Title: Sommets fortement critiques d'un tournoi indécomposable
Abstract: Let $T=(V,A)$ be a tournament. For $X\subseteq V$, the subtournament of $T$ induced by $X$ is denoted by $T[X]$. A subset $I$ of $V$ is an interval of $T$ provided that for every $a,b\in I$ and $x\in V\setminus I$, $(a,x)\in A$ if and only if $(b,x)\in A$. For example, $\varnothing $, ${x}$ ($x \in V$) and $V$ are intervals of $T$, called trivial intervals. The tournament $T$ is indecomposable if all its intervals are trivial, otherwise, it is decomposable. A critical tournament is an indecomposable tournament $T$ of cardinality $\geqslant 5$ such that every vertex $x$ of $T$ is critical, i.e., the subtournament $T[V(T)\setminus\{x\}]$ is decomposable. Given an indecomposable tournament $T$, a vertex $x$ of $T$ is strongly critical, if for every $X\subseteq V(T)$ such that $x\in X$, $\vert X\vert \geqslant 5$ and $T[X]$ is indecomposable, $x$ is a critical vertex of $T[X]$. Let $T$ be an indecomposable tournament and let $\mathscr{C}(T)$ be the set of the strongly critical vertices of $T$. We prove that, if $T$ is non-critical, then $f(T):=\vert \mathscr{C}(T)\vert \leqslant 4$, and that the correspondence $f(T)$ is decreasing from the class of indecomposable and non-critical tournaments (defined by means of embedding) to $\{0,1,2,3,4\}$. By giving examples, we also verify that the bounds 0 and 4 are optimal. This article is an extract from my master's thesis \cite{mon mast\`ere}.


Title: Construction of spherical cubature formulas using lattices
Abstract: We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked out on the sphere of dimension n-1 for n=4, 8, 12, 14, 16, 20, 23, and 24, and the sizes of the cubature formulas we obtain are compared with the lower bounds given by Linear Programming.


Title: Classification of algebras of level two in the variety of nilpotent algebras and Leibniz algebras
Abstract: This paper is devoted to the description of complex finite-dimensional algebras of level two. We obtain the classification of algebras of level two in the variety of Leibniz algebras. It is shown that, up to isomorphism, there exist three Leibniz algebras of level two, one of which is solvable, and two of which are nilpotent. Moreover we describe all algebras of level two in the variety of nilpotent algebras.


Title: Integrability of moduli and regularity of Denjoy counterexamples
Abstract: We study the regularity of exceptional actions of groups by $C^{1,\alpha}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $\alpha$. Let $G$ be a finitely generated group admitting a $C^{1,\alpha}$ action $\rho$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/\alpha^d$ is integrable near zero, then under some mild technical assumptions on $\alpha$, there is a sequence of exceptional $C^{1,\alpha}$ actions of $G$ which converge to $\rho$ in the $C^1$ topology. As a consequence for a single diffeomorphism, we obtain that if the function $1/\alpha$ is integrable near zero, then there exists a $C^{1,\alpha}$ exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus $\alpha$. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $C^1$ diffeomorphisms of the circle.


Title: Bagchi's Theorem for families of automorphic forms
Abstract: We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.


Title: Real-and-Present: Investigating the Use of Life-Size 2D Video Avatars in HMD-Based AR Teleconferencing
Abstract: Augmented Reality (AR) teleconferencing allows separately located users to interact with each other in 3D through agents in their own physical environments. Existing methods leveraging volumetric capturing and reconstruction can provide a high-fidelity experience but are often too complex and expensive for everyday usage. Other solutions target mobile and effortless-to-setup teleconferencing on AR Head Mounted Displays (HMD). They directly transplant the conventional video conferencing onto an AR-HMD platform or use avatars to represent remote participants. However, they can only support either a high fidelity or a high level of co-presence. Moreover, the limited Field of View (FoV) of HMDs could further influence users' immersive experience. To achieve a balance between fidelity and co-presence, we explore using life-size 2D video-based avatars (video avatars for short) in AR teleconferencing. Specifically, with the potential effect of FoV on users' perception of proximity, we first conduct a pilot study to explore the local-user-centered optimal placement of video avatars in small-group AR conversations. With the placement results, we then implement a proof-of-concept prototype of video-avatar-based teleconferencing. We conduct user evaluations with the prototype to verify its effectiveness in balancing fidelity and co-presence. Following the indication in the pilot study, we further quantitatively explore the effect of FoV size on the video avatar's optimal placement through a user study involving more FoV conditions in a VR-simulated environment. We regress placement models to serve as references for computationally determining video avatar placements in such teleconferencing applications on various existing AR HMDs and future ones with bigger FoVs.


Title: CLASS-M: Adaptive stain separation-based contrastive learning with pseudo-labeling for histopathological image classification
Abstract: Histopathological image classification is an important task in medical image analysis. Recent approaches generally rely on weakly supervised learning due to the ease of acquiring case-level labels from pathology reports. However, patch-level classification is preferable in applications where only a limited number of cases are available or when local prediction accuracy is critical. On the other hand, acquiring extensive datasets with localized labels for training is not feasible. In this paper, we propose a semi-supervised patch-level histopathological image classification model, named CLASS-M, that does not require extensively labeled datasets. CLASS-M is formed by two main parts: a contrastive learning module that uses separated Hematoxylin and Eosin images generated through an adaptive stain separation process, and a module with pseudo-labels using MixUp. We compare our model with other state-of-the-art models on two clear cell renal cell carcinoma datasets. We demonstrate that our CLASS-M model has the best performance on both datasets. Our code is available at github.com/BzhangURU/Paper_CLASS-M/tree/main


Title: Weyl modules and Levi subalgebras
Abstract: For a simple complex Lie algebra of finite rank and classical type, we fix a triangular decomposition and consider the simple Levi subalgebras associated to closed subsets of roots. We study the restriction of global and local Weyl modules of current algebras to this Levi subalgebra. We identify necessary and sufficient conditions on a pair of a Levi subalgebra and a dominant integral weight, such that the restricted module is a global (resp. a local) Weyl module.


Title: KK-duality for the Cuntz-Pimsner algebras of Temperley-Lieb subproduct systems
Abstract: We prove that the Cuntz-Pimsner algebra of every Temperley-Lieb subproduct system is KK-self-dual. We show also that every such Cuntz-Pimsner algebra has a canonical KMS-state, which we use to construct a Fredholm module representative for the fundamental class of the duality. This allows us to describe the K-homology of the Cuntz-Pimsner algebras by explicit Fredholm modules. Both the construction of the dual class and the proof of duality rely in a crucial way on quantum symmetries of Temperley-Lieb subproduct systems. In the simplest case of Arveson's $2$-shift our work establishes $U(2)$-equivariant KK-self-duality of $S^3$.


Title: From transient elastic linkages to friction: a complete study of a penalized fourth order equation with delay
Abstract: In this paper we consider a fourth order nonlinear parabolic delayed problem modelling a quasi-instantaneous turn-over of linkages in the context of cell-motility. The model depends on a small parameter $\epsilon$ which represents a typical time scale of the memory effect. We first prove global existence and uniqueness of solutions for $\epsilon$ fixed. This is achieved by combining suitable fixed-point and energy arguments and by uncovering a nonlocal in time, integral conserved quantity. After giving a complete classification of steady states in terms of elliptic functions, we next show that every solution converges to a steady state as $t \to \infty$. When $\epsilon \to 0$, we then establish convergence results on finite time intervals, showing that the solution tends in a suitable sense towards the solution of a parabolic problem without delay. Moreover, we establish the convergence of energies as $\epsilon \to 0$, which enables us to show that, for $\epsilon$ small enough, the $\epsilon$-dependent problem inherits part of the large time asymptotics of the limiting parabolic problem.


Title: A PDE approach for solving the characteristic function of the generalised signature process
Abstract: The signature of a path, as a fundamental object in Rough path theory, serves as a generating function for non-communicative monomials on path space. It transforms the path into a grouplike element in the tensor algebra space, summarising the path faithfully up to a generalised form of re-parameterisation (a negligible equivalence class in this context). Our paper concerns stochastic processes and studies the characteristic function of the path signature of the stochastic process. In contrast to the expected signature, it determines the law on the random signatures without any regularity condition. The computation of the characteristic function of the random signature offers potential applications in stochastic analysis and machine learning, where the expected signature plays an important role. In this paper, we focus on a time-homogeneous It\^o diffusion process, and adopt a PDE approach to derive the characteristic function of its signature defined at any fixed time horizon. A key ingredient of our approach is the introduction of the generalised-signature process. This lifting enables us to establish the Feynman-Kac-type theorem for the characteristic function of the generalised-signature process by following the martingale approach. Moreover, as an application of our results, we present a novel derivation of the joint characteristic function of Brownian motion coupled with the L\'evy area, leveraging the structure theorem of anti-symmetric matrices.


Title: Solving Fokker-Planck equations using the zeros of Fokker-Planck operators and the Feynman-Kac formula
Abstract: First we show that physics-informed neural networks are not suitable for a large class of parabolic partial differential equations including the Fokker-Planck equation. Then we devise an algorithm to compute solutions of the Fokker-Planck equation using the zeros of Fokker-Planck operator and the Feynman-Kac formula. The resulting algorithm is mesh-free, highly parallelizable and able to compute solutions pointwise, thus mitigating the curse of dimensionality in a practical sense. We analyze various nuances of this algorithm that are determined by the drift term in the Fokker-Planck equation. We work with problems ranging in dimensions from 2 to 10. We demonstrate that this algorithm requires orders of magnitude fewer trajectories for each point in space when compared to Monte-Carlo. We also prove that under suitable conditions the error that is caused by letting some trajectories (associated with the Feynman-Kac expectation) escape our domain of knowledge is proportional to the fraction of trajectories that escape.


