We prove an operator-theoretic reconstruction of the Beilinson-Bloch regulator for compact Riemann surfaces, using loop operators and the Connes-Karoubi character for Fredholm modules. The proof includes a new computation of the Connes-Karoubi character for Steinberg symbols of the circle, which relies on the Helton-Howe determinant theory, but not on the Carey-Pincus theory of joint torsion.
A key computation in our work is the evaluation of the Connes--Karoubi character @math on Steinberg symbols @math . This is known to equal the joint torsion of Toeplitz operators [Prop. 1] Carey-Pincus:1999 , thanks to a combination of recent results of Kaad @cite_8 and Migler @cite_26 that enable the joint torsion, in this case, to be matched with the Connes--Karoubi character. (For details, see the remark at the end of .) However, since our work does not concern joint torsion, we choose to execute a direct, and considerably shorter, computation of @math , which employs the determinant theory of Helton and Howe @cite_4 , but is otherwise fairly elementary. The validity of the reconstruction formula for the full @math -group is, to the best of our knowledge, a new result.
@cite_26 @cite_4 @cite_8
1687736749 2075162834
A determinant in algebraic @math -theory is associated to any two almost commuting Fredholm operators. On the other hand, one can calculate a homologically defined invariant known as joint torsion. We answer in the affirmative a conjecture of Richard Carey and Joel Pincus, namely that these two invariants agree. In particular, this implies that joint torsion is norm continuous, depends only on the images of the operators modulo trace class, and satisfies the expected Steinberg relations. Moreover, we show that the determinant invariant of two commuting operators can be computed simply as a determinant on a finite dimensional vector space. In the context of 2-summable Fredholm modules, we prove that the Connes-Karoubi multiplicative character coincides with Brown's determinant invariant on algebraic K -theory. We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and @math -adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class- @math -nilpotent groups obtained from three infinite families of group schemes, generalizing the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type @math . This allows us to establish a formula for the joint distribution of three functions, or statistics'', on such Weyl groups. Finally, we compare our explicit formulae to @math -adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces. We introduce a new method to compute explicit formulae for various zeta functions associated to groups and rings. The specific form of these formulae enables us to deduce local functional equations. More precisely, we prove local functional equations for the subring zeta functions associated to rings, the subgroup, conjugacy and representation zeta functions of finitely generated, torsion-free nilpotent (or T -)groups, and the normal zeta functions of T -groups of class 2. In particular we solve the two problems posed in [9, Section 5]. We deduce our theorems from a ‘blueprint result’ on certain p-adic integrals which generalises work of Denef and others on Igusa’s local zeta function. The Malcev correspondence and a Kirillov-type theory developed by Howe are used to ‘linearise’ the problems of counting subgroups and representations in T -groups, respectively. In this article, we study @math -adic torus periods for certain @math -adic valued functions on Shimura curves coming from classical origin. We prove a @math -adic Waldspurger formula for these periods, generalizing the recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic @math -adic @math -function of Rankin-Selberg type. At a character of positive weight, the @math -adic @math -function interpolates the central critical value of the complex Rankin-Selberg @math -function. Its value at a Dirichlet character, which is outside the range of interpolation, essentially computes the corresponding @math -adic torus period. We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artin's Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex. We present techniques to parallelize membership tests for Deterministic Finite Automata (DFAs). Our method searches arbitrary regular expressions by matching multiple bytes in parallel using speculation. We partition the input string into chunks, match chunks in parallel, and combine the matching results. Our parallel matching algorithm exploits structural DFA properties to minimize the speculative overhead. Unlike previous approaches, our speculation is failure-free, i.e., (1) sequential semantics are maintained, and (2) speed-downs are avoided altogether. On architectures with a SIMD gather-operation for indexed memory loads, our matching operation is fully vectorized. The proposed load-balancing scheme uses an off-line profiling step to determine the matching capacity of each participating processor. Based on matching capacities, DFA matches are load-balanced on inhomogeneous parallel architectures such as cloud computing environments. We evaluated our speculative DFA membership test for a representative set of benchmarks from the Perl-compatible Regular Expression (PCRE) library and the PROSITE protein database. Evaluation was conducted on a 4 CPU (40 cores) shared-memory node of the Intel Academic Program Manycore Testing Lab (Intel MTL), on the Intel AVX2 SDE simulator for 8-way fully vectorized SIMD execution, and on a 20-node (288 cores) cluster on the Amazon EC2 computing cloud. Obtained speedups are on the order of @math O 1 + | P | - 1 | Q | · ? , where @math | P | denotes the number of processors or SIMD units, @math | Q | denotes the number of DFA states, and @math 0 < ? ≤ 1 represents a statically computed DFA property. For all observed cases, we found that @math 0.02 < ? < 0.47 . Actual speedups range from 2.3 @math × to 38.8 @math × for up to 512 DFA states for PCRE, and between 1.3 @math × and 19.9 @math × for up to 1,288 DFA states for PROSITE on a 40-core MTL node. Speedups on the EC2 computing cloud range from 5.0 @math × to 65.8 @math × for PCRE, and from 5.0 @math × to 138.5 @math × for PROSITE. Speedups of our C-based DFA matcher over the Perl-based ScanProsite scan tool range from 559.3 @math × to 15079.7 @math × on a 40-core MTL node. We show the scalability of our approach for input-sizes of up to 10 GB. Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ( @math ) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex @math of dimension greater than @math , the boundary matrix @math is totally unimodular if and only if @math is torsion-free for all pure subcomplexes @math in @math of dimensions @math and @math , respectively, where @math . Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under @math coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal @math -cycle in a given homology class for any triangulation of an orientable compact @math -manifold or for any finite simplicial complex embedded in @math . Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity. The choice of modeling units is critical to automatic speech recognition (ASR) tasks. Conventional ASR systems typically choose context-dependent states (CD-states) or context-dependent phonemes (CD-phonemes) as their modeling units. However, it has been challenged by sequence-to-sequence attention-based models, which integrate an acoustic, pronunciation and language model into a single neural network. On English ASR tasks, previous attempts have already shown that the modeling unit of graphemes can outperform that of phonemes by sequence-to-sequence attention-based model. In this paper, we are concerned with modeling units on Mandarin Chinese ASR tasks using sequence-to-sequence attention-based models with the Transformer. Five modeling units are explored including context-independent phonemes (CI-phonemes), syllables, words, sub-words and characters. Experiments on HKUST datasets demonstrate that the lexicon free modeling units can outperform lexicon related modeling units in terms of character error rate (CER). Among five modeling units, character based model performs best and establishes a new state-of-the-art CER of @math on HKUST datasets without a hand-designed lexicon and an extra language model integration, which corresponds to a @math relative improvement over the existing best CER of @math by the joint CTC-attention based encoder-decoder network. The authors propose a novel control method based on two observers to suppress the vibration of a flexible joint, even under fast motion. One observer is a disturbance observer and is used for the realization of an acceleration controller of the motor axis. The other observer estimates the link velocity and is effective for the suppression of torsional vibration. The disturbance observer estimates the total sum of the disturbance torques which are imposed on the motor axis. Feedback compensation by this estimated disturbance torque increases the servo stiffness and the robustness of the motor portion. The feedback of the estimated link velocity totally suppresses the torsional vibration of the flexible joint even under fast motion. The proposed control was implemented in a 16-b microprocessor with a digital signal processor for a laboratory test. > A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the @math -dimensional setting, however, it requires the inversion of an @math covariance matrix, @math , as well as the evaluation of its determinant, @math . In many cases, such as regression using Gaussian processes, the covariance matrix is of the form @math , where @math is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix @math is typically dense, causing standard direct methods for inversion and determinant evaluation to require @math work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix @math can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an @math algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant @math , permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining @math . Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels. We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sln(Qp). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions. In this work, we address the problem of joint modeling of text and citations in the topic modeling framework. We present two different models called the Pairwise-Link-LDA and the Link-PLSA-LDA models. The Pairwise-Link-LDA model combines the ideas of LDA [4] and Mixed Membership Block Stochastic Models [1] and allows modeling arbitrary link structure. However, the model is computationally expensive, since it involves modeling the presence or absence of a citation (link) between every pair of documents. The second model solves this problem by assuming that the link structure is a bipartite graph. As the name indicates, Link-PLSA-LDA model combines the LDA and PLSA models into a single graphical model. Our experiments on a subset of Citeseer data show that both these models are able to predict unseen data better than the baseline model of Erosheva and Lafferty [8], by capturing the notion of topical similarity between the contents of the cited and citing documents. Our experiments on two different data sets on the link prediction task show that the Link-PLSA-LDA model performs the best on the citation prediction task, while also remaining highly scalable. In addition, we also present some interesting visualizations generated by each of the models. Symmetric spaces over local fields, and the harmonic analysis of their class one representations, arise as the local calculation of Jacquet's theory of the relative trace formula. There is an extensive literature at the real place, but few general results for p-adic fields are known. The objective here is to carry over to symmetric spaces as much as possible of Queens and the prerequisite results of Howe, and to provide counterexamples for those things which do not generalize. We derive the Weyl integration formula, local constancy of the spherical character on the θ-regular set, Howe's conjecture for θ-groups, the germ expansion for spherical characters at the origin, and a spherical version of Howe's Kirillov theory for compact p-adic groups. We find that the density property of regular orbital integrals fails. Some of the basic ideas are nascent in Hakim's thesis (written under Herve Jacquet). A central theme in distributed network algorithms concerns understanding and coping with the issue of locality . Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for . In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard @math model of computation and define @math (for local decision ) as the class of decision problems that can be solved in @math communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class @math , containing all languages for which there exists a randomized algorithm that runs in @math rounds, accepts correct instances with probability at least @math and rejects incorrect ones with probability at least @math . We show that @math is a threshold for the containment of @math in @math . More precisely, we show that there exists a language that does not belong to @math for any @math but does belong to @math for any @math such that @math . On the other hand, we show that, restricted to hereditary languages, @math , for any function @math and any @math such that @math . In addition, we investigate the impact of non-determinism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that non-determinism does help, but that this help is limited, as there exist languages that cannot be decided non-deterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with non-determinism that enables to decide languages . Finally, we introduce the notion of local reduction, and establish some completeness results. In the recent past, there has been considerable progress in devising algorithms for the all-pairs shortest paths (APSP) problem running in time significantly smaller than the obvious time bound of O(n3). Unfortunately, all the new algorithms are based on fast matrix multiplication algorithms that are notoriously impractical. Our work is motivated by the goal of devising purely combinatorial algorithms that match these improved running times. Our results come close to achieving this goal, in that we present algorithms with a small additive error in the length of the paths obtained. Our algorithms are easy to implement, have the desired property of being combinatorial in nature, and the hidden constants in the running time bound are fairly small. Our main result is an algorithm which solves the APSP problem in unweighted, undirected graphs with an additive error of 2 in time @math . This algorithm returns actual paths and not just the distances. In addition, we give more efficient algorithms with running time @math for the case where we are only required to determine shortest paths between k specified pairs of vertices rather than all pairs of vertices. The starting point for all our results is an @math algorithm for distinguishing between graphs of diameter 2 and 4, and this is later extended to obtaining a ratio 2 3 approximation to the diameter in time @math . Unlike in the case of APSP, our results for approximate diameter computation can be extended to the case of directed graphs with arbitrary positive real weights on the edges. A citing sentence is one that appears in a scientific article and cites previous work. Citing sentences have been studied and used in many applications. For example, they have been used in scientific paper summarization, automatic survey generation, paraphrase identification, and citation function classification. Citing sentences that cite multiple papers are common in scientific writing. This observation should be taken into consideration when using citing sentences in applications. For instance, when a citing sentence is used in a summary of a scientific paper, only the fragments of the sentence that are relevant to the summarized paper should be included in the summary. In this paper, we present and compare three different approaches for identifying the fragments of a citing sentence that are related to a given target reference. Our methods are: word classification, sequence labeling, and segment classification. Our experiments show that segment classification achieves the best results. Re-Pair is an efficient grammar compressor that operates by recursively replacing high-frequency character pairs with new grammar symbols. The most space-efficient linear-time algorithm computing Re-Pair uses @math words on top of the re-writable text (of length @math and stored in @math words), for any constant @math ; in practice however, this solution uses complex sub-procedures preventing it from being practical. In this paper, we present an implementation of the above-mentioned result making use of more practical solutions; our tool further improves the working space to @math words (text included), for some small constant @math . As a second contribution, we focus on compact representations of the output grammar. The lower bound for storing a grammar with @math rules is @math bits, and the most efficient encoding algorithm in the literature uses at most @math bits and runs in @math time. We describe a linear-time heuristic maximizing the compressibility of the output Re-Pair grammar. On real datasets, our grammar encoding uses---on average---only @math more bits than the information-theoretic minimum. In half of the tested cases, our compressor improves the output size of 7-Zip with maximum compression rate turned on. Any algorithm for finding a pattern of length k in a string of length n must examine at least @math of the characters of the string in the worst case. By considering the pattern @math , we prove that this is the best possible result. Therefore there do not exist pattern matching algorithms whose worst-case behavior is “sublinear” in n (that is, linear with constant less than one), in contrast with the situation for average behavior (the Boyer-Moore algorithm is known to be sublinear on the average).