text
stringlengths 137
2.43k
| field
stringclasses 2
values | label
int64 0
1
|
|---|---|---|
Title: Affine Metrics and Associated Algebroid Structures: Application to General Relativity
Abstract: In this paper, algebroid bundle associated to affine metrics provide an structure for unification of gravity and electromagnetism and, geometrization of matter. | math | 1 |
Title: Fourier-based schemes for computing the mechanical response of composites with accurate local fields
Abstract: We modify the Green operator involved in Fourier-based computational schemes in elasticity, in 2D and 3D. The new operator is derived by expressing continuum mechanics in terms of centered differences on a rotated grid. Use of the modified Green operator leads, in all systems investigated, to more accurate strain and stress fields than using the discretizations proposed by Moulinec and Suquet (1994) or Willot and Pellegrini (2008). Moreover, we compared the convergence rates of the "direct" and "accelerated" FFT schemes with the different discretizations. The discretization method proposed in this work allows for much faster FFT schemes with respect to two criteria: stress equilibrium and effective elastic moduli. | math | 1 |
Title: Physics-informed Generalizable Wireless Channel Modeling with Segmentation and Deep Learning: Fundamentals, Methodologies, and Challenges
Abstract: Channel modeling is fundamental in advancing wireless systems and has thus attracted considerable research focus. Recent trends have seen a growing reliance on data-driven techniques to facilitate the modeling process and yield accurate channel predictions. In this work, we first provide a concise overview of data-driven channel modeling methods, highlighting their limitations. Subsequently, we introduce the concept and advantages of physics-informed neural network (PINN)-based modeling and a summary of recent contributions in this area. Our findings demonstrate that PINN-based approaches in channel modeling exhibit promising attributes such as generalizability, interpretability, and robustness. We offer a comprehensive architecture for PINN methodology, designed to inform and inspire future model development. A case-study of our recent work on precise indoor channel prediction with semantic segmentation and deep learning is presented. The study concludes by addressing the challenges faced and suggesting potential research directions in this field. | cs | 0 |
Title: Gland Segmentation in Colon Histology Images: The GlaS Challenge Contest
Abstract: Colorectal adenocarcinoma originating in intestinal glandular structures is the most common form of colon cancer. In clinical practice, the morphology of intestinal glands, including architectural appearance and glandular formation, is used by pathologists to inform prognosis and plan the treatment of individual patients. However, achieving good inter-observer as well as intra-observer reproducibility of cancer grading is still a major challenge in modern pathology. An automated approach which quantifies the morphology of glands is a solution to the problem. This paper provides an overview to the Gland Segmentation in Colon Histology Images Challenge Contest (GlaS) held at MICCAI'2015. Details of the challenge, including organization, dataset and evaluation criteria, are presented, along with the method descriptions and evaluation results from the top performing methods. | cs | 1 |
Title: Real-Time FJ/MAC PDE Solvers via Tensorized, Back-Propagation-Free Optical PINN Training
Abstract: Solving partial differential equations (PDEs) numerically often requires huge computing time, energy cost, and hardware resources in practical applications. This has limited their applications in many scenarios (e.g., autonomous systems, supersonic flows) that have a limited energy budget and require near real-time response. Leveraging optical computing, this paper develops an on-chip training framework for physics-informed neural networks (PINNs), aiming to solve high-dimensional PDEs with fJ/MAC photonic power consumption and ultra-low latency. Despite the ultra-high speed of optical neural networks, training a PINN on an optical chip is hard due to (1) the large size of photonic devices, and (2) the lack of scalable optical memory devices to store the intermediate results of back-propagation (BP). To enable realistic optical PINN training, this paper presents a scalable method to avoid the BP process. We also employ a tensor-compressed approach to improve the convergence and scalability of our optical PINN training. This training framework is designed with tensorized optical neural networks (TONN) for scalable inference acceleration and MZI phase-domain tuning for \textit{in-situ} optimization. Our simulation results of a 20-dim HJB PDE show that our photonic accelerator can reduce the number of MZIs by a factor of $1.17\times 10^3$, with only $1.36$ J and $1.15$ s to solve this equation. This is the first real-size optical PINN training framework that can be applied to solve high-dimensional PDEs. | cs | 0 |
Title: Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds
Abstract: The Polynomial-Time Hierarchy ($\mathsf{PH}$) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers. Quantumly, however, despite the fact that at least \emph{four} definitions of quantum $\mathsf{PH}$ exist, it has been challenging to prove analogues for these of even basic facts from $\mathsf{PH}$. This work studies three quantum-verifier based generalizations of $\mathsf{PH}$, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings ($\mathsf{QCPH}$) and quantum mixed states ($\mathsf{QPH}$) as proofs, and one of which is new to this work, utilizing quantum pure states ($\mathsf{pureQPH}$) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for $\mathsf{QCPH}$. Then, for our new class $\mathsf{pureQPH}$, we show one-sided error reduction for $\mathsf{pureQPH}$, as well as the first bounds relating these quantum variants of $\mathsf{PH}$, namely $\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}$. | cs | 0 |
Title: Improved Online Algorithm for Weighted Flow Time
Abstract: We discuss one of the most fundamental scheduling problem of processing jobs on a single machine to minimize the weighted flow time (weighted response time). Our main result is a $O(\log P)$-competitive algorithm, where $P$ is the maximum-to-minimum processing time ratio, improving upon the $O(\log^{2}P)$-competitive algorithm of Chekuri, Khanna and Zhu (STOC 2001). We also design a $O(\log D)$-competitive algorithm, where $D$ is the maximum-to-minimum density ratio of jobs. Finally, we show how to combine these results with the result of Bansal and Dhamdhere (SODA 2003) to achieve a $O(\log(\min(P,D,W)))$-competitive algorithm (where $W$ is the maximum-to-minimum weight ratio), without knowing $P,D,W$ in advance. As shown by Bansal and Chan (SODA 2009), no constant-competitive algorithm is achievable for this problem. | cs | 1 |
Title: General runner removal and the Mullineux map
Abstract: We prove a new `runner removal theorem' for $q$-decomposition numbers of the level 1 Fock space of type $A^{(1)}_{e-1}$, generalising earlier theorems of James--Mathas and the author. By combining this with another theorem relating to the Mullineux map, we show that the problem of finding all $q$-decomposition numbers indexed by partitions of a given weight is a finite computation. | math | 1 |
Title: Geometric structures of late points of a two-dimensional simple random walk
Abstract: We consider the problem, as suggested by Dembo ($2003$, $2006$), of late points of a simple random walk in two dimensions. It has been shown that the exponents for the numbers of pairs of late points coincide with those of nearly favorite points and high points in the Gaussian free field, whose exact values are known. We estimate the exponents for the numbers of a multipoint set of late points in average. While there have been observed certain similarities between among three classes of points, our results exhibit a difference. | math | 1 |
Title: The novel Tauberian conditions associated with the $(\overline{N},p,q)$ summability of double sequences
Abstract: In this paper, our primary objective is to provide a fresh perspective on the relationship between the $(\overline{N},p,q)$ method, which is a product of relevant one-dimensional summability methods, and $P$-convergence for double sequences. To accomplish this objective, we establish certain Tauberian conditions that control the behavior of a double sequence in terms of both $O_L$-oscillation and $O$-oscillation in certain senses, building a bridge between $(\overline{N},p,q)$ summability and $P$-convergence, while imposing certain restrictions on the weight sequences. As special circumstances of our findings, we demonstrate that Landau-type $O_L$ condition with respect to $(P_m)$ and $(Q_n),$ as well as Hardy-type $O$ condition with respect to $(P_m)$ and $(Q_n),$ serve as Tauberian conditions for $(\overline{N},p,q)$ summability under particular additional conditions. Consequently, these results encompass all classical Tauberian theorems, including conditions such as slow decrease or slow oscillation in certain senses. | math | 0 |
Title: Penalty Parameter Selection in Deconvolution by Estimating the Risk for a Smaller Sample Size
Abstract: We address the choice of penalty parameter in the Smoothness-Penalized Deconvolution (SPeD) method of estimating a probability density under additive measurement error. Cross-validation gives an unbiased estimate of the risk (for the present sample size n) with a given penalty parameter, and this function can be minimized as a function of the penalty parameter. Least-squares cross-validation, which has been proposed for the similar Deconvoluting Kernel Density Estimator (DKDE), performs quite poorly for SPeD. We instead estimate the risk function for a smaller sample size n_1 < n with a given penalty parameter, using this to choose the penalty parameter for sample size n_1, and then use the asymptotics of the optimal penalty parameter to choose for sample size n. In a simulation study, we find that this has dramatically better performance than cross-validation, is an improvement over a SURE-type method previously proposed for this estimator, and compares favorably to the classic DKDE with its recommended plug-in method. We prove that the maximum error in estimating the risk function is of smaller order than its optimal rate of convergence. | math | 0 |
Title: On the safe set of Cartesian product of two complete graphs
Abstract: For a connected graph $G$, a vertex subset $S$ of $V(G)$ is a safe set if for every component $C$ of the subgraph of $G$ induced by $S$, $|C| \ge |D|$ holds for every component $D$ of $G-S$ such that there exists an edge between $C$ and $D$, and, in particular, if the subgraph induced by $S$ is connected, then $S$ is called a connected safe set. For a connected graph $G$, the safe number and the connected safe number of $G$ are the minimum among sizes of the safe sets and the minimum among sizes of the connected safe sets, respectively, of $G$. Fujita et al. introduced these notions in connection with a variation of the facility location problem. In this paper, we study the safe number and the connected safe number of Cartesian product of two complete graphs. Figuring out a way to reduce the number of components to two without changing the size of safe set makes it sufficient to consider only partitions of an integer into two parts without which it would be much more complicated to take care of all the partitions. In this way, we could show that the safe number and the connected safe number of Cartesian product of two complete graphs are equal and present a polynomial-time algorithm to compute them. Especially, in the case where one of complete components has order at most four, we precisely formulate those numbers. | math | 1 |
