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Title: Absence of Lavrentiev's gap for anisotropic functionals
Abstract: We establish the absence of the Lavrentiev gap between Sobolev and smooth maps for a non-autonomous variational problem of a general structure, where the integrand is assumed to be controlled by a function which is convex and anisotropic with respect to the last variable. This fact results from new results on good approximation properties of the natural underlying unconventional function space. Scalar and vector-valued problems are studied. | math | 1 |
Title: Efficient Second Order Online Learning by Sketching
Abstract: We propose Sketched Online Newton (SON), an online second order learning algorithm that enjoys substantially improved regret guarantees for ill-conditioned data. SON is an enhanced version of the Online Newton Step, which, via sketching techniques enjoys a running time linear in the dimension and sketch size. We further develop sparse forms of the sketching methods (such as Oja's rule), making the computation linear in the sparsity of features. Together, the algorithm eliminates all computational obstacles in previous second order online learning approaches. | cs | 1 |
Title: New multivariate Gini's indices
Abstract: The Gini's mean difference was defined as the expected absolute difference between a random variable and its independent copy. The corresponding normalized version, namely Gini's index, denotes two times the area between the egalitarian line and the Lorenz curve. Both are dispersion indices because they quantify how far a random variable and its independent copy are. Aiming to measure dispersion in the multivariate case, we define and study new Gini's indices. For the bivariate case we provide several results and we point out that they are "dependence-dispersion" indices. Covariance representations are exhibited, with an interpretation also in terms of conditional distributions. Further results, bounds and illustrative examples are discussed too. Multivariate extensions are defined, aiming to apply both indices in more general settings. Then, we define efficiency Gini's indices for any semi-coherent system and we discuss about their interpretation. Empirical versions are considered in order as well to apply multivariate Gini's indices to data. | math | 0 |
Title: A Stochastic Alternating Direction Method of Multipliers for Non-smooth and Non-convex Optimization
Abstract: Alternating direction method of multipliers (ADMM) is a popular first-order method owing to its simplicity and efficiency. However, similar to other proximal splitting methods, the performance of ADMM degrades significantly when the scale of the optimization problems to solve becomes large. In this paper, we consider combining ADMM with a class of stochastic gradient with variance reduction for solving large-scale non-convex and non-smooth optimization problems. Global convergence of the generated sequence is established under the extra additional assumption that the object function satisfies Kurdyka-Lojasiewicz (KL) property. Numerical experiments on graph-guided fused Lasso and computed tomography are presented to demonstrate the performance of the proposed methods. | math | 1 |
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. | math | 1 |
Title: Hierarchical Aligned Multimodal Learning for NER on Tweet Posts
Abstract: Mining structured knowledge from tweets using named entity recognition (NER) can be beneficial for many down stream applications such as recommendation and intention understanding. With tweet posts tending to be multimodal, multimodal named entity recognition (MNER) has attracted more attention. In this paper, we propose a novel approach, which can dynamically align the image and text sequence and achieve the multi-level cross-modal learning to augment textual word representation for MNER improvement. To be specific, our framework can be split into three main stages: the first stage focuses on intra-modality representation learning to derive the implicit global and local knowledge of each modality, the second evaluates the relevance between the text and its accompanying image and integrates different grained visual information based on the relevance, the third enforces semantic refinement via iterative cross-modal interactions and co-attention. We conduct experiments on two open datasets, and the results and detailed analysis demonstrate the advantage of our model. | cs | 0 |
Title: Prescribed graphon symmetries and flavors of rigidity
Abstract: We prove that an arbitrary compact metrizable group can be realized as the automorphism group of a graphing; this is a continuous analogue to Frucht's theorem recovering arbitrary finite groups are automorphism groups of finite graphs. The paper also contains a number of results the persistence of transitivity of a compact-group action upon passing to a limit of graphons. Call a compact group $\mathbb{G}$ graphon-rigid if, whenever it acts transitively on each member $\Gamma_n$ of a convergent sequence of graphons, it also acts transitively on the limit $\lim_n \Gamma$. We show that for a compact Lie group $\mathbb{G}$ graphon rigidity is equivalent to the identity component $\mathbb{G}_0$ being semisimple; as a partial converse to a result of Lov\'{a}sz and Szegedy, this is also equivalent to weak randomness: the property that the group have only finitely many irreducible representations in each dimension. Similarly, call a compact group $\mathbb{G}$ image-rigid if for every compact Lie group $\mathbb{H}$ the images of morphisms $\mathbb{G}\to \mathbb{H}$ form a closed set (of closed subgroups, in the natural topology). We prove that graphon rigidity implies image rigidity for compact groups that are either connected or profinite, and the two conditions are equivalent (and also equivalent to being torsion) for profinite abelian groups. | math | 1 |
Title: Indecomposable motivic cycles on K3 surfaces of degree 2
Abstract: In this paper we construct new indecomposable motivic cycles in the group $H^3_{\mathcal M}(X,{\mathds Q}(2))$ where X is a degree 2 K3 surface. This generalizes our construction in [Sre22] for Kummer surfaces of Abelian surfaces as well as the recent work of Ma and Sato [MS23] on degree 2 K3 surfaces. | math | 0 |
Title: Poisson catenarity in Poisson nilpotent algebras
Abstract: We prove that for the iterated Poisson polynomial rings known as Poisson nilpotent algebras (or Poisson-CGL extensions), the Poisson prime spectrum is catenary, i.e., all saturated chains of inclusions of Poisson prime ideals between any two given Poisson prime ideals have the same length. | math | 1 |
Title: Limit trees for free group automorphisms: universality
Abstract: To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism's dominating exponential dynamics. As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn--Handel's recognition theorem for atoroidal outer automorphisms. | math | 1 |
Title: Analytic Linear Lie rack Structures on Leibniz Algebras
Abstract: A linear Lie rack structure on a finite dimensional vector space $V$ is a Lie rack operation $(x,y)\mapsto x\rhd y$ pointed at the origin and such that for any $x$, the left translation $\mathrm{L}_x:y\mapsto \mathrm{L}_x(y)= x\rhd y$ is linear. A linear Lie rack operation $\rhd$ is called analytic if for any $x,y\in V$, \[ x\rhd y=y+\sum_{n=1}^\infty A_{n,1}(x,\ldots,x,y), \]where $A_{n,1}:V\times\ldots\times V\Leftarrow V$ is an $n+1$-multilinear map symmetric in the $n$ first arguments. In this case, $A_{1,1}$ is exactly the left Leibniz product associated to $\rhd$. Any left Leibniz algebra $(\mathfrak{h},[\;,\;])$ has a canonical analytic linear Lie rack structure given by $x\stackrel{c}{\rhd} y=\exp(\mathrm{ad}_x)(y)$, where $\mathrm{ad}_x(y)=[x,y]$. In this paper, we show that a sequence $(A_{n,1})_{n\geq1}$ of $n+1$-multilinear maps on a vector space $V$ defines an analytic linear Lie rack structure if and only if $[\;,\;]:=A_{1,1}$ is a left Leibniz bracket, the $A_{n,1}$ are invariant for $(V,[\;,\;]:)$ and satisfy a sequence of multilinear equations. Some of these equations have a cohomological interpretation and can be solved when the zero and the 1-cohomology of the left Leibniz algebra $(V,[\;,\;])$ are trivial. On the other hand, given a left Leibniz algebra $(\mathfrak{h},[\;,\;])$, we show that there is a large class of (analytic) linear Lie rack structures on $(\mathfrak{h},[\;,\;])$ which can be built from the canonical one and invariant multilinear symmetric maps on $\mathfrak{h}$. A left Leibniz algebra on which all the analytic linear Lie rack structures are build in this way will be called rigid. We use our characterizations of analytic linear Lie rack structures to show that $\mathfrak{sl}_2(\mathbb{R})$ and $\mathfrak{so}(3)$ are rigid. We conjecture that any simple Lie algebra is rigid as a left Leibniz algebra. | math | 1 |
Title: Counting double cosets with application to generic 3-manifolds
Abstract: We study the growth of double cosets in the class of groups with contracting elements, including relatively hyperbolic groups, CAT(0) groups and mapping class groups among others. Generalizing a recent work of Gitik and Rips about hyperbolic groups, we prove that the double coset growth of two Morse subgroups of infinite index is comparable with the orbital growth function. The same result is further obtained for a more general class of subgroups whose limit sets are proper subsets in the entire limit set of the ambient group. As an application, we confirm a conjecture of Maher that hyperbolic 3-manifolds are exponentially generic in the set of 3-manifolds built from Heegaard splitting using complexity in Teichm\"{u}ller metric. | math | 0 |
Title: StROL: Stabilized and Robust Online Learning from Humans
