Title: A New Foundation for Finitary Corecursion
Abstract: This paper contributes to a theory of the behaviour of "finite-state" systems that is generic in the system type. We propose that such systems are modeled as coalgebras with a finitely generated carrier for an endofunctor on a locally finitely presentable category. Their behaviour gives rise to a new fixpoint of the coalgebraic type functor called locally finite fixpoint (LFF). We prove that if the given endofunctor preserves monomorphisms then the LFF always exists and is a subcoalgebra of the final coalgebra (unlike the rational fixpoint previously studied by Ad\'amek, Milius and Velebil). Moreover, we show that the LFF is characterized by two universal properties: 1. as the final locally finitely generated coalgebra, and 2. as the initial fg-iterative algebra. As instances of the LFF we first obtain the known instances of the rational fixpoint, e.g. regular languages, rational streams and formal power-series, regular trees etc. And we obtain a number of new examples, e.g. (realtime deterministic resp. non-deterministic) context-free languages, constructively S-algebraic formal power-series (and any other instance of the generalized powerset construction by Silva, Bonchi, Bonsangue, and Rutten) and the monad of Courcelle's algebraic trees.


Title: On the completeness of root function system of the $2\times 2$ Dirac operators with non-regular boundary conditions
Abstract: The paper is concerned with the completeness property of root functions of the $2\times 2$ Dirac operator with summable complex-valued potential and non-regular boundary conditions. Sufficient conditions for the completeness of the root function system of the operator under consideration are established.


Title: Evolutionary Alternating Direction Method of Multipliers for Constrained Multi-Objective Optimization with Unknown Constraints
Abstract: Constrained multi-objective optimization problems (CMOPs) pervade real-world applications in science, engineering, and design. Constraint violation has been a building block in designing evolutionary multi-objective optimization algorithms for solving constrained multi-objective optimization problems. However, in certain scenarios, constraint functions might be unknown or inadequately defined, making constraint violation unattainable and potentially misleading for conventional constrained evolutionary multi-objective optimization algorithms. To address this issue, we present the first of its kind evolutionary optimization framework, inspired by the principles of the alternating direction method of multipliers that decouples objective and constraint functions. This framework tackles CMOPs with unknown constraints by reformulating the original problem into an additive form of two subproblems, each of which is allotted a dedicated evolutionary population. Notably, these two populations operate towards complementary evolutionary directions during their optimization processes. In order to minimize discrepancy, their evolutionary directions alternate, aiding the discovery of feasible solutions. Comparative experiments conducted against five state-of-the-art constrained evolutionary multi-objective optimization algorithms, on 120 benchmark test problem instances with varying properties, as well as two real-world engineering optimization problems, demonstrate the effectiveness and superiority of our proposed framework. Its salient features include faster convergence and enhanced resilience to various Pareto front shapes.


Title: Asymptotic approximations of the continuous Hahn polynomials and their zeros
Abstract: Asymptotic approximations for the continuous Hahn polynomials and their zeros as the degree grows to infinity are established via their three-term recurrence relation. The methods are based on the uniform asymptotic expansions for difference equations developed by Wang and Wong (\textit{Numer. Math.}, 2003) and the matching technique in the complex plane developed by Wang (\textit{J. Approx. Theory}, 2014).


Title: Unique equilibrium states for some intermediate beta transformations
Abstract: We prove uniqueness of equilibrium states for subshifts corresponding to intermediate beta transformations with $\beta > 2$ having the property that the orbit of 0 is bounded away from 1.


Title: A quick probability-oriented introduction to operator splitting methods
Abstract: This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.


Title: On Higher-Order Extensions of the Weighted Projection Body Operator
Abstract: For a convex body $K$ in $\mathbb{R}^n$, the inequalities of Rogers-Shephard and Zhang, written succinctly, are $\text{vol}_n(DK)\leq \binom{2n}{n} \text{vol}_n(K) \leq \text{vol}_n(n\text{vol}_n(K)\Pi^\circ K).$ Here, $DK=\{x\in\mathbb{R}^n:K\cap(K+x)\neq \emptyset\}$ is the difference body of $K$, and $\Pi^\circ K$ is the polar projection body of $K$. There is equality in either if, and only if, $K$ is a $n$-dimensional simplex. In fact, there exists a collection of convex bodies, the so-called radial mean bodies $R_p K$ introduced by Gardner and Zhang, which continuously interpolates between $DK$ and $\Pi^\circ K$. Schneider defined the higher-order difference body as, for $m\in\mathbb{N}$, $$D^m(K)=\{(x_1,\dots,x_m)\in\mathbb{R}^{nm}:K\cap_{i=1}^m(K+x_i)\neq \emptyset\}\subset \mathbb{R}^{nm}$$ and proved a higher-order version of the Rogers-Shephard inequality. In a prequel to this work, the authors, working with Haddad, extended the higher-order concept to the radial mean bodies and the polar projection body, establishing the associated Zhang-type inequality. In this work, we introduce weighted versions of the above-mentioned operators by replacing the Lebesgue measure with measures that have density. The weighted version of these operators in the $m=1$ case was first done by Roysdon (difference body), Langharst-Roysdon-Zvavitch (polar projection body) and Langharst-Putterman (radial mean bodies). This work can be seen as a sequel to all those works, generalizing them to the higher-order setting. In the last section, we extend many of these ideas to the setting of generalized volume, first introduced by Gardner-Hug-Weil-Xing-Ye.


Title: Sheaves of Probability
Abstract: What does it mean for multiple agents' credence functions to be consistent with each other, if the agents have distinct but overlapping sets of evidence? Mathematical philosopher Michael Titelbaum's rule, called Generalized Conditionalization (GC), sensibly requires each pair of agents to acquire identical credences if they updated on each other's evidence. However, GC allows for paradoxical arrangements of agent credences that we would not like to call consistent. We interpret GC as a gluing condition in the context of sheaf theory, and show that if we further assume that the agents' evidence is logically consistent then the sheaf condition is satisfied and the paradoxes are resolved.


Title: Casson towers and slice links
Abstract: We prove that a Casson tower of height 4 contains a flat embedded disc bounded by the attaching circle, and we prove disc embedding results for height 2 and 3 Casson towers which are embedded into a 4-manifold, with some additional fundamental group assumptions. In the proofs we create a capped grope from a Casson tower and use a refined height raising argument to establish the existence of a symmetric grope which has two layers of caps, data which is sufficient for a topological disc to exist, with the desired boundary. As applications, we present new slice knots and links by giving direct geometric constructions of slicing discs. In particular we construct a family of slice knots which are potential counterexamples to the homotopy ribbon slice conjecture.


Title: A Preliminary Exploration of Floating Point Grammatical Evolution
Abstract: Current GP frameworks are highly effective on a range of real and simulated benchmarks. However, due to the high dimensionality of the genotypes for GP, the task of visualising the fitness landscape for GP search can be difficult. This paper describes a new framework: Floating Point Grammatical Evolution (FP-GE) which uses a single floating point genotype to encode an individual program. This encoding permits easier visualisation of the fitness landscape arbitrary problems by providing a way to map fitness against a single dimension. The new framework also makes it trivially easy to apply continuous search algorithms, such as Differential Evolution, to the search problem. In this work, the FP-GE framework is tested against several regression problems, visualising the search landscape for these and comparing different search meta-heuristics.


Title: Emotionally Numb or Empathetic? Evaluating How LLMs Feel Using EmotionBench
Abstract: Evaluating Large Language Models' (LLMs) anthropomorphic capabilities has become increasingly important in contemporary discourse. Utilizing the emotion appraisal theory from psychology, we propose to evaluate the empathy ability of LLMs, i.e., how their feelings change when presented with specific situations. After a careful and comprehensive survey, we collect a dataset containing over 400 situations that have proven effective in eliciting the eight emotions central to our study. Categorizing the situations into 36 factors, we conduct a human evaluation involving more than 1,200 subjects worldwide. With the human evaluation results as references, our evaluation includes five LLMs, covering both commercial and open-source models, including variations in model sizes, featuring the latest iterations, such as GPT-4 and LLaMA-2. We find that, despite several misalignments, LLMs can generally respond appropriately to certain situations. Nevertheless, they fall short in alignment with the emotional behaviors of human beings and cannot establish connections between similar situations. Our collected dataset of situations, the human evaluation results, and the code of our testing framework, dubbed EmotionBench, is made openly accessible via https://github.com/CUHK-ARISE/EmotionBench. We aspire to contribute to the advancement of LLMs regarding better alignment with the emotional behaviors of human beings, thereby enhancing their utility and applicability as intelligent assistants.


Title: On a Navier-Stokes-Fourier-like system capturing transitions between viscous and inviscid fluid regimes and between no-slip and perfect-slip boundary conditions
Abstract: We study a generalization of the Navier-Stokes-Fourier system for an incompressible fluid where the deviatoric part of the Cauchy stress tensor is related to the symmetric part of the velocity gradient via a maximal monotone 2-graph that is continuously parametrized by the temperature. As such, the considered fluid may go through transitions between three of the following regimes: it can flow as a Bingham fluid for a specific value of the temperature, while it can behave as the Navier-Stokes fluid for another value of the temperature or, for yet another temperature, it can respond as the Euler fluid until a certain activation initiates the response of the Navier-Stokes fluid. At the same time, we regard a generalized threshold slip on the boundary that also may go through various regimes continuously with the temperature. All material coefficients like the dynamic viscosity, friction or activation coefficients are assumed to be temperature-dependent. We establish the large-data and long-time existence of weak solutions, applying the $L^{\infty}$-truncation technique to approximate the velocity field.