Title: Integrated Sensing and Communication with Massive MIMO: A Unified Tensor Approach for Channel and Target Parameter Estimation
Abstract: Benefitting from the vast spatial degrees of freedom, the amalgamation of integrated sensing and communication (ISAC) and massive multiple-input multiple-output (MIMO) is expected to simultaneously improve spectral and energy efficiencies as well as the sensing capability. However, a large number of antennas deployed in massive MIMO-ISAC raises critical challenges in acquiring both accurate channel state information and target parameter information. To overcome these two challenges with a unified framework, we first analyze their underlying system models and then propose a novel tensor-based approach that addresses both the channel estimation and target sensing problems. Specifically, by parameterizing the high-dimensional communication channel exploiting a small number of physical parameters, we associate the channel state information with the sensing parameters of targets in terms of angular, delay, and Doppler dimensions. Then, we propose a shared training pattern adopting the same time-frequency resources such that both the channel estimation and target parameter estimation can be formulated as a canonical polyadic decomposition problem with a similar mathematical expression. On this basis, we first investigate the uniqueness condition of the tensor factorization and the maximum number of resolvable targets by utilizing the specific Vandermonde | cs | 0 |
Title: On the existence of analytic families of stable lattices in trianguline representations and their reductions
Abstract: In this article, we prove the existence of rigid analytic families of $G$-stable lattices with locally constant reductions inside families of representations of a topologically compact group $G$, extending a result of Hellman obtained in the semi-simple residual case. Implementing this generalization in the context of Galois representations, we prove a local constancy result for reductions modulo prime powers of trianguline representations of generic dimension $d$. Moreover, we present two explicit applications. First, in dimension two, we extend to a prime power setting and to the whole rigid projective line a recent result of Bergdall, Levin and Liu concerning reductions of semi-stable representations of $\text{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ with fixed Hodge-Tate weights and large $\mathcal{L}$-invariant. Second, in dimension $d$, let $V_n$ be a sequence of crystalline representations converging in a certain geometric sense to a crystalline representation $V$. We show that for any refined version $(V, \sigma)$ of $V$ (or equivalently for any chosen triangulation of its attached $(\varphi, \Gamma)$-module $D_{\text{rig}} (V)$ over the Robba ring), there exists a sequence of refinement $\sigma_n$ of each of the $V_n$ such that the limit as refined representations $(V_n , \sigma_n )$ converges to the $(V, \sigma)$. This result does not hold under the weaker assumption that $V_n$ converges only uniformly $p$-adically to $V$ (in the sense of Chenevier, Khare and Larsen). | math | 0 |
Title: The Stanley Conjecture Revisited
Abstract: In the seminal work of Stanley, several conjectures were made on the structure of Littlewood-Richardson coefficients for the multiplication of Jack symmetric functions. Motivated by recent results of Alexandersson and the present author, we postulate that a certain 'windowing' property holds for all such Jack L-R coefficients. This property provides a vast set of relations between these coefficients and allows for their direct computation in certain novel cases. We demonstrate compatibility between our windowing conjecture and the conjectures of Stanley, with the hope of illuminating the structures within. | math | 0 |
Title: The Effect of Noise on the Emergence of Continuous Norms and its Evolutionary Dynamics
Abstract: We examine the effect of noise on societies of agents using an agent-based model of evolutionary norm emergence. Generally, we see that noisy societies are more selfish, smaller and discontent, and are caught in rounds of perpetual punishment preventing them from flourishing. Surprisingly, despite the effect of noise on the population, it does not seem to evolve away. We carry out further analysis and provide reasons for why this may be the case. Furthermore, we claim that our framework that evolves the noise/ambiguity of norms may be a new way to model the tight/loose framework of norms, suggesting that despite ambiguous norms detrimental effect on society, evolution does not favour clarity. | cs | 0 |
Title: Locally dualizable modules abound
Abstract: It is proved that given any prime ideal $\mathfrak{p}$ of height at least 2 in a countable commutative noetherian ring $A$, there are uncountably many more dualizable objects in the $\mathfrak{p}$-local $\mathfrak{p}$-torsion stratum of the derived category of $A$ than those that are obtained as retracts of images of perfect $A$-complexes. An analogous result is established dealing with the stable module category of the group algebra, over a countable field of positive characteristic $p$, of an elementary abelian $p$-group of rank at least 3. | math | 0 |
Title: On Brown-York mass and compactly conformal deformations of scalar curvature
Abstract: In this article, we found a connection between Brown-York mass and the first Dirichlet Eigenvalue of a Schr\"odingier type operator. In particular, we proved a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigated compactly conformal deformations which either increase or decrease scalar curvature. We found local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed manifolds with positive scalar curvature. We also constructed such kind of deformations on a type of product manifolds that either increase or decrease their scalar curvature compactly and conformally. These results together answered a natural question arises in \cite{Corvino, Lohkamp}. | math | 1 |
Title: Singularity-agnostic incomplete U-statistics for testing polynomial constraints in Gaussian covariance matrices
Abstract: Testing the goodness-of-fit of a model with its defining functional constraints in the parameters could date back to Spearman (1927), who analyzed the famous "tetrad" polynomial in the covariance matrix of the observed variables in a single-factor model. Despite its long history, the Wald test typically employed to operationalize this approach could produce very inaccurate test sizes in many situations, even when the regular conditions for the classical normal asymptotics are met and a very large sample is available. Focusing on testing a polynomial constraint in a Gaussian covariance matrix, we obtained a new understanding of this baffling phenomenon: When the null hypothesis is true but "near-singular", the standardized Wald test exhibits slow weak convergence, owing to the sophisticated dependency structure inherent to the underlying U-statistic that ultimately drives its limiting distribution; this can also be rigorously explained by a key ratio of moments encoded in the Berry-Esseen bound quantifying the normal approximation error involved. As an alternative, we advocate the use of an incomplete U-statistic to mildly tone down the dependence thereof and render the speed of convergence agnostic to the singularity status of the hypothesis. In parallel, we develop a Berry-Esseen bound that is mathematically descriptive of the singularity-agnostic nature of our standardized incomplete U-statistic, using some of the finest exponential-type inequalities in the literature. | math | 0 |
Title: Characterizing BigBench queries, Hive, and Spark in multi-cloud environments
Abstract: BigBench is the new standard (TPCx-BB) for benchmarking and testing Big Data systems. The TPCx-BB specification describes several business use cases -- queries -- which require a broad combination of data extraction techniques including SQL, Map/Reduce (M/R), user code (UDF), and Machine Learning to fulfill them. However, currently, there is no widespread knowledge of the different resource requirements and expected performance of each query, as is the case to more established benchmarks. At the same time, cloud providers currently offer convenient on-demand managed big data clusters (PaaS) with a pay-as-you-go model. In PaaS, analytical engines such as Hive and Spark come ready to use, with a general-purpose configuration and upgrade management. The study characterizes both the BigBench queries and the out-of-the-box performance of Spark and Hive versions in the cloud. At the same time, comparing popular PaaS offerings in terms of reliability, data scalability (1GB to 10TB), versions, and settings from Azure HDinsight, Amazon Web Services EMR, and Google Cloud Dataproc. The query characterization highlights the similarities and differences in Hive an Spark frameworks, and which queries are the most resource consuming according to CPU, memory, and I/O. Scalability results show how there is a need for configuration tuning in most cloud providers as data scale grows, especially with Sparks memory usage. These results can help practitioners to quickly test systems by picking a subset of the queries which stresses each of the categories. At the same time, results show how Hive and Spark compare and what performance can be expected of each in PaaS. | cs | 1 |
Title: Randomness Requirements and Asymmetries in Nash Equilibria
Abstract: In general, Nash equilibria in normal-form games may require players to play (probabilistically) mixed strategies. We define a measure of the complexity of finite probability distributions and study the complexity required to play NEs in finite two player $n\times n$ games with rational payoffs. Our central results show that there exist games in which there is an exponential vs. linear gap in the complexity of the mixed distributions that the two players play at (the unique) NE. This gap induces gaps in the amount of space required to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. We also establish upper and lower bounds on the complexity of any NE in the games that we study. These results highlight (i) the nontriviality of the assumption that players can any mixed strategy and (ii) the disparities in resources that players may require to play NEs in the games that we study. | cs | 0 |
Title: Hilbert modular forms and Galois representations
Abstract: In this expository article, we present a brief introduction to the theory of Hilbert modular forms and Galois representations, and describe what it means to attach a compatible system of Galois representations to a Hilbert modular form. | math | 0 |
Title: Section Rings of $\mathbb{Q}$-Divisors on Genus-$1$ Curves
Abstract: We compute generators and relations for the section ring of a rational divisor on an elliptic curve. Our technique is a generalization of \cite{O'Dorney} and \cite{VZB} that accounts for the additional subtlety that genus one curves pose: their group structure. We give explicit minimal presentations for section rings of divisors supported at one point and for section rings of effective divisors supported at two points. Our results for one-point divisors are quite similar to the corresponding case in genus zero from \cite{O'Dorney}, and are a combination of the one-point cases in genera one and zero for two-point effective divisors. | math | 0 |
Title: Multifractality and intermittency in the limit evolution of polygonal vortex filaments
