Physics-Informed Neural Networks (PINNs) have gained significant attention for their simplicity and flexibility in engineering and scientific computing. In this study, we introduce a normalized PINN (NPINN) framework to solve a class of wave propagation equations in non-unitized domains over extended time ranges. This is achieved through a normalization technique that involves either spatial or temporal variable normalization. To enhance the capability of NPINN in solving wave equations, we integrate a Fourier-induced deep neural network as the solver, leading to a novel architecture termed NFPINN. Furthermore, we explore different normalization strategies for spatial and temporal variables and identify the optimal normalization approach for our method. To assess the effectiveness and robustness of the proposed NFPINN, we present numerical experiments in both two-dimensional and three-dimensional Euclidean spaces, considering regular and irregular domains. The results confirm the accuracy and stability of our approach.

A mathematical model for crack-tip fields is proposed in this paper for the response of a three-dimensional (3-D) porous elastic solid whose material moduli are dependent on the density. Such a description wherein the generalized Lam\`e coefficients are nonlinear functions of material stiffness is more realistic because most engineering materials are porous, and their material properties depend on porosity and density. The governing boundary value problem for the static equilibrium state in a 3-D, homogeneous, isotropic material is obtained as a second-order, quasilinear partial-differential-equation system with a classical traction-free crack-surface boundary condition. The numerical solution is obtained from a continuous trilinear Galerkin-type finite element discretization. A Picard-type linearization is utilized to handle the nonlinearities in the discrete problem. The proposed model can describe the state of stress and strain in various materials, including recovering the classical singularities in the linearized model. The role of \textit{tensile stress}, \textit{stress intensity factor} (SIF), and \textit{strain energy density} are examined. The results indicate that the maximum values of all these quantities occur directly before the crack-tip, consistent with the observation made in the canonical problem for the linearized elastic fracture mechanics. One can use the same classical local fracture criterion, like the maximum of SIF, to study crack tips' quasi-static and dynamic evolution within the framework described in this article.

This study presents an aposteriori error analysis of adaptive finite element approximations of parabolic boundary control problems with bilateral box constraints that act on a Neumann boundary. The control problem is discretized using the symmetric interior penalty Galerkin (SIPG) technique. We derive both reliable and efficient type residual-based error estimators coupling with the data oscillations. The implementation of these error estimators serves as a guide for the adaptive mesh refinement process, indicating whether or not more refinement is required. Although the control error estimator effectively captured control approximation errors, it had limitations in guiding refinement localization in critical cases. To overcome this, an alternative control indicator was used in numerical tests. The results demonstrated the clear superiority of adaptive refinements over uniform refinements, confirming the proposed approach's effectiveness in achieving accurate solutions while optimizing computational efficiency. numerical experiment showcases the effectiveness of the derived error estimators.

Accurate interpolation of functions and derivatives is crucial in solving partial differential equations. Hermite Radial Basis Function (HRBF) methods improve accuracy by incorporating derivative information but suffer from ill-conditioning at low to moderate shape parameters for infinitely smooth kernels. This work proposes a Modified HRBF (MHRBF) method that introduces an additional polynomial to balance kernel behavior, improving accuracy while maintaining or lowering computational cost. The numerical results show that MHRBF achieves lower $L_{\infty}$--errors with fewer unknowns compared with the original HRBF, making it a robust alternative for stable and accurate RBF-based interpolation.

This Master's thesis examines the properties of large degree vertices in random recursive directed acyclic graphs (RRDAGs), a generalization of the well-studied random recursive tree (RRT) model. Using a novel adaptation of Kingman's coalescent, we extend results from RRTs to RRDAGs, focusing on different vertex properties. For large degrees, we establish the asymptotic joint distribution of the degree of multiple uniform vertices, proving that they follow a multivariate geometric distribution, and obtain results on maximal and near-maximal degree vertices. In addition, we consider a version of vertex depth that we call ungreedy depth and describe its asymptotic behavior, along with the labels, of single uniform vertices with a given large degree. Finally, we extend this analysis to multiple uniform vertices by deriving the asymptotic behavior of their labels conditional on large degrees.

We show that smooth polarized Fano fibrations have no nontrivial finite covers. Using results by Sun-Zhang and Wylie, it follows that shrinking K\"ahler-Ricci solitons are simply-connected.

The low Mach number limit for the compressible viscous diffusion approximation model arising in radiation hydrodynamics is rigorously justified. For the 3-D Cauchy problem, the solutions in an equilibrium diffusion regime are shown to converge to the solutions of an incompressible Navier-Stokes equations locally and globally in time as Mach number goes to zero, when the effect of the small temperature variation upon the limit is taken into account.

We first study subextensions of m-subharmonic functions in weighted energy classes with given boundary values. The results are used to approximate an m-subharmonic function in weighted energy classes with given boundary values by an increasing sequence of m-subharmonic functions defined on larger domains.

We introduce for non-uniform messages a novel hybrid universal network coding cryptosystem (NU-HUNCC) in the finite blocklength regime that provides Post-Quantum (PQ) security at high communication rates. Recently, hybrid cryptosystems offered PQ security by premixing the data using secure linear coding schemes and encrypting only a small portion of it. The data is assumed to be uniformly distributed, an assumption that is often challenging to enforce. Standard fixed-length lossless source coding and compression schemes guarantee a uniform output in normalized divergence. Yet, this is not sufficient to guarantee security. We consider an efficient compression scheme uniform in non-normalized variational distance for the proposed hybrid cryptosystem, that by utilizing a uniform sub-linear shared seed, guarantees PQ security. Specifically, for the proposed PQ cryptosystem, first, we provide an end-to-end practical coding scheme, NU-HUNCC, for non-uniform messages. Second, we show that NU-HUNCC is information-theoretic individually secured (IS) against an eavesdropper with access to any subset of the links and provide a converse proof against such an eavesdropper. Third, we introduce a modified security definition, individual semantic security under a chosen ciphertext attack (ISS-CCA1), and show that against an all-observing eavesdropper, NU-HUNCC satisfies its conditions. Finally, we provide an analysis of NU-HUNCC's high data rate, low computational complexity, and the negligibility of the shared seed size.

This paper considers the linear objective function optimization with respect to a more general class of bipolar fuzzy relational equations, where the fuzzy compositions are defined by an arbitrary continuous Archimedean t-norm. In addition, a faster method for finding a global optimum is proposed that, unlike the previous work, does not require obtaining all local optimal solutions and classifying the constraints. Analytical concepts and properties of the Archimedean bipolar fuzzy equations are investigated and two necessary conditions are presented to conceptualize the feasibility of the problem. It is shown that the feasible solution set can be resulted by a union of the finite number of compact sets, where each compact set is obtained by a function. Moreover, to accelerate identification of the mentioned compact sets (and therefore, to speed up solution finding), four simplification techniques are presented, which are based on either omitting redundant constraints and/or eliminating unknowns by assigning them a fixed value. Also, three additional simplification techniques are given to reduce the search domain by removing some parts of the feasible region that do not contain optimal solutions. Subsequently, a method is proposed to find an optimal solution for the current linear optimization problems. The proposed method consists of two accelerative strategies that are used during the problem solving process. By the first strategy, the method neglects some candidate solutions that are not optimal, by considering only a subset of admissible functions. As for the second strategy, a branch-and-bound method is used to delete non-optimal branches. Then, the method is summarized in an algorithm that represents all essential steps of the solution and finally, the whole method is applied in an example that has been chosen in such a way that the various situations are illustrated.

This paper is concerned with the absolute stability analysis of discrete-time feedback systems with slope-restricted nonlinearities. By employing static O'Shea-Zames-Falb multipliers in the framework of integral quadratic constraints, we can obtain a certificate for the absolute stability in the form of a linear matrix inequality (LMI). However, since this LMI certificate is only a sufficient condition, we cannot draw any definite conclusion if the LMI turns out to be infeasible. To address this issue, we focus on the dual LMI that is feasible if and only if the original (primal) LMI is infeasible. As the main result, if the dual solution satisfies a certain rank condition, we prove that we can detect a destabilizing nonlinearity within the assumed class of slope-restricted nonlinearities as well as a non-zero equilibrium point of the resulting feedback system, thereby we can conclude that the system of interest is never absolutely stable. The effectiveness of the technical results is demonstrated through numerical examples.

Risk-sensitive control balances performance with resilience to unlikely events in uncertain systems. This paper introduces ergodic-risk criteria, which capture long-term cumulative risks through probabilistic limit theorems. By ensuring the dynamics exhibit strong ergodicity, we demonstrate that the time-correlated terms in these limiting criteria converge even with potentially heavy-tailed process noises as long as the noise has a finite fourth moment. Building upon this, we proposed the ergodic-risk constrained policy optimization which incorporates an ergodic-risk constraint to the classical Linear Quadratic Regulation (LQR) framework. We then propose a primal-dual policy optimization method that optimizes the average performance while satisfying the ergodic-risk constraints. Numerical results demonstrate that the new risk-constrained LQR not only optimizes average performance but also limits the asymptotic variance associated with the ergodic-risk criterion, making the closed-loop system more robust against sporadic large fluctuations in process noise.