Title: Prompt Decoupling for Text-to-Image Person Re-identification
Abstract: Text-to-image person re-identification (TIReID) aims to retrieve the target person from an image gallery via a textual description query. Recently, pre-trained vision-language models like CLIP have attracted significant attention and have been widely utilized for this task due to their robust capacity for semantic concept learning and rich multi-modal knowledge. However, recent CLIP-based TIReID methods commonly rely on direct fine-tuning of the entire network to adapt the CLIP model for the TIReID task. Although these methods show competitive performance on this topic, they are suboptimal as they necessitate simultaneous domain adaptation and task adaptation. To address this issue, we attempt to decouple these two processes during the training stage. Specifically, we introduce the prompt tuning strategy to enable domain adaptation and propose a two-stage training approach to disentangle domain adaptation from task adaptation. In the first stage, we freeze the two encoders from CLIP and solely focus on optimizing the prompts to alleviate domain gap between the original training data of CLIP and downstream tasks. In the second stage, we maintain the fixed prompts and fine-tune the CLIP model to prioritize capturing fine-grained information, which is more suitable for TIReID task. Finally, we evaluate the effectiveness of our method on three widely used datasets. Compared to the directly fine-tuned approach, our method achieves significant improvements.


Title: Stress and Adaptation: Applying Anna Karenina Principle in Deep Learning for Image Classification
Abstract: Image classification with deep neural networks has reached state-of-art with high accuracy. This success is attributed to good internal representation features that bypasses the difficulties of the non-convex optimization problems. We have little understanding of these internal representations, let alone quantifying them. Recent research efforts have focused on alternative theories and explanations of the generalizability of these deep networks. We propose the alternative perturbation of deep models during their training induces changes that lead to transitions to different families. The result is an Anna Karenina Principle AKP for deep learning, in which less generalizable models unhappy families vary more in their representation than more generalizable models happy families paralleling Leo Tolstoy dictum that all happy families look alike, each unhappy family is unhappy in its own way. Anna Karenina principle has been found in systems in a wide range: from the surface of endangered corals exposed to harsh weather to the lungs of patients suffering from fatal diseases of AIDs. In our paper, we have generated artificial perturbations to our model by hot-swapping the activation and loss functions during the training. In this paper, we build a model to classify cancer cells from non-cancer ones. We give theoretical proof that the internal representations of generalizable happy models are similar in the asymptotic limit. Our experiments verify similar representations of generalizable models.


Title: The Security and Privacy of Mobile Edge Computing: An Artificial Intelligence Perspective
Abstract: Mobile Edge Computing (MEC) is a new computing paradigm that enables cloud computing and information technology (IT) services to be delivered at the network's edge. By shifting the load of cloud computing to individual local servers, MEC helps meet the requirements of ultralow latency, localized data processing, and extends the potential of Internet of Things (IoT) for end-users. However, the crosscutting nature of MEC and the multidisciplinary components necessary for its deployment have presented additional security and privacy concerns. Fortunately, Artificial Intelligence (AI) algorithms can cope with excessively unpredictable and complex data, which offers a distinct advantage in dealing with sophisticated and developing adversaries in the security industry. Hence, in this paper we comprehensively provide a survey of security and privacy in MEC from the perspective of AI. On the one hand, we use European Telecommunications Standards Institute (ETSI) MEC reference architecture as our based framework while merging the Software Defined Network (SDN) and Network Function Virtualization (NFV) to better illustrate a serviceable platform of MEC. On the other hand, we focus on new security and privacy issues, as well as potential solutions from the viewpoints of AI. Finally, we comprehensively discuss the opportunities and challenges associated with applying AI to MEC security and privacy as possible future research directions.


Title: Lifespan of Solution to MHD Boundary Layer Equations with Analytic Perturbation of General Shear Flow
Abstract: In this paper, we consider the lifespan of solution to the MHD boundary layer system as an analytic perturbation of general shear flow. By using the cancellation mechanism in the system observed in \cite{LXY1}, the lifespan of solution is shown to have a lower bound in the order of $\varepsilon^{-2+}$ if the strength of the perturbation is of the order of $\varepsilon$. Since there is no restriction on the strength of the shear flow and the lifespan estimate is larger than the one obtained for the classical Prandtl system in this setting, it reveals the stabilizing effect of the magnetic field on the electrically conducting fluid near the boundary.


Title: Generating synthetic data for neural operators
Abstract: Numerous developments in the recent literature show the promising potential of deep learning in obtaining numerical solutions to partial differential equations (PDEs) beyond the reach of current numerical solvers. However, data-driven neural operators all suffer from the same problem: the data needed to train a network depends on classical numerical solvers such as finite difference or finite element, among others. In this paper, we propose a new approach to generating synthetic functional training data that does not require solving a PDE numerically. The way we do this is simple: we draw a large number $N$ of independent and identically distributed `random functions' $u_j$ from the underlying solution space (e.g., $H_0^1(\Omega)$) in which we know the solution lies according to classical theory. We then plug each such random candidate solution into the equation and get a corresponding right-hand side function $f_j$ for the equation, and consider $(f_j, u_j)_{j=1}^N$ as supervised training data for learning the underlying inverse problem $f \rightarrow u$. This `backwards' approach to generating training data only requires derivative computations, in contrast to standard `forward' approaches, which require a numerical PDE solver, enabling us to generate a large number of such data points quickly and efficiently. While the idea is simple, we hope that this method will expand the potential for developing neural PDE solvers that do not depend on classical numerical solvers.


Title: Gradient-Based Optimization of Lattice Quantizers
Abstract: Lattices with minimal normalized second moments are designed using a new numerical optimization algorithm. Starting from a random lower-triangular generator matrix and applying stochastic gradient descent, all elements are updated towards the negative gradient, which makes it the most efficient algorithm proposed so far for this purpose. A graphical illustration of the theta series, called theta image, is introduced and shown to be a powerful tool for converting numerical lattice representations into their underlying exact forms. As a proof of concept, optimized lattices are designed in dimensions up to 16. In all dimensions, the algorithm converges to either the previously best known lattice or a better one. The dual of the 15-dimensional laminated lattice is conjectured to be optimal in its dimension.


Title: Posets, Tensor Products and Schur positivity
Abstract: Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then $P(\lambda, k)/\sim$ coincides with the set of S_k-orbits in $P(\lambda,k)$, where S_k acts on $P(\lambda, k)$ as the permutations of components. If g is of type A_n and k=2, we show that the S_2-orbit of the row shuffle defined by Fomin et al is the unique maximal element in the poset. Given an element of $P(\lambda, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, \lambda, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when \lambda is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (\lambda is general), there exists an inclusion of tensor products of g-modules along with the partial order. In particular, if g is of type A_n, this means that the difference of the characters is Schur positive.


Title: Towards the Atiyah-Sutcliffe conjectures for coplanar hyperbolic points
Abstract: The Atiyah-Sutcliffe normalized determinant function $D$ is a smooth complex-valued function on $C_n(H^3)$, where $C_n(H^3)$ denotes the configuration space of $n$ distinct points in hyperbolic $3$-space $H^3$. The hyperbolic version of the Atiyah-Sutcliffe conjecture $1$ (AS conjecture $1$) states that $D$ is nowhere vanishing. AS conjecture $2$ (hyperbolic version) is the stronger statement that $|D(\mathbf{x})| \geq 1$ for any $\mathbf{x} \in C_n(H^3)$. In this short article, we prove AS conjecture $2$ for hyperbolic convex coplanar quadrilaterals, that is for configurations of $4$ points in $H^2$ with none of the points in the configuration lying in the convex hull of the other three. We also obtain Y. Zhang and J. Ma's result, namely AS conjecture $1$ for non-convex quadrilaterals in $H^2$. Finally, we find an explicit lower bound for $|D|$ depending on $n$ only for the natural ``star-based'' variant of the AS problem, for convex coplanar hyperbolic configurations. The latter result holds for any $n \geq 2$. The proofs for $n=4$ make use of the symbolic library of Python. The proof of the general result follows from a general formula for the determinant. In all these cases, $D$ can be expanded as a linear combination of non-negative rational functions with positive coefficients.


Title: Towards a Solution to Bongard Problems: A Causal Approach
Abstract: Even though AI has advanced rapidly in recent years displaying success in solving highly complex problems, the class of Bongard Problems (BPs) yet remain largely unsolved by modern ML techniques. In this paper, we propose a new approach in an attempt to not only solve BPs but also extract meaning out of learned representations. This includes the reformulation of the classical BP into a reinforcement learning (RL) setting which will allow the model to gain access to counterfactuals to guide its decisions but also explain its decisions. Since learning meaningful representations in BPs is an essential sub-problem, we further make use of contrastive learning for the extraction of low level features from pixel data. Several experiments have been conducted for analyzing the general BP-RL setup, feature extraction methods and using the best combination for the feature space analysis and its interpretation.