Abstract: With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann's non-differentiable functions \begin{equation} R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2\pi i ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{equation} These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When $x_0$ is rational, we show that $R_{x_0}$ is multifractal and intermittent by completely determining the spectrum of singularities of $R_{x_0}$ and computing the $L^p$ norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that $R_{x_0}$ has a multifractal behavior also when $x_0$ is irrational. The proofs rely on a careful design of Diophantine sets that depend on $x_0$, which we study by crucially using the Duffin-Schaeffer theorem and the Mass Transference Principle. | math | 0 |
Title: DIALIGHT: Lightweight Multilingual Development and Evaluation of Task-Oriented Dialogue Systems with Large Language Models
Abstract: We present DIALIGHT, a toolkit for developing and evaluating multilingual Task-Oriented Dialogue (ToD) systems which facilitates systematic evaluations and comparisons between ToD systems using fine-tuning of Pretrained Language Models (PLMs) and those utilising the zero-shot and in-context learning capabilities of Large Language Models (LLMs). In addition to automatic evaluation, this toolkit features (i) a secure, user-friendly web interface for fine-grained human evaluation at both local utterance level and global dialogue level, and (ii) a microservice-based backend, improving efficiency and scalability. Our evaluations reveal that while PLM fine-tuning leads to higher accuracy and coherence, LLM-based systems excel in producing diverse and likeable responses. However, we also identify significant challenges of LLMs in adherence to task-specific instructions and generating outputs in multiple languages, highlighting areas for future research. We hope this open-sourced toolkit will serve as a valuable resource for researchers aiming to develop and properly evaluate multilingual ToD systems and will lower, currently still high, entry barriers in the field. | cs | 0 |
Title: Rational trigonometry via projective geometric algebra
Abstract: We show that main results of rational trigonometry (as developed by NJ Wildberger, "Divine Proportions", 2005) can be succinctly expressed using projective geometric algebra (PGA). In fact, the PGA representation exhibits distinct advantages over the original vector-based approach. These include the advantages intrinsic to geometric algebra: it is coordinate-free, treats lines and points in a unified framework, and handles many special cases in a uniform and seamless fashion. It also reveals structural patterns not visible in the original formulation, for example, the exact duality of spread and quadrance. The current article handles only a representative (euclidean) subset of the full content of Wildberger's work, but enough to establish the value of this approach for further development. The metric-neutral framework of PGA makes it especially promising also to handle universal geometry, which extends rational trigonometry to the hyperbolic plane. | math | 1 |
Title: Non-Wire Alternatives to Capacity Expansion
Abstract: Distributed energy resources (DERs) can serve as non-wire alternatives to capacity expansion by managing peak load to avoid or defer traditional expansion projects. In this paper, we study a planning problem that co-optimizes DERs investment and operation (e.g., energy efficiency, energy storage, demand response, solar photovoltaic) and the timing of capacity expansion. We formulate the problem as a large scale (in the order of millions of variables because we model the operation of DERs over a period of decades) non-convex optimization problem. Despite its non-convexities, we find its optimal solution by decomposing it using the Dantzig-Wolfe Decomposition Algorithm and solving a series of small linear problems. Finally, we present a real planning problem at the University of Washington Seattle Campus. | math | 1 |
Title: Deep Learning Based Superposition Coded Modulation for Hierarchical Semantic Communications over Broadcast Channels
Abstract: We consider multi-user semantic communications over broadcast channels. While most existing works consider that each receiver requires either the same or independent semantic information, this paper explores the scenario where the semantic information desired by different receivers is different but correlated. In particular, we investigate semantic communications over Gaussian broadcast channels where the transmitter has a common observable source but the receivers wish to recover hierarchical semantic information in adaptation to their channel conditions. Inspired by the capacity achieving property of superposition codes, we propose a deep learning based superposition coded modulation (DeepSCM) scheme. Specifically, the hierarchical semantic information is first extracted and encoded into basic and enhanced feature vectors. A linear minimum mean square error (LMMSE) decorrelator is then developed to obtain a refinement from the enhanced features that is uncorrelated with the basic features. Finally, the basic features and their refinement are superposed for broadcasting after probabilistic modulation. Experiments are conducted for two-receiver image semantic broadcasting with coarse and fine classification as hierarchical semantic tasks. DeepSCM outperforms the benchmarking coded-modulation scheme without a superposition structure, especially with large channel disparity and high order modulation. It also approaches the performance upperbound as if there were only one receiver. | cs | 0 |
Title: Analytic problems for elliptic curves
Abstract: We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and of the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (Note : This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv). | math | 1 |
Title: Fiedler Linearizations of Rectangular Rational Matrix Functions
Abstract: Linearization is a standard approach in the computation of eigenvalues, eigenvectors and invariant subspaces of matrix polynomials and rational matrix value functions. An important source of linearizations are the so called Fiedler linearizations, which are generalizations of the classical companion forms. In this paper the concept of Fiedler linearization is extended from square regular to rectangular rational matrix valued functions. The approach is applied to Rosenbrock functions arising in mathematical system theory. | math | 1 |
Title: On Borkar and Young Relaxed Control Topologies and Continuous Dependence of Invariant Measures on Control Policy
Abstract: In deterministic and stochastic control theory, relaxed or randomized control policies allow for versatile mathematical analysis (on continuity, compactness, convexity and approximations) to be applicable with no artificial restrictions on the classes of control policies considered, leading to very general existence results on optimal measurable policies under various setups and information structures. On relaxed controls, two studied topologies are the Young and Borkar (weak$^*$) topologies on spaces of functions from a state/measurement space to the space of probability measures on control action spaces; the former via a weak convergence topology on probability measures on a product space with a fixed marginal on the input (state) space, and the latter via a weak$^*$ topology on randomized policies viewed as maps from states/measurements to the space of signed measures with bounded variation. We establish implication and equivalence conditions between the Young and Borkar topologies on control policies. We then show that, under some conditions, for a controlled Markov chain with standard Borel spaces the invariant measure is weakly continuous on the space of stationary control policies defined by either of these topologies. An implication is near optimality of quantized stationary policies in state and actions or continuous stationary and deterministic policies for average cost control under two sets of continuity conditions (with either weak continuity in the state-action pair or strong continuity in the action for each state) on transition kernels. | math | 0 |
Title: Maximal polarization for periodic configurations on the real line
Abstract: We prove that among all 1-periodic configurations $\Gamma$ of points on the real line $\mathbb{R}$ the quantities $$ \min_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \pi \alpha (x - \gamma)^2} \quad \text{and} \quad \max_{x \in \mathbb{R}} \sum_{\gamma \in \Gamma} e^{- \pi \alpha (x - \gamma)^2}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $n$ per period is sufficiently large (depending on $\alpha$). This solves the polarization problem for periodic configurations with a Gaussian weight on $\mathbb{R}$ for large $n$. The first result is shown using Fourier series. The second result follows from work of Cohn and Kumar on universal optimality and holds for all $n$ (independent of $\alpha$). | math | 0 |
Title: The Power of Training: How Different Neural Network Setups Influence the Energy Demand
Abstract: This work examines the effects of variations in machine learning training regimes and learning paradigms on the corresponding energy consumption. While increasing data availability and innovation in high-performance hardware fuels the training of sophisticated models, it also supports the fading perception of energy consumption and carbon emission. Therefore, the goal of this work is to create awareness about the energy impact of general training parameters and processes, from learning rate over batch size to knowledge transfer. Multiple setups with different hyperparameter initializations are evaluated on two different hardware configurations to obtain meaningful results. Experiments on pretraining and multitask training are conducted on top of the baseline results to determine their potential towards sustainable machine learning. | cs | 0 |
Title: Entropy-based Probing Beam Selection and Beam Prediction via Deep Learning
Abstract: Hierarchical beam search in mmWave communications incurs substantial training overhead, necessitating deep learning-enabled beam predictions to effectively leverage channel priors and mitigate this overhead. In this study, we introduce a comprehensive probabilistic model of power distribution in beamspace, and formulate the joint optimization problem of probing beam selection and probabilistic beam prediction as an entropy minimization problem. Then, we propose a greedy scheme to iteratively and alternately solve this problem, where a transformer-based beam predictor is trained to estimate the conditional power distribution based on the probing beams and user location within each iteration, and the trained predictor selects an unmeasured beam that minimizes the entropy of remaining beams. To further reduce the number of interactions and the computational complexity of the iterative scheme, we propose a two-stage probing beam selection scheme. Firstly, probing beams are selected from a location-specific codebook designed by an entropy-based criterion, and predictions are made with corresponding feedback. Secondly, the optimal beam is identified using additional probing beams with the highest predicted power values. Simulation results demonstrate the superiority of the proposed schemes compared to hierarchical beam search and beam prediction with uniform probing beams. | cs | 0 |
Title: Equivariant Morse theory for Lie algebra actions on Riemannian foliations
Abstract: Consider the transverse isometric action of a finite dimensional Lie algebra g on a Riemannian foliation. This paper studies the equivariant Morse-Bott theory on the leaf space of the Riemannian foliations in this setting. Among other things, we establish a foliated version of the Morse-Bott lemma for a g-invariant basic Morse-Bott function, and a foliated version of the usual handle presentation theorem. In the non-equivariant case, we apply these results to present a new proof of the Morse inequalities on Riemannian foliations. In the equivariant case, we apply these results to study Hamiltonian action of an abelian Lie algebra on a presymplectic manifold whose underlying foliation is also Riemannian, and extend the Kirwan surjectivity and injectivity theorem in equivariant symplectic geometry to this situation. Among other things, this implies the Kirwan surjectivity and injectivity hold for Hamiltonian torus actions on symplectic orbifolds. | math | 0 |