Over an algebraically closed ffeld F of characteristic p>0, the restricted twisted Heisenberg Lie algebras are studied. We use the Hochschild-Serre spectral sequence relative to its Heisenberg ideal to compute the trivial cohomology. The ordinary 1- and 2-cohomology spaces are used to compute the restricted 1- and 2-cohomology spaces and describe the restricted 1-dimensional central extensions, including explicit formulas for the Lie brackets and [p]-operators.

Likelihood inference for max-stable random fields is in general impossible because their finite-dimensional probability density functions are unknown or cannot be computed efficiently. The weighted composite likelihood approach that utilizes lower dimensional marginal likelihoods (typically pairs or triples of sites that are not too distant) is rather favored. In this paper, we consider the family of spatial max-stable Brown-Resnick random fields associated with isotropic fractional Brownian fields. We assume that the sites are given by only one realization of a homogeneous Poisson point process restricted to $\mathbf{C}=(-1/2,1/2]^{2}$ and that the random field is observed at these sites. As the intensity increases, we study the asymptotic properties of the composite likelihood estimators of the scale and Hurst parameters of the fractional Brownian fields using different weighting strategies: we exclude either pairs that are not edges of the Delaunay triangulation or triples that are not vertices of triangles.

In an era where data-driven decision-making and computational efficiency are paramount, optimization plays a foundational role in advancing fields such as mathematics, computer science, operations research, machine learning, and beyond. From refining machine learning models to improving resource allocation and designing efficient algorithms, optimization techniques serve as essential tools for tackling complex problems. This book aims to provide both an introductory guide and a comprehensive reference, equipping readers with the necessary knowledge to understand and apply optimization methods within their respective fields. Our primary goal is to demystify the inner workings of optimization algorithms, including black-box and stochastic optimizers, by offering both formal and intuitive explanations. Starting from fundamental mathematical principles, we derive key results to ensure that readers not only learn how these techniques work but also understand when and why to apply them effectively. By striking a careful balance between theoretical depth and practical application, this book serves a broad audience, from students and researchers to practitioners seeking robust optimization strategies.

We present a straightforward energy stable weak implementation procedure of open boundary conditions for nonlinear initial boundary value problems. It simplifies previous work and its practical implementation.

Batchelor predicted that a passive scalar $\psi^\nu$ with diffusivity $\nu$, advected by a smooth fluid velocity, should typically have Fourier mass distributed as $|\hat \psi^\nu|^2(k) \approx |k|^{-d}$ for $|k| \ll \nu^{-1/2}$. For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball.

In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u = \lambda\frac{a(x)}{u^p} + \frac{\alpha}{\alpha+\beta}\theta |u|^{\alpha - 2}u|v|^{\beta}, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N,\\ (-\Delta)^sv +V_2(x)v= \lambda \frac{b(x)}{v^q}+ \frac{\beta}{\alpha+\beta}\theta |u|^{\alpha}|v|^{\beta-2}v, \,\,\, \mbox{in} \,\,\, \mathbb{R}^N, \end{array}\right. \;\;\;(u, v) \in H^s(\mathbb{R}^N) \times H^s(\mathbb{R}^N),$$ where $ 0<p \leq q < 1<\;\alpha, \beta \;,\;2<\alpha + \beta < 2^*_s$, $\theta > 0, \lambda > 0, N > 2s$, and $s \in (0,1)$. The potentials $V_1, V_2: \mathbb{R}^N \to \mathbb{R}$ are continuous functions which are bounded from below. Under our assumptions, we prove that there exists the largest positive number $\lambda^* > 0$ such that our main problem admits at least two positive solutions for each $\lambda \in (0, \lambda^*)$. Here we apply the nonlinear Rayleigh quotient together with the Nehari method. The main feature is to minimize the energy functional in Nehari set which allows us to prove our results without any restriction on the size of parameter $\theta > 0$. Moreover, we shall consider the multiplicity of solutions for the case $\lambda = \lambda^*$ where degenerated points are allowed.

In the present work, we establish the existence and multiplicity of positive solutions for the singular elliptic equations with a double weighted nonlocal interaction term defined in the whole space $\mathbb{R}^N$. The nonlocal term and the fact that the energy functional is not differentiable are the main difficulties for this kind of problem. We apply the Nehari method and the nonlinear Rayleigh quotient to prove that our main problem has at least two positive weak solutions. Furthermore, we prove a nonexistence result related to the extreme $\lambda^*> 0$ given by the nonlinear Rayleigh quotient.

We show the existence of a solution to the Ricci flow with a compact length space of bounded curvature, i.e., a space that has curvature bounded above and below in the sense of Alexandrov, as its initial condition. We show that this flow converges in the $C^{1,\alpha}$-sense to a $C^{1,\alpha}$-continuous Riemannian manifold which is isometric to the original metric space. Moreover, we prove that the flow is uniquely determined by the initial condition, up to isometry.

An equidistant set in the Euclidean space consists of points having equal distances to both members of a given pair of sets, called focal sets. Having no effective formulas to compute the distance of a point and a set, it is hard to determine the points of an equidistant set in general. Special classes of equidistant sets allow us to approximate the equidistant points in more complicated cases. In the paper we have a hyperplane corresponding to the first order (linear) approximation for one of the focal sets and the second one is considered as the epigraph of a positive-valued continuous function. In the first part of the paper we prove that the equidistant points having equal distances to the epigraph of a positive-valued continuous function and its domain form the graph of a multivariate function. Therefore such an equidistant set is called a multivariate equidistant function. We also prove that the equidistant function one of whose focal sets is constituted by the pointwise minima of finitely many positive-valued continuous functions is given by the pointwise minima of the corresponding equidistant functions. In the second part of the paper we consider equidistant functions belonging to the epigraph of a convex function under some smoothness conditions. Independently of the dimension of the space we present a special parameterization for the equidistant points based on the closest point property of the epigraph as a convex set and we give the characterization of the equidistant functions as well. An example is also presented with a hyperboloid of revolution as one of the focal sets.

We consider local weak solutions to the fractional $p$-Poisson equation of order $s$, i.e. $\left( - \Delta_p\right)^s u = f$. In the range $p>1$ and $s\in \big(\frac{p-1}{p},1\big)$ we prove Calder\'on & Zygmund type estimates at the gradient level. More precisely, we show for any $q>1$ that \begin{equation*} f\in L^{\frac{qp}{p-1}}_{\rm loc} \quad\Longrightarrow\quad \nabla u\in L^{qp}_{\rm loc}. \end{equation*} The qualitative result is accompanied by a local quantitative estimate.

This paper investigates the optimal control problem for a class of parabolic equations where the diffusion coefficient is influenced by a control function acting nonlocally. Specifically, we consider the optimization of a cost functional that incorporates a controlled probability density evolving under a Fokker-Planck equation with state-dependent drift and diffusion terms. The control variable is subject to spatial convolution through a kernel, inducing nonlocal interactions in both drift and diffusion terms. We establish the existence of optimal controls under appropriate convexity and regularity conditions, leveraging compactness arguments in function spaces. A maximum principle is derived to characterize the optimal control explicitly, revealing its dependence on the adjoint state and the nonlocal structure of the system. We further provide a rigorous financial application in the context of mean-variance portfolio optimization, where both the asset drift and volatility are controlled nonlocally, leading to an integral representation of the optimal investment strategy. The results offer a mathematically rigorous framework for optimizing diffusion-driven systems with spatially distributed control effects, broadening the applicability of nonlocal control methods to stochastic optimization and financial engineering.

Hybridizable discretizations allow for the elimination of local degrees-of-freedom leading to reduced linear systems. In this paper, we determine and analyse an approach to construct parameter-robust preconditioners for these reduced systems. Using the framework of Mardal and Winther (Numer. Linear Algebra Appl., 18(1):1--40, 2011) we first determine a parameter-robust preconditioner for the full system. We then eliminate the local degrees-of-freedom of this preconditioner to obtain a preconditioner for the reduced system. However, not all reduced preconditioners obtained in this way are automatically robust. We therefore present conditions that must be satisfied for the reduced preconditioner to be robust. To demonstrate our approach, we determine preconditioners for the reduced systems obtained from hybridizable discretizations of the Darcy and Stokes equations. Our analysis is verified by numerical examples in two and three dimensions.

We provide a complete characterization of compactness of Sobolev embeddings of radially symmetric functions on the entire space $\mathbb{R}^n$ in the general framework of rearrangement-invariant function spaces. We avoid any unnecessary restrictions and cover also embeddings of higher order, providing a complete picture within this framework. To achieve this, we need to develop new techniques because the usual techniques used in the study of compactness of Sobolev embeddings in the general framework of rearrangement-invariant function spaces are limited to domains of finite measure, which is essential for them to work. Furthermore, we also study certain weighted Sobolev embeddings of radially symmetric functions on balls. We completely characterize their compactness and also describe optimal target rearrangement-invariant function spaces in these weighted Sobolev embeddings.

We generalize the enhanced power graph by replacing elements with conjugacy classes. The main result of this paper is to determine when this graph is triangle-free.

