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Amplitude Factorization in the Electroweak Standard Model ; We lay out the basis of factorization at the amplitude level for processes involving the entire Standard Model. The factorization appears in a generalized eikonal approximation in which we expand around a quasisoft limit for massive gauge bosons, fermions, and scalars. We use the chiralityflow formalism together with a flow basis for isospin to express loop exchanges or emissions as operators in chirality and isospin flow. This forms the basis for amplitude evolution with parton exchange and branching in the full Standard Model, including the electroweak sector.

cyberaCTIve a STIXbased Tool for Cyber Threat Intelligence in Complex Models ; Cyber threat intelligence CTI is practical realworld information that is collected with the purpose of assessing threats in cyberphysical systems CPS. A practical notation for sharing CTI is STIX. STIX offers facilities to create, visualise and share models; however, even a moderately simple project can be represented in STIX as a quite complex graph, suggesting to spread CTI across multiple simpler subprojects. Our tool aims to enhance the STIXbased modelling task in contexts when such simplifications are infeasible. Examples can be the microgrid and, more in general, the smart grid.

Phenotypic heterogeneity in a model of tumor growth existence of solutions and incompressible limit ; We consider a degenerate crossdiffusion model of tumor growth structured by phenotypic trait. We prove the existence of weak solutions and the incompressible limit as the pressure becomes stiff extending methods recently introduced in the context of twospecies crossdiffusion systems. In the stiffpressure limit, the compressible model generates a free boundary problem of HeleShaw type. Moreover, we prove a new L4bound on the pressure gradient.

The dynamics of scalarfield Quintom cosmological models ; We shall present a complete compactified dynamical systems analysis of the Quintom model comprised of an interacting quintessence scalar field and a phantom. We find a range for the model parameters kappa, lambda such that there are expanding Quintom cosmologies that undergo two inflationary periods, and this behaviour is not destabilized by spatial curvature. We also discuss a class of bouncing cosmologies. Finally, the linear cosmological perturbations are studied.

Observability Analysis of VisualInertial Odometry with Online Calibration of VelocityControl Based Kinematic Motion Models ; In this paper, we analyze the observability of the visualinertial odometry VIO using stereo cameras with a velocitycontrol based kinematic motion model. Previous work shows that in general case the global position and yaw are unobservable in VIO system, additionally the roll and pitch become also unobservable if there is no rotation. We prove that by integrating a planar motion constraint roll and pitch become observable. We also show that the parameters of the motion model are observable.

Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids ; We show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated to the double of the underlying Lie bialgebroid. This fieldtheoretic result follows from a Lietheoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of CrainicMarcut to a particular gauge fixing of the 3d theory.

On the Role of Pretrained Language Models in Word Ordering A Case Study with BART ; Word ordering is a constrained language generation task taking unordered words as input. Existing work uses linear models and neural networks for the task, yet pretrained language models have not been studied in word ordering, let alone why they help. We use BART as an instance and show its effectiveness in the task. To explain why BART helps word ordering, we extend analysis with probing and empirically identify that syntactic dependency knowledge in BART is a reliable explanation. We also report performance gains with BART in the related partial tree linearization task, which readily extends our analysis.

Beyond L1 Faster and Better Sparse Models with skglm ; We propose a new fast algorithm to estimate any sparse generalized linear model with convex or nonconvex separable penalties. Our algorithm is able to solve problems with millions of samples and features in seconds, by relying on coordinate descent, working sets and Anderson acceleration. It handles previously unaddressed models, and is extensively shown to improve stateofart algorithms. We provide a flexible, scikitlearn compatible package, which easily handles customized datafits and penalties.

Energetically Consistent Model Reduction for Metriplectic Systems ; The metriplectic formalism is useful for describing complete dynamical systems which conserve energy and produce entropy. This creates challenges for model reduction, as the elimination of highfrequency information will generally not preserve the metriplectic structure which governs longterm stability of the system. Based on proper orthogonal decomposition, a provably convergent metriplectic reducedorder model is formulated which is guaranteed to maintain the algebraic structure necessary for energy conservation and entropy formation. Numerical results on benchmark problems show that the proposed method is remarkably stable, leading to improved accuracy over long time scales at a moderate increase in cost over naive methods.

CVA in fractional and rough volatility models ; In this work we present a general representation formula for the price of a vulnerable European option, and the related CVA in stochastic either rough or not volatility models for the underlying's price, when admitting correlation with the default event. We specialize it for some volatility models and we provide price approximations, based on the representation formula. We study numerically their accuracy, comparing the results with Monte Carlo simulations, and we run a theoretical study of the error. We also introduce a seminal study of roughness influence on the claim's price.

Nondeterminsitic algebraic rewriting as adjunction ; We develop a general model theoretic semantics to rewriting beyond the usual confluence and termination assumptions. This is based on preordered algebra which is a model theory that extends many sorted algebra. In this framework we characterise rewriting in arbitrary algebras rather than term algebras called algebraic rewriting as a persistent adjunction and use this result, on the one hand for proving the soundness and the completeness of an abstract computational model of rewriting that underlies the nondeterministic programming with Maude and CafeOBJ, and on the other hand for developing a compositionality result for algebraic rewriting in the context of the pushoutbased modularisation technique.

Nonergodic delocalized paramagnetic states in quantum neural networks ; Typically, it is assumed that a highenergy eigenstate of a generic interacting quantum manybody Hamiltonian is thermal and obeys the eigenstate thermalization hypothesis. In this work, we show that the paramagnetic phase of a quantum Hopfield neural network model is delocalized but nonergodic. The combination of permutational symmetry and frustration in this model organize its highenergy eigenstates into clusters, which can each be considered a large quantum spin and has no correlation with others. This model provides another ergodicitybreaking mechanism in quantum manybody systems.

Some cosmological features of 4D GaussBonnet gravity with varying cosmological constant ; We explore some cosmological features of the newly suggested 4D GaussBonnet gravity through two different models assuming a varying cosmological constant. Observational constraints, such as the cosmic transit and the flat curvature, have been considered in constructing the models. The cosmology in the current work has been probed using a given scale factor derived from the desired cosmic behavior which is the inverse of the usual viewpoint. The stability and cosmography have been studied for the two models.