Title: Order of uniform approximation by polynomial interpolation in the complex plane and beyond
Abstract: For Lagrange polynomial interpolation on open arcs $X=\gamma$ in $\mathbb{C}$, it is well-known that the Lebesgue constant for the family of Chebyshev points ${\bf{x}}_n:=\{x_{n,j}\}^{n}_{j=0}$ on $[-1,1]\subset \mathbb{R}$ has growth order of $O(log(n))$. The same growth order was shown in [45] for the Lebesgue constant of the family ${\bf {z^{**}_n}}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of some properly adjusted Fej\'er points on a rectifiable smooth open arc $\gamma\subset \mathbb{C}$. On the other hand, in our recent work [15], it was observed that if the smooth open arc $\gamma$ is replaced by an $L$-shape arc $\gamma_0 \subset \mathbb{C}$ consisting of two line segments, numerical experiments suggest that the Marcinkiewicz-Zygmund inequalities are no longer valid for the family of Fej\'er points ${\bf z}_n^{*}:=\{z_{n,j}^{*}\}^{n}_{j=0}$ on $\gamma$, and that the rate of growth for the corresponding Lebesgue constant $L_{{\bf {z}}^{*}_n}$ is as fast as $c\,log^2(n)$ for some constant $c>0$. The main objective of the present paper is 3-fold: firstly, it will be shown that for the special case of the $L$-shape arc $\gamma_0$ consisting of two line segments of the same length that meet at the angle of $\pi/2$, the growth rate of the Lebesgue constant $L_{{\bf {z}}_n^{*}}$ is at least as fast as $O(Log^2(n))$, with $\lim\sup \frac{L_{{\bf {z}}_n^{*}}}{log^2(n)} = \infty$; secondly, the corresponding (modified) Marcinkiewicz-Zygmund inequalities fail to hold; and thirdly, a proper adjustment ${\bf z}_n^{**}:=\{z_{n,j}^{**}\}^{n}_{j=0}$ of the Fej\'er points on $\gamma$ will be described to assure the growth rate of $L_{{\bf z}_n^{**}}$ to be exactly $O(Log^2(n))$. | math | 1 |
Title: Periodic solutions of one-dimensional cellular automata with random rules
Abstract: We study cellular automata with randomly selected rules. Our setting are two-neighbor rules with a large number $n$ of states. The main quantity we analyze is the asymptotic probability, as $n \to \infty$, that the random rule has a periodic solution with given spatial and temporal periods. We prove that this limiting probability is non-trivial when the spatial and temporal periods are confined to a finite range. The main tool we use is the Chen-Stein method for Poisson approximation. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented. | math | 1 |
Title: Graphon games: A statistical framework for network games and interventions
Abstract: In this paper, we present a unifying framework for analyzing equilibria and designing interventions for large network games sampled from a stochastic network formation process represented by a graphon. We first introduce a new class of infinite population games, termed graphon games, where a continuum of heterogeneous agents interact according to a graphon. After studying properties of equilibria in graphon games, we show that graphon equilibria can approximate equilibria of large network games sampled from the graphon. We next show that, under some regularity assumptions, the graphon approach enables the design of asymptotically optimal interventions via the solution of an optimization problem with much lower dimension than the one based on the entire network structure. We illustrate our framework on a synthetic dataset of rural villages and show that the graphon intervention can be computed efficiently and based solely on aggregated relational data. | cs | 1 |
Title: A new approach to convergence analysis of iterative models with optimal error bounds
Abstract: In this paper, we study a new approach related to the convergence analysis of Ishikawa-type iterative models to a common fixed point of two non-expansive mappings in Banach spaces. The main novelty of our contribution lies in the so-called \emph{optimal error bounds}, which established some necessary and sufficient conditions for convergence and derived both the error estimates and bounds on the convergence rates for iterative schemes. Although a special interest here is devoted to the Ishikawa and modified Ishikawa iterative sequences, the theory of \emph{optimal error bounds} proposed in this paper can also be favorably applied to various types of iterative models to approximate common fixed points of non-expansive mappings. | math | 0 |
Title: Towards a Foundation Purchasing Model: Pretrained Generative Autoregression on Transaction Sequences
Abstract: Machine learning models underpin many modern financial systems for use cases such as fraud detection and churn prediction. Most are based on supervised learning with hand-engineered features, which relies heavily on the availability of labelled data. Large self-supervised generative models have shown tremendous success in natural language processing and computer vision, yet so far they haven't been adapted to multivariate time series of financial transactions. In this paper, we present a generative pretraining method that can be used to obtain contextualised embeddings of financial transactions. Benchmarks on public datasets demonstrate that it outperforms state-of-the-art self-supervised methods on a range of downstream tasks. We additionally perform large-scale pretraining of an embedding model using a corpus of data from 180 issuing banks containing 5.1 billion transactions and apply it to the card fraud detection problem on hold-out datasets. The embedding model significantly improves value detection rate at high precision thresholds and transfers well to out-of-domain distributions. | cs | 0 |
Title: Cluster algebras and monotone Lagrangian tori
Abstract: Motivated by recent developments in the construction of Newton--Okounkov bodies and toric degenerations via cluster algebras in [GHKK18, FO20], we consider a family of Newton--Okounkov polytopes of a complex smooth projective variety $X$ related by a composition of tropicalized cluster mutations. According to the work of [HK15], the toric degeneration associated with each Newton--Okounkov polytope $\Delta$ in the family produces a Lagrangian torus fibration of $X$ over $\Delta$. We investigate circumstances in which each Lagrangian torus fibration possesses a monotone Lagrangian torus fiber. We provide a sufficient condition, based on the data of tropical integer points and exchange matrices, for the family of constructed monotone Lagrangian tori to contain infinitely many monotone Lagrangian tori, no two of which are related by any symplectomorphisms. By employing this criterion and exploiting the correspondence between the tropical integer points and the dual canonical basis elements, we generate infinitely many distinct monotone Lagrangian tori on flag manifolds of arbitrary type except in a few cases. | math | 0 |
Title: A sample iterated small cancellation theory for groups of Burnside type
Abstract: We develop yet another technique to present the free Burnside group $B(m,n)$ of odd exponent $n$ with $m\ge2$ generators as a group satisfying a certain iterated small cancellation condition. Using the approach, we provide a reasonably accessible proof that $B(m,n)$ is infinite with a moderate bound $n > 2000$ on the odd exponent $n$. | math | 1 |
Title: Uncovering the Disentanglement Capability in Text-to-Image Diffusion Models
Abstract: Generative models have been widely studied in computer vision. Recently, diffusion models have drawn substantial attention due to the high quality of their generated images. A key desired property of image generative models is the ability to disentangle different attributes, which should enable modification towards a style without changing the semantic content, and the modification parameters should generalize to different images. Previous studies have found that generative adversarial networks (GANs) are inherently endowed with such disentanglement capability, so they can perform disentangled image editing without re-training or fine-tuning the network. In this work, we explore whether diffusion models are also inherently equipped with such a capability. Our finding is that for stable diffusion models, by partially changing the input text embedding from a neutral description (e.g., "a photo of person") to one with style (e.g., "a photo of person with smile") while fixing all the Gaussian random noises introduced during the denoising process, the generated images can be modified towards the target style without changing the semantic content. Based on this finding, we further propose a simple, light-weight image editing algorithm where the mixing weights of the two text embeddings are optimized for style matching and content preservation. This entire process only involves optimizing over around 50 parameters and does not fine-tune the diffusion model itself. Experiments show that the proposed method can modify a wide range of attributes, with the performance outperforming diffusion-model-based image-editing algorithms that require fine-tuning. The optimized weights generalize well to different images. Our code is publicly available at https://github.com/UCSB-NLP-Chang/DiffusionDisentanglement. | cs | 1 |
Title: Decay rates for cubic and higher order nonlinear wave equations on asymptotically flat spacetimes
Abstract: In this paper, we prove pointwise decay rates for cubic and higher order nonlinear wave equations, including quasilinear wave equations, on asymptotically flat and time-dependent spacetimes. We assume that the solution to the linear equation (rather than the nonlinear equation) satisfies a weaker form of the standard integrated local energy decay, or Morawetz, estimate. For nonlinearities with a total derivative structure, we prove better pointwise decay rates. | math | 1 |
Title: JPEG XT Image Compression with Hue Compensation for Two-Layer HDR Coding
Abstract: We propose a novel JPEG XT image compression with hue compensation for two-layer HDR coding. LDR images produced from JPEG XT bitstreams have some distortion in hue due to tone mapping operations. In order to suppress the color distortion, we apply a novel hue compensation method based on the maximally saturated colors. Moreover, the bitstreams generated by using the proposed method are fully compatible with the JPEG XT standard. In an experiment, the proposed method is demonstrated not only to produce images with small hue degradation but also to maintain well-mapped luminance, in terms of three kinds of criterion: TMQI, hue value in CIEDE2000, and the maximally saturated color on the constant-hue plane. | cs | 1 |
Title: The second largest component in the supercritical 2D Hamming graph
Abstract: The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices $(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability $p$, so that the average degree $2(n-1)p=1+\epsilon$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component has size $\sim 2\epsilon n$. Here we show that the second largest component has size close to $\epsilon^{-2}$, so that the dominant component has emerged. This result also suggests that a {\it discrete duality principle} might hold, whereby, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime. | math | 1 |
Title: Thermodynamic Consistent Neural Networks for Learning Material Interfacial Mechanics