In this paper, we introduce a streamlined procedure for constructing small symplectic 4-manifolds via contact gluing, based on a technique invented by David Gay around 2000. We also give a few applications, ranging from embeddings of singular Lagrangian RP^2s to realizing an infinite family of lens spaces as hypersurfaces of contact type in a symplectic Hirzebruch surface. Furthermore, our investigation on S^1-invariant contact structures also suggests there should be a universal upper bound for the self-intersection of a rational unicuspidal curve with one Puiseux pair in any algebraic surface (the bound depends only on the singularity).

Proto-exact and parabelian categories serve as non-additive analogues of exact and quasi-abelian categories, respectively. They give rise to algebraic K-theory and Hall algebras similarly to the additive setting. We show that every parabelian category admits a canonical proto-exact structure and we study several classes of parabelian categories, including categories of normed and Euclidean vector spaces, pointed closure spaces and pointed matroids, Hermitian vector bundles over rings of integers. We also examine finitary algebraic categories arising in Arakelov geometry and provide a criterion for determining when such a category is parabelian. In particular, we prove that the categories of pointed convex spaces and absolutely convex spaces are parabelian.

This work proposes a new method to select the augmentation parameters in the operator splitting quadratic program (OSQP) algorithm so as to reduce the computation time of overall algorithm. The selection is based upon the information of conjugate directions of the coefficient matrix of a linear system of equations present in the algorithm. This selection makes it possible to cache these conjugate directions, instead of computing them at each iteration, resulting in faster computation of the solution of the linear system thus reducing the overall computation time. This reduction is demonstrated by a numerical example.

A parametrized Yang-Baxter equation is a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. For the six-vertex model, there are two main regimes of the Yang-Baxter equation: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group GL(2)xGL(1). For non-free-fermionic six-vertex matrices, there are also parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that does reflect all possible Yang-Baxter equations in the six-vertex model.

We prove that the (hermitian) rank of $QP^d$ is bounded from below by the rank of $P^d$ whenever $Q$ is not identically zero and real-analytic in a neighborhood of some point on the zero set of $P$ in $\mathbb{C}^n$ and $P$ is a polynomial of bidegree at most $(1,1)$. This result generalizes the theorem of D'Angelo and the second author which assumed that $P$ was bihomogeneous. Examples show that no hypothesis can be dropped.

A survey of some results and open questions related to the following algebraic invariants of compact complex manifolds, that can be obtained from differential forms: cohomology groups, Chern classes, rational homotopy groups, and higher operations.

Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by timescale separation, conservation laws, and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods to DAE systems with unknown constraints and time scales. We introduce Sparse Optimization for Differential-Algebraic Systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems and has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves numerical stability when handling high correlations between library terms -- caused by near-perfect algebraic relationships -- by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical, and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.

We consider a two-speed branching random walk, which consists of two macroscopic stages with different reproduction laws. We prove that the centered maximum converges in law to a Gumbel variable with a random shift and the extremal process converges in law to a randomly shifted decorated Poisson point process, which can be viewed as a discrete analog for the corresponding results for the two-speed branching Brownian motion, previously established by Bovier and Hartung [12].

We investigate the classification of quasihomogeneous polynomials in two variables with real coefficients under semialgebraic bi-Lipschitz equivalence in a neighborhood of the origin in ${\mathbb R}^2$. Building on the work of Birbrair, Fernandes, and Panazzolo, our approach is based on reducing the problem to the Lipschitz classification of associated single-variable polynomial functions, called height functions. We establish conditions under which semialgebraic bi-Lipschitz equivalence of quasihomogeneous polynomials corresponds to the Lipschitz equivalence of their height functions. To achieve this, we develop the theory of $\beta$-transforms and inverse $\beta$-transforms. As an application, we examine a family of quasihomogeneous polynomials previously used by Henry and Parusi\'nski to show that the bi-Lipschitz equivalence of analytic function germs $({\mathbb R}^2,0)\rightarrow({\mathbb R},0)$ admits continuous moduli. Our results show that semialgebraic bi-Lipschitz equivalence of real quasihomogeneous polynomials in two variables also admits continuous moduli.

Given a regular function $\phi$ on a smooth stack, and a $(-1)$-shifted Lagrangian $M$ on the derived critical locus of $\phi$, under fairly general hypotheses, we construct a pullback map from the Grothendieck group of coherent matrix factorizations of $\phi$ to that of coherent sheaves on $M$. This map satisfies a functoriality property with respect to the composition of Lagrangian correspondences, as well as the usual bivariance and base-change properties. We provide three applications of the construction, one in the definition of quantum $K$-theory of critical loci (Landau-Ginzburg models), paving the way to generalize works of Okounkov school from Nakajima quiver varieties to quivers with potentials, one in establishing a degeneration formula for $K$-theoretic Donaldson-Thomas theory of local Calabi-Yau 4-folds, the other in confirming a $K$-theoretic version of Joyce-Safronov conjecture.

Non-attacking fillings are combinatorial objects central to the theory of Macdonald polynomials. A probabilistic bijection for partition-shaped non-attacking fillings was introduced by Mandelshtam (2024) to prove a compact formula for symmetric Macdonald polynomials. In this work, we generalize this probabilistic bijection to composition-shaped non-attacking fillings. As an application, we provide a bijective proof to extend a symmetry theorem for permuted-basement Macdonald polynomials established by Alexandersson (2019), proving a version with fewer assumptions.

Recent approaches to the theory of dynamic programming view dynamic programs as families of policy operators acting on partially ordered sets. In this paper, we extend these ideas by shifting from arbitrary partially ordered sets to ordered vector space. The advantage of working in this setting is that ordered vector spaces have well integrated algebric and order structure, which leads to sharper fixed point results. These fixed point results can then be exploited to obtain strong optimality properties. We illustrate our results through a range of applications, including new findings for several useful models.

In this paper, we prove a rigidity theorem for Poincar\'e-Einstein manifolds whose conformal infinity is a flat Euclidean space. The proof relies on analyzing the propagation of curvature tensors over the level sets of an adapted boundary defining function. Additionally, we provide examples of Poincar\'e-Einstein manifolds with non-compact conformal infinities. Furthermore, we draw analogies with Ricci-flat manifolds exhibiting Euclidean volume growth, particularly when the compactified metric has non-negative scalar curvature.

In this paper, we give a short and entirely elementary proof of the proposition ``For any positive integer $ N $, there exists a real number $ L $ such that for any real number $ x \geqq L $, there are at least $ N $ primes in the interval $ [kx, (k+1)x] $'' for $ k \leqq 15 $. Our proof is based on the idea of the proof by Erd\"{o}s for $ k=1 $ and its improvement by Hitotsumatsu and by Sainose for $ k=2 $. In the case of $ k=3 $ and $ k=4 $, the method is very similar to the case of $ k=2 $, however, in the case of $ k \geqq 5 $, we need new idea to complete the proof.

The reconfigurability of fluid antenna systems (FASs) and reconfigurable intelligent surfaces (RISs) provides significant flexibility in optimizing channel conditions by jointly adjusting the positions of fluid antennas and the phase shifts of RISs. However, it is challenging to acquire the instantaneous channel state information (CSI) for both fluid antennas and RISs, while frequent adjustment of antenna positions and phase shifts will significantly increase the system complexity. To tackle this issue, this paper investigates the two-timescale design for FAS-RIS multi-user systems with linear precoding, where only the linear precoder design requires instantaneous CSI of the end-to-end channel, while the FAS and RIS optimization relies on statistical CSI. The main challenge comes from the complex structure of channel and inverse operations in linear precoding, such as regularized zero-forcing (RZF) and zero-forcing (ZF). Leveraging on random matrix theory (RMT), we first investigate the fundamental limits of FAS-RIS systems with RZF/ZF precoding by deriving the ergodic sum rate (ESR). This result is utilized to determine the minimum number of activated antennas to achieve a given ESR. Based on the evaluation result, we propose an algorithm to jointly optimize the antenna selection, regularization factor of RZF, and phase shifts at the RIS. Numerical results validate the accuracy of performance evaluation and demonstrate that the performance gain brought by joint FAS and RIS design is more pronounced with a larger number of users.

We provide a new proof of the fractional version of the De Giorgi conjecture for the Allen-Cahn equation in $\mathbb{R}^2$ for the full range of exponents. Our proof combines a method introduced by A. Farina in 2003 with the $s$-harmonic extension of the fractional Laplacian in the half-space $\mathbb{R}^{3}_+$ introduced by L. Caffarelli and L. Silvestre in 2007. We also provide a representation formula for finite-energy weak solutions of a class of weighted elliptic partial differential equations in the half-space $\mathbb{R}^{n+1}_+$ under Neumann boundary conditions. This generalizes the $s$-harmonic extension of the fractional Laplacian and allows us to relate a general problem in the extended space with a nonlocal problem on the trace.