Large N fractons ; We consider theories of fractons with N fields. These theories have exotic spacetime symmetries, including a conserved dipole moment. Using collective fields we solve these models to leading order in large N. The large N solution reveals that these models are strongly correlated, and that interactions generate a momentumdependent selfenergy. Dipole symmetry is spontaneously broken throughout the phase diagram of these models, leading to a lowenergy Goldstone description of what we dub dipole superfluids.

Phononmodulatedhopping Polarons Xrepresentation Technique ; Motivated by the problem of polaron effect due to phononmodulated hopping, we formulate a generic Monte Carlo technique for solving it in the coordinate representation for both the particle and atomic displacements. The method applies to a broad class of models; the only condition is that the hopping amplitude be signpositive. A dramatic simplification of the scheme, with the corresponding efficiency gain, takes place in models with dispersionless phonons. Our study sheds important light on the nature and universality of the most striking qualitative and quantitative effects demonstrated by the standard PeierlsSuSchriefferHeeger model based on the linearized displacementmodulated hopping.

Integrating Structural and ReducedForm Methods in Empirical Finance ; I discuss various ways in which inference based on the estimation of the parameters of statistical models reducedform estimation can be combined with inference based on the estimation of the parameters of economic models structural estimation. I discuss five basic categories of integration directly combining the two methods, using statistical models to simplify structural estimation, using structural estimation to extend the validity of reducedform results, using reducedform techniques to assess the external validity of structural estimations, and using structural estimation as a sample selection remedy. I illustrate each of these methods with examples from corporate finance, banking, and personal finance.

Asymptotic comparison of identifying constraints for BradleyTerry models ; The BradleyTerry model is widely used for pairwise comparison data analysis. In this paper, we analyze the asymptotic behavior of the maximum likelihood estimator of the BradleyTerry model in its logistic parameterization, under a general class of linear identifiability constraints. We show that the constraint requiring the BradleyTerry scores for all compared objects to sum to zero minimizes the sum of the variances of the estimated scores, and recommend using this constraint in practice.

Largetime asymptotic behaviors for linear Blackstock's model of thermoviscous flow ; In the classical theory of acoustic waves, Blackstock's model was proposed in 1963 to characterize the propagation of sound in thermoviscous fluids. In this paper, we investigate largetime asymptotic behaviors of the linear Cauchy problem for general Blackstock's model that is, without Becker's assumption on monatomic perfect gases. We derive first and secondorder asymptotic profiles of solution as tgg1 by applying refined WKB analysis and Fourier analysis. Our results not only improve optimal estimates in ChenIkehataPalmieri, emphIndiana Univ. Math. J. 2023 for lower dimensional cases, but also illustrate the optimal leading term and novel secondorder profiles of solution with additional weighted L1 data.

Magnetized tori around a uniformly accelerating black hole ; We generalise the relativistic accretion thick disc model to the background of a spinning charged accelerating black hole described by the Cmetric to study the effects of this background on the disc model. We show the properties of this accretion disc model and its dependence on the initial parameters. This background can be distinguishable from the Kerr spacetime by analysing the observing features of accretion discs.

Model Agnostic Local Explanations of Reject ; The application of machine learning based decision making systems in safety critical areas requires reliable high certainty predictions. Reject options are a common way of ensuring a sufficiently high certainty of predictions made by the system. While being able to reject uncertain samples is important, it is also of importance to be able to explain why a particular sample was rejected. However, explaining general reject options is still an open problem. We propose a model agnostic method for locally explaining arbitrary reject options by means of interpretable models and counterfactual explanations.

Full Counting Statistics and Fluctuation Theorem for the Currents in the Discrete Model of Feynman's Ratchet ; We provide a detailed investigation on the fluctuations of the currents in the discrete model of Feynman's ratchet proposed by Jarzynski and Mazonka in 1999. Two macroscopic currents are identified, with the corresponding affinities determined using Schnakenberg's graph analysis. We also investigate full counting statistics of the two currents and show that fluctuation theorem holds for their joint probability distribution. Moreover, fluctuationdissipation relation, Onsager reciprocal relation and their nonlinear generalizations are numerically shown to be satisfied in this model.

Neutrinoless double beta decay and Sterile dark matter in extended left right symmetric model ; We have studied a flavor symmetrybased extended leftright symmetric modelLRSM with a dominant typeII seesaw mechanism and have explored the associated neutrino phenomenology. The particle content of the model includes usual quarks, and leptons along with additional sterile fermion per generation in the fermion sector while the scalar content contains Higgs doublets and scalar bidoublet. Realization of this extension of LRSM has been done by using A4times Z4 discrete symmetries. In this work, we have also included the study of sterile neutrino dark matterDM phenomenology along with neutrinoless double beta decay within the framework.

Spatial frequency of unstable eigenfunction of the coreperiphery model with transport cost of differentiated agriculture ; The coreperiphery model with transport cost of differentiated agriculture is extended to continuously multiregional space, and the stability of a homogeneous stationary solution of the model on a periodic space is investigated. When a spatial disturbance around the homogeneous stationary solution is decomposed into eigenfunctions, an unstable area, defined as the area of parameters where each eigenfunction is unstable, is observed to generally expand in its size as the frequency number of each eigenfunction increases.

On Some Properties of the Beta Inverse Rayleigh Distribution ; We study with some details a lifetime model of the class of beta generalized models, called the beta inverse Rayleigh distribution, which is a special case of the Beta Fr'echet distribution. We provide a better foundation for some properties including quantile function, moments, mean deviations, Bonferroni and Lorenz curves, R'enyi and Shannon entropies and order statistics. We fit the proposed model using maximum likelihood estimation to a real data set to illustrate its flexibility and potentiality.

Relativistic Mean Field Study of Neutron Stars and Hyperon Stars ; This thesis focuses on a variety of active research topics, such as nuclear matter, neutron stars, and phase transition within the framework of the RMF model. We use the previously successful effective field theorydriven Relativistic Mean Field RMF and densitydependent RMF DDRMFformalisms for analyzing hadron matter to examine the infinite nuclear matter and neutron stars. The presence of exotic phases such as quarks has been investigated using the MIT Bag model and its variants, such as the vBag model, at various bag constants. The other exotic phases, such as hyperons, have also been studied under the influence of a strong magnetic field.