Abstract: Robots often need to learn the human's reward function online, during the current interaction. This real-time learning requires fast but approximate learning rules: when the human's behavior is noisy or suboptimal, current approximations can result in unstable robot learning. Accordingly, in this paper we seek to enhance the robustness and convergence properties of gradient descent learning rules when inferring the human's reward parameters. We model the robot's learning algorithm as a dynamical system over the human preference parameters, where the human's true (but unknown) preferences are the equilibrium point. This enables us to perform Lyapunov stability analysis to derive the conditions under which the robot's learning dynamics converge. Our proposed algorithm (StROL) uses these conditions to learn robust-by-design learning rules: given the original learning dynamics, StROL outputs a modified learning rule that now converges to the human's true parameters under a larger set of human inputs. In practice, these autonomously generated learning rules can correctly infer what the human is trying to convey, even when the human is noisy, biased, and suboptimal. Across simulations and a user study we find that StROL results in a more accurate estimate and less regret than state-of-the-art approaches for online reward learning. See videos and code here: https://github.com/VT-Collab/StROL_RAL | cs | 0 |
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. | math | 0 |
Title: Asymptotic characterizations of strong pseudoconvexity on pseudoconvex domains of finite type in $\mathbb{C}^2$
Abstract: In this paper, we provide some characterizations of strong pseudoconvexity by the boundary behavior of intrinsic invariants for smoothly bounded pseudoconvex domains of finite type in $\mathbb{C}^2$. As a consequence, if such domain is biholomorphically equivalent to a quotient of the unit ball, then it is strongly pseudoconvex. | math | 0 |
Title: Weak Solutions of SPDEs in the space of Tempered distributions
Abstract: In this article, we construct weak solutions for a class of Stochastic PDEs in the space of tempered distributions via Girsanov's theorem. It is to be noted that our drift and diffusion coefficients $(L,A)$ of the considered Stochastic PDE satisfy a Monotonicity type inequality, rather than Lipschitz conditions. As such, we can not follow the usual infinite dimensional analysis as described in \cite[sections 10.2 and 10.3]{MR3236753}. Instead, we exploit related SDEs to obtain our desired result, and we point out an important observation that the same Novikov condition is used in changing the Brownian motion in both the SDEs and the Stochastic PDEs. | math | 0 |
Title: Smoothing Methods for Automatic Differentiation Across Conditional Branches
Abstract: Programs involving discontinuities introduced by control flow constructs such as conditional branches pose challenges to mathematical optimization methods that assume a degree of smoothness in the objective function's response surface. Smooth interpretation (SI) is a form of abstract interpretation that approximates the convolution of a program's output with a Gaussian kernel, thus smoothing its output in a principled manner. Here, we combine SI with automatic differentiation (AD) to efficiently compute gradients of smoothed programs. In contrast to AD across a regular program execution, these gradients also capture the effects of alternative control flow paths. The combination of SI with AD enables the direct gradient-based parameter synthesis for branching programs, allowing for instance the calibration of simulation models or their combination with neural network models in machine learning pipelines. We detail the effects of the approximations made for tractability in SI and propose a novel Monte Carlo estimator that avoids the underlying assumptions by estimating the smoothed programs' gradients through a combination of AD and sampling. Using DiscoGrad, our tool for automatically translating simple C++ programs to a smooth differentiable form, we perform an extensive evaluation. We compare the combination of SI with AD and our Monte Carlo estimator to existing gradient-free and stochastic methods on four non-trivial and originally discontinuous problems ranging from classical simulation-based optimization to neural network-driven control. While the optimization progress with the SI-based estimator depends on the complexity of the program's control flow, our Monte Carlo estimator is competitive in all problems, exhibiting the fastest convergence by a substantial margin in our highest-dimensional problem. | cs | 0 |
Title: Shayona@SMM4H23: COVID-19 Self diagnosis classification using BERT and LightGBM models
Abstract: This paper describes approaches and results for shared Task 1 and 4 of SMMH4-23 by Team Shayona. Shared Task-1 was binary classification of english tweets self-reporting a COVID-19 diagnosis, and Shared Task-4 was Binary classification of English Reddit posts self-reporting a social anxiety disorder diagnosis. Our team has achieved the highest f1-score 0.94 in Task-1 among all participants. We have leveraged the Transformer model (BERT) in combination with the LightGBM model for both tasks. | cs | 0 |
Title: Perfect precise colorings of plane semiregular tilings
Abstract: A coloring of a planar semiregular tiling $\mathcal{T}$ is an assignment of a unique color to each tile of $\mathcal{T}$. If $G$ is the symmetry group of $\mathcal{T}$, we say that the coloring is perfect if every element of $G$ induces a permutation on the finite set of colors. If $\mathcal{T}$ is $k$-valent, then a coloring of $\mathcal{T}$ with $k$ colors is said to be precise if no two tiles of $\mathcal{T}$ sharing the same vertex have the same color. In this work, we obtain perfect precise colorings of some families of $k$-valent semiregular tilings in the plane, where $k\leq 6$. | math | 0 |
Title: Pointwise estimates for rough operators with applications to Sobolev inequalities
Abstract: We investigate Sobolev inequalities for several rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev-type inequalities with an operator on the left-hand side. | math | 0 |
Title: Minimal surfaces with symmetries
Abstract: Let $G$ be a finite group acting on a connected open Riemann surface $X$ by holomorphic automorphisms and acting on a Euclidean space $\mathbb R^n$ $(n\ge 3)$ by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a $G$-equivariant conformal minimal immersion $F:X\to\mathbb R^n$. We show in particular that such a map $F$ always exists if $G$ acts without fixed points on $X$. Furthermore, every finite group $G$ arises in this way for some open Riemann surface $X$ and $n=2|G|$. We obtain an analogous result for minimal surfaces having complete ends with finite total Gaussian curvature, and for discrete infinite groups acting on $X$ properly discontinuously and acting on $\mathbb R^n$ by rigid transformations. | math | 0 |
Title: The impact of the limit $q$-Durrmeyer operator on continuous functions
Abstract: The limit $q$-Durrmeyer operator, $D_{\infty,q},$ was introduced and its approximation properties were investigated by V. Gupta in 2008 during a study of $q$-analogues for the Bernstein-Durrmeyer operator. In the present work, this operator is investigated from a different perspective. More precisely, the growth estimates are derived for the entire functions comprising the range of $D_{\infty,q}$. The interrelation between the analytic properties of a function $f$ and the rate of growth for $D_{\infty,q}f$ are established, and the sharpness of the obtained results are demonstrated. | math | 0 |
Title: On the modulus of continuity of solutions to nonlocal parabolic equations
Abstract: A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. H\"older modulus of continuity is then deduced under a slightly stronger tail condition. These regularity estimates are demonstrated under the framework of nonlocal $p$-Laplacian with measurable kernels. | math | 0 |
Title: Raphtory: The temporal graph engine for Rust and Python
Abstract: Raphtory is a platform for building and analysing temporal networks. The library includes methods for creating networks from a variety of data sources; algorithms to explore their structure and evolution; and an extensible GraphQL server for deployment of applications built on top. Raphtory's core engine is built in Rust, for efficiency, with Python interfaces, for ease of use. Raphtory is developed by network scientists, with a background in Physics, Applied Mathematics, Engineering and Computer Science, for use across academia and industry. | cs | 0 |
Title: Answers to Two Questions on the DP Color Function
Abstract: DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvo\v{r}\'{a}k and Postle. The chromatic polynomial of a graph is a notion that has been extensively studied since the early 20th century. The chromatic polynomial of graph $G$ is denoted $P(G,m)$, and it is equal to the number of proper $m$-colorings of $G$. In 2019, Kaul and Mudrock introduced an analogue of the chromatic polynomial for DP-coloring; specifically, the DP color function of graph $G$ is denoted $P_{DP}(G,m)$. Two fundamental questions posed by Kaul and Mudrock are: (1) For any graph $G$ with $n$ vertices, is it the case that $P(G,m)-P_{DP}(G,m) = O(m^{n-3})$ as $m \rightarrow \infty$? and (2) For every graph $G$, does there exist $p,N \in \mathbb{N}$ such that $P_{DP}(K_p \vee G, m) = P(K_p \vee G, m)$ whenever $m \geq N$? We show that the answer to both these questions is yes. In fact, we show the answer to (2) is yes even if we require $p=1$. | math | 1 |
Title: Provably Powerful Graph Neural Networks for Directed Multigraphs
Abstract: This paper analyses a set of simple adaptations that transform standard message-passing Graph Neural Networks (GNN) into provably powerful directed multigraph neural networks. The adaptations include multigraph port numbering, ego IDs, and reverse message passing. We prove that the combination of these theoretically enables the detection of any directed subgraph pattern. To validate the effectiveness of our proposed adaptations in practice, we conduct experiments on synthetic subgraph detection tasks, which demonstrate outstanding performance with almost perfect results. Moreover, we apply our proposed adaptations to two financial crime analysis tasks. We observe dramatic improvements in detecting money laundering transactions, improving the minority-class F1 score of a standard message-passing GNN by up to 30%, and closely matching or outperforming tree-based and GNN baselines. Similarly impressive results are observed on a real-world phishing detection dataset, boosting three standard GNNs' F1 scores by around 15% and outperforming all baselines. | cs | 0 |
Title: Self-Stabilizing Indulgent Zero-degrading Binary Consensus
