orthogonal to the KI-baseline, reflects the shape and binary sensitivity of the point spread function of visibility datasets, based on one baseline. At each point in the FoV, we plot color-coded the [*brightest*]{} companion flux ratio allowed for a companion at that location. For instance, the blue spots in the plot of DM Tau indicate where we found binary solutions with a flux ratio close to unity. [ The bright companion solutions are all grouped around a $PA$ of about $20\,\pm\,20~^\circ$. This is quantified in the position angle histograms of the solutions in Fig. \[fig:82\]. At a confidence level of 67 %, this is true down to a $FR$ of 0.05 (Fig. \[fig:83\]). This $FR$ of 0.05 is the sensitivity limit of our observations, because now binary solutions start to appear all over the FoV (right panel of Fig. \[fig:81\]&b). On the other hand, for $FR$ brighter than 0.08, we typically rule out more than 99 % of the probed FoV.]{} In Table \[tab:2\], we report the fraction of the FoV without binary solutions. [ Although we cannot rule out completely the probed binary parameter range for DM Tau, GM Aur, LkCa 15, and UX Tau A individually, we can strongly rule out the possibility that the majority of our target stars are binaries. For instance, if two (three) of these stars are binaries, the probability that the two (three) binaries would have position angles aligned with $20\,\pm\,20\,^\circ$ is about 3 % (0.4 %) only [^6]. This statistical argument is further strengthened by a similar baseline-dependent constraint of the solutions to separations of multiples of the KI resolution only. Thus, at the 99.7 % (67 %) confidence level of rejecting binary models, it is very unlikely that more than one out of our five targets have a binary companion of 0.08 (0.05) brightness of the primary.]{} The masses that such flux limits represent depend on both the luminosity of the primary and the assumed age of the system. Based on the PMS evolutionary tracks of @1999ApJ...525..772P and an age of 1 Myr, flux ratios of 0.1 would refer to companion masses of about 1/4 of the primary mass. The fact that we can rule out binaries for the major part of the tested FoV, and for a brightness ratio down to 0.05 is mostly related to the measured small [*differential*]{} visibility variations. In contrast, the [*absolute*]{} level constrains binary parameters to a lesser extent only, but is rather linked to the amount of (over-)resolved emission. An equal brightness binary would result in large ($\gtrsim\,0.2$) squared visibility variations over the probed range of spatial frequencies of the KI baseline. This amplitude of the visibility variation decreases with decreasing companion brightness (see the left panels in Fig. \[fig:80\]). Our binary search is limited to $FR\,\gtrsim\,0.05$ because even lower $FR$ would result in differential visibility variations compatible with models of a single resolved, extended emission structure, as discussed in the next section. Fitting binary models to such data of low differential visibility variation is very ambiguous. The visibilities of GM Aur show a statistically significant differential visibility signal of $\sim\,0.1$, different from the other four targets, but our analysis shows that this still small change in visibility does not appreciably alter the range of allowed binary solutions. We discuss this visibility trend further below in Sect. \[sec:111\]. The set of binary parameters, excluded by our data, does not significantly depend on the fact that so far we probed the data against simple, dust-free binary models only. It is possible that some extended, circumbinary dust emission has been resolved out by the interferometer, resulting in the visibilities below unity. This raises the concern that the putative binary signature in the visibility variation is not recovered by the analysis because the absolute visibility level has changed with respect to the calculated one, due to the missing over-resolved dust emission in the correlated flux. To probe if this scenario would have an effect on the findings of the above binary analysis, we simulate the contribution of over-resolved dust emission by dividing the measured visibilities by their maximum. Then, the same binary analysis can be applied to the modified data set. The results are very similar. Binarity of RY Tau is still ruled out within the searched parameter range, due to the large $u,v$-coverage of the combined KI & PTI data. The modified visibilities of DM Tau, GM Aur, and LkCa 15 are now all close to unity, and thus are similar to the originally measured data of UX Tau A. Therefore the same argument holds as above. Binary solutions along the KI baseline position angle cannot be ruled out completely, but it is very improbable that all targets [ harbor]{} a similarly oriented binary. Summarizing, our observations - [ are sensitive to binary companions with flux ratios comparable to the near-IR excess fraction observed for these sources]{} - exclude almost all of the probed binary parameter range, down to companion flux ratios of 0.05 and 2.5 mas (3.5 AU) separations - reject stellar binarity as the dominant mechanism in creating a transitional disk appearance [ @2008ApJ...678L..59I searched with $K$-band Keck aperture masking interferometry for inner binaries in DM Tau, GM Aur, LkCa 15, and UX Tau A, and excluded companions with mass ratios $>\, 0.1$ over $20\,-\,160\,{\rm mas}$ separation range. Thus, we extend this search at similar sensitivity down to 2.5 mas due to the superior resolution of the KI baseline.]{} Modeling the disk emission -------------------------- For all targets but DM Tau, previous SED and high resolution studies concluded that significantly more than 5% of the $K$-band flux is not emitted by the central stellar object but rather is expected to come from some kind of circumstellar material (see also our individual target discussions below, and the references therein). [ In the following, we focus on analyzing the geometric constraints derived from the interferometric data. We do not aim at differentiating between candidates for this emission. Therefore, we simply speak of hot dust as emitter, keeping in mind that other flux contributions from circumstellar gas and scattering of stellar flux are possible as well.]{} As we have established in the previous section, we can exclude nearly all possible explanations which invoke a binary companion with flux ratio $FR > 0.05$. Furthermore, the fact that we resolve all targets with the interferometer indicates a significant $K$-band flux contribution from extended hot circumstellar matter. Thus it appears to be rather unlikely that the inner regions of these transitional disks are totally clear of dust. Instead, a gap must separate the innermost hot dust (visible at $2~\mu {\rm m}$) from the cooler dust which dominates the SED at wavelengths longer than about 8 $\mu {\rm m}$. This correlates well with the fact that all our targets have been classified as [*classical*]{} TTauri stars, with indications for ongoing mass accretion, which requires mass to be located close to the star. If there [*is*]{} circumstellar dust radiating at 2 $\mu {\rm m}$, the expected 3d-morphology is a ring at distances where the dust grain equilibrium temperature is comparable to sublimation temperatures of 1000-1500 $K$. Since our observations are limited to the $K$-band (at about 2 $\mu$m), we are not sensitive to cooler dust [ outside of a few AU]{}. The limited [*u,v*]{}-coverage inhibits a model-free interpretation of the visibility data, but without many additional assumptions, we can evaluate the following simple disk scenario, chosen to minimize the number of model parameters. We fit to our data a central point source plus a face-on, centro-symmetric, narrow ring, leaving only two degrees of freedom: the flux ratio between star and ring, and the ring radius. If, in fact, the simple model is reasonably close to the astrophysical reality, the fitted radii would represent the approximate, order of magnitude, location of the radiating material. Only if the emission morphology departed significantly from circular symmetry (jet-like, edge-on disk), the radii derived here would be less meaningful. Fitting more realistic [*inclined*]{} rings or disks would add at least two more parameters (position and inclination angles), and would essentially require measurements of an orthogonal baseline to obtain good constraints. Previous measurements do not show a very large inclination ($>\,60^\circ$) for any of the targets (Table \[tab:1\]), so in keeping with our small [*u,v*]{}-coverage, we do not explore this scenario. We did test a three-parameter model which lets the radial thickness of the bright ring also vary, but we found no significant improvement over the two-parameter model described here. The results are shown in Fig. \[fig:61\]-\[fig:65\]. For each target, the left panel shows the reduced $\chi^2$ of the given model parameters; red contours indicate the $\chi^2$ contour corresponding to 99.7% confidence, as described above. For GM Aur, LkCa 15, UX Tau A, and RY Tau, we have also overplotted the disk-to-star flux ratio and disk inner radius estimates based on previous SED fits (Table 1). The horizontal dashed green lines in each plot represent a typical error $\pm 5$% for such photometry based NIR-excess estimation. Similarly, we over-plotted for DM Tau an upper limit of 5%$\,K_{\rm excess}/K_{\rm total}$. @2005ApJ...630L.185C find for GM Aur dust inside 5 AU, and the horizontal dashed lines represent the respective dust excess. But they cannot further constrain the location of this inner dust. Appropriately, no vertical line is given in the figure for this target. Qualitatively the ’allowed’ regions of the $\chi^2$-plots in the left panel with $\chi^2$ close to unity show two features: a diagonal feature for small ring radii where we truly have the angular resolution [*and*]{} sensitivity to resolve the ring, and a wavy horizontal feature at larger radii, where the ring is [*over*]{}-resolved. The flux ratio of this feature is connected to the average visibility level, and refers to how much of the total flux is over-resolved. In the right panel of Fig. \[fig:61\]&\[fig:65\], we plot the minimum $\chi^2$ per ring radius, from the parameter range within the SED-model estimated flux ratios (horizontal dashed lines in the left panels, also Table \[tab:1\]). The fact, that these minimum $\chi^2$ are often significantly below unity, does not indicate that our used conservative $V^2$ uncertainty estimates including instrumental and observational systematics, are too large for the presented data. However, it indicates that these systematic biases do not necessarily change from data point to data point, and that the measurements are not completely independent at this accuracy level. We then derive our model radius for GM Aur, LkCa 15, and RY Tau as the radius of the minimum $\chi ^2$, for the disk-to-star flux ratios derived from the SEDs. For DM Tau and UX Tau A, we derive a lower, and a upper limit for the radius, respectively, as discussed below. Table \[tab:2\] lists the best-fit model radius and $\chi^2$ for each target, derived using the constraint of the disk-to-star flux ratios from SEDs. The vertical dashed lines in the right panels of Fig. \[fig:61\]&\[fig:65\] show again the radii where dust emission is expected, as derived from SED models. There is a reasonable coincidence with the results from our visibility analysis for all targets. Details for the individual targets are given in the following sections, but here we summarize the findings qualitatively. - Our data typically require a minimum flux contribution from the inner disk. Since this is $K$-band flux, it cannot come from very large radii, where cooler dust resides (peaking at longer wavelengths). Thus, our data reject the scenario of a totally cleared inner disk. Scenarii for transitional disks, with some hot dust very close to the star and / or a transition zone with no or optically thin dust emission inside of the cooler, outer optically thick dust appear consistent with the data. - Our two-parameter star/ring model fits the data well within our confidence limits, but the radius is not well constrained by the imaging information from our KI data alone. Unambiguous model radii can be derived if we add the $K$-band excess flux the $K$-band excess flux estimation to the visibility analysis. [ These radii are comparable to the 0.1 AU size scales of the innermost dust, inferred from state-of-the-art SED disk models.]{} [ As discussed in Sect.\[sec:115\], DM Tau might be the only target without dust such close to the star. ]{} Discussion of Individual Targets -------------------------------- If not cited otherwise, the background information discussed in this section, in particular photometry and the Spitzer spectroscopy, is published or referenced in the survey articles of @2006ApJS..165..568F [2009]. All our five targets show the typical mid-IR excess at wavelengths longer than about 8 $\mu {\rm m}$, and Spitzer IRS spectra reveal the 10 $\mu {\rm m}$ silicate emission feature, generated by optically thin dust in the surface layer of optically thick disks, or in gaps void of optically thick dust. The strength of the silicate feature varies between targets, and in particular UX Tau A shows very weak silicate emission, potentially due to larger grains in the inner disk from grain growth during disk evolution. We discuss the stars below in increasing order of the expected disk contribution to the $K$-band flux as derived from SED-models. ### DM Tau, expected $K_{\rm inner\,dust}/K_{\rm total}$: $\lesssim\,5\,\%$ {#sec:115} DM Tau does not show any significant emission excess at wavelengths shorter than 8 $\mu {\rm m}$. @2005ApJ...630L.185C fit a typical dust disk model to the optical to mid-IR SED, including an IRS spectrum. They find no significant excess from dust closer than 3 AU to the star, although their model allows for small amounts of optically thin dust in the inner region. [ Their estimation of the upper limit for the optically thin dust mass is 2$\,\cdot10^{-11}\,M_{\rm \odot}$. This resembles an upper limit on the disk excess of about 5 %.]{} However, our visibility measurements clearly show that at least about 5% of the $K$-band flux is not compact photosheric emission (see the horizontal feature of good solutions along the 5% line in Fig. \[fig:61\]). Without an a priori star-to-extended flux ratio, we cannot completely constrain the average stellocentric location of this inner emission, as we do for the other targets. But, assuming a 5% upper limit for $K_{\rm excess}/K_{\rm total}$, we can [*exclude*]{} radii smaller than 0.2 AU, based on our ring-model. Given this lower limit, and the small $K$-band excess, the extended emission in DM Tau might be scattered star light alone, without hot dust emission from very small stellocentric radii. [ Our measurement demonstrates that even a single calibrated KI data point is more sensitive to a small contribution from extended flux at the few percent level than spatially unresolved SED fits. ]{} [ A recent spectro-photometric study finds neither a significant continuum excess nor circumstellar CO emission at 5 $\mu$m in DM Tau [@2009ApJ...699..330S]. ]{} The fact that there is very little 2-5$\,\mu$m circumstellar excess emission might be due to low dust masses and an increased grain size. [ However, @2009ApJ...699..330S confirm the detection of mass accretion onto the star, traced by HI recombination lines. ]{} ### \[sec:111\]GM Aur, expected $K_{\rm inner\,dust}/K_{\rm total}$: 12% @2005ApJ...630L.185C fitted the same type of model to GM Aur as was used for DM Tau, using optically thin inner and optically thick outer dust. They find for GM Aur a dust disk distribution which is not completely devoid of inner dust: the inner, optically thin dust appears to not extend to radii larger then 5 AU, and the outer optically thick dust is outside of about 24 AU. Thus, they inferred a gap between the inner and outer dust distributions. With their given flux ratio, the KI data constrains the emission location of the bulk of the inner dust to be much further in, at radii of $\sim 1$ mas, (0.15AU). @2005ApJ...635.1173A_KI report a slightly larger ring radius fitted to their single KI-data point ($1.58\pm0.6$ mas), which is fully consistent with our findings due to our now increased [*u,v*]{}-coverage. Due to the simplicity of the face-on ring model, the resulting radius is an order-of-magnitude estimation, but the resolution advantage of the interferometer over SED models alone remains apparent. The visibility data constrains the location of the $K$-band excess emission significantly better. We did observe a significant trend of increasing visibilities with increasing hour angle and baseline length (Fig. \[fig:2\]). This could indicate that we observe an inclined disk, in which case our fit radius is a lower limit to the true radius. In particular, the published data point from [@2005ApJ...635.1173A_KI] shows a [*lower*]{} visibility, i.e. a larger size, at a shorter baseline, i.e. at [*lower*]{} angular resolution. This cannot be explained with a circular-symmetric structure. Indeed, @2000ApJ...545.1034S find with mm interferometry a disk inclination of 56 degrees for the cool outer disk around GM Aur. This (outer) disk inclination is confirmed by [Nicmos]{} images of scattered light [@2003AJ....125.1467S]. The circular ring model hits with a disk inclination of 60 $^\circ$ a validity limit. Even higher inclinations could lead to a significant under estimation of the disk size, if modeled by a circular ring, but our sample does not include stars with disk inclinations beyond this limit. Our limited [*u,v*]{}-coverage does not allow us to reliably fit an inclined disk / ring model, but a broad range of inclined disk models are consistent with the observed visibility trend. A combined mm/NIR modeling approach might be worthwhile but bears some caveats. The inner hot disk might have a different orientation than the cool outer disk, and it is also possible that we observe a very close binary system instead of an inner disk, outside the fit range of Sect. \[sec:51\]. ### LkCa 15, expected $K_{\rm inner\,dust}/K_{\rm total}$: 23% Although its visibilities are lower than for GM Aur, we fit a smaller ring radius of about 0.85 mas to the data of LkCa 15, due to the larger flux contribution of the inner disk. @2000ApJ...545.1034S fit similar inclination and position angles to the cool mm-disk of LkCa 15 and GM Aur. Given the increased disk dominance, a trend of the visibility versus hour angle should be stronger in LkCa 15, if the orientation angles are comparable. However, we do not observe such a trend within our uncertainties, which might be due to a combination of the interpretation caveats given in the previous section. Based on near infrared spectroscopy, @2008ApJ...682L.125E_LkCa found that LkCa 15 has a gap between inner and outer optically thick dust, placing it as a member of the so-called pre-transitional disks. Such SED-models suggest that LkCa 15’s optically thick inner disk is located between 0.12 and 0.15 AU [@2007ApJ...670L.135E_LkUX; @2008ApJ...682L.125E_LkCa]. Our high angular resolution data support these findings by confining the bulk of the inner dust radiation to within 1 mas (0.15 AU) from the star. Indeed, our best-fit ring radius is $0.85 \pm 0.05$ mas = $0.12\pm0.01$ AU. Note that the fact that our radius is slightly below the SED model estimation might be due to modelling an inclined inner disk (Table \[tab:1\]) with a face-on ring. ### UX Tau A, expected $K_{\rm inner\,dust}/K_{\rm total}$: 32% UX Tau A’s properties, as measured by the KI, differ from our other targets. It is less spatially resolved. In fact the data are marginally consistent with a point source. Also, the Spitzer IRS spectrum shows a significantly weaker 10 $\mu {\rm m}$ silicate emission feature than in any other transitional disk. Both properties are surprising at first glance given the relatively large inner disk contribution as derived from SED models. The proposed explanation is an optically thick inner disk containing primarily large dust grains [@2007ApJ...670L.135E_LkUX]. Those authors find characteristic radii for UX Tau A’s optically thick inner and outer disks of 0.16 AU and 56 AU respectively. At the given flux ratio, our modeling suggests a ring radius smaller than 0.7 mas (0.1 AU) for a face-on disk. This roughly agrees with the numbers from the SED models, given the order-of-magnitude quality of our face-on ring models. In particular, the radius solutions of our circular ring model are likely slightly too small due the 60 $^\circ$ inclination of the system [@2007ApJ...670L.135E_LkUX]. ### RY Tau, expected $K_{\rm inner\,dust}/K_{\rm total}$: 73% In RY Tau, the NIR excess over the photospheric radiation is much more prominent than in the other targets. This is likely related to RY Tau’s earlier spectral type, and thus hotter photosphere. The higher luminosity may cause a puffed-up inner disk rim and increased NIR excess. Its visibilities are much lower than any other target (mean $V^2 = 0.28$), confirming that in the $K$ band, this system’s light is dominated by the extended disk. For RY Tau we took only three new KI data points, and added them to the measurements of an earlier PTI experiment [@2005ApJ...622..440A_PTI]. The availability of several baselines leads to the complete exclusion of the entire binary parameter range probed, as discussed in Sect. \[sec:51\], assuming that the brightness distribution did not change significantly during the few years between the PTI observations and our KI experiment. For the sake of comparison, we show in Fig. \[fig:65\] our simple star plus face-on ring fit to the data, which shows the same two qualitative features as before: the diagonal feature at low separation and the horizontal feature at larger separations–marking the resolved and over-resolved ring radii. The face-on ring model fits the data fairly well, although due to the much greater number of baselines probed, the reduced $\chi^2$ significance limit is much reduced compared to our other targets, and no models for $K_{\rm inner\,dust}/K_{\rm total}$=73% fit within our 99.7% confidence level. This most likely suggests that we have moved beyond the applicability of our simplistic face-on ring model. Due to the addition of the PTI baselines, we are sensitive to the inclination of RY Tau. @2005ApJ...622..440A_PTI estimated an inclination of $25\pm 3$ degrees for a ring model, consistent with our data. Our inferred ring radius of $\sim 1.5~{\rm mas}$ agrees with the earlier findings within the uncertainties. Note that @2003ApJ...597L.149M reports a spectro-photometric model fit for RY Tau with a high disk inclination, which is not supported by the interferometric data [see the discussion in @2005ApJ...622..440A_PTI]. This demonstrates the [ ambiguity of such model fits without including high angular resolution imaging data]{}. Comparison with CoKu Tau/4 {#sec:52} -------------------------- As noted above, recent high-resolution aperture-masking by @2008ApJ...678L..59I has revealed that the supposedly transitional disk around CoKu Tau/4 is instead a circumbinary disk around a near-equal flux binary with a (projected) star-star separation of 53 mas. Thus, there is no need to invoke planet formation to produce its inferred disk clearing within $\sim 10$ AU. But for all other known and well-studied transitional disks in Taurus, we have probed the binary separation range that could be responsible for creating the inner holes implied by the target SEDs, and concluded that the interferometric data rule out binarity as the predominant cause for the lack of hot dust emission in these systems. CoKu Tau/4 appears to be the exception, not the rule. Is there any systematic difference between CoKu Tau/4 and other transitional disks which might let us distinguish between these two classes of objects based on other criteria? [ Table \[tab:1\] lists the physical properties of the CoKu Tau/4 system as compared to our target stars. Despite of their similar age and spectral type, CoKu Tau/4 is about twice as bright as DM Tau. This hints already to the existence of an equal-mass binary, as found in the diffraction limited Keck images described above, and supports at the same time our negative result of the binary search, given that our program stars do not show such stellar over-luminosity.]{} @2008ApJ...678L..59I note further, that one of the main differences is that only CoKu Tau/4 is a weak-lined TTS: its spectrum does not show significant signs of mass accretion [@2005ApJ...621..461D]. However all five targets of our sample are mass-accreting classical TTS, with typical accretion rates of $10^{-9..-8}\,{\rm M_{\odot}\,yr^{-1}}$ [@2005ApJ...630L.185C]. This matches with the theoretical work of @1994ApJ...421..651A [and references therein], which suggests that accretion of a circum-binary disk onto the central stars is inhibited by the resonant torques at the circumbinary dust disk inner edge. Thus, the binary in the CoKu Tau/4 system may be the cause not only of the disk hole, but also for the insignifcant accretion of the disk material onto the stars. Furthermore, our data confirm the presence of hot dust in the inner regions of transitional disks. This suggests a scenario in which true transitional disks will in general retain some small amount of hot, radiating dust inside of the outer, partially cleared optically thick disk. They also will have signs of accretion onto the central star. AU-scale binaries, however, may clear disks in a way that they mimic transitional disk SEDs, but lack ongoing accretion of material on sub-AU scales, inside the binary, and also inside the outer optically thick dust, cleared by dynamical interaction with the binary. [lccccc]{} & DM Tau & GM Aur & LkCa 15 & UX Tau A & RY Tau\ % FoV without companion $>0.05$ & 95.1 & 94.8 & 97.2 & 94.4 & 100.\ % FoV without companion $>0.08$ & 99.4 & 99.3 & 99.9 & 98.0 & 100.\ Best-fit ring radius $\rho$ (mas)& $>\,1.5$ & 1 & 0.85 & $<\,0.7$ & 1.4\ Best-fit ring radius $\rho$ (AU) & $>\,0.2$ & 0.14 & 0.12 & $<\,0.1$ & 0.2\ $\chi_{\rm red}^2$-limit (99.7 %) & 3.6 & 4.6 & 4.0 & 5.8 & 2.2\ Conclusions {#sec:6} =========== Within a single night of repeated observations, our experiment was sensitive to close binaries with 2.5 - 30 mas separation and flux ratios down to 0.05. We were able to rule out nearly all possible binary companions within that parameter range. [ For four of our five targets, we cannot entirely rule out all possible companions. The remaining solutions cover $\lesssim\,2\%,\,(\lesssim\,6\%)$ of the probed FoV for companion flux ratios larger then 0.08 (0.05) of the primary’s brightness. Those solutions are preferentially aligned with the baseline of the Keck Interferometer, where our sparse sampling of the $u,v$-plane renders our measurements insensitive. Observations along an orthogonal baseline (e.g. from CHARA or VLTI) should suffice to completely rule out the remaining part of parameter space, as was the case for RY Tau. However, already now a simple statistical analysis reveals that it is very unlikely that more than one of our target stars would indeed harbor a binary of separations larger than the minimum binary separation probed. These interferometric observations extend the finding of similar binary searches, based on diffraction limited 10 m class telescope imaging, down to ten times smaller projected separations. We conclude that, unlike in the CoKu Tau/4 system, binarity is in general not responsible for either clearing disk holes to produce transitional-disk-type SEDs, or for the near-infrared excess over the photospheric emission. ]{} Instead, we spatially resolve a fraction of the $K$-band emission in all five stars, ranging between $\sim5-70\%$ of the total $K$-band flux. By fitting a toy disk model to the data, we find that this inner disk emission typically comes from radii of about 1 mas (0.15 AU), consistent with previous work. In particular, these findings are consistent with recent disk models fitting spatially unresolved spectro-photometric data of transitional disks. A next step for studying transitional disks at high angular resolution could be to fit more realistic models simultaneously to the observed visibilities and the SED, as has been recently done for a few systems (Tannirkulam et al. 2008, Pinte et al. 2008). We confirm that the transitional disk phase is often characterized by several distinct dust zones: an inner (of order 0.1 AU), and an outer part (of order $>10$ AU) which are not smoothly connected by a continuous distribtion of optically thick material. This supports the general hypothesis that these young objects are indeed in a transitional evolutionary state between primordial optically thick disks and optically thin disks. In each of our targets, the habitable zones are devoid of optically thick dust emission. The fact that we resolve excess emission very close to the star in a transitional disk, which lacks such emission further out, underlines that the evolution of a primordial disk is not as simple as a clearing from the inside out due to photoevaporation. In fact, the presence of gaps suggests that we may see the effect of planet formation on disks. MDP was supported by a NSF Astronomy & Astrophysics Postdoctoral Fellowship. We are grateful to the team at WMKO and NExScI for making these observations a success. We thank R. Akeson for providing the reduced data of the previously published PTI and KI measurements of some of our sources. The data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. The Keck Interferometer is funded by the National Aeronautics and Space Administration as part of its Navigator program. This work has made use of services produced by the NASA Exoplanet Science Institute at the California Institute of Technology. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. 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This $\chi^2$ modeling assumes errors that are statistically independent, when in fact calibration systematics result in correlated errors, slightly modifying the derived probability levels. We do not attempt to model this in detail, but our choice of a conservative estimate of 0.03 for the errors including such calibration systematics implies that true rejection confidence levels are most likely more stringent than the formal 99.7% level. [^6]: Due to the point-symmetry of visibility amplitude data, these probabilities calculate as $(40/180)^N\,*\,(5-N)/5$ where N is the number of binaries out of our five targets, and (5-N)/5 is the probability to chose one of the single stars for the PTI observations.

--- abstract: 'The optical/near-IR stellar continuum carries unique information about the stellar population in a galaxy, its mass function and star-formation history. Star-forming regions display rich emission-line spectra from which we can derive the dust and gas distribution, map velocity fields, metallicities and young massive stars and locate shocks and stellar winds. All this information is very useful in the dissection of the starburst phenomenon. We discuss a few of the advantages and limitations of observations in the optical/near-IR region and focus on some results. Special attention is given to the role of interactions and mergers and observations of the relatively dust-free starburst dwarfs. In the future we expect new and refined diagnostic tools to provide us with more detailed information about the IMF, strength and duration of the burst and its triggering mechanisms.' author: - 'Nils Bergvall, Thomas Marquart, Göran Östlin, Erik Zackrisson' --- galaxies:dwarfs, galaxies:evolution, galaxies:interactions, galaxies:starburst, infrared:galaxies Introduction ============ Optical/near-IR broadband photometry of a starburst galaxy gives a first indication of burst strength, age and distribution of the young and old populations and their basic morphological structure parameters. Model based spectrophotometric tools are provided for more detailed analysis. A rich set of emission lines are used for analysis of kinematics, chemical abundance, shocks, stellar upper mass limit and distribution of dust and molecular gas. Absorption line indices provide estimates of age and IMF of the evolved population. Fig. \[specfig\] shows a synthetic spectrum a mixture of a young and old population with a mass ratio 2:1. A general review of the diagnostic tools and the limitations of the photoionization models used in the analysis is discussed by Schaerer (2001). {width="65.00000%"} \[specfig\] Heavy dust obscuration, in particular in LIRGs and ULIRGs, has been a problem in the optical/near-IR. Here we will therefore focus on starbursts in low-extinction regions, notably starburst dwarf galaxies. First, however, we will discuss a widely debated issue where optical data originally had a strong impact, namely the importance of gravitational interactions as a starburst triggering mechanism. Starbursts and tidal interaction ================================ It is clear from the properties of ULIRGs that mergers are required to trigger major starbursts. But is it a sufficient requirement? How often do mergers and close encounters generate starbursts? To answer this question it is common to compare two galaxy samples - interacting/merging galaxies (IGs) or pairs, and non-interacting galaxies (NIGs). A problem with the comparison is that NIGs and IGs have evolved in different environments where e.g. mergers, ram pressure, harassment and gas infall have different influence. Integrated broadband photometry and H$\alpha$ emission are the most widely used tools in this context. In the classical paper by Larson & Tinsley (1978) the authors claim, based on UBV data, that interactions frequently trigger a major SF increase involving as much as 5% of the total mass. Many follow-up studies seem to confirm the result but are often influenced by strong selection effects, non-matching morphological type distribution NIGs/IGs and are focusing on the most dramatic cases. Studies based on more well constrained samples (Bergvall et al. 2002, Brosch et al. 2004) do not confirm these results but find that tidal interactions have an insignificant influence on the SF history of galaxies in the local universe. There seems to be an agreement however, of a correlation between interaction and increased SF within the central kpc (first discussed by Keel et al. 1985). [*Galaxy pairs*]{} with small separations show similar trends as seen in H$\alpha$ (Barton et al. 2000, Lambas et al. 2003, Nikolic et al. 2004). The mean increase is in both cases is quite moderate however, and few cases are qualified to be called ’nuclear starbursts’. Bergvall et al. (2000) and Varela et al. (2004) find that masses of perturbed galaxies are higher than NIGs of similar morphology indicating that they experience mergers more frequently. This may lead to a steady inflow of gas that can explain part of the increased SF in the centre. Varela et al. also find a [*higher frequency of bars in disturbed systems*]{}, in accordance with related studies in the past (see Knapen 2004). Bars are known to generate mass inflows. Thus it is not clear what is the main triggering mechanism of the central increase in SF. The conclusion must be that there is [*no strong support that tidal interactions generate starburst activity that significantly affects the SF history of galaxies in the local universe*]{}. Estimates give room for major starbursts among less than a few % of the IGs. Blue compact galaxies ===================== Blue compact galaxies (BCGs) is a not well defined type as the galaxies are selected either from spectroscopic or photometric critera. The general properties are high surface brightness, low chemical abundance and a high gas mass fraction. They have a wide range of morphologies (Loose & Thuan 1986). Are they bursting? Fig. \[mbhilb\] shows L$_B$/$\cal M_{\rm HI}$ vs. M$_B$ of different types of gas rich galaxies. The BCG sample is incomplete but constitutes a representative part of the nearby sample of starburst dwarfs (Mrk, UM, Tololo etc.). We see that there is a continuous distribution towards high L$_B$/$\cal M_{\rm HI}$ but that the properties of most BCGs are similar to dIrr and late type spirals of similar luminosity, i.e. they are probably not bursting. The high surface brightness of the burst could be due to a high column density (and a small scalelength, cf. Papaderos et al. 1996 and Salzer et al. 2002), perhaps caused by a low angular momentum. Since their gas mass often constitutes a major fraction of the total mass (Salzer et al. 2002), the diagram shows that starbursts in these galaxies are either shortlived or rare. {width="60.00000%"} \[mbhilb\] Some BCGs have a $\sim$ tenfold global increase in SFR, i.e they are true starbursts. What are their specific properties? There is no strong indication of a correlation between SF activity and tidal interactions (Brosch 2004, Hunter and Elmegreen 2004). On the other hand BCGs appear to be involved in mergers with intense SF more frequently than other dwarfish galaxies (e.g. Gil de Paz 2004). It could indicate that mergers are important triggers and morphologically shortlived. The gas consumption rates are typically shorter than 100 Myr, i.e. similar to the dynamical timescale of a merger. ### Ages and masses Dynamical mass estimates of BCGs are difficult since the kinematics sometimes are quite chaotic due to the mass motions that cause the burst and because of the SN winds. To overcome the problem with the stellar winds it becomes necessary to use stellar absorption features. The only useful lines for this purpose are the Ca II triplet lines at about 8500 Å. Not until quite recently has this option become accessible (Östlin et al. 2004). The results are very promising and will soon help to solve the question regarding the coupling between gas and stars and facilitate the detailed analysis of velocity fields based on H$\alpha$ (e.g. Marquart et al. 2004). Age and SFR are often estimated from the H$\alpha$ flux, the H$\alpha$ equivalent width (EW(H$\alpha$)) and broadband photometry. From this the ’photometric mass’ is obtained assuming that the SFR is constant. The age is however difficult to determine, even if we assume that the SFR is constant. In such a case, EW(H$\alpha$) is a function of the IMF and age. The IMF slope in starbursts seems to be well constrained in the intermediate stellar mass range (Elmegreen 2004) but not so well for high masses. Fig \[ewfig\] shows the predicted EW(H$\alpha$) for two values of the upper mass limit, 40 and 120 solar masses. It can be seen that the predicted ages differ with a factor of 5-10 over a large age range. There is also an observational problem in that intense starbursts may have huge Strömgrenspheres from which the H$\alpha$ emission may be lost due to a limited aperture size. The uncertainty in the determination of the widely used b parameter (b = SFR/$<$SFR$>$) obviously must be quite high, in particular if we consider the poorly constrained SF history. For BCGs there seems to be a simple way to account for the SF history reasonably well. It is based on a two component model of the galaxy consisting of a starburst superposed on a host galaxy with an exponential luminosity profile. If photometric masses are applied to this model we find that there a fairly tight correlation between mass and central velocity dispersion (Östlin et al. 2001), indicating that this simple model is quite successful. {width="50.00000%"} \[ewfig\] A very useful method to determine the past starburst activity in a galaxy is based on its rich system of super star clusters and globular clusters (GCs). The GC IMF is Salpeter-like and their stellar content is coeval. This makes them quite reliable as standard clocks and optical/near-IR photometry and spectroscopy can be used to determine their ages and trace past star formation history thereby identifying bursts (e.g. Östlin et al. 2003; de Grijs et al. 2004). The best method to derive the ages is from colour-magnitude diagrams of the stellar population, but most starburst galaxies are to distant to make this method feasible. The few results available give no support for strong shortlived bursts separated by quiescent periods (Annibali et al. 2003, Schulte-Ladbeck et al. 2001). A similar conclusion was reached by Westera et al. (2004) in a study of 200 HII galaxies based on stellar absorption features. Taking the previous discussion into account, these observations indicate that true starbursts are rare rather than shortlived or that they are shortlived but change morphological type at or after the burst. ### The starburst host It is well established that the luminosity profile of most BCGs can be characterised by a burst superposed on a host galaxy with (mostly) red colours, typical of an old stellar population, and a morphology resembling an early type galaxy (e.g. Papaderos et al. 1996, Gil de Paz et al. 2004). An attractive scenario is that a dE is merging with a gas rich galaxy that triggers the burst. Recently it was found that the optical/near-IR colours of the host of luminous BCGs at very faint levels has a red excess, difficult to explain with a normal IMF and a low metallicity (Bergvall & Östlin 2002). This problem is discussed in the paper by Zackrisson et al. (2004). It could indicate that a host galaxy of special properties is needed to trigger a true starburst. 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--- abstract: 'We give a short proof of a recent result by Bernik, Mastnak, and Radjavi, stating that an irreducible group of complex matrices with nonnegative diagonal entries is diagonally similar to a group of nonnegative monomial matrices. We also explore the problem when an irreducible matrix semigroup in which each member is diagonally similar to a nonnegative matrix is diagonally similar to a semigroup of nonnegative matrices.' address: 'Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia' author: - Grega Cigler - Roman Drnovšek title: On semigroups of matrices with nonnegative diagonals --- matrices ,semigroups ,nonnegative matrices ,cones ,irreducibility 15B48 ,20M20 ,47D03 6.8mm Introduction ============ Multiplicative semigroups of matrices with nonnegative diagonal entries have been studied in the papers [@BMR] and [@SWG]. Their authors considered the general question under which additional assumptions such a semigroup is simultaneously similar to a semigroup of nonnegative matrices. The main result of [@BMR] is that every irreducible group of complex matrices with nonnegative diagonal entries is diagonally similar to a group of nonnegative monomial matrices. In Section 2 we give a short proof of this result. Our proof is more geometric and less group-theoretic than the proof in [@BMR]. Multiple authors of the paper [@SWG] provided several examples showing that it is impossible to extend this result from groups to semigroups. So, to obtain similarity to a semigroup of nonnegative matrices, stronger assumptions on a given semigroup must be imposed. In Section 3 we explore the problem when an irreducible matrix semigroup in which each member is diagonally similar to a nonnegative matrix is necessarily diagonally similar to a semigroup of nonnegative matrices. We now recall some definitions and basic facts. The set of all nonnegative real numbers is denoted by ${{\mathbb{R}}}_+$. A convex set $K \subseteq {{\mathbb{R}}}^n$ is said to be a [*cone*]{} if $r K \subseteq K$ for all $r \in {{\mathbb{R}}}_+$. A cone $K \subseteq {{\mathbb{R}}}^n$ is [*proper*]{} if it is closed, [*pointed*]{} ($K \cap (-K) = \{0\}$), and [*solid*]{} (the interior of $K$ is nonempty). The most natural example of a proper cone is the [*nonnegative orthant*]{} ${{\mathbb{R}}}_+^n$. A cone $K \subseteq {{\mathbb{R}}}^n$ is [*reproducing*]{} if $K - K = {{\mathbb{R}}}^n$. It is well-known that a closed cone is solid if and only if it is reproducing. Let $K$ be a closed cone in ${{\mathbb{R}}}^n$. A vector $x \in K$ is an [*extremal vector*]{} of $K$ if $y \in K$ and $x-y \in K$ imply that $y$ is a nonnegative multiple of $x$. By ${{\rm Ext \,}}(K)$ we denote the set of all extremal vectors of $K$. By the Krein-Milman theorem, $K$ is the convex hull of ${{\rm Ext \,}}(K)$. The angle $\phi \in [0, \pi]$ between non-zero vectors $x$, $y \in {{\mathbb{R}}}^n$ is determined by the equality  $x^T y = \|x\| \, \|y\| \, \cos \phi$. If $F$ is a subset of complex numbers, then $M_n(F)$ denotes the set of all $n \times n$ matrices with entries in $F$. If ${{\cal C}}\subseteq M_n({{\mathbb{C}}})$ is a collection of complex matrices, then $\overline{{{\cal C}}}$ denotes its closure in the Euclidean topology, and ${{\mathbb{R}}}_+ {{\cal C}}$ denotes its [*homogenization*]{}, i.e., ${{\mathbb{R}}}_+ {{\cal C}}= \{r C: r \in {{\mathbb{R}}}_+, C \in {{\cal C}}\}$. We say that a matrix has a [*nonnegative diagonal*]{} if all of its diagonal entries are nonnegative. A matrix is called [*monomial*]{} if it has the same nonzero pattern as a permutation matrix, i.e., there is exactly one nonzero entry in each row and in each column. A collection ${{\cal C}}\subseteq M_n({{\mathbb{C}}})$ (where $n \ge 2$) is [*reducible*]{} if there exists a common invariant subspace other than the trivial ones $\{0\}$ and ${{\mathbb{C}}}^n$, or equivalently, there exists an invertible matrix $S \in M_n({{\mathbb{C}}})$ such that the collection $S {{\cal C}}S^{-1}$ has a block upper-triangular form; otherwise, the collection ${{\cal C}}$ is said to be [*irreducible*]{}. If the matrix $S$ can be chosen to be a permutation matrix, then the collection ${{\cal C}}$ is said to be [*decomposable*]{}; otherwise, it is called [*indecomposable*]{} (or [*ideal-irreducible*]{}). Groups of matrices with nonnegative diagonals ============================================= The study of semigroups of matrices having nonnegative diagonals was initiated by the authors of [@BMR]. They started their discussion by the following result (see [@BMR Theorem 4.1]). \[rank-one\] Let ${{\cal S}}\subseteq M_n({{\mathbb{C}}})$ be an irreducible semigroup of matrices of rank at most one having nonnegative diagonals. If $\overline{{{\mathbb{R}}}_+ {{\cal S}}} = {{\cal S}}$, then, after a diagonal similarity, ${{\cal S}}= X Y^T$ for some subsets $X$ and $Y$ of ${{\mathbb{R}}}_+^n$ each of which spans ${{\mathbb{C}}}^n$. Using the Haar measure one can prove the following assertion (see [@BMR Proposition 4.3]). \[positive-valued\] Let ${{\cal S}}\subseteq M_n({{\mathbb{C}}})$ be an irreducible semigroup of matrices. Suppose that $\overline{{{\mathbb{R}}}_+ {{\cal S}}} = {{\cal S}}$ and that there exists a non-zero functional $\varphi: M_n({{\mathbb{C}}}) \to {{\mathbb{C}}}$ such that $\varphi(S) \in {{\mathbb{R}}}_+$ for all $S \in {{\cal S}}$. Then ${{\cal S}}$ has members of rank one. The following theorem is the main result of [@BMR Theorem 5.5]. We provide a short proof that is more geometric and less group-theoretic than the original one. \[positivediaggroup\] If ${{\cal G}}\subset M_n({{\mathbb{C}}})$ is an irreducible group of matrices with nonnegative diagonals, then, up to a diagonal similarity, ${{\cal G}}$ is a group in $M_n({{\mathbb{R}}}_+)$. Therefore, each member of the group ${{\cal G}}$ is a nonnegative monomial matrix. With no loss of generality we may assume that $t G \in {{\cal G}}$ for all $t > 0$ and $G \in {{\cal G}}$. Let ${{\cal S}}= \overline{{{\cal G}}}$. Applying Proposition \[positive-valued\] for the trace functional, we conclude that ${{\cal S}}$ contains elements of rank one. The semigroup ideal ${{\cal S}}_1$ of all elements of rank at most one in ${{\cal S}}$ is irreducible (see [@RR]). By Theorem \[rank-one\], we can assume that, after a diagonal similarity, ${{\cal S}}_1 = X Y^T$ for some subsets $X$ and $Y$ of ${{\mathbb{R}}}_+^n$ each of which spans ${{\mathbb{C}}}^n$. We can also assume that ${{\mathbb{R}}}_+ X = X$ and ${{\mathbb{R}}}_+ Y = Y$. The cone $\widehat{X}$ generated by $X$ is closed, and it is invariant under any $S \in {{\cal S}}$, since $ (Sx) y^T = S (x y^T) \in {{\cal S}}_1$ for every $x \in X$ and $y \in Y$. Similarly, it follows from $x (S^T y)^T = (x y^T) S \in {{\cal S}}_1$ that $Y$ is invariant under $S^T$. The dual cone $$Y^d = \{ z \in {{\mathbb{R}}}^n : z^T y \ge 0 \textrm{ for all } y \in Y \}$$ of the set $Y$ obviously contains ${{\mathbb{R}}}_{+}^n$, and it is invariant under any $S \in {{\cal S}}$, as $(Sz)^T y = z (S^T y) \ge 0$ for all $y \in Y$ and $z \in Y^d$. It follows that every $G \in {{\cal G}}$ is a bijective mapping on both $\widehat{X}$ and $Y^d$, implying that every $G \in {{\cal G}}$ maps ${{\rm Ext \,}}(\widehat{X})$ to itself, and the same holds for the cone $Y^d$. We want to show that the inclusions $\widehat{X} \subseteq {{\mathbb{R}}}_+^n \subseteq Y^d$ are in fact equalities. Assume, if possible, that $\widehat{X} \neq Y^d$. Then there exists a unit vector $x \in X \setminus Y^d$ which is extremal for the cone $\widehat{X}$. Since the cone $Y^d$ is closed, the distance between $x$ and $Y^d$ is strictly positive. It follows that there is a number $\phi \in (0, \pi/2)$ such that, for each $z \in {{\rm Ext \,}}(Y^d)$, the angle between $z$ and $x$ is at least $\phi$. Since $x \in X$ and the set $Y$ is spanning, there is a vector $y \in Y$ such that $P = x y^T \in {{\cal S}}$ with $y^T x > 0$. We can assume that $y^T x = 1$, so that $P x = x$. Choose any $\epsilon > 0$. Since ${{\cal S}}= \overline{{{\cal G}}}$, there is a matrix $G \in {{\cal G}}$ such that $\|G - P\| < \epsilon$. Now, for any $z \in {{\rm Ext \,}}(Y^d)$ with norm $1$, we have $$\epsilon^2 > \|G z - P z\|^2 = \| G z - (y^T z) x \|^2 = \|G z\|^2 + (y^T z)^2 - 2 (y^T z) \|G z\| \cos \phi_z ,$$ where $\phi_z$ is the angle between the vector $x$ and the vector $G z \in {{\rm Ext \,}}(Y^d)$. Since $y^T z \in {{\mathbb{R}}}_+$ and $\phi_z \geq \phi$, we conclude that $$\epsilon^2 > \|G z\|^2 + (y^T z)^2 - 2 (y^T z) \|Gz\| \cos \phi = (y^T z - \|G z\| \cos \phi)^2 + \|G z\|^2 \sin^2 \phi .$$ It follows that $$\|G z\| \sin \phi < \epsilon \ \ \textrm{and} \ \ \ \left|y^T z - \|G z\| \cos \phi \right| < \epsilon ,$$ and so $$0 \le y^T z < \epsilon + \|G z\| \cos \phi < \epsilon + \frac{\epsilon}{\sin \phi} \cos \phi .$$ Since $\epsilon>0$ is arbitrary, we obtain that $y^T z = 0$ for all vectors $z \in {{\rm Ext \,}}(Y^d)$, implying that $y = 0$. This contradiction completes the proof of the equality $\widehat{X} = Y^d = {{\mathbb{R}}}_+^n$. Consequently, the inclusion ${{\cal G}}\subset M_n({{\mathbb{R}}}_+)$ holds, as asserted. Since the map associated to any matrix $G \in {{\cal G}}$ maps ${{\rm Ext \,}}({{\mathbb{R}}}_+^n)$ to itself and it is invertible, the matrix $G$ must be monomial, and so the proof is complete. Semigroups of matrices diagonally similar to nonnegative ones ============================================================= Let ${{\cal S}}\subseteq M_n({{\mathbb{C}}})$ be a semigroup in which each member $A\in {{\cal S}}$ is diagonally similar to a nonnegative matrix. In this section we are looking for additional assumptions under which the whole semigroup ${{\cal S}}$ is diagonally similar to a semigroup of nonnegative matrices. We first show that it does not suffice to assume that the semigroup ${{\cal S}}$ is indecomposable. Define $n \times n$ matrices $A=a a^T$ and $B=b b^T$, where $n \ge 2$, $a=[1,1,\ldots,1]^T$ and $b=[1,1,\ldots,1,1-n]^T$. Then every nonzero member of the semigroup ${{\cal S}}$ generated by $A$ and $B$ is an indecomposable matrix of rank one that is diagonally similar to a nonnegative matrix. However, the whole semigroup ${{\cal S}}$ is not diagonally similar to a semigroup of nonnegative matrices. Note that $A^k=n^{k-1} A$ and $B^k=(n(n-1))^{k-1} B$ for all $k\in{{\mathbb{N}}}$, while $AB=BA=0$. Therefore, ${{\cal S}}$ is contained in the semigroup ${{\mathbb{R}}}_+ A \cup {{\mathbb{R}}}_+ B$. If $D$ is the diagonal matrix with diagonal $(1,1,\ldots,1,-1)$, then the matrix $DBD^{-1}$ is nonnegative, and therefore each matrix from ${{\cal S}}$ is diagonally similar to a nonnegative matrix. Since the matrices $A$ and $B$ are indecomposable, every nonzero member of ${{\cal S}}$ is indecomposable as well. It is easy to verify that the whole semigroup ${{\cal S}}$ is not diagonally similar to a semigroup of nonnegative matrices. In the rest of the paper we explore the case when the semigroup ${{\cal S}}$ is irreducible. We first show that, with no loss of generality, we may assume that ${{\cal S}}$ is a closed set. \[closurelemma\] Let ${{\cal C}}\subset M_n({{\mathbb{C}}})$ be a collection in which each member $A\in {{\cal C}}$ is diagonally similar to a nonnegative matrix. Then the closure $\overline{{{\mathbb{R}}}_+{{\cal C}}}$ also consists of matrices which are diagonally similar to nonnegative matrices. Clearly, we may assume that ${{\mathbb{R}}}_+ {{\cal C}}= {{\cal C}}$. If $A \in \overline{{{\cal C}}}$, then there is a sequence $\{A_k\}_{k \in {{\mathbb{N}}}}$ in ${{\cal C}}$ converging to the matrix $A$. For each $k\in{{\mathbb{N}}}$, let $D_k$ be a diagonal matrix such that $D_k A_k D_k^{-1}$ is a nonnegative matrix. We may assume that each diagonal entry of $D_k$ has absolute value one. Since the sequence $\{D_k\}_{k \in {{\mathbb{N}}}}$ is bounded, it has a convergent subsequence $\{D_{k_m}\}_{m \in {{\mathbb{N}}}}$ converging to some diagonal matrix $D$. Since $DAD^{-1} =\lim_{m\to\infty} D_{k_m}A_{k_m}D_{k_m}^{-1}$, the matrix $DAD^{-1}$ is nonnegative, and so $A$ is also diagonally similar to a nonnegative matrix. This completes the proof. We continue with a reduction of the problem to the real setting. \[backtoreality\] Let ${{\cal S}}= \overline{{{\mathbb{R}}}_+{{\cal S}}} \subseteq M_n({{\mathbb{C}}})$ be an irreducible semigroup such that each member $A\in {{\cal S}}$ is diagonally similar to a nonnegative matrix. Then there exists an invertible diagonal matrix $D \in M_n({{\mathbb{C}}})$ such that the semigroup $D {{\cal S}}D^{-1}$ consists of real matrices, and there exist two sets $X,Y \subseteq {{\mathbb{R}}}_+^n$, each of which spans ${{\mathbb{C}}}^n$, such that $$D{{\cal S}}_1D^{-1}=(D{{\cal S}}D^{-1})_1=XY^T ,$$ where ${{\cal S}}_1$ is the ideal of ${{\cal S}}$ consisting of members of rank at most one. Furthermore, the subcone of ${{\mathbb{R}}}_+^{n}$ generated by $X$ is a proper cone invariant under every member of ${{\cal S}}$. Our assumption implies in particular that all diagonal elements of any member of ${{\cal S}}$ must be nonnegative. By Proposition \[positive-valued\], the ideal ${{\cal S}}_1$ of all members of ${{\cal S}}$ with rank at most one is nonzero. Since ${{\cal S}}$ is an irreducible semigroup, it is also necessarily irreducible (see [@RR]). Then by Theorem \[rank-one\] we can find an invertible diagonal matrix $D$ and two sets $X,Y\subset {{\mathbb{R}}}_+^n$, each of which spans ${{\mathbb{C}}}^n$, such that $D {{\cal S}}_1 D^{-1} = X Y^T$. As we are interested in diagonal similarities, we can assume that $D$ is the identity, so that ${{\cal S}}_1=XY^T$. To prove the inclusion ${{\cal S}}\subset M_n({{\mathbb{R}}})$, pick any $A\in {{\cal S}}$ and $x\in X$. Since for any nonzero vector $y\in Y$ the matrix $A(xy^T)=(Ax)y^T$ belongs to ${{\cal S}}_1$, we conclude that $Ax \in X \subseteq {{\mathbb{R}}}_+^n$. It follows that the cone of ${{\mathbb{R}}}_+^{n}$ generated by $X$ is a proper cone invariant under $A$. Since the set $X$ spans ${{\mathbb{C}}}^n$, it follows that $A ({{\mathbb{R}}}^n) \subseteq {{\mathbb{R}}}^n$, and therefore $A \in M_n({{\mathbb{R}}})$. This completes the proof. From now on we consider real matrices. If a real matrix $A$ is diagonally similar to a nonnegative matrix via diagonal matrix $D$, we clearly may assume that each diagonal entry of $D$ is either $1$ or $-1$. In this case we say that $D$ is a [*$\pm 1$-diagonal*]{} matrix. \[indecompobstruction\] Let $A \in M_n({{\mathbb{R}}})$ be an indecomposable matrix and $D$ a $\pm 1$-diagonal matrix such that $A'=DAD$ is a nonnegative matrix. If there exists a proper cone $K$ such that $A(K)\subseteq K$ and $K\subseteq {{\mathbb{R}}}^n_+$, then $D=\pm I$ and $A$ itself is a nonnegative matrix. By the Perron-Frobenius Theorem, the spectral radius $\rho(A')=\rho(A)$ of the indecomposable matrix $A'$ is a simple eigenvalue having exactly one (up to a scalar multiplication) strictly positive eigenvector $e$. On the other hand, since the proper cone $K$ is invariant under $A$, the extension of the Perron-Frobenius Theorem (see [@BP Theorem 3.2]) ensures that there is a non-zero vector $x \in K$ such that $A x = \rho(A) x$. However, $A' D x= D A x =\rho(A) D x$, and so the vectors $Dx$ and $e$ are collinear. It follows that either $De$ or $-De$ belongs to $K \subseteq {{\mathbb{R}}}_+^{n}$, and this implies that $D=\pm I$ and $A$ itself is a nonnegative matrix. The following simple example shows that in Lemma \[indecompobstruction\] we cannot omit the assumption that the cone $K$ is proper. Let $n \ge 2$, $a=[1,1, \ldots,1,1-n]^T$ and $K={{\mathbb{R}}}_+ [1,1, \ldots,1]^T$. The matrix $A=a a^T$ is indecomposable, and the cone $K$ is invariant under $A$, while $DAD$ is a nonnegative matrix for the diagonal matrix $D$ with diagonal $(1,1, \ldots,1,-1)$. For $n\ge 2$ we say that a matrix $A\in M_n({{\mathbb{R}}})$ is $1$-[*decomposable*]{} if there is a permutation matrix $P$ such that $$PAP^T=\left[\matrix {A_1 & B \cr 0 & A_2 }\right],$$ where each of $A_1$ and $A_2$ is either an indecomposable (square) matrix or a $1 \times 1$ block. The following assertion is crucial for the proof of the main result. \[obstruction\] Let $A \in M_n({{\mathbb{R}}})$ be a $1$-decomposable matrix that is diagonally similar to a nonnegative matrix. Let $K$ and $L$ be proper cones of ${{\mathbb{R}}}^n_+$ that are invariant under $A$ and $A^T$, respectively. Then $A$ is a nonnegative matrix. Let $P$ be a permutation matrix such that the matrix $PAP^T$ has the block form $$PAP^T=\left[\matrix {A_1 & B \cr 0 & A_2 }\right]$$ with respect to the decomposition ${{\mathbb{R}}}^n={{\mathbb{R}}}^k\oplus{{\mathbb{R}}}^l$, where $1 \le k < n$, $l = n - k$, and each of $A_1$ and $A_2$ is either an indecomposable (square) matrix or a $1 \times 1$ block. We first prove that the diagonal blocks $A_1$ and $A_2$ are nonnegative matrices. If $DAD$ is a nonnegative matrix for a suitable $\pm 1$-diagonal matrix $D$, then $E=PDP^T$ is a $\pm 1$-diagonal matrix such that $E(PAP^T) E$ is a nonnegative matrix. It follows that matrix $PAP^T$ satisfies our assumptions provided that the cones $K$ and $L$ are replaced by the cones $P(K)$ and $P(L)$. We can therefore assume that $A$ itself is of the block form $$A=\left[\matrix {A_1 & B \cr 0 & A_2 }\right] \ .$$ Let $\Pi_1:{{\mathbb{R}}}^n\to{{\mathbb{R}}}^k$ and $\Pi_2:{{\mathbb{R}}}^n\to{{\mathbb{R}}}^l$ be the corresponding projections, and let $C\subseteq {{\mathbb{R}}}^n_+$ be a proper cone. As $C\subseteq\Pi_1(C)+\Pi_2(C)$ and $\Pi_1(C)$ contains at most $k$ linearly independent vectors, it follows that $\Pi_2(C)$ contains at least $n-k=l$ linearly independent vectors. Consequently, $\Pi_2(C)$ contains exactly $l$ linearly independent vectors, so that $\Pi_2(C)$ is a generating cone of ${{\mathbb{R}}}^l$. Similarly, $\Pi_1(C)$ is a generating cone of ${{\mathbb{R}}}^k$. Since $C\subseteq {{\mathbb{R}}}^n_+$, both $\Pi_1(C)$ and $\Pi_2(C)$ are pointed and therefore proper cones. Assume now that the cone $C$ is invariant under $A$. If $x_2\in \Pi_2(C)$, then $x_2=\Pi_2(x)$ for some $x\in C$, and so $A_2(x_2)=A_2(\Pi_2(x))=\Pi_2(Ax)\in\Pi_2(C)$, since $A(C)\subseteq C$. Therefore, the cone $\Pi_2(C)$ is invariant under $A_2$. This means that $\Pi_2(K)\subseteq {{\mathbb{R}}}^l_+$ is a proper cone invariant under $A_2$. Since the indecomposable matrix $A_2$ is diagonally similar to a nonnegative matrix, we can apply Lemma \[indecompobstruction\] to conclude that $A_2$ is a nonnegative matrix. In order to show that $A_1$ is also a nonnegative matrix, we consider the transposed matrix $A^T$. The proper cone $L\subseteq {{\mathbb{R}}}^n_+$ is invariant under $A^T$. Then the cone $\Pi_1(L)$ is a proper cone invariant under $A_1^T$. Since $A_1$ is indecomposable, $A_1^T$ is indecomposable and again by Lemma \[indecompobstruction\] we conclude that $A_1$ must be a nonnegative matrix. It remains to prove that the block $B$ is nonnegative. Suppose to the contrary that $B$ has some strictly negative entries. If $D=D_1\oplus D_2$ is a $\pm 1$-diagonal matrix such that $DAD$ is a nonnegative matrix, then $D_iA_iD_i$ for $i=1,2$ and $D_1BD_2$ are nonnegative matrices. Using Lemma \[indecompobstruction\] we conclude that $D_i=\pm I$ for $i=1,2$ and $D_1BD_2=\pm B$. Since $B$ contains some strictly negative entries, the matrix $-B$ must be nonnegative. Since we can add the identity matrix to the matrix $A$, without loss of generality we can assume that the matrices $A_1$ and $A_2$ are both primitive, i.e., the spectral radius $\rho(A_i)$ is the only point in the peripheral spectrum of $A_i$, $i=1,2$. For $k\in{{\mathbb{N}}}$ we have $$A^k=\left[\matrix {A_1^k & B_k\cr 0 & A_2^k }\right] ,$$ where $$B_k=\sum_{l=0}^{k-1}A_1^{k-1-l}BA_2^{l}.$$ If we multiply the matrix $A$ by a suitable positive scalar, we can assume that $\rho(A) = \max\{\rho(A_1), \rho(A_2)\}=1$. We must consider the following three cases: \(1) $\rho(A_1)=\rho(A_2)=1$: By Perron-Frobenius theory, the limits $$\lim_{k\to\infty} A_1^k=E_1 \textrm{ and } \lim_{k\to\infty} A_2^k=E_2$$ are strictly positive idempotents of rank $1$. In particular, there is a constant $C>0$ such that $\|A_1^k\|,\|A_2^k\|\le C$ for all $k\in {{\mathbb{N}}}$. Then we have, for any $m\in{{\mathbb{N}}}$, $$\|B_{4m}\| = \left\| \sum_{l=0}^{4m-1} A_1^{4m-1-l}BA_2^{l} \right\| \le \sum_{l=0}^{4m-1}\|A_1^{4m-1-l}\|\|B\|\|A_2^{l}\|\le 4m \, C^2 \|B\| ,$$ and so the sequence $\{\frac{1}{4m} B_{4m}\}_{m \in {{\mathbb{N}}}}$ is bounded. It follows that some subsequence $\{\frac 1{4m_k}A^{4m_k}\}_{k \in {{\mathbb{N}}}}$ of the sequence $\{\frac{1}{4m} A_{4m}\}_{m \in {{\mathbb{N}}}}$ converges to the matrix of the form $$A_\infty= \lim_{k\to \infty}\frac 1{4m_k}A^{4m_k}=\left[\matrix {0 & B_\infty\cr 0 & 0 }\right].$$ Choose $m \in {{\mathbb{N}}}$ such that $\frac 12 E_i \le A_i^{l}$ for $i=1,2$ and all $l \ge m$. As $-B$ is a nonnegative matrix, we obtain that $A_1^{4m-1-l}BA_2^{l} \le \frac 14 E_1BE_2$ for all $l = m, m+1, m+2, \ldots, 3m-1$. Since the matrices $- A_1^{4m-1-l} B A_2^{l}$ are nonnegative, we have $$B_{4m} = \sum_{l=0}^{4m-1}A_1^{4m-1-l}BA_2^{l} \le \sum_{l=m}^{3m-1}A_1^{4m-1-l}BA_2^{l}\le \frac 14\sum_{l=m}^{3m-1}E_1BE_2 .$$ It follows that $$B_\infty \le\lim_{m\to\infty} \frac 1{4m}\left(\frac 14\sum_{l=m}^{3m-1}E_1BE_2\right)=\frac 18 E_1BE_2 ,$$ and so $B_\infty$ is a matrix with some strictly negative entries. Therefore, there is a strictly positive vector $e \in K$ such that the vector $A_\infty e$ is not in ${{\mathbb{R}}}^n_+$. As the cone $K$ is closed and invariant under all powers of $A$, it has to be invariant under $A_\infty$, so that $A_\infty e \in K \subseteq {{\mathbb{R}}}^n_+$. This contradiction completes the proof in this case. \(2) $1=\rho(A_1)>\rho(A_2)$: As before, the limit $\lim_{k\to\infty} A_1^k=E_1$ is a strictly positive idempotent of rank $1$. Since $L \subseteq {{\mathbb{R}}}^n_+$ is a proper cone invariant under $A^T$, we can find a strictly positive vector $e\in L$ such that for all $k\in{{\mathbb{N}}}$ we have $(A^T)^k e \in L \subseteq {{\mathbb{R}}}^n_+$. If $k$ is large enough, we have $A_1^{k-1}\ge \frac 12E_1$ and therefore $B_k\le A_1^{k-1}B\le \frac 12E_1 B$. Writing $e=e_1\oplus e_2$ with respect to the given decomposition, we get $B_k^T e_1 \le \frac 12 (E_1 B)^T e_1 = \frac 12 B^T E_1^T e_1$. Since the vector $B^T E_1^T e_1$ has at least one strictly negative component, the same holds for $B_k^T e_1$. Since $\lim_{k\to \infty} A_2^k=0$, there is some power $k$ such that the vector $(A^T)^k e =((A_1^T)^k e_1)\oplus (B_k^T e_1 +(A_2^T)^k e_2)$ has at least one strictly negative component. This is a contradiction with $(A^T)^k e \in L \subseteq {{\mathbb{R}}}^n_+$. \(3) $\rho(A_1)<\rho(A_2)=1$: This case can be handled in a way similar to the case (2); we get the contradiction with the assumption that $K$ is a proper cone invariant under $A$. The next example shows that in Proposition \[obstruction\] none of the cones $K$ and $L$ can be omitted. The proper cone $K=\{(x,y)\ | \ x\ge y\ge 0\}\subset {{\mathbb{R}}}_+^2$ is invariant under the matrix $$A=\left[\matrix {1 & -1 \cr 0 & 0 }\right],$$ which is diagonally similar to a nonnegative matrix, but it is not nonnegative itself. Therefore, the cone $L$ cannot be omitted in Proposition \[obstruction\]. By duality, the cone $K$ cannot be omitted as well. The following is the main result of the paper. \[semi n&gt;2\] Let ${{\cal S}}\subset M_n({{\mathbb{C}}})$ be an irreducible semigroup such that each member of ${{\cal S}}$ is diagonally similar to a nonnegative matrix. Suppose that every member of rank at least $2$ is either indecomposable or $1$-decomposable. Then ${{\cal S}}$ is (simultaneously) diagonally similar to a semigroup of nonnegative matrices. By Lemma \[closurelemma\], we can assume that ${{\cal S}}=\overline{{{\mathbb{R}}}_+{{\cal S}}}$. Then, by Lemma \[backtoreality\], we can assume that ${{\cal S}}\subset M_n({{\mathbb{R}}})$ and that there are spanning sets $X,Y\subseteq {{\mathbb{R}}}_+^{n}$ such that ${{\cal S}}_1=XY^T$. We can also assume that $X={{\mathbb{R}}}_+ X$ and $Y={{\mathbb{R}}}_+ Y$. Denote by $\widehat{X}$ and $\widehat{Y}$ the cones generated by $X$ and $Y$, respectively. Since $X$ and $Y$ are spanning sets, the cones $\widehat{X},\widehat{Y} \subseteq {{\mathbb{R}}}^n_+$ are proper. Choose any member $A\in {{\cal S}}$ of rank at least $2$. Then, for all $x\in X$ and $y\in Y$, the matrices $Axy^T=(Ax)y^T$ and $xy^TA=x(A^Ty)^T$ belong to ${{\cal S}}_1=XY^T$. It follows that $Ax\in X$ and $A^Ty \in Y$, and therefore the proper cone $\widehat{X}$ is invariant under $A$, while the proper cone $\widehat{Y}$ is invariant under $A^T$. Since the matrix $A$ is either indecomposable or $1$-decomposable, we now apply either Lemma \[indecompobstruction\] or Proposition \[obstruction\] to conclude that $A$ is nonnegative. This completes the proof. \[semi n=2\] Let ${{\cal S}}\subset M_2({{\mathbb{C}}})$ be an irreducible semigroup such that each member of ${{\cal S}}$ is diagonally similar to a nonnegative matrix. Then ${{\cal S}}$ is (simultaneously) diagonally similar to a semigroup of nonnegative matrices. We conclude the paper with the following example showing that the (in)decomposability assumptions in Proposition \[obstruction\] and Theorem \[semi n&gt;2\] cannot be omitted. Define the matrix $$A_3 = \left[\matrix {1 & 0 & 1 \cr 0 & 1 & -1 \cr 0 & 0 & 0 }\right]$$ and the proper cones $K_3=\{(x,y,z)\in{{\mathbb{R}}}^3\ | \ x\ge 0\,,\ y\ge z\ge 0\}\subset{{\mathbb{R}}}^3_+$ and $L_3=\{(x,y,z)\in{{\mathbb{R}}}^3\ | \ x\ge y\ge 0\,,z\ge 0\}\subset{{\mathbb{R}}}^3_+$. It is easy to see that $K_3$ is invariant under $A_3$, while $L_3$ is invariant under $A_3^T$. For $n\ge 3$ we define the proper cones $K_n=K_3\oplus {{\mathbb{R}}}^{n-3}_+$ and $L_n=L_3\oplus {{\mathbb{R}}}^{n-3}_+$. Now we define an irreducible semigroup ${{\cal S}}_1=K_n L_n^T$, consisting of matrices of rank at most $1$. We extend the matrix $A_3$ with a zero block to get a matrix $A_n=A_3\oplus 0 \in M_n({{\mathbb{R}}})$. As $K_3$ is invariant under $A_3$ and $L_3$ is invariant under $A_3^T$, it is clear that the cones $K_n$ and $L_n$ are invariant under $A_n$ and $A_n^T$, respectively. Since $A_n^2 = A_n$, ${{\cal S}}={{\cal S}}_1 \cup \{A_n\}$ is an irreducible semigroup in which each member is diagonally similar to a nonnegative matrix, while the whole semigroup is not diagonally similar to a semigroup of nonnegative matrices. [99]{} A. Berman, R. J. Plemmons, Nonnegative matrices in the mathematical sciences, revised reprint of the 1979 original, Classics in Applied Mathematics 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. J. Bernik, M. Mastnak, H. Radjavi, [*Positivity and matrix semigroups*]{}, Linear Algebra Appl. [**434**]{} (2011), No. 3, 801–812. H. Radjavi and P. Rosenthal, [*Simultaneous triangularization*]{}, Springer-Verlag, Berlin, Heidelberg, New York (2000). Semigroups Working Group at LAW’08, Kranjska Gora (H. Radjavi, R. Drnovšek, J. Bernik, G. Cigler, A.A. Jafarian, D. Kokol Bukovšek, T. Košir, M. Kramar Fijavž, G. Kudryavtseva, T. Laffey, L. Livshits, G.W. MacDonald, M. Omladič, P. Rosenthal), Semigroups of operators with nonnegative diagonals, Linear Algebra Appl. [**433**]{} (2010), No. 11-12, 2080–2087.

--- abstract: | In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), have been employed to give approximate solutions of nonlinear fractional-integro differential equations (FIDEs). Comparing with the exact solution, the PIA produces reliable and accurate results for FIDEs. **Keywords:** Fractional-integro differential equations, Caputo fractional derivative, Initial value problems, Perturbation-Iteration Algorithm. author: - | Mehmet ŞENOL and İ. Timuçin DOLAPCİ\ Nevşehir Haci Bektaş Veli University, Department of Mathematics, Nevşehir, Turkey\ Celal Bayar University, Department of Mechanical Engineering,\ Manisa, Turkey\ e-mail:[email protected], [email protected] title: 'On the Numerical Solution of Nonlinear Fractional-Integro Differential Equations' --- Introduction ============ Scientists has been interested in fractional order calculus as long as it has been in classical integer order analysis. However, for many years it could not find practical applications in physical sciences. Recently, fractional calculus has been used in applied mathematics, viscoelasticity [@1], control [@2], electrochemistry [@3], electromagnetic [@4]. Developments in symbolic computation capabilities is one of the driving forces behind this rise. Different multidisciplinary problems can be handled with fractional derivatives and integrals. [@5] and [@6] are studies that describe the fundamentals of fractional calculus give some applications. Existence and uniqueness of the solutions are also studied in [@7]. Similar to the studies in physical sciences, fractional order integro differential equations (FIDEs) also gave scientists the opportunity of describing and modeling many important and useful physical problems. In this manner, a remarkable effort has been expended to propose numerical methods for solving FIDEs, in recent years. Fractional variational iteration method [@8; @9], homotopy analysis method [10,11]{}, Adomian decomposition method [@12; @13] and fractional differential transform method [@14; @15; @16] are among these methods. In our study, we use the previously developed method PIA, to obtain approximate solutions of some FIDEs. This method can be applied to a wide range of problems without requiring any special assumptions and restrictions. A few fractional derivative definitions of an arbitrary order exists in the literature. Two most used of them are the Riemann-Liouville and Caputo fractional derivatives. The two definitions are quite similar but have different order of evaluation of derivation. The Riemann-Liouville fractional integral of order $\alpha $ is described by:$$J^{\alpha }u(x)=\frac{1}{\Gamma (\alpha )}\int_{0}^{x}(x-t)^{\alpha -1}u(t)dt,\quad \alpha >0,\quad x>0. \label{1}$$ The Riemann-Liouville and Caputo fractional derivatives of an arbitrary order are defined as the following, respectively$$D^{\alpha }u(x)=\frac{d^{m}}{dx^{m}}\left( J^{m-\alpha }u(x)\right) \label{2}$$$$D_{\ast }^{\alpha }u(x)=J^{m-\alpha }\left( \frac{d^{m}}{dx^{m}}u(x)\right) . \label{3}$$where $m-1<\alpha \leqslant m$ and $m\in \mathbb{N} .$ Due to the appropriateness of the initial conditions, fractional definition of Caputo is often used in recent years. The Caputo fractional derivative of a function $u(x)$ is defined as$$D_{\ast }^{\alpha }u(x)=\left\{ \begin{array}{cc} \frac{1}{\Gamma (m-\alpha )}\int_{0}^{x}(x-t)^{m-\alpha -1}u^{(m)}(t)dt, & m-1<\alpha \leqslant m \\ \frac{d^{m}u(x)}{dx^{m}} & \alpha =m\end{array}\right. \label{4}$$for $m-1<\alpha \leqslant m,$ $m\in \mathbb{N} ,$ $x>0,$ $u\in C_{-1}^{m}.$ Following lemma gives the two main properties of Caputo fractional derivative. For $m-1<\alpha \leqslant m,$ $u\in C_{\mu }^{m},$ $\mu \geqslant -1$ and $m\in \mathbb{N} $ then $$D_{\ast }^{\alpha }J^{\alpha }u(x)=u(x) \label{5}$$and $$J^{\alpha }D_{\ast }^{\alpha }u(x)=u(x)-\sum_{k=0}^{m-1}u^{(k)}(0^{+})\frac{x^{k}}{k!},\quad x>0. \label{6}$$ After this introductory section, Section 2 is reserved to a brief review of the Perturbation-Iteration Algorithm PIA, in Section 3 some examples are illustrated to show the simplicity and effectiveness of the algorithm. Finally the paper ends with a conclusion in Section 4. Analysis of the PIA =================== Differential equations are naturally used to describe problems in engineering and other applied sciences. Searching approximate solutions for complicated equations has always attracted attention. Many different methods and frameworks exist for this purpose and perturbation techniques [@17; @18; @19] are among them. These techniques can be used to find approximate solutions for both ordinary and partial differential equations. Requirement of a small parameter in the equation that is sometimes artificially inserted is a major limitation of perturbation techniques that renders them valid only in a limited range. Therefore, to overcome the disadvantages come with the perturbation techniques, several methods have been proposed by authors [@20; @21; @22; @23; @24; @25; @26; @27; @28; @29]. Parallel to these attempts, a perturbation-iteration method has been proposed by Aksoy, Pakdemirli and their co-workers [@33; @34; @35] previously. A primary effort of producing root finding algorithms for algebraic equations [@30; @31; @32], finally guided to obtain formulae for differential equations also [@33; @34; @35]. In the new technique, an iterative algorithm is constructed on the perturbation expansion. The present method has been tested on Bratu-type differential equations [@33] and first order differential equations [@34] with success. Then the algorithms were applied to nonlinear heat equations also [@35]. Finally, the solutions of the Volterra and Fredholm type integral equations [@36] and ordinary differential equation systems [@37] have been presented by the developed method. This new algorithm have not been used for any fractional integro differential equations yet. To obtain the approximate solutions of FIDEs, the most basic perturbation-iteration algorithm PIA(1,1) is employed by taking one correction term in the perturbation expansion and correction terms of only first derivatives in the Taylor series expansion. [@33; @34; @35]. Take the fractional-integro differential equation. $$F\left( u^{(\alpha )},u,\int_{0}^{t}{g\left( t,s,u(s)\right) ds},\varepsilon \right) =0 \label{7}$$ where $u=u(t)$ and $\varepsilon $ is a small parameter. The perturbation expansions with only one correction term is $$\begin{aligned} u_{n+1} &=&u_{n}+\varepsilon {\left( u_{c}\right) }_{n}\ \notag \\ u_{n+1}^{\prime } &=&u_{n}^{\prime }+\varepsilon {\left( u_{c}^{\prime }\right) }_{n}\ \label{8}\end{aligned}$$ Replacing Eq.$(\ref{8})$ into Eq.$(\ref{7})$ and writing in the Taylor series expansion for only first order derivatives gives $$\begin{aligned} &&F\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \notag \\ &&+F_{u}\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \varepsilon {\left( u_{c}\right) }_{n} \notag \\ &&+F_{u^{\left( \alpha \right) }}\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \varepsilon {\left( u_{c}^{(\alpha )}\right) }_{n} \notag \\ &&+F_{\int {u}}\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \varepsilon \int {{\left( u_{c}\right) }_{n}} \notag \\ &&+F_{\varepsilon }\left( u_{n}^{\left( \alpha \right) },u_{n},\int_{0}^{t}{g\left( t,s,u_{n}(s)\right) ds},0\right) \varepsilon =0 \label{9}\end{aligned}$$ or $${\left( u_{c}^{(\alpha )}\right) }_{n}\frac{\partial F}{\partial u^{(\alpha )}}+{\left( u_{c}\right) }_{n}\frac{\partial F}{\partial u}+\left( \int {{\left( u_{c}\right) }_{n}}\right) \frac{\partial F}{\partial (\int {u})}+\frac{\partial F}{\partial \varepsilon }+\frac{F}{\varepsilon }=0 \label{10}$$ Here $(.)^{\prime }$ represents the derivative according to the independent variable and $$F_{\varepsilon }=\frac{\partial F}{\partial \varepsilon },~F_{u}=\frac{\partial F}{\partial u},~F_{u^{\prime }}=\frac{\partial F}{\partial u^{\prime }},\ldots \label{11}$$ The derivatives in the expansion are evaluated at $\varepsilon =0$. Beginning with an initial function $u_{0}(t)$, first ${\left( u_{c}\right) }_{0}(t)$ is calculated by the help of $(\ref{10})$ and then substituted into Eq.$(\ref{8})$ to calculate $u_{1}(t)$. Iteration procedure is continued using $(\ref{10}) $ and $(\ref{8})$ until obtaining a reasonable solution. Applications ============ Consider the following nonlinear fractional-integro differential equation [@38]: $$\frac{d^{\alpha }u(t)}{{dt}^{\alpha }}-\int_{0}^{1}{ts({u(s))}^{2}ds}=1-\frac{t}{4},\ \ \ t>0,\ \ \ 0\leq t<1,\ \ \ 0<\alpha \leq 1 \label{12}$$ with the initial condition $u\left( 0\right) =0$ and the known exact solution for $\alpha =1$ is $$u\left( t\right) =t \label{13}$$ Before iteration process rewriting Eq.$(\ref{12})$ with adding and subtracting $u^{\prime }(t)$ to the equation gives $$\varepsilon \frac{d^{\alpha }u(t)}{{dt}^{\alpha }}-u^{^{{\prime }}}\left( t\right) +{\varepsilon u}^{^{{\prime }}}\left( t\right) -{\varepsilon }\int_{0}^{1}{ts({u(s))}^{2}ds}-1+\frac{t}{4}=0 \label{14}$$ In this case for $$F\left( u^{^{{\prime }}},u,\varepsilon \right) =\frac{1}{\Gamma (1-\alpha )}\varepsilon \int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha }}ds-u_{n}^{^{{\prime }}}\left( t\right) +\varepsilon u_{n}^{^{{\prime }}}\left( t\right) -\varepsilon \int_{0}^{1}{ts({u_{n}(s))}^{2}ds}-1+\frac{t}{4}} \label{15}$$ and the iteration formula$$u^{^{{\prime }}}(t)+\frac{F_{u}}{F_{u^{\prime }}}u\left( t\right) =-\frac{F_{\varepsilon }+\frac{F}{\varepsilon }}{F_{u^{\prime }}} \label{16}$$ the terms that will be replaced in, are $$\begin{aligned} F &=&{u_{n}^{^{{\prime }}}\left( t\right) }-1+\frac{t}{4} \notag \\ F_{u} &=&0 \notag \\ F_{u^{\prime }} &=&1 \notag \\ F_{\varepsilon } &=&-{u_{n}^{^{{\prime }}}\left( t\right) }+\frac{1}{\Gamma (1-\alpha )}\int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha }}ds}-\int_{0}^{1}{ts({u(s))}^{2}ds} \label{17}\end{aligned}$$ After substitution the differential equation for this problem, Eq.$(\ref{10})$ becomes $$\frac{\int_{0}^{t}{{\left( -s+t\right) }^{-\alpha }{u_{n}}^{^{{\prime }}}(s)ds}}{\Gamma (1-\alpha )}+{\left( {u}_{c}^{\prime }(t)\right) }_{n}=\int_{0}^{1}{st{\left( u_{n}\left( s\right) \right) }^{2}ds}+\frac{4-t+4\left( -1+\varepsilon \right) {u}_{n}^{\prime }\left( t\right) }{4\varepsilon } \label{18}$$ Appropriate to the initial conditions, chosen $u_{0}\left( t\right) =0$ and, solving Eq.$(\ref{18})$ for $n=0$ gives $${{(u}_{c}(t))}_{0}=t-\frac{t^{2}}{8}+C_{1} \label{19}$$ This expression written in $$u_{1}=u_{0}+\varepsilon {{(u}_{c}(t))}_{0} \label{20}$$ gives $$u_{1}\left( x,t\right) =u_{0}\left( x,t\right) +\varepsilon (t-\frac{t^{2}}{8}+C_{1}) \label{21}$$ or $$u_{1}\left( x,t\right) =\varepsilon (t-\frac{t^{2}}{8}+C_{1}) \label{22}$$ Solving this equation for $$u_{1}\left( 0\right) =0 \label{23}$$ we obtain $$C_{1}=0 \label{24}$$ For this value and $\varepsilon =1$ reorganizing $u_{1}(t)$ $$u_{1}\left( t\right) =t-\frac{t^{2}}{8} \label{25}$$ gives the first iteration result. If the iteration procedure is continued in a similar way, we obtain the following iterations. $$u_{2}(t)=2t-\frac{571t^{2}}{3840}+\frac{t^{2-\alpha }(t+4(-3+\alpha ))}{4\Gamma (4-\alpha )}\ \label{26}$$ $$\begin{aligned} u_{3}\left( t\right) &=&3t+\frac{29844889t^{2}}{176947200}-\frac{t^{3-2\alpha }\left( t+8\left( -2+\alpha \right) \right) }{4\Gamma \left( 5-2\alpha \right) } \notag \\ &&+\frac{t^{2}\left( 3379230+8t^{-\alpha }\left( 1051t+5760\left( -3+\alpha \right) \right) \left( -7+\alpha \right) \left( -6+\alpha \right) \left( -5+\alpha \right) \right) }{15360\left( -7+\alpha \right) \left( -6+\alpha \right) \left( -5+\alpha \right) \Gamma \left( 4-\alpha \right) } \notag \\ &&-\frac{2240277\alpha +\left( 450151-28436\alpha \right) \alpha ^{2}}{15360\left( -7+\alpha \right) \left( -6+\alpha \right) \left( -5+\alpha \right) \Gamma \left( 4-\alpha \right) } \notag \\ &&-\frac{t^{2}\left( -4+\alpha \right) \left( -1159+2\alpha \left( 529+16\left( -10+\alpha \right) \alpha \right) \right) }{64\left( -7+2\alpha \right) {\Gamma \left( 5-\alpha \right) }^{2}} \label{27}\end{aligned}$$ The other iterations contain large inputs and are not given. A computational software program could help to calculate the other iterations up to any order. In Table 1. some of the PIA iteration results are compared with the exact solution. The results express that the present method gives highly approximate solutions. Also in Figure 1. the obtained results are illustrated graphically. ------- ---------- ---------- ---------- ---------- ---------------- ---------------- t $u_{2}$ $u_{3}$ $u_{4}$ $u_{5}$ Exact Solution Absolute Error $0.0$ 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 $0.1$ 0.099763 0.099953 0.099990 0.099981 0.100000 1.872712E-6 $0.2$ 0.199052 0.199812 0.199962 0.199992 0.200000 7.490848E-6 $0.3$ 0.297867 0.299577 0.299915 0.299983 0.300000 1.685440E-5 $0.4$ 0.396208 0.399249 0.399850 0.399970 0.400000 2.996339E-5 $0.5$ 0.494075 0.498826 0.499765 0.499953 0.500000 4.681780E-5 $0.6$ 0.591468 0.598310 0.599662 0.599932 0.600000 6.741763E-5 $0.7$ 0.688388 0.697700 0.699541 0.699908 0.700000 9.176289E-5 $0.8$ 0.784833 0.796996 0.799400 0.799880 0.800000 1.198535E-4 $0.9$ 0.880804 0.896198 0.899241 0.899848 0.900000 1.516896E-4 $1.0$ 0.976302 0.995307 0.999063 0.999812 1.000000 1.872712E-4 ------- ---------- ---------- ---------- ---------- ---------------- ---------------- : Numerical results of Example 3.1. for different $u$ values when $\protect\alpha =1$ {width="3.50in"} Consider the following system of nonlinear fractional-integro differential equations [@39]: $$\begin{aligned} \frac{d^{\alpha _{1}}u(t)}{{dt}^{\alpha _{1}}} &=&1-\frac{1}{2}{\left( k^{^{{\prime }}}\left( t\right) \right) }^{2}+\int_{0}^{t}{\left( \left( t-s\right) k\left( s\right) +u\left( s\right) k\left( s\right) \right) ds} \notag \\ \frac{d^{\alpha _{2}}k(t)}{{dt}^{2}} &=&2t+\int_{0}^{t}{\left( \left( t-s\right) u\left( s\right) -k^{2}\left( s\right) +u^{2}\left( s\right) \right) ds}\ \ \ \ 0<\alpha _{1},\alpha _{2}\leq 1 \label{28}\end{aligned}$$ Given with $u\left( 0\right) =0,\ \ k(0)=1$ as initial conditions. The exact solution for $\alpha _{1} =\alpha _{2} =1$ is $$\begin{aligned} u\left( t\right) &=&sinht \notag \\ k(t) &=&cosht \label{29}\end{aligned}$$ Rewriting Eq.$(\ref{28})$ in the following for with adding and subtracting $u^{\prime }(t)$ and $k^{\prime }(t)$ to the equation respectively gives $$\begin{aligned} &&\varepsilon \frac{d^{\alpha _{1}}u(t)}{{dt}^{\alpha _{1}}}+u^{\prime }\left( t\right) -\varepsilon u^{\prime }(t)-1+\frac{1}{2}{\left( k^{\prime }\left( t\right) \right) }^{2}-\varepsilon \int_{0}^{t}{\left( \left( t-s\right) k\left( s\right) -u\left( s\right) k\left( s\right) \right) ds\ } \notag \\ &&\varepsilon \frac{d^{\alpha _{2}}u(t)}{{dt}^{\alpha _{2}}}+k^{\prime }\left( t\right) -\varepsilon k^{\prime }\left( t\right) -2t-\varepsilon \int_{0}^{t}{\left( \left( t-s\right) u\left( s\right) +{k}^{2}\left( s\right) -{u}^{2}\left( s\right) \right) ds} \label{30}\end{aligned}$$In this case for $$\begin{aligned} F\left( u^{\prime },u,\varepsilon \right) &=&\frac{1}{\Gamma (1-\alpha _{1})}\varepsilon \int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha _{1}}}ds-\varepsilon \int_{0}^{t}{\left( \left( t-s\right) k\left( s\right) +u\left( s\right) k\left( s\right) \right) ds}-1+\frac{1}{2}{\left( k^{\prime }\left( t\right) \right) }^{2}} \notag \\ F\left( k^{\prime },k,\varepsilon \right) &=&\frac{1}{\Gamma (1-\alpha _{2})}\varepsilon \int_{0}^{t}{\frac{u^{\prime }(s)}{{(t-s)}^{\alpha _{2}}}ds-\varepsilon \int_{0}^{t}{\left( \left( t-s\right) u\left( s\right) -k^{2}\left( s\right) +u^{2}\left( s\right) \right) ds}}-2t \label{31}\end{aligned}$$ and the iteration formula $$u^{\prime }\left( t\right) +\frac{Fu}{Fu^{\prime }}u\left( t\right) =-\frac{F_{\varepsilon }+\frac{F}{\varepsilon }}{Fu^{\prime }} \label{32}$$ the terms that will be replaced in, are $$\begin{aligned} F &=&u_{n}^{\prime }(t)-1+\frac{{k_{n}^{\prime }(t)}^{2}}{2} \notag \\ F_{u} &=&0 \notag \\ F_{u^{\prime }} &=&1 \notag \\ F_{\varepsilon } &=&-u_{n}^{\prime }(t)+\frac{1}{\Gamma (1-\alpha _{1})}\int_{0}^{t}{\frac{u_{n}^{\prime }(s)}{{(t-s)}^{\alpha _{1}}}ds}-\int_{0}^{t}{((t-s)k_{n}(s)+u_{n}(s)k_{n}(s))ds} \label{33}\end{aligned}$$ and the iteration formula$$k^{\prime }\left( t\right) +\frac{F_{k}}{F_{k^{\prime }}}k\left( t\right) =-\frac{F_{\varepsilon }+\frac{F}{\varepsilon }}{F_{k^{\prime }}} \label{34}$$ the terms that will be replaced in, are $$\begin{aligned} F &=&k_{n}^{\prime }\left( t\right) -2t\ \notag \\ F_{k} &=&0 \notag \\ F_{k^{\prime }} &=&1 \notag \\ F_{\varepsilon } &=&-k_{n}^{\prime }(t)+\frac{1}{\Gamma (1-\alpha _{2})}\int_{0}^{t}{\frac{k_{n}^{\prime }(s)}{{(t-s)}^{\alpha _{2}}}ds}-\int_{0}^{t}{((t-s)u_{n}(s)-{k_{n}(s)}^{2}+{u_{n}(s)}^{2})ds} \label{35}\end{aligned}$$ After substitution, the system of differential equations for this problem become $$\frac{1}{\Gamma (1-\alpha _{1})}\int_{0}^{t}{{(-s+t)}^{-\alpha _{1}}{u}}^{\prime }{{_{n}}(s)ds}+{\left( {u^{\prime }}_{c}(t)\right) }_{n}+\frac{-1+\frac{1}{2}{k}^{\prime }{{_{n}}(t)}^{2}+{u}^{\prime }{_{n}}(t)}{\varepsilon }=\int_{0}^{t}{k_{n}(s)(-s+t+u_{n}(s))ds}+{u}^{\prime }{_{n}}(t)$$ $$\frac{1}{\Gamma (1-\alpha _{2})}\int_{0}^{t}{{(-s+t)}^{-\alpha _{2}}{k}}^{\prime }{{_{n}}(s)ds}+{\left( {k^{\prime }}_{c}(t)\right) }_{n}=\int_{0}^{t}{(-{k_{n}(s)}^{2}+u_{n}(s)(-s+t+u_{n}(s)))ds}+\frac{2t+(-1+\varepsilon ){k}_{n}^{\prime }(t)}{\varepsilon } \label{36}$$ Appropriate to the initial conditions, chosen $u_{0}\left( t\right) =0$ and $k_{0}\left( t\right) =1$ and solving Eq.$(\ref{36})$ for $n=0,1,2,3,...$ the successive iterations are $$u_{1}(t)=\frac{1}{6}(6t+t^{3}) \label{37}$$ $$k_{1}(t)=1+\frac{t^{2}}{2} \label{38}$$ $$u_{2}\left( t\right) =\frac{1}{504}t\left( 1008+168t^{2}+21t^{4}+t^{6}\right) -\frac{t^{2-\alpha _{1}}\left( 12+t^{2}+\left( -7+\alpha _{1}\right) \alpha _{1}\right) }{\Gamma (5-\alpha _{1})} \label{39}$$ $$k_{2}\left( t\right) =1+t^{2}+\frac{t^{4}}{24}+\frac{t^{6}}{240}+\frac{t^{8}}{2016}-\frac{t^{3-\alpha _{2}}}{\Gamma (4-\alpha _{2})} \label{40}$$ Following in this manner the third iteration results, $u_{3}(t)$ and $k_{3}(t),$ are calculated. Again Table 2, Figure 2 and Figure 3 prove that PIA give remarkably approximate results. We can say that the higher iterations would give closer results. ------- --------------- ---------------- ---------------- --------------- ---------------- ---------------- t PIA $(u_{3})$ Exact Solution Absolute Error PIA $(k_{3})$ Exact Solution Absolute Error $0.0$ 0.000000 0.000000 0.000000 1.000000 1000000. 0.000000 $0.1$ 0.100166 0.100166 1.591577E-10 1.005004 1.005004 1.191735E-11 $0.2$ 0.201335 0.201336 2.053723E-8 1.020066 1.020066 3.060393E-9 $0.3$ 0.304519 0.304520 3.556439E-7 1.045338 1.045338 7.884730E-8 $0.4$ 0.410749 0.410752 2.714842E-6 1.081073 1.081072 7.934216E-7 $0.5$ 0.521082 0.521095 1.326132E-5 1.127630 1.127625 4.774578E-6 $0.6$ 0.636604 0.636653 4.893639E-5 1.185485 1.185465 2.077300E-5 $0.7$ 0.758434 0.758583 1.490491E-4 1.255241 1.255169 7.230620E-5 $0.8$ 0.887710 0.888105 3.950285E-4 1.337648 1.337434 2.139083E-4 $0.9$ 0.025574 0.026516 9.426045E-4 1.433645 1.433086 5.592545E-4 $1.0$ 0.173128 0.175201 2.072716E-3 1.544407 1.543080 1.327116E-3 ------- --------------- ---------------- ---------------- --------------- ---------------- ---------------- : Numerical results of Example 3.2. for $u_{3}$ and $k_{3}$ values when $\alpha _{1} =\alpha _{2} =1$ {width="3.50in"} {width="3.50in"} Conclusion ========== In this study, Perturbation-Iteration Algorithm was introduced for some Factional Differential Equations. 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--- author: - | Robert M Corless$^1$, Robert HC Moir$^1$, Marc Moreno Maza$^1$, Ning Xie$^2$\ [$^1$Ontario Research Center for Computer Algebra,\ University of Western Ontario, Canada]{}\ [$^2$Huawei Technologies Corporation, Markham, ON]{} bibliography: - 'symbint.bib' title: 'Symbolic-Numeric Integration of Rational Functions' ---

--- abstract: 'We consider the Kalman-filtering problem with multiple sensors which are connected through a communication network. If all measurements are delivered to one place called fusion center and processed together, we call the process centralized Kalman-filtering (CKF). When there is no fusion center, each sensor can also solve the problem by using local measurements and exchanging information with its neighboring sensors, which is called distributed Kalman-filtering (DKF). Noting that CKF problem is a maximum likelihood estimation problem, which is a quadratic optimization problem, we reformulate DKF problem as a consensus optimization problem, resulting in that DKF problem can be solved by many existing distributed optimization algorithms. A new DKF algorithm employing the distributed dual ascent method is provided and its performance is evaluated through numerical experiments.' author: - 'Kunhee Ryu and Juhoon Back${}^*$[^1] [^2]' bibliography: - 'mybib.bib' title: '**Distributed Kalman-filtering: Distributed optimization viewpoint**' --- Introduction ============ It goes without saying that the Kalman-filter, an optimal state estimator for dynamic systems, has had a huge impact on various fields such as engineering, science, economics, etc. [@Welch1995; @Bell1993TAC; @Humpherys2010CSM; @Thrun2005Book]. Basically, the filter predicts the expectation of the system state and its covariance based on the dynamic model and the statistical information on the model uncertainty or process noise, and then correct them using new measurement, sensor model, and the information on measurement noise. When multiple sensors possibly different types are available, we can just combine the sensor models to process the measurements altogether. Thanks to the rapid development of sensor devices and communication technology, we are now able to monitor large scale systems or environments such as traffic network, plants, forest, sea, etc. In those systems, sensors are geometrically distributed, may have different types, and usually not synchronized. To process the measurements, the basic idea would be to deliver all the data to one place, usually called fusion center, and do the correction step as in the case of multiple sensors. This is called the centralized Kalman-filtering (CKF). As expected, CKF requires a powerful computing device to handle a large number of measurements and sensor models, is exposed to a single point of failure, and is difficult to scale up. In order to overcome these drawbacks, researchers developed the distributed Kalam-filtering (DKF) in which each sensor in the network solves the problem by using local measurements and communicating with its neighbors. Compared with CKF, DKF has advantageous in terms of the scalability, robustness to component loss, computational cost, and thus the literature on this topic is expanding rapidly [@Olfati2007CDC; @Olfati2009CDC; @Bai2011ACC; @Carli2008SAC; @Khan2008TSP; @Kim2016CDC; @Wu2016IFAC; @WU2018Aut]. For more details on DKF, see the survey [@Mahmoud2013Survey] and references therein. Some relevant results are summarized as follows. In [@Olfati2007CDC], the author proposed scalable distributed Kalman-Bucy filtering algorithms in which each node only communicates with its neighbors. An algorithm with average consensus filters using the internal models of signals being exchanged is proposed in [@Bai2011ACC]. It is noted that the algorithm works in a single-time scale. In the work [@Wu2016IFAC], the authors proposed a continuous-time algorithm that makes each norm of all local error covariance matrices be bounded, thus overcomes a major drawback of [@Olfati2007CDC]. In [@Kim2016CDC], an algorithm with a high gain coupling term in the error covariance matrix is introduced and it is shown that the local error covariance matrix approximately converges to that of the steady-state centralized Kalman-filter. An in-depth discussion on distributed Kalman-filtering problem has been provided in [@Battistelli2015TAC; @Battistelli2016Aut], and the algorithms that exchange the measurements themselves, or exchange certain signals instead of the measurements are proposed, respectively. Although each of the existing algorithms has own novel ideas and advantages, to the best of the authors’ knowledge, we do not have a unified viewpoint for DKF problem. Motivated by this, it is the aim of this paper to provide a framework for the problem from the perspective of distributed optimization. We start by observing that the [*[correction]{}*]{} step of Kalman-filtering is basically an optimization problem [@Bell1993TAC; @Humpherys2010CSM; @Thrun2005Book], and then formulate DKF problem as a consensus optimization problem, which provides a fresh look at the problem. This results in that DKF problem can be solved by many existing distributed optimization algorithms [@Boyd+2011FTML; @Nedic+2009TAC; @Nedic+2010TAC; @Zhang2018CDC; @Dorfler2017], expecting various DKF algorithms to be derived. As an instance, a new DKF algorithm employing the [*[dual ascent method]{}*]{} [@Dorfler2017], one of the basic algorithms for distributed optimization problems, is provided in this paper. This paper is organized as follows. In Section \[Sec:ProblemSetting\], we recall CKF problem from the optimization perspective, and connects DKF problem to a distributed optimization problem. A new DKF algorithm based on [*[dual ascent method]{}*]{} is proposed in Section \[Sec:DKF-DA\], and numerical experiments evaluating the proposed algorithm is conducted in Section \[Sec:NE\]. [**Notation**]{}: For matrices $A_1$, …, $A_n$, $\operatorname{\text{diag}}(A_1,\dots,A_n)$ denotes the block diagonal matrix composed of $A_1$ to $A_n$. For scalars $a_1$,…, $a_n$, $[a_1;\dots;a_n] := [a_1^\top,\dots,a_n^\top]^\top$, and $[A_1;\dots;A_n]$ with matrices $A_i$’s is defined similarly. $1_n \in \mathbb{R}^n$ denotes the vector whose components are all 1, and $I_n$ is the identity matrix whose dimension is $n \times n$. The maximum and minimum eigenvalue of a matrix $A$ are denoted by $\sigma_{\max}(A)$ and $\sigma_{\min}(A)$, respectively. For a random variable $x$, $x \sim \mathsf{N}(\mu, \sigma^2)$ denotes $x$ is normally distributed with the mean $\mu$ and the variance $\sigma^2$, and $\mathbb{E}\{ x\}$ denotes the [*[expected value]{}*]{} of a random variable $x$, [*[i.e.,]{}*]{} $\mathbb{E}\{ x\} = \mu$. The half vectorization of a symmetric matrix $M \in \mathbb{R}^{n \times n}$ is denoted by ${\text{vec}_h({M})} \in \mathbb{R}^{n(n+1)/2}$, whose elements are filled in Column-major order. $i.e., {\text{vec}_h({M})} := [M_{1,1}; \dots; M_{1,n}; M_{2,2}; \dots; M_{2,n}; \dots;$ $ M_{n-1,n-1};M_{n-1,n};M_{n,n}]$ where $M_{i,j}$ is $i,j$ element of $M$, and ${\text{vec}_h^{-1}({\cdot})}$ denotes the inverse function of ${\text{vec}_h({\cdot})}$, $i.e., {\text{vec}_h^{-1}({{\text{vec}_h({M})}})} = M$. For a function $f(x, y): \mathbb{R}^{n}\times \mathbb{R}^m \rightarrow \mathbb{R}$, $\nabla_{x} f(x,y)$ denotes the gradient vector $\frac{\partial f(x,y)}{\partial x} = [\frac{\partial f(x,y)}{\partial x_1}; \dots;\frac{\partial f(x,y)}{\partial x_n}]$. [**Graph theory**]{}: For a network consisting of $N$ nodes, the communication among nodes is modeled by a graph $\mathcal{G}$. Let ${\mathcal{A}} = [a_{ij}] \in {\mathbb{R}}^{N \times N}$ be an adjacency matrix associated to ${\mathcal{G}}$ where $a_{ij}$ is a weight of an edge between nodes $i$ and $j$. If node $i$ communicates to node $j$ then, $a_{ij} > 0$, or if not $a_{ij} = 0$. Assume there is not self edge, [*i.e.*]{}, $a_{ii} = 0$. The Laplacian matrix associated to the graph $\mathcal{G}$, denoted by $L$ is a $N \times N$ matrix such that $l_{ij, i \neq j} = -a_{ij}$, and $l_{ii} = \sum_{j=1}^N a_{ij}$. ${\mathcal{N}}_i$ is a set of nodes communicating with node $i$, [*i.e.*]{}, ${\mathcal{N}}_i = \{j | a_{ij}>0 \}$. Distributed Kalman-filtering and Its Connection to Consensus Optimization {#Sec:ProblemSetting} ========================================================================= In this section, we recall CKF problem in terms of optimization, which is the maximum likelihood estimation[@Bell1993TAC], and establish a connection between DFK and distributed optimization. Consider a discrete-time linear system with $N$ sensors described by \[eq:System\] $$\begin{aligned} x_{k+1} &= Fx_k + w_k\\ y_k &= H x_k + v_k = \begin{bmatrix} H_1 \\ H_2 \\ \vdots \\ H_N \end{bmatrix} x_k + \begin{bmatrix} v_{1,k} \\ v_{2,k} \\ \vdots \\ v_{N,k} \end{bmatrix}\end{aligned}$$ where $x_{k} \in {\mathbb{R}}^{n}$ is the state vector of the dynamic system, $y_k := [y_{1,k};\dots;y_{N,k}] \in \mathbb{R}^{m}$ is the output vector, and $y_{i,k} \in \mathbb{R}^{m_i}$ is the output associated to sensor $i$. $m_i$’s satisfy $\sum^N_{i=1} m_i = m$. $F$ is the system matrix and $H$ is the output matrix consisting of $H_i \in \mathbb{R}^{m_i \times n}$ which is the output matrix associated to sensor $i$. $w_k\in {\mathbb{R}}^{n}$ with $w_k \sim \mathsf{N}(0, Q)$ is the process noise, $v_{i,k} \sim \mathsf{N}(0, R_i)$ is the measurement noise on sensor $i$, and $v_k :=[v_{1,k};\dots;v_{N,k}] \in {\mathbb{R}}^{m}$ with $v_k \sim \mathsf{N}(0, \operatorname{\text{diag}}(R_1, \dots, R_N))$. Assume that the pair $(F, H)$ is observable, and each $v_{i,k}$ is uncorrelated to $v_{{j}, k}$ for $j \neq i$. Centralized Kalman-filtering problem from the optimization perspective ---------------------------------------------------------------------- If all the measurements from $N$ sensors are collected and processed altogether, the problem can be seen as the one with a imaginary sensor that measures $y_k$ with complete knowledge on $H$, thus called centralized Kalman-filtering.The filtering consists of two steps, [*[prediction]{}*]{} and [*[correction]{}*]{}. In the prediction step, the predicted estimate $\hat{x}_{k|k-1}$ and error covariance matrix $P_{k|k-1}$ are obtained based on the previous estimate, error covariance matrix, and the system dynamics. The update rules are given by $$\begin{aligned} \hat{x}_{k|k-1} &= F \hat{x}_{k-1}\\ P_{k|k-1} &= {\mathbb{E}}\{ e_{k|k-1}e_{k|k-1}^\top \}\nonumber\\ &= F {\mathbb{E}}\{ e_{k-1}e_{k-1}^\top \}F^{\top} + {\mathbb{E}}\{ w_k w_k^\top \} \nonumber\\ &= F P_{k-1} F^\top + Q\end{aligned}$$ where $\hat{x}_{k-1}$ and $P_{k-1}$ are estimate and error covariance matrix in previous time, respectively, and $e_{k|k-1} := x_k - \hat{x}_{k|k-1}$, $e_{k} := x_k - \hat{x}_{k}$. Assume that $P_k$ is initialized as a positive definite matrix ($P_0>0$, usually set as $Q$). In the correction step, the predicted estimate and the error covariance matrix are updated based on the current measurements containing the measurement noise. The correction step can be regarded as a process to find the optimal parameter (estimate) from the predicted estimate $\hat{x}_{k|k-1}$, error covariance $P_{k|k-1}$, and the observation $y_k$. In fact, it is known that this step is an optimization problem ([*[maximum likelihood estimation, MLE]{}*]{}[@Bell1993TAC]) and we recall the details below. Let ${z}_k = [y_k;\hat{x}_{k|k-1}] \in \mathbb{R}^{m+n}$ and $\bar{H}_c = [H;I_n] \in \mathbb{R}^{(m+n) \times n}$. Then, ${z}_k \sim \mathsf{N} ({\bar{H}_c} x_k, S_k)$ where $S_k =\text{diag}\{R, P_{k|k-1}\}$. For the random variable ${z}_k$, the likelihood function is given by $$\begin{aligned} \mathfrak{L}(\xi_c) = \frac{1}{\sqrt{(2\pi)^{(m+n)}|S_k|}} e^{-\frac{1}{2}({z}_k-{\bar{H}_c} \xi_c)^{\top} S_k^{-1}({z}_k- {\bar{H}_c}\xi_c)}\end{aligned}$$ where the right-hand side is nothing but the probability density function of $z_k$ with the free variable $\xi_c \in \mathbb{R}^n$. Now, the maximum likelihood estimate $\hat{x}_{k}$ is defined as $$\begin{aligned} \hat{x}_{k} := \operatorname*{argmax}_{\xi_c} (\mathfrak{L}(\xi_c)).\end{aligned}$$ Since $\mathfrak{L}(\xi_c)$ is a monotonically decreasing function with respect to $f_{\text{c}}(\xi_c) := \frac{1}{2}(z_k-\bar{H}_c \xi_c)^{\top}S^{-1} (z_k-\bar{H}_c\xi_c)$, $\hat{x}_k$ can also be obtained by $$\begin{aligned} \label{eq:CKFUpdate} \begin{split} \hat{x}_{k} &= \operatorname*{argmin}_{\xi_c} (f_c(\xi_c))\\ &= \hat{x}_{k|k-1} + K (y_k - H\hat{x}_{k|k-1}) \end{split}\end{aligned}$$ where $K = (H^\top R^{-1} H + P^{-1}_{k|k-1})^{-1} H^\top R^{-1}$. With the [*[matrix inversion lemma]{}*]{}, the Kalman-gain $K$ can be written as $K = P_{k|k-1}H^\top(H P_{k|k-1} H^\top + R)^{-1}$, which appears in the standard Kalman-filtering. On the other hand, by the definition of $P_{k} := {\mathbb{E}} \{ (\hat{x}_k - x_k)(\hat{x}_k - x_k)^\top \}$, the update rule of the error covariance matrix $P_{k}$ of CKF is given by $$\begin{aligned} \label{eq:ECovUpdateCKF} P_k &= (\bar{H}_c^\top S^{-1} \bar{H}_c)^{-1} = (H^\top R^{-1} H + P^{-1}_{k|k-1})^{-1}\\ &= P_{k|k-1} - (H^\top R^{-1} H + P^{-1}_{k|k-1})^{-1} H^{\top} R^{-1} H P_{k|k-1}. \nonumber\end{aligned}$$ For more details, see [@Thrun2005Book; @Bell1993TAC; @Humpherys2010CSM]. Derivation of distributed Kalman-filtering problem {#Sec:DerivationDKF} -------------------------------------------------- Now, we consider a sensor network which consists of $N$ sensors and suppose that each sensor runs an estimator without the fusion center. Each estimator in the network tries to find the optimal estimate by processing the local measurement and exchanging information with its neighbors through communication network. The communication network among estimators is modeled by a graph $\mathcal{G}$ and the Laplacian matrix associated with $\mathcal{G}$ is denoted by $L \in \mathbb{R}^{N \times N}$. Under the setting (\[eq:System\]), estimator $i$ measures only the local measurement $y_{k,i}$, and the parameters $H_i$ and $R_i$ are kept private to estimator $i$. It is noted that the pair $(F, H_i)$ is not necessarily observable. We assume that the graph is connected and undirected [*[i.e.,]{}*]{} $L = L^\top$, and $F$ and $Q$ are open to all estimators. Similar to CKF, DKF has two steps, [*[local prediction]{}*]{} and [*[distributed correction]{}*]{}. In the local prediction step, each estimator predicts $$\begin{aligned} \hat{x}_{i, k|k-1} &= F \hat{x}_{i, k-1}\\ P_{i, k|k-1} &= F P_{i, k-1} F^\top + Q.\end{aligned}$$ where $\hat{x}_{i, k|k-1}$ and $P_{i, k|k-1} $ are local estimates of $\hat{x}_{k|k-1}$ and $P_{k|k-1} $, respectively, that estimator $i$ holds. In the distributed correction step, each estimator solves the maximum likelihood estimation in a distributed manner. The objective function of CKF $f_c (\xi)$ can be rewritten as $$\begin{aligned} \sum_{i=1}^N f_i(\xi_c) = \sum_{i=1}^N \frac{1}{2} (\bar{z}_{i,k} - \bar{H}_i \xi_c)^\top \bar{S}_{i,k}^{-1} (\bar{z}_{i,k} - \bar{H}_i \xi_c)\end{aligned}$$ where $\bar{z}_{i,k} = [y_{i,k};\hat{x}_{i, k|k-1}]$, $\bar{H}_i = [H_i; I_n]$, $\bar{S}_{i,k} = \operatorname{\text{diag}}(R_{i}, N P_{i, k|k-1})$. We assume that $\hat{x}_{i, k|k-1}=\hat{x}_{k|k-1}$ and $P_{i, k|k-1}=P_{k|k-1}$. This makes sense when the each sensor reached a consensus on $ \hat{x}_{i, k-1}$ and $P_{i, k-1}$ in the previous correction step. Assuming that each estimator holds its own optimization variable $\xi_i \in \mathbb{R}^{n}$ for $\xi_c$, DKF problem is written as the following consensus optimization problem. \[eq:DKFP\] $$\begin{aligned} \text{minimize}& \quad \sum_{i=1}^N f_i(\xi_i)\label{eq:DKFP_obj}\\ \text{subject to}& \quad \xi_1 = \cdots = \xi_N.\label{eq:DKFP_const}\end{aligned}$$ If there exists a distributed algorithm that finds a minimizer $(\xi_1^*, \dots, \xi_N^*)$, we say that the algorithm solves DKF problem. Since the kernel of Laplacian $L$ is $\text{span} \{1_N\}$, the constraints (\[eq:DKFP\_const\]) can be written with as $(L \otimes I_n) \xi = 0$ where $\xi = [\xi_1;\dots;\xi_N]$. To proceed, we define the Lagrangian to solve the problem (\[eq:DKFP\]) as $$\begin{aligned} \label{eq:LagrangianDKFP} {\mathcal{L}}(\xi, \lambda) &= \sum_{i=1}^N f_i(\xi_i) + \lambda^\top \bar{L} \xi\end{aligned}$$ where $\lambda \in \mathbb{R}^{Nn}$ is the Lagrange multipliers (dual variable) associated with (\[eq:DKFP\_const\]) and $\bar{L} = (L \otimes I_n)$. We decompose the Lagrangian into local ones defined by $$\begin{aligned} \label{eq:LagrangianDKFPLocal} {\mathcal{L}}_i(\xi_i, \lambda_i) &= f_i(\xi_i) + \lambda_i^\top \sum_{j \in {\mathcal{N}}_i} a_{ij} (\xi_i - \xi_j).\end{aligned}$$ For the Lagrangian (\[eq:LagrangianDKFP\]), the partial derivatives over $\xi$ and $\lambda$ are given by $$\begin{aligned} \nabla_{\xi} \mathcal{L}(\xi, \lambda) &= -\bar{H}^{\top} \bar{S}_k^{-1}(\bar{z}_k - \bar{H} \xi) + \bar{L} \lambda \\ \nabla_{\lambda} \mathcal{L}(\xi, \lambda) &= \bar{L} \xi,\end{aligned}$$ where $\bar{z}_k = [\bar{z}_{1,k};\dots;\bar{z}_{N,k}]$, $\bar{H}=\operatorname{\text{diag}}(\bar{H}_1, \dots, \bar{H}_N)$ and $\bar{S}_k = \operatorname{\text{diag}}(\bar{S}_{1,k}, \dots, \bar{S}_{N,k})$. Then, the optimality condition for ($\xi^*$, $\lambda^*$) becomes the following saddle point equation (KKT conditions), namely $$\begin{aligned} \label{eq:SPE} \begin{bmatrix} -\bar{H}^\top \bar{S}_k^{-1} \bar{H} & -\bar{L} \\ \bar{L} & {{0}} \end{bmatrix} \begin{bmatrix} \xi^* \\ \lambda^* \end{bmatrix} = \begin{bmatrix} -\bar{H}^\top \bar{S}_k^{-1} \bar{z}_k\\ 0 \end{bmatrix}\end{aligned}$$ where $\xi^* := [\xi_1^*; \dots; \xi_N^*]$ and $\lambda^* := [\lambda_1^*; \dots; \lambda_N^*]$. \[Lemma:DKFConvergence\] The solutions to DKF problem are parameterized as $(\xi^*, \lambda^*)=((1_N\otimes I_n) \xi^\dagger, (1_N\otimes I_n) \tilde \lambda + \bar \lambda)$ where $\xi^\dagger\in \mathbb R^n$ and $\bar \lambda \in \mathbb R^{Nn}$ are unique vectors and $\tilde \lambda\in \mathbb R^n$ is an arbitrary vector. If $(\xi^*, \lambda^*)$ is an optimal solution to DKF problem, then $\xi_i^*$ is the optimal solution to CFK problem. $\diamond$ By multiplying $1_N^\top \otimes I_n$ to the dual feasibility equation in (\[eq:SPE\]), one can obtain $$\begin{aligned} \label{eq:proofLemma1} (1_N^\top \otimes I_n) \bar{H}^\top \bar{S}_k^{-1} \bar{H} \xi^* = (1_N^\top \otimes I_n) \bar{H}^\top \bar{S}_k^{-1} \bar{z}_k.\end{aligned}$$ The primal feasibility equation in (\[eq:SPE\]) implies that $\xi^*=(1_N \otimes I_n) \xi^\dagger$, hence (\[eq:proofLemma1\]) becomes $$\begin{aligned} (1_N^\top \otimes I_n) \bar{H}^\top \bar{S}_k^{-1} \bar{H} (1_N \otimes I_n) \xi^\dagger = (1_N^\top \otimes I_n) \bar{H}^\top \bar{S}_k^{-1} \bar{z}_k.\end{aligned}$$ From $\bar{H}^\top \bar{S}_k^{-1} \bar{H} = \operatorname{\text{diag}}(\frac{1}{N}P^{-1}_{k|k-1} + H_1^\top R_1^{-1} H_1, \dots,$ $\frac{1}{N}P^{-1}_{k|k-1} + H_N^\top R_N^{-1} H_N)$, one has $$\begin{aligned} \Big (P^{-1}_{k|k-1} &+ \sum_{i=1}^N H_i^{\top} R_i^{-1} H_i\Big) \xi^\dagger \\ &= P^{-1}_{k|k-1} \hat{x}_{k|k-1} + \sum_{i=1}^N H_i^\top R_i^{-1} y_{i,k}.\end{aligned}$$ Since $\sum_{i=1}^N H_i^\top R_i^{-1} y_{i,k} = H^\top R^{-1} y_k$ and $\sum_{i=1}^N H_i^\top R_i^{-1} H_i = H^\top R^{-1} H$, it follows that $$\begin{aligned} \xi^\dagger = \hat{x}_{k|k-1} + K_k (y_k - H\hat{x}_{k|k-1})\end{aligned}$$ where $K_k = (P^{-1}_{k|k-1} + H^\top R^{-1} H)^{-1} H^\top R^{-1}$ and by the matrix inversion lemma, we have $K_k = P_{k|k-1} H^\top (R + H P_{k|k-1} H^\top)^{-1}$. From the fact that the right-hand side of above equation is the same with the update rule (\[eq:CKFUpdate\]) of CKF, it follows that $\xi_i^{*}=\xi^\dagger$ is the optimal estimate of CKF $\hat{x}_k$. On the other hand, one can observe that the optimal dual variable $\lambda^*$ is not unique since the dual feasibility equation $$\begin{aligned} \label{eq:lambda_star} (L \otimes I_n) \lambda^* = \bar{H}^\top \bar{S}_k^{-1} (\bar{z}_k - \bar{H} (1_N \otimes I_n) \xi^\dagger)\end{aligned}$$ is singular. To find $\eta^*$, consider the orthonormal matrix $U = [U_1 ~ \bar{U}]$ such that $LU = U \Lambda$ where $U_1 = \frac{1}{\sqrt{N}} 1_N$, $\bar{U}$ consists of the eigenvectors associated with the non-zero eigenvalues of $L$, denoted by $\sigma_2, $…$, \sigma_N$, and $\Lambda = \operatorname{\text{diag}}({0, \sigma_2, \dots, \sigma_N})$. Left multiplying $U^\top\otimes I_n$ to the equation yields $$\begin{aligned} \left( \begin{bmatrix} 0 & \\ & \bar{\Lambda} \end{bmatrix} \otimes I_n \right) \left( \begin{bmatrix} U_1^\top \\ \bar{U}^\top \end{bmatrix} \otimes I_n \right) \lambda^* = \left( \begin{bmatrix} U_1^\top \\ \bar{U}^\top \end{bmatrix} \otimes I_n \right) b\end{aligned}$$ where $\bar{\Lambda} = \operatorname{\text{diag}}(\sigma_2, \dots, \sigma_N)$ and $b = \bar{H}^\top \bar{S}_k^{-1} (\bar{z}_k - \bar{H} (1_N \otimes I_n) \xi^\dagger)$. Hence, the optimal dual variable $\lambda^*$ becomes $\lambda^* = (U\otimes I_n)\begin{bmatrix} \tilde{\lambda}^* ; (\bar{\Lambda}^{-1} \bar{U}^\top \otimes I_n)b \end{bmatrix}$ where $\tilde{\lambda}^* \in \mathbb{R}^{n}$ is an arbitrary vector. This completes the proof. Information form of DKF problem ------------------------------- It is well known that the dual of the Kalman-filter is the [*[Information filter]{}*]{} which uses the [*[canonical parameterization]{}*]{} to represent the normal (Gaussian) distribution [@Thrun2005Book]. With the canonical parameterization, DKF problem (\[eq:DKFP\]) can also be written in information form. Let $\eta_i = (H_i^\top R^{-1}_i H_i + \frac{1}{N} \Omega_{i, k|k-1})\xi_i$, $\Omega_{i, k|k-1} = P_{i, k|k-1}^{-1}$ and $\tau_{i, k|k-1} = P_{i, k|k-1}^{-1}\hat{x}_{i, k|k-1}$ which are the local decision variable for the information vector of the estimator $i$, the locally predicted information matrix and information vector, respectively. With these transformations, we rewrite the problem (\[eq:DKFP\]) as \[eq:DIFP\] $$\begin{aligned} \text{minimize}& \quad \sum_{i=1}^N h_i(\eta_i)\label{eq:DIFP_obj}\\ \text{subject to}& \quad \eta_1 = \cdots = \eta_N\label{eq:DIFP_const}\end{aligned}$$ where $$\begin{aligned} h_i(\eta_i) = \frac{1}{2} &\Big( \eta_i^\top \Phi_i^{-1} \eta_i - \eta_i^\top \Phi_i^{-1} (H_i^\top R_i^{-1} y_i + \frac{1}{N}\tau_{i, k|k-1}) \\ & \hspace{0.9cm} + y_i^\top R_i^{-1}y_i + \frac{1}{N}\tau_{i, k|k-1}^\top \Omega_{i, k|k-1}^{-1} \tau_{i, k|k-1} \Big)\end{aligned}$$ and $\Phi_i = H_i^\top R^{-1}_i H_i + \frac{1}{N} \Omega_{i, k|k-1}$. For the distributed problem (\[eq:DIFP\]), the Lagrangian is given by $$\begin{aligned} {\mathcal{L}}_{\eta}(\eta, \lambda) &= \sum_{i=1}^N h_i(\eta_i) + \nu^\top \bar{L} \eta\end{aligned}$$ where $\eta := [\eta_1;\dots;\eta_N]$ and $\nu$ is the Lagrange multipliers. The associated saddle point equation becomes $$\begin{aligned} \begin{bmatrix} -(\bar{H}^\top \tilde{S}_k^{-1} \bar{H})^{-1} & -\bar{L} \\ \bar{L} & {0} \end{bmatrix} \begin{bmatrix} \eta^* \\ \nu^* \end{bmatrix} = \begin{bmatrix} -\bar{H}^\top \tilde{S}_k^{-1} \tilde{z}_k \\ 0 \end{bmatrix}\end{aligned}$$ where $\tilde{z}_k = [\tilde{z}_{1,k};\dots;\tilde{z}_{N,k}]$, $\tilde{S}_k = \operatorname{\text{diag}}(\tilde{S}_{1,k}, \dots, \tilde{S}_{N,k})$, $\tilde{z}_{i,k} = [y_{i, k}; \tau_{i,k|k-1}]$ and $\tilde{S}_{i,k} = \operatorname{\text{diag}}(R_i, N\Omega_{i, k|k-1}^{-1})$. Interpretations of existing DKF algorithm from the optimization perspective {#Ssec:DiscussionOnOthers} --------------------------------------------------------------------------- One of the recent DKF algorithms, [*[Consensus on Information]{}*]{} (CI) [@Battistelli2015TAC; @Battistelli2016Aut] can be interpreted in the provided framework. CI consists of three steps, [*[prediction]{}*]{}, [*[local correction]{}*]{}, and [*[consensus]{}*]{}. In the prediction step, each estimator predicts the estimate based on the system dynamics and previous estimate similar to the standard information filter algorithm. Each estimator also updates the estimate with local measurements and output matrix in the local correction step. After that, the estimators find the agreed estimate by averaging the local estimates in the consensus step. In the provided framework, CI can be viewed as the algorithm which solves the problem through the two steps, the local correction step and the consensus step. In the former step, each of estimators finds the local minimizer (estimate) of the local objective function $h_i(\cdot)$. Since the partial derivative of $h_i(\eta_i)$ becomes $$\begin{aligned} \nabla_{\eta_i} h_i(\eta_i) = \Phi_i^{-1}\eta_i - \Phi_i^{-1} (H_i^\top R_i^{-1} y_i + \frac{1}{N}\tau_{i, k|k-1})\end{aligned}$$ and the local minimizer $\eta_i^*$ can be obtained by $\eta^*_i = H_i^\top R_i^{-1} y_i + \frac{1}{N}\tau_{i, k|k-1}$, which is the local update rule of CI[^3]. The local minimizer, however, can be different among estimators, since it minimizes only the local objective function $h_i(\cdot)$, which violates the constraint (\[eq:DIFP\_const\]). The consensus step of CI performs a role to find an agreed (average) value of the local estimates, using the doubly stochastic matrix, and the results of the consensus step satisfy the constraint (\[eq:DIFP\_const\]). The agreed estimate, however, may not be the global minimizer of (\[eq:DIFP\]), which means that the consensus step cannot guarantee the convergence of the estimates to that of CKF. A Solution to DKF Problem {#Sec:DKF-DA} ========================= One can observe that (\[eq:LagrangianDKFP\]) is strictly convex, differentiable, and the local objective function $f_i(\cdot)$ is a quadratic function, hence [*[strong duality]{}*]{} holds. In addition, from the fact $\bar{H}^\top \bar{S}_k^{-1} \bar{H}$ is a nonsingular and block diagonal matrix, the optimal conditions (\[eq:SPE\]) are already in a distributed form. This implies that the minimizer $\xi^*$ can be obtained in a distributed manner as long as $\lambda^*$ is given, [*[i.e.,]{}*]{} $\xi_i^* = (\bar{H}_i^\top \bar{S}_{i,k}^{-1} \bar{H}_i)^{-1} (\bar{H}_i^\top \bar{S}_{i,k}^{-1} \bar{z}_{i,k} - \sum_{j \in \mathcal{N}_i} a_{ij} (\lambda^*_{i} - \lambda^*_{j}))$. Based on the above discussion, we see that one possible algorithm solving (\[eq:DKFP\]), guaranteeing the asymptotic convergence to the global minimizer $\xi^*$, is the dual ascent method [@Boyd+2011FTML; @Dorfler2017] which is given by \[eq:UpdateDAGlobal\] $$\begin{aligned} \xi_{l+1} &= (\bar{H}^\top \bar{S}_k^{-1} \bar{H})^{-1} (\bar{H}^{\top} \bar{S}_k^{-1} \bar{z}_k - \bar{L} \lambda_l) \label{eq:PrimalUpdateGlobal}\\ \lambda_{l+1} &= \lambda_l + \alpha_\lambda \bar{L} \xi_{l+1}\label{eq:DualUpdateGlobal}\end{aligned}$$ where $\alpha_\lambda > 0$ is a step size. The update rule can be written locally as \[eq:DKF\_DA\] $$\begin{aligned} \xi_{i,l+1} &= \hat{x}_{i, k|k-1} + K_{i,k} (y_{i,k} - H_i \hat{x}_{i, k|k-1}) - \psi_{i,l}\\ \lambda_{i,l+1} &= \lambda_{i,l} + \alpha_\lambda \sum_{j \in \mathcal{N}_i} a_{ij} (\xi_{i,l+1} - \xi_{j,l+1}).\end{aligned}$$ where $K_{i,k} = (H_i^{\top} R_i^{{-1}} H_i + \frac{1}{N} P_{i, k|k-1}^{{-1}})^{-1} H_i^{\top} R_i^{-1}$, $\psi_{i,l} = (H_i^\top R_i^{-1} H_i + \frac{1}{N} P_{i, k|k-1}^{-1})^{-1} \sum_{j \in \mathcal{N}_i} a_{ij} (\lambda_{i,l} - \lambda_{j,l})$, and $l$ is the iteration index to find the minimizer. Regarding the convergence of the update rule , we have the following result. \[lem:convergence\] Assume that the network $\mathcal{G}$ is undirected and connected. Then, the sequence $\{ \xi_{i,l} \}$ generated by the dual ascent method (\[eq:DKF\_DA\]) converges to $\hat{x}_k$ of CKF problem (\[eq:CKFUpdate\]), as $l$ goes to infinity, provided that the step size $\alpha_\lambda > 0$ is chosen such that $$\label{eq:bound_step_size} \alpha_\lambda < \frac{2}{\sigma_N^2 \max_{i} \{ \| (\bar{H}_i^\top S_{i,k}^{-1} \bar{H}_i)^{-1}\| \}}$$ where $\sigma_N$ is the maximum eigenvalue of $L$. Moreover, the sequence $\{\lambda_{i,l} \}$ converges to a vector which is uniquely determined by the initial conditions of $\lambda_i$’s. $\diamond$ Substituting the dual feasibility equation to the primal feasibility equation of (\[eq:SPE\]) yields $$\begin{aligned} \label{eq:DualOpt} \bar{L} (\bar{H}^\top \bar{S}_k^{-1} \bar{H})^{-1} \bar{L} \lambda^* = \bar{L} (\bar{H}^\top \bar{S}_k^{-1} \bar{H})^{-1} \bar{H}^\top \bar{S}_k^{-1} \bar{z}_k.\end{aligned}$$ Now let $e^\lambda_{l} = \lambda_{l} - \lambda^*$. Then, one obtains $$\begin{aligned} e^\lambda_{l+1} &= \lambda_l + \alpha_\lambda \bar{L} \xi_{l+1} - \lambda^*\\ &= \lambda_l + \alpha_\lambda \bar{L}(\bar{H}^\top \bar{S}_k^{-1} \bar{H})^{-1}(\bar{H}^\top \bar{S}_k^{-1} \bar{z}_k - \bar{L}\lambda_l) - \lambda^*.\end{aligned}$$ From the identity (\[eq:DualOpt\]), we have $$\begin{aligned} \label{eq:DualErrorDynamics} \begin{split} e^\lambda_{l+1} &= (I - \alpha_\lambda \bar{L} (\bar{H}^\top \bar{S}_k^{-1} \bar{H})^{-1} \bar{L}) e^{\lambda}_l\\ &:= (I - \alpha_\lambda \tilde{A}_\lambda) e^{\lambda}_l. \end{split}\end{aligned}$$ Here, $\tilde{A}_\lambda$ is a symmetric positive semi-definite matrix which has $n$ simple zero eigenvalues, and it holds that $I - \alpha_\lambda \sigma_{\max}(\tilde{A}_\lambda)I \leq I - \alpha_\lambda \tilde{A}_\lambda \leq I - \alpha_\lambda \sigma_{\min}(\tilde{A}_\lambda)I$. Since $\sigma_{\min}(\tilde{A}_\lambda)$ is zero, it follows that if $\alpha_\lambda > 0$ is chosen such that $\alpha_\lambda \sigma_{\max}(\tilde A_{\lambda}) < 2$, all eigenvalues of $I - \alpha_\lambda \tilde{A}_\lambda$, except $1$, are located inside the unit circle. The bound ensures this. Regarding the convergence of $\lambda_l$, we proceed as follows. With the orthonormal matrix $U$ used in [*[Lemma]{}*]{} \[Lemma:DKFConvergence\], $\tilde{A}_\lambda$ can be written as $$\begin{aligned} \tilde{A}_\lambda &= (U \Lambda U^\top \otimes I_n) (\bar{H}^\top \bar{S}_k^{-1} \bar{H})^{-1} (U \Lambda U \otimes I_n) \\ &= (U \otimes I_n) \operatorname{\text{diag}}(0_n, M_{\text{sub}}) (U^\top \otimes I_n)\end{aligned}$$ where ${M}_{\text{sub}} \in \mathbb{R}^{(N-1)n \times (N-1)n} $ is a submatrix with the first $n$ rows and first $n$ columns removed. In the new coordinates $\bar{e}^\lambda_l$, defined by $\bar{e}^\lambda_l= (U^\top \otimes I_n )e^\lambda_l $, the error dynamics of the dual variable can be expressed as $$\begin{aligned} \bar{e}^\lambda_{l+1} &= \operatorname{\text{diag}}(I, I - \alpha_\lambda {M}_{\text{sub}}) \bar{e}^\lambda_{l}.\end{aligned}$$ From this equation, we know that the first $n$ components of $\bar{e}^\lambda_{l}$, denoted by $\tilde{e}^\lambda_{l}$, remains the same for any $l$, [*i.e.*]{}, $\tilde e_l^\lambda = \tilde e_0^\lambda$, $\forall l\ge 0$, meaning that $(U_1^\top \otimes I_n) e^\lambda_l =\tilde e_0^\lambda, \ \forall l\ge 0$, which means that $\tilde e_0^\lambda= (U_1^\top \otimes I_n) e^\lambda_0$. Moreover, with $\alpha_\lambda$ chosen as , which guarantees that the matrix $I - \alpha_\lambda {M}_{\text{sub}}$ has all its eigenvalues except 1 inside the unit circle, we have $\lim_{l \rightarrow \infty} \bar{e}^\lambda_{l} = \begin{bmatrix} \tilde{e}^\lambda_{0}; 0 \end{bmatrix}$, from which it follows that $$\label{eq:Limit_e_lambda} \lim_{l \rightarrow \infty} e^\lambda_l =(U\otimes I_n) \begin{bmatrix} \tilde{e}^\lambda_{0}; 0 \end{bmatrix}= (U_1\otimes I_n)(U_1^\top \otimes I_n) e^\lambda_0.$$ Recalling that $e^\lambda_l := \lambda_l - \lambda^*$, we have from $$\begin{aligned} \lim_{l \rightarrow \infty} \lambda_l &= \lambda^* + (U_1 U_1^\top \otimes I_n) (\lambda_0 - \lambda^*).\end{aligned}$$ Applying $\lambda^* = (U_1 \otimes I_n) \tilde{\lambda}^* + (\bar U \bar{\Lambda}^{-1} \bar{U}^\top \otimes I_n)b$ (for $\tilde \lambda^*$ and $b$, see the proof of [*Lemma*]{} \[Lemma:DKFConvergence\]), we have $$\begin{aligned} \lim_{l \rightarrow \infty} \lambda_l &=(\bar U \bar{\Lambda}^{-1} \bar{U}^\top \otimes I_n)b + (1_N \otimes I_n) \text{avg}(\lambda_{i, 0})\end{aligned}$$ where $\text{avg}(\lambda_{i, 0}) = \frac{1}{N} \sum_{i=1}^N \lambda_{i, 0}$, and this completes the proof. Now, we derive an update rule of the error covariance matrix. With the information matrix $\Omega_k := P_k^{-1}$, the error covariance update rule (\[eq:ECovUpdateCKF\]) can be written as $$\begin{aligned} \Omega_k &= H^\top R^{-1} H + \Omega_{k|k-1}\\ &= \frac{1}{N} \sum^N_{i=1} (NH_i^{\top} R_i^{-1} H_i + \Omega_{k|k-1}).\end{aligned}$$ Define $\Omega_{i,k} := H^\top_i R^{-1}_i H_i + \frac{1}{N}\Omega_{i, k|k-1}$. Then, the updated information matrix of CKF can be obtained by solving the following distributed optimization problem \[eq:ECovProblem\] $$\begin{aligned} \text{minimize}& \quad \sum_{i=1}^N (\zeta_i - {\text{vec}_h({N \Omega_{i, k}})} )^2 \label{eq:ECovObj}\\ \text{subject to}& \quad \zeta_1 = \cdots = \zeta_N\label{eq:DKFPECov_const}\end{aligned}$$ where $\zeta_i \in \mathbb{R}^{n(n+1)/2}$ is the decision variable. Note that the minimizer $\zeta^* := [\zeta_i^{*};\dots;\zeta_N^{*}] \in \mathbb{R}^{Nn(n+1)/2}$ of the above optimization problem is nothing but the average of all $\text{vec}(N \Omega_{i, k})$, which corresponds to $\Omega_k$. Define the Lagrangian for the problem (\[eq:ECovProblem\]) as $$\begin{aligned} \label{eq:Lagrangian_ECov} {\mathcal{L}}_{\Omega}(\zeta, \mu) &= \sum_{i=1}^N (\zeta_i - {\text{vec}_h({N\Omega_{i,k}})} )^2 + \mu^\top (L \otimes I) \zeta\end{aligned}$$ where $\mu \in \mathbb{R}^{Nn(n+1)/2}$ is the dual variable. The saddle point equation for (\[eq:Lagrangian\_ECov\]) is given by \[eq:OptConditionECov\] $$\begin{aligned} \begin{bmatrix} -I & -L \otimes I\\ L \otimes I & 0 \end{bmatrix} \begin{bmatrix} \zeta^* \\ \mu^* \end{bmatrix} = \begin{bmatrix} -\bar{z}_{\Omega, k} \\ 0 \end{bmatrix}\end{aligned}$$ where $\bar{z}_{\Omega, k} := [{\text{vec}_h({N\Omega_{1, k}})};\dots;{\text{vec}_h({N \Omega_{N, k}})}]$, and $\mu^*$ is the dual variable of the optimal point. From the similar arguments in the proof of [*Lemma*]{} \[Lemma:DKFConvergence\], we have $$\begin{aligned} \zeta^* &= (1_{N} \otimes I) \frac{1}{N} \sum^N_{i=1} {\text{vec}_h({N \Omega_{i, k}})}\\ &= (1_{N} \otimes I) ( {\text{vec}_h({H^\top R^{-1} H})} + \frac{1}{N} \sum^N_{i=1} {\text{vec}_h({\Omega_{i,k|k-1}})}).\end{aligned}$$ This implies that the optimal solution $\zeta^*_i$ is the half vectorization of the average of the locally predicted information matrix corrected by the global information $H^\top R^{-1} H$. Based on the above arguments, we propose a dual ascent type update rule for the error covariance matrix as \[eq:ECov\_DA\] $$\begin{aligned} \zeta_{i, l+1} &= {\text{vec}_h({\Omega_{i, k}})} - \sum_{j \in \mathcal{N}_i} a_{ij} (\mu_{i, l} - \mu_{j, l})\\ \mu_{i, l+1} &= \mu_{i, l} + \alpha_\mu \sum_{j \in \mathcal{N}_i} a_{ij} (\zeta_{i, l+1} - \zeta_{j, l+1}).\end{aligned}$$ where $\alpha_\mu$ is a step size such that $0 < \alpha_\mu < 2/\sigma_N^2$, which obtained by the similar arguments in the proof of [*Lemma*]{} \[lem:convergence\]. Putting all pieces together, we propose a DKF algorithm described in Algorithm \[Algo:DKF\_DA\]. $\hat{x}_{i, k|k-1} = A_d \hat{x}_{i, k-1}$ $P_{i, k|k-1} = A_d P_{i, k-1} A_d^\top + Q$, $\Omega_{i, k|k-1} = P^{-1}_{i, k|k-1}$ $\lambda_{i, 0}, \mu_{i, 0} = 0$ $l= 0,\dots, l^*-1$, [**[do]{}**]{} (\[eq:DKF\_DA\]) $\xi_{i, l+1} = \hat{x}_{i, k|k-1} + K_{i, k} (y_{i, k} - H_i \hat{x}_{i, k|k-1}) - \psi_{i, l}$ $\lambda_{i, l+1} = \lambda_{i, l} + \alpha_\lambda \sum_{j \in \mathcal{N}_i} a_{ij} (\xi_{i, l+1} - \xi_{j, l+1})$ (\[eq:ECov\_DA\]) $\zeta_{i, l+1} = {\text{vec}_h({\Omega_{i, k}})} - \sum_{j \in \mathcal{N}_i} a_{ij} (\mu_{i, l} - \mu_{j, l})$ $\mu_{i, l+1} = \mu_{i, l} + \alpha_\mu \sum_{j \in \mathcal{N}_i} a_{ij} (\zeta_{i, l+1} - \zeta_{j, l+1})$ $\hat{x}_{i, k} = \xi_{i, l^*}$, $P_{i, k} = ({\text{vec}_h^{-1}({\zeta_{i, l^*}})})^{-1}$ In the structural point of view, the algorithm consists of [*[local prediction]{}*]{} step and [*[distributed correction]{}*]{} step as in CKF. In the local prediction step, each estimator locally predicts the estimate and the corresponding covariance matrix. In the distributed correction step, each estimator finds the optimal points for the state estimate and its error covariance matrix, iteratively, by using the local measurement information and exchanging information with its neighbors. With sufficiently large $l^*$, locally updated $\xi_{i, l^*}$ and $P_{i, k}$ converge to those of CKF with tunable size of errors. Numerical Experiments {#Sec:NE} ===================== We have two examples for the developed theory. The first one is a simple academic example, while the second one is more practical one. \[example\] Consider a system given by $$\begin{aligned} x_{k+1} &= \begin{bmatrix} 0.4& 0.9& 0& 0\\ -0.9& 0.4& 0& 0\\0& 0& 0.5& 0.8\\0& 0& -0.8& 0.5\\ \end{bmatrix}x_k + w_k\\ y_k &= \begin{bmatrix} 1& 0& 0& 0\\ 1& 1& 0& 0\\0& 0& 1& 1\\0& 0& 1& 0\\ \end{bmatrix} x_k + v_k\end{aligned}$$ and $Q = 0.1$, $R = \operatorname{\text{diag}}(0.1, 0.2, 0.3, 0.1)$, and suppose that $4$ estimators are connected through a communication network whose Laplacian matrix is given by $$\begin{aligned} L = \begin{bmatrix} 3& 0& -1& -2\\0& 2& -2& 0\\-1& -2& 4& -1\\-2& 0& -1& 3 \end{bmatrix}.\end{aligned}$$ The step sizes for the algorithm are chosen as $\alpha_\lambda, \alpha_\mu = 0.01$. Figure \[fig:ENorm\_A1\] shows that the average error norm defined by $\text{avg}(\| e_{i, k} \|) = \frac{1}{N} \sum_{i=1}^{N} \|\hat{x}_{i, k} - x_k\|$ decreases more rapidly as $l^*$ increases. Figure \[fig:PNorm\_A1\] also shows that as $l^*$ increases, the average error covariance norm defined by $\text{avg}(\| P_{i, k}\|) := \frac{1}{N} \sum_{i=1}^N \| P_{i, k} \|$ approaches $\| P_k \|$ which is the norm of the error covariance matrix of CKF. It is seen that, when $l^* = 50$, there is very little difference between $\text{avg}(\| P_{i, k}\|)$ and $\| P_k \|$ of CKF. $\square$ [0.24]{}  [0.24]{}  In this example, we evaluate DA-DKF with a network consisting of 50 estimators to estimate the state of a target system. The dynamics of the target system is described by $$\begin{aligned} x_{k+1} = e^{A} x_k + w_k, \quad A = \begin{bmatrix} 0 & 0.5 & 0 & 0\\-0.5 & 0 & 0 & 0\\0 & 0 & 0 & -0.5\\0 & 0 & 0.5 & 0 \end{bmatrix}\end{aligned}$$ where $w_k \sim \mathsf{N}(0, Q)$ and $Q = 0.1$. The first and the third components of $x_k$ represent the $x$-axis position and $y$-axis position in the plane, respectively . The estimator $i$ knows $e^A$, $Q$, $H_i \in \mathbb{R}^{1 \times 4}$ and $R_i>0$, and each $H_i$ and $R_i$ is randomly chosen. The connections among estimators are also randomly selected and the weight is $1$ when connected, and the maximum eigenvalue of $L$ is $18.5$. For all $i$, $P_{i,0} = Q$ and each component of the initial estimate $\hat{x}_{i, 0}$ is randomly chosen within $(-15, 15)$ as shown in Figure \[fig:k=0\]. The parameters for DA-DKF were chosen as $\alpha, \beta = 10^{-5}$, $l^* = 10$. Figure \[fig:50DKF\] shows four snapshots of the target system’s position (black cross) and the each estimator’s estimate (red circles). The blue line is the trajectory of the target system. As time goes by (as $k$ increases), the estimates of the distributed Kalman-filters converge to the vicinity of the position of the target system. $\square$ [0.237]{} ![A sensor network with $50$ distributed Kalman-filters tracking a moving target using DA-DKF.[]{data-label="fig:50DKF"}](simdata/LargeScaleSim/k=0.eps "fig:") [0.237]{} ![A sensor network with $50$ distributed Kalman-filters tracking a moving target using DA-DKF.[]{data-label="fig:50DKF"}](simdata/LargeScaleSim/k=10.eps "fig:") [0.238]{} ![A sensor network with $50$ distributed Kalman-filters tracking a moving target using DA-DKF.[]{data-label="fig:50DKF"}](simdata/LargeScaleSim/k=20.eps "fig:") [0.238]{} ![A sensor network with $50$ distributed Kalman-filters tracking a moving target using DA-DKF.[]{data-label="fig:50DKF"}](simdata/LargeScaleSim/k=50.eps "fig:") Conclusions and Future Work =========================== This paper dealt with DKF from the optimization perspective. By observing that the correction step of Kalman-filtering is basically an optimization problem, we formulated DKF problem from the centralized one. The formulated problem is a quadratic consensus optimization problem. One of the recent DKF algorithms, Consensus on Information [@Battistelli2015TAC] was reinterpreted from the distributed optimization perspective. In addition, various DKF algorithms can be derived, by employing many existing distributed optimization methods to DKF problem. As an instance, DA-DKF has been presented, employing the distributed dual ascent method, and the algorithm has been validated with numerical experiments. For the future work, we plan to analyze the effect of the residuals of the previous iteration $k$, especially how the residuals affect the convergence. In addition, researches on more practical obstacles, such as considering the time-varying network topology, reducing communication loads, will be conducted. [^1]: ${}^*$Corresponding author [^2]: K. Ryu and J. Back are with School of Robotics, Kwangwoon University, Seoul, Republic of Korea [{ryuhhh, backhoon}@kw.ac.kr]{} [^3]: In the CI, the scalar $\frac{1}{N}$ is neglected [@Battistelli2015TAC].