Spectral determinant of the twophoton quantum Rabi model ; The various generalized spectral determinants Gfunctions of the twophoton quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only the Gfunction proposed by Chen et al. in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. We derive this function rigorously employing the mathbbZ4symmetry of the model and show that its zeros correspond to the complete regular spectrum.

Analytic solution and Noether symmetries for the hyperbolic inflationary model in the Jordan frame ; The Noether symmetry analysis is applied for the study of a multifield cosmological model in a spatially flat FLRW background geometry. The gravitational Action Integral consists of two scalar fields, the BransDicke field and a second scalar field minimally coupled to gravity. However, the two scalar fields interact in the kinetic terms. This multifield has been found to describe the equivalent of hyperbolic inflation in the Jordan frame. The application of Noether's theorems constrain the free parameters of the model that conservation laws exist. We find that the field equations form an integrable dynamical system and the analytic solution is derived.

Geometric Minimization of SoftlyBroken Potentials ; We study the minimization of multiHiggs models with symmetries that are softlybroken. The powerful method of geometric minimization enables analytic minimization of multiHiggs models with large symmetries. When these symmetries are softlybroken, the method needs to be adapted. We propose a useful generalization that considers the effect of the softbreaking terms to the quadratic part of the potential, by applying the procedure to restricted orbit spaces. We exemplify our novel methodology by finding and classifying the minima for an S4 multiHiggs model that is softlybroken with specific terms.

Examples of exact exponential cosmological solutions with three isotropic subspaces in the EinsteinGaussBonnet gravity ; We consider 1 8 and 110dimensional EinsteinGaussBonnet models with the cosmological Lambdaterm. Some new examples of exact solutions with three constant Hubblelike parameters in this model are obtained, governed by three noncoinciding Hubblelike parameters H neq 0, h1 and h2, obeying m H k1 h1 k2 h2 neq 0, corresponding to factor spaces of dimensions mgeqslant 3, k1 1 and k2geqslant 1. In this case, the multidimensional cosmological model deals with three factor spaces the external 3dimensional our world and internal subspaces with dimensions m3, k1 and k2.

Lifting integrable models and longrange interactions ; In this paper we discuss a constructive approach to check whether a constant Hamiltonian is YangBaxter integrable. We then apply our method to longrange interactions and find the Lax operator and Rmatrix of the twoloop SU2 sector in N4 SYM. We show that all known integrable longrange deformations of the 6vertex models of this type can be obtained from a Lax operator and an Rmatrix. Finally we discuss what happens at higher loops and highlight some general structures that these models seem to exhibit.

Axial vectors in DarkCast ; In this work, we explore new spin1 states with axial couplings to the standard model fermions. We develop a datadriven method to estimate their hadronic decay rates based on data from tau decays and using SU3rm flavor symmetry. We derive the current and future experimental constraints for several benchmark models. Our framework is generic and can be used for models with arbitrary vectorial and axial couplings to quarks. We have made our calculations publicly available by incorporating them into the DarkCast package, see httpsgitlab.comdarkcastreleases.

Maximally modular structure of growing hyperbolic networks ; Hyperbolic models are remarkably good at reproducing the scalefree, highly clustered and smallworld properties of networks representing real complex systems in a very simple framework. Here we show that for the popularitysimilarity optimization model from this family, the generated networks become also extremely modular in the thermodynamic limit, in spite of lacking any explicit community formation mechanism in the model definition. According to our analytical results supported by numerical simulations, when the system size is increased, the modularity approaches one surprisingly fast.

Parameter Estimation Methods of Required Rate of Return on Stock ; In this study, we introduce new estimation methods for the required rate of return of the stochastic dividend discount model DDM and the private company valuation model, which will appear below. To estimate the required rate of return, we use the maximum likelihood method, the Bayesian method, and the Kalman filtering. We apply the model to a set of firms from the SP 500 index using historical dividend and price data over a 32year period. Overall, suggested methods can be used to estimate the required rate of return.

A new solution of Einstein's field equations in isotropic coordinates ; In this work, an exact solution of Einstein's field equations in isotropic coordinates for anisotropic matter distribution is obtained by considering a particular metric choice of metric potential grr. To check the feasibility of the model, we have investigated all the physical characteristics of a realistic star. It is found that the model is potentially stable, and the adiabatic index is greater than frac43. The model have been analysed for Compact star textbf4U 153852.

Particles of negative and zero energy in black holes and cosmological models ; Particles with negative energies are considered for three different cases inside of horizon of Schwarzschild black hole, Milne's coordinates in flat Minkowski spacetime Milne's universe using nonsynchronous coordinates and in cosmological Godel model of the rotating universe. It is shown that differently from the Godel model with nondiagonal term where it occurs that negative energies are impossible they are present in all other cases considered in the paper. Particles with zero energy are also possible in first two cases.

Affinity Classification Problem by Stochastic Cellular Automata ; This work introduces a new problem, named as, affinity classification problem which is a generalization of the density classification problem. To solve this problem, we introduce temporally stochastic cellular automata where two rules are stochastically applied in each step on all cells of the automata. Our model is defined on 2dimensional grid having affection capability. We show that this model can be used in several applications like modeling selfhealing systems.

Accelerated Kinetic Monte Carlo methods for general nonlocal traffic flow models ; This paper presents a class of onedimensional cellular automata CA models on traffic flows, featuring nonlocal lookahead interactions. We develop kinetic Monte Carlo KMC algorithms to simulate the dynamics. The standard KMC method can be inefficient for models with global interactions. We design an accelerated KMC method to reduce the computational complexity in the evaluation of the nonlocal transition rates. We investigate several numerical experiments to demonstrate the efficiency of the accelerated algorithm, and obtain the fundamental diagrams of the dynamics under various parameter settings.

Parameter estimation of a two state delay differential equation modeling the human respiratory system ; We study parameter estimation for the two state model which describes the balance equation for carbon dioxide and oxygen in human respiratory system. These are nonlinear parameter dependent and because of the transport delay in the respiratory control system, they are modeled with delay differential equation. Numerically simulated noisy data are generated and several examples are studied with LevenbergMarquardt and Trustregion algorithms to determine the values of unknown parameters.