Title: On grids in point-line arrangements in the plane
Abstract: The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for point-line incidence. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be two sets of $t$ lines in the plane and let $P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\}$ be the set of intersection points between $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $(P, \mathcal{L}_1 \cup \mathcal{L}_2)$ forms a \emph{natural $t\times t$ grid} if $|P| =t^2$, and $conv(P)$ does not contain the intersection point of some two lines in $\mathcal{L}_i,$ for $i = 1,2.$ For fixed $t > 1$, we show that any arrangement of $n$ points and $n$ lines in the plane that does not contain a natural $t\times t$ grid determines $O(n^{\frac{4}{3}- \varepsilon})$ incidences, where $\varepsilon = \varepsilon(t)$. We also provide a construction of $n$ points and $n$ lines in the plane that does not contain a natural $2 \times 2$ grid and determines at least $\Omega({n^{1+\frac{1}{14}}})$ incidences.


Title: Approximation to multifractional Riemann-Liouville Brownian sheet
Abstract: In this paper, we first introduce multifrational Riemann-Liouville Brownian sheets. Then, we show a result of approximation in law of the multifractional Riemann-Liouville Brownian sheet. The construction of these approximations is based on a sequence of I.I.D random variables.


Title: Probabilistic representation for mild solution of the Navier-Stokes equations
Abstract: This paper is based on a formulation of the Navier-Stokes equations developed by Iyer and Constantin \cite{Cont} , where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. Our contribution is to establish this probabilistic representation formula for mild solutions of the Navier-Stokes equations on $\mathbb{R}^{d} $.


Title: Optimal Hardy-weights for the $(p,A)$-Laplacian with a potential term
Abstract: We construct new optimal $L^p$ Hardy-type inequalities for elliptic Schr\"odinger-type operators


Title: Tarski's Least Fixed Point Theorem: A Type Theoretic Formulation
Abstract: We translate Giovanni Curi's predicative least fixed point theorem into type theory. There are multiple benefits of having a type theoretic formulation apart from the potential for routine formalization. By taking advantage of (higher) inductive types, we have skirted the painstaking set theoretic constructions and as a result believe our presentation is conceptually clearer. Additionally, due the predicative admissibility of (higher) inductive types we take a step towards the \say{system independent} derivation that Curi calls for in his conclusion. We also explore restrictions on monotone maps that guarantee they are \say{generated} in a sense we make precise. This allows for an alternative statement of the least fixed point theorem which goes beyond the version found in Curi's work.


Title: Attacks in Adversarial Machine Learning: A Systematic Survey from the Life-cycle Perspective
Abstract: Adversarial machine learning (AML) studies the adversarial phenomenon of machine learning, which may make inconsistent or unexpected predictions with humans. Some paradigms have been recently developed to explore this adversarial phenomenon occurring at different stages of a machine learning system, such as backdoor attack occurring at the pre-training, in-training and inference stage; weight attack occurring at the post-training, deployment and inference stage; adversarial attack occurring at the inference stage. However, although these adversarial paradigms share a common goal, their developments are almost independent, and there is still no big picture of AML. In this work, we aim to provide a unified perspective to the AML community to systematically review the overall progress of this field. We firstly provide a general definition about AML, and then propose a unified mathematical framework to covering existing attack paradigms. According to the proposed unified framework, we build a full taxonomy to systematically categorize and review existing representative methods for each paradigm. Besides, using this unified framework, it is easy to figure out the connections and differences among different attack paradigms, which may inspire future researchers to develop more advanced attack paradigms. Finally, to facilitate the viewing of the built taxonomy and the related literature in adversarial machine learning, we further provide a website, \ie, \url{http://adversarial-ml.com}, where the taxonomies and literature will be continuously updated.


Title: Free constructions of geometries of Coxeter type
Abstract: We establish two free constructions of geometries of Coxeter type. The first construction deals with any Coxeter diagram having no subdiagram of type A_3, the second one with diagrams of type C_n and H_4.


Title: Incentivizing Massive Unknown Workers for Budget-Limited Crowdsensing: From Off-Line and On-Line Perspectives
Abstract: How to incentivize strategic workers using limited budget is a very fundamental problem for crowdsensing systems; nevertheless, since the sensing abilities of the workers may not always be known as prior knowledge due to the diversities of their sensor devices and behaviors, it is difficult to properly select and pay the unknown workers. Although the uncertainties of the workers can be addressed by the standard Combinatorial Multi-Armed Bandit (CMAB) framework in existing proposals through a trade-off between exploration and exploitation, we may not have sufficient budget to enable the trade-off among the individual workers, especially when the number of the workers is huge while the budget is limited. Moreover, the standard CMAB usually assumes the workers always stay in the system, whereas the workers may join in or depart from the system over time, such that what we have learnt for an individual worker cannot be applied after the worker leaves. To address the above challenging issues, in this paper, we first propose an off-line Context-Aware CMAB-based Incentive (CACI) mechanism. We innovate in leveraging the exploration-exploitation trade-off in an elaborately partitioned context space instead of the individual workers, to effectively incentivize the massive unknown workers with a very limited budget. We also extend the above basic idea to the on-line setting where unknown workers may join in or depart from the systems dynamically, and propose an on-line version of the CACI mechanism. We perform rigorous theoretical analysis to reveal the upper bounds on the regrets of our CACI mechanisms and to prove their truthfulness and individual rationality, respectively. Extensive experiments on both synthetic and real datasets are also conducted to verify the efficacy of our mechanisms.


Title: Factorial Moments of the Geometric Distribution of Order $k$
Abstract: We derive a simple expression for the $r^{th}$ factorial moment $\mu_{(r)}$ of the geometric distribution of order $k$ with success parameter $p\in(0,1)$ (and $q=1-p$) in terms of its probability mass function $f_k(n)$. Specifically, $\mu_{(r)} = r!f_k((r+1)k+r)/((qp^k)^{r+1})$.


Title: Equidistribution from the Chinese Remainder Theorem
Abstract: We prove the equidistribution of subsets of $(\Rr/\Zz)^n$ defined by fractional parts of subsets of~$(\Zz/q\Zz)^n$ that are constructed using the Chinese Remainder Theorem.


Title: An experimental sorting method for improving metagenomic data encoding
Abstract: Minimizing data storage poses a significant challenge in large-scale metagenomic projects. In this paper, we present a new method for improving the encoding of FASTQ files generated by metagenomic sequencing. This method incorporates metagenomic classification followed by a recursive filter for clustering reads by DNA sequence similarity to improve the overall reference-free compression. In the results, we show an overall improvement in the compression of several datasets. As hypothesized, we show a progressive compression gain for higher coverage depth and number of identified species. Additionally, we provide an implementation that is freely available at https://github.com/cobilab/mizar and can be customized to work with other FASTQ compression tools.


Title: The Weyl law for algebraic tori
Abstract: We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus $T$ with bounded analytic conductor. The analytic conductor which we use is defined via the local Langlands correspondence for tori by choosing a finite dimensional complex algebraic representation of the $L$-group of $T$. Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor.


Title: Logit-Q Dynamics for Efficient Learning in Stochastic Teams
Abstract: We present two logit-Q learning dynamics combining the classical and independent log-linear learning updates with an on-policy value iteration update for efficient learning in stochastic games. We show that the logit-Q dynamics presented reach (near) efficient equilibrium in stochastic teams. We quantify a bound on the approximation error. We also show the rationality of the logit-Q dynamics against agents following pure stationary strategies and the convergence of the dynamics in stochastic games where the reward functions induce potential games, yet only a single agent controls the state transitions beyond stochastic teams. The key idea is to approximate the dynamics with a fictional scenario where the Q-function estimates are stationary over finite-length epochs only for analysis. We then couple the dynamics in the main and fictional scenarios to show that these two scenarios become more and more similar across epochs due to the vanishing step size.


Title: Infinite Eulerian trails are computable on graphs with vertices of infinite degree
Abstract: The Erd\H{o}s, Gr\"unwald and Weiszfeld theorem provides a characterization of infinite graphs which are Eulerian. That is, infinite graphs which admit infinite Eulerian trails. In this article we complement this theorem with a characterization of those finite trails that can be extended to infinite Eulerian trails. This allows us to prove an effective version of the Erd\H{o}s, Gr\"unwald and Weiszfeld theorem for a class of graphs that includes non locally finite ones, generalizing a theorem of D.Bean.


Title: Twinless articulation points and some related problems
Abstract: Let $G=(V,E)$ be a twinless strongly connected graph. a vertex $v\in V$ is a twinless articulation point if the subrgraph obtained from $G$ by removing the vertex $v$ is not twinless strongly connected. An edge $e\in E$ is a twinless bridge if the subgraph obtained from $G$ by deleting $e$ is not twiless strongly connected graph. In this paper we study twinless articulation points and twinless bridges. We also study the problem of finding a minimum cardinality edge subset $E_{1} \subseteq E$ such that the subgraph $(V,E_{1})$ is twinless strongly connected. Moreover, we present an algorithm for computing the $2$-vertex-twinless connected components of $G$.