Abstract: For multilayer materials in thin substrate systems, interfacial failure is one of the most challenges. The traction-separation relations (TSR) quantitatively describe the mechanical behavior of a material interface undergoing openings, which is critical to understand and predict interfacial failures under complex loadings. However, existing theoretical models have limitations on enough complexity and flexibility to well learn the real-world TSR from experimental observations. A neural network can fit well along with the loading paths but often fails to obey the laws of physics, due to a lack of experimental data and understanding of the hidden physical mechanism. In this paper, we propose a thermodynamic consistent neural network (TCNN) approach to build a data-driven model of the TSR with sparse experimental data. The TCNN leverages recent advances in physics-informed neural networks (PINN) that encode prior physical information into the loss function and efficiently train the neural networks using automatic differentiation. We investigate three thermodynamic consistent principles, i.e., positive energy dissipation, steepest energy dissipation gradient, and energy conservative loading path. All of them are mathematically formulated and embedded into a neural network model with a novel defined loss function. A real-world experiment demonstrates the superior performance of TCNN, and we find that TCNN provides an accurate prediction of the whole TSR surface and significantly reduces the violated prediction against the laws of physics. | cs | 1 |
Title: Learning Safe, Generalizable Perception-based Hybrid Control with Certificates
Abstract: Many robotic tasks require high-dimensional sensors such as cameras and Lidar to navigate complex environments, but developing certifiably safe feedback controllers around these sensors remains a challenging open problem, particularly when learning is involved. Previous works have proved the safety of perception-feedback controllers by separating the perception and control subsystems and making strong assumptions on the abilities of the perception subsystem. In this work, we introduce a novel learning-enabled perception-feedback hybrid controller, where we use Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) to show the safety and liveness of a full-stack perception-feedback controller. We use neural networks to learn a CBF and CLF for the full-stack system directly in the observation space of the robot, without the need to assume a separate perception-based state estimator. Our hybrid controller, called LOCUS (Learning-enabled Observation-feedback Control Using Switching), can safely navigate unknown environments, consistently reach its goal, and generalizes safely to environments outside of the training dataset. We demonstrate LOCUS in experiments both in simulation and in hardware, where it successfully navigates a changing environment using feedback from a Lidar sensor. | cs | 1 |
Title: Squared chromatic number without claws or large cliques
Abstract: Let $G$ be a claw-free graph on $n$ vertices with clique number $\omega$, and consider the chromatic number $\chi(G^2)$ of the square $G^2$ of $G$. Writing $\chi'_s(d)$ for the supremum of $\chi(L^2)$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\chi(G^2)\le \chi'_s(\omega)$ for $\omega \in \{3,4\}$. For $\omega=3$, this implies the sharp bound $\chi(G^2) \leq 10$. For $\omega=4$, this implies $\chi(G^2)\leq 22$, which is within $2$ of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il. | math | 1 |
Title: Notes on exotic and perverse-coherent sheaves
Abstract: Exotic sheaves are certain complexes of coherent sheaves on the cotangent bundle of the flag variety of a reductive group. They are closely related to perverse-coherent sheaves on the nilpotent cone. This expository article includes the definitions of these two categories, applications, and some structure theory, as well as detailed calculations for SL(2). | math | 1 |
Title: Local limit theorem for time-inhomogeneous functions of Markov processes
Abstract: In this paper, we consider a continuous-time Markov process and prove a local limit theorem for the integral of a time-inhomogeneous function of the process. One application is in the study of the fast-oscillating perturbations of linear dynamical systems. | math | 0 |
Title: GridFormer: Point-Grid Transformer for Surface Reconstruction
Abstract: Implicit neural networks have emerged as a crucial technology in 3D surface reconstruction. To reconstruct continuous surfaces from discrete point clouds, encoding the input points into regular grid features (plane or volume) has been commonly employed in existing approaches. However, these methods typically use the grid as an index for uniformly scattering point features. Compared with the irregular point features, the regular grid features may sacrifice some reconstruction details but improve efficiency. To take full advantage of these two types of features, we introduce a novel and high-efficiency attention mechanism between the grid and point features named Point-Grid Transformer (GridFormer). This mechanism treats the grid as a transfer point connecting the space and point cloud. Our method maximizes the spatial expressiveness of grid features and maintains computational efficiency. Furthermore, optimizing predictions over the entire space could potentially result in blurred boundaries. To address this issue, we further propose a boundary optimization strategy incorporating margin binary cross-entropy loss and boundary sampling. This approach enables us to achieve a more precise representation of the object structure. Our experiments validate that our method is effective and outperforms the state-of-the-art approaches under widely used benchmarks by producing more precise geometry reconstructions. The code is available at https://github.com/list17/GridFormer. | cs | 0 |
Title: Luna's fundamental lemma for diagonalizable groups
Abstract: We study relatively affine actions of a diagonalizable group $G$ on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a $G$-equivariant morphism $f: X'\to X$ to be $strongly\ equivariant$, namely the base change of the morphism $f/\!/G$ of quotient schemes, and establish descent criteria for $f/\!/G$ to be an open embedding, \'etale, smooth, regular, syntomic, or lci. | math | 1 |
Title: Existence and rigidity of quantum isometry groups for compact metric spaces
Abstract: We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative $C^*$-algebra of continuous functions on the Riemannian isometry group. | math | 1 |
Title: On an inverse Robin spectral problem
Abstract: We consider the problem of the recovery of a Robin coefficient on a part $\gamma \subset \partial \Omega$ of the boundary of a bounded domain $\Omega$ from the principal eigenvalue and the boundary values of the normal derivative of the principal eigenfunction of the Laplace operator with Dirichlet boundary condition on $\partial \Omega \setminus \gamma$. We prove uniqueness, as well as local Lipschitz stability of the inverse problem. Moreover, we present an iterative reconstruction algorithm with numerical computations in two dimensions showing the accuracy of the method. | math | 1 |
Title: Randomization can be as helpful as a glimpse of the future in online computation
Abstract: We provide simple but surprisingly useful direct product theorems for proving lower bounds on online algorithms with a limited amount of advice about the future. As a consequence, we are able to translate decades of research on randomized online algorithms to the advice complexity model. Doing so improves significantly on the previous best advice complexity lower bounds for many online problems, or provides the first known lower bounds. For example, if $n$ is the number of requests, we show that: (1) A paging algorithm needs $\Omega(n)$ bits of advice to achieve a competitive ratio better than $H_k=\Omega(\log k)$, where $k$ is the cache size. Previously, it was only known that $\Omega(n)$ bits of advice were necessary to achieve a constant competitive ratio smaller than $5/4$. (2) Every $O(n^{1-\varepsilon})$-competitive vertex coloring algorithm must use $\Omega(n\log n)$ bits of advice. Previously, it was only known that $\Omega(n\log n)$ bits of advice were necessary to be optimal. For certain online problems, including the MTS, $k$-server, paging, list update, and dynamic binary search tree problem, our results imply that randomization and sublinear advice are equally powerful (if the underlying metric space or node set is finite). This means that several long-standing open questions regarding randomized online algorithms can be equivalently stated as questions regarding online algorithms with sublinear advice. For example, we show that there exists a deterministic $O(\log k)$-competitive $k$-server algorithm with advice complexity $o(n)$ if and only if there exists a randomized $O(\log k)$-competitive $k$-server algorithm without advice. Technically, our main direct product theorem is obtained by extending an information theoretical lower bound technique due to Emek, Fraigniaud, Korman, and Ros\'en [ICALP'09]. | cs | 1 |
Title: Root multiplicities for Borcherds algebras and graph coloring
Abstract: We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed formula for certain root multiplicities. Using this connection we give a second interpretation, namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph (depending on the root). Finally, using the combinatorics of Lyndon words we construct a basis for the root spaces corresponding to these roots and determine the Hilbert series in the case when all simple roots are imaginary. As an application we give a Lie theoretic proof of Stanley's reciprocity theorem of chromatic polynomials. | math | 1 |
Title: Perspective Plane Program Induction from a Single Image
Abstract: We study the inverse graphics problem of inferring a holistic representation for natural images. Given an input image, our goal is to induce a neuro-symbolic, program-like representation that jointly models camera poses, object locations, and global scene structures. Such high-level, holistic scene representations further facilitate low-level image manipulation tasks such as inpainting. We formulate this problem as jointly finding the camera pose and scene structure that best describe the input image. The benefits of such joint inference are two-fold: scene regularity serves as a new cue for perspective correction, and in turn, correct perspective correction leads to a simplified scene structure, similar to how the correct shape leads to the most regular texture in shape from texture. Our proposed framework, Perspective Plane Program Induction (P3I), combines search-based and gradient-based algorithms to efficiently solve the problem. P3I outperforms a set of baselines on a collection of Internet images, across tasks including camera pose estimation, global structure inference, and down-stream image manipulation tasks. | cs | 1 |
Title: Eulerian partitions for configurations of skew lines
Abstract: In this paper, which is a complement of \cite{BG}, we study a few elementary invariants for configurations of skew lines, as introduced and analyzed first by Viro and his collaborators. We slightly simplify the exposition of some known invariants and use them to define a natural partition of the lines in a skew configuration. We also describe an algorithm which constructs a spindle-permutation for a given switching class, or proves non-existence of such a spindle-permutation. | math | 1 |
Title: STAS: Spatial-Temporal Return Decomposition for Multi-agent Reinforcement Learning