Exact Holeinduced ResonatingValenceBond Ground State in Certain Uinfty Hubbard Models ; We prove that the motion of a single hole induces the nearestneighbor resonatingvalencebond ground state in the Uinfty Hubbard model on a triangular cactus a treelike variant of a kagome lattice. The result can be easily generalized to tJ models with antiferromagnetic interactions Jgeq 0 on the same graphs. This is a weak converse of Nagaoka's theorem of ferromagnetism on a bipartite lattice.

LAD Language Models as Data for ZeroShot Dialog ; To facilitate zeroshot generalization in taskoriented dialog, this paper proposes Language Models as Data LAD. LAD is a paradigm for creating diverse and accurate synthetic data which conveys the necessary structural constraints and can be used to train a downstream neural dialog model. LAD leverages GPT3 to induce linguistic diversity. LAD achieves significant performance gains in zeroshot settings on intent prediction 15, slot filling 31.4 F1 and next action prediction 11 F1. Furthermore, an interactive human evaluation shows that training with LAD is competitive with training on human dialogs. LAD is opensourced, with the code and data available at httpsgithub.comShikiblad.

Simple models predict behavior at least as well as behavioral scientists ; How accurately can behavioral scientists predict behavior To answer this question, we analyzed data from five studies in which 640 professional behavioral scientists predicted the results of one or more behavioral science experiments. We compared the behavioral scientists' predictions to random chance, linear models, and simple heuristics like behavioral interventions have no effect and all published psychology research is false. We find that behavioral scientists are consistently no better than and often worse than these simple heuristics and models. Behavioral scientists' predictions are not only noisy but also biased. They systematically overestimate how well behavioral science works overestimating the effectiveness of behavioral interventions, the impact of psychological phenomena like time discounting, and the replicability of published psychology research.

Coherence without Rationality at the Zero Lower Bound ; Standard rational expectations RE models with an occasionally binding zero lower bound ZLB constraint either admit no solutions incoherence or multiple solutions incompleteness. This paper shows that deviations from fullinformation RE mitigate concerns about incoherence and incompleteness. Models with no RE equilibria admit selfconfirming equilibria involving the use of simple misspecified forecasting models. Completeness and coherence is restored if expectations are adaptive or if agents are less forwardlooking due to some information or behavioral friction. In the case of incompleteness, the Estability criterion selects an equilibrium.

Nonuniqueness of weak solutions to the dissipative AwRascle model ; We prove nonuniqueness of weak solutions to multidimensional generalisation of the AwRascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.

Learning Diverse Document Representations with Deep Query Interactions for Dense Retrieval ; In this paper, we propose a new dense retrieval model which learns diverse document representations with deep query interactions. Our model encodes each document with a set of generated pseudoqueries to get queryinformed, multiview document representations. It not only enjoys high inference efficiency like the vanilla dualencoder models, but also enables deep querydocument interactions in document encoding and provides multifaceted representations to better match different queries. Experiments on several benchmarks demonstrate the effectiveness of the proposed method, outperforming strong dual encoder baselines.The code is available at urlhttpsgithub.comjordane95dualcrossencoder

RandomSCM interpretable ensembles of sparse classifiers tailored for omics data ; Background Understanding the relationship between the Omics and the phenotype is a central problem in precision medicine. The high dimensionality of metabolomics data challenges learning algorithms in terms of scalability and generalization. Most learning algorithms do not produce interpretable models Method We propose an ensemble learning algorithm based on conjunctions or disjunctions of decision rules. Results Applications on metabolomics data shows that it produces models that achieves high predictive performances. The interpretability of the models makes them useful for biomarker discovery and patterns discovery in high dimensional data.

Neural Embeddings for Text ; We propose a new kind of embedding for natural language text that deeply represents semantic meaning. Standard text embeddings use the outputs from hidden layers of a pretrained language model. In our method, we let a language model learn from the text and then literally pick its brain, taking the actual weights of the model's neurons to generate a vector. We call this representation of the text a neural embedding. We confirm the ability of this representation to reflect semantics of the text by an analysis of its behavior on several datasets, and by a comparison of neural embedding with state of the art sentence embeddings.

Uncharged and charged anisotropic likeDurgapal stellar model with vanishing complexity ; In this work we use the vanishing complexity factor as a supplementary condition to construct uncharged and charged likeDurgapal models. We provide the gtt component of the metric of the wellknown Durgapal IV and V solutions and a particular form for the anisotropy, related to the electric charge, to close the system of differential equations. The physical acceptance of the models is discussed.

Taskspecific Pretraining and Prompt Decomposition for Knowledge Graph Population with Language Models ; We present a system for knowledge graph population with Language Models, evaluated on the Knowledge Base Construction from Pretrained Language Models LMKBC challenge at ISWC 2022. Our system involves taskspecific pretraining to improve LM representation of the masked object tokens, prompt decomposition for progressive generation of candidate objects, among other methods for higherquality retrieval. Our system is the winner of track 1 of the LMKBC challenge, based on BERT LM; it achieves 55.0 F1 score on the hidden test set of the challenge.

Lower bounds for the scalar curvatures of Ricci flow singularity models ; In a series of papers, Bamler Bam20a,Bam20b,Bam20c further developed the highdimensional theory of Hamilton's Ricci flow to include new monotonicity formulas, a completely general compactness theorem, and a longsought partial regularity theory analogous to CheegerColding theory. In this paper we give an application of his theory to lower bounds for the scalar curvatures of singularity models for Ricci flow. In the case of 4dimensional nonRicciflat steady soliton singularity models, we obtain as a consequence a quadratic decay lower bound for the scalar curvature.

Spontaneous symmetry breaking in the BFSS model Analytical results using the Gaussian expansion method ; We apply the Gaussian expansion method to the BFSS matrix model in the high temperature limit. When the Euclidean BFSS action is expanded about a Gaussian ansatz, it is shown that the SO9 symmetry is spontaneously broken, analogous to what happens in the IKKT model. The analysis of the free energy, using the set of gap equations which determines the width of the Gaussian terms, is sufficient to show that this symmetry breaking happens only when the fermionic terms are included and is absent in the bosonic case.

Dynamics of the CDM model of the universe from the aspect of the dynamical systems theory ; In this paper we exploit the theory of the dynamical systems to study the dynamics of the standard cosmological model of the universe, which is known as the LambdaCDM model. We assume that the matter content in our universe consists of barotropic perfect fluids without mutual interaction. Furthermore, we present the appropriate physical interpretation, as well as new dependencies between the scale expansion factor of the universe and cosmological density parameters.