Title: Can poachers find animals from public camera trap images?
Abstract: To protect the location of camera trap data containing sensitive, high-target species, many ecologists randomly obfuscate the latitude and longitude of the camera when publishing their data. For example, they may publish a random location within a 1km radius of the true camera location for each camera in their network. In this paper, we investigate the robustness of geo-obfuscation for maintaining camera trap location privacy, and show via a case study that a few simple, intuitive heuristics and publicly available satellite rasters can be used to reduce the area likely to contain the camera by 87% (assuming random obfuscation within 1km), demonstrating that geo-obfuscation may be less effective than previously believed.


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


Title: A characteristic-index inequality for closed embeddings of locally compact groups
Abstract: The characteristic index of a locally compact connected group $G$ is the non-negative integer $d$ for which we have a homeomorphism $G\cong K\times \mathbb{R}^d$ with $K\le G$ maximal compact. We prove that the characteristic indices of closed connected subgroups are dominated by those of the ambient groups.


Title: Chiplet Cloud: Building AI Supercomputers for Serving Large Generative Language Models
Abstract: Large language models (LLMs) such as ChatGPT have demonstrated unprecedented capabilities in multiple AI tasks. However, hardware inefficiencies have become a significant factor limiting the democratization of LLMs. We propose Chiplet Cloud, an ASIC supercomputer architecture that optimizes total cost of ownership (TCO) per token for serving generative LLMs. Chiplet Cloud fits all model parameters inside the on-chip SRAMs to eliminate bandwidth limitations while moderating the die size to improve system costs while leveraging software mappings to overcome data communication overhead. We propose a comprehensive design methodology that accurately explores a spectrum of major design trade-offs in the joint space of hardware-software and generates a detailed performance-cost analysis on all valid design points. We evaluate Chiplet Cloud on four popular LLMs. Compared to GPU and TPU, our architecture can achieve up to 94x and 15x improvement in TCO/Token respectively, significantly reducing the cost for realistically serving modern LLMs.


Title: New obstructions to symplectic embeddings
Abstract: In this paper we establish new restrictions on symplectic embeddings of certain convex domains into symplectic vector spaces. These restrictions are stronger than those implied by the Ekeland-Hofer capacities. By refining an embedding technique due to Guth, we also show that they are sharp.


Title: Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold
Abstract: For a Fano manifold, We consider the geometric quantization of the K\"ahler-Ricci flow and the associated entropy functional. Convergence to the original flow and entropy is established. It is also possible to formulate the finite-dimensional analogue of the optimal degeneration for the anti-canonical polarization.


Title: Numerical Analysis for Dirichlet Optimal Control Problems on Convex Polyhedral Domains
Abstract: In this paper error analysis for finite element discretizations of Dirichlet boundary control problems is developed. For the first time, optimal discretization error estimates are established in the case of three dimensional polyhedral and convex domains. The convergence rates solely depend on the size of largest interior edge angle. These results are comparable to those for the two dimensional case. However, the approaches from the two dimensional setting are not directly extendable such that new techniques have to be used. The theoretical results are confirmed by numerical experiments.


Title: Ravnest: Decentralized Asynchronous Training on Heterogeneous Devices
Abstract: Modern deep learning models, growing larger and more complex, have demonstrated exceptional generalization and accuracy due to training on huge datasets. This trend is expected to continue. However, the increasing size of these models poses challenges in training, as traditional centralized methods are limited by memory constraints at such scales. This paper proposes an asynchronous decentralized training paradigm for large modern deep learning models that harnesses the compute power of regular heterogeneous PCs with limited resources connected across the internet to achieve favourable performance metrics. Ravnest facilitates decentralized training by efficiently organizing compute nodes into clusters with similar data transfer rates and compute capabilities, without necessitating that each node hosts the entire model. These clusters engage in $\textit{Zero-Bubble Asynchronous Model Parallel}$ training, and a $\textit{Parallel Multi-Ring All-Reduce}$ method is employed to effectively execute global parameter averaging across all clusters. We have framed our asynchronous SGD loss function as a block structured optimization problem with delayed updates and derived an optimal convergence rate of $O\left(\frac{1}{\sqrt{K}}\right)$. We further discuss linear speedup with respect to the number of participating clusters and the bound on the staleness parameter.


Title: On the Constructor-Blocker Game
Abstract: In the Constructor-Blocker game, two players, Constructor and Blocker, alternatively claim unclaimed edges of the complete graph $K_n$. For given graphs $F$ and $H$, Constructor can only claim edges that leave her graph $F$-free, while Blocker has no restrictions. Constructor's goal is to build as many copies of $H$ as she can, while Blocker attempts to stop this. The game ends once there are no more edges that Constructor can claim. The score $g(n,H,F)$ of the game is the number of copies of $H$ in Constructor's graph at the end of the game, when both players play optimally and Constructor plays first. In this paper, we extend results of Patk\'os, Stojakovi\'c and Vizer on $g(n, H, F)$ to many pairs of $H$ and $F$: We determine $g(n, H, F)$ when $H=K_r$ and $\chi(F)>r$, also when both $H$ and $F$ are odd cycles, using Szemer\'edi's Regularity Lemma. We also obtain bounds of $g(n, H, F)$ when $H=K_3$ and $F=K_{2,2}$.


Title: Estimation of the incubation time distribution in the singly and doubly interval censored model
Abstract: We analyze nonparametric estimators for the distribution function of the incubation time in the singly and doubly interval censoring model. The classical approach is to use parametric families like Weibull, log-normal or gamma distributions in the estimation procedure. We propose nonparametric estimates which stay closer to the data than the classical parametric methods. We also give explicit limit distributions for discrete versions of the models and apply this to compute confidence intervals. The methods complement the analysis of the continuous model. R scripts for computation of the estimates are provided on https://github.com/pietg/incubationtime.


Title: Generalized Modularity Embedding: a General Framework for Network Embedding
Abstract: The network embedding problem aims to map nodes that are similar to each other to vectors in a Euclidean space that are close to each other. Like centrality analysis (ranking) and community detection, network embedding is in general considered as an ill-posed problem, and its solution may depend on a person's view on this problem. In this book chapter, we adopt the framework of sampled graphs that treat a person's view as a sampling method for a network. The modularity for a sampled graph, called the generalized modularity in the book chapter, is a similarity matrix that has a specific probabilistic interpretation. One of the main contributions of this book chapter is to propose using the generalized modularity matrix for network embedding and show that the network embedding problem can be treated as a trace maximization problem like the community detection problem. Our generalized modularity embedding approach is very general and flexible. In particular, we show that the Laplacian eigenmaps is a special case of our generalized modularity embedding approach. Also, we show that dimensionality reduction can be done by using a particular sampled graph. Various experiments are conducted on real datasets to illustrate the effectiveness of our approach.


Title: Large deviation principle for a two-time-scale McKean-Vlasov model with jumps
Abstract: This work focus on the large deviation principle for a two-time scale McKean-Vlasov system with jumps. Based on the variational framework of the McKean-Vlasov system with jumps, it is turned into weak convergence for the controlled system. Unlike general two-time scale system, the controlled McKean-Vlasov system is related to the law of the original system, which causes difficulties in qualitative analysis. In solving this problem, employing asymptotics of the original system and a Khasminskii-type averaging principle together is efficient. Finally, it is shown that the limit is related to the Dirac measure of the solution to the ordinary differential equation.


Title: On an open problem about a class of optimal ternary cyclic codes
Abstract: Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth. Let $C_{(1,e)}$ be a cyclic code of length $3^m-1$ over GF(3) with two nonzeros $\alpha$ and $\alpha^e$, where $\alpha$ is a generator of $GF(3^m)^*$ and e is a given integer. It is shown that $C_{(1,e)}$ is optimal with parameters $[3^m-1,3^m-1-2m,4]$ if one of the following conditions is met. 1) $m\equiv0(\mathrm{mod}~ 4)$, $m\geq 4$, and $e=3^\frac{m}{2}+5$. 2) $m\equiv2(\mathrm{mod}~ 4)$, $m\geq 6$, and $e=3^\frac{m+2}{2}+5$.


Title: Suppression of blow-up in 3-D Keller-Segel model via Couette flow in whole space
Abstract: In this paper, we study the 3-D parabolic-parabolic and parabolic-elliptic Keller-Segel models with Couette flow in $\mathbb{R}^3$. We prove that the blow-up phenomenon of solution can be suppressed by enhanced dissipation of large Couette flows. Here we develop Green's function method to describe the enhanced dissipation via a more precise space-time structure and obtain the global existence together with pointwise estimates of the solutions. The result of this paper shows that the enhanced dissipation exists for all frequencies in the case of whole space and it is reason that we obtain global existence for 3-D Keller-Segel models here. It is totally different from the case with the periodic spatial variable $x$ in [2,10]. This paper provides a new methodology to capture dissipation enhancement and also a surprising result which shows a totally new mechanism.


Title: Efficient tensor tomography in fan-beam coordinates
Abstract: We propose a thorough analysis of the tensor tomography problem on the Euclidean unit disk parameterized in fan-beam coordinates. This includes, for the inversion of the Radon transform over functions, using another range characterization first appearing in [Pestov-Uhlmann, IMRN 2004] to enforce in a fast way classical moment conditions at all orders. When considering direction-dependent integrands (e.g., tensors), a problem where injectivity no longer holds, we propose a suitable representative (other than the traditionally sought-after solenoidal candidate) to be reconstructed, as well as an efficient procedure to do so. Numerical examples illustrating the method are provided at the end.