Abstract: Guerraoui proposed an indulgent solution for the binary consensus problem. Namely, he showed that an arbitrary behavior of the failure detector never violates safety requirements even if it compromises liveness. Consensus implementations are often used in a repeated manner. Dutta and Guerraoui proposed a zero-degrading solution, \ie during system runs in which the failure detector behaves perfectly, a node failure during one consensus instance has no impact on the performance of future instances. Our study, which focuses on indulgent zero-degrading binary consensus, aims at the design of an even more robust communication abstraction. We do so through the lenses of self-stabilization - a very strong notion of fault-tolerance. In addition to node and communication failures, self-stabilizing algorithms can recover after the occurrence of arbitrary transient faults; these faults represent any violation of the assumptions according to which the system was designed to operate (as long as the algorithm code stays intact). This work proposes the first, to the best of our knowledge, self-stabilizing algorithm for indulgent zero-degrading binary consensus for time-free message-passing systems prone to detectable process failures. The proposed algorithm has an O(1) stabilization time (in terms of asynchronous cycles) from arbitrary transient faults. Since the proposed solution uses an {\Omega} failure detector, we also present the first, to the best of our knowledge, self-stabilizing asynchronous {\Omega} failure detector, which is a variation on the one by Most\'efaoui, Mourgaya, and Raynal. | cs | 1 |
Title: Near Optimal Bounds for Collision in Pollard Rho for Discrete Log
Abstract: We analyze a fairly standard idealization of Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G. It is found that, with high probability, a collision occurs in $O(\sqrt{|G|\log |G| \log \log |G|})$ steps, not far from the widely conjectured value of $\Theta(\sqrt{|G|})$. This improves upon a recent result of Miller--Venkatesan which showed an upper bound of $O(\sqrt{|G|}\log^3 |G|)$. Our proof is based on analyzing an appropriate nonreversible, non-lazy random walk on a discrete cycle of (odd) length |G|, and showing that the mixing time of the corresponding walk is $O(\log |G| \log \log |G|)$. | math | 1 |
Title: A Bregman Proximal Stochastic Gradient Method with Extrapolation for Nonconvex Nonsmooth Problems
Abstract: In this paper, we explore a specific optimization problem that involves the combination of a differentiable nonconvex function and a nondifferentiable function. The differentiable component lacks a global Lipschitz continuous gradient, posing challenges for optimization. To address this issue and accelerate the convergence, we propose a Bregman proximal stochastic gradient method with extrapolation (BPSGE), which only requires smooth adaptivity of the differentiable part. Under the variance reduction framework, we not only analyze the subsequential and global convergence of the proposed algorithm under certain conditions, but also analyze the sublinear convergence rate of the subsequence, and the complexity of the algorithm, revealing that the BPSGE algorithm requires at most O(epsilon\^\,(-2)) iterations in expectation to attain an epsilon-stationary point. To validate the effectiveness of our proposed algorithm, we conduct numerical experiments on three real-world applications: graph regularized nonnegative matrix factorization (NMF), matrix factorization with weakly-convex regularization, and NMF with nonconvex sparsity constraints. These experiments demonstrate that BPSGE is faster than the baselines without extrapolation. | math | 0 |
Title: Estimating Categorical Counterfactuals via Deep Twin Networks
Abstract: Counterfactual inference is a powerful tool, capable of solving challenging problems in high-profile sectors. To perform counterfactual inference, one requires knowledge of the underlying causal mechanisms. However, causal mechanisms cannot be uniquely determined from observations and interventions alone. This raises the question of how to choose the causal mechanisms so that resulting counterfactual inference is trustworthy in a given domain. This question has been addressed in causal models with binary variables, but the case of categorical variables remains unanswered. We address this challenge by introducing for causal models with categorical variables the notion of counterfactual ordering, a principle that posits desirable properties causal mechanisms should posses, and prove that it is equivalent to specific functional constraints on the causal mechanisms. To learn causal mechanisms satisfying these constraints, and perform counterfactual inference with them, we introduce deep twin networks. These are deep neural networks that, when trained, are capable of twin network counterfactual inference -- an alternative to the abduction, action, & prediction method. We empirically test our approach on diverse real-world and semi-synthetic data from medicine, epidemiology, and finance, reporting accurate estimation of counterfactual probabilities while demonstrating the issues that arise with counterfactual reasoning when counterfactual ordering is not enforced. | cs | 1 |
Title: On the real zeros of depth 1 quasimodular forms
Abstract: We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth $1$ for $\mathrm{PSL}_2(\mathbb Z)$. In particular, we consider the derivatives of the unique weight $k$ modular forms $f_k$ with the maximal number of consecutive zero Fourier coefficients following the constant $1$. Our main results state that (1) every zero of a depth $1$ quasimodular form near the derivative of the Eisenstein series in the standard fundamental domain lies on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$, and (2) more than half of zeros of $f_k$ in the standard fundamental domain lie on the geodesic segment $\{z \in \mathbb H: \Re(z)=1/2\}$ for large enough $k$ with $k\equiv 0 \pmod{12}$. | math | 0 |
Title: A new metric on the contactomorphism group of orderable contact manifolds
Abstract: We introduce a pseudo-metric on the contactomorphism group of any contact manifold $(M,\xi)$ with a cooriented contact structure $\xi$. It is the contact analogue of a corresponding semi-norm in Hofer's geometry, and on certain classes of contact manifolds, its lift to the universal cover can be viewed as a continuous version of the integer valued bi-invariant metric introduced by Fraser, Polterovich, and Rosen. We show that it is non-degenerate if and only if $(M,\xi)$ is strongly orderable and that its metric topology agrees with the interval topology introduced by Chernov and Nemirovski. In particular, the interval topology is Hausdorff whenever it is non-trivial, which answers a question of Chernov and Nemirovski. We discuss analogous results for isotopy classes of Legendrians and universal covers. | math | 0 |
Title: Searching for (sharp) thresholds in random structures: where are we now?
Abstract: We survey the current state of affairs in the study of thresholds and sharp thresholds in random structures on the occasion of the recent proof of the Kahn--Kalai Conjecture by Park and Pham and the fairly recent proof of the satisfiability conjecture for large $k$ by Ding, Sly, and Sun. Random discrete structures appear as fundamental objects of study in many scientific and mathematical fields including statistical physics, combinatorics, algorithms and complexity, social choice theory, coding theory, and statistics. While the models and properties of interest in these fields vary widely, much progress has been made through the development of general tools applicable to large families of models and properties all at once. Historically these tools originated to solve or make progress on specific, difficult conjectures in the areas mentioned above. We will survey recent progress on some of these hard problems and describe some challenges for the future. | math | 0 |
Title: On Zeros of q-Entire Functions
Abstract: In this work we first give a upper bound for the modulus of q-transcendental entire functions, then prove certain sums associated with their zeros are convergent, and derive the asymptotic behaviors of their associated heat kernels. | math | 0 |
Title: A Multi-Criteria Metaheuristic Algorithm for Distributed Optimization of Electric Energy Storage
Abstract: The distributed schedule optimization of energy storage constitutes a challenge. Such algorithms often expect an input set containing all feasible schedules or respectively require to efficiently search the schedule space. It is hardly possible to accomplish this with energy storage due to its high flexibility. In this paper, the problem is introduced in detail and addressed by a metaheuristic algorithm, which generates a preselection of schedules. Three contributions are presented to achieve this goal: First, an extension for a distributed schedule optimization allowing a simultaneous optimization is developed. Second, an evolutionary algorithm is designed to generate optimized schedules. Third, the algorithm is extended to include an arbitrary local criterion. It is shown that the presented approach is suitable to schedule electric energy storage in real households and industries with different generator and storage types. | cs | 1 |
Title: Re-evaluating the Memory-balanced Pipeline Parallelism: BPipe
Abstract: Pipeline parallelism is an essential technique in the training of large-scale Transformer models. However, it suffers from imbalanced memory consumption, leading to insufficient memory utilization. The BPipe technique was proposed to address this issue and has proven effective in the GPT-3 model. Nevertheless, our experiments have not yielded similar benefits for LLaMA training. Additionally, BPipe only yields negligible benefits for GPT-3 training when applying flash attention. We analyze the underlying causes of the divergent performance of BPipe on GPT-3 and LLaMA. Furthermore, we introduce a novel method to estimate the performance of BPipe. | cs | 0 |
Title: Unsupervised learning from videos using temporal coherency deep networks