A fastconvolution based spacetime Chebyshev spectral method for peridynamic models ; Peridynamics is a nonlocal generalization of continuum mechanics theory which adresses discontinuous problems without using partial derivatives and replacing its by an integral operator. As a consequence, it finds applications in the framework of the development and evolution of fractures and damages in elastic materials. In this paper we consider a onedimensional nonlinear model of peridynamics and propose a suitable twodimensional fastconvolution spectral method based on Chebyshev polynomials to solve the model. This choice allows us to gain the same accuracy both in space and time. We show the convergence of the method and perform several simulations to study the performance of the spectral scheme.

Instrument Separation of Symbolic Music by Explicitly Guided Diffusion Model ; Similar to colorization in computer vision, instrument separation is to assign instrument labels e.g. piano, guitar... to notes from unlabeled mixtures which contain only performance information. To address the problem, we adopt diffusion models and explicitly guide them to preserve consistency between mixtures and music. The quantitative results show that our proposed model can generate highfidelity samples for multitrack symbolic music with creativity.

AILABUdineSMM4H 22 Limits of Transformers and BERT Ensembles ; This paper describes the models developed by the AILABUdine team for the SMM4H 22 Shared Task. We explored the limits of Transformer based models on text classification, entity extraction and entity normalization, tackling Tasks 1, 2, 5, 6 and 10. The main takeaways we got from participating in different tasks are the overwhelming positive effects of combining different architectures when using ensemble learning, and the great potential of generative models for term normalization.

Analyzing Linear DSGE models the Method of Undetermined Markov States ; I show that a class of Linear DSGE models with one endogenous state variable can be represented as a threestate Markov chain. I develop a new analytical solution method based on this representation, which amounts to solving for a vector of Markov states and one transition probability. These two objects constitute sufficient statistics to compute in closed form objects that have routinely been computed numerically impulse response function, cumulative sum, present discount value multiplier. I apply the method to a standard New Keynesian model that features optimal monetary policy with commitment.

CP violation in b and c quark decays ; Generically, nonStandardModel particles contribute to processes with orderone chargeparity CP violating phases, as CP is not a fundamental symmetry of nature. The exploration of CP violation becomes therefore a sensitive search for nonStandardModel physics. We briefly review the current status of CPviolation studies in bottom and charmquark transitions focusing on those quantities that are most sensitive to nonStandardModel contributions and discuss opportunities and challenges for the next decade and beyond.

Supergravity Application in Particle Physics ; We provide a pedagogical introduction to N1 supergravitysupersymmetry in relation to particle physics. The various steps in the construction of a generic N1 supergravity model are briefly described, and we focus on its low energy supersymmetric limit. The conditions for supersymmetry and supergravity breaking are investigated, and realistic mechanisms suitable for particle physics identified. We then study the modelbuilding aspects of softlybroken' supersymmetric extensions of the Standard Model and discuss several of their phenomenological features.

GIDN A Lightweight Graph Inception Diffusion Network for Highefficient Link Prediction ; In this paper, we propose a Graph Inception Diffusion NetworksGIDN model. This model generalizes graph diffusion in different feature spaces, and uses the inception module to avoid the large amount of computations caused by complex network structures. We evaluate GIDN model on Open Graph BenchmarkOGB datasets, reached an 11 higher performance than AGDN on ogblcollab dataset.

Unit Selection Case Study and Comparison with AB Test Heuristic ; The unit selection problem defined by Li and Pearl identifies individuals who have desired counterfactual behavior patterns, for example, individuals who would respond positively if encouraged and would not otherwise. Li and Pearl showed by example that their unit selection model is beyond the AB test heuristics. In this paper, we reveal the essence of the AB test heuristics, which are exceptional cases of the benefit function defined by Li and Pearl. Furthermore, We provided more simulated use cases of LiPearl's unit selection model to help decisionmakers apply their model correctly, explaining that AB test heuristics are generally problematic.

Constructing local models for general measurements on bosonic Gaussian states ; We derive a simple sufficient criterion for the locality of correlations obtained from given measurements on a Gaussian quantum state. The criterion is based on the construction of a localhiddenvariable model which works by passing part of the inherent Gaussian noise of the state onto the measurements. We illustrate our result in the setting of displaced photodetection on a twomode squeezed state. Here, our criterion exhibits the existence of a localhiddenvariable model for a range of parameters where the state is still entangled.

Chirality in an E8 model of elementary particles ; We show how chirality emerges naturally from an embedding of the standard model of particle physics into E824. The wellknown argument that there is no chiral theory of fundamental physics in E8 is avoided by implementing chirality not as a property of the complexified Lorentz group, but as a property of the complex representations of the real Lorentz group, combined with a real scalar. This avoids the problems of complexification, and ensures that the model is completely contained in the real Lie group.

textttRGESolver a textttC library to perform Renormalization Group evolution in the Standard Model Effective Theory ; Renormalization group evolution above the electroweak scale is a crucial ingredient in the phenomenology of the Standard Model Effective Theory. The RGESolver opensource C library performs the evolution at leading order for dimensionsix operators in the most general flavour scenario assuming lepton and baryon number conservation. Given its efficiency, RGESolver can be used to include the effects of renormalization group evolution in extensive phenomenological analyses in the framework of the Standard Model Effective Theory.

Deep learning waveform anomaly detector for numerical relativity catalogs ; Numerical Relativity has been of fundamental importance for studying compact binary coalescence dynamics, waveform modelling, and eventually for gravitational waves observations. As the sensitivity of the detector network improves, more precise template modelling will be necessary to guarantee a more accurate estimation of astrophysical parameters. To help improving the accuracy of numerical relativity catalogs, we developed a deep learning model capable of detecting anomalous waveforms. We analyzed 1341 binary black hole simulations from the SXS catalog with various massratios and spins, considering waveform dominant and higher modes. In the set of waveform analyzed, we found and categorised seven types of anomalies appearing close to the merger phase.

Perturbation and bifurcation analysis of a gonorrhoea dynamics model with control ; A model for the transmission dynamics of gonorrhoea with control incorporating passive immunity is formulated. We show that introduction of treatment or control parameters leads to transcritical bifurcation. The backward bifurcation coefficients were calculated and their numerical perturbation results to different forms of equilibria. The calculated effective reproduction number of the model with control is sufficiently small. This implies asymptotically stability of the solution, thus, the disease can be controlled in a limited time.