--- abstract: 'We present new K- and L$''$-band imaging of a representative sample of members of the young 3$-$5Myr old $\sigma$Orionis cluster. We identified objects with $(K-L'')$ excess by analysing colour-colour diagrams and comparing the observations with empirical main-sequence colours. The derived disk frequency depends on the method used: (54$\pm$15)% if measured directly from the $JHKL''$ colour-colour diagram; or (46$\pm$14)% if excesses are computed with respect to predicted photospheric colours (according to the objects spectral types, 2-$\sigma$ excess detections). We compare the $(K-L'')$ excess with other indicators and show that this is a robust and reliable disk indicator. We also compare the derived disk frequency with similarly aged clusters and discuss possible implications for disk lifetimes. The computed age of the $\sigma$Ori cluster is very important: a cluster age of 3Myr would support the overall disk lifetime of 6Myr proposed in the literature, while an age $>$4Myr would point to a slower disk destruction rate.' author: - | J.M. Oliveira[^1], R.D. Jeffries and J.Th. van Loon\ School of Chemistry and Physics, Keele University, Keele, Staffordshire ST5 5BG, UK title: 'An L$''$-band survey for circumstellar disks around low-mass stars in the young $\sigma$Orionis cluster' --- \[firstpage\] circumstellar matter – infrared: stars – star: pre-main-sequence – stars: late type – open clusters and associations: individual ($\sigma$ Orionis) stars Introduction ============ Disk-like structures are believed to be ubiquitous around young protostars. These disks are dissipated very early in pre-main-sequence (PMS) evolution, perhaps by powerful stellar jets/outflows or photodissociation by the far-ultraviolet flux from nearby massive OB stars. Despite their short lives, the timescales and mass dependence of disk dissipation have far reaching consequences in astrophysics: the efficiency of disk depletion could be the strongest factor in determining the timescales on which planets form in a particular stellar system [@haisch01], or whether they form at all [@brandner00]. Disks probably play a significant role in early angular momentum regulation and the dissipation timescale is thought to control the spread in rotation rates of young stars [@sills00]. Stars may accrete a significant fraction of their final mass from a circumstellar disk, so the timescale and mass dependence of that accretion influences PMS evolution and thus attempts to estimate ages and masses from evolutionary PMS models [@comeron03]. The mass dependence of disk frequencies can provide a stern test for low-mass stellar and brown dwarf formation theories. For instance, models involving competitive accretion and subsequent ejection of brown dwarfs from protostellar aggregates [@reipurth00; @bate03] may imply shorter disk dissipation times for the lower mass fragments. Observed disk frequencies in samples of young stars with different ages, masses and environments provide an empirical determination of disk lifetimes. Judging from L-band excesses, young clusters exhibit high disk frequencies ($\ga$80%, e.g. the Trapezium cluster: @lada00) up to ages of $\sim$1.5Myr, which then decrease rapidly with age: at $\sim$3Myr, 50% of disks have been dissipated, and the timescale for all cluster members to lose their disks may be as short as $\sim$6Myr [@haisch01]. Such timescales have been questioned by a high disk frequency in the 9Myr $\eta$Chamaeleontis cluster, a sparsely populated cluster with no massive stars [@lyo03]. $\sigma$Orionis is a Trapezium-like system with an O9.5V primary. The population of low-mass stars spatially clustered around this system was discovered as bright X-ray sources in ROSAT images, and follow-up optical spectroscopy confirmed most sources as PMS stars [@wolk96; @walter97]. This association is young, nearby and affected by low reddening, making it an ideal target to analyse the PMS population even down to brown dwarfs [e.g. @bejar01; @barrado03; @kenyon03] and isolated planetary mass objects [@osorio00]. Furthermore, at an age of 3$-$5Myr [e.g. @oliveira02; @osorio02; @jayawardhana03], the $\sigma$Orionis cluster is at a crucial stage in terms of disk evolution and it is therefore a key case to better constrain disk dissipation timescales. Recently, a possible proto-planetary disk, apparently on the process of being dissipated, has been discovered very close to $\sigma$Ori [@loon03]. The $K_{\rm s}$-excess disk frequency is 5$-12$% for the low-mass and brown dwarf members of the $\sigma$Orionis cluster [@oliveira02; @barrado03]. On the other hand, the presence of strong H$\alpha$ emission suggests accretion disk frequencies as high as 30% [@osorio02]. However, the most reliable method to determine the disk frequency in a low-mass population is by measuring the $(K-L)$ colours and deriving colour excesses [e.g. @wood02]. @jayawardhana03 have obtained L-band observations of 6 $\sigma$Ori cluster members, finding two with a $(K-L)$ excess. The significance of this result is obviously limited by the size of the sample. We have performed L$'$-band (3.8$\mu$m) observations of a representative sample of 28 cluster members, using the newly installed imager UIST at the United Kingdom Infrared Telescope (UKIRT). Young stars are well known for their variability across the spectrum including infrared (IR) wavelengths [@carpenteretal01; @carpenter02], therefore we have obtained nearly simultaneous K-band observations for all our targets. In this paper, we describe the results of this survey, and discuss our derived disk frequency within the framework of disk destruction timescales by comparing with similar surveys in other young clusters. Cluster members and properties ============================== Sample of cluster members ------------------------- We have an on-going program to observe in the L-band $\sigma$Ori cluster members identified at optical wavelengths. We describe here the observations of 28 of the brightest cluster members: their positions, $I_{\rm c}$ magnitudes, 2MASS (Two Micron All Sky Survey) $J, H$ and $K_{\rm s}$ magnitudes, the new K- and L$'$-band magnitudes and identifications are listed in Table\[obs\_table\]. Some sources were first identified as ROSAT X-ray sources and photometric cluster candidates by @wolk96 [ W96] while other objects are photometric candidates identified by @bejar01 [ B01]. @osorio02 [ ZO02] have spectroscopically confirmed cluster membership for both these sets of objects. The remaining objects are spectroscopic cluster members found by @kenyon03 [ K03]. The I-band magnitudes are from either @bejar01 or @kenyon03. The brighter objects in the sample are mostly from the X-ray selected sample [@wolk96] while the fainter objects were photometrically and spectroscopically selected. In Sect.5.1 we discuss the effects of possible selection biases on our results. Searching for circumstellar disks around these 24 objects is the main goal of these observations. To this sample we have added 4 objects that are known IRAS sources in the region (no reference entry in Table\[obs\_table\]). They have been confirmed by @oliveira03b (see also @oliveira03a) as mid-infrared sources with spectral energy distributions (SEDs) consistent with them being young stars with dusty circumstellar disks; thus, based on their location, youth and infrared excesses they are also likely to be members of the $\sigma$Ori cluster. --- -------------------- -------------------- ------- ------- ------- ------- -------- ------- ------ ------ -------------------- ----------- ra dec ($^{h\,\,m\,\,s}$) ($^{d\,\,m\,\,s}$) (mag) (mag) (mag) (mag) 1 5 38 33.68 $-$2 44 14.2 10.13 9.28 8.66 8.600 0.00 7.71 0.02 TXOri 2 5 38 48.04 $-$2 27 14.2 12.08 10.16 9.46 9.19 9.208 0.00 8.65 0.08 4771-899 W96, ZO02 3 5 38 27.26 $-$2 45 09.7 12.82 11.96 10.79 9.94 9.729 0.00 8.64 0.04 4771-41, V505Ori W96, ZO02 4 5 40 08.89 $-$2 33 33.7 11.50 10.55 9.91 9.812 0.00 8.76 0.06 Haro5-39 5 5 39 39.83 $-$2 33 16.0 12.22 10.96 10.07 0.387 0.00 9.09 0.06 V603Ori 6 5 39 39.82 $-$2 31 21.8 11.84 10.90 10.22 9.831 0.00 8.48 0.07 V510Ori 7 5 38 38.23 $-$2 36 38.4 12.37 11.16 10.46 10.31 0.232 0.00 0.18 0.03 r053838$-$0236 W96,ZO02 8 5 38 31.58 $-$2 35 14.9 13.52 11.52 10.70 10.35 0.565 0.00 9.63 0.05 r053831$-$0235 W96,ZO02 9 5 38 40.27 $-$2 30 18.5 12.80 11.51 10.76 10.40 0.350 0.00 9.75 0.07 r053840$-$0230 W96,ZO02 0 5 38 49.17 $-$2 38 22.2 12.88 11.39 10.66 0.51 10.456 0.009 0.22 0.04 r053849$-$0238 W96,ZO02 1 5 39 05.41 $-$2 32 30.3 12.66 11.55 10.86 0.67 10.618 0.009 0.44 0.05 4771-1075 W96,ZO02 2 5 39 11.63 $-$2 36 02.9 12.78 11.62 10.97 0.75 10.72 0.01 0.53 0.09 4771-1038 W96,ZO02 3 5 39 20.44 $-$2 27 36.8 13.51 12.15 11.42 1.17 11.12 0.01 0.91 0.06 J053920.5$-$022737 B01,ZO02 4 5 40 01.96 $-$2 21 32.6 14.32 12.34 11.58 1.25 11.18 0.01 0.67 0.05 J054001.8$-$022133 B01,ZO02 5 5 38 47.55 $-$2 27 12.0 14.46 12.14 11.50 1.27 11.20 0.01 0.83 0.05 J053847.5$-$022711 B01,ZO02 6 5 38 20.21 $-$2 38 01.6 14.41 12.58 11.86 1.61 11.51 0.02 1.16 0.11 J053820.1$-$023802 B01,ZO02 7 5 39 51.73 $-$2 22 47.2 14.59 12.60 12.01 1.68 11.65 0.02 1.02 0.04 J053951.6$-$022248 B01,ZO02 8 5 39 40.98 $-$2 16 24.4 15.84 12.87 12.15 1.74 11.72 0.02 1.00 0.05 J053941.0$-$021624 K03 9 5 38 27.51 $-$2 35 04.2 14.50 12.83 12.11 1.86 11.72 0.02 1.18 0.05 J053827.4$-$023504 B01,ZO02 0 5 39 58.26 $-$2 26 18.8 14.19 12.78 12.05 1.83 11.77 0.02 1.60 0.05 J053958.1$-$022619 B01,ZO02 1 5 39 22.87 $-$2 33 33.1 14.16 12.83 12.13 1.87 11.82 0.02 1.55 0.04 r053923$-$0233 W96,ZO02 2 5 39 07.59 $-$2 28 23.4 14.33 12.88 12.14 1.96 12.12 0.03 1.61 0.06 r053907$-$0228 W96,ZO02 3 5 40 09.33 $-$2 25 06.7 15.11 13.15 12.50 2.15 12.06 0.03 1.55 0.04 J054009.3$-$022507 K03 4 5 40 01.01 $-$2 19 59.8 15.02 13.10 12.50 2.25 12.18 0.02 1.90 0.03 J054551.0$-$021960 K03 5 5 39 14.47 $-$2 28 33.4 14.75 13.34 12.65 2.34 12.32 0.03 1.79 0.10 J053914.5$-$022834 B01,ZO02 6 5 38 17.47 $-$2 09 23.6 14.94 13.28 12.66 2.36 12.46 0.03 2.14 0.04 J053817.5$-$020924 K03 7 5 37 54.53 $-$2 58 26.5 15.51 13.31 12.71 2.41 12.29 0.03 1.70 0.10 J053754.5$-$025827 K03 8 5 37 58.40 $-$2 41 26.2 15.36 13.29 12.70 2.42 12.39 0.03 1.62 0.11 J053758.4$-$024126 K03 --- -------------------- -------------------- ------- ------- ------- ------- -------- ------- ------ ------ -------------------- ----------- Cluster age ----------- ![$I/(I-J)$ colour-magnitude diagram of the $\sigma$Ori cluster. All the stars in this diagram have been spectroscopically confirmed as cluster members [@osorio02; @barrado03; @kenyon03]. The filled symbols are the objects we have observed in the K- and L$'$-band. We overplotted evolutionary tracks (dotted lines) and isochrones (solid lines) from @baraffe98 [ see text]. []{data-label="ij_cmd"}](MD837rv_fig1.eps) The age determination for clusters at very young ages is always an uncertain affair. In OB associations, several methods can be used: the properties of the massive O-stars, low-mass stellar isochrone fitting, and lithium abundance evolution in the association. The age of the multiple system $\sigma$Ori is estimated to be 1.7$-$7Myr based on stellar properties and its membership of the OrionOB1b group [@brown94 and references therein]. Using isochrones from several authors, @bejar99 [ and references therein] determined a low-mass star cluster age between 1$-$5Myr; using a different sample of low-mass cluster members, @oliveira02 determined an isochronal median age of 4.2$^{+2.7}_{-1.5}$Myr, where the quoted errors are due to the uncertainties of the Hipparcos distance to $\sigma$Ori. @osorio02 compared the lithium abundance in low-mass cluster members with theoretical predictions from several authors. They found no evidence of appreciable lithium destruction and they inferred an upper limit to the cluster age of 8Myr. In Fig.\[ij\_cmd\] we plot the $I/(I-J)$ colour-magnitude diagram for the present sample. We use the @baraffe98 evolutionary models with the mixing length parameter set to 1.0 pressure scale height for lower mass objects and to 1.9 for $M > 0.62$M$_{\sun}$. We have opted to use these models because they incorporate model atmospheres and predict PMS magnitudes without the use of empirical effective temperature-colour relations. We have computed the age and mass of each object; two objects (from the 24 with I-band magnitudes) fall outside the Baraffe grid of models — they appear to be very young (age$<$1Myr). The objects have masses in the range 1.0$-$0.13M$_{\sun}$. The sample has a median age of 3.6Myr (at the Hipparcos distance of 352pc), consistent with previous age determinations (see above). The conservatively large uncertainty in the Hipparcos distance (352$^{+166}_{-85}$pc) produces median age uncertainties as computed by @oliveira02. The most obvious thing from this colour-magnitude diagram is the large apparent age spread ($<$1 to 20Myr). @hartmann01 has analysed many possible sources of uncertainty in the determined stellar ages in star forming regions. He concluded that observational errors probably account for a large fraction of the observed age spread in such regions. Therefore, it is still not firmly established whether observed age spreads are real effects. The main sources of uncertainty in the case of the present data set are photometric variability and unresolved binaries. Unresolved binaries would tend to make objects appear brighter and redder, making them look younger than isolated cluster siblings. This is one reason why we consider the true age of the cluster likely to be older than the median age of the cluster (3.6Myr, see above). Another effect, seldom considered, is described by @comeron03: they propose that underluminous cluster members in the Lupus3 dark cloud provide evidence for accretion-modified evolution. These objects, if not [*a priori*]{} excluded from a cluster sample, would appear older than the rest of the cluster members. The effect that we consider to be the major cause of the large age spread we observe in the $\sigma$Ori cluster is variability. We make use of 2MASS J-band magnitudes and I-band magnitudes from the literature. PMS stars are known to be variable in $J, H$ and $K$ [@carpenteretal01; @carpenter01] and we found evidence of K-band variability in this sample (Sect.3.2). They found a dispersion among variable stars of about 0.1mag in $J$; if the effect in the I-band is of similar amplitude and the variability in the two bands is independent, an uncertainty in the $(I-J)$ colour of more than 0.1mag would not be surprising. If variability is related to the presence of a circumstellar disk, then the uncertainty could be even larger [@carpenteretal01; @carpenter01]. Indeed, from the 6 objects we found to be variable in the K-band (Sect.3.2), two objects have no I-band measurements; one object falls outside the model grid i.e. appears too young; of the remaining 3 objects, 2 have computed ages of 16Myr and 20Myr. Although limited by small number statistics, 3 out of 4 K-band variable objects have “anomalous” ages, compatible with the spread being caused by variability. Therefore, we believe that most of the spread we observe in Fig.\[ij\_cmd\] is not a real age spread, and that the likely age of the cluster is larger than 3.6Myr with an upper limit of 8Myr imposed by @osorio02. We will take into account the age uncertainty in our subsequent analysis. This short summary of the problems in determining the age of young clusters highlights the importance of using consistent age determinations, for instance when trying to infer an overall disk destruction timescale (Sect.5.3). Reddening --------- ![$IJH$ colour-colour diagram for the target sample. The solid lines are the empirical loci for main-sequence and giant stars (to spectral type M5) and the dashed lines are reddening band [from @bessell88 converted to the appropriate photometric systems]. Isochrones for 3Myr (upper dotted line) and 5Myr (lower dotted line) from @baraffe98 are also plotted. The double-circle symbols are objects that are found to exhibit $(K-L')$ excess in the $IJKL'$ colour-colour diagram (see Sect.4.1). The object with $(J-H)\sim 1.2$ is a bright cluster member and it is the only object with a large $(H-K_{\rm s})$ excess (Fig.\[colour\_colour\]). This diagram shows no evidence for significant reddening towards these cluster members.[]{data-label="reddening"}](MD837rv_fig2.eps) The average reddening towards the O-star $\sigma$Ori is quite low: $E(B-V)=0.05$ [e.g. @brown94]. However, a superficial analysis (see below) of traditional colour-colour diagrams seems to hint at some amount of reddening towards cluster members. Therefore, we have decided to investigate the question of reddening for the target sample. Fig.\[reddening\] shows the $IJH$ colour-colour diagram for the target sample. This diagram has been used with success to compute individual reddening to young stars in clusters [@lucas00; @thompson03]. The solid lines are the empirical loci for main-sequence and giant stars (from @bessell88 converted to 2MASS magnitudes using the transformations from @carpenter02) and the dashed lines are the reddening bands [@rieke85]. Traditionally, one attempts to de-redden individual objects with respect to main-sequence colours. But PMS stars have lower gravities than field dwarfs and thus populate the region between the main-sequence and giant loci. The dotted lines are isochrones from @baraffe98 for 3Myr and 5Myr; the target colours in the diagram are quite well represented by the isochrones. According to I. Baraffe (private communication) we should be cautious when using H-band magnitudes produced by their models because of possible shortcomings in the water line-list used (the H-band includes several strong water absorption bands). Nevertheless, very few objects show any evidence of significant reddening. The few objects that seem reddened actually are found to exhibit a $(K-L')$ excess that indicates the presence of a circumstellar disk (see Sect.4). We could use the reddening measured towards $\sigma$Ori, but it is so small that its effect is negligible — especially at IR wavelengths. Therefore, the results described in this paper were obtained without de-reddening the target colours and magnitudes. Infrared Observations ===================== K and L$'$-band imaging ----------------------- In order to compute meaningful $(K-L')$ excesses, it is essential that we are able to detect the stellar photospheres both in the K- and L$'$-bands. We used the 2MASS $K_{\rm s}$ magnitudes of the targets to estimate exposure times in the K-band and use the @baraffe98 evolutionary models (see previous section) to estimate L’-band magnitudes. Every target was detected in both bands. The observations in the two filters were consecutive, in order to minimise the effect of variability in the measured $(K-L')$ colours. The observations were performed at the UKIRT on the 17th and 19th January 2003, with the newly installed imager-spectrometer UIST. We observed with the broad-band K and L$'$ filters on the Mauna Kea Observatory Near-Infrared (MKO-NIR) system [@tokunaga02]. For both K- and L$'$-band observations, we used the 0.06$\arcsec$ pixel scale and windowed the detector to 512$\times$512 pixels to reduce overheads, resulting in a field of view of 31$\arcsec \times 31\arcsec$. Typically this results in no more than one cluster member per field, so that exposure times are tailored for each target. Adverse weather conditions meant that we only could observe for about 30% of the observing time allocated to this project; from our original target list we observed the 28 brightest (in $K_{\rm s}$) targets. On the first night conditions were stable and dry, with typical seeing of the order of 0.7arcsecond (measured as the full-width at half-maximum of the K-band images); in the second night conditions were less stable, with seeing of about 1 arcsecond. In the L$'$-band, on-target individual exposure times ranged from 1.5min to 15min; the brightest targets were observed on a 4-point dither pattern and the fainter targets on an 8-point dither pattern. At each dither position, 27 (sky-limited) exposures of 0.81s were co-added. An 8-point dither pattern is complete in approximately 3min (on-target) and the cycle was repeated until the requested exposure time was reached for each target. The frames were reduced and combined using ORAC-DR (the UKIRT data reduction and high level instrument control software), with dedicated recipes for the reduction of thermal imaging data. Sky background was removed by subtracting a median-average of the closest-in-time exposures at the different dither positions; flat-field frames were created by combining normalised object frames using the median at each pixel. Aperture photometry was performed on the mosaiced frames using the Starlink package GAIA (Graphical Astronomy and Image Analysis Tool), with apertures of 1.2$\arcsec$ and 1.8$\arcsec$ for the first and second nights respectively. The statistical signal-to-noise ratio of the L$'$-band photometry ranges from 50 to 9. To calibrate the observations we used the following standards [see @leggett03]: HD40335 ($L'=6.45$mag) and SAO112626 ($L'=8.56$mag), respectively for the 4-point and 8-point dither patterns. Standards were observed every one to two hours. In the K-band the targets are rather bright (Table\[obs\_table\]) so the exposure time was set to 10s. These observations were performed in a 5-point dither pattern. The K-band images were also reduced and analysed within ORAC-DR and GAIA. These observations were calibrated using the following standards: FS11 ($K=11.252$mag) and FS121 ($K=11.302$mag) from the list of UKIRT Faint Standards. The signal-to-noise ratio of the K-band photometry exceeds 30 and is more precise than the corresponding $L'$ measurement in all cases. K-band variability ------------------ For the targets in Table\[obs\_table\] we have two independent measurements of K-band magnitude: $K_{\rm s}$ from the 2MASS All-Sky Catalog of Point Sources [@cutri03] — data taken in October 1998 — and the $K$ from the present observations. To convert the 2MASS photometry to the MKO system we first converted from 2MASS to the old UKIRT system using the transformation in @carpenter01 and then from this system to the new MKO system using the transformation provided in @hawarden01. ![Histogram of the difference between MKO $K$ and 2MASS $K_{\rm s}$ measurements. The 2MASS magnitudes were converted to the MKO photometric system (see text). The typical 1-$\sigma$ error-bar for these differences is indicated; the gaussian error distribution highlights how many objects are likely to be variable. There is a systematic shift of about 0.04mag between the 2 sets of measurements. Most objects do not show evidence of significant variability, with only 3 objects varying by more than 0.2mag.[]{data-label="k_band"}](MD837rv_fig3.eps) Fig.\[k\_band\] shows the histogram of the difference between the two sets of measurements in the MKO system. We can see that there is a systematic effect in the $K$ magnitude of the order of 0.04mag, that is acceptable within the measurement uncertainties — it is not possible to say which of the steps of the transformation (2MASS to UKIRT or UKIRT to MKO) introduces this effect. Small differences can be easily explained by uncertainties in the calibrations and photometric system transformations. Nevertheless some objects show differences that clearly hint at K-band variability, namely targets 5, 6 and 8 and more marginally targets 3, 22 and 26. PMS stars are known to be variable at near-infrared wavelengths. @carpenteretal01 [@carpenter02] found that $\sim$30% of objects in Orion A and Chamaeleon I molecular clouds are variable. For more than 70% of variable PMS stars, variability is characterised by small-amplitude ($\sim$0.1mag), essentially colourless, fluctuations that can be attributed to rotational modulation by cool spots. However, when the observed fluctuations are of larger amplitude and occur both in magnitudes and colours, then a combination of effects related with the presence of circumstellar disks is a likely cause: hot spots, variable circumstellar extinction and varying disk properties (accretion rate and/or inner disk temperature). Objects 5, 6 and 8 could fall in this latter category (they also have large $(K-L')$ excesses) and objects 3, 22 and 26 in the former but our data really does not allow us to say much more. Analysis of the $(K-L')$ colours of the cluster members ======================================================= Colour-colour diagrams ---------------------- The most immediate method to search for circumstellar disks is to look for stars with $(H-K)$ or $(K-L)$ excesses in the $JHK$ or $JHKL$ colour-colour diagrams, that would indicate the presence of circumstellar dust material. Near-IR colours trace the warmer dust and the magnitude of a near-infrared excess depends critically on the position and temperature of the inner disk boundary. Therefore, disk frequencies determined from K-band excesses tend to be lower limits to the true frequency. At longer wavelengths, IR excesses grow rapidly and disk properties interfere less with the ability to detect disks. In particular for the L-band, disks remain optically thick over a wide range of disk masses, resulting in measurable $(K-L)$ excess down to very low disk masses [@wood02]. Therefore, a $(K-L)$ excess is a very reliable disk diagnostic [@haisch01; @wood02].  Fig.\[colour\_colour\] shows several colour-colour diagrams for the 28 cluster members. As a clarification, throughout this paper $(J-H)$ and $(H-K_{\rm s})$ colours are always from 2MASS, while $(K-L')$ is always from our UKIRT observations. The IRAS sources are known to possess disks so they will be excluded from the discussion on disk frequencies (so as not to introduce any bias), but they are shown here to illustrate the position in the diagrams of classical T Tauri stars. The diagrams show the position of the objects with respect to the main-sequence and giant star loci and their reddening bands (see Fig.\[reddening\]), converted to the 2MASS and MKO systems using transformations from @bessell88, @hawarden01 and @carpenter01. When analysing such diagrams, stars are considered to exhibit an excess when they appear to the right of the reddening vector for late-type stars. From the $JHK_{\rm s}$ diagram (top left) it can be seen that (with the exception of the IRAS sources) only one object exhibits a considerable K-band excess. This scenario changes dramatically when looking at the $JHKL'$ diagram (top right): maybe as many as 13/24 objects (or 54%) seem to have an L$'$-band excess. Simply counting stars with excess in such diagrams may still underestimate the true disk frequency: the reddening bands are rather broad, meaning that early-type stars with an excess might not be distinguishable from later-type stars with small reddening and no excess. The two diagrams at the bottom of Fig.\[colour\_colour\] largely remove this ambiguity. In the $HK_{\rm s}KL'$ and $IJKL'$ diagrams, the reddening bands are narrow and almost parallel to the empirical loci, so that stars with an excess in the L-band are clearly above these bands. The @baraffe98 isochrones for 3Myr and 5Myr predict which region of the diagrams would be populated by diskless PMS stars. The $IJKL'$ diagram (bottom right) is particularly clear, with two populations of stars: one with colours in between the giant and main-sequence loci and one of objects with colours more than 0.1mag above the empirical main-sequence locus. A legitimate question in the light of our concern about variability, is whether we can make use of $(I-J)$ colours that mix magnitudes obtained at different epochs ($J$ magnitudes are from 2MASS and $I$ magnitudes are from sources in the literature). The strength of this diagram, however, is that variability in $(I-J)$ would displace objects horizontally across the diagram (the same way as reddening) while an excess moves the objects almost vertically, so these effects act orthogonally. We can see from this diagram that, with respect to empirical main-sequence colours, 11/24 stars show no excess, 13/24 stars (or 54%) show excesses $\ga 0.1$mag and 8/24 stars (or 33%) show excesses $\ga 0.2$mag. $(K-L')$ excesses ----------------- --- --------- ----- ------ ----------- ------ ------ ----- ----- ----- spT H$\alpha$ ref. bs. exc. ror obs. exc. ror (Å) 1 4 .63 0.49 .05 0.86 0.80 .04 h98 2 7 .28 0.13 .05 0.55 0.44 .08 3.1 O02 3 7 .86 0.70 .05 1.09 0.98 .05 3.5 O02 4 – .62 0.22 1.05 0.59 5 – .86 0.46 1.30 0.84 6 8 .67 0.49 .05 1.35 1.22 .07 h98 7 8 .17 0.01 .05 0.05 0.08 .04 2.9 O02 8 0 .36 0.16 .05 0.93 0.78 .06 4.5 O02 9 0 .37 0.18 .05 0.60 0.45 .07 6.7 O02 0 0.5 .18 0.03 .05 0.24 0.07 .05 2.6 O02 1 7 .20 0.04 .05 0.18 0.07 .05 0.7 O02 2 8 .22 0.05 .05 0.18 0.06 .10 2.0 O02 3 2 .26 0.02 .05 0.21 0.01 .06 3.2 O02 4 4 .32 0.02 .05 0.51 0.17 .06 6.5 O02 5 5 .21 0.14 .05 0.37 0.02 .06 7.8 O02 6 4 .24 0.06 .05 0.35 0.01 .11 9.6 O02 7 5.5 .33 0.04 .05 0.63 0.21 .05 0.0 O02 8 6$\dag$ .40 0.01 .07 0.72 0.28 .08 9 3.5 .30 0.01 .05 0.54 0.22 .06 1.2 O02 0 3 .21 0.06 .05 0.16 0.12 .06 4.0 O02 1 2 .26 0.02 .05 0.27 0.06 .06 4.1 O02 2 3 .19 0.09 .05 0.51 0.22 .07 3.6 O02 3 5$\dag$ .35 0.01 .07 0.52 0.12 .07 4 4$\dag$ .26 0.04 .07 0.30 0.04 .06 5 3.5 .31 0.02 .05 0.52 0.21 .10 4.2 O02 6 4$\dag$ .30 0.01 .07 0.33 0.01 .08 7 5$\dag$ .30 0.05 .07 0.59 0.20 .11 8 4$\dag$ .27 0.03 .07 0.77 0.43 .12 --- --------- ----- ------ ----------- ------ ------ ----- ----- ----- : $(H-K_{\rm s})$ and $(K-L')$ excesses for the target sample. Column 2 gives the spectral types from the literature or, if none is available, estimated from photometric colours (see text). Columns 3 and 6 are the measured $(H-K_{\rm s})$ and $(K-L')$ colours; columns 4$-$5 and 7$-$8 are excesses with respect to photospheric colours and respective uncertainties. Column 9 is the measured EW\[H$\alpha$\] (in Å). Column 10 provides the references for the spectral types and/or EW\[H$\alpha$\] (ZO02; @kholopov98, Kh98).[]{data-label="excess_table"} [$\dag$]{} objects with spectral types estimated from their colours.\ PMS objects are young and have lower surface gravities than the field dwarfs used to calibrate the empirical relations used in the previous section [@luhman99]. We investigated if theoretical models (see Sect.2.2) could be used to predict K- and L$'$-band magnitudes for our sample. If, in the $IJKL'$ diagram (Fig.\[colour\_colour\]), we compare the colours of objects with no apparent excess with the theoretical isochrones of @baraffe98, the model $(K-L')$ colours seem to be too blue (even bluer than corresponding giant colours), especially for earlier spectral types ($(I-J) < 1.7$mag). In terms of surface gravity, we would expect PMS stars to have colours between main-sequence and giant stars, therefore we opt to use empirical spectral type calibrations to compute $(K-L')$ excesses. Furthermore, a disk frequency based on the comparison with model colours would not be readily comparable to other surveys that tend to use the main-sequence calibration. If model colours were used, the $IJKL'$ diagrams implies that a larger fraction ($\sim$80%) of objects would have an $(K-L')$ excess. ![$(K-L')$ and $(H-K_{\rm s})$ excesses. These excesses were computed with respect to the targets’ predicted colours from their spectral types. The filled circles are objects with known spectral types [@osorio02] whereas open circles represent objects with spectral types estimated from their $(I-J)$ and $(H-K_{\rm s})$ colours. The triangles represent the IRAS objects: filled for objects with known spectral type and open for lower limits to the excess (see text). The errors are the uncertainties in the measured and intrinsic $(K-L')$ combined in quadrature (see text); the error bars in $K$ are typically the size of the plotting symbol.[]{data-label="excess_spt"}](MD837rv_fig5.eps){height="16cm"} Spectral types are known for 18 objects in the sample . Spectral types are also known for 2 of the IRAS sources. For the 6 objects identified by @kenyon03 spectral types are not known. Their $(I-J)$ and $(H-K_{\rm s})$ colours may be used to estimate spectral types, as none of these objects show considerable excess in the K-band. We use tabulated relations between spectral type and colours (again from @bessell88) to define linear relations between spectral type and each colour (these relations are only valid for spectral types later than M2). These are then interpolated and averaged to obtain the spectral type for the sources. The results are in column2 in Table\[excess\_table\] (indicated by $\dag$). We apply the same procedure to the stars with known spectral types in order to test its validity. For the 10 sources with known spectral type later than M2, we obtain the following: for 6/10 stars we compute the correct spectral type within 0.5subclass, for 3/10 within 1subclass and for 1/10 within 1.5subclass. So we are confident that our computed spectral types are mostly correct to within 1subclass. Even though these may not be the most accurate spectral type determinations, they are easily adequate for the purposes of computing $(K-L')$ excesses. Between spectral types M2 and M6, one subclass is equivalent to at most a variation of 0.055mag in photospheric $(K-L')$ colour and 0.05mag in $(H-K_{\rm s})$. We adopt these values as estimates of the errors in the excesses introduced by this procedure; spectral types from the literature are uncertain by 0.5 subclass. For each target we have now spectral types and we have computed $(H-K_{\rm s})$ and $(K-L')$ excesses that are listed in columns4&7 in Table\[excess\_table\]. For the 2 IRAS sources without known spectral types, we compute lower limits for their excesses by assigning them spectral type M6 (even though their brightness clearly indicates earlier spectral type) and photospheric colours of respectively $(K-L')=0.46$mag and $(H-K_{\rm s})=0.40$mag. In Fig.\[excess\_spt\] we plot the computed excesses against K-band magnitude for both $(K-L')$ and $(H-K_{\rm s})$. The error bars are obtained by adding in quadrature the photometric uncertainty in the colours with the uncertainty introduced by the spectral type determination. As expected all 4 IRAS objects have very large excesses in the L$'$-band, consistent with their mid-IR colours [@oliveira03b]. We consider that an object has a circumstellar disk if it exhibits an excess detected at a 2-$\sigma$ level. Thus 11/24 (or 46%) objects show a $(K-L')$ excess. If we arbitrarily adjust for the systematic effect discussed on Sect.3.2, then the computed excesses would be larger by about 0.04mag and 14/24 objects would have a significant excess. Discussion ========== How representative is this sample? ---------------------------------- The sample we discuss here is not complete, but we believe it is representative. As we discuss below, there is no serious bias either for or against the detection of disks in this sample (apart from the 4 IRAS sources). ### X-ray selection bias The equivalent width (EW) of the H$\alpha$ emission line has traditionally been used to classify T Tauri stars (TTS): classical T Tauri stars (CTTS) are defined as having large EW\[H$\alpha$\] as evidence of circumstellar accretion and thus of the presence of a circumstellar disk; weak-lined T Tauri stars (WTTS) have chromospheric EW\[H$\alpha$\] thus showing no signatures of accretion. CTTS are found to be underluminous in X-rays when compared with WTTS in the same star-forming regions [e.g. @neuhauser95; @flaccomio03]. This would imply that surveys for circumstellar disks in X-ray selected samples may be biased towards objects with no accretion disk signatures, depending on the sensitivity of the X-ray surveys. But many PMS stars classified as WTTS based on their H$\alpha$ emission were actually found to have circumstellar disks. @haisch01b found that a large fraction of WTTS in IC348 have considerable $(K-L)$ excesses, i.e. WTTS are not necessarily naked TTS. Furthermore, @preibisch02 provided statistically significant evidence that accreting stars in IC348 have lower X-ray luminosity when compared with non-accreting objects, but found [*no evidence*]{} for a difference between stars with or without $(K-L)$ excess. This suggests that X-ray selection will not impair the detection of circumstellar disks in the L-band, even though it might be biased against strongly accreting objects. In fact, as we discuss in the next section, about half of the X-ray selected cluster members have large $(K-L')$ excesses, similar to the overall disk fraction. ### Spectroscopic selection bias Another potential source of bias has to do with the spectroscopic identifications of cluster members. The presence of the Li[i]{} 6708Å feature in stellar spectra is a sure sign of youth at least in the mass range we are considering here. However, photospheric spectral lines in CTTS can be heavily veiled at optical wavelengths [e.g. @gullbring98]. In particular, the hot continuum attributed to disk accretion can fill-in the Li[i]{} 6708Å line [e.g. @magazzu92]. The sample was selected on the basis of the detection of the lithium feature, therefore our estimated disk frequencies could be regarded as a lower limit. @kenyon03 found a few objects in their sample of cluster candidates that show no lithium in their spectra but have radial velocities and Na[i]{} doublet strengths consistent with cluster membership. However, their measured H$\alpha$ line widths are not consistent with accreting PMS objects [@white03]. No evidence has been found for heavily-veiled strongly-accreting cluster members, therefore we consider that the spectroscopic selection of cluster members is not a significant source of bias. {height="8cm"} L$'$-band excess versus other disk indicators --------------------------------------------- In this section we compare $(K-L')$ excesses with other disk indicators. @osorio02 describe spectra of 18 of the stars in our sample (Table\[obs\_table\]); in particular they have measured equivalent widths for the H$\alpha$ emission and several forbidden emission lines. A classification criterion for PMS stars based on EW\[H$\alpha$\] has to take into account the different levels of chromospheric H$\alpha$ emission observed for different spectral types [@martin98]. @white03 have proposed the following (empirical and spectral-type dependent) classification: a PMS star is classified as a CTTS if EW\[H$\alpha$\]$>$3Å for K0$-$K5 stars, EW\[H$\alpha$\]$>$10Å for K7$-$M2.5 stars, EW\[H$\alpha$\]$>$20Å for M3$-$M5.5 stars and EW\[H$\alpha$\]$>$40Å for M6$-$M7.5 stars. This classification was devised using high-resolution spectra, so when applying it to intermediate- and low-resolution spectra (as is the case here) we have to take into account that EW\[H$\alpha$\] is probably overestimated due to blending with a nearby TiO feature. The presence of forbidden lines (e.g. \[O[i]{}\], \[N[ii]{}\] and \[S[ii]{}\]) in the spectrum of a PMS star is evidence of jets and outflows that are also related with accretion processes and circumstellar disks [e.g. @edwards87]. In Fig.\[halpha\] (left), we plot EW\[H$\alpha$\] and spectral types for the objects from @osorio02; on the right we plot EW\[H$\alpha$\] against $(K-L')$ excesses. We can see that 4 objects show EW\[H$\alpha$\] in excess of what is expected for their spectral types; these 4 objects have $(K-L')$ excesses of the order of 0.17$-$0.98mag, consistent with the presence of circumstellar disks (objects 3, 14, 17 and 19 in Tables\[obs\_table\] & \[excess\_table\]). Of these 4 objects, 3 exhibit spectra with forbidden line emission (objects 3, 14 and 19). Of the 10 objects that were X-ray selected, only one object has a large EW\[H$\alpha$\]; half of these objects have a $(K-L')$ excess. The IRAS sources have, as expected, large $(K-L')$ excesses, as they also have large mid-IR excesses [@oliveira03b]. There are also 3 objects that have $(K-L')$ excesses larger than 0.4mag, but do not show large EW\[H$\alpha$\]. This again illustrates that $(K-L')$ is the most efficient and robust disk indicator. H$\alpha$ emission is known to be extremely variable [@guenther97] and objects that have $(K-L')$ excesses indicative of circumstellar disks could alternate from episodes of high accretion activity (CTTS state) to episodes of undetected accretion activity (WTTS). More interestingly, if we interpret this as an evolutionary sequence (i.e. the disk-star system evolves from an accreting system, through a weakly or non-accreting system, to a naked PMS star), this could be an indication that circumstellar disks can survive for some time after accretion onto the stellar surface has stopped. The $\sigma$Ori cluster is at a crucial stage of disk destruction (see next section) and we would indeed expect to find a mixture of objects that are still accreting and others where accretion has strongly diminished or ceased. Comparison with other young clusters ------------------------------------ The most complete analysis of clusters’ disk frequencies is described in @haisch01 [ and references therein]. They compile the results of several cluster surveys in the K- and L-bands. These authors have consistently used the same method to determine which objects possess a $(K-L)$ excess: they place cluster members in a $JHKL$ colour-magnitude diagram — only those objects with $K$ magnitudes brighter than the completeness limit of the $L$ survey are considered, to assure that stellar photospheres are detected in both bands; they count the stars that lie to the right of the reddening vector that passes through the position in this diagram of an M5 main-sequence star. We can apply exactly the same procedure in Fig.\[colour\_colour\] (top right): 13/24 stars (54$\pm$15)%[^2] are to the right of the reddening band. This result depends on the adopted reddening law — the reddening law defines the position of the boundary between objects with and without excesses — but the different determinations are within the quoted statistical errors [@haisch01]. Depending on the spectral type distribution of the sample, this method might underestimate the true disk frequency particularly for earlier spectral types [@lada00]. According to this technique, the $\sigma$Ori cluster has a (54$\pm$15)% $JHKL'$-excess disk frequency, at an age of 3$-$8Myr. How does this compare with other clusters? Such comparison can be done by placing the $\sigma$Ori cluster in Fig.1 from @haisch01 that plots the fraction of $JHKL$ excess objects against stellar age for several young clusters and associations. In terms of its age, the $\sigma$Ori cluster can be compared with NGC2264 (age$\sim$3.2Myr, disk frequency 52%$\pm$4%) and NGC2362 (age$\sim$5Myr, disk frequency 12%$\pm$10%). Crucially, the measurements for these two clusters — particularly NGC2362 — allowed the authors to estimate that the overall disk lifetime is about 6Myr, with 50% of the disks dispersed by about 3Myr. In this scenario, the $\sigma$Ori cluster can play a key role to better constrain disk destruction timescales. In @haisch01 analysis, the ages of the clusters were determined using different PMS models and these authors estimate that this introduces an overall systematic uncertainty in the ages of the order of 1.2Myr. The age of $\sigma$Ori is also not well established (see discussion Sect.2.2): 3$-$5Myr is favoured by several authors (@bejar99 [@oliveira02; @jayawardhana03]; this work) with an upper limit of 8Myr [@osorio02]. If the age of the cluster is $\la$4Myr then our estimate of disk frequency is consistent with the above mentioned timescales. However, if the cluster is older than 4Myr, then our result suggests a slower disk destruction, i.e. a longer overall disk lifetime. The NGC2264 and NGC2362 surveys are for objects more massive than $\sim$0.85M$_{\sun}$ (i.e. earlier spectral types) while our sample populates the range 0.13$-$1.0M$_{\sun}$. As mentioned above, this technique might be at fault for earlier spectral types. Furthermore, if disk-destruction timescales are mass-dependent then one has to be careful when comparing these results. @lyo03 analysed $(K-L)$ excesses in the $\sim$9Myr, sparsely-populated $\eta$Chamaeleontis cluster. They found that of the 12 late-type stars in the central part of the cluster 7 objects have $(K-L)$ excess. Their mass range is comparable to the mass range of our sample, but even so the disk frequency in $\eta$Cha is remarkably high for its age. Environmental effects might explain such result: disk destruction timescales might be controlled also by the stellar density in the star-forming region and by the photoevaporation power of nearby O-stars. If this is the case, then $\eta$Cha cannot be straight forwardly compared with the other clusters. Comparison with disk model predictions -------------------------------------- @wood02 investigates the observational signatures of circumstellar disks in a simple evolutionary scenario by which disk mass (of small particles) decreases with time (for a stellar effective temperature of 4000K, spectral type K7-M0). Except for high disk masses, the main contribution to the spectral energy distribution (SED) is from reprocessing of starlight (or disk irradiation), as inferred from observations [@hartmann98]. For the $(K-L)$ excess (or $\Delta(K-L)$) their circumstellar disk model predicts that: [*i*]{}) large excesses ($\Delta(K-L) \ga$ 0.7mag) can only be achieved with the contribution of accretion luminosity from massive disks; [*ii*]{}) for passive disks, $\Delta(K-L)$ is insensitive to disk mass over several orders of magnitude; [*iii*]{}) for disk masses lower than 10$^{-7}$M$_{\sun}$, $\Delta(K-L)$ decreases rapidly. In our sample, 9 of the objects have spectral types K7$-$M0.5 and 3 of these objects have large excesses ($\ga$ 0.7mag) that can hint at a contribution from accretion luminosity to the SEDs. Only one of these objects shows signatures of accretion in its spectrum (Sect.5.2) but variability likely plays an important role. The objects with later spectral types tend to have modest excesses ($\la$ 0.4mag), consistent with disk irradiation. Summary ======= In this paper we present new K- and L$'$-band photometry for a representative sample of $\sigma$Orionis cluster members. Different methods and disk-excess criteria will provide somewhat different disk frequencies. We computed a disk frequency for this cluster of (54$\pm$15)%, if measured directly from the $JHKL'$ colour-colour diagram, or (46$\pm$14)%, if excesses are computed with respect to predicted photospheric colours (according to the objects spectral types, 2-$\sigma$ excess detections). Therefore, when trying to determine a disk destruction timescale, one has to be consistent in the methods used. An overall disk lifetime of 6Myr has been proposed, but this relies heavily on the age and disk frequency derived from NGC2362 (age$\sim$5Myr and disk-frequency$\sim$12%, @haisch01). The disk frequency for the $\sigma$Ori cluster could increase support for this timescale, but we find that the crucial factor is not so much the derived disk frequency, but the [*age determination*]{} of the cluster. As is the case for most young clusters [@hartmann01], the age of the $\sigma$Ori cluster is uncertain, with a likely age in the range 3$-$5Myr [@oliveira02; @osorio02; @jayawardhana03]. If the cluster age is about 3Myr then it lends support to the above mentioned disk lifetimes. However, if the cluster is older than 4Myr then it points towards slower disk destruction timescales. If disk dissipation timescales are mass dependent then the disk frequencies derived for the different clusters are not so readily comparable. Another factor that is probably relevant is the environment of the star forming region, in particular the rates of close stellar encounters and the amount of ionizing radiation produced by massive (O-type) cluster members. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the staff of the United Kingdom Infrared Telescope (UKIRT) for their support during the observing run. 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--- abstract: 'In this paper we are interested in parallels to the classical notions of special subsets in ${\mathbb{R}}$ defined in the generalized Cantor and Baire spaces ($2^\kappa$ and $\kappa^\kappa$). We consider generalizations of well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, $\sigma$-sets, $\gamma$-sets, sets with Menger, Rothberger or Hurewicz property, but also of some less-know classes like $X$-small sets, meagre additive sets, Ramsey null sets, $T''$-sets, Marczewski, Silver, Miller and Laver-null sets. We also show some relations between those classes.' author: - 'Michał Korch, Tomasz Weiss' title: 'Special subsets of the generalized Cantor space $2^\kappa$ and generalized Baire space $\kappa^\kappa$' --- Introduction and preliminaries ============================== Many classical notions of special subsets of $2^\omega$ can be generalized to the case of the generalized Cantor space $2^\kappa$. In this paper we study those classes of sets in this setting. In some cases, when it seems more appropriate, we also study such classes in the generalized Baire space $\kappa^\kappa$. It turns out that many the properties of subsets of $2^\omega$ or of $\omega^\omega$ can be easily proved in $2^\kappa$ or $\kappa^\kappa$, although sometimes one has to use some additional set-theoretic assumptions. Next we deal with less common classes of small sets in $2^\kappa$ and $\kappa^\kappa$. Special subsets of the real line {#intro-special} -------------------------------- In theory of special subsets of the real line we deal with sets which are very small. We recall below some notions which will be generalized later in this paper. ### Special subsets related to category Among classes of special subsets of the real line, the class of perfectly meager sets plays an important role. A set is [[**perfectly meager**]{}]{} if it is meager relative to any perfect set, and we denote it by ${\boldsymbol{\text{P}{{\mathcal{M}}}}}$ (this concept first appeared in [@nl:pb]). A set $A$ is called [[**strongly null**]{}]{} (strongly of measure zero) if for any sequence of positive $\varepsilon_{n}>0$, there exists a sequence of open sets $\left<A_{n}\right>_{n\in\omega}$, with ${\text{diam}}A_{n}<{\varepsilon}_{n}$ for $n\in\omega$, and such that $A\subseteq\bigcup_{n\in\omega}A_{n}$. We denote the class of such sets by ${\boldsymbol{\text{S}{{\mathcal{N}}}}}$. The idea was introduced for the first time in [@eb:cemn], and [[**Borel conjectured**]{}]{} that all ${\boldsymbol{\text{S}{{\mathcal{N}}}}}$ sets are countable. This hypothesis turned out to be independent from ZFC (see [@rl:cbc]). It is easy to see that a set $A$ is strongly null if and only if for any sequence of positive $\varepsilon_{n}>0$, there exists a sequence of open sets $\left<A_{n}\right>_{n\in\omega}$, with ${\text{diam}}A_{n}<{\varepsilon}_{n}$ for $n\in\omega$, and such that $$A\subseteq\bigcap_{m\in\omega}\bigcup_{n>m}A_{n}.$$ Galvin, Mycielski and Solovay (in [@fgjmrs:smzs]) proved that a set $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}$ (in $2^{\omega}$) if and only if for any meagre set $B$, there exists $t\in 2^{\omega}$ such that $A\cap(B+t)={\varnothing}$. We shall say that a set $L\subseteq 2^{\omega}$ is a [[**$\kappa$-Lusin set**]{}]{} (for $\omega<\kappa\leq 2^\omega$) if for any meagre set $X$, $|L\cap X|<\kappa$, but $|L|\geq \kappa$. An $\aleph_1$-Lusin set is simply called a [[**Lusin set**]{}]{}. This idea was introduced independently in [@nl:pb] and [@pm:tkm]. The existence of a Lusin set is independent from ZFC. It is easy to see that under CH such a set exists. Indeed, enumerate all closed nowhere dense sets and inductively take a point form a complement of each such set distinct from all the points chosen so far. The same can be easily done if ${\text{cov}}({\boldsymbol{{{\mathcal{M}}}}})={\text{cof}}({\boldsymbol{{{\mathcal{M}}}}})=\aleph_1$ (see e.g. [@lb:srl]). A set $A$ is called [[**meagre-additive**]{}]{} ($A\in {\boldsymbol{{{\mathcal{M}}}}}^{*}$) if for any meagre set $X$, $A+X$ is meagre (see e.g. [@tw:manascs] and [@tbhj:stsrl]). The following [[**characterization of meagre-additive sets**]{}]{} is well-known. A set $X\in {\boldsymbol{{{\mathcal{M}}}}}^*$ ([@tbhj:stsrl]\[Theorem 2.7.17\]) if and only if for every increasing $f\in \omega^\omega$, there exists $g\in \omega^\omega$ and $y\in 2^\omega$ such that for all $x\in X$, there exists $m\in \omega$ such that for every $n>m$, there exists $k_n\in\omega$ with $g(n)\leq f(k_n)<f(k_n+1)\leq g(n+1)$ and such that $$x{\mathord{\upharpoonright}}[f(k_n), f(k_n+1))=y{\mathord{\upharpoonright}}[f(k_n), f(k_n+1)).$$ ### Trees Fix any set $A$ and an ordinal number $\xi$. Given a sequence $t\in 2^\alpha$ with $\alpha<\xi$, we denote $\alpha={\text{len}}(t)$. A set $T\subseteq A^{<\xi}$ will be called a [[**tree**]{}]{} if for all $t\in T$ and $\alpha< {\text{len}}(t)$, $t{\mathord{\upharpoonright}}\alpha\in T$ as well. A branch in a tree is a maximal chain in it. For a tree $T\subseteq A^{<\xi}$, let $$[T]=\{x\in A^\xi\colon \forall_{\alpha<\xi} x{\mathord{\upharpoonright}}\alpha\in T\}.$$ A node $s\in T\subseteq A^{<\xi}$ is called a [[**branching point**]{}]{} of $T$ if $s^\frown a,s^\frown b\in T$ for some distinct $a,b\in A$. The set of all branching points of a tree $T$ is denoted by ${\text{Split}}(T)$. For $\alpha<\xi$, $t\in {\text{Split}}_\alpha(T)$ if $\langle\{s\subsetneq t\colon s\in {\text{Split}}(T)\},\subseteq\rangle$ is order isomorphic with $\alpha$. A tree $T\subseteq A^{<\xi}$ is [[**perfect**]{}]{} if for any $t\in T$, there exists $s\in T$ such that $t\subseteq s$ and $s\in{\text{Split}}(T)$. Tree $T\subseteq A^\xi$ is pruned, if its every maximal chain is of length $\xi$. Notice that if $T\subseteq A^\omega$, then a set $C\subseteq A^\omega$ is closed if an only if $C=[T]$ for a pruned tree $T$. We denote such tree by $T_C$. Moreover, a set $P\subseteq 2^\omega$ is perfect if and only if $T_P$ is a perfect tree. Notice also that a closed set $C\subseteq \omega^\omega$ is compact if and only if there exists a sequence $\langle n_i\rangle_{i\in\omega}$ such that if $x\in C$, then $x(i)<n_i$ for all $i\in\omega$. A perfect tree $T\subseteq A^\omega$ is called a [[**Silver perfect tree**]{}]{} if $$\forall_{w,v\in T}\left( {\text{len}}(v)={\text{len}}(w)\Rightarrow\forall_{j\in A}(w^\frown j\in T\Rightarrow v^\frown j\in T)\right).$$ A perfect tree $T\subseteq \omega^{<\omega}$ is called a [[**Laver perfect tree**]{}]{} if there exists $s\in T$ such that for all $t\in T$, either $t\subseteq s$, or $$\left|\left\{n\in\omega\colon t^\frown n\in T\right\}\right|=\aleph_0.$$ Similarly, a perfect tree $T\subseteq \omega^{<\omega}$ is called a [[**Miller perfect tree**]{}]{} if for every $s\in T$ there exists $t\in T$ such that $s\subseteq t$, and $$\left|\left\{n\in\omega\colon t^\frown n\in T\right\}\right|=\aleph_0.$$ A set $P\subseteq 2^\omega$ is called [[**Silver perfect set**]{}]{} if $T_P$ is a Silver perfect tree. Analogously, a set $P\subseteq \omega^\omega$ is called [[**Laver (respectively, Miller) perfect set**]{}]{} if $T_P$ is a Laver (respectively, Miller) perfect tree. ### Other notions of special subsets An open cover ${{\mathcal{U}}}$ of a topological space $A$ is [[**proper**]{}]{} if $A\notin {{\mathcal{U}}}$. From now on we assume that all considered covers are proper. An open cover ${{\mathcal{U}}}$ of a set $A$ such that for any $C\in [A]^{<\omega}$ there exists $U\in{{\mathcal{U}}}$ such that $C\subseteq U$, is called an [[**$\omega$-cover**]{}]{}, and we call it a [[**$\gamma$-cover**]{}]{} if $$A\subseteq \bigcup_{n\in\omega}\bigcap_{m\geq n} U_m.$$ The family of all $\omega$-covers (respectively, $\gamma$-covers) of $A$ is denoted by $\Omega(A)$ (respectively, $\Gamma(A)$). The family of all open covers of $A$ is denoted by ${{\mathcal{O}}}(A)$. The underlying set is often omitted in this notation if it is clear from the context. In this paper we consider the analogues of the following special subsets of the real line (or the Cantor space) (see [@am:ssrl] and [@lb:srl]): set concentrated on a set $C$ : i.e. a set $A$ such that $A\setminus U$ is countable for every open $U$ with $C\subseteq U$ ([@fr:epcdpl]). Notice that every set concentrated on a countable set is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}$, $\lambda$-set : i.e. a set $A$ such that every countable $B\subseteq A$ is a relative $G_\delta$-set ([@kk:fes]). Every $\lambda$-set is perfectly meagre, $\lambda'$-set : i.e. a set $A$ such that for every countable $B$, $A\cup B$ is a $\lambda$-set. Obviously, every $\lambda'$-set is a $\lambda$-set, $\gamma$-set : i.e. a set $A$ such that if for every open $\omega$-cover ${{\mathcal{U}}}$, there exists ${{\mathcal{V}}}\subseteq{{\mathcal{U}}}$ which is a $\gamma$-cover ([@jgzn:spc]), Ramsey null set : i.e a set $A$ such that for any $n\in\omega$, $s\in 2^n$ and $S\in [\omega\setminus n]^\omega$, there exists $S'\in [S]^\omega$ such that $[s,S']\cap A={\varnothing}$, where if $s\in 2^n$, $n\in\omega$ and $S\in [\omega\setminus n]^\omega$, then $$[s,S]=\{x\in 2^\omega\colon s^{-1}[\{1\}]\subseteq x^{-1}[\{1\}]\subseteq s^{-1}[\{1\}]\cup S\land |x^{-1}[\{1\}]\cap S|=\omega\}$$ (see [@sp:crs]), T’-set : i.e. a set $A$ such that there exists a sequence $\langle l_n\rangle_{n\in\omega}\in\omega^\omega$ such that for every increasing sequence $\langle d_n\rangle_{n\in\omega}\in\omega^\omega$ with $d_0=0$, there exists a sequence $\langle e_n\rangle_{n\in\omega}\in \omega^\omega$, and $$H_{n}\in \left[2^{d_{e_n+1}\setminus d_{e_n}}\right]^{\leq l_{e_n}},$$ for all $n\in\omega$ such that $$A\subseteq \left\{x\in 2^\omega\colon \forall_{m\in\omega }\exists_{n>m} x{\mathord{\upharpoonright}}(d_{e_n+1}\setminus d_{e_n})\in H_n \right\}$$ (defined in [@antw:rpsssas] and also introduced in different context in [@mr:fptts]), $s_0$-set : i.e. a set $A\subseteq 2^\omega$ such that for any perfect set $P$ there exists a perfect set $Q\subseteq P$ such that $A\cap Q={\varnothing}$ ([@em:cfscce]), $v_0$-set : i.e. a set $A\subseteq 2^\omega$ such that for every Silver perfect set $P$, there exists a Silver perfect set $Q\subseteq P$ such that $Q\cap A={\varnothing}$ (see [@mkantw:ssrtfn]), $l_0$-set : i.e. a set $A\subseteq \omega^\omega$ such that for every Laver perfect set $P$, there exists a  Laver perfect set $Q\subseteq P$ such that $Q\cap A={\varnothing}$ (see [@mktw:ssrtfn]), $m_0$-set : i.e. a set $A\subseteq \omega^\omega$ such that for every Miller perfect set $P$, there exists a  Miller perfect set $Q\subseteq P$ such that $Q\cap A={\varnothing}$ (see [@mktw:ssrtfn]). ### Selection principles If ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are families of covers of a topological space $X$, then $X$ has [[**$S_1({{\mathcal{A}}},{{\mathcal{B}}})$ principle**]{}]{} if for every sequence $\langle {{\mathcal{U}}}_n\rangle_{n\in\omega}\in{{\mathcal{A}}}^\omega$, there exists ${{\mathcal{U}}}=\{ U_n\colon n\in\omega\}$ with $U_n\in{{\mathcal{U}}}_n,$ for all $n\in\omega$ such that ${{\mathcal{U}}}\in{{\mathcal{B}}}$. $X$ has [[**$U_{<\omega}({{\mathcal{A}}},{{\mathcal{B}}})$ principle**]{}]{} if for every sequence $\langle {{\mathcal{U}}}_n\rangle_{n\in\omega}\in{{\mathcal{A}}}^\omega$ such that for every $n\in\omega$ if ${{\mathcal{W}}}\subseteq {{\mathcal{U}}}_n$ is finite, then ${{\mathcal{W}}}$ is not a cover, there exists $\langle {{\mathcal{U}}}_n\rangle_{n\in\omega}$ such that ${{\mathcal{U}}}_n\in [{{\mathcal{U}}}]^{<\omega}$, and $\{\bigcup {{\mathcal{U}}}_n\colon n\in\omega\}\in{{\mathcal{B}}}$. The covering principles were first systematically studied in [@ms:cocrt]. It can be proven that a set $X$ is a $\gamma$-set if and only if $X$ satisfies $S_1(\Omega,\Gamma)$. A set $X$ is said to have the [[**Menger property**]{}]{} ([@km:up]) if it satisfies $U_{<\omega}({{\mathcal{O}}},{{\mathcal{O}}})$. It has the [[**Hurewicz property**]{}]{} ([@wh:fsf]) if it satisfies $U_{<\omega}({{\mathcal{O}}},\Gamma)$. Finally, it has the [[**Rothberger property**]{}]{} ([@fr:vec]) if it satisfies $S_1({{\mathcal{O}}},{{\mathcal{O}}})$. Introducing the generalized Cantor space $2^\kappa$ and the generalized Baire space $\kappa^\kappa$ {#intro-gen} --------------------------------------------------------------------------------------------------- In this paper we consider the generalized Cantor space $2^\kappa$ and generalized Baire space $\kappa^\kappa$ for an infinite cardinal $\kappa>\omega$ and study special subsets of these spaces. In the recent years the theory of the generalized Cantor and Baire spaces was extensively developed (see, e.g. [@pllmps:hdgbs], [@sfthvk:gdstct], [@sfgl:nii], [@gl:afnrprl], [@gl:gssm], [@ss:pnii], [@sssc:grrfic], [@sf:hdsti], [@sf:idst], [@sf:ccu] and many others). An important part of the research in this subject is an attempt to transfer the results in set theory of the real line to $2^\kappa$ and $\kappa^\kappa$ (the list of open questions can be found in [@glblis:qgbs]). Despite the rapid development in this theory, the authors are not aware of any thorough research in the subject of special subsets in $2^\kappa$ or $\kappa^\kappa$. Known results are related mainly to the ideal of strongly null sets (see [@ah:nsgbs] and [@ahss:smzs]). Throughout this paper, unless it is stated otherwise, we assume that $\kappa$ is an uncountable regular cardinal number and $\kappa>\omega$. ### Preliminaries We consider the space $2^\kappa$, called [[**$\kappa$-Cantor space (or the generalized Cantor space)**]{}]{}, endowed with so called bounded topology with a base $\{[x]\colon x\in 2^{<\kappa}\}$, where for $x\in 2^{<\kappa}$, $$[x]=\{f\in 2^\kappa\colon f{\mathord{\upharpoonright}}{\text{dom}}x= x\}.$$ Similarly, the space $\kappa^\kappa$ along with bounded topology with a base $\{[x]\colon x\in \kappa^{<\kappa}\}$, where for $x\in \kappa^{<\kappa}$, $$[x]=\{f\in \kappa^\kappa\colon f{\mathord{\upharpoonright}}{\text{dom}}x= x\}.$$ is called [[**$\kappa$-Baire space (or the generalized Baire space)**]{}]{}. Throughout this paper let ${{\mathfrak{K}}}\in\{2,\kappa\}$. Therefore, ${{\mathfrak{K}}}^\kappa$ denotes the generalized Cantor space or the generalized Baire space. If we additionally assume that $\kappa^{<\kappa}=\kappa$, the above base has cardinality $\kappa$. This assumption proves to be very convenient when considering the generalized Cantor space and the generalized Baire space, and is assumed throughout this paper, unless stated otherwise (see e.g. [@sfthvk:gdstct]). The space $2^\kappa$ will also be treated as a vector space over ${\mathbb{Z}}_{2}$. In particular, for $A,B\subseteq 2^{\kappa}$, let $A+B=\{t+s\colon t\in A, s\in B\}$. Let ${\boldsymbol{0}}\in 2^\kappa$ be such that ${\boldsymbol{0}}(\alpha)=0$ for all $\alpha<\kappa$, let ${\boldsymbol{1}}\in 2^\kappa$ be such that ${\boldsymbol{1}}(\alpha)=1$ for all $\alpha<\kappa$, and let ${\boldsymbol{Q}}=\{x\in 2^\kappa\colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa} x(\beta)=0\}$. Similarly, if $x,y\in \kappa^\kappa$, then $x+y\in \kappa^\kappa$ is such that $x(\alpha)+y(\alpha)=(x+y)(\alpha)$, for all $\alpha<\kappa$. Notice that if $x\in {{\mathfrak{K}}}^\alpha$, with $\alpha<\kappa$, then $${{\mathfrak{K}}}^\kappa\setminus [x] = \bigcup_{\beta<\alpha}\bigcup_{a\in {{\mathfrak{K}}}\setminus \{x(\beta)\}} \left[x{\mathord{\upharpoonright}}\beta\,^\frown a \right].$$ So, $2^\kappa\setminus [x]$ is also open. Therefore, the bases defined above consist of clopen sets. Notice also that an intersection of less than $\kappa$ of basic sets is a basic set or an empty set. Therefore, an intersection of less than $\kappa$ open sets is still open. Notice also that there are $2^{\kappa}$ closed sets in those spaces. Additionally, (under the assumption $\kappa^{<\kappa}=\kappa$) there exists a family ${{\mathcal{F}}}$ of subsets of $\kappa$ such that $|{{\mathcal{F}}}|=2^\kappa$, and for all $A,B\in{{\mathcal{F}}}$, $|A\cap B|<\kappa$ if $A\neq B$. Indeed, let $b\colon 2^{<\kappa}\to \kappa$ be a bijection. Then $${{\mathcal{F}}}=\left\{b\left[\{x{\mathord{\upharpoonright}}\alpha\colon\alpha<\kappa\}\right]\colon x\in 2^\kappa\right\}$$ is such a family. A $T_1$ topological space is said to be [[**$\kappa$-additive**]{}]{} if for any $\alpha<\kappa$, an intersection of an $\alpha$-sequence of open subsets of this space is open. Various properties of $\kappa$-additive spaces were considered by R. Sikorski in [@rs:rstshp]. The generalized Cantor and Baire spaces are examples of $\kappa$-additive spaces. It is also easy to see that every $\kappa$-additive topological space $X$ with clopen base of cardinality $\kappa$ is homeomorphic to a subset of $2^\kappa$. Therefore, the generalized Cantor space is a zero-dimensional $\kappa$-additive space which is completely normal. The character, density and weight of $2^\kappa$ equal $\kappa$ (the assumption $\kappa^{<\kappa}=\kappa$ is needed in the case of density and weight). It is easy to see that $A\subseteq 2^\kappa$ is closed if and only if $A=[T]$ for some tree $T\subseteq 2^{<\kappa}$. Indeed, if $A=[T]$ and $T$ is a tree, then if $x\notin A$, there exists $\alpha<\kappa$ such that $x{\mathord{\upharpoonright}}\alpha\notin T$. Therefore $[x{\mathord{\upharpoonright}}\alpha]\subseteq 2^\kappa\setminus A$, so $A$ is closed. On the other hand, if $A$ is closed, let $T=\{x{\mathord{\upharpoonright}}\alpha\colon x\in A,\alpha<\kappa\}$. Then, if $a\in 2^\kappa$, and $a{\mathord{\upharpoonright}}\alpha \in T$ for all $\alpha<\kappa$, we have that $a\in A$, since $A$ is closed. A similar fact is also true in the generalized Baire space. For a closed $A\subseteq {{\mathfrak{K}}}^\kappa$, a tree $T\subseteq {{\mathfrak{K}}}^{<\kappa}$ such that $A=[T]$ is denoted by $T_A$. The family of [[**$\kappa$-Borel sets**]{}]{} is the smallest family of subsets of ${{\mathfrak{K}}}^\kappa$ containing all open sets and closed under complementation and under taking intersections of size $\kappa$. The family of such sets is denoted here by ${{\mathcal{B}}}_\kappa$. We say that a set is [[**$\kappa$-meagre**]{}]{} if it is a union of at most $\kappa$ nowhere dense (in the bounded topology) sets. Notice also that the generalization of the Baire category theorem holds, namely $2^\kappa$ is not $\kappa$-meagre (see [@rs:rstshp Theorem xv]), and neither is $\kappa^\kappa$. The family of all $\kappa$-meagre sets in $2^\kappa$ or $\kappa^\kappa$ is denoted by ${\boldsymbol{{{\mathcal{M}}}}}_\kappa$ (the underlying space will be clear from the context). Also let $${\text{cof}}({{\mathcal{M}}}_\kappa)=\min\left\{|{{\mathcal{A}}}|\colon {{\mathcal{A}}}\subseteq {{\mathcal{M}}}_\kappa\land \forall_{A\in{{\mathcal{M}}}_\kappa} \exists_{B\in {{\mathcal{A}}}} A\subseteq B\right\},$$ and $${\text{cov}}({{\mathcal{M}}}_\kappa)=\min\left\{|{{\mathcal{A}}}|\colon {{\mathcal{A}}}\subseteq {{\mathcal{M}}}_\kappa\land \bigcup {{\mathcal{A}}}= {{\mathfrak{K}}}^\kappa\right\}.$$ Notice also that if $\langle x_\alpha\rangle_{\alpha<\kappa}\in ({{\mathfrak{K}}}^\kappa)^\kappa$ is a sequence of points in ${{\mathfrak{K}}}^\kappa$ such that for all $\xi<\kappa$, there exists $\delta_\xi<\kappa$ such that for all $\delta_\xi\leq\alpha,\beta<\kappa$, $x_\alpha{\mathord{\upharpoonright}}\xi=x_\beta{\mathord{\upharpoonright}}\xi$, then there exists $x\in {{\mathfrak{K}}}^\kappa$ which is a (topological) limit of $\langle x_\alpha\rangle_{\alpha<\kappa}$ (i.e. for every open set $U$ with $x\in U$, there exists $\xi<\kappa$ such that for all $\xi<\alpha<\kappa$, $x_\alpha\in U$). Indeed, take $$x=\bigcup_{\xi<\kappa}x_{\delta_\xi}{\mathord{\upharpoonright}}\xi.$$ Obviously, if $C\subseteq {{\mathfrak{K}}}^\kappa$ is closed, and $\langle x_\alpha\rangle_{\alpha<\kappa}\in ({{\mathfrak{K}}}^\kappa)^\kappa$ is a sequence of points of $C$ with limit $x\in {{\mathfrak{K}}}^\kappa$, then $x\in C$ as well. Therefore, if $\langle C_\alpha\rangle_{\alpha\in\kappa}$ is a sequence of non-empty closed sets such that $C_\beta\subseteq C_\alpha$, when $\alpha<\beta<\kappa$, and such that there exists an increasing sequence $\langle \xi_\alpha\rangle_{\alpha\in \kappa}\in\kappa^\kappa$ and $\langle s_\alpha\rangle_{\alpha\in \kappa}\in ({{\mathfrak{K}}}^{<\kappa})^\kappa$ such that $C_\alpha\subseteq [s_\alpha]$ and $s_\alpha\in {{\mathfrak{K}}}^{\xi_\alpha}$, then there exists $x\in {{\mathfrak{K}}}^\kappa$ such that $$\bigcap_{\alpha<\kappa} C_\alpha=\{x\}.$$ Indeed, let $\langle x_\alpha\rangle_{\alpha<\kappa}\in ({{\mathfrak{K}}}^\kappa)^\kappa$ be any sequence of points such that $x_\alpha\in C_\alpha$, for any $\alpha\in \kappa$, then there exists a limit of this sequence $x$. But $x\in C_\alpha$ for any $\alpha<\kappa$, because $\langle x_{\beta}\rangle_{\alpha\leq\beta<\kappa}$ is a sequence of points in $C_\alpha$. Obviously, spaces $2^\kappa\times 2^\kappa$ and $2^\kappa$ are homeomorphic, and the canonical homeomorphism between them is given by the canonical well-order of $2\times\kappa$, $g\colon 2 \times\kappa\to \kappa$. ### Cardinal coefficients in $2^\kappa$ A statement $2^\kappa=\kappa^+$ is the [[**Continuum Hypothesis for $\kappa$**]{}]{} and is denoted by $CH_\kappa$. Recall that $\diamondsuit_\kappa(E)$ for $E\subseteq \kappa$ is the following principle: there exists a sequence $\langle S_\alpha\rangle_{\alpha\in E}$ such that $S_\alpha\subseteq \alpha$ for all $\alpha\in E$, and the set $$\left\{\alpha\in E\colon X\cap \alpha=S_\alpha\right\}$$ is a stationary subset of $\kappa$ for every $X\subseteq \kappa$ (see e.g. [@tj:st]\[Chapter 23\]). The principle $\diamondsuit_\kappa(\kappa)$ is simply denoted by $\diamondsuit_\kappa$ (and called the [[**diamond principle for $\kappa$**]{}]{}). If $f,g\in \kappa^\kappa$, then we write $f\leq^\kappa g$ if there exists $\alpha<\kappa$ such that for all $\beta<\kappa$ if $\beta>\alpha$, then $f(\beta)\leq g(\beta)$. In this case we say that $f$ is [[**eventually dominated**]{}]{} by $g$. Analogously as in the case of $\omega^\omega$, one can define cardinals related to the order $\leq^\kappa$. The two following cardinals play an important role: $${{\mathfrak{b}}}_\kappa=\min\{|{{\mathcal{A}}}|\colon {{\mathcal{A}}}\subseteq \kappa^\kappa\land \lnot\exists_{f\in \kappa^\kappa}\forall_{g\in A}g\leq^\kappa f\},$$ and $${{\mathfrak{d}}}_\kappa=\min\{|{{\mathcal{A}}}|\colon {{\mathcal{A}}}\subseteq \kappa^\kappa\land \forall_{f\in \kappa^\kappa}\exists_{g\in A}f\leq^\kappa g\},$$ which are called the [[**bounding and dominating number for $\kappa$**]{}]{}, respectively. Obviously, $\kappa<{{\mathfrak{b}}}_\kappa\leq{{\mathfrak{d}}}_\kappa\leq 2^\kappa$. ### $\kappa$-Compactness Not all the results of theory of the real line can be easily generalized to the case of $2^\kappa$. One of the main obstacles is the notion of compactness. We shall say that a topological space $X$ is [[**$\kappa$-compact**]{}]{} (or $\kappa$-Lindelöf) if every open cover of $X$ has a subcover of cardinality less than $\kappa$ (see [@dmds:asret], [@hhsn:sh]). Obviously, the Cantor space $2^\omega$ is $\omega$-compact (i.e. compact in the traditional sense). But it is not always the case that $2^\kappa$ is $\kappa$-compact. Recall that a cardinal number $\kappa$ is weakly compact if it is uncountable and for every two-colour colouring of the set of all two-element subsets of $\kappa$, there exists a set $H\subseteq \kappa$ of cardinality $\kappa$, which is homogeneous (every two-element subset of $H$ have the same colour in the considered colouring) (see [@tj:st]). Recall also that every weakly compact cardinal is strongly inaccessible. Actually, the generalized Cantor space $2^\kappa$ is $\kappa$-compact if and only if $\kappa$ is a weakly compact cardinal (see [@dmds:asret]). And there is even more to that. If $\kappa$ is not weakly compact, then all reasonable $\kappa$-additive spaces are homeomorphic. Precisely, if $\kappa$ is not weakly compact, then every completely regular $\kappa$-additive topological space $X$ without isolated points such that there exists a family of open sets ${{\mathcal{B}}}$ in $X$ such that: (1) the family of all intersections of less than $\kappa$ sets from ${{\mathcal{B}}}$ is a base of the topology of $X$, (2) if ${{\mathcal{C}}}\subseteq {{\mathcal{B}}}$ is such that for any $n\in\omega$ and any $C_0,C_1,\ldots C_n\in {{\mathcal{C}}}$, $\bigcap_{i=0}^n C_n\neq{\varnothing}$, then $\bigcap {{\mathcal{C}}}\neq {\varnothing}$, (3) $|{{\mathcal{B}}}|\leq 2^{<\kappa}$, (4) ${{\mathcal{B}}}=\bigcup_{\alpha<\kappa}{{\mathcal{B}}}_\alpha$, where for any $\alpha<\kappa$, ${{\mathcal{B}}}_{\alpha}$ is a partition of $X$ into open sets, is homeomorphic to $2^\kappa$ (see [@hhsn:sh2 Theorem 2.3] and [@hh:apgshs]). On the other hand, if $\kappa$ is weakly compact, then a completely regular $\kappa$-additive space $X$ without isolated points is homeomorphic to $2^\kappa$ if and only if there exists a family of open sets ${{\mathcal{B}}}$ in $X$ satisfying conditions (1)-(3) and also: 1. ${{\mathcal{B}}}=\bigcup_{\alpha<\kappa}{{\mathcal{B}}}_\alpha$, where for any $\alpha<\kappa$, ${{\mathcal{B}}}_{\alpha}$ is a partition of $X$ into open sets, and $|{{\mathcal{B}}}_\alpha|<\kappa$. We will refer to the above theorem as the [[**Hung-Negrepontis characterization**]{}]{}. In particular, the generalized Cantor space $2^\kappa$ and the generalized Baire spaces $\kappa^\kappa$ are homeomorphic if and only if $\kappa$ is not a weakly compact cardinal. Also notice that every $\kappa$-additive regular space is zero-dimensional (see [@rs:rstshp]). Indeed, if $\left<G_n\right>_{n\in\omega}$ is a sequence of open sets such that ${{\text{cl}}}G_{n+1}\subseteq G_n$, for all $n\in\omega$, then $\bigcap_{n\in\omega} G_n$ is a clopen set. ### Perfect sets in ${{\mathfrak{K}}}^\kappa$ A set $P\subseteq {{\mathfrak{K}}}^\kappa$ is a [[**perfect set**]{}]{} if it is closed and has no isolated points. Notice that a set $P\subseteq {{\mathfrak{K}}}^\kappa$ is perfect if and only if $T_P$ is a perfect tree. A perfect tree $T$ will be called [[**$\kappa$-perfect**]{}]{} if for every limit $\beta<\kappa$, and $t\in {{\mathfrak{K}}}^\beta$ such that for all $\alpha<\beta$, $t{\mathord{\upharpoonright}}\alpha\in T$, we have $t\in T$. Notice that every $\kappa$-perfect tree $T\subseteq 2^{<\kappa}$ is order-isomorphic with $2^{<\kappa}$. A set $P\subseteq {{\mathfrak{K}}}^\kappa$ is [[**$\kappa$-perfect**]{}]{} if $P=[T]$ for a $\kappa$-perfect tree $T$. Obviously, every $\kappa$-perfect set is perfect. On the other hand, the converse does not hold. Notice that if $x\in [T]$, and $T$ is a $\kappa$-perfect tree, then for all $\alpha<\kappa$, $\{x{\mathord{\upharpoonright}}\beta\colon\beta<\kappa\}\cap {\text{Split}}_\alpha(T)\neq {\varnothing}$. For example, if $s\in 2^\omega$ is such that $s(n)=0$ for all $n\in\omega$, then $2^\kappa\setminus [s]$ is a perfect set but is not $\kappa$-perfect. Another major difference between $2^\kappa$ and $2^\omega$ is the perfect set property of analytic set. In $2^\omega$ every uncountable analytic set contains a perfect set. On the other hand, the generalization of this theorem for $2^\kappa$ may not be true even for closed sets. There may even exist a perfect set which does not contain a $\kappa$-perfect set. Recall that a tree $T\subseteq 2^{<\kappa}$ is a [[**$\kappa$-Kurepa**]{}]{} tree if: (1) $|[T]|>\kappa$, (2) if $\alpha$ is uncountable, then $|T\cap 2^\alpha|\leq |\alpha|$. If $T$ is a $\kappa$-Kurepa tree, then $[T]$ is an example of a closed set of cardinality larger than $\kappa$, with no $\kappa$-perfect subsets (see e.g. [@glblis:qgbs; @sf:hdsti]). Fortunately, one can see that every $\kappa$-comeagre set contains a $\kappa$-perfect set. Indeed, if $G=\bigcup_{\alpha<\kappa} G_\alpha$ with $G_\alpha$ nowhere dense, we choose by induction $\left<t_s\right>_{s\in {{\mathfrak{K}}}^{<\kappa}}$ such that $t_s \in {{\mathfrak{K}}}^{<\kappa}$, and for $s,s'\in {{\mathfrak{K}}}^{<\kappa}$, $s\subsetneq s'$ if and only if $t_s\subsetneq t_{s'}$. Indeed, let $t_{\varnothing}$ be such that $[t_{\varnothing}]\cap G_0={\varnothing}$. Then, given $t_s$, $s\in {{\mathfrak{K}}}^\alpha$, let $t'_s\supsetneq t_s$ be such that $[t'_s]\cap G_{\alpha+1}= {\varnothing}$. For $\xi\in{{\mathfrak{K}}}$, set $t_{s^\frown \xi}=t_s^{\prime\frown}\xi$. For limit $\beta<\kappa$, and $s\in {{\mathfrak{K}}}^\beta$, let $t'_s=\bigcup_{\alpha<\beta}t_{s{\mathord{\upharpoonright}}\alpha}$. Let $t_s\supsetneq t'_s$ be such that $[t'_s]\cap G_\beta={\varnothing}$. Finally, let $$T=\bigcup_{\alpha<\kappa}\{t\in {{\mathfrak{K}}}^{<\kappa}\colon t\subseteq t_{s}, s\in {{\mathfrak{K}}}^{\alpha}\}.$$ Obviously, $T$ is a $\kappa$-perfect tree, so $P=[T]_{\kappa}$ is a $\kappa$-perfect subset of ${{\mathfrak{K}}}^\kappa\setminus G$. Special subsets of $2^\kappa$ and $\kappa^\kappa$: simple generalizations {#chsimple} ========================================================================= The aim of this section is to generalize to the case of $2^\kappa$ or $\kappa^\kappa$ certain notions of special subsets defined for $2^\omega$ (see Section \[intro-special\]), and to check their properties and relations between them. Most of the results presented here have their counterparts in the standard case of $2^\omega$, and if so we give a reference in the form ($\omega$: \[n\]). The results presented in this section consist of relatively simple generalizations of some results summarized in [@am:ssrl] and [@lb:srl] to the case of $2^\kappa$. Lusin sets for $\kappa$ ----------------------- Let $\kappa<\lambda\leq 2^\kappa$. A set $L\subseteq {{\mathfrak{K}}}^\kappa$ such that $|L|\geq \lambda$, and if $X\subseteq {{\mathfrak{K}}}^\kappa$ is any $\kappa$-meager set, then $|X\cap L|< \lambda$ will be called a [[**$\lambda$-$\kappa$-Lusin set**]{}]{}. A $\kappa^+$-$\kappa$-Lusin set is simply called a [[**Lusin set for $\kappa$**]{}]{}. \[p-lusin\] If $\lambda={\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)={\text{cof}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$, then there exists a $\lambda$-$\kappa$-Lusin set $L\subseteq 2^\kappa$. Proof: The proof is straightforward as in the case $\kappa=\omega$. Let $\left<A_\alpha\colon \alpha<\lambda\right>$ be a sequence of $\kappa$-meagre sets such that for every $\kappa$-meagre set $A$, there exists $\alpha<\kappa$ such that $A\subseteq A_\alpha$. Inductively, for $\alpha<\lambda$, choose $$x_\alpha\in 2^\kappa\setminus \left(\{x_\beta\colon\beta<\alpha\}\cup \bigcup_{\beta<\alpha}A_\beta\right).$$ The above is always possible because a complement of a union of $<\lambda$ $\kappa$-meagre sets is always not empty and even of cardinality $\geq \lambda$, because for every $x\in 2^\kappa$, $\{x\}$ is $\kappa$-meagre. Now, set $L=\{x_\alpha\colon \alpha<\lambda\}$ to get a $\lambda$-$\kappa$-Lusin set.  $\square$ Obviously, since $2^\kappa\subseteq \kappa^\kappa$, we get that under the above conditions there exists a exists a $\lambda$-$\kappa$-Lusin set $L\subseteq \kappa^\kappa$. Also, immediately we get the following corollary. Assume $CH_\kappa$. Then there exists a Lusin set for $\kappa$ in ${{\mathfrak{K}}}^\kappa$.  $\square$ On the other hand, the existence of a $\lambda$-$\kappa$-Lusin set constrains the value of ${\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$. Assume that $\lambda$ is a regular cardinal and $\kappa<\lambda\leq 2^\kappa$. If $L$ is a $\lambda$-$\kappa$-Lusin set, then $|L|\leq {\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$. Proof: Let $L$ be a $\lambda$-$\kappa$-Lusin set, and let $\langle A_\alpha\rangle_{\alpha<{\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)}$ be a sequence of $\kappa$-meagre sets such that $\bigcup_{\alpha<{\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)} A_\alpha= 2^\kappa$. Notice that $$L=\bigcup_{\alpha<{\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)}\left(A_\alpha\cap L\right).$$ But for any $\alpha<{\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$, $|L\cap A_\alpha|<\lambda$. Since $\lambda\leq |L|$, we get that $|L|\leq {\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$. $\square$ Assume that $\lambda$ is a regular cardinal, $\kappa<\lambda\leq 2^\kappa$, and that there exists a $\lambda$-$\kappa$-Lusin set $L$. Then ${\text{non}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)\leq \lambda\leq {\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$.  $\square$ Sets of $\kappa$-strong measure zero ------------------------------------ A set $A\subseteq {{\mathfrak{K}}}^\kappa$ will be called [[**$\kappa$-strongly measure zero**]{}]{} (${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$) if for every $\langle \xi_\alpha \rangle_{\alpha<\kappa}\in \kappa^\kappa$, there exists $\left<x_\alpha\right>_{\alpha< \kappa}$ such that $x_\alpha\in {{\mathfrak{K}}}^{\xi_\alpha}$, $\alpha<\kappa$ and $A\subseteq \bigcup_{\alpha< \kappa} [x_\alpha]$ (see also [@ah:nsgbs] and [@ahss:smzs]). Obviously if $A\in [{{\mathfrak{K}}}^\kappa]^{\leq\kappa}$, then $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. The well-known characterization of strongly null sets can be generalized to ${{\mathfrak{K}}}^\kappa$. If $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, and $\langle \xi_\alpha \rangle_{\alpha<\kappa}\in \kappa^\kappa$, there exists $\left<x_\alpha\right>_{\alpha< \kappa}\in({{\mathfrak{K}}}^\kappa)^\kappa$ such that $x_\alpha\in {{\mathfrak{K}}}^{\xi_\alpha}$ for all $\alpha<\kappa$, and $$A\subseteq \bigcap_{\alpha<\kappa}\bigcup_{\alpha<\beta< \kappa} [x_\beta].$$ Proof: Let $\langle X_\alpha\rangle_{\alpha<\kappa}\in ([\kappa]^\kappa)^{\kappa}$ be a sequence of pairwise disjoint sets such that $\bigcup_{\alpha<\kappa} X_\kappa=\kappa$. Since $A\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, for all $\alpha<\kappa$, there exist $\langle x_\beta\rangle_{\beta\in X_\alpha}\in({{\mathfrak{K}}}^\kappa)^{X_\alpha}$ such that $A\subseteq\bigcup_{\beta\in X_\alpha}[x_\beta{\mathord{\upharpoonright}}\xi_\beta]$. Then $$A\subseteq \bigcap_{\alpha<\kappa}\bigcup_{\alpha<\beta< \kappa} [x_\beta{\mathord{\upharpoonright}}\xi_\beta].$$ $\square$ In particular, the family of ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ sets forms a $\kappa^+$-complete ideal. Notice also that ${{\mathfrak{K}}}^\kappa\notin {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Indeed, assume otherwise, that $2^\kappa\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, and take $\langle a_\alpha\rangle_{\alpha<\kappa}\in (2^\kappa)^\kappa$ such that $2^\kappa=\bigcup_{\alpha<\kappa}[a_\alpha{\mathord{\upharpoonright}}\alpha+1]$. Let $x\in 2^\kappa$ be such that $x(\alpha)=a_\alpha(\alpha)+1$. Then $$x\in 2^\kappa\setminus \bigcup_{\alpha<\kappa}[a_\alpha{\mathord{\upharpoonright}}\alpha+1],$$ which is a contradiction. The [[**Generalized Borel Conjecture for $\kappa$ (GBC($\kappa$))**]{}]{} states that $${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa=[{{\mathfrak{K}}}^\kappa]^{\leq\kappa}.$$ Some properties of this class of sets were considered in [@ahss:smzs]. In particular, it is proven that if $\kappa$ is a successor cardinal, then ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ is a ${{\mathfrak{b}}}_\kappa$-additive ideal. Under Generalized Martin Axiom for $\kappa$ (GMA($\kappa$), see [@ss:wgmahc]), ${{\mathfrak{b}}}_\kappa=2^\kappa$, so then ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ is $2^\kappa$-additive. Finally, it is proven that GBC($\kappa$) fails for successor $\kappa$. We study some other properties of $\kappa$-strong measure zero sets. Assume that $\kappa$ is a strongly inaccessible cardinal. Then the family of all closed subsets of ${{\mathfrak{K}}}^\kappa$ which are not ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ does not satisfy $2^\kappa$-chain condition. Proof: Let $X\in [\kappa]^\kappa$, and let $$A_X=\left\{x\in {{\mathfrak{K}}}^\kappa \colon \forall_{\alpha\in \kappa\setminus X} x(\alpha)=0\right\}.$$ Notice that $A_X$ is a closed set in ${{\mathfrak{K}}}^\kappa$, and moreover $A_X\notin{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Indeed, consider $X'=\{\alpha+1\colon \alpha\in X\}$. Let $\left<x_\alpha\right>_{\alpha\in X}\in \left({{\mathfrak{K}}}^\kappa\right)^X$ be any sequence, and let $x\in 2^\kappa$ be such that $$x(\alpha)=\begin{cases}x_\alpha(\alpha)+1, & \text{ if } \alpha\in X, \\ 0, & \text{ otherwise.} \end{cases}$$ Then $$x\in A_X\setminus \bigcup_{\alpha\in X}[x_\alpha{\mathord{\upharpoonright}}\alpha+1],$$ so $A_X\notin{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Since $\kappa^{<\kappa}=\kappa$, we can take a family ${{\mathcal{F}}}$ of subsets of $\kappa$ such that $|{{\mathcal{F}}}|=2^\kappa$, and for all $X,Y\in{{\mathcal{F}}}$, $|X\cap Y|<\kappa$ if $X\neq Y$ (see Section \[intro-gen\]). Consider the family ${{\mathcal{A}}}=\{A_X\colon X\in{{\mathcal{F}}}\}$. If $X,Y\in {{\mathcal{F}}}$ are such that $X\neq Y$, then $$A_X\cap A_Y= \left\{x\in {{\mathfrak{K}}}^\kappa \colon \forall_{\alpha\in \kappa\setminus (X\cap Y)} x(\alpha)=0\right\}.$$ so $|A_X\cap A_Y|={{\mathfrak{K}}}^\lambda<\kappa$ ($\kappa$ is strongly inaccessible), for some $\lambda<\kappa$, because $|X\cap Y|<\kappa$. Thus $A_X\cap A_Y\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, and ${{\mathcal{A}}}$ is an antichain of size $2^\kappa$ in the family of all closed subsets of ${{\mathfrak{K}}}^\kappa$ which are not ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. $\square$ Assume $CH_\kappa$. Then there exists a Lusin set for $\kappa$ $L\subseteq 2^\kappa$ such that $L\times L\notin {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: Let $\left<X_\alpha\colon \alpha<\kappa^+\right>$ be an enumeration of all closed nowhere dense sets, and let $\{y_{\alpha}\colon \alpha<\kappa^+\}=2^{\kappa}$. Inductively, for $\alpha<\kappa^+$, choose $$x_\alpha, x'_{\alpha}\in 2^\kappa\setminus \left(\{x_\beta\colon\beta<\alpha\}\cup\{x'_\beta\colon\beta<\alpha\}\cup \bigcup_{\beta<\alpha}X_\beta\right)$$ such that $x_\alpha+x'_\alpha=y_\alpha$. This is possible, since $$F_{\alpha}=\{x_\beta\colon\beta<\alpha\}\cup\{x'_\beta\colon\beta<\alpha\}\cup \bigcup_{\beta<\alpha}X_\beta$$ is $\kappa$-meagre, so $(y_{\alpha}+ F_{\alpha})\cup F_{\alpha}$ is also $\kappa$-meagre. Thus, there exists $x_{\alpha}\notin (y_{\alpha}+ F_{\alpha})\cup F_{\alpha}$. Let $x'_{\alpha}=x_{\alpha}+y_{\alpha}$. Then also $x'_{\alpha}\notin F_{\alpha}$. Obviously $L$ is a Lusin set for $\kappa$. Nevertheless, $L\times L$ is not a ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ set. Indeed, let $f\colon 2^{\kappa}\times 2^{\kappa}\to 2^\kappa$ be given by $f(x,x')=x+x'$. Notice that if $\alpha<\kappa$ is a limit ordinal, then $g(0,\alpha)=\alpha$, where $g$ is the canonical well-order of $2\times \kappa$. Therefore, if $x\in 2^\beta$, for $\omega\leq \beta<\kappa$, then $[x]$ when considered as a subset of $2^\kappa\times 2^\kappa$ is contained in $[y]\times [z]$, where $y,z\in 2^\alpha$ with $\alpha$ a limit ordinal such that $\alpha\leq \beta <\alpha+\omega$. This implies that $f[[x]]\subseteq [y+z]$, and thus if $X\subseteq 2^\kappa\times 2^\kappa$ is $\kappa$-strongly null, then $f[X]$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ as well. But $f[L]=2^\kappa$, so $L\notin {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$.  $\square$ Next, we study the possibility of generalization of Galvin, Mycielski and Solovay ([@fgjmrs:smzs]) characterization of strongly null sets. One of the implications can be proved under no additional assumptions. Before finalization of this paper, the authors became aware that results of Proposition \[gms1\], Lemma \[lem-gms\], Theorem \[gms\] had appeared ealier in [@ww:ssrnvbc]. Nevertheless, we present those results with proofs here for the sake of completeness. \[gms1\] Let $A\subseteq 2^\kappa$ be such that for any nowhere dense set $F$, there exists $x\in 2^\kappa$ such that $(x+A)\cap F={\varnothing}$. Then, $A$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: Let $\{\xi_{\alpha}\colon \alpha<\kappa\}\in [\kappa]^\kappa$. Fix an enumeration $$\{x_{\alpha}\colon \alpha<\kappa\}={\boldsymbol{Q}},$$ and let $$F=2^{\kappa}\setminus \bigcup_{\alpha<\kappa}[x_{\alpha}{\mathord{\upharpoonright}}\xi_{\alpha}].$$ Since $F$ is nowhere dense, there exists $x\in 2^\kappa$ such that $(x+A)\cap F={\varnothing}$. Therefore, $$A\subseteq \bigcup_{\alpha<\kappa}[(x_{\alpha}+x){\mathord{\upharpoonright}}\xi_{\alpha}].$$  $\square$ The reversed implication can be generalized provided $\kappa$ is a weakly compact cardinal. \[lem-gms\] Assume that $\kappa$ is weakly compact. For any closed nowhere dense set $C\subseteq 2^\kappa$ and $s\in 2^{<\kappa}$, there exists $\xi<\kappa$ and $F\subseteq \{s'\in 2^{<\kappa}\colon s\subsetneq s'\}$ with $|F|<\kappa$ such that for any $t\in 2^\xi$, there exists $s'\in F$ such that $$([s']+[t])\cap C={\varnothing}.$$ Proof: Let $x\in 2^\kappa$. Since $x+C$ is nowhere dense, we can find $s_x\supsetneq s$ such that $[s_x]\cap (x+C)={\varnothing}$. Let $\alpha_x={\text{len}}(s_x)$. Then $$\left([x{\mathord{\upharpoonright}}\alpha_x]+[s_x]\right)\cap C={\varnothing}.$$ The family $\{[x{\mathord{\upharpoonright}}\alpha_x]\colon x\in 2^\kappa\}$ is an open covering of $2^\kappa$, and since $\kappa$ is weakly compact, there exists $\lambda<\kappa$ and a sequence $\langle x_\alpha\rangle_{\alpha<\lambda}$ such that $\{[x_\alpha{\mathord{\upharpoonright}}\alpha_{x_\alpha}]\colon \alpha<\lambda\}$ covers $2^\kappa$. Let $F=\{s_{x_\alpha}\colon \alpha<\lambda\},$ and $\xi=\bigcup_{\alpha<\lambda}\alpha_{x_\alpha}<\kappa$. If $t\in 2^\xi$, then there exists $\alpha<\lambda$ such that $x_\alpha{\mathord{\upharpoonright}}\alpha_{x_\alpha}\subseteq t$, so $[t]\subseteq [x_\alpha{\mathord{\upharpoonright}}\alpha_{x_\alpha}]$. Therefore, $$\left([s_{x_\alpha}]+[t]\right)\cap C={\varnothing}.$$  $\square$ \[gms\] Assume that $\kappa$ is a weakly compact cardinal, and $A\subseteq 2^\kappa$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Then for any $\kappa$-meagre set $F$, there exists $x\in 2^\kappa$ such that $(x+A)\cap F={\varnothing}$. Proof: Let $F=\bigcup_{\alpha<\kappa} C_\alpha$ with $C_\alpha$ closed nowhere dense sets. We can assume that $C_{\alpha}\subseteq C_\beta$ if $\alpha<\beta$. We construct inductively a tree $T\subseteq \kappa^\kappa$, along with sequences $\langle \delta_u\rangle_{u\in T},\langle \xi_u\rangle_{u\in T}\in \kappa^T$, and $\left<s_u\right>_{u\in T}\in \left(2^{<\kappa}\right)^T$ such that: (a) if $u\in T\cap\kappa^\beta$, $\beta<\kappa$, then $\{u'\in T\cap\kappa^{\beta+1}\colon u\subseteq u'\}=\{u^\frown\alpha\colon \alpha<\delta_u\}$, (b) for any $u,u'\in T$ if $u\subsetneq u'$, then $s_u\subsetneq s_{u'}$, (c) for any $u\in T\cap \kappa^\beta$, $\beta<\kappa$, and $t\in 2^{\xi_u}$, there exists $\alpha<\delta_u$ such that $$\left([s_{u^\frown \alpha}]+[t]\right)\cap C_{\beta}={\varnothing}.$$ Precisely, let $s_{\varnothing}={\varnothing}$. If $u\in T\cap \kappa^\beta$, $\beta<\kappa$, apply Lemma \[lem-gms\] to $C_\beta$ and $s_u$ to get $\xi_u<\kappa$ and $F_u\subseteq \{s'\in 2^{<\kappa}\colon s\subseteq s'\}$ with $|F|=\delta_{u}<\kappa$ such that for any $t\in 2^{\xi_u}$, there exists $s'\in F_u$, so that $([s']+[t])\cap C_\beta={\varnothing}$. Fix an enumeration $F_u=\{s'_{u,\alpha}\colon \alpha<\delta_u\}$, and put $\{u'\in T\cap\kappa^{\beta+1}\colon u\subseteq u'\}=\{u^\frown\alpha\colon \alpha<\delta_u\}$. For all $\alpha<\delta_u$, let $s_{u^\frown\alpha}=s'_{u,\alpha}$. If $\beta<\kappa$ is a limit ordinal, let $$T\cap \kappa^\beta=\{u\in \kappa^\beta\colon \forall_{\alpha<\beta} u{\mathord{\upharpoonright}}\alpha \in T\}.$$ Also, for $u\in T\cap \kappa^\beta$, let $s_u=\bigcup_{\alpha<\beta} s_{u{\mathord{\upharpoonright}}\alpha}$. Next, define $\langle \delta_\alpha\rangle_{\alpha<\kappa},\langle \xi_\alpha\rangle_{\alpha<\kappa}$ in the following way. For $\alpha<\kappa$, let $$\delta_\alpha=\bigcup_{u\in T\cap \kappa^\alpha}\delta_u,$$ and $$\xi_\alpha=\bigcup_{u\in T\cap \kappa^\alpha}\xi_u.$$ Notice that for all $\alpha<\kappa$, $\delta_\alpha, \xi_\alpha<\kappa$. Indeed, if it is the case for $\alpha<\kappa$, then $|T\cap\kappa^{\alpha+1}|=\delta_\alpha<\kappa$, so $\delta_{\alpha+1}, \xi_{\alpha+1}<\kappa$ since $\kappa$ is regular. If $\alpha$ is a limit ordinal, then $T\cap \kappa^\alpha\subseteq \delta^\alpha$ with $\delta=\bigcup_{\beta<\alpha}\delta_{\beta}<\kappa$. And $\delta^\alpha<\kappa$, because $\kappa$ is strongly inaccessible (every weakly compact cardinal is strongly inaccessible). $A$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Therefore, there exists $\left<x_\alpha\right>_{\alpha\in \kappa}$ such that $x_\alpha\in 2^{\xi_\alpha}$, $\alpha\in \kappa$ and $$A\subseteq \bigcap_{\beta<\kappa}\bigcup_{\beta<\alpha<\kappa} [x_\alpha].$$ By induction construct $y\in\kappa^\kappa$ such that: (a) for all $\alpha<\kappa$, $y{\mathord{\upharpoonright}}\alpha \in T$, (b) for all $\alpha<\kappa$, $\left([s_{y{\mathord{\upharpoonright}}(\alpha+1)}]+[x_\alpha]\right)\cap C_\alpha={\varnothing}$. Precisely, let $y(\alpha)<\delta_{y{\mathord{\upharpoonright}}\alpha}$ be such that $$\left([s_{y{\mathord{\upharpoonright}}\alpha^\frown y(\alpha)}]+[x_\alpha]\right)\cap C_\alpha={\varnothing}.$$ Notice that if $\alpha$ is a limit ordinal, then $$y{\mathord{\upharpoonright}}\alpha=\bigcup_{\beta<\alpha} y{\mathord{\upharpoonright}}\beta\in T.$$ Finally, let $$x=\bigcup_{\alpha<\kappa} s_{y{\mathord{\upharpoonright}}\alpha}\in 2^\kappa.$$ Notice that for all $\beta\leq\alpha<\kappa$, we get $(x+[x_\alpha])\cap C_\beta={\varnothing}$. Therefore, $$(x+A)\cap F={\varnothing}.$$  $\square$ The above propositions imply the following corollaries (see [@lb:srl Corollary 8.14]). Assume that $\kappa$ is weakly compact, and $A,B\subseteq 2^\kappa$ are such that $|A|<{\text{add}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$ and $B\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Then $A\cup B\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: As in the proof of [@lb:srl Corollary 8.14], assume that ${\boldsymbol{0}}\in A\cap B$. Let $F$ be $\kappa$-meagre. Then $(A\cup B)+F\subseteq B+A+F\neq 2^\kappa$, by Theorem \[gms\], because $A+F$ is $\kappa$-meagre. Thus, $A\cup B$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ by Proposition \[gms1\].  $\square$ \[covsn\] If $A\subseteq 2^\kappa$, and $|A|<{\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$, then $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: Indeed, if $F$ is $\kappa$-meagre, then $A+F=\bigcup_{a\in A}a+F\neq 2^\kappa$. Therefore by Proposition \[gms1\], $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$.  $\square$ $\kappa^+$-Concentrated sets ---------------------------- Furthermore, a set $A\subseteq {{\mathfrak{K}}}^\kappa$ will be called [[**$\lambda$-concentrated on a set $B\subseteq {{\mathfrak{K}}}^\kappa$**]{}]{} (for $\kappa<\lambda\leq 2^\kappa$) if for any open set $G$ such that $B\subseteq G$, we have $|A\setminus G|<\lambda$. The relation between concentrated sets, Lusin sets, and strongly null sets can be easily generalized to $\kappa$. \[lusin-con\] A set $A\subseteq {{\mathfrak{K}}}^\kappa$ is a Lusin set for $\kappa$ if and only if $|A|>\kappa$ and is $\kappa^+$-concentrated on every dense set $D\subseteq {{\mathfrak{K}}}^\kappa$ with $|D|=\kappa$. Proof: Indeed, if $A$ is a Lusin set for $\kappa$, then $|A|>\kappa$ and moreover, if $D\subseteq {{\mathfrak{K}}}^\kappa$ is dense with $|D|=\kappa$, and $G\supseteq D$ is open, then $G$ is a dense open set, so $|A\setminus G|=|({{\mathfrak{K}}}^\kappa\setminus G)\cap A|\leq \kappa$. On the other hand, let $A\subseteq {{\mathfrak{K}}}^\kappa$ with $|A|>\kappa$ be a set $\kappa^+$-concentrated on every dense set $D\subseteq {{\mathfrak{K}}}^\kappa$ with $|D|=\kappa$ and let $X\subseteq {{\mathfrak{K}}}^\kappa$ be a nowhere dense set. Since $X$ is contained in a closed nowhere dense set, ${{\mathfrak{K}}}^\kappa\setminus X\supseteq G$, where $G$ is a dense open set. But there exists a dense set $D\subseteq G$ with $|D|=\kappa$, and hence $A$ is $\kappa^+$-concentrated on $D$. Thus, $|A\setminus G|\leq\kappa$, so $A$ is a Lusin set for $\kappa$. $\square$ \[consn\] If a set $A\subseteq {{\mathfrak{K}}}^\kappa$ is $\kappa^+$-concentrated on a set $B$ such that $|B|\leq \kappa$, then $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: Fix an enumeration of $B$, $B=\{b_\alpha\colon \alpha<\kappa\}$. Let $I=\{\xi_\alpha\colon \alpha<\kappa\}\in[\kappa]^\kappa$, and let $f\colon \kappa\times\{0,1\}\to \kappa$ be a bijection. Moreover, let $$G=\bigcup_{\alpha<\kappa} \left[b_\alpha{\mathord{\upharpoonright}}\xi_{f(\alpha,0)}\right].$$ Then $|A\setminus G|\leq\kappa$, so let $A\setminus G=\{c_\alpha\colon \alpha<\kappa\}$. Therefore, $$A\subseteq \bigcup_{\alpha<\kappa} \left[b_\alpha{\mathord{\upharpoonright}}\xi_{f(\alpha,0)}\right]\cup \bigcup_{\alpha<\kappa} \left[c_\alpha{\mathord{\upharpoonright}}\xi_{f(\alpha,1)}\right],$$ which proves that $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. $\square$ Every Lusin set for $\kappa$ in ${{\mathfrak{K}}}^\kappa$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$  $\square$ On the other hand, we get the following. Assume $CH_\kappa$. Then there exists a set $A\subseteq 2^\kappa$ such that $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, but $A$ is not $\kappa^+$-concentrated on any $B\subseteq 2^\kappa$ with $|B|\leq \kappa$. Proof: Let $\left<X_\alpha\colon \alpha<\kappa^+\right>$ be an enumeration of all closed nowhere dense sets. Inductively, for $\alpha<\kappa^+$, choose a perfect nowhere dense set $P_\alpha$ such that $$P_\alpha\cap \left(\bigcup_{\beta<\alpha} P_\beta \cup \bigcup_{\beta<\alpha} X_\alpha\right)={\varnothing}.$$ Choosing such a set is possible since every co-meagre set contains a $\kappa$-perfect set (see Section \[intro-gen\]). Therefore, for any $\alpha<\beta<\kappa^+$, $P_\alpha$ is a perfect nowhere dense set, and $P_\alpha\cap P_\beta={\varnothing}$. Moreover, if $X$ is $\kappa$-meagre, then there exists $\xi<\kappa^+$ such that $$X\cap\bigcup_{\xi<\beta<\kappa^+} P_\beta ={\varnothing}.$$ Let $I=\{\xi_\alpha\colon \alpha<\kappa\}\in[\kappa]^\kappa$, and let $$f\colon [\kappa]^{<\kappa}\times\{0\}\cup \kappa\times\kappa \times \{1\}\to \kappa$$ be a bijection. For $s\in [\kappa]^{<\kappa}$, let $\chi_s\in 2^\kappa$ be the characteristic function of $s$, and let $$G=\bigcup_{s\in [\kappa]^{<\kappa}}\left[\chi_s{\mathord{\upharpoonright}}\xi_{f(s,0)}\right].$$ Notice that $G$ is open and dense, and therefore there exists $\xi<\kappa^{+}$ such that $$\bigcup_{\xi<\beta<\kappa^+} P_\beta\subseteq G.$$ Let $L_\alpha\subseteq P_\alpha$ be a Lusin set relativized to $P_\alpha$, $\alpha<\kappa^+$, and let $A=\bigcup_{\alpha<\kappa^+} L_\alpha$. Let $g\colon \xi+1\to\kappa$ be an injection. Since for all $\beta<\kappa^+$, we have $L_\beta\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, let $\left<x_{\alpha,\beta}\in 2^{<\kappa}\colon \alpha<\kappa, \beta\leq \xi\right>$, be such that for all $\beta\leq \xi$, $$L_\beta\subseteq \bigcup_{\alpha<\kappa}\left[x_{\alpha,\beta}{\mathord{\upharpoonright}}\xi_{f\left(\alpha,g(\beta),1\right)}\right].$$ Then $$A\subseteq \bigcup_{\xi<\beta<\kappa^+} P_\beta \cup \bigcup_{\beta\leq\xi} L_\beta \subseteq \bigcup_{s\in [\kappa]^{<\kappa}}\left[\chi_s{\mathord{\upharpoonright}}\xi_{f(s,0)}\right] \cup \bigcup_{\beta\leq \xi}\bigcup_{\alpha<\kappa}\left[x_{\alpha,\beta}{\mathord{\upharpoonright}}\xi_{f\left(\alpha,g(\beta),1\right)}\right],$$ so $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. On the other hand, if $B\subseteq 2^\kappa$ with $|B|\leq \kappa$, then there exists $\alpha<\kappa^+$ such that $P_\alpha\cap B={\varnothing}$. Therefore, $G= 2^\kappa\setminus P_\alpha$ is an open set such that $B\subseteq G$, but $$A\setminus G=A\cap P_\alpha = L_\alpha,$$ and $|L_\alpha|>\kappa$. $\square$ If $A\subseteq 2^\kappa$ is ${\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$-concentrated on an ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ set, then $A$ is also ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: Let $f\colon 2\times\kappa \to\kappa$ be a bijection and $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$. Let $A\subseteq 2^\kappa$ be ${\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$-concentrated on an ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ set $B$. There exists a sequence $\langle a_\alpha\rangle_{\alpha<\kappa}\in (2^\kappa)^\kappa$ such that $$B\subseteq G= \bigcup_{\alpha<\kappa} [a_{\alpha}{\mathord{\upharpoonright}}\xi_{f(0,\alpha)}].$$ $G$ is open, therefore $|A\setminus G|<{\text{cov}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)$. By Proposition \[covsn\], $A\setminus G\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, so there exists a sequence $\langle b_\alpha\rangle_{\alpha<\kappa}\in (2^\kappa)^\kappa$ such that $$A\setminus G\subseteq \bigcup_{\alpha<\kappa} [b_\alpha{\mathord{\upharpoonright}}\xi_{f(1,\alpha)}].$$ Therefore, $$A\subseteq \bigcup_{\alpha<\kappa} [a_\alpha{\mathord{\upharpoonright}}\xi_{f(0,\alpha)}] \cup \bigcup_{\alpha<\kappa} [b_\alpha{\mathord{\upharpoonright}}\xi_{f(1,\alpha)}].$$  $\square$ Perfectly $\kappa$-meagre sets and $\kappa$-$\lambda$-sets ---------------------------------------------------------- A set $A\subseteq {{\mathfrak{K}}}^\kappa$ is a [[**$\kappa$-$\lambda$-set**]{}]{} if for any $B\subseteq A$ with $|B|\leq \kappa$ there exists a sequence $\left<B_\alpha\right>_{\alpha<\kappa}$, where $B_\alpha\subseteq {{\mathfrak{K}}}^{\kappa}$ are open, and $\bigcap_{\alpha<\kappa} B_\alpha \cap A=B$. Furthermore, a set $A\subseteq {{\mathfrak{K}}}^\kappa$ will be called [[**perfectly $\kappa$-meagre**]{}]{} (${\boldsymbol{\text{P}{{\mathcal{M}}}}}_\kappa$) if for every perfect $P\subseteq {{\mathfrak{K}}}^\kappa$, $A\cap P$ is $\kappa$-meagre relatively to $P$. Additionally, a set $A\subseteq {{\mathfrak{K}}}^\kappa$ will be called [[**$\kappa$-perfectly $\kappa$-meagre**]{}]{} (${\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$) if for every $\kappa$-perfect $P\subseteq {{\mathfrak{K}}}^\kappa$, $A\cap P$ is $\kappa$-meagre relatively to $P$. Obviously, if $A\in {\boldsymbol{\text{P}{{\mathcal{M}}}}}_\kappa$, then $A\in {\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$. \[lambdapm\] Every $\kappa$-$\lambda$-set $A\subseteq {{\mathfrak{K}}}^\kappa$ is perfectly $\kappa$-meagre. Proof: Let $P\subseteq {{\mathfrak{K}}}^\kappa$ be a perfect set and $A\cap P\neq {\varnothing}$. Since there exists a base of size $\kappa$, we can find a set $B\subseteq P\cap A$ with $|B|\leq \kappa$ which is dense in $P\cap A$. Let $\left<B_\alpha\right>_{\alpha<\kappa}$ be a sequence of open sets such that $\bigcap_{\alpha<\kappa} B_\alpha \cap A=B$. Therefore, $$P\cap A\subseteq B \cup \bigcup_{\alpha<\kappa} (P\cap A\setminus B_{\alpha})$$ is $\kappa$-meagre in $P$.  $\square$ On the other hand, since not every $\kappa$-analytic subset of ${{\mathfrak{K}}}^\kappa$ has to have $\kappa$-Baire property (see e.g. [@sf:idst]), it is not clear whether there always exists a ${\boldsymbol{\text{P}{{\mathcal{M}}}}}_\kappa$ set of cardinality greater then $\kappa$. \[pnkprob\] Is there a set $A\subseteq {{\mathfrak{K}}}^\kappa$ such that $|A|=\kappa^+$ and $A\in{\boldsymbol{\text{P}{{\mathcal{M}}}}}_\kappa$ in every model of ZFC? A set $A$ will be called a [[**$\kappa$-$\lambda'$-set**]{}]{} if for any $F$ such that $|F|\leq \kappa$, $A\cup F$ is a $\kappa$-$\lambda$-set. A union of $\kappa$ many $\kappa$-$\lambda'$-sets is a $\kappa$-$\lambda'$-set. Proof: Indeed, let $\langle A_\alpha\rangle_{\alpha<\kappa}$ be a sequence of $\kappa$-$\lambda'$-sets, and let $F$ be such that $|F|\leq\kappa$. Then, let $\left<G_{\alpha,\beta}\right>_{\alpha,\beta<\kappa}$ be a collection of open sets such that $$F=(A_\alpha \cup F)\cap \bigcap_{\beta<\kappa} G_{\alpha,\beta},$$ for any $\alpha<\kappa$. We have that $$F=\left(F\cup \bigcup_{\alpha<\kappa} A_\alpha\right)\cap\bigcap_{\alpha,\beta<\kappa} G_{\alpha,\beta}.$$  $\square$ If $X,Y\subseteq 2^\kappa$ are $\kappa$-$\lambda$ sets, then $X\times Y$ is also a $\kappa$-$\lambda$ set. Proof: Let $F\subseteq X\times Y$ be such that $|F|\leq\kappa$. Then $F_1=\pi_1[F]$ and $F_2=\pi_2[F]$ are also at most of cardinality $\kappa$. Let $\left<G_{\alpha,1}\right>_{\alpha<\kappa}$ and $\left<G_{\alpha,2}\right>_{\alpha<\kappa}$ be such that $$F_1=X\cap \bigcap_{\alpha<\kappa} G_{\alpha,1}$$ and $$F_2=Y\cap \bigcap_{\alpha<\kappa} G_{\alpha,2}.$$ We obtain $$F=X\times Y \cap \bigcap_{\alpha<\kappa} G_{\alpha,1}\times 2^\kappa \cap \bigcap_{\alpha<\kappa} 2^\kappa\times G_{\alpha,2}\cap \bigcap_{x\in F_1\times F_2\setminus F} (2^\kappa\times 2^\kappa \setminus\{x\}).$$  $\square$ The above proposition can be proven analogously for $\kappa$-$\lambda'$ sets. A set $A\subseteq {{\mathfrak{K}}}^\kappa$ is a [[**$\kappa$-$s_0$-set**]{}]{} if for any $\kappa$-perfect set $P\subseteq {{\mathfrak{K}}}^\kappa$, there exists a $\kappa$-perfect set $Q\subseteq P$ such that $Q\cap A={\varnothing}$. Every ${\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$ subset of $2^\kappa$ is a $\kappa$-$s_0$-set. Proof: Let $P$ be $\kappa$-perfect, and $A\in{\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$. There exists a homeomorphism $h\colon P\to 2^\kappa$. Then $h[A\cap P]$ is $\kappa$-meagre, so there exists a $\kappa$-perfect set $Q'\subseteq 2^\kappa\setminus h[A]$. Then $Q=h^{-1}[Q']$ is a $\kappa$-perfect set included in $P\setminus A$.  $\square$ Similar proposition can be proven for ${\boldsymbol{\text{P}{{\mathcal{M}}}}}_\kappa$ sets. Every ${\boldsymbol{\text{P}{{\mathcal{M}}}}}_\kappa$ subset of $2^\kappa$ is an $s_0$-set. Proof: If $G=\bigcup_{\alpha<\kappa} G_\alpha\subseteq P$ with $G_\alpha$ nowhere dense in $P$, we construct by induction a partial function $F\colon 2^{<\kappa}\to T_P$ such that for $s,s'\in {\text{dom}}F$, $s\subsetneq s'$ if and only if $F(s)\subsetneq F(s')$. Indeed, let $F({\varnothing})$ be such that $[F({\varnothing})]\cap G_0={\varnothing}$. Then, given $F(s)$, $s\in 2^\alpha\cap {\text{dom}}F$, let $t_s\supsetneq F(s)$ be such that $[t_s]\cap G_{\alpha+1}= {\varnothing}$ and $t_s\in {\text{Split}}(T)$. Set $F(s^\frown 0)=t_s\,^{\frown}0$ and $F(s^\frown 1)=t_s\,^{\frown}1$. For limit $\beta<\kappa$, and $s\in 2^\beta$ such that $s{\mathord{\upharpoonright}}\alpha\in {\text{dom}}F$ for all $\alpha<\beta$, let $t_s=\bigcup_{\alpha<\beta} F(s{\mathord{\upharpoonright}}\alpha)$. If $t_s\in T_P$, then let $F(s)\supsetneq t_s$ be such that $F(s)\cap G_\beta={\varnothing}$. Otherwise, $s\notin {\text{dom}}F$. Notice that since $G_\alpha$ is nowhere dense for all $\alpha<\kappa$, for any $s\in 2^{<\beta}\cap {\text{dom}}F$ there exists $s'\in 2^\beta\cap {\text{dom}}F$ such that $s\subseteq s'$. Finally, let $$T_Q=\{t\in 2^{<\kappa}\colon t\subseteq F(s), s\in {\text{dom}}F\}.$$ Obviously, $T_Q\subseteq T_P$ is a perfect tree, so $Q=[T_Q]$ is a perfect subset of $P\setminus G$.   $\square$ Notice that a set having only $\kappa$-meagre homeomorphic images may not be perfectly $\kappa$-meagre. There exists a set $A\subseteq 2^\kappa$ which is not ${\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$, but its every homeomorphic image is $\kappa$-meagre. Proof: Let $P\subseteq 2^\kappa$ be a $\kappa$-meagre $\kappa$-perfect set, e.g. $$P=\{x\in 2^\kappa\colon \forall_{\alpha<\kappa} x(\alpha+1)=0\}.$$ Let $\left<P_\xi\right>_{\xi<2^\kappa}$ be an enumeration of all $\kappa$-perfect subsets of $P$. Find inductively $\langle x_\xi\rangle_{\xi<2^\kappa}$ and $\langle y_\xi\rangle_{\xi<2^\kappa}$ such that $x_\xi\neq y_\xi$, and $$x_{\xi}, y_{\xi}\in P_{\xi}\setminus \bigcup_{\eta<\xi}\{x_\eta,y_\eta\},$$ for all $\xi< 2^\kappa$. Finally, let $$A={\boldsymbol{Q}}\cup\bigcup_{\xi<2^\kappa} \{x_\xi\}.$$ Notice that $A$ is not ${\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$, as it is not a $\kappa$-$s_0$-set. Indeed, there is no $\kappa$-perfect $Q\subseteq P$ such that $Q\cap A={\varnothing}$. But if $s\in 2^{<\kappa}$, then $[s^\frown 1]\cap P={\varnothing}$, so every open set contains an open subset $U$ such that $|U\cap A|\leq\kappa$. Therefore if $h$ is a homeomorphism, then $h[A]$ has also this property. In particular, for $s\in 2^{<\kappa}$ let $t_s\in 2^{<\kappa}$ be such that $s\subseteq t_s$, and $|h[A]\cap [t_s]|\leq \kappa$. Then $$A'=\bigcup_{s\in 2^{<\kappa}}(h[A]\cap [t_s])$$ is of cardinality at most $\kappa$, and $h[A]\setminus A'$ is nowhere dense.  $\square$ On the other hand, for $\kappa$-$\lambda$-sets we get the following. Let $A,B\subseteq {{\mathfrak{K}}}^\kappa$, and assume that $f\colon A\to B$ is a one-to-one continuous map. If $B$ is a $\kappa$-$\lambda$ set, then $A$ is also a $\kappa$-$\lambda$-set. Proof: Indeed, let $C\subseteq A$ and $|C|\leq\kappa$. then $f[C]\subseteq B$ is also of cardinality at most $\kappa$, and there exists a sequence of open sets $\langle G_\alpha\rangle_{\alpha<\kappa}$ such that $$B\cap \bigcap_{\alpha<\kappa} G_\alpha=f[C].$$ But since $f$ is one-to-one, we get $$A\cap \bigcap_{\alpha<\kappa} f^{-1}[G_\alpha]=C.$$  $\square$ A similar statement can be proven for $\kappa$-$\lambda'$-sets. Let $X,Y\subseteq {{\mathfrak{K}}}^\kappa$, and assume that $f\colon X\to Y$ is a continuous map. Let $A\subseteq X$ and $B\subseteq Y$ be such that $B$ is a $\kappa$-$\lambda'$-set, and $f{\mathord{\upharpoonright}}A$ is one-to-one onto $B$. Then $A$ is also a $\kappa$-$\lambda'$ set. Proof: The proof is similar to the proof in the case $\kappa=\omega$. Namely, let $C\subseteq X$ with $|C|<\kappa$. Then there exists a sequence of open sets $\langle G_\alpha\rangle_{\alpha<\kappa}$ such that $(B\cup f[C])\cap G=f[C]$, where $G=\bigcap_{\alpha<\kappa} G_\alpha$. Therefore, $$f^{-1}[G]=f^{-1}[B\cap f[C]]\cup f^{-1}[G\setminus B]= (A\cap C)\cup f^{-1}[G\setminus B],$$ because $f$ is one-to-one on $A$. This implies that $$f^{-1}[G]\cap (A\cup C)= (A\cap C)\cup (f^{-1}[G]\cap C)=C.$$  $\square$ $\kappa$-$\sigma$-Sets ---------------------- A set $A\subseteq {{\mathfrak{K}}}^\kappa$ will be called [[**$\kappa$-$\sigma$-set**]{}]{} if for any sequence of closed sets $\langle F_\alpha\rangle_{\alpha< \kappa}$, there exists a sequence of open sets $\langle G_\alpha\rangle_{\alpha< \kappa}$ such that $$A\cap \bigcup_{\alpha<\kappa} F_{\alpha}=A\cap \bigcap_{\alpha<\kappa} G_\alpha.$$ \[sigmapm\] Every $\kappa$-$\sigma$-set is ${\boldsymbol{\text{P}{{\mathcal{M}}}}}_\kappa$. Proof: Let $A$ be a $\kappa$-$\sigma$ set, and let $P\subseteq {{\mathfrak{K}}}^\kappa$ be a perfect set, and assume that $P\cap A\neq {\varnothing}$. Let $C\in [A\cap P]^{\leq\kappa}$ be such that for all $s\in {{\mathfrak{K}}}^{<\kappa}$ if $[s]\cap P\cap A\neq {\varnothing}$, then $[s]\cap C\neq {\varnothing}$. There exists a sequence of open sets $\langle G_\alpha\rangle_{\alpha< \kappa}$ such that $$C = A\cap \bigcap_{\alpha<\kappa} G_\alpha.$$ Therefore $C\subseteq G_\alpha$, for any $\alpha<\kappa$. Thus, for all $\alpha<\kappa$, $A\setminus G_\alpha$ is nowhere dense in $P$. We have that $$A=C\cup (A\setminus C)=C\cup \left(A\setminus \bigcap_{\alpha<\kappa} G_\alpha\right)= C\cup\bigcup_{\alpha<\kappa} \left(A\setminus G_\alpha\right)$$ is $\kappa$-meagre in $P$.  $\square$ Cover selection principles in $2^\kappa$ ---------------------------------------- In this section we study analogues of cover selection properties for subsets of ${{\mathfrak{K}}}^\kappa$. ### $\kappa$-$\gamma$-Sets A family of open subsets ${{\mathcal{U}}}$ of a topological space $X$ will be called a [[**$\kappa$-cover**]{}]{} of $X$ if for any $A\in[X]^{<\kappa}$ there exists $U\in{{\mathcal{U}}}$ such that $A\subseteq U$. It is a [[**$\gamma$-$\kappa$-cover**]{}]{} if ${{\mathcal{U}}}=\{U_\alpha\colon \alpha<\kappa\}$, and $$X\subseteq \bigcup_{\alpha<\kappa}\bigcap_{\alpha<\beta<\kappa} U_\beta.$$ Notice that every subsequence of length $\kappa$ of a $\kappa$-$\gamma$-cover is still a $\kappa$-$\gamma$-cover. The family of all $\kappa$-covers of $X$ will be denoted by $\Omega_\kappa (X)$, and the family of all $\kappa$-$\gamma$-covers will be denoted by $\Gamma_\kappa(X)$. The family of all open covers of size $\kappa$ of $X$, is denoted by ${{\mathcal{O}}}_\kappa(X)$. The underlying set can be omitted in this notation if it is apparent from the context. We always assume that the covers which are considered in this article are proper, i.e. the set itself is never an element of its cover. $X\subseteq {{\mathfrak{K}}}^\kappa$ will be called a [[**$\kappa$-$\gamma$-set**]{}]{} if for every open $\kappa$-cover ${{\mathcal{U}}}$ of $X$ there exists a sequence $\langle U_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{U}}}^\kappa$ such that $\{U_\alpha\colon \alpha<\kappa\}$ is a $\kappa$-$\gamma$-cover. If ${{\mathcal{A}}},{{\mathcal{B}}}$ are families of open covers of a set $X$, we shall say that it has [[**$S^\kappa_1({{\mathcal{A}}},{{\mathcal{B}}})$ property**]{}]{} if for every sequence $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha<\kappa}\in{{\mathcal{A}}}^\kappa$, there exists a sequence $\langle U_\alpha\rangle_{\alpha<\kappa}$ such that $U_\alpha\in {{\mathcal{U}}}_\alpha$, for all $\alpha<\kappa$, and $\{U_\alpha\colon \alpha<\kappa\}\in {{\mathcal{B}}}$. We aim to prove that similarly to the case $\kappa=\omega$, $\kappa$-$\gamma$-sets can be characterized in terms of selection principles. First we need the following easy observation. \[s1refinement\] Let $X$ be a subset of a $\kappa$-additive topological space, and ${{\mathcal{A}}},{{\mathcal{B}}}$ be any families of open covers of cardinality $\kappa$ of $X$ such that: (a) if ${{\mathcal{V}}}\in {{\mathcal{B}}}$ is a refinement of an open cover ${{\mathcal{U}}}$, then there exists ${{\mathcal{U}}}'\subseteq {{\mathcal{U}}}$ with ${{\mathcal{U}}}'\in {{\mathcal{B}}}$, (b) if $\beta<\kappa$, and $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha<\beta}\in {{\mathcal{A}}}^\beta$, then there exists ${{\mathcal{U}}}\in {{\mathcal{A}}}$ such that ${{\mathcal{U}}}$ is a refinement of ${{\mathcal{U}}}_\alpha$ for every $\alpha<\beta$, (c) if $\{U_\alpha\colon \alpha<\kappa\}\in {{\mathcal{B}}}$, and ${{\mathcal{V}}}_{\beta}=\{V_{\alpha,\beta}\colon \alpha<\gamma_{\beta}\}$ for $\beta<\kappa$ and $\langle \gamma_\beta\rangle_{\beta\in \kappa}\in \kappa^\kappa$ are such that $U_\beta\subseteq V_{\alpha,\beta}$ for all $\beta<\kappa, \alpha<\gamma_\beta$, then $\bigcup_{\beta<\kappa}{{\mathcal{V}}}_\beta\in{{\mathcal{B}}}$. Then $X$ satisfies $S^\kappa_1({{\mathcal{A}}},{{\mathcal{B}}})$ if and only if for every $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{A}}}^\kappa$ such that ${{\mathcal{U}}}_\beta$ is a refinement of ${{\mathcal{U}}}_\alpha$, for all $\alpha<\beta<\kappa$, there exists $\langle U_\alpha\rangle_{\alpha<\kappa}$ with $\{U_\alpha\colon \alpha<\kappa\}\in{{\mathcal{B}}}$, and $U_\alpha\in{{\mathcal{U}}}_\alpha$ for all $\alpha<\kappa$. Proof: Let $X$ be a set satisfying the premise of the Lemma \[s1refinement\] along with families ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ and such that for every $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{A}}}^\kappa$ such that ${{\mathcal{U}}}_\beta$ is a refinement of ${{\mathcal{U}}}_\alpha$, for all $\alpha<\beta<\kappa$, there exists $\langle U_\alpha\rangle_{\alpha<\kappa}$ with $\{U_\alpha\colon \alpha<\kappa\}\in{{\mathcal{B}}}$, and $U_\alpha\in{{\mathcal{U}}}_\alpha$ for all $\alpha<\kappa$. Let $\langle {{\mathcal{W}}}_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{A}}}^\kappa$ be arbitrary. By induction we construct $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{A}}}^\kappa$ such that ${{\mathcal{U}}}_\beta$ is a refinement of ${{\mathcal{U}}}_\alpha$ and ${{\mathcal{W}}}_\alpha$, for all $\alpha<\beta<\kappa$. Hence, there exists $\langle U_\alpha\rangle_{\alpha<\kappa}$ such that $\{U_\alpha\colon \alpha<\kappa\}\in{{\mathcal{B}}}$, and $U_\alpha\in{{\mathcal{U}}}_\alpha$ for all $\alpha<\kappa$. For all $\alpha<\kappa$, let $O_\alpha\in {{\mathcal{W}}}_\alpha$ be such that $U_\alpha\subseteq O_\alpha$. Then $\{U_\alpha\colon \alpha<\kappa\}$ is a refinement of $\{O_\alpha\colon \alpha<\kappa\}$ thus there exists $A\subseteq \kappa$ such that $\{O_\alpha\colon \alpha\in A\}\in {{\mathcal{B}}}$. Now, choose $\langle V_\alpha\rangle_{\alpha<\kappa}$ such that $V_\alpha=O_\alpha$ if $\alpha\in A$, and $V_\alpha\in {{\mathcal{W}}}_\alpha$ be such that $O_\beta\subseteq V_\alpha$ for $\beta=\min (A\setminus \alpha)$. Then $\{V_\alpha\colon \alpha<\kappa\}\in {{\mathcal{B}}}$, and for any $\alpha<\kappa$, $V_\alpha\in {{\mathcal{W}}}_{\alpha}$.    $\square$ \[lemgamma\] If $X\subseteq {{\mathfrak{K}}}^\kappa$, then ${{\mathcal{A}}}=\Omega_\kappa(X)$ and ${{\mathcal{B}}}=\Gamma_\kappa(X)$ satisfy the premise of Lemma \[s1refinement\]. Proof: Recall that an intersection of less than $\kappa$ open sets in ${{\mathfrak{K}}}^\kappa$ is still open. The rest of the proof is obvious.$\square$ \[gamma1\] A set $X\subseteq {{\mathfrak{K}}}^\kappa$, with $|X|\geq\kappa$ is a $\kappa$-$\gamma$-set if and only if it has property $S_1^\kappa(\Omega_\kappa,\Gamma_\kappa)$. Proof: As in the case $\kappa=\omega$, choose a sequence of distinct points $\langle x_\alpha\rangle_{\alpha<\kappa}\in X^\kappa$. Assume that $\langle{{\mathcal{W}}}_{\alpha}\rangle_{\alpha<\kappa}\in (\Omega_{\kappa}(X))^\kappa$ is a sequence of covers such that for $\alpha<\beta$, ${{\mathcal{W}}}_{\beta}$ is a refinement of ${{\mathcal{W}}}_\alpha$. Let $${{\mathcal{U}}}=\{U\setminus \{x_{\alpha}\}\colon U\in {{\mathcal{W}}}_\alpha, \alpha\in\kappa\}.$$ Notice that ${{\mathcal{U}}}$ is a $\kappa$-cover of $X$. Since $X$ is a $\kappa$-$\gamma$-set, there exists a $\kappa$-$\gamma$-cover ${{\mathcal{V}}}\subseteq {{\mathcal{U}}}$. Let ${{\mathcal{V}}}=\{V_\alpha\colon \alpha<\kappa\}$, and let $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be such that $V_\alpha=U_{\alpha}\setminus \{x_{\xi_\alpha}\}$ with $U_\alpha\in{{\mathcal{W}}}_{\xi_\alpha}$. Notice that $|\{\xi_\alpha\colon \alpha<\kappa\}|=\kappa$. Indeed, if this is not the case, an ordinal $\gamma<\kappa$ occurs cofinitely many times in the sequence $\langle \xi_\alpha\rangle_{\alpha\in\kappa}$, thus $$x_{\gamma}\notin \bigcup_{\alpha<\kappa}\bigcap_{\alpha<\beta<\kappa} V_\beta.$$ Therefore, let $\langle\delta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ be such that $\langle \xi_{\delta_\alpha}\rangle_{\alpha<\kappa}$ is a strictly increasing sequence. Notice that $\langle V_{\delta_\alpha}\rangle_{\alpha<\kappa}$ is also a $\kappa$-$\gamma$-cover of $X$. Let $A=\{\xi_{\delta_\alpha}\colon \alpha<\kappa\}$, and choose $\langle W_\alpha\rangle_{\alpha<\kappa}$ such that $W_{\xi_\alpha}=V_\alpha$ if $\xi_\alpha\in A$, and otherwise choose $W_\alpha\in {{\mathcal{W}}}_\alpha$ such that $V_\beta\subseteq W_\alpha$ for $\beta=\min (A\setminus \alpha)$. Then $\{W_\alpha\colon \alpha<\kappa\}\in \Gamma_\kappa$, and for any $\alpha<\kappa$, $W_\alpha\in {{\mathcal{W}}}_{\alpha}$. Therefore, by Lemmas \[s1refinement\] and \[lemgamma\], $X$ satisfies $S_1^\kappa(\Omega_\kappa,\Gamma_\kappa)$. $\square$ Every $\kappa$-$\gamma$-set satisfies $S_1^\kappa(\Gamma_\kappa,\Gamma_\kappa)$. Proof: Obviously, every $\kappa$-$\gamma$-cover is a $\kappa$-cover. $\square$ Finally, we prove that every union of $\kappa$ many closed subsets of $\kappa$-$\gamma$-set is $\kappa$-$\gamma$-set as well. \[cl-gamma\] A $\kappa$-union of closed subsets of a $\kappa$-$\gamma$-set is a $\kappa$-$\gamma$-set. Proof: Let $F=\bigcup_{\alpha<\kappa} F_\alpha$ with $F_\alpha\subseteq X$, where $X$ is a $\kappa$-$\gamma$-set and $F_\alpha$ are closed in $X$. Assume that for $\alpha<\beta<\kappa$, $F_\alpha\subseteq F_\beta$, and let ${{\mathcal{U}}}$ be a $\kappa$-cover of $F$. For any $\alpha<\kappa$, $${{\mathcal{U}}}_\alpha=\{U\cup (X\setminus F_\alpha)\colon U\in{{\mathcal{U}}}\}$$ is a $\kappa$-cover of $X$. Thus, by Theorem \[gamma1\], there exists a sequence $\langle U_{\alpha}\rangle_{\beta<\kappa}$ such that $U_\alpha\in {{\mathcal{U}}}_\alpha$, and $$X\subseteq \bigcup_{\gamma<\kappa}\bigcap_{\gamma<\beta<\kappa} U_{\beta}.$$ Let $\langle V_{\alpha}\rangle_{\alpha,\beta<\kappa}\in {{\mathcal{U}}}^\kappa$ be such that $U_{\alpha}=V_{\alpha}\cup (X\setminus F_\alpha)$. Then $$F\subseteq \bigcup_{\alpha<\kappa}\bigcap_{\alpha<\beta<\kappa} V_\beta,$$ because if $x\in F$, there exists $\alpha<\kappa$ such that $x\notin X\setminus F_\beta$ for all $\beta<\kappa$ with $\alpha<\beta$. Thus, $$x\in \bigcap_{\alpha<\beta<\kappa} V_\beta.$$ $\square$ ### $\kappa$-Hurewicz property A cover ${{\mathcal{U}}}$ of a set $X$ is [[**essentially of size $\kappa$**]{}]{} if for every ${{\mathcal{V}}}\in[{{\mathcal{U}}}]^{<\kappa}$, $X\setminus \bigcup{{\mathcal{V}}}\neq {\varnothing}$. We will say that a set $X$ satisfies [[**$U_{<\kappa}^\kappa({{\mathcal{A}}},{{\mathcal{B}}})$ principle**]{}]{} if for every sequence $\langle{{\mathcal{U}}}_{\alpha}\rangle_{\alpha<\kappa}\in {{\mathcal{A}}}^\kappa$ of covers essentially of size $\kappa$, there exists $\langle{{\mathcal{V}}}_\alpha\rangle_{\alpha<\kappa}$ such that ${{\mathcal{V}}}_\alpha\in[{{\mathcal{U}}}_\alpha]^{<\kappa}$ for all $\alpha<\kappa$, and $\{\bigcup{{\mathcal{V}}}_\alpha\colon \alpha<\kappa\}\in {{\mathcal{B}}}$. A set $X$ has [[**$\kappa$-Hurewicz property**]{}]{} if it satisfies $U_{<\kappa}^\kappa({{\mathcal{O}}}_\kappa,\Gamma_\kappa)$ principle. \[hurewicz1\] If $X\subseteq {{\mathfrak{K}}}^\kappa$ satisfies $S^\kappa_1(\Gamma_\kappa,\Gamma_\kappa)$, then it has $\kappa$-Hurewicz property. Proof: Assume that $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha\in\kappa}$ is a sequence of open covers of $X$ which are essentially of size $\kappa$. Let ${{\mathcal{U}}}_\beta=\{U_{\beta,\alpha}\colon \alpha<\kappa\}$, for all $\beta<\kappa$, and let $V_{\beta,\alpha}=\bigcup_{\gamma<\alpha} U_{\beta, \gamma}$ for all $\alpha,\beta<\kappa$. Notice that, for any $\beta<\kappa$, $\langle V_{\beta,\alpha}\rangle_{\alpha<\kappa}$ is a $\kappa$-$\gamma$-cover of $X$. Indeed, if there exists $$x\in X\setminus\bigcup_{\alpha<\kappa}\bigcap_{\alpha<\gamma<\kappa} V_{\beta,\gamma}=X\setminus\bigcup_{\alpha<\kappa}\bigcap_{\alpha<\gamma<\kappa} \bigcup_{\delta<\gamma} U_{\beta, \delta},$$ then $x\notin U_{\beta,\delta}$ for all $\delta<\kappa$. Thus, there exists a sequence $\langle\xi_{\alpha}\rangle_{\alpha\in\kappa}\in \kappa^\kappa$ such that $\{V_{\alpha,\xi_\alpha}\colon \alpha<\kappa\}$ is a $\kappa$-$\gamma$-cover. For $\alpha<\kappa$, let ${{\mathcal{V}}}_{\alpha}=\{U_{\alpha,\beta}\colon \beta<\xi_{\alpha}\}$. Then $$\left\{\bigcup{{\mathcal{V}}}_\alpha\colon \alpha<\kappa\right\}=\{V_{\alpha,\xi_\alpha}\colon \alpha<\kappa\}$$ is the desired $\kappa$-$\gamma$-cover. $\square$ \[hurewicz2\] If $X$ is a $\kappa$-$\gamma$-set, then it has $\kappa$-Hurewicz property. $\square$ On the other had, no Lusin set for $\kappa$ can have $\kappa$-Hurewicz property. Indeed, we get the following lemma. If $A\subseteq {{\mathfrak{K}}}^\kappa$ with an empty interior has $\kappa$-Hurewicz property, then $A$ is $\kappa$-meagre. Proof: Let $\{s_\alpha\colon \alpha<\kappa\}={{\mathfrak{K}}}^{<\kappa}$, and let $\{x_\alpha\colon \alpha<\kappa\}$ be such that $x_\alpha\in [s_\alpha]\setminus A$ for all $\alpha<\kappa$, and let $U_{\alpha,\beta}={{\mathfrak{K}}}^\kappa\setminus [x_{\alpha}{\mathord{\upharpoonright}}\beta]$. Finally, let ${{\mathcal{U}}}_\alpha=\{U_{\alpha,\beta}\colon \beta<\kappa\}$ for $\alpha<\kappa$. For $\alpha<\kappa$, ${{\mathcal{U}}}_\alpha$ is an increasing open cover of $A$, which is essentially of size $\kappa$. Since $A$ has $\kappa$-Hurewicz property, there exists $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ such that $$\{U_{\alpha,\xi_\alpha}\colon \alpha<\kappa\}$$ is a $\kappa$-$\gamma$-cover of $A$. In other words, $$A=\bigcup_{\alpha<\kappa}\bigcap_{\alpha<\beta<\kappa}U_{\beta,\xi_\beta}.$$ Obviously, $$\bigcap_{\alpha<\beta<\kappa}U_{\beta,\xi_\beta}=\bigcap_{\alpha<\beta<\kappa} \left({{\mathfrak{K}}}^\kappa\setminus [x_{\beta}{\mathord{\upharpoonright}}\xi_{\beta}]\right)={{\mathfrak{K}}}^\kappa\setminus \bigcup_{\alpha<\beta<\kappa}[x_{\beta}{\mathord{\upharpoonright}}\xi_{\beta}]$$ is a nowhere dense set for any $\alpha<\kappa$. Hence, $A$ is $\kappa$-meagre.$\square$ \[lusin-hur\] If $\kappa<\lambda\leq 2^\kappa$, and $L\subseteq {{\mathfrak{K}}}^\kappa$ is a $\lambda$-$\kappa$-Lusin set, then $L$ does not have $\kappa$-Hurewicz property. $\square$ ### $\kappa$-Menger property A set has [[**$\kappa$-Menger property**]{}]{} if it satisfies $U_{<\kappa}^\kappa({{\mathcal{O}}}_\kappa,{{\mathcal{O}}}_\kappa)$ principle. Despite that every Lusin set for $\kappa$ lacks $\kappa$-Hurewicz property (see Corollary \[lusin-hur\]), it has $\kappa$-Menger property. \[lus-men\] Let $L\subseteq {{\mathfrak{K}}}^\kappa$ be a Lusin set for $\kappa$. Then $L$ has $\kappa$-Menger property. Proof: Let $\{s_\alpha\colon \alpha<\kappa\}=\{s\in {{\mathfrak{K}}}^{<\kappa}\colon [s]\cap L\neq 0\}$, and let $\{x_\alpha\colon \alpha<\kappa\}$ be such that $x_\alpha\in [s_\alpha]\cap L$ for all $\alpha<\kappa$. Let $\langle{{\mathcal{U}}}_\alpha\rangle_{\alpha<\kappa}$ be a sequence of open covers essentially of size $\kappa$. For $\alpha<\kappa$, let $U_\alpha\in{{\mathcal{U}}}_\kappa$ be such that $x_\alpha\in U_\alpha$. Then, $L\setminus \bigcup_{\alpha<\kappa}U_\alpha$ is nowhere dense, hence $|L\setminus \bigcup_{\alpha<\kappa}U_\alpha|\leq\kappa$. Thus let $$L\setminus \bigcup_{\alpha<\kappa}U_\alpha=\{y_\alpha\colon \alpha<\kappa\}.$$ For all $\alpha<\kappa$, let $V_\alpha\in {{\mathcal{U}}}_\alpha$ be such that $y_\alpha\in V_\alpha$. Let ${{\mathcal{V}}}_\alpha=\{U_\alpha,V_\alpha\}$, for $\alpha<\kappa$. Then $\{\bigcup_{\alpha<\kappa} {{\mathcal{V}}}_\alpha\colon \alpha<\kappa\}$ is an open cover of $L$. $\square$ ### $\kappa$-Rothberger property A set has [[**$\kappa$-Rothberger property**]{}]{} if it satisfies $S^\kappa_1({{\mathcal{O}}}_\kappa,{{\mathcal{O}}}_\kappa)$ principle. Obviously, this property implies $\kappa$-Menger property. \[roth-sn\] If $A\subseteq {{\mathfrak{K}}}^\kappa$ has $\kappa$-Rothberger property, then $A\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: Let $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ be a sequence of ordinals. For $\alpha<\kappa$, let $${{\mathcal{U}}}_\alpha=\left\{[s]\colon s\in 2^{\xi_{\alpha}}\right\}.$$ Since $A$ has $\kappa$-Rothberger property, we get that there exists a sequence $\langle s_\alpha\rangle_{\alpha<\kappa}$ such that $s_\alpha\in {{\mathfrak{K}}}^{\xi_\alpha}$ for all $\alpha<\kappa$, and $\{[s_{\xi_\alpha}]\colon \alpha<\kappa\}$ is a cover of $A$. $\square$ The generalized Cantor space $2^\kappa$ and the generalized Baire space $\kappa^\kappa$ do not have $\kappa$-Rothberger property. $\square$ Proposition \[consn\] can be formulated in a stronger form. \[con-roth\] If $A\subseteq {{\mathfrak{K}}}^\kappa$ is $\kappa^+$-concentrated on a set $B\subseteq {{\mathfrak{K}}}^\kappa$ with $|B|\leq \kappa$, then $A$ has $\kappa$-Rothberger property. Proof: We modify the proof of Proposition \[consn\]. Fix an enumeration of $B$, $B=\{b_\alpha\colon \alpha<\kappa\}$. Let $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha<\kappa}\in({{\mathcal{O}}}_\kappa)^\kappa$ be a sequence of open covers of size $\kappa$, and let $f\colon \kappa\times\{0,1\}\to \kappa$ be a bijection. For all $\alpha<\kappa$, let ${{\mathcal{U}}}_\alpha=\{U_{\alpha,\beta}\colon \beta<\kappa\}$. Let $\langle\xi_\alpha\rangle_{\alpha\in\kappa}\in\kappa^\kappa$ be such that $b_\alpha\in U_{f(\alpha,0),\xi_\alpha}$ for all $\alpha<\kappa$. Moreover, let $G=\bigcup_{\alpha<\kappa} U_{f(\alpha,0),\xi_\alpha}$. Then $|A\setminus G|\leq\kappa$, so let $A\setminus G=\{c_\alpha\colon \alpha<\kappa\}$. Find $\langle\delta_\alpha\rangle_{\alpha\in\kappa}\in\kappa^\kappa$ such that $c_\alpha \in U_{f(\alpha,1),\delta_\alpha}$ for all $\alpha<\kappa$. Then, $$A\subseteq \bigcup_{\alpha<\kappa} U_{f(\alpha,0),\xi_\alpha} \cup \bigcup_{\alpha<\kappa} U_{f(\alpha,1),\delta_\alpha}.$$ $\square$ This allows us to formulate a stronger version of Proposition \[lus-men\]. Every Lusin set for $\kappa$ has $\kappa$-Rothberger property. Proof: By Proposition \[lusin-con\], every Lusin set for $\kappa$ satisfies the premise of Proposition \[con-roth\]. $\square$ \[lemoo\] If $X\subseteq {{\mathfrak{K}}}^\kappa$, then ${{\mathcal{A}}}={{\mathcal{O}}}_\kappa(X)$ and ${{\mathcal{B}}}={{\mathcal{O}}}_\kappa(X)$ satisfy the premise of Lemma \[s1refinement\]. $\square$ \[rothgamma\] Every $\kappa$-$\gamma$-set of cardinality $\geq \kappa$ has $\kappa$-Rothberger property. Proof: Assume that $X\subseteq {{\mathfrak{K}}}^\kappa$ is a $\kappa$-$\gamma$-set, and let $\langle {{\mathcal{U}}}_\alpha\rangle_{\alpha\in\kappa}$ be a sequence of open covers of $X$ of size $\kappa$ such that ${{\mathcal{U}}}_{\beta}$ is a refinement of ${{\mathcal{U}}}_\alpha$ for all $\alpha<\beta$. Let $\langle a_\alpha\rangle_{\alpha<\kappa}\in X^\kappa$ be a sequence of distinct points. Let $b\colon \langle\bigcup_{\alpha<\kappa}\{\alpha\}\times\alpha,\leq_{{\text{lex}}}\rangle \to \kappa$ be the order isomorphism. For $\alpha<\kappa$, let $${{\mathcal{V}}}_\alpha=\left\{\bigcup_{\beta<\alpha} U_\beta\setminus\{a_\alpha\}\colon \langle U_\beta\rangle_{\beta<\alpha}\text{ such that }\forall_{\beta<\alpha}U_\beta\in {{\mathcal{U}}}_{b(\alpha,\beta)}\right\},$$ and let ${{\mathcal{V}}}=\bigcup_{\alpha<\kappa}{{\mathcal{V}}}_\alpha$. Notice that if $B\subseteq X$ is such that $|B|=\lambda<\kappa$, then there exists $\alpha<\kappa$ such that $\lambda<\alpha$, and $a_\alpha\notin B$. Let $B=\{b_\beta\colon \beta<\lambda\}$. For $\beta<\lambda$, let $U_\beta\in {{\mathcal{U}}}_{b(\alpha,\beta)}$ be such that $b_\beta\in U_\beta$, and for $\lambda\leq\beta<\alpha$, let $U_\beta\in {{\mathcal{U}}}_{b(\alpha,\beta)}$ be arbitrary. Then $$B\subseteq \bigcup_{\beta<\alpha} U_\beta\setminus\{a_\alpha\}\in {{\mathcal{V}}}_\alpha\subseteq {{\mathcal{V}}}.$$ Thus, ${{\mathcal{V}}}$ is a $\kappa$-cover of $X$. Since $X$ is a $\kappa$-$\gamma$-set, there exist a $\kappa$-$\gamma$-cover $\langle V_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{V}}}^\kappa$. Let $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ be such that $V_\alpha\in {{\mathcal{V}}}_{\xi_\alpha}$. Notice that $|\{\xi_\alpha\colon \alpha<\kappa\}|=\kappa$, because for all $\alpha<\kappa$, $a_\alpha\notin\bigcup {{\mathcal{V}}}_\alpha$. Therefore, there exists an increasing sequence $\langle \delta_\alpha\rangle_{\alpha\in\kappa}$ such that $\langle\xi_{\delta_\alpha}\rangle_{\alpha\in \kappa}$ is strictly increasing. Then $\langle V_{\delta_\alpha}\rangle_{\alpha<\kappa}$ is a $\kappa$-$\gamma$-cover as well. For $\alpha<\kappa$, let $\langle U_{\alpha,\beta}\rangle_{\beta<\xi_{\delta_\alpha}}$ be such that $U_{\alpha,\beta}\in {{\mathcal{U}}}_{b(\xi_{\delta_\alpha},\beta)}$, for $\beta<\xi_{\delta_\alpha}$, and $$V_{\delta_\alpha}=\bigcup_{\beta<\xi_{\delta_\alpha}} U_{\alpha,\beta}\setminus\{a_{\xi_{\delta_\alpha}}\}.$$ Let $A=\{b(\xi_{\delta_\alpha},\beta)\colon \alpha<\kappa, \beta<\xi_{\delta_\alpha}\}$, and choose $\langle W_\alpha\rangle_{\alpha<\kappa}$ such that $W_\alpha=U_{\beta,\gamma}\in{{\mathcal{U}}}_{\alpha}$ if $\alpha\in A$ and $\alpha=b(\beta,\gamma)$. If $\alpha\notin A$, choose $W_\alpha\in {{\mathcal{U}}}_\alpha$ be such that $W_\alpha\supseteq W_\beta$ for $\beta=\min (A\setminus \alpha)$. Then $\{W_\alpha\colon \alpha<\kappa\}\in {{\mathcal{O}}}_\kappa$, and for any $\alpha<\kappa$, $W_\alpha\in {{\mathcal{U}}}_{\alpha}$. Therefore, by Lemmas \[s1refinement\] and \[lemoo\], $X$ satisfies $S_1^\kappa({{\mathcal{O}}}_\kappa,{{\mathcal{O}}}_\kappa)$. $\square$ Every $\kappa$-$\gamma$-set is $\kappa$-strongly null. Proof: Follows by Corollary \[roth-sn\]. $\square$ The generalized Cantor space $2^\kappa$ and the generalized Baire space $\kappa^\kappa$ are not $\kappa$-$\gamma$-sets. $\square$ Thus, no $\kappa$-perfect subset of $2^\kappa$ is a $\kappa$-$\gamma$-set. Nevertheless, the following question remains unanswered. Is there a closed subset of $2^\kappa$ which is a $\kappa$-$\gamma$-set? We finish by proving a lemma which becomes useful in the next section. \[lem-gamma\] Assume that $\kappa$ is a weakly inaccessible cardinal. Let $A\subseteq 2^\kappa$ be a $\kappa$-$\gamma$ set which is not closed. Then there exists $B\in [\kappa]^\kappa$ such that for all $C\in [B]^\kappa$, $\chi_C\notin A$. Proof: Let $A\subseteq 2^\kappa$ be a $\kappa$-$\gamma$ set, and let $b\colon\bigcup_{\alpha<\kappa}\{\alpha\}\times \alpha\to \kappa$ be a bijection. Notice that $2^\kappa\setminus A$ is not an open set. Therefore, there exists $y\in 2^\kappa\setminus A$ such that $A\cap[y{\mathord{\upharpoonright}}\alpha]\neq{\varnothing}$, for any $\alpha<\kappa$. Choose inductively a sequence $\langle x_\alpha\rangle_{\alpha<\kappa}\in A^\kappa$ such that if for $\alpha,\beta<\kappa$, $x_\alpha=x_\beta$ only if $\alpha=\beta$, and for every $\gamma<\kappa$ there exists $\alpha<\kappa$ such that $y{\mathord{\upharpoonright}}\gamma = x_\alpha{\mathord{\upharpoonright}}\gamma$. To achieve this, take any $x_0\in A$, and for $\alpha<\kappa$, let $$\xi = \bigcup_{\beta<\alpha}\bigcup\{\gamma<\kappa\colon y{\mathord{\upharpoonright}}\gamma = x_\beta{\mathord{\upharpoonright}}\gamma\}.$$ Let $x_\alpha\in A\cap [y{\mathord{\upharpoonright}}\xi+1]$. If $I\subseteq \kappa$ and $s\in 2^I$, let $[s]$ denote $\{x\in 2^\kappa\colon x{\mathord{\upharpoonright}}I=s\}$. For $\alpha<\kappa$, let $${{\mathcal{U}}}_{\alpha}=\left\{\bigcup_{s\in S} [s]\cap A\setminus \bigcup_{\alpha\leq \beta<\kappa} \{x_{\beta}\}\colon S\in \left[2^{b[\{\alpha\}\times \alpha]}\right]^{<|\alpha|}\right\},$$ and let ${{\mathcal{U}}}=\bigcup_{\alpha<\kappa}{{\mathcal{U}}}_\alpha$. Notice that ${{\mathcal{U}}}$ is a $\kappa$-cover of $A$, because $\kappa$ is weakly inaccessible. Therefore, we have $\langle U_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{U}}}^\kappa$ such that $$A\subseteq \bigcup_{\alpha<\kappa}\bigcap_{\alpha<\beta<\kappa} U_\beta.$$ But since $$x_\alpha\notin \bigcup_{\beta<\alpha}\bigcup{{\mathcal{U}}}_\beta,$$ for all $\alpha< \kappa$, we get that for any $\alpha<\kappa$, there exists $\xi<\kappa$ such that for all $\xi<\beta<\kappa$, there exists $\alpha<\gamma<\kappa$ such that $U_\beta\in{{\mathcal{U}}}_\gamma$. Therefore, we can choose inductively increasing sequences $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $\langle\delta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that $U_{\xi_\alpha}\in {{\mathcal{U}}}_{\delta_\alpha}$, for any $\alpha<\kappa$. Fix $\alpha<\kappa$, and let $S_\alpha\in\left[2^{b[\{\delta_\alpha\}\times \delta_\alpha]}\right]^{<|\delta_\alpha|}$ be such that $$U_{\xi_\alpha}=\bigcup_{s\in S_\alpha}[s]\cap A\setminus \bigcup_{\delta_\alpha\leq \beta<\kappa} \{x_{\beta}\}.$$ There exists $\eta_\alpha<\delta_\alpha$ such that $\chi_{a}\notin S_\alpha$ for any $a\supseteq \{b(\delta_\alpha,\eta_\alpha)\}$. Let $$B=\{b(\delta_\alpha,\eta_\alpha)\colon \alpha<\kappa\}.$$ Then, for all $C\in [B]^\kappa$, $\chi_C\notin A$. Indeed, if $C\in [B]^\kappa$, then for every $\alpha<\kappa$, there is $\alpha<\beta<\kappa$ such that for $$a=C\cap b[\{\delta_\beta\}\times \delta_\beta\}]=\{b(\delta_\beta,\eta_\beta)\}$$ we get that $\chi_a\notin S_\beta$. For such $\beta$, $\chi_C\notin U_{\xi_\beta}$, therefore for all $\alpha<\kappa$, $$\chi_C\notin \bigcap_{\alpha<\beta<\kappa}U_{\xi_\beta},$$ and hence $$\chi_C\notin \bigcup_{\alpha<\kappa}\bigcap_{\alpha<\beta<\kappa} U_\beta \supseteq A.$$  $\square$ Generalization of other notions of small sets in $2^\kappa$ and $\kappa^\kappa$ {#chother} =============================================================================== In this section we present generalizations of some less common notions of small sets. Some of the results presented here have their counterparts in the standard case of $2^\omega$ (or $\omega_1^{\omega_1}$), and if so, we give a reference in the form ($\omega$: \[n\]) (or ($\omega_1$: \[n\])). In this section we use notation and notions described in Sections \[intro-special\], \[intro-gen\], and Section \[chsimple\]. $X$-small sets -------------- In this section we present some parallels of the results from [@ah:nsgbs Chapter 4]. If $X\subseteq \kappa$, then a set $A\subseteq {{\mathfrak{K}}}^\kappa$ will be called [[**$X$-small**]{}]{} if there exists $\langle a_\alpha\rangle_{\alpha\in X}\in ({{\mathfrak{K}}}^\kappa)^X$ such that $$A\subseteq \bigcup_{\alpha\in X}[a_\alpha{\mathord{\upharpoonright}}\alpha].$$ Notice that $A$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ if it is $X$-small for any $X\in [\kappa]^\kappa$. Consider the following order on $[\kappa]^\kappa$. For $X,Y\subseteq [\kappa]^\kappa$, let $X<Y$ (respectively, $X\leq Y$) if and only if there exists a bijection $F\colon X\to Y$ such that for all $\alpha\in X$, $\alpha< F(\alpha)$ (respectively, $\alpha\leq F(\alpha)$). Let $X+1=\{\alpha+1\colon \alpha\in X\}$. Notice that if $X<Y$, then $X+1\leq Y$. Let $X,Y\in [\kappa]^\kappa$ be such that $X<Y$. Then, the family of $Y$-small sets is a proper subfamily of $X$-small sets (see [@ah:nsgbs]). Indeed, it is sufficient to prove that there exists a $X$-small set which is not a $(X+1)$-small. Assume that $A\subseteq\bigcup_{\alpha\in X}[a_\alpha{\mathord{\upharpoonright}}\alpha]$ with $\langle a_\alpha\rangle_{\alpha\in X}\in ({{\mathfrak{K}}}^\kappa)^X$. We can assume that if $\beta,\alpha\in X$ with $\beta>\alpha$, then $a_\beta\notin [a_\alpha{\mathord{\upharpoonright}}\alpha]$. To obtain a contradiction assume that $$A\subseteq B=\bigcup_{\alpha\in X}[b_\alpha{\mathord{\upharpoonright}}\alpha+1]$$ with $\langle b_\alpha\rangle_{\alpha\in X}\in ({{\mathfrak{K}}}^\kappa)^X$. Then consider $x\in {{\mathfrak{K}}}^\kappa$ such that $$x(\alpha)=\begin{cases} a_{\min X}(\alpha), & \text{ if } \alpha<\min X,\\ b_\alpha(\alpha)+1,& \text{ if } \alpha\in X, \\ 0, & \text{ otherwise.} \end{cases}$$ Notice that $x\in [a_{\min X}{\mathord{\upharpoonright}}\min X]\subseteq A$, but $x\notin B$, which is a contradiction. Let $\lambda<\kappa$. We say that a set $A\subseteq {{\mathfrak{K}}}^\kappa$ is [[**$\lambda$-$X$-small**]{}]{} for $X\subseteq \kappa$ if there exists $\langle a_{\alpha,\beta}\rangle_{\alpha\in X, \beta<\lambda}\in \left(({{\mathfrak{K}}}^\kappa)^X\right)^\lambda$ such that $$A\subseteq \bigcup_{\alpha\in X}\bigcup_{\beta<\lambda}[a_{\alpha,\beta}{\mathord{\upharpoonright}}\alpha].$$ $A\subseteq {{\mathfrak{K}}}^\kappa$ is [[**${{\mathcal{X}}}$-null**]{}]{} for ${{\mathcal{X}}}\subseteq [\kappa]^{\leq\kappa}$ if for all $X\in {{\mathcal{X}}}$, $A$ is $X$-small, and [[**$\lambda$-${{\mathcal{X}}}$-null**]{}]{} if for all $X\in {{\mathcal{X}}}$, $A$ is $\lambda$-$X$-small. Obviously, $A$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ if and only if $A$ is $[\kappa]^\kappa$-null. The notion of $\lambda$-${{\mathcal{X}}}$-null sets for ${{\mathcal{X}}}\subseteq [\kappa]^\lambda$ does not depend precisely on ${{\mathcal{X}}}$. Indeed, we get the following proposition. \[null\] Let $\lambda<\kappa$. A set $A\subseteq {{\mathfrak{K}}}^\kappa$ is $\lambda$-$\{\{\alpha\}\colon \alpha<\kappa\}$-null in ${{\mathfrak{K}}}^\kappa$ if and only if it is $[\kappa]^\lambda$-null. Proof: Let $\lambda<\kappa$, and assume that $A$ is $\lambda$-$\{\{\alpha\}\colon \alpha<\kappa\}$-null. Let $X=\{\xi_\beta\colon\beta<\lambda\}\in [\kappa]^\lambda$ and $\alpha=\bigcup X$. Obviously, $\alpha<\kappa$. Therefore, there exists a sequence $\langle a_\beta\rangle_{\beta<\lambda}$ such that $$A\subseteq \bigcup_{\beta<\lambda} [a_{\beta}{\mathord{\upharpoonright}}\alpha]\subseteq \bigcup_{\beta<\lambda} [a_{\beta}{\mathord{\upharpoonright}}\xi_{\beta}],$$ so $A$ is $X$-small. On the other hand, assume that $A$ is $[\kappa]^\lambda$-null and $\alpha<\kappa$. Then let $X=\{\alpha+\beta\colon \beta<\lambda\}\in [\kappa]^\lambda$. There exists a sequence $\langle a_\beta\rangle_{\beta<\lambda}$ such that $$A\subseteq \bigcup_{\beta<\lambda} [a_{\beta}{\mathord{\upharpoonright}}\alpha+\beta]\subseteq \bigcup_{\beta<\lambda} [a_{\beta}{\mathord{\upharpoonright}}\alpha],$$ so $A$ is $\lambda$-$\{\alpha\}$-small.  $\square$ A set $A\subseteq {{\mathfrak{K}}}^\kappa$ will be called [[**small in ${{\mathfrak{K}}}^\kappa$**]{}]{} if there exists $\lambda<\kappa$ such that $A$ is $\lambda$-$\{\{\alpha\}\colon \alpha<\kappa\}$-null. Obviously, every $A\subseteq {{\mathfrak{K}}}^\kappa$ with $|A|<\kappa$ is small in ${{\mathfrak{K}}}^\kappa$. Notice that every small set in ${{\mathfrak{K}}}^\kappa$ is $\kappa$-strongly null. \[smallsn\] Let $A\subseteq {{\mathfrak{K}}}^\kappa$ be small in ${{\mathfrak{K}}}^\kappa$. Then $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Proof: Let $\lambda<\kappa$ be such that $A$ is $\lambda$-$\{\{\alpha\}\colon \alpha<\kappa\}$-null. Therefore, by Proposition \[null\], $A$ is $[\kappa]^\lambda$-null. Let $X=\{\xi_{\alpha}\colon \alpha<\kappa\}\in [\kappa]^\kappa$. There exists a sequence $\langle a_{\alpha}\rangle_{\alpha<\lambda}\in ({{\mathfrak{K}}}^\kappa)^\lambda$ such that $A\subseteq \bigcup_{\alpha<\lambda}[a_{\alpha}{\mathord{\upharpoonright}}\xi_\alpha]$. For $\lambda\leq \alpha<\kappa$ set $a_\alpha={\boldsymbol{0}}$. We get that $A\subseteq \bigcup_{\alpha<\kappa}[a_{\alpha}{\mathord{\upharpoonright}}\xi_\alpha]$.  $\square$ A set $A\subseteq {{\mathfrak{K}}}^\kappa$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$ if and only if there exists $\lambda<\kappa$ such that $A$ is $\lambda$-$[\kappa]^\kappa$-null. Proof: If $A\subseteq {{\mathfrak{K}}}^\kappa$ is ${\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$, it is obviously $\lambda$-$[\kappa]^\kappa$-null for all $\lambda<\kappa$. Assume that $\lambda<\kappa$, and $A\subseteq {{\mathfrak{K}}}^\kappa$ is $\lambda$-$[\kappa]^\kappa$-null. Let $X=\{\xi_\alpha\colon \alpha<\kappa\}\in [\kappa]^\kappa$. Let $b\colon \lambda\times\kappa\to\kappa$ be a bijection, and for all $\alpha<\kappa$, let $X_\alpha=\{\xi_{b(\beta,\alpha)}\colon\beta<\lambda\}\in [\kappa]^\lambda$. Let $\delta_\alpha=\bigcup X_\alpha$, for $\alpha<\kappa$. Finally, let $Y=\{\delta_\alpha\colon \alpha<\kappa\}\in [\kappa]^\kappa$. We can find $\langle x_{\alpha,\beta}\rangle_{\alpha<\kappa,\beta<\lambda}\in ({{\mathfrak{K}}}^\kappa)^{\kappa\times\lambda}$ such that $$A\subseteq \bigcup_{\alpha<\kappa}\bigcup_{\beta<\lambda}[x_{\alpha,\beta}{\mathord{\upharpoonright}}\delta_\alpha].$$ For $\alpha<\kappa$, let $z_{\alpha}=x_{b^{-1}(\alpha)}$. Then $$A\subseteq \bigcup_{\alpha<\kappa}[z_{\alpha}{\mathord{\upharpoonright}}\delta_{\pi_2(b^{-1}(\alpha))}]\subseteq \bigcup_{\alpha<\kappa}[z_{\alpha}{\mathord{\upharpoonright}}\xi_{\alpha}].$$ $\square$ Let $X\subseteq\kappa$ be such that $0\notin X$ and $X\cap{\text{Lim}}={\varnothing}$. If $A\subseteq 2^\kappa$ is $X$-small, then $|2^\kappa\setminus A|=2^\kappa$. Proof: Let $\langle x_\alpha\rangle_{\alpha\in X}\in \left(2^\kappa\right)^X$ be such that $A\subseteq\bigcup_{\alpha\in X}[x_\alpha{\mathord{\upharpoonright}}\alpha]$. Consider the set $$B=\left\{x\in 2^\kappa\colon \forall_{\alpha<\kappa}\left(\alpha+1\in X\Rightarrow x(\alpha)=x_{\alpha+1}(\alpha)+1\right)\right\}.$$ Then for all $\alpha\in X$, $B\cap [x_\alpha{\mathord{\upharpoonright}}\alpha]={\varnothing}$. Thus, $B\cap A={\varnothing}$. Furthermore, $B$ contains a set homeomorphic to $2^\kappa$, so $|2^\kappa\setminus A|=2^\kappa$. $\square$ Next we study a connection between the diamond principle for $\kappa$ (see section \[intro-gen\]) and the notion of $C$-smallness for closed unbounded or stationary sets $C\subseteq \kappa$. For $E\subseteq \kappa$, $A\subseteq 2^\kappa$, ${{\mathcal{I}}}\subseteq [\kappa]^{\leq\kappa}$, let $\diamondsuit_\kappa(E,A,{{\mathcal{I}}})$ denote the following principle: there exists a sequence $\langle s_\alpha\rangle_{\alpha<\kappa}\in \left(2^{<\kappa}\right)^\kappa$ such that for all $x\in A$, $$\{\alpha\in E\colon x{\mathord{\upharpoonright}}\alpha=s_{\alpha}\}\notin {{\mathcal{I}}}.$$ Notice the following easy observation. If $A\subseteq 2^\kappa$, and $E\subseteq \kappa$, then $\diamondsuit_\kappa(E,A,\{{\varnothing}\})$ if and only if $A$ is $E$-small. Proof: Indeed, $\diamondsuit_\kappa(E,A,\{{\varnothing}\})$ if and only if for all $x\in A$, $$\{\alpha\in E\colon x{\mathord{\upharpoonright}}\alpha=s_{\alpha}\}\neq{\varnothing}.$$ $\square$ \[diamond\] Let $E\subseteq \kappa$. The principle $\diamondsuit_\kappa(E,2^\kappa,{\boldsymbol{{{\mathcal{N}}}{{\mathcal{S}}}}}_\kappa)$ holds if and only if $\diamondsuit_\kappa(E)$ holds. Proof: Let $\langle s_\alpha\rangle_{\alpha\in E}\in \left(2^{<\kappa}\right)^\kappa$ be such that for all $x\in 2^\kappa$, $\{\alpha\in E\colon x{\mathord{\upharpoonright}}\alpha=s_{\alpha}\}$ is stationary in $\kappa$. Let $S_\alpha=s_\alpha^{-1}[\{1\}]\cap \alpha$, for $\alpha\in E$, and let $X\subseteq \kappa$. Then $$\left\{\alpha\in E\colon X\cap \alpha=S_\alpha\right\}=\left\{\alpha\in E\colon \chi_X{\mathord{\upharpoonright}}\alpha =s_\alpha\right\}$$ is a stationary subset of $\kappa$, so $\diamondsuit_\kappa(E)$ holds. Similarly, if $\langle S_\alpha\rangle_{\alpha<\kappa}\in \left([\kappa]^{<\kappa}\right)^\kappa$ is such that for any $X\subseteq \kappa$, $\left\{\alpha\in E\colon X\cap \alpha=S_\alpha\right\}$ is stationary, let $s_\alpha=\chi_{S_\alpha\cap \alpha}$, for $\alpha<\kappa$. This sequence witnesses $\diamondsuit_\kappa(E,2^\kappa,{\boldsymbol{{{\mathcal{N}}}{{\mathcal{S}}}}}_\kappa)$. $\square$ \[clubsmall\] Assume $\diamondsuit_\kappa$. If $C$ is a closed unbounded set in $\kappa$, then $2^\kappa$ is $C$-small. Proof: By Proposition \[diamond\], there exists a sequence $\langle s_\alpha\rangle_{\alpha<\kappa}\in \left(2^{<\kappa}\right)^\kappa$ such that for all $x\in 2^\kappa$, $\{\alpha\in \kappa\colon x{\mathord{\upharpoonright}}\alpha=s_{\alpha}\}$ is stationary in $\kappa$. Therefore, if $C$ is a closed unbounded set in $\kappa$, then $\{\alpha\in C\colon x{\mathord{\upharpoonright}}\alpha=s_{\alpha}\}$ is stationary, thus non-empty for all $x\in 2^\kappa$. Therefore, $2^\kappa=\bigcup_{\alpha\in C} [s_\alpha]$. $\square$ \[diamondEsmall\] Let $E\subseteq\kappa$, and assume $\diamondsuit_\kappa(E)$. Then $2^\kappa$ is $E$-small. Proof: By Proposition \[diamond\], there exists a sequence $\langle s_\alpha\rangle_{\alpha<\kappa}\in \left(2^{<\kappa}\right)^\kappa$ such that for all $x\in 2^\kappa$, $\{\alpha\in E\colon x{\mathord{\upharpoonright}}\alpha=s_{\alpha}\}$ is stationary in $\kappa$. So it is not empty, and $2^\kappa=\bigcup_{\alpha\in E} [s_\alpha]$. $\square$ \[vlsmall\] Assume $V=L$. Then $2^\kappa$ is $X$-small for every stationary set $X\subseteq\kappa$. Proof: Recall that $V=L$ implies $\diamondsuit_\kappa(X)$ for every stationary set $X\subseteq \kappa$ (see [@kk:stiip Exercise VI.14]). Therefore, by Proposition \[diamondEsmall\], $2^\kappa$ is small for every stationary $X\subseteq \kappa$. $\square$ The whole space $2^\kappa$ can be presented as a union of a $\kappa$-meagre set, and a ${{\mathcal{X}}}$-null set for ${{\mathcal{X}}}\in \left[[\kappa]^\kappa\right]^\kappa$. Let ${{\mathcal{X}}}\in \left[[\kappa]^\kappa\right]^\kappa$. There exist $A,B\subseteq 2^\kappa$ such that $A$ is ${{\mathcal{X}}}$-null and $B$ is $\kappa$-meagre, and $A\cup B=2^\kappa$. Proof: Let ${{\mathcal{X}}}\in \left[[\kappa]^\kappa\right]^\kappa$. Let ${\boldsymbol{Q}}=\{q_\alpha\colon \alpha<\kappa\}$ and let ${{\mathcal{X}}}=\{X_\alpha\colon \alpha<\kappa\}$, and $X_\alpha=\{x_{\alpha,\beta}\colon\beta<\kappa\}$ be enumerations. For $\alpha<\kappa$, put $$A_{\alpha}=\bigcup_{\beta<\kappa}[q_{\beta}{\mathord{\upharpoonright}}x_{\alpha,\beta}].$$ Notice that $2^\kappa\setminus A_\alpha$ is nowhere dense, therefore, if $A=\bigcap_{\alpha<\kappa} A_\alpha$, then $2^\kappa\setminus A$ is $\kappa$-meagre. Obviously, $A$ is ${{\mathcal{X}}}$-null. $\square$ On the other hand, we have the following. \[smnd\] Every small set in ${{\mathfrak{K}}}^\kappa$ is nowhere dense. Proof: Let $\lambda<\kappa$ be such that $A\subseteq 2^\kappa$ is $\lambda-\{\{\alpha\}\colon \alpha<\kappa\}$-null. Let $s\in 2^\beta$ with $\beta<\kappa$, and let $\xi=\beta+\lambda$. There exists $\langle x_\alpha\rangle_{\alpha<\lambda}\in ({{\mathfrak{K}}}^\kappa)^\lambda$ such that $$A\subseteq \bigcup_{\alpha<\lambda}[x_\alpha{\mathord{\upharpoonright}}\xi].$$ But $|\{x{\mathord{\upharpoonright}}\xi\colon x\in [s]\}|=2^\lambda$, thus there exists $t\in 2^\xi$ such that $s\subseteq t$, and $[t]\cap A={\varnothing}$. $\square$ But not every nowhere dense set in ${{\mathfrak{K}}}^\kappa$ is small in ${{\mathfrak{K}}}^\kappa$. \[ndnsm\] There exists a nowhere dense set $A\subseteq {{\mathfrak{K}}}^\kappa$ which is not $\kappa$-strongly null. Proof: Let $\langle \xi_\alpha\rangle\in \kappa^\kappa$ be an increasing sequence of limit ordinals. Let $$A=\left\{x\in {{\mathfrak{K}}}^\kappa\colon \forall_{\alpha<\kappa}x(\xi_\alpha)=0\right\}.$$ Obviously, $A$ is nowhere dense. Assume that $A\in {\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Then there exists $\langle x_\alpha\rangle_{\alpha<\kappa}\in ({{\mathfrak{K}}}^\kappa)^\kappa$ such that $A\subseteq \bigcup_{\alpha<\kappa} [x_\alpha{\mathord{\upharpoonright}}\xi_\alpha+1]$. Let $x\in 2^\kappa$ be such that $x(\xi_\alpha)=x_\alpha(\xi_\alpha)+1$ for all $\alpha<\kappa$, and $x(\beta)=0$ for $\beta\notin\{\xi_\alpha\colon \alpha\in \kappa\}$. Then $x\in A$, but $x\notin \bigcup_{\alpha<\kappa} [x_\alpha{\mathord{\upharpoonright}}\xi_\alpha+1]$, which is a contradiction. $\square$ $\kappa$-Meagre additive sets ----------------------------- In this section we present some generalizations of results concerning meagre additive sets. We start by generalizing the combinatorial characterization of meagre sets (see [@tbhj:stsrl Theorem 2.2.4]). \[char-meagre\] Assume that $\kappa$ is strongly inaccessible, and $A\subseteq 2^\kappa$ is a $\kappa$-meagre set. Then there exist $y\in 2^\kappa$ and an increasing sequence $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \exists_{\beta<\kappa}\forall_{\beta<\gamma<\kappa} \exists_{\xi_{\gamma}\leq\xi<\xi_{\gamma+1}} x(\xi)\neq y(\xi)\right\}.$$ Proof: Let $A\subseteq\bigcup_{\alpha<\kappa} F_\alpha$ with $F_\alpha$ closed nowhere dense for all $\alpha<\kappa$. Additionally, we assume that if $\alpha<\beta<\kappa$, then $F_\alpha\subseteq F_\beta$. We define $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $y\in 2^\kappa$ by induction. Let $\xi_0=0$. Assume that $\eta<\kappa$, and $\xi_{\eta}$ and $y{\mathord{\upharpoonright}}\eta$ are defined. Let $\langle t_{\alpha,\eta}\rangle_{\alpha<\delta_\eta}$ be an enumeration of $2^\eta$. Notice that $\delta_\eta<\kappa$, since $\kappa$ is assumed to be strongly inaccessible. Define inductively $\langle s_{\alpha,\eta}\rangle_{\alpha<\delta_\eta}$ such that (a) if $\alpha\leq\beta<\delta_\eta$, then $s_{\alpha,\eta}\subseteq s_{\beta,\eta}$, (b) $[t_{\alpha,\eta}\,^\frown s_{\alpha,\eta}]\cap F_\eta={\varnothing}$. Let $s_\eta=\bigcup_{\alpha<\delta_\eta}s_{\alpha,\eta}$, and let ${\text{len}}(s_\eta)=\gamma_\eta$. Obviously, $\gamma_\eta<\kappa$. Set $\xi_{\eta+1}=\xi_\eta+\gamma_\eta$ and $y(\xi_{\eta}+\alpha)=s_\eta(\alpha)$ for $\alpha<\gamma_\eta$. If $\eta<\kappa$ is a limit ordinal set $\xi_\eta=\bigcup_{\alpha<\eta} \xi_\alpha$. It follows that if $x\in 2^\kappa$, and the set of all $\gamma<\kappa$ such that for all $\xi$ such that $\xi_{\gamma}\leq\xi<\xi_{\gamma+1}$, we have $x(\xi)= y(\xi)$, is cofinal in $\kappa$, then for all $\alpha<\kappa$, there exists $\gamma<\kappa$ with $\gamma\geq \alpha$, and $x\notin F_\gamma$. Therefore, $x\notin \bigcup_{\alpha<\kappa} F_\alpha\supseteq A$.  $\square$ A set $A\subseteq 2^\kappa$ will be called [[**$\kappa$-meagre additive**]{}]{} if for any $\kappa$-meagre set $F$, $A+F$ is $\kappa$-meagre. The family of all $\kappa$-meagre additive sets we denote by ${\boldsymbol{{{\mathcal{M}}}}}_{\kappa}^*$. By Proposition \[gms1\], we immediately get the following Corollary. Every $\kappa$-meagre additive set is $\kappa$-strongly null.  $\square$ The following theorem is a generalization of the characterization of meagre-additive sets ([@tbhj:stsrl Theorem 2.7.17], see also Section \[intro-special\]). \[madditive\] Assume that $\kappa$ is strongly inaccessible, and $X\subseteq 2^\kappa$. Then $X\in {\boldsymbol{{{\mathcal{M}}}}}_\kappa^*$ if and only if for every increasing sequence $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ there exist a sequence $\langle\eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $z\in 2^\kappa$ such that $$X\subseteq \left\{ x\in 2^\kappa \colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa}\exists_{\gamma<\kappa} \left(\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta)\right)\right\}.$$ Proof: Assume that $X\in {\boldsymbol{{{\mathcal{M}}}}}_\kappa^*$, and $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$. Let $$B=\left\{y\in 2^\kappa\colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa} \exists_{\xi_\beta\leq \delta<\xi_{\beta+1}} y(\delta)\neq 0\right\}.$$ Obviously, $B$ is $\kappa$-meagre, so $X+B$ is also $\kappa$-meagre, and $X+B=\bigcup_{x\in X}B_x$, where $$B_x=\left\{y\in 2^\kappa\colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa} \exists_{\xi_\beta\leq \delta<\xi_{\beta+1}} y(\delta)\neq x(\delta)\right\}.$$ By Proposition \[char-meagre\], there exist a sequence $\langle\eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $z\in 2^\kappa$ such that $$X+B\subseteq C=\left\{a\in 2^\kappa\colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa} \exists_{\eta_{\beta}\leq\delta<\eta_{\beta+1}} a(\delta)\neq z(\delta)\right\}.$$ Therefore, for any $x\in X$, $B_x\subseteq C$. Similarly to [@tbhj:stsrl Lemma 2.7.5], we prove that there exists $\alpha<\kappa$ such that for all $\alpha<\beta<\kappa$, there exists $\gamma<\kappa$ such that $\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}$ and for all $\xi_\gamma\leq \delta< \xi_{\gamma+1}$, we get $x(\delta)=z(\delta)$. Indeed, let $$S=\left\{\beta<\kappa\colon \lnot\exists_{\gamma<\kappa}\left(\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1} \land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta) \right)\right\}.$$ To obtain a contradiction, assume that for all $\alpha<\kappa$, $S\setminus\alpha\neq{\varnothing}$. Let $S=\{\sigma_\alpha\colon\alpha<\kappa\}$, and let $S'=\{\sigma_\alpha\colon \alpha<\kappa \land \alpha\text{ is a~limit ordinal}\}$. Finally, let $$D=\left\{\alpha<\kappa\colon \exists_{\beta\in S'}\, \eta_\beta\leq \alpha <\eta_{\beta+1}\right\}.$$ Notice that if for $\beta<\kappa$, $\{\delta<\kappa\colon \xi_\beta\leq \delta<\xi_{\beta+1}\}\subseteq D$, then there exists $\xi_\beta\leq \delta<\xi_{\beta+1}$ such that $x(\delta)\neq z(\delta)$. Let $y\in 2^\kappa$ be such that $$y(\delta)=\begin{cases}z(\delta),&\text{ if }\delta\in D,\\ x(\delta)+1,& \text{ otherwise}.\end{cases}$$ Then $y\in B_x$, but $y\notin C$, which is a contradiction. Therefore, $$X\subseteq \left\{ x\in 2^\kappa \colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa}\exists_{\gamma<\kappa} \left(\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta)\right)\right\}.$$ Conversely, assume that $X\subseteq 2^\kappa$ is such that for every sequence $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, there exist a sequence $\langle\eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $z\in 2^\kappa$ such that $$X\subseteq \left\{ x\in 2^\kappa \colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa}\exists_{\gamma<\kappa} \left(\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta)\right)\right\}.$$ Let $F$ be $\kappa$-meagre. Then, by Proposition \[char-meagre\] we get a sequence $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $y\in 2^\kappa$ such that $$F\subseteq F'=\left\{a\in 2^\kappa\colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa} \exists_{\xi_{\beta}\leq\delta<\xi_{\beta+1}} a(\delta)\neq y(\delta)\right\}.$$ Let $\langle\eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $z\in 2^\kappa$ be such that $$X\subseteq \left\{ x\in 2^\kappa \colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa}\exists_{\gamma<\kappa} \left(\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta)\right)\right\}.$$ Then $$X+F\subseteq X+F'\subseteq \left\{a\in 2^\kappa\colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa} \exists_{\eta_{\beta}\leq\delta<\eta_{\beta+1}} a(\delta)\neq y(\delta)+z(\delta)\right\},$$ which is a $\kappa$-meagre set. Therefore, $X\in{\boldsymbol{{{\mathcal{M}}}}}_\kappa^*$.  $\square$ Notice that this implies that under the same assumption every $\kappa$-meagre additive set is ${\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$. \[mapkmk\] Assume that $\kappa$ is a strongly inaccessible cardinal. Then every $\kappa$-meagre additive set is ${\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$. Proof: Let $A\in{\boldsymbol{{{\mathcal{M}}}}}_\kappa^*$, and let $P\subseteq 2^\kappa$ be a $\kappa$-perfect set. By induction we construct a sequence $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that $\xi_0=0$, and for $\alpha<\kappa$, $$\xi_{\alpha+1}=\bigcup_{t\in T_P\cap 2^{\xi_\alpha}}\min\left\{{\text{len}}(s)\colon t\subseteq s\in {\text{Split}}(T_P)\right\}+1.$$ Finally, for limit $\alpha<\kappa$, let $\xi_\alpha=\bigcup_{\beta<\alpha} \xi_\beta$. By Proposition \[madditive\], we can find a sequence $\langle\eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $z\in 2^\kappa$ such that $$A\subseteq \bigcup_{\alpha<\kappa}\left\{ x\in 2^\kappa \colon \forall_{\alpha<\beta<\kappa}\exists_{\gamma<\kappa} \left(\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta)\right)\right\}.$$ Let $\alpha<\kappa$, and let $s\in T_P$. Fix $s'\in T_P$ such that $s\subseteq s'$, and for some $\beta>\alpha$, ${\text{len}}(s')=\eta_\beta$. Let $$\gamma_0=\min\left\{\gamma<\kappa\colon\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\right\}$$ and $$\gamma_1= \bigcup\left\{\gamma<\kappa\colon \eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\right\}+1.$$ Inductively, we construct a sequence $\langle t_{\delta}\rangle_{\gamma_0\leq\delta\leq\gamma_1}$ such that for all $\gamma_0\leq\delta\leq\delta'\leq\gamma_1$, $t_{\delta}\in T_P\cap 2^{\xi_\delta}$, $t_\delta\subseteq t_{\delta'}$, and $\exists_{\xi_\delta\leq \xi< \xi_{\delta+1}} t_{\delta+1}(\xi)\neq z(\xi)$. Indeed, let $t_{\gamma_0}\in T_P$ be such that $s\subseteq t_{\gamma_0}$, and ${\text{len}}(t_{\gamma_0})=\xi_{\gamma_0}$. Given $t_\delta$, by definition of $\langle \xi_\alpha\rangle_{\alpha<\kappa}$, one can find $t_{\delta+1}\supseteq t_\delta$ such that $\exists_{\xi_\delta\leq \xi< \xi_{\delta+1}} t_{\delta+1}(\xi)\neq z(\xi)$, because $|\{t\in T_P\cap 2^{\xi_{\delta+1}}\colon t\supseteq t_{\delta}\}|\geq 2$. For limit $\delta<\kappa$, set any $t_\delta\supseteq \bigcup_{\gamma_0\leq\xi<\delta} t_\xi$ such that ${\text{len}}(t_\delta)=\xi_\delta$. Then, $$[t_{\gamma_1}]\cap P\cap \left\{ x\in 2^\kappa \colon \forall_{\alpha<\beta<\kappa}\exists_{\gamma<\kappa} \left(\eta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \eta_{\beta+1}\land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta)\right)\right\}$$ is empty, and hence $A$ is $\kappa$-meagre in $P$. $\square$ $\kappa$-Ramsey null sets ------------------------- In this section we generalize some results presented in [@antw:rpsssas]. For $\alpha<\kappa$, $s\in 2^\alpha$ and $S\in [\kappa\setminus \alpha]^\kappa$, let $$[s,S]=\{x\in 2^\kappa\colon s^{-1}[\{1\}]\subseteq x^{-1}[\{1\}]\subseteq s^{-1}[\{1\}]\cup S \land |x^{-1}[\{1\}]\cap S|=\kappa\}.$$ A set $A\subseteq 2^\kappa$ will be called [[**$\kappa$-Ramsey null ($\kappa-CR_0$)**]{}]{} if for any $\alpha<\kappa$, $s\in 2^\alpha$ and $S\in [\kappa\setminus \alpha]^\kappa$, there exists $S'\in [S]^\kappa$ such that $[s,S']\cap A={\varnothing}$. It is a well-known fact that the ideal of Ramsey null subsets of $2^\omega$ is a $\sigma$-ideal (see e.g. [@lh:cst]). We do not know whether the analogue holds for $\kappa$-Ramsey null sets. \[qrams\] Is the ideal of $\kappa$-Ramsey null subsets of $2^\kappa$ $\kappa^+$-complete? \[gammarams\] Assume that $\kappa$ is a weakly inaccessible cardinal. Then every $\kappa$-$\gamma$-set which is not closed in $2^\kappa$ is $\kappa$-Ramsey null. Proof: The proof is similar to the proof of [@antw:rpsssas Theorem 2.1]. Namely, let $A\subseteq 2^\kappa$ be a $\kappa$-$\gamma$-set, and $\delta<\kappa$, $s\in 2^\delta$ and $$S=\{\xi_\alpha\colon \alpha<\kappa\}\in [\kappa\setminus \delta]^\kappa.$$ Let $$E=\{x\in 2^\kappa\colon s^{-1}[\{1\}]\subseteq x^{-1}[\{1\}]\subseteq s^{-1}[\{1\}]\cup S\}=s_0+S_0,$$ where $s_0=s\cup\{\langle\beta,0\rangle\colon \beta\in\kappa\setminus\delta\}$ and $S_0=\{f\cup\{\langle\beta,0\rangle\colon \beta\notin S\}\colon f\in 2^S\}$. Notice that $S_0$ is a closed set in $2^\kappa$, and so is $E$. Moreover, $\varphi\colon 2^\kappa\to E$ given by the following expression $$\varphi(x)=s_0+\chi_{\{\xi_\alpha\colon x(\alpha)=1\land\alpha<\kappa\}}$$ is a homeomorphism. By Proposition \[cl-gamma\], $E\cap A$ is a $\kappa$-$\gamma$ set, and therefore so is $\varphi^{-1}[E\cap A]$. By Lemma \[lem-gamma\], there exists $B\in [\kappa]^\kappa$ such that for all $C\in [B]^\kappa$, $\chi_C\notin \varphi^{-1}[E\cap A]$, which means that $\varphi(\chi_C)\notin A$. Let $S'=\{\xi_\alpha\colon \alpha\in B\}$. Then $S'\in [S]^\kappa$, and $[s,S']=\{\varphi(\chi_C)\colon C\in [B]^\kappa\}$. Thus, $[s,S']\cap A={\varnothing}$. $\square$ \[lem-mm\] If $A,B\subseteq 2^\kappa$, then $$2^\kappa\setminus\left(A+(2^\kappa\setminus B)\right)=\{x\in 2^\kappa\colon x+A\subseteq B\}.$$ Proof: The proof of [@antw:rpsssas]\[Lemma 4.1\] is valid for any vector space over ${\mathbb{Z}}_2$.$\square$ \[mmrn\] Assume that $\kappa$ is strongly inaccessible, and $A\subseteq 2^\kappa$ is a $\kappa$-meagre set. Then there exists a $\kappa$-meagre set $B\subseteq 2^\kappa$ such that $A+(2^\kappa\setminus B)$ is $\kappa$-Ramsey null. Proof: By Proposition \[char-meagre\], we get $z\in 2^\kappa$ and a sequence $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that $A\subseteq A'$, where $$A'= \left\{x\in 2^\kappa\colon \exists_{\beta<\kappa}\forall_{\beta<\gamma<\kappa} \exists_{\xi_{\gamma}\leq\xi<\xi_{\gamma+1}} z(\xi)\neq x(\xi)\right\}.$$ Fix a bijection $b\colon \kappa\times 2\to \kappa$. Let $$B=\left\{x\in 2^\kappa\colon \exists_{\beta<\kappa}\forall_{\beta<\alpha<\kappa} \exists_{\gamma\in \left(\left(\xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}\right)\cup \left(\xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}\right)\right)}x(\gamma) \neq z(\gamma)\right\}$$ Let $\eta<\kappa$, $s\in 2^\eta$ and $S\in [\kappa\setminus \eta]^\kappa$. We shall find $S'\subseteq S$ such that $$[s,S']\cap (A'+(2^\omega\setminus B))={\varnothing}.$$ Let $S'\in [S]^\kappa$ be such that for all $\alpha<\kappa$ such that $\xi_{b(\alpha,0)},\xi_{b(\alpha,1)}>\alpha$, $$\left|\left( \left(\xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}\right)\cup \left(\xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}\right) \right)\cap S'\right|\leq 1.$$ Let $v\in [s,S']$, and assume that $v=a+b$ for some $a\in A'$, $b\in 2^\omega\setminus B$. Thus, (a) there exists $\xi<\kappa$ such that for all $\xi<\alpha<\kappa$, there exists $\gamma_0\in \xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}$ and $\gamma_1\in \xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}$ such that $a(\gamma_0)\neq z(\gamma_0)$ and $a(\gamma_1)\neq z(\gamma_1)$, (b) for every $\delta<\kappa$, there exists $\delta<\alpha<\kappa$ such that for all $\beta\in \left(\xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}\right)\cup \left(\xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}\right)$, $b(\beta)=z(\beta)$. Hence, there exists $\alpha<\kappa$ such that (i) there exists at most one $\eta\in \left(\xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}\right)\cup \left(\xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}\right)$ such that $v(\eta)=1$, (ii) there exists $\gamma_0\in \xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}$ and $\gamma_1\in \xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}$ such that $a(\gamma_0)\neq z(\gamma_0)$ and $a(\gamma_1)\neq z(\gamma_1)$, (iii) for all $\beta\in \left(\xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}\right)\cup \left(\xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}\right)$, $b(\beta)=z(\beta)$. Then, either for all $\beta\in \xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}$, $v(\eta)=0$, or for all $\beta\in \xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}$, $v(\eta)=0$. Hence, either for all $\beta\in \xi_{b(\alpha,0)+1}\setminus\xi_{b(\alpha,0)}$, $a(\eta)=b(\eta)$, or for all $\beta\in \xi_{b(\alpha,1)+1}\setminus\xi_{b(\alpha,1)}$, $a(\eta)=b(\eta)$. This is a contradiction, thus, $$[s,S']\subseteq 2^\kappa\setminus\left(A'+(2^\kappa\setminus B)\right).$$ Hence, $A+(2^\kappa\setminus B)\subseteq A'+(2^\kappa\setminus B)$ is $\kappa$-Ramsey null. $\square$ We get the following theorem. \[manrams\] Assume that $\kappa$ is strongly inaccessible, ${\text{cov}}(\kappa-CR_0)\geq 2^\kappa$, and ${\text{add}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)=2^\kappa$. Then there exists a $\kappa$-meagre additive set which is not $\kappa$-Ramsey null. Proof: Let $\{F_\alpha\colon\alpha<2^\kappa\}$ be an enumeration of all closed nowhere dense sets in $2^\kappa$, and $[\kappa]^\kappa=\{X_\alpha\colon \alpha<2^\kappa\}$. We construct a sequence $\langle x_\alpha\rangle_{\alpha<2^\kappa}\in (2^\kappa)^{2^\kappa}$ by induction. For $\alpha<2^\kappa$, using Proposition \[mmrn\], choose a $\kappa$-meagre set $B_\alpha\subseteq 2^\kappa$ such that $F_\alpha+(2^\omega\setminus B_\alpha)$ is $\kappa$-Ramsey null. Choose any $$x_\alpha\in \left\{\chi_S\colon S\in [X_\alpha]^\kappa\right\}\setminus \bigcup_{\beta<\alpha} \left(F_\beta+(2^\omega\setminus B_\beta)\right).$$ Such $x_\alpha$ exists, because ${\text{cov}}(\kappa-CR_0)\geq 2^\kappa$. Let $A=\{x_\alpha\colon \alpha<2^\kappa\}$. Obviously, $A$ is not $\kappa$-Ramsey null, because for all $S\in [\kappa]^\kappa$, there exists $S'\in [S]^\kappa$ such that $\chi_{S'}\in A$. Moreover, if $F$ is nowhere dense, then let $\alpha<2^\kappa$ be such that $F\subseteq F_\alpha$. For every $\beta>\alpha$, $$x_\beta\notin F_\alpha+(2^\omega\setminus B_\alpha),$$ thus by Lemma \[lem-mm\], $$x_\beta+F_\alpha\subseteq B_\alpha.$$ Hence, $$A+F\subseteq A+F_\alpha=\bigcup_{\beta\leq\alpha} (x_\beta+F_\alpha)\cup \bigcup_{\alpha<\beta<2^\kappa} (x_\beta+F_\alpha)=\bigcup_{\beta\leq\alpha} (x_\beta+F_\alpha)\cup B_\alpha,$$ which is $\kappa$-meagre, since ${\text{add}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)=2^\kappa$. $\square$ $\kappa$-T’-sets ---------------- A definition of a T’-set was given in [@antw:rpsssas] (see also section \[intro-special\]). We provide a generalization of this notion in case of $2^\kappa$. A set $A\subseteq 2^\kappa$ is here called [[**$\kappa$-T’-set**]{}]{} if there exists a sequence of cardinal numbers $\langle \lambda_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ such that for every increasing sequence $\langle \delta_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ with $\delta_0=0$, and $\delta_\alpha=\bigcup_{\beta<\alpha} \delta_\beta$ for limit $\alpha$, there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$H_{\alpha}\in \left[2^{\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha}}\right]^{\leq\lambda_{\eta_\alpha}},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\in H_\alpha \right\}.$$ Similarly to [@antw:rpsssas] we prove some equivalent characterizations of this class of sets. We start by an easy observation. A set $A\subseteq 2^\kappa$ is here called [[**$\kappa$-T’-set**]{}]{} if there exists a sequence of cardinal numbers $\langle \lambda_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ such that for every increasing sequence $\langle \delta'_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$H'_{\alpha}\in \left[2^{\delta'_{\eta_\alpha+1}\setminus \delta'_{\eta_\alpha}}\right]^{\leq\lambda_{\eta_\alpha}},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta'_{\eta_\alpha+1}\setminus \delta'_{\eta_\alpha})\in H_\alpha \right\}.$$ Proof: Obviously, every set which satisfies the premise is a $\kappa$-$T'$-set. Conversely, let $\langle \delta'_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be an increasing sequence. Notice, that it is sufficient to take sequence $\langle\delta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that $$\delta_\alpha=\begin{cases}\bigcup_{\beta<\alpha}\delta'_\beta & \text{if }\alpha\text{ is a limit ordinal,}\\ \delta'_\alpha & \text{otherwise,}\end{cases}$$ for $\alpha<\kappa$. Then we get a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$H_{\alpha}\in \left[2^{\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha}}\right]^{\leq\lambda_{\eta_\alpha}},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\in H_\alpha \right\}.$$ Take $$H'_\alpha=\left\{f{\mathord{\upharpoonright}}(\delta'_{\eta_\alpha+1}\setminus \delta'_{\eta_\alpha})\colon h\in H_\alpha\right\}.$$ $\square$ \[tprym1\] A set $A\subseteq 2^\kappa$ is a $\kappa$-T’-set if and only if there exists a sequence of cardinal numbers $\langle \lambda_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ such that for every increasing sequences $\langle \delta_{0,\alpha}\rangle_{\alpha<\kappa}, \langle \delta_{1,\alpha}\rangle_{\alpha<\kappa}\in\kappa^\kappa$, with $\delta_{0,\alpha}<\delta_{1,\alpha}\leq \delta_{0,\alpha+1}$ for all $\alpha<\kappa$, there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$I_{\alpha}\in \left[2^{\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha}}\right]^{\leq\lambda_{\eta_\alpha}},$$ for all $\alpha<\kappa$, so that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha})\in I_\alpha \right\}.$$ Proof: Obviously, a set which fulfils the above condition is a $\kappa$-T’-set. On the other hand, if $A\subseteq 2^\kappa$ is a $\kappa$-T’-set, then let $\langle \lambda_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be a sequence of cardinals given by the definition of a $\kappa$-T’-set. Let $$\delta_{\alpha}=\bigcup_{\beta<\alpha}\delta_{1,\beta},$$ for $\alpha<\kappa$. There exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$H_{\alpha}\in \left[2^{\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha}}\right]^{\leq\lambda_{\eta_\alpha}},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\in H_\alpha \right\}.$$ Notice that $(\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha})\subseteq (\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})$. Let $$I_\alpha=\left\{f{\mathord{\upharpoonright}}(\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha})\colon f\in H_\alpha\right\},$$ for $\alpha<\kappa$. Obviously, $|I_\alpha|\leq |H_\alpha|\leq \lambda_{\eta_\alpha}$, and $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha})\in I_\alpha \right\}.$$  $\square$ \[tprym2\] Assume that $\kappa$ is a weakly inaccessible cardinal. A set $A\subseteq 2^\kappa$ is a $\kappa$-T’-set if and only if for every increasing sequence $\langle \delta_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ such that $\delta_0=0$, and $\delta_\alpha=\bigcup_{\beta<\alpha} \delta_\beta$ for limit $\alpha<\kappa$, there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that for limit $\beta<\kappa$, $\eta_\beta=\bigcup_{\alpha<\beta}\eta_\alpha$, and $$J_{\alpha}\in \left[2^{\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha}}\right]^{\leq \left|\eta_\alpha\right|},$$ for all $\alpha<\kappa$, so that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\in J_\alpha \right\}.$$ Proof: Obviously, a set which fulfils the above condition is $\kappa$-T’-set. On the other hand, if $A\subseteq 2^\kappa$ is $\kappa$-T’-set, then let $\langle \lambda_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be a sequence of cardinals given by the definition of a $\kappa$-T’-set. Since $\kappa$ is weakly inaccessible, we can assume that $\langle\lambda_\alpha\rangle_{\alpha<\kappa}$ is strictly increasing and $\bigcup_{\alpha<\beta}\lambda_\alpha=\lambda_\beta$ for limit $\beta<\kappa$. Let $\langle \delta_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be an increasing sequence. Let $\langle\delta'_{\alpha}\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be the following sequence: $\delta'_{0}=0$, $$\delta'_{\alpha+1}=\delta_{\lambda_{\alpha}+1},$$ and $\delta'_{\alpha}=\bigcup_{\beta<\alpha}\delta'_\beta$, when $\alpha$ is a limit ordinal. There exists a sequence $\langle \eta'_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$H_{\alpha}\in \left[2^{\delta'_{\eta'_\alpha+1}\setminus \delta'_{\eta'_\alpha}}\right]^{\leq\lambda_{\eta'_\alpha}},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta'_{\eta'_\alpha+1}\setminus \delta'_{\eta'_\alpha})\in H_\alpha \right\}.$$ One can also assume that $\eta'_\beta=\bigcup_{\alpha<\beta}\eta'_\alpha$ for all limit $\beta<\kappa$. Let $\eta_\alpha=\lambda_{\eta'_\alpha}$. Notice that $\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha}\subseteq \delta'_{\eta'_\alpha+1}\setminus \delta'_{\eta'_\alpha}$. Thus, let $$J_{\alpha}=\left\{f{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\colon f\in H_\alpha\right\}.$$ We get that $$|J_{\alpha}|\leq |H_\alpha|\leq \lambda_{\eta'_\alpha}=\eta_\alpha,$$ and $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\in J_\alpha \right\}.$$  $\square$ \[tprym\] Assume that $\kappa$ is a weakly inaccessible cardinal. A set $A\subseteq 2^\kappa$ is a $\kappa$-T’-set if and only if for every increasing sequences $\langle \delta_{0,\alpha}\rangle_{\alpha<\kappa}, \langle \delta_{1,\alpha}\rangle_{\alpha<\kappa}\in\kappa^\kappa$ such that $\delta_{0,\alpha}<\delta_{1,\alpha}\leq \delta_{0,\alpha+1}$ for all $\alpha<\kappa$, there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that for limit $\beta<\kappa$, $\bigcup_{\alpha<\beta}\eta_\alpha=\eta_\beta$, and $$I_{\alpha}\in \left[2^{\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha}}\right]^{\leq|\eta_\alpha|},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha})\in I_\alpha \right\}.$$  $\square$ \[tprymideal\] Assume that $\kappa$ is a weakly inaccessible cardinal. The class of $\kappa$-T’-sets forms a $\kappa^+$-complete ideal of subsets of $2^\kappa$. Proof: Let $\langle A_\alpha\rangle_{\alpha<\kappa}$ be a sequence of $\kappa$-T’-sets, and let sequences $\langle \delta_{0,\alpha}\rangle_{\alpha<\kappa}, \langle \delta_{1,\alpha}\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be increasing sequences such that $\delta_{0,\alpha}<\delta_{1,\alpha}\leq \delta_{0,\alpha+1}$ for all $\alpha<\kappa$. Inductively construct sequences $\langle \eta_{\alpha,\beta}\rangle_{\alpha,\beta<\kappa}\in \kappa^{\kappa\times\kappa}$ and $\langle J_{\alpha,\beta}\rangle_{\alpha,\beta<\kappa}$ such that: (a) if $\beta_1<\beta_2<\kappa$, then $\{\eta_{\alpha,\beta_2}\colon \alpha<\kappa \}\subseteq \{\eta_{\alpha,\beta_1}\colon \alpha<\kappa \}$, (b) $\{\eta_{\alpha,\beta}\colon \alpha<\kappa\}$ is a closed unbounded set in $\kappa$ for every $\beta<\kappa$, (c) $J_{\alpha,\beta}\in \left[2^{\delta_{1,\eta_{\alpha, \beta}}\setminus \delta_{0,\eta_{\alpha,\beta}}}\right]^{\leq \left|\eta_{\alpha,\beta}\right|},$ for all $\alpha,\beta<\kappa$, (d) $A_\beta\subseteq \left\{x\in 2^\kappa\colon \forall_{\gamma<\kappa}\exists_{\gamma<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{1,\eta_{\alpha,\beta}}\setminus \delta_{0,\eta_{\alpha,\beta}})\in J_{\alpha,\beta}\right\}$, for all $\beta<\kappa$. To obtain the above, inductively construct a sequence $\left<I_\alpha\right>_{\alpha<\kappa}\in \left([\kappa]^\kappa\right)^\kappa$ such that $I_0=\kappa$, and let $I_{\beta+1}=\{\eta_{\alpha, \beta}\colon \alpha<\kappa\}$. Moreover, for limit $\alpha<\kappa$, let $I_\alpha=\bigcap_{\beta<\alpha} I_\beta$. Obviously, $I_\alpha$ is closed unbounded. Now, for $\beta<\kappa$, by Corollary \[tprym\], we can get $\langle\eta'_{\alpha,\beta}\rangle_{\alpha< \kappa}$ and $\langle J_{\alpha,\beta}\rangle_{\alpha<\kappa}$ for sequences $\langle\delta_{0,\zeta_{\alpha,\beta}}\rangle_{\alpha\in \kappa}$ and $\langle\delta_{0,\zeta_{\alpha,\beta}}\rangle_{\alpha\in \kappa}$, where $\{\zeta_{\alpha,\beta}\colon \alpha<\kappa\}=I_\beta$ is the increasing enumeration, i.e. such that $$J_{\alpha,\beta}\in \left[2^{\delta_{1,\zeta_{\eta'_{\alpha, \beta},\beta}}\setminus \delta_{0,\zeta_{\eta'_{\alpha,\beta},\beta}}}\right]^{\leq \left|\eta'_{\alpha,\beta}\right|},$$ for all $\alpha<\kappa$, and $$A_\beta\subseteq \left\{x\in 2^\kappa\colon \forall_{\gamma<\kappa}\exists_{\gamma<\alpha<\kappa} x{\mathord{\upharpoonright}}\left(\delta_{1,\zeta_{\eta'_{\alpha,\beta},\beta}}\setminus \delta_{0,\zeta_{\eta'_{\alpha,\beta},\beta}}\right)\in J_{\alpha,\beta}\right\}.$$ and $\eta'_{\beta}=\bigcup_{\alpha<\beta}\eta'_\alpha$ for all limit $\beta<\kappa$. Now, let $\eta_{\alpha,\beta}=\zeta_{\eta'_{\alpha,\beta},\beta}$, for $\alpha<\kappa$. We get $$\left|J_{\alpha,\beta}\right|\leq \left|\eta'_{\alpha,\beta}\right|\leq \left|\eta_{\alpha,\beta}\right|,$$ for all $\alpha<\kappa$. Let $$I=\bigcup_{\beta<\kappa}\{\zeta_{\alpha,\beta}\colon\alpha<\beta\}.$$ Notice that $|I|=\kappa$, and for all $\beta<\kappa$ there exists $\gamma<\kappa$ such that $I\setminus\gamma\subseteq I_\beta$. Let $\{\zeta_\alpha\colon\alpha<\kappa\}=I$ be the increasing enumeration of $I$. Let $$J_\alpha=\bigcup_{\beta<\alpha} \left\{g\in 2^{\delta_{1,\zeta_{\alpha}}\setminus \delta_{0,\zeta_{\alpha}}}\colon\exists_{f\in J_{\gamma,\beta}} g{\mathord{\upharpoonright}}{\text{dom}}f= f \land \zeta_\alpha=\eta_{\gamma,\beta}\land\gamma<\kappa\right\},$$ for $\alpha<\kappa$. Notice that $J_\alpha\subseteq 2^{\delta_{1,\zeta_{\alpha}}\setminus \delta_{0,\zeta_{\alpha}}}$, and $$|J_\alpha|\leq |\alpha|\cdot|\zeta_\alpha|= |\zeta_\alpha|,$$ for $\omega\leq\alpha<\kappa$. Finally, notice that $$\bigcup_{\alpha<\kappa} A_\alpha\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}\left(\delta_{1,\zeta_\alpha}\setminus \delta_{0,\zeta_\alpha}\right)\in J_\alpha \right\}.$$ Thus, by Corollary \[tprym\], $\bigcup_{\alpha<\kappa} A_\alpha$ is a $\kappa$-T’-set.$\square$ \[tprymsum\] If $A,B\subseteq 2^\kappa$ are $\kappa$-T’-sets, then $A+B$ is also a $\kappa$-T’-set. Proof: Let $\langle \lambda^A_\alpha\rangle_{\alpha<\kappa}, \langle \lambda^B_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be sequences of cardinals given by the definition of $\kappa$-T’-sets for $A$ and $B$, respectively. Let $$\lambda_\alpha=\max\{\lambda^A_\alpha,\lambda^B_\alpha,\aleph_0\},$$ for $\alpha<\kappa$. Let $\langle \delta_{0,\alpha}\rangle_{\alpha<\kappa}, \langle \delta_{1,\alpha}\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be sequences such that $\delta_{0,\alpha}<\delta_{1,\alpha}\leq \delta_{0,\alpha+1}$ for all $\alpha<\kappa$. By Proposition \[tprym1\], we get a sequence $\langle \eta^A_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$I^A_{\alpha}\in \left[2^{\delta_{1,\eta^A_\alpha}\setminus \delta_{0,\eta^A_\alpha}}\right]^{\leq\lambda^A_{\eta^A_\alpha}},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{1,\eta^A_\alpha}\setminus \delta_{0,\eta^B_\alpha})\in I^A_\alpha \right\}.$$ Let $\delta^B_{0,\alpha}=\delta_{0,\eta^A_\alpha}$, and $\delta^B_{1,\alpha}=\delta_{1,\eta^A_\alpha}$, for $\alpha<\kappa$. Again, by Proposition \[tprym1\], we get a sequence $\langle \eta^B_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$I^B_{\alpha}\in \left[2^{\delta^B_{1,\eta^B_\alpha}\setminus \delta^B_{0,\eta^B_\alpha}}\right]^{\leq\lambda^B_{\eta^B_\alpha}},$$ for all $\alpha<\kappa$ such that $$B\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}\left(\delta^B_{1,\eta^B_\alpha}\setminus \delta^B_{0,\eta^B_\alpha}\right)\in I^B_\alpha \right\}.$$ Let $\eta_\alpha=\eta^A_{\eta^B_\alpha}$, for $\alpha<\kappa$, and let $$I_{\alpha}=I^A_{\eta^B_\alpha}+I^B_{\alpha}\subseteq 2^{\delta^B_{1,\eta^B_\alpha}\setminus \delta^B_{0,\eta^B_\alpha}}=2^{\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha}},$$ for $\alpha<\kappa$. Notice that $|I_\alpha|\leq \lambda_\alpha$, for all $\alpha<\kappa$, and $$A+B\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}\left(\delta_{1,\eta_\alpha}\setminus \delta_{0,\eta_\alpha}\right)\in I_\alpha \right\},$$ so by Proposition \[tprym1\], $A+B$ is a $\kappa$-T’-set. $\square$ \[gammatprym\] Assume that $\kappa$ is a strongly inaccessible cardinal. Then every $\kappa$-$\gamma$-set is a $\kappa$-T’-set. Proof: Assume that $A\subseteq 2^\kappa$ is a $\kappa$-$\gamma$-set, and let $\langle \delta_\alpha\rangle_{\alpha<\kappa}\in\kappa^\kappa$ be a sequence such that $\delta_0=0$, and $\delta_\alpha=\bigcup_{\beta<\alpha} \delta_\beta$ for limit $\alpha$. Let $$I_{\alpha}=\left[2^{\delta_{\alpha+1}\setminus \delta_{\alpha}}\right]^{\leq |\alpha|},$$ for $\alpha<\kappa$, and let $$U_{\alpha,S}=\left\{x\in 2^\kappa\colon x{\mathord{\upharpoonright}}\left(\delta_{\alpha+1}\setminus \delta_{\alpha}\right) \in S\right\},$$ for $\alpha<\kappa$, and $S\in I_\alpha$. Obviously, ${{\mathcal{U}}}=\{U_{\alpha,S}\colon \alpha<\kappa\land S\in I_\alpha\}$ is an open $\kappa$-cover of $2^\kappa$. Therefore, there exists a sequence $\langle V_\alpha\rangle_{\alpha<\kappa}\in {{\mathcal{U}}}^\kappa$ such that $$A\subseteq \bigcup_{\alpha<\kappa}\bigcap_{\alpha<\beta<\kappa} V_\beta.$$ Since $\kappa$ is strongly inaccessible, for every $\beta,\gamma<\kappa$, there exist $\gamma\leq\alpha<\kappa$ and $\beta\leq\delta<\kappa$ such that $V_\alpha=U_{\delta,S}$, with $S\in I_\delta$. Therefore, there exist increasing sequences $\langle\xi_{\alpha}\rangle_{\alpha<\kappa},\langle\eta_{\alpha}\rangle_{\alpha<\kappa}$ such that $V_{\xi_\alpha}=U_{\eta_\alpha, S_\alpha}$, where $S_\alpha\in I_{\eta_\alpha}$. Thus, $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\in S_\alpha \right\}.$$ Hence, $A$ is a $\kappa$-T’-set. $\square$ \[tprymma\] Assume that $\kappa$ is a strongly inaccessible cardinal. Then every $\kappa$-T’-set is $\kappa$-meagre additive. Proof: Let $A\subseteq 2^\kappa$ be a $\kappa$-T’-set, and let $\langle\xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ be an increasing sequence. Let $\langle\zeta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ be a sequence such that $\zeta_0=0$, $\zeta_{\alpha+1}=\zeta_{\alpha}+\alpha$, and $\zeta_\alpha=\bigcup_{\beta<\alpha}\zeta_\beta$, for limit $\alpha<\kappa$. Let $\delta_{\alpha}=\xi_{\zeta_\alpha}$. By Proposition \[tprym2\], there exists a sequence $\langle \eta_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$, and $$J_{\alpha}\in \left[2^{\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha}}\right]^{\leq \left|\eta_\alpha\right|},$$ for all $\alpha<\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \forall_{\beta<\kappa}\exists_{\beta<\alpha<\kappa} x{\mathord{\upharpoonright}}(\delta_{\eta_\alpha+1}\setminus \delta_{\eta_\alpha})\in J_\alpha \right\}.$$ For $\beta<\kappa$ let $\{j_{\alpha,\beta},\alpha<\eta_\alpha\}=J_\beta$ be an enumeration. Let $z\in 2^\kappa$ be the following: $$z(\gamma)=\begin{cases} j_{\alpha,\beta}(\gamma), & \text{ if }\xi_{\zeta_{\eta_\beta}+\alpha}\leq \gamma < \xi_{\zeta_{\eta_\beta}+\alpha+1}, \alpha,\beta<\kappa,\\ 0, & \text{ otherwise.}\end{cases}$$ We have that $$A\subseteq \left\{ x\in 2^\kappa \colon \exists_{\alpha<\kappa}\forall_{\alpha<\beta<\kappa}\exists_{\gamma<\kappa} \left(\delta_\beta\leq \xi_\gamma <\xi_{\gamma+1}\leq \delta_{\beta+1}\land \forall_{\xi_\gamma\leq \delta< \xi_{\gamma+1}} x(\delta)=z(\delta)\right)\right\}.$$ Thus, by Proposition \[madditive\], $A$ is $\kappa$-meagre additive. $\square$ Therefore, we get the following. \[gammama\] Assume that $\kappa$ is a strongly inaccessible cardinal. Then every $\kappa$-$\gamma$-set is $\kappa$-meagre additive.  $\square$ On the other hand, recall that if $\kappa$ is strongly inaccessible, ${\text{cov}}(\kappa-CR_0)\geq 2^\kappa$, and ${\text{add}}({\boldsymbol{{{\mathcal{M}}}}}_\kappa)=2^\kappa$, then there exists a $\kappa$-meagre additive set which is not $\kappa$-Ramsey null (Theorem \[manrams\]), but ever $\kappa$-$\gamma$-set is $\kappa$-Ramsey-null (Theorem \[gammarams\]). Thus, under those conditions the above implication cannot be reversed. $\kappa$-$v_0$-Sets ------------------- A $\kappa$-perfect set $P\subseteq 2^\kappa$ is a [[**$\kappa$-Silver perfect**]{}]{} if for all $\alpha<\kappa$ and any $i\in\{0,1\}$, $$\exists_{s\in 2^\alpha\cap T_P} s^\frown i\in T_P\Rightarrow \forall_{s\in 2^\alpha\cap T_P} s^\frown i\in T_P.$$ A set $A\subseteq 2^\kappa$ is a [[**$\kappa$-$v_0$-set**]{}]{} if for all $\kappa$-Silver perfect set $P\subseteq 2^\kappa$, there exists a $\kappa$-Silver perfect set $Q\subseteq P$ such that $A\cap Q={\varnothing}$. The notion of $\kappa$-$v_0$ sets was considered in [@gl:gssm]. We study the relation between this notion and other notions of special subsets of $2^\kappa$. \[comv0\] Assume that $\kappa$ is a strongly inaccessible cardinal. Then every $\kappa$-comeagre subset of $2^\kappa$ contains a $\kappa$-Silver perfect set. Proof: Let $A\subseteq 2^\kappa$ be $\kappa$-meagre, and by Proposition \[char-meagre\], we get $z\in 2^\kappa$ and a sequence $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ such that $$A\subseteq \left\{x\in 2^\kappa\colon \exists_{\beta<\kappa}\forall_{\beta<\gamma<\kappa} \exists_{\xi_{\gamma}\leq\xi<\xi_{\gamma+1}} z(\xi)\neq x(\xi)\right\}.$$ Let $$Q=\{x\in 2^\kappa \colon \forall_{\alpha\in {\text{Lim}}} \forall_{\xi_{\alpha}\leq\xi<\xi_{\alpha+1}} x(\xi)=z(\xi)\}.$$ Then $Q\subseteq 2^\kappa\setminus A$, and $Q$ is a $\kappa$-Silver perfect set. $\square$ \[pkmkv0\] Assume that $\kappa$ is a strongly inaccessible cardinal. Then every $\kappa$-perfectly $\kappa$-meagre set in $2^\kappa$ is a $\kappa$-$v_0$-set. Proof: Notice that for every $\kappa$-Silver perfect set $P\subseteq 2^\kappa$, there exists a natural homeomorphism $h\colon P\to 2^\kappa$ such that $Q\subseteq 2^\kappa$ is a $\kappa$-Silver perfect set if an only if $h^{-1}[Q]$ is $\kappa$-Silver perfect. The corollary follows from Proposition \[comv0\].  $\square$ \[snv0\] Every $\kappa$-strongly null set in $2^\kappa$ is a $\kappa$-$v_0$-set. Proof: Let $P\subseteq 2^\kappa$ be a $\kappa$-Silver perfect set, and $A\in{\boldsymbol{\text{S}{{\mathcal{N}}}}}_\kappa$. Let $$S=\{{\text{len}}(s)\colon s\in {\text{Split}}(T_P)\}.$$ Let $b\colon \kappa\times \{0,1\}\to S$ be a bijection, and let $X=f[\kappa\times\{0\}]$. Let $\langle x_\alpha\rangle_{\alpha\in X}\in (2^\kappa)^X$ be such that $$A\subseteq \bigcup_{\alpha\in X} [x_\alpha{\mathord{\upharpoonright}}\alpha+1].$$ Then $$Q=\left\{x\in P\colon \forall_{\alpha\in X} x(\alpha)=x_\alpha(\alpha)+1\right\}$$ is a $\kappa$-Silver perfect set such that $Q\subseteq P$, and $Q\cap A={\varnothing}$. $\square$ $\kappa$-$l_0$-Sets ------------------- A perfect tree $T\subseteq \kappa^{<\kappa}$ is called a [[**$\kappa$-Laver perfect tree**]{}]{} if there exists $s\in T$ such that for all $t\in T$, either $t\subseteq s$, or $$\left|\left\{\alpha<\kappa\colon t^\frown \alpha\in T\right\}\right|=\kappa.$$ A $\kappa$-perfect set $P$ is [[**$\kappa$-Laver**]{}]{}, if $T_P$ is a $\kappa$-Laver perfect tree. A set $A\subseteq \kappa^\kappa$ is [[**$\kappa$-$l_0$-set**]{}]{} if for every $\kappa$-Laver $\kappa$-perfect set $P$, there exists a $\kappa$-Laver $\kappa$-perfect set $Q\subseteq P$ such that $Q\cap A={\varnothing}$. Every $\kappa$-strongly null set in $\kappa^\kappa$ is a $\kappa$-$l_0$-set. Proof: Let $T\subseteq \kappa^{<\kappa}$ be a $\kappa$-perfect $\kappa$-Laver tree, and let $A\subseteq \kappa^\kappa$ be a $\kappa$-strongly null set. Let $t_0\in T$ be such that for every $s\in T$ such that $t_0\subseteq s$, $$\left|\left\{\alpha<\kappa\colon t_0\,^\frown \alpha\in T\right\}\right|=\kappa.$$ Let $I=\{\alpha<\kappa\colon {\text{len}}(t)<\alpha\}$, and let $\langle s_{\alpha}\rangle_{\alpha\in I}$ be such that $$A=\bigcup_{\alpha\in I} [s_\alpha]$$ and for all $\alpha\in I$, $s_\alpha\in 2^\alpha$. We construct tree $T'\subseteq \kappa^{<\kappa}$ in the following way. Let $T'\cap \kappa^{\leq{\text{len}}(t_0)}=T\cap \kappa^{\leq{\text{len}}(t_0)}$, and assume that $\alpha<\kappa$ is such that $\alpha> {\text{len}}(t_0)$, and $t\in T\cap \kappa^\alpha$. Then let $$T'\cap \{s\in \kappa^{\alpha+1}\colon t\subseteq s\}= T \cap \{s\in \kappa^{\alpha+1}\colon t\subseteq s\}\setminus \{s_\alpha\}.$$ For limit $\beta<\kappa$ with $\beta>{\text{len}}(t_0)$, let $t\in T'$ if and only if $t{\mathord{\upharpoonright}}\alpha\in T'$ for every $\alpha<\beta$. Since $\kappa$ is regular, $T'$ is a $\kappa$-perfect $\kappa$-Laver tree, and $[T']\subseteq [T]\setminus A$. $\square$ $\kappa$-$m_0$-Sets ------------------- A perfect tree $T\subseteq \kappa^{<\kappa}$ is called a [[**$\kappa$-Miller perfect tree**]{}]{} if for every $s\in T$ there exists $t\in T$ such that $s\subseteq t$, and $$\left|\left\{\alpha<\kappa\colon t^\frown \alpha\in T\right\}\right|=\kappa.$$ A $\kappa$-perfect set $P$ is [[**$\kappa$-Miller**]{}]{}, if $T_P$ is a $\kappa$-Miller perfect tree. A set $A\subseteq 2^\kappa$ is [[**$\kappa$-$m_0$-set**]{}]{} if for every $\kappa$-Miller $\kappa$-perfect set $P$, there exists a  $\kappa$-Miller $\kappa$-perfect set $Q\subseteq P$ such that $Q\cap A={\varnothing}$. Every $\kappa$-perfectly $\kappa$-meagre set in $\kappa^\kappa$ is a $\kappa$-$m_0$-set. Proof: Let $P$ be a $\kappa$-perfect $\kappa$-Miller set, and $A\in{\boldsymbol{\text{P}_\kappa{{\mathcal{M}}}}}_\kappa$. There exists a homeomorphism $h\colon P\to \kappa^\kappa$. Notice also that under this homeomorphism, if $Q\subseteq \kappa^\kappa$ is a $\kappa$-perfect $\kappa$-Miller set, then $h^{-1}[Q]$ is a $\kappa$-Miller $\kappa$-perfect set as well. Then $B=h[A\cap P]$ is $\kappa$-meagre. Let $B=\bigcup_{\alpha<\kappa} G_\alpha$ with $G_\alpha$ nowhere dense closed for every $\alpha<\kappa$. We choose by induction $\left<t_s\right>_{s\in \kappa^{<\kappa}}$ such that $t_s \in \kappa^{<\kappa}$, for every $s\in \kappa^\kappa$, $$\left|\{\alpha<\kappa \colon \exists_{s'\in \kappa^{\kappa}}\colon t_s\,^\frown \alpha\subseteq t_{s'}\}\right|=\kappa,$$ and for $s,s'\in \kappa^{<\kappa}$, $s\subsetneq s'$ if and only if $t_s\subsetneq t_{s'}$. Indeed, let $t_{\varnothing}$ be such that $[t_{\varnothing}]\cap G_0={\varnothing}$ Then, given $t_s$, $s\in \kappa^\alpha$, let $t'_s\supsetneq t_s$ be such that $[t'_s]\cap G_{\alpha+1}= {\varnothing}$. Set $t_{s^\frown \xi}=t_s^{\prime\frown}\xi$, for all $\xi<\kappa$. For limit $\beta<\kappa$, and $s\in \kappa^\beta$, let $t'_s=\bigcup_{\alpha<\beta}t_{s{\mathord{\upharpoonright}}\alpha}$. Let $t_s\supsetneq t'_s$ be such that $[t_s]\cap G_\beta={\varnothing}$. Finally, let $$T=\bigcup_{\alpha<\kappa}\{t\in \kappa^{<\kappa}\colon t\subseteq t_{s}, s\in \kappa^{\alpha}\}.$$ Obviously, $T$ is a $\kappa$-perfect $\kappa$-Miller tree, so $P'=[T]_{\kappa}$ is a $\kappa$-perfect $\kappa$-Miller subset of $\kappa^\kappa\setminus B$. Thus, there exists a $\kappa$-perfect $\kappa$-Miller $Q\subseteq P\setminus A$.  $\square$ Assume that $\kappa$ is a strongly inaccessible cardinal. Then every $\kappa$-strongly null set in $\kappa^\kappa$ is a $\kappa$-$m_0$-set. Proof: Let $A\subseteq 2^\kappa$ be $\kappa$-strong measure zero, and let $T$ be a $\kappa$-Miller tree. Let $$S=\left\{s\in T \colon |\{\alpha<\kappa\colon s^\frown \alpha\in T\}|=\kappa\right\},$$ and let $s_0\in S$. By induction we define a sequence $\langle \xi_\alpha\rangle_{\alpha<\kappa}\in \kappa^\kappa$ and $\langle A_\alpha\rangle_{\alpha<\kappa}\in([\kappa^{<\kappa}]^{<\kappa})^\kappa$ such that (a) $A_0=s_0$, $\alpha_0={\text{len}}(s_0)$, (b) for every $\alpha<\kappa$, if $s\in A_\alpha$, then there exists $t\in A_{\alpha+1}\setminus A_\alpha$ such that $s\subseteq t$, $t\in S$, ${\text{len}}(t)>\xi_\alpha$, (c) for every $\alpha<\kappa$, and for all $s\in A_\alpha$, ${\text{len}}(s)\leq\xi_\alpha$, (d) for every $\alpha<\beta<\kappa$, $A_\alpha\subseteq A_\beta$, and $\xi_\alpha<\xi_\beta$, (e) for every $\alpha<\kappa$, and every limit ordinal $\beta\leq\xi_\alpha$, for every $t\in T\cap \kappa^\beta$ such that for all $\gamma<\beta$ there exists $s\in A_\alpha$ such that $t{\mathord{\upharpoonright}}\gamma\subseteq s$, there exists $u\in A_{\alpha+1}\setminus A_\alpha$ such that $t\subseteq u$, $u\in S$, ${\text{len}}(u)>\xi_\alpha$. This is possible, since $\kappa$ is strongly inaccessible. Indeed, for any $\alpha<\kappa$, find $B\in [\kappa^{<\kappa}]^{<\kappa}$ such that $A_{\alpha+1}=A_\alpha+B$ fulfils conditions (b) and (e), and fix $$\xi_{\alpha+1}=\bigcup A_{\alpha+1}<\kappa.$$ If $\beta<\kappa$ is a limit ordinal, let $$\xi_\beta=\bigcup_{\alpha<\beta}\xi _\alpha$$ and $$A_\beta=\bigcup_{\alpha<\kappa}A_\alpha.$$ Now let $\langle s_\alpha\rangle_{\alpha<\kappa}\in (2^{<\kappa})^\kappa$ such that for all $\alpha<\kappa$, $s_\alpha\in \kappa^{\xi_\alpha+1}$, and $$A\subseteq \bigcup_{\alpha<\kappa}[s_\alpha].$$ Let now $$T'=\left\{s\in T\colon \exists_{x\in \kappa^\kappa} s\subseteq x \land \forall_{\alpha<\kappa} x\notin [s_\alpha]\right\}.$$ Observe that there exists a $\kappa$-Miller $\kappa$-perfect tree $T''\subseteq T'$. Indeed notice that that $s_0\in T'$. Assume now that there exists $\alpha<\kappa$ and $s\in A_\alpha\cap T'$ such that $$|\{\alpha<\kappa\colon s^\frown\alpha\in T'\}|<\kappa.$$ Let $\alpha_0$ be such that $s^\frown\alpha_0\in T'$. But (b) and $$|\{\alpha<\kappa\colon s^\frown\alpha\in T'\}|<\kappa.$$ imply that there exists $\beta<\alpha$ such that for all $\alpha<\gamma<\kappa$, $s^\frown\alpha_0\not\subseteq s_\gamma$, thus there exists $\kappa$-Miller tree $T''\subseteq [s^\frown \alpha_0]$. Similarly (e) implies that we can find such a $\kappa$-perfect tree. On the other hand, if for all $\alpha<\kappa$, for all $s\in A_\alpha\cap T'$, $$|\{\alpha<\kappa\colon s^\frown\alpha\in T'\}|=\kappa,$$ the existence of such a tree is clear. But $[T'']\subseteq [T]\setminus A$, so $A$ is a $\kappa$-$m_0$-set. $\square$ <span style="font-variant:small-caps;"></span> \[1\][\#1]{} [\[10\]]{} <span style="font-variant:small-caps;">N. Lusin</span>, Sur un problème de [M]{}. 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[EUROPEAN LABORATORY FOR PARTICLE PHYSICS ]{} CERN-EP/98-091\ 4 June 1998 [ **Inclusive Production of Charged Hadrons and $\ks$ Mesons in Photon-Photon Collisions** ]{} [The OPAL Collaboration ]{} [Abstract]{} The production of charged hadrons and $\ks$ mesons in the collisions of quasi-real photons has been measured using the OPAL detector at LEP. The data were taken at $\ee$ centre-of-mass energies of $161$ and $172$ GeV. The differential cross-sections as a function of the transverse momentum and the pseudorapidity of the charged hadrons and $\ks$ mesons have been compared to the leading order Monte Carlo simulations of PHOJET and PYTHIA and to perturbative next-to-leading order (NLO) QCD calculations. The distributions have been measured in the range $10<W<125$ GeV of the hadronic invariant mass $W$. By comparing the transverse momentum distribution of charged hadrons measured in $\gg$ interactions with $\gamma$-proton and meson-proton data we find evidence for hard photon interactions in addition to the purely hadronic photon interactions. [(submitted to European Physics Journal C) ]{} [The OPAL Collaboration ]{} [ K.Ackerstaff$^{ 8}$, G.Alexander$^{ 23}$, J.Allison$^{ 16}$, N.Altekamp$^{ 5}$, K.J.Anderson$^{ 9}$, S.Anderson$^{ 12}$, S.Arcelli$^{ 2}$, S.Asai$^{ 24}$, S.F.Ashby$^{ 1}$, D.Axen$^{ 29}$, G.Azuelos$^{ 18, a}$, A.H.Ball$^{ 17}$, E.Barberio$^{ 8}$, R.J.Barlow$^{ 16}$, R.Bartoldus$^{ 3}$, J.R.Batley$^{ 5}$, S.Baumann$^{ 3}$, J.Bechtluft$^{ 14}$, T.Behnke$^{ 8}$, K.W.Bell$^{ 20}$, G.Bella$^{ 23}$, S.Bentvelsen$^{ 8}$, S.Bethke$^{ 14}$, S.Betts$^{ 15}$, O.Biebel$^{ 14}$, A.Biguzzi$^{ 5}$, S.D.Bird$^{ 16}$, V.Blobel$^{ 27}$, I.J.Bloodworth$^{ 1}$, M.Bobinski$^{ 10}$, P.Bock$^{ 11}$, J.Böhme$^{ 14}$, M.Boutemeur$^{ 34}$, S.Braibant$^{ 8}$, P.Bright-Thomas$^{ 1}$, R.M.Brown$^{ 20}$, H.J.Burckhart$^{ 8}$, C.Burgard$^{ 8}$, R.Bürgin$^{ 10}$, P.Capiluppi$^{ 2}$, R.K.Carnegie$^{ 6}$, A.A.Carter$^{ 13}$, J.R.Carter$^{ 5}$, C.Y.Chang$^{ 17}$, D.G.Charlton$^{ 1, b}$, D.Chrisman$^{ 4}$, C.Ciocca$^{ 2}$, P.E.L.Clarke$^{ 15}$, E.Clay$^{ 15}$, I.Cohen$^{ 23}$, J.E.Conboy$^{ 15}$, O.C.Cooke$^{ 8}$, C.Couyoumtzelis$^{ 13}$, R.L.Coxe$^{ 9}$, M.Cuffiani$^{ 2}$, S.Dado$^{ 22}$, G.M.Dallavalle$^{ 2}$, R.Davis$^{ 30}$, S.De Jong$^{ 12}$, L.A.del Pozo$^{ 4}$, A.de Roeck$^{ 8}$, K.Desch$^{ 8}$, B.Dienes$^{ 33, d}$, M.S.Dixit$^{ 7}$, M.Doucet$^{ 18}$, J.Dubbert$^{ 34}$, E.Duchovni$^{ 26}$, G.Duckeck$^{ 34}$, I.P.Duerdoth$^{ 16}$, D.Eatough$^{ 16}$, P.G.Estabrooks$^{ 6}$, E.Etzion$^{ 23}$, H.G.Evans$^{ 9}$, F.Fabbri$^{ 2}$, A.Fanfani$^{ 2}$, M.Fanti$^{ 2}$, A.A.Faust$^{ 30}$, F.Fiedler$^{ 27}$, M.Fierro$^{ 2}$, H.M.Fischer$^{ 3}$, I.Fleck$^{ 8}$, R.Folman$^{ 26}$, A.Fürtjes$^{ 8}$, D.I.Futyan$^{ 16}$, P.Gagnon$^{ 7}$, J.W.Gary$^{ 4}$, J.Gascon$^{ 18}$, S.M.Gascon-Shotkin$^{ 17}$, C.Geich-Gimbel$^{ 3}$, T.Geralis$^{ 20}$, G.Giacomelli$^{ 2}$, P.Giacomelli$^{ 2}$, V.Gibson$^{ 5}$, W.R.Gibson$^{ 13}$, D.M.Gingrich$^{ 30, a}$, D.Glenzinski$^{ 9}$, J.Goldberg$^{ 22}$, W.Gorn$^{ 4}$, C.Grandi$^{ 2}$, E.Gross$^{ 26}$, J.Grunhaus$^{ 23}$, M.Gruwé$^{ 27}$, G.G.Hanson$^{ 12}$, M.Hansroul$^{ 8}$, M.Hapke$^{ 13}$, C.K.Hargrove$^{ 7}$, C.Hartmann$^{ 3}$, M.Hauschild$^{ 8}$, C.M.Hawkes$^{ 5}$, R.Hawkings$^{ 27}$, R.J.Hemingway$^{ 6}$, M.Herndon$^{ 17}$, G.Herten$^{ 10}$, R.D.Heuer$^{ 8}$, M.D.Hildreth$^{ 8}$, J.C.Hill$^{ 5}$, S.J.Hillier$^{ 1}$, P.R.Hobson$^{ 25}$, A.Hocker$^{ 9}$, R.J.Homer$^{ 1}$, A.K.Honma$^{ 28, a}$, D.Horváth$^{ 32, c}$, K.R.Hossain$^{ 30}$, R.Howard$^{ 29}$, P.Hüntemeyer$^{ 27}$, P.Igo-Kemenes$^{ 11}$, D.C.Imrie$^{ 25}$, K.Ishii$^{ 24}$, F.R.Jacob$^{ 20}$, A.Jawahery$^{ 17}$, H.Jeremie$^{ 18}$, M.Jimack$^{ 1}$, A.Joly$^{ 18}$, C.R.Jones$^{ 5}$, P.Jovanovic$^{ 1}$, T.R.Junk$^{ 8}$, D.Karlen$^{ 6}$, V.Kartvelishvili$^{ 16}$, K.Kawagoe$^{ 24}$, T.Kawamoto$^{ 24}$, P.I.Kayal$^{ 30}$, R.K.Keeler$^{ 28}$, R.G.Kellogg$^{ 17}$, B.W.Kennedy$^{ 20}$, A.Klier$^{ 26}$, S.Kluth$^{ 8}$, T.Kobayashi$^{ 24}$, M.Kobel$^{ 3, e}$, D.S.Koetke$^{ 6}$, T.P.Kokott$^{ 3}$, M.Kolrep$^{ 10}$, S.Komamiya$^{ 24}$, R.V.Kowalewski$^{ 28}$, T.Kress$^{ 11}$, P.Krieger$^{ 6}$, J.von Krogh$^{ 11}$, P.Kyberd$^{ 13}$, G.D.Lafferty$^{ 16}$, D.Lanske$^{ 14}$, J.Lauber$^{ 15}$, S.R.Lautenschlager$^{ 31}$, I.Lawson$^{ 28}$, J.G.Layter$^{ 4}$, D.Lazic$^{ 22}$, A.M.Lee$^{ 31}$, E.Lefebvre$^{ 18}$, D.Lellouch$^{ 26}$, J.Letts$^{ 12}$, L.Levinson$^{ 26}$, R.Liebisch$^{ 11}$, B.List$^{ 8}$, C.Littlewood$^{ 5}$, A.W.Lloyd$^{ 1}$, S.L.Lloyd$^{ 13}$, F.K.Loebinger$^{ 16}$, G.D.Long$^{ 28}$, M.J.Losty$^{ 7}$, J.Ludwig$^{ 10}$, D.Lui$^{ 12}$, A.Macchiolo$^{ 2}$, A.Macpherson$^{ 30}$, M.Mannelli$^{ 8}$, S.Marcellini$^{ 2}$, C.Markopoulos$^{ 13}$, A.J.Martin$^{ 13}$, J.P.Martin$^{ 18}$, G.Martinez$^{ 17}$, T.Mashimo$^{ 24}$, P.Mättig$^{ 26}$, W.J.McDonald$^{ 30}$, J.McKenna$^{ 29}$, E.A.Mckigney$^{ 15}$, T.J.McMahon$^{ 1}$, R.A.McPherson$^{ 28}$, F.Meijers$^{ 8}$, S.Menke$^{ 3}$, F.S.Merritt$^{ 9}$, H.Mes$^{ 7}$, J.Meyer$^{ 27}$, A.Michelini$^{ 2}$, S.Mihara$^{ 24}$, G.Mikenberg$^{ 26}$, D.J.Miller$^{ 15}$, R.Mir$^{ 26}$, W.Mohr$^{ 10}$, A.Montanari$^{ 2}$, T.Mori$^{ 24}$, K.Nagai$^{ 26}$, I.Nakamura$^{ 24}$, H.A.Neal$^{ 12}$, B.Nellen$^{ 3}$, R.Nisius$^{ 8}$, S.W.O’Neale$^{ 1}$, F.G.Oakham$^{ 7}$, F.Odorici$^{ 2}$, H.O.Ogren$^{ 12}$, M.J.Oreglia$^{ 9}$, S.Orito$^{ 24}$, J.Pálinkás$^{ 33, d}$, G.Pásztor$^{ 32}$, J.R.Pater$^{ 16}$, G.N.Patrick$^{ 20}$, J.Patt$^{ 10}$, R.Perez-Ochoa$^{ 8}$, S.Petzold$^{ 27}$, P.Pfeifenschneider$^{ 14}$, J.E.Pilcher$^{ 9}$, J.Pinfold$^{ 30}$, D.E.Plane$^{ 8}$, P.Poffenberger$^{ 28}$, B.Poli$^{ 2}$, J.Polok$^{ 8}$, M.Przybycień$^{ 8}$, C.Rembser$^{ 8}$, H.Rick$^{ 8}$, S.Robertson$^{ 28}$, S.A.Robins$^{ 22}$, N.Rodning$^{ 30}$, J.M.Roney$^{ 28}$, K.Roscoe$^{ 16}$, A.M.Rossi$^{ 2}$, Y.Rozen$^{ 22}$, K.Runge$^{ 10}$, O.Runolfsson$^{ 8}$, D.R.Rust$^{ 12}$, K.Sachs$^{ 10}$, T.Saeki$^{ 24}$, O.Sahr$^{ 34}$, W.M.Sang$^{ 25}$, E.K.G.Sarkisyan$^{ 23}$, C.Sbarra$^{ 29}$, A.D.Schaile$^{ 34}$, O.Schaile$^{ 34}$, F.Scharf$^{ 3}$, P.Scharff-Hansen$^{ 8}$, J.Schieck$^{ 11}$, B.Schmitt$^{ 8}$, S.Schmitt$^{ 11}$, A.Schöning$^{ 8}$, T.Schorner$^{ 34}$, M.Schröder$^{ 8}$, M.Schumacher$^{ 3}$, C.Schwick$^{ 8}$, W.G.Scott$^{ 20}$, R.Seuster$^{ 14}$, T.G.Shears$^{ 8}$, B.C.Shen$^{ 4}$, C.H.Shepherd-Themistocleous$^{ 8}$, P.Sherwood$^{ 15}$, G.P.Siroli$^{ 2}$, A.Sittler$^{ 27}$, A.Skuja$^{ 17}$, A.M.Smith$^{ 8}$, G.A.Snow$^{ 17}$, R.Sobie$^{ 28}$, S.Söldner-Rembold$^{ 10}$, M.Sproston$^{ 20}$, A.Stahl$^{ 3}$, K.Stephens$^{ 16}$, J.Steuerer$^{ 27}$, K.Stoll$^{ 10}$, D.Strom$^{ 19}$, R.Ströhmer$^{ 34}$, R.Tafirout$^{ 18}$, S.D.Talbot$^{ 1}$, S.Tanaka$^{ 24}$, P.Taras$^{ 18}$, S.Tarem$^{ 22}$, R.Teuscher$^{ 8}$, M.Thiergen$^{ 10}$, M.A.Thomson$^{ 8}$, E.von Törne$^{ 3}$, E.Torrence$^{ 8}$, S.Towers$^{ 6}$, I.Trigger$^{ 18}$, Z.Trócsányi$^{ 33}$, E.Tsur$^{ 23}$, A.S.Turcot$^{ 9}$, M.F.Turner-Watson$^{ 8}$, R.Van Kooten$^{ 12}$, P.Vannerem$^{ 10}$, M.Verzocchi$^{ 10}$, P.Vikas$^{ 18}$, H.Voss$^{ 3}$, F.Wäckerle$^{ 10}$, A.Wagner$^{ 27}$, C.P.Ward$^{ 5}$, D.R.Ward$^{ 5}$, P.M.Watkins$^{ 1}$, A.T.Watson$^{ 1}$, N.K.Watson$^{ 1}$, P.S.Wells$^{ 8}$, N.Wermes$^{ 3}$, J.S.White$^{ 28}$, T.Wiesler$^{ 10}$, G.W.Wilson$^{ 14}$, J.A.Wilson$^{ 1}$, T.R.Wyatt$^{ 16}$, S.Yamashita$^{ 24}$, G.Yekutieli$^{ 26}$, V.Zacek$^{ 18}$, D.Zer-Zion$^{ 8}$ ]{} $^{ 1}$School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK $^{ 2}$Dipartimento di Fisica dell’ Università di Bologna and INFN, I-40126 Bologna, Italy $^{ 3}$Physikalisches Institut, Universität Bonn, D-53115 Bonn, Germany $^{ 4}$Department of Physics, University of California, Riverside CA 92521, USA $^{ 5}$Cavendish Laboratory, Cambridge CB3 0HE, UK $^{ 6}$Ottawa-Carleton Institute for Physics, Department of Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada $^{ 7}$Centre for Research in Particle Physics, Carleton University, Ottawa, Ontario K1S 5B6, Canada $^{ 8}$CERN, European Organisation for Particle Physics, CH-1211 Geneva 23, Switzerland $^{ 9}$Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago IL 60637, USA $^{ 10}$Fakultät für Physik, Albert-Ludwigs-Universität, D-79104 Freiburg, Germany $^{ 11}$Physikalisches Institut, Universität Heidelberg, D-69120 Heidelberg, Germany $^{ 12}$Indiana University, Department of Physics, Swain Hall West 117, Bloomington IN 47405, USA $^{ 13}$Queen Mary and Westfield College, University of London, London E1 4NS, UK $^{ 14}$Technische Hochschule Aachen, III Physikalisches Institut, Sommerfeldstrasse 26-28, D-52056 Aachen, Germany $^{ 15}$University College London, London WC1E 6BT, UK $^{ 16}$Department of Physics, Schuster Laboratory, The University, Manchester M13 9PL, UK $^{ 17}$Department of Physics, University of Maryland, College Park, MD 20742, USA $^{ 18}$Laboratoire de Physique Nucléaire, Université de Montréal, Montréal, Quebec H3C 3J7, Canada $^{ 19}$University of Oregon, Department of Physics, Eugene OR 97403, USA $^{ 20}$Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX, UK $^{ 22}$Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel $^{ 23}$Department of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel $^{ 24}$International Centre for Elementary Particle Physics and Department of Physics, University of Tokyo, Tokyo 113, and Kobe University, Kobe 657, Japan $^{ 25}$Institute of Physical and Environmental Sciences, Brunel University, Uxbridge, Middlesex UB8 3PH, UK $^{ 26}$Particle Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel $^{ 27}$Universität Hamburg/DESY, II Institut für Experimental Physik, Notkestrasse 85, D-22607 Hamburg, Germany $^{ 28}$University of Victoria, Department of Physics, P O Box 3055, Victoria BC V8W 3P6, Canada $^{ 29}$University of British Columbia, Department of Physics, Vancouver BC V6T 1Z1, Canada $^{ 30}$University of Alberta, Department of Physics, Edmonton AB T6G 2J1, Canada $^{ 31}$Duke University, Dept of Physics, Durham, NC 27708-0305, USA $^{ 32}$Research Institute for Particle and Nuclear Physics, H-1525 Budapest, P O Box 49, Hungary $^{ 33}$Institute of Nuclear Research, H-4001 Debrecen, P O Box 51, Hungary $^{ 34}$Ludwigs-Maximilians-Universität München, Sektion Physik, Am Coulombwall 1, D-85748 Garching, Germany $^{ a}$ and at TRIUMF, Vancouver, Canada V6T 2A3 $^{ b}$ and Royal Society University Research Fellow $^{ c}$ and Institute of Nuclear Research, Debrecen, Hungary $^{ d}$ and Department of Experimental Physics, Lajos Kossuth University, Debrecen, Hungary $^{ e}$ on leave of absence from the University of Freiburg Introduction ============ Inclusive hadron production in collisions of quasi-real photons can be used to study the structure of photon interactions complementing similar studies of jet production in $\gg$ collisions [@bib-opalgg]. The photons are radiated by the beam electrons[^1] carrying only small negative squared four-momenta $Q^2$. They can therefore be considered to be quasi-real ($Q^2 \approx 0$) if the electrons are scattered at very small angles where they are not detected. For the “anti-tagged” event sample, events are rejected if one or both scattered electrons have been detected. The interactions of the photons can be modelled by assuming that each photon can either interact directly or appear resolved through its fluctuations into hadronic states. In leading order Quantum Chromodynamics (QCD) this model leads to three different event classes for the $\gamma\gamma$ interactions: direct, single-resolved and double-resolved. In resolved events partons (quarks or gluons) from the hadronic fluctuation of the photon take part in the hard interaction. The probability to find a parton in the photon carrying a certain momentum fraction of the photon is parametrised by parton density functions. We measure differential production cross-sections as a function of the transverse momentum and the pseudorapidity of charged hadrons and neutral $\ks$ mesons. Since the distributions are fully corrected for losses due to event and track selection cuts, the acceptance and the resolution of the detector, they are directly comparable to leading order Monte Carlo models and to next-to-leading order (NLO) perturbative QCD calculations by Binnewies, Kniehl and Kramer [@bib-binnewies]. Until now, transverse momentum distributions of charged hadrons have only been measured for single-tagged events by TASSO [@bib-tasso] and MARK II [@bib-mark2] at an average $\langle Q^2 \rangle$ of 0.35 GeV$^2$ and 0.5 GeV$^2$, respectively. We present the first measurement in anti-tagged collisions of quasi-real photons. Furthermore, the transverse momentum distributions in $\gg$ interactions are expected to have a harder component than in photon-proton or meson-proton interactions due to the direct photon interactions. This will be demonstrated by comparing our data to the photo- and hadroproduction data measured by WA69 [@bib-wa69]. At large transverse momenta (after crossing the charm threshold) the production of $\ks$ mesons in photon-photon collisions is sensitive to the direct production of primary charm quarks in addition to the production of primary strange quarks, since the photon couples to the quark charge. $\ks$ production in anti-tagged $\gg$ collisions has previously been measured by TOPAZ [@bib-topaz] and in single-tagged events by MARK II [@bib-mark2]. In this paper, charged hadron and $\ks$ production are studied using the full data sample taken in 1996 at $\ee$ centre-of-mass energies of 161 and 172 GeV corresponding to an integrated luminosity of about 20 pb$^{-1}$. The OPAL detector {#sec-dec} ================= A detailed description of the OPAL detector can be found in Ref. [@opaltechnicalpaper], and therefore only a brief account of the main features relevant to the present analysis will be given here. The central tracking system is located inside a solenoidal magnet which provides a uniform axial magnetic field of 0.435 T along the beam axis[^2]. The detection efficiency for charged particles is close to 100 $\%$ within the polar angle range $|\cos\theta|<0.92$. The magnet is surrounded in the barrel region ($|\cos\theta|<0.82$) by a lead glass electromagnetic calorimeter (ECAL) and a hadronic sampling calorimeter (HCAL). Outside the HCAL, the detector is surrounded by muon chambers. There are similar layers of detectors in the endcaps ($0.81<|\cos\theta|<0.98$). The small angle region from 47 to 140 mrad around the beam pipe on both sides of the interaction point is covered by the forward calorimeters (FD) and the region from 25 to 59 mrad by the silicon tungsten luminometers (SW). From 1996 onwards, including the data presented in this paper, the lower boundary of the acceptance has been increased to 33 mrad following the installation of a low angle shield to protect the central detector against possible synchrotron radiation. Starting with the innermost components, the tracking system consists of a high precision silicon microvertex detector, a vertex drift chamber, a large volume jet chamber with 159 layers of axial anode wires and a set of $z$ chambers measuring the track coordinates along the beam direction. The transverse momenta $\pt$ of tracks are measured with a precision parametrised by $\sigma_{\pt}/\pt=\sqrt{0.02^2+(0.0015\cdot \pt)^2}$ ($\pt$ in GeV/$c$) in the central region. In this paper “transverse” is always defined with respect to the $z$ axis. The jet chamber also provides measurements of the energy loss, ${ \rm d} E/ {\rm d}x$, which are used for particle identification [@opaltechnicalpaper]. The barrel and endcap sections of the ECAL are both constructed from lead glass blocks with a depth of $24.6$ radiation lengths in the barrel region and more than $22$ radiation lengths in the endcaps. The FD consist of cylindrical lead-scintillator calorimeters with a depth of 24 radiation lengths divided azimuthally into 16 segments. The electromagnetic energy resolution is about $18\%/\sqrt{E}$, where $E$ is in GeV. The SW detectors [@bib-siw] consist of 19 layers of silicon detectors and 18 layers of tungsten, corresponding to a total of 22 radiation lengths. Each silicon layer consists of 16 wedge shaped silicon detectors. The electromagnetic energy resolution is about $25\%/\sqrt{E}$ ($E$ in GeV). Kinematics and Monte Carlo simulation ===================================== The properties of the two interacting photons ($i=1,2$) are described by their negative four-momentum transfers $Q_{i}^2$. Each $Q_i^2$ is related to the electron scattering angle $\theta'_i$ relative to the beam direction by $$Q_i^2=-(p_i-p'_i)^2\approx 2E_i E'_i(1-\cos\theta'_i), \label{eq-q2}$$ where $p_i$ and $p'_i$ are the four-momenta of the beam electrons and the scattered electrons, respectively, and $E_i$ and $E'_i$ are their energies. Events with detected scattered electrons (single-tagged or double-tagged events) are excluded from the analysis. This anti-tagging condition is met when the scattering angle $\theta'$ of the electron is less than 33 mrad between the beam axis and the inner edge of the SW detector. It defines an effective upper limit, $Q^2_{\rm max}$, on the values of $Q_{i}^2$ for both photons. The hadronic final state is described by its invariant mass $W$. The spectrum of photons with an energy fraction $y$ of the electron beam may be obtained by the Equivalent Photon Approximation (EPA) [@bib-wwa]: $$\fg(y)= \frac{\alpha}{2\pi}\left(\frac{1+(1-y)^2}{y} \log\frac{\qmax}{\qmin} -2m^2_{\rm e} y\left( \frac{1}{\qmin}-\frac{1}{\qmax} \right)\right),$$ with $\alpha$ being the electromagnetic coupling constant. The minimum kinematically allowed negative squared four-momentum transfer $\qmin$ is determined by the electron mass $m_{\rm e}$: $$\qmin=\frac{m_{\rm e}^2y^2}{1-y}.$$ The Monte Carlo generators PYTHIA [@bib-pythia] and PHOJET [@bib-phojet] have been used to simulate quasi-real photon-photon interactions. More details about the event generation can be found in Ref. [@bib-opalgg]. All possible hard interactions relevant to photon-photon interactions are included. The fragmentation is handled by JETSET [@bib-pythia]. PYTHIA uses the SaS-1D parametrisation [@bib-sas1d] for the parton densities of the photon and PHOJET uses the GRV parametrisation [@bib-grv]. An approximation is used for the processes with primary charm quarks, i.e. where the charm quark is produced in the hard interaction. These processes are simulated using the matrix elements for light quarks. Subsequently the charm quarks are put on the mass-shell. Event selection and background {#sec-evsel} ============================== The production of charged hadrons and $\ks$ mesons was studied using the data taken at $\ee$ centre-of-mass energies, $\sqee$, of 161 and 172 GeV with an integrated luminosity of about 9.9 pb$^{-1}$ and 10.0 pb$^{-1}$, respectively. Photon-photon events are selected with the following set of cuts: - The sum of all energy deposits in the ECAL and the HCAL has to be less than 45 GeV. - The visible invariant hadronic mass, $\WECAL$, calculated from the position and the energy of the clusters measured in the ECAL, has to be greater than 3 GeV. - The missing transverse energy of the event measured in the ECAL and the forward calorimeters has to be less than 5 GeV. - At least 3 tracks must have been found in the tracking chambers. A track is required to have a minimum transverse momentum of 120 MeV/$c$, at least 20 hits in the central jet chamber, and the innermost hit of the track must be within a radius of 60 cm with respect to the $z$ axis. The distance of the point of closest approach to the origin in the r$\phi$ plane must be less than 30 cm in the $z$ direction and less than 2 cm in the $r\phi$ plane. Tracks with a momentum error larger than the momentum itself are rejected if they have fewer than 80 hits. The number of measured hits in the jet chamber must be more than half of the number of possible hits. The number of possible hits is calculated from the polar angle $\cos\theta$ of the track, assuming that the track has no curvature. - To remove events with scattered electrons in the FD or SW, the total energy measured in the FD has to be less than 50 GeV and the total energy measured in the SW has to be less than 35 GeV. These cuts also reduce contamination from multihadronic events with their thrust axis close to the beam direction. - To reduce the background due to beam-gas and beam-wall interactions, $|\langle z_{0}\rangle|$ must be smaller than 10 cm where $\langle z_{0}\rangle$ is the error-weighted average of the track’s $z$ coordinates at the point of closest approach to the origin in the $r\phi$ plane. Beam-wall events with a vertex in the beam-pipe are rejected by requiring the radial position of the primary vertex in the $r\phi$ plane to be less than 3 cm. After all cuts 56732 events remain. All relevant background processes apart from beam-gas and beam-wall events were studied using Monte Carlo generators. Multihadronic events ($\ee\rightarrow \qqbar(\gamma)$) were simulated with PYTHIA 5.722 [@bib-pythia]. KORALZ 4.02 [@bib-koralz] was used to generate the process $\ee\rightarrow\tau^+\tau^-(\gamma)$ and BHWIDE [@bib-bhwide] to generate the Bhabha process $\ee\rightarrow\ee(\gamma)$. Processes with four fermions in the final state, including W pair production, were simulated with grc4f [@bib-grc4f], EXCALIBUR [@bib-excal], VERMASEREN [@bib-vermaseren] and FERMISV [@bib-fermisv]. All signal and background Monte Carlo samples were generated with a full simulation of the OPAL detector [@bib-gopal]. They were analysed using the same reconstruction algorithms as for the data. The main background processes are multi-hadronic $\ee$ annihilation events and $\ee\rightarrow\ee\tau^+\tau^-$ events. Other background processes are found to be negligible. The multihadronic background is mainly reduced by the cut on the sum of the energy measured by the HCAL and the ECAL and by the cut on the missing transverse energy. The background from all these processes after the selection cuts amounts to less than $1\%$. The cut on the energy in SW and FD rejects photon-photon events with electrons scattered at angles $\theta'$ larger than 33 mrad and with an energy greater than 35 GeV in the SW or greater than 50 GeV in the FD. From the Monte Carlo, the rate of events with $\theta'>33$ mrad and energies less than 50 GeV is estimated to be negligible. The effective anti-tagging condition is therefore $\theta'<33$ mrad. Analysis ======== Correction procedure -------------------- The measured transverse momentum and pseudorapidity distributions of the charged hadrons and the $\ks$ mesons have to be corrected for losses due to the event and track selection cuts, for the acceptance and for the resolution of the detector. This is done with Monte Carlo events which were generated with PYTHIA 5.722 and PHOJET 1.05c. The data are corrected by multiplying the experimental distribution, e.g. of the transverse momentum $\pT$, with correction factors which are calculated as the bin-by-bin ratio of the generated and the reconstructed Monte Carlo distributions: $$\left(\frac{{\rm d}\sigma}{{\rm d} \pT} \right)_{\rm corrected}= \frac{\left(\frac{{\rm d}\sigma}{{\rm d} \pT} \right)^{\rm MC}_{\rm generated~~~~}}{ \left(\frac{{\rm d}\sigma}{{\rm d} \pT} \right)^{\rm MC}_{\rm reconstructed}} \left(\frac{{\rm d}\sigma}{{\rm d} \pT} \right)_{\rm measured}. \label{eq1}$$ As a correction factor the mean value from PYTHIA and PHOJET is used. The distributions of the pseudorapidity $\eta=-\ln\tan(\theta/2)$ are corrected in the same way. This method only yields reliable results if the migration between bins due to the finite resolution is small. The bins of the $\pt$ and $|\eta|$ distributions have therefore been chosen to be significantly larger than the resolution expected from the Monte Carlo simulation. The average transverse momentum, $\langle \pt \rangle$, and the average pseudorapidity, $\langle |\eta| \rangle$, in each bin is calculated directly from the data, since detector corrections are small compared to the statistical errors. The visible invariant mass, $W_{\rm vis}$, is determined from all tracks and calorimeter clusters, including the FD and the SW detectors. An algorithm is applied to avoid double-counting of particle momenta in the central tracking system and the calorimeters [@bib-opalgg]. All distributions are shown for $10<W<125$ GeV where $W$ is the hadronic invariant mass corrected for detector effects. To minimize migration effects when using Eq. \[eq1\] for the detector correction, the bins in $W$ must be larger than the experimental resolution and the average reconstructed hadronic invariant mass, $\langle W_{\rm rec}\rangle$, should be approximately equal to the average generated hadronic invariant mass, $\langle W_{\rm gen}\rangle$. The average $\langle W_{\rm vis}\rangle $ and the resolution on $W_{\rm vis}$ as a function of the generated hadronic invariant mass $W_{\rm gen}$ are therefore shown in Fig. \[fig-wcorr\]a, where the vertical bars show the standard deviation (resolution) in each bin. The average $\langle W_{\rm gen}\rangle $ as a function of $W_{\rm vis}$ is plotted in Fig. \[fig-wcorr\]b, where the vertical bars give the error on the mean. This plot is used to determine a correction function so that $\langle W_{\rm gen}\rangle/W_{\rm rec} \approx 1$. The value of $W_{\rm vis}$ measured in the detector is on average significantly smaller than $W_{\rm gen}$. The relation between $W_{\rm gen}$ and $W_{\rm vis}$ shown in Fig. \[fig-wcorr\]b is almost independent of the beam energy and the Monte Carlo generator used. A single polynomial is therefore used to calculate $W_{\rm rec}$ from $W_{\rm vis}$. The polynomial is obtained from the fit shown in Fig. \[fig-wcorr\]b. It is applied to the data and the Monte Carlo. The efficiency to reconstruct photon-photon events in the detector, estimated by the Monte Carlo, is greater than $20\%$ for $W_{\rm gen}>10$ GeV and greater than $60\%$ for $W_{\rm gen}>50$ GeV. The trigger efficiency is defined as the ratio of the number of selected and triggered events to the number of selected events. It was studied using data samples which were obtained using nearly independent sets of triggers. On average the trigger efficiency for the lowest $W$ range, $10<W<30$ GeV, is greater than $97\%$ and it approaches $100\%$ for larger values of $W$. Only lower limits on the trigger efficiency can be determined with this method and therefore no correction factor is applied. Charged hadron production ------------------------- For the charged hadron analysis only particles with a proper lifetime $\tau>0.3$ ns are used to define the primary charged hadronic multiplicity in the Monte Carlo. The primary charged hadrons originate either directly from the primary interaction or from the decay of particles with a lifetime $\tau<0.3$ ns including $\Lambda$ and $\ks$ decay products. The track selection criteria are defined as in Section \[sec-evsel\]. In order to avoid regions where the detector has little or no acceptance, all measurements of charged hadrons were restricted to the range $|\eta|<1.5~(|\cos\theta|{\raisebox{-.6ex}{${\textstyle\stackrel{<}{\sim}}$}}~0.9)$. In this range, the resolution on $\pt$ is given by $\sigma_{\pt}/\pt \approx 0.02$ (see Section \[sec-dec\]) and the resolution on $\eta$ by $\sigma_{\eta}\approx 0.02$. For the $\pt$ distribution in the range $10<W<125$ GeV the correction factors as defined in Eq. \[eq1\] decrease from about 1.7 for $\pt>120$ MeV/$c$ to about $1.1-1.4$ for $\pt>2$ GeV/$c$. The correction factor of about 1.6 for the $\eta$ distribution is nearly constant for $|\eta|<1.5$. The PHOJET and PYTHIA correction factors differ by about $3-10~\%$. K$^0_{\rm S}$ production ------------------------ The $\ks$ mesons are reconstructed using the decay channel $\ks\rightarrow \pi^+\pi^-$ which has a branching ratio of about 69$\%$ [@bib-pdg]. The reconstruction procedure is similar to the procedure described in Ref. [@bib-z01995]. It has been optimised to increase the efficiency for finding $\ks$ mesons in photon-photon events. Tracks of opposite charge are paired together. In addition to other quality cuts the tracks must have a minimum transverse momentum of 120 MeV/$c$ and at least 20 jet chamber hits. The intersection of the tracks in the $r\phi$ plane is considered as a secondary vertex candidate if it satisfies the following criteria: - the radial distance between primary vertex and the intersection point must be greater than 0.5 cm and less than 150 cm. For events with at least 6 tracks the primary vertex is fitted and for events with less than 6 tracks the beam spot reconstructed from tracks collected from many consecutive $\ee$ events during a LEP fill is taken as primary vertex [@bib-prim]. - the difference between the radial coordinate of the secondary vertex and the radial coordinate of the first jet-chamber hit associated with either of the two tracks has to be less than 10 cm; - the radial coordinate of the tracks at the point of closest approach to the primary vertex has to be greater than 0.2 cm; - the angle between the direction of flight from primary to secondary vertex and the combined momentum vector of the two tracks at the intersection point has to be less than 5$^{\circ}$. In addition, a fit was performed for track pairs passing all these cuts, constraining them to originate from a common vertex. A correction procedure was used to compensate for the energy loss of the pions in the inactive material of the detector. All secondary vertices satisfying $|M(\pi^+\pi^-)-0.4977\mbox{~GeV}/c^2|< 0.02$ GeV/$c^2$ are considered to be $\ks$ decay vertices, where the mass $M$ is calculated assuming that both tracks are pions. Finally, the residual background is reduced by requiring at least 20 ${\rm d}E/{\rm d}x$ hits. The two tracks are identified as pions if the ${\mathrm d}E/{\mathrm d}x$ probability for the pion hypothesis, that is the probability that the specific ionisation energy loss in the jet chamber $({\rm d}E/{\rm d}x)$ is compatible with that expected for a pion, exceeds $5\%$. In Fig. \[fig-mpp\] the $\pi^+\pi^-$ invariant mass $M$ is shown for all identified secondary vertices in the selected events before and after applying the ${ \rm d} E/ {\rm d}x$ cuts. After all cuts the reconstruction efficiency for $\ks\rightarrow\pi^+\pi^-$ decays is about $35.5\%$ and the purity is about $95.5\%$ for $\pt(\ks)>1$ GeV/$c$, $|\eta(\ks)|<1.5$ and $10<W<125$ GeV. Systematic errors ================= The following systematic errors, common to the charged hadron and $\ks$ measurements, are taken into account: - The correction factors are obtained using PHOJET and PYTHIA, separately. The resulting distributions are averaged to get the final result. The differences between the two distributions are used to define the systematic error. - The lower limit on the trigger efficiency is taken into account by an additional systematic error of $3\%$ on the cross-section in the range $10<W<30$ GeV. - Systematic errors due to the modelling of the detector resolution for the measurement of tracks were found to be negligibly small in comparison to the other errors. The systematic error due to the uncertainty in the energy scale of the electromagnetic calorimeter was estimated by varying the reconstructed ECAL energy in the Monte Carlo by $\pm 5\%$. - The limited statistics of the Monte Carlo samples, especially at large transverse momenta $\pt$, is also included in the systematic error. - The systematic error of the luminosity measurement is negligible compared to the other systematic errors. The systematic error of the Monte Carlo modelling and of the ECAL energy scale and, for the low $W$ region, the error from the trigger efficiency contribute about equally to the total systematic error. In the $\Ks$ reconstruction additional systematic errors were studied by varying the parameters of the secondary vertex finder and the ${ \rm d} E/ {\rm d}x$ cuts. The full difference between the results is used to estimate the contribution to the total systematic error from the $\Ks$ reconstruction, the Monte Carlo model dependence and the ECAL energy scale. The systematic error affecting the $\Ks$ reconstruction and the error from comparing the PHOJET and PYTHIA correction factors are of similar magnitude. The total systematic error was obtained by adding all systematic errors in quadrature. The total systematic errors are highly correlated from bin to bin. Results {#sec-results} ======= The differential inclusive cross-section $\dspt$ for charged hadrons in the region $|\eta|<1.5$ is shown in Fig. \[fig-pt1\] for different corrected $W$ ranges together with the statistical and systematical errors. The corrected cross-sections are given in Tables \[tab-pt1a\] and \[tab-pt1b\]. The measured differential cross-sections are compared to NLO calculations by Binnewies, Kniehl and Kramer [@bib-binnewies]. The cross-sections are calculated using the QCD partonic cross-sections to NLO for direct, single- and double-resolved processes. The hadronic cross-section is a convolution of the Weizsäcker-Williams effective photon distribution, the parton distribution functions and the fragmentation functions of Ref. [@bib-bkk] which are obtained from a fit to $\ee$ data from TPC and ALEPH. The NLO GRV parametrisation of the parton densities of the photon [@bib-grv] is used with $\Lambda^{(5)}_{\overline{\rm MS}}=131$ MeV and $m_{\rm c}=1.5$ GeV$/c^2$. The renormalization and factorization scales in the calculation are set equal to $\xi\pt$ with $\xi=1$. The change in slope around $\pt=3$ GeV/$c$ in the NLO calculation is due to the charm threshold, below which the charm distribution in the resolved photon and the charm fragmentation functions are set to zero. The cross-section calculation was repeated for the kinematic conditions of the data presented here at an average $\ee$ centre-of-mass energy $\sqee=166.5$ GeV and for scattering angles $\theta'<33$ mrad. For the differential cross-section $\dspt$ a minimum $p_{\rm T}$ of 1 GeV/$c$ is required to ensure the validity of the perturbative QCD calculation. For the same reason the differential cross-section $\dseta$ is restricted to the region $\pt>1.5$ GeV/$c$. The scale dependence of the NLO calculation was studied by setting $\xi=0.5$ and 2. This leads to a variation of the cross-section of about $30\%$ at $\pt=1$ GeV/$c$ and of about $10\%$ for $\pt>5$ GeV/$c$. The NLO calculations lie significantly below the data for $W<30$ GeV for $\dspt$ and $\dseta$. The agreement with the data improves in the higher $W$ bins. The NLO calculation is shown separately for double-resolved, single-resolved and direct interactions. At large $\pt$ the direct interactions dominate. It should be noted that these classifications are scale dependent in NLO. The $p_{\rm T}$ distribution for $10<W<30$ GeV is compared in Fig. \[fig-wa69\] to $p_{\rm T}$ distributions in $\gamma$p and hp (h$=\pi,$K) interactions measured by the experiment WA69 [@bib-wa69]. The hp data are weighted by WA69 in such a way that they contain $60\%$ $\pi$p and $40\%$ Kp data to match the expected mixture of non-strange and strange quarks in the photon beam of the $\gamma$p data. The WA69 data is normalised to the $\gg$ data in the low $p_{\rm T}$ region at $\pt\approx 200$ MeV/$c$ using the same factor for the hp and the $\gamma$p data. The $p_{\rm T}$ distribution of WA69 has been measured in the Feynman-$x$ range $0.0<x_{\rm F}<1.0$. The hadronic invariant mass of the hp data is $W=16$ GeV and the average $\langle W\rangle$ is of similar size for the $\gamma$p data. In the $\gg$ Monte Carlo the average $\langle W \rangle$ is about 17 GeV in the range $10<W<30$ GeV, i.e. the average values of $W$ in the different data samples are approximately the same. Whereas only a small increase is observed in the $\gamma$p data compared to the $\pi$p and K$\pi$ data at large $\pt$, there is a significant increase of the relative rate in the range $\pt>2$ GeV/$c$ for $\gg$ interactions due to the direct process. A clear deviation is seen at large $\pt$ from the exponential fall-off expected for purely hadronic interactions. The differential cross-section $\dseta$ is compared to the predictions of the Monte Carlo generators PHOJET 1.10 and PYTHIA 5.722 in Fig. \[fig-eta1\] taking into account the anti-tagging condition $\theta'<33$ mrad. In PHOJET the $Q^2$ suppression of the total $\gg$ cross-section is parametrised using Generalised Vector Meson Dominance (GVMD) and a model for the change of soft hadron production and diffraction with increasing photon virtuality $Q^2$ is also included. The photon-photon mode of PYTHIA only simulates the interactions of real photons with $Q^2=0$. The virtuality of the photons defined by $Q^2$ enters only through the equivalent photon approximation in the generation of the photon energy spectrum, but the electrons are scattered at zero angle. This model is not expected to be correct for larger values of $Q^2$. We have therefore simulated events with $Q^2<1$ GeV$^2$ with the photon-photon mode of PYTHIA and events with $Q^2>1$ GeV$^2$ and $\theta'<33$ mrad with the electron-photon mode of PYTHIA. The differential cross-section $\dseta$ shown in Fig. \[fig-eta1\] is nearly independent of $|\eta|$ in the measured range. The $|\eta|$ distribution is reasonably well described by PYTHIA and PHOJET for $\pt>120$ MeV$/c$, apart from the high $W$ region where PHOJET appears to be below the data. For transverse momenta $\pt>1.5$ GeV$/c$ (Fig. \[fig-eta2\]) both Monte Carlo models underestimate the data significantly. The same behaviour is observed for the NLO calculation at low $W$, but the agreement of the NLO calculation with the data improves in the high $W$ bins. The corrected cross-sections are given in Tables \[tab-eta2\] and \[tab-eta1\]. The differential inclusive cross-sections $\dspt$ and $\dseta$ have been measured for $\ks$ mesons with $\pt(\ks)>1$ GeV$/c$ and $|\eta(\ks)|<1.5$. The $\pt$ and $\eta$ dependent cross-sections are presented in the $W$ range $10~<~W~<~125$ GeV (Fig. \[fig-ks1\] and Tables \[tab-ks1\]–\[tab-ks2\]). In addition the $\pt$ distribution is shown for two separate $W$ ranges (Fig. \[fig-ks2\] and Table \[tab-ks3\]). The results are compared to PHOJET and PYTHIA. Based on the PYTHIA simulation using SaS-1D, about half of the $\ks$ are expected to be produced from charm quarks at large $\pt$ where direct processes are dominant. Both Monte Carlo models significantly underestimate the $\ks$ production cross-section in the low $\pt$ region where most $\ks$ are expected to originate from primary strange quarks. The distributions are reasonably well described by the NLO calculations which use the $\ks$ fragmentation function fitted to MARK II [@bib-markk0] and ALEPH [@bib-alephk0] data in Ref. [@bib-fk0]. The change in slope between $\pt=2$ and $\pt=3$ GeV/$c$ in the NLO calculation is again due to the charm threshold. The variation of the calculated cross-section as a function of $\pt$ for different choices of scales, $\xi=0.5$ and $2$, is largest around the charm threshold, about $30-40\%$, and $10-20\%$ elsewhere. Conclusions =========== We present measurements of differential cross-sections as a function of transverse momentum and pseudorapidity for charged hadrons and $\Ks$ mesons produced in photon-photon collisions at LEP. The data were taken at $\ee$ centre-of-mass energies of 161 and 172 GeV. The differential cross-section $\dspt$ for charged hadrons is compared to NLO calculations. In the range $10<W<30$ GeV more charged hadrons are found at large $\pt$ than predicted. Good agreement between the NLO calculation and the data is found in the highest $W$ range, $55<W<125$ GeV. The Monte Carlo models PYTHIA and PHOJET both underestimate the cross-section for tracks with $\pt>1.5$ GeV and $|\eta|<1.5$ in all $W$ ranges. The shape of the differential cross-section $\dseta$ is well reproduced by the NLO calculations and the Monte Carlo models. A comparison of the $\pt$ distributions of the $\gg$ data to $\pt$ distributions measured in $\gamma$p and ($\pi$,K)p processes at similar invariant masses shows the relative increase of hard interactions in $\gg$ processes due to the direct component. The transverse momentum and pseudorapidity distributions of the $\ks$ mesons are reasonably well reproduced by the NLO calculations, but they are significantly underestimated by the Monte Carlo models PHOJET and PYTHIA. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Janko Binnewies, Bernd Kniehl and Gustav Kramer for providing the NLO calculations and for many useful discussions.\ We particularly wish to thank the SL Division for the efficient operation of the LEP accelerator at all energies and for their continuing close cooperation with our experimental group. We thank our colleagues from CEA, DAPNIA/SPP, CE-Saclay for their efforts over the years on the time-of-flight and trigger systems which we continue to use. 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J. Binnewies, B.A. Kniehl and G. Kramer, Phys. Rev. D53 (1996) 3573. ------------------- ----------------------------------- ----------------------------------------- ----------------------------------- ------------------------------------------ $\pt$ \[GeV/$c$\] $\langle \pt \rangle$ \[GeV/$c$\] $\langle \pt \rangle$ \[GeV/$c$\] 0.12-0.28 0.20 (3.11$\pm$0.01$\pm$0.24)$\times 10^{4}$ 0.20 (1.06$\pm$0.01$\pm$0.08)$\times 10^{4}$ 0.28-0.44 0.36 (2.70$\pm$0.01$\pm$0.20)$\times 10^{4}$ 0.35 (8.80$\pm$0.06$\pm$0.58)$\times 10^{3}$ 0.44-0.60 0.51 (1.60$\pm$0.01$\pm$0.10)$\times 10^{4}$ 0.51 (5.31$\pm$0.04$\pm$0.32)$\times 10^{3}$ 0.60-0.80 0.69 (8.00$\pm$0.06$\pm$0.46)$\times 10^{3}$ 0.69 (2.67$\pm$0.03$\pm$0.14)$\times 10^{3}$ 0.80-1.00 0.89 (3.38$\pm$0.04$\pm$0.17)$\times 10^{3}$ 0.89 (1.26$\pm$0.02$\pm$0.06)$\times 10^{3}$ 1.00-1.20 1.09 (1.48$\pm$0.02$\pm$0.08)$\times 10^{3}$ 1.09 (5.95$\pm$0.12$\pm$0.22)$\times 10^{2}$ 1.20-1.40 1.29 (6.64$\pm$0.16$\pm$0.40)$\times 10^{2}$ 1.29 (2.92$\pm$0.09$\pm$0.12)$\times 10^{2}$ 1.40-1.60 1.49 (3.29$\pm$0.11$\pm$0.21)$\times 10^{2}$ 1.49 (1.55$\pm$0.07$\pm$0.06)$\times 10^{2}$ 1.60-1.80 1.69 (1.75$\pm$0.08$\pm$0.11)$\times 10^{2}$ 1.69 (9.01$\pm$0.49$\pm$0.55)$\times 10^{1}$ 1.80-2.00 1.89 (1.00$\pm$0.06$\pm$0.07)$\times 10^{2}$ 1.89 (5.96$\pm$0.39$\pm$0.38)$\times 10^{1}$ 2.00-2.20 2.10 (6.04$\pm$0.48$\pm$0.37)$\times 10^{1}$ 2.09 (3.29$\pm$0.29$\pm$0.13)$\times 10^{1}$ 2.20-2.40 2.30 (4.18$\pm$0.39$\pm$0.27)$\times 10^{1}$ 2.29 (2.15$\pm$0.23$\pm$0.20)$\times 10^{1}$ 2.40-2.60 2.50 (2.06$\pm$0.30$\pm$0.08)$\times 10^{1}$ 2.50 (1.64$\pm$0.20$\pm$0.07)$\times 10^{1}$ 2.60-2.80 2.68 (2.04$\pm$0.31$\pm$0.07)$\times 10^{1}$ 2.70 (1.01$\pm$0.16$\pm$0.08)$\times 10^{1}$ 2.80-3.00 2.90 (1.12$\pm$0.25$\pm$0.05)$\times 10^{1}$ 2.88 (9.18$\pm$1.55$\pm$1.10) 3.00-3.50 3.22 (1.09$\pm$0.16$\pm$0.11)$\times 10^{1}$ 3.21 (4.26$\pm$0.65$\pm$0.68) 3.50-4.00 3.71$\pm$0.01 (6.03$\pm$1.35$\pm$0.21) 3.74$\pm$0.01 (3.58$\pm$0.62$\pm$0.32) 4.00-5.00 4.36$\pm$0.01 (2.99$\pm$0.82$\pm$0.33) 4.38$\pm$0.02 (1.07$\pm$0.25$\pm$0.04) 5.00-6.00 5.55$\pm$0.03 (1.40$\pm$0.64$\pm$0.07) 5.50$\pm$0.04 (7.12$\pm$2.38$\pm$0.44)$\times 10^{-1}$ 6.00-8.00 – – 6.94$\pm$0.07 (2.74$\pm$1.22$\pm$0.15)$\times 10^{-1}$ 8.00-15.0 – – 9.51$\pm$0.32 (1.11$\pm$0.73$\pm$0.36)$\times 10^{-1}$ ------------------- ----------------------------------- ----------------------------------------- ----------------------------------- ------------------------------------------ : Differential inclusive charged hadron production cross-sections $\dspt$ for $|\eta|<1.5$ and in the $W$ ranges $10<W<30$ GeV and $30<W<55$ GeV. The first error is statistical and the second is systematic. No value is given if the error on $\langle \pt \rangle$ is less than 0.01.[]{data-label="tab-pt1a"} ------------------- ----------------------------------- ------------------------------------------ ----------------------------------- ------------------------------------------ $\pt$ \[GeV/$c$\] $\langle \pt \rangle$ \[GeV/$c$\] $\langle \pt \rangle$ \[GeV/$c$\] 0.12-0.28 0.20 (5.85$\pm$0.05$\pm$0.58)$\times 10^{3}$ 0.20 (4.78$\pm$0.02$\pm$0.38)$\times 10^{4}$ 0.28-0.44 0.35 (4.75$\pm$0.04$\pm$0.46)$\times 10^{3}$ 0.36 (4.05$\pm$0.01$\pm$0.31)$\times 10^{4}$ 0.44-0.60 0.51 (2.92$\pm$0.03$\pm$0.28)$\times 10^{3}$ 0.51 (2.42$\pm$0.01$\pm$0.17)$\times 10^{4}$ 0.60-0.80 0.69 (1.53$\pm$0.02$\pm$0.12)$\times 10^{3}$ 0.69 (1.21$\pm$0.01$\pm$0.07)$\times 10^{4}$ 0.80-1.00 0.89 (7.25$\pm$0.15$\pm$0.47)$\times 10^{2}$ 0.89 (5.33$\pm$0.04$\pm$0.27)$\times 10^{3}$ 1.00-1.20 1.09 (3.89$\pm$0.10$\pm$0.20)$\times 10^{2}$ 1.09 (2.45$\pm$0.03$\pm$0.12)$\times 10^{3}$ 1.20-1.40 1.29 (1.91$\pm$0.07$\pm$0.09)$\times 10^{2}$ 1.29 (1.14$\pm$0.02$\pm$0.06)$\times 10^{3}$ 1.40-1.60 1.49 (1.04$\pm$0.06$\pm$0.04)$\times 10^{2}$ 1.49 (5.87$\pm$0.13$\pm$0.31)$\times 10^{2}$ 1.60-1.80 1.69 (6.20$\pm$0.41$\pm$0.18)$\times 10^{1}$ 1.69 (3.28$\pm$0.10$\pm$0.19)$\times 10^{2}$ 1.80-2.00 1.89 (3.78$\pm$0.33$\pm$0.20)$\times 10^{1}$ 1.89 (2.00$\pm$0.08$\pm$0.11)$\times 10^{2}$ 2.00-2.20 2.09 (2.53$\pm$0.27$\pm$0.06)$\times 10^{1}$ 2.09 (1.20$\pm$0.06$\pm$0.06)$\times 10^{2}$ 2.20-2.40 2.28 (1.72$\pm$0.22$\pm$0.04)$\times 10^{1}$ 2.29 (8.10$\pm$0.49$\pm$0.57)$\times 10^{1}$ 2.40-2.60 2.50 (1.10$\pm$0.18$\pm$0.02)$\times 10^{1}$ 2.50 (5.00$\pm$0.38$\pm$0.20)$\times 10^{1}$ 2.60-2.80 2.69 (7.89$\pm$1.49$\pm$0.23) 2.69 (3.81$\pm$0.34$\pm$0.18)$\times 10^{1}$ 2.80-3.00 2.92 (5.09$\pm$1.15$\pm$0.34) 2.90 (2.70$\pm$0.29$\pm$0.11)$\times 10^{1}$ 3.00-3.50 3.23 (3.18$\pm$0.59$\pm$0.04) 3.22 (1.77$\pm$0.16$\pm$0.07)$\times 10^{1}$ 3.50-4.00 3.70$\pm$0.01 (2.07$\pm$0.48$\pm$0.10) 3.72 (1.16$\pm$0.13$\pm$0.04)$\times 10^{1}$ 4.00-5.00 4.46$\pm$0.02 (9.30$\pm$2.44$\pm$0.58)$\times 10^{-1}$ 4.40$\pm$0.01 (4.32$\pm$0.61$\pm$0.19) 5.00-6.00 5.38$\pm$0.04 (3.30$\pm$1.67$\pm$0.64)$\times 10^{-1}$ 5.48$\pm$0.01 (1.95$\pm$0.45$\pm$0.09) 6.00-8.00 6.81$\pm$0.09 (1.24$\pm$0.76$\pm$0.05)$\times 10^{-1}$ 6.92$\pm$0.03 (5.97$\pm$2.01$\pm$0.40)$\times 10^{-1}$ 8.00-15.0 9.91$\pm$0.34 (2.64$\pm$1.76$\pm$0.23)$\times 10^{-2}$ 9.69$\pm$0.18 (1.10$\pm$0.46$\pm$0.07)$\times 10^{-1}$ ------------------- ----------------------------------- ------------------------------------------ ----------------------------------- ------------------------------------------ : Differential inclusive charged hadron production cross-sections $\dspt$ for $|\eta|<1.5$ and in the $W$ range $55<W<125$ GeV and for all $W$ ($10<W<125$ GeV). The first error is the statistical error and the second error is the systematic error. No value is given if the error on $\langle \pt \rangle$ is less than 0.01.[]{data-label="tab-pt1b"} ----------- -------------------------- ------------------------- ---------------------------- ------------------------- $|\eta|$ $\langle |\eta| \rangle$ $\langle |\eta | \rangle$ 0.00-0.30 0.15 9.91$\pm$0.06$\pm$0.70 0.15 3.27$\pm$0.03$\pm$0.20 0.30-0.60 0.45 9.98$\pm$0.06$\pm$0.71 0.45 3.33$\pm$0.03$\pm$0.19 0.60-0.90 0.75 10.05$\pm$0.06$\pm$0.71 0.75 3.41$\pm$0.03$\pm$0.19 0.90-1.20 1.05 10.01$\pm$0.06$\pm$0.71 1.05 3.43$\pm$0.03$\pm$0.20 1.20-1.50 1.35 9.54$\pm$0.06$\pm$0.67 1.35 3.33$\pm$0.03$\pm$0.19 $|\eta|$ $\langle |\eta| \rangle$ $\langle |\eta | \rangle$ 0.00-0.30 0.15 1.80$\pm$0.02$\pm$0.16 0.15 14.97$\pm$0.06$\pm$1.12 0.30-0.60 0.45 1.85$\pm$0.02$\pm$0.17 0.45 15.16$\pm$0.07$\pm$1.13 0.60-0.90 0.75 1.88$\pm$0.02$\pm$0.17 0.75 15.33$\pm$0.07$\pm$1.14 0.90-1.20 1.05 1.93$\pm$0.02$\pm$0.16 1.05 15.35$\pm$0.07$\pm$1.15 1.20-1.50 1.35 1.90$\pm$0.02$\pm$0.16 1.35 14.75$\pm$0.06$\pm$1.08 ----------- -------------------------- ------------------------- ---------------------------- ------------------------- : Differential inclusive charged hadron production cross-sections $\dseta$ for $\pt>120$ MeV$/c$ and in the $W$ ranges $10<W<30$ GeV, $30<W<55$ GeV, $55<W<125$ GeV and for all $W$ ($10<W<125$ GeV). The first error is the statistical error and the second error is the systematic error.[]{data-label="tab-eta2"} ----------- -------------------------- ------------------------------------------ ---------------------------- ------------------------------------------ $|\eta|$ $\langle |\eta| \rangle$ $\langle |\eta | \rangle$ 0.00-0.30 0.15 (8.26$\pm$0.47$\pm$0.37)$\times 10^{-2}$ 0.14 (4.51$\pm$0.27$\pm$0.20)$\times 10^{-2}$ 0.30-0.60 0.45 (9.33$\pm$0.49$\pm$0.49)$\times 10^{-2}$ 0.45 (4.36$\pm$0.27$\pm$0.09)$\times 10^{-2}$ 0.60-0.90 0.75 (7.72$\pm$0.45$\pm$0.24)$\times 10^{-2}$ 0.76 (4.66$\pm$0.28$\pm$0.44)$\times 10^{-2}$ 0.90-1.20 1.05 (7.95$\pm$0.47$\pm$0.35)$\times 10^{-2}$ 1.05 (4.39$\pm$0.29$\pm$0.27)$\times 10^{-2}$ 1.20-1.50 1.34 (8.17$\pm$0.47$\pm$0.31)$\times 10^{-2}$ 1.35 (4.56$\pm$0.30$\pm$0.19)$\times 10^{-2}$ $|\eta|$ $\langle |\eta| \rangle$ $\langle |\eta | \rangle$ 0.00-0.30 0.15 (3.07$\pm$0.24$\pm$0.07)$\times 10^{-2}$ 0.15 (1.62$\pm$0.06$\pm$0.09)$\times 10^{-1}$ 0.30-0.60 0.45 (2.91$\pm$0.23$\pm$0.09)$\times 10^{-2}$ 0.45 (1.65$\pm$0.06$\pm$0.08)$\times 10^{-1}$ 0.60-0.90 0.75 (3.21$\pm$0.25$\pm$0.07)$\times 10^{-2}$ 0.75 (1.59$\pm$0.06$\pm$0.09)$\times 10^{-1}$ 0.90-1.20 1.05 (3.14$\pm$0.25$\pm$0.12)$\times 10^{-2}$ 1.05 (1.55$\pm$0.06$\pm$0.07)$\times 10^{-1}$ 1.20-1.50 1.35 (3.35$\pm$0.26$\pm$0.07)$\times 10^{-2}$ 1.35 (1.60$\pm$0.06$\pm$0.07)$\times 10^{-1}$ ----------- -------------------------- ------------------------------------------ ---------------------------- ------------------------------------------ : Differential inclusive charged hadron production cross-sections $\dseta$ for $\pt>1.5$ GeV$/c$ and in the $W$ ranges $10<W<30$ GeV, $30<W<55$ GeV, $55<W<125$ GeV and for all $W$ ($10<W<125$ GeV). The first error is the statistical error and the second error is the systematic error.[]{data-label="tab-eta1"} ------------------- ----------------------------------- --------------------------------- $\pt$ \[GeV$/c$\] $\langle \pt \rangle$ \[GeV$/c$\] d$\sigma$/d$\pt$ \[pb/GeV$/c$\] 1.0–1.2 1.09 $\pm$0.01 206.2 $\pm$ 17.4 $\pm$ 16.1 1.2–1.5 1.33 $\pm$0.01 100.2 $\pm$ 8.9 $\pm$ 8.5 1.5–1.9 1.66 $\pm$0.01 32.9 $\pm$ 4.5 $\pm$ 3.7 1.9–2.4 2.11 $\pm$0.02 13.5 $\pm$ 2.6 $\pm$ 1.3 2.4–3.0 2.65 $\pm$0.04 5.2 $\pm$ 1.3 $\pm$ 0.7 3.0–4.0 3.37 $\pm$0.08 2.0 $\pm$ 0.7 $\pm$ 0.2 4.0–5.5 4.56 $\pm$0.15 0.4 $\pm$ 0.3 $\pm$ 0.1 ------------------- ----------------------------------- --------------------------------- : Differential inclusive $\ks$ production cross-sections $\dspt$ for $\pt(\ks)>1$ GeV$/c$ and $|\eta(\ks)|<1.5$ in the $W$ range $10<W<125$ GeV. The first error is the statistical error and the second error is the systematic error.[]{data-label="tab-ks1"} $|\eta|$ $\langle |\eta| \rangle$ d$\sigma$/d$|\eta|$ \[pb\] ---------- -------------------------- ---------------------------- 0.0–0.3 0.15 $\pm$ 0.01 61.7 $\pm$ 7.2 $\pm$ 5.2 0.3–0.6 0.45 $\pm$ 0.01 63.1 $\pm$ 7.3 $\pm$ 5.4 0.6–0.9 0.76 $\pm$ 0.01 72.4 $\pm$ 7.8 $\pm$ 6.1 0.9–1.2 1.05 $\pm$ 0.01 70.2 $\pm$ 8.0 $\pm$ 4.7 1.2–1.5 1.34 $\pm$ 0.01 58.0 $\pm$ 7.5 $\pm$ 4.0 : Differential inclusive $\ks$ production cross-sections $\dseta$ for $\pt(\ks)>1$ GeV$/c$ and $|\eta(\ks)|<1.5$ in the $W$ range $10<W<125$ GeV. The first error is the statistical error and the second error is the systematic error.[]{data-label="tab-ks2"} ----------------- --------------------------------- ------------------------------- --------------------------------- ------------------------------- $\pt$ \[GeV/c\] $\langle \pt \rangle$ \[GeV/c\] d$\sigma$/d$\pt$ \[pb/GeV/c\] $\langle \pt \rangle$ \[GeV/c\] d$\sigma$/d$\pt$ \[pb/GeV/c\] 1.0–1.2 1.08 $\pm$0.01 127.3 $\pm$ 14.6 $\pm$ 12.6 1.10 $\pm$0.01 75.0 $\pm$ 9.5 $\pm$ 5.1 1.2–1.5 1.33 $\pm$0.01 73.6 $\pm$ 8.4 $\pm$ 6.6 1.33 $\pm$0.01 30.0 $\pm$ 4.3 $\pm$ 2.3 1.5–1.9 1.66 $\pm$0.02 19.8 $\pm$ 3.4 $\pm$ 2.5 1.67 $\pm$0.02 13.1 $\pm$ 3.0 $\pm$ 2.3 1.9–2.4 2.11 $\pm$0.03 9.8 $\pm$ 2.8 $\pm$ 1.3 2.11 $\pm$0.03 4.4 $\pm$ 1.2 $\pm$ 0.4 2.4–3.0 2.72 $\pm$0.04 1.7 $\pm$ 0.9 $\pm$ 0.2 2.62 $\pm$0.05 2.9 $\pm$ 0.8 $\pm$ 0.6 3.0–4.0 3.37 $\pm$0.15 0.9 $\pm$ 0.4 $\pm$ 0.3 3.38 $\pm$0.09 1.0 $\pm$ 0.6 $\pm$ 0.2 4.0–5.5 4.56 $\pm$0.15 0.3 $\pm$ 0.3 $\pm$ 0.1 ----------------- --------------------------------- ------------------------------- --------------------------------- ------------------------------- : Differential inclusive $\ks$ production cross-sections $\dspt$ for $\pt(\ks)>1$ GeV$/c$ and $|\eta(\ks)|<1.5$ in the $W$ ranges $10<W<35$ GeV and $35<W<125$ GeV. The first error is the statistical error and the second error is the systematic error.[]{data-label="tab-ks3"} -- -- -- -- -- -- -- -- [^1]: Positrons are also referred to as electrons [^2]: In the OPAL coordinate system the $z$ axis points in the direction of the e$^-$ beam. The polar angle $\theta$, the azimuthal angle $\phi$ and the radius $r$ denote the usual spherical coordinates.

--- abstract: 'We exploit a slightly noncollinear second-harmonic cross-correlation scheme to map the 3D space-time intensity distribution of an unknown complex-shaped ultrashort optical pulse. We show the capability of the technique to reconstruct both the amplitude and the phase of the field through the coherence of the nonlinear interaction down to a resolution of 10 $\mu$m in space and 200 fs in time. This implies that the concept of second-harmonic holography can be employed down to the sub-ps time scale, and used to discuss the features of the technique in terms of the reconstructed fields.' address: - | Department of Physics “Aldo Pontremoli”, University of Milan and\ Istituto Nazionale Fisica della Materia, Via Celoria 16, I-20133 Milano, Italy - 'Instituto di Ciencias Fotonicas c/Jordi Girona, 29 - NEXUS II E-08034 Barcelona, Spain' - 'Istitito Nazionale Fisica della Materia, Dipartimento di Scienze Chimiche, Fisiche, Matematiche, Università dell’Insubria, Via Valleggio 11, I-22100 Como, Italy' - 'Department of Quantum Electronics, Vilnius University, Sauletekio 9, building III, LT-2040 Vilnius, Lithuania' author: - 'Marco A.C. Potenza' - Stefano Minardi - Jose Trull - Gianni Blasi - Domenico Salerno - Paolo Di Trapani - Arunas Varanavičius - Algis Piskarskas title: Three dimensional imaging of short pulses --- $^{\dagger}$ Holography; Non-linear optics; Cross-correlation of ultrashort pulses; Introduction ============ The study of ultrafast phenomena has been a major scientific priority during last decades covering different topics such as the study of radiation-matter interactions [@hullier], transient response of molecules and atoms [@zewail], coherent control of chemical reactions [@Baumert] or communication and information technology [@stegeman]. The growth of this field relies upon the development of sources of femtosecond radiation and of appropriate techniques able to provide time domain information in the femtosecond scale. However, during the interaction of short optical pulses with a nonlinear medium, different mechanisms can lead to its reshaping into complex [*spatio-temporal*]{} structures with non-trivial light distribution [@PDT00]. As a consequence, their complete characterization requires a method capable of acquiring a snap-shot of their intensity distribution in the whole 3-dimensional (3D; $x,y,t-z/c$) comoving frame. Most of the available methods for pulse diagnostic provide information of the WP characteristics in a space of reduced dimensionality. The use of frequency resolved autocorrelation techniques (i.e. FROG, SPIDER) allows for the recovery of the temporal intensity and phase profile of a given pulse but assumes uniform transverse spatial distribution [@Trebino; @Jaconis]. On the contrary, the characterization of transversally localized beams often relies upon the optical imaging onto CCD cameras, therefore the temporal information is lost because of their integration times unavoidably larger than the optical pulse duration. Recently, a space-time characterization method based on extended SPIDER technique has been developed capable of resolving electric field characteristics in time and along one spatial coordinate [@dorrer]. A quite direct way of obtaining spatio-temporal intensity profiles of a WP is to perform measurements with a streak camera, which allows a temporal resolution up to fractions of ps [@streak]. This technique allowed the investigation of the dynamics of the breakup along the pulse envelope of a large elliptical beam propagating into a saturable Kerr nonlinear medium [@Lantz02a; @Lantz02b]. However, also in this case, the space-time maps are intrinsically two dimensional (1 spatial + temporal dimensions). A different approach to the problem considers the retrieval of the pulse shape through an all-optical processing by means of spatially resolved detection systems combined with gating techniques. The principle of the method is that of characterizing with spatial resolution an optical field that is proportional to the product $E_O({\mathbf x},t)E_R({\mathbf x},t)$, where $E_O({\mathbf x},t)$ is the object to be measured and $E_R({\mathbf x},t)$ is a suitable reference pulse. Since the product is different from zero only on the intersection of the support of both fields[^1], by translating the reference with respect to the object, we get the possibility of recording information from different parts of the object. Among the linear time gating techniques, light–in–flight holographic recording has been the first technique which permitted the recording of dynamically evolving light fields during propagation [@Denisyuk69; @Abramson83; @Abramson89]. Recently, this technique has been adapted to study the propagation of a 3 ps long pulse in linear media [@Kubota02]. Linear probing techniques were also exploited in order to obtain time resolved imaging, like the probing of the birefringence properties of plasma by means of a delayed, spatially-extended 100 fs pulses to investigate the dynamics of laser pulse focusing in air [@Fujimoto99]. Nonlinear processes have been employed since long ago to resolve in time the evolution of ultrafast phenomena. Among them, the quadratic nonlinearity has been proved to be particularly versatile due to the fact that it provides easily terms containing the product of two optical fields. Recently, a type II degenerate parametric amplification scheme has been employed to obtain time resolved 2D images of a ps-pulse hitting a diffusing screen with 35 ps resolution [@Devaux95], thus yielding a 3D imaging. The same technique was later used to image an object embedded in a thick diffusing sample [@Devaux99]. Althought our setup is actually an improved version of that described in [@Devaux99], we point out that our conceptual approach is different from the study of the propagation of a wave front. In fact, in our case the propagation variable is fixed. Our goal in this article is to demonstrate the potentiality of the optical gating technique to acquire a high resolution space-time map of short, focused WPs in their comoving reference frame. Furthermore, we devise the capability of the technique to reconstruct both the amplitude and the phase of the WP thanks to the coherence of the nonlinear interaction. We propose a method that is based on quadratic type I interaction in a sum-frequency generation scheme either by a non-collinear second harmonic generation or by a collinear sum-frequency scheme. The latter has been used in [@jose]. Here we discuss the first option, showing that if the interaction angle between the two interacting fields is small enough, then a reliable space-time map of the object pulse can be obtained. This can be achieved if the duration of the gate is much smaller than that of the object to be imaged. A holographic interpretation of the method permits to gain insight into the process of up-conversion of the space-time slices of the object into the SF field, and to prove that the coherence of the SF process is able to reconstruct the wavefront in both amplitude and phase. Our results confirm this possibility. The theoretical discussion of the method is followed by section \[experiment\], where we present the set-up and the experimentally reconstructed space-time intensity profiles of a parametric spatial soliton excited by a 1 ps light pulse. For our setting, we estimate a mapping resolution of 200 fs in time and about 10$\mu$m in space. The features of the technique are presented in section \[features\], pointing out the limitations that may arise and discussing the possible implementations in each case. In the last section the main conclusions are presented. Description of the technique: intensity and field reconstruction {#crosscorrelation} ================================================================ In this section we explicitly show how a non-collinear sum-frequency (SF) scheme can be exploited to get high resolution space-time intensity maps of an unknown light wave packet with a space-time structure (object wave). The recovery of a 3D intensity map is obtained by means of a short reference pulse which provides a time gating inside a nonlinear (NL) crystal, and generates a SF signal containing the information about a set of 2D slices of the object obtained by changing the reference delay. We first discuss the case for the reconstruction of the object intensity profile, and then we show how the intrinsic coherence of the SF process allows in fact for a truly holographic recording of the unknown object. 3D Intensity profile mapping ---------------------------- Let us denote the object ($\bar{E}_O$) and reference ($\bar{E}_R$) wave packets as follows: $$\begin{aligned} \bar{E}_O&=& E_O(x,y,z,t)e^{i[\omega_1 t-k_z(\omega_1)z-k_x(\omega_1)x]}+c.c.\\ \bar{E}_R&=& E_R(x,y,z,t)e^{i[\omega_2 t-k_z(\omega_2)z+k_x(\omega_2)x]}+c.c.\end{aligned}$$ where the complex functions $E_O(x,y,t,z)$ and $E_R(x,y,t,z)$ are the slowly varying envelopes of two waves with frequencies $\omega_1$ and $\omega_2$. Note that in this form the equations describe two wavepackets propagating in the positive $z$ direction and colliding at an angle $\theta = 2 \arctan ( k_x / k_z)$ in the $x-z$ plane (here $k = \sqrt{k_x^2 + k_z^2} = 2 \pi / \lambda_0$). For a SF generation process occurring inside a quadratic nonlinear crystal, the polarization source giving rise to the SF can be written as: $$\begin{aligned} P_{SF} \propto 2 E_O E _R e^{i [\omega_3 t - k_z(\omega_3) z ]} + c.c. \label{polarizzazione}\end{aligned}$$ where the phase and energy matching conditions $k_z(\omega_3) = k_z(\omega_1) + k_z(\omega_2) $, $k_x(\omega_1) = - k_x(\omega_2)$ and $\omega_3 = \omega_1 + \omega_2$ have been used. The SF field propagates along $z$ direction and has a slowly varying envelope function that we will indicate by $E_{SF}(x,y,t,z)$. Now we introduce the following assumptions: 1) small depletion of both the $E_O$ and $E_R$ fields; 2) negligible diffraction and dispersion within the crystal; 3) equal group velocities of the object, reference and sum-frequency pulses, namely $u_O$, $u_R$ and $u_{SF}$, that is $u_O=u_R=u_{SF}=u$. Note that all these assumptions approximatively hold as long as the thickness of the crystal $\Delta z$ is small compared to the characteristic lenghts of the system (nonlinear length, dispersion and diffraction lengths, pulse walkoff length). These assumptions allow to find a travelling reference frame for all the propagating pulses by introducing the retarded time $\tau=t-z/u$. The general partial differential equations describing the interaction process then reduces to an ordinay differential equation for the envelope $E_{SF}$. If we also introduce a time delay $\tau_i$ on the reference wavefront, the equation takes the form: $$\frac{d E_{SF}(x,y,\tau,z)}{d z} = i2\sigma E_O(x,y,\tau,z)E_R(x,y,\tau-\tau_i,z) \label{sumfrequency}$$ where $\sigma$ is the nonlinear coupling term. The last equation is readily integrated and, if the mixing crystal is placed at position $z_0$, it reads: $$E_{SF}(x,y,\tau,z_0)=i\sigma 2 E_O(x,y,\tau,z_0)E_R(x,y,\tau-\tau_i,z_0)\Delta z \label{campi}$$ A deeper discussion about the meaning of this expression will be given in the following subsection. Here we just point out how the intensity profile of the SF field can be used to retrieve the 3D intensity map of the object. More precisely, for a given lag time $\tau_i$, the spatially dependent SF fluence profile (the CCD signal) $S(x,y,\tau_i,z_0)$ recorded just at the exit face of the mixing crystal is given by: $$S(x,y,\tau_i,z_0) \simeq (\sigma\Delta z)^2\int^{+\infty}_{-\infty}|E_O(x,y,\tau,z_0)|^2 |E_R(x,y,\tau-\tau_i,z_0)|^2 d\tau \label{convolution}$$ This expression provides the convolution between the intensity profiles of the object and reference wave packets, lagged in time by $\tau_i$, and shows that the signal recorded by a CCD sensor is a [*linear*]{} function of the intensity of both the object and reference wavepackets. In the particular case in which the reference wave is spatially homogeneous in the transverse $x-y$ plane, and temporally much shorter than the object, we can write expression \[convolution\] in the following form: $$S(x,y,\tau_i,z_0) \propto (\sigma\Delta z)^2 |E_R|^2 I_O(x,y,\tau_i,z_0) \label{intensity}$$ where $I_O = |E_O|^2$. By imposing a set of delays ($\tau_i$, $i = 1...n$) to the reference WP with respect to the object, a reliable 3D reconstruction of the WP structure can be achieved by the collection of the $n$ images $S(x,y,\tau_i,z_0)$. By changing the plane $z_0$, the temporal evolution of the WP can also be obtained. Notice that the model described above does not take into account the dispersion of the mixing crystal. Therefore, the model as it is predicts no limits for the resolution of the map as long as arbitrary short reference pulses are available. Actually the real mixing crystal has a finite bandwidth that limits the spatiotemporal resolution of the obtainable maps. Therefore a careful evaluation of the dispersion characteristics of the mixing crystal have to be gauged as ultrashort pulses are either investigated or used as a reference. More details are discussed in section \[features\]. 3D field reconstruction ----------------------- The method described above is not limited to the intensity reconstruction, but can be also implemented for the field reconstruction as can be stated from equation \[campi\]. This point can also be explained in terms of a holographic description of the process, thus bringing to a deeper understanding of the imaging reconstruction process. As stated in [@Denisyuk99; @Denisyuk00; @staselko], for a plane wave reference the generated wavefront through the NL interaction behaves as a conventional hologram recorded at frequency $\omega$ and illuminated with a radiation at $2\omega$. As a consequence, the position, scale, resolution and all the other properties of transformation of the reconstructed image can be predicted by means of the ordinary laws of holography. This hologram is recorded and reconstructed at a time, and exists only when light propagates inside the crystal. Yet we also point out that we obtain the 3D map of our pulse by collecting a set of independent 2D holograms by slicing the object pulse at different delays. Nevertheless, according to the holographic properties of the SF process, the slicing can be done with the mixing crystal at any $z$ from the real object to be recovered and the information recorded is enough to reconstruct the slice. In order to clarify this point, let’s assume to perform an experiment in which we reconstruct a slice of an object WP (Fig \[holoscheme\].a). At any distance $z$ (Fig \[holoscheme\].b) the reconstructed WP will correspond to the propagated version of the one at $z_0$. Let us consider the case when the object WP has a bandwidth small enough with respect to the SF bandwidth that our simplified model applies, the pulse does not have appreciable angular dispersion[^2] and the spatial and the temporal evolution can be separated. Under these assumptions, any slice recorded in the confocal configuration (see Fig. \[holoscheme\].a) could also be reconstructed when the mixing crystal is displaced at position $z$ and the field to be converted is the propagated one. This is possible by exploiting the holographic properties, provided that the detecting system is set to reconstruct the virtual image of the slice. This image is placed at a distance 2$z$ far from the mixing crystal (see Fig. \[holoscheme\].c), its transverse size being identical to that of the object (see [@Denisyuk00], and also note that we are working in the degenerate case, $\omega_1 = \omega_2$). The possibility to recover the intensity profile of a virtual image comes from the complete wavefront ([*field*]{}) reconstruction arising from the SF process coherence and contained in Eq. \[sumfrequency\])[^3]. This important property also allows to get intensity maps with a wider dynamical range. For example, in the cases when the object is so intense so that the undepleted pump approximation does not hold, we can get the WP profile by displacing the NL crystal to a plane where the intensity is reduced and then recover the intensity profile at the desired plane $z_0$ by suitably moving the imaging system (Fig \[holoscheme\].c)). Experimental results {#experiment} ==================== We prepared several experiments in order to prove the possibility to recover 3D maps of short WPs and to show their holographic properties.The experimental set-up is sketched in Fig. \[STimaging\]. The 1 ps pulses of a Nd:glass laser source (TWINKLE, Light Conversion, wavelength 1055 nm) are splitted on two lines by means of a beam splitter. In the first line, the laser pulse is frequency doubled in a KDP crystal and focused on a 15mm-long litium triborate (LBO) crystal. For pulses about 1 $\mu$J in energy , spatial solitons are formed in this crystal because of the optical parametric generation process [@DiTrapani98]. The gaussian temporal profile causes the spatial soliton to be formed only in the central part of the pulses where the intensity is higher [@Barthelemy02]. This leads to a non-trivial space-time structure in the output pump wave packet [@Minardi03], which we will consider as the object. Furthermore, the object has been magnified 4 times by means of a two-lens telescope imaging the LBO exit face into the NL mixing crystal ( a 100 $\mu$m thick, $\beta$ barium borate crystal, BBO). The beam expansion has been necessary to: [*i*]{}) reduce the beam intensity and therefore fulfill the undepleted mixing requirement; [*ii*]{}) to avoid information loss related to the finite spatial resolution of the mixing process. In the second line, a 200-fs reference pulse is produced at the wavelength of 527.5 nm by means of a second-harmonic pulse compressor [@Stabinis91] and expanded 3 times by means of a telescope. Both lines are recombined in the thin BBO crystal, cut and oriented to generate the non-collinear SF from the object and reference beams. An external incidence angle of $\sim6.5^\circ$ between the propagation directions has been chosen. The delay between the two pulses can be varied by means of a suitable delay line placed on the reference pulse line. The non-collinear SF radiation is spatially selected by an aperture, then selected in frequency by means of coloured filters and the plane of the BBO is imaged onto the CCD sensor (PULNIX TM6CN). Finally, both the BBO crystal and the imaging system could move independently on a rail along the optical axis. At first we focused the object inside the BBO crystal and adjusted the position of the imaging system as in Fig. \[holoscheme\].a). By recording the spatial profile of the SF radiation as a function of the reference pulse time lag (steps of $\Delta \tau_i = 66.6$ fs has been used, where $\Delta \tau_i = \tau_i - \tau_{i-1}$), we have retrieved the 3D isointensity maps of the object pulse at the exit face of the LBO crystal. Fig. \[isointensity\].a shows three different intensity levels of the pulse in the $(x,y,\tau)$ space, while the corresponding contour plot of the $(x,\tau)$ plane section is depicted in Fig. \[isointensity\].b. The space-time maps clearly show that the pulse is formed by a spatially focused structure followed by a diffracted tail. The accuracy of this reconstruction has been successfully tested by comparing the experimental plots with the results of a 3D-numerical simulation of the formation process of the object pulse (see [@Minardi03]). In order to check for the validity of the holographic interpretation in this process, we have reconstructed the same WP of Figure \[isointensity\] by moving the NL crystal a distance z far from the previous position and by scanning the position of the imaging system (CCD+lens) along the SF propagation direction in order to find out the position of the (virtual) reconstructed image. First we fixed the time lag between the object and the reference and selected a slice of the object corresponding to a narrow focused spot of $\sim 40 \mu$m in diameter inside the BBO crystal (corresponding to case a) in Fig. \[holoscheme\]). Furthermore, the displacements $z$ of the BBO crystal have been chosen large enough to ensure that the propagated wavefronts had lost the transverse structure. As a rough estimate, for the focused part of the object, the Rayleigh range is about 7 mm long, while we spanned distances from -15 mm to 15 mm. In Fig. \[holoprop\], data show the position of the virtual image against the position of the real object from the crystal (see Fig. 1.c)), showing a remarkable fit to a straight line which slope is close to the value of 2, according to what discussed in the precedent section. The error bars shown in the figure indicate the estimated uncertainties in the reconstructed image plane position, measured by scanning the whole Rayleigh range and by extracting the position of the central point. In the same figure the two insets show two isointensity maps recovered for the case when the imaging system is focusing directly the BBO plane (z=0) and when the BBO crystal has been moved a distance z=15 mm (the maximum propagation distance we imposed to the BBO). The agreement between the two intensity profiles proves the reliability of our method to work with virtual images in recovering 2D slices of WPs like those we used here. In order to verify that we have always been operating in the undepleted regime, we have measured the dependence of the generated SF field as a function of the object-pulse energy at $z = 0$. The results are presented in Fig. \[linearity\], where the peak fluence (as registered from the CCD images) is plotted vs the transmission of the neutral-density filters that attenuate the object. The costant slope confirms the absence of any saturation in the SF process. Data also indicate that a slight overestimate of the background has been done during the profile acquisition (zero SF field is found for 15 % filter transmission). Features of the technique {#features} ========================= In the previous sections we have proved that the described technique is able to perform the 3D mapping of objects similar to those we used for our measurements. Anyway, one can expect that in general the reliability of the technique to recover the slices could be affected by particular effects dealing with the interaction geometry, the mixing crystal, and the features of the interacting pulses, in particular for broad-band, ultra-short and chirped pulses. Therefore we discuss here three main features of our technique, namely, the duplication bandwidth of the mixing crystal, the interaction angle and the reference pulse shape, that bring to devise some possible limitations to the fidelity of the technique. Space-time resolution --------------------- As we briefly mentioned above, the interaction model we describe does not take into account the actual finite bandwidth of the mixing crystal, limiting the spatiotemporal resolution of the maps. In fact, in our experiment the BBO crystal has been chosen thin enough that the converted bandwidth was large compared to the object one. As a matter of fact, the spatio–temporal resolution of the slices is dictated by the maximum range of angles and frequencies over which the conversion is effective (for very broad–band objects the angular blurring that could arise from the phase matching condition should be considered). The key parameter here is the maximum phase mismatch between the fundamental and the SF waves at which the conversion efficiency vanishes, $\Delta k_{max}$. On the basis of the existing literature [@niko], we have estimated the maximum bandwidth converted by the crystal as the FWHM of the efficiency curve, yielding to a temporal bandwidth of about $630$ cm$^{-1}$ and an angular bandwidth of about $510$ cm$^{-1}$. By considering only the angles for which the conversion efficiency is higher than 1/e of the maximum value [@devaux], we obtain the remarkable resolution of details approximately 10 $\mu$m in size (100 lines/mm). We can notice that the holographic interpretation of the wavefront reconstruction leads to an easy description of the imaging process. Furthermore, our optical system could be used in principle to achieve a microscopy of ultrashort objects shorter than the one we used here [@jose], although the limit of very shot WP is detrimental for the holographic reconstruction of the phases. Interaction angle ----------------- A point to be discussed concerns the influence of the noncollinear geometry and pulse chirp on the reconstruction of the hologram. In principle the holographic interpretation is strictly valid for collinear geometry only (when $k_2 = 2 k_1$) [@Denisyuk00]. Although this condition is not strictly fulfilled in our experiment, we have maintained the interaction angle small enough to make this disturbance negligible. However, a temporal chirp in the object pulse could give rise to a spatial phase distortion of the SF wavefront [@Stabinis91b]. Although this does not affect the results obtained with the confocal configuration (see Fig. \[holoscheme\].a), it could distort the maps obtained in the holographic one (Fig. \[holoscheme\].c). We expect that this effect is really rampant only when complicated ultrashort pulses with strong chirp are considered, or large interaction angle are employed. We are confident that our system was operated far from this limit, since, as we checked from the data, our experimental holograhic maps do not show any appreciable enhancement of the astigmatism. As it is well known, the non-collinear SF scheme is largely used to get single-shot autocorrelation traces of ultrashort pulses [@Gyuzalian79]. In fact, because of the geometry of the interaction, it is easy to show that the time of the interaction between the object and reference pulses depends on the transverse coordinate of the intersecting planes (see [@zewail pag. 426-428]). Therefore, if the interacting angle is large, we expect that distortions of the spatial profile along this direction could affect the space-time maps. However, it turns out that in the confocal configuration (*i.e.* as in Fig. \[holoscheme\].a) the effect of the interaction angle is merely that of skewing the intensity map, so that the actual time-axis of the object pulse is not parallel to that measured on the delay line. This is evident in the map depicted in Fig. \[skewed\], obtained with an external crossing angle of $\sim 16^\circ$. In the paraxial approximation, the slope of the object-pulse time-axis on the space-time map is given by: $$\tan\beta=\frac{c}{n}\sin\frac{\gamma}{2} \label{angoli}$$ where $\gamma$ is the angle between the propagation directions of the object and the reference wavepackets inside the crystal, and $n$ is the refractive index of the medium. It is easy to recognize this formula as the calibration relation of the non-collinear, single-shot intensity autocorrelator [@Gyuzalian79; @zewail]. Reference pulse shape --------------------- Finally the last point to be discussed concerns the shape of the reference pulse. In our experiments it was only slightly affected by the presence of satellites because of the generation [*via*]{} the pulse compressor [@Stabinis91], the amplitude of these structures being small enough to be considered negligible to our aim. Actually, the recovered intensity profile of the object would be distorted in case the reference pulse has a structure more complicated than the single peak envelope, as Eq. \[convolution\] clearly points out. This suggests that a careful measurement of the reference shape (for example carried out by means of an autocorrelation technique) can be used to retrieve the real intensity profile by means of a deconvolution procedure. We point out, however, that any deconvolution unavoidably introduces an extra spurious noise which could degrade the final quality of the mapping [@deconvolution]). Conclusions =========== We have shown that the 3D intensity maps of optical WPs with a complex space-time structure can be retrived by an optical gating technique. The method allows the reconstruction of the WP in its comoving reference frame and, by exploiting the holographic properties of a slightly non-collinear degenerate, sum-frequency process, we have shown that a complete amplitude and phase reconstruction is actually obtained. The maps are obtained by suitably imaging the second harmonic radiation obtained by cross-correlating an object pulse with a much shorter plane wave-packet delayed in time. The holographic properties of the generated radiation have been carefully tested experimentally, and the distorsions of the intensity maps introduced by the non-collinear geometry have been discussed in detail. Finally, theoretical considerations point out that the ultimate resolution of our set-up is in the order of 100 lines/mm in space, and of a few fs in time. However, the choice of the reference pulse limits the actual time resolution to about 200 fs. We forsee that the developed technique will benefit all the fields where a space-time mapping of light pulses is relevant, such as the investigations on the reshaping of ultrashort pulses propagating in non-linear materials [@Lantz02a; @Lantz02b; @PDT00; @Minardi03; @jose]. Moreover, the holographic features of the technique might be exploited to fully reconstruct the field of an object pulse and its evolution during the propagation. This work was partially supported by MIUR (COFIN01 and FIRB01), the European Commission EC-CEBIOLA project (ICA1-CT-2000-70027) and DGI BFM2002-04369-C04-03 (Spain). 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Comm.*]{} 105 (1994) 67-72 R.N. Gyuzalian, S.B. Sogomonian, and Z. Gy. Horvath [*Opt. Comm.*]{} 29 (1979) 239-242 See for example P.A. Jansson Ed., [*Deconvolution*]{}, Academic Press inc., Orlando 1984 [^1]: Rigorously, the support of a function is the set of points where its value is different from zero. For realistic optical fields that have exponentially decaying tails we can define a “practical” support, defined as the set of points in the space-time in which the field amplitude is larger than $1/e$ times the peak value. [^2]: In the case of angular dispersion, the diffraction drives an effective group velocity dispersion also in the vacuum, then the spatial evolution cannot be separated from the temporal one (see for example [@sonajalg], [@zozyula]) [^3]: We suggest for example the possibility to use an interferometric or heterodyne device in order to obtain a complete charaterization of the field. By measuring the field instead of the intensity, a remarkable increment of the dynamical range is also possible.

--- abstract: 'Vacuum instability of the strong electromagnetic field has been discussed since long time ago. The instability of the strong electric field due to creation of electron pairs is one of the examples, which is known as Schwinger process. What matters are the coupling of particles to the electromagnetic field and the mass of the particle to be produced. The critical electric field for electrons in the minimal coupling is $E_c \sim \frac{m^2}{e} $. Spin 1/2 neutral particles but with magnetic dipole moments can interact with the electromagnetic field through Pauli coupling. The instability of the particular vacuum under the strong magnetic field can be formulated as the emergence of imaginary parts of the effective potential. In this talk, the development of the imaginary part in the effective potential as a function of the magnetic field strength is discussed for the configurations of the uniform magnetic field and the inhomogeneous magnetic field. Neutrinos are the lightest particle(if not photon or gluon) in the “standard model", of which electromagnetic property is poorly known experimentally. Recently the observation of neutrino oscillation shows the necessity of neutrino masses. It implies that the standard model is subjected to be modified such that non-trivial electromagnetic structure of neutrino should be reconsidered although they are assigned to be neutral. And the possibility of anomalous electromagnetic form factor is an open question theoretically and experimentally. In this talk, the implication of non-vanishing magnetic dipole moment of neutrinos is also discussed: the instability of the strong magnetic field and the enhancement of neutrino production in high energy collider experiments.' address: | Department of Physics, Hanyang University University\ Seoul, 04763, Korea\ $^*$E-mail: [email protected] author: - 'Hyun Kyu Lee$^*$' title: Instability of strong magnetic field and neutrino magnetic dipole moment --- Introduction ============ There are various indications of the ultrastrong magnetic fields , $B > B_c \sim 10^{13}$ G, from astrophysical observations and terrestrial accelerator experiments and there have been continuous efforts in finding their definite evidence in Nature. Among the examples considered so far are strong magnetic fields captured on magnetar and gamma ray bursts(GRB) central engine and also created in noncentral heavy-ion collisions. Magnetars[@magnetar] are considered to be neutron stars, where strong magnetic fields of typically $10^{13} \sim 10^{15}$ G are the main source of energy. Even the stronger field strength is expected inside magnetars. One of the viable models of GRB central engine is to make use the idea of tapping the rotational energy of black hole by the strong magnetic field[@LBW]. It is estimated to be $\sim 10^{15}$ G particularly for long and energetic bursts. In noncentral heavy-ion collisions, strong magnetic fields can also be created by the two electric currents in opposite directions generated by two colliding nuclei. it is expected that the magnetic fields in RHIC Au + Au collisions and LHC Pb + Pb collisions can be as large as $10^{19}$ G and $10^{20}$ G respectively[@Huang]. These field strengths are much stronger than the critical magnetic field and offer an interesting opportunity to study the effect of super-strong electromagnetic fields beyond the classical electrodynamics and beyond the standard model of electro-weak interaction. One of the interesting questions with given such strong electromagnetic fields is the stability of the field configuration or equivalently whether there is any instability, which leads to the particle production. Vacuum instability of the strong electromagnetic field has been discussed since long time ago. The instability of the strong electric field due to the creation of electron pairs is one of the examples, which is known as Schwinger process[@schwinger; @Kim]. What matters are the coupling of particles to the electromagnetic field and the mass of the particle to be produced. The critical electric field for electron is $E_c \sim \frac{m^2}{e} $. If it is possible to imagine a charged particle lighter than the electron, the critical field can be lowered down to the field strength produced in the future high intensity laser and will be subjected to be tested in the laboratory. The pair production of fermions in a purely magnetic field configuration is shown to be absent[@dunnehall]. Therefore, the pair production of minimally interacting particles is considered to be a purely electric effect. However, while the minimal coupling derived by the local gauge invariance is of fundamental nature, there appear also non-minimal couplings as well in the form of the effective theory. Pauli introduced a non-minimal coupling of spin-1/2 particles with electromagnetic fields, which can be interpreted as an effective interaction of fermions with an anomalous magnetic moment[@Pauli]. Hence for the neutral fermions, which have no minimal coupling to electromagnetic fields, the nonvanishing magnetic moments may be the primary window through which the electromagnetic interaction of neutral fermions can be probed with the Pauli interaction. It is known the spatial inhomogeneity of the magnetic field exerts force on the magnetic dipole moment through the Pauli interaction. It plays a similar role analogous to the electric field for the creation of charged particle pairs with the minimal coupling. The possibility of pair production of the neutral fermions in a purely magnetic field configuration with spatial inhomogeneity has been demonstrated in 2+1 dimension[@lin]. The production rate in 3+1 dimension has been calculated explicitly for the magnetic fields with a spatial inhomogeneity[@LY1; @Gitman], which can be approximated as m\^4 e\^[-a m\^2/|B’|]{} analogues to the Schwinger process. The instability of the particular vacuum under the strong magnetic field can be formulated as the emergence of imaginary parts of the effective potential. For uniform magnetic fields which interact with spin-1/2 fermions through the Pauli interaction[@LY2], it is found that the non-vanishing imaginary part develops for a magnetic field stronger than the critical field $B_c$, whose strength is the ratio of the fermion mass to its magnetic moment, $B_c = \frac{m}{\mu}$, (V\_[eff]{}) = ( -1)\^3 ( +3). In section 2, the calculations of the effective potential and vacuum decay rates for the neutral fermion with Pauli coupling to electromagnetic field will be reviewed. The implication of non-vanishing magnetic dipole moment of neutrinos, the instability of the strong magnetic field and the neutrino production through the Pauli coupling in high energy collider experiment, is discussed in section 3. Imaginary part of effective potential and pair production ========================================================= The instability of the electro-weak vacuum for a strong magnetic field was discussed long time ago[@AHN]. The one-loop effective potential considering the weak-boson($W$) loop is found to have imaginary part under the pure magnetic background, (V\_[eff]{}) = B\^2(1-)(B - ) \[ImVW\] where $m_W$ is the mass of $W$-boson . In the limit of $m_W \rightarrow 0$, eq.(\[ImVW\]) agrees with the effective potential given by Nielson and Olesen [@NO]. It was argued that the instability could be avoided if the condensation of $W$ and $Z$ bosons[@AHN] appears with the strong magnetic field. The basic reason for the emergence of an imaginary part for $B > \frac{m_W^2}{e^2}$ is that the energy eigenvalue crosses zero at $B =\frac{m_W^2}{e^2}$ because of the anomalous magnetic moment of $W$ boson. One can see that there is also a level crossing of energy eigenvalue of a neutrino with Pauli interaction for a strong enough magnetic field. For a uniform magnetic field, the energy eigenvalues of the Hamiltonian are given by E = \[Epauli\] where $p_l$ and $p_t$ are respectively the longitudinal and the transversal momentum to the magnetic field direction. One can see that for a magnetic field stronger than the critical field $Bc = m/\mu$ , the ground state with $p_l = p_t = 0$ crosses the zero energy state. This indicates the possible instability of the magnetic field configuration beyond critical field strength as in electro-weak instability [@LY3]. For a uniform magnetic field, the imaginary part of the one-loop effective action is calculated explicitly[@LY2] (V\_[eff]{}) = ( -1)\^3 ( +3)(|B| - m). which takes the similar form as in eq. (\[ImVW\]). It is interesting to note that the development of the imaginary part associated with a level crossing has been also demonstrated in the different contexts[@GF]. The state occupied in the negative energy sea becomes a particle state and the vacant positive energy state plunges into the negative sea to make an antiparticle state. The instability can be interpreted as the pair creation at the expense of the magnetic field strength. Then the particle production rate is given approximately by \~2 [I]{}(V\_[eff]{}) = ( -1)\^3 ( +3)(|B| - m).\[rate\] For a nonuniform magnetic field, the inhomogeneity of the magnetic field coupled directly to the magnetic dipole moment plays an interesting role analogous to the electric field for a charged particle: The non-zero gradient of the magnetic field can exert a force on a magnetic dipole moment. Then the vacuum production of neutral fermions with a non-zero magnetic moment in an inhomogeneous magnetic field is possible. As a simple example, a static magnetic field configuration with a constant gradient, $B'= dB_z/dx$, along x-direction, B\_z(x) = B\_0 + B’x. has been considered. It is not necessary to consider an infinitely extended ever-increasing magnetic field to meet the linear magnetic field configuration. Because the particle production rate density is a local quantity, it is sufficient to have a uniform gradient magnetic field in the Compton wavelength scale of the particle. The imaginary part of the one-loop effective potential is calculated in the integral form[@LY2], (V\_[eff]{}) &=& - \^\_0 \[F( v, v) - v - v\]\ & & - \^\_0 \[(v v)\^[3/2]{}- 1 - \] v, where v &=& s B’,    = ,    = ,\ F(a,b) &=& \^1\_0 d(1-)(a\^2-b) Numerically it is found that it can be fitted to an analytic expression[@LY2], (V\_[eff]{}) m\^4 e\^[- b ]{}, which is decreasing exponentially with respect to the inverse of the field gradient. The qualitative feature of this result can be compared with the expression obtained recently by Gavrilov and Gitman[@Gitman]. Neutrino magnetic dipole moment and its implications ==================================================== Neutrinos are the lightest particles(if not photon or gluon) in the “standard model", of which electromagnetic property is poorly known experimentally. Moreover the observation of neutrino oscillation shows the necessity of neutrino masses. It implies that the standard model is subjected to be modified such that non-trivial electromagnetic structure of neutrino should be reconsidered although they are assigned to be neutral. While there is no observation which is not consistent with the electric-charge neutrality of neutrinos, the possibility of anomalous electromagnetic form factor is an open question theoretically and experimentally[@GS]. So far there is no experimental evidence of the magnetic dipole moment of the neutrinos but one can not simply rule out the magnetic moment just because of the electrical neutrality. For example, in a minimal extension of the standard model to incorporate the neutrino mass the anomalous magnetic moment of a neutrino is known to be developed in one loop calculation [@fujikawa], $\mu_\nu = \frac{3eG_F}{8\sqrt{2}\pi^2} m_\nu $. Here one can notice that the non-zero mass is essential to get a non-vanishing magnetic moment. There are strong evidences from the neutrino oscillation observations that neutrinos have nonzero-masses [@numass]. However, the simple extensions of the standard model can give only much smaller magnetic moment, $ \sim 10^{-20}\mu_B$ if we take the neutrino mass as 1 eV , than the experimental bound. GEMMA collaboration[@gemma] observed that the neutrino magnetic moment is bounded from above by the value $\mu_\nu < 2.9 \times 10^{-11} \mu_B $ and the bound obtained from Borexino[@borex] data gave $\mu_\nu \leq 3.1 \times 10^{-11}\mu_B$ . Since a wide range of neutrino magnetic moments is possible up to the current laboratory upper limit beyond standard model, it is interesting to investigate the phenomenological consequences of the nonvanishing magnetic moment up to the current bounds, which may constrain the bound tighter than the direct measurements of the neutrino magnetic dipole moment. For the vacuum instability, the typical field strength(or critical field strength) is given by $B_c = m_\nu/\mu_{\nu}$. Taking the possible magnetic moment to be as large as the experimental upper bound $\mu_\nu = 10^{-11}\mu_B$ and the mass of the neutrino $m_\nu \sim 10^{-2} eV$ constrained by the neutrino oscillations, the critical field strength is estimated to be $B_c \sim 10^{17}$ G, which is not far from the field strength inferred on the magnetar surface and might be possible inside the magnetar. With vacuum instability, the particle production rate per unit time per unit volume is typically of order $m_\nu^4$ for the critical field strength. As a possible environment, let us consider the pair creation of neutrinos with non-zero magnetic dipole moments from the very strongly magnetized compact objects with $B = B_c$ and radius $R_B$. With $R^3_B$ as an effective volume of the magnetosphere, the production rate can be estimated[@LY4] as \_\~10\^[39]{}( )\^4( )\^4()\^3 / s. Using the currently estimated masses of neutrinos the corresponding luminosity is estimated to be $L \sim 10^{25}$ erg/s assuming $E_\nu \sim m_\nu$, which is much weaker than the X-ray luminosity of magnetars, $L_X \sim 10^{35}$erg/s. However, since the time scale is much longer than the Schwinger process by a factor of $(m_e/m_\nu)^4$, it can be a continuous source of neutrinos. A larger luminosity for observation can be easily obtained with a massive particle but the magnetic moment should be increasing to keep the critical field strength, which significantly constrains models for particles involved. For the vacuum instability beyond critical field strength in case of the uniform magnetic field, the interpretation of vacuum instability as the neutrino pair production may not be the whole story. It is because there may be a possibility that the instability could be avoided if a nontrivial condensation[@AHN] develops along with the critical magnetic field, which is subjected to further discussion. The pair production of neutrinos in collider experiments can be enhanced if endowed with the nonvanishing magnetic dipole moment. The pair production of neutrinos are expected from the annihilation of charged particles in colliding experiments through photon channel[@klpy] in addition to the weak boson channel in the standard model. One of the interesting features of Pauli coupling is that the angular distribution of pair creation peaks at $\theta = \pi/2$. It is compared to the angular distribution of the standard model process, minimum at $\theta = \pi/2$ but maximum for $\theta = 0$ and $\pi$. For the Majorana neutrinos, the pairs with different flavors can be produced due to the transit magnetic moment couplings. The the angular distribution peaks at $\theta=\pi/2$ with respect to the beam direction, which is the enhancement of the neutrino production at right angle in contrast to weak Z-channel. It is interesting to note that the flavor mixing angle can be inferred if the flavor correlation in pairs can be measured in the future experiments[@L]. The detectors located around the right angle to the beam direction can measure the back-to-back correlations in the pair production, where the production rate is supposed to be maximum. Most of the detectors are using electromagnetic triggers at the end stations, which implies corresponding charged leptons($l_\alpha$ and $l_\beta$), for example electrons or muons, are those to be detected finally. The Majorana pairs in mass eigenstates, $\nu_1$ and $\nu_2$, are produced through the Pauli coupling and interact weakly with other particles to produce charged leptons to be detected. To simplify the situation let us consider only two mass eigenstates which are mixed states of weak eigenstates, $\nu_\alpha$ $\nu_\beta$, \_1 =   \_+   \_,    \_2 = -  \_+   \_where $\delta$ is the mixing angle. Now consider two targets A and B placed at the opposite side of the center of mass at the right angle to the beam direction. At the targets the neutrinos can produce the corresponding charged leptons via the charged current weak interaction(W-boson). The back-to-back correlation, R, defined by the product of the detection rates of $\alpha$ type leptons at the target A and $\beta$ type leptons at the target B, can be given by R = ( 1 + \^2 2). Since the correlation $R$ turns out to be dependent only on the mixing angle, $\delta$, it may serve as a clean measurement of mixing angles in the collider experiments. For example the maximum value, $R= 1$, is obtained for the cases of no-mixing($\delta=0$), $R=1/2$ for the maximal mixing($\delta=\pi/4$). These features are quite different aspect from standard model, which can be clearly distinguished in the high energy collider experiment[@L]. Sine the strength of Pauli coupling to the photon is determined by the neutrino magnetic dipole moment, which is very small, the pair creation cross section through photon channel by the Pauli coupling is smaller that that of weak boson($Z$) channel until the center of mass energy of the colliding particle is sufficiently high. To get some idea of the energy scale for which the Pauli coupling becomes dominant, we can define the energy scale, $E_{0.1}$, for which the total cross section becomes $10 \%$ of the standard model, E\_[0.1]{} = 10\^3() TeV. It is a bit higher than LHC energy, $E_{CM} \sim 10$ TeV and also higher than the GZK cut-off energy of ultra high energy cosmic rays, $E_{CM} \sim 100$ TeV. So to see the distinguished feature from standard model, the higher energy collider experiments than present LHC seem to be needed. The magnetic field is expected be created by the two electric currents in opposite directions generated by two colliding nuclei. It turns out the field strength may be very large as $10^{19} - 10^{20}$G at RHIC and LHC. It gives $10^4$ times stronger Pauli coupling than expected at the surface of the magnetars . The magnetic vacuum instability due to magnetic dipole moment of neutrino can be much more enhanced. How to identify experimental observables corresponding to the vacuum instability and particle productions that are sensitive to the neutrino magnetic dipole moment may be an interesting and challenging subject to be discussed. Acknowledgments =============== The author would like to thank Dmitry Gitman, Sang Pyo Kim and Yongsung Yoon for useful comments and discussions. [0]{} S.Mereghetti, J.Pons, A. Melaos, Space Science Review, 191, 1315(2015); McGill Catalogue,http://www.physics.mcgill.ca/$\sim$ pulsar/magnetar/main.html H.K. Lee, R. Wijers and G.E. Brown , Phys. Rep. 325, 83 (1999). X-G. Huang, Rep. Prog. Phys. 79, 076302 (2016). J. Schwinger, Phys. Rev. 82, 664(1951). See for example, G. Dunne and T.M. Hall, Phys. Lett. B419, 322(1998); Phys. Rev. D60, 065002(1999). W. Pauli, Rev. Mod. Phys. 13, 203(1941). Q.-G. Lin, J. Phys. G. 25, 1793(1999). H.K. Lee and Y.S. Yoon, JHEP 0603, 078(2006). S.P. Gavrilov and D.M. Gitman Phys. Rev. D 87, 125025 (2013). S.P. Kim, H.K. Lee and Y.S. Yoon, Phys.Rev.D82,025015(2010) and references therein. H.K. Lee and Y.S. Yoon, JHEP 0703,086 (2007). Y.M. Koh, H.K. Lee, W-G. Paeng and Y. Yoon, J. Korean Phys. Soc. 56, 1884 (2010). C. Giunti and A. Studenkin, Rev. Mod. Phys. 87, 531(2015). H.K.Lee, Phys. Rev. D84, 077302 (2011). J. Ambjorn, R.J. Hughes and N.K. Nielson, Ann. Phys.(NY), 150, 92(1983):J. Ambjorn and P. Olesen, Nucl.Phys.B 330, 193(1990) and references therein. N.K. Nielson and P. Olesen, Nucl. Phys. B 144, 376(1978). N. Graham and R.L. Jaffe, Phys.Lett. B435, 145 (1998). H.K. Lee and Y. Yoon, Mod. Phys. Lett. A 22, 2081(2007). K. Fujikawa and R. Shrock, Phys. Rev. Lett. 45, 963 (1980). See for example, D. Forero, M. Tortola, J. Valle, Phys. Rev. D 90, 093006(2014). A.G. Beda, V.B. Brudanin, V.G.Egorov, D.V. Medvedev et al., Adv. High Energy Phys. 2012, 350150(2012). B.C. Canas, O.G. Miranda,A. Parada , M. Tortola and J.W.F. Valle, Physics Letters B 753, 191(2016) H.K. Lee and Y. Yoon, Mod. Phys. Lett. A 23, 17(2008).

--- address: | Université de Paris Sud, Laboratoire de l’Accélérateur Linéaire, Bât. 200, B.P. 34, FR-91898 ORSAY CEDEX\ E-mail: [email protected] author: - 'C. BOURDARIOS' title: 'STUDY OF D$^{**}$ AND D$^{*''}$ PRODUCTION IN B AND C JETS, WITH THE DELPHI DETECTOR' --- Introduction ============ For mesons containing heavy and light quarks (Q$\bar q$), and in the limit where the heavy quark mass is much larger than the typical QCD scale, the spin $\overrightarrow{s_Q}$ of the heavy quark decouples from other degrees of freedom. Thus, for strong decays, the total (spin+orbital) angular momentum $\overrightarrow{j_q} = \overrightarrow{s_q} + \overrightarrow{L}$ of the light component is conserved. This heavy quark symmetry, together with quark potential models used for lower mass mesons, allows the masses and decay widths of heavy mesons to be predicted [@HQET]. The present knowledge of charmed meson spectroscopy is summarized in Figure \[fig:spectro\]. The well established D and D$^*$ mesons [@PDG] correspond to the two degenerate levels of the (L=0, $j_q$ = 1/2) state. The two (L=1, $j_q$ = 3/2)states have been clearly observed [@PDG], because they have narrow decay widths of about 20 [MeV/]{}$c^2$. The measured masses of the $D^0_1$(2420) and $D^{*0}_2$(2460) agree within 20 with the prediction of the models. Section 3 presents a measurement of their production rate in [ ]{}and [ ]{}jets. The (L=1, $j_q$ = 1/2) states decay through a S wave and are expected to have large decay widths. Up to now, they have not been observed directly, but their total production rate is measured using B meson semi-leptonic decays (section 4). In addition to these orbital excitations, radial excitations of heavy mesons are foreseen. The D$^{'}$ and D$^{*'}$ are expected to have masses of 2.58 and 2.64 respectively, with a 10-25 uncertainty on the mass predictions [@thdstar]. They are expected to decay, in S wave, into $D^{(*)}\pi\pi$. Section 5 presents the first evidence for the D$^{*'}$ meson, observed in the decay mode ($D^* \pi \pi$). D$^{**}$ and D$^{*'}$ reconstruction ==================================== DELPHI [@delphi] is a multipurpose LEP detector, with special emphasis on precise vertex and charged tracks momentum reconstruction, and particle identification. The micro-vertex detector provides 3 R$\phi$ and 2 Z hits per track, with intrinsic resolutions of 7.6 and 9 $\mu$m. For muons of 45 momentum, a resolution of $\sigma(p)/p$ of $\pm$ 3% is obtained, and the precision of the track extrapolation to the beam collision point is 26 $\pm$ 2 $\mu$m. Kaon and pion identification is performed using a Ring Imaging CHerenkov detector, and the ionisation loss in the TPC, which is the main tracking device. A total of 3.4 million hadronic events is obtained from the 1992-1995 data, at center-of-mass energies close to the Z$^0$ mass. D$^*$ reconstruction -------------------- All the decay channels considered here involve the $D^{*+} \rightarrow D^0 \pi^+_*$ decay, followed by $D^0 \rightarrow (K^-\pi^+)$ or $D^0 \rightarrow (K^-\pi^+\pi^-\pi^+)$. [^1] To reconstruct the $D^0$ decay final state, all ($K^-\pi^+$) and ($K^-\pi^+\pi^-\pi^+$) combinations are tried to fit a secondary vertex in space. Kinematical and track selection cuts are described in detail in [@n483]. Kaon candidates are considered if they have a momentum larger than 1 and, in the $K 3 \pi$ channel, a loose kaon identification is required. The D$^0$ momentum and invariant mass are computed from the momenta of the decay products. Then, all charged particles with momentum between 0.4 and 4.5 and charge opposite to that of the kaon candidate are used as pion candidates for the $D^{*+} \rightarrow D^0 \pi^+_*$ decay. In the $K \pi$ ($K 3 \pi$) channel, events are selected if the mass difference $(M_{K \pi \pi_*} - M_{K \pi}$) (resp. $(M_{K3\pi\pi_*} - M_{K3\pi})$) is within $\pm$ 2 ($\pm$ 1 ) of the nominal value ($M_{D^*}-M_{D^0}$). The D$^*$ candidates must have an energy fraction $X_E(D^*) = E(D^*)/E_{beam}$ greater than 0.25. Figure \[fig:d0\] shows the distribution of the M($K\pi$) and M($K3\pi$) invariant masses for the selected events. The fitted $D^0$ masses and widths are 1868 $\pm$ 1 (1869 $\pm$ 1) and 19 $\pm$ 1 (12 $\pm$ 2) . The reconstructed D$^0$ mass is required to lie within $\pm$ 40 ($\pm$ 30) of the nominal D$^0$ mass: 4661 $\pm$ 88 (2164 $\pm$ 65) D$^*$ candidates are selected in the $K \pi$ ( $K 3 \pi$) channels. The selection efficiency is estimated, using the simulation, to be 21% (8%). $D^0_1$, $D^{*0}_2$ and D$^{*'}$ reconstuction ---------------------------------------------- Similar selection criteria and vertex reconstruction are used to reconstruct narrow orbitally and radially excited states. In the case of $D^0_1$ and $D^{*0}_2$ decaying into $D^{*+}\pi^-$, a pion with a charge opposite that of the D$^{*+}$ is added, and the $D^0 \pi^+_* \pi^-$ vertex is fitted. All combinations are tried, provided the pion candidate has a momentum larger than 1.0 (1.5) in the $K\pi$ ($K3\pi$) channel. The reconstruction efficiency is 14% (6%) in the $K \pi$ ( $K 3 \pi$) channels. In the case of D$^{*'}$ decaying into D$^{*+}\pi^+\pi^-$, all pairs of oppositely charged pions are used to fit a $D^0 \pi^+ \pi^-$ vertex. The pion candidates are required to have a momentum larger than 0.6(1.0) [GeV/c]{}, and those compatible with a kaon according to particle identification are rejected. For a signal of mass 2640 [MeV/]{}$c^2$, the reconstruction efficiency is 4% (2%) in the $K \pi$ ( $K 3 \pi$) channels. In both cases, the precision on the invariant mass reconstruction is improved by correcting for a 4 shift observed in the D$^0$ mass, by using: $$\begin{array}{rcl} M(D^*\pi) = M_{(D^0\pi_*\pi)} - M_{(D^0\pi_*)} + m_{D^*} \\ M(D^*\pi\pi) = M_{(D^0\pi_*\pi\pi)} - M_{(D^0\pi_*)} + m_{D^*} \end{array} \label{eq:mass}$$ where $m_{D^*}$ is the nominal $D^{*+}$ mass. The simulation predicts a resolution of about 6 on the mass reconstruction, for both radial and orbital excitations. Selection of [ ]{}and [ ]{}samples ---------------------------------- Due to the relatively long lifetimes of charmed and bottomed particles, heavy flavour events are characterized by the presence of secondary vertices. The probability $\mathcal{P}$ that all tracks detected in the event come from the primary vertex is small: for [ ]{}events, a purity of 90% is archieved, with an efficiency of 60%, by requiring $\mathcal{P} \le 10^{-2}$. Charmed mesons from $Z^0 \rightarrow b \bar b$ events are distinguished from those in [ ]{}events by considering both their energy and lifetime informations. Bottom quarks fragment into a B hadron, which subsequently decays into a D$^{*+}$ meson, whereas in [ ]{}events charmed mesons are directly produced in the fragmentation process. This difference in the hadronization leads to a smaller energy fraction of $X_E(D^*)$ for [ ]{}events. Also, due to the b quark lifetime, the apparent flight of the $D^0$ meson is greater than the true decay length. Its measured proper time distribution is larger than the mean B meson lifetime, 1.6 ps, compared to a true $D^0$ lifetime of 0.4 ps. By combining these variables, [ ]{}and [ ]{}samples are selected, with high purities: 92 % for [ ]{}, 89 % for $c\bar c$. In the [ ]{}sample, the combinatorial background is higher, but is reduced by 50% using the kaon identification, and also by asking that the impact parameter of the additional pion is positive, i.e. that the intersection of the pion and $D^*$ directions is on the same side of the primary vertex as the $D^*$ vertex. As a consequence, the ratio of efficiencies $\epsilon(D^* \pi) / \epsilon(D^*)$ is 52% for [ ]{}, compared to 62% for [ ]{}. Study of narrow orbital excitations =================================== Figure \[fig:d1d2\] shows the $M(D^*\pi)$ invariant mass distribution obtained for the sum of the [ ]{}and [ ]{}samples [@n240]. A clear excess of ($D^{*+}\pi^-$) pairs is observed between 2.4 and 2.5 [GeV/]{}$c^2$, corresponding to the two overlapping contributions of the $D^0_1$ and $D^{*0}_2$. They are fitted by two Breit-Wigner functions, whose widths are fixed to the measured world average [@PDG], convoluted with the experimental resolution. A total signal of (361 $\pm$ 58) $D^0_1$ + $D^{*0}_2$ events is fitted, out of which (65 $\pm$ 10) % is assigned to $D^0_1$. The masses are left free in the fit, and the result is $M_{D^0_1} = 2425 \pm 3 (stat)$ and $M_{D^{*0}_2} = 2461 \pm 6 (stat)$ [MeV/]{}$c^2$, i.e. consistent with the world averages [@PDG]. The helicity distributions are consistent with the production of a $J^p=1^+$ and $J^p=2^+$ states. Figure \[fig:ccbb\] shows the same mass distribution, but for the [ ]{}and [ ]{}samples separately. The same fit is performed, but both $D^0_1$ and $D^{*0}_2$ masses and widths are fixed to the world average. The result of the fit is (97 $\pm$ 26) $D^0_1$ and (69 $\pm$ 27) $D^{*0}_2$ in the [ ]{}sample,(141 $\pm$ 26) $D^0_1$ and (104 $\pm$ 26) $D^{*0}_2$ in the [ ]{}sample. In order to measure the $D^0_1$ and $D^{*0}_2$ production rates, these results are unfolded from the reconstruction efficiencies, signal purities, $D^0_1$ and $D^{*0}_2$ decay widths into $D^{*+}\pi^-$. The errors quoted below are statistical only. Systematic errors are still under study, but smaller than the statistical errors. The [ ]{}sample provides direct information on the charm fragmentation. Results are: $$\begin{array}{rcl} f(c \rightarrow D^0_1) = 1.9 \pm 0.4~(stat)~\% \\ f(c \rightarrow D^{*0}_2) = 4.3 \pm 1.3~(stat)~\% \end{array} \label{eq:rescc}$$ Both results are in agreement with previous LEP and CLEO measurements. For the $D^0_1$, the result is also in agreement with theoretical calculations [@becatini], which predicted 1.7%. For the $D^{*0}_2$, the result is high compared to the expectation (2.4 %), but has large errors. A more precise measurement would need to use the $D^{*0}_2 \rightarrow D^+ \pi^-$ channel, which is forbidden for the $D^0_1$. For the [ ]{}sample, results are: $$\begin{array}{rcl} f(b \rightarrow D^0_1) = 2.2 \pm 0.6 ~(stat)~\% \\ f(b \rightarrow D^{*0}_2) = 4.8 \pm 2.0 ~(stat) ~\% \end{array} \label{eq:resbb}$$ This shows that the charm fragmentation properties are similar, although in a different environment. Study of broad orbital excitations in B meson semileptonic decays ================================================================= In B meson semileptonic decays, only 60% to 70% of the final states are described by $D\ell\bar\nu_{\ell}$ and $D^*\ell\bar\nu_{\ell}$. The remaining contribution is attributed to $D^{**}$. The total production rate, including broad orbital excitations, can be measured using the impact parameter of the pion, denoted $\pi_{**}$, emitted in the decay chain $B \rightarrow D^{**} X \rightarrow (D^* \pi_{**}) X'$. Events are selected if a lepton with momentum larger than 3 is identified, and if its transverse momentum relative to the $D^{*+}$ is larger than 0.5 [GeV/]{}$c^2$. The kaon candidates in the $D^0$ decay must have the same charge as the lepton. 459 $\pm$25 (288 $\pm$ 19) events are selected in the $K\pi$($K3\pi$) channel. All remaining tracks, of charge opposite to that of the $D^{*+}$, are $\pi_{**}$ candidates. Kinematical and selection cuts are described in [@n239]. The background due to fake $D^{*+}$ associated to a true lepton $\ell^-$ is subtracted by using events in the tail of the $D^{*}$ invariant mass distribution. The contribution of true $D^{*+}$ associated to a fake lepton is subtracted using $D^{*} \ell$ pairs with the wrong sign combination. The remaining 111 $\pm$ 16 events are due to true b semileptonic decays into $D^{*+}\ell^-X$ final state, associated with a $\pi_{**}$ candidate either from $D^{**}$ decay, or from jet fragmentation. The shape of the two contributions are shown in figure \[fig:fitmc\]. They are used to fit the $\pi_{**}$ impact parameter distribution shown in figure \[fig:fitrd\]. From the result of the fit, the following branching ratio is obtained: $$\begin{array}{rcl} BR(B^- \rightarrow (D^{*+} \pi^-) \ell^- \bar \nu X ) \\ = 1.15 \pm 0.17 (stat) \pm 0.14 (syst) ~\% \end{array} \label{eq:resd2}$$ This result significantly improves a previous DELPHI measurement, and is in agreement with other LEP measurements. The ($D^{*+}\pi_{**}$) invariant mass is also reconstructed and used to fit, in the way described in the previous section, the $D^0_1$ and $D^{*0}_2$ narrow resonances. A signal of 26.7 $\pm$ 8.2 $D^0_1$ is fitted, and the corresponding production rate is: $$\begin{array}{rcl} BR(B^- \rightarrow D^0_1 \ell^- \bar \nu X) \\ = 0.72 \pm 0.22 (stat) \pm 0.13 (syst) ~\% \end{array} \label{eq:resd1}$$ For the $D^{*0}_2$ state, 14.8 $\pm$ 7.7 events are fitted, i.e. a signal significance smaller than 2 $\sigma$. More data would be necessary to estimate the corresponding branching ratio. Evidence for a narrow radial excitation ======================================= Figure \[fig:dstar\] shows the invariant mass distribution obtained when two pions of opposite charges are added to the D$^*$ candidate, and using the sum of the two [ ]{}and [ ]{}samples. An excess of 66 $\pm$ 14 (stat) events is observed in the $(D^{*+}\pi^+\pi^-)$ combination. The signal is fitted by a Gaussian distribution of free parameters: the $\chi^2$ per degree of freedom is 60/59, and would be 91/62 if the Gaussian was removed. About (57 $\pm$ 10)% of the signal is selected in the [ ]{}sample. The fitted mass is 2637 $\pm$ 2 (stat) $\pm$ 6 (syst) [MeV/]{}$c^2$. It is thus consistent with the predictions for the D$^{*'}$ radial excitation [@thdstar]: 2640 [MeV/]{}$c^2$. Other L=2 states are predicted, but with masses higher by at least 50 . The width of the fitted Gaussian is 7 $\pm$ 2 [MeV/]{}$c^2$, i.e. compatible with the detector resolution. Therefore, only an upper limit is derived: the full width of the signal is smaller than 15 at 95 % C.L. There is no natural explanation of such a small value, neither for the D$^{*'}$ nor for higher orbital excitations [@pene]. Various checks were performed. Varying the background shape and the kinematical cuts has no effect within statistics. No peculiar double counting was noticed, and the signal is stable when the $\pi^*$ is added to the $D^0 \pi^+\pi^-$ tracks in the vertex fit. As explained above, mass shifts are studied using $D^0_1$ and $D^{*0}_2$ narrow states, and a conservative systematic error of 6 is attached to the mass measurement. The production rate of this signal can be compared with that of the $D^0_1$ and $D^{*0}_2$ narrow states: $$\begin{array}{rcl} \frac { < N_{D^{*'}} > \times Br ( D^{*'} \rightarrow D^* \pi^+ \pi^- ) } {\sum_{J=1,2} < N_{D^{(*)}_J} > \times Br ( D^{(*)}_J \rightarrow D^* \pi ) } \\ = 0.49 \pm 0.18 (stat) \pm 0.10 (syst) \end{array} \label{eq:ratio}$$ Most of the systematic uncertainties cancel in this ratio. The quoted systematics is due to the Monte-Carlo statistics, and to the uncertainties on widths and on the kaon rejection. This result is compatible, within its large errors, with the value obtained using the thermodynamical models already mentioned for orbital states [@n483; @becatini]. Conclusion ========== Using about 7000 exclusively reconstructed $D^*$ mesons, the $D^0_1$ and $D^{*0}_2$ multiplicities are measured in [ ]{}events, and found to be consistent with theoretical calculations. The measured multiplicities in [ ]{}events are consistent with the ones in [ ]{}events, both for the $D^0_1$ and $D^{*0}_2$ . The total D$^{**}$ production rate, involving a D$^{*+}$ in the final state, is measured in B meson semileptonic decays. A narrow signal is observed in the $(D^{*+}\pi^+\pi^-)$ final state, at the mass M = 2637 $\pm$ 2 (stat) $\pm$ 6 (syst) [MeV/]{}$c^2$, interpreted as the first evidence of the predicted D$^{*'}$ meson. Acknowledgements {#acknowledgements .unnumbered} ================ I am very grateful to D. Bloch and P. Roudeau for their help while preparing this talk, and to the DELPHI collaboration for choosing me to give it. References {#references .unnumbered} ========== [99]{} N. Isgur and M.B. Wise, Phys. Rev. Lett. [**66**]{}(1991)1130 and ref. therein. Particle Data Group, “Review of Particle Properties”, Euro. Phys. J. [**C3**]{}(1998)1. S. Godfrey and N. Isgur, Phys. Rev. [**D32**]{}(1985)189. D. Ebert, R.N. Faustov and V.O. Galkin, “Mass spectrum of orbitally and radially excited heavy-light mesons in the relativistic quark model”, preprint HUB-EP-97/90, hep-ph/9712318(1997). DELPHI collab., P. Aarnio et al., Nucl. Instr. Meth. [**A303**]{}(1991)233. DELPHI collab., P. Aarnio et al., Nucl. Instr. Meth. [**A378**]{}(1996)57. DELPHI collab., P. Abreu et al., Phys. Lett. [**B426**]{}(1998)231-242 and contribution to ICHEP’98 number 483. Contribution to ICHEP’98 number 240, DELPHI 98-128 CONF 189. F. Becattini, J. Phys. [**G23**]{}(1997)1933. Contribution to ICHEP’98 number 239, DELPHI 98-119 CONF 180. D. Melikhov and O. Péne, HEP-PH/9809308. [^1]: Throughout this paper, charge-conjugate states are always implied, and the $\pi$ from D$^*$ decay is denoted as $\pi_*$.