Abstract: In this work we address the challenging problem of unsupervised learning from videos. Existing methods utilize the spatio-temporal continuity in contiguous video frames as regularization for the learning process. Typically, this temporal coherence of close frames is used as a free form of annotation, encouraging the learned representations to exhibit small differences between these frames. But this type of approach fails to capture the dissimilarity between videos with different content, hence learning less discriminative features. We here propose two Siamese architectures for Convolutional Neural Networks, and their corresponding novel loss functions, to learn from unlabeled videos, which jointly exploit the local temporal coherence between contiguous frames, and a global discriminative margin used to separate representations of different videos. An extensive experimental evaluation is presented, where we validate the proposed models on various tasks. First, we show how the learned features can be used to discover actions and scenes in video collections. Second, we show the benefits of such an unsupervised learning from just unlabeled videos, which can be directly used as a prior for the supervised recognition tasks of actions and objects in images, where our results further show that our features can even surpass a traditional and heavily supervised pre-training plus fine-tunning strategy. | cs | 1 |
Title: Discriminants of stable rank two sheaves on some general type surfaces
Abstract: We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. The key technical ingredient is to study them as torsion sheaves in projective space via tilt stability in the derived category. We then proceed to describe the surface itself as a moduli space of rank two vector bundles on it. Lastly, we give a proof of the Bogomolov inequality for semistable rank two sheaves on integral surfaces in three-dimensional projective space in all characteristics. | math | 1 |
Title: Infinite-Time Singularities of the Lagrangian Mean Curvature Flow
Abstract: In this paper, we exhibit examples of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as $t\to\infty$. This result shows that infinite-time singularities can form in the Thomas--Yau `semi-stable' situation. Our work is a parabolic analogue of the results of Dominic Joyce and Yng-Ing Lee regarding desingularisation of special Lagrangians with conical singularities. The gluing construction that we employ is inspired by the work of Simon Brendle and Nikolaos Kapouleas regarding ancient solutions of the Ricci flow. | math | 0 |
Title: Sparsity and spectral properties of dual frames
Abstract: We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than $n^2$ non-vanishing entries, where $n$ denotes the ambient dimension, and that for most frames no sparser dual is possible. Moreover, we derive an expression for the exact sparsity level of the sparsest dual for any given finite frame using a generalized notion of spark. We then study the spectral properties of dual frames in terms of singular values of the synthesis operator. We provide a complete characterization for which spectral patterns of dual frames are possible for a fixed frame. For many cases, we provide simple explicit constructions for dual frames with a given spectrum, in particular, if the constraint on the dual is that it be tight. | math | 1 |
Title: Uncertainty in Minimum Cost Multicuts for Image and Motion Segmentation
Abstract: The minimum cost lifted multicut approach has proven practically good performance in a wide range of applications such as image decomposition, mesh segmentation, multiple object tracking, and motion segmentation. It addresses such problems in a graph-based model, where real-valued costs are assigned to the edges between entities such that the minimum cut decomposes the graph into an optimal number of segments. Driven by a probabilistic formulation of minimum cost multicuts, we provide a measure for the uncertainties of the decisions made during the optimization. We argue that access to such uncertainties is crucial for many practical applications and conduct an evaluation by means of sparsifications on three different, widely used datasets in the context of image decomposition (BSDS-500) and motion segmentation (DAVIS2016 and FBMS59) in terms of variation of information (VI) and Rand index (RI). | cs | 1 |
Title: Entropy-based Discovery of Summary Causal Graphs in Time Series
Abstract: This study addresses the problem of learning a summary causal graph on time series with potentially different sampling rates. To do so, we first propose a new causal temporal mutual information measure for time series. We then show how this measure relates to an entropy reduction principle that can be seen as a special case of the probability raising principle. We finally combine these two ingredients in PC-like and FCI-like algorithms to construct the summary causal graph. There algorithm are evaluated on several datasets, which shows both their efficacy and efficiency. | cs | 1 |
Title: Counting symmetric and non-symmetric peaks in a set partition
Abstract: The aim of this paper is to derive explicit formulas for two distinct values. The first is the total number of symmetric peaks in a set partition of $[n]$ with exactly $k$ blocks, and the second one is the total number of non-symmetric peaks in a set partition of $[n]$ with exactly $k$ blocks. We represent these results in two ways. First by using the theory of generating functions, and the second by using combinatorial tools. | math | 0 |
Title: A Bayesian Model for Discovering Typological Implications
Abstract: A standard form of analysis for linguistic typology is the universal implication. These implications state facts about the range of extant languages, such as ``if objects come after verbs, then adjectives come after nouns.'' Such implications are typically discovered by painstaking hand analysis over a small sample of languages. We propose a computational model for assisting at this process. Our model is able to discover both well-known implications as well as some novel implications that deserve further study. Moreover, through a careful application of hierarchical analysis, we are able to cope with the well-known sampling problem: languages are not independent. | cs | 1 |
Title: On Dyck Path Expansion Formulas for Rank 2 Cluster Variables
Abstract: In this paper, we simplify and generalize formulas for the expansion of rank 2 cluster variables. In particular, we prove an equivalent, but simpler, description of the colored Dyck subpaths framework introduced by Lee and Schiffler. We then prove the conjectured bijectivity of a map constructed by Feiyang Lin between collections of colored Dyck subpaths and compatible pairs, objects introduced by Lee, Li, and Zelevinsky to study the greedy basis. We use this bijection along with Rupel's expansion formula for quantum greedy basis elements, which sums over compatible pairs, to provide a quantum generalization of Lee and Schiffler's colored Dyck subpaths formula. | math | 1 |
Title: Towards understanding the Pierce-Birkhoff conjecture via MV-algebras
Abstract: Our main issue was to understand the connection between \L ukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of \textit{f}MV-algebras, which are MV-algebras endowed with both an internal binary product and a scalar product with scalars from $[0,1]$. The proper quasi-variety generated by $[0,1]$, with both products interpreted as the real product, provides the desired framework: the normal form theorem of its corresponding logical system can be seen as a local version of the Pierce-Birkhoff conjecture. | math | 1 |
Title: Allison-Benkart-Gao functor and the free non-unital alternative algebras
Abstract: Let $k$ be a field of characteristic $0$. We introduce a pair of adjoint functors, Allison-Benkart-Gao functor $\mathcal{ABG}$ and Berman-Moody functor $\mathcal{BM}$, between the category of non-unital alternative algebras over $k$ and the category ${\text{\bf Lie}_{\text{R}}}$ of Lie algebras with appropriate $sl_3(k)$-module structures. Surprisingly, when $A$ is a non-unital alternative algebra, the Allison-Gao Lie algebra $\mathcal{ABG}(A)$ is different from the more well-known Steinberg Lie algebra $st_3(A)$. Next, let $A(D)$ be the free (non-unit) alternative algebra generated by $D$ elements and $\text{Inner} A(D)$ the inner derivation algebra of $A(D)$. A conjecture on the homology of $H_r(\mathcal{ABG}(A(D)))$ is proposed. Let $A(D)_n$(resp. $\text{Inner} A(D)_n$) be the degree $n$ component of $A(D)_n$(resp. $\text{Inner} A(D)_n$). The previous conjecture implies another conjecture on the dimensions on $A(D)_n$ and $\text{Inner} A(D)_n$. We also give some evidences to support the these conjectures. | math | 0 |
Title: A characterisation of elementary fibrations
Abstract: Grothendieck fibrations provide a unifying algebraic framework that underlies the treatment of various form of logics, such as first order logic, higher order logics and dependent type theories. In the categorical approach to logic proposed by Lawvere, which systematically uses adjoints to describe the logical operations, equality is presented in the form of a left adjoint to reindexing along a diagonal arrows in the base. Taking advantage of the modular perspective provided by category theory, one can look at those Grothendieck fibrations which sustain just the structure of equality, the so-called elementary fibrations, aka fibrations with equality. The present paper provides a characterisation of elementary fibrations based on particular structures in the fibres, called transporters. The characterisation is a substantial generalisation of the one already available for faithful fibrations. There is a close resemblance between transporters and the structures used in the semantics of the identity type of Martin-L\"of type theory. We close the paper by comparing the two. | math | 1 |
Title: Stability over a predicate and prime closure
Abstract: We prove that in a theory $T$ stable over a predicate $P$, for any $\lambda > |T|$, there is a $\lambda$-prime model over any complete set A with a $\lambda$-saturated $P$-part. | math | 0 |
Title: An Effective Baseline for Robustness to Distributional Shift