Title: Continuous Credit Networks and Layer 2 Blockchains: Monotonicity and Sampling
Abstract: To improve transaction rates, many cryptocurrencies have implemented so-called ''Layer-2'' transaction protocols, where payments are routed across networks of private payment channels. However, for a given transaction, not every network state provides a feasible route to perform the payment; in this case, the transaction must be put on the public ledger. The payment channel network thus multiplies the transaction rate of the overall system; the less frequently it fails, the higher the multiplier. We build on earlier work on credit networks and show that this network liquidity problem is connected to the combinatorics of graphical matroids. Earlier work could only analyze the (unnatural) scenario where transactions had discrete sizes. Superficially, it might seem like the continuous case would be harder to examine. However, removing this assumption lets us make progress in two important directions. First, we give a partial answer to the ``monotonicity conjecture'' that previous work left open. This conjecture asks that the network's performance not degrade as capacity on any edge increases. And second, we construct here a network state sampling procedure with much faster asymptotic performance than off-the-shelf Markov chains ($O(\vert E\vert \beta(\vert E\vert))$, where $\beta(x)$ is the complexity of solving a linear program on $x$ constraints.) We obtain our results by mapping the underlying graphs to convex bodies and then showing that the liquidity and sampling problems reduce to bounding and computing the volumes of these bodies. The transformation relies crucially on the combinatorial properties of the underlying graphic matroid, as do the proofs of monotonicity and fast sampling.


Title: An analysis of training and generalization errors in shallow and deep networks
Abstract: This paper is motivated by an open problem around deep networks, namely, the apparent absence of over-fitting despite large over-parametrization which allows perfect fitting of the training data. In this paper, we analyze this phenomenon in the case of regression problems when each unit evaluates a periodic activation function. We argue that the minimal expected value of the square loss is inappropriate to measure the generalization error in approximation of compositional functions in order to take full advantage of the compositional structure. Instead, we measure the generalization error in the sense of maximum loss, and sometimes, as a pointwise error. We give estimates on exactly how many parameters ensure both zero training error as well as a good generalization error. We prove that a solution of a regularization problem is guaranteed to yield a good training error as well as a good generalization error and estimate how much error to expect at which test data.


Title: Fusion Categories over Non-Algebriacally Closed Fields
Abstract: Several complications arise when attempting to work with fusion categories over arbitrary fields. Here we describe some of the new phenomena that occur when the field is not algebraically closed, and we adapt tools such as the Frobenius-Perron dimension in order to accommodate these new effects.


Title: Sphere-like isoparametric hypersurfaces in Damek-Ricci spaces
Abstract: Locally harmonic manifolds are Riemannian manifolds in which small geodesic spheres are isoparametric hypersurfaces, i.e., hypersurfaces whose nearby parallel hypersurfaces are of constant mean curvature. Flat and rank one symmetric spaces are examples of harmonic manifolds. Damek-Ricci spaces are non-compact harmonic manifolds, most of which are non-symmetric. Taking the limit of an "inflating" sphere through a point $p$ in a Damek-Ricci space as the center of the sphere runs out to infinity along a geodesic half-line $\gamma$ starting from $p$, we get a horosphere. Similarly to spheres, horospheres are also isoparametric hypersurfaces. In this paper, we define the sphere-like hypersurfaces obtained by "overinflating the horospheres" by pushing the center of the sphere beyond the point at infinity of $\gamma$ along a virtual prolongation of $\gamma$. They give a new family of isoparametric hypersurfaces in Damek-Ricci spaces connecting geodesic spheres to some of the isoparametric hypersurfaces constructed by J. C. D\'iaz-Ramos and M. Dom\'inguez-V\'azquez [arXiv:1111.0264] in Damek-Ricci spaces. We study the geometric properties of these isoparametric hypersurfaces, in particular their homogeneity and the totally geodesic condition for their focal varieties.


Title: On a reduction map for Drinfeld modules
Abstract: In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\widehat\varphi}={\phi}_{1}^{e_1}\times \dots \times {\phi}_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\mathbb F}_{q}[t].$ We assume that the ${\mathrm{rank}}(\phi)_{i})=d_{i}$ and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. ${\mathrm{End}}({\phi}_{i})=A$ for $ i=1,\dots , t.$ Our main result is the following numeric criterion. Let ${N}={N}_{1}^{e_1}\times\dots\times {N}_{t}^{e_t}$ be a finitely generated $A$ submodule of the Mordell-Weil group ${\widehat\varphi}({\cal O}_{K})={\phi}_{1}({\cal O}_{K})^{e_{1}}\times\dots\times {\phi}_{t}({\cal O}_{K})^{{e}_{t}},$ and let ${\Lambda}\subset N$ be an $A$ - submodule. If we assume $d_{i}\geq e_{i}$ and $P\in N$ such that $r_{\cal W}(P)\in r_{\cal W}({\Lambda}) $ for almost all primes ${\cal W}$ of ${\cal O}_{K},$ then $P\in {\Lambda}+N_{tor}.$ We also build on the recent results of S.Bara{\'n}czuk \cite{b17} concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned $t$-modules.


Title: Unique Triangulated 1-Planar Graphs
Abstract: It is well-known that every 3-connected planar graph has a unique planar embedding on the sphere. We study the extension to triangulated 1-planar graphs, T1P graphs for short, which admit an embedding in which each edge is crossed at most once and each face is a triangle, and obtain an algorithmic solution by a cubic time recognition algorithm that also counts the number of T1P embeddings. In particular, we show that every triangulated planar graph has a unique T1P embedding, although it may admit many 1-planar embeddings, and that any 6-connected T1P graph has a unique 1-planar embedding, except for full generalized two-stars that admit two or eight 1-planar embeddings. Our algorithm extends, refines, and corrects a previous recognition algorithm by Chen, Grigni and Papadimitiou (``Recognizing Hole-Free 4-Map Graphs in Cubic Time'', Algorithmica 45 (2006)).


Title: Improved uncertainty quantification for neural networks with Bayesian last layer
Abstract: Uncertainty quantification is an important task in machine learning - a task in which standardneural networks (NNs) have traditionally not excelled. This can be a limitation for safety-critical applications, where uncertainty-aware methods like Gaussian processes or Bayesian linear regression are often preferred. Bayesian neural networks are an approach to address this limitation. They assume probability distributions for all parameters and yield distributed predictions. However, training and inference are typically intractable and approximations must be employed. A promising approximation is NNs with Bayesian last layer (BLL). They assume distributed weights only in the linear output layer and yield a normally distributed prediction. To approximate the intractable Bayesian neural network, point estimates of the distributed weights in all but the last layer should be obtained by maximizing the marginal likelihood. This has previously been challenging, as the marginal likelihood is expensive to evaluate in this setting. We present a reformulation of the log-marginal likelihood of a NN with BLL which allows for efficient training using backpropagation. Furthermore, we address the challenge of uncertainty quantification for extrapolation points. We provide a metric to quantify the degree of extrapolation and derive a method to improve the uncertainty quantification for these points. Our methods are derived for the multivariate case and demonstrated in a simulation study. In comparison to Bayesian linear regression with fixed features, and a Bayesian neural network trained with variational inference, our proposed method achieves the highest log-predictive density on test data.


Title: Hochschild cohomology of the second kind: Koszul duality and Morita invariance
Abstract: We define Hochschild cohomology of the second kind for differential graded (dg) or curved algebras as a derived functor in a compactly generated derived category of the second kind, and show that it is invariant under Morita equivalence of the second kind. A bimodule version of Koszul duality is constructed and used to show that Hochschild cohomology of the second kind is preserved under (nonconilpotent) Koszul duality. Hochschild cohomology of the second kind of an algebra often computes the ordinary Hochschild cohomology of geometrically meaningful dg categories. Examples include the category of infinity local systems on a topological space, the bounded derived category of a complex algebraic manifold and the category of matrix factorizations.


Title: Coverage Explorer: Coverage-guided Test Generation for Cyber Physical Systems
Abstract: Given the safety-critical functions of autonomous cyber-physical systems (CPS) across diverse domains, testing these systems is essential. While conventional software and hardware testing methodologies offer partial insights, they frequently do not provide adequate coverage in a CPS. In this study, we introduce a testing framework designed to systematically formulate test cases, effectively exploring the state space of CPS. This framework introduces a coverage-centric sampling technique, coupled with a cluster-based methodology for training a surrogate model. The framework then uses model predictive control within the surrogate model to generates test cases tailored to CPS specifications. To evaluate the efficacy of the framework, we applied it on several benchmarks, spanning from a kinematic car to systems like an unmanned aircraft collision avoidance system (ACAS XU) and automatic transmission system. Comparative analyses were conducted against alternative test generation strategies, including randomized testing, as well as falsification using S-TaLiRo.


Title: Analyzing Misinformation Claims During the 2022 Brazilian General Election on WhatsApp, Twitter, and Kwai
Abstract: This study analyzes misinformation from WhatsApp, Twitter, and Kwai during the 2022 Brazilian general election. Given the democratic importance of accurate information during elections, multiple fact-checking organizations collaborated to identify and respond to misinformation via WhatsApp tiplines and power a fact-checking feature within a chatbot operated by Brazil's election authority, the TSE. WhatsApp is installed on over 99% of smartphones in Brazil, and the TSE chatbot was used by millions of citizens in the run-up to the elections. During the same period, we collected social media data from Twitter (now X) and Kwai (a popular video-sharing app similar to TikTok). Using the WhatsApp, Kwai, and Twitter data along with fact-checks from three Brazilian fact-checking organizations, we find unique claims on each platform. Even when the same claims are present on different platforms, they often differ in format, detail, length, or other characteristics. Our research highlights the limitations of current claim matching algorithms to match claims across platforms with such differences and identifies areas for further algorithmic development. Finally, we perform a descriptive analysis examining the formats (image, video, audio, text) and content themes of popular misinformation claims.