Constantroll Inflation in Brane Induced Gravity Cosmology ; In this article we study a constantroll inflationary model in the context of the DGP braneworld cosmology caused by a quintessence scalar field. We determine an analytical solution for the Friedman equation coupled to the equation of motion of the scaler field. The evolution of the primordial scalar and tensor perturbations is also studied. To check the viability of the model we use numerical approaches and plot some figures. Our results for the scalar spectral index and the tensor to scaler ratio show good consistency with observations for given values of model parameters.

Cosmological constraints on scaledependent cosmology ; This paper examines a cosmological model of scaledependent gravity. The gravitational action is taken to be the EinsteinHilbert term supplemented with a cosmological constant, where the couplings run with the energy scale. The model is deeply analyzed, confronting its predictions with recent observational data from Hz, muz, and BAOCMB. The viability of the model is shown, obtaining the bestfit parameters and the maximum likelihood contours for these observables. Finally, a joint analysis is performed.

A comment about the cosmology on a bubble wall ; The interface between a big bubble of true AdSD vacuum expanding inside a false AdSD vacuum is a model of an inflating D1 dimensional universe. It looks like an interesting setup to study fundamentals of inflation. A recent computation shows that the prediction of this model for the wavefunction of the universe disagrees with that of Hartle and Hawking. We show that this discrepancy is because the effective D1 dimensional description of the model is spin0 Nordstrom gravity rather than spin2 Einstein gravity.

Deep Latent Mixture Model for Recommendation ; Recent advances in neural networks have been successfully applied to many tasks in online recommendation applications. We propose a new framework called cone latent mixture model which makes use of handcrafted state being able to factor distinct dependencies among multiple related documents. Specifically, it uses discriminative optimization techniques in order to generate effective multilevel knowledge bases, and uses online discriminative learning techniques in order to leverage these features. And for this joint model which uses confidence estimates for each topic and is able to learn a discriminatively trained jointly to automatically extracted salient features where discriminative training is only uses features and then is able to accurately trained.

Change of measure in a HestonHawkes stochastic volatility model ; We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the wellknown Heston model. A Hawkes process is a selfexciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology and other fields. We prove a general result on the existence of a family of equivalent local martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed.

The Adiabatic WignerWeisskopf Model ; We consider a slowly varying time dependent dlevel atom interacting with a photon field. Restricted to the single excitation atomfield sector, the model is a timedependent generalization of the WignerWeisskopf model describing spontaneous emission of an atomic excitation into the radiation field. We analyze the dynamics of the atom and of the radiation field in the adiabatic and small coupling approximations, in various regimes. In particular, starting with an excited atomic state, we provide a description of both the radiative decay of the atom and of the buildup of the photon excitation in the field.

Sudakov effects and the dipole amplitude ; In this study we incorporate the Sudakov form factor into the dipole factorization formula, where the hard scale of the former is provided by the photon virtuality Q2. We obtain a general formula which we then apply to the wellknown GBW and BGK saturation models. Parameters of the above Sudakovimproved models are successfully fitted to the F2 data from HERA. We observe, in particular, that inclusion of the Sudakov factor on top of the GBW model improves description of data at large Q2.

Six lectures on model theory and differentialalgebraic geometry ; This is a writeup of some lectures I gave in the Fall of 2021 at the Fields Institute in Toronto, as part of the Thematic Programme on Trends in Pure and Applied Model Theory. The goal of the module was to give a quick introduction to the model theory of differential fields that puts differentialalgebraic geometry at the center. I focus here on the birational geometry of algebraic vector fields and more generally Dvarieties in the sense of Buium.

Gap Labels for Zeros of the Partition Function of the 1D Ising Model via the Schwartzman Homomorphism ; Inspired by the 1995 paper of BaakeGrimmPisani, we aim to explain the empirical observation that the distribution of LeeYang zeros corresponding to a onedimensional Ising model appears to follow the gap labelling theorem. This follows by combining two main ingredients first, the relation between the transfer matrix formalism for 1D Ising model and an ostensibly unrelated matrix formalism generating the SzegHo recursion for orthogonal polynomials on the unit circle, and second, the gap labelling theorem for CMV matrices.

Logdensity gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods ; A metric tensor for Riemann manifold Monte Carlo particularly suited for nonlinear Bayesian hierarchical models is proposed. The metric tensor is built from here proposed symmetric positive semidefinite logdensity gradient covariance LGC matrices. The LGCs measure the joint information content and dependence structure of both a random variable and the parameters of said variable. The proposed methodology is highly automatic and allows for exploitation of any sparsity associated with the model in question. When implemented in conjunction with a Riemann manifold variant of the recently proposed numerical generalized randomized Hamiltonian Monte Carlo processes, the proposed methodology is highly competitive, in particular for the more challenging target distributions associated with Bayesian hierarchical models.

Quark generalized TMDs at skewness and Wigner Distribution in boost invariant longitudinal space ; The boostinvariant longitudinal space, defined by the parameter sigmafrac12bP can be studied from the Fourier transformation of distributions over the conjugate variable skewness xi. We investigate quark Wigner distributions in the sigma space in dressed quark model and found diffraction patterns that are analogous to the single slit experiment of light in optics. The width of the central maxima varies with energy transfer to the system and essentially xi behaves like a slitwidth. Qualitatively similar diffraction pattern is reported recently in other models. In this model, we compute all the leading twist GTMDs with nonzero skewness for quarks which provides Wigner distributions under Fourier transformation.

Strictly BreadthFirst AMR Parsing ; AMR parsing is the task that maps a sentence to an AMR semantic graph automatically. We focus on the breadthfirst strategy of this task, which was proposed recently and achieved better performance than other strategies. However, current models under this strategy only emphencourage the model to produce the AMR graph in breadthfirst order, but emphcannot guarantee this. To solve this problem, we propose a new architecture that emphguarantees that the parsing will strictly follow the breadthfirst order. In each parsing step, we introduce a textbffocused parent vertex and use this vertex to guide the generation. With the help of this new architecture and some other improvements in the sentence and graph encoder, our model obtains better performance on both the AMR 1.0 and 2.0 dataset.