Abstract: Refraining from confidently predicting when faced with categories of inputs different from those seen during training is an important requirement for the safe deployment of deep learning systems. While simple to state, this has been a particularly challenging problem in deep learning, where models often end up making overconfident predictions in such situations. In this work we present a simple, but highly effective approach to deal with out-of-distribution detection that uses the principle of abstention: when encountering a sample from an unseen class, the desired behavior is to abstain from predicting. Our approach uses a network with an extra abstention class and is trained on a dataset that is augmented with an uncurated set that consists of a large number of out-of-distribution (OoD) samples that are assigned the label of the abstention class; the model is then trained to learn an effective discriminator between in and out-of-distribution samples. We compare this relatively simple approach against a wide variety of more complex methods that have been proposed both for out-of-distribution detection as well as uncertainty modeling in deep learning, and empirically demonstrate its effectiveness on a wide variety of of benchmarks and deep architectures for image recognition and text classification, often outperforming existing approaches by significant margins. Given the simplicity and effectiveness of this method, we propose that this approach be used as a new additional baseline for future work in this domain. | cs | 1 |
Title: Big pictures of motivic and classical homotopy theories
Abstract: Motivic homotopy theory is meant to play the role of algebraic topology, in particular homotopy theory, in the context of algebraic geometry. As proved by Oliver Rondigs and Paul Arne Ostvaer, this theory is closely connected to Voevodsky's triangulated category of motives. A connection that is the motivic analogue of the connection between algebraic topology and homological algebra. In this paper, we try to understand the big picture of motivic homotopy theory and its connection to Voevodsky's motives by comparison to the classical counterpart. | math | 0 |
Title: An almost linear time algorithm testing whether the Markoff graph modulo $p$ is connected
Abstract: The Markoff graph modulo $p$ is known to be connected for all but finitely many primes $p$ (see Eddy, Fuchs, Litman, Martin, Tripeny, and Vanyo [arxiv:2308.07579]), and it is conjectured that these graphs are connected for all primes. In this paper, we provide an algorithmic realization of the process introduced by Bourgain, Gamburd, and Sarnak [arxiv:1607.01530] to test whether a Markoff graph modulo $p$ is connected for arbitrary primes. Our algorithm runs in $o(p^{1 + \epsilon})$ time for every $\epsilon > 0$. We demonstrate this algorithm by confirming that the Markoff graph modulo $p$ is connected for all primes less than one million. | math | 0 |
Title: Sorting by Reversals and the Theory of 4-Regular Graphs
Abstract: We show that the theory of sorting by reversals fits into the well-established theory of circuit partitions of 4-regular multigraphs (which also involves the combinatorial structures of circle graphs and delta-matroids). In this way, we expose strong connections between the two theories that have not been fully appreciated before. We also discuss a generalization of sorting by reversals involving the double-cut-and-join (DCJ) operation. Finally, we also show that the theory of sorting by reversals is closely related to that of gene assembly in ciliates. | cs | 1 |
Title: Harmonic curvature in dimension four
Abstract: We provide a step towards classifying Riemannian four-manifolds in which the curvature tensor has zero divergence, or -- equivalently -- the Ricci tensor Ric satisfies the Codazzi equation. Every known compact manifold of this type belongs to one of five otherwise-familiar classes of examples. The main result consists in showing that, if such a manifold (not necessarily compact or even complete) lies outside of the five classes -- a non-vacuous assumption -- then, at all points of a dense open subset, Ric has four distinct eigenvalues, while suitable local coordinates simultaneously diagonalize Ric, the metric and, in a natural sense, also the curvature tensor. Furthermore, in a local orthonormal frame formed by Ricci eigenvectors, the connection form (or, curvature tensor) has just twelve (or, respectively, six) possibly-nonzero components, which together satisfy a specific system, not depending on the point, of homogeneous polynomial equations. A part of the classification problem is thus reduced to a question in real algebraic geometry. | math | 0 |
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. | cs | 1 |
Title: The Hochschild Cohomology of Uniform Roe Algebras
Abstract: In Rufus Willett's and the authors paper "Bounded Derivations on Uniform Roe Algebras" we showed that all bounded derivations on a uniform Roe algebra $C^*_u(X)$ associated to a bounded geometry metric space $X$ are inner. This naturally leads to the question of whether or not the higher dimensional Hochschild cohomology groups of the uniform Roe algebra vanish also. While we cannot answer this question completely, we are able to give necessary and sufficient conditions for the vanishing of $H^n_c(C_u^*(X),C_u^*(X))$. Lastly, we show that if the norm continuous Hochschild cohomology of a uniform Roe algebra vanishes in all dimensions then the ultraweak-weak* continuous Hochschild cohomology of that uniform Roe algebra vanishes also. | math | 1 |
Title: Isometric Dilations for Representations of Product Systems
Abstract: We discuss representations of product systems (of $W^*$-correspondences) over the semigroup $\mathbb{Z}^n_+$ and show that, under certain pureness and Szego positivity conditions, a completely contractive representation can be dilated to an isometric representation. For $n=1,2$ this is known to hold in general (without assuming the conditions) but, for $n\geq 3$, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria and Sarkar. Our dilation is explicitly constructed and we present some applications. | math | 0 |
Title: The Knothe-Rosenblatt distance and its induced topology
Abstract: A basic and natural coupling between two probabilities on $\mathbb R^N$ is given by the Knothe-Rosenblatt coupling. It represents a multiperiod extension of the quantile coupling and is simple to calculate numerically. We consider the distance on $\mathcal P (\mathbb R^N)$ that is induced by considering the transport costs associated to the Knothe-Rosenblatt coupling. We show that this Knothe-Rosenblatt distance metrizes the adapted weak topology which is a stochastic process version of the usual weak topology and plays an important role, e.g. concerning questions on stability of stochastic control and probabilistic operations. We also establish that the Knothe-Rosenblatt distance is a geodesic distance, give a Skorokhod representation theorem for the adapted weak topology, and provide multi-dimensional versions of our results. | math | 0 |
Title: Green functions for GJMS operators on spheres, Gegenbauer polynomials and rigidity theorems
Abstract: We derive explicit representation formulae of Green functions for GJMS operators on $n$-spheres, including the fractional ones. These formulae not only have natural geometric interpretations concerning the extrinsic geometry of the round sphere, but also reflect the spherical rigidity among closed embedded hypersurfaces in $\mathbb{R}^{n+1}$. | math | 0 |
Title: Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity
Abstract: In this paper, we establish the asymptotic expansion at infinity of gradient graph in dimension 2 with vanishing mean curvature at infinity. This corresponds to our previous results in higher dimensions and generalizes the results for minimal gradient graph on exterior domain in dimension 2. Different from the strategies for higher dimensions, instead of the equivalence of Green's function on unbounded domains, we apply a version of iteration methods from Bao--Li--Zhang [Calc.Var PDE, 52(2015), pp. 39-63] that is refined by spherical harmonic expansions to provide a more explicit asymptotic behavior than known results. | math | 1 |
Title: Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian
Abstract: For $N\geq 5$ and $0<\mu<N-4$, we first show a non-degenerate result of the extremal for the following Rellich-Sobolev type inequality \begin{align*} & \int_{\mathbb{R}^N}|\Delta u|^2 \mathrm{d}x -C_{\mu,1}\int_{\mathbb{R}^N}\frac{|\nabla u|^2}{|x|^2} \mathrm{d}x +C_{\mu,2}\int_{\mathbb{R}^N}\frac{u^2}{|x|^4} \mathrm{d}x \geq \mathcal{S}_\mu\left(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-4}} \mathrm{d}x\right)^\frac{N-4}{N},\quad u\in C^\infty_0(\mathbb{R}^N), \end{align*} where $C_{\mu,1}$, $C_{\mu,1}$ and $\mathcal{S}_\mu$ are constants depending on $\mu$, furthermore equality only holds for some radial functions, which is a key ingredient in analyzing the blow-up phenomena of solutions to various elliptic equations on bounded or unbounded domain. Moreover, by using spectral estimate combined with a compactness argument, we give the remainder term of above inequality. | math | 0 |
Title: Jacobi-Jordan conformal algebras: Basics, Constructions and related structures
Abstract: The main purpose of this paper is to introduce and investigate the notion of Jacobi-Jordan conformal algebra. They are a generalization of Jacobi-Jordan algebras which correspond to the case in which the formal parameter lambda equals 0. We consider some related structures such as conformal modules, corresponding representations and O-operators. Therefore, conformal derivations from Jacobi-Jordan conformal algebras to their conformal modules are used to describe conformal derivations of Jacobi-Jordan conformal algebras of semidirect product type. Moreover, we study a class of Jacobi-Jordan conformal algebras called quadratic Jacobi-Jordan conformal algebras, which are characterized by mock-Gel'fand Dorfman bialgebras. Finally, the C[delta]-split extending structures problem for Jacobi-Jordan conformal algebras is studied. Furthermore, we introduce an unified product of a given Jacobi-Jordan conformal algebra $J$ and a given C[delta]-module K. This product includes some other interesting products of Jacobi-Jordan conformal algebras such as twisted product or crossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the C[delta]-split extending structures problem. | math | 0 |
Title: Uncertainty-Aware Deep Attention Recurrent Neural Network for Heterogeneous Time Series Imputation