Given a graph $G=(V,E)$, a set $S\subseteq V$ is said to be a monitoring edge-geodetic set if the deletion of any edge in the graph results in a change in the distance between at least one pair of vertices in $S$. The minimum size of such a set in $G$ is called the monitoring edge-geodetic number of $G$ and is denoted by $meg(G)$. In this work, we compute the monitoring edge-geodetic number efficiently for the following graph classes: distance-hereditary graphs, $P_4$-sparse graphs, bipartite permutation graphs, and strongly chordal graphs. The algorithms follow from structural characterizations of the optimal monitoring edge-geodetic sets for these graph classes in terms of \emph{mandatory vertices} (those that need to be in every solution). This extends previous results from the literature for cographs, interval graphs and block graphs.

Denote the Tutte polynomial of a graph $G$ and a matroid $M$ by $T_G(x,y)$ and $T_M(x,y)$ respectively. $T_G(x,1)$ and $T_G(1,y)$ were generalized to hypergraphs and further extended to integer polymatroids by K\'{a}lm\'{a}n \cite{Kalman} in 2013, called interior and exterior polynomials respectively. Let $G$ be a $(k+1)$-edge connected graph of order $n$ and size $m$, and let $g=m-n+1$. Guan et al. (2023) \cite{Guan} obtained the coefficients of $T_G(1,y)$: \[[y^j]T_G(1,y)=\binom{m-j-1}{n-2} \text{ for } g-k\leq j\leq g,\] which was deduced from coefficients of the exterior polynomial of polymatroids. Recently, Chen and Guo (2025) \cite{Chen} further obtained \[[y^j]T_G(1,y)=\binom{m-j-1}{n-2}-\sum_{i=k+1}^{g-j}\binom{m-j-i-1}{n-2}|\mathcal{EC}_i(G)|\] for $g-3(k+1)/2< j\leq g$, where $\mathcal{EC}_i(G)$ denotes the set of all minimal edge cuts with $i$ edges. In this paper, for any matroid $M=(X,rk)$ we first obtain \[[y^j]T_M(1,y)=\sum_{t=j}^{|X|-r}(-1)^{t-j}\binom{t}{j}\sigma_{r+t}(M),\] where $\sigma_{r+t}(M)$ denotes the number of spanning sets with $r+t$ elements in $M$ and $r=rk(M)$. Moveover, the expression of $[x^i]T_M(x,1)$ is obtained immediately from the duality of the Tutte polynomial. As applications of our results, we generalize the two aforementioned results on graphs to the setting of matroids. This not only resolves two open problems posed by Chen and Guo in \cite{Chen} but also provides a purely combinatorial proof that is significantly simpler than their original proofs.

We show that for any positive integer $h$, a knot surgered elliptic surface $E(n)_{T(2,2h+1)}$ for a $(2,2h+1)$-torus knot $T(2,2h+1)$ and the elliptic surface $E(1)_{2,2h+1}$ admit handle decompositions without 1- and 3-handles using the Kirby diagrams ``on surfaces" derived from Lefschetz fibrations on them.

In this article, we prove a lower bound for the Hausdorff dimension of the set of exactly $\psi$-approximable vectors with values in a local field of positive characteristic. This is the analogue of the corresponding theorem of Bandi and de Saxc'e on reals \cite{bandi2023hausdorff} and is a higher-dimensional version of a theorem of Zhang \cite{MR2834892}.

In this note, we discuss the uniform ergodicity of a diffusion process given by an It\^o stochastic differential equation. We present an integral condition in terms of the drift and diffusion coefficients that ensures the uniform ergodicity of the corresponding transition kernel with respect to the total variation distance. Applications of the obtained results to a class of subordinate diffusion processes are also presented.

We perform a complete analysis of the limiting behaviour of a class of quasilinear problems with Dirichlet boundary data g. We show that the Lipschitz constant of g plays a role in controlling the Gamma-convergence of the natural energies. However the solutions converge uniformly to solution of a limiting equation irrelevant to the Lipschitz constant of g. The limiting equation has no coercivity in u. We prove that the limiting equation admits a weak comparison principle and has a unique viscosity solution. We also obtain a Poincare inequality in the Sobolev-Orlicz space for discontinuous operator, which paves the way for our study of an extremal problem where its operator becomes unbounded in a subdomain. Upon giving proper meaning to its solution, we show that the extremal problem has a unique solution. It turns out the solution has sufficient continuity, although operator is discontinuous. In the appendix we provide some technical inequalities which play crucial roles in the proof of uniqueness and we believe will be of independent interest.

We consider the evolution of contact lines for thermal convection of viscous fluids in a 2D open-top vessel. The domain is bounded above by a free moving boundary and otherwise by the solid wall of a vessel. The dynamics of the fluid are governed by the incompressible Boussinesq approximation under the influence of gravity, and the interface between fluid and air is under the effect of capillary forces. Motivated by energy-dissipation structure in [Guo-Tice, J. Eur. Math. Soc, 2024], we develop global well posedness theory in the framework of nonlinear energy methods for the initial data sufficiently close to equilibrium. Moreover, the solutions decay to equilibrium at an exponential rate. Our methods are mainly based on the construction of solutions to convected heat equation and a priori estimates of a geometric formulation of the Boussinesq equations.

We introduce a new PL invariant, called the balanced genus, for balanced normal $d$-pseudomanifolds. As a key result, we establish that for any 3-manifold $M$ that is not a sphere, the balanced genus satisfies the lower bound $\mathcal{G}_M \geq m+3$, where $m$ is the rank of its fundamental group. Furthermore, we prove that a 3-manifold $M$ is homeomorphic to the 3-sphere if and only if its balanced genus $\mathcal{G}_M$ is at most 3. For 4-manifolds, we establish a similar characterization: if $M$ is not homeomorphic to a sphere, then its balanced genus is bounded below by $\mathcal{G}_M \geq 2\chi(M) + 5m + 11$, where $m$ is the rank of $\pi_1(M)$. Additionally, we prove that a 4-manifold $M$ is PL-homeomorphic to the 4-sphere if and only if its balanced genus satisfies $\mathcal{G}_M \leq 2\chi(M) + 10$. We believe that the balanced genus provides a new perspective in combinatorial topology and will inspire further developments in the field. To this end, we outline several research directions for future exploration.

The Lyapunov rank of a cone is the dimension of the Lie algebra of its automorphism group. It is invariant under linear isomorphism and in general not unique - two or more non-isomorphic cones can share the same Lyapunov rank. It is therefore not possible in general to identify cones using Lyapunov rank. But suppose we look only among symmetric cones. Are there any that can be uniquely identified (up to isomorphism) by their Lyapunov ranks? We provide a complete answer for irreducible cones and make some progress in the general case.

Generative AI (GenAI) is driving the intelligence of wireless communications. Due to data limitations, random generation, and dynamic environments, GenAI may generate channel information or optimization strategies that violate physical laws or deviate from actual real-world requirements. We refer to this phenomenon as wireless hallucination, which results in invalid channel information, spectrum wastage, and low communication reliability but remains underexplored. To address this gap, this article provides a comprehensive concept of wireless hallucinations in GenAI-driven communications, focusing on hallucination mitigation. Specifically, we first introduce the fundamental, analyze its causes based on the GenAI workflow, and propose mitigation solutions at the data, model, and post-generation levels. Then, we systematically examines representative hallucination scenarios in GenAI-enabled communications and their corresponding solutions. Finally, we propose a novel integrated mitigation solution for GenAI-based channel estimation. At the data level, we establish a channel estimation hallucination dataset and employ generative adversarial networks (GANs)-based data augmentation. Additionally, we incorporate attention mechanisms and large language models (LLMs) to enhance both training and inference performance. Experimental results demonstrate that the proposed hybrid solutions reduce the normalized mean square error (NMSE) by 0.19, effectively reducing wireless hallucinations.

Motivated by asymptotic phenomena of moduli spaces of higher rank stable sheaves on algebraic surfaces, we study the Picard number of the moduli space of one-dimensional stable sheaves supported in a sufficiently positive divisor class on a surface. We give an asymptotic lower bound of the Picard number in general. In some special cases, we show that this lower bound is attained based on the geometry of moduli spaces of stable pairs and relative Hilbert schemes of points. Additionally, we discuss several related questions and provide examples where the asymptotic irreducibility of the moduli space fails, highlighting a notable distinction from the higher rank case.

In this paper we characterise absolutely norm attaining quasi*paranormal weighted shifts on directed trees and give some examples. Moreover we give some examples which show that the spectrum of a positive absolutely norm attaining operator containing more than one eigenvalue with infinite multiplicity.

Multiplicative convolution $\mu \ast \nu$ of two finite signed measures $\mu$ and $\nu$ on $\mathbb{R}^n$ and a related product $\mu \circledast \nu$ on the sphere $S^{n-1}$ are studied. For fixed $\mu$ the injectivity in $\nu$ of both operations is characterised given an arbitrary group of reflections along the coordinate axes. The results for the sphere yield generalised versions of the theorems in Molchanov and Nagel (2021) about convex bodies.