Title: Geometries arising from trilinear forms on low-dimensional vector spaces
Abstract: Let ${\mathcal G}_k(V)$ be the $k$-Grassmannian of a vector space $V$ with $\dim V=n$. Given a hyperplane $H$ of ${\mathcal G}_k(V)$, we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating $k$-linear forms, J. Algebraic Combin. doi:10.1007/s10801-016-0730-6] a point-line subgeometry of ${\mathrm{PG}}(V)$ called the {\it geometry of poles of $H$}. In the present paper, exploiting the classification of alternating trilinear forms in low dimension, we characterize the possible geometries of poles arising for $k=3$ and $n\leq 7$ and propose some new constructions. We also extend a result of [J.Draisma, R. Shaw, Singular lines of trilinear forms, Linear Algebra Appl. doi:10.1016/j.laa.2010.03.040] regarding the existence of line spreads of ${\mathrm{PG}}(5,{\mathbb K})$ arising from hyperplanes of ${\mathcal G}_3(V).$


Title: k-Winners-Take-All Ensemble Neural Network
Abstract: Ensembling is one approach that improves the performance of a neural network by combining a number of independent neural networks, usually by either averaging or summing up their individual outputs. We modify this ensembling approach by training the sub-networks concurrently instead of independently. This concurrent training of sub-networks leads them to cooperate with each other, and we refer to them as "cooperative ensemble". Meanwhile, the mixture-of-experts approach improves a neural network performance by dividing up a given dataset to its sub-networks. It then uses a gating network that assigns a specialization to each of its sub-networks called "experts". We improve on these aforementioned ways for combining a group of neural networks by using a k-Winners-Take-All (kWTA) activation function, that acts as the combination method for the outputs of each sub-network in the ensemble. We refer to this proposed model as "kWTA ensemble neural networks" (kWTA-ENN). With the kWTA activation function, the losing neurons of the sub-networks are inhibited while the winning neurons are retained. This results in sub-networks having some form of specialization but also sharing knowledge with one another. We compare our approach with the cooperative ensemble and mixture-of-experts, where we used a feed-forward neural network with one hidden layer having 100 neurons as the sub-network architecture. Our approach yields a better performance compared to the baseline models, reaching the following test accuracies on benchmark datasets: 98.34% on MNIST, 88.06% on Fashion-MNIST, 91.56% on KMNIST, and 95.97% on WDBC.


Title: A spectral result for Hardy inequalities
Abstract: Let P be a linear, second order, elliptic operator satisfying a Hardy inequality with potential W (i.e. $P-W\geq0$) and best constant $\alpha$. We give conditions so that the spectrum of $W^{-1}P$ is $[\alpha,\infty)$. We apply this to several well-known Hardy inequalities: (improved) Hardy inequalities on a bounded convex domain with potential involving the distance to the boundary, and Hardy inequalities for minimal submanifolds of the Euclidean space.


Title: Perfect matchings and Hamilton cycles in uniform attachment graphs
Abstract: We study Hamilton cycles and perfect matchings in a uniform attachment graph. In this random graph, vertices are added sequentially, and when a vertex $t$ is created, it makes $k$ independent and uniform choices from $\{1,\dots,t-1\}$ and attaches itself to these vertices. Improving the results of Frieze, P\'erez-Gim\'enez, Pra\l{}at and Reiniger (2019), we show that, with probability approaching 1 as $n$ tends to infinity, a uniform attachment graph on $n$ vertices has a perfect matching for $k \ge 5$ and a Hamilton cycle for $k\ge 13$. One of the ingredients in our proofs is the identification of a subset of vertices that is least likely to expand, which provides us with better expansion rates than the existing ones.


Title: Influence Maximization in Ising Models
Abstract: Given a complex high-dimensional distribution over $\{\pm 1\}^n$, what is the best way to increase the expected number of $+1$'s by controlling the values of only a small number of variables? Such a problem is known as influence maximization and has been widely studied in social networks, biology, and computer science. In this paper, we consider influence maximization on the Ising model which is a prototypical example of undirected graphical models and has wide applications in many real-world problems. We establish a sharp computational phase transition for influence maximization on sparse Ising models under a bounded budget: In the high-temperature regime, we give a linear-time algorithm for finding a small subset of variables and their values which achieve nearly optimal influence; In the low-temperature regime, we show that the influence maximization problem cannot be solved in polynomial time under commonly-believed complexity assumption. The critical temperature coincides with the tree uniqueness/non-uniqueness threshold for Ising models which is also a critical point for other computational problems including approximate sampling and counting.


Title: Characterizations and Constructions of Linear Intersection Pairs of Cyclic Codes over Finite Fields
Abstract: Linear intersection pairs of linear codes have become of interest due to their nice algebraic properties and wide applications. In this paper, we focus on linear intersection pairs of cyclic codes over finite fields. Some properties of cyclotomic cosets in cyclic groups are presented as key tools in the study of such linear intersection pairs. Characterization and constructions of two cyclic codes of a fixed intersecting dimension are given in terms of their generator polynomials and cyclotomic cosets. In some cases, constructions of two cyclic codes of a fixed intersecting subcode are presented as well. Based on the theoretical characterization, some numerical examples of linear intersection pairs of cyclic codes with good parameters are illustrated.


Title: The ZFC analogue of the Halpern-Levy theorem
Abstract: Here we present ZFC theorems yielding the Halpern-L\a"uchli theorem and avoiding metamathematical notions in their formulations.


Title: A determinantal formula for orthosymplectic Schur functions
Abstract: We prove a new determinantal formula for the characters of irreducible representations of orthosymplectic Lie superalgebras analogous to the formula developed by Moens and Jeugt (J. Algebraic Combin., 2003) for general linear Lie superalgebras. Our proof uses the Jacobi--Trudi type formulas for orthosymplectic characters. As a consequence, we show that the odd symplectic characters introduced by Proctor (Invent. Math., 1988) are the same as the orthosymplectic characters with some specialized indeterminates. We also give a generalization of an odd symplectic character identity due to Brent, Krattenthaler and Warnaar (J. Combin. Theory Ser. A, 2016).


Title: Non-conforming FEM for the quasi-static contact problem
Abstract: In this article, we addressed the numerical solution of a non-linear evolutionary variational inequality, which is encountered in the investigation of quasi-static contact problems. Our study encompasses both the semi-discrete and fully-discrete schemes, where we employ the backward Euler method for time discretization and utilize the lowest order Crouzeix-Raviart nonconforming finite element method for spatial discretization. By assuming appropriate regularity conditions on the solution, we establish \emph{a priori} error analysis for these schemes, achieving the optimal convergence order for linear elements. To illustrate the numerical convergence rates, we provide numerical results on a two-dimensional test problem.


Title: Mid-point embedding of Hamiltonian systems and variational integrators
Abstract: Following the discrete embedding formalism, we give a new derivation of the mid-point variational integrators as developed by J.M. Wendlandt and J.E. Marsden by defining an adapted order two discrete differential and integral calculus. This allows us to obtain a clearer correspondence between the discrete and continuous case. We also discuss the corresponding definition of a discrete Hamiltonian system. A complete comparaison with the results of J.M. Wendlandt and J.E. Marsden is provided.


Title: Faster Projection-Free Augmented Lagrangian Methods via Weak Proximal Oracle
Abstract: This paper considers a convex composite optimization problem with affine constraints, which includes problems that take the form of minimizing a smooth convex objective function over the intersection of (simple) convex sets, or regularized with multiple (simple) functions. Motivated by high-dimensional applications in which exact projection/proximal computations are not tractable, we propose a \textit{projection-free} augmented Lagrangian-based method, in which primal updates are carried out using a \textit{weak proximal oracle} (WPO). In an earlier work, WPO was shown to be more powerful than the standard \textit{linear minimization oracle} (LMO) that underlies conditional gradient-based methods (aka Frank-Wolfe methods). Moreover, WPO is computationally tractable for many high-dimensional problems of interest, including those motivated by recovery of low-rank matrices and tensors, and optimization over polytopes which admit efficient LMOs. The main result of this paper shows that under a certain curvature assumption (which is weaker than strong convexity), our WPO-based algorithm achieves an ergodic rate of convergence of $O(1/T)$ for both the objective residual and feasibility gap. This result, to the best of our knowledge, improves upon the $O(1/\sqrt{T})$ rate for existing LMO-based projection-free methods for this class of problems. Empirical experiments on a low-rank and sparse covariance matrix estimation task and the Max Cut semidefinite relaxation demonstrate that of our method can outperform state-of-the-art LMO-based Lagrangian-based methods.


Title: Two dimensional dimers beyond planarity
Abstract: We consider a generalisation of the double dimer model which includes several models of interest, such as the monomer double dimer model, the dimer model, the Spin O(N) model, and it is related to the loop O(N) model. We prove that on two-dimension like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices converge to zero as the distance between such vertices gets large. Our analysis is by introducing a new (complex) spin representation for all models belonging to this class, and by deriving a new proof of the Mermin-Wagner theorem which does not require the positivity of the Gibbs measure. Even for the well studied dimer and double dimer model our results are new since - not relying on solvability and Kasteleyn's theorem - they hold beyond the framework of planar graphs.