--- abstract: 'Let $G$ be a finite $p$-group and let $\operatorname{Aut}(G)$ denote the full automorphism group of $G$. In the recent past, there has been interest in finding necessary and sufficient conditions on $G$ such that certain subgroups of $\operatorname{Aut}(G)$ are equal. We prove a technical lemma and, as a consequence, obtain some new results and short and alternate proofs of some known results of this type.' author: - | **Deepak Gumber[^1] and Hemant Kalra\ *School of Mathematics and Computer Applications\ *Thapar University, Patiala - 147 004, India\ **** title: '**Equality of certain automorphism groups of finite $p$-groups**' --- [**2010 Mathematics Subject Classification:**]{} 20D45, 20D15. [**Keywords:**]{} Central automorphism, IA-automorphism, $p$-group. Introduction ============ Let $G$ be a finite non-abelian $p$-group and let $M_1,M_2,N_1,N_2$ be normal subgroups of $G$. For normal subgroups $X$ and $Y$ of $G$, let $\operatorname{Aut}^X(G)$ and $\operatorname{Aut}_Y(G)$ denote the subgroups of $\operatorname{Aut}(G)$ centralizing $G/X$ and $Y$ respectively. We denote the intersection $\operatorname{Aut}^X(G)\cap\operatorname{Aut}_Y(G)$ by $\operatorname{Aut}_Y^X(G)$. In the recent past, many results have been proved which give necessary and sufficient conditions on $G$ such that $\operatorname{Aut}_Y^X(G)=\operatorname{Inn}(G),Z(\operatorname{Inn}(G))$; or $\operatorname{Aut}_{N_1}^{M_1}(G)=\operatorname{Aut}_{N_2}^{M_2}(G)$ with particular choices of $M_i$ and $N_i$ (see e.g. \[3-5, 7-13\]). Quite recently, Azhdari and Malayeri [@azhmal Theorem B, Corollary C] have found conditions on certain $M_i$ and $N_i$ so that $\operatorname{Aut}_{N_1}^{M_1}(G)=\operatorname{Aut}_{N_2}^{M_2}(G)$. We prove a short technical lemma, Lemma 2.2, and as a consequence, obtain very short and easy proofs of these main results of Azhdari and Malayeri. Subsequently, we also obtain some new results of this type and alternate proofs of main results of Attar [@att3 Theorem A], Jafari [@jaf Theorem] and Rai [@rai Theorem B(1)]. Notations are mostly standard. By $\operatorname{Hom}(G,A)$ we denote the group of all homomorphisms of $G$ into an abelian group $A$ and by $C_n$ we denote the cyclic group of order $n$. The rank, exponent and nilpotence class of $G$ are respectively denoted as $d(G)$, $\exp(G)$ and $cl(G)$. A non-abelian group $G$ that has no non-trivial abelian direct factor is said to be purely non-abelian. An automorphism $\alpha$ of $G$ is called a central automorphism if it centralizes $G/Z(G)$, or equivalently, $x^{-1}\alpha(x)\in Z(G)$ for all $x\in G$. By $\operatorname{Aut}_c(G)$ we denote the group of all central automorphisms of $G$, and by $C^*$ we denote the group of all those central automorphisms of $G$ which fix $Z(G)$ element-wise. An automorphism $\alpha$ of $G$ is called an IA-automorphism if it centralizes the abelianized group $G/G'$. The group of all IA-automorphisms is denoted as $\operatorname{IA}(G)$ and the group of all those IA-automorphisms which fix $Z(G)$ element-wise is denoted as $\operatorname{IA}(G)^*$. Automorphism groups of $G$ ========================== While proving the equality of different automorphism groups of $G$, the foremost tool has been to express the group $\operatorname{Aut}^{X}_{Y}(G)$ in the form $\operatorname{Hom}(A,B)$ for suitable subgroups or quotient groups $A$ and $B$ of $G$. The trick is very well-known since old days. Our next lemma is a little modification of arguments of Alperin [@alp Lemma 3] and Fournelle [@fou Section 2]. \[ML\] Let $G$ be any group and $X$ be a central subgroup of $G$ contained in a normal subgroup $Y$ of $G$. Then $\operatorname{Aut}^{X}_{Y}(G)\simeq\operatorname{Hom}(G/Y,X)$. Let $X$ and $Y$ be two finite abelian $p$-groups and let $X\simeq C_{p^{x_1}}\times C_{p^{x_2}}\times\ldots\times C_{p^{x_h}}$ and $Y\simeq C_{p^{y_1}}\times C_{p^{y_2}}\times\ldots\times C_{p^{y_k}}$ be the cyclic decompositions of $X$ and $Y$, where $x_i\ge x_{i+1}$ and $y_i\ge y_{i+1}$ are positive integers. If either $X$ is a subgroup or is a quotient group of $Y$, then $h\le k$ and $x_i\le y_i$ for $1\le i\le h$. Consider the situation when $d(X)=d(Y)$ and $X$ is a proper subgroup or a proper quotient group of $Y$. In these circumstances, $h=k$ and there certainly exists an $r,\;1\le r\le h$, such that $x_r<y_r$ and $x_j=y_j$ for $r+1\le j\le h$. For this unique fixed $r$, let $var(X,Y)=p^{x_r}$. In other words, $var(X,Y)$ denotes the order of the last cyclic factor of $X$ whose order is less than that of corresponding cyclic factor of $Y$. \[MT\] Let $A,B,C$ and $D$ be finite abelian $p$-groups with $B$ a subgroup of $C$ and $D$ a quotient group of $A$. Then 1. $\operatorname{Hom}(A,B)= \operatorname{Hom}(A,C)$ if and only if either $B=C$ or $d(B)=d(C)$ and $\exp(A)\le var(B,C)$, 2. $|\operatorname{Hom}(D,B)|=|\operatorname{Hom}(A,B)|$ if and only if either $D=A$ or $d(D)=d(A)$ and $\exp(B)\le var(D,A)$. We prove only $(i)$ as the proof is similar for $(ii)$. Let $$\begin{array}{rcl} A &\simeq & C_{p^{\alpha_1}}\times C_{p^{\alpha_2}}\times\ldots \times C_{p^{\alpha_l}},\\ B & \simeq & C_{p^{\beta_1}}\times C_{p^{\beta_2}}\times\ldots \times C_{p^{\beta_m}}, \;\mbox{and}\\ C &\simeq & C_{p^{\gamma_1}}\times C_{p^{\gamma_2}}\times\ldots \times C_{p^{\gamma_n}} \end{array}$$ be the cyclic decompositions of $A$, $B$ and $C$, where $\alpha_i\geq\alpha_{i+1},\;\beta_i\geq\beta_{i+1},$ and $\gamma_i\geq\gamma_{i+1}$ are positive integers. First suppose that $\operatorname{Hom}(A,B)= \operatorname{Hom}(A,C)$ and $B<C$. Then $$\displaystyle\prod_{i=1}^l\displaystyle\prod_{j=1}^mp^{\mathrm {min}\{\alpha_i, \beta_j\}}=\displaystyle\prod_{i=1}^l\displaystyle\prod_{k=1}^np^{\mathrm {min}\{\alpha_i,\gamma_k\}}.$$ Since $m\le n$ and $\beta_j\le \gamma_j$ for each $1\le j\le m$, $\mathrm{min}\lbrace\alpha_i,\beta_j\rbrace\le \mathrm{min}\lbrace\alpha_i,\gamma_j\rbrace$. If $m<n$, then $|\operatorname{Hom}(A,B)|<|\operatorname{Hom}(A,C)|$, which is not so. Thus $m=n$ and $\mathrm{min}\lbrace\alpha_i,\beta_j\rbrace= \mathrm{min}\lbrace\alpha_i,\gamma_j\rbrace$ for all $i$ and $j$. Let $var(B,C)=p^{\beta_r}$, $1\le r\le m$. Observe that $\exp(A)\le var(B,C)$, because if $p^{\alpha_1}>p^{\beta_r}$, then $\beta_r=\mathrm{min}\lbrace\alpha_1,\beta_r\rbrace=\mathrm{min}\lbrace\alpha_1,\gamma_r\rbrace>\beta_r$, a contradiction. Conversely, if $B=C$, then result holds trivially. We therefore suppose that $B<C$, $d(B)=d(C)=m$ and $\exp(A)\le var(B,C)$. Then $\alpha_i\le \alpha_1\le \beta_r<\gamma_r$ for $1\le i\le l$, and $\beta_j=\gamma_j$ for $r+1\le j\le m$. It follows that $$|\operatorname{Hom}(A,B)|=\prod_{i=1}^l\prod_{j=1}^mp^{\mathrm {min}\{\alpha_i, \beta_j\}}=p^{r(\alpha_1+\cdots+\alpha_l)}\prod_{i=1}^l\prod_{j=r+1}^mp^{\mathrm {min}\{\alpha_i,\beta_j\}}$$ and $$|\operatorname{Hom}(A,C)|=\prod_{i=1}^l\prod_{j=1}^mp^{\mathrm {min}\{\alpha_i, \gamma_j\}}=p^{r(\alpha_1+\cdots+\alpha_l)}\prod_{i=1}^l\prod_{j=r+1}^mp^{\mathrm {min}\{\alpha_i, \beta_j\}}.$$ Thus $|\operatorname{Hom}(A,B)|=|\operatorname{Hom}(A,C)|$ and hence $\operatorname{Hom}(A,B)=\operatorname{Hom}(A,C),$ because $\operatorname{Hom}(A,B)$ is a subgroup of $\operatorname{Hom}(A,C)$. Let $G$ be a finite non-abelian $p$-group. Let $M_1$, $M_2$, $N_1$ and $N_2$ be normal subgroups of $G$ such that $M_i\le Z(G)\cap N_i$ for $i=1,2$, $M_1\le M_2$ and $N_2\le N_1$. Then $\operatorname{Aut}_{N_1}^{M_1}(G)=\operatorname{Aut}_{N_2}^{M_2}(G)$ if and only if one of the following conditions holds: 1. $M_1=M_2$ and either $G/G'N_1=G/G'N_2$ or $d(G/G'N_1)=d(G/G'N_2)$ and $\exp(M_1)\le var(G/G'N_1,G/G'N_2)$. 2. $G/G'N_1=G/G'N_2$ and either $M_1=M_2$ or $d(M_1)=d(M_2)$ and $\exp(G/G'N_1)\le var(M_1,M_2)$. It follows from Lemma \[ML\] that for $i=1$ and 2, $$\operatorname{Aut}_{N_i}^{M_i}(G)\simeq\operatorname{Hom}(G/N_i, M_i)\simeq\operatorname{Hom}(G/G'N_i, M_i).$$ First suppose that $\operatorname{Aut}_{N_1}^{M_1}(G)=\operatorname{Aut}_{N_2}^{M_2}(G)$. Then $$|\operatorname{Hom}(G/G'N_1, M_1)|=|\operatorname{Hom}(G/G'N_2, M_2)|,$$ and therefore either $M_1=M_2$ or $G/G'N_1=G/G'N_2$ by [@curmac Lemma D]. Assume that $M_1=M_2$ and $G/G'N_1$ is a proper quotient group of $G/G'N_2$. Then proof of $(i)$ follows from Lemma \[MT\]$(ii)$ by taking $D=G/G'N_1, \;A=G/G'N_2$ and $B=M_1=M_2$. Next assume that $G/G'N_1=G/G'N_2$ and $M_1<M_2$. Then it follows from (1) that $$\operatorname{Hom}(G/G'N_1, M_1)=\operatorname{Hom}(G/G'N_2, M_2).$$ The proof of $(ii)$ now follows from Lemma 2.2$(i)$ by taking $A=G/G'N_1=G/G'N_2$, $B=M_1$ and $C=M_2$. Conversely, first assume that condition $(ii)$ holds. Then, taking $A=G/G'N_1=G/G'N_2$, $B=M_1$ and $C=M_2$ in Lemma 2.2$(i)$, we get $\operatorname{Hom}(G/G'N_1, M_1)=\operatorname{Hom}(G/G'N_2, M_2)$. It follows that $|\operatorname{Aut}_{N_1}^{M_1}(G)|=|\operatorname{Aut}_{N_2}^{M_2}(G)|$ and hence $\operatorname{Aut}_{N_1}^{M_1}(G)=\operatorname{Aut}_{N_2}^{M_2}(G)$ because $\operatorname{Aut}_{N_1}^{M_1}(G)\le\operatorname{Aut}_{N_2}^{M_2}(G)$. Now assume that condition $(i)$ holds. Then, taking $D=G/G'N_1$, $A=G/G'N_2$ and $B=M_1=M_2$ in Lemma 2.2$(ii)$, we get $|\operatorname{Hom}(G/G'N_1, M_1)|=|\operatorname{Hom}(G/G'N_2, M_2)|$ and hence $\operatorname{Aut}_{N_1}^{M_1}(G)=\operatorname{Aut}_{N_2}^{M_2}(G)$. Taking $M_1=M, N_1=N$ and $M_2=N_2=Z(G)$ in the above corollary, we get the following. Let $G$ be a finite non-abelian $p$-group. Let $M$ and $N$ be normal subgroups of $G$ such that $M\le Z(G)\le N$. Then $\operatorname{Aut}{_{N}^{M}}(G)=C^*$ if and only if one of the following conditions hold: 1. $M=Z(G)$ and either $G/G'N=G/G'Z(G)$ or $d(G/G'N)=d(G/G'Z(G))$ and $\exp(M)\le var(G/G'N,G/G'Z(G))$. 2. $G/G'N=G/G'Z(G)$ and either $M=Z(G)$ or $d(M)=d(Z(G))$ and $\exp(G/G'N)\le var(M,Z(G))$. \[[*cf.*]{} [[@azhmal Corollary C(ii)]]{}\] Let $G$ be a finite non-abelian $p$-group. Let $M$ and $N$ be normal subgroups of $G$ such that $M\le Z(G)\le N$. Then $\operatorname{Aut}_{N}^{M}(G)=\operatorname{Aut}_c(G)$ if and only if one of the following conditions hold: 1. $M=Z(G)$ and either $N\le G'$ or $d(G/G'N)=d(G/G')$ and $\exp(M)\le var(G/G'N,G/G')$. 2. $N\le G'$ and either $M=Z(G)$ or $d(M)=d(Z(G))$ and $\exp(G/G')\le var(M,Z(G))$. Observe that $\operatorname{Aut}_{N}^{M}(G)\simeq \operatorname{Hom}(G/G'N,M)$. First suppose that $\operatorname{Aut}_{N}^{M}(G)=\operatorname{Aut}_c(G)$. Then $\operatorname{Aut}_c(G)=C^*$ because $M\le Z(G)\le N$. Thus $G$ is purely non-abelian by [@yad Lemma 2.4] and hence $|\operatorname{Aut}_c(G)|=|\operatorname{Hom}(G/G', Z(G))|$ by [@adnyen Theorem 1]. It follows that $$|\operatorname{Hom}(G/G'N,M)|=|\operatorname{Hom}(G/G', Z(G))|,$$ and therefore either $N\le G'$ or $M=Z(G)$ by [@curmac Lemma D]. The proof now follows as in Corollary 2.3. Conversely, if condition $(ii)$ holds, then $Z(G)\le G'$; and if condition $(i)$ holds, then $d(G/G'N)=d(G/G')$ implies that $G'N$ and hence $Z(G)$ does not contain any generating element of $G$. In either of the cases, $G$ is purely non-abelian and hence $|\operatorname{Aut}_c(G)|=|\operatorname{Hom}(G/G', Z(G))|$ by [@adnyen Theorem 1]. The proof now follows as in Corollary 2.3. Some particular cases of this corollary give the following results of Rai [@rai], Attar[@att3] and Jafari [@jaf]. \[[[@rai Theorem B(1)]]{}\] Let $G$ be a finite non-abelian $p$-group. Then $\operatorname{IA}(G)^*=\operatorname{Aut}_c(G)$ if and only if $G'=Z(G)$. It is easy to see that if $G'=Z(G)$, then $\operatorname{IA}(G)^*=\operatorname{Aut}_c(G)$. Conversely, if $\operatorname{IA}(G)^*=\operatorname{Aut}_c(G)$, then $cl(G)=2$ and thus $\operatorname{IA}(G)^*=\operatorname{Aut}_{Z(G)}^{G'}(G)=\operatorname{Aut}_c(G).$ The result now follows by taking $M=G'$ and $N=Z(G)$ in Corollary 2.5. Let $G$ be a finite non-abelian $p$-group. Then $\operatorname{Aut}_c(G)=C^*$ if and only if either $Z(G)\le G'$ or $d(G/G'Z(G))=d(G/G')$ and $\exp(Z(G))\le var(G/G'Z(G),G/G')$. The proof follows by taking $M=N=Z(G)$ in Corollary 2.5. We next obtain some new results which are immediate consequences of Lemma 2.2. Let $G$ be a finite $p$-group of nilpotence class $2$. Then $IA(G)=\operatorname{IA}(G)^*$ if and only if either $G'=Z(G)$ or $d(G/Z(G))=d(G/G')$ and $\exp(G')\le var(G/Z(G),G/G')$. It follows from Lemma 2.1 that $\operatorname{IA}(G)\simeq \operatorname{Hom}(G/G',G')$ and $\operatorname{IA}(G)^*\simeq\operatorname{Hom}(G/Z(G),G').$ The result now follows from Lemma \[MT\]$(ii)$ by taking $D=G/Z(G)$, $A=G/G'$ and $B=G'$. Let $G$ be a finite non-abelian $p$-group. Then $\operatorname{IA}(G)^*=C^*$ if and only if either $G'=Z(G)$ or $G'<Z(G)$, $d(G')=d(Z(G))$ and $\exp(G/Z(G))\le var(G',Z(G))$. Observe that if $\operatorname{IA}(G)^*=C^*$, then nilpotence class of $G$ is 2 and hence $\operatorname{IA}(G)^*\simeq\operatorname{Hom}(G/Z(G),G')$ and $C^*\simeq\operatorname{Hom}(G/Z(G),Z(G))$ by Lemma 2.1. The result now follows from Lemma \[MT\]$(i)$ by taking $A=G/Z(G)$, $B=G'$ and $C=Z(G)$. Let $G$ be a finite non-abelian $p$-group. Then $\operatorname{IA}(G)=C^*$ if and only if either $G'=Z(G)$ or $G'<Z(G)$, $d(G')=d(Z(G))$, $d(G/Z(G))=d(G/G')$ and $\exp(G')=var(G/Z(G),G/G')=\exp(G/Z(G))= var(G',Z(G))$. That the conditions are sufficient follows from Corollaries 2.8 and 2.9. Conversely suppose that $\operatorname{IA}(G)=C^*$. Then nilpotence class of $G$ is 2 and $\operatorname{IA}(G)=\operatorname{IA}(G)^*=C^*$. It follows from Corollaries 2.8 and 2.9 that either $G'=Z(G)$ or $G'<Z(G)$, $d(G')=d(Z(G))$, $d(G/Z(G))=d(G/G')$ and $\exp(G')\le var(G/Z(G),G/G')\le \exp(G/Z(G))\le var(G',Z(G))\le \exp(G')$. Thus $\exp(G')=var(G/Z(G),G/G')=\exp(G/Z(G))= var(G',Z(G))$ and hence the result. As another application of Lemma 2.2, we next find necessary and sufficient conditions on a finite non-abelian $p$-group $G$ such that $\operatorname{IA}(G)=\operatorname{Aut}_c(G)$. We start with the following lemma. Let $G$ be a finite nilpotent group of class $2$ such that $d(G')=d(Z(G))$. Then $G$ is purely non-abelian. On the contrary, suppose that $G=H\times K$, where $H$ is a non-trivial abelian and $K$ is a purely non-abelian subgroup of $G$. Then $Z(G)=H\times Z(K)$ and $G'=K'\le Z(K)$. It follows that $d(G')\le d(Z(K))<d(Z(G))$, because $H$ is non-trivial. This is a contradiction and hence $G$ is purely non-abelian. Let $G$ be a finite non-abelian $p$-group. Then $\operatorname{IA}(G)=\operatorname{Aut}_c(G)$ if and only if either $G'=Z(G)$ or $G'<Z(G)$, $d(G')=d(Z(G))$ and $\exp(G/G')\le var(G',Z(G))$. First suppose that $\operatorname{IA}(G)=\operatorname{Aut}_c(G)$. Then nilpotence class of $G$ is 2 and $\operatorname{IA}(G)\simeq \operatorname{Hom}(G/G',G')$. We prove that $G$ is purely non-abelian. Assume that $G=A\times B$, where $A$ is abelian and $B$ is purely non-abelian. For $1\ne f\in\operatorname{Hom}(B, A)$, define $f^*:G\rightarrow G$ by $f^*(ab)=abf(b)$, $a\in A,\;b\in B$. It is easy to see that $f^*$ is a central but not an IA-automorphism of $G$. This is against the assumption and thus $G$ is purely non-abelian. It now follows from [@adnyen Theorem 1] that $|\operatorname{Aut_c}(G)|=|\operatorname{Hom}(G/G',Z(G))|$. Thus $|\operatorname{Hom}(G/G',G')|=|\operatorname{Hom}(G/G',Z(G))|$ and hence $\operatorname{Hom}(G/G',G')=\operatorname{Hom}(G/G',Z(G))$, because $\operatorname{Hom}(G/G',G')$ is a subgroup of $\operatorname{Hom}(G/G',Z(G))$. The result now follows from Lemma \[MT\] $(i)$ on replacing $A$ by $G/G'$, $B$ by $G'$ and $C$ by $Z(G)$. Conversely, if $G'=Z(G)$, then trivially $\operatorname{IA}(G)=\operatorname{Aut}_c(G)$. Therefore suppose that $G'<Z(G)$, $d(G')=d(Z(G))$ and $\exp(G/G')\le var(G',Z(G))$. By Lemma 2.11, $G$ is purely non-abelian. It now follows by [@adnyen Theorem 1], Lemma \[ML\] and Lemma \[MT\]$(i)$ that $|\operatorname{IA}(G)|=|\operatorname{Hom}(G/G',G')|= |\operatorname{Hom}(G/G',Z(G))|=|\operatorname{Aut}_c(G)|$ and hence $\operatorname{IA}(G)=\operatorname{Aut}_c(G)$, because $\operatorname{IA}(G)\le\operatorname{Aut}_c(G)$. [88]{}J. E. Adney and T. Yen, [*Automorphisms of a $p$-group,*]{} Illinois J. Math., [**9**]{} (1965), 137-143. J. L. Alperin, [*Groups with finitely many automorphisms*]{}, Pacific J. Math., [**12**]{} (1962), 1-5. Z. Azhdari and M. Akhavan-Malayeri, [*On automorphisms fixing certain groups,*]{} J. Algebra. Appl. [**12**]{}(2) (2013), 1250163. M. J. Curran [*Finite groups with central automorphism group of minimal order*]{}, Math. Proc. Roy. Irish Acad. [**104A**]{}(2) (2004), 223-229. M. J. Curran and D. J. McCaughan, [*Central automorphisms that are almost inner*]{}, Comm. Algebra, [**29**]{} (2001), 2081-2087. T. A. 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Algebra, [**37**]{} (2009), 4325-4331. [^1]: The research of first author is supported by University Grants Commission of India under the Research Award Scheme

--- abstract: 'We consider the latency minimization problem in a task-offloading scenario, where multiple servers are available to the user equipment for outsourcing computational tasks. To account for the temporally dynamic nature of the wireless links and the availability of the computing resources, we model the server selection as a multi-armed bandit (MAB) problem. In the considered MAB framework, rewards are characterized in terms of the end-to-end latency. We propose a novel online learning algorithm based on the principle of optimism in the face of uncertainty, which outperforms the state-of-the-art algorithms by up to $\sim$1 s. Our results highlight the significance of heavily discounting the past rewards in dynamic environments.' author: - | Aniq Ur Rahman$^{{\href{https://orcid.org/0000-0003-3685-7201}{\mbox{\scalerel*{ \begin{tikzpicture}[yscale=-1,transform shape] \pic{orcidlogo}; \end{tikzpicture} }{|}}}}}$,  Gourab Ghatak$^{{\href{https://orcid.org/0000-0002-8240-4038}{\mbox{\scalerel*{ \begin{tikzpicture}[yscale=-1,transform shape] \pic{orcidlogo}; \end{tikzpicture} }{|}}}}}$, and Antonio De Domenico$^{{\href{https://orcid.org/0000-0003-1229-4045}{\mbox{\scalerel*{ \begin{tikzpicture}[yscale=-1,transform shape] \pic{orcidlogo}; \end{tikzpicture} }{|}}}}}$ [^1] [^2] [^3] [^4] bibliography: - 'bare\_jrnl.bib' title: 'An Online Algorithm for Computation Offloading in Non-Stationary Environments' --- Mobile Edge Computing, Online Learning, Computation Offloading, Multi-armed Bandit. Introduction ============ future mobile networks will be characterized by ubiquitous coverage, ultra-low latency services, quasi-deterministic communications, and the need for extremely high data rates. In this context, a radical change consists of empowering mobile devices and with data processing and storage capabilities, thereby reducing the end-to-end latency of the mobile services. This paradigm is called  [@Mao2017], also known as mobile edge computing. In networks, small cells integrate computing capabilities and local cache memories to the standard . Consequently, a can request a small cell to run a computational assignment on its behalf, resulting in a reduced effective latency and an increased battery-life. This procedure is called *task* or *computation offloading* [@Barbarossa2014]. Additionally, the -enabled small cells can implement proactive caching strategies to satisfy the ever growing demand for downloadable multimedia content in the mobile networks, thereby limiting the load on the transport network [@Bastug2014]. The resources are often divided into three categories: communication, computing, and caching [@Wang2017]. In [@elbamby2019wireless] the authors have provided a detailed overview of technology and its use-cases, particularly focusing on the services requiring low-latency and highly-reliable communications. Several researchers have investigated policies to determine when computation offloading is more efficient than local processing. For instance, Elbamby [*et al.*]{} [@elbamby2017proactive] have studied the task-offloading problem formulated as a matching game, subject to latency and reliability constraints. More recently, computation offloading was also extended to more realistic scenarios, where system dynamics and information uncertainty is taken into consideration. For example, Liao [*et al.*]{} [@icc_fog] have proposed a robust two-stage task offloading algorithm that integrates contract theory with computational intelligence to minimize the long-term delay of task assignment. On the same lines, is an online framework that can be used to find an optimal policy when the reward distribution of the actions is not *a priori* known [@lattimore2018bandit]. In particular, we focus on the case where the system characteristics, i.e., the resource availability and the wireless channel are [*non-stationary*]{}[^5]. It must be noted that, in non-stationary scenarios, off-the-shelf algorithms may indeed be sub-optimal due to the usage of outdated information. Therefore, it becomes necessary to [*forget*]{} past rewards and rapidly update the reward distribution based on recent information. However, selecting the policy refresh rate is challenging since the *agent* is typically not aware of the temporal behaviour of the system. Earlier, researchers have come up with the idea of *discounting* the past rewards, to make the system adaptive to the dynamic changes and introduced the discounted variants [@raj2017taming; @garivier2008upper] of classical algorithms. Garivier and Moulines [@garivier2008upper] considered a scenario where the distribution of the rewards remain constant over epochs and change at unknown time instants (i.e., abrupt changes). They analyzed the theoretical upper bounds of the regret for the discounted upper confidence bound (UCB) and sliding window UCB. Gupta [*et al.*]{} [@gupta2011thompson], extended this idea to Bayesian methods, and proposed the [Dynamic Thompson Sampling (Dynamic TS)]{}. Hartland [*et al.*]{} [@hartland2006multi] considered dynamic bandits with [abrupt changes]{} in the reward generation process, and proposed an algorithm called [Adapt-EvE]{}. Slivkins and Upfal. [@slivkins2008adapting] considered a dynamic bandit setting where the [reward evolves as Brownian motion]{} or a random walk, and provided results of regret linear in time horizon. Sana [*et al.*]{} [@sana2019multi] have solved the problem of optimizing the UE-BS association by employing Deep Reinforcement Learning. Liao [*et al.*]{} [@liao2019learning] have maximized the long-term throughput for a machine type device (MTD) subject to energy and data-size constraints in a learning-based channel selection framework. The learning algorithm proposed is a variant of UCB. However, these works do not take into account, the abrupt changes at unknown times. In this paper, we model the server selection problem as the exploration-exploitation dilemma of a restless framework with non-stationary rewards. For this problem, we propose an online learning algorithm *Sisyphus* that [is model-free and is based]{} on the principle of optimism in the face of uncertainty. In particular, we selectively retain the knowledge of the past rewards so as to keep up with the dynamic environment. We show that Sisyphus achieves the lowest normalized regret as compared to the other algorithms proposed for the non-stationary bandit problem, namely, Thompson sampling (TS), discounted TS, discounted optimistic TS, and discounted UCB. Consequently, Sisyphus is shown to reduce the end-to-end latency by up to 1 s under the considered test environment. System Model ============ [We focus on]{} a offloading its computational task to [a nearby]{} server $s_i \in \mathcal{S}$, where $\mathcal{S}$ represents the set of all servers. We assume that one task is offloaded by the UE in each time-step $t \in \{1, 2, 3, ..., T \}$ of duration $\delta$. The aim of the is to select the server which results in a minimum delay, while taking into account the task execution and signal propagation delays. The server $s_i$ performs the task with intensity $\kappa$, which denotes the CPU cycles required to process a byte of task, using its available computing resources, which evolves over time [@dandachi2019artificial]. Unlike the centralized architecture in [@liang2019multiuser], we consider a distributed system where each user selects an MEC server independently of the other users’ decision. Specifically, the link between the and the server is assumed to be affected by dynamic blockages, where the probability of blockage of the server $s_i$ is denoted by $p_{B,i}$. In addition, we model the servers as the arms in an framework, where the resource availability $a_i(t)$, varies with time in a *doubly-stochastic* manner. The computing resources available at time-step $t$ is expressed as $a_i(t)c_i$, where $c_i$ is the maximum computing capacity[^6] of the server and $a_i(t) \in (0,1)$ is the fraction of the computing capacity available at time $t$. We refer to this quantity $a_i(t)$ as *resource availability*. We assume that the number of UEs associated with a server changes after certain number of time-steps, which in turn impacts the resource availability. This set of consecutive time-steps constitute an **epoch**. If the probability that the number of UEs connected $v(t)$ to a server $s_i$ changes in a single time-step is $p = \text{Pr}\{v(t) \neq v(t-1)\}$, then the probability that it remains unchanged for $\Delta$ consecutive time-steps, is given by the geometric distribution [@vaseghi1995state]: $$\Pi_{l=1}^{\Delta} (1-p) = (1-p)^{\Delta}.$$ We set $p = \frac{1}{\Lambda_i}$ where $\Lambda_i$ is the mean value of epoch duration. The $j^{\rm th}$ epoch size $\Delta_i^j$ can then be drawn from the distribution: $$p_{\Delta_i^j}(\Delta_i^j = \Delta; \Lambda_i) = \bigg( 1-\frac{1}{\Lambda_i} \bigg)^{\Delta},$$ where the expected value $\mathbb{E}\{ \Delta_i^j \} = \Lambda_i$. The instantaneous resource availability of an MEC server $a_i(t)$ is a function of the associated UEs. If server $s_i$ can accept upto $N$ users at a time, and $q$ users offload their tasks to it, then, $a_i(t)=1-\frac{q}{N}$. Now, we derive the probability that $q$ UEs offload their tasks to the server at a given time-step. The considered scenario is as follows: (i) there are $w$ UEs in communication range of the MEC server, (ii) for the $j^{\rm th}$ epoch, out of these $w$ UEs, $v_j$ are connected to the small cell hosting the server, (iii) at a given time-step $t$ within the $j^{\rm th}$ epoch, out of these $v_j$ UEs, only $q_{t,j}$ UEs offload their computation tasks.\ The probability that $v_j$ UEs out of $w$ are connected to the server follows a binomial distribution: $$p_{v_j}(v_j=v) = {w \choose v} \psi_0^v (1-\psi_0)^{w-v}, \label{v}$$ where $\psi_0$ is the probability of a single in-range UE to be connected to the server. The value $\psi_0$ is specific for a server $s_i$ because of the radio characteristics of the environment surrounding $s_i$ (e.g., blockages). Out of these $v_j$ UEs, only a fraction of the UEs offload their task to the server [e.g., depending on the task computational complexity]{}. Therefore, we denote with $\psi_1$ the probability that a connected UE decides to offload a task. Then, $q_{t,j}$ follows a binomial distribution: $$p_{q_{t,j}}(q_{t,j}=q) = {v_j \choose q} \psi_1^q (1-\psi_1)^{v_j-q}. \label{q}$$ For a given server $s_i$, the resource availability at time-step $t$ in the $j^{\rm th}$ epoch is then expressed as: $a{_i}(t) = 1 - \frac{q_{t,j}}{N}$. Therefore, the dynamic resource availability characteristics of a server $s_i \in \mathcal{S}$ can be controlled through the parameters $\{\psi_0, \psi_1, w, N, \Lambda\}_i$. Let us assume that the amount of uplink data [related to the task to be offloaded]{} be given by [$L_U$]{} bytes. The downlink data [size, after the MEC server processing, is denoted as $L_D$ and is related to the uplink data as: $L_D = \Omega L_U, \Omega \in \mathbb{R}^+$]{}. Furthermore, let $\gamma$ denote the path-loss exponent of the transmissions, which varies depending on the blockage conditions, i.e., whether the channel visibility state is in or . Additionally, let the reference uplink at 1 m be denoted as $\mathcal{P}_U$. [Similarly, the downlink at 1 m is denoted as $\mathcal{P}_D$. The uplink and downlink bandwidths are denoted as $B_U$ and $B_D$ respectively]{}. Thus, the total transmission delay $\tau_i(t)$ [when the distance between the UE and server is $r_i$,]{} can be written as: $$\tau_i(t) = \sum_{Z \in \{U, D\}} \frac{L_Z}{B_Z \log_2(1 + \mathcal{P}_Z\cdot r_i^{-\gamma})};$$ For the processing phase, the computation delay $\eta_i(t)$ is defined as the time taken by the server $s_i$ to process the data and generate the output, [ which is expressed mathematically as:]{} $$\eta_i(t) = \frac{\kappa L}{c_i a_i(t)}.$$ Then, the total delay is the sum of transmission and computation delays: $D_i(t) = \tau_i(t) + \eta_i(t)$. Finally. the reward associated with server $s_i$ at time-step $t$ is denoted by $\rho_i(t)$. [Let $D_{\max}$ be the latency requirement of the task that the UE wants to offload; then, we can define]{} the reward $\rho_i(t)$ as: $$\rho_i(t) = \mathds{1}_{ \{ D_i \leq D_{\max} \} },$$ which ensures that the reward is positive and bounded by 1. The UE follows a policy $\pi$ (see Section \[sec: Algo\]) to select an arm at each time-step. Let $\rho_j({t})$ be the reward of the arm chosen at time-step $t$ and $\max \rho_i({t})$ denote the highest reward among all arms’ reward; then, the *time-normalized cumulative regret* $R_{\alpha}({T})$ for ${T}$ time-steps is defined as the cumulative sum of the difference between the rewards of the best arm and the chosen arm (according to $\pi$) divided by the count of time-steps ${T}$. We refer to it as the normalized regret, given by: $$R_{\pi}({T}) = \frac{1}{{T}}\sum_{{t=1}}^{{T}} \left(\max\limits_{s_i \in \mathcal{S}} (\rho_i({t})) - \rho_j({t}) \right).$$ The objective of the framework is to design the policy $\pi$ so as to minimize $R_{\pi}({T})$. In the next section, we propose one such policy which outperforms the state-of-the-art algorithms. Proposed Online Learning Algorithm {#sec: Algo} ================================== We consider an $|\mathcal{S}|$-armed bandit, where the UE, at each time-step, plays the arm (i.e., selects the server) which has the highest expected reward, based on the past experiences of playing the arms. [Specifically, for each server $s_i \in \mathcal{S}$]{}, the UE tracks the total number of times each arm has been played, denoted by $k_i$ [and]{} maintains a score $\mu_i$, as described below. [On playing the arm $s_i$ for the $k_i^{\rm th}$ time, we obtain a **reward** $\rho_i(k_i)$, then the **score** $\mu_i(k_i)$ for that arm is updated as:]{} $$\mu_i(k_i) = \frac{1- \alpha^{k_i -1}}{2 -\alpha - \alpha^{k_i}} \mu_i(k_i -1) + \frac{1- \alpha}{2 -\alpha - \alpha^{k_i}}\rho_i(k_i), \label{learneq}$$ where $\alpha\in [0,1)$ is the **retention rate**. The parameter $\alpha$ controls the amount of memory in the framework. Two extreme states can be determined [in the system]{}, by substituting the value of $\alpha=0$ and $\alpha \rightarrow 1$. If $\alpha$ is set to zero, the UE gives equal weight-age to the new reward compared to the weighted sum of previous rewards. For $\alpha \rightarrow 1$, past rewards have [a larger]{} effect on the current score and thereby influence more the UE’s decision. In essence, lower the value of $\alpha$, the lesser memory the system has about the past rewards. The score assigned to arm $s_i$ can be expressed as a weighted sum of rewards, where $\phi_{\alpha}(k_i, m)$ denotes the **memory weight** for the reward when the arm $s_i$ is played for the $m^{\rm th}$ time: $$\mu_i(k_i)= \sum_{m=1}^{k_i} \phi_{\alpha}(k_i, m)\cdot \rho_i(m); \quad k_i > 0. \label{learn_weight}$$ [ $$\phi_{\alpha}(k_i, m) = \frac{1- \alpha}{2 - \alpha -\alpha^m} \cdot \left( \prod_{j = m+1}^{k_i} \frac{1- \alpha^{j-1}}{2 - \alpha -\alpha^j} \right). \label{learn_weight}$$ ]{} ![Memory weight across different retention rates.[]{data-label="fig:phi_3d"}](figures/Figure_1.eps){width="0.8\columnwidth"} After a certain [play-count]{} $k_i$, the reward at the $m^{th}$ [play]{}, becomes negligible [($m < k_i$)]{}. This is bound to happen, as the recorded reward successively fades, until it no longer affects the score of that arm. This is more intuitive than resetting the previous rewards to zero at regular predefined intervals (e.g., see [@aniq_wmlc]), since smooth transitions allow to take care of abrupt changes in the reward distribution. Refreshing the score to zero at fixed intervals may either reset it too early, or too late, resulting in sub-optimal performance. In essence, [the UE gives]{} importance to the score of an arm and the number of times it has been played. This prevents us from getting biased by the performance of an arm in a few trials. This is the optimistic approach, where we expect that a poorly performing arm might perform well in the future draws owing to the uncertain behaviour of the arms. We have depicted the concept of memory weights graphically in Fig. \[fig:phi\_3d\]. For smaller values of retention rate $\alpha$, the reward recorded for an arm fades quickly as it is played more number of times ($k$). On the other hand, for values of $\alpha$ close to 1, the reward fades slowly in comparison. The [proposed]{} algorithm **Sisyphus** (SSPH) is described in Algorithm 1. The scores $\mu$ and counts $k$ for all the arms are initialized to zero in step 1 and step 2 respectively. A time loop starts in step 3 which is terminated in step 9 within which, the following operations are performed sequentially: an expected reward $\theta$ is drawn from the normal distribution (step 4) and the arm with the maximum expected reward is chosen to be played (step 5). The [play-]{}count of that arm (which tracks the number of times the arm has been played) is incremented by 1 (step 6). When the selected arm is played, the actual reward is revealed, after which we update the score of the chosen arm in step 7 and that of [the set of]{} the never-played arms [$\mathcal{S}^0$]{} in step 8. {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} [Retention rate $\alpha$]{} $\mu_i(0) \gets 0; \quad \forall s_i \in \mathcal{S}$ $k_i \gets 0; \quad \forall s_i \in \mathcal{S}$ $\theta_i \sim \mathcal{N}(\mu_i(k_i), \sigma^2); \quad \forall s_i \in \mathcal{S}$ $s_j(t) \gets \arg \max_{s_i \in \mathcal{S}} \theta_i$ $k_j \gets k_j +1$ $\mu_j(k_j) \gets \frac{1- \alpha^{k_j -1}}{2 -\alpha - \alpha^{k_j}} \mu_j(k_j -1) + \frac{1- \alpha}{2 -\alpha - \alpha^{k_j}}\rho_j(k_j)$ $\mu_u(0) \gets \frac{1}{|\mathcal{S} \setminus \mathcal{S}^0|}\sum_{s_i \in \mathcal{S} \setminus \mathcal{S}^0} \mu_i(k_i); \quad \forall s_u \in \mathcal{S}^0 $ The algorithm is based on the principle of optimism in the face of uncertainty[^7] [@lattimore2018bandit]. We first assign the score of zero to each arm and then draw the [expected]{} reward from a normal distribution with mean equal to the score $\mu_i(k_i)$ and variance[^8] equal to $\sigma^2$. This is a Bayesian approach [@poupart2006analytic] and allows us to look for expected rewards in the neighborhood of the recorded score $\mu_i(k_i)$, since it is not wise to make decisions by comparing the scores of the arms directly, in a non-stationary environment. This enables us to predict values which would otherwise be ignored in a greedy technique [@wunder2010classes]. As we play, we update the score of the arms that have never been sampled as the average of the scores of the played arms. This boosts the probability of exploration of the unexploited arms. In contrast to the classical MAB algorithms, e.g., UCB, which add specific terms to facilitate exploration, the proposed scheme is a randomized algorithm in which the exploration-exploitation trade-off is based on a Bayesian framework. In the following section, we show several numerical results that compares our algorithm with other state-of-the-art algorithms. Simulation Results ================== To assess the proposed online learning algorithm, we define five classes of servers $\{s_1, s_2, s_3, s_4, s_5 \} \in \mathcal{S}$, whose characteristics are described in Table \[tab:servers\]. In our simulations, the $j^{\rm th}$ server is assigned to one of these classes of servers as: $s_j \gets s_{j \pmod 5}; \quad j>5$, where $\mod$ denotes the modulus operation. [| &gt;p[1.5cm]{}| &gt;p[0.8cm]{} | &gt;p[0.8cm]{}| &gt;p[0.8cm]{} | &gt;p[0.8cm]{}| &gt;p[0.8cm]{} |]{} **Parameter** & $s_1$ & $s_2$ & $s_3$ & $s_4$ & $s_5$\ ${\psi_0}$ & 0.7 & 0.6 & 0.5 & 0.4 & 0.3\ $\Lambda_i$ & 100 & 150 & 100 & 100 & 50\ $r_i$ \[m\] & 7 & 10 & 12 & 14 & 16\ $p_{B,i}$ & 0.3 & 0.4 & 0.5 & 0.6 & 0.7\ $c_i$ \[GHz\] & 5 & 3.3 & 3.3 & 3.3 & 5\ Additional simulation parameters are: [$L_U = 20$ MB [@qi2016quantifying], $\Omega=1$, $B_D = B_U = 500$ MHz, $\mathcal{P}_U= 20$ dBm, $\mathcal{P}_D= 40$ dBm,]{} $D_{\max} = {1}$ s, $\kappa = 10$ cycles/byte, $\gamma_{\text{LOS}} = 2, \gamma_{\text{NLOS}} = 4$ [$w=N=100$, $\psi_1 = 0.5 \,\forall s_i \in \mathcal{S}$, [and $\delta = 1$ s]{}]{}. We compare the performance of Sisyphus (SSPH) with the following algorithms: Thompson Sampling (TS) [@thompson1933likelihood], Discounted Thompson Sampling (dTS) [@raj2017taming], Discounted Optimistic Thompson Sampling (dOTS) [@raj2017taming] and Discounted UCB (D-UCB) [@garivier2008upper]. It is important to note that the last three algorithms (dTS, dOTS, and D-UCB) are designed to tackle the issue of dynamically changing environments in the framework. The value of $\alpha$ is set to $0.6$ for Sisyphus. The discounting factor of the benchmark algorithms are chosen for their best performance: dTS ($0.8$), dOTS ($0.7$) and D-UCB ($0.5$). Normalized Regret ----------------- In Fig. \[fig:regret\], we plot the temporal evolution of the normalized regret for the different algorithms. Here, the solid lines represent the mean of the normalized regret, and the shaded region represents the variance. The proposed algorithm SSPH evidently outperforms all the other algorithms and has a much lower normalized regret [$(\sim 0.32)$]{} compared to the other algorithms [$(> 0.37)$]{}. Interestingly, we observe that SSPH also has a considerably lower variance, which indicates that it is more robust than the other contending algorithms. Latency ------- Naturally, the reduced normalized regret will be reflected on the latency performance with different algorithms. To validate this, we plot the variation of time-normalized latency for various algorithms in Fig. \[fig:latency\]. We observe that as the temporal process evolves, the latency of most of the contending algorithms increases gradually and settle into a higher value [$> 2.5$ s]{}. On the contrary, the latency of the proposed algorithm is considerably lower [($\sim 1.5$ s)]{}. Parameter Tuning ---------------- Indeed the performance of SSPH will depend on the agility of the environment change, and the corresponding choice of $\alpha$. However, the algorithm developed is model-free, and takes the rewards as input at each time-step to update its score for the respective arm. The performance can be tweaked by tuning the parameters $\alpha$ which denotes how strongly the algorithm retains the past rewards and the variance $\sigma^2$ which controls the degree of exploration. For a highly dynamic system, the past rewards need to be forgotten quickly and in an environment with less number of arms, the exploration factor can be kept low. In our work, for all the algorithms, the corresponding retention parameters are tweaked to obtain the best performance. In Fig. \[fig:alpha\_scatter\], we show how the normalized regret varies with varying $\alpha$ for SSPH with 5 classes of servers. Here, the [blue]{} scattered points are observations, black solid line is mean of the observations, [red]{} lines are the standard deviation around the mean value. It can be observed that the mean of the scattered point remains reasonably flat, i.e., ranging within [$[0.3, 0.35]$ for $\alpha \in [0.1, 0.6]$]{}. This indicates that a fairly robust selection of $\alpha$ can be made for deploying SSPH in the UE. Scalability ----------- Next, in Fig. \[fig:algo\_comp\], we vary the number of arms (i.e., the number of MEC servers) $|\mathcal{S}|$ and compare the mean normalized regret of the different algorithms. [The normalized regret for TS, DTS and DOTS increases with increase in $|\mathcal{S}|$. On the other hand the normalized regret of SSPH and D-UCB does not change significantly with increase in $|\mathcal{S}|$. It must be noted that]{} SSPH maintains the minimum value of mean normalized regret among the contenders. ![Mean Normalized Regret across various algorithms for different number of arms.[]{data-label="fig:algo_comp"}](figures/Figure_5.eps){width="0.8\columnwidth"} To capture these nuances of SSPH in a more concrete manner, currently we are investigating the theoretical regret bounds of the proposed algorithm and testing it for other online learning use cases. Conclusion ========== In this paper, we proposed an online learning algorithm for the MAB framework with an objective to minimize the end-to-end latency in offloading computation tasks to MEC servers. In particular, we showed that selective retention of past rewards is necessary to tackle temporally varying environments. The proposed algorithm (Sisyphus) works on the principle of optimism in the face of uncertainty, and outperforms the other state-of-the-art algorithms for non-stationary MAB frameworks. We show that the proposed algorithm, in the test environment achieves a latency which is [at least $\sim 1$ s]{} lower than the other benchmark algorithms. [^1]: A. U. Rahman is with the Department of Electrical Communication Engineering, Indian Institute of Science, 560012 Bangalore, India. Email: [email protected]; [^2]: G. Ghatak is with the Department of Electronics and Communication Engineering, Indraprastha Institute of Information Technology Delhi, India.Email: [email protected]; [^3]: A. De Domenico is with Huawei Technologies, Paris Research Center, 20 quai du Point du Jour, Boulogne Billancourt, France. Email: [email protected]. [^4]: Manuscript received XX XX, 2020; revised XX XX, 2020. [^5]: [This refers to a random process whose probability distribution changes in time or space [@besbes2014stochastic].]{} [^6]: Computing capacity refers to the frequency of the processor clock, i.e., number of cycles per second, typically measured in GHz. [^7]: The optimism in the face of uncertainty principle states that the actions should be chosen assuming the environment to be as nice as plausibly possible. [^8]: The appropriate value of $\sigma^2$ can be tuned based on empirical history.

--- abstract: 'We consider an inverse $N$-body scattering problem of determining two potentials—an external potential acting on all particles and a pair interaction potential—from the scattering particles. This paper finds that the time-dependent Hartree-Fock approximation for a three-dimensional inverse $N$-body scattering in quantum mechanics enables us to recover the two potentials from the scattering states with high-velocity initial states. The main ingredient of mathematical analysis in this paper is based on the asymptotic analysis of the scattering operator defined in terms of a scattering solution to the Hartree-Fock equation at high energies. We show that the leading part of the asymptotic expansion of the scattering operator uniquely reconstructs the Fourier transform of the pair interaction, and the second term of the expansion uniquely reconstructs the $X$-ray transform of the external potential.' author: - | Michiyuki Watanabe\ Faculty of Education\ Niigata University\ Niigata, Japan\ `[email protected]`\ bibliography: - 'michiyukirefs2.bib' title: 'Inverse $N$-body scattering with the time-dependent Hartree-Fock approximation' --- Introduction ============ Problem and result ------------------ Consider the quantum $N$-body systems of identical particles interacting pairwise by the two-body potential under an external potential acting on all particles. A typical example is $N$ electrons in an atom with proton number $Z$ at the nucleus. In that case, the external potential is the nucleus-electron attraction, and the two-body potential is the electron to electron repulsion. Inverse $N$-body scattering problems ask to determine the interaction potential and the external potential from the scattering states of particles. Such inverse problems have been extensively studied for $N$-body Schrödinger equations with no external potentials (Enss and Weder [@Enss-Weder1995]; Novikov [@Novikov]; Wang [@Wang; @Wang1996]; Vasy [@Vasy]; Uhlmann and Vasy [@Uhlmann-Vasy; @Uhlmann-Vasy2003; @Uhlmann-Vasy2004]). The inverse scattering for the $N$-body Schrödinger equation in an external constant electric field was investigated by Valencia and Weder [@Valencia-Weder2012]. Differently, Lemm and Uhlig [@Lemm-Uhlig2000] have investigated an inverse $N$-body problem by using Bayesian approach with the Hartree-Fock approximation. They gave a computationally feasible method of reconstructing an interaction potential from data by solutions to a stationary Hartree-Fock equation. Their work indicates that the Hartree-Fock approximation is also extremely useful as a way to investigate the inverse $N$-body problems. The above mentioned works have focused only on recovering interactions. Since $N$-body systems is generally described by a non-relativistic Hamiltonian consisting of a one-body term with the kinetic energy and an external potential, and a two-body interaction term, the inverse problems of determining both the interaction potential and the external potential should be also investigated. However, little has been reported on the determination both the interaction potential and the external potential in the quantum $N$-body systems. In this paper, we find that the time-dependent Hartree-Fock approximation for the inverse $N$-body scattering in quantum mechanics enables us to recover two potentials—an external potential acting on all particles and a pair interaction potential—from the scattering states with high-velocity initial states. This paper also propose a new reconstruction procedure of recovering the two potentials. Let us formulate our inverse problem and state our main result. We first recall that the $n$-dimensional $N$-body Schrödinger equation has the form: $$\begin{aligned} & i\frac{{\partial}}{{\partial}t}\Psi(t) = \widetilde{H}_N \Psi (t), \\ & \widetilde{H}_N = \sum_{j=1}^N \left[ \frac{1}{2}\left( -i \nabla_{\mathbf{x}_j} \right)^2 +V_{ext}(\mathbf{x}_j)\right] + \sum_{j<k}^N V_{int}(\mathbf{x}_j - \mathbf{x}_k),\end{aligned}$$ where $i=\sqrt{-1}$, $\mathbf{x}_j\in {\mathbb{R}}^n$, $V_{ext}(\mathbf{x}_j)$ is an external potential and $V_{int}(\mathbf{x}_j)$ is an interaction potential with $V_{int}(\mathbf{x}_j)=V_{int}(-\mathbf{x}_j)$. The Hartree-Fock approximation is known as the simplest one-body approximation. Writing the $N$-body wave function $\Psi(t)=\Psi (t, \mathbf{x}_1, \cdots , \mathbf{x}_N)$ with the Slater determinant $$\Psi (t, \mathbf{x}_1, \cdots , \mathbf{x}_N) = (N!)^{-1/2} {\rm det} \left( u_j (t, \mathbf{x}_k) \right)_{1\le j, k \le N}$$ yields the one-body Schrödinger equation: $$\begin{aligned} \label{eqn:1-1} i \frac{{\partial}u_j }{{\partial}t} &= H(u_k) u_j, \\ H(u_k)u_j &= \left[ H_0 + V_{ext} + Q_H(x,{\mbox{\boldmath ${u}$}}) \right] u_j + \int_{{\mathbb{R}}^n} Q_F (x,y, {\mbox{\boldmath ${u}$}}) u_j (t, y) \, dy \qquad \text{for $1\le j \le N$}, \notag\end{aligned}$$ where $H_0= -\dfrac{1}{2} \Delta= -\dfrac{1}{2}\sum_{j=1}^n \frac{{\partial}^2}{{\partial}x_j^2}$ and ${\mbox{\boldmath ${u}$}}={\mbox{\boldmath ${u}$}}(t,x)=( u_j(t,x) )_{1\le j\le N}$ is an unknown function in $(t,x) \in {\mathbb{R}}\times {\mathbb{R}}^n$, and $$\begin{aligned} Q_H (x, {\mbox{\boldmath ${u}$}}) & = \int_{{\mathbb{R}}^n} V_{int}(x-y) \sum_{\substack{k=1 \\ k \not=j}}^N | u_{k}(t, y) |^2 \, dy, \\ &= V_{int}* \sum_{\substack{k=1 \\k\not=j}}^N | u_k (t, \cdot)|^2, \\ Q_F(x,y, {\mbox{\boldmath ${u}$}}) &= - V_{int}(x-y) \sum_{\substack{k=1 \\k\not=j}}^N \overline{u_k} (t, y) u_k (t, x). \end{aligned}$$ The non-linear Schrödinger equation we study in this paper is called the Hartree-Fock equation (HF equation). The terms $Q_H( x,{\mbox{\boldmath ${u}$}} ) u_j (t,x)$ and $\int Q_F(x,y,{\mbox{\boldmath ${u}$}}) u_j (t,y) dy$ are called the Hartree term and the Fock term, respectively. Next, we introduce some notations and assumptions on the potentials. Let $W^{k,p}({\mathbb{R}}^n)$ be the usual Sobolev space in $L^p({\mathbb{R}}^n)$. We abbreviate $W^{k,2}({\mathbb{R}}^n)$ as $H^k({\mathbb{R}}^n)$. The weighted $L^2$-space is denoted as $$L^{2,s}({\mathbb{R}}^n) =\left\{ u(x)\, : \, (1+|x|^2)^{s/2}u(x) \in L^2({\mathbb{R}}^n) , \, s\in {\mathbb{R}}\right\}.$$ Let $C_0^{\infty}({\mathbb{R}}^n)$ be the set of compactly supported smooth functions and ${\ensuremath{\mathcal{S}}}({\mathbb{R}}^n)$ be the set of rapidly decreasing functions on ${\mathbb{R}}^n$. The Fourier transform is denoted as $$\left( {\ensuremath{\mathcal{F}}}u \right)(\xi) = \widehat{u}(\xi) = \frac{1}{(2\pi)^{n/2}}\int_{{\mathbb{R}}^n} e^{-ix\cdot \xi} u(x) \, dx.$$ We define a function space ${\ensuremath{\mathcal{S}}}_0({\mathbb{R}}^n)$ as $${\ensuremath{\mathcal{S}}}_0 ({\mathbb{R}}^n) = \left\{ f\in {\ensuremath{\mathcal{S}}}({\mathbb{R}}^n)\, ; \, \widehat{f} \in C_0^{\infty}({\mathbb{R}}^n)\right\}.$$ The multiplication operator with a fixed function $V(x)$ is denoted as $V$. The unitary group of the self-adjoint operator $H_0$ with a domain $H^1({\mathbb{R}}^n)$ is denoted as $U_0(t)$ or $e^{-itH_0}$. Then, solutions of the free Schrödinger equation $i{\partial}_t v = H_0 v$ with initial data $v(0)=f$ is written as $v(t)=U_0(t)f = e^{-itH_0}f$. Consider solutions to the equation with $u_j(t) \longrightarrow U_0 (t) f_j^{\pm}$ as $t\to \pm \infty$ in some function space. We term the solutions scattering solution and $f_j^{\pm}$ scattering states. The scattering operator $S$ assigns the free state $U_0(t) f_j^-$ at $t=-\infty$ to the free state $U_0(t)f_j^+$ at $t=+\infty$, or equivalently $S: f_j^- \to f_j^+$. Our goal is to recover the external potential $V_{ext}(x)$ and the interaction potential $V_{int}(x)$ from the scattering operator $S$. Although we consider the three-dimensional inverse problem, to make it easy to explain a proof of our theorem, we will denote the spatial dimension by $n$ throughout this paper. Let potentials satisfy the following conditions. \[ass:interaction-2\] Let $n=3$. We assume that the real-valued function $V_{int}(x)$ has the following conditions: 1. $V_{int}(x) \ge 0$ and $$| V_{int}(x)| \le C | x |^{-2}, \qquad \text{or} \quad V_{int}\in L^{n/2}({\mathbb{R}}^n).$$ 2. $\nabla V_{int} \in L^{n/2}({\mathbb{R}}^n)$. 3. $V_{int}\in L^q ({\mathbb{R}}^n) + L^{\infty}({\mathbb{R}}^n)$ with $1\le q$. 4. $x\cdot \nabla V_{int}\in L^{\delta} ({\mathbb{R}}^n)+ L^{\infty}({\mathbb{R}}^n)$ with $1\le \delta$. 5. $V_{int}(-x)=V_{int}(x)$. 6. $|x|^2 V_{int}(x)$ is a non-increasing function of $|x|$. 7. $\sup_{x\in{\mathbb{R}}^n}(1+|x|)^{1+s}|V_{int}(x)|<\infty$ for $s>n/2$. \[ass:external\] Let $n=3$. We assume that the real-valued function $V_{ext}(x)$ has the following conditions: 1. $V_{ext}(x)\ge 0$. 2. $V_{ext}(x)$ is a homogeneous function of degree $-\gamma$: $$V_{ext}(\alpha x) = \alpha^{-\gamma}V_{ext}(x), \qquad \text{for $\alpha >0$ and $\gamma \ge 1$}.$$ 3. $|x|^2 V_{ext}(x)$ is a non-increasing function of $|x|$. 4. Zero is not an eigenvalue of the Schrödinger operator $H=H_0 + V_{ext}$. 5. $\nabla V_{ext}\in L^{\infty}({\mathbb{R}}^n)$ and $\Delta V_{ext} \in L^n({\mathbb{R}}^n)$. 6. $V_{ext}\in L^p ({\mathbb{R}}^n) + L^{\infty}({\mathbb{R}}^n)$ with $1\le p$. 7. $x\cdot \nabla V_{ext}\in L^{\delta} ({\mathbb{R}}^n) + L^{\infty}({\mathbb{R}}^n)$ with $1\le \beta$. 8. Let $\ell \ge 0$ be an arbitrary fixed integer. For $\delta > 3n/2 +1$, $p_0 > n/2$ and multi-indices $\alpha$ with $|\alpha|\le \ell$, $$\sup_{x\in{\mathbb{R}}^n}(1+|x|^2)^{\delta/2} \left( \int_{|x-y|\le 1} \left| D^{\alpha} V_{ext}(y) \right|^{p_0} \, dy \right)^{1/p_0} <\infty.$$ Here we have denoted the integer part of $x$ by $[x]$ and $D^{\alpha}=D_1^{\alpha_1}\cdots D_n^{\alpha_n}$, $D_j=-i\frac{{\partial}}{{\partial}x_j}$. A proof of the unique existence theorem on the scattering solution requires the $L^p$-decay of solutions of the Cauchy problems for time-dependent Schrödinger equations: $i\frac{{\partial}u}{{\partial}t}= Hu$. The condition $V_{ext}(x)\ge 0$ causes an absence of zero resonance for the Schrödinger operator $H$. Then, under the conditions 1, 4 and 8 in Assumption \[ass:external\], the $W^{k, p}({\mathbb{R}}^n)$-continuity of wave operators for the Schrödinger operator $H$ for any $k=0,1,\cdots, \ell$ and $1\le p \le \infty$ follows from the result developed by Yajima [@Yajima1995-2], which implies the $L^p$-decay of the solutions. The Assumption \[ass:external\] is rather complicated condition. We therefore give another conditions on $V_{ext}(x)$ simpler than Assumption \[ass:external\]. \[ass:external-2\] Let $n=3$. We assume that the real-valued function $V_{ext}(x)$ has the following conditions: 1. $V_{ext}(x)\ge 0$. 2. $V_{ext}(x)$ is a continuously differentiable function and a homogeneous function of degree $-\gamma$ with $\gamma \ge 2$. 3. Zero is not an eigenvalue of the Schrödinger operator $H=H_0 + V_{ext}$. 4. For $|\alpha | \le 2$ and $\kappa >(3n)/2 +3$, $$|D^{\alpha} V_{ext}(x) | \le \frac{C}{(1+|x|)^{\kappa}}.$$ The Assumption \[ass:external\] includes the Assumption \[ass:external-2\]. Indeed, letting $r=|x|$, we have $$V+\frac{1}{2} x\cdot \nabla V_{ext} = V+\frac{1}{2}r {\partial}_r V_{ext} = \frac{1}{2r} {\partial}_r (r^2 V_{ext}).$$ This identity means that the condition $V+\frac{1}{2}x\cdot \nabla V_{ext}\le 0$ replaces the condition 3 in Assumption \[ass:external\]. Recall the Euler’s homogeneous function theorem: if the function $V(x)$ on ${\mathbb{R}}^n$ is continuously differentiable function, then $V(x)$ is a positively homogeneous of degree $\gamma$ if and only if $V(x)$ satisfies $x\cdot \nabla V(x)=\gamma V(x)$. Then, in view of the Euler’s homogeneous function theorem, the condition 2 in Assumption \[ass:external\] gives $x\cdot V_{ext}=-\gamma V_{ext}\le 0$, which implies $$V_{ext}+\frac{1}{2} x \cdot \nabla V_{ext}=V_{ext}-\frac{\gamma}{2}V_{ext} \le 0$$ for $\gamma \ge 2$. Direct computations show that the function $V_{ext}(x)$ satisfies conditions 5-8 in Assumption \[ass:external\]. As it turns out in Section \[sec:2\], under the Assumption \[ass:interaction-2\] and the Assumption \[ass:external\], there exists a unique scattering solution $u_j(t,x)$ of with a condition $u_j(t,x)\to e^{-itH_0}\varphi_{j}$ as $t\to -\infty$ in $L^2({\mathbb{R}}^n)$ for any $\varphi_{j}\in {\ensuremath{\mathcal{S}}}_0$ , $j=1,\cdots , N$ sufficiently close to zero function. Put $$\begin{aligned} \psi_j(x)= (S{\mbox{\boldmath ${\varphi}$}})_{j} (x) & := \varphi_{j} (x) + \frac{1}{i} \int_{{\mathbb{R}}} e^{itH_0} P_j(x,{\mbox{\boldmath ${u}$}}) \, dt, \qquad \text{$ j= 1, 2, \cdots, N$} \label{eqn:1-2}\\ P_j(x,{\mbox{\boldmath ${u}$}}) &= \left( Q_{H}(x,{\mbox{\boldmath ${u}$}}) + V_{ext}(x) \right) u_j(t,x) + \int_{{\mathbb{R}}^n}Q_{F}(x,y,{\mbox{\boldmath ${u}$}}) u_j (t,y)\, dy, \notag\end{aligned}$$ where ${\mbox{\boldmath ${u}$}} (t,x)$ is the scattering solution to . Then it will be shown that $u_j(x,t) \to e^{-itH_0}\psi_j(x)$ as $t\to +\infty$ in $L^2({\mathbb{R}}^n)$. Therefore, the operator $S$ defined as represents a scattering operator for the HF equation . The inverse problem considered in this paper is to determine the interaction and the external potentials from the scattering operator defined in terms of the scattering solution to the Hartree-Fock equation . Our main result is \[thm:1-1\] Let $n=3$. Assume that $V_{int}(x)$ and $V_{ext}(x)$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Then the potentials $V_{int}$, $V_{ext}$ are uniquely determined by $S$. We remark that our proof gives an explicit way to reconstruct the interaction and the external potentials from the asymptotic behavior of the function $<(S-I){\mbox{\boldmath ${\Psi}$}}_v, {\mbox{\boldmath ${\Psi}$}}_v>_{L^2}$ at $|v|\to \infty $, where ${\mbox{\boldmath ${\Psi}$}}_v(x)=e^{iv\cdot x}{\mbox{\boldmath ${\varphi}$}}(x)$ and $<\, ,\, >_{L^2}$ is the inner product in $L^2({\mathbb{R}}^n)$. Methods ------- Because the high velocity limit (HVL) of the scattering operator (Enss and Weder [@Enss-Weder1995]) makes it possible to recover the Schrödinger operator with the potentials, it has become an important tools for studying the inverse scattering problems for time-dependent Schrödinger equations (Weder [@Weder1997-2]; Valencia and Weder [@Valencia-Weder2012], and references therein; Adachi and Maehara [@Adachi-Maehara2007]; Adachi, Fujiwara and Ishida [@Adachi-Fujiwara-Ishida2013]; Adachi et al. [@Adachi.et2011]; Ishida [@Ishida2019]). According to recent researches for inverse nonlinear scattering ([@Watanabe2018] and [@Watanabe2019]), the method of the HVL also make it possible to recover nonlinearities. On the other hand, the small amplitude limit (SAL) of the scattering operator (Weder [@Weder1997; @Weder1999; @Weder1999_2; @Weder2000_1; @Weder2000_2; @Weder2000_3; @Weder2001_1; @Weder2001_2; @Weder2002; @Weder2003]) make it possible to recover both the potential and nonlinearities. This method of the SAL, however, fails to reconstruct the general interaction potential in the HF equation . Details of the method of SAL are described briefly as follows. Because the HF equation is a non-linear equation, our inverse problem is a non-linear inverse problem of recovering the linear part—zero-order coefficient— and non-linear part. Such non-linear inverse scattering problem have been extensively studied by Weder. It was proved that the SAL of the scattering operator uniquely determines coefficients—the coefficient of the linear part and coefficients of the power type non-linearity. In other words, this method of the SAL is an asymptotic analysis of the scattering operator $S(\varepsilon \varphi)$ as $\varepsilon \to 0$. In particular, Weder [@Weder1997] showed that the Fréchet derivative of the scattering operator $S$ uniquely determines the linear scattering operator for the Schrödinger operator $H=H_0+V_{ext}$. Thus, the non-linear inverse scattering problem of recovering the potential $V_{ext}$ is reduce to the problem of recovering the Schrödinger operator $H$ from the linear scattering operator. This method of Weder was applied to inverse problems for Hartree equations ([@Watanabe0; @Watanabe2007-1]). Here, we briefly review the method to recover the coefficient function $V_{ext}$ of the linear term in the case of $N=2$ and $u_1=u_2$ to the equation . Following Weder [@Weder1997], the scattering operator $S$ is defined in terms of wave operators $W_{\pm}=\lim_{t\to \pm\infty}e^{itH} e^{-itH_0}$ for the Schrödinger operator $H=H_0+V_{ext}$: $$\begin{aligned} S & = W_+^* S_N W_-, \\ (S_N {\mbox{\boldmath ${\varphi}$}})_j(x)& = \varphi_j(x) +\frac{1}{i} \int_{{\mathbb{R}}} e^{itH}N_j(x, {\mbox{\boldmath ${u}$}})\, dt, \\ N_j(x, {\mbox{\boldmath ${u}$}}) &= Q_H (x, {\mbox{\boldmath ${u}$}})u_j(t,x) + \int_{{\mathbb{R}}^n} Q_F (x,y, {\mbox{\boldmath ${u}$}})u_j(t,y)\, dy, \end{aligned}$$ and the scattering operator $S$ has expansion $$S(\varepsilon{\mbox{\boldmath ${\varphi}$}}) = \varepsilon S_{V_{ext}} {\mbox{\boldmath ${\varphi}$}} + O(\varepsilon^3)$$ as $\varepsilon \to 0$ in $H^1({\mathbb{R}}^n)$, where $S_{V_{ext}}$ denotes the scattering operator for the Schrödinger operator $H$. We shall term this expansion “$\varepsilon$-expansion”. This ${\varepsilon}$-expansion indicates that the scattering operator $S$ uniquely determines the scattering operator $S_{V_{ext}}$. As is well-known (see, e.g., [@Enss-Weder1995]), the operator $S_{V_{ext}}$ uniquely determines the external potential $V_{ext}$. Then we can construct wave operators $W_+=\lim_{t\to \infty}e^{itH} e^{-itH_0}$ and $W^*_-$. Defining $S_F$ as $S_F = W_+ S W_-^*$, the small amplitude limit of the function $\frac{1}{\varepsilon^3}(S_F-I)(\varepsilon{\mbox{\boldmath ${\varphi}$}})$ uniquely determines the interaction potential of the form $V_{int}(x)= \lambda |x|^{-\sigma}$ (see [@Watanabe2007]). This method fails to reconstruct the general interaction potential due to the difficulty of analysis to the operator $e^{itH}$. In order to overcome this difficulty, we give another representation of the scattering operator with no operator $H$. Recently, the general interaction potential is successfully reconstructed by means of the method of the HVL of the scattering operator in the case of the Hartree-Fock equation with no external potential $V_{ext}$ ([@Watanabe2019]). Our representation permits the high-velocity analysis of the scattering operator to reconstruct both the interaction potential and the external potential. More precisely, analyzing the asymptotic expansion of the function $<(S-I){\mbox{\boldmath ${\Psi}$}}_v, {\mbox{\boldmath ${\Psi}$}}_v>_{L^2}$ at $|v|\to \infty$ ($v$-expansion) shows that the leading term of the expansion uniquely determines the Fourier transform of the interaction potential $V_{int}$ and the second term of the expansion uniquely determines the $X$-ray transform of the external potential $V_{ext}$. This method of the $v$-expansion dose not require construction of wave operators $W_{\pm}$ to reconstruct the potentials. Thus the method of the $v$-expansion is simpler and less complicated than the method of the ${\varepsilon}$-expansion. In order to prove that the operator defined as is a scattering operator to the equation , we require a time-decay $L^{\infty}$-estimate on the solution to the Cauchy problem of the equation . Such estimate on the solution to the Hartree type equation has been extensively investigated for the case where the potential has the form $\lambda |x|^{-\gamma}$ for some constants $\lambda$ and $\gamma$ (see, e.g., Wada [@Wada]; Hayashi and Naumkin [@Hayashi-Naumkin98]; Hayashi and Ozawa [@Hayashi-Ozawa1988]). Few researchers have addressed the problem of the time-decay $L^{\infty}$-estimate on the solution to the Hartree-Fock equation with general interaction and external potentials. Making use of the pseudo-conformal conservation law and the Gagliardo-Nirenberg inequality, with the help of assumptions on the potentials, we give a $L^{\infty}$-estimate on the solution to the equation . This paper is organized as follows: Section 2 proves that the operator $S$ defined as is the scattering operator for the Hartree-Fock equation , after proving a time-decay $L^{\infty}$-estimate on the solution to the Cauchy problem for the equation . We give the asymptotic expansion of the scattering operator with the high-velocity initial states in Section 3. Section 4 is devoted to reconstructions of the interaction potential and the external potential. Representation of the scattering operator {#sec:2} ========================================= This section shows that the scattering operator for the Hartree-Fock equation has a representation . We first recall that the unique existence of the scattering solution of Hartree equations with the external potential are studied in [@Watanabe0]. This result can be easily applicable to the Hartree-Fock equation . Assume that potentials $V_{ext}$ and $V_{int}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Then there exists $\varepsilon_0 >0$ such that the equation satisfying the condition $u_j(t)\to e^{-itH_0}\varphi_j$ as $t\to -\infty$ in $L^2({\mathbb{R}}^n)$ for $\varphi_j \in L^2({\mathbb{R}}^n)$ with $\| \varphi_j \|_{L^2}\le \varepsilon_0$, $j=1,2, \cdots, N,$ has a unique solution $$u_j \in {\ensuremath{\mathcal{W}}} = L^3({\mathbb{R}}: L^q({\mathbb{R}}^n)) \cap L^{\infty}({\mathbb{R}}: L^2({\mathbb{R}}^n)), \qquad \frac{1}{q}=\frac{1}{2}-\frac{2}{3n}.$$ Moreover, there exists a unique $\psi_j\in L^2({\mathbb{R}}^n)$ such that $u_j(t)\to e^{-itH_0}\psi_j$ as $t\to \infty$ in $L^2({\mathbb{R}}^n)$. In what follows, because we are interested in the inverse scattering problem, we consider the scattering for high-velocity initial states. Let $\varphi\in {\ensuremath{\mathcal{S}}}_0({\mathbb{R}}^n)$. Then the function $\Phi_v(x)=e^{iv\cdot x}\varphi(x)$ has a compact velocity support in the momentum space around $v$, due to the fact that $\widehat{\Phi}_v (\xi)=\widehat{\varphi}(\xi-v)$. The main result of this section is \[thm:2-1\] Let $n=3$. Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Let $u_j$, $j=1,2, \cdots, N$, be the scattering solutions to with initial scattering states $\varphi_j\in {\ensuremath{\mathcal{S}}}_0$. Put $$\begin{aligned} \psi_j=(S{\mbox{\boldmath ${\varphi}$}})_j (x) & = \varphi_j (x) + \frac{1}{i} \int_{{\mathbb{R}}} e^{itH_0} P_j(x,{\mbox{\boldmath ${u}$}}) \, dt, \label{eqn:2-1}\\ P_j(x,{\mbox{\boldmath ${u}$}}) &= \left( Q_{H}(x,{\mbox{\boldmath ${u}$}}) + V_{ext}(x) \right) u_j(t,x) + \int_{{\mathbb{R}}^n}Q_{F}(x,y,{\mbox{\boldmath ${u}$}}) u_j (t,y)\, dy. \notag\end{aligned}$$ Then we have $$\| u_j(t) - e^{-itH_0}\psi_j \|_{L^2} \longrightarrow 0 \qquad \text{as $t\to \infty$}.$$ In Subsection \[subsec:2-1\], we prepare some lemmas to prove Theorem \[thm:2-1\]. Subsection \[subsec:2-2\] is devoted to state a $L^{\infty}$ estimate on the solution to the equation and its proof. Theorem \[thm:2-1\] is proved in Subsection \[subsec:2-3\]. Preliminary lemmas {#subsec:2-1} ------------------ \[lem:Gagliardo\] Let $q,r$ be any number satisfying $1\le q,r \le \infty$ and let $j,m$ be any integers satisfying $0\le j <m$. Then for any $u\in W^{m,r}({\mathbb{R}}^n) \cap L^q({\mathbb{R}}^n)$, we have $$\label{eqn:Gagliardo} \sum_{|\alpha |=j} \left\| D^{\alpha} u \right\|_{L^p} \le M \sum_{|\beta |=m} \left\| D^{\beta} u \right\|_{L^r}^a \left\| u \right\|_{L^q}^{1-a},$$ where $1/p= j/m + a(1/r-m/n)+(1-a)/q$ for all $a\in[j/m, 1]$ with the following exception: if $m-j-(n/r)$ is a non-negative integer, then is asserted for $a=j/m$, and where $M$ is a positive constant depending only on $n,m,j,q,r,a$. The proof of Lemma \[lem:Gagliardo\] will be found in Friedman [@Friedman1969]. \[lem:2-2\] Let $1/q=1/2-2/(3n)$. Assume that the potential $V_{int}$ satisfies Assumption \[ass:interaction-2\]. Then for any $u_j\in L^{q}({\mathbb{R}}^n)$, $j=1,2,\cdots, N$, we have $$\| Q_H(\cdot, {\mbox{\boldmath ${u}$}})u_j \|_{L^2}+ \left\| \int_{{\mathbb{R}}^n} Q_F(\cdot,y, {\mbox{\boldmath ${u}$}})u_j(y,t)\, dy \right\|_{L^2} \le C \| {\mbox{\boldmath ${u}$}} \|^3_{L^{q}},$$ where $C$ is a positive constant. Following Mochizuki [@Mochizuki Lemma 4.6.], we obtain $$\begin{aligned} \| Q_H(\cdot, {\mbox{\boldmath ${u}$}})u_j \|_{L^2} & \le C \sum_{k=1, k\not=j}^N \|u_k\|_{L^{2a}} \| u_k\|_{L^{2b}} \|u_j\|_{L^{2h}}, \\ \left\| \int_{{\mathbb{R}}^n} Q_F (\cdot, y,{\mbox{\boldmath ${u}$}}) u_j(y,t)\, dy \right\|_{L^2} & \le C \sum_{k=1, k\not=j}^N \left\| (V_{int}* u_j \overline{u_k})u_k \right\|_{L^2} \\ & \le \sum_{k=1, k\not=j}^N \|u_j\|_{L^{2a}} \| \overline{u_k} \|_{L^{2b}} \|u_k\|_{L^{2h}}, \\ \end{aligned}$$ where the positive constants $a, b, h$ satisfy $2a=2b=2h=q$, due to the Hölder’s inequality, and the Hardy-Littlewood-Sobolev inequality in the case where $V_{int}$ satisfies $|V_{int}(x)|\le C|x|^{-2}$, or the Young’s inequality in the case where $V_{int}$ satisfies $V_{int}\in L^{n/2}({\mathbb{R}}^n)$. In view of the inequality $\alpha^2 \beta < 2/3 ( \alpha^3+\beta^3)$, one has $$\begin{aligned} \| Q_H(\cdot, {\mbox{\boldmath ${u}$}})u_j \|_{L^2}+ \left\| \int_{{\mathbb{R}}^n} Q_F(\cdot,y)u_j(y,t)\, dy \right\|_{L^2} & \le C \left( \sum_{k=1, k\not=j}^N \|u_k\|_{L^{q}}^{3} + \|u_j\|_{L^{q}}^3 \right) = C \| {\mbox{\boldmath ${u}$}} \|^3_{L^{q}}.\end{aligned}$$ This completes the proof. \[lem:Enss-Weder\] Let $n\ge 2$ and $s>1$ and put $\Phi_v (x)=e^{iv\cdot x}\varphi(x)$. Assume that $q$ is a compact operator from $L^2({\mathbb{R}}^n)$ to $L^{2,s}({\mathbb{R}}^n)$. Then for any $\varphi \in {\ensuremath{\mathcal{S}}}_0$, there exist a positive constant $C$ such that $$\int_{-\infty}^{\infty} \| q U_0(t) \Phi_v \|_{L^2} \, dt \le \frac{C}{|v|}$$ for $|v|$ large enough. The proof of Lemma \[lem:Enss-Weder\] will be found in [@Enss-Weder1995 Lemma 2.2] and its proof. Let ${\mbox{\boldmath ${u}$}}(t)$ be the scattering solution and let $\Omega_-$ be the wave operator which assigns the free state $U_0(t){\mbox{\boldmath ${f}$}}^-$ to the interacting state ${\mbox{\boldmath ${u}$}}(t)=U(t){\mbox{\boldmath ${\psi}$}}$ (see, e.g., Strauss [@Strauss1974]). In particular, $\Omega_- \, : \, H^1 \ni {\mbox{\boldmath ${f}$}}^-\to {\mbox{\boldmath ${u}$}}(0)\in H^1$. \[lem:2-4\] Let $n= 3$ and ${\mbox{\boldmath ${\Phi}$}}_v(x)=(e^{iv\cdot x}\varphi_j (x))_{1\le j\le N}$, $j=1,2, \cdots, N$. Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Then for any $\varphi_j \in {\ensuremath{\mathcal{S}}}_0$, we have $$\| ((\Omega_- -I)U_0(t){\mbox{\boldmath ${\Phi}$}}_v)_j \|_{L^2}=O(|v|^{-1})$$ as $|v|\to \infty$ uniformly in $t\in {\mathbb{R}}$. The proof of Lemma \[lem:2-4\] is quite the same as [@Watanabe2019 Lemma 2.3]. Time-decay estimate of solutions {#subsec:2-2} -------------------------------- \[prop:2-1\] Let $n=3$ and $a=([n/2]+1)^{-1}(n/2)=3/4$. Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Then the solutions $u_j(t)$, $j=1,2, \cdots, N$ of with initial states $\varphi_j \in {\ensuremath{\mathcal{S}}}_0$ satisfies $$\| u_j (t) \|_{L^{\infty}} \le C t^{-n/2}(\log t)^{a},$$ for $t\ge e$, where $C>0$. Let us prepare for proving the Proposition \[prop:2-1\]. We denote $V(tx)$ by $V^t(x)$. Putting $$v_j(t)=(it)^{n/2} e^{-it|x|^2/2} u_j(t, tx)$$ gives a equation (see e.g., Wada [@Wada]) $$\begin{aligned} i\frac{{\partial}v_j}{{\partial}t} &= -\frac{1}{2t^2}\Delta v_j + f_j (t, {\mbox{\boldmath ${v}$}}), \label{eqn:2-2-1} \\ f_j(t, {\mbox{\boldmath ${v}$}})&=V^t_{ext}(x)v_j(t) + \sum_{k=1}^N \left\{ (V_{int}^t * |v_k|^2)v_j - (V_{int}^t*v_j \overline{v_k})v_k \right\}. \label{eqn:2-2-2}\end{aligned}$$ We note that the $L^2$-conservation law for $v_j(t)$ holds (see Isozaki [@Isozaki1983]): $$\label{eqn:2-2-3} \| v_j(t) \|_{L^2} = \| u_j (t) \|_{L^2} = \| u_j (0) \|_{L^2}, \qquad j=1,2, \cdots , N.$$ Thanks to the Gagliardo-Nirenberg inequality (Lemma \[lem:Gagliardo\]) and , we have $$\begin{aligned} \| u_j (t) \|_{L^{\infty}} & \le t^{-n/2} \| v_j(t) \|_{L^{\infty}} \notag \\ & \le C t^{-n/2} \|v_j(t)\|_{L^2}^{1-a} \| \Delta v_j(t) \|_{L^2}^{a} \notag \\ & = C t^{-n/2} \| \Delta v_j (t) \|_{L^2}^{a} \label{eqn:2-2-4}\end{aligned}$$ for some $C>0$. Therefore, estimating $\| \Delta v_j (t) \|_{L^2}$ gives the proof of Proposition \[prop:2-1\]. In order to estimate $\| \Delta v_j (t) \|_{L^2}$, we need \[lem:2-2-1\] Let $n=3$. Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Then $v_j(t)$ satisfies $$\sum_{j=1}^N \| \nabla v_j (t) \|_{L^2}^2 \le C$$ for some $C>0$ and for any $t>0$. We first note that the pseudo-conformal conservation laws for $v_j(t)$ (see, e.g., Cazenave [@Cazenave2003 Section 7.2]) holds: $$\sum_{j=1}^N \left\| \nabla v_j (t) \right\|_{L^2}^2 + t^2 G(t, {\mbox{\boldmath ${v}$}}) = \sum_{j=1}^N \left\| x u_j(0) \right\|_{L^2}^2 + \int_0^t s \Theta (s, {\mbox{\boldmath ${v}$}})\, ds, \label{eqn:2-2-5}$$ where $$\begin{aligned} G(t, {\mbox{\boldmath ${v}$}}) & = \sum_{j=1}^N \int_{{\mathbb{R}}^n} \frac{1}{2} V^t_{ext}(x) |v_j(t)|^2\, dx + \sum_{j,k=1}^N \frac{1}{4} \int_{{\mathbb{R}}^n} (V^t_{int}*|v_k(t)|^2)|v_j(t)|^2\,dx \\ & \hspace{1em} - \sum_{j,k=1}^N \frac{1}{4} \int_{{\mathbb{R}}^n} (V^t_{int}*v_j(t)\overline{v_k}(t))v_k(t) \overline{v_j}(t)\,dx, \\ \Theta(t, {\mbox{\boldmath ${v}$}}) &= \sum_{j=1}^N \int_{{\mathbb{R}}^n} \left( V^t_{ext}(x) +\frac{1}{2} x\cdot (\nabla V^t_{ext})(x) \right) |v_j(t)|^2\, dx \\ & \hspace{1em} + \sum_{j,k=1}^N \int_{{\mathbb{R}}^n} \left( \left(V^t_{int}+\frac{1}{2}x\cdot \nabla V^t_{int} \right)*|v_k(t)|^2\right)|v_j(t)|^2\,dx \\ & \hspace{1em} - \sum_{j,k=1}^N \int_{{\mathbb{R}}^n} \left( \left(V^t_{int}+\frac{1}{2}x\cdot \nabla V^t_{int} \right)*v_j(t)\overline{v_k}(t)\right)v_k(t) \overline{v_j}(t)\,dx. \end{aligned}$$ In view of the Assumption \[ass:external\], Assumption \[ass:interaction-2\] and the Cauchy-Schwarz inequality, we find that $G(t, {\mbox{\boldmath ${v}$}})\ge 0$ and $\Theta (t, {\mbox{\boldmath ${v}$}}) \le 0$ for $t \ge 0$. It therefore follows from that $$\frac{d}{dt} \left( \sum_{j=1}^N \| \nabla v_j(t) \|_{L^2}^2 + t^2 G(t, {\mbox{\boldmath ${v}$}}) \right) = t \Theta (t, {\mbox{\boldmath ${v}$}}) \le 0$$ for $t\ge 0$, which implies that $\sum_{j=1}^N \| \nabla v_j(t) \|_{L^2}^2 \le C$. [*Proof of Proposition \[prop:2-1\].*]{} We are now in a position to prove Proposition \[prop:2-1\]. In the proof, we abbreviate the $L^p$-norm of a function $f$ as $\| f \|_p$. Applying $\Delta$ to the equation , one has $$\label{eqn:2-2-6} i\frac{{\partial}}{{\partial}t}\Delta v_j(t) = -\frac{1}{2t^2}\Delta^2 v_j(t) + \Delta f_j (t, {\mbox{\boldmath ${v}$}}).$$ Multiplying by $\Delta \overline{v_j}$ and integrating the imaginary part over ${\mathbb{R}}^n$, we have $$\frac{1}{2} \frac{d}{dt} \left\| \Delta v_j (t) \right\|_{L^2}^2 = {\rm Im} \int_{{\mathbb{R}}^n} \Delta f_j(t,{\mbox{\boldmath ${v}$}}) \Delta \overline{v_j} (t) \, dx.$$ Due to the fact that integrals $$\begin{aligned} &{\rm Im} \int_{{\mathbb{R}}^n} V^t_{ext}(x) | \Delta v_j(t) |^2 \, dx, \quad {\rm Im} \sum_{j,k=1}^N\int_{{\mathbb{R}}^n} (V^t_{int}*|v_k(t)|^2) | \Delta v_j(t) |^2 \, dx, \\ &{\rm Im} \sum_{j,k=1}^N\int_{{\mathbb{R}}^n} (V^t_{int}*v_j(t) \overline{v_k}(t)) \Delta v_k(t) \Delta \overline{v_j}(t) \, dx\end{aligned}$$ vanish, one gets $$\frac{1}{2} \sum_{j=1}^N \frac{d}{dt} \left\| \Delta v_j (t) \right\|_{L^2}^2 = \sum_{\ell=1}^6 I_{\ell}(t),$$ where $$\begin{aligned} I_1(t) &= {\rm Im} \sum_{j=1}^N \int_{{\mathbb{R}}^n} (\Delta V^t_{ext})(x) v_j(t) \Delta \overline{v_j}(t) \, dx, \\ I_2(t) &= 2 {\rm Im} \sum_{j=1}^N \int_{{\mathbb{R}}^n} (\nabla V^t_{ext})(x)\cdot \nabla v_j(t) \Delta \overline{v_j}(t) \, dx, \\ I_3(t) &= {\rm Im} \sum_{j,k=1}^N \int_{{\mathbb{R}}^n} v_j(t) \Delta \overline{v_j}(t) \left( \Delta ( V^t_{int}*|v_k(t)|^2)\right)(x) \, dx, \\ I_4(t) &= 2 {\rm Im} \sum_{j,k=1}^N \int_{{\mathbb{R}}^n} \Delta \overline{v_j}(t) \left( \nabla v_j(t)\cdot \nabla ( V^t_{int}*|v_k(t)|^2)\right)(x) \, dx, \\ I_5(t) &= - {\rm Im} \sum_{j,k=1}^N \int_{{\mathbb{R}}^n} v_k(t) \Delta \overline{v_j}(t) \left( \Delta ( V^t_{int}*v_j(t) \overline{v_k}(t))\right)(x) \, dx, \\ I_6(t) &= -2 {\rm Im} \sum_{j,k=1}^N \int_{{\mathbb{R}}^n} \Delta \overline{v_j}(t) \left( \nabla v_k (t)\cdot \nabla ( V^t_{int}*v_j(t)\overline{v_k}(t)) \right)(x) \, dx.\end{aligned}$$ We shall show that $|I_1|, |I_2| \le C t^{-\gamma} \sum_{j=1}^N \| \Delta v_j(t) \|_{L^2}$. Due to the Assumption \[ass:external\] that $V_{ext}(x)$ is a homogeneous function of degree $-\gamma$, one has $$(\nabla V^t_{ext})(x)=t^{-\gamma} (\nabla V_{ext})(x), \qquad (\Delta V^t_{ext}) (x) = t^{-\gamma} (\Delta V_{ext})(x).$$ By using the Schwartz inequality, Hölder inequality, Gagliardo-Nirenberg inequality (Lemma \[lem:Gagliardo\]) and Lemma \[lem:2-2-1\], we obtain $$\begin{aligned} | I_1 (t) | & \le t^{-\gamma} \sum_{j=1}^N \| (\Delta V_{ext}) v_j(t)\|_{2} \| \Delta v_j (t) \|_{2} \\ & \le t^{-\gamma} \| \Delta V_{ext} \|_n \sum_{j=1}^N \| v_j(t)\|_{\frac{2n}{n-2}} \| \Delta v_j (t) \|_2 \\ & \le C t^{-\gamma} \| \Delta V_{ext} \|_n \sum_{j=1}^N \| \nabla v_j(t)\|_{2} \| \Delta v_j (t) \|_2 \\ & \le C t^{-\gamma} \sum_{j=1}^N \| \Delta v_j (t) \|_2\end{aligned}$$ and $$\begin{aligned} | I_2 (t) | & \le t^{-\gamma} \| \nabla V_{ext} \|_{\infty} \sum_{j=1}^N \| \nabla v_j (t) \|_2 \| \Delta v_j (t) \|_{2} \\ & \le C t^{-\gamma} \sum_{j=1}^N \| \Delta v_j (t) \|_2.\end{aligned}$$ We next claim that $| I_{\ell} (t) | \le C t^{-1} \sum_{j=1}^N \| \Delta v_j (t)\|_2$, $\ell = 3,4,5,6$. It is easy to check that $$\begin{aligned} \int_{{\mathbb{R}}^n} v_j (t) \Delta \overline{v_j}(t) \left( \Delta ( V^t_{int}*|v_k(t)|^2 )\right)(x)\, dx &= \int_{{\mathbb{R}}^n} v_j (t) \Delta \overline{v_j}(t) \sum_{m=1}^3 ( {\partial}_{x_m}V^t_{int}* \overline{v_k}(t) {\partial}_{x_m}v_k(t))(x)\, dx \\ & \hspace{1em} + \int_{{\mathbb{R}}^n} v_j (t) \Delta \overline{v_j}(t) \sum_{m=1}^3 ( {\partial}_{x_m}V^t_{int}* v_k(t) {\partial}_{x_m}\overline{v_k}(t))(x)\, dx.\end{aligned}$$ By using the Schwartz inequality, Hölder inequality, Young inequality and Gagliardo-Nirenberg inequality (Lemma \[lem:Gagliardo\]), we have $$\begin{aligned} &\left| \int_{{\mathbb{R}}^n} v_j (t) \Delta \overline{v_j}(t) \sum_{m=1}^3 ( {\partial}_{x_m}V^t_{int}* \overline{v_k}(t) {\partial}_{x_m}v_k(t))(x)\, dx \right| \le \sum_{m=1}^3 \|\left\{ {\partial}_{x_m}V^t_{int}*\overline{v_k}(t) {\partial}_{x_m}v_k (t)\right\}v_j(t) \|_2 \| \Delta \overline{v_j} (t) \|_2 \\ & \hspace{3em} \le \| \Delta \overline{v_j}(t) \|_2 \| v_j(t) \|_{\frac{2n}{n-2}} \sum_{m=1}^3 \| {\partial}_{x_m}V^t_{int}*\overline{v_k}(t) {\partial}_{x_m}v_k (t) \|_n \\ & \hspace{3em} \le C \| \Delta \overline{v_j}(t) \|_2 \| v_j(t) \|_{\frac{2n}{n-2}} \sum_{m=1}^3 \| {\partial}_{x_m}V^t_{int} \|_{\frac{n}{2}} \| \overline{v_k}(t) {\partial}_{x_m}v_k (t) \|_{\frac{n}{n-1}} \\ & \hspace{3em} \le C \| \Delta \overline{v_j}(t) \|_2 \| v_j(t) \|_{\frac{2n}{n-2}} \sum_{m=1}^3 \| {\partial}_{x_m}V^t_{int} \|_{\frac{n}{2}} \| \overline{v_k}(t) \|_{\frac{2n}{n-2}} \| {\partial}_{x_m}v_k (t) \|_{2} \\ & \hspace{3em} \le C t^{-1} \| \Delta \overline{v_j}(t) \|_2 \| \nabla v_j (t) \|_{2} \| \nabla \overline{v_j}(t) \|_2 \sum_{m=1}^3 \| {\partial}_{x_m}V_{int} \|_{\frac{n}{2}} \| {\partial}_{x_m}v_k (t) \|_2 \\ & \hspace{3em} \le C t^{-1} \| \Delta v_j (t) \|_2,\end{aligned}$$ which implies that $|I_3 (t) |, |I_5 (t) | \le C t^{-1}\sum_{j=1}^N \| \Delta v_j (t) \|_2$. Here we have used the fact that $\|V^t\|_p = t^{-n/p}\|V\|_p$. Similarly, one has $$\begin{aligned} &\left| \int_{{\mathbb{R}}^n} \Delta \overline{v_j}(t) \nabla v_j (t) \cdot \nabla \left( ( V^t_{int}* |v_k(t)|^2 ) \right)(x) \, dx \right| \le \int_{{\mathbb{R}}^n} \left| \Delta \overline{v_j}(t) \nabla v_j \cdot (\nabla V^t_{int}*v_k (t) \overline{v_k}(t))(x) \right| \, dx \\ & \hspace{3em} \le \| \Delta \overline{v_j}(t) \|_2 \| \nabla v_j(t) \|_{2} \| \nabla V^t_{int}* v_k (t) \overline{v_k}(t) \|_{\infty} \\ & \hspace{3em} \le C \| \Delta \overline{v_j}(t) \|_2 \| \nabla v_j(t) \|_{2} \| \nabla V^t_{int}\|_{\frac{n}{2}} \| v_k (t) \overline{v_k}(t) \|_{\frac{n}{n-2}} \\ & \hspace{3em} \le t^{-1} \| \Delta \overline{v_j}(t) \|_2 \| \nabla v_j(t) \|_{2} \| \nabla V_{int} \|_{\frac{n}{2}} \| v_k (t) \|_{\frac{2}{n-2}}^2 \\ & \hspace{3em} \le t^{-1} \| \Delta \overline{v_j}(t) \|_2 \| \nabla v_j(t) \|_{2} \| \nabla V_{int} \|_{\frac{n}{2}} \| \nabla v_k (t) \|_{2}^2, \\\end{aligned}$$ which implies that $|I_4 (t) |, |I_6 (t) | \le C t^{-1}\sum_{j=1}^N \| \Delta v_j (t) \|_2$. In view of the inequalities for $|I_{\ell}(t) |$, $\ell=1,2, \cdots 6$, we obtain a differential inequality: $$\frac{1}{2} \sum_{j=1}^N \frac{d}{dt} \| \Delta v_j (t) \|_2^2 \le C(t^{-\gamma}+t^{-1}) \left( \sum_{j=1}^N \| \Delta v_j (t) \|_2^2 \right)^{1/2}$$ for $t>0$, where $C>0$ and $\gamma \ge 1$. This differential inequality implies that $$\sum_{j=1}^N \| \Delta v_j(t) \|_2 \le C ( \log t +1) \le C \log t$$ for $t\ge e$. Hence, from , we obtain the desired estimate. $\Box$ Proof of Theorem \[thm:2-1\] {#subsec:2-3} ---------------------------- Let $u_j \in {\ensuremath{\mathcal{W}}}$ be a scattering solution to with initial states $e^{-itH_0}\varphi_j$ at $t=-\infty$. Because the scattering solution satisfies the integral equation $$u_j(t)=e^{-itH_0} \varphi_{j}+\frac{1}{i} \int_{-\infty}^{t} e^{-i(t-\tau)H_0}P_{j}(x, {\mbox{\boldmath ${u}$}} ), d\tau,$$ we obtain $$\begin{aligned} \left( e^{-itH_0}\psi_j\right)(x) &= \left( e^{-itH_0}\varphi_j\right)(x) + \frac{1}{i} \int_{{\mathbb{R}}} e^{-i(t-\tau)H_0}P_j(x,{\mbox{\boldmath ${u}$}})\, d\tau \\ &= u_j(t,x) + \frac{1}{i} \int_{t}^{\infty}e^{-i(t-\tau)H_0}P_j(x,{\mbox{\boldmath ${u}$}})\, d\tau .\end{aligned}$$ Thanks to Proposition \[prop:2-1\] and Lemma \[lem:2-2\], for $t\ge e$, one has $$\begin{aligned} \| u_j(t) - e^{-itH_0}\psi_j \|_{L^2} & \le \int_t^{\infty} \| e^{-i(t-\tau)H_0} V_{ext}u_j(\tau) \|_{L^2} \, d\tau \\ & \hspace{1em} + \int_t^{\infty} \left\| e^{-i(t-\tau)H_0}\left( Q_H(\cdot, u)u_j(\tau) + \int_{{\mathbb{R}}^n} Q_F(x,y,{\mbox{\boldmath ${u}$}})u_j(y,\tau)\, dy \right) \right\|_{L^2} \, d\tau \\ & \le \int_t^{\infty} \| V_{ext} u_j(\tau) \|_{L^2}\, d\tau + C\int_{t}^{\infty}\| u_j(\tau) \|_{L^{q}}^3\, d\tau \\ & \le C_1 \| V_{ext} \|_{L^2} \int_t^{\infty} \tau^{-3/2}(\log \tau )^{3/4} \, d\tau + C_2 \int_{t}^{\infty} \| u_j( \tau ) \|_{L^{q}}^3 \, d\tau \\ & \longrightarrow 0 \qquad \text{as $t\to \infty$}\end{aligned}$$ for some $C_1$, $C_2>0$, due to the fact that $u_j \in L^3({\mathbb{R}}; L^{q})$ and $$\int_{t}^{\infty} \tau^{-3/2}(\log \tau )^{3/4}\, d\tau < 2^{7/4}\int_0^{\infty}e^{-\mu} \mu^{3/4}\, d\mu =2^{7/4} \Gamma\left( \frac{7}{4} \right),$$ where $\Gamma(s)$ is the Gamma function. This completes the proof. $\Box$ Asymptotics of the scattering operator {#sec:3} ====================================== Let ${\mbox{\boldmath ${\Phi}$}}_v (x) = e^{iv\cdot x}{\mbox{\boldmath ${\varphi}$}} (x)$. The components of the vector ${\mbox{\boldmath ${\Phi}$}}_v$ is denoted as $({\mbox{\boldmath ${\Phi}$}}_v)_j$ $(j=1,2,\cdots, N)$. Put $I_j(v)=i< ((S-I){\mbox{\boldmath ${\Phi}$}}_v)_j, ({\mbox{\boldmath ${\Phi}$}}_v)_j>_{L^2}$. We consider the asymptotic behavior of the function $I_j(v)$ as $|v|\to \infty$. The $X$-ray transform of a function $f$ is defined to be $$(Xf)(x, \theta)=\widetilde{f}(x,\theta)=\int_{-\infty}^{\infty} f(x+\theta t) \, dt,$$ where $x\in {\mathbb{R}}^n$ and $\theta \in {\mathbb{S}}^{n-1}$. \[thm:3-1\] Let $ n=3$. Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Then for $|v|$ sufficiently large and for any $\varphi_j\in {\ensuremath{\mathcal{S}}}_0$, $j=1,2,\cdots, N$, the function $I_j(v)$ has the expansion $$I_j(v) = \int_{{\mathbb{R}}^n} \widehat{V_{int}}(\xi) H_j(\xi) \, d\xi + \frac{1}{|v|} \left< \widetilde{V_{ext}}(\cdot, \widehat{v}) \varphi_j, \varphi_j\right>_{L^2} + O(|v|^{-2})$$ as $|v|\to \infty$, where $\widehat{v}=v/|v|\in {\mathbb{S}}^{n-1}$ and $$\begin{aligned} H_j(\xi) &= \sum_{k=1}^N \int_{{\mathbb{R}}} {\ensuremath{\mathcal{F}}} \left( |U_0 (t) \varphi_k \right|^2 ) (\xi) \overline{{\ensuremath{\mathcal{F}}} \left( |U_0 (t) \varphi_j \right|^2 ) (\xi)} \, dt \\ &\hspace{2em}- \sum_{k=1}^N \int_{{\mathbb{R}}} \left| {\ensuremath{\mathcal{F}}} \left( \left(U_0(t) \varphi_j \right) \overline{ \left( U_0(t) \varphi_k \right) } \right) (\xi) \right|^2 \, dt. \end{aligned}$$ In view of the representation of the scattering operator , we break the function $I_j(v)$ in two parts: $$I_j(v) = I_j^{(0)}(v) +I^{(1)}_j(v),$$ where $$\begin{aligned} I_j^{(0)}(v) &= \int_{{\mathbb{R}}} \left< N_j(\cdot,{\mbox{\boldmath ${u}$}}), U_0(s) ({\mbox{\boldmath ${\Phi}$}}_v)_j \right>_{L^2}\, ds, \\ I_j^{(1)}(v) &= \int_{{\mathbb{R}}} \left< V_{ext} u_j(s), U_0(s) ({\mbox{\boldmath ${\Phi}$}}_v)_j \right>_{L^2}\, ds.\end{aligned}$$ We know (see [@Watanabe2019 subsection 2.2]) that the function $I^{(0)}_j(v)$ can be expanded as $$I_j^{(0)}(v)=\int_{{\mathbb{R}}^n} \widehat{V_{int}} H_j(\xi)\, d\xi + R_1(v)$$ with the estimate $|R_1(v)|\le C |v|^{-2}$ for some $C>0$ and for $|v|$ sufficient large. We will claim that $$I_j^{(1)}(v) = \frac{1}{|v|} \left< \widetilde{V_{ext}}(\cdot, \widehat{v})\, \varphi_j, \varphi_j \right>_{L^2} + O(|v|^{-2})$$ as $|v|\to \infty$. Let $\Omega_-$ be a wave operator. Then we have $$\begin{aligned} I_j^{(1)}(v) &= \left< \int_{{\mathbb{R}}} U_0(-t) V_{ext} \left\{ u_j(t)-U_0(t)({\mbox{\boldmath ${\Phi}$}}_v)_j\right\} \, dt, ({\mbox{\boldmath ${\Phi}$}}_v)_j \right>_{L^2} + \left<\int_{{\mathbb{R}}} U_0(-t) V_{ext} U_0(t) ({\mbox{\boldmath ${\Phi}$}}_v)_j\, dt, ({\mbox{\boldmath ${\Phi}$}}_v)_j \right>_{L^2} \notag \\ &= \left< \int_{{\mathbb{R}}} [(\Omega_- -I)(U_0(t){\mbox{\boldmath ${\Phi}$}}_v)]_j\, dt, V_{ext}U_0(t)({\mbox{\boldmath ${\Phi}$}}_v)_j \right>_{L^2} \notag \\ & \hspace{1em} + \frac{1}{|v|} \left< \widetilde{V_{ext}}(\cdot ,\widehat{v}) U_0(\tau/v) \varphi_j, \, U_0(\tau /v)\varphi_j \right>_{L^2} \notag \\ & \le \| [( \Omega_- - I)U_0(t){\mbox{\boldmath ${\Phi}$}}_v]_j\|_{L^2} \left\| \int_{{\mathbb{R}}}V_{ext}U_0(t)({\mbox{\boldmath ${\Phi}$}}_v)_j\, dt \right\|_{L^2} \notag \\ & \hspace{1em} + \frac{1}{|v|} \left< \widetilde{V_{ext}} (\cdot , \widehat{v}) U_0(\tau /v)\varphi_j, \, U_0(\tau/v)\varphi_j \right>_{L^2}. \label{eqn:3-1}\end{aligned}$$ Thanks to Lemma \[lem:Enss-Weder\] and Lemma \[lem:2-4\], the first term in is estimated as $O(|v|^{-2})$ for $|v|$ sufficiently large. We know (see [@Enss-Weder1995]) that the second term in is equal to $$\frac{1}{|v|} \left< \widetilde{V_{ext}} (\cdot , \widehat{v}) \varphi_j, \, \varphi_j \right>_{L^2} + O(|v|^{-2})$$ for $|v|$ sufficiently large. The proof is completed. Reconstructions {#sec:4} =============== We complete the proof of Theorem \[thm:1-1\] and give reconstruction formulas. Reconstruction of the interaction potential ------------------------------------------- Let $\Gamma \subset {\mathbb{R}}$ be a the compact set and ${\mbox{\boldmath ${\Phi}$}}_{v}(x, \lambda)= e^{iv\cdot x}{\mbox{\boldmath ${\varphi}$}}((\lambda+1)x)$. Put $$S_{j}^{lim}(\lambda) = \lim_{|v|\to \infty} i \left< ((S-I){\mbox{\boldmath ${\Phi}$}}_v(\cdot, \lambda))_j, ({\mbox{\boldmath ${\Phi}$}}_v)_j(\cdot, \lambda) \right>_{L^2}, \qquad j=1,2,\cdots, N.$$ In view of Theorem \[thm:3-1\], we have $$\label{eqn:4-1} S^{lim}_{j}(\lambda)=\int_{{\mathbb{R}}^n} \widehat{V_{int}}(\xi) H_j(\xi, \lambda)\, d\xi$$ for any $\varphi_j\in {\ensuremath{\mathcal{S}}}_0$. Due to the fact (see [@Watanabe2019 Theorem 1.12]) that the equation is an integral equation of the first kind with a compact operator from $H^k({\mathbb{R}}^n)$ to $L^2(\Gamma)$ for $k>n/2$, we can reconstruct $\widehat{V_{int}}$ from the scattering operator by using the theory of integral equations (see e.g., [@Kress Section 15.4]) or approximate techniques (see e.g., [@Morse-Feshbach Section 8.3]). For example, the Picard’s theorem allows us to obtain a reconstruction formula of $\widehat{V_{int}}$. Let $X$ and $Y$ be Hilbert space, ${\ensuremath{\mathcal{A}}}\, :\, X\to Y$ be a compact linear operator, and ${\ensuremath{\mathcal{A}}}^* \, : \, Y \to X$ be its adjoint. Singular values of ${\ensuremath{\mathcal{A}}}$ is the non-negative square roots of the eigenvalue of non-negative self-adjoint compact operator ${\ensuremath{\mathcal{A}}}^* {\ensuremath{\mathcal{A}}} \, : \, X \to X$. The singular system of ${\ensuremath{\mathcal{A}}}$ is the system $\{ \mu_n, \varphi_n, g_n\}$, $n\in \mathbb{N}$, where $\varphi_n \in X$ and $g_n\in Y$ are orthonormal sequences such that ${\ensuremath{\mathcal{A}}}\phi_n = \mu_n g_n$ and ${\ensuremath{\mathcal{A}}}^*g_n = \mu_n \phi_n$ for all $n\in \mathbb{N}$. We denote the null-space of the operator $T$ by ${\ensuremath{\mathcal{N}}}(T)$. Let $n=3$. Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\], respectively. Then for any $\varphi_j \in {\ensuremath{\mathcal{S}}}_0$, $j=1,2, \cdots, N$ the function $S^{lim}_j (\lambda)$ is the $L^2$-function on a compact set $\Gamma \subset {\mathbb{R}}$. Moreover, letting $\{ \mu_n, \phi_n, g_n\}$, $n\in \mathbb{N}$ be a singular system of the integral operator $T$: $$(Tf)(\lambda):= \int_{{\mathbb{R}}^n} f(\xi) H_j(\xi, \lambda)\, d\xi,$$ the Fourier transform of the interaction potential is reconstructed by the formula: $$\widehat{V_{int}}(\xi)= \sum_{n=1}^{\infty} \frac{1}{\mu_n} \left< S^{lim}_j , g_n \right>_{L^2(\Gamma)} \phi_n$$ if and only if $S^{lim}_j\in {\ensuremath{\mathcal{N}}}(T^*)^{\perp}$ and satisfies $$\sum_{n=1}^{\infty} \frac{1}{\mu_n^2} \left| \left<S^{lim}_j , g_n \right>_{L^2(\Gamma)} \right|^2 <\infty.$$ The procedure of constructing the singular system is as follows: Due to the fact that the operator $T^*T$ is a self-adjoint compact operator on $H^k({\mathbb{R}}^n)$ for $k >n/2$, the operator $T^*T$ has at least one eigenvalues different from zero and at most a countable set of eigenvalues accumulating only at zero. Let $(\phi_n)$ denote an orthonormal sequences such that $T^*T\phi_n = \mu_n^2 \phi_n$. Then we can define $g_n$ as $g_n = \mu_n^{-1} T^{*}\phi_n$. Uniqueness of identifying $\widehat{V_{int}}$ follows from [@Watanabe2019 Theorem 1.14]. Thus we conclude that one can uniquely determine $\widehat{V_{int}}$ from $S$. Reconstruction of the external potential ---------------------------------------- Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\]. In addition, suppose that $$\label{eqn:4-2-1} \int_{K}^{\infty} (1+R) \| V_{ext}(x)F(|x|\ge R)\|_{L^{\infty}({\mathbb{R}}^n)} \, dR <\infty, \qquad K>0,$$ where $F(A)$ is the characteristic function of $A\subset {\mathbb{R}}^n$. We note that the Assumption \[ass:external-2\] includes the condition . Hence, the following also holds under the Assumption \[ass:interaction-2\] and Assumption \[ass:external-2\]. In view of Theorem \[thm:3-1\] with the help of Takiguchi [@Takiguchi1998 Proposition 3.2.], it is easy to verify that $$\lim_{|v|\to \infty} |v| \left< i \left( e^{-iv\cdot x}(S-I){\mbox{\boldmath ${\Phi}$}}_{v}\right)_j - \int_{{\mathbb{R}}} U_0(-t) N_{j}(x, U_0(t){\mbox{\boldmath ${\varphi}$}})\, dt, \psi \right>_{L^2} =\left< \widetilde{V_{ext}}(\cdot, \widehat{v}) \varphi_j, \psi \right>_{L^2}$$ for any $\varphi_j \in {\ensuremath{\mathcal{S}}}_0$ and for any $\psi \in {\ensuremath{\mathcal{S}}}$. From Theorem \[thm:4-2\], $\widehat{V_{int}}(\xi)$ is the known function. Therefore, $N_{j}(x, U_0(t){\mbox{\boldmath ${\varphi}$}})$ is known function. Hence, the above identity shows that one can determine the $X$-ray transform of $V_{ext}$ from $S$ in the sense of the tempered distribution ${\ensuremath{\mathcal{S}}}'$. By using the inversion formula for the $X$-ray transform, we obtain a reconstruction formula of $V_{ext}$. More precisely, define the operator $I^a$, which is called the Riesz potential, as $$I^{a}f := {\ensuremath{\mathcal{F}}}^{-1} ( |\xi|^{-a}\widehat{f}(\xi)), \qquad a<n.$$ We denote a hyperplane passing through the origin and orthogonal to $\alpha \in {\mathbb{S}}^{n-1}$ by $\alpha^{\perp}$. Let $I^a_{\alpha^{\perp}}$ be the $(n-1)$-dimensional Riesz potential acting on the hyperplane $\alpha^{\perp}$. The adjoint of the X-ray transform is denoted as $X^*$: $$(X^{*}g)(x)=\int_{{\mathbb{S}}^{n-1}}g(\theta, x-(\theta\cdot x)\theta)\, d\sigma,$$ where $d\sigma$ is the Lebesgue measure on the unit sphere ${\mathbb{S}}^{n-1}$ in ${\mathbb{R}}^n$. We know (see, e.g., Ramm-Katsevich [@Ramm-Katsevich1996 Theorem 2.6.2.]) that letting $f\in {\ensuremath{\mathcal{S}}}({\mathbb{R}}^n)$ and $g=Xf$, one has for any $|a|<n$ $$f=\frac{1}{2\pi |{\mathbb{S}}^{n-2}|}I^{-a}X^* I_{\alpha^{\perp}}^{a-1}g,$$ where $|{\mathbb{S}}^{n-2}|$ is the surface area of ${\mathbb{S}}^{n-2}$. This inversion formula also holds for the tempered distribution $f\in {\ensuremath{\mathcal{S}}}'({\mathbb{R}}^n)$ and $g=Xf \in {\ensuremath{\mathcal{S}}}'(T)$, where $T=\alpha^{\perp} \times {\mathbb{S}}^{n-1}$ (see Takiguchi [@Takiguchi1998]). Thus, we obtain \[thm:4-2\] Let $ n =3 $ and $|a|<n $. Assume that potentials $V_{int}$ and $V_{ext}$ satisfy Assumption \[ass:interaction-2\] and Assumption \[ass:external\] with , respectively. Then for any $\varphi_j \in {\ensuremath{\mathcal{S}}}_0$, we have $$V_{ext}=\frac{I^{-a} X^* I^{a-1}_{\alpha^{\perp}}}{2\pi |{\mathbb{S}}^{n-2}|} \frac{1}{\varphi_j} \lim_{|v|\to \infty}|v| \left\{ i \left( e^{-iv\cdot x}(S-I){\mbox{\boldmath ${\Phi}$}}_{v}\right)_j - \int_{{\mathbb{R}}} U_0(-t) N_{j}(x, U_0(t){\mbox{\boldmath ${\varphi}$}})\, dt \right\}$$ in ${\ensuremath{\mathcal{S}}}'$. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported by JSPS KAKENHI Grant Number 19K03617.

--- abstract: | Am 14.Dezember des Jahres 1900 berichtete Max Planck der Deutschen Physikalischen Gesellschaft “uber seine physikalische Interpretation einer harmlos aussehenden, von ihn selbst zuvor aufgestellten Formel, die das spektrale Verhalten der sogenannten W”armestrahlung beschreibt. Ma“sgeblich durch das Eingreifen Albert Einsteins entwickelte sich daraus im folgenden Vierteljahrhundert eine fundamentale Krise der Physik, die dann in einer wissenschaftlichen Revolution gr”o“sten Ausma”ses m"undete: der Quantentheorie. Die Quantentheorie entwickelte sich von Anfang an diametral gegen die Intentionen ihrer Sch“opfer. F”ur Planck bedeutete sie – trotz gr“o”ster “au”serer Anerkennungen – das vollst“andige Scheitern eines langj”ahrigen Forschungsprogramms, f“ur Einstein letztlich eine Absage an seine wissenschaftlichen Grund”uberzeugungen. Wir schildern die Hintergr“unde dieser seltsamen Entwicklung und beleuchten damit die begriffliche Seite physikalischer (und allgemein naturwissenschaftlicher) Forschung, die gemeinhin stark untersch”atzt wird. author: - | Domenico Giulini\ Universit“at Freiburg\ Physikalisches Institut\ Hermann-Herder-Stra”se 3\ 79104 Freiburg title: | **,,*Es lebe die Unverfrorenheit!*“**\ Albert Einstein und die Begr"undung der Quantentheorie[^1] --- Um die Rolle zu verstehen, die Albert Einstein bei der Entwicklung der Quantentheorie gespielt hat, m“ussen wir uns zun”achst die vorangegangenen Leistungen Plancks vergegenw“artigen, die ihn zur Aufstellung seiner ber”uhmten Strahlungsformel gef“uhrt haben. Mit dieser gelang ihm die vollst”andige *quantitative* Aufkl“arung des Ph”anomens der *W"armestrahlung*, die ihm den Nobelpreis des Jahres 1918 einbrachte: “als Anerkennung des Verdienstes, das er sich durch seine Quantentheorie um die Entwicklung der Physik erworben hat”. Zu diesem Zeitpunkt lag die eigentliche Tat schon mehr als 17 Jahre zur"uck. Genauer ist sie auf den 14.Dezember des Jahres 1900 zu datieren. Davon wird weiter unten die Rede sein. Etwas weniger bekannt ist die Tatsache, da“s diese wissenschaftliche Gro”stat Plancks gleichzeitig auch die restlose Zerschlagung seines langj“ahrigen, akribisch vorbereiteten und meisterhaft durchgef”uhrten Forschungsprogramms bedeutete, das in einer tief anti-atomistischen, an absoluten Gesetzm“a”sigkeiten sich orientierenden Naturauffassung wurzelt. In der Verfolgung dieser Ideale legt Planck den Grundstein zur Quantentheorie, die dem konsequenten Atomismus zum endg“ultigen Durchbruch verhilft und dem Element des Zufalls eine fundamentale Bedeutung innerhalb des Gef”uges physikalischer Gesetzm“a”sigkeiten zuweist. Hauptmotor dieser Entwicklung, die den Planckschen Vorstellungen diametral entgegenlief, war Albert Einstein. Hartn“ackig und mitunter unverfroren[^2] bestand er auf der restlosen Kl”arung der begrifflichen Grundlagen und Konsequenzen der Planckschen Theorie. Mit seiner Lichtquantenhypothese erkl“arte er nicht nur den photoelektrischen Effekt, sondern legte den eigentlich revolution”aren Kern dieser Theorie frei und provozierte so ma“sgeblich eine tiefe Krise, die 20 Jahre sp”ater in der Formulierung der Quantenmechanik m“undete. Etwas ”ubertreibend, aber im Kern doch zutreffend, kann man sagen, da“s Einstein der einzige war, der die Plancksche Theorie wirklich ernst nahm – so ernst, da”s die Konsequenzen sich schlie“slich auch gegen seine Grund”uberzeugungen richteten. Plancks Programm ================ Planck hatte sich schon in jungen Jahren ein ehrgeiziges Forschungsprogramm zurechtgelegt. Er wollte den sogenannten 2.Hauptsatz[^3] der Thermodynamik mit Hilfe der Theorie elektromagnetischer Vorg“ange streng begr”unden. Dies geschah aus einer Opposition zu den Vertretern des Atomismus, die in den Gesetzen der Thermodynamik lediglich statistische Gesetzm“a”sigkeiten einer sonst regellosen Bewegung sehr vieler Molek“ule sehen wollten, w”ahrend Planck fest an eine strenge Gesetzlichkeit ohne statistische Ausnahmen glaubte. In einer Jugendarbeit aus dem Jahre 1884 schreibt der 24-j“ahrige selbstbewu”st ([@Planck-GW], BandI, Dokument Nr.4, pp.162-163): > “Der zweite Hauptsatz der mechanischen W“armetheorie consequent durchgef”uhrt, ist unvertr“aglich mit der Annahme endlicher Atome. Es ist daher vorauszusehen, da”s es im Laufe der weiteren Entwicklung der Theorie zu einem Kampfe zwischen diesen beiden Theorien kommen wird, der einer von ihnen das Leben kostet.” Zwei Zeilen weiter l“a”st er wenig Zweifel dar“uber, welche der Theorien seiner Meinung und Hoffnung nach das Leben wird lassen m”ussen: > “... indessen scheinen mir augenblicklich verschiedenartige Anzeichen darauf hinzudeuten, da“s man trotz der bisherigen Erfolge der atomistischen Theorie sich schlie”slich doch einmal zu einer Aufgabe derselben und zur Annahme einer continuierlichen Materie wird entschlie“sen m”ussen.” Zu dieser Zeit war der junge Planck ein erkl“arter Anti-Atomist. Sein Plan war, zu versuchen, die thermodynamischen Gesetze nicht ”uber eine Mechanik elementarer Konstituenten (Atome, Molek“ule) zu begr”unden, sondern mit Hilfe der Gesetze der Elektrodynamik, die mit rein kontinuierlichen, im Raum verteilten Gr“o”sen operiert. In seiner Antrittsrede anl“a”slich seiner Aufnahme in die Preu“sische Akademie der Wissenschaften im Jahre 1894 erkl”arte er ([@Planck-GW], Band III, Dokument Nr.122, p.3): > “Es hat sich neuerdings in der physikalischen Forschung auch das Bestreben Bahn gebrochen, den Zusammenhang der Erscheinungen “uberhaupt gar nicht in der Mechanik zu suchen \[..\]. Ebenso steht zu hoffen, da”s wir auch “uber diejenigen elektrodynamischen Prozesse, welche direkt durch die Temperatur bedingt sind, wie sie sich namentlich in der W”armestrahlung “au”sern, n“ahere Aufkl”arung erfahren k“onnen, ohne erst den m”uhsamen Umweg durch die mechanische Deutung der Elektrizit“at nehmen zu m”ussen.” Planck glaubte also an die M“oglichkeit, die Gesetze der Thermodynamik, namentlich den 2.Hauptsatz, als strenge Folge bekannter elektromagnetischer Gesetze zu verstehen.[^4] Dieser sollte aus allgemeinsten Prinzipien ableitbar sein, entsprechend seiner wissenschaftlichen Disposition, die er in seinem sp”aten, pers“onlich gehaltenen Artikel ”‘Zur Geschichte der Auffindung des physikalischen Wirkungsquantums"’ aus dem Jahre 1943 so charakterisierte ([@Planck-GW], BandIII, Dokument141, p.255): > "\`Was mich in der Physik von jeher vor allem interessierte, waren die gro“sen allgemeinen Gesetze, die f”ur s“amtliche Naturvorg”ange Bedeutung besitzen, unabh“angig von den Eigenschaften der an den Vorg”angen beteiligten K“orper.”’ Fr"uhe Strahlungstheorie ======================== Man denke sich einen Hohlraum, der vollst“andig durch W”ande umschlossen ist, etwa das Innere eines Ofens. Bringt man die W“ande auf eine konstante Temperatur[^5] $T$, so wird sich nach einiger Zeit im Hohlraum eine bestimmte Konfiguration elektromagnetischer Strahlung einstellen, die sogenannte W”armestrahlung. Diese wird aus elektromagnetischen Wellen aller Frequenzen mit unterschiedlichen Intensit“aten bestehen. Zwischen Strahlung und der die W”ande bildenden Materie wird nach einiger Zeit ein thermodynamisches Gleichgewicht bestehen. Einzig wesentliche Voraussetzung f“ur die Existenz eines stabilen Gleichgewichtszustandes ist die Annahme, da”s die Materie (oder zumindest Anteile davon) in *allen* Frequenzbereichen mit der Strahlung wechselwirkt, also Strahlung aller Frequenzen emittieren und absorbieren kann. Mit Hilfe dieser Annahme folgerte Gustav Kirchhoff bereits 1859 die Existenz einer *universellen* Funktion $\rho(\nu,T)$ f“ur die spektrale Energieverteilung der Strahlung. Diese gibt an, wieviel Energie in Form von elektromagnetischen Wellen der Frequenz $\nu$ (genauer: in einem kleinen Frequenzintervall $[\nu,\nu+d\nu]$ um den Wert $\nu$) in einem Einheitsvolumen (z.B. Kubikzentimeter) des Hohlraumes enthalten ist, wenn die W”ande auf die Temperatur $T$ aufgeheizt wurden. Da“s diese Funktion ”‘universell“’ ist, bedeutet, da”s sie *nicht* von der genaueren Beschaffenheit der W“ande abh”angt, also nicht von ihrer Form oder ihrem Material. Egal, ob die W“ande aus Kupfer, Uran, Keramik oder sonstwas bestehen, immer wird sich bei vorgegebener Temperatur ein und dieselbe spektrale Energieverteilung von W”armestrahlung einstellen. Darin liegt die nichttriviale Einsicht Kirchhoffs. Daraus entsteht nun die theoretische Aufgabe, diese universelle Funktion aus den bekannten Gesetzen der Thermodynamik und Elektrodynamik zu bestimmen. Man beachte, da“s diese Aufgabe nur wegen der Universalit”at l“osbar erscheint, da dadurch die Kenntnis komplizierter Materialeigenschaften sowie deren (zum damaligen Zeitpunkt gr”o“stenteils unbekannter) Einfl”usse auf die Wechselwirkung zwischen Material und Strahlung nicht vorausgesetzt werden m"ussen. Durch weitere, raffiniertere thermodynamische “Uberlegungen konnte Wilhelm Wien 1893 zeigen, da”s die Funktion $\rho(\nu,T)$ aus dem Produkt der dritten Potenz der Frequenz $\nu$ und einer Funktion $f$ bestehen mu“s, die jetzt nur noch von *einer* Variablen abh”angt, n“amlich dem Quotienten der Frequenz und der Temperatur[^6]. Es mu”s also gelten: $$\label{eq:Wien1} \rho(\nu,T)=\nu^3f(\nu/T)\,.$$ Der Fortschritt dieser Einsicht Wiens besteht also in der Reduktion des Problems auf die Bestimmung einer Funktion mit nur *einer* anstatt zwei unabh“angigen Variablen. Bestimmt man $f$, so ist damit nach (\[eq:Wien1\]) auch $\rho(\nu,T)$ bekannt. Au”serdem folgen aus (\[eq:Wien1\]) auch ohne Kenntnis der Funktion $f$ bereits erste, experimentell pr“ufbare Konsequenzen, die gl”anzend best“atigt wurden. So ergibt sich einerseits das sogenannte Wiensche Verschiebungsgesetz, welches besagt, da”s die Frequenz, bei der die spektrale Energieverteilung ihr Maximum hat, proportional mit der Temperatur w“achst. Ebenso ergibt sich, da”s die gesamte, “uber alle Frequenzen summierte Energieabstrahlung mit der vierten Potenz der Temperatur anw”achst. Dies bezeichnet man als das Stefan-Boltzmannsche Gesetz. Wie gesagt, bestand die eigentliche Aufgabe nun in der Bestimmung der einen Funktion $f$. Durch weitere Anwendung fundamentaler Prinzipien sollte dies schlie“slich ohne allzu gro”sen Aufwand gelingen – so dachten die Physiker zwischen 1893 und 1900. Doch erwies sich diese Aufgabe “uberraschenderweise als fast unl”osbar. R"uckschauend aus dem Jahre 1913 charakterisierte Einstein die Situation so ([@Einstein-CW], Band4, Dokument Nr.23, p.562): > "\`Es w“are erhebend, wenn wir die Gehirnsubstanz auf eine Waage legen k”onnten, die von den theoretischen Physikern auf dem Altar dieser universellen Funktion $f$ hingeopfert wurde; und es ist diesen grausamen Opfers kein Ende abzusehen! Noch mehr: auch die klassische Mechanik fiel ihr zum Opfer, und es ist nicht abzusehen, ob Maxwells Gleichungen der Elektrodynamik die Krisis “uberdauern werden, welche diese Funktion $f$ mit sich gebracht hat.”’ Doch zur“uck zum Geschehen. Aus ”Uberlegungen, die man eher als “educated guessing” bezeichnen kann, schl“agt Wien eine einfache Exponentialfunktion f”ur $f$ vor, die dann im Verbund mit (\[eq:Wien1\]) zum sogenannten *Wienschen Strahlungsgesetz* f"uhrt ($\exp$ bezeichnet im folgenden die Exponentialfunktion): $$\label{eq:Wien2} \rho(\nu,T)=a\nu^3\exp\left[-\frac{b\nu}{T}\right]\,,$$ wobei $a$ und $b$ Konstanten sind, die es noch zu bestimmen gilt. Zahlreiche Experimente schienen ausnahmslos diese Form der spektralen Energieverteilung zu best“atigen (dies blieb der Fall bis etwa Mitte 1900). ”Uberzeugt von seiner Richtigkeit stellt sich daher Planck die Aufgabe, das Wiensche Strahlungsgesetz aus ersten Prinzipien theoretisch abzuleiten. Als “Prinzipienlieferant” akzeptiert er vornehmlich die Elektrodynamik und die Thermodynamik und hier an erster Stelle den 2.Hauptsatz “uber die Zunahme der Entropie. Nach langen M”uhen gelingt ihm schlie“slich im Jahre 1899 eine Ableitung von (\[eq:Wien2\]). Er res”umiert stolz ([@Planck-GW], BandI, Dokument Nr.34, p.597): > "\`Ich glaube hieraus schlie“sen zu m”ussen, da“s die gegebene Definition der Strahlungsentropie und damit auch das Wiensche Energieverteilungsgesetz eine notwendige Folge der Anwendung des Principes der Vermehrung der Entropie auf die elektromagnetische Strahlungstheorie ist und da”s daher die Grenzen der G“ultigkeit dieses Gesetzes, falls solche ”uberhaupt existieren, mit denen des zweiten Hauptsatzes der W“armetheorie zusammenfallen.”’ Ironischerweise sind es Experimentalphysiker (Lummer und Pringsheim), die den Theoretiker Planck in einer Ver“offentlichung des gleichen Jahres, die der experimentellen ”Uberpr“ufung des Wienschen Strahlungsgesetzes gewidmet ist, h”oflich darauf hinweisen, da“s hier ein logisch unzul”assiger Umkehrschlu"s vorliegt ([@Lummer-Pringsheim], p.225): > “Herr Planck spricht es aus, da“s dieses \[d.h. (\[eq:Wien2\])\] Gesetz eine nothwendige Folge der Anwendung des Principes der Vermehrung der Entropie auf die elektromagnetische Strahlung ist, und da”s daher die Grenzen seiner G“ultigkeit, falls solche ”uberhaupt existieren, mit denen des zweiten Hauptsatzes der W“armetheorie zusammenfallen. Soviel uns scheint, w”are die Planck’sche Theorie erst zwingend, wenn wirklich nachgewiesen werden kann, da“s *jede* von obiger Gleichung abweichende Form zu einem Ausdruck der Entropie f”uhrt, der dem Entropiegesetz widerspricht.” Planck hatte n“amlich keineswegs gezeigt, da”s das Wiensche Gesetz eine logische Folge des 2.Hauptsatzes der Thermodynamik ist, sondern nur, da“s es dem 2.Hauptsatz nicht widerspricht. Trotz dieses logischen Lapsus’ ist die von Planck verwendete Methode bemerkenswert. Da sie charakteristisch f”ur das Vorgehen eines theoretischen Physikers ist, soll sie hier etwas ausf"uhrlicher beschrieben werden. Das n"ahere Vorgehen Plancks ============================ Planck st“utzt sich auf Kirchhoff, der ja einwandfrei argumentiert hatte, da”s im thermodynamischen Gleichgewicht die spektrale Energieverteilung $\rho(\nu,T)$ eine *universelle* Funktion ist, also von der Form des Hohlraums und der Beschaffenheit der W“ande g”anzlich unabh“angig ist. Die geniale, aber in den meisten Darstellungen wenig hervorgehobene Idee Plancks ist nun folgende (vorgetragen in [@Planck-GW], BandI, Dokument Nr.34, pp.592-593): wegen der Unabh”angigkeit der spektralen Energieverteilung von der physikalischen Beschaffenheit der Wand darf man sich *zum Zwecke der theoretischen Bestimmung* der Funktion $\rho(\nu,T)$ die Wand auch aus einem hypothetischen, der theoretischen Beschreibung leicht zug“anglichen Material ersetzt denken. Dabei ist es ganz unwesentlich, ob dies hypothetische Medium in der realen Welt tats”achlich existiert, sondern wesentlich ist nur, da“s es den bekannten Gesetzen der Physik gen”ugt, also in diesem Sinne existieren *k"onnte*. Die Kirchhoffsche “Uberlegung versichert dann, da”s die spektrale Energieverteilung, die sich (theoretisch) im Hohlraum des hypothetischen Mediums einstellt, dieselbe ist wie die im Hohlraum eines tats"achlich existierenden Materials. Planck w“ahlt als hypothetisches Medium eine Art Gitter von kleinen elektrischen Ladungen, die mit einer kleinen Feder elastisch an eine Ruhelage befestigt sind. Planck nennt diese Gebilde ”‘Resonatoren“’, denn sie sollen f”ahig sein, kleine Schwingungen mit einer festen Frequenz $\nu$ (der sogenannten “Eigenfrequenz”) auszuf“uhren, wenn sie von einer elektromagnetischen Welle dieser Frequenz getroffen werden. Dieses sehr vereinfachte Modell einer ”‘Wand“’ ist nun durch die damals bekannten Gesetze der Elektrodynamik und Mechanik vollst”andig zu erfassen – ganz im Gegensatz zu einer realistischen Wand, deren mikroskopischer Aufbau und vor allem deren komplizierte Wechselwirkung mit auftreffenden Lichtstrahlen zum damaligen Zeitpunkt noch ganz unverstanden waren. Aus der selbstverst“andlichen Bedingung, da”s im thermodynamischen Gleichgewicht jeder dieser elementaren Resonatoren genauso viel elektromagnetische Energie emittiert wie absorbiert, leitete Planck die folgende Bedingung zwischen spektraler Energiedichte $\rho(\nu,T)$ und mittlerer Energie ${\bar{E}}(\nu,T)$ eines einzelnen Resonators der Schwingungsfrequenz $\nu$ bei der Temperatur $T$ ab: $$\label{eq:Planck1} \rho(\nu,T)=\frac{8\pi\nu^2}{c^3}{\bar{E}}(\nu,T)\,.$$ Es mu“s hier nochmals betont werden, da”s diese Gleichung eine unzweideutige Folge der Gesetze der klassischen Physik (Mechanik und Elektrodynamik) ist. H“atte Planck die damals bereits von seinem wissenschaftlichen Widersacher Ludwig Boltzmann (1844-1906) ausgearbeitete statistische Mechanik akzeptiert, so h”atte er sofort einen Ausdruck f“ur ${\bar{E}}(\nu,T)$ angeben k”onnen. Aus dem sogenannten “Aquipartitionsgesetz der statistischen Mechanik folgt n”amlich, da“s $$\label{eq:Aequipartition} {\bar{E}}(\nu,T)=\frac{R}{N_A}T\,,$$ wobei $R$ die sogenannte universelle Gaskonstante ist (durch Messungen gut bekannt) und $N_A$ die Avogadro-Zahl, also die Zahl der in einem Mol Gas enthaltenen Molek”ule. Er w“are damit zum sogenannten Rayleigh-Jeans-Gesetz gelangt: $$\label{eq:Rayleigh-Jeans} \rho(\nu,T)= \frac{8\pi\,\nu^2}{c^3}\frac{R}{N_A}T\,,$$ das – obwohl eine ebenso unzweideutige Folge der klassischen Physik – ganz unsinnige Aussagen macht. Zum Beispiel besagt es, da”s bei fester Temperatur $T$ die in elektromagnetischen Wellen der Frequenz $\nu$ abgestrahlte Energie quadratisch in $\nu$ w“achst, insgesamt also unendlich viel Energie abgestrahlt wird, wenn man ”uber alle Frequenzen summiert. Auch hinsichtlich der Abh“angigkeit von $T$ geht der Ausdruck (\[eq:Rayleigh-Jeans\]) v”ollig fehl. Das direkt proportionale Ansteigen der Strahlungsenergie mit der Temperatur h“atte zum Beispiel zur Folge, da”s bei jeder Frequenz die Energieabstrahlung bei Raumtemperatur – $T$ etwa gleich 290Grad Kelvin – immerhin noch ein Sechstel der Abstrahlung bei der Temperatur von 1700 Grad Kelvin w“are. Letztere entspricht etwa der Temperatur schmelzenden (d.h. wei”sgl“uhenden) Stahls. Dies ist offensichtlich eine groteske ”Ubersch“atzung der Abstrahlung bei Raumtemperatur. Doch Planck erw”ahnt diese katastrophale Folge mit keinem Wort. Erst Einstein wird in seiner Nobelpreisarbeit von 1905 darauf beharren, da“s die klassische Physik notwendig zum inakzeptablen Rayleigh-Jeans-Gesetz f”uhrt und deswegen fundamental nicht richtig sein kann. Planck geht v“ollig andere, recht seltsame Wege, um die jetzt noch fehlende Funktion $\rho(\nu,T)$ zu bestimmen. In der Annahme der Richtigkeit des Wienschen Gesetzes kennt er das Ziel und wei”s daher, welchen Ausdruck f“ur $\rho(\nu,T)$ er ”‘herbeiargumentieren“’ mu”s, um (\[eq:Wien2\]) aus (\[eq:Planck1\]) folgen zu lassen. An dieser Stelle bringt er nun den 2.Hauptsatz der Thermodynamik ins Spiel: Statt die Energie ${\bar{E}}(\nu,T)$ des einzelnen Resonators zu bestimmen – wof“ur er keine direkte Methode hat –, geht er den Umweg ”uber die Entropie $S(\nu,T)$, denn diese sollte sich aus den Forderungen des 2.Hauptsatzes ergeben. Aus einer allgemein g“ultigen thermodynamischen Relation, nach der die Ableitung der Entropie nach der Energie die inverse Temperatur ist (siehe Gleichung (\[eq:Entropie-Energie-Temp\]) im Anhang\[sec:AnhangPlanck\]), w”urde sich dann auch die Funktion ${\bar{E}}(\nu,T)$ ergeben. Planck gibt dann tats“achlich einen Entropieausdruck an, von dem er zeigen kann, da”s er allen Anforderungen des 2.Hauptsatzes gen“ugt und der direkt zum Wienschen Gesetz f”uhrt. Entgegen seiner obigen Aussagen zeigt er aber nicht, da“s dieser Ausdruck eindeutig ist. Es k”onnte also durchaus andere, ebenfalls mit dem 2.Hauptsatz formal vertr“agliche Strahlungsgesetze geben (was sich sp”ater auch als tats"achlich gegeben herausstellt). Der Widerspruch =============== Experimentelle Messungen an der Physikalisch-Technischen Reichsanstalt in Berlin im Jahre 1899 ergaben systematische Abweichungen vom Wienschen Strahlungsgesetz im Bereich niederer Frequenzen (d.h. gro“ser Wellenl”angen) [@Lummer-Pringsheim; @Rubens-Kurlbaum]. Die gemessenen Energien lagen bei kleinen Frequenzen systematisch oberhalb der Wienschen Kurve. Dazu m“u”sten erst neue Me“smethoden entwickelt werden, um den niederfrequenten Anteil des Spektrums m”oglichst sauber zu isolieren. Diese “Divergenzen von erheblicher Natur” (Planck) wurden in der Sitzung der Deutschen Physikalischen Gesellschaft am 19.Oktober mitgeteilt. Es ist bekannt, da“s Planck bereits am 7.Oktober – einem Sonntag – von Heinrich Rubens, einem der Experimentatoren, aufgesucht und von den neuen experimentellen Befunden unterrichtet wurde. Noch am gleichen Abend fand Planck durch geschicktes Probieren (im Anhang\[sec:AnhangEnergiefluktuationen\] erl”autert) eine neue, von der Wienschen leicht abweichende Strahlungsformel, die die neuen Resultate befriedigend wiederzugeben vermochte. Diese teilte er dann ebenfalls am 19.Oktober im Anschlu“s an das Referat des Experimentalphysikers Kurlbaum der Deutschen Physikalischen Gesellschaft mit. Damit war die Plancksche Strahlungsformel geboren: $$\rho(\nu,T)=\frac{a\nu^3}{\exp\bigl(b\nu/T\bigr)-1}\,. \label{eq:Planck}$$ Sie unterscheidet sich von der Wienschen Formel (\[eq:Wien2\]) lediglich durch die -1 im Nenner, so da”s f“ur hohe Frequenzen und/oder kleine Temperaturen beide Ausdr”ucke approximativ gleich sind. F“ur kleine Verh”altnisse $\nu/T$ verl“auft die Plancksche Kurve aber systematisch *oberhalb* der Wienschen, sagt also bei gegebener Temperatur eine merklich h”ohere Energiedichte im Bereich kleiner Frequenzen (d.h. gr“o”serer Wellenl"angen) voraus. Dies ist in Abbildung\[fig:PlanckWien\] dargestellt. (6,9)[$\nu$]{} (-250,165)[$\rho(\nu,T)$]{} (-212,80)[$-$Wien]{} (-236,130)[Planck$-$]{} Experimentell wurden diese Abweichungen von der Wienschen Formel bei gro“sen Wellenl”angen zuerst von Otto Lummer und Ernst Pringsheim [@Lummer-Pringsheim] gemessen und sofort darauf von Heinrich Rubens und Ferdinand Kurlbaum [@Rubens-Kurlbaum] noch eindr“ucklicher best”atigt. Beide Gruppen arbeiteten zu dieser Zeit an der Physikalisch-Technischen Reichsanstalt in Berlin-Charlottenburg. Die Me“skurve aus der Originalver”offentlichung von Lummer und Pringsheim ist als Abbildung\[fig:SpekLP\] im Anhang\[sec:AnhangLP\] wiedergegeben. Zum Schlu“s dieses Abschnitts erw”ahnen wir noch, da“s in moderner Schreibweise die Konstanten $a$ und $b$ in (\[eq:Planck\]) durch andere Konstanten ausgedr”uckt werden, n“amlich die Lichtgeschwindigkeit $c$, die Boltzmann-Konstante $k=R/N_A$ und das Plancksche Wirkungsquantum $h$: $$\label{eq:KonstanteRel} a=\frac{8\pi h}{c^3}\,,\qquad b=\frac{h}{k}\,.$$ Dieser Zusammenhang wird allerdings erst durch die theoretische Begr”undung der Planckschen Strahlungsformel verst"andlich werden. Intermezzo: Einsteins Bestimmung der Avogadro-Zahl ================================================== Zu Beginn seiner ber“uhmten Arbeit ”uber Lichtquanten aus dem Jahre 1905 ([@Einstein-CW], Band2, Dokument14) macht Einstein eine wichtige Bemerkung, die man etwa so zusammenfassen kann: Fordert man, da“s das Gesetz (\[eq:Rayleigh-Jeans\]), was eine notwendige Folge der klassischen Physik ist, als Grenzgesetz in der als ph”anomenologisch g“ultig angesehenen Planckschen Formel enthalten ist, so ergibt sich eine von jeder *theoretischen Begr"undung* der Planckschen Formel *unabh"angige* Methode zur Bestimmung der Avogadro-Zahl $N_A$. Entwickelt man die Exponentialfunktion im Nenner von (\[eq:Planck\]) bis zu linearer Ordnung in $b\nu/T$, so ergibt sich das Gesetz (\[eq:Rayleigh-Jeans\]) genau dann, wenn die Avogadro-Zahl $N_A$ mit den Konstanten $a,b$ des Planckschen Gesetzes in folgender Beziehung steht: $$N_A=\frac{b}{a}\cdot\frac{8\pi R}{c^3}\,. \label{Einstein}$$ Da $R$ und $c$ gut bekannt sind, liefert jede Bestimmung von $a$ und $b$ durch Strahlungsmessungen auch einen Wert f”ur $N_A$. Einstein erhielt so den Wert $N_A=6,17\cdot 10^{23}$. Zu dieser Zeit war dies der mit Abstand genaueste Wert der Avogadro-Zahl (vgl. Kapitel 5 in [@Pais]). Aber man konnte noch weiter schlie“sen: Aus der Kenntnis der Faradaykonstante (elektrische Ladung eines Mols einwertiger Ionen), die aus Elektrolysedaten gut bekannt war, erh”alt man nach Division durch $N_A$ den Wert der elektrischen Elementarladung $e$. Die Elementarladung (Betrag der Ladung eines Elektrons) lie“s sich also aus Strahlungsmessungen mit Hilfe der Planckschen Formel gewinnen, wobei sich ein weit besserer Wert als jemals zuvor ergab. Dies geht eindr”ucklich aus folgendem Vergleich der damals diskutierten Werte mit dem heutigen Wert der “Particle Data Group” (PDG) hervor (in Einheiten von $10^{-10}\,esu$, wo $1esu=0{,}1\, Ampere\times meter/c$ die Ladungseinheit “electrostatic units” bezeichnet): ------------------------- ------------------ Richarz (1894): 1.29 J.J. Thomson (1898): 6.50 Planck/Einstein (1901): 4.69 PDG (2000): 4.803 204 20(19) ------------------------- ------------------ Tats“achlich hatte bereits Planck 1901 die hier angegebenen Werte f”ur die Avogadro-Zahl und die Elementarladung mit Hilfe seiner Formel und den Ergebnissen von Strahlungsmessungen ausgerechnet ([@Planck-GW], BandI, Dokument Nr.44, pp.717-727). Aber erst Einstein sah, da“s dieses Vorgehen weitgehend unabh”angig von Plancks theoretischer Begr“undung seiner Strahlungsformel gerechtfertigt werden kann, wenn man nur die Forderung nach dem klassischen Limes stellt.[^7] Die Ironie dieser Episode ist, da”s diese Pr“azisionsbestimmung einer fundamental atomistischen Gr”o“se ausgerechnet durch den damaligen Anti-Atomisten Planck erm”oglicht wurde. Der “Akt der Verzweiflung” ========================== Wie sollte nun Planck nach all seinen M“uhen, das Wiensche Gesetz theoretisch zu begr”unden, eine Ableitung des neuen Gesetzes (\[eq:Planck\]) herzaubern? Hatte er nicht noch gerade argumentiert, da“s der 2.Hauptsatz notwendig zum Wienschen Gesetz f”uhre? Immerhin blieb er seiner “klassischen” Formel (\[eq:Planck1\]) treu und seiner Strategie, die mittlere Resonatorenergie ${\bar{E}}(\nu,T)$ aus der Entropie zu bestimmen. Er erkannte jetzt endg“ultig, da”s der Ausdruck f“ur letztere, den er vorher nach vielen M”uhen erhalten hatte und der ihm scheinbar unausweichlich zum Wienschen Gesetz f“uhrte, nicht der formal einzig m”ogliche sein konnte. So sehr sich Planck aber auch abm“uhte, eine Begr”undung des erforderlichen neuen Ausdrucks zu liefern, es wollte ihm einfach nicht gelingen. In seinem Ringen um das Auffinden allgemeiner Methoden, die es erlauben w“urden, die Entropie eines Resonators im Strahlungsfeld zu berechnen, verfiel er schlie”slich auf den verzweifelten Ausweg, ausgerechnet die von ihm bisher vehement bek“ampfte Methode der statistischen Interpretation der Entropie seines Widersachers Boltzmann zu verwenden. Danach ist die Entropie eine rein kombinatorische Gr”o“se, die bekannt ist, wenn man die Anzahl der M”oglichkeiten kennt, eine feste Energiemenge auf eine feste Anzahl von Resonatoren zu verteilen. Diese Anzahl w“are unendlich – und damit die Entropie unbestimmt –, wenn jeder Resonator Energie in kontinuierlichen Mengen aufnehmen k”onnte. Damit die Entropie endlich herauskommt, mu“s Planck annehmen, da”s die Gesamtenergie nur in ganzzahligen Vielfachen einer bestimmten Grundeinheit “uber die Resonatoren verteilt werden kann. Aus dem allgemeinen Gesetz (\[eq:Wien1\]) ergibt sich, da”s diese Grundeinheit proportional zur Eigenfrequenz $\nu$ des Resonators sein mu“s. Diese Proportionalit”atskonstante nennt man heute das Plancksche Wirkungsquantum $h$. F“ur die Energie-Grundeinheit $\varepsilon$ gilt also die Plancksche Formel $$\label{eq:hnu} \varepsilon=h\nu\,.$$ F”ur Planck war dies eine rein formale Annahme von h“ochstens heuristischer Bedeutung, die er hoffte, sp”ater durch ein physikalisches Argument eliminieren zu k“onnen. Immerhin f”uhrte sie ihn zu einer Ableitung, die er der Deutschen Physikalischen Gesellschaft in der Sitzung am 14.Dezember des Jahres 1900 mitteilte. Dieses Datum gilt bis heute als die Geburtsstunde der Quantentheorie. “Uber die ihm so seltsam aufgezwungene Annahme der Energiequantelung schrieb Planck r”uckschauend in einem Brief aus dem Jahre 1931 ([@Planck-Brief-1931]): > “Das war eine rein formale Annahme \[Energiequantelung\], und ich dachte mir eigentlich nicht viel dabei, sondern eben nur das, da“s ich unter allen Umst”anden, koste es, was es wolle, ein positives Resultat herbeif“uhren m”u“ste. \[...\] Kurz zusammengefa”st kann ich die ganze Tat als einen Akt der Verzweiflung bezeichnen. Denn von Natur bin ich friedlich und bedenklichen Abenteuern abgeneigt.” Im Anhang\[sec:AnhangPlanck\] ist Plancks “Akt der Verzweiflung” nochmals etwas genauer beschrieben. Einsteins Kritik ================ Einstein war mit Plancks theoretischer Begr"undung der Strahlungsformel (\[eq:Planck\]) zutiefst unzufrieden. Dabei brachte er im wesentlichen zwei Hauptkritikpunkte vor: - Planck benutzt wesentlich Gleichung (\[eq:Planck1\]), die mit Hilfe der Maxwellschen Theorie abgeleitet ist und voraussetzt, da“s der Energieaustausch zwischen Planckschen Resonatoren und Strahlungsfeld kontinuierlich verl”auft, *im Gegensatz* zur Quantisierungsannahme (\[eq:hnu\]). Zwar kann man zun“achst argumentieren, da”s (\[eq:Planck1\]) ja nur f“ur den statistischen Mittelwert der Resonatorenergie G”ultigkeit beansprucht und somit vielleicht auch unter einem gequantelten Energieaustausch, zumindest in guter N“aherung, g”ultig bleibt.[^8] Doch w“are das nur dann zu erwarten, wenn die mittlere Energie des einzelnen Resonators ${\bar{E}}(\nu,T)$ sehr gro”s gegen die Energieportionen (\[eq:hnu\]) ist. Aus (\[eq:Planck1\]) und der Planckschen Formel (\[eq:Planck\]) kann man aber ${\bar{E}}(\nu,T)$ direkt ablesen. Mit den Bezeichnungen (\[eq:KonstanteRel\]) ergibt sich $$\label{eq:ResEnergieMittel} {\bar{E}}(\nu,T)=\frac{h\nu}{\exp(h\nu/kT)-1}\,.$$ Demnach ist sogar umgekehrt ${\bar{E}}(\nu,T)$ sehr viel kleiner als $h\nu$, falls $h\nu$ viel gr“o”ser als $kT$ ist. Dies ist genau im Geltungsbereich des Wienschen Gesetzes der Fall, in dem die Annahme von (\[eq:Planck1\]) mit der Quantisierungsvorschrift ($h\nu\gg kT$) also unvertr"aglich zu sein scheint. - Zur Berechnung von ${\bar{E}}(\nu,T)$ “uber die Entropie verwendet Planck die Boltzmannsche Entropiedefinition (\[eq:Boltzmann-Entropie\]). Die zun”achst nur durch formales Abz“ahlen bestimmte mikroskopische Multiplizit”at eines makroskopischen Zustands ist jedoch nur dann proportional seiner physikalischen Wahrscheinlichkeit, wenn die Mikrozust“ande im Sinne der tats”achlich gegebenen Dynamik des Systems auch *physikalisch gleich wahrscheinlich* sind, soll hei“sen: im Laufe einer langen Zeit mit gleichen relativen Zeitdauern eingenommen werden. Da Planck f”ur den Resonator die klassische Dynamik als richtig annimmt (z.B. in der Ableitung der Gleichung (\[eq:Planck1\])), w“urde die Boltzmannsche Entropiedefinition korrekt angewendet notwendig zur Rayleigh-Jeans-Formel (\[eq:Rayleigh-Jeans\]) f”uhren. Diesbez"uglich kommentiert Einstein 1909 in der ihm eigenen charmant-frechen Weise ([@Einstein-CW], Band2, Dokument56, pp.544-545): > "\`So sehr sich jeder Physiker dar“uber freuen mu”s, da“s sich Herr Planck in so gl”ucklicher Weise “uber diese Forderung hinwegsetzte, so wenig w”are es angebracht, zu vergessen, da“s die Plancksche Strahlungsformel mit der theoretischen Grundlage, von welcher Herr Planck ausgegangen ist, unvereinbar ist.”’ Diese Bedenken tr“agt Einstein u.a. auch in seinem umfassenden Bericht ”‘“Uber die Entwicklung unserer Anschauungen ”uber das Wesen und die Konstitution der Strahlung“’ auf der 81. Versammlung Deutscher Naturforscher und ”Arzte 1909 in Salzburg vor, wo der 30-J“ahrige seinen ersten gr”o“seren ”offentlichen Auftritt hatte. In der sich anschlie“senden Diskussion erl”autert Planck nochmals seine Sichtweise der Quantisierungsannahme (\[eq:hnu\]), die er dezidiert aufgefa“st wissen wollte als Ausdruck eines noch unverstandenen Mechanismus, der lediglich die *Wechselwirkung* von Strahlung und Materie betraf. Materie war eben nur in der Lage, Energie in gewissen endlichen Portionen an das Strahlungsfeld anzugeben oder aus dem Strahlungsfeld aufzunehmen. Weder hatte Planck im Sinn, damit eine grunds”atzliche Modifikation der Dynamik des Resonators auszusprechen und schon gar nicht eine Quantisierung des Strahlungsfeldes selbst zu postulieren. Er hoffte auf jeden Fall, die Maxwellsche Theorie des Elektromagnetismus, die durchweg von der Vorstellung kontinuierlicher Prozesse in Raum und Zeit ausgeht, zumindest im wechselwirkungsfreien Fall beizubehalten. W"ortlich sagte Planck: ([@Einstein-CW], Band2, Dokument Nr.61, pp.585-586): > "\`Jedenfalls meine ich, man m“u”ste zun“achst versuchen, die ganze Schwierigkeit der Quantentheorie zu verlegen in das Gebiet der *Wechselwirkung* zwischen der Materie und der strahlenden Energie; die Vorg”ange im reinen Vakuum k“onnte man dann vorl”aufig noch mit den Maxwellschen Gleichungen erkl“aren.”’ Dem gegen"uber steht Einsteins Resumee seines Vortrages: ([@Einstein-CW], Band2, Dokument Nr.60, pp.576-577): > "\`Die Plancksche Theorie annehmen hei“st nach meiner Meinung geradezu die Grundlagen unserer Strahlungstheorie verwerfen.”’ W“ahrend Plancks ablehnende Haltung gegen”uber einer Modifikation der Maxwellschen Theorie des freien Strahlungsfeldes aus seinen Schriften ganz offenbar wird (was sich auch in seiner Kritik der Lichtquantenhypothese “au”sert), ist seine Haltung gegen“uber einer Modifikation der mechanischen Gesetze, hier im Zusammenhang mit den Resonatoren, etwas umstritten. Diesbez”uglich hat sich in j“ungerer Zeit sogar ein sogenannter ”‘Historikerstreit“’ entz”undet (vgl. [@Darrigol]), der mir aber etwas “ubertrieben scheint. In seiner urspr”unglichen Ableitung macht Planck in der Tat die formale Annahme (\[eq:hnu\]), ohne eine Modifikation der mechanischen Gesetze zu erw“ahnen. 1906 gibt Einstein eine Ableitung des Planckschen Gesetzes mit Hilfe der von ihm selbst entwickelten allgemeinen Methoden der statistischen Mechanik ([@Einstein-CW], Band2, Dokument34). Dort zeigt er, da”s man konsistent (d.h. unter Vermeidung der oben unter Punkt2 ge“au”serten Kritik) zu (\[eq:ResEnergieMittel\]) gelangt, wenn man annimmt, da“s die Resonatoren selbst nur ganzzahlige Vielfache der Energie $h\nu$ annehmen k”onnen und formuliert dies als eigentliche, der Planckschen Ableitung zugrundeliegende Annahme. Auf die Verallgemeinerung dieser Annahme auf jedes schwingungsf“ahige Gebilde in einem Festk”orper st“utzt Einstein kurz darauf seine Quantentheorie der spezifischen W”arme ([@Einstein-CW], Band2, Dokument38). Planck ist damit aber nicht einverstanden und versucht sp“ater (1911-12) sogar, eine Ableitung seines Strahlungsgesetzes zu geben, in der nur der Proze”s der Emission, nicht jedoch der Proze“s der Absorption ”‘gequantelt“’ ist ([@Planck-GW], Band2, Dokumente73,74,75).[^9] Die Unterscheidung dieser ”‘neuen Strahlungshypothese“’ Plancks von der urspr”unglichen ist aber nur dann sinnvoll, wenn man annimmt, da“s die Resonatorenergien grunds”atzlich kontinuierliche Werte annehmen k“onnen. Daraus mu”s man m.E. schlie“sen, da”s zwar Einstein, aber nicht Planck die Gleichung (\[eq:hnu\]) im heutigen quantenmechanischen Sinne verstanden haben wollte, n“amlich als allgemeine Quantisierungsbedingung materieller schwingungsf”ahiger Systeme. In seiner “Lichtquantenhypothese” erweiterte Einstein diese Quantisierungsbedingung dann auch auf das freie Strahlungsfeld, was der heutigen Sichtweise der Quantenelektrodynamik entspricht. Einsteins Lichtquantenhypothese =============================== In den uns vorliegenden schriftlichen Dokumenten Einsteins kennzeichnet er nur eine einzige seiner wissenschaftlichen Ideen als "\`sehr revolution“ar”’ ([@Einstein-CW], Band5, Dokument Nr.27, p.31)[^10], n“amlich die Lichtquantenhypothese. Diese ver”offentlicht er im Jahre 1905, 26-j“ahrig, im gleichen Zeitschriftenband wie seine spezielle Relativit”atstheorie und die Theorie der Brownschen Bewegung. Die Arbeit tr“agt den Titel ”‘“Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt”’. Dieser “heuristische Gesichtspunkt” besteht in einer v“ollig anderen Interpretation der Planckschen Quantisierungsbedingung (\[eq:hnu\]), n”amlich als Eigenschaft des Strahlungsfeldes selbst. Er schreibt ([@Einstein-CW], Band2, Dokument Nr.14, p.151): > "\`Nach der hier ins Auge zu fassenden Annahme ist bei Ausbreitung eines von einem Punkte ausgehenden Lichtstrahles die Energie nicht kontinuierlich auf gr“o”ser und gr“o”ser werdende verteilt, sondern es besteht dieselbe aus einer endlichen Zahl von in Raumpunkten lokalisierten Energiequanten, welche sich bewegen, ohne sich zu teilen und nur als Ganze absorbiert und erzeugt werden k“onnen.”’ Diese scheinbare R“uckkehr zur l”angst “uberkommenen Partikelvorstellung des Lichts, die zwar noch Newton vertreten hatte, die aber dann im fr”uhen 19.Jahrhundert durch den Siegeszug der Wellentheorie geradezu hinweggefegt wurde, mu“ste auf die Zeitgenossen Einsteins als eine Mischung aus naiv und provokant gewirkt haben, eben geradezu unverfroren. ”Au“serungen dazu werden uns weiter unten begegnen. Und doch war Einsteins Sichtweise, wie die seiner Gegner, nicht unbegr”undet. Durch eine scharfe Analyse des Strahlungsgesetzes, insbesondere des ihm immer suspekt erschienenen Wienschen Bereichs, zeigt er, da"s ([@Einstein-CW], Band2, Dokument Nr.14, p.161): > "\`Monochromatische Strahlung von geringer Dichte (innerhalb des G“ultigkeitsbereiches der Wienschen Strahlungsformel) verh”alt sich in w“armetheoretischer Beziehung so, wie wenn sie aus voneinander unabh”angigen Energiequanten von der Gr“o”se $h\nu$ best“unde”’.\[zit:EinsteinLQ\] Die genauere Argumentation Einsteins ist in Anhang\[sec:AnhangEinstein\] erl"autert. Einstein ist klar, da“s sich diese Vorstellung auch an der Erkl”arung bekannter Ph“anomene wird behaupten m”ussen, namentlich solcher, die die noch unverstandenen Prozesse bei der Wechselwirkung von Licht mit Materie betreffen. Einer dieser Prozesse ist der sogenannte “Photoelektrische Effekt”, bei dem durch Bestrahlung einer Metallplatte mit Licht Elektronen aus dem Material herausgel“ost werden. Die Energie des ankommenden Lichtes wird also durch irgendeinen Proze”s dazu verwandt, das Elektron aus dem Atomverband herauszul“osen, wozu eine nur vom Material abh”angige Energie $P$ aufzuwenden ist. Die “ubersch”ussige Energie des ankommenden Lichtes wird dann in die Bewegungsenergie $E_{\text{kin}}$ des austretenden Elektrons investiert. Gem“a”s der traditionellen Wellentheorie des Lichtes erfolgt dessen Ausbreitung stetig “uber alle Raumbereiche. Da die Energie des Lichtes dann proportional zu seiner Intensit”at ist, m“u”ste z.B. die Energie der herausgel“osten Elektronen mit dem Abstand der Lichtquelle von der Metallplatte fallen, da mit dem Abstand auch die Intensit”at abnimmt. Was aber durch den Experimentalphysiker Philipp Lenard (1862-1947, Nobelpreis 1905) im Jahre 1900 tats“achlich beobachtet wurde, ist, da”s zwar die Anzahl der herausgel“osten Elektronen mit fallender Intensit”at abnimmt, nicht aber deren individuelle Energien, die sich als *von der Intensit“at des eingestrahlten Lichtes unabh”angig* ergaben. Auf das einzelne Elektron wird also eine immer gleiche Energie “ubertragen. Dieser Tatbestand pa”st nun “uberhaupt nicht zur Wellentheorie des Lichtes, wird aber sofort plausibel bei Zugrundelegung der Lichtquantenhypothese. Nach dieser wird jedes der einzelnen Elektronen durch ein ganzes, unteilbares Lichtquant der Energie $h\nu$ herausgel”ost und mit einer Bewegungsenergie $E_{\text{kin}}$ heraustreten, die der Differenz der Energie des Lichtquants zur Abl“osungsenergie $P$ entspricht: $$\label{eq:Photoeffekt} E_{\text{kin}}=h\nu-P.$$ Diese sogenannte ”‘Einsteinsche Gleichung“’ zum Photoeffekt wurde teilweise durch Lenard und sp”ater vor allem durch den amerikanischen Experimentalphysiker Robert Millikan (1868-1953, Nobelpreis 1923) vollauf best“atigt, was sogar mit ein Grund f”ur die Vergabe des Nobelpreises war: “for his work on the elementary charge of electricity and on the photoelectric effect”. Somit schien der Photoeffekt mit einem Schlag eine v“ollig nat”urliche Erkl"arung zu finden – vorausgesetzt, man akzeptierte die Lichtquantenhypothese! Kritik an der Lichtquantenhypothese =================================== Trotzdem war aber allen Beteiligten klar, da“s die Annahme der Einsteinschen Vorstellung der Lichtquanten v”ollig unvereinbar sein w“urde mit der g”angigen (Maxwellschen) Theorie des Elektromagnetismus, die wenige Jahre zuvor durch die Aufsehen erregenden Versuche Heinrich Hertz’ scheinbar so gl“anzend best”atigt wurde und auf die Planck die Ableitung seiner Ausgangsgleichung (\[eq:Planck1\]) wesentlich gest“utzt hatte. Die aus der Planckschen Strahlungsformel in gewisser Weise ableitbare Lichtquantenhypothese anzunehmen, hie”se dann also gleichzeitig, der theoretischen Begr“undung dieser Formel den Boden zu entziehen. Das genau war die Kritik Einsteins, die er ”uber viele Jahre hinweg in mannigfacher Variation immer wieder vorbrachte. Wenig “uberraschend ist es daher, da”s die Einsteinsche Lichtquantenhypothese vor allem bei Planck, aber auch bei anderen Physikern auf starke Ablehnung stie“s, darunter auch solche, die Einstein wissenschaftlich und pers”onlich sehr nahe standen (worunter man sonst auch Planck z“ahlen mu”s, aber eben mit Ausnahme dieses einen Punktes betreffend die Lichtquantenhypothese). So beginnt z.B. der Theoretiker Arnold Sommerfeld, einer der besten Kenner der Materie und von Einstein sehr geachtet, im Jahre 1911 seinen l“angeren Vortrag auf der 83. Versammlung der Gesellschaft Deutscher Naturforscher und ”Arzte so ([@Sommerfeld:1911], p.31) : > "\`Als der wissenschaftliche Ausschu“s unserer Gesellschaft an mich die Aufforderung richtete, dieser Versammlung einen Bericht ”uber die Relativit“atstheorie zu erstatten, erlaubte ich mir dagegen geltend zu machen, da”s das Relativit“atsprinzip kaum mehr zu den eigentlich aktuellen Fragen der Physik geh”ore. Obwohl erst 6 Jahre alt – Einsteins Arbeit erschien 1905 – scheint es schon in den gesicherten Besitz der Physik “ubergegangen zu sein. Ganz anders aktuell und problematisch ist die Theorie der *Energiequanten* \[...\]. Hier sind die Grundbegriffe noch im Flu”s und die Probleme ungez“ahlt.”’ Und f"ahrt kurz darauf fort: > “Einstein zog aus der Planckschen Entdeckung die weitestgehenden Folgen \[...\] und “ubertrug das Quantenhafte von dem Emissions- und Absorptionsvorgang auf die Struktur der Lichtenergie im Raume, ohne, wie ich glaube, seinen damaligen Standpunkt heute noch in seiner ganzen K”uhnheit aufrecht zu erhalten.” Und selbst der bereits erw“ahnte gro”se amerikanische Experimentalphysiker Robert Millikan, der 10 Jahre seines Forscherlebens der experimentellen “Uberpr”ufung der Einsteinschen Formel (\[eq:Photoeffekt\]) f“ur den Photoeffekt widmete und dadurch auch die ersten Pr”azisionsmessungen des Planckschen Wirkungsquantums $h$ realisierte (siehe Anhang\[sec:AnhangMillikan\]), schrieb 1916 in einem langen, zusammenfassenden Artikel “uber die gerade von ihm so gl”anzend best"atigte Einsteinsche Formel ([@Millikan:1916], p.384): > “Despite then the apparently complete success of the Einstein equation, the physical theory of which it was designed to be the symbolic expression is found so untenable that Einstein himself, I believe, no longer holds to it.” Die Lichtquantenhypothese im allgemeinen kommentiert Millikan bereits auf der ersten Seite seines Artikels wie folgt ([@Millikan:1916], p.355): > “This hypothesis may well be called reckless, first because an electromagnetic disturbance which remains localized in space seems a violation of the very conception of an electromagnetic disturbance, and second because it flies in the face of the thoroughly established facts of interference.” Kurz zuvor, im Jahre 1913, als Einstein die Ehre zuteil wird, in die Preu"sische Akademie der Wissenschaften aufgenommen zu werden, verfassen Planck, Nernst, Rubens und Warburg ein Empfehlungsschreiben, das mit folgenden Worten endet ([@Einstein-CW], Band5, Dokument Nr. 445, p.527): > "\`Zusammenfassend kann man sagen, da“s es unter den gro”sen Problemen, an denen die moderne Physik so reich ist, kaum eines gibt, zu dem nicht Einstein in bemerkenswerter Weise Stellung genommen h“atte. Da”s er in seinen Spekulationen gelegentlich auch einmal “uber das Ziel hinausgeschossen haben mag, wie z.B. in seiner Hypothese der Lichtquanten, wird man ihm nicht allzuschwer anrechnen d”urfen; denn ohne ein Risiko zu wagen, l“a”st sich auch in der exaktesten Naturwissenschaft keinerlei wirkliche Neuerung einf“uhren.”’ Acht Jahre sp“ater, 1921, wird Einstein f”ur die Erkl“arung des Photoelektrischen Effektes mit Hilfe der Lichtquantenhypothese der Nobelpreis f”ur Physik zuerkannt. Aber auch danach verklingen die Zweifel noch nicht. Ein Jahr nach Einstein bekommt Niels Bohr den Nobelpreis. In seiner Nobel-Vorlesung mit dem Titel “The Structure of the Atom” schl“agt Bohr ganz ”ahnliche T“one an wie sechs Jahre zuvor Millikan. Unter anderem findet sich in der Niederschrift von Bohrs Vorlesung folgender eindr”ucklicher Passus [@Bohr-Nobel]: > “This phenomenon \[des Photoelektrischen Effekts\], which had been entirely unexplainable on the classical theory, was thereby placed in quite a different light, and the predictions of Einstein’s theory have received such exact experimental confirmation in recent years, that perhaps the most exact determination of Planck’s constant is afforded by measurements on the photoelectric effect. In spite of this heuristic value, however, the hypothesis of light-quanta, which is irreconcilable with so-called interference phenomena, is not able to throw light on the nature of radiation. I need only recall that these interference phenomena constitute our only means of investigating the properties of radiation and therefore of assigning any closer meaning to the frequency which in Einstein’s theory fixes the magnitude of the light-quantum.” Wie bereits erw“ahnt, folgte auf Bohr Millikan als Physik-Nobelpreistr”ager des Jahres 1923. In seiner Nobel-Vorlesung mit dem Titel “ The electron and the light-quant from the experimental point of view” “au”sert auch er sich nochmals kritisch, wenn auch mittlerweile in etwas abgeschw"achter Form (die Hervorhebungen sind seine) [@MillikanNobel]: > “ In view of this methods and experiments the general validity of Einstein’s equation \[gemeint ist Gleichung (\[eq:Photoeffekt\])\] is, I think, now universally conceded, and *to that extent the reality of Einstein’s light-quanta may be considered as experimentally established.* But the conception of *localized* light-quanta out of which Einstein got his equation must still be regarded as far from being established. Whether the mechanism of interaction between ether waves and electrons has its seat in the unknown conditions and laws existing within the atom, or is to be looked for primarily in the essentially corpuscular Thomas-Planck-Einstein conception as to the nature of radiant energy is the all-absorbing uncertainty upon the frontiers of modern Physics.” Ein letztes dramatisches Aufb“aumen der Kritiker ”au“serte sich 1924 in einer damals sehr viel Aufsehen erregenden Arbeit von Bohr, Kramers und Slater [@BKS], in der eine statistische Theorie der Wechselwirkung zwischen Strahlung und Materie formuliert wird mit dem erkl”arten Ziel, g“anzlich ohne die Lichtquanten auszukommen. Als Preis daf”ur sollte man hinnehmen, da“s die Erhaltungss”atze von Energie und Impuls zwar im statistischen Mittel, nicht jedoch f“ur den individuellen Elementarproze”s g“ultig seien. Dabei hatte gerade ein Jahr zuvor Arthur Compton (1892-1962, geteilter Nobelpreis 1927) die klassisch unverst”andlichen[^11] Eigenschaften der Streuung von R“ontgenstrahlen an Materie mit der Annahme erkl”art, da“s es sich dabei um individuelle St”o“se von Lichtquanten mit einzelnen Elektronen handle, wobei f”ur jeden Sto“s individuell Energie- und Impulserhaltung gelten [@Compton1] (sogenannter Comptoneffekt). Dies mag andeuten, wie verzweifelt der Vorschlag von Bohr, Kramers und Slater damals war, die nun argumentierten mu”sten, da“s die Ph”anomene auch mit einer nur im statistischen Mittel g“ultigen Energieerhaltung vertr”aglich w“aren, was aber schon kurz darauf durch zahlreichen Experimente widerlegt wurde (z.B. auch wieder durch Compton; siehe [@Compton2]). Erst ab 1925, das auch das Geburtsjahr der Quantenmechanik ist, kann man also davon sprechen, da”s sich Einsteins Lichtquantenhypothese in den ma"sgebenden Fachkreisen wirklich durchgesetzt hatte. Zusammenfassung und Ausblick ============================ Plancks gr“o”ste wissenschaftliche Leistung ist auf ironisch und fast tragische Weise erkauft mit dem Scheitern seines gro“s angelegten Planes, dessen Ziel es war, den 2.Hauptsatz als streng kausales Gesetz aus den Gesetzen der Elektrodynamik zu begr”unden. Auf seinem Weg dorthin findet er stattdessen ein neues, experimentell gl“anzend best”atigtes Strahlungsgesetz unter Zugrundelegung der von ihm sonst vehement bek“ampften statistischen Entropiedefinition. Die theoretischen Implikationen dieses Gesetzes, namentlich die Lichtquantenhypothese Einsteins, entziehen Planck geradezu die gesamte Grundlage, von der aus er urspr”unglich startete. Wie kein anderer f“orderte Einstein in dieser Zeit durch hartn”ackiges Hinterfragen der Grundlagen der Planckschen Strahlungstheorie den endg“ultigen Bruch mit der klassischen Physik. So wurde Planck *durch Einstein* zum Revolution”ar wider Willen. Aber auch Einstein selbst bleibt dieses Schicksal nicht erspart. Noch 1916 gibt er eine wunderbar einfache Ableitung der Planckschen Strahlungsformel, die nun vollst“andig auf den Gebrauch der Beziehung (\[eq:Planck1\]) verzichtet.[^12]. Dazu betrachtet Einstein die Absorption und Emission von Licht als statistische Prozesse, m”oglicherweise in der Hoffnung, sie sp“ater doch noch deterministisch verstehen zu k”onnen. Interessanterweise mu“s er, um zur Planckschen Formel zu gelangen, neben dem Proze”s der *spontanen* Emission auch einen bis dahin unbekannten Proze“s der *induzierten* Emission postulieren (ohne ihn w”are er formal zum Wienschen Gesetz gelangt), der sp“ater die Grundlage des Funktionsprinzips des Lasers werden sollte. Bez”uglich der statistischen Natur dieser Prozesse schreibt er am Ende dieser Arbeit ([@Einstein-CW], Band6, Dokument Nr.38, p.396): > "\`Die Schw“ache der Theorie liegt einerseits darin, da”s sie uns dem Anschlu“s an die Undulationstheorie \[d.h. Wellentheorie\] nicht n”aher bringt, andererseits darin, da“s sie Zeit und Richtung der Elementarprozesse \[der Lichtabsorption und Emission\] dem ‘Zufall’ ”uberl“a”st; trotzdem hege ich das volle Vertrauen in die Zuverl“assigkeit des eingeschlagenen Weges”’. Doch f“uhrte eben dieser eingeschlagene Weg nach weiteren 10 Jahren geradewegs zur heutigen Quantenmechanik (1925-26) und Quantenelektrodynamik (1928), die Einstein mit seinen wissenschaftlichen Grund”uberzeugungen nicht in Einklang bringen konnte – insbesondere deshalb, weil in ihnen der Zufall als irreduzibler Bestandteil der Naturerkl“arung auftritt. Doch das genauer zu erl”autern, bed"urfte eines weiteren Vortrags. Betrachtet man r“uckblickend die fr”uhe Entstehungsgeschichte der Quantentheorie, so h“atte sie beim besten Willen ironischer nicht sein k”onnen. Wir erinnern uns, da“s ihr Ausgangspunkt die gemessenen Abweichungen vom Wienschen Gesetz waren, welches zu diesem Zeitpunkt (f”alschlicherweise) als strenge Konsequenz der klassischen Physik angenommen wurde, ma“sgeblich durch die Arbeiten von Planck. Wie Einstein in seiner Lichtquantenarbeit aber gezeigt hatte, repr”asentiert das Wiensche Gesetz gerade den typisch quantentheoretischen Teilchenaspekt der Strahlung. Die von Lummer und Pringsheim gemessenen Abweichungen vom Wienschen Gesetz liegen im langwelligen Bereich, in dem das Rayleigh-Jeans-Gesetz ann“ahernd g”ultig ist, das nun tats"achlich eine unabweisbare Konsequenz der klassischen Physik ist, wie Einstein ebenfalls zeigte, und dem Wellenbild der Strahlung entspricht. *Etwas “uberspitzt kann man im Nachhinein also sagen, da”s die Quantentheorie aus Messungen klassischer Korrekturen an einem gl“ucklich erratenen Quantengesetz entstand, das irrt”umlich f"ur ein Gesetz der klassischen Physik gehalten wurde.* Als Ausblick sei zum Schlu“s noch erw”ahnt, da“s nicht nur in der Wissenschaft vom Kleinsten, sondern auch in den gr”o“sten uns heute zug”anglichen Dimensionen, in der Kosmologie, die Plancksche Strahlungsformel eine zentrale Rolle spielt. So ist ja unser gesamtes Universum ein einziger Strahlungshohlraum, der erf“ullt ist von einer elektromagnetischen Strahlung der Temperatur von etwas unter 3 Grad Kelvin (etwa -270 Grad Celsius). Diese Strahlung entstand etwa 30000 Jahre nach dem Urknall, als sich aus zun”achst gegenseitig ungebundenen Elektronen und Atomkernen stabile Atome bildeten. Zu diesem Zeitpunkt betrug die Temperatur etwa $100\,000$ Grad Kelvin. Wegen der best“andigen Ausdehnung des Universums k”uhlt sich die Strahlung stetig ab und hat zur gegenw"artigen Epoche den eben genannten Wert. Seit einigen Jahren werden charakteristische Eigenschaften dieser sogenannten “Kosmischen Hintergrundstrahlung” durch Satelliten vermessen, da diese eine reiche F“ulle von Informationen ”uber Entwicklung und Zusammensetzung unseres physikalischen Universums verraten. Nat“urlich wurde dabei auch die spektrale Energieverteilung gemessen und mit der Planckschen Formel verglichen. Das Resultat ist in Abbildung\[fig:CMB\] dargestellt, in der die Fehlerbalken auf unnat”urliche 400 (!) Standardabweichungen vergr“o”sert wurden, damit sie “uberhaupt sichtbar sind. Normale Fehlerbalken von wenigen Standardabweichungen w”aren weniger hoch als die Strichdicke der Kurve. Damit ist dies die pr"azisest vermessene Planckkurve bis zum heutigen Tag. [**ANH"ANGE**]{} N“aheres zu Plancks ”‘Akt der Verzweiflung"’ {#sec:AnhangPlanck} ============================================ Es wurde beschrieben, da“s Planck bei der theoretischen Begr”undung sowohl des Wienschen als auch seines eigenen Strahlungsgesetzes stets von der Beziehung (\[eq:Planck1\]) ausging und da“s er die darin auftretende Funktion ${\bar{E}}(\nu,T)$, die die mittlere Energie eines Resonators der Eigenfrequenz $\nu$ im Strahlungsfeld der Temperatur $T$ angibt, durch die Entropiefunktion $S(\nu,T)$ dieses Resonators zu bestimmen suchte. So ging er auch bei der theoretischen Begr”undung seines Gesetzes (\[eq:Planck\]) am 19.Dezember 1900 vor. Dazu wandte er die statistische Definition der Entropie von Ludwig Boltzmann an. Diese besagt, da“s die Entropie eines Systems proportional zum nat”urlichen Logarithmus des statistischen Gewichtes dieses Zustandes ist. Letzteres ist definiert als die Anzahl $W$ der M“oglichkeiten, den (makroskopisch definierten) Zustand auf verschiedene mikroskopische Arten zu realisieren. Dies dr”uckt folgende Formel aus (genannt die “Boltzmannsche”, die aber erst Planck so hinschrieb), die man noch heute auf der Grabplatte Boltzmanns auf dem Wiener Zentralfriedhof bewundern kann: $$\label{eq:Boltzmann-Entropie} S=k\, \ln(W)\,.$$ Dabei ist eben $k$ die Proportionalit“atskonstante zwischen Entropie und Logarithmus des statistischen Gewichtes. Man kann zeigen, da”s diese Konstante gerade gleich ist dem Quotienten aus zwei uns bereits bekannten Gr“o”sen, n"amlich der universellen Gaskonstante $R$ und der Avogadro-Zahl $N_A$. In seiner Verzweiflung, endlich eine theoretische Begr“undung seiner bisher nur gl”ucklich erratenen Strahlungsformel (\[eq:Planck\]) liefern zu m“ussen, verfiel Planck auf den Ausweg, die Boltzmannsche Gleichung (\[eq:Boltzmann-Entropie\]) als Definition der Entropie zu akzeptieren und sie zur Berechnung der Entropie eines Resonators im Strahlungsfeld zu verwenden. Dazu ging Planck so vor: Angenommen, es gibt $n$ Resonatoren der Eigenfrequenz $\nu$, die zusammengenommen in einem Zustand der Energie $E_{\text{total}}$ sind. Dann ist das statistische Gewicht $W$ dieses Zustandes definiert durch die Anzahl der M”oglichkeiten, die Energie $E_{\text{total}}$ auf die $n$ Resonatoren zu verteilen. Physikalisch geht hier die oft nicht explizit genannte, aber dennoch sehr wichtige Hypothese ein, da“s jede dieser Verteilungen gem”a“s der Dynamik des Systems im Laufe der Zeit gleich h”aufig vorkommt. W“aren die Resonatoren in der Lage, kontinuierliche Mengen von Energie aufzunehmen und abzugeben, so w”are das statistische Gewicht unendlich und Formel (\[eq:Boltzmann-Entropie\]) erg“abe ebenfalls keinen endlichen Wert. Diesen Schlu”s kann man durch die Annahme umgehen, da“s jeder der Resonatoren seine Energie nur portionsweise in Einheiten einer festen Grundmenge aufnehmen und abgeben kann. Sei diese Grundmenge $\varepsilon$, so gibt es also insgesamt $n=E/\varepsilon$ Energieportionen zu verteilen. Es ist nun eine elementare kombinatorische Aufgabe, zu berechnen, wie viele M”oglichkeiten es gibt, $n$ Portionen Energie auf $N$ Resonatoren zu verteilen. Die Antwort ist $$\label{eq:Kombinatorik} W=\frac{(n+N-1)!}{n!(N-1)!}\,.$$ Daraus erh“alt man mit (\[eq:Boltzmann-Entropie\]) die Entropie des Zustandes aller Resonatoren der Eigenfrequenz $\nu$ und nach weiterer Division durch die Anzahl $N$ dieser Resonatoren die gesuchte Entropie eines einzelnen Resonators der Eigenfrequenz $\nu$. Das Ergebnis kann man ausdr”ucken[^13] durch die mittlere Energie $E=E_{\text{total}}/N$ eines Resonators und des noch unbekannten “Energiequantums” $\varepsilon$: $$\label{eq:Entropie-Osz} S= \bigl(1+E/\varepsilon\bigr)\cdot \ln\bigl(1+E/\varepsilon\bigr)- \bigl(E/\varepsilon\bigr)\cdot\ln\bigl(E/\varepsilon\bigr)\,.$$ Damit ist die Aufgabe fast gel“ost. Denn es gilt in der Thermodynamik immer (unabh”angig davon, ob man die statistische Interpretation der Entropie zugrundelegt), da“s die Ableitung der Entropie nach der Energie gleich dem Kehrwert der Temperatur ist: $$\label{eq:Entropie-Energie-Temp} \frac{dS}{dE}=\frac{1}{T}.$$ Wendet man dies auf (\[eq:Entropie-Osz\]) an, so kann man sofort $E$ als Funktion von $\varepsilon$ und $T$ berechnen, was dann eingesetzt in (\[eq:Planck1\]) f”ur das Strahlungsgesetz liefert: $$\label{eq:Osz-Energie} \rho(\nu, T) =\frac{8\pi\nu^2}{c^3}\frac{\varepsilon}{\exp(\varepsilon/kT)-1}\,.$$ Damit dies dann dem Planckschen Strahlungsgesetz (\[eq:Planck\]) gleicht, mu“s man eine Annahme ”uber die tats“achliche Gr”o“se der ”‘Energiequanten“’ $\varepsilon$ machen, was ja bisher noch nicht geschehen ist. Schon aus einem direkten Vergleich von (\[eq:Osz-Energie\]) mit der allgemein g”ultigen Gleichung (\[eq:Wien1\]) ergibt sich, da“s $\varepsilon$ proportional zu $\nu$ sein mu”s. Nennt man die Proportionalit“atskonstante $h$, die die Dimension einer Wirkung haben mu”s, so hat man gerade (\[eq:hnu\]), und es ergibt sich die Plancksche Formel. Denkt man sich die Planckschen Energieportionen als Lichtquanten, d.h. im Raum lokalisierte Energiepakete, so entspricht die durch (\[eq:Kombinatorik\]) ausgedr“uckte Abz”ahlung der sogenannten *Bose-Einstein-Statistik*. An dieser ist bemerkenswert, da“s die Lichtquanten als *ununterscheidbare* Entit”aten behandelt werden, d.h. es ist egal, welche der $N$ Lichtquanten den individuellen Resonator besetzen, wichtig ist nur ihre Anzahl. F“ur Planck war (\[eq:Kombinatorik\]) jedoch nicht Ausdruck einer irgendwie ungew”ohnlichen Statistik, da er nicht im Bild der Lichtquanten argumentierte. So bekommt man etwa (\[eq:Kombinatorik\]) auch als Antwort auf die Frage, wieviel M"oglichkeiten es gibt, $N$ Kellen Suppe auf $n$ (unterscheidbare) Teller zu verteilen. Der Planckschen Quantisierungsannahme entspricht hier lediglich die Regel, immer nur ganze Kellen an Suppe zu verteilen. N"aheres zu Einsteins Lichtquantenhypothese {#sec:AnhangEinstein} =========================================== In diesem Anhang wollen wir etwas n“aher ausf”uhren, durch welche mathematische Schlu“skette Einstein zu seiner Lichtquantenhypothese gef”uhrt wurde. Grundlage ist wieder das Boltzmannsche Prinzip (\[eq:Boltzmann-Entropie\]). In diesem d“urfen wir $W$ auch durch die Wahrscheinlichkeit des Makrozustands ersetzen, denn diese ist proportional zur Anzahl $W$ seiner mikroskopischen Realisierungen. Das Ersetzen von $W$ durch einen dazu proportionalen Ausdruck unter dem Logarithmus f”uhrt aber zu einer additiven Konstanten zur Entropie, die in den nachfolgenden “Uberlegungen herausf”allt, da stets nur Entropie*differenzen* eine Rolle spielen. Als Vorbereitung betrachte man ein Gas aus $N$ Atomen in einem Volumen $V$ bei fester Temperatur $T$. Hinsichtlich der Dynamik der Atome wird nur vorausgesetzt, da“s ihre Aufenthaltswahrscheinlichkeit im Volumen konstant ist, da”s also Teilvolumina gleichen Inhalts auch mit gleicher Wahrscheinlichkeit von einem Atom besetzt werden. Die Wahrscheinlichkeit daf“ur, da”s sich alle Atome in einem Teilvolumen $V_0\subset V$ befinden, ist dann gegeben durch $(V_0/V)^N<1$. Entsprechend hat dieser Zustand eine um einen Betrag $\Delta S$ geringere Entropie als der “uber ganz $V$ gleichverteilte Zustand, wobei $$\label{eq:GasEntropie} \Delta S=S-S_0=k\cdot\ln\left(\frac{V}{V_0}\right)^N\,.$$ Eine analoge ”Uberlegung stellt Einstein nun mit W“armestrahlung an, ebenfalls im Volumen $V$ bei der Temperatur $T$. Dazu mu”s er aber den Ausdruck f“ur die Strahlungsentropie berechnen. Diesen erh”alt er so: Sei $\rho(\nu,T)$ die spektrale Dichte der Energie (hier als bekannt vorausgesetzt) und $\varphi(\nu,T)$ der (zu bestimmende) Ausdruck f“ur die spektrale Dichte der Entropie. Das hei”st, da“s der auf das Volumen $V$ und das Frequenzintervall $[\nu\,,\,\nu+d\nu]$ entfallende Anteil der Strahlungsenergie durch $\rho(\nu,T)V\,d\nu$ und der Anteil der Strahlungsentropie durch $\varphi(\nu,T)V\,d\nu$ gegeben ist. Ganz allgemein gilt in der Thermodynamik, da”s die Ableitung der Entropie nach der Energie das Inverse der absoluten Temperatur $T$ ist. Das gilt auch f“ur die spektralen Verteilungen. Also hat man $$\label{eq:EinLicht1} \frac{\partial \rho}{\partial\varphi}=\frac{1}{T}\,.$$ Kennt man das Strahlungsgesetz, d.h. die Funktion $\rho(\nu,T)$, so kann man damit auf der rechten Seite $1/T$ als Funktion von $\nu$ und $\rho$ ausdr”ucken und die Gleichung integrieren, wodurch man $\varphi$ als Funktion von $\nu$ und $\rho$ erh"alt. Einstein benutzt nun nicht das Plancksche, sondern das Wiensche Gesetz (vgl. (\[eq:Wien2\]), das sich im Grenzfall hoher Frequenzen und/oder kleiner Temperaturen aus ersterem ergibt. L“ost man dieses nach $1/T$ auf, setzt es auf der rechten Seite von (\[eq:EinLicht1\]) ein und integriert einmal nach $\rho$, so erh”alt man $$\label{eq:EinLicht2} \varphi(\nu,T)=-\,\frac{\rho}{b\nu}\cdot \left[\ln\left(\frac{\rho}{a\nu^3}\right)-1\right]+konst.$$ Setzt man f“ur die im Volumen $V$ und Frequenzintervall $[\nu\,,\,\nu+d\nu]$ enthaltene Energie $E=\rho V\,d\nu$ und Entropie $S=\varphi V\,d\nu$, so kann man dies auch so schreiben: $$\label{eq:EinLicht3} S(E,\nu)=-\,\frac{E}{b\nu}\cdot \left[\ln\left(\frac{E}{Va\nu^3\,d\nu}\right)-1\right]+konst.$$ Betrachtet man bei konstantem $E$ und $\nu$ die Differenz der Entropien der Strahlung, einmal im Volumen $V$ und einmal im Volumen $V_0$, so erh”alt man $$\label{eq:EinLicht4} \Delta S=S-S_0=\frac{E}{b\nu}\cdot\ln\left(\frac{V}{V_0}\right) =k\cdot\ln\left(\frac{V}{V_0}\right)^{E/b k\nu}\,.$$ Dies vergleicht Einstein mit (\[eq:GasEntropie\]) und kommt zu dem Schlu“s, da”s sich W“armestrahlung im G”ultigkeitsbereich des Wienschen Strahlungsgesetzes entropisch gesehen so verh“alt, wie ein Gas aus $N=E/b k\nu$ Atomen (siehe das Zitat Einsteins auf Seite). Die ”‘Atome“’ des Lichts hei”sen *Lichtquanten*. Sie sind (im G“ultigkeitsbereich des Wienschen Gesetzes!) als r”aumlich lokalisiert zu denken und haben die Energie $$\label{eq:EinLicht5} \varepsilon=E/N=b k\nu=h\nu\,,$$ wobei wir noch ausgenutzt haben, da“s die Konstante $b$ des Wienschen Gesetzes mit der Planckschen Konstante $h$ ”uber die Boltzmann-Konstante $k$ gem“a”s $h=bk$ verbunden ist. Erster experimenteller Hinweis auf die Quantentheorie {#sec:AnhangLP} ===================================================== Millikans Messungen zum Photoeffekt und seine Pr"azisionsbestimmung des Planckschen Wirkungsquantums {#sec:AnhangMillikan} ==================================================================================================== Energiefluktuationen {#sec:AnhangEnergiefluktuationen} ==================== Als Planck seine Formel zum ersten Mal niederschrieb, geschah dies ohne Kenntnis der erst sp“ater von ihm erdachten Ableitung. Vielmehr erhielt er sie, indem er eine gewisse thermodynamische Gr”o“se f”ur die Wiensche und die Rayleigh-Jeanssche Formel einfach addierte. Eine physikalische Interpretation dieses formalen Vorgehens hatte Planck nicht – wie er sp“ater selber zugab. Erst Einstein hat diese Interpretation sp”ater geliefert, die einen interessanten Aspekt des “Welle-Teichen-Dualismus” darstellt. Setzen wir zur Abk“urzung $\beta:=1/(kT)$ und sei ${\bar{E}}$ wieder die mittlere Energie eines Resonators, jetzt aufgefa”st als Funktion von $\beta$ (statt $T$) und $\nu$, so ist die von Planck betrachtete Gr“o”se gegeben durch die Ableitung $-\,d{\bar{E}}/d\beta$. Was bedeutet Sie? Bevor wir dies kl“aren, wollen wir sie f”ur die aus den drei Strahlungsgesetzen (Rayleigh-Jeans, Wien, Planck) folgenden Ausdr“ucke f”ur ${\bar{E}}$ berechnen. Es ist $$\label{eq:Flukt1} {\bar{E}}= \begin{cases} 1/\beta &\quad\text{Rayleigh-Jeans}\\ h\nu\,\exp(-\beta h\nu) &\quad\text{Wien} \\ \frac{h\nu}{\exp(\beta h\nu)-1} &\quad\text{Planck}\,, \end{cases}$$ also gilt $$\label{eq:Flukt2} -\,\frac{d{\bar{E}}}{d\beta}= \begin{cases} {\bar{E}}^2 &\quad\text{Rayleigh-Jeans}\\ h\nu\,{\bar{E}}&\quad\text{Wien}\\ {\bar{E}}^2+h\nu\,{\bar{E}}&\quad\text{Planck}\,. \end{cases}$$ Somit ist in der Tat f“ur das Plancksche Gesetz diese Gr”o“se (als Funktion von ${\bar{E}}$) additiv aus den entsprechenden Ausdr”ucken des Rayleigh-Jeansschen und Wienschen Gesetzes zusammengesetzt. Einsteins “Uberlegungen sind nun statistischer Natur – genauer gesagt betrachtet er statistische Fluktuationen der Energie des Strahlungsfeldes, was man wegen (\[eq:Planck1\]) auch auf die Resonatoren ”ubertragen kann. Dabei ist seine zentrale Idee, die Boltzmannsche Gleichung (\[eq:Boltzmann-Entropie\]) umgekehrt zu lesen, d.h. das statistische Gewicht als Funktion der Entropie auszudr“ucken. Dr”uckt man die Entropie (bei fester Temperatur) als Funktion der Energie aus und entwickelt um das dem Gleichgewicht entsprechende lokale Maximum bei ${\bar{E}}= {\bar{E}}_0$, so kann man daraus in quadratischer Ordnung die normierte Wahrscheinlichkeitsverteilung f“ur eine Energiefluktuation $\epsilon={\bar{E}}-{\bar{E}}_0$ ableiten: $$\label{eq:Flukt3} P(\epsilon)=\sqrt{\frac{\gamma}{2\pi}}\, \exp\bigl(-\tfrac{1}{2}\gamma\epsilon^2\bigr)\,.$$ Hier ist $$\label{eq:Flukt4} \gamma:=-k^{-1}\cdot\frac{d^2}{d{\bar{E}}^2}\Big\vert_{{\bar{E}}={\bar{E}}_0} =-\frac{d\beta}{d{\bar{E}}}\Big\vert_{{\bar{E}}={\bar{E}}_0}\,,$$ wobei die zweite Gleichheit aus der allgemein g”ultigen thermodynamischen Relation (\[eq:Entropie-Energie-Temp\]) folgt. Also ist das mittlere Schwankungsquadrat der Energie gegeben durch: $$\label{eq:Flukt5} \langle\epsilon^2\rangle:=\int_{-\infty}^\infty P(\epsilon)\epsilon^2\,d\epsilon=\gamma^{-1}=-\,\frac{d{\bar{E}}}{d\beta}\,.$$ Damit ist die Gr“o”se, die Planck seiner formalen Interpolation zugrundelegte, als das mittlere Schwankungsquadrat der Energie erkannt. Dieses verh“alt sich bei der Planckschen Formel so, als ob es zwei statistisch unabh”angige Ursachen h“atte: die Energieschwankungen der Rayleigh-Jeans-Formel, die man mit dem klassischen Wellenbild erkl”aren kann, und die der Wienschen Formel, die dem Teilchenbild der Lichtquanten entspricht. In dieser Hinsicht vereinigt die Plancksche Formel beide Aspekte in gleichberechtigter Weise. Stellt man das Gesagte konsequent im Teilchenbild (Lichtquanten) dar, so kann man statt von Resonatorenergien von Besetzungszahlen $n$ sprechen, indem man jede Energie durch $h\nu$ dividiert. Aus der letzten Zeile in (\[eq:Flukt2\]) erh“alt man dann einen einfachen Ausdruck f”ur das Schwankungsquadrat der Besetzungszahl: $$\label{eq:Flukt6} \langle (n-\bar n)^2\rangle=\bar n +\bar n^2\,.$$ Der erste Term w“are alleine vorhanden, wenn es sich um klassisch unabh”angige Teilchen handelte, wie man leicht nachpr"uft.[^14] [99]{} Niels Bohr: “The Structure of the Atom”. Nobel-Vorlesung vom 11. Dezember 1922 (Nobelpreis 1922).\ Online unter $\langle$nobelprize.org/physics/laureates/1922/bohr-lecture.html$\rangle$. Niels Bohr, Hendrik Kramers und J. Slater: "\`“Uber die Quantentheorie der Strahlung”’. Zeitschrift f“ur Physik, 24 (1924) pp.69-87. Arthur H. Compton: ”‘A quantum theory of the scattering of X-rays by light elements.“’ Physical Review 21 (1923) pp.483-502. Arthur H. Compton: ”‘Directed quanta of scattered X-rays“’. Physical Review 26 (1925) pp.289-299. Oliver Darrigol: ”‘The historians disagreement over the meaning of Planck’s quantum“’. Centaurus, 43 (2001) 219-239.\ Online unter $\langle$www.mpiwg-berlin.mpg.de/de/forschung/preprints.html$\rangle$ als Preprint Nr.150 verf”ugbar. Albert Einstein: Collected Works (Princeton University Press). Siehe auch die Internetseite des “Einstein Papers Project”: $\langle$www.einstein.caltech.edu$\rangle$. Gerald Grawert: “Quantenmechanik” (Akademische Verlagsgesellschaft Wiesbaden 1977). Otto Lummer und Ernst Pringsheim: "\`Die Vertheilung der Energie im Spectrum des schwarzen K“orpers und des blanken Platins”’. Verhandlungen der Deutschen Physikalischen Gesellschaft im Jahre 1899, erster Jahrgang, pp.23-41. Herausgegeben von Arthur K“onig, Verlag von Johann Ambrosius Barth, Leipzig 1899. Robert Millikan: ”‘A Direct Photoelectric Determination of Planck’s ‘$h$’.“’ Physical Review, Vol.7 (1916), pp.355-388. Robert Millikan: ”‘The electron and the light-quant from the experimental point of view“’. Nobel-Vorlesung vom 23. Mai 1924 (Nobelpreis 1923).\ Online unter $\langle$nobelprize.org/physics/laureates/1923/millikan-lecture.pdf$\rangle$. Abraham Pais: ”‘Raffiniert ist der Herrgott...“’ Albert Einstein. Eine wissenschaftliche Biographie (Vieweg & Sohn, Braunschweig, 1986). Max Planck: Physikalische Abhandlungen und Vortr”age, Bd. I-III (Vieweg & Sohn, Braunschweig, 1958). Max Planck: Brief an Robert Williams Wood von 1931. Wiedergegeben in "\`Fr“uhgeschichte der Quantentheorie”’, p. 31, von A. Hermann (Physik Verlag, Mosbach 1969). Ernst Pringsheim: “Einfache Herleitung des Kirchhoff’schen Gesetzes”. Verhandlungen der Deutschen Physikalischen Gesellschaft in Jahre 1901, dritter Jahrgang, pp.81-84. Herausgegeben von Arthur K“onig, Verlag von Johann Ambrosius Barth, Leipzig 1901. Heinrich Rubens und Ferdinand Kurlbaum: ”‘“Uber die Emission langwelliger W”armestrahlen durch den schwarzen K“orper bei verschiedenen Temperaturen”’. Sitzungsberichte der Preu“sischen Akademie der Wissenschaften 1900, Gesamtsitzung vom 25.Oktober, pp.929-941. Arnold Sommerfeld: ”‘Das Plancksche Wirkungsquantum und seine allgemeine Bedeutung f“ur die Molek”ulphysik“’. Verhandlungen der Gesellschaft Deutscher Naturforscher und ”Arzte, 83. Versammlung zu Karlsruhe 1911, zweiter Teil, pp.31-50. [^1]: Erschienen in: Herbert Hunziker (Hrsg.) *Der jugendliche Einstein und Aarau* (Birkh"auser Verlag, Basel, 2005) [^2]: Das Zitat des Titels entstammt einem Brief ([@Einstein-CW], Band1, Dokument127) Einsteins vom 12.Dezember 1901 an seine damalige Freundin und sp“atere Frau Mileva Maric, in dem er seine Courage in einer privaten Angelegenheit zu einer Art Lebensmotto erhob und kommentierte: ”‘Es lebe die Unverfrorenheit! Sie ist mein Schutzengel in dieser Welt."’ [^3]: Der 1.Hauptsatz ist der Satz “uber die Erhaltung der Energie. Der 2. Hauptsatz betrifft nicht die Energie, sondern eine andere Zustandsgr”o“se, genannt *Entropie*. Er besagt in der ”alteren, Planck n“aherliegenden Formulierung, da”s die Entropie zeitlich nicht abnimmt. In der modernen, von Planck zun“achst bek”ampften statistischen Interpretation der Entropie, ist diese ein Ma“s f”ur die “Unordnung”. Genauer gesagt ist die Entropie ein (logarithmisches) Ma“s f”ur die Anzahl der Mikrozust“ande, die einen makroskopisch definierten Zustand realisieren (siehe dazu Anhang\[sec:AnhangPlanck\]). Der 2.Hauptsatz besagt in dieser Interpretation, da”s die Entropie *im Zeitmittel* nicht abnimmt (statistische Schwankungen, in denen die Entropie vor“ubergehend kurz abnimmt, sind also erlaubt). Der 2.Hauptsatz regelt die Irreversibilit”at gewisser Prozesse. Das sind dann solche, bei denen die Entropie zunimmt. [^4]: Aus heutiger Sicht ist diese Hoffnung schwer verst“andlich, da die Gesetze der Elektrodynamik genauso wie die Gesetze der Mechanik *invariant unter Bewegungsumkehr* sind. Das bedeutet, da”s mit jeder den Gesetzen gen“ugenden Bewegung die entsprechend zeitlich r”uckl“aufige Bewegung wieder eine m”ogliche Bewegung im Sinne der Gesetze ist. Aus dieser mathematischen Tatsache folgt zwingend die Unm“oglichkeit eines Beweises ”uber die ausnahmslose zeitliche Zunahme einer Zustandsgr“o”se, wie etwa der Entropie. Nur unter *zus"atzlichen* Annahmen, die immer Einschr“ankungen an die Anfangsbedingungen beinhalten, k”onnen solche Beweise funktionieren. Auch Planck wird sp“ater bei seiner ‘Ableitung’ der Wienschen Strahlungsformel eine solche Annahme in etwas versteckter Form machen (durch seine ”‘Hypothese der nat“urlichen Strahlung”’), was f“ur die hier zu besprechenden Entwicklungen aber nicht weiter relevant ist. Noch schwerer verst”andlich wird das Festhalten Plancks an dieser Hoffnung durch den Hinweis, da“s Planck das eben skizzierte Argument sicherlich kannte, n”amlich durch den Mathematiker Ernst Zermelo, der in den Jahren 1894-1897 sein Assistent war und dar"uber einiges publiziert hat. [^5]: Aus bestimmten Gr“unden benutzen Physiker lieber die sogenannte absolute Temperaturskala, auf der die Temperatur nicht in Grad Celsius, sondern in Grad Kelvin angegeben wird. Beide Skalen unterscheiden sich um den konstanten Betrag von $273{,}15$, d.h. $X$ Grad Celsius entsprechen $X+273{,}15$ Grad Kelvin. Null Grad Kelvin, also $-273{,}15$ Grad Celsius, bildet eine absolute untere Grenze f”ur alle erreichbaren Temperaturen, die unter keinen Umst"anden unterschritten werden kann. [^6]: Zwar treten im Argument der Funktion sowohl die Frequenz als auch die Temperatur $T$ auf, aber nur als Quotient $\nu/T$. Dieser Quotient ist die *eine* Variable, von der $f$ alleine abh"angt. [^7]: Dies ist meines Wissens die erste Formulierung eines “Korrespondenzprinzips”, gem“a”s dem die klassische Physik in einem geeigneten “klassischen Limes” aus der Quantentheorie folgen soll. Erst sp"ater hat Niels Bohr diese Forderung zu einem allgemeinen Prinzip erhoben. [^8]: Dies war stets Plancks Haltung, die er noch 1910 "offentlich vertritt; siehe [@Planck-GW], Band2, Dokument71. [^9]: Dadurch erh“alt er eine Modifikation seines fr”uheren Ausdrucks (\[eq:ResEnergieMittel\]) f“ur die mittlere Energie eines Resonators um einen additiven Term $h\nu/2$. Dies markiert das erste Auftreten der heute aus der Quantenmechanik wohlbekannten ”‘Nullpunktsenergie"’. [^10]: Bei diesem Dokument handelt es sich um einen Brief Einsteins an seinen Freund Conrad Habicht vom Mai 1905, dem Einstein vier wissenschaftliche Arbeiten mit folgenden Worten ank“undigt: ”‘Ich verspreche Ihnen vier Arbeiten daf“ur, von denen ich die erste in B”alde schicken k“onnte, da ich die Freiexemplare baldigst erhalten werde. Sie handelt ”uber die Strahlung und die energetischen Eigenschaften des Lichtes und ist sehr revolution“ar, wie Sie sehen werden, wenn Sie mir Ihre Arbeit *vorher* schicken. \[...\] Die vierte Arbeit liegt erst im Konzept vor und ist eine Elektrodynamik bewegter K”orper unter Ben“utzung einer Modifikation der Lehre von Raum und Zeit; der rein kinematische Teil dieser Arbeit wird Sie interessieren”’. Die zuletzt, eher lapidar angek“undigte Arbeit, ist die spezielle Relativit”atstheorie. [^11]: Man beobachtet z.B. eine Zunahme der Wellenl“ange des gestreuten R”ontgenlichts, ganz im Gegensatz zur wellentheoretischen Streutheorie (nach J.J. Thomson). Im Bild der Lichtquanten entspricht diese einfach der Abgabe von Energie des Lichtquants an das als ruhend (bzw. hinreichend langsam) angenommene Elektron. [^12]: Diese Beziehung, die von Planck auf rein klassischem Wege abgeleitet wurde, kann tats“achlich auch durch die Quantenmechanik und Quantenelektrodynamik begr”undet werden; siehe z.B. Kap.15 in [@Grawert] f“ur eine instruktive ”‘halbklassische“’ Ableitung. Im wesentlichen mu”s das Verh“altnis der Wahrscheinlichkeiten f”ur die spontane und induzierte Emission berechnet werden. [^13]: Man verwendet dazu die N“aherungsformel $\ln(N!)\approx N\ln(N)-N$, die f”ur gro“se $N$ g”ultig ist. Nach Planck werden an dieser Stelle sowohl $N$ als auch $n$ als gro“s angenommen. Letzteres ist tats”achlich nicht immer korrekt, was Einstein Planck sp"ater vorwirft. [^14]: Die Wahrscheinlichkeit, von $N$ unterscheidbaren Teilchen irgendwelche $n$ in einem Zustand der Wahrscheinlichkeit $p$ anzutreffen (z.B. im Teilvolumen $V_0\subset V$ zu sein, wobei $V_0/V=p$), ist $W(n)=\binom{N}{n}p^n(1-p)^{N-n}$. Man berechnet nun leicht $\bar{n}=\langle n\rangle:=\bar n=\sum_{n=0}^NnW(n)=Np$ und $\langle n(n-1)\rangle:=\sum_{n=0}^Nn(n-1)W(n)=N(N-1)p^2$. Aus beiden zusammen ergibt sich $\langle(n-\bar{n})^2\rangle=\bar{n}(1-p)$, was f“ur kleine $p$ in $\bar{n}$ ”ubergeht.

--- abstract: | A new two-component system with cubic nonlinearity and linear dispersion: $$\begin{aligned} \left\{\begin{array}{l} m_t=bu_{x}+\frac{1}{2}[m(uv-u_xv_x)]_x-\frac{1}{2}m(uv_x-u_xv), \\ n_t=bv_{x}+\frac{1}{2}[ n(uv-u_xv_x)]_x+\frac{1}{2} n(uv_x-u_xv), \\m=u-u_{xx},~~ n=v-v_{xx}, \end{array}\right. $$ where $b$ is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. Geometrically, this system describes a nontrivial one-parameter family of pseudo-spherical surfaces. In the case of $b=0$, the peaked soliton (peakon) and multi-peakon solutions are studied. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. In the case of $b\neq0$, the weak kink solution is discussed. In addition, a new integrable nonlinear Schrödinger type equation $$\begin{aligned} m_t=bu_{x}+\frac{1}{2}[m(|u|^2-|u_x|^2)]_x-\frac{1}{2}m(uu^\ast_x-u_xu^\ast), \quad m=u-u_{xx},\end{aligned}$$ is obtained by imposing the complex conjugate reduction $v=u^\ast$ to the two-component system. The complex valued $N$-peakon solution and weak kink solution of this nonlinear Schrödinger type equation are also derived. [**Keywords:**]{}Integrable system, Lax pair, Peakon, Weak kink. [**PACS:**]{}02.30.Ik, 04.20.Jb. author: - | Baoqiang Xia$^{1}$[^1],   Zhijun Qiao$^{2}$[^2]\ $^{1}$School of Mathematics and Statistics, Jiangsu Normal University,\ Xuzhou, Jiangsu 221116, P. R. China\ $^2$Department of Mathematics, University of Texas-Pan American,\ Edinburg, Texas 78541, USA title: ' A new two-component integrable system with peakon and weak kink solutions' --- Introduction ============= In recent years, the Camassa-Holm (CH) equation [@CH] $$\begin{aligned} m_t-bu_x+2m u_x+m_xu=0, \quad m=u-u_{xx}, \label{CH}\end{aligned}$$ where $b$ is an arbitrary constant, derived by Camassa and Holm [@CH] as a shallow water wave model, has attracted much attention in the theory of soliton and integrable system. The CH equation was implied in the work of Fuchssteiner and Fokas on hereditary symmetries as a very special case [@FF1]. Since the work of Camassa and Holm [@CH], more diverse studies on this equation have been remarkably developed [@CH2]-[@CGI]. The most interesting feature of the CH equation (\[CH\]) is that it admits peaked soliton (peakon) solutions in the case of $b=0$. A peakon is a weak solution in some Sobolev space with corner at its crest. The stability and interaction of peakons were discussed in several references [@CS1]-[@JR]. In addition to the CH equation, other integrable models with peakon solutions have been found [@DP1]-[@NV1]. Among these models, there are two integrable peakon equations with cubic nonlinearity, which are $$\begin{aligned} m_t=bu_x+\left[ m(u^2-u^2_x)\right]_x, \quad m=u-u_{xx},\label{cCHQ}\end{aligned}$$ and $$\begin{aligned} m_t=u^2m_x+3uu_xm, \quad m=u-u_{xx}.\label{cCHN}\end{aligned}$$ Equation (\[cCHQ\]) was proposed independently by Fokas (1995) [@Fo], Fuchssteiner (1996) [@Fu], Olver and Rosenau (1996) [@OR], and Qiao (2006) [@Q1] where the Lax pair and peaked/cusped solitons are presented. Equation (\[cCHQ\]) is the first cubic nonlinear integrable system possessing peakon solutions. Recently, the peakon stability of equation (\[cCHQ\]) with $b=0$ was worked out by Gui, Liu, Olver and Qu [@GLOQ]. In 2009, Novikov [@NV1] derived another cubic equation, which is equation (\[cCHN\]), from the symmetry approach, and Hone and Wang [@HW1] gave its Lax pair, bi-Hamiltonian structure, and peakon solutions. Very recently [@QXL], we derived the Lax pair, bi-Hamiltonian structure, peakons, weak kinks, kink-peakon interactional and smooth soliton solutions for the following integrable equation with both quadratic and cubic nonlinearity: $$\begin{aligned} m_t=bu_x+\frac{1}{2}k_1\left[ m(u^2-u^2_x)\right]_x+\frac{1}{2}k_2(2 m u_x+ m_xu), \quad m=u-u_{xx},\label{gCH}\end{aligned}$$ where $b$, $k_1$, and $k_2$ are three arbitrary constants. It is very interesting for us to study the multi-component integrable generalizations of peakon equations. For example, in [@OR; @CLZ; @Fa], the authors proposed the two-component generalizations of the CH equation (\[CH\]) with $b=0$, and in [@GX; @SQQ], the authors present the two-component extensions of the cubic nonlinear equation (\[cCHN\]) and equation (\[cCHQ\]) with $b=0$. In this paper, we propose the following two-component system with cubic nonlinearity and linear dispersion $$\begin{aligned} \left\{\begin{array}{l} m_t=bu_{x}+\frac{1}{2}[m(uv-u_xv_x)]_x-\frac{1}{2}m(uv_x-u_xv), \\ n_t=bv_{x}+\frac{1}{2}[ n(uv-u_xv_x)]_x+\frac{1}{2} n(uv_x-u_xv), \\m=u-u_{xx},~~ n=v-v_{xx}, \end{array}\right. \label{eq}\end{aligned}$$ where $b$ is an arbitrary real constant. This system is reduced to the CH equation (\[CH\]), the cubic CH equation (\[cCHQ\]), and the generalized CH equation (\[gCH\]) as $v=-2$, $v=2u$, and $v=k_1u+k_2$, respectively. Thus it is a kind of two-component extensions of equation (\[CH\]), (\[cCHQ\]) and (\[gCH\]) with a linear dispersive term. We prove integrability of system (\[eq\]) by providing its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. Geometrically system (\[eq\]) describes pseudo-spherical surfaces and thus it is also integrable in the sense of geometry. In the case of $b=0$ (dispersionless case), we show that this system admits the single-peakon of traveling wave solution as well as multi-peakon solutions. In particular, the two-peakon dynamic system is explicitly solved and their interactions are investigated in details. In the case of $b\neq0$ (dispersion case), we find that the two-component system (\[eq\]) possesses the weak kink solution. Moreover, by imposing the complex conjugate reduction $v=u^\ast$ to system (\[eq\]), we obtain a new integrable nonlinear Schrödinger type equation $$\begin{aligned} m_t=bu_{x}+\frac{1}{2}[m(|u|^2-|u_x|^2)]_x-\frac{1}{2}m(uu^\ast_x-u_xu^\ast), \quad m=u-u_{xx}, \label{nlseq}\end{aligned}$$ where the symbol $^\ast$ denotes the complex conjugate of a potential. The complex valued $N$-peakon solution and weak kink solution for this nonlinear Schrödinger type system are also proposed. The whole paper is organized as follows. In section 2, a Lax pair, bi-Hamiltonian structure as well as infinitely many conservation laws of equation (\[eq\]) are presented. In section 3, the geometric integrability of equation (\[eq\]) are studied. In section 4, the single-peakon, multi-peakon, and two-peakon dynamics are discussed for the case of $b=0$. Section 5 shows that equation (\[eq\]) possesses the weak kink solution for the case of $b\neq0$. Section 6 derives the peakon and kink solutions of the nonlinear Schrödinger type equation (\[nlseq\]). Some conclusions and open problems are described in section 7. Lax pair, bi-Hamiltonian structure and conservation laws ======================================================== Let us consider a pair of linear spectral problems $$\begin{aligned} \left(\begin{array}{c}\phi_{1}\\\phi_{2} \end{array}\right)_x&=& U\left(\begin{array}{c} \phi_{1}\\\phi_{2} \end{array}\right),\quad U=\frac{1}{2}\left( \begin{array}{cc} -\alpha & \lambda m\\ -\lambda n & \alpha \\ \end{array} \right), \label{LPS}\\ \left(\begin{array}{c}\phi_{1}\\\phi_{2} \end{array}\right)_t&=& V\left(\begin{array}{c} \phi_{1}\\\phi_{2} \end{array}\right),\quad V=-\frac{1}{2}\left( \begin{array}{cc} A & B \\ C & -A \\ \end{array} \right), \label{LPT}\end{aligned}$$ where $m=u-u_{xx}, \ n=v-v_{xx}$, $b$ is an arbitrary constant, $\lambda$ is a spectral parameter, $\alpha=\sqrt{1-\lambda^2b}$, and $$\begin{aligned} \begin{split} A=& \lambda^{-2}\alpha+\frac{\alpha}{2}(uv-u_xv_x)+\frac{1}{2}(uv_x-u_xv), \\ B=& -\lambda^{-1}(u-\alpha u_x)-\frac{1}{2}\lambda m(uv-u_xv_x), \\ C=&\lambda^{-1}(v+\alpha v_x)+\frac{1}{2}\lambda n(uv-u_xv_x). \end{split}\end{aligned}$$ The compatibility condition of (\[LPS\]) and (\[LPT\]) generates $$\begin{aligned} U_t-V_x+[U,V]=0.\label{cc}\end{aligned}$$ Substituting the expressions of $U$ and $V$ given by (\[LPS\]) and (\[LPT\]) into (\[cc\]), we find that (\[cc\]) is nothing but equation (\[eq\]). Hence, (\[LPS\]) and (\[LPT\]) exactly give the Lax pair of (\[eq\]). Let $$\begin{aligned} \begin{split} K&=\left( \begin{array}{cc} 0 & \partial^2-1 \\ 1-\partial^2 & 0 \\ \end{array} \right), \\J&=\left( \begin{array}{cc} \partial m\partial^{-1}m\partial-m\partial^{-1}m & \partial m\partial^{-1} n\partial+m\partial^{-1} n+2b\partial \\ \partial n\partial^{-1} m\partial+ n\partial^{-1} m+2b\partial & \partial n\partial^{-1} n\partial- n\partial^{-1} n \\ \end{array} \right). \label{JK} \end{split}\end{aligned}$$ $J$ and $K$ are Hamiltonian operators. [**Proof**]{} It is obvious that $K$ is a Hamiltonian operator. It is easy to check $J$ is skew-symmetric. We need to prove Jacobi identity $$\begin{aligned} \langle \alpha, J'[J\beta]\gamma\rangle+\langle \beta, J'[J\gamma]\alpha\rangle+\langle \gamma, J'[J\alpha]\beta\rangle=0, \label{Jacb}\end{aligned}$$ where $$\begin{aligned} \alpha=(\alpha_1,\alpha_2)^T,\quad \beta=(\beta_1,\beta_2)^T, \quad \gamma=(\gamma_1,\gamma_2)^T. \label{alpha}\end{aligned}$$ Introduce $$\begin{aligned} \begin{split} \tilde{A}&=\partial^{-1}(m\alpha_{1,x}+n\alpha_{2,x}),\quad \tilde{B}=\partial^{-1}(m\beta_{1,x}+n\beta_{2,x}),\quad \tilde{C}=\partial^{-1}(m\gamma_{1,x}+n\gamma_{2,x}), \\ A&=\partial^{-1}(m\alpha_{1}-n\alpha_{2}),\quad B=\partial^{-1}(m\beta_{1}-n\beta_{2}),\quad C=\partial^{-1}(m\gamma_{1}-n\gamma_{2}). \end{split}\end{aligned}$$ By direct calculations, we have $$\begin{aligned} \begin{split} \langle \alpha, J'[J\beta]\gamma\rangle=&\int[(\gamma_{1,x}m_x+\gamma_{2,x}n_x)\tilde{B}\tilde{A}-(\alpha_{1,x}m_x+\alpha_{2,x}n_x)\tilde{B}\tilde{C} +\tilde{C_x}\tilde{B_x}\tilde{A}-\tilde{A_x}\tilde{B_x}\tilde{C}]dx \\&+\int[(\alpha_{1,x}m-\alpha_{2,x}n)(B\tilde{C}+C\tilde{B})-(\gamma_{1,x}m-\gamma_{2,x}n)(B\tilde{A}+A\tilde{B})]dx \\&+\int[(\alpha_{1}m+\alpha_{2}n)BC-(\gamma_{1}m+\gamma_{2}n)BA]dx \\&-2b\int[(\alpha_{1,x}\beta_{2,x}+\alpha_{2,x}\beta_{1,x})\tilde{C}-(\beta_{2,x}\gamma_{1,x}+\beta_{1,x}\gamma_{2,x})\tilde{A}]dx \\&+2b\int[(\alpha_{2}\beta_{1,x}-\alpha_{1}\beta_{2,x})C-(\beta_{1,x}\gamma_{2}-\beta_{2,x}\gamma_{1})A]dx. \end{split} \label{Jacb1}\end{aligned}$$ Based on (\[Jacb1\]), we may verify (\[Jacb\]) directly. The following relation holds $$\begin{aligned} \langle \alpha, J'[K\beta]\gamma\rangle+\langle \beta, J'[K\gamma]\alpha\rangle+\langle \gamma, J'[K\alpha]\beta\rangle +\langle \alpha, K'[J\beta]\gamma\rangle+\langle \beta, K'[J\gamma]\alpha\rangle+\langle \gamma, K'[J\alpha]\beta\rangle=0. \label{CHO}\end{aligned}$$ [**Proof**]{} Direct calculations yield that $$\begin{aligned} \begin{split} \langle \alpha, J'[K\beta]\gamma\rangle=&-\int[(\alpha_{2,x}\beta_1-\alpha_{1,x}\beta_2)\tilde{C}-(\gamma_{2,x}\beta_1-\gamma_{1,x}\beta_2)\tilde{A}]dx \\&-\int[(\alpha_{1,x}\beta_{2,xx}-\alpha_{2,x}\beta_{1,xx})\tilde{C}-(\beta_{2,xx}\gamma_{1,x}-\beta_{1,xx}\gamma_{2,x})\tilde{A}]dx \\&+\int[(\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1})C-(\beta_{1}\gamma_{2}+\beta_{2}\gamma_{1})A]dx \\&-\int[(\alpha_{1}\beta_{2,xx}+\alpha_{2}\beta_{1,xx})C-(\beta_{1,xx}\gamma_{2}+\beta_{2,xx}\gamma_{1})A]dx. \end{split} \label{CHO1}\end{aligned}$$ Formula (\[CHO\]) may be verified based on (\[CHO1\]). From Lemmas 1 and 2, we immediately obtain $J$ and $K$ are compatible Hamiltonian operators. Furthermore, we have Equation (\[eq\]) can be rewritten as the following bi-Hamiltonian form $$\begin{aligned} \left(m_t,~ n_t\right)^{T}=J \left(\frac{\delta H_1}{\delta m},~\frac{\delta H_1}{\delta n}\right)^{T}=K \left(\frac{\delta H_2}{\delta m},~\frac{\delta H_2}{\delta n}\right)^{T},\label{BH}\end{aligned}$$ where $J$ and $K$ are given by (\[JK\]), and $$\begin{aligned} \begin{split} H_1&=\frac{1}{2}\int_{-\infty}^{+\infty}(uv+u_xv_x)dx, \\ H_2&=\frac{1}{4}\int_{-\infty}^{+\infty}[(u^2v_x+u_x^2v_x-2uu_xv)n+2b(uv_x-u_xv)]dx. \end{split} \label{H}\end{aligned}$$ Let us now construct conservation laws of equation (\[eq\]). Let $\omega=\frac{\phi_2}{\phi_1}$, where $\phi_1$ and $\phi_2$ are determined through equations (\[LPS\]) and (\[LPT\]). From (\[LPS\]), one can easily verify that $\omega$ satisfies the following Riccati equation $$\begin{aligned} \omega_x=-\frac{1}{2}\lambda m \omega^2+ \alpha\omega-\frac{1}{2}\lambda n. \label{ric}\end{aligned}$$ Equations (\[LPS\]) and (\[LPT\]) give rise to $$\begin{aligned} (\ln \phi_1)_x=-\frac{\alpha}{2}+\frac{1}{2}\lambda m\omega, \quad (\ln \phi_1)_t=-\frac{1}{2}\left(A+B\omega\right), \label{lnp}\end{aligned}$$ which yields conservation law of equation (\[eq\]): $$\begin{aligned} \rho_t=F_x, \label{CL}\end{aligned}$$ where $$\begin{aligned} \begin{split} \rho&=m\omega, \\F&=\lambda^{-2}(u-\alpha u_x)\omega-\frac{1}{2}\lambda^{-1}\left(\alpha uv-\alpha u_xv_x+uv_x-u_xv\right)+\frac{1}{2}m(uv-u_xv_x)\omega. \end{split} \label{rj}\end{aligned}$$ Usually $\rho$ and $F$ are called a conserved density and an associated flux, respectively. To derive the explicit form of conservation densities in the case of $b=0$, we expand $\omega$ in terms of negative powers of $\lambda$ as follows: $$\omega=\sum_{j=0}^{\infty}\omega_j\lambda^{-j}.\label{oe1}$$ Substituting (\[oe1\]) into (\[ric\]) and equating the coefficients of powers of $\lambda$, we arrive at $$\begin{aligned} \omega_{0}&=\sqrt{-\frac{n}{m}}, \qquad \omega_{1}=\frac{mn_x-m_xn-2mn}{2m^2n}, \label{w1}\end{aligned}$$ and the recursion relation for $\omega_{j}$: $$\begin{aligned} \omega_{j+1}&=\frac{1}{m\omega_0}\left[\omega_j-\omega_{j,x}-\frac{1}{2}m\sum_{i+k=j+1,~i,k\geq 1}\omega_i\omega_k\right],\quad j\geq 1. \label{wj}\end{aligned}$$ Inserting (\[oe1\]), (\[w1\]) and (\[wj\]) into (\[rj\]),we finally get the following infinitely many conserved densities and the associated fluxes of equation (\[eq\]): $$\begin{aligned} \begin{split} \rho_{0}&=\sqrt{-mn}, ~~~~ F_0=\frac{1}{2}\sqrt{-mn}(uv-u_xv_x), \\ \rho_{1}&=\frac{mn_x-m_xn-2mn}{2mn}, ~~~~ F_1=-\frac{1}{2}(uv-u_xv_x+uv_x-u_xv)+\frac{1}{2}\rho_1(uv-u_xv_x), \\ \rho_{j}&=m\omega_j, ~~~~F_{j}=(u-u_x)\omega_{j-2}+\frac{1}{2}\rho_j(uv-u_xv_x),\quad j\geq 2, \end{split} \label{rjj}\end{aligned}$$ where $\omega_j$ is given by (\[w1\]) and (\[wj\]). Geometric integrability ======================= Based on the work of Chern and Tenenblat [@CT] and the subsequent works [@EGR; @EGR2], a differential equation for a real valued function $u(x,t)$ is said to describe pseudo-spherical surfaces if it is the necessary and sufficient condition for the existence of smooth functions $f_{\alpha\beta}$, $\alpha=1,~2,~3$, $\beta=1,~2$, depending on $x$, $t$, $u$ and its derivatives, such that the one-forms $\omega_{\alpha}=f_{\alpha 1}dx+f_{\alpha 2}dt$ satisfy the structure equations of a surface of constant Gaussian curvature equal to $-1$ with metric $\omega_{1}^2+\omega_{2}^2$ and connection one-form $\omega_{3}$, namely $$\begin{aligned} \begin{split} d\omega_1&=\omega_3\wedge\omega_2, \\ d\omega_2&=\omega_1\wedge\omega_3, \\ d\omega_3&=\omega_1\wedge\omega_2. \end{split} \label{se}\end{aligned}$$ Let us consider $$\begin{aligned} \begin{split} f_{11}&=-\frac{1}{2}\lambda[e^{(\alpha-\lambda)x}m-e^{(\lambda-\alpha)x}n], \\ f_{12}&=\frac{1}{2}\lambda^{-1}[e^{(\lambda-\alpha)x}(v+\alpha v_x)-e^{(\alpha-\lambda)x}(u-\alpha u_x)]+\frac{1}{4}\lambda[e^{(\lambda-\alpha)x}n-e^{(\alpha-\lambda)x}m](uv-u_xv_x), \\ f_{21}&=\lambda, \\ f_{22}&=\lambda^{-2}\alpha+\frac{\alpha}{2}(uv-u_xv_x)+\frac{1}{2}(uv_x-u_xv), \\ f_{31}&=-\frac{1}{2}\lambda[e^{(\alpha-\lambda)x}m+e^{(\lambda-\alpha)x}n], \\ f_{32}&=-\frac{1}{2}\lambda^{-1}[e^{(\lambda-\alpha)x}(v+\alpha v_x)+e^{(\alpha-\lambda)x}(u-\alpha u_x)]-\frac{1}{4}\lambda[e^{(\lambda-\alpha)x}n+e^{(\alpha-\lambda)x}m](uv-u_xv_x), \end{split} \label{f}\end{aligned}$$ and introduce the following three one-forms $$\begin{aligned} \begin{split} \omega_1&=f_{11}dx+f_{12}dt, \\ \omega_2&=f_{21}dx+f_{22}dt, \\ \omega_3&=f_{31}dx+f_{32}dt. \end{split} \label{omg}\end{aligned}$$ Through a direct computation, we find that the structure equations (\[se\]) hold whenever $u(x,t)$ and $v(x,t)$ are solutions of system (\[eq\]). Thus we have System (\[eq\]) describes pseudo-spherical surfaces. Recall that a differential equation is geometrically integrable if it describes a nontrivial one-parameter family of pseudo-spherical surfaces. It follows that System (\[eq\]) is geometrically integrable. According to [@CT]-[@RS], we have the following fact A geometrically integrable equation with associated one-forms $\omega_{\alpha}$, $\alpha=1, 2, 3$, is the integrability condition of a one-parameter family of $sl(2,R)$-valued linear problem $$\begin{aligned} dv=\Omega v, \label{dv}\end{aligned}$$ where $\Omega$ is the matrix-valued one-form $$\begin{aligned} \Omega=Xdx+Tdt=\frac{1}{2}\left( \begin{array}{cc} \omega_2 & \omega_1-\omega_3\\ \omega_1+\omega_3 & -\omega_2 \\ \end{array} \right). \label{Omega}\end{aligned}$$ Therefore, the one-forms (\[omg\]) and (\[dv\]) yield an $sl(2,R)$-valued linear problem $v_x=Xv$ and $v_t=Tv$, whose integrability condition is the two-component system (\[eq\]). The expression (\[Omega\]) implies that the matrices $X$ and $T$ are $$\begin{aligned} \begin{split} X&=\frac{1}{2}\left( \begin{array}{cc} \lambda & \lambda e^{(\lambda-\alpha)x} n\\ -\lambda e^{(\alpha-\lambda)x} m & -\lambda \\ \end{array} \right), \\ T&=\frac{1}{2}\left( \begin{array}{cc} \lambda^{-2}\alpha+\frac{\alpha}{2}(uv-u_xv_x)+\frac{1}{2}(uv_x-u_xv) & [\lambda^{-1}(v+\alpha v_x) +\frac{\lambda}{2}n(uv-u_xv_x)]e^{(\lambda-\alpha)x}\\ -[\lambda^{-1}(u-\alpha u_x)+\frac{\lambda}{2} m(uv-u_xv_x)]e^{(\alpha-\lambda)x} & -\lambda^{-2}\alpha-\frac{\alpha}{2}(uv-u_xv_x)-\frac{1}{2}(uv_x-u_xv) \\ \end{array} \right). \end{split} \label{XT}\end{aligned}$$ Peakon solutions of (\[eq\]) in the case of $b=0$ ================================================= Let us suppose that the single peakon solution of (\[eq\]) with $b=0$ is of the following form $$\begin{aligned} u=c_1e^{-\mid x-ct\mid},\quad v=c_2e^{-\mid x-ct\mid}, \label{ocp}\end{aligned}$$ where two constants $c_1$ and $c_2$ are to be determined. From the expressions of $u$ and $v$ in (\[ocp\]), we see that their first order derivatives are discontinuous at $x=ct$. Thus (\[ocp\]) can not be a solution of equation (\[eq\]) with $b=0$ in the classical sense. However, with the help of distribution theory we are able to write out $u_x$, $m$ and $v_x$, $n$ as follows $$\begin{aligned} \begin{split} u_x&=-c_1sgn(x-ct)e^{-\mid x-ct\mid}, \quad m=2c_1\delta(x-ct), \\ v_x&=-c_2sgn(x-ct)e^{-\mid x-ct\mid}, \quad n=2c_2\delta(x-ct). \end{split} \label{ocpd}\end{aligned}$$ Substituting (\[ocp\]) and (\[ocpd\]) into (\[eq\]) with $b=0$ and integrating in the distribution sense, one can readily see that $c_1$ and $c_2$ should satisfy $$\begin{aligned} c_1c_2=-3c. \label{C1}\end{aligned}$$ In particular, for $c_1=c_2$, we recover the single peakon solution $u=\pm \sqrt{-3c}e^{-\mid x-ct\mid}$ of the cubic CH equation (\[cCHQ\]) with $b=0$ [@GLOQ; @QXL]. Let us now assume the two-peakon solution as follows:$$\begin{aligned} u=p_1(t)e^{-\mid x-q_1(t)\mid}+p_2(t)e^{-\mid x-q_2(t)\mid}, \quad v=r_1(t)e^{-\mid x-q_1(t)\mid}+r_2(t)e^{-\mid x-q_2(t)\mid}.\label{tp}\end{aligned}$$ In the sense of distribution, we have $$\begin{aligned} \begin{split} u_x&=-p_1sgn(x-q_1)e^{-\mid x-q_1\mid}-p_2sgn(x-q_2)e^{-\mid x-q_2\mid}, \quad m=2p_1\delta(x-q_1)+2p_2\delta(x-q_2), \\ v_x&=-r_1sgn(x-q_1)e^{-\mid x-q_1\mid}-r_2sgn(x-q_2)e^{-\mid x-q_2\mid}, \quad n=2r_1\delta(x-q_1)+2r_2\delta(x-q_2). \end{split} \label{tcpd}\end{aligned}$$ Substituting (\[tp\]) and (\[tcpd\]) into (\[eq\]) with $b=0$ and integrating through test functions yield the following dynamic system: $$\begin{aligned} \left\{\begin{array}{l} p_{1,t}=\frac{1}{2}p_1(p_1r_2-p_2r_1) sgn(q_1-q_2)e^{ -\mid q_1-q_2\mid},\\ p_{2,t}=\frac{1}{2}p_2(p_2r_1-p_1r_2) sgn(q_2-q_1)e^{ -\mid q_2-q_1\mid},\\ q_{1,t}=-\frac{1}{3}p_1r_1-\frac{1}{2}\left(p_1r_2+p_2r_1\right)e^{ -\mid q_1-q_2\mid},\\ q_{2,t}=-\frac{1}{3}p_2r_2-\frac{1}{2}\left(p_1r_2+p_2r_1\right)e^{ -\mid q_2-q_1\mid},\\ r_{1,t}=-\frac{1}{2}r_1(p_1r_2-p_2r_1) sgn(q_1-q_2)e^{ -\mid q_1-q_2\mid},\\ r_{2,t}=-\frac{1}{2}r_2(p_2r_1-p_1r_2) sgn(q_2-q_1)e^{ -\mid q_2-q_1\mid}.\\ \end{array}\right. \label{tpode}\end{aligned}$$ Guided by the above equations, we may conclude the following relations: $$\begin{aligned} p_1=Dp_2, ~~p_1r_1=A_1, ~~p_2r_2=A_2, \label{rtp}\end{aligned}$$ where $D$, $A_1$ and $A_2$ are three arbitrary integration constants. [**If $A_1=A_2$**]{}, we arrive at the following solution of (\[tpode\]): $$\begin{aligned} \begin{split} p_1(t)&=Be^{\frac{1}{2D}(D^2A_1-A_1)sgn(C_1)e^{-\mid C_1\mid}t},\\ p_2(t)&=\frac{p_1}{D},\\ r_1(t)&=\frac{A_1}{p_1},\\ r_2(t)&=\frac{A_1}{p_2},\\ q_{1}(t)&=-\left[\frac{1}{3}A_1+\frac{1}{2D}(D^2A_1+A_1)e^{-\mid C_1\mid}\right]t+\frac{1}{2}C_1,\\ q_{2}(t)&=q_{1}(t)-C_1, \end{split} \label{tps1}\end{aligned}$$ where $B$ and $C_1$ are two arbitrary non-zero constants. In this case, the collision between two peakons will not happen since $q_{2}(t)=q_{1}(t)-C_1$. In particular, as $ A_1=B=D=1$, $C_1=2$, (\[tps1\]) is reduced to $$\begin{aligned} \begin{split} p_1(t)&=p_2(t)=r_1(t)=r_2(t)=1, \\ q_{1}(t)&=-\left(\frac{1}{3}+e^{-2}\right)t+1, \qquad q_{2}(t)=-\left(\frac{1}{3}+e^{-2}\right)t-1. \end{split}\end{aligned}$$ Thus, the associated solution of (\[eq\]) with $b=0$ becomes $$\begin{aligned} u(x,t)=v(x,t)=e^{-\left|x+\left(\frac{1}{3}+e^{-2}\right)t-1\right|}+e^{-\left|x+\left(\frac{1}{3}+e^{-2}\right)t+1\right|}, \label{tps11}\end{aligned}$$ which has two peaks, and looks like a M-shape soliton solution [@Q1], but not, and see Figure \[fmp\] for this M-shape two-peakon solution. As $ A_1=-B=-D=1$, $C_1=2$, the associated solution of (\[eq\]) with $b=0$ becomes $$\begin{aligned} u(x,t)=v(x,t)=-e^{-\left|x+\left(\frac{1}{3}-e^{-2}\right)t-1\right|}+ e^{-\left|x+\left(\frac{1}{3}-e^{-2}\right)t+1\right|}, \label{tps12}\end{aligned}$$ which has one peak and one trough and looks like N-shape soliton solution. See Figure \[f12\] for this N-shape two-peakon (peakon-antipeakon interaction) solution. ![[]{data-label="f12"}](F11.eps){width="2.2in"} ![[]{data-label="f12"}](F12.eps){width="2.2in"} As $B=2D=1$, $A_1=C_1=2$, (\[tps1\]) becomes $$\begin{aligned} \begin{split} p_1(t)&=\frac{1}{2}p_2(t)=e^{-\frac{3}{2}e^{-2}t}, \quad r_1(t)=2r_2(t)=2e^{\frac{3}{2}e^{-2}t}, \\ q_{1}(t)&=-\left(\frac{2}{3}+\frac{5}{2}e^{-2}\right)t+1, \qquad q_{2}(t)=-\left(\frac{2}{3}+\frac{5}{2}e^{-2}\right)t-1. \end{split} \label{f13pq}\end{aligned}$$ and the associated solution of (\[eq\]) with $b=0$ becomes $$\begin{aligned} \left\{\begin{array}{l} u(x,t)=e^{-\frac{3}{2}e^{-2}t}\left(e^{- \left| x+(\frac{2}{3}+\frac{5}{2}e^{-2})t-1 \right|} +2e^{- \left| x+(\frac{2}{3}+\frac{5}{2}e^{-2})t+1 \right|}\right), \\ v(x,t)=e^{\frac{3}{2}e^{-2}t}\left(2e^{- \left| x+(\frac{2}{3}+\frac{5}{2}e^{-2})t-1 \right|} +e^{- \left| x+(\frac{2}{3}+\frac{5}{2}e^{-2})t+1 \right|}\right). \end{array} \right. \label{tps13}\end{aligned}$$ From (\[f13pq\]), one can easily see that the amplitudes $p_1(t)$ and $p_2(t)$ of potential $u(x,t)$ are two monotonically decreasing functions of $t$, while the amplitudes $r_1(t)$ and $r_2(t)$ of potential $v(x,t)$ are two monotonically increasing functions of $t$. Figures \[f13u\] and \[f13v\] show the profiles of the potentials $u(x,t)$ and $v(x,t)$. ![[]{data-label="f13v"}](F13u.eps){width="2.2in"} ![[]{data-label="f13v"}](F13v.eps){width="2.2in"} [**If $A_1\neq A_2$**]{}, we may obtain the following solution of (\[tpode\]): $$\begin{aligned} \begin{split} p_1(t)&=Be^{\frac{3(A_2D^2-A_1)}{2D(A_1-A_2)}e^{- \frac{1}{3}\mid(A_1-A_2)t\mid}}, \\ p_2(t)&=\frac{p_1}{D},\\ r_1(t)&=\frac{A_1}{p_1}, \\ r_2(t)&=\frac{A_2}{p_2},\\ q_{1}(t)&=-\frac{1}{3}A_1t+\frac{3(A_2D^2+A_1)}{2D(A_1-A_2)}sgn[(A_1-A_2)t]\left(e^{- \frac{1}{3}\mid(A_1-A_2)t\mid}-1\right),\\ q_{2}(t)&=-\frac{1}{3}A_2t+\frac{3(A_2D^2+A_1)}{2D(A_1-A_2)}sgn[(A_1-A_2)t]\left(e^{- \frac{1}{3}\mid(A_1-A_2)t\mid}-1\right), \end{split} \label{tps2}\end{aligned}$$ where $B$ is an arbitrary integration constant. Let us study the following special cases of this solution. [Case 1]{}. Let $A_1=1$, $A_2=4$, $B=1$, $D=\frac{1}{2}$, then $$\begin{aligned} \left\{\begin{array}{l} p_1(t)=r_1(t)=1, ~~p_2=r_2(t)=2,\\ q_{1}(t)=-\frac{1}{3}t+2sgn(t)\left(e^{- \mid t\mid}-1\right),\\ q_{2}(t)=-\frac{4}{3}t+2sgn(t)\left(e^{- \mid t\mid}-1\right). \end{array}\right. \label{case1}\end{aligned}$$ The associated two-peakon solution of (\[eq\]) becomes $$\begin{aligned} u(x,t)=v(x,t)=e^{-\left|x+\frac{1}{3}t-2sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}+2e^{-\left|x+\frac{4}{3}t-2sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}. \label{tps21}\end{aligned}$$ As $t<0$ and $t$ is going to $0$, the tall peakon with amplitude $2$ and at peak position $q_2$ chases after the short peakon with the amplitude $1$ and at peak position $q_1$. The two-peakon collides at the moment of $t=0$. After the collision ($t>0$), the peaks separate (the tall peakon surpasses the short one) and develop on their own way. See Figure \[f21\] for the detailed development of this kind of two-peakon. . ![[]{data-label="f22u"}](F1u.eps){width="2.2in"} ![[]{data-label="f22u"}](F2u.eps){width="2.2in"} [Case 2]{}. Let $A_1=1$, $A_2=4$, $B=1$, $D=1$, then we have $$\begin{aligned} \left\{\begin{array}{l} p_1(t)=p_2(t)=e^{-\frac{3}{2}e^{- \mid t\mid}}, \\ r_1(t)=e^{\frac{3}{2}e^{- \mid t\mid}},~~r_2(t)=4e^{\frac{3}{2}e^{- \mid t\mid}},\\ q_{1}(t)=-\frac{1}{3}t+\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right),\\ q_{2}(t)=-\frac{4}{3}t+\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right). \end{array}\right. \label{case2}\end{aligned}$$ The associated two-peakon solution of (\[eq\]) becomes $$\begin{aligned} \left\{\begin{array}{l} u(x,t)=e^{-\frac{3}{2}e^{- \mid t\mid}}\left(e^{-\left|x+\frac{1}{3}t-\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}+e^{-\left|x+\frac{4}{3}t-\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}\right),\\ v(x,t)=e^{\frac{3}{2}e^{- \mid t\mid}}\left(e^{-\left|x+\frac{1}{3}t-\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}+4e^{-\left|x+\frac{4}{3}t-\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}\right). \end{array}\right. \label{tps22}\end{aligned}$$ For the potential $u(x,t)$, the two-peakon solution possesses the same amplitude $e^{-\frac{3}{2}e^{- \mid t\mid}}$, which reaches the minimum value at the moment of collision ($t=0$). Figure \[f22u\] shows the profile of the two-peakon dynamics for the potential $u(x,t)$. For the potential $v(x,t)$, the two-peakon solution with the amplitudes $e^{\frac{3}{2}e^{- \mid t\mid}}$ and $4e^{\frac{3}{2}e^{- \mid t\mid}}$ collides at the moment of $t=0$. At this moment, the amplitudes attain the maximum value and the two-peakon overlaps into one peakon $5e^{\frac{3}{2}}e^{- \mid x \mid}$, which is much higher than other moments. It is interesting that this wave looks somewhat like the so-called rogue wave or monster wave [@GL; @OOS]. See Figures \[f22v\] and \[f22v3d\] for the 2-dimensional and 3-dimensional graphs of the two-peakon dynamics for the potential $v(x,t)$. ![[]{data-label="f22v3d"}](F2v.eps){width="2.2in"} ![[]{data-label="f22v3d"}](F2v3d.eps){width="2.2in"} [Case 3]{}. Let $A_1=1$, $A_2=4$, $B=1$, $D=-1$, then we have $$\begin{aligned} \left\{\begin{array}{l} p_1(t)=-p_2(t)=e^{\frac{3}{2}e^{- \mid t\mid}}, \\ r_1(t)=e^{-\frac{3}{2}e^{- \mid t\mid}},~~r_2(t)=-4e^{-\frac{3}{2}e^{- \mid t\mid}},\\ q_{1}(t)=-\frac{1}{3}t-\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right),\\ q_{2}(t)=-\frac{4}{3}t-\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right). \end{array}\right. \label{case3}\end{aligned}$$ The associated two-peakon solution of (\[eq\]) becomes $$\begin{aligned} \left\{\begin{array}{l} u(x,t)=e^{\frac{3}{2}e^{- \mid t\mid}}\left(e^{-\left|x+\frac{1}{3}t+\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}-e^{-\left|x+\frac{4}{3}t+\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}\right),\\ v(x,t)=e^{-\frac{3}{2}e^{- \mid t\mid}}\left(e^{-\left|x+\frac{1}{3}t+\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}-4e^{-\left|x+\frac{4}{3}t+\frac{5}{2}sgn(t)\left(e^{- \mid t\mid}-1\right)\right|}\right). \end{array}\right. \label{tps23}\end{aligned}$$ For the potential $u(x,t)$, the peakon-antipeakon collides and vanishes at the moment of $t=0$. After the collision, the peakon and antipeakon reemerge and separate. For the potential $v(x,t)$, the peakon and trough overlap at the moment of $t=0$, and then they separate. Figures \[f23u\] and \[f23v\] show the peakon-antipeakon dynamics for the potentials $u(x,t)$ and $v(x,t)$. ![[]{data-label="f23v"}](F3u.eps){width="2.2in"} ![[]{data-label="f23v"}](F3v.eps){width="2.2in"} [In general]{}, we suppose $N$-peakon solution has the following form: $$\begin{aligned} u(x,t)=\sum_{j=1}^N p_j(t)e^{-\mid x-q_j(t)\mid}, ~~v(x,t)=\sum_{j=1}^N r_j(t)e^{-\mid x-q_j(t)\mid}. \label{NP}\end{aligned}$$ Substituting (\[NP\]) into (\[eq\]) with $b=0$ and integrating through test functions, we obtain the $N$-peakon dynamic system as follows: $$\begin{aligned} \left\{\begin{array}{l} p_{j,t}=\frac{1}{2}p_j\sum_{i,k=1}^N p_ir_k \left(sgn(q_j-q_k)-sgn(q_j-q_i)\right)e^{ -\mid q_j-q_k\mid-\mid q_j-q_i\mid},\\ q_{j,t}=\frac{1}{6}p_jr_j-\frac{1}{2}\sum_{i,k=1}^Np_ir_k\left(1-sgn(q_j-q_i)sgn(q_j-q_k)\right)e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid},\\ r_{j,t}=-\frac{1}{2}r_j\sum_{i,k=1}^N p_ir_k \left(sgn(q_j-q_k)-sgn(q_j-q_i)\right)e^{ -\mid q_j-q_k\mid-\mid q_j-q_i\mid}. \end{array}\right. \label{dNcp}\end{aligned}$$ It is interesting to study whether the above system is able to be rewritten as an integrable Hamiltonian system by introducing a Poisson bracket. We will investigate this in the future. Weak kink solution of (\[eq\]) in the case of $b\neq0$ ====================================================== In this section, we show that equation (\[eq\]) possesses a weak kink solution in the case of $b\neq0$. We assume equation (\[eq\]) has the following kink wave solution: $$\begin{aligned} u=C_1sgn(x-ct)\left(e^{-\mid x-ct\mid}-1\right), \quad v=C_2sgn(x-ct)\left(e^{-\mid x-ct\mid}-1\right), \label{kink1}\end{aligned}$$ where the constant $c$ is the wave speed, and $C_1$ and $C_2$ are two constants to be determined. In fact, if $C_1\neq0$ and $C_2\neq0$, the potentials $u$ and $v$ in (\[kink1\]) are kink wave solutions due to $$\begin{aligned} \begin{array}{l} \lim_{x\rightarrow +\infty} u=-\lim_{x\rightarrow -\infty} u=-C_1, \\ \lim_{x\rightarrow +\infty} v=-\lim_{x\rightarrow -\infty} v=-C_2. \end{array} \label{reasonkink}\end{aligned}$$ One may easily check that the first order partial derivatives of (\[kink1\]) read $$\begin{aligned} \begin{array}{l} u_x=-C_1e^{-\mid x-ct\mid}, \quad u_t=cC_1e^{-\mid x-ct\mid}, \\ v_x=-C_2e^{-\mid x-ct\mid}, \quad v_t=cC_2e^{-\mid x-ct\mid}. \end{array} \label{dkink1}\end{aligned}$$ But, unfortunately, the second and higher order partial derivatives of (\[kink1\]) do not exist at the point $x=ct$. Thus, like the case of peakon solutions, the kink wave solution in the form of (\[kink1\]) should also be understood in the distribution sense, and therefor we call (\[kink1\]) the weak kink solution to equation (\[eq\]) with $b\neq0$. Substituting (\[kink1\]) and (\[dkink1\]) into (\[eq\]) and integrating through test functions, we may arrive at $$\begin{aligned} \left\{\begin{array}{l} c=-\frac{1}{2}b,\\ C_1C_2=-b. \end{array}\right. \label{ikss}\end{aligned}$$ In the above formula, $c=-\frac{1}{2}b$ means that the kink wave speed is exactly $-\frac{1}{2}b$. In particular, we take $b=-2$ and $C_1=1$, then the corresponding weak kink solution is cast to $$\begin{aligned} u=sgn(x-t)\left(e^{-\mid x-t\mid}-1\right), \quad v=2sgn(x-t)\left(e^{-\mid x-t\mid}-1\right). \label{kink1s}\end{aligned}$$ See Figure \[figkink\] for the profile of the kink wave solution. ![[]{data-label="figkink"}](Fkink.eps){width="2.2in"} Usually, multi-peakon solutions take the form of superpositions of the single-peakon solutions. But, we need to point out that equation (\[eq\]) with $b\neq0$ does not allow the multi-kink solution in the form of the superpositions of single-kink solutions like this: $$\begin{aligned} u=\sum_{j=1}^N p_j(t)sgn(x-q_j(t))\left(e^{-\mid x-q_j(t)\mid}-1\right), \quad v=\sum_{j=1}^N r_j(t)sgn(x-q_j(t))\left(e^{-\mid x-q_j(t)\mid}-1\right). \label{mk}\end{aligned}$$ In fact, substituting (\[mk\]) into (\[eq\]) and integrating through test functions, we find that the solution assumed in the form (\[mk\]) is reduced to zero or single-kink solution (\[kink1\]). In paper [@QXL], we proposed the peakon-kink interactional solutions in the form of $$\begin{aligned} u=p_0(t)sgn(x-q_0(t))\left(e^{-\mid x-q_0(t)\mid}-1\right)+\sum_{j=1}^N p_j(t)e^{-\mid x-q_j(t)\mid},\end{aligned}$$ for the cubic CH equation (\[cCHQ\]) (see section 4.2 in [@QXL]). Here, we naturally hope to seek the peakon-kink interactional solutions for the two-component cubic equation (\[eq\]) with $b\neq0$ in the form of $$\begin{aligned} \left\{\begin{array}{l} u=p_0(t)sgn(x-q_0(t))\left(e^{-\mid x-q_0(t)\mid}-1\right)+\sum_{j=1}^N p_j(t)e^{-\mid x-q_j(t)\mid}, \\ v=r_0(t)sgn(x-q_0(t))\left(e^{-\mid x-q_0(t)\mid}-1\right)+\sum_{j=1}^N r_j(t)e^{-\mid x-q_j(t)\mid}. \end{array} \right. \label{pki}\end{aligned}$$ Unfortunately, by direct calculations, we find that such peakon-kink interactional solutions in the form (\[pki\]) exist only when the potentials $u$ and $v$ satisfy the reduction condition $u=kv$, which reduces the two-component cubic system (\[eq\]) to the cubic CH equation (\[cCHQ\]). In other words, we have not yet found multi-kink wave solutions and peakon-kink interactional solutions to the two-component cubic CH system (\[eq\]) for the general case: $u\not=kv$. This is an interesting and challenging topic, and we will make a further study elsewhere. Solutions of the nonlinear Schrödinger type equation ==================================================== As we mentioned above, system (\[eq\]) is cast into the new nonlinear Schrödinger type equation (\[nlseq\]) under the complex conjugate reduction $v=u^\ast$. Thus the nonlinear Schrödinger type equation (\[nlseq\]) possesses the following Lax pair $$\begin{aligned} \left(\begin{array}{c}\phi_{1}\\\phi_{2} \end{array}\right)_x&=& U\left(\begin{array}{c} \phi_{1}\\\phi_{2} \end{array}\right),\quad U=\frac{1}{2}\left( \begin{array}{cc} -\alpha & \lambda m\\ -\lambda m^\ast & \alpha \\ \end{array} \right), \label{LPS2}\\ \left(\begin{array}{c}\phi_{1}\\\phi_{2} \end{array}\right)_t&=& V\left(\begin{array}{c} \phi_{1}\\\phi_{2} \end{array}\right),\quad V=-\frac{1}{2}\left( \begin{array}{cc} A & B \\ C & -A \\ \end{array} \right), \label{LPT2}\end{aligned}$$ with $\alpha=\sqrt{1-\lambda^2b}$, and $$\begin{aligned} \begin{split} A=& \lambda^{-2}\alpha+\frac{\alpha}{2}(|u|^2-|u_x|^2)+\frac{1}{2}(uu_x^*-u^*u_x), \\ B=& -\lambda^{-1}(u-\alpha u_x)-\frac{1}{2}\lambda m(|u|^2-|u_x|^2), \\ C=&\lambda^{-1}(v+\alpha v_x)+\frac{1}{2}\lambda n(|u|^2-|u_x|^2). \end{split} \label{ABC2}\end{aligned}$$ Next we show that the dispersionless version of equation (\[nlseq\]) (equation (\[nlseq\]) with $b=0$) admits the complex valued $N$-peakon solution, and the dispersion version of equation (\[nlseq\]) (equation (\[nlseq\]) with $b\neq0$) allows the complex valued kink solution. Complex valued peakon solution of (\[nlseq\]) with $b=0$ -------------------------------------------------------- Let us assume the complex valued $N$-peakon solution of (\[nlseq\]) with $b=0$ as the form of $$\begin{aligned} u=\sum_{j=1}^N \left(p_j(t)+ir_j(t)\right)e^{-\mid x-q_j(t)\mid}, \label{NP2}\end{aligned}$$ where $i$ is the imaginary unit $i=\sqrt{-1}$, $p_j(t)$, $r_j(t)$ and $q_j(t)$ are real valued functions. Substituting (\[NP2\]) into (\[nlseq\]) with $b=0$ and integrating through real valued test functions, and separating the real part and imaginary part, we finally obtain that $p_j(t)$, $r_j(t)$ and $q_j(t)$ evolve according to the dynamical system $$\begin{aligned} \left\{ \begin{split} p_{j,t}=&r_j\sum_{l,k=1}^N p_lr_k \left(sgn(q_j-q_k)-sgn(q_j-q_l)\right)e^{ -\mid q_j-q_k\mid-\mid q_j-q_l\mid},\\ r_{j,t}=&p_j\sum_{l,k=1}^N p_lr_k \left(sgn(q_j-q_l)-sgn(q_j-q_k)\right)e^{ -\mid q_j-q_k\mid-\mid q_j-q_l\mid},\\ q_{j,t}=&\frac{1}{6}(p_j^2+r_j^2)-\frac{1}{2}\sum_{l,k=1}^N(p_lp_k+r_lr_k)\left(1-sgn(q_j-q_l)sgn(q_j-q_k)\right)e^{ -\mid q_j-q_l\mid-\mid q_j-q_k\mid}. \end{split} \right.\label{dNcp2}\end{aligned}$$ For $N=1$, (\[dNcp2\]) becomes $$\begin{aligned} p_{1,t}=0,\quad r_{1,t}=0,\quad q_{1,t}=-\frac{1}{3}(p_1^2+r_1^2), \label{dNcp21}\end{aligned}$$ which gives $$\begin{aligned} p_{1}=c_1,\quad r_{1}=c_2,\quad q_{1}=-\frac{1}{3}(c_1^2+c_2^2)t, \label{dNcp21s}\end{aligned}$$ where $c_{1}$ and $c_{2}$ are real valued integration constants. Thus we arrive at the single-peakon solution $$\begin{aligned} u=(c_1+ic_{2})e^{-\mid x+\frac{c_1^2+c_2^2}{3}t\mid}=ce^{-\mid x+\frac{1}{3}|c|^2t\mid},\label{ocpnp2}\end{aligned}$$ where $c=c_{1}+ic_{2}$ and $|c|$ is the modulus of $c$. For $N=2$, we may solve (\[dNcp2\]) as $$\begin{aligned} \left\{\begin{array}{l} q_{1}(t)=-\frac{1}{3}A_1^2t+\Gamma_{1}(t),\\ q_{2}(t)=-\frac{1}{3}A_2^2t+\Gamma_{1}(t),\\ p_{1}(t)=A_1\sin(\Gamma_{2}(t)+A_3),\\ p_{2}(t)=A_2\sin(\Gamma_{2}(t)+A_4),\\ r_{1}(t)=A_1\cos(\Gamma_{2}(t)+A_3),\\ r_{2}(t)=A_2\cos(\Gamma_{2}(t)+A_4), \end{array}\right. \label{2pq}\end{aligned}$$ where $$\begin{aligned} \begin{split} \Gamma_{1}(t)&=\frac{3A_1A_2\cos(A_3-A_4)}{|A_1^2-A_2^2|}sgn(t)\left(e^{-\frac{1}{3}\mid(A_1^2-A_2^2) t\mid}-1\right),\\ \Gamma_{2}(t)&=\frac{3A_1A_2\sin(A_3-A_4)}{A_1^2-A_2^2}e^{-\frac{1}{3}\mid(A_1^2-A_2^2) t\mid}, \end{split} \label{gma}\end{aligned}$$ and $A_1$, $\cdots$, $A_4$ are real valued integration constants. Hence the two-peakon solution reads $$\begin{aligned} u=A_1ie^{-i(\Gamma_{2}(t)+A_3)}e^{-\mid x+\frac{1}{3}A_1^2t-\Gamma_{1}(t)\mid}+A_2ie^{-i(\Gamma_{2}(t)+A_4)}e^{-\mid x+\frac{1}{3}A_2^2t-\Gamma_{1}(t)\mid}, \label{2su}\end{aligned}$$ where the Euler formula $e^{ix}=cosx+isinx$ is employed. Complex valued kink solution of (\[nlseq\]) with $b\neq0$ --------------------------------------------------------- Suppose the complex valued kink wave solution of equation (\[nlseq\]) with $b\neq0$ as the form of $$\begin{aligned} u=\left(C_1+iC_2\right)sgn(x-ct)\left(e^{-\mid x-ct\mid}-1\right), \label{kink12}\end{aligned}$$ where the real constant $c$ is the wave speed, and $C_1$ and $C_2$ are two real constants to be determined. Substituting (\[kink12\]) into equation (\[nlseq\]) with $b\neq0$ and integrating through real valued test functions, and separating its real part and imaginary part, we finally arrive at $$\begin{aligned} \left\{\begin{array}{l} c=-\frac{1}{2}b,\\ C_1^2+C_2^2=-b. \end{array}\right. \label{ikss}\end{aligned}$$ This formula implies that the wave speed is exactly $-\frac{1}{2}b$, where $b<0$ is the coefficient of the linear dispersive term. Thus the complex valued weak kink solution becomes $$\begin{aligned} u=Csgn(x+\frac{1}{2}b t)\left(e^{-\mid x+\frac{1}{2}bt\mid}-1\right), \label{kink12s}\end{aligned}$$ where $C=C_1+iC_2$ and $|C|^2=-b$, $b<0$. It is very interesting and important to study whether equation (\[nlseq\]) possesses the soliton solutions like the form of the soliton solution of the standard nonlinear Schrödinger equation. We will make a further study for these topics. Conclusions and discussions =========================== In our paper, we present a two-component integrable system (\[eq\]) and an integrable nonlinear Schrödinger type equation (\[nlseq\]) with cubic nonlinearity and linear dispersion. For the dispersionless version of the two equations, we derive their $N$-peakon solutions. For the dispersion version of the two equations, we show they admit weak kink solutions. In particular, the two-peakon dynamical system is explicitly solved. The M-shape and N-shape peakons are analytically obtained, and the interactions between two peakons are discussed and shown through their graphs. In [@SQQ], the authors introduced a two-component integrable extension of the dispersionless version of cubic nonlinear equation (\[cCHQ\]) $$\begin{aligned} \left\{\begin{array}{l} m_t=-[m(uv-u_xv_x+u_xv-uv_x)]_x, \\ n_t=-[ n(uv-u_xv_x+u_xv-uv_x)]_x, \\m=u-u_{xx},~~ n=v-v_{xx}. \end{array}\right. \label{SQQ}\end{aligned}$$ We remark that the dispersionless version of our two-component system (system (\[eq\]) with $b=0$) is not equivalent to system (\[SQQ\]). System (\[eq\]) in our paper is able to be reduced to the CH equation, but system (\[SQQ\]) is not, that also implies that these two equations are not equivalent from this aspect. In fact, system (\[eq\]) with $b=0$ and system (\[SQQ\]) belong to some more general negative flows in a hierarchy. For the details of this topic, one may see our very recent paper [@XQZ]. It is an interesting task to study whether there have other new features in the structure of solutions for our two-component system and, especially, for our nonlinear Schrödinger type system with a linear dispersive term. Also other topics, such as smooth soliton solutions [@MY1], cuspons, peakon stability, and algebra-geometric solutions, remain to be developed for our system (\[eq\]) and (\[nlseq\]). ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== The authors Xia was supported by the National Natural Science Foundation of China (Grant Nos. 11301229 and 11271168), the Natural Science Foundation of the Jiangsu Province (Grant No. BK20130224) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB110009). The author Qiao was supported by the Texas Norman Hackerman Advanced Research Program (Grant No. 003599-0001-2009). [^1]: E-mail address: [email protected] [^2]: E-mail address: [email protected]