The mean-variance (MV) model is the core of modern portfolio theory. Nevertheless, it suffers from the over-fitting problem due to the estimation errors of model parameters. We consider the $\ell_{1}$ regularized MV model, which adds an $\ell_{1}$ regularization term in the objective to prevent over-fitting and promote sparsity of solutions. By investigating the relationship between sample size and over-fitting, we propose an initial regularization parameter scheme in the $\ell_{1}$ regularized MV model. Then we propose an adaptive parameter tuning strategy to control the amount of short sales. ADMM is a well established algorithm whose performance is affected by the penalty parameter. In this paper, a penalty parameter scheme based on regularized Barzilai-Borwein step size is proposed, and the modified ADMM is used to solve the $\ell_{1}$ regularized MV problem. Numerical results verify the effectiveness of the two types of parameters proposed in this paper.

In 1970, Schneider introduced the $m$th-order extension of the difference body $DK$ of a convex body $K\subset\mathbb R^n$, the convex body $D^m(K)$ in $\mathbb R^{nm}$. He conjectured that its volume is minimized for ellipsoids when the volume of $K$ is fixed. In this note we solve a dual version of this problem: we show that the volume of the polar body of $D^m(K)$ is maximized precisely by ellipsoids. For $m=1$ this recovers the symmetric case of the celebrated Blaschke-Santal\'o inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically \'a la Bourgain-Milman. We also consider a functional version.

We consider the classical geometric problem of prescribing the scalar and the boundary mean curvature in the unit ball endowed with the standard Euclidean metric. We will deal with the case of negative scalar curvature showing the existence of infinitely many non-radial positive solutions when the dimension is larger or equal to 5. This is the first result of existence of solutions in the case of negative prescribed scalar curvature problem in higher dimensions.

Let $p$ be a prime number. The $p$-power cyclic resultant of a polynomial is the determinant of the Sylvester matrix of $t^{p^n}-1$ and the polynomial. It is known that the sequence of $p$-power cyclic resultants and its non-$p$-parts converge in $\mathbb{Z}_p$. This article shows the $p$-adic convergence of the iterated $p$-power cyclic resultants of multivariable polynomials. As an application, we show the $p$-adic convergence of the torsion numbers of $\mathbb{Z}_p^d$-coverings of links. We also explicitly calculate the $p$-adic limits for the twisted Whitehead links as concrete examples. Moreover, in a specific case, we show that our $p$-adic limit of torsion numbers coincides with the $p$-adic torsion, which is a homotopy invariant of a CW-complex introduced by S. Kionke.

We describe five ideal triangulations of the 3-cusped hyperbolic `magic manifold' that are each compatible with well-established techniques for triangulating Dehn fillings. Using these techniques, we construct low-complexity triangulations for all partial fillings of the magic manifold, and in particular, recover minimal triangulations for 229 of the hyperbolic census knots. Along the way, these census knots are sorted into 42 families related by twisting that can be extended indefinitely, with each member of each infinite family inheriting an upper bound on its triangulation complexity. These triangulations are conjectured to be minimal for all 42 families.

The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. This problem was formulated by Caro and Ruiz and motivated by the data-driven prediction problem in quantum mechanics. Caro and Ruiz analysed the question of uniqueness for Hamiltonians of the form $-\Delta + V$ with an electric potential $V = V(\mathrm{t}, \mathrm{x})$ that depends on the time and space variables. In this context, they proved that uniqueness holds in dimension $n \geq 2$ whenever the potentials are bounded and have super-exponential decay at infinity. Although their result does not seem to be optimal, one would expect at least some degree of exponential decay to be necessary for the potentials. However, in this paper, we show that by restricting the analysis to Hamiltonians with time-independent electric potentials, namely $V = V(\mathrm{x})$, uniqueness can be established for bounded integrable potentials exhibiting only super-linear decay at infinity, in any dimension $n \geq 2$. This surprising improvement is possible because, unlike Caro and Ruiz's approach, our argument avoids the use of complex geometrical optics (CGO). Instead, we rely on the construction of stationary states at different energies -- this is possible because the potential does not depend on time. These states will have an explicit leading term, given by a Herglotz wave, plus a correction term that will vanish as the energy grows. Besides the significant relaxation of decay assumptions on the potential, the avoidance of CGO solutions is important in its own right, since such solutions are not readily available in more complicated geometric settings.

This paper presents a novel variant of the Broyden quasi-Newton secant-type method aimed at solving constrained mixed generalized equations, which can include functions that are not necessarily differentiable. The proposed method integrates the classical secant approach with techniques inspired by the Conditional Gradient method to handle constraints effectively. We establish local convergence results by applying the contraction mapping principle. Specifically, under assumptions of Lipschitz continuity, a modified Broyden update for derivative approximation, and the metric regularity property, we show that the algorithm generates a well-defined sequence that converges locally at a Q-linear rate.

Let $\chi$ be a non-principal Dirichlet character of modulus $q$ with associated \textit{L}-function $L(s,\chi)$. We prove that $$|L(1,\chi)|\le\left(\frac{1}{2}+O\Big(\frac{\log\log q}{\log q}\Big)\right)\frac{\varphi(q)}{q}\log q\,,$$ where $\varphi(\cdot)$ is Euler's phi function. This refines known bounds of the form $(c+o(1))\log q $ or $(c+O(\frac{1}{\log q}))\log q $ and is relevant for prime-rich moduli. It follows from Mertens' third theorem and the prime number theorem that $\inf_{q>2}\max_{\chi\ne\chi_0\,(\mod q)}\frac{|L(1,\chi)|}{\log q/\log\log q}\le\frac{1}{2}e^{-\gamma}$.

In this paper, we investigate the isometric structure of the $L^p$-spaces associated with an infinite-dimensional analogue of the Lebesgue measure $(\mathbb{R}^{\mathbb{N}}, \mu)$. Our primary result establishes that under the Continuum Hypothesis (CH), $L^p(\mu)$ is isometrically isomorphic to $\ell^p(\mathfrak{c}, L^p[0,1])$, where $\mathfrak{c}$ denotes the cardinality of the continuum. In the general case, without assuming CH, we prove that $L^p(\mu)$ contains an isometric and complemented copy of $\ell^p(\mathfrak{c}, L^p[0,1])$. Furthermore, we characterize isometries between $L^p$-spaces and establish necessary and sufficient conditions for an isometric isomorphism of the form $L^p(\nu) \cong \ell^p(\kappa, L^p[0,1])$ for some cardinal $\kappa$. In particular, we classify all possible isometric isomorphisms between such spaces.

Let $\mathbf{O}(\mathbb{F})$ be the split octonion algebra over an algebraically closed field $\mathbb{F}$. For positive integers $k_1, k_2\geq 2$, we study surjectivity of the map $A_1(x^{k_1}) + A_2(y^{k_2}) \in \mathbf{O}(\mathbb{F})\langle x, y\rangle$ on $\mathbf{O}(\mathbb{F})$. For this, we use the orbit representatives of the ${G}_2(\mathbb{F})$-action on $\mathbf{O}(\mathbb{F}) \times \mathbf{O}(\mathbb{F}) $ for the tuple $(A_1, A_2)$, and characterize the ones which give a surjective map.

The orbit method in its quantitative form due to Nelson and Venkatesh has played a central role in recent advances in the analytic theory of higher rank $L$-functions. The goal of this note is to explain how the method can be applied to the sup-norm problem for automorphic forms on $\operatorname{PGL}(2)$. Doing so, we prove a new hybrid bound for newforms $\varphi$ of large prime-power level $N = p^{4n}$ and large eigenvalue $\lambda$. It states that $\| \varphi \|_\infty \ll_p (\lambda N)^{5/24 + \varepsilon}$, recovering the result of Iwaniec and Sarnak spectrally and improving the local bound in the depth aspect for the first time in this non-compact setting. We also provide an exposition of the microlocal tools used, illustrating and motivating the theory through the classical case of $\operatorname{PGL}(2)$, following notes and lectures of Nelson and Venkatesh.

In this paper, we consider the set of partitions $ped(n)$ which counts the number of partitions of $n$ wherein the even parts are distinct (and the odd parts are unrestricted). Using an algorithm developed by Radu, we prove congruences modulo 192 which were conjectured by Nath. Further, we prove a few infinite families of congruences modulo 24 by using a result of Newman. Also, we prove that $ped(9n+7)$ is lacunary modulo $2^{k+2}\cdot 3$ and $3^{k+1}\cdot 4$ for all positive integers $k\geq0$. We further prove an infinite family of congruences for $ped(n)$ modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.

In line with known results on groups, we give some characterizations of radicals in Lie rings with minimal condition on centralizers. In particular, we prove the existence of the nilpotent radical and we deduce a form of Engel theorem if the characteristic is zero.

Arcs and caps are fundamental structures in finite projective spaces. They can be generalised. Here, a survey is given of some important results on these objects, in particular on generalised ovals and generalised ovoids. The paper also contains recent results and several open problems.

For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.

We compare the integral category O of shifted affine quantum groups of symmetric and non symmetric types. To do so we compute the K-theoretic analog of the Coulomb branches with symmetrizers introduced by Nakajima and Weekes. This yields an equivalence of the category O with a module category over a new type of quiver Hecke algebras. At the decategorified level, this establishes a connection between the Grothendieck group of O and a finite-dimensional module over a simple Lie algebra of unfolded symmetric type. We compute this module in certain cases and give a combinatorial rule for its crystal.