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


Title: Spatiotemporal Monitoring of Epidemics via Solution of a Coefficient Inverse Problem
Abstract: Let S,I and R be susceptible, infected and recovered populations in a city affected by an epidemic. The SIR model of Lee, Liu, Tembine, Li and Osher, \emph{SIAM J. Appl. Math.},~81, 190--207, 2021 of the spatiotemoral spread of epidemics is considered. This model consists of a system of three nonlinear coupled parabolic Partial Differential Equations with respect to the space and time dependent functions S,I and R. For the first time, a Coefficient Inverse Problem (CIP) for this system is posed. The so-called \textquotedblleft convexification" numerical method for this inverse problem is constructed. The presence of the Carleman Weight Function (CWF) in the resulting regularization functional ensures the global convergence of the gradient descent method of the minimization of this functional to the true solution of the CIP, as long as the noise level tends to zero. The CWF is the function, which is used as the weight in the Carleman estimate for the corresponding Partial Differential Operator. Numerical studies demonstrate an accurate reconstruction of unknown coefficients as well as S,I,R functions inside of that city. As a by-product, uniqueness theorem for this CIP is proven. Since the minimal measured input data are required, then the proposed methodology has a potential of a significant decrease of the cost of monitoring of epidemics.


Title: Hochschild and cyclic Homologies with bounded poles
Abstract: We show that the classical Hochschild homology and (periodic and negative) cyclic homology groups are representable in the category of motives with modulus. We do this by constructing Hochschild homology and (periodic and negative) cyclic homologies for modulus pairs. We show a modulus version of HKR theorem, that is, there exists an isomorphism between modulus Hochschild homology and modulus K\"ahler differentials for affine normal crossing modulus pairs. By using the representability of modulus Hodge sheaves in the category of motives with modulus, we construct an object of the category of motives with modulus which represents modulus Hochschild homology. Similarly, We compare modulus de Rham cohomology and modulus cyclic homologies and obtain a representability of modulus cyclic homologies.


Title: Category-Level 6D Object Pose Estimation with Flexible Vector-Based Rotation Representation
Abstract: In this paper, we propose a novel 3D graph convolution based pipeline for category-level 6D pose and size estimation from monocular RGB-D images. The proposed method leverages an efficient 3D data augmentation and a novel vector-based decoupled rotation representation. Specifically, we first design an orientation-aware autoencoder with 3D graph convolution for latent feature learning. The learned latent feature is insensitive to point shift and size thanks to the shift and scale-invariance properties of the 3D graph convolution. Then, to efficiently decode the rotation information from the latent feature, we design a novel flexible vector-based decomposable rotation representation that employs two decoders to complementarily access the rotation information. The proposed rotation representation has two major advantages: 1) decoupled characteristic that makes the rotation estimation easier; 2) flexible length and rotated angle of the vectors allow us to find a more suitable vector representation for specific pose estimation task. Finally, we propose a 3D deformation mechanism to increase the generalization ability of the pipeline. Extensive experiments show that the proposed pipeline achieves state-of-the-art performance on category-level tasks. Further, the experiments demonstrate that the proposed rotation representation is more suitable for the pose estimation tasks than other rotation representations.


Title: The ghosts of departed quantities in switches and transitions
Abstract: Transitions between steady dynamical regimes in diverse applications are often modelled using discontinuities, but doing so introduces problems of uniqueness. No matter how quickly a transition occurs, its inner workings can affect the dynamics of the system significantly. Here we discuss the way transitions can be reduced to discontinuities without trivializing them, by preserving so-called hidden terms. We review the fundamental methodology, its motivations, and where their study seems to be heading. We derive a prototype for piecewise smooth models from the asymptotics of systems with rapid transitions, sharpening Filippov's convex combinations by encoding the tails of asymptotic series into nonlinear dependence on a switching parameter. We present a few examples that illustrate the impact of these on our standard picture of smooth or only piecewise smooth dynamics.


Title: On the Effects of Battery Imperfections in an Energy Harvesting Device
Abstract: Energy Harvesting allows the devices in a Wireless Sensor Network to recharge their batteries through environmental energy sources. While in the literature the main focus is on devices with ideal batteries, in reality several inefficiencies have to be considered to correctly design the operating regimes of an Energy Harvesting Device (EHD). In this work we describe how the throughput optimization problem changes under \emph{real battery} constraints in an EHD. In particular, we consider imperfect knowledge of the state of charge of the battery and storage inefficiencies, \emph{i.e.}, part of the harvested energy is wasted in the battery recharging process. We formulate the problem as a Markov Decision Process, basing our model on some realistic observations about transmission, consumption and harvesting power. We find the performance upper bound with a real battery and numerically discuss the novelty introduced by the real battery effects. We show that using the \emph{old} policies obtained without considering the real battery effects is strongly sub-optimal and may even result in zero throughput.


Title: CBD: A Certified Backdoor Detector Based on Local Dominant Probability
Abstract: Backdoor attack is a common threat to deep neural networks. During testing, samples embedded with a backdoor trigger will be misclassified as an adversarial target by a backdoored model, while samples without the backdoor trigger will be correctly classified. In this paper, we present the first certified backdoor detector (CBD), which is based on a novel, adjustable conformal prediction scheme based on our proposed statistic local dominant probability. For any classifier under inspection, CBD provides 1) a detection inference, 2) the condition under which the attacks are guaranteed to be detectable for the same classification domain, and 3) a probabilistic upper bound for the false positive rate. Our theoretical results show that attacks with triggers that are more resilient to test-time noise and have smaller perturbation magnitudes are more likely to be detected with guarantees. Moreover, we conduct extensive experiments on four benchmark datasets considering various backdoor types, such as BadNet, CB, and Blend. CBD achieves comparable or even higher detection accuracy than state-of-the-art detectors, and it in addition provides detection certification. Notably, for backdoor attacks with random perturbation triggers bounded by $\ell_2\leq0.75$ which achieves more than 90\% attack success rate, CBD achieves 100\% (98\%), 100\% (84\%), 98\% (98\%), and 72\% (40\%) empirical (certified) detection true positive rates on the four benchmark datasets GTSRB, SVHN, CIFAR-10, and TinyImageNet, respectively, with low false positive rates.


Title: Starling: An I/O-Efficient Disk-Resident Graph Index Framework for High-Dimensional Vector Similarity Search on Data Segment
Abstract: High-dimensional vector similarity search (HVSS) is receiving a spotlight as a powerful tool for various data science and AI applications. As vector data grows larger, in-memory indexes become extremely expensive because they necessitate substantial expansion of main memory resources. One possible solution is to use disk-based implementation, which stores and searches vector data in high-performance devices like NVMe SSDs. However, HVSS for data segments is still challenging in vector databases, where one machine has multiple segments for system features (like scaling) purposes. In this setting, each segment has limited memory and disk space, so HVSS on the data segment needs to balance accuracy, efficiency, and space cost. Existing disk-based methods are sub-optimal because they do not consider all these requirements together. In this paper, we present Starling, an I/O-efficient disk-resident graph index framework that optimizes data layout and search strategy in the segment. It has two main components: (1) a data layout that includes an in-memory navigation graph and a reordered disk-based graph with locality enhancement, which reduces the search path length and disk bandwidth wastage; and (2) a block search strategy that minimizes expensive disk I/Os when executing a vector query. We conduct extensive experiments to verify Starling's effectiveness, efficiency, and scalability. On a data segment with 2GB memory and 10GB disk capacity, Starling can maintain up to 33 million vectors in 128 dimensions, and serve HVSS with more than 0.9 average precision and top-10 recall rate, and latency of under 1 millisecond. The results show that Starling exhibits 43.9$\times$ higher throughput with 98% lower query latency than state-of-the-art methods under the same accuracy.


Title: $L^p$-spectral theory for the Laplacian on forms
Abstract: In this article, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the $L^p$-spectrum of the Laplacian on $k$-forms, and also prove the decomposition of the $L^p$-spectrum depending on the order of the forms. We then show that the resolvent set of an operator such as the Laplacian on $L^p$ lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. We conclude by providing a detailed description of the $L^p$ spectrum of the Laplacian on $k$-forms over hyperbolic space.


Title: On the number of perfect matchings in random lifts
Abstract: Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let X_G be the number of perfect matchings in a random lift of G. We study the distribution of X_G in the limit as n tends to infinity, using the small subgraph conditioning method. We present several results including an asymptotic formula for the expectation of X_G when G is d-regular, d\geq 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of X_G, with full details given for two example multigraphs, including the complete graph K_4. To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.


Title: Low level definability above large cardinals
Abstract: We study some connections between definability in generalized descriptive set theory and large cardinals, particularly measurable cardinals and limits thereof, working in ZFC. We show that if $\kappa$ is a limit of measurable cardinals then there is no $\Sigma_1(H_\kappa\cup\mathrm{OR})$ wellorder of a subset of $P(\kappa)$ of length $\geq\kappa^+$; this answers a question of L\"ucke and M\"uller. However, in $M_1$, the minimal proper class mouse with a Woodin cardinal, for every uncountable cardinal $\kappa$ which is not a limit of measurables, there is a $\Sigma_1(H_\kappa\cup\{\kappa\})$ good wellorder of $H_{\kappa^+}$. If $\kappa$ is a limit of measurables then there is no $\Sigma_1(H_\kappa\cup\mathrm{OR})$ mad family $F\subseteq P(\kappa)$ of cardinality $>\kappa$, and if also $\mathrm{cof}(\kappa)>\omega$ then there is no $\Sigma_1(H_\kappa\cup\mathrm{OR})$ almost disjoint family $F\subseteq P(\kappa)$ of cardinality $>\kappa$. However, relative to the consistency of large cardinals, $\Pi_1(\{\kappa\})$ mad families and maximal independent families $F\subseteq P(\kappa)$ can exist, when $\kappa$ is a limit of measurables, and even more. We also examine some of the features of $L[U]$, and answer another question of L\"ucke and M\"uller, showing that if $\kappa$ is a weakly compact cardinal such that every $\Sigma_1(H_\kappa\cup\{\kappa\})$ subset of $P(\kappa)$ of cardinality $>\kappa$ has a subset which is the range of a perfect function, then there is an inner model satisfying "there is a weakly compact limit of measurable cardinals".