Abstract: Centralized Training with Decentralized Execution (CTDE) has been proven to be an effective paradigm in cooperative multi-agent reinforcement learning (MARL). One of the major challenges is credit assignment, which aims to credit agents by their contributions. While prior studies have shown great success, their methods typically fail to work in episodic reinforcement learning scenarios where global rewards are revealed only at the end of the episode. They lack the functionality to model complicated relations of the delayed global reward in the temporal dimension and suffer from inefficiencies. To tackle this, we introduce Spatial-Temporal Attention with Shapley (STAS), a novel method that learns credit assignment in both temporal and spatial dimensions. It first decomposes the global return back to each time step, then utilizes the Shapley Value to redistribute the individual payoff from the decomposed global reward. To mitigate the computational complexity of the Shapley Value, we introduce an approximation of marginal contribution and utilize Monte Carlo sampling to estimate it. We evaluate our method on an Alice & Bob example and MPE environments across different scenarios. Our results demonstrate that our method effectively assigns spatial-temporal credit, outperforming all state-of-the-art baselines. | cs | 0 |
Title: Coverage Explorer: Coverage-guided Test Generation for Cyber Physical Systems
Abstract: Given the safety-critical functions of autonomous cyber-physical systems (CPS) across diverse domains, testing these systems is essential. While conventional software and hardware testing methodologies offer partial insights, they frequently do not provide adequate coverage in a CPS. In this study, we introduce a testing framework designed to systematically formulate test cases, effectively exploring the state space of CPS. This framework introduces a coverage-centric sampling technique, coupled with a cluster-based methodology for training a surrogate model. The framework then uses model predictive control within the surrogate model to generates test cases tailored to CPS specifications. To evaluate the efficacy of the framework, we applied it on several benchmarks, spanning from a kinematic car to systems like an unmanned aircraft collision avoidance system (ACAS XU) and automatic transmission system. Comparative analyses were conducted against alternative test generation strategies, including randomized testing, as well as falsification using S-TaLiRo. | cs | 0 |
Title: A Simple Generalization of a Result for Random Matrices with Independent Sub-Gaussian Rows
Abstract: In this short note, we give a very simple but useful generalization of a result of Vershynin (Theorem 5.39 of [1]) for a random matrix with independent sub-Gaussian rows. We also explain with an example where our generalization is useful. | math | 1 |
Title: A load balanced chemistry model with analytical Jacobian for faster reactive simulations in OpenFOAM
Abstract: In this study, we introduce a novel open-source chemistry model for OpenFOAM to speed-up the reactive computational fluid dynamics (CFD) simulations using finite-rate chemistry. First, a dynamic load balancing model called DLBFoam is introduced to balance the chemistry load during runtime in parallel simulations. In addition, the solution of the cell-based chemistry problem is improved by utilizing an analytical Jacobian using an open-source library called pyJac and an efficient linear algebra library LAPACK. Combination of the aforementioned efforts yields a speed-up factor 200 for a high-fidelity large-eddy simulation spray combustion case compared to the standard OpenFOAM implementation. It is worth noting that the present implementation does not compromise the solution accuracy. | cs | 1 |
Title: Uncertainty Estimates for Ordinal Embeddings
Abstract: To investigate objects without a describable notion of distance, one can gather ordinal information by asking triplet comparisons of the form "Is object $x$ closer to $y$ or is $x$ closer to $z$?" In order to learn from such data, the objects are typically embedded in a Euclidean space while satisfying as many triplet comparisons as possible. In this paper, we introduce empirical uncertainty estimates for standard embedding algorithms when few noisy triplets are available, using a bootstrap and a Bayesian approach. In particular, simulations show that these estimates are well calibrated and can serve to select embedding parameters or to quantify uncertainty in scientific applications. | cs | 1 |
Title: On metrically complete Bruhat-Tits buildings
Abstract: Metrical completeness for Bruhat-Tits buildings is a natural and useful condition. In this paper we determine which Bruhat-Tits buildings are metrically complete up to certain cases involving infinite-dimensionality and residue characteristic two. | math | 1 |
Title: Chordal graphs, even-hole-free graphs and sparse obstructions to bounded treewidth
Abstract: We present and study the following conjecture: for an integer $t\geq 4$ and a graph $H$, every even-hole-free graph of large enough treewidth has an induced subgraph isomorphic to either $K_t$ or $H$, if (and only if) $H$ is a $K_4$-free chordal graph. The ``only if'' part follows from the properties of the so-called layered wheels due to Sintiari and Trotignon. Alecu, Chudnovsky, Spirkl and the author recently proved the conjecture in two special cases: (a) when $t=4$; and (b) when $H=cone (F)$ for some forest $F$; that is, $H$ is obtained from $F$ by adding a universal vertex. Our first result is a common strengthening: for an integer $t\geq 4$ and graphs $F$ and $H$, (even-hole, $cone(cone (F))$, $H$, $K_t$)-free graphs have bounded treewidth if and only if $F$ is a forest and $H$ is a $K_4$-free chordal graph. For general $t\geq 4$, we push the current state of the art further than (b) by settling the conjecture for the smallest choices of $H$ that are not coned forests. This follows from our second result: we prove the conjecture when $H$ is a crystal; that is, a graph obtained from several coned double stars by gluing them together along the middle edges of the double stars. We also propose another conjecture motivated by the following observation: except for complete graphs and complete bipartite graphs, all the constructions of graphs with unbounded treewidth which have been discovered so far are $2$-degenerate. Specifically, we conjecture that for every $t\geq 1$, every graph of sufficiently large treewidth has an induced subgraph of treewidth $t$ which is either complete, complete bipartite, or $2$-degenerate. This conjecture is the first to predict a grid-type theorem for induced subgraphs, and what makes it even more relevant is somewhat serendipitous: if true, it would imply our former conjecture by reducing it to (a). | math | 0 |
Title: Graph Neural Networks for Tabular Data Learning: A Survey with Taxonomy and Directions
Abstract: In this survey, we dive into Tabular Data Learning (TDL) using Graph Neural Networks (GNNs), a domain where deep learning-based approaches have increasingly shown superior performance in both classification and regression tasks compared to traditional methods. The survey highlights a critical gap in deep neural TDL methods: the underrepresentation of latent correlations among data instances and feature values. GNNs, with their innate capability to model intricate relationships and interactions between diverse elements of tabular data, have garnered significant interest and application across various TDL domains. Our survey provides a systematic review of the methods involved in designing and implementing GNNs for TDL (GNN4TDL). It encompasses a detailed investigation into the foundational aspects and an overview of GNN-based TDL methods, offering insights into their evolving landscape. We present a comprehensive taxonomy focused on constructing graph structures and representation learning within GNN-based TDL methods. In addition, the survey examines various training plans, emphasizing the integration of auxiliary tasks to enhance the effectiveness of instance representations. A critical part of our discussion is dedicated to the practical application of GNNs across a spectrum of GNN4TDL scenarios, demonstrating their versatility and impact. Lastly, we discuss the limitations and propose future research directions, aiming to spur advancements in GNN4TDL. This survey serves as a resource for researchers and practitioners, offering a thorough understanding of GNNs' role in revolutionizing TDL and pointing towards future innovations in this promising area. | cs | 0 |
Title: Leveraging SAM for Single-Source Domain Generalization in Medical Image Segmentation
Abstract: Domain Generalization (DG) aims to reduce domain shifts between domains to achieve promising performance on the unseen target domain, which has been widely practiced in medical image segmentation. Single-source domain generalization (SDG) is the most challenging setting that trains on only one source domain. Although existing methods have made considerable progress on SDG of medical image segmentation, the performances are still far from the applicable standards when faced with a relatively large domain shift. In this paper, we leverage the Segment Anything Model (SAM) to SDG to greatly improve the ability of generalization. Specifically, we introduce a parallel framework, the source images are sent into the SAM module and normal segmentation module respectively. To reduce the calculation resources, we apply a merging strategy before sending images to the SAM module. We extract the bounding boxes from the segmentation module and send the refined version as prompts to the SAM module. We evaluate our model on a classic DG dataset and achieve competitive results compared to other state-of-the-art DG methods. Furthermore, We conducted a series of ablation experiments to prove the effectiveness of the proposed method. The code is publicly available at https://github.com/SARIHUST/SAMMed. | cs | 0 |
Title: Virtual rigid motives of semi-algebraic sets
Abstract: Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan's motivic integration and Ayoub's equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber. | math | 1 |
Title: Word-Representability of Graphs with respect to Split Recomposition
Abstract: In this work, we show that the class of word-representable graphs is closed under split recomposition and determine the representation number of the graph obtained by recomposing two word-representable graphs. Accordingly, we show that the class of parity graphs is word-representable. Further, we obtain a characteristic property by which the recomposition of comparability graphs is a comparability graph. Consequently, we also establish the permutation-representation number (prn) of the resulting comparability graph. We also introduce a subclass of comparability graphs, called prn-irreducible graphs. We provide a criterion such that the split recomposition of two prn-irreducible graphs is a comparability graph and determine the prn of the resultant graph. | cs | 0 |
Title: Counterexamples to the B-spline conjecture for Gabor frames
Abstract: The frame set conjecture for B-splines $B_n$, $n \ge 2$, states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form $ab=r$, where $r$ is a rational number smaller than one and $a$ and $b$ denote the sampling and modulation rates, respectively, has infinitely many pieces, located around $b=2,3,\dots$, \emph{not} belonging to the frame set of the $n$th order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines $B_n$, $n \ge 2$. | math | 1 |
Title: A Simple Construction of Tournaments with Finite and Uncountable Dichromatic Number
Abstract: The dichromatic number $\chi(\vec{G})$ of a digraph $\vec{G}$ is the minimum number of colors needed to color the vertices $V(\vec{G})$ in such a way that no monochromatic directed cycle is obtained. In this note, for any $k\in \mathbb{N}$, we give a simple construction of tournaments with dichromatic number exactly equal to $k$. The proofs are based on a combinatorial lemma on partitioning a checkerboard which may be of independent interest. We also generalize our finite construction to give an elementary construction of a complete digraph of cardinality equal to the cardinality of $\mathbb{R}$ and having an uncountable dichromatic number. Furthermore, we also construct an oriented balanced complete $n$-partite graph $\vec{K}^{(m)}_n$, such that the minimum number of colors needed to color its vertices such that there is no monochromatic directed triangle is greater than or equal to $nm/(n+2m-2)$. | math | 0 |