We present the fundamental properties of the K-theory groups of complex vector bundles endowed with actions of magnetic groups. In this work we show that the magnetic equivariant K-theory groups define an equivariant cohomology theory, we determine its coefficients, we show Bott's, Thom's and the degree shift isomorphism, we present the Atiyah-Hirzeburh spectral sequence, and we explicitly calculate two magnetic equivariant K-theory groups in order to showcase its applicability. These magnetic equivariant K-theory groups are relevant in condensed matter physics since they provide topological invariants of gapped Hamiltonians in magnetic crystals.

Hopf braces are the quantum analogues of skew braces and, as such, their cocommutative counterparts provide solutions to the quantum Yang-Baxter equation. We investigate various properties of categories related to Hopf braces. In particular, we prove that the category of Hopf braces is accessible while the category of cocommutative Hopf braces is even locally presentable. We also show that functors forgetting multiple antipodes and/or multiplications down to coalgebras are monadic. Colimits in the category of cocommutative Hopf braces are described explicitly and a free cocommutative Hopf brace on an arbitrary cocommutative Hopf algebra is constructed.

In this article, we provide a modification to the Bregman Golden Ratio Algorithm (B-GRAAL). We analyze the B-GRAAL algorithm with a new step size rule, where the step size increases after a certain number of iterations and does not require prior knowledge of the global Lipschitz constant of the cost operator. Under suitable assumptions, we establish the global iterate convergence as well as the R-linear rate of convergence of the modified algorithm. The numerical performance of the proposed approach is validated for the matrix game problem and the sparse logistic regression problem in machine learning.

In this paper we investigate classifications of all (transposed) Poisson algebras of the associated associative null-filiform algebra

We study an inviscid limit problem for a class of Navier-Stokes equations with vanishing measurable viscous coefficients in 3-dimensional spatial domains whose boundaries are oscillatory, depending on a small parameter, and become flat when the parameter converges to zero. Under some sufficient conditions on the anisotropic vanishing rates of the eigenvalues of the matrices of the viscous coefficients and the oscillatory parameter, we show that Leray-Hopf weak solutions of the Navier-Stokes equations with no slip boundary condition converge to solutions of the Euler equations in the upper half space. To prove the result, we apply a change of variables to flatten the boundaries of the spatial domains for the Navier-Stokes equations, and then construct the boundary layer terms. As the Navier-Stokes equations and the Euler equations are originally written in two different domains, additional boundary layer terms are constructed and their estimates are obtained.

In [MV] the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial example of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [VV1] on the differential KZ equations to the difference qKZ equations.

In this paper, we introduce and study a class of monoids, called Layered Catalan Monoids (\( {LC}_n \)), which satisfy the structural conditions for $\ll$-smoothness as defined in~\cite{Sha-Det2}. These monoids are defined by specific identities inspired by Catalan monoids. We establish their canonical forms and compute their determinant, proving that it is non-zero for \(1 \leq n \leq 7\) but vanishes for \(n \geq 8\).

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic properties. The motivation comes from the current lack of circle packing algorithms in spherical geometry, and the discrete Schwarzian derivative may provide for new approaches. A companion localized notion called an intrinsic schwarzian is also investigated. The main concrete results of the paper are limited to circle packing flowers. A parameterization by intrinsic schwarzians is established, providing an essential packing criterion for flowers. The paper closes with the study of special classes of flowers that occur in the circle packing literature. As usual in circle packing, there are pleasant surprises at nearly every turn, so those not interested in circle packing theory may still enjoy the new and elementary geometry seen in these flowers.

We introduce and investigate a novel notion of transversely affine foliation, comparing and contrasting it to the previous ones in the literature. We then use it to give an extension of the classic Hadamard's theorem from Riemannian geometry to this setting. Our main result is a transversely affine version of a well-known "Hadamard-like" theorem by J. Hebda for Riemannian foliations. Alternatively, our result can be viewed as a foliation-theoretic analogue of the Hadamard's theorem for affine manifolds proven by Beem and Parker. Namely, we show that under the transverse analogs of pseudoconvexity and disprisonment for the family of geodesics in the transverse affine geometry, together with an absence of transverse conjugate points, the universal cover of a manifold endowed with a transversely affine foliation whose leaves are compact and with finite holonomy is diffeomorphic to the product of a contractible manifold with the universal cover of a leaf. This also leads to a Beem-Parker-type Hadamard-like theorem for affine orbifolds.

The convergence of stochastic integrals is essential to stochastic analysis, especially in applications to mathematical finance, where they model the gains associated with a self-financing strategy. However, Fatou convergence of $(X^{n})_{n=1}^{\infty}$ $\unicode{x2014}$a notion introduced for its amenability to compactness principles$\unicode{x2014}$implies little about the sequence of It\^o integrals $\left(\int_{0}^{\cdot}YdX^{n}\right)_{n=1}^{\infty}$ for a fixed integrand $Y$. Under a boundedness condition, we find convex combinations $(\widetilde{X}^{n})_{n=1}^{\infty}$ of $(X^{n})_{n=1}^{\infty}$ with Fatou limit $\widetilde{X}$, such that $\left(\int_{0}^{\cdot}Yd\widetilde{X}^{n}\right)_{n=1}^{\infty}$ converges in a Fatou-like sense to $\int_{0}^{\cdot}Yd\widetilde{X}$ for all continuous semimartingales $Y$. The result is sharp, in the sense that continuity of $Y$ cannot be relaxed to being the left limits process of a semimartingale.

Let $R$ be a commutative noetherian ring, and let $\mathscr{S}$(resp. $\mathscr{L}$) be a Serre(resp. localizing) subcategory of the category of $R$-modules. If $\Bbb F$ is an unbounded complex of $R$-modules Tor-perpendicular to $\mathscr{S}$ and $d$ is an integer, then $\HH{i\geqslant d}{S\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each $R$-module $S$ in $\mathscr{S}$ if and only if $\HH{i\geqslant d}{k(\fp)\otimes_R \Bbb F}$ is in $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$, where $k(\fp)$ is the residue field at $\fp$. As an application, we show that for any $R$-module $M$, $\Tor_{i\geqslant 0}^R(k(\fp),M)$ is in $\mathscr{L}$ for each prime ideal $\fp$ such that $R/\fp$ is in $\mathscr{S}$ if and only if $\Ext^{i \geqslant 0}_R(S,M)$ is in $\mathscr{L}$ for each cyclic $R$-module $S$ in $\mathscr{S}$. We also obtain some new characterizations of regular and Gorenstein rings in the case of $\mathscr{S}$ consists of finite modules with supports in a specialization-closed subset $V(I)$ of $\Spec R$.

In 2023, Greaves et~al.\ constructed several sets of 57 equiangular lines in dimension 18. Using the concept of switching root introduced by Cao et~al.\ in 2021, these sets of equiangular lines are embedded in a lattice of rank 19 spanned by norm 3 vectors together with a switching root. We characterize this lattice as an overlattice of the root lattice $A_9\oplus A_9\oplus A_1$, and show that there are at least $246896$ sets of 57 equiangular lines in dimension $18$ arising in this way, up to isometry. Additionally, we prove that all of these sets of equiangular lines are strongly maximal. Here, a set of equiangular lines is said to be strongly maximal if there is no set of equiangular lines properly containing it even if the dimension of the underlying space is increased. Among these sets, there are ones with only six distinct Seidel eigenvalues.

Let $G$ be a finite group and $p$ be a prime. We denote by $C_p(G)$ the poset of all cosets of $p$-subgroups of $G$. We characterize the homotopy type of the geometric realization $|\Delta C_p(G)|$ for $p$-closed groups $G$, which is motivated by K.S.Brown's Question. We will further demonstrate that $\chi(C_{p}(G)) \equiv |G|_{p'} (\text{mod} p)$ for any finite group $G$ and any prime $p$.

The one-dimensional toy models proposed for the three-dimensional electron magnetohydrodynamics in our previous work share some similarities with the original dynamics under certain symmetry. We continue to study the well-posedness issue and explore the potential singularity formation scenario for these models.

Crystal bases are powerful combinatorial tools in the representation theory of quantum groups $U_q(\mathfrak{g})$ for a symmetrizable Kac-Moody algebras $\mathfrak{g}$. The polyhedral realizations are combinatorial descriptions of the crystal base $B(\infty)$ for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when $\mathfrak{g}$ is finite dimensional simple. It is a fundamental and natural problem to find explicit forms of the polyhedral cone. The monomial realization expresses crystal bases $B(\lambda)$ of integrable highest weight representations as Laurent monomials with double indexed variables. In this paper, we give a conjecture between explicit forms of the polyhedral cones and monomial realizations. We prove the conjecture is true when $\mathfrak{g}$ is a classical Lie algebra, a rank $2$ Kac-Moody algebra or a classical affine Lie algebra.

In 2007, J. P. Cossey conjectured that if $G$ is a finite $p$-solvable group and $\varphi$ is an irreducible Brauer character of $G$ with vertex $Q$, then the number of lifts of $\varphi$ is at most $|Q:Q'|$. In this paper we revisited Cossey's conjecture for $p=2$ from the perspective of Navarro vertices and obtained a new way to count the number of lifts of $\varphi$. Some applications were given.