Title: Twisted restricted conformal blocks of vertex operator algebras I: $g$-twisted correlation functions and fusion rules
Abstract: In this paper, we introduce a notion of $g$-twisted restricted conformal block on the three-pointed twisted projective line $\mathfrak{x}\colon\overline{C}\to\mathbb{P^1}$ associated with an untwisted module $M^1$ and the bottom levels of two $g$-twisted modules $M^2$ and $M^3$ over a vertex operator algebra $V$. We show that the space of twisted restricted conformal blocks is isomorphic to the space of $g$-twisted (restricted) correlation functions defined by the same datum and to the space of intertwining operators among these twisted modules. As an application, we derive a twisted version of the Fusion Rules Theorem.


Title: Optimal jump set in hyperbolic conservation laws
Abstract: This paper deals with some qualitative properties of entropy solutions to hyperbolic conservation laws. In [11] the jump set of entropy solution to conservation laws has been introduced. We find an entropy solution to scalar conservation laws for which the jump set is not closed, in particular, it is dense in a space-time domain. In the later part of this article, we obtain a similar result for the hyperbolic system. We give two different approaches for scalar conservation laws and hyperbolic system to obtain the results. For the scalar case, obtained solutions are more explicitly calculated.


Title: Marginal Debiased Network for Fair Visual Recognition
Abstract: Deep neural networks (DNNs) are often prone to learn the spurious correlations between target classes and bias attributes, like gender and race, inherent in a major portion of training data (bias-aligned samples), thus showing unfair behavior and arising controversy in the modern pluralistic and egalitarian society. In this paper, we propose a novel marginal debiased network (MDN) to learn debiased representations. More specifically, a marginal softmax loss (MSL) is designed by introducing the idea of margin penalty into the fairness problem, which assigns a larger margin for bias-conflicting samples (data without spurious correlations) than for bias-aligned ones, so as to deemphasize the spurious correlations and improve generalization on unbiased test criteria. To determine the margins, our MDN is optimized through a meta learning framework. We propose a meta equalized loss (MEL) to perceive the model fairness, and adaptively update the margin parameters by metaoptimization which requires the trained model guided by the optimal margins should minimize MEL computed on an unbiased meta-validation set. Extensive experiments on BiasedMNIST, Corrupted CIFAR-10, CelebA and UTK-Face datasets demonstrate that our MDN can achieve a remarkable performance on under-represented samples and obtain superior debiased results against the previous approaches.


Title: Wasserstein gradient flows from large deviations of thermodynamic limits
Abstract: We study the Fokker-Planck equation as the hydrodynamic limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the rate functional, that characterizes the large deviations from the hydrodynamic limit, in a way that the free energy appears explicitly. Next we use this formulation via the contraction principle to prove that the discreet time rate functional is asymptotically equivalent in the Gamma-convergence sense to the functional derived from the Wasserstein gradient discretization scheme.


Title: The quantitative behaviour of polynomial orbits on nilmanifolds
Abstract: A theorem of Leibman asserts that a polynomial orbit $(g(1),g(2),g(3),\ldots)$ on a nilmanifold $G/\Gamma$ is always equidistributed in a union of closed sub-nilmanifolds of $G/\Gamma$. In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit $(g(1),\ldots,g(N))$ in a nilmanifold. More specifically we show that there is a factorization $g = \epsilon g'\gamma$, where $\epsilon(n)$ is "smooth", $\gamma(n)$ is periodic and "rational", and $(g'(a),g'(a+d),\ldots,g'(a + d(l-1)))$ is uniformly distributed (up to a specified error $\delta$) inside some subnilmanifold $G'/\Gamma'$ of $G/\Gamma$, for all sufficiently dense arithmetic progressions $a,a+d,\ldots,a+d(l-1)$ inside $\{1,..,N\}$. Our bounds are uniform in $N$ and are polynomial in the error tolerance delta. In a subsequent paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from our earlier paper "Linear equations in primes".


Title: Selection Collider Bias in Large Language Models
Abstract: In this paper we motivate the causal mechanisms behind sample selection induced collider bias (selection collider bias) that can cause Large Language Models (LLMs) to learn unconditional dependence between entities that are unconditionally independent in the real world. We show that selection collider bias can become amplified in underspecified learning tasks, and although difficult to overcome, we describe a method to exploit the resulting spurious correlations for determination of when a model may be uncertain about its prediction. We demonstrate an uncertainty metric that matches human uncertainty in tasks with gender pronoun underspecification on an extended version of the Winogender Schemas evaluation set, and we provide an online demo where users can apply our uncertainty metric to their own texts and models.


Title: On the local density problem for graphs of given odd-girth
Abstract: Erd\H{o}s conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n/2\rfloor$ vertices that spans at most $n^2/50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs homomorphic to so-called Andr\'asfai graphs. As a consequence, Erd\H{o}s' conjecture holds for every triangle-free graph $G$ with minimum degree $\delta (G)>10n/29$ and if $\chi (G)\leq 3$ the degree condition can be relaxed to $\delta (G)> n/3$. In fact, we obtain a more general result for graphs of higher odd-girth.


Title: Moonshot: Optimizing Chain-Based Rotating Leader BFT via Optimistic Proposals
Abstract: Existing chain-based rotating leader BFT SMR protocols for the partially synchronous network model that commit blocks with $O(1)$ minimum latency have block periods of at least $2\delta$ (where $\delta$ is the message transmission latency). While a protocol with a block period of $\delta$ exists under the synchronous model, its minimum commit latency is linear in the size of the system. To close this gap, we present the first chain-based BFT SMR protocols with best-case delays of $\delta$ between the proposals of distinct honest leaders, and minimum commit latencies of $3\delta$. We present three protocols for the partially synchronous network model under different notions of optimistic responsiveness, two of which implement pipelining and one of which does not. All of our protocols achieve reorg resilience and two have short view lengths; properties that many existing chain-based BFT SMR protocols lack. We experimentally evaluate our protocols and show that they achieve significant increases in throughput and reductions in latency compared to the state-of-the-art, Jolteon. Our results also demonstrate that techniques commonly employed to reduce communication complexity$\unicode{x2014}$such as vote-pipelining and the use of designated vote-aggregators$\unicode{x2014}$actually reduce practical performance in many settings.


Title: Quantum Geometry, Integrability, and Opers
Abstract: This review article discusses recent progress in understanding of various families of integrable models in terms of algebraic geometry, representation theory, and physics. In particular, we address the connections between soluble many-body systems of Calogero-Ruijsenaars type, quantum spin chains, spaces of opers, representations of double affine Hecke algebras, enumerative counts to quiver varieties, to name just a few. We formulate several conjectures and open problems. This is a contribution to the proceedings of the conference on Elliptic Integrable Systems and Representation Theory, which was held in August 2023 at University of Tokyo.


Title: Lengths of three simple periodic geodesics on a Riemannian $2$-sphere
Abstract: Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$. We also present an upper bound that depends only on the area and diameter for the lengths of the three simple periodic geodesics with positive indices that appear as minimax critical values in the classical proofs of the Lyusternik-Shnirelman theorem. Finally, we present better bounds for these three lengths for "thin" spheres, when the area $A$ is much less than $d^2$, where the bounds for the lengths of the first two simple periodic geodesics are asymptotically optimal, when ${A\over d^2}\longrightarrow 0$.


Title: The distance signatures of the incidence graphs of affine resolvable designs
Abstract: In this note, we determined the distance signatures of the incidence matrices of affine resolvable designs. This proves a conjecture by Kohei Yamada.


Title: REDriver: Runtime Enforcement for Autonomous Vehicles
Abstract: Autonomous driving systems (ADSs) integrate sensing, perception, drive control, and several other critical tasks in autonomous vehicles, motivating research into techniques for assessing their safety. While there are several approaches for testing and analysing them in high-fidelity simulators, ADSs may still encounter additional critical scenarios beyond those covered once they are deployed on real roads. An additional level of confidence can be established by monitoring and enforcing critical properties when the ADS is running. Existing work, however, is only able to monitor simple safety properties (e.g., avoidance of collisions) and is limited to blunt enforcement mechanisms such as hitting the emergency brakes. In this work, we propose REDriver, a general and modular approach to runtime enforcement, in which users can specify a broad range of properties (e.g., national traffic laws) in a specification language based on signal temporal logic (STL). REDriver monitors the planned trajectory of the ADS based on a quantitative semantics of STL, and uses a gradient-driven algorithm to repair the trajectory when a violation of the specification is likely. We implemented REDriver for two versions of Apollo (i.e., a popular ADS), and subjected it to a benchmark of violations of Chinese traffic laws. The results show that REDriver significantly improves Apollo's conformance to the specification with minimal overhead.