Abstract: Missingness is ubiquitous in multivariate time series and poses an obstacle to reliable downstream analysis. Although recurrent network imputation achieved the SOTA, existing models do not scale to deep architectures that can potentially alleviate issues arising in complex data. Moreover, imputation carries the risk of biased estimations of the ground truth. Yet, confidence in the imputed values is always unmeasured or computed post hoc from model output. We propose DEep Attention Recurrent Imputation (DEARI), which jointly estimates missing values and their associated uncertainty in heterogeneous multivariate time series. By jointly representing feature-wise correlations and temporal dynamics, we adopt a self attention mechanism, along with an effective residual component, to achieve a deep recurrent neural network with good imputation performance and stable convergence. We also leverage self-supervised metric learning to boost performance by optimizing sample similarity. Finally, we transform DEARI into a Bayesian neural network through a novel Bayesian marginalization strategy to produce stochastic DEARI, which outperforms its deterministic equivalent. Experiments show that DEARI surpasses the SOTA in diverse imputation tasks using real-world datasets, namely air quality control, healthcare and traffic. | cs | 0 |
Title: LLM4TS: Aligning Pre-Trained LLMs as Data-Efficient Time-Series Forecasters
Abstract: Multivariate time-series forecasting is vital in various domains, e.g., economic planning and weather prediction. Deep train-from-scratch models have exhibited effective performance yet require large amounts of data, which limits real-world applicability. Recently, researchers have explored pre-trained Large Language Models (LLMs) for limited non-linguistic datasets. However, incorporating LLMs with time-series data presents challenges of limited adaptation due to different compositions between time-series and linguistic data, and the inability to process multi-scale temporal information. To tackle these challenges, we propose LLM4TS, a framework for time-series forecasting with pre-trained LLMs. LLM4TS consists of a two-stage fine-tuning strategy: the time-series alignment stage to align LLMs with the nuances of time-series data, and the forecasting fine-tuning stage, which is specifically designed for time-series forecasting tasks. Furthermore, our framework features a novel two-level aggregation method that integrates multi-scale temporal data within pre-trained LLMs, enhancing their ability to interpret time-specific information. In experiments across 7 time-series forecasting datasets, LLM4TS is superior to existing state-of-the-art methods, including those trained from scratch, in full-shot scenarios, and also achieves an average improvement of 6.84% in MSE in few-shot scenarios. In addition, evaluations compared with different self-supervised learning approaches highlight LLM4TS's effectiveness with representation learning in forecasting scenarios. | cs | 0 |
Title: An optimization-based reformulation of the classical displacement approach for state update of non-linear material models
Abstract: In this paper, we build on recent work using a mathematical programming approach for incremental state update in analysis of non-linear mechanics models. In particular, we consider quasi-static analysis of continuum problems in the linearized kinematics regime, with non-linear material models described using convex energy functions. We find in this case that the classical displacement-based nested approach for incremental state update can be reformulated as solving a reduced dual optimization problem. This reformulation provides insights into the working of the algorithm, and eliminates the need for some heuristics. An important purpose of this paper is to further illustrate the unifying nature of the mathematical programming approach. We therefore present relationships with several of these types of algorithms recently presented in the literature for incremental state update. | math | 1 |
Title: A class of finite $p$-groups and the normalized unit groups of group algebras
Abstract: Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in $\mathbb{F}_pG$. Suppose that $G$ is a finite $p$-group given by a central extension of the form $$1\longrightarrow \mathbb{Z}_{p^n}\times \mathbb{Z}_{p^m} \longrightarrow G \longrightarrow \mathbb{Z}_p\times \cdots\times \mathbb{Z}_p \longrightarrow 1$$ and $G'\cong \mathbb{Z}_p$, $n, m\geq 1$ and $p$ is odd. In this paper, the structure of $G$ is determined. And the relations of $V(\mathbb{F}_pG)^{p^l}$ and $G^{p^l}$, $\Omega_l(V(\mathbb{F}_pG))$ and $\Omega_l(G)$ are given. Furthermore, there is a direct proof for $V(\mathbb{F}_pG)^p\bigcap G=G^p$. | math | 0 |
Title: Perceptual Musical Features for Interpretable Audio Tagging
Abstract: In the age of music streaming platforms, the task of automatically tagging music audio has garnered significant attention, driving researchers to devise methods aimed at enhancing performance metrics on standard datasets. Most recent approaches rely on deep neural networks, which, despite their impressive performance, possess opacity, making it challenging to elucidate their output for a given input. While the issue of interpretability has been emphasized in other fields like medicine, it has not received attention in music-related tasks. In this study, we explored the relevance of interpretability in the context of automatic music tagging. We constructed a workflow that incorporates three different information extraction techniques: a) leveraging symbolic knowledge, b) utilizing auxiliary deep neural networks, and c) employing signal processing to extract perceptual features from audio files. These features were subsequently used to train an interpretable machine-learning model for tag prediction. We conducted experiments on two datasets, namely the MTG-Jamendo dataset and the GTZAN dataset. Our method surpassed the performance of baseline models in both tasks and, in certain instances, demonstrated competitiveness with the current state-of-the-art. We conclude that there are use cases where the deterioration in performance is outweighed by the value of interpretability. | cs | 0 |
Title: On the hardness of learning under symmetries
Abstract: We study the problem of learning equivariant neural networks via gradient descent. The incorporation of known symmetries ("equivariance") into neural nets has empirically improved the performance of learning pipelines, in domains ranging from biology to computer vision. However, a rich yet separate line of learning theoretic research has demonstrated that actually learning shallow, fully-connected (i.e. non-symmetric) networks has exponential complexity in the correlational statistical query (CSQ) model, a framework encompassing gradient descent. In this work, we ask: are known problem symmetries sufficient to alleviate the fundamental hardness of learning neural nets with gradient descent? We answer this question in the negative. In particular, we give lower bounds for shallow graph neural networks, convolutional networks, invariant polynomials, and frame-averaged networks for permutation subgroups, which all scale either superpolynomially or exponentially in the relevant input dimension. Therefore, in spite of the significant inductive bias imparted via symmetry, actually learning the complete classes of functions represented by equivariant neural networks via gradient descent remains hard. | cs | 0 |
Title: Topological aspects of Boolean functions
Abstract: We discuss ways in which tools from topology can be used to derive lower bounds for the circuit complexity of Boolean functions. | math | 1 |
Title: On the unicity of types in special linear groups
Abstract: Let $F$ be a non-archimedean local field. We show that any representation of a maximal compact subgroup of $\mathbf{SL}_N(F)$ which is typical for an essentially tame supercuspidal representation must be induced from a Bushnell--Kutzko maximal simple type. From this, we explicitly count and describe the conjugacy classes of such typical representations, and give an explicit description of an inertial Langlands correspondence for essentially tame irreducible $N$-dimensional projective representations of the Weil group of $F$. | math | 1 |
Title: Verifiable Computation with Massively Parallel Interactive Proofs
Abstract: As the cloud computing paradigm has gained prominence, the need for verifiable computation has grown increasingly urgent. The concept of verifiable computation enables a weak client to outsource difficult computations to a powerful, but untrusted, server. Protocols for verifiable computation aim to provide the client with a guarantee that the server performed the requested computations correctly, without requiring the client to perform the computations herself. By design, these protocols impose a minimal computational burden on the client. However, existing protocols require the server to perform a large amount of extra bookkeeping in order to enable a client to easily verify the results. Verifiable computation has thus remained a theoretical curiosity, and protocols for it have not been implemented in real cloud computing systems. Our goal is to leverage GPUs to reduce the server-side slowdown for verifiable computation. To this end, we identify abundant data parallelism in a state-of-the-art general-purpose protocol for verifiable computation, originally due to Goldwasser, Kalai, and Rothblum, and recently extended by Cormode, Mitzenmacher, and Thaler. We implement this protocol on the GPU, obtaining 40-120x server-side speedups relative to a state-of-the-art sequential implementation. For benchmark problems, our implementation reduces the slowdown of the server to factors of 100-500x relative to the original computations requested by the client. Furthermore, we reduce the already small runtime of the client by 100x. Similarly, we obtain 20-50x server-side and client-side speedups for related protocols targeted at specific streaming problems. We believe our results demonstrate the immediate practicality of using GPUs for verifiable computation, and more generally that protocols for verifiable computation have become sufficiently mature to deploy in real cloud computing systems. | cs | 1 |
Title: Hilbert Poincaré series and kernels for products of $L$-functions
Abstract: We study Hilbert Poincar\'e series associated to general seed functions and construct Cohen's kernels and double Eisenstein series as series of Hilbert Poincar\'e series. Then we calculate the Rankin-Cohen brackets of Hilbert Poincar\'e series and Hilbert modular forms and extend Zagier's kernel formula to totally real number fields. Finally, we show that the Rankin-Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant. | math | 0 |
Title: Computationally Assisted Quality Control for Public Health Data Streams