We prove the Lusztig's $a$-function is bounded for any Coxeter group of finite rank.

The global null controllability of stochastic semilinear parabolic equations with globally Lipschitz nonlinearities has been addressed in recent literature. However, there are no results concerning their numerical approximation and the behavior of discrete controls when the discretization parameter goes to zero. This paper is intended to studying the null controllability for semi-discrete stochastic semilinear parabolic equations, where the spatial variable is discretized with finite difference scheme and the time is kept as a continuous variable. The proof is based on a new refined semi-discrete Carleman estimate and Banach fixed point method. The main novelty here is that the Carleman parameters and discretization parameter are made explicit and are then used in a Banach fixed point method.

The volumes of Kostka polytopes appear naturally in questions of random matrix theory in the context of the randomized Schur-Horn problem, i.e., evaluating the probability density that a random Hermitian matrix with fixed spectrum has a given diagonal. We give a polynomial-time deterministic algorithm for approximating the volume of a ($\Omega(n^2)$ dimensional) Kostka polytope $\mathrm{GT}(\lambda, \mu)$ to within a multiplicative factor of $\exp(O(n\log n))$, when $\lambda$ is an integral partition with $n$ parts, with entries bounded above by a polynomial in $n$, and $\mu$ is an integer vector lying in the interior of the Schur-Horn polytope associated to $\lambda$. The algorithm thus gives asymptotically correct estimates of the log-volume of Kostka polytopes corresponding to such $(\lambda, \mu)$. Our approach is based on a partition function interpretation of the continuous analogue of Schur polynomials, and an associated maximum entropy principle.

The Cage Problem requires for a given pair $k \geq 3, g \geq 3$ of integers the determination of the order of a smallest $k$-regular graph of girth $g$. We address a more general version of this problem and look for the $(k,g)$-spectrum of orders of $(k,g)$-graphs: the (infinite) list of all orders of $(k,g)$-graphs. By establishing these spectra we aim to gain a better understanding of the structure and properties of $(k,g)$-graphs and hope to use the acquired knowledge in both determining new orders of smallest $k$-regular graphs of girth $g$ as well as developing a set of tools suitable for constructions of extremal graphs with additional requirements. We combine theoretical results with computer-based searches, and determine or determine up to a finite list of unresolved cases the $(k,g)$-spectra for parameter pairs for which the orders of the corresponding cages have already been established.

In this article, we extend our previous work (Applicable Analysis, 2024, pp. 1-25) on the steepest descent method for uncertain multiobjective optimization problems. While that study established local convergence, it did not address global convergence and the rate of convergence of the steepest descent algorithm. To bridge this gap, we provide rigorous proofs for both global convergence and the linear convergence rate of the steepest descent algorithm. Global convergence analysis strengthens the theoretical foundation of the steepest descent method for uncertain multiobjective optimization problems, offering deeper insights into its efficiency and robustness across a broader class of optimization problems. These findings enhance the method's practical applicability and contribute to the advancement of robust optimization techniques.

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$ and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq {V\brack k}$ is called $k$-uniform $r$-wise $t$-intersecting if for any $F_1, F_2, \dots, F_r \in \mathcal{F}$, we have $\dim\left(\bigcap_{i=1}^r F_i \right) \geq t$. An $r$-wise $t$-intersecting family $\{X_1, X_2, \dots, X_{r+1}\}$ is called a $(r+1,t)$-simplex if $\dim\left(\bigcap_{i=1}^{r+1} X_i \right) < t$, denoted by $\Delta_{r+1,t}$. Notice that it is usually called triangle when $r=2$ and $t=1$. For $k \geq t \geq 1$, $r \geq 2$ and $n \geq 3kr^2 + 3krt$, we prove that the maximal number of $\Delta_{r+1,t}$ in a $k$-uniform $r$-wise $t$-intersecting subspace family of $V$ is at most $n_{t+r,k}$, and we describe all the extreme families. Furthermore, we have the extremal structure of $k$-uniform intersecting families maximizing the number of triangles for $n\geq 2k+9$ as a corollary.

In this paper, we prove a fluctuation theorem for the occupation time of the multi-species stirring process on a lattice starting from a stationary distribution. Our result shows that the occupation times of different species interact with each other at the level of equilibrium fluctuation. The proof of our result utilizes the resolvent strategy introduced in \cite{Kipnis1987}. A coupling relationship between the multi-species stirring process and an auxiliary process and a graphical representation of the auxiliary process play the key roles in the proof.

Numerous real-world applications of uncertain multiobjective optimization problems (UMOPs) can be found in science, engineering, business, and management. To handle the solution of uncertain optimization problems, robust optimization is a relatively new field. An extended version of the projected gradient method (PGM) for a deterministic smooth multiobjective optimization problem (MOP) is presented in the current study as a PGM for UMOP. An objective-wise worst-case cost (OWWC) type robust counterpart is considered, and the PGM is used to solve a UMOP by using OWWC. A projected gradient descent algorithm is created using theoretical findings. It is demonstrated that the projected gradient descent algorithm's generated sequence converges to the robust counterpart's weak Pareto optimal solution, which will be the robust weak Pareto optimal solution for UMOP. Under a few reasonable presumptions, the projected gradient descent algorithm's full convergent behavior is also justified. Finally, numerical tests are presented to validate the proposed method.

This work is dedicated to uniqueness and numerical algorithms for determining the point sources of the biharmonic wave equation using scattered fields at sparse sensors. We first show that the point sources in both $\mathbb{R}^2$ and $\mathbb{R}^3$ can be uniquely determined from the multifrequency sparse scattered fields. In particular, to deal with the challenges arising from the fundamental solution of the biharmonic wave equation in $\mathbb{R}^2$, we present an innovative approach that leverages the Fourier transform and Funk-Hecke formula. Such a technique can also be applied for identifying the point sources of the Helmholtz equation. Moreover, we present the uniqueness results for identifying multiple point sources in $\mathbb{R}^3$ from the scattered fields at sparse sensors with finitely many frequencies. Based on the constructive uniqueness proofs, we propose three numerical algorithms for identifying the point sources by using multifrequency sparse scattered fields. The numerical experiments are presented to verify the effectiveness and robustness of the algorithms.

Robust adaptive beamforming (RAB) based on interference-plus-noise covariance (IPNC) matrix reconstruction can experience serious performance degradation in the presence of look direction and array geometry mismatches, particularly when the input signal-to-noise ratio (SNR) is large. In this work, we present a RAB technique to address covariance matrix reconstruction problems. The proposed method involves IPNC matrix reconstruction using a low-complexity spatial sampling process (LCSSP) and employs a virtual received array vector. In particular, we devise a power spectrum sampling strategy based on a projection matrix computed in a higher dimension. A key feature of the proposed LCSSP technique is to avoid reconstruction of the IPNC matrix by integrating over the angular sector of the interference-plus-noise region. Simulation results are shown and discussed to verify the effectiveness of the proposed LCSSP method against existing approaches.

This paper introduces a Heisenberg picture approach to Matrix Product States (MPS), offering a rigorous yet intuitive framework to explore their structure and classification. MPS efficiently represent ground states of quantum many-body systems, with infinite MPS (iMPS) capturing long-range correlations and thermodynamic behavior. We classify MPS into projective and non-projective types, distinguishing those with finite correlation structures from those requiring ergodic quantum channels to define a meaningful limit. Using the Markov-Dobrushin inequality, we establish conditions for infinite-volume states and introduce ergodic and mixing MPS. As an application, we analyze the depolarizing MPS, highlighting its lack of finite correlations and the need for an alternative ergodic description. This work deepens the mathematical foundations of MPS and iMPS, providing new insights into entanglement, phase transitions, and quantum dynamics.

We consider the Wigner minor process, i.e. the eigenvalues of an $N\times N$ Wigner matrix $H^{(N)}$ together with the eigenvalues of all its $n\times n$ minors, $H^{(n)}$, $n\le N$. The top eigenvalues of $H^{(N)}$ and those of its immediate minor $H^{(N-1)}$ are very strongly correlated, but this correlation becomes weaker for smaller minors $H^{(N-k)}$ as $k$ increases. For the GUE minor process the critical transition regime around $k\sim N^{2/3}$ was analyzed by Forrester and Nagao (J. Stat. Mech.: Theory and Experiment, 2011) providing an explicit formula for the nontrivial joint correlation function. We prove that this formula is universal, i.e. it holds for the Wigner minor process. Moreover, we give a complete analysis of the sub- and supercritical regimes both for eigenvalues and for the corresponding eigenvector overlaps, thus we prove the decorrelation transition in full generality.

In this paper, we develop a modified nonlinear dynamic diffusion (DD) finite element method for convection-diffusion-reaction equations. This method is free of stabilization parameters and is capable of precluding spurious oscillations. We prove existence and, under an assumption of small mesh size, uniqueness of the discrete solution, and derive the optimal first order convergence rate of the approximation error in the energy norm plus a dissipation term. Numerical examples are provided to verify the theoretical analysis.