Title: Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity
Abstract: We address the persistence of regularity for the 2D $\alpha$-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for $(u_{0}, \rho_{0}) \in W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$, where $s> 1$ and $q \in (2, \infty)$. We prove that the solution $(u(t), \rho(t))$ exists and belongs to $W^{s,q}(\mathbb R^2) \times W^{s,q}(\mathbb R^2)$ for all positive time $t$ for $q>2$, where $\alpha\in(1,2)$ is arbitrary.


Title: Fast Certification of Vision-Language Models Using Incremental Randomized Smoothing
Abstract: A key benefit of deep vision-language models such as CLIP is that they enable zero-shot open vocabulary classification; the user has the ability to define novel class labels via natural language prompts at inference time. However, while CLIP-based zero-shot classifiers have demonstrated competitive performance across a range of domain shifts, they remain highly vulnerable to adversarial attacks. Therefore, ensuring the robustness of such models is crucial for their reliable deployment in the wild. In this work, we introduce Open Vocabulary Certification (OVC), a fast certification method designed for open-vocabulary models like CLIP via randomized smoothing techniques. Given a base "training" set of prompts and their corresponding certified CLIP classifiers, OVC relies on the observation that a classifier with a novel prompt can be viewed as a perturbed version of nearby classifiers in the base training set. Therefore, OVC can rapidly certify the novel classifier using a variation of incremental randomized smoothing. By using a caching trick, we achieve approximately two orders of magnitude acceleration in the certification process for novel prompts. To achieve further (heuristic) speedups, OVC approximates the embedding space at a given input using a multivariate normal distribution bypassing the need for sampling via forward passes through the vision backbone. We demonstrate the effectiveness of OVC on through experimental evaluation using multiple vision-language backbones on the CIFAR-10 and ImageNet test datasets.


Title: Transformers in Action Recognition: A Review on Temporal Modeling
Abstract: In vision-based action recognition, spatio-temporal features from different modalities are used for recognizing activities. Temporal modeling is a long challenge of action recognition. However, there are limited methods such as pre-computed motion features, three-dimensional (3D) filters, and recurrent neural networks (RNN) for modeling motion information in deep-based approaches. Recently, transformers success in modeling long-range dependencies in natural language processing (NLP) tasks has gotten great attention from other domains; including speech, image, and video, to rely entirely on self-attention without using sequence-aligned RNNs or convolutions. Although the application of transformers to action recognition is relatively new, the amount of research proposed on this topic within the last few years is astounding. This paper especially reviews recent progress in deep learning methods for modeling temporal variations. It focuses on action recognition methods that use transformers for temporal modeling, discussing their main features, used modalities, and identifying opportunities and challenges for future research.


Title: Analyticity of Entropy Rate of Hidden Markov Chains
Abstract: We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An example is given to estimate the radius of convergence for the entropy rate. We then show that the positivity assumptions can be relaxed, and examples are given for the relaxed conditions. We study a special class of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol, and we give necessary and sufficient conditions for analyticity of the entropy rate for this case. Finally, we show that under the positivity assumptions the hidden Markov chain {\em itself} varies analytically, in a strong sense, as a function of the underlying Markov chain parameters.


Title: Approximate Distance and Shortest-Path Oracles for Fault-Tolerant Geometric Spanners
Abstract: In this paper, we present approximate distance and shortest-path oracles for fault-tolerant Euclidean spanners motivated by the routing problem in real-world road networks. An $f$-fault-tolerant Euclidean $t$-spanner for a set $V$ of $n$ points in $\mathbb{R}^d$ is a graph $G=(V,E)$ where, for any two points $p$ and $q$ in $V$ and a set $F$ of $f$ vertices of $V$, the distance between $p$ and $q$ in $G-F$ is at most $t$ times their Euclidean distance. Given an $f$-fault-tolerant Euclidean $t$-spanner $G$ with $O(n)$ edges and a constant $\varepsilon$, our data structure has size $O_{t,f}(n\log n)$, and this allows us to compute an $(1+\varepsilon)$-approximate distance in $G-F$ between $s$ and $s'$ can be computed in constant time for any two vertices $s$ and $s'$ and a set $F$ of $f$ failed vertices. Also, with a data structure of size $O_{t,f}(n\log n\log\log n)$, we can compute an $(1+\varepsilon)$-approximate shortest path in $G-F$ between $s$ and $s'$ in $O_{t,f}(\log^2 n\log\log n+\textsf{sol})$ time for any two vertices $s$ and $s'$ and a set $F$ of failed vertices, where $\textsf{sol}$ denotes the number of vertices in the returned path.


Title: Unleashing the Emergent Cognitive Synergy in Large Language Models: A Task-Solving Agent through Multi-Persona Self-Collaboration
Abstract: Human intelligence thrives on cognitive synergy, where collaboration among different minds yield superior outcomes compared to isolated individuals. In this work, we propose Solo Performance Prompting (SPP), which transforms a single LLM into a cognitive synergist by engaging in multi-turn self-collaboration with multiple personas. A cognitive synergist is an intelligent agent that collaboratively combines multiple minds' strengths and knowledge to enhance problem-solving in complex tasks. By dynamically identifying and simulating different personas based on task inputs, SPP unleashes the potential of cognitive synergy in LLMs. Our in-depth analysis shows that assigning multiple fine-grained personas in LLMs improves problem-solving abilities compared to using a single or fixed number of personas. We evaluate SPP on three challenging tasks: Trivia Creative Writing, Codenames Collaborative, and Logic Grid Puzzle, encompassing both knowledge-intensive and reasoning-intensive types. Unlike previous works, such as Chain-of-Thought, that solely enhance the reasoning abilities in LLMs, experimental results demonstrate that SPP effectively reduces factual hallucination, and maintains strong reasoning capabilities. Additionally, comparative experiments show that cognitive synergy only emerges in GPT-4 and does not appear in less capable models, such as GPT-3.5-turbo and Llama2-13b-chat, which draws an interesting analogy to human development. Code, data, and prompts can be found at: https://github.com/MikeWangWZHL/Solo-Performance-Prompting.git.


Title: The Implicit Bias of Benign Overfitting
Abstract: The phenomenon of benign overfitting, where a predictor perfectly fits noisy training data while attaining near-optimal expected loss, has received much attention in recent years, but still remains not fully understood beyond well-specified linear regression setups. In this paper, we provide several new results on when one can or cannot expect benign overfitting to occur, for both regression and classification tasks. We consider a prototypical and rather generic data model for benign overfitting of linear predictors, where an arbitrary input distribution of some fixed dimension $k$ is concatenated with a high-dimensional distribution. For linear regression which is not necessarily well-specified, we show that the minimum-norm interpolating predictor (that standard training methods converge to) is biased towards an inconsistent solution in general, hence benign overfitting will generally not occur. Moreover, we show how this can be extended beyond standard linear regression, by an argument proving how the existence of benign overfitting on some regression problems precludes its existence on other regression problems. We then turn to classification problems, and show that the situation there is much more favorable. Specifically, we prove that the max-margin predictor (to which standard training methods are known to converge in direction) is asymptotically biased towards minimizing a weighted \emph{squared hinge loss}. This allows us to reduce the question of benign overfitting in classification to the simpler question of whether this loss is a good surrogate for the misclassification error, and use it to show benign overfitting in some new settings.


Title: Development of An Autonomous Bridge Deck Inspection Robotic System
Abstract: The threat to safety of aging bridges has been recognized as a critical concern to the general public due to the poor condition of many bridges in the U.S. Currently, the bridge inspection is conducted manually, and it is not efficient to identify bridge condition deterioration in order to facilitate implementation of appropriate maintenance or rehabilitation procedures. In this paper, we report a new development of the autonomous mobile robotic system for bridge deck inspection and evaluation. The robot is integrated with several nondestructive evaluation (NDE) sensors and a navigation control algorithm to allow it to accurately and autonomously maneuver on the bridge deck to collect visual images and conduct NDE measurements. The developed robotic system can reduce the cost and time of the bridge deck data collection and inspection. For efficient bridge deck monitoring, the crack detection algorithm to build the deck crack map is presented in detail. The impact-echo (IE), ultrasonic surface waves (USW) and electrical resistivity (ER) data collected by the robot are analyzed to generate the delamination, concrete elastic modulus, corrosion maps of the bridge deck, respectively. The presented robotic system has been successfully deployed to inspect numerous bridges in more than ten different states in the U.S.


Title: Toda brackets in n-angulated categories
Abstract: We introduce Toda brackets for n-angulated categories and show that the various definitions of Toda brackets coincide. We prove juggling formulas for these Toda brackets generalizing the triangulated case. Following that, we generalize a theorem due to Heller in the triangulated setting to the setting of n-angulated categories. We also provide several examples of computing Toda brackets for n-angulated categories. Finally, for an n-angulated category sitting in a triangulated category as in the setup of Geiss, Keller and Oppermann, we show that Toda brackets in the n-angulated sense coincide with n-fold Toda brackets in the triangulated sense up to an explicit sign.