--- abstract: | The authors in their previous papers obtained compact, arbitrarily accurate expressions for two-center one- and two-electron relativistic molecular integrals expressed over Slater-type orbitals. In this present study, the accuracy limits of given expressions is examined for three-center nuclear attraction integrals, which are the first integral set do not have analytically closed form relations. They are expressed through new molecular auxiliary functions obtained via Neumann expansion of Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrarily values of orbital parameters, quantum numbers. Several methods, such as Laplace expansion of Coulomb interaction, single-center expansion, Fourier transformation method, have been performed in order to evaluate these integrals considering the values of principal quantum numbers in the set of positive integer numbers. This is the first attempts to study the three-center integrals without any restrictions on quantum numbers and in all ranges of orbital parameters. Keywords : PACS numbers : ... . author: - 'A. Ba[ğ]{}c[i]{}' - 'P. E. Hoggan' title: 'Benchmark values for molecular three-center integrals arising in the Dirac equation' --- \[sec:intro\]Introduction ========================= The LCAO-SCF [@Roothaan1951] method is generally employed for molecules, in which molecular wave functions taken to be linear combinations of atomic basis functions whose should possess the cusps condition at the nuclei [@Kato1957] and decay exponentially for large distances [@Agmon1982]. This approach leads to use, namely, Slater-type orbitals [@Slater1930; @Parr1957], $$\begin{aligned} \label{eq:STSOs} \chi_{nlm} \left(\zeta,\vec{r}\right)= \frac{\left(2\zeta \right)^{n+1/2}}{\sqrt{\Gamma(2n+1)}}r^{n-1}e^{-\zeta r}Y_{lm}(\theta,\phi),\end{aligned}$$ here, $Y_{lm}$ are complex or real spherical harmonics $(Y^{*}_{lm}=Y_{l-m}; Y_{lm} \equiv S_{lm})$ differs from the Condon$-$Shortley phases by sign factor $(-1)^{m}$ [@CS1935; @Steinborn1978; @Blanco1997], $\Gamma(z)$ are gamma functions [@Abramowitz1972], $\left\lbrace n, l, m \right\rbrace$ are the principal, orbital, magnetic quantum numbers with, $n \in \mathbb{R}^{+}$, $0\leq l \leq \lfloor n \rfloor$, $-l \leq m \leq l$ and $\lfloor n \rfloor$ stands for the integer part of $n$, respectively, in one$-$ and two$-$electron multi$-$center molecular integrals. These integrals needs to be calculated in spectroscopic accuracy in order to meaningful discussions on basis-set expansion methods, Born-Oppenheimer energy, vibrational frequency calculations. The difficulty of finding analytically closed form relations, however, for molecular integrals have more than two-center referred to as *The bottleneck of quantum chemistry* [@Mulliken1959], have been greatest obstacle since Slater-type orbitals have no simple addition theorem; relations for products of two Slater-type orbitals centered on different positions not available in compact form [@Bouferguene1998]. The Slater-type orbitals are obtained by simplification of Laguerre functions in hydrogen$-$like orbitals [@Willock2009] by keeping only the term of the highest power of $r$, for integer values of principal quantum number $n$ (ISTOs), where $n \in \mathbb{N}^{+}$, $\Gamma(2n+1)=(2n)!$ and it has been proved that they provide extra flexibility for closer variational description of atoms and molecules by considering the values of $n$ in more general set of number, namely positive real numbers (NSTOs), where $n \in \mathbb{R}^{+}$. The studies on the evaluation of molecular integrals, thus, are performed in two main group: those restrict the principal quantum number with integer values, which are practically used in nonrelativistic molecular electronic structure calculations [@Bouferguene1996; @Rico2001] and those free them from any specification but also reduce the area of applications only to investigation of atoms [@Koga1997-1; @Koga1997-2; @Koga1997-3; @Koga1998; @Koga2000; @Erturk2015]. The multi$-$center molecular integrals over ISTOs can be evaluated by expansion of Slater-type orbitals through complete orthonormal basis functions to a new origin [@Barnett1951; @Harris1965; @Guseinov1978; @Guseinov2001; @Bouferguene2005] (see also references therein), $$\begin{gathered} \label{eq:WAVEEXPTHEO} \chi_{nlm}(\zeta,\vec r_{A})\\ =\lim_{N_{e} \to \infty}\sum_{n'l'm'}^{N_{e}} V_{nlm,n'l'm'}^{N_{e}}(\zeta,\vec R_{AB})\chi_{n'l'm'}(\zeta,\vec r_{B}).\end{gathered}$$ or by expressing them as a finite linear combination of $B$ functions through Fourier transform [@Filter1978-1; @Filter1978-2; @Weniger1983; @Grotendorst1985; @Steinborn1992; @Homeier1992]. However, infinite series representation formulas arising in expansion method require increasing upper limit of summation as much as possible to converge to exact values with sufficient decimals (the choice adopted as threshold for the total energy in nonrelativistic variational energy calculation is of order E$-$03 atomic units, therefore, constitute matrix elements should be accurate to E$-$10 atomic units) and presence of spherical Bessel functions brings computational difficulties in Fourier transform method since they provoke an oscillation [@Safouhi1999; @Safouhi2000; @Safouhi2003-1; @Safouhi2003-2]. (Origin) at (0,0,0); (0,0,0) – (7,0,0) node\[anchor=north east\][$Y$]{}; (0,0,0) – (0,7,0) node\[anchor=north west\][$Z$]{}; (0,0,0) – (0,0,7) node\[anchor=south\][$X$]{}; \(A) at (2,5,2); (B) at (7,4,2); (C) at (2.5,2.7,2.5); (E) at (4.5,7,2); (Origin) circle node \[left\] [O]{}; (A) circle (3pt) node \[below left=-0.3cm and 0.00 of A\] [A]{}; (B) circle (3pt) node \[right\] [B]{}; (C) circle (3pt) node \[below=0.1cm and 0.00 of C\] [C]{}; (E) circle (2pt) node \[right\] [$\vec{r}$]{}; (2,5,2) – (7,4,2); (2,5,2) – (2.5,2.7,2.5); (7,4,2) – (2.5,2.7,2.5); (0,0,0) – (2,5,2) node \[black, pos=0.6, left\] [$\vec{R}_{A}$]{}; (0,0,0) – (7,4,2)node \[black,pos=0.6, below\] [$\vec{R}_{B}$]{}; (0,0,0) – (2.5,2.7,2.5) node \[black,pos=0.75, below\] [$\vec{R}_{C}$]{}; (0,0,0) – (4.5,7,2) node \[black,pos=0.55, left\] ; (2,5,2) – (4.5,7,2) node \[black,pos=0.6, left\] [$\vec{r}_{A}$]{}; (7,4,2) – (4.5,7,2) node \[black,pos=0.6, below\] [$\vec{r}_{B}$]{}; (2.5,2.7,2.5) – (4.5,7,2) node \[black,pos=0.7, right\] [$\vec{r}_{C}$]{}; (OriginA) at (2,5,2); (OriginB) at (7,4,2); (OriginC) at (2.5,2.7,2.5); The problem of multi$-$center integrals evaluation by the use of NSTOs even much more through insurmountable. The Slater type orbitals with noninteger principal quantum numbers do not have infinite series representation formulas; they can not be expanded via complete orthonormal basis functions since power series for a function such as $z^\rho$, $z \in \mathbb{C}$ and $\rho \in \mathbb{R}/\mathbb{N}_{0}$ are not analytic at the origin [@Weniger2008; @Weniger2012], where the symbols $\mathbb{C}$, $\mathbb{R}$, $\mathbb{N}_{0}$ used to denote the sets of complex, real and natural numbers, respectively. It should be noted that, this also eliminates possibility of applying binomial expansion theorem in order to evaluate the two-center integrals, those are analytically closed form relations may obtain. Therefore, in mathematical point of view evaluation of multi$-$center molecular integrals using noninteger principal quantum numbers in Slater-type orbitals is an open question. It is far more than better representation of electronic wave$-$function in nonrelativistic electronic structure calculations it is also directly related with solution of the Dirac equation in algebraic approximation. The basis functions to be used in solution of matrix form of the Dirac equation are obtained analogously to L-spinors [@Grant2000; @Grant2007] which are related to the Dirac hydrogenic solutions. Their explicit form include power functions $r^{\gamma}$, $$\begin{aligned} \gamma=\sqrt{\kappa^2-\frac{Z^2}{c^2}}, \end{aligned}$$ with, $Z$ is nuclear charge, $c$ is speed of light, $\kappa=\pm1,\pm2, \pm2, ...$, respectively. They can only be represent by finite summation of Slater-type orbitals with noninteger principal quantum numbers. In particular, the three$-$center integrals are the first set of multi$-$center integrals do not have analytically closed form relations. They have a fundamental importance in the study of molecular systems through $ab-$initio and density functional theory. They are central to the understanding of multi-center integrals. They have been commonly studied with methods presented above. They can also be evaluated through Neumann [@Guseinov1976; @Rico1992; @Harris2002; @Peuker2008] and Laplace expansion [@Roberts1969; @Rico1989; @Rico1991] of Coulomb interaction in prolate spheroidal coordinates. In a new approach the two$-$center integrals have been calculated by the authors for arbitrary values of parameters and quantum numbers via numerical integration techniques [@Bagci2014; @Bagci2015]. The new relativistic molecular auxiliary functions in prolate spheroidal coordinates are presented [@Bagci2015]. They are used to obtain compact form relations for two-electron integrals. Afterwards this idea adapted to calculate overlap integrals via Fourier transform formulas [@Silverstone2014]. The same accuracy, 36-digits, is achieved in both methods. These are so far only known precise calculations for molecular integrals over NSTOs. Hence, they are used in this paper to produce benchmark values for three-center one-electron molecular Dirac integrals as a first time in the literature. The [*Mathematica*]{} programming language [@Mathematica] is utilized for both analytical and numerical calculations. \[sec:Threecenter\]Three-center nuclear attraction integrals ============================================================ Taking into account Fig. \[fig:GeometryFig\], where depiction of coordinates are given for one electron in a triangular conformation, the three-center nuclear attraction integrals are defined as follows, $$\begin{gathered} \label{eq:THREECENTERIDEF} I_{nlm,n'l'm'}(\zeta,\zeta',\vec{R}_{AB},\vec{R}_{AC})\\ =\int \chi_{nlm}^{*} \left(\zeta,\vec{r}\right) \frac{1}{\vert \vec{r}-\vec{R}_{AC} \vert} \chi_{n'l'm'} \left(\zeta',\vec{r}-\vec{R}_{AB}\right)dV,\end{gathered}$$ with, $A,B,C$ are three arbitrary points of the euclidian space, $\vec{R}_{AB}=\vec{AB}$, $\vec{R}_{AC}=\vec{AC}$. The Neumann expansion for $1/{\vert \vec{r}-\vec{R}_{AC} \vert}$ in prolate spheroidal coordinates ($\xi, \nu, \phi$), where $1\leq\xi\leq\infty$, $-1\leq\nu\leq1$, $0\leq\phi\leq2\pi$ [@Ruedenberg1951], $$\begin{gathered} \label{eq:NEUMANNEX} \frac{1}{\vert \vec{r}-\vec{R}_{AC} \vert} =\frac{8\pi}{R_{AB}}\sum_{LM}(-1)^{M}\frac{(L-\vert M \vert)!}{(L+\vert M \vert)!}\\ \times\mathcal{P}_{L}^{\vert M \vert}(\xi_{<})\mathcal{Q}_{L}^{\vert M \vert}(\xi_{>})\\ \times Y_{L}^{M}(\nu_{C},\phi_{C})Y_{L}^{M}(\nu,\phi)^{*},\end{gathered}$$ here, $\mathcal{P}_{L}^{\vert M \vert}(\xi)$, $\mathcal{Q}_{L}^{\vert M \vert}(\xi)$ are first and second kind associated Legendre functions [@Abramowitz1972], $\left\lbrace \xi_<, \xi_> \right\rbrace$ refers to lesser and greater of $\left\lbrace \xi_c, \xi \right\rbrace$, respectively, is utilized in order to obtain expressions for the three-center nuclear attraction integrals over Slater-type orbitals, where the principal quantum numbers are free from specifications; $$\begin{gathered} \label{eq:METHREECENTER} I_{nlm,n'l'm'}(\zeta,\zeta',\vec{R}_{AB},\vec{R}_{AC})\\ =\frac{4\sqrt{2\pi}}{R_{AB}}\mathcal{N}_{nn'}(\zeta,\zeta',R_{AB})\sum_{LM}(-1)^{M} \frac{(L-\vert M \vert)!}{(L+\vert M \vert)!} A_{mm'}^{M}\\ \times Y_{L}^{M}(\nu_{C},\phi_{C}) \left\lbrace \mathcal{Q}_{L}^{\vert M \vert}(\xi_{C}) \mathcal{J}_{nlm,n'l'm'}^{LM}(\zeta,\zeta',R_{AB},\xi_{C}) \right. \\ + \left. \mathcal{P}_{L}^{\vert M \vert}(\xi_{C}) \mathcal{K}_{nlm,n'l'm'}^{LM}(\zeta,\zeta',R_{AB},\xi_{C})\right\rbrace,\end{gathered}$$ here, $$\begin{gathered} \label{eq:NORMCOEF} \mathcal{N}_{nn'}(\zeta,\zeta',R)=\\ \frac{\left(2\zeta \right)^{n+1/2}\left(2\zeta' \right)^{n'+1/2}}{\left[\Gamma(2n+1)\Gamma(2n'+1) \right]^{1/2}}\left( \frac{R}{2}\right)^{n+n'+1}\end{gathered}$$ are the normalization constants and $A^{M}$ coefficients [@Guseinov1970] $$\begin{gathered} \label{eq:NORMCOEF} A_{mm'}^{M}=\frac{1}{\sqrt{2}}\left(2-\vert \eta_{mm'}^{m-m'} \vert \right)^{1/2}\delta_{M,\epsilon\vert m-m' \vert}\\ +\frac{1}{\sqrt{2}}\eta_{mm'}^{m+m'}\delta_{M,\epsilon\vert m+m' \vert},\end{gathered}$$ are the integration over azimuthal angle. The symbol $\epsilon$ may have the value $\pm 1$ and is determined by the product of the signs $m$ and $m'$ (the sign of zero is regarded as positive). The symbols $\eta_{mm'}^{m\pm m'}$ may have the values $\pm 1$ and 0: if among the indices $m$, $m'$ and $m \pm m'$ there occurs a value equal to zero, then $\eta_{mm'}^{m\pm m'}$ is also zero; if all the indices differ from zero, $\eta_{mm'}^{m\pm m'}= \pm 1$ and the sign is determined by product of the signs $m$, $m'$, $m \pm m'$. Thus, the coefficients $A^{M}$ differ from zero only with the values $\vert M \vert =\vert m-m' \vert$, $\vert M \vert =\vert m+m' \vert$.\ Since it is assumed axes of prolate spheroidal coordinate system centered on A, B substitutions in Eqs. (\[eq:THREECENTERIDEF\], \[eq:NEUMANNEX\]) can be written as follows [@Gordadse1935], $$\begin{aligned} \xi=\frac{r_{A}+r_{B}}{R_{AB}}; \hspace{5mm}\nu=\frac{r_{A}-r_{B}}{R_{AB}},\end{aligned}$$ $$\begin{aligned} r_{A}=\frac{R_{AB}}{2}\left(\xi +\nu \right); \hspace{5mm}r_{B}=\frac{R_{AB}}{2}\left(\xi -\nu \right),\end{aligned}$$ $$\begin{aligned} \xi_{C}=\frac{R_{AC}+R_{BC}}{R_{AB}}; \hspace{5mm}\nu_{C}=\frac{R_{AC}-R_{BC}}{R_{AB}}.\end{aligned}$$ The $\mathcal{J}^{LM}, \mathcal{K}^{LM}$ integrals are the auxiliary functions and they are defined as, $$\begin{gathered} \label{eq:JGENAUXILIARY} \mathcal{J}_{nlm,n'l'm'}^{LM}(\zeta,\zeta',R_{AB},\xi_{C})=\int_{1}^{\xi_C}\int_{-1}^{+1}(\xi+\nu)^{n}(\xi-\nu)^{n'}\\ \times e^{-\xi\left[\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB} \right]-\nu\left[\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB} \right]}\\ \times\overline{\mathcal{P}}_{lm } \left( \frac{1+\xi\nu}{\xi+\nu}\right)\overline{\mathcal{P}}_{l'm'} \left( \frac{1-\xi\nu}{\xi-\nu}\right)\\ \times P_{L}^{\vert M \vert}\left(\xi \right)\overline{\mathcal{P}}_{LM}\left(\nu \right)d\xi d\nu\end{gathered}$$ $$\begin{gathered} \label{eq:KGENAUXILIARY} \mathcal{K}_{nlm,n'l'm'}^{LM}(\zeta,\zeta',R_{AB},\xi_{C})=\int_{\xi_C}^{\infty}\int_{-1}^{+1}(\xi+\nu)^{n}(\xi-\nu)^{n'}\\ \times e^{-\xi\left[\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB} \right]-\nu\left[\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB} \right]}\\ \times\overline{\mathcal{P}}_{lm } \left( \frac{1+\xi\nu}{\xi+\nu}\right)\overline{\mathcal{P}}_{l'm'} \left( \frac{1-\xi\nu}{\xi-\nu}\right)\\ \times Q_{L}^{\vert M \vert}\left(\xi \right)\overline{\mathcal{P}}_{LM}\left(\nu \right)d\xi d\nu,\end{gathered}$$ where, $\overline{\mathcal{P}}_{l \vert m \vert}(x)$ are the normalized associated Legendre functions. The auxiliary functions $\mathcal{J}^{LM}, \mathcal{K}^{LM}$ can be defined in a simpler form through relation given for product of two normalized associated Legendre functions centered on points $A$, $B$ in prolate spheroidal coordinates [@Guseinov1976], $$\begin{gathered} \label{eq:PTWOANLEGENDRE} \left[(\xi^2-1)(1-\nu^2) \right]^{\Lambda-\frac{\lambda+\lambda'}{2}}\overline{\mathcal{P}}_{l\lambda }(\cos \theta _{A}) \overline{\mathcal{P}}_{l'\lambda'}(\cos \theta _{B})\\ =\sum _{\alpha =-(2\Lambda-\lambda)}^{l}\sum_{\beta =\lambda'}^{l'} \sum _{q=0}^{\alpha +\beta-2\Lambda-\lambda-\lambda'}{g_{\alpha \beta}^{q}(l\lambda ,l'\lambda';\Lambda)}\\ \times{\left[\frac{({\xi \nu })^{q}} {(\xi +\nu)^{\alpha }(\xi -\nu )^{\beta }}\right]}.\end{gathered}$$ which is obtained in explicit form by taking advantage of binomial expansion theorem, $$\begin{gathered} \label{eq:BINOMXP} (x+a)^{N_{1}}(x-a)^{N_2} \\ =\sum_{s=0}^{N_1+N_2} F_{s}(N_{1},N_{2})x^{N_{1}+N_{2}-s}a^{s},\end{gathered}$$ with, $$\begin{aligned} \cos \theta_{A}=\frac{1+\xi \nu }{\xi +\nu };\hspace{5mm}\cos \theta _{B}=\frac{1-\xi \nu }{\xi -\nu }.\end{aligned}$$ The coefficients $g_{\alpha\beta}^{q}$ occurring in Eq. (\[eq:PTWOANLEGENDRE\]) are determined by, $$\begin{aligned} \label{eq:GABC1} g_{\alpha\beta}^{q}(l\lambda,l'\lambda)=g_{\alpha\beta}^{0}(l\lambda,l'\lambda)F_{q}(\alpha+\lambda,\beta-\lambda)\end{aligned}$$ $$\begin{aligned} \label{eq:GABC1} g_{\alpha\beta}^{0}(l\lambda,l'\lambda)=\sum_{s=0}^{\lambda}(-1)^{s}F_{s}(\lambda)D_{\alpha+2\lambda-2s}^{l\lambda}D_{\beta}^{l'\lambda},\end{aligned}$$ $$\begin{gathered} \label{eq:GAUNT} D_{\beta}^{l\lambda}=\frac{1}{2^l}(-1)^{(l-\beta)/2}\left[\frac{2l+1}{2}\frac{F_{l}(l+\lambda)}{F_{\lambda}(l)} \right]^{1/2}\\ \times F_{(l-\beta)/2}(l)F_{\beta-\lambda}(l+\beta),\end{gathered}$$ where, $\lambda= \vert m \vert$, $\lambda'= \vert m' \vert$ and the quantities $F_{s}(N,N')$ are the generalized binomial coefficients. They are given as, $$\begin{aligned} \label{eq:GENBINOM} F_{s}(N,N')=\sum_{s'}(-1)^{s'}F_{s-s'}(N)F_{s'}(N')\end{aligned}$$ with, $\frac{1}{2}\left[(s-N)+\vert s-N \vert \right]\leq s' \leq min(s,N)$ and $F_{s}(N)$ are binomial coefficients indexed by $N$ and $s$ is usually written $\left(\begin{array}{cc}N\\s\end{array} \right)$, respectively. The Eqs. (\[eq:JGENAUXILIARY\], \[eq:KGENAUXILIARY\]) are, therefore, obtained as follows, $$\begin{gathered} \label{eq:JKGENAUXILIARY} \left[\begin{array}{cc} \mathcal{J}^{LM}_{nlm,n'l'm'}\left(\zeta,\zeta',R_{AB},\xi_{C} \right) \\ \mathcal{K}^{LM}_{nlm,n'l'm'}\left(\zeta,\zeta',R_{AB},\xi_{C} \right) \end{array} \right]=\sum_{\alpha\beta q}g_{\alpha \beta}^{q}(l\lambda ,l'\lambda';\Lambda)\\ \times \left[\begin{array}{cc} \mathcal{J}^{LM,q}_{n-\alpha, n'-\beta}\left(\zeta,\zeta',R_{AB},\xi_{C} \right) \\ \mathcal{K}^{LM,q}_{n-\alpha, n'-\beta}\left(\zeta,\zeta',R_{AB},\xi_{C} \right) \end{array} \right],\end{gathered}$$ with, $$\begin{gathered} \label{eq:JKREDAUXILIARY} \left[\begin{array}{cc} \mathcal{J}^{L\Lambda,q}_{n-\alpha,n'-\beta}\left(\zeta,\zeta',R_{AB},\xi_{C} \right) \\ \mathcal{K}^{L\Lambda,q}_{n-\alpha,n'-\beta}\left(\zeta,\zeta',R_{AB},\xi_{C} \right) \end{array} \right]\\ =\int_{{\tiny \left[\begin{array}{cc} 1 \\ \xi_{C} \end{array} \right]}}^{{\tiny \left[\begin{array}{cc} \xi_{C} \\ \infty \end{array} \right]}}\int_{-1}^{1}{\left(\xi\nu \right)^{q}\left(\xi+\nu \right)^{n-\alpha}\left(\xi-\nu \right)^{n'-\beta}}\\ \times e^{-\xi\left[\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB} \right]-\nu\left[\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB} \right]}\\ \times \begin{bmatrix} \mathcal{P}_{L}^{\vert \Lambda \vert} \left(\xi\right) \\ \mathcal{Q}_{L}^{\vert \Lambda \vert} \left(\xi\right)] \end{bmatrix}\overline{\mathcal{P}}_{L \vert \Lambda \vert}\left(\nu \right)d\xi d\nu.\end{gathered}$$ There are no known convergent series representation formulas, free from specifications on parameters for power functions such as $(\xi+\nu)^{N_{1}}, (\xi-\nu)^{N_{2}}$, $\left\lbrace N_{1}, N_{2} \right\rbrace \in \mathbb{R}$, yet and that poses an obstacle to analytically reduce the $\mathcal{J}^{L\Lambda,q}$, $\mathcal{K}^{L\Lambda,q}$ auxiliary functions to one variable $w_{\mu}^{q}, L_{\mu}^{q}, k_{\mu}^{q}$ auxiliary functions introduced in [@Harris2002]. Thus, the solution should be obtained on the basis of numerical methods. Note that, taking advantage of binomial expansion method for terms containing the angular part of Slater-type orbitals in order to simplify the expressions increases the number of integrals should be numerically calculated. In [*Mathematica*]{} programming language instead of using Eq. (\[eq:JKGENAUXILIARY\]), direct computation of Eqs. (\[eq:JGENAUXILIARY\], \[eq:KGENAUXILIARY\]) are faster. The given relations for auxiliary functions in Eq. (\[eq:JKREDAUXILIARY\]) are calculated by using different expressions of Legendre polynomials and compared according to computational time in Fig. (\[fig:CPUT\]) as sample. The discussions on results are made in the next section. \[sec:ResDis\]Results and discussions ===================================== The literature currently, lack of benchmark values for multi$-$center integrals when Slater-type orbitals are used. Recently, a robust numerical Global$-$adaptive strategy with Gauss$-$Kronrod extension has been applied for two$-$center integrals through prolate spheroidal coordinates and fourier transform method in [@Bagci2014; @Bagci2015; @Silverstone2014]. Benchmark results have been presented for them. The hermitian properties are thus, represented correctly free from specification on quantum numbers, orbital parameters and internuclear distances. In this study it is extended for solution of three$-$center integrals. The algorithm described in [@Bagci2014] has been incorporated into a computer program written in the [*Mathematica*]{} programming language with the included numerical computation packages for solving Eqs. (\[eq:METHREECENTER\], \[eq:JGENAUXILIARY\], \[eq:KGENAUXILIARY\]). The [*Mathematica*]{} programming language can handle approximate real numbers with any number of digits and it is suitable for benchmark evaluation. It is also provides a uniquely integrated and automated environment for parallel computing. It is allows us to compute the formulas including summations using all cores of PC effectively via **ParallelSum** command instead of **Sum**. Note that, in this study all results are given in atomic units (a.u.). [cccccccc]{} $L$ & $\Lambda$ & $q$ & $N_{1}$ & $N_{1}$ & $p_{1}$ & $p_{2}$ & Results\ $0$ & $0$ & $0$ & $1.0$ & $1.0$ & $1.5$ & $1.5$ & ---------------------------------- 3.62319 79582 17897 45490 E$-$01 1.75859 65139 47296 72718 E$-$01 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $1$ & $0$ & $2$ & $3.0$ & $2.0$ & $4.5$ & $0.5$ & ---------------------------------- 1.02307 57195 13525 65648 E$-$03 1.20185 35031 71549 79713 E$-$03 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $2$ & $2$ & $2$ & $3.0$ & $2.0$ & $4.5$ & $0.5$ & ---------------------------------- 1.26482 46553 60927 73809 E$-$02 3.05572 36905 83528 19812 E$-$04 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $3$ & $2$ & $5$ & $9.0$ & $4.0$ & $22.5$ & $0.1$ & ---------------------------------- 1.54156 95966 91532 03328 E$-$11 9.77333 04203 13413 96225 E$-$18 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $5$ & $4$ & $6$ & $15.0$ & $3.0$ & $4.0$ & $0.1$ & ---------------------------------- 7.45864 92397 67729 51682 E$+$06 1.65225 27526 05586 45258 E$+$05 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $0$ & $0$ & $0$ & $1.2$ & $1.5$ & $1.5$ & $1.5$ & ---------------------------------- 4.93516 66112 08595 80377 E$-$01 3.72444 04752 38238 19870 E$-$01 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $1$ & $0$ & $2$ & $3.3$ & $2.4$ & $4.5$ & $0.5$ & ---------------------------------- 7.02522 66592 59862 42272 E$-$04 4.07093 04267 92761 66995 E$-$05 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $2$ & $2$ & $2$ & $3.5$ & $2.5$ & $4.5$ & $0.5$ & ---------------------------------- 2.01392 20090 23026 29191 E$-$02 6.86518 34228 55977 92648 E$-$04 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $3$ & $2$ & $5$ & $9.9$ & $4.1$ & $22.5$ & $0.1$ & ---------------------------------- 2.74437 86677 61624 01320 E$-$11 2.53854 67628 59329 46820 E$-$17 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $5$ & $4$ & $6$ & $15.2$ & $3.6$ & $4.0$ & $0.1$ & ---------------------------------- 9.75947 26622 39909 40431 E$+$06 4.76137 54661 40175 09867 E$+$05 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $5$ & $4$ & $6$ & $15.2$ & $3.6$ & $0.1$ & $4.0$ & ---------------------------------- 9.01557 46105 79752 64049 E$+$08 9.49361 57667 79094 34667 E$+$37 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. \ $5$ & $4$ & $6$ & $15.2$ & $3.6$ & $0.1$ & $4.0$ & ---------------------------------- 6.67509 82662 40505 33704 E$+$08 1.34769 33998 41265 16644 E$+$36 ---------------------------------- : \[tab:Spectrum1\] The values for auxiliary functions defined in Eq.(\[eq:JKREDAUXILIARY\]) with $p_{1}=\frac{1}{2}\left(\zeta+\zeta' \right)R_{AB}$, $p_{2}=\frac{1}{2}\left(\zeta-\zeta' \right)R_{AB}$ and $N_{1},N_{2} \in \mathbb{R}^{+}$. The calculation results are presented in Tables \[tab:Spectrum1\]$-$\[tab:Threecenter4\] and Fig.\[fig:CPUT\] for arbitrary values of quantum numbers, orbital parameters and internuclear distances. The comparisons are made with expansion methods which are given for expansion of wave function in Eq. (\[eq:WAVEEXPTHEO\]) and for charge density expansion to same center by following formula [@Guseinov2000; @Guseinov2002-1; @Guseinov2007] $$\begin{gathered} \label{eq:CHARSEXPTHEO} \rho_{nlm,n'l'm'}(\zeta,\vec r;\zeta',\vec r)\\ =\sum_{l''=\vert l-l'\vert}^{l+l'}\sum_{m''=-l''}^{l''} W_{nlm,n'l'm',n+n'-1l''m''}(\zeta,\zeta',z)\\ \times\chi_{n+n'-1l''m''}(z,\vec r).\end{gathered}$$ They are useful to reduce the three$-$center integrals to basic nuclear attraction integrals, $$\begin{aligned} \label{eq:BASICNUCATTRACT} J_{\kappa\lambda\tau}(z,\vec{R}_{BC})=\frac{1}{\sqrt{4\pi}}\int \chi_{\kappa\lambda\tau}^{*}(z,\vec{r}_{B})\frac{1}{r_{C}}dv_{1}.\end{aligned}$$ The Eqs. (\[eq:WAVEEXPTHEO\], \[eq:CHARSEXPTHEO\]) are used to transform the wave function centered at $A$ to a wave function centered at $B$ then, to transform the charge density centered on same positions to a single wave function, respectively. Here, $z=\zeta+\zeta'$, $\vec{R}_{BC}=\vec{AC}$. The resulting basic nuclear attraction integrals are calculated by the following formula, [@Guseinov2002-2], $$\begin{gathered} \label{eq:BASICNUCATTRACTEVAL} J_{\kappa\lambda\tau}(z,\vec{R})=\\ \frac{2^{\kappa}}{2\lambda+1}\sqrt{\frac{2}{z}}\frac{\Gamma(\kappa+\lambda+2)}{\sqrt{\Gamma(2\kappa+1)}}\frac{1}{(zR)^{\lambda+1}}\\ \times \left(1-\frac{\Gamma(\kappa+\lambda+2,zR)}{\Gamma(\kappa+\lambda+2)}+\frac{(zR)^{2\lambda+1}\Gamma(\kappa-\lambda+1,zR)}{\Gamma(\kappa+\lambda+2)} \right)\\ \times Y_{\lambda,\tau}(\theta,\phi),\end{gathered}$$ where, $\Gamma(n,m)$ are incomplete gamma functions [@Abramowitz1972]. In Fig. \[fig:CPUT\] the Eq. (\[eq:JKREDAUXILIARY\]) is investigated according to computational time in [*Mathematica*]{} programming language. [*Mathematica*]{} includes all the common special functions of mathematical physics. It also provides easy way of computing them precisely. Here, explicit formula (EF) [@Guseinov1995], $$\begin{aligned} \label{eq:ANALLEG} \overline{\mathcal{P}}_{l\lambda}(x)=\left(1-x^2 \right)^{\frac{\lambda}{2}}\sum_{k}b_{l\lambda}^{k}x^{l-\lambda-2k},\end{aligned}$$ $$\begin{gathered} \label{eq:ANALLEGCOFF} b_{l\lambda}^{k}=\frac{1}{2^l}\left[\frac{2l+1}{2F_{\lambda}(l)F_{\lambda}(l+\lambda)} \right]^{\frac{1}{2}}\\ \times(-1)^{k}F_{k}(\lambda+k)F_{l-k}(2l-2k)F_{l-\lambda-2k}(l-k),\end{gathered}$$ where, $0 \leq k \leq E\left[\frac{l-\lambda}{2} \right]$, recurrence relation formula (RF) [@NIST2014] of Legendre polynomials are compared with [*Mathematica*]{} built-function (MF) **LegendreP\[n,m,x\]**. They are presented with red, blue, green lines in Fig. \[fig:CPUT\], respectively. It can be seen from this figure, the direct use of [*Mathematica*]{} buit-function, given for computing of Legendre polynomials, in numerical integration of Eq. (\[eq:JKREDAUXILIARY\]) provide the results faster then explicit or recurrence relation formulas.\ The results for calculation of Eq. (\[eq:JKREDAUXILIARY\]) are also presented in Table \[tab:Spectrum1\] with integer and noninteger values of principal quantum numbers. The first, second rows are obtained from calculation $\mathcal{J}^{L\Lambda,q}_{N_1N_2}$ and $\mathcal{K}^{L\Lambda,q}_{N_1N_2}$ functions, respectively. The auxiliary functions $w_{\mu}^{q}, L_{\mu}^{q}, k_{\mu}^{q}$ defined in [@Harris2002] for three-center integrals are special case of Eq. (\[eq:JKREDAUXILIARY\]). Hence, we believe an importance of present the results for general form of Eq. (\[eq:JKREDAUXILIARY\]). ![\[fig:CPUT\] CPU time for computation of $\mathcal{J}^{L\Lambda,q}_{N_1N_2}$ auxiliary function in Eq.(\[eq:JKREDAUXILIARY\]) according to methods used for calculation of the Legendre polynomials, where, [*Mathematica*]{} built-function (MF), recurrence relation formula (RF), explicit formula (EF) and, $L=3$, $\Lambda=1$, $q=0$, $N_{1}=3$, $N_{2}=2$, $p_{1}=2.5$, $p_{2}=1.5$.](CPUT.eps){width="50.00000%" height="0.22\textheight"} \ The results for Eqs. (\[eq:JGENAUXILIARY\], \[eq:KGENAUXILIARY\]) and Eq. (\[eq:JKGENAUXILIARY\]) are presented in Tables \[tab:Spectrum2\], \[tab:Spectrum3\]. They are given in first, second rows for $\mathcal{J}^{LM}_{nlm,n'l'm'}$ and $ \mathcal{K}^{LM}_{nlm,n'l'm'}$ auxiliary functions, respectively. Note that, the Eq. (\[eq:JKGENAUXILIARY\]) is only differ from Eqs. (\[eq:JGENAUXILIARY\], \[eq:KGENAUXILIARY\]) in that, the normalized associated Legendre polynomials on right-hand side are expanded via Eq. (\[eq:PTWOANLEGENDRE\]) and the numerical global adaptive method is performed to remaining parts. Performing the calculations in [*Mathematica*]{} programming language for such formulas contain summations is disadvantageous in terms of calculation time. The results in Tables \[tab:Spectrum2\], \[tab:Spectrum3\] shows that, numerical Global adaptive method with Gauss-Kronrod extension can be used for computation of Eqs. (\[eq:JGENAUXILIARY\], \[eq:KGENAUXILIARY\]) which is eliminate necessity applying binomial expansion theorem. In Tables \[tab:Threecenter1\], \[tab:Threecenter2\], \[tab:Threecenter3\] the results obtained for three-center integrals are presented for upper limit of summation $L$ is $L=30$. The correct digits are underlined. The digits in bold indicate the convergence property of used method. In first and second rows benchmark results obtained from numerical global adaptive method and results those found in the literature are presented, respectively. Later rows are the results obtained from expansion of the wave-function method and they are in complete agreement with ones from E. Şahin (personal communication), where the upper imit of summations $N_{e}$ are given in parenthesis. The expansion of the wave-function method is tested up to upper limit of summation $N_{e}$, $N_{e}=160$. It is observed that, the results hardly convergent with 10$-$digits for given quantum numbers and orbital parameters in table \[tab:Threecenter2\]. In other tables the convergence remains between 5$-$digits and 10$-$digits. Note that, it is necessary to take into account eight summation and four of them should be infinite in expansion of the wave-function method if NSTOs are used. The values presented in Tables \[tab:Threecenter1\], \[tab:Threecenter2\], \[tab:Threecenter3\] for STOs clearly demonstrate pointlessness of such an attempt.\ On the other hand in our previous papers [@Bagci2014; @Bagci2015] it have been proved that the numerical Global adaptive method with Gauss-Kronrod extension is able to give benchmark values. In particular for Table \[tab:Threecenter4\] the results are presented for different upper limit of summation $L$ appears in Eq. (\[eq:METHREECENTER\]). They differs from upper limit of summation $N_{e}$ used in expansion of STOs in that they are given in brackets. Convergence property of Eq. (\[eq:METHREECENTER\]) is examined in this table. It is found that, by increasing the upper limit of summation the results are convergence to exact values. The results obtained up to upper limit of summation $L$ is $L=40$. It is achieved to 25$-$digits accuracy by determining the upper limit of summation $L$ is $L=30$ accordingly, there is no necessity of performing calculations with upper limit of summation higher than $L=30$ unless more precise results required for a given values of parameters. It should be point out that, the summation appears in Eq. (\[eq:METHREECENTER\]) should not be regarded as having same characteristic with summation arising in expansion of STOs. It is based on expansion of spherical harmonics which have form a complete set of orthonormal functions. Any square-integrable function can be expanded as a linear combination of spherical harmonics. The convergence problems arising in expansion NSTOs can not exist in our method. Without any computational difficulty by increasing the upper limit of summation $L$ can be achieved to desired accuracy rapidly. 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(\[eq:JKGENAUXILIARY\])\ $0$ & $0$ & $1.0$ & $0$ & $0$ & $1.0$ & $0$ & $0$ & $1.5$ & $2.5$ & ---------------------------------------------- 2.90399 07988 62162 00938 68706 03684 E$-$01 1.47689 95097 54804 95414 54214 62847 E$-$01 ---------------------------------------------- \ $2$ & $0$ & $3.0$ & $0$ & $0$ & $2.0$ & $0$ & $0$ & $0.01$ & $2.5$ & ---------------------------------------------- 3.42865 83511 73345 68718 16357 54166 E$-$00 8.97033 20343 34376 52929 54799 07851 E$+$05 ---------------------------------------------- \ $5$ & $0$ & $9.0$ & $0$ & $0$ & $4.0$ & $0$ & $0$ & $22.5$ & $0.1$ & ---------------------------------------------- 6.61192 17510 24844 44964 35310 16794 E$-$12 1.27733 49592 55934 35439 28200 66327 E$-$23 ---------------------------------------------- \ $10$ & $0$ & $12.0$ & $0$ & $0$ & $8.0$ & $0$ & $0$ & $40.0$ & $0.001$ & ---------------------------------------------- 1.73471 85107 61691 61997 46481 63897 E$-$20 1.87470 00384 84104 39018 32142 95122 E$-$41 ---------------------------------------------- \ $1$ & $0$ & $3.0$ & $1$ & $0$ & $2.0$ & $1$ & $0$ & $2.3$ & $4.5$ & ---------------------------------------------- 1.06208 64400 83910 27727 15664 09620 E$-$00 2.45315 16908 16648 21895 84639 84955 E$-$01 ---------------------------------------------- \ $6$ & $0$ & $6.0$ & $4$ & $0$ & $5.0$ & $3$ & $0$ & $0.001$ & $0.001$ & ---------------------------------------------- 2.25821 42942 16571 75095 60318 57985 E$+$46 2.48928 81670 28192 48883 41015 71291 E$+$56 ---------------------------------------------- \ $1$ & $0$ & $2.0$ & $1$ & $1$ & $2.0$ & $1$ & $1$ & $0.01$ & $0.01$ & ---------------------------------------------- 1.39224 44783 13768 61934 60782 72901 E$-$02 1.63293 63668 41775 85565 65259 43851 E$+$03 ---------------------------------------------- \ $3$ & $0$ & $6.0$ & $4$ & $3$ & $5.0$ & $3$ & $3$ & $8.5$ & $0.1$ & ---------------------------------------------- 1.22009 58147 71182 05449 05454 44137 E$-$04 1.63709 36934 93917 83173 07173 88885 E$-$08 ---------------------------------------------- \ $3$ & $2$ & $5.0$ & $3$ & $3$ & $3.0$ & $2$ & $1$ & $1.5$ & $1.5$ & ---------------------------------------------- 8.36879 35046 74242 89056 27574 95673 E$+$01 2.18611 99801 01738 31430 63739 74869 E$-$00 ---------------------------------------------- \ $3$ & $2$ & $10.0$ & $4$ & $4$ & $8.0$ & $2$ & $2$ & $0.1$ & $9.0$ & ---------------------------------------------- 8.73612 92750 76127 55419 06451 97990 E$+$06 1.89278 99267 51317 93832 38020 79768 E$+$06 ---------------------------------------------- [ccccccccccc]{} $L$ & $M$ & $n$ & $l$ & $m$ & $n'$ & $l'$ & $m'$ & $p_{1}$ & $p_{2}$ & Eq.(16,17) and Eq.(19)\ $0$ & $0$ & $1.1$ & $0$ & $0$ & $1.5$ & $0$ & $0$ & $1.5$ & $1.5$ & ---------------------------------------------- 4.09565 63231 31448 87039 55147 40491 E$-$01 2.93614 23789 04133 04860 14608 03840 E$-$01 ---------------------------------------------- \ $1$ & $0$ & $1.3$ & $0$ & $0$ & $1.8$ & $0$ & $0$ & $0.5$ & $0.1$ & ---------------------------------------------- 3.33243 32418 64279 67130 02542 67866 E$-$01 1.02388 22819 29861 91096 69560 06751 E$-$02 ---------------------------------------------- \ $2$ & $0$ & $3.2$ & $0$ & $0$ & $2.4$ & $0$ & $0$ & $0.01$ & $2.5$ & ---------------------------------------------- 5.49415 76314 41209 10218 23695 48480 E$-$00 7.89858 88444 51044 77570 43625 77546 E$+$06 ---------------------------------------------- \ $5$ & $0$ & $9.5$ & $0$ & $0$ & $4.8$ & $0$ & $0$ & $22.5$ & $0.1$ & ---------------------------------------------- 8.85039 89771 63772 65903 44204 90792 E$-$12 9.74443 14137 20230 92395 49657 75617 E$-$23 ---------------------------------------------- \ $10$ & $0$ & $12.5$ & $0$ & $0$ & $8.9$ & $0$ & $0$ & $40.0$ & $0.001$ & ---------------------------------------------- 1.17276 22934 30019 21937 27274 20564 E$-$20 5.51171 88865 58272 19013 29428 46796 E$-$41 ---------------------------------------------- \ $1$ & $0$ & $3.3$ & $1$ & $0$ & $2.6$ & $1$ & $0$ & $2.3$ & $4.5$ & ---------------------------------------------- 1.87161 11340 32480 14502 70364 65188 E$-$00 6.65100 22118 49834 63694 23667 20358 E$-$01 ---------------------------------------------- \ $6$ & $0$ & $5.9$ & $4$ & $0$ & $6.1$ & $3$ & $0$ & $0.001$ & $0.001$ & ---------------------------------------------- 7.38511 70805 47763 29724 30676 68341 E$+$46 1.24500 66118 34410 00379 04215 62074 E$+$60 ---------------------------------------------- \ $1$ & $0$ & $2.3$ & $1$ & $1$ & $2.1$ & $1$ & $1$ & $0.01$ & $0.01$ & ---------------------------------------------- 1.85417 21104 44330 08693 85023 65883 E$-$01 1.40783 85008 42084 72060 75228 55754 E$+$04 ---------------------------------------------- \ $3$ & $0$ & $5.9$ & $4$ & $3$ & $5.1$ & $3$ & $3$ & $8.5$ & $0.1$ & ---------------------------------------------- 1.13313 80933 64382 91558 05780 65710 E$-$04 1.56774 69588 19924 39645 11990 75063 E$-$08 ---------------------------------------------- \ $3$ & $0$ & $8.8$ & $4$ & $3$ & $10.3$ & $3$ & $3$ & $0.01$ & $0.01$ & ---------------------------------------------- 4.97973 96013 63878 34830 92006 86281 E$+$04 7.95070 98832 40529 88293 62402 30270 E$+$42 ---------------------------------------------- \ $3$ & $2$ & $4.5$ & $3$ & $3$ & $3.5$ & $2$ & $1$ & $1.5$ & $1.5$ & ---------------------------------------------- 1.12197 95987 12589 22596 64533 39225 E$+$02 2.51548 44999 37750 08034 41787 62496 E$-$00 ---------------------------------------------- \ $3$ & $2$ & $10.5$ & $4$ & $4$ & $8.1$ & $2$ & $2$ & $0.1$ & $9.0$ & ---------------------------------------------- 1.10758 39108 61744 40668 98531 33050 E$+$07 1.32690 39575 82905 48879 15023 14344 E$+$29 ---------------------------------------------- [ccccccccc]{} $n$ & $l$ & $m$ & $\zeta$ & $n'$ & $l'$ & $m'$ & $\zeta'$ & Results\ $1.0$ & $0$ & $0$ & $1.24$ & $1.0$ & $0$ & $0$ & $5.67$ & ---------------------------------------------- E$-$02 E$-$02 **23**67 19340 54421 10253 45246 E$-$02 (5) **54**20 14999 28792 85854 63190 E$-$02 (10) **69** 30497 61154 50027 23145 E$-$02 (20) **53**848 92871 61006 54907 E$-$02 (40) **58**796 89501 79906 76961 E$-$02 (80) **2** 13114 00159 34423 E$-$02 (160) ---------------------------------------------- \ $1.1$ & $0$ & $0$ & $1.24$ & $1.1$ & $0$ & $0$ & $5.67$ & -------- E$-$02 -------- \ $1.2$ & $0$ & $0$ & $1.24$ & $1.2$ & $0$ & $0$ & $5.67$ & -------- E$-$02 -------- \ $1.3$ & $0$ & $0$ & $1.24$ & $1.3$ & $0$ & $0$ & $5.67$ & -------- E$-$02 -------- \ $1.0$ & $0$ & $0$ & $1.24$ & $2.0$ & $0$ & $0$ & $1.61$ & ----------------------------------------------- E$-$01 78 E$-$01 .**57**770 82722 80008 33479 04172 E$-$01 (5) **08**4 83584 29304 06752 43872 E$-$01 (10) **75**3 33306 97032 54324 80156 E$-$01 (20) **73** 68838 68511 26148 86494 E$-$01 (40) **59** 82475 67392 07711 26316 E$-$01 (80) **10**673 25488 09376 70768 E$-$01 (160) ----------------------------------------------- \ $1.1$ & $0$ & $0$ & $1.24$ & $2.1$ & $0$ & $0$ & $1.61$ & -------- E$-$01 -------- \ $1.2$ & $0$ & $0$ & $1.24$ & $2.2$ & $0$ & $0$ & $1.61$ & -------- E$-$01 -------- \ $1.3$ & $0$ & $0$ & $1.24$ & $2.3$ & $0$ & $0$ & $1.61$ & -------- E$-$01 -------- [ccccccc]{} $n$ & $l$ & $m$ & $n'$ & $l'$ & $m'$ & Results\ $2.0$ & $0$ & $0$ & $2.0$ & $0$ & $0$ & ----------------------------------------------- E$-$04 5 E$-$04 15138 130 E$-$04 .**49**978 10366 11814 81692 67046 E$-$04 (5) **1** 93448 02014 78421 78747 E$-$04 (10) **06** 71569 59704 90361 E$-$04 (20) **3** 47449 39243 55321 E$-$04 (40) **98**380 66714 63012 E$-$04 (80) **56**735 48522 57686 E$-$04 (160) ----------------------------------------------- \ $2.1$ & $0$ & $0$ & $2.1$ & $0$ & $0$ & -------- E$-$04 -------- \ $2.2$ & $0$ & $0$ & $2.2$ & $0$ & $0$ & -------- E$-$04 -------- \ $2.3$ & $0$ & $0$ & $2.3$ & $0$ & $0$ & -------- E$-$04 -------- \ $2.0$ & $1$ & $0$ & $2.0$ & $0$ & $0$ & ---------- E$-$04 1 E$-$04 ---------- \ $2.1$ & $1$ & $0$ & $2.1$ & $0$ & $0$ & -------- E$-$04 -------- \ $2.2$ & $1$ & $0$ & $2.2$ & $0$ & $0$ & -------- E$-$04 -------- \ $2.3$ & $1$ & $0$ & $2.3$ & $0$ & $0$ & -------- E$-$03 -------- \ $2.0$ & $1$ & $0$ & $2.0$ & $1$ & $0$ & ----------- E$-$04 77 E$-$04 ----------- \ $2.1$ & $1$ & $0$ & $2.1$ & $1$ & $0$ & -------- E$-$04 -------- \ $2.2$ & $1$ & $0$ & $2.2$ & $1$ & $0$ & -------- E$-$04 -------- \ $2.3$ & $1$ & $0$ & $2.3$ & $1$ & $0$ & -------- E$-$03 -------- [ccccccc]{} $n$ & $l$ & $m$ & $n'$ & $l'$ & $m'$ & Results\ $2.0$ & $0$ & $0$ & $2.0$ & $0$ & $0$ & ----------------------------------------------- E$-$02 E$-$02 E$-$02 .**39**466 24791 10067 86601 96990 E$-$02 (5) **95** 39598 23811 62963 42293 E$-$02 (10) **4** 64572 83043 94945 93805 E$-$02 (20) **8** 73240 73721 74843 59312 E$-$02 (40) **22**349 71134 02044 81338 E$-$02 (80) **51**272 77100 02097 74777 E$-$02 (160) ----------------------------------------------- \ $2.1$ & $0$ & $0$ & $2.1$ & $0$ & $0$ & -------- E$-$02 -------- \ $2.2$ & $0$ & $0$ & $2.2$ & $0$ & $0$ & -------- E$-$02 -------- \ $2.3$ & $0$ & $0$ & $2.3$ & $0$ & $0$ & -------- E$-$02 -------- \ $2.0$ & $1$ & $0$ & $2.0$ & $0$ & $0$ & -------- E$-$02 E$-$02 -------- \ $2.1$ & $1$ & $0$ & $2.1$ & $0$ & $0$ & -------- E$-$02 -------- \ $2.2$ & $1$ & $0$ & $2.2$ & $0$ & $0$ & -------- E$-$02 -------- \ $2.3$ & $1$ & $0$ & $2.3$ & $0$ & $0$ & -------- E$-$02 -------- \ $2.0$ & $1$ & $0$ & $2.0$ & $1$ & $0$ & ---------- E$-$02 6 E$-$02 ---------- \ $2.1$ & $1$ & $0$ & $2.1$ & $1$ & $0$ & -------- E$-$02 -------- \ $2.2$ & $1$ & $0$ & $2.2$ & $1$ & $0$ & -------- E$-$02 -------- \ $2.3$ & $1$ & $0$ & $2.3$ & $1$ & $0$ & -------- E$-$02 -------- [ccccccc]{} $n$ & $l$ & $m$ & $n'$ & $l'$ & $m'$ & Results\ $2.0$ & $0$ & $0$ & $2.0$ & $0$ & $0$ & ------------------------------------------------- .**72**309 91701 60088 20662 71607 E$-$02 \[1\] **6** 01923 98151 89286 78890 E$-$02 \[5\] **21**9 79709 38269 45472 E$-$02 \[10\] **87** 20151 76256 32122 E$-$02 \[11\] **89** 88329 48522 85282 E$-$02 \[12\] **86**2 41008 43703 E$-$02 \[15\] **6** 45442 E$-$02 \[20\] E$-$02 \[30\] E$-$02 \[40\] 89 8 E$-$02 2 E$-$02 ------------------------------------------------- \ $2.1$ & $0$ & $0$ & $2.1$ & $0$ & $0$ & ---------------------------------------- **94** 03007 59134 38353 E$-$02 \[10\] **18** 15620 66646 E$-$02 \[15\] E$-$02 \[30\] E$-$02 \[40\] ---------------------------------------- \ $2.2$ & $0$ & $0$ & $2.2$ & $0$ & $0$ & ----------------------------------------- **72**4 25497 07263 17130 E$-$02 \[10\] **29** 39385 11310 E$-$02 \[15\] **70**0 E$-$02 \[30\] E$-$02 \[40\] ----------------------------------------- \ $2.3$ & $0$ & $0$ & $2.3$ & $0$ & $0$ & ---------------------------------------- **50** 29555 79331 66393 E$-$02 \[10\] **82** 87274 34304 E$-$02 \[15\] E$-$02 \[30\] E$-$02 \[40\] ---------------------------------------- \ $2.0$ & $1$ & $0$ & $2.0$ & $0$ & $0$ & ------------------------------------------ **90**80 46322 59289 24910 E$-$02 \[10\] **62**6 90451 70130 E$-$02 \[15\] E$-$02 \[30\] E$-$02 \[40\] 3 E$-$02 ------------------------------------------ \ $2.1$ & $1$ & $0$ & $2.1$ & $0$ & $0$ & ------------------------------------------ **91**11 80051 67464 97075 E$-$02 \[10\] **65** 69293 67888 E$-$02 \[15\] **85** E$-$02 \[30\] E$-$02 \[40\] ------------------------------------------ \ $2.2$ & $1$ & $0$ & $2.2$ & $0$ & $0$ & ----------------------------------------- **46**2 42909 94539 77789 E$-$02 \[10\] **53**8 25384 87321 E$-$02 \[15\] **80**30 E$-$02 \[30\] E$-$02 \[40\] ----------------------------------------- \ $2.3$ & $1$ & $0$ & $2.3$ & $0$ & $0$ & ---------------------------------------- **27** 44329 75185 31334 E$-$02 \[10\] **70**36 51507 08634 E$-$02 \[15\] **123** E$-$02 \[30\] E$-$02 \[40\] ---------------------------------------- \ $2.0$ & $1$ & $0$ & $2.0$ & $1$ & $0$ & ----------------------------------------- **91**8 27546 98165 04057 E$-$02 \[10\] **83**02 55100 75237 E$-$02 \[15\] E$-$02 \[30\] E$-$02 \[40\] 5 E$-$02 ----------------------------------------- \ $2.1$ & $1$ & $0$ & $2.1$ & $1$ & $0$ & ----------------------------------------- **55**7 31327 16654 23236 E$-$02 \[10\] **24**8 40191 32912 E$-$02 \[15\] **22**7 E$-$02 \[30\] E$-$02 \[40\] ----------------------------------------- \ $2.2$ & $1$ & $0$ & $2.2$ & $1$ & $0$ & ----------------------------------------- **79**0 03807 01164 82993 E$-$02 \[10\] **11**20 86583 47853 E$-$02 \[15\] **51**1 E$-$02 \[30\] E$-$02 \[40\] ----------------------------------------- \ $2.3$ & $1$ & $0$ & $2.3$ & $1$ & $0$ & ------------------------------------------ **31**43 98419 21348 89833 E$-$02 \[10\] **97**50 51077 50463 E$-$02 \[15\] **42**0 E$-$02 \[30\] E$-$02 \[40\] ------------------------------------------

--- abstract: 'In order to more effectively cope with the real-world problems of vagueness, impreciseness, and subjectivity, fuzzy discrete event systems (FDESs) were proposed recently. Notably, FDESs have been applied to biomedical control for HIV/AIDS treatment planning and sensory information processing for robotic control. Qiu, Cao and Ying independently developed supervisory control theory of FDESs. We note that the controllability of events in Qiu’s work is fuzzy but the observability of events is crisp, and, the observability of events in Cao and Ying’s work is also crisp although the controllability is not completely crisp since the controllable events can be disabled with any degrees. Motivated by the necessity to consider the situation that the events may be observed or controlled with some membership degrees, in this paper, we establish the supervisory control theory of FDESs with partial observations, in which both the observability and controllability of events are fuzzy instead. We formalize the notions of fuzzy controllability condition and fuzzy observability condition. And Controllability and Observability Theorem of FDESs is set up in a more generic framework. In particular, we present a detailed computing flow to verify whether the controllability and observability conditions hold. Thus, this result can decide the existence of supervisors. Also, we use this computing method to check the existence of supervisors in the Controllability and Observability Theorem of classical discrete event systems (DESs), which is a new method and different from classical case. A number of examples are elaborated on to illustrate the presented results.' author: - 'Daowen Qiu and Fuchun Liu[^1][^2][^3][^4]' title: 'Fuzzy Discrete Event Systems under Fuzzy Observability and a Test-Algorithm' --- Discrete event systems, fuzzy logic, observability, supervisory control, fuzzy finite automata. Introduction ============ ISCRETE event systems (DESs) are dynamical systems whose evolution in time is governed by the abrupt occurrence of physical events at possibly irregular time intervals. Event though DESs are quite different from traditional continuous variable dynamical systems, they clearly involve objectives of control and optimization. A fundamental issue of supervisory control for DESs is how to design a controller (or supervisor), whose task is to enable and disable the controllable events such that the resulting closed-loop system obeys some prespecified operating rules \[1\]. Up to now, the supervisory control theory of DESs has been significantly applied to many technological and engineering systems such as automated manufacturing systems, interaction telecommunication networks and protocol verification in communication networks \[2-9\]. In most of engineering applications, the states of a DES are crisp. However, this is not the case in many other applications in complex systems such as biomedical systems and economic systems, in which vagueness, impreciseness, and subjectivity are typical features. For example, it is vague when a man’s condition of the body is said to be “good". Moreover, it is imprecise to say at what point exactly a man has changed from state “good" to state “poor". It is well known that the fuzzy set theory first proposed by Zadeh \[10\] is a good tool to cope with those problems. Indeed, up to now, fuzzy control systems have been well developed by many authors, and we may refer to \[11\] (and these references therein) regarding a survey on model-based fuzzy control systems. Notably, Lin and Ying \[12, 13\] recently initiated significantly the study of [*fuzzy discrete event systems*]{} (FDESs) by combining fuzzy set theory \[14\] with classical DESs. Notably, FDESs have been applied to biomedical control for HIV/AIDS treatment planning \[15, 16\] and decision making \[17\]. More recently, R. Huq [*et al*]{} \[18, 19\] have proposed an intelligent sensory information processing technique using FDESs for robotic control in the field of mobile robot navigation. Just as Lin and Ying \[13\] pointed out, a comprehensive theory of FDESs still needs to be set up, including many important concepts, methods and theorems, such as controllability, observability, and optimal control. These issues have been partially investigated in \[20-23\]. It is worthy to mention that Qiu \[20\], Cao and Ying \[21\] independently developed the supervisory control theory of FDESs. The similarity between the two theories is that the fuzzy systems considered in both \[20\] and \[21\] are modeled by max-min automata instead of max-product automata adopted in \[13\], and the controllability theorem was established in their respective frameworks. However, there are great differences between them. For the purpose of control, the set of events in \[21\] is partitioned into two disjoint subsets of controllable and uncontrollable events, as usually done in classical DESs, but the controllability of events is not completely crisp since the controllable events can be disabled by supervisors with any degrees. In contrast with \[21\], the controllable set and uncontrollable set of events in \[20\] are two [*fuzzy subsets*]{} of the set of events. That is, each event not only belongs to the uncontrollable set but also belongs to the controllable set; only its degree of belonging to those sets may be different. In particular, Qiu \[20\] presented an algorithm to check the existence of fuzzy supervisors for FDESs. As a continuation of the supervisory control under full observations \[20, 21\], this paper is to deal with the supervisory control of FDESs with fuzzy observations (generalizing partial observations). We notice that the observability in Qiu’s work \[20\] and Cao and Ying’s work \[21-23\] is [*crisp*]{}, that is, each fuzzy event is either completely observable or completely unobservable, although the controllability is fuzzy in \[20\] and not completely crisp in \[21-23\] where the controllable events can be disabled with any degrees. However, in real-life situation, each event generally has a certain degree to be observable and unobservable, and, also, has a certain degree to be controllable and uncontrollable. In fact, this idea of fuzziness of observability and controllability was originally proposed by Lin and Ying \[13\], and Qiu \[20\], and then it has been subsequently applied to robot sensory information processing by Huq [*et al*]{} \[18, 19\]. For example, in the cure process for a patient having cancer via either operation or drug therapy \[24\], some treatments (events) can be clearly seen by supervisors (viewed as a group of physicians), while some therapies (such as some operations) may not completely be observed by supervisors. For another example, in order to provide state-based decision making for a physical agent in mobile robot control, Huq [*et al*]{} \[18, 19\] introduced the concept of state-based observability to interpret the degree of reliability of the sensory information used in constructing fuzzy event matrices. Motivated by the necessity to consider the situation that the events may be observed or controlled with some membership degrees, in this paper, we establish the supervisory control theory of FDESs with partial observations, in which both the observability and the controllability of events are [*fuzzy*]{} instead. We formalize the notions of fuzzy controllability condition and fuzzy observability condition. A Controllability and Observability Theorem of FDESs is set up in a more generic framework. In particular, we present a computing flow to verify whether the controllability and observability conditions hold, which can decide the existence of supervisors. Also, we apply this computing method to testing the existence of supervisors in the Controllability and Observability Theorem of classical DESs \[1\], which is a different method from classical case \[1\]. The remainder of the paper is organized as follows. In the interest of readability, in Section II, we recall related notation and notions in supervisory control theory of FDESs. In Section III, we establish a Controllability and Observability Theorem of FDESs. Section IV deals with the realization of supervisors in the theorem; we present a computing flow for testing the existence of supervisors. Also, we elaborate on a number of related examples to illustrate the presented results. Preliminaries ============= Firstly we give some notation. ${\cal P}(X)$ denotes the power set of set $X$. A fuzzy subset of set $X$ is defined as a mapping from $X$ to $ [0,1]$. The set of all fuzzy subsets over $X$ is denoted as ${\cal F}(X)$. For two fuzzy subsets $\widetilde{A}$ and $\widetilde{B}$, $\widetilde{A}\subseteq \widetilde{B}$ stands for $\widetilde{A}(x) \leq \widetilde{B}(x)$ for any element $x$ of domain. A nondeterministic finite automaton \[25\] is a system described by $ G=(Q,E,\delta,q_{0},Q_{m})$, where $Q$ is the finite set of states with the initial state $q_{0}$, $E$ is the finite set of events, $\delta: Q\times E\rightarrow {\cal P}(Q)$ is the transition relation, and $Q_{m}\subseteq Q$ is called the set of marked states. Each sequence over $E$ is called a [*string*]{}. $E^{*}$ denotes the set of all finite strings over $E$. For $u\in E^{*}$, $|u|$ denotes the length of $u$; if $|u|=0$, then $u$ is an empty string, denoted by $\epsilon$. A subset of $E^{*}$ is called a [*language*]{}. In the setting of FDESs, states are fuzzy subsets of the crisp state set $Q$, which are called [*fuzzy states*]{}. If the crisp state set $Q=\{q_0, q_1,\dots, q_{n-1}\}$, then each fuzzy state $\widetilde{q}$ can be written as a vector $[a_{0}\hskip 1mm a_{1}\hskip 1mm\cdots\hskip 1mm a_{n-1}]$, where $a_{i}\in[0,1]$ represents the possibility of the current state being $q_{i}$. Similarly, a [*fuzzy event*]{} $\widetilde{\sigma}$ is denoted by a matrix $[a_{ij}]_{n\times n}$, in which every entry $a_{ij}$ belongs to $[0,1]$ and means the possibility of system transforming from the current state $q_{i}$ to state $q_{j}$ when event $\sigma$ occurs. [*Definition 1:*]{} A fuzzy finite automaton is a max-min system $$\widetilde{G}=(\widetilde{Q},\widetilde{E},\widetilde{\delta},\widetilde{q}_{0}, \widetilde{Q}_{m}),$$ where $\widetilde{Q}$ is a set of fuzzy states; $\widetilde{E}$ consists of fuzzy events; $\widetilde{q}_{0}$ is the initial state; $\widetilde{Q}_m\subseteq \widetilde{Q}$ is the set of marking states; the state transition relation $\widetilde{\delta}: \widetilde{Q}\times \widetilde{E}\rightarrow \widetilde{Q}$ is defined as $\widetilde{\delta}(\widetilde{q},\widetilde{\sigma})= \widetilde{q}\odot \widetilde{\sigma}$. Note that $\odot$ is [*max-min operation*]{} introduced in fuzzy set theory \[26\]: for matrix $A=[a_{ij}]_{n\times m}$ and matrix $B=[b_{ij}]_{m\times k}$, define $A\odot B=[c_{ij}]_{n\times k}$, where $c_{ij}=\max_{l=1}^{m} \min\{a_{il}, b_{lj}\}$. The fuzzy languages generated and marked by $\widetilde{G}$, denoted by ${\cal L}_{\widetilde{G}}$ and ${\cal L}_{\widetilde{G},m}$, respectively, are defined as two functions from $\widetilde{E}^{*}$ to \[0,1\] as follows: for any $ \widetilde{\sigma}_{1}\cdots\widetilde{\sigma}_{k} \in \widetilde{E}^{*}$, $${\cal L}_{\widetilde{G}} (\widetilde{\sigma}_{1}\cdots\widetilde{\sigma}_{k}) =\max_{i=1}^{n}\widetilde{q}_{0}\odot \widetilde{\sigma}_{1}\odot\cdots\odot\widetilde{\sigma}_{k}\odot \overline{s}_{i}^{{\rm T}},$$ $${\cal L}_{\widetilde{G},m}(\widetilde{\sigma}_{1}\cdots\widetilde{\sigma}_{k}) =\max_{\widetilde{q}\in \widetilde{Q}_{m}} \widetilde{q}_{0}\odot \widetilde{\sigma}_{1}\odot\cdots\odot\widetilde{\sigma}_{k}\odot \widetilde{q}^{{\rm T}},$$ where ${\rm T}$ is transpose operation and $\overline{s}_{i}=[0\cdots1\cdots0]$ where 1 is in the $i$th place. The following property is obtained in \[20\]: for any $ \widetilde{s} \in \widetilde{E}^{*}$ and any $ \widetilde{\sigma} \in \widetilde{E}$, $${\cal L}_{\widetilde{G},m}( \widetilde{s} \widetilde{\sigma} )\leq {\cal L}_{\widetilde{G}}( \widetilde{s} \widetilde{\sigma} )\leq {\cal L}_{\widetilde{G}}( \widetilde{s} ).$$ [*Remark 1:*]{} The framework of this paper is based on \[20-23\] in which the set of fuzzy events is a finite set. Indeed, the above definition of fuzzy finite automaton is similar to the fuzzy automaton defined by Steimann and Adlassning \[27\] for dealing with an application of clinical monitoring. Furthermore, we would like to consider max-min automata usually in practical applications since the set of fuzzy states $\{\widetilde{q}_{0}\odot \widetilde{s}:\widetilde{s}\in \widetilde{E}^{*}\}$ in any max-min automaton is clearly finite \[27\]. For fuzzy automata theory and related applications, we can refer to \[28-31\]. We further need some notions. A [*sublanguage*]{} of ${\cal L}_{\widetilde{G}}$ is represented as $\widetilde{K}\in {\cal F}(\widetilde{E})$ satisfying $\widetilde{K}\subseteq {\cal L}_{\widetilde{G}}$. For $\widetilde{s}\in\widetilde{E}^{*}$, symbol $pr(\widetilde{s})$ represents all of the prefix substrings of $\widetilde{s}$. And for any fuzzy language ${\cal L}$ over $\widetilde{E}$, its prefix-closure fuzzy language $pr({\cal L}):\widetilde{E}^{*}\rightarrow [0,1]$ is defined as $$pr({\cal L})(\widetilde{s})=\sup_{\widetilde{s}\in pr(\widetilde{t})}{\cal L}(\widetilde{t}),$$ which denotes the possibility of string $\widetilde{s}$ belonging to the prefix-closure of ${\cal L}$. Controllability and Observability Theorem ========================================= Let $\widetilde{G}=(\widetilde{Q},\widetilde{E},\widetilde{\delta},\widetilde{q}_{0}, \widetilde{Q}_{m})$ be a fuzzy finite automaton. As mentioned in Section I, each fuzzy event may be observable or controllable with a certain membership degree. Thus, the uncontrollable set $\widetilde{\Sigma}_{uc}$ and controllable set $\widetilde{\Sigma}_{c}$, as well as, the unobservable set $\widetilde{\Sigma}_{uo}$ and observable set $\widetilde{\Sigma}_{o}$, are thought of as four fuzzy subsets of $\widetilde{E}$, which are defined formally as follows. [*Definition 2:*]{} The uncontrollable set $\widetilde{\Sigma}_{uc}\in {\cal F}(\widetilde{E})$ and controllable set $\widetilde{\Sigma}_{c}\in {\cal F}(\widetilde{E})$ are respectively defined as a function $\widetilde{\Sigma}_{uc}: \widetilde{E}\rightarrow [0,1]$ and a function $\widetilde{\Sigma}_{c}: \widetilde{E}\rightarrow [0,1]$ which satisfy: for any $\widetilde{\sigma} \in \widetilde{E}$, $$\widetilde{\Sigma}_{uc}(\widetilde{\sigma})+\widetilde{\Sigma}_{c}(\widetilde{\sigma})=1.$$ Similarly, the unobservable set $\widetilde{\Sigma}_{uo}\in {\cal F}(\widetilde{E})$ and observable set $\widetilde{\Sigma}_{o}\in {\cal F}(\widetilde{E})$ are respectively defined as $\widetilde{\Sigma}_{uo}: \widetilde{E}\rightarrow [0,1]$ and $\widetilde{\Sigma}_{o}: \widetilde{E}\rightarrow [0,1]$ which satisfy: for any $\widetilde{\sigma} \in \widetilde{E}$, $$\widetilde{\Sigma}_{uo}(\widetilde{\sigma})+\widetilde{\Sigma}_{o}(\widetilde{\sigma})=1.$$ [*Remark 2:*]{} The degrees of observability and unobservability for FDESs were originally proposed by Lin and Ying (\[13\], pp. 412), and the degrees of controllability and uncontrollability were introduced by Qiu (\[20\], pp. 76). Intuitively, $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})$ and $\widetilde{\Sigma}_{c}(\widetilde{\sigma})$ represent the degree of fuzzy event $\widetilde{\sigma}$ to be uncontrollable and the degree of $\widetilde{\sigma}$ to be controllable, respectively. And, $\widetilde{\Sigma}_{uo}(\widetilde{\sigma})$ and $\widetilde{\Sigma}_{o}(\widetilde{\sigma})$ represent the degree of $\widetilde{\sigma}$ to be unobservable and the degree of $\widetilde{\sigma}$ to be observable, respectively. [*Definition 3:*]{} The projection $P:\widetilde{E}\rightarrow \widetilde{E}$ is defined as: $$P(\widetilde{\sigma})=\left\{\begin{array}{ll} \widetilde{\sigma},& {\rm if}\hskip 2mm \widetilde{\Sigma}_{o}(\widetilde{\sigma})>0,\\ \epsilon, & {\rm otherwise}. \end{array}\right.$$ And it can be extended to $\widetilde{E}^{*}$ by $P(\epsilon)=\epsilon$ and $P(\widetilde{s}\widetilde{\sigma})=P(\widetilde{s}) P(\widetilde{\sigma})$ for $\widetilde{s}\in\widetilde{E}^{*}$ and $\widetilde{\sigma}\in \widetilde{E}$. [*Remark 3:*]{} The purpose of projection is to erase the completely unobservable fuzzy events in the strings. In order to emphasize the observability degree of fuzzy event strings by means of projection $P$ associated with the fuzzy observable subset $\widetilde{\Sigma}_{o}$, we define the [*factor of observable projection*]{} $\widetilde{D}$ as a fuzzy subset of $P(\widetilde{E})$: for any $\widetilde{\sigma}\in P(\widetilde{E})$, $\widetilde{D}(\widetilde{\sigma})=\widetilde{\Sigma}_{o}(\widetilde{\sigma})$, and $$\widetilde{D}(\widetilde{\sigma}_{1}\widetilde{\sigma}_{2} \cdots\widetilde{\sigma}_{n})= \min\{\widetilde{D}(\widetilde{\sigma}_{i}): i=1,2,\cdots,n\},$$ where $\widetilde{\sigma}_{i}\in P(\widetilde{E})$, $i=1,2,\cdots,n$. Especially, $\widetilde{D}(\epsilon)=0$. Intuitively, $\widetilde{D}(P(\widetilde{s}))\cdot{\cal L}_{\widetilde{G}}(\widetilde{s})$ represents the possibility for the fuzzy event string $\widetilde{s}\in\widetilde{E}^{*}$ being possible under the effect of observable projection. And, $\widetilde{D}(P(\widetilde{s}\widetilde{\sigma}))\cdot\widetilde{\Sigma}_{uc}(\widetilde{\sigma})$ and $\widetilde{D}(P(\widetilde{s}))\cdot pr(\widetilde{K})(\widetilde{s})$ respectively denote the degree of $\widetilde{\sigma}\in\widetilde{E}$, as a continuation of the string $\widetilde{s}$, being uncontrollable, and the possibility of string $\widetilde{s}$ belonging to the prefix-closure of sublanguage $\widetilde{K}$ under the effect of observable projection. Furthermore, for the sake of convenience, in what follows we use the following notation: $${\cal L}_{\widetilde{G}}^{f}(\widetilde{s})=\left\{\begin{array}{ll} 1,& {\rm if}\hskip 2mm \widetilde{s}=\epsilon,\\ \widetilde{D}(P(\widetilde{s}))\cdot{\cal L}_{\widetilde{G}}(\widetilde{s}), & {\rm otherwise}; \end{array}\right.$$ $$pr(\widetilde{K})^{f}(\widetilde{s})=\left\{\begin{array}{ll} 1,& {\rm if}\hskip 2mm \widetilde{s}=\epsilon,\\ \widetilde{D}(P(\widetilde{s}))\cdot pr(\widetilde{K})(\widetilde{s}), & {\rm otherwise}; \end{array}\right.$$ $$\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}) =\widetilde{D}(P(\widetilde{s}\widetilde{\sigma})) \cdot\widetilde{\Sigma}_{uc}(\widetilde{\sigma}),$$ where $\widetilde{\sigma}$ is the continuation of string $\widetilde{s}$. [*Definition 4:*]{} For any FDES $\widetilde{G}$, a supervisor under the projection $P$ is said a [*fuzzy supervisor*]{}, denoted by $\widetilde{S}_{P}$, that is formally defined as a function $$\widetilde{S}_{P}:P(\widetilde{E}^{*})\rightarrow {\cal F}(\widetilde{E})$$ where for each $\widetilde{s}\in\widetilde{E}^{*}$ and $\widetilde{\sigma}\in \widetilde{E}$, $\widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})$ represents the possibility of fuzzy event $\widetilde{\sigma}$ being enabled after the occurrence of the string $P(\widetilde{s})$. The supervisors $\widetilde{S}_{P}$ are usually required to satisfy the following admissibility condition. [*Definition 5:*]{} The [*fuzzy admissibility condition*]{} for fuzzy supervisor $\widetilde{S}_{P}$ is characterized as follows: for each $\widetilde{s}\in\widetilde{E}^{*}$ and each continuation $\widetilde{\sigma}\in \widetilde{E}$, the following inequality holds $$\min\{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 2mm{\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma})\}\leq \widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma}).$$ Intuitively, the fuzzy admissibility condition (10) means that, under the effect of observable projection, the degree of any fuzzy event $\widetilde{\sigma}$ following any fuzzy event string $\widetilde{s}$ being possible together with $\widetilde{\sigma}$ being uncontrollable is not larger than the possibility for $\widetilde{\sigma}$ being enabled by the fuzzy supervisor $\widetilde{S}_{P}$ after string $P(\widetilde{s})$ occurring. The fuzzy controlled system by means of $\widetilde{S}_{P}$, denoted by $ \widetilde{S}_{P}/\widetilde{G} $, is an FDES, and, the behavior of $ \widetilde{S}_{P}/\widetilde{G} $ when $\widetilde{S}_{P}$ is controlling $\widetilde{G}$ is defined as follows. [*Definition 6:*]{} The fuzzy languages $ {\cal L}_{\widetilde{S}_{P}/\widetilde{G}} $ and ${\cal L}_{\widetilde{S}_{P}/\widetilde{G},m}$ generated and marked by $\widetilde{S}_{P}/\widetilde{G}$, respectively, are defined as follows: for any $\widetilde{s}\in\widetilde{E}^{*}$ and any $\widetilde{\sigma}\in \widetilde{E}$, 1\) $ {\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\epsilon)=1$; 2\) $ {\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})= \min\{{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}), \hskip 1mm{\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm\widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})\}$; 3\) $ {\cal L}_{\widetilde{S}_{P}/\widetilde{G},m} ={\cal L}_{\widetilde{S}_{P}/\widetilde{G}} \widetilde{\cap} {\cal L}_{\widetilde{G},m}$,\ where symbol $\widetilde{\cap}$ denotes Zadeh fuzzy AND operator, i.e., $(\widetilde{A}\widetilde{\cap} \widetilde{B})(x)=\min\{\widetilde{A}(x),\widetilde{B}(x)\}$. Definition 6 indicates that the degree of $\widetilde{s}\widetilde{\sigma}$ being physically possible in the controlled system $\widetilde{S}_{P}/\widetilde{G}$ is the smallest one among the degree of $\widetilde{s}$ being possible in $\widetilde{S}_{P}/\widetilde{G}$, the degree of $\widetilde{s}\widetilde{\sigma}$ being possible in $\widetilde{G}$ under the effect of observable projection, and the possibility of $\widetilde{\sigma}$ being enabled by the supervisor after the occurrence of $P(\widetilde{s})$. It is clear that Definition 6 generalizes the corresponding concepts from full observations (\[20\], pp. 6) to partial observations. In supervisory control of DESs, nonblockingness is usually required, and it means that the controlled system does not produce deadlocks \[1, 20\]. [*Definition 7:*]{} A fuzzy supervisor $\widetilde{S}_{P}$ of $\widetilde{G}$ is said to be [*nonblocking*]{}, if for any $\widetilde{s}\in\widetilde{E}^{*}$, the following equation holds: $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s})=\left\{\begin{array}{ll} 1,& {\rm if}\hskip 2mm \widetilde{s}=\epsilon,\\ \widetilde{D}(P(\widetilde{s}))\cdot pr({\cal L}_{\widetilde{S}_{P}/\widetilde{G},m})(\widetilde{s}), & {\rm otherwise}. \end{array}\right.$$ Intuitively, if $\widetilde{S}_{P}$ is nonblocking, then for any string $\widetilde{s}$, the possibility that $\widetilde{s}$ is one of the behaviors of the supervised fuzzy system $\widetilde{S}_{P}/\widetilde{G}$ equals the degree of $\widetilde{s}$ belonging to the prefix-closure of the fuzzy language marked by the supervised fuzzy system $\widetilde{S}_{P}/\widetilde{G}$ under the effect of observable projection. [*Definition 8:*]{} A fuzzy sublanguage $\widetilde{K}$ is said to be [*${\cal L}_{\widetilde{G},m}$-closed*]{}, if for any $\widetilde{s}\in\widetilde{E}^{*}$, $$\widetilde{K}(\widetilde{s})=\left\{\begin{array}{ll} 1,& {\rm if}\hskip 2mm \widetilde{s}=\epsilon,\\ \min\{pr(\widetilde{K})^{f}(\widetilde{s}), \hskip 1mm{\cal L}_{\widetilde{G},m}(\widetilde{s})\}, & {\rm otherwise}. \end{array}\right.$$ Obviously, if all fuzzy events can be observed fully \[20\], that is to say, $\widetilde{\Sigma}_{o}(\widetilde{\sigma})=1$ for any fuzzy event $\widetilde{\sigma}$, then Eq. (12) reduces to $K=pr(K)$ introduced in \[1, 4, 7, 20\], where all events are supposed to be observable. [*Definition 9:*]{} Let $\widetilde{K}\subseteq {\cal L}_{\widetilde{G}}$. If for any $\widetilde{s}\in\widetilde{E}^{*}$ and its continuation $\widetilde{\sigma}\in\widetilde{E}$, the following inequality holds: $$\begin{aligned} \min\{pr(K)^{f}(\widetilde{s}),\hskip 1mm \widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 1mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma})\}\leq pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma}),\end{aligned}$$ then we call $\widetilde{K}$ satisfying [*fuzzy controllability condition with respect to $\widetilde{G}$, $P$ and $\widetilde{\Sigma}_{uc}$*]{}. Intuitively, (13) means that under the effect of observable projection, the degree to which any fuzzy event string $\widetilde{s}$ belongs to the prefix-closure of $\widetilde{K}$ and fuzzy event $\widetilde{\sigma}$ following string $\widetilde{s}$ is physically possible together with $\widetilde{\sigma}$ being uncontrollable, is not larger than the possibility of string $\widetilde{s}\widetilde{\sigma}$ belonging to the prefix-closure of $\widetilde{K}$. [*Remark 4:*]{} Definition 9 generalizes the corresponding concepts concerning controllability in \[1, 20\]. If all fuzzy events can be observed fully, then Ineq. (13) reduces to the fuzzy controllability condition introduced in \[20\]. If we further assume that the events and states are crisp, then it reduces to the controllability condition introduced in \[1\]. To illustrate the application of fuzzy controllability condition, we provide an example. [*Example 1.*]{} Consider a fuzzy automaton $\widetilde{G}=(\widetilde{Q}_{1},\widetilde{E},\widetilde{\delta}, \widetilde{q}_{0})$, where $\widetilde{E}=\{\widetilde{a},\widetilde{b}, \widetilde{c}\}$, $\widetilde{q}_{0}$=$[0.8, 0]$, and $$\widetilde{a}=\left[ \begin{array}{cc} 0.8 &\ 0.2\\ 0&\ 0.2 \end{array} \right], \widetilde{b}=\left[ \begin{array}{cc} 0.2 &\ 0.8\\ 0&\ 0.2 \end{array} \right], \widetilde{c}=\left[ \begin{array}{cc} 0.2&\ 0\\ 0.8&\ 0.2 \end{array} \right].$$ Let $pr(\widetilde{K})$ be generated by a fuzzy automaton $\widetilde{H}=(\widetilde{Q}_{2},\widetilde{E},\widetilde{\delta},\widetilde{p}_{0})$, where $\widetilde{p}_{0}$=$[0.5,0]$, $\widetilde{E}=\{\widetilde{a},\widetilde{b}, \widetilde{c}\}$, and $\widetilde{a},\widetilde{b}$ are the same as those in $\widetilde{G}$ , but $\widetilde{c}$ is changed as follows: $$\widetilde{c}=\left[ \begin{array}{cc} 0.1&\ 0\\ 0.4&\ 0.1 \end{array} \right].$$ Suppose that $\widetilde{\Sigma}_{uc}$ and $\widetilde{\Sigma}_{o}$ are defined as: $\widetilde{\Sigma}_{uc}(\widetilde{a})=0.3$, $\widetilde{\Sigma}_{uc}(\widetilde{b})=0.5$, and $\widetilde{\Sigma}_{uc}(\widetilde{c})= 0.8$; $\widetilde{\Sigma}_{o}(\widetilde{a})=\widetilde{\Sigma}_{o}(\widetilde{b})=0.7$, and $\widetilde{\Sigma}_{o}(\widetilde{c})= 0.5$. In the following, we show that $\widetilde{K}$ is not fuzzy controllable. Take $\widetilde{s}=\widetilde{b}$ and $ \widetilde{\sigma}=\widetilde{c}$. Then $$\begin{aligned} &&\min\{pr(K)^{f}(\widetilde{s}),\hskip 2mm \widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 2mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma})\}\\ &=&\min\left\{0.7\times0.5,\hskip 1mm 0.5\times0.8, \hskip 2mm0.5\times0.8\right\}=0.35.\end{aligned}$$ However, $$pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma}) =0.5\times0.4=0.2.$$ Therefore, the fuzzy controllability condition does not hold. If $\widetilde{\Sigma}_{uc}$ is changed into $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})\leq 0.05$ for any $\widetilde{\sigma}\in \widetilde{E}$, then we can check that the fuzzy controllability condition holds. Before setting up the Controllability and Observability Theorem of FDESs, we need a characterization of the observability of fuzzy sublanguage. [*Definition 10:*]{} Let $\widetilde{K}\subseteq {\cal L}_{\widetilde{G}}$. If for any $\widetilde{s}\in\widetilde{E}^{*}$ and $\widetilde{\sigma}\in\widetilde{E}$, the following inequality holds: $$\begin{aligned} \min\{pr(K)^{f}(\widetilde{s}), pr(\widetilde{K})^{f}(\widetilde{t}\widetilde{\sigma}), {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma})\}\leq pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})\end{aligned}$$ for any $\widetilde{t}\in \widetilde{\Sigma}^{*}$, where $P(\widetilde{s})=P(\widetilde{t})$, then $\widetilde{K}$ is said satisfying [*fuzzy observability condition with respect to $\widetilde{G}$ and $P$*]{}. Intuitively, (14) means that if there is another string $\widetilde{t}$ possessing the same projection as $\widetilde{s}$, then under the effect of observable projection, the degree to which string $\widetilde{s}$ belongs to the prefix-closure of $\widetilde{K}$ and fuzzy event $\widetilde{\sigma}$ following $\widetilde{s}$ is physically possible together with $\widetilde{t}\widetilde{\sigma}$ belonging to the prefix-closure of $\widetilde{K}$, is not larger than the possibility of string $\widetilde{s}\widetilde{\sigma}$ belonging to the prefix-closure of $\widetilde{K}$. [*Example 2.*]{} Consider a fuzzy automaton $\widetilde{G}=(\widetilde{Q}_{1},\widetilde{E}, \widetilde{\delta},\widetilde{q}_{0})$, where $\widetilde{E}=\{\widetilde{a},\widetilde{b}, \widetilde{c}, \widetilde{d}\}$, $\widetilde{q}_{0}$=$[0.8, 0]$, and $$\widetilde{a}=\left[ \begin{array}{cc} 0.8 &\ 0.2\\ 0&\ 0.2 \end{array} \right], \hskip 3mm \widetilde{b}=\left[ \begin{array}{cc} 0.2 &\ 0.8\\ 0&\ 0.2 \end{array} \right],$$$$\widetilde{c}=\left[ \begin{array}{cc} 0.5 &\ 0\\ 0.4&\ 0.5 \end{array} \right], \hskip 3mm \widetilde{d}=\left[ \begin{array}{cc} 0.2&\ 0\\ 0.8&\ 0.2 \end{array} \right].$$ Let $pr(\widetilde{K})$ be generated by $\widetilde{H}=(\widetilde{Q}_{2},\widetilde{E},\widetilde{\delta},\widetilde{p}_{0})$, where $\widetilde{p}_{0}$=$[0.5,0]$, $\widetilde{E}=\{\widetilde{a},\widetilde{b}, \widetilde{c},\widetilde{d}\}$, and $\widetilde{a},\widetilde{b}$ are the same as those in $\widetilde{G}$ , but $\widetilde{c}$ and $\widetilde{d}$ are changed as follows: $$\widetilde{c}=\left[ \begin{array}{cc} 0.3 &\ 0\\ 0.4&\ 0.3 \end{array} \right], \hskip 3mm \widetilde{d}=\left[ \begin{array}{cc} 0.2&\ 0\\ 0.4&\ 0.2 \end{array} \right].$$ Suppose that $\widetilde{\Sigma}_{o}$ is defined as: $\widetilde{\Sigma}_{o}(\widetilde{a})=0.5$, $\widetilde{\Sigma}_{o}(\widetilde{b})=0.7$, $\widetilde{\Sigma}_{o}(\widetilde{c})= 0.4$, and $\widetilde{\Sigma}_{o}(\widetilde{d})=0$. If we take $\widetilde{s}=\widetilde{b}\widetilde{d}$, $ \widetilde{\sigma}=\widetilde{c}$ and $\widetilde{t}=\widetilde{b}$, then $P(\widetilde{s})=P(\widetilde{t})$, and $$\begin{aligned} &&\min\left\{pr(K)^{f}(\widetilde{s}),\hskip 2mm pr(\widetilde{K})^{f}(\widetilde{t}\widetilde{\sigma}),\hskip 2mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma})\right\}\\ &=&\min\left\{0.7\times0.4,\hskip 2mm 0.4\times0.4, \hskip 2mm0.4\times0.5\right\}=0.16.\end{aligned}$$ However, $$pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma}) =0.4\times0.3=0.12.$$ Therefore, the fuzzy observability condition does not hold. On the basis of the preliminaries, we are ready to present the main theorem of the paper. [*Theorem 1:*]{} ([*Controllability and Observability Theorem of FDESs*]{}). Let $\widetilde{G}=(\widetilde{Q},\widetilde{E}, \widetilde{\delta},\widetilde{q}_{0},\widetilde{Q}_{m})$ be a fuzzy automaton with a projection $P$. Suppose that fuzzy language $\widetilde{K}\subseteq {\cal L}_{\widetilde{G},m}$ satisfies $\widetilde{K}(\epsilon)=1$ and $pr(\widetilde{K})\subseteq {\cal L}_{\widetilde{G},m}$. Then there exists a nonblocking fuzzy supervisor $\widetilde{S}_{P}:P(\widetilde{E}^{*})\rightarrow {\cal F}(\widetilde{E})$, such that $\widetilde{S}_{P}$ satisfies the fuzzy admissibility condition, and $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s})= pr(\widetilde{K})^{f}(\widetilde{s}) \hskip 7mm {\rm and} \hskip 7mm {\cal L}_{\widetilde{S}_{P}/\widetilde{G},m}(\widetilde{s})= \widetilde{K}(\widetilde{s})$$ for any $\widetilde{s}\in \widetilde{E}^{*}$, if, and only if the following conditions hold:\ 1. $\widetilde{K}$ satisfies fuzzy controllability condition w.r.t. $\widetilde{G}$, $P$ and $\widetilde{\Sigma}_{uc}$.\ 2. $\widetilde{K}$ satisfies fuzzy observability condition w.r.t. $\widetilde{G}$ and $P$.\ 3. $\widetilde{K}$ is ${\cal L}_{\widetilde{G},m}$-closed. See Appendix. Realization of Supervisors in Controllability and Observability Theorem of FDESs ================================================================================ In this section, we present a detailed computing method to verify the controllability and observability conditions. Thus, this method can decide the existence of supervisors in Controllability and Observability Theorem of FDESs. As applications, two examples are elaborated to illustrate that this computing method is suitable to check the existence of supervisors not only for FDESs but also for classical DESs. Method of Checking the Existence of Supervisors for FDESs --------------------------------------------------------- Clearly, the existence of supervisor is associated with both fuzzy controllability condition and fuzzy observability condition. Therefore, testing the two conditions described by Ineqs. (13, 14) is of great importance. In classical DESs, for a given automaton $G$ and a language $K$, the controllability condition is checked by comparing the active event set of each state of $H\times G$ with the active event set of each state of $G$, where automaton $H$ generates $pr(K)$. And the observability condition is checked by building an observer of an automaton with unobservable events at each site \[1\]. For FDESs, a computing method of checking the fuzzy controllability condition was given by Qiu \[20\]. Based on the main idea of the finiteness of fuzzy states in FDESs modeled by max-min automata, we present a detailed approach for testing the fuzzy observability condition by means of [*computing trees*]{}. Let $\widetilde{G}=(\widetilde{Q},\widetilde{E},\widetilde{\delta}, \widetilde{q}_{0},\widetilde{Q}_{m})$ be a fuzzy automaton with partial observations and $\widetilde{E}=\{\widetilde{a}_{1},\widetilde{a}_{2},\cdots,\widetilde{a}_{n}\}$. Assume that the prefix-closure of fuzzy language $\widetilde{K}\subseteq {\cal L}_{\widetilde{G},m}$ is generated by a fuzzy automaton $\widetilde{H}=(\widetilde{Q}_{1},\widetilde{E},\widetilde{\delta},\widetilde{p}_{0})$. We describe the computing process via three steps as follows. The first step gives a computing tree for deriving the set of all fuzzy states reachable from the initial state $\widetilde{q}_{0}$, and the sets of strings respectively corresponding to each accessible fuzzy state are also obtained. The basic idea is based on the following two points: - $\widetilde{p}_{0}\odot \widetilde{s}=\widetilde{p}_{0}\odot \widetilde{s}\odot (\widetilde{t})^{k}$ for any $k\geq 0$ if $\widetilde{p}_{0}\odot \widetilde{s}=\widetilde{p}_{0}\odot \widetilde{s}\odot \widetilde{t}$ for $ \widetilde{t}\in \widetilde{E}^{*}$, where $(\widetilde{t})^{k}$ denotes the $\odot$ product of $k$’s $\widetilde{t}$. - The set of fuzzy states $\{\widetilde{p}_{0}\odot \widetilde{s}: s\in \widetilde{E}^{*}\}$ is always finite since $\widetilde{E}$ is finite \[20\]. Without loss of generality, we present the computing tree for $\widetilde{E}=\{\widetilde{a}_{1},\widetilde{a}_{2}\}$ of two fuzzy events via Fig. 1, and the case of more than two fuzzy events is analogous. [*Step 1:*]{} For a fuzzy automaton $\widetilde{H}=(\widetilde{Q}_{1},\widetilde{E},\widetilde{\delta},\widetilde{p}_{0})$, we search for all possible fuzzy states $\widetilde{r}_{i}$ reachable from $\widetilde{p}_{0}$ in $\widetilde{H}$, $i=1,2,\ldots,m_{1}$; also, we can obtain the sets $C(\widetilde{r}_{i})$ of all fuzzy event strings whose inputs lead $\widetilde{p}_{0}$ to $\widetilde{r}_{i}$, $i=1,2,\ldots,m_{1}$. This process can be realized by the finite computing tree that is visualized by Fig. 1 as follows. In the computing tree, the initial fuzzy state $\widetilde{p}_{0}$ is its root; each vertex, say $\widetilde{p}_{0}\odot \widetilde{s}$, may produce $n$’s sons, i.e., $\widetilde{p}_{0}\odot \widetilde{s}\odot \widetilde{a}_{i}$, $i=1,2,\ldots,n$. However, if $\widetilde{p}_{0}\odot \widetilde{s}\odot \widetilde{a}_{i}$ equals some its father, then $\widetilde{p}_{0}\odot \widetilde{s}\odot \widetilde{a}_{i}$ is a leaf, that is marked by a underline. The computing ends with a leaf at the end of each branch. (60,47) (25,40)[(10,5)\[c\][[Begin]{}]{}]{} (30,41)[(0,-1)[2]{}]{} (25,34)[(10,5)\[c\][[$\widetilde{p}_{0}$]{}]{}]{} (30,34)[(0,-1)[2]{}]{} (25,32)[(1,0)[10]{}]{} (25,32)[(0,-1)[5]{}]{}(35,32)[(0,-1)[5]{}]{} (20,22)[(10,5)\[c\][[$\widetilde{p}_{0}\odot\widetilde{a}_1$]{}]{}]{}(30,22)[(10,5)\[c\][[$\widetilde{p}_{0}\odot\widetilde{a}_2$]{}]{}]{} (25,22)[(0,-1)[8]{}]{}(8,19)[(1,0)[17]{}]{} (8,19)[(0,-1)[5]{}]{} (35,22)[(0,-1)[8]{}]{}(35,19)[(1,0)[17]{}]{} (52,19)[(0,-1)[5]{}]{} (2,9)[(10,5)\[c\][[$\widetilde{p}_{0}\odot\widetilde{a}_1\odot\widetilde{a}_1$]{}]{}]{} (20,9)[(10,5)\[c\][[$\widetilde{p}_{0}\odot\widetilde{a}_1\odot\widetilde{a}_2$]{}]{}]{} (34,9)[(10,5)\[c\][[$\widetilde{p}_{0}\odot\widetilde{a}_2\odot\widetilde{a}_1$]{}]{}]{} (50,9)[(10,5)\[c\][[$\widetilde{p}_{0}\odot\widetilde{a}_2\odot\widetilde{a}_2$]{}]{}]{} (4,6)[(10,5)\[c\]]{} (20,6)[(10,5)\[c\]]{} (33,6)[(10,5)\[c\]]{} (47,6)[(10,5)\[c\]]{} (20,27)[(5,5)\[c\][[$\widetilde{a}_1$]{}]{}]{}(35,22)[(5,15)\[c\][[$\widetilde{a}_2$]{}]{}]{} (3,9)[(5,15)\[c\][[$\widetilde{a}_1$]{}]{}]{}(20,9)[(5,15)\[c\][[$\widetilde{a}_2$]{}]{}]{} (35,9)[(5,15)\[c\][[$\widetilde{a}_1$]{}]{}]{}(52,9)[(5,15)\[c\][[$\widetilde{a}_2$]{}]{}]{} (23,0)[(20,5)\[c\][[Fig. 1. Computing tree of all states reachable from $\widetilde{p}_{0}$. ]{}]{}]{} For two fuzzy automata $\widetilde{G}$ and $\widetilde{H}$, our purpose is to search for the all different fuzzy state pairs reachable from the initial fuzzy state pair $(\widetilde{q}_{0},\hskip 2mm\widetilde{p}_{0})$. The method is similar to Step 1, which is also carried out by a computing tree. In this computing tree, the root is labelled with pair $(\widetilde{q}_{0},\hskip 1mm\widetilde{p}_{0})$, and each vertex, say $(\widetilde{q}_{0}\odot \widetilde{s},\hskip 2mm\widetilde{p}_{0}\odot \widetilde{s})$ for $\widetilde{s}\in \widetilde{E}^{*}$, may produce $n$’s sons, i.e., $(\widetilde{q}_{0}\odot \widetilde{s}\odot \widetilde{a}_{i},\hskip 2mm\widetilde{p}_{0}\odot \widetilde{s}\odot \widetilde{a}_{i})$, $i=1,2,\ldots,n$. But if a pair $(\widetilde{q}_{0}\odot \widetilde{s}\odot \widetilde{a}_{i},\hskip 2mm\widetilde{p}_{0}\odot \widetilde{s}\odot \widetilde{a}_{i})$ is the same as one of its fathers, then this pair will be treated as a leaf, that is marked with a underline. Such a computing tree is depicted by Fig. 2. Since the set of all fuzzy state pairs is finite, the computing tree ends with a leaf at the end of each branch. [*Step 2:*]{} For fuzzy automata $\widetilde{G}$ and $\widetilde{H}$, we search for all possible pairs of fuzzy states $(\widetilde{q}_{i},\widetilde{p}_{i})$, $i=1,2,\ldots,m_{2}$, reachable from $(\widetilde{q}_{0},\widetilde{p}_{0})$ by a finite computing tree (Fig. 2), and, in the same time, we can decide the sets $C(\widetilde{q}_{i},\widetilde{p}_{i})$ of all fuzzy event strings each of which makes $(\widetilde{q}_{0},\widetilde{p}_{0})$ become $(\widetilde{q}_{i},\widetilde{p}_{i})$, $i=1,2,\ldots,m_{2}$. (60,58) (28,50)[(10,5)\[c\][[Begin]{}]{}]{} (33,51)[(0,-1)[3]{}]{} (28,44)[(10,5)\[c\][[($\widetilde{q}_{0}$, $\widetilde{p}_{0}$)]{}]{}]{} (33,44)[(0,-1)[3]{}]{} (23,41)[(1,0)[20]{}]{} (23,41)[(0,-1)[5]{}]{}(43,41)[(0,-1)[5]{}]{} (13,31)[(10,5)\[c\][[($\widetilde{q}_{0}\odot \widetilde{a}_1$, $\widetilde{p}_{0}\odot\widetilde{a}_1$)]{} ]{}]{}(43,31)[(10,5)\[c\][[($\widetilde{q}_{0}\odot\widetilde{a}_2$, $\widetilde{p}_{0}\odot\widetilde{a}_2$)]{}]{}]{} (23,31)[(0,-1)[16]{}]{} (6,28)[(1,0)[17]{}]{} (6,28)[(0,-1)[5]{}]{} (43,31)[(0,-1)[8]{}]{} (43,28)[(1,0)[15]{}]{} (58,28)[(0,-1)[13]{}]{} (3,18)[(10,5)\[c\][[($\widetilde{q}_{0}\odot\widetilde{a}_1^{2}$, $\widetilde{p}_{0}\odot\widetilde{a}_1^{2}$)]{} ]{}]{}(17,10)[(10,5)\[c\][[($\widetilde{q}_{0}\odot\widetilde{a}_1\odot\widetilde{a}_2$, $\widetilde{p}_{0}\odot\widetilde{a}_1\odot\widetilde{a}_2$)]{}]{}]{} (38,18)[(10,5)\[c\][[($\widetilde{q}_{0}\odot\widetilde{a}_2\odot\widetilde{a}_1$, $\widetilde{p}_{0}\odot\widetilde{a}_2\odot\widetilde{a}_1$)]{} ]{}]{}(51,10)[(10,5)\[c\][[($\widetilde{q}_{0}\odot\widetilde{a}_2^{2}$, $\widetilde{p}_{0}\odot\widetilde{a}_2^{2}$)]{}]{}]{} (18,36)[(5,5)\[c\][[$\widetilde{a}_1$]{}]{}]{}(38,36)[(5,5)\[c\][[$\widetilde{a}_2$]{}]{}]{} (1,23)[(5,5)\[c\][[$\widetilde{a}_1$]{}]{}]{}(23,23)[(5,5)\[c\][[$\widetilde{a}_2$]{}]{}]{} (38,23)[(5,5)\[c\][[$\widetilde{a}_1$]{}]{}]{}(52,23)[(5,5)\[c\][[$\widetilde{a}_2$]{}]{}]{} (1,14)[(10,5)\[c\]]{} (16,7)[(10,5)\[c\]]{} (38,14)[(10,5)\[c\]]{} (53,7)[(10,5)\[c\]]{} (23,1)[(20,5)\[c\][[Fig. 2. Computing tree of all state pairs reachable from $(\widetilde{q}_{0},\widetilde{p}_{0})$.]{} ]{}]{} We now present Step 3, and, following that, we will give a proposition to further show the feasibility of this step. [*Step 3:*]{} Set $P(\widetilde{q}_{i},\widetilde{p}_{i})=\{\widetilde{s^{'}}| P(\widetilde{s^{'}})=P(\widetilde{s}),\widetilde{s}\in C(\widetilde{q}_{i},\widetilde{p}_{i})\}$, $i=1,2,\ldots,m_{2}$, and further set $$R_{j}(\widetilde{q}_{i},\widetilde{p}_{i})=\{\widetilde{t}| P(\widetilde{q}_{i},\widetilde{p}_{i})\cap C(\widetilde{r}_{j})\not=\emptyset, \widetilde{p}_{0}\odot \widetilde{t}=\widetilde{r}_{j}\},$$ for $i=1,2,\ldots,m_{2}$, and $j=1,2,\ldots,m_{1}$. If $R_{j}(\widetilde{q}_{i},\widetilde{p}_{i})\not=\emptyset$, we arbitrarily choose a string, say $\widetilde{t}_{ij}\in R_{j}(\widetilde{q}_{i},\widetilde{p}_{i})$ (usually, we try to choose a shorter string, and this will decrease our computing complexity in what follows). Given any $i\in\{1,2,\ldots,m_{2}\}$, by $FR(\widetilde{q}_{i},\widetilde{p}_{i})$ we mean the set of all strings $\widetilde{t}_{ij}$ we have chosen, say $$FR(\widetilde{q}_{i},\widetilde{p}_{i})= \{\widetilde{t}_{i1},\widetilde{t}_{i2},\ldots,\widetilde{t}_{ik_{i}}\}.$$ If Ineq. (14) holds for each $\widetilde{s}_{i}\in C(\widetilde{q}_{i},\widetilde{p}_{i})$ and each $\widetilde{t}_{ij}\in FR(\widetilde{q}_{i},\widetilde{p}_{i})$ where $i\in\{1,2,\ldots,m_{2}\}$ and $j\in\{1,2,\ldots,k_{i}\}$, then the fuzzy observability condition (14) holds; otherwise it does not hold. This is further verified by the following Proposition 2. [*Proposition 2:*]{} Let $\widetilde{G}=(\widetilde{Q},\widetilde{E},\widetilde{\delta}, \widetilde{q}_{0},\widetilde{Q}_{m})$ and $\widetilde{H}=(\widetilde{Q}_{1},\widetilde{E},\widetilde{\delta},\widetilde{p}_{0})$ be two fuzzy automata. Suppose that fuzzy sublanguage $\widetilde{K}$ satisfies $pr(\widetilde{K})={\cal L}_{\widetilde{H}}\subseteq {\cal L}_{\widetilde{G},m}$. If for any $i=1,2,\ldots,m_{2}$, there exist $\widetilde{s}_{i}\in C(\widetilde{q}_{i},\widetilde{p}_{i})$ such that for any $\widetilde{r}\in FR(\widetilde{q}_{i},\widetilde{p}_{i})$ and any $\widetilde{\sigma}\in\widetilde{E}$, Ineq. (14) holds, then the fuzzy observability condition described by Ineq. (14) holds. For any $\widetilde{t}\in\widetilde{E}^{*}$, without loss of generality, suppose that $\widetilde{t}\in C(\widetilde{q}_{i_{0}},\widetilde{p}_{i_{0}})$ for some $i_{0}\in \{1,2,\ldots,m_{2}\}$, since $\widetilde{E}^{*}= \bigcup_{i=1}^{m_{2}}C(\widetilde{q}_{i},\widetilde{p}_{i})$. For any $\widetilde{t^{'}}\in \widetilde{E}^{*}$ satisfying $P(\widetilde{t^{'}})=P(\widetilde{t})$, then $\widetilde{t^{'}}\in P(\widetilde{q}_{i_{0}},\widetilde{p}_{i_{0}})$, and we can further assume $\widetilde{t^{'}}\in C(\widetilde{r}_{j_{0}})$ for some $j_{0}\in \{1,2,\ldots,m_{1}\}$, due to $\widetilde{E}^{*}=\bigcup_{j=1}^{m_{1}}C(\widetilde{r}_{j})$. Therefore, there is $\widetilde{t}_{i_{0}j_{0}}\in FR(\widetilde{q}_{i_{0}},\widetilde{p}_{i_{0}})$. Now we have the following relations: $$pr(\widetilde{K})(\widetilde{t})=[\widetilde{p}_{0}\odot \widetilde{t}]=[\widetilde{p}_{0}\odot \widetilde{s}_{i_{0}}],$$ $$pr(\widetilde{K})(\widetilde{t^{'}}\widetilde{\sigma})=[\widetilde{p}_{0}\odot \widetilde{t^{'}}\odot\widetilde{\sigma}]=[\widetilde{p}_{0}\odot \widetilde{t}_{i_{0}j_{0}}\odot\widetilde{\sigma}],$$ $$pr(\widetilde{K})(\widetilde{t}\widetilde{\sigma})=[\widetilde{p}_{0}\odot \widetilde{t}\odot\widetilde{\sigma}]=[\widetilde{p}_{0}\odot \widetilde{s}_{i_{0}}\odot\widetilde{\sigma}],$$ $${\cal L}_{\widetilde{G}} (\widetilde{t}\widetilde{\sigma})=[\widetilde{q}_{0}\odot \widetilde{t}\odot\widetilde{\sigma}]=[\widetilde{q}_{0}\odot \widetilde{s}_{i_{0}}\odot\widetilde{\sigma}].$$ By means of the existing condition in this proposition, we know that $$\begin{aligned} \begin{array}{lll} &&\min\{pr(\widetilde{K})^{f}(\widetilde{s}_{i_{0}}), pr(\widetilde{K})^{f}(\widetilde{t}_{i_{0}j_{0}}\widetilde{\sigma}),{\cal L}_{\widetilde{G}}^{f}(\widetilde{s}_{i_{0}}\widetilde{\sigma})\}\\ &\leq& pr(\widetilde{K})^{f}(\widetilde{s}_{i_{0}}\widetilde{\sigma}). \end{array}\end{aligned}$$ In terms of Eqs. (16-19) and Ineq. (20) we therefore obtain $$\begin{aligned} \begin{array}{lll} && \min\{pr(\widetilde{K})^{f}(\widetilde{t}), pr(\widetilde{K})^{f}(\widetilde{t^{'}}\widetilde{\sigma}),{\cal L}_{\widetilde{G}}^{f}(\widetilde{t}\widetilde{\sigma})\}\\ &\leq& pr(\widetilde{K})^{f}(\widetilde{t}\widetilde{\sigma}), \end{array}\end{aligned}$$ and this completes the proof of proposition. Based on the above Proposition 2, we can check the fuzzy observability condition described by Ineq. (14) by the above computing flow (Steps 1–3). Furthermore, the fuzzy controllability condition described Ineq. (13) also can be clearly tested by similar computing flow with slight changes ($pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})$ is replaced by $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})$), besides using the approach proposed by Qiu \[20\]. [*Remark 6.*]{} To conclude this section, we roughly analyze the complexity of the above computing flow. Suppose that the number of all different fuzzy states $\{\widetilde{p}_{0}\odot \widetilde{s}: \widetilde{s}\in \widetilde{E}^{*}\}$ is $m_{1}$, and the number of all different fuzzy state pairs reachable from the initial fuzzy state pair $(\widetilde{q}_{0},\widetilde{p}_{0})$, namely, $\{(\widetilde{q}_{0}\odot \widetilde{s},\widetilde{p}_{0}\odot \widetilde{s}): \widetilde{s}\in \widetilde{E}^{*}\}$, is $m_{2}$. Then, in [*Step 1*]{}, by means of Figure 1 the number of computing steps is $O(m_{1})$ and, in [*Step 2*]{}, in terms of Figure 2 the number of computing steps is $O(m_{2})$. As to [*Step 3*]{}, we can see that the computing complexity is $O(m_{1}m_{2}|\widetilde{E}|)$, where $|\widetilde{E}|$ is the cardinal number of alphabet $\widetilde{E}$. Thus, if we avoid the cost regarding the operation $\odot$, the computing steps of the above flow is $O(m_{1}m_{2}|\widetilde{E}|)$. Applications to Supervisory Control of Classical DESs and FDESs --------------------------------------------------------------- In this subsection, we present two examples to illustrate the applications of the supervisory control theory for FDESs presented above. Example 3 will indicate that the computing approach given in Section IV-A can be applied to check the existence of supervisors for classical DESs. Example 4 arising from a medical treatment will describe a detailed computing processing for FDESs, which may be viewed as an applicable background of supervisory control of FDESs under partial observations. We first recall some notions of classical DESs. Let $G$ be a classical DES. Suppose that $\Sigma_{c}$ and $\Sigma_{o}$ are designated as controllable and observable event sets, respectively. $P$ is the corresponding projection. A language $K$ is said to be observable with respect to $G$ and $P$, if for all $s$, $t\in pr(K)$ and all $\sigma\in \Sigma_{c}$, if $P(s)=P(t)$, then $$s\sigma\in {\cal L}_{G},\hskip 3mm t\sigma\in pr(K) \hskip 4mm \Rightarrow \hskip 4mm s\sigma\in pr(K).$$ In classical DESs \[1\], for a given automaton $G$ and a language $K$, the controllability condition is checked by comparing the active event set of each state of $H\times G$ with the active event set of each state of $G$, where automaton $H$ generates $pr(K)$. And the observability condition is checked by building an observer of an automaton with unobservable events at each site \[1\]. [*Example 3:*]{} Consider the example presented in Section 3.7 of \[1\] (Example 3.18, page 196) to illustrate the method of testing the observability condition (22). $G$ and $H$ are two automata of classical DESs with crisp state set $E=\{u, b\}$ shown in Fig. 3. Language $K$ satisfies $pr(K)={\cal L}_{H}$. Assume that $\Sigma_{o}=\{b\}$ and $\Sigma_{c}=\{u, b\}$. In order to test $K$ being unobservable, an observer automaton $H_{obs}$ is constructed in \[1\]. In fact, the observability condition (22) cannot be satisfied when $s=\epsilon$, $t=u$ and $\sigma=b$. (150,45) (10,28)[(-10,0)[$0$]{}]{} (30,36)[(-10,0)[$1$]{}]{} (50,28)[(-10,0)[$2$]{}]{} (30,20)[(-10,0)[$3$]{}]{} (2.5,28)[(1,0)[5]{}]{}(12.5,29)[(2,1)[15]{}]{} (12.5,27)[(2,-1)[15]{}]{} (32.5,36)[(2,-1)[15]{}]{} (32.5,20)[(2,1)[15]{}]{} (19,35)[(0,0)\[c\][$u$]{}]{} (19,21)[(0,0)\[c\][$b$]{}]{} (40,35)[(0,0)\[c\][$b$]{}]{} (40,21)[(0,0)\[c\][$u$]{}]{} (20,11)[(20,1)\[c\][[(1) Automaton $G$]{}]{}]{} (70,26)[(-10,0)[$0$]{}]{} (87,34)[(-10,0)[$1$]{}]{} (105,26)[(-10,0)[$2$]{}]{} (62.5,26)[(1,0)[5]{}]{}(72.5,27)[(2,1)[12]{}]{} (90,34)[(2,-1)[13]{}]{} (79,33)[(0,0)\[c\][$u$]{}]{} (100,33)[(0,0)\[c\][$b$]{}]{} (78,11)[(20,1)\[c\][[(2) Automaton $H$]{}]{}]{} (50,3)[(20,1)\[c\][[Fig. 3. Automata $G$ and $H$ of classical DESs in Example 3.]{}]{}]{} In the following, we verify the above conclusion by means of the computing method we presented in Section IV-A. Firstly, classical DES $G$ can be viewed as a fuzzy automaton $\widetilde{G}=(\widetilde{Q}_{1},\widetilde{E}, \widetilde{\delta},\widetilde{q}_{0})$, where the fuzzy states are $$\widetilde{q}_{0}=[1,0,0,0],\hskip 4mm\widetilde{q}_{1}=[0,1,0,0],$$$$\widetilde{q}_{2}=[0,0,1,0],\hskip 4mm\widetilde{q}_{3}=[0,0,0,1],$$ and the fuzzy events are $$\widetilde{u}=\left[ \begin{array}{cccc} 0 &\ 1 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 1\\ 0 &\ 0 &\ 0 &\ 0\\ \end{array} \right], \hskip 4mm \widetilde{b}=\left[ \begin{array}{cccc} 0 &\ 0 &\ 1 &\ 0\\ 0 &\ 0 &\ 0 &\ 1\\ 0 &\ 0 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 0\\ \end{array} \right].$$\ Similarly, the automaton $H$ can be viewed as a fuzzy automaton $\widetilde{H}=(\widetilde{Q}_{2},\widetilde{E}, \widetilde{\delta},\widetilde{p}_{0})$, where the fuzzy states are $$\widetilde{p}_{0}=[1,0,0,0],\hskip 4mm\widetilde{p}_{1}=[0,1,0,0],\hskip 4mm \widetilde{p}_{2}=[0,0,1,0],$$ and the fuzzy events are $$\widetilde{u}=\left[ \begin{array}{cccc} 0 &\ 1 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 0\\ \end{array} \right], \hskip 4mm \widetilde{b}=\left[ \begin{array}{cccc} 0 &\ 0 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 1\\ 0 &\ 0 &\ 0 &\ 0\\ 0 &\ 0 &\ 0 &\ 0\\ \end{array} \right].$$ The fuzzy subsets $\widetilde{\Sigma}_{o}$ and $\widetilde{\Sigma}_{c}$ are determined by $\Sigma_{o}=\{b\}$ and $\Sigma_{c}=\{u, b\}$, which are listed as follows: $$\widetilde{\Sigma}_{o}(\widetilde{u})=0, \hskip 4mm\widetilde{\Sigma}_{o}(\widetilde{b})=1; \hskip 4mm\widetilde{\Sigma}_{c}(\widetilde{u})=\widetilde{\Sigma}_{c}(\widetilde{b})=1.$$ By constructing the computing trees of $\widetilde{H}$, $\widetilde{G}$ and $\widetilde{H}$, we know that there are three fuzzy states $\widetilde{p}_{0}$, $\widetilde{p}_{1}$, $\widetilde{p}_{2}$ reachable from $\widetilde{p}_{0}$, and three fuzzy states pairs $(\widetilde{q}_{0},\widetilde{p}_{0})$, $(\widetilde{q}_{1},\widetilde{p}_{1})$, $(\widetilde{q}_{2},\widetilde{p}_{2})$ reachable from $(\widetilde{q}_{0},\widetilde{p}_{0})$, and the corresponding fuzzy event strings are $\epsilon$, $\widetilde{u}$, and $\widetilde{u}\widetilde{b}$. Therefore, we should necessarily check the fuzzy observability condition in term of whether the all elements in the rightmost column of the following Table I are “T" (True) when $\widetilde{s}=\epsilon$, $\widetilde{s}=\widetilde{u}$, and $\widetilde{s}=\widetilde{u}\widetilde{b}$, where - $x_{1}=[\widetilde{p}_0\odot \widetilde{s}]$, $x_{2}=[\widetilde{p}_0\odot \widetilde{t}\odot\widetilde{\sigma}]$, $x_{3}=[\widetilde{q}_0\odot \widetilde{s}\odot\widetilde{\sigma}]$, - $y=[\widetilde{p}_0\odot \widetilde{s}\odot\widetilde{\sigma}]$, - $V=\min\{pr(K)^{f}(\widetilde{s}), pr(\widetilde{K})^{f}(\widetilde{t}\widetilde{\sigma}), {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma})\}$, - $W=pr(K)^{f}(\widetilde{s}\widetilde{\sigma})$. $\widetilde{s}$ $\widetilde{t}$ $\widetilde{\sigma}$ $x_{1}$ $x_{2}$ $x_{3}$ $y$ $V$ $ W $ $V\leq W $ ------------------------------ ------------------------------ ---------------------- --------- --------- --------- ----- ----- ------- ------------ $\epsilon$ $\widetilde{u}$ 1 1 1 1 0 0 T $\epsilon$ $\widetilde{b}$ 1 0 1 0 0 0 T $\widetilde{u}$ $\widetilde{u}$ 1 0 1 1 0 0 T $\widetilde{b}$ 1 1 1 0 1 0 F $\epsilon$ $\widetilde{u}$ 1 1 0 0 0 0 T $\widetilde{u}$ $\widetilde{b}$ 1 0 1 1 0 1 T $\widetilde{u}$ $\widetilde{u}$ 1 0 0 0 0 0 T $\widetilde{b}$ 1 1 1 1 0 1 T $\widetilde{b}$ $\widetilde{u}$ 1 0 0 0 0 0 T $\widetilde{b}$ 1 0 0 0 0 0 T $\widetilde{u}\widetilde{b}$ $\widetilde{b}\widetilde{u}$ $\widetilde{u}$ 1 0 0 0 0 0 T $\widetilde{b}$ 1 0 0 0 0 0 T $\widetilde{u}\widetilde{b}$ $\widetilde{u}$ 1 0 0 0 0 0 T $\widetilde{b}$ 1 0 0 0 0 0 T : Testing the fuzzy observability condition in Example 3 From Table I we see that the fuzzy observability condition does not hold since an “F" (False) has been found out in the rightmost column when $\widetilde{s}=\epsilon$, $\widetilde{t}=\widetilde{u}$ and $\widetilde{\sigma}=\widetilde{b}$. Example 3 indicates that our method also can be applied to testing the existence of supervisors for classical DESs \[1\]. Next we apply our results to an applicable example arising from a medical treatment problem. [*Example 4:*]{} Suppose that there is a patient sickening for a new disease. For simplicity, it is assumed that the doctors consider roughly the patient’s condition to be two states, say “poor" and “good". For the new disease, the doctors have no complete knowledge about it, but they believe by their experience that these drugs such as theophylline, Erythromycin Ethylsuccinate and dopamine may be useful to the disease. As mentioned in Introduction, considering the features of vagueness, patient’s condition can simultaneously belong to “poor" and “good" with respective memberships; also, an event occurring (i.e., treatment) may lead a state to multistates with respective degrees. Therefore, the patient’s conditions and their changes after the treatments can be modeled by an FDES $\widetilde{G}=(\widetilde{Q}_{1},\widetilde{E}, \widetilde{\delta},\widetilde{q}_{0},\widetilde{Q}_{m})$, in which each fuzzy state, denoted as a two-dimensional vector $\widetilde{q}=[a_{1}, a_{2}]$, is represented as the possibility distribution of the patient’s condition over the two crisp states “poor" and “good"; each fuzzy event, denoted as a $2\times 2$ matrix $\widetilde{\sigma}=[a_{ij}]_{2\times 2}$, means the possibility for patient’s condition to transfer from one crisp state to another crisp state when a certain drug treatment is adopted. Suppose that the patient’s initial condition is $\widetilde{q}_{0}=[0.9, \hskip 1mm 0]$. The drug events $\widetilde{a},\widetilde{b}, \widetilde{c}$, namely, theophylline, Erythromycin Ethylsuccinate and dopamine, respectively, may be evaluated according to doctors’ experience as follows: $$\widetilde{a}=\left[ \begin{array}{cc} 0.9 &\ 0.4\\ 0&\ 0.4 \end{array} \right], \widetilde{b}=\left[ \begin{array}{cc} 0.4 &\ 0.9\\ 0&\ 0.4 \end{array} \right],\widetilde{c}=\left[ \begin{array}{cc} 0.4&\ 0 \\ 0.4&\ 0.9 \end{array} \right].$$ We specify a fuzzy set of control specifications $\widetilde{K}$ that are desired for the doctors. For the sake of simplicity, it is assumed that $\widetilde{K}$ is ${\cal L}_{\widetilde{G},m}$-closed. As usual, let $pr(\widetilde{K})$ be generated by a fuzzy automaton $\widetilde{H}=(\widetilde{Q}_{2},\widetilde{E}, \widetilde{\delta},\widetilde{p}_{0})$, where $\widetilde{p}_{0}$=$[0.9,0]$, $\widetilde{E}=\{\widetilde{a},\widetilde{b},\widetilde{c}\}$, with $\widetilde{a},\widetilde{b}$ being the same as those in $\widetilde{G}$, except that $\widetilde{c}$ is changed as follows: $$\widetilde{c}=\left[ \begin{array}{cc} 0.2&\ 0 \\ 0.2&\ 0.9 \end{array} \right].$$ For these drug events, some effects such as headache disappears are clearly observed, but some effects may be observed only by means of medical instruments; also, some effects such as alleviation of pain can be controlled, but some potential side effects may be uncontrolled. Therefore, each drug event may be observed or controlled with some membership degrees. Suppose that $\widetilde{\Sigma}_{uc}$ and $\widetilde{\Sigma}_{o}$ are defined as follows: $$\widetilde{\Sigma}_{uc}(\widetilde{a})=\widetilde{\Sigma}_{uc}(\widetilde{b})=0.1,\hskip 2mm \widetilde{\Sigma}_{uc}(\widetilde{c})= 0.2;$$$$\widetilde{\Sigma}_{o}(\widetilde{a})=0.4,\hskip 2mm \widetilde{\Sigma}_{o}(\widetilde{b})=0.6,\hskip 2mm \widetilde{\Sigma}_{o}(\widetilde{c})=0.$$ In supervisory control of FDESs, the purpose of nonblocking fuzzy supervisors is to disable the fuzzy events with respective degrees such that the generated and marked behaviors of the supervised system satisfy some prespecified specifications, and the controlled system does not produce deadlocks. Therefore, for this example, the problem is whether there exists such a nonblocking fuzzy supervisor $\widetilde{S}_{P}: P(\widetilde{E}^{*})\rightarrow {\cal F}(\widetilde{E})$. In the following, we will answer the problem by proving $\widetilde{K}$ to be fuzzy controllable and fuzzy observable by means of computing approach presented in Section IV-A. For $\widetilde{G}$ and $\widetilde{H}$, we search for all possible fuzzy state pairs $(\widetilde{q}_{i},\widetilde{p}_{i})$ reachable from $(\widetilde{q}_{0},\widetilde{p}_{0})$ by the finite computing tree shown in Fig.4, which is followed by the other three subtrees visualized by Figs. 5, 6, 7, respectively. (40,29) (26,26)[(10,5)\[c\][[$(\widetilde{q}_0,\widetilde{p}_0)=([0.9,0],$ $[0.9,0]$)]{}]{}]{} (31,26)[(0,-1)[2]{}]{} (11,24)[(1,0)[40]{}]{} (11,24)[(0,-1)[5]{}]{}(31,24)[(0,-1)[5]{}]{}(51,24)[(0,-1)[5]{}]{} (6,14)[(10,5)\[c\][[($ [0.9,0.4],$ $[0.9,0.4]$)]{} ]{}]{}(26,14)[(10,5)\[c\][[($ [0.4,0.9],$ $[0.4,0.9]$)]{}]{}]{} (46,14)[(10,5)\[c\][[($ [0.4,0],$ $[0.2,0]$)]{} ]{}]{} (11,19)[(5,5)\[c\][[$\widetilde{a}$]{}]{}]{}(26,19)[(5,5)\[c\][[$\widetilde{b}$]{}]{}]{} (51,19)[(5,5)\[c\][[$\widetilde{c}$]{}]{}]{} (11,14)[(0,-1)[4]{}]{} (31,14)[(0,-1)[4]{}]{} (51,14)[(0,-1)[4]{}]{} (8,5)[(5,5)\[c\][[Subtree $T_{1} $]{}]{}]{}(29,5)[(5,5)\[c\][[Subtree $ T_{2}$]{}]{}]{} (49,5)[(5,5)\[c\][[Subtree $T_{3} $]{}]{}]{} (2,0)[(5,5)\[l\][[Fig. 4. Computing tree of all state pairs reachable from $(\widetilde{q}_0,\widetilde{p}_0)$.]{} ]{}]{} (40,35) (25,27)[(10,5)\[c\][[$([0.9,0.4],$ $[0.9,0.4]$)]{}]{}]{} (30,28)[(0,-1)[3]{}]{} (10,25)[(1,0)[40]{}]{} (10,25)[(0,-1)[5]{}]{}(30,25)[(0,-1)[5]{}]{}(50,25)[(0,-1)[5]{}]{} (5,15)[(10,5)\[c\][[()]{} ]{}]{}(25,15)[(10,5)\[c\][[($ [0.4,0.9],$ $[0.4,0.9]$)]{}]{}]{} (47,15)[(10,5)\[c\][[($ [0.4,0.4],$ $[0.2,0.4]$)]{} ]{}]{} (10,20)[(5,5)\[c\][[$\widetilde{a}$]{}]{}]{}(25,20)[(5,5)\[c\][[$\widetilde{b}$]{}]{}]{} (50,20)[(5,5)\[c\][[$\widetilde{c}$]{}]{}]{} (30,15)[(0,-1)[5]{}]{} (50,15)[(0,-1)[5]{}]{} (27,5)[(5,5)\[c\][[Subtree $ T_{2}$]{}]{}]{} (47,5)[(10,5)\[c\][[()]{} ]{}]{} (53,10)[(5,5)\[c\][[$\widetilde{a}$ or $\widetilde{b}$ or $\widetilde{c}$]{}]{}]{} (20,0)[(5,5)\[l\][[Fig. 5. Subtree $T_{1}$.]{}]{}]{} (60,55) (25,48)[(10,5)\[c\][[($[0.4,0.9],$ $[0.4,0.9])$]{}]{}]{} (30,49)[(0,-1)[2]{}]{} (25,47)[(1,0)[10]{}]{} (25,47)[(0,-1)[4]{}]{}(35,47)[(0,-1)[4]{}]{} (13,38)[(10,5)\[c\][[($ [0.4,0.4],$ $[0.4,0.4])$]{}]{}]{} (36,38)[(10,5)\[c\][[($ [0.4,0.9],$ $[0.2,0.9])$]{}]{}]{} (25,38)[(0,-1)[14]{}]{}(8,35)[(1,0)[17]{}]{} (8,35)[(0,-1)[5]{}]{} (35,38)[(0,-1)[14]{}]{}(35,35)[(1,0)[17]{}]{} (52,35)[(0,-1)[5]{}]{} (5,25)[(10,5)\[c\][[(]{}]{}]{} (14,19)[(10,5)\[c\][[($ [0.4,0.4],$ $[0.2,0.4])$]{}]{}]{} (37,19)[(10,5)\[c\][[($ [0.4,0.4],$ $[0.2,0.4])$]{}]{}]{}(48,25)[(10,5)\[c\][[(]{}]{}]{} (25,19)[(0,-1)[5]{}]{} (35,19)[(0,-1)[5]{}]{} (16,9)[(6,5)\[c\][[(]{}]{}]{} (37,9)[(10,5)\[c\]]{} (17,43)[(5,5)\[c\][[$\widetilde{a}$ or $\widetilde{b}$]{}]{}]{}(25,38)[(25,15)\[c\][[$\widetilde{c}$]{}]{}]{} (0,25)[(25,15)\[c\][[$\widetilde{a}$ or $\widetilde{b}$]{}]{}]{}(10,24)[(25,15)\[c\][[$\widetilde{c}$]{}]{}]{} (27,24)[(25,15)\[c\][[$\widetilde{a}$ or $\widetilde{b}$]{}]{}]{}(42,25)[(25,15)\[c\][[$\widetilde{c}$]{}]{}]{} (18,9)[(45,15)\[c\][[$\widetilde{a}$ or $\widetilde{b}$ or $\widetilde{c}$]{}]{}]{} (7,14)[(25,5)\[c\][[$\widetilde{a}$ or $\widetilde{b}$ or $\widetilde{c}$]{}]{}]{} (20,2)[(5,5)\[l\][[Fig. 6. Subtree $T_{2}$.]{}]{}]{} (60,33) (25,28)[(10,5)\[c\][[($[0.4,0],$ $[0.2,0])$]{}]{}]{} (30,29)[(0,-1)[3]{}]{} (25,26)[(1,0)[10]{}]{} (25,26)[(0,-1)[4]{}]{}(35,26)[(0,-1)[4]{}]{} (13,17)[(10,5)\[c\][[($[0.4,0.4],$ $[0.2,0.2])$]{}]{}]{} (35,17)[(10,5)\[c\]]{} (25,17)[(0,-1)[4]{}]{} (13,8)[(10,5)\[c\][[(]{}]{}]{} (17,22)[(5,5)\[c\][[$\widetilde{a}$ or $\widetilde{b}$]{}]{}]{}(25,17)[(25,15)\[c\][[$\widetilde{c}$]{}]{}]{} (18,8)[(25,15)\[c\][[$\widetilde{a}$ or $\widetilde{b}$ or $\widetilde{c}$]{}]{}]{} (20,2)[(5,5)\[l\][[Fig. 7. Subtree $T_{3}$.]{}]{}]{} From above computing trees, it follows that there are only eight different fuzzy state pairs and eight different fuzzy states reachable from $(\widetilde{q}_{0},\widetilde{p}_{0})$ and $\widetilde{p}_{0}$, respectively, which together with the corresponding fuzzy event strings are listed in Table II. Therefore, we should necessarily check the fuzzy observability condition only when $\widetilde{s}=\epsilon$, or $\widetilde{a}$, or $\widetilde{b}$, or $\widetilde{c}$, or $\widetilde{b}\widetilde{a}\widetilde{c}$, or $\widetilde{b}\widetilde{a}$, or $\widetilde{b}\widetilde{c}$, or $\widetilde{c}\widetilde{a}$. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- $\widetilde{s}$ $(\widetilde{q}_{0}\odot \widetilde{s}$, $\widetilde{p}_{0}\odot \widetilde{s}$) $\widetilde{s}$ $(\widetilde{q}_{0}\odot \widetilde{s}$, $\widetilde{p}_{0}\odot \widetilde{s}$) ----------------- ---------------------------------------------------------------------------------- ------------------------------------------- ---------------------------------------------------------------------------------- $\epsilon$ ($[0.9, 0], [0.9, 0]$) $\widetilde{b}\widetilde{a}\widetilde{c}$ ($[0.4, 0.4], [0.2, 0.4]$) $\widetilde{a}$ ($[0.9, 0.4], [0.9, 0.4]$) $\widetilde{b}\widetilde{a}$ ($[0.4, 0.4], [0.4, 0.4]$) $\widetilde{b}$ ($[0.4, 0.9], [0.4, 0.9]$) $\widetilde{b}\widetilde{c}$ ($[0.4, 0.9], [0.2, 0.9]$) $\widetilde{c}$ ($[0.4, 0], [0.2, 0]$) $\widetilde{c}\widetilde{a}$ ($[0.4, 0.4], [0.2, 0.2]$) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Eight different state pairs reachable from $(\widetilde{q}_{0},\widetilde{p}_{0})$ \(1) If $\widetilde{s}=\epsilon$, from Fig. 4 we know that $[\widetilde{p}_{0}\odot\widetilde{\sigma}] =[\widetilde{q}_{0}\odot\widetilde{\sigma}]$ for $\widetilde{\sigma}=\widetilde{a}$ and $\widetilde{\sigma}=\widetilde{b}$, so the fuzzy observability condition holds for $\widetilde{\sigma}=\widetilde{a}$ and $\widetilde{\sigma}=\widetilde{b}$. For $\widetilde{\sigma}=\widetilde{c}$, it is clear to know that the fuzzy observability condition holds since $\widetilde{D}(P(\widetilde{s}\widetilde{\sigma}))=0$. \(2) If $\widetilde{s}=\widetilde{a}$, or $\widetilde{b}$, or $\widetilde{b}\widetilde{a}\widetilde{c}$, or $\widetilde{b}\widetilde{a}$, or $\widetilde{b}\widetilde{c}$, we check the fuzzy observability condition via Figs. 4, 5, 6. The fuzzy observability condition holds obviously since in subtrees $T_{1}$ and $T_{2}$, for any $\widetilde{s}$ and any $\widetilde{\sigma}$, $[\widetilde{p}_{0}\odot \widetilde{s}\odot\widetilde{\sigma}] =[\widetilde{q}_{0}\odot \widetilde{s}\odot\widetilde{\sigma}]$. \(3) We consider the last cases of $\widetilde{s}=\widetilde{c}$, or $\widetilde{s}=\widetilde{c}\widetilde{a}$. If $\widetilde{s}=\widetilde{c}$, then $\widetilde{t}=\epsilon$, or $\widetilde{t}=\widetilde{c}$ such that $P(\widetilde{s})=P(\widetilde{t})$. If $\widetilde{s}=\widetilde{c}\widetilde{a}$, then $\widetilde{t}=\widetilde{a}$, or $\widetilde{t}=\widetilde{a}\widetilde{c}$, or $\widetilde{t}=\widetilde{c}\widetilde{a}$ such that $P(\widetilde{s})=P(\widetilde{t})$. According to Fig. 7, we can test that the fuzzy observability condition holds when $s\in\{\widetilde{c},\widetilde{c}\widetilde{a}\}$ by means of the following Table III. In light of the above computing process, we have verified that $\widetilde{K}$ satisfies the fuzzy observability condition. On the other hand, we notice that $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})\leq 0.2$ and $pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})\geq 0.2$ for any $\widetilde{\sigma}\in \widetilde{E}$ and any $\widetilde{s}\in \widetilde{E}^{*}$, so $\widetilde{K}$ satisfies the fuzzy controllability condition clearly. Therefore, from $\widetilde{K}$ being fuzzy observable and fuzzy controllable together with the assumption of $\widetilde{K}$ being ${\cal L}_{\widetilde{G},m}$-closed, by Theorem 1, we know that there exists a nonblocking fuzzy supervisor $\widetilde{S}_{P}: P(\widetilde{E}^{*})\rightarrow {\cal F}(\widetilde{E})$ that can disable the fuzzy events with respective degrees such that $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s})= pr(\widetilde{K})^{f}(\widetilde{s}) \hskip 7mm {\rm and} \hskip 7mm {\cal L}_{\widetilde{S}_{P}/\widetilde{G},m}(\widetilde{s})= \widetilde{K}(\widetilde{s}).$$ In fact, $\widetilde{S}_{P}$ may be constructed as the proof of Theorem 1 in Appendix. $\widetilde{s}$ $\widetilde{t}$ $\widetilde{\sigma}$ $x_{1}$ $x_{2}$ $x_{3}$ $y$ $V$ $ W $ $V\leq W $ ------------------------------ ------------------------------ ---------------------- --------- --------- --------- ----- ------ ------- ------------ $\widetilde{c}$ $\epsilon$ $\widetilde{a}$ 0.2 0.9 0.4 0.2 0 0.08 T $\widetilde{b}$ 0.2 0.9 0.4 0.2 0 0.12 T $\widetilde{c}$ 0.2 0.2 0.4 0.2 0 0 T $\widetilde{c}$ $\widetilde{a}$ 0.2 0.2 0.4 0.2 0 0.08 T $\widetilde{b}$ 0.2 0.2 0.4 0.2 0 0.12 T $\widetilde{c}$ 0.2 0.2 0.4 0.2 0 0 T $\widetilde{c}\widetilde{a}$ $\widetilde{a}$ $\widetilde{a}$ 0.2 0.9 0.4 0.2 0.08 0.08 T $\widetilde{b}$ 0.2 0.9 0.4 0.2 0.08 0.08 T $\widetilde{c}$ 0.2 0.4 0.4 0.2 0.08 0.08 T $\widetilde{a}\widetilde{c}$ $\widetilde{a}$ 0.2 0.4 0.4 0.2 0.08 0.08 T $\widetilde{b}$ 0.2 0.4 0.4 0.2 0.08 0.08 T $\widetilde{c}$ 0.2 0.4 0.4 0.2 0.08 0.08 T $\widetilde{c}\widetilde{a}$ $\widetilde{a}$ 0.2 0.2 0.4 0.2 0.08 0.08 T $\widetilde{b}$ 0.2 0.2 0.4 0.2 0.08 0.08 T $\widetilde{c}$ 0.2 0.2 0.4 0.2 0.08 0.08 T : Testing the fuzzy observability condition for $\widetilde{c}$ and $\widetilde{c}\widetilde{a}$ Concluding Remarks ================== Since FDES was introduced by Lin and Ying \[12, 13\], it has been successfully applied to biomedical control for HIV/AIDS treatment planning \[15, 16\], decision making \[17\] and intelligent sensory information processing for robotic control \[18, 19\]. In view of the impreciseness for some events being observable and controllable in practice, in this paper we dealt with Controllability and Observability Theorem, in which both the observability and the controllability of events are considered to be fuzzy. In particular, we have presented a computing method for deciding whether or not the fuzzy observability and controllability conditions hold, and thus, this can further test the existence of supervisors in Controllability and Observability Theorem of FDESs. As some examples (Example 3) presented show, this computing method is clearly applied to testing the existence of supervisors in the Controllability and Observability Theorem of classical DESs \[1\], and this is a different method from classical case \[1\]. As pointed out in \[1\], in supervisory control theory there are three fundamental theorems: Controllability Theorem, Nonblocking Controllability Theorem, and Controllability and Observability Theorem. This paper, together with \[20-23\], has primarily established supervisory control theory of FDESs. An further issue is regarding the diagnosis of FDESs, as the diagnoses of classical and probabilistic DESs \[32, 33\]. Also, it is worth further considering to apply the supervisory control theory of FDESs to practical control issues, particularly in biomedical systems and traffic control systems \[34, 35\]. Moreover, dealing with FDESs modelled by fuzzy petri nets \[36\] is of interest, as the issue of DESs modelled by Petri nets \[37-39\]. Proof of Theorem 1 ================== We construct a fuzzy supervisor $\widetilde{S}_{P}:P(\widetilde{E}^{*})\rightarrow {\cal F}(\widetilde{E})$ as follows: $ \widetilde{S}_{P}(\epsilon)(\widetilde{\sigma}) =pr(\widetilde{K})^{f}(\widetilde{\sigma})$, and for $\widetilde{s}\in\widetilde{E}^{*}$, $\widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})$ is defined by the following two cases: [*Case 1:*]{} If there exists another string $\widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=P(\widetilde{s^{'}})$, then $$\begin{aligned} \begin{array}{ll} \widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})=\\ \left\{ \begin{array}{ll} \max \{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\min\{ pr(\widetilde{K})^{f}(\widetilde{s^{'}}\widetilde{\sigma}),{\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma})\}\},&\\ \hskip 15mm{\rm if}\hskip 4mm pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})\leq pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma});&\\ \max \{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}), pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})\}, & \\ \hskip 15mm{\rm if}\hskip 4mm pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma}). \end{array} \right. \end{array}\end{aligned}$$ [*Case 2:*]{} If there does not exist another string $\widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=P(\widetilde{s^{'}})$, then $$\begin{array}{ll} \widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})=\\ \left\{ \begin{array}{ll} \min \{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}), {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}) \}, &{\rm if}\ pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})\leq \widetilde{\Sigma}_{uc}(\widetilde{\sigma}),\\ pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma}), &{\rm if}\ pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})>\widetilde{\Sigma}_{uc}(\widetilde{\sigma}). \end{array} \right. \end{array}$$ Firstly we prove the sufficiency. 1\. We check the fuzzy admissibility condition. Let $\widetilde{s}\in\widetilde{E}^{*}$ and $\widetilde{\sigma}\in \widetilde{E}$. If $\widetilde{s}=\epsilon$, then by the fuzzy controllability condition, we have $$\min\{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 1mm{\cal L}_{\widetilde{G}}(\widetilde{\sigma})\} \leq pr(\widetilde{K})^{f}(\widetilde{\sigma})= \widetilde{S}_{P}(\epsilon)(\widetilde{\sigma}).$$ Therefore, the fuzzy admissibility condition holds when $\widetilde{s}=\epsilon$. For $\widetilde{s}\neq\epsilon$, we check the fuzzy admissibility condition by the following two cases. (i) If there exists $\widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=P(\widetilde{s^{'}})$, then from (23), we have $$\min\{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 1mm{\cal L}_{\widetilde{G}}(\widetilde{s}\widetilde{\sigma})\} \leq \widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma})\leq \widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma}).$$ (ii) If there does not exist $\widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=P(\widetilde{s^{'}})$, then from (24), we have $$\min\{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 1mm{\cal L}_{\widetilde{G}}(\widetilde{s}\widetilde{\sigma})\}= \widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})$$ when $pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})\leq \widetilde{\Sigma}_{uc}(\widetilde{\sigma})$, and $$\min\{\widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 1mm{\cal L}_{\widetilde{G}}(\widetilde{s}\widetilde{\sigma})\} <pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})= \widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})$$ when $pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})> \widetilde{\Sigma}_{uc}(\widetilde{\sigma})$. 2\. We check ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s})= pr(\widetilde{K})^{f}(\widetilde{s})$ for any $\widetilde{s}\in \widetilde{E}^{*}$, where $\widetilde{\Sigma}_{o}(\widetilde{s})>0$. We proceed by induction on the length of $\widetilde{s}$. If $\mid \widetilde{s}\mid=1$, by Definition 6, $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{\sigma})= \min\{{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\epsilon), \hskip 1mm{\cal L}_{\widetilde{G}}^{f}(\widetilde{\sigma}),\hskip 1mm\widetilde{S}_{P}(\epsilon)(\widetilde{\sigma})\}.$$ Notice that $\widetilde{S}_{P}(\epsilon)(\widetilde{\sigma}) =pr(\widetilde{K})^{f}(\widetilde{\sigma})$ and $\widetilde{K}\subseteq {\cal L}_{\widetilde{G},m}$, we have that ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{\sigma})= pr(\widetilde{K})^{f}(\widetilde{\sigma})$. Suppose ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s})= pr(\widetilde{K})^{f}(\widetilde{s})$ holds for $\mid \widetilde{s}\mid\leq k-1$ where $\widetilde{\Sigma}_{o}(\widetilde{s})>0$. The following is to verify the equality for any $\widetilde{s}\widetilde{\sigma}$ where $\mid \widetilde{s}\mid= k-1$. By Definition 6, and the assumption of induction, we have $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})= \min\{pr(K)^{f}(\widetilde{s}),\hskip 1mm{\cal L}_{G}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm S_{P}(P(\widetilde{s}))(\widetilde{\sigma})\}.$$ Next we divide it into three cases. \(1) If there exists another string $\widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=P(\widetilde{s^{'}})$, and $ pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})\leq pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$, then with the definition of $\widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})$, we have $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma}) =\min\{pr(K)^{f}(\widetilde{s}),\hskip 1mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm \widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma})\}$$ when ${\cal L}_{\widetilde{G}}(\widetilde{s}\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$ and $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma});$ and $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma}) =\min\{pr(K)^{f}(\widetilde{s}),\hskip 1mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm pr(K)^{f}(\widetilde{s^{'}}\widetilde{\sigma})\}$$ when ${\cal L}_{\widetilde{G}}(\widetilde{s}\widetilde{\sigma})\leq pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$ or $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})\leq pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$. By the fuzzy controllability condition and fuzzy observability condition, we obtain that ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})\leq pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})$. On the other hand, it is clear that $pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})\leq {\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})$. \(2) If there exists another string $\widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=p(\widetilde{s^{'}})$, but $ pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$, then from (23), we have $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})=\min\{{\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})\}$$ when $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})\leq pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})$; and $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma}) =\min\{pr(K)^{f}(\widetilde{s}),\hskip 1mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm \widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma})\}$$ when $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})$. Due to the fuzzy controllability condition and the assumption $ pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$, we have ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})\leq pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})$. And the inverse ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})\geq pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})$ holds clearly. \(3) If there does not exist another string $ \widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=P(\widetilde{s^{'}})$, then with the definition of $\widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma})$ (i.e., Eq. (24)), we obtain that $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})= \min\{pr(K)^{f}(\widetilde{s}),\hskip 1mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm \widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma})\}$$ when $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})\leq pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})$; and $${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})= \min\{{\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 1mm pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})\}$$ when $\widetilde{\Sigma}_{uc}(\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})$. We can analogously verify ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma})= pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma})$ from the fuzzy controllability condition. 3\. We show that ${\cal L}_{\widetilde{S}_{P}/\widetilde{G},m}= \widetilde{K}$ and $\widetilde{S}_{P}$ is nonblocking as follows. Since $\widetilde{K}$ is ${\cal L}_{\widetilde{G},m}$-closed and ${\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s})= pr(\widetilde{K})^{f}(\widetilde{s})$ has been proved above, by Definition 6, $$\begin{aligned} \begin{array}{lll} &&{\cal L}_{\widetilde{S}_{P}/\widetilde{G},m}(\widetilde{s}) =\min\{{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}), \hskip 1mm {\cal L}_{\widetilde{G},m}(\widetilde{s})\} \\&=&\min\{pr(\widetilde{K})^{f}(\widetilde{s}), \hskip 1mm {\cal L}_{\widetilde{G},m}(\widetilde{s})\}=\widetilde{K}(\widetilde{s}). \end{array}\end{aligned}$$ Furthermore, $$\begin{aligned} \begin{array}{lll} &&\widetilde{D}(P(\widetilde{s}))\cdot pr({\cal L}_{S_{P}/ G,m})(\widetilde{s})\\&=&\widetilde{D}(P(\widetilde{s}))\cdot pr(K)(\widetilde{s})=pr(\widetilde{K})^{f}(\widetilde{s})={\cal L}_{S_{P}/ G}(\widetilde{s}).\end{array}\end{aligned}$$ We have completed the proof of [*sufficiency*]{}. The remainder is to demonstrate the [*necessity*]{}. 1\. We prove that $\widetilde{K}$ satisfies the fuzzy controllability condition. Obviously, the fuzzy controllability condition holds for $\widetilde{s}=\epsilon$. For any $\widetilde{s}\in\widetilde{E}^{*}$, by the fuzzy admissibility condition, we have $$\begin{aligned} && \min\{pr(\widetilde{K})^{f}(\widetilde{s}),\hskip 1mm \widetilde{\Sigma}_{uc}^{f}(\widetilde{\sigma}),\hskip 1mm {\cal L}_{G}^{f}(\widetilde{s}\widetilde{\sigma})\}\\ &\leq& \min\{{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}),\hskip 1mm \widetilde{S}_{P}(P(\widetilde{s}))(\widetilde{\sigma}),\hskip 1mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma} )\}\\ &=&{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}( \widetilde{s}\widetilde{\sigma})=pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma}).\end{aligned}$$ 2\. $\widetilde{K}$ is ${\cal L}_{\widetilde{G},m}$-closed obviously. In fact, from ${\cal L}_{\widetilde{S}_{P}/\widetilde{G},m}=\widetilde{K}$, we have $$\begin{aligned} \begin{array}{lll}&&\widetilde{K}(\widetilde{s})={\cal L}_{\widetilde{S}_{P}/\widetilde{G},m}(\widetilde{s})\\ &=&\min\{{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}), \hskip 1mm{\cal L}_{\widetilde{G},m}(\widetilde{s})\}\\ &=&\min\{pr(\widetilde{K})^{f}(\widetilde{s}),\hskip 1mm {\cal L}_{\widetilde{G},m}(\widetilde{s})\}.\end{array}\end{aligned}$$ 3\. We check that $\widetilde{K}$ satisfies the fuzzy observability condition. For any $\widetilde{s}\in\widetilde{E}^{*}$ and $\widetilde{\sigma}\in \widetilde{E}$, if there exists another string $\widetilde{s^{'}}\in\widetilde{E}^{*}$ such that $P(\widetilde{s})=P(\widetilde{s^{'}})$, then the fuzzy observability condition holds obviously if $ pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})> pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$. If $ pr(\widetilde{K})(\widetilde{s}\widetilde{\sigma})\leq pr(\widetilde{K})(\widetilde{s^{'}}\widetilde{\sigma})$, we have $$\begin{aligned} &&\min\{pr(\widetilde{K})^{f}(\widetilde{s}),\hskip 2mm pr(\widetilde{K})^{f}(\widetilde{s^{'}}\widetilde{\sigma}),\hskip 2mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma} )\}\\ &=& \min\{pr(\widetilde{K})^{f}(\widetilde{s}), \hskip 2mm{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s^{'}}\widetilde{\sigma}),\hskip 2mm {\cal L}_{\widetilde{G}}^{f}( \widetilde{s}\widetilde{\sigma} )\}\\ &=& \min\{pr(\widetilde{K})^{f}(\widetilde{s}),\hskip 1mm{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s^{'}}),\hskip 1mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s^{'}}\widetilde{\sigma}),\\&& \widetilde{S}_{P}(P(\widetilde{s^{'}})(\widetilde{\sigma}),\hskip 1mm{\cal L}_{\widetilde{G}}^{f}( \widetilde{s}\widetilde{\sigma} )\}\\ &\leq& \min\{pr(\widetilde{K})^{f}(\widetilde{s}),\hskip 2mm {\cal L}_{\widetilde{G}}^{f}(\widetilde{s}\widetilde{\sigma}),\hskip 2mm \widetilde{S}_{P}(P(\widetilde{s^{'}}))(\widetilde{\sigma})\}\\ &=&{\cal L}_{\widetilde{S}_{P}/\widetilde{G}}(\widetilde{s}\widetilde{\sigma} )=pr(\widetilde{K})^{f}(\widetilde{s}\widetilde{\sigma}).\end{aligned}$$ Therefore, $\widetilde{K}$ satisfies the fuzzy observability condition. 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[^1]: This work was supported in part by the National Natural Science Foundation under Grant 90303024 and Grant 60573006, and the Research Foundation for the Doctorial Program of Higher School of Ministry of Education under Grant 20050558015, of China. [^2]: D.W. Qiu is with Department of Computer Science, Zhongshan University, Guangzhou, 510275, China (e-mail: [email protected]). [^3]: F.C. Liu is with 1) Department of Computer Science, Zhongshan University, Guangzhou, 510275, China; 2) Faculty of Computer, Guangdong University of Technology, Guangzhou, 510090, China (e-mail: [email protected]). [^4]:

--- abstract: 'Approximate inference algorithm is one of the fundamental research fields in machine learning. The two dominant theoretical inference frameworks in machine learning are variational inference (VI) and Markov chain Monte Carlo (MCMC). However, because of the fundamental limitation in the theory, it is very challenging to improve existing VI and MCMC methods on both the computational scalability and statistical efficiency. To overcome this obstacle, we propose a new theoretical inference framework called ergodic Inference based on the fundamental property of ergodic transformations. The key contribution of this work is to establish the theoretical foundation of ergodic inference for the development of practical algorithms in future work.' bibliography: - 'database.bib' nocite: '[@langley00]' --- Introduction {#sec:intro} ============ Statistical inference is the cornerstone of probabilistic modelling in machine learning. The research on inference algorithms always attracts a great attention in the research community, because it is the fundamentally important in the computation of Bayesian inference, deep generative models. The majority of research is focused on algorithmic development in two theoretical frameworks: variational inference (VI) and Markov chain Monte Carlo (MCMC). These two methods are significantly different. VI is an optimisation-based approach, in particular, which fits a simple distribution to a given target. In contrast, MCMC is a simulation-based approach, which sequentially generates asymptotically unbiased samples of arbitrary target. Unfortunately, both VI and MCMC suffer from fundamental limitations. VI methods are in general biased because the density function of approximate distribution must be in closed-form. MCMC methods are also biased in practice because the Markov property limits the sample simulation in a local sample space close to previous samples. However, VI is in general more scalable in computation. Optimising variational distribution and simulating samples in VI are computationally efficient and can be accelerated by parallelization on GPU. In contrast, simulating Markov chains is computationally inefficient and, more importantly, asynchronized parallel simulation of multiple Markov chains has no effect on reducing sample correlations but multiplies the computation. Ergodic Measure preserving flow (EMPF), introduced by [@DBLP:journals/corr/abs-1805-10377], is a recent novel optimisation-based inference method that overcomes the limitations of both MCMC and VI. However, there is no theoretical proof of the validity of EMPF. In this work, we will generalize EMPF to a novel inference framework called ergodic inference. In particular, the purpose of this work is to establish the theoretical foundation of ergodic inference. We list the key contribution of this work as following - The mathematical foundation of ergodic inference. (Section \[sec:ei\_principle\] and \[sec:ergodic\_transformation\]) - A tractable loss of ergodic inference and the proof of the validity of the loss. (Section \[sec:ergodic\_loss\]) - An ergodic inference model: deep ergodic inference networks (Section \[sec:deins\]) - Clarification of differences between ergodic inference, MCMC and VI (Section \[sec:deins\]) The background {#sec:background} ============== Convergence of probability measures is the foundation of statistical inference. Distance metric between probability measures is critical in the study of convergence. We will review the basics of distance metrics between probability measures and connect these metrics to theoretical foundation of inference methods. Distance Metric of Probability Measures --------------------------------------- Total variation distance is fundamentally important in probability theory, because it defines the strongest convergence of probability measure. Let $(\Omega, {\mathcal{F}})$ be a measure space, where $\Omega$ denotes the sample space and ${\mathcal{F}}$ denotes the collection of measurable subsets of $\Omega$. Given two probability measure $P$ and $Q$ defined on $(\Omega, {\mathcal{F}})$, the TV distance between $Q$ and $P$ is defined as $$\begin{aligned} {D_{\text{TV}}}(Q, P) = \sup_{A \in {\mathcal{F}}} \vert Q(A) - P(A) \vert.\end{aligned}$$ Convergence in TV, that is ${D_{\text{TV}}}(Q, P)=0$, means $Q$ and $P$ cannot be distinguished on any measurable set. The Kullback-Leibler (KL) divergence is an important measure of difference between probability measures in statistical methods. For a continuous sample space $\Omega$, the KL divergence is defined as $$\begin{aligned} {D_{\text{KL}}}(Q \vert \vert P) = \int_{\Omega} dQ\log \frac{dQ}{dP},\end{aligned}$$ where $dP$ denote the density of probability measure. Approximate Monte Carlo Inference --------------------------------- Monte Carlo method is the most popular simulation based inference technique in probabilistic modelling. For example, to fit a probabilistic model ${\pi}$ by maximum likelihood estimation, it is essential to compute the gradient of the partition function $Z(\theta) = \int {\pi^*}(z)dz$. Given the unnormalised density function $\log{\pi^*}(z)$, computing the gradient becomes a problem of expectation estimation $$\partial_{\theta}Z(\theta) = {\mathbf{E}}_{{\pi}(z)}[\partial_{\theta}\log{\pi^*}(z)].$$ Monte Carlo methods allow us to construct unbiased estimator of expectation as $${\mathbf{E}}_{{\pi}(z)}[f(z)] = \lim_{N \rightarrow \infty}\frac{1}{N} \sum_{i=1}^N f(z_i),$$ where $z_i$ denotes samples from ${\pi}$. Unfortunately, it is intractable to generate samples from complex distributions, like the posterior distributions in model parameters or latent variables. Because of this challenge, approximate Monte Carlo Inference is fundamentally important. We will review the theoretical foundation of two important inference methods: variational Inference (VI) and Markov chain Monte Carlo (MCMC) in the next two sections. Variational Inference {#sec:vi} --------------------- The theoretic foundation of VI is Pinsker’s inequality. Pinsker’s inequality states that the KL divergence is a upper bound of TV distance $$\begin{aligned} {D_{\text{TV}}}(Q, P) \le {D_{\text{KL}}}(Q \vert \vert P).\end{aligned}$$ Given a parametric distribution $Q$ and the target distribution ${\pi}$, minimising the KL divergence ${D_{\text{KL}}}(Q \vert \vert {\pi})$ implies the less TV distance ${D_{\text{TV}}}(Q, {\pi})$. The key challenge of VI is how to construct the parametric family ${\mathcal{Q}}$ so that the estimation of the KL divergence is tractable and family ${\mathcal{Q}}$ is expressive to approximate complex target. This forces most VI methods to choose $Q$ with closed-form density function. Otherwise, the estimation of entropy term $\text{H}(Q)=-\int Q(dz) \log q(z)$ becomes challenging. In practice, the approximation family ${\mathcal{Q}}$ in most VI methods are rather simple, like Gaussian distribution, so the approximation bias due to oversimplified $Q$ is the key issue of VI. However, simple approximate family gives VI methods great computational advantage in practice. First, the main loss function in VI is known as the evidence lower bound (ELBO) $$\begin{aligned} L_{\text{ELBO}} = \int_{\Omega} dQ\log \frac{d{\pi^*}}{dQ} \le \log \int d{\pi^*}.\end{aligned}$$ With analytic form of the entropy of $Q$, ELBO can be efficiently computed and optimized using standard gradient descent algorithm. Second, simulating i.i.d. samples from a simple variational family $Q$ is straightforward and very efficient. Markov Chain Monte Carlo {#sec:mcmc} ------------------------ The theoretical foundation of Markov chain Monte Carlo (MCMC) is ergodic theorem. Ergodic theorem states that, given an ergodic Markov chain $(Z_n)$ with a stationary distribution ${\pi}$, the average cross states of chain is equivalent to the average in state space of the chain, that is $${\mathbf{E}}_{{\pi}}[f] = \lim_{m \rightarrow \infty}\frac{1}{M} f(Z_{\infty}^{m}) = \lim_{n \rightarrow \infty}\frac{1}{N} f(Z_n),$$ where $Z_{\infty}^m$ denotes the sample of a well-mixed Markov chains after infinitely long transitions. Ergodic theorem implies that we can generate unbiased samples from every Markov transition without waiting forever for the chains to reach stationary state. Therefore, we can trade computational efficiency with a bias that may decrease in a long time. The key challenge of MCMC methods is to define ergodic Markov chains with any given stationary distribution ${\pi}$. This challenge was solved first by Metropolis-Hastings algorithm. We will discuss in detail in Section \[sec:mh\]. Ergodic Markov chains enjoy strong stability. Irrespective of the distribution of initial state $\mu(z_0)$ and the parameter of Markov kernel $K(\cdot, \cdot)$, the distribution of the state of the chain is guaranteed to converge to the stationary distribution in total variation after every transition. Formally, that means the reduce of TV distance to stationary for all $L \ge 0$ $$\begin{aligned} {D_{\text{TV}}}\left(Q_{L+1}, {\pi}\right) < {D_{\text{TV}}}\left(Q_{L}, {\pi}\right)\end{aligned}$$ where $q_{L}$ denotes the marginal distribution of the $L$-th state and $$\begin{aligned} q_{L}(dz') = \int K(z, d z')q_{L}(dz).\end{aligned}$$ As $L$ increases, the distribution $q_L$ converges to a unique stationary distribution ${\pi}$ $$\lim_{l \rightarrow \infty}{D_{\text{TV}}}(Q_l, {\pi}) =0.$$ In spite of the theoretical convergence property, the convergence of MCMC chains is not guaranteed in practice. Because the burn-in stage cannot be infinite long, the samples from MCMC methods are often biased. The problem is that there is no reliable measurement of such a sampling bias related to TV distance or KL divergence. The iterative simulation of Markov chain is another limitation in computational efficiency. Each sample from MCMC methods requires one simulation of Markov transition and this can only be executed in a sequential manner due to the nature of Markov chain. Therefore, the sampling time of MCMC grows linearly with the number of samples. Ergodic Inference Principle {#sec:ei_principle} =========================== In this section, we present the mathematical foundation of ergodic inference principle. Motivation ---------- First, we would like to propose the the following properties of ideal inference method: - Parallelizable: the simulation of each sample is computationally independent; - Statistically efficient: there is zero correlation between samples; - Asymptotic unbiased: more computational power guarantees diminishing of simulation bias. The bias can be eliminated in theory with sufficient computation. Both MCMC and VI fail to have all the properties above. For this reason, there are existing works on a hybrid methods that combine MCMC and VI, for example, accelerate the burn-in of MCMC using variational approximation in [@pmlr-v70-hoffman17a] or optimise ELBO based on tractable density function of MCMC kernel in [@salimans2015markov]. To some extend, such algorithmic hybrid approach can be useful in practice. However, the limitation in theoretical foundation of MCMC and VI cannot be eliminated by algorithmic modification. To achieve an ideal inference method, it is necessary to have a new mathematical theoretical foundation. The Theoretical Foundation {#sec:foundation} -------------------------- Different from Pinsker’s inequality and ergodic theorem, the theoretical motivation of the proposed inference is the characteristic property of ergodic Markov transition: *there is a unique invariant distribution for every ergodic Markov Kernel*. Formally, let $K_{{\pi}}$ be an ergodic Markov transition kernel with an invariant distribution ${\pi}$. By construction of $K_{{\pi}}$, ${\pi}$ is guaranteed to be the only distribution satisfies the condition ${\pi}(d{\mathbf{z}}') = \int K_{{\pi}}({\mathbf{z}}, d{\mathbf{z}}') {\pi}(d{\mathbf{z}})$. Based on the property of ergodic Markov kernel, we construct the following criteria to verify if a distribution is equivalent to the stationary distribution of the kernel. Given a distribution $q$, the distribution of $q$ after one Markov transition by $K$ is given by $$\begin{aligned} q_1({\mathbf{z}}') = \int K_{\pi}({\mathbf{z}}, {\mathbf{z}}') q(d{\mathbf{z}}). \label{eq:q_1}\end{aligned}$$ We say the distribution $q$ is preserved by $K_{\pi}$ if $$\begin{aligned} {D_{\text{TV}}}(q_1, q) = 0. \label{eq:db_L2}\end{aligned}$$ By the uniqueness of the invariant distribution of ergodic kernel $K_{\pi}$, the preservation of $q$ by $K_{\pi}$ as implies ${D_{\text{TV}}}(q, {\pi}) = 0$. This motivates the following loss function. Given a Markov kernel $K_{\pi}({\mathbf{z}}, {\mathbf{z}}')$ that is ergodic w.r.t. a distribution ${\pi}$, the ergodic loss of a distribution $q$ is defined as $$L^*(q, K_{{\pi}}) = {D_{\text{TV}}}\left(\int K_{{\pi}}({\mathbf{z}}, \cdot)q(d{\mathbf{z}}), q(\cdot) \right).$$ As mentioned earlier, the loss $L^*(q, K_{{\pi}}) $ is equal to 0 if and only if ${D_{\text{TV}}}(q, {\pi})$ is equal to 0. Let ${\pi}$ be the target distribution and $q$ be the approximate distribution in a parametric family ${\mathcal{Q}}$. Given an ergodic Markov kernel $K_{\pi}$, the closest $q \in {\mathcal{Q}}$ to the target ${\pi}$ can be identified by the parameter $\phi^*$ optimising the ergodic loss $L^*(\cdot, K_{\pi})$ $$\phi^* = \operatorname*{arg\,min}_{\phi}L^*(q_{\phi}, K_{\pi}).$$ If the target distribution is in ${\mathcal{Q}}$, then the optimal parameter $\phi^*$ should have the loss $$L^*(q_{\phi^*}, K_{{\pi}}) =0,$$ otherwise the $L^2$ norm of the gradient of the loss should vanish $$\vert\vert \partial_{\phi^*} L^*(q_{\phi^*}, K_{{\pi}}) \vert\vert_{2}^2 = 0.$$ Technical Challenges -------------------- There are two technical challenges of ergodic inference methods in practice. First, we need a tractable estimation of a loss function equivalent to ${D_{\text{TV}}}(q_1, q)$. The estimation of the gradient of the loss should also be tractable for the optimisation of the parameter $\phi$. Second, we need a general parametric family ${\mathcal{Q}}$ that can approximate any target distribution up to a certain amount of error. More specific, the error can be controlled and even eliminated by increase the complexity of approximation family of ${\mathcal{Q}}$, i.e. the number of parameters of ${\mathcal{Q}}$ is unlimited. The computational cost of optimisation is associated with the complexity of ${\mathcal{Q}}$. We will present the solution to the first challenge in Section \[sec:ergodic\_loss\] and the solution to the second challenge in Section \[sec:deins\]. Ergodic Transformations {#sec:ergodic_transformation} ======================= The key of solving the technical challenges in ergodic inference is the reparameterization of the ergodic Markov kernel. This is important in both algorithmic development and theoretical analysis. Ergodic Transformations and Markov Kernels ------------------------------------------ Ergodic Markov kernels are essentially conditional distributions, which can be reparameterized by deterministic transformations known as measure preserving transformations (MPTs). Given a probability measure $\mu$, a deterministic transformation $T$ preserves $\mu$ if for any measurable subset of sample space $A$, $\mu(T^{-1}(A)) = \mu(A)$. The shear transformation $T(x, y)=(x+y, x)$, which preserves Lebesgue measure, is a classic example of MPT [@Bill86]. The following conditions are often used in the literature MCMC theory for verification of ergodic property: 1. Irreducibility: $T(A) \ne A\,, \forall A \in {\mathcal{F}}$ except $\emptyset$ and $\Omega$. 2. Density preservation: ${\pi}(T({\mathbf{z}})) = {\pi}({\mathbf{z}})$. 3. Lebesgue preservation: the determinant of the Jacobian of $T$ is equal to 1. Formally, we define the reparameterisation of Ergodic Markov chains as following. \[def:ergodic\_reparam\] (Ergodic Reparameterisation of MCMC) Given a target distribution ${\pi}({\mathbf{z}})$, a MCMC kernel $K({\mathbf{z}}, {\mathbf{z}}')$ with invariant ${\pi}$ can be reformed as two steps: 1. Simulate an auxiliary variable ${\mathbf{r}}$ with distribution $\mu({\mathbf{r}})$ 2. Deterministic transformation $({\mathbf{z}}', {\mathbf{r}}') = T_{{\pi}\mu}({\mathbf{z}}, {\mathbf{r}})$, where $T_{{\pi}\mu}$ is an ergodic transformation that preserves the probability measure ${\pi}({\mathbf{z}})\mu({\mathbf{r}})$. The transformation $T_{{\pi}\mu}$ in ergodic reparameterisation is fundamentally different from volume preserving transformation $V({\mathbf{z}})$ in the sample space of ${\mathbf{z}}$ for two reasons. - $T_{{\pi}\mu}({\mathbf{z}}, {\mathbf{r}})$ does not preserve the volume/entropy in the sample space of ${\mathbf{z}}$, but $V({\mathbf{z}})$ must preserves the volume/entropy in the space of ${\mathbf{z}}$. - $T_{{\pi}\mu}({\mathbf{z}}, {\mathbf{r}})$ preserves the probability measure ${\pi}({\mathbf{z}})$, but $V({\mathbf{z}})$ does not preserve ${\pi}({\mathbf{z}})$ in general. Ergodic transformations also allow us to form the expectation under Markov transition as composition of functions, that is not used in classic MCMC literature. Formally, this is given by the following proposition. \[proposition:mpt\] Given an ergodic transformation $T_{\pi}$ w.r.t. $\pi$, the expectation is preserved by the transformation, which means, for any function $f$ $$\int_{\Omega} f({\mathbf{z}}) \pi(d{\mathbf{z}}) = \int_{\Omega} f \circ T_{\pi}({\mathbf{z}}) \pi(d{\mathbf{z}}) = \int_{\Omega'} f({\mathbf{z}}') T_{\pi *}\pi(d{\mathbf{z}}'),$$ where $\Omega'$ is the image of $\Omega$ under $T_{\pi}$ and $T_{\pi *}\pi(\cdot)$ denotes the pushforward probability measure of $\pi$ under $T_{\pi}$. Because $T_{\pi}$ preserves ${\pi}$, $\Omega' = \Omega$ and ${D_{\text{TV}}}(\pi, T_{\pi *}\pi) = 0$. In the next two sections, we will demonstrate the ergodic reparameterization with two well-known MCMC kernels. Metropolis-Hastings Transformations {#sec:mh} ----------------------------------- Metropolis-Hastings (MH) algorithm is the first and most well-known MCMC methods. We will show that it is straightforward to form the MH transition kernel as an ergodic transformation. Given a target distribution $\pi({\mathbf{z}})$ and a transition proposal distribution $q({\mathbf{r}}\vert {\mathbf{z}})$, MH kernel in most text books is described as following two steps: 1. Sample ${\mathbf{r}}$ from $q(\cdot \vert {\mathbf{z}})$. 2. Return the new state of the chain as ${\mathbf{r}}$ with probability $$\begin{aligned} p_{MH} = \min\left\{1, \frac{\pi({\mathbf{r}})q({\mathbf{z}}\, \vert\, {\mathbf{r}})}{ \pi({\mathbf{z}})q({\mathbf{r}}\, \vert\, {\mathbf{z}})}\right\},\end{aligned}$$ otherwise the state remains as ${\mathbf{z}}$. It is straightforward to verify that MH transition kernel preserves the density function as $$\begin{aligned} &\pi({\mathbf{z}})\left[q({\mathbf{r}}\vert {\mathbf{z}})\min\left\{1, \frac{\pi({\mathbf{r}})q({\mathbf{z}}\, \vert\, {\mathbf{r}})}{ \pi({\mathbf{z}})q({\mathbf{r}}\, \vert\, {\mathbf{z}})}\right\}\right]\\ =&\min\left\{\pi({\mathbf{z}})q({\mathbf{r}}\, \vert\, {\mathbf{z}}), \pi({\mathbf{r}})q({\mathbf{z}}\, \vert\, {\mathbf{r}})\right\}\\ =&\pi({\mathbf{r}})\left[ q({\mathbf{z}}\vert {\mathbf{r}})\min\left\{1, \frac{ \pi({\mathbf{z}})q({\mathbf{r}}\, \vert\, {\mathbf{z}})}{\pi({\mathbf{r}})q({\mathbf{z}}\, \vert\, {\mathbf{r}})} \right\} \right],\end{aligned}$$ where the MH transition kernel $K_{MH}(\cdot, \cdot)$ is in squared rackets. This verification of stationary distribution is known as detailed balance. It is important because it proves the existence of stationary distribution. Now we consider an alternative representation of MH kernel. In particular, we define a stationary distribution as the joint distribution of all random variables involved in the target $\pi$ and MH kernel $K_{MH}$, that is $\pi({\mathbf{z}}, {\mathbf{r}}, u) = \pi({\mathbf{z}}) q({\mathbf{r}}\vert {\mathbf{z}})\nu(u)$, where $\nu(u)$ denotes uniform distribution between $[0, 1]$. Following the ergodic reparameterization (Definition \[def:ergodic\_reparam\]), we can rewrite the MH algorithm as 1. Resample ${\mathbf{r}}$ from $q(\cdot \vert {\mathbf{z}})$ and $u$ from $\nu(\cdot)$. 2. Return the next state $({\mathbf{z}}', {\mathbf{r}}', u') = T_{MH}({\mathbf{z}}, {\mathbf{r}}, u)$ defined as $$\begin{aligned} T_{MH}({\mathbf{z}}, {\mathbf{r}}, u) &= ({\mathbf{z}}, {\mathbf{r}}, u)\delta(u>p_{MH}) \nonumber \\ &+ ({\mathbf{r}}, {\mathbf{z}}, u)\delta(u<p_{MH}), \label{eq:mh_func}\end{aligned}$$ where $\delta(\cdot)$ denotes indicator function. Notice that the transformation $T_{MH}({\mathbf{z}}, {\mathbf{r}}, u)$ above is a deterministic function. It is obvious that resampling ${\mathbf{r}}$ and $u$ from their conditional distribution leaves $\pi({\mathbf{z}}, {\mathbf{r}}, u)$ invariant. Then, it is straightforward to show the preservation of density function $$\pi({\mathbf{s}})\delta({\mathbf{s}}' = T_{MH}({\mathbf{s}})) = \pi({\mathbf{s}}')\delta({\mathbf{s}}= T_{MH}({\mathbf{s}}')),$$ where ${\mathbf{s}}$ denote the triple $({\mathbf{z}}, {\mathbf{r}}, u)$. It is also easy to verify that the determinate of Jacobian of $\partial_{({\mathbf{z}}, {\mathbf{r}}, u)}T_{MH}({\mathbf{z}}, {\mathbf{r}}, u)$ is always equal to 1. Hamiltonian Measure Preserving Transformations ---------------------------------------------- Hamiltonian Monte Carlo (HMC), originally known as Hybrid Monte Carlo, is an important MCMC method. Originally, HMC is considered as a hybrid method, because its combines both deterministic and stochastic simulation. The deterministic simulation in HMC essentially refers to any dynamics that generalize the classic Hamiltonian dynamics in physics. Hamiltonian system in physics is a system of moving particles in an energy field and the energy of the system is constant over time. Given $n$ particles, the state of Hamiltonian system is defined by the position ${\mathbf{z}}\in {\mathbb{R}}^n$ and the momenta ${\mathbf{r}}\in {\mathbb{R}}^n$. The position ${\mathbf{z}}$ is associated with potential energy $U:{\mathbb{R}}^n \rightarrow {\mathbb{R}}$ and the momentum ${\mathbf{r}}$ is associated with kinetic energy $K:{\mathbb{R}}^n \rightarrow {\mathbb{R}}$. The state $({\mathbf{z}}, {\mathbf{r}})$ evolves over time $t$, according to Hamilton’s equations: $$\begin{aligned} \dot {\mathbf{z}}(t) = \partial_{{\mathbf{r}}}K({\mathbf{r}});\, \dot {\mathbf{r}}(t) = -\partial_{{\mathbf{r}}}U({\mathbf{z}}),\end{aligned}$$ where $\dot {\mathbf{z}}$ denotes the derivative of ${\mathbf{z}}$ w.r.t. time $t$. It is straightforward to verify that the total energy $H = U + K$ does not change over time $$\dot H({\mathbf{z}}, {\mathbf{r}}) = \left(\partial_{{\mathbf{r}}}U({\mathbf{z}})\right)^T\partial_{{\mathbf{r}}}K({\mathbf{r}}) - \left(\partial_{{\mathbf{r}}}U({\mathbf{z}})\right)^T\partial_{{\mathbf{r}}}K({\mathbf{r}}) = 0.$$ Given an initial condition $({\mathbf{z}}, {\mathbf{r}})$, the solution of Hamiltonian dynamics is a function of time $t$ $$({\mathbf{z}}(t), {\mathbf{r}}(t)) = T_{H}(t, {\mathbf{z}}, {\mathbf{r}}).$$ Given a fixed time $t$, the solution $T_{H}$ becomes a map $T_{H, t}: {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n}$ between two states $({\mathbf{z}}, {\mathbf{r}})$ and $({\mathbf{z}}', {\mathbf{r}}')$ with the same total energy $H$. Intuitively, ${\mathbf{z}}(t)$ forms a trajectory of particle traversing in a $n$-dimensional space and the velocity of the particle is given by $\dot {\mathbf{z}}(t) = \partial_{{\mathbf{r}}}K({\mathbf{r}}(t))$. It is well-known in MCMC literature that $T_{H, t}$ is essentially a family of measure preserving transformations with any parameter $t \in {\mathbb{R}}\ne 0$. It is clear that $T_{H, t}$ is irreducible if $t \ne 0$ and density preserving w.r.t. $\exp(-H)$. The volume preservation property of Hamiltonian dynamics in the state space $({\mathbf{z}}, {\mathbf{r}})$ is a well-known result of Liouville’s Theorem. Therefore, we know that $T_{H, t}({\mathbf{z}}, {\mathbf{r}})$ with any $t \ne 0 $ is an ergodic transformation w.r.t. the distribution $\pi({\mathbf{z}})\mu({\mathbf{r}}) \propto \exp(-H({\mathbf{z}}, {\mathbf{r}}))$. This implies $T_{H, t}$ also preserves $\pi \propto \exp(-U)$ by the definition of marginal distribution. In practice, Hamiltonian dynamics do not have closed-form solutions. Fortunately, there is a rich literature on the numeric simulation of Hamiltonian dynamics. The most known approximate approach in HMC is Leapfrog algorithm, which is constructed as a sequential of shear transformations. Leapfrog algorithm enjoys strong stability and good approximation error is around squared discretized step size. See more detailed analysis in [@radford2010; @leimkuhler2004simulating]. Ergodic Loss {#sec:ergodic_loss} ============ ${\pi}$-Ergodic Loss Function ----------------------------- By the definition of TV distance, we know that $q$ is the stationary distribution of $K$ if and only if for all function $f(\cdot)$ with ${\mathbf{E}}_{\pi}[f({\mathbf{z}})] < \infty$, $$\begin{aligned} {\mathbf{E}}_{q_1}[ f({\mathbf{z}}) ] = {\mathbf{E}}_{q}[ f({\mathbf{z}})]. \label{eq:db_L}\end{aligned}$$ However, it is impossible to compare the expectation of all possible function $f$, but given specific function $f$ it is possible to estimate $$\begin{aligned} L_{K, f}(\phi) = \vert {\mathbf{E}}_{q_1}[f({\mathbf{z}})] - {\mathbf{E}}_{q}[ f({\mathbf{z}})] \vert \,. \label{eq:f-ergodic_Loss}\end{aligned}$$ With the optimal choice of function $f$ and certain condition, we can claim that $L_{K, f}(\phi) = 0$ implies ${D_{\text{TV}}}(q, {\pi}) = 0$. The log density function is an intuitive choice, because we can identify a distribution by its density function. Therefore, we define the following ${\pi}$-ergodic loss. (Ergodic Loss Function) $$\begin{aligned} L_{K, {\pi}}(\phi) = \vert {\mathbf{E}}_{q_1}[ \log {\pi}({\mathbf{z}}) ] - {\mathbf{E}}_{q}[ \log {\pi}({\mathbf{z}}) ] \vert\,. \label{eq:db_Loss_1_LL}\end{aligned}$$ (Ergodic Loss Convergence Theorem)\[theorem:loss\_convergence\] Given the ergodic Markov kernel $K_{{\pi}}$ with invariant distribution ${\pi}$, the loss $L_{K, {\pi}}(\phi) = 0$ if and only if ${\mathbf{E}}_{{\pi}}[ \log \pi({\mathbf{z}}) ] = {\mathbf{E}}_{q}[ \log \pi({\mathbf{z}}) ] $. The convergence of loss $L_{K, {\pi}}(\phi)=0$ implies $$\begin{aligned} {\mathbf{E}}_{q_1({\mathbf{z}})\mu({\mathbf{r}})}[ \log {\pi}({\mathbf{z}}) ] = {\mathbf{E}}_{q({\mathbf{z}})\mu({\mathbf{r}})}[ \log {\pi}({\mathbf{z}}) ]\,, \label{eq:theorem1_1}\end{aligned}$$ where $q_1({\mathbf{z}})$ is given by . Notice that $q_1$ is essentially the marginal of the pushforward of $q({\mathbf{z}})\mu({\mathbf{r}})$ under the measure preserving transformation $T_{{\pi}\mu}$. By Proposition \[proposition:mpt\], the expectations in can be written as following $$\begin{aligned} {\mathbf{E}}_{q_1({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ] \overset{\Delta}{=} \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu}\, d(q\mu) = \int_{\Omega} \log{\pi}\, d(q\mu), \label{eq:theorem1_10}\end{aligned}$$ where $d(q\mu)$ is the shorthand notations for $q({\mathbf{z}})\mu({\mathbf{r}})d{\mathbf{z}}d{\mathbf{r}}$. Replacing $q\mu$ on both sides in with any distribution, the equality still holds. If we replace $q\mu$ in with with the pushforward probability measure of $q({\mathbf{z}})\mu({\mathbf{r}})$ under $T_{{\pi}\mu}$, denoted by $T_{{\pi}\mu*} (q\mu)$, we have $$\begin{aligned} \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu} \circ d(T_{{\pi}\mu*}(q\mu)) = \int_{\Omega} \log{\pi}\, \circ d(T_{{\pi}\mu*}(q\mu)),\end{aligned}$$ which can be rewritten as $$\begin{aligned} \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu}^1\circ T_{{\pi}\mu}\, d(q\mu^1)\, = \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu}\, d(q\mu), \label{eq:theorem1_11}\end{aligned}$$ where $T_{{\pi}\mu}^1$ denotes $T_{{\pi}\mu} = ({\mathbf{z}}, {\mathbf{r}}_1)$ and $d\mu^1$ denotes $\mu(d{\mathbf{r}}_1)$. Notice that the LHS of is an expectation under the distribution of ${\mathbf{z}}$ after two ergodic Markov transitions from $q$, that is ${\mathbf{E}}_{q_2({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ]$. Therefore, by and , we have $$\begin{aligned} {\mathbf{E}}_{q_2({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ] &\overset{\Delta}{=} \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu}^1\circ T_{{\pi}\mu}\, d(q\mu^1)\, \nonumber \\ &= \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu}\, d(q\mu) \nonumber \\ &= {\mathbf{E}}_{q({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ]. \label{eq:theorem1_12}\end{aligned}$$ By induction, we know the expectation of ${\mathbf{E}}_{q_n}[\log {\pi}]$ does not change after any number of measure preserving transformation $T_{{\pi}\mu}$, that gives $$\begin{aligned} {\mathbf{E}}_{q_{\infty}({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ] = {\mathbf{E}}_{q({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ]. \label{eq:theorem1_13}\end{aligned}$$ By , we know if we simulate infinitely long ergodic Markov chain by kernel $K_{\pi}$, then the expectation ${\mathbf{E}}_{q_{\infty}({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ]$ is the same as the initial expectation ${\mathbf{E}}_{q({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ]$. Because an ergodic Markov chain has unique invariant distribution, implies $$\begin{aligned} {\mathbf{E}}_{{\pi}({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ] = {\mathbf{E}}_{q({\mathbf{z}})}[ \log {\pi}({\mathbf{z}}) ]. \label{eq:theorem1_14}\end{aligned}$$ Recall that the convergence of loss $L_{K, {\pi^*}}(\phi)$ cannot be sufficient for the convergence of the TV distance ${D_{\text{TV}}}(q, {\pi})=0$. Fortunately, under some reasonable condition, the loss $L_{K, {\pi^*}}(\phi) = 0$ implies the convergence in TV distance. Formally, this is given by the following theorem. (Ergodic Measure Convergence Theorem)\[theorem:ergodic\_fundamental\_theorem\] Let $K_{\pi}$ be an ergodic Markov kernel with invariant distribution ${\pi}$. Assume that the entropy of $Q$ is not less than the entropy of ${\pi}$, that is $\text{H}(Q) \ge \text{H}({\pi})$, the loss $L_{K, {\pi^*}}(\phi) = 0$ if and only if ${D_{\text{TV}}}(q, {\pi}) = 0$. By the definition of the KL divergence, we have $$\begin{aligned} {D_{\text{KL}}}(q \vert \vert {\pi}) = {\mathbf{E}}_q[\log q] - {\mathbf{E}}_q[\log {\pi}].\end{aligned}$$ By Theorem \[theorem:loss\_convergence\], we have $$\begin{aligned} {D_{\text{KL}}}(q \vert \vert {\pi}) = {\mathbf{E}}_q[\log q] - {\mathbf{E}}_{\pi}[\log {\pi}],\end{aligned}$$ which is equivalent to $${D_{\text{KL}}}(q({\mathbf{z}}) \vert \vert {\pi}) = H({\pi}) - H(Q).$$ Because the KL divergence is never less than 0, we have $$H({\pi}) \ge H(Q).$$ Finally, by the assumption $H({\pi}) \le H(Q)$, we know $H({\pi}) = H(Q)$, so we know $0\le {D_{\text{TV}}}(q, {\pi}) \le {D_{\text{KL}}}(q \vert \vert {\pi}) = 0,$ which implies ${D_{\text{TV}}}(q, {\pi}) = 0$. By the monotonic convergence in TV distance of ergodic Markov chain, it is straightforward to show that \[prop:loss\] Given a smooth ergodic transformations w.r.t. the probability measure ${\pi}({\mathbf{z}})$, if ${\mathbf{E}}_{q}[\log{\pi}] < {\mathbf{E}}_{{\pi}}[\log{\pi}]\,$, the loss $$\begin{aligned} {\mathbf{E}}_{q}[\log {\pi^*}({\mathbf{z}})] - {\mathbf{E}}_{q_1}[ \log {\pi^*}({\mathbf{z}}) ] > 0. \label{eq:final_objective2}\end{aligned}$$ Assume that ${\mathbf{E}}_{q}[\log{\pi}] < {\mathbf{E}}_{{\pi}}[\log{\pi}]\,$, we have $$\begin{aligned} L_{K, {\pi^*}}^*(\phi) = {\mathbf{E}}_{q}[\vert \log \pi^*({\mathbf{z}}) \vert ] - {\mathbf{E}}_{q_1}[\vert \log \pi^*({\mathbf{z}}) \vert ] \,, \label{eq:db_Loss_2_LL}\end{aligned}$$ Optimising ${\pi^*}$-Ergodic Loss {#sec:optimising_ergodic_loss} --------------------------------- Let $q_{01}({\mathbf{z}}, {\mathbf{z}}_1)$ be the joint distribution $q({\mathbf{z}})K({\mathbf{z}}, {\mathbf{z}}_1)$. Then, we can rewrite as $$\begin{aligned} L_{K, {\pi^*}}^*(\phi) = {\mathbf{E}}_{q_{01}}[\log \pi^*({\mathbf{z}}) - \log \pi^*({\mathbf{z}}_1) ] \,, \label{eq:db_Loss_3_LL}\end{aligned}$$ which can be estimated by samples of $({\mathbf{z}}, {\mathbf{z}}_1)$. To optimise the loss , we need to compute the gradient $\partial_{\phi}L_{K, {\pi^*}}^*(\phi)$. Notice that the ${\mathbf{z}}$ and ${\mathbf{z}}_1$ are coupled by the kernel $K$ and the density function of most MCMC kernels, which makes the computation of the gradient $\partial_{\phi}L_{K, {\pi^*}}^*(\phi)$ unstable. To avoid this, we reparameterize both $q(\cdot)$ and the ergodic Markov kernel $K({\mathbf{z}}, \cdot)$ by a transformation $T_{\phi}$ and a measure preserving transformation $T_{{\pi}}$ respectively. This allows us to transform some simple random variable ${\mathbf{r}}$ and ${\mathbf{r}}_1$, that is independent of $\phi$, into $({\mathbf{z}}, {\mathbf{z}}_1)$ as $$\begin{aligned} {\mathbf{z}}= T_{\phi}({\mathbf{r}}),\quad {\mathbf{z}}_1 = T_{{\pi}}({\mathbf{z}}, {\mathbf{r}}_1). \label{eq:db_Loss_3_LL_part_2}\end{aligned}$$ Therefore, we can compute the loss with following reformulation $$\begin{aligned} L_{K, {\pi^*}}^*(\phi) = {\mathbf{E}}_{\mu({\mathbf{r}})\mu_1({\mathbf{r}}_1)}[L_{{\pi^*}, T_{\phi}, T_{{\pi}}}({\mathbf{r}}, {\mathbf{r}}_1) ] \,, \label{eq:db_Loss_3_LL}\end{aligned}$$ where $L_{{\pi^*}, T_{\phi}, T_{{\pi}}} = \log {\pi^*}({\mathbf{z}}) - \log {\pi^*}({\mathbf{z}}_1)$ and $({\mathbf{z}}, {\mathbf{z}}_1) = T_{\phi, \pi}$ as . As discussed above, the only requirement of approximate family ${\mathcal{Q}}$ in ergodic inference is the transformation $T_{\phi}$ is known and it is a measurable function. It is an important advantage over VI, where the density function of ${\mathcal{Q}}$ must be in closed form. Deep Ergodic Inference Model {#sec:deins} ============================ Ergodic transformations are not only fundamentally important in the ergodic loss, they are also powerful tools for constructing flexible approximation family ${\mathcal{Q}}$. In this section, we will present how to construct and optimise the approximation family ${\mathcal{Q}}$ by stacking multiple layers of ergodic transformations. Definition ---------- Let $\{K_1, K_2, \dots, K_N\}$ be $N$ ergodic transition kernel with independent parameters $\{\phi_1, \phi_2, \dots, \phi_N\}$. Let $q_0$ be the distribution of initial state also has parameter $\phi_0$. By ergodic reparameterization, we reform each ergodic Markov kernel $K_n({\mathbf{z}}, {\mathbf{z}}')$ as a transformation ${\mathbf{z}}_{n} = T_{n}({\mathbf{z}}_{n-1}, {\mathbf{r}})$, where $T_n$ is a deterministic function depends on the kernel parameter $\phi_n$ and ${\mathbf{r}}$ is sampled from a standard distribution $\mu_n$. We also reparameterize the initial distribution $q$ from a simple distribution $\mu_0$ by a transformation $T_0$. Then, we can generate samples of ${\mathbf{z}}_n$ by transforming samples of $({\mathbf{r}}_0, {\mathbf{r}}_1,\dots,{\mathbf{r}}_{N-1})$ from $\mu(\cdot) = \prod_{i=0}^{N-1} \mu_i(\cdot)$ as $$\begin{aligned} {\mathbf{z}}_n = T_{{\mathbf{r}}_{N-1}} \circ \cdots \circ T_{{\mathbf{r}}_{1}} \circ T_0({\mathbf{r}}_0), \label{eq:dein_def}\end{aligned}$$ where $T_{{\mathbf{r}}_{n}}(\cdot)$ denotes $T_{n}(\cdot, {\mathbf{r}}_n)$. We call this multiple layer ergodic transformation $T_{{\mathbf{r}}_N-1} \circ T_{{\mathbf{r}}_1} \circ T_0(\cdot)$ deep ergodic inference network (DEIN). The expectation of $q_N$ can be reformed as $${\mathbf{E}}_{q_N}[f({\mathbf{z}}_N)] = {\mathbf{E}}_{\mu}[f\circ T_{{\mathbf{r}}_{N-1}} \circ \cdots \circ T_{{\mathbf{r}}_{1}} \circ T_0({\mathbf{r}}_0)],$$ which allows us to estimate the gradient of any function by Monte Carlo method $$\partial_{{\boldsymbol \phi}} {\mathbf{E}}_{q_N}[f({\mathbf{z}}_N)] \approx \frac{1}{M} \sum_{i=1}^M\partial_{{\boldsymbol \phi}} f\circ T_{{\mathbf{r}}_{N-1}^i} \circ \cdots \circ T_{{\mathbf{r}}_{1}^i} \circ T_0({\mathbf{r}}_0^{i})\,.$$ Optimisation and Convergence of DEINs ------------------------------------- This is a non-parametric model because the number of parameters of this model grows with the number of transformations. Different from deep neural networks, DEIN has strong stability by the natural of ergodicity. In particular, DEINs can be arbitrarily deep and the stability and simulation quality is guaranteed to improve with the depth. First, we define a loss for each transition $K_n$ as $$\begin{aligned} L^n(\phi_n) = {\mathbf{E}}_{q_n}[\log \pi^*({\mathbf{z}})] - {\mathbf{E}}_{q_{n-1}}[\log \pi^*({\mathbf{z}})] \,,\end{aligned}$$ where $q_N$ denotes the marginal of the last state $$\begin{aligned} q_n({\mathbf{z}}; {\boldsymbol \phi}_{0:n}) = \int K({\mathbf{z}}_{n-1}, {\mathbf{z}}_n) q_{n-1}({\mathbf{z}}_{n-1}; {\boldsymbol \phi}_{0:n-1}). \label{eq:Dein_density}\end{aligned}$$ \[prop:loss\_dein\] Assume that $ {\mathbf{E}}_{q_0}[\log \pi({\mathbf{z}})] < {\mathbf{E}}_{{\pi}}[\log \pi({\mathbf{z}})]$, minimizing the ergodic loss $L^*_{K, {\pi^*}}$ in with $q_N$ of deep ergodic Inference network is equivalent to maximizing the total ergodic loss $\sum_{n=1}^N L^n(\phi_n)$ $$\begin{aligned} L_N({\boldsymbol \phi}) = {\mathbf{E}}_{q_N}[\log \pi^*({\mathbf{z}})] - {\mathbf{E}}_{q_0}[\log \pi^*({\mathbf{z}})]\,. \label{eq:Loss_all}\end{aligned}$$ which is equivalent to $$\begin{aligned} L_N({\boldsymbol \phi}; \phi_0) = {\mathbf{E}}_{q_N}[\log \pi^*({\mathbf{z}})]. \label{eq:Loss_empf}\end{aligned}$$ when the parameter of $q_0$ is fixed. The total loss is consistent with the loss proposed by [@DBLP:journals/corr/abs-1805-10377] in ergodic measure preserving flows. By Proposition \[prop:loss\], it is straightforward to show that DEINs enjoy incremental improvement as the depth grows. (Incremental Convergence of DEIN) \[theorem:incremental\_convergence\] Given a $N$-layer DEIN defined as , the optimal total ergodic loss $L_N({\boldsymbol \phi}^*) = \max_{{\boldsymbol \phi}}L_N({\boldsymbol \phi})$ increases monotonically as $N$ increases. Similar to the convergence of ergodic Markov chains, we have the asymptotic unbiased convergence of DEINs as following. (Asymptotic Unbiased Convergence of DEINs) \[theorem:incremental\_convergence\] For arbitrarily small $\epsilon>0$, there always exists a DEIN with finite number of layer $N$, so that with the optimal distribution $q_N^*$ has the ergodic loss $L_{K, {\pi}}^* = {D_{\text{TV}}}\left(\int K({\mathbf{z}}, \cdot)dq_N^*, q_N^*(\cdot)\right) \le \epsilon$. Comparison with Auto-Tuning MCMC -------------------------------- From an algorithmic perspective, auto-tuning MCMC (AMCMC) and DEIN are very similar, because both methods simulate ergodic Markov chains and optimise the parameters of the kernel w.r.t. a loss. This may give a false impression of that AMCMC and DEIN share the same theoretical foundation. To clear this impression, we will discuss the fundamental difference between DEINs and AMCMC. First of all, AMCMC is essentially a class of MCMC methods with auto-tuning strategy of kernel parameters. In particular, the purpose of auto-tuning is to boost the statistical power of samples from MCMC by encouraging distant jump between states in Euclidean space, which is inspired by the work of [@pasarica2010adaptively] on reducing sample correlation of MCMC. In contrast, as a parametric family in ergodic inference methods. The parameters in DEINs is optimised w.r.t. the ergodic loss, which is based on the ergodic inference principle in Section \[sec:foundation\]. The fundamental difference have two important effects in practice. The first effect is on the sample correlation. By the nature of Markov property, optimising the auto-tuning loss can never eliminate the correlation of samples from MCMC. In contrast, the samples from DEINs are generated by deterministic transformation of i.i.d. samples from initial distribution, which is still i.i.d. samples. The second consequence is on the MH-correction. In particular, MH correction is optional for DEINs for three reasons. First, DEIN is a parametric approximate family ${\mathcal{Q}}$ rather than unbiased simulation procedure. Second, by optimising the ergodic loss, DEINs guarantee the convergence towards the target in TV distance. Finally, even with approximate ergodic transformations, the existence of a stationary distribution (not necessarily the target) is guaranteed by measure preserving property, in particularly with the depth of DEIN is always finite. In contrast, the convergence of AMCMC chains is only guaranteed with MH correction. In particular, without MH correction, the existence of a stationary distribution of MCMC chains becomes questionable. With unlimited number of recurrent Markov transitions, Markov chains are not guaranteed to converge to any distribution. The existence of stationary distribution is the necessary condition of ergodic theorem [@Robert:2005:MCS:1051451]. Therefore, without MH-correction (implicitly proved by detailed balance condition), the bias of samples from MCMC may not be bounded. This is particularly true when the Markov kernel parameter is tuned to maximize the jumping distance between states. Comparison with Normalising Flows --------------------------------- Normalizing Flow (NF), introduced by [@Rezende:2015:VIN:3045118.3045281], is a recent variational inference framework, where the variational parametric distribution is defined in an iterative procedure. The fundamental idea of NF is to define an expressive parametric family by a sequence of deterministic transformations with closed-form Jacobian. Let ${\mathbf{z}}_0$ be a random variable from a simple distribution $\mu$, like Gaussian, and $f_1\dots, f_M$ be $M$ deterministic functions from ${\mathbb{R}}^n$ to ${\mathbb{R}}^n$. We define a sequence of random variable ${{\mathbf{z}}_1\dots{\mathbf{z}}_M}$ as $${\mathbf{z}}_M = f_M \circ\dots \circ f_1({\mathbf{z}}_0).$$ By the rule of changing variables, the density function of ${\mathbf{z}}_M$ is given by $$\log p(d{\mathbf{z}}_M) = \log q(d{\mathbf{z}}_0) - \sum_{i=1} \log \left\vert \det \partial_{{\mathbf{z}}_i} f_i({\mathbf{z}}_i)\right \vert.$$ There are three important difference between DEINs and NFs. First, without manually engineering ergodic transformations, DEINs have theoretical guarantee of better performance with more transformations (Theorem \[theorem:incremental\_convergence\]). In contrast, the transformations $f_i$ in NFs is predefined based on heuristics and experimental evidence. Second, ergodic transformations $T_{{\pi}}$ has no closed form solutions, but the transformations $f_i$ in NFs is limited to simple functions with tractable Jacobian. Finally, the distribution of DEINs is very expressive, which may not even have a closed form as . More importantly, there is no need to compute the density for optimising the parameters. It is the opposite for NFs. In particular, the transformations in NFs are often restricted to simple functions to have closed-form Jacobian. The computation of the Jacobian is also one of computational bottlenecks in optimisation. Comparison Overview ------------------- The key difference between ergodic inference, AMCMC and VI is highlighted in the following table. - TV-Loss: Optimising the loss function leads to the convergence in TV distance. - Independent samples: computationally and statistically independent sample simulation. - Implicit Simulation Density: no closed-form density function of simulation distribution is required in training. Related Works ============= Hamiltonian variational inference (HVI), introduced by [@salimans2015markov], is an interesting variational framework using MCMC kernel as variational parametric distribution. The motivation of HVI is that the joint density function of all the states of HMC chains is tractable to compute. Unfortunately, the variational lower bound is still intractable to compute, because the reverse probability of HMC chain given the last state is intractable. To overcome this problem, they propose to approximate the reverse density function using neural network. Although HVI shows improvement in performance over VAEs, the additional approximation limits the potential of this method. However, optimising the HMC kernel parameters w.r.t. ELBO is still an attractive feature of HVI. [@pmlr-v70-hoffman17a [@pmlr-v70-hoffman17a]]{} proposed another hybrid method based on VI and HMC without auxiliary approximation. The idea is to use a Monte Carlo estimation of the marginal likelihood by averaging over samples from HMC chains, that are initialized by variational distribution. In [@han2017alternating] a very similar framework is proposed using Metropolis-adjusted Langevin dynamics. This idea is very similar to contrastive divergence in [@Hinton02]. The main disadvantage of this methods is that the HMC parameters are manually pretuned. Especially, As mentioned by [@pmlr-v70-hoffman17a], No-U-turn Sampler (NUTS), an adaptive HMC, is not appliable due to engineering difficulties. [@radford2010] pointed out that HMC is very sensitive to the choice of Leapfrog step size and number of leaps. Stein Variational Gradient Descent (SVGD) is a recent particle based dynamical inference method proposed by [@NIPS2017_6904]. In SVGD, the approximation distribution is a set point mass $q$ generated by transforming a set of points sampled from a distribution $\mu$ using a perturbation function $T(x) = x + \phi(x)$, where $\phi$ is in a function space with boundary norm. With this setup, the optimisation of $T$ w.r.t. the KL divergence between $q$ and the target ${\pi}$ is transformed into a stochastic optimisation in the kernel space of $\phi$. The theoretical foundation of convergence of SVDG is sound and appealing. However, this method faces two practical challenges. First, the optimisation complexity grows quadratically with the number of particles. Second, it is very difficult to approximate high dimensional distribution well with a limited number of point mass approximation. Summary {#sec:summary} ======= I proposed a new generic inference method based on optimization and ergodic deterministic transformations. This work provides us the very foundation of ergodic inference including: the fundamental ergodic inference principle; tractable estimation of ergodic loss and the its gradient; a generic construction of approximation family.

--- abstract: 'We describe <span style="font-variant:small-caps;">Artemis</span> (Annotation methodology for Rich, Tractable, Extractive, Multi-domain, Indicative Summarization), a novel hierarchical annotation process that produces indicative summaries for documents from multiple domains. Current summarization evaluation datasets are single-domain and focused on a few domains for which naturally occurring summaries can be easily found, such as news and scientific articles. These are not sufficient for training and evaluation of summarization models for use in document management and information retrieval systems, which need to deal with documents from multiple domains. Compared to other annotation methods such as Relative Utility and Pyramid, <span style="font-variant:small-caps;">Artemis</span> is more tractable because judges don’t need to look at all the sentences in a document when making an importance judgment for one of the sentences, while providing similarly rich sentence importance annotations. We describe the annotation process in detail and compare it with other similar evaluation systems. We also present analysis and experimental results over a sample set of 532 annotated documents.' author: - | \ [**Rahul Jha**]{}$^\star$, [**Keping Bi**]{}$^\dag$, [**Yang Li**]{}$^{\star}$, [**Mahdi Pakdaman**]{}$^{\star}$\ [**Asli Celikyilmaz**]{}$^\star$, [**Ivan Zhiboedov**]{}$^\ddag$, [**Kieran McDonald**]{}$^{\star}$\ $^\star$ Microsoft Corporation\ $^\dag$ Umass Amherst\ $^\ddag$ Facebook Inc bibliography: - 'refs.bib' title: '<span style="font-variant:small-caps;">Artemis</span>: A Novel Annotation Methodology for Indicative Single Document Summarization' --- Introduction {#sec:intro} ============ Annotation Methodology {#sec:method} ====================== Related Work {#sec:rel} ============ Annotated Data Analysis {#sec:analysis} ======================= Experiments {#sec:experiments} =========== Concluding Remarks {#sec:conclusion} ==================

--- abstract: 'Out-of-plane screening (OPS) is expected to occur generally in metal-semiconductor interfaces but this aspect has been overlooked in previous studies. In this paper we study the effect of OPS in electron-hole bilayer (EHBL) systems. The validity of the dipolar interaction induced by OPS is justified with a RPA calculation. Effect of OPS in electron-hole liquid with close-by screening layers is studied. We find that OPS affects the electronic properties in low density and long wavelength regime. The corresponding zero-temperature phase diagram is obtained within a mean field treatment. We argue that our result is in general relevant to other heterostrucutures. The case of strongly correlated EHBL is also discussed.' author: - Cheung Chan - 'T. K. Ng' title: Out of plane screening and dipolar interactions in heterostructures --- introduction ============ Modern micro-electronics relies to a large degree on surface science, which concerns the material properties near a surface or interface. To enhance the performance of such devices, knowledge of the electronic states near the interfaces is required. Near a surface or interface, electronic reconstruction may alter three key factors - interaction strengths, bandwidths and electron densities [@AMillis] which determine electronic states and their properties. In this paper, we consider another factor - the modification in form of interaction between electrons. For instance, in an insulator-semiconductor-insulator superstructure, if the dielectric constant of the semiconductor is sizably larger than that of insulator (barrier layer), the image charges induced at semiconductor-insulator interface can substantially enhance the binding energy of the excitons confined in the semiconductor layer [@Xconfine95; @Xconfine92]. In this case, the electrons and holes do not interact via usual Coulomb potential after the effect of the image charges at the semiconductor-insulator interface is taken into account. Recently, Huang *et al.* observed non-activated electronic conductivity of a two-dimensional (2D) low density hole system in a heterojunction insulated-gate field-effect transistor [@Nonact-transport]. Such non-activated conductivity is unexpected as at low charge density strong Coulomb interaction is expected to crystallize the system (Wigner crystal), which is then pinned by disorder resulting in insulating behavior and activated conductivity. Huang *et al.* attribute the behavior to the screening of Coulomb interactions by the metallic gate, which leads to destruction of the Wigner crystal phase. Physically, the metallic gate which is located at a distance away from the 2D hole gas, provides an out of plane screening (OPS) to the hole-hole interaction, resulting in effective dipolar interaction between holes. Microscopically, when a charge is placed near a metal surface, an image charge of opposite sign will be induced at the surface to screen out the (static) electric field from the charge. From elementary electrostatics, the system can be described equivalently as a dipole formed by the charge and its image charge and the interaction between two charges located near the interface changes from a Coulomb potential $\sim1/r$ to a dipolar potential $\sim1/r^{3}$. This modified interaction, which is generally expected to exist in metal-semiconductor heterostructures, can change the electronic properties near the interface. Unexpectedly, there has been no detailed theoretical study of this effect on electronic properties until recently [@HoLH]. The neglect of OPS might be due to dynamical screening of in-plane charges [@HoLH]. For high charge density, the screening can effectively reduce both Coulomb and dipolar interactions to short range interactions. However for low charge density electronic liquids in-plane screening is less effective and OPS can lead to a difference, as is observed by Huang *et al.* [@Nonact-transport]. In this paper, we study how OPS affects the electronic properties in systems with two-layer of charges of opposite sign, i.e. the 2D electron-hole bilayer (EHBL) system. We shall study how OPS affects Wigner crystalization and exciton condensate in the system [@Nonact-transport; @exciton-cond] and will also comment on the effect of OPS in interfaces between metals and strongly correlated electron systems [@thin_film_on_metal; @YBCO_metal_interface1; @YBCO_metal_interface2]. OPS and effective interaction between charges ============================================= ![\[fig:EHBL-OPS\] (a) EHBL system separated by distance $b$. (b) EHBL with OPS by metallic plates in both layers. Dotted line represents metallic interface, separated from the main layer by a distance of $a/2$. (c) Similar to (b) but with only one OPS layer. (d) Charges (black dots) and screening charge response at the metal interface (grey patches). (e) Effective image charge (grey dots) and effective interactions $V^{\text{intra}}$ and $V^{x}$. (f) Charges attract when they are aligned while repel when they are not. This behavior is different from the Coulomb potential which is always attractive for a pair of electron and hole. The repulsive behavior inhibits exciton pairing.](EHBL-OPS){width="0.9\columnwidth"} In this section we provide the details for the EHBL systems we study and the corresponding OPS effective interaction. We shall assume that the only effect of the metallic screening layers is to provide an image charge for point charges sitting close to it and the effective interaction between charges will be derived from the image charge picture. The validity of this approximation is bounded by the plasma frequency $\omega_{p}^{(s)}$ of the screening layer, above which the screening layer cannot respond rapidly to the charge fluctuations. Thus our approximation is valid when the plasma frequency of the EHBL layer $\omega_{p}$ is much less than $\omega_{p}^{(s)}$, or that the screening layer has density of electric charge much larger than the charge density of the EHBL layers we consider. The image charge picture can be justified by a Random Phase Approximation (RPA) calculation which is shown in the Appendix. Starting with a EHBL system (Fig.\[fig:EHBL-OPS\](a)), two metallic screening layers can be added as shown in Fig.\[fig:EHBL-OPS\](b), or a single metallic screening layer can be added as shown in Fig.\[fig:EHBL-OPS\](c). We first consider the two-layer case (b). Fig.\[fig:EHBL-OPS\](d) depicts the charge response in the metallic layer to a nearby charge. The charge response is assumed to be an image charge, which carries opposite charge of the same magnitude and is centered at distance $a$ from the point charge. Thus the point charge and the screening charge together form a dipole. We have assumed that the distance between the two layers of charges $b$ is sufficiently larger than $a$ ($b\gg a$) such that the presence of the other screening layer does not affect the simple dipole picture. In this case, the intralayer interaction between two charges located in an OPS layer (Fig.\[fig:EHBL-OPS\](e)) is in real space $$V^{\mathrm{intra}}(\vec{r})=\frac{e^{2}}{\epsilon_{e,h}}\left(\frac{1}{r}-\frac{1}{\sqrt{r^{2}+a^{2}}}\right)\;,$$ where $r$ is the charge-charge distance within the charge plane. It is easy to see that for $r\gg a$, $V^{\mathrm{intra}}$ scales as $1/r^{3}$ while for $r\ll a$ it follows the usual Coulomb scaling $1/r$. By using 2D Fourier transform $\frac{1}{\sqrt{r^{2}+a^{2}}}\overset{\mathrm{2D}\mathcal{F}}{\longrightarrow}\frac{2\pi}{k}e^{-ka}$, the Fourier transformed interaction is $$V^{\mathrm{intra}}(\vec{k})=\frac{2\pi e^{2}}{\epsilon_{e,h}k}\left(1-e^{-ka}\right)\;.\label{v2}$$ For an electron and a hole sitting in different layers, the interlayer interaction is $$\begin{aligned} V^{x,2}(\vec{r}) & = & -\frac{e^{2}}{\epsilon_{x}}\left(\frac{1}{\sqrt{r^{2}+b^{2}}}\right.\\ & & \left.-\frac{2}{\sqrt{r^{2}+(a+b)^{2}}}+\frac{1}{\sqrt{r^{2}+(2a+b)^{2}}}\right)\end{aligned}$$ and its Fourier counterpart is $$V^{x,2}(\vec{k})=-\frac{2\pi e^{2}}{\epsilon_{x}k}e^{-kb}(1-e^{-ka})^{2}\;.\label{v3}$$ $\epsilon_{e,h}$ and $\epsilon_{x}$ are the intra-layer and inter-layer dielectric constants, respectively. Next we consider EHBL with only one metallic screening layer (see Fig.\[fig:EHBL-OPS\](c)). In this case the two layers of charges have distance $a/2+b$ (layer $1$) and $a/2$ (layer $2$) from the screening layer, respectively. The intralayer interactions are thus $$\begin{aligned} V_{1}^{\mathrm{intra}}(\vec{r}) & = & \frac{e^{2}}{\epsilon_{1}}\left(\frac{1}{r}-\frac{1}{\sqrt{r^{2}+(a+2b)^{2}}}\right)\;,\\ V_{2}^{\mathrm{intra}}(\vec{r}) & = & \frac{e^{2}}{\epsilon_{2}}\left(\frac{1}{r}-\frac{1}{\sqrt{r^{2}+a^{2}}}\right)\;.\end{aligned}$$ with corresponding Fourier transforms $$\begin{aligned} V_{1}^{\mathrm{intra}}(\vec{k}) & = & \frac{2\pi e^{2}}{\epsilon_{1}k}\left(1-e^{-k(2b+a)}\right)\;,\label{v4}\\ V_{2}^{\mathrm{intra}}(\vec{k}) & = & \frac{2\pi e^{2}}{\epsilon_{2}k}\left(1-e^{-ka}\right)\;.\nonumber \end{aligned}$$ The corresponding intralayer interaction is given by $$V^{x,1}(\vec{r})=-\frac{e^{2}}{\epsilon_{x}}\left(\frac{1}{\sqrt{r^{2}+b^{2}}}-\frac{1}{\sqrt{r^{2}+\left(a+b\right)^{2}}}\right)$$ and $$V^{x,1}(\vec{k})=-\frac{2\pi e^{2}}{\epsilon_{x}k}e^{-kb}\left(1-e^{-ka}\right)\;.\label{v5}$$ Collective density responses ============================ In this section we study the collective density responses of the EHBL systems we considered. For a two component electronic system, the density-density response of the system is described by a $2\times2$ matrix $\chi_{ij}(q,\omega)$ with $i,j=1,2$. The density-density response matrix is given in RPA by $$\left(\begin{array}{cc} \chi_{11} & \chi_{12}\\ \chi_{21} & \chi_{22}\end{array}\right)=\frac{1}{\kappa}\left(\begin{array}{cc} (1-\chi_{02}V_{22})\chi_{01} & \chi_{01}V_{12}\chi_{02}\\ \chi_{02}V_{21}\chi_{01} & (1-\chi_{01}V_{11})\chi_{02}\end{array}\right)\label{dmatrix}$$ where $$\begin{aligned} \kappa(q,\omega) & = & (1-\chi_{01}(q,\omega)V_{11}(q))(1-\chi_{02}(q,\omega)V_{22}(q))\nonumber \\ & & -\chi_{01}(q,\omega)V_{12}(q)\chi_{02}(q,\omega)V_{21}(q)\;,\label{pole}\end{aligned}$$ $V_{ij}(q)$ is the “bare” interaction between $i^{th}$ and $j^{th}$ components of the electronic liquid and $$\chi_{0i}(q,\omega)=g_{s}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{n_{F}\left(\varepsilon_{k}^{(i)}\right)-n_{F}\left(\varepsilon_{k+q}^{(i)}\right)}{\hbar\omega+\varepsilon_{k}^{(i)}-\varepsilon_{k+q}^{(i)}}\;,\label{eq:responsefunc}$$ where $\varepsilon_{k}^{(i)}\sim k^{2}/2m^{(i)}$ is kinetic energy of species $i$ particles (of mass $m^{(i)}$), $n_{F}$ is the Fermi-Dirac distribution function and $g_{s}=2$ is spin degeneracy. In the case of two screening layers the interactions $V_{11(22)}$ and $V_{12}=V_{21}$ are given by $V^{\text{intra}}(q)$ (eq.) and $V^{x,2}(q)$ (eq.), respectively whereas they are given by $V_{1(2)}^{\mathrm{intra}}(q)$ (eq.) and $V^{x,1}(q)$ (eq.), respectively if there is only one screening layer. Next we study the collective excitations (i.e. plasmons) in the system. The dispersion of the collective excitations are given by the equation $$\kappa(q,\omega(q))=0\;.\label{eq:plasmoneq}$$ We shall first consider the long wavelength limit ($q\rightarrow0$) where the equation can be studied analytically. In this limit it is easy to show that $$\chi_{0}\left(q,\omega\right)=\frac{n}{m}\left(\frac{q}{\omega}\right)^{2}+\mathcal{O}\left(\left(\frac{q}{\omega}\right)^{4}\right)\;,$$ where $n=\frac{g_{s}(\pi k_{F}^{2})}{(2\pi)^{2}}$ is carrier density. We have neglected the component index $i$ for brevity. We begin with the Coulomb case (no screening layer). The interactions are respectively $V_{1,2}(q)=\frac{2\pi e^{2}}{\epsilon_{1,2}q}$ and $V_{x}(q)=-\frac{2\pi e^{2}}{\epsilon_{x}q}e^{-qb}\sim-\frac{2\pi e^{2}}{\epsilon_{x}q}$ for $q\ll b^{-1}$. The plasmon equation in $q\rightarrow0$ limit reads $$\begin{aligned} 1-2\pi e^{2}\left(\frac{n_{1}}{m_{1}\epsilon_{1}}+\frac{n_{2}}{m_{2}\epsilon_{2}}\right)\frac{q}{\omega^{2}}+\left(2\pi e^{2}\right)^{2}\nonumber \\ \times\frac{n_{1}n_{2}}{m_{1}m_{2}}\left[\frac{1}{\epsilon_{1}\epsilon_{2}}-\frac{1}{\epsilon_{x}^{2}}\right]\left(\frac{q}{\omega^{2}}\right)^{2} & = & 0\;.\label{eq:Coul plasmon}\end{aligned}$$ We first consider the case $\epsilon_{x}^{2}=\epsilon_{1}\epsilon_{2}$ such that the term in the square bracket is zero. In this case we need to expand the interlayer interaction to one order higher in $q$. As a result the last term in eq. is replaced by a term of order $\frac{q^{3}}{\omega^{4}}$ and the plasmon equation at long wavelength limit yields two solutions, which are the out-of-phase mode ($\omega\sim q$) and in-phase mode ($\omega\sim\sqrt{q}$). Indeed this occurs usually in a 2D electronic systems with both conduction and valence bands where the same dielectric constant $\epsilon_{x}^{2}=\epsilon_{1}\epsilon_{2}$ is found for all interactions. In the more general case $\epsilon_{x}^{2}\neq\epsilon_{1}\epsilon_{2}$, which arises quite naturally in the complex environment of EHBL heterostructures, we can easily see from eq. that the plasmon frequency scales as $\omega\sim\sqrt{q}$. There are two modes of plasmons. For the OPS case with two screening layers, the interactions are respectively $V_{1,2}=\frac{2\pi e^{2}}{\epsilon_{1,2}q}(1-e^{-qa})\sim2\pi e^{2}/\epsilon_{1,2}\left(a-a^{2}q/2\right)$ and $V_{x}=-\frac{2\pi e^{2}}{\epsilon_{x}q}e^{-qb}(1-e^{-qa})^{2}\sim-2\pi a^{2}e^{2}q/\epsilon_{x}$ for $q\ll a^{-1}$. Notice the removal of the $1/q$ singularity in the interactions by OPS. We then obtain after solving the equation the collective modes (up to order $q^{2}$) $$\omega_{1,2}=\sqrt{\frac{2\pi ae^{2}n_{1,2}}{m_{1,2}\epsilon_{1,2}}}\left(q-\frac{aq^{2}}{4}\right)\;.$$ Notice that OPS effectively reduced the long-ranged Coulomb interaction into short-ranged interactions resulting in two collective modes scaling linearly with $q$. The collective modes represent separate collective motion of the two layers because $\left(V^{x}\right)^{2}$ is of higher order in $q$ than $V_{1}V_{2}$, and the inter-layer interaction appears only to order $q^{3}$. For completeness, we have computed numerically the collective modes spectrums at finite $q$ as shown in Fig.\[fig:Plasmon-d-d\]. ![\[fig:Plasmon-d-d\]Plasmon excitations in EHBL with two screening layers computed numerically. The spectrum is computed by solving eq. numerically with the full expression of $\chi_{0}(q,\omega)$ . The solid lines are the plasmons excitations and the dash line represents the boundary of the particle-hole continuum. The wave-number $q$ and frequency $\omega$ are normalized with respect to Fermi momentum $k_{F}$ and Fermi energy $\epsilon_{F}$, respectively. We set $a=1$, $b=15$, $m_{1}=1$, $m_{2}=1.25$, $k_{F}=10$ and all $\epsilon=1$ in the calculation.](dipole){width="1\columnwidth"} ![\[fig:singleplasmon\]Plasmon excitations in EHBL with only one screening layer. At small $q$, both plasmon modes scale linearly with $q$ with a larger slope $\propto\sqrt{2b+a}$ for $\omega_{1}$. We set $a=1$, $b=15$, $m_{1}=1$, $m_{2}=1.25$, $k_{F}=10$ and all $\epsilon=1$ in the calculation.](D+C){width="1\columnwidth"} With only one screening layer, the interactions are $V_{1}(q)\sim\frac{2\pi e^{2}}{\epsilon_{1}}\left((a+2b)-(a+2b)^{2}q/2\right)$, $V_{2}(q)\sim\frac{2\pi e^{2}}{\epsilon_{2}}\left(a-a^{2}q/2\right)$ and $V_{x}(q)=-\frac{2\pi e^{2}}{\epsilon_{x}q}e^{-qb}(1-e^{-qa})\sim-2\pi ae^{2}/\epsilon_{x}$, respectively at small $q$. The collective modes are given by (up to order $q^{2}$) $$\begin{aligned} \omega_{1} & = & \sqrt{\frac{2\pi(a+2b)e^{2}n_{1}}{m_{1}\epsilon_{1}}}\left(q-\frac{(a+2b)q^{2}}{4}\right)\nonumber \\ \omega_{2} & = & \sqrt{\frac{2\pi ae^{2}n_{2}}{m_{2}\epsilon_{2}}}\left(q-\frac{aq^{2}}{4}\right)\;.\end{aligned}$$ Again there are two linear plasmon modes and effect of $V^{x}$ does not enter until $q^{3}$. The main difference is that the electron-hole layer separation $b$ enters the slope of $\omega_{1}$ mode ($\propto\sqrt{2b+a}$). The numerically calculated plasmon spectrums are depicted in Fig.\[fig:singleplasmon\]. Exciton Condensation and Wigner Crystalization ============================================== In this section we study exciton condensation and Wigner crystalization in an electron-hole liquid with OPS. The system without OPS has been extensively studied for the search of exciton condensation. We shall consider exciton condensation in a BCS type mean-field theory where the exciton condensation is described by the order parameter $\left\langle c_{1k\uparrow}c_{2\bar{k}\downarrow}\right\rangle $ (1,2 are layer indices). For simplicity we assume the layers are doped with equal amount of charges (with opposite signs) and the electrons and holes are spin-polarized. Singlet pairing of excitons is implicitly assumed. The EHBL Hamiltonian in momentum representation is $$\begin{aligned} H & = & \sum_{\alpha k}\xi_{k}^{\alpha}c_{\alpha k}^{\dagger}c_{\alpha k}+\sum_{pqk}V^{x}(k)c_{1p+k}^{\dagger}c_{2q-k}^{\dagger}c_{2q}c_{1p}\nonumber \\ & & +\frac{1}{2}\sum_{\alpha pqk}V^{\alpha}(k)c_{\alpha p+k}^{\dagger}c_{\alpha q-k}^{\dagger}c_{\alpha q}c_{\alpha p}\;,\end{aligned}$$ where $\alpha=1,\!2$ is the layer index; $c_{k}$ ($c_{k}^{\dagger}$) is the momentum $k$ fermion annihilation (creation) operator, $\xi_{k}^{\alpha}=\frac{k^{2}}{2m_{\alpha}}-\mu_{\alpha}$ is the electron or hole dispersion and $V^{\alpha}(k)$ ($V^{x}(k)$) is the intralayer (interlayer) OPS effective interaction. Next we employ the standard Hartree-Fock-Bogoliubov method [@HartreeFock] to derive the mean field equations for exciton condensate. The Hartree-Fock terms $\Sigma_{k}^{\alpha}=\sum_{q}V^{\alpha}(p-q)\left\langle c_{\alpha k}^{\dagger}c_{\alpha k}\right\rangle $ modify the particle dispersions $\xi_{k}^{\alpha}\rightarrow\xi_{k}^{\alpha}-\Sigma_{k}^{\alpha}$ and need to be solved self-consistently. Here we concentrate on the effect of exciton binding on the Fermi surface and shall assume that the self-energy can be captured by introducing effective masses $m_{\alpha}^{*}\left(\epsilon_{\alpha}\right)$ and renormalized chemical potentials $\mu_{\alpha}^{*}\left(\epsilon_{\alpha}\right)$, i.e. $\xi_{k}^{\alpha}-\Sigma_{k}^{\alpha}\sim\frac{k^{2}}{2m_{\alpha}^{*}}-\mu_{\alpha}^{*}$. With this approximation, we obtain the mean field Bogoliubov Hamiltonian $$H_{\text{MF}}=\sum_{k\sigma}\left(\begin{array}{cc} c_{1k}^{\dagger} & c_{2\bar{k}}\end{array}\right)\left(\begin{array}{cc} \xi_{k}^{1} & -\Delta_{k}\\ -\Delta_{k} & -\xi_{k}^{2}\end{array}\right)\left(\begin{array}{c} c_{1k}\\ c_{2\bar{k}}^{\dagger}\end{array}\right)\;,$$ where $$\Delta_{k}=-\sum_{q}V^{x}(k-q)\left\langle c_{1k}c_{2\bar{k}}\right\rangle \label{eq:gapeq}$$ is the exciton order parameter. $H_{\text{MF}}$ can be diagonalized easily by the Bogoliubov transformation $$\left(\begin{array}{c} c_{1k}\\ c_{2\bar{k}}^{\dagger}\end{array}\right)=\left(\begin{array}{cc} u_{k} & \upsilon_{k}\\ -\upsilon_{k} & u_{k}\end{array}\right)\left(\begin{array}{c} \gamma_{1k}\\ \gamma_{2\bar{k}}^{\dagger}\end{array}\right)\;,$$ $$\left\{ \begin{array}{rcl} u_{k}^{2} & = & \frac{1}{2}\left(1+\frac{\bar{\xi}_{k}}{E_{k}}\right)\;,\\ \upsilon_{k}^{2} & = & \frac{1}{2}\left(1-\frac{\bar{\xi}_{k}}{E_{k}}\right)\;,\end{array}\right.$$ where $E_{k}=\sqrt{\left(\bar{\xi}_{k}\right)^{2}+\Delta_{k}^{2}}$, $\bar{\xi}_{k}=\frac{1}{2}\left(\xi_{k}^{1}+\xi_{k}^{2}\right)\equiv\frac{k^{2}}{2m_{\text{eff}}}-\mu$, where $m_{\text{eff}}^{-1}=(m_{1}^{*-1}+m_{2}^{*-1})/2$ and $\mu=(\mu_{1}^{*}+\mu_{2}^{*})/2$. The ground state wavefunction is $$\left|\psi_{G}\right\rangle =\prod_{k}\left(u_{k}+\upsilon_{k}c_{1k}^{\dagger}c_{2\bar{k}}^{\dagger}\right)\left|0\right\rangle \;.$$ where $\Delta_{k}$ is determined by the self-consistent equation $$\Delta_{k}=-\frac{1}{2}\sum_{q}V^{x}(k-q)\frac{\Delta_{q}}{E_{q}}\;.\label{eq:self1}$$ The equation is to be solved with the particle number constraint $$n=\sum_{k}\upsilon_{k}^{2}\;,\label{eq:self2}$$ where $\upsilon_{k}^{2}$ is the probability of finding an electron-hole pair in state $k$ at the ground state. A zero-temperature phase diagram can be determined by numerically solving eqs. and . To simplify calculation we assume further that exciton gap is momentum independent $\Delta_{k}=\Delta$ and $\Delta$ is determined by minimizing the ground state energy. We note that we are considering a band structure with isotropic dispersion and the electron and hole Fermi surfaces are perfectly nested. In this case, the exciton pairing gap $\Delta$ is always non-zero in the mean-field theory, although its value can be very small. In reality the mean-field gap will be destroyed by quantum fluctuations when it’s magnitude is small [@exction-cond2], but this is not reflected in a mean field theory. To capture this physics qualitatively, we assume that the transition from the exciton condensed state to the normal state occurs at $\Delta=10^{-5}\mu$. Although quantitatively unreliable, this procedure allows us to examine the effect of screening on the phase diagram semi-quantitatively as we shall see below. With the above criteria, the phase diagram for different average particle-particle separation $r_{s}=\frac{1}{a_{B}}\sqrt{\frac{1}{\pi n}}$ ($n$ is particle/hole density; $a_{B}=\epsilon_{x}\hbar^{2}/m_{\text{eff}}e^{2}$ is the effective Bohr radius of electron-hole pair) and transition layer separation $b_{c}(r_{s})$ can be determined by solving the self-consistent equations and . We first consider EHBL with two screening layers. The result of calculation is depicted in Fig.\[fig:d-vs-rs\] for different separation between the electron/hole and its image charge $a$ (filled symbols). ![\[fig:d-vs-rs\] The phase diagram for EHBL with two- and one- screening layers for varied image charge separation $a$. $b$ is the bilayer separation and $r_{s}\sim\frac{1}{k_{F}}$ is the dimensionless average particle-particle separation in unit of the effective Bohr radius $a_{B}=\epsilon_{x}\hbar^{2}/m_{\text{eff}}e^{2}$. Below the transition lines $b_{c}(r_{s})$ an exciton gap $\Delta$ of magnitude larger than $10^{-5}\mu$ is formed.](d-vs-rs){width="1\columnwidth"} In the small $r_{s}$ (high density) regime, kinetic energy dominates over potential energy and the exciton pairing gap goes to zero as $r_{s}\rightarrow0$. In large $r_{s}$ or low density limit, the exciton pairing is diminished due to the repulsive nature of the interlayer OPS potential at short distance (see Fig.\[fig:EHBL-OPS\](f)). This leads to a linear dependence of $V^{x}(k)$ versus $k$ at small $k$ (see eq.). In this case, the gap equation eq. is of the form $\int_{0}^{k_{F}}\frac{k}{\sqrt{\xi_{k}^{2}+\Delta^{2}}}d^{2}k=\text{constant}$ for small gap $\Delta$, where $k_{F}\sim1/r_{s}$ and larger $r_{s}$ (smaller $k_{F}$) implies a smaller $\Delta$ to satisfy the equation. The electrons and holes need to be placed closer to each other to produce a large enough $\Delta$ and leads to the drop of $b_{c}(r_{s})$ at large $r_{s}$. As the screening separation $a$ increases, the transition line shifts upward as the interlayer OPS potential is strengthened which enhances pairing. For $a=25$ (comparable with $b$), the image charge effect becomes negligible and potential becomes essentially Coulomb-like which permits exciton formation for all $r_{s}$ we considered (cf. Fig.1 in Ref.[@exction-cond2]). The main effect of OPS potential is to suppress exciton pairing at low density. Previous numerical study of the same EHBL with no screening layer [@exction-cond2] reveals also an excitonic Wigner crystal phase at large $r_{s}$. Wigner crystal is commonly formed in low density (i.e. large $r_{s}$) electron liquid because of domination of Coulomb repulsive potential energy ($\sim1/r_{s}$) over kinetic energy ($\sim1/r_{s}^{2}$). To minimize the potential energy the electron wavefunction “crystallizes” to ensure maximum separation between electrons which yields the Wigner crystal phase. Here we argue that OPS suppresses the Wigner crystal phase in two ways. Firstly, as shown above, exciton formation is suppressed at large $r_{s}$ and thus the *excitonic* Wigner crystal is unlikely to form. On the other hand, electronic Wigner crystals in separated layers are also prohibited since introduction of OPS reduces the (intralayer) potential energy and changes its scaling form to $\sim1/r_{s}^{3}$ (dipolar interaction, see eq.) at large particle separation $r\gg a$. In this case kinetic energy again dominates at large $r_{s}$ and an usual electron/hole liquid phase should occur. The situation is similar to the case as found in Ref.[@Nonact-transport] where the electronic Wigner crystal phase is destroyed by screening. We note, however that our simple study cannot rule out the possibility of having a Wigner crystal phase at some intermediate values of $r_{s}$ where the kinetic and potential energies are of comparable magnitudes. We now consider the situation of EHBL with only one screening layer which may be easier to realize experimentally (Fig.\[fig:EHBL-OPS\](c)). In this case we adopt eq. for interlayer interaction, where $V^{x}(k)$ scales as constant at small $k$. We can again consider the gap equation and argue similarly that the exciton phase boundary would also drop at large $r_{s}$, as in the two-layer screening case. Indeed we have solved the gap equations and find that the phase diagram is qualitatively the same as the two OPS layer case except that the area under the phase boundary $b_{c}(r_{s})$ is larger (see Fig.\[fig:d-vs-rs\] (open symbols)). For the Wigner crystal phase, the “asymmetric” OPS introduces some complications. First we note that an excitonic Wigner crystal phase is also unlikely to occur at large $r_s$. However the system may form a hybrid phase where a Wigner crystal is formed at layer $1$ and electron/hole liquid phase remains for layer $2$ because screening mainly affects layer $2$. To examine this possibility we check the effective intralayer interaction after taking into account the screening effect of the other charged layer (see eq. in Appendix and discussions thereafter). We see that the effective intralayer interaction is mainly dominated by $V_{1,2}^{\text{intra}}(q)$, and screening from the other layer is not important. Therefore, we expect that at large $r_s$ kinetic energy again dominates and the both layers are in the electron/hole liquid phase. Notice, however that $V_{1}^{\text{intra}}(q)$ has a dipolar form only when $r_s\sim r/a_B \gg b/a_B$ for layer 1. Thus for some large enough $b/a_B$, a hybrid phase (Wigner crystal at layer $1$, electron/hole liquid at layer $2$) may still occur at some intermediate densities $b/a_B\gg r_s\gg 1$. We see that OPS becomes important for low density electronic systems due to change in scaling of the potential energy. Generally speaking, for heterostructures, insulating behavior resulting from low carrier density can be avoided by addition of metallic screening layers [@Nonact-transport]. This method may be preferred over other methods like increasing carrier density by dopants since dopants act like impurities and introduce unnecessary scattering at low temperature. Strongly Correlated EHBL ======================== In strongly correlated materials, the basic electronic properties are determined by the bandwidth, the on-site Coulomb interactions $U$ and the charge transfer energy $E_{c}$. If such a ultra-thin film, originally a Mott insulator, is placed close to a metal surface, $U$ and $E_{c}$ can be strongly reduced by OPS [@thin_film_on_metal]. When the bandwidth exceeds the suppressed $U$ and $E_{c}$, the insulating film can undergo an insulator-metal phase transition. Furthermore, if a heterostructure is formed, structural relaxation and local electronic states may exist at the interfaces. For instance, in an interface formed by $\mathrm{YBa_{2}Cu_{3}O_{7}}$ (YBCO) cuprate and metal [@YBCO_metal_interface1; @YBCO_metal_interface2], the $\mathrm{CuO_{2}}$ plane near the interface (depletion layer) is intrinsically doped by electronic reconstruction resulting in a strongly correlated electron system with OPS interaction induced by the metal. We shall consider here how OPS would affect the properties of this system. The mean field analysis on effect of OPS can also be performed for strongly correlated EHBL systems [@SLI-1; @SLI-2] with a two-layer *t-J* type model. We assume here that the suppression of $U$ and $E_c$ induced by OPS are not strong enough to destroy strong correlation, otherwise we can simply apply the usual electron-hole liquid picture described in previous section. Therefore the setting is similar to that shown in Fig.\[fig:EHBL-OPS\](b) except that the electron-hole liquid is replaced by a strongly correlated EHBL with holons and doublons and the excitons are formed by holon-doublon pairs instead of electron-hole pairs. A mean field calculation similar to that of Ref.[@SLI-1] can be carried out by applying the slave-boson mean field theory to the two-layer *t-J* model. The main difference is that the on-site interlayer interaction $V_{0}\sum_{i}b_{1i}^{\dagger}b_{1i}b_{2i}^{\dagger}b_{2i}$ is replaced by the OPS effective interaction $\sum_{ij}V_{ij}^{x}b_{1i}^{\dagger}b_{1i}b_{2j}^{\dagger}b_{2j}$, where $b_{\alpha i}$ ($b_{\alpha i}^{\dagger}$) is the bosonic holon $(\alpha=1)$ or doublon $(\alpha=2)$ annihilation (creation) operator of layer $\alpha$ at site $i$. The OPS interaction is then decoupled as $$\begin{aligned} \sum_{ij}V_{ij}^{x}b_{1i}^{\dagger}b_{1i}b_{2j}^{\dagger}b_{2j} & \mathcal{\overset{F}{\longrightarrow}} & \sum_{pqk}V_{k}^{x}b_{1p}^{\dagger}b_{2q}^{\dagger}b_{2q+k}b_{1p-k}\\ & \approx & \sum_{p}\Delta_{p}^{b}\left(b_{1p}b_{2\bar{p}}+b_{2\bar{p}}^{\dagger}b_{1p}^{\dagger}\right)\\ & & -\sum_{p}\Delta_{p}^{b}\left\langle b_{1p}b_{2\bar{p}}\right\rangle \;,\end{aligned}$$ where $\Delta_{p}^{b}=\sum_{q}V_{p-q}^{x}\left\langle b_{1q}b_{2\bar{q}}\right\rangle $ is the exciton pairing. Assuming that $\Delta_{p}^{b}=\Delta^{b}\delta(p)$ is homogeneous in space, we obtain a mean field Hamiltonian which is of the same form as in previous study [@SLI-1] for on-site interaction $V_{0}$ with $\left\langle b_{1i}b_{2i}\right\rangle \sim\sum_{k}\left\langle b_{1k}b_{2\bar{k}}\right\rangle $. Since in terms of exciton pairing the attract-repel behavior renders the interlayer OPS interaction resembling an on-site interaction (see Fig.\[fig:EHBL-OPS\](f)), we expect that the mean field phase diagrams in both case are qualitatively the same. The introduction of OPS interaction solely shifts the exciton phase boundary due to a reduction of interaction strength, as in the case of usual electron-hole liquid. We next comment on the possibility of forming spatially inhomogeneous phases. One example of such inhomogeneity is charge corrugation in the form of stripes. By applying mean field theory to *t-J* model with long range Coulomb interaction $V_{c}\sum_{i\neq j}\frac{1}{r_{ij}}n_{i}n_{j}$, it is shown that stripes are preferred to minimize the exchange $J$ term [@LeeDH-stripes]. In particular, it is the decoupling of the exchange term into the anti-ferromagnetic channel $m_{i}$ that drives the stripe formation, while the Coulomb interaction controls the spacing evolution of stripes with doping. Moreover, the stripes spacing increases as the doping $\delta$ decreases. The effect of OPS on stripes is two-fold. Firstly, OPS weakens the on-site repulsion $U$ [@Inc-J] and thus the superexchange $J\sim t^{2}/U$ term is enhanced (assuming that strong correlation is still intact). Consequently, the stripes phase is strengthened. On the other hand, Coulomb interaction tends to smooth out the charge density, while a dipolar interaction ($V\sim1/r^{3}$ for large $r$) would be less effective and a more inhomogeneous phase would be preferred. Notice that extreme charge inhomogeneity like phase separation [@phase_sep] is not likely since the OPS interaction scales like $1/r$ for small $r$ and still suppresses phase separation. Summary ======= We have constructed a dipolar interaction for OPS effect of metallic layer in heterostructures and have justified the construction by a RPA calculation. The OPS interaction is expected to be present rather generally at interfaces with metallic layers. We apply the OPS interaction to EHBL system and find that OPS mainly affects the electronic properties in the low density regime. Our conclusion is not restricted to EHBL since the behavior is mainly due to the modification of the interaction scaling from $1/r$ (Coulomb) to $1/r^{3}$ (dipole) at distance of large $r$. OPS might be employed to eliminate Wigner-crystal like behavior at low temperatures. For strongly correlated electron systems, OPS mainly affects the magnetic channel by reducing the Hubbard $U$ and charge transfer energy $E_{c}$. The reduction of $U$ may drive the system into usual electron liquid. Furthermore, the reduction in interaction range may drive the system into an inhomogeneous state. We acknowledge Prof. P. A. Lee and Prof. N. Nagaosa for insightful comments and Dr. Y. Zhou, Dr. X. Y. Feng, C. K. Chan and Z. X. Liu for helpful discussions. Justification of OPS interactions by RPA ======================================== In this appendix, we employ RPA to justify the image-charge picture of OPS interactions. The RPA method enables us to obtain an effective interaction by “integrating out” the screening layers. First we consider a charged layer $2$ with a metallic screening layer $s$ separated from layer $2$ by distance $a/2$, as shown in Fig.\[fig:EHBL-OPS\](c). We can write down the effective intralayer interaction of layer $2$ after taking into account the effect of screening by the metallic layer (see Fig.\[fig:Feyn\]) $$\begin{aligned} V^{\text{intra}}(q) & = & V_{2}(q)+\frac{V_{2s}(q)\chi_{0s}V_{s2}(q)}{1-\chi_{0s}V_{ss}(q)}\\ & = & 2\pi e^{2}\left(\frac{1}{q}-\frac{e^{-aq}}{q}\frac{2\pi e^{2}N_{F}}{q+2\pi e^{2}N_{F}}\right)\\ & \approx & \frac{2\pi e^{2}}{q}\left(1-e^{-aq}\right)\;,\end{aligned}$$ where $V_{2}(q)=V_{ss}(q)=2\pi e^{2}/q$ are the bare Coulomb interactions of layer $2$ and screening layer $s$, $V_{2s}(q)=V_{s2}(q)=2\pi e^{2}e^{-aq/2}/q$ is the interlayer Coulomb interaction between the layers, and $\chi_{0s}=\chi_{0s}(q\rightarrow0,\omega=0)=-N_{F}$ is the $q\rightarrow0$ static density-density response function of layer $s$, $N_{F}$ is density of states at the Fermi surface. Notice we have assumed that $q$ is small (long wavelength limit) in writing down the interactions and therefore $$F_{0}\equiv\frac{2\pi e^{2}N_{F}}{q+2\pi e^{2}N_{F}}\approx1\;.$$ This approximation is valid if the charge density $n_{2}$ of layer 2 is much less than the density of the screening layer $n_{s}$ and $q\lesssim\sqrt{n_{2}}\ll\sqrt{n_{s}}$. We shall take the same limit in the following derivations. This gives eq.. ![\[fig:Feyn\]Diagram for construction of OPS interactions. Thin lines are bare interactions, bubble is the density-density response function $\chi_{0s}$ and the thick line is the resulting effective interaction.](feyn){width="0.8\columnwidth"} Similarly we can construct the interlayer interaction eq.: $$\begin{aligned} V^{x,1}(q) & = & V_{12}(q)+\frac{V_{1s}(q)\chi_{0s}V_{s2}(q)}{1-\chi_{0s}V_{ss}(q)}\\ & = & 2\pi e^{2}\left(\frac{e^{-bq}}{q}-\frac{e^{-(a+b)q}}{q}F_{0}\right)\\ & \approx & 2\pi e^{2}\frac{e^{-bq}}{q}\left(1-e^{-aq}\right)\;,\end{aligned}$$ where $V_{12}(q)=2\pi e^{2}e^{-bq}/q$, $V_{1s}(q)=2\pi e^{2}e^{-(a/2+b)q}/q$ and $V_{s2}(q)=2\pi e^{2}e^{-aq/2}/q$ are the bare interlayer interactions between the pair of layers ($1$,$2$), $(1,s)$ and $(s,2)$ respectively. For two screening layers (Fig.\[fig:EHBL-OPS\](b)), we assume that layer $2$ and $s$ form an effective system $2^{\prime}$ and thus we can adopt $V^{x,1}$ as the “bare” interlayer interactions in the following: $$\begin{aligned} V^{x,2}(q) & = & V^{x,1}(q)+\frac{V_{1s}(q)\chi_{0s}V_{s2^{\prime}}(q)}{1-\chi_{0s}V_{ss}(q)}\\ & = & 2\pi e^{2}\left(1-e^{-aq}\right)\left(\frac{e^{-bq}}{q}-\frac{e^{-(a+b)q}}{q}F_{0}\right)\\ & \approx & 2\pi e^{2}\frac{e^{-bq}}{q}\left(1-e^{-aq}\right)^{2}\;,\end{aligned}$$ where $V_{s2^{\prime}}(q)=V^{x,1}(b\rightarrow b+\frac{a}{2})=\frac{1}{q}e^{-(\frac{a}{2}+b)q}\left(1-e^{-aq}\right)$. The validity of assuming the effective system $2^{\prime}$ is based on the choice of $b\gg a$. This gives eq.. Here we derive the effective intralayer interaction $V_{1,\text{eff}}^{\text{intra}}(q)$ with two-layer OPS (Fig.\[fig:EHBL-OPS\](b)) taking into account the screening of system $2^{\prime}$ (i.e. integrated out all the screening by $2$, $s1$ and $s2$): $$V_{1,\text{eff}}^{\text{intra}}(q)=V_{1}^{\text{intra}}(q)+\frac{\chi_{02}\left(V^{x,2}(q)\right)^{2}}{1-V_{2}^{\text{intra}}(q)\chi_{02}}\;.\label{eq:effintra2}$$ In the small $q$ limit, $V^{x,2}$ and $V_{2}^{\text{intra}}$ scale as $q$ and constant respectively. The second term due to screening is of higher order in $q$ and thus it cannot alter the scaling of the $V_{1}^{\text{intra}}(q)$ term ($\sim\text{constant}$). One can repeat the analysis for the one-layer OPS case (see Fig.\[fig:EHBL-OPS\](c)) and the scaling of the effective intralayer interaction $V_{1,2}^{\text{intra}}(q)$ in the lowest order of $q$ is not affected by screening of the opposite charged layer. [17]{} S. Okamoto and A. J. Millis, Nature **428**, 630 (2004). 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--- abstract: 'We establish Hardy–Littlewood inequalities for the Heckman–Opdam transform associated to a general root datum $({\mathfrak{a}},\Sigma,m)$ that generalizes an analogous result for the spherical Fourier transform on a Riemannian symmetric space of the non-compact type due to Eguchi and Kumahara. In particular we obtain a more precise Hausdorff–Young inequality that generalizes a recent result due to Narayanan, Pasquale, and Pusti.' address: | Mathematisches Seminar\ Chr.-Albrechts–Universität zu Kiel\ Ludewig-Mey–Str. 4, DE-24098 Kiel\ Germany author: - Troels Roussau Johansen title: 'Hardy–Littlewood inequalities for the Heckman–Opdam transform' --- Introduction ============ The classical Hausdorff–Young inequality $\|{\hat{f}}\|_q\leq c_p\|f\|_p$, $1\leq p\leq 2$, $\frac{1}{p}+\frac{1}{q}=1$, for the Euclidean Fourier transform can be viewed as a partial extension of the Plancherel theorem to $L^p$-functions. More generally, the Fourier transform extends to a continuous mapping from $L^p({\mathbb{R}}^n)$ into the Lorentz space $L^{p',p}({\mathbb{R}}^n)$, a result that is due to Paley. A variation on this theme is provided by the Hardy–Littlewood inequality which may be stated as follows: Let $f$ be a measurable function on ${\mathbb{R}}^n$ such that $x\mapsto f(x)\|x\|^{n(1-2/q)}$ belongs to $L^q({\mathbb{R}}^n)$, where $q\geq 2$. Then $f$ has a well-defined Fourier transform ${\hat{f}}$ in $L^q({\mathbb{R}}^n)$ and there exists a positive constant $A_q$ independent of $f$ such that $$\label{HL-ineq1} \Bigl(\int_{{\mathbb{R}}^n}\vert{\hat{f}}(\xi)\vert^qd\xi\Bigr)^{1/q}\leq A_q\Bigl(\int_{{\mathbb{R}}^n}\vert f(x)\vert^q\|x\|^{n(q-2)}dx\Bigr)^{1/q}.$$ By duality and general properties of the Fourier transform, one has the following equivalent formulation: For every $p\in(1,2)$ there exists a positive constant $B_p$ independent of $f$ such that $$\label{HL-ineq2} \Bigl(\int_{{\mathbb{R}}^n}|{\hat{f}}(\xi)|^p|\xi|^{n(p-2)}\,d\xi\Bigr)^{1/p}\leq B_p\Bigl(\int_{{\mathbb{R}}^n}|f(x)|^p\,dx\Bigr)^{1/p}.$$ An analogue of for the spherical transform on a Riemannian symmetric space $G/K$ was obtained by Eguchi and Kumahara in [@Eguchi-Kumahara Theorem 1, Section 5]: \[thm.1\] Let $q\geq 2$. The spherical Fourier transform can be defined for $K$-invariant functions $f$ on $G/K$ with the property that $f\cdot\sigma^{n(1-2/q)}\Omega^{1-2/q}$ belongs to $L^q(K\setminus G/K)$, and there exists a positive constant $A_q$ that is independent of $f$ such that $$\label{ineq-HL-GK} \Bigl(\frac{1}{\vert W\vert}\int_{{\mathfrak{a}}^*}\vert\widetilde{f}(\lambda)\vert^q\,\vert\mathbf{c}(\lambda)\vert^{-2}\,d\lambda\Bigr)^{1/q} \leq A_q\Bigl(\int_G\vert f(x)\vert^q\sigma(x)^{n(q-2)}\Omega(x)^{q-2}\,dx\Bigr)^{1/q}$$ for all $f\in\mathcal{S}(K\setminus G/K)$. Here $\sigma(x)=\left<X,X\right>^{1/2}$ where $\left<\cdot,\cdot\right>$ is the Cartan–Killing form and $G\ni x=k\exp X\in K\times\mathfrak{p}$, and $\Omega(\exp H)=c\prod_{\alpha\in\Sigma}\vert\sinh\alpha(H)\vert^{m(\alpha)}$, $H\in\mathfrak{a}$, the usual weight and $\mathcal{S}(K\setminus G/K)$ an $L^2$-based Schwartz space of $K$-invariant functions on $G/K$. An interpolation argument leads to an analogous statement for exponents below $2$: \[thm.2\] Let $p\in(1,2]$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $r\in[p,q]$ and set $\mu=\frac{1}{r}+\frac{1}{q}-1=\frac{1}{r}-\frac{1}{p}$. Then there exists a positive constant $B_r$ independent of $f$ such that $$\Bigl(\frac{1}{|W|}\int_{{\mathfrak{a}}^*}\vert\widetilde{f}(\lambda)\vert^q\vert\mathbf{c}(\lambda)\vert^{-2}\,d\lambda\Bigr)^{1/q} \leq B_r\Bigl(\int_G\vert f(x)\vert^r\sigma(x)^{-n\mu r}\Omega(x)^{-\mu r}\,dx\Bigr)^{1/r}$$ for all $f$ satisfying $f\cdot\sigma^{-n\mu}\Omega^{-\mu}\in L^r(K\setminus G/K)$. It was remarked in the MathSciNet review by Michael Cowling that one could simplify the proof of Eguchi and Kumahara by means of more refined interpolation techniques. These were later incorporated in [@Mohanty-II] where the authors established an analogue of for the Helgason–Fourier transform on a noncompact Riemannian symmetric space of rank one: It holds that $$\label{eqn.MRSS} \int_{{\mathfrak{a}}^*} \|\widetilde{f}(\lambda,\cdot)\|_{L^1(K)}|\lambda|^{p-2}(1+|\lambda|)^{-(m_\gamma+m_{2\gamma})}|\mathbf{c}(\lambda)|^{-2}\,d\lambda\leq C\|f\|_p^p$$ for $1<p<2$. According to [@Mohanty-II Remark 4.6], their method also works for higher rank spaces. While we share this sentiment, it turns out to be slightly involved to fill in the necessary details. One may also object that the appearance of the average over $K$ is not natural. A different version was recently obtained in [@Ray-Sarkar-trans], to which we shall return later. A further drawback of is that for $p=2$ it does not resemble the Parseval identity, and section \[sec.HY-ineqs\] opens with the observation that the analogue of for the Heckman–Opdam transform, or even just the Jacobi transform in rank one, does not hold for arbitrary non-negative root multiplicities. We also wish to emphasize a quantitative difference between and : In the first inequality a weight is introduced on the function-side, whereas the second inequality incorporates a weight on the Fourier transform side. Theorem \[thm.1\] and theorem \[thm.2\] therefore resemble , whereas resembles . It is the purpose of the present paper to obtain analogues of and for the Heckman–Opdam transform associated to a triple $({\mathfrak{a}},\Sigma,m)$, where ${\mathfrak{a}}$ is an Euclidean $n$-dimensional vector space, $\Sigma$ a root system in ${\mathfrak{a}}^*$ and $m$ a positive multiplicity function. In order to place the contributions of the present paper in perspective, the reader is reminded that some classical aspects of the $L^2$-theory for hypergeometric Fourier analysis in root systems (that is, Plancherel and Paley–Wiener theorems and an inversion formula) was already obtained in [@Opdam-acta], whereas the $L^p$-analysis is much more recent. As far as we can ascertain, the first decisive contribution was given in the recent publication [@Narayanan-Pasquale-Pusti], and the results we obtain should be seen as natural contributions to the general theme of classical harmonic analysis in a root system framework, The details pertaining to harmonic analysis in root systems will be presented in section \[sec.root1\]. There are several standard references but we follow closely the presentation in [@Narayanan-Pasquale-Pusti] as far as the Heckman–Opdam theory is concerned. Section \[sec.root1\] also summarizes the interpolation theorems for Lorentz spaces. An immediate consequence is a generalized Hausdorff–Young inequality of Paley-type. Section \[sec.HY-ineqs\] presents several versions of the Hardy–Littlewood inequality for the Heckman–Opdam transforms, corresponding to different weights. The last section briefly outlines a generalization of the Eguchi–Kumahara result for the Cartan motion groups. On can introduce a ‘flat’ Heckman–Opdam transform $\mathcal{F}_0$ in analogy with generalized Bessel transform on the flat space $G_0/K$, and we obtain Hardy–Littlewood inequalities for $\mathcal{F}_0$ as well. This involves generalized Bessel-type functions associated with root systems that were already considered by Opdam in [@Opdam-Bessel Section 6]. The connection to spherical functions on the Cartan motion group was explicitly indicated in [@Opdam-Bessel Remark 6.12] and later established formally in [@deJeu-PW]. Harmonic analysis in root systems {#sec.root1} ================================= Let ${\mathfrak{a}}$ be an $n$-dimensional real Euclidean vector space with inner product $\langle\cdot,\cdot\rangle$ and let ${\mathfrak{a}}^*$ denote the linear dual of ${\mathfrak{a}}$. For $\lambda\in{\mathfrak{a}}^*$ let $x_\lambda$ be the unique vector in ${\mathfrak{a}}$ such that $\lambda(x)=\langle x,x_\lambda\rangle$ for every $x\in{\mathfrak{a}}$. Define an inner product on ${\mathfrak{a}}^*$ via $\langle \lambda,\mu\rangle=\langle x_\lambda,x_\mu\rangle$, and let ${\mathfrak{a}}_{\mathbb{C}}$ and ${\mathfrak{a}}_{\mathbb{C}}^*$ denote the complexifications of ${\mathfrak{a}}$ and ${\mathfrak{a}}^*$. The inner products on ${\mathfrak{a}}$ and ${\mathfrak{a}}^*$ extend by ${\mathbb{C}}$-linearity to inner products on ${\mathfrak{a}}_{\mathbb{C}}$ and ${\mathfrak{a}}_{\mathbb{C}}^*$ that will denoted by the same symbol. Set $\lambda_\alpha=\frac{\langle\lambda,\alpha\rangle}{\langle\alpha,\alpha\rangle}$ and $|x|=\langle x,x\rangle^{1/2}$ for $x\in{\mathfrak{a}}$. Let $\Sigma$ be a root system in ${\mathfrak{a}}^*$ and let $W$ denote the associated Weyl group generated by the root reflections $r_\alpha:\lambda\mapsto\lambda -2\lambda_\alpha\alpha$ for $\alpha\in\Sigma$. Fix a compatible set $\Sigma^+$ of positive roots in $\Sigma$ and let $\Pi= \{\alpha_1,\ldots,\alpha_n\}\subset\Sigma^+$ be the associated set of simple roots. Let $\Sigma_0$ denote the set of roots in $\Sigma$ that are indivisible, in the sense that if $\alpha$ belongs to $\Sigma_0$, then $\alpha/2$ is not a root. A (strictly) positive multiplicity function is a $W$-invariant function $m:\Sigma\to(0,\infty)$. We often write $m_\alpha=m(\alpha)$. By $W$-invariance it holds that $m_{w\alpha}=m_\alpha$ for all $\alpha\in\Sigma$ and $w\in W$. We adhere to the conventions in [@Narayanan-Pasquale-Pusti] rather than Heckman and Opdam: Their root system $\mathfrak{R}$ and multiplicity function $k$ are related to $\Sigma$ and $m$ above by the identities $\mathfrak{R}=\{2\alpha\,:\,\alpha\in\Sigma\}$, $k_{2\alpha}=m_\alpha/2$ for $\alpha\in\Sigma$. Set $\Sigma_0^+=\Sigma^+\cap\Sigma_0$. The complexification ${\mathfrak{a}}_{\mathbb{C}}$ of ${\mathfrak{a}}$ may be viewed as the Lie algebra of the complex torus $A_{\mathbb{C}}={\mathfrak{a}}_{\mathbb{C}}/\{2\pi ix_\alpha/\langle\alpha,\alpha\rangle\,:\,\alpha\in\Sigma\}{\mathbb{Z}}$. Let $\exp:{\mathfrak{a}}_{\mathbb{C}}\to A_{\mathbb{C}}$ be the exponential map. The real form $A=\exp{\mathfrak{a}}$ of $A_{\mathbb{C}}$ is an abelian subgroup of $A_{\mathbb{C}}$ with Lie algebra ${\mathfrak{a}}$ such that $\exp:{\mathfrak{a}}\to A$ is a diffeomorphism, by means of which we shall often identify ${\mathfrak{a}}$ with $A$. The $W$-action extends to ${\mathfrak{a}}$ by duality, and to ${\mathfrak{a}}_{\mathbb{C}}^*$ and ${\mathfrak{a}}_{\mathbb{C}}$ by ${\mathbb{C}}$-linearity. Moreover $W$ acts via the left regular representation of functions on either one of these. The positive Weyl chamber ${\mathfrak{a}}^+$ is defined as the set of elements $x\in{\mathfrak{a}}$ for which $\alpha(x)>0$ for all $\alpha\in\Sigma^+$. Let $P=\{\lambda\in{\mathfrak{a}}^*\,:\,\lambda_\alpha\in{\mathbb{Z}}\text{ for all }\alpha\in\Sigma\}$ denote the restricted weight lattice. To $\lambda\in P$ is associated the single-valued exponential $e^\lambda:A_{\mathbb{C}}\to{\mathbb{C}}$ given by $e^\lambda(h)=e^{\lambda(\log h)}$, and these are the characters of $A_{\mathbb{C}}$. It can be seen that $\mathrm{span}_{\mathbb{C}}\{e^\lambda\}$ is isomorphic to the ring ${\mathbb{C}}[A_{\mathbb{C}}]$ of regular functions on $A_{\mathbb{C}}$, the latter viewed as an algebraic variety, and $W$ acts on it by $w(e^\lambda):=e^{w\lambda}$ (which is well-defined since $P$ is $W$-invariant). The set of regular points for the $W$-action on $A_{\mathbb{C}}$ coincides with the set $A_{\mathbb{C}}^{\mathrm{reg}}=\{h\in{\mathbb{C}}\,:\,e^{2\alpha(\log h)}\neq 1\text{ for all }\alpha\in\Sigma\}$, and the algebra ${\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]$ of regular functions on $A_{\mathbb{C}}^{\mathrm{reg}}$ is the subalgebra of the quotient field of ${\mathbb{C}}[A_{\mathbb{C}}]$ generated by ${\mathbb{C}}[A_{\mathbb{C}}]$ and $\{\frac{1}{1-e^{-2\alpha}}\,:\,\alpha\in\Sigma^+\}$. We denote by ${\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]^W$ the subalgebra of $W$-invariant elements. Let $S({\mathfrak{a}}_{\mathbb{C}})$ denote the symmetric algebra over ${\mathfrak{a}}_{\mathbb{C}}$ and let $S({\mathfrak{a}}_{\mathbb{C}})^W$ denote the subalgebra of its $W$-invariant elements. An element $p\in S({\mathfrak{a}}_{\mathbb{C}})$ gives rise to a constant coefficient differential operator $\partial(p)$ acting on functions $f$ on $A_{\mathbb{C}}$ such that $\partial(x)$ is the directional derivative in the direction of $x$ for every $x\in{\mathfrak{a}}$. We shall denote the algebra $\{\partial(p)\,:\,p\in S({\mathfrak{a}}_{\mathbb{C}})\}$ by the symbol $S({\mathfrak{a}}_{\mathbb{C}})$, which is justified since $p\mapsto\partial(p)$ is an algebra isomorphism. Let $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})={\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]\otimes S({\mathfrak{a}}_{\mathbb{C}})$ denote the algebra of differential operators on $A_{\mathbb{C}}$ with coefficients in ${\mathbb{C}}[A_{\mathbb{C}}^{\mathrm{reg}}]$ and $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})^W$ its subalgebra of $W$-invariant elements, where $W$ acts by $w(\varphi\otimes\partial(p))=(w\varphi)\otimes\partial(wp)$. We introduce an associative algebra structure on $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ via $(D_1\otimes w_1)\cdot(D_2\otimes w_2)=D_1w_1(D_2)\otimes w_1w_2$, where $(wD)(wf):=w(Df)$. Elements of $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ are the differential-reflection operators on $A_{\mathbb{C}}^{\mathrm{reg}}$, and they act on functions $f$ on $A_{\mathbb{C}}^{\mathrm{reg}}$ by $(D\otimes w)f=D(wf)$. The *Cherednik operator* $T_x\in\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ associated with $x\in{\mathfrak{a}}$ is defined by $$T_x=\partial_x-\rho(x)+\sum_{\alpha\in\Sigma^+}m_\alpha\alpha(x)(1-e^{-2\alpha})^{-1}\otimes(1-r_\alpha),$$ where $2\rho=\sum_{\alpha\in\Sigma^+}m_\alpha\alpha\in{\mathfrak{a}}^*$. It is a deep result that the operators $T_x$ commute, an important consequence of which is that the map $x\mapsto T_x$ extends uniquely to an algebra homomorphism $p\mapsto T_p$ of $S({\mathfrak{a}}_{\mathbb{C}})$ into $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$. Define $\Upsilon:\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]\to\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})$ by $\Upsilon(\sum_j D_j\otimes w_j)=\sum_jD_j$. Then $\Upsilon(P)f=Pf$ for every $P\in\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})\otimes{\mathbb{C}}[W]$ and every $W$-invariant function $f$ on $A_{\mathbb{C}}^{\mathrm{reg}}$. In particular we can define $D_p:=\Upsilon(T_p)$ for every $p\in S({\mathfrak{a}}_{\mathbb{C}})$. One can show that $D_p$ belongs to $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})^W$ whenever $p\in S({\mathfrak{a}}_{\mathbb{C}})^W$, and that $\mathbb{D}:=\{D_p\,:\,p\in S({\mathfrak{a}}_{\mathbb{C}})^W\}$ is a commutative subalgebra of $\mathbb{D}(A_{\mathbb{C}}^{\mathrm{reg}})^W$, see [@Narayanan-Pasquale-Pusti p. 232] for details. In the geometric case where $({\mathfrak{a}},\Sigma,m)$ corresponds to the root datum of a Riemannian symmetric space, $\mathbb{D}$ is the algebra of radial components along $A$ of $G$-invariant differential operators on $G/K$. The analogue of the radial component of the Laplace–Beltrami operator is the operator $D_{p_L}$, where $p_L\in S({\mathfrak{a}}_{\mathbb{C}})^W$ is the polynomial $p_L(\lambda)=\langle\lambda,\lambda\rangle$. Then $D_{p_L}=L+\langle\rho,\rho\rangle$ where $$L=L_{\mathfrak{a}}+\sum_{\alpha\in\Sigma^+}m_\alpha\coth \alpha \partial(x_\alpha);$$ here $L_{\mathfrak{a}}$ is the Euclidean Laplace operator on ${\mathfrak{a}}$, and $\coth \alpha=\frac{1+e^{-2\alpha}}{1-e^{-2\alpha}}$. Let $\lambda\in{\mathfrak{a}}_{\mathbb{C}}^*$ be fixed. The *hypergeometric function with spectral parameter* $\lambda\in{\mathfrak{a}}_{\mathbb{C}}^*$ is the unique analytic $W$-invariant function $\varphi_\lambda$ on ${\mathfrak{a}}$ that satisfies the system of differential equations $$D_p\varphi = p(\lambda)\varphi,\quad p\in S({\mathfrak{a}}_{\mathbb{C}})^W$$ and is normalized by $\varphi_\lambda(0)=1$. In the case $n=1$, $\Sigma^+$ consists of at most two elements, $\alpha$ and $2\alpha$. Identify ${\mathfrak{a}}$ and ${\mathfrak{a}}^*$ with ${\mathbb{R}}$ by setting $x_\alpha/2\equiv 1$ and $\alpha\equiv 1$. Then ${\mathfrak{a}}_+\simeq (0,\infty)$, and $W=\{-1,1\}$ acts on ${\mathbb{R}}$ and ${\mathbb{C}}$ by multiplication. The algebra $\mathbb{D}$ is generated by a single element, for example the operator $D_{\rho_L}=L+\rho^2$, where $\rho=m_\alpha/2+m_{2\alpha}$. The hypergeometric system of differential equations used to define $\varphi$ reduces to the sdifferential equation $$\frac{d^2\varphi}{dz^2}+(m_\alpha\coth z+m_{2\alpha}\coth(2z))\frac{d\varphi}{dz}=(\lambda^2-\rho^2)\varphi$$ which may be transformed into the hypergeometric differential equation $$\zeta(1-\zeta)\frac{d^2\psi}{d\zeta^2}+(c-(1+a+b)\zeta)\frac{d\psi}{d\zeta}-ab\zeta=0$$ where $\zeta=\frac{1}{2}(1-\cosh z)$, $a=\frac{\lambda+\rho}{2}$, $b=\frac{-\lambda+\rho}{2}$, and $c=\frac{m_\alpha+m_{2\alpha}+1}{2}$. The solution $\varphi_\lambda$ is therefore the Jacobi functions $$\varphi_\lambda(t)={_2}F_1\Bigl(\frac{m_\alpha/2+m_{2\alpha}+\lambda}{2},\frac{m_\alpha/2+m_{2\alpha}-\lambda}{2};\frac{m_\alpha+m_{2\alpha}+1}{2};-\sinh^2t\Bigr)$$ which are well known to describe the elementary spherical functions on a rank one Riemannian symmetric space $G/K$. Existence, uniqueness and regularity properties of $\varphi_\lambda$ were investigated in several publications of Heckman and Opdam, later sharpened by Schapira [@Schapira] (where the functions are denoted $F_\lambda$ and their non-symmetric versions $G_\lambda$) and most recently by Narayanan, Pasquale, and Pusti [@Narayanan-Pasquale-Pusti]. Since we do not need to estimate the functions $\varphi_\lambda$ and the associated Harish-Chandra series expansions in the results that follow, we merely refer the reader to [@Narayanan-Pasquale-Pusti Sections 2–4] for the details. The *Heckman–Opdam* transform of a function $f\in C_c^\infty({\mathfrak{a}})^W$ is defined by $$\mathcal{F}f(\lambda)=\int_{\mathfrak{a}}f(x)\varphi_\lambda(x)\,d\mu(x).$$ Often $\mathcal{F}$ is called the hypergeometric Fourier transform, since the functions $\varphi_\lambda$ can be seen as generalized hypergeometric functions. In some literature, such as [@Schapira], these are denoted by $F_\lambda$. Their non-symmetric version appear as $G_\lambda$, in terms of which one can define a hypergeometric Fourier transform of a function $f\in C_c^\infty({\mathfrak{a}})$ that is not necessarily $W$-invariant. The terminology ‘hypergeometric’ was certainly used by Delorme in [@Delorme-hyper] but the transform itself was studied earlier by Cherednik from a different perspective, and by Opdam in [@Opdam-acta]. When $\mathcal{F}$ acts on functions that might not be $W$-invariant, we therefore talk about the Cherednik–Opdam transform or the hypergeometric transform. Its restriction to $W$-invariant functions is the transform that Heckman and Opdam studied, hence the name. A convenient analogy is to think of the Heckman–Opdam transform as a root space generalization of the spherical Fourier transform associated with a Riemannian symmetric space. The more general Cherednik–Opdam transform is not related to the Helgason–Fourier transform on a symmetric space, however. The ${\mathbf{c}}$-function associated with $({\mathfrak{a}},\Sigma,m)$ is defined by $${\mathbf{c}}(\lambda)=c\prod_{\alpha\in\Sigma_0^+}{\mathbf{c}}_\alpha(\lambda),\quad \text{with}\quad {\mathbf{c}}_\alpha(\lambda)=\frac{2^{-\lambda_\alpha}}{\Gamma(\frac{\lambda_\alpha}{2}+\frac{m_\alpha}{4}+\frac{1}{2})\Gamma(\frac{\lambda_\alpha}{2}+\frac{m_\alpha}{4}+\frac{m_{2\alpha}}{2})},$$ where $c$ is a normalizing constant that is chosen so that ${\mathbf{c}}(\rho)=1$. It is known that $|{\mathbf{c}}_\alpha(\lambda)|^{-2}\asymp|\!\left<\lambda,\alpha\right>\!|(1+\|\lambda\|)^{m_\alpha+m_{2\alpha}-2}$, so by the Cauchy–Schwarz inequality in ${\mathfrak{a}}$, $$\label{eqn.c-est1} |{\mathbf{c}}_\alpha(\lambda)|^{-2}\asymp \begin{cases} \|\lambda\|^{2}(1+\|\lambda\|)^{m_\alpha+m_{2\alpha}-2}&\text{for }\lambda\in i{\mathfrak{a}}^*\text{ with } \|\lambda\| \text{ large},\\ \|\lambda\|^{2}&\text{for } \lambda\in i{\mathfrak{a}}^*\text{ with } \|\lambda\|\lesssim 1\end{cases}.$$ In particular, $$|{\mathbf{c}}(\lambda)|^{-2}\asymp \prod_{\alpha\in\Sigma_0^+}|\!\left<\lambda,\alpha\right>\!|^2(1+|\!\left<\lambda,\alpha\right>\!|)^{m_\alpha+m_{2\alpha}-2} \asymp \|\lambda\|^{2|\Sigma_0^+|}(1+\|\lambda\|)^{\beta-2|\Sigma_0^+|},$$ where $\beta=\sum_{\alpha\in\Sigma_0^+}(m_\alpha+m_{2\alpha})$ and where $|\Sigma_0^+|$ is the cardinality of $\Sigma_0^+$. Let $dx$ denote a fixed normalization of the Haar measure on the abelian group ${\mathfrak{a}}$, and associate to $({\mathfrak{a}},\Sigma,m)$ the weighted measure $d\mu(x)=J(x)\,dx$ on ${\mathfrak{a}}$, where $$J(x)=\prod_{\alpha\in\Sigma^+}|e^{\alpha(x)}-e^{-\alpha(x)}|^{m_\alpha}.$$ It is known, cf. [@Narayanan-Pasquale-Pusti Theorem 1.13] and the references to the literature, that there exists a suitable normalization of the measure $d\lambda$ on $i{\mathfrak{a}}^*$ such that the transform $\mathcal{F}$ extends to an isometric isomorphism from $L^2({\mathfrak{a}},d\mu)^W$ onto $L^2(i{\mathfrak{a}}^*,|{\mathbf{c}}(\lambda)|^{-2}d\lambda)^W$. Moveover, for $f\in C_c^\infty({\mathfrak{a}})^W$, $$f(x)=\int_{i{\mathfrak{a}}^*}\mathcal{F}f(\lambda)\varphi_{-\lambda}(x)|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda$$ for all $x\in{\mathfrak{a}}$. For $p\in(0,2]$, set $\epsilon_p=\frac{2}{p}-1$. Let $C(\epsilon_p)$ be the convex hull in ${\mathfrak{a}}^*$ of the set $\{\epsilon_p w\rho\,:\,w\in W\}$, and let ${\mathfrak{a}}^*_{\epsilon_p}=C(\epsilon_p\rho)+i{\mathfrak{a}}^*$. The following two versions of the Hausdorff–Young theorem for the Heckman–Opdam transform were recently established by Narayanan, Pasquale, and Pusti, cf. [@Narayanan-Pasquale-Pusti Lemma 5.2, Lemma 5.3]. The first proof involves Riesz–Thorin interpolation, whereas the second uses interpolation with an analytic family of operators. Let $f\in L^p(A,d\mu)^W$. Then the following properties hold. 1. The hypergeometric transform $\mathcal{F}f(\lambda)$ is well defined for all $\lambda$ in the interior of ${\mathfrak{a}}^*_{\epsilon_p}$ and defines a holomorphic function. 2. Let $p,q$ be so that $1<p<2$ and $\frac{1}{p}+\frac{1}{q}=1$. Then there exists a positive constant $c_p$ independent of $f$ so that $$\label{ineq-HY-root-1} \|\mathcal{F}f\|_q = \Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^q\,d\nu(\lambda)\Bigr)^{1/q}\leq c_p\|f\|_p.$$ A more precise formulation is given as follows. Let $f\in L^1({\mathfrak{a}},d\mu)^W\cap L^2({\mathfrak{a}},d\mu)^W$. If $1\leq p\leq 2$, $q=\frac{p}{p-1}$, then $\|\mathcal{F}f\|_{q}\leq c_p\|f\|_p$. Since $L^p$ is dense in $L^1\cap L^2$ for $1\leq p\leq 2$, the Heckman–Opdam transform $\mathscr{F}_pf$ can be defined uniquely for all $f\in L^p({\mathfrak{a}},d\mu)^W$, $1\leq p\leq 2$, so that $\mathscr{F}_p:L^p({\mathfrak{a}},d\mu)^W\to L^q(i{\mathfrak{a}}^*,d\nu)^W$ is a linear contraction with $\mathscr{F}_pf=\mathcal{F}f$ for all $f\in L^1({\mathfrak{a}},d\mu)^W\cap L^2({\mathfrak{a}},d\mu)^W$. It is known from [@Narayanan-Pasquale-Pusti Theorem 5.4] that $\mathscr{F}_p$ is injective on $L^p({\mathfrak{a}},d\mu)^W$ whenever $p\in[1,2]$. \[lemma.NPP-2\] Let $f\in L^p({\mathfrak{a}},d\mu)^W$ for some $p\in(1,2)$ and let $\eta$ be in the interior of $C(\epsilon_p\rho)$. Then the following properties hold: 1. Assume $\frac{1}{p}+\frac{1}{q}=1$. There exists a positive constant $C_{p,\eta}$ such that for all $f\in L^p({\mathfrak{a}},d\mu)^W$, $$\label{ineq-HY-root-2} \Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^q|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)^{1/q}\leq C_{p,\eta}\|f\|_p.$$ 2. It holds that $\sup_{\lambda\in i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|\leq C_{p,\eta}\|f\|_p$ and $$\lim_{\lambda\in{\mathfrak{a}}^*_{\epsilon_p},|\Im\lambda|\to\infty} |\mathcal{F}f(\lambda)|=0.$$ An important ingredient in the proof of both results is a characterization of the set of spectral parameters $\lambda$ for which $\varphi_\lambda$ is bounded. This description was obtained in [@Narayanan-Pasquale-Pusti Theorem 4.2]: $\varphi_\lambda$ is bounded if and only if $\lambda\in C(\rho)+i{\mathfrak{a}}^*$, in which case $|\varphi_\lambda(x)|\leq 1$ for every $x\in{\mathfrak{a}}$. Note that $C(\epsilon_p\rho)\subset C(\rho)$ for $p\geq 2$. Also note that $C(\epsilon_2\rho)=\{0\}$. \[lemma.p1p2\] 1. If $f$ belongs to $(L^{p_1}\cap L^{p_2})({\mathfrak{a}},d\mu)^W$ for some $p_1,p_2\in[1,2]$, then $\mathscr{F}_{p_1}f=\mathscr{F}_{p_2}f$ $\nu$-almost everywhere on $i{\mathfrak{a}}^*$. 2. If $h$ belongs to $(L^{q_1}\cap L^{q_2})(i{\mathfrak{a}}^*,d\nu)^W$ for some $q_1,q_2\in[1,2]$, then $\mathscr{I}_{q_1}h=\mathscr{I}_{q_2}h$ $\mu$-almost everywhere on ${\mathfrak{a}}$. <!-- --> 1. Choose a sequence $\{g_n\}_{n=1}^\infty$ of simple $W$-invariant functions on ${\mathfrak{a}}$ such that $$\lim_{n\to\infty}\|f-g_n\|_{p_1}=\lim_{n\to\infty}\|f-g_n\|_{p_2}=0.$$ Each function $\mathcal{F}g_n$ belongs to $(L^{p_1^\prime}\cap L^{p_2^\prime})(i{\mathfrak{a}}^*,d\nu)^W$ by the Hausdorff–Young inequality , and $$\lim_{n\to\infty}\|\mathscr{F}_{p_1}f-\mathcal{F}g_n\|_{p_1^\prime}=\lim_{n\to\infty}\|\mathscr{F}_{p_2}f\mathcal{F}g_n\|_{p_2^\prime}=0.$$ One can therefore extract subsequences $\{\mathcal{F}g_{n_k}\}_{k=1}^\infty$ and $\{\mathcal{F}g_{n_l}\}_{l=1}^\infty$ of $\{\mathcal{F}g_n\}_{n=1}^\infty$ such that $\mathcal{F}g_{n_k}\to\mathscr{F}_{p_1}f$ and $\mathcal{F}g_{n_l}\to\mathscr{F}_{p_2}f$ $\nu$-almost everywhere on $i{\mathfrak{a}}^*$, from which it follows that $\mathscr{F}_{p_1}f=\mathscr{F}_{p_2}f$ $\nu$-almost everywhere on $i{\mathfrak{a}}^*$ as claimed. 2. Since $\mathscr{I}_{q_j}h\in L^{q_j^\prime}({\mathfrak{a}},d\mu)^W$ for $j=1,2$, the claim follows from the injectivity of $\mathscr{F}_p$ for $p\in(1,2]$ and (i). The remainder of the section is concerned with interpolation results in Lorentz spaces that will be needed in our proof of the Hardy–Littlewood inequality. The interested reader may consult [@Stein-Weiss-analysis Chapter V] for detailed proofs and historical remarks. Let $(X,\mu)$ be a $\sigma$-finite measure space and let $p\in(1,\infty)$. Define $$\|f\|^*_{p,q}=\begin{cases}\displaystyle \Bigl(\frac{q}{p}\int_0^\infty t^{q/p-1}f^*(t)^q\,dt\Bigr)^{1/q}&\text{if } q<\infty\\ \displaystyle \sup_{t>0} t\lambda_f(t)^{1/p}&\text{when } q=\infty\end{cases}$$ where $\lambda_f$ is the distribution function of $f$ and $f^*$ the non-increasing rearrangement of $f$, that is $$\lambda_f(s) =\mu(\{x\in X\,:\, |f(x)|>s\}) \quad\text{and}\quad f^*(t)=\inf \{s\,:\,\lambda_f(s)\leq t\}.$$ By definition, the Lorentz space $L^{p,q}(X)$ consists of measurable functions $f$ on $X$ for which $\|f\|^*_{p,q}<\infty$. Let $(X,d\mu)$ and $(Y,d\overline{\nu})$ be $\sigma$-finite measure spaces. A linear operator $T:L^p(X,d\mu)\to L^q(Y,d\overline{\nu})$ is *strong type $(p,q)$* if it is continuous on $L^p(X,d\mu)$. Moreover, $T$ is *weak type $(p,q)$* if there exists a positive constant $K$ independent of $f$ such that for all $f\in L^p(X,d\mu)$ and all $t>0$, $$\mu\bigl(\bigl\{y\in Y\,:\, |Tf(y)|>t\bigr\}\bigr)\leq \Bigl(\frac{K}{s}\|f\|_{L^p(X,d\mu)}\Bigr)^q.$$ The infimum if such $K$ is the weak type $(p,q)$ norm of $T$. For $1\leq p,q\leq\infty$, $L^{p,p}(X,d\mu)=L^p(X,d\mu)$, and $\|f\|^*_{p,q_2}\leq\|f\|^*_{p,q_1}$ whenever $q_1\leq q_2$. The following version of Hölder’s inequality for Lorentz spaces is well-known. Let $0<p,q,r\leq\infty$, $0<s_1,s_2\leq\infty$. Then $$\|f\cdot g\|_{r,s}^* \leq C_{p,q,s_1,s_2}\|f\|^*_{p,s_1}\|g\|^*_{q,s_2}$$ where $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $\frac{1}{s_1}+\frac{1}{s_2}=\frac{1}{s}$. The dual of $L^{p,q}(X,d\mu)$ is the space $L^{p^\prime,q^\prime}(X,d\mu)$, where $\frac{1}{p}+\frac{1}{p^\prime}=1=\frac{1}{q}+\frac{1}{q^prime}$, and the dual of $L^{1,q}(X,d\mu)$ is $\{0\}$ when $1<q<\infty$. The following interpolation theorem is classical and can be found as Theorem 3.15 in [@Stein-Weiss-analysis Chapter V]. It subsumes the interpolation theorem of Marcinkiewicz, for example. \[thm.lorentz-inter\] Suppose $T$ is a subadditive operator of (restricted) weak types $(r_j,p_j)$, $j=0,1$, with $r_0<r_1$ and $p_0\neq p_1$, then there exists a constant $B=B_\theta$ such that $\|T\|^*_{p,q}\leq B\|f\|^*_{r,q}$ for all $f$ belonging to the domain of $T$ and to $L^{r,q}$, where $1\leq q\leq\infty$, $$\label{eqn.interpolate-indices} \frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1},\quad \frac{1}{r}=\frac{1-\theta}{r_0}+\frac{\theta}{r_1}\quad\text{and}\quad 0<\theta<1.$$ \[cor.HY-Lorentz\] If $f\in L^p({\mathbb{R}}^n)$, $1<p\leq 2$, then its Fourier transform ${\hat{f}}$ belongs to $L^{p',p}({\mathbb{R}}^n)$ and there exists a constant $B=B_p$ independent of $f$ such that $\|{\hat{f}}\|_{p',p}^*\leq B_p\|f\|_p$, where $\frac{1}{p}+\frac{1}{p'}=1$. In particular the Fourier transform is a continuous linear mapping from $L^p({\mathbb{R}}^n)$ to the Lorentz space $L^{p',p}({\mathbb{R}}^n)$ for $1<p<2$. Taking $(r_0,p_0)=(1,\infty)$, $(r_1,p_1)=(2,2)$ in theorem \[thm.lorentz-inter\], the conditions in translate into $\frac{1}{p}=\frac{\theta}{2}$ and $\frac{1}{r}=1-\frac{\theta}{2}$, that is, $r=p'$. Furthermore take $q=r$. Since $\theta\in(0,1)$ in the hypothesis of theorem \[eqn.interpolate-indices\], the role of $p$ and $p'$ must be exchanged when we consider the setup in the present corollary. (Since $\frac{2}{p}=\theta\in(0,1)$ if and only if $p>2$). With this adjustment in mind, the conclusion to theorem \[eqn.interpolate-indices\] becomes $\|{\hat{f}}\|^*_{p',p}\leq B\|f\|^*_{p,p}=B\|f\|_p$. As in the proof of corollary \[cor.HY-Lorentz\], we obtain the following extension immediately from the interpolation theorem \[thm.lorentz-inter\]. The Heckman–Opdam transform is a continuous mapping from $L^p(A,d\mu)^W$ to $L^{p',p}(i{\mathfrak{a}}^*,d\nu)^W$ whenever $1<p<2$. The preceding two corollaries are stronger than their respective standard forms since $L^{p^\prime,p}$ is continuously and properly embedded in $L^{p^\prime}$. The last result on Lorentz spaces that we will need is due to R. O’Neil, [@Oneil-convolution], and concerns the pointwise product of two functions. \[thm.Oneil\] Let $q\in(2,\infty)$ and set $r=\frac{q}{q-2}$. For $g\in L^q(X)$ and $h\in L^{r,\infty}(X)$ it holds that $gh$ belongs to $L^{q',q}(X)$ with $\|gh\|^*_{q',q}\leq\|g\|_q\|h\|_{r,\infty}^*$. The Hardy–Littlewood inequalities {#sec.HY-ineqs} ================================= The first part of the present section generalizes the inequality . We have decided to treat the rank one case separately as an illustrative example of the interpolation arguments that will be used throughout the section. Assume $\mathrm{dim}\,{\mathfrak{a}}=1$, $m_\alpha+m_{2\alpha}\geq 1$, and define $Tf(\lambda)=|\lambda|^2\mathcal{F}f(\lambda)$. Since $-m_\alpha-m_{2\alpha}+1\leq 0$, it follows that $(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}\leq 1$ for all $\lambda\in i{\mathfrak{a}}^*$. Moreover, $$\begin{split} \|Tf\|_2^2 &=\int_{i{\mathfrak{a}}^*}|Tf(\lambda)|^2|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ &= |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ &\leq |W|^{-1}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda = \|f\|_2^2, \end{split}$$ so $T$ is of strong type $(2,2)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W=L^2(i{\mathfrak{a}}^*,|W|^{-1}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}d\nu(\lambda))^W$. This is no longer true when $m_\alpha+m_{2\alpha}<1$, in which case one would have to employ a different type of weight and/or modify the measure $d\overline{\nu}$. Note that $|\mathcal{F}f(\lambda)|\leq C\|f\|_1$ for all $\lambda\in i{\mathfrak{a}}^*$ and $f\in L^1(A,d\mu)$.. For $t>0$ and $0\neq f\in L^1(A,d\mu)^W$, define $$E_t(f)=\{\lambda\,:\,|Tf(\lambda)|>t\},\quad A_t(f)=\Bigl\{\lambda\,:\,|\lambda|>\Bigl(\frac{t}{C\|f\|_1}\Bigr)^{1/2}\Bigr\},\quad\text{and}\quad a_t =\Bigl(\frac{t}{C\|f\|_1}\Bigr)^{1/2}.$$ It follows from the definition of $T$ that $E_t(f)\subset A_t(f)$ for all $t>0$, hence $|E_t(f)|\leq|A_t(f)|$, where $$\begin{split} |A_t(f)| &= \frac{1}{|W|}\int_{A_t(f)} (1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\\ &\asymp \frac{1}{|W|}\int_{A_t(f)} (1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}|\lambda|^2(1+|\lambda|)^{m_\alpha+m_{2\alpha}-2}\,d\lambda\\ &=\frac{1}{|W|}\int_{A_t(f)}|\lambda|^{-2}(1+|\lambda|)^{-1}\,d\lambda=\frac{1}{|W|}\int_{a_t}^\infty |\lambda|^{-2}(1+|\lambda|)^{-1}\,d\lambda\\ &\approx \frac{1}{|W|}\int_{a_t}^\infty\frac{d\lambda}{\lambda^3} = Ca_t^{-2}=C'\frac{\|f\|_1}{t} \end{split}$$ This shows that $T$ is also of weak type $(1,1)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\overline{\nu})^W$. It follows from the Marcinkiewicz interpolation theorem that $T$ is of strong type $(p,p)$, for $p\in(1,2)$, that is $$\label{eqn.inter-ineq} \frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|Tf(\lambda)|^p(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}|\lambda|^{-4}\,d\nu(\lambda)\leq C\|f\|_p^p.$$ Since $1+|\lambda|\geq|\lambda|^{2-p}$ for all $\lambda$, it holds that $(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})+1}\geq|\lambda|^{2-p}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})}$, which allows us to rewrite as $$\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^p|\lambda|^{p-2}(1+|\lambda|)^{-(m_\alpha+m_{2\alpha})}\,d\nu(\lambda)\leq C\|f\|_p^p.$$ Consider the more natural weighted measure $d\widetilde{\nu}(\lambda)=|W|^{-1}|\lambda|^{-4}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda$. The operator $T$ is then of strong type $(2,2)$ as an operator from $L^2(A,d\mu)^W$ into $L^2(i{\mathfrak{a}}^*,d\widetilde{\nu})^W$ *for all* $m_\alpha,m_{2\alpha}$ but it is no clear if $T$ is of weak type $(1,1)$. A good choice of weights is therefore essential. It is also possible to consider weighted measures of the form $d{\hat{\mu}}(\lambda)=\psi(\lambda)|{\mathbf{c}}(\lambda)|^{-2}d\lambda$ where $$\psi(\lambda)=\begin{cases}\psi_1(\lambda)&\text{for }\|\lambda\|\leq 1\\ \psi_2(\lambda)&\text{for } \|\lambda\|>1\end{cases}$$ for suitable choices of $\psi_1,\psi_2$, but this leads to so much freedom that one should no longer speak of Hausdorff–Young inequalities. It would, however, allow one to treat the case $0\leq m_\alpha+m_{2\alpha}<1$ as well. The next result may be seen as a weighted Hardy–Littlewood inequality, from which a natural analogue of the Hardy–Littlewood inequality in [@Mohanty-II] will follow. It is important to allow a certain freedom in the choice of weights, since it would otherwise be difficult to ‘guess’ the correct formulation in higher rank. The parameter constraints arise from having to be able to find an elementary proof of the required weak type $(1,1)$ estimate. Recall that $\beta=\sum_{\alpha\in\Sigma_0^+}(m_\alpha+m_{2\alpha})$, where $m:\Sigma\to[0,\infty)$ is a non-negative multiplicity function, and $m_\alpha:=m(\alpha)$. \[prop.abstract-interpolate\] Assume $\beta+n>0$ and $1<p<2$, and consider the operator $T$ defined on $L^2(X,d\mu)$, $X=A/W$, by $Tf(\lambda)=\|\lambda\|^{k+n}\mathcal{F}f(\lambda)$, where $k\geq 0$. Moreover let $$(Y,d\overline{\nu})=\Bigl(i{\mathfrak{a}}^*, \frac{1}{\vert W\vert}\|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)$$ where the parameters $k,a,b$ satisfy the conditions 1. $a+b\leq \frac{2}{3}(n-\beta)$ 2. $a+b+\beta+n=-(k+n)$ Then $T$ is of strong type $(2,2)$ and weak type $(1,1)$ as an operator from $L^2(X,d\mu)$ into $L^2(Y,d\overline{\nu})$, and therefore of strong type $(p,p)$. More precisely, there exists a positive constant $C_p$ independent of $f$ such that $$\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}\vert\mathcal{F}f(\lambda)\vert^p \|\lambda\|^{(k+n)p+a}(1+\|\lambda\|)^b\,d\nu(\lambda)\leq C_p\|f\|_p^p$$ for every $f\in L^p(A,d\mu)^W$. Consider the measure spaces $(X,d\mu)=(A/W,d\mu)$ and $$(Y,d\overline{\nu})=\Bigl(i{\mathfrak{a}}^*, \frac{1}{\vert W\vert}\|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr).$$ Let $Tf(\lambda)=\|\lambda\|^{k+n}\mathcal{F}f(\lambda)$, where $k$ is to be determined shortly. As in the rank one calculation that preceded the present theorem, the crux of the proof will be to verify that $T$ is strong type $(2,2)$ and weak type $(1,1)$ as an operator from $(X,d\mu)$ into $(Y,d\overline{\nu})$. As for the first property, it follows from Plancherel’s theorem that $$\|Tf\|_2^2 =\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^2\|\lambda\|^{2(k+n)}\|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda \simeq \|f\|_2^2$$ *provided* $\psi(\lambda)=\|\lambda\|^{2(k+n)}\|\lambda\|^a(1+\|\lambda\|)^b\asymp 1$. This holds whenever $2(k+n)+a+b\leq 0$ (the factor $\psi$ stays bounded for $\|\lambda\|\gg 1$) and at the same time $2(k+n)+a\geq 0$ (so that $\psi$ is bounded near $\lambda=0$). For $t>0$ and $f\in L^1\cap L^2$, $\|f\|_1\neq 0$, consider the sets $E_t(f)=\{\lambda\,:\,\vert Tf(\lambda)\vert>t\}$ and $A_t(f)=\{\lambda\,:\,\|\lambda\|>(\frac{t}{C\|f\|_1})^{\frac{1}{k+n}}\}$, where $C$ is the constant coming from the estimate $\vert\mathcal{F}f(\lambda)|\leq C\|f\|_1$. By definition of $T$ it follows that $E_t(f)\subset A_t(f)$, whereby $$\begin{split} |E_t(f)|& \leq |A_t(f)| =\frac{1}{|W|}\int_{A_t(f)} \|\lambda\|^a(1+\|\lambda\|)^b|{\mathbf{c}}(\lambda)\|^{-2}\,d\lambda\\ &\simeq \int_{A_t(f)} \|\lambda\|^{a+2\vert\Sigma_0^+\vert}(1+\|\lambda\|)^{b+\beta-2\vert\Sigma_0^+\vert}\,d\lambda\simeq \int_{A_t(f)}\|\lambda\|^{a+b+\beta}\,d\lambda \\ &= C'\int_{a_t}^\infty s^{a+b+\beta}s^{n-1}\,ds=C'\Bigl(\Bigl(\frac{t}{C\|f\|_1}\Bigr)^{\frac{1}{k+n}}\Bigr)^{a+b+\beta+n} = C''\frac{\|f\|_1}{t} \end{split}$$ since $a+b+\beta+n=-(k+n)$ by construction[^1]. The extension to the case $p>2$ utilizes the stronger interpolation result theorem \[thm.lorentz-inter\] and is motivated by the rank one statement in [@Mohanty-II Theorem 4.5]. A similar argument leads to a generalization of [@Anker-besov Lemma §4.1] but we leave it to the interested reader to write down the details. A *Young function* is a measurable function $\psi:{\mathfrak{a}}_+\to{\mathbb{R}}$ with the property that $\mu(\{x\in{\mathfrak{a}}\,:\,\vert\psi(x)\vert\leq t\})\lesssim t$ for all $t>0$. \[example.young\] In ${\mathbb{R}}^n$, the function $\psi(x)=\|x\|^m$ is a Young function in ${\mathbb{R}}^n$ if and only if $m=n$, since $|\{x\in{\mathbb{R}}^n\,:\,\|x\|^m<t\}|=|B(0,t^{1/m})|=Ct^{n/m}$. Since norms on ${\mathbb{R}}^n$ are equivalent, the $n$.th power of *any* norm on ${\mathbb{R}}^n$ gives rise to a Young function. It is easy to construct Young functions associated with $({\mathfrak{a}},\Sigma,m)$. An infinite family of examples is given by $\psi(x)=h(x)J(x)$, where $h(x)=h_0(\|x\|)$ is radial and satisfies $\int_0^\infty \frac{s^n-1}{h_0(s)}ds<\infty$. In this case, $$\label{ineq.sublevel} \mu(\{x\,:\,\vert\psi(x)\vert\leq t\})\leq t\int_{{\mathfrak{a}}}\frac{dx}{h(x)} =Ct\int_0^\infty\frac{s^{n-1}}{h_0(s)}\,ds = C't$$ for every $t>0$. For many purposes the estimate in is too crude, however. The norm power $\|\cdot\|^n$ would not meet the requirement that $\int_0^\infty s^{n-1}/s\,ds$ be finite, for example, so the estimate is only sensible when the measure of the sublevel sets $\{x\,:\,|\psi(x)|<t\}$ cannot be estimated directly. One such example is the following. The *hyperbolic* Young function associated with $({\mathfrak{a}},\Sigma,m)$ is the function $\psi_h(x)=\cosh(\|x\|)J(x)$. This is a root system analogue of the Young function considered in [@Mohanty-II Lemma 4.4]. Let $\psi$ be a Young function for $({\mathfrak{a}},\Sigma,m)$ and $q>2$. The space $L^{(q)}_\psi({\mathfrak{a}}_+)$ consists of all measurable $W$-invariant functions $f:{\mathfrak{a}}\to{\mathbb{C}}$ such that $$\|f\|_{(q),\psi}:=\Bigl(\int_{{\mathfrak{a}}_+}\vert f(x)\vert^q\psi(x)^{q-2}\,d\mu(x)\Bigr)^{1/q}<\infty.$$ In other words, $f$ belongs to $L^{(q)}_\psi$ if and only if $f\cdot\psi^{1-\frac{2}{q}}$ belongs to $L^q$. The next result is a generalization of theorem \[thm.1\] in the introduction. \[thm.HY-qlarge\] Let $q>2$ and $f\in L^{(q)}_{\psi_h}({\mathfrak{a}}_+)$. There exists a positive constant $D_q$ independent of $f$ such that $$\frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^q\,d\nu(\lambda)\leq D_q^q\|f\|^q_{(q),\psi}.$$ Let $f$ be a simple function on $A$ and let $Tf(\lambda)=\mathcal{F}f(\lambda)$ (we do not need to add weights to the operator that enters the interpolation argument). Then $\|Tf\|^*_{\infty,\infty}=\|Tf\|_\infty\leq C\|f\|_1=\|f\|^*_{1,1}$, and by the Plancherel theorem it furthermore holds that $\|Tf\|_{2,\infty}^*\leq\|Tf\|^*_{2,2}\leq\|f\|_2\leq\|f\|^*_{2,1}$. By interpolation (cf. theorem \[thm.lorentz-inter\]) it follows that $\|Tf\|^*_{q,q}\leq \|f\|^*_{q',q}$. Now define $g(x)=f(x)\psi_h(x)^{1-\frac{2}{q}}$, where $\psi_h(x)=\cosh(\|x\|)J(x)$. Then $g$ belongs to $L^q(A,d\mu)^W$ by hypothesis, since $$\|g\|^q_q=\int_A\vert f(x)\vert^q\vert\psi_h(x)\vert^{q-2}\,d\mu(x) = \|f\|^q_{(q),\psi_h}$$ It follows from the sublevel set estimate implied by $\psi_h$ being a Young function for $({\mathfrak{a}},\Sigma,m)$ that $$\mu\bigl(\{x\,:\,\vert\psi_h(x)\vert^{\frac{2}{q}-1}>t\}\bigr) = \mu\Bigl(\Bigl\{x\,:\,\vert\psi_h(x)\vert^{1-\frac{2}{q}}<\frac{1}{t}\Bigr\}\Bigr)\leq Ct^{-\frac{q}{q-2}},$$ whence $\psi^{\frac{2}{q}-1}$ belongs to $L^{r,\infty}(A,d\mu)$, where $r=\frac{q}{q-2}$. By an application of O’Neil’s theorem \[thm.Oneil\] it is seen that $$\begin{split} \frac{1}{|W|}\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda)|^q\,d\nu(\lambda)&\leq \|f\|^*_{p,q}\leq \|g\|_q\|\psi_h\|^*_{t,\infty}\\ & \leq C\int_A|f(x)|^q|\psi_h(x)|^{q-2}J(x)\,dx = c\|f\|^p_{(q),\psi_h} \end{split}$$ which was the desired conclusion for simple functions. The extension to general functions in $L^{(q)}_{\psi_h}({\mathfrak{a}}_+)$ now follows by standard density arguments. As briefly mentioned in the introduction, Ray and Sarkar were able to obtain a different version of the Hardy–Littlewood inequality for the Helgason–Fourier transform. They might have been motivated by the complex version of the Hausdorff–Young inequality, cf. . where the transform is extended holomorphically into a certain domain in the complex plane. At the same time Ray and Sarkar used slightly different weights in their interpolation argument, the result being the following theorem, cf. [@Ray-Sarkar-trans Theorem 4.11] (in their notation $S$ denotes a Damek–Ricci space, but the reader may replace it with a hyperbolic space). \[thm.Ray-Sarkar\] 1. Let $1<q\leq 2$ be fixed. Then for $f\in L^p(S)$, $1<p\leq q$, $$\Bigl(\int_{\mathbb{R}}\|\widetilde{f}(\lambda\pm i\gamma_q\rho,\cdot)\|_{L^q(N)}^r (|\lambda||{\mathbf{c}}(\lambda)|^{-2})^{r/p'-1}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)^{1/r}\leq C_{\pm,p,q}\|f\|_p$$ where $\frac{1}{r}=1-\frac{q'-1}{p'}$. 2. Let $2\leq q<\infty$ be fixed. Then for $f\in L^{(p)}(S)$ with $q\leq p<\infty$, $$\Bigl(\int_{\mathbb{R}}\|\widetilde{f}(\lambda\pm i\gamma_{q'}\rho,\cdot)\|^p_{L^{q'}(N)}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\Bigr)^{1/p}\leq C_{\pm,p,q}\|f\|_{(p)}.$$ Here $\|f\|_{(p)}=\Bigl(\int_S|f(x)|^pJ(x)^{p-2}\,dx\Bigr)^{1/p}$, where $J(x)=(\sinh\frac{r(x)}{2})^m(\sinh r(x))^l$ is essentially the Jacobian associated with polar coordinates in $S$. There are subtle technical issues pertaining to the proper domain of definition of the Helgason–Fourier transform that make the proof of theorem \[thm.Ray-Sarkar\] more involved than what might be expected, but the strategy of proof is still to use interpolation. Indeed the main task is again to identify two suitable measure spaces and a sublinear operator in such a way that the abstract interpolation machinery produces the desired inequality. In statement (i), one considers measure spaces $(S,dx)$ and $({\mathbb{R}}^\times,d\overline{\mu}(\lambda))$, where $d\overline{\mu}(\lambda)=|\lambda|^{-q}|{\mathbf{c}}(\lambda)|^{-2(1-q)}\,d\lambda$, and a sublinear operator $T$ defined for $f\in L^1(S)+L^q(S)$ by $Tf(\lambda)=\|\widetilde{f}(\lambda+i\gamma_q\rho,\cdot)\|_{L^q(N)}(|\lambda||{\mathbf{c}}(\lambda)|^{-2})^{q/q'}$. It can be verified that $T$ is of strong type $(q,q')$ (which follows from a Hausdorff–Young inequality and the Plancherel theorem) and weak type $(1,1)$ (which requires more work). An interpolation argument yields the conclusion in (i). In (ii), it is convenient to consider measure spaces $(S,dx)$ and $({\mathbb{R}},|{\mathbf{c}}(\lambda)|^{-2}d\lambda)$ and a sublinear operator $T$ defined for $f\in L^1(S)+L^{p',1}(S)$ by $Tf(\lambda)=\|\widetilde{f}(\lambda+i\gamma_{q'}\rho,\cdot)\|_{L^{q'}(N)}$. While we shall not give the details of their interpolation argument, it also uses interpolation between Lorentz spaces, showing that for $q\leq<\infty$ and $1\leq s\leq\infty$, it holds that $\|Tf\|_{p,s}^*\leq C_{p,q}\|f\|^*_{p',s}$, cf. [@Ray-Sarkar-trans Eqn. (4.27)]. An important difference is that Ray and Sarkar use the function $J$ as Young function, which also dictates their definition of $\|f\|_{(p)}$. Let $u=\frac{p}{p-2}$ and $g(x)=f(x)J(x)^{1/u}$. Then $g\in L^p(S)$ and $\|g\|_p=\|f\|_{(p)}$. Moreover $m(\{x\in S\|:\| J(x)\leq t\})\leq Ct$ for all $t>0$, for some constant $C$, where $m$ is Haar measure on $S$ (so $J$ is indeed a Young function in our terminology). Consequently, $J(x)^{-1/u}\in L^{u,\infty}(S)$. It follows by Hölder’s inequality that $\|f\|^*_{p',p}\leq C_p\|g\|_p\|J^{-1/u}\|_{u,\infty}$, and therefore (taking $s=p$) $\|Tf\|_{p,p}^*\leq C_{p,q}\|g\|_p$, from which (ii) follows. In order to generalize theorem \[thm.Ray-Sarkar\], we must therefore choose suitable measure spaces, define convenient sublinear operators $T$ and finally choose a good Young function. \[lemma-J\] Let $({\mathfrak{a}},\Sigma,m)$ be a fixed root datum. The function $$\psi(x)=J(x)=\prod_{\alpha\in\Sigma^+}|e^{\alpha(x)}-e^{-\alpha(x)}|^{m_\alpha}$$ is a Young function. Since $|e^{\alpha(x)}-e^{-\alpha(x)}|\leq 2e^{\|\alpha\|\|x\|}$ for every $\alpha\in\Sigma^+$, $x\in{\mathfrak{a}}$, it follows that $|J(x)|\leq C e^{2\|\rho\|\|x\|}$ for all $x\in{\mathfrak{a}}$, where $\rho=\frac{1}{2}\sum_{\alpha\in\Sigma^+}m_\alpha\alpha$. Since $\int_0^\infty s^{n-1}e^{-2\|\rho\|s}ds<\infty$ for all $n\in{\mathbb{N}}$, it follows as in the discussion succeeding example \[example.young\] that $\psi$ is a Young function. The following result is a direct analogue of [@Ray-Sarkar-trans Theorem 4.11], and our proof follows theirs closely. The main addition is that we once again have to choose the underlying measure spaces and the operator $T$ in such a way that the weak type $(1,1)$ estimate – now with respect to a weighted measure in the $n$-dimensional vector space $i{\mathfrak{a}}^*$. As in the proof of theorem \[thm.HY-qlarge\] this is essentially achieved by working our way backwards from the desired weak type $(1,1)$ estimate, modifying $T$ accordingly. We shall not work with an arbitrary Young function but rather the choice in lemma \[lemma-J\]. Accordingly $L^{(q)}({\mathfrak{a}})$, $q>2$ denotes the space of measurable $W$-invariant functions $f:{\mathfrak{a}}\to{\mathbb{C}}$ for which $$\|f\|_{(q)}:=\Bigl(\int_{\mathfrak{a}}|f(x)|^qJ(x)^{q-2}d\mu(x)\Bigr)^{1/q}<\infty.$$ \[thm.HL-ver3\] 1. Let $1<q\leq 2$ be fixed. For $f\in L^p({\mathfrak{a}},d\mu)^W$ with $1<p\leq q$, $$\Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^r(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{r/p'-1}\,d\nu(\lambda)\Bigr)^{1/r}\leq C_{p,q,\eta}\|f\|_p$$ for every $\eta$ in the interior of $C(\epsilon_p\rho)$, where $\frac{1}{r}=1-\frac{q'-1}{p'}$. 2. Let $2\leq q<\infty$ be fixed. For $f\in L^{(p)}({\mathfrak{a}})$ with $q\leq p<\infty$, $$\Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^p\,d\nu(\lambda)\Bigr)^{1/p}\leq C_{p,q,\eta}\|f\|_{(p)}$$ for every $\eta$ in the interior of $C(\epsilon_p\rho)$. Note that the special case $p=q=r=2$ recovers the a special case of the Hausdorff–Young inequality in lemma \[HL-ineq2\] and the Plancherel formula as an inequality in the case $\eta=0$. Fix $q\in(1,2]$ and consider the measure spaces $({\mathfrak{a}}_+,d\mu)$ and $(i{\mathfrak{a}}^+,d\overline{\nu}(\lambda)$, where $d\overline{\nu}(\lambda)=\|\lambda\|^{-nq}|{\mathbf{c}}(\lambda)|^{-2(1-nq)}\,d\lambda$. Define $Tf$, $f\in L^p({\mathfrak{a}},d\mu)^W$, by $Tf(\lambda)=|\mathcal{F}f(\lambda+\eta)|(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{\frac{q\cdot n}{q'}}$. It then follows from lemma \[lemma.NPP-2\] that $$\|Tf\|_{q'}^{q'} =\int_{i{\mathfrak{a}}^*} |Tf(\lambda)|^{q'}\|\lambda\|^{-nq}|{\mathbf{c}}(\lambda)|^{-2(1-nq)}\,d\lambda= \int_{i{\mathfrak{a}}^*}|\mathcal{F}f(\lambda+\eta)|^{q'}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda \leq C_{p,q,\eta}\|f\|_q^{q'},$$ so $T$ is of strong type $(q,q')$. The operator $T$ is furthermore of weak type $(1,1)$, as we shall now prove. The argument is nearly the same as in the proof of theorem \[thm.HY-qlarge\] but relies on a clever trick employed in the proof of [@Ray-Sarkar-trans Theorem 4.11]. For $t>0$ define the set $E_t(f)=\{\lambda\in i{\mathfrak{a}}^*\,:\,|Tf(\lambda)>t\}$, that is, $$E_t(f)=\{\lambda\in i{\mathfrak{a}}^*\,:\,(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{nq/q'}|\mathcal{F}f(\lambda+\eta)|>t\}.$$ According to lemma \[lemma.NPP-2\](b), $E_t(f)$ is contained in the set $$\begin{split} A_t(f)&:=\{\lambda\in i{\mathfrak{a}}^*\,:\, c_p(\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{nq/q'}\|f\|_1>t\}\\ &=\{\lambda\in i{\mathfrak{a}}^*\,:\, \|\lambda\||{\mathbf{c}}(\lambda)|^{-2}>a_t\},\quad a_t=\Bigl(\frac{t}{c_p\|f\|_1}\Bigr)^{\frac{q'}{nq}}, \end{split}$$ We can now invoke the trick of Ray and Sarkar: Noting that $$\|\lambda\||{\mathbf{c}}(\lambda)|^{-2}\asymp \|\lambda\|^{2\vert\Sigma_0^+|+1}(1+\|\lambda\|)^{\beta-2|\Sigma_0^+|}=G(\|\lambda\|),$$ where $G(s):=s^{2|\Sigma_0^+|+1}(1+s)^{\beta-2|\Sigma_0^+|}$, it it seen that for $s\geq 0$, $G'(\|\lambda\|)\asymp|{\mathbf{c}}(\lambda)|^{-2}$. It follows that $$\begin{split} |A_t(f)|&=\int_{i{\mathfrak{a}}^*}\mathbf{1}_{A_t(f)}(\lambda) (\|\lambda\||{\mathbf{c}}(\lambda)|^{-2})^{-nq}|{\mathbf{c}}(\lambda)|^{-2}\,d\lambda\lesssim \int_{i{\mathfrak{a}}^*} G(\|\lambda\|)^{-q}G'(\|\lambda\|)\,d\lambda\\ &=C\int_{a_t}^\infty s^{-nq}s^{n-1}\,ds\quad\text{ by passage to polar coordinates in }{\mathfrak{a}}\\ &=C'a_t^{n-nq} = C''\Bigl(\frac{t}{\|f\|_1}\Bigr)^{\frac{q'}{nq}(n-nq)}= C'''\frac{\|f\|_1}{t} \end{split}$$ This proves that $T$ is of weak type $(1,1)$. In particular, it follows by interpolation that $T$ is of strong type $(p,r)$ whenever $p$ satisfies the identity $\frac{1}{p}=\frac{1-t}{1}+\frac{t}{q}$ for some $t\in(0,1)$. With $p$ being given in the hypothesis of (i), this identity holds precisely when $\frac{1}{r}=1-t+\frac{t}{q'}=\frac{p'q-q'}{p'q}$, which establishes the asserted inequality in (i). Now consider the measure spaces $({\mathfrak{a}}/W,d\mu)$ and $(i{\mathfrak{a}}^*/W,d\nu(\lambda))$, together with the operator $Tf(\lambda)=|\mathcal{F}f(\lambda+\eta)|$, where $\eta\in C(\epsilon_p\rho)^\circ$ is fixed. As in the proof of [@Ray-Sarkar-trans Theorem 4.11] we shall use lemma \[lemma.NPP-2\] and $J(x)$ as Young function in a double interpolation argument as follows. Fix a function $f\in L^{(p)}({\mathfrak{a}})$ where $p\geq q\geq 2$ and observe that $$\label{eqn.interpolate1} \|Tf\|^+_{\infty,\infty}\leq\|f\|^*_{1,1}.$$ Indeed, it holds more generally that $\|Tf\|^*_{\infty,\infty}=\|Tf\|_{\infty}\leq\|f\|_1=\|f\|_{1,1}^*$, which is clear for $\eta=0$ (where it amounts to the Riemann–Lebesgue lemma) and follows in general from the estimate $|\varphi_\lambda(x)|\leq 1$ for $x\in{\mathfrak{a}}$ and $\eta\in C(\epsilon_p\rho)\subset C(\epsilon_1\rho)=C(\rho)$. In addition $$\label{eqn.intermediate} \|Tf\|_q\leq C\|f\|_{q'}\quad\text{ for } q\geq 2$$ which follows from the Plancherel theorem in the case $q=2$ (in which case $\eta\in C(\epsilon_2\rho)=\{0\}$) and for $q>2$ from lemma \[lemma.NPP-2\](a) (since in this case $q'<2$). Consequently $$\|Tf\|^*_{q,\infty}\leq\|Tf\|^*_{q,q}=\|Tf\|_q\leq C_q\|f\|_{q^\prime}\leq C_q\|f\|^*_{q^\prime,1},$$ that is $$\label{eqn.interpolate2} \|Tf\|^*_{q,\infty}\leq C_q\|f\|^*_{q^\prime,1}.$$ It follows from the interpolation theorem \[thm.lorentz-inter\] that $$\label{eqn.interpolate3} \|Tf\|^*_{p,s}\leq C_{p,q}\|f\|^*_{p^\prime,s}$$ for $q\leq p\leq \infty$ and $1\leq s\leq \infty$. Note that the function $g$ defined by $g(x)=f(x)J(x)^\frac{p-2}{2}$ belongs to $L^p({\mathfrak{a}},d\mu)^W$ with norm $\|g\|_{p}=\|f\|_{(p)}$. Since $J$ is a Young function according to lemma \[lemma-J\], it is seen that $\mu(\{x\in{\mathfrak{a}}\,:\,J(x)^{-\frac{p-2}{p}}>t\})\lesssim t^{-\frac{p}{p-2}}$, whence $J^{-\frac{p-2}{p}}$ belongs to the Lorentz space $L^{\frac{p}{p-2},\infty}({\mathfrak{a}},d\mu)^W$. By Hölder’s inequality for Lorentz spaces, $\|f\|^*_{p,p^\prime}\leq C_p\|g\|_p\|J^{-\frac{p-2}{p}}\|_{\frac{p}{p-2},\infty}$. Combined with in the special case $s=p$, we conclude that $$\|Tf\|^*_{p,p}\leq C_{p,q}\|g\|_p = C_{p,q}\Bigl(\int_{\mathfrak{a}}|f(x)|^pJ(x)^{p-2}\,d\mu(x)\Bigr)^{1/p}$$ which completes the proof of (ii). A remark on the flat analogues {#section.flat} ============================== Eguchi and Kumahara also obtained a Hardy–Littlewood inequality for the ‘flat’ spherical transform. Consider the Cartan motion group $G_0=\mathfrak{p}\rtimes K$ and identify the flat Riemannian symmetric space $G_0/K$ with ${\mathfrak{p}}$, which is isomorphic to the tangent space of $G/K$ at the origin $eK$. Spherical analysis on $G_0/K$ was developed by Helgason in [@Helgason-dualityIII], by means of which the Hardy–Littlewood inequality may be stated as follows. For $q\geq 2$ there exists a positive constant $A_{0,q}$ such that $$\Bigl(\frac{1}{|W}\int_{{\mathfrak{a}}^*}|\widetilde{f}(v)|^qJ(v)\,dv\Bigr)^{1/q}\leq A_{0,q}\Bigl(\int_{G_0}|f(x)|^q\sigma_0(x)^{n(q-2)}\Omega_0(x)^{q-2}\,dx\Bigr)^{1/q}$$ for every $f\in C_c(K\setminus G_0/K)$. Here $\widetilde{f}$ is the generalized Bessel transform (the ’flat’ spherical transform) of $f$, $\sigma_0(x)=\|X\|$ for $x=(X,k)\in G_0$, and $\Omega_0(x)=|W|^{-1}(2\pi)^{n/2}\mathrm{vol}(K/M)J(H)$ for $x=k(H,1)k'$, and $J(H)=\prod_{\alpha\in\Sigma}|\alpha(H)|^{m_\alpha}$. Notice the formal similarity with . We should like to mention a natural extension of these results to the present root system framework. Contrary to the case of the symmetric space $G/K$ being contracted to the flat space $G_0/K$, the ground space ${\mathfrak{a}}$ remains the same. Instead Ben Saïd and Ørsted [@BSO-Bessel], and de Jeu [@deJeu-PW], consider a limit transition of the the hypergeometric functions $\varphi_\lambda$, namely the functions $\psi(x)=\lim_{\epsilon\to 0} F_{\lambda/\epsilon}(\epsilon x)$. In the case of rank one symmetric spaces, the function $\psi$ *is* indeed a Bessel function, which is explained as follows. We already know that $$\varphi_\lambda(t)={_2}F_1\Bigl(\frac{i\lambda+\rho}{2},\frac{-i\lambda+\rho}{2};\frac{m_\alpha+m_{2\alpha}+1}{2};-\sinh^2t\Bigr).$$ It can be proved on the basis of the asymptotic estimate $$\frac{\Gamma(z+a)}{\Gamma(z+b)} = z^{a-b}\Bigl(1+\frac{(a-b)(a+b-1)}{2z}+O(z^{-2})\Bigr)\text{ as } z\to\infty$$ that $$\psi(\lambda,t)=\Gamma\Bigl(\frac{m_\alpha+m_{2\alpha}+1}{2}\Bigr)\Bigl(\frac{\lambda t}{2}\Bigr)^{-\frac{m_\alpha+m_{2\alpha}-1}{2}}J_{\frac{m_\alpha+m_{2\alpha}-1}{2}}(\lambda t),$$ where $J_\nu$ is the standard Bessel function of the first kind. It is therefore seen that the generalized Bessel functions $\psi$ on the flat symmetric space $G_0/K$ *are* the spherical Bessel functions in the rank one case. At least in the case of $\operatorname{SO}_e(1,n)/\operatorname{SO}(n)$, the associated integral transform is known as the Fourier–Bessel transform (or Hankel transform) in the literature. The ‘flat’ Heckman–Opdam transform $\mathcal{F}_0$ is therefore defined as the integral transform arising by integrating a suitable $W$-invariant function on ${\mathfrak{a}}$ against the generalized Bessel function $\psi$. The details can be found in [@BSO-Bessel] and [@deJeu-PW Section 4] but the main point is that the generalized Bessel transform of Ben Saïd and Ørsted coincides with the symmetric Dunkl transform on ${\mathbb{R}}^n$, cf. [@deJeu-PW Theorem 4.15] (it is seen by inspecting the proof of [@BSO-Bessel Theorem 3.15] that the measures involved in the respective integral transforms coincide as well). It is interesting to note that one can also ‘contract’ the Helgason–Fourier transform on $G/K$ to an integral transform on $G_0/K$. Helgason introduced in [@Helgason-flat-horocycle] the so-called *flat horocycle transform* as follows. Let $G/K$ be a Riemannian symmetric space of dimension $n$ and rank $\ell$, and let $X_0=G_0/K\simeq \mathfrak{p}$ denote the tangent space to $G/K$ at the origin $eK$. Let $\Xi_0$ denote the $n$-dimensional manifold consisting of $(n-\ell)$-dimensional affine hyperplanes in $X_0$. The flat horocycle transform is a map that assigns to a function $f$ on $X_0$ the function $\widetilde{f}$ on $\Xi_0$ that is defined by $\widetilde{f}(\xi)=\int_\xi f(Y)\,dm(Y)$, where $dm(Y)$ is the standard Euclidean measure on $\xi$. Its analogue on $G/K$ is then the Helgason–Fourier transform, and it seems likely that the results from [@Mohanty-II Section 4] have natural analogues for the transform $f\mapsto \widetilde{f}$ acting on functions living on ${\mathfrak{p}}$. It is an advantage of the approach by Ben Saïd and Ørsted that one also obtains that the relevant measures $d\mu_{0}$ and $d\nu_{0}$ for a Plancherel theorem for the flat transform by means of the limit procedure both coincide with the measure $\omega_m(x)dx$, where $\omega_m(x)=\prod_{\alpha\in\Sigma^+}|\langle\alpha,x\rangle |^{m_\alpha}$ is the standard Plancherel weight for the Dunkl transform $\mathcal{T}_m$. Since the flat Heckman–Opdam transform $\mathcal{F}_0$ is the symmetrized Dunkl transform, the the following Hardy–Littlewood inequality follows from [@Anker-besov Lemma 4.1]. If $f\in L^p(A,d\mu_{0})^W$ for some $p\in (1,2)$, then $$\Bigl(\int_{i{\mathfrak{a}}^*}\|\lambda\|^{2(\rho+\frac{d}{2})(p-2)}\|\mathcal{F}_{0}f(\lambda)|^{p}\,d\nu_{0}(\lambda)\Bigr)^{1/p}\leq C_{0,q}\Bigl(\int_{\mathfrak{a}}|f(x)|^p\,d\mu_{0}(x)\Bigr)^{1/p}.$$ Although the analogue of the strengthened Hausdorff–Young lemma \[HL-ineq2\] is false for the Dunkl transform except for $\eta=0$, one can still use the interpolation techniques in the proof of theorem \[thm.HL-ver3\] to prove the following result, which resembles theorem \[thm.HY-qlarge\] and is new for the Dunkl transform. Let $L^{(p)}({\mathbb{R}}^n,\omega_m)$ denote the space of measurable functions $f$ on ${\mathbb{R}}^n$ for which $$\|f\|_{m,(p)}:=\Bigl(\int_{{\mathbb{R}}^n}|f(x)|^p\omega_{m}(x)^{p-2}\omega_m(x)\,dx\Bigr)^{1/p}<\infty.$$ \[prop.Dunkl-RS\] 1. Let $1<q\leq 2$ be fixed. For $f\in L^p({\mathbb{R}}^n,\omega_m)$ with $1<p\leq q$, $$\Bigl(\int_{{\mathbb{R}}^n}|\mathcal{T}_mf(x)|^r(\|x\|\omega_m(x))^{r/p'-1}\omega_m(x)\,dx\Bigr)^{1/r}\leq C_{p,q}\|f\|_{L^p({\mathbb{R}}^n,\omega_m)}$$ where $\frac{1}{r}=1-\frac{q'-1}{p'}$. 2. Let $2\leq q<\infty$ be fixed. For $f\in L^{(p)}({\mathbb{R}}^n,\omega_m)$ with $q\leq p<\infty$, $$\Bigl(\int_{{\mathbb{R}}^n}|\mathcal{T}_mf(x)|^p\omega_k(x)\,dx\Bigr)^{1/p}\leq C_{p,q}\|f\|_{m,(p)}.$$ For the first part, with $q\in (1,2]$ fixed, consider the measure spaces $({\mathbb{R}}^n,d\mu_m)$ and $({\mathbb{R}}^n,d\overline{\mu}_m)$, where $d\mu_m(x)=\omega_m(x)\,dx$ and $d\overline{\mu}_m(x)=\|x\|^{-nq}\omega_k(x)^{1-nq}\,dx$. Define $Tf(x)=|\mathcal{T}_mf(x)|(\|x\|\omega_m(x))^{\frac{nq}{q'}}$. Then $T$ is of strong type $(q,q')$. The operator $T$ is also of weak type $(1,1)$ but the details are different. One uses that $\|x\|\omega_m(x)\asymp C\|x\|^{2\rho+1}$, instead of the polynomial estimates for $|{\mathbf{c}}(\lambda)|^{-2}$. For the second statement one uses that $\omega_m$ is a Young function on ${\mathbb{R}}^n$ with respect to the weighted measure $\omega_m(x)dx$. In particular, the same inequality holds for $\mathcal{F}_0$ acting on $L^p({\mathfrak{a}},d\mu_{0})^W$: 1. Let $1<q\leq 2$ be fixed. For $f\in L^p({\mathfrak{a}},d\mu_{0})^W$ with $1<p\leq q$, $$\Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}_0f(\lambda)|^r(\|\lambda\|\omega_m(\lambda))^{r/p'-1}d\nu_{0}(\lambda)\Bigr)^{1/r} \leq C_{p,q}\|f\|_p$$ where $\frac{1}{r}=1-\frac{q'-1}{p'}$. 2. Let $2\leq q<\infty$ be fixed. For $f\in L^{(p)}({\mathfrak{a}},d\mu_{0})^W$ with $q\leq p<\infty$, $$\Bigl(\int_{i{\mathfrak{a}}^*}|\mathcal{F}_0f(\lambda)|^p\omega_m(\lambda)\,d\lambda\Bigr)^{1/p}\leq C_{p,q}\|f\|_{0,(p)},$$ where $$\|f\|_{0,(p)}:=\Bigl(\int_{{\mathfrak{a}}}|f(x)|^p\omega_m(x)^{p-2}\,d\mu_{0}(x)\Bigr)^{1/p}.$$ We obtain a more familiar form of the Hausdorff–Young inequality in proposition \[prop.Dunkl-RS\] by choosing as Young function a power of the Euclidean norm instead of the density $\omega_m$, that is $\psi(x)=\|x\|^k$ for some $k\geq 0$. As in example \[example.young\], $\psi$ is a Young function for a unique choice of $k$. Since $$\mu_m(\{x\in{\mathbb{R}}^n\,:\,\psi(x)\leq t\}) =\mu_m(B(0,t^{1/k})) = \frac{c_m^{-1}}{2^{\gamma+\frac{n}{2}-1}\Gamma(\gamma+\frac{n}{2})}\int_0^{t^{1/k}} r^{2\rho+n-1}\,dr = Ct^{\frac{2\rho+n}{m}}$$ for all $t>0$, it follows that $\psi$ is a Young function if and only if $k=2\rho+n$, which agrees with example \[example.young\]. The space $L^{(p)}({\mathbb{R}}^n,\omega_m)$ now consists of all measurable functions $f:{\mathbb{R}}^n\to{\mathbb{C}}$ for which $$\|f\|_{m,(p)}:=\Bigl(\int_{{\mathbb{R}}^n}|f(x)|^p\|x\|^{p-2}\omega_m(x)\,dx\Bigr)^{1/p}<\infty.$$ For $f\in L^{(p)}({\mathbb{R}}^n,\omega_m)$ with $2\leq p<\infty$ it holds that $$\Bigl(\int_{{\mathbb{R}}^n}|\mathcal{T}_mf(x)|^p\omega_m(x)\,dx\Bigr)^{1/p}\leq C_p\Bigl(\int_{{\mathbb{R}}^n}|f(x)|^p\|x\|^{2(\rho+\frac{n}{2})(p-2)}\omega_m(x)\,dx\Bigr)^{1/p},$$ which is the ‘dual’ form of the Hardy–Littlewood inequality for the Dunkl transform obtained in [@Anker-besov Lemma 4.1]. A complete extension of and its dual form for the Dunkl transform $\mathcal{T}_m$ and the flat Heckman–Opdam transform $\mathcal{F}_0$ has thereby been obtained. The Hausdorff–Young and Hardy–Littlewood inequalities for $\mathcal{F}$ and $\mathcal{F}_0$ are formally all but identical. This begs the question: Is it possible to obtain the inequalities for $\mathcal{F}_0$ *directly* from the analogous inequalities for $\mathcal{F}$? To be more precise, the limit transition defining the generalized Bessel functions $\psi$ gives rise to a family of intermediate integral transforms $\mathcal{F}_\epsilon$ that interpolate between $\mathcal{F}$ and $\mathcal{F}_0$. One can establish, say, a Hausdorff–Young inequality for $\mathcal{F}_\epsilon$, $\epsilon\in(0,1]$ that formally interpolates between the Hausdorff–Young inequalities for $\mathcal{F}$ and $\mathcal{F}_0$, respectively, so it is tempting to let $\epsilon$ tend to zero in this $\epsilon$-parametrized Hausdorff–Young inequality and recover the inequality for $\mathcal{F}_0$. In turn this technique would allow one to ‘generate’ a host of new inequalities for the flat transform $\mathcal{F}_0$ from known results for $\mathcal{F}$. There are many technically sound versions of this heuristic principle in classical harmonic analysis, referred to as transference or restriction principles, but it is not yet clear if the techniques can be extended to the setting of root systems. In the case of a Hausdorff–Young inequality for $\mathcal{F}_\epsilon$, for example, one would need to know that $\limsup_{\epsilon\to 0} C_{\epsilon,p}$ is finite and at the same time take heed of the fact that measures used in the definition of $\mathcal{F}_\epsilon$ also change with $\epsilon$. Although the Hausdorff–Young and Hardy–Littlewood inequalities for $\mathcal{F}_0$ were easy to prove we still find such a philosophy promising and plan to investigate it at length in the near future. [MRSS04]{} C. Abdelkefi, J.-P. Anker, F. Sassi, and M. Sifi, *Besov-type spaces on [$\Bbb R^d$]{} and integrability for the [D]{}unkl transform*, SIGMA Symmetry Integrability Geom. Methods Appl. **5** (2009), Paper 019, 15. 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--- author: - Tianxing Ma - Fan Yang - Zhongbing Huang - 'Hai-Qing Lin' title: 'Triplet $p$-wave pairing correlation in low-doped zigzag graphene nanoribbons' --- Introduction {#introduction .unnumbered} ============ Triplet superconductivity (SC) has been a focus of modern condensed matter physics because of its possible connection to topological quantum information and computation[@Kitaev2001; @Alicea2012; @PhysRevLett.107.217001; @Mourik1003; @Deng2012; @Rokhinson2012; @PhysRevLett.109.056803; @Anindya2012; @PhysRevB.87.241401; @PhysRevA.82.053611]. It has been proposed that a gapless Majorana bound state would localize at the end of the one-dimensional spinless $p-$wave superconductor[@Kitaev2001], which could be used to practically realize topological quantum computation[@Kitaev2003; @RevModPhys.80.1083]. To realize such a Majorana bound state in real material, the superconducting proximity effect was proposed [@PhysRevLett.105.077001; @PhysRevLett.106.127001; @PhysRevLett.105.227003], and experimental evidence of its existence was recently reported[@Perge602]. Here, we explore the possibility of intrinsic triplet SC, which is generated by an electronic correlation. In this paper, we reveal a possible edge-spin triplet $p$-wave superconducting pairing correlation in slightly doped zigzag graphene nanoribbons with appropriate interactions. Graphene, a single layer of carbon, has generated immense interest ever since its experimental discovery[@Novoselov666; @RevModPhys.81.109]. Recently, experimental advances in doping methods have made it possible to change the type of carriers (electrons or holes)[@PhysRevLett.104.136803; @PhysRevLett.105.256805], opening the doors for exotic phases, such as SC and magnetism induced by repulsive interactions. For instance, it was shown by the two-stage renormalization-group calculation that unconventional SC is induced by weak repulsive interactions in honeycomb Hubbard models that are away from half-filling[@PhysRevB.81.224505], and that a topological $d+id$ SC is induced in a heavily doped system[@PhysRevB.75.134512; @PhysRevB.78.205431; @PhysRevB.81.085431; @PhysRevB.86.020507; @PhysRevB.84.121410; @PhysRevB.85.035414; @Nandkishore2012]. At graphene edges the density of states may be peaked due to the presence of edge-localized states close to the Fermi level[@PhysRevLett.106.226401]. Especially at extended zigzag edges this leads to a phenomenon called edge magnetism, for which various theories [@PhysRevB.80.245436; @PhysRevB.91.075410; @LiJPCM2016] predict ferromagnetic (FM) intraedge and antiferromagnetic (AFM) interedge correlations. [ The proposed magnetism is similar to the flat-band ferromagnetism appearing in the orbital-active optical honeycomb lattice[@PhysRevA.82.053618], where the band flatness dramatically amplifies the interaction effect, driving the ferromagnetic transition even with a relatively weak repulsive interaction]{}. From these discoveries, a question which naturally arises: is there is triplet SC mediated by the FM spin correlations on each edge in the doped zigzag graphene nanoribbons? ![(Color online) A piece of a honeycomb lattice displaying zigzag edges with $L_y=4$ which defines the width of the ribbon in the transverse direction and $L_x$=12, which defines the length in the longitudinal direction. The lattice sites at zigzag edge are much larger than the sites in the bulk, indicating that the charge carriers are moving along the edge. []{data-label="Fig:Structure"}](Fig1) ![(Color online) The carrier distribution (a) as a function of the site index at $U=2.0t$ and (b) from edge $\rightarrow$ bulk $\rightarrow$ edge with different $U$. It is clear to see that most charge carriers are distributed along the edge. []{data-label="Fig:ndistrubution"}](Fig2) ![(Color online) Band structure (a) and DOS (b) of a six-chain nanoribbon system. Note that the flat band bottom, located at approximately $-0.2t$ in (a), leads to the DOS peak in (b). The Fermi level of the half-filled system is marked by the red dashed lines in both figures. []{data-label="Fig:band"}](Fig3) In the present work, we establish the $p$-wave superconducting pairing correlation at the edges of zigzag graphene nanoribbons by using combined random-phase approximation (RPA)[@RevModPhys.84.1383; @PhysRevB.69.104504; @JPSJ; @Graser2009; @PhysRevB.75.224509; @PhysRevLett.101.087004; @PhysRevLett.111.066804; @Srep0820], the finite-temperature determinant quantum Monte Carlo (DQMC)[@PhysRevD.24.2278; @PhysRevB.31.4403; @MaAPL2010; @PhysRevLett.110.107002; @PhysRevB.94.075106] and the ground-state constrained-path quantum Monte Carlo (CPQMC)[@PhysRevLett.74.3652; @PhysRevB.55.7464; @PhysRevB.84.121410; @WuEPL2013; @MaEPL2015] methods. Our unbiased results show that both the ferromagnetic spin correlation and the effective $p$-wave superconducting pairing correlation are greatly enhanced as the interaction increases. Results {#results .unnumbered} ======= The ribbon geometry considered here is depicted in Fig. \[Fig:Structure\], in which the blue and white circles represent sublattices A and B, respectively, and the transverse integer index $1,2, . . . ,L_y$ defines the width of the ribbon while $1,2, . . . ,L_x$ at the zigzag edge defines the length. Assuming the ribbon to be infinite in the $x$ direction but finite in the $y$ direction, we produce a graphene nanoribbon with zigzag edges. In the following studies, the interaction $U$ is introduced through the standard Hubbard model. In Fig.\[Fig:ndistrubution\], the carrier distribution (a) as a function of the site index at $U=2.0t$ and (b) from edge $\rightarrow$ bulk $\rightarrow$ edge with different interactions is shown. It is clear to see that most charge carriers distribute along the edge, and the increasing interaction pushes many more charge carriers to the edges. The band structure of a six-chain nanoribbon system is shown in Fig. \[Fig:band\](a). Here, as the system is periodic in the $x$-direction, the momentum $k_x$ is a good quantum number. From Fig. \[Fig:band\](a), one finds a flat band bottom with energies located near the Fermi level ($\approx-0.2t$) of the half-filled system. Physically, such a flat band bottom is caused by the edge states, which leads to the DOS peak at approximately $-0.2t$ shown in Fig. \[Fig:band\](b). ![(Color online) (a) The largest eigenvalue $\chi(q_x)$ of the susceptibility matrix $\chi^{(0)}_{l,m}\left(q_x\right)$ as a function of $q_x$ for three different dopings, i.e., $\mu=-0.195t$ ($\delta=3.6\%$), $\mu=-0.2t$ ($\delta=3.0\%$) and $\mu=-0.205t$ ($\delta=0.8\%$) for the 6-chain system near half-filling. (b) Sketch of the pattern of the dominating spin fluctuations for $\mu=-0.2t$, as determined by the eigenvector of $\chi^{(0)}_{l,m}\left(q_x=0\right)$ corresponding to its largest eigenvalue. []{data-label="Fig:magnetic"}](Fig4) RPA study {#rpa-study .unnumbered} --------- Guided by the idea that triplet SC may be mediated by the strong FM spin fluctuations in the system, we performed an RPA-based study on the possible pairing symmetries of the system. The multi-orbital RPA approach[@RevModPhys.84.1383; @PhysRevB.69.104504; @JPSJ; @Graser2009; @PhysRevB.75.224509; @PhysRevLett.101.087004; @PhysRevLett.111.066804; @Srep0820], which is a standard and effective approach for the case of the weak coupling limit, is applied in our study for small $U$($<0.01t$). Various bare susceptibilities of this system are defined as $$\begin{aligned} \chi^{(0)l_{1},l_{2}}_{l_{3},l_{4}}\left(q_x,\tau\right)\equiv \frac{1}{N}\sum_{k_{1},k_{2}}\left<T_{\tau}c^{\dagger}_{l_{1}}(k_{1},\tau) c_{l_{2}}(k_{1}+q_x,\tau)c^{+}_{l_{3}}(k_{2}+q_x,0)c_{l_{4}}(k_{2},0)\right>_0,\label{free_sus} \end{aligned}$$ where $l_{i}$ $(i=1,2L_y)$ denote orbital (sublattice) indices. The largest eigenvalue $\chi(q_x)$ of the susceptibility matrix $\chi^{(0)}_{l,m}\left(q_x\right)\equiv \chi^{(0)l,l}_{m,m}\left(q_x,i\nu=0\right)$ as function of $q_x$ is shown in Fig. \[Fig:magnetic\](a) for three different dopings near half-filling. As a result, the susceptibility for the doping $\delta=3\%$ with chemical potential $\mu=-0.2t$ peaks at zero momentum, which suggests strong FM intra-sublattice spin fluctuations in the system. Further more, from the eigenvector of the susceptibility matrix, one can obtain the pattern of the dominating spin fluctuation in the system, which is shown in Fig. \[Fig:magnetic\](b). Obviously, the dominating spin fluctuation, which is mainly located on the two edges, is FM on each edge and AFM between the two edges. When $\mu$ deviates from $-0.2t$, the susceptibility peaks deviate from zero, as shown in Fig. \[Fig:magnetic\](a), suggesting weaker FM spin fluctuations in the system. With weak-Hubbard $U$, the spin ($\chi^{s}$) and charge ($\chi^{c}$) susceptibilities in the RPA level are given by $$\chi^{s\left(c\right)}\left(\mathbf{q},i\nu\right)= \left[I\mp\chi^{(0)}\left(\mathbf{q},i\nu\right)\bar U\right]^{-1}\chi^{(0)} \left(\mathbf{q},i\nu\right),\label{RPA}$$ where $\bar U^{\mu\nu}_{\mu'\nu'}$ ($\mu,\nu=1,\cdots, 2L_y$) is a $4L_{y}^2\times4L_{y}^2$ matrix, whose nonzero elements are $\bar U^{\mu\mu}_{\mu\mu}=U$ ($\mu=1,\cdots,2L_y$). Clearly, the repulsive $U$ suppresses $\chi^{c}$ but enhances $\chi^{s}$. Thus, the spin fluctuations take the main role of mediating the Cooper pairing in the system. In the RPA level, via exchanging the spin fluctuations represented by the spin susceptibilities, the Cooper pairs near the FS acquire an effective interaction $V_\textrm{eff}$[@RevModPhys.84.1383; @PhysRevLett.111.066804], from which one solves the linearized gap equation to obtain the leading pairing symmetry and its critical temperature $T_c$. Focusing on the low-doping regime in which the chemical potential $\mu$ is located within the flat band bottom, we obtained the largest pairing eigenvalues $\lambda$ for the singlet and triplet pairings as functions of $\mu$ for a 6-chain ribbon with weak interaction $U=0.001t$, as shown in Fig. \[Fig:SC\](a). Interestingly, while both pairings attain their largest eigenvalues at $\mu=-0.2t$ (3% doping) due to the DOS peak there (as shown in Fig. \[Fig:band\](b)), the triplet pairing wins over the singlet one in the low doping regime at the flat band bottom. Physically, the triplet pairing in this regime is mediated by the FM spin fluctuations shown in Fig. \[Fig:magnetic\]. In Fig. \[Fig:SC\](b), the results for $U=0.005t$ are shown. Comparing (a) and (b), it’s obvious that stronger interaction leads to pairing correlations that are qualitatively the same as but quantitatively stronger than weak interaction. In Fig. \[Fig:SC\](c) and (d), the results for a 4-chain ribbon and 8-chain ribbon are shown with $U=0.001t$. The results for all these cases are qualitatively similar. Note that we have chosen a very weak $U$ in our RPA calculations, since for $U>U_c\approx 0.007t$ (for $\mu=-0.2t$), the divergence of the spin-susceptibility invalidates our calculations. Physically, such a spin susceptibility divergence will not lead to a magnetically ordered state since the Mermin and Wagner’s theorem prohibits a one-dimensional system from forming long-range order. Instead, short-ranged FM spin correlations here might mediate triplet superconducting pairing correlations. We leave the study of the case of $U>U_c$ to the following DQMC and CPQMC approaches, which are suitable for strong coupling problems. ![(Color online) Doping dependence of the largest eigenvalues $\lambda$ of singlet and triplet pairings for (a) $U=0.001t$, and (b) $U=0.005t$ for the 6-chain system, (c) $U=0.005t$ for the 4-chain system and (d) $U=0.001t$ for the 8-chain system.[]{data-label="Fig:SC"}](Fig5) QMC Result {#qmc-result .unnumbered} ---------- As FM fluctuations play an essential role, we first study the magnetic correlations. In Fig. \[Fig:Spin\] (a), the edge spin susceptibility $\chi$ is shown as a function of temperature with different $U$ at $\delta$=0.02. The edge $\chi$ is calculated by summing over the sites on the edge, such as those marked as larger circles in Fig. \[Fig:Structure\]. It is interesting to see that $\chi$ increases as the temperature decreases, which indicates a dominant FM fluctuations on the zigzag edge. Additionally, $\chi$ increases as $U$ increases, indicating that the electronic correlation is important for the magnetic excitation in such a system. The uniform spin susceptibility $\chi_B$ for the whole system is also shown, which decreases slightly as the temperature decreases. To further reveal the FM correlation on the zigzag edge, the spin-spin correlation along the edge is shown in Fig. \[Fig:Spin\] (b). One can see that the spin correlation $S_{1i}(i=2,3,\cdots)$ along the edge is always positive, suggesting FM correlation. One may also see that the spin correlation is weakened as the system is doped away from half filling, which is in agreement with the results indicated by RPA shown in Fig. \[Fig:magnetic\](a). ![(Color online) (a) The edge $\chi$ as a function of temperature at $\delta$=0.02 for different $U$, and the uniform $\chi_B$ for $U=2.0t$ is also present. (b) The spin correlation as a function of the site index along the edge for $U=2.0t$ at $\delta$=0.02 and $\delta$=0.04, and $U=1.0t$ at $\delta$=0.02. []{data-label="Fig:Spin"}](Fig6) In Fig. \[Fig:Pe\], we plot the effective pairing interaction $P_\alpha$ as a function of temperature for different $U$ and electron fillings on a lattice with $2 \times 4 \times 12$ sites. Clearly in Fig. \[Fig:Pe\], the intrinsic pairing interaction $P_\alpha$ is positive and increases with the lowering of temperature. Such a temperature dependence of $P_\alpha$ suggests that effective attractions are generated between electrons and that there is instability towards SC in the system at low temperatures. Moreover, Fig. \[Fig:Pe\] shows that the intrinsic pairing interaction for $p$-wave symmetry is enhanced for larger $U$, indicating that the pairing strength increases with the enhancement of the electron correlations. For another extensive-$s$ pairing symmetry, our DQMC results yield negative intrinsic pairing interactions (not shown here). Numerical approaches, such as DQMC, however, have their own difficulties as follows: they are typically being limited to small sizes and high temperatures, and experience the infamous fermion sign problem, which cause exponential growth in the variance of the computed results and hence an exponential growth in computational time as the lattice size is increased and the temperature is lowered[@PhysRevD.24.2278]. In general, to determine the superconducting pairing symmetry by numerical calculation for models of finite size, we have to look at the distance-dependent pair-correlation function at zero temperature. To shed light on this critical issue, it is important to discuss the results obtained by using the CPQMC method[@PhysRevLett.74.3652; @PhysRevB.55.7464] on a larger lattice. In a variety of benchmarking calculations, the CPQMC method has yielded very accurate results on the ground-state energy and many other ground-state observables for large systems[@PhysRevB.55.7464]. ![(Color online) The effective $p$-type pairing interaction as a function of temperature on a lattice with $2\times 4 \times 12$ sites for different $U$ at $\delta=0.03$. []{data-label="Fig:Pe"}](Fig7) ![(Color online) The $p$-wave superconducting pairing correlation as a function of the distance $r$ on a lattice with $2\times 6 \times 24$ sites. Inset: the doping-dependent pairing correlation at $r=12$. []{data-label="Fig:Pdr"}](Fig8) In Fig. \[Fig:Pdr\], we compare the pairing correlations on lattices with $2\times 6 \times 24$ sites for different electron fillings at $U=2.0t$. Here, the simulations are performed for the closed-shell cases. The distance-dependent pairing correlations for $\delta=3/144\simeq 0.021$ (dark triangle), $\delta=5/144\simeq 0.035$ (red circles ), and $\delta=7/144\simeq 0.049$ (blue square) are shown. One can readily see that $C_{p}(r)$ decreases as the distance increases, and the decay of the distance-dependent pairing correlations is different for different dopings. In the inset of Fig.\[Fig:Pdr\], the pairing correlation $C_{p}(r=12)$ at the largest distance is shown as a function of the doping. In the filling range that we investigated, $C_{p}(r=12)$ is not a monotonic function of the doping and there exists an “optimal“ doping (approximately 0.035 electron/site) at which the magnitudes of $C_{p}(r=12)$ are maximized. This result is consistent with that of RPA, where the doping-dependent pairing correlation bears some similarity to the famous superconducting ”dome" in the phase diagram of high-temperature superconductors[@RevModPhys.78.17], while the spin correlation at the edge is weakened as the system is doped away from half filling. Discussion {#discussion .unnumbered} ========== We performed a combined RPA and quantum Monte Carlo study of the magnetic and pairing correlations at the edges in low-doped zigzag graphene nanoribbons. Our studies show that the triplet edge $p$-wave SC occurs as the ground state of our model system. The optimal doping is approximately 0.03, which can be easily understood as the DOS peaks at this doping level, and this doping level is currently achievable experimentally capability for graphene-based material. Our accurate numerical results establish the properties of the $p$-wave superconducting correlation in zigzag graphene nanoribbons, and will be important for any experimental scheme aimed at detecting the $p$-type superconducting state, as such a scheme will likely be based on the distinctive properties of the edge. Methods {#methods .unnumbered} ======= The electronic and magnetic properties of the studied system can be well described by the following Hubbard model[@RevModPhys.81.109] $$\begin{aligned} H&=&-t\sum_{\left\langle i,j\right\rangle}c^{\dagger}_{i\sigma}c_{j\sigma}-t'\sum_{\left\langle\langle i,j\right\rangle\rangle}c^{\dagger}_{i\sigma}c_{j\sigma}+U\sum_{i}n_{i\uparrow}n_{i\downarrow} +\mu\sum_{i\sigma}n_{i\sigma }\label{H}\end{aligned}$$ where $c^\dag_{i\sigma}$ is the electron-creation operator at site $i$ with spin polarization $\sigma=\A,\V$, $U$ denotes the on-site repulsive interaction, and $\mu$ is the chemical potential. Here, the $t$ and $t'$ terms describe the nearest-neighbor (NN) and next nearest-neighbor (NNN) hoppings, respectively. In the following study, we adopted $t^{\prime}=-0.1t$, which is consistent with experiments[@PhysRevB.88.165427]. In our calculation, we employ periodic boundary conditions in the $x$ direction and open boundary conditions at the zigzag edge. Specifically, we compute the spin correlation $S_{i,j}=\langle S_{i}\cdot S_{j}\rangle$ in the $z$ direction, and define the uniform spin susceptibility at zero frequency, $$\begin{aligned} \chi= \frac{1}{N_s}\int_{0}^{\beta}d\tau \sum_{d,d'=a,b} \sum_{i,j} \langle\textrm{m}_{i_{d}}(\tau) \cdot \textrm{m}_{j_{d'}}(0)\rangle\end{aligned}$$ To investigate the SC property, we compute the effective pairing interaction and study the distance dependent pairing correlation. The effective pairing interaction is extracted from the pairing susceptibility, $$p_{\alpha}=\frac{1}{N_s}\sum_{i,j}\int_{0}^{\beta }d\tau \langle \Delta _{\alpha }^{\dagger }(i,\tau)\Delta _{\alpha }^{\phantom{\dagger}}(j,0)\rangle,\label{sus}$$ with $$\begin{aligned} \Delta_{\alpha }^{\dagger }(i)\ =\sum_{l}f_{\alpha}^{\dagger} (\delta_{l})(c_{{i}\uparrow }c_{{i+\delta_{l}}\downarrow }- c_{{i}\downarrow}c_{{i+\delta_{l}}\uparrow })^{\dagger}.\end{aligned}$$ Here, $\alpha$ stands for the pairing symmetry, $f_{\alpha}(\bf{\delta}_{l})$ is the form factor of the pairing function, and the vectors $\bf{\delta_{l}}$ $(l=1,2)$ denote the NNN sites along the edge. To extract the effective pairing interaction, the bubble contribution $\widetilde{p} _{\alpha }(i,j)$ indicating $\langle c_{{i}\downarrow }^{\dag }c_{{j}\downarrow }c_{i+\delta_{l}\uparrow}^{\dag} c_{j+\delta_{l'}\uparrow}\rangle $ in Eq. (\[sus\]) is being replaced by $\langle c_{{i}\downarrow }^{\dag }c_{{j}\downarrow }\rangle \langle c_{i+\delta_{l}\uparrow }^{\dag } c_{j+\delta_{l'}\uparrow }\rangle $. Thus we have obtained an effective pairing interaction $P_\alpha=p_{\alpha}-\widetilde{p}_\alpha$. 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Acknowledgements {#acknowledgements .unnumbered} ================ Ma T. thanks CAEP for partial financial support. This work is supported in part by NSCFs (Grant Nos. 11374034, 11274041 and 11334012). Yang F. is supported by the NCET program under the grant No. NCET-12-0038. Lin H.-Q. acknowledges support from NSAF U1530401 and computational resource from the Beijing Computational Science Research Center. We also acknowledge the support from the HSCC of Beijing Normal University, and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase). Author contributions statement {#author-contributions-statement .unnumbered} ============================== Ma T. provided the QMC data, Yang F. provided the RPA results. Ma T., Yang F, Huang Z., and Lin H.-Q. provided the theoretical understanding and wrote the paper together. Additional information {#additional-information .unnumbered} ====================== Competing financial interests {#competing-financial-interests .unnumbered} ----------------------------- The authors declare no competing financial interests.