Foundation Models for Semantic Novelty in Reinforcement Learning ; Effectively exploring the environment is a key challenge in reinforcement learning RL. We address this challenge by defining a novel intrinsic reward based on a foundation model, such as contrastive language image pretraining CLIP, which can encode a wealth of domainindependent semantic visuallanguage knowledge about the world. Specifically, our intrinsic reward is defined based on pretrained CLIP embeddings without any finetuning or learning on the target RL task. We demonstrate that CLIPbased intrinsic rewards can drive exploration towards semantically meaningful states and outperform stateoftheart methods in challenging sparsereward procedurallygenerated environments.

Rational Homotopy Type Of Complements Of Submanifold Arrangements ; We will provide an explicit cdga controlling the rational homotopy type of the complement to a smooth arrangement Xcupi Zi in a smooth compact algebraic variety X over mathbbC. This generalizes the corresponding result of Morgan in case of a divisor with normal crossings to arbitrary smooth arrangements. The model is given in terms of the arrangement Zi and agrees with a model introduced by ChenLuWu for computing the cohomology. As an application we reprove a formality theorem due to FeichtnerYuzvinksy. Then we show that the KritzTotaro model computes the rational homotopy type in case of chromatic configuration spaces of smooth compact algebraic varieties.

Single and pair J production in the Improved Color Evaporation Model using the Parton Reggeization Approach ; In the article, we study single and pair Jpsi hadroproduction in the Improved Color Evaporation Model via the Parton Reggeization Approach. The last one is based on kTfactorization of hard processes in multiRegge kinematics, the KimberMartinRyskinWatt model for unintegrated parton distribution functions, and the effective field theory of Reggezied gluons and quarks, suggested by L.N. Lipatov. We compare contributions from the single and double parton scattering mechanisms in the pair Jpsi production. The numerical calculations are realized using the MonteCarlo event generator KaTie.

An Empirical Analysis of the Advantages of Finite v.s. InfiniteWidth Bayesian Neural Networks ; Comparing Bayesian neural networks BNNs with different widths is challenging because, as the width increases, multiple model properties change simultaneously, and, inference in the finitewidth case is intractable. In this work, we empirically compare finite and infinitewidth BNNs, and provide quantitative and qualitative explanations for their performance difference. We find that when the model is misspecified, increasing width can hurt BNN performance. In these cases, we provide evidence that finitewidth BNNs generalize better partially due to the properties of their frequency spectrum that allows them to adapt under model mismatch.

Enhancing CrisisRelated Tweet Classification with EntityMasked Language Modeling and MultiTask Learning ; Social media has become an important information source for crisis management and provides quick access to ongoing developments and critical information. However, classification models suffer from eventrelated biases and highly imbalanced label distributions which still poses a challenging task. To address these challenges, we propose a combination of entitymasked language modeling and hierarchical multilabel classification as a multitask learning problem. We evaluate our method on tweets from the TRECIS dataset and show an absolute performance gain w.r.t. F1score of up to 10 for actionable information types. Moreover, we found that entitymasking reduces the effect of overfitting to indomain events and enables improvements in crossevent generalization.

MultiDirectional Subspace Editing in StyleSpace ; This paper describes a new technique for finding disentangled semantic directions in the latent space of StyleGAN. Our method identifies meaningful orthogonal subspaces that allow editing of one human face attribute, while minimizing undesired changes in other attributes. Our model is capable of editing a single attribute in multiple directions, resulting in a range of possible generated images. We compare our scheme with three stateoftheart models and show that our method outperforms them in terms of face editing and disentanglement capabilities. Additionally, we suggest quantitative measures for evaluating attribute separation and disentanglement, and exhibit the superiority of our model with respect to those measures.

Inverse Solvability and Security with Applications to Federated Learning ; We introduce the concepts of inverse solvability and security for a generic linear forward model and demonstrate how they can be applied to models used in federated learning. We provide examples of such models which differ in the resulting inverse solvability and security as defined in this paper. We also show how the large number of users participating in a given iteration of federated learning can be leveraged to increase both solvability and security. Finally, we discuss possible extensions of the presented concepts including the nonlinear case.

Deterministic Chaos in Integrable Models ; In this work we present analytical and numerical evidences that classical integrable models possessing infinitely many degrees of freedom exhibit some features that are typical of chaotic systems. By studying how the conserved charges change under a small deformation of the initial conditions, we conclude that the inverse scattering map is responsible for this chaotic behavior, in spite of the system being integrable. We investigate this phenomenon in the explicit examples of the KdV equation and the sineGordon model and further provide general arguments supporting this statement.

Learning Coherent Clusters in WeaklyConnected Network Systems ; We propose a structurepreserving modelreduction methodology for largescale dynamic networks with tightlyconnected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the network feedback. Then, a reduced network is built, where each node represents the aggregate dynamics of each coherent group, and the reduced network captures the dynamic coupling between the groups. We provide an upper bound on the approximation error when the network graph is randomly generated from a weight stochastic block model. Finally, numerical experiments align with and validate our theoretical findings.

Probabilistic Modelling of Signal Mixtures with Differentiable Dictionaries ; We introduce a novel way to incorporate prior information into semi supervised nonnegative matrix factorization, which we call differentiable dictionary search. It enables general, highly flexible and principled modelling of mixtures where nonlinear sources are linearly mixed. We study its behavior on an audio decomposition task, and conduct an extensive, highly controlled study of its modelling capabilities.

Frequency Domain Gaussian Process Models for Hinfty Uncertainties ; Complexvalued Gaussian processes are used in Bayesian frequencydomain system identification as prior models for regression. If each realization of such a process were an Hinfty function with probability one, then the same model could be used for probabilistic robust control, allowing for robustly safe learning. We investigate sufficient conditions for a general complexdomain Gaussian process to have this property. For the special case of processes whose Hermitian covariance is stationary, we provide an explicit parameterization of the covariance structure in terms of a summable sequence of nonnegative numbers.

Hijack Vertical Federated Learning Models with Adversarial Embedding ; Vertical federated learning VFL is an emerging paradigm that enables collaborators to build machine learning models together in a distributed fashion. In general, these parties have a group of users in common but own different features. Existing VFL frameworks use cryptographic techniques to provide data privacy and security guarantees, leading to a line of works studying computing efficiency and fast implementation. However, the security of VFL's model remains underexplored.