Title: Beyond Extraction: Contextualising Tabular Data for Efficient Summarisation by Language Models
Abstract: The conventional use of the Retrieval-Augmented Generation (RAG) architecture has proven effective for retrieving information from diverse documents. However, challenges arise in handling complex table queries, especially within PDF documents containing intricate tabular structures.This research introduces an innovative approach to enhance the accuracy of complex table queries in RAG-based systems. Our methodology involves storing PDFs in the retrieval database and extracting tabular content separately. The extracted tables undergo a process of context enrichment, concatenating headers with corresponding values. To ensure a comprehensive understanding of the enriched data, we employ a fine-tuned version of the Llama-2-chat language model for summarisation within the RAG architecture. Furthermore, we augment the tabular data with contextual sense using the ChatGPT 3.5 API through a one-shot prompt. This enriched data is then fed into the retrieval database alongside other PDFs. Our approach aims to significantly improve the precision of complex table queries, offering a promising solution to a longstanding challenge in information retrieval.


Title: Ramsey numbers for bipartite graphs with small bandwidth
Abstract: We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.


Title: Analysis of Twisted Supercharge Families on Product Manifolds
Abstract: Twisted supercharge families on product manifolds $\mathbb{T} \times M$ have been applied in the analysis of the odd twisted K-theory. We shall suspend these families to the even twisted K-theory and solve their twisted families index problem. This is applied to give analytic representatives of the twisted K-theory classes on tori - including all the torsion classes.


Title: On Picard groups and Jacobians of directed graphs
Abstract: The Picard group of an undirected graph is a finitely generated abelian group, and the Jacobian is the torsion subgroup of the Picard group. These groups can be computed by using the Smith normal form of the Laplacian matrix of the graph or by using chip-firing games associated with the graph. One may consider its generalization to directed graphs based on the Laplacian matrix. We compute Picard groups and Jacobians for several classes of directed trees, cycles, wheel, and multipartite graphs.


Title: Mitigating Procrastination in Crowdsourcing Via Efficient Scheduling Algorithm
Abstract: Several works related to crowdsourcing have been proposed in the direction where the task executors are to perform the tasks within the stipulated deadlines. Though the deadlines are set, it may be a practical scenario that majority of the task executors submit the tasks as late as possible. This situation where the task executors may delay their task submission is termed as procrastination in behavioural economics. In many applications, these late submission of tasks may be problematic for task requesters. In literature, how to prevent this procrastination within the deadline is not addressed in crowdsourcing scenario. However, in a bipartite graph setting one procrastination aware scheduling is proposed but balanced job distribution in different slots (also termed as schedules) is not considered there. In this paper, a procrastination aware scheduling of jobs is proliferated by proposing an (randomized) algorithm in crowdsourcing scenario (also applicable in mobile and spatial crowdsourcing). Our algorithm ensures that balancing of jobs in different schedules are maintained. Our scheme is compared with the existing algorithm through extensive simulation and in terms of balancing effect, our proposed algorithm outperforms the existing one. Analytically it is shown that our proposed algorithm maintains the balanced distribution.


Title: Extended Special Linear group $ESL_2(\mathbb{F})$ and square roots in matrix groups $SL_2(\mathbb{F})$, $SL_2(\mathbb{Z})$, $ESL_2(\mathbb{F})$, $GL_2(\mathbb{F}_p)$
Abstract: First time, we introduce Extended special linear group $ESL_2(F)$, which is generalization of matrix group $SL_2(F)$ over arbitrary field $F$. Extended special linear group $ESL_2(k)$, where $k$ is arbitrary perfect field, is storage of all square matrix roots from $ESL_2(k)$. The analytical formulas of roots of 2-nd, 3-rd, 4-th and $n$-th powers in $ SL_2(\mathbb{F}_p)$ are found by us. Also for roots in $ SL_2(\mathbb{Z})$, $ ESL_2(\mathbb{Z})$ and in $ SL_2({k})$ as well as in $ESL_2({k})$, where $k$ is arbitrary perfect field, is found by us. New linear group which is storage of square roots from $ SL_2{\mathbb{F}_p}$ is found and investigated by us. The criterion of roots existing for different classes of matrix -- simple and semisimple matrixes from $ SL_2({\mathbb{F}_p})$, $ SL_2({\mathbb{Z}})$ are established. The problems of square root from group element existing in $SL_2(F_p)$, $SL_2(F_p)$ and $GL_2(F_p)$ for arbitrary prime $p$ are solved in this paper. The similar goal of root finding was reached in the GM algorithm adjoining an $n$-th root of a generator \cite{GM} results in a discrete group for group $SL(2,R)$, but we consider this question over finite field $F_p$. Over method gives answer about existing $\sqrt{ M^n}$ without exponenting $M$ to $n$-th power. We only use the trace of $M$ or only eigenvalues of $M$. In \cite{Amit} only the Anisotropic case of group $SL_1(Q)$, where $Q$ is a quaternion division algebra over $k$ was considered. The authors of \cite{Amit} considered criterion to be square only for the case $F_p$ is a field of characteristics not equal 2. We solve this problem even for fields $F_2$ and $F_{2^n}$. The criterion to $g \in SL_2 (F_2)$ be square in $SL_2(F_2)$ was not found by them what was declared in a separate sentence in \cite{Amit}. We consider more general case \cite{SkSq} consisting in whole group $G= SL_2(F_q)$.


Title: Application of the Cartier Operator in Coding Theory
Abstract: The $a$-number is an invariant of the isomorphism class of the $p$-torsion group scheme. We use the Cartier operator on $H^0(\mathcal{A}_2,\Omega^1)$ to find a closed formula for the $a$-number of the form $\mathcal{A}_2 = v(Y^{\sqrt{q}}+Y-x^{\frac{\sqrt{q}+1}{2}})$ where $q=p^s$ over the finite field $\mathbb{F}_{q^2}$. The application of the computed $a$-number in coding theory is illustrated by the relationship between the algebraic properties of the curve and the parameters of codes that are supported by it.


Title: The continuum limit of higher-order Follow-the-Leader models
Abstract: We study a generalized Follow-the-Leader model where the driver considers the position of an arbitrary but finite number of vehicles ahead, as well as the position of the vehicle directly behind the driver. It is proved that this model converges to the classical Lighthill-Whitham-Richards model for traffic flow when traffic becomes dense. This also underscores the robustness of the Lighthill-Whitham-Richards model.


Title: Multi-scale Discriminant Saliency with Wavelet-based Hidden Markov Tree Modelling
Abstract: The bottom-up saliency, an early stage of humans' visual attention, can be considered as a binary classification problem between centre and surround classes. Discriminant power of features for the classification is measured as mutual information between distributions of image features and corresponding classes . As the estimated discrepancy very much depends on considered scale level, multi-scale structure and discriminant power are integrated by employing discrete wavelet features and Hidden Markov Tree (HMT). With wavelet coefficients and Hidden Markov Tree parameters, quad-tree like label structures are constructed and utilized in maximum a posterior probability (MAP) of hidden class variables at corresponding dyadic sub-squares. Then, a saliency value for each square block at each scale level is computed with discriminant power principle. Finally, across multiple scales is integrated the final saliency map by an information maximization rule. Both standard quantitative tools such as NSS, LCC, AUC and qualitative assessments are used for evaluating the proposed multi-scale discriminant saliency (MDIS) method against the well-know information based approach AIM on its released image collection with eye-tracking data. Simulation results are presented and analysed to verify the validity of MDIS as well as point out its limitation for further research direction.


Title: Stationary Distributions for Two-Dimensional Sticky Brownian Motions: Exact Tail Asymptotics and Extreme Value Distributions
Abstract: In this paper, we consider a two-dimensional sticky Brownian motion. Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. For example, a sticky Brownian motion can be used to model a storage system.with exceptional services. In this paper, we focus on stationary distributions for sticky Brownian motions. The main results obtained here include tail asymptotic properties in boundary stationary distributions, marginal distributions, and joint distributions. The kernel method, copula concept and extreme value theory are main tools used in our analysis.


Title: Non-unital polygraphs form a presheaf category
Abstract: We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf category. More generally we develop a new criterion for proving that certain classes of polygraphs are presheaf categories. This criterion also applies to the larger class of polygraphs where only the source of each generator is not an identity, and to the class of "many-to-one polygraphs", producing a new, more direct, proof that this is a presheaf category. The criterion itself seems to be extendable to more general type of operads over possibly different combinatorics, but we leave this question for future work. In an appendix we explain why this result is relevant if one wants to fix the arguments of a famous paper of M.Kapranov and V.Voevodsky and make them into a proof of C.Simpson's semi-strictification conjecture. We present a program aiming at proving this conjecture, which will be continued in subsequent papers.


Title: On the center of near-group fusion category of type $\mathbb{Z}_3+6$
Abstract: Let $\mathcal{A}$ be a near-group fusion category of type $\mathbb{Z}_3+6$. We show that there is a modular tensor equivalence $\mathcal{Z}(\mathcal{A})\cong\mathcal{C}(\mathbb{Z}_3,\eta)\boxtimes\mathcal{C}(\mathfrak{sl}_3,9)_{\mathbb{Z}_3}^0$. Moreover, we construct two non-trivial faithful extensions of $\mathcal{A}$ explicitly, whose Drinfeld centers can also be obtained from representation categories quantum groups at root of unity.