Title: A Bag of Visual Words Approach for Symbols-Based Coarse-Grained Ancient Coin Classification
Abstract: The field of Numismatics provides the names and descriptions of the symbols minted on the ancient coins. Classification of the ancient coins aims at assigning a given coin to its issuer. Various issuers used various symbols for their coins. We propose to use these symbols for a framework that will coarsely classify the ancient coins. Bag of visual words (BoVWs) is a well established visual recognition technique applied to various problems in computer vision like object and scene recognition. Improvements have been made by incorporating the spatial information to this technique. We apply the BoVWs technique to our problem and use three symbols for coarse-grained classification. We use rectangular tiling, log-polar tiling and circular tiling to incorporate spatial information to BoVWs. Experimental results show that the circular tiling proves superior to the rest of the methods for our problem.


Title: The Unreasonable Effectiveness of Deep Evidential Regression
Abstract: There is a significant need for principled uncertainty reasoning in machine learning systems as they are increasingly deployed in safety-critical domains. A new approach with uncertainty-aware regression-based neural networks (NNs), based on learning evidential distributions for aleatoric and epistemic uncertainties, shows promise over traditional deterministic methods and typical Bayesian NNs, notably with the capabilities to disentangle aleatoric and epistemic uncertainties. Despite some empirical success of Deep Evidential Regression (DER), there are important gaps in the mathematical foundation that raise the question of why the proposed technique seemingly works. We detail the theoretical shortcomings and analyze the performance on synthetic and real-world data sets, showing that Deep Evidential Regression is a heuristic rather than an exact uncertainty quantification. We go on to discuss corrections and redefinitions of how aleatoric and epistemic uncertainties should be extracted from NNs.


Title: An Experimental Study of ILP Formulations for the Longest Induced Path Problem
Abstract: Given a graph $G=(V,E)$, the longest induced path problem asks for a maximum cardinality node subset $W\subseteq V$ such that the graph induced by $W$ is a path. It is a long established problem with applications, e.g., in network analysis. We propose novel integer linear programming (ILP) formulations for the problem and discuss efficient implementations thereof. Comparing them with known formulations from literature, we prove that they are beneficial in theory, yielding stronger relaxations. Moreover, our experiments show their practical superiority.


Title: Abacus models for parabolic quotients of affine Weyl groups
Abstract: We introduce abacus diagrams that describe minimal length coset representatives in affine Weyl groups of types B, C, and D. These abacus diagrams use a realization of the affine Weyl group of type C due to Eriksson to generalize a construction of James for the symmetric group. We also describe several combinatorial models for these parabolic quotients that generalize classical results in affine type A related to core partitions.


Title: The stochastic fractional Strichartz estimate and blow-up for Schrödinger equation
Abstract: We establish the stochastic Strichartz estimate for the fractional Schr\"odinger equation with multiplicative noise. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to the stochastic fractional nonlinear Schr\"odinger equation in $L^2(\mathbb{R}^n)$. In addition, we also prove a general blow up result by deriving a localized virial estimate and the generalized Strauss inequality with a restricted class of initial data.


Title: Robust Physics Informed Neural Networks
Abstract: We introduce a Robust version of the Physics-Informed Neural Networks (RPINNs) to approximate the Partial Differential Equations (PDEs) solution. Standard Physics Informed Neural Networks (PINN) takes into account the governing physical laws described by PDE during the learning process. The network is trained on a data set that consists of randomly selected points in the physical domain and its boundary. PINNs have been successfully applied to solve various problems described by PDEs with boundary conditions. The loss function in traditional PINNs is based on the strong residuals of the PDEs. This loss function in PINNs is generally not robust with respect to the true error. The loss function in PINNs can be far from the true error, which makes the training process more difficult. In particular, we do not know if the training process has already converged to the solution with the required accuracy. This is especially true if we do not know the exact solution, so we cannot estimate the true error during the training. This paper introduces a different way of defining the loss function. It incorporates the residual and the inverse of the Gram matrix, computed using the energy norm. We test our RPINN algorithm on two Laplace problems and one advection-diffusion problem in two spatial dimensions. We conclude that RPINN is a robust method. The proposed loss coincides well with the true error of the solution, as measured in the energy norm. Thus, we know if our training process goes well, and we know when to stop the training to obtain the neural network approximation of the solution of the PDE with the true error of required accuracy.


Title: Offline Policy Optimization with Eligible Actions
Abstract: Offline policy optimization could have a large impact on many real-world decision-making problems, as online learning may be infeasible in many applications. Importance sampling and its variants are a commonly used type of estimator in offline policy evaluation, and such estimators typically do not require assumptions on the properties and representational capabilities of value function or decision process model function classes. In this paper, we identify an important overfitting phenomenon in optimizing the importance weighted return, in which it may be possible for the learned policy to essentially avoid making aligned decisions for part of the initial state space. We propose an algorithm to avoid this overfitting through a new per-state-neighborhood normalization constraint, and provide a theoretical justification of the proposed algorithm. We also show the limitations of previous attempts to this approach. We test our algorithm in a healthcare-inspired simulator, a logged dataset collected from real hospitals and continuous control tasks. These experiments show the proposed method yields less overfitting and better test performance compared to state-of-the-art batch reinforcement learning algorithms.


Title: Transformer Neural Autoregressive Flows
Abstract: Density estimation, a central problem in machine learning, can be performed using Normalizing Flows (NFs). NFs comprise a sequence of invertible transformations, that turn a complex target distribution into a simple one, by exploiting the change of variables theorem. Neural Autoregressive Flows (NAFs) and Block Neural Autoregressive Flows (B-NAFs) are arguably the most perfomant members of the NF family. However, they suffer scalability issues and training instability due to the constraints imposed on the network structure. In this paper, we propose a novel solution to these challenges by exploiting transformers to define a new class of neural flows called Transformer Neural Autoregressive Flows (T-NAFs). T-NAFs treat each dimension of a random variable as a separate input token, using attention masking to enforce an autoregressive constraint. We take an amortization-inspired approach where the transformer outputs the parameters of an invertible transformation. The experimental results demonstrate that T-NAFs consistently match or outperform NAFs and B-NAFs across multiple datasets from the UCI benchmark. Remarkably, T-NAFs achieve these results using an order of magnitude fewer parameters than previous approaches, without composing multiple flows.


Title: EPA: Neural Collapse Inspired Robust Out-of-Distribution Detector
Abstract: Out-of-distribution (OOD) detection plays a crucial role in ensuring the security of neural networks. Existing works have leveraged the fact that In-distribution (ID) samples form a subspace in the feature space, achieving state-of-the-art (SOTA) performance. However, the comprehensive characteristics of the ID subspace still leave under-explored. Recently, the discovery of Neural Collapse ($\mathcal{NC}$) sheds light on novel properties of the ID subspace. Leveraging insight from $\mathcal{NC}$, we observe that the Principal Angle between the features and the ID feature subspace forms a superior representation for measuring the likelihood of OOD. Building upon this observation, we propose a novel $\mathcal{NC}$-inspired OOD scoring function, named Entropy-enhanced Principal Angle (EPA), which integrates both the global characteristic of the ID subspace and its inner property. We experimentally compare EPA with various SOTA approaches, validating its superior performance and robustness across different network architectures and OOD datasets.


Title: SwitchTab: Switched Autoencoders Are Effective Tabular Learners
Abstract: Self-supervised representation learning methods have achieved significant success in computer vision and natural language processing, where data samples exhibit explicit spatial or semantic dependencies. However, applying these methods to tabular data is challenging due to the less pronounced dependencies among data samples. In this paper, we address this limitation by introducing SwitchTab, a novel self-supervised method specifically designed to capture latent dependencies in tabular data. SwitchTab leverages an asymmetric encoder-decoder framework to decouple mutual and salient features among data pairs, resulting in more representative embeddings. These embeddings, in turn, contribute to better decision boundaries and lead to improved results in downstream tasks. To validate the effectiveness of SwitchTab, we conduct extensive experiments across various domains involving tabular data. The results showcase superior performance in end-to-end prediction tasks with fine-tuning. Moreover, we demonstrate that pre-trained salient embeddings can be utilized as plug-and-play features to enhance the performance of various traditional classification methods (e.g., Logistic Regression, XGBoost, etc.). Lastly, we highlight the capability of SwitchTab to create explainable representations through visualization of decoupled mutual and salient features in the latent space.


Title: Simplicial $*$-modules and mild actions
Abstract: We develop an analogue of the theory of $*$-modules in the world of simplicial sets, based on actions of a certain simplicial monoid $E\mathcal M$ originally appearing in the construction of global algebraic $K$-theory. As our main results, we show that strictly commutative monoids with respect to a certain box product on these simplicial $*$-modules yield models of equivariantly and globally coherently commutative monoids, and we give a characterization of simplicial $*$-modules in terms of a certain mildness condition on the $E\mathcal M$-action, relaxing the notion of tameness previously investigated by Sagave-Schwede and the first author.


Title: Free Brownian motion and free convolution semigroups: multiplicative case
Abstract: We consider a pair of probability measures $\mu,\nu$ on the unit circle such that $\Sigma_{\lambda}(\eta_{\nu}(z))=z/\eta_{\mu}(z)$. We prove that the same type of equation holds for any $t\geq 0$ when we replace $\nu$ by $\nu\boxtimes\lambda_t$ and $\mu$ by $\mathbb{M}_t(\mu)$, where $\lambda_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subordination functions in multiplicative free Brownian motion and multiplicative free convolution semigroups. We use the modified $\mathcal{S}$-transform introduced by Raj Rao and Speicher to deal with the case that $\nu$ has mean zero. The same type of the result holds for convolutions on the positive real line. We also obtain some regularity properties for the free multiplicative analogue of the normal distributions.