Autoencoding heterotic orbifolds with arbitrary geometry ; Artificial neural networks have become important to improve the search for admissible string compactifications and characterize them. In this paper, by scanning a large set of configurations, we construct a general deep autoencoder to study heterotic orbifold models arising from various Abelian orbifold geometries. In particular, we show that our autoencoder is capable to compress with good accuracy the large parameter space of two promising orbifold geometries in just three parameters. Most orbifold models with phenomenologically appealing features appear in bounded regions of this small space. Our contribution hints towards a possible simplification of the classification of promising heterotic orbifold models.

Intersection theory of the complex quartic Kontsevich model ; We expand correlation functions of the LangmannSzaboZarembo LSZ model in terms of intersection numbers on the moduli space of complex curves. This provides an explicit, physically motivated example for the expansion of correlation functions generated by ChekhovEynardOrantin topological recursion. To this end, we unify notation as well as different conventions present in the literature and use a set of moduli of the spectral curve adapted to the physically motivated model. The presentation focuses on an illustrative, stepbystep comprehension of the work.

Parametric Modal Regression with Error in Covariates ; An inference procedure is proposed to provide consistent estimators of parameters in a modal regression model with a covariate prone to measurement error. A scorebased diagnostic tool exploiting parametric bootstrap is developed to assess adequacy of parametric assumptions imposed on the regression model. The proposed estimation method and diagnostic tool are applied to synthetic data generated from simulation experiments and data from realworld applications to demonstrate their implementation and performance. These empirical examples illustrate the importance of adequately accounting for measurement error in the errorprone covariate when inferring the association between a response and covariates based on a modal regression model that is especially suitable for skewed and heavytailed response data.

Semisupervised Learning with Robust Loss in Brain Segmentation ; In this work, we used a semisupervised learning method to train deep learning model that can segment the brain MRI images. The semisupervised model uses less labeled data, and the performance is competitive with the supervised model with full labeled data. This framework could reduce the cost of labeling MRI images. We also introduced robust loss to reduce the noise effects of inaccurate labels generated in semisupervised learning.

Robust Speech Recognition via LargeScale Weak Supervision ; We study the capabilities of speech processing systems trained simply to predict large amounts of transcripts of audio on the internet. When scaled to 680,000 hours of multilingual and multitask supervision, the resulting models generalize well to standard benchmarks and are often competitive with prior fully supervised results but in a zeroshot transfer setting without the need for any finetuning. When compared to humans, the models approach their accuracy and robustness. We are releasing models and inference code to serve as a foundation for further work on robust speech processing.

Secondorder force scheme for lattice Boltzmann method ; We present an a priori derivation of the force scheme for lattice Boltzmann method based on kinetic theoretical formulation. We show that the discrete lattice effect, previously eliminated a posteriori in BGK collision model, is due to firstorder spacetime discretization and can be eliminated generically for a wide range of collision models with secondorder spacetime discretization. Particularly, the force scheme for the recently developed spectral multiplerelaxationtime SMRT collision model is obtained and numerically verified.

How flexibility can enhance catalysis ; Conformational changes are observed in many enzymes, but their role in catalysis is highly controversial. Here we present a theoretical model that illustrates how rigid catalysts can be fundamentally limited and how a conformational change induced by substrate binding can overcome this limitation, ultimately enabling barrierfree catalysis. The model is deliberately minimal, but the principle it illustrates is general and consistent with unique features of proteins as well as with previous informal proposals to explain the superiority of enzymes over other classes of catalysts. Implementing the discriminative switch suggested by the model could help overcome limitations currently encountered in the design of artificial catalysts.

Concentration of Equilibria and Relative Instability in Disordered NonRelaxational Dynamics ; We consider a system of random autonomous ODEs introduced by Cugliandolo et al. 22, which serves as a nonrelaxational analog of the gradient flow for the spherical pspin model. The asymptotics for the expected number of equilibria in this model was recently computed by Fyodorov 32 in the highdimensional limit, followed a similar computation for the expected number of stable equilibria by Garcia 38. We show that for p 9 the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni 61, 64 in the relaxational case. In particular, we confirm that this model undergoes a transition from relative to absolute instability, in the sense of Ben Arous, Fyodorov, and Khoruzhenko 11.

Scaling limits of nonlinear functions of random grain model, with application to Burgers' equation ; We study scaling limits of nonlinear functions G of random grain model X on mathbbRd with longrange dependence and marginal Poisson distribution. Following Kaj et al 2007 we assume that the intensity M of the underlying Poisson process of grains increases together with the scaling parameter lambda as M lambdagamma , for some gamma 0. The results are applicable to the Boolean model and exponential G and rely on an expansion of G in Charlier polynomials and a generalization of Mehler's formula. Application to solution of Burgers' equation with initial aggregated random grain data is discussed.

Self Meta Pseudo Labels Meta Pseudo Labels Without The Teacher ; We present Self Meta Pseudo Labels, a novel semisupervised learning method similar to Meta Pseudo Labels but without the teacher model. We introduce a novel way to use a single model for both generating pseudo labels and classification, allowing us to store only one model in memory instead of two. Our method attains similar performance to the Meta Pseudo Labels method while drastically reducing memory usage.

Hidden Symmetries, Rapid Turns and Cosmic Acceleration ; Hidden symmetries provide a powerful tool for finding exact solutions in multifield cosmological models. We review how, using such symmetries, one can find inflationary solutions in twofield models, which lead to the generation of primordial black holes. We also discuss an exact solution in a twofield cosmological model, which describes dark energy. This solution is obtained with the use of a hidden symmetry, although the latter is broken by a constant term in the scalar potential. All of the above solutions are characterized by fieldspace trajectories with rapid turns.

Friction Laws and Numerical Modeling of the Seismic Cycle ; Earthquakes rank among the most destructive manifestations of the Earth's dynamics. Can they be predicted This is often the first question students ask. To answer that right away no, at present it is not possible to anticipate the date, site and magnitude of future seismic events. However, there does exist a general framework to describe observations related to earthquakes and understand the processes that lead to their occurrence the seismic cycle. This chapter introduces the reader to the friction laws from a historical to state of the art perspective. It then deals with mechanical modelling of the seismic cycle through simple analog models and finally presents some open questions and directions for future research.