Title: On the Structure of Boolean Functions with Small Spectral Norm
Abstract: In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of f is $\|\hat{f}\|_1=\sum_{\alpha}|\hat{f}(\alpha)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n \to \{0,1\}$ with $\|\hat{f}\|_1=A$. 1. There is a subspace $V$ of co-dimension at most $A^2$ such that $f|_V$ is constant. 2. f can be computed by a parity decision tree of size $2^{A^2}n^{2A}$. (a parity decision tree is a decision tree whose nodes are labeled with arbitrary linear functions.) 3. If in addition f has at most s nonzero Fourier coefficients, then f can be computed by a parity decision tree of depth $A^2 \log s$. 4. For every $0<\epsilon$ there is a parity decision tree of depth $O(A^2 + \log(1/\epsilon))$ and size $2^{O(A^2)} \cdot \min\{1/\epsilon^2,O(\log(1/\epsilon))^{2A}\}$ that \epsilon-approximates f. Furthermore, this tree can be learned, with probability $1-\delta$, using $\poly(n,\exp(A^2),1/\epsilon,\log(1/\delta))$ membership queries. All the results above also hold (with a slight change in parameters) to functions $f:Z_p^n\to \{0,1\}$. | cs | 1 |
Title: Boundary regularity of stochastic PDEs
Abstract: The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $\alpha>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $\alpha$-H\"older continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $C^1$ domains are proved to be $\alpha$-H\"older continuous up to the boundary with some $\alpha>0$. | math | 1 |
Title: Biharmonic and harmonic homomorphisms between Riemannian three dimensional unimodular Lie groups
Abstract: We classify biharmonic and harmonic homomorphisms $f:(G,g_1)\rightarrow(G,g_2)$ where $G$ is a connected and simply connected three-dimensional unimodular Lie group and $g_1$ and $g_2$ are left invariant Riemannian metrics. | math | 1 |
Title: Current Trends in Digital Twin Development, Maintenance, and Operation: An Interview Study
Abstract: Digital twins (DT) are often defined as a pairing of a physical entity and a corresponding virtual entity (VE), mimicking certain aspects of the former depending on the use-case. In recent years, this concept has facilitated numerous use-cases ranging from design to validation and predictive maintenance of large and small high-tech systems. Various heterogeneous cross-domain models are essential for such systems and model-driven engineering plays a pivotal role in the design, development, and maintenance of these models. We believe models and model-driven engineering play a similarly crucial role in the context of a VE of a DT. Due to the rapidly growing popularity of DTs and their use in diverse domains and use-cases, the methodologies, tools, and practices for designing, developing, and maintaining the corresponding VEs differ vastly. To better understand these differences and similarities, we performed a semi-structured interview research with 19 professionals from industry and academia who are closely associated with different lifecycle stages of digital twins. In this paper, we present our analysis and findings from this study, which is based on seven research questions. In general, we identified an overall lack of uniformity in terms of the understanding of digital twins and used tools, techniques, and methodologies for the development and maintenance of the corresponding VEs. Furthermore, considering that digital twins are software intensive systems, we recognize a significant growth potential for adopting more software engineering practices, processes, and expertise in various stages of a digital twin's lifecycle. | cs | 0 |
Title: On the Notion of Equal Figures in Euclid
Abstract: Euclid uses an undefined notion of "equal figures", to which he applies the common notions about equals added to equals or subtracted from equals. When (in previous work) we formalized Euclid Book~I for computer proof-checking, we had to add fifteen axioms about undefined relations "equal triangles" and "equal quadrilaterals" to replace Euclid's use of the common notions. In this paper, we offer definitions of "equal triangles" and "equal quadrilaterals", that Euclid could have given, and prove that they have the required properties. This removes the need for adding new axioms. The proof uses the theory of proportions. Hence we also discuss the "early theory of proportions", which has a long history. | math | 1 |
Title: Mean Value Theorems for Binary Egyptian Fractions
Abstract: In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary. | math | 1 |
Title: Several new classes of MDS symbol-pair codes derived from matrix-product codes
Abstract: In order to correct the pair-errors generated during the transmission of modern high-density data storage that the outputs of the channels consist of overlapping pairs of symbols, a new coding scheme named symbol-pair code is proposed. The error-correcting capability of the symbol-pair code is determined by its minimum symbol-pair distance. For such codes, the larger the minimum symbol-pair distance, the better. It is a challenging task to construct symbol-pair codes with optimal parameters, especially, maximum-distance-separable (MDS) symbol-pair codes. In this paper, the permutation equivalence codes of matrix-product codes with underlying matrixes of orders 3 and 4 are used to extend the minimum symbol-pair distance, and six new classes of MDS symbol-pair codes are derived. | cs | 0 |
Title: Backward propagation of warped product structures and asymptotically conical shrinkers
Abstract: We establish sufficient conditions which ensure that a locally-warped product structure propagates backward in time under the Ricci flow. As an application, we prove that if an asymptotically conical gradient shrinking soliton is asymptotic to a cone whose cross-section is a product of Einstein manifolds, the soliton must itself be a multiply-warped product over the same manifolds. | math | 0 |
Title: Subgroups of bounded rank in hyperbolic 3-manifold groups
Abstract: We prove a finiteness theorem for subgroups of bounded rank in hyperbolic $3$-manifold groups. As a consequence, we show that every bounded rank covering tower of closed hyperbolic $3$-manifolds is a tower of finite covers associated to a fibration over a $1$-orbifold. | math | 0 |
Title: Two improvements in Brauer's theorem on forms
Abstract: Let $k$ be a Brauer field, that is, a field over which every diagonal form in sufficiently many variables has a nonzero solution; for instance, $k$ could be an imaginary quadratic number field. Brauer proved that if $f_1, \ldots, f_r$ are homogeneous polynomials on a $k$-vector space $V$ of degrees $d_1, \ldots, d_r$, then the variety $Z$ defined by the $f_i$'s has a non-trivial $k$-point, provided that $\dim{V}$ is sufficiently large compared to the $d_i$'s and $k$. We offer two improvements to this theorem, assuming $k$ is infinite. First, we show that the Zariski closure of the set $Z(k)$ of $k$-points has codimension $<C$, where $C$ is a constant depending only on the $d_i$'s and $k$. And second, we show that if the strength of the $f_i$'s is sufficiently large in terms of the $d_i$'s and $k$, then $Z(k)$ is actually Zariski dense in $Z$. The proofs rely on recent work of Ananyan and Hochster on high strength polynomials. | math | 0 |
Title: Natural real algebraic maps of non-positive codimensions with prescribed images whose boundaries consist of non-singular real algebraic hypersurfaces intersecting with transversality
Abstract: We present new real algebraic maps of non-positive codimensions with prescribed images whose boundaries consist of explicit non-singular real algebraic hypersurfaces intersecting with so-called "transversality". We also have the maps with explicit information on important real polynomials. This gives new construction of explicit real algebraic maps of non-positive codimensions with explicit information on images, preimages and important real polynomials. Thanks to some celebrated theory of Nash, smooth closed manifolds are {\it non-singular} real algebraic manifolds and the zero sets of some real polynomials. In considerable cases, we can approximate smooth functions or more generally, maps, by real algebraic ones. It is in general difficult to have explicit examples. We have constructed some previously. We have ones of some new type with explicit information on global structures of the maps. | math | 0 |
Title: Free constructions of geometries of Coxeter type
Abstract: We establish two free constructions of geometries of Coxeter type. The first construction deals with any Coxeter diagram having no subdiagram of type A_3, the second one with diagrams of type C_n and H_4. | math | 1 |
Title: Approximate Distance and Shortest-Path Oracles for Fault-Tolerant Geometric Spanners
Abstract: In this paper, we present approximate distance and shortest-path oracles for fault-tolerant Euclidean spanners motivated by the routing problem in real-world road networks. An $f$-fault-tolerant Euclidean $t$-spanner for a set $V$ of $n$ points in $\mathbb{R}^d$ is a graph $G=(V,E)$ where, for any two points $p$ and $q$ in $V$ and a set $F$ of $f$ vertices of $V$, the distance between $p$ and $q$ in $G-F$ is at most $t$ times their Euclidean distance. Given an $f$-fault-tolerant Euclidean $t$-spanner $G$ with $O(n)$ edges and a constant $\varepsilon$, our data structure has size $O_{t,f}(n\log n)$, and this allows us to compute an $(1+\varepsilon)$-approximate distance in $G-F$ between $s$ and $s'$ can be computed in constant time for any two vertices $s$ and $s'$ and a set $F$ of $f$ failed vertices. Also, with a data structure of size $O_{t,f}(n\log n\log\log n)$, we can compute an $(1+\varepsilon)$-approximate shortest path in $G-F$ between $s$ and $s'$ in $O_{t,f}(\log^2 n\log\log n+\textsf{sol})$ time for any two vertices $s$ and $s'$ and a set $F$ of failed vertices, where $\textsf{sol}$ denotes the number of vertices in the returned path. | cs | 0 |
Title: KCES: A Workflow Containerization Scheduling Scheme Under Cloud-Edge Collaboration Framework
Abstract: As more IoT applications gradually move towards the cloud-edge collaborative mode, the containerized scheduling of workflows extends from the cloud to the edge. However, given the high delay of the communication network, loose coupling of structure, and resource heterogeneity between cloud and edge, workflow containerization scheduling in the cloud-edge scenarios faces the difficulty of resource coordination and application collaboration management. To address these two issues, we propose a KubeEdge-Cloud-Edge-Scheduling scheme named KCES, a workflow containerization scheduling scheme for the KubeEdge cloud-edge framework. The KCES includes a cloud-edge workflow scheduling engine for KubeEdge and workflow scheduling strategies for task horizontal roaming and vertical offloading. Considering the scheduling optimization of cloud-edge workflows, this paper proposes a cloud-edge workflow scheduling model and cloud-edge node model and designs a cloud-edge workflow scheduling engine to maximize cloud-edge resource utilization under the constraint of workflow task delay. A cloud-edge resource hybrid management technology is used to design the cloud-edge resource evaluation and resource allocation algorithms to achieve cloud-edge resource collaboration. Based on the ideas of distributed functional roles and the hierarchical division of computing power, the horizontal roaming among the edges and vertical offloading strategies between the cloud and edges for workflow tasks are designed to realize the cloud-edge application collaboration. Through a customized IoT application workflow instance, experimental results show that KCES is superior to the baseline in total workflow duration, average workflow duration, and resource usage and has the capabilities of horizontal roaming and vertical offloading for workflow tasks. | cs | 0 |