--- abstract: | We prove some normality criteria for a family of meromorphic functions under a condition on differential polynomials generated by the members of the family. 0.15cm *Keywords:* Meromorphic function, Normal family, Nevanlinna theory. 0.15cm Mathematics Subject Classification 2010: 30D35. --- $\qquad$ SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS and Nguyen Van Thin$^c$ [$^{a}$ Université de Brest, LMBA, UMR CNRS 6205,\ 6, avenue Le Gorgeu - C.S. 93837, 29238 Brest Cedex 3, France ]{} [$ ^{b}$ Department of Mathematics, Hanoi National University of Education,\ 136 Xuan Thuy Street, Cau Giay, Hanoi, Vietnam]{} [$ ^{c}$ Department of Mathematics, Thai Nguyen University of Education,\ Luong Ngoc Quyen Street, Thai Nguyen City, Vietnam]{} 0.15cm Introduction ============ Let $D$ be a domain in the complex plane $\C$ and $\mathcal F$ be a family of meromorphic functions in $D.$ The family $\mathcal F$ is said to be normal in $D,$ in the sense of Montel, if for any sequence $\{f_v\}\subset \mathcal F,$ there exists a subsequence $\{f_{v_i}\}$ such that $\{f_{v_i}\}$ converges spherically locally uniformly in $D,$ to a meromorphic function or $\infty.$\ In 1989, Schwick proved: [**Theorem A** ]{}([@Sch], Theorem 3.1)[**.**]{} [*Let $k, n$ be positive integers such that $n\geq k+3.$ Let $\mathcal F$ be a family of meromorphic functions in a complex domain $D$ such that for every $f\in\mathcal F,$ $(f^n)^{(k)}(z)\ne 1$ for all $z\in D.$ Then $\mathcal F$ is normal on $D.$*]{} [**Theorem B** ]{}([@Sch], Theorem 3.2)[**.**]{} [*Let $k, n$ be positive integers such that $n\geq k+1.$ Let $\mathcal F$ be a family of entire functions in a complex domain $D$ such that for every $f\in\mathcal F,$ $(f^n)^{(k)}(z)\ne 1$ for all $z\in D.$ Then $\mathcal F$ is normal on $D.$*]{} The following normality criterion was established by Pang and Zalcman [@PZ] in 1999: [**Theorem C**]{} ([@PZ])[**.**]{} [*Let $n$ and $k$ be natural numbers and $\mathcal F$ be a family of holomorphic functions in a domain $D$ all of whose zeros have multiplicity at least $k.$ Assume that $f^nf^{(k)}-1$ is non-vanishing for each $f\in\mathcal F.$ Then $\mathcal F$ is normal in D.*]{}\ The main purpose of this paper is to establish some normality criteria for the case of more general differential polynomials. Our main results are as follows: \[Th1\] Take $q \;(q\geq1)$ distinct nonzero complex values $a_1,\dots,a_q,$ and $q$ positive integers (or $+\infty$) $\ell_1,\dots\ell_q.$ Let $n$ be a nonnegative integer, and let $n_1,\dots,n_k, t_1,\dots,t_k$ be positive integers ($k\geq 1$). Let $\mathcal F$ be a family of meromorphic functions in a complex domain $D$ such that for every $f\in\mathcal F$ and for every $m\in\{1,\dots,q\},$ all zeros of $f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}-a_m$ have multiplicity at least $\ell_m.$ Assume that $ a)\quad n_j\geq t_j \text{\;for all \;} 1\leqslant j\leqslant k,\;\text{ and\;} \ell_i\geq 2 \text{\;for all\;} 1\leqslant i\leqslant q,$\ $ b)\quad \sum_{i=1}^q\frac{1}{\ell_i}<\frac{ qn-2+\sum_{j=1}^kq(n_j-t_j)}{n+\sum_{j=1}^k(n_j+t_j)}.$\ Then $\mathcal F$ is a normal family. Take $q=1$ and $\ell_1=+\infty,$ we get the following corollary of Theorem \[Th1\]: \[H1\] Let $a$ be a nonzero complex value, let $n$ be a nonnegative integer, and $n_1,\dots,n_k,t_1,\dots,t_k$ be positive integers. Let $\mathcal F$ be a family of meromorphic functions in a complex domain $D$ such that for every $f\in\mathcal F,$ $f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}-a$ is nowhere vanishing on $D.$ Assume that $a)$ $n_j\geq t_j \text{\;for all \;} 1\leqslant j\leqslant k,$ $b)$ $n+\sum_{j=1}^kn_j\geq 3+\sum_{j=1}^kt_j.$\ Then $\mathcal F$ is normal on $D.$ We remark that in the case where $n\geq 3,$ condition $a)$ in the above corollary implies condition $b);$ and in the case where $n=0$ and $k=1,$ Corollary \[H1\] gives Theorem A. For the case of entire functions, we shall prove the following result: \[Th2\] Take $q \;(q\geq1)$ distinct nonzero complex values $a_1,\dots,a_q,$ and $q$ positive integers $($or $+\infty)$ $\ell_1,\dots\ell_q.$ Let $n$ be a nonnegative integer, and let $n_1,\dots,n_k, t_1,\dots,t_k$ be positive integers $(k\geq 1).$ Let $\mathcal F$ be a family of holomorphic functions in a complex domain $D$ such that for every $f\in\mathcal F$ and for every $m\in\{1,\dots,q\},$ all zeros of $f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}-a_m$ have multiplicity at least $\ell_m.$ Assume that $ a)\quad n_j\geq t_j \text{\;for all \;} 1\leqslant j\leqslant k,\;\text{ and\;} \ell_i\geq 2 \text{\;for all\;} 1\leqslant i\leqslant q,$\ $ b)\quad \sum_{i=1}^q\frac{1}{\ell_i}<\frac{ qn-1+\sum_{j=1}^kq(n_j-t_j)}{n+\sum_{j=1}^kn_j}.$\ Then $\mathcal F$ is a normal family. Take $q=1$ and $\ell_1=+\infty,$ Theorem \[Th2\] gives the following generalization of Theorem B, except for the case $n=k+1$. So for the latter case, we add a new proof of Theorem B in the Appendix which is slightly simpler than the original one. \[H2\] Let $a$ be a nonzero complex value, let $n$ be a nonnegative integer, and $n_1,\dots,n_k,t_1,\dots,t_k$ be positive integers. Let $\mathcal F$ be a family of holomorphic functions in a complex domain $D$ such that for every $f\in\mathcal F,$ $f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}-a$ is nowhere vanishing on $D.$ Assume that $a)$ $n_j\geq t_j \text{\;for all \;} 1\leqslant j\leqslant k,$ $b)$ $n+\sum_{j=1}^kn_j\geq 2+\sum_{j=1}^kt_j.$\ Then $\mathcal F$ is normal on $D.$ In the case where $n\geq 2,$ condition $a)$ in the above corollary implies condition $b).$ \[Re\] Our above results remain valid if the monomial $f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}$ is replaced by the following polynomial $$\begin{aligned} f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}+\sum_{I}c_If^{n_I}(f^{n_{1I}})^{(t_{1I})}\cdots(f^{n_{kI}})^{(t_{kI})},\end{aligned}$$ where $c_I$ is a holomorphic function on $D,$ and $n_I,n_{jI},t_{jI}$ are nonnegative integers satisfying $$\alpha_I:=\frac{ \sum_{j=1}t_{jI}}{n_I+\sum_{j=1}^kn_{jI}}<\alpha:=\frac{ \sum_{j=1}t_j}{n+\sum_{j=1}^kn_j}.$$ Some notations and results of Nevanlinna theory =============================================== Let $\nu$ be a divisor on $\C.$ The counting function of $\nu$ is defined by $$N (r,\nu) = \int\limits_1^r \frac{n (t)}{t} dt \ \ (r>1), \text{\ where\ } n(t) =\sum_{\vert z\vert \le t} \nu (z).$$ For a meromorphic function $f$ on $\C$ with $f\not\equiv\infty,$ denote by $\nu_f$ the pole divisor of $f,$ and the divisor $\overline{\nu}_f$ is defined by $\overline{\nu}_f(z):=\min\{\nu_f(z),1\}.$ Set $N(r, f):=N (r,\nu_f)$ and $\overline N(r, f):=N (r,\overline{\nu}_f).$ The proximity function of $f$ is defined by $$m(r, f) =\frac {1} {2\pi} \int\limits_0^{2\pi} \log^+ \big\vert f (re^{i\theta}) \big\vert d\theta,$$ where $\log^+ x =\max \{\log \, x, 0 \} \ \ \text { for } x \ge 0.$ The characteristic function of $f$ is defined by $$T(r,f):=m(r,f)+N(r,f).$$ We state the Lemma on Logarithmic Derivative, the First and Second Main Theorems of Nevanlinna theory. [Lemma on Logarithmic Derivative.]{} [*Let $f$ be a nonconstant meromorphic function on $\C,$ and let $k$ be a positive integer. Then the equality $$m(r,\frac{f^{(k)}}{f})=o(T(r,f))$$ holds for all $r \in [1,\infty)$ excluding a set of finite Lebesgue measure.*]{} [First Main Theorem.]{} [*Let $f$ be a meromorphic functions on $\C$ and $a$ be a complex number. Then $$T(r,\dfrac{1}{f-a})=T(r,f)+O(1).$$*]{} [Second Main Theorem.]{} [*Let $f$ be a nonconstant meromorphic function on $\C$. Let $a_{1},\ldots ,a_{q}$ be $q$ distinct values in $\C$. Then $$(q-1)T(r,f)\leqslant \overline N(r,f) +\sum\limits_{i=1}^q \overline{N}(r, \frac{1}{f-a_i}) +o(T(r,f)),$$ for all $r \in [1,\infty)$ excluding a set of finite Lebesgue measure.*]{} Proof of our results ==================== To prove our results, we need the following lemmas: \[L1\] Let $\mathcal F$ be a family of meromorphic functions defined in the unit disc $\bigtriangleup.$ Then if $\mathcal F$ is not normal at a point $z_0\in\bigtriangleup,$ there exist, for each real number $\alpha$ satisfying $-1<\alpha<1,$ $1)$ a real number $r,\;0<r<1,$ $2)$ points $z_n,\;|z_n|<r,$ $z_n\to z_0,$ $3)$ positive numbers $\rho_n,\rho_n\to 0^+,$ $4)$ functions $f_n,\;f_n\in\mathcal F$ such that $$g_n(\xi)=\frac{f_n(z_n+\rho_n\xi)}{\rho_n^\alpha}\to g(\xi)$$ spherically uniformly on compact subsets of $\C,$ where $g(\xi)$ is a non-constant meromorphic function and $g^{\#}(\xi)\leqslant g^{\#}(0)=1.$ Moreover, the order of $g$ is not greater than $2.$ Here, as usual, $g^\#(z)=\frac{|g'(z)|}{1+|g(z)|^2}$ is the spherical derivative. \[L2\] Let $g$ be a entire function and $M$ is a positive constant. If $g^{\#}(\xi)\leqslant M$ for all $\xi\in\C,$ then $g$ has order at most one. \[R1\] In Lemma \[L1\], if $\mathcal F$ is a family of holomorphic functions, then by Hurwitz theorem, $g$ is a holomorphic function. Therefore, by Lemma \[L2\], the order of $g$ is not greater than $1.$ We consider a nonconstant meromorphic function $g$ in the complex plane $\C,$ and its first $p$ derivatives. A differential polynomial $P$ of $g$ is defined by $$P(z):=\sum_{i=1}^n\alpha_i(z)\prod_{j=0}^p(g^{(j)}(z))^{S_{ij}},$$ where $S_{ij} \;(1\leqslant i\leqslant n, \:0\leqslant j\leqslant p )$ are nonnegative integers, and $\alpha_i\not\equiv 0 \;(1\leqslant i\leqslant n)$ are small (with respect to $g$) meromorphic functions. Set $$d(P):=\min_{1\leqslant i\leqslant n}\sum_{j=0}^pS_{ij}\;\text{and}\; \theta(P):=\max_{1\leqslant i\leqslant n}\sum_{j=0}^pjS_{ij}.$$ In 2002, J. Hinchliffe [@Hi] generalized theorems of Hayman [@Ha] and Chuang [@Ch] and obtained the following result: Let $g$ be a transcendental meromorphic function, let P(z) be a non-constant differential polynomial in $g$ with $d(P)\geq 2.$ Then $$\begin{aligned} T(r,g)\leqslant\frac{\theta(P)+1}{d(P)-1}\overline{N}(r,\frac{1}{g})+\frac{1}{d(P)-1}\overline{N}(r,\frac{1}{P-1})+o(T(r,g)),\end{aligned}$$ for all $r\in[1,+\infty)$ excluding a set of finite Lebesgues measure. In order to prove our results, we now give the following generalization of the above result: \[L3\] Let $a_1,\dots,a_q$ be distinct nonzero complex numbers. Let $g$ be a nonconstant meromorphic function, let P(z) be a nonconstant differential polynomial in $g$ with $d(P)\geq 2.$ Then $$\begin{aligned} T(r,g)\leqslant\frac{q\theta(P)+1}{qd(P)-1}\overline{N}(r,\frac{1}{g})+\frac{1}{qd(P)-1}\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})+o(T(r,g)),\end{aligned}$$ for all $r\in[1,+\infty)$ excluding a set of finite Lebesgues measure. Moreover, in the case where $g$ is a entire function, we have $$\begin{aligned} T(r,g)\leqslant\frac{q\theta(P)+1}{qd(P)}\overline{N}(r,\frac{1}{g})+\frac{1}{qd(P)}\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})+o(T(r,g)),\end{aligned}$$ for all $r\in[1,+\infty)$ excluding a set of finite Lebesgue measure. For any $z$ such that $|g(z)|\leqslant 1,$ since $\sum_{j=0}^pS_{ij}\geq d(P)\;(1\leqslant i\leqslant n),$ we have $$\begin{aligned} \frac{1}{|g(z)|^{d(P)}}&=\frac{1}{|P(z)|}\cdot\frac{|P(z)|}{|g(z)|^{d(P)}}\\ &\leqslant\frac{1}{|P(z)|}\cdot\sum_{i=1}^n\big(|\alpha_i(z)|\prod_{j=0}^p\big|\frac{g^{(j)}(z)}{g(z)}\big|^{S_{ij}}\big).\end{aligned}$$ This implies that for all $z\in\C,$ $$\begin{aligned} \log^+\frac{1}{|g(z)|^{d(P)}}\leqslant \log^+\big(\frac{1}{|P(z)|}\cdot\sum_{i=1}^n\big(|\alpha_i(z)|\prod_{j=0}^p\big|\frac{g^{(j)}(z)}{g(z)}\big|^{S_{ij}}\big)\big).\end{aligned}$$ Therefore, by the Lemma on Logarithmic Derivative and by the First Main Theorem, we have $$\begin{aligned} d(P) m(r,\frac{1}{g})\leqslant m(r,\frac{1}{P})+o(T(r,g))&=T(r,\frac{1}{P})-N(r,\frac{1}{P})+o(T(r,g))\\ &=T(r,P)-N(r,\frac{1}{P})+o(T(r,g)).\end{aligned}$$ On the other hand, by the Second Main Theorem (used with the $q+1$ different values $0, a_1,...,a_q$) we have $$\begin{aligned} qT(r,P)\leqslant\overline{N}(r,P)+\overline{N}(r,\frac{1}{P})+\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})+o(T(r,g)),\end{aligned}$$ Hence, $$\begin{aligned} d(P)m(r,\frac{1}{g})\leqslant\frac{1}{q}\big(\overline{N}(r,P)+\overline{N}(r,\frac{1}{P})&+\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})\big)\\ &-N(r,\frac{1}{P})+o(T(r,g)).\end{aligned}$$ Therefore, by the First Main Theorem, we have $$\begin{aligned} \label{1} d(P)T(r, g)&=d(P)T(r,\frac{1}{g})+O(1)\notag\\ &=d(P)m(r,\frac{1}{g})+d(P)N(r,\frac{1}{g})+O(1)\notag\\ &\leqslant \frac{1}{q}\big(\overline{N}(r,P)+\overline{N}(r,\frac{1}{P})+\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})\big)\notag\\ &\;\;\;\;\;\;\;\;\;\;\;+d(P)N(r,\frac{1}{g})-N(r,\frac{1}{P})+o(T(r,g)).\end{aligned}$$ We have $$\begin{aligned} \frac{1}{g^{d(P)}}=\frac{1}{P(z)}\sum_{i=1}^n\big(\alpha_ig^{(\sum_{j=0}^pS_{ij})-d(P)}\prod_{j=0}^p(\frac{g^{(j)}}{g})^{S_{ij}}\big).\end{aligned}$$ (note that $(\sum_{j=0}^pS_{ij})-d(P)\geq 0).$ Therefore, $$\begin{aligned} d(P)\nu_{\frac{1}{g}}&\leqslant \nu_{\frac{1}{P}}+\max_{1\leqslant i\leqslant n}\{\nu_{\alpha_i}+\sum_{j=0}^pjS_{ij}\overline{\nu}_{\frac{1}{g}}\}\\ &\leqslant \nu_{\frac{1}{P}}+\sum_{i=1}^n\nu_{\alpha_i}+\theta(P)\overline{\nu}_{\frac{1}{g}},\end{aligned}$$ where $\nu_\phi$ is the pole divisor of the meromorphic $\phi$ and $\overline{\nu}_\phi:=\min\{\nu_\phi,1\}.$ This implies, $$d(P)\nu_{\frac{1}{g}}-\nu_{\frac{1}{P}}+\frac{1}{q}\overline{\nu}_{\frac{1}{P}}\leqslant (\theta(P)+\frac{1}{q})\overline{\nu}_{\frac{1}{g}}+\sum_{i=1}^n\nu_{\alpha_i},$$ (note that for any $z_0,$ if $\nu_{\frac{1}{g}}(z_0)=0$ then $d(P)\nu_{\frac{1}{g}}(z_0)-\nu_{\frac{1}{P}}(z_0)+\frac{1}{q}\overline{\nu}_{\frac{1}{P}}(z_0)\leqslant 0).$ Then, $$\begin{aligned} d(P)N(r,\frac{1}{g})-N(r,\frac{1}{P})+\frac{1}{q}\overline{N}(r,\frac{1}{P})&\leqslant (\theta(P)+\frac{1}{q})\overline{N}(r, \frac{1}{g})+\sum_{i=1}^nN(r,\alpha_i)\\ &=(\theta(P)+\frac{1}{q})\overline{N}(r, \frac{1}{g})+o(T(r,g)).\end{aligned}$$ Combining with (\[1\]), we have $$\begin{aligned} d(P)T(r, g)\leqslant \frac{1}{q}\big(\overline{N}(r,P)+\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})\big)+(\theta(P)+\frac{1}{q})\overline{N}(r, \frac{1}{g})+o(T(r,g)).\end{aligned}$$ On the other hand, by the definition of the differential polynomial $P,$ Pole$(P)\subset\cup_{i=1}^n$ Pole$(\alpha_i)\cup$ Pole$(g).$ Hence (since $\overline{N}(r,\alpha_i) \leq T(r, \alpha_i) = o(T(r,g)$ for $i=1,...,n$), we get $$\begin{aligned} \label{2} d(P)T(r, g)\leqslant \frac{1}{q}\big(\overline{N}(r,g)+\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})\big)+(\theta(P)+\frac{1}{q})\overline{N}(r, \frac{1}{g})+o(T(r,g))\notag\\ \leqslant\frac{1}{q}\big(T(r,g)+\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})\big)+(\theta(P)+\frac{1}{q})\overline{N}(r, \frac{1}{g})+o(T(r,g)).\end{aligned}$$ Therefore, $$\begin{aligned} T(r,g)\leqslant\frac{q\theta(P)+1}{qd(P)-1}\overline{N}(r,\frac{1}{g})+\frac{1}{qd(P)-1}\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})+o(T(r,g)).\end{aligned}$$ In the case where $g$ is an entire function, the first inequality in $(3.2)$ becomes $$\begin{aligned} d(P)T(r, g)\leqslant \frac{1}{q}\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})+(\theta(P)+\frac{1}{q})\overline{N}(r, \frac{1}{g})+o(T(r,g)).\end{aligned}$$ This implies that $$\begin{aligned} T(r, g)\leqslant \frac{\theta(P)q+1}{qd(P)})\overline{N}(r, \frac{1}{g})+\frac{1}{qd(P)}\sum_{j=1}^q\overline{N}(r,\frac{1}{P-a_j})+o(T(r,g)).\end{aligned}$$ We have completed the proof of Lemma \[L3\]. [**Proof of Theorem \[Th1\].**]{} Without loss the generality, we may asssume that $D$ is the unit disc. Suppose that $\mathcal F$ is not normal at $z_0\in D.$ By Lemma \[L1\], for $\alpha =\frac{\sum_{j=1}^kt_j}{n+\sum_{j=1}^kn_j}$ there exist $1)$ a real number $r,\;0<r<1,$ $2)$ points $z_v,\;|z_v|<r,$ $z_v\to z_0,$ $3)$ positive numbers $\rho_v,\rho_v\to 0^+,$ $4)$ functions $f_v,\;f_v\in\mathcal F$ such that $$\begin{aligned} \label{ad1} g_v(\xi)=\frac{f_v(z_v+\rho_v\xi)}{\rho_v^\alpha}\to g(\xi)\end{aligned}$$ spherically uniformly on compact subsets of $\C,$ where $g(\xi)$ is a non-constant meromorphic function and $g^{\#}(\xi)\leqslant g^{\#}(0)=1.$ On the other hand, $$\begin{aligned} \big(g_v^{n_j}(\xi)\big)^{(t_j)}&=\big((\frac{f_v(z_v+\rho_v\xi)}{\rho_v^\alpha})^{n_j}\big)^{(t_j)}\\ &=\frac{1}{\rho_v^{n_j\alpha-t_j}}(f_v^{n_j})^{(t_j)}(z_v+\rho_v\xi).\end{aligned}$$ Therefore, by the definition of $\alpha$ and by (\[ad1\]), we have $$\begin{aligned} \label{ad2} f_v^{n}(z_v&+\rho_v\xi)(f_v^{n_1})^{(t_1)}(z_v+\rho_v\xi)\cdots(f_v^{n_k})^{(t_k)}(z_v+\rho_v\xi)\notag\\ &=g_v^n(\xi)(g_v^{n_1}(\xi))^{(t_1)}\dots(g_v^{n_k}(\xi))^{(t_k)}\to g^n(\xi)(g^{n_1}(\xi))^{(t_1)}\dots(g^{n_k}(\xi))^{(t_k)}\end{aligned}$$ spherically uniformly on compact subsets of $\C.$ Now, we prove the following claim: [**Claim:**]{} [*$g^n(\xi)(g^{n_1}(\xi))^{(t_1)}\dots(g^{n_k}(\xi))^{(t_k)}$ is non-contstant.* ]{} Since $g$ is non-constant and $n_j\geq t_j \;(j=1,\dots,k),$ it easy to see that $(g^{n_j}(\xi))^{(t_j)}\not\equiv 0,$ for all $j\in\{1,\dots,k\}.$ Hence, $g^n(\xi)(g^{n_1}(\xi))^{(t_1)}\dots(g^{n_k}(\xi))^{(t_k)}\not\equiv 0.$ Suppose that $g^n(\xi)(g^{n_1}(\xi))^{(t_1)}\dots(g^{n_k}(\xi))^{(t_k)}\equiv a,$ $a\in\C\setminus \{0\}.$ We first remark that, from conditions $a),b),$ we have that in the case $n=0,$ there exists $i\in\{1,\dots,k\}$ such that $n_i>t_i.$ Therefore, in both cases ($n=0$ and $n \not= 0$), since $a\ne 0,$ it is easy to see that $g$ is entire having no zero. So, by Lemma \[L2\], $g(\xi)=e^{c\xi+d},\; c\ne 0.$ Then $$\begin{aligned} g^n(\xi)(g^{n_1}(\xi))^{(t_1)}\cdots(g^{n_k}(\xi))^{(t_k)}&=e^{nc\xi+nd}(e^{n_1c\xi+n_1d})^{(t_1)}\cdots (e^{n_kc\xi+n_kd})^{(t_k)}\\ &=(n_1c)^{t_1}\cdots (n_kc)^{t_k}e^{(n+\sum_{j=1}^kn_j)c\xi+(n+\sum_{j=1}^kn_j)d}.\end{aligned}$$ Then $(n_1c)^{t_1}\cdots (n_kc)^{t_k}e^{(n+\sum_{j=1}^kn_j)c\xi+(n+\sum_{j=1}^kn_j)d}\equiv a,$ which is impossible. So, $g^n(\xi)(g^{n_1}(\xi))^{(t_1)}\dots(g^{n_k}(\xi))^{(t_k)}$ is nonconstant, which proves the claim. By the assumption of Theorem \[Th1\] and by Hurwitz’s theorem, for every $m\in\{1,\dots,q\},$ all zeros of $g(\xi)^n(g^{n_1}(\xi))^{(t_1)}\cdots(g^{n_k}(\xi))^{(t_k)}-a_m$ have multiplicity at least $\ell_m.$ For any $j\in\{1,\cdots,k\},$ we have that $(g^{n_j}(\xi))^{(t_j)}$ is nonconstant. Indeed, if $(g^{n_j}(\xi))^{(t_j)}$ is constant for some $j\in\{1,\dots,k\},$ then since $n_j\geq t_j,$ and since $g$ is nonconstant, we get that $n_j=t_j$ and $g(\xi)=a\xi+b,$ where $a,b$ are constants, $a\ne 0.$ Thus, we can write$$g(\xi)^n(g^{n_1}(\xi))^{(t_1)}\cdots(g^{n_k}(\xi))^{(t_k)}=c(a\xi+b)^{n+\sum_{j=1}^k(n_j-t_j)},$$ where $c$ is a nonzero constant. This contradicts to the fact that all zeros of $g(\xi)^n(g^{n_1}(\xi))^{(t_1)}\cdots(g^{n_k}(\xi))^{(t_k)}-a_m$ have multiplicity at least $\ell_m\geq2$ (note that $a_m\ne 0$, and that, by condition b) of Theorem \[Th1\], $n+ \sum_{j=1}^k (n_j-t_j) >0$). Thus, $(g^{n_j}(\xi))^{(t_j)}$ is nonconstant, for all $j\in\{1,\cdots,k\}.$ On the other hand, we can write $${(g^{n_j})}^{(t_j)}=\sum c_{m_0,m_1,...,m_{t_j}}g^{m_0}{(g')}^{m_1}\dots{(g^{(t_j)})}^{m_{t_j}},$$ $c_{m_0,m_1,...,m_{t_j}}$ are constants, and $m_0, m_1,\dots, m_{t_j}$ are nonnegative integers such that $ m_0+\dots+m_{t_j}=n_j,\sum_{j=1}^{t_j}{jm_j}=t_j.$ Thus, by an easy computation, we get that $d(P)=n+\sum_{j=1}^kn_j, \theta(P)=\sum_{j=1}^kt_j.$ Now, we apply Lemma \[L3\] for the differential polynomial $$P=g(\xi)^n(g^{n_1}(\xi))^{(t_1)}\cdots(g^{n_k}(\xi))^{(t_k)}.$$ By Lemma \[L3\], we have (note that, by condition b) of Theorem \[Th1\], $n+ \sum_{j=1}^k n_j \geq 2$) $$\begin{aligned} \label{ad3} T(r,g)&\leqslant\frac{q\sum_{j=1}^kt_j+1}{qn+q\sum_{j=1}^kn_j-1}\overline{N}(r,\frac{1}{g})\notag\\ &\quad\quad+\frac{1}{qn+q\sum_{j=1}^kn_j-1}\sum_{m=1}^q\overline{N}(r,\frac{1}{P-a_m})+o(T(r,g)).\end{aligned}$$ For any $m\in\{1,\dots,q\},$ we have, by the First Main Theorem, $$\begin{aligned} \label{ad4} \overline{N}(r,\frac{1}{P-a_m})&=\overline{N}(r,\frac{1}{g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)}-a_m})\notag\\ &\leqslant\frac{1}{\ell_m}N(r,\frac{1}{g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)}-a_m})\notag\\ &\leqslant\frac{1}{\ell_m}T(r,g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)})+O(1)\notag\\ &=\frac{1}{\ell_m}m(r,g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)})\notag\\ &\quad\quad+\frac{1}{\ell_m}N(r,g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)})+O(1).\end{aligned}$$ By the Lemma on Logarithmic Derivative and by the First Main Theorem, $$\begin{aligned} \label{ad5} m(r,&g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)})+N(r,g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)})\notag\\ &\leqslant m(r,\frac{g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)}}{g^ng^{n_1}\cdots g^{n_k}})+m(r,g^ng^{n_1}\cdots g^{n_k})\notag\\ &\quad\quad\quad\quad+N(r,g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)})\notag\\ &\leqslant (n+\sum_{j=1}^kn_j)m(r,g)+N(r,g^n(g^{n_1})^{(t_1)}\cdots(g^{n_k})^{(t_k)})+o(T(r,g))\notag\\ &= (n+\sum_{j=1}^kn_j)m(r,g)+(n+\sum_{j=1}^kn_j)N(r,g)+(\sum_{j=1}^k t_j)\overline{N}(r,g)+o(T(r,g))\notag\\ &\leqslant (n+\sum_{j=1}^kn_j)T(r,g)+ (\sum_{j=1}^kt_j)\overline{N}(r,g)+o(T(r,g)).\end{aligned}$$ Combining with (\[ad4\]), for all $m\in\{1,\dots,q\}$ we have $$\begin{aligned} \label{ad6} \overline{N}(r,\frac{1}{P-a_m})&\leqslant\frac{1}{\ell_m}(n+\sum_{j=1}^kn_j)T(r,g)+\frac{1}{\ell_m}(\sum_{j=1}^k t_j)\overline{N}(r,g)+o(T(r,g))\notag\\ &\leq\frac{1}{\ell_m}(n+\sum_{j=1}^kn_j+\sum_{j=1}^kt_j)T(r,g)+o(T(r,g)). \end{aligned}$$ Therefore, by (\[ad3\]) and by the First Main Theorem, we have $$\begin{aligned} (qn+q\sum_{j=1}^kn_j-1)T(r,g)\leqslant(q\sum_{j=1}^kt_j+1)\overline{N}(r,\frac{1}{g})+\sum_{m=1}^q\overline{N}(r,\frac{1}{P-a_m})+o(T(r,g))\\ \leqslant(q\sum_{j=1}^kt_j+1)T(r,g)+(n+\sum_{j=1}^kn_j+\sum_{j=1}^kt_j)(\sum_{m=1}^q\frac{1}{\ell_m})T(r,g)+o(T(r,g).\end{aligned}$$ This implies that $$\begin{aligned} \frac{ qn+\sum_{j=1}^kq(n_j-t_j)-2}{n+\sum_{j=1}^k(n_j+t_j)}T(r,g)\leqslant \sum_{m=1}^q\frac{1}{\ell_m}T(r,g)+o(T(r,g)).\end{aligned}$$ Combining with assumption $b)$ we get that $g$ is constant. This is a contradiction. Hence $\mathcal F$ is a normal family. We have completed the proof of Theorem \[Th1\]. $\Box$ We can obtain Theorem \[Th2\] by an argument similar to the the proof of Theorem \[Th1\]: We first remark that although condition b) of Theorem \[Th2\] is different from condition b) of Theorem \[Th1\], whereever it has been used in the proof of Theorem \[Th1\] before equation (\[ad3\]), the condition b) of Theorem \[Th2\] still allows the same conclusion. And from equation (\[ad3\]) on we modify as follows : Since $\mathcal F$ is a family of holomorphic functions and by Remark \[R1\], $g$ is an entire functions. So, similarly to (\[ad3\]), by Lemma \[L3\], we have $$\begin{aligned} \label{ad3a} T(r,g)\leqslant\frac{q\sum_{j=1}^kt_j+1}{qn+q\sum_{j=1}^kn_j}\overline{N}(r,\frac{1}{g})+\frac{1}{q(n+\sum_{j=1}^kn_j)}\sum_{m=1}^q\overline{N}(r,\frac{1}{P-a_m})+o(T(r,g))\notag\\ \leqslant\frac{q\sum_{j=1}^kt_j+1}{qn+q\sum_{j=1}^kn_j}T(r,g)+\frac{1}{q(n+\sum_{j=1}^kn_j)}\sum_{m=1}^q\overline{N}(r,\frac{1}{P-a_m})+o(T(r,g)).\end{aligned}$$ Since g is a holomorphic function, $\overline{N}(r,g)=0$. Therefore, by (\[ad4\]) and (\[ad5\]) (which remain unchanged), we have $$\begin{aligned} \label{ad5a} \overline{N}(r,\frac{1}{P-a_m})\leqslant\frac{1}{\ell_m}(n+\sum_{j=1}^kn_j)T(r,g)+o(T(r,g)). \end{aligned}$$ By (\[ad3a\]), (\[ad5a\]), we have $$\begin{aligned} \frac{ qn+\sum_{j=1}^kq(n_j-t_j)-1}{n+\sum_{j=1}^kn_j}T(r,g)\leqslant \sum_{m=1}^q\frac{1}{\ell_m}T(r,g)+o(T(r,g)).\end{aligned}$$ Combining with assumption $b)$ of Theorem \[Th2\], we get that $g$ is constant. This is a contradiction. We have completed the proof of Theorem \[Th2\]. $\Box$\ In connection with Remark \[Re\], we note that the proofs of Theorem \[Th1\] and Theorem \[Th2\] remain valid for the case where the monomial $f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}$ is replaced by the following polynomial $$\begin{aligned} f^n(f^{n_1})^{(t_1)}\cdots(f^{n_k})^{(t_k)}+\sum_{I}c_If^{n_I}(f^{n_{1I}})^{(t_{1I})}\cdots(f^{n_{kI}})^{(t_{kI})},\end{aligned}$$ where $c_I$ is a holomorphic function on $D,$ and $n_I,n_{jI},t_{jI}$ are nonnegative integers satisfying $$\alpha_I:=\frac{ \sum_{j=1}t_{jI}}{n_I+\sum_{j=1}^kn_{jI}}<\alpha:=\frac{ \sum_{j=1}t_j}{n+\sum_{j=1}^kn_j}.$$ In fact, since $\alpha_I<\alpha$ and by (\[ad1\]), we get $$\begin{aligned} {g_I}_v(\xi):=\frac{f_v(z_v+\rho_v\xi)}{\rho_v^{\alpha_I}}=\rho_v^{\alpha-\alpha_I}g_v(\xi)\to 0,\end{aligned}$$ spherically uniformly on compact subsets of $\C.$ Therefore, similarly to (\[ad2\]) $$\begin{aligned} c_I(z_v+\rho_v\xi)f_v^{n_I}(z_v&+\rho_v\xi)(f_v^{n_{1I}})^{(t_{1I})}(z_v+\rho_v\xi)\cdots(f_v^{n_{kI}})^{(t_{kI})}(z_v+\rho_v\xi)\notag\\ &=c_I(z_v+\rho_v\xi){g_I}_v^{n_I}(\xi)(g_v^{n_{1I}}(\xi))^{(t_{1I})}\dots({g_I}_v^{n_{Ik}}(\xi))^{(t_{kI})}\to 0,\end{aligned}$$ spherically uniformly on compact subsets of $\C.$ This implies that $$\begin{aligned} \label{ad2a} f_v^{n}(z_v&+\rho_v\xi)(f_v^{n_1})^{(t_1)}(z_v+\rho_v\xi)\cdots(f_v^{n_k})^{(t_k)}(z_v+\rho_v\xi)\notag\\ &+\sum_{I}c_I(z_v+\rho_v\xi)f_v^{n_I}(z_v+\rho_v\xi)(f_v^{n_{1I}})^{(t_{1I})}(z_v+\rho_v\xi)\cdots(f_v^{n_{kI}})^{(t_{kI})}(z_v+\rho_v\xi)\notag\\ &=g_v^n(\xi)(g_v^{n_1}(\xi))^{(t_1)}\dots(g_v^{n_k}(\xi))^{(t_k)}\notag\\ &\quad\quad+\sum_{I}c_I(z_v+\rho_v\xi){g_I}_v^{n_I}(\xi)({g_I}_v^{n_{1I}}(\xi))^{(t_{1I})}\dots({g_I}_v^{n_{Ik}}(\xi))^{(t_{kI})}\notag\\ &\to g^n(\xi)(g^{n_1}(\xi))^{(t_1)}\dots(g^{n_k}(\xi))^{(t_k)}.\end{aligned}$$ spherically uniformly on compact subsets of $\C.$ We use again the proofs of Theorem \[Th1\] and Theorem \[Th2\] for the general case above after changing (\[ad2\]) by (\[ad2a\]).$\Box$ Appendix ======== Using our methods above, we give a slightly simpler proof of the case of Theorem B above which did not follow from our Corollary \[H2\]: \[add\] Let $k$ be a positive integer and $a$ be a nonzero constant. Let $\mathcal F$ be a family of entire functions in a complex domain $D$ such that for every $f\in\mathcal F,$ $(f^{k+1})^{(k)}(z)\ne a$ for all $z\in D.$ Then $\mathcal F$ is normal on $D.$ In order to prove the above theorem we need the following lemma: \[Hen\] Let g be a transcendental holomorphic function on the complex plane $\C,$ and $k$ be a positive integer. Then $(g^{k+1})^{(k)}$ assumes every nonzero value infinitely often. [**Proof of Theorem \[add\].**]{} Without loss the generality, we may assume that $D$ is the unit disc. Suppose that $\mathcal F$ is not normal at $z_0\in D.$ Then, by Lemma \[L1\], for $\alpha =\frac{k}{k+1}$ there exist $1)$ a real number $r,\;0<r<1,$ $2)$ points $z_v,\;|z_v|<r,$ $z_v\to z_0,$ $3)$ positive numbers $\rho_v,\rho_v\to 0^+,$ $4)$ functions $f_v,\;f_v\in\mathcal F$ such that $$\begin{aligned} \label{ad1} g_v(\xi)=\frac{f_v(z_v+\rho_v\xi)}{\rho_v^\alpha}\to g(\xi)\end{aligned}$$ spherically uniformly on compact subsets of $\C,$ where $g(\xi)$ is a non-constant holomorphic function and $g^{\#}(\xi)\leqslant g^{\#}(0)=1.$ Therefore $$\begin{aligned} (f_v^{k+1})^{(k)}(z_v+\rho_v\xi)=\big((\frac{f_v(z_v+\rho_v\xi)}{\rho_v^\alpha})^{k+1}\big)^{(k)}\\ =\big(g_v^{k+1}(\xi)\big)^{(k)}\to(g^{k+1}(\xi))^{(k)}\end{aligned}$$ spherically uniformly on compact subsets of $\C.$ By Hurwitz’s theorem either $(g^{k+1})^{(k)}\equiv a$, either $(g^{k+1})^{(k)}\ne a.$ On the other hand, it is easy to see that there exists $z_0$ such that $(g^{k+1})^{(k)}(z_0)= a$ (the case where $g$ is a nonconstant polynomial is trivial and the case where $g$ is transcendental follows from Lemma \[Hen\]). Hence, $(g^{k+1})^{(k)}\equiv a.$ Therefore $g$ has no zero point. Hence, by Lemma \[L2\], $g(\xi)=e^{c\xi+d},\; c\ne 0.$ Then $a\equiv (g^{k+1})^{(k)}(\xi)\equiv ((k+1)c)^ke^{(k+1)(c\xi+d)},$ which is impossible.$\Box$\ [99]{} C. T. Chuang, On differential polynomials. In: Analysis of One Complex Variable, World Sci. Publishing, Singapore, 1987, 12-32. J. Clunie and W. K. Hayman, *The spherical derivative of integral and meromorphic functions,* Comm. Math. Helv., **40** (1966), 117-148. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964. W. Hennekemper, *Über die Werteverteilung von $(f^{k+1})^{(k)}$ ,* Math. Z., **177** (1981), 375-380. J. D. Hinchliffe, *On a result of Chuang related to Hayman’s Alternative,* Comput. Method. Funct. Theory, **2** (2002), 293-297. W. Schwick, *Normal criteria for families of meromorphic function,* J. Anal. Math., **52** (1989), 241-289. X. C. Pang and L. Zalcman, *On theorems of Hayman and Clunie,* New Zealand J. Math., **28** (1999), 71-75. L. Zalcman, *Normal families: New perspective,* Bull. Amer. Math. Soc., **35** (1998), 215-230.

--- abstract: 'The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg [@BBS08] proved a law of large numbers for this quantity. Recently, Limic and Talarczyk [@LiT15] proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-time branching processes.' author: - 'Yier Lin[^1] and Bastien Mallein[^2]' title: Second order behavior of the block counting process of beta coalescents --- Introduction {#sec:intro} ============ A coalescent process is a stochastic model for the genealogy of an infinite haploid population, built backward in time. In such a model, an individual is represented by an integer $n \in {\mathbb{N}}$. At each time $t$, we denote by $\Pi(t)$ the partition of ${\mathbb{N}}$ such that two individuals $i$ and $j$ belong to the same set in $\Pi(t)$ (that we call “bloc” from now on) if they share a common ancestor less than $t$ units of time in the past. In particular, we always assume that $\Pi(0) = \{ \{ 1\},\{2\}, \ldots \}$ is the partition in singletons. We construct $(\Pi(t), t \geq 0)$ as a Markov process on the set of partitions, that gets coarser over time. Let $\Lambda$ be a probability measure on $[0,1]$. The $\Lambda$-coalescent is a coalescent process such that given there are $b$ distinct blocs in $\Pi(t)$, any particular set of $k$ blocs merge at rate $$\lambda_{b,k} = \int_0^1 x^{k-2}(1-x)^{b-k} \Lambda(dx).$$ The $\Lambda$-coalescent has been introduced independently by Pitman [@Pit99] and Sagitov [@Sag99]. In this process, several blocs may merge at once, but at most one such coalescing event may occur at a given time. For any $t \geq 0$, we denote by $N(t)$ the number of blocs in $\Pi(t)$. We have in particular $N(0) = +\infty$. We say that the $\Lambda$-coalescent comes down from infinity if almost surely $N(t) < +\infty$ for any $t > 0$. Pitman [@Pit99] proved that if $\Lambda(\{1\})=0$, either the $\Lambda$-coalescent comes down from infinity, or $N(t)= +\infty$ for any $t >0$ a.s. In the rest of the article, we always assume that $\Lambda$ has no atom at 1. Schweinsberg [@Sch00] obtained a necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity, that Bertoin and Le Gall [@BeG06] proved equivalent to $$\label{eqn:defPsi} \int_1^{+\infty} \frac{dq}{\psi(q)} < +\infty, \quad \text{where } \psi(q) = \int_0^1 (e^{-qx} - 1 + q x) x^{-2} \Lambda(dx).$$ Berestycki, Berestycki and Limic [@BBL10] obtained the almost sure behaviour for the number of blocs $N(t)$ as $t$ goes to 0, which they called the speed of coming down from infinity. More precisely, setting $v_\psi(t) = \inf\{ s > 0 : \int_s^{+\infty} \frac{dq}{\psi(q)} \leq t \}$, they proved that for a $\Lambda$-coalescent that comes down from infinity, $$\label{eqn:speedCDI} \lim_{t \to 0} \frac{N(t)}{v_\psi(t)} = 1 \quad \text{a.s.}$$ In this article, we consider the one parameter family of coalescent processes called Beta-coalescents. For any $\alpha \in (0,2)$, we consider the $\Lambda$-coalescent such that the measure $\Lambda$ is ${\mathrm{Beta}}(2-\alpha,\alpha)$, i.e. $$\Lambda(dx) = \frac{1}{\Gamma(\alpha)\Gamma(2-\alpha)} x^{1-\alpha} (1-x)^{\alpha - 1} dx.$$ The Beta-coalescents have a number of interesting properties (see e.g. [@BBC+05; @BBS08] and references therein). In particular, if $\alpha \in (1,2)$, it can be constructed as the genealogy of an $\alpha$-stable continuous state branching process. We observe that thanks to , $\alpha \in (1,2)$ is a necessary and sufficient condition for the Beta-coalescent to come down from infinity. Moreover, can be restated as $$\lim_{t \to 0} t^\frac{1}{\alpha-1} N(t) = (\alpha \Gamma(\alpha))^\frac{1}{\alpha - 1} \quad \text{a.s.}$$ The speed of coming down from infinity for the Beta coalescent can also be found in [@BBS08]. The main result of this article is a central limit theorem for the number of blocs, as $t \to 0$. \[thm:main\] Let $\alpha \in (1,2)$ we set $(\Pi(t),t \geq 0)$ the ${\mathrm{Beta}}(2-\alpha,\alpha)$-coalescent and $N(t) = \# \Pi(t)$ the number of blocs at time $t$, we have $$\lim_{t \to 0} t^{\frac{1}{\alpha(\alpha-1)}} \left(N(t) - \left(\frac{\alpha \Gamma(\alpha)}{t}\right)^{\frac{1}{\alpha -1}} \right) = - D_\alpha X \quad \text{in law,}$$ where $D_\alpha = \left(\Gamma(\alpha)\alpha \right)^{\frac{1}{\alpha(\alpha-1)}}(\alpha-1)^{-\frac{1}{\alpha}}$, $X=\int_0^1 Y(t) dt$ and $(Y(t), t\geq 0)$ is a Lévy process satisfying $\operatorname{\mathbb{E}}(e^{-\lambda Y_t}) = e^{t \lambda^\alpha}$. Note that a more precise functional central limit theorem has been obtained by [@LiT15] for any $\Lambda$-coalescent with a regularly varying density in a neighbourhood of $0$. However, our proof follows from simple coupling arguments, that might be of independent interest. We observe that the random variable $X$ defined in Theorem \[thm:main\] is an $\alpha$-stable random variable, that satisfies $$\operatorname{\mathbb{E}}(e^{-\lambda X}) = \exp\left( \frac{\lambda^\alpha}{\alpha + 1} \right).$$ In Section \[sec:csbp\], we use [@BBS08] to couple the Beta-coalescent with a stable continuous state branching process, and link the small times behaviour of the number of blocs with the small times behaviour of the continuous-state branching process. In Section \[sec:lamperti\], we use the so-called Lamperti transform to transfer the computations into the small times asymptotic of an $\alpha$-stable Lévy process, and use scaling properties to conclude. Continuous state branching process {#sec:csbp} ================================== A continous-state branching process (or CSBP for short) is a càdlàg (right-continuous with left limits at each point) Markov process $(Z(t), t \geq 0)$ on ${\mathbb{R}}_+$ that satisfies the so-called branching property: For any $x, y \geq 0$, if $(Z_x(t), t \geq 0)$ and $(Z_y(t), t \geq 0)$ are two independent versions of $Z$ starting from $x$ and $y$ respectively, then the process $(Z_x(t) + Z_y(t), t \geq 0)$ is also a version of $Z$ starting from $x+y$. The study of CSBP started with the seminal work of [@Jir58]. As observed in [@Lam67; @Sil67], there exists a deep connexion between CSBP and Lévy processes. In effect, we observe that for any $x,t, \lambda \geq 0$, the Laplace transform of the CSBP $Z$ satisfies $$\operatorname{\mathbb{E}}\left( \exp(-\lambda Z_x(t) \right) = \exp(- x u_t(\lambda)),$$ where $u$ is the solution of the following differential equation $$\label{eqn:csbpLevy} \partial_t u_t(\lambda) = \phi(u_t(\lambda)), \quad \text{with } u_0(\lambda) = \lambda,$$ and $\phi$ is the Lévy-Khinchine exponent of a spectrally positive Lévy process (i.e. a Lévy process with no negative jump). The function $\phi$ is called the branching mechanism of the CSBP. If $\phi : \lambda \mapsto \lambda^\alpha$ with $\alpha \in (1,2)$, we call $Z$ the $\alpha$-stable CSBP. Let $\alpha \in (1,2)$. Berestycki, Berestycki and Schweinsberg gave in [@BBS08] a coupling between the $\alpha$-stable CSBP and the ${\mathrm{Beta}}(2-\alpha,\alpha)$-coalescent, that we recall here. Let $(Z_a(t), t \geq 0, a \in [0,1])$ be a random field, càdlàg in $t$ and $a$, such that for any $a < b$, the process $(Z_b(t)-Z_a(t), t \geq 0)$ is the $\alpha$-stable CSBP starting from $b-a$, and is independent with $(Z_c(t), t \geq 0, c < a)$. For any $t > 0$, the function $a \mapsto Z_a(t)$ is a.s. increasing, and we set $$\label{eqn:defD} D(t) = \# \left\{ a \in (0,1) : Z_{a-}(t) < Z_a(t) \right\}$$ the number of atoms in the measure $\mu_t$ satisfying $\mu_t([0,a]) = Z_a(t)$ a.s. We also introduce $R(t) = C_\alpha \int_0^t Z_1(s)^{1-\alpha} dt$, where $C_\alpha = \alpha(\alpha-1)\Gamma(\alpha)$, as well as its generalized inverse $$\label{eqn:defRm1} R^{-1}(t) = \inf\left\{ s \geq 0 : R(s) > t \right\}.$$ The coupling between the CSBP and the Beta-coalescent is obtained as a straightforward combination of Lemmas 2.1 and 2.2 in [@BBS08]. \[lem:coupling\] For any $t > 0$, we have $N(t) {{\overset{(d)}{=}}}D(R^{-1}(t))$. Using this result, to compute the small times behaviour of $N(t)$, it is enough to study the asymptotic behaviour of $D(r)$ and $R^{-1}(t)$ separately. We first provide a straightforward estimate on the asymptotic behaviour of $D$. \[thm:poisson\] For any $\alpha \in (1,2)$, for any $\epsilon>0$, we have $$\lim_{r \to 0} \frac{D(r) - ((\alpha-1)r)^{-\frac{1}{\alpha -1}}}{r^{-\frac{1}{2(\alpha-1)}-\epsilon}} = 0 \quad \text{a.s.}$$ We note that $(D(r), r > 0)$ is decreasing. Moreover, for any $r \geq 0$, $D(r)$ is a Poisson random variable with parameter $\theta_r = ((\alpha-1)r)^{-\frac{1}{\alpha-1}}$, by Lemma 2.2 of [@BBS08]. Therefore, by a deterministic change of variables, it is enough to observe that for any increasing process $(P(t), t \geq 0)$ such that $P(t)$ is a Poisson random variable with parameter $t$, we have $$\lim_{t \to +\infty} \frac{P(t)-t}{t^{\frac{1}{2}+\epsilon}} = 0 \quad \text{a.s.}$$ Using the exponential Markov inequality, for any $\lambda > 0$ we have $${\mathbb{P}}(P(t) - t > t^{\frac{1}{2} + \epsilon}) \leq e^{-\lambda t^{\frac{1}{2}+\epsilon}} \operatorname{\mathbb{E}}\left( e^{\lambda(P(t)-t)} \right) = \exp\left( t (e^{\lambda}-1-\lambda) - \lambda t^{\frac{1}{2} + \epsilon} \right).$$ Applying this inequality with $\lambda = t^{-1/2}$, there exists $C_\epsilon>0$ such that for any $t \geq 1$, ${\mathbb{P}}(P(t) - t > t^{\frac{1}{2} + \epsilon}) \leq C_\epsilon e^{-t^\epsilon}$. With similar computations, we have $${\mathbb{P}}(P(t) - t < - t^{\frac{1}{2} + \epsilon}) \leq C_\epsilon e^{-t^{\epsilon}}.$$ We apply the Borel-Cantelli lemma, yielding $\limsup_{n \to + \infty} \frac{|P(n)-n|}{n^{\frac{1}{2}+\epsilon}} \leq 1$ a.s. As $P$ is increasing, we obtain that for any $\epsilon>0$, $\lim_{t \to +\infty} \frac{P(t)-t}{t^{\frac{1}{2}+\epsilon}} = 0$ a.s. concluding the proof. The Lamperti transform {#sec:lamperti} ====================== The connexion between CSBP and spectrally positive Lévy processes observed in can be strengthen. In [@Lam67], Lamperti observed that a CSBP with branching mechanism $\phi$ could be constructed as a random time change of a Lévy process with Lévy-Khinchine exponent $\phi$. A proof of this result can be found in [@CLUB]. More precisely, let $(Y(t), t \geq 0)$ be a spectrally positive Lévy process starting from $a$, such that $\operatorname{\mathbb{E}}(e^{-\lambda Y(t)}) = e^{-a\lambda+t \phi(\lambda)}$. We set $T = \inf\{ s \geq 0 : Y(s) \leq 0 \}$ and $$U(t) = \inf\left\{ s \geq 0 : \int_0^s \frac{dr}{Y(r \wedge T)} > t \right\}.$$ The Lamperti transform states that for $Z$ a CSBP with branching mechanism $\phi$ such that $Z(0)=a$, we have $$\label{eqn:lampertiTransform} \left( Z(t), t \geq 0 \right) {{\overset{(d)}{=}}}\left( Y(U(t)), t \geq 0 \right)$$ In the rest of the section, we denote by $(Y(t), t \geq 0)$ a Lévy process with Lévy-Khinchine exponent $\phi(\lambda) = \lambda^\alpha$ such that $Y(0)=1$ a.s. We also set $Y_0(t) = Y(t)-1$. We write $T = \inf\left\{ s \geq 0 : Y(s) \leq 0 \right\}$ and $$U(t) = \inf\left\{ s \geq 0 : \int_0^s \frac{du}{Y(u \wedge T)} \geq t \right\}.$$ Using , the process defined in satisfies $$\label{eqn:applLamperti} \left(R^{-1}(t), t \geq 0 \right) {{\overset{(d)}{=}}}\left( \inf\left\{ s \geq 0 : C_\alpha\int_0^s Y(U(u))^{1-\alpha} du \geq t\right\}, t \geq 0 \right).$$ Therefore, up to a slight abuse of notation, we write $$\label{eqn:defR} R(t) = C_\alpha\int_0^t Y(U(s))^{1-\alpha} ds = C_\alpha \int_0^{U(t)} Y(u)^{-\alpha} du,$$ by change of variable, and again $R^{-1}(t) = \inf\left\{ s \geq 0 : R(s) \geq t \right\}$. We first prove a central limit theorem for the asymptotic behaviour of $R(t)$ as $t \to 0$. \[thm:asymptoticR\] We denote by $X = \int_0^1 Y_0(s) ds$. We have $$\lim_{t \to 0} \frac{R(t) - C_\alpha t}{t^{1 + \frac{1}{\alpha}}} = (1-\alpha)C_\alpha X \quad \text{in law}.$$ For any $\epsilon>0$ and $t > 0$, we write ${\mathcal{A}}_{t,\epsilon} = \{ |Y(s)-1| \leq \epsilon, s \leq 2t \}$ the event such that $Y$ stays in an $\epsilon$ neighbourhood of 1 until time $2t$. As observed in [@BBS08 Lemma 4.2], there exists $C>0$ such that ${\mathbb{P}}({\mathcal{A}}_{t,\epsilon}^c) \leq C t \epsilon^{-\alpha}$. We first prove that $\lim_{t \to 0} \frac{U(t)}{t} = 1$ and $\lim_{t \to 0} \frac{R(t)}{t} = C_\alpha$ a.s. Let $\epsilon<1/2$, observe that on the event ${\mathcal{A}}_{t,\epsilon}$, we have $T > 2t$, therefore for any $s \leq t$, we have $$U(s) = \inf\left\{ r \geq 0 : \int_0^r \frac{du}{Y(u)} \geq s \right\} \in \left[ \tfrac{s}{1+\epsilon}, \tfrac{s}{1-\epsilon}\right].$$ In particular, letting $t \to 0$ we obtain $$\frac{1}{1+\epsilon} \leq \liminf_{s \to 0} \frac{U(s)}{s} \leq \limsup_{s \to 0} \frac{U(s)}{s} \leq \frac{1}{1-\epsilon} \quad \text{a.s.}$$ Letting $\epsilon \to 0$, this yields $\lim_{t \to 0} \frac{U(s)}{s} = 1$ a.s. Similarly, by we have $$\frac{1}{(1+\epsilon)^{1+\alpha}} \leq \liminf_{s \to 0} \frac{R(s)}{C_\alpha s} \leq \limsup_{s \to 0} \frac{R(s)}{C_\alpha s} \leq \frac{1}{(1-\epsilon)^{1+\alpha}},$$ yielding $\lim_{t \to 0} \frac{R(t)}{t} = C_\alpha$ a.s. We set $\tilde{R}(t) = R(t) - C_\alpha t$, we have $$\tilde{R}(t) = C_\alpha \int_0^{U(t)} \left(Y(s)^{-\alpha} - \frac{1}{Y(s)}\right) ds = C_\alpha \int_0^{U(t)} \frac{(1 + Y_0(s))^{1-\alpha} - 1}{1+Y_0(s)}ds.$$ As a consequence, we have $$\label{eqn:firstDecomposition} \tilde{R}(t) = C_\alpha(1-\alpha) \int_0^{U(t)} Y_0(s) ds + \Delta(t),$$ where $\Delta(t) = C_\alpha \int_0^{U(t)} \frac{(1 + Y_0(s))^{1-\alpha}-1 -(1-\alpha)Y_0(s) - (1-\alpha)Y_0(s)^2}{1 + Y_0(s)} ds$. Note that as $Y_0$ is an $\alpha$-stable Lévy process, the following scaling property holds for any $\lambda > 0$: $$\label{eqn:scaling} \left(Y_0(t),t \geq 0 \right) {{\overset{(d)}{=}}}\left( \lambda^\frac{1}{\alpha} Y_0(t/\lambda), t \geq 0 \right).$$ We first prove that $\lim_{t \to 0} \frac{\Delta(t)}{t^{1 + \frac{1}{\alpha}}} = 0$ in probability. There exists $K_\alpha > 0$ such that $|(1 + x)^{1-\alpha}-1 -(1-\alpha)x - (1-\alpha)x^2| \leq K_\alpha x^2$ for any $x \in (0,1)$. Therefore, on the event ${\mathcal{A}}_{t,\epsilon}$, for any $s \leq t$, we have $$\begin{aligned} |\Delta(s)| &\leq \int_0^{U(s)} \frac{\left|(1 + Y_0(r))^{1-\alpha}-1 -(1-\alpha)Y_0(r) - (1-\alpha)Y_0(r)^2 \right|}{Y(r)}dr\\ &\leq \frac{K_\alpha}{1-\epsilon} \int_0^{(1+\epsilon)s} Y_0(r)^2 dr. \end{aligned}$$ Using with $\lambda=t$, for any $\delta > 0$, we have $$\begin{aligned} {\mathbb{P}}( |\Delta(t)| \geq \delta t^{1 + \frac{1}{\alpha}} ) &\leq {\mathbb{P}}( {\mathcal{A}}_{t,\epsilon}^c) + {\mathbb{P}}\left( \frac{K_\alpha t^{1 + \frac{2}{\alpha}}}{1-\epsilon} \int_0^{1+\epsilon} Y_0(r)^2 \geq \delta t^{1 + \frac{1}{\alpha}} \right)\\ &\leq C t \epsilon^{-\alpha} + {\mathbb{P}}\left( \frac{K_\alpha}{1-\epsilon} \int_0^{1+\epsilon} Y_0(r)^2 \geq \delta t^{-\frac{1}{\alpha}}\right).\end{aligned}$$ Letting $t \to 0$, we have $\lim_{t \to 0} t^{-1 - \frac{1}{\alpha}} \Delta(t) = 0$ in probability. We now study the asymptotic behaviour of $t^{-1-\frac{1}{\alpha}}\int_0^{U(t)} Y_0(s)ds$. First observe that for any $\delta,\eta > 0$, we have $$\begin{aligned} &{\mathbb{P}}\left( \left| \int_t^{U(t)} Y_0(s) ds \right| \geq \eta t^{1+\frac{1}{\alpha}} \right)\\ \leq &{\mathbb{P}}\left( |U(t)-t| \geq \delta t \right) + {\mathbb{P}}\left( \int_{(1-\delta)t}^{(1+\delta)t} |Y_0(s)| ds \geq \eta t^{1+\frac{1}{\alpha}} \right)\\ \leq &{\mathbb{P}}\left( \left| \frac{U(t)}{t} - 1 \right| \geq \delta \right) + {\mathbb{P}}\left( \int_{1-\delta}^{1+\delta} |Y_0(s)| ds \geq \eta \right),\end{aligned}$$ using . As $\lim_{t \to 0} \frac{U(t)}{t} = 1$ a.s, letting $t \to 0$ then $\delta \to 0$, we conclude that $$\lim_{t \to 0} \int_t^{U(t)} Y_0(s) ds = 0 \quad \text{in probability}.$$ Finally, using again, we have $t^{-1-\frac{1}{\alpha}} \int_0^t Y_0(s) ds {{\overset{(d)}{=}}}\int_0^1 Y_0(s) ds = X$ for any $t >0$. As a conclusion, yields $\displaystyle \lim_{t \to 0} t^{-1-\frac{1}{\alpha}} \tilde{R}_t = (1-\alpha)C_\alpha X$ in law. As a straightforward consequence of Theorem \[thm:asymptoticR\], we obtain the asymptotic behaviour of $R^{-1}$ at small times. \[cor:Rm1\] We have $\lim_{t \to 0} \frac{R^{-1}(t) - \frac{t}{C_\alpha}}{t^{1 + \frac{1}{\alpha}}} = \frac{(\alpha - 1)}{C_\alpha^{1 + \frac{1}{\alpha}}} X$ in law. Let $x \in {\mathbb{R}}$ and $t \geq 0$, we observe that $${\mathbb{P}}\left( R^{-1}(t) - \frac{t}{C_\alpha} > t^{1 + \frac{1}{\alpha}}x \right) = {\mathbb{P}}\left( R(\tau_{x,t}) < t \right),$$ where we set $\tau_{x,t} = \frac{t}{C_\alpha} + t^{1 + \frac{1}{\alpha}} x$. Observe that for any fixed $x \in {\mathbb{R}}$, we have $$t = C_\alpha \tau_{x,t} - x C_\alpha^{2 + \frac{1}{\alpha}} \tau_{x,t}^{1 + \frac{1}{\alpha}} + o(\tau_{x,t}^{1 + \frac{1}{\alpha}}),$$ as $t \to 0$. Therefore, by Theorem \[thm:asymptoticR\], we obtain $$\begin{aligned} \qquad \qquad \lim_{t \to 0} {\mathbb{P}}\left( R^{-1}(t) - \frac{t}{C_\alpha} > t^{1 + \frac{1}{\alpha}}x \right) &= {\mathbb{P}}\left( (1-\alpha)C_\alpha X < - x C_\alpha^{2 + \frac{1}{\alpha}} \right)\\ &= {\mathbb{P}}\left(\frac{(\alpha-1)X}{C_\alpha^{1 + \frac{1}{\alpha}}} > x \right). \qquad \qquad \text{\qedhere}\end{aligned}$$ Using this result, we now compute the asymptotic behaviour of $R^{-1}(t)^{-\frac{1}{\alpha - 1}}$, which is used to prove Theorem \[thm:main\]. \[lem:estimateRpower\] We denote by $D_\alpha = \frac{(\alpha \Gamma(\alpha))^{\frac{1}{\alpha(\alpha-1)}}}{(\alpha-1)^\frac{1}{\alpha}}$, we have $$\lim_{t \to 0} t^{\frac{1}{\alpha(\alpha-1)}} \left( \left((\alpha - 1) R^{-1}(t)\right)^{-\frac{1}{\alpha-1}} - \left(\alpha \Gamma(\alpha)/t\right)^\frac{1}{\alpha - 1} \right) = -D_\alpha X \quad \text{in law.}$$ The proof follows the same lines as Corollary \[cor:Rm1\]. For any $x \in {\mathbb{R}}$, for any $t > 0$ small enough we have $$\begin{aligned} &{\mathbb{P}}\left( \left((\alpha - 1) R^{-1}(t)\right)^{-\frac{1}{\alpha-1}} - \left(\alpha \Gamma(\alpha)/t\right)^\frac{1}{\alpha - 1} > xt^{-\frac{1}{\alpha(\alpha-1)}} \right)\\ &= {\mathbb{P}}\left( (\alpha-1)R^{-1}(t) < \left( \left(\alpha \Gamma(\alpha)/t\right)^\frac{1}{\alpha - 1} + xt^{-\frac{1}{\alpha(\alpha-1)}}\right)^{1-\alpha} \right)\\ &= {\mathbb{P}}\left( (\alpha-1)R^{-1}(t) < \frac{t}{\alpha \Gamma(\alpha)} + \frac{(1-\alpha)x}{(\alpha \Gamma(\alpha))^{\frac{\alpha}{\alpha-1}}}t^{1+\frac{1}{\alpha}} + o(t^{1+\frac{1}{\alpha}}) \right).\end{aligned}$$ Therefore, using Corollary \[cor:Rm1\], we obtain for any $x \in {\mathbb{R}}$ $$\lim_{t \to 0} {\mathbb{P}}\left( \left((\alpha - 1) R^{-1}(t)\right)^{-\frac{1}{\alpha-1}} - \left(\alpha \Gamma(\alpha)/t\right)^\frac{1}{\alpha - 1} > xt^{-\frac{1}{\alpha(\alpha-1)}} \right)={\mathbb{P}}(D_\alpha X < - x),$$ which concludes the proof. By Lemma \[lem:coupling\], the asymptotic behaviours of the number of blocs $N(t)$ and $D(R^{-1}(t))$ are the same. Therefore, we only have to prove that $$\lim_{t \to 0} t^\frac{1}{\alpha(\alpha - 1)} \left(D(R^{-1}(t)) - \left(\alpha \Gamma(\alpha)/t\right)^{\frac{1}{\alpha - 1}}\right) = -D_\alpha X \quad \text{in law}.$$ Observe that by Corollary \[cor:Rm1\], we have $\lim_{t \to 0} C_\alpha R^{-1}(t)/t = 1$ in probability. Moreover, as $\alpha \in (1,2)$, we have $\frac{1}{\alpha(\alpha-1)} > \frac{1}{2(\alpha-1)}$, thus $$\lim_{\tau \to 0} \frac{D(\tau) - \left((\alpha - 1) \tau\right)^{\frac{-1}{\alpha - 1}}}{\tau^{\frac{-1}{\alpha (\alpha - 1)}}} = 0 \quad \text{ a.s.}$$ by Theorem \[thm:poisson\]. We conclude that $$\lim_{t \to 0} t^{\frac{1}{\alpha (\alpha - 1)}}\left(D(R^{-1}(t))- \left((\alpha - 1) R^{-1}(t)\right)^{\frac{-1}{\alpha - 1}}\right) = 0 \quad \text{in probability.}$$ Therefore, using Lemma \[lem:estimateRpower\], we have $$\begin{aligned} &\lim_{t \to 0} t^\frac{1}{\alpha(\alpha - 1)} \left(D(R^{-1}(t)) - \left(\alpha \Gamma(\alpha)/t\right)^{\frac{1}{\alpha - 1}}\right)\\ = &\lim_{t \to 0} t^\frac{1}{\alpha(\alpha - 1)} \left( \left((\alpha - 1) R^{-1}(t)\right)^\frac{-1}{\alpha - 1} - \left(\alpha \Gamma(\alpha)/t\right)^{\frac{1}{\alpha - 1}}\right) = -D_\alpha X \quad \text{in law.}\end{aligned}$$ [10]{} J. Berestycki, N. Berestycki, and V. Limic. The [$\Lambda$]{}-coalescent speed of coming down from infinity. , 38(1):207–233, 2010. J. Berestycki, N. Berestycki, and J. Schweinsberg. Small-time behavior of beta coalescents. , 44(2):214–238, 2008. J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. [III]{}. [L]{}imit theorems. , 50(1-4):147–181 (electronic), 2006. M. Birkner, J. Blath, M. Capaldo, A. Etheridge, M. M[ö]{}hle, J. Schweinsberg, and A. Wakolbinger. Alpha-stable branching and beta-coalescents. , 10:no. 9, 303–325 (electronic), 2005. Ma. Emilia Caballero, Amaury Lambert, and Ger[ó]{}nimo Uribe Bravo. Proof(s) of the [L]{}amperti representation of continuous-state branching processes. , 6:62–89, 2009. M. Ji[ř]{}ina. Stochastic branching processes with continuous state space. , 8 (83):292–313, 1958. J. F. C. Kingman. The coalescent. , 13(3):235–248, 1982. J. Lamperti. Continuous state branching processes. , 73:382–386, 1967. V. Limic and A. Talarczyk. Second-order asymptotics for the block counting process in a class of regularly varying [$\Lambda$]{}-coalescents. , 43(3):1419–1455, 2015. J. Pitman. Coalescents with multiple collisions. , 27(4):1870–1902, 1999. S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. , 36(4):1116–1125, 1999. J. Schweinsberg. A necessary and sufficient condition for the [$\Lambda$]{}-coalescent to come down from infinity. , 5:1–11 (electronic), 2000. M. L. Silverstein. A new approach to local times. , 17:1023–1054, 1967/1968. [^1]: The author would like to express his sincere thank to ’Tsinghua Xue Tang Program’, which provides him funds and opportunity to do research in ENS. [^2]: DMA, ENS.

--- abstract: 'We have calculated a complete set of primary fission fragment mass yields, $Y(A)$, for heavy nuclei across the chart of nuclides, including those of particular relevance to the rapid neutron capture process ($r$ process) of nucleosynthesis. We assume that the nuclear shape dynamics are strongly damped which allows for a description of the fission process via Brownian shape motion across nuclear potential-energy surfaces. The macroscopic energy of the potential was obtained with the Finite-Range Liquid-Drop Model (FRLDM), while the microscopic terms were extracted from the single-particle level spectra in the fissioning system by the Strutinsky procedure for the shell energies and the BCS treatment for the pairing energies. For each nucleus considered, the fission fragment mass yield, $Y(A)$, is obtained from 50,000 – 500,000 random walks on the appropriate potential-energy surface. The full mass and charge yield, $Y(Z,A)$, is then calculated by invoking the Wahl systematics. With this method, we have calculated a comprehensive set of fission-fragment yields from over 3,800 nuclides bounded by $80\leq Z \leq 130$ and $A\leq330$; these yields are provided as an ASCII formatted database in the supplemental material. We compare our yields to known data and discuss general trends that emerge in low-energy fission yields across the chart of nuclides.' author: - 'M. R. Mumpower' - 'P. Jaffke' - 'M. Verriere' - 'J. Randrup' bibliography: - 'refs.bib' title: Primary fission fragment mass yields across the chart of nuclides --- \#1 \ This paper is dedicated to the memory of our friend and colleague Arnie J. Sierk who contributed significantly to the development and application of macroscopic-microscopic nuclear fission theory throughout his career. \ \ \ Introduction ============ The description of nuclear fission has presented exceptional challenges to the theoretical modeling of heavy nuclei since its discovery in the late 1930’s [@Hahn+39]. One way to view this complicated physical process is to consider the evolution of the nuclear shape as it progresses from a compact form through increasingly deformed shapes until the division into two fragments occurs at the scission configuration [@Meitner+39; @Bohr+39], as illustrated in Fig. \[fig:schema\]. This general picture naturally leads to the description of the fission process in terms of a potential-energy surface (PES) as a function of the nuclear shape. The accumulation of many fission events provides the primary fission fragment yield whose appearance is sensitive to the structure of the nuclear system. In this description, much is still uncertain about the evolution of the nuclear shape and, consequently, about the extracted fission yields. For example, what are the most probable trajectories through the shape configuration space? How do these paths depend upon the dissipative coupling of the shape to the remainder of the system? And, which microscopic properties impact the division of the nucleus at scission? Questions like these drive the current research in fission dynamics. Our ability to calculate fission fragment yields across the chart of nuclides has wide reaching implications for a variety of applications, from nuclear security and reactor operations to our understanding of the cosmos in astrophysical explosions [@Andreyev+13; @Eichler+15; @Talou+18; @Jaffke+18; @Horowitz+18; @Holmbeck+19; @Fotiades+19]. Many methods have been proposed for calculating fission fragment yields. Phenomenological approaches [@Kodama+75; @Wahl+80; @Brosa+86; @Wahl+88; @Brosa+90; @Brosa+99; @Benlliure+98; @Schmidt+16] typically consist of simple models with fitted parameters with varying degrees of refinement. The parameters of these models are determined by comparisons to mass or charge yields or other fission observables in the actinide region. Simple, yet insightful descriptions of observed phenomena can arise, such as in the case with the unchanged charge distribution of Ref. [@Wahl+02]. These approaches can reproduce experimental or evaluated data when it is known, but the applicability across the chart of nuclides outside the narrow fitting region is still in question. In contrast, microscopic models for the description of fission are built upon the consideration of an effective energy density functional (EDF), minimized in a chosen trial subspace of the full many-body Fock space while subject to external constraints on the density distribution (e.g. the quadrupole moment $Q_2$ which governs the overall distortion away from a sphere or the octupole moment $Q_3$ which influences the reflection asymmetry of the system) [@Schunck+16]. The self-consistent Hartree-Fock (HF) equations arise from the minimization of the EDF by assuming a system of independent nucleons, with the trial space taken to be the set of Slater determinants of the constituent nucleons. Pairing can be included self-consistently by extending the trial space to quasi-particle Slater determinants, leading to the Hartree-Fock-Bogoliubov (HFB) model [@Goutte+05; @Giuliani+19]. These treatments make it possible to calculate the nuclear PES as a function of the constraints employed ($Q_2$, $Q_3$, ..), and they have been widely used in fission studies [@Berger+89; @Goriely+07; @Minato+09; @Regnier+16]. However, the required computational effort is considerable which imposes a practical limit on the number of constraints that can be included, currently up to just two or three [@Schunck+15; @Regnier+16; @Regnier+17]. As a consequence, the resulting energy surfaces may exhibit spurious discontinuities and, importantly, the fission barrier heights cannot be determined with confidence [@Myers+96; @Moller+09; @Dubray+12; @Schunck+14]. Although methods exist for remedying this inherent problem [@Dubray+12], the required computational cost is prohibitive. The microscopic approach, at the present time, is therefore best suited for studies of specific nuclei, but is not adequate for large-scale, global studies of fission yields and their trends across the chart of nuclides. A recent review covering the progress of this approach can be found in Ref. [@Schunck+16]. ![\[fig:schema\] A schematic illustration of the fission process: The lower panel shows the potential energy of the nuclear system along its most probable path, while the upper panel shows the appearance of the system at four stages along that path. The nuclear shape, which is initially located near that of the ground state, is strongly coupled to the internal microscopic degrees of freedom and, as a result, it executes a Brownian-like random walk on the multidimensional potential-energy surface. After passing over the various saddle points, generally after multiple attempts, the system eventually acquires a binary shape and reaches a necked-in scission configuration where it divides into two fission fragments. The shown potential-energy profile is representative of known actinides, and may be differ qualitatively for nuclei in other regions. ](fission-schematic.pdf){width="\columnwidth"} The macroscopic-microscopic approach offers a simpler and very effective framework for calculating the nuclear PES [@Nix+65]. This method was originally developed for the calculation of nuclear masses because purely microscopic calculations tend to have difficulty obtaining accurate absolute energies due to the small but significant role played by many-body correlations which are hard to treat. Nuclear masses exhibit smoothly varying macroscopic trends, reflecting the energetics of a charged droplet, overlaid with small-amplitude deviations reflecting the microscopic nuclear structure [@Gustafsson+71; @Brack+72; @Bolsterli+72]. The nuclear potential-energy surface is therefore considered to consist of a [*macroscopic*]{} liquid-drop like energy functional, whose parameters (volume energy, surface tension, ...) are determined by global fitting to the measured masses, and a [*microscopic*]{} contribution expressing the shell [@Strutinsky+63] and pairing corrections [@Nogami+64], which can be calculated from the neutron and proton level spectra in the deformed effective potential well. This approach makes it possible to calculate the potential energy of any nuclear system with $Z$ protons and $N$ neutrons, $(Z,N)$, as a function of its shape (as well as its angular momentum). The above approaches can be used to not only provide the static nuclear PES but also to obtain the temporal evolution of the fissioning system. The HF and HFB Hamiltonians naturally lead to the time-dependent Hartree-Fock (TD-HF) and time-dependent Hartree-Fock-Bogoliubov equations (TD-HFB) [@Negele+78; @Bulgac+16; @Scamps+18; @Bulgac+18]. However, these methods are not well suited for processes that generate qualitatively different final configuration, such as fission, because of the restriction to a single Slater determinant. A more general approach considers the time-dependent state as a superposition of many microscopic states having time-dependent weights, leading to the time-dependent generator coordinate method (TDGCM) [@Verriere+17; @Regnier+18]. A recent attempt has been made to couple TD-HF methods with TDGCM [@Berger+91; @Goutte+04; @Regnier+18]. An alternative approach is to treat the evolution of the shape degrees of freedom (whether the multipole constraints $Q_\lambda$ used in the microscopic models or the shape parameters ${{\mbox{\boldmath $\chi$}}}$ used in the macroscopic-microscopic treatment) by means of classical transport theory [@Pomorski+81]. The most complete transport treatment is provided by the Langevin equation [@Wada+92; @Nadtochy+07; @Sierk+17] which, in addition to the PES, also requires the associated collective inertial-mass tensor as well as the dissipation tensor describing the coupling of the collective variables to the remaining system. Because the nuclear shape evolution is strongly dissipative [@Blocki+78; @Bulgac+18], substantial simplification may be obtained by considering the strongly damped limit in which the evolution is governed by the balancing of the driving force from the PES and the dissipative force. The collective coordinates then exhibit a Brownian-like motion which can be simulated numerically as a random walk [@Randrup+11a]. The simplicity of this approach together with its remarkable agreement with known data make it suitable for global studies of fission yields [@Randrup+11b; @Randrup+13]. We therefore employ this treatment of fission dynamics in the present work. The present study requires several successive steps and we discuss them in turn below. First, in Sect. \[sec:shapes\], we introduce the adopted nuclear shape family and then, in Sect. \[sec:PES\], we describe the calculation of the associated potential-energy surfaces. The key assumptions about the shape evolution are reviewed in Sect. \[sec:evo\] and the details of the fragment yield calculations are provided in Sect. \[sec:calcy\]. Building upon past work [@Moller+15a], we calculate, in Sect. \[sec:results\], the primary fragment mass yields for the entire region of nuclides bounded by $80\leq Z \leq 130$ and $A\leq330$ and discuss the emerging global trends. The supplemental material provides the calculated fission yields in tabulated ASCII format. Throughout we make several remarks on the possible implications of these yields for the astrophysical rapid neutron capture process ($r$-process) of nucleosynthesis. Nuclear shapes {#sec:shapes} ============== An important lesson from the fission studies in recent years is that it is critical to consider a sufficiently rich shape family to allow the fissioning system to exploit the detailed topographic features of the associated multi-dimensional potential-energy surface, such as the height and character of the barrier and the shell effects in the emerging fragments. As covered in detail in Ref. [@Moller+09], fallacies in finding saddle points can ensue if only a limited number of shape degrees of freedom are considered. As first pointed out by Nix [@Nix+65], it appears that a minimum of five shape degrees of freedom are required for an adequate description of low-energy fission, namely a measure of the overall elongation of the system, the degree of indentation between the two emerging fragments, the individual deformations of these fragments, and the overall reflection asymmetry. A well suited parameterization is provided by the three-quadratic-surface (3QS) shape family [@Nix+69] in which those shape characteristics are given respectively by the overall quadrupole moment $Q_2$, the neck radius $c$, the deformation parameters $\varepsilon_{\rm f1}$ and $\varepsilon_{\rm f2}$, of the two spheroidal endcaps, and the geometrical mass asymmetry $\alpha_{\rm g}$. The shapes included range from compact (even oblate) configurations (including ground-state shapes) over intermediate shapes (such as saddle and isomeric configurations) to the binary configurations near (and at) scission and beyond. This parameterization has been employed extensively, see Ref. [@Sierk+17] and references therein. In particular, it has been used to calculate potential-energy landscapes from which binding energies [@Moller+16], fission barriers  [@Moller+09; @Moller+15c], and other properties have been derived and benchmarked against available data throughout the nuclear chart. Several alternative shape parameterizations exist to 3QS, e.g. see [@Stavinsky+68; @Hasse+88; @Ivanyuk+14] and references therein. The difficulty for any parameterization in describing fission happens where very distorted shapes appear and where the microscopic effects in the fledging fragments are essential. Other frequently used shape parameterizations, for example, those used in early studies by Nilsson [[*et al.*]{}]{} [@Nilsson+69] employed perturbed spheroids, but while these are well suited for shapes near the ground state (see, for example, Ref. [@Moller+09]), they generally grow ever more inadequate (or impractical) for large deformations. In their seminal work [@Brack+72], Brack and collaborators introduced a three-dimensional shape family that has been employed in numerous studies ever since, but it lacks sufficient flexibility and is more appropriate at higher energies where the microscopic effects are minimal. Finally, the frequently employed multipole expansion of the nuclear radius does not provide a unique representation of the nuclear potential-energy surface in the region of large deformations relevant to fission [@Rohozinski+97]. In the present work we shall therefore employ the 3QS shape family. Accordingly, a particular shape is then characterized by the five-dimensional shape coordinate ${{\mbox{\boldmath $\chi$}}}=(Q_2,c,\varepsilon_{\rm f1},\varepsilon_{\rm f2},\alpha_{\rm g})$ and a corresponding Cartesian lattice was constructed in Ref. [@Moller+09]. We employ a similar discrete Cartesian lattice in the five-dimensional shape parameter space and use the indices $(I,J,K,L,M)$ to identify the sites. The index $I=1,\dots,45$ represents values of the quadrupole moment, $Q_2$; the index $J=1,\dots,15$ represents the neck radius, $c$; the indices $K,L=1,\dots,15$ span endcap deformations $\varepsilon$ ranging from -0.2 to 0.5; and the index $M=-33,\dots,33$ spans a sufficiently wide range of asymmetries. The values of the quantities corresponding to the indices $I,J,K,L$ are not necessarily equidistant. This lattice contains over ten million sites but, due to the fact that the site $(I,J,K,L,M)$ represents the same physical shape as the site $(I,J,L,K,-M)$, except for an overall reflection, there is no need to tabulate potential energies for negative $M$ values. The step size of $\alpha_{\rm g}$ gives a fragment mass resolution of $\Delta A = 2.4$ for $^{240}$Pu and $\Delta A \approx 3$ for the heaviest nuclei considered in this work having a fissioning mass number of $A = 330$. This tolerance is roughly similar to the current experimental fragment mass resolution. Potential-energy surfaces {#sec:PES} ========================= The potential energy of an arbitrarily shaped nuclear system, $U({\mbox{\boldmath $S$}})$, represents the lowest possible energy the system can have at the specified *geometric* shape. This function can be conveniently calculated by means of the macroscopic-microscopic method, according to which the energy is a sum of a smoothly varying liquid-drop like macroscopic term and an undulating microscopic term that accounts for the shell and pairing energies, $$\label{eqn:pes} U(\boldsymbol S) = E_{\rm macro}({\mbox{\boldmath $S$}}) + E_{\rm micro}({\mbox{\boldmath $S$}}),$$ where ${\mbox{\boldmath $S$}}$ denotes the specified shape. When  replaces ${\mbox{\boldmath $S$}}$, this signals that a given choice of shape parameters, in our case from the 3QS shape parameterization, has been used to describe the geometry of the nuclear shape. At a given total energy, $E$, the local excitation energy (i.e. the excitation energy of the nucleus at a specified shape ) is given by $E^*({\mbox{\boldmath $S$}})=E-U({\mbox{\boldmath $S$}})$. As the total energy is increased (by increasing the kinetic energy of the incoming neutron in (n,f) reactions), the local excitation energies increase correspondingly and, genereally, the microscopic contributions to the potential energy decrease. As a result the effective potential energy surface experienced by the evolving shape is modified, approaching $U_{\rm macro}({\mbox{\boldmath $S$}})$ at high energies. This effect will be taken into account by multiplying $U_{\rm micro}({\mbox{\boldmath $S$}})$ by the suppression factor ${\cal S}(E^*({\mbox{\boldmath $S$}}))$ suggested in Ref. [@Randrup+13] and thus using $$U_E({\mbox{\boldmath $S$}}) = E_{\rm macro}({\mbox{\boldmath $S$}}) + E_{\rm micro}({\mbox{\boldmath $S$}})\,{\cal S}(E^*({\mbox{\boldmath $S$}}))\ .$$ This method has been extensively benchmarked and widely applied in the context of fission studies [@Moller+09; @Randrup+11a; @Randrup+11b; @Randrup+13; @Moller+15a; @Sierk+17]. When applied to studies of ground state properties via the Finite-Range Droplet Model (FRDM), it also yields a very good overall reproduction of measured nuclear masses throughout the nuclear chart [@Moller+95; @Moller+12; @Moller+16]. This is an incredible triumph of this methodology, given that the parameters have varied very little over the years, and the predictability with respect to new measurements has remained rather constant. The construction of the nuclear PES proceeds as follows: 1. Specification of a nuclear shape parameterization (in this work the five shape coordinates). 2. Calculate the macroscopic energy terms as outlined in the next section. 3. Calculate the single-particle levels using a folded-Yukawa potential as in Ref. [@Moller+16]. 4. Calculate microscopic shell and pairing corrections. 5. Add the macroscopic and microscopic correction terms together using Eq. (\[eqn:pes\]). A collection of all the possible distinct nuclear shapes between the ground state and scission configurations defines the complete PES for the specified choice of shape parameterization. The large choice of grid space can lead to some shape combinations that may produce unphysical results. We handle this issue, as in past work, by making those points of the PES very large (inaccessible) relative to the physical points. We now review the macroscopic and microscopic terms that comprise the PES. Macroscopic energy {#Emacro} ------------------ For the macroscopic energy, we adopt the Finite-Range Liquid-Drop Model (FRLDM), $E_{\rm macro}({\mbox{\boldmath $S$}})=E_{\rm FRLDM}({\mbox{\boldmath $S$}})$. While this model was described in Ref. [@Moller+95], the actual parameter values employed in Refs. [@Moller+04; @Moller+09] have not appeared in an individual publication. We therefore assemble here the different formulas and parameter values involved in the model for completeness. $$\begin{aligned} E_{\rm FRLDM}(\boldsymbol S) &= M_{\rm H} Z + M_{\rm n} N & \text{mass excess} \\ &- a_{\rm v}E_{\rm V}(\boldsymbol S) & \text{volume energy} \\ &+ a_{\rm s}E_{\rm S}(\boldsymbol S) & \text{surface energy} \\ &+ a_0A^0B_{\rm W}(\boldsymbol S) & \text{A\textsuperscript{0} energy} \\ &+ c_1 \frac{Z^2}{A^{1/3}}B_3(\boldsymbol S) & \text{Coulomb energy} \\ &- c_4\frac{Z^{4/3}}{A^{1/3}} & \text{Coul. exchange corr.} \\ &+ f(k_{\rm f}r_{\rm p})\frac{Z^2}{A} & \text{prot. form-factor corr.} \\ &- c_{\rm a}(N-Z) & \text{charge-asym. energy} \\ &+ WE_{\rm W}(\boldsymbol S) & \text{Wigner energy} \\ &+ \bar{\Delta} & \text{avg. pairing energy} \\ &- a_{\rm el} Z^{2.39} \ , & \text{bound electrons} \end{aligned}$$ where we have $$\begin{aligned} E_{\rm V}(\boldsymbol S) &= \left(1 - \kappa_{\rm v}I^2\right)A \ ,\\ E_{\rm S}(\boldsymbol S) &= \left(1 - \kappa_{\rm s}I^2\right)B_{1}(\boldsymbol S)A^{2/3} \ ,\\ E_{\rm W}(\boldsymbol S) &= |I|B_{\rm W}(\boldsymbol S) + \begin{cases} \frac{1}{A} & \text{$Z$ and $N$ odd and equal\ ,} \\ 0 & \text{otherwise\ ,} \end{cases} \\ \bar{\Delta} &= \begin{cases} + \bar{\Delta}_{\rm p} + \bar{\Delta}_{\rm n} - \delta_{\rm np} & \text{$Z$ and $N$ odd\ ,} \\ + \bar{\Delta}_{\rm p} & \text{$Z$ odd, $N$ even\ ,} \\ + \bar{\Delta}_{\rm n} & \text{$Z$ even, $N$ odd\ ,} \\ + 0 & \text{$Z$ and $N$ even\ ,} \end{cases} \\ I &= \frac{N-Z}{A} \ , \\ c_1 &= \frac{3}{5}\frac{e^2}{r_0} \ , \\ c_4 &= \frac{5}{4}\left(\frac{3}{2\pi}\right)^{\frac{2}{3}}c_1 \ , \\ f(x) &= -\frac{r_{\rm p}^2e^2}{8r_0^3}\left(\frac{145}{48} - \frac{327}{2880}x^2 + \frac{1527}{1209600}x^4\right) \ , \\ k_{\rm f} &= \left(\frac{9\pi Z}{4A}\right)^{\frac{1}{3}}\frac{1}{r_0} \ , \\ \bar{\Delta}_{\rm n} &= \frac{r_{\rm mac}B_{\rm s}(\boldsymbol S)}{N^{1/3}} \ , \\ \bar{\Delta}_{\rm p} &= \frac{r_{\rm mac}B_{\rm s}(\boldsymbol S)}{Z^{1/3}} \ , \\ \delta_{\rm np} &= \frac{h}{B_{\rm s}(\boldsymbol S)A^{2/3}}\ .\end{aligned}$$ The shape-dependent coefficients are the relative surface energy $B_{\rm s}(\boldsymbol S)$, the relative generalized surface energy $B_{1}(\boldsymbol S)$, the relative Coulomb energy $B_3(\boldsymbol S)$ and the relative Wigner energy $B_{\rm W}(\boldsymbol S)$. These quantities are defined by integrals over the geometry of the nuclear shape: $$\begin{aligned} B_{\rm s}(\boldsymbol S) &= \frac{A^{-2/3}}{4\pi r_0^2}\int_S {\rm d}S \ , \\ B_1(\boldsymbol S) &= \frac{A^{-2/3}}{8\pi^2 r_0^2 a^4}\iint_V\left(2 - \frac{\sigma}{a}\right)\frac{e^{-\sigma/a}}{\sigma/a}{\rm d}^3r{\rm d}^3r' \ , \\ B_3(\boldsymbol S) &= \frac{15A^{-5/3}}{32\pi^2 r_0^5} \iint_V \frac{{\rm d}^3r{\rm d}^3r'}{\sigma} \left[1 - \Big(1+\frac{\sigma}{2a_{\rm den}}\Big)e^{-\frac{\sigma}{a_{\rm den}}}\right] \ , \\ B_{\rm W}(\boldsymbol S) &= \begin{cases} \left(1 - \frac{S_3(\boldsymbol S)}{S_1(\boldsymbol S)}\right)^2a_{\rm d} + 1 & \sigma_2 \leq 0 \ , \\ 1 & \sigma_2 \geq 0 \ . \end{cases}\end{aligned}$$ In these expressions, $\sigma = |r - r'|$, $S_1(\boldsymbol S)$ is the area of the maximum cross section of the smaller one of the end bodies and $S_3(\boldsymbol S)$ is the area of the geometric shape $\boldsymbol S$ at the neck location. The model parameters involved in these expressions are decomposed into four categories. The first category corresponds to the fundamental constants and contains $$\begin{array}{rcrl} M_{\rm H} &=& 7.289034 & {\rm MeV} \ , \\ M_{\rm n} &=& 8.071431 & {\rm MeV} \ , \\ e^2 &=& 1.4399764 & {\rm MeV fm}\ . \end{array}$$ The second category is the set of parameters not constrained by atomic masses (e.g. with comparison to evaluated data [@Huang+17; @Wang+17]) $$\begin{array}{rcrl} a_{\rm el} &=& 1.433 \times 10^{-5} & {\rm MeV}\ , \\ r_{\rm p} &=& 0.80 & {\rm fm}\ , \\ r_0 &=& 1.16 & {\rm fm}\ , \\ a &=& 0.68 & {\rm fm}\ , \\ a_{\rm den} &=& 0.70 & {\rm fm}\ . \\ \end{array}$$ The value of the parameters included in these two categories are unchanged since `FRLDM1993` [@Moller+95]. The third category corresponds to the parameters that are chosen from consideration of odd-even mass differences. Their values are $$\begin{array}{rcrl} r_{\rm mac} &=& 4.80 & {\rm MeV}\ , \\ h &=& 6.6 & {\rm MeV}\ , \\ W &=& 0.68 & {\rm MeV}\ , \\ a_{\rm d} &=& 0.9\ . \end{array}$$ In this category, only the value of the Wigner damping constant $a_{\rm d}$ has been modified from $a_{\rm d} = 0$ (`FRLDM1993`) to $a_{\rm d} = 0.9$ (`FRLDM2002`). The nonzero value of the $a_{\rm d}$ parameter plays a role in the preference of asymmetric shape configurations near scission. The fourth and last category contains the parameters that are adjusted on evaluated masses $$\begin{array}{rcrl} a_{\rm v} &=& 16.02500 & {\rm MeV}\ , \\ \kappa_{\rm v} &=& 1.93200 & {\rm MeV}\ , \\ a_{\rm s} &=& 21.33000 & {\rm MeV}\ , \\ \kappa_{\rm s} &=& 2.378 & {\rm MeV}\ , \\ a_{0} &=& 2.04000 & {\rm MeV}\ , \\ c_{\rm a} &=& 0.09700 & {\rm MeV}\ . \end{array}$$ Microscopic energy ------------------ The calculation of the shape-dependent microscopic energy term, $E_{\rm micro}({\mbox{\boldmath $S$}})$, is as described in [FRLDM1993]{} [@Moller+95], including the details of the shell correction and Lipkin-Nogami pairing along with all the associated parameter values. Once a shape family has been adopted, the shape parameter  can be regarded as specifying a sharp generating density, $\hat{\rho}_{{\mbox{\boldmath\scriptsize $\chi$}}}({\mbox{\boldmath $r$}})$, from which the corresponding diffuse effective neutron and proton potentials can be generated by a convolution procedure, using a kernel of Yukawa form; spin-orbit and Coulomb potentials are subsequently added. The Schr[ö]{}dinger equation then yields the associated single-particle level spectra from which the shell energy is obtained by the Strutinsky subtraction procedure and the pairing energy is obtained by means of the BCS treatment [@Bolsterli+72]. The resulting microscopic energy then has an additive form in both the constituent neutrons and protons, $$\begin{aligned} &~& E_{\rm micro}({\mbox{\boldmath $S$}})\ =\ E_{\rm shell}({\mbox{\boldmath $S$}}) + E_{\rm pair}({\mbox{\boldmath $S$}})\\ \nonumber &=& E_{\rm shell}^{({\rm n})}({\mbox{\boldmath $S$}}) +E_{\rm shell}^{({\rm p})}({\mbox{\boldmath $S$}}) + E_{\rm pair}^{({\rm n})}({\mbox{\boldmath $S$}}) + E_{\rm pair}^{({\rm p})}({\mbox{\boldmath $S$}})\ .\end{aligned}$$ It is worth keeping in mind that in the macroscopic-microscopic approach, the effective single-particle potentials as well as the neutron and proton density distributions obtained from the corresponding wave functions generally have multipole moments that differ slightly from those of the specified generating density as well as from one another. Typical features of potential-energy surfaces {#sec:features} --------------------------------------------- Because it is difficult to visualize the features of the five-dimensional potential energy surface, a reduction to two dimensions is often performed and these can be very instructive for the analysis of fission yields. In order to illustrate the typical character of the energy landscape, we show in Fig. \[fig:fispes\] reduced landscapes in the $Q_2-\alpha_{\rm g}$ plane for three widely studied cases, namely (a) $\prescript{236}{92}{\text{U}}$, (b) $\prescript{240}{94}{\text{Pu}}$, and (c) $\prescript{234}{96}{\text{Cm}}$. These two-dimensional visualizations have been obtained by minimizing $U(I,J,K,L,M)$ over $J,K,L$ for each combination $(I,M)$. {width="\textwidth"} The left two panels show that both $^{236}$U and $^{240}$Pu exhibit a distinct barrier ridge ($\gtrsim$ 9 MeV) that inhibits symmetric fission at low energies. These nuclei have their largest barriers in scission trajectories on the order of $5$ MeV. The higher barriers shown here represents the variation in the potential that can arise from the choice of other shape coordinates when projecting down to two dimensions; a key point to remember in our further discussions. Turning to panel (c), a symmetric fission mode is likely in $\prescript{234}{96}{\text{Cm}}$ as the fission path along $\alpha_g=0$ shows no major hills to climb while a slight ridge between $(Q_2/\rm{b})^{\frac{1}{2}} \sim 6$ and $8$ discourages more asymmetric paths. The differences in the topography of these potential-energy surfaces illustrates the importance of the microscopic effects in determining the character of the resulting fission fragment yields. A pedagogical example comes from the study of major actinides, for which the heavy fragment tends to be near the closed neutron shell at $N=82$ and, consequently, tends to have a spherical shape, while the lighter partner is moderately deformed. Shape evolution {#sec:evo} =============== In the treatment of the fission dynamics, the shape parameter ${{\mbox{\boldmath $\chi$}}}=(Q_2,c,\varepsilon_{\rm f1},\varepsilon_{\rm f2},\alpha_{\rm g})$, is regarded as a classical variable. Accordingly, its evolution may be described within the framework of standard transport theory. The most important physical ingredient in this treatment is the potential energy landscape, $U({{\mbox{\boldmath $\chi$}}})$, which provides the driving force acting on the shape coordinate . The resulting ‘acceleration’ of  will endow the system with a collective kinetic energy and it is therefore generally necessary to also know the associated collective inertial-mass tensor, ${\mbox{\boldmath $M$}}({{\mbox{\boldmath $\chi$}}})$. Furthermore, the macroscopic degrees of freedom associated with the nuclear shape are coupled to the remaining, microscopic, degrees of freedom which can be regarded as a thermal reservoir. As a consequence, the shape parameter  continually receives impulses whose effect can be described by means of the collective dissipation tensor, ${\mbox{\boldmath $\gamma$}}({{\mbox{\boldmath $\chi$}}})$. As of now, a microscopic calculation of the inertia tensor ${\mbox{\boldmath $M$}}({{\mbox{\boldmath $\chi$}}})$ would involve the inversion of the QRPA matrix, see Ref. [@Schunck+16]. Standard approximations (e.g. the cranking model) used to avoid calculating the full tensor are not well understood [@Ivanyuk+97]. Therefore, more complete dynamical treatments, such as those based on the classical or quantal Langevin equation, most often employ a fluid-dynamical mass tensor calculated under the assumption of incompressible irrotational flow, even though the resulting tensor is known to be incorrect, both quantitatively and even qualitatively. In our present treatment, we avoid this problem by working in the limit of strong dissipation where the collective motion is so slow that the inertia plays no role for the shape evolution [@Randrup+11a; @Randrup+11b]. Strongly damped limit --------------------- If the coupling of the considered shape degrees of freedom  to the residual nuclear system is sufficiently strong then the resulting shape motion is so slow that the inertial effects are negligible [@Abe+96]. In this limit, the general Langevin equation reduces to the Smoluchowski equation which expresses the balancing of the driving force and the dissipative force, $$\label{eqn:smolu} {\mbox{\boldmath $F$}}_{\rm pot}({{\mbox{\boldmath $\chi$}}})+{\mbox{\boldmath $F$}}_{\rm diss}({{\mbox{\boldmath $\chi$}}},\dot{{{\mbox{\boldmath $\chi$}}}})={\mbox{\boldmath $0$}}\ ,$$ where $\dot{{{\mbox{\boldmath $\chi$}}}}$ is the time derivative of the collective shape variables. The driving force ${\mbox{\boldmath $F$}}_{\rm pot}=-\partial U({{\mbox{\boldmath $\chi$}}})/\partial{{\mbox{\boldmath $\chi$}}}$ seeks to lower the potential energy. The dissipative force ${\mbox{\boldmath $F$}}_{\rm diss}$ arises from the coupling of  to the remaining part of the system and it has a stochastic character, so that it is necessary to consider an entire ensemble of possible evolutions. The average of ${\mbox{\boldmath $F$}}_{\rm diss}$ is the friction force, ${\mbox{\boldmath $F$}}_{\rm fric}=-{\mbox{\boldmath $\gamma$}}\cdot\dot{{{\mbox{\boldmath $\chi$}}}}$, which damps the shape motion, while the residual part of ${\mbox{\boldmath $F$}}_{\rm diss}$ causes the evolution to also be diffusive. A general formal framework for treating the ensemble of evolutions, generated by the Smoluchowski equation is provided by the Fokker-Planck equation which governs the time evolution of $P({{\mbox{\boldmath $\chi$}}})$, probability distribution for the system to have the shape ${{\mbox{\boldmath $\chi$}}}=\{\chi_i\}$, $${\partial\over\partial t} P({{\mbox{\boldmath $\chi$}}},t)= -\sum_{i=1}^N{\partial\over\partial\chi_i}V_i P +\sum_{i,j=1}^N{\partial\over\partial\chi_i}{\partial\over\partial\chi_j} D_{ij} P \ .$$ The drift coefficient, a tensor of rank one, ${\mbox{\boldmath $V$}}({{\mbox{\boldmath $\chi$}}})=\{V_i({{\mbox{\boldmath $\chi$}}})\}$, determines the average evolution, while the diffusion coefficient, a tensor of rank two, ${\mbox{\boldmath $D$}}({{\mbox{\boldmath $\chi$}}})=\{D_{ij}({{\mbox{\boldmath $\chi$}}})\}$, governs the growth of correlated fluctuations. These roles of the transport coefficients are most clearly brought out when one starts from a sharply peaked distribution, $P(\chi,t=0)\sim\delta(\chi-\chi_0)$, in which case the mean shape parameters, $\{\bar{\chi}_i(t)\}\equiv\{\int\chi_iP({{\mbox{\boldmath $\chi$}}},t)d{{\mbox{\boldmath $\chi$}}}\}$, and their covariances, $\{\sigma_{ij}\}\equiv\{\int\chi_i\chi_jP({{\mbox{\boldmath $\chi$}}},t)d{{\mbox{\boldmath $\chi$}}}-\bar{\chi}_i(t)\bar{\chi}_j(t)\}$, evolve initially as follows, $${\partial\over\partial t}\bar{\chi}_i = \langle V_i({{\mbox{\boldmath $\chi$}}})\rangle\ ,\, {\partial\over\partial t}\sigma_{ij} = 2\langle D_{\ij}({{\mbox{\boldmath $\chi$}}})\rangle\ .$$ The basic transition rates for the shape changes must satisfy detailed balance. Thus the rate for the change ${{\mbox{\boldmath $\chi$}}}\to{{\mbox{\boldmath $\chi$}}}'$ and the rate for the reverse change ${{\mbox{\boldmath $\chi$}}}'\to{{\mbox{\boldmath $\chi$}}}$ must have a ratio equal to that of the corresponding final-state level densities, $\rho({{\mbox{\boldmath $\chi$}}}')$ and $\rho({{\mbox{\boldmath $\chi$}}})$, respectively, $$\label{eqn:db} \lambda({{\mbox{\boldmath $\chi$}}}\to{{\mbox{\boldmath $\chi$}}}') \rho({{\mbox{\boldmath $\chi$}}}) = \lambda({{\mbox{\boldmath $\chi$}}}'\to{{\mbox{\boldmath $\chi$}}}) \rho({{\mbox{\boldmath $\chi$}}}')\ .$$ In the approximation where the shape-dependent level density $\rho({{\mbox{\boldmath $\chi$}}})$ depends only on the local nuclear excitation energy $E^*({{\mbox{\boldmath $\chi$}}})$, [*i.e.*]{} $\rho({{\mbox{\boldmath $\chi$}}})=\tilde{\rho}(E^*({{\mbox{\boldmath $\chi$}}}))$, the transport coefficients are given by $$\label{eqn:VD} {\mbox{\boldmath $V$}}({{\mbox{\boldmath $\chi$}}})={\mbox{\boldmath $\mu$}}({{\mbox{\boldmath $\chi$}}})\cdot{\mbox{\boldmath $F$}}_{\rm pot}({{\mbox{\boldmath $\chi$}}})\ ,\,\,\ {\mbox{\boldmath $D$}}({{\mbox{\boldmath $\chi$}}})={\mbox{\boldmath $\mu$}}({{\mbox{\boldmath $\chi$}}})\,T({{\mbox{\boldmath $\chi$}}})\ ,$$ where the mobility tensor, ${\mbox{\boldmath $\mu$}}$, is the inverse of the dissipation tensor ${\mbox{\boldmath $\gamma$}}$ and $T=1/[\partial\ln\rho(E^*)/\partial E^*]$ is the local temperature (see below). The relation (\[eqn:VD\]) is consistent with the fluctuation-dissipation theorem often referred to as the Einstein relation. When the parameter space is multi-dimensional (in the present case,  is five-dimensional), it is often impractical to solve the Fokker-Planck equation, as both space and time requirements grow overwhelming. Instead, it is preferable to represent $P({{\mbox{\boldmath $\chi$}}},t)$ by a sample of dynamical trajectories, $\{{{\mbox{\boldmath $\chi$}}}^{(n)}(t)\}$ whose evolutions are simulated directly. Any desired observable can then be readily extracted from these as easily as from $P({{\mbox{\boldmath $\chi$}}},t)$. Brownian motion on the shape lattice ------------------------------------ When the Smoluchowski equation is simulated directly, the shape parameter  executes a generalized Brownian motion. Each change in  consists of a deterministic term, caused by the driving force from the PES (and resisted by the friction force, see Eq. \[eqn:smolu\]), and a random term resulting from the residual part of ${\mbox{\boldmath $F$}}_{\rm diss}$. The change in  accumulated over a small time interval, $\Delta t$, can be computed by diagonalizing the mobility tensor, ${\mbox{\boldmath $\mu$}}$. In this basis, it reads, $$\label{brown} \Delta{{\mbox{\boldmath $\chi$}}}= \sum_{n=1}^5 {\mbox{\boldmath $e$}}^{(n)} \left[\Delta t\,{\mbox{\boldmath $e$}}^{(n)}\cdot{\mbox{\boldmath $F$}}_{\rm pot} +\sqrt{2T\Delta t}\,\xi_n\right] \ ,$$ where $\{{\mbox{\boldmath $e$}}^{(n)}({{\mbox{\boldmath $\chi$}}})\}$ are the five eigenvectors of the mobility tensor, ${\mbox{\boldmath $\mu$}}=\sum_n{\mbox{\boldmath $e$}}^{(n)}{\mbox{\boldmath $e$}}^{(n)}$, and $\{\xi_n\}$ are five random numbers sampled from a distribution having zero mean and unit variance (such as a normal distribution) [@Randrup+11b]. The above propagation procedure applies when the potential energy and the mobility tensor are known as functions of the shape parameter . Further,  is considered as a continuous variable. However, in the present study we know these quantities only on the discrete shape lattice described in Sec. \[sec:shapes\]. We therefore wish to replace the above continuous Brownian motion governed by (\[brown\]) with a random walk on the lattice sites. This is a difficult task because the mobility tensor is generally not aligned with the lattice directions. However, it was argued in Ref. [@Randrup+11a] that the outcome of the strongly damped nuclear shape evolution is not so sensitive to the specific structure of ${\mbox{\boldmath $\mu$}}$, so that it may be replaced by an isotropic tensor, $\mu_{nn'}\sim\delta_{n,n'}$. This expectation was supported by the subsequent studies in Ref. [@Randrup+11b] and we shall adopt this approximation in our present study even though we must expect it to be occasionally less accurate. It is then elementary to show [@Randrup+11a; @Randrup+11b] that the Brownian shape evolution (\[brown\]) can be performed on the lattice by the simple Metropolis procedure [@Metropolis+53] according to which the shape is moved from the current lattice site ${\mbox{\boldmath $X$}}=(I,J,K,L,M)$ to a randomly chosen neighboring one ${\mbox{\boldmath $X$}}'=(I',J',K',L',M')$ with the probability $$\label{eqn:P} P({\mbox{\boldmath $X$}}\to{\mbox{\boldmath $X$}}')\ =\ \rho({\mbox{\boldmath $X$}}')/\rho({\mbox{\boldmath $X$}})\ ,$$ where $\rho({\mbox{\boldmath $X$}})$ is the local level density at the shape corresponding to the lattice site . The above relation (\[eqn:P\]) should be understood to mean that the proposed shape change happens with certainty whenever $\rho({\mbox{\boldmath $X$}}')\geq\rho({\mbox{\boldmath $X$}})$. In the present study, we employ the simplified Fermi-gas level density, $\rho({{\mbox{\boldmath $\chi$}}})_{\rm FG}(E^*({{\mbox{\boldmath $\chi$}}}))\sim\exp(2\sqrt{aE^*})$, for which the shape dependence enters only via the shape dependence of the local excitation energy, $E^*({{\mbox{\boldmath $\chi$}}})=E_{\rm tot}-U({{\mbox{\boldmath $\chi$}}})$, which is of the form leading to Eq. (\[eqn:VD\]). The parameter $a=A/8$ is the typical constant level density for a system with $A$ nucleons. The spacing of the employed shape lattice is sufficiently fine to allow a first-order expansion [@Randrup+11b], so the Metropolis criterion (\[eqn:P\]) then simplifies, $$\label{eqn:simple} {\rho({{\mbox{\boldmath $\chi$}}}')\over\rho({{\mbox{\boldmath $\chi$}}})}\ \approx\ \exp\left[{\partial\ln\rho_{\rm FG}\over\partial E^*}\, {\partial E^*\over\partial{{\mbox{\boldmath $\chi$}}}}\cdot\Delta{{\mbox{\boldmath $\chi$}}}\right]\ \approx\ {\rm e}^{-\Delta U/T} \ ,$$ where $\Delta{{\mbox{\boldmath $\chi$}}}={{\mbox{\boldmath $\chi$}}}'-{{\mbox{\boldmath $\chi$}}}$ is the proposed shape change and $\Delta U=U({{\mbox{\boldmath $\chi$}}}')-U({{\mbox{\boldmath $\chi$}}})\approx-{\mbox{\boldmath $F$}}_{\rm pot}\cdot\Delta{{\mbox{\boldmath $\chi$}}}$ is the associated change in the potential energy, where the driving force is ${\mbox{\boldmath $F$}}_{\rm pot}=\partial E^*({{\mbox{\boldmath $\chi$}}})/\partial{{\mbox{\boldmath $\chi$}}}=-\partial U({{\mbox{\boldmath $\chi$}}})/\partial{{\mbox{\boldmath $\chi$}}}$. The above expression (\[eqn:simple\]) is the form used in the original [@Randrup+11a] and subsequent work. We shall employ it here as well. Features of the shape evolution {#sec:evo_features} ------------------------------- We discuss here the most interesting features of the Brownian shape motion using the evolution across the $\prescript{236}{92}{\text{U}}$ PES as an example. We set the excitation energy to just above $\sim 5$ MeV, which is slightly higher than the highest fission barrier. {width="150mm"} Four distinct stages of the Metropolis implementation of Brownian motion for $\prescript{236}{92}{\text{U}}$ are shown in Fig. \[fig:fisstages\]. This calculation begins in the ground state, panel (a), and proceeds via stochastic steps towards scission, panel (d). Typically in fission calculations, the bulk of the computational effort is taken up attempting to move out of the ground state minimum and beyond the first major saddle point. However, along the path, the density of the Monte Carlo steps is highest in the fission isomer minimum between panels (b) and (c). The reason for this is a biased potential employed between the ground state and maximum saddle, which has been used extensively in past work and is discussed further in Sec. \[sec:modelassump\]. Once beyond the outer saddle, between panels (c) and (d), the system quickly proceeds downhill with relatively few steps required to reach scission. Note the appearance of an asymmetric fission valley only once the trajectory comes close to scission in panel (d). The larger energy scale in this figure relative to Fig. \[fig:fispes\] arises due to the projection of the PES using the shape variables ($c$, $\epsilon_{\rm f1}$, $\epsilon_{\rm f2}$) that are held fixed given the state of the system, denoted by an orange circle, in each panel. The large mountains of $60$ MeV seen in Fig \[fig:fisstages\] are never reached in any stochastic random walks for the low-energy fission considered here. To emphasize this point, we consider the ensemble of many scission trajectories in Figure \[fig:pathpes\]. The averaged path across the PES never reaches above the fission barrier around $5$ MeV. The ‘funneling’ that occurs near this saddle point is also evident with a decrease in variance of the energy along the path. For the majority of the random walk, the fission path width is roughly on the order of an MeV. Beyond the outer saddle the variation in the fission path ranges several MeV with the most probable fragments generated by trajectories near the averaged path. ![\[fig:pathpes\] (Color Online) The averaged projected potential energy of $\prescript{236}{92}{\text{U}}$ as a function of elongation for a set of trajectories. ](effbh_mcmc.pdf){width="90mm"} This type of calculation provides an alternative to immersion methods for finding the most interesting PES features along trajectories [@Moller+09]. The main limitation of this procedure is the number of events required to build sufficient statistics. Figure \[fig:pathpes\] was constructed with 5000 trajectories. In contrast, the full yields are typically evaluated with ten to one hundred times more statistics. In this procedure, one does not obtain information about the full shape configuration space, since the calculation only requires the most commonly traveled PES points confined to scission trajectories. It can be argued that this last point is in fact a motivation for using the new method, since it optimizes the time-consuming PES calculations on the configuration space to only what is absolutely necessary. This procedure may be particularly useful for fission calculations which do not rely on discretization of the shape lattice space or pre-calculated PES. Fragment mass and charge yields {#sec:calcy} =============================== We now describe the calculation of the fragment mass, $Y(A)$, fragment charge, $Y(Z)$, and full fission fragment yields, $Y(Z,A)$. Mass distribution {#sec:mass} ----------------- The ensemble of scission events such as the one illustrated in Fig. \[fig:fisstages\], is the basis for the creation of the mass yields, $Y(A)$ for a fissioning system with $Z_{0}$ protons and $A_{0}$ nucleons. We use a geometric property for the calculation of $Y(A)$. Specifically, for each scission configuration we tally the mass asymmetry, $\alpha_g$, and convert this quantity to nucleon number, $A$, via $A=A_{0}(1-|\alpha_g|)/2$ for the lighter fragment. Due to the symmetry of primary mass yields, the heavier fragment nucleon number can be computed via the conservation of nucleon number. Naturally, this calculation does not produce integral values of $A$. We therefore evaluate the yield, $Y(\alpha_g)$ using linear interpolation to construct $Y(A)$. The resultant fragment yields are over discrete integer values of $A$ and we normalize such that $\sum_{Z,A} Y(Z,A)=2$, implicitly assuming that there is no ternary fission. This choice provides consistency checks on preserving the fissioning system nucleon, $A_{0}=\sum_{Z,A} Y(Z,A)\times A$, and proton, $Z_{0}=\sum_{Z,A} Y(Z,A)\times Z$, numbers. The construction of the full yield, $Y(Z,A)$, in both mass and charge is discussed in Sec. \[sec:charge\]. Charge distribution {#sec:charge} ------------------- The charge asymmetry is notably absent from the assumed five-dimensional shape degrees of freedom. Our current calculations therefore only support the calculation of mass yields, $Y(A)$. This is not a limitation of the FRLDM model, rather, it comes from an attempt to save storage space on a computer with the grided approach to the nuclear PES. An additional shape degree of freedom could be added to produce the full fragment yields in both charge and mass, $Y(Z,A)$ [@Moller+15b]. Conversely, other methods exist in the literature to obtain the splitting configurations at scission [@Verriere+19]. We plan to explore these methods in detail in future work. For now, we apply the following technique to obtain the full fragment yields. We assume that the results of our Markov Chain Monte Carlo provides the mass yield, $Y(A)$. We use the unchanged charge distribution, which is a scaling factor, $\eta = Z_{0} / A_{0}$, where $Z_{0}$ and $A_{0}$ are the charge and mass of the fissioning nucleus, to translate between the calculated mass and charge yields, $Y(Z)$. Following the procedure of Wahl [@Wahl+02], this description assumes a Gaussian form for the charge yield as a function of $A$, $$Y(Z|A) = \frac{1}{\sqrt{2\pi\sigma_Z^2}}\exp\bigg[-[Z - Z_p(A)]^2/2\sigma_Z^2\bigg] \ ,$$ where the mean $Z_p(A)$ is given by $Z_p(A) = A\times \eta$ for a given fragment mass $A$. To obtain the full yields we perform an iterative procedure to determine the variance parameter, $\sigma_Z$. The variance $\sigma_Z$ is determined for each fission system at a specified excitation energy. We do not consider the systematics of $\sigma_Z$ with excitation energy in this work as we are evaluating the yields at a single energy, as discussed in the next section. A trial full fragment yield $Y_\text{t}(Z,A) = Y(A) \times Y(Z|A)$ is used with an initial guess for $\sigma_Z$ until an appropriate threshold is reached. The fit constraint for $\sigma_Z$ satisfies the minimization of $Y(Z)=\sum_{A}{Y_\text{t}(Z,A)}$. The full fragment yields are then given by $Y(Z,A) = Y(A) \times Y(Z|A)$ using the optimal $\sigma_Z$. Figure \[fig:ffd\_236U\] shows the fragment yield, $Y(Z,A)$, for ${}^{236}$U calculated with the above procedure. Experimental data of Refs. [@Straede+87; @Zeynalov+06] are shown for reference in the top panel. The assumption of unchanged charge distribution is employed using a value of $\sigma_Z=0.435$ for this nucleus, leading to the distribution of daughter fragments in the bottom panel. From the bottom panel, one can draw a single straight line in the $NZ$ plane that pierces through the center of the distribution, from bottom left to top right. In experimental data, an offset is often seen between the light fragment and heavy fragment lobes in relation to this line due to charge polarization [@Naik+97]. Our description using the Metropolis method to obtain $Y(A)$ does not include this effect, nor odd-even staggering often seen in charge yields. Despite this shortcoming, as we shall see, the method does very well when compared to known data. ![\[fig:ffd\_236U\] (Color Online) (a) The primary mass yield of ${}^{236}$U. (b) The distribution of daughter fragments in the chart of nuclides given the assumption of unchanged charge distribution (UCD). Solid black denotes stable nuclei with light gray showing the extent of bound nuclei using FRDM2012 masses. ](yield_frldm_frdm2012_92_236.pdf){width="90mm"} Additional model assumptions {#sec:modelassump} ---------------------------- With the evolution of the Markov chains and calculation of the fragment yields described in the previous sections, we now detail several choices that may have leverage on the Brownian-shape motion calculations. ### Starting shape The starting shape configuration for the random walk is an open, but critical choice in terms of computational cost of the yield calculations. Possible starting positions include the ground state, near the fission isomer-minimum or beyond the outermost saddle point [@Sierk+17]. For all nuclei, we chose to start as close to the ground state as possible given the 3QS shape parameterization. The reason for this is that we can always isolate this position in the PES for every nucleus across the chart of nuclides. The choice of the other starting positions could be difficult to define, for instance, there may be only a single saddle point along the fission path or the PES could be smooth and featureless in the region of moderate elongation, thus making the choice of starting at the fission isomer-minimum impossible. The choice of starting beyond the outermost saddle point is also not without its problems, as one must artificially produce a spread in the trajectories, which arises from starting at a more compact configuration, recall Fig. \[fig:pathpes\]. These issues are discussed throughout Refs. [@Randrup+11b; @Randrup+13; @Moller+15a; @Sierk+17]. ### Bias potential The choice of starting near the ground state configuration is not itself without a drawback. The wall clock time of the yields becomes substantially longer the closer to the ground state the calculations are initialized. This unfortunate computational circumstance reflects the physical nature of fission. In previous work, e.g. Ref. [@Moller+15a], a bias potential was used to speed up the calculation of the stochastic random walk from the ground state to roughly the first fission-isomer minimum. For the limited nuclei studied in the previous work, this was a good assumption as most always the maximum saddle point was the first saddled point encountered in a scission trajectory. However, systems with larger neutron excess may have more complicated potentials. We have therefore replaced the bias potential appearing in previous work with a quadratic form, $$\begin{aligned} \label{eqn:biased} E_\text{bias}(Q_{2}) &= \begin{cases} E_\text{tilt} \Big(\frac{Q_{2}-Q^{\text{sa}}_{2}}{Q^{\text{gs}}_{2}-Q^{\text{sa}}_{2}}\Big)^{2} & Q_{2} \leq Q^{\text{sa}}_{2} \ , \\ 0 & Q_{2} > Q^{\text{sa}}_{2} \ , \end{cases}\end{aligned}$$ where $Q_{2}$ is the current elongation between the ground state, $Q^{\text{gs}}_{2}$, and maximum saddle, $Q^{\text{sa}}_{2}$, and the tilt parameter, $E_\text{tilt}$, is dependent on the height of the maximum saddle allowing for a smooth connection between the biased and nuclear potentials. For elongations after $Q^{\text{sa}}_{2}$, the two potentials are exactly equal, resulting in no modification to a given trajectory after this point. This functional form serves to reduce the necessary height of the biased potential, that was in previous works typically set around $60$ MeV, and minimize the variation of trajectories through the maximum saddle point. In this work, the maximum coefficient of the biased potential considered for any nucleus is $10$ MeV. The impact of the bias potential is shown in Fig. \[fig:etilt\] for four example nuclei. These nuclei were chosen based off the distinct nature of their mass yields. The nucleus $\prescript{227}{90}{\text{Th}}$ in panel (a) exhibits mostly an asymmetric split with some tendency for a symmetric split depending on the exact choice of $E_\text{tilt}$. The dependence of the bias potential here is the strongest amongst the four nuclei because of the possible opening and closing of the symmetric channel. The uranium isotope shown in panel (b) has no symmetric mode at low excitation energy while $\prescript{260}{101}{\text{Md}}$ in panel (c) shows a preference for asymmetric fission along with an open symmetric path. An extreme neutron-rich No isotope, panel (d), with a rather broad yield displays very little dependence on the bias potential. ![\[fig:etilt\] (Color Online) The impact of the choice of biased potential for (a) $\prescript{227}{90}{\text{Th}}$ (b) $\prescript{236}{92}{\text{U}}$ (c) $\prescript{260}{101}{\text{Md}}$ and (d) $\prescript{277}{102}{\text{No}}$. The coefficient for the biased potential in this work is limited to $E_{\rm tilt}=10$ MeV or less. ](etiltdep_4panel.pdf){width="90mm"} ### End configurations The random walks continue until reaching a specified critical neck radius $c$ where the mass partition is assumed to be frozen in. This happens well before actual scission where the two emerging fragments are fully formed and separate. Figure \[fig:scicnfg\] shows the variation in the mass yield for various values of the critical neck radius. Some nuclei exhibit a relatively weak dependence on the neck radius, as shown in panels (c) and (d). Other nuclei, such as $\prescript{227}{90}{\text{Th}}$, show a much stronger dependence that even shifts the peak of the yield distribution multiple units in mass number. Mass yields of certain nuclei therefore may benefit from a variation of this number. However, it is not instructive to change this number in an ad hoc manner without physical motivation when considering a global calculation of mass yields across the chart of nuclides. We therefore adopt the critical neck radius to be $c=2.5$ fm as the standard criterion for extracting the mass yield. This value is based on the result of matching select yield distributions of major actinides. ![\[fig:scicnfg\] (Color Online) The variation in mass yield upon choice of scission configuration for (a) $\prescript{227}{90}{\text{Th}}$ (b) $\prescript{236}{92}{\text{U}}$ (c) $\prescript{260}{101}{\text{Md}}$ and (d) $\prescript{277}{102}{\text{No}}$. ](scicnfg_4panel.pdf){width="90mm"} ### Nuclear temperature The random walk depends on the local temperature, $T({{\mbox{\boldmath $\chi$}}})$, appearing in the Metropolis step criterion (\[eqn:simple\]). As in past work, we relate $T$ to the local excitation energy $E^*$ by the simple Fermi-gas relation $E^*=aT^{2}$. Accordingly, the temperature for a given shape  is thus given by $T({\mbox{\boldmath $\chi$}}) = [(E-U({{\mbox{\boldmath $\chi$}}}))/a_A]^{1/2}$. Thus, the local temperature is initially relatively large while the shape explores the region around the ground state, it is smallest as the shape passes through the barrier region, beyond which it increases steadily as the system drifts down the outer barrier. Figure \[fig:tdep\] shows mass yields calculated for various constant values of the temperature. This simple example illustrates the importance of the nuclear temperature in determining the fragment yields. More recent treatments, e.g. that of Ward *et al.* [@Ward+17], have refined the treatment of the shape evolution by employing shape-dependent microscopic level densities, which account for pairing correlations and shell effects. ![\[fig:tdep\] (Color Online) Mass yields calculated for various constant values of the temperature, $T=0.5, 1.0, 1.5\,{\rm MeV}$, for $\prescript{227}{90}{\text{Th}}$ (a), $\prescript{236}{92}{\text{U}}$ (b), $\prescript{260}{101}{\text{Md}}$ (c), and $\prescript{277}{102}{\text{No}}$ (d). Also shown are the yields obtained by calculating the local temperature on the basis of the macroscopic potential alone.](tdep_4panel.pdf){width="90mm"} ### Excitation energy The nuclear shape evolution depends on the total energy of the system, $E$, which in turn depends on how the fissioning nucleus is being prepared. For low-energy neutron-induced fission, the initial compound nucleus is the result of the target nucleus absorbing a neutron. If the incident neutron has kinetic energy $\epsilon$, the resulting compound nucleus excitation energy is $E_0*=S_{\rm n}+\epsilon$, where $S_{\rm n}$ is its neutron separation energy, and the total energy is $E=E_0^*+M_0c^2$, where $M_0$ is the ground-state mass of the compound nucleus. Many measurements have shown that fission yields are energy dependent [@Akimov+71; @Zoller+91; @Hambsch+00; @Vives+00; @Duke+14]. For the major actinides, an increase of the excitation energy leads to an gradual change from asymmetric to symmetric fission. This general feature is a result of the fact that the microscopic (shell and pairing) effects diminish as the temperature grows. Recalling Fig. \[fig:tdep\], we can interpret the higher constant temperature evolutions as washing out the shell effects. Our calculations utilize the shell suppression term, $S$, to estimate the energy dependence of fragment yields as in Ref. [@Randrup+13]. To provide a complete set of yields across the chart of nuclides, we set the excitation energy as close as possible to the maximum saddle height. An additional amount of excitation energy, ranging from $0-2$ MeV, is needed for some nuclei to achieve sufficient statistics. With this initial excitation energy we can roughly approximate near-thermal incident neutron energies for the actinides. For nuclei with extreme neutron-excess, this choice may produce excitation energies that tend to be rather high for neutron-induced fission and rather low for $\beta$-delayed fission. It is therefore important to note that the calculated fission yields exhibit a rather weak energy dependence in the range of astrophysical interest. Thus, for low-energy applications that include those in astrophysics, this choice of excitation energy appears to be suitable. On the other hand, because we consider energies above the fission barrier and cannot enter the classically forbidden regions, the calculated fragment yields may not be appropriate for spontaneous fission. Future work will study the systematics of our fission yields as a function of excitation energy across the chart of nuclides. ![\[fig:tess\] (Color Online) (a) A uniform tessellation versus a (b) non-uniform tessellation of the plane given two coordinate variables. Both methods are applicable to nuclear potential-energy surfaces so long as the notion of distance, $d_{ij}$, between different configurations is well defined. ](tessellation.pdf){width="\columnwidth"} ### Space discretization We end this section by revisiting the notion of the discretization of the shape configuration space. The lattice structure used in the present work has been introduced at the end of Sec. \[sec:shapes\]. This grid consists of a uniform tessellation of the canonical shape degrees of freedom with unique spacing in each of these parameters. The original motivation for introducing a lattice structure was to minimize the computer storage required, which is substantial when thousands of nuclei are being considered. The specific choice of lattice does, however, play a role for both the computational effort required and the physical assumptions made when performing a Metropolis procedure using a discrete random walk. If the grid is too dense it may take many steps to proceed in a given direction relative to another, while offering little to no improvement in predictive capability. In our case, one variable that could benefit from grid refinement would be the mass asymmetry, $\alpha_{\rm g}$, whose step size controls the mass resolution when using the macroscopic nuclear geometry condition for scission. On the other hand, sparsity of lattice spacing for a given variable may cause a miss of important features and could prevent the Metropolis procedure from ever selecting the step if the corresponding jump in the potential is too high. It is clear that a proper notion of distance, $d_{ij}$, between lattice sites is needed when considering the tessellation of the nuclear PES, as illustrated in Fig. \[fig:tess\]. Non-uniform tessellations, such as those constructed by a Voronoi diagram or Delaunay triangulation, are useful for constraining the parameter space to physically relevant points with variable grid density. This method coupled with the technique discussed at the end of Sec. \[sec:evo\_features\] could have significant ramifications for the computational tractability of fission. Another aspect of using a discrete random walk is that, inherently, the choice of relative lattice step sizing encodes a principal physical assumption regarding the isotropic nature of the mobility tensor. In our case, the particular griding results in equal probability for being the next candidate step in the random walk, as discussed by Randrup and colleagues [@Randrup+11b]. It is important to remember in a discretized approach, such as the one implemented here, the likelihood of movement between two lattice sites is distinct from the candidate shape choice probability. The probability of movement between two lattice sites is controlled by the Metropolis procedure and dependent on the difference in potential energy of the two sites. For the next lattice candidate there are three choices for each canonical shape degree of freedom: to remain at the same location, to move forward in the grid to the nearest neighbor, or to move backward to a grid point with lower index. With five shape degrees of freedom, this results in $3^5-1=242$ equally possible candidate points for the current step in the random walk, where the possibility of staying at the exact same grid point has been removed. Previous studies have shown that the exact choice of the candidate shape choice probability has a minor impact on the calculations [@Randrup+11b; @Moller+15a; @Sierk+17]. Studies that commission a continuous shape space, for example, those based off Smoluchowski or Langevin, bypass these considerations all together as they remove the direct dependence on the candidate shape choice probability. In summary, when using a discretization approach a balance must be struck between adequate tessellation, physical assumptions, and computational tractability. The adopted lattice reasonably satisfies all of these considerations. The study of non-uniform tessellation procedures will be the subject of future work. Results {#sec:results} ======= We begin the discussion of our results by starting with a comparison of our yield calculations to relevant data of several actinides. We follow with a study of the global trends that arise in the fragment yields across the chart of nuclides. Comparison of yields with data ------------------------------ We benchmark our Discrete Random Walk (DRW) code (version 1.0) with comparison to experimental data. Caution must be issued here as model output is not exactly a one-to-one comparison with experiment nor evaluation. It is so-called independent fission product data (i.e. after prompt neutron and $\gamma$-ray emission) that is generally measured in an experiment, due to the fast timescale of prompt particle emission. Often times, this data is transformed back to a state of fragment mass yields (prior to prompt particle emission) which is suitable for comparison with the output of our random walk with the caveat that a model has been used to construct such data. One could imagine comparing product mass yields directly, however, this introduces several more theoretical models in calculating the de-excitation of the nascent fragments [@Talou+11; @Becker+13; @Talou+14]. Chiefly among the concerns is the calculation of average fragment kinetic energy and the degree of excitation energy of each individual fragment which significantly affect the number of neutrons evaporated and hence the mass number of the resulting product nucleus. Current fission event generators obtain these quantities from phenomenological parameterizations [@Vogt+14; @Litaize+15]. A comparison of charge yields would not suffer from this problem because the neutron emission leaves the charge number unaffected. But the charge yields exhibit significant odd-even staggering and this effect has not yet been included in current shape evolution treatments which provide charge yields by a simple rescaling of the mass yields. An additional problem arises due to the high excitation energies associated with such experiments that may open multi-chance fission channels [@Moller+17]. Other experimental setups that ascertain charge yields may have low resolution of the excitation energy, resulting in yields that depend on a spread of energies [@Martin+15]. For a review of experimental fission methods see Ref. [@Schmidt+18]. With these caveats established, we proceed with comparing fragment yields in both charge and mass. {width="\textwidth"} Figure \[fig:ya\_compare\] compares the calculated mass yields to measured yields for several (n$_{\rm th}$,f) cases. The data for these comparisons can be found from (a) Ref. [@Unik+73] (b) Ref. [@Hambsch+00] (c) Ref. [@Schillebeeckx+92] and (d) Ref. [@Unik+73]. For these actinides, the mass yields are asymmetric. The positions of the asymmetric peaks are reproduced to a satisfactory degree. The widths of the distributions are also well reproduced, except for the last case where the peaks come out somewhat too wide. {width="\textwidth"} Several years ago, the Metroplos-walk method was successfully benchmarked against 70 measured fragment charge yields in Ref. [@Moller+15a]. To demonstrate that the present slightly modified treatment does equally well, we show in Fig. \[fig:yz\_compare\] similar comparisons for eight typical cases selected from the entire range. The agreement with the experimental data across this range is remarkable. In particular, the calculations reproduce the transition from symmetric fission below $Z\sim90$ to asymmetric fission above $Z\sim90$. The experimental conditions were such that a range of excitation energies are combined and the calculations were carried out using the reported average excitation, $E^*\approx11\,{\rm MeV}$. At these energies the pairing effects giving rise to an odd-even staggering in the charge yields have been largely been damped out and are still visible in only a few of the cases. Further comparisons of calculated fragment yields to experimental data have been made in previous work [@Randrup+11a; @Randrup+11b; @Randrup+13; @Moller+15a]. More recent comparisons of independent yields have also been undertaken for some actinides [@Jaffke+18; @Okumura+18]. ![\[fig:ya\_compare2\] (Color Online) Comparisons of our model mass fragment yield predictions (red) with the JENDL-4.0 evaluation of independent yields [@Shibata+11a; @Shibata+11b] (top panel) and experiment [@John+71] (bottom panel) for the sharp transition between the asymmetric distribution of (a) $^{256}$Fm and symmetric distribution of (b) $^{258}$Fm.](ya_compare_frldm2.pdf){width="80mm"} Here we add several comparisons for nuclei with $Z=100$, shown in Fig. \[fig:ya\_compare2\]. The nuclei are so short-lived that only spontaneous fission can be observed, whereas the calculations were carried out at excitations within 2 MeV above the barrier. The two fermium cases, $^{256,258}$Fm, are well-known examples where model calculations deviate from experimental data [@Flynn+72; @Hoffman+89; @Gonnenwein+99]. The origin of this transition has long been debated [@Warda+02; @Bonneau+06]. Improvements to model calculations can be made by increasing the smoothing range of the Strutinsky shell-correction procedure [@Albertsson+2019a; @Albertsson+2019b] or by applying Langevin dynamics [@Usang+19]. Global mass yield trends ------------------------ Understanding the trends of fission yields across the chart of nuclides is of particular interest to the astrophysical $r$-process of nucleosynthesis [@Cote+18; @Giuliani+18]. To this end, we introduce in what follows three key metrics to classify a given mass yield, $Y(A)$. [**1: Number of peaks, $N_{\rm p}$.**]{}The fission fragment mass number distribution $Y(A)$ is always symmetric around the midpoint, $\mbox{$1\over2$}A_0$, due to nucleon number conservation $A_{\rm L}+A_{\rm H}=A_0$, but it may exhibit any number of peaks. Purely symmetric fission leads to a single centrally located peak, while single-mode asymmetric fission leads to two peaks located at opposite sides of the midpoint. Bimodal fission also occurs. For example, $^{226}$Th(n,f) exhibits both symmetric and asymmetric components (with comparable peak heights), $^{235}$U(n,f) has two nearly coinciding asymmetric components, in addition to an increasingly prominent symmetric component as the energy is increased, and some nuclei are predicted to have two widely different asymmetric components. In order to assign the value of the peak index $N_{\rm p}$, we proceed as follows. (1) We first spline interpolate the $Y(A)$ curve, creating $Y_\textrm{s}(A)$, to smooth out any minor bumps that may exist which can be misinterpreted as a peak. (2) Next, we count the maxima by computing the first derivative, $\dot{Y}_\textrm{s}(A)=0$ and second derivative $\ddot{Y}_\textrm{s}(A)<0$. (3) Since large features are typically spread out in $A$, we prevent the algorithm from finding major peaks within 10 mass units of another feature. The procedure yields a reasonable result for most nuclei, but when there are several subtle inflections in the yield curve it may lead to a too high value of $N_{\rm p}$. Fortunately, this problem is limited to a relatively small subset of the nuclei under consideration. [**2: Degree of asymmetry, $S_\textrm{f}$.**]{}  A second property of the mass yield curve is the degree of asymmetry, $S_\textrm{f}$. This quantity indicates how many units the mass number of the maximum in the mass yield, $A_{\rm max}$ differs from symmetry, $S_{\rm f}=|A_{\rm max}-\mbox{$1\over2$}A_0|$. Because we are considering primary fragment yields, we can use either the heavy or light fragment mass peak. Super-asymmetric mass yields will have large values of $S_\textrm{f}$, while those centered at $\mbox{$1\over2$}A_0$ will have $S_\textrm{f}=0$. For example, $S_{\rm f}(^{236}{\rm U})=18$ and $S_{\rm f}(^{240}{\rm Pu})=17$. [**3: Overall width, $W_{\rm d}$.**]{}  A third useful characteristic of a mass yield $Y(A)$ is its overall width, $W_\textrm{d}$. The calculation of this quantity requires some care. The full width at half maximum (FWHM) is useful only for single-peak distributions, while the standard definition of FWHM may not yield meaningful results for multi-peak yield functions. We therefore employ a simple definition of the width, $W_\textrm{d}=\sum_A\theta(Y(A)-0.01)$, i.e. the width is the number of $A$ values for which $Y(A)$ exceeds 0.01. Thus, the larger the value of $W_\textrm{d}$ is, the more spread out is the mass yield. For example, $W_{\rm d}(^{236}{\rm U})=40$ and $W_{\rm d}(^{240}{\rm Pu})=48$, while very heavy nuclei may have $W_{\rm d}>100$. Values above 100 are attainable because the yield is normalized to two, $\sum_AY(A)=2$. {width="\textwidth"} An overall impression of the calculated mass yields can be gained from Fig. \[fig:3panel\] showing the three classification metrics for each nucleus in the $NZ$ region considered. A striking structure emerges as one moves across the nuclear chart and we now discuss several particularly interesting features. An inspection of the number of yield peaks, shown in panel (a), suggests that the bulk of the nuclei situated between $N\approx140$ and $N\approx180$ undergo asymmetric fission. This is confirmed by comparison with the degree of asymmetry, shown in panel (b). The width of these yield functions tends to grow with both increasing $Z$ and $N$. The reason for this comes from a preferential flattening of the potential energy surfaces after the last saddle point as more nucleons are put into the system. This in turn spreads our random walk calculations in the asymmetry coordinate, making a wide range of splitting configurations comparatively favorable. A transition from predominantly asymmetric to predominantly symmetric yields occurs around $N\approx170$ for $Z\approx90$. This region is of interest for $\beta$-delayed and neutron-induced fission channels in nucleosynthesis simulations of the $r$ process [@Mumpower+18; @Vassh+19]. We find a diverse set of mass yields in this region, especially for $Z\approx85$ to $Z\approx95$ and $N\approx170$ to $N\approx190$. One consistent feature of these mass yields is that they are all relatively wide with $W_{\rm d}\approx70$ (whereas major actinides typically have $W_{\rm d}\approx45$). Thus, the exact division into an asymmetric or symmetric fission branch could be less important for $r$-process simulations as these features will be washed out due to the wide nature of the fragment yields. The yields with the largest width, $W_\textrm{d}\gtrsim100$, in our model occur near $A\sim315$. In early liquid drop-based theoretical studies of fission it was suggested that a correlation exists between the mass asymmetry and the parameter $Z^{2}_{0}/A_{0}$ [@Swiatecki+55]. The variation in panel (b) of Fig. \[fig:3panel\] clearly dispels this suggestion, showing that across the chart of nuclides, the details of microscopic effects are more important in shaping the yield functions. {width="\textwidth"} In Fig. \[fig:mass\_dump\] we show the placement of the peak of the fragment distribution in $A$ for either the symmetric or heavy fragment peak. Generally, this quantity is increasing with notable exception when yields transition from symmetric to asymmetric or vice versa. Fission Q-values ---------------- Effective fission Q-values may be estimated from the fragment yields via the relation, $$Q_\textrm{fiss} \approx M^{*}(Z_0,A_0) - \sum_{Z,A}Y(Z,A)\, M(Z,A)\ ,$$ where $M^{*}(Z_0,A_0)$ is the mass of the fissioning nucleus (including excitation energy), $M(Z,A)$ is the mass of a fragment , and $Y(Z,A)$ the yield of this fragment species. This relation is exact for spontaneous fission, when the nucleus is not excited and fission occurs in the ground state. It is only slightly modified for neutron-induced or $\beta$-delayed fission due to the existence of additional particles or change in target (parent) nucleus. Figure \[fig:fissQ\] shows this effective fission Q-value along isotopic chains where the fragment yields have been calculated. The flat trend along each isotopic chain indicates that the dependence of $N_{0}$ is rather weak, while the spacing between the isotopic chains reveals a stronger dependence on $Z_{0}$. A sudden jump along an isotopic chain may arise when a yield function exhibits a substantial change relative to those of the neighboring isotopes. For this calculation, the nuclear binding energies were obtained from the latest (2012) version of FRDM [@Moller+12; @Moller+16]. {width="\textwidth"} Summary {#sec:summary} ======= We have used the well established Finite-Range Liquid-Drop Model (FRLDM) to explore fission fragment yields across the chart of nuclides bounded by the region between $80\leq Z \leq 130$ and $A\leq330$. The fragment yield of each fissioning system are calculated using a discrete random walk across a static potential energy surface under the assumption of strong dissipation. Our procedure produces over 3800 fission yields at excitation energies suitable for possible applications of neutron-induced and $\beta$-delayed fission. We find that individual fragment yields exhibit a prominent behavior with both the mass ($A_{0}$) and charge ($Z_{0}$) of the fissioning system, indicating the importance of including microscopic effects in the calculation of this quantity. The size of fragment distributions show a propensity to expand with increasing neutron-excess of the fissioning system. For these super-heavy systems, the difference between splitting symmetrically versus asymmetrically is not as crucial as the spread of fragments across a large mass region in the NZ-plane. This result is likely to have significant consequences for the formation of the heavy elements in the astrophysical rapid neutron capture process. Our yields also permit the estimation of fission Q-values across the chart of nuclides. A visible flat trend arises across isotopic chains indicating a primary dependence on the charge of fissioning system. The authors look forward to the use of these yields in applications including the study of astrophysical phenomena and note that further model enhancements of FRLDM are underway at Los Alamos that seek to improve the microscopic description of fission. The authors would like to thank Peter M[ö]{}ller for providing the PES used in this work [@Moller+09]. The authors additionally thank Nicolas Schunck, Toshihiko Kawano, Ionel Stetcu and Patrick Talou for their insights and valuable discussions related to nuclear fission. This work was supported by the US Department of Energy through Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). MM, PJ and MV were supported in part by the U.S. Department of Energy (DOE) under Contract No. DE-AC52-07NA27344 for the topical collaboration Fission In R-process Elements (FIRE). Supplemental data ================= We provide the calculated fragment yields in individual ASCII formatted files. The ASCII filenames list the proton number, $Z_{0}$, and nucleon number, $A_{0}$ of the fissioning system. Therefore, for use in neutron-induced fission, the target nucleus would be ($Z_{0}$,$A_{0}-1$). When using the yields for $\beta$-delayed fission, the parent is ($Z_{0}-1$,$A_{0}$). The ASCII files themselves are formatted in three columns: fragment proton number ($Z$), fragment nucleon number ($A$) and fragment yield $Y(Z,A)$. The yields are on an integer grid and normalized such that $\sum_{Z,A} Y(Z,A)=2$, providing consistency checks for preservation of the fissioning system nucleon, $A_{0}=\sum_{Z,A} Y(Z,A)\times A$, and proton, $Z_{0}=\sum_{Z,A} Y(Z,A)\times Z$, numbers. Additional FRLDM-based calculations, for instance, yields at a given excitation energy, are available by request.

--- author: - | Nesime Tatbul [^1]\ Intel Labs and MIT\ `[email protected]`\ Tae Jun Lee $^*$\ Microsoft\ `[email protected]`\ Stan Zdonik\ Brown University\ `[email protected]`\ Mejbah Alam\ Intel Labs\ `[email protected]`\ Justin Gottschlich\ Intel Labs\ `[email protected]`\ bibliography: - 'main.bib' title: Precision and Recall for Time Series --- [^1]: Lead authors.

--- abstract: 'Federated learning obtains a central model on the server by aggregating models trained locally on clients. As a result, federated learning does not require clients to upload their data to the server, thereby preserving the data privacy of the clients. One challenge in federated learning is to reduce the client-server communication since the end devices typically have very limited communication bandwidth. This paper presents an enhanced federated learning technique by proposing a synchronous learning strategy on the clients and a temporally weighted aggregation of the local models on the server. In the asynchronous learning strategy, different layers of the deep neural networks are categorized into shallow and deeps layers and the parameters of the deep layers are updated less frequently than those of the shallow layers. Furthermore, a temporally weighted aggregation strategy is introduced on the server to make use of the previously trained local models, thereby enhancing the accuracy and convergence of the central model. The proposed algorithm is empirically on two datasets with different deep neural networks. Our results demonstrate that the proposed asynchronous federated deep learning outperforms the baseline algorithm both in terms of communication cost and model accuracy.' author: - 'Yang Chen, Xiaoyan Sun, Yaochu Jin, [^1] [^2] [^3]' title: 'Communication-Efficient Federated Deep Learning with Asynchronous Model Update and Temporally Weighted Aggregation' --- Federated learning, Deep neural network, aggregation, asynchronous learning, temporally weighted aggregation INTRODUCTION {#sec1} ============ Smart phones, wearable gadgets, and distributed wireless sensors usually generate huge volumes of privacy sensitive data. In many cases, service providers are interested in mining information from these data to provide personalized services, for example, to make more relevant recommendations to clients. However, the clients are usually not willing to allow the service provider to access the data for privacy reasons. Federated learning is a recently proposed privacy-preserving machine learning framework [@mcmahan2017communication]. The main idea is to train local models on the clients, send the model parameters to the server, and then aggregate the local models on the server. Since all local models are trained upon data that are locally stored in clients, the data privacy can be perserved. The whole process of the typical federated learning is divided into communication rounds, in which the local models on the clients are trained on the local datasets. For the $k$-th client, where $k \in S$, and $S$ refers to the participating subset of $m$ clients, its training samples are denoted as $\mathcal{P}_k$ and the trained local model is represented by the model parameter vector $\omega^k$. In each communication round, only models of the clients belonging to the subset $S$ will download the parameters of the central model from the server ans use them as the initial values of the local models. Once the local training is completed, the participating clients send the updated parameters back to the server. Consequently, the central model can be updated by aggregating the updated local models, i.e. $\omega=Agg(\omega^k)$ [@Konecny2015; @Konecny2016; @mcmahan2017communication]. In this setting, the local models of each client can be any type of machine learning models, which can be chosen according to the task to be accomplished. In most existing work on federated learning [@mcmahan2017communication], deep neural networks (DNNs), e.g., long short-term memory (LSTM), are employed to conduct text-word/text-character prediction tasks. In recent years, DNNs have been successfully applied to many complex problem-solvings, including text classification, image classification, and speech recognition [@lecun2015deep; @Shin2016Deep; @Greff2015LSTM]. Therefore, DNNs are widely adopted as the local model in federated learning, and the stochastic gradient descent (SGD) is the most popular learning algorithm for training the local models. As aforementioned, one communication round includes parameter download (on clients), local training (on clients), trained parameter upload (on clients), and model aggregation (on the server). Such a framework appears to be similar to distributed machine learning algorithm [@ma2017distributed; @reddi2016aide; @shamir2014communication; @zhang2015disco; @chilimbi2014project; @dean2012large]. In federated learning, however, only the models’ parameters are uploaded and downloaded between the clients and server, and the data of local clients are not uploaded to the server or exchanged between the clients. Accordingly, the data privacy of each client can be preserved. Compared with other machine leanring paradiagms, federated learning are subject to the following challenges [@mcmahan2017communication; @konevcny2016federated]: 1. **Unbalanced data**: The data amount on different clients may be highly imbalanced because there are light and heavy users. 2. **Non-IID data**: The data on the clients may be strongly non-IID because of different preferences of different users. As a result, local datasets are not able to represent the overall data distribution, and the local distributions are different from each other, too. The IID assumption in distributed learning that training data are distributed over local clients uniformly at random [@Boyd2011a] usually does not hold in federated learning. 3. **Massively distributed data**: The number of clients is large. For example, the clients may be mobile phone users [@Konecny2015], which can be enormous. 4. **Unreliable participating clients**: It is common that a large portion of participating clients are often offline or on unreliable connections. Again in case the clients are mobile phone users, their communication state can vary a lot and thus cannot ensure their participation in each round of learning [@mcmahan2017communication]. Apart from the above challenges, the total communication cost is often used as an overall performance indicator of federated learning due to the limited bandwidth and battery capacity of mobile phones. Of course, like other learning algorithms, the learning accuracy, which is mainly determined by the local training and the aggregation strategy, is also of great importance. Accordingly, the motivation of our work is to reduce the communication cost and improve the accuracy of the central model, assuming that DNNs are used as the local learning models. Inspired by interesting observations in fine-tuning of DNNs [@yosinski2014transferable], an asynchronous strategy for local model updating and aggregation is proposed to improve the communication efficiency in each round. The main contributions of the present work are as follows. First, an asynchronous strategy that aggregates and updates the parameters in the shallow and deep layers of DNNs at different frequencies is proposed to reduce the number of parameters to be communicated between the server and clients. Second, a temporally weighted aggregation strategy is suggested to more efficiently integrate information of the previously trained local models in model aggregation to enhance the learning performance. The remainder of the paper is organized as follows. In Section \[sec2\], related work is briefly reviewed. The detail of the proposed algorithm, especially the asynchronous strategy, the temporally weighted aggregation and the overall framework are described in Section \[sec3\]. Section \[sec4\] presents the experimental results and discussions. Finally, conclusions are drawn in Section \[sec5\]. Related Work {#sec2} ============ Kone[č]{}n[ý]{} et al. developed the first framework of federated learning and also experimentally proved that existing machine learning algorithms are not suited for this setting [@Konecny2015]. In [@Konecny2016], Kone[č]{}n[ý]{} et al. proposed two ways to reduce the uplink communication costs, i.e., structured updates and sketched updates, using data compression/reconstruction techniques. A more recent version of federated learning, FedAVG for short, was reported in [@mcmahan2017communication], which was developed for obtaining a central prediction model of Google’s Gboard APP and can be embedded in a mobile phone to protect the user’s privacy. The pseudo code of FedAVG is provided in Algorithm 1. initialize $w_{0}$ $m \gets$ max($C\cdot K, 1$) $S_{t} \gets$ (random set of $m$ clients) $w_{t+1}^k \gets$ ClientUpdate($k, w_t$) $w_{t+1} \gets \sum_{k=1}^K \frac{n_k}{n} w_{t+1}^k$\ $\mathcal{B} \gets$ (split $\mathcal{P}_k$ into batches of size $B$) $w \gets$ $w- \eta \bigtriangledown \ell(w;b)$) return $w$ to server In the following, we briefly explain the main components of FedAVG: 1. **Server Execution** consists of the *initialization* and *communication rounds*. 1. *Initialization:* Line 2 initializes parameter $\omega_0$. 2. *Communication Rounds:* Line 4 obtains $m$, the number of participating clients; $K$ indicates the number of local clients, and $C$ corresponds to the fraction of participating clients per round, according to which line 5 randomly selects participating subset $S_t$. In lines 6-8, sub-function $Client Update$ is called in parallel to get $\omega_{t+1}^k$. Line 9 executes aggregation to update $\omega_{t+1}$. 2. **Client Update** The sub-function gets $k$ and $\omega$. $B$ and $E$ are the local mini-batch size and the number of local epochs, respectively. Line 14 splits data into batches. Lines 15-19 execute the local SGD on batches. Line 20 returns the local parameters. The equation in line 9, $\omega_{t+1} \gets \sum_{k=1}^K \frac{n_k}{n}*\omega_{t+1}^k$ described the aggregation strategy, where $n_k$ is the sample size of the $k$-th client and $n$ is the total number of training samples. $\omega_{t+1}^k$ comes from client $k$ in round $t+1$; however, it is not always updated in the current communication round. For clients that do not participate, their models remain unchanged until they are chosen to participate. In model aggregation, the parameters uploaded from clients in the current round and those in previous ones contribute equally to the central model. Apart from reducing communication costs, other studies of federated learning have focused on protocol security. To tackle differential attacks, for example, Gayer et al. [@geyer2017differentially] proposed an algorithm incorporating a preserving mechanism into federated learning for client sided differential privacy. In [@bonawitz2017practical], Bonawitz et al. designed an efficient and robust transfer protocol for secure aggregation of high-dimensional data. While privacy preserving and reduction of communication costs are two important aspects to take into account in federated learning, the main concern remains to be the enhancement of learning performance of the central model on all data. The loss function of federated learning is defined as $ F\left( \omega \right){\text{ = }}\sum\limits_{k = 1}^{K} {\frac{{{n_k}}}{n}{f_k}\left( \omega \right)} $, where ${f_k}\left( \omega \right)$ is the loss function of the $k$-th client model. Clearly, the performance of federated learning heavily depends on the model aggregation strategy. Not much work has been reported on reducing the communication cost by reducing the number of parameters to be uploaded and downloaded between clients and the server except for some recent work reported most recently [@Zhu2018]. In this work, we present an asynchronous model learning mode that updates only part of the model parameters to reduce communication and a temporally weighted aggregation strategy that gives a higher weight on more recent models in aggregation to enhance learning performance. Asynchronous Model Update and Temporally Weighted Aggregation {#sec3} ============================================================= Asynchronous Model Update {#sec3A} ------------------------- The most intrinsic requirement for decreasing the communication cost of federated learning is to upload/download as little data as possible without deteriorating the performance of the central model. To this end, we present in this work an asynchronous model update strategy that updates only part of the local model parameters to reduce the amount of data to be uploaded or downloaded. Our idea was primarily inspired from the following interesting observations made in fine tuning deep neural networks [@krizhevsky2014one; @yosinski2014transferable]: 1. Shallow layers in a DNN learn the general features that are applicable to different tasks and datasets, meaning that a relatively small part of the parameters in DNNs (those in the shallow layers) represent features general to different data sets. 2. By contrast, deep layers in a deep neural network learn ad hoc features related to specific data sets and a large number of parameters focus on learning features in specific data. These above observations indicate that the relatively smaller number of parameters of in the shallow layers are more pivotal for the performance of the central model in federated learning. Accordingly, parameters of the shallow layers should be updated more frequently than those parameters in the deep layers. Therefore, the parameters in the shallow and deep layers in the models can be updated asynchronously, thereby reducing the amount of data to be sent between the server and clients. We term this asynchronous model update. The DNN employed for the local models on the clients is shown in Fig. \[fig\_1\]. As illustrated in the figure, it can be separated into shallow layers for learning general features and deep layers for learning specific-feature layers features, which are denoted as $\omega _\text{g}$ and $\omega _\text{s}$, respectively. The sizes of $\omega _\text{g}$ and $\omega _\text{s}$ are denoted as $S_\text{g}$ and $S_\text{s}$, respectively, and typically, $S_\text{g} \ll S_\text{s}$. In the proposed asynchronous learning strategy, $\omega _\text{g}$ will be updated and uploaded/downloaded more frequently than $\omega _\text{s}$. Assume that the whole federated process is divided into loops and each loop has $T$ rounds of model updates. In each loop, $\omega _\text{g}$ will be updated and communicated in every round, while $\omega _\text{s}$ will be updated and communicated in only $fe$ rounds, where $fe < T$. As a result, the number of parameters to be exchanged between the server and clients will be $(T-fe)*S_\text{s}$. This way, the communication cost can be significantly reduced, since $S_\text{s}$ is usually very large in DNNs. ![Illustration of shallow and deep layers of a deep neural network.[]{data-label="fig_1"}](F1){width="3.59in"} An example is given in Fig. \[fig\_2\] to illustrate the asynchronous learning strategy. The abscissa and ordinate denote the communication round and the local client, respectively. In this example, there are five local devices, i.e., $\{A, B, C, D, E\}$ and a server. Point ($A, t$) indicates that client $A$ is participating in updating the global model in round $t$. Fig. \[fig\_2\] (a) provides an illustration of a conventional synchronous aggregation strategy, where both $\omega _{{\text{g}}}$ and $\omega _{{\text{s}}}$ are uploaded/downloaded in each round. The grey rounded rectangles in the bottom represent the aggregation, in which both shallow and deep layers participate in all rounds. By contrast, Fig. \[fig\_2\] (b) illustrates the proposed asynchronous learning strategy. In this example, there are six computing rounds ($t-5, t-4, ..., t$) in the loop, and the parameters of deep layers are exchanged in rounds $t-1$ and $t$ only. As a result, the number of reduced parameters to be communicated is $ 2/3* S_{\text{s}}$. Temporally Weighted Aggregation {#sec3B} ------------------------------- In federated learning, the aggregation strategy (Line 9 in Algorithm \[AlgoFedAVG\]) usually weights the importance of each local model based on the size of the training data on the corresponding client. That is, the larger $n_k$ is, the more the local on client $k$ will contribute to the central model, as illustrated in Fig. \[fig\_3\]. In that example, the blue diamonds represent the previously trained local models, which do not take part in the $t+1$-th update of the central model and only the orange dots are the most recently updated local model on each client and will participate in the current round of aggregation. All participating local models (orange dots in Fig. \[fig\_3\]) will be weighted by their data size only regardless the computing round in which these models are updated. In other words, local models updated in round $t-p$ are as important as those updated in round $t$, which might not be reasonable. ![Conventional aggregation strategy.[]{data-label="fig_3"}](F4a.pdf){height="2.39in"} In federated learning, however, the training data on each participating client are changing in each round and therefore, the most recently updated model should have a higher weight in the aggregation. Accordingly, the following model aggregation method taking into account of timeliness of the local models is proposed. ![Illustration of the temporally weighted aggregation.[]{data-label="fig_4"}](F5){height="2.36in"} $$\label{eq4}{ \omega_{t+1} \gets \sum_{k=1}^K \frac{n_k}{n}*(e/2)^{-(t-timestamp^k)}*\omega^k }$$ where $e$ is the natural logarithm used to depict the time effect, $t$ means the current round, and $timestamp^k$ is the round in which the newest $\omega^k$ was updated. The proposed temporally weighted aggregation is illustrated in Fig. \[fig\_4\]. Similar to the setting used in Fig. \[fig\_3\], all clients participating in the aggregation are denoted by an orange dot, while others are represented by blue diamonds. Furthermore, the depth of the brown color indicates the timeliness and the deeper the color, the higher weight this local model will have in aggregation. Framework {#sec3C} --------- {height="3.69in"} The framework of the proposed federated learning with an asynchronous model update and temporally weighted aggregation is illustrated in Fig. \[fig\_5\]. A detailed description of the main components will be given in the following subsection. Temporally Weighted Asynchronous Federated Learning {#sec3D} --------------------------------------------------- initialize $\omega_{0}$ $timestamp^k_\text{g} \gets 0$ $timestamp^k_\text{s} \gets 0$ $^\S$ $flag \gets$ True $flag \gets$ False $m \gets$ max($C * K, 1$) $S_{t} \gets$ (random set of $m$ clients) $\omega^k \gets$ ClientUpdate($k, \omega_t, flag$) $timestamp^k_\text{g} \gets t$ $timestamp^k_\text{s} \gets t$ $\omega^k_\text{g} \gets$ ClientUpdate($k, \omega_{g,t}, flag$) $timestamp^k_\text{g} \gets t$ $\omega_{g,t+1} \gets \sum_{k=1}^K\frac{n_k}{n}*f_\text{g}(t,k)*\omega^k_\text{g}$ $^\dagger$ $\omega_{s,t+1} \gets \sum_{k=1}^K\frac{n_k}{n}*f_\text{s}(t,k)*\omega^k_\text{s}$ $^\ddagger$ $\S$ $rounds\_in\_loop=15$ and $set_{ES}=\{11, 12, 13, 14, 0\}$ $\dagger$ $f_\text{g}(t,k)=a^{-(t-timestamp^k_\text{g})}$ $\ddagger$ $f_\text{s}(t,k)=a^{-(t-timestamp^k_\text{s})}$ The pseudo code for the two main components of the proposed temporally weighted aggregation asynchronous federated learning (TWAFL in short), one implemented on the server and the other on the clients is given in Algorithm \[AlgoTWAFL\_S\] and Algorithm \[AlgoTWAFL\_C\], respectively. $\mathcal{B} \gets$ (split $\mathcal{P}_k$ into batches of size $B$) $\omega \gets \omega$ $\omega_\text{s} \gets \omega$ $\omega \gets$ $\omega - \eta * \bigtriangledown \ell(w;b)$ return $\omega$ to server return $\omega_\text{s}$ to server The part to be implemented on the server consists of an initialization step followed by a number of communication rounds. In initialization (Algorithm \[AlgoTWAFL\_S\], Lines 2-6 ), the central model $\omega_0$, timestamps $timestamp_\text{g}$ and $timestamp_\text{s}$ are initialized. Timestamps are stored and to be used to weight the timeliness of corresponding parameters in aggregation. The training process is divided into loops. Lines 8-12 in Algorithm \[AlgoTWAFL\_S\] set $flag$ to be true in the last $1/freq$ rounds in each loop. Assume there are $rounds\_in\_loop$ rounds in each loop. Lines 13-14 randomly select a participating subset $S_t$ of the clients according to $C$, which is the fraction of participating clients per round. In lines 15-24, sub-function $Client Update$ is called in parallel to get $\omega^k$/$\omega_\text{g}^k$, and the corresponding timestamps are updated. $flag$specified whether all layers or the shallow layers only will be updated and communicated. Then in lines 25-28, the aggregation is performed to update $\omega_\text{g}$. Note that compared with Equation \[eq4\], a parameter $a$ is introduced into the weighting function in line 25 or line 27 to examine the influence of different weightings in the experiments. In this work, $a$ is set to $e$ or $e/2$. The implementation of the local model update (Algorithm \[AlgoTWAFL\_C\]) is controlled by three parameters, $k$, $\omega$, and $flag$, where $k$ is the index of the selected client, $flag$ indicates whether all layers or the shallow layers will be updated. $B$ and $E$ denote the local mini-batch size and the local epoch, respectively. In Algorithm \[AlgoTWAFL\_C\], Line 2 splits data into batches, whereas Lines 3-7 set all layers or shallow layers of the local model to be downloaded according to $flag$. In lines 8-12, local SGD is performed. Lines 13-17 return local parameters. Experimental Results and Analyses {#sec4} ================================= Experimental Design {#sec4A} ------------------- We perform a set of experiments using two popular deep neural network models, i.e., convolution neural networks (CNNs) for image processing and long short-term memory (LSTM) for human activity recognition, respectively, to empirically examine the learning peformance and communication cost of the prposed TWAFL. A CNN with two stacked convolution layers and one fully connected layer [@krizhevsky2012imagenet; @lecun1998gradient] is applied on the MNIST handwritten digit recognition data set, and a LSTM with two stacked LSTM layers and one fully connected layer is chosen to accomplish the human activity recognition task [@anguita2013public; @gers1999learning]. Both MNIST and the human activity recognition datasets are adapted to test the performance of the proposed federated learning framework for different real-world scenarios, e.g., non-IID distribution, unbalanced amount, and massively decentralized datasets, which will be discussed in detail in the next subsection. The federated averaging (FedAVG) [@mcmahan2017communication] is selected as the baseline algorithm since it is the state-of-the-art approach. The proposed TWAFL is also compared with two variants, namely, TWFL that adopts temporally weighted aggregation without asynchronous model update, and AFL that employs the asynchronous model update without using temporally weighted aggregation. Thus, four algorithms, i.e., FedAVG, TWAFL, TEFL and AFL will be compared in the following experiments. The most important parameters in the proposed algorithm are listed in Table \[Table1\]. Parameter $freq$ controls the frequency for updating and exchanging the parameters in the deep layers $\omega_\text{s}$ between the server and the local clients in a loop. For instance, $freq = 5/15$ means that only in the last five of the 15 rounds, the parameters in the deep layers $\omega_\text{s}$ will be uploaded/downloaded between the server and clients. $a$ is a parameter for adjusting the time effect in model aggregation. $K$ and $m$ are environmental parameters controlling the scale or complexity of the experiments, $K$ denotes the number of local clients, and $m$ is the number of participating clients per round. [@p[3.6cm]{}&lt;p[3.6cm]{}&lt;@]{} **Notion** & **Parameter Range**\ $freq$ & {3/15, $5/15^*$, 7/15}\ $a$ & {$e/2^*$, e}\ $K$ & {10, $20^*$}\ $m$ & {1, $2^*$}\ Default setting Settings on Datasets {#sec4B} -------------------- As discussed in Section I, the federated learning framework has its particular challenges, such as non-IID, unbalanced, and massively decentralized datasets. Therefore, the datasets used in our experiments should be designed to reflect these challenges. The generation of the client dataset is described in detail in Algorithm \[AlgoG\_localData\], which is controlled by four parameters $Labels$, $N_c$, $S_{min}$, and $S_{max}$, where $N_c$ controls the number of classes in each local dataset, $S_{min}$ and $S_{max}$ specify the range of the size of the local data, and $Labels$ indicates the names of classes involved in the corresponding tasks. $classes \gets$ Choices($Labels$,$N_C$) $L \gets $ Len($Labels$) $weights \gets$ Zeros($L$) $\mathcal{P} \gets$ Zeros($L$) $weights_{class} \gets$ Random(0,1) $sum \gets \sum_{class=1}^{L} weights_{class}$ $num \gets $ Random($S_{min},S_{max}$) $\mathcal{P}_{class} \gets \frac{weights_{class}}{sum} \times num $ $\mathcal{P}_{k} \gets \mathcal{P}$ ### Handwritten Digit Recognition Using CNN {#sec4B1} The MNIST dataset has ten different kinds of digits and one digit is a 28\*28-pixel gray-scale image. To partition the data over local clients, we first sort them by their digit labels and divide them into ten shards, i.e., 0, 1, ..., 9. Then, Algorithm \[AlgoG\_localData\] is performed to compute $\mathcal{P}_k$, which is the $k$-th client’s partition coefficient corresponding to these shards. In this task, $Labels = \{0, 1, 2, ..., 9\}$; $N_c$ is randomly chosen from $\{2, 3\}$, given $K = 20$, $S_{min} = 1000$ and $S_{max} = 1600$. For the sake of easy analyses, five partitions/local datasets, namely 1@MNIST, 2@MNIST, ..., 5@MNIST are predefined. Their corresponding 3-D column charts are plotted in Fig. \[fig\_6abcde\]. The architecture of the CNN used for the MNIST task has two $5\times5$ convolution layers (the first one has 32 channels and the other has 64) and $2\times2$ max pooling layers. Then a fully connected layer with 512 units and ReLu activation is followed. The output layer is a softmax unit. The parameter settings for the CNN are listed in Table \[Table2\]. [@p[3.6cm]{}&lt;p[3.6cm]{}&lt;@]{} **Layer** & **Shapes**\ conv2d\_1 & $5\times5\times1\times32$\ conv2d\_1 & $32$\ conv2d\_2 & $5\times5\times32\times64$\ conv2d\_2 & $64$\ dense\_1 & $1024\times512$\ dense\_1 & $512$\ dense\_2 & $512\times10$\ dense\_2& $10$\ ### Human Activity Recognition using LSTM {#sec4B2} In the Human Activity Recognition (HAR) dataset, each data is a sequence of images with a label out of six activities. Similar operations are applied on the HAR dataset to divide the dataset over local clients. Here, $Labels = \{0, 1, 2, ..., 5\}$; $N_c$ is randomly chosen from $\{2, 3\}$, given $K = 20$, $S_{min} = 250$ and $S_{max} = 500$. The architecture of the LSTM used in this study has two $5\times5$ LSTM layers (the first one with $cell\_size = 20$ and $time\_steps = 128$ and the other with $cell\_size = 10$ and the same $time\_steps$), a fully connected layer with 128 units and the ReLu activation, and a softmax output layer. The corresponding parameters of the LSTM is given in Table \[Table3\]. [@p[3.6cm]{}&lt;p[3.6cm]{}&lt;@]{} **Layer** & **Shapes**\ lstm\_1 & $9\times100$\ lstm\_1 & $25\times100$\ lstm\_1 & $100$\ lstm\_2 & $25\times100$\ lstm\_2 & $25\times100$\ lstm\_2 & $100$\ dense\_1 & $25\times256$\ dense\_1& $256$\ dense\_2 & $256\times6$\ dense\_2& $6$\ Results and analysis {#sec4C} -------------------- Two sets of experiments are performed in this subsection. The first set of experiments examines the influence of the most important parameters, $freq$, $a$, $K$ and $m$, on the performance of the two strategies using the CNN for the 1@MNIST dataset. The second set of experiments compares four algorithms in terms of the communication cost and learning accuracy using the LSTM on the HAR dataset, since the HAR is believed to be more challenging than the MNIST dataset. ### Effect of the Parameters {#sec4c1} The experiments here are not meant for a detailed sensitivity analysis. These experiments and related discussions aim to offer a basic understanding of parameter settings that may be helpful in practice. We mainly conduct research on the parameters listed in Table \[Table1\]. In investigating the influence of a particular parameter, all others are set to be their default value. Experiment on $freq$ are carried out for $freq=\{3/15, 5/15, 7/15\}$, given $a = e/2$, $K = 20$, and $m = 2$. Two metrics are adopted in this work for measuring the performance of the compared algorithms. One is the best accuracy of the central model within 200 rounds and the other is to the required rounds before the central model’s accuracy reaches 95.0%. Note that the same computing rounds means the same communication cost. All experiments are independently run for ten times, and their average (AVG) and standard deviation (STDEV) values are presented in Tables \[Table5\], \[Table6\], and \[Table8\]. In the tables, the average value is listed before the standard deviation in parenthesis. [@p[2.4cm]{}&lt;p[2.4cm]{}&lt;p[2.4cm]{}&lt;@]{} ***freq*** & **Accuracy** & **Round**\ 3/15 & 96.72% (0.0037) & 115.74 (32.19)\ 5/15 & 97.08% (0.0037) & 87.82 (20.03)\ 7/15 & 97.12% (0.0046) & 95.43 (14.64)\ AVG (STDEV) of best overall accuracy within 200 rounds. TWAFL on 1@MNIST reaches the accuracy 95.0% within 200 rounds. [@p[2.4cm]{}&lt;p[2.4cm]{}&lt;p[2.4cm]{}&lt;@]{} ***a*** & **Accuracy** & **Round**\ e & 96.92% (0.0041) & 75.26 (2.09)\ e/2 & 97.08% (0.0037) & 87.82 (20.03)\ AVG (STDEV) of best overall accuracy within 200 rounds. TWAFL on 1@MNIST reaches the accuracy 95.0% within 200 rounds. e $\approx$ 2.72 Based on the results in Tables \[Table5\], \[Table6\], and \[Table8\], the following observations can be made. - **Analysis on *freq*:** From the results presented in Table \[Table5\], we can conclude that the lower the exchange frequency is, the less communication costs will be required, which is very much expected. However, a too low $freq$ will deteriorate the accuracy of the central model. - **Analysis on *a*:** $a$ is a parameter that controls the influence of time effect on the model aggregation. When $a$ is set as $e$, the more recently updated local models are more heavily weighted in the aggregated model, and $a$ takes the value $e/2$, the previously updated local models will have a greater impact on the central model. The results in Table \[Table6\] indicate that $e/2$ is a better option for the CNN on the 1@MNIST dataset. When $a=1$, the algorithm is reduced to AFL, meaning that the parameters uploaded in different rounds will be of equal importance in the aggregation. Both parameters $K$ and $m$ leverage the scalability of federated learning. The AVG and STDEV values are calculated based on the recognition accuracy of the CNN on the five predefined datasets. Different combinations of $K$ and $m$ are separately assessed. Based on the results presented in Table \[Table8\], the following three conclusions can be made. First, a larger number of involved clients (a larger $m$) leads to a higher recognition accuracy. Second, TWAFL outperforms FedAVG when the active client fraction is $C = 0.1$, as indicated in the results in rows one and two in the table. Third, FedAVG is slightly better when the total number of clients is smaller and $C$ is higher, as shown by the results listed in the last row of the table. This implies that the advantage of the proposed algorithm over the traditional federated learning will become more apparent as the number of clients increases. This is encouraging since in most real-world problems, the number of clients is typically very large. [@p[2.4cm]{}&lt;p[2.4cm]{}&lt;p[2.4cm]{}&lt;@]{} [@c@]{} **Scalability**\ --------- --------- ***K*** ***m*** --------- --------- : Experimental results on the scalability of the algorithm.[]{data-label="Table8"} & **FedAVG** & **TWAFL**\ ---- --- 20 2 ---- --- : Experimental results on the scalability of the algorithm.[]{data-label="Table8"} & 96.83% (0.0097) & 97.16% (0.0096)\ ---- --- 10 1 ---- --- : Experimental results on the scalability of the algorithm.[]{data-label="Table8"} & 94.26% (0.0083) & 94.76% (0.0102)\ ---- --- 10 2 ---- --- : Experimental results on the scalability of the algorithm.[]{data-label="Table8"} & 96.16% (0.0167) & 96.11% (0.0909)\ AVG (STDEV) of best overall accuracy within 200 rounds. ### Comparison on Accuracy and Communication Cost {#sec4c2} The following experiments are conducted for testing the overall performance of the algorithms under comparison using the default values of the parameters. Five local datasets are predefined for MNIST and HAR, respectively. For instance, local dataset 1 of task MNIST are termed as 1@MNIST. The importance of the temporal weighting is first demonstrated by comparing the changes of the accuracy over the computing rounds. The baseline FedAVG and TEFL without the asynchronous update are compared and the results on the MNIST and HAR datasets are shown in Fig.\[fig\_9\] and Fig. \[fig\_10\], respectively. The following conclusions can be reached from the results in the two figures. First, the proposed temporally weighted aggregation helps the central model to converge to an acceptable accuracy. On the 1@MNIST and 2@MNIST datasets, TEFL needs about 30 communication rounds to achieve an accuracy reaches to 95.0%, while the traditional FedAVG needs about 75 rounds to achieve a similar accuracy, leading to a reduce of 40% communication cost. Similar conclusions can also be drawn on datasets 3@MNIST, 4@MNIST and 5@MNIST, although the accuracy of TEFL becomes more fluctuating. Second, on the HAR dataset, a more difficult task, TEFL can mostly achieve a higher accuracy than FedAVG except on 5@HAR. Notably, TEFL only needs about 75 communication rounds to achieve an accuracy of 90%, while FedAVG requires about 750 rounds, resulting in a nearly 90% reduction of the communication cost. Even on 5@HAR, TEFL shows a much faster convergence than FedAVG in the early stage. Finally, the temporally weighted aggregation may result in some fluctuations on the learning performance, which may be attributed to the fact that the contributions of some high-quality local models are less weighted in some communication rounds. [@ccccc@]{} **Dataset\_ID@Task** & [@c@]{} **FedAVG**\ ------------------------ --------------- ***Round (Accuracy)*** ***C. Cost*** ------------------------ --------------- & [@c@]{} **TEFL**\ ------------------------ --------------- ***Round (Accuracy)*** ***C. Cost*** ------------------------ --------------- & [@c@]{} **TWAFL**\ ------------------------ --------------- ***Round (Accuracy)*** ***C. Cost*** ------------------------ --------------- & [@c@]{} **AFL**\ ------------------------ --------------- ***Round (Accuracy)*** ***C. Cost*** ------------------------ --------------- \ 1@MNIST & ------------ ------ 75 (97.2%) 6.16 ------------ ------ & ---------------- ------ **31 (97.9%)** 0.74 ---------------- ------ & ------------- --- 106 (97.7%) 1 ------------- --- & ------------- ------ 175 (95.4%) 2.46 ------------- ------ \ 2@MNIST & ------------ ------ 85 (97.2%) 6.76 ------------ ------ & ---------------- ------ **32 (98.5%)** 1.33 ---------------- ------ & ------------ ------- 61 (98.1%) **1** ------------ ------- & ------------------- ------------ —(94.8%)$\dagger$ —$\dagger$ ------------------- ------------ \ 3@MNIST & ------------ ------ 73 (97.7%) 5.99 ------------ ------ & ---------------- ------ **31 (98.7%)** 1.13 ---------------- ------ & ------------ ------- 70 (97.9%) **1** ------------ ------- & ------------------- ------------ —(94.9%)$\dagger$ —$\dagger$ ------------------- ------------ \ 4@MNIST & ------------- ------ 196 (95.2%) 8.18 ------------- ------ & ---------------- ------ **61 (98.0%)** 1.14 ---------------- ------ & ------------- ------- 136 (96.1%) **1** ------------- ------- & ------------------- ------------ —(93.1%)$\dagger$ —$\dagger$ ------------------- ------------ \ 5@MNIST & ------------ ------ 98 (97.1%) 3.28 ------------ ------ & ---------------- ------ **76 (97.8%)** 3.17 ---------------- ------ & ------------ ------- 61 (96.1%) **1** ------------ ------- & ------------- ------ 109 (96.7%) 2.66 ------------- ------ \ 1@HAR & ------------- ------ 526 (92.3%) 4.72 ------------- ------ & ----------------- ---------- **166** (94.0%) **0.69** ----------------- ---------- & ----------------- --- 358 **(94.6%)** 1 ----------------- --- & --------------------- ------------- —(89.2%) $\ddagger$ —$\ddagger$ --------------------- ------------- \ 2@HAR & ------------- ------ 451 (94.7%) 5.65 ------------- ------ & ----------------- ---------- **119** (95.4%) **0.57** ----------------- ---------- & ----------------- --- 313 **(95.9%)** 1 ----------------- --- & --------------------- ------------- —(82.9%) $\ddagger$ —$\ddagger$ --------------------- ------------- \ 3@HAR & --------------------- ------------- —(87.3%) $\ddagger$ —$\ddagger$ --------------------- ------------- & ----------------- ---------- **174 (94.4%)** **0.69** ----------------- ---------- & ------------- --- 376 (93.2%) 1 ------------- --- & --------------------- ------------- —(88.5%) $\ddagger$ —$\ddagger$ --------------------- ------------- \ 4@HAR & ------------- ------ 856 (90.2%) 7.05 ------------- ------ & ----------------- ------ **181 (95.1%)** 1.28 ----------------- ------ & ------------- ------- 211 (94.1%) **1** ------------- ------- & ------------- ------ 751 (91.2%) 5.30 ------------- ------ \ 5@HAR & ------------- ------ 571 (92.6%) 5.49 ------------- ------ & ----------------- ---------- **155 (93.6%)** **0.57** ----------------- ---------- & ------------- --- 404 (93.1%) 1 ------------- --- & ------------- ------ 646 (92.5%) 2.38 ------------- ------ \ Cells are filled when the accuracy reaches 95% within 200 rounds. Cells are filled when the accuracy reaches 90% within 1000 rounds. Cells are marked when it can not reach 95% within 200 rounds. Cells are marked when it can not reach 90% within 1000 rounds. Rounds are needed to reach a certain accuracy, after which the best accuracy (within 200/1000 rounds) is listed in parenthesis. Total communication cost; the C. cost of TWAFL is termed as 1. Finally, the comparative results of the four algorithms on ten test cases generated from MNIST and HAR tasks are given in Table \[Table10\]. The listed metrics include the number of rounds needed, the classification accuracy (listed in parenthesis), and total communication cost (C. Cost for short). From these results, the following observations can be made: - Both TWAFL and TEFL outperform FedAVG on most cases in terms of the total number of rounds, the best accuracy, and the total communication cost. - TEFL achieves the best performance on most tasks in terms of the total number of rounds and the best accuracy. The temporally weighted aggregation strategy accelerates the convergence of the learning and improved the learning performance. - TWAFL performs slightly better than TEFL on MNIST in terms of the total communication cost, while TEFL works better than TWAFL on the HAR datasets. The asynchronous model update strategy significantly contributes to reducing the communication cost per round. - AFL performs the worst among the four compared algorithms. When comparing the performance of the AFL only adopting the asynchronous strategy and the TWAFL using both of them, the asynchronous one always needs the help of the other strategy. CONCLUSIONS AND FUTURE WORK {#sec5} =========================== This paper aims to reduce the communication costs and improve the learning performance of federated learning by suggesting an synchronous model update strategy and a temporally weighted aggregation method. Empirical studies comparing the performance and communication costs of the canonical federated learning and the proposed federated learning on the MNIST and human action recognition datasets demonstrate that the proposed asynchronous federated learning with temporally weighted aggregation outperforms the canonical one in terms of both learning performance and communication costs. This study follows the assumption that all local models adopt the same neural networks architecture and share same hyper parameters such as the learning rate of SGD. In future research, we are going to develop new federated learning algorithms allowing clients to evolve their local models to further improve the learning performance and reduce the communication costs. Acknowledgment {#acknowledgment .unnumbered} ============== This work is supported by the National Natural Science Foundation of China with Grant No.61473298 and 61876184. [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[ l@\#1 =l@\#1 \#2]{}]{} B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” in *Artificial Intelligence and Statistics*, 2017, pp. 1273–1282. J. Kone[č]{}n[ý]{}, B. McMahan, and D. Ramage, “[Federated Optimization: Distributed Optimization Beyond the Datacenter]{},” *arXiv Prepr. arXiv1511.03575*, no. 1, pp. 1–5, 2015. J. Konecn[ý]{}, H. B. McMahan, F. X. Yu, P. Richt[á]{}rik, A. T. 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Gers, J. Schmidhuber, and F. Cummins, “Learning to forget: Continual prediction with lstm,” 1999. [^1]: This work is supported by the National Natural Science Foundation of China with Grant No.61473298 and 61876184. (*Corresponding author: Yaochu Jin*) [^2]: Y. Chen and Xi. Sun are with the School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221116, China. Y. Chen and X. Sun contributed equally to this work and are co-first authors.(e-mail: [email protected]; [email protected]) [^3]: Y. Jin is with the Department of Computer Science, University of Surrey, Guildford, GU2 7XH, United Kingdom. (Email: [email protected])