Title: Quantum Team Logic and Bell's Inequalities
Abstract: A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this end we generalise the team semantics of dependence logic [7] first to probabilistic team semantics, and then to what we call quantum team semantics.


Title: On knots that divide ribbon knotted surfaces
Abstract: We define a knot to be half ribbon if it is the cross-section of a ribbon 2-knot, and observe that ribbon implies half ribbon implies slice. We introduce the half ribbon genus of a knot K, the minimum genus of a ribbon knotted surface of which K is a cross-section. We compute this genus for all prime knots up to 12 crossings, and many 13-crossing knots. The same approach yields new computations of the doubly slice genus. We also introduce the half fusion number of a knot K, that measures the complexity of ribbon 2-knots of which K is a cross-section. We show that it is bounded from below by the Levine-Tristram signatures, and differs from the standard fusion number by an arbitrarily large amount.


Title: Optimizing Information Freshness in Uplink Multiuser MIMO Networks with Partial Observations
Abstract: This paper investigates a multiuser scheduling problem within an uplink multiple-input multi-output (MIMO) status update network, consisting of a multi-antenna base station (BS) and multiple single-antenna devices. The presence of multiple antennas at the BS introduces spatial degrees-of-freedom, enabling concurrent transmission of status updates from multiple devices in each time slot. Our objective is to optimize network-wide information freshness, quantified by the age of information (AoI) metric, by determining how the BS can best schedule device transmissions, while taking into account the random arrival of status updates at the device side.To address this decision-making problem, we model it as a partially observable Markov decision process (POMDP) and establish that the evolution of belief states for different devices is independent.We also prove that feasible belief states can be described by finite-dimensional vectors. Building on these observations, we develop a dynamic scheduling (DS) policy to solve the POMDP, and then derive an upper bound of its AoI performance, which is used to optimize the parameter configuration. To gain more design insights, we investigate a symmetric network, and put forth a fixed scheduling (FS) policy with lower computational complexity. An action space reduction strategy is applied to further reduce the computational complexity of both DS and FS policies. Our numerical results validate our analyses and indicate that the DS policy with the reduced action space performs almost identically to the original DS policy, and both outperform the baseline policies.


Title: Dynamic Local Regret for Non-convex Online Forecasting
Abstract: We consider online forecasting problems for non-convex machine learning models. Forecasting introduces several challenges such as (i) frequent updates are necessary to deal with concept drift issues since the dynamics of the environment change over time, and (ii) the state of the art models are non-convex models. We address these challenges with a novel regret framework. Standard regret measures commonly do not consider both dynamic environment and non-convex models. We introduce a local regret for non-convex models in a dynamic environment. We present an update rule incurring a cost, according to our proposed local regret, which is sublinear in time T. Our update uses time-smoothed gradients. Using a real-world dataset we show that our time-smoothed approach yields several benefits when compared with state-of-the-art competitors: results are more stable against new data; training is more robust to hyperparameter selection; and our approach is more computationally efficient than the alternatives.


Title: Separable homology of graphs and the Whitehead complex
Abstract: We introduce the Whitehead complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of $\mathbf{F}_n$. The Whitehead complex admits an action of $\mathrm{Out}(\mathbf{F}_n)$ by isometries if the associated cover corresponds to a characteristic subgroup of $\mathbf{F}_n$. We prove that the Whitehead complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex.


Title: The spectrality of self-affine measure under the similarity transformation of $GL_n(p)$
Abstract: Let $\mu_{M,D}$ be the self-affine measure generated by an expanding integer matrix $M\in M_n(\mathbb{Z})$ and a finite digit set $D\subset\mathbb{Z}^n$. It is well known that the two measures $\mu_{M,D}$ and $\mu_{\tilde{M},\tilde{D}}$ have the same spectrality if $\tilde{M}=B^{-1}MB$ and $\tilde{D}=B^{-1}D$, where $B\in M_n(\mathbb{R})$ is a nonsingular matrix. This fact is usually used to simplify the digit set $D$ or the expanding matrix $M$. However, it often transforms integer digit set $D$ or expanding matrix $M$ into real, which brings many difficulties to study the spectrality of $\mu_{\tilde{M},\tilde{D}}$. In this paper, we introduce a similarity transformation of general linear group $GL_n(p)$ for some self-affine measures, and discuss their spectrality. This kind of similarity transformation can keep the integer properties of $D$ and $M$ simultaneously, which leads to many advantages in discussing the spectrality of self-affine measures. As an application, we extend some well-known spectral self-affine measures to more general forms.


Title: Reference-less Measure of Faithfulness for Grammatical Error Correction
Abstract: We propose USim, a semantic measure for Grammatical Error Correction (GEC) that measures the semantic faithfulness of the output to the source, thereby complementing existing reference-less measures (RLMs) for measuring the output's grammaticality. USim operates by comparing the semantic symbolic structure of the source and the correction, without relying on manually-curated references. Our experiments establish the validity of USim, by showing that (1) semantic annotation can be consistently applied to ungrammatical text; (2) valid corrections obtain a high USim similarity score to the source; and (3) invalid corrections obtain a lower score.


Title: Support theorem for the transverse ray transform of tensor fields of rank 2
Abstract: Let (M, g) be a simple, real analytic, Riemannian manifold with boundary and of dimension n>=3. In this work, we prove a support theorem for the transverse ray transform of tensor fields of rank 2 defined over such manifolds. More specifically, given a symmetric tensor field f of rank 2, we show that if the transverse ray transform of f vanishes over an appropriate open set of maximal geodesics of M , then the support of f vanishes on the points of M that lie on the union of the aforementioned open set of geodesics.


Title: Mod-$p$ isogeny classes on Shimura varieties with parahoric level structure
Abstract: We study the special fiber of the integral models for Shimura varieties of Hodge type with parahoric level structure constructed by Kisin and Pappas in [KP]. We show that when the group is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands Rapoport conjecture in [LR]. We also verify most of the He-Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are non-empty for these models.


Title: The limiting shape of a full mailbox
Abstract: We study a model for email communication due to Gabrielli and Caldarelli, where someone receives and answers emails at the times of independent Poisson processes with intensities $\lambda_{\rm in}>\lambda_{\rm out}$. The receiver assigns i.i.d. priorities to incoming emails according to some atomless law and always answers the email in the mailbox with the highest priority. Since the frequency of incoming emails is higher than the frequency of answering, below a critical priority, the mailbox fills up ad infinitum. We prove a theorem about the limiting shape of the mailbox just above the critical point, linking it to the convex hull of Brownian motion. We conjecture that this limiting shape is universal in a class of similar models, including a model for the evolution of an order book due to Stigler and Luckock.


Title: On two open questions for extension bundles
Abstract: In this paper we give positive answers for two open questions on extension bundles over weighted projective lines, raised by Kussin, Lenzing and Meltzer in the paper ``Triangle singularities, ADE-chains and weighted projective lines''.


Title: Evolution of Retweet Rates in Twitter User Careers: Analysis and Model
Abstract: We study the evolution of the number of retweets received by Twitter users over the course of their "careers" on the platform. We find that on average the number of retweets received by users tends to increase over time. This is partly expected because users tend to gradually accumulate followers. Normalizing by the number of followers, however, reveals that the relative, per-follower retweet rate tends to be non-monotonic, maximized at a "peak age" after which it does not increase, or even decreases. We develop a simple mathematical model of the process behind this phenomenon, which assumes a constantly growing number of followers, each of whom loses interest over time. We show that this model is sufficient to explain the non-monotonic nature of per-follower retweet rates, without any assumptions about the quality of content posted at different times.


Title: b-articulation points and b-bridges in strongly biconnected directed graphs
Abstract: A directed graph $G=(V,E)$ is called strongly biconnected if $G$ is strongly connected and the underlying graph of $G$ is biconnected. This class of directed graphs was first introduced by Wu and Grumbach. Let $G=(V,E)$ be a strongly biconnected directed graph. An edge $e\in E$ is a b-bridge if the subgraph $G\setminus \left\lbrace e\right\rbrace =(V,E\setminus \left\lbrace e\right\rbrace) $ is not strongly biconnected. A vertex $w\in V$ is a b-articulation point if $G\setminus \left\lbrace w\right\rbrace$ is not strongly biconnected, where $G\setminus \left\lbrace w\right\rbrace$ is the subgraph obtained from $G$ by removing $w$. In this paper we study b-articulation points and b-bridges.


Title: On Error and Compression Rates for Prototype Rules
Abstract: We study the close interplay between error and compression in the non-parametric multiclass classification setting in terms of prototype learning rules. We focus in particular on a recently proposed compression-based learning rule termed OptiNet (Kontorovich, Sabato, and Urner 2016; Kontorovich, Sabato, and Weiss 2017; Hanneke et al. 2021). Beyond its computational merits, this rule has been recently shown to be universally consistent in any metric instance space that admits a universally consistent rule--the first learning algorithm known to enjoy this property. However, its error and compression rates have been left open. Here we derive such rates in the case where instances reside in Euclidean space under commonly posed smoothness and tail conditions on the data distribution. We first show that OptiNet achieves non-trivial compression rates while enjoying near minimax-optimal error rates. We then proceed to study a novel general compression scheme for further compressing prototype rules that locally adapts to the noise level without sacrificing accuracy. Applying it to OptiNet, we show that under a geometric margin condition, further gain in the compression rate is achieved. Experimental results comparing the performance of the various methods are presented.