Title: Oscillator representations of quantum affine orthosymplectic superalgebras
Abstract: We introduce a category of $q$-oscillator representations over the quantum affine superalgebras of type $D$ and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible $q$-oscillator representations of type $X_n^{(1)}$ and the finite-dimensional irreducible representations of type $Y_n^{(1)}$ for $(X,Y)=(C,D),(D,C)$ under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from Howe's reductive dual pairs $(\mathfrak{g},G)$, where $\mathfrak{g}=\mathfrak{sp}_{2n}, \mathfrak{so}_{2n}$ and $G=O_\ell, Sp_{2\ell}$. | math | 0 |
Title: A new combinatorial approach for edge universality of Wigner matrices
Abstract: In this paper we introduce a new combinatorial approach to analyze the trace of large powers of Wigner matrices. Our approach is motivated from the paper by \citet{sosh}. However the counting approach is different. We start with classical word sentence approach similar to \citet{AZ05} and take the motivation from \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} to encode the words to objects similar to Dyck paths. To be precise the map takes a word to a Dyck path with some edges removed from it. Using this new counting we prove edge universality for large Wigner matrices with sub-Gaussian entries. One novelty of this approach is unlike \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} we do not need to assume the entries of the matrices are symmetrically distributed around $0$. The main technical contribution of this paper is two folded. Firstly we produce an encoding of the ``contributing words" (for definition one might look at Section \ref{sec:word}) of the Wigner matrix which retrieves the edge universality. Hence this is the best one can do. We hope this method will be applicable to many other scenarios in random matrices. Secondly in course of the paper we give a combinatorial description of the GOE Tracy Widom law. The explanation for GUE is very similar. This explanation might be important for the models where exact calculations are not available but some combinatorial structures are present. | math | 1 |
Title: Lower bounds for bulk deviations for the simple random walk on $\mathbb{Z}^d$, $d\geq 3$
Abstract: This article investigates the behavior of the continuous-time simple random walk on $\mathbb{Z}^d$, $d \geq 3$. We derive an asymptotic lower bound on the principal exponential rate of decay for the probability that the average value over a large box of some non-decreasing local function of the field of occupation times of the walk exceeds a given positive value. This bound matches at leading order the corresponding upper bound derived by Sznitman in arXiv:1906.05809, and is given in terms of a certain constrained minimum of the Dirichlet energy of functions on $\mathbb{R}^d$ decaying at infinity. Our proof utilizes a version of tilted random walks, a model originally constructed by Li in arXiv:1412.3959 to derive lower bounds on the probability of the event that the trace of a simple random walk disconnects a macroscopic set from an enclosing box. | math | 0 |
Title: Linear stability of compact shrinking Ricci solitons
Abstract: In this paper, we continue to investigate the second variation of Perelman's $\nu$-entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in "H.-D. Cao and M. Zhu, Math. Ann. 353 (2012), No. 3, 747-763" and the more recent work in "M. Mansour and R. Asadollah, arXiv:2104.08343" and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in "arXiv:math.DG/0404165" (see also "H.-D. Cao and C. He, J. Reine Angew. Math. 2015 (2015), no. 709, 229-246.") for positive Einstein manifolds to the compact shrinking Ricci soliton case. | math | 0 |
Title: Holonomic D-modules on abelian varieties
Abstract: We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology support loci of a holonomic D-module are finite unions of linear subvarieties, which go through points of finite order for objects of geometric origin; that the standard t-structure on the derived category of holonomic complexes corresponds, under the Fourier-Mukai transform, to a certain perverse coherent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov; and that Fourier-Mukai transforms of simple holonomic D-modules are intersection complexes in this t-structure. This supports the conjecture that Fourier-Mukai transforms of holonomic D-modules are "hyperk\"ahler perverse sheaves". | math | 1 |
Title: The Yule-$Λ$ Nested Coalescent: Distribution of the Number of Lineages
Abstract: We study a model of a population with individuals sampled from different species. The Yule-$\Lambda$ nested coalescent describes the genealogy of the sample when each species merges with another randomly chosen species with a constant rate $c$ and the mergers of individuals in each species follow the $\Lambda$-coalescent. For the Yule-$\Lambda$ nested coalescent with $c<\int_0^1x^{-1}\Lambda(dx)<\infty$, where $\Lambda$ is the measure that characterizes the $\Lambda$-coalescent, we show that under some initial conditions, the distribution of the number of individual lineages belonging to one species converges weakly to the distribution $\nu_c^*$, which is the solution to some recursive distributional equation (RDE) with finite mean. In addition, we show that for some values of $c$, the RDE has another solution with infinite mean. | math | 0 |
Title: Singular degree of a rational matrix pseudodifferential operator
Abstract: In our previous work we studied minimal fractional decompositions of a rational matrix pseudodifferential operator: H=A/B, where A and B are matrix differential operators, and B is non-degenerate of minimal possible degree deg(B). In the present paper we introduce the singular degree sdeg(H)=deg(B), and show that for an arbitrary rational expression H=sum_a (A^a_1)/(B^a_1)...(A^a_n)/(B^a_n), we have that sdeg(H) is less than or equal to sum_{a,i} deg(B^a_i). If the equality holds, we call such an expression minimal. We study the properties of the singular degree and of minimal rational expressions. These results are important for the computations involved in the Lenard-Magri scheme of integrability. | math | 1 |
Title: Demystifying $μ$
Abstract: We develop the theory of illfounded and cyclic proof systems in the context of the modal $\mu$-calculus. A fine analysis of provability and admissibility bridges the finitary, cyclic and illfounded notions of proof for this logic and re-enforces the subtlety of two important normal form theorems: guardedness and disjunctiveness. | math | 0 |
Title: Splitting of Uniform bundles on generalized Grassmannians and Kumar's conjecture
Abstract: Let $E$ be a uniform bundle on an arbitrary generalised Grassmannian $X$. We show that if the rank of $E$ is smaller than $e.d.(\mathrm{VMRT})$, then $E$ is necessarily splitting. For some generalised Grassmannians, we prove that the upper bound $e.d.(\mathrm{VMRT})$ is optimal. On the other hand, Kumar's conjecture predicts that if the minss rank of $G'/P'$ is bigger that the maxss rank of $G/P$, then any morphism $f:G'/P'\rightarrow G/P$ is constant. We prove some partially affirmative results about this conjecture. | math | 0 |
Title: Asymptotic expansion and optimal symmetry of minimal gradient graph equations in dimension 2
Abstract: In this paper, we study asymptotic expansion at infinity and symmetry of zero mean curvature equations of gradient graph in dimension 2, which include the Monge--Amp\`ere equation, inverse harmonic Hessian equation and the special Lagrangian equation. This refines the research of asymptotic behavior, gives the precise gap between exterior minimal gradient graph and the entire case, and extends the classification results of Monge--Amp\`ere equations. | math | 1 |
Title: Analysis of the embedded cell method for the numerical homogenization of metal-ceramic composite materials
Abstract: In this paper, we analyze the embedding cell method, an algorithm which has been developed for the numerical homogenization of metal-ceramic composite materials. We show the convergence of the iteration scheme of this algorithm and the coincidence of the material properties predicted by the limit with the effective material properties provided by the analytical homogenization theory in three situations, namely for a one dimensional linear elasticity model, a simple one dimensional plasticity model and a two dimensional model of linear hyperelastic isotropic materials with constant shear modulus and slightly varying first Lam\'e parameter. | math | 1 |
Title: On the homology of locally finite graphs
Abstract: We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension~1 captures precisely the topological cycle space of graphs but works in any dimension. | math | 1 |
Title: On positivity of the two-parameter bivariate kernel built of q-ultraspherical polynomials and other Lancaster-type expansions of bivariate distributions
Abstract: Our most important result concerns the positivity of certain kernels built of the so-called $q-$ultraspherical polynomials. Since this result appears at first sight as primarily important for those who are working in orthogonal polynomials, $q-$series theory and the so-called quantum polynomials, it might have a limited number of interested researchers. That is why, we put our result into a broader context. We recall the theory of Hilbert-Schmidt operators, Lancaster expansions and their applications in Mathematical statistics, or bivariate distributions absolutely continuous with respect to the product of their marginal distributions leading to the generation of Markov process with polynomial conditional moments (the main representative of such processes is a famous Wiener process). | math | 0 |
Title: Reciprocity obstructions in semigroup orbits in SL(2, Z)
Abstract: We study orbits of semigroups of $\text{SL}(2,\mathbb{Z})$, and demonstrate reciprocity obstructions: we show that certain such orbits avoid squares, but not as a consequence of such obstructions on the Zariski closure, and not as a consequence of congruence obstructions. This is in analogy to the reciprocity obstructions recently used to disprove the Apollonian local-global conjecture. We give an example of such an orbit which is known exactly, and misses all squares together with an explicit finite list of sporadic values: the corresponding semigroup is not thin, but its Zariski closure does not miss squares. We also demonstrate thin semigroups with reciprocity obstructions, including semigroups associated to continued fractions formed from finite alphabets. Zaremba's conjecture states that for continued fractions with coefficients chosen from $\{1,\ldots,5\}$, every positive integer appears as a denominator. Bourgain and Kontorovich proposed a generalization of Zaremba's conjecture in the context of semigroups associated to finite alphabets. We disprove their conjecture. In particular, we demonstrate classes of finite continued fraction expansions which never represent rationals with square denominator, but not as a consequence of congruence obstructions, and for which the limit set has Hausdorff dimension exceeding $1/2$. An example of such a class is continued fractions of the form $[0; a_1, a_2, \ldots, a_n,1,1,2]$, where the $a_i$ are chosen from the set $\{4,8,12,\ldots,128\}$. The object at the heart of these results is a semigroup $\Psi\subseteq\Gamma_1(4)$ which preserves Kronecker symbols. | math | 0 |