Title: Affine Symmetries of Orbit Polytopes
Abstract: An orbit polytope is the convex hull of an orbit under a finite group $G \leq \operatorname{GL}(d,\mathbb{R})$. We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups. We prove that the symmetry group of a generic orbit polytope is again $G$ if $G$ is itself the affine symmetry group of some orbit polytope, or if $G$ is absolutely irreducible. On the other hand, we describe some general cases where the affine symmetry group grows. We apply our theory to representation polytopes (the convex hull of a finite matrix group) and show that their affine symmetries can be computed effectively from a certain character. We use this to construct counterexamples to a conjecture of Baumeister et~al.\ on permutation polytopes [Advances in Math. 222 (2009), 431--452, Conjecture~5.4].


Title: Stability Conditions and Semiorthogonal Decompositions I: Quasi-convergence
Abstract: We develop a framework relating semiorthogonal decompositions of a triangulated category $\mathcal{C}$ to paths in its space of stability conditions. We prove that when $\mathcal{C}$ is the homotopy category of a smooth and proper pre-triangulated dg-category, every semiorthogonal decomposition whose semiorthogonal factors admit a Bridgeland stability condition can be obtained from our framework.


Title: Vietnamese Poem Generation & The Prospect Of Cross-Language Poem-To-Poem Translation
Abstract: Poetry generation has been a challenging task in the field of Natural Language Processing, as it requires the model to understand the nuances of language, sentiment, and style. In this paper, we propose using Large Language Models to generate Vietnamese poems of various genres from natural language prompts, thereby facilitating an intuitive process with enhanced content control. Our most efficacious model, the GPT-3 Babbage variant, achieves a custom evaluation score of 0.8, specifically tailored to the "luc bat" genre of Vietnamese poetry. Furthermore, we also explore the idea of paraphrasing poems into normal text prompts and yield a relatively high score of 0.781 in the "luc bat" genre. This experiment presents the potential for cross-Language poem-to-poem translation with translated poems as the inputs while concurrently maintaining complete control over the generated content.


Title: Covariant Dirac Operators on Quantum Groups
Abstract: We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a $q$-deformation of the classical Clifford algebra. The tensor space $ U_q(\g) \otimes \mathrm{cl}_q(\g)$ is given a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and $K$-homology cycles. This work generalizes the operator introduced by Bibikov and Kulish in \cite{BK}.


Title: Sub-Poissonian estimates for exponential moments of additive functionals over pairs of particles with respect to determinantal and symplectic Pfaffian point processes governed by entire functions
Abstract: The aim of this note is to estimate the tail of the distribution of the number of particles in an interval under determinantal and Pfaffian point processes. The main result of the note is that the square of the number of particles under the determinantal point process whose correlation kernel is an entire function of finite order has sub-Poissonian tails. The same result also holds in the symplectic Pfaffian case. As a corollary, sub-Poissonian estimates are also obtained for exponential moments of additive functionals over pairs of particles.


Title: Law of large numbers and fluctuations in the sub-critical and $L^2$ regions for SHE and KPZ equation in dimension $d\geq 3$
Abstract: There have been recently several works studying the regularized stochastic heat equation (SHE) and Kardar-Parisi-Zhang (KPZ) equation in dimension $d\geq 3$ as the smoothing parameter is switched off, but most of the results did not hold in the full temperature regions where they should. Inspired by martingale techniques coming from the directed polymers literature, we first extend the law of large numbers for SHE obtained in [MSZ16] to the full weak disorder region of the associated polymer model and to more general initial conditions. We further extend the Edwards-Wilkinson regime of the SHE and KPZ equation studied in [GRZ18,MU17,DGRZ20] to the full $L^2$-region, along with multidimensional convergence and general initial conditions for the KPZ equation (and SHE), which were not proven before. To do so, we rely on a martingale CLT combined with a refinement of the local limit theorem for polymers.


Title: Towards Higher Spectral Efficiency: Rate-2 Full-Diversity Complex Space-Time Block Codes
Abstract: The upcoming 5G networks demand high-speed and high spectral-efficiency communications to keep up with the proliferating traffic demands. To this end, Massive multiple-input multiple-output (MIMO) techniques have gained significant traction owing to its ability to achieve these without increasing bandwidth or density of base stations. The preexisting space-time block code (STBC) designs cannot achieve a rate of more than 1 for more than two transmit antennas while preserving the orthogonality and full diversity conditions. In this paper, we present Jagannath codes - a novel complex modulation STBC, that achieves a very high rate of 2 for three and four transmit antennas. The presented designs achieve full diversity and overcome the previously achieved rates with the three and four antenna MIMO systems. We present a detailed account of the code construction of the proposed designs, orthogonality and full diversity analysis, transceiver model and conditional maximum likelihood (ML) decoding. In an effort to showcase the improvement achieved with the presented designs, we compare the rates and delays of some of the known STBCs with the proposed designs. The effective spectral efficiency and coding gain of the presented designs are compared to the Asymmetric Coordinate Interleaved design (ACIOD) and Jafarkhani code. We presented an effective spectral efficiency improvement by a factor of 2 with the proposed Jagannath codes. Owing to the full diversity of the presented designs, we demonstrate significant coding gains (6 dB and 12 dB) with the proposed designs.


Title: Concurrent Brainstorming & Hypothesis Satisfying: An Iterative Framework for Enhanced Retrieval-Augmented Generation (R2CBR3H-SR)
Abstract: Addressing the complexity of comprehensive information retrieval, this study introduces an innovative, iterative retrieval-augmented generation system. Our approach uniquely integrates a vector-space driven re-ranking mechanism with concurrent brainstorming to expedite the retrieval of highly relevant documents, thereby streamlining the generation of potential queries. This sets the stage for our novel hybrid process, which synergistically combines hypothesis formulation with satisfying decision-making strategy to determine content adequacy, leveraging a chain of thought-based prompting technique. This unified hypothesize-satisfied phase intelligently distills information to ascertain whether user queries have been satisfactorily addressed. Upon reaching this criterion, the system refines its output into a concise representation, maximizing conceptual density with minimal verbosity. The iterative nature of the workflow enhances process efficiency and accuracy. Crucially, the concurrency within the brainstorming phase significantly accelerates recursive operations, facilitating rapid convergence to solution satisfaction. Compared to conventional methods, our system demonstrates a marked improvement in computational time and cost-effectiveness. This research advances the state-of-the-art in intelligent retrieval systems, setting a new benchmark for resource-efficient information extraction and abstraction in knowledge-intensive applications.


Title: Complementing Model Learning with Mutation-Based Fuzzing
Abstract: An ongoing challenge for learning algorithms formulated in the Minimally Adequate Teacher framework is to efficiently obtain counterexamples. In this paper we compare and combine conformance testing and mutation-based fuzzing methods for obtaining counterexamples when learning finite state machine models for the reactive software systems of the Rigorous Exampination of Reactive Systems (RERS) challenge. We have found that for the LTL problems of the challenge the fuzzer provided an independent confirmation that the learning process had been successful, since no additional counterexamples were found. For the reachability problems of the challenge, however, the fuzzer discovered more reachable error states than the learner and tester, albeit in some cases the learner and tester found some that were not discovered by the fuzzer. This leads us to believe that these orthogonal approaches are complementary in the context of model learning.


Title: A conjecture of Stanley on alternating permutations
Abstract: We give two simple proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.


Title: Frequency Domain Nuances Mining for Visible-Infrared Person Re-identification
Abstract: The key of visible-infrared person re-identification (VIReID) lies in how to minimize the modality discrepancy between visible and infrared images. Existing methods mainly exploit the spatial information while ignoring the discriminative frequency information. To address this issue, this paper aims to reduce the modality discrepancy from the frequency domain perspective. Specifically, we propose a novel Frequency Domain Nuances Mining (FDNM) method to explore the cross-modality frequency domain information, which mainly includes an amplitude guided phase (AGP) module and an amplitude nuances mining (ANM) module. These two modules are mutually beneficial to jointly explore frequency domain visible-infrared nuances, thereby effectively reducing the modality discrepancy in the frequency domain. Besides, we propose a center-guided nuances mining loss to encourage the ANM module to preserve discriminative identity information while discovering diverse cross-modality nuances. To the best of our knowledge, this is the first work that explores the potential frequency information for VIReID research. Extensive experiments show that the proposed FDNM has significant advantages in improving the performance of VIReID. Specifically, our method outperforms the second-best method by 5.2\% in Rank-1 accuracy and 5.8\% in mAP on the SYSU-MM01 dataset under the indoor search mode, respectively. Besides, we also validate the effectiveness and generalization of our method on the challenging visible-infrared face recognition task. \textcolor{magenta}{The code will be available.}


Title: Riemann surface of the Riemann zeta function
Abstract: In this paper we treat the classical Riemann zeta function as a function of three variables: one is the usual complex $\adyn$-dimensional, customly denoted as $s$, another two are complex infinite dimensional, we denote it as $\b = \{b_n\}_{n=1}^{\infty}$ and $\z =\{z_n\}_{n=1}^{\infty}$. When $\b = \{1\}_{n=1}^{\infty}$ and $\z = \{\frac{1}{n}\}_{n=1}^{\infty}$ one gets the usual Riemann zeta function. Our goal in this paper is to study the meromorphic continuation of $\zeta (\b , \z ,s)$ as a function of the triple $(\a , \z , s)$. Minor corrections, to appear in the Journal of Mathematical Analysis and Applications.