--- author: - 'G. L. Litvinov' title: Hypergroups and hypergroup algebras --- [^1] The survey contains a brief description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. Representations of hypergroups are considered, being treated as continuous representations of topological hyper-group algebras. 1. Introduction {#introduction .unnumbered} =============== [**1.1.**]{} The important role which group-theoretic methods play in analysis and its applications, in particular in applications to theoretical physics, are well known. Such basic mathematical concepts as translation operator, convolution, periodic function, almost periodic function, positive definite function, etc. are formulated in group-theoretic terms. One can get far-reaching generalizations of the fundamental principles and results, connected with the concepts indicated, in the framework of the theory of hypergroups. Essential fragments of this theory became familiar as the theory of the Delsarte – B. M. Levitan generalized translation operators, the theory of Yu. M. Berezanskii – S. G. Krein of hypercomplex systems with continuous basis, the theory of convolution algebras, etc. Roughly speaking, a hypergroup is a topological space or manifold with a supplementary structure, which permits one to construct a Banach or topological algebra of the type of a group algebra – a hypergroup algebra. Thus, in the theory of hypergroups, just as in the theory of supergroups, the object of generalization is not so much a group as a group algebra (coalgebra). The ideas and methods of the theory of group representations carries over to the case of hypergroups, while it is convenient to treat representations of hypergroups as representations of the corresponding hypergroup algebras. With the help of the theory of representations for hypergroups one can generalize the duality principle of L. S. Pontryagin, construct an analog of the Fourier transform, get the Plancherel theorem and the inversion formula. It turns out that the converse result is also valid: the existence of a transformation of the type of the Fourier transform, for which the Plancherel theorem and inversion formula are valid, is necessarily connected with the existence of a certain hypergroup. This result explains the appearance of hypergroup structures in various problems of harmonic analysis. For some classes of hypergroups results on a connection with infinitesimal objects in the spirit of the Lie theory are found. Not only Lie algebras but also algebras generated by commutation relations of a more general kind can appear as such objects. The present survey contains a short description of the ideas, constructions, results, and prospects of the theory of hypergroups and generalized translation operators. In composing it material of the papers \[17, 57, 61, 62\] has been used in part. Separate aspects of the theory are considered in detail in the following works of monograph or survey type: \[3, 7, 37, 51, 55, 56, 66, 93, 97, 102, 103, 131, 132, 136, 160, 169, 183\], in which one can find additional information and references to the literature. An extremely large literature is devoted to hypergroups of special form – topological groups and semigroups, and also their representations (cf. in particular, \[31, 32, 40, 43, 59, 69, 72, 73, 86, 92, 156-158, 174, 177\]), the systematic analysis of which would leave the framework of the present survey. The literature cited in the present survey does not pretend to exhaustive completeness; the works included in this list contain additional bibliography.\ [**1.2.**]{} The concept of hypergroup arose originally as a generalization of the concept of abstract group. An abstract “algebraic” hypergroup is a set $H$ with a binary multiplication operation $a , b \mapsto ab$ which associates with any pair of elements of $H$ a nonempty subset of $H$. The multiplication is assumed to be associative in the sense that the sets $(ab)c$ and $a(bc)$ coincide; here $(ab)c$ denotes the union of the sets $dc$ for all $d \in (ab)$, and the product $a(bc)$ is defined analogously. A hypergroup $H$ has an identity $e \in H$ if $a \in ea \cap ae$ for all $a \in H$. The standard examples of hypergroups are connected with sets of cosets and conjugacy classes of elements in groups, with sets of points in certain geometries. However, it is more convenient to start with the analysis of an example, which, at first glance, is of a different kind. Let $G$ be a compact group, $\widehat{G}$ be the set of all irreducible linear (finite-dimensional) representations of the group $G$, considered up to equivalence. For any irreducible representations $\alpha$ and $\beta$ of the group $G$, their tensor product $\alpha \otimes \beta$ decomposes uniquely into a direct sum of primary representations $$\alpha \otimes \beta = \sum^n_{i=1} m_i \pi_i, \eqno(1.1)$$ where $\pi_i \in \widehat{G}$ and $m_i$ is the multiplicity with which the irreducible representation $\pi_i$ occurs in the tensor product $\alpha \otimes \beta$. If the product of the elements $\alpha$ and $\beta$ in $\widehat{G}$ is defined as the set $\{ \pi_1 , \pi_2 \ldots ,\pi_n \}$ of irreducible representations contained in $\alpha \otimes \beta$, then $\widehat{G}$ gets the structure of a hypergroup. The example given was considered by Helgason in \[130\], which is devoted to lacunary Fourier series on compact groups. In order to take into account the multiplicities with which the irreducible representations occur in the decomposition (1.1), Helgason defined the product $\alpha \beta$ as a finite measure on $\widehat{G}$. The support of this measure coincides with the set $\{ \pi_1 , \pi_2 \ldots ,\pi_n \}$ of elements of $\widehat{G}$ which occur in the decomposition (1.1), and the measure of the point $\pi_i$ is the integer $m_i$. If one identifies each element $\pi \in G$ with the unit measure $\delta \pi$, concentrated at the point $\pi$ (the delta-function), then one can consider the measure $\alpha \beta$ as the result of a convolution type operation over the measures $\delta_{\alpha}$ and $\delta_{\beta}$. Since any measure on $\widehat{G}$ is a linear combination of delta-functions, one can extend this operation linearly to the space $\mathscr{M} (\widehat{G})$ of all finite complex measures with finite support and even to the space $\mathscr{M}^b (\widehat{G})$ of all bounded measures on $\widehat{G}$. As a result, $\mathscr{M} (\widehat{G})$ and $\mathscr{M}^b (\widehat{G})$ are turned into associative algebras with identities (hypergroup algebras).\ [**1.3.**]{} It is easy to modify the definition of multiplication – convolution of measures – in the example considered (replacing the measure $\delta_{\pi}$ by $\delta_{\pi} / \dim \pi$) so that the product of delta-functions appears as a probability measure, i.e., a positive measure with unit volume. Constructions of this kind were studied particularly intensively after the appearance of the papers of Dunkl (112, 113\], Jewett \[136\], Spector \[169\], and the beautiful survey of Ross \[160\]. Following these papers, in the modern literature hypergroup usually means a locally compact space $H$, such that on the set $\mathscr{M}^b (H)$ of bounded Radon measures there is given an associative bilinear operation $\mu_1 , \mu_2 \mapsto \mu_1 \ast \mu_2$ called (generalized) convolution, where the result of the convolution of any probability measures is again a probability measure. It is required that the convolution turn $\mathscr{M}^b (H)$ into a Banach algebra with identity, whose role is played by the delta-function $\delta_e$, concentrated at some point $e \in H$. Moreover, it is required that the convolution be compatible with some involution in $H$ and that additional conditions of the type of continuity hold. In particular, if $H$ is a locally compact group, then the operation in $\mathscr{M}^b (H)$ coincides with the usual convolution of measures, and $\mathscr{M}^b (H)$ coincides with the group algebra. In this case $\delta_a \ast \delta_b = \delta_c$, where $c$ is the product of the elements $a$ and $b$ in $H$. In general, $\delta_a \ast \delta_b$ is a probability measure which can be considered as the distribution of a random variable with values in $H$. Hence one can say that in the hypergroup $H$ there is defined an associative product of elements, but it is defined “randomly” and its result is a “random” element in $H$. In what follows, hypergroups in the sense of \[112, 136, 169\] will be called p-hypergroups, so as to distinguish them from the objects of a more general kind introduced by Delsarte.\ [**1.4.**]{} Although in Ross’ survey \[160\] it is indicated that Helgason \[130\] was the first analyst to use the term “hypergroup,” this term was used in a broader sense in the important (and earlier) paper of Delsarte \[108\] in connection with various analytic applications (differential operators, mean-periodic functions, almost periodic functions). Hypergroup in the sense of \[108\] is a concept which is essentially equivalent with the concept of generalized translation operators (g.t.o. for the sake of brevity) introduced by Delsarte at the end of the thirties, by the axiomatization of the properties of group translations. Important ideas and a number of original results on the theory of generalized translation operators are due to him \[105-109\]. The systematic construction of the theory is given mainly in the papers of B. M. Levitan, including various questions of abstract harmonic analysis, elements of the theory of finite-dimensional linear representations, generalization of Lie theory, applications to the theory of differential operators and questions of decomposition in eigenfunctions, generalization of the theory of almost periodic functions, etc.; cf., e.g., \[47-56\]. In 1950 Yu. M. Berezanskii and S. G. Krein, developing the theory of generalized translation operators (g.t.o.), introduced the concept of hypercomplex system (h.s.) with continuous basis; this basis is a locally compact space, provided with a structure which turns it into a hypergroup. Later, harmonic analysis was constructed for h.s. At the first stage basically commutative h.s. with compact and discrete basis were studied and the case of a general locally compact basis was to a large extent only noted; cf., e.g., \[1, 2, 6-8\]. The results of this stage are summarized in \[7\]. Recently, in \[3-5, 35\] a systematic construction of the theory of commutative h.s. with locally compact basis is given; \[14, 15\] are devoted to similar questions. Interest in this circle of questions was reanimated under the influence of a stream of papers devoted to objects similar to $p$-hypergroups (cf. above). We note that in the framework of the theory of p-hypergroups, many results previously found in the theory of generalized shift operators and h.s. with continuous basis were proved again. Convolution algebras are similar to p-hypergroups and h.s. in their nature \[135, 149, 150, 156, 157, 178, 20-22, 121, 122, 174\].\ [**1.5.**]{} In the study of hypergroups the apparatus of the theory of Banach algebras is widely applied. Even in the early papers of B. M. Levitan \[47-50\] Banach “hypergroup” algebras are constructed; analogs of the Plancherel theorem, Bochner’s theorem on the representation of positive definite functions, and the Pontryagin duality law are found for a wide class of commutative families of g.t.o. However in certain important cases, instead of Banach algebras one should consider hypergroup algebras of a different kind. Even for topological groups and Lie groups some class or other of topological group algebras arises naturally, depending on the version of the theory of representations and harmonic analysis being investigated, cf., e.g., \[32, 59\]. For hypergroups the situation is made more complex in that one is not always able to construct the Banach hypergroup algebra; a familiar example is the family of g.t.o., generated by a Sturm – Liouville operator with rapidly increasing potential, cf. \[69, Sec. 31, point 11\]. Topological hypergroup algebras generated by Sturm – Liouville operators and consisting of generalized functions (or measures) with compact support on the line were considered in \[134\]. A quite general theory of commutative g.t.o. in spaces of basic and generalized functions is actually constructed by L. Ehrenpreis \[117\]. Topological algebras of analytic functionals, connected with certain g.t.o. on the complex line are described in \[41\]. For g.t.o. of general type, topological hypergroup algebras and the theory of their representations are considered in \[61\], in which the problems of spectral analysis and synthesis are treated as problems of studying ideals of hypergroup algebras. Hilbert hypergroup algebras are considered in \[17\] in connection with the generalization of the Plancherel theorem to the case of (not necessarily commutative) hypergroups. Infinitesimal hypergroup algebras arise in generalizing Lie theory and play the same role as universal enveloping algebras of Lie algebras and associative upper-envelope Lie algebras in the sense of P. K. Rashevskii \[77\] for Lie groups. B. M. Levitan proved that one can associate with each Lie algebra not only a Lie group, but also a differentiable family of g.t.o., cf. \[52-56\], and also \[24, 25, 126\]; thus enveloping algebras of Lie algebras can appear as infinitesimal objects not only for Lie groups, but also for hypergroups. Recently, hypergroup algebras generated by non-Lie commutation relations have been investigated intensively, cf. \[28-30, 36, 37, 65-67\]. We note the application of the techniques of hypergroup algebras (infinitesimal and Banach), generated by commutation relations, to solving problems of mathematical physics in the framework of the operator method of V. P. Maslov \[37, 65, 66\].\ [**1.6.**]{} A characteristic singularity of the contemporary stage of development of the theory of hypergroups is the intensive interaction of this theory with the theories of supermanifolds, supergroups, and Lie super-algebras (cf. \[9, 11, 58, 64, 67, 71, 139, 162\]), ringed groups (Kac algebras, cf. \[38, 16, 120, 163, 164\]), algebras generated by commutation relations (cf. above and \[19, 33, 34, 79-81, 133, 155\]), operator and topological algebras (cf. \[31, 61, 62, 69, 70, 82, 120, 137, 172\]), mean-periodic functions, and convolution equations (cf. \[104, 116, 148, 165, 26, 27, 41, 44-46, 59, 61, 70, 75, 76, 78, 83, 87, 91, 109, 118, 119\]). The spectrum of applications of the theory of hypergroups continues to expand. Of the relatively new domains of application of the theory we note algebraic topology (cf. \[12,13, 88\]) and the theory of probability, in particular, the theory of probability measures on groups, hypergroups, and homogeneous spaces (cf. \[131, 132\] for a surveying account, cf. also \[90, 93-95, 111, 124, 125, 139, 145, 146\]). At the same time active work continues on such traditional domains of application of the theory of generalized translation operators as differential operators and equations (cf. \[7, 25, 27, 37, 511 55, 56, 65, 66, 90, 97, 98, 106, 134, 136, 145\]), expansions in orthogonal systems of functions and special functions (cf. \[2, 7, 51, 55, 56, 114, 131, 136, 142-144, 183\]), spectral decomposition of operators (cf. \[3, 5, 51, 55, 56, 135, 149, 150, 178\]), duality theory (cf. \[3, 7, 15, 18, 39, 50, 96, 127, 151, 152, 169\]), questions of harmonic analysis (cf. \[3, 7, 14, 17, 35, 51, 55, 56, 76, 78, 87, 93, 96, 99-101, 107, 136, 140-144, 167-170, 175, 177, 179-183\]), etc.\ [**1.7.**]{} The author considers it his pleasant duty to thank V. M. Bukhshtaber, D. I. Gurevich, B. M. Levitan, L. I. Vainerman, A. M. Vershik, and D. P. Zhelobenko for valuable suggestions, assistance, and guidance. 2. Hypergroups and Generalized Translation Operators {#hypergroups-and-generalized-translation-operators .unnumbered} ==================================================== Let $H$ be an arbitrary set, $\Phi$ be some linear space of complex-valued functions, defined on $H$. Let us assume that to each element $x \in H$ there is assigned a linear operator $R^x$ in $\Phi$, where for any fixed $y \in H$ the function $\psi (x) =R^x \varphi (y)$ is contained in $\Phi$ for all $\varphi \in \Phi$. We denote the linear operator $\varphi (x) \to \psi (x) = R^x \varphi (y)$ in $\Phi$ by $L^x$. The set $H$ is called a [*hypergroup*]{}, if the following conditions (axioms) hold: 1\) Associativity Axiom. For any elements $x, y \in H$ one has $$R^x L^y = L^y R^x .$$ 2) There exists in $H$ an element $e$, called neutral (or the identity), such that $R^e = I$, where $I$ is the identity operator in $\Phi$. In this case one says that the operators $R^x$ form a family of generalized translation operators (g.t.o.); thus the concepts of g.t.o. and hypergroup are considered as equivalent. The operators $R^x$ are often called (generalized) right translation operators, in which case the $L^y$ are called left translation operators. G.t.o. obviously arise in any linear subspace of functions on an arbitrary group or semigroup with identity which is invariant with respect to translations. We set $R^x \colon \varphi (h) \mapsto \varphi (hx)$ where $hx$ is the product of the elements $h$ and $x$ in the semigroup. It is clear that $L^y \varphi (h) = \varphi (yh)$ and the axiom of associativity reduces to the associativity of multiplication in the semigroup, and the neutral element is the identity in the semigroup. In the present case the operators $L^y$ also form a family of g.t.o. However in general the $L^y$ do not form g.t.o., since the operator $L^e$ is not necessarily the identity. It is easy to verify that this operator is a projector; its image $\widetilde{\Phi}$ is called the [*basic subspace*]{} of $\Phi$. A basic subspace is invariant with respect to left and right translations, and in $\widetilde{\Phi}$ the operators $L^y$ form a family of g.t.o., so that the symmetry between left and right translations is reestablished. Often the second axiom is strengthened by requiring that $L^e = I$, i.e., $\Phi = \widetilde{\Phi}$, or weakened, for example, by requiring the existence of only an approximate identity (in some sense or other). Conditions 1) and 2) are the most general axioms for a hypergroup. By adding additional conditions, one can single out narrower classes of hypergroups. For example, if $$R^x R^y = R^y R^x \eqno (2.1)$$ for all $x, y \in H$, then the hypergroup $H$ is said to be [*commutative*]{} (respectively, the g.t.o. $R^x$ are called [*commutative*]{}). In this case $R^x \varphi = L^x \varphi$ for any function $\varphi$ from the basic subspace $\widetilde{\Phi}$. If $H$ is a locally compact space with a positive measure $m$, then it is usually required that the operators $L^x$ and $R^x$ act compatibly on the space $C(H)$ of continuous functions on $H$ and on the spaces $L^p (H,m)$ for $p \geq 1$, while on $R^x$ and $L^x$ one imposes additional conditions of the type of continuity; if $H$ is a smooth manifold, then one imposes conditions of the type of differentiability, etc. Cf. \[3, 7, 14, 17, 18, 47, 51, 55, 56, 61, 65, 105, 108, 112, 113, 136, 160, 169, 156, 157\] for various versions of the axiomatics of hypergroups and g.t.o.\ Let $\Phi$ be a finite h.s., i.e., a finite-dimensional associative algebra with fixed basis $H = \{ h_1 ,\\ \ldots , h_n \}$. We identify $\Phi$ with the space of functions on the finite set $H$, by assigning to the function $\varphi$ the element with coordinates $\varphi (h_1) , \ldots ,\\ \varphi(h_n)$, i.e., the element $\sum^n_{i=1} \varphi (h_i) h_i \in \Phi$. We set $R^x \varphi (h) = \sum^n_{i=1} \varphi (h_i) h_i \ast x$ where $h_i \ast x$ is the product of the elements $h$ and $x$ in the algebra $\Phi$. It is easy to verify that the operators $R^x$ form a family of g.t.o. (so that $H$ is a hypergroup) if and only if one of the elements of the basis of $H$ is a right identity in the algebra $\Phi$ (a two-sided identity if one requires that the basic subspace $\widetilde{\Phi}$ coincide with $\Phi$). In the way indicated one establishes a correspondence between finite hypergroups and finite h.s. Thus, the concept of hypergroup (as well as that of g.t.o.) can be considered as a far-reaching generalization of the classical concept of hypercomplex system. Many examples of infinite hypergroups which are naturally treated as h.s. with countable or continuous basis are considered in \[1-8, 51, 55, 56\] and in the papers devoted to p-hypergroups (cf. the Introduction). Some of these examples are considered below.\ Let $G$ be a topological group, $K$ be a compact group of automorphisms of the group $G, dk$ be an invariant measure on $K$, and $\int dk = 1$. In the space $\Phi = C(G)$ of continuous functions on $G$, we define a g.t.o. $R^x$ with the help of the equation: $$R^x \varphi (g) = \int_K \varphi (g \cdot k (x)) dk, \eqno(2.2)$$ where $\varphi \in \Phi, k(x)$ is the image of the element $x \in G$ under the automorphism $k \in K, g \cdot k(x)$ is the product of the elements $g$ and $k(x)$ in $G$. It is easy to verify that both hypergroup axioms hold, where the neutral element is the identity of the group $G$. The hypergroup constructed is commutative, if the group $G$ is commutative. In the present case the basic subspace $\widetilde{\Phi}$ consists of all functions which are constant on orbits with respect to the action of the group $K$, and the operators $R^x$ and $R^y$ coincide in $\widetilde{\Phi}$, if $x$ and $y$ lie on the same orbit. Hence the space of orbits $H$ can also carry the structure of a hypergroup, identifying $\widetilde{\Phi}$ with the space $C(H)$ of continuous functions on $H$ (with respect to the quotient toplogy) and setting $R^h = R^x$, where $x$ is an arbitrary element of the orbit $h$. The process described, of constructing a hypergroup consisting of orbits, is called reduction. In general, reduction consists of replacing $\Phi$ by $\widetilde{\Phi}$ and identifying elements of the original hypergroup if the values of any function on $\widetilde{\Phi}$ on these elements coincide. For the reduced hypergroup one has a strengthened form of the axiom of existence of an identity: $R^e = L^e = I$. On the other hand, as a result of reduction, a smooth manifold can be turned, for example, into a manifold with boundary (cf. below point 2.4). A generalization of Delsarte’s construction, which lets one get a large class of hypergroups, is given in the important paper \[28\].\ We consider a special case of the construction described in point 2.3, when $G$ coincides with the real axis ${\bf R}$, and the group of automorphisms consists of two elements (reflection with respect to zero and the identity map). In this case (2.2) takes the form $$R^x \varphi (t) = \frac{1}{2} [\varphi (t + x)+ \varphi (t-x)] . \eqno(2.3)$$ $x, t \in {\bf R}$. The basic subspace consists of all even functions. The operator of left generalized translation has the form $$L^y \varphi (t) = \frac{1}{2} [ \varphi (t+y)+ \varphi (-t+y)] .$$ It is clear that the operator $L^y$ for any value of the parameter $y$ carries an arbitrary function into an even one. It is easy to verify that the given hypergroup is commutative and on the basic subspace the right generalized translations coincide with the left ones. Reduction also provides the orbit space, which can be identified with the half-line $0 \leq t < \infty$.\ Another interesting special case of Delsarte’s example we have for $G = K$ and $k(x) = k^{-1} xk$; here the basic subspace consists of all central functions on $G$, i.e., functions which are constant on conjugacy classes of elements. The reduced hypergroup $H$ is commutative and is made up of conjugacy classes of elements of the compact group $G$. This hypergroup is closely connected with representations of the group $G$ and is dual with respect to the commutative hypergroup $\widehat{G}$, considered above in point 1.2. The hypergroup algebra for $H$ coincides with the center of the group algebra $G$. From this point of view the hypergroup of conjugacy classes of elements of a compact group was studied in \[1\]. Cf. \[160, 161, 129\] for various generalizations. For example, if a compact group of automorphisms $K$ of an arbitrary locally compact group $G$ includes all inner automorphisms, then the orbit space in $G$ with respect to $K$ is provided with a commutative hypergroup structure.\ Let $G$ be a locally compact group with a compact subgroup $K$, and $H = K\setminus G/K$ be the space of double cosets with respect to the subgroup $K$ (such a coset contains along with the element $g \in G$ all elements of the form $k_1 gk_2$, where $k_1 , k_2 \in K$). If $K$ is a normal subgroup of $G$, then $H$ coincides with the quotient group $G/K$. Let $\Phi$ be the space consisting of all those continuous functions $\varphi (g)$ on $G$, such that $\varphi (k_1 g k_2) = \varphi (g)$ for any elements $k_1 , k_2 \in K, g \in G$. The family $R^x$ of right generalized translations in $\Phi$ is defined by $$R^x \varphi (g) = \int_K \varphi (gkx) dk . \eqno(2.4)$$ The space $\Phi$ can be identified with the space $C(H)$ of continuous functions on $H$, and arguing just as in Delsarte’s example (cf. above point 2.3), one can provide $H$ with a hypergroup structure. We note that (2.4) defines a g.t.o. in the space of continuous functions on $G$, invariant with respect to the action of the subgroup $K$ on the right, also, i.e., on the space of continuous functions on the homogeneous space $G/K$ of left cosets. The hypergroup $H$ is constructed from the hypergroup $G/K$ by reduction, while elements in $H$ correspond to orbits in $G/K$ with respect to the action of the stationary subgroup $K$. For example, if $G$ is the group of Euclidean motions of the plane, and $K$ is the subgroup of rotations about a fixed point, then $G/K$ coincides with the plane, the orbits in $G/K$ with respect to the action of $K$ form a family of concentric circles and an element of the hypergroup $H = K\smallsetminus G/K$ is determined by the radius of the corresponding circle. Thus, in the present case H can be identified with the half-line $[0, \infty )$. This hypergroup is closely connected with the Helmholtz equation on the plane and the Bessel equation (cf., e.g., \[136\]). Starting from different considerations, this same example was actually already considered by Delsarte \[105\]. Hypergroups of this type were studied in the frameworks of the theory of hypercomplex systems with continuous basis and the theory of p-hypergroups (cf., e.g., \[7, 131, 136, 169, 146\]).\ \ Let the hypergroup $H$ be a locally compact space, on which there is given a positive Borel measure $m$. Let us assume that the action of the g.t.o. $R^x$ and $L^x$ is defined not only on the original space $\Phi$, but also on the Hilbert space $L^2 (H, m)$. We denote by $\tilde{R^x}$ the adjoint of the operator $R^x$, defined by $$\int_H R^x \varphi (y) \cdot \bar{\psi} (y) dm(y) = \int_H \varphi (y) \cdot \overline{\widetilde{R^x} \psi (y)} dx (y) , \eqno(2.5)$$ where $\varphi, \psi \in L^2 (H, m)$ and $\psi \mapsto \bar{\psi}$ is complex conjugation. The operators $\widetilde{R^x}$ form a family of adjoint (right) g.t.o.; one defines adjoint operators of left translation $\widetilde{L^x}$ analogously. If $G$ is a locally compact group with right invariant measure $m$, then the adjoint of the right translation $\varphi (g) \mapsto \varphi (gx)$ is the operator $\varphi (g) \mapsto \varphi (g x^{-1})$; if the measure m is left-invariant, then the operator adjoint to left translation by $x$ has the form $\varphi (g) \mapsto \varphi (x^{-1} g)$. B. M. Levitan formulated a condition on the measure $m$ of the type of invariance of this measure with respect to generalized translations in terms of adjoints of g.t.o. (cf. \[47, 51, 55, 56\]). In particular, for commutative g.t.o. the corresponding condition has the form $$\widetilde{R^x} R^y = R^y \widetilde{R^x} , \eqno(2.6)$$ i.e., the g.t.o. and the adjoint g.t.o commute; if (2.6) holds, then the hypergroup is called normal. Let us assume that there exists an involutive homeomorphism $x \mapsto x^*$ mapping the hypergroup $H$ to itself, while in $\Phi$ there is induced an involution $\varphi (x) \mapsto \tilde{\varphi} (x) = \varphi (x^*)$. We call $H$ a hypergroup with involution, if for all $x, y \in H, \varphi \in \Phi$ one has $$(R^{x^*} \varphi ) (y^*) = \overline{R^y (\Tilde{\Tilde \varphi} (x))} . \eqno(2.7)$$ If the g.t.o. $R^x$ are real, i.e., commute with complex conjugation, then (2.7) assumes the form $$(R^{x^*} \varphi ) (y^*) = R^y (\tilde \varphi (x)) . \eqno(2.8)$$ (2.7) and (2.8) mean that an involution is also induced on the hypergroup algebra (cf. below, point 3.6). An involution on a hypergroup is a substitute for the map $x \mapsto x^{-1}$ in a group. For a hypergroup with involution the space $\Phi$ coincides with its basic subspace $\widetilde{\Phi}$. We call a measure $m$ on $H$ right-invariant, if the adjoint right translation operator $\widetilde{R^x}$ coincides with the right translation $R^{x*}$ , defined by the element $x^* \in H$; similarly a measure is left-invariant, if $\widetilde{L^x} = L^{x*}$. Under additional conditions, which reduce to the fact that the function $\varepsilon (x) \equiv 1$ (as well as other constants) is unchanged under the action of the g.t.o., a right-invariant (left-invariant) measure is unchanged under the action of right (left) translations. In fact, formally substituting the function $\varepsilon$ into (2.5) in place of $\psi$ and considering that $\widetilde{R^{x*}} \varepsilon = R^{x*} \varepsilon \equiv 1$ we get $\int R^x \varphi (y) dm (y) = \int \varphi (y) dm (y)$ which is what was required. With the help of the involution a left-invariant measure can be transformed into a right-invariant one and conversely. The reduced hypergroups described above in points 2.3 – 2.6 are hypergroups with involutions. If $g$ is an element of the orbit (coset) $x$, then $x^*$ coincides with the orbit (coset) of the element $g^{-1}$. The g.t.o. arising here are real. The hypergroups indicated have invariant measures, induced by the invariant measures on the corresponding groups. A hypergroup with involution is called [*Hermitian*]{}, if the involution reduces to the identity map. For example, the reduced hypergroup described in point 2.4 is Hermitian. A Hermitian hypergroup is automatically commutative, if it generates real g.t.o. 3. Hypergroup Algebras {#hypergroup-algebras .unnumbered} ====================== [**3.1.**]{} If V is a locally convex space, and $V'$ is the dual space (consisting of all continuous linear functionals on $V$), then the value of the functional $v' \in V'$ on the element $v \in R$ will be denoted by $\langle v', v \rangle$. We recall that the weak topology $\sigma (V', V)$ on $V'$ is defined by the seminorms $\upsilon ' \mapsto \vert \langle \upsilon ', \upsilon \rangle \vert $ , where $v$ runs through $V$; analogously, the seminorms $\upsilon ' \mapsto \vert \langle \upsilon ', \upsilon \rangle \vert $, where $v' \in V'$, define the weak topology on $V$. The strongest locally convex topology on $V$, admitting $V'$ as the space of all continuous linear functionals, is called the [*Mackey topology*]{} and is denoted by $\tau (V, V')$; one defines the [*Mackey topology*]{} $\tau (V', V)$ on $V'$ analogously. The space $V$ is called a [*Mackey space*]{} if its topology is the Mackey topology $\tau (V, V')$. All the barreled spaces (in particular, all the Banach spaces and all the complete metrizable spaces) are Mackey spaces. In what follows, if nothing is said to the contrary, then dual spaces are provided with the Mackey topologies. If $V$ is a Mackey space, then passage to the Mackey-dual space is reflexive in the sense that $(V')' = V$. We denote by $S(V)$ the algebra of all weakly continuous linear operators on $V$, provided with the weak operator topology; this topology is defined by the seminorms $A \mapsto \vert \langle \upsilon ', A \upsilon \rangle \vert$, where $v$ and $v'$ run through $V$ and $V'$ respectively. If $V$ is a Mackey space, then any operator from $S(V)$ is continuous. Elements of $S(V)$ are denoted below either by capital Latin or by lower-case Greek letters. Let $\Phi$ be a locally convex space, consisting of functions $\varphi (x)$, defined on some set $H$. The value $\langle f, \varphi \rangle$ of the functional $f \in \Phi '$ on the function $\varphi \in \Phi$ will sometimes be denoted by $\int \varphi (x) d f (x)$. If the operator-valued function $x \mapsto A^x$ maps $H$ into $S(V)$ and the function $\varphi (x) = \langle v', A^x v \rangle$ is contained in $\Phi$ for all $v \in V, v' \in V'$, then we denote by $A (f) = \int A^x df (x)$ the linear operator on $V$, defined by $$\langle \upsilon ', A(f) \upsilon \rangle = \int \langle \upsilon ', A^x \upsilon \rangle df(x) , \eqno(3.1)$$ where $v$ and $v'$ run respectively through $V$ and $V'$ (the existence of the operator $A(f)$ in each concrete case requires a proof). The notation $\int \pi_x df(x)$ has analogous meaning, where $x \mapsto \pi_x$ is an operator-valued function. The information on locally convex spaces needed for what follows can be found, for example, in \[89\]. The proofs of the basic assertions on hypergroup algebras formulated in points 3.2 – 3.6 are easily extracted from \[61, 62\]; cf. also \[156, 157\].\ We consider g.t.o. acting on the space $C(H)$ of continuous functions on the locally compact space $H$, where $C(H)$ is provided with the topology of uniform convergence on compacta. Let us assume that the right translation and left translation operators $R^x$ and $L^x$ are continuous and depend weakly continuously on the parameter $x$; in this case we shall say that the given family of g.t.o. is [*continuous*]{} and generates in $H$ a [*locally compact hypergroup structure*]{}. For this it is sufficient that the correspondence $\varphi (x) \mapsto \psi (x,y) = R^x \varphi (y)$ be a continuous map $C(H) \to C(H \times H)$. Let $\mathscr{M} (H)$ be the dual space to $C(H)$, consisting of complex Radon measures with compact support. For any element $f \in \mathscr{M} (H)$, with the help of equations of the type of (3.1), one defines continuous linear operators on $C(H)$: $$R(f) = \int R^x df(x), L(f) = \int L^x df(x) . \eqno(3.2)$$ By the generalized convolution (or simply convolution) of the elements $f, g \in \mathscr{M} (H)$ is meant the functional $f \ast g$ defined by $$\langle f \ast g, \varphi \rangle = \langle f, R(g) \varphi \rangle . \eqno(3.3)$$ Since the operator $R(g)$ is continuous, $f \ast g \in \mathscr{M} (H)$. One can verify that one also has $$\langle f \ast g, \varphi \rangle = \langle g, L(f) \varphi \rangle . \eqno(3.4)$$ (3.3) and (3.4) show that the operator $R(g)$ is adjoint to the operator $f \mapsto f \ast g$ on $\mathscr{M} (H)$ and the operator $L(f)$ is adjoint to the operator $g \mapsto f \ast g$. It follows from this and from the continuity of the operators $L(f), R(g)$, that convolution is separately continuous in the weak topology $\sigma (\mathscr{M} (H), C(H))$ and in the Mackey topology $\tau (\mathscr{M} (H), C(H))$, i.e., gives a separately continuous bilinear map $\mathscr{M} (H) \times \mathscr{M} (H) \to \mathscr{M} (H)$. It is easy to prove that the operators $L(f)$ and $R(g)$ commute for all $f, g \in \mathscr{M} (H)$ from which it follows that convolution is an associative operation. Thus, $\mathscr{M} (H)$ is a topological algebra (or locally convex algebra), i.e., an associative algebra, provided with a locally convex topology such that the product of elements is separately continuous. We denote by $\delta_x$ the Dirac measure, concentrated at the point $x \in H$ (“$\delta$ -function”), i.e., the functional from $\mathscr{M} (H)$ such that $\langle \delta_x , \varphi \rangle = \varphi (x)$ for all $\varphi \in C(H)$. It is well known that the element $f \in \mathscr{M} (H)$ coincides with some Dirac measure $\delta_x$,if and only if it is a multiplicative functional on $C(H)$, i.e., when $\langle f, \varphi \psi \rangle = \langle f, \varphi \rangle \langle f. \psi \rangle$ where $\varphi \psi$ is the usual product of functions $\varphi , \psi \in C(H)$. It is clear from (3.2) that $R(\delta_x ) = R^x$ and $L( \delta_x ) = L^x$. If $e$ is the neutral element of $H$, then $\langle f \ast \delta_e , \varphi \rangle = \langle f, R^e \varphi \rangle = \langle f, \varphi \rangle$ for all $f \in \mathscr{M} (H) , \varphi \in C(H)$, i.e., $f \ast \delta_e = f$ and $\delta_e$ is a right identity in $\mathscr{M} (H)$. Thus, the following result is valid.\ For any locally compact hypergroup $H$ the operation of generalized convolution turns the space $\mathscr{M} (H)$ provided with the Mackey topology (or the weak topology) into a topological algebra with right identity, where this identity is a multiplicative functional on $C(H)$.\ The converse theorem is also valid.\ Let $H$ be a locally compact space and suppose given in $\mathscr{M} (H)$ the structure of a topological algebra with right identity, while this identity is a multiplicative functional on $C(H)$, and the topology in $\mathscr{M} (H)$ is compatible with the duality between $\mathscr{M} (H)$ and $C(H)$ (for example, coincides with the Mackey or with the weak topology). Then there exists a unique locally compact hypergroup structure on $H$ such that the multiplication in $\mathscr{M} (H)$ coincides with the corresponding convolution.\ If there is given in $\mathscr{M} (H)$ the structure of a topological algebra in accord with the hypothesis of Theorem 2, then one can define the operator of right translation $R^x$ in $G(H)$ as the adjoint of the operator of multiplication by $\delta_x$ on the right in $\mathscr{M} (H)$. Respectively, $L^x$ is adjoint to the operator of multiplication by $\delta_x$ on the left. The adjoint to the projector $L^e$ to the basic subspace $\widetilde{C} (H)$ in $C(H)$ (cf. above, point 2.1) is the projector $\tau \mapsto \delta_e \ast f$ in $\mathscr{M} (H)$. The image of this operator will be denoted by $\widetilde{\mathscr{M}} (H)$. It is easy to prove\ $\widetilde{\mathscr{M}} (H)$ is a closed subalgebra in $\mathscr{M} (H)$ while $\delta_e$ is a two-sided identity in $\widetilde{\mathscr{M}} (H)$. One can identify the subalgebra $\widetilde{\mathscr{M}} (H)$ with the quotient-algebra (provided with the quotient-topology) of the algebra $\mathscr{M} (H)$ by the ideal orthogonal to the basic subspace $\widetilde{C} (H)$. Thus, $\widetilde{\mathscr{M}} (H)$ coincides with the dual space to $\widetilde{C}(H)$. The algebra $\widetilde{\mathscr{M}} (H)$ is called the basic subalgebra of $\mathscr{M} (H)$. This algebra is the hypergroup algebra of the reduced hypergroup (cf. above, point 2.3). We note that the hypergroup algebra $\mathscr{M} (H)$ is noncommutative if $\widetilde{C} (H)$ does not coincide with $C(H)$, even when the hypergroup $H$ is commutative. On the other hand, in this case the basic subalgebra $\widetilde{\mathscr{M}} (H)$ is commutative. The constructions and results on hypergroup algebras described can be considered as the formalization of the heuristic considerations expressed by B.M. Levitan (cf., e.g., \[54, Sec. 2, point 1\]). These constructions and results can also be carried over to the case when instead of $C(H)$ one considers the Banach space $C_0 (H)$, consisting of all continuous functions on $H$, which vanish at infinity. In other words, $\varphi (x) \in C_0 (H)$ if for any $\varepsilon > 0$ one can find a compact set $K \subset H$ such that $\vert \varphi (x) \vert < \varepsilon$ for $x \notin K$ and the norm is determined by the equation $\| \varphi \| = \max_{x \in H} | \varphi (x)|$. The dual space $\mathscr{M}^{b} (H)$ of $C_0 (H)$ consists of all bounded complex Radon measures. If in $C_0 (H)$ there are defined continuous left and right generalized translation operators $L^x$ and $R^x$, where the functions $x \mapsto \langle f, R^x \varphi \rangle$ and $x \mapsto \langle f, L^x \varphi \rangle$ are contained in $C_0 (H)$ for all $F \in \mathscr{M}^{b} (H), \varphi \in C_0 (H)$, then $\mathscr{M}^{b} (H)$ can be provided with the structure of a topological hypergroup algebra and even a Banach algebra (with respect to to the norm of the Banach space dual of $G_0 (H)$). Structures of the type indicated on $\mathscr{M}^{b} (H)$ were apparently first considered for a sufficiently general case in \[156, 157\] in connection with the theories of semigroups and semigroup algebras. The algebra $\mathscr{M}^{b} (H)$ is a basic object in the theory of $p$-hypergroups (cf., e.g., \[112, 113, 136, 169, 160, 93, 131, 132\]).\ In Delsarte’s example, described in point 2.3, the hypergroup algebra $\mathscr{M} (H)$ coincides with the subalgebra of the group algebra $\mathscr{M} (G)$ consisting of measures, invariant with respect to the action of the compact group $K$. For the special case of Delsarte’s example described in point 2.4, $\widetilde{\mathscr{M}} (R)$ consists of all even measures, while for these measures the generalized convolution coincides with the ordinary one. For the hypergroup of classes of conjugate elements, described in point 2.5, the hypergroup algebra consists of central measures on $G$. Finally, if $H$ is the hypergroup of double cosets, described in point 2.6, then $\mathscr{M} (H)$ is the subalgebra of the group algebra $\mathscr{M} (G)$ consisting of measures, invariant with respect to left and right translations by elements of the compact subgroup $K$. The hypergroup algebras of bounded measures admit an analogous description in all the examples considered.\ Let $H$ be an infinitely differentiable manifold, $C^{\infty} (H)$ be the space of all infinitely differentiable functions on $H$, provided with the topology of uniform convergence along with derivatives on compact subsets of $H$. The dual space $\mathscr{D} (H)$ of $C^{\infty} (H)$ consists of all generalized functions with compact support on $H$. The space $C^{\infty} (H)$ is reflexive (and is a Mackey space); hence the Mackey topology on $\mathscr{D} (H)$ coincides with the strong topology of uniform convergence on bounded sets from $C^{\infty} (H)$. A family of g.t.o. $R^x$ on $C^{\infty} (H)$ is called infinitely differentiable (respectively, $H$ is called an infinitely differentiable hypergroup), if the correspondence $\varphi (x) \mapsto \psi (x,y) = R^y \varphi (x)$ is a continuous map $C^{\infty} (H) \to C^{\infty} (H \times H)$. It follows from this that the operators $R^x$ and $L^x$ are continuous, and the functions $x \mapsto \langle f, R^x \varphi \rangle$ and $x \mapsto \langle f, L^x \varphi \rangle$ are infinitely differentiable for any $ \varphi \in C^{\infty} (H), f \in \mathscr{D} (H)$. Arguing as in point 2.2, one can define a generalized convolution in $\mathscr{D} (H)$. Strengthened analogs of Theorems 1 and 2 are valid here.\ The generalized convolution turns $\mathscr{D} (H)$ into an associative topological algebra with continuous (and not only separately continuous) multiplication and with a right identity which is a multiplicative functional on $C^{\infty} (H)$.\ Suppose given in $\mathscr{D} (H)$ an associative algebra structure and a multiplicative functional which is a right identity in $\mathscr{D} (H)$. If the multiplication in $\mathscr{D} (H)$ is separately continuous in any locally convex topology, which is compatible with the duality between $\mathscr{D} (H)$ and $C^{\infty} (H)$, then this multiplication is also continuous in the Mackey topology $\tau (\mathscr{D} (H), C^{\infty} (H))$, which coincides with the strong topology of the dual space to $C^{\infty} (H)$. In this case there exists a unique structure of infinitely differentiable hypergroup on $H$, such that the multiplication in $\mathscr{D} (H)$ coincides with the corresponding convolution.\ The analog of Proposition 1 is also valid, so that one can define the basic subalgebra of $\mathscr{D} (H)$ and it is natural to consider this subalgebra as the hypergroup algebra of the reduced hypergroup. Let us now assume that $H$ is a complex analytic manifold. We denote by $\mathscr{H} (H)$ the space of all holomorphic functions on $H$ with the topology of uniform convergence on compact subsets, and by $\mathscr{A} (H)$ the dual space, consisting of analytic functionals. In $H$ there is given a structure of [*holomorphic (or complex-analytic) hypergroup*]{} if on $\mathscr{H} (H)$ there act g.t.o. $R^x$ such that the map $\varphi (x) \mapsto \psi (x,y) = R^y \varphi (x)$ is a continuous map $\mathscr{H} (H) \mapsto \mathscr{H} (H \times H)$. In this case $\mathscr{A} (H)$ is turned into a hypergroup algebra with respect to the generalized convolution and the analogs of Theorems 1 and 2 and Proposition 1 are valid. Finally, if the g.t.o. act on the space of real analytic functions on a real analytic manifold, then (if the standard conditions hold) one can endow the space of hyperfunctions with compact support with the structure of a topological hypergroup algebra.\ The results formulated above admit further formalization which clarifies their nature. Let $\Phi$ be a topological algebra (associative and with separately continuous map) and let the dual space $\mathscr{F}$ of $\Phi$ also be endowed with the structure of a topological algebra, where $\Phi$ and $\mathscr{F}$ are given the Mackey topologies $\tau (\Phi , \mathscr{F} )$ and $\tau (\mathscr{F} , \Phi )$. We call the pair $\mathfrak{A} = (\Phi , \mathscr{F} )$ a [*double topological algebra*]{} (for short d.t.a.). Wanting to indicate that the algebra $\Phi$ has a certain property (for example, that it is commutative or has an identity), we shall say that the d.t.a. $\mathfrak{A}$ has this property; now if we want to indicate that the algebra $\mathscr{F}$ has some property, then we shall use the prefix “co.” For example, if $H$ is a locally compact hypergroup, then the pair $\mathfrak{A} = (C(H), \mathscr{M} (H))$ where the multiplication in $C(H)$ coincides with the multiplication of functions, and in $\mathscr{M} (H)$ with the generalized convolution, forms a commutative d.t.a. with identity and right coidentity. Analogously, one defines d.t.a. connected with infinitely differentiable and holomorphic hypergroups. In the theory of hypergroups, d.t.a. play the same kind of role as double Hilbert algebras do in the theory of ringed groups (Kac algebras), cf. \[38, 16, 120, 163, 164\]. If the algebra $\Phi$ is commutative, then we denote by $H = H(\Phi )$ the subset of $\mathscr{F}$ consisting of all multiplicative functionals on $\Phi$ and endowed with the induced topology. The Gelfand transformation $\varphi \mapsto \varphi (x) = \langle x, \varphi \rangle$ maps $\Phi$ into the space $C(H)$ of continuous functions on $H$. We call the algebra $\Phi$ semisimple, if the Gelfand transformation has no kernel; in this case we identify $\Phi$ with a linear subspace of $C(H)$. If $x \in H$, then by $R^x$ (respectively, $L^x$) we denote the operator on $\Phi$, adjoint to multiplication by $x$ on the right (left) in $\mathscr{F}$.\ The concept of d.t.a. is closely related with the modern concept of quantum hypergroup (and quantum group).\ If the d.t.a. $\mathfrak{A}$ is commutative, semisimple, and has a right coidentity, which is a multiplicative functional, then the $R^x$ are g.t.o. and multiplication in $\mathscr{F}$ coincides with generalized convolution; here the $L^x$ are left translation operators. Thus $H$ is endowed with the structure of a hypergroup, and $\mathscr{F}$ is a hypergroup algebra.\ If the condition of Proposition 2 holds, then all the operators $R^x$ are endomorphisms of the algebra $\Phi$ if and only if the multiplication in $\mathscr{F}$ induces a semigroup structure on $H$; here $R^y \varphi (x) = \varphi (xy)$. The semigroup indicated is a group, if $\mathfrak{A}$ has a two-sided coidentity $e \in H$ and there exists an involutive automorphism $S \colon \Phi \to \Phi$, such that $\langle x, S R^x \varphi \rangle = \langle e, \varphi \rangle$; in this case $(S \varphi ) (x) = \varphi (x^{-1})$. The group $H$ is topological (i.e., the multiplication in $H$ is not only separately continuous, but is also continuous), if the space $H$ is locally compact.\ We call the d.t.a. $\mathfrak{A}' = (\mathscr{F} , \Phi )$ dual to the d.t.a. $\mathfrak{A} = (\Phi , \mathscr{F} )$. It is clear that $(\mathfrak{A}' )' = \mathfrak{A}$. If the d.t.a. $\mathfrak{A}$ is commutative and cocommutative, semisimple and cosemisimple, has an identity and a coidentity, which are multiplicative functionals, then corresponding to Proposition 2 the d.t.a. $\mathfrak{A}$ generates a hypergroup $H$, and the d.t.a. $\mathfrak{A}'$ generates a hypergroup $H'$, which is naturally considered as dual to $H$. The hypergroups $H$ and $H'$ are commutative and $(H')' = H$. The present version of the duality theory is not a direct generalization of Pontryagin duality. For example, if $\mathfrak{A} = (C^{\infty} ({\bf R}), \mathscr{D} ({\bf R}))$ where $\mathscr{D} ({\bf R})$ is considered as the group algebra of the group $R$, then the differentiable hypergroup $H$ coincides with this group, and the dual hypergroup $H'$ coincides with the group of complex numbers ${\bf C}$. The Gel’fand transform (which in the present case coincides with the Fourier–Laplace transform) maps $\mathscr{D} (R)$ to some algebra consisting of entire analytic functions on ${\bf C}$ according to the Payley–Wiener–Schwartz theorem. The situation becomes especially transparent if $\Phi$ is a nuclear $F$-space (for example, $C^{\infty} (H)$ or $\mathscr{H} (H))$ or a complete nuclear $DF$-space (for example, $\mathscr{D} (H)$ or $\mathscr{A} (H))$. Then $\Phi$ is reflexive and $\mathscr{F} = \Phi'$ is a complete nuclear $DF$-space or nuclear $F$-space respectively. The multiplication in $\Phi$ is automatically continuous and extends to a continuous map of the completed tensor product $\Phi \hat{\otimes} \Phi$ into $\Phi$. The dual map $\Phi' \to (\Phi \hat{\otimes} \Phi )' = \Phi' \hat{\otimes} \Phi'$ is called the comultiplication on $\Phi'$. Analogously, the continuous linear map $\Phi \to \Phi \hat{\otimes} \Phi$ is called a comultiplication on $\Phi$ if the dual map induces a topological algebra structure on $\Phi'$. One can define comultiplication in the standard way in the language of commutative diagrams, since passage to dual spaces and maps leads to “reversal of arrows” in diagrams and the tensor products $\Phi \hat{\otimes} \Phi$ and $\Phi' \hat{\otimes} \Phi'$ in the present case go into one another. Thus, the diagram expressing the associativity of multiplication in $\Phi'$ goes into the diagram expressing the associativity of the comultiplication in $\Phi$, etc. If there are given in $\Phi$ a continuous multiplication and comultiplication, then the pair $\mathfrak{A} = (\Phi , \Phi' )$ is a d.t.a. and conversely. For example, if $H$ is an infinitely differentiable hypergroup, then the map $\varphi (x) \mapsto R^y \varphi (x)$ is a comultiplication $C^{\infty} (H) \to C^{\infty} (H) \hat{\otimes} C^{\infty} (H) = C^{\infty} (H \times H)$, and the multiplication of functions in $C^{\infty} (H)$ generates a comultiplication in $\mathscr{D} (H)$. If the algebras $\Phi$ and $\Phi'$ have identities and the multiplications in $\Phi$ and $\Phi'$ are compatible in the sense that the comultiplication $\Phi \to \Phi \hat{\otimes} \Phi$ is a homomorphism of algebras (Hopf condition), then $\Phi'$ is called a topological Hopf algebra (t.H.a.). In this case $\Phi'$ also has the structure of a t.H.a. In the situation of Propositions 2 and 3 the Hopf condition means that $H$ is a semigroup. In the presence of an involution $S$ this semigroup is a topological group (local compactness is not required).\ Let $H$ be a hypergroup with hypergroup algebra $\mathscr{F}$. If there is given in $\mathscr{F}$ a map $f \mapsto f^*$ such that $(f^*)^* = f$ and $(f \ast g)^* = g^* \ast f^*$ for all $f, g \in H$, then $\mathscr{F}$ is an [*algebra with involution*]{}; if $\mathscr{F}$ is a topological algebra, then the map $f \mapsto f^*$ is assumed to be continuous. If $\mathscr{F}$ is an algebra with involution, then the right identity in $\mathscr{F}$ is also a two-sided identity, while the involution carries this element into itself. It follows from this that $\mathscr{F}$ coincides with its basic subalgebra, and the left and right generalized translations are completely equivalent. There is a special interest in the case when the involution in $\mathscr{F}$ is induced by the involution $x \mapsto x^*$ in the hypergroup $H$ (cf. above, point 2.7). In this case in the space $\Phi$, where the g.t.o. act, there arises a map $\varphi (x) \mapsto \tilde{\varphi} (x) = \varphi (x^*)$ Combining this map with complex conjugation, one can define an involution in $\mathscr{F}$ with the help of the equation $$\langle f^* , \varphi \rangle = \langle \overline{f, \tilde{\tilde {\varphi}}} \rangle . \eqno(3.5)$$ (2.7) of point 2.7 means precisely that the involution defined by (3.5) is an antiautomorphism of the algebra $\mathscr{F}$, i.e., $( f \ast g)^* = g^* \ast f^*$ for all $f, g \in \mathscr{F}$. In particular, in group algebras there is a canonical involution, generated by passage to the inverse element in the group. As indicated in point 2.7, reduced hypergroups, described in points 2.3-2.6, are hypergroups with involution. Their hypergroup algebras are described in point 3.3 as subalgebras of group algebras. It is easy to see that the involution in each of these hypergroup algebras is induced by the canonical involution in the corresponding group algebra.\ In order to define the generalized convolution of functions on the hypergroup $H$, it is necessary to fix a measure on $H$. We return to the situation described in point 2.7: the hypergroup $H$ is a locally compact space with a positive measure $m$. With any function $a(x) \in L^1 (H, m)$ one can associate the bounded measure $a(x)m$ and define the convolution of functions as the convolution of the corresponding measures. If $H$ is a hypergroup with involution, the measure $m$ is left-invariant, and the space $L^1 (H, m)$ is invariant with respect to real left generalized translations $L^x$, then the convolution $(a \ast b)(x)$ of the functions $a(x)$ and $b(x)$ from $L^1 (H, m)$ is given by $$(a \ast b)(x) = \int a (y) \cdot L^{y^{*}} b(x)dm(y) . \eqno(3.6)$$ It is clear that the corresponding analogous formula is valid for right-invariant measures and right generalized translation operators. Let the measure $m$ be left-invariant and the involution in $H$ carry this measure into the equivalent right-invariant measure $m^* (x) = m(x^*)$. We denote $\Delta (x)$ the modular function, i.e., the Radon–Nikodym derivative $dm(x) / dm^*(x)$. The involution $a(x) \mapsto a^*(x)$ induced in $L^1 (H, m)$ by the imbedding $L^1 (H, m) \to \mathscr{M}^b (H)$ described is given by $$a^* (x) = \Delta (x^*) \overline{a(x^*)} . \eqno(3.7)$$ To study hypergroup algebras of the type of $L^1 (H, m)$ one can use the powerful apparatus of the theory of Banach algebras with involutions (cf. Introduction). By an analogous scheme one also constructs other hypergroup algebras consisting of functions (for example, finite ones). One can define a hypergroup $C^*$-algebra as the enveloping $C^*$-algebra of the algebra $L^1 (H, m)$. In \[17\], in connection with the proof of the Plancherel theorem for hypergroups, hypergroup left Hilbert algebras were considered (cf., e.g., \[82\] on left Hilbert algebras). The scalar product in this Hilbert algebra is induced by the imbedding in $L^2 (H, m)$, and the multiplication (convolution) and involution are defined in terms of the adjoint g.t.o. For hypergroups with involution and real g.t.o. these definitions reduce to (3.6) and (3.7). 4. Infinitesimal Hypergroup Algebras and Generalized Lie Theory {#infinitesimal-hypergroup-algebras-and-generalized-lie-theory .unnumbered} =============================================================== Let $\Phi$ be a space of functions (germs, formal power series) of the variables $x_1, \ldots , x_n$, containing exponentials of the form $\exp (\zeta_1 x_1 + \ldots + \zeta_n x_n)$. By the [*Laplace transform*]{} of the functional $\int \colon~ \varphi \mapsto \langle f, \varphi \rangle$ defined on $\Phi$ is meant the function $f(\zeta_1 , \ldots , \zeta_n ) = \langle f, e^{\zeta_1 x_1 , \ldots , \zeta_n x_n} \rangle$. If $\Phi$ coincides with the space $\mathscr{H}_0$ of germs of analytic functions at the point $x_1 = \ldots = x_n =0$, endowed with the standard toplogy, then the dual space $\mathscr{F}_0$ of $\mathscr{H}_0$ can be identified with the help of the Laplace transform with the space of entire functions, which grow slower than $\exp [\varepsilon (| \zeta_1 | + \ldots | \zeta_n | )]$ for arbitrarily small $\varepsilon > 0$. For all $f \in \mathscr{F}_0 , \varphi \in \mathscr{H}_0$ one has $$\langle f, \varphi \rangle = \sum_{s_1 , \ldots , s_n} \frac{f_{s_1 , \ldots , s_n} \varphi_{s_1 , \ldots , s_n}}{s_1 ! s_2 ! \ldots s_n ! }, \eqno(4.1)$$ where $\varphi = \sum_{s_1 , \ldots , s_n} \varphi_{s_1 , \ldots , s_n} \frac{x_1^{s_1} \ldots x_n^{s_n}}{s_1 ! \ldots s_n !} , f(\zeta_1 , \ldots , \zeta_n ) = \sum_{s_1 , \ldots , s_n} f_{s_1 , \ldots , s_n} \frac{\zeta_1^{s_1} \ldots \zeta_n^{s_n}}{s_1 ! \ldots s_n !}$ are the expansion in a Taylor series of the germ of $\varphi$ and the Laplace transform of the functional $f$. If $\Phi$ coincides with the space of germs at zero of infinitely differentiable functions of the real variables $x_1 ,\ldots x_n$ (endowed with the standard topology of uniform convergence along with derivatives on some neighborhood of zero), then the dual space consists of all generalized functions with support at zero. The Laplace transform identifies this space with the space ${\bf C} [\zeta_1 , \ldots , \zeta_n ]$ of all polynomials of $\zeta_1 , \ldots , \zeta_n$; (4.1) remains valid. Finally, let $\Phi$ coincide with the space ${\bf C}[[x_1 , \ldots , x_n ]]$ of formal power series of $x_1 ,\ldots , x_n$. If we endow $\Phi$ with the strongest locally convex topology, then $\Phi'$ consists of all linear functionals on $\Phi$. The Laplace transform identifies $\Phi'$ with the space of polynomials ${\bf C} [ \zeta_1 , \ldots , \zeta_n ]$, and the duality between ${\bf C} [[x_1 , \ldots , x_n ]$ and ${\bf C}[\zeta_1 , \ldots , \zeta_n ]$ is given by (4.1).\ We consider a double topological algebra $\mathfrak{A} =(\mathscr{H}_0 , \mathscr{F}_0)$ where $\mathscr{H}_0$ is the algebra of germs of analytic functions at zero (cf. above) with respect to multiplication, $\mathscr{F}_0$ is the dual space of $\mathscr{H}_0$. If the multiplicative functional $\varphi \mapsto \varphi (0)$ on $\mathscr{H}_0$ is a right identity in the algebra $\mathscr{F}_0$, then by analogy with the results of points 3.4 and 3.5 it is natural to consider $\mathscr{F}_0$ as the hypergroup algebra of a “local hypergroup”. In the one-dimensional case such algebras were studied in \[42\]. Algebras of the type indicated arise naturally as subalgebras of hypergroup algebras of analytic functionals. Associative ultra-envelopes of Lie algebras in the sense of P. K. Rashevskii \[77\] correspond to the group case.\ \ By virtue of the duality between ${\bf C}[[x_1 , \ldots , x_n]]$ and ${\bf C}[\zeta_1 , \ldots , \zeta_n ]$ the associative comultiplication in ${\bf C}[[x_1 , \ldots , x_n ]]$ induces an associative multiplication in ${\bf C}[\zeta_1 , \ldots , \zeta_n ]$ (which in general does not coincide with the usual multiplication of polynomials), so that a dual algebra arises. If it has a right coidentity, which is a multiplicative functional (i.e., the polynomial $\varepsilon \equiv 1$ is a right identity with respect to the induced multiplication), then by analogy with the results of points 3.4 and 3.5, it is natural to consider that the comultiplication in ${\bf C}[[x_1 , \ldots , x_n]]$ defines the structure of a formal hypergroup, and the dual algebra is a “hypergroup” one. It is also natural to require that the hypergroup multiplication be compatible with the natural filtration in ${\bf C}[\zeta_1 , \ldots , \zeta_n ]$ and that the identity in ${\bf C}[[x_1 , \ldots , x_n ]]$ be a multiplicative functional on ${\bf C}[\zeta_1 , \ldots , \zeta_n ]$. Obvious generalizations are connected with replacement of the field $C$ by other fields and rings, passage to algebraic extensions of algebras of power series, the introduction of anticommuting variables, etc. Projectors generated by the right identity in the hypergroup algebra single out the basic subalgebra in this algebra and the basic subspace in ${\bf C}[[x_1 , \ldots , x_n ]]$ by analogy with the constructions of points 2.3 and 3.2. The dual algebra which arises defines a reduced formal hypergroup. Formal hypergroups and generalizations of them from various points of view arise in many questions of topology, analysis, and mathematical physics, cf., e.g., \[9, 11-13, 19, 28-30, 33, 34, 36, 37, 52-56, 63, 65-67, 79-81, 88, 108, 162\]. In the group case the scheme indicated leads to universal enveloping algebras of Lie algebras. Let $\mathfrak{g}$ be a Lie algebra with basis $X_1 , \ldots , X_n$. The enveloping algebra $U (\mathfrak{g})$ is the algebra of polynomials with complex coefficients of the generators $X_1 , \ldots , X_n$, connected by the relations $X_i X_j = X_j X_i + \sum_{k=1}^n C^k_{ij} X_k$, where $C^k_{ij}$ are the structural constants of the Lie algebra $\mathfrak{g}$. The Poincare–Birkhoff–Witt theorem (for short PBW) lets one construct a one-to-one linear correspondence $\Lambda \colon U(\mathfrak{g} ) \to {\bf C} [\zeta_1 , \ldots , \zeta_n ]$ compatible with the filtrations in these algebras. As a result the structure of associative algebra can be carried over from $U(\mathfrak{g} )$ to ${\bf C}[\zeta_1 , \ldots , \zeta_n ]$, and by duality there arises in ${\bf C} [[x_1, \ldots , x_n]]$ a (group) multiplication. The construction of the map $\Lambda$ is connected with the choice of local coordinates $x_1 , \ldots , x_n$ in the Lie group $G$ with the given Lie algebra $\mathfrak{g}$. By L. Schwartz theorem $U(\mathfrak{g})$ can be identified with the subalgebra of the group algebra $\mathscr{D} (G)$ consisting of all generalized functions with support at the identity of the group. The Laplace transform in canonical coordinates of the second kind connected with the basis $X_1, \ldots , X_n$ in $\mathfrak{g}$ carries the element $\sum f_{s_1 , \ldots , s_n} X_1^{s_1} \ldots X_n^{s_n} \in U(\mathfrak{g})$ (more precisely, its image in $\mathscr{D}(G)$) into the polynomial $\sum f_{s_1 , \ldots , s_n} \zeta_1^{s_1} \ldots \zeta_n^{s_n}$; the Laplace transform in canonical coordinates of the first kind carries the element $\sum f_{s_1 , \ldots , s_n} ((X_1^{s_1} \ldots X_n^{s_n}))$, where $((X_1^{s_1} \ldots X_n^{s_n}))$ is the result of symmetrization of the monomial $X_1^{s_1} \ldots X_n^{s_n}$, into this polynomial (cf. \[59, Chap. III\]). For many associative algebras with identity generated by a finite collection of generators $X_1 ,\dots , X_m$ (and classes of such algebras), it is proved (or assumed) that the analogs of the PBW theorem are valid (cf. \[19, 28-30, 33, 34, 36, 37, 65, 66, 80, 81\]); one can also give analogs of the Jacobi identity, guaranteeing the validity of a PBW type theorem in this case or that. For algebras generated by quadratic commutation relations, possible generalizations of the PBW theorem are analyzed in \[19\]. Algebras for which analogs of the PBW theorem are valid are natural candidates for the role of formal and infinitesimal hypergroup algebras.\ Let $H$ be an infinitely differentiable hypergroup with hypergroup algebra $\mathscr{D}(H)$ consisting of generalized functions with compact support (cf. above, point 3.4). By the infinitesimal hypergroup algebra of the hypergroup $H$, we mean the subalgebra $\mathscr{D}_e (H)$ generated by generalized functions with support at the identity $e \in H$. If the support of any functional from $\mathscr{D}_e (H)$ coincides with $e$, then the Laplace transform in any local coordinates maps $\mathscr{D}_e (H)$ one-to-one and linearly onto the space of polynomials, and we return to the situation considered in point 4.3. The correspondence $f \mapsto R(f)$ is a representation of the algebra $\mathscr{D}_e (H)$ on the space $C^{\infty} (H)$ and maps this algebra onto the algebra of infinitesimal right translations $\mathscr{D}_e^R (H)$; analogously, the antirepresentation $f \mapsto L(f)$ maps $\mathscr{D}_e (H)$ onto the algebra of infinitesimal left translations $\mathscr{D}_e^L (H)$. The algebras $\mathscr{D}_e (H)$ and $\mathscr{D}_e^L (H)$ are isomorphic. If $\delta_c$ is a two-sided identity in $\mathscr{D}_e (H)$ then the algebras $\mathscr{D}_e (H)$ and $\mathscr{D}_e^R (H)$ are isomorphic. In general the algebra $\mathscr{D}_e^R (H)$ is isomorphic to the intersection of $\mathscr{D}_e (H)$ with the basic subalgebra of $\mathscr{D}_e (H)$; hence $\mathscr{D}_e^R (H)$ should be considered as the infinitesimal hypergroup algebra of the reduced hypergroup (cf. above points 2.3, 3.2, and 3.4), although the reduced hypergroup is not necessarily a smooth manifold. The construction of infinitesimal hypergroup algebras and the reconstruction of “global” hypergroups from given “infinitesimal” objects is a natural subject of Lie theory. The construction of such a theory is far from to be completed, although important results in this direction were already found in the first papers devoted to g.t.o.\ Let $H$ be an infinitely differentiable hypergroup, so that $u(x, y) = R^x \varphi (y)$ is a smooth function on $H \times H$ for all $\varphi \in C^{\infty} (H)$. We denote by $x_1 , \ldots , x_n$ local coordinates of a point $x \in H$ and we choose a coordinate system such that the neutral element has zero coordinates. By generators of right translation of order $n$ we mean linear operators of the form $$\left. R_{s_1 , \ldots , s_n ;h}~ \colon~ \varphi (h) \mapsto \frac{\partial^s u (x,h)}{\partial x^{s_1}_1 \ldots \partial x^{s_n}_n} \right|_{x=0} , \eqno(4.2)$$ where $u(x, h) =R^x \varphi (h), s = s_1 + \ldots + s_n$. Analogously, generators of left translation have the form: $$\left. L_{s_1 , \ldots , s_n ;h}~ \colon~ \varphi (h) \mapsto \frac{\partial^s u (x,h)}{\partial x^{s_1}_1 \ldots \partial x^{s_n}_n} \right|_{x=0} . \eqno(4.3)$$ In exactly the same way one can define generators for complex local coordinates on a holomorphic hypergroup for $\varphi \in \mathscr{H} (H)$. It follows from the associativity axiom that any generator of left translation commutes with any generator of right translation (just as with the operators $R^x$). Differentiating the associativity condition $R^x L^y \varphi (h) = L^y R^x \varphi (h)$ with respect to the variables $h_1 , \ldots , h_n$ the corresponding number of times, and then letting $h_1 = \ldots = h_n =0$, we get the system of equations $$L_{s_1 , \ldots , s_n ;x} (u) = R_{s_1 , \ldots , s_n ;y} (u) , \eqno(4.4)$$ where $u(x, y) = R^y \varphi (y)$. One should consider the system (4.4) as a generalization of the first Lie theorem \[52, 54-56\]. For Delsarte hypergroups this result is found in \[108\]. For the hypergroup described in point 2.6, where $G$ is a semisimple linear group and $K$ is its maximal compact subgroup, a complete collection of generators is actually calculated in \[23, 10\]. It is not necessary to use all the equations of the system (4.4) to determine $u(x, y)$. For example for a translation on a Lie group, the first order generators already determine the function u uniquely (i.e., the group multiplication). In general some generators of lower orders may degenerate (for example, to multiplication by a constant), so that the corresponding equations of the system (4.4) do not contain useful information. Hence an important problem arises: to select the minimal number of equations from the system (4.4), which uniquely determine the g.t.o. Here the degenerated generators enlarge the number of initial conditions. If a finite system of the form (4.4) under certain initial conditions including the condition $u_{1 x=0} = \varphi (y)$ uniquely determines the solution $u(x, y) = R^x \varphi (y)$, where the operators on the left sides of the system commute with all the operators on the right sides, then $R^x$ are g.t.o. (i.e., $H$ is a hypergroup). This assertion is an analog of the first Lie inverse theorem \[53-56\]. For a certain class of g.t.o. B.M. Levitan proved analogs of the second and third (direct and inverse) Lie theorems. In particular, in the space of smooth functions of n variables, g.t.o. are constructed for which the generators of right (left) translation generate any given n-dimensional Lie algebra \[52 56\]. In \[25, 126\] a complete explicit description of these generators is found in the form of second order integrodifferential operators. With the help of an analogous technique generators of any order acting on the space of entire analytic functions of $n$ variables and generating any n-dimensional real Lie algebra are constructed in \[24\]; from these generators one can reconstruct the hypergroup. One can construct hypergroups starting from commutation relations of more general type \[53, 65-67, 36, 37, 28 30\]. We note that already in Delsarte’s first paper \[105\] g.t.o. on the line were constructed, starting from an explicitly given second order generator, with the help of a series analogously to the way in wich an ordinary translation decomposes into a series in powers of the differentiation operator (Taylor series). B. M. Levitan and A. Ya. Povzner studied commutative g.t.o. on the line with second order generator of Sturm – Liouville type (cf. \[51 55 56\]); a complete classification of one-dimensional infinitely differentiable hypergroups with Sturm – Liouville type generators (including noncommutative ones) is given in \[27\]. A vast literature is devoted to g.t.o. on the line generated by a differential operator, cf., e.g., \[7, 51, 55, 56, 83, 90, 97, 98, 109, 134, 146\]. It is easy to verify that the generators (4.2) and (4.3) can be represented in the form $R(f)$ and $L(f)$ respectively, where the element $f \in \mathscr{D} (H)$ has the form $$\left. f \ \colon \ \varphi \mapsto \langle f, \varphi \rangle = \frac{\partial^s \varphi (x)}{\partial x^{s_1}_1 \ldots \partial x^{s_n}_n} \right|_{x_1 = x_2 =\ldots = x_n = 0} . \eqno(4.5)$$ Since any generalized function with support at the point $e \in H$ is a finite linear combination of the element $\delta_e$ and generalized functions of the form (4.5), the following assertion is valid.\ Generalized functions of the form (4.5) and the delta-function $\delta_e$ generate the infinitesimal hypergroup algebra $\mathscr{D}_e (H)$, and the generators of left (right) translation and the operator $L^e$ (respectively $R^e =\mathfrak{D}$ generate the algebra of infinitesimal left (right) translations $\mathscr{D}_e^L (H)$ (respectively $\mathscr{D}_e^R (H)$).\ We return to the example described in point 2.4. Differentiating (2.3), it is easy to calculate the generators $R_n$ and $L_n$ of left and right translation of the n-th order: $R_1 = R_3 = \ldots = R_{2k+1} = \ldots = 0; R_2 = \frac{d^2}{dt^2} , R_{2k} = (R_2)^k; L_n = L_0 \frac{d^n}{dt^n}$ where $L_0= L^e \varphi (t) \mapsto [\varphi (t) + \varphi (-t)] /2$. The algebra of infinitesimal right translations $\mathscr{D}^R_0$, coincides with the algebra of polynomials in the operator $R_2 = d^2 / dt^2$ and is isomorphic to the subalgebra, consisting of even generalized functions with support at zero, of the infinitesimal hypergroup algebra $\mathscr{D}_0$. For the elements of this subalgebra, the generalized convolution coincides with the ordinary one; the Laplace transform $f \mapsto f(\zeta )$ carries the indicated subalgebra into the algebra of polynomials of $\zeta^2$ with ordinary multiplication. The algebra of infinitesimal left translations $\mathscr{D}^L_0$ and the algebra $\mathscr{D}_0$, antiisomorphic to it, have a more complex nature. The operators $L_n = L_0 \cdot (d^n / dt^n)$ form a basis in $\mathscr{D}^L_0$ while $L_{2k+1} L_n = 0$ and $L_{2k} L_n = L_{2k+n}$ for all $n, k = 0, 1, 2, 3, \ldots$; it follows from this that $L_{2k} = (L_2 )^k$ and $L_{2k+1} = (L_2 )^k L_1$. Thus, the elements $L_0$ (left identity), $L_1$ and $L_2$ generate the algebra $\mathscr{D}^L_0$ (here $L_1 L_0 = L^2_1 = L_1 L_2 =0, L_2 L_0 =L_2 , L_2 L_1 = L_3 , L_2^2 =L_4$), and the analog of the Poincare–Birkhoff–Witt theorem obviously holds. The structure of the algebra $\mathscr{D}_0$ is easily determined by virtue of the antiisomorphism between $\mathscr{D}_0$ and $\mathscr{D}_0^L$. The system of equations (4.4) in the present case reduces to the wave equation $\partial^2 u / \partial x^2 = \partial^2 u / \partial t^2$ and the function $u(x, t) = R^x \varphi (t) = [\varphi (t+x)+ \varphi (t-x)] /2$ is a solution of this equation with initial conditions $u(0, t) = \varphi (t)$ and $u'_x (0, t) = 0$. On the reduced hypergroup, i.e., on the half-line $0 \leq t < \infty$, one can define generalized translation as a solution of the wave equation for $t \geq 0$ with the same initial conditions. If the generator $d^2 / dt^2$ is replaced by a Sturm–Liouville operator with potential $v(t)$, then the corresponding generalized translation on the half-line $0 \leq t < \infty$ can be defined as a solution of the equation $$\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial t^2} + v(x) - v(t) = 0$$ with the initial conditions indicated above.\ We consider the group $G = SU(2)$, and in the Lie algebra of this group we choose a standard basis $X, Y, Z$, such that $[X, Y] = Z,~ [Y, Z] =X,~ [Z, X] = Y$. Let $a$ and $b$ be automorphisms of the group $G$, carrying the basis $(X, Y, Z)$ respectively into $(-X, Y, -Z)$ and $(X, -Y, -Z)$. The equation $$R^x \varphi (t) = \frac{1}{2} [\varphi (t \cdot x) +\varphi (b(t) \cdot x) + \varphi (t \cdot a(x)) - \varphi (b(t) \cdot a(x))]$$ defines a g.t.o. in $C^{\infty} (G)$, and the first order generators of right translation $X, Y$, and $Z$ satisfy the relations $XY + YX =Z,~ YZ + ZY =-X,~ ZX + XZ =-Y$ (cf. \[28\]). It is easty to verify that the algebra with identity generated by these generators is isomorphic to the infinitesimal hypergroup algebra and the analog of the PBW theorem is valid. 5. Special Classes of Hypergroups {#special-classes-of-hypergroups .unnumbered} ================================= The contents of Section 3 (in particular, Theorem 2 and its analogs) show that hypergroups form an arbitrarily large class, that to get meaningful results it is reasonable to include additional restrictions and to consider special classes of hypergroups. For example, one can single out classes of hypergroups in terms of generalized Lie theory (cf. above). It is also natural to require that the comultiplication $\varphi (x) \mapsto \psi (x,y) = R^y \varphi (x)$ be compatible to some degree or other with ordinary multiplication of functions. A relatively weak requirement is that the space $\Phi$, where the g.t.o. act, is a (commutative) algebra with identity, while the identity in $\Phi$ is a multiplicative functional on the hypergroup algebra $\mathscr{F} = \Phi'$ and the identity in $\mathscr{F}$ is a multiplicative functional on $\Phi$. The stronger Hopf condition says that the hypergroup reduces to a semigroup (cf. above, point 3.5). However even for hypergroups which do not reduce to semigroups, one is sometimes able to give $\Phi$ the structure of a (noncommutative) algebra in such a way that the comultiplication is a homomorphism $\Phi \to \Phi \widehat{\otimes} \Phi$. Infinitesimal and formal objects of this kind, called generalized Lie algebras and groups, are introduced in \[28, 30\]; here the concept of tensor product is generalized in the spirit of MacLane \[147\] with the help of a symmetry operator (cf. also \[63, 67\]). Generalized Lie groups and algebras are closely connected with Lie supergroups and superalgebras, and also with the Yang–Baxter equation which is popular in theoretical physics. The hypergroup described in point 4.7 is an example of a (global) generalized Lie group; one can construct a new associative operation in $C^{\infty} (G)$ by combining multiplication of functions with the action of the automorphisms $a$ and $b$. Some other conditions on the comultiplication, replacing the Hopf condition, are considered below.\ Let $h$ be a hypergroup and for any elements $x, y \in H$ let the generalized convolution of the delta-functions $\delta_x$ and $\delta_y$ have the form $$\delta_x \ast \delta_y = \sum^n_{i=1} a_i \delta_{x_i} ,$$ where $a_i$ are numerical coefficients depending on $x$ and $y$, and $x_i$ are elements of $H$, depending on $x$ and $y$. In this case, following \[29\] (where differentiable and holomorphic hypergroups are considered), we call $H$ a metamultivalued group; if $a_1 = a_2 = \ldots = a_n = 1/n$, then $H$ is called an n-valued group (the connection with the definition of abstract hypergroup in point 1.2 is obvious). An example of a two-valued hypergroup is described in point 2.4, and one of a metamultivalued group is described in point 4.7. In \[29\] regular methods of construction of hypergroups of this type are given. One-dimensional formal multivalued groups are introduced by V. M. Bukhshtaber in connection with topological applications, cf. \[12\] for a surveying account; there is further development along these lines in \[13, 88\], for example.\ An enormous literature is devoted to hypergroups with involution, for which the Hopf condition is replaced by the requirement that the generalized convolution have certain positivity properties. Of this class are p-hypergroups and hypercomplex systems (h.s.) with continuous bases (cf. Introduction). Similar objects (of more general nature) such as dual algebras with involution and coinvolution, identity and coidentity, and with additional positivity conditions were considered by A. M. Vershik \[18\] in connection with the geometric theory of $C^*$-algebras and duality in representation theory; cf. also \[39\]. At the present time the use of p-hypergroups (usually called simply hypergroups) is of the greatest popularity. Let $H$ be a locally compact hypergroup with involution (cf. above, points 2.7, 3.2, and 3.6), where the space $C_0 (H)$ of continuous functions which vanish at infinity is invariant with respect to the g.t.o. and the involution, so that not only the space $\mathscr{M} (H)$ of measures with compact support but also the space $\mathscr{M}^b (H)$ of bounded measures is endowed with the structure of a hypergroup algebra with involution. We denote by $\mathscr{M}^b_{+} (H)$ the set of all nonnegative measures from $\mathscr{M}^b (H)$ endowed with the weakest topology for which the maps $f \mapsto f (H) = \int_H df$ and $f \mapsto \langle f, \varphi \rangle = \int \varphi df$ are continuous for all nonnegative continuous functions $\varphi$ with compact support in $H$. The hypergroup $H$ is a $p$-hypergroup if the following conditions hold: 1) the generalized convolution of nonnegative measures is a nonnegative measure, and the convolution induces a continuous map $\mathscr{M}^b_+ (H) \times \mathscr{M}^b_+ (H) \to \mathscr{M}^b_+ (H)$; 2) for any $x, y \in H$ the convolution of the Dirac measure (delta-functions) $\delta_x$ and $\delta_y$ is a probability measure, so that $\delta_x \ast \delta_y (H) = \int d(\delta_x \ast \delta_y )$; 3) the support supp$(\delta_x \ast \delta_y )$ of the measure $\delta_x \ast \delta_y$ contains the neutral element (identity) of the hypergroup if and only if $x = y^*$; 4) the map $(x, y) \mapsto$ supp$(\delta_x \ast \delta_y )$ is a continuous map from $H \times H$ to the space $\mathfrak {R} (H)$ consisting of compact subsets of $H$ and endowed with the Michael topology \[154\]; this topology is generated by the subbase ${K \in \mathfrak{R} (H) \colon K \cap U \ne \varnothing , K \subseteq V}$, where $U$ and $V$ run through the collection of open subsets of $H$. The present definition is a slight modification of the definition formulated in \[136\]; for other versions of the axiomatics of p-hypergroups, cf. \[112, 113, 128, 169, 170, 160\]; in \[128\] even non-Hausdorff hypergroups, which arise as dual objects to certain groups, are considered. The hypergroups described in points 1.2, 1.3, 2.3-2.6 (cf. also points 2.7, 3.3, and 3.6) are examples of p-hypergroups. Other interesting examples are described, for example, in \[94, 115, 127-132, 142-144, 160, 161, 169, 182, 183\]. These hypergroups can also be treated as h.s. with locally compact basis. Objects of this type are similar to locally compact groups in their properties to a large degree. The theory of commutative h.s. and $p$-hypergroups is especially well developed. The basic results of harmonic analysis, including Pontryagin duality and refined questions of spectral analysis and synthesis, generalize to this case (cf. Introduction). On commutative h.s. and p-hypergroups one can construct the nuclear space of “basic” finite functions with the invariance property \[4\]. The existence is proved of a positive invariant measure on commutative p-hypergroups \[170\]; cf. \[8, 35\] for the corresponding result for h.s. (existence of a multiplicative measure). The existence of an invariant measure is proved for compact and discrete $p$-hypergroups \[136, 169\]; the question is open for the general case [^2]. If an invariant measure m exists, then one can construct the hypergroup algebra $L^1 (H, m)$ (cf. above, point 3.7) and use the methods of the theory of Banach algebras with involution. By no means all interesting hypergroups with involution are $p$-hyper-groups or h.s. with locally compact bases. A standard example is the hypergroup generated by a Sturm – Liouville operator on the half-line with rapidly growing potential (cf. above, points 1.5 and 4.6). Even in the compact case passage to the dual hypergroup leaves the class of $p$-hypergroups (h.s.) (for example, cf. \[3, 7, 112, 136, 169, 182\]) due to the loss of the positivity condition. Nevertheless the class of $p$-hypergroups (not necessarily commutative) can be immersed in a larger class (consisting of objects of the type of Kac algebras), for which a satisfactory duality theory is constructed, cf. \[138\].\ Let $H = K \setminus G / K$ be the hypergroup of double cosets of the locally compact group $G$ with respect to its compact subgroup $K$ (cf. above, points 2.6, 2.7, 3.3, and 3.6). If the hypergroup $H$ is commutative, then the pair $(G, K)$ is called a [*Gelfand pair.*]{} These objects were studied intensively (in particular, in the framework of the theory of p-hypergroups and h.s.), among others in connection with the theory of spherical functions and the theories of probability measures on groups and hypergroups (for example, in connection with the generalization of the central limit theorem to the case of convolution of probability measures), cf. \[23, 7, 10, 94, 96, 110, 111, 125, 131, 132, 136, 137, 145, 151, 167-169, 175, 196\]. The following result goes back to Gel’fand’s work \[23\], cf. also \[131\].\ Let the group $G$ be unimodular, and the compact subgroup $K$ coincide with the set of fixed points of the involution automorphism $\sigma$. If any element $x \in G$ admits a decomposition $x = k \cdot y$, where $k \in K, \sigma (y) = y^{-1}$. then $(G, K)$ is a Gelfand pair.\ It is easy to see that to prove this theorem it suffices to establish the commutativity of the hypergroup algebra $L^1 (H)$. We identify $L^1 (H)$ with a subalgebra of the group algebra $L^1 (G)$ (cf. points 3.6 and 3.7); this subalgebra consists of all K-biinvariant functions in $L^1 (G)$. Let $f(x)$ be such a function. We set $f^{\sigma} (x) = f(\sigma (x)), \tilde{f}(x) = f(x^{-1})$. Then $f^{\sigma} (x) = f(\sigma (x)) = f(\sigma (k \cdot y)) = f(\sigma (k) \cdot \sigma (y)) = f(k \cdot y^{-1}) = f(y^{-1} \cdot k^{-1}) = f (x^{-1}) = \tilde{f} (x)$. The map $f \mapsto f^{\sigma}$ is an automorphism of the algebra $L^1 (H)$ and preserves the order of the factors in the product of elements of $L^1 (H)$; the map $f \mapsto \tilde{f}$ is an antiautomorphism and changes the places of the factors in the product of two elements of $L^1 (H)$. Since these maps coincide, the algebra $L^1 (H)$ is commutative, which is what was needed. The hypothesis of Gelfand’s theorem holds, if $G$ is a semisimple linear group or group of motions, and $K$ is a maximal subgroup of it, and also if $(G, K)$ is a Riemannian symmetric pair in the sense of \[84, Chap. VI, Sec. 1\]. 6. Representations of Hypergroups and Harmonic Analysis {#representations-of-hypergroups-and-harmonic-analysis .unnumbered} ======================================================= The theory of representations of hypergroups is to a large extent analogous to the theory of representations of groups. This analogy extends quite far. For example, with the help of such concepts as generator and infinitesimal hypergroup algebra, one can study representations of differentiable and holomorphic hypergroups by an infinitesimal method just as is done for Lie groups; cf. \[108, 55, 56, 24\], in which finite-dimensional representations of Delsarte hypergroups (cf. above, point 2.3) and hypergroups generated by generators of the second order, generating the Lie algebra (cf. point 4.5) are considered. It is convenient to treat representations of hypergroups as representations of hypergroup algebras. Just as in the case of groups, to different types of hypergroup algebras correspond different versions of the theory of representations and harmonic analysis (cf. \[32, 59, 61\]). Let $V$ be an arbitrary locally convex space, $S(V)$ be the algebra of all weakly continuous linear operators on $V$, endowed with the weak operator topology (cf. above, point 3.1), $\mathscr{F}$ be an arbitrary topological algebra (cf. point 3.2). By a continuous representation of the algebra $\mathscr{F}$ on the space $V$ we shall mean a continuous map $\mathscr{F} \to S(V)$ which is a homomorphism of algebras; if $\mathscr{F}$ has a two-sided identity, we shall assume (without special mention) that the identity goes into the identity operator. Let $H$ be a hypergroup, where the corresponding g.t.o. $R^x$ and $L^x$ act on the locally convex space $\Phi$. By a representation of the hypergroup $H$ on the space $V$ we mean a map $x \mapsto \pi_x$ of the space $H$ into $S(V)$, such that the following conditions hold: 1\) for any elements $v \in V, v' \in V'$ the function $\varphi_{v, v'} (x) = \langle v', \pi_x v \rangle$, called a [*matrix element of the representation*]{}, is contained in $\Phi$ and $R^y \varphi_{v,v'} (x) = \langle v', \pi_x \pi_y v \rangle$ for all $x, y \in H$; 2\) for any element $f$ of the dual space $\Phi'$ to $\Phi$ there exists an operator $\pi (f) = \int \pi_x df (x) \in S(V)$. It follows from the definition of the integral $\int \pi_x df(x)$ (cf. point 3.1), Eq. (3.1), and the fact that the matrix elements of the representation $\pi$ are contained in $\Phi$, that the map $f \mapsto \pi (f) = \int \pi_x df(x)$ of the space $\Phi'$ into $S(V)$ is continuous in the weak topology $\sigma (\Phi' , \Phi)$ and consequently in the Mackey topology $\tau (\Phi' , \Phi)$ and in any topology compatible with the duality between $\Phi'$ and $\Phi$. If the generalized convolution turns $\Phi'$ into a topological hypergroup algebra, then the map $f \mapsto \pi (f)$ is a continuous representation of the algebra $\Phi'$. Condition 2) holds automatically, if $V$ is a complete or quasicomplete barreled space (for example, a reflexive or complete metrizable space), and $\Phi$ coincides with the space $C(H)$ of continuous functions ($C^{\infty} (H)$ or $\mathscr{H} (H)$ of infinitely differentiable or holomorphic functions) on the locally compact (respectively infinitely differentiable or holomorphic) hypergroup $H$, cf. \[61\]. We call a representation $\pi$ of the hypergroup $H$ [*nondegenerate*]{}, if the operator $\pi_e$ (where $e$ is the neutral element in $H$) is the identity operator. If $\Phi$ coincides with its basic subspace (cf. point 2.1), then we shall assume representations of the hypergroup $H$ to be nondegenerate without any special mention. Let $H$ be a locally compact hypergroup. Any representation $x \mapsto \pi_x$ of it extends to a continuous representation $f \mapsto \pi (f)$ of the hypergroup algebra $\mathscr{M} (H)$ (cf. above). Conversely, if we have a continuous representation $f \mapsto \pi (f)$ of the algebra $\mathscr{M} (H)$, then the correspondence $x \mapsto \pi_x = \pi (\delta_x)$ is (as is easy to verify, cf. \[61\]) a representation of the hypergroup $H$. It is easy to derive the following theorem from this.\ The representations of a locally compact hypergroup $H$ are in one-to-one correspondence with continuous representations of the topological hypergroup algebra $\mathscr{M} (H)$ . If a representation of the hypergroup $H$ is nondegenerate, then the corresponding representation of $\mathscr{M} (H)$ is the composition of a continuous representation of the basic subalgebra $\widetilde{\mathscr{M}} (H)$ and the canonical projection $\mathscr{M} (H) \to \widetilde{\mathscr{M}} (H)$. Thus a one-to-one correspondence is established between nondegenerate representations of the hypergroup $H$ and continuous representations of the algebra $\widetilde{\mathscr{M}} (H)$.\ Analogous results are valid for infinitely differentiable and holomorphic hypergroups. To representations with infinitely differentiable (holomorphic) matrix elements correspond continuous representations of hypergroup algebras of generalized functions with compact support (algebras of analytic functionals), while to non-degenerate representations of hypergroups correspond representations of basic subalgebras, cf. \[61\]. The analog of the theory of unitary representations of groups is the theory of symmetric (Hermitian) representations of hypergroup algebras with involution on Hilbert spaces (the representation $\pi$ is [*symmetric*]{} if the operator $\pi (f^*)$ is adjoint to the operator $\pi (f)$ where $f \mapsto f^*$ is the involution in the hypergroup algebra). Representations of this type were studied (primarily for the commutative case) by B. M. Levitan, Yu. M. Berezanskii, S. G. Krein, and other authors, cf., e.g., \[3, 5, 7, 8, 14, 17, 47-51, 93, 96, 112, 129, 131, 132, 135, 136, 142, 144, 149, 150, 169, 180\]. In particular, in the theory of $p$-hypergroups, by a representation of the hypergroup $H$ on the Hilbert space $V$ is meant a symmetric homomorphism $f \mapsto \pi (f)$, which does not decrease the norm, of the Banach algebra $\mathscr{M}^b (H)$ of bounded measures to the algebra of bounded operators on $V$, such that the induced map $\mathscr{M}^b_+ S(V)$ is continuous (with respect to the weak operator topology on $S(V)$ and the topology on $\mathscr{M}^b_+ (H)$ described in point 5.3). The matrix elements of this representation are bounded continuous functions on $H$ (cf., e.g., \[136\]). The irreducible (symmetric) representations of an arbitrary p-hypergroup $H$ separate points in $H$ so that there are sufficiently many such representations \[180\]. The basic concepts, constructions, and results of the theory of unitary representations of groups generalize to the case of hypergroup algebras with involution; for example, analogs of the theorem on the connection of representations with positive definite functions, Bochner’s theorem, Plancherel’s theorem, etc. are valid.\ Let $\mathscr{F}$ be a topological algebra, $\Phi$ be the dual space to $\mathscr{F}$, endowed with a locally convex topology compatible with the duality between $\Phi$ and $\mathscr{F}$, for example, the Mackey topology $\tau (\Phi , \mathscr{F} )$. We denote by $R(f)$ the operator on $\Phi$ adjoint to the operator $g \mapsto gf$ and by $L(f)$ the operator adjoint to the operator $g \mapsto fg$. The map $f \mapsto R(f)$ is a continuous representation of the algebra $\mathscr{F}$. We call this representation [*right-regular*]{}. We note that the correspondence $f \mapsto L(f)$ is an antirepresentation of the algebra $\mathscr{F}$. If $\mathscr{F}$ coincides with the hypergroup algebra of the hypergroup $H$, while the g.t.o. $R^x$ and $L^x$ act on the space $\Phi$, then $R(f) = \int R^x df(x)$ and $L(f) = \int L^x df(x)$. Hence the correspondence $x \mapsto R^x$ is a nondegenerate representation of the hypergroup $H$ and extends to a right-regular representation of the algebra $\mathscr{F}$. The representation $x \mapsto R^x$ is called [*the right-regular representation of the hypergroup $H$*]{}. If $H$ is a hypergroup with involution, then the correspondence $x \mapsto L^{x^*}$ is also a representation of the hypergroup $H$. This representation is called [*left-regular*]{} and extends to the [*left-regular representation*]{} $f \mapsto L(f^*)$ of the algebra with involution $\mathscr{F}$. Let us assume that the action of the operators $R^x, L^x, R(f), L(f)$ is defined not only on the space $\Phi$, but also in the Hilbert space $L^2 (H, m)$, where m is a right (left) invariant measure on $H$, cf. point 2.7. In this case there arises on $L^2 (H, m)$ a symmetric representation $f \mapsto R(f)$ (respectively $f \mapsto L(f^*)$). Representations of this type are also called regular.\ As in point 6.2 let $\mathscr{F}$ be a topological algebra, $\Phi$ be a locally convex space dual to $\mathscr{F}$. By a [*matrix element*]{} of an arbitrary continuous representation $f \mapsto \pi (f)$ of the algebra $\mathscr{F}$ on the space $V$ is meant an element $\varphi \in \Phi$ such that $\langle f, \varphi \rangle = \langle v', \pi (f)v \rangle$, where $v \in V, v' \in V'$. The space $\Phi (\pi )$ in $\Phi$, spanned by matrix elements of the representation $\pi$, will be called the [*space of matrix elements of this representation*]{}. We denote by Ker $\pi$ the kernel of the representation $\pi$, i.e., the closed two-sided ideal $\{ f \in \mathscr{F} \colon \pi (f) = 0 \}$ in $\mathscr{F}$. Continuous representations $\pi_1$ and $\pi_2$ of the algebra $\mathscr{F}$ are called [*isomorphic*]{} if Ker $\pi_1 =$ Ker$ \pi_2$. For symmetric representations of $C^*$-algebras the concept of isomorphism was introduced by M. A. Naimark \[69\]; the general case was considered in \[59-61\]. For completely irreducible representations isomorphism coincides with the [*Fell equivalence*]{} \[123\] (or [*weak equivalence*]{}), for irreducible finite-dimensional representations it consides with ordinary equivalence, defined by a similarity operator. Conditions are given in \[123\] and \[32\] under which Fell equivalence (for irreducible representations) coincides with Naimark equivalence, which is popular in the theory of representations of semisimple groups. We shall call a subspace of $\Phi$, which is invariant with respect to all the operators $R(f)$ (respectively $L(f)$), [*right-(left-)invariant*]{}. The next proposition follows from the definition of the operators $R(f)$ and $L(f)$.\ The subspace $\Phi_1$ of $\Phi$ is right-(left-)invariant if and only if its orthogonal complement $\{f \in \mathscr{F} \colon \langle f, \varphi \rangle =0$ for all $\varphi \in \Phi_1 \}$ is a right (left) ideal in the algebra $\mathscr{F}$.\ Thus to the subrepresentations of the right-regular representation (i.e., to its restrictions to invariant subspaces of $\Phi$) correspond closed right ideals in $\mathscr{F}$ and by the Hahn-Banach theorem this correspondence is one-to-one. It is clear that to maximal closed right ideals of $\mathscr{f}$ correspond topologically irreducible subrepresentations .\ Closed two-sided ideals of $\mathscr{f}$ are in one-to-one correspondence with subspaces of $\Phi$, which are simultaneously right- and left-invariant.\ The orthogonal complement in $\mathscr{F}$ of the space of matrix elements $\Phi (\pi)$ of the continuous representation $\pi$ coincides with the kernel of this representation Ker $\pi$.\ Continuous representations of the algebra $\mathscr{f}$ are isomorphic if and only if their spaces of matrix elements coincide.\ Let $\delta$ be a fixed right identity in $\mathscr{F}$; we call the subalgebra $\widetilde{\mathscr{F}} = \{f \in \mathscr{F} \colon~ \delta f = f \}$ the [*basic subalgebra*]{} of $\mathscr{F}$ and the subspace $\widetilde{\Phi} = \{ \varphi \in \Phi \colon~ L(\delta ) \varphi =\varphi \}$ the [*basic subspace*]{} of $\Phi$. We call the representation $\pi \colon \mathscr{F} \to S(V)$ [*non-degenerate*]{}, if the operator $\pi (\delta )$ is the identity operator. This terminology agrees with the definitions introduced above for hypergroup algebras. The algebra $\tilde{\mathscr{F}}$ has a two-sided identity $\delta$ and is isomorphic to the quotient-algebra $\mathscr{F} / J$, where $J$ is the orthogonal complement of $\widetilde{\Phi}$. The nondegenerate representations of the algebra $\mathscr{F}$ are in one-to-one correspondence with representations of the algebra $\tilde{\mathscr{F}}$, as in Theorem 3. The closed ideals (right, left, two-sided) of $\tilde{\mathscr{F}}$ are in one-to-one correspondence with the invariant subspaces of $\widetilde{\Phi}$ (respectively, right-, left-, or bilaterally invariant). In particular, if $\mathscr{F} = \mathscr{M} (H)$ where $H$ is a locally compact hypergroup, then the space of matrix elements of any nondegenerate representation of $H$ is contained in the basic subspace $\widetilde{C} (H)$, which coincides with the space of matrix elements of the right-regular representation. We note that for an arbitrary hypergroup it is impossible to define the tensor product of representations. Hence the collection of all matrix elements will not necessarily be an algebra with respect to multiplication of functions. In what follows, when we speak of representations of hypergroups having certain properties or others (for example, being irreducible, equivalent, etc.), we shall have in mind that the corresponding representations of hypergroup algebras have these properties.\ Reduction to hypergroup algebras lets one carry over the concept of the character of a representation to the case of hypergroups, where it is treated as a linear functional on an ideal in the hypergroup algebra $\mathscr{F}$ (or on an $\mathscr{F}$-bimodule). If the character $\chi$ of a topologically irreducible representation $\pi$ is defined on a dense ideal in $\mathscr{F}$ and $\chi (f)$ coincides with the trace of a nuclear operator $\pi (f)$, then the representation $\pi$ is completely irreducible and is determined by its character $\chi$ up to Fell equivalence \[63\]. Cf. \[60\] for a more general approach and other conditions under which a representation is determined by its character up to isomorphism. As in the case of Lie groups, under certain conditions the character can be considered as a generalized function on a hypergroup. In what follows, by a [*character of a hypergroup*]{} $H$ we shall mean a nonzero one-dimensional representation (which can be identified with its character in the sense considered above or with a matrix element). Thus, a character can be considered as a function $\chi (x)$ on $H$, satisfying the equation $$R^x \chi (x) = \chi (y) \chi (x). \eqno(6.1)$$ If $H$ is a hypergroup with involution and the representation $x \mapsto \chi (x)$ is symmetric, then $\chi (x^*) = \bar{\chi} (x)$; in this case the character $\chi$ is called [*Hermitian*]{}. Characters of commutative hypergroups were studied by many authors, cf., e.g., \[3, 7, 41, 48-51, 55, 56, 61, 112, 117, 136, 169\].\ Let $H$ be the hypergroup generated by the Gelfand pair $(G, K)$, cf. above, points 2.6 and 5.4. The characters of this hypergroup correspond to continuous functions $\varphi (x)$ on $G$, satisfying the relation $$\int_K \varphi (xky) dk = \varphi (x) \varphi (y) \eqno(6.2)$$ for all $x, y \in K$ and the normalized Haar measure dk. (6.2), which follows from (2.4) and (6.1), means that $\varphi (x)$ is a spherical function on $G$, cf., e.g., \[23, 10, 69, 84, 85\].\ Let $H$ be a hypergroup with a hypergroup algebra $\mathscr{F}$ and suppose that with each element $\zeta$ of some set $Z$ there is associated a representation $\pi_{\zeta}$ of the hypergroup $H$. By the [*generalized Fourier transform*]{} we mean the linear operator, which carries the functional $f \in \mathscr{F}$ into the function $f(\zeta ) = \pi_{\zeta} (f)$ assuming operator values. It is clear that here the generalized convolution goes into the product of the operator-valued functions. Such a construction is standard in the representation theory. If the set $Z$ consists of characters of the hypergroup $H$, then $f(\zeta )$ is a scalar function and the generalized convolution goes into ordinary multiplication of functions. Here the basic subalgebra $\widetilde{\mathscr{F}}$ is mapped isomorphic-ally onto some algebra of functions if and only if linear combinations of characters from $Z$ are dense in the basic subspace $\widetilde{\Phi}$, where $\Phi = \mathscr{F}'$. For example, if with each complex number $\zeta$ one associates the character $x \mapsto e^{ix \zeta}$ of the group ${\bf R}$ of real numbers, then for the algebra $\mathscr{D} ({\bf R})$ the construction indicated leads to the ordinary Fourier-Laplace transform of generalized functions with compact support. If the hypergroup $H$ is generated by a Gelfand pair and $Z$ consists of Hermitian characters of this hypergroup, then the generalized Fourier transform reduces to the spherical Fourier transform, cf., e.g., \[7, 10, 84, 85, 93, 131, 136, 146, 169\]. Other examples are given below.\ It is clear that the description of continuous representations of the topological algebra $\mathscr{F}$ up to isomorphism reduces to the description of closed two-sided ideals of $\mathscr{F}$. In what follows we shall consider only continuous representations and closed two-sided ideals. We are particularly interested in (nontrivial) maximal, primitive and primary ideals. [*Primitive*]{} ideals are kernels of irreducible representations, and an ideal which is contained in a unique maximal ideal is called [*primary*]{} (we note that this definition, which is customary in the theory of topological algebras, does not agree with the definition of a primary ideal in abstract algebra; sometimes an ideal which is contained in a unique primitive ideal is called a primary ideal in a topological algebra). A representation $\pi$ is called [*prime*]{} (respectively [*primary*]{}), if its kernel is a maximal (primary) ideal. A finite-dimensional representation is prime if and only if it is isomorphic to an irreducible finite-dimensional representation; cf. \[60\] on prime representations. We shall say that the representation $\pi_1$ is [*contained in the representation*]{} $\pi_2$, if Ker $\pi_1 \supset$ Ker $\pi_2$. In this case the space of matrix elements of the representation $\pi_2$ contains the space of matrix elements of the representation $\pi_1$. If $\pi_1$ is a subrepresentation of $\pi_2$, then clearly $\pi_1$ is contained in $\pi_2$; the converse assertion is true only “up to isomorphism.” It is clear that a representation is primary if it contains a unique prime representation. We fix some collection $\Gamma = \{ I_{\gamma} \}$ of ideals in $\mathscr{F}$ and for any ideal $I_{\gamma} \in \Gamma$ we denote by $\pi_{\gamma}$ a representation such that $I_{\gamma} =$ Ker $\pi_{\gamma}$. It is appropriate to choose $\Gamma$ so that this collection is visible, consists of ideals of relatively simple structure, and other ideals can be described conveniently as intersections of ideals of $\Gamma$ in the spirit of the famous theorem of Lasker-Noether. We shall say that the ideal $I$ admits [*spectral synthesis*]{} (with respect to $\Gamma$), if $I$ coincides with the intersection of those ideals of $\Gamma$ which contain it; a [*representation $\pi$ admits spectral synthesis*]{} if the ideal Ker $\pi$ admits it. The next proposition follows from Propositions 5 and 6 and the Hahn-Banach theorem.\ The representation $\pi$ admits spectral synthesis if and only if any matrix element of this representation can be approximated in $\mathscr{F}'$ by matrix elements of those representations $\pi_{\gamma}$ which are contained in $\pi$.\ For example, for representations of a locally compact hypergroup one is concerned with approximation of matrix elements in the topology of uniform convergence on compact subsets of $H$. It is natural to treat the problem of [*spectral analysis*]{} as the problem of describing ideals of $\Gamma$, containing a given ideal, or representations $\pi_{\gamma}$ contained in a given representation. The scheme described of one of the versions of harmonic analysis, connected with the Delsarte-Schwartz theory of mean-periodic functions and the theory of differential equations with constant coefficients and convolution equations, allows one to consider results which at first glance have little in common between them, from a single point of view. A vast literature is devoted to other questions of harmonic analysis on hypergroups (the theory of almost periodic functions, expansions in orthogonal systems of functions, etc., cf., points 1.6, 5.3, 6.1, 6.8).\ The theory of representations of compact groups gives the classical example of spectral synthesis. If $G$ is a compact group, then in the topological group algebra $\mathscr{M} (G)$ consisting of measures with compact support, any primary ideal is a maximal ideal of finite codimension. The Peter-Weyl theorem is equivalent to the assertion that any ideal in $\mathscr{M} (G)$ is the intersection of the maximal ideals containing it. A generalization of the Peter-Weyl theorem to the case of compact hypergroups is given by B. M. Levitan (cf. \[55, 56\]; the Peter-Weyl theory and harmonic analysis on p-hypergroups were considered in \[136, 180\]; cf. also \[3, 8, 7, 14\]). If there is given on the compact hypergroup $H$ a finite measure, satisfying certain additional conditions, then any continuous function from the basic subspace $\widetilde{C} (H)$ can be approximated in the topology of uniform convergence by matrix elements of finite-dimensional irreducible representations. One can deduce from this that any ideal in the basic subalgebra $\tilde{\mathscr{M}} (H)$ is the intersection of the maximal ideals containing it. Analogous results are also valid for other hypergroup algebras, for example, for the Hilbert algebra $L^2 (H)$. For the group $G$ of conformal transformations of the unit disc, which is locally isomorphic to the group $SL(2, R)$, Ehrenpreis and Mautner \[119\] described the ideals in the group algebra $\mathscr{D} (G)$ (we note that for any Lie group $G$ the ideals in $\mathscr{D} (G)$ are in one-to-one correspondence with the ideals in $\mathscr{M} (G)$). It turns out that there are ideals which are not intersections of primary ideals, and sufficient conditions are given in \[119\] under which an ideal admits spectral synthesis. Using the concept of g.t.o., P. K. Rashevskii found exhaustive results on spectral analysis and synthesis for other function spaces on $SL(2, R)$. Cf. \[76, 87\] for the further development of problems of this kind for other semisimple groups of rank 1. For groups of arbitrary rank additional difficulties arise, which are familiar from the “multidimensional” theory of mean-periodic functions. We consider the topological algebras $\mathscr{M} ({\bf R}^n ), \mathscr{D} ({\bf R}^n ), \mathscr{A} ({\bf C}^n)$ connected with ordinary translations in an n-dimensional vector space. The multiplication in these algebras coincides with the usual convolution of the corresponding measures, generalized functions, or analytic functionals. The problem of describing the ideals in these topological algebras and of the spectral synthesis for such ideals is equivalent with the basic problem of the theory of mean-periodic functions. For $n = 1$, this problem is solved in \[165\]. For any complex number $\zeta$ and integer $k \geq 0$ we denote by $C(\zeta , k)$ the subspace of $C({\bf R})$ spanned by the quasimonomials $e^{\zeta t} , te^{\zeta t} , \ldots , t^k e^{\zeta t}$ and by $I(\zeta , k)$ we denote the ideal in $\mathscr{M} ({\bf R})$ orthogonal to $C(\zeta , k)$. Ideals of this type (and only these) are primary. Any nontrivial ideal in $\mathscr{M} ({\bf R})$ is the intersection of a no more than countable family of primary ideals, and each nontrivial closed subspace of $C({\bf R})$, which is invariant with respect to translations, is spanned by a no more than countable set of quasimonomials. Analogous results are valid for the algebras $\mathscr{D} ({\bf R})$ and $\mathscr{A} ({\bf C})$. For $n > 1$ the situation is more complex. Schwartz assumed that in the multidimensional case also the invariant subspaces are spanned by quasimonomials. In this case the corresponding ideals admit spectral synthesis. Although a counterexample has been constructed to Schwartz’ conjecture \[26\], this conjecture is proved for many important special cases, cf., e.g., \[148, 116, 118, 75, 70\]. For spaces of solutions of homogeneous systems of linear partial differential equations with constant coefficients, stronger results have been found which can be interpreted as the decomposition of nonunitary representations into the direct integral of primary components \[75, 118\]. Cf. \[44-46\] for generalizations of these results, for example, to the case of symmetric spaces. For certain one-dimensional hypergroups, the basic results of the theory of mean-periodic functions (including the primary synthesis) are found in \[41\]. Analogous results are found in \[91\] for the hypergroup $H = K\smallsetminus G/K$ where $G$ is a semisimple linear Lie group and $K$ is a maximal compact subgroup of it. The following example is described in \[61\]. Let the real line ${\bf R}$ be endowed with the structure of the hypergroup described in point 2.4, and let $\mathscr{M}$ be the corresponding topological hypergroup algebra, consisting of measures with compact support. The basic subspace $\widetilde{C} ({\bf R})$ consists of all even continuous functions, and the basic subalgebra $\tilde{\mathscr{M}}$ consists of all even measures with compact support. Equation (6.1), which defines the characters, has, in the present case, the form $\chi (x+y) + \chi (x-y) = 2\chi (x) \chi (y)$ and the characters are the functions $\chi_{\zeta} (x) = cos (\zeta x)$ where $\zeta$ is any complex number. The generalized Fourier transform $f \mapsto f(\zeta ) = \int \cos (\zeta x ) df(x)$ maps $\tilde{\mathscr{M}}$ one-to-one to the algebra consisting of even entire analytic functions of exponential growth (not all). Any primary ideal in $\tilde{\mathscr{M}}$ has the form $\{ f \in \tilde{\mathscr{M}} \colon f (\zeta ) = 0, f' (\zeta ) = 0, \ldots , f^{(k)} (\zeta ) = 0 \}$ where the complex number $\zeta$ and the integer $k \geq 0$ are fixed. Any nontrivial ideal in $\tilde{\mathscr{M}}$ is the intersection of a no more than countable collection of primary ideals. Any nontrivial subspace of $\widetilde{C} ({\bf R})$ which is invariant with respect to g.t.o., is spanned by a no more than a countable set of functions of the form $\cos (\zeta x ), x^{2n - 1} \sin (\zeta x), x^{2n} \cos (\zeta x )$ where $\zeta \in {\bf C}, n= 1, 2, \ldots$. Analogous results are valid for smooth and holomorphic functions. The more general case of g.t.o. with Sturm – Liouville generator is considered in \[27\], where examples of nonprimary synthesis are given. One can extract additional examples, for example from \[59, 61, 83, 109\].\ Suppose given on certain spaces $H$ and $\widehat{H}$ positive measures $m$ and $\mu$ respectively. Let us assume that with the help of the function $u(x, \chi )$ defined on $H \times \widehat{H}$, there is given an “abstract Fourier transform” $$\varphi (x) \mapsto \hat{\varphi} (\chi ) = \int \varphi (x) \overline{u(x, \chi )} dm(x), \eqno(6.3)$$ which defines an isomorphism of the Hilbert spaces $L^2 (H, m)$ and $L^2 (\widehat{H} , \mu )$, so that the “Plancherei formula” is valid: $$\int \varphi (x) \overline{\psi (x)} dm(x) = \int \hat{\varphi} (\chi) \overline{\hat{\psi} (\chi )} d\mu (\chi ) .\eqno(6.4)$$ Let us assume in addition that the [*inversion formula*]{} $$\varphi (x) = \int \hat{\varphi} (\chi ) u (x, \chi ) d\mu (\chi) \eqno(6.5)$$ is valid, which one can get by formal substituting the delta-function for $\psi$ in (6.4) and considering the form of the transformation (6.3). If the measure $\mu$ is discrete, then (6.5) gives the expansion of the function $\varphi$ in an “abstract Fourier series.” It turns out (cf. \[51, 55, 56\]) that $H$ has the structure of a commutative hypergroup, if for some point $e \in H$ and for all $\chi \in \widehat{H}$ one has $u(e, \chi ) = 1$ (the element $e$ is the identity in $H$; sometimes one is naturally restricted by the requirement of the existence of an approximate identity). Here the g.t.o. are defined by $$R^y \varphi (x) = \int \hat{\varphi} (\chi ) u(y, \chi ) u(x, \chi ) d\mu (\chi ) .$$ Analogously one defines a hypergroup structure on $H$. Hence hypergroups and g.t.o. arise naturally in problems connected with expansion in orthogonal systems of functions, the spectral theory of operators, etc., which guarantees a large circle of applications of the theory of hypergroups and g.t.o. For a large class of hypergroups with invariant measure the transformation (6.3) can be defined as a [*generalized Fourier*]{} transform (cf. point 6.5), applied to the measure $\varphi (x)m$, to give explicitly the space $\widehat{H}$ and the measure $\mu$ (called the [*Plancherel measure*]{}) and to prove the Plancherel formula and the inversion formula. In the commutative case $\widehat{H}$ consists of Hermitian characters (or of one-dimensional representations of the corresponding hypergroup algebra with involution) and $u(x, \chi ) = \chi (x)$, where $\chi$ is a character from $\widehat{H}$. The Plancherel theorem and inversion formula for commutative g.t.o. were first proved by B. M. Levitan \[47-50\], cf. also \[3, 7, 136, 169\]. In \[17\] a generalization of the Plancherel theorem and inversion formula to a certain class of (generally) noncommutative hypergroups is given. This class includes, for example, all p-hypergroups with invariant measure and all locally compact groups. Thus, this result generalizes the theorem proved by Segal \[166\] and Tatsuuma \[173\] for locally compact groups. We note that the existence of the inversion formula is a distinctive property of hypergroups, while the Plancherel theorem is also valid for objects of a more general nature, connected with Hilbert algebras, for example, for groupoids with measure, cf. \[159\]. 7. Conclusion {#conclusion .unnumbered} ============= The theory of hypergroups is a relatively new domain of mathematics, comparable, in breadth of content of its material, interest, and richness of applications, with such classical domains as the theory of Lie groups or the theory of Hilbert spaces, but considerably less developed. The Lie theory for hypergroups (especially in relation to procedures for constructing global objects from given infinitesimal objects, for example, commutation relations) is far from to be complete. Comparatively little concrete material has been accumulated in the theory of representations of noncommutative hypergroups. In particular, there is interest in the calculation of concrete Plancherel measures and their supports, and in “nonunitary” theory, in description of collections of ideals in hypergroup algebras and invariant subspaces in function spaces on hypergroups, guaranteeing the meaningfulness of spectral synthesis. One can hope that the contemporary theory of hypergroups and g.t.o. will enter as a fragment or the skeleton of a broader and more general theory, which will include, for example, the theory of supergroups also. Generalizations of the concept of hypergroup arise naturally in Lie theory, duality theory, group representation theory. The direction of further development of the theory of hypergroups will apparently be determined to a considerable degree by impulses connected with the applications of this theory to mathematical physics. [999]{} Yu. M. 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--- abstract: 'Let $M_R$ be a module and $\sigma$ an endomorphism of $R$. Let $m\in M$ and $a\in R$, we say that $M_R$ satisfies the condition $\mathcal{C}_1$ (respectively, $\mathcal{C}_2$), if $ma=0$ implies $m\sigma(a)=0$ (respectively, $m\sigma(a)=0$ implies $ma=0$). We show that if $M_R$ is p.q.-Baer then so is $M[x;\sigma]_{R[x;\sigma]}$ whenever $M_R$ satisfies the condition $\mathcal{C}_2$, and the converse holds when $M_R$ satisfies the condition $\mathcal{C}_1$. Also, if $M_R$ satisfies $\mathcal{C}_2$ and $\sigma$-skew Armendariz, then $M_R$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a p.p.-module. Many generalizations are obtained and more results are found when $M_R$ is a semicommutative module.' --- **[Mohamed Louzari]{}** Department of mathematics Abdelmalek Essaadi University B.P. 2121 Tetouan, Morocco [email protected] \[section\] \[Theorem\][Definition]{} \[Theorem\][Proposition]{} \[Theorem\][Corollary]{} \[Theorem\][Lemma]{} \[Theorem\][Example]{} \[Theorem\][Remark]{} This work is dedicated to my Professor El Amin Kaidi Lhachmi from University of Almería on the occasion of his 62nd birthday. [**Mathematics Subject Classification:**]{} 16S36, 16D80, 16W80\ [**Keywords:**]{} Semicommutative modules, p.q.-Baer modules, p.p.-modules. Introduction ============ In this paper, $R$ denotes an associative ring with unity and modules are unitary. We write $M_R$ to mean that $M$ is a right module. Throughout, $\sigma$ is an endomorphism of $R$ (unless specified otherwise), that is, $\sigma\colon R\rightarrow R$ is a ring homomorphism with $\sigma(1)=1$. The set of all endomorphisms (respectively, automorphisms) of $R$ is denoted by $End(R)$ (respectively, Aut(R)). In [@Kaplansky], Kaplansky introduced Baer rings as rings in which the right (left) annihilator of every nonempty subset is generated by an idempotent. According to Clark [@clark], a ring $R$ is said to be [*quasi-Baer*]{} if the right annihilator of each right ideal of $R$ is generated (as a right ideal) by an idempotent. These definitions are left-right symmetric. Recently, Birkenmeier et al. [@birk/pqBaer] called a ring $R$ a [*right*]{} $($respectively, [*left$)$ principally quasi-Baer*]{} (or simply [*right*]{} $($respectively, [*left$)$ p.q.-Baer*]{}) if the right (respectively, left) annihilator of a principally right (respectively, left) ideal of $R$ is generated by an idempotent. $R$ is called a [*p.q.-Baer*]{} ring if it is both right and left p.q.-Baer. A ring $R$ is a right (respectively, left) [*p.p.-ring*]{} if the right (respectively, left) annihilator of an element of $R$ is generated by an idempotent. $R$ is called a [*p.p.-ring*]{} if it is both right and left p.p.-ring. Lee-Zhou [@lee/zhou] introduced Baer, quasi-Baer and p.p.-modules as follows: $(1)$ $M_R$ is called [*Baer*]{} if, for any subset $X$ of $M$, $r_R(X)=eR$ where $e^2=e\in R$. $(2)$ $M_R$ is called [*quasi-Baer*]{} if, for any submodule $N$ of $M$, $r_R(N)=eR$ where $e^2=e\in R$. $(3)$ $M_R$ is called [*p.p.*]{} if, for any $m\in M$, $r_R(m)=eR$ where $e^2=e\in R$. In [@baser2007], a module $M_R$ is called [*principally quasi Baer*]{} (p.q.-Baer for short) if, for any $m\in M$, $r_R(mR)=eR$ where $e^2=e\in R$. It is clear that $R$ is a right p.q.-Baer ring if and only if $R_R$ is a p.q.-Baer module. If $R$ is a p.q.-Baer ring, then for any right ideal $I$ of $R$, $I_R$ is a p.q.-Baer module. Every submodule of a p.q.-Baer module is p.q.-Baer module. Moreover, every quasi-Baer module is p.q.-Baer, and every Baer module is quasi-Baer module. A ring $R$ is called [*semicommutative*]{} if for every $a\in R$, $r_R(a)$ is an ideal of $R$ (equivalently, for any $a,b\in R$, $ab=0$ implies $aRb=0$). In [@rege2002], a module $M_R$ is semicommutative, if for any $m\in M$ and $a\in R$, $ma=0$ implies $mRa=0$. Let $\sigma$ an endomorphism of $R$, $M_R$ is called $\sigma$-semicommutative module [@zhang/chen] if, for any $m\in M$ and $a\in R$, $ma=0$ implies $mR\sigma(a)=0$. According to Annin [@annin], a module $M_R$ is $\sigma$-[*compatible*]{}, if for any $m\in M$ and $a\in R$, $ma=0$ if and only if $m\sigma(a)=0$. In [@lee/zhou], Lee-Zhou introduced the following notations. For a module $M_R$, we consider $M[x;\sigma]:={\left\{\sum_{i=0}^sm_ix^i:s\geq 0,m_i\in M\right\}},$ $M[[x;\sigma]]:={\left\{\sum_{i=0}^\infty m_ix^i:m_i\in M\right\}},$ $M[x,x^{-1};\sigma]:={\left\{\sum_{i=-s}^tm_ix^i:\;t\geq 0,s\geq 0,m_i\in M\right\}},$ $M[[x,x^{-1};\sigma]]:={\left\{\sum_{i=-s}^\infty m_ix^i:s\geq 0,m_i\in M\right\}}.$ Each of these is an Abelian group under an obvious addition operation. Moreover $M[x;\sigma]$ becomes a module over $R[x;\sigma]$ under the following scalar product operation: For $m(x)=\sum_{i=0}^n m_ix^i\in M[x;\sigma]$ and $f(x)=\sum_{j=0}^m a_jx^j\in R[x;\sigma]$ $$m(x)f(x)=\sum_{k=0}^{n+m}{\left(\sum_{k=i+j}m_i\sigma^i(a_j)\right)}x^k\eqno(*)$$ Similarly, $M[[x;\sigma]]$ is a module over $R[[x;\sigma]]$. The modules $M[x;\sigma]$ and $M[[x;\sigma]]$ are called the [*skew polynomial extension*]{} and the [*skew power series extension of $M$*]{}, respectively. If $\sigma\in Aut(R)$, then with a scalar product similar to $(*)$ , $M[x,x^{-1};\sigma]$ (respectively, $M[[x,x^{-1};\sigma]]$) becomes a module over $R[x,x^{-1};\sigma]$ (respectively, $R[[x,x^{-1};\sigma]]$). The modules $M[x,x^{-1};\sigma]$ and $M[[x,x^{-1};\sigma]]$ are called the [*skew Laurent polynomial extension*]{} and the [*skew Laurent power series extension*]{} of $M$, respectively. In [@zhang/chen], a module $M_R$ is called $\sigma$-[*skew Armendariz*]{}, if $m(x)f(x)=0$ where $m(x)=\sum_{i=0}^nm_ix^i\in M[x;\sigma]$ and $f(x)=\sum_{j=0}^ma_jx^j\in R[x;\sigma]$ implies $m_i\sigma^i(a_j)=0$ for all $i$ and $j$. According to Lee-Zhou [@lee/zhou], $M_R$ is called $\sigma$-[*Armendariz*]{}, if it is $\sigma$-compatible and $\sigma$-skew Armendariz. In this paper, we show that if $M_R$ is p.q.-Baer then so is $M[x;\sigma]_{R[x;\sigma]}$ whenever $M_R$ satisfies the condition $\mathcal{C}_2$, and the converse holds when $M_R$ satisfies the condition $\mathcal{C}_1$ (Proposition \[prop pqbaer\]). Also, if $M_R$ satisfies $\mathcal{C}_2$ and $\sigma$-skew Armendariz, then $M_R$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ ($\sigma\in Aut(R)$) is a p.p.-module (Proposition \[prop pp\]). As a consequence, if $M_R$ is semicommutative and $\sigma$-compatible then: $M_R$ is a p.p.-module $\Leftrightarrow$ $M_R$ is a p.q.-Baer module $\Leftrightarrow$ $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module $\Leftrightarrow$ $M[x;\sigma]_{R[x;\sigma]}$ is a p.q.-Baer module (Theorem \[theo2\]). Moreover, we obtain a generalization of some results in [@baser2007; @Baser2; @birk/OnpolyExt; @lee/zhou]. Skew polynomials over p.q.-Baer modules ======================================= We start with the next definition. \[df1\]Let $m\in M$ and $a\in R$. We say that $M_R$ satisfies the condition $\mathcal{C}_1$ $($respectively, $\mathcal{C}_2$$)$, if $ma=0$ implies $m\sigma(a)=0$ $($respectively, $m\sigma(a)=0$ implies $ma=0$$)$. Note that $M_R$ is $\sigma$-compatible if and only if it satisfies $\mathcal{C}_1$ and $\mathcal{C}_2$. Let $M_R$ be a module and $\sigma\in End(R)$. \[lemma idempo\]If $M_R$ satisfies $\mathcal{C}_1$ or $\mathcal{C}_2$, then $me=m\sigma(e)$ for any $m\in M$ and any $e^2=e\in R$. Suppose $\mathcal{C}_2$, from $m\sigma(e)(1-\sigma(e))=0$, we have $0=m\sigma(e)(1-e)=m\sigma(e)-m\sigma(e)e$, so $m\sigma(e)e=m\sigma(e)$. From $m(1-\sigma(e))\sigma(e)=0$, we have $0=m(1-\sigma(e))e=me-m\sigma(e)e$, so $m\sigma(e)=m\sigma(e)e=me$. The same for $\mathcal{C}_1$. \[prop pqbaer\] Let $M_R$ be a module and $\sigma\in End(R)$. $(1)$ If $M_R$ is a p.q.-Baer module then so is $M[x;\sigma]_{R[x;\sigma]}$, whenever $M_R$ satisfies the condition $\mathcal{C}_2$. $(2)$ If $M[x;\sigma]_{R[x;\sigma]}$ or $M[[x;\sigma]]_{R[[x;\sigma]]}$ is a p.q.-Baer module then so is $M_R$, whenever $M_R$ satisfies the condition $\mathcal{C}_1$. $(1)$ Let $m(x)=m_0+m_1x+\cdots+m_nx^n\in M[x;\sigma]$. Then $r_R(m_iR)=e_iR$, for some idempotents $e_i\in R\;(0\leq i\leq n)$. Let $e=e_0e_1\cdots e_n$, then $eR=\cap_{i=0}^nr_R(m_iR)$. We show that $r_{R[x;\sigma]}(m(x)R[x;\sigma])=eR[x;\sigma]$. Let $\phi(x)=a_0+a_1x+a_2x^2+\cdots+a_px^p\in r_{R[x;\sigma]}(m(x)R[x;\sigma])$. Since $m(x)R\phi(x)=0$, we have $m(x)b\phi(x)=0$ for all $b\in R$. Then $$m(x)b\phi(x)=\sum_{\ell=0}^{n+p}{\left(\sum_{\ell=i+j}m_i\sigma^i(ba_j)\right)}x^{\ell}=0.$$ - $\ell=0$ implies $m_0ba_0=0$ then $a_0\in r_R(m_0R)=e_0R$. - $\ell=1$ implies $$m_0ba_1+m_1\sigma(ba_0)=0\eqno(1)$$ Let $s\in R$ and take $b=se_0$, so $m_0se_0a_1+m_1\sigma(se_0a_0)=0$, since $m_0se_0=0$ we have $m_1\sigma(se_0a_0)=m_1\sigma(sa_0)=0$, so $m_1sa_0=0$, thus $a_0\in r_R(m_1R)=e_1R$. In equation $(1)$, $m_1\sigma(ba_0)=m_1\sigma(be_1a_0)=m_1\sigma(b)e_1\sigma(a_0)=0$, by Lemma \[lemma idempo\]. Then equation (1) gives $m_0ba_1=0$, so $a_1\in e_0R$. - $\ell=2$ implies $$m_0ba_2+m_1\sigma(ba_1)+m_2\sigma^2(ba_0)=0\eqno(2)$$ Let $s\in R$ and take $b=se_0e_1$, so $m_0se_0e_1a_2+m_1\sigma(s)e_0e_1\sigma(a_1)+m_2\sigma^2(se_0e_1a_0)=0$, but $m_0se_0e_1a_2=m_1\sigma(s)e_0e_1\sigma(a_1)=0$ we have $m_2\sigma^2(se_0e_1a_0)=0$, since $e_0e_1a_0=a_0$ we have $m_2\sigma^2(sa_0)=0$ and so $m_2sa_0=0$ for all $s\in R$. Hence $a_0\in e_2R$ (thus, $a_0\in e_0e_1e_2R$). Equation $(2)$, becomes $m_0ba_2+m_1\sigma(ba_1)+m_2\sigma^2(b)e_0e_1e_2\sigma^2(a_0)=0$, which gives $$m_0ba_2+m_1\sigma(ba_1)=0\eqno(2')$$ Take $b=se_0$ in equation $(2')$, we have $m_0se_0a_2+m_1\sigma(se_0a_1)=0$, but $m_0se_0a_2=0$ so $m_1\sigma(se_0a_1)=m_1\sigma(sa_1)=0$ and thus $m_1sa_1=0$, hence $a_1\in e_1R$ (so, $a_1\in e_0e_1R$). Equation $(2')$ gives $m_0ba_2=0$, so $a_2\in e_0R$. At this point, we have $a_0\in e_0e_1e_2R,\;a_1\in e_1e_2R$ and $a_2\in e_0R$. Continuing this procedure yields $a_i\in eR$ ($0\leq i\leq n$). Hence $\phi(x)\in eR[x;\sigma]$. Consequently, $r_{R[x;\sigma]}(m(x)R[x;\sigma])\subseteq eR[x;\sigma]$. Conversely, let $\varphi(x)=b_0+b_1x+b_2x^2+\cdots+b_px^p\in R[x;\sigma]$. Then $$m(x)\varphi(x)e=\sum_{\ell=0}^{n+p}{\left(\sum_{\ell=i+j}m_i\sigma^i(b_j)\sigma^{\ell}(e)\right)}x^{\ell}= \sum_{\ell=0}^{n+p}{\left(\sum_{\ell=i+j} m_i\sigma^i(b_j)e\right)}x^{\ell}.$$ Since $e\in\bigcap_{i=0}^nr_R(m_iR)$, then $m_iRe=0$ ($0\leq i\leq n$). Thus $m(x)\varphi(x)e=0$, hence $eR[x;\sigma]\subseteq r_{R[x;\sigma]}(m(x)R[x;\sigma])$. Thus $r_{R[x;\sigma]}(m(x)R[x;\sigma])=eR[x;\sigma]$, therefore $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer. $(2)$ Let $0\neq m\in M$. We have $r_{R[x;\sigma]}(mR[x;\sigma])=e{R[x;\sigma]}$ for some idempotent $e=\sum_{i=0}^ne_ix^i\in R[x;\sigma]$. We have $r_{R[x;\sigma]}(mR[x;\sigma])\cap R=e_0R$. On other hand, we show that $r_{R[x;\sigma]}(mR[x;\sigma])\cap R=r_R(mR)$. Let $a\in r_R(mR)$ then $mRa=0$, so $mR\sigma^i(a)=0$ for all $i\geq 1$. So $mR[x;\sigma]a=0$. Therefore $a\in r_{R[x;\sigma]}(mR[x;\sigma])\cap R$. Conversely, let $a\in r_{R[x;\sigma]}(mR[x;\sigma])\cap R$, then $mR[x;\sigma]a=0$, in particular $mRa=0$, so $a\in r_R(mR)$. Thus $a\in r_R(mR)=e_0R$, with $e_0^2=e_0\in R$. So $M_R$ is p.q.-Baer. The same method for $M[[x;\sigma]]$. \[cor pqbaer2\] $M_R$ is p.q.-Baer if and only if $M[x]_{R[x]}$ is p.q.-Baer. \[cor pqbaer1\] $R$ is right p.q.-Baer if and only if $R[x]$ is right p.q.-Baer. $M_R$ is called $\sigma$-[*reduced module*]{} by Lee-Zhou [@lee/zhou], if for any $m\in M$ and $a\in R$: $(1)$ $ma=0$ implies $mR\cap Ma=0$, $(2)$ $ma=0$ if and only if $m\sigma(a)=0$. \[cor pqbaer3\]Let $M_R$ a $\sigma$-compatible module. Then the following hold: $(1)$ If $M[x;\sigma]_{R[x;\sigma]}$ is a p.q.-Baer module then so is $M_R$. The converse holds if in addition $M_R$ is $\sigma$-reduced. $(2)$ If $M[[x;\sigma]]_{R[[x;\sigma]]}$ is a p.q.-Baer module then so is $M_R$. \[cor pqbaer4\]Let $M_R$ be a $\sigma$-compatible module. Then $M_R$ is p.q.-Baer if and only if $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer. Skew polynomials over p.p.-modules ================================== Let $M_R$ be an $\sigma$-Armendariz module, if $me=0$ where $e^2=e\in R$ and $m\in M$, then $mfe=0$ for any $f^2=f\in R$ (by [@lee/zhou Lemma 2.10]). This result still true if we replace the condition “$M_R$ is $\sigma$-Armendariz” by “$M_R$ is $\sigma$-skew Armendariz satisfying $\mathcal{C}_2$”. \[prop pp\]Let $M_R$ be a $\sigma$-skew Armendariz module which satisfies the condition $\mathcal{C}_2$. The following statements hold: $(1)$ $M_R$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module, $(2)$ Let $\sigma\in Aut(R)$, then $M_R$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ is a p.p.-module. $(1)$$(\Leftarrow)$ Is clear by [@lee/zhou Theorem 2.11]. $(\Rightarrow)$ Let $m(x)=m_0+m_1x+\cdots+m_nx^n\in M[x;\sigma]$, then $r_R(m_i)=e_iR$, for some idempotents $e_i\in R\;(0\leq i\leq n)$. Let $e=e_0e_1\cdots e_n$, then $m_ie=0$ for all $0\leq i\leq n$ ([@lee/zhou Lemma 2.10]) and by Lemma \[lemma idempo\], we have $m_i\sigma^j(e)=0$ for all $0\leq i\leq n$ and $j\geq 0$. Therefore $e\in r_{R[x;\sigma]}(m(x))$, so $eR[x;\sigma]\subseteq r_{R[x;\sigma]}(m(x))$. Conversely, let $\phi(x)=a_0+a_1x+\cdots+a_px^p\in r_{R[x;\sigma]}(m(x))$, then $m(x)\phi(x)=0$. Since $M_R$ is $\sigma$-skew Armendariz, we have $m_i\sigma^i(a_j)=0$ for all $i,j$ and with the condition $\mathcal{C}_2$ we have $m_ia_j=0$ for all $i,j$. So $a_j\in r_R(m_i)=e_iR$ for all $i,j$. Thus $a_j\in\cap_{i=0}^n r_R(m_i)=eR$ for each $j$. Then $\phi(x)\in e{R[x;\sigma]}$, therefore $r_{R[x;\sigma]}(m(x))=eR[x;\sigma]$. With the same method, we can prove $(2)$. If $M_R$ is $\sigma$-Armendariz. Then: $(1)$ $M_R$ is a p.p.-module if and only if $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module, $(2)$ Let $\sigma\in Aut(R)$, then $M_R$ is a p.p.-module if and only if $M[x,x^{-1};\sigma]_{R[x,x^{-1};\sigma]}$ is a p.p.-module. If $M_R$ is a semicommutative module such that, $m\sigma(a)a=0$ implies $m\sigma(a)=0$ for any $m\in M$ and $a\in R$. Then $M_R$ is $\sigma$-semicommutative and hence it satisfies the condition $\mathcal{C}_1$. To see this, suppose that $ma=0$ then $mRa=0$, in particular $mr\sigma(a)a=0$ for all $r\in R$. By the above condition, $mr\sigma(a)=0$ for all $r\in R$. Thus $M_R$ is $\sigma$-semicommutative. \[lemma1\]If $M_R$ is a semicommutative module such that $m\sigma(a)a=0$ implies $m\sigma(a)=0$ for any $m\in M$ and $a\in R$. Then $M_R$ is $\sigma$-skew Armendariz. Let $m(x)=m_0+m_1x+\cdots+m_nx^n\in M[x;\sigma]$ and $f(x)=a_0+a_1x+\cdots+a_px^p\in R[x;\sigma]$. From $m(x)f(x)=0$, we have $\sum_{i+j=k}m_i\sigma^i(a_j)=0$, for $0\leq k\leq n+p$. So, $m_0a_0=0$. Assume that $s\geq 0$ and $m_i\sigma^i(a_j)=0$ for all $i,j$ with $i+j\leq s$. Note that for $s+1$, we have $$m_0a_{s+1}+m_1\sigma(a_s)+\cdots+m_s\sigma^s(a_1)+a_{s+1}\sigma^{s+1}(a_0)=0\eqno(1)$$ Multiplying $(1)$ by $\sigma^s(a_0)$ from the right hand, we obtain $$m_0a_{s+1}\sigma^s(a_0)+m_1\sigma(a_s)\sigma^s(a_0)+\cdots+m_s\sigma^s(a_1)\sigma^s(a_0)+a_{s+1}\sigma^{s+1}(a_0) \sigma^s(a_0)=0,$$ we have $m_0a_0=0$, then $m_0\sigma^s(a_0)=0$ because $M_R$ is $\sigma$-semicommutative, and so $m_0a_{s+1}\sigma^s(a_0)=0$. Also, $m_1\sigma(a_0)=0$ then $m_1\sigma^s(a_0)=0$, thus $m_1\sigma(a_s)\sigma^s(a_0)=0$. Continuing this process until the step $s$, $m_s\sigma^s(a_0)=0$ then $m_s\sigma^s(a_1)$ $\sigma^s(a_0)=0$. Therefore $m_{s+1}\sigma^{s+1}(a_0)\sigma^s(a_0)=0$. But $$m_{s+1}\sigma^{s+1}(a_0)\sigma^s(a_0)=m_{s+1}\sigma[\sigma^{s}(a_0)]\sigma^s(a_0)=0.$$ So $m_{s+1}\sigma^{s+1}(a_0)=0$ . Therefore, equation $(1)$, becomes $$m_0a_{s+1}+m_1\sigma(a_s)+\cdots+m_s\sigma^s(a_1)=0\eqno(2)$$ Multiplying $(2)$, by $\sigma^{s-1}(a_1)$ from the right hand to obtain $m_s\sigma^s(a_1)=0$. Continuing this procedure yields $$m_0a_{s+1}=m_1\sigma(a_s)=\cdots=m_s\sigma^s(a_1)=a_{s+1}\sigma^{s+1}(a_0)=0.$$ A simple induction shows that $m_i\sigma^i(a_j)=0$, for all $i,j$. \[prop1\]Let $M_R$ be a module such that $m\sigma(a)a=0$ implies $m\sigma(a)=0$ for any $m\in M$ and $a\in R$. If $M_R$ is semicommutative then $M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are semicommutative. Let $m(x)=\sum_{i=0}^n m_ix^i\in M[x;\sigma]$, $f(x)=\sum_{j=0}^q a_jx^j\in R[x;\sigma]$ and $\phi(x)=\sum_{k=0}^p b_kx^k\in R[x;\sigma]$. Suppose that $m(x)f(x)=0$. The coefficients of $m(x)\phi(x)f(x)$ are of the form $$\sum_{u+v=w}m_u\sigma^u{\left(\sum_{i+j=v}b_i\sigma^i(a_j)\right)}=\sum_{u+v=w}{\left(\sum_{i+j=v}m_u\sigma^u(b_i)\sigma^{u+i}(a_j)\right)}.$$ By Lemma \[lemma1\], $m_u\sigma^u(a_j)=0$, for all $u,j$ and by $\mathcal{C}_1$, $m_u\sigma^{u+i}(a_j)=0$, for all $i,j,u$. Since $M_R$ is semicommutative then $m_u\sigma^u(b_i) \sigma^{u+i}(a_j)=0$, therefore $$\sum_{u+v=w}m_u\sigma^u{\left(\sum_{i+j=v}b_i\sigma^i(a_j)\right)}=0.$$ So $m(x)\phi(x)f(x)=0$, then $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative. The same for $M[[x;\sigma]]_{R[[x;\sigma]]}$. According to Baser and Harmanci [@baser2007], a module $M_R$ is [*reduced*]{} if for any $m\in M$ and $a\in R$, $ma^2=0$ implies $mR\cap Ma=0$. By [@Baser1 Lemma 2.11], if $M_R$ is semicommutative p.p. or semicommutative p.q.-Baer then it’s reduced. \[prop2\]Let $M_R$ be a semicommutative module satisfying the condition $\mathcal{C}_1$, if $M_R$ is p.q.-Baer or p.p. then $M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are semicommutative. Let $a\in R$ and $m\in M$ such that $m\sigma(a)a=0$, then $m(\sigma(a))^2=0$ (by $\mathcal{C}_1$), since $M_R$ is reduced we have $m\sigma(a)=0$. By Proposition \[prop1\], $M[x;\sigma]_{R[x;\sigma]}$ and $M[[x;\sigma]]_{R[[x;\sigma]]}$ are semicommutative. \[theo2\]If $M_R$ is semicommutative and $\sigma$-compatible. Then the following are equivalent: $(1)$ $M_R$ is p.p. $(2)$ $M_R$ is p.q.-Baer, $(3)$ $M[x;\sigma]_{R[x;\sigma]}$ is p.p., $(4)$ $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer, $(1)\Leftrightarrow (2)$ By [@Baser1 Proposition 2.7]. $(2)\Leftrightarrow (4)$ By Proposition \[prop pqbaer\]. $(3)\Rightarrow (4)$ Since $M_R$ is a p.p.-module, then Corollary \[prop2\] implies that $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative. Therefore $M[x;\sigma]_{R[x;\sigma]}$ is p.q.-Baer by [@Baser1 Proposition 2.7]. $(4)\Rightarrow (3)$ By Proposition \[prop pqbaer\], $M_R$ is p.q.-Baer, since $M_R$ is semicommutative then $M[x;\sigma]_{R[x;\sigma]}$ is semicommutative, and so $M[x;\sigma]_{R[x;\sigma]}$ is a p.p.-module. Let $M_R$ be a semicommutative module. Then the following are equivalent: $(1)$ $M_R$ is p.p. $(2)$ $M_R$ is p.q.-Baer, $(3)$ $M[x]_{R[x]}$ is p.p., $(4)$ $M[x]_{R[x]}$ is p.q.-Baer, If $M_R$ is a reduced module. Then the following are equivalent: $(1)$ $M_R$ is p.p. $(2)$ $M_R$ is p.q.-Baer, $(3)$ $M[x]_{R[x]}$ is p.p., $(4)$ $M[x]_{R[x]}$ is p.q.-Baer, Every reduced module is semicommutative by [@lee/zhou Lemma 1.2]. [20]{} S. Annin, [*Associated primes over skew polynomials rings*]{}, Comm. Algebra, [**30**]{} (2002), 2511-2528 M. Baser and N. Agayev, [*On reduced and semicommutative modules*]{}, Turk. J. Math., [**30**]{} (2006), 285-291. M. Baser and A. Harmanci, [*reduced and p.q.-Baer modules*]{}, Taiwanese J. Math., [**2**]{} (1) (2007), 267-275. M. Baser and A. Harmanci, [*On quasi-Baer and p.q.-Baer modules*]{}, Kyungpook Math. Journal, [**49**]{} (2009), 255-263. M. Baser and M.T. Kosan, [*On quasi-Armendariz modules*]{}, Taiwanese J. Math., [**12**]{} (3) (2008), 573-582. G.F. Birkenmeier, J.Y. Kim and J.K. Park, [*On polynomial extensions of principally quasi-Baer rings*]{}, Kyungpook Math. journal, [**40**]{} (2) (2000), 247-253. G.F. Birkenmeier, J.Y. Kim and J.K. Park, [*Principally quasi-Baer rings*]{}, Comm. Algebra, [**29**]{} (2) (2001), 639-660. A.M. Buhphang and M.B. Rege, [*semicommutative modules and Armendariz modules*]{}, Arab J. Math. Sciences, [**8**]{} (2002), 53-65. W.E. Clark, [*Twisted matrix units semigroup algebras*]{}, Duke Math. Soc., [**35**]{} (1967), 417-424. I. Kaplansky, [*Rings of operators*]{}, Math. Lecture Notes series, Benjamin, New York, 1965. T.K. Kwak, [*Extensions of extended symmetric rings*]{}, Bull. Korean Math. Soc., [**44**]{} (2007), 777-788. T. K. Lee and Y. Lee, [*Reduced Modules*]{}, Rings, modules, algebras and abelian groups, 365-377, Lecture Notes in Pure and App. Math., [**236**]{}, Dekker, New york, (2004). C.P Zhang and J.L. Chen, [*$\sigma$-skew Armendariz modules and $\sigma$-semicommutative modules*]{}, Taiwanese J. Math., [**12**]{} (2) (2008), 473-486.

--- author: - 'Mari Carmen Bañuls,' - 'Krzysztof Cichy,' - Karl Jansen - and Hana Saito bibliography: - 'MPSSchwinger.bib' title: Chiral condensate in the Schwinger model with Matrix Product Operators --- Introduction {#sec:intro} ============ Investigations of gauge field theories within the Hamiltonian approach have progressed substantially in the last years with the help of tensor network (TN) techniques [@verstraete08algo; @cirac09rg; @orus2014review]. Taking the example of the Schwinger model, numerical calculations have been performed to investigate ground state properties [@Byrnes:2002nv; @Cichy:2012rw; @Banuls:2013jaa; @Banuls:2013zva; @Rico:2013qya; @Buyens:2015dkc], to demonstrate real-time dynamics [@Buyens:2013yza; @Buyens:2014pga] and to address the phenomenon of string breaking [@Pichler:2015yqa; @Buyens:2015tea], which has also been explored in non-Abelian models [@Kuhn:2015zqa]. In Refs. [@Banuls:2015sta; @Saito:2014bda; @Saito:2015ryj], thermal properties of the Schwinger model were studied for massless fermions. From a more conceptual point of view, TN have been developed that incorporate the gauge symmetry by construction, and constitute ground states of gauge invariant lattice models [@Tagliacozzo:2014bta; @Silvi:2014pta; @haegeman15gauging; @zohar2015peps]. Yet a different line of work is the study of potential quantum simulations of these models, using ultracold atoms, see Refs. [@wiese2013review; @Zohar:2015hwa; @Dalmonte:2016alw] for a review. Also in this field, TN techniques can play a determinant role to study the feasibility of the proposals [@kuehn2014schwinger]. The last numerical developments go beyond standard Markov Chain Monte Carlo (MC-MC) methods. At zero temperature, the Hamiltonian approach allows us to go substantially closer to the continuum limit and reach a much improved accuracy compared to MC-MC. When temperature is switched on, a broad and very large set of non-zero temperature points can be evaluated, ranging from very high to almost zero temperature. In the string breaking calculation, a nice picture of the string breaking phenomenon and the emergence of the hadron states can be demonstrated. Finally, real-time simulations are not even possible in principle with MC-MC methods. The key to this success is the employment of tensor network states and, in the case of one spatial dimension, as for the Schwinger model, the Matrix Product States (MPS). In this approach, which is closely linked to the Density Matrix Renormalization Group (DMRG) [@white92dmrg], the problem, which has an exponentially large dimension in terms of the system size, is reduced to an –admittedly– sophisticated variational solution which can be encoded in substantially smaller $D\times D$ matrices. The ansatz can represent arbitrary states in the Hilbert space if $D$ is large enough (exponential in the system size). Instead in numerical applications, usually an approximation is found to the desired state within the set of MPS with fixed $D$. By varying $D$, an extrapolation of results to $D\to \infty$ can be performed allowing thus to reach the solution of the real system under consideration. A different approach also using tensor network techniques was applied to the Schwinger model with a topological $\theta$-term in Refs. [@Shimizu:2014uva; @Shimizu:2014fsa], where the exact partition function on the lattice was expressed as a two dimensional tensor network and approximately contracted using the Tensor Renormalization Group (TRG). The application of the MPS technique discussed in the present paper is concerned with non-zero temperature properties of the Schwinger model. In Refs. [@Banuls:2015sta; @Saito:2014bda; @Saito:2015ryj], we have for the first time investigated the thermal evolution of the chiral condensate in the Schwinger model. In the first paper, where we only studied the massless case, we could demonstrate that the MPS technique can be successfully used to compute such a thermal evolution from very high to almost zero temperature. For massless fermions, the results from our MPS calculation could be confronted with the analytical solution of Ref. [@Sachs:1991en] and a very nice agreement was found demonstrating the correctness and the power of the MPS approach. In the present paper, we will extend our calculations of the thermal evolution of the chiral condensate to the case of non-vanishing fermion masses. Here, no exact results exist anymore, but only approximate solutions are available [@Hosotani:1998za] which can be tested against our results. For our work at zero fermion mass, we also introduced a truncation of the charge sector [@Banuls:2015sta] which was necessary to obtain precise results at high temperature. Here, we will employ this truncation method, too. It needs to be stressed that the calculations with MPS, as performed here, have a number of systematic uncertainties which are very important to control. This concerns in particular: - an estimate of results for infinite bond dimension; [^1] - an extrapolation to zero step size in the thermal evolution process; - a study of the truncation in the charge sector of the model; - an infinite volume extrapolation; - and a careful analysis of the continuum limit employing various extrapolation functions with different orders in the lattice spacing. Controlling these systematic effects renders the calculations with MPS demanding, but it is absolutely necessary to obtain precise and trustworthy results. We have therefore made a significant effort to perform the above extrapolations and we will provide various examples in this paper for the studies of systematic effects carried through here. The Schwinger model and chiral symmetry breaking {#sec:schwinger} ================================================ The one-flavour Schwinger model [@schwinger62], i.e. Quantum Electrodynamics in 1+1 dimensions, is one of the simplest gauge theories and a toy model allowing for studies of new lattice techniques before employing them to real theories of interest, like Quantum Chromodynamics (QCD). Despite its apparent simplicity, it has a non-perturbatively generated mass gap and shares some features with QCD, such as confinement and chiral symmetry breaking, although the mechanism of the latter is different than in QCD – it is not spontaneous, but results from the chiral anomaly. We start with the Hamiltonian of the Schwinger model in the staggered discretization, derived and discussed in Ref. [@Banks:1975gq]: $$\begin{aligned} \label{eq:H} H &=& x \displaystyle \sum_{n=0}^{N-2} \left[ \sigma_n^+ \sigma_{n+1}^- + \sigma_n^- \sigma_{n+1}^+ \right] +\frac{\mu}{2} \sum_{n=0}^{N-1} \Big[ 1+ (-1)^n \sigma_n^z \Big] + \sum_{n=0}^{N-2} \left[ L(n) \right] ^2\\ &\equiv& H_{hop} + H_m + H_g,\nonumber \end{aligned}$$ where $n$ is the site index, $x=1/g^2a^2$, $a$ is the lattice spacing, $g$ is the coupling, and $\mu=2m/g^2a$ with $m$ denoting the fermion mass and $N$ the number of lattice sites. We use open boundary conditions (OBC). The gauge field, $L(n)$, can be integrated out using the Gauss law: $$L(n+1) - L(n) = \frac{1}{2} \left[ (-1)^{n+1} + \sigma_{n+1}^z \right]. \label{eq:Gausslaw}$$ Thus, only $L(n)$ at one of the boundaries is an independent parameter and we take $L(0)=0$, i.e. no background electric field. We work with the following basis for our numerical computations: $\left| s_0 s_1 \cdots \right\rangle$  [@Banuls:2013jaa], where $s_n=\{\downarrow,\uparrow\}$ is the spin state at site $n$ and all the gauge degrees of freedom have been integrated out. In this paper, we are interested in the chiral symmetry breaking ($\chi$SB) in the Schwinger model, both at zero and non-zero temperature. The order parameter of $\chi$SB is the chiral condensate $\Sigma=\left\langle {\bar \psi}\psi \right\rangle$, which can be written in terms of spin operators as $\Sigma = \frac{g\sqrt{x}}{N} \sum_n (-1)^n \frac{1+\sigma_n^z}{2}$. The ground state and thermal expectation values of the chiral condensate diverge logarithmically in the continuum limit for non-zero fermion mass [@deForcrand98; @duerr05scaling; @Christian:2005yp]. This divergence is present even in the non-interacting case, where the theory is exactly solvable and the Hamiltonian (\[eq:H\]) reduces to the XY spin model in a staggered magnetic field. The ground state energy of this model (with OBC) reads: $\frac{E_0}{g}=\frac{\mu}{2}N-\sum_{q=1}^{N/2} \sqrt{\mu^2+4 x^2 \cos^2\frac{q \pi}{N+1}}$. The ground state expectation value of $\Sigma$ can then be computed from the derivative $\frac{d E_0}{d\mu}$: $$\Sigma_{\mathrm{free}}(\mu,x,N)=\frac{g\sqrt{x}}{N}\sum_{q=1}^{N/2}\frac{\mu}{\sqrt{\mu^2+4 x^2 \cos^2 \frac{q \pi}{N+1}}}. \label{eq:freecond}$$ The free condensate value computed from this formula can be used to subtract the divergence in the interacting case at a finite lattice size $N$, a finite lattice spacing $1/\sqrt{x}$ and a given fermion mass $m/g$. However, one can exactly evaluate the infinite volume limit of the free condensate first, yielding: $$\Sigma_{\mathrm{free}}(m/g,x)=\frac{m}{\pi}\frac{1}{\sqrt{1+\frac{m^2}{g^2 x}}} \mathrm{K}\left(\frac{1}{1+\frac{m^2}{g^2x}}\right), \label{eq:freecondbulk}$$ where $\mathrm{K}(u)$ is the complete elliptic integral of the first kind [@abramowitz]. Note that by expanding this expression in the limit $x\to\infty$, the divergent logarithmic term $\frac{1}{2\pi}\frac{m}{g}\log x$ is indeed seen already in the free case. In this way, we can extrapolate our lattice interacting condensate $\Sigma(m/g,x,N)$ first to infinite volume limit, $\Sigma(m/g,x)$, at a finite $x$ and a given $m/g$ and then subtract the infinite volume free condensate ($\Sigma_{\mathrm{free}}(m/g,x)$) given by Eq. (\[eq:freecondbulk\]): $$\label{eq:subtr} \Sigma_{\rm subtr}(m/g,x)=\Sigma(m/g,x)-\Sigma_{\mathrm{free}}(m/g,x),$$ obtaining finally the subtracted condensate $\Sigma_{\rm subtr}(m/g,x)$, which can then be extrapolated to the continuum limit $x\rightarrow\infty$. Note that a non-zero temperature does not bring any further divergence, hence the above renormalization scheme, subtracting the zero temperature free condensate in the infinite volume limit, can be applied for any $T$. Actually, one can equivalently subtract the free condensate at any finite $T$. This defines an alternative renormalization scheme that we can also implement. Both options would lead to the correct value at $T=0$, i.e. compatible with the one directly obtained from the ground state calculations, but in order to compare to other results in the literature, we adopt in the following the $T=0$ renormalization scheme for all temperatures. In the massless case, the temperature dependence of the chiral condensate was computed analytically by Sachs and Wipf [@Sachs:1991en]: $$\begin{aligned} \left\langle {\bar \psi}\psi \right\rangle &=& \frac{m_{\gamma}}{2\pi} e^{\gamma} e^{2I(m_{\gamma}/T)} = \left\{ \begin{array}{cl} \frac{m_{\gamma}}{2\pi} e^{\gamma} & \hspace{1cm} {\rm for} \hspace{1cm} T\rightarrow 0 \\ 2T e^{-\pi T/m_{\gamma}} & \hspace{1cm} {\rm for} \hspace{1cm} T\rightarrow \infty, \end{array} \right.\end{aligned}$$ where $I(x) = \int_0^{\infty} \frac{dt}{1-e^{x\cosh(t)}}$, $\gamma\approx0.577216$ is the Euler-Mascheroni constant and $m_{\gamma}=g/\sqrt{\pi}$ is the non-perturbatively generated mass of the lowest lying boson (the vector boson). According to the above formula, chiral symmetry is broken at any finite temperature (zero or non-zero) and it gets restored ($\Sigma=0$) only at infinite temperature. There is no phase transition, i.e. chiral symmetry restoration is smooth. In the massive case, there is no analytical formula describing the temperature dependence of the condensate. However, the massive model was treated by Hosotani and Rodriguez with a generalized Hartree-Fock approach in Ref. [@Hosotani:1998za], yielding an approximate thermal dependence of $\Sigma$. In the following, we will confront our results with ones from this approximation and thus conclude about its validity. Tensor network approach {#sec:TN} ======================= In this work, we make use of two different applications of tensor network ansatzes. In order to obtain the results at zero temperature, we approximate variationally the ground state of the Schwinger model Hamiltonian (\[eq:H\]) on a finite lattice using a MPS. For the temperature dependence, we employ the matrix product operator (MPO) to describe the thermal equilibrium states at finite temperatures. Although the details of these ansatzes and the basic algorithms involved can be found in the literature, for completeness we compile in this section the fundamental ideas of both approaches, with special emphasis on the particularities associated to the problem at hand. Given a system of $N$ sites, a MPS [@vidal03eff; @verstraete04dmrg; @schollwoeck11age] is a state of the form $$|\Psi\rangle = \sum_{i_0,\ldots i_{N-1}=0}^{d-1} \mathrm{tr}(A_0^{i_0}\ldots A_{N-1}^{i_{N-1}}) |i_0 \ldots i_{N-1}\rangle, \label{eq:MPS}$$ where $d$ is the dimension of the local Hilbert space for each site. For two-level quantum systems, as in the case we are studying, $d=2$. The state is parametrized by the $dN$ matrices, $A_k^{i}$, which have dimension $D\times D$, except for the ones at the edges, $A_0^{i}$ and $A_{N-1}^{i}$, which, for the open boundary conditions we consider, are $D$-dimensional vectors. The parameter $D$ is called the bond dimension, and determines the number of variational parameters in the ansatz. The MPS can efficiently approximate ground states of local gapped Hamiltonians in one spatial dimension, and the ansatz lies at the basis of the success of the Density Matrix Renormalization Group (DMRG) method [@white92dmrg; @schollwoeck11age]. In practice, they have been successfully applied to much more general problems, including long range interactions and two dimensional systems. Different algorithms exist to find an MPS approximation to the ground state of a certain Hamiltonian. We use a variational search [@verstraete04dmrg; @schollwoeck11age], in which the energy is minimized over the set of MPS with a given bond dimension, $D$, by successively optimizing over one of the tensors, while keeping the rest fixed. The procedure is repeated, while sweeping over all the tensors, until convergence is attained in the value of the energy, to a certain relative precision, $\varepsilon_{\mathrm{tol}}$, ultimately limited by machine precision. The computational cost of this procedure scales as $\mathcal{O}(d D^3)$ with the dimensions of the tensors. The effect of running the algorithm with a limited bond dimension is to suffer a truncation error. By running the algorithm with increasing values of $D$, we can estimate the magnitude of this error and extrapolate to the $1/D \to 0$ limit, as discussed in detail in Sec. \[sec:results\]. Our previous works [@Banuls:2013jaa; @Banuls:2013zva] demonstrated that the MPS ansatz provides very good approximations to the ground state and lower lying excited states of the Schwinger model. The MPS ansatz can be extended to the description of operators, and in particular density matrices [@verstraete04mpdo; @zwolak04mpo; @pirvu10mpo]. A matrix product operator (MPO) is thus of the form $$\begin{aligned} \rho &=& \sum_{\{i_k,j_k\}}{\rm Tr} \left( M[0]^{i_0j_0} \cdots M[N-1]^{i_{N-1}j_{N-1}} \right) |i_0\ldots i_{N-1}\rangle\langle j_0\ldots j_{N-1}|. \label{eq:MPO}\end{aligned}$$ While any MPS (\[eq:MPS\]) can represent a valid physical state, as far as it is normalized, in order to describe a physical density operator, the MPO needs in addition to be positive. This condition cannot be guaranteed locally for generic tensors $M[k]$. However, it is possible to ensure the positivity of a MPO using the *purification ansatz* [@verstraete04mpdo; @delascuevas2013], in which each tensor of the MPO has the form $M[k]_{{\tilde \ell} {\tilde{r}}}^{ij}=\sum_p {A_{\ell' r'}^{ip}}^* A_{\ell r}^{jp}$. This corresponds to a (pure state) MPS ansatz for an extended chain, with one ancillary system per site, such that $\rho$ is the reduced state for the original system, obtained by tracing out the ancillas. It has been shown that thermal equilibrium states of local Hamiltonians can be well approximated by this kind of ansatz [@hastings06gapped; @molnar15gibbs] in arbitrary dimensions. In the case of finite temperature, a MPO approximation can be constructed for the Gibbs state via imaginary time evolution of the identity operator [@verstraete04mpdo], $\rho(\beta)\propto e^{-\beta H}=e^{-\frac{\beta}{2} H} \Id e^{-\frac{\beta}{2} H}$, where $\beta\equiv1/T$ is the inverse temperature. To achieve this, we apply a second order Suzuki-Trotter expansion [@trotter59; @suzuki90] to the exponential, and approximate every step of width $\delta=\beta/M$ by a product of five terms, $$\begin{aligned} e^{-\beta {H}}\approx \left[ e^{- \frac{\delta}{2} {H}_e} e^{- \frac{\delta}{2} {H}_z} e^{-\delta {H}_o} e^{- \frac{\delta}{2} {H}_z} e^{-\frac{\delta}{2} {H}_e} \right]^M, \label{eq:trotter}\end{aligned}$$ where $H_z=H_m+H_g$ is diagonal in the $z$ basis, and the hopping term is split in two sums $H_{hop}=H_e+H_o$, with the $H_{e}$ ($H_{o}$) term containing the two-body terms that act on each even-odd (odd-even) pair of sites. If each of the exponential terms can be exactly computed, the error of this approximation scales as $O(\delta^2)$. The exponentials of $H_e$ and $H_o$ have indeed an exact MPO expression with constant bond dimension $4$. The term $H_z$ contains long range interactions, but its structure allows us to also write it exactly as a MPO, with bond dimension $(N+1)$, as detailed in Ref. [@Banuls:2015sta]. The only non-vanishing elements of the tensors specifying the MPO are $(M_n^{ii})_{L_{n-1} L_n}=e^{-\delta h_n}$, for $L_{n}=L_{n-1}+\frac{1}{2}[(-1)^n+(\sigma_n^z)_{ii}]$, where $h_n=\frac{\mu}{2} \left[ 1+ (-1)^n \sigma_n^z \right] +L_n^2$ for $n<N-1$, and $h_{N-1}=\frac{\mu}{2} \left[ 1+ (-1)^{N-1} \sigma_{N-1}^z \right]$. The virtual bond then carries the information about the electric flux on each link, which can assume values $L_n\in[-N/2,N/2]$. Instead of working with the exact exponential of $H_z$, which has a bond dimension $N+1$, we find it convenient, given the large system sizes we want to study, to truncate the dimension of the MPO, by defining a maximum value the virtual bond can attain, $|L_n| \leq L_{\mathrm{cut}}$. This is equivalent to truncating the physical space to those states where the electric flux on a link cannot exceed $L_{\mathrm{cut}}$ and is thus related to approaches where one explicitly truncates the maximum allowed occupation number of the bosonic gauge degrees of freedom [@Buyens:2013yza]. Starting with the identity operator, $\rho(0)$, which has a trivial expression as a MPO with bond dimension one, we successively apply steps of the evolution, using the approximation above, and approximate the result by a MPO with the desired maximum bond dimension. This is achieved with the help of a Choi isomorphism [@choi], $|i\rangle\langle j| \rightarrow |i\rangle\otimes |j\rangle$, to vectorize the density operators, such that the MPO is transformed in a MPS, [with physical dimension per site $d^2$]{}, on which the evolution steps act linearly. The approximated effect of the evolution is then found by minimizing the Euclidean distance between the original and final MPS. The procedure can be repeated until inverse temperature $\beta/2$ is reached. Then we construct $\rho(\beta)\propto\rho(\beta/2)^{\dagger} \rho(\beta/2)$ (up to normalization) such that the purification ansatz is realized and we ensure a positive thermal equilibrium state. The computational cost of this calculation is the same as that of time evolution of a MPS state, with the increased physical dimension, i.e. it scales as $\mathcal{O}(d^2D^3)$. Using the MPO ansatz with limited bond dimension induces also a truncation error in the $T>0$ case, which is not equivalent to the one described for $T=0$. First of all, different ansatzes are used for both cases, and while the MPS truncation in the pure state case can be related to the entanglement in the state, the same is not true for the MPO ansatz in the case of mixed states. [^2] Moreover, the distinct numerical algorithms used in both cases also mean that errors are introduced in different ways. In the thermal algorithm, each application of one of the exponential factors in (\[eq:trotter\]) potentially increases the bond dimension of the resulting MPO. Hence, after every step, the ansatz needs to be truncated to the maximum desired value of the bond dimension. In practice, this is achieved by minimizing a cost function that corresponds to the Frobenius norm of the difference to the true evolved operator. As in the ground state search, this optimization is done by an alternating least squares (ALS) scheme, in which all tensors but one are fixed, and repeated sweeping is performed over the chain. Also in this case, we use a tolerance parameter, $\varepsilon_{\mathrm{tol}}$, to decide about the convergence of the iteration, but now the value bounds the relative change in the cost function during the sweeping that follows the application of each single exponential factor. This procedure leads to errors accumulating along the thermal evolution, and while at $\beta=0$ the state can be exactly written as a MPO with $D=1$, the largest truncation errors will occur for the lowest temperatures. Thus, recovering zero temperature results from such a procedure is a non-trivial check that the method is working correctly. The $T=0$ calculation, in contrast, does not suffer from this effect, as it directly targets the ground state variationally. Additionally, the Suzuki-Trotter expansion (\[eq:trotter\]) introduces another systematic error in the thermal evolution, by using a finite step width $\delta$, which we need to extrapolate to $\delta \to 0$, and another one in the form of the truncation of the physical subspace to a maximum $L_{\mathrm{cut}}$, described above. All these factors need to be taken into account when performing the extrapolations required to extract the continuum values of the observables under study (see Sec. \[sec:results\] for details). Results {#sec:results} ======= Zero temperature {#sec:zero} ---------------- We begin with our results for the ground state chiral condensate for various fermion masses. For the massless case, an analytical result can be obtained, $\Sigma/g=e^\gamma/2\pi^{3/2}\approx0.159929$. We are able to reproduce this number with great accuracy and also obtain results in the massive case, where no analytical results exist. Our numerical procedure consists in computing several sets of data points corresponding to different values of ($D$, $N$, $x$) and extrapolating in the way described below.\ **Infinite bond dimension ($D\rightarrow\infty$) extrapolation.** We use several values of $D\in[40,160]$ to check the effects from changing the bond dimension. Our final value is taken as the condensate corresponding to the largest computed value of $D=160$ and its error as the difference between the value for $D=160$ and $D=140$. The lower values of $D$ serve to ensure that the two highest bond dimensions are large enough, such that it can be argued that the difference between $D=\infty$ and $D=160$ is smaller than the one between $D=160$ and $D=140$, which makes our error estimate valid. ![\[fig:D200\]Examples of the $D$-dependence of the ground state chiral condensate for $m/g=0.125$, $x=200$ and five system sizes (left). The right plot shows a zoom into the region $D\in[80,160]$ for $N=368$. The red band represents the uncertainty related to the bond dimension, taken as explained in the text.](cond_mg0.125_x200_N.ps "fig:"){width="34.50000%"} ![\[fig:D200\]Examples of the $D$-dependence of the ground state chiral condensate for $m/g=0.125$, $x=200$ and five system sizes (left). The right plot shows a zoom into the region $D\in[80,160]$ for $N=368$. The red band represents the uncertainty related to the bond dimension, taken as explained in the text.](cond_mg0.125_x200_N368.ps "fig:"){width="34.50000%"} ![\[fig:D10\]Examples of the $D$-dependence of the ground state chiral condensate for $m/g=0.125$, $x=10$ and five system sizes (left). The right plot shows a zoom into the region $D\in[80,160]$ for $N=84$. See comments in the text about the irregular approach to the $1/D=0$ limit. The red band represents the uncertainty related to the bond dimension, taken as explained in the text. ](cond_mg0.125_x10_N.ps "fig:"){width="34.50000%"} ![\[fig:D10\]Examples of the $D$-dependence of the ground state chiral condensate for $m/g=0.125$, $x=10$ and five system sizes (left). The right plot shows a zoom into the region $D\in[80,160]$ for $N=84$. See comments in the text about the irregular approach to the $1/D=0$ limit. The red band represents the uncertainty related to the bond dimension, taken as explained in the text. ](cond_mg0.125_x10_N84.ps "fig:"){width="34.50000%"} A typical example of such extrapolation is shown in Fig. \[fig:D200\] for $x=200$ and in Fig. \[fig:D10\] for $x=10$, at $m/g=0.125$. In both cases, we observe very good convergence towards the $1/D=0$ limit, with the above defined error from this step being of $\mathcal{O}(10^{-9})$ for the former and $\mathcal{O}(10^{-12})$ for the latter. This error is represented by a red band. Note that despite going to $D=160$, the convergence in bond dimension is so good that actually even with $D=40$ we would already obtain the result with an outstanding precision, of $\mathcal{O}(10^{-8})$ for $x=200$ (i.e. only an order of magnitude worse than with $D=160$) or even of $\mathcal{O}(10^{-12})$ for $x=10$ (i.e. the same as with $D=160$). The $x=10$ case illustrates that in some cases the convergence in $D$ is so good that our uncertainty comes from issues with the numerical precision. The MPS optimization procedure is considered to be converged when the relative change in the ground state energy in subsequent sweeps falls below a certain tolerance parameter, taken to be $\varepsilon_{\mathrm{tol}}=10^{-12}$ in our case. Notice, however, that this precision refers to the ground state energy, which typically converges better than other observables, so it will correspond to a somewhat worse precision in the chiral condensate, which we estimate to be in the $10^{-11}-10^{-10}$ region. In the $x=10$ case, the variation of $\Sigma/g$ values for different $D$ becomes smaller than this, which explains the irregular behaviour of the $D$-dependence for this case (left plot of Fig. \[fig:D10\]), compared to the apparently regular convergence for the case $x=200$. We account for this bias (that happens only for our smallest $x$ values) in our next step, the infinite volume extrapolation. We emphasize that this is definitely not a drawback of the method, but even better precision could be attained for certain parameter ranges with the same $D$ values, by adopting a more demanding convergence criterion. On the other hand, since the ultimate limit of machine precision, which we label by $\epsilon_{\mathrm{num}}$, affects the optimization of individual tensors, so that after one sweep over the whole chain, it may affect the value of the energy in $\mathcal{O}(N \epsilon_{\mathrm{num}}).$ This means that for chains of hundreds of sites, as required for the largest values of $x$ we explore, $\varepsilon_{\mathrm{tol}}=10^{-12}$ is the best allowed by double precision numerics. ![\[fig:N0\]Examples of the infinite volume extrapolations of the $T=0$ chiral condensate for $x=10$ (left) and $x=200$, at $m/g=0.125$. Lines are fits of Eq. (\[eq:fitN\]). The points have error bars, but they are too small to be seen. ](cond_mg0.125_x10.ps "fig:"){width="34.50000%"} ![\[fig:N0\]Examples of the infinite volume extrapolations of the $T=0$ chiral condensate for $x=10$ (left) and $x=200$, at $m/g=0.125$. Lines are fits of Eq. (\[eq:fitN\]). The points have error bars, but they are too small to be seen. ](cond_mg0.125_x200.ps "fig:"){width="34.50000%"} **Infinite volume ($N\rightarrow\infty$) extrapolation.** The results corresponding to our estimates of the $D\rightarrow\infty$ limit can then be extrapolated to infinite volume by using a linear fitting ansatz: $$\label{eq:fitN} \Sigma(m/g,x,N)=\Sigma(m/g,x)+\frac{\alpha(m/g,x)}{N},$$ where $\Sigma(m/g,x,N)$ is the infinite-$D$ condensate for a fixed fermion mass, volume and lattice spacing. The fitting parameters are $\Sigma(m/g,x)$ (infinite volume condensate at a given lattice spacing and fermion mass) and the mass and lattice spacing-dependent slope of the finite volume $1/N$ correction, $\alpha(m/g,x)$. We show an example of such extrapolation in Fig. \[fig:N0\], again for $x=10$ (left) and $x=200$ (right), at $m/g=0.125$. We always choose the volumes to be large enough, such that the above linear fitting ansatz yields a good description of data. We have found that this holds when the volumes used are scaled proportionally to $\sqrt{x}$ and we take $N=\{22\sqrt{x},\,26\sqrt{x},\,30\sqrt{x},\,32\sqrt{x},\,34\sqrt{x}\}$. Indeed, in all cases where no issues with machine precision are observed, this leads to very good fits. The resulting error of the fitting coefficient $\Sigma(m/g,x)$ is the propagated error from the $D$-extrapolation. For very small values of $x$ (lower than approx. 30), we need to deal with the numerical precision bias. The errors from the $D$-extrapolation are in such case underestimated, since they do not take into account the finite numerical precision. This leads to $\chi^2/{\rm dof}$ values of $\mathcal{O}(10-100)$. However, we know from the analysis for large values of $x$ that the linear fitting ansatz (\[eq:fitN\]) yields an excellent description of data, with $\chi^2/{\rm dof}$ usually much smaller than 1. Hence, we account for the bias by inflating the $D$-extrapolation errors to such levels that $\chi^2/{\rm dof}=1$ by construction. In this way, the final error after the infinite volume extrapolation step is properly rescaled and becomes comparable to the one at larger $x$ (e.g. approx. $8.2\cdot10^{-10}$ for $x=10$ and $6.3\cdot10^{-9}$ for $x=200$). In the end, all our errors of infinite volume condensates, $\Sigma(m/g,x)$, differ by less than an order of magnitude in the whole considered range of $x$ and for all fermion masses.\ **Continuum limit ($x\rightarrow\infty$) extrapolation.** Finally, the infinite volume results $\Sigma(m/g,x)$ can be extrapolated to the continuum limit. First, we subtract the infinite volume free condensate according to Eq. (\[eq:subtr\]), obtaining the subtracted condensate $\Sigma_{\rm subtr}(m/g,x)$. Then, we apply the following fitting ansatz: $$\label{eq:cont} \Sigma_{\rm subtr}(m/g,x)=\Sigma_{\rm subtr}(m/g)+\frac{a(m/g)}{\sqrt{x}}\log(x)+\frac{b(m/g)}{\sqrt{x}}+\frac{c(m/g)}{x},$$ with fitting parameters $\Sigma_{\rm subtr}(m/g)$ (the continuum condensate for a given fermion mass), $a(m/g)$, $b(m/g)$ and $c(m/g)$. This is a fitting ansatz quadratic in the lattice spacing (the role of the lattice spacing is played by $1/\sqrt{x}$), with logarithmic corrections. The latter appear already in the free theory, where their presence can be shown analytically (see Sec. \[sec:schwinger\]). Note that the final result obtained from this procedure will, to some extent, depend on the fitting range in $1/\sqrt{x}$. To quote final values independent from such choices, we adopt a systematic procedure analogous to the one we used in our spectrum investigation in the Schwinger model, described in detail in the appendix of Ref. [@Banuls:2013jaa]. In short, this consists in performing fits in different possible fitting ranges by varying the minimal and maximal values of $x$ entering the fits. The number of fits that we obtain in this way is of $\mathcal{O}(100)$ and allows us to build a distribution of the continuum values, weighted with $\exp(-\chi^2/{\rm dof})$ of the fits. The final value that we quote is the median of the distribution and the systematic error from the choice of the fitting range comes from the 68.3% confidence interval (such that in the limit of infinite number of fits it corresponds to the width of a resulting Gaussian distribution). This error is then combined in quadrature with our propagated error from $D$- and $N$-extrapolations, which we take as the error of one selected fit, taken to be the one in the interval $x\in[20,600]$. ![\[fig:cont0\]Continuum limit extrapolations of the $T=0$ chiral condensate for all our fermion masses. Shown are the data for the subtracted infinite volume condensate and the fits of Eq. (\[eq:cont\]) in the interval $1/\sqrt{x}\in[0,0.23]$ ($x\in[20,600]$, i.e. left of the dashed line). These fits are example fits that enter our procedure of extraction of the systematic uncertainty related to the choice of the fitting range. Note that the rightmost two points are not included in the fit, but are still rather well described by it. ](cond_m0_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont0\]Continuum limit extrapolations of the $T=0$ chiral condensate for all our fermion masses. Shown are the data for the subtracted infinite volume condensate and the fits of Eq. (\[eq:cont\]) in the interval $1/\sqrt{x}\in[0,0.23]$ ($x\in[20,600]$, i.e. left of the dashed line). These fits are example fits that enter our procedure of extraction of the systematic uncertainty related to the choice of the fitting range. Note that the rightmost two points are not included in the fit, but are still rather well described by it. ](cond_m0.0625_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont0\]Continuum limit extrapolations of the $T=0$ chiral condensate for all our fermion masses. Shown are the data for the subtracted infinite volume condensate and the fits of Eq. (\[eq:cont\]) in the interval $1/\sqrt{x}\in[0,0.23]$ ($x\in[20,600]$, i.e. left of the dashed line). These fits are example fits that enter our procedure of extraction of the systematic uncertainty related to the choice of the fitting range. Note that the rightmost two points are not included in the fit, but are still rather well described by it. ](cond_m0.125_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont0\]Continuum limit extrapolations of the $T=0$ chiral condensate for all our fermion masses. Shown are the data for the subtracted infinite volume condensate and the fits of Eq. (\[eq:cont\]) in the interval $1/\sqrt{x}\in[0,0.23]$ ($x\in[20,600]$, i.e. left of the dashed line). These fits are example fits that enter our procedure of extraction of the systematic uncertainty related to the choice of the fitting range. Note that the rightmost two points are not included in the fit, but are still rather well described by it. ](cond_m0.25_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont0\]Continuum limit extrapolations of the $T=0$ chiral condensate for all our fermion masses. Shown are the data for the subtracted infinite volume condensate and the fits of Eq. (\[eq:cont\]) in the interval $1/\sqrt{x}\in[0,0.23]$ ($x\in[20,600]$, i.e. left of the dashed line). These fits are example fits that enter our procedure of extraction of the systematic uncertainty related to the choice of the fitting range. Note that the rightmost two points are not included in the fit, but are still rather well described by it. ](cond_m0.5_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont0\]Continuum limit extrapolations of the $T=0$ chiral condensate for all our fermion masses. Shown are the data for the subtracted infinite volume condensate and the fits of Eq. (\[eq:cont\]) in the interval $1/\sqrt{x}\in[0,0.23]$ ($x\in[20,600]$, i.e. left of the dashed line). These fits are example fits that enter our procedure of extraction of the systematic uncertainty related to the choice of the fitting range. Note that the rightmost two points are not included in the fit, but are still rather well described by it. ](cond_m1.0_paper.ps "fig:"){width="34.50000%"} ------------- ------------------ ------------------------------ ------------------------------------ \*[$m/g$]{} \*[Our result]{} \*[Ref. [@Buyens:2014pga]]{} Exact ($m=0$) or Ref. [@Hosotani:1998za] ($m>0$) 0 0.159929(7) 0.159929(1) 0.159929 0.0625 0.1139657(8) – 0.1314 0.125 0.0920205(5) 0.092019(2) 0.1088 0.25 0.0666457(3) 0.066647(4) 0.0775 0.5 0.0423492(20) 0.042349(2) 0.0464 1.0 0.0238535(28) 0.023851(8) 0.0247 ------------- ------------------ ------------------------------ ------------------------------------ : \[tab:T0\] Final continuum values of the $T=0$ chiral condensate (in units of $g$) for the used fermion masses. We compare with results from Ref. [@Buyens:2014pga] and with the analytical result in the massless case or the approximated result from Ref. [@Hosotani:1998za] in the massive case. Our continuum limit extrapolations are shown in Fig. \[fig:cont0\] for all fermion masses that we considered. We show in these plots the fit from which we estimated our propagated error from earlier extrapolations ($x\in[20,600]$), i.e. one of the fits that enter the distribution built to assess our final values and their uncertainties. The final values for each fermion mass are summarized in Tab. \[tab:T0\]. We compare to the result of a similar calculation in Ref. [@Buyens:2014pga] and to the exact result in the massless case or the approximation of Ref. [@Hosotani:1998za]. For the former, we observe perfect agreement, which is quite remarkable given the precision of both results being at the $\mathcal{O}(0.01\%-0.001\%)$ level. Similarly good is the agreement with the analytical result at $m=0$. We will comment more on the agreement with Ref. [@Hosotani:1998za] in the next subsection. Thermal evolution {#sec:thermal} ----------------- In our previous papers [@Saito:2014bda; @Saito:2015ryj; @Banuls:2015sta], we showed results for the temperature dependence of the chiral condensate in the massless case. We employed a method without any truncations in the gauge sector and found that it is numerically very demanding to achieve lattice spacings small enough to reliably extrapolate to the continuum at high temperatures. This led us to the method of introducing a finite cut-off, $L_{\rm cut}$, in the gauge sector and we showed that this method works very well in the massless case, allowing for good precision of results for the whole range of temperatures. In the present paper, we test the method, explained in detail in Sec. \[sec:TN\], in the massive case. Although this method is different from the one used for $T=0$, the analysis procedure at a given temperature is rather similar to the one described in the previous subsection. We begin by shortly outlying the new parts of the analysis in the thermal case. In the following, we typically express the temperature with its inverse, $\beta\equiv1/T$. ![\[fig:D9\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$, $N=60$ and $\delta=0.0020833$ (upper left), $\delta=0.0041667$ (upper right) and $\delta=0.0083333$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$ and three values of $N=48,\,60,\,72$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_D_mg0.25_x9_gb0.5_N60_dlt.00208333333333333333.ps "fig:"){width="34.50000%"} ![\[fig:D9\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$, $N=60$ and $\delta=0.0020833$ (upper left), $\delta=0.0041667$ (upper right) and $\delta=0.0083333$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$ and three values of $N=48,\,60,\,72$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_D_mg0.25_x9_gb0.5_N60_dlt.00416666666666666666.ps "fig:"){width="34.50000%"} ![\[fig:D9\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$, $N=60$ and $\delta=0.0020833$ (upper left), $\delta=0.0041667$ (upper right) and $\delta=0.0083333$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$ and three values of $N=48,\,60,\,72$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_D_mg0.25_x9_gb0.5_N60_dlt.00833333333333333333.ps "fig:"){width="34.50000%"} ![\[fig:D9\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$, $N=60$ and $\delta=0.0020833$ (upper left), $\delta=0.0041667$ (upper right) and $\delta=0.0083333$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=9$ and three values of $N=48,\,60,\,72$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_dlt_mg0.25_x9_gb0.5_N.ps "fig:"){width="34.50000%"} ![\[fig:D1024\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$, $N=640$ and $\delta=0.00019531$ (upper left), $\delta=0.00039063$ (upper right) and $\delta=0.00078125$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$ and three values of $N=512,\,640,\,768$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_D_mg0.25_x1024_gb0.5_N640_dlt.00019531250000000000.ps "fig:"){width="34.50000%"} ![\[fig:D1024\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$, $N=640$ and $\delta=0.00019531$ (upper left), $\delta=0.00039063$ (upper right) and $\delta=0.00078125$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$ and three values of $N=512,\,640,\,768$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_D_mg0.25_x1024_gb0.5_N640_dlt.00039062500000000000.ps "fig:"){width="34.50000%"} ![\[fig:D1024\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$, $N=640$ and $\delta=0.00019531$ (upper left), $\delta=0.00039063$ (upper right) and $\delta=0.00078125$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$ and three values of $N=512,\,640,\,768$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_D_mg0.25_x1024_gb0.5_N640_dlt.00078125000000000000.ps "fig:"){width="34.50000%"} ![\[fig:D1024\]Examples of the $D$-dependence of the chiral condensate for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$, $N=640$ and $\delta=0.00019531$ (upper left), $\delta=0.00039063$ (upper right) and $\delta=0.00078125$ (lower left). The red bands represent the uncertainty related to the bond dimension, taken as explained in the text. Example of a $\delta$-extrapolation for $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$, $x=1024$ and three values of $N=512,\,640,\,768$ (lower right). Lines are fits of Eq. (\[eq:fitdelta\]). The data points have error bars, but they are too small to be seen.](ex_dlt_mg0.25_x1024_gb0.5_N.ps "fig:"){width="34.50000%"} There are two new parameters with respect to $T=0$ computations, apart from the bond dimension, $D$, the system size, $N$, and the inverse coupling, $x$ — the $L_{\rm cut}$ parameter describing the cut-off in the gauge sector and the step width, $\delta$. Thus, our sequence of extrapolations follows the order given below. **Infinite bond dimension ($D\rightarrow\infty$) extrapolation.** This extrapolation is done as in the $T=0$ case and we again take the result at our largest $D$ as the central value and the difference between this value and one at $(D-20)$ as the estimate of the uncertainty from the finite bond dimension. Examples of such extrapolations are shown in Figs. \[fig:D9\] and \[fig:D1024\], for $x=9$ and $x=1024$, respectively (both at $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$). They illustrate a general feature in the $D$-dependence of the chiral condensate — the convergence becomes worse towards the continuum limit. However, this convergence is in all cases good — the difference between our two largest bond dimensions (140 and 160) is of $\mathcal{O}(10^{-9})$ at $x=9$ and of $\mathcal{O}(5\cdot10^{-5})$ at $x=1024$. This difference also depends on the temperature — since lower temperatures are reached by increasing $g\beta$, the error from the finite bond dimension also increases at increasing $g\beta$, approximately linearly. Note that in the thermal case, the convergence in $D$ is somewhat worse than at $T=0$ and we do not observe issues with insufficient machine precision (cf. Sec. \[sec:zero\] and the comments about double precision as not enough for certain parameter ranges). Finally, there is little dependence on the value of $\delta$, the volume and on the fermion mass.\ **Zero step width ($\delta\rightarrow0$) extrapolation.** We denote the results from the previous step as $\Sigma(m/g,x,N,\delta)$ and they differ from the $\delta=0$ limit by $\mathcal{O}(\delta^2)$. Hence, we extrapolate to $\delta=0$ with: $$\label{eq:fitdelta} \Sigma(m/g,x,N,\delta)=\Sigma(m/g,x,N)+r(m/g,x,N)\delta^2,$$ with the fitting parameters $\Sigma(m/g,x,N)$ and $r(m/g,x,N)$. We always use three values of $\delta$ for each $(m/g,\,x,\,N)$, which allows us to verify that a fitting ansatz linear in $\delta^2$ is proper. Since we want to access inverse temperatures $g\beta\in[0,8]$ with a step of $\Delta g\beta=0.1$, we use values of $\delta$ small enough such that this is possible. Examples are shown in the lower right plots of Figs. \[fig:D9\] and \[fig:D1024\], for $x=9$ and $x=1024$, respectively (again at $m/g=0.25$, $g\beta=0.5$, $L_{\rm cut}=10$), and three volumes that are later used for infinite volume extrapolation. Since the resulting errors are the propagated errors from the $D$-extrapolation, one again observes similar parameter dependences for the error obtained at this step. We also note that the linear ansatz (\[eq:fitdelta\]) works very well. ![\[fig:N\]Examples of the infinite volume extrapolations of the chiral condensate for $g\beta=0.5$ (left) and $g\beta=4$ and five values of $x$, with $L_{\rm cut}=10$. Lines are fits of Eq. (\[eq:fitN\]). ](ex_N_mg0.25_x_gb0.5.ps "fig:"){width="34.50000%"} ![\[fig:N\]Examples of the infinite volume extrapolations of the chiral condensate for $g\beta=0.5$ (left) and $g\beta=4$ and five values of $x$, with $L_{\rm cut}=10$. Lines are fits of Eq. (\[eq:fitN\]). ](ex_N_mg0.25_x_gb4.0.ps "fig:"){width="34.50000%"} \ **Infinite volume ($N\rightarrow\infty$) extrapolation.** The results corresponding to our estimates of the $D\rightarrow\infty$ and $\delta\rightarrow0$ limits are extrapolated to infinite volume by using the same kind of linear fitting ansatz as in the $T=0$ case, i.e. Eq. (\[eq:fitN\]), and volumes $N=\{16\sqrt{x},\,20\sqrt{x},\,24\sqrt{x}\}$. An example extrapolation is shown in Fig. \[fig:N\], for $m/g=0.25$, $L_{\rm cut}=10$, five values of the lattice spacing and two temperatures: $g\beta=0.5$ (left) and $g\beta=4$ (right). As in the $T=0$ case, we observe that the fitting ansatz gives very good description of our data.\ **Removing the $L_{\rm cut}$ cut-off ($L_{\rm cut}\rightarrow\infty$ extrapolation).** The physical results have to be independent of the used gauge sector cut-off. We found empirically that for all ranges of our parameters, $L_{\rm cut}=10$ always yields results compatible with $L_{\rm cut}=8$ and $L_{\rm cut}=12$. Hence, this value of $L_{\rm cut}$ is effectively $L_{\rm cut}=\infty$ and no explicit extrapolation is needed (see also Ref. [@Banuls:2015sta]).\ **Continuum limit ($x\rightarrow\infty$) extrapolation.** As our final step, we perform the continuum limit extrapolation of the infinite volume results $\Sigma(m/g,x)$. Before this is done, we subtract the infinite volume free condensate according to Eq. (\[eq:subtr\]) and obtain the subtracted condensate $\Sigma_{\rm subtr}(m/g,x)$. We consider the following three fitting ansatzes: $$\label{eq:cont1} \Sigma_{\rm subtr}(m/g,x)=\Sigma_{\rm subtr}^{(1)}(m/g)+\frac{a^{(1)}(m/g)}{\sqrt{x}}\log(x)+\frac{b^{(1)}(m/g)}{\sqrt{x}},$$ $$\label{eq:cont2} \Sigma_{\rm subtr}(m/g,x)=\Sigma_{\rm subtr}^{(2)}(m/g)+\frac{a^{(2)}(m/g)}{\sqrt{x}}\log(x)+\frac{b^{(2)}(m/g)}{\sqrt{x}}+\frac{c^{(2)}(m/g)}{x},$$ $$\label{eq:cont3} \Sigma_{\rm subtr}(m/g,x)=\Sigma_{\rm subtr}^{(3)}(m/g)+\frac{a^{(3)}(m/g)}{\sqrt{x}}\log(x)+\frac{b^{(3)}(m/g)}{\sqrt{x}}+\frac{c^{(3)}(m/g)}{x}+\frac{d^{(3)}(m/g)}{x^{3/2}},$$ which differ by the order of the polynomial in $1/\sqrt{x}$. We refer to them as linear+log, quadratic+log and cubic+log, respectively. We observe that the discretization effects are very different at different temperatures, in particular these effects become very strong at high temperatures and a polynomial cubic in $1/\sqrt{x}$ is needed to obtain a good description of data. We adopt a modified procedure to obtain the systematic error from the choice of the fitting range and the fitting ansatz. The procedure used to analyze the $T=0$ data is inappropriate here, because of the large dependence of the uncertainty from the $D$-extrapolation on the lattice spacing. This uncertainty at a fine lattice spacing ($x=100-500$) is up to four orders of magnitude larger than the one for our coarsest lattice spacings. Hence, the analogue of the weighted histogram built at $T=0$ is no longer reliable, as it contains fits with very large uncertainties. This does not happen at $T=0$, where the fine lattice spacings have only slightly larger uncertainties from the $D$ and $N$-extrapolations than the coarse lattice spacings. This reflects the difference in strategies used to approximate thermal and ground states as tensor networks. In practice, it translates into a somewhat different manner the truncation errors are accumulated in the thermal evolution with respect to the $T=0$ algorithm. At large $g\beta$, i.e. after several steps of imaginary time evolution, the truncation errors are much larger than in the ground state. As a consequence, the $T=0$ procedure of obtaining the systematic error does not make sense in the $T>0$ case, since only one or two fits dominate the weighted histogram. For this reason, the procedure to extract the fitting range/ansatz uncertainty is the following. It is performed separately for each temperature $g\beta$ at a given fermion mass $m/g$. We fix the maximum $x$ entering each fit to be the one corresponding to the finest lattice spacing. Then, we build all possible fits of Eqs. (\[eq:cont1\])-(\[eq:cont3\]) changing only the minimal entering $x$ ($x_{\rm min}$). We take as the central value $\Sigma_{\rm subtr}^{(i)}(m/g)$ that corresponds to the smallest uncertainty propagated through $D$, $\delta$ and $N$-extrapolations, but one that satisfies the condition $\chi^2/{\rm dof}\leq1$ and has all its fitting coefficients statistically significant. We denote it by $\Sigma_{\rm subtr}(m/g)$ and its error by $\Delta\Sigma_{\rm subtr}(m/g)$. We combine this uncertainty quadratically with the uncertainty from the choice of the fitting interval, $\Delta^{\rm interval}\Sigma_{\rm subtr}(m/g)$, and from the choice of the fitting ansatz, $\Delta^{\rm ansatz}\Sigma_{\rm subtr}(m/g)$. The former is defined as the difference between $\Sigma_{\rm subtr}(m/g)$ and the most outlying $\Sigma_{\rm subtr}^{(i)}(m/g)$ (corresponding to the same $(i)$, i.e. the same functional form of the fitting ansatz) which has still all the fitting coefficients statistically significant. The latter is taken to be the difference between $\Sigma_{\rm subtr}(m/g)$ and the most outlying $\Sigma_{\rm subtr}^{(j)}(m/g)$ (where $(j)\neq(i)$, i.e. from another fitting ansatz) which has again statistically significant fitting coefficients. Below, we illustrate this procedure with a few examples at the fermion mass $m/g=0.25$ (Fig. \[fig:cont\]). We start with a low temperature, $g\beta=6$, effectively corresponding to $T=0$ (after a certain $m$-dependent $g\beta$, the continuum result does not change any more — in the case of $m/g=0.25$, zero temperature is reached around $g\beta=6$). Here, taking the linear+log fitting ansatz and $x_{\rm min}=9$ yields a good fit, with $\chi^2/{\rm dof}\approx0.07$. It can be compared to only two other fits, both of them linear+log, with $x_{\rm min}=16$ and $x_{\rm min}=25$. Increasing $x_{\rm min}$ further or changing the fit form to quadratic+log or cubic+log leads to at least one of the fitting coefficients becoming statistically insignificant. Hence, our final result for this temperature and fermion mass is $\Sigma_{\rm subtr}(\Delta)(\Delta^{\rm interval})(\Delta^{\rm ansatz})=0.0657(3)(43)(0)$ and is dominated by the uncertainty from the choice of the fitting interval. The error from the choice of the fitting ansatz is zero, since no quadratic+log or cubic+log fit produces a significant result. Since $g\beta=6$ is effectively $T=0$, this result can be compared to our $T=0$ result at this fermion mass in Tab. \[tab:T0\]. We observe full consistency, although the precision of the thermal computation is four orders of magnitude worse than of the ground state one. This is hardly surprising, as thermal evolution is definitely not the best method to investigate ground state properties. ![\[fig:cont\]Examples of the continuum limit extrapolations of the chiral condensate for $m/g=0.25$ and temperatures $g\beta=6.0$ (upper left), $g\beta=2.0$ (upper right), $g\beta=0.4$ (lower left) and $g\beta=0.2$ (lower right). ](ex_x_syst_mg0.25_gb6.0_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont\]Examples of the continuum limit extrapolations of the chiral condensate for $m/g=0.25$ and temperatures $g\beta=6.0$ (upper left), $g\beta=2.0$ (upper right), $g\beta=0.4$ (lower left) and $g\beta=0.2$ (lower right). ](ex_x_syst_mg0.25_gb2.0_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont\]Examples of the continuum limit extrapolations of the chiral condensate for $m/g=0.25$ and temperatures $g\beta=6.0$ (upper left), $g\beta=2.0$ (upper right), $g\beta=0.4$ (lower left) and $g\beta=0.2$ (lower right). ](ex_x_syst_mg0.25_gb0.4_paper.ps "fig:"){width="34.50000%"} ![\[fig:cont\]Examples of the continuum limit extrapolations of the chiral condensate for $m/g=0.25$ and temperatures $g\beta=6.0$ (upper left), $g\beta=2.0$ (upper right), $g\beta=0.4$ (lower left) and $g\beta=0.2$ (lower right). ](ex_x_syst_mg0.25_gb0.2_paper.ps "fig:"){width="34.50000%"} Another example continuum extrapolation is shown for $g\beta=2$ (upper right plot of Fig. \[fig:cont\]). In this case, the central value comes from a linear+log fit with $x_{\rm min}=16$ and it is compared to the same functional form of the fit with $x_{\rm min}=49$ as well as to a quadratic+log fit with $x_{\rm min}=16$. Finally, we get $\Sigma_{\rm subtr}(\Delta)(\Delta^{\rm interval})(\Delta^{\rm ansatz})=-0.0078(0.3)(36)(38)$. Towards higher temperatures, cut-off effects become increasingly important, in the sense that one needs higher order polynomials in $1/\sqrt{x}$. For $g\beta=0.4$ (lower left of Fig. \[fig:cont\]), the central value that we take comes from a quadratic+log fit with $x_{\rm min}=25$, compared to $x_{\rm min}=121$ and a cubic+log fit with $x_{\rm min}=81$. This leads to $\Sigma_{\rm subtr}(\Delta)(\Delta^{\rm interval})(\Delta^{\rm ansatz})=-0.229(0.05)(6)(11)$. Our final example is $g\beta=0.2$ (lower right of Fig. \[fig:cont\]). Here, the central value comes from a cubic+log fit with $x_{\rm min}=64$, compared to $x_{\rm min}=144$ and a quadratic+log fit with $x_{\rm min}=144$. We get $\Sigma_{\rm subtr}(\Delta)(\Delta^{\rm interval})(\Delta^{\rm ansatz})=-0.297(0.14)(3)(7)$. In all these cases, the error is dominated by the uncertainty from the choice of the fitting interval and ansatz. Nevertheless, with the adopted systematic error estimation procedure, one can have these uncertainties reliably under control. ![\[fig:gbeta\]Inverse temperature dependence of the continuum limit extrapolated chiral condensate for all our fermion masses. Shown is also the result obtained at $T=0$ and the result of the approximation of Ref. [@Hosotani:1998za].](gbeta_mg0.0625-syst-hosotaniHF.ps "fig:"){width="34.50000%"} ![\[fig:gbeta\]Inverse temperature dependence of the continuum limit extrapolated chiral condensate for all our fermion masses. Shown is also the result obtained at $T=0$ and the result of the approximation of Ref. [@Hosotani:1998za].](gbeta_mg0.125-syst-hosotaniHF.ps "fig:"){width="34.50000%"} ![\[fig:gbeta\]Inverse temperature dependence of the continuum limit extrapolated chiral condensate for all our fermion masses. Shown is also the result obtained at $T=0$ and the result of the approximation of Ref. [@Hosotani:1998za].](gbeta_mg0.25-syst-hosotaniHF.ps "fig:"){width="34.50000%"} ![\[fig:gbeta\]Inverse temperature dependence of the continuum limit extrapolated chiral condensate for all our fermion masses. Shown is also the result obtained at $T=0$ and the result of the approximation of Ref. [@Hosotani:1998za].](gbeta_mg0.5-syst-hosotaniHF.ps "fig:"){width="34.50000%"} ![\[fig:gbeta\]Inverse temperature dependence of the continuum limit extrapolated chiral condensate for all our fermion masses. Shown is also the result obtained at $T=0$ and the result of the approximation of Ref. [@Hosotani:1998za].](gbeta_mg1.0-syst-hosotaniHF.ps "fig:"){width="34.50000%"} We repeat the analysis steps for all our fermion masses and we summarize the continuum limit results in Fig. \[fig:gbeta\], where we show results up to $g\beta=8$ ($m/g=0.0625$ and $0.25$) or $g\beta=6$ ($m/g=0.125,\,0.5,\,1$). The most important feature confirming the validity of our results is that we always reproduce the $T=0$ result within our errors — actually the difference between our central values at large enough $g\beta$ and the $T=0$ MPS result is much smaller than our errors, suggesting that the error estimation procedure is rather conservative. We also note that our systematic error procedure makes the final errors strongly dependent on temperature — with sometimes irregular jumps of the error caused by some other fitting interval or fitting ansatz entering the procedure at certain $g\beta$ values[^3]. Apart from the agreement with the $T=0$ result, we observe that the approach to this result is faster for higher fermion masses — for $m/g=1$, $g\beta=3$ is already effectively zero temperature, while for our lowest mass, $m/g=0.0625$, we have small changes of the central value even above $g\beta=6$. Concerning the agreement with the approximation of Ref. [@Hosotani:1998za] (referred to as “Hosotani HF” in the plot), the latter provides good qualitative description of the temperature dependence of the chiral condensate. However, the quantitative agreement is not perfect, with typical deviations of 10-20%. It is known that the approximation becomes exact in the massless limit and indeed, e.g. Hosotani’s $T=0$ result at $m/g=0.0625$ is relatively closer to the MPS result than the one at $m/g=0.125$. On the other hand, the approximation of Ref. [@Hosotani:1998za] also approaches the analytical result of zero at infinite fermion mass and $T=0$ — hence one also expects an increasing agreement in this regime. Indeed, the relative difference at $m/g=1$ is the smallest from among all our considered masses. However, when we consider the slope of the $g\beta$-dependence, we clearly observe that the agreement between Hosotani HF and our computation becomes better towards small fermion masses, with both curves being almost parallel for $m/g=0.0625$. Summary and prospects {#sec:summary} ===================== In this paper, we have performed a study of the temperature dependence of the chiral condensate for the one-flavour Schwinger model using a Hamiltonian approach. We emphasize that while for zero temperature we employ a matrix product [*state*]{} (MPS) ansatz, for non-vanishing temperature we use a matrix product [*operator*]{} (MPO) ansatz. In addition, for the non-zero temperature calculation, we have to perform a thermal evolution by starting from a well defined infinite temperature state and evolve the system in incremental inverse temperature steps towards zero temperature using a density operator. Thus, non-zero temperature calculations within the Hamiltonian approach are rather different from the so far carried out zero temperature ones and hence non-zero temperature computations for gauge theories are novel and need to be tested. While in Ref. [@Banuls:2015sta] we have initiated such non-zero temperature computations for massless fermions, in this paper we went substantially beyond this work by studying the system at various fermion masses. In addition, we employed consistently a truncation of the gauge sector. This allowed us to reach very large system sizes and, keeping the physical extent of the model fixed, very small values of the lattice spacing. Within our calculation of the chiral condensate, we carried out a substantial and challenging effort to control the systematic effects. To this end, we performed extrapolations to zero thermal evolution step size, infinite bond dimension, infinite volume and zero lattice spacing. In addition, we tested that our cut parameter for the gauge sector truncation has been sufficiently large. The final non-trivial check of the validity of our approach has been to recover the zero temperature result of the chiral condensate after the long thermal evolution performed. As a result of our work, we could compute the chiral condensate over a broad temperature range from infinite to almost zero temperature with controlled errors. This has been done for zero, light and heavy fermion masses. For zero fermion mass, we found excellent agreement with the analytical results of Ref. [@Sachs:1991en]. Moving to non-zero fermion masses, a comparison to Ref. [@Hosotani:1998za] did not lead to a clear conclusion, see Fig. \[fig:gbeta\]. Although qualitatively the temperature dependence of the chiral condensate shows a comparable behaviour between the analytical result of Ref. [@Hosotani:1998za] and our data, there does not seem to be an agreement on the quantitative level. This is presumably due to the fact that the approximations made in Ref. [@Hosotani:1998za] are too rough to reach a satisfactory quantitative agreement. We consider the here performed work, besides of the clear interest in its own, as a necessary step towards investigating the Schwinger model when adding a chemical potential. This setup leads to the infamous sign problem and it would be very reassuring to see whether the here used MPS and MPO approaches can lead to a successful application for this very hard problem, which is very difficult, if not impossible to solve by standard Markov chain Monte Carlo methods. We thank J. I. Cirac for discussions. This work was partially funded by the EU through SIQS grant (FP7 600645). K.C. was supported in part by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse and in part by the Deutsche Forschungsgemeinschaft (DFG), project nr. CI 236/1-1 (Sachbeihilfe). Calculations for this work were performed on the LOEWE-CSC high-performance computer of Johann Wolfgang Goethe-University Frankfurt am Main and in the computing centers of DESY Zeuthen and RZG Garching. [^1]: For a given system size, $N$, exact results would actually be attained with finite bond dimension, $D=2^{N/2}$[@verstraete04dmrg], which is many orders of magnitude larger than the largest one we use in the simulations. [^2]: In the case of operators one should instead talk about operator space entanglement entropy, a measure related to truncation error in the MPO that was introduced in Ref. [@PhysRevA.76.032316]. [^3]: For example, at $m/g=0.25$ all the quadratic+log fits have at least one fitting coefficient statistically insignificant above $g\beta=3.5$ and at this temperature and higher (smaller $g\beta$), quadratic+log fits become statistically significant and thus enlarge our error.

--- abstract: 'We examine the quantum tunneling process in Bose condensates of two interacting species trapped in a double well configuration. We discover the condition under which particles of different species can tunnel as pairs through the potential barrier between two wells in opposition directions. This novel form of tunneling is due to the interspecies interaction that eliminates the self- trapping effect. The correlated motion of tunneling atoms leads to the generation of quantum entanglement between two macroscopically coherent systems.' address: | [Department of Physics, The Chinese University of Hong Kong,]{}\ [Shatin, NT, Hong Kong, China]{} author: - 'H. T. Ng, C. K. Law, and P. T. Leung' title: 'Entangled quantum tunneling of two-component Bose-Einstein condensates' --- Quantum tunneling of macroscopically coherent systems is an intriguing phenomena well known in the context of Josephson junction effects in superconducting electronic systems. For superfluids consisting of neutral particles, detailed investigations of tunneling is aided by the recent realization Bose-Einstein condensation of atomic vapor in a well controllable environment. Indeed, recent experiments have successfully demonstrated quantum tunneling for condensates confined in an array of optical potentials [@kasevich1; @burger]. One prominent feature of tunneling in Bose condensates is the nonlinear dynamics arising from the interaction between atoms. Quite remarkably, for single-component condensates trapped in double-well configurations, previous studies have indicated that a self-trapping mechanism can suppress the tunneling rate significantly by increasing the atom-atom interaction strength [@shenoy; @walls; @raghavan; @williams]. An interesting extension of the tunneling problem involves Bose condensates of two interacting species (Fig. 1). The main issue is how the interspecies interaction affects the tunneling process, and particularly the quantum coherence as the two condensates mix together. Previous studies of the general properties of two-component Bose condensates have emphasized the important role of the interspecies interaction, which leads to novel features, such as the components separation [@phase1; @phase2], cancellation of the mean field energy shift [@pu], and the suppression of quantum phase diffusion [@law]. However, the investigation of the influence of interspecies interaction on tunneling dynamics has only just begun [@lobo; @pu2]. In this paper we present a novel tunneling mechanism for a two-component condensate trapped in a double-well (see Fig. 1). The atoms of the component $A(B)$ are initially prepared in the left (right) potential well. We discover the condition under which the interspecies interaction can eliminate the self-trapping effect and thus enhances the tunneling significantly. Such an enhanced tunneling originates from the correlated quantized motion of the two condensates. We also show that atoms of different species tunnel through the barrier as correlated pairs in opposition directions, i.e., a form of [*quantum entangled tunneling*]{}. Therefore tunneling serves as a mechanism to build up a strong correlation among atoms of different species, and this leads to the generation of quantum entanglement between two multi-particle systems. The configuration of our double-well system is sketched in Fig. 1. Our focus in this paper is the quantum dynamics beyond the mean field description. An exact many-body description is difficult even for single-component condensate problems. The usual method to capture the essential physics is based on the two-mode approximation in which the evolution is confined by the left and right localized mode functions associated with the respective potential wells [@shenoy; @walls; @raghavan; @williams; @juha]. Such an approximation is valid when each potential well is sufficiently deep so that higher modes of the wells essentially do not participate in the dynamics. =2.5in In the two-mode approximation, the system is modeled by the Hamiltonian $(\hbar =1)$, $$\begin{aligned} H &=& \frac{\Omega}{2} ({\aL}a_{R}+{\aR}a_{L}+{\bL}b_{R}+{\bR}b_{L}) \nonumber \\ && +\frac{\kappa_{a}}{2}\left[ ({\aL}a_{L})^{2}+({\aR}a_{R})^{2}\right] \nonumber \\ && + \frac{\kappa_{b}}{2}\left[({\bL}b_{L})^{2}+({\bR}b_{R})^{2}\right] \nonumber \\ && +\kappa ({\aL}a_{L}{\bL}b_{L}+{\aR}a_{R}{\bR}b_{R}). \label{Hamiltonian}\end{aligned}$$ Here the subscripts $L$ and $R$ respectively denote the localized modes in the left and right potential wells. Since there are two modes available for each component, the model in fact consists of four mode operators. We use $a^{\dag}_j$ and $b^{\dag}_j$ $(j=L,R)$ to denote the creation operators of the component $A$ and $B$ respectively. The parameters $\Omega$, $\kappa_a(\kappa_b)$ and $\kappa$ describe the tunneling rate, self-interaction strength of the component $A(B)$ and the interspecies interaction strength respectively. To gain insight of the quantum correlation developing in the tunneling process, we first consider the exactly solvable case with only one $A$ atom in the left well and one $B$ atom in the right well. In this case the system is spanned by four basis vectors: $|1,0\rangle_{A}|1,0\rangle_{B}$, $|1,0\rangle_{A}|0,1\rangle_{B}$, $|0,1\rangle_{A}|1,0\rangle_{B}$ and $|0,1\rangle_{A}|0,1\rangle_{B}$, where $|p,q\rangle_{S}$ denotes the state with $p$ atoms of species $S$ $(S=A,B)$ in the left well and $q$ atoms of species $S$ in the right well. The eigenvalues and eigenvectors of $H$ can be found straightforwardly. In the regime where the interspecies interaction is sufficiently strong such that $\kappa \gg \Omega$, the state vector evolves as $$\begin{aligned} |\Psi(t)\rangle &=& e^{-i[(\kappa_{a}+\kappa_{b})/2-\omega_0]{t}} [\cos{\omega_{0}{t}}|1,0\rangle_{A}|0,1\rangle_{B} \nonumber \\ && + i\sin{\omega_{0}{t}}|0,1\rangle_{A}|1,0\rangle_{B}] + O (\Omega / \kappa) . \label{2atom-state}\end{aligned}$$ In writing Eq. (\[2atom-state\]) we have defined $\omega_0 = \Omega^2/2 \kappa$ as an effective tunneling frequency. Because of the strong interaction between the atoms, the probability of finding both particles in the same well at any time $t$ is negligible (of order $\Omega^2/\kappa^2$). The tunneling motion of the two atoms are anti-correlated in the sense that the atom $A$ and the atom $B$ always move in opposite directions. Such an anti-correlated tunneling motion gives rise to quantum entanglement between the two atoms. At time $t=(n+1/4)\pi / \omega_0$, ($n=$ integers), the state is a form of Bell’s state that is maximally entangled in the two-particle two-mode subspace. Now we examine the multiple atoms case. In order to facilitate the discussion, we assume the number of particles are the same for the two components, i.e., $N_a=N_b=N$, and the condensates shares the same interaction strength i.e., $\kappa_a=\kappa_b=\kappa$. The latter condition is a good approximation to $^{87}$[Rb]{} condensate of atoms in hyperfine spins states $|F=2,m_{f}=1{\rangle}$ and $|F=1,m_{f}=-1{\rangle}$, which share similar scattering lengths [@phase2]. However, we emphasize that these assumptions are not crucial, we shall relax these conditions later in the paper. We shall limit our study to the $4 \kappa {\gg} N\Omega$ regime where the nonlinear interaction is dominant. As before we consider the initial condition in which all atoms in the component $A(B)$ are localized in the left (right) potential well. The general form of the state vector at time $t$ is given by: $|{\Psi}(t){\rangle}= e^{-i{\kappa}N^{2}t} \sum_{n=0}^{N}\sum_{m=0}^{N} {c}_{n,m}(t) |n,N-n{\rangle}_A|m,N-m{\rangle}_B$. The amplitudes ${c}_{n,m} (t)$ are governed by the Schrödinger equation according to the Hamiltonian (\[Hamiltonian\]): $$\begin{aligned} \label{qpamp} i\dot{{c}}_{n,m}&=&\frac{\Omega}{2} \left[ \sqrt{(n+1)(N-n)}{{c}}_{n+1,m} \right. \nonumber \\ && \left. \ \ \ \ \ \ + \sqrt{n(N-n+1)}{{c}}_{n-1,m}\right] \nonumber\\ && + \frac{\Omega}{2}\left[\sqrt{(m+1)(N-m)}{{c}}_{n,m+1} \right. \nonumber \\ && \left. \ \ \ \ \ \ + \sqrt{m(N-m+1)}{{c}}_{n,m-1}\right] \nonumber \\ && + {\kappa}\left[(n+m-N)^{2}\right]{{c}}_{n,m}\end{aligned}$$ with the initial condition $c_{n,m} (0)= \delta_{n,N} \delta_{m,0}$. Let us first present the typical results obtained from the numerical solutions of Eq. (3). In Fig. 2, we show the particle number difference $W \equiv \langle {a_L^{\dag} a_L - a_R^{\dag} a_R } \rangle$ of species $A$ between the two wells as a function of time. The occurrence of tunneling is revealed by the decrease of $W$. At longer times $W$ approaches zero, therefore the numbers of $A$ atoms in the two potential wells are roughly equalized. We emphasize that the nonzero interspecies interaction is responsible for the tunneling to occur. If the two species do not interact with each other (i.e., $\kappa =0$), then a sufficiently strong self-interaction $\kappa_j> N \Omega$ $(j=a,b)$ can suppress the tunneling almost completely by the self-trapping effect [@shenoy]. =2.8in To understand the mechanism of the two-component tunneling, we note that the state vector (3) is a superposition of $(N+1)^2$ states of the form $|n ,N-n {\rangle}_A|m,N-m{\rangle}_B$. However, only a few number of them are actively involved. This is intuitively understood because $|n ,N-n {\rangle}_A |m,N-m{\rangle}_B$ have different energies for different values of $n$ and $m$, and only those with energies closer to that of the initial state are more accessible. The difference in energies is significant in the regime $4 \kappa \gg N \Omega$ considered here. We find that the states $$|\phi_q \rangle =|q ,N-q {\rangle}_A |N-q,q {\rangle}_B \ \ \ \ (q=0,1,2,...,N) \label{degenerate-state}$$ are approximately degenerate energy eigenvectors of $H$ (to zero order in $\Omega$), and the energies of states other than $|\phi_q \rangle$ are higher than that of $|\phi_q \rangle$ by an amount of order of $\kappa$ or higher. Since the initial state $|\Psi(0){\rangle}$ is $|\phi_0 \rangle$, $|\Psi(t){\rangle}$ is mainly a superposition of $|\phi_q \rangle$ at any time $t$ according to the energy argument above. We find that this is indeed the case. To provide a numerical evidence of our finding, we show in the inset of Fig. 2 the overlap probability defined by $P(t) = \sum\limits_{q = 0}^N {} |\left\langle {{\phi _q }} \mathrel{\left | {\vphantom {{\phi _q } \Psi }} \right. \kern-\nulldelimiterspace} {\Psi (t) } \right\rangle |^2$. For the parameters used in this figure, the set of $|\phi_q \rangle$ contributes more than 90$\%$ of $|\Psi(t){\rangle}$. Our further numerical tests suggest that $P \to 1$ in the limit $N\Omega / \kappa \to 0$. We remark that all $|\phi_q \rangle$ have the same number of particles (component $A$ plus component $B$) in the left potential well, and the same also holds for the right well. Therefore $P(t) \approx 1$ implies small fluctuations of total particle number in the each potential well. In the case of Fig. 2, we find that the fluctuation of total particle number in the left well $\left\langle {\Delta N_L } \right\rangle$ is much smaller than $ \left\langle {N_L } \right\rangle ^{1/2}$, i.e., a sub-Poissonian distribution. The time-dependent problem now is significantly simplified because the evolution of the condensates mainly involves the set of degenerate states $|\phi_q \rangle$. For those states that do not have the same energy as $|\phi_q \rangle$, they act as intermediate states that are rarely populated. We may eliminate such intermediate states with the adiabatic following method which is well known in quantum optics. The last line in Eq. (3) indicates that the states with $n+m=N+r$ ($r=$integer) has the diagonal term $r^2\kappa$ that can be interpreted as the energy of the states if the tunneling interaction $\Omega$ were switched off. Since $\kappa$ is a large parameter here, the states with $n+m=N \pm 1$ have a much different energy than that of the states $|\phi_q \rangle$ with $n+m=N$. Any transition (due to $\Omega$) from the manifold $n+m=N$ to the manifolds $n+m=N \pm 1$ must quickly return to $n+m=N$. In other words, the states with $n+m=N \pm 1$ are intermediate states that are hardly occupied at any time. The amplitudes associated with the $n+m=N \pm 1$ manifold are: ${c}_{n+1,N-n}$, ${c}_{n-1,N-n}$, ${c}_{n,N-n+1}$ and ${c}_{n,N-n-1}$, and they can be found approximately by adiabatic following under the condition $4\kappa{\gg}{N}\Omega$. With these approximate amplitudes, the Schrödinger equation of ${c}_{n,N-n}$ is given by: $$\begin{aligned} i\dot{{c}}_{n,N-n}&{\approx}&-\frac{\Omega^{2}}{2\kappa} (n+1)(N-n){c}_{n+1,N-n-1} \nonumber \\ && - \frac{\Omega^{2}}{2\kappa}n(N-n+1) {c}_{n-1,N-n+1} \nonumber\\ &&-\frac{\Omega^{2}}{\kappa}[n(N-n)+N] {c}_{n,N-n}. \label{adabatic-eqn}\end{aligned}$$ This corresponds to the Schrödinger equation governed by the effective Hamiltonian: $$\begin{aligned} H_{\rm{eff}}&=&-\frac{\Omega^{2}}{2\kappa} \left( {\aL}a_{R}b_{L} {\bR}+a_{L}{\aR}{\bL}b_{R}+{\aL}a_{L}{\aR}a_{R} \right. \nonumber \\ && \left. \ \ \ \ \ \ \ \ +{\bL}b_{L}{\bR}b_{R} \right), \label{effective-Hamiltonian1}\end{aligned}$$ apart from a constant term proportional to the total number of particles. The effective Hamiltonian $H_{\rm{eff}}$ is an approximation that captures the essential tunneling mechanism in the $4 \kappa \gg N \Omega$ limit. We have tested the validity of the effective Hamiltonian numerically. For example, in Fig. 2 the dashed line, which is obtained from the evolution based on $H_{\rm{eff}}$, agrees well with the exact numerical solution. The physical picture becomes transparent according to $H_{\rm{eff}}$. The interaction term ${\aR}{\bL}a_{L}b_{R}$ suggests that every time when an atom $A$ moves from the left well to the right well there must be an atom $B$ moves from the right well to the left well. This explains why the small fluctuations in the total particle number in each potential well. The reverse process is described by ${\aL}{\bR}a_{R}b_{L}$ in $H_{\rm{eff}}$. In other words, the atoms $A$ and $B$ must be moving in pair in opposite directions during the tunneling process. It is worth noting that $H_{\rm{eff}}$ can be cast in the form $$H_{\rm{eff}}=-\frac{\Omega^{2}}{4\kappa}\left( { K_{+}K_{-}+K_{-}K_{+}} \right) \label{effective-Hamiltonian2}$$ where $K_{+}= a^{\dag}_{L}a_{R}+ b^{\dag}_{L}b_{R} $ and $K_{-}= a_{L}a^{\dag}_{R}+ b_{L}b^{\dag}_{R}$ satisfy the angular momentum commutation relation: $ \left[ {K_ + ,K_ - } \right] = 2K_z $, $ \left[ {K_z ,K_ \pm } \right] = \pm K_ \pm$, where $K_z = (a_L^{\dag} a_L + b_L^{\dag} b_L - a_R^{\dag} a_R - b_R^{\dag} b_R)/2$. Therefore $K_{\pm}$ and $K_z$ are analogous to collective spin operators, and the eigenvectors and eigenvalues of $H_{\rm{eff}}$ can be solved analytically using angular momentum algebra. This shares a common feature in spinor condensates where the spin mixing problem can be solved in a similar fashion [@spin]. The nonlinear interaction between collective spins is the key to generate nonclassical correlations between different spin components, and particularly, the notion of multi-particle entanglement has been discussed in the context of squeezed spin states [@duan; @multi; @you]. We show here that quantum entanglement between two multi-particle subsystems ($A$ and $B$) can be achieved in the double-well tunneling process. To this end we quantify the degree of entanglement between the two species based on entanglement entropy: $E=-\mbox{tr}(\rho_{A}\mbox{ln}\rho_{A})=-\mbox{tr}(\rho_{B} \mbox{ln}\rho_{B})$, where $\rho_{A}$ and $\rho_{B}$ are reduced density matrices of the respective subsystems, i.e., $\rho_{A}=\mbox{tr}_{B}{\rho_{AB}}$ and $\rho_{B}=\mbox{tr}_{A}{\rho_{AB}}$ with $ \rho_{AB} =\left| {\Psi (t)} \right\rangle \left\langle {\Psi (t)} \right|$ being the density matrix of the whole system. A disentangled state (for example the initial state above) has zero entanglement entropy. The more entangled the systems, the larger the value of $E$ is. As an illustration, we show in Fig. 3 how the entanglement is established in time for various particle numbers. As the time increases the value of $E$ increases until a saturated value is reached. Since there are $N+1$ degenerate states $|\phi_q \rangle$ mainly involved in the evolution, $E\approx \ln{(N+1)}$ is a maximum if all $|\phi_q \rangle$ have equal contributions to the state of the system. In the case of Fig. 3, the value of $E$ can reach as high as 90% of the value $\ln (N+1)$. Finally we would like to address the conditions for the entangled tunneling to occur. Our discussion above has been restricted to the simplest symmetric situation: $N_a=N_b=N$ and $\kappa_a=\kappa_b=\kappa$, in order to illustrate the essential mechanism under the condition $4\kappa \gg N \Omega$. The same analysis can be performed to examine the general situations. We have examined the system with unequal particle numbers $(N_b-N_a) \equiv D >1$ and unequal coupling strengths $\kappa_a=\kappa+\delta$, $\kappa_b=\kappa-\delta$ with $|\delta |\ll \kappa$. We find that if the tunneling strength $\Omega$ is sufficiently weak or the self interaction strength $\kappa$ is sufficiently strong such that $$4|\kappa (D-1)-[N_a(D-2)|\delta|]/2 | \gg \Omega N_b. \label{general-condition}$$ then the system mainly evolves among the states $|n ,N_a-n {\rangle}_A |N_a-n,N_b-N_a+n{\rangle}_B$, where $n=0,1,2,...N_a$. In other words, the tunneling under condition (8) is characterized by entanglement generation term ${\aR}{\bL}a_{L}b_{R}$ as before. However, we point out that the tunneling are generally less efficient for the cases with non-zero $\delta$ and $D$. This is because the states $|n ,N_a-n {\rangle}_A |N_a-n,N_b-N_a+n{\rangle}_B$ are not as degenerate as that in the symmetric case with $\delta$ and $D$ are both zero. We also remark that the condition (8) does not apply to the special case $N_b-N_a = \pm 1$ in which some of the states in the $n+m=N$, and $n+m=N \pm 1$ manifolds are accidentally degenerated. =2.8in To conclude, we have presented a novel mechanism of double-well tunneling involving Bose condensates of two interacting components, based on the two-mode approximation model in the strong coupling regime. We find that the interplay of intraspecies and interspecies interactions permits a set of energy degenerate states, a small tunneling coupling can push the system to ‘explore’ through these degenerate states and thus result in a substantial tunneling not limited by the self trapping effect. The most interesting feature is the strongly correlated tunneling motion. 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**Semi-Finite Forms of** Bilateral Basic Hypergeometric Series [ William Y. C. Chen]{}$^{1}$ and $^{2}$ Center for Combinatorics, LPMC\ Nankai University, Tianjin 300071, P.R. China Email: $^[email protected], $^[email protected] [**Abstract.**]{} We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan’s $_1\psi_1$ summation, Bailey’s $_2\psi_2$ transformations, and Bailey’s $_6\psi_6$ summation. [**Corresponding Author:**]{} William Y. C. Chen, Email: [email protected] [**AMS Classification:**]{} 33D15 [**Keywords:**]{} Bilateral hypergeometric summation, semi-finite forms, Ramanujan’s ${}_1\psi_1$ summation, Bailey’s ${}_2\psi_2$ transformations, Bailey’s ${}_6\psi_6$ summation. Introduction ============ We follow the terminology for basic hypergeometric series in [@GR]. Assuming $|q|<1$, let $$(a;q)_\infty = (1-a) (1-aq) (1-aq^2) \cdots .$$ For any integer $n$, the $q$-shifted factorial $(a;q)_n$ is given by $$(a;q)_n = { (a;q)_\infty \over (aq^n;q)_\infty}.$$ For $n\geq 0$, we have the following relation which is crucial for this paper: $$\label{Defi} (a;q)_{-n}= \frac{1}{(aq^{-n};q)_n}={ (-q/a)^{n} q^{\binom{n}{2} }\over (q/a;q)_{n}} .$$ For convenience, we employ the following usual notation: $$(a_1, a_2, \ldots, a_m;q)_n=(a_1;q)_n(a_2;q)_n\ldots(a_m;q)_n.$$ The (unilateral) basic hypergeometric series $_{r+1}\phi_r$ is defined by $$\begin{aligned} \label{Hype} _{r+1}\phi_r\left[ \begin{array}{c} a_1, a_2, \cdots, a_{r+1}\\ b_1, b_2, \cdots, b_{r} \end{array};q, z \right]=\sum_{k=0}^{\infty}A(k),\end{aligned}$$ where $$A(k)=\frac{(a_1, a_2, \cdots, a_{r+1};q)_k}{(b_1, b_2, \cdots, b_r,q;q)_k}z^k.$$ The bilateral basic hypergeometric series $_s\psi_s$ is defined as follows, $$\begin{aligned} \label{Bila} _s\psi_s\left[ \begin{array}{c} a_1, a_2, \cdots, a_{s}\\ b_1, b_2, \cdots, b_{s} \end{array};q, z \right]=\sum_{k=-\infty}^{\infty} B(k),\end{aligned}$$ where $$B(k)=\frac{(a_1, a_2, \cdots, a_{s};q)_k}{(b_1, b_2, \cdots, b_s;q)_k}z^k.$$ In this paper, we propose the following method of deriving bilateral summation and transformation formulas using [*semi-finite forms*]{}. For a bilateral series $_s\psi_s$ as given in (\[Bila\]), we construct a summand $G(k,m)$ which implies a unilateral series $_{r+s+1}\phi_{r+s}$, where $r$ is a nonnegative integer, such that $$\lim _{m \rightarrow \infty }G(k,m)=B(k)$$ for all $k$, and the summation $$\label{gn0} \sum_{k=-m}^\infty G(k,m)$$ can be easily accomplished as a Laurent extension of the summation $$\label{laurants} \sum_{k=0}^\infty G(k-m, m)= G(-m,m)\sum_{k=0}^\infty A(k),$$ where $G(k,m)$ can be written as $$G(k-m, m) = G(-m,m) A(k)$$ for some $A(k)$. The bilateral series (\[Bila\]) is then obtained from (\[gn0\]) as $m\to\infty$, subject to suitable convergence conditions. We apply this procedure to derive bilateral series identities from suitable unilateral ones. The above summation (\[gn0\]) is called the [ *semi-finite form*]{} of the bilateral summation (\[Bila\]). A method similar to ours was recently used by Schlosser [@SCHL], and Jouhet and Schlosser [@SCHL04], who derived summations for bilateral series from [*finite forms*]{}. We also note that another method, which uses a similar factorization as above, for deriving bilateral series identities from unilateral ones was used by Ismail [@Ismail], and Askey and Ismail [@AsIs]. Rather than taking limits, they apply analytic continuation as the main ingredient. In this paper, we present semi-finite forms of several classical bilateral summation and transformation formulas such as Ramanujan’s $_{1}\psi_1$ formula, Bailey’s $_2\psi_2$ transformations, and Bailey’s $_6\psi_6$ summation. From $_2\phi_1$ to $_1\psi_1$ ============================= Using the well known Gauss summation formula $$\label{Gauss} _2\phi_1\left[ \begin{array}{c} a,b\\ c \end{array};q, c/ab \right]=\frac{(c/a,c/b;q)_{\infty}}{(c,c/ab;q)_{\infty}},$$ where $|c/ab|<1$, we get a semi-finite form of Ramanujan’s summation of the general $_1\psi_1$, $$\label{Ran} _{1}\psi_1 \left[ \begin{array}{l} a\\ b \end{array};q,z\right]=\sum_{k=-\infty}^{\infty}\frac{(a;q)_k}{(b;q)_k}z^k= \frac{(q;q)_{\infty}(b/a;q)_{\infty}(az;q)_{\infty}(q/az;q)_{\infty}} {(b;q)_{\infty}(q/a;q)_{\infty}(z;q)_{\infty}(b/az;q)_{\infty}},$$ where $|b/a|<|z|<1$. For $|z|<1$, the following identity holds: \[theo\] $$\label{r-f} \sum_{k=-m}^{\infty}\frac{(a;q)_k(bq^{m}/az;q)_k}{(q^{1+m};q)_k(b;q)_k}z^k =\frac{(q;q)_m(q/az;q)_m}{(q/a;q)_m(b/az;q)_m} \frac{(b/a;q)_{\infty}(az;q)_{\infty}}{(b;q)_{\infty}(z;q)_{\infty}}.$$ [*Proof.*]{} The left hand side of (\[r-f\]) can be rewritten as $$\begin{aligned} \lefteqn{\sum_{k=0}^{\infty}\frac{(a;q)_{k-m}(bq^{m}/az;q)_{k-m}} {(q^{1+m};q)_{k-m}(b;q)_{k-m}}z^{k-m}}\\[6pt] &=&z^{-m}\frac{(a;q)_{-m}(bq^{m}/az;q)_{-m}}{(q^{1+m};q)_{-m}(b;q)_{-m}} \sum_{k=0}^{\infty}\frac{(aq^{-m};q)_k(b/az;q)_k}{(q;q)_k(bq^{-m};q)_k}z^k\\[6pt] &\overset{(\ref{Gauss})}{=}&z^{-m}\frac{(a;q)_{-m}(bq^{m}/az;q)_{-m}}{(q^{1+m};q)_{-m}(b;q)_{-m}} \frac{(b/a;q)_{\infty}(azq^{-m};q)_{\infty}}{(bq^{-m};q)_{\infty}(z;q)_{\infty}} \\[6pt] &\overset{(\ref{Defi})}{=}&z^{-m}\frac{(q;q)_{m}(azq^{-m};q)_m}{(aq^{-m};q)_m (b/az;q)_m}\frac{(az;q)_{\infty}(b/a;q)_{\infty}} {(b;q)_{\infty}(z;q)_{\infty}},\end{aligned}$$ which equals the right hand side of (\[r-f\]). ------------------------------------------------------------------------ Taking the limit $m \rightarrow \infty$ in Proposition \[theo\] while assuming $|b/az|<1$, we immediately obtain (\[Ran\]). We remark that our method is different from the method of M. Jackson’s elementary proof of (\[Ran\]) (see the exposition of Schlosser [@SCHL]) in the sense that Jackson’s proof does not give a semi-finite form although the Gauss summation is also the basic ingredient. We should also note that a finite form of Ramanujan’s $_1\psi_1$ summation has been given by Schlosser [@Schl03b] using the terminating $q$-Pfaff-Saalschütz summation. From $_3\phi_2$ to $_2\psi_2$ ============================= In this section, we use two $_3\phi_2$ summation and transformation formulas to give the semi-finite forms of $_2\psi_2$ formulas due to Bailey. We begin with the following $_2\psi_2$ transformation formula [@GR Ex. 5.20(i)] valid for $|z|, |cd/abz|,|d/a|,|c/b|<1$: $$\begin{aligned} \label{22} _{2}\psi_{2}\left[ \begin{array}{l} a,b\\ c,d \end{array};q,z \right]=\frac{(az,d/a,c/b,dq/abz;q)_{\infty}}{(z,d,q/b,cd/abz;q)_{\infty}} \, {_{2}}\psi_{2}\left[ \begin{array}{l} a,abz/d\\ az,c \end{array};q,\frac{d}{a} \right].\end{aligned}$$ Using a $q$-analogue of the Kummer-Thomae-Whipple formula [@GR Eq. (3.2.7)]: $$\begin{aligned} \label{Kummer} _{3}\phi_2\left[ \begin{array}{l} a, b, c\\ d,e \end{array};q, \frac{de}{abc} \right]=\frac{(e/a,de/bc;q)_{\infty}}{(e,de/abc;q)_{\infty}}{_{3}}\phi_2\left[ \begin{array}{l} a, d/b, d/c\\ d,de/bc \end{array};q, \frac{e}{a} \right],\end{aligned}$$ where $|de/abc|<1$ and $|e/a|<1$, we get a semi-finite form of (\[22\]). \[Thm1\] For $|z|<1$ and $|d/a|<1$, we have $$\begin{aligned} \sum_{k=-m}^{\infty}\frac{(a,b;q)_k(cdq^m/abz;q)_k}{(c,d;q)_k(q^{1+m};q)_k}z^k&=& \frac{(az,d/a;q)_{\infty}}{(z,d;q)_{\infty}}\frac{(c/b,dq/abz;q)_{m}} {(q/b,cd/abz;q)_{m}} \nonumber \\\label{xy} && \;\; \; \cdot \sum_{k=-m}^{\infty}\frac{(a,cq^m/b,abz/d;q)_{k}} {(c,q^{1+m},az;q)_{k}}(d/a)^{k}.\end{aligned}$$ The left hand side of (\[xy\]) equals $$\begin{aligned} \lefteqn{z^{-m}\frac{(a,b,cdq^m/abz;q)_{-m}}{(c,d,q^{1+m};q)_{-m}}\sum_{k=0}^{\infty}\frac{(aq^{-m},bq^{-m},cd/abz;q)_k} {(cq^{-m},dq^{-m},q;q)_k}z^k} \\ &\overset{(\ref{Kummer})}{=}&z^{-m}\frac{(a,b,cdq^m/abz;q)_{-m}}{(c,d,q^{1+m};q)_{-m}}\frac{(d/a,azq^{-m};q)_{\infty}}{(dq^{-m},z;q)_{\infty}}\\ && \;\;\; \cdot \sum_{k=0}^{\infty}\frac{(aq^{-m},c/b, abzq^{-m}/d;q)_k}{(q,cq^{-m},azq^{-m};q)_k}\left(\frac{d}{a}\right)^k\\ &\overset{(\ref{Defi})}{=}&\frac{(d/a,az;q)_{\infty}}{(d,z;q)_{\infty}}\frac{(c/b,abzq^{-m}/d;q)_m} {(bq^{-m}, cd/abz;q)_m}\left(\frac{d}{az}\right)^m\\ & & \;\;\; \cdot \sum_{k=0}^{\infty}\frac{(a,cq^m/b,abz/d;q)_{k-m}} {(c,q^{1+m},az;q)_{k-m}}(d/a)^{k-m},\end{aligned}$$ which can be rewritten in the form of the right hand side of (\[xy\]). ------------------------------------------------------------------------ The next $_2\psi_2$ transformation formula we consider is the following [@GR Ex. 5.20(ii)]: $$\begin{aligned} \label{b2} _{2}\psi_{2}\left[ \begin{array}{c} a,b\\ c,d \end{array};q,z \right]=\frac{(az,bz,cq/abz,dq/abz;q)_{\infty}}{(q/a,q/b,c,d;q)_{\infty}} \, {_{2}}\psi_{2}\left[ \begin{array}{l} abz/c,abz/d\\ az,bz \end{array};q,\frac{cd}{abz} \right] .\end{aligned}$$ Using a summation of Hall [@GR Eq. (3.2.10)]: $$\begin{aligned} \label{Hall} _{3}\phi_2\left[ \begin{array}{l} a, b, c\\ d,e \end{array};q, \frac{de}{abc} \right]=\frac{(b,de/ab,de/bc;q)_{\infty}}{(d,e,de/abc;q)_{\infty}}{_{3}}\phi_2\left[ \begin{array}{l} d/b,e/b, de/abc\\ de/ab,de/bc \end{array};q, b \right],\end{aligned}$$ where $|de/abc|<1$ and $|b|<1$, we obtain the following semi-finite form of (\[b2\]). For $|z|<1$ and $|cd/abz|<1$, we have $$\begin{aligned} \sum_{k=-m}^{\infty}\frac{(a,b;q)_k(cdq^m/abz;q)_k}{(c,d;q)_k(q^{1+m};q)_k}z^k&=& \frac{(az,bz,cd/abz;q)_{\infty}}{(c,d,z;q)_{\infty}}\frac{(cq/abz,dq/abz,z;q)_m}{(q/a,q/b,cd/abz;q)_m}\\ && \;\;\; \cdot \sum_{k=-m}^{\infty}\frac{(abz/c,abz/d,zq^m;q)_{k}}{(az,bz,q^{1+m};q)_{k}}(cd/abz)^{k}.\end{aligned}$$ From nonterminating $_8\phi_7$ to $_6\psi_6$ ============================================ In this section, we give a semi-finite form of Bailey’s $_6\psi_6$ summation formula by using Bailey’s $3$-term transformation formula for a nonterminating very-well-poised $_8\phi_7$ series [@GR Eq. (2.11.1)]: $$\begin{aligned} \label{Watson2} \lefteqn{_{8}\phi_7\left[ \begin{array}{c} a, qa^{1\over 2},-qa^{1\over 2},b,c, d, e, f\\ a^{1\over 2},-a^{1\over 2},aq/b,aq/c,aq/d,aq/e,aq/f \end{array};q, \frac{a^2q^2}{bcdef} \right] }\nonumber\\ && \, =\frac{(aq,aq/de,aq/df,aq/ef,eq/c,fq/c,b/a,bef/a;q)_{\infty}} {(aq/d,aq/e,aq/f,aq/def,q/c,efq/c,be/a,bf/a;q)_{\infty}}\nonumber\\ && \qquad \cdot_{8}\phi_7\left[ \begin{array}{c} ef/c, q(ef/c)^{1\over 2},-q(ef/c)^{1\over 2},aq/bc,aq/cd, ef/a, e, f\\ (ef/c)^{1\over 2},-(ef/c)^{1\over 2}, bef/a,def/a,aq/c,fq/c,eq/c \end{array};q, \frac{bd}{a} \right] \nonumber\\ &&\qquad +\frac{b}{a}\frac{(aq,bq/a,bq/c,bq/d,bq/e,bq/f,d,e,f;q)_{\infty}} {(aq/b,aq/c,aq/d,aq/e,aq/f,bd/a,be/a,bf/a,def/a;q)_{\infty}}\nonumber\\ &&\qquad \cdot \frac{(aq/bc,bdef/a^2,a^2q/bdef;q)_{\infty}}{(aq/def,q/c,b^2q/a;q)_{\infty}}\nonumber\\ && \qquad \cdot _{8}\phi_7\left[ \begin{array}{c} b^2/a, qba^{-{1\over 2}},-qba^{-{1\over 2}},b,bc/a, bd/a, be/a, bf/a\\ ba^{-{1\over 2}},-ba^{-{1\over 2}},bq/a,bq/c,bq/d,bq/e,bq/f \end{array};q, \frac{a^2q^2}{bcdef} \right],\end{aligned}$$ where $|bd/a|<1$ and $|a^2q^2/bcdef|<1$. When $|bd/a|<1$ and $|a^2q^2/bcdef|<1$, we have $$\begin{aligned} \label{66} \lefteqn{\sum_{k=-m}^{\infty}\frac{(q^{m-k+1}/a,fq^m;q)_{k}}{(q^{m-k}f/a,q^{1+m};q)_k} \frac{(qa^{1\over 2},-qa^{1\over 2},b,c,d,e;q)_k}{(a^{1\over 2},-a^{1\over 2},aq/b,aq/c,aq/d,aq/e;q)_k} \left(\frac{qa^2}{bcde}\right)^k}\nonumber\\ &=&\frac{1-efq^{2m}/c}{1-efq^m/c}\frac{(q/a,df/a,ef/a,aq/bc,aq/cd,efq^m/a;q)_m} {(f/a,q/b,q/c,q/d,def/a,fq^{1+m}/c;q)_m}\nonumber\\ && \times\frac{(aq,aq/de,aq/df,aq/ef,eq^{1+m}/c,fq^{1+m}/c,b/a,befq^m/a;q)_{\infty}} {(aq/d,aq/e,aq/f,aq/def,q^{1+m}/c,efq^{1+m}/c,be/a,bfq^m/a;q)_{\infty}}\nonumber\\ && \times\sum_{k=-m}^{\infty}\frac{(efq^m/c,q^{1+m}(ef/c)^{1\over 2},-q^{1+m}(ef/c)^{1\over 2},aq^{1+m}/bc;q)_k} {(q^{1+m},q^{m}(ef/c)^{1\over 2},-q^{m}(ef/c)^{1\over 2},befq^m/a;q)_k}\nonumber\\ && \qquad \cdot\frac{(aq^{1+m}/cd,efq^{2m}/a,e,fq^m;q)_k}{(defq^m/a,aq/c,fq^{1+2m}/c,eq^{1+m}/c;q)_k} \left(\frac{bd}{a}\right)^k\nonumber\\ &&+\frac{b}{a}\frac{1-b^2q^{2m}/a}{1-b^2q^m/a}\left(\frac{a^2q}{bcde}\right)^m \frac{(q/a,bc/a;q)_{m}}{(f/a;q)_m}\frac{(aq,bq^{1+2m}/a,bq^{1+m}/c;q)_\infty} {(aq/b,aq/c,aq/d;q)_\infty}\nonumber\\ && \times \frac{(bq^{1+m}/d,bq^{1+m}/e,bq/f,d,e,fq^m,aq/bc,bdef/a^2,a^2q/bdef;q)_{\infty}} {(aq/e,aq/f,bdq^m/a,beq^m/a,bfq^{2m}/a,def/a,aq/def,q/c,b^2q^{1+m}/a;q)_{\infty}}\nonumber\\ && \times \sum_{k=-m}^{\infty}\frac{(b^2q^m/a,q^{1+m}ba^{-{1\over 2}},-q^{1+m}ba^{-{1\over 2}};q)_k} {(q^{1+m},q^{m}ba^{-{1 \over 2}},-q^mba^{-{1 \over 2}};q)_k}\nonumber\\ &&\qquad \cdot\frac{(b,bcq^m/a,bdq^m/a,beq^m/a,bfq^{2m}/a;q)_k}{(bq^{1+2m}/a,bq^{1+m}/c,bq^{1+m}/d,bq^{1+m}/e,bq/f;q)_k} \left(\frac{a^2q^2}{bcdef}\right)^k.\end{aligned}$$ The left hand side of (\[66\]) equals $$\begin{aligned} \lefteqn{\sum_{k=-m}^{\infty}\frac{(aq^{-m},fq^m;q)_{k}}{(aq^{1-m}/f,q^{1+m};q)_k} \frac{(qa^{1\over 2},-qa^{1\over 2},b,c,d,e;q)_k}{(a^{1\over 2},-a^{1\over 2},aq/b,aq/c,aq/d,aq/e;q)_k} \left(\frac{q^2a^2}{bcdef}\right)^k} \\ &\overset{(\ref{laurants})}{=}& \frac{(aq^{-m},fq^m,qa^{1\over 2},-qa^{1\over 2},b,c,d,e;q)_{-m}}{(aq^{1-m}/f,q^{1+m},a^{1\over 2},-a^{1\over 2},aq/b,aq/c,aq/d,aq/e;q)_{-m}} \left(\frac{q^2a^2}{bcdef}\right)^{-m}\\ &&\times \sum_{k=0}^{\infty}\frac{(aq^{-2m}, q^{1-m}a^{1\over 2},-q^{1-m}a^{1\over 2};q)_{k}}{(q,q^{-m}a^{1\over 2},-q^{-m}a^{1\over 2};q)_k}\\ && \qquad \, \cdot \frac{(bq^{-m},cq^{-m},dq^{-m},eq^{-m},f;q)_k}{(aq^{1-m}/b,aq^{1-m}/c,aq^{1-m}/d,aq^{1-m}/e,aq^{1-2m}/f;q)_k} \left(\frac{q^2a^2}{bcdef}\right)^k\\ &\overset{(\ref{Watson2})}{=}&\frac{(aq^{-m},fq^m,qa^{1\over 2},-qa^{1\over 2},b,c,d,e;q)_{-m}}{(aq^{1-m}/f,q^{1+m},a^{1\over 2},-a^{1\over 2},aq/b,aq/c,aq/d,aq/e;q)_{-m}} \left(\frac{q^2a^2}{bcdef}\right)^{-m}\\ && \times \frac{(aq^{1-2m},aq/de,aq^{1-m}/df,aq^{1-m}/ef,eq/c,fq^{1+m}/c,bq^m/a,bef/a;q)_{\infty}} {(aq^{1-m}/d,aq^{1-m}/e,aq^{1-2m}/f,aq/def,q^{1+m}/c,efq/c,be/a,bfq^m/a;q)_{\infty}}\\ && \times _{8}\phi_7\left[ \begin{array}{c} ef/c, q(ef/c)^{1\over 2},-q(ef/c)^{1\over 2},aq/bc,aq/cd, efq^m/a, eq^{-m}, f\\ (ef/c)^{1\over 2},-(ef/c)^{1\over 2}, bef/a,def/a,aq^{1-m}/c,fq^{1+m}/c,eq/c \end{array};q, \frac{bd}{a} \right]\\ && +\frac{(aq^{-m},fq^m,qa^{1\over 2},-qa^{1\over 2},b,c,d,e;q)_{-m}}{(aq^{1-m}/f,q^{1+m},a^{1\over 2},-a^{1\over 2},aq/b,aq/c,aq/d,aq/e;q)_{-m}} \left(\frac{q^2a^2}{bcdef}\right)^{-m}\\ && \times \frac{bq^m}{a}\frac{(aq^{1-2m},bq^{1+m}/a,bq/c,bq/d,bq/e,bq^{1-m}/f,dq^{-m};q)_{\infty}} {(aq^{1-m}/b,aq^{1-m}/c,aq^{1-m}/d,aq^{1-m}/e,aq^{1-2m}/f,bd/a,be/a;q)_{\infty}}\\ && \times \frac{(eq^{-m},f,aq/bc,bdefq^m/a^2,a^2q^{1-m}/bdef;q)_{\infty}} {(bfq^m/a,def/a,aq/def,q^{1+m}/c,b^2q/a;q)_\infty}\\ && \times _{8}\phi_7\left[ \begin{array}{c} b^2/a, qba^{- {1\over 2}},-qba^{- {1\over 2}},bq^{-m},bc/a,bd/a,be/a, bfq^m/a\\ ba^{-{ 1\over 2}},-ba^{-{1\over 2}}, bq^{1+m}/a,bq/c,bq/d,bq/e,bq^{1-m}/f \end{array};q, \frac{a^2q^2}{bcdef} \right],\end{aligned}$$ which equals to the right hand side of (\[66\]). ------------------------------------------------------------------------ The above proposition can be viewed as a semi-finite form of Bailey’s $_6\psi_6$ summation formula. By taking $f=b$ and $m \rightarrow \infty$ in (\[66\]) while assuming $|a^2q/bcde|< 1$, we get $$\begin{aligned} \lefteqn{_{6}\psi_6\left[ \begin{array}{c} qa^{1\over 2},-qa^{1\over 2},b,c, d, e\\ a^{1\over 2},-a^{1\over 2},aq/b,aq/c,aq/d,aq/d,aq/e \end{array};q, \frac{a^2q}{bcde} \right] }\\ &=&\frac{(q/a,bd/a,aq/bc,aq/cd,aq, aq/de, aq/bd, aq/be;q)_{\infty}}{(q/b,q/c,q/d,aq/b, aq/d,aq/e,aq/bde,bde/a;q)_{\infty}}\\ && \times \sum_{k=-\infty}^{\infty}\frac{(e;q)_k}{(aq/c;q)_k}(bd/a)^k\\ &\overset{(\ref{Ran})}{=}&\frac{(q/a,bd/a,aq/bc,aq/cd,aq, aq/de, aq/bd, aq/be;q)_{\infty}}{(q/b,q/c,q/d,aq/b, aq/d,aq/e,aq/bde,bde/a;q)_{\infty}}\\ && \times \frac{(q,aq/ce,bde/a,aq/bde;q)_{\infty}} {(aq/c,q/e,bd/a,a^2q/bcde;q)_{\infty}}\\ &=&\frac{(aq,aq/bc,aq/bd,aq/be,aq/ce,aq/cd,aq/de,q,q/a;q)_{\infty}} {(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^2/bcde;q)_{\infty}}.\end{aligned}$$ Many proofs of above identity have been found, see, for example, Slater and Lakin [@SlLa], Andrews [@Andrews], Askey and Ismail [@AsIs], Askey [@Askey], Chen and Liu [@ChenLiu], Schlosser [@SCHL] and Jouhet and Schlosser [@SCHL04]. Our proof shows that the semi-finite form of the ${}_6\psi_6$ summation is in essence a shifted version of the $_8\phi_7$ summation. This proof utilizes Ramanujan’s ${}_1\psi_1$ summation (\[Ran\]). It would be interesting to find a proof using a semi-finite (or even finite) form which yields Bailey’s ${}_6\psi_6$ summation in a direct limit, without the need to invoke another summation formula as above. [**Acknowledgments.**]{} We thank R. Askey, Wenchang Chu and M. E. H. Ismail for their valuable comments. In particular, we are indebted to M. Schlosser for crucial suggestions leading to considerable improvement of an earlier version. This work was done under the auspices of the 973 Project on Mathematical Mechanization, the Ministry of Education, the Ministry of Science and Technology, and the National Science Foundation of China. [99]{} G. E. Andrews, Applications of basic hypergeometric functions, SIAM, Rev. 16 (1974), 441-484. R. Askey and M. E. H. Ismail, The very well poised $_6\psi_6$, Proc. Amer. Math. Soc., 77 (1979), 218-222. R. Askey, The very well poised $_6\psi_6$ II, Proc. Amer. Math. Soc., 90 (1984), 575-579. W. N. Bailey, Series of hyerpergeometric type which are infinite in both directions, Quart. J. Math., 7 (1936), 105-115. W. Y. C. Chen and Z. G. Liu, Parameter augmentation for basic hypergeometric series I, Mathematical Essays in Honor of Gian-Carlo Rota, Eds., B. E. Sagan and R. P. Stanley, Birkhäuser, Boston, 1998, pp. 111-129. G. Gasper and M. Rahman, Basic Hypergeometric Series, $2^{{\mathrm{nd}}}$ ed., Encyclopedia of Mathematics and Its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004. M. E. H. Ismail, A simple proof of Ramanujan’s $_{1}\psi_1$ sum, Proc. Amer. Math. Soc., 63 (1977) 185-186. F. Jouhet and M. Schlosser, Another proof of Bailey’s $_6\psi_6$ summation, Aequationes Math., to appear. M. Schlosser, A simple proof of Bailey’s very-well-poised $_6\psi_6$ summation, Proc. Amer. Math. Soc., 130 (2002), 1113-1123. M. Schlosser, Abel-Rothe type generalizations of Jacobi’s triple product identity, in “Theory and Applications of Special Functions, A Volume Dedicated to Mizan Rahman" (M. E. H. Ismail and E. Koelink, eds.), Dev. Math., to appear. L. J. Slater and A. Lakin, Two proofs of the $_6\psi_6$ summation theorem, Proc. Edin. Math. Soc., 9 (1956), 116-121.