Abstract: Irregularities in public health data streams (like COVID-19 Cases) hamper data-driven decision-making for public health stakeholders. A real-time, computer-generated list of the most important, outlying data points from thousands of daily-updated public health data streams could assist an expert reviewer in identifying these irregularities. However, existing outlier detection frameworks perform poorly on this task because they do not account for the data volume or for the statistical properties of public health streams. Accordingly, we developed FlaSH (Flagging Streams in public Health), a practical outlier detection framework for public health data users that uses simple, scalable models to capture these statistical properties explicitly. In an experiment where human experts evaluate FlaSH and existing methods (including deep learning approaches), FlaSH scales to the data volume of this task, matches or exceeds these other methods in mean accuracy, and identifies the outlier points that users empirically rate as more helpful. Based on these results, FlaSH has been deployed on data streams used by public health stakeholders. | cs | 0 |
Title: An example for Kuznetsov-Shinder conjecture
Abstract: We give an example for the Kuznetsov-Shinder conjecture, with infinitely many non-isomorphic D-equivalent and L-equivalent varieties. | math | 0 |
Title: A new characterization of $E_8 (p)$ via its vanishing elements
Abstract: Let $G$ be a finite group, and $g \in G$. Then $g$ is said to be a vanishing element of $G$, if there exists an irreducible character $\chi$ of $G$ such that $\chi (g)=0$. Denote by ${\rm Vo} (G)$ the set of the orders of vanishing elements of $G$. We say a non-abelian group $G$ is V-recognizable, if any group $N$ with ${\rm Vo} (N) = {\rm Vo} (G)$ is isomorphic to $G$. In this paper, we investigate the V-recognizability of $E_8 (p)$, where $p$ is a prime number. As an application, among the 610 primes $p$ with $p<10000$ and $p \equiv 0,1,4\,(\!\!\!\mod 5)$, we obtain that the method is always valid for confirming the V-recognizability of $E_8 (p)$ for all such $p$ but $ 919,1289,1931,3911,4691,5381$ and $7589 $. | math | 0 |
Title: Representing topological full groups in Steinberg algebras and C*-algebras
Abstract: We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group. | math | 0 |
Title: LLM-SAP: Large Language Model Situational Awareness Based Planning
Abstract: This work pioneers evaluating emergent planning capabilities based on situational awareness in large language models. We contribute (i) novel benchmarks and metrics for standardized assessment; (ii) a unique dataset to spur progress; and (iii) demonstrations that prompting and multi-agent schemes significantly enhance planning performance in context-sensitive planning tasks. Positioning this within a situated agent and automated planning research, we highlight inherent reliability challenges--efficiently mapping world states to actions without environmental guidance remains open despite simulated domain advances. Although out-of-scope, limitations around validation methodology and data availability indicate exciting directions, including fine-tuning on expanded planning corpora and optimizations for triggering fast latent planning. By conclusively demonstrating current methods' promise and limitations via rigorous comparison, we catalyze investigating reliable goal-directed reasoning for situated agents. | cs | 0 |
Title: 2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry
Abstract: For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than $2\pi$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists. | math | 1 |
Title: Graph Neural Network Based Access Point Selection for Cell-Free Massive MIMO Systems
Abstract: A graph neural network (GNN) based access point (AP) selection algorithm for cell-free massive multiple-input multiple-output (MIMO) systems is proposed. Two graphs, a homogeneous graph which includes only AP nodes representing the structure of the APs in the network, and a heterogeneous graph which includes both AP nodes and user equipment (UE) nodes are constructed to represent a cell-free massive MIMO network. A GNN based on the inductive graph learning framework GraphSAGE is used to obtain the embeddings which are then used to predict the links between the nodes. The numerical results show that compared to the proximity-based AP selection algorithms, the proposed GNN based algorithm predicts the potential APs with more accuracy. Compared to the large scale fading coefficient based AP selection algorithms, the proposed algorithm does not require measured and sorted signal strengths of all the neighbouring APs. Furthermore, the proposed algorithm is scalable in terms of the number of users in the cell-free system. | cs | 1 |
Title: Novikov homology and noncommutative Alexander polynomials
Abstract: In the early 2000's Cochran and Harvey introduced non-commutative Alexander polynomials for 3-manifolds. Their degrees give strong lower bounds on the Thurston norm. In this paper we make the case that the vanishing of a certain Novikov-Sikorav homology module is the correct notion of a monic non-commutative Alexander polynomial. Furthermore we will use the opportunity to give new proofs of several statements about Novikov-Sikorav homology in the three-dimensional context. | math | 1 |
Title: Singular metrics and a conjecture by Campana and Peternell
Abstract: A conjecture by Campana and Peternell says that if a positive multiple of $K_X$ is linearly equivalent to an effective divisor $D$ plus a pseudo-effective divisor, then the Kodaira dimension of $X$ should be at least as big as the Iitaka dimension of $D$. This is a very useful generalization of the non-vanishing conjecture (which is the case $D = 0$). We use recent work about singular metrics on pluri-adjoint bundles to show that the Campana-Peternell conjecture is almost equivalent to the non-vanishing conjecture. | math | 1 |
Title: On the expected L2-discrepancy of stratified samples from parallel lines
Abstract: We study the expected $\mathcal{L}_2$-discrepancy of stratified samples generated from special equi-volume partitions of the unit square. The partitions are defined via parallel lines that are all orthogonal to the diagonal of the square. It is shown that the expected discrepancy of stratified samples derived from these partitions is a factor 2 smaller than the expected discrepancy of the same number of i.i.d uniformly distributed random points in the unit square. We conjecture that this is best possible among all partitions generated from parallel lines. | math | 0 |
Title: The Alexander module of a trigonal curve
Abstract: We describe the Alexander modules and Alexander polynomials (both over $\Q$ and over finite fields $\FF{p}$) of generalized trigonal curves. The rational case is closed completely; in the case of characteristic $p>0$, a few points remain open. | math | 1 |
Title: Deep AUC Maximization for Medical Image Classification: Challenges and Opportunities
Abstract: In this extended abstract, we will present and discuss opportunities and challenges brought about by a new deep learning method by AUC maximization (aka \underline{\bf D}eep \underline{\bf A}UC \underline{\bf M}aximization or {\bf DAM}) for medical image classification. Since AUC (aka area under ROC curve) is a standard performance measure for medical image classification, hence directly optimizing AUC could achieve a better performance for learning a deep neural network than minimizing a traditional loss function (e.g., cross-entropy loss). Recently, there emerges a trend of using deep AUC maximization for large-scale medical image classification. In this paper, we will discuss these recent results by highlighting (i) the advancements brought by stochastic non-convex optimization algorithms for DAM; (ii) the promising results on various medical image classification problems. Then, we will discuss challenges and opportunities of DAM for medical image classification from three perspectives, feature learning, large-scale optimization, and learning trustworthy AI models. | cs | 1 |
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. | math | 1 |
Title: CardiGraphormer: Unveiling the Power of Self-Supervised Learning in Revolutionizing Drug Discovery
Abstract: In the expansive realm of drug discovery, with approximately 15,000 known drugs and only around 4,200 approved, the combinatorial nature of the chemical space presents a formidable challenge. While Artificial Intelligence (AI) has emerged as a powerful ally, traditional AI frameworks face significant hurdles. This manuscript introduces CardiGraphormer, a groundbreaking approach that synergizes self-supervised learning (SSL), Graph Neural Networks (GNNs), and Cardinality Preserving Attention to revolutionize drug discovery. CardiGraphormer, a novel combination of Graphormer and Cardinality Preserving Attention, leverages SSL to learn potent molecular representations and employs GNNs to extract molecular fingerprints, enhancing predictive performance and interpretability while reducing computation time. It excels in handling complex data like molecular structures and performs tasks associated with nodes, pairs of nodes, subgraphs, or entire graph structures. CardiGraphormer's potential applications in drug discovery and drug interactions are vast, from identifying new drug targets to predicting drug-to-drug interactions and enabling novel drug discovery. This innovative approach provides an AI-enhanced methodology in drug development, utilizing SSL combined with GNNs to overcome existing limitations and pave the way for a richer exploration of the vast combinatorial chemical space in drug discovery. | cs | 0 |
Title: Introduction to twisted commutative algebras
Abstract: This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The second part introduces a related category and develops its basic properties. The third part develops some basic properties of twisted commutative algebras from the perspective of classical commutative algebra and summarizes some of the results of the authors. We have tried to keep the prerequisites to this article at a minimum. The article is aimed at graduate students interested in commutative algebra, algebraic combinatorics, or representation theory, and the interactions between these subjects. | math | 1 |
Title: How Descriptive are GMRES Convergence Bounds?