Title: Autonomous Reinforcement Learning via Subgoal Curricula
Abstract: Reinforcement learning (RL) promises to enable autonomous acquisition of complex behaviors for diverse agents. However, the success of current reinforcement learning algorithms is predicated on an often under-emphasised requirement -- each trial needs to start from a fixed initial state distribution. Unfortunately, resetting the environment to its initial state after each trial requires substantial amount of human supervision and extensive instrumentation of the environment which defeats the goal of autonomous acquisition of complex behaviors. In this work, we propose Value-accelerated Persistent Reinforcement Learning (VaPRL), which generates a curriculum of initial states such that the agent can bootstrap on the success of easier tasks to efficiently learn harder tasks. The agent also learns to reach the initial states proposed by the curriculum, minimizing the reliance on human interventions into the learning. We observe that VaPRL reduces the interventions required by three orders of magnitude compared to episodic RL while outperforming prior state-of-the art methods for reset-free RL both in terms of sample efficiency and asymptotic performance on a variety of simulated robotics problems.


Title: Not all Minorities are Equal: Empty-Class-Aware Distillation for Heterogeneous Federated Learning
Abstract: Data heterogeneity, characterized by disparities in local data distribution across clients, poses a significant challenge in federated learning. Substantial efforts have been devoted to addressing the heterogeneity in local label distribution. As minority classes suffer from worse accuracy due to overfitting on local imbalanced data, prior methods often incorporate class-balanced learning techniques during local training. Despite the improved mean accuracy across all classes, we observe that empty classes-referring to categories absent from a client's data distribution-are still not well recognized. This paper introduces FedED, a novel approach in heterogeneous federated learning that integrates both empty-class distillation and logit suppression simultaneously. Specifically, empty-class distillation leverages knowledge distillation during local training on each client to retain essential information related to empty classes from the global model. Moreover, logit suppression directly penalizes network logits for non-label classes, effectively addressing misclassifications in minority classes that may be biased toward majority classes. Extensive experiments validate the efficacy of FedED, surpassing previous state-of-the-art methods across diverse datasets with varying degrees of label distribution shift.


Title: Test ideals in mixed characteristic: a unified theory up to perturbation
Abstract: Let $X$ be an integral scheme of finite type over a complete DVR of mixed characteristic. We provide a definition of a test ideal which agrees with the multiplier ideal after inverting $p$, can be computed from a sufficiently large alteration, agrees with previous mixed characteristic BCM test ideals after localizing and completing at any point of residue characteristic $p$ (up to small perturbation), and which satisfies the full suite of expected properties of a multiplier or test ideal. This object is obtained via the $p$-adic Riemann-Hilbert functor.


Title: Slot-guided Volumetric Object Radiance Fields
Abstract: We present a novel framework for 3D object-centric representation learning. Our approach effectively decomposes complex scenes into individual objects from a single image in an unsupervised fashion. This method, called slot-guided Volumetric Object Radiance Fields (sVORF), composes volumetric object radiance fields with object slots as a guidance to implement unsupervised 3D scene decomposition. Specifically, sVORF obtains object slots from a single image via a transformer module, maps these slots to volumetric object radiance fields with a hypernetwork and composes object radiance fields with the guidance of object slots at a 3D location. Moreover, sVORF significantly reduces memory requirement due to small-sized pixel rendering during training. We demonstrate the effectiveness of our approach by showing top results in scene decomposition and generation tasks of complex synthetic datasets (e.g., Room-Diverse). Furthermore, we also confirm the potential of sVORF to segment objects in real-world scenes (e.g., the LLFF dataset). We hope our approach can provide preliminary understanding of the physical world and help ease future research in 3D object-centric representation learning.


Title: Mining Fine-Grained Image-Text Alignment for Zero-Shot Captioning via Text-Only Training
Abstract: Image captioning aims at generating descriptive and meaningful textual descriptions of images, enabling a broad range of vision-language applications. Prior works have demonstrated that harnessing the power of Contrastive Image Language Pre-training (CLIP) offers a promising approach to achieving zero-shot captioning, eliminating the need for expensive caption annotations. However, the widely observed modality gap in the latent space of CLIP harms the performance of zero-shot captioning by breaking the alignment between paired image-text features. To address this issue, we conduct an analysis on the CLIP latent space which leads to two findings. Firstly, we observe that the CLIP's visual feature of image subregions can achieve closer proximity to the paired caption due to the inherent information loss in text descriptions. In addition, we show that the modality gap between a paired image-text can be empirically modeled as a zero-mean Gaussian distribution. Motivated by the findings, we propose a novel zero-shot image captioning framework with text-only training to reduce the modality gap. In particular, we introduce a subregion feature aggregation to leverage local region information, which produces a compact visual representation for matching text representation. Moreover, we incorporate a noise injection and CLIP reranking strategy to boost captioning performance. We also extend our framework to build a zero-shot VQA pipeline, demonstrating its generality. Through extensive experiments on common captioning and VQA datasets such as MSCOCO, Flickr30k and VQAV2, we show that our method achieves remarkable performance improvements. Code is available at https://github.com/Artanic30/MacCap.


Title: The six-vertex model on random planar maps revisited
Abstract: We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov in 2000 using matrix integral techniques. We convert Kostov's work to a combinatorial argument involving functional equations coming from recursive decompositions of the maps, which we solve rigorously using complex analysis. We then investigate modular properties of the solution, which lead to simplifications in certain special cases. In particular, in two special cases of combinatorial interest we rederive the formulae discovered by Bousquet-M\'elou and the first author.


Title: Learning Discretized Neural Networks under Ricci Flow
Abstract: In this paper, we study Discretized Neural Networks (DNNs) composed of low-precision weights and activations, which suffer from either infinite or zero gradients due to the non-differentiable discrete function during training. Most training-based DNNs in such scenarios employ the standard Straight-Through Estimator (STE) to approximate the gradient w.r.t. discrete values. However, the use of STE introduces the problem of gradient mismatch, arising from perturbations in the approximated gradient. To address this problem, this paper reveals that this mismatch can be interpreted as a metric perturbation in a Riemannian manifold, viewed through the lens of duality theory. Building on information geometry, we construct the Linearly Nearly Euclidean (LNE) manifold for DNNs, providing a background for addressing perturbations. By introducing a partial differential equation on metrics, i.e., the Ricci flow, we establish the dynamical stability and convergence of the LNE metric with the $L^2$-norm perturbation. In contrast to previous perturbation theories with convergence rates in fractional powers, the metric perturbation under the Ricci flow exhibits exponential decay in the LNE manifold. Experimental results across various datasets demonstrate that our method achieves superior and more stable performance for DNNs compared to other representative training-based methods.


Title: A note on concentration for polynomials in the Ising model
Abstract: We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Lata{\l}a. In particular, for quadratic forms we obtain a Hanson-Wright type inequality. We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.


Title: Sieve in discrete groups, especially sparse
Abstract: We survey the recent applications and developments of sieve methods related to discrete groups, especially in the case of infinite index subgroups of arithmetic groups.


Title: Radio Map-Based Spectrum Sharing for Joint Communication and Sensing
Abstract: The sixth-generation (6G) network is expected to provide both communication and sensing (C&S) services. However, spectrum scarcity poses a major challenge to the harmonious coexistence of C&S systems. Without effective cooperation, the interference resulting from spectrum sharing impairs the performance of both systems. This paper addresses C&S interference within a distributed network. Different from traditional schemes that require pilot-based high-frequency interactions between C&S systems, we introduce a third party named the radio map to provide the large-scale channel state information (CSI). With large-scale CSI, we optimize the transmit power of C&S systems to maximize the signal-to-interference-plus-noise ratio (SINR) for the radar detection, while meeting the ergodic rate requirement of the interfered user. Given the non-convexity of both the objective and constraint, we employ the techniques of auxiliary-function-based scaling and fraction programming for simplification. Subsequently, we propose an iterative algorithm to solve this problem. Simulation results collaborate our idea that the extrinsic information, i.e., positions and surroundings, is effective to decouple C&S interference.


Title: Generative Optimization Networks for Memory Efficient Data Generation
Abstract: In standard generative deep learning models, such as autoencoders or GANs, the size of the parameter set is proportional to the complexity of the generated data distribution. A significant challenge is to deploy resource-hungry deep learning models in devices with limited memory to prevent system upgrade costs. To combat this, we propose a novel framework called generative optimization networks (GON) that is similar to GANs, but does not use a generator, significantly reducing its memory footprint. GONs use a single discriminator network and run optimization in the input space to generate new data samples, achieving an effective compromise between training time and memory consumption. GONs are most suited for data generation problems in limited memory settings. Here we illustrate their use for the problem of anomaly detection in memory-constrained edge devices arising from attacks or intrusion events. Specifically, we use a GON to calculate a reconstruction-based anomaly score for input time-series windows. Experiments on a Raspberry-Pi testbed with two existing and a new suite of datasets show that our framework gives up to 32% higher detection F1 scores and 58% lower memory consumption, with only 5% higher training overheads compared to the state-of-the-art.


Title: Clique number of Xor products of Kneser graphs
Abstract: In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in "A general 2-part Erd\H{o}s-Ko-Rado theorem" by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xor product of two isomorphic $KG(N,k)$ Kneser graphs. Denote this number with $f(k,N)$. We give lower and upper bounds on $f(k,N)$, and we solve the problem up to a constant deviation depending only on $k$, and find the exact value for $f(2,N)$ if $N$ is large enough. We also compute that $f(k,k^2)$ is asymptotically equivalent to $k^2$.