--- abstract: | Let $q = p^s$ be a power of a prime number $p$ and let ${\mathbb{F}_q}$ be the finite field with $q$ elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ where both $r \geq 3$ and $q$ are odd, $\gcd(q,r) = 1,$ and $n\in \mathbb{N}.$ Previously, only the special cases when $r = 1,\ 3,\ 5,$ had been achieved. For this we make the assumption that the explicit factorization of $\Phi_r$ over ${\mathbb{F}_q}$ is given to us as a known. Let $n = p_1^{e_1}p_2^{e_2} \cdots p_s^{e_s}$ be the factorization of $n \in \mathbb{N}$ into powers of distinct primes $p_i,\ 1\leq i \leq s.$ In the case that the orders of $q$ modulo all these prime powers $p_i^{e_i}$ are pairwise coprime we show how to obtain the explicit factors of $\Phi_{n}$ from the factors of each $\Phi_{p_i^{e_i}}.$ We also demonstrate how to obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),{\operatorname{ord}}_n(q)) = 1.$ Here $\phi$ is the Euler’s totient function, and ${\operatorname{ord}}_n(q)$ denotes the multiplicative order of $q$ modulo $n.$ Moreover, we present the construction of a new class of irreducible polynomials over ${\mathbb{F}_q}$ and generalize a result due to Varshamov (1984) [@Varshamov]. address: - 'School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada.' - 'School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada.' author: - Aleksandr Tuxanidy - Qiang Wang title: Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields --- [^1] Introduction ============ Composed Products and Applications ---------------------------------- Let $q = p^s$ be a power of a prime $p,$ and ${\mathbb{F}_q}$ be a finite field with $q$ elements. The multiplicative version of composed products of two polynomials $f,\ g \in {\mathbb{F}_q}[x]$ (or composed multiplication for short) defined by $$(f \odot g)(x) = \prod_{\alpha}\prod_{\beta} (x - \alpha \beta)$$ where the product $\prod_\alpha \prod_{\beta}$ runs over all roots $\alpha,\ \beta$ of $f,\ g$ respectively, was first introduced by Selmer (1966) [@Selmer] for the purposes of studying linear recurrence sequences (LRS). Informally, LRS’s are sequences whose terms depend linearly on a finite number of its predecessors; thus a famous example of a LRS is the Fibonacci sequence whose terms are the sum of the previous two terms. Let $k$ be a positive integer and let $a,a_0,\dots,a_{k-1}$ be given elements in ${\mathbb{F}_q}.$ Then a sequence $S = \{s_0,s_1,\dots\}$ of elements $s_i \in {\mathbb{F}_q}$ satisfying the relation $$s_{n+k} = a_{k-1}s_{n+k-1} + a_{k-2}s_{n+k -2} + \dots + a_0s_n + a,{\hspace*{2em}}n=0,1,\dots$$ is a LRS. If $a = 0,$ then $S$ is called a *homogeneous* LRS. If we let $k = 2,\ a = 0,\ a_0 = a_1 = 1$ and $s_0 = 0,\ s_1 = 1$ then $S$ becomes the (homogeneous) Fibonacci sequence. LRS’s have applications in coding theory, cryptography, and other areas of electrical engineering where electric switching circuits such as linear feedback shift registers (LFSR) are used to generate them. See Chapter 8 in [@Lidl] for this and a general introduction. In particular, the matter of the linear complexity of a LRS, and more generally, the linear complexity of the component wise multiplication of LRS’s, is of great importance in stream cipher theory, a branch in cryptography; here a higher complexity is preferred. See [@Gao] for instance and the references contained therein. Since the linear complexity of a LRS is given by the degree of the minimal polynomial of the LRS, minimal polynomials with higher degrees are therefore preferred. The polynomial $$f(x) = x^k -a_{k-1}x^{k-1} - a_{k-2}x^{k-2} - \dots - a \in {\mathbb{F}_q}[x]$$ is called the *characteristic polynomial of S* (see [@Lidl]). In 1973, Zierler and Mills [@Zierler] showed that the characteristic polynomial of a component wise multiplication of homogeneous LRS’s is the composed multiplication of the characteristic polynomials of the respective LRS’s. That is, if $S_1,S_2,\dots,S_r$ are homogeneous LRS’s with respective characteristic polynomials $f_1,f_2,\dots, f_r,$ then the characteristic polynomial of $S_1S_2 \cdots S_r,$ with component wise multiplication, is given by $f_1 \odot f_2 \odot \dots \odot f_r.$ We refer the reader to page 433-435 in [@Lidl] as well. Note that since the required minimal polynomials are factors of the characteristic polynomials $f_1 \odot f_2 \odot \dots \odot f_r$ of LRS’s, the study of factorizations of composed products has an important significance. Thus composed products have applications in stream cipher theory, LFSR, and LRS in general. Similarly, the *composed sum* of $f, g \in {\mathbb{F}_q}[x]$ is defined by $$(f \oplus g)(x) = \prod_\alpha \prod_\beta (x - (\alpha + \beta))$$ where the product runs over all the roots $\alpha$ of $f$ and $\beta$ of $g,$ including multiplicities. In 1987, Brawley and Carlitz [@Brawley; @and; @Carlitz] generalized composed multiplications and composed sums in the following. [**[@Brawley; @and; @Carlitz] (Composed Product)**]{} Let $G$ be a non-empty subset of the algebraic closure $\Gamma_q$ of ${\mathbb{F}_q}$ with the property that $G$ is invariant under the Frobenius automorphism $\alpha \mapsto \sigma(\alpha) = \alpha^q$ (i.e., if $\alpha \in G,$ then $\sigma(\alpha) \in G$). Suppose a binary operation $\diamond$ is defined on $G$ satisfying $\sigma(\alpha \diamond \beta) = \sigma(\alpha)\diamond \sigma(\beta)$ for all $\alpha,\beta \in G.$ Then the *composed product* of $f$ and $g,$ denoted by $f \diamond g,$ is the polynomial defined by $$(f \diamond g)(x) = \prod_\alpha \prod_\beta (x - (\alpha \diamond \beta)),$$ where the $\diamond$-products run over all roots $\alpha$ of $f$ and $\beta$ of $g.$ Observe that $\deg (f \diamond g) = (\deg f)(\deg g)$ clearly. Moreover, in [@Brawley; @and; @Carlitz] it is noted that when $G = {\Gamma_q}-\{0\}$ (respectively, ${\Gamma_q}$) and $\diamond$ is the usual multiplication (respectively, addition) then $f \diamond g$ becomes $f \odot g$ (respectively, $f \oplus g,$). Other less common examples are \(i) $G = {\Gamma_q},\ \alpha \diamond \beta = \alpha + \beta - c$ where $c \in {\mathbb{F}_q}$ is fixed. \(ii) $G = {\Gamma_q}- \{1\},\ \alpha \diamond \beta = \alpha + \beta - \alpha\beta$ (sometimes called the circle product), and \(iii) $G =$ any $\sigma$-invariant subset of ${\Gamma_q}, \alpha \diamond \beta = f(\alpha,\beta)$ where $f(x,y)$ is any fixed polynomial in ${\mathbb{F}_q}[x,y]$ such that $f(\alpha,\beta) \in G$ for all $\alpha, \beta \in G.$ Let $M_G[q,x]$ be the set of all monic polynomials over ${\mathbb{F}_q}$ of degree $\geq 1$ whose roots lie in $G.$ It is also shown in [@Brawley; @and; @Carlitz] that the condition $\sigma(\alpha \diamond \beta) = \sigma(\alpha)\diamond \sigma(\beta)$ implies that $f \diamond g \in {\mathbb{F}_q}[x].$ Moreover, if $\diamond$ is an associative (respectively, commutative) product on $G,$ the composed product is associative (respectively, commutative) on $M_G[q,x].$ In particular, composed multiplications and sums of polynomials are associative and commutative in ${\mathbb{F}_q}[x].$ In fact, $(G, \diamond)$ is an abelian group for composed multiplication, composed addition, and the example in (i) or (ii). Irreducible Constructions ------------------------- The construction of irreducible polynomials over finite fields is currently a strong subject of interest with important applications in coding theory, cryptography and complexity theory ([@Cohen], [@Cohen; @2005], [@Kyuregyan], [@Lidl], [@Varshamov]). One of the most popular methods of construction is the method of composition of polynomials (not to be confused with composed products) where an irreducible polynomial of a higher degree is produced from a given irreducible polynomial of lower degree by applying a substitution operator. For a recent survey of previous results up to the year 2005 on this subject see [@Cohen; @2005]. Perhaps one of the most applicable results in this area is the following. \[cohen thm\] Let $f$ and $g$ be two non-zero relatively prime irreducible polynomials over ${\mathbb{F}_q}$ and $P$ be an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n >0.$ Then the composition $$F(x) = g(x)^nP\left(f(x)/g(x)\right)$$ is irreducible over ${\mathbb{F}_q}$ if and only if $f - \alpha g$ is irreducible over $\mathbb{F}_{q^n}$ for some root $\alpha \in \mathbb{F}_{q^n}$ of $P.$ Note that Theorem \[cohen thm\] has been used extensively in the past by several authors in order to produce iterative constructions of irreducible polynomials. See [@Cohen; @2005] for instance and the references there. Recently, Kyuregyan-Kyureghyan provides another proof of Theorem \[cohen thm\] in [@Kyuregyan] using the idea of composing factors of irreducible polynomials over extension fields. Suppose $f$ is an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n$ and $g(x) = \sum_{i=0}^{n/d} g_i x^i \in \mathbb{F}_{q^d}[x]$ is a factor of $f.$ Then all the remaining factors are $$g^{(u)}(x) = \sum_{i=0}^{n/d} g_i^{q^u} x^i,$$ where $1\leq u \leq d-1.$ We denote $g = g^{(0)},$ and thus $f = \prod_{u=0}^{d-1} g^{(u)}.$ Conversely, given an irreducible polynomial $g$ of degree $n/d$ over $\mathbb{F}_{q^d},$ we can form the product $f = \prod_{u=0}^{d-1} g^{(u)}.$ However, $f$ is not always an irreducible polynomial over ${\mathbb{F}_q}.$ It is an irreducible polynomial only when $\mathbb{F}_{q^d}$ is the smallest extension field of ${\mathbb{F}_q}$ containing the coefficients of $g,$ i.e., when ${\mathbb{F}_q}(g_0, \dots, g_k) = \mathbb{F}_{q^d}.$ In particular, they obtain the following. \[kk thm\] Let $k > 1$, $\gcd(k, d) =1$, and $f$ be an irreducible polynomial of degree $k$ over ${\mathbb{F}_q}$. Further let $\alpha \neq 0$ and $\beta$ be elements of $\mathbb{F}_{q^d}$. Set $g(x) := f(\alpha x + \beta)$. Then the polynomial $$F = \prod_{u=0}^{d-1} g^{(u)}$$ of degree $n = dk$ is irreducible over ${\mathbb{F}_q}$ if and only if ${\mathbb{F}_q}(\alpha, \beta) = \mathbb{F}_{q^d}$. We note that besides the above results there are others that are, perhaps, equally applicable in this area. In particular, a result due to Brawley and Carlitz (1987) [@Brawley; @and; @Carlitz], is also instrumental in the construction of irreducible polynomials of relatively higher degree from given polynomials of relatively lower degrees. \[thm 1x\] Suppose that $(G,\diamond)$ is a group and let $f,g \in M_G[q,x]$ with $\deg f = m$ and $\deg g = n.$ Then the composed product $f \diamond g$ is irreducible if and only if $f$ and $g$ are both irreducible with $\gcd(m,n) = 1.$ In Section 2 we construct irreducible polynomials through the use of composed products. First, we show that for some choices of $\alpha,\ \beta,$ the product of irreducible polynomials in Theorem \[kk thm\], $F,$ is in fact a composed product, and therefore can be derived from Theorem \[thm 1x\]. Moreover, we obtain several concrete constructions of irreducible polynomials (Theorem \[thm 2\] and Theorem \[thm 3\]) where Theorem \[thm 3\] generalizes a classical result due to Varshamov [@Varshamov] (see also Theorem 3 [@Kyuregyan]) and both Theorems \[thm 2\], \[thm 3\], use cyclotomic polynomials as one of two inputs of composed products. Factorization of Cyclotomic Polynomials --------------------------------------- Let $\Phi_n$ denote the $n$-th cyclotomic polynomial $$\Phi_n(x) = \prod_{0 < j \leq n,\ (j,n) = 1}\left(x-\xi_n^j\right)$$ where $\xi_n$ is a primitive $n$-th root of unity. Clearly, $x^n-1 = \prod_{d \mid n}\Phi_{d}(x)$ and the Mobius Inversion Formula gives $\Phi_n(x) = \prod_{d \mid n}(x^d-1)^{\mu(n/d)}$ where $\mu$ is the Mobius function. Cyclotomic polynomials have been studied extensively since they first appeared in the 18th century works of Euler, Lagrange, Gauss, and others, and to this day continue to be a strong subject of interest in Mathematics ([@Bamunoba], [@Sury], [@Washington]). This is a class of polynomials which naturally arise in the classical 2000 year old Greek problem of Cyclotomy which concerns the division of the circumference of the unit circle into $n$ equal parts, a problem that was finally solved by Gauss at the turn of the 19th century. It is well known the fact that all cyclotomic polynomials are irreducible over the field of rational numbers; this is not the case over finite fields. In fact, $\Phi_n$ decomposes into $\phi(n)/d$ irreducibles over ${\mathbb{F}_q}$ of the same degree $d = {\operatorname{ord}}_n(q)$ (see Theorem 2.47 in [@Lidl]). The first steps in the factorization of cyclotomic polynomials over finite fields were made in the 19th century by Gauss, Pellet and others who restricted their studies to the prime fields $\mathbb{F}_p$ (p. 77, [@Lidl]). More recently, Fitzgerald and Yucas (2005) [@Fitzgerald; @2005] discovered a relationship between the factorization of cyclotomic polynomials and that of Dickson polynomials of the first and second kind. This provides us with an alternative method to factor a Dickson polynomial when we know the factorization of the corresponding cyclotomic polynomial. However, the problem of the explicit factorization of cyclotomic polynomials over finite fields still remains open. We now give a brief survey of some of the past accomplishments regarding the factorization of cyclotomic polynomials over finite fields; these are especially related to our quest to factor $\Phi_{2^nr}.$ The factorization of $\Phi_{2^n}$ over ${\mathbb{F}_q}$ when $q \equiv 1 \pmod{4}$ can be found for example in [@Lidl] and is stated here in Theorem \[2\^n and q = 1 mod 4\]; the more difficult case when $q \equiv 3 \pmod{4}$ was achieved in 1996 by Meyn [@Meyn]. More recently, Fitzgerald and Yucas (2007) [@Fitzgerald; @2007] gave the factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ for the special cases where $r$ is an odd prime and $q \equiv \pm 1 \pmod{r}$ is odd. As a result, the factorizations over ${\mathbb{F}_q}$ of $\Phi_{2^n3},$ and the Dickson polynomials of the first and second kind $D_{2^n3},\ E_{2^n3-1},$ respectively, are thus obtained. In 2011, L. Wang and Q. Wang [@Prof] went a step further and gave the factorization of $\Phi_{2^n5}$ over ${\mathbb{F}_q}.$ In this paper we obtain the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ for arbitrary $r \geq 3$ odd and $q$ odd such that $\gcd(q,r) = 1.$ Thus, we generalize the results in [@Fitzgerald; @2007] and [@Prof]. We make the assumption that the explicit factorization of $\Phi_r$ is given to us as a known. When $q = p$ and $r$ is an odd prime (distinct from $p$) one may use for instance the results due to Stein (2001) [@Stein] to compute the factors of $\Phi_r$ efficiently. We achieve our result by applying the theory of composed products as well as by using, and refining in some cases, some of the techniques and results in [@Prof] now generalized for arbitrary odd number $r > 1.$ In particular, we refine the following result of theirs. Let $v_2(k)$ denote the highest power of $2$ dividing $k.$ \[L\] Let $q = p^s$ be a power of an odd prime $p,$ let $r \geq 3$ be any odd number such that $\gcd(r,q) = 1,$ and let $L:= L_{\phi(r)} = v_2\left(q^{\phi(r)} - 1\right)$ be the highest power of $2$ dividing $q^{\phi(r)} - 1.$ For any $n \geq L$ and any irreducible factor $f$ of $\Phi_{2^nr}$ over ${\mathbb{F}_q},$ $f(x^{2^{n-L}})$ is also irreducible over ${\mathbb{F}_q}.$ Moreover, all irreducible factors of $\Phi_{2^nr}$ are obtained in this way. This result implies that if the factorization of $\Phi_{2^Lr}$ is known, then for $n > L$ we can obtain the factorization of $\Phi_{2^nr}$ by simply applying the substitution $x \rightarrow x^{2^{n-L}}$ to each of the irreducible factors of $\Phi_{2^Lr}.$ Thus it only remains to factor $\Phi_{2^nr}$ when $1 \leq n \leq L.$ We improve the result stated above by giving a smaller bound $K = v_2(q^{d_r} - 1) \leq L,$ when $d_r = {\operatorname{ord}}_r(q)$ is even or $q \equiv 1 \pmod{4};$ here $K$ has the same properties as $L$ just described, i.e. if the factorization of $\Phi_{2^Kr}$ is known, then for $n > K$ we obtain the factorization of $\Phi_{2^nr}$ by applying the substitution $x \rightarrow x^{2^{n-K}}$ to each of the irreducible factors of $\Phi_{2^Kr}.$ In the case $d_r$ is odd and $q \equiv 3 \pmod{4},$ we show that the corresponding bound is $v_2(q+1) < L.$ Consequently, it only remains to factor $\Phi_{2^nr}$ when $1 \leq n \leq K $ (or $v_2(q+1)$) $\leq L.$ Moreover, we show that $K$ and $v_2(q+1)$ are the smallest such bounds can be in these cases. In order to obtain the irreducible factors when $1 \leq n \leq L,$ the authors of [@Prof] employed the properties $\Phi_{2r}(x) = \Phi_{r}(-x),$ and $\Phi_{2^{n}r}(x) = \Phi_{2^{n-1}r}(x^2),\ n > 1,$ of cyclotomic polynomials, together with an iteration of $L$ steps that consists of the following strategy:\ \ 1. Obtain the factorization for $n = 0,1.$\ 2. For $1< n \leq L$ and each irreducible factor $h_{n-1}(x)$ of $\Phi_{2^{n-1}r}(x),$ factor $h_{n-1}(x^2)$ into irreducibles; these are all the irreducible factors of $\Phi_{2^{n}r}(x).$\ If $n = L,$ stop.\ First, note that since $q > 1$ is odd, we may write $q = 2^Am \pm 1,$ for some $A \geq 2,$ and some $m$ odd. Some of our improvements to the above are as follows: In the case $n \leq A$ or $d_r = {\operatorname{ord}}_r(q)$ odd, we give the explicit factorization of $\Phi_{2^nr}$ without the need of any iterations. On the other hand, in the case $d_r$ is even and $n > A,$ we use a similar strategy to step 2, where we replace $L$ by $K.$ We show that in the case $d_r$ even it is enough to iterate for at most $v_2(d_r) < L$ steps starting at $n = A.$ This is quite significant as $L = A + v_2(\phi(r)),$ and so if $A$ is large, say when $q = 2^A - 1$ is a large Mersenne prime, then $L$ is large. However, as discussed, we only need to iterate for at most $v_2(d_r)$ steps which is relatively much smaller. We remark that, similarly as done in [@Prof], whenever $d_r$ is even or $q \equiv 1 \pmod{4}$ the factorization of $\Phi_{2^nr}$ can also be formulated in terms of a system of non-linear recurrence relations for $n \leq K.$ For small finite fields and small $d_r,$ this can be computed fairly fast. As the reader can infer from the previous discussion on the properties of the bounds $K$ and $v_2(q+1)$, the irreducible factors of these cyclotomic polynomials $\Phi_{2^nr}$ are sparse polynomials with a relatively small fixed amount of non-zero coefficients and a relatively much higher (as high as needed) degree. For applications of sparse polynomials in LRS, efficient implementation of LFSR, and in finite field arithmetic, see for instance [@Berlekamp], [@Golomb], and [@Blake]. Moreover, as another consequence to our factorization, we obtain infinite families of irreducible polynomials. We show in Section 3.1 that cyclotomic polynomials are composed multiplications of other cyclotomic polynomials of lower order. In particular, $\Phi_{2^nr} = \Phi_{2^n} \odot \Phi_r.$ As a result, we now have at our disposal additional tools such as the results due to Brawley and Carlitz (1987) [@Brawley; @and; @Carlitz] which we quote in Section 2.1; these are instrumental to our results. We remark that none of the previous authors listed above in our survey considered this insight. Let $n = p_1^{e_1}p_2^{e_2} \cdots p_s^{e_s}$ be the factorization of $n \in \mathbb{N}$ into powers of distinct primes $p_i,\ 1\leq i \leq s.$ In the case that the orders of $q$ modulo all these prime powers $p_i^{e_i}$ are pairwise coprime, in Theorem \[cyclotomics are composed\] we show how to obtain the factorization of $\Phi_{n}$ from the factorizations of each $\Phi_{p_i^{e_i}}.$ In Theorem \[cyclo and minimal\] we demonstrate how to obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),{\operatorname{ord}}_n(q)) = 1.$ Note that if $S = \{s_k\},\ T = \{t_k\},$ are homogeneous LRS’s with characteristic polynomials $\Phi_{2^n},\ \Phi_r,$ respectively, then the characteristic polynomial of $ST = \{s_k t_k\}$ is $\Phi_{2^nr} = \Phi_{2^n} \odot \Phi_r$ by our previous discussion on composed products. We obtain that for $n$ strictly greater than the corresponding bound $K$ or $v_2(q+1),$ the linear complexity of such $ST$ is of the form $2^{z(n)}d_r$ where $z(n) = n-K$ or $z(n) = n - v_2(q+1) + 1,$ respectively. Thus by letting $n \rightarrow \infty,$ the LRS $ST$ will have a linear complexity approaching infinity. As previously discussed, this is very desirable in stream cipher theory. The rest of this paper goes as follows. In Section 2.1 we discuss a few more properties of composed products and show that some cases of the Kyuregyan-Kyureghyan’s construction are composed products. In Section 2.2 we give some results regarding the constructions of irreducible polynomials; for this we made use of a theorem on the irreducibility of composed products, due to Brawley and Carlitz (1987). We consider Theorem \[thm 3\] our main result in this section. As a corollary, this generalizes a result due to Varshamov (1984). As another consequence to Theorem \[thm 3\], in Theorem \[cyclo and minimal\] we show how to obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),{\operatorname{ord}}_n(q)) = 1.$ In Section 3.1 we give a number of results which we later use in order to obtain the factorization of $\Phi_{2^nr}.$ In Sections 3.2 and 3.3 we give the factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ when $q \equiv 1 \pmod{4}$ and $q \equiv 3 \pmod{4},$ respectively. Finally in Appendix A we give a table of examples for Theorem \[thm 3\] and two tables of examples in Appendix B testing the recurrence relations in Theorems \[2\^nr and q = 1 mod 4\] and \[2\^nr and q = 3 mod 4\]. Irreducible Composed Products and Cyclotomic Polynomials {#section:irredCon} ======================================================== In this section we apply Theorem \[thm 1\], due to Brawley and Carlitz [@Brawley; @and; @Carlitz], in the construction of new classes of irreducible polynomials of higher degrees from irreducible polynomials of lower degrees. We devote most of our attention to polynomials of the form $f \odot \Phi_n.$ We consider Theorem \[thm 3\] our main result in this section. As a corollary, this generalizes a result due to Varshamov (1984) [@Varshamov]. As another consequence to Theorem \[thm 3\] we show in Theorem \[cyclo and minimal\] how to obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),{\operatorname{ord}}_n(q)) = 1.$ First, in Section 2.1 we give a number of known results in the theory of composed products which are instrumental. Composed Products ----------------- We need the following known results regarding composed products. \[comput\] Let $f,\ g \in {\mathbb{F}_q}[x].$ Then $$\left(f\odot g\right)(x) = \prod_\alpha \alpha^{n}g\left(\alpha^{-1}x\right)$$ and $$\left(f\oplus g\right)(x) = \prod_\alpha g\left(x - \alpha\right)$$ where the products $\prod_\alpha$ run over all the roots $\alpha$ of $f.$ $$\begin{aligned} \left(f\odot g\right)(x) &=& \prod_\alpha \prod_\beta \left(x - \alpha \beta\right) = \prod_\alpha \prod_\beta \alpha\left(\alpha^{-1}x - \beta\right) = \prod_\alpha \alpha^{n} g\left(\alpha^{-1}x\right).\\\\ \left(f \oplus g\right)(x) &=& \prod_\alpha \prod_\beta \left(x - \left(\alpha + \beta\right)\right) = \prod_\alpha \prod_\beta \left(\left(x - \alpha \right) - \beta\right) = \prod_\alpha g\left(x - \alpha\right).\qedhere\end{aligned}$$ \[product of composed\] Let $f_i,\ 1\leq i \leq s,\ g_j,\ 1 \leq j \leq t,$ be polynomials over ${\mathbb{F}_q}.$ Then $$\left(\prod_i f_i \odot \prod_j g_j\right) = \prod_i \prod_j \left(f_i \odot g_j\right).$$ As we remarked earlier, $(G, \diamond)$ is an abelian group when $\diamond$ is the composed multiplication $\odot$, composed sum $\oplus$, or circle product $\otimes.$ Theorem \[thm 1x\] therefore deduces the following consequence. \[thm 1\] Let $f,\ g \in {\mathbb{F}_q}[x]$ of degree $m,\ n,$ respectively. Then $f \odot g$, $f \oplus g$, $f \otimes g$ are irreducible over ${\mathbb{F}_q}$ if and only if $f,\ g$ are irreducible over ${\mathbb{F}_q}$ and $\gcd(m,n) = 1.$ Now we show that some cases of the construction in Theorem \[kk thm\] are in fact composed products and therefore consequences of Theorem \[thm 1\]. Let $\gcd(k, d) =1$, and $f$ be an irreducible polynomial of degree $k$ over ${\mathbb{F}_q}$. Further let $\alpha \neq 0$ and $\beta$ be elements of $\mathbb{F}_{q^d}$. Set $g(x) := f(\alpha x + \beta)$ and let $$F = \prod_{u=0}^{d-1} g^{(u)}$$ be a polynomial over ${\mathbb{F}_q}$ of degree $n = dk$. Then \(i) if $\alpha \in {\mathbb{F}_q}$ and ${\mathbb{F}_q}(\beta) = \mathbb{F}_{q^d}$, then $F$ is a composed sum of two irreducible polynomials with degrees $k$ and $d$ respectively, hence irreducible. \(ii) if $\beta \in {\mathbb{F}_q}$ and ${\mathbb{F}_q}(\alpha) = \mathbb{F}_{q^d}$, then $F$ is a composed multiplication of two irreducible polynomials with degrees $k$ and $d$ respectively, hence irreducible. \(iii) if ${\mathbb{F}_q}(\alpha) = \mathbb{F}_{q^d}$ and $\beta = c \alpha,$ where $c \in {\mathbb{F}_q},$ then $F$ is the result of a linear substitution operation $x \rightarrow (x + c)$ applied to an irreducible composed multiplication, and hence irreducible. \(iv) if $\alpha = -\beta + 1$ and ${\mathbb{F}_q}(\alpha, \beta) = \mathbb{F}_{q^d},$ then $F$ is the circle product of two irreducible polynomials with degrees $k$ and $s$ respectively, where $s\mid d,$ hence irreducible. \(v) if $\alpha = \beta + 1$ and ${\mathbb{F}_q}(\alpha, \beta) = \mathbb{F}_{q^d},$ then $F$ is the composed product of two irreducible polynomials with degrees $k$ and $s$ respectively, where $s\mid d,$ hence irreducible. \(i) Because $\alpha \in {\mathbb{F}_q}$, we write $\bar{f}(x) = f(\alpha x)$. So $\bar{f} (x)$ is also an irreducible polynomial of degree $k$ over ${\mathbb{F}_q}$. Therefore, by Proposition \[comput\], $$F(x) = \prod_{u=0}^{d-1} f^{(u)}(\alpha x + \beta) = \prod_{u=0}^{d-1} \bar{f}^{(u)}( x + \alpha^{-1} \beta)$$ is the composed sum of $\bar{f}$ and the minimal polynomial of $\alpha^{-1} \beta$ (an irreducible polynomial of degree $d$). \(ii) In this case, let $\bar{f}(x) = f(x + \beta)$. So $\bar{f} (x)$ is also an irreducible polynomial of degree $k$ over ${\mathbb{F}_q}$. $$F(x) = \prod_{u=0}^{d-1} f^{(u)}(\alpha x + \beta) = \prod_{u=0}^{d-1} \bar{f}^{(u)}( \alpha x).$$ Hence all the roots of $F$ are the product of roots of $\bar{f}$ and roots of the minimal polynomial of $\alpha^{-1};$ moreover, both are irreducible polynomials over ${\mathbb{F}_q}.$ Therefore $F$ is the irreducible composed multiplication of $\bar{f}$ and the minimal polynomial of $\alpha^{-1}$ (both have coprime degrees). \(iii) Note that $\prod_{u=0}^{d-1} \alpha^{-kq^u}f\left(\alpha^{q^u} x\right)$ is an irreducible composed multiplication over ${\mathbb{F}_q}.$ Thus, since $\prod_{u=0}^{d-1}\alpha^{-kq^u} \in {\mathbb{F}_q}^*,$ it must be that $$H(x) = \prod_{u=0}^{d-1} f\left(\alpha^{q^u} x\right) = \prod_{u=0}^{d-1} f^{(u)}(\alpha x)$$ is irreducible as well over ${\mathbb{F}_q}.$ But then $$H(x + c) = \prod_{u=0}^{d-1} f^{(u)}(\alpha(x + c)) = \prod_{u=0}^{d-1} f^{(u)}(\alpha x + \beta) = F(x)$$ is irreducible over ${\mathbb{F}_q}.$ \(iv) Let $h$ be the minimal polynomial of $-\alpha^{-1} + 1$. Because ${\mathbb{F}_q}(\alpha, \beta) = \mathbb{F}_{q^d}$, there are $s \mid d$ distinct conjugates of $- \alpha^{-1} + 1$ and thus the degree of $h$ is $s$. We denote an arbitrary root of $f$ and $h$ by $\alpha_f$ and $\alpha_h$ respectively. Then an arbitrary root of $ F(x) = \prod_{u=0}^{d-1} f^{(u)}(\alpha x + \beta) $ can be written as $$\alpha^{-1} (\alpha_f - \beta) = \alpha^{-1} (\alpha_f + \alpha - 1) = \alpha^{-1} \alpha_f + 1 - \alpha^{-1} = (1 - \alpha_h) \alpha_f + \alpha_h = \alpha_f + \alpha_h - \alpha_f \alpha_h.$$ Because $h$ has degree $s \mid d$ as a consequence of ${\mathbb{F}_q}(\alpha, \beta) = \mathbb{F}_{q^d},$ the polynomial $F$ is the composed product of two irreducible polynomials of coprime degrees, and hence irreducible. \(v) Here we define the composed product $\diamond$ for $G = \Gamma_{q} - \{ -1\}$ by $a \diamond b = a + b + ab,$ which forms an abelian group similar to the group corresponding to the circle product. Similarly, let $h$ be the minimal polynomial of $\alpha^{-1} - 1$ and denote an arbitrary root of $f$ and $h$ by $\alpha_f$ and $\alpha_h$ respectively. Then an arbitrary root of $ F(x) = \prod_{u=0}^{d-1} f^{(u)}(\alpha x + \beta) $ can be written as $$\alpha^{-1} (\alpha_f - \beta) = \alpha^{-1} (\alpha_f - \alpha +1) = \alpha^{-1} \alpha_f - 1 + \alpha^{-1} = (1+\alpha_h) \alpha_f + \alpha_h = \alpha_f + \alpha_h + \alpha_f \alpha_h.$$ Because $h$ has degree $s \mid d$ as a consequence of ${\mathbb{F}_q}(\alpha, \beta) = \mathbb{F}_{q^d},$ the polynomial $F$ is the composed product of two irreducible polynomials of coprime degrees, and hence irreducible. Irreducible Constructions ------------------------- In this subsection we use the composed multiplication to construct some new classes of irreducible polynomials. \[lem 1\] Let $f$ be an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n$ belonging to order $t$, and let $r$ be a positive integer. Then $f(x)\mid f(x^r)$ implies $r \equiv q^i \pmod{t}$ for some $i \in [0,n-1]$. Furthermore, let $\alpha$ be a root of $f$ and assume $r \equiv q^i \pmod{t}$ as above. Then the sets $$R = \{\alpha^{r^kq^u};\ 0\leq u \leq n-1\},\ F = \{\alpha^{q^u};\ 0\leq u \leq n-1\}$$ are equal for any $k\geq 0$. Recall that the roots of $f$ are $\alpha^{q^u}$, $0\leq u\leq n-1$, and $q^n \equiv 1 \pmod{t}$ because $t\mid q^n-1$. Moreover, note that $q^n \equiv 1 \pmod{t}$ implies that for any $m \geq 0$ there exists an $s \in [0,n-1]$ such that $q^m \equiv q^s \pmod{t}$. We have: $f(x)\mid f(x^r)$ implies $f(\alpha^{q^ur}) = 0$ for all $u \in [0,n-1]$ giving $\alpha^{q^ur} = \alpha^{q^j}$, some $j \in [0,n-1];$ hence $q^ur \equiv q^j \pmod{t}$ and so $r \equiv q^{n+j-u} \equiv q^i \pmod{t},$ some $i \in [0,n-1].$ Next, assume $r \equiv q^i \pmod{t}$ for some $i\in [0,n-1]$. We show that $R = F$. Clearly, $r^kq^u \equiv q^{ik+u} \equiv q^j \pmod{t}$ for some $j\in [0,n-1].$ Thus, $\alpha^{r^kq^u} = \alpha^{q^j} \in F;$ hence $R \subseteq F.$ Now let $\alpha^{q^u} \in F$. Note that $r\equiv q^i\pmod{t}$ implies $r^k \equiv q^l \pmod{t}$ for some $l\in [0,n-1].$ If $u \geq l$, then $r^kq^{u-l} \equiv q^u \pmod{t}$, so $\alpha^{q^u} = \alpha^{r^kq^{u-l}} \in R$. If $u < l$, write $r^k \equiv q^{u+s}\pmod{t},$ where $0 < s = l - u \leq n-1$. Then $r^kq^{n-s} \equiv q^{u+s+(n-s)} \equiv q^u\pmod{t},$ and hence $\alpha^{q^u} = \alpha^{r^kq^{n-s}} \in R$. Therefore $R = F.$ \[lem 2\] Let $r$ be an odd prime number and $q$ a prime power. Suppose that $q$ is a primitive root modulo $r$ and $r^2\nmid \left(q^{r-1}-1\right)$. Then the polynomial $$\Phi_r(x^{r^k}) = x^{(r-1)r^k} + x^{(r-2)r^k} + \dots + x^{r^k} + 1$$ is irreducible over ${\mathbb{F}_q}$ for each $k \geq 0$. First, recall that the hypotheses imply that $q$ is a primitive root modulo $r^k$, $k\geq 1$. Then $\Phi_{r^{k+1}},\ k\geq 0$, is irreducible over ${\mathbb{F}_q}$. Thus, if we show $\Phi_{r^{k+1}}(x) = \Phi_r(x^{r^k}),$ the result is achieved. Indeed, $$\Phi_{r^{k+1}}(x) = \prod_{d\mid r^{k+1}}\left(x^{r^{k+1}/d}-1\right)^{\mu(d)} = \frac{x^{r^{k+1}}-1}{x^{r^k}-1} = \Phi_r\left(x^{r^k}\right). \qedhere$$ The following result is the construction of a new infinite family of irreducible polynomials over ${\mathbb{F}_q}.$ \[thm 2\] Let $r$ be a prime number and let $f$ be an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n$ such that ------- ------------------------------------ (i) $f(x)\mid f(x^r)$ (ii) $q$ is a primitive root modulo $r$ (iii) $\gcd(n,r-1) = 1$. ------- ------------------------------------ \ \ We have:\ (a) The polynomial $F(x) = f(x^r)\left(f(x)\right)^{-1} = \left(f\odot \Phi_r\right)(x)$ is irreducible over ${\mathbb{F}_q}$ of degree $n(r-1)$.\ (b) If $r$ is an odd prime such that $r^2\nmid \left(q^{r-1}-1\right)$ and $\gcd(n,r(r-1)) = 1,$ then $F\left(x^{r^k}\right) = \left(f\odot \Phi_{r^{k+1}}\right)(x),\ k \geq 0,$ is an irreducible polynomial over ${\mathbb{F}_q}$ of degree $nr^k(r-1).$ \(a) Condition (i) and Lemma \[lem 1\] imply that $$f(x) = \prod_{u=0}^{n-1}\left(x-\alpha^{q^u}\right) = \prod_{u=0}^{n-1}\left(x-\alpha^{rq^u}\right).$$ As a result, $$F(x) = \frac{f(x^r)}{f(x)} = \prod_{u=0}^{n-1}\left(\frac{x^r-\alpha^{rq^u}}{x-\alpha^{q^u}}\right).$$ Note that $$\frac{x^r-\alpha^{rq^u}}{x-\alpha^{q^u}} = x^{r-1} +\alpha^{q^u} x^{r-2} + \dots +\alpha^{(r-1)q^u} = \alpha^{(r-1)q^u}\Phi_r\left(\alpha^{-q^u}x\right).$$ Condition (ii) implies that $\Phi_r$ is irreducible over ${\mathbb{F}_q}$ of degree $r-1$ which is coprime to $n$ by condition (iii). It only remains to observe that $$F(x) = \prod_{u=0}^{n-1}\alpha^{(r-1)q^u}\Phi_r\left(\alpha^{-q^u}x\right) = \left(f\odot \Phi_r\right)(x)$$ by Proposition \[comput\]. Now Theorem \[thm 1\] completes the proof of (a).\ We now prove (b): Lemma \[lem 2\] gives $\Phi_r\left(x^{r^k}\right) = \Phi_{r^{k+1}}(x)$ is irreducible over ${\mathbb{F}_q}$ of degree $r^k(r-1)$ which is coprime to $n$ by assumption. By condition (i), Lemma \[lem 1\], and Proposition \[comput\], we obtain $$\begin{aligned} F\left(x^{r^k}\right) &=& \prod_{u=0}^{n-1}\alpha^{(r-1)q^u}\Phi_r\left(\alpha^{-q^u}x^{r^k}\right) = \prod_{u=0}^{n-1}\alpha^{r^k(r-1)q^u}\Phi_r\left(\alpha^{-r^kq^u}x^{r^k}\right)\\ &=& \prod_{u=0}^{n-1}\alpha^{r^k(r-1)q^u}\Phi_{r^{k+1}}\left(\alpha^{-q^u}x\right) = \left(f\odot \Phi_{r^{k+1}}\right)(x). \end{aligned}$$ Noting that $f\odot \Phi_{r^{k+1}}$ is irreducible over ${\mathbb{F}_q}$ of degree $nr^k(r-1)$ by Theorem \[thm 1\], we thus obtain the result. We give an example where conditions (i), (ii), (iii) are satisfied. As shown in Lemma \[lem 1\], if $f(x)\mid f\left(x^r\right)$, then $r \equiv q^i\pmod{t}$ for some $i \in [0,n-1],$ where $t$ is the order of $f.$ Moreover, we need ${\operatorname{ord}}_t(q) = n$ (see Lemma \[lem 3\]), ${\operatorname{ord}}_r(q) = \phi(r)$ and $\gcd(n,\phi(r)) = 1.$ The reader can verify that when $\left(q,n,t,r,f(x)\right) = \left(2,3,7,11,x^3+x^2+1\right)$ all the conditions are met. Furthermore, $11^2\nmid\left(2^{10}-1\right)$ and $\gcd(3,11 \cdot 10) = 1,$ so part (b) also holds in this case. We generalize the last result further in the following theorem. This also generalizes a result due to Varshamov (1984) which we state in Corollary \[varshamov\]. We need the following well known fact. \[lem 3\] Let $f$ be an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n$ belonging to order $t$. Then the multiplicative order of $q$ modulo $t$ is $n$. \[thm 3\] Let $m \in \mathbb{N}$ and assume that $q$ is a primitive root modulo $m$. Let $f$ be an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n$ such that $\gcd(n,\phi(m)) = 1$ with $f$ belonging to order $t$. If $m$ and $t$ are even, further assume that $n$ is the multiplicative order of $q$ modulo $t/2.$ For each positive divisor $d$ of $m$ define the polynomials $R_d$, $\Psi_d $ over ${\mathbb{F}_q}$ as follows: Set $x^d \equiv R_d(x)\pmod {f(x)}$, and $\Psi_{d}(x) = \sum_{i=0}^{n}\Psi_{d,i}x^i$, where $\Psi_{d}$ is the non-zero polynomial of minimal degree satisfying the congruence $$\sum_{i=0}^{n}\Psi_{d,i}\left(R_d(x)\right)^i\equiv 0 \pmod {f(x)}.$$ Then the polynomials $\Psi_d,\ d \mid m,$ are irreducible over ${\mathbb{F}_q}$ of degree $n$. Furthermore, $$F_m(x) = \prod_{d \mid m}\Psi_d\left(x^d\right)^{\mu(m/d)} = \left(f\odot \Phi_m\right)(x)$$ is an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n\phi(m)$ belonging to order ${\operatorname{lcm}}(t,m)$. We first prove that for each positive divisor $d$ of $m,\ \Psi_d$ is irreducible over ${\mathbb{F}_q}$ of degree $n$. Now, let $\alpha \in {\mathbb{F}_{q^n}}$ be a root of $f.$ Then the congruence relations $\sum_{i=0}^{n}\Psi_{d,i}\left(R_d(x)\right)^i \equiv 0 \pmod {f(x)}$ and $x^d \equiv R_d(x) \pmod {f(x)}$ imply that $R_d(\alpha) = \alpha^d$ is a root of $\Psi_d.$ Thus, by the assumption of the minimality of the degree of $\Psi_d$ we deduce that $\Psi_d$ is the minimal polynomial of $\alpha^d$ over ${\mathbb{F}_q}.$ As a result, $\Psi_d$ is irreducible over ${\mathbb{F}_q}.$ We now prove $\deg \left(\Psi_d\right) = n.$ Suppose $\deg \left(\Psi_d\right) = s_d \leq n.$ Note that ${\operatorname{ord}}\left(\Psi_d\right) = {\operatorname{ord}}\left(\alpha^d\right) = t/\gcd(d,t).$ Then by Lemma \[lem 3\] we have ${\operatorname{ord}}_t(q) = n,$ and ${\operatorname{ord}}_{t/\gcd(d,t)}(q) = s_d.$ Since $q$ is a primitive root modulo $m,$ then $m$ must be either $1,\ 2,\ 4,\ r^k,$ or $2r^k$ for some odd prime $r$ and some $k\geq 1.$ We show that in all these cases $s_d = n$ for each $1\leq d \mid m.$ Observe that $\Psi_1$ is the minimal polynomial of $\alpha$ which is $f$; hence $\Psi_1 = f$ and $s_1 = n.$ Suppose $d = 2\mid m.$ If $\gcd(d,t) = 1,$ then $s_2 = {\operatorname{ord}}_{t/\gcd(2,t)}(q) = {\operatorname{ord}}_{t}(q) = n.$ Otherwise $t$ is even and so $s_2 = {\operatorname{ord}}_{t/\gcd(2,t)}(q) = {\operatorname{ord}}_{t/2}(q) = n$ by the hypothesis for $m$ even. Note that whenever $m > 2$ we can’t have $\gcd(m,t) = m$ otherwise $q^n \equiv 1 \pmod{t}$ gives $q^n \equiv 1 \pmod{m}$ implying $\phi(m)\mid n$ contrary to $\gcd(n,\phi(m)) = 1$ and $\phi(m) > 1.$ Thus whenever $m = 4$ we must have either $\gcd(m,t) = 1$ or $\gcd(m,t) = 2.$ In both cases we obtain $s_4 = {\operatorname{ord}}_{t/\gcd(4,t)}(q) = {\operatorname{ord}}_{t}(q) = n$ or $s_4 = {\operatorname{ord}}_{t/2}(q) = n$ also by the hypothesis for $m$ even. Consider the cases $m = r^k,\ 2r^k,$ for some odd prime $r,$ some $k\geq 1.$ Let $d = r^j\mid m,\ 1\leq j \leq k.$ Either $r\mid \gcd \left(r^j,t\right)$ or $\gcd \left(r^j,t\right) = 1.$ Suppose $r\mid \gcd \left(r^j,t\right).$ In particular, $r\mid t.$ Note that $\phi(m) > 1$ is even and so the assumption $\gcd(n,\phi(m)) = 1$ implies $n$ is odd. Moreover, because $q$ is a primitive root modulo $m = r^k$ or $2r^k,$ then $q$ is a primitive root modulo $r.$ Now, $q^n \equiv 1 \pmod{t}$ gives $q^n \equiv 1 \pmod{r}$ implying $\phi(r) = r-1\mid n.$ But $n$ is odd and $r-1$ is even because $r$ is odd. Thus we have reached a contradiction and so we must have $\gcd\left(r^j,t\right) = 1$. As a result we obtain $s_{r^j} = {\operatorname{ord}}_{t/\gcd\left(r^j,t\right)}(q) = {\operatorname{ord}}_{t}(q) = n.$ At this point we have accounted for all possible positive divisors $d$ of $m$ and we thus conclude $s_d = n$ for each $1\leq d \mid m;$ therefore $$\Psi_d(x) = \prod_{u=0}^{n-1}\left(x - \alpha^{dq^u}\right).$$ Now, we know that $\Phi_m$ is irreducible over ${\mathbb{F}_q}$ since $q$ is a primitive root modulo $m.$ Moreover, $\deg \left(\Phi_m\right) = \phi(m)$ is coprime to $n$ by assumption. Thus, by Theorem \[thm 1\], $f\odot \Phi_m$ is irreducible over ${\mathbb{F}_q}$ of degree $n\phi(m).$ Furthermore, because the roots $\{\xi_m\}$ of $\Phi_m$ are the primitive $m$-th roots of unity, i.e., $m$ is the least positive integer $l$ such that $\xi_m^l = 1,$ then ${\operatorname{ord}}\left(\xi_m\right) = m.$ Hence, ${\operatorname{ord}}\left(f\odot \Phi_m\right) = {\operatorname{ord}}\left(\alpha \xi_m\right) = {\operatorname{lcm}}(t,m).$ In conclusion, if we show $F_m = f\odot \Phi_m,$ the proof will be complete. First, recall $$x^m-1 = \prod_{k=0}^{m-1}\left(x - \xi_m^k\right) = \prod_{d \mid m}\Phi_d(x) = \prod_{d \mid m}\prod_{\substack{k=0\\ \gcd(k,d)=1}}^{d-1}\left(x-\xi_d^k\right).$$ We have $$\begin{aligned} \Psi_m\left(x^m\right) &=& \prod_{u=0}^{n-1}\left(x^m-\alpha^{mq^u}\right) = \prod_{u=0}^{n-1}\prod_{k=0}^{m-1}\left(x-\alpha^{q^u}\xi_m^k\right) = \left(f\odot \left(x^m-1\right) \right)(x) = \left(f\odot \prod_{d \mid m} \Phi_d\right)(x)\\ &=& \prod_{u=0}^{n-1}\prod_{d \mid m}\prod_{\substack{k=0\\ \gcd(k,d)=1}}^{d-1}\left(x-\alpha^{q^u}\xi_d^k\right) = \prod_{d \mid m}\prod_{u=0}^{n-1}\prod_{\substack{k=0\\ \gcd(k,d)=1}}^{d-1}\left(x-\alpha^{q^u}\xi_d^k\right)\\ &=& \prod_{d \mid m}\left(f\odot \Phi_d\right)(x).\end{aligned}$$ By applying the Mobius Inversion Formula now we obtain the desired result. Whenever the hypotheses in Theorem \[thm 3\] are true, the proof shows, in particular, that the characteristic polynomial of each $\alpha^d,\ 1\leq d \mid m,$ is its minimal polynomial, and thus it is irreducible. Note that the condition “If $m$ and $t$ are even, further assume that $n$ is the multiplicative order of $q$ modulo $t/2$” is necessary to ensure that for any even positive divisor $d$ of $m,$ the characteristic polynomial of $\alpha^d$ is irreducible; this is true in most cases here. However, the reader can observe from the proof that if we define $\Psi_d$ as the characteristic polynomial of $\alpha^d$ instead, $F_m$ will still be irreducible. Note that since $m$ is either of $1,\ 2,\ 4,\ r^k,\ 2r^k,$ and $\mu(c) = 0$ whenever there exists some prime $p$ such that $p^2\mid c,$ then any $F_m$ must be a product and division of at most four minimal polynomials $\Psi_d$ evaluated at $x^d.$ Since one of these must be the given $\Psi_1 = f,$ we only need to compute at most three minimal (or characteristic, see above) polynomials. Thus, this may provide an alternative more efficient way to compute $f \odot \Phi_m$ versus other known general methods for computing composed products. See [@Brawley; @et; @al] for known methods of computing composed products efficiently. We further remark that our formula $F_m = f \odot \Phi_m$ holds even if $\gcd(n,\phi(m)) \neq 1,$ although $F_m$ is not irreducible in this case. \[rem 3\] Theorem \[thm 2\] (a) is a corollary of Theorem \[thm 3\]. Indeed, $$F(x) = \frac{f\left(x^r\right)}{f(x)} = \left(f\odot \Phi_{r}\right)(x) = F_r(x).$$ Theorem \[thm 3\] generalizes a result due to Varshamov (1984) which was given without a proof. For an independent proof of Corollary \[varshamov\] we refer the reader to Theorem 3 in [@Kyuregyan]. \[varshamov\] Let $r$ be an odd prime number which does not divide $q$ and $r-1$ be the order of $q$ modulo $r.$ Further, let $n \in \mathbb{N}$ such that $\gcd(n,r-1) = 1,$ and let $f$ be an irreducible polynomial of degree $n$ over ${\mathbb{F}_q}$ belonging to order $t.$ Define the polynomials $R$ and $\psi$ over ${\mathbb{F}_q}$ as follows: Set $x^r \equiv R(x)\pmod {f(x)}$ and $\psi(x) = \sum_{u=0}^{n}\psi_ux^u,$ where $\psi$ is the nonzero polynomial of minimal degree satisfying the congruence $$\sum_{u=0}^{n}\psi_u\left(R(x)\right)^u \equiv 0 \pmod {f(x)}.$$ Then the polynomial $\psi$ is an irreducible polynomial of degree $n$ over ${\mathbb{F}_q}$ and $$F(x) = \left(f(x)\right)^{-1}\psi\left(x^r\right)$$ is an irreducible polynomial of degree $(r-1)n$ over ${\mathbb{F}_q}.$ Moreover, $F$ belongs to order $rt.$ In Theorem \[thm 3\], let $m = r.$ Then $F_r$ is an irreducible polynomial over ${\mathbb{F}_q}$ of degree $\phi(r)n = (r-1)n$ belonging to order ${\operatorname{lcm}}(r,t).$ Recall from the proof of Theorem \[thm 3\] that if an odd prime $r$ divides $m,$ then $\gcd(r,t) = 1.$ Thus $F_r$ belongs to order ${\operatorname{lcm}}(r,t) = rt.$ Let $\alpha$ be a root of $f$. The definition of $\psi$ implies it is the minimal polynomial of $\alpha^r$ which is $\Psi_r;$ thus $\psi = \Psi_r$ and so $\psi$ is irreducible over ${\mathbb{F}_q}$ of degree $n$. It only remains to observe $$F_r(x) = \prod_{d\mid r}\Psi_d\left(x^d\right)^{\mu(r/d)} = \frac{\Psi_r\left(x^r\right)}{\Psi_1(x)} = \frac{\psi\left(x^r\right)}{f(x)} = F(x).\qedhere$$ Let $r$ be an odd prime and assume $q$ is a primitive root modulo $r$ such that $r^2\nmid \left(q^{r-1}-1\right).$ Let $f$ be an irreducible polynomial over ${\mathbb{F}_q}$ of degree $n$ such that $f(x)\mid f\left(x^r\right)$ and $\gcd\left(n,r(r-1)\right) = 1$. Then for $k \geq 0,$ $$F_r\left(x^{r^k}\right) = F_{r^{k+1}}(x)$$ is an irreducible polynomial over ${\mathbb{F}_q}$ of degree $nr^k(r-1).$ Let $F(x) = \left(f(x)\right)^{-1}f\left(x^r\right) = \left(f\odot\Phi_r\right)(x)$ as in Theorem \[thm 2\]. Then $F\left(x^{r^k}\right)$ is irreducible over ${\mathbb{F}_q}$ of degree $nr^k(r-1)$ by Theorem \[thm 2\] (b). It only suffices to note that by Remark \[rem 3\] and Theorem \[thm 2\] (b) we have $$F_r\left(x^{r^k}\right) = F\left(x^{r^k}\right) = \left(f\odot \Phi_{r^{k+1}}\right)(x) = F_{r^{k+1}}(x). \qedhere$$ Explicit Factorization of the Cyclotomic Polynomial $\Phi_{2^nr}$ {#cyclo fact} ================================================================= In this section we present new results, Theorems \[thm 5\], \[2\^nr and q = 1 mod 4\], \[2\^nr and q = 3 mod 4\], of the explicit factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ where $q$ is odd, $n \in \mathbb{N},$ and $r \geq 3$ is any odd number such that $\gcd(q,r) = 1$. Previously, only $\Phi_{2^n3}$ and $\Phi_{2^n5}$ had been factored in [@Fitzgerald; @2007] and [@Prof], respectively. We also show how to obtain the factorization of $\Phi_n$ in a special case in Theorem \[cyclotomics are composed\], and how to obtain the factorization of $\Phi_{mn}$ from the given factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),{\operatorname{ord}}_n(q)) = 1.$ Preliminaries ------------- The following result shows that cyclotomic polynomials are in fact composed multiplications of other cyclotomic polynomials. Moreover, it shows how we may obtain the factorization of $\Phi_n$ in a special case. \[cyclotomics are composed\] Let $n = p_1^{e_1}p_2^{e_2}\dots p_s^{e_s}$ be the complete factorization of $n \in \mathbb{N}.$ Let $\Phi_{p_1^{e_1}} = \prod_i f_{1_i},\ \Phi_{p_2^{e_2}} = \prod_j f_{2_j}, \dots, \ \Phi_{p_s^{e_s}} = \prod_k f_{s_k}$ be the corresponding factorizations over ${\mathbb{F}_q}.$ Then $$\begin{aligned} \Phi_n &=& \Phi_{p_1^{e_1}} \odot \Phi_{p_2^{e_2}} \odot \dots \odot \Phi_{p_s^{e_s}}\\ &=& \prod_i \prod_j \cdots \prod_k \left(f_{1_i}\odot f_{2_j} \odot \cdots \odot f_{s_k}\right).\end{aligned}$$ Moreover, if the multiplicative orders of $q$ modulo all these primes powers $p_i^{e_i}$ are pairwise coprime, then this is the complete factorization of $\Phi_{n}$ over ${\mathbb{F}_q}.$ For brevity’s sake, let $F = \Phi_{p_1^{e_1}} \odot \cdots \odot \Phi_{p_s^{e_s}}.$ By definition, $$F(x) = \prod_{\xi_{p_1^{e_1}}} \cdots \prod_{\xi_{p_s^{e_s}}}(x - \xi_{p_1^{e_1}} \cdots \xi_{p_s^{e_s}})$$ where the products $\prod_{\xi_{p_i^{e_i}}}$ run over all primitive $p_i^{e_i}$-th roots of unity $\xi_{p_i^{e_i}}$. Note that each $\xi_{p_1^{e_1}}\xi_{p_2^{e_2}} \cdots \xi_{p_s^{e_s}}$ is a root of $\Phi_n.$ Indeed, ${\operatorname{ord}}(\xi_{p_1^{e_1}} \cdots \xi_{p_s^{e_s}}) = p_1^{e_1} \cdots p_s^{e_s} = n$ as ${\operatorname{ord}}(\xi_{p_i^{e_i}}) = p_i^{e_i}$ and the $p_i$’s are coprime; thus each $\xi_{p_1^{e_1}} \cdots \xi_{p_s^{e_s}}$ is a primitive $n$-th root of unity, and hence a root of $\Phi_n.$ Furthermore, both polynomials are monic and $\deg (F) = \prod_{i=1}^s \phi(p_i^{e_i}) = \phi(\prod_{i=1}^s p_i^{e_i}) = \phi(n) = \deg \Phi_n.$ Now, recall that all the roots of a cyclotomic polynomial are distinct. If we show that all roots $\xi_{p_1^{e_1}} \xi_{p_2^{e_2}}\cdots \xi_{p_s^{e_s}}$ of $F$ are distinct, the desired result $\Phi_n = F$ must then follow. Suppose $\xi_{p_1^{e_1}}^{i_1} \cdots \xi_{p_s^{e_s}}^{i_s} = \xi_{p_1^{e_1}}^{j_1} \cdots \xi_{p_s^{e_s}}^{j_s}$ is a root of $F.$ Then $\xi_{p_1^{e_1}}^{i_1-j_1} \cdots \xi_{p_{s-1}^{e_{s-1}}}^{i_{s-1}-j_{s-1}} = \xi_{p_s^{e_s}}^{j_s-i_s}.$ In particular, ${\operatorname{ord}}(\xi_{p_1^{e_1}}^{i_1-j_1} \cdots \xi_{p_{s-1}^{e_{s-1}}}^{i_{s-1}-j_{s-1}})$ = ${\operatorname{ord}}(\xi_{p_s^{e_s}}^{j_s-i_s}).$ Moreover, ${\operatorname{ord}}(\xi_{p_1^{e_1}}^{i_1-j_1} \cdots \xi_{p_{s-1}^{e_{s-1}}}^{i_{s-1}-j_{s-1}}) \mid p_1^{e_1} \cdots p_{s-1}^{e_{s-1}}$ and ${\operatorname{ord}}(\xi_{p_s^{e_s}}^{j_s-i_s}) \mid p_s^{e_s}.$ But then, as $\gcd(p_1^{e_1} \cdots p_{s-1}^{e_{s-1}},\ p_s^{e_s}) = 1,$ we must have $\xi_{p_s^{e_s}}^{j_s-i_s} = 1.$ Since $p_s^{e_s} > 1$ and $0 < i_s, j_s < p_s^{e_s},$ necessarily $i_s = j_s.$ Similarly, by induction we can show $i_k = j_k,\ 1 \leq k \leq s.$ Thus, $\Phi_n = F.$ The second statement of the theorem follows from Proposition \[product of composed\], the associativity of composed multiplications, and Theorem \[thm 1\] combined with the fact that the degrees of the irreducible factors $f_i$ of $\Phi_{p_i^{e_i}}$ are ${\operatorname{ord}}_{p_i^{e_i}}(q).$ Let $q = 11,\ n = 595 = 5\cdot 7 \cdot 17.$ As ${\operatorname{ord}}_5(q) = 1,\ {\operatorname{ord}}_7(q) = 3,\ {\operatorname{ord}}_{17}(q) = 16$ are pairwise coprime, then by Theorem \[cyclotomics are composed\] the complete factorization of $\Phi_{595}$ over $\mathbb{F}_{11}$ is given by $$\Phi_{595} = \prod_i \prod_j \prod_k (f_i \odot g_j \odot h_k)$$ where the $f_i,\ g_j,\ h_k$ are the irreducible factors of $\Phi_5,\ \Phi_7,\ \Phi_{17},$ respectively, over $\mathbb{F}_{11}.$ We have the following corollary to Theorem \[cyclotomics are composed\]. \[Phi\_mn\] Let $m,\ n \in \mathbb{N}$ be coprime. Then $\Phi_{mn} = \Phi_m\odot \Phi_n$. Further, let $\Phi_{m} = \prod_i f_i, \ \Phi_n = \prod_j g_j$ be the respective factorizations over ${\mathbb{F}_q}$. Then $$\Phi_{mn} = \prod_i \prod_j\left(f_i\odot g_j\right).$$ Moreover, if $\gcd({\operatorname{ord}}_m(q),{\operatorname{ord}}_n(q)) = 1,$ then this is the complete factorization of $\Phi_{mn}$ over ${\mathbb{F}_q}$. The result is clear if $m = 1$ or $n = 1.$ Assume $m = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k},\ n = p_{k+1}^{e_{k+1}} p_{k+2}^{e_{k+2}} \cdots p_s^{e_s}$ are complete factorizations of $m,\ n$ over $\mathbb{N}.$ Then by Theorem \[cyclotomics are composed\] we have $$\Phi_{m} = \Phi_{p_1^{e_1}} \odot \cdots \odot \Phi_{p_k^{e_k}}, {\hspace*{2em}}\Phi_n = \Phi_{p_{k+1}^{e_{k+1}}} \odot \cdots \odot \Phi_{p_s^{e_s}},$$ giving $$\Phi_m \odot \Phi_n = (\Phi_{p_1^{e_1}} \odot \cdots \odot \Phi_{p_k^{e_k}}) \odot (\Phi_{p_{k+1}^{e_{k+1}}} \odot \cdots \odot \Phi_{p_s^{e_s}}) = \Phi_{p_1^{e_1}} \odot \cdots \odot \Phi_{p_s^{e_s}} = \Phi_{mn}.$$ The second statement follows immediately from Proposition \[product of composed\] and Theorem \[thm 1\] combined with the fact that the degrees of the irreducible factors $f_i,\ g_j$ are ${\operatorname{ord}}_m(q),\ {\operatorname{ord}}_n(q)$ respectively. In particular, whenever $r$ is odd we have $\Phi_{2^nr} = \Phi_{2^n} \odot \Phi_r.$ Thus whenever the factorizations of $\Phi_m,\ \Phi_n$ are known, and $\gcd(m,n) = \gcd({\operatorname{ord}}_m(q),{\operatorname{ord}}_n(q)) = 1,$ we can obtain all the irreducible factors of $\Phi_{mn}$ by computing each $f_i\odot g_j.$ This is a significant tool in the factorization of polynomials which we will use frequently in order to obtain some of the following results. The following result shows how we may obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ whenever $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),{\operatorname{ord}}_n(q)) = 1.$ Recall that $\Phi_n$ decomposes into $\phi(n)/{\operatorname{ord}}_n(q)$ irreducible factors over ${\mathbb{F}_q}$ of the same degree ${\operatorname{ord}}_n(q)$ whenever $\gcd(q,n) = 1.$ \[cyclo and minimal\] Let $m,\ n \in \mathbb{N},\ \gcd(m,n) = \gcd(\phi(m),d_n) = 1,$ where $d_n = {\operatorname{ord}}_n(q)$. Assume $q$ is a primitive root modulo $m.$ Let $\Phi_n = \prod_{i=1}^{\phi(n)/d_n} f_i$ be the corresponding factorization over ${\mathbb{F}_q}.$ Then the factorization of $\Phi_{mn}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{mn}(x) = \prod_{i=1}^{\phi(n)/d_n} \left(\prod_{d \mid m}\Psi_{i,d}\left(x^d\right)^{\mu(m/d)}\right)$$ where each $\Psi_{i,d}$ is the minimal polynomial of $\xi_{n,i}^d$ with $\xi_{n,i}$ a root of $f_i.$ Since $q$ is a primitive root modulo $m,$ $\Phi_m$ is irreducible over ${\mathbb{F}_q}.$ Note $\gcd(d_n,\phi(m)) = 1$ implies each polynomial $f_i\odot \Phi_m$ is irreducible over ${\mathbb{F}_q}$ by Theorem \[thm 1\]. Then by Corollary \[Phi\_mn\] and Theorem \[thm 3\] the complete factorization of $\Phi_{mn}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{mn}(x) = \prod_{i=1}^{\phi(n)/d_n}\left(f_i\odot \Phi_m\right)(x) = \prod_{i=1}^{\phi(n)/d_n}\left(\prod_{d \mid m}\Psi_{i,d}\left(x^d\right)^{\mu(m/d)}\right)$$ as required. Note that the irreducible factors of $\Phi_{mn}$ are expressed in terms of the minimal polynomials $\Psi_{i,d}$ over ${\mathbb{F}_q}$ of $\xi_{n,i}^d,$ where the root $\xi_{n,i}$ of $f_i$ is a primitive $n$-th root of unity. We remark that it is not necessary to compute the minimal polynomials: Since $\gcd(m,n) = 1,$ then $\gcd(d,n) = 1$ for each $d \mid m;$ hence $\xi_{n,i}^d$ is a primitive $n$-th root of unity, and so it must be a root of some irreducible factor $f_j$ of $\Phi_n.$ But then $\Psi_{i,d} = f_j.$ As a particular consequence, we can now let $\Phi_n$ be as in Theorems \[2\^n and q = 1 mod 4\], \[2\^nr and q = 1 mod 4\], \[2\^n and q = 3 mod 4\], \[2\^nr and q = 3 mod 4\], etc, and then use the respective factorizations $\prod_i f_i$ given there to factor $\Phi_{mn}.$ This is now merely a matter of computation. On the other hand, in the case that we do not know the factorization of $\Phi_n,$ we can let $S = \{\xi_{n_i}\}_{i=1}^{\phi(n)/{d_n}}$ be a set of pairwise non-conjugate primitive $n$-th roots of unity $\xi_n.$ Then we can write the complete factorization of $\Phi_{mn}$ over ${\mathbb{F}_q}$ as $$\Phi_{mn}(x) = \prod_{i=1}^{\phi(n)/d_n} \left(\prod_{d \mid m}\Psi_{i,d}\left(x^d\right)^{\mu(m/d)}\right) = \prod_{\xi_{n_i}\in S} \left(\prod_{d \mid m}\Psi_{i,d}\left(x^d\right)^{\mu(m/d)}\right)$$ where $\Psi_{i,d}$ is the minimal polynomial of $\xi_{n_i}^d.$ Indeed, $\xi_{n_i}$ is a root of $\Psi_{i,1} = f_i,$ and for non-conjugates $\xi_{n_i},\ \xi_{n_j},$ we have $f_i \neq f_j;$ finally, there are $ |S| = \phi(n)/d_n$ irreducible factors $f_i$ of $\Phi_n.$ \[composition\] Let $f_1,\ f_2,\dots,\ f_N$ be all distinct monic irreducible polynomials in ${\mathbb{F}_q}[x]$ of degree $m$ and order $e,$ and let $t\geq 2$ be an integer whose prime factors divide $e$ but not $\left(q^{m}-1\right)/e.$ Assume also that $q^m \equiv 1 \pmod{4}$ if $t \equiv 0 \pmod{4}.$ Then $f_1\left(x^t\right),\ f_2\left(x^t\right),\dots,\ f_N\left(x^t\right)$ are all distinct monic irreducible polynomials in ${\mathbb{F}_q}[x]$ of degree $mt$ and order $et.$ \[tricks\][ ]{}\ (a) $\Phi_{2n}(x) = \Phi_n(-x)$ for $n\geq 3$ and $n$ odd.\ (b) $\Phi_{mt}(x) = \Phi_m\left(x^t\right)$ for all positive integers $m$ that are divisible by the prime $t.$\ (c) $\Phi_{mt^k}(x) = \Phi_{mt}\left(x^{t^{k-1}}\right)$ if $t$ is a prime and $m,\ k$ are arbitrary positive integers. Note that Lemma \[tricks\] implies that, in particular, for $n\geq 2,$ $\Phi_{2^nr}(x) = \Phi_{2^{n-1}r}(x^2).$ Observe that if $\Phi_{2^{n-1}r} = \prod_i h_i$ is the corresponding factorization, then $\Phi_{2^nr}(x) = \Phi_{2^{n-1}r}\left(x^2\right) = \prod_i h_i\left(x^2\right).$ *This means that we can obtain all the irreducible factors of $\Phi_{2^{n}r}$ by factoring each $h_i\left(x^2\right).$* Let $v_2(k)$ denote the highest power of $2$ dividing $k.$ \[v2\] For $i\geq 1,$ $$\begin{aligned} v_2\left(q^i - 1\right) &=& v_2(q-1) + v_2\left(q^{i-1} + q^{i-2} +\dots + 1\right)\\ &=& \begin{cases} v_2(q-1) + v_2(i) + v_2(q+1) - 1, & \mbox{if } i\mbox{ is even} \\ v_2(q-1), & \mbox{if } i\mbox{ is odd.} \end{cases}\end{aligned}$$ \[lem 7\] Let $q = p^s$ be a power of an odd prime $p,$ let $r\geq 3$ be any odd number coprime to $q,$ and let $d_r = {\operatorname{ord}}_r(q).$ If $q \equiv 1\pmod{4},$ write $q = 2^Am + 1,\ A\geq 2,\ m$ odd. Otherwise if $q \equiv 3\pmod{4},$ write $q = 2^Am - 1,\ A\geq 2,\ m$ odd. Set $K:=v_2\left(q^{d_r} - 1\right).$ Then if $d_r$ is even, in both cases cases of $q$ we have $K = A + v_2(d_r) > A \geq 2.$ If $d_r$ is odd and $q\equiv 1 \pmod{4},$ then $K = A.$ If $d_r$ is odd and $q\equiv 3 \pmod{4},$ then $K = 1.$ First assume $d_r$ is even. Then $v_2\left(d_r\right) > 0,$ and so $A + v_2(d_r) > A \geq 2.$ If $q\equiv 1\pmod{4},$ we have $q - 1 = 2^Am$ and $q+1 = 2\left(2^{A-1}m + 1\right) = 2m',$ where $m'$ is odd. Thus $v_2(q-1) = A,$ and $v_2(q+1) = 1.$ Hence, $K = v_2(q-1) + v_2(d_r) + v_2(q+1) - 1 = A + v_2(d_r).$ If $q\equiv 3\pmod{4},$ we have $q - 1 = 2\left(2^{A-1}m - 1\right)$ and $q+1 = 2^Am.$ Thus $v_2(q-1) = 1$ and $v_2(q+1) = A.$ Hence, $K = v_2(q-1) + v_2(d_r) + v_2(q+1) - 1 = A + v_2(d_r).$ Now if $d_r$ is odd, by Lemma \[v2\], $K = v_2(q-1).$ If $q\equiv 1 \pmod{4},$ then $K = A.$ Otherwise, if $q\equiv 3 \pmod{4},$ then $K = 1.$ The following result represents an improvement over Theorem \[L\] in [@Prof]. Later on we use it often in the following sections. \[thm 5\] Let $q = p^s$ be a power of an odd prime $p,$ let $r\geq 3$ be any odd number such that $\gcd(r,q) = 1.$ Let $d_r = {\operatorname{ord}}_r(q).$ If $d_r$ is odd, further assume $q \equiv 1 \pmod{4}.$ Set $K:= v_2\left(q^{d_r}-1\right).$ Then for $n\leq K$ and any irreducible factor $h_n$ of $\Phi_{2^nr},$ we have $\deg (h_n) = d_r.$ Furthermore, if $0 < n < K$ strictly, then $h_n\left(x^2\right)$ decomposes into precisely two irreducible factors of degree $d_r$ which are irreducible factors of $\Phi_{2^{n+1}r}.$ On the other hand, for $n>K,$ and any irreducible factor $h_K$ of $\Phi_{2^Kr}$ over ${\mathbb{F}_q},$ $h_K\left(x^{2^{n-K}}\right)$ is also irreducible over ${\mathbb{F}_q}.$ Moreover, all irreducible factors of $\Phi_{2^nr}$ are obtained in this way. Since $q^{d_r} \equiv 1 \pmod{r}$ and $K = v_2\left(q^{d_r} - 1\right),$ we have $q^{d_r} \equiv 1 \pmod{2^Kr}.$ Let $n\leq K.$ It is true that $q^{d_r} \equiv 1 \pmod{2^nr}.$ Let $d_n = {\operatorname{ord}}_{2^nr}(q).$ Then $d_n\mid d_r.$ On the other hand, $q^{d_n} \equiv 1 \pmod{2^nr}$ gives $q^{d_n} \equiv 1 \pmod{r}$ implying $d_r \mid d_n.$ Consequently, $d_n = d_r.$ Recalling that the degree of each irreducible factor of $\Phi_{2^nr}$ is ${\operatorname{ord}}_{2^nr}(q) = d_n,$ we conclude that for $n\leq K,$ each irreducible factor of $\Phi_{2^nr}$ has degree $d_r.$ For $0< n < K,$ let $h_n$ be an irreducible factor (of degree $d_r$) of $\Phi_{2^nr}.$ Then $h_n\left(x^2\right)$ has degree $2d_r$ and is a factor of $\Phi_{2^{n+1}r}$ clearly. Because $n+1\leq K,$ then $h_n\left(x^2\right)$ must decompose into an amount $z$ of irreducibles of degree $d_r.$ But this is possible only if $z = 2.$ Note $e = 2^Kr$ is the order of $\Phi_{2^{K}r}$ and thus the order of any irreducible factor $h_K$ of it. By definition, $2^{K+1}\nmid \left(q^{d_r} - 1\right).$ Hence, $2\nmid \left(q^{d_r} - 1\right)/e,$ and by Lemma \[composition\], $h_K\left(x^2\right)$ is irreducible over ${\mathbb{F}_q}.$ If $d_r$ is even, then $K > 2$ by Lemma \[lem 7\]. If $d_r$ is odd, then $q\equiv 1 \pmod{4}$ by assumption, and so $K = A \geq 2$ by Lemma \[lem 7\]. Then $2^2 = 4\mid \left(q^{d_r}-1\right).$ As a result, for $n>K,$ Lemma \[composition\] gives $h_K\left(x^{2^{n-K}}\right)$ is irreducible over ${\mathbb{F}_q}.$ Because $$\Phi_{2^nr}(x) = \Phi_{2^Kr}\left(x^{2^{n-K}}\right) = \prod_i h_{K_i}\left(x^{2^{n-K}}\right),$$ where $\Phi_{2^Kr} = \prod_i h_{K_i}$ is the corresponding factorization, the factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is complete. Thus we can obtain all irreducible factors of $\Phi_{2^nr}$ in this way. Whenever $d_r$ is even, or $q \equiv 1 \pmod{4},$ the bound $K = v_2\left(q^{d_r}-1\right)$ in Theorem \[thm 5\] represents an improvement over the bound $L = v_2\left(q^{\phi(r)}-1\right)$ of Theorem \[L\] due to L. Wang and Q. Wang [@Prof]. This is because $K \leq L$ as $\left(q^{d_r}-1\right)\mid \left(q^{\phi(r)}-1\right).$ Moreover, it is clear that $K$ is the smallest bound with the property that $\Phi_{2^nr}(x) = \prod_i h_{K_i}\left(x^{2^{n-K}}\right)$ is the corresponding factorization over ${\mathbb{F}_q}$ for $n > K.$ In Theorem \[2\^nr and q = 3 mod 4\] we will show that, in particular, when $d_r$ is odd and $q \equiv 3 \pmod{4},$ the corresponding bound is $v_2(q+1) = A.$ That is, if $\Phi_{2^Ar} = \prod_i h_{A_i}$ is the corresponding factorization, then for $n > A$ the factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $\Phi_{2^nr}(x) = \prod_i h_{A_i}\left(x^{2^{n-A}}\right).$ Before we move on to the following sections we need the following notations. Let $\Omega(r)$ be the set of $r$-th primitive roots of unity and let $U_n$ be the set of the $2^n$-th primitive roots of unity. Similarly as done in [@Prof] we let the expression $$\prod_{a\in A}\dots \prod_{b\in B}f_i(x,a,\dots,b)$$ denote the product of *distinct* polynomials $f_i(x,a,\dots,b)$ satisfying conditions $a\in A,\dots,b\in B.$ For example, if we let $g_w$ be an irreducible factor of $\Phi_r$ with root $w,$ say in $\mathbb{F}_{q^{d_r}},$ then in the product $\prod_{w\in \Omega(r)} g_w$ we take $g_w$ and not any of $g_{w^{q^i}}$ as $g_w = g_{w^{q^i}}$ in this case. Recall the *elementary symmetric polynomials* $S_i$ defined by $$S_i(x_1,x_2,\dots,x_n) = \sum_{k_1<k_2<\dots <k_i} x_{k_1}x_{k_2}\dots x_{k_i}$$ for any $i=1,2,\dots,n,$ with $S_0 = 1.$ The following proposition is a well known fact. Write $S_i = S_i(x_1,x_2,\dots,x_n)$ for $1\leq i\leq n.$ Then $$\prod_{i=1}^n \left(x-x_i\right) = \sum_{i=0}^n\left(-1\right)^iS_ix^{n-i}.$$ From now on for any proper element $w\in {\mathbb{F}_{q^n}},$ i.e. ${\mathbb{F}_q}(w) = {\mathbb{F}_{q^n}},$ we use the notation $S_{i,w} = S_i\left(w,w^q,\dots,w^{q^{n-1}}\right).$ Factorization of $\Phi_{2^nr}$ when $q \equiv 1 \pmod{4}$ --------------------------------------------------------- In this section and the following we make the assumption that the explicit factorization of $\Phi_r$ is given to us as a known. One may use for instance the results due to Stein (2001) to compute the factors of $\Phi_r$ efficiently when $q = p$ and $r$ is an odd prime distinct to $p.$ First, we need the following well known theorem concerning the factorization of $\Phi_{2^n}$ when $q \equiv 1 \pmod{4}$ which follows from Theorems 2.47 and 3.35 in [@Lidl]. \[2\^n and q = 1 mod 4\] Let $q \equiv 1 \pmod{4},$ i.e. $q = 2^Am+1,\ A \geq 2,\ m$ odd. Let $U_n$ denote the set of primitive $2^n$-th roots of unity. \(a) If $1\leq n\leq A,$ then ${\operatorname{ord}}_{2^n}(q) = 1$ and $\Phi_{2^n}$ is the product of $2^{n-1}$ irreducible linear factors over ${\mathbb{F}_q}:$ $$\Phi_{2^n}(x) = \prod_{u\in U_n}\left(x+u\right).$$ \(b) If $n>A,$ then ${\operatorname{ord}}_{2^n}(q) = 2^{n-A}$ and $\Phi_{2^n}$ is the product of $2^{A-1}$ irreducible binomials over ${\mathbb{F}_q}$ of degree $2^{n-A}:$ $$\Phi_{2^n}(x) = \prod_{u\in U_A}\left(x^{2^{n-A}} + u\right).$$ First recall that whenever $\gcd(q,n) = 1$, $\Phi_n$ decomposes into $\phi(n)/{\operatorname{ord}}_n(q)$ irreducibles over ${\mathbb{F}_q}$ of degree ${\operatorname{ord}}_n(q)$ (Theorem 2.47, [@Lidl]). In particular, $\Phi_r$ decomposes into irreducibles of degree $d_r = {\operatorname{ord}}_r(q)$ over ${\mathbb{F}_q}$ when $q,\ r$ are coprime. We now give the factorization of $\Phi_{2^nr}$ when $q \equiv 1 \pmod{4}.$ \[2\^nr and q = 1 mod 4\] Let $q \equiv 1 \pmod{4},$ say $q = 2^Am+1,\ A \geq 2,\ m$ odd. Let $r \geq 3$ be odd such that $\gcd(q,r) = 1,$ and let $d_r = {\operatorname{ord}}_r(q).$\ 1. If $1\leq n\leq A,$ then $$\Phi_{2^nr}(x) = \prod_{u\in U_n}\prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r} u^i S_{i,w}x^{d_r-i}\right)$$ is the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}.$\ 2. If $n>A,$ we have:\ (a) If $d_r$ is odd, then $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r} u^i S_{i,w}x^{2^{n-A}\left(d_r-i\right)}\right).$$ is the complete factorization of $\Phi_{2^nr},\ n>A,$ over ${\mathbb{F}_q}.$\ (b) If $d_r$ is even, then: \(i) For $A < n \leq K,$ the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(x^{d_r}+ \sum_{i=1}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where each $a_{n_i},\ 1 \leq i\leq d_r,$ satisfies the following system of non-linear recurrence relations $$\Bigg\{\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j} = a_{(n-1)_k}, {\hspace*{2em}}1\leq k\leq d_r \Bigg\}$$ with initial values $a_{A_k} = u^kS_{k,w},\ 1\leq k\leq d_r.$ \(ii) For $n > K,$ the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(x^{2^{n-K}d_r}+ \sum_{i=1}^{d_r}a_{K_i}x^{2^{n-K}(d_r-i)}\right)$$ where each $a_{K_i},\ 1\leq i\leq d_r,$ is as obtained in (i). Let $$\Phi_r(x) = \prod_{w\in \Omega(r)}g_w(x) = \prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r}(-1)^i S_{i,w}x^{d_r-i}\right)$$ be the factorization of $\Phi_r$ over ${\mathbb{F}_q}.$ 1\. By Theorem \[2\^n and q = 1 mod 4\] (a) and Corollary \[Phi\_mn\] we have $$\Phi_{2^nr}(x) = \left(\Phi_{2^n}\odot \Phi_r\right)(x) = \prod_{u\in U_n}\prod_{w\in \Omega(r)}\left((x+u)\odot g_w\right)(x).$$ By Proposition \[comput\], $$\begin{aligned} \left((x+u)\odot g_w\right)(x) &=& (-u)^{d_r}g_w\left((-u)^{-1}x\right) = (-u)^{d_r}\sum_{i=0}^{d_r}(-1)^iS_{i,w}(-u)^{i-d_r}x^{d_r-i}\\ &=& \sum_{i=0}^{d_r}S_{i,w}u^ix^{d_r-i}.\end{aligned}$$ Noting that each $(x+u) \odot g_w$ is irreducible over ${\mathbb{F}_q}$ by Theorem \[thm 1\], these factors give us a complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ for $1 \leq n\leq A.$\ \ 2 (a): Since $q \equiv 1 \pmod{4}$ and $d_r$ is odd, Lemma \[lem 7\] gives $K = A;$ consequently if $\Phi_{2^Ar} = \prod_i h_{A_i}$ is the corresponding factorization over ${\mathbb{F}_q}$, then Theorem \[thm 5\] gives that for $n > A,$ the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $\Phi_{2^nr}(x) = \prod_i h_{A_i}\left(x^{2^{n-A}}\right).$ Thus it only remains to make the substitution $x \rightarrow x^{2^{n-A}}$ in each irreducible factor $h_{A_i}$ obtained in Part 1 as the statement in the theorem shows. \(b) (i) ($A<n \leq K$ and $d_r$ even): Let $h_{n-1}$ be an irreducible factor of $\Phi_{2^{n-1}r}.$ By Theorem \[thm 5\], $\deg (h_{n-1}) = d_r$ and $h_{n-1}\left(x^2\right)$ decomposes into two irreducibles of degree $d_r$ which are irreducible factors of $\Phi_{2^nr}.$ Let $h_{n-1}\left(x^2\right) = f_n(x) g_n(x)$ be the corresponding factorization. First, we show $g_n(x) = f_n(-x).$ Let $\alpha$ be a root of $f_n.$ We claim that $-\alpha$ is not a root of $f_n.$ On the contrary, suppose $f_n(-\alpha) = 0.$ Then $-\alpha = \alpha^{q^i}$ for some $i \in [0,d_r-1]$ implies $-1 = \alpha^{q^i-1}$ and $1 = \alpha^{2\left(q^i-1\right)}.$ But then ${\operatorname{ord}}(\alpha) = 2^nr \mid 2\left(q^i-1\right)$ and so $r \mid \left(q^i-1\right).$ However, this contradicts ${\operatorname{ord}}_r(q) = d_r > i.$ Therefore $f_n(-\alpha) \neq 0.$ Now, we have $$f_n(-\alpha)g_n(-\alpha) = h_{n-1}\left((-\alpha)^2 \right) = h_{n-1}\left( \alpha^2 \right) = f_n(\alpha) g_n(\alpha) = 0.$$ As $f_n(-\alpha) \neq 0,$ necessarily $g_n(-\alpha) = 0.$ Thus both $f_n(-x),\ g_n(x)$ have $-\alpha$ as a root. But then since both $f_n(-x),\ g_n(x)$ are monic irreducible polynomials over ${\mathbb{F}_q}$ of degree $d_r,$ it must be that $g_n(x) = f_n(-x).$ Therefore $h_{n-1}(x^2) = f_n(x)f_n(-x)$ is the corresponding factorization. We may write $$h_{n-1}(x) = x^{d_r}+ \sum_{k=1}^{d_r}a_{(n-1)_k}x^{d_r-k}$$ and $$f_n(x) = x^{d_r} + \sum_{i=1}^{d_r}a_{n_i}x^{d_r - i}$$ for some coefficients $a_{(n-1)_k},\ a_{n_i} \in {\mathbb{F}_q}.$ Now, $h_{n-1}(x^2) = f_n(x)f_n(-x)$ gives $$\begin{aligned} x^{2d_r}+ \sum_{k=1}^{d_r}a_{(n-1)_k}x^{2(d_r-k)} &=&\left(x^{d_r}+\sum_{i=1}^{d_r}a_{n_i}x^{d_r-i}\right)\left(x^{d_r}+\sum_{j=1}^{d_r}a_{n_j}(-1)^jx^{d_r-j}\right)\\ &=& x^{2d_r} + \sum_{k=1}^{2d_r}\sum_{i+j=k}(-1)^ja_{n_i}a_{n_j}x^{2d_r-k}\\ &=& x^{2d_r} + \sum_{k=1}^{d_r}\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j}x^{2(d_r-k)}.\end{aligned}$$ The last equality followed from the fact that the coefficients of odd powers of $x$ in $h_{n-1}(x^2)$ are $0.$ Comparing coefficients on each side we see that each $a_{n_i},\ 1\leq i\leq d_r,$ satisfies the following system of non-linear equations $$\Bigg\{\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j} = a_{(n-1)_k},{\hspace*{2em}}1\leq k\leq d_r \Bigg\}.$$ We know the system must have a solution, otherwise $h_{n-1}(x^2) \neq f_n(x)f_n(-x)$ contrary to the previous arguments. Moreover, the solution must be unique by the uniqueness of factorizations. Furthermore, the reader can see that we can obtain the coefficients of $f_n,$ and hence of $f_n(-x),$ by a recursion where the initial values are the coefficients $a_{A_k} = u^kS_{k,w},\ 1\leq k\leq d_r$ of an irreducible factor of $\Phi_{2^Ar}$ which we already know from Part 1. Next, we show that we can obtain all the irreducible factors of $\Phi_{2^nr}$ in this way. We claim that for any two distinct initial-value sets $I = \{u_i^kS_{k,w} \}, \ J = \{u_j^kS_{k,w} \},$ all the irreducible factors generated by $I$ and $J$ are distinct. By induction on $n$ where $A < n \leq K$: Let $g_A,\ h_A$ be the distinct irreducible factors of $\Phi_{2^Ar}$ corresponding to $I$ and $J.$ Then in particular $g_A(x^2) \neq h_A(x^2).$ As each of these decomposes into two irreducible factors of the form $f_{A+1}(x),\ f_{A+1}(-x),$ then all four irreducible factors must be distinct. Otherwise if they share an irreducible factor, say $f_{A+1}(-x),$ then necessarily they must share $f_{A+1}(x)$ resulting in $g_A(x^2) = h_A(x^2),$ a contradiction. Similarly one can show that the inductive step follows from the inductive hypothesis. The claim now follows. Consequently, if we let $s = n - A,$ then each initial-value set $\{u^kS_{k,w}\}$ corresponding to an irreducible factor $g_A$ of $\Phi_{2^Ar}$ will generate a total of $2^s$ distinct irreducible factors of $\Phi_{2^nr}.$ Since there are $\phi(2^Ar)/d_r$ irreducible factors of $\Phi_{2^Ar},$ the initial-value sets generate a total of $2^s\phi(2^Ar)/d_r = 2^{s+A-1}\phi(r)/d_r = 2^{n-1}\phi(r)/d_r = \phi(2^nr)/d_r$ distinct irreducible factors of $\Phi_{2^nr},$ as desired. The factorization is complete. \(ii) ($n > K$): If $\Phi_{2^Kr} = \prod_i h_{K_i}$ is the corresponding factorization, then by Theorem \[thm 5\], for $n > K,$ we obtain $\Phi_{2^nr}(x) = \prod_i h_{K_i}\left(x^{2^{n-K}}\right)$ as its complete factorization. Since each $h_{K_i}$ is already known from Part (i), it only remains to make the substitution $x\rightarrow x^{2^{n-K}}$ in each $h_{K_i}$ to obtain each irreducible factor of $\Phi_{2^nr},$ as the statement in the theorem shows. The proof of (ii) is complete. \[rem 2.1\] In order to obtain each irreducible factor of $\Phi_{2^nr},$ for any $n\in \mathbb{N},$ we require at most $v_2(d_r)$ iterations of the system of non-linear recurrence relations in (i): For $n\leq A,$ the explicit factorization is already given in Part 1. However, for $A <n \leq K$ and $d_r$ even, the system of non-linear recurrence relations in (i) must iterate for $n-A$ steps. In the case $A< n = K,$ the system will iterate for the maximum number of steps $K-A.$ By Lemma \[lem 7\], this equals $v_2(d_r).$ \[rem 2.2\] We can also formulate the factorization of $\Phi_{2^nr},\ 1\leq n \leq K,$ in terms of the non-linear recurrence relation in (i) with initial values corresponding to $n = 1.$ For small finite fields and small $d_r,$ this can be computed fairly fast. \[rem 2.3\] Let $n > K,$ let $S = \{s_k\},\ T = \{t_k\}$ be homogeneous LRS’s with characteristic polynomials $\Phi_{2^n},\ \Phi_{r}$ respectively. Then as discussed earlier, the characteristic polynomial of $ST = \{s_k t_k \}$ is $\Phi_{2^nr} = \Phi_{2^n} \odot \Phi_r.$ Since all irreducible factors of $\Phi_{2^nr},\ n > K,$ have degree $2^{n-K}d_r,$ the minimal polynomial of $ST$ must have degree $2^{n-K}d_r.$ This is the linear complexity of $ST.$ Note that if we let $n \rightarrow \infty,$ the linear complexity of the corresponding LRS $ST$ approaches infinity. For the subcases $q \equiv 1 \pmod{4}$ with $q \equiv \pm 1 \pmod{r}$ and thus $d_r = 1,\ 2,$ where $r$ is an odd prime, Theorem \[2\^nr and q = 1 mod 4\] becomes Theorem 1, Parts 2 and 4 in Fitzgerald and Yucas (2007) [@Fitzgerald; @2007]. Factorization of $\Phi_{2^nr}$ when $q \equiv 3 \pmod{4}$ --------------------------------------------------------- We need the following result due to Meyn (1996) [@Meyn]. \[2\^n and q = 3 mod 4\] Let $q \equiv 3 \pmod{4},$ i.e. $q = 2^Am - 1,\ A\geq 2,\ m$ odd. Let $n\geq 2.$ \(a) If $n\leq A,$ then $\Phi_{2^n}$ is the product of $2^{n-2}$ irreducible trinomials over ${\mathbb{F}_q}:$ $$\Phi_{2^n}(x) = \prod_{u\in U_n}\left(x^2 + \left(u + u^{-1}\right)x + 1\right).$$ \(b) If $n > A,$ then $\Phi_{2^n}$ is the product of $2^{A-2}$ irreducible trinomials over ${\mathbb{F}_q}:$ $$\Phi_{2^n}(x) = \prod_{u\in U_A}\left(x^{2^{n-A+1}}+\left(u-u^{-1}\right)x^{2^{n-A}}-1\right).$$ We are now ready to give the factorization of $\Phi_{2^nr}$ when $q \equiv 3 \pmod{4}.$ \[2\^nr and q = 3 mod 4\] Let $q \equiv 3 \pmod{4},$ i.e. $q = 2^Am - 1,\ A\geq 2,\ m$ odd. Let $r \geq 3$ be odd such that $\gcd(q,r) = 1,$ and let $d_r = {\operatorname{ord}}_r(q).$\ 1. If $n = 1,$ then $$\Phi_{2r}(x) = \prod_{w\in\Omega(r)}\left(\sum_{i=0}^{d_r}S_{i,w}x^{d_r - i}\right)$$ is the complete factorization of $\Phi_{2r}$ over ${\mathbb{F}_q}.$\ 2. If $2\leq n\leq A,$ we have: \(i) If $d_r$ is odd, the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_n}\prod_{w\in \Omega(r)}\left(\sum_{k=0}^{2d_r}\sum_{i+j=k}S_{i,w}S_{j,w}u^{i-j}x^{2d_r-k} \right).$$ \(ii) If $d_r$ is even, $\Phi_{2^nr}$ decomposes into irreducibles of degree $d_r$ over ${\mathbb{F}_q}$ so that $$\Phi_{2^nr}(x) = \prod_{u\in U_n}\prod_{w\in \Omega(r)}\Bigg[\left(x^{d_r}+ \sum_{i=1}^{d_r}a_{n_i}x^{d_r-i} \right) \left(x^{d_r}+ \sum_{j=1}^{d_r}b_{n_j}x^{d_r-j} \right)\Bigg]$$ is the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q},$ where each $a_{n_i},\ b_{n_j} \in {\mathbb{F}_q},\ 1 \leq i,\ j \leq d_r,$ satisfies the following system of equations $$\Bigg\{\sum_{i+j = k}a_{n_i}b_{n_j} = \sum_{i+j = k} S_{i,w}S_{j,w}u^{i-j}, {\hspace*{2em}}1\leq k\leq 2d_r \Bigg\}.$$\ 3. If $d_r$ is odd, then for $n> A$ the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(\sum_{k=0}^{d_r}\sum_{i+j=k}u^{i-j}S_{i,w}S_{j,w}x^{2^{n-A}(2d_r-k)}\right).$$\ 4. If $d_r$ is even, we have: \(iii) For $A < n \leq K,$ the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(x^{d_r}+ \sum_{i=1}^{d_r}a_{n_i}x^{d_r-i}\right)$$ where each $a_{n_i},\ 1 \leq i\leq d_r,$ satisfies the following system of non-linear recurrence relations $$\Bigg\{\sum_{i+j=2k}(-1)^ja_{n_i}a_{n_j} = a_{(n-1)_k}, {\hspace*{2em}}1\leq k\leq d_r \Bigg\}$$ with initial values $a_{A_k},\ 1\leq k\leq d_r,$ as obtained in (ii). \(iv) For $n > K,$ the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(x^{2^{n-K}d_r}+ \sum_{i=1}^{d_r}a_{K_i}x^{2^{n-K}(d_r-i)}\right)$$ where each $a_{K_i},\ 1\leq i\leq d_r,$ is as obtained in (iii). Let $$\Phi_r(x) = \prod_{w\in \Omega(r)}g_w(x) = \prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r}(-1)^i S_{i,w}x^{d_r - i}\right)$$ be the factorization of $\Phi_r$ over ${\mathbb{F}_q}.$\ 1. $(n=1):$ Because $g_w$ is irreducible over ${\mathbb{F}_q}$, $g_w(-x)$ is irreducible over ${\mathbb{F}_q}.$ By Theorem \[tricks\], $$\begin{aligned} \Phi_{2r}(x) &=& \Phi_r(-x) = \prod_{w\in \Omega(r)}g_w(-x) = \prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r}(-1)^{d_r} S_{i,w}x^{d_r - i}\right).\end{aligned}$$ Note that in the case $d_r$ is odd the number of irreducible factors of $\Phi_{2r},$ which is $\phi(r)/d_r,$ is even. Thus, it follows that we may write the factorization above as $$\Phi_{2r}(x) = \prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r}S_{i,w}x^{d_r - i}\right).$$ The factorization is complete.\ 2. $\left(2\leq n\leq A\right):$ By Theorem \[2\^n and q = 3 mod 4\] (a) we have $$\begin{aligned} \Phi_{2^nr}(x) &=& \prod_{u\in U_n}\prod_{w\in \Omega(r)}\left(\left(x^2 + \left(u + u^{-1}\right)x + 1\right) \odot g_w\right)(x)\\ &=& \prod_{u\in U_n}\prod_{w\in \Omega(r)}(-u)^{d_r}g\left((-u)^{-1}x\right)(-u)^{-d_r}g\left(-ux\right)\\ &=& \prod_{u\in U_n}\prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r}(-1)^iS_{i,w}(-u)^{i-d_r}x^{d_r-i}\right)\left(\sum_{j=0}^{d_r}(-1)^jS_{j,w}(-u)^{d_r-j}x^{d_r-j}\right)\\ &=& \prod_{u\in U_n}\prod_{w\in \Omega(r)}\left(\sum_{k=0}^{2d_r}\sum_{i+j = k}S_{i,w}S_{j,w}u^{i-j}x^{2d_r-k}\right). {\hspace*{2em}}{\hspace*{2em}}(*)\\ \end{aligned}$$ First, note that these factors in $(*)$ are over ${\mathbb{F}_q}$ as the composed product of polynomials over ${\mathbb{F}_q}$ are polynomials over ${\mathbb{F}_q}.$ We have: \(i) If $d_r$ is odd, then $\gcd(2,d_r) = 1$ and so each factor $\left(x^2 + \left(u + u^{-1}\right)x + 1\right) \odot g_w$ is irreducible by Theorem \[thm 1\]; hence the factorization is complete. \(ii) If $d_r$ is even, then in particular $A < A + v_2(d_r) = K.$ Then by Theorem \[thm 5\] each factor in $(*)$ of $\Phi_{2^nr}$ must decompose into two irreducibles of degree $d_r.$ Thus, for some coefficients $a_{n_i},\ b_{n_j} \in {\mathbb{F}_q}$ we must have $$\begin{aligned} \sum_{k=0}^{d_r}\sum_{i+j = k}S_{i,w}S_{j,w}u^{i-j}x^{2d_r-k} &=& \left(x^{d_r}+ \sum_{i=1}^{d_r}a_{n_i}x^{d_r-i} \right) \left(x^{d_r}+ \sum_{j=1}^{d_r}b_{n_j}x^{d_r-j} \right)\\ &=& x^{2d_r} + \sum_{k=1}^{2d_r}\sum_{i+j=k}a_{n_i}b_{n_j}x^{2d_r-k}.\end{aligned}$$ Comparing coefficients on each side we see that each $a_{n_i},\ b_{n_j}, \ 1\leq i,\ j\leq d_r,$ satisfies the following system of equations $$\Bigg\{\sum_{i+j = k}a_{n_i}b_{n_j} = \sum_{i+j = k}S_{i,w}S_{j,w}u^{i-j},{\hspace*{2em}}1\leq k\leq 2d_r \Bigg\}$$ which has a solution. We stress that the solution must be unique by the uniqueness of factorizations. Hence the result follows.\ \ 3. ($n > A$ and $d_r$ odd): Since $\gcd\left(2^{n-A+1},d_r\right) = 1,$ the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(\left(x^{2^{n-A+1}}+\left(u-u^{-1}\right)x^{2^{n-A}}-1\right) \odot g_w\right)(x).$$ Since the computation of the composed product above is somewhat more involved this time, we proceed as follows: First note that for $n > A$ all irreducible factors of $\Phi_{2^nr}$ have degree $2^{n-A+1}d_r.$ It then follows that if a factor of $\Phi_{2^nr}$ has degree $2^{n-A+1}d_r,$ it must be an irreducible factor. Because $q = 2^Am-1,$ we know that $2^A \mid (q+1)$ and $q^2 - 1 = (q+1)(q-1)$ imply that if $u\in U_A,$ then $u^{q+1} = 1$ and so $u\in \mathbb{F}_{q^2}.$ Note that since $q \equiv 3 \pmod{4},$ then $q^2 \equiv 1 \pmod{4}.$ Then by Theorem \[2\^nr and q = 1 mod 4\], Part 2 (a), the complete factorization of $\Phi_{2^nr}$ over $\mathbb{F}_{q^2}$ is given by $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left(\sum_{i=0}^{d_r}u^i S_{i,w}x^{2^{n-A}(d_r-i)}\right). {\hspace*{2em}}{\hspace*{2em}}(**)$$ Let $Z_u(x) = \sum_{i=0}^{d_r}u^i S_{i,w}x^{2^{n-A}(d_r-i)}$ above, and since $u^q = u^{-1},$ consider its conjugate $$\overline{Z}_u(x) = \sum_{j=0}^{d_r}u^{-j} S_{j,w}x^{2^{n-A}(d_r-j)}.$$ First, note that $u^{-1} \in U_A$ and $(**)$ imply $\overline{Z}_u$ is an irreducible factor of $\Phi_{2^nr}$ over $\mathbb{F}_{q^2}.$ Moreover, $Z_u \neq \overline{Z}_u.$ Indeed, ovserve that $u^{d_r} \neq u^{-d_r},$ otherwise $u^{2d_r} = 1,$ and so ${\operatorname{ord}}(u) = 2^A$ gives $2^A \mid 2d_r$ contrary to $A \geq 2$ and $d_r$ odd. Then $u^{d_r}S_{d_r,w} \neq u^{-d_r}S_{d_r,w}.$ As these are the coefficients of $x^0$ in $Z_u(x),\ \overline{Z}_u(x),$ respectively, necessarily $Z_u \neq \overline{Z}_u.$ We have $$Z_u(x)\overline{Z}_u(x) = \sum_{k=0}^{2d_r}\sum_{i+j=k}u^{i-j}S_{i,w}S_{j,w}x^{2^{n-A}(2d_r-k)}.$$ Note from Part 2 and $(*)$ above that for $u\in U_A$ we have $\sum_{i+j=k}u^{i-j}S_{i,w}S_{j,w} \in {\mathbb{F}_q}$ (since the composed products of polynomials over ${\mathbb{F}_q}$ are polynomials over ${\mathbb{F}_q}$). Thus $Z_u\overline{Z}_u \in {\mathbb{F}_q}[x],$ it has degree $2^{n-A+1}d_r,$ and is a factor of $\Phi_{2^nr}$ clearly. But then $Z_u\overline{Z}_u$ must be irreducible over ${\mathbb{F}_q};$ hence the complete factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ must be $$\Phi_{2^nr}(x) = \prod_{u\in U_A}\prod_{w\in \Omega(r)}\left( \sum_{k=0}^{2d_r}\sum_{i+j=k}u^{i-j}S_{i,w}S_{j,w}x^{2^{n-A}(2d_r-k)} \right)$$ as required.\ 4. (iii) Similar to the proof of (i) in Theorem \[2\^nr and q = 1 mod 4\]. \(iv) Similar to the proof of (ii) in Theorem \[2\^nr and q = 1 mod 4\]. See Remark \[rem 2.1\] after Theorem \[2\^nr and q = 1 mod 4\]. Furthermore, comparing the factorizations in Parts 2 (i) and 3, we see that the factors in Part 3 can be obtained from the factors in Part 2 (i) by the substitution $x \rightarrow x^{2^{n-A}}.$ Thus, for $n > A = v_2(q+1),$ if $\Phi_{2^Ar} = \prod_k h_{A_k}$ is the corresponding factorization, then $\Phi_{2^nr}(x) = \prod_k h_{A_k}(x^{2^{n-A}})$ is the complete factorization over ${\mathbb{F}_q}.$ Moreover, it is easy to see that $A = v_2(q+1)$ is the smallest such bound with this property. In the case $d_r$ is even, see Remarks \[rem 2.2\] and \[rem 2.3\] after Theorem \[2\^nr and q = 1 mod 4\]. Let $n > A,$ let $S = \{s_k\},\ T = \{t_k\}$ be homogeneous LRS’s with characteristic polynomials $\Phi_{2^n},\ \Phi_{r}$ respectively. Then as discussed earlier, the characteristic polynomial of $ST = \{s_k t_k \}$ is $\Phi_{2^nr} = \Phi_{2^n} \odot \Phi_r.$ Suppose $d_r$ is odd. Since all irreducible factors of $\Phi_{2^nr},\ n > A,$ have degree $2^{n-A+1}d_r,$ the minimal polynomial of $ST$ must have degree $2^{n-A+1}d_r.$ This is the linear complexity of $ST.$ Note that if we let $n \rightarrow \infty,$ the linear complexity of the corresponding LRS $ST$ approaches infinity. For the subcases $q \equiv 3 \pmod{4}$ with $q \equiv \pm 1 \pmod{r},$ and thus $d_r = 1,\ 2,$ where $r$ is an odd prime, Theorem \[2\^nr and q = 3 mod 4\] becomes Theorem 1, Parts 1 and 3 in Fitzgerald and Yucas (2007) [@Fitzgerald; @2007]. Conclusion ========== In this paper we gave the factorization of the cyclotomic polynomial $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ where both $r \geq 3, \ q$ are odd and $\gcd(q,r) = 1.$ Previously, only $\Phi_{2^n3}$ and $\Phi_{2^n5}$ had been factored in [@Fitzgerald; @2007] and [@Prof], respectively. As a result we have obtained infinite families of irreducible sparse polynomials from these factors. Furthermore, we showed how to obtain the factorization of $\Phi_n$ in a special case (see Theorem \[cyclotomics are composed\]). We also showed in Theorem \[cyclo and minimal\] how to obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),{\operatorname{ord}}_n(q)) = 1.$ The factorization of $\Phi_{2^n}$ was already given in [@Lidl] when $q \equiv 1 \pmod{4}$ and in [@Meyn] when $q \equiv 3 \pmod{4}.$ It is natural to consider the factorization of $\Phi_{3^n}.$ We then wonder if some of the techniques used in Section 3 could be applied to factor $\Phi_{3^nr} = \Phi_{3^n}\odot \Phi_r.$ In particular, it would be desirable to generalize Theorem \[thm 5\] to allow for other cases (besides $2^n$). It is expected that these irreducible factors will be sparse as well. Note that we can allow $q$ to be even in this case by forcing $r$ to be odd. This is significant as the fields $\mathbb{F}_{2^m}$ are the most commonly used in modern engineering. In Section 2 we considered irreducible composed products of the form $f \odot \Phi_m.$ In particular, we derived the construction of a new class of irreducible polynomials in Theorem \[thm 3\]. It is natural to consider other classes of polynomials and substitute them for $\Phi_m$ and see what the result may be. We also gave formulas for the linear complexity of $ST$ when $\Phi_{2^n},\ \Phi_r$ are characteristic polynomials of the homogeneous LRS’s $S,\ T,$ respectively. We showed that by letting $n \rightarrow \infty,$ the linear complexity of $ST$ will approach infinity. Another matter of interest is the factorization of composed products. Since the minimal polynomial of a LRS, say $ST,$ is an irreducible factor of some composed product, this has applications in stream cipher theory, LFSR and LRS in general. D. Mills (2001) [@Mills] had already studied the factorization of arbitrary composed products. In particular, if $\deg f = m$ and $\deg g = n$ with $f,\ g$ irreducible over ${\mathbb{F}_q},$ Mills gave $d = \gcd(m,n)$ as an upper bound for the number of irreducible factors that $f \diamond g$ could decompose into. He also gave the possible degrees that these irreducible factors may attain. As a result, we now know the possible linear complexities that $ST$ could attain. On the other hand his work was generalized for two arbitrary irreducible polynomials $f$ and $g.$ In the case that at least one of these polynomials belongs to a certain class of polynomials with well defined properties, we wonder if it could be possible to obtain more precise information regarding the number of irreducible factors and their degrees. For instance, in the case of $f \odot \Phi_m,$ can we know precisely the degrees of the irreducible factors? Can we know precisely in how many irreducible factors does $f \odot \Phi_m$ decompose into? Note that the subject of the factorization of composed products is one for which very little research has been done. Currently, the authors were able to find only one paper [@Mills] on this matter and they feel this is a topic that has been somewhat neglected. [99]{} T. M. Apostol (1976). [*Introduction to Analytic Number Theory*]{}, Springer-Verlag New York Inc. A. S. Bamunoba (2010). [*Cyclotomic Polynomials*]{}, African Institute for Mathematical Sciences, Stellenbosch University, South Africa. Available at: www.users.aims.ac.za/$\sim$bamunoba/bamunoba.pdf E. R. Berlekamp (1982). [*Bit-serial Reed-Solomon Encoders*]{}, IEEE Trans. Info. Theory [**28**]{}, 869-874. F. R. Beyl (1977). [*Cyclic Subgroups of the Prime Residue Group*]{}, Amer. Math. Monthly [**84**]{}, 46-68. I. Blake, S. Gao and D. Mills (1991). [*Factorization of Polynomials of the type $f\left(x^t\right)$*]{}, presented at the International Conference on Finite Fields, Coding Theory, and Advances in Comm. and Computing, Las Vegas. J. V. Brawley and L. Carlitz (1987). [*Irreducibles and the Composed Product for Polynomials over a Finite Field*]{}, Discrete Math., [**65**]{}, 115-139. J. V. Brawley, S. Gao and D. Mills (1997). [*Computing Composed Products of Polynomials*]{}, Finite fields: theory, applications, and algorithms (Waterloo, ON, 1997), 1-15, Contemp. Math., [**225**]{}, Amer. Math. Soc., Providence, RI. S. D. Cohen (1969). [*On Irreducible Polynomials of certain types in Finite Fields*]{}, Proc. Cambridge Philos. Soc. [**66**]{}, 335-344. S. D. Cohen (2005). [*Explicit Theorems on Generator Polynomials over Finite Fields*]{}, Finite Fields Appl. [**11**]{}, 337-357. R. W. Fitzgerald and J. L. Yucas (2005). [*Factors of Dickson Polynomials over Finite Fields*]{}, Finite Fields Appl. [**11**]{}, no. 4, 724-737. R. W. Fitzgerald and J. L. Yucas (2007). [*Explicit Factorization of Cyclotomic and Dickson Polynomials over Finite Fields*]{}, Arithmetic of Finite Fields, [*Lecture Notes in Comput. Sci.*]{} [**4547**]{}, Springer, Berlin, 1-10. Z. Gao and F. Fu (2009). [*The Minimal Polynomial over ${\mathbb{F}_q}$ of Linear Recurring Sequence over $\mathbb{F}_{q^m}$*]{}. Finite Fields Appl. [**15**]{}, no. 6, 774-784. S. Golomb and G. Gong (2005). [*Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar*]{}, Cambridge University Press. M. K. Kyuregyan and G. H. Kyureghyan (2011). [*Irreducible Compositions of Polynomials over Finite Fields*]{}, Designs, Codes and Cryptography, [**61**]{}, no. 3, 301-314. R. Lidl and H. Niederreiter (1997). [*Finite Fields*]{}, in Encyclopedia of Mathematics and its Applications, 2nd ed., vol 20, Cambridge University Press, Cambridge. H. Meyn (1996). [*Factorization of the Cyclotomic Polynomials $x^{2^n} + 1$ over Finite Fields*]{}, Finite Fields Appl. [**2**]{}, 439-442. D. Mills (2001). [*Factorizations of Root-based Polynomial Compositions*]{}, Discrete Math. [**240**]{}, no. 1-3, 161-173. W. K. Nicholson (1999). [*Introduction to Abstract Algebra*]{}, 2nd ed., John Wiley & Sons, Inc. E. S. Selmer (1966). [*Linear Recurrence Relations over Finite Fields*]{}, Univ. of Bergen. G. Stein (2001). [*Using the Theory of Cyclotomy to factor Cyclotomic Polynomials over Finite Fields*]{}, Math. Comp. [**70**]{}, no. 235, 1237-1251. B. Sury (1999). [*Cyclotomy and Cyclotomic Polynomials: The story of how Gauss narrowly missed becoming a philologist*]{}, RESONANCE, 41-53. Available at: www.springerlink.com/index/h44xt1p4p42m3987.pdf A. Tuxanidy (2011). [*Composed Products and Factorization of Cyclotomic Polynomials over Finite Fields*]{}, Honours Project, Carleton University. R. Varshamov (1984). [*A General Method of Synthesizing Irreducible Polynomials over Galois Fields*]{}, Soviet Math. Dokl., [**29**]{}, 334-336. Z. Wan (2003). [*Lectures on Finite Fields and Galois Rings*]{}, World Scientific Publishing Co. Pte. Ltd. M. Wang and I. F. Blake (1989). [*Bit-serial Multiplication in Finite Fields*]{}, IEEE Trans. Comput. [**38**]{}, 1457-1460. L. Wang and Q. Wang (2011). [*On Explicit Factors of Cyclotomic Polynomials over Finite Fields*]{}, Designs, Codes and Cryptography, Springer Netherlands. L. C. Washington (1982). [*Introduction to Cyclotomic Fields*]{}, Springer-Verlag New York Inc. N. Zierler and W. H. Mills (1973). [*Products of Linear Recurring Sequences*]{}, J. Algebra [**27**]{}, 147-157. Samples of Irreducible Polynomials $F_m$ ======================================== We provide a table of examples for Theorem \[thm 3\]. MAPLE software was used in the computations.\ Table 1. Table of (irreducible) samples of $F_m$ from Theorem \[thm 3\] outputed on inputs $(m,q,n)$ and $f.$ --------------------------------------------------------------------------------------------------------------------------------------------------------------- $\left(m,q,n\right)$ $f(x)$ $F_m(x)$ ------------------------ ------------------------------------ ------------------------------------------------------------------------------------------------- $(2,3,6)$ $x^6+ 2x^4 + x^3 +2x+1 $ $x^6 + x^5+2x^4+x^3+x+2$ $(2,5,5)$ $x^5+3x^4+4x^3+4x+2$ $x^5+2x^4+4x^3+4x+3$ $(4,3,9)$ $x^9+x^7+x^6+x+1$ $x^{18}+x^{16}+x^{14}+x^{12}+2x^{10}+x^8+x^6+x^2+1$ $(4,7,3)$ $x^3+4x^2+1$ $x^6+2x^4+6x^2+1$ $\left(3^2,5,5\right)$ $x^5+3x^4+4x^2+x+1$ $x^{30}+3x^{27}+3x^{24}+3x^{21}+3x^{18}+x^{15}+2x^9+4x^6 + 2x^3+1$ $\left(7^2,3,5\right)$ $x^5+x^4+x^2+2x+2$ $x^{210} + 2x^{203}+ \dots + 1$ $(6,5,9)$ $x^9+4x^8+3x^7+x^5+3x^4+4x^2+2x+3$ $x^{18}+4x^{17}+3x^{16}+2x^{15}+3x^{14}+x^{11}+x^{10}+2x^9+4x^8+x^7+x^6+x^5+2x^4+3x^3+2x^2+x+4$ $(10,3,5)$ $x^5+x^3+x^2+2x+2$ $x^{20}+2x^{18}+x^{17}+2x^{16}+x^{15}+x^{14}+x^{12}+2x^{10}+2x^8+x^7+2x^3+2x^2+x+1$ $\left(3^2,2,5\right)$ $x^5+x^2+1$ $x^{30} + x^{27}+ x^{21}+ x^6+1$ $\left(3^3,2,5\right)$ $x^5+x^2+1$ $x^{90}+x^{81}+x^{72}+x^{45}+x^{27}+x^9+1$ --------------------------------------------------------------------------------------------------------------------------------------------------------------- Recursive Computations ====================== We provide the following tables of examples for Theorems $\ref{2^nr and q = 1 mod 4}$ (i) and $\ref{2^nr and q = 3 mod 4}$ (iii). The coefficients $(a_{n_1},a_{n_2},\dots , a_{n_6})$ are the coefficients of the irreducible factors of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ for $q = 5,\ 19,\ r = 7,\ n\leq K = 3,$ calculated by using the recurrence relations in Theorems $\ref{2^nr and q = 1 mod 4}$ (i) and $\ref{2^nr and q = 3 mod 4}$ (iii). In particular, the tables show that these recursive relations, now with initial values corresponding to $n = 1,$ may be used to obtain the factors of $\Phi_{2^nr}$ when $n \leq A$ as well. MAPLE software was used in the computations.\ Table 2. Factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ where $r = 7,\ q = 5,\ n \leq K= 3$ $n$ 1 2 3 ------------------------------------- -------------------- -------------------- -------------------- $(a_{n_1},a_{n_2},\dots , a_{n_6})$ (4, 1, 4, 1, 4, 1) (2, 4, 3, 1, 2, 4) (1, 4, 3, 2, 4, 2) (3, 4, 2, 1, 3, 4) (4, 4, 2, 2, 1, 2) (2, 1, 4, 2, 3, 3) (3, 1, 1, 2, 2, 3) \ \ Table 3. Factorization of $\Phi_{2^nr}$ over ${\mathbb{F}_q}$ where $r = 7,\ q = 19,\ n \leq K = 3$ $n$ 1 2 3 ------------------------------------- ----------------------- ----------------------- ------------------------ $(a_{n_1},a_{n_2},\dots , a_{n_6})$ (18, 1, 18, 1, 18, 1) (8, 3, 8, 3, 8, 1) (2, 6, 10, 13, 2, 18) (11, 3, 11, 3, 11, 1) (17, 6, 9, 13, 17, 18) (8, 9, 18, 10, 8, 18) (11, 9, 1, 10, 11, 18) [^1]: Aleksandr Tuxanidy wishes to dedicate his work here to Dr. E. Lorin and Dr. Q. Wang for their support and guidance throughout the years 2010, 2011. In particular, they made him believe in himself as a student once more. The research of Qiang Wang is partially supported by NSERC of Canada.

--- abstract: 'Turbulent Rayleigh-Bénard convection displays a large-scale order in the form of rolls and cells on lengths larger than the layer height once the fluctuations of temperature and velocity are removed. These turbulent superstructures are reminiscent of the patterns close to the onset of convection. They are analyzed by numerical simulations of turbulent convection in fluids at different Prandtl number ranging from 0.005 to 70 and for Rayleigh numbers up to $10^7$. For each case, we identify characteristic scales and times that separate the fast, small-scale turbulent fluctuations from the gradually changing large-scale superstructures. The characteristic scales of the large-scale patterns, which change with Prandtl and Rayleigh number, are also found to be correlated with the boundary layer dynamics, and in particular the clustering of thermal plumes at the top and bottom plates. Our analysis suggests a scale separation and thus the existence of a simplified description of the turbulent superstructures in geo- and astrophysical settings.' author: - Ambrish Pandey - 'Janet D. Scheel' - Jörg Schumacher title: 'Turbulent superstructures in Rayleigh-Bénard convection' --- Large temperature differences across a horizontally extended fluid layer induce a turbulent convective fluid motion which is relevant in numerous geo- and astrophysical systems [@Kadanoff2001]. These flows are typically highly turbulent with very large Rayleigh numbers $Ra$, the parameter that quantifies the intensity of the thermal driving in convection. From the classical perspective of turbulence one would expect a chaotic, irregular motion of differently sized vortices and thermal plumes. Rather than such a featureless stochastic fluid motion, some turbulent flows in nature display an organization into prominent and regular flow patterns that persist for times long compared to an eddy turnover time and extend over lengths which are larger than the height scale. Examples are cloud streets in the atmosphere [@Markson1975] or granulation networks at the solar surface [@Nordlund2009] and other stars [@Michel2008]. This large-scale order will be termed a turbulent superstructure. It is observed in turbulent convection flows with very different molecular dissipation properties. The Prandtl number $Pr=\nu/\kappa$, another dimensionless parameter which relates kinematic viscosity $\nu$ to temperature diffusivity $\kappa$, is for example very small for stellar convection, $Pr\lesssim 10^{-3}$ [@Spiegel1962; @Thual1992; @Hanasoge2016]. It is 0.7 for atmospheric flows and 7.0 for heat transport in the oceans. Rayleigh-Bénard convection (RBC) is the simplest turbulent convection flow evolving in a planar fluid layer of height $H$ that is uniformly heated with a temperature $T=T_b$ from below and cooled from above with $T=T_t$ such that $T_b-T_t=\Delta T>0$. The Rayleigh number is given by $Ra=g\alpha \Delta T H^3/(\nu\kappa)$ with $g$ being the acceleration due to gravity and $\alpha$ the thermal expansion coefficient. RBC can be considered as a paradigm for many applications [@Ahlers2009; @Chilla2012] that usually contain further physical processes, such as radiation [@Christensen1996] and phase changes [@Stevens2005; @Pauluis2011], and additional fields such as magnetic fields [@Aurnou2010]. Numerical simulations of convection [@Hartlep2003; @Hartlep2005; @Rincon2005; @Hardenberg2008; @Bailon2010; @Emran2015] have enabled researchers to access the large-scale structure formation in turbulent convection flows. Long-term investigations at very small Prandtl numbers $Pr\ll 0.1$ require simulations on massively parallel supercomputers in order to resolve the highly inertial turbulence properly. Such simulations have not been done before and this is a central motivation for the present study. At the onset of convection, $Ra_c=1708$, straight convection rolls have a unique and Prandtl-number-independent wavelength, $\lambda_c\approx 2H$ [@Jeffreys1928; @Chandrasekhar1961]. For $Ra\gtrsim Ra_c$, these rolls become susceptible to secondary linear instabilities causing modulations, such as Eckhaus, zig-zag or oscillatory patterns [@Busse1978; @Cross1993; @Bodenschatz2000]. These secondary instabilities depend strongly on the Prandtl number of the working fluid and the wavenumber range of the plane-wave perturbation to the convection straight rolls in the layer [@Busse1978]. Dependencies on Rayleigh and Prandtl numbers of the pattern wavelength for $Ra>Ra_c$ have been studied systematically in RBC experiments in air, water and silicone oil by Willis et al. [@Willis1972]. Average roll widths tend to increase with $Ra$, which the authors attributed to increasingly unsteady three-dimensional motions. The trend with growing $Pr$ is less systematic [@Hartlep2003] and accompanied by hystereses at $Pr\gg 1$ [@Willis1972]. Roll and cell patterns of the velocity field in a [*turbulent*]{} RBC for $Ra\gtrsim 10^5$ that are reminiscent of the flow structures in the weakly nonlinear regime at $Ra \lesssim 5\times 10^3$ have been observed in recent DNS at $Pr\gtrsim 1$ [@Bailon2010; @Emran2015]. Their detection requires an averaging over a time interval that should be long enough to remove the turbulent fluctuations in the fields effectively and yet short enough to not wash away the large-scale structures [@Emran2015]. A sliding time average with an appropriate time window width should thus be able to separate the fast, small-scale turbulent fluctuations of velocity and temperature from the gradual variation of the large-scale superstructure patterns. Physically, this time window should be connected with the turnover time of fluid parcels in the superstructure rolls and cells. The determination of this averaging time scale as a function of $Ra$ and $Pr$ is a second motivation for the present study. In the present work, we report an analysis of the characteristic spatial and temporal scales of turbulent superstructures in RBC by means of three-dimensional direct numerical simulations (DNS) spanning more than four orders of magnitude in $Pr$ and more than three orders in $Ra$. All simulations reported here are of the Boussinesq equations of motion and performed in an extended closed square cell of aspect ratio of 25:25:1. We identify the characteristic averaging time scales, $\tau(Ra, Pr)$, which will be connected with a characteristic spatial scale (or wavelength) that can be determined by a spectral analysis of the turbulent superstructures. Our study of large-aspect-ratio turbulent RBC extends to very small Prandtl numbers with values significantly below 0.1, which have not been obtained before. The gradual evolution of the patterns at all Prandtl numbers is confirmed by radially averaged, azimuthal power spectra that reveal a gradual switching of the orientation of the superstructures which is reminiscent of cross-roll or skewed varicose instabilities that are well-known from the weakly nonlinear regime of RBC. Furthermore, we compare the characteristic pattern scale in the bulk of the RBC flow to the scales of plumes and plume clusters that are present in the boundary layers in the vicinity of the top and bottom walls. The temperature patterns in the bulk are found to be correlated with the most prominent ridges in the vertical temperature field derivative at the bottom and top plates which in turn are correlated with the wall stresses of the advecting velocity. Our analysis provides characteristic separation time and length scales for turbulent convection flows in extended domains and thus opens the possibility to describe the superstructure patterns in turbulent convection by effective and reduced models that separate the fast, small scales from the slow, large scales. These reduced models can advance our understanding of a variety of turbulent systems that exhibit large-scale pattern formation, including mesoscale convection and solar granulation. Results {#results .unnumbered} ======= [**Superstructures for different Rayleigh and Prandtl numbers.**]{} Figure \[fig0\] shows the velocity field lines (top row) and the corresponding temperature contours in the midplane (bottom row) for a simulation at one of the lowest Prandtl numbers in our simulations. While the instantaneous pictures display the expected irregularity of a turbulent flow as visible for example by the streamline tangle in panel (a), the averaged data reveal a much more ordered pattern. We also see that the superstructure patterns are more easily discerned in temperature field snapshots than in those of the velocity field. Figure \[fig2\] confirms this observation. Here, we plot the root mean square (rms) values of the vertical velocity component $u_z$ and the temperature $T$. In agreement with Fig. \[fig0\], we split both fields into contributions coming from the time average over the time interval $\tau$ and the fluctuations, $$\begin{aligned} u_z({\bm x},t)&=U({\bm x})+u_z^{\prime}({\bm x},t)\,,\\ T({\bm x},t)&=\Theta({\bm x})+T^{\prime}({\bm x},t)\,.\end{aligned}$$ The averaging volume $\tilde{V}$ is a slab around the midplane. See Eqns. (\[uvf\]) and (\[tvf\]) later in the text for definitions of $U$ and $\Theta$. It can be seen that the rms values of the total and time averaged temperature are always close together when Prandtl and Rayleigh number are varied. This is in contrast to the vertical velocity component. Fluctuations dominate here when the Prandtl numbers are low and the Rayleigh numbers are sufficiently high. An averaging with respect to time is thus necessary to reveal the patterns for both turbulent fields. Figure \[fig1\] displays velocity field lines and temperature contours of time-averaged turbulent RBC flows at Prandtl number ranging from $Pr=0.005$ to 70 at $Ra=10^5$ and at Rayleigh number ranging from $Ra=5\times 10^3$ to $10^7$ for convection in air at $Pr=0.7$. All runs are turbulent and thus beyond the weakly nonlinear regime, except the runs in panel (e) at $Pr=70$, panel (f) at $Ra=5000$, and panel (g) at $Ra=10^4$ respectively. For the non-turbulent cases the time averaged data does not deviate significantly from the instantaneous snapshots. If we look at the trends for all runs, we see that the velocity field lines form curved rolls for the lower $Pr$ and cell-like patterns for $Pr\ge 7$. These structures fill the whole layer and are reminiscent of patterns at the onset of convection at much smaller Rayleigh numbers [@Bodenschatz2000]. The corresponding temperature averages in the midplane show alternating ridges of cold downwelling and hot upwelling fluid which are coarser for the lowest Prandtl numbers and the highest Rayleigh numbers, respectively. For $Pr=0.005$ and 0.021, this is due to the highly diffusive temperature field that is in conjunction with an inertia-dominated fluid turbulence [@Schumacher2015; @Schumacher2016; @Scheel2016]. In case of the highest Prandtl number, $Pr=70$ at $Ra=10^5$, the amplitude of the turbulent velocity field fluctuations is significantly smaller and the temperature field displays much finer filaments. Coarser temperature patterns can also be observed for the highest Rayleigh number at $Ra=10^7$. In the Supplementary Material, we plot additional vertical profiles of the velocity fluctuations as well as list further details for all simulation runs and in the Methods section the characteristic units are given which we use to formulate the Boussinesq model in dimensionless form. While low-Prandtl-number convection transports momentum very efficiently, the heat transport becomes significantly larger at the higher Prandtl numbers. Figure \[fig1\] also demonstrates that the characteristic mean width of the rolls and spirals varies with $Pr$ and $Ra$. {width="90.00000%"} [**Characteristic times and scales of superstructures.**]{} The free-fall time $T_f=(H/g\alpha\Delta T)^{1/2}$ is a characteristic convective time unit that stands for the (relatively) fast dynamics of thermal plumes and larger vortices in a turbulent convection flow. A slower time unit in the turbulent flow is either a vertical viscous ($Pr<1$) or a vertical diffusive ($Pr>1$) time composing an effective dissipative time by $T_d=\max(t_{\kappa}, t_{\nu})$ with $t_{\kappa}=H^2/\kappa$ and $t_{\nu}=H^2/\nu$. A complete removal of the large-scale patterns would require an averaging period on the order of $\Gamma^2T_d$ (with $\Gamma$ being the aspect ratio of the domain) which is $\gg 10^3 -10^4 T_f$, i.e., times which are not accessible in our massively parallel turbulence simulations. Thus, the averaging time $\tau$ that separates small-scale turbulence and superstrucutres should be bounded by $$T_f \ll \tau(Ra, Pr) \ll T_d\,.$$ This time $\tau$ should be considered as a representative value of a finite range of times rather than an exact time and is expected to show a dependence on our two system parameters $Ra$ and $Pr$. In the Supplementary Material it is shown for two different Prandtl numbers how the patterns change when the averaging time is varied. On the one hand, $\tau$ should be long enough to remove all small-scale fluctuations and to reveal the superstructures, in particular of velocity. On the other hand, $\tau$ has to be short enough such that the large-scale patterns are not removed completely. Hence we define $\tau$ as the characteristic turnover time of fluid parcels in the circulation rolls or cells, the latter of which extend across the whole layer from bottom to top and are considered as the building blocks of the superstructure velocity patterns. In order to proceed, we decompose the RBC fields into a fast changing and gradually evolving contribution. This is inspired by asymptotic expansions that are developed for constrained turbulence, e.g., fast rotation or strong magnetic fields [@Julien2007; @Klein2010; @Malecha2014]. Furthermore, we substitute the full temperature field, $T({\bm x},t)$, by its deviation from the linear diffusive equilibrium profile, $\theta({\bm x},t)=T({\bm x},t)-T_{\text{lin}}(z)$. Our focus is on the horizontal patterns in the system. Therefore, the subsequent superstructure analysis is focussed on the symmetry plane at $z=1/2$ where the patterns are identified by upwelling hot and downwelling cold fluid (see Fig. \[fig3\](a)). The gradually varying fields are given by the following sliding time average with respect to $\tau$ $$\begin{aligned} \label{uvf} U(x,y;\tau,t_0) &=\frac{1}{\tau}\int^{t_0+\tau/2}_{t_0-\tau/2} u_z(x,y,z=1/2,t^{\prime})\,dt^{\prime}\,,\\ \label{tvf} \Theta(x,y;\tau,t_0) &=\frac{1}{\tau}\int^{t_0+\tau/2}_{t_0-\tau/2} \theta(x,y,z=1/2,t^{\prime})\, dt^{\prime}\,.\end{aligned}$$ Snapshot data is output periodically and $t_0$ is the time scale for this output interval (see the Supplementary Material for more details). Both fields are transformed onto a polar wavevector grid in Fourier space giving $\hat{U}(k,k_{\phi};\tau, t_0)$ and $\hat{\Theta}(k,k_\phi;\tau, t_0)$. Azimuthally averaged Fourier spectra (see Fig. \[fig3\](b)) are given by $$E_{\omega}(k;\tau, t_0)=\frac{1}{2\pi} \int_0^{2\pi} |\hat\omega(k,k_\phi;\tau, t_0)|^2 \,dk_\phi\,,$$ with $\hat\omega=\{\hat U,\hat \Theta\}$. All spectra $E_{\omega}(k;\tau, t_0)$ show a global maximum. An additional average over all $t_0$ yields a unique maximum wavenumber $k^\ast_{U,\Theta}=2\pi/\hat\lambda_{U,\Theta}$ which depends on $Ra$ and $Pr$ as shown in Figs. \[fig3\] (c,d). The wavelength $\hat\lambda_{U,\Theta}(Ra,Pr)/2$ is the characteristic mean width of the superstructure rolls as sketched in panel (a) of Fig. \[fig3\]. We note that the spectra $E_{\omega}(k;\tau, t_0)$ do not vary significantly with $t_0$, in particular in respect to the maximum wavenumber $k^{\ast}$. The characteristic wavelengths in Figs. \[fig3\](c, d) are larger than the critical wavelength $\lambda_c=2\pi/k_c \approx 2$ at the onset of convection with $Ra_c=1708$ [@Chandrasekhar1961]. It is seen that the wavelength grows with $Ra$ at fixed $Pr$. The dependence on the Prandtl at fixed Rayleigh number in our data indicates a growth up to $Pr\sim 10$ and a subsequent decrease for even higher values which is in agreement with [@Hartlep2003] for smaller $\Gamma$. In the Supplementary Material we demonstrate that nearly the same scales can be obtained by an analysis of the two-point correlation functions in physical space. {width="75.00000%"} Interestingly, Fig. \[fig3\] also shows that $\lambda_{\Theta} \gtrsim \lambda_{U}$. At the onset of convection, both wavelengths are exactly the same since both fields are perfectly synchronized in the midplane. Hot fluid is advected upwards ($\theta, u_z>0$) while cold fluid is brought downwards ($\theta,u_z<0$). This perfect synchronicity breaks down with increasing $Ra$ since the temperature field is not only advected by vertical velocity component across the midplane, but also by rising horizontal velocity fluctuations. They expand the temperature patterns compared to those of the vertical velocity component which manifests in a somewhat larger wavelength $\lambda_{\Theta}$. We quantified this effect by the calculation of a horizontal Péclet number $Pe_h=v_h H/\kappa$ based on a horizontal root mean square velocity in the midplane, $v_h=(\langle u_x^2\rangle+\langle u_y^2\rangle)^{1/2}$. The Péclet number is always larger than 10 which underlines a dominance of convection in comparison to diffusion. With the characteristic width of the superstructure rolls (or cells) of $\hat{\lambda}_U/2$ determined, we can now define the characteristic turnover time for a fluid parcel. We estimate this time scale by an elliptical circumference, $\ell\approx \pi(a+b)$ with $a$ and $b$ (see again Fig. \[fig3\](a)) being the half-axes, and root mean square velocity of the turbulent flow. The characteristic time scale of the turbulent superstructures, beyond which the gradual evolution of the large-scale patterns proceeds, is given by $$\tau(Ra,Pr)\approx 3\frac{\ell}{u_{rms}} \approx 3\frac{\pi \left(\frac{1}{4}\lambda_U+\frac{1}{2}H\right) } {\langle u_x^2+u_y^2+u_z^2\rangle_{V,t}^{1/2}} \,. \label{tau}$$ Figures \[fig3\](e, f) display these computed times as a function of $Ra$ and $Pr$. The prefactor of 3 in Eq. (\[tau\]) accounts for the fact that an individual fluid parcel is not perfectly circulating around in such a roll when the flow is turbulent. We tested that different prefactors of same order of magnitude do not change the results qualitatively (see also Supplementary Material). The characteristic time $\tau$ is found to be nearly unchanged at the fixed Prandtl number. It increases with $Pr$ at fixed $Ra$, remaining however always well below the upper bound, the dissipation time scale $T_d$ (see the table in the Supplementary Material). [**Radially averaged power spectra for slow superstructure evolution.**]{} On time scales larger than $\tau$ the turbulent superstructure patterns are found to evolve by slow changes in orientation and topology. This can be quantified by an angular spectral analysis [@Zhong1992]. We take the radially averaged power spectrum of temperature $\Theta$ which is given by $$E_{\Theta}(k_\phi;\tau,t_0)=\frac{1}{k_m} \int_0^{k_m} |\hat\Theta(k,k_\phi;\tau,t_0)|^2\,dk\,, \label{radpower}$$ and plot the spectra in Fig. \[fig4\] versus time $\tau$. The wavenumber $k_m$ in Eq. (\[radpower\]) denotes a cutoff with $k_m \gg k^{\ast}_{\Theta}$. Local maxima in this spectrum indicate now a preferential orientation of parallel rolls. The slow evolution of the turbulent superstructures becomes visible by the slow variation of the local maxima in the spectrum in all presented runs. We can identify in all cases a small number of local maxima that grow and then decay with time. As the old maxima decay, new ones set in that are shifted by discrete angles from the old ones. This suggests secondary modulations of the dominant roll pattern. We also see that for the highest $Pr$ the maxima persist for a very long period while they switch more rapidly in case of the lower $Pr$. This behaviour is reminiscent of cross-roll or skewed varicose instabilities that have been studied in detail in weakly nonlinear convection above onset [@Bodenschatz2000]. {width="97.00000%"} [**Connection of superstructures to boundary layers.**]{} Figures \[fig3\](c, d) show that the characteristic scale of the superstructures varies with $Pr$ and $Ra$. For example, the growth of the wavelength with $Ra$ can be attributed to the increasingly erratic variations of the temperature filaments which in turn cause an effective increase of the size of the time-averaged structures in the Rayleigh number range that is monitored here. The trend with Prandtl number at fixed Rayleigh number is less obvious. Therefore Fig. \[fig5\] compares the instantaneous temperature field structure in the boundaries at the top and bottom plates with that in the midplane at $z=1/2$ for two different Prandtl numbers. We display the vertical temperature derivative $\partial T/\partial z$ at $z=0, 1$ in panels (a, e, f, j) of Fig. \[fig5\]. This field is one way to highlight the thermal plume ridges [@Shishkina2005]. As expected, thin filaments and subfilaments are observed for a higher $Pr$ while the derivative contours appear somewhat blurred and coarse grained for the lowest Prandtl number. Panels (b, d, g, i) of Fig. \[fig5\] display a zoom of the same data together with the field lines of the skin friction field ${\bm s}=(\partial_z u_x, \partial_z u_y)$ at the plates. This two-dimensional vector field is composed of the two non-vanishing components of the velocity gradient tensor at $z=0,1$. It contains sources and sinks and is fully determined by its critical points, ${\bm s}=0$ [@Chong2012; @Bandaru2015]. These critical points are either unstable nodes, stable nodes or saddles, and much less frequently unstable and stable foci. Groups of saddles and stable nodes are correlated with local regions of the formation of dominant plumes while unstable nodes are mostly found where colder (hotter) fluid impacts the bottom (top) plate. This is very clearly visible for $Pr=7$ in the bottom row of the figure, but does also hold for the low-Prandtl-number data displayed here. The structures at the top plate display the same plume ridges, but are shifted by a roll-length when compared with those at the bottom plate, as is expected for a system of parallel rolls (see also Supplementary Material). The panels (c, h) of Fig. \[fig5\] show the instantaneous temperature field $T$ in the midplane with local maxima and minima exactly where the hot and cold plume ridges are present at the plates, respectively. These dominant ridges are the ones that persist as the superstructures once the time-averaging over $\tau$ is performed. Figure \[fig6\] demonstrates also that the turbulent superstructures are directly connected to the strongest thermal plumes in the boundary layers. This plume formation process is determined by two aspects: (i) the molecular diffusivity of the temperature field (and the resulting differences in the thicknesses of thermal and viscous boundary layers) and (ii) the typical variation scale of the horizontal velocity field near the walls that forms the plume ridges by temperature field advection. While the first aspect will affect the shape of the plume ridges and thus the characteristic thickness scale of the local temperature maxima and minima in the midplane, the second one is directly connected to the spacing of the dominant temperature structures in the midplane and thus the width of the large-scale circulation rolls and cells that fill the layer. The divergence of the skin friction field which is given at the bottom plate by $$\mbox{div} {\bm s}=\frac{\partial^2 u_x}{\partial x\,\partial z}\Bigg|_{z=0}+ \frac{\partial^2 u_y}{\partial y\,\partial z}\Bigg|_{z=0} =-\frac{\partial u_z^2}{\partial z^2}\Bigg|_{z=0} \,, \label{divs}$$ can be considered as a blueprint of the alternating impact (source with $\mbox{div} {\bm s}>0$) and ridge formation (sink with $\mbox{div} {\bm s}< 0$) regions. The skin friction field is thus a key to understanding the clustering of thermal plumes near the wall, a phenomenon which has been reported for example in [@Parodi2004]. The same picture holds at the top plate. ![[**Scale correlations of bulk and boundary layer.**]{} Spectra of temperature $\Theta$ in midplane (a), of vertical temperature derivative $\partial_z T$ at bottom and top plates (b), and of skin friction field divergence at bottom and top plates (c) are compared for five different Prandtl numbers as indicated by the legend. The vertical dashed lines denote the maximum wavenumbers $k^{\ast}_{\Theta}$ and are replotted in panels (b) and (c). The spectra are averaged over all snapshots. The dashed lines for $Pr=0.021$ and 70 (blue and cyan) collapse.[]{data-label="fig6"}](Figure6){width="45.00000%"} Figure \[fig6\] underlines this correlation by means of the power spectra of the temperature in the midplane, the vertical temperature derivative at the plates and the divergence of the skin friction field. We have applied again the sliding time average over $\tau$. All three spectra are found to peak at the same scale (except $Pr\ge 7$ where the scales however are still comparable). Our result is thus robust with respect to $Pr$ and underlines that the same dynamical processes are at work for all Prandtl numbers. As seen in Fig. \[fig6\], the characteristic scale of the skin friction divergence is expected to decrease when the Prandtl number gets smaller. It is documented in refs. [@Schumacher2016; @Scheel2016] that the Reynolds number increases significantly when $Pr$ decreases at constant $Ra$ thus indicating a much more vigorous fluid turbulence, both in the bulk and in the boundary layers (see also Supplementary Material). Thus the spatially extended advection patches of the horizontal velocity field, as visible in the magnification for the case for $Pr=7$ in Fig. \[fig5\] (g, i), will not persist for low-Prandtl-number convection. Discussion {#discussion .unnumbered} ========== Our main motivation was to study the large-scale patterns in turbulent convection which are termed turbulent superstructures. We then analysed the characteristic length and time scales associated with these turbulent superstructures as a function of Rayleigh and Prandtl numbers and found a separation between large-scale, slowly evolving structures and small-scale, rapidly turning vortices and filaments. The system that we have chosen is the simplest setting for a turbulent flow that is initiated by temperature differences, a Rayleigh-Bénard convection flow between uniformly heated and cooled plates. This flow has already been studied intensively with respect to pattern formation in the weakly nonlinear regime above the onset of convection at $Ra=Ra_c$, as documented in the cited reviews [@Busse1978; @Cross1993; @Bodenschatz2000]. Our study shows that patterns of rolls and cells continue to exist into the fully turbulent and time-dependent flow regime once the small-scale fluctuations of the temperature and velocity fields are removed. Prandtl numbers that vary here over more than four orders of magnitude change the character of convective turbulence drastically from a highly inertia-dominated Kolmogorov-type turbulence at the lowest $Pr$ to a fine-structured convection at the highest $Pr$. This results in a strong dependence of the characteristic spatial and temporal separation scales that are necessary to describe the gradual large-scale evolution of the flow at hand. These spatial separation scales are found to continuously increase up to $Pr\lesssim 10$ and to decay for $Pr\gtrsim 10$ for the parameter values that we were able to cover here which is in agreement with [@Hartlep2003]. A saturation of the characteristic scale might occur for the opposite limit, $Pr\to 0$. Our data indicate such a behavior which is supported by previous studies at zero-Prandtl convection by Thual [@Thual1992]. There they found only small differences between $Pr=0.025$ and the singular limit $Pr=0$. However these former studies have been conducted in much smaller boxes at significantly smaller spectral resolutions. A further interesting observation that was made in the present study is the connection between the mean scales of the turbulent superstructure patterns analysed in the midplane and those of the near-wall flows. Our analysis suggests that the characteristic scales of large-scale superstructures are correlated with the thermal plume ridges in the boundary layers. We showed for all $Pr$ that the maximum wavenumber of the temperature spectrum in the midplane $k^{\ast}_{\Theta}$ nearly perfectly coincides with the wavenumber at which the power spectrum of the divergence of the skin friction field peaks. The latter wavenumber characterizes the mean distance of impact (div ${\bm s} >0$) and ejection (div ${\bm s} <0$) regions at the walls. It is thus the characteristic variation scale of the horizontal velocity field that advects the hot (cold) fluid together at the bottom (top) boundary to form prominent thermal plume ridges. The interplay between the thermal and viscous boundary layers of different thicknesses could thus be responsible for the variation of the characteristic superstructure scale with growing $Pr$. The viscous boundary layer becomes ever thicker as $Pr$ increases and velocity fluctuations decrease thus generating more coherent advection patterns. Competing boundary layers that control transport and structure formation in convection flows have been discussed in other settings, for example in ref. [@King2009] for rapidly rotating convection. The characteristic superstructure scales which we have detected in the present work suggest a scale separation for convective turbulence. There is the fast convective motion below the characteristic width of individual circulation rolls or cells on times smaller than several tens of free-fall times. Then after the removal of the small-scale turbulence, the large-scale patterns of rolls are revealed and these fill the whole layer and vary slowly on time scales larger than a few hundreds of free-fall times. The latter dynamic processes can be of interest for a global effective description of mesoscale convection phenomena in atmospheric turbulence [@Randall2003] or of pattern formation in a scale range between solar granulation and supergranulation [@Rincon2017]. In contrast to rapidly rotating convection flows or magnetoconvection in the presence of strong external magnetic fields, the present RBC flow permits a mathematically rigorous asymptotic expansion that generates simplified equations for the dynamics of these patterns (see e.g. [@Stellmach2014]). The unresolved dynamics at the fine and fast scales below $\lambda_{\Theta,U}$ and $\tau$ will be modeled empirically. This is being further investigated and will be reported elsewhere. Methods {#methods .unnumbered} ======= [**Boussinesq equations and numerical method.**]{} We solve the coupled three-dimensional equations of motion for velocity field $u_i$ and temperature field $T$ in the Boussinesq approximation of thermal convection: $$\begin{aligned} \label{ceq} \frac{\partial u_i}{\partial x_i}&=0\,,\\ \label{nseq} \frac{\partial u_i}{\partial t}+u_j \frac{\partial u_i}{\partial x_j} &=-\frac{\partial p}{\partial x_i}+\sqrt{\frac{Pr}{Ra}} \frac{\partial^2 u_i}{\partial x_j^2}+ T \delta_{i3}\,,\\ \frac{\partial T}{\partial t}+u_j \frac{\partial T}{\partial x_j} &=\frac{1}{\sqrt{Ra Pr}} \frac{\partial^2 T}{\partial x_j^2}\,, \label{pseq}\end{aligned}$$ with Rayleigh number $Ra=g\alpha\Delta T H^3/(\nu\kappa)$ and Prandtl number $Pr=\nu/\kappa$. The equations are made dimensionless by cell height $H$, free-fall velocity $U_f=\sqrt{g \alpha \Delta T H}$ and the imposed temperature difference $\Delta T$ between bottom and top plates. The aspect ratio $\Gamma=L/H=25$ with the cell length $L$. The variable $g$ stands for the acceleration due to gravity, $\alpha$ is the thermal expansion coefficient, $\nu$ is the kinematic viscosity, and $\kappa$ is the thermal diffusivity. No-slip boundary conditions for the fluid are applied at all walls. The sidewalls are thermally insulated and the top and bottom plates are held at constant dimensionless temperatures $T=0$ and 1, respectively. The equations are numerically solved by the Nek5000 spectral element method package [@nek5000]. We have two series of direct numerical simulations: six runs at $Pr=0.005$, 0.0021, 0.7, 7, 35, and 70 for $Ra=10^5$ and six runs at $Pr=0.7$ for $Ra=2.3\times 10^3, 5\times 10^3, 10^4, 10^5, 10^6$ and $10^7$. Details on the size of the simulations as well as some characteristic parameters of the simulations can be found in a comprehensive table in the Supplementary Material. Acknowledgements. {#acknowledgements. .unnumbered} ================= AP and JDS acknowledge support by the Deutsche Forschungsgemeinschaft within the Priority Programme Turbulent Superstructures under Grant No. SPP 1881. We acknowledge supercomputing time at the Blue Gene/Q JUQUEEN at the Jülich Supercomputing Centre by large-scale project HIL12 of the John von Neumann Institute for Computing and at the SuperMUC Cluster at the Leibniz Supercomputing Centre Garching by large-scale project pr62se. Author contributions {#author-contributions .unnumbered} ==================== All three authors made significant contributions to this work. All authors designed the numerical experiments and analysed the data. AP and JS ran the production simulations at the supercomputing sites in Garching and Jülich. All authors discussed the results and wrote the paper together. [1]{} Kadanoff, L. P., Turbulent heat flow: Structures and scaling, [*Phys. Today*]{} [**54**]{}(8), 34–39 (2001). 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--- abstract: 'We comment on zero- and low-temperature structural phase transitions, expecting that these comments might be relevant not only for this structural case. We first consider a textbook model whose classical version is the only model for which the Landau theory of phase transitions and the concept of “soft mode” introduced by Ginzburg are exact. Within this model, we reveal the effects of quantum fluctuations and thermal ones at low temperatures. To do so, the knowledge of the dynamics of the model is needed. However, as already was emphasized by Ginzburg [*et al.*]{} in eighties, a realistic theory for such a dynamics at high temperatures is lacking, what also seems to be the case in the low temperature regime. Consequently, some theoretical conclusions turn out to be dependent on the assumptions on this dynamics. We illustrate this point with the low-temperature phase diagram, and discuss some unexpected shortcomings of the continuous medium approaches.' author: - 'A. Cano' - 'A. P. Levanyuk' title: 'On low-temperature structural phase transitions' --- Introduction ============ Zero- and low-temperature ($T$) phase transitions are nowadays a subject of great interest (see, e.g., Refs. [@Sondhi97; @Kvyatkovskii01; @Vojta03; @Belitz05; @Sachdev00] for recent reviews). The special case of structural phase transitions deserves, in our opinion, a special attention. First, it is very convenient when introducing the topic of low-$T$ phase transitions although, to the best of our knowledge, this pedagogical facet of the structural case has not been developed in the literature. One of the purposes of the present paper is just to develop this facet. Second, the discussion of structural transitions allows to reveal some unsolved problems which might have a fairly broad interest. It is worth mentioning that our study will be restricted to the region of small fluctuations (not very close to the phase-transition point). This region normally is not the region of main interest in the aforementioned papers, but the main specific features of the phase transition anomalies are clearly seen already in this region, not to mention that for interpretation of the experimental data this region is quite often the most relevant one. A considerable part of the theory of low-$T$ structural phase transitions is very simple. Its formulation uses elementary formulas of quantum and statistical mechanics, and its development involves a fairly simple mathematics. Nevertheless, this elementary theory suffices to discuss some points of general interest such as the validity of the Landau theory, the soft-mode concept, the role of quantum fluctuations in defining the phase-transition point, the specific features of the low-$T$ phase diagram, etc. This constitutes the first part of the paper where, because of pedagogical considerations, we use a very simple model. Nevertheless, even within this elementary treatment, there arise some questions as well as not completely justified assumptions which will be discussed in the second part of the paper. These questions and assumptions refer to the character of the dynamics of the order parameter near the zero- and low-$T$ phase transitions. This character has not been successfully explained for high-$T$ phase transitions: the origin of the so-called central peak in the soft mode spectrum is understood only qualitatively [@Ginzburg80]. For zero- and low-$T$ structural transitions this question has not been studied at all, although the dynamics of the order parameter is much more important here. Indeed, according to the classical statistical mechanics the static properties of the system do not depend on its dynamics. This is because (gaussian) integration over momenta simply gives a factor in the corresponding partition function. But the situation is different when quantum effects play a role. In this case, the partition function does not factorize because momenta and coordinates, now operators, do not commute with each other [@note1]. Therefore, a lack of exact knowledge of the dynamics impedes obtaining definite results for, e.g., such a “static” property as dependence of the phase transition temperature on a control parameter (e.g., strain or pressure) in the low-$T$ region. Given this situation, we will discuss several possibilities without proposing a finite conclusion about which of them corresponds to the reality. For this discussion we need no model at all, and the system is considered in this second part as a continuous medium. The single-ion model ==================== The so-called single-ion model (see, e.g., Ref. [@Strukov_Levanyuk]) is very convenient when illustrating a zero-$T$ structural phase transition. Within this model one assumes, first of all, that the crystal is composed by two types of atoms, say $A$ and $B$. Our aim is to describe “active” $A$-atoms in the simplest way, so we further assume that i) the sublattice of $B$-atoms can only be deformed homogeneously and ii) the interaction between $A$ atoms is a nearest-neighbor interaction mediated by springs. Additionally, there is an interaction between $A$ and $B$ atoms which is responsible for the relative position of the corresponding sublattices. Restricting ourselves to the orthorhombic case, let us choose the unit cell with $B$-atoms placed at the apices of the corresponding cell (see Fig. \[fig:1\]). Thus, the potential acting on $A$-atoms due to the $B$ ones has to be symmetric with respect to the center of this cell. This is so if this potential has i) a minimum in the center of the unit cell or ii) two symmetric out-of-center minima. In the following we shall assume that i) is the case when the crystal is strongly compressed and then, along the $z$-axis, it turns into case ii) with diminishing the compression (see Fig. \[fig:1\]). This makes possible a change in the mean position of $A$-atoms, i.e. a phase transition, in a fairly simple way. The potential energy of the system then can be written as $$\begin{aligned} U= U_0 +\sum_{\bm R}\left({a\over 2}u_{\bm R}^2 + {b\over 4}u_{\bm R}^4\right) +\sideset{}{'}\sum_{\bm R,\bm R'}{c\over 2} (u_{\bm R}-u_{\bm R'})^2, \label{potential}\end{aligned}$$ where $u_{\bm R}$ represents the displacement of the $A$-atom along the $z$-axis in the $\bm R$th unit cell. The first sum in this expression represents the effective potential acting on $A$-atoms due to $B$ ones. Let us characterize the compression of the system by the magnitude $w= (V_0 - V)/V_0$, where $V $ is the volume of the system and $V_0$ is this volume at zero pressure for the (nonequilibrium) configuration in which all $A$-atoms are maintained in the center of the corresponding unit cells (i.e., $u_{\bm R}=0$). Thus, by taking $a=\alpha(w-w_0)$, with $\alpha>0$, and $b$ as a positive constant; $w_0$ gives the strain at which the form of this potential change from one-well to two-well. \[The usually small difference between $V$ and $V_0$ ($|w|,|w_0|\ll 1$) turns out to be relevant for the change in the sign of the coefficient $a$ only, so we shall not distinguish between $V$ and $V_0$ anywhere but here.\] The second sum in Eq. is the interaction potential between $A$-atoms, where $c$ is the stiffness coefficient of springs linking pairs of $A$-atoms (see Fig. \[fig:1\]) and summation is carried out over nearest-neighbors only. Static properties: A classical zero-$T$ transition -------------------------------------------------- Let us suppose at this point that the mass of $A$-atoms is infinite, so they can be treated as classical particles. Consequently, the configuration of the system will be the one which simply minimizes the potential energy. The static properties of the system will be in accordance with this configuration, so let us proceed to determine it. ![The model: unit cell, effective potential acting on $A$-atoms due to $B$ ones, and an illustration of the interaction between $A$-atoms.[]{data-label="fig:1"}](SingleIonModel_a){width=".1\textwidth"} ![The model: unit cell, effective potential acting on $A$-atoms due to $B$ ones, and an illustration of the interaction between $A$-atoms.[]{data-label="fig:1"}](SingleIonModel_b "fig:"){width=".175\textwidth"} ![The model: unit cell, effective potential acting on $A$-atoms due to $B$ ones, and an illustration of the interaction between $A$-atoms.[]{data-label="fig:1"}](SingleIonModel_c "fig:"){width=".275\textwidth"} It is clear that the minimum of the potential energy corresponds to the configuration in which the springs linking $A$-atoms do not experience any deformation. So all the atoms will be located in the same minimum of the effective potential created by $B$-atoms: $u_{\bm R}=u_0$. Eq. then reduces to $$\begin{aligned} U= U_0 + N \Big( {a\over 2}u_{0}^2 + {b\over 4}u_{0}^4 \Big), \label{}\end{aligned}$$ where $N$ is the number unit cells. Minimizing this potential we find the equilibrium value of $u_0$: $$\begin{aligned} u_{0,\text{eq}}^2=\begin{cases} 0& (w>w_0),\\ -{\displaystyle a\over \displaystyle b}&(w<w_0), \end{cases} \label{eq_u}\end{aligned}$$ and the corresponding value of the potential energy: $$\begin{aligned} U_{\text{eq}}= \begin{cases} U_0&(w>w_0),\\ U_0 -N{\displaystyle a^2\over \displaystyle 4b}&(w<w_0). \end{cases}\label{}\end{aligned}$$ In accordance with these formulas, the change in the form of the effective potential acting on $A$-atoms and the phase transition take place simultaneously at $w_0$. ### Phase transition anomalies As we have mentioned, the sublattice of $B$-atoms, and therefore the system as a whole, can be compressed homogeneously by applying pressure. Let us see how the corresponding stiffness of the system changes as a result of the phase transition. To this end, we present the potential energy of the system with $A$-atoms at the center of the corresponding unit cell as $$\begin{aligned} U_0\simeq U_{0}^\circ + {U_0 ' \over 2}w^2 + \dots \end{aligned}$$ (the term linear in $w$ is absent by virtue of the definition of $w$), and further take into account the energy of the mechanism applying the pressure: $-P (V_0 - V)=-PV_0w$. Minimizing the total energy with respect to $w$ we find this pressure: $$\begin{aligned} P= \begin{cases} \zeta_0 w&(w>w_0),\\ \zeta_0 w - n{\alpha^2 \over 2b} (w - w_0) &(w<w_0), \end{cases}\end{aligned}$$ where $\zeta_0 = U_0' /V $ and $n = N /V$. As a result, the stiffness is $$\begin{aligned} \zeta = {dP\over dw }= \begin{cases} \zeta _0&(w>w_0),\\ \zeta_0 - n{\alpha ^2 \over 2b}&(w<w_0). \label{compressibility}\end{cases}\end{aligned}$$ Its behavior is illustrated in Fig. \[fig:AnomaliasLandau\], which is just the standard behavior for an anomaly described within the Landau theory of second-order phase transitions. ### Exactness of the Landau theory for the model It is worth noticing that we have no fluctuations within this model, neither thermal (as long as $T=0$), nor quantum (as long as the masses are infinite). This makes Landau theory [@Landau37; @Landau_SP; @Strukov_Levanyuk] to be an exact theory here. It is frequently said that the Landau theory is a mean-field approximation, but this example shows that this statement is highly inappropriate. The possibility of having a phase transition with no fluctuations questions another frequent saying: “zero-$T$ phase transitions are due to quantum fluctuations.” Why? We had no fluctuations but we had a transition. Equally unjustified is to say that high-$T$ phase transitions are due to thermal fluctuations. Thermal fluctuations give rise to some contribution, of course, but as a rule they are not the unique reason of the transition. The origin these sayings may be in the fact that phase transitions are frequently illustrated using the Ising model. For this model, it is true that the transition is due to the thermal fluctuations only. But while the Ising model played an important role in the theory of phase transitions, it is very specific, reflecting (as any model) no more than an aspect of much a more many-faceted reality. Dynamics -------- Let us now consider that the masses of $A$-atoms are finite, although very large; so that the motion of these atoms is possible, but the corresponding dynamics can still be considered as classical. Let us study some specific features of such a dynamics. To do so, it is convenient to introduce the new variables $$\begin{aligned} u_{\bm k} = {1\over N}\sum_{\bm R}u_{\bm R}e^{ - i \bm k \cdot \bm R}\end{aligned}$$ having the meaning of the Fourier components of the displacement field: $$\begin{aligned} u_{\bm R} = \sum_{\bm k}u_{\bm k}e^{i \bm k \cdot \bm R}.\end{aligned}$$ ![\[fig:AnomaliasLandau\]](AnomaliasLandau_z){width=".3\textwidth"} In terms of these variables the potential energy Eq. has the form $$\begin{aligned} U &= U_0 + N\sum_{\bm k} {a(\bm k)\over 2}|u_{\bm k}|^2 \nonumber \\ &\quad +N{b\over 4}\sum_{\bm k, \bm k ' , \bm k '' } u_{\bm k} u_{\bm k'}u_{\bm k''}u_{- \bm k - \bm k' - \bm k''},\end{aligned}$$ where $a(\bm k)=a + 4c (\sin ^2 {k_x l \over 2} +\sin ^2 {k_y l \over 2} + \sin ^2 {k_z l \over 2})$, with $l$ being the cell parameter (here it has been taken into account that $u_{-\bm k}= u_{\bm k}^*$). By separating the zero Fourier component of the displacement field: $u_0=N^{-1}\sum _{\bm R} u_{\bm R}$, which evidently has the meaning of the mean value of the displacement field, the above expression for the potential energy can be rewritten as $$\begin{aligned} U &= U_0 + N \Big( {a\over 2}u_0^2 + {b\over 4}u_0^4 \Big) + N \sum_{\bm k \not =0} {a(\bm k, u_0)\over 2}|u_{\bm k}|^2 \nonumber \\&\quad + N{b}u_0 \sum_{\bm k, \bm k ' \not =0} u_{\bm k} u_{\bm k'}u_{- \bm k - \bm k' } \nonumber \\&\quad +N{ b\over 4}\sum_{\bm k, \bm k ' , \bm k '' \not =0} u_{\bm k} u_{\bm k'}u_{\bm k''}u_{- \bm k - \bm k' - \bm k''}. \label{Full_U}\end{aligned}$$ where $a(\bm k,u_0) = a(\bm k) + 3bu_0^2$. Below we shall consider the case of small-amplitude oscillations such that the contribution of the last two sums in this expression can be neglected. It is worth noticing that this does not mean to neglect all the anharmonicity of the system as long as the coefficient $b$ is still present in the remaining terms: $$\begin{aligned} U &\simeq U_0 + N \Big({a\over 2}u_0^2 + {b\over 4}u_0^4 \Big) +N\sum_{\bm k \not =0} {a(\bm k, u_0)\over 2}|u_{\bm k}|^2 . \label{U_WeakAnharmonicity}\end{aligned}$$ In the simplest case, the equation of motion for the Fourier components of the displacement field can be written as $$\begin{aligned} m \ddot u_{\bm k} + a(\bm k,u_0) u_{\bm k} =0, \label{motion_phonon}\end{aligned}$$ where $m$ is the mass of the $A$-atoms. We then have an optical branch with the dispersion law $\omega_c^2(\bm k, u_0) = a(\bm k,u_0)/ m $ (see Fig. \[fig:Dispersion\]). For small wavevectors: $$\begin{aligned} \omega_c^2(\bm k, u_0) = (a + 3b u_0^2 + \widetilde c k ^2)/m, \label{omega_c}\end{aligned}$$ where $\widetilde c = c l^2$. Substituting here the equilibrium value of $u_0$, Eq. , we obtain the normal frequencies of the system as a function of the control parameter $w$. For $\bm k = 0$ that is $$\begin{aligned} \omega_c^2 (0, u_{0,\text{eq}}) = \begin{cases} {\alpha |w - w_0|\over m}& (w>w_0),\\ {2\alpha |w - w_0|\over m}& (w<w_0).\\ \end{cases}\end{aligned}$$ This behavior is illustrated in Fig. \[fig:Dispersion\]. It is worth noticing that, at $w = w_0$, this optic branch has the same dispersion law as the acoustic one: $\omega \propto k$. As a result of this behavior, first noticed by Ginzburg [@Ginzburg49], this $\bm k = 0$ mode is usually termed as the soft mode associated with the transition. Zero-$\bm T$ transition: Quantum effects ---------------------------------------- Let us now consider that the masses of $A$-atoms are “normal” (not specially large). In this case, if we consider $A$-atoms as immobile atoms when studying the thermodynamic properties of the system we are wrong. Even at $T=0$ these atoms are moving exhibiting what is called quantum fluctuations, and we have to take into account this quantum effects. ![\[fig:Dispersion\]](Dispersion){width=".3\textwidth"} In a first step, the system can be considered as a set of harmonic oscillators \[see Eq. \]. For a fixed value of $u_0$, the ground state energy of the system then can be written as $$\begin{aligned} E = U_0 + N \Big({a \over 2}u_0^2 + {b\over 4}u_0^4\Big) +\sum _{\bm k} {\hbar \omega_c(\bm k, u_0)\over 2} \label{FreeEnergy_Phonons}\end{aligned}$$ (recall that $T=0$). As a result of the possible displacements of $A$-atoms along the $z$-axis, here we have the classical (macroscopic) contribution considered in previous section plus a quantum term accounting for the ground-state energy of the corresponding oscillators with normal frequencies $\omega_c(\bm k, u_0)$. In further calculations it is convenient to take the continuous-medium limit of our model (i.e., to replace summation by integration $\sum_{\bm k} \to V \int {d\bm k\over (2 \pi )^3}$): $$\begin{aligned} E = U_0 + N \Big({a \over 2}u_0^2 + {b\over 4}u_0^4\Big) + V\int{d \bm k\over (2\pi)^3}{\hbar \omega_c(\bm k, u_0)\over 2}. \label{FreeEnergy_PhononsContinuous}\end{aligned}$$ ### The phase-transition point Because of the dependence on $u_0$ of the quantum term ($\propto \hbar$) in Eq. , it can be said that there effectively is a quantum renormalization of the function $E(u_0)$. By expanding the last term in Eq. in power series of $u_0$ we can obtain, for example, the quantum correction to the coefficient $a$. The resulting coefficient, $a^*$, is of particular interest to us: when $a^*=0$ the system losses its stability with respect to nonvanishing values of $u_0$, what just defines the phase-transition point. Taking into account that $$\begin{gathered} \left.{\partial \omega_c (\bm k,u_0)\over \partial u_0}\right|_{u_0=0}=0,\\ \left.{\partial^2 \omega_c (\bm k,u_0)\over \partial u_0^2}\right|_{u_0=0}={3b\over m\omega_c(\bm k,0)},\end{gathered}$$ we find that $$\begin{aligned} a^* = a + {3 \hbar b v\over 2 m}\int {d \bm k\over (2\pi)^3}{1\over \omega_c(\bm k,0)}. \label{renormalized_a}\end{aligned}$$ where $v=V/N=l^{3}$ is the volume of the unit cell. When trying to determine the phase-transition point one realizes that this formula is, however, somewhat contradictory. The coefficient $a$ has to be negative in order to further obtain $a^{\ast }=0$. But in this case, in accordance with Eq. , there is a region of wavevectors for which $\omega_{c}^{2}< 0$. This difficulty can be circumvented if this region is much smaller than the whole Brillouin zone. When this possibility takes place the system can be labeled as displacive and, when calculating the integral in Eq. , one can put $a=0$ quite safely. As a result one finds that the phase-transition strain is given by $w_{c}=w_{0}+\delta w$, where $\delta w$ is the contribution due to quantum effects (zero-point quantum fluctuations) which can be estimated as $$\begin{aligned} \delta w \simeq - {3\hbar b v\over (2 \pi)^2 \alpha m^{1/2}}\int_0^{2\pi/l} {k^2 d k\over \sqrt{a + \widetilde c k^2} } \simeq - {3 \hbar b \over 2 \alpha \sqrt{m c}}.\end{aligned}$$ Alternatively, by realizing that the frequency of the soft mode should vanish at the “experimental” value of phase transition strain $w_c$, one could replace $a \to a^{\ast }$ in the integral in Eq. . This is equivalent to take into account, in an effective way, higher order corrections in Eq. (see below). Although the problem seems to be overcame, it is instructive to discuss this “overcoming” in more detail. What we have done is to “correct” the dynamical moduli of the system in a meaningful way. This is quite similar to what is done when calculating, e.g., the phase transition temperature of a classical displacive system: here they are the static moduli what are corrected in a proper way. By realizing that our system is in fact an anharmonic system, as indicated by the last two sums in Eq. , it can be expected that going beyond the approximation of decoupled oscillators such a correction (renormalization) will be obtained. The possibility of using a perturbation theory to account for this anharmonicity is therefore quite attractive, and valuable irrespective of the character, classical or quantum, of the problem in question. However, the quantum case is somewhat more complex than the classic one as long as dynamics plays a key role. One can realize that we did two things in fact: i) assumed that dynamics is still well-described in the same terms (i.e., within a soft-mode scenario) and then ii) corrected the corresponding dynamical moduli. The former has not to be necessarily true in a real case. Indeed, one of the well known shortcomings of the theory of high-$T$ structural phase transitions is that the soft mode behavior has never been observed experimentally to a full extent: the “soft mode”, at best, diminishes its frequency considerably; but this frequency never goes to zero at the phase transition point. At the same time, close to the phase transition point there appears the so-called “central peak” in the order parameter fluctuation spectrum, and the increase of the order parameter fluctuations turns out to be comprised within this peak. Such a peak is a natural feature of the so-called order-disorder systems, for which the order-parameter dynamics is relaxational instead of phonon-like as we have considered so far. Though there is no consistent analytical theory of the central peak, its appearance is not surprising at all [@Ginzburg80]. Any system with a phase transition is inherently anharmonic. So the phonon-like (“soft mode”) picture is not the full picture; there is another part, essentially an anharmonic one, which reveals itself in the central peak. Similar effects are quite possible at low $T$s: the presentation of the motion as a set of normal vibrations, even followed by taking into account the anharmonism within a perturbation theory, may fail to describe accurately the dynamics of the system close to the corresponding phase-transition point. Consequently, one can do nothing but assume Eq. with caution and hope that it is qualitatively correct. The importance of the quantum corrections to phase-transition strain can be estimated as follows. The very possibility of observing the transition implies that $|w_0|$ is much less than the atomic value of this magnitude $|w_\text{at}| \sim 1$. The other constants in the model, however, can have their “atomic” values. Introducing the atomic (binding) energy $\varepsilon_\text{at}\sim \hbar^2/(m_e l^2)$, where $m_e$ is the electron mass and $l\sim 1 \text{\AA}$ the atomic distance, we then have $$\begin{aligned} \alpha \sim {\varepsilon_\text{at}\over l^2 w_\text{at}},\quad b\sim {\varepsilon_\text{at}\over l^4},\quad c\sim {\varepsilon_\text{at}\over l^2}. \label{ForTheEstimates}\end{aligned}$$ Thus we find that $$\begin{aligned} \left|{\delta w \over w_0}\right|\sim \left({m_e\over m}\right)^{1/2} \left|{w_\text{at}\over w_0}\right| .\end{aligned}$$ As long as $m_e \ll m$, but ${|w_\text{at}|\gg |w_0|}$, here we see that the correction to the classical phase-transition strain can be quite important: it may give rise to a magnitude $w_c$ comparable, or even greater, than $w_0$. ### Order-disorder (spin-like) limit We have discussed the displacive limit but what happens if the value of $|a|$ at the phase-transition point is so large that $\omega_{c}^2<0$ in most of the Brillouin zone? In this case, one simply has to realize that the starting point was not chosen appropriately. Looking back we can see that this case corresponds to an effective potential acting on $A$-atoms due to $B$ ones with two profound wells even in the symmetric phase. Under these circumstances, instead of characterizing the motion of $A$-atoms by their displacement from the center of the corresponding unit cell (what $u_{\bm R}$ really means), it makes more sense to assume, first of all, the atoms to be confined inside one of the two wells and then take into account the possibility of delocallization by, e.g., quantum tunneling. This can be done by associating the two initial positions of $A$-atoms with the two possible orientations of (pseudo-)spins $1/2$ and accounting for the presence of a transversal field in order to reproduce the tunneling. Thus our model becomes a spin model exhibiting a zero-$T$ phase transition (Ref. [@Sachdev00; @Vojta03]). The corresponding excitations (spin waves) are similar to the optical vibrations considered above in the sense that there is a “soft spin-wave” whose frequency goes to zero at the phase-transition point. In this sense, when it comes to zero-$T$ transitions, displacive and order-disorder limiting cases are not so different one another. ### Phase-transition anomalies Let us now calculate the quantum contribution to the anomaly in the stiffness of the system within the displacive scenario. The main purpose of such a calculation is to estimate the region in which the Landau theory is applicable, so it suffices to consider the symmetric phase ($u_{0}=0$). From Eqs. and , we find that $$\begin{aligned} \zeta &= {1 \over V}{\partial ^2 F \over \partial w^2 }= \zeta _0 -{\hbar \alpha^2 \over 8 m^2} \int {d{\bm k } \over (2\pi )^3}{1 \over \omega_c^{3/2} (\bm k,0)} \nonumber \\& \simeq \zeta_0 - {\hbar n \alpha^2 \over 32 \pi^2 m^{1/2} c^{3/2}} \ln \left( { 4 \pi ^2 c \over \alpha (w-w_0)}\right), \label{compressibility_quantum_corrections}\end{aligned}$$ where $\zeta_0$ is in accordance with Eq. . Let us analyze this result in some detail. At first sight, it seems that the stiffness of the system changes its sign and diverges at $w=w_{0}$. However, as long as we are considering the lowest order correction only, we simply can say that it diminishes close to $w_{0}$. In any case, why close to $w_0?$ An anomaly at the phase-transition point would be not surprising but, in accordance with Eq. , the phase-transition point is not $w_{0}$. What happens? The answer is that here have the same kind of inconsistency that we have found before \[compare the integral in Eqs. and \]. So the replacement of $w_0$ by the actual value associated with the phase-transition strain, which can be extracted from the experiments, solves it. An estimate of the region of applicability of the Landau theory in our case can be made by demanding that the (quantum) contribution to the stiffness of the system is much smaller than the discontinuity obtained within the Landau theory itself. That is $$\begin{aligned} {\hbar n \alpha^2 \over 32 \pi^2 m^{1/2}c^{3/2}} \left| \ln \left( { 4 \pi ^2 c \over \alpha (w-w_c)}\right) \right| \ll n{\alpha ^2 \over 2 b} \label{}\end{aligned}$$ \[see Eq. \]. In accordance with previous estimates of the coefficients appearing in this expression \[see Eq. \], this gives $$\begin{aligned} \left|\ln \left( { w - w_c\over w_\text{at}} \right)\right| \ll 10^2 \left( {m \over m_e} \right)^{1/2}\sim 10^4.\end{aligned}$$ As wee see, it is hardly possible to abandon this region in real experiments. It is worth mentioning that the behavior of the stiffness of our model in the scaling region has been known since long ago [@Rechester71]. Roughly speaking, it is such that one has to replace $\ln $ by $\ln ^{1/3}$ in Eq. . As we see, both the first-order approximation and the theory for the scaling region provide qualitatively the same results (see Fig. \[AnomaliasPrimerOrden\]). However, as a rule, the first-order approximation gives rise to an overestimate of the relevant magnitudes in the very vicinity of the phase-transition point: in our case, by extrapolating the increase of the stiffness obtained within the first-order approximation to the scaling region one overestimates such an increase. ### Thermal effects Thermal effects can be revealed by following the same considerations as above. At finite temperature, the free energy of the system can be written as $$\begin{aligned} F &= F_0 + N \Big({a \over 2}u_0^2 + {b\over 4}u_0^4\Big) + V\int {d\bm k\over (2\pi)^3}{\hbar \omega_c(\bm k, u_0)\over 2} \nonumber \\&\quad + {T }V\int {d\bm k\over (2\pi)^3}\ln \left\{ 1 - \exp[- \hbar \omega_c(\bm k, u_0)/T] \right\}. \label{FreeEnergy_PhononsT}\end{aligned}$$ (Notice that this expression is nothing but the free energy of a set of harmonic oscillators, see, e.g., Ref. [@Landau_SP].) Therefore, the coefficient at the term quadratic in $u_0$ is $$\begin{aligned} a^* &= a + {3 \hbar b v\over 2 m}\int {d \bm k\over (2\pi)^3}{1\over \omega_c(\bm k,0)} \nonumber \\ &\quad + {3 \hbar b v\over 4 m}\int {d \bm k\over (2\pi)^3}{n [\omega_c(\bm k,0)] \over \omega_c(\bm k,0)}, \label{renormalized_aT}\end{aligned}$$ where $n(\omega) = [\exp(\hbar \omega /T) - 1]^{-1}$ is the Bose-Einstein distribution function. Here we have the quantum contribution computed in previous section \[see Eq. \] plus a thermal one (the latter integral). Similar to what we have done when discussing the quantum contribution to $w_{c}$, it is reasonable to replace $a \to a^{\ast }$ in the expression of $\omega _{c}$ entering in Eq. . Let us discuss the phase diagram. The border between the symmetric and non-symmetric phases in the ($T,w$)-plane is defined by the condition $a^{\ast }=0$. This gives the line $$\begin{aligned} w-w_{c} =-\alpha^{-1} f\left(T_c\right),\end{aligned}$$ in the ($T,w$)-plane, where $f\left( T_c\right) $ is the last integral in Eq. evaluated at $a^*=0$. At low temperatures \[$\hbar \omega(0,0)\ll T \ll \hbar \omega(\bm k_\text{max},0)$\], this integral can be estimated as $$\begin{aligned} &\int {d \bm k\over (2\pi)^3}{n [\omega_c(\bm k,0)] \over \omega_c(\bm k,0)} ={1\over 2\pi^2}\int {n[\omega_c(k,0)]\over \omega_c(k,0)}k^2 d k \nonumber \\&\quad\simeq {1\over 2\pi ^2}\int_0^{k_\text{max}(T)} {T\over \hbar \omega_c^2(k,0)}k^2dk ={m^{3/2}\over 2\pi^2 \hbar ^2 c^{3/2}v}T^2. %\nonumber \end{aligned}$$ taking into account that the Bose-Einstein distribution function decreases very strongly within the integration interval \[it vanishes for $k \gtrsim k_\text{max}(T)= {m^{1/2} \over \hbar \widetilde c^{1/2}}T$\]. This gives $$\begin{aligned} w-w_c \propto T^2_c. \label{}\end{aligned}$$ ![\[AnomaliasPrimerOrden\]](AnomaliasPrimerOrden){width=".3\textwidth"} At high temperatures \[$ \hbar \omega_c(\bm k,0)\ll T $\], the Bose-Einstein distribution can safely be replaced by its classical limit: $$\begin{aligned} \int {d \bm k\over (2\pi)^3}{n [\omega_c(\bm k,0)] \over \omega_c(\bm k,0)}\simeq \int {k^2 d k\over 2\pi^2}{T \over \hbar \omega_c^2(\bm k,0)}\simeq {m \over \pi \hbar c v}T. \label{}\end{aligned}$$ We then have $$\begin{aligned} w-w_c\propto T_c. \label{}\end{aligned}$$ Continuous medium approach ========================== At this point, it is worth making a comparison between origin of the quantum and thermal contributions in $a^*$ obtained previously \[see Eqs. and \]. While the quantum contribution comes from the ground-state energy of all the optical phonons, the thermal contribution is obtained from the optical phonons with small wavevectors only. In consequence, the actual value of phase-transition strain at zero temperature ($w_{c}$) is sensitive to the microscopic details of the system, whereas the function $T_{c}(w)$ defining the phase transition temperature as a function of strain is not. This can be taken as a justification to use a continuous medium approach when studying thermal dependencies. Just in this sense one speaks about an universal low-$T$ behavior of the systems, exemplified by the temperature dependence of specific heat of solids (the Debye law): it comes from the contribution of small-wavevector acoustic phonons and, therefore, is characterized by macroscopic quantities such as the velocity of sound. However, in our case, this universality is not so evident. It can be expected when the soft-mode scenario takes place, as we were able to argue for a classical zero-$T$ transition. But as soon as we make the model more realistic, taking into account real values of the ion masses and, therefore, the quantum effects, as well as considering other degrees of freedom, in particular, acoustic modes, the dynamics of the system becomes far more complicated. This has not been studied to a full extent, for which there is a reasonable explanation: difficulties similar to those found for the high-$T$ order-parameter dynamics are indeed expected. As we have mentioned, the soft-mode scenario is not completely applicable to high-$T$ phase transitions: in both real and numerical experiments the central-peak phenomenon appears time and again, and this lacks a theoretical explanation beyond “hand-waving” arguments [@Ginzburg80]. The situation for zero- and low-$T$ phase transitions is unknown, but one may suspect that it is similar. In consequence it is quite reasonable to consider, within the continuous medium approach, both hypothetical cases: that of the soft-mode (phonon-like dynamics) scenario and the central-peak-like (relaxational dynamics) one. As we shall see, the aforementioned universality is lost in the second case for which the continuous medium approach becomes no more than a model. Phonon-like dynamics\[S:Phonon-like dynamics\] ---------------------------------------------- Our arguments about the universality were referred just for the phonon-like case. In this case, the theory can be made somewhat more realistic, even model-independent, than the one presented above where only one (“active”) optical branch is taken into account. It is straightforward, in particular, to take into account the acoustic branches (e.g., to account for the motion of $B$-atoms as well). Indeed one may expect that these acoustic phonons give rise to a significant contribution to $a^{\ast }$: they are low-frequency excitations of the system. Consider the free energy $$\begin{aligned} F &= \widetilde F_0 + N \Big({a^*(T)\over 2}u_0^2 + {b\over 4}u_0^4\Big) + 3V\int {d\bm k\over (2\pi)^3}{\hbar \omega_\text{ac}(k, u_0)\over 2} \nonumber \\&\quad + 3{T }V\int {d\bm k\over (2\pi)^3}\ln \left\{ 1 - \exp[- \hbar \omega_\text{ac}(k, u_0)/T] \right\}.\label{}\end{aligned}$$ where $a^*(T)$ is given by Eq. . This is nothing but Eq. where the contribution of the acoustic phonons has been added explicitly. This contribution depends on $u_0$ because of the corresponding dependence of the velocity of sound: $$\begin{aligned} \omega_\text{ac}(k, u_0)= c(u_0)k,\end{aligned}$$ which can be taken as $c(u_0)= c_0(1 + {\alpha \over 2}u_0 ^2)$. (This velocity has to be understood as an averaged velocity of sound in the same sense as in Ref. [@Landau_SP].) Thus the new coefficient of the free energy at the quadratic term in $u_0$ can be written as $$\begin{aligned} a^{**}(T)&= a^*(T) \nonumber \\ &\quad + {3\hbar \alpha v \over 2 \pi^2} \int \left\{ {1\over 2} + n [\omega_\text{ac}(k,0)] \right\} \omega_\text{ac}(k,0){ k^2 d k}. \label{renormalized_aT_acoustic}\end{aligned}$$ As we see in this expression, we have a new contribution to the phase-transition strain $w_c$ which is not surprising (the part of the integral which does not vanish at $T=0$). But we also have a thermal one which can be estimated as $$\begin{aligned} {3\hbar \alpha v\over 2\pi^2 } \int {n[\omega_\text{ac}(k,0)]\omega_\text{ac}(k,0)}k^2 d k \simeq {\hbar c_0\alpha v\over 2\pi^2 } \left( {T\over \hbar c_0}\right)^4. %\nonumber \end{aligned}$$ Having in mind that the thermal activation of the optical phonons obtained above is exponentially suppressed with moving away the phase-transition point \[see Eq. \], one realizes that this thermal contribution due to acoustic phonons may be the most important one in a significant region of the ($T,w$)-plane. However, this is not the case close enough to the phase transition point where $a^*(T)- a^*(0) \propto T^2$. Consequently, the form of the phase diagram does not change from the one expounded above. These changes take place if, for instance, there exist long-range interactions in the system. In this case, complicated dispersion relations may arise such that the dimensionality of the integrals in Eq. effectively increase. In the case of uniaxial ferroelectrics, for instance, dipolar interactions lead to $\omega_c^2(\bm k, 0)= (a + \widetilde c k^2 + 4\pi \cos^2 \theta)/m$ for small wavevectors. This further gives $a^*(T)- a^*(0) \propto T^3$ (see, e.g., Refs. [@Khmelnitskii71; @Kvyatkovskii01]). The linear coupling between the order parameter and one of the component of the strain tensor, i.e., the piezoelectric effect in the case of ferroelectrics, gives rise a peculiar dependence $a^*(T)- a^*(0) \propto T^{5/2}$ [@Cano04b]. These long-range interactions, however, do not modify the high-$T$ behavior. Dissipative dynamics -------------------- Let us now consider that there is a central peak in the order-parameter fluctuation spectrum, at least close to the phase-transition point. Such an assumption is far from being trivial: it implies that there is an essential frequency dispersion in the order-parameter response function and, therefore, a dissipative dynamics of the order parameter even at $T=0$. In a number of the systems, e.g., for perfect crystals without phase transitions, this is certainly not the case. But the situation with such an inherently anharmonic system as the one we are considering here (recall that $a<0$ in our case) is unclear. What seems to be clear, however, is that for crystals with some defects the dissipative dynamics of the type we are supposing here takes place even at $T=0$ [@Cano04a]. Therefore, even if our consideration would prove to be irrelevant for pure systems, it might still be relevant for systems with some defects. A dissipative dynamics for a set of variables means that these variables are not coordinates of a closed system but there is a “reservoir” where the system in question transfers its energy to. In our case, the long-wave acoustic phonons could play the role of such a reservoir. When calculating the contribution of the set of dissipative variables one should, of course, not forget the contribution of the reservoir. Only in the case when the latter contribution is irrelevant it makes sense to consider a system of dissipative variables as the main contributor (see, e.g., Ref. [@Weiss]). Let us mention that even apart from possibility of the central peak, an account of a phonon damping at $T=0$ due to e.g. defects is of some interest for zero- and low-$T$ transitions. We shall begin with this case by postulating the equation of motion $$\begin{aligned} m \ddot u_{\bm k} + \widetilde \gamma \dot u_{\bm k} + a(\bm k,u_0) u_{\bm k} =0, \label{dissipative_motion}\end{aligned}$$ where the viscosity coefficient $\widetilde{\gamma}$ accounts for the dissipation. ### Thermal effects Making use of some results obtained by previous authors (see, e.g., Ref. [@Weiss]), we can study the case in which the motion of the “active”-atoms is governed by Eq. . In this case, the free energy can be written as [@Weiss] $$\begin{aligned} F &= \widetilde F_0 + N \Big({a \over 2}u_0^2 + {b\over 4}u_0^4 \Big) + {T}V\int {d\bm k\over (2\pi)^3}\left[\ln \Big({2 \pi [\lambda_1(\bm k, u_0)\lambda_2(\bm k, u_0)]^{1/2}\over \nu}\Big) -\ln \Gamma\Big(1 + {\lambda_1(\bm k, u_0)\over \nu}\Big)-\ln \Gamma\Big(1 + {\lambda_2(\bm k, u_0)\over \nu}\Big) \right], \label{FreeEnergy_DissipativeT}\end{aligned}$$ where $\nu = 2 \pi T/\hbar$ and $\{\lambda_i(\bm k, u_0)\}$ are the roots of the equation $\lambda ^2(\bm k, u_0) - \gamma \lambda(\bm k, u_0) + \omega_c^2(\bm k, u_0) =0$, satisfying the relations $$\begin{gathered} \lambda_1 (\bm k, u_0)+ \lambda_2 (\bm k, u_0)= \gamma, \\ \lambda_1 (\bm k, u_0)\lambda_2(\bm k, u_0) = \omega_c ^2(\bm k, u_0). \label{}\end{gathered}$$ It is worth mentioning that the first term $\widetilde F_0$ in Eq. now includes the contribution due to the reservoir. In principle, this contribution could exhibit a nontrivial behavior close to the transition point as a result of the coupling between the active atoms (which clearly “feel” the transition) and the degrees of freedom in the reservoir (which also “feel” the transition because of this coupling). Nevertheless, if these degrees of freedom forming the reservoir are low-frequency acoustic phonons, which would be quite natural, it can be shown that their contribution i) is not substantially modified because of damping [@Cano04a] and ii) is not the most important one close to the (low-$T$) phase-transition point as we have seen in Sec. \[S:Phonon-like dynamics\] [@Cano04b]. The above formulas allow us to compute the relevant magnitudes close to the phase transition by following the same procedure as before (see Ref. [@Cano04b]). The coefficient at the quadratic term in $u_0$ in the free energy, for instance, is found to be $$\begin{aligned} a^*(T) &= a + {3b v\over m k_BT} \int {d{\bm k} \over (2 \pi)^3}\bigg({1\over 2\omega_c^2(\bm k,0)} \nonumber \\ &\quad + {\psi[1+\lambda_1(\bm k,0)/\nu]-\psi[1+\lambda_2(\bm k,0)/\nu]\over \nu [\lambda_1(\bm k,0) - \lambda_2(\bm k,0)]} \bigg). \label{chi_simple}\end{aligned}$$ Close to the phase-transition point, this further gives $$\begin{aligned} a^*(T) - a^*(0)\propto T^2, \label{}\end{aligned}$$ in the relaxation limit ($m\to 0$). It is worth mentioning that this behavior is found irrespective of long-range forces [@Cano04b], which is related to the fact that the thermal activation of the elementary excitations of the system is substantially modified as a result of the damping \[notice that the Bose-Einstein distribution function in Eq. has been replaced by the psi functions in Eq. \]. In fact, the most important conclusion here is that, as a result of damping, the paradigm: low-$T$ behaviors $=$ small wavevectors, is not longer valid. This is because all degrees of freedom may give a contribution to the corresponding thermal behavior as a result of the damping \[if $m\to 0$, in Eq. there is no temperature-dependent cutoff analogous to that in Eq. \]. In some sense, this is similar to what happens in the case of the phase transition strain: one has to go beyond the continuous media approach to calculate this strain. The situation is new, however, in the sense that this happens even for the thermal behavior. Conclusions =========== The subject of low-temperature phase transitions has been examined, dealing with the structural case and focusing on the region of small fluctuations. Within an oversimplified but illustrative model we have discussed some points of general interest, such as the validity of the Landau theory, the role of quantum fluctuations in defining the phase-transition point and the specific features of the low-temperature phase diagram. We have pointed out also that a profound study of dynamics, both experimental and theoretical, is still needed to get fully justified results about the phase diagrams and anomalies at structural and may be other low-temperature phase transitions. We thank C. Barcel' o and F.F. L' opez-Ruiz for helpful comments on the manuscript. [0]{} S.L. Sondhi [*et al.*]{}, Rev. Mod. Phys. [**69**]{}, 315 (1997). S. Sachdev, Science [**288**]{}, 475 (2000). O.E. Kvyatkovskii, Fiz. Tverd. Tela (Leningrad) [**43**]{} (8), 1401 (2001) \[Sov. Phys.– Solid State [**43**]{} (3), 1362 (2001)\]. M. Vojta, Rep. Prog. Phys. [**66**]{}, 2069 (2003). D. Belitz, T.R. Kirkpatrick and T. Vojta, Rev. Mod. Phys. [**77**]{}, 579 (2005). V.L. Ginzburg, A. Levanyuk and A.A. Sobyanin, Phys. Rep. [**57**]{}, 151 (1980). This difference is even more apparent within the path integral formalism: the domain of integration over the (imaginary) time variable turns out to be infinitesimal in the classical case (high-$T$s), while it is not in the quantum one (low-$T$s). B.A. Strukov and A. P. Levanyuk, [*Ferroelectric Phenomena in Crystals*]{} (Springer-Verlag, Berlin, 1998). L.D. Landau and E.M. Lifshitz, [*Statistical Physics*]{} (Pergamon, Oxford, 1980). L.D. Landau, Zh. Eksp. Teor. Fiz. [**7**]{}, 19 (1937); 627 (1937); translated in [*Collected Papers of L.D. Landau*]{}, ed. D. ter Haar (Gordon and Breach, New York, 1965). V.L. Ginzburg, Zh. Eksp. Teor. Fiz. [**19**]{}, 36 (1949). A.B. Rechester, Zh. Eksp. Teor. Fiz. [**60**]{} (2), 782 (1971) \[Sov. Phys. JETP [**33**]{}, 423 (1971)\]. D.E. Khmel’nitskii and V.L. Shneerson, Fiz. Tverd. Tela (Leningrad) [**13**]{} (3), 832 (1971) \[Sov. 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--- abstract: 'We establish a Galois-theoretic interpretation of cohomology in semi-abelian categories: cohomology with trivial coefficients classifies central extensions, also in arbitrarily high degrees. This allows us to obtain a duality, in a certain sense, between “internal” homology and “external” cohomology in semi-abelian categories. These results depend on a geometric viewpoint of the concept of a higher central extension, as well as the algebraic one in terms of commutators.' address: - 'Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005–139 Faro, Portugal' - 'Centro de Matemática, Universidade de Coimbra, 3001–454 Coimbra, Portugal' - 'Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, chemin du cyclotron 2 bte L7.01.02, 1348 Louvain-la-Neuve, Belgium' author: - Diana Rodelo - Tim Van der Linden title: | Higher central extensions\ and cohomology --- [^1] [^2] Introduction {#introduction .unnumbered} ============ This article exposes a hidden duality between “internal” homology and “external” cohomology for certain group-like structures: we prove that cohomology with trivial coefficients classifies (higher) central extensions. Together with the work in low dimensions and with several closely related results in homology theory, this reveals a deep connection between Galois theory and cohomology, and a close link with homology which has been invisible so far. The context in which we work is sufficiently general to cover cohomology of, say, groups, crossed modules, Lie algebras and non-unitary rings, as well as the Yoneda $\operatorname{Ext}$ groups in the abelian case, and many new examples can easily be added to the list. In fact, almost any semi-abelian category would do, as long as it satisfies a certain commutator condition which occurs naturally in this setting—see below. This interpretation of cohomology is part of a bigger programme which intends to understand homological algebra in a non-abelian environment from the viewpoint of (categorical) Galois theory. Related results include, for instance, higher Hopf formulae for homology in semi-abelian categories [@EGVdL], higher-dimensional torsion theories [@Everaert-Gran-TT], a theory of satellites for homology without projectives [@GVdL2], and higher-dimensional commutator theory based on a notion of higher centrality [@EverVdL4; @EverVdLRCT]. Higher centrality {#higher-centrality .unnumbered} ----------------- The key novelty in the present approach to (co)homology of non-abelian algebraic objects is the concept of *higher centrality*. It allows us to express in an abstract but simple way the commutator conditions which we have to deal with. Following the ideas of Janelidze [@Janelidze:Double; @Janelidze:Hopf-talk], the formal theory of (not necessarily central) *higher (cubic) extensions* was first developed in [@EGVdL] in order to provide a general setting for the Brown–Ellis–Hopf formulae [@Brown-Ellis; @Donadze-Inassaridze-Porter]. The notion of *centrality* in the sense of categorical Galois theory [@Borceux-Janelidze; @Janelidze:Pure; @Janelidze-Kelly] depends on a Galois structure, and accordingly, centrality of higher extensions is defined using a tower of Galois structures. Let us make this explicit with a concrete example. Consider the category ${\ensuremath{\mathsf{Gp}}}$ of all groups and its (reflective) subcategory ${\ensuremath{\mathsf{Nil}}}_{2}$ determined by all groups of nilpotency class at most $2$. The induced reflector ${\ensuremath{\mathsf{nil}}}_{2}\colon{{\ensuremath{\mathsf{Gp}}}\to {\ensuremath{\mathsf{Nil}}}_{2}}$, left adjoint to the inclusion functor, takes a group $G$ and sends it to its $2$-nilpotent quotient $G/[[G,G],G]$. This situation—${\ensuremath{\mathsf{Gp}}}$ being a variety of algebras over ${\ensuremath{\mathsf{Set}}}$, and ${\ensuremath{\mathsf{Nil}}}_{2}$ a subvariety of it—admits a canonical homology theory: Barr–Beck comonadic homology [@Barr-Beck] with coefficients in the reflector ${\ensuremath{\mathsf{nil}}}_{2}$. Now for any group $Z$, the induced third homology group ${\mathrm{H}}_{3}(Z,{\ensuremath{\mathsf{nil}}}_{2})$ of $Z$ may be expressed by a Hopf formula, namely the quotient [@EGVdL Theorem 9.3] $$\frac{K_{0}\cap K_{1}\cap [[X,X],X]}{[[K_{0}\cap K_{1},X],X][[K_{0}, K_{1}],X][[K_{0},X],K_{1}][[X,K_{0}],K_{1}][[X,X],K_{0}\cap K_{1}]}.$$ Here the objects $K_{0}={\operatorname{Ker}(c)}$ and $K_{1}={\operatorname{Ker}(d)}$ are the kernels of $c$ and $d$, for any *two-cubic presentation* $$\label{Double-Extension-Intro} \vcenter{\xymatrix{X \ar@{ >>}[r]^-{c} \ar@{ >>}[d]_-{d} & C \ar@{ >>}[d] \\ D \ar@{ >>}[r] & Z}}$$ of $Z$. This means that the objects $C$, $D$ and $X$ are projective (= free) groups, and furthermore this commutative square is a *two-cubic extension* of $Z$: all its arrows, as well as the induced arrow to the pullback ${\lgroup}d,c{\rgroup}\colon{X\to D\times_{Z}C}$, are surjections. The denominator in the formula is a generalised commutator: a two-cubic extension of groups such as  is central (with respect to ${\ensuremath{\mathsf{Nil}}}_{2}$) precisely when this denominator is zero. The concept of *centrality* of two-cubic extensions is given by the Galois structure $\Gamma_{1}$ in the “tower” consisting of $$\Gamma_{0}=({\ensuremath{\mathsf{Gp}}},{\ensuremath{\mathsf{Nil}}}_{2},{\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{F}}},{\ensuremath{\mathsf{nil}}}_{2},\subseteq)$$ and $$\Gamma_{1}=({\ensuremath{\mathsf{Ext}}}({\ensuremath{\mathsf{Gp}}}),{\ensuremath{\mathsf{CExt}}}_{{\ensuremath{\mathsf{Nil}}}_{2}}({\ensuremath{\mathsf{Gp}}}),{\ensuremath{\mathcal{E}}}^{1},{\ensuremath{\mathcal{F}}}^{1},({\ensuremath{\mathsf{nil}}}_{2})_{1},\subseteq),$$ where ${\ensuremath{\mathcal{E}}}$, ${\ensuremath{\mathcal{F}}}$ are the classes of surjections and ${\ensuremath{\mathcal{E}}}^{1}$, ${\ensuremath{\mathcal{F}}}^{1}$ are the classes of two-cubic extensions in ${\ensuremath{\mathsf{Gp}}}$ and in ${\ensuremath{\mathsf{Nil}}}_{2}$, respectively. Here $\Gamma_{1}$ is induced by $\Gamma_{0}$ through its one-cubic central extensions, which are the objects of the full reflective subcategory ${\ensuremath{\mathsf{CExt}}}_{{\ensuremath{\mathsf{Nil}}}_{2}}({\ensuremath{\mathsf{Gp}}})$ with reflector $({\ensuremath{\mathsf{nil}}}_{2})_{1}$ of the category ${\ensuremath{\mathsf{Ext}}}({\ensuremath{\mathsf{Gp}}})$ of one-cubic extensions in ${\ensuremath{\mathsf{Gp}}}$. It is not hard to construct a two-cubic presentation of an object, certainly not in the varietal case, since a truncation of any simplicial projective resolution will do. As is apparent from the formula, the main difficulty in making it explicit lies in characterising the (two-cubic) central extensions corresponding to the functor which is being derived (in this case, ${\ensuremath{\mathsf{nil}}}_{2}$). Higher cubic central extensions are defined by induction; let us explain how this is done for lowest degrees (more details can be found in the following sections and in the articles [@EverHopf; @EGoeVdL; @EGVdL], amongst others). A [semi-abelian]{} category [@Janelidze-Marki-Tholen; @Borceux-Bourn] is pointed, Barr-exact [@Barr] and Bourn-protomodular [@Bourn1991] with binary sums. Let ${\ensuremath{\mathcal{X}}}$ be a semi-abelian category and ${\ensuremath{\mathcal{B}}}$ a [Birkhoff subcategory]{} [@Janelidze-Kelly] of ${\ensuremath{\mathcal{X}}}$—full, reflective and closed under subobjects and regular quotients, so that a Birkhoff subcategory of a variety is nothing but a subvariety. Let $$\label{Adjunction-1} \vcenter{\xymatrix{{{\ensuremath{\mathcal{X}}}} \ar@<1ex>[r]^-{I} \ar@{}[r]|-{\perp} & {\ensuremath{\mathcal{B}}}\ar@<1ex>[l]^-{\supset}}}$$ denote the induced adjunction, with $I$ the reflector and $\eta\colon{1_{{\ensuremath{\mathcal{X}}}}{\Rightarrow}I}$ the unit. In this context, a [cubic extension]{} $f\colon{X\to Z}$ is defined as a regular epimorphism, and an [extension]{} $(k,f)$ as a short exact sequence $$\xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r]^-{k} & X \ar@{ >>}[r]^-{f} & Z \ar[r] & 0.}$$ Together with the classes of cubic extensions in ${\ensuremath{\mathcal{X}}}$ and in ${\ensuremath{\mathcal{B}}}$, the adjunction  forms a *Galois structure* in the sense of Janelidze [@Borceux-Janelidze; @Janelidze:Pure]. Central (cubic) extensions are defined with respect to such a Galois structure, as follows. A cubic extension $f$ is called [trivial]{} when the induced naturality square $$\vcenter{\xymatrix{X \ar@{ >>}[r]^-{f} \ar@{ >>}[d]_-{\eta_{X}} & Z \ar@{ >>}[d]^{\eta_{Z}} \\ IX \ar@{ >>}[r]_-{If} & IZ}}$$ is a pullback; $f\colon{X\to Z}$ is [central]{} when either of the kernel pair projections $\operatorname{pr}_{0}$, $\operatorname{pr}_{1}\colon {\operatorname{Eq}(f)}=X\times_Z X\to X$ is trivial [@Janelidze-Kelly]. An extension $(k,f)$ is said to be [trivial]{} or [central]{} whenever so is the underlying cubic extension $f$. $\vcenter{\xymatrix@!0@=3.5em{& 0 \ar@{.>}[d] & 0 \ar@{->}[d] & 0 \ar@{->}[d]\\ 0 \ar@{->}[r] & A \ar@{{ |>}->}[r] \ar@{{ |>}.>}[d] & \cdot \ar@{{ |>}->}[d] \ar@{-{ >>}}[r] & \cdot \ar@{{ |>}->}[d] \ar@{->}[r] & 0\\ 0 \ar@{->}[r] & \cdot \ar@{.>}[rd] \ar@{{ |>}->}[r] \ar@{-{ >>}}[d] & F_{2} \ar@{-{ >>}}[r]_-{f_{0}} \ar@{-{ >>}}[d]^-{f_{1}} & \cdot \ar@{-{ >>}}[d] \ar@{->}[r] & 0\\ 0 \ar@{->}[r] & \cdot \ar@{{ |>}->}[r] \ar[d] & \cdot \ar@{.{ >>}}[r] \ar@{->}[d] & Z \ar@{->}[d] \ar@{.>}[r] & 0\\ & 0 & 0 & 0 }}$ It turns out that the cubic central extensions relative to ${\ensuremath{\mathcal{B}}}$ determine a reflective subcategory ${\ensuremath{\mathsf{CExt}}}_{\ensuremath{\mathcal{B}}}({\ensuremath{\mathcal{X}}})$ of the category ${\ensuremath{\mathsf{Ext}}}({\ensuremath{\mathcal{X}}})$ of cubic extensions in ${\ensuremath{\mathcal{X}}}$, so we have an adjunction $$\xymatrix{{{\ensuremath{\mathsf{Ext}}}({\ensuremath{\mathcal{X}}})} \ar@<1ex>[r]^-{I_{1}} \ar@{}[r]|-{\perp} & {\ensuremath{\mathsf{CExt}}}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}}).\ar@<1ex>[l]^-{\supset}}$$ Together with classes of two-cubic extensions, defined as in the case of groups above, this adjunction forms a Galois structure, and thus we acquire the notion of *two-cubic central extension* with respect to ${\ensuremath{\mathcal{B}}}$. This construction may be repeated ad infinitum, so that notions of *$n$-cubic extension* (special $n$-dimensional cubes in ${\ensuremath{\mathcal{X}}}$) and *$n$-cubic central extension* are obtained. An [$n$-fold (central) extension]{} will be a special diagram of short exact sequences: a $3^{n}$-diagram, which is essentially an $n$-cubic extension with chosen kernels. For instance, a two-fold extension is a short exact sequence of short exact sequences—a $3\times 3$-diagram as in Figure \[3x3 diag\], where the bottom right square is the underlying two-cubic extension. The dotted arrows in this diagram form a *Yoneda extension* [@Yoneda-Exact-Sequences] from $A$ to $Z$, which in the abelian case allows to reconstruct the entire $3\times 3$-diagram up to equivalence. In general, though, the diagram may not be thus reduced without loss of information. See Figure \[Figure-Direction\] below for a picture in dimension three. Of course, whether or not an $n$-cubic extension is central with respect to some chosen Birkhoff subcategory depends on this subcategory more than anything else. In many cases (like, for instance, the case of groups vs. $2$-nilpotent groups) there are explicit descriptions of the central extensions in some, or in all, degrees (see, for instance, [@CKLVdL; @Everaert-Gran-TT; @Everaert-Gran-nGroupoids; @EverVdL3]). Knowing, in a given case, what the central extensions are, gives a complete description of the corresponding homology objects as higher Hopf formulae: this is the content of Theorem 8.1 in [@EGVdL]. In this article we only consider cubic extensions which are central with respect to the Birkhoff subcategory ${{\ensuremath{\mathcal{B}}}={\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})}$ of all *abelian* objects in ${\ensuremath{\mathcal{X}}}$, the objects which admit an internal abelian group structure; that is to say, they are central with respect to abelianisation. The reason for this constraint is that we only treat cohomology with trivial coefficients—coefficients in trivial modules, which are precisely the internal abelian group objects. In the non-trivial case, where the theory involves Birkhoff subcategories of ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$, the situation becomes significantly more complicated, and forms the subject of current work-in-progress. The Hopf formulae now take the following shape [@EGVdL]: $$\label{General-Hopf} {\mathrm{H}}_{n+1}(Z,{\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}}))\cong\dfrac{\bigcap_{i\in n}{\operatorname{Ker}(f_{i})}\cap {\langle}F_{n}{\rangle}}{L_{n}[F]}$$ for any $n$-presentation $F$ of $Z$. Here $F_{n}$ is the “initial object” of the $n$-cubic extension $F$ and the $f_{i}$ are the “initial arrows” (see the solid part of Figure \[Figure-Direction\] for a picture in degree three). The brackets ${\langle}-{\rangle}$ in the formula give the *zero-dimensional commutator* of $F_{n}$ determined by its abelianisation: for any object $X$ of ${\ensuremath{\mathcal{X}}}$ there is a short exact sequence $$\label{Angular Bracket} \xymatrix@1{0 \ar[r] & {\langle}X{\rangle}\ar@{{ |>}->}[r] & X \ar@{-{ >>}}[r]^-{\eta_{X}} & {\ensuremath{\mathsf{ab}}}X \ar[r] & 0,}$$ so ${\langle}X{\rangle}=[X,X]$, the Huq commutator [@BG; @Huq] of $X$ with itself, giving a functor ${\langle}- {\rangle}\colon {{\ensuremath{\mathcal{X}}}\to {\ensuremath{\mathcal{X}}}}$. The object in the denominator of the Hopf formula is the smallest normal subobject of $F_{n}$ which, when divided out, makes $F$ central; in other words, an $n$-cubic extension $F$ is central if and only if $L_{n}[F]=0$. In many cases (see Section \[Section-Commutator-Assumption\]) this “abstract higher-dimensional commutator” may be computed as a join of binary Huq commutators [@EGVdL; @RVdL3]. On the other hand, the use of higher (central) extensions is not at all limited to homology and Hopf formulae. The concept of higher (cubic) extension is quite interesting in its own right [@EGoeVdL] while centrality may, for instance, be used to model more exotic commutator theories [@EverVdL4; @EverVdLRCT]. The present article is meant to clarify the connection with cohomology and obtain a higher-dimensional counterpart of the the low-dimensional work which has been done in this context [@Bourn1999; @Bourn-Janelidze:Torsors; @Gran-VdL; @RVdL]. Cohomology and centrality {#cohomology-and-centrality .unnumbered} ------------------------- The current development starts with the long-established interpretation of the second cohomology group ${\mathrm{H}}^{2}(Z,A)$ of a group $Z$ with coefficients in an abelian group $A$ in terms of central extensions of $Z$ by $A$ (see for instance [@MacLane:Homology]). A central extension $(k,f)$ of $Z$ by $A$ is a short exact sequence of groups $$\label{1-fold extension} \xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r]^-{k} & X \ar@{ >>}[r]^-{f} & Z \ar[r] & 0,}$$ so $k=\ker f$ and $f=\operatorname{coker}k$, such that the commutator $[A,X]$ is trivial: $$\text{$axa^{-1}x^{-1}=1$ for all $a\in A$ and~${x\in X}$.}$$ Two extensions $(k,f)\colon {A\to X\to Z}$ and $(k',f')\colon{A\to X'\to Z}$ of $Z$ by $A$ are said to be equivalent if and only if there exists a group (iso)morphism $i\colon{X\to X'}$ satisfying ${f'{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}i=f}$ and $i{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}k=k'$. The corresponding equivalence classes, together with the so-called Baer sum, form an abelian group $\operatorname{Centr}^{1}(Z,A)$, and this group is isomorphic to ${\mathrm{H}}^{2}(Z,A)$. In this article we generalise the isomorphism $\operatorname{Centr}^{1}(Z,A)\cong {\mathrm{H}}^{2}(Z,A)$ in two ways: first of all, we also consider higher cohomology groups; secondly, we replace the concrete context of groups by an abstract context of a semi-abelian category satisfying an additional axiom which holds in many important examples of semi-abelian categories—in particular, it holds in the category of groups and in any abelian category. It was proved in [@Gran-VdL], see also [@Bourn:Baer-Sums-2007] and [@Bourn-Janelidze:Torsors], that the classical interpretation of group cohomology via central extensions may be extended from the context of groups to semi-abelian categories. Here the concept of centrality is the one coming from Galois theory, using the Birkhoff subcategory of all abelian objects [@Bourn-Gran]—that is, we use centrality relative to abelianisation—and the cohomology is comonadic cohomology [@Barr-Beck]. Thus the well-known similar results for Lie algebras over a field, commutative algebras, non-unitary rings, (pre)crossed modules, etc. could be included in a general theory, and new examples could be studied. When $A$ is a $Z$-module with a non-trivial action, of course the above concept of central extension does no longer suffice to capture cohomology with coefficients in $A$. Nevertheless, there is good hope that something can be done in general and in higher degrees which extends both the present work and the results in [@Bourn-Janelidze:Torsors]. In the current paper we limit ourselves to the case of trivial module coefficients essentially for the sake of simplicity. In contrast with this potential extension of the theory, when $A$ is not even an abelian object—so when we enter the realm of true non-abelian cohomology—it is not clear at all how the current approach could form the basis of a new theory. The next step, an interpretation of the third cohomology group in similar terms, turned out to be quite hard to take. The reason is that one needs a theory of higher central extensions for this—which until recently was unavailable. The problem was finally solved in [@RVdL], where the characterisation of two-cubic central extensions given in [@Janelidze:Double; @Gran-Rossi] is extended to semi-abelian categories and an isomorphism $${\mathrm{H}}^{3}(Z,A)\cong \operatorname{Centr}^{2}(Z,A)$$ is constructed. The abelian group $\operatorname{Centr}^{2}(Z,A)$ consists of equivalence classes of two-fold central extensions of an object $Z$ by an abelian object $A$ as in Figure \[3x3 diag\], equipped with a canonical addition induced by the internal group structure of $A$. It must be mentioned that the cohomology theory used in [@RVdL]—the *directions approach*, using internal $n$-fold groupoids, introduced by Bourn in [@Bourn1999; @Bourn2002b; @Bourn:Baer-Sums-2007] and further worked out by Bourn and Rodelo in [@Bourn-Rodelo; @Rodelo-Directions]—is less classical than the one we shall be using here, or at least is not obviously related to it in higher degrees. In this paper we use a different interpretation of cohomology which is based on higher torsors [@Duskin; @Duskin-Torsors; @Glenn]. $\vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& 0 \ar@{}[d]|(.67){a} \ar[rd]^-{f_{0}(a)=0}\\ 0 \ar[ru]^-{f_{2}(a)=0} \ar[rr]_-{f_{1}(a)=0} && 0}}$ A key ingredient here is the concept of [direction]{} of a higher (central) extension, which is the initial object of this extension $E$, when $E$ is considered as a diagram of short exact sequences in ${\ensuremath{\mathcal{X}}}$: the objects $A$ in Figure \[3x3 diag\] and Figure \[Figure-Direction\]. From the point of view of the $n$-cubic extension $F$ underlying $E$, the direction is an intersection of (chosen) kernels. For a one-fold extension such as , the direction is the kernel $A={\operatorname{Ker}(f)}$ of the underlying one-cubic extension $f\colon{X\to Z}$, while for a two-cubic extension as in Figure \[3x3 diag\] it is the intersection of the kernels ${{\operatorname{Ker}(f_{0})}\cap {\operatorname{Ker}(f_{1})}}$. Considering the underlying two-cubic extension $F$ as an arrow in the category of arrows in ${\ensuremath{\mathcal{X}}}$, the kernel of $F$ is a one-cubic extension in ${\ensuremath{\mathcal{X}}}$, whose kernel is isomorphic to the direction of $E$; so we write it as ${\operatorname{Ker}^2(F)}$. In higher degrees a similar (inductive) analysis makes sense: given an $n$-fold extension $E$ with underlying $n$-cubic extension $F$, its direction is ${\operatorname{Ker}^n(F)}=\bigcap_{i\in n}{\operatorname{Ker}(f_{i})}$, which is an abelian object of ${\ensuremath{\mathcal{X}}}$ when $F$ is central (compare with the Hopf formula ). Figure \[Figure-Direction\] gives a picture in degree $3$. The different but equivalent ways in which the direction may be obtained as a kernel come from the several ways in which a three-cubic extension may be considered as an arrow between two-cubic extensions, etc. An element of $F_{3}$ should be viewed as a (directed) triangle with faces given by $f_{0}$, $f_{1}$ and $f_{2}$, and such a triangle $a$ is in the direction $A$ if and only if all its faces (edges) are zero. We write ${\ensuremath{\mathsf{CExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ for the category of $n$-fold central extensions over $Z$. Thus, for each $n\geq 1$ and any object $Z$ in ${\ensuremath{\mathcal{X}}}$, we obtain a functor $${\ensuremath{\mathsf{D}}}_{(n,Z)}\colon{\ensuremath{\mathsf{CExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$$ that sends an $n$-fold central extension $E$ of $Z$ to its direction $A$. Given any object $Z$ in ${\ensuremath{\mathcal{X}}}$ and any abelian object $A$, an [$n$-fold central extension of $Z$ by $A$]{} is an $n$-fold central extension $F$ of $Z$ with direction $A$, an object of the fibre ${\ensuremath{\mathsf{D}}}_{(n,Z)}^{-1}A$. Taking connected components gives us the (possibly large) set $$\operatorname{Centr}^{n}(Z,A)=\pi_{0}({\ensuremath{\mathsf{D}}}_{(n,Z)}^{-1}A)$$ which admits a canonical abelian group structure, since, as established in Proposition \[Proposition-Centr\^[n]{}(Z,-)\], the assignment $A\mapsto \operatorname{Centr}^{n}(Z,A)$ gives rise to a product-preserving functor ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}}) \to {\ensuremath{\mathsf{Set}}}$. Now the question remains whether these groups have any cohomological meaning. The main body of this article explains that, indeed, they have: we shall prove that, under the commutator condition (CC), they agree with the interpretation of comonadic cohomology in terms of higher torsors. Cohomology via higher torsors {#cohomology-via-higher-torsors .unnumbered} ----------------------------- One could say that Duskin and Glenn’s *higher torsors* [@Duskin; @Duskin-Torsors; @Glenn] are to central extensions what truncations of simplicial resolutions are to cubic extensions, or what groupoids are to pregroupoids: $$\frac{\text{torsor}}{\text{central extension}}=\frac{\text{truncation of simplicial resolution}}{\text{cubic extension}}=\frac{\text{groupoid}}{\text{pregroupoid}}.$$ In a groupoid $$\xymatrix{G_{1} \ar@<-1ex>[r]_-{{\partial}_{0}} \ar@<1ex>[r]^-{{\partial}_{1}} & G_{0} \ar[l]|-{\sigma_{0}}}$$ there are identities (given by $\sigma_{0}$) and a composition $m$ $$\vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\beta}\\ {\cdot} \ar[ru]^-{\alpha} \ar@{.>}[rr]_-{\gamma} && {\cdot}}} \qquad\qquad m(\beta,\alpha)=\gamma$$ which is associative, admits inverses and is compatible with the identities; there is only one object of objects, $G_{0}$. On the other hand, a pregroupoid [@Kock-Pregroupoids] $$\xymatrix@!0@=3em{& G_{1} \ar[dl]_-{{\partial}_{0}} \ar[dr]^-{{\partial}_{1}} \\ G_{0} && G'_{0}}$$ has two objects of objects, $G_{0}$ and $G_{0}'$. Consequently, it has no identities, and instead of a composition, there is an (associative) Mal’tsev operation $p$ $$\vcenter{\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\gamma} \ar@{.>}[rd]_-{\delta} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\alpha}\\ &{\cdot}}} \qquad\qquad p(\alpha,\beta,\gamma)=\delta$$ satisfying $p(\alpha,\alpha,\gamma)=\gamma$ and $p(\alpha,\gamma,\gamma)=\alpha$. In the present context, associativity is automatic. In dimension $3$ now, truncating a simplicial object ${\ensuremath{\mathbb{X}}}$ at degree $2$, we obtain a diagram as on the left $$\vcenter{\xymatrix@1@!0@=45pt{{\ensuremath{\mathbb{X}}}_{2} \ar[r]|-{{\partial}_{1}} \ar@<2ex>[r]^-{{\partial}_{2}} \ar@<-2ex>[r]_-{{\partial}_{0}} & {\ensuremath{\mathbb{X}}}_{1} \ar@<-1ex>[l] \ar@<1ex>[l] \ar@<1ex>[r]^-{{\partial}_{1}} \ar@<-1ex>[r]_-{{\partial}_{0}} & {\ensuremath{\mathbb{X}}}_{0} \ar[r]^-{{\partial}_{0}} \ar[l]|-{\sigma_{0}} & {\ensuremath{\mathbb{X}}}_{-1}}} \qquad\qquad \vcenter{\xymatrix@1@!0@=35pt{ & {\ensuremath{\mathbb{X}}}_{2} \ar[rr]^-{{\partial}_{0}} \ar[dd]|(.25){{\partial}_{2}}|-{\hole} \ar[ld]_-{{\partial}_{1}} && {\ensuremath{\mathbb{X}}}_{1} \ar[dd]^-{{\partial}_{1}} \ar[ld]|-{{\partial}_{0}} \\ {\ensuremath{\mathbb{X}}}_{1} \ar[rr]^(.75){{\partial}_{0}} \ar[dd]_-{{\partial}_{1}} && {\ensuremath{\mathbb{X}}}_{0} \ar[dd]^(.25){{\partial}_{0}} \\ & {\ensuremath{\mathbb{X}}}_{1} \ar[ld]_-{{\partial}_{1}} \ar[rr]^(.25){{\partial}_{0}}|-{\hole} && {\ensuremath{\mathbb{X}}}_{0} \ar[ld]^-{{\partial}_{0}} \\ {\ensuremath{\mathbb{X}}}_{0} \ar[rr]_-{{\partial}_{0}} && {\ensuremath{\mathbb{X}}}_{-1}}}$$ which may be “unfolded” to a commutative cube as on the right. The extension property of this cube corresponds to acyclicity of the given simplicial object ${\ensuremath{\mathbb{X}}}$ (its being a resolution) up to degree $2$. Note that this cube is special, because certain objects in it occur several times; on the other hand, the cube does not capture the degeneracies present in the simplicial object. Groupoids (and torsors) live in the *simplicial* world, whereas pregroupoids belong to the *cubical* world of $n$-cubic (central) extensions. As we shall explain in Subsection \[Maltsev\], higher central extensions may be considered as *higher-dimensional pregroupoids* in some precise sense. Given an object $Z$ and an abelian object $A$ in a semi-abelian category ${\ensuremath{\mathcal{X}}}$, we consider the augmented simplicial object ${\ensuremath{\mathbb{K}}}(Z,A,n)$ determined by $$\resizebox{\textwidth}{!}{ $ \vcenter{\xymatrix@R=20pt@C=45pt{\scriptstyle{n+1} & \scriptstyle{n} & \scriptstyle{n-1} & \scriptstyle{n-2} \ar@{}[r]|-{\cdots} & \scriptstyle{0} & \scriptstyle{-1}\\ A^{n+1} \times Z \ar@<2.33ex>[r]^-{{\partial}_{n+1}\times 1_{Z}} \ar@<1.16ex>[r]|-{\operatorname{pr}_{n}\times 1_{Z}} \ar@<-2.33ex>[r]_-{\operatorname{pr}_{0}\times 1_{Z}}^-{\vdots} & A \times Z \ar@<1.75ex>[r]^-{\operatorname{pr}_{Z}} \ar@<-1.75ex>[r]_-{\operatorname{pr}_{Z}}^-{\vdots} & Z \ar@{=}@<1.75ex>[r] \ar@{=}@<-1.75ex>[r]^-{\vdots} & Z \ar@{}[r]|-{\cdots} & Z \ar@{=}[r] & Z}} $}$$ with ${\partial}_{n+1}=(-1)^{n}\sum^{n}_{i=0}(-1)^{i}\operatorname{pr}_{i}$. An [$n$-torsor of $Z$ by $A$]{} is an augmented simplicial object ${\ensuremath{\mathbb{T}}}$ equipped with a simplicial morphism ${\ensuremath{\mathbb{t}}}\colon {{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(Z,A,n)}$ such that 1. ${\ensuremath{\mathbb{t}}}$ is a fibration which is exact from degree $n$ on; 2. ${\ensuremath{\mathbb{T}}}\cong {\ensuremath{\mathsf{Cosk}}}_{n-1}{\ensuremath{\mathbb{T}}}$, the $(n-1)$-coskeleton of ${\ensuremath{\mathbb{T}}}$; 3. ${\ensuremath{\mathbb{T}}}$ is a resolution. Axiom (T2) means that ${\ensuremath{\mathbb{T}}}$ does not contain information above level $n-1$, which together with (T3) amounts to the $(n-1)$-truncation $T$ of ${\ensuremath{\mathbb{T}}}$, considered as an $n$-cube, being an $n$-cubic extension. The fibration property in (T1) is (almost) automatic, while the exactness tells us that, for all $i$, $$\label{Torsor-Iso} {\triangle}({\ensuremath{\mathbb{T}}},n)\cong A\times {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n).$$ Here $A=\bigcap_{i\in n}{\operatorname{Ker}({\partial}_{i})}$ is the direction of $T$, the object ${\triangle}({\ensuremath{\mathbb{T}}},n)$ consists of all $n$" cycles in ${\ensuremath{\mathbb{T}}}$ and ${\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$ is the object of $(n,i)$-horns in ${\ensuremath{\mathbb{T}}}$. In degree two, for instance, we obtain the following picture: $$\label{Simplicial-Dimension-Two} \begin{matrix}{\triangle}({\ensuremath{\mathbb{T}}},2) & \cong & A & \times & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{1}({\ensuremath{\mathbb{T}}},2) \\ \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\beta}\\ {\cdot} \ar[ru]^-{\alpha} \ar[rr]_-{\gamma} && {\cdot}}} && \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{{0} \ar[rr]_-{a} && {0}}} && \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\beta}\\ {\cdot} \ar[ru]^-{\alpha} && {\cdot}}} \end{matrix}$$ Given $a$, $\alpha$ and $\beta$, there is a unique arrow $\gamma=\mu^{1}(a,\beta,\alpha)$ such that the projection $\operatorname{pr}_{A}(\beta, \gamma, \alpha)$ on $A$ gives back $a$. In some sense $a=0$ if and only if the triangle on the left “commutes”, and taking $\gamma=\mu^{1}(0,\beta,\alpha)=m^{1}(\beta,\alpha)$ as a composite of $\beta$ and $\alpha$ really does define a groupoid structure $m^{1}$ on $T$. Let ${\ensuremath{\mathsf{s}}}^{+}({\ensuremath{\mathcal{X}}})$ denote the category of augmented simplicial objects in ${\ensuremath{\mathcal{X}}}$. The full subcategory of the slice ${\ensuremath{\mathsf{s}}}^{+}({\ensuremath{\mathcal{X}}})/{\ensuremath{\mathbb{K}}}(Z,A,n)$ determined by the $n$-torsors of $Z$ by $A$ is written $\operatorname{Tors}^{n}(Z,A)$. Taking connected components we obtain the set $$\operatorname{Tors}^{n}[Z,A]=\pi_{0}\operatorname{Tors}^{n}(Z,A)$$ of equivalence classes of $n$-torsors of $Z$ by $A$. It is, in fact, an abelian group [@Duskin-Torsors]. Duskin explains in [@Duskin; @Duskin-Torsors] that the group $\operatorname{Tors}^{n}[Z,A]$ may be considered as a cohomology group ${\mathrm{H}}^{n+1}(Z,A)$ of $Z$ with coefficients in the trivial module $A$, and that under certain conditions this cohomology coincides with other known cohomology theories. For instance, when ${\ensuremath{\mathcal{X}}}$ is monadic over ${\ensuremath{\mathsf{Set}}}$, we obtain Barr–Beck cohomology [@Duskin-Torsors Theorem 5.2]. If ${\ensuremath{\mathbb{G}}}$ is the comonad induced by the forgetful/free adjunction of ${\ensuremath{\mathcal{X}}}$ to ${\ensuremath{\mathsf{Set}}}$, if $Z$ is an object of ${\ensuremath{\mathcal{X}}}$ and $A$ an abelian object, then for any natural number $n$, $${\mathrm{H}}^{n+1} (Z,A)_{{\ensuremath{\mathbb{G}}}}={\mathrm{H}}^{n}\operatorname{hom}({\ensuremath{\mathsf{ab}}}{\ensuremath{\mathbb{G}}}Z,A)$$ is the $(n+1)$-th cohomology group of $Z$ with coefficients in $A$, relative to the comonad ${\ensuremath{\mathbb{G}}}$ [@Barr-Beck]. This defines a functor ${\mathrm{H}}^{n+1} (-,A)\colon {\ensuremath{\mathcal{X}}}\to {\ensuremath{\mathsf{Ab}}}$, for any ${n\geq 0}$. As mentioned in the previous paragraph, Duskin obtains an isomorphism $${\mathrm{H}}^{n+1} (Z,A)_{{\ensuremath{\mathbb{G}}}}\cong {\mathrm{H}}^{n+1} (Z,A),$$ where the latter cohomology group is $\operatorname{Tors}^{n}[Z,A]$ by definition. The geometry of higher central extensions {#the-geometry-of-higher-central-extensions .unnumbered} ----------------------------------------- In Section \[Section-Geometry\] we analyse higher central extensions from a geometrical point of view so that we can compare them with higher torsors. We work towards Proposition \[Proposition-Central-then-Torsor\] which says that an augmented simplicial object carries a (unique) structure of $n$-torsor as soon as its underlying $n$-fold arrow is an $n$-cubic central extension. This result is based on Theorem \[Theorem-Higher-Centrality\] which gives a new characterisation of higher central extensions: an $n$-cubic extension $F$ is central if and only if its direction $A$ is abelian and there is a canonical isomorphism $$\label{Iso-Box} \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\cong A\times \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$$ for any (hence, all) $I\subseteq n$. (Compare with the isomorphism .) The precise definition of the objects $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$ and $\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$ will be presented in Section \[Section-Geometry\], but we can already explain the meaning of this characterisation in some low-dimensional cases and give the main idea. When $n=1$ this characterisation of centrality becomes the well-known result (see [@Bourn-Gran; @Bourn-Gran-Maltsev]) that an extension $(k,f)\colon{A\to X\to Z}$ is central if and only if the kernel (direction) $A$ of $f$ is abelian and the kernel pair of $f$ may be decomposed into a product as ${\operatorname{Eq}(f)}\cong A\times X$. When $F$ is the two-cubic extension  the isomorphism becomes $${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}\cong A\times ({\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}),$$ where the direction $A$ is given by $A={\operatorname{Ker}(d)}\cap {\operatorname{Ker}(c)}$. As explained in Subsection \[Degree two\], this isomorphism can be obtained as a consequence of the analysis of two-cubic extensions carried out in [@RVdL]. Recall [@Janelidze-Pedicchio; @Borceux-Bourn] that ${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}$ contains *diamonds* (as on the left) $$\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\gamma} \ar[rd]_-{\delta} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\alpha}\\ &{\cdot}} \qquad\qquad \xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\gamma} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\alpha}\\ &{\cdot}}$$ in $X$, so that the object ${\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}$, which is an instance of the pullback  on page , contains *diamonds with one face missing* (as on the right above) and $$\pi\colon {{\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}\to {\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}}$$ is the projection which forgets $\delta$. The analogy with  is clear and not accidental: the missing $\delta$ corresponds to a unique element $a$ of $A$; on the other hand, given any diamond (including $\delta$), the corresponding element $a$ of $A$ measures how far the diamond is from being “commutative” (in which case one may think of $\delta$ as a composite $\alpha\beta^{-1}\gamma$). Note that instead of forgetting $\delta$, we could have chosen to forget $\alpha$, $\beta$ or $\gamma$; each of those choices determines a different pullback ${{\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}}$ which, for the sake of clarity, could be written as ${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxdot}}}^{I} {\operatorname{Eq}(c)}$ where the index ${I\subseteq 2}$ determines the chosen projection (indeed there are four options). In general, given an $n$-cubic extension $F$, the object $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$ contains what we call “$n$-dimensional diamonds” in $F_{n}$ (see Figure \[Figure-Diamond\] on page  for an illustration in dimension $3$) and $\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$ contains “$n$-dimensional diamonds” with one face (determined by the index $I\subseteq n$) missing. The cubic extension $F$ is central when its direction $A$ is abelian and the canonical projection $$\pi^{I}\colon\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$$ induces the isomorphism ; this means that a missing face in any $n$-fold diamond in $F_{n}$ is completely determined by an element in $A$. We also obtain an explicit formula for the splitting $\operatorname{pr}_{A}\colon {\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to A}$ of the kernel of $\pi^{I}$, the projection on $A$ which gives us a “measure of commutativity” for $n$-fold diamonds: Proposition \[Centrality-Sum\] states that the lifting of $$\sum_{J\subseteq n}(-1)^{|J|}\eta_{F_{n}}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{J}$$ over $A$, where $\operatorname{pr}_{J}\colon{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to F_{n}}$ sends a diamond to its $J$-face, is $\operatorname{pr}_{A}$. Using this geometrical interpretation of centrality we can compare torsors and central extensions. Any $n$-cycle may be “completed” into an $n$-fold diamond by adding well-chosen degeneracies, and thus restricting the isomorphism  to an isomorphism  we may prove that any augmented simplicial object of which the underlying $n$-fold arrow is a central extension is in fact an $n$-torsor. The converse, however, needs more, since in general it is not clear how an isomorphism on the simplicial level may be extended to an isomorphism on the level of higher-dimensional diamonds. For this implication we pass via an interpretation of centrality in terms of commutators. The commutator condition {#the-commutator-condition .unnumbered} ------------------------ In order to complete the equivalence between torsors and higher central extensions, we shall assume that centrality may be characterised in terms of binary Huq commutators. We call this assumption the [commutator condition (CC)]{} on higher central extensions [@RVdL3]: it holds when, for all $n\geq 1$, an $n$-cubic extension $F$ is central if and only if $$\displaystyle\Bigl[\bigcap_{i\in I}{\operatorname{Ker}(f_{i})},\bigcap_{i\in n\setminus I}{\operatorname{Ker}(f_{i})}\Bigr]=0$$ for all $I\subseteq n$. Following [@RVdL3], an $n$-cubic extension which satisfies this commutator condition is called [H-central]{}. If we name the concept of centrality coming from Galois theory [categorical centrality]{}, then (CC) says that *H-central and categorically central extensions are the same*. The condition (CC) amounts to asking that the Hopf formula for higher homology  becomes a quotient of binary Huq commutators: its denominator $L_{n}[F]$ is then equal to the join $\bigcup_{I\subseteq n}\bigl[\bigcap_{i\in I}{\operatorname{Ker}(f_{i})},\bigcap_{i\in n\setminus I}{\operatorname{Ker}(f_{i})}\bigr]$, so that $${\mathrm{H}}_{n+1}(Z,{\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}}))\cong\dfrac{\bigcap_{i\in n}{\operatorname{Ker}(f_{i})}\cap [F_{n},F_{n}]}{\bigcup_{I\subseteq n}\bigl[\bigcap_{i\in I}{\operatorname{Ker}(f_{i})},\bigcap_{i\in n\setminus I}{\operatorname{Ker}(f_{i})}\bigr]}$$ for any $n$-cubic presentation $F$ of any object $Z$ and for any $n\geq 1$. We shall, however, focus on the cohomological meaning of this condition rather than on Hopf formulae. It is certain that many categories satisfy (CC), although thus far no explicit characterisation is known; in the article [@EGVdL] the categories of groups, Lie algebras and non-unitary rings are given as examples, and it is not difficult to add new examples to the list by using the technique explained there. A wide range of (generally non-trivial) examples are those semi-abelian categories with a protoadditive abelianisation functor [@Everaert-Gran-nGroupoids; @Everaert-Gran-TT], of which two extreme special cases are all semi-abelian arithmetical categories, such as the categories of von Neumann regular rings, Boolean rings and Heyting semilattices (where the cohomology theory becomes trivial) on the one hand, and all abelian categories (where, via a version of the Dold–Kan correspondence [@Dold-Puppe], the theory gives us the Yoneda $\operatorname{Ext}$ groups) on the other. More recently it was shown in [@RVdL3] that all semi-abelian categories with the *Smith is Huq* [@MFVdL] property satisfy (CC), while the categories of loops and of commutative loops do not. So, *action representative* semi-abelian categories [@BJK2; @Borceux-Bourn-SEC], *action accessible* categories [@BJ07] (which makes all *categories of interest* [@Orzech] examples [@Montoli]), *strongly semi-abelian* [@Bourn2004] and *Moore* categories [@Gerstenhaber; @Rodelo:Moore] are all examples of categories with the *Smith is Huq* property, thus satisfy (CC). For instance, so do the categories of associative and non-associative algebras and of (pre)crossed modules, and all *varieties of groups* in the sense of [@Neumann]. In any case, every semi-abelian category satisfies (CC) for ${n=1}$ (see [@Gran-Alg-Cent; @Gran-VdL]). Proposition \[Proposition-Torsor-then-Central\] now tells us that in a semi-abelian category with (CC), the $n$" cubic extension underlying an $n$-torsor is always central. Hence for a truncated augmented simplicial object, the two concepts are equivalent (Theorem \[Theorem-Torsor-Equivalence\]): indeed, given a simplicial object ${\ensuremath{\mathbb{T}}}$, when it exists, a morphism ${\ensuremath{\mathbb{t}}}$ making $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ a torsor is necessarily unique; furthermore, any morphism of $n$-fold central extensions of $Z$ by $A$ which restricts to the truncation of a simplicial morphism, extends uniquely to a morphism of $n$-torsors of $Z$ by $A$—see Section \[Section-Torsors-and-Centrality\], in particular Proposition \[Proposition-Full\]. The main theorem {#the-main-theorem .unnumbered} ---------------- Proposition \[Proposition-Simplification-of-Central-Extension\] tells us that, as soon as enough projectives exist, any $n$-fold central extension of an object $Z$ by an abelian object $A$ is connected to an $n$-fold central extension of $Z$ by $A$ of which the underlying $n$-cube is a truncation of an augmented simplicial object. Thus under (CC), any $n$-cubic central extension is connected to the simplicial-object part of an $n$-torsor. Since, furthermore, this process is compatible with directions, and we acquire an isomorphism $$\operatorname{Tors}^{n}[Z,A]=\pi_{0}\operatorname{Tors}^{n}(Z,A)\cong \pi_{0}({\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A)=\operatorname{Centr}^{n}(Z,A),$$ natural in $A$. As a consequence we obtain the main result of this article, Theorem \[Main-Theorem\]: if $Z$ is an object and $A$ an abelian object in a semi-abelian category with enough projectives satisfying the commutator condition (CC), then for every $n\geq 1$ we have that $${\mathrm{H}}^{n+1}(Z,A)\cong\operatorname{Centr}^{n}(Z,A)$$ as abelian groups. This establishes the result conjectured in [@RVdL], though for a different definition of cohomology, and has several other interesting implications. For instance, from [@Duskin] it follows that there is a long exact sequence for $\operatorname{Centr}^{n}(Z,-)$. “Duality” between “internal” homology and “external” cohomology {#duality-between-internal-homology-and-external-cohomology .unnumbered} --------------------------------------------------------------- We call a (co)homology theory [internal]{} when the (co)homology object is an actual (abelian) object in the ground semi-abelian category, and [external]{} if it is an (abelian) group and hence, in general, is an object outside the ground category. For instance, the approach to homology in terms of higher Hopf formulae discussed above is internal, while the approach to cohomology via higher central extensions is external. An example of internal *co*homology is developed in Gray’s Ph.D. thesis [@GrayPhD]. Combined with the main result of the article [@GVdL2], our present interpretation of external cohomology gives an answer to the following somewhat naive question: > *In which sense are internal homology\ > and external cohomology “dual” to each other?* The word “dual” here should not be read in its formal categorical sense, but similarly to the way we read “dual of a vector space”. It is true that there is a kind of “duality”, or at least a strong symmetry, in the definitions of homology and cohomology when one uses, for instance, the comonadic Barr–Beck approach. Nevertheless, so far there was no meaningful connection at all between the *interpretations* of internal homology (using Hopf formulae, say) and external cohomology (many different approaches here), at least not for non-abelian algebraic objects. Following [@Tim-Missing-Link], we claim that the hidden connection is the concept of *direction for higher central extensions* and the analysis of both homology (internal, relative to abelianisation) and cohomology (external, with trivial coefficients) in these terms. The theory of *satellites* [@GVdL2; @Janelidze-Satellites] makes it possible to replace Hopf formulae for homology with (possibly large) limits, so that homology objects may also be computed in contexts where not enough projective objects are available. The results in [@GVdL2] are again based on higher central extensions in semi-abelian categories, and the article’s Corollary 4.10 tells us that, for any Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of a semi-abelian category ${\ensuremath{\mathcal{X}}}$, for any object $Z$ of ${\ensuremath{\mathcal{X}}}$ and any integer ${n\geq 1}$, the homology object ${\mathrm{H}}_{n+1}(Z,{\ensuremath{\mathcal{B}}})$ is the limit of the diagram ${\ensuremath{\mathsf{D}}}_{(n,Z)}\colon{{\ensuremath{\mathsf{CExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathcal{B}}}}$. That is to say, in the case of abelianisation, all the internal homological and external cohomological information on an object $Z$ at a given level $n$ is contained in one and the same functor $${\ensuremath{\mathsf{D}}}_{(n,Z)}\colon{\ensuremath{\mathsf{CExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})\colon E\mapsto {\ensuremath{\mathsf{D}}}_{(n,Z)}E=\bigcap_{i\in n}{\operatorname{Ker}(f_{i})}={\operatorname{Ker}^n(F)}$$ (where $F=E|_{2^{n}}$) in two “opposite” ways, $${\mathrm{H}}_{n+1}(Z,{\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}}))=\lim {\ensuremath{\mathsf{D}}}_{(n,Z)}\qquad\text{and}\qquad {\mathrm{H}}^{n+1}(Z,A)=\pi_{0}({\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A);$$ homology is a limit of ${\ensuremath{\mathsf{D}}}_{(n,Z)}$ while cohomology consists of connected components of a fibre of ${\ensuremath{\mathsf{D}}}_{(n,Z)}$. So on the one end we have the limit of all possible directions and, on the other, all classes of all central extensions with one given and fixed direction—again, see Figure \[Figure-Direction\]. We consider this “duality” (Theorem \[Duality-Theorem\]) to be a major point of the present article. In the article [@PVdL1] it is analysed from the point of view of the Yoneda lemma, which deals with precisely this kind of contrast or “duality” between “internal” and “external”. Structure of the text {#structure-of-the-text .unnumbered} --------------------- In Section \[Section-Preliminaries\] we recall some basic definitions and results which we need later on: semi-abelian categories, simplicial objects, higher extensions and higher torsors. Section \[Section-Central-Extensions\] contains all the theory needed to introduce the groups $\operatorname{Centr}^{n}(Z,A)$. In Section \[Section-Geometry\] we give a geometric interpretation of the concept of higher central extension (Theorem \[Theorem-Higher-Centrality\] and Proposition \[Centrality-Sum\]), used in the next section where we analyse torsors in terms of this geometry. The most important result here is Proposition \[Proposition-Central-then-Torsor\] which says that a truncation of an augmented simplicial object, considered as a higher extension, is a torsor as soon as it is a central extension. The other implication in the equivalence between torsors and central extensions is obtained in Section \[Section-Commutator-Assumption\] (Proposition \[Proposition-Torsor-then-Central\] and Theorem \[Theorem-Torsor-Equivalence\]). However, to make it work, we have to strengthen the context of semi-abelian categories with the additional commutator condition (CC). The short last Section \[Section-Main-Theorem\] explains how to suitably transform an $n$-cubic central extension (which need not be a truncation of anything simplicial) into an $n$-cubic central extension underlying a torsor, so that we may conclude with Theorem \[Main-Theorem\], the isomorphism ${\mathrm{H}}^{n+1}(Z,A)\cong\operatorname{Centr}^{n}(Z,A)$ for all $n\geq 1$, and Theorem \[Duality-Theorem\], the “duality” between internal homology and external cohomology. Preliminaries {#Section-Preliminaries} ============= We sketch the context in which we shall be working: homological and semi-abelian categories for all general results, with the approach to external cohomology in Barr-exact categories due to Duskin [@Duskin; @Duskin-Torsors] and Glenn [@Glenn]. We also recall the definition of higher extensions and the relation with simplicial resolutions [@EGVdL; @EGoeVdL]. Barr-exact, homological and semi-abelian categories --------------------------------------------------- For the sake of clarity, the results in this article will be presented in the context of semi-abelian categories. Although this is an extremely convenient environment to work in, it is probably not the most general context in which the theory may be developed. Nevertheless, we believe that in this first approach it is better not to cloud our results in technical subtleties concerning the surrounding category, but rather to focus on their intrinsic meaning and their correctness. The only disadvantage this added transparency may possibly have is the potential loss of some more elaborate examples; such examples can always be recovered later on. We recall the main definitions and properties of Barr-exact [@Barr], homological [@Borceux-Bourn] and semi-abelian categories [@Janelidze-Marki-Tholen]. Recall that a [regular epimorphism]{} is the coequaliser of some pair of morphisms. A finitely complete category endowed with a pullback-stable (regular epimorphism, monomorphism)-factorisation system is called [regular]{}. Regular categories provide a natural context for working with relations. We denote the kernel relation (= kernel pair) of a morphism $f$, the pullback of $f$ along itself, by $({\operatorname{Eq}(f)},\operatorname{pr}_{0},\operatorname{pr}_{1})$ or $({\operatorname{Eq}(f)},f_{0},f_{1})$, depending on the situation. A regular category is said to be [Barr-exact]{} when every equivalence relation is the kernel pair of some morphism [@Barr]. A [pointed]{} category (that is, with a [zero object]{}, an initial object that is also terminal) that admits pullbacks is called [Bourn-protomodular]{} [@Bourn1991] when the Split Short Five Lemma holds. Moreover, if the pointed category is regular, then protomodularity is equivalent to the (Regular) Short Five Lemma: given a commutative diagram $$\label{Short-Five-Lemma} \vcenter{\xymatrix{0 \ar[r] & {\operatorname{Ker}(f')} \ar[d]_{k} \ar@{{ |>}->}[r]^-{\ker f'} & X' \ar[d]_-{x} \ar@{-{ >>}}[r]^-{f'} & Y' \ar[d]^-{y} \ar[r] & 0\\ 0 \ar[r] & {\operatorname{Ker}(f)} \ar@{{ |>}->}[r]_-{\ker f} & X \ar@{-{ >>}}[r]_-{f} & Y \ar[r] & 0}}$$ with regular epimorphisms $f$, $f'$ and their kernels, if $k$ and $y$ are isomorphisms then also $x$ is an isomorphism. We usually denote the kernel of a morphism $f$ by $({\operatorname{Ker}(f)}, \operatorname{ker}f)$. A pointed, regular and protomodular category is called [homological]{} [@Borceux-Bourn]. This is a context where many of the basic diagram lemmas of homological algebra hold. In particular, here the notion of *(short) exact sequence* has its full meaning: a regular (= normal) epimorphism with its kernel. In order for commutator theory to work flawlessly, the context should be finitely cocomplete and Mal’tsev. By definition, a [Mal’tsev]{} category [@Carboni-Lambek-Pedicchio; @CPP] is finitely complete and such that every reflexive relation is necessarily an equivalence relation. It is well known that any finitely complete protomodular category is necessarily a Mal’tsev category [@Bourn1996]. Joining all these conditions brings us to the notion of a [semi-abelian]{} category which can be defined as a pointed, Barr-exact and protomodular category that admits binary coproducts. This definition unifies many older approaches towards a suitable categorical context for the study of homological properties of non-abelian categories such as the categories of groups, Lie algebras, etc. In the founding article [@Janelidze-Marki-Tholen] which introduces the concept, it is explained how this solves the problem of finding the right axioms to be added to Barr-exactness in order that the resulting context is equivalent with the contexts obtained in terms of “old-style axioms” such as, for instance, the one introduced in [@Huq]. Examples of semi-abelian categories include all varieties of $\Omega$-groups [@Higgins], such as groups and non-unitary rings, precrossed and crossed modules, and categories of non-unitary algebras such as associative algebras and Leibniz and Lie $n$-algebras; then there are non-unitary $C^{*}$-algebras and loops; also any abelian category is an example, as is the dual of the category of pointed objects in any elementary topos. [@Bourn-Janelidze:Semidirect; @Bourn2001]\[Lemma-Iso-Pullback\] In a semi-abelian category, given a commutative diagram with short exact rows such as  above, $k$ is an isomorphism if and only if the right-hand square is a pullback. [@Bourn-Gran-Normal-Sections Theorem 4.9]\[Lemma-Split-Kernel-Product\] In a semi-abelian category, given a short exact sequence $$\vcenter{\xymatrix{0 \ar[r] & {\operatorname{Ker}(f)} \ar@<-.5ex>@{{ |>}->}[r]_-{\ker f} & X \ar@{-{ >>}}[r]^-{f} \ar@<-.5ex>@{-{ >>}}[l]_-{p} & Y \ar[r] & 0}}$$ in which the kernel of $f$ is split by a morphism $p$, the object $X$ is a product of which the projections are $p$ and $f$. Applying Lemma \[Lemma-Iso-Pullback\] to the diagram $$\vcenter{\xymatrix{0 \ar[r] & {\operatorname{Ker}(f)} \ar@{=}[d] \ar@{{ |>}->}[r]^-{\ker f} & X \ar[d]_-{p} \ar@{-{ >>}}[r]^-{f} & Y \ar[d] \ar[r] & 0\\ 0 \ar[r] & {\operatorname{Ker}(f)} \ar@{=}[r] & {\operatorname{Ker}(f)} \ar@{-{ >>}}[r] & 0}}$$ shows that its right hand square is a pullback. The Huq commutator and the Smith/Pedicchio commutator {#Commutators} ----------------------------------------------------- We work in a semi-abelian category ${\ensuremath{\mathcal{X}}}$. A coterminal pair $$\label{Cospan} \vcenter{\xymatrix{K \ar[r]^-{k} & X & L \ar[l]_-{l}}}$$ of morphisms in ${\ensuremath{\mathcal{X}}}$ [(Huq-)commutes]{} [@BG; @Huq] when there is a (necessarily unique) morphism $\varphi_{k,l}$ such that the diagram $$\xymatrix@!0@=3em{ & K \ar[ld]_{{\lgroup}1_{K},0{\rgroup}} \ar[rd]^-{k} \\ K\times L \ar@{.>}[rr]|-{\varphi_{k,l}} && X\\ & L \ar[lu]^{{\lgroup}0,1_{L}{\rgroup}} \ar[ru]_-{l}}$$ is commutative. We shall only consider the case where $k$ and $l$ are normal monomorphisms (kernels). The [Huq commutator $[k,l]\colon {[K,L]\to X}$ of $k$ and $l$]{} is the smallest normal subobject of $X$ which should be divided out to make $k$ and $l$ commute, so that they do commute if and only if $[K,L]=0$. It may be obtained through the colimit $Q$ of the outer square above, as the kernel of the (normal epi)morphism ${X\to Q}$. The commutator $[K,L]$ becomes the ordinary commutator of normal subgroups $K$ and $L$ in the case of groups, the ideal generated by $KL+LK$ in the case of non-unitary rings, the Lie bracket in the case of Lie algebras, and so on. Consider a pair of equivalence relations $$\label{Category-RG} \xymatrix@!0@=6em{R \ar@<1.5ex>[r]^-{r_{0}} \ar@<-1.5ex>[r]_-{r_{1}} & X \ar[l]|-{{\lgroup}1_{X},1_{X}{\rgroup}} \ar[r]|-{{\lgroup}1_{X},1_{X}{\rgroup}} & S \ar@<1.5ex>[l]^-{s_{0}} \ar@<-1.5ex>[l]_-{s_{1}}}$$ on a common object $X$ and consider the induced pullback of $r_{1}$ and $s_{0}$: $$\label{Pullback-Smith} \vcenter{\xymatrix@!0@=4em{R\times_{X}S \pullback \ar[r]^-{\pi_{S}} \ar[d]_-{\pi_{R}} & S \ar[d]^-{s_{0}} \\ R \ar[r]_-{r_{1}} & X}}$$ The pair $(R,S)$ [(Smith/Pedicchio-)commutes]{} [@Smith; @Pedicchio; @BG] when there is a (necessarily unique) morphism $\theta$ such that the diagram $$\xymatrix@!0@=3em{ & R \ar[ld]_{{\lgroup}1_{R},{\lgroup}1_{X},1_{X}{\rgroup}r_{1}{\rgroup}} \ar[rd]^-{r_{0}} \\ R\times_{X}S \ar@{.>}[rr]|-{\theta} && X\\ & S \ar[lu]^{{\lgroup}{\lgroup}1_{X},1_{X}{\rgroup}s_{0},1_{S}{\rgroup}} \ar[ru]_-{s_{1}}}$$ is commutative. As for the Huq commutator, the [Smith/Pedicchio commutator]{} is the smallest equivalence relation $[R,S]$ on $X$ which, divided out of $X$, makes $R$ and $S$ commute. It can be obtained through a colimit, similarly to the situation above. Thus $R$ and $S$ commute if and only if $[R,S]=\Delta_{X}$, where $\Delta_{X}$ is the smallest equivalence relation on $X$. We say that $R$ is a [central]{} equivalence relation when it commutes with $\nabla_X$, the largest equivalence relation on $X$, so that $[R,\nabla_X]=\Delta_X$. Abelian objects, Beck modules {#Subsection-Abelian} ----------------------------- In a semi-abelian category ${\ensuremath{\mathcal{X}}}$, an object $A$ is said to be [abelian]{} when $[A,A]=0$. The abelian objects of ${\ensuremath{\mathcal{X}}}$ determine a full and reflective subcategory which is denoted ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$. Given any object $X$ of ${\ensuremath{\mathcal{X}}}$, we shall write ${{\langle}X{\rangle}=[X,X]}$, so that we obtain a short exact sequence $$\xymatrix@1{0 \ar[r] & {\langle}X{\rangle}\ar@{{ |>}->}[r] & X \ar@{-{ >>}}[r]^-{\eta_{X}} & {\ensuremath{\mathsf{ab}}}X=X/[X,X] \ar[r] & 0}$$ where $\eta_{X}$ is the $X$-component of the unit $\eta$ of the adjunction $$\xymatrix{{{\ensuremath{\mathcal{X}}}} \ar@<1ex>[r]^-{{\ensuremath{\mathsf{ab}}}} \ar@{}[r]|-{\perp} & {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}}). \ar@<1ex>[l]^-{\supset}}$$ An object in a semi-abelian category is abelian precisely when it admits a (necessarily unique) internal abelian group structure. In fact, ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$ may be viewed as the abelian category of internal abelian groups in ${\ensuremath{\mathcal{X}}}$. For instance, an abelian object in the category of groups is an abelian group, and an abelian associative algebra over a field is a vector space (equipped with a trivial multiplication). Given an object $Z$ of ${\ensuremath{\mathcal{X}}}$, a [$Z$-module]{} or [Beck module over $Z$]{} is an abelian group in the slice category ${\ensuremath{\mathcal{X}}}/Z$. Thus a $Z$-module $(f,m,s)$ consists of a morphism $f\colon {X\to Z}$ in ${\ensuremath{\mathcal{X}}}$, equipped with a multiplication $m$ and a unit $s$ as in the diagrams $$\vcenter{\xymatrix@!0@R=3em@C=2em{{\operatorname{Eq}(f)} \ar[rr]^-{m} \ar[rd] && X \ar[ld]^-{f} \\ & Z}} \qquad \qquad \vcenter{\xymatrix@!0@R=3em@C=2em{Z \ar[rr]^-{s} \ar@{=}[rd] && X \ar[ld]^-{f} \\ & Z}}$$ satisfying the usual axioms. In particular we obtain a split short exact sequence $$\xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r]^-{\operatorname{ker}f} & X \ar@<-.5ex>@{ >>}[r]_-{f} & Z \ar[r] \ar@<-.5ex>[l]_-{s} & 0}$$ where $A$ is an abelian object in ${\ensuremath{\mathcal{X}}}$ and $f$ is split by $s$. Furthermore, the morphism $f$ satisfies $[{\operatorname{Eq}(f )},{\operatorname{Eq}(f)}]=\Delta_{X}$. Conversely, given the splitting $s$ of $f$, this latter condition makes it possible to recover the multiplication $m$. Hence, for split epimorphisms in a semi-abelian category, “being a Beck module” is a property; the entire module structure is contained in the splitting. Using the equivalence between split epimorphisms and internal actions [@Bourn-Janelidze:Semidirect], we can replace $X$ with a semi-direct product ${(A,\xi)\rtimes Z}$. By the above, modules are “abelian actions”. For simplicity, we denote a $Z$-module by its induced $Z$-algebra $(A,\xi)$. For us, the most important case arises when the $Z$-module structure on $A$ is the trivial one, denoted $(A,\tau)$: then $A$ is just an abelian object, the semidirect product ${(A,\tau)\rtimes Z}$ is $A\times Z$ and $f$ is the product projection $\operatorname{pr}_{Z}\colon {A\times Z\to Z}$. Connected components {#Connected-components} -------------------- In a category ${\ensuremath{\mathcal{X}}}$, two objects are [connected]{} when there exists a (finite) zigzag of morphisms between them. This defines an equivalence relation between the objects of ${\ensuremath{\mathcal{X}}}$, of which the equivalence classes form the set $\pi_{0}({\ensuremath{\mathcal{X}}})$ of [connected components]{} of ${\ensuremath{\mathcal{X}}}$. In general $\pi_{0}({\ensuremath{\mathcal{X}}})$ may not be a small set, and even in the two situations where we shall use this construction (Subsection \[Torsors\] and Definition \[Definition-Centr\]) it will a priori not be clear whether or not the result is not a proper class. In fact, even when it *is* a proper class, this has no significant effect at all on the theory we develop, so we decided not to go into this question any further. Additionally, in the monadic case the smallness of the cohomology groups follows from the interpretation in terms of Barr–Beck cohomology. A lemma on double split epimorphisms ------------------------------------ By a result in [@Bourn1996], a finitely complete category is [naturally Mal’tsev]{} [@Johnstone:Maltsev] when, given a split epimorphism of split epimorphisms as in $$\label{Double-Split-Epi} \vcenter{\xymatrix@=3em{A_{1} \ar@<-.5ex>[r]_-{f_{1}} \ar@<-.5ex>[d]_-{a} & B_1 \ar@<-.5ex>[d]_-{b} \ar@<-.5ex>[l]_-{\overline{f_{1}}}\\ A_{0} \ar@<-.5ex>[u]_-{\overline{a}} \ar@<-.5ex>[r]_-{f_{0}} & B_{0} \ar@<-.5ex>[u]_-{\overline{b}} \ar@<-.5ex>[l]_-{\overline{f_{0}}}}}$$ (all squares commute), if the square is a (down-right) pullback of split epimorphisms, then it is an (up-left) pushout of split monomorphisms. As a consequence we obtain the following lemma (see also [@MFVdL2] and [@EGoeVdL]). \[Lemma-Naturally-Maltsev\] In a naturally Mal’tsev category, given a double split epimorphism such as , the universally induced comparison morphism $${\lgroup}a,f_{1}{\rgroup}\colon A_{1}\to A_{0}\times_{B_{0}}B_{1}$$ to the pullback of $f_{0}$ and $b$ is a split epimorphism, with a unique splitting $$\nu\colon {A_{0}\times_{B_{0}}B_{1}\to A_{1}}$$ such that $\overline{a}=\nu {\lgroup}1_{A_{0}},\overline{b}f_{0}{\rgroup}$ and $\overline{f_{1}}=\nu{\lgroup}\overline{f_{0}}b,1_{B_{1}}{\rgroup}$. It is well known that every additive category is naturally Mal’tsev. In particular, for any semi-abelian category ${\ensuremath{\mathcal{X}}}$, the above lemma is valid in the abelian category ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$. The von Neumann construction of the finite ordinals --------------------------------------------------- We shall write ${0=\emptyset}$ and $n=\{0,\dots, n-1\}$ for $n\geq 1$. We also write $2^{n}$ for the power-set of $n$, considered as a category of which an object is a subset of $n$, and an arrow ${I\to J}$ is an inclusion $I\subseteq J$. Higher arrows {#HDA} ------------- Let ${\ensuremath{\mathcal{X}}}$ be any category. The category ${\ensuremath{\mathsf{Arr}^{n}}}({\ensuremath{\mathcal{X}}})$ consists of [$n$-fold arrows]{} in ${\ensuremath{\mathcal{X}}}$: ${\ensuremath{\mathsf{Arr}}}^{0}({\ensuremath{\mathcal{X}}})={\ensuremath{\mathcal{X}}}$, while ${\ensuremath{\mathsf{Arr}}}^{1}({\ensuremath{\mathcal{X}}})={\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathcal{X}}})$ is the category of arrows in ${\ensuremath{\mathcal{X}}}$ and ${\ensuremath{\mathsf{Arr}^{n+1}}}({\ensuremath{\mathcal{X}}})={\ensuremath{\mathsf{Arr}}}({\ensuremath{\mathsf{Arr}^{n}}}({\ensuremath{\mathcal{X}}}))$. The category of arrows in ${\ensuremath{\mathcal{X}}}$ is the functor category ${{\ensuremath{\mathsf{Fun}}}(2^{\operatorname{op}},{\ensuremath{\mathcal{X}}})={\ensuremath{\mathcal{X}}}^{2^{\operatorname{op}}}}$. Similarly, any $n$-fold arrow $F$ in ${\ensuremath{\mathcal{X}}}$ may be viewed as an “$n$-fold cube with chosen directions”, a functor $F\colon{(2^{n})^{\operatorname{op}}\to {\ensuremath{\mathcal{X}}}}$, and any morphism of $n$-fold arrows as a natural transformation between such functors. If $F$ is an $n$-fold arrow and $I$ and $J$ are subsets of $n$ such that $I\subseteq J$, we shall write $F_{I}=F(I)$ for the value of $F$ in $I$ and $f^{J}_{I}\colon{F_{J}\to F_{I}}$ for the value of $F$ in the morphism induced by the inclusion $I\subseteq J$. When $I=J\setminus \{i\}$ we write $f_{i}\colon{F_{J}\to F_{I}}$ for $f^{J}_{I}$. An $n$-fold arrow given as a functor $F\colon{(2^{n})^{\operatorname{op}}\to {\ensuremath{\mathcal{X}}}}$ can be seen as an arrow between $(n-1)$-fold arrows $F\colon \operatorname{dom}F \to \operatorname{cod}F$, where its domain $\operatorname{dom}F$ is determined by the restriction of $F$ to all $I\subseteq n$ which contain $n-1$, and its codomain $\operatorname{cod}F$ by the restriction of $F$ to all $I\subseteq n$ which do not contain $n-1$. Thus, if ${n\geq 2}$, we may see $F$ as a commutative square $$\label{Double-Extension} \vcenter{\xymatrix{X \ar[r]^-{c} \ar[d]_-{d} & C \ar[d]^-{g}\\ D \ar[r]_-{f} & Z}}$$ in ${\ensuremath{\mathsf{Arr}}}^{n-2}({\ensuremath{\mathcal{X}}})$ or, equivalently, a morphism $(c,f)\colon d\to g$ of ${\ensuremath{\mathsf{Arr}}}^{n-1}({\ensuremath{\mathcal{X}}})$. Given an $n$-fold arrow $F\colon{(2^{n})^{\operatorname{op}}\to {\ensuremath{\mathcal{X}}}}$, we can always consider the restriction of this diagram to the subcategory $2^{n}\setminus \{n\}$; it is the $n$-fold cube $F$ without its “initial object” $F_{n}$. When it exists, write $({\ensuremath{\mathsf{L}}}F,(\operatorname{pr}_{i})_{i\in n})$ for the limit of this diagram, and $$l_{F}={\lgroup}f_{0},\dots,f_{n-1}{\rgroup}\colon{F_{n}\to {\ensuremath{\mathsf{L}}}F}$$ for the universally induced comparison morphism. \[Lemma-L-pullbacks\] Given $n\geq 2$, if $F$ is an $n$-fold arrow considered as a square  of $(n-1)$-fold arrows, then ${\ensuremath{\mathsf{L}}}F$ may be obtained as ${\ensuremath{\mathsf{L}}}G$, where the $(n-1)$-fold arrow $G$ is ${\lgroup}d,c{\rgroup}\colon X\to D\times_{Z}C$, induced by the pullback of $f$ and $g$. Furthermore, $l_{F}=l_{G}$. This is part of the proof of Proposition 1.16 in [@EGoeVdL]. Higher cubic extensions {#Extensions} ----------------------- Let ${\ensuremath{\mathcal{X}}}$ be a semi-abelian category. A [zero-cubic extension]{} in ${\ensuremath{\mathcal{X}}}$ is an object of ${\ensuremath{\mathcal{X}}}$ and a [one-cubic extension]{} is a regular epimorphism in ${\ensuremath{\mathcal{X}}}$. For $n\geq 2$, an [$n$-cubic extension]{} is a commutative square  in ${\ensuremath{\mathsf{Arr}}}^{n-2}({\ensuremath{\mathcal{X}}})$ such that the morphisms $c$, $d$, $f$, $g$ and the universally induced comparison morphism ${\lgroup}d,c{\rgroup}\colon{X\to D\times_Z C}$ to the pullback of $f$ with $g$ are $(n-1)$-cubic extensions. The $n$-cubic extensions determine a full subcategory ${\ensuremath{\mathsf{Ext}^{n}}}({\ensuremath{\mathcal{X}}})$ of ${\ensuremath{\mathsf{Arr}^{n}}}({\ensuremath{\mathcal{X}}})$, and ${\ensuremath{\mathsf{Ext}}}({\ensuremath{\mathcal{X}}})={\ensuremath{\mathsf{Ext}}}^{1}({\ensuremath{\mathcal{X}}})$. [@EGoeVdL]\[Limit-Characterisation-Extensions\] Given any $n$-fold arrow $F$ in a regular category, the following are equivalent: 1. $F$ is an $n$-cubic extension; 2. for all $\emptyset\neq I\subseteq n$, the morphism ${F_{I}\to\lim_{J\subsetneq I} F_{J}}$ is a regular epimorphism. In particular, the induced comparison $l_{F}={\lgroup}f_{0},\dots,f_{n-1}{\rgroup}\colon{F_{n}\to {\ensuremath{\mathsf{L}}}F}$ is regular epimorphic. In a Mal’tsev category, a double split epimorphism such as  above is always a two-cubic extension. That is to say, the induced comparison morphism ${\lgroup}a,f_{1}{\rgroup}$ may not be a split epimorphism as in Lemma \[Lemma-Naturally-Maltsev\], but it will certainly be a regular epimorphism. More generally, any split epimorphism between one-cubic extensions is a two-cubic extension, as follows from [@Carboni-Kelly-Pedicchio Theorem 5.7]. Extensions as diagrams of short exact sequences {#3^n diagrams} ----------------------------------------------- In what follows we view higher extensions slightly differently: as diagrams of short exact sequences, such as the one displayed in Figure \[3x3 diag\] on page  and in Figure \[Figure-Direction\] on page . Consider the ordinal $3$ as a category $0\to1\to 2$ and, for $n\geq 1$, its $n$-th power $3^{n}=3\times \cdots \times 3$. The category $ 3^n$ has initial object $ i_n= (0,\dots,0)$ and terminal object $ t_n= (2,\dots ,2)$. Moreover, it has an embedding $$\alpha_{e,i}\colon 3\to 3^n\colon k\mapsto (e_{1},\dots,e_{i-1},k,e_{i+1},\dots, e_{n})$$ parallel to the $i$-th coordinate axis, for each object $e$ of $3^{n}$. Now, given objects $Z$ and $A$ in ${\ensuremath{\mathcal{X}}}$, an [$n$-fold extension]{} ([under $A$ and over $Z$]{}, or [of $Z$ by $A$]{}) in ${\ensuremath{\mathcal{X}}}$ is a functor $E\colon (3^n)^{\operatorname{op}}\to {\ensuremath{\mathcal{X}}}$ which sends $i_n$ to $Z$ and $t_n$ to $A$, and such that each composite $$\xymatrix@R=5ex@C=3em{ 3^{\operatorname{op}} \ar[r]^-{\alpha_{e,i}^{\operatorname{op}}} & (3^n)^{\operatorname{op}} \ar[r]^-{E} & {\ensuremath{\mathcal{X}}}}$$ is a short exact sequence. For example, a one-fold extension under $A$ and over $Z$ is just a short exact sequence $ A=E_2 \to E_1 \to E_0=Z$. A two-fold (or [double]{}) extension under $A$ and over $Z$ is a [$3\times 3$-diagram]{} [@Bourn2001] as in Figure \[3x3 diag\], in which each row and column is short exact: $$\xymatrix@R=5ex@C=3em{ A=E_{2,2} \ar@{{ |>}->}[r] \ar@{{ |>}->}[d] & E_{1,2} \ar@{-{ >>}}[r] \ar@{{ |>}->}[d] & E_{0,2} \ar@{{ |>}->}[d] \\ E_{2,1} \ar@{{ |>}->}[r] \ar@{-{ >>}}[d] & E_{1,1} \ar@{-{ >>}}[r] \ar@{-{ >>}}[d] & E_{0,1} \ar@{-{ >>}}[d] \\ E_{2,0} \ar@{{ |>}->}[r] & E_{1,0} \ar@{-{ >>}}[r] & Z=E_{0,0} }$$ Figure \[Figure-Direction\] displays a $3$-fold extension as a [$3\times 3\times 3$-diagram]{}. In general, an $n$-fold extension is the same thing as a [$3^{n}$-diagram]{} in ${\ensuremath{\mathcal{X}}}$. We write ${\ensuremath{3^{n}\text{-}\mathsf{Diag}}}({\ensuremath{\mathcal{X}}})$ for the category of $n$-fold extensions in ${\ensuremath{\mathcal{X}}}$, considered as a full subcategory of ${\ensuremath{\mathsf{Fun}}}((3^{n})^{\operatorname{op}},{\ensuremath{\mathcal{X}}})$. The natural embedding $2^{n}\to 3^{n}$ induces a forgetful functor $$(-)|_{2^{n}}\colon{\ensuremath{3^{n}\text{-}\mathsf{Diag}}}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ext}}}^{n}({\ensuremath{\mathcal{X}}})\colon E\mapsto F=E|_{2^{n}}.$$ We shall, however, always think of an $n$-cubic extension as being part of some $3^{n}$-diagram. \[3x3 vs ext\] For any $n\geq 1$, the forgetful functor ${\ensuremath{3^{n}\text{-}\mathsf{Diag}}}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ext}}}^{n}({\ensuremath{\mathcal{X}}})$ is an equivalence of categories. By induction on $n$, we prove that an $n$-fold arrow underlies a $3^{n}$-diagram if and only if it is an $n$-cubic extension. This then shows that the above functor is well defined and (essentially) surjective. The case $n=1$ is clear. Suppose now that $n\geq 1$. Then a $3^{n+1}$-diagram, being a short exact sequence of $3^{n}$-diagrams, corresponds to a short exact sequence of $n$-cubic extensions by the induction hypothesis. By Proposition 3.9 in [@EGVdL], the exactness of this sequence is equivalent to its cokernel piece being an $(n+1)$-cubic extension. Moreover, the functor is fully faithful, because any morphism between the $n$-cubic extensions underlying two given $3^{n}$-diagrams extends uniquely to their chosen kernels. Depending on the situation, we may prove categorical properties of ${\ensuremath{3^{n}\text{-}\mathsf{Diag}}}({\ensuremath{\mathcal{X}}})$ for ${\ensuremath{\mathsf{Ext}}}^{n}({\ensuremath{\mathcal{X}}})$ and vice versa. Augmented simplicial objects ---------------------------- Recall that the [augmented simplicial category]{} $\Delta^{+}$ has finite ordinals $n\geq 0$ for objects and order preserving functions for morphisms. The category ${\ensuremath{\mathsf{s}}}^{+}({\ensuremath{\mathcal{X}}})$ of [augmented simplicial objects]{} and augmented simplicial morphisms in a category ${\ensuremath{\mathcal{X}}}$ is the functor category ${\ensuremath{\mathsf{Fun}}}((\Delta^{+})^{\operatorname{op}},{\ensuremath{\mathcal{X}}})$. An augmented simplicial object ${\ensuremath{\mathbb{X}}}\colon {(\Delta^{+})^{\operatorname{op}}\to {\ensuremath{\mathcal{X}}}}$ is usually considered as a sequence of objects $({\ensuremath{\mathbb{X}}}_{n})_{n\geq -1}$, with [face operators]{} ${\partial}_{i}\colon {{\ensuremath{\mathbb{X}}}_{n}\to {\ensuremath{\mathbb{X}}}_{n-1}}$ and [degeneracy operators]{} $\sigma_{i}\colon {{\ensuremath{\mathbb{X}}}_{n}\to {\ensuremath{\mathbb{X}}}_{n+1}}$ for ${n\geq i\geq 0}$, subject to the simplicial identities $$\begin{aligned} {\partial}_{i}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{j} &={\partial}_{j-1}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{i}\quad \text{if $i<j$}\\ \sigma_{i}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\sigma_{j} &= \sigma_{j+1}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\sigma_{i}\quad \text{if $i\leq j$} \end{aligned} \qquad\text{and}\qquad {\partial}_{i}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\sigma_{j}=\begin{cases}\sigma_{j-1}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{i} & \text{if $i<j$} \\ 1 & \text{if $i=j$ or $i=j+1$}\\ \sigma_{j}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{i-1} & \text{if $i>j+1$.}\end{cases}$$ All simplicial objects we shall be considering in this text will come equipped with some augmentation, even when we occasionally drop the word “augmented”. Truncations and coskeleta {#Truncations} ------------------------- For $n\geq 0$, let $\Delta^{+}_{n}$ denote the full subcategory of $\Delta^{+}$ determined by the ordinals $i\leq n$. The functor category $${\ensuremath{\mathsf{SArr}}}^{n}({\ensuremath{\mathcal{X}}})={\ensuremath{\mathsf{Fun}}}((\Delta^{+}_{n})^{\operatorname{op}},{\ensuremath{\mathcal{X}}})$$ is the category of [$(n-1)$-truncated simplicial objects]{} in ${\ensuremath{\mathcal{X}}}$. Indeed, as soon as ${\ensuremath{\mathcal{X}}}$ is finitely complete, there is the adjunction $$\xymatrix@C=50pt{{{\ensuremath{\mathsf{s}}}^{+}({\ensuremath{\mathcal{X}}})} \ar@<1ex>[r]^-{{\ensuremath{\mathsf{tr}}}_{n-1}} \ar@{}[r]|-{\perp} & {\ensuremath{\mathsf{SArr}}}^{n}({\ensuremath{\mathcal{X}}}), \ar@<1ex>[l]^-{{\ensuremath{\mathsf{cosk}}}_{n-1}}}$$ where the truncation functor ${\ensuremath{\mathsf{tr}}}_{n-1}$ is given by composition of a simplicial object with the inclusion $\Delta^{+}_{n}\subseteq \Delta^{+}$, and its right adjoint ${\ensuremath{\mathsf{cosk}}}_{n-1}$ by right Kan extension along this functor. More explicitly, a coskeleton of an $(n-1)$-truncated simplicial object may be computed using iterated simplicial kernels (see the next subsection). Clearly, ${\ensuremath{\mathsf{tr}}}_{n-1}{\ensuremath{\mathsf{cosk}}}_{n-1}=1_{{\ensuremath{\mathsf{SArr}}}^{n}({\ensuremath{\mathcal{X}}})}$. Conversely, a coskeleton of an $(n-1)$-truncated simplicial object contains no information above simplicial degree $n-1$; given any simplicial object ${\ensuremath{\mathbb{X}}}$, we can remove all higher-dimensional information by applying the functor ${\ensuremath{\mathsf{Cosk}}}_{n-1}={\ensuremath{\mathsf{cosk}}}_{n-1}{\ensuremath{\mathsf{tr}}}_{n-1}\colon{{\ensuremath{\mathsf{s}}}^{+}({\ensuremath{\mathcal{X}}})}\to {{\ensuremath{\mathsf{s}}}^{+}({\ensuremath{\mathcal{X}}})}$ to it. Any $(n-1)$-truncated simplicial object may be considered as an $n$-fold arrow, through composition with the functor $${\ensuremath{\mathsf{a}}}_{n}\colon 2^{n}\to \Delta^{+}_{n}$$ which maps a set $I\subseteq n$ to the associated ordinal $|I|$, and an inclusion $I\subseteq J$ to the corresponding order-preserving map ${|I|\to |J|}$. This defines a faithful functor $${\ensuremath{\mathsf{arr}}}_{n}={\ensuremath{\mathsf{Fun}}}(-,{\ensuremath{\mathsf{a}}}_{n})\colon{{\ensuremath{\mathsf{SArr}}}^{n}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Arr}}}^{n}({\ensuremath{\mathcal{X}}})}.$$ (An ${(n-1)}$-truncated simplicial object has the additional structure of the degeneracies: a morphism of $n$-fold arrows between two given $(n-1)$-truncated simplicial objects need not commute with the degeneracy operators, and furthermore its components at two given sets of the same size need not coincide.) Hence, if $X$ denotes the $n$-fold arrow underlying the $(n-1)$-truncation of a simplicial object ${\ensuremath{\mathbb{X}}}$, then ${X_{I}={\ensuremath{\mathbb{X}}}(|I|)={\ensuremath{\mathbb{X}}}_{|I|-1}}$ and, in particular, $X_{n}={\ensuremath{\mathbb{X}}}_{n-1}$. Note how the difference in font style allows to distinguish between the absolute degree $n$ and the simplicial degree $n-1$. In presence of enough projectives, we may now characterise higher extensions as follows. \[Projective-Characterisation-Extensions\] Given any $n$-fold arrow $F$ in a regular category with enough projectives, the following are equivalent: 1. $F$ is an $n$-cubic extension; 2. for any ${(n-1)}$-truncated degreewise projective simplicial object ${\ensuremath{\mathbb{X}}}$, any collection of arrows ${(X_{J}\to F_{J})_{|J|\leq i}}$ satisfying the conditions of a morphism of $n$-fold arrows up to absolute degree $i\in n$ extends to an actual morphism of $n$-fold arrows ${X\to F}$. We first use induction on $i$ to prove that (i) implies (ii). Suppose a collection ${(X_{J}\to F_{J})_{|J|\leq i}}$ like in the statement is given and let $I\subseteq n$ be such that $|I|=i+1$. Then we obtain the needed morphism ${X_{I}\to F_{I}}$ as the dotted lifting in the diagram $$\xymatrix{&&F_{I} \ar@{-{ >>}}[d]\\ X_{I} \ar@{.>}[rru] \ar[r] & \lim_{J\subsetneq I} X_{J} \ar[r] & \lim_{J\subsetneq I} F_{J}}$$ —which exists because $X_{I}$ is projective by assumption, while the right hand side vertical arrow is a regular epimorphism by Proposition \[Limit-Characterisation-Extensions\]. To see that (ii) implies (i) we again use Proposition \[Limit-Characterisation-Extensions\]. This time we have to show that any morphism ${P\to \lim_{J\subsetneq I} F_{J}}$ for $P$ projective lifts to a morphism ${P\to F_{I}}$. We simply use the $(n-1)$-truncation of the constant simplicial object $P$, and extend the collection of arrows induced by the given arrow to a morphism of $n$-fold arrows to obtain the needed lifting. Simplicial kernels ------------------ Let $$(f_i\colon {X\to Y})_{i\in n}$$ be a sequence of $n$ morphisms in a finitely complete category ${\ensuremath{\mathcal{X}}}$. A [simplicial kernel]{} of $(f_0,\ldots,f_{n-1})$ is a sequence $$(k_i\colon {K\to X})_{i\in n+1}$$ of $n+1$ morphisms in ${\ensuremath{\mathcal{X}}}$ satisfying $f_ik_j=f_{j-1}k_i$ for $0\leq i<j\leq n$, which is universal with respect to this property. It may be computed as a limit in ${\ensuremath{\mathcal{X}}}$. We need the following lemma, which is probably well known: \[Lemma coskeleton pullback\] Let $(f_i\colon {X\to Y})_{i\in n}$ and $(f'_i\colon {X'\to Y'})_{i\in n}$ be two sequences of $n$ morphisms in a finitely complete category ${\ensuremath{\mathcal{X}}}$, and consider morphisms $\chi\colon {X\to X'}$ and $\upsilon\colon {Y\to Y'}$ for which all squares in any diagram $$\vcenter{\xymatrix{ X \ar[r]^-{f_{i}} \ar[d]_-{\chi} & Y \ar[d]^-{\upsilon}\\ X' \ar[r]_-{f'_{i}} & Y'}}$$ are pullbacks. Then all of the induced squares between the respective simplicial kernels of $(f_{i})_{i\in n}$ and $(f'_{i})_{i\in n}$ are pullbacks as well. It suffices to give a formal proof in ${\ensuremath{\mathsf{Set}}}$. Consider $x_{m}$ in $X$ and $(x'_{0},\dots,x'_{n})$ in the simplicial kernel $K'$ of $(f'_{i})_{i\in n}$ such that $\chi(x_{m})=x'_{m}$. If $(x_{0},\dots,x_{n})$ is an element of the simplicial kernel $K$ of $(f_{i})_{i\in n}$, then $x_{j}$ necessarily satisfies $$\upsilon(f_{m}(x_{j}))=\upsilon(f_{j-1}(x_{m}))=f'_{j-1}(\chi(x_{m}))=f'_{j-1}(x'_{m})=f'_{m}(x'_{j})$$ in case $m<j$, and $$\upsilon(f_{m-1}(x_{j}))=\upsilon(f_{j}(x_{m}))=f'_{j}(\chi(x_{m}))=f'_{j}(x'_{m})=f'_{m-1}(x'_{j})$$ when $m>j$. This completely determines $(x_{0},\dots,x_{n})$ via the pullback property which we assume to hold. It is also clear that any tuple $(x_{0},\dots,x_{n})$ thus obtained is indeed an element of $K$. In fact, it turns out to be precisely the needed unique element for which $\chi(x_{0},\dots,x_{n})=(x'_{0},\dots,x'_{n})$ and $k_{m}(x_{0},\dots,x_{n})=x_{m}$. When ${\ensuremath{\mathbb{X}}}$ is a simplicial object and $n\geq 0$, we write $$({\partial}_{i}\colon{\triangle}({\ensuremath{\mathbb{X}}},n)\to {\ensuremath{\mathbb{X}}}_{n-1})_{i\in n+1}$$ for the simplicial kernel of the faces $({\partial}_{i}\colon {{\ensuremath{\mathbb{X}}}_{n-1}\to {\ensuremath{\mathbb{X}}}_{n-2}})_{i\in n}$. The object ${\triangle}({\ensuremath{\mathbb{X}}},n)$ consists of [$n$-cycles]{} in ${\ensuremath{\mathbb{X}}}$. For instance, the object ${\triangle}({\ensuremath{\mathbb{X}}},2)$ of $2$-cycles in ${\ensuremath{\mathbb{X}}}$ contains empty triangles: $$\vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\beta}\\ {\cdot} \ar[ru]^-{\alpha} \ar[rr]_-{\gamma} && {\cdot}}}$$ Note that ${\triangle}({\ensuremath{\mathbb{X}}},n)={\ensuremath{\mathsf{L}}}({\ensuremath{\mathsf{tr}}}_{n}{\ensuremath{\mathbb{X}}})$. Clearly ${{\triangle}({\ensuremath{\mathbb{X}}},1)={\operatorname{Eq}({\partial}_0)}}$; we also write ${\triangle}({\ensuremath{\mathbb{X}}},0)$ for ${\ensuremath{\mathbb{X}}}_{-1}$. As mentioned in Subsection \[Truncations\], any ${(n-1)}$-truncated simplicial object $X$ in ${\ensuremath{\mathcal{X}}}$ may be universally extended to an $n$-truncated simplicial object. Its initial object and morphisms, in (absolute!) degree $n+1$, are given by the simplicial kernel $(k_{i}\colon{K\to X_{n}})_{i\in n+1}$ of the initial morphisms ${(x_{i}\colon {X_{n}\to X_{n-1}})_{i\in n}}$ of $X$. The degeneracies ${(\sigma_{j}\colon X_{n}\to K)_{j\in n}}$ are induced by the simplicial identities $$k_{i}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\sigma_{j}=\begin{cases}\sigma_{j-1}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}x_{i} & \text{if $i<j$} \\ 1_{X_{n}} & \text{if $i=j$ or $i=j+1$}\\ \sigma_{j}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}x_{i-1} & \text{if $i>j+1$}\end{cases}$$ of $X$ and the universal property of the simplicial kernel. Repeating this construction indefinitely gives the $(n-1)$-coskeleton of $X$. Resolutions {#Resolutions} ----------- An augmented simplicial object ${\ensuremath{\mathbb{X}}}$ in a regular category is called [acyclic]{} or [a resolution (of ${\ensuremath{\mathbb{X}}}_{-1}$)]{} when for every $n\geq 0$, the comparison morphism $${\lgroup}{\partial}_{i}{\rgroup}_{i}\colon {\ensuremath{\mathbb{X}}}_{n}\to {\triangle}({\ensuremath{\mathbb{X}}},n)$$ is a regular epimorphism. (Every $n$-cycle is a boundary of an $n$-simplex.) As explained in [@EGoeVdL], in a semi-abelian category this is the case precisely when all the truncations of ${\ensuremath{\mathbb{X}}}$, considered as higher arrows, are extensions. For this reason we may sometimes also call a truncated simplicial resolution an extension. The simplicial objects ${\ensuremath{\mathbb{K}}}(A,n)$ and ${\ensuremath{\mathbb{K}}}(Z,A,n)$ ---------------------------------------------------------------------------------------------- Let $A$ be an abelian group in a Barr-exact category ${\ensuremath{\mathcal{X}}}$ and take $n\geq 1$. The augmented simplicial object ${\ensuremath{\mathbb{K}}}(A,n)$ is the coskeleton of the $(n+1)$-truncated simplicial object $$\vcenter{\xymatrix@R=20pt@C=35pt{\scriptstyle{n+1} & \scriptstyle{n} & \scriptstyle{n-1} & \scriptstyle{n-2} \ar@{}[r]|-{\cdots} & \scriptstyle{0} & \scriptstyle{-1}\\ A^{n+1} \ar@<2.33ex>[r]^-{{\partial}_{n+1}} \ar@<1.16ex>[r]|-{\operatorname{pr}_{n}} \ar@<-2.33ex>[r]_-{\operatorname{pr}_{0}}^-{\vdots} & A \ar@<1.75ex>[r]^-{!} \ar@<-1.75ex>[r]_-{!}^-{\vdots} & 1 \ar@{=}@<1.75ex>[r] \ar@{=}@<-1.75ex>[r]^-{\vdots} & 1 \ar@{}[r]|-{\cdots} & 1 \ar@{=}[r] & 1}}$$ with the $A$ in simplicial degree $n$ (in absolute degree $n+1$), where the degeneracies ${1\to A}$ are determined by the neutral element $0$ of $A$ and ${\partial}_{n+1}$ is equal to $$(-1)^{n}\sum^{n}_{i=0}(-1)^{i}\operatorname{pr}_{i}.$$ When the category is a slice ${\ensuremath{\mathcal{X}}}/Z$ over an object $Z$ in a semi-abelian category ${\ensuremath{\mathcal{X}}}$ and $(A,\xi)$ is a $Z$-module, the simplicial object ${\ensuremath{\mathbb{K}}}((A,\xi),n)$, considered as a diagram in ${\ensuremath{\mathcal{X}}}$, takes the following shape: $$\resizebox{\textwidth}{!}{ $\vcenter{\xymatrix@R=20pt@C=50pt{\scriptstyle{n+1} & \scriptstyle{n} & \scriptstyle{n-1} & \scriptstyle{n-2} \ar@{}[r]|-{\cdots} & \scriptstyle{0} & \scriptstyle{-1}\\ (A,\xi)^{n+1} \rtimes Z \ar@<2.33ex>[r]^-{{\partial}_{n+1}\rtimes 1_Z} \ar@<1.16ex>[r]|-{\operatorname{pr}_{n}\rtimes 1_Z} \ar@<-2.33ex>[r]_-{\operatorname{pr}_{0}\rtimes 1_Z}^-{\vdots} & (A,\xi) \rtimes Z \ar@<1.75ex>[r]^-{f} \ar@<-1.75ex>[r]_-{f}^-{\vdots} & Z \ar@{=}@<1.75ex>[r] \ar@{=}@<-1.75ex>[r]^-{\vdots} & Z \ar@{}[r]|-{\cdots} & Z \ar@{=}[r] & Z}} $}$$ In case $\xi$ is the trivial module structure $\tau$, we obtain $$\resizebox{\textwidth}{!}{ $\vcenter{\xymatrix@C=50pt{\scriptstyle{n+1} & \scriptstyle{n} & \scriptstyle{n-1} & \scriptstyle{n-2} \ar@{}[r]|-{\cdots} & \scriptstyle{0} & \scriptstyle{-1}\\ A^{n+1} \times Z \ar@<2.33ex>[r]^-{{\partial}_{n+1}\times 1_{Z}} \ar@<1.16ex>[r]|-{\operatorname{pr}_{n}\times 1_{Z}} \ar@<-2.33ex>[r]_-{\operatorname{pr}_{0}\times 1_{Z}}^-{\vdots} & A \times Z \ar@<1.75ex>[r]^-{\operatorname{pr}_{Z}} \ar@<-1.75ex>[r]_-{\operatorname{pr}_{Z}}^-{\vdots} & Z \ar@{=}@<1.75ex>[r] \ar@{=}@<-1.75ex>[r]^-{\vdots} & Z \ar@{}[r]|-{\cdots} & Z \ar@{=}[r] & Z}} $}$$ with ${\partial}_{n+1}$ as above and degeneracies ${\lgroup}0,1_{Z}{\rgroup}\colon{Z\to A\times Z}$. Given any object $Z$ and any abelian object $A$, we shall write ${\ensuremath{\mathbb{K}}}(Z,A,n)$ for this simplicial object in ${\ensuremath{\mathcal{X}}}$. In particular, ${\ensuremath{\mathbb{K}}}(0,A,n)={\ensuremath{\mathbb{K}}}(A,n)$. (Exact) fibrations ------------------ Let ${\ensuremath{\mathbb{X}}}$ be a simplicial object in a finitely complete category ${\ensuremath{\mathcal{X}}}$ and consider $n\geq 2$ and $0\leq i\leq n$. The [object of $(n,i)$-horns in ${\ensuremath{\mathbb{X}}}$]{} is an object ${\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n)$ together with morphisms $x_{j}\colon{{\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n)\to {\ensuremath{\mathbb{X}}}_{n-1}}$ for $i\neq j\in n+1$ satisfying $$\text{${\partial}_{j}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}x_{k}={\partial}_{k-1}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}x_{j}$ for all $j<k$ with $j$, $k\neq i$}$$ which is universal with respect to this property; also ${\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{0}({\ensuremath{\mathbb{X}}},1)={\ensuremath{\mathbb{X}}}_{0}={\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{1}({\ensuremath{\mathbb{X}}},1)$. For instance, the object ${\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{1}({\ensuremath{\mathbb{X}}},2)$ of $(2,1)$-horns in ${\ensuremath{\mathbb{X}}}$ $$\vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\beta}\\ {\cdot} \ar[ru]^-{\alpha} && {\cdot}}}$$ contains “composable pairs of arrows”. We write $$\widehat f_{i}={\lgroup}f_j{\rgroup}_{i\neq j\in n+1}\colon {W\to {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n)}$$ for the morphism induced by a family $(f_{j}\colon W\to {\ensuremath{\mathbb{X}}}_{n-1})_{i\neq j\in n+1}$ in which the morphism $f_{i}$ is missing. Now suppose that ${\ensuremath{\mathcal{X}}}$ is a regular category. A simplicial morphism ${\ensuremath{\mathbb{f}}}\colon{{\ensuremath{\mathbb{X}}}\to {\ensuremath{\mathbb{Y}}}}$ satisfies the [Kan condition]{} (respectively satisfies the Kan condition [exactly]{}) in degree $n$ for $i$ when the morphism $${\lgroup}\widehat{\partial}_{i}, {\ensuremath{\mathbb{f}}}_{n}{\rgroup}\colon {\ensuremath{\mathbb{X}}}_{n}\to {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n)\times_{\smallhorn^{i}({\ensuremath{\mathbb{Y}}},n)}{\ensuremath{\mathbb{Y}}}_{n}$$ universally induced by the square $$\vcenter{\xymatrix{{\ensuremath{\mathbb{X}}}_{n} \ar[d]_-{\widehat{\partial}_{i}} \ar[r]^-{{\ensuremath{\mathbb{f}}}_{n}} & {\ensuremath{\mathbb{Y}}}_{n} \ar[d]^-{\widehat{\partial}_{i}}\\ {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n) \ar[r]_{\smallhorn^{i}({\ensuremath{\mathbb{f}}},n)} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{Y}}},n)}} \qquad\qquad$$ is a regular epimorphism (respectively an isomorphism). The morphism ${\ensuremath{\mathbb{f}}}$ is called a [fibration]{} when it satisfies the Kan condition for all $n\geq 1$ and all $i$. A fibration is [exact]{} in degrees larger than $n$ when the Kan condition is satisfied exactly in simplicial degrees larger than $n$ for all $i$. A regular category is Mal’tsev if and only if every simplicial object is Kan: every morphism ${{\ensuremath{\mathbb{X}}}\to {\ensuremath{\mathbb{1}}}}$ is a fibration [@Carboni-Kelly-Pedicchio Theorem 4.2]. Furthermore, a regular epimorphism of simplicial objects in a regular Mal’tsev category is always a fibration [@EverVdL2 Proposition 4.4]. The Kan property for simplicial objects may also be expressed in terms of higher extensions: in a semi-abelian category, a simplicial object ${\ensuremath{\mathbb{X}}}$ is Kan if and only if all of its truncations, considered as higher arrows in all possible directions, have a domain which is an extension [@EGoeVdL]. \[Lemma-DeltaLambda-Square\] In a finitely complete category, given $n\geq 1$, $i\in n$, and an augmented simplicial object ${\ensuremath{\mathbb{X}}}$, the square $$\vcenter{\xymatrix{{\triangle}({\ensuremath{\mathbb{X}}},n) \ar[r]^-{\widehat{\partial}_{i}} \ar[d]_-{{\partial}_{i}} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n) \ar[d]^-{{\partial}_{i-1}^{i}\times {\partial}_{i}^{n-i}} \\ {\ensuremath{\mathbb{X}}}_{n-1} \ar[r]_-{{\lgroup}{\partial}_{j}{\rgroup}_{j}} & {\triangle}({\ensuremath{\mathbb{X}}},n-1)}}$$ is a pullback, where the arrow on the right is the restriction of $${\partial}_{i-1}^{i}\times {\partial}_{i}^{n-i}=\underbrace{{\partial}_{i-1}\times \cdots\times {\partial}_{i-1}}_{\text{$i$ times}} \times \underbrace{{\partial}_{i}\times \cdots \times {\partial}_{i}}_{\text{$n-i$ times}}\colon {\ensuremath{\mathbb{X}}}_{n-1}^{n}\to {\ensuremath{\mathbb{X}}}_{n-2}^{n}$$ to a morphism ${\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n)\to {\triangle}({\ensuremath{\mathbb{X}}},n-1)$. Here is a picture in degree $n=2$ for $i=1$: $$\begin{matrix} \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\beta}\\ {\cdot} \ar[ru]^-{\alpha} \ar[rr]_-{\gamma} && {\cdot}}} & \mapsto & \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\beta}\\ {\cdot} \ar[ru]^-{\alpha} && {\cdot}}} \\ {\rotatebox{270}{$\mapsto$}}&& {\rotatebox{270}{$\mapsto$}}\\ \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\hole}\\ {\cdot} \ar[rr]_-{\gamma} && {\cdot}}} & \mapsto & \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\hole}\\ {\cdot} && {\cdot}}} \end{matrix}$$ We again only need to give a formal proof in ${\ensuremath{\mathsf{Set}}}$, where it suffices to compare the set of couples $$(x_{i},(x_{0},\dots,\widehat x_{i},\dots,x_{n}))\in {\ensuremath{\mathbb{X}}}_{n-1}\times {\ensuremath{\mathbb{X}}}_{n-1}^{n}$$ which satisfy $$\text{${\partial}_{j}(x_{k})={\partial}_{k-1}(x_{j})$ for all $j<k$ with $j$, $k\neq i$}$$ and $${\partial}_{j}(x_{i})= \begin{cases} {\partial}_{i-1}(x_{j}) & \text{if $j<i$}\\ {\partial}_{i}(x_{j+1}) & \text{if $i\leq j<n$} \end{cases}$$ with the set $$\{(x_{0},\dots,x_{n})\in {\ensuremath{\mathbb{X}}}_{n-1}^{n+1}\mid \text{${\partial}_{i}(x_{j})={\partial}_{j-1}(x_{i})$ for $0\leq i<j\leq n$}\}.$$ These sets are clearly isomorphic, which finishes the proof. Higher-dimensional torsors {#Torsors} -------------------------- Let $A$ be an abelian group in a Barr-exact category ${\ensuremath{\mathcal{X}}}$ and consider $n\geq 1$. A [${\ensuremath{\mathbb{K}}}(A,n)$-torsor]{} is an augmented simplicial object ${\ensuremath{\mathbb{T}}}$ equipped with a simplicial morphism ${\ensuremath{\mathbb{t}}}\colon {{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(A,n)}$ such that 1. ${\ensuremath{\mathbb{t}}}$ is a fibration which is exact from degree $n$ on; 2. ${\ensuremath{\mathbb{T}}}\cong {\ensuremath{\mathsf{Cosk}}}_{n-1}{\ensuremath{\mathbb{T}}}$; 3. ${\ensuremath{\mathbb{T}}}$ is a resolution. Let $Z$ be an object of a semi-abelian category ${\ensuremath{\mathcal{X}}}$ and $(A,\xi)$ a $Z$-module. An [$n$" torsor of $Z$ by $(A,\xi)$]{} is a ${\ensuremath{\mathbb{K}}}((A,\xi),n)$-torsor in the category ${\ensuremath{\mathcal{X}}}/Z$. Morphisms of ${\ensuremath{\mathbb{K}}}(A,n)$-torsors are defined as in the slice over ${\ensuremath{\mathbb{K}}}(A,n)$, and thus we obtain the category $\operatorname{Tors}^{n}({\ensuremath{\mathcal{X}}},A)$ of ${\ensuremath{\mathbb{K}}}(A,n)$-torsors in ${\ensuremath{\mathcal{X}}}$ as a full subcategory of ${\ensuremath{\mathsf{s}}}^{+}({\ensuremath{\mathcal{X}}})/{\ensuremath{\mathbb{K}}}(A,n)$. When the action $\xi$ is trivial, we call $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ an [$n$-torsor of $Z$ by $A$]{}, and obtain the following picture: $$\resizebox{\textwidth}{!}{\mbox{$ \vcenter{\xymatrix@=45pt{ {\triangle}({\ensuremath{\mathbb{T}}},n+1) \ar[d]_{\cdots\quad{\lgroup}{\lgroup}\varsigma\circ{\partial}_{i}{\rgroup}_{i},{\partial}_{0}^{n+2}{\rgroup}} \ar@<2.33ex>[r] \ar@<1.16ex>[r] \ar@<-2.33ex>[r]^-{\vdots} & {\triangle}({\ensuremath{\mathbb{T}}},n) \ar[d]^{{\lgroup}\varsigma,{\partial}_{0}^{n+1}{\rgroup}} \ar@<1.75ex>[r] \ar@<-1.75ex>[r]^-{\vdots} & {\ensuremath{\mathbb{T}}}_{n-1} \ar[d]^-{{\partial}_{0}^{n}} \ar@<1.75ex>[r] \ar@<-1.75ex>[r]^-{\vdots} & {\ensuremath{\mathbb{T}}}_{n-2} \ar[d]^-{{\partial}_{0}^{n-1}} \ar@{}[r]|-{\cdots} & {\ensuremath{\mathbb{T}}}_{0} \ar[d]^-{{\partial}_{0}} \ar[r]^-{{\partial}_{0}} & {\ensuremath{\mathbb{T}}}_{-1} \ar@{=}[d]\\ A^{n+1} \times Z \ar@<2.33ex>[r]^-{{\partial}_{n+1}\times 1_{Z}} \ar@<1.16ex>[r]|-{\operatorname{pr}_{n}\times 1_{Z}} \ar@<-2.33ex>[r]_-{\operatorname{pr}_{0}\times 1_{Z}}^-{\vdots} & A \times Z \ar@<1.75ex>[r]^-{\operatorname{pr}_{Z}} \ar@<-1.75ex>[r]_-{\operatorname{pr}_{Z}}^-{\vdots} & Z \ar@{=}@<1.75ex>[r] \ar@{=}@<-1.75ex>[r]^-{\vdots} & Z \ar@{}[r]|-{\cdots} & Z \ar@{=}[r] & Z}} $}}$$ When $Z$ is an object of a semi-abelian category ${\ensuremath{\mathcal{X}}}$ and $A$ is an abelian object in ${\ensuremath{\mathcal{X}}}$ considered as a trivial $Z$-module $(A,\tau)$, we write $\operatorname{Tors}^{n}(Z,A)$ for the category $\operatorname{Tors}^{n}({\ensuremath{\mathcal{X}}}/Z,(A,\tau))$. Taking connected components we obtain the set $$\operatorname{Tors}^{n}[Z,A]=\pi_{0}\operatorname{Tors}^{n}(Z,A)$$ of equivalence classes of $n$-torsors of $Z$ by $A$ which is, in fact, an abelian group [@Duskin-Torsors]. We shall further analyse the concept of torsor in Section \[Section-Torsors-and-Centrality\]; for now it suffices to understand their cohomological meaning. The $(n+1)$-th cohomology group ------------------------------- It follows from [@Duskin-Torsors Theorem 5.2] that, when ${\ensuremath{\mathcal{X}}}$ is a Barr-exact category and $${\ensuremath{\mathbb{G}}}=(G\colon{\ensuremath{\mathcal{X}}}\to{\ensuremath{\mathcal{X}}},\,\delta\colon G{\Rightarrow}G^{2},\,\epsilon\colon G{\Rightarrow}1_{{\ensuremath{\mathcal{X}}}})$$ is a comonad on ${\ensuremath{\mathcal{X}}}$ such that the ${\ensuremath{\mathbb{G}}}$-projectives coincide with the regular projectives in ${\ensuremath{\mathcal{X}}}$, then $${\mathrm{H}}^{n+1}(1,A)_{{\ensuremath{\mathbb{G}}}}\cong \pi_{0}\operatorname{Tors}^{n}({\ensuremath{\mathcal{X}}},A)$$ where $A$ is an internal abelian group in ${\ensuremath{\mathcal{X}}}$ and $1$ is the terminal object. If now $Z$ is an object of ${\ensuremath{\mathcal{X}}}$ then ${\ensuremath{\mathbb{G}}}$ induces a comonad ${\ensuremath{\mathbb{G}}}/Z=(G^{Z},\delta^{Z},\epsilon^{Z})$ on ${\ensuremath{\mathcal{X}}}/Z$ via $$\delta^{Z}_{f}=\left(\vcenter{\xymatrix@!0@R=3em@C=2em{GX \ar[rr]^-{\delta_{X}} \ar[rd]_-{G^{Z}f=f\circ\epsilon_{X}} && GGX \ar[ld]^-{G^{Z}G^{Z}f=f\circ\epsilon_{X}\circ\epsilon_{GX}} \\ & Z}}\right) \quad\text{and}\quad \epsilon^{Z}_{f}=\left(\vcenter{\xymatrix@!0@R=3em@C=2em{GX \ar[rr]^-{\epsilon_{X}} \ar[rd]_-{G^{Z}f=f\circ\epsilon_{X}} && X \ar[ld]^-{f} \\ & Z}}\right)$$ for all $f\colon{X\to Z}$. Hence when, in a semi-abelian category ${\ensuremath{\mathcal{X}}}$, we consider an abelian object $A$ as a trivial $Z$-module, we see that $${\mathrm{H}}^{n+1}(1_{Z},(A,\tau))_{{\ensuremath{\mathbb{G}}}/Z}\cong \pi_{0}\operatorname{Tors}^{n}({\ensuremath{\mathcal{X}}}/Z,(A,\tau))$$ and $${\mathrm{H}}^{n+1}(Z,A)_{{\ensuremath{\mathbb{G}}}}\cong\operatorname{Tors}^{n}[Z,A].$$ For instance, ${\ensuremath{\mathcal{X}}}$ may be chosen to be a variety of algebras over ${\ensuremath{\mathsf{Set}}}$, so that ${\ensuremath{\mathbb{G}}}$ is canonically induced by the forgetful/free adjunction. In any case, $\operatorname{Tors}^{n}[Z,A]$ does indeed carry an abelian group structure. Moreover, this defines an additive functor $$\operatorname{Tors}^{n}[Z,-]\colon {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}.$$ The groups of equivalence classes of higher central extensions {#Section-Central-Extensions} ============================================================== We work towards a definition of the group $\operatorname{Centr}^{n}(Z,A)$ of equivalence classes of $n$-fold central extensions of $Z$ by $A$, extending the definition of $\operatorname{Centr}^{2}(Z,A)$ given in Section 4 of [@RVdL]. We start with some basic theory of (higher-dimensional) central extensions, first recalling known results and then proving some new ones. Central extensions {#Central extensions} ------------------ We first consider some general definitions and results valid in a homological category with a chosen strongly Birkhoff subcategory. Here we follow [@EGVdL]. A [Galois structure]{} [@Janelidze:Precategories] $\Gamma=({\ensuremath{\mathcal{X}}},{\ensuremath{\mathcal{B}}},{\ensuremath{\mathcal{E}}},{\ensuremath{\mathcal{F}}},I,H)$ consists of categories ${\ensuremath{\mathcal{X}}}$ and ${\ensuremath{\mathcal{B}}}$, an adjunction $$\xymatrix{{\ensuremath{\mathcal{X}}}\ar@<1 ex>[r]^-{I} \ar@{}[r]|-{} & {\ensuremath{\mathcal{B}}}, \ar@<1 ex>[l]^-H \ar@{}[l]|{\perp}}$$ and classes ${\ensuremath{\mathcal{E}}}$ and ${\ensuremath{\mathcal{F}}}$ of morphisms of ${\ensuremath{\mathcal{X}}}$ and ${\ensuremath{\mathcal{B}}}$ respectively, such that: 1. ${\ensuremath{\mathcal{X}}}$ has pullbacks along morphisms in ${\ensuremath{\mathcal{E}}}$; 2. ${\ensuremath{\mathcal{E}}}$ and ${\ensuremath{\mathcal{F}}}$ contain all isomorphisms, are closed under composition and are pullback-stable; 3. $I({\ensuremath{\mathcal{E}}}) \subseteq {\ensuremath{\mathcal{F}}}$; 4. $H({\ensuremath{\mathcal{F}}}) \subseteq {\ensuremath{\mathcal{E}}}$. An element of ${\ensuremath{\mathcal{E}}}$ is called an [${\ensuremath{\mathcal{E}}}$-extension]{}. We shall only consider Galois structures where ${\ensuremath{\mathcal{X}}}$ is (at least) a homological category, all ${\ensuremath{\mathcal{E}}}$-extensions are regular epimorphisms, and ${\ensuremath{\mathcal{B}}}$ is a full replete ${\ensuremath{\mathcal{E}}}$-reflective subcategory of ${\ensuremath{\mathcal{X}}}$. We shall never write its inclusion $H$. Such a subcategory is called [strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff]{} when for every ${\ensuremath{\mathcal{E}}}$-extension $f\colon{X\to Z}$ the induced naturality square $$\label{Birkhoff-Square} \vcenter{\xymatrix{X \ar@{ >>}[r]^-{f} \ar@{ >>}[d]_-{\eta_{X}} & Z \ar@{ >>}[d]^-{\eta_{Z}} \\ IX \ar@{ >>}[r]_-{If} & IZ}}$$ is a two-cubic ${\ensuremath{\mathcal{E}}}$-extension. (The universally induced morphism to the pullback must be in ${\ensuremath{\mathcal{E}}}$.) From now on we shall always assume this to be the case. If ${\ensuremath{\mathcal{X}}}$ is an exact Mal’tsev category and ${\ensuremath{\mathcal{E}}}$ consists of all regular epimorphisms, a strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategory of ${\ensuremath{\mathcal{X}}}$ is precisely a [Birkhoff subcategory]{}: full, reflective and closed under subobjects and regular quotients in ${\ensuremath{\mathcal{X}}}$, see [@Janelidze-Kelly]. A Birkhoff subcategory of a variety of algebras is the same thing as a subvariety. Outside the exact Mal’tsev context, however, when ${\ensuremath{\mathcal{E}}}$ is the class of regular epimorphisms, the strong ${\ensuremath{\mathcal{E}}}$-Birkhoff property is generally stronger than the Birkhoff property, since not every pushout of extensions needs to be a two-cubic extension. It is well known that, in any semi-abelian category ${\ensuremath{\mathcal{X}}}$, the full subcategory ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$ determined by the abelian objects is Birkhoff. This is the situation which we shall be most interested in here, in particular from Subsection \[Directions\] on. An ${\ensuremath{\mathcal{E}}}$-extension $f\colon{X\to Z}$ in ${\ensuremath{\mathcal{X}}}$ is [trivial]{} when the induced square  is a pullback. Of course, if $X$ and $Z$ lie in ${\ensuremath{\mathcal{B}}}$ then $f$ is a trivial ${\ensuremath{\mathcal{E}}}$-extension. The ${\ensuremath{\mathcal{E}}}$-extension $f$ is said to be [normal]{} when both projections $\operatorname{pr}_{0}$, $\operatorname{pr}_{1}$ in the kernel pair $({\operatorname{Eq}(f)},\operatorname{pr}_{0},\operatorname{pr}_{1})$ of $f$ are trivial. Finally, $f$ is [central]{} when there exists an ${\ensuremath{\mathcal{E}}}$" extension $g\colon{Y\to Z}$ such that the pullback of $f$ along $g$ is trivial. It is clear that every trivial ${\ensuremath{\mathcal{E}}}$-extension is central. Moreover, every normal ${\ensuremath{\mathcal{E}}}$" extension is central; in the present context, also the converse holds (via Theorem 4.8 of [@Janelidze-Kelly] or Proposition 2.6 in [@EGVdL]). Hence the concepts of normality and centrality coincide. It follows immediately from the definition that pullbacks of ${\ensuremath{\mathcal{E}}}$-extensions along ${\ensuremath{\mathcal{E}}}$-extensions reflect centrality. Furthermore, in the present context, Proposition 4.1 and 4.3 in [@Janelidze-Kelly] may be modified to prove that both the classes of trivial and of central ${\ensuremath{\mathcal{E}}}$-extensions are pullback-stable. It is also well known that a split epimorphic central ${\ensuremath{\mathcal{E}}}$-extension is always trivial. The following important result (see [@Gran-Alg-Cent; @EGVdL]) will be used in Section \[Section-Geometry\]. \[Lemma-Pullbacks\] When ${\ensuremath{\mathcal{X}}}$ is a homological category and ${\ensuremath{\mathcal{B}}}$ is a strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff subcategory of ${\ensuremath{\mathcal{X}}}$, the reflector $I\colon{{\ensuremath{\mathcal{X}}}\to {\ensuremath{\mathcal{B}}}}$ preserves pullbacks of ${\ensuremath{\mathcal{E}}}$-extensions along split epimorphisms. The tower of Galois structures for cubic central extensions {#Subsection-Tower} ----------------------------------------------------------- Now we describe the Galois structures for centrality of $n$-cubic extensions introduced in [@EGVdL]. We start with a semi-abelian category ${\ensuremath{\mathcal{X}}}$ and a Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ of ${\ensuremath{\mathcal{X}}}$. Choosing ${\ensuremath{\mathcal{E}}}$ and ${\ensuremath{\mathcal{F}}}$ to be the classes of regular epimorphisms in ${\ensuremath{\mathcal{X}}}$ and ${\ensuremath{\mathcal{B}}}$, we obtain a Galois structure $\Gamma$ as above—${\ensuremath{\mathcal{B}}}$ is strongly ${\ensuremath{\mathcal{E}}}$-Birkhoff. We may now drop the prefix ${\ensuremath{\mathcal{E}}}$; the elements of this class are the one-cubic extensions of Subsection \[Extensions\]. Let us view the objects of ${\ensuremath{\mathcal{X}}}$ as zero-cubic extensions, and the objects of ${\ensuremath{\mathcal{B}}}$ as zero-cubic central extensions. With respect to the Galois structure $\Gamma_{0}=\Gamma$, there is the notion of central extension, and it is such that the full subcategory ${\ensuremath{\mathsf{CExt}}}^{1}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})$ of ${\ensuremath{\mathsf{Ext}}}^{1}({\ensuremath{\mathcal{X}}})$ determined by those one-cubic central extensions is again reflective. Its reflector $I_{1}\colon{{\ensuremath{\mathsf{Ext}}}^{1}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{CExt}}}^{1}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})}$, together with the classes ${\ensuremath{\mathcal{E}}}^{1}$ and ${\ensuremath{\mathcal{F}}}^{1}$ of one-cubic extensions in ${\ensuremath{\mathsf{Ext}}}^{1}({\ensuremath{\mathcal{X}}})$ and in ${\ensuremath{\mathsf{CExt}}}^{1}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})$ (which we choose to be two-cubic extensions in ${\ensuremath{\mathcal{X}}}$, and two-cubic extensions with central domain and codomain), in turn determines a Galois structure $\Gamma_{1}$. This Galois structure is again “nice” in that ${\ensuremath{\mathsf{CExt}}}^{1}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})$ is again strongly ${\ensuremath{\mathcal{E}}}^{1}$-Birkhoff in the homological category ${\ensuremath{\mathsf{Ext}}}^{1}({\ensuremath{\mathcal{X}}})$. Inductively, this defines a family of Galois structures $(\Gamma_{n})_{n\geq 0}$: $$\Gamma_{n}=({\ensuremath{\mathsf{Ext}^{n}}}({\ensuremath{\mathcal{X}}}),{\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}}),{\ensuremath{\mathcal{E}}}^{n},{\ensuremath{\mathcal{F}}}^{n},I_{n},\subseteq),$$ each of which gives rise to a notion of $(n+1)$-cubic central extension which determines the next structure [@EGVdL Theorem 4.6]. (Here ${\ensuremath{\mathcal{E}}}^0={\ensuremath{\mathcal{E}}}$, ${\ensuremath{\mathcal{F}}}^0={\ensuremath{\mathcal{F}}}$ and $I_0=I$.) In particular, for every $n\geq 1$ we obtain a reflector (the centralisation functor) $$I_{n}\colon{{\ensuremath{\mathsf{Ext}^{n}}}({\ensuremath{\mathcal{X}}})\to{\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}}),}$$ left adjoint to the inclusion ${\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})\subseteq{\ensuremath{\mathsf{Ext}^{n}}}({\ensuremath{\mathcal{X}}})$. For any $n\geq 1$, the $n$-cubic extension ${\langle}F{\rangle}_{{\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})}$ in the short exact sequence $$\xymatrix{0 \ar[r] & {\langle}F{\rangle}_{{\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})} \ar@{{ |>}->}[r]^-{\mu^{n}_{F}} & F \ar@{-{ >>}}[r]^-{\eta^{n}_{F}} & I_{n}F \ar[r] & 0}$$ induced by the centralisation of an $n$-cubic extension $F$ is zero everywhere except in its initial object ${\langle}F{\rangle}^{n}=({\langle}F{\rangle}_{{\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})})_{n}$, because centralisation keeps all objects in an $n$-cubic extension fixed except the initial object. So, restricting to initial objects, we obtain a short exact sequence $$\label{brackets} \xymatrix{0 \ar[r] & {\langle}F{\rangle}^{n} \ar@{{ |>}->}[r] & F_{n} \ar@{-{ >>}}[r] & I_{n}[F] \ar[r] & 0}$$ in ${\ensuremath{\mathcal{X}}}$. In parallel with the case ${n=0}$ considered in Subsection \[Subsection-Abelian\], this object ${\langle}F{\rangle}^{n}$ acts like an $n$-dimensional commutator which may be computed as the kernel of the restriction of the kernel pair projection $(\operatorname{pr}_{0})_{n-1}\colon{{\operatorname{Eq}(F)}_{n-1}\to \operatorname{dom}F_{n-1}}$ to a morphism $${\langle}\operatorname{pr}_{0}{\rangle}^{n-1}\colon {{\langle}{\operatorname{Eq}(F)}{\rangle}^{n-1}\to {\langle}\operatorname{dom}F{\rangle}^{n-1}}$$ in ${\ensuremath{\mathcal{X}}}$. Furthermore, an $n$-cubic extension $F$ is central if and only if the induced morphisms $${\langle}\operatorname{pr}_{0}{\rangle}^{n-1}, {\langle}\operatorname{pr}_{1}{\rangle}^{n-1}\colon{{\langle}{\operatorname{Eq}(F)}{\rangle}^{n-1}\to {\langle}\operatorname{dom}F{\rangle}^{n-1}}$$ are isomorphisms—which happens precisely when they coincide; see [@EverHopf; @EGVdL] for more details. The notation ${\langle}F{\rangle}^{n}$ not mentioning the Birkhoff subcategory ${\ensuremath{\mathcal{B}}}$ need not lead to confusion, because the only case which we shall use it in is ${\ensuremath{\mathcal{B}}}={\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$; keeping this in mind, we also write ${\langle}X{\rangle}^{0}={\langle}X{\rangle}$ for the kernel of $\eta_{X}\colon {X\to {\ensuremath{\mathsf{ab}}}X}$ when $X$ is an object of ${\ensuremath{\mathcal{X}}}$. \[def n-fold central extension\] An [$n$-fold central extension]{} is an $n$-fold extension of which the underlying $n$-cubic extension is central. \[Example-K(A,n)-as-Extension\] Given any integer $n\geq 1$, any object $Z$ and any abelian object $A$ in ${\ensuremath{\mathcal{X}}}$, the $(n+1)$-cubic extension underlying ${\ensuremath{\mathbb{K}}}(Z,A,n)$ is always trivial with respect to abelianisation. This follows by induction from the fact that both its domain and its codomain are $n$-cubic trivial extensions. Note, however, that the ${(n+2)}$-fold arrow underlying ${\ensuremath{\mathbb{K}}}(Z,A,n)$ is not even a cubic extension! \[Example-Dimension-One\] Recall that a surjective group homomorphism $f\colon{X\to Z}$ is central (with respect to ${\ensuremath{\mathsf{Ab}}}$) if and only if $[{\operatorname{Ker}(f)},X]=0$. This result was adapted to a semi-abelian context in [@Bourn-Gran; @Gran-Alg-Cent]: when ${\ensuremath{\mathcal{X}}}$ is a semi-abelian category and ${\ensuremath{\mathcal{B}}}={\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$ is the Birkhoff subcategory determined by all abelian objects in ${\ensuremath{\mathcal{X}}}$, the one-cubic central extensions induced by the Galois structure (the “categorically central” ones) are the central extensions in the algebraic sense. These may be characterised through the Smith/Pedicchio commutator of equivalence relations as those $f\colon{X\to Z}$ such that ${[{\operatorname{Eq}(f)},\nabla_{X}]=\Delta_{X}}$, which means that the kernel pair of the arrow $f$ is a central equivalence relation (Subsection \[Commutators\]). A characterisation closer to the group case appears in [@Gran-VdL] where the condition is reformulated in terms of the Huq commutator of normal subobjects so that it becomes $[{\operatorname{Ker}(f)},X]=0$. \[Example-Dimension-Two\] One level up, the double central extensions of groups vs. abelian groups were first characterised in [@Janelidze:Double]: a two-cubic extension such as  above is central if and only if $$[{\operatorname{Ker}(d)},{\operatorname{Ker}(c)}]=0=[{\operatorname{Ker}(d)}\cap {\operatorname{Ker}(c)},X].$$ General versions of this characterisation were given in [@Gran-Rossi] for Mal’tsev varieties, then in [@RVdL] for semi-abelian categories and finally in [@EverVdL3] for exact Mal’tsev categories: the two-cubic extension  is central (with respect to abelianisation) if and only if $$\label{Double-Central} [{\operatorname{Eq}(d)},{\operatorname{Eq}(c)}]=\Delta_{X}=[{\operatorname{Eq}(d)}\cap {\operatorname{Eq}(c)},\nabla_{X}].$$ This means that the span $(X,d,c)$ is a special kind of pregroupoid in the slice category ${\ensuremath{\mathcal{X}}}/Z$. The main technical problem here is that later on, we will use the Huq commutator of normal monomorphisms rather than the Smith/Pedicchio commutator of equivalence relations—and the correspondence between the two which exists in level one is no longer there when we go up in degree. In fact, it is well known and easily verified that if the Smith/Pedicchio commutator of two equivalence relations is trivial, then the Huq commutator of their normalisations is also trivial [@BG]. But, in general, the converse is false; in [@Borceux-Bourn; @Bourn2004] a counterexample is given in the category of digroups, which is a semi-abelian variety, even a variety of $\Omega$-groups [@Higgins]. The equivalence of these commutators is known as the [*Smith is Huq* condition (SH)]{} and it is shown in [@MFVdL] that, for a semi-abelian category, this condition holds if and only if every star-multiplicative graph is an internal groupoid, which is important in the study of internal crossed modules [@Janelidze]. Moreover, the *Smith is Huq* condition is also known to hold for pointed strongly protomodular categories [@BG] (in particular, for any Moore category [@Rodelo:Moore]) and in action accessible categories [@BJ07] (in particular, for any category of interest [@Montoli; @Orzech]). The condition (SH) also implies that every action of an object on an abelian object is a module: here, the equality $[{\operatorname{Eq}(f)},{\operatorname{Eq}(f)}]=\Delta_{X}$ in Subsection \[Subsection-Abelian\] follows from $[{\operatorname{Ker}(f)},{\operatorname{Ker}(f)}]=[A,A]=0$. Two lemmas on higher centrality ------------------------------- The centrality of a cubic extension implies that certain induced lower-dimensional cubic extensions are also central. The present proof of Lemma \[Lemma-Direction-Limit\] was kindly offered to us by Everaert and Gran; it is more general and more elegant than our original proof. In the case of abelianisation, it also follows easily from Theorem \[Theorem-Higher-Centrality\]. We first recall a well-known result (essentially Proposition 4.3 in [@Janelidze-Kelly]): \[Lemma-Kernel-Central-Extension\] In a semi-abelian category with a chosen Birkhoff subcategory, let $f\colon{X\to Y}$ be an $n$-cubic central extension considered as an arrow between $(n-1)$-cubic extensions $X$ and $Y$. Then its kernel $K$ is an $(n-1)$-cubic central extension. The $(n-1)$-cubic extension $K$ may be obtained as the kernel of the $n$" cubic trivial extension $f_{0}\colon{\operatorname{Eq}(f)\to X}$, hence also as the kernel of the $n$-cubic extension $I_{n-1}f_{0}\colon{I_{n-1}\operatorname{Eq}(f)\to I_{n-1}X}$ in ${\ensuremath{\mathsf{CExt}}}_{{\ensuremath{\mathcal{B}}}}^{n-1}({\ensuremath{\mathcal{X}}})$. \[Lemma-Direction-Limit\] Let $F$ be an $n$-cubic central extension in a semi-abelian category with a chosen Birkhoff subcategory. Then the one-cubic extension $l_{F}\colon{F_{n}\to {\ensuremath{\mathsf{L}}}F}$ induced by $F$ is always central. The case $n=1$ is clear (because then $F=l_{F}$), so take $n\geq 2$. We shall prove that for an $n$-cubic central extension, considered as a square  of $(n-1)$-cubic extensions, the induced comparison ${\lgroup}d,c{\rgroup}\colon{X\to D\times_{Z}C}$ is an $(n-1)$-cubic central extension $G$. Then the claim follows by induction, because $l_{F}=l_{G}$ by Lemma \[Lemma-L-pullbacks\]. Since ${\lgroup}d,c{\rgroup}$ is an $(n-1)$-cubic extension by definition, we just have to show its centrality. First we may reduce the situation to trivial extensions. Indeed, taking kernel pairs to the left, we obtain the diagram $$\vcenter{\xymatrix{{\operatorname{Eq}(c)} \ar@{-{ >>}}[d]_{r} \ar@{-{ >>}}@<.5ex>[r]^-{c_{1}} \ar@{-{ >>}}@<-.5ex>[r]_-{c_{0}} & X \ar@{-{ >>}}[r]^{c} \ar@{-{ >>}}[d]_-{d} & C \ar@{-{ >>}}[d]^{g}\\ {\operatorname{Eq}(f)} \ar@<.5ex>@{-{ >>}}[r]^-{f_{1}} \ar@{-{ >>}}@<-.5ex>[r]_-{f_{0}} & D \ar@{-{ >>}}[r]_-{f} & Z.}}$$ It is not hard to see that the induced comparison ${\lgroup}r,c_{0}{\rgroup}\colon{{\operatorname{Eq}(c)}\to {\operatorname{Eq}(f)}\times_{D}X}$ is a pullback of the cubic extension ${\lgroup}d,c{\rgroup}$: the diagram $$\vcenter{\xymatrix@1@!0@=35pt{ \operatorname{Eq}(c) \pullbackdots \ar[rr]^-{{\lgroup}r,c_{0}{\rgroup}} \ar[dd]_-{c_{0}} && \operatorname{Eq}(f)\times_{D}X \pullback \skewpullback \ar[rr]^-{d^{*}f_{1}} \ar[dd]|(.5){\hole} \ar[ld]_(.6){f_{1}^{*}d} && X \ar[dd]^-{c} \ar[ld]_-{d} \\ &\operatorname{Eq}(f) \pullback \ar[rr]^(.75){f_{1}} \ar[dd]_(.25){f_{0}} && D \ar[dd]^(.25){f} \\ X \ar[rr]_(.25){{\lgroup}d,c{\rgroup}}|(.5){\hole} && D\times_{Z}C \skewpullback \ar[ld]|-{f^{*}g} \ar[rr]|(.5){\hole}|(.7){g^{*}f} && C \ar[ld]^-{g} \\ &D \ar[rr]_-{f} && Z}}$$ shows how the back pullback rectangle decomposes as a composite of pullback squares. Hence if ${\lgroup}r,c_{0}{\rgroup}$ is central then so is ${\lgroup}d,c{\rgroup}$, because pulling back reflects centrality. Now we reduce to $n$-cubic extensions between $(n-1)$-cubic central extensions. Suppose that the square , viewed as an arrow from $d$ to $g$, is an $n$-cubic trivial extension. Consider the following cube, which displays the centralisation of $d$ and of $g$ using the notation of : $$\vcenter{\xymatrix@!0@=2.5em{&& {I_{n-1}[d]} \ar@{~{ >>}}[rd] \ar@{-{ >>}}[rrrr] \ar@{.{ >>}}[dddd]|-{\hole} &&&& {I_{n-1}[g]} \ar@{-{ >>}}[dddd]^-{I_{n-1}g} \\ &&& {\cdot} \ar@{.{ >>}}[lddd] \ar@{.{ >>}}[rrru] \pullback \\ {X} \ar@{~{ >>}}[rd] \ar@{-{ >>}}[rrrr] \ar@{-{ >>}}[dddd]_-{d} \ar@{~{ >>}}[rruu] &&&& {C} \ar@{-{ >>}}[dddd]^(.25){g} \ar@{-{ >>}}[rruu]\\ & {\cdot} \ar@{~{ >>}}[rruu]|-{\hole} \ar@{-{ >>}}[lddd] \ar@{-{ >>}}[rrru] \pullback \\ && {D} \ar@{.{ >>}}[rrrr] &&&& {Z} \\\\ {D} \ar@{-{ >>}}@<-.5ex>[rrrr] \ar@{:}[rruu] &&&& {Z} \ar@{=}[rruu]}}$$ Recall from Subsection \[Subsection-Tower\] that the centralisation functor only changes the domain of a cubic extension, which explains the two identity morphisms in the cube. Since the front square is a trivial extension, the top square is a pullback. By pullback cancellation, the top square of the prism between the front and back pullbacks is also a pullback, and it follows that the square of wiggly arrows is a pullback too. This completes the reduction, since cubic central extensions are pullback-stable. Finally, for an $n$-cubic extension  between $(n-1)$-cubic central extensions $d$ and $g$ the claim that ${\lgroup}d,c{\rgroup}\colon{X\to D\times_{Z}C}$ is an $(n-1)$-cubic central extension holds, since ${\lgroup}d,c{\rgroup}$ is a subobject of the cubic extension $d$ in the category of $(n-1)$-cubic central extensions—indeed, a monomorphism of cubic extensions is a square of which the top map is a monomorphism—and cubic central extensions are closed under subobjects. (Central) extensions over a fixed base object --------------------------------------------- Let $Z$ be an object of ${\ensuremath{\mathcal{X}}}$ and $n\geq 1$. Denote by ${\ensuremath{\mathsf{Ext}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ the category of [$n$-fold extensions of $Z$]{} or [over $Z$]{}, defined as the fibre over $Z$ (the pre-image of the identity $1_Z$) of the functor $$(-)_{0,\dots,0}\colon{\ensuremath{3^{n}\text{-}\mathsf{Diag}}}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathcal{X}}}\colon E\mapsto E_{0,\dots,0}$$ which projects an $n$-fold extension on its terminal object—see Subsection \[3\^n diagrams\]. Thus the objects of ${\ensuremath{\mathsf{Ext}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ are $3^{n}$-diagrams with “terminal object” $Z$, and the morphisms are those morphisms in ${\ensuremath{3^{n}\text{-}\mathsf{Diag}}}({\ensuremath{\mathcal{X}}})$ which restrict to the identity on $Z$ under the functor $(-)_{0,\dots,0}$. Similarly ${\ensuremath{\mathsf{CExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ is the full subcategory of ${\ensuremath{\mathsf{Ext}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ determined by the $n$-fold extensions of $Z$ that are central with respect to ${\ensuremath{\mathcal{B}}}$ as in Definition \[def n-fold central extension\]. (The index ${\ensuremath{\mathcal{B}}}$ being dropped here is not really problematic, since we shall take ${\ensuremath{\mathcal{B}}}$ equal to ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$ anyway from Subsection \[Directions\] on.) Sending an $n$" fold (central) extension to its underlying $n$-cubic (central) extension, we obtain an equivalence with the category of [$n$-cubic (central) extensions over $Z$]{}, the fibre over $Z$ of the functor $$\operatorname{cod}^n=\underbrace{\operatorname{cod}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\cdots{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{cod}}_{\text{$n$ times}}=(-)_{0}\colon {{\ensuremath{\mathsf{Ext}}}^{n}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathcal{X}}}\colon F\mapsto F_{0}}$$ or, in the case of central extensions, its restriction to ${\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})$. \[Lemma-Product-Over-Z\] Consider a semi-abelian category ${\ensuremath{\mathcal{X}}}$ with a chosen Birkhoff subcategory. Let $Z$ be an object of ${\ensuremath{\mathcal{X}}}$ and $n\geq 1$. Then ${\ensuremath{\mathsf{Ext}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ and ${\ensuremath{\mathsf{CExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ have binary products: the product of two $n$-fold (central) extensions $F$ and $G$ over $Z$ is an $n$-fold (central) extension over $Z$. Moreover, $l_{F\times G}=l_{F}\times_{Z}l_{G}$. Given two $n$-cubic extensions $F$ and $G$ over $Z$, their product $F\times G$ in the category ${\ensuremath{\mathsf{Ext}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ is given pointwise by pullbacks in ${\ensuremath{\mathcal{X}}}$: $$(F\times G)_{I}=F_{I}\times_{Z}G_{I}$$ for $I\subseteq n$. To see that it is the product as $n$-fold arrows over $Z$, it suffices to verify the universal property. This $n$-fold arrow is indeed an $n$-cubic extension by Proposition \[Limit-Characterisation-Extensions\], since $$\begin{aligned} {(F\times G)_{I}\to\lim_{J\subsetneq I} (F\times G)_{J}} &= {(F_{I}\times_{Z} G_{I})\to\lim_{J\subsetneq I} (F_{J}\times_{Z} G_{J})}\\ &= {(F_{I}\times_{Z} G_{I})\to(\lim_{J\subsetneq I} F_{J}\times_{Z} \lim_{J\subsetneq I} G_{J})}\\ &= {(F_{I}\to\lim_{J\subsetneq I} F_{J})\times_{Z} (G_{I}\to \lim_{J\subsetneq I} G_{J})}\end{aligned}$$ for all $\emptyset\neq I\subseteq n$. Note that in particular, $$l_{F\times G}=l_{F}\times_{Z}l_{G}\colon{(F_{n}\to\lim_{J\subsetneq n} F_{J})\times_{Z} (G_{n}\to \lim_{J\subsetneq n} G_{J}).}$$ The $n$-cubic extension $F\times G$ is central by the Birkhoff property of ${\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})$, it being a subobject in ${\ensuremath{\mathsf{Ext}}}^{n}({\ensuremath{\mathcal{X}}})$ of an $n$-cubic central extension. Indeed it is a subobject of the product $(F_{I}\times G_{I})_{I\subseteq n}$ of $F$ and $G$ in ${\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})$ which, since ${\ensuremath{\mathsf{CExt}}}^{n}_{{\ensuremath{\mathcal{B}}}}({\ensuremath{\mathcal{X}}})$ is a reflective subcategory, is computed pointwise as in ${\ensuremath{\mathsf{Ext}}}^{n}({\ensuremath{\mathcal{X}}})$. The direction of a higher (central) extension {#Directions} --------------------------------------------- From now on we assume that ${\ensuremath{\mathcal{X}}}$ is a semi-abelian category and ${\ensuremath{\mathcal{B}}}={\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$ is the Birkhoff subcategory determined by the abelian objects of ${\ensuremath{\mathcal{X}}}$. We introduce the concept of *direction* for $n$-fold (central) extensions in ${\ensuremath{\mathcal{X}}}$, which is crucial in the definition and in the study of the groups $\operatorname{Centr}^{n}(Z,A)$. As explained in [@RVdL], this notion is based on Bourn’s concept of direction for internal groupoids [@Bourn2002b]. \[Definition-Centr\] The [direction]{} of an $n$-fold extension $E$ is its initial object $E_{2,\dots,2}$ (see Subsection \[3\^n diagrams\]). From the point of view of the underlying $n$-cubic extension $F$, it is the object ${\operatorname{Ker}^n(F)}$, obtained by taking kernels $n$ times—each time considering a $(k+1)$-cubic extension as an arrow between $k$-cubic extensions—in the way determined by the extension $E$. If $F$ is an $n$-cubic extension, then its kernel ${\operatorname{Ker}(F)}$ is an $(n-1)$-cubic extension, whose kernel is an $(n-2)$-cubic extension ${\operatorname{Ker}^2(F)}$, and so on. By taking kernels $n$ times we obtain a $0$-cubic extension, so an object ${\operatorname{Ker}^n(F)}$. If $E$ is central then the direction of $E$ is an abelian object of ${\ensuremath{\mathcal{X}}}$ by Lemma \[Lemma-Kernel-Central-Extension\] and the convention regarding zero-cubic central extensions. Given any object $Z$ of ${\ensuremath{\mathcal{X}}}$, this defines the [direction functor]{} $${\ensuremath{\mathsf{D}}}_{(n,Z)} \colon{{\ensuremath{\mathsf{CExt}}}^n_Z({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}}).}$$ The fibre ${\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A$ of this functor over an abelian object $A$ is the category of [$n$" fold central extensions of $Z$ by $A$]{}, which are special $3^{n}$-diagrams under $A$ and over $Z$. Two $n$-fold central extensions of $Z$ by $A$ which are connected by a zigzag in ${\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A$ are called [equivalent]{}. As explained in Subsection \[Connected-components\], the equivalence classes, which we shall denote $[E]$ for $E$ an $n$-fold central extension of $Z$ by $A$, form the set $$\operatorname{Centr}^{n}(Z,A)=\pi_0({\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A)$$ of connected components of the category ${\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A\subseteq {\ensuremath{\mathsf{CExt}}}^n_Z({\ensuremath{\mathcal{X}}})$. Abusing terminology, when this does not lead to confusion, we sometimes talk about *the direction of an $n$-cubic extension*—which is only determined up to isomorphism, since this $n$-cubic extension may be part of many $n$-fold extensions. \[Direction-as-Kernel\] For any $n$-fold central extension $E$ with underlying $n$-cubic extension $F$ we have $${\ensuremath{\mathsf{D}}}_{(n,Z)} E={\operatorname{Ker}(l_{F})}=\bigcap_{i\in n} {\operatorname{Ker}(f_{i})}$$ where $l_{F}$ and the morphisms $f_{i}$ are as in \[HDA\]. The chain $${\operatorname{Ker}(l_{F})}={\operatorname{Ker}(l_{{\operatorname{Ker}(F)}})}=\cdots={\operatorname{Ker}(l_{{\operatorname{Ker}^{n-1}(F)}})}={\operatorname{Ker}({\operatorname{Ker}^{n-1}(F)})}$$ gives us the first equality; the second is immediate from the definition. For an $n$-cubic extension $F$ underlying an $n$-fold extension $E$, an “element” $x$ of $F_{n}$ is an $n$-dimensional hyper-tetrahedron with faces $x_{i}=f_{i}(x)$. Such a tetrahedron is in the direction of $E$ precisely when all its faces $x_{i}$ are zero—see Figure \[Figure-Direction\] on page  for the case ${n=3}$. The group structure on $\operatorname{Centr}^{n}(Z,A)$ ------------------------------------------------------ We are now ready to show that the set $\operatorname{Centr}^{n}(Z,A)$ of equivalence classes of $n$-fold central extensions of $Z$ by $A$ carries a canonical abelian group structure (Corollary \[Corollary-Centr\^[n]{}(Z,-)\]). \[Lemma-Product-and-Direction\] For any object $Z$ of a semi-abelian category ${\ensuremath{\mathcal{X}}}$ and any $n\geq 1$, the direction functor ${\ensuremath{\mathsf{D}}}_{(n,Z)} \colon{{\ensuremath{\mathsf{CExt}}}^n_Z({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})}$ preserves finite products. The terminal object $1$ of ${\ensuremath{\mathsf{CExt}}}^n_Z({\ensuremath{\mathcal{X}}})$ is determined by the “constant” $n$-cubic central extension of $Z$ formed out of the identities $1_Z$; it is clear that the direction of $1$ is zero. Given two $n$-fold central extensions with respective underlying $n$-cubic extensions $F$ and $G$ over $Z$ and directions $A$ and $B$, we have to prove that their product over $Z$ has direction ${A\times B}$. Lemma \[Lemma-Product-Over-Z\] tells us that the product in question does indeed exist. While lemmas \[Direction-as-Kernel\] and \[Lemma-Product-Over-Z\] give us the direction: the kernel of ${l_{F\times G}=l_{F}\times_{Z}l_{G}}$ is $A\times_Z B=A\times B$, since the morphisms from $A$ and $B$ to $Z$ are null. \[Proposition-Centr\^[n]{}(Z,-)\] Let $Z$ be an object of a semi-abelian category ${\ensuremath{\mathcal{X}}}$. Mapping any abelian object $A$ of ${\ensuremath{\mathcal{X}}}$ to the set $\operatorname{Centr}^{n}(Z,A)$ of equivalence classes of $n$-fold central extensions of $Z$ by $A$ gives a finite product-preserving functor $$\operatorname{Centr}^{n}(Z,-)\colon {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Set}}}.$$ We explain how the functoriality of $\operatorname{Centr}^{n}(Z,-)$ follows from the functoriality of $\operatorname{Centr}^{1}({\ensuremath{\mathsf{L}}}F,-)$, which is an instance of Proposition 6.1 in [@Gran-VdL]. Given an $n$-fold central extension $E$ of $Z$ by $A$ with underlying $n$-cubic extension $F$, we have an induced one-fold central extension $$\xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r]^-{k_{F}} & F_{n} \ar@{ >>}[r]^{l_{F}} & {\ensuremath{\mathsf{L}}}F \ar[r] & 0}$$ by Lemma \[Lemma-Direction-Limit\] and Lemma \[Direction-as-Kernel\]. Now let $a\colon {A\to B}$ be a morphism of abelian objects in ${\ensuremath{\mathcal{X}}}$. Then, applying the function $\operatorname{Centr}^{1}({\ensuremath{\mathsf{L}}}F,a)$ to $[l_{F}]$, we obtain an element $[l_{F'}]$ of $\operatorname{Centr}^{1}({\ensuremath{\mathsf{L}}}F,B)$ through the following construction. $$\xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r]^-{k_{F}} \ar[d]_-{{\lgroup}1_{A},0{\rgroup}} & F_{n} \ar@{ >>}[r]^-{l_{F}} \ar[d]^-{{\lgroup}1_{F_{n}},0{\rgroup}} & {\ensuremath{\mathsf{L}}}F \ar@{=}[d] \ar[r] & 0\\ 0 \ar[r] & A\oplus B \ar@{{ |>}->}[r]^-{k_{F}\times 1_B} \ar@{ >>}[d]_-{\left{\lgroup}\begin{smallmatrix} a& 1_B\end{smallmatrix}\right{\rgroup}} & F_{n}\times B \ar@{ >>}[d] \ar@{ >>}[r] & {\ensuremath{\mathsf{L}}}F \ar@{=}[d] \ar[r] & 0 \\ 0 \ar[r] & B \ar@{{ |>}->}[r] & F'_{n} \pushout \ar@{ >>}[r]_-{l_{F'}} & {\ensuremath{\mathsf{L}}}F \ar[r] & 0}$$ (Here $A\oplus B$ is the biproduct of $A$ and $B$ in ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$, which may be computed as their product $A\times B$ in ${\ensuremath{\mathcal{X}}}$, and $F_{n}'$ is the pushout of $\left{\lgroup}\begin{smallmatrix} a& 1_B\end{smallmatrix}\right{\rgroup}$ and $k_{F}\times 1_{B}$.) We define $\operatorname{Centr}^n(Z,a)[E]=[E']$, where $E'$ is determined by the $n$-cubic extension $F'$ with initial object $F_n'$, with initial morphisms $f_i'=\operatorname{pr}_{i}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}l_{F'}$ for $i\in n$, and with $F'_I=F_I$ for all $I\subsetneq n$. The centrality of $F'$ is a consequence of $F$ being central, since the extension $F'$ is a quotient of $F\times {\ensuremath{\mathbb{K}}}(B,n-1)$, which is central as a product of central extensions (see Example \[Example-K(A,n)-as-Extension\]). The functoriality of $\operatorname{Centr}^n(Z,-)$ is now an immediate consequence of the functoriality of $\operatorname{Centr}^1({\ensuremath{\mathsf{L}}}F,-)$. The functor $\operatorname{Centr}^{n}(Z,-)$ preserves terminal objects: indeed, $\operatorname{Centr}^{n}(Z,0)$ is a singleton, because the terminal object of ${\ensuremath{\mathsf{CExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ has direction $0$ by Lemma \[Lemma-Product-and-Direction\]; if $E$ is an $n$-fold central extension of $Z$ by $0$, there is the unique morphism ${E\to 1}$ to testify that $[E]=[1]$. As for binary products, we must define an inverse to the map $$\xymatrix@=10em{\operatorname{Centr}^n(Z, A\times B) \ar@<-.5ex>[r]_-{{\lgroup}\operatorname{Centr}^n(Z,\operatorname{pr}_A), \operatorname{Centr}^n(Z,\operatorname{pr}_B){\rgroup}} & \operatorname{Centr}^n(Z,A)\times \operatorname{Centr}^n(Z,B). \ar@<-.5ex>@{.>}[l]}$$ This inverse takes a couple $([E],[E'])$ and sends it to $[E\times E']$, where the product is taken over $Z$: Lemma \[Lemma-Product-and-Direction\] insures that the direction of $E\times E'$ is $A\times B$, and the two morphisms are easily seen to compose to the respective identities. Indeed, for any couple $([E],[E'])$, the $n$-fold extension $E$ is an element of $\operatorname{Centr}^n(Z,\operatorname{pr}_A)[E\times E']$, while $E'$ is an element of $\operatorname{Centr}^n(Z,\operatorname{pr}_B)[E\times E']$. This proves that the dotted arrow is a section. On the other hand, by the universal property of pullbacks, any $n$-fold extension $H$ of $Z$ by $A\times B$ is connected to $E\times E'$ when $([E],[E'])={\lgroup}\operatorname{Centr}^n(Z,\operatorname{pr}_A), \operatorname{Centr}^n(Z,\operatorname{pr}_B){\rgroup}[H]$. Hence the dotted arrow is a retraction. \[Corollary-Centr\^[n]{}(Z,-)\] When ${\ensuremath{\mathcal{X}}}$ is a semi-abelian category, the functor $\operatorname{Centr}^{n}(Z,-)$ lifts uniquely over the forgetful functor ${{\ensuremath{\mathsf{Ab}}}\to{\ensuremath{\mathsf{Set}}}}$ to yield a functor $$\operatorname{Centr}^{n}(Z,-)\colon {{\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}}.$$ In particular, any $\operatorname{Centr}^{n}(Z,A)$ carries a canonical abelian group structure. The geometry of higher central extensions {#Section-Geometry} ========================================= We give a geometrical interpretation of the concept of higher central extension, essentially a higher-dimensional version of Bourn and Gran’s result [@Bourn-Gran] that a one-cubic extension $f\colon {X\to Z}$ is central if and only if its kernel $A$ is abelian and its kernel pair $(\operatorname{Eq}(f),f_{0},f_{1})$ is the product $A\times X$ with $f_{0}=\operatorname{pr}_{X}$ and $f_{1}=\varphi_{\ker f,1_{X}}$ as in Subsection \[Commutators\]. Our Theorem \[Theorem-Higher-Centrality\] in essence says that an $n$-cubic extension $F$ is central if and only if 1. the direction of $F$ is abelian, and 2. any face in any $n$-fold diamond in $F$ is uniquely determined by an element of the direction of $F$. In the following sections this will lead to an equivalence between torsors and central extensions, Theorem \[Theorem-Torsor-Equivalence\], which in turn will lead to our main result on cohomology, Theorem \[Main-Theorem\]. Higher equivalence relations ---------------------------- Recall that a [double equivalence relation]{} is an equivalence relation of equivalence relations: given two (internal) equivalence relations $R_{0}$ and $R_{1}$ on an object $X$, it is an equivalence relation ${R\rightrightarrows R_{1}}$ on the relation ${R_{0}\rightrightarrows X}$ as in the diagram below: $$\xymatrix@=3em{R \ar@<-.5ex>[d]_{\operatorname{pr}^{0}_0} \ar@<.5ex>[d]^{\operatorname{pr}^{0}_1} \ar@<-.5ex>[r]_{\operatorname{pr}^{1}_0} \ar@<.5ex>[r]^{\operatorname{pr}^{1}_1} & R_{1} \ar@<-.5ex>[d]_-{r^{1}_{0}} \ar@<.5ex>[d]^-{r^{1}_{1}}\\ R_{0} \ar@<-.5ex>[r]_-{r^{0}_{0}} \ar@<.5ex>[r]^-{r^{0}_{1}} & X.}$$ That is, each of the four pairs of parallel morphisms on this diagram represents an equivalence relation, and these relations are compatible in an obvious sense. For instance, $R_{1} {\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}R_{0}$ denotes the largest double equivalence relation on $R_{0}$ and $R_{1}$, a two-dimensional version of $\nabla_{X}$; see [@CPP; @Smith; @Borceux-Bourn; @Bourn2003; @Janelidze-Pedicchio]. It “consists of” all quadruples $(\alpha, \beta, \gamma, \delta)$ in $X^{4}$ in the configuration $$\vcenter{\xymatrix@1@!0@=3.5em{\gamma \ar@{.}[r] \ar@{.}[d]|-{1} & \beta \ar@{.}[d] \\ \delta \ar@{.}[r]|-{0} & \alpha,}}$$ a $2\times 2$ matrix where $(\delta,\alpha)$, $(\gamma,\beta)\in R_{0}$ and $(\alpha,\beta)$, $(\delta,\gamma)\in R_{1}$. We shall be especially interested in the particular case where $R$ is induced by a two-cubic extension $F$ as in Diagram , as follows: $R_{0}={\operatorname{Eq}(c)}$ is the kernel pair of $c$, the relation $R_{1}={\operatorname{Eq}(d)}$ is the kernel pair of $d$ and $R={\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}$. It is easily seen that then the rows and columns of the induced diagram $$\label{Blokske} \vcenter{\xymatrix{{\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)} \ar@<.5ex>[r]^-{p_1} \ar@<-.5ex>[r]_-{p_0} \ar@<.5ex>[d]^-{r_1} \ar@<-.5ex>[d]_-{r_0} & {\operatorname{Eq}(d)} \ar[r]^-{p} \ar@<.5ex>[d]^-{d_1} \ar@<-.5ex>[d]_-{d_0} & {\operatorname{Eq}(g)} \ar@<.5ex>[d]^-{g_{1}} \ar@<-.5ex>[d]_-{g_{0}} \\ {\operatorname{Eq}(c)} \ar@<.5ex>[r]^-{c_1} \ar@<-.5ex>[r]_-{c_0} \ar[d]_-{r} & X \ar[r]^-{c} \ar[d]_-{d} & C \ar[d]^-{g}\\ {\operatorname{Eq}(f)} \ar@<.5ex>[r]^-{f_{1}} \ar@<-.5ex>[r]_-{f_{0}} & D \ar[r]_-{f} & Z}}$$ are exact forks, so consist of (effective) equivalence relations with their coequalisers; it is a denormalised $3\times 3$ diagram as studied in [@Bourn2003]. Since the “elements” of $X$ may now be viewed as arrows with a domain in $D$ and a codomain in $C$, any “element” of ${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}$ corresponds to a [(two-fold) diamond]{} [@Janelidze-Pedicchio] in the two-cubic extension $F$: $$\label{Two-Diamond} \vcenter{\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\gamma} \ar[rd]_-{\delta} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\alpha}\\ &{\cdot}}}\qquad\qquad \vcenter{\xymatrix@1@!0@=2em{\gamma \ar@{.}[rr] \ar@{.}[dd] & {\cdot} & \beta \ar@{.}[dd] \\ {\cdot} \ar[ru] \ar[rd] && {\cdot} \ar[lu] \ar[ld]\\ \delta \ar@{.}[rr] & \cdot & \alpha}}\qquad\qquad \vcenter{\xymatrix@1@!0@=4em{\gamma \ar@{.}[r] \ar@{.}[d]|-{1} & \beta \ar@{.}[d] \\ \delta \ar@{.}[r]|-{0} & \alpha}}$$ Note the geometrical duality here, which at this level is almost invisible since the dual of a square is a square. This will become more manifest in higher degrees. In some sense ${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}$ is a kind of *denormalised direction* of $F$ (where the kernels are replaced by kernel pairs), also in that ${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}$ may be considered as ${\operatorname{Eq}^2(F)}$—see Diagram  and compare with Definition \[Definition-Centr\] for $n=2$. Inductively, an [$n$-fold equivalence relation]{} may be defined as an equivalence relation of $(n-1)$-fold equivalence relations. Considered as a diagram in the base category ${\ensuremath{\mathcal{X}}}$, it has $n$ underlying equivalence relations $R_{0}$, …, $R_{n-1}$ on a common object $X$. An internal $n$-fold equivalence relation is the same thing as an internal $n$" fold groupoid ($n$-cat-group in the case of groups [@Loday]; double categories appear in [@Benabou:Bicategories; @Ehresmann], for instance) in which all pairs of projections are jointly monomorphic. The *largest* $n$-fold equivalence relation on $n$ given equivalence relations $R_{0}$, …, $R_{n-1}$ on an object $X$—meaning that it contains all $n$-fold equivalence relations on those relations—is denoted $$\label{blokske Ri} \bigboxvoid_{i\in n}R_{i}.$$ It has projections $\operatorname{pr}^{i}_{0}$ and $\operatorname{pr}^{i}_{1}$ to $R_{i}$, for all $i\in n$, and thus consists of $2^{n}$ commutative cubes of projections, one for each choice of projection (either $\operatorname{pr}^{i}_{0}$ or $\operatorname{pr}^{i}_{1}$) in each direction $i\in n$. This largest $n$-fold equivalence relation on $R_{0}$, …, $R_{n-1}$ does indeed exist; the elements of $\bigboxvoid_{i\in n}R_{i}$ are $n$-dimensional matrices in $X$, in fact matrices of order $$\underbrace{2\times \cdots \times 2}_{n}.$$ In the $i$-th direction of the matrix (counting from $0$ to $n-1$) the elements are related by the equivalence relation $R_{i}$. In Subsection \[constructing blokske\] we give a two-step formal construction. In practice the $n$-fold equivalence relation will be induced by an $n$-cubic extension $F$, by taking $R_{i}={\operatorname{Eq}(f_{i})}$. The induced object ${\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}={\operatorname{Eq}^n(F)}}$ may then be considered as a denormalised direction of $F$. Its elements are called [ diamonds in $F$]{} because of their shape in the lower dimensions. When $F$ is a three-cubic extension (see Figure \[Figure-Blokske\]) such a diamond is a hollow octahedron (see Figure \[Figure-Diamond\]) of which the faces are elements of $F_{3}$. We name the faces of the octahedron by the vertices of a cube which is formally a three-dimensional matrix where ${\operatorname{Eq}(f_{0})}$ is the left-right relation, ${\operatorname{Eq}(f_{1})}$ is bottom-top and ${\operatorname{Eq}(f_{2})}$ is front-back. Note how the geometrical duality between the octahedron and the cube is explicit here. Formal construction of $\bigboxvoid_{i\in n}R_{i}$ {#constructing blokske} -------------------------------------------------- *Starting with an $n$-fold arrow.* Given an $n$-fold arrow $F$, first consider it as an arrow $\operatorname{dom}F\to \operatorname{cod}F$ between $(n-1)$-fold arrows, and then take its kernel pair ${\operatorname{Eq}(F)\rightrightarrows \operatorname{dom}F}$. By Corollary 3.10 in [@EGVdL], both projections are $n$-cubic extensions in ${\ensuremath{\mathcal{X}}}$ if such is $F$. Then consider those $(n-1)$-fold arrows as (vertical) arrows between $(n-2)$-fold arrows and take kernel pairs, obtaining a double equivalence relation $$\xymatrix{{\operatorname{Eq}^2(F)} \ar@<.5ex>[r] \ar@<-.5ex>[r] \ar@<.5ex>[d] \ar@<-.5ex>[d] & {\operatorname{Eq}(\operatorname{dom}F)} \ar@<.5ex>[d] \ar@<-.5ex>[d] \\ \operatorname{dom}{\operatorname{Eq}(F)} \ar@<.5ex>[r] \ar@<-.5ex>[r] & \operatorname{dom}^{2}F}$$ of $(n-2)$-fold arrows. All commutative squares in it are $n$-cubic extensions in ${\ensuremath{\mathcal{X}}}$ if such is $F$, again by [@EGVdL Corollary 3.10]. Repeat the process until an $n$-fold equivalence relation in ${\ensuremath{\mathcal{X}}}$ is obtained. The object ${\operatorname{Eq}^n(F)}$ is $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$. *Starting with equivalence relations $(R_{i})_{i\in n}$.* Take coequalisers $f_{i}$ so that each $R_{i}$ is ${\operatorname{Eq}(f_{i})}$. Take pushouts of the $f_{i}$ along each other and along their pushouts until an $n$-fold arrow $F$ is obtained. Now we can apply the above construction to obtain $\bigboxvoid_{i\in n}R_{i}=\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$. Indexing the elements of $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$ {#indexing} ------------------------------------------------------------------------- Consider an $n$-cubic extension $F$. An element of $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$ being an $n$-dimensional matrix, its entries are indexed by the elements of $2^{n}$, the subsets of the ordinal $n$. An entry $x_{I}$ in a matrix ${x\in\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}}$ finds itself in the first entry of the $i$-th direction when ${i\not \in I}$ and in the second entry of the $i$-th direction when $i\in I$. Hence the entry $x_{I}=\operatorname{pr}_{I}(x)$ is $$\label{diagonal} (\operatorname{pr}^{0}_{\delta_{I}(0)}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}^{1}_{\delta_{I}(1)}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\cdots{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}^{n-1}_{\delta_{I}(n-1)})(x)$$ where $$\delta_{I}(i)=\begin{cases} 0 & \text{if $i\not\in I$}\\ 1 & \text{if $i\in I$} \end{cases}$$ and $\operatorname{pr}^{i}_{0}$ and $\operatorname{pr}^{i}_{1}$ are the first and second projection of ${\operatorname{Eq}(f_{i})}$, extended to morphisms $$\bigboxvoid_{j\in k}{\operatorname{Eq}(f_{j})}\to \bigboxvoid_{j\in k\setminus \{i\}}{\operatorname{Eq}(f_{j})}$$ for all $i<k\leq n$ (see Figure \[Figure-Blokske\]). Two entries $x_{I}$ and $x_{J}$ are related by ${\operatorname{Eq}(f_{i})}$ when the only difference between $I$ and $J$ is that one does, and the other does not, contain $i$. So $(x_{I},x_{J})\in {\operatorname{Eq}(f_{i})}$ when $J=I\cup\{i\}$ or $I=J\cup\{i\}$. For instance, in Figure \[Figure-Diamond\], the face $\beta$ corresponds to the entry $x_2$: the set ${2\subseteq 3}$ contains $0$ and $1$ but it doesn’t contain $2$. The induced $n$-cubes {#Chosen-cubes} --------------------- As explained in the paragraph following  above, given an $n$-cubic extension $F$, any choice of a set $I\subseteq n$ corresponds to one of the commutative $n$-cubes in the $n$-fold equivalence relation $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$, namely the cube whose diagonal  “picks” the $I$’th entry of any given $n$-fold diamond. We shall denote it ${\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I)$. In fact, it again forms an $n$-cubic extension in ${\ensuremath{\mathcal{X}}}$, and its initial morphisms are the $$\operatorname{pr}^{i}_{\delta_{I}(i)}\colon\bigboxvoid_{j\in n}{\operatorname{Eq}(f_{j})}\to \bigboxvoid_{j\in n\setminus \{i\}}{\operatorname{Eq}(f_{j})}.$$ The property of being an extension follows, for instance, from the fact that all its morphisms are compatibly split (by the reflexivity of all the equivalence relations involved). Since no confusion with the other arrows is possible (compare with the notation introduced in Subsection \[HDA\]), we shall denote such a “composed splitting” or “composed projection” $${\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I)^{J}_{K}\colon{{\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I)_{J}\to {\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I)_{K}}$$ when $J\subseteq K\in 2^{n}$ or $K\subseteq J\in 2^{n}$, respectively. The objects $\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$ --------------------------------------------------------------- Given an $n$-cubic extension $F$ and $I\subseteq n$, the elements of the object $\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$ are diamonds in $F$ with the $I$-face missing, or equivalently, $n$-dimensional matrices (of order $2\times\cdots\times 2$) with the $I$-entry left out; it is the limit ${\ensuremath{\mathsf{L}}}({\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,n\setminus I))$ from Subsection \[HDA\] determined by the $n$-cubic extension ${\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,n\setminus I)$. Indeed, removing an $I$’th vertex from a cube is the same thing as considering only those faces which contain the complementary $(n\setminus I)$’th vertex. Let $$\pi^{I}=l_{\boxvoid(F,n\setminus I)}\colon \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$$ denote the canonical projection which forgets the $I$-face, then clearly the kernel of $\pi^{I}$ is isomorphic to the direction of $F$. (If a diamond is in the kernel of $\pi^{I}$ then all faces but one in this diamond are zero, and of course this face has its boundary zero.) In fact, this gives us a version of Lemma \[Lemma-DeltaLambda-Square\], valid for higher extensions: \[Lemma-Diamond-Pullback\] For any $n$-cubic extension $F$, the square $$\label{Diamond-Pullback} \vcenter{\xymatrix{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar[d]_{\operatorname{pr}_{I}} \ar[r]^-{\pi^{I}} & \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})} \ar[d] \\ F_{n} \ar[r]_-{{\lgroup}f_{i}{\rgroup}_{i}} & {\ensuremath{\mathsf{L}}}F}}$$ is a pullback. This follows from Lemma \[Lemma-Iso-Pullback\] since ${\operatorname{Ker}({\lgroup}f_{i}{\rgroup}_{i})} = {\ensuremath{\mathsf{D}}}_{(n,Z)} F={\operatorname{Ker}(\pi^{I})}$ as explained above. For instance, in $\bigboxdot_{i\in 3}^{2}{\operatorname{Eq}(f_{i})}$ we have $3$-fold diamonds as in Figure \[Figure-Diamond\] in which the face $\beta=x_{2}$ is missing. In degree two the pullback ${\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}$ (mentioned in the introduction, and computed as in Diagram ) that contains two-fold diamonds in which the face $\delta$ is missing, is nothing but ${{\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxdot}}}^{\varnothing} {\operatorname{Eq}(c)}}$, and the projection $\pi$ is $\pi^{\varnothing}$. Analysis of centrality in degree two {#Degree two} ------------------------------------ As explained in [@RVdL] (and, in full generality, in the proof of Theorem \[Theorem-Higher-Centrality\] below), the two-cubic extension $F$ from Diagram  is central if and only if in the diagram $$\xymatrix{& {\langle}{\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}{\rangle}\ar@{ >>}[r]^-{{\langle}\pi{\rangle}} \ar@{{ |>}->}[d] & {\langle}{\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}{\rangle}\ar@{{ |>}->}[d] \\ A \ar@{{ |>}->}[r] \ar@{ >>}[d] & {\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)} \ar@{}[rd]|-{\texttt{(i)}} \ar@<-.5ex>@{ >>}[r]_-{\pi} \ar@{ >>}[d] & {\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)} \ar@{.>}@<-.5ex>[l]_-{\iota} \ar@{ >>}[d]\\ {\operatorname{Ker}({\ensuremath{\mathsf{ab}}}\pi)} \ar@{{ |>}->}[r] & {\ensuremath{\mathsf{ab}}}({\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}) \ar@{ >>}@<-.5ex>[r]_-{{\ensuremath{\mathsf{ab}}}\pi} & {\ensuremath{\mathsf{ab}}}({\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}) \ar@<-.5ex>[l]_-{\nu}}$$ the morphism ${\langle}\pi{\rangle}$ is an isomorphism. (Recall the bracket notation from .) By Lemma \[Lemma-Iso-Pullback\], this occurs when the square $\texttt{(i)}$ is a pullback, which is precisely saying that $\pi$ is a one-cubic trivial extension. (Indeed $\pi$ is a one-cubic extension, because it is the comparison to the pullback in a two-cubic extension, in fact in a double split epimorphism.) Note that ${\ensuremath{\mathsf{ab}}}\pi$ is a split epimorphism by Lemma \[Lemma-Naturally-Maltsev\], because ${\ensuremath{\mathsf{ab}}}$ preserves the pullback ${\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}$: in fact, it preserves all pullbacks of split epimorphisms along split epimorphisms, or even all pullbacks of split epimorphisms along cubic extensions (Lemma \[Lemma-Pullbacks\]). This makes ${\ensuremath{\mathsf{ab}}}\pi$ a product projection. Further recall that the kernel of $\pi$ is the direction of $F$. Finally, note that the splitting $\nu$ commutes with the sections in the double equivalence relation ${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}$ and the ones induced to ${\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}$. In fact, it is uniquely determined by this property (Lemma \[Lemma-Naturally-Maltsev\]). Hence if $F$ is central then $\pi$ is a split epimorphism, in fact a product projection (since product projections are stable under pullbacks), and thus we see that $$\label{Iso-Dimension-Two'} {\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}\cong A\times ({\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)})$$ where $A$ is the direction of $F$, an abelian object. Conversely, whenever $A$ is abelian and $\pi$ is the projection in the product , the extension $\pi$ is trivial, so that the square $\texttt{(i)}$ is a pullback, and $F$ is a two-cubic central extension. The inclusion $\iota$ of ${\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}$ into ${\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}$ is *compatible with degeneracies* in the following sense. The conditions which determine $\nu$ uniquely in Lemma \[Lemma-Naturally-Maltsev\] extend to the splitting $\iota$ of $\pi$, so that $\iota$ maps $$\vcenter{\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\gamma} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\beta}\\ &{\cdot}}}\qquad \text{to}\qquad \vcenter{\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\gamma} \ar[rd]_-{\gamma} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\beta}\\ &{\cdot}}} \qquad\text{and}\qquad \vcenter{\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\beta} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\alpha}\\ &{\cdot}}}\qquad\text{to}\qquad \vcenter{\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\beta} \ar[rd]_-{\alpha} && {\cdot} \ar[lu]_-{\beta} \ar[ld]^-{\alpha}\\ &{\cdot}}}.$$ Note how the conditions satisfied by a Mal’tsev operation appear here. We may also view this slightly differently: by Lemma 3.3 in [@Bourn-Gran-Maltsev], the condition $[{\operatorname{Eq}(d)},{\operatorname{Eq}(c)}]=\Delta_{X}$ in  is equivalent to the morphism $$\pi\colon{{\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}\to {\operatorname{Eq}(d)}\times_{X} {\operatorname{Eq}(c)}}$$ being a split epimorphism, compatible with certain splittings as in Lemma \[Lemma-Naturally-Maltsev\]. Also $\Delta_{X}=[{\operatorname{Eq}(d)}\cap {\operatorname{Eq}(c)},\nabla_{X}]$ if and only if $\pi$ is central [@Gran-Alg-Cent]. Now a split epimorphism is a one-cubic central extension if and only if it is a one-cubic trivial extension, so $\pi$ is trivial—the square $\texttt{(i)}$ is a pullback—when the commutators $[{\operatorname{Eq}(d)},{\operatorname{Eq}(c)}]$ and $[{\operatorname{Eq}(d)}\cap {\operatorname{Eq}(c)},\nabla_{X}]$ vanish. Higher degrees -------------- This characterisation of centrality goes up to higher dimensions. The basic idea is to show by induction that an $n$-cubic extension $F$ is central if and only if the morphisms $${\langle}\pi^{I}{\rangle}\colon {\langle}\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}{\rangle}\to {\langle}\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}{\rangle}$$ are isomorphisms. As we shall see, this then amounts to an isomorphism $$\label{Iso-General-n} \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\cong A\times \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$$ where $A$ is the direction of $F$: any missing face in an $n$-fold diamond is completely determined by an element in $A$. \[Lemma-ab-of-Square\] When ${\ensuremath{\mathcal{X}}}$ is a semi-abelian category, the functor ${\ensuremath{\mathsf{ab}}}\colon{{\ensuremath{\mathcal{X}}}\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})}$ preserves any limit $\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$ induced by any $n$-cubic extension $F$ and $I\subseteq n$. Furthermore, the comparison morphism $${\ensuremath{\mathsf{ab}}}\pi^{I}\colon{\ensuremath{\mathsf{ab}}}(\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}) \to \bigboxdot_{i\in n}^{I}{\ensuremath{\mathsf{ab}}}({\operatorname{Eq}(f_{i})})$$ admits a splitting. This splitting is uniquely determined by the property that it commutes with the sections in the $n$-cubic extension ${\ensuremath{\mathsf{ab}}}({\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,n\setminus I))$ and the induced sections to the limit object $\bigboxdot_{i\in n}^{I}{\ensuremath{\mathsf{ab}}}({\operatorname{Eq}(f_{i})})$. The first part follows from Lemma \[Lemma-Pullbacks\], because by Lemma \[Lemma-L-pullbacks\], the limit ${\ensuremath{\mathsf{L}}}({\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,n\setminus I))$ may be computed by repeated pullbacks of regular epimorphisms along split epimorphisms. Indeed, all arrows in the cube ${\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,n\setminus I)$ are (compatibly) split, so that the pullback in the statement of Lemma \[Lemma-L-pullbacks\] is a pullback of a split epimorphism along a split epimorphism. Now Lemma \[Lemma-L-pullbacks\] is applied to the induced $(n-1)$-cubic extension $G$, in which all morphisms are (compatibly) split, except in the direction ${\lgroup}d,c{\rgroup}$. We now take a pullback of an extension along a split epimorphism. This procedure is repeated until nothing but a pullback of a regular epimorphism along a split epimorphism is left. Again, by Lemma \[Lemma-Pullbacks\] all those pullbacks are preserved. In the abelian category ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathsf{Arr}}}^{n-1}({\ensuremath{\mathcal{X}}}))$, the reflection ${\lgroup}{\ensuremath{\mathsf{ab}}}d,{\ensuremath{\mathsf{ab}}}c{\rgroup}$ of ${\lgroup}d,c{\rgroup}$ is a split epimorphism by Lemma \[Lemma-Naturally-Maltsev\]. There is actually a unique splitting, compatible with the sections in the square , coming from the fact that the ${\operatorname{Eq}(f_{i})}$ are equivalence relations. Also at each further stage of the above proof we may now apply Lemma \[Lemma-Naturally-Maltsev\], so that at every stage a pullback of a split epimorphism along a split epimorphism is taken, and eventually the needed morphism $\nu^{I}$ is obtained. The requirement that at each stage the chosen splitting commutes with the given sections determines $\nu^{I}$ uniquely. Recall from Subsection \[Subsection-Tower\] the notation ${\langle}F{\rangle}^{n}$ for the initial object of the kernel of the unit of the centralisation of an $n$-cubic extension $F$, which vanishes if and only if $F$ is central. It determines a functor ${\langle}-{\rangle}^{n}\colon {{\ensuremath{\mathsf{Ext}}}^{n}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathcal{X}}}}$. In particular, ${\langle}-{\rangle}={\langle}-{\rangle}^{0}\colon {\ensuremath{\mathcal{X}}}\to {\ensuremath{\mathcal{X}}}$ is the kernel of the unit of the abelianisation functor as in . \[Theorem-Higher-Centrality\] In a semi-abelian category, let $F$ be an $n$-cubic extension with direction $A$. Then the following are equivalent: 1. $F$ is central; 2. the $n$-cubic extension ${\langle}{\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I){\rangle}$ is a limit $n$-cube; 3. the morphism ${\langle}\pi^{I}{\rangle}\colon {\langle}\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}{\rangle}\to {\langle}\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}{\rangle}$ is an isomorphism; 4. $A$ is abelian and $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\cong A\times \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$: we have a short exact sequence $$\xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r] & \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar@{ >>}[r]^-{\pi^{I}} & \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})} \ar[r] & 0}$$ where $\pi^{I}$ is a product projection; for any, hence for all, ${I\subseteq n}$. Furthermore, when these conditions are satisfied, there is a unique splitting $$\iota^{I}\colon \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}\to \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$$ of $\pi^{I}$ which commutes with the given sections in the $n$-cubic extension ${\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,n\setminus I)$ and the induced sections to the object $\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$. First we show that (i) and (ii) are equivalent. The $n$-cubic extension $F$, considered as a morphism ${\operatorname{dom}F\to \operatorname{cod}F}$, is central if and only if either one of the projections ${{\operatorname{Eq}(F)}\to \operatorname{dom}F}$ is trivial. This, by definition of trivial extensions, occurs when the morphisms $$\xymatrix{{\langle}{\operatorname{Eq}(F)}{\rangle}^{n-1} \ar@<.5ex>[r] \ar@<-.5ex>[r] & {\langle}\operatorname{dom}F{\rangle}^{n-1}}$$ are isomorphisms (see Subsection \[Subsection-Tower\] for more details). By Lemma \[Lemma-Iso-Pullback\] this happens when either one of the commutative squares in $$\xymatrix{{\langle}{\operatorname{Eq}^2(F)}{\rangle}^{n-2} \ar@<.5ex>[r] \ar@<-.5ex>[r] \ar@<.5ex>[d] \ar@<-.5ex>[d] & {\langle}{\operatorname{Eq}(\operatorname{dom}F)}{\rangle}^{n-2} \ar@<.5ex>[d] \ar@<-.5ex>[d] \\ {\langle}\operatorname{dom}{\operatorname{Eq}(F)}{\rangle}^{n-2} \ar@<.5ex>[r] \ar@<-.5ex>[r] & {\langle}\operatorname{dom}^{2}F{\rangle}^{n-2}}$$ is a pullback. This, in turn, is equivalent to either one of the commutative cubes in $$\xymatrix@!0@R=4em@C=6em{& {\langle}{\operatorname{Eq}^3(F)}{\rangle}^{n-2} \ar@{.>}@<.5ex>[dd] \ar@{.>}@<-.5ex>[dd] \ar@<.5ex>[rr] \ar@<-.5ex>[rr] \ar@<.5ex>[dl] \ar@<-.5ex>[dl] && {\langle}{\operatorname{Eq}^2(\operatorname{dom}F)}{\rangle}^{n-2} \ar@<.5ex>[dd] \ar@<-.5ex>[dd] \ar@<.5ex>[dl] \ar@<-.5ex>[dl] \\ {\langle}{\operatorname{Eq}(\operatorname{dom}{\operatorname{Eq}(F)})}{\rangle}^{n-2} \ar@<.5ex>[dd] \ar@<-.5ex>[dd] \ar@<.5ex>[rr] \ar@<-.5ex>[rr] && {\langle}{\operatorname{Eq}(\operatorname{dom}^{2}F)}{\rangle}^{n-2} \ar@<.5ex>[dd] \ar@<-.5ex>[dd]\\ & {\langle}\operatorname{dom}{\operatorname{Eq}^2(F)}{\rangle}^{n-2} \ar@{.>}@<.5ex>[rr] \ar@{.>}@<-.5ex>[rr] \ar@{.>}@<.5ex>[dl] \ar@{.>}@<-.5ex>[dl] && {\langle}\operatorname{dom}{\operatorname{Eq}(\operatorname{dom}F)}{\rangle}^{n-2} \ar@<.5ex>[dl] \ar@<-.5ex>[dl] \\ {\langle}\operatorname{dom}^{2} {\operatorname{Eq}(F)}{\rangle}^{n-2} \ar@<.5ex>[rr] \ar@<-.5ex>[rr] && {\langle}\operatorname{dom}^{3}F{\rangle}^{n-2}}$$ being a limit cube. This process continues until we obtain a cube of dimension $n$ whose vertices are brackets ${\langle}-{\rangle}$ and whose edges are parallel pairs of arrows as in the diagrams above. This cube is precisely the $n$-fold equivalence relation ${\langle}\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}{\rangle}$, considered as a diagram in ${\ensuremath{\mathcal{X}}}$—compare with the construction in Subsection \[constructing blokske\]. As in Subsection \[Chosen-cubes\], a choice of ${I\subseteq n}$ picks one of any two parallel arrows in this diagram in such a way that we obtain the $n$-cubic extension ${\langle}{\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I){\rangle}$. The equivalence between (ii) and (iii) is clear because ${\langle}{\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I){\rangle}$ is nothing but one of the cubes induced by choosing an $n$-fold arrow (making a choice of projections) in the $n$-fold equivalence relation ${\langle}\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}{\rangle}$ as in Subsection \[Chosen-cubes\]; so ${\langle}\pi^{I}{\rangle}$ is an isomorphism if and only if this cube is a limit. The functor ${\langle}-{\rangle}$ does indeed preserve the limit $\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})}$, since so does ${\ensuremath{\mathsf{ab}}}$ by Lemma \[Lemma-ab-of-Square\]. Now we prove the equivalence between (iii) and (iv). Condition (iii) is equivalent to the square $$\label{Square-Blokskes} \vcenter{\xymatrix{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar@{ >>}[r]^-{\pi^{I}} \ar@{ >>}[d] & \bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})} \ar@{ >>}[d]\\ {\ensuremath{\mathsf{ab}}}(\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}) \ar@{ >>}[r]_-{{\ensuremath{\mathsf{ab}}}\pi^{I}} & {\ensuremath{\mathsf{ab}}}(\bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})})}}$$ being a pullback, which means that $\pi^{I}$ is a one-cubic trivial extension. Since its kernel is the abelian object $A$, the extension $\pi^{I}$ is a product projection if and only if it is a split epimorphism. By Lemma \[Lemma-ab-of-Square\], the latter condition does indeed hold. The final statement is again a consequence of Lemma \[Lemma-ab-of-Square\]: the needed morphism $\iota^{I}$ is induced by the pullback and the splitting $\nu^{I}$ of ${\ensuremath{\mathsf{ab}}}\pi^{I}$ given by the lemma. In what follows we shall use this result to obtain one half of the equivalence between torsors and central extensions. \[Remark-Full-3\] Note that the splitting $\iota^{I}$ of $\pi^{I}$ constructed in the proof above is natural in $F$, so that also the product decompositions (iv) are natural in the extension considered. \[Remark-Tomas-Centrality\] The proof of Theorem \[Theorem-Higher-Centrality\] shows that an $n$-cubic extension $F$ is central precisely when, for any $I\subseteq n$, the induced $(n+1)$-cubic extension $${{\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I)\to {\ensuremath{\mathsf{ab}}}({\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I))}$$ is a limit $(n+1)$-cube. In fact, these $(n+1)$-cubic extensions are part of the regular epimorphism of $n$-fold groupoids $$\eta_{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}}\colon{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to {\ensuremath{\mathsf{ab}}}(\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})})},$$ which therefore is a discrete fibration if and only if $F$ is central. (The concept of [discrete fibration]{} between higher-dimensional internal groupoids is the obvious extension of the one-fold groupoid case: any of its induced $n$-fold arrows must be a pullback. In the situation at hand this gives precisely the condition on the $(n+1)$-cubes ${{\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I)\to {\ensuremath{\mathsf{ab}}}({\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,I))}$ mentioned above.) In the article [@Everaert-Gran-nGroupoids], the authors study the Galois structure for $n$-fold groupoids in a semi-abelian category ($n$-cat-groups in ${\ensuremath{\mathsf{Gp}}}$, for instance [@Loday]) induced by the reflection $$\vcenter{\xymatrix{{{\ensuremath{\mathsf{Gpd}}}^{n}({\ensuremath{\mathcal{X}}})} \ar@<1ex>[r]^-{\Pi^{n}_{0}} \ar@{}[r]|-{\perp} & {\ensuremath{\mathsf{Dis}}}^{n}({\ensuremath{\mathcal{X}}})\simeq {\ensuremath{\mathcal{X}}}\ar@<1ex>[l]^-{\supset}}}$$ to ${\ensuremath{\mathcal{X}}}$ via the “connected components” functor to discrete $n$-fold groupoids. It turns out [@Everaert-Gran-nGroupoids Proposition 2.9] that the central extensions with respect to this reflection are again the regular epimorphisms of internal $n$-fold groupoids which are discrete fibrations. Hence an $n$-cubic extension $F$ in ${\ensuremath{\mathcal{X}}}$ is central relative to ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$ if and only if the induced extension of $n$-fold groupoids $\eta_{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}}$ is central relative to ${\ensuremath{\mathcal{X}}}$. Higher central extensions as higher-dimensional pregroupoids {#Maltsev} ------------------------------------------------------------ The isomorphisms  determine “multiplications” or “compositions” of $(n-1)$-dimensional hyper-tetrahedra (or $n$-dimensional hyper-triangles) in an $n$-cubic central extension, in the sense that any aggregation of hyper-tetrahedra in the shape of an $n$-fold diamond with a face missing “composes” to the missing face. That is to say, the composite morphism $$\xymatrix{p^{I}\colon\bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})} \ar[r]^-{{\lgroup}0, 1{\rgroup}} & A\times \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})} \ar[r]^-{\cong} & \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar[r]^-{\operatorname{pr}_{I}} & F_{n}}$$ acts as a *higher-dimensional Mal’tsev operation* or, more precisely, as a *higher-dimensional pregroupoid structure* on $F$. Indeed, Proposition \[Centrality-Sum\] below implies that $p^{I}$ satisfies certain conditions which we could call *higher-dimensional Mal’tsev laws*. In higher degrees those algebraic properties of the $p^{I}$ still have to be further studied—for instance, what about associativity?—but we may already give a few examples. In the two-dimensional case, $\delta=p^{\emptyset}(\alpha,\beta,\gamma)$ is the unique choice of $\delta$ such that the projection $a=\operatorname{pr}_{A}(\alpha,\beta,\gamma,\delta)$ of the diamond $(\alpha,\beta,\gamma,\delta)$ on the direction $A$ is zero. In this case we may think of $\delta$ as a composite $\gamma\beta^{-1}\alpha$. Furthermore, $p^{\emptyset}(\alpha,\alpha,\gamma)=\gamma$, since once $\alpha=\beta$ we have to take $\delta=\gamma$, as already explained in Subsection \[Degree two\]. Proposition \[Centrality-Sum\] below gives us an alternative argument: there is no other choice possible for $\delta$ because $\operatorname{pr}_{A}(\alpha,\alpha,\gamma,\delta)$ has to be zero, and $\delta=\gamma$ is a *valid* choice, since $\operatorname{pr}_{A}(\alpha,\alpha,\gamma,\gamma)=0$, so it is the *uniquely valid* one. Similarly, writing $\delta=p^{\emptyset}(a,b,c,d,\alpha,\beta,\gamma)$ for a configuration such as in Figure \[Figure-Diamond\], we find $$\begin{cases} d=p^{\emptyset}(a,b,c,d,a,b,c)\\ \alpha=p^{\emptyset}(a,b,b,a,\alpha,\beta,\beta)\\ \gamma=p^{\emptyset}(b,b,c,c,\beta,\beta,\gamma). \end{cases}$$ \[Centrality-Sum\] In a semi-abelian category, let $F$ be an $n$-cubic central extension with direction $A$. Then in any product diagram $$\xymatrix{0 \ar[r] & A \ar@<-.5ex>@{{ |>}->}[r]_-{\ker\pi^{I}} & \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar@<-.5ex>@{ >>}[r]_-{\pi^{I}} \ar@{.>}@<-.5ex>[l]_-{\operatorname{pr}_{A}} & \bigboxdot_{i\in n}^{I}{\operatorname{Eq}(f_{i})} \ar[r] \ar@{.>}@<-.5ex>[l]_-{\iota^{I}} & 0}$$ induced by Theorem \[Theorem-Higher-Centrality\], the projection $\operatorname{pr}_{A}$ is an alternating sum $$\label{Sum-Formula} \sum_{J\subseteq n}(-1)^{|J|}\eta_{F_{n}}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{J}$$ where $\operatorname{pr}_{J}\colon{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to F_{n}}$ sends a diamond to its $J$-face. The idea behind the proof may be illustrated as follows in dimension two. (Here we let $F$ be the two-cubic extension from Diagram  to simplify notations.) When the calculation $$\vcenter{\xymatrix@1@!0@=4em{\gamma \ar@{.}[r] \ar@{.}[d]|-{1} & \beta \ar@{.}[d] \\ \delta \ar@{.}[r]|-{0} & \alpha}} - \vcenter{\xymatrix@1@!0@=4em{\gamma \ar@{.}[r] \ar@{.}[d]|-{1} & \beta \ar@{.}[d] \\ \gamma \ar@{.}[r]|-{0} & \beta}} + \vcenter{\xymatrix@1@!0@=4em{\beta \ar@{.}[r] \ar@{.}[d]|-{1} & \beta \ar@{.}[d] \\ \beta \ar@{.}[r]|-{0} & \beta}} - \vcenter{\xymatrix@1@!0@=4em{\beta \ar@{.}[r] \ar@{.}[d]|-{1} & \beta \ar@{.}[d] \\ \alpha \ar@{.}[r]|-{0} & \alpha}} = \vcenter{\xymatrix@1@!0@R=4em@C=6em{0 \ar@{.}[r] \ar@{.}[d]|-{1} & 0 \ar@{.}[d] \\ \delta-\gamma+\beta-\alpha \ar@{.}[r]|-{0} & 0,}}$$ in which we denote the equivalence classes in the quotient by representative elements, is made in the abelian object ${\ensuremath{\mathsf{ab}}}({\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)})$, we see that the result belongs to the kernel $A$ of the projection ${\ensuremath{\mathsf{ab}}}\pi^{\emptyset}$. Indeed, the pullback ${\operatorname{Eq}(d)}\times_{X}{\operatorname{Eq}(c)}$ is preserved by the functor ${\ensuremath{\mathsf{ab}}}$, and the projections to ${\ensuremath{\mathsf{ab}}}{\operatorname{Eq}(d)}$ and ${\ensuremath{\mathsf{ab}}}{\operatorname{Eq}(c)}$ send the above sum to zero. Writing $\eta_{\boxvoid}=\eta_{{\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}}$, this gives us the morphism $$\eta_{\boxvoid}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{\emptyset}\;-\;\eta_{\boxvoid}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{\{1\}}\;+\;\eta_{\boxvoid}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{2}\;-\;\eta_{\boxvoid}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{1}\colon{{\operatorname{Eq}(d)}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}(c)}\to A},$$ clearly a splitting for $\ker \pi^{\emptyset}$; hence by Lemma \[Lemma-Split-Kernel-Product\] it is the needed product projection. Note that the terms in this sum are obtained by projecting a diamond $(\alpha,\beta,\gamma,\delta)$ to a certain subdiamond, and then considering it again as a two-fold diamond via reflexivity. In the first term we do not project at all, in the second term we project to $(\gamma,\beta)$ in ${\operatorname{Eq}(f_{0})}$, in the fourth we project to $(\alpha,\beta)$ in ${\operatorname{Eq}(f_{1})}$, and in the third term we project all the way to $F_{2}$. For general $n$, let us again consider the commutative square —which is a pullback by centrality of $F$—and the induced kernels: $$\xymatrix{A \ar@{=}[d] \ar@{{ |>}->}[r] & \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar@{ >>}[r]^-{\pi^{I}} \ar@{ >>}[d]_-{\eta_{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}}} \pullback & \bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})} \ar@{ >>}[d]^-{\eta_{\bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})}}}\\ A \ar@{{ |>}->}[r] & {\ensuremath{\mathsf{ab}}}(\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}) \ar@{ >>}[r]_-{{\ensuremath{\mathsf{ab}}}\pi^{I}} & {\ensuremath{\mathsf{ab}}}(\bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})})}$$ Since ${\ensuremath{\mathsf{ab}}}(\bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})})=\bigboxdot^{I}_{i\in n}{\ensuremath{\mathsf{ab}}}{\operatorname{Eq}(f_{i})}$ by Lemma \[Lemma-ab-of-Square\], the abelian object $A$ being the kernel of ${\ensuremath{\mathsf{ab}}}\pi^{I}$ implies that it is the direction of ${\ensuremath{\mathsf{ab}}}({\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,n\setminus I))$, which means ${A=\bigcap_{i\in n}{\operatorname{Ker}({\ensuremath{\mathsf{ab}}}\operatorname{pr}^{i}_{\delta_{n\setminus I}(i)})}}$. In order to define a morphism with codomain $A$, we now only need to define a morphism with codomain ${\ensuremath{\mathsf{ab}}}(\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})})$ which becomes zero when composed with the $${\ensuremath{\mathsf{ab}}}\operatorname{pr}^{i}_{\delta_{n\setminus I}(i)}\colon{\ensuremath{\mathsf{ab}}}(\bigboxvoid_{j\in n}{\operatorname{Eq}(f_{j})})\to {\ensuremath{\mathsf{ab}}}(\bigboxvoid_{j\in n\setminus \{i\}}{\operatorname{Eq}(f_{j})}).$$ We shall use this procedure to define a splitting for $\ker\pi^{I}$ as an alternating sum, which will then automatically be the needed product projection by Lemma \[Lemma-Split-Kernel-Product\]. Recall the notation introduced in Subsection \[Chosen-cubes\]. Then, for any $J\subseteq n$, write $I\ominus J$ for the symmetric difference $(I\cup J)\setminus (I\cap J)$ of $I$ and $J$, and put $$\xi_{(F,I,J)}={\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,J)^{n\setminus (I\ominus J)}_{n}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\raisebox{.4mm}{\ensuremath{\boxvoid}}}(F,J)^{n}_{n\setminus (I\ominus J)}\colon{\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}}.$$ This formalises the process of “projecting to a subdiamond, then including again via reflexivity”. Now note that, given any element $x$ of $\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}$, the $I$-entry of $\xi_{(F,I,J)}(x)$ is $x_{J}$. Furthermore, after projecting in any direction $i\in n$ onto ${\operatorname{Eq}(f_{i})}$, every morphism $\operatorname{pr}^{i}_{\delta_{n\setminus I}(i)}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\xi_{(F,I,J)}$ occurs twice: indeed $$\operatorname{pr}^{i}_{\delta_{n\setminus I}(i)}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\xi_{(F,I,J)}=\operatorname{pr}^{i}_{\delta_{n\setminus I}(i)}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\xi_{(F,I,J\cup \{i\})}$$ when $i\not\in J$. Two such terms will cancel each other when the alternating sum below is composed with ${\ensuremath{\mathsf{ab}}}\pi^{I}$. Hence the induced morphism $$\sum_{J\subseteq n}(-1)^{|J|}\eta_{(\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})})}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\xi_{(F,I,J)}\colon\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})}\to {\ensuremath{\mathsf{ab}}}(\bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})})$$ satisfies the conditions required to lift over $A$—that is to say, it becomes zero when we compose it with ${\ensuremath{\mathsf{ab}}}\pi^{I}$. Its $I$-entry is precisely the needed formula , while the other entries are zero. In particular, the morphism  does indeed split the kernel of $\pi^{I}$. Note that the formula  for the projection $\operatorname{pr}_{A}$ is independent of the chosen index $I\subseteq n$. When $n=1$, Proposition \[Centrality-Sum\] reduces to a well-known property of (one-cubic) central extensions (see [@Bourn-Gran]): if $f\colon{X\to Z}$ is central and $x_{0}$, $x_{1}\colon {W\to X}$ are such that $f{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}x_{0}=f{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}x_{1}$, then they induce a unique morphism ${x_{1}-x_{0}\colon{W\to A}}$ to the kernel $A$ of $f$ such that $x_{0}$ and ${x_{1}-x_{0}}$ together determine $x_{1}$. Torsors and centrality {#Section-Torsors-and-Centrality} ====================== We analyse the concept of torsor from the point of view of centrality of higher extensions. We prove that a truncated simplicial resolution of an object $Z$ is a torsor of $Z$ by an abelian object $A$ if and only if the underlying extension is central with direction $A$ (Theorem \[Theorem-Torsor-Equivalence\]; one implication is Proposition \[Proposition-Central-then-Torsor\], the other Proposition \[Proposition-Torsor-then-Central\]). Let $Z$ be an object and $(A,\xi)$ a $Z$-module in a semi-abelian category ${\ensuremath{\mathcal{X}}}$. Recall from Subsection \[Torsors\] that an $n$-torsor of $Z$ by $(A,\xi)$ is an augmented simplicial object ${\ensuremath{\mathbb{T}}}$ together with a simplicial morphism ${\ensuremath{\mathbb{t}}}\colon {{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}((A,\xi),n)}$ such that 1. ${\ensuremath{\mathbb{t}}}$ is a fibration which is exact from degree $n$ on; 2. ${\ensuremath{\mathbb{T}}}\cong {\ensuremath{\mathsf{Cosk}}}_{n-1}{\ensuremath{\mathbb{T}}}$; 3. ${\ensuremath{\mathbb{T}}}$ is a resolution. Why extensions? --------------- Condition (T2) in the definition of $n$-torsor means that (the simplicial object-part ${\ensuremath{\mathbb{T}}}$ of) an $n$-torsor $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ *is* the ${(n-1)}$-truncated simplicial object $T={\ensuremath{\mathsf{tr}}}_{n-1} {\ensuremath{\mathbb{T}}}$ (Subsection \[Truncations\]), in the sense that this is the only information ${\ensuremath{\mathbb{T}}}$ contains. Its initial object is $T_{n}={\ensuremath{\mathbb{T}}}(n)={\ensuremath{\mathbb{T}}}_{n-1}$. Condition (T3) means that the underlying $n$-fold arrow of $T$ is an extension (Subsection \[Resolutions\]). Why trivial actions? -------------------- We shall prove that for an $n$-torsor $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ of an object $Z$ by a $Z$-module $(A,\xi)$ in a semi-abelian category, the action $\xi$ is trivial if and only if the induced one-cubic extension $${\lgroup}{\partial}_{i}{\rgroup}_{i}=l_{T}\colon {T_{n}={\ensuremath{\mathbb{T}}}_{n-1}\to {\triangle}({\ensuremath{\mathbb{T}}},n-1)={\ensuremath{\mathsf{L}}}T}$$ is central with respect to abelianisation. In other words, an $n$-torsor $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ has a trivial action if and only if $$\Bigl[\bigcap_{i\in n} {\operatorname{Eq}({\partial}_{i})},\nabla_{T_{n}}\Bigr]=\Delta_{T_{n}} \qquad \text{or, equivalently,} \qquad \Bigl[\bigcap_{i\in n} {\operatorname{Ker}({\partial}_{i})},T_{n}\Bigr]=0;$$ see Example \[Example-Dimension-One\]. This extends Proposition 3.3 in [@RVdL] to higher dimensions. It also explains why only cohomology *with trivial coefficients* can ever classify higher central extensions: this commutator condition is part of the centrality by Lemma \[Lemma-Direction-Limit\]. \[Proposition-Trivial-Action\] In a semi-abelian category, consider an object $Z$ and a $Z$" module $(A,\xi)$. For any $n$-torsor $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ of $Z$ by $(A,\xi)$, the kernel of $${\lgroup}{\partial}_{i}{\rgroup}_{i}=l_{T}\colon {T_{n}\to {\triangle}({\ensuremath{\mathbb{T}}},n-1)={\ensuremath{\mathsf{L}}}T}$$ is $A$, and the following conditions are equivalent: 1. the action $\xi$ is trivial; 2. the one-cubic extension $l_{T}$ is central; 3. for all $i\in n$ we have ${\triangle}({\ensuremath{\mathbb{T}}},n)\cong A\times {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$; more precisely, $$\widehat{\partial}_{i}\colon {\triangle}({\ensuremath{\mathbb{T}}},n)\to {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$$ is a product projection with kernel $A$. For any $i\in n$, Lemma \[Lemma-DeltaLambda-Square\] tells us that the square $$\xymatrix{{\triangle}({\ensuremath{\mathbb{T}}},n) \ar[r]^-{\widehat{\partial}_{i}} \ar[d]_-{{\partial}_{i}} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n) \ar[d] \\ T_{n} \ar[r]_-{{\lgroup}{\partial}_{i}{\rgroup}_{i}} & {\triangle}({\ensuremath{\mathbb{T}}},n-1)}$$ is a pullback. Note that all its arrows are regular epimorphisms: the morphism ${\partial}_{i}$ as any split epimorphism; ${\lgroup}{\partial}_{i}{\rgroup}_{i}$ since ${\ensuremath{\mathbb{T}}}$ is a resolution; and $\widehat{\partial}_{i}$ either by the Kan property, which all simplicial objects in a semi-abelian category have, or as a pullback of ${\lgroup}{\partial}_{i}{\rgroup}_{i}$. We see that the kernel of ${\lgroup}{\partial}_{i}{\rgroup}_{i}$ is isomorphic to the kernel of $\widehat{\partial}_{i}$ (Lemma \[Lemma-Iso-Pullback\]), and furthermore ${\lgroup}{\partial}_{i}{\rgroup}_{i}$ is central if and only if so is $\widehat{\partial}_{i}$—indeed, central extensions are preserved and reflected by pullbacks of extensions along extensions. Since $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ is an $n$-torsor, also the square $$\xymatrix{{\triangle}({\ensuremath{\mathbb{T}}},n) \ar@<-.5ex>[r]_-{\widehat{\partial}_{i}} \ar[d]_-{{\lgroup}\varsigma,{\partial}_{0}^{n+1}{\rgroup}} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n) \ar@{.>}@<-.5ex>[l] \ar[d]^{{\partial}_{0}^{n}} \\ (A,\xi) \rtimes Z \ar@<-.5ex>[r]_-{p} & Z \ar@<-.5ex>[l]_-{s}}$$ is a pullback, by the exact fibration property. This already proves that the kernel of ${\lgroup}{\partial}_{i}{\rgroup}_{i}$ is $A$ (again Lemma \[Lemma-Iso-Pullback\]). Note that a split epimorphism with abelian kernel represents a trivial action if and only if it is a product projection, if and only if it is a trivial extension, if and only if it is a central extension. Again using that central extensions are preserved and reflected by pullbacks of extensions along extensions we obtain the claimed result. Hence, from now on, we shall only have to consider torsors of $Z$ by a trivial module $(A,\tau)$—we called them *$n$-torsors of $Z$ by $A$* in Subsection \[Torsors\]—and restrict our cohomology theory accordingly. In this case, a torsor “looks” as follows: $$\label{torsordiagram} \resizebox{.9\textwidth}{!}{\mbox{$ \vcenter{\xymatrix@=45pt{ {\triangle}({\ensuremath{\mathbb{T}}},n+1) \ar[d]_{\cdots\quad{\lgroup}{\lgroup}\varsigma\circ{\partial}_{i}{\rgroup}_{i},{\partial}_{0}^{n+2}{\rgroup}} \ar@<2.33ex>[r] \ar@<1.16ex>[r] \ar@<-2.33ex>[r]^-{\vdots} & {\triangle}({\ensuremath{\mathbb{T}}},n) \ar[d]^{{\lgroup}\varsigma,{\partial}_{0}^{n+1}{\rgroup}} \ar@<1.75ex>[r] \ar@<-1.75ex>[r]^-{\vdots} & {\ensuremath{\mathbb{T}}}_{n-1} \ar[d]^-{{\partial}_{0}^{n}} \ar@<1.75ex>[r] \ar@<-1.75ex>[r]^-{\vdots} & {\ensuremath{\mathbb{T}}}_{n-2} \ar[d]^-{{\partial}_{0}^{n-1}} \ar@{}[r]|-{\cdots} & {\ensuremath{\mathbb{T}}}_{0} \ar[d]^-{{\partial}_{0}} \ar[r]^-{{\partial}_{0}} & {\ensuremath{\mathbb{T}}}_{-1} \ar@{=}[d]\\ A^{n+1} \times Z \ar@<2.33ex>[r]^-{{\partial}_{n+1}\times 1_{Z}} \ar@<1.16ex>[r]|-{\operatorname{pr}_{n}\times 1_{Z}} \ar@<-2.33ex>[r]_-{\operatorname{pr}_{0}\times 1_{Z}}^-{\vdots} & A \times Z \ar@<1.75ex>[r]^-{\operatorname{pr}_{Z}} \ar@<-1.75ex>[r]_-{\operatorname{pr}_{Z}}^-{\vdots} & Z \ar@{=}@<1.75ex>[r] \ar@{=}@<-1.75ex>[r]^-{\vdots} & Z \ar@{}[r]|-{\cdots} & Z \ar@{=}[r] & Z}} $}}$$ \[Naturality-Product-Decomposition\] It is clear from the proof that the product decomposition (iii) is natural in $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$, so that any morphism of torsors is compatible with the induced product decompositions. \[Lemma-Direction-Cycle\] For any simplicial resolution ${\ensuremath{\mathbb{X}}}$, the kernel of any induced regular epimorphism $\widehat{\partial}_{i}\colon{{\triangle}({\ensuremath{\mathbb{X}}},n)\to {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n)}$ is the direction $A$ of the underlying $n$-cubic extension $X$. If an $(n,i)$-horn $\widehat x_{i}$ of an $n$-cycle $x$ in ${\ensuremath{\mathbb{X}}}$ is zero, then the $i$-face $x_{i}$ which is missing in the horn must have boundary zero, so that $x_{i}$ belongs to $A$. Conversely, the inclusion of $A$ into ${\triangle}({\ensuremath{\mathbb{X}}},n)$ takes an element $a$ of $A$ and sends it to the $n$-cycle in ${\ensuremath{\mathbb{X}}}$ which is zero everywhere—except in its $i$-entry, where it is $a$. This $n$-cycle is sent to zero by $\widehat{\partial}_{i}$. More formally, this also follows from Lemma \[Direction-as-Kernel\] combined with Lemma \[Lemma-DeltaLambda-Square\], since ${{\lgroup}{\partial}_i{\rgroup}_i = l_X\colon {\ensuremath{\mathbb{X}}}_{n-1}\to {\triangle}({\ensuremath{\mathbb{X}}},n-1)}$: the kernel of $\widehat{\partial}_{i}$ coincides with the kernel of $l_{X}$ since the square in Lemma \[Lemma-DeltaLambda-Square\] is a pullback, and the latter kernel is $A$ by Lemma \[Direction-as-Kernel\]. Multiplying simplices in a torsor {#Multiplication} --------------------------------- As explained in [@Duskin], given an $n$-torsor $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ of $Z$ by $A$ and an integer $i\in n$, the isomorphism $${\triangle}({\ensuremath{\mathbb{T}}},n)\cong A\times {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$$ induces a multiplication or composition of the simplices in a horn to the “missing face” such that the thus completed $n$-cycle “commutes”, in the sense that its projection on $A$ is zero. So a horn may be considered as a *composable aggregation of simplices*—compare with the higher Mal’tsev structures $p^{I}$ from Subsection \[Maltsev\]. Indeed, we may simply use the morphism $$m^{i}\colon\xymatrix@1{{\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n) \ar[r]^-{{\lgroup}0,1{\rgroup}} & A\times {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n) \ar[r]^-{\cong} & {\triangle}({\ensuremath{\mathbb{T}}},n) \ar[r]^-{{\partial}_{i}} & T_{n}.}$$ This composition of $(n,i)$-horns satisfies certain additional properties [@Duskin], of which for us the most important one is compatibility with degeneracies. From the axioms of torsor (the requirement that ${\ensuremath{\mathbb{t}}}\colon{{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(Z,A,n)}$ be a simplicial morphism) it follows that a degenerate $n$-cycle commutes. Hence any $(n,i)$-horn in ${\ensuremath{\mathbb{T}}}$ which *may be* completed to a degenerate $n$-cycle *has to be* completed this way, and hence composes to the $i$-face of this degenerate $n$-cycle. For instance, in degree two, the left hand side $(2,1)$-horn $$\vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{\sigma_{0}{\partial}_{0}\alpha}\\ {\cdot} \ar[ru]^-{\alpha} && {\cdot}}} \qquad\qquad \vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar@{}[d]|(.7){\sigma_{1}\alpha} \ar[rd]^-{\sigma_{0}{\partial}_{0}\alpha}\\ {\cdot} \ar[ru]^-{\alpha} \ar@{.>}[rr]_-{\alpha} && {\cdot}}}$$ fits into the right hand side degenerate $2$-simplex $\sigma_{1}\alpha$. It follows by uniqueness that $m^{1}(\sigma_{0}{\partial}_{0}\alpha,\alpha)=\alpha$. Likewise, $m^{0}(\alpha,\alpha)=\sigma_{0}{\partial}_{0}\alpha$, etc. The exact fibration property ---------------------------- Most of the fibration property (T1) of a torsor comes for free, since a regular epimorphism of simplicial objects in a regular Mal’tsev category is always a fibration [@EverVdL2 Proposition 4.4]. Given a simplicial morphism ${\ensuremath{\mathbb{t}}}\colon{{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(Z,A,n)}$ satisfying (T2) and (T3), already the ${{\ensuremath{\mathbb{t}}}_{i}={\partial}_0^{i+1}}$ are regular epimorphisms for all $i\in n$, so it suffices to check the regularity of ${\ensuremath{\mathbb{t}}}_{n}$ and ${\ensuremath{\mathbb{t}}}_{n+1}$. Then there is the exactness, but this reduces to one square being a pullback—Diagram  for any $i\in n$—which in turn corresponds to a direction property. \[Isos-are-Free\] Suppose that $Z$ is an object and $A$ is an abelian object in a semi-abelian category. Let ${\ensuremath{\mathbb{t}}}\colon{{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(Z,A,n)}$ be as in the definition of torsors, satisfying conditions [(T2)]{} and [(T3)]{}. Then for every $i$ the square $$\label{Fundamental-Square} \vcenter{\xymatrix{{\triangle}({\ensuremath{\mathbb{T}}},n) \ar[r]^-{\widehat{\partial}_{i}} \ar[d]_-{{\lgroup}\varsigma,{\partial}_{0}^{n+1}{\rgroup}} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n) \ar[d]^{{\partial}_{0}^{n}} \\ A \times Z \ar[r]_-{\operatorname{pr}_{Z}} & Z}}$$ is a pullback if and only if the induced morphism ${\bigcap_{i}{\operatorname{Ker}({\partial}_{i})}\to A}$ is an isomorphism. When this is the case, the simplicial morphism ${\ensuremath{\mathbb{t}}}$ is a fibration, exact from degree $n$ on, so that $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ is an $n$-torsor of $Z$ by $A$. Again, $\widehat{\partial}_{i}$ is a regular epimorphism by the Kan property. As in the proof of Lemma \[Lemma-Direction-Cycle\], the kernel of $\widehat{\partial}_{i}$ is ${\operatorname{Ker}({\lgroup}{\partial}_{i}{\rgroup}_{i})}=\bigcap_{i}{\operatorname{Ker}({\partial}_{i})}$. Via Lemma \[Lemma-Iso-Pullback\] this already proves the equivalence. Recall that every regular epimorphism of simplicial objects in a semi-abelian category is a fibration. When the above square  is a pullback (for any ${i\in n}$), the morphism ${{\partial}_{0}^{n}}$ being regular epimorphic implies that also ${\lgroup}\varsigma,{\partial}_{0}^{n+1}{\rgroup}$ is a regular epimorphism. One degree up, the corresponding squares are automatically pullbacks: indeed, any comparison ${{\triangle}({\ensuremath{\mathbb{T}}},n+1)\to {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n+1)}$ is an isomorphism by the axiom (T2) which tells us that every $n$-simplex in ${\ensuremath{\mathbb{T}}}$ is an $n$-cycle, as is any morphism $$\widehat{\partial}_{i}\colon A^{n+1}\times Z \to {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{K}}}(Z,A,n),n+1)=A^{n+1}\times Z.$$ In higher degrees there is nothing to be checked because ${\ensuremath{\mathbb{t}}}\colon{{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(Z,A,n)}$ is completely determined by the coskeleton construction. This implies that ${\ensuremath{\mathbb{t}}}$ is a regular epimorphism in all degrees, hence it is a fibration. This fibration is exact in degree $n$ since is a pullback for every $i$, and in higher degrees since both its domain and its codomain are constructed as a coskeleton, so that we can apply Lemma \[Lemma coskeleton pullback\]. Thus we see that an $n$-torsor $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ of $Z$ by $A$ has an underlying $n$-cubic extension of $Z$ of which the direction is $A$. Furthermore, the squares  are pullbacks, which means that ${\triangle}({\ensuremath{\mathbb{T}}},n)\cong A\times {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$. Note that the projection on ${\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$ is $\widehat{\partial}_{i}$ and the projection on $A$ is $\varsigma$. In what follows we shall prove that this condition is equivalent to the centrality of the underlying $n$-cubic extension. Given an $n$-cubic central extension $T$ of $Z$ by $A$, we construct a simplicial morphism ${\ensuremath{\mathbb{t}}}\colon{{\ensuremath{\mathbb{T}}}={\ensuremath{\mathsf{cosk}}}_{n-1}T\to {\ensuremath{\mathbb{K}}}(Z,A,n)}$ such that the squares  are all pullbacks. As explained above, this is enough for $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ to be an $n$-torsor. Furthermore, Proposition \[Proposition-Full\] tells us that such a simplicial morphism ${\ensuremath{\mathbb{t}}}$ is uniquely determined, so that its existence is a *property* of $T$, not additional structure—as it should be, because centrality is also a property. The other implication (which says that the underlying $n$-cubic extension of an $n$-torsor is always central) will be treated in the following section. Embedding cycles into diamonds ------------------------------ Up to symmetry of the diamond, there is a unique way a cycle may be embedded into a diamond using degeneracies to fill up missing faces. In degree two there is the morphism $$s_{2}({\ensuremath{\mathbb{X}}})\colon{\triangle}({\ensuremath{\mathbb{X}}},2)\to {\operatorname{Eq}({\partial}_{1})}{\raisebox{.4mm}{\,\ensuremath{\boxvoid}}}{\operatorname{Eq}({\partial}_{0})}\colon {\lgroup}x_{0},x_{1},x_{2}{\rgroup}\mapsto\vcenter{\xymatrix@1@!0@=3.5em{\sigma_{0}{\partial}_{1}x_{0} \ar@{.}[r] \ar@{.}[d]|-{1} & x_{2} \ar@{.}[d] \\ x_{0} \ar@{.}[r]|-{0} & x_{1}}}$$ which sends the left hand side (empty) triangle $$\vcenter{\xymatrix@1@!0@R=2.4495em@C=1.4142em{& {\cdot} \ar[rd]^-{x_{0}}\\ {\cdot} \ar[ru]^-{x_{2}} \ar[rr]_-{x_{1}} && {\cdot}}} \qquad\qquad \vcenter{\xymatrix@1@!0@=2em{& {\cdot}\\ {\cdot} \ar[ru]^-{\sigma_{0}{\partial}_{1}x_{0}} \ar[rd]_-{x_{0}} && {\cdot} \ar[lu]_-{x_{2}} \ar[ld]^-{x_{1}}\\ &{\cdot}}}$$ to the right hand side diamond. In degree three we have $$s_{3}({\ensuremath{\mathbb{X}}})\colon{\triangle}({\ensuremath{\mathbb{X}}},3)\to \bigboxvoid_{i\in 3}{\operatorname{Eq}({\partial}_{i})}\colon {\lgroup}x_{0},x_{1},x_{2},x_{3}{\rgroup}\mapsto\vcenter{\xymatrix@1@!0@=2.5em{& \sigma_{0}{\partial}_{2}x_{0} \ar@{.}[rr] \ar@{.}[dd] \ar@{.}[dl] && x_{3} \ar@{.}[dd] \ar@{.}[dl]\\ \sigma_{0}{\partial}_{1}x_{0} \ar@{.}[rr] \ar@{.}[dd]|-{1} && x_{2} \ar@{.}[dd] \\ & \sigma_{1}{\partial}_{2}x_{0} \ar@{.}[rr] \ar@{.}[dl]|-{2} && \sigma_{1}{\partial}_{2}x_{1} \ar@{.}[dl] \\ x_{0} \ar@{.}[rr]|-{0} && x_{1}}}$$ and in general we have an inductive formula, as follows. Let ${}^{-}{\ensuremath{\mathbb{X}}}$ denote the [décalage]{} of ${\ensuremath{\mathbb{X}}}$, the augmented simplicial object constructed out of ${\ensuremath{\mathbb{X}}}$ by forgetting the lowest degree ${\ensuremath{\mathbb{X}}}_{-1}$ and the last face operators ${\partial}_{n}\colon{{\ensuremath{\mathbb{X}}}_{n}\to {\ensuremath{\mathbb{X}}}_{n-1}}$, so that ${}^{-}{\ensuremath{\mathbb{X}}}_{n}={\ensuremath{\mathbb{X}}}_{n+1}$. We obtain a morphism of simplicial objects ${\ensuremath{\mathbb{d}}}\colon{{}^{-}{\ensuremath{\mathbb{X}}}\to {\ensuremath{\mathbb{X}}}}$ by ${\ensuremath{\mathbb{d}}}_{n}={\partial}_{n+1}\colon {{}^{-}{\ensuremath{\mathbb{X}}}_{n}={\ensuremath{\mathbb{X}}}_{n+1}\to {\ensuremath{\mathbb{X}}}_{n}}$. \[Proposition-Inclusion\] For any simplicial object ${\ensuremath{\mathbb{X}}}$ in a semi-abelian category and any ${n\geq 2}$ there is a canonical natural inclusion $$s_{n}({\ensuremath{\mathbb{X}}})\colon{\triangle}({\ensuremath{\mathbb{X}}},n)\to \bigboxvoid_{i\in n}{\operatorname{Eq}({\partial}_{i})}.$$ We give a proof by induction; the base step is explained above. Suppose $s_{n}({\ensuremath{\mathbb{X}}})$ is defined for every ${\ensuremath{\mathbb{X}}}$ and natural in ${\ensuremath{\mathbb{X}}}$; we then construct a morphism $s_{n+1}({\ensuremath{\mathbb{X}}})$, natural in ${\ensuremath{\mathbb{X}}}$. Given an $(n+1)$-cycle $$x={\lgroup}x_{0},\dots,x_{n},x_{n+1}{\rgroup}\in{\triangle}({\ensuremath{\mathbb{X}}},n+1),$$ note that both $\widehat x_{n+1}={\lgroup}x_{0},\dots,x_{n}{\rgroup}$ and $$\widehat y_{n+1}={\lgroup}\sigma_{n-1}{\partial}_{0}x_{n+1},\dots,\sigma_{n-1}{\partial}_{n-1}x_{n+1},x_{n+1}{\rgroup},$$ where $$y=\sigma_{n}x_{n+1}={\lgroup}\sigma_{n-1}{\partial}_{0}x_{n+1},\dots,\sigma_{n-1}{\partial}_{n-1}x_{n+1},x_{n+1},x_{n+1}{\rgroup},$$ are in ${\triangle}({}^{-}{\ensuremath{\mathbb{X}}},n)$. The induction hypothesis gives us a pair of diamonds, and we define $$s_{n+1}({\ensuremath{\mathbb{X}}})(x)={\lgroup}s_{n}({}^{-}{\ensuremath{\mathbb{X}}})(\widehat x_{n+1}),s_{n}({}^{-}{\ensuremath{\mathbb{X}}})(\widehat y_{n+1}){\rgroup}\in\bigboxvoid_{i\in n}{\operatorname{Eq}({}^{-}{\partial}_{i})}\times \bigboxvoid_{i\in n}{\operatorname{Eq}({}^{-}{\partial}_{i})}.$$ Now we only have to show that this pair does belong to $\bigboxvoid_{i\in n+1}{\operatorname{Eq}({\partial}_{i})}$, which means that ${\partial}_{n}(s_{n}({}^{-}{\ensuremath{\mathbb{X}}})(\widehat x_{n+1}))={\partial}_{n}(s_{n}({}^{-}{\ensuremath{\mathbb{X}}})(\widehat y_{n+1}))$. This equality follows from the naturality of $s_{n}$, which makes the square $$\xymatrix{{\triangle}({}^{-}{\ensuremath{\mathbb{X}}},n) \ar[r]^-{{\triangle}({\ensuremath{\mathbb{d}}},n)} \ar[d]_-{s_{n}({}^{-}{\ensuremath{\mathbb{X}}})} & {\triangle}({\ensuremath{\mathbb{X}}},n) \ar[d]^{s_{n}({\ensuremath{\mathbb{X}}})} \\ \bigboxvoid_{i\in n}{\operatorname{Eq}({}^{-}{\partial}_{i})} \ar[r]_-{{\ensuremath{\mathbb{d}}}} & \bigboxvoid_{i\in n}{\operatorname{Eq}({\partial}_{i})}}$$ commute, and the fact that ${\triangle}({\ensuremath{\mathbb{d}}},n)(\widehat x_{n+1})$ is equal to ${\triangle}({\ensuremath{\mathbb{d}}},n)(\widehat y_{n+1})$. Indeed, we have ${\partial}_{n}x_{n}={\partial}_{n}x_{n+1}$ and $${\partial}_{n}x_{i}={\partial}_{i}x_{n+1}={\partial}_{n}\sigma_{n-1}{\partial}_{i}x_{n+1}$$ for every $i\in n$, so that the latter equality holds. This completes the construction of $s_{n+1}({\ensuremath{\mathbb{X}}})$, which is evidently natural in ${\ensuremath{\mathbb{X}}}$. The morphism $s_{n}({\ensuremath{\mathbb{X}}})$ constructed above takes an “element” $x={\lgroup}x_{0},\dots,x_{n}{\rgroup}$ of the object ${\triangle}({\ensuremath{\mathbb{X}}},n)$ and maps it to the diamond $s_{n}({\ensuremath{\mathbb{X}}})(x)$ which has $x_{i}$ on its $i$" entry and degeneracies elsewhere (see Subsection \[indexing\]). Clearly, $s_{n}({\ensuremath{\mathbb{X}}})$ restricts to morphisms $$\dot s_{n}^{i}({\ensuremath{\mathbb{X}}})\colon{\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n)\to \bigboxdot^{i}_{j\in n}{\operatorname{Eq}({\partial}_{j})},$$ natural in ${\ensuremath{\mathbb{X}}}$. When we say that an ${(n-1)}$-truncated simplicial resolution is [central]{}, we mean that such is the underlying $n$-cubic extension. We write ${\ensuremath{\mathsf{SCExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$ for the (non-full) subcategory of ${\ensuremath{3^{n}\text{-}\mathsf{Diag}}}({\ensuremath{\mathcal{X}}})$ consisting of those $3^{n}$-diagrams with an underlying $n$-cubic extension which is a central ${(n-1)}$-truncated simplicial resolution, with morphisms between such which restrict to simplicial morphisms. We write $$\label{Dd} {\ensuremath{\mathsf{d}}}_{(n,Z)} \colon{{\ensuremath{\mathsf{SCExt}}}^n_Z({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})}$$ for the restriction of ${\ensuremath{\mathsf{D}}}_{(n,Z)}$ to this category. \[Proposition-Central-then-Torsor\] If, in a semi-abelian category, an $(n-1)$-truncated simplicial resolution is central, then it is an $n$-torsor. Let ${\ensuremath{\mathbb{T}}}$ be a simplicial resolution and let $A$ be the direction of $T={\ensuremath{\mathsf{tr}}}_{n-1}{\ensuremath{\mathbb{T}}}$, considered as a trivial $Z$-module. We have to define a morphism of augmented simplicial objects ${\ensuremath{\mathbb{t}}}\colon{{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(Z, A, n)}$ as in . Such a simplicial morphism is completely determined by the choice of a suitable morphism $\varsigma\colon{{\triangle}({\ensuremath{\mathbb{T}}},n)\to A}$. Consider, for $i\in n+1$, the commutative square of solid arrows $$\xymatrix{0 \ar@{.>}[r] & A \ar@{:}[d] \ar@{.>}@<-.5ex>[r] & {\triangle}({\ensuremath{\mathbb{T}}},n) \pullbackdots \ar@{.>}@<-.5ex>[l]_-{\varsigma} \ar@<-.5ex>[r]_-{\widehat{\partial}_{i}} \ar[d]_-{s_{n}({\ensuremath{\mathbb{T}}})} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n) \ar@{.>}@<-.5ex>[l] \ar[d]^{\dot s_{n}^{i}({\ensuremath{\mathbb{T}}})} \ar@{.>}[r] & 0 \\ 0 \ar@{.>}[r] & A \ar@{.>}@<-.5ex>[r] & \bigboxvoid_{j\in n}{\operatorname{Eq}({\partial}_{j})} \ar@<-.5ex>[r]_{\pi^{i}} \ar@{.>}@<-.5ex>[l]_-{\operatorname{pr}_{A}} & \bigboxdot_{j\in n}^{i}{\operatorname{Eq}({\partial}_{j})} \ar@{.>}@<-.5ex>[l] \ar@{.>}[r] & 0}$$ which embeds cycles into diamonds. By assumption, the kernel of $\pi^{i}$ is $A$; moreover, by Theorem \[Theorem-Higher-Centrality\], $$\bigboxvoid_{j\in n}{\operatorname{Eq}({\partial}_{j})}\cong A\times \bigboxdot_{j\in n}^{i}{\operatorname{Eq}({\partial}_{j})}$$ with $\pi^{i}$ the projection on $\bigboxdot_{j\in n}^{i}{\operatorname{Eq}({\partial}_{j})}$. The square above is a pullback as a consequence of Lemma \[Lemma-Iso-Pullback\], since $\widehat{\partial}_{i}$ is a regular epimorphism by the extension property of $T$, and since the kernel of $\widehat{\partial}_{i}$ is $A$ by Lemma \[Lemma-Direction-Cycle\]. This implies that $${\triangle}({\ensuremath{\mathbb{T}}},n)\cong A\times {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$$ with $\widehat{\partial}_{i}$ the projection on ${\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)$. We may now complete the square with the dotted arrows. We choose $\varsigma$ to be $\operatorname{pr}_{A}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}s_{n}({\ensuremath{\mathbb{T}}})\colon{{\triangle}({\ensuremath{\mathbb{T}}},n)\to A}$, the projection of ${\triangle}({\ensuremath{\mathbb{T}}},n)$ on $A$. We must prove that this does indeed give us a genuine morphism ${\ensuremath{\mathbb{t}}}\colon{{\ensuremath{\mathbb{T}}}\to {\ensuremath{\mathbb{K}}}(Z, A, n)}$; then the exact fibration property holds by Proposition \[Isos-are-Free\], so that $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ is an $n$-torsor. For this, we only need to check that all the squares in the diagram $$\xymatrix@=45pt{ {\triangle}({\ensuremath{\mathbb{T}}},n+1) \ar[d]_{{\lgroup}{\lgroup}\varsigma\circ{\partial}_{i}{\rgroup}_{i},{\partial}_{0}^{n+2}{\rgroup}} \ar@<2.33ex>[r]^-{{\partial}_{n+1}} \ar@<1.16ex>[r] \ar@<-2.33ex>[r]^-{\vdots} & {\triangle}({\ensuremath{\mathbb{T}}},n) \ar[d]^{{\lgroup}\varsigma,{\partial}_{0}^{n+1}{\rgroup}}\\ A^{n+1} \times Z \ar@<2.33ex>[r]^-{{\partial}_{n+1}\times 1_{Z}} \ar@<1.16ex>[r]|-{\operatorname{pr}_{n}\times 1_{Z}} \ar@<-2.33ex>[r]_-{\operatorname{pr}_{0}\times 1_{Z}}^-{\vdots} & A \times Z}$$ commute. This condition reduces to the commutativity of just one square, the one “on top”: $$\label{Sum-condition} \varsigma{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{n+1}=(-1)^{n}\sum^{n}_{i=0}(-1)^{i}\varsigma{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{i}.$$ In fact the morphism ${\lgroup}{\lgroup}\varsigma{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{i}{\rgroup}_{i},{\partial}_{0}^{n+2}{\rgroup}$ is already the unique one that makes all the other squares commute. But this equality follows from Proposition \[Centrality-Sum\], which tells us that the morphism $\varsigma$ itself may be considered as an alternating sum, $$\varsigma=\sum_{J\subseteq n}(-1)^{|J|}\eta_{{\ensuremath{\mathbb{T}}}_{n-1}}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{J}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}s_{n}({\ensuremath{\mathbb{T}}}).$$ Using that the alternating sum $$\sum_{i=0}^{n+1}(-1)^{i}\eta_{{\triangle}({\ensuremath{\mathbb{T}}},n)}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\partial}_{i}$$ is zero by the simplicial identities, the equality  may now be obtained via a direct calculation in the abelian object $A$. \[Proposition-Full\] Given $f\colon X\to Y$ in ${\ensuremath{\mathsf{SCExt}}}^{n}_{Z}({\ensuremath{\mathcal{X}}})$, let $({\ensuremath{\mathbb{X}}},{\ensuremath{\mathbb{x}}})$ and $({\ensuremath{\mathbb{Y}}},{\ensuremath{\mathbb{y}}})$ be the $n$-torsors corresponding to $X$ and $Y$ and ${\ensuremath{\mathbb{f}}}\colon{{\ensuremath{\mathbb{X}}}\to {\ensuremath{\mathbb{Y}}}}$ the simplicial morphism corresponding to $f$. If $f$ keeps the direction fixed, that is to say, if $${\ensuremath{\mathsf{d}}}_{(n,Z)}f=1\colon {\ensuremath{\mathsf{d}}}_{(n,Z)}X\to {\ensuremath{\mathsf{d}}}_{(n,Z)}Y,$$ then ${\ensuremath{\mathbb{y}}}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\ensuremath{\mathbb{f}}}={\ensuremath{\mathbb{x}}}$ so that ${\ensuremath{\mathbb{f}}}$ is a morphism of torsors. In other words, we have a functor $${\ensuremath{\mathsf{d}}}_{(n,Z)}^{-1}A\to \operatorname{Tors}^{n}(Z,A).$$ Furthermore, this functor is fully faithful. For any morphism of central truncated simplicial objects which keeps the terminal object and the direction fixed, the projections to the directions are compatible with it. To see this we only need to consider the diagram $$\xymatrix{A \ar@{=}[d] \ar@<-.5ex>@{{ |>}->}[r] & {\triangle}({\ensuremath{\mathbb{X}}},n) \ar[d]^-{{\triangle}({\ensuremath{\mathbb{f}}},n)} \ar@{ >>}@<-.5ex>[l]_-{\varsigma_{{\ensuremath{\mathbb{X}}}}} \ar@{ >>}[r] & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{X}}},n) \ar[d]^-{\smallhorn^{i}({\ensuremath{\mathbb{f}}},n)}\\ A \ar@<-.5ex>@{{ |>}->}[r] & {\triangle}({\ensuremath{\mathbb{Y}}},n) \ar@{ >>}@<-.5ex>[l]_-{\varsigma_{{\ensuremath{\mathbb{Y}}}}} \ar@{ >>}[r] & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{Y}}},n)}$$ and note that $\varsigma_{{\ensuremath{\mathbb{Y}}}}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}{\triangle}({\ensuremath{\mathbb{f}}},n)=\varsigma_{{\ensuremath{\mathbb{X}}}}$ by naturality of the product decompositions *induced by centrality*—see Remark \[Remark-Full-3\] or the above proof. This shows that ${\ensuremath{\mathbb{f}}}$ is a morphism of torsors. This immediately shows that the functor ${\ensuremath{\mathsf{d}}}_{(n,Z)}^{-1}A\to \operatorname{Tors}^{n}(Z,A)$ is fully faithful as claimed: after all, a morphism in $\operatorname{Tors}^{n}(Z,A)$ is nothing but a morphism in ${\ensuremath{\mathsf{d}}}_{(n,Z)}^{-1}A$ satisfying an additional condition—but we just proved that this condition always holds. The commutator condition {#Section-Commutator-Assumption} ======================== In general it is not clear how an isomorphism on the simplicial level may be extended to an isomorphism on the level of higher-dimensional diamonds. Therefore, to prove that every $n$-torsor is an $n$-cubic central extension, we shall add an assumption on the base category: we ask that higher central extensions may be characterised in terms of binary Huq commutators. This happens in many cases, but thus far we have no precise characterisation of the categories which satisfy this condition. It is proved in Section 9.1 of [@EGVdL] that an $n$-cubic extension of groups $F$ is central with respect to ${\ensuremath{\mathsf{Ab}}}$ if and only if $\bigl[\bigcap_{i\in I}{\operatorname{Ker}(f_{i})},\bigcap_{i\in n\setminus I}{\operatorname{Ker}(f_{i})}\bigr]=0$ for all $I\subseteq n$. The theory which we develop depends crucially on a similar characterisation of higher central extensions, valid in a sufficiently general context. \[Commutator-assumption\] [@RVdL3] We say that an $n$-cubic extension $F$ in a semi-abelian category ${\ensuremath{\mathcal{X}}}$ is [H-central]{} when $$\Bigl[\bigcap_{i\in I}{\operatorname{Ker}(f_{i})},\bigcap_{i\in n\setminus I}{\operatorname{Ker}(f_{i})}\Bigr]=0$$ for all $I\subseteq n$. The category ${\ensuremath{\mathcal{X}}}$ satisfies the [commutator condition on $n$-cubic central extensions]{} when the H-central $n$-cubic extensions in ${\ensuremath{\mathcal{X}}}$ coincide with the [categorically central]{} ones, namely those which are central with respect to ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$ in the Galois-theoretic sense used throughout the rest of the paper. We say that ${\ensuremath{\mathcal{X}}}$ satisfies the [commutator condition (CC)]{} when it satisfies the commutator condition on $n$-cubic central extensions for all $n$. The cases $n=1$ and $n=2$ ------------------------- As explained in the Introduction and in Example \[Example-Dimension-One\], every semi-abelian category satisfies the commutator condition for one-cubic central extensions. From the Introduction and Example \[Example-Dimension-Two\], it follows that in a semi-abelian category, the commutator condition on two-cubic central extensions is weaker than the *Smith is Huq* condition [@MFVdL]. Some examples ------------- It is shown in [@EGVdL] that, next to the category of groups, also the categories Lie algebras and non-unitary rings have (CC). The examples of Leibniz and Lie $n$-algebras were treated in [@CKLVdL]. Moreover, from [@RVdL3] we know that any semi-abelian category with the *Smith is Huq* condition has (CC), while the categories of loops and of commutative loops do not satisfy this condition. As examples we have *action representative* semi-abelian categories [@BJK2; @Borceux-Bourn-SEC], *action accessible* categories [@BJ07]—in particular all *categories of interest* [@Orzech; @Montoli], so also all *varieties of groups* [@Neumann]—next to all *strongly semi-abelian* [@Bourn2004] and *Moore* categories [@Gerstenhaber; @Rodelo:Moore]. For instance, the categories of associative and non-associative algebras and of (pre)crossed modules satisfy (CC). A general context where many examples may be found is given by those semi-abelian categories for which the abelianisation functor is protoadditive [@Everaert-Gran-nGroupoids; @Everaert-Gran-TT], as considered below. This example gives two extreme special cases: semi-abelian arithmetical categories recalled in Example \[Arithmetical categories\], such as the categories of von Neumann regular rings, Boolean rings and Heyting semilattices (where the cohomology theory becomes trivial) on the one hand, and abelian categories recalled in Example \[Abelian categories\] (where, via a version of the Dold–Kan correspondence [@Dold-Puppe], the theory gives us the Yoneda $\operatorname{Ext}$ groups) on the other. Recall from [@Everaert-Gran-nGroupoids] that a functor between semi-abelian categories is [protoadditive]{} when it preserves split short exact sequences $$\vcenter{\xymatrix{0 \ar[r] & K \ar@{{ |>}->}[r]^-{k} & X \ar@<-.5ex>@{-{ >>}}[r]_-{f} & Y \ar[r] \ar@<-.5ex>[l]_-{s} & 0}}$$ (the cokernel $f$ is split by some morphism $s$). It is explained in [@Everaert-Gran-TT] that, when ${\ensuremath{\mathcal{X}}}$ is semi-abelian and the abelianisation functor ${\ensuremath{\mathsf{ab}}}\colon{{\ensuremath{\mathcal{X}}}\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})}$ is protoadditive, the Huq commutator $[K,L]$ of two normal subobjects $K$, $L$ of an object $X$ is ${\langle}K\cap L{\rangle}=[K\cap L,K\cap L]$. This gives us $$\Bigl[\bigcap_{i\in I}{\operatorname{Ker}(f_{i})},\bigcap_{i\in n\setminus I}{\operatorname{Ker}(f_{i})}\Bigr]=\Bigl[\bigcap_{i\in n}{\operatorname{Ker}(f_{i})},\bigcap_{i\in n}{\operatorname{Ker}(f_{i})}\Bigr]={\langle}{\ensuremath{\mathsf{D}}}_{(n,Z)}F{\rangle}$$ for any $n$-cubic extension $F$ of $Z$ and any $I\subseteq n$. Furthermore, by another result in [@Everaert-Gran-TT], an $n$-cubic extension is categorically central if and only if its direction is abelian; hence the commutator condition (CC) holds. In fact, this argument extends easily to a proof that all semi-abelian arithmetical categories satisfy (SH). A non-trivial instance of this situation, mentioned in [@Everaert-Gran-TT], is the variety of non-unitary rings that satisfy the law $abab=ab$. We now explain another special case, one which is less interesting from a cohomological point of view, but which does give a class of extreme examples. \[Arithmetical categories\] Recall from [@Pedicchio2] that an exact Mal’tsev category is [arithmetical]{} when every internal groupoid is an equivalence relation. We restrict ourselves to semi-abelian arithmetical categories, examples of which are the dual of the category of pointed sets, more generally, the dual of the category of pointed objects in any topos, and also the categories of von Neumann regular rings, Boolean rings and Heyting semi-lattices [@Borceux-Bourn]. Since in such a category all abelian objects are trivial, the abelianisation functor is protoadditive, so that the commutator condition (CC) holds. (By the above, as shown in [@MFVdL2], every arithmetical category moreover satisfies (SH).) Here an $n$-cubic extension is categorically central if and only if its direction is zero, which means that the extension is a limit $n$-cube (or an isomorphism, when $n=1$). Hence the interpretation of cohomology in terms of higher central extensions (Theorem \[Main-Theorem\]) just means that any two $n$-cubic central extensions of an object $Z$, so limit $n$-cubes over $Z$, are connected, because $\operatorname{Centr}^{n}(Z,0)\cong {\mathrm{H}}^{n+1}(Z,0)$ is trivial—which is, however, not difficult to prove directly. At the other end of the spectrum we find the context of abelian categories where (CC) also holds, and where the cohomology theory reduces to Yoneda’s interpretation of $\operatorname{Ext}^{n}(Z,A)$. \[Abelian categories\] In an abelian category all Huq commutators are zero while all extensions are categorically central, which already gives us (CC). On top of that the abelianisation functor is an identity, so that it is trivially protoadditive. Via the Dold–Kan theorem [@Dold-Puppe], an $(n-1)$-truncated simplicial resolution, considered as an $n$-fold extension of $Z$ by $A$, corresponds to an exact sequence $$\xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r] & C_{n-1} \ar[r] & C_{n-2} \ar[r] & {\cdots} \ar[r] & C_{0} \ar@{ >>}[r] & Z \ar[r] & 0}$$ of length $n$. (See Figure \[3x3 diag\] on page  for a picture when $n=2$.) As a consequence of the results in Section \[Section-Main-Theorem\] we get $$\operatorname{Ext}^{n}(Z,A)\cong \operatorname{Centr}^{n}(Z,A)\cong {\mathrm{H}}^{n+1}(Z,A),$$ the equivalence between Yoneda’s cohomology via $\operatorname{Ext}$ groups [@Yoneda-Exact-Sequences; @MacLane:Homology] and comonadic cohomology which was first established in [@Beck]. (See also [@Barr-Beck]. A proof of the same result via torsor theory is given in [@Glenn].) The dimension shift is there because our numbering of the cohomology objects agrees with the classical non-abelian examples (groups, Lie algebras, etc.) rather than with the abelian case. From torsors to central extensions ---------------------------------- We are now ready to prove the equivalence between torsors and central extensions we need for our cohomological interpretation of higher central extensions. \[Proposition-Torsor-then-Central\] In a semi-abelian category, the underlying $n$-cubic extension of an $n$-torsor is H-central. Let $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ be an $n$-torsor of $Z$ by $A$ with underlying $n$-cubic extension $T$. Then already the commutator $[T_{n},A]$ is zero: by Proposition \[Proposition-Trivial-Action\], since $A$ is a trivial $Z$-module, and by Example \[Example-Dimension-One\]. Now suppose that $\emptyset\neq I\subsetneq n$ and consider $x\colon {X=\bigcap_{j\in I}{\operatorname{Ker}({\partial}_{j})}\to T_{n}}$ and $y\colon {Y=\bigcap_{j\in n\setminus I}{\operatorname{Ker}({\partial}_{j})}\to T_{n}}$. We are to show that $x$ and $y$ Huq-commute (see Subsection \[Commutators\]), so that $T$ is H-central. Without any loss of generality we may assume that ${\partial}_{0}y=0$. (If not, reverse the roles of $x$ and $y$.) Let $i$ be the smallest element of $I$. Then ${\partial}_{i}x=0$ and ${\partial}_{j}y=0$ for all $j<i$, and the boundaries $${\partial}\sigma_{i-1}x={\lgroup}\sigma_{i-2}{\partial}_{0}x,\dots,\sigma_{i-2}{\partial}_{i-2}x,x,x,0,\sigma_{i-1}{\partial}_{i+1}x,\dots,\sigma_{i-1}{\partial}_{n-1}x{\rgroup}$$ and $${\partial}\sigma_{i}y={\lgroup}0,\dots,0,0,y,y,\sigma_{i}{\partial}_{i+1}y,\dots,\sigma_{i}{\partial}_{n-1}y{\rgroup}$$ of $\sigma_{i-1}x$ and $\sigma_{i}y$ determine $(n,i)$-horns $$\overline{x}=\widehat{({\partial}\sigma_{i-1}x)}_{i}={\lgroup}\sigma_{i-2}{\partial}_{0}x,\dots,\sigma_{i-2}{\partial}_{i-2}x,x,0,\sigma_{i-1}{\partial}_{i+1}x,\dots,\sigma_{i-1}{\partial}_{n-1}x{\rgroup}$$ and $$\overline{y}=\widehat{({\partial}\sigma_{i}y)}_{i}={\lgroup}0,\dots,0,0,y,\sigma_{i}{\partial}_{i+1}y,\dots,\sigma_{i}{\partial}_{n-1}y{\rgroup}$$ in ${\ensuremath{\mathbb{T}}}$ which Huq-commute with each other. The first $i+1$ components do so because anything commutes with zero. The others commute for the same reason, one of ${\partial}_{j}x$ or ${\partial}_{j}y$ being trivial by assumption on $x$ and $y$. In other words, there is a morphism $\widehat\varphi_{i}$ such that the diagram $$\xymatrix{X \ar[rd]|-{{\lgroup}1_{X},0{\rgroup}} \ar@/^/[rrd]^-{\overline{x}} \\ & X\times Y \ar@{.>}[r]|-{\widehat\varphi_{i}} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n)\\ Y \ar[ru]|-{{\lgroup}0,1_{Y}{\rgroup}} \ar@/_/[rru]_-{\overline{y}}}$$ is commutative, namely, the morphism determined by the family $$\varphi_{j}=\begin{cases} \overline{x}_{j}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{X} & \text{if $j<i$ (so that $j\not\in I$)}\\ \overline{x}_{j}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{X} & \text{if $j>i$ and $j-1\not\in I$}\\ \overline{y}_{j}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\operatorname{pr}_{Y} & \text{if $j>i$ and $j-1\in I$;} \end{cases}$$ note that indeed ${\partial}_{k}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\varphi_{j}={\partial}_{j-1}{\raisebox{0.2mm}{\ensuremath{\scriptstyle{\circ}}}}\varphi_{k}$ for all $k<j$ such that $i\not\in\{j,k\}$. Furthermore, being induced by degeneracies, $\overline{x}$ and $\overline{y}$ compose to the face left out—see Subsection \[Multiplication\]—so that the diagram $$\xymatrix@C=3em{X \ar[rd]|-{{\lgroup}1_{X},0{\rgroup}} \ar@/^/[rrd]_-{\overline{x}} \ar@/^/[rrrd]^-{x} \\ & X\times Y \ar[r]|-{\widehat\varphi_{i}} & {\raisebox{.145mm}{\scalebox{1.3}[1.15]{$\smallhorn$}}\!}^{i}({\ensuremath{\mathbb{T}}},n) \ar[r]|(.6){m^{i}} & T_{n}\\ Y \ar[ru]|-{{\lgroup}0,1_{Y}{\rgroup}} \ar@/_/[rru]^-{\overline{y}} \ar@/_/[rrru]_-{y}}$$ is commutative, and $x$ and $y$ Huq-commute. Thus we proved that, in a semi-abelian category, any categorically central truncated simplicial resolution gives a torsor (Proposition \[Proposition-Central-then-Torsor\]) and any torsor gives an H-central truncated simplicial resolution (Proposition \[Proposition-Torsor-then-Central\]). To complete the circle, what we need is precisely the commutator condition (CC). \[Theorem-Torsor-Equivalence\] In a semi-abelian category which satisfies the commutator condition [(CC)]{}, an augmented simplicial object ${\ensuremath{\mathbb{T}}}$ is part of an $n$-torsor $({\ensuremath{\mathbb{T}}},{\ensuremath{\mathbb{t}}})$ if and only if its underlying $n$-fold arrow is an $n$-cubic central extension. \[Corollary-Torsor-Equivalence\] Under [(CC)]{}, the functor ${\ensuremath{\mathsf{d}}}_{(n,Z)}^{-1}A\to \operatorname{Tors}^{n}(Z,A)$ described in Proposition \[Proposition-Full\] is an equivalence of categories. Theorem \[Theorem-Torsor-Equivalence\] tells us that this functor is essentially surjective, while it is fully faithful by Proposition \[Proposition-Full\]. Cohomology classifies higher central extensions {#Section-Main-Theorem} =============================================== In this last section we prove our main result, Theorem \[Main-Theorem\], which states that, for any object $Z$, any abelian object $A$, and any $n\geq 1$, we have a natural group isomorphism $${\mathrm{H}}^{n+1}(Z,A)\cong\operatorname{Centr}^{n}(Z,A).$$ To do so, we only need to use the results of the previous sections and establish a natural bijection between the underlying sets. We already know that, for truncated simplicial resolutions, being a torsor is equivalent to being central. Now we have to explain how any (central) extension may be approximated by a truncated augmented simplicial object so that the two types of objects may be compared. In fact, any equivalence class of central extensions of $Z$ by $A$ contains a truncated simplicial object. Simplicial approximation of higher-dimensional arrows ----------------------------------------------------- Using a classical Kan extension argument, every $n$-dimensional arrow may be universally approximated by an ${(n-1)}$-truncated simplicial object. Indeed, the functor from Subsection \[Truncations\] $${\ensuremath{\mathsf{arr}}}_{n}={\ensuremath{\mathsf{Fun}}}(-,{\ensuremath{\mathsf{a}}}_{n})\colon{{\ensuremath{\mathsf{SArr}}}^{n}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Arr}}}^{n}({\ensuremath{\mathcal{X}}})}$$ has a right adjoint $${\ensuremath{\mathsf{s}}}_{n}={\ensuremath{\mathsf{Ran}}}_{{\ensuremath{\mathsf{a}}}_{n}}(-)\colon{{\ensuremath{\mathsf{Arr}}}^{n}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{SArr}}}^{n}({\ensuremath{\mathcal{X}}})}$$ which takes an $n$-fold arrow $F\colon{(2^{n})^{\operatorname{op}}\to {\ensuremath{\mathcal{X}}}}$ and maps it to the right Kan extension $${\ensuremath{\mathsf{Ran}}}_{{\ensuremath{\mathsf{a}}}_{n}}F\colon{(\Delta^{+}_{n})^{\operatorname{op}}\to {\ensuremath{\mathcal{X}}}}$$ of $F$ along the functor ${\ensuremath{\mathsf{a}}}_{n}\colon 2^{n}\to \Delta^{+}_{n}$. \[Proposition-Simplification-of-Extension\] Let ${\ensuremath{\mathcal{X}}}$ be a regular category with enough projectives. Then for all $n\geq 1$, the functors ${\ensuremath{\mathsf{arr}}}_{n}$ and ${\ensuremath{\mathsf{s}}}_{n}$ preserve $n$-cubic extensions. Since an ${(n-1)}$-truncated simplicial object is by definition an $n$-cubic extension if and only if so is its underlying $n$-fold arrow, the functor ${\ensuremath{\mathsf{arr}}}_{n}$ preserves and reflects $n$-cubic extensions. The case of ${\ensuremath{\mathsf{s}}}_{n}$ for $n\geq 2$ is more complicated: given an $n$-fold arrow $F$, the Kan extension ${\ensuremath{\mathsf{s}}}_{n}F={\ensuremath{\mathsf{Ran}}}_{{\ensuremath{\mathsf{a}}}_{n}}F$ is computed pointwise as a limit (see for instance [@MacLane]). For example, a two-cubic extension such as  has $$\xymatrix{{\operatorname{Eq}(d)}\times_{X}{\operatorname{Eq}(c)} \ar@<-1ex>[r] \ar@<1ex>[r] & X \ar[l] \ar[r] & Z}$$ as its simplicial approximation. It is easily seen that also in general, ${\ensuremath{\mathsf{s}}}_{n}F$ has the morphism ${F_{n}\to F_{0}}$ as its augmentation. In order to obtain a quick formal proof in higher degrees we assume that ${\ensuremath{\mathcal{X}}}$ has enough projectives and use Proposition \[Projective-Characterisation-Extensions\]. Let ${\ensuremath{\mathbb{X}}}$ be an $(n-1)$-truncated degreewise projective simplicial object and consider a collection of arrows ${(X_{J}\to ({\ensuremath{\mathsf{s}}}_{n}F)_{J})_{|J|\leq i}}$ as in the proposition. Composing with the counit $u\colon{{\ensuremath{\mathsf{arr}}}_{n}{\ensuremath{\mathsf{s}}}_{n}F\to F}$ of the adjunction at $F$ we obtain a collection of arrows to the $n$-cubic extension $F$, which extends to a morphism of $n$-fold arrows ${X\to F}$. By adjointness we now obtain the needed morphism of $n$-fold arrows ${X\to {\ensuremath{\mathsf{s}}}_{n}F}$, extending the given collection of arrows. This proves that ${\ensuremath{\mathsf{s}}}_{n}F$ is an $n$-cubic extension. We now return to the semi-abelian context and prove that then these adjunctions also preserve centrality. These two results together extend Proposition 5.1 in [@RVdL] to higher degrees and beyond the case of central extensions. \[Lemma Counit-square\] Suppose $F$ is an $n$-cubic extension, $G={\ensuremath{\mathsf{arr}}}_{n}{\ensuremath{\mathsf{s}}}_{n}F$ and $u\colon{G\to F}$ is the counit of the adjunction at $F$. Let ${\ensuremath{\mathsf{L}}}F$ (respectively ${\ensuremath{\mathsf{L}}}G$) be the limit described in Subsection \[HDA\] and $l_{F}$ (respectively $l_{G}$) the comparison morphism. Then the square $$\label{Counit-square} \vcenter{\xymatrix{G_{n} \ar[r]^-{u_{n}} \ar@{ >>}[d]_-{l_{G}} & F_{n} \ar@{ >>}[d]^-{l_{F}}\\ {\ensuremath{\mathsf{L}}}G \ar[r]_-{{\ensuremath{\mathsf{L}}}u} & {\ensuremath{\mathsf{L}}}F}}$$ is a pullback. We prove that the pullback ${\ensuremath{\mathsf{L}}}G\times_{{\ensuremath{\mathsf{L}}}F}F_{n}$ is isomorphic to $G_{n}$ by showing that the $(n-1)$-truncated simplicial object $H$ which is equal to $G$ everywhere, except in level $n-1$ where it is ${\ensuremath{\mathsf{L}}}G\times_{{\ensuremath{\mathsf{L}}}F}F_{n}$, is actually isomorphic to the simplicial approximation of $F$. The $n$-cubic extension $H$ is indeed an ${(n-1)}$-truncated simplicial object: the degeneracies are induced by composition of the degeneracies of $G$ with the comparison morphism ${G_{n}\to {\ensuremath{\mathsf{L}}}G\times_{{\ensuremath{\mathsf{L}}}F}F_{n}}$. This comparison morphism is part of a morphism of $(n-1)$-truncated simplicial objects ${G\to H}$. Its inverse ${H\to G}$ is now induced by the universal property of $G$. \[Proposition-Simplification-of-Central-Extension\] Let ${\ensuremath{\mathcal{X}}}$ be a semi-abelian category with enough projectives. For all $n\geq 1$, the functors ${\ensuremath{\mathsf{arr}}}_{n}$ and ${\ensuremath{\mathsf{s}}}_{n}$ preserve centrality. Furthermore, both functors preserve the direction of a central extension. Let $F$ be an $n$-cubic central extension. Then the direction $A=\bigcap_{i\in n}{\operatorname{Ker}(f_{i})}$ of $F$ (Lemma \[Direction-as-Kernel\]) is part of the short exact sequence $$\xymatrix{0 \ar[r] & A \ar@{{ |>}->}[r] & F_{n} \ar@{ >>}[r]^-{l_{F}} & {\ensuremath{\mathsf{L}}}F \ar[r] & 0.}$$ By Lemma \[Lemma Counit-square\] we have that the square  is a pullback. Via Lemma \[Lemma-Direction-Limit\], this already implies that $l_{G}$ is a central extension when $F$ is central; moreover, we have $A=\bigcap_{i\in n}{\operatorname{Ker}(g_{i})}$ by Lemma \[Lemma-Iso-Pullback\], so that the functor ${\ensuremath{\mathsf{s}}}_{n}$ preserves directions. Since pullbacks preserve product projections, by Theorem \[Theorem-Higher-Centrality\] we only need to prove that any square $$\xymatrix{\bigboxvoid_{i\in n}{\operatorname{Eq}(g_{i})} \ar[r] \ar[d]_-{\pi^{I}_{G}} & \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar[d]^-{\pi^{I}_{F}} \\ \bigboxdot^{I}_{i\in n}{\operatorname{Eq}(g_{i})} \ar[r] & \bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})}}$$ is a pullback. To see this, consider the commutative cube $$\xymatrix@!0@R=3em@C=4em{& \bigboxvoid_{i\in n}{\operatorname{Eq}(g_{i})} \ar[ld] \ar[rr] \ar@{.>}[dd]^(.25){\pi^{I}_{G}} && \bigboxvoid_{i\in n}{\operatorname{Eq}(f_{i})} \ar[dd]^-{\pi^{I}_{F}} \ar[ld]\\ G_{n} \ar[rr]_(.25){u_{n}} \ar[dd]_-{l_{G}} && F_{n} \ar[dd]^(.25){l_{F}}\\ & \bigboxdot^{I}_{i\in n}{\operatorname{Eq}(g_{i})} \ar@{.>}[rr] \ar@{.>}[ld] && \bigboxdot^{I}_{i\in n}{\operatorname{Eq}(f_{i})} \ar[ld]\\ {\ensuremath{\mathsf{L}}}G \ar[rr]_-{{\ensuremath{\mathsf{L}}}u} && {\ensuremath{\mathsf{L}}}F}$$ in which the left and right hand side faces are the ones of Lemma \[Lemma-Diamond-Pullback\]. These squares are pullbacks, and since we already proved that the front face  is a pullback as well, the claim follows. The equivalence between central extensions and torsors ------------------------------------------------------ By Proposition \[Proposition-Simplification-of-Central-Extension\] the functors ${\ensuremath{\mathsf{arr}}}_{n}$ and ${\ensuremath{\mathsf{s}}}_{n}$ not only preserve central extensions, but also the directions of those central extensions. Hence for any object $Z$ and any abelian object $A$, these functors (co)restrict to an adjunction $$\xymatrix@1@=4em{{{\ensuremath{\mathsf{d}}}^{-1}_{(n,Z)}A} \ar@<1ex>[r]^-{{\ensuremath{\mathsf{arr}}}_{n}} \ar@{}[r]|-{\perp} & {{\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A} \ar@<1ex>[l]^-{{\ensuremath{\mathsf{s}}}_{n}}}$$ where the functor  $${\ensuremath{\mathsf{d}}}_{(n,Z)} \colon{{\ensuremath{\mathsf{SCExt}}}^n_Z({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})}$$ is the restriction of ${\ensuremath{\mathsf{D}}}_{(n,Z)}$ to those $3^{n}$-diagrams which, as an $n$-cubic central extension, are an $(n-1)$-truncated simplicial object. Taking connected components gives a bijection of sets (see Remark 5.2 in [@RVdL]). By Corollary \[Corollary-Torsor-Equivalence\] this bijection takes the shape $$\label{bijection} \pi_{0}\operatorname{Tors}^{n}(Z,A)\cong\pi_{0}({\ensuremath{\mathsf{d}}}^{-1}_{(n,Z)}A)\cong\pi_{0}({\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A)$$ when also the commutator condition (CC) holds. The bijection  is natural in $A$. In [@Duskin-Torsors Section 4] the functor $\pi_{0}\operatorname{Tors}^{n}(Z,-)$ is defined as follows: given $f\colon{A\to B}$ in ${\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})$, an $n$-torsor $({\ensuremath{\mathbb{X}}},{\ensuremath{\mathbb{x}}})$ of $Z$ by $A$ universally induces an $n$-torsor $f_{*}({\ensuremath{\mathbb{X}}},{\ensuremath{\mathbb{x}}})$ of $Z$ by $B$. The construction in Proposition \[Proposition-Centr\^[n]{}(Z,-)\] gives us another $n$-torsor $({\ensuremath{\mathbb{Y}}},{\ensuremath{\mathbb{y}}})$ of $Z$ by $B$ together with a simplicial morphism ${\ensuremath{\mathbb{f}}}\colon {{\ensuremath{\mathbb{X}}}\to{\ensuremath{\mathbb{Y}}}}$ over ${\ensuremath{\mathbb{K}}}(f,n)$ which, by the universal property defining $f_{*}({\ensuremath{\mathbb{X}}},{\ensuremath{\mathbb{x}}})$, yields a morphism $f_{*}({\ensuremath{\mathbb{X}}},{\ensuremath{\mathbb{x}}})\to ({\ensuremath{\mathbb{Y}}},{\ensuremath{\mathbb{y}}})$ of $n$-torsors of $Z$ by $B$. Thus $f_{*}({\ensuremath{\mathbb{X}}},{\ensuremath{\mathbb{x}}})$ and $({\ensuremath{\mathbb{Y}}},{\ensuremath{\mathbb{y}}})$ end up in the same equivalence class, so that the bijection  is natural in $A$. Thus we see that the underlying sets of the abelian groups $${\mathrm{H}}^{n+1}(Z,A) = \operatorname{Tors}^{n}[Z,A] = \pi_{0}\operatorname{Tors}^{n}(Z,A)$$ and $\operatorname{Centr}^{n}(Z,A)=\pi_{0}({\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A)$ are naturally isomorphic. Since both $${\mathrm{H}}^{n+1}(Z,-)\quad\text{and}\quad \operatorname{Centr}^{n}(Z,-)\colon {\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})\to {\ensuremath{\mathsf{Set}}}$$ are product-preserving functors (Proposition \[Proposition-Centr\^[n]{}(Z,-)\]), they lift to naturally isomorphic functors ${{\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}})\to{\ensuremath{\mathsf{Ab}}}}$, which gives us \[Main-Theorem\] Let $Z$ be an object and $A$ an abelian object in a semi-abelian category with enough projectives satisfying the commutator condition [(CC)]{}. Then for every $n\geq 1$ we have an isomorphism ${\mathrm{H}}^{n+1}(Z,A)\cong\operatorname{Centr}^{n}(Z,A)$, natural in $A$. Thus we obtain the claimed “duality” between internal homology and external cohomology. \[Duality-Theorem\] Consider $n\geq 1$ and let $Z$ be an object in a semi-abelian category with enough projectives ${\ensuremath{\mathcal{X}}}$ which satisfies the commutator condition [(CC)]{}. Then $${\mathrm{H}}_{n+1}(Z,{\ensuremath{\mathsf{Ab}}}({\ensuremath{\mathcal{X}}}))=\lim {\ensuremath{\mathsf{D}}}_{(n,Z)}\qquad\text{and}\qquad {\mathrm{H}}^{n+1}(Z,A)\cong\pi_{0}({\ensuremath{\mathsf{D}}}^{-1}_{(n,Z)}A),$$ where $A$ is any abelian object in ${\ensuremath{\mathcal{X}}}$. When, in particular, ${\ensuremath{\mathcal{X}}}$ is monadic over ${\ensuremath{\mathsf{Set}}}$, the homology and the cohomology are comonadic Barr–Beck (co)homology with respect to the canonical comonad on ${\ensuremath{\mathcal{X}}}$. The equality holds by definition, while the isomorphism is Theorem \[Main-Theorem\]. 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He was supported by Centro de Matemática da Universidade de Coimbra (CMUC) and by FCT–Fundação para a Ciência e a Tecnologia (under grant number SFRH/BPD/38797/2007) and wishes to thank the Janelidze family for its warm hospitality during his stay in Cape Town