Title: Weighted Proportional Allocations of Indivisible Goods and Chores: Insights via Matchings
Abstract: We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods.


Title: On the standing waves of the NLS-log equation with point interaction on a star graph
Abstract: We study a nonlinear Schr\"odinger equation with logarithmic nonlinearity on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate orbital stability and spectral instability of the standing wave solutions $e^{i\omega t}\mathbf{\Phi}(x)$ to the equation when the profile $\mathbf\Phi(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory.


Title: Boolean TQFTs with accumulating defects, sofic systems, and automata for infinite words
Abstract: Any finite state automaton gives rise to a Boolean one-dimensional TQFT with defects and inner endpoints of cobordisms. This paper extends the correspondence to Boolean TQFTs where defects accumulate toward inner endpoints, relating such TQFTs and topological theories to sofic systems and $\omega$-automata.


Title: An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems
Abstract: We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is efficient and robust for solving large-scale CMOT problems. The experiments on the discrete tomography problem also highlight the potential modeling power of our model.


Title: Towards Fully Decoupled End-to-End Person Search
Abstract: End-to-end person search aims to jointly detect and re-identify a target person in raw scene images with a unified model. The detection task unifies all persons while the re-id task discriminates different identities, resulting in conflict optimal objectives. Existing works proposed to decouple end-to-end person search to alleviate such conflict. Yet these methods are still sub-optimal on one or two of the sub-tasks due to their partially decoupled models, which limits the overall person search performance. In this paper, we propose to fully decouple person search towards optimal person search. A task-incremental person search network is proposed to incrementally construct an end-to-end model for the detection and re-id sub-task, which decouples the model architecture for the two sub-tasks. The proposed task-incremental network allows task-incremental training for the two conflicting tasks. This enables independent learning for different objectives thus fully decoupled the model for persons earch. Comprehensive experimental evaluations demonstrate the effectiveness of the proposed fully decoupled models for end-to-end person search.


Title: Radical subgroups of finite reductive groups
Abstract: Radical subgroups play an important role in both group theory and representation theory. In this paper we present a strategy of classifying radical subgroups of finite reductive groups. As an application, we complete the proof of the inductive blockwise Alperin weight condition for the Chevalley groups $\F_4(q)$, contributing to the program to prove the Alperin weight conjecture by verifying its inductive condition for simple groups.


Title: Blow up of solutions for a Parabolic-Elliptic Chemotaxis System with gradient dependent chemotactic coefficient
Abstract: We consider a Parabolic-Elliptic system of PDE's with a chemotactic term in a $N$-dimensional unit ball describing the behavior of the density of a biological species "$u$" and a chemical stimulus "$v$". The system includes a nonlinear chemotactic coefficient depending of ``$\nabla v$", i.e. the chemotactic term is given in the form $$- div (\chi u |\nabla v|^{p-2} \nabla v), \qquad \mbox{ for } \ p \in ( \frac{N}{N-1},2), \qquad N >2 $$ for a positive constant $\chi$ when $v$ satisfies the poisson equation $$- \Delta v = u - \frac{1}{|\Omega|} \int_{\Omega} u_0dx.$$ We study the radially symmetric solutions under the assumption in the initial mass $$ \frac{1}{|\Omega|} \int_{\Omega} u_0dx>6.$$ For $\chi$ large enough, we present conditions in the initial data, such that any regular solution of the problem blows up at finite time.


Title: The extension problem in free harmonic analysis
Abstract: This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls of finite radius to the entire group. Previous work showed that such extensions always exist and we study the problem of simultaneous extension of multiple positive definite functions. More specifically, we define a concept of 'relative energy' which measures the proximity between a pair of positive definite functions, and show that a pair of positive definite functions on a finite ball can be extended to the entire group without increasing their relative energy. The proof is analytic, involving differentiation of noncommutative Szego parameters.


Title: Characterization of commuting graphs of finite groups having small genus
Abstract: In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only $K_8 \sqcup 9K_1$, $K_8 \sqcup 5K_2$, $K_8 \sqcup 3K_4$, $K_8 \sqcup 9K_3$, $K_8\sqcup 9(K_1 \vee 3K_2)$, $3K_6$ and $3K_6 \sqcup 4K_4 \sqcup 6K_2$ can be realized as commuting graphs of finite groups. As consequences of our results we also show that for any finite non-abelian group $G$ if the commuting graph of $G$ (denoted by $\Gamma_c(G)$) is double-toroidal or triple-toroidal then $\Gamma_c(G)$ and its complement satisfy Hansen-Vuki{\v{c}}evi{\'c} Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group $(\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8$, that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs.


Title: The Rotation of Eigenspaces of Perturbed Matrix Pairs
Abstract: We revisit the relative perturbation theory for invariant subspaces of positive definite matrix pairs. As a prototype model problem for our results we consider parameter dependent families of eigenvalue problems. We show that new estimates are a natural way to obtain sharp --- as functions of the parameter indexing the family of matrix pairs --- estimates for the rotation of spectral subspaces.


Title: Beck modules and alternative algebras
Abstract: We set out the general theory of "Beck modules" in a variety of algebras and describe them as modules over suitable "universal enveloping" unital associative algebras. We pay particular attention to the somewhat nonstandard case of "alternative algebras," defined by a restricted associative law, and determine the Poincar\'e polynomial of the universal enveloping algebra in the homogenous case.


Title: Backward propagation of warped product structures and asymptotically conical shrinkers
Abstract: We establish sufficient conditions which ensure that a locally-warped product structure propagates backward in time under the Ricci flow. As an application, we prove that if an asymptotically conical gradient shrinking soliton is asymptotic to a cone whose cross-section is a product of Einstein manifolds, the soliton must itself be a multiply-warped product over the same manifolds.


Title: Bordered and Framed Toeplitz and Hankel Determinants with Applications to Integrable Probability
Abstract: Bordered and framed Toeplitz/Hankel determinants have the same structure as Toeplitz/Hankel determinants except in small number of matrix rows and/or columns. We review these structured determinants and their connections to orthogonal polynomials, collecting well-known and perhaps less well-known results. We present some applications for these structured determinants to ensembles of non-intersecting paths and the six-vertex model, with an eye towards asymptotic analysis. We also prove some asymptotic formulae for the probability of non-intersection for an ensemble of continuous time random walks for certain choices of starting and ending points as the number of random walkers tends to infinity.


Title: Towards dense object tracking in a 2D honeybee hive
Abstract: From human crowds to cells in tissue, the detection and efficient tracking of multiple objects in dense configurations is an important and unsolved problem. In the past, limitations of image analysis have restricted studies of dense groups to tracking a single or subset of marked individuals, or to coarse-grained group-level dynamics, all of which yield incomplete information. Here, we combine convolutional neural networks (CNNs) with the model environment of a honeybee hive to automatically recognize all individuals in a dense group from raw image data. We create new, adapted individual labeling and use the segmentation architecture U-Net with a loss function dependent on both object identity and orientation. We additionally exploit temporal regularities of the video recording in a recurrent manner and achieve near human-level performance while reducing the network size by 94% compared to the original U-Net architecture. Given our novel application of CNNs, we generate extensive problem-specific image data in which labeled examples are produced through a custom interface with Amazon Mechanical Turk. This dataset contains over 375,000 labeled bee instances across 720 video frames at 2 FPS, representing an extensive resource for the development and testing of tracking methods. We correctly detect 96% of individuals with a location error of ~7% of a typical body dimension, and orientation error of 12 degrees, approximating the variability of human raters. Our results provide an important step towards efficient image-based dense object tracking by allowing for the accurate determination of object location and orientation across time-series image data efficiently within one network architecture.


Title: Convergence of boundary layers of chemotaxis models with physical boundary conditions~I: degenerate initial data
Abstract: The celebrated experiment of Tuval et al. \cite{tuval2005bacterial} showed that the bacteria living a water drop can form a thin layer near the air-water interface, where a so-called chemotaxis-fluid system with physical boundary conditions was proposed to interpret the mechanism underlying the pattern formation alongside numerical simulations. However, the rigorous proof for the existence and convergence of the boundary layer solutions to the proposed model still remains open. This paper shows that the model with physical boundary conditions proposed in \cite{tuval2005bacterial} in one dimension can generate boundary layer solution as the oxygen diffusion rate $\varepsilon>0$ is small. Specifically, we show that the solution of the model with $\varepsilon>0$ will converge to the solution with $\varepsilon=0$ (outer-layer solution) plus the boundary layer profiles (inner-layer solution) with a sharp transition near the boundary as $ \varepsilon \rightarrow 0$. There are two major difficulties in our analysis. First, the global well-posedness of the model is hard to prove since the Dirichlet boundary condition can not contribute to the gradient estimates needed for the cross-diffusion structure in the model. Resorting to the technique of taking anti-derivative, we remove the cross-diffusion structure such that the Dirichlet boundary condition can facilitate the needed estimates. Second, the outer-layer profile of bacterial density is required to be degenerate at the boundary as $ t \rightarrow 0 ^{+}$, which makes the traditional cancellation technique incapable. Here we employ the Hardy inequality and delicate weighted energy estimates to overcome this obstacle and derive the requisite uniform-in-$\varepsilon$ estimates allowing us to pass the limit $\varepsilon \to 0$ to achieve our results.