Abstract: GMRES is a popular Krylov subspace method for solving linear systems of equations involving a general non-Hermitian coefficient matrix. The conventional bounds on GMRES convergence involve polynomial approximation problems in the complex plane. Three popular approaches pose this approximation problem on the spectrum, the field of values, or pseudospectra of the coefficient matrix. We analyze and compare these bounds, illustrating with six examples the success and failure of each. When the matrix departs from normality due only to a low-dimensional invariant subspace, we discuss how these bounds can be adapted to exploit this structure. Since the Arnoldi process that underpins GMRES provides approximations to the pseudospectra, one can estimate the GMRES convergence bounds as an iteration proceeds. | math | 1 |
Title: Discovery of Causal Additive Models in the Presence of Unobserved Variables
Abstract: Causal discovery from data affected by unobserved variables is an important but difficult problem to solve. The effects that unobserved variables have on the relationships between observed variables are more complex in nonlinear cases than in linear cases. In this study, we focus on causal additive models in the presence of unobserved variables. Causal additive models exhibit structural equations that are additive in the variables and error terms. We take into account the presence of not only unobserved common causes but also unobserved intermediate variables. Our theoretical results show that, when the causal relationships are nonlinear and there are unobserved variables, it is not possible to identify all the causal relationships between observed variables through regression and independence tests. However, our theoretical results also show that it is possible to avoid incorrect inferences. We propose a method to identify all the causal relationships that are theoretically possible to identify without being biased by unobserved variables. The empirical results using artificial data and simulated functional magnetic resonance imaging (fMRI) data show that our method effectively infers causal structures in the presence of unobserved variables. | cs | 1 |
Title: Semi-supervised Contrastive Outlier removal for Pseudo Expectation Maximization (SCOPE)
Abstract: Semi-supervised learning is the problem of training an accurate predictive model by combining a small labeled dataset with a presumably much larger unlabeled dataset. Many methods for semi-supervised deep learning have been developed, including pseudolabeling, consistency regularization, and contrastive learning techniques. Pseudolabeling methods however are highly susceptible to confounding, in which erroneous pseudolabels are assumed to be true labels in early iterations, thereby causing the model to reinforce its prior biases and thereby fail to generalize to strong predictive performance. We present a new approach to suppress confounding errors through a method we describe as Semi-supervised Contrastive Outlier removal for Pseudo Expectation Maximization (SCOPE). Like basic pseudolabeling, SCOPE is related to Expectation Maximization (EM), a latent variable framework which can be extended toward understanding cluster-assumption deep semi-supervised algorithms. However, unlike basic pseudolabeling which fails to adequately take into account the probability of the unlabeled samples given the model, SCOPE introduces an outlier suppression term designed to improve the behavior of EM iteration given a discrimination DNN backbone in the presence of outliers. Our results show that SCOPE greatly improves semi-supervised classification accuracy over a baseline, and furthermore when combined with consistency regularization achieves the highest reported accuracy for the semi-supervised CIFAR-10 classification task using 250 and 4000 labeled samples. Moreover, we show that SCOPE reduces the prevalence of confounding errors during pseudolabeling iterations by pruning erroneous high-confidence pseudolabeled samples that would otherwise contaminate the labeled set in subsequent retraining iterations. | cs | 1 |
Title: State Space Decomposition and Subgoal Creation for Transfer in Deep Reinforcement Learning
Abstract: Typical reinforcement learning (RL) agents learn to complete tasks specified by reward functions tailored to their domain. As such, the policies they learn do not generalize even to similar domains. To address this issue, we develop a framework through which a deep RL agent learns to generalize policies from smaller, simpler domains to more complex ones using a recurrent attention mechanism. The task is presented to the agent as an image and an instruction specifying the goal. This meta-controller guides the agent towards its goal by designing a sequence of smaller subtasks on the part of the state space within the attention, effectively decomposing it. As a baseline, we consider a setup without attention as well. Our experiments show that the meta-controller learns to create subgoals within the attention. | cs | 1 |
Title: Adversarial Data Poisoning for Fake News Detection: How to Make a Model Misclassify a Target News without Modifying It
Abstract: Fake news detection models are critical to countering disinformation but can be manipulated through adversarial attacks. In this position paper, we analyze how an attacker can compromise the performance of an online learning detector on specific news content without being able to manipulate the original target news. In some contexts, such as social networks, where the attacker cannot exert complete control over all the information, this scenario can indeed be quite plausible. Therefore, we show how an attacker could potentially introduce poisoning data into the training data to manipulate the behavior of an online learning method. Our initial findings reveal varying susceptibility of logistic regression models based on complexity and attack type. | cs | 0 |
Title: Biologically Plausible Learning of Text Representation with Spiking Neural Networks
Abstract: This study proposes a novel biologically plausible mechanism for generating low-dimensional spike-based text representation. First, we demonstrate how to transform documents into series of spikes spike trains which are subsequently used as input in the training process of a spiking neural network (SNN). The network is composed of biologically plausible elements, and trained according to the unsupervised Hebbian learning rule, Spike-Timing-Dependent Plasticity (STDP). After training, the SNN can be used to generate low-dimensional spike-based text representation suitable for text/document classification. Empirical results demonstrate that the generated text representation may be effectively used in text classification leading to an accuracy of $80.19\%$ on the bydate version of the 20 newsgroups data set, which is a leading result amongst approaches that rely on low-dimensional text representations. | cs | 1 |
Title: Error Forward-Propagation: Reusing Feedforward Connections to Propagate Errors in Deep Learning
Abstract: We introduce Error Forward-Propagation, a biologically plausible mechanism to propagate error feedback forward through the network. Architectural constraints on connectivity are virtually eliminated for error feedback in the brain; systematic backward connectivity is not used or needed to deliver error feedback. Feedback as a means of assigning credit to neurons earlier in the forward pathway for their contribution to the final output is thought to be used in learning in the brain. How the brain solves the credit assignment problem is unclear. In machine learning, error backpropagation is a highly successful mechanism for credit assignment in deep multilayered networks. Backpropagation requires symmetric reciprocal connectivity for every neuron. From a biological perspective, there is no evidence of such an architectural constraint, which makes backpropagation implausible for learning in the brain. This architectural constraint is reduced with the use of random feedback weights. Models using random feedback weights require backward connectivity patterns for every neuron, but avoid symmetric weights and reciprocal connections. In this paper, we practically remove this architectural constraint, requiring only a backward loop connection for effective error feedback. We propose reusing the forward connections to deliver the error feedback by feeding the outputs into the input receiving layer. This mechanism, Error Forward-Propagation, is a plausible basis for how error feedback occurs deep in the brain independent of and yet in support of the functionality underlying intricate network architectures. We show experimentally that recurrent neural networks with two and three hidden layers can be trained using Error Forward-Propagation on the MNIST and Fashion MNIST datasets, achieving $1.90\%$ and $11\%$ generalization errors respectively. | cs | 1 |
Title: Hereditary $n$-exangulated categories
Abstract: Herschend-Liu-Nakaoka introduced the concept of $n$-exangulated categories as higher-dimensional analogues of extriangulated categories defined by Nakaoka-Palu. The class of $n$-exangulated categories contains $n$-exact categories and $(n+2)$-angulated categories as specific examples. In this article, we introduce the notion of hereditary $n$-exangulated categories, which generalize hereditary extriangulated categories. We provide two classes of hereditary $n$-exangulated categories through closed subfunctors. Additionally, we define the concept of $0$-Auslander $n$-exangulated categories and discuss the circumstances under which these two classes of hereditary $n$-exangulated categories become $0$-Auslander. | math | 0 |
Title: Algebraic boundary of matrices of nonnegative rank at most three
Abstract: The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Grobner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author. | math | 1 |
Title: Irreducibility and periodicity in $\mathbb{Z}^{2}$ symbolic systems
Abstract: We show that there exist $\mathbb{Z}^{2}$ symbolic systems that are strongly irreducible and have no (fully) periodic points | math | 0 |
Title: Measurable functions on charge spaces
Abstract: The concept of measurability of functions on a charge space is generalised for functions taking values in a uniform space. Several existing forms of measurability generalise naturally in this context, and new forms of measurability are proposed. Conditions under which the various forms of measurability are logically equivalent are identified. Applying these concepts to real-valued functions, some recent characterisations of measurable functions on a bounded charge space are generalised to the unbounded case. | math | 0 |
Title: Maximum principal ratio of the signless Laplacian of graphs
Abstract: Let $G$ be a connected graph and $Q(G)$ be the signless Laplacian of $G$. The principal ratio $\gamma(G)$ of $Q(G)$ is the ratio of the maximum and minimum entries of the Perron vector of $Q(G)$. In this paper, we consider the maximum principal ratio $\gamma(G)$ among all connected graphs of order $n$, and show that for sufficiently large $n$ the extremal graph is a kite graph obtained by identifying an end vertex of a path to any vertex of a complete graph. | math | 1 |
Title: Efficiency, Fairness, and Stability in Non-Commercial Peer-to-Peer Ridesharing
Abstract: Unlike commercial ridesharing, non-commercial peer-to-peer (P2P) ridesharing has been subject to limited research -- although it can promote viable solutions in non-urban communities. This paper focuses on the core problem in P2P ridesharing: the matching of riders and drivers. We elevate users' preferences as a first-order concern and introduce novel notions of fairness and stability in P2P ridesharing. We propose algorithms for efficient matching while considering user-centric factors, including users' preferred departure time, fairness, and stability. Results suggest that fair and stable solutions can be obtained in reasonable computational times and can improve baseline outcomes based on system-wide efficiency exclusively. | cs | 1 |