--- author: - Simone Secchi subtitle: 'Ph.D. Thesis' title: Nonlinear differential equations on noncompact domains ---

--- abstract: 'The motion of colloids in the flow field of a viscous liquid is investigated. The small colloid size compare to the macroscopical scale of the flow allow to calculate their velocity relative to that of the liquid. If the inner colloid density is larger then the density of the liquid the flow field has the domains where the colloid velocity is close to the liquid velocity. But in the domains with a strong braking of the liquid velocity the colloids are accelerated relative to the liquid. This effect is used for the qualitative explanation of the drag reduction in the flow around macroscopical bodies and in the pipes.' author: - | S.V.Iordanski\ Landau Institute for Theoretical Physics RAS\ 142432 Russia, Chernogolovka title: The flow around a macroscopical body by a colloid solution and the drag crisis --- More then 60 years ago [@1] it was discovered that a small concentration of polymers in liquid solution essentially decrease the drag in pipes. This effect is used in the oil transportation. There is a lot of theoretical and experimental publications on this subject. However there is no accepted qualitative interpretation of the physical origin for the observed drug reduction. Rather detailed paper[@2] use a complicate theory of the polymer deformation and its dependence on the inner strain but does not state its connection with the liquid flow. The recent work [@3] shows a bad agreement of the performed experiment with existing theories. The large emount of publications is devoted to the rheological properties of the concentrated polymer solutions see e.g. the review [@4], or the book [@5]. We shall not discuss this complicate subject suggesting that in the dilute polymer solutions the main problem is connected with the interaction of the separate polymer and the flow field of the liquid. The more subtle effects of the polymer deformation may be important for more exact quantitative description. In this note we considier first the more investigated problem of the flow around an immobile macroscopical body by Newtonian viscous liquid and its modification due the dilute solution of comparatively large polymer molecules. The description of the large polymer having thousands of the connected links is well developed (see e.g.[@7] or [@8]). The equilibrium state is represented by a coil having on average a spherical form with the radius $l=\sqrt{\frac{Na^2}{6}}$. Where $N$ is the number of the links, and $a$ is their length. The molecular weight of such a coil is much more then the molecular weight of the solvent. Therefore it is possible to neglect the Brownian motion because the polymer thermal velocity is small to the considering flow velocities. To simplify the problem further we shall treat the polymer as a large spherical colloid weakly compressible. The largest scale of the motion is given by the size of the macroscopic body $L$ which is much more then the average distance between the nearest colloids $c^{-1/3}$ where $c$ is the small volume concentration of the collids. This distance is large compare to the size of the colloid $$L\gg c^{-1/3}\gg l$$ {width="5cm"} The motion of the colloid relative to the liquid ================================================ At the low colloid concentration it is possible to use the linear approximation (see e.g [@9]). Let us consider one colloid in the flow field. At the large distances compare to the colloid size $l$ the liquid flow can be considered as uniform. The equations of the motion have the form of the standard equations for an incompressible viscous liquid. We assume that the colloid is incompressible also. At the colloid boundary the liquid velocities and that of the colloid surface are equal and the tensor of the momentum transfer is continuous. The macroscopic motion connected with the large scale $L$ is a kind of an external force acting on the separate colloid and we can use the well known result (see e.g. [@6]) for the calculation of the force acting on the body (colloid) immersed in the liquid. If the body is moved with the liquid this force is equal to $\rho^lV_0\frac{dv_i^l}{dt}$, where $\rho^l$ is the density of the liquid, $V_0=\frac{4\pi l^3}{3}$ is the volume of the colloid. But really one must take into account the relative motion and add $-m_{ik}\frac{d}{dt}(w_i^p-v_i^l)$, where $m_{i,k}$ is the tensor of the associated masses. The relative motion gives also Stokes friction force $-6\pi\eta l(w_i^p-v_i^l)$. Summing the various contributions one get the force acting on the colloid $$\label{2} \rho^pV_0\frac{dw_{i}^p}{dt}=\rho^lV_0\frac{dv_i^l}{dt}-m_{ik}\frac{d}{dt}(w_k^p-v_k^l)- 6\pi\eta l(w_i^p-v_i^l)$$ Using Stokes law suggests that the time of the “viscous” relaxation $\frac{l^2\rho^l}{\eta}$ is small compare to the “hydrodynamical” time $L/U$, where $U$ is the velocity of the macroscopic body relative to the liquid. Stokes force is proportional to the first power of the colloid size and therefore the terms with the accelerations are comparatively small. Therefore the zero approximation gives $w_i^p=v_i^l$ where $v_i^l$ is the local fluid velocity at a small distance $l$ from the colloid. The next approximation can be obtained by introducing the zero approximation in that gives $$\label{3} w_i^p-v_i^l=-(\rho^p-\rho^l)\frac{2l^2}{9\eta}\frac{dv_i^l}{dt}$$ We shall use this approximation for the relative velocity of the colloid. This local “ microscopic” motion gives the contribution to the averaged equations of motion on the distances of the order $c^{-1/3}$. On the contrary neglecting this contribution the colloids have no effect on the averaged equations of motion. In Stokes approximation the force acting on unit area of the spherical colloids surface is constant $F_i=-\frac{3\eta}{l}(w_i^p-v_i^l)$ (see e.g. [@6]). Therefore the collid deformations are absent. In order to find the deformations one needs to consider Oseens corrections connected with the nonlinear inertial terms. The procedure to find the corrections in Reynolds number $$Re=\frac{|\vec{w}^p-\vec{v}^l|}{\eta}\rho^l$$ made in [@10; @11] can be found in [@6]. It is necessary to investigate the solution of the equation $$(\vec{u}\vec{\nabla})\vec{v}^l=-\frac{1}{\rho^l}\vec{\nabla}p+\nu\Delta\vec{v}^l$$ where $\nu=\frac{\eta}{\rho^l}$ is the kinematic viscosity, $\vec{u}$ is the relative velocity equal to its constant value far from the coil. In the external flow field there are two domains: the near one at $r\ll l/Re$ and the far at $r\gg l$, overlapping at $l/{Re}\gg r\gg l$. In the near domain the starting approximation coincides with Stokes solution, in the far domain the starting is Oseen approximation $\vec{u}=const$. The sewing of the appropriate solutions in the overlapping region gives the corrections to Stokes solution $\vec{v}^{(1)}$ $$\begin{aligned} v_r^{(2)}=\frac{3Re}{8}v_r^{(1)}+\frac{3Re}{32}\left(1-\frac{1}{r'}\right)\left(2+\frac{1}{r'}+\frac{1}{(r')^2}\right)(1-3cos^2\vartheta)\\ v_\vartheta^{(2)}=\frac{3Re}{8}v_{\vartheta}^{(1)}+\frac{3Re}{32}\left(1-\frac{1}{r'}\right) \left(4+\frac{1}{r'}+\frac{1}{(r')^2}+\frac{2}{(r')^2}\right)sin\vartheta cos\vartheta\end{aligned}$$ Here the spherical coordinates with the polar ax along the relative velocity are used and the non dimensional quantities are introduced:$r'$ in the units of the colloid radius $l$ and the velocities in the units of the relative velocity $u$. The calculations are simplified at small velocities and give at the colloid surface the pressure $$p^{(2)}=-(1-3cos^2\vartheta)\frac{3}{8l}\eta Re|\vec{u}^l-\vec{w}^p|$$ and the tangential strain $$\label{9} \sigma_{r\theta}=\frac{3\eta Re}{8l}|\vec{u}^l-\vec{w}^p|sin\vartheta cos\vartheta$$ At the picture (1) it is shown some section of the colloid and the points on its surface with the maximal stresses and their directions. There is the compression along the relative velocity direction and the elongation along the meridians with zeros at the poles and the equator. These stresses produce the deformation of the colloid but the calculation require a definite model for the connection of deformations and stresses inside the colloid. There is the general statement belonging to Maxwell that the rapid (short time) stresses correspond to the elasticity theory but the slow (long time) stresses correspond to the viscous liquid. For the case of colloid solutions at high Reynolds numbers the colloids have the velocity close to the velocity of the solvent that gives the short time action on the separate colloid in the domain with the large acceleration and most probable is the validity of the elasticity theory. As a result the colloid is compressed in moving direction and is elongated in the perpendicular directions. That gives the increased colloid cross section and it moves in the direction with the largest drag. Therefore the motion instability is possible. However this question has not direct relation to the drug reduction at the flow around a macroscopic body. The flow around a macroscopic body. =================================== The averaged hydrodynamical equations for the incompressible liquid with the solution of weakly compressible colloids at the small concentration contain the equation $$div\vec{v}^l=0$$ and the conservation law of the colloid number $$\frac{\partial c}{\partial t}+div(c\vec{w}^p)=0$$ The colloid velocity is close to that of the liquid $\vec{w}^p\approx \vec{v}^l$. Therefore it follows from the last two equations that $c=const $ is the solution. The last equation is the conservation law of the total momentum for the system of the liquid and colloids $$\label{12} \frac{\partial}{\partial t}\left[<\rho>v_i^l+cm(w_i^p-v_i^l)\right] = -\frac{\partial}{\partial x_k} \left[ \Pi_{ik}+ cm(w_i^p-v_i^l)v_k^l \right]$$ where $<\rho>=\rho^l+cm$, $\Pi_{ik}=<rho>v_i^l v_k^l +p\delta_{ik}-\eta\frac{\partial v_i^l}{\partial x_k}$, $m=\frac{4\pi l^3}{3}\rho^p$. Stokes drag is absent due to the conservation of the total momentum. We consider first the absence of the colloids. The general picture of the liquid flow around an immobile macroscopic body is well investigated. For large Reynolds number $Re=\frac{UL}{\nu}$ at the far domain on the distances large compare to the boundary layer thickness the flow is vortex free. At the boundary layer approximate external surface the normal to the body velocity is almost zero. The boundary layer begins in the forward critical point with a finite thickness. Moving in the flow direction one get the slow increase of the tangent velocity and the boundary layer thickness till the maximum of the tangent velocity is achieved. After that the tangent velocity is lowering and the laminar layer become unstable . Its thickness has a sharp increase with a sharp decrease of the tangent velocity in the same region. This phenomenon is known as “the tearing off” the boundary layer from the body surface and the creation of the turbulent “stagnation” domain for the flow adjoining to the second critical point as it is shown on fig.2. {width="5cm"} If one has the colloid solution in the liquid then the large laminar part of the boundary layer has a small liquid acceleration and the colloid velocity is quite close to the liquid velocity. Therefore they do not affect on the average flow according to . However in the region where the boundary layer is “tearing off” one has the large acceleration (“braking”) at large Reynolds number. That gives the strong colloid stream in the initial flow direction if the colloid density is larger then that of the liquid according to . The drag at large Reynolds numbers in the colloid absence is given ([@6]) by formula $$F=C(Re)1/2 U^2\rho^lS$$ where $S$ is the body cross section, $U$ is the body velocity relative to the liquid, $C(Re)$ is a constant depending on Reynolds number only. From the experiment it is known that at the large Reynolds number $Re*$ the function $C(Re)$ essentially changes its form giving a sharp drag reduction at comparatively small increase of Reynolds number $\delta Re*\ll Re*$ known as “a drag crisis”. This phenomenon is defined as the moving the“ tearing off” domain down the flow and the strong decrease of the “effective” body cross section. According to the influence of the colloids in the domain for “tearing off” is quite close to the enlargement of Reynolds number because the effective momentum stream is increased $$\Pi_{ik}^{ef}=<\rho>v_i^l v_k^l+p\delta_{ik}-\eta[\frac{\partial v_{i}^l}{\partial x_k}+ \frac{\partial v_k^l}{\partial x_i}+cm\frac{2l^2}{9\eta}\frac{dv_i^l}{dt}v_k^l (\rho^p-\rho^l)$$ The colloid contribution can be written as $\delta \Pi_{ik}=\rho^l v_k^l\delta v_i^l$ where $$\delta v_i^l=cV_0\left(\frac{\rho^p}{\rho^l}\right)^2(1-\frac{\rho^l}{\rho^p})\frac{2l^2}{9\nu}v_n^l\frac{\partial v_i^l}{\partial x_n}$$ The contribution to the integral can be estimated as $$\int\frac{\delta v_i^l}{\nu}dx\sim cV_0\left(\frac{\rho^p}{\rho^l}\right)^2\left(1-\frac{\rho^l}{\rho^p}\right)\frac{2l^2|v_l|^2}{9\nu^2}$$ This expression gives the estimate of the colloid contribution to Reynolds number. If this quantity is equal to the experimental $\delta Re*>0$ then one get the drug reduction like the drug crisis without the polymer solution. The condition of the drug reduction by the polymer solution define the coil concentration and the sign of the difference $\rho^p-\rho^l>0$ . This condition can be rewritten in the form $$\delta Re*=cV_0\left(\frac{\rho^p}{\rho^l}\right)^2\left(1-\frac{\rho^l}{\rho^p}\right)\frac{2l^2}{9L^2}\left(Re*\right)^2$$ By suggestion $cV_0\ll 1$ therefore the requirement $\delta Re* \frac{L^2}{(Re*)^2}\ll l^2$ for the colloid size must be fulfilled. This consideration does not give the difficult numerical estimate. The details of the turbulent flows in a long pipes at large Reynolds numbers are not so well investigated . Partially phenomenological Karman-Prandtl theory is based (as was shown by L.D.Landau) on the dimensional analysis and the experimental determination of some empirical constants. The main statement [@6] is the existence of the central area in the pipe cross section with a weak dependence of the mean velocity on the radius (fig.3 the shaded area) and the viscous sublayer close to the wall of the tube with the linear decrease of the liquid velocity to zero at the wall. The velocity derivative in the viscous sublayer is large compare to that in the central area. If the colloids are present the flow in the central domain is weakly affected because the mean colloid velocities are very close to that of the liquid. However near the transition boundary colloids can achieve an essential velocity according to if $\rho^p\>\rho^l$. Therefore the central domain with the large mean velocity must increase due to the colloids motion. The numerical estimate is difficult because one must know the mean value of the acceleration $\frac{dv_i^l}{dt}$ for the turbulent flow in the transition region, and the unknown empirical constants defining the central and side domains. {width="5cm"} Author express his gratitude to E.Kats, I.Kolokolov and V.Lebedev for the numerous discussions on the subject. The work is partially supported by the Ministry of Science and Education RF (Russian Federal Targeted Programs “S&SPPIP”and “I&DPFS&T”. [99]{} B.A.Toms(1949) Proc.Int.Rheological congress,Holland (1948) p.135 I.Procaccia,V.Lvov, R.Benzi,arXiv:nlin/0702034v1\[nlin.CD\]15 Feb.2007 You.Burnishev,V.Steinberg,EPL 100(2012)24001,doi:10.1209/0295-5075/100/24001 Suzanne M.Fielding,Softmatter, 2007,3,1262-1279 R.G.Larson,Constitutive equations for polimer melts,Butterworth series in chemical engineering (1988) L.D.Landau,Е.М.Lifshits, HydroDynamics,Moscow ,Physmathlit(2003) M.Kleman,О.Lavrentovich, Soft Matter Physics,Springer 2003 Москва,Физматлит 2007, М.Doi,S.F.Edwards ,The theory of polymer dynamics, Мир (1986),Clarendon Press-Oxford С.В.Иорданский,А.Г.Куликовский, Механика жидкости и газа ,(Известия АН СССР) №4,1977,р12 S.Kaplun,P.A.Lagerstrom, J.Math. and Phys.6(1957),p.585-593 I.Proudman,J.R.A.Pearson, J.Fluid Mech.2(1957)p. 237-262

--- abstract: 'We explore the long-time dynamics of Rabi model in a driven-dissipative setting and show that, as the atom-cavity coupling strength becomes larger than the cavity frequency, a new time scale emerges. This time scale, much larger than the natural relaxation time of the atom and the cavity, leads to long-lived metastable states susceptible to being observed experimentally. By applying a Floquet-Liouville approach to the time-dependent master equation, we systematically investigate the set of possible metastable states. We find that the properties of the metastable states can differ drastically from those of the steady state and relate these properties to the energy spectrum of the Rabi Hamiltonian.' author: - 'Alexandre Le Boité, Myung-Joong Hwang, and Martin B. Plenio' title: 'Metastability in the driven-dissipative Rabi model' --- Introduction ============ In the context of cavity quantum electrodynamics (QED), a common way to probe the quantum nature of the interaction between light and matter is to drive the system with a classical light field and record the statistics of the photons emitted from the cavity. For example, a sub-Poissonian statistics of output photons is an important evidence of effective photon-photon interactions induced by the atom-cavity coupling [@Walls:2007]. Such genuine quantum effects have been observed in a variety of systems, in the so-called strong-coupling regime of cavity QED, when the atom-cavity coupling strength is larger than any dissipation rate [@Rempe:1987; @Reithmaier:2004; @Wallraff:2004; @Peter:2005]. Recently, experimental progress in tailoring the light-matter interaction has made it possible to achieve a coupling strength that is comparable or even larger than the cavity frequency $\omega_c$ [@Devoret:2007; @Bourassa:2009; @Todorov:2010; @Niemczyk:2010; @Forn-Diaz:2010; @Nataf:2011; @Forn-Diaz:2016; @Yoshihara:2016; @Forn-Diaz:2016b]. From a theoretical perspective, the possibility of exploring this so-called ultrastrong coupling regime has stimulated numerous studies on the quantum Rabi model that takes into account the counter-rotating terms in the atom-cavity interaction [@Irish:2007; @Ashhab:2010; @Hwang:2010; @Casanova:2010; @Braak:2011; @Hwang:2015; @Wang:2016]. Since dissipation also plays a crucial role in most quantum optical setups, a meaningful description in this context involves a driven-dissipative scenario [@Ciuti:2006; @DeLiberato:2009; @Beaudoin:2011; @Ridolfo:2012; @Henriet:2014], in which the interplay between cavity losses and the external field drives the system into a steady state. In such a driven-dissipative setting of the Rabi model, it has been shown recently in Ref. [@LeBoite:2016] that as the coupling strength increases from $0.1\omega_c$ to $3\omega_c$, a series of transitions occurs in the output photon statistics, leading to a breakdown and revival of the so-called photon blockade effect and to a reversion to non-interacting photons. It demonstrates that the intricate interplay among the ultrastrong light-matter coupling, the external coherent driving and the dissipation stabilizes the system into a steady state exhibiting a rich quantum optical phenomenology. In this paper, going beyond the study of steady-state properties, we investigate the transient dynamics of the driven-dissipative Rabi model and show that it exhibits metastability in the ultrastrong coupling regime. Namely, we find that the convergence to the steady state is governed by a time scale significantly larger than the decay times of the atom and the cavity, giving rise to long-lived metastable states. When the atom-cavity coupling is much smaller than the cavity frequency, the time dependency of the Liouvillian can be eliminated by a change of reference frame [@Walls:2007]. All the information on the dynamics and metastable states is then encoded in the eigenvalues and eigenfunctions of the time-independent Liouvillian [@Risken:1987; @Vogel:1988; @Risken:1988; @Vogel:1989; @Casteels:2016; @Macieszczak:2016]. The break-down of the rotating-wave approximation in the ultrastrong coupling do not allow for such a simple transformation and the master equation remains time-dependent [@Ridolfo:2012; @LeBoite:2016]. To circumvent this issue we employ a Floquet-Liouville approach [@Ho:1986; @Grifoni:1998]: By applying Floquet theory to the Linblad master equation we reduce the time-dependent master equation to a time-independent eigenvalue problem in an enlarged Hilbert space. Within this theoretical framework, we compute the long-time dynamics in the weak-excitation regime, for a driving field resonant with the second available transition. We find that the corresponding Liouvillian gap becomes significantly smaller than the natural decay rates as one increases the atom-cavity coupling strength and relate this feature to the dressed-state properties of the Rabi Hamiltonian. More specifically, a central role is played by a parity shift occurring in the spectrum, resulting in the existence of two distinct decay channels. Metastability stems from the interplay between the two different time scales involved in these two channels. The Floquet-Liouville formalism also allows us to derive analytical expressions for the set of all possible metastable sates in terms of eigenvectors of the Floquet-Liouvillian and set bounds on the deviations from the steady state. Finally, we discuss practical implications of our analysis for future experiments probing the steady-state properties of the driven-dissipative Rabi model. The paper is organized as follows: The model is introduced in Sec. \[sec:model\]. The first numerical evidence of a separation of time scales in the dynamics and the emergence of metastable states are presented in Sec. \[sec:longtime\]. Section \[sec:floquet\] is devoted to the Floquet-Liouville formalism which is applied in Sec. \[sec:meta\] to a more thorough and systematic analysis of metastability. In Sec. \[sec:noise\] we evaluate the robustness of our findings when pure dephasing noise is included in the model and we conclude in Sec. \[sec:conclu\]. More details on Floquet theory are presented in Appendix \[app:floquet\] and the proofs of some spectral properties of the Floquet-Liouville operator are provided in Appendix \[app:meta\]. The model {#sec:model} ========= We consider a single cavity mode coupled to a two-level atom described by the Rabi Hamiltonian, $$\label{Hamilto_r} H_r = \omega_c a^{\dagger}a + \omega_a\sigma_+\sigma_- -g(a+a^{\dagger})\sigma_x,$$ where we have introduced the photon annihilation operator $a$, and the Pauli matrices $\sigma_x$, $\sigma_y$ (with $\sigma_{\pm} = \frac{1}{2}(\sigma_x \pm i\sigma_y)$). Here, $\omega_c$ is the cavity frequency, $\omega_a$ the atomic transition frequency, and $g$ the atom-cavity coupling strength. In the following we will focus on a resonant case, i.e., $\omega_c = \omega_a$. Note that there is no general explicit expression for the eigenstates and eigenvalues of the Rabi model. In the following, it will be convenient to label them by using an important symmetry property of the Hamiltonian, namely that the parity of the total number of excitations, $\Pi=\exp[i\pi(a^\dagger a +\sigma_+\sigma_-)]$, is a conserved quantity. We will denote by $|\Psi_j ^{\pm}\rangle$ the $j^{th}$ eigenstate ($j=0,1,..$) of the $\pm$ parity subspace and by $E_{j}^{\pm}$ the corresponding energy. With these notations, the ground state of $H_r$ is the state $|\Psi_0^{+}\rangle$, which is the lowest energy state of the $+$ parity subspace; while the first excited state of $H_r$, which corresponds to the lowest energy state of the $-$ parity subspace, is $|\Psi_0^{-}\rangle$. We focus in this paper on a driven-dissipative scenario where the cavity is driven by a monochromatic coherent field and both the cavity and the atom are coupled to their environments, leading to dissipation. The total time-dependent Hamiltonian of the system is $$\label{Hamilto} H(t) = H_r + F\cos(\omega_dt)(a+a^{\dagger}),$$ where $F$ is the intensity of the driving field and $\omega_d$ its frequency. The time evolution of the density matrix $\rho(t)$ is governed by a master equation of the form, $$\label{ME} \partial_t \rho = i[\rho,H(t)] +\mathcal{L}_a\rho+\mathcal{L}_{\sigma}\rho,$$ where the term $\mathcal{L}_a\rho+\mathcal{L}_{\sigma}\rho$ describes the dissipation of the system excitations into the environment. In the ultrastrong coupling regime, it is crucial to take fully into account the coupling between the atom and the cavity in the derivation of the master equation [@DeLiberato:2009; @Beaudoin:2011]. In particular, the atom and the cavity can no longer be regarded as being independently coupled to their own environment and the jump operators must involve transitions between eigenstates of the total atom-cavity Hamiltonian [@Beaudoin:2011]. A natural basis to express the correct master equation is therefore the dressed-state basis $\{|\Psi_j^{p}\rangle \}$ with $p=\pm$, in which the Hamiltonian (without driving) is diagonal. In this basis, the dissipative part reads, $$\begin{aligned} \label{MEdiss} \mathcal{L}_a\rho+\mathcal{L}_{\sigma}\rho=\sum_{p=\pm}\sum_{k,j}\Theta(\Delta_{jk}^{p\bar{p}})\left(\Gamma_{jk}^{p\bar{p}}+K_{jk}^{p\bar{p}}\right)\mathcal{D}[|\Psi_j^{p}\rangle\langle \Psi_k^{\bar{p}}|],\end{aligned}$$ where $\Theta(x)$ is a step function, i.e., $\Theta(x)=0$ for $x\leq0$ and $\Theta(x)=1$ for $x>0$, and $\bar{p}=-p$. We have also introduced the following notation, $\mathcal{D}[\mathcal{O}] = \mathcal{O}\rho\mathcal{O}^{\dagger} - \frac{1}{2}(\rho\mathcal{O}^{\dagger}\mathcal{O} + \mathcal{O}^{\dagger}\mathcal{O}\rho)$. The quantities $\Gamma_{jk}^{p\bar{p}}$ and $K_{jk}^{p\bar{p}}$ denote the rates of transition from a dressed-state $|\Psi_k^{\bar{p}}\rangle$ to $|\Psi_j^{p}\rangle$ due to the atomic and cavity decay, respectively; the transition rates are defined as [@Beaudoin:2011; @Ridolfo:2012] $$\begin{aligned} \label{trans_rates} \Gamma_{jk}^{p\bar p} &= \gamma\frac{\Delta_{jk}^{p \bar p}}{\omega_c}|\langle \Psi_j^{p}|(a - a^{\dagger})|\Psi_k^{\bar p}\rangle|^2, \nonumber\\ K_{jk}^{p\bar p} &= \kappa\frac{\Delta_{jk}^{p\bar p}}{\omega_c}|\langle \Psi_j^{p}|(\sigma_- - \sigma_+)|\Psi_k^{\bar p}\rangle|^2,\end{aligned}$$ where $\Delta_{jk}^{p\bar p} = E_k^{\bar p} - E_j^{p}$ is the transition frequency and $\gamma$, $\kappa$ are respectively the cavity and the atom decay rates. Note that the transition between states belonging to the same parity space is forbidden because both operators $a - a^{\dagger}$ and $\sigma^- - \sigma^+$ change the parity of the state. In Eqs (\[MEdiss\]) and (\[trans\_rates\]), the usual quantum optical master equation in which the jump operators are simply $a$ and $\sigma^-$ is recovered when the coupling strength is much smaller than the cavity frequency. In the following, we will be interested in the long time dynamics of Eq. (\[ME\]). As in most quantum optical setups, the relevant observables to characterize the system are correlation functions of the output field. As shown in Ref. [@Ridolfo:2012], the output field in the ultrastrong coupling is proportional to an operator $\dot X^+$, defined in the dressed-state basis as: $$\dot{X}^+ = \sum_{p=\pm}\sum_{k,j}\Theta(\Delta_{jk}^{p\bar{p}}) \Delta_{jk}^{p\bar{p}}|\Psi_j^{p}\rangle \langle \Psi_j^{p}|i(a^\dagger-a)|\Psi_k^{\bar{p}}\rangle \langle \Psi_k^{\bar{p}}|.$$ The two main correlation functions that we will consider are the intensity of the emitted photons, which is proportional to $I_{out}=\langle \dot X^-\dot X^+\rangle$, and the second-order correlation function, which reads $$g^{(2)}(0) = \frac{\langle \dot X^-\dot X^-\dot X^+ \dot X^+\rangle}{\langle \dot X^-\dot X^+\rangle^2}.$$ Note that except for a sufficiently small $g$, where the rotating approximation on qubit-cavity coupling can be applied, Eq. (\[ME\]) generally does not have a particular rotating-frame where the equation becomes time-independent. Therefore, the solution has a residual oscillation at the driving frequency $\omega_d$ even in the $t\rightarrow\infty$ limit. The steady-state properties are then obtained by averaging the solution over several driving periods, which corresponds to a time integrated measurement in an actual experiment [@Ridolfo:2012]. Long time dynamics and separation of time scales {#sec:longtime} ================================================ In Ref. [@LeBoite:2016] we have shown that in terms of output photon statistics, the most interesting properties are obtained when driving the second available transition, $|\Psi_0^+\rangle\to|\Psi_1^-\rangle $ (See Fig. \[fig:spec\]). We will therefore also focus on this driving scenario in all that follows. One of the main characteristic of the steady state is then that the $g^{(2)}(0)$ function exhibits a nonmonotonic behavior as a function of the coupling strength. More precisely, four different phases of photon emission can be identified: The photon blockade effect that is well-known to occur in the strong coupling regime \[$\gamma/\omega_c,\kappa/\omega_c\ll g/\omega_c \ll 1$\] persists up to a coupling strength $g/\omega_c\sim0.45$. It is then followed by a break-down and revival of the photon blockade effect (for $0.45\lesssim g/\omega_c\lesssim1$ and $1\lesssim g/\omega_c\lesssim2.5$ respectively), and a transition to a noninteracting regime (for $ g/\omega_c \gtrsim 2.5$). These results are summarized in Fig. \[fig:timeinteg\], where the blue solid line shows the output intensity $I_{out}$ and $g^{(2)}(0)$ in the steady state as a function of the coupling strength $g/\omega_c$. The intensity of the driving field and the dissipation rates are chosen such that the system stays in a weak-excitation regime: $\gamma = \kappa = 10^{-2}\omega_c$ and $F/\gamma = 0.1$. Figure \[fig:timeinteg\] also shows the same quantities obtained for long but finite simulation times $\tau$ (where the system is assumed to be in the ground-state at $t = 0$). For both finite-time and steady-state values, fast oscillations are eliminated by averaging over one period of the driving frequency (a time much smaller than the decay time) [@Ridolfo:2012; @LeBoite:2016]. Surprisingly, we observe that the long-time dynamics in the regime where the revival of the photon-blockade occurs, i.e., $1\lesssim g/\omega_c\lesssim2.5$, sharply stands out from other coupling strengths: The output intensity and the correlation function are far from having reached their steady-state values even after a time significantly longer than the natural relaxation time, i.e., $\tau = 1000/\gamma$, while for both $g<1$ and $g>2.5$ the steady-state values are already reached for $\gamma\tau = 10$. These unexpected, large discrepancies between the exact steady-state values and the finite-time simulations in the ultrastrong coupling regime suggest the emergence of a new relaxation time scale. To explore this further, we compute numerically the exact long-time dynamics of the output intensity for different values of $g/\omega_c$. In Fig. \[fig:timedyn\] (a) $I_{out}$ is shown as a function of time $\tau$, for times up to $\tau\gamma = 10^5$, and for $g/\omega_c = 1$, 1.2 and 1.5. The driving and dissipation parameters are the same as in Fig. \[fig:timeinteg\]. For $g/\omega_c = 1$ (blue dashed-dotted line), there is only one time scale in the transient dynamics and the steady-state value is reached for $1< \tau\gamma < 10$. This is a common feature for any coupling strength $g<1$. For $g = 1.2$, (dashed red lines), this simple picture is significantly modified. The steady-state value is only reached for $\tau\gamma > 10^3$ and two distinct phases in the transient dynamics are visible: a first evolution leads the system to an intermediate state for $\tau\gamma \approx 10$, followed by a slower decay to the steady state. This separation of time scales in the dynamics is greatly amplified for $g/\omega_c = 1.5$ (solid yellow line). In this case, the intermediate state is a long-lived metastable state. The output intensity is quasi-constant for a large time interval $10 \lesssim \tau\gamma \lesssim 10^3$ and reaches its asymptotic value only for $\tau\gamma \approx 10^5$. The transient dynamics is thus characterized by a gap between the two time scales for fast and slow decay processes, giving rise to metastable states. The numerical results presented in Figs. \[fig:timeinteg\] and \[fig:timedyn\] are one of the main findings of the present paper. They will be of significant experimental relevance for any setup in which the time scale of the the experiment is shorter than the time necessary to reach the steady state. In this case, the measured properties of the system in the long time limit would be that of metastable states and not of the true steady state. The principal aim of the remaining part of the paper is to establish a proper understanding of our numerical observations and explore the metastability in the driven-dissipative Rabi model in a systematic fashion. In the case of *time-indepedent* master equation, the time-scale of the transient dynamics and the properties of the metastable states can be understood in terms of spectral properties of the Liouvillian governing the time-evolution [@Macieszczak:2016]. To tap into this existing framework and investigate metastability in our *time-dependent* setting, the master equation in Eq. (\[ME\]) should therefore be cast into a time-independent form. However, due to the presence of the counter-rotating terms, there does not exist a reference frame where the time dependency is eliminated. Instead, as we will see in the next section, a time-independent formulation can be established by employing a Floquet-Liouville approach [@Ho:1986]. In this framework, eigenvalues of a Floquet-Liouvillian operator will play the same role as those of the usual Liouvillian. To illustrate this idea and motivate further the use of Floquet theory, we anticipate on what will follow and show on Fig. \[fig:timedyn\] (b) the quantity $\delta I_{out} = |I_{out}(\tau)-I_{out}(\infty)|/I_{out}(\infty)$ as a function of time. The different values of the coupling strength and the other parameters correspond to that of Fig. \[fig:timedyn\] (a). For each values of the $g/\omega_c$, the black dotted lines show an exponential fit with the corresponding eigenvalue $\Omega$ of the Floquet-Liouvillian operator, which will be introduced in the following section. The perfect agreement in the long-time limit is consistent with the separation of time scale described previously; after a sufficiently long time, only one slow-decaying component remains. \ Floquet-Liouville approach {#sec:floquet} =========================== Floquet theory applies to linear differential equations with periodic coefficients [@Floquet:1883] and, in the present context, can be used to reduce the time-dependent master equation to a time-independent eigenvalue problem in an enlarged Hilbert space. Although this so-called Floquet-Liouville approach is known and has found applications in various fields [@Grifoni:1998; @Chu:2004], it has not, to the best of our knowledge, been directly applied to the current setting of the driven and dissipative Rabi model. We therefore find it useful to present in this section the general formalism that lies at the core of our analysis. Further details on Floquet theory have also been included in Appendix \[app:floquet\]. As a useful comparison we refer to Ref. [@Hausinger:2011] where Floquet theory is applied to a *closed* Rabi model under strong driving. The master equation given in Eq. (\[ME\]) can be written as $$\label{MEshort} \partial_t \rho = \mathscr{L}(t) \rho,$$ where $\mathscr{L}$ is a periodic linear superoperator acting on the density matrix $\rho$ and satisfying $\mathscr{L}(t+T)=\mathscr{L}(t)$, where $T = 2\pi/\omega_d$. In the following we will denote by $\mathcal{H}$ the Hilbert space of the system. ($\rho$ is then an element of $\mathcal{H}^2$.) The Floquet theorem states that there exist solutions of Eq. (\[MEshort\]) of the form $$\label{floquetSol} \rho(t) = \sum_{\alpha}c_{\alpha}e^{\Omega_{\alpha}t}R_{\alpha}(t).$$ Here, $R_{\alpha}(t)$ is a periodic function of period $T$ and $\Omega_{\alpha}$ is a complex number, which are eigenfunctions and eigenvalues, respectively, of the following operator $$\label{eqPeriodic_main} (\mathscr{L}(t)-\partial_t)R_{\alpha}(t) = \Omega_{\alpha}R_{\alpha}(t).$$ Note that this last equation does not define a unique set of eigenvalues and eigenfunctions $\{\Omega_\alpha,R_\alpha\}$, the following transformation $$\begin{aligned} \Omega_{\alpha} &\to \Omega_{\alpha} -ik\omega_d, \\ R_{\alpha}(t) &\to e^{ik\omega_d} R_{\alpha}(t) \label{relEig},\end{aligned}$$ with $k\in \mathbb{Z}$, gives exactly the same solution for $\rho(t)$. In the remainder of this section we will therefore label the eigenvalues and eigenfunctions with two indices $\alpha$ and $k$, the sets $\{\Omega_{\alpha,0}R_{\alpha,0}\}$ and $\{\Omega_{\alpha,k},R_{\alpha,k}\}$ being linked by the above transformation. The key element in Eq. (\[eqPeriodic\_main\]) is that all the functions appearing in it are periodic. The problem can therefore be made time-independent by applying a Fourier transform. Equation becomes $$\label{floquetEig} \sum_{m = -\infty}^{\infty} \mathscr{L}^{(n-m)}R^{(m)}_{\alpha,k} +in\omega_dR_{\alpha,k}^{(n)} = \Omega_{\alpha}R_{\alpha,k}^{(n)},$$ where we have used the following convention for the Fourier series, $R_{\alpha,k}(t) = \sum_{n = -\infty}^{\infty} R_{\alpha,k}^{(n)}e^{-i n\omega_d t}$, $\mathscr{L}(t) = \sum_{n = -\infty}^{\infty} \mathscr{L}^{(n)}e^{-i n\omega_d t}$. Equation (\[floquetEig\]) is an eigenvalue problem in an enlarged Hilbert space and is sufficient, in this formulation, to find the expression of $\rho(t)$. For practical purposes, it is useful to go one step further and make the structure of the enlarged Hilbert space more explicit. This Hilbert space, sometimes called Floquet space is the space of $T$-periodic matrices on $\mathcal{H}^2$. Formally, it is the tensor product $\mathcal{H}^2\otimes \mathcal{T}$, where $\mathcal{T}$ denotes the Hilbert space of $T$-periodic functions. As a basis for the space $\mathcal{T}$, a natural choice is obviously the functions $\phi_n(t) = e^{-in\omega_dt}$. Following Refs. [@Grifoni:1998; @Hausinger:2010], we will denote $\phi_n$ by $|n)$ and write $\phi_n(t) = (t|n)$. With these notations, we represent the periodic matrix $R_{\alpha,k}(t)$ by a vector $|R_{\alpha,k}\rangle \rangle$ in $\mathcal{H}^2\otimes\mathcal{T}$ , defined as $$\label{FloNot} |R_{\alpha,k}\rangle \rangle = \sum_{n= -\infty}^{\infty} R_{\alpha,k}^{(n)}\otimes|n),$$ and we have $R_{\alpha,k}(t)=(t|R_{\alpha,k}\rangle \rangle$ by definition. This equation can therefore be seen as another way of writing the Fourier series of a periodic function. Within this framework, the eigenvalue problem of Eq. (\[floquetEig\]), can be written as $$\label{eigProbFlo} \tilde{\mathscr{L}}|R_{\alpha,k}\rangle \rangle = \Omega_{\alpha,k}|R_{\alpha,k}\rangle \rangle.$$ where the operator $\tilde{\mathscr{L}}$ acts on element of $\mathcal{H}^2\otimes \mathcal{T}$. As $\tilde{\mathscr{L}}$ is not Hermitian, it is necessary to distinguish the right eigenvectors defined above from the left eigenvectors obeying $$\begin{aligned} \tilde{\mathscr{L}}^{\dagger}|L_{\alpha , k} \rangle \rangle &= \Omega^*_{\alpha , k}|L_{\alpha , k} \rangle \rangle.\end{aligned}$$ We also introduce a scalar product on $\mathcal{H}^2\otimes \mathcal{T}$, $$\label{scalarProd} \langle \langle A|B\rangle\rangle = \sum_n \mathrm{Tr}[A^{(n) \dagger}B^{(n)}],$$ which derives from the usual scalar product on $\mathcal{T}$, $(f|g) = \frac{1}{T}\int_{0}^T f^*(t)g(t)\mathrm{d}t$ and the scalar product on $\mathcal{H}^2$, $\langle A|B\rangle = \mathrm{Tr[A^{\dagger}B}]$. Putting all this together, we can finally express the time evolution of the density matrix, i.e., the solution of Eq. (\[MEshort\]), in terms of the eigenvalues and the left and right eigenfunctions of the Floquet-Liouville operator $\tilde{\mathscr{L}}$. The first step is to express an initial density matrix of the system $\rho_0$ in Floquet space, e.g., $|\rho_0\rangle\rangle =\rho_0\otimes|0)$, and then decompose it in terms of eigenfunctions of $\tilde{\mathscr{L}}$, $$|\rho_0\rangle\rangle = \sum_{\alpha,k}c_{\alpha,k}|R_{\alpha,k}\rangle \rangle,$$ with $c_{\alpha,k} = \langle\langle L_{\alpha,k}|\rho_0\rangle\rangle$. Note that for a given initial density matrix $\rho_0$, the choice of the $|\rho_0\rangle\rangle$ is not unique, but this arbitrariness has no influence on the dynamics (see Appendix \[app:floquet\] for a proof of this statement). The time-evolution of this initial state then immediately follows as $$|\rho(t)\rangle\rangle = \sum_{\alpha,k}c_{\alpha,k}e^{\Omega_{\alpha,k}t}|R_{\alpha,k}\rangle \rangle,$$ which is the solution of Eq. (\[MEshort\]) expressed in the Floquet space. In this expression, the non-periodic part of the dynamics appears explicitly in $e^{\Omega_{\alpha,k}t}$, while the periodic part of the dynamics is implicitly encoded in $|R_{\alpha,k}\rangle \rangle$. As a final step, the solution can be expressed in the original Hilbert space using $\rho(t)=(t|\rho(t)\rangle\rangle$ and $R_{\alpha,k}(t)=(t|R_{\alpha,k}\rangle \rangle$, that is, $$\label{rhoDyn} \rho(t) = \sum_{\alpha,k}c_{\alpha,k}e^{\Omega_{\alpha,k}t}R_{\alpha,k}(t).$$ Note that in Eq. (\[rhoDyn\]), the summation is performed over both indices $\alpha$ and $k$, while the Floquet theorem as expressed in Eq. (\[floquetSol\]) involves only a sum over $\alpha$. The sum over $k$ can be suppressed by using Eq. (\[relEig\]) and writing Eq. (\[rhoDyn\]) in terms of eigenvalues and eigenvectors belonging only to the “first Brillouin zone”, $\Omega_{\alpha,0}$ and $|R_{\alpha,0}\rangle\rangle$. The final expression is then strictly equivalent to Eq. (\[floquetSol\]) and reads $$\begin{aligned} \label{rhoDynFin} \rho(t) = \sum_{\alpha}c_{\alpha}e^{\Omega_{\alpha,0}t}R_{\alpha,0}(t),\end{aligned}$$ where we have introduced the more compact notations $c_{\alpha} = \sum_n c_{\alpha,n}$. The structure of $\mathscr{L}$ guaranties that one of the eigenvalues, e.g. $\Omega_{0,0}$, is equal to zero [@Ho:1986]. The other eigenvalues are complex with a negative real part that determine the different time scales of the transient dynamics. Taking the limit $t \to + \infty$ in Eq. (\[rhoDynFin\]), we also see that the asymptotic density matrix is periodic and given by $\rho_{\infty}(t) = c_{0}R_{0,0}(t)$. In addition, the condition $\mathrm{Tr}[\rho_{\infty}(t)] = 1$ implies that the coefficient $c_0$ does not depend on the initial state and is simply a normalization constant. Absorbing it in the definition of $R_0(t)$, we can write $\rho_{\infty}$ as $$\label{rhoInf} \rho_{\infty}(t) = R_{0,0}(t).$$ Equations (\[rhoDynFin\]) and (\[rhoInf\]) show that the theory presented in this section gives the appropriate framework for investigating long-time properties of the system. It provides a direct access to the time scales involved and an efficient way to compute the time evolution of $\rho(t)$ for arbitrary long times without having to perform any time integration of the master equation. In the next section, we use these results to systematically investigate the long time dynamics and metastability in the driven-dissipative Rabi model. Metastable states {#sec:meta} ================== To apply the results of the previous section to our specific setting, let us first give a more explicit expression for the Liouville-Floquet operator corresponding to Eq. (\[ME\]). Making use of the notation introduced in Sec. \[sec:model\], the matrix elements of $\rho$, are expressed in the dressed state basis as $\langle \Psi_{i}^p|\rho|\Psi_{i'}^{p'}\rangle$, where $i,i' \in \mathbb{N}$ and $p,p' \in \{\pm\}$, and are therefore labeled by a set of four indices $\{i,p,i',p'\}$. To simplify the notation in the corresponding Floquet space we will denote by a single greek letter such a set of indices. Using also the basis $|n)$ introduced in the previous section for periodic functions, we deduce from Eq. (\[floquetEig\]) that the matrix elements of the Floquet-Liouville operator $\tilde{\mathscr{L}}$ read $$\label{floquetMatelem} \langle \langle \eta,n|\tilde{\mathscr{L}}|\beta,m\rangle\rangle = \mathscr{L}_{\eta \beta}^{(n-m)} +in\omega_d\delta_{nm}\delta_{\eta \beta},$$ where here $\mathscr{L} = i[\cdot, H] + \mathcal{L}_a+ \mathcal{L}_\sigma$. As in Sec. \[sec:floquet\], $\mathscr{L}^{(k)}$ refers to the $k$th Fourier component of $\mathscr{L}$. Note that the driving frequency appears explicitly in $\tilde{\mathscr{L}}$ in the form of a diagonal term. Moreover, since the time-dependency of the driving field is expressed through a cosine function, only matrix elements of $\tilde{\mathscr{L}}$ with $n-m = 0$ or $\pm 1$ are nonvanishing. All the numerical results presented in this paper have been obtained by diagonalizing $\tilde{\mathscr{L}}$ as expressed in Eq. (\[floquetMatelem\]) and computing the dynamics through Eq. (\[rhoDynFin\]). Within this framework, the results of Fig. \[fig:timedyn\] are straightforward to interpret. In particular, in the long-time limit, the reported exponential decay is governed by the eigenvalue $\Omega_{\alpha,0}$ of $\tilde{\mathscr{L}}$ that satisfies $ \mathrm{Re}[\Omega_{\alpha,0}] \neq 0$ and that has the smallest absolute real part. More importantly, we can now define a general criteria for the appearance of metastability in the system: metastable states exist if there is at least one non-zero eigenvalue $\Omega_{\alpha,0}$ of $\tilde{\mathscr{L}}$ satisfying $|\mathrm{Re}[\Omega_{\alpha,0}]|\ll \gamma$. For convenience, let us label the eigenvalues of $\tilde{\mathscr{L}}$ in such a way that $|\mathrm{Re}[\Omega_{\alpha,0}]|<|\mathrm{Re}\Omega_{\alpha+1,0}|$. We show in Fig. \[fig:liouvgap\] the real part of the first three non-zero eigenvalues, $\Omega_{1,0}$, $\Omega_{2,0}$ and $\Omega_{3,0}$ as a function of $g/\omega_c$. Remarkably, $|\mathrm{Re}[\Omega_{1,0}]|$ (blue dashed line) decreases sharply for $1\lesssim g/\omega_c \lesssim 2$, and reaches $10^{-6}\gamma$ while $|\mathrm{Re}[\Omega_{3,0}]|$ and $|\mathrm{Re}[\Omega_{2,0}]|$ rapidly saturate around $\gamma$ and $0.01\gamma$ respectively. This predicts that metastable states are likely to be observed for $g\gtrsim1$, and it is in good agreement with our previous numerical observation shown in Fig. \[fig:timeinteg\] To go further, it is important to keep in mind that unlike the steady state, metastable states are not unique; the one observed in an experiment will depend on the initial state. A natural task is then to determine the set of all possible metastable states and their properties. Once again, the Floquet-Liouville formalism will prove to be the appropriate tool. Let us begin the discussion by recalling two general results that can be deduced from the structure of the master equation. These results are a generalization to Floquet-Liouville formalism of metastability theory as presented, e. g., in Ref [@Macieszczak:2016]. i) If $\Omega$ is an eigenvalue of Eq. (\[eqPeriodic\_main\]) and $R(t)$ a corresponding eigenfunction, then $R^{\dagger}(t)$ is also an eigenfunction, and the associated eigenvalue is $\Omega^*$. ii) If $\Omega \in \mathbb{R}$, the left and right eigenfunctions $R(t)$ and $L(t)$ can be chosen Hermitian. In terms of Fourier component, this translates into $R^{(-n)} = R^{(n)\dagger}$. Proofs of these results are provided in Appendix \[app:meta\]. A first consequence is that the matrix $R_{0,0}$ appearing in Eq. (\[rhoInf\]) is Hermitian. To find the general expression for the metastable states, we will rely on an additional property of $\Omega_{1,0}$ visible on Fig. \[fig:liouvgap\]: for $g/\omega_c \gtrsim 1.3$, $\Omega_{1,0}$ not only satisfies $|\mathrm{Re}[\Omega_{1,0}]|\ll \gamma$, but also $|\mathrm{Re}[\Omega_{1,0}]|\ll |\mathrm{Re}[\Omega_{\alpha,0}]| $ for $\alpha > 1$. This means that after a sufficiently long time, the density matrix will take the form $$\label{metaGen} \rho(t) \approx R_{0,0}(t) + c_1R_{1,0}(t).$$ Another important feature of $\Omega_{1,0}$ is that it is pure real. The eigenfunction $R_{1,0}(t)$ can therefore be chosen Hermitian. Moreover, we know from Eq. (\[rhoInf\]) that $\mathrm{Tr}[R_{0,0}(t)] = 1$ for every time $t$, which in turn implies that $\mathrm{Tr}[R_{1,0}(t)] = 0$. Since $R_{1,0}(t)$ is Hermitian, we have also $c_1\in \mathbb{R}$. Conversely, any matrix taking the form of Eq. (\[metaGen\]) with $c_1\in \mathbb{R}$ and satisfying the positivity requirement of the density matrix is a possible metastable state. In particular, the set $\mathcal{M}$ of metastable states is a convex subset of the set of density matrices $\mathcal{D}$. Furthermore, $\mathcal{M}$ is parametrized by a single real coefficient. The set of all possible values of $c_1$ is therefore a segment $[c_{\mathrm{min}},c_{\mathrm{max}}]\subset \mathbb{R}$. To find $c_{\mathrm{max}}$ and $c_{\mathrm{min}}$, let us go back to the general expression for the coefficients $c_{\alpha} = \sum_k c_{\alpha,k}$. Using Eq. (\[scalarProd\]) for the scalar product defining $c_{\alpha,k }$ and assuming that the initial state $|\rho_0\rangle\rangle$ is of the form $\rho_0 \otimes |0)$, the coefficients $c_\alpha$ can be written as $$c_{\alpha} = \sum_{k} \mathrm{Tr}[L_{\alpha,k}^{(0)\dagger}\rho_0]$$ As previously, it is more convenient to express every quantity in terms of eigenfunctions $L_{\alpha,0}$ only. It is possible through the relation $L^{(0)}_{\alpha,k} = L^{(k)}_{\alpha,0}$, which is equivalent to Eq. (\[relEig\]). We find $$c_{\alpha} = \sum_{k} \mathrm{Tr}[L_{\alpha,0}^{(k)\dagger}\rho_0] = \mathrm{Tr}[L_{\alpha,0}(t= 0)\rho_0],$$ where the last equality follows from the definition of $L_{\alpha,0}(t)$ and the fact that $\rho_0$ is Hermitian. Applying this last result to $L_{1,0}$, we find that $c_{\mathrm{min}}$ is given by $c_{\mathrm{min}} = \min_{\rho \in \mathcal{D}}\mathrm{Tr[L_{1,0}(t=0)\rho]} $. A similar expression holds for $c_{\mathrm{max}}$. Given the positivity of $\rho$, the minimum is simply the smallest eigenvalue of $L_{1,0}(t=0)$ (which exist and is real since $L_{1,0}(t)$ is Hermitian). We have therefore the final result $$\begin{aligned} c_{\mathrm{min}} = \min \mathrm{Sp}[L_{1,0}(t=0)],\\ c_{\mathrm{max}} = \max\mathrm{Sp}[L_{1,0}(t=0)],\end{aligned}$$ where Sp denotes the spectrum. Any metastable state will then be a convex combination of two extremal states $$\begin{aligned} \rho_{\mathrm{min}} = R_0(t)+ c_{\mathrm{min}}R_1(t),\\ \rho_{\mathrm{max}} = R_0(t)+ c_{\mathrm{max}}R_1(t).\end{aligned}$$ Note that the results presented above are valid when $\Omega_{1,0}$ satisfies $|\mathrm{Re}[\Omega_{1,0}]|\ll |\mathrm{Re}[\Omega_{\alpha,0}]| $ for $\alpha > 1$. Figure \[fig:liouvgap\] shows that it is not the case for $g/\omega_c \sim 1$. Indeed, around this value of the coupling strength, the three eigenvalues $\Omega_{1,0}$, $\Omega_{2,0}$ and $\Omega^*_{2,0}$ are of the same order of magnitude and are all much smaller than $\gamma$. Hence, the general form of the metastable states in this regime of parameters is $\rho(t) \approx R_{0,0}(t) + c_1R_{1,0}(t)+c_2R_{2,0}(t)+ c_2^*R^{\dagger}_{2,0}(t)$. However, numerical simulations show that the eigenvalues of $L_{2,0}(t=0)$ are always much smaller than those of $L_{1,0}(t=0)$ and thus $c_{2},c_{2}^*\ll c_1$. Therefore, $R_{2,0}(t)$ and $R^{\dagger}_{2,0}(t)$ do not contribute significantly to the dynamics and the analysis of metastable states based on Eq. (\[metaGen\]) remains valid. An overview of the properties of the metastable states is given in Fig. \[fig:metamanifold\]. The output intensity and $g^{(2)}(0)$ in $\rho_{\infty}$, $\rho_{\mathrm{min}}$ and $\rho_{\mathrm{max}}$ are plotted as a function of the coupling strength. Note that, by definition of the extremal states, all the information on the set of metastable states is contained in $\rho_{\mathrm{min}}$ and $\rho_{\mathrm{max}}$. The values shown on Fig. \[fig:metamanifold\] set bounds on the deviation from the true steady-state value that can be observed in an experiment. As expected from our previous results, it is for $1\lesssim g/\omega_c\lesssim 2$ that the differences between these three states in terms of observables are the highest. In particular the photon statistics differs radically, being sub-Poissonian for $\rho_{\mathrm{min}}$ and strongly super-Poissonian for $\rho_{\mathrm{max}}$. Although metastable states also exist for higher values of $g$ ($g\gtrsim2$), the value of $I_{out} $ and $g^{(2)}(0)$ converge to the steady-state value in this case. Comparing the results of Fig. \[fig:timeinteg\] and Fig. \[fig:metamanifold\], we find that the metastable state observed when the system is in its ground state at $t= 0$ is very close to the state $\rho_{\mathrm{min}}$. Conversly, a metastable state close to $\rho_{\mathrm{max}}$ is obtained when the initial state is the first excited state $|\Psi_0^-\rangle$ (not shown). \ A qualitative explanation for the difference in photon statistics for $\rho_{\mathrm{min}}$ and $\rho_{\mathrm{max}}$ can be drawn from the dressed state properties of the Rabi model and the competing decay processes at play. As shown in Fig. \[fig:spec\](a), when the transition $|\Psi_0^+\rangle \to |\Psi_1^-\rangle $ is driven, there appear two decay channels for $g/\omega_c \gtrsim 0.45$, after a parity shift in the spectrum has occurred [@LeBoite:2016]. The first decay channel involves the direct transition $|\Psi_1^-\rangle \to |\Psi_0^+\rangle $, while the second one involves the cascaded transition $|\Psi_1^-\rangle \to |\Psi_1^+\rangle \to |\Psi_0^-\rangle \to |\Psi_0^+\rangle$. Because the direct transition leads to sub-Poissonian and the cascaded transition to super-Poissonian statistics of the output photons [@LeBoite:2016], we can expect that the competition between these two decay processes will ultimately determine the output photon statistics. More precisely, numerical simulations show that the metastable state $\rho_{\mathrm{min}}$ is mainly a statistical mixture of $|\Psi_0^+\rangle$ and $|\Psi_1^-\rangle$, namely $\rho_{\mathrm{min}}\approx \lambda_0|\Psi_0^+\rangle\langle\Psi_0^+| + \lambda_3|\Psi_1^-\rangle\langle\Psi_1^-|$, with $\lambda_3\ll \lambda_0$. The metastable state $\rho_{\mathrm{max}}$ on the other hand is a statistical mixture of $|\Psi_0^-\rangle$ and $|\Psi_1^+\rangle$, $\rho_{\mathrm{max}}\approx \lambda_1|\Psi_0^-\rangle\langle\Psi_0^-| + \lambda_2|\Psi_1^+\rangle\langle\Psi_1^+|$, with $\lambda_3/\lambda_0 \approx \lambda_2/\lambda_1$. This means that $\rho_{\mathrm{min}}$ and $\rho_{\mathrm{max}}$ can be reached when the dominant relaxation process is the direct transition or the cascaded transition, respectively. Therefore, $\rho_{\mathrm{min}}$ leads to a pronounced photon blockade that can be even stronger than in the steady state while, in contrast, $\rho_{\mathrm{max}}$ shows photon bunching (see Fig. \[fig:metamanifold\] (b)). The observed metastable state depends sensitively on the initial state. For example, when the initial state is the ground state, the eigenstates $|\Psi_0^-\rangle$ and $|\Psi_1^+\rangle$ can be populated only through the cascaded transition. We show in Fig. \[fig:spec\] (b) that the transition rates $\chi_{00}^{+-}$ and $\chi_{11}^{+-}$ for $|\Psi_0^-\rangle\to|\Psi_0^+\rangle$ and $|\Psi_1^-\rangle\to|\Psi_1^+\rangle$ respectively, drop sharply for $g/\omega_c\gtrsim1$, while the transition rates $\chi_{01}^{+-}$ and $\chi_{01}^{-+}$ for $|\Psi_1^-\rangle \to |\Psi_0^+\rangle$ and $|\Psi_1^+\rangle \to |\Psi_0^-\rangle$ are much higher and satisfy $\chi_{01}^{+-}\sim \chi_{01}^{-+}$. Therefore, the processes leading to the system being in the subspace $\{|\Psi_0^-\rangle, |\Psi_1^+\rangle\}$ take place at a much slower rate. Hence, on the relatively short time scale on which metastability is observed, this subspace does not play a significant role in the dynamics and the metastable state is very close to $\rho_{\mathrm{min}}$ \[Fig. \[fig:timeinteg\] (b)\]. To summarize, the general physical picture is the following: the steady state is reached when the pumping mechanisms exactly compensate the losses induced by the different decay channels. In the Rabi model, the parity shift occurring in the Hamiltonian for $g/\omega_c \approx 0.45$ leads to the existence of two distinct decay channels \[Fig. \[fig:spec\] (a)\]. Furthermore, the time scales involved in these two channels become widely separated as the coupling strength becomes larger than the cavity frequency ($g/\omega_c \gtrsim1$) \[Fig. \[fig:spec\] (b)\]. As a result, there exists an intermediate time scale in which losses from the fast decay channel are already compensated by the driving field while the other channel has not yet come into play. In such a time interval, which is long enough to be observed experimentally, the system is in a metastable state whose properties can differ radically from those of the true steady state. Discrepancies between metastable states and the steady state are particularly sharp in the regime of coupling strength where the revival of the photon blockade takes place \[$1\lesssim g/\omega_c\lesssim 2$\], since in this regime the two decay channels have opposite effects on the photon statistics: the fast one favors the photon blockade effect, while the slower one destroys it by inducing additional fluctuations. This picture however breaks down for $g\gg1$ where the energy spectrum of the Rabi model becomes quasi-linear [@Hwang:2016]; in this case, the states $|\Psi_j^+\rangle$ and $|\Psi_j^-\rangle$ are quasi-degenerate and the relaxation processes also involve transitions between higher-energy states. The decay channels are now two distinct “ladders”: $|\Psi_j^-\rangle \to |\Psi_{j-1}^+\rangle \to \dots \to |\Psi_1^-\rangle \to |\Psi_{0}^+\rangle$ when the initial state is the ground state, and $|\Psi_j^+\rangle \to |\Psi_{j-1}^-\rangle \to \dots \to |\Psi_1^+\rangle \to |\Psi_0^-\rangle$ when the system is initially in its first excited state. A separation of time scales still exists in this regime; it stems from the very low probability of transition between the two ladders through processes such as $|\Psi_{j}^+\rangle \to |\Psi_{j}^-\rangle$. However, the two channels both lead to a quasi-coherent statistics, explaining the convergence of the metastable-states properties to those of the steady state. Effect of pure dephasing noise {#sec:noise} ============================== In this section we evaluate the robustness of our findings against pure dephasing noise, inevitably present in any experimental setup. Following Ref. [@Beaudoin:2011] we model the dephasing noise by including an additional term in the Liouvillian. Its general form is, $$\begin{aligned} \mathcal{L_{\phi}}\rho &= \mathcal{D}\left[\sum_{p=\pm}\sum_{k}\Phi^p_k|\Psi_k^p\rangle\langle \Psi_k^p|\right] \\ &+ \sum_{p=\pm}\sum_{k,j}\Theta(\Delta_{jk}^{pp})\Phi_{jk}^{pp}\mathcal{D}[|\Psi_j^{p}\rangle\langle \Psi_k^{p}|]. \end{aligned}$$ For this type of noise, the transition rates depend on the matrix elements of the operator $\sigma_z$ in the dressed-state basis and are given by $$\begin{aligned} \Phi^p_k &= \sqrt{\frac{\gamma_{\phi}(0)}{2}}\langle \Psi_k^p|\sigma_z|\Psi_k^p\rangle, \\ \Phi_{jk}^{p p} &= \frac{\gamma_{\phi}(\Delta_{jk}^{p p})}{2}|\langle \Psi_j^{p}|\sigma_z|\Psi_k^{ p}\rangle|^2.\end{aligned}$$ These coefficients depend on the spectral density of the bath at the different transition frequencies $\Delta_{jk}^{p p}$, expressed by the function $\gamma_{\phi}(\Delta_{jk}^{p p})$. Just as in the case of the other dissipative terms, we assumed that the spectral density of the bath vanishes at negative frequency, since the system is in thermal equilibrium at zero temperature. Note that in contrast with the operators $a$ and $\sigma_-$, the operator $\sigma_z$ can induce transitions only between states of the same parity. In principle, the additional transitions between dressed-states induced by the dephasing noise can affect the transient regime and reduce the life time of the metastable states. We show in Fig. \[fig:liouvgapdeph\], numerical simulations of the Floquet-Liouvillian eigenvalues for a white dephasing noise, whose rate is comparable to the other noise sources \[$\gamma_{\phi} = \gamma = \kappa$\]. Globally, the real part of the eigenvalues is larger, which means that the time to reach state is indeed reduced compared to the results of Fig. (\[fig:liouvgap\]). However, the clear separation of time scales is still visible and the life time of the metastable states is long enough to allow for experimental observation. Hence, there is no qualitative change and our results remain valid even when this additional noise channel is included in the model. Conclusion {#sec:conclu} ========== In this paper, we have investigated the long-time dynamics and metastability of the driven-dissipative Rabi model in the ultrastrong coupling regime within the framework of Floquet-Liouville theory. In the ultrastrong coupling regime, the counter-rotating terms make the master equation for the driven Rabi model explicitly time-dependent, and the Floquet-Liouville theory allows one to eliminate this explicit time-dependence by considering the time evolution in an enlarged Hilbert space of periodic matrices. Our work demonstrates that the use of Floquet-Liouville theory in the driven-dissipative Rabi model not only makes an efficient calculation of arbitrarily long time-evolution possible, but also enables one to obtain analytical results and a qualitative understanding. More specifically, we have considered a driving scenario in which the external field is resonant with the second available transition and have shown that, as the atom-cavity coupling strength becomes larger than the cavity frequency, $g/\omega_c \gtrsim 1$, the time necessary to reach the steady state becomes much larger that the natural relaxation time $1/\gamma$. Within the framework of Floquet-Liouville theory, the different time scales of the transient dynamics are understood by investigating the eigenvalues of the time-independent Floquet-Liouvillian operator. For $g/\omega_c>1$, one non-zero eigenvalue with zero imaginary part (purely dissipative mode) was found to be several orders of magnitude smaller than all the other eigenvalues, explaining the emergence of long-lived metastable states. We attributed this feature of the Floquet-Liouvillian to the existence of two decay channels for the system with different transition rates. In particular, the transition rates for the first and third part of the cascaded transition $|\Psi_1^-\rangle \to |\Psi_1^+\rangle \to |\Psi_0^-\rangle \to |\Psi_0^+\rangle$ go to zero as $g/\omega_c$ increases. As a result, this decay channel starts to play a significant role only in the long-time dynamics. During the large time interval for which the other decay channel, $|\Psi_1^-\rangle \to |\Psi_0^+\rangle $ dominates, the system reaches a metastable state, which eventually decays into the true steady state when the second channel comes into play. By extending the recently developed metastability theory [@Macieszczak:2016] to our time-dependent setting through the Floquet-Liouville approach, we also derived analytical expressions for the set of all possible metastable states in terms of eigenvectors of the Floquet-Liouvillian. This enabled us to set bounds on the deviation from the true steady state that could be observed in an experiment. More specifically, we showed that for $1\lesssim g/\omega_c\lesssim 2$ the photon statistics in the metastable states can differ drastically from that of the steady state ; it can either show an enhanced anti-bunching or, conversely, strong bunching. All these results were derived by considering dissipation coming from the coupling of the cavity and the atom to the environment at zero temperature. We have also performed additional simulations including pure dephasing noise and have shown that our findings remain unchanged when this etra noise channel is included in the model. In a circuit QED experiment with a typical cavity frequency $\omega_c$ of the order of the GHz and dissipation rates similar to the one considered here \[$\kappa = \gamma = 10^{-2}\omega_c$\], the time scale on which metastability will be observed is of the order of 0.1 millisecond, a time sufficiently short to be reached experimentally. This work was supported by the EU STREPs DIADEMS and EQUAM, the ERC Synergy Grant BioQ as well as the DFG via the SFB TRR/21 and SPP 1601. Floquet theory and dynamics in Floquet space {#app:floquet} ============================================ We give in this appendix a more detailed and self-contained presentation of Floquet theory and its formulation in the Floquet space introduced in the main text. To simplify the notations, we consider the case of a usual Schrödinger equation on a Hilbert space $\mathcal{H}$ of finite dimension $N$, $$\label{eqDiffFlo} i\partial_t |X\rangle = A(t)|X\rangle,$$ where $A$ is a periodic matrix of period $T$ and $X$ a vector in $\mathcal{H}$. The Floquet theorem states that there exist solutions of the form $$\label{floquetSolApp} |X_{\alpha}(t)\rangle = e^{-i\epsilon_{\alpha} t} |p_{\alpha}(t)\rangle,$$ with $|p_{\alpha}(t)\rangle$ periodic, of period $T$, and $\epsilon_{\alpha}$ a complex number. The functions $|p_{\alpha}(t)\rangle$ are eigenfunctions of the following operator $$\label{eqPeriodic} (A(t)-i\partial_t)|p_\alpha(t)\rangle = \epsilon_{\alpha}|p_{\alpha}(t)\rangle.$$ Since all the functions that appear in Eq. are periodic, this equation translates the original problem into an eigenvalue problem in a space of periodic functions. Let us therefore introduce the space $\mathcal{F} = \mathcal{H}\otimes T$ of periodic functions on $\mathcal{H}$. This space is a Hilbert space whose scalar product derives for the one defined on $\mathcal{H}$ and $\mathcal{T}$. Following the notations of Ref. [@Grifoni:1998; @Hausinger:2010], we define the scalar product on $\mathcal{T}$ as $$(f|g) = \frac{1}{T}\int_0^T f^*(t)g(t)dt,$$ and the scalar product on $\mathcal{H}\otimes \mathcal{T}$ as $$\langle \langle \cdot |\cdot \rangle \rangle = \frac{1}{T}\int_0^T \langle \cdot |\cdot \rangle dt.$$ This definition coincides with the usual definition of the scalar product on a tensor-product space. Indeed, for two factorized states $|\Psi_1\rangle \rangle = f_1(t) |\phi_1\rangle $ and $|\Psi_2\rangle \rangle =f_2(t) |\phi_2\rangle $, with $f_1, f_2 \in \mathcal{T}$ and $|\phi_1\rangle, |\phi_2\rangle$ time-independent, we have: $$\langle \langle \Psi_1|\Psi_2 \rangle \rangle = \langle \phi_1|\phi_2\rangle \frac{1}{T}\int_0^T f_1^*(t)f_2(t) dt = \langle \phi_1|\phi_2\rangle (f_1|f_2).$$ A natural basis on $\mathcal T$ is obviously $\phi_n(t) = e^{-in\omega_dt}$, for which we use the notation $|n)$. By analogy with usual Dirac notations, we will also write $\phi_n(t) = (t|n)$. Let $\{|\mu\rangle\}$ denote a basis of $\mathcal{H}$, the vectors $|\mu,n\rangle\rangle = |\mu\rangle\otimes|n)$ then form a basis of $\mathcal{F}$ and the projection on this basis coincides with the Fourier transform. In other words, with these notations, any periodic state vector $|\psi(t)\rangle $ of $\mathcal{H}$ is represented in $\mathcal{F}$ by a vector $|\psi\rangle \rangle$ whose components are given by $$\langle \langle \mu,n|\psi\rangle \rangle = \frac{1}{T}\int_{0}^Te^{in\omega_pt} \langle \mu|\psi(t)\rangle = \langle \mu|\psi^{(n)}\rangle.$$ where $|\psi^{(n)}\rangle$ is the $n^{\mathrm{th}}$ Fourier component. Coming back to the eigenvalue problem of Eq. (\[eqPeriodic\]), it has a time-independent formulation in $\mathcal{F}$ and can be written as $$\label{floquetEigFSpace} \tilde{A}|p_{\alpha}\rangle \rangle = \epsilon_{\alpha}|p_{\alpha}\rangle \rangle,$$ In the basis introduced above, the matrix elements of the operator $\tilde{A}$ are given by $$\langle \langle \alpha,n|\tilde{A}|\beta, m \rangle \rangle = A^{(n-m)}_{\alpha \beta} - n\omega_d\delta_{nm}\delta_{\alpha \beta}.$$ If $\tilde{A}$ is diagonalizable, we can find a basis of eigenvector in $\mathcal{F}$. Since $\mathcal{F}$ is infinite dimensional, let us label the eigenvalues and eigenvectors of Eq. (\[floquetEigFSpace\]) with a double index, $\{|p_{\alpha,k}\rangle\rangle, \epsilon_{\alpha,k}\}$, where $1\leq \alpha \leq N$ and $k\in \mathbb{Z}$. In principle, for every such eigenvector and eigenvalue, one can define a solution of Eq. (\[eqDiffFlo\]) given by $$|X_{\alpha,k}(t)\rangle = e^{\-i\epsilon_{\alpha,k}t}(t|p_{\alpha,k}\rangle\rangle.$$ However, we know from the theory of ordinary differential equations that only $N$ such functions are linearly independent. This is reflected in the following relation between eigenvalues and eigenvectors in $\mathcal{F}$ : let $p_{\alpha,0}$ denote the eigenfunctions whose eigenvalue satisfies $|\epsilon_{\alpha,0}| < \omega_d/2$, the other eigenvalues and eigenvectors are given by $$\begin{aligned} \epsilon_{\alpha,k} &= \epsilon_{\alpha,0}+ k\omega_d,\\ |p_{\alpha,k}\rangle \rangle &= \sum_{n= -\infty}^{\infty} |p_{\alpha,0}^{(k+n)}\rangle\otimes|n),\end{aligned}$$ or equivalently, $$\label{relP} (t|p_{\alpha,k}\rangle\rangle = e^{ik\omega_pt}(t|p_{\alpha,0}\rangle\rangle.$$ This simply means that for any $k\in \mathbb{Z}$, $|X_{\alpha,k}(t)\rangle = |X_{\alpha,0}(t)\rangle$. The advantage of introducing the Floquet space is that Eq. (\[floquetEigFSpace\]) is time-independent. The dynamics in $\mathcal{H}$ can therefore be computed in the following way: let $|X_0\rangle\rangle$ denote a periodic function satisfying $(t|X_0\rangle\rangle|_{t= 0} = |X(0)\rangle$ (a possible choice is the constant function $|X(0)\rangle\otimes |0))$. The time evolution of $|X\rangle$ is then given by, $$\label{dynaFlo} |X(t)\rangle = (t|e^{-it\tilde{A}}|X_0\rangle \rangle.$$ The freedom in the choice of $|X_0\rangle\rangle$ comes from the infinite dimension of $\mathcal{F}$. Let us prove that it has no consequence on the dynamics in $\mathcal{H}$. For any initial vector $|X_0\rangle\rangle$ we can introduce the following decomposition $$|X_0\rangle \rangle = \sum_{\alpha,k} \lambda_{\alpha,k}|p_{\alpha,k}\rangle \rangle.$$ The initial condition then reads, $$|X(0) \rangle = \sum_{\alpha,k} \lambda_{\alpha,n}(t|p_{\alpha,n}\rangle \rangle|_{t=0}.$$ Using Eq. (\[relP\]), we find $$|X(0)\rangle = \sum_{\alpha= 1}^N \lambda_{\alpha}(t|p_{\alpha,0}\rangle\rangle_{t=0}$$ with $ \lambda_{\alpha}= \sum_{n = -\infty}^{\infty} \lambda_{\alpha,n}$. This last decomposition is unique since the functions $e^{-i\epsilon_{\alpha,0}t}(t|p_{\alpha,0}\rangle\rangle$ form a basis of solutions of Eq. (\[eqDiffFlo\]). Therefore, the coefficients $\lambda_{\alpha}$ do not depend on the choice of $|X_0\rangle\rangle$. Moreover, they completely determine the dynamics. Indeed, using again Eq.(\[relP\]) we can write $$\label{dyna} |X(t)\rangle = \sum_{\alpha = 1}^N\lambda_{\alpha}e^{-i\epsilon_{\alpha,0}t}(t|p_{\alpha,0}\rangle\rangle.$$ Similarly, Eq.(\[dynaFlo\]) can be extended to any initial time $t'$, $$\label{evolFloGen} |X(t)\rangle = (t|e^{-i(t-t')\tilde{A}}|X(t'),0\rangle \rangle,$$ where we have use the notation $|X(t'),0\rangle \rangle = |X(t')\rangle\otimes|0)$. Equation (\[evolFloGen\]) thus defines the propagator $U(t,t')$ such that $|X(t)\rangle = U(t,t')|X(t')\rangle$. The matrix elements of $U(t,t')$ in the basis $\{|\mu\rangle\}$ then read, $$U_{\mu,\nu}(t,t') = \sum_{\alpha,n,m} \langle \langle \mu,m |p_{\alpha,n}\rangle\rangle \langle \langle p_{\alpha,n}|\nu,0\rangle \rangle e^{-i\epsilon_{\alpha,n}(t-t')-im\omega_dt}.$$ Spectral properties of the Floquet-Liouville operator {#app:meta} ===================================================== In this appendix we prove the following properties of the periodic functions $R_{\alpha,k}(t)$ and $L_{\alpha,k}(t)$ introduced in the main text as left and right eigenfunctions of the operator $\mathscr{L}(t)-\partial_t$: 1. if $\Omega_{\alpha,k}$ is an eigenvalue such that $\mathrm{Re}[\Omega_{\alpha,k}]\neq 0$, then $\mathrm{Tr}[R_{\alpha,k}(t)] = 0$ for all $t$. 2. if $\Omega_{\alpha,k}$ in an eigenvalue, $\Omega_{\alpha,k}^*$ is also an eigenvalue and the corresponding eigenfunction is $R_{\alpha,k}^{\dagger}(t)$. 3. if $\Omega_{\alpha,k}$ is a real eigenvalue, $R_{\alpha,k}(t)$ and $L_{\alpha,k}(t)$ can both be chosen Hermitian. We assume that the operator $\mathscr{L}(t)$ is of Lindblad form, i.e. $\mathscr{L}(t)\rho = i[H(t),\rho] +1/2\sum_j (2C_i\rho C_i^{\dagger} - \rho C_iC^{\dagger}_i- C_iC^{\dagger}_i\rho) $, for some jump operators $C_i$. *Proof of 1.* This property follows from the fact that $\mathscr{L}$ is trace preserving: for any time $t$ and any matrix $\rho$, we have $\mathrm{Tr}[\mathscr{L}(t)\rho] = 0$. Injecting this relation into the eigenvalue equation satisfied by $R_{\alpha,k}(t)$ we find $$\label{traceEq} \partial_t \mathrm{Tr}[R_{\alpha,k}(t)] = -\Omega_{\alpha,k} \mathrm{Tr}[R_{\alpha,k}(t)].$$ In addition, $\mathrm{Tr}[R_{\alpha,k}(t)]$ must be periodic, (just as $R_{\alpha,k}(t)$). As a result, if $\mathrm{Re}[\Omega_{\alpha,k}]\neq 0$, the only solution to Eq. (\[traceEq\]) satisfying this condition is $\mathrm{Tr}[R_{\alpha,k}(t)] = 0$. *Proof of 2.* Due to the Linblad structure, the operator $\mathscr{L}(t)$ is invariant under Hermitian conjugation. More precisely, for any matrix $\rho$ we have $$(\mathscr{L}(t)\rho)^{\dagger} = \mathscr{L}(t)\rho^{\dagger}.$$ The result then follows by taking the Hermitian conjugate of the equation obeyed by $R_{\alpha,k}(t)$. We directly find $$(\mathscr{L}(t)-\partial_t)R^{\dagger}_{\alpha}(t) = \Omega^*_{\alpha}R^{\dagger}_{\alpha}(t).$$ *Proof of 3.* Let $\Omega_{\alpha,k}$ be a real eigenvalue and $R_{\alpha,k}$ a corresponding eigenfunction. We deduce from Prop. 2. that $R^{\dagger}_{\alpha,k}(t)$ is also an eigenfunction with the same eigenvalue. Hence, if $R'_{\alpha,k} = 1/2(R_{\alpha,k}(t)+R^{\dagger}_{\alpha,k}(t))$ is not constant and equal to zero, then it is a proper Hermitian eigenfunction. In the case were $R'_{\alpha,k}(t) = 0$, then $iR_{\alpha,k}(t)$ is an Hermitian eigenfunction. Suppose now that $R_{\alpha,k}(t)$ is Hermitian. In terms of Fourier components, this is equivalent to $R_{\alpha,k}^{(-n)} = R^{(n)\dagger}_{\alpha,k}$. Let us show that the corresponding left eigenfunction $L_{\alpha,k}(t)$ is also Hermitian. Given its expression in Floquet space, $L_{\alpha,k}(t)$ is uniquely defined by the following set of relations involving its Fourier components, $$\begin{aligned} \sum_n\mathrm{Tr}[L^{(n)\dagger}_{\alpha,k}R^{(n)}_{\beta,l}] &= 0 \quad \mathrm{for }\quad \beta\neq\alpha, l\neq k , \\ \sum_n\mathrm{Tr}[L^{(n)\dagger}_{\alpha,k}R^{(n)}_{\alpha,k}] &= 1.\end{aligned}$$ From the fact that for every $\beta$ and $l$, $R^{\dagger}_{\beta,l}$ is also an eigenfunction, we find that $$\begin{aligned} \sum_n\mathrm{Tr}[L^{(n)\dagger}_{\alpha,k}R^{(-n)\dagger}_{\beta,l}] &= \sum_n\mathrm{Tr}[L^{(-n)}_{\alpha,k}R^{(n)}_{\beta,l}] &= 0.\end{aligned}$$ Similarly, using the relation $R_{\alpha,k}^{(-n)} = R^{(n)\dagger}_{\alpha,k}$, we have $$\begin{aligned} \sum_n\mathrm{Tr}[L^{(n)\dagger}_{\alpha,k}R^{(-n\dagger)}_{\alpha,k}] &= \sum_n\mathrm{Tr}[L^{(-n)}_{\alpha,k}R^{(n)}_{\alpha,k}] &= 1.\end{aligned}$$ Combining these last two results, we see that the function $L(t)^{\dagger}_{\alpha,k}$, defined in terms of Fourier components by $(L^{\dagger}_{\alpha,k})^{(n)} = L^{(-n)\dagger}_{\alpha,k}$, satisfies the same set of relation as $L_{\alpha,k}(t)$. 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--- author: - Leigh Martin bibliography: - 'LeighBibliography.bib' title: Quantum feedback for measurement and control --- To my parents, my step parents, and my amazing sister Willow. The fact that research is not an isolated endeavor is one of the most important lessons I learned from my time at Berkeley. The quality of one’s research, and the joy of doing it are a direct result of one’s collaborators, colleagues and friends. I feel incredibly fortunate to have worked with such a talented, curious and kind group of people. I first wish to thank my advisors, Irfan Siddiqi and Birgitta Whaley for taking a risk on me by supporting my joint work in theory and experiment. The chance to work with both research groups has been an amazing and irreplaceable opportunity. I wish to thank both of them for providing the perfect balance of encouragement and critique, and guidance and freedom. In my experimental work, I am deeply grateful for the guidance of and collaboration with Shay Hacohen-Gourgy and Emmanuel Flurin, who showed me the ropes of experimental work, helped me discover my flaws and strengths, and never turned down an chance to discuss a crazy idea (no matter how sure they were that it was wrong!). I am also indebted to Mollie Schwartz, who helped give me the opportunity to work in Irfan’s group and introduced the field to me. Her warmth and encouragement made an enormous difference in embarking on a new path in research. In my theory work, I am indebted to the guidance and sharp intuition of Felix Motzoi, whose initial suggestion for a project carried me through a PhD’s worth of theory research, as well as Mohan Sarovar, whose mentorship gave me confidence and stability. I also wish to thank Mahrud Sayrafi, Sissi Wang, Yitian Chen, Song Zhang and Yuxiao Jiang for the meetings all over Berkeley while we carried out our joint projects. I looked forward and enjoyed each and every one of these discussions, which consistently took us in exciting and unexpected directions. In my experimental work, I especially wish to thank Vinay Ramasesh, who taught me the importance of cordial competition and open communication, and William Livingston, who always helped me see the light in the darkness of challenge or my own stubbornness. Many of the general insights that I attempt to convey in this thesis are of their making. I greatly appreciate Machiel Blok’s support, and all of the late afternoons spent tossing around ideas (also my apologies to Esther Blok for the countless times that I made Machiel late). I also wish to thank Sydney Schreppler, Kevin O’Brien, John Mark Kreikebaum, Andrew Eddins, David Toyli and Norman Yao for their support, collaborations and friendship. Finally, an enormous thank you to my parents, step parents, sister and friends in Berkeley, whose support was everything.

--- abstract: 'Using the numerical data of MHD simulation for AGN jets based on our “sweeping magnetic twist model”, we calculated the Faraday rotation measure (FRM) and the Stokes parameters to compare with observations. We propose that the FRM distribution can be used to discuss the 3-dimensional structure of magnetic field around jets, together with the projected magnetic field derived from the Stokes parameters. In the present paper, we supposed the basic straight part of AGN jet, and used the data of axisymmetric simulation. The FRM distribution we derived has a general tendency to have gradient across the jet axis, which is due to the toroidal component of the helical magnetic field generated by the rotation of the accretion disk. This kind of gradient in the FRM distribution is actually observed in some AGN jets (e.g. Asada et al. 2002), which suggests helical magnetic field around the jets and thus supports our MHD model. Following this success, we are now extending our numerical observation to the wiggled part of the jets using the data of 3-dimensional simulation based on our model in the following paper.' author: - 'Yutaka Uchida, Hiromitsu Kigure, Shigenobu Hirose, Masanori Nakamura, and Robert Cameron' title: | Distribution of Faraday Rotation Measure\ in Jets from Active Galactic Nuclei\ I. Prediction from our Sweeping Magnetic Twist Model --- Introduction ============ To explain the formation of active galactic nucleus (AGN) jets and other astrophysical jets, various models have been proposed. Among them, magnetohydorodynamic (MHD) model is one of the most promising models, since it can explain both the acceleration and the collimation of the jets. Lovelace (1976) and Blandford (1976) first proposed the magnetically driven jet from accretion disks, and Blandford & Payne (1982) discussed magneto-centrifugally driven outflow from a Keplerian disk in steady, axisymmetric and self-similar situation. Uchida & Shibata (1985) performed a time-dependent, two-dimensional axisymmetric simulation in the case of star-forming outflows. They pointed out that large amplitude torsional Alfvén waves (TAW’s) generated by the interaction between the accretion disk and a large scale magnetic field play an important role (detail is described in section \[sec:review-model\]). In this paper, we refer this model as “sweeping magnetic twist model”. Uchida & Shibata (1986) extended the treatment to the case of AGN jets. After this work, many authors have performed time-dependent, two-dimensional axisymmetric simulations (e.g. Stone & Norman 1994, Ustyugova et al. 1995, Matsumoto et al. 1996, Ouyed & Pudritz 1997, Kudoh, Matsumoto, & Shibata 1998). Acceleration mechanisms in MHD model were studied in detail by 1.5-dimensional MHD equations (Kudoh & Shibata 1997a, 1997b). Using the numerical data of MHD model, observational quantities such as the Faraday rotation measure (FRM) or the Stokes parameters have been derived to compare with observations of AGN jets: Laing (1981) computed the total intensity, the linear polarization, and the projected magnetic field distributions, assuming some simple magnetic field configurations and high energy particle distributions in the cylindrical jet. Clarke, Norman, & Burns (1989) performed two dimensional MHD simulations in which a supersonic jet with a dynamically passive helical magnetic field was computed, and derived distributions of the total intensity, the projected electric field, and the linear polarization. Hardee & Rosen (1999) calculated the total intensity and the projected magnetic field distributions, using 3-dimensional MHD simulations of strongly magnetized conical jets. Hardee & Rosen (2002) calculated the FRM distribution and discussed that the radio source 3C465 in Abell cluster A2634 (Eilek & Owen 2002) suggests helical twisting of the flow. The FRM is given by the integral of $n_e B_\parallel$ along the line-of-sight between the emitter and the observer (where $B_\parallel$ is the line-of-sight component of the magnetic field, and $n_e$ is the electron density there). It is, in principle, not possible to specify which part on the line-of-sight the contribution comes from. However, in recent high-resolution radio observations (e.g. Eilek & Owen 2002, Asada et al. 2002), the FRM distribution seems to have good correlation with the configuration of the jet; this suggests that the FRM variation is due to the magnetized thermal plasma surrounding the emitting part of the jet. In fact, sharp FRM gradients seen in 3C273 can not be produced by a foreground Faraday screen (Taylor 1998, Asada et al. 2002). If this is the case, we can get a new information, that is, the line-of-sight component of the magnetic field, and thus can predict the 3-dimensional configuration of the magnetic field around the jet, together with the projected magnetic field. In this paper, we calculate the FRM, projected magnetic field, and total intensity from the numerical data of MHD simulation based on our “sweeping magnetic twist model”, and discuss these model counterparts comparing with some observations. Here we consider the straight part of the jet, and thus use the data of axisymmetric simulation. In section 2, we review the physics of our “sweeping magnetic twist model”. We introduce the method to calculate model counterparts of observational quantities in section 3, and show the results in section 4. Comparisons of model counterparts with some observations are discussed in section 5. Brief Review of Our “Sweeping Magnetic Twist Model” {#sec:review-model} =================================================== In this section, we briefly review the results of 2.5-dimensional MHD simulations based on our “sweeping magnetic twist model” to discuss the magnetic field around the straight part of jets. In the following paper, we will extend our treatment to the wiggled part of jets, which we have given an interpretation using a 3-dimensional MHD simulation based on our model (Nakamura, Uchida, & Hirose 2001). In the original MHD model (Uchida & Shibata 1985) for bipolar outflows in star-forming regions, they considered a gravitational contraction of magnetized gas to form a star (plus an accretion disk). They attributed the large scale magnetic field to the weak field in the Galactic arms. It is strengthened in the process of gravitational contraction of the interstellar gas to the star-forming core, and plays a critical role. The toroidal field is continuously produced from the poloidal field by the rotation of the accretion disk. This causes magnetic braking to the disk material, and the material which loses angular momentum falls gradually toward the central gravitator, and releases the gravitational energy. A part of the released gravitational energy is supplied to the jets along the magnetic field. The produced toroidal magnetic field propagates into two directions along the bunched large scale magnetic field as large amplitude TAW’s. These TAW’s serve to collimate the large scale poloidal field into the shape of a slender jet by dynamically pinching it in the propagation (“sweeping pinch effect”). This process, verified in the simulation, was proposed by Uchida & Shibata (1985) as a generic magnetic effect operating in the formation of astrophysical jets utilizing gravitational energy. The mechanism was applied to the case of AGN jets (Uchida & Shibata 1986) by supposing that a large scale intergalactic magnetic field plays the role in the case of the formation of a protogalaxy and a giant black hole at its core. They argued that the same process as in the star formation case is applicable to the AGN jet cases with more or less similar set up (having accretion disk around the central gravitator etc.), due to the similarity of the basic equation system. One of the possible differences between AGN jets and the star formation jets may be the relativistic effects. The effect of general relativity will be appreciable very close to the central giant black hole comparable to the Schwarzschild radius (Koide, Shibata, & Kudoh 1998). There are regions in which the special relativity should be taken into account when the Alfvén velocity estimated in the classical definition is close to or exceed the velocity of light. Here in this paper, we concentrate ourselves on the essential physical process in the production and collimation of the jet in the non-relativistic range. The problem was treated with the non-linear system of MHD equations in a time dependent way for the first time when they proposed this model in 1985. The numerical approach was so-called axial 2.5-dimensional approximation, where the quantities are axisymmetric, but the azimuthal components of vectors are included to allow them play very essential roles such as centrifugal effect or pinch effect. Thus the authors were able to deal with the physical driving and collimating mechanism they proposed to be in operation for astrophysical jets. Figure \[FIG01\] shows the time development in the 2.5-dimensional MHD simulation based on our model. The rotating gas pulls the magnetic field gradually inward, which twists up the magnetic field because the rotational velocity is faster as close to the center (Figure \[FIG02\]). This continuously supplies large amplitude TAW’s (Poynting flux) along the external magnetic field, which pinch the poloidal magnetic field into the shape of a slender jet as discussed in the above. The gas in the surface of the torus is swirled out into two directions along the axis, both by the magnetic pressure gradient and the centrifugal effect. Thus the propagation of the torsional Alfvén wave accelerates the gas in the surface of the disk into the spinning jets. It is noted that the accretion toward the central gravitator takes place in the form of avalanches from upper and lower surfaces of the geometrically thick torus (Matsumoto et al. 1996), because the transfer of angular momentum to the external magnetic field is most efficient there. The magnetic fields of opposite polarity, brought with the accretion flows avalanching on the surfaces of the disk, make reconnection at the innermost edge of the disk in the equatorial plane (Figure \[FIG01\]). This process will contribute to the supply of the “seed high energy particles” into the jet. Such particles will be re-accelerated through the Fermi-I acceleration process when two TAW fronts trapping the particles in between them approach to each other, for example, as the foregoing one is decelerated due to an encounter with high density gas blob remaining in the collapse (Uchida et al. 1999). Method of Calculation of Model Counterparts =========================================== Using the numerical data of the 2.5-dimensional simulation explained in the previous section, we computed the distributions of the FRM, the projected magnetic field, and the total intensity with some viewing angles. We computed the FRM distribution by integrating $n_e B_\parallel$ along the line-of-sight (Hardee & Rosen 2002). To calculate the Stokes parameters, we assume the following: (1)radiation process is synchrotron radiation, (2)synchrotron self absorption is negligible, (3)the spectral index, $\alpha$, is equal to unity, and (4)the projected magnetic field is perpendicular to the projected electric field. The emissivity of the synchrotron radiation is given by $\epsilon = p |B \sin \psi|^{\alpha + 1}$, where $B$ is the local magnetic field strength, $\psi$ is the angle between the local magnetic field and the line-of-sight, and $p$ is the gas pressure. In our simulation the relativistic particles are not explicitly tracked, therefore we assume that the energy and number densities of the relativistic particles are proportional to the energy and number densities of the thermal fluid (Clarke et al. 1989, Hardee & Rosen 1999, 2002). The total intensity is then given by the integration of the emissivity along the line-of-sight as $I = \int \epsilon ds $. Other Stokes parameters are given by $Q = \int \epsilon \cos 2 \chi^{'} ds$ and $U = \int \epsilon \sin 2 \chi^{'} ds$, where the local polarization angle $\chi^{'}$ is determined by the direction of the local magnetic field and the direction of the line-of-sight. Using these $U$ and $Q$, the polarization angle $\chi$ is given by $\chi = (1/2)\tan^{-1}(U/Q)$. Finally the projected magnetic field is determined from the polarization angle $\chi$ and the polarization intensity $\sqrt{Q^2 + U^2}$. Here we separate the Faraday rotation screen and the emitting region, and we performed the integrations only in the emitting region for the Stokes parameters, and only in the Faraday rotation screen for the FRM. We assumed this separation on the basis of the fact that linear dependence of the observed polarization angle on wavelength-squared holds in some observations (Perley, Bridle, & Willis 1984, Feretti et al. 1999, Asada et al. 2002); this would not be the case if the Faraday rotation is caused in the emitting region (Burn 1966). Figure \[FIG03\] shows the emitting region (the region in the box of dashed lines) and the Faraday rotation screen (the region in the box of dotted lines) assumed in our calculations. The cylindrical shell outside the emitting region is assumed to play the main role of the Faraday rotation screen, because the temperature is lower and the toroidal field is stronger there (Figure \[FIG03\]) compared with those in the tenuous clouds in the intergalactic space. We consider two types of the emitting regions; one is the layer type (type L: high energy particles exist only in the skin part of the dashed box in Figure \[FIG03\](a)), and the other is the column type (type C: high energy particles fill the whole region of the dashed box). The former corresponds to the idea that the high energy particles are injected into the inner skin part of the jet due to the magnetic reconnection at the inner edge of the accretion disk as described in the previous section. The latter may happen if the high energy particles come from the pair plasma creation in the black hole magnetospheres. Results of Numerical Observations ================================= Faraday Rotation Measure ------------------------ The model counterparts of the FRM distribution with different viewing angles are shown in Figure \[FIG04\]. When we see the jet from the direction perpendicular to its axis ($\theta=90^\circ$, $\theta$ is the angle between the jet axis and the line-of-sight), we see only the toroidal component of the helical field and thus the value of the FRM is almost [*antisymmetric*]{} with respect to the axis (it is not perfectly antisymmetric because the radial component of the magnetic field is not equal to zero). The distribution of the FRM is distorted from antisymmetry as the viewing angle varies, but it [*always shows gradient across the jet axis*]{} (Figure \[FIG04\]-2); this gradient across the jet axis can be interpreted as the sum of the antisymmetric part due to the toroidal component of the helical field and a base-value due to the longitudinal component. When $|\theta-90^\circ|$ is larger than the pitch angle of the helical field, this longitudinal component dominates (Figure \[FIG04\]-1). Projected Magnetic Field and Total Intensity -------------------------------------------- Figure \[FIG05\] shows the distributions of the projected magnetic field and the total intensity with different viewing angles in the case of the type L emitting region. When $\theta$ is equal to $90^\circ$, the total intensity is nearly constant, but has an edge-brightening; this is because the emissivity becomes smaller away from the axis and the integration depth of the emitter has a maximum at edges. In other cases, the distribution of the total intensity is asymmetric, since the emissivity changes as $|\sin \left( \theta - \zeta \right)|^{\alpha+1}$ at the left edge and $|\sin \left( \theta + \zeta \right)|^{\alpha+1}$ at the right edge, where $\zeta$ is the pitch angle of the clockwise (seen from the jet origin) helical field.[^1] Therefore, for example, it becomes dark at the left edge when $\theta$ is nearly equal to $\zeta$. As for the projected magnetic field, it is perpendicular to the jet axis in the almost entire region, which does not depend on the viewing angle. This is because the the toroidal magnetic field is dominant in the emitting region. Figure \[FIG06\] shows the distributions in the case of the type C emitting region. In this case, the intensity has the peak on the axis when $\theta=90^\circ$, because the integration depth has a maximum at the center of the jet. When the viewing angle is not so small, the projected magnetic field is parallel to the jet axis, which corresponds to the poloidal magnetic field in the emitting region. When the viewing angle is small (e.g. $\theta=15^\circ$), the distributions of the total intensity and the projected magnetic field are almost same as those in the type L. This is because the fraction of the magnetic field perpendicular to the line-of-sight, which contributes to the synchrotron radiation, is small in both cases. Summary and Discussion ====================== We calculated the FRM and the Stokes parameters using numerical data of 2.5-dimensional MHD simulation based on our “sweeping magnetic twist model”. The distribution of the FRM always has a gradient across the jet axis, which is caused by the toroidal magnetic field generated by the disk rotation. In calculating the Stokes parameters, we assumed two types of emitting regions, the column type and the layer type. In the former, the projected magnetic field tends to be parallel to the jet axis and the total intensity has the peak at the jet center. On the other hand, in the latter, the projected magnetic field is perpendicular to the axis; the total intensity has “edge-brightening”, which is observed in Centaurus A (Clarke, Burns, & Feigelson 1986), M87 (Biretta, Owen, & Hardee 1983, Junor, Biretta, & Livio 1999). In the following, we discuss the 3-dimensional magnetic field structure around the observed jets using these results of our numerical observations, especially focusing on the FRM distribution. Figure \[FIG07\] shows a recent observational result for 3C273 jet obtained by Asada et al. (2002) by using the VLBA Archive Data. In this case, the FRM distribution has a systematic gradient in the direction perpendicular to the jet axis, which can be interpreted as the sum of an antisymmetric distribution and a base value (a constant over the source). It is not likely that the foreground magnetized cloud at a large distance from the jet has a very sharp variation of either magnetic field or the density, just along the projected very thin jet (Taylor 1998, Asada et al. 2002). One likely interpretation may be that the antisymmetric contribution is due to the toroidal component of the helical magnetic field as stated in the above. The base-value may come either from the foreground large scale magnetized clouds at large distances, or from the longitudinal component of the helical magnetic field. The latest observation of the 3C273 jet shows that the systematic FRM gradient persists along a more significant length of the jet (Asada, private communication), which would be a strong evidence for the above idea. The projected magnetic field in 3C273 is generally tilted somewhat from the axis of the jet, and the tilt angle becomes large at the blob called “anomaly”. Asada et al. (2002) attributed this to the shocks created on encountering non-uniformity. Another possibility is that the jet [*is bent*]{} at that point so that the jet axis has less angle from the line-of-sight; the change of the FRM value can be explained with our model counterparts of different viewing angles (see the change, for example, from (c) to (b) in Figure \[FIG04\](1)). The observation of the jet from the AGN core of NGC6251 (Perley et al. 1984) is shown in Figure \[FIG08\]. This may be a bit weaker evidence, but still considerably positive evidence for the systematic twist in the magnetic field. In this case, the contour lines of the FRM are nearly parallel in the central region up to 40; therefore the distribution of the FRM has a systematic antisymmetry with a gradient in the direction perpendicular to the jet axis, if we subtract the possible contribution from the spherically condensed gas cloud probably associated with the central part of the host galaxy. The projected magnetic field is aligned and this would be consistent with the propagation of the torsional Alfvén waves. The systematic and gradual change of the FRM value across the jet axis as seen in the above examples might suggest the existence of helical magnetic field. In the MHD models, it is naturally explained as the results of the interaction between the large-scale magnetic field and the disk rotation. Non-magnetic models may not be able to explain the FRM gradient. Although hydrodynamic models can include magnetic field as a passive ingredient, the magnetic field in such a case is expected to be carried around passively by the motions of non-magnetic gas dynamics. Such passively distorted magnetic field will produce the Faraday depolarization, rather than showing clear systematic FRM distribution. The FRM distribution, when the Faraday rotation occurs in the external medium around the jet, can be useful to determine the 3-dimensional magnetic field structure around the jet, together with the projected magnetic field. We have demonstrated that the characteristic distribution of the FRM (systematic gradient perpendicular to the jet axis) in the straight part of some AGN jets can be explained by our “sweeping magnetic twist model”. On the basis of this success, we are extending our numerical observation to the wiggled part of jets, which can be explained as a structural helix produced by MHD kink instability in the regime of our model (Nakamura et al. 2001). The results will be reported in the following paper. We acknowledge Mr. K. Asada for providing their precious results of the FRM distribution, and Prof. M. Inoue for discussion. We hope that more observations of the FRM distribution in AGN jets will be performed in the near future, since they are very important in determining the correct theoretical model. Numerical computations were carried out on VPP5000 at the Astronomical Data Analysis Center of the National Astronomical Observatory, Japan, which is an interuniversity research institute of astronomy operated by the Ministry of Education, Culture, Sports, Science, and Technology. [9999]{} Asada, K., Inoue, M., Uchida, Y., Kameno, S., Fujisawa, K., Iguchi, S., & Mutoh, M. 2002, , 54, L39 Biretta, J. 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