In this paper, we propose a condition on the coefficients of a cohomology-valued power series, which we call ``asymptotically Mittag-Leffler''. We show that if the $J$-function of a Fano manifold is asymptotically Mittag-Leffler, then it has the exponential growth as $t\to +\infty$. This provides an alternative method to compute the principal asymptotic class of a Fano manifold using the coefficients of $J$-function. We also verify that the $J$-function of the projective space is asymptotically Mittag-Leffler, and the property of having an asymptotically Mittag-Leffler $J$-function is preserved when taking product and hypersurface.

In the first part of the paper, we establish an homotopical version of the snail lemma (which is a generalization of the classical snake lemma). In the second part, we introduce the category $\mathbf{Seq}(\mathcal A)$ of sequentiable families of arrows in a category $\mathcal A$ and we compare it with the category of chain complexes in $\mathcal A.$ We apply the homotopy snail lemma to a morphism in $\mathbf{Seq}(\mathcal A)$ obtaining first a six-term exact sequence in $\mathbf{Seq}(\mathcal A)$ and then, unrolling the sequence in $\mathbf{Seq}(\mathcal A),$ a long exact sequence in $\mathcal A.$ When $\mathcal A$ is abelian, this sequence subsumes the usual long homology sequence obtained from an extension of chain complexes.

This work is dedicated to a novel sampling method for accurately reconstructing elastic and electromagnetic sources from the far field patterns. We show that the proposed indicators in the form of integrals with full far field patterns are exactly the source functions. These facts not only give constructive uniqueness proofs of the inverse source problems, but also establish the theoretical basis of the proposed sampling methods. Furthermore, we derive the stability estimates for the corresponding discrete indicators using the far field patterns with finitely many observations and frequencies. We have also proposed the indicators with partial far field patterns and proved their validity for providing the derivative information of the unknown sources. Numerical examples are presented to verify the accuracy and stability of the proposed quantitative sampling method.

Let $\mathcal{F}$ be a one dimensional holomorphic foliation with isolated singularities, leaving invariant a singular complete interserction $V = \bigcap_{\lambda =1}^kD_{\lambda}$ of $k$ hypersurfaces $D_{\lambda}$. We proved formulas to calculate the GSV and Schwartz indices of singularities of $\mathcal{F}$ and we find bounds for these indices. As applications, we provide some results related to Poincar\'e's problem for foliations on the prejective space $\mathbb{P}$.

This work highlights a peculiar phenomenon of interval exchange. Considering a real number beta less than -1, the negative beta-shift is coded if and only if its absolute value is greater than the golden ratio. We study an increasing sequence of algebraic integers with limit-1 and the absolute value of the first term equals to the golden ratio such that for a base x taken in the interval of consicutive terms of this sequence, the measure of the maximal entropy is carried by the image of a beta-shift, with the golden ratio les than the absolute value of beta, under the mapping of an injective substitution.

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs $(q,n)$ for which this is the case. This continues a recent publication by Cohen-Danchev et al. in Turk. J. Math. (2024) in which the tripotent version was examined in-depth as well as it extends recent results of this branch established by Abyzov-Tapkin in Sib. Math. J. (2024).

$F$-invariant for a pair of good elements (e.g. cluster monomials) in cluster algebras is introduced by the author in a previous work. A key feature of $F$-invariant is that it is a coordinate-free invariant, that is, it is mutation invariant under the initial seed mutations. $E$-invariant for a pair of decorated representations of quivers with potentials is introduced by Derksen, Weyman and Zelevinsky, which is also a coordinate-free invariant. The strategies used to show the mutation-invariance of $F$-invariant and $E$-invariant are totally different. In this paper, we give a new proof of the mutation-invariance of $F$-invariant following the strategy used by Derksen, Weyman and Zelevinsky. As a result, we prove that $F$-invariant coincides with $E$-invariant on cluster monomials. We also give a proof of Reading's conjecture, which says that the non-compatible cluster variables in cluster algebras can be separated by the sign-coherence of $g$-vectors.

We prove that any complex local representation of the flat virtual braid group, $FVB_2$, into $GL_2(\mathbb{C})$, for $n\geq 2$, has one of the types $\lambda_i: FVB_2 \rightarrow GL_2(\mathbb{C})$, $1\leq i\leq 12$. We find necessary and sufficient conditions that guarantee the irreducibility of representations of type $\lambda_i$, $1\leq i\leq 5$, and we prove that representations of type $\lambda_i$, $6\leq i\leq 12$, are reducible. Regarding faithfulness, we find necessary and sufficient conditions for representations of type $\lambda_6$ or $\lambda_7$ to be faithful. Moreover, we give sufficient conditions for representations of type $\lambda_1$, $\lambda_2$, or $\lambda_4$ to be unfaithful, and we show that representations of type $\lambda_i$, $i=3, 5, 8, 9, 10, 11, 12$ are unfaithful. We prove that any complex homogeneous local representations of the flat virtual braid group, $FVB_n$, into $GL_{n}(\mathbb{C})$, for $n\geq 2$, has one of the types $\gamma_i: FVB_n \rightarrow GL_n(\mathbb{C})$, $i=1, 2$. We then prove that representations of type $\gamma_1: FVB_n \rightarrow GL_n(\mathbb{C})$ are reducible for $n\geq 6$, while representations of type $\gamma_2: FVB_n \rightarrow GL_n(\mathbb{C})$ are reducible for $n\geq 3$. Then, we show that representations of type $\gamma_1$ are unfaithful for $n\geq 3$ and that representations of type $\gamma_2$ are unfaithful if $y=b$. Furthermore, we prove that any complex homogeneous local representation of the flat virtual braid group, $FVB_n$, into $GL_{n+1}(\mathbb{C})$, for all $n\geq 4$, has one of the types $\delta_i: FVB_n \rightarrow GL_{n+1}(\mathbb{C})$, $1\leq i\leq 8$. We prove that these representations are reducible for $n\geq 10$. Then, we show that representations of types $\delta_i$, $i\neq 5, 6$, are unfaithful, while representations of types $\delta_5$ or $\delta_6$ are unfaithful if $x=y$.

We study special $\mathbb{G}_m$-equivariant degenerations of a smooth del Pezzo surface $X$ induced by valuations that are log canonical places of $(X,C)$ for a nodal anti-canonical curve $C$. We show that the space of special valuations in the dual complex of $(X,C)$ is connected and admits a locally finite partition into sub-intervals, each associated to a $\mathbb{G}_m$-equivariant degeneration of $X$. This result is an example of higher rank degenerations of log Fano varieties studied by Liu-Xu-Zhuang, and verifies a global analog of a conjecture on Koll\'ar valuations raised by Liu-Xu. For del Pezzo surfaces with quotient singularities, we obtain a weaker statement about the space of special valuations associated to a normal crossing complement.

This paper investigates contractive projections on closed subspaces $X$ of $L^p$ with $0<p<\infty$. One of the main results states that, subject to certain mild conditions, every contractive projection $P$ on $X$ preserving constants coincides with a conditional expectation on $L^\infty \cap P^{-1}(L^\infty)$. It results in some interesting applications concerning contractive idempotent coefficient multipliers for analytic function spaces and translation-invariant subspaces of $L^p(G),$ where $G$ is a compact Abelian group. Focusing specifically on descriptions of boundedness and contractivity of conditional expectations on the Hardy space $H^p(\mathbb{T})$ with $0<p<1$, we give a complete characterization of contractive idempotent coefficient multipliers for $H^p(\mathbb{T}^d)$ with $0<p<1$, which complements a remarkable result due to Brevig, Ortega-Cerd\`{a}, and Seip characterizing such multipliers on $H^p(\mathbb{T}^d)$ for $1\leq p \leq \infty$.

Let Y be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study the probabilistic degenerate poly-Bell polynomials associated with the random variable Y, arising from the degenerate polyexponential functions, which are probabilistic extensions of degenerate versions of the poly-Bell polynomials. We derive several explicit expressions and some related identities for them. In addition, we consider the special cases that Y is the Bernoulli random variable with probability of success p or the gamma random variable with parameters 1,1.

Rough stochastic differential equations (RSDEs) are common generalisations of Ito SDEs and Lyons RDEs and have emerged as new tool in several areas of applied probability, including non-linear stochastic filtering, pathwise stochastic optimal control, volatility modelling in finance and mean-fields analysis of common noise system. We here take a unified perspective on rough Ito processes and discuss in particular when and how they become, upon randomisation, "doubly stochastic" Ito processes, and what can be said about their conditional laws.

First-order methods for minimization and saddle point (min-max) problems are one of the cornerstones of modern ML. The majority of works obtain favorable complexity guarantees of such methods assuming that exact gradient information is available. At the same time, even the use floating-point representation of real numbers already leads to relative error in all the computations. Relative errors arise also in such applications as bilevel optimization, inverse problems, derivative-free optimization, inexact proximal methods. This paper answers several theoretical open questions on first-order optimization methods under relative errors. We propose an explicit single-loop accelerated gradient method that preserves optimal convergence rate under maximal possible relative error in the gradient and explore the tradeoff between the relative error and deterioration in the linear convergence rate. We further explore similar questions for saddle point problems showing that a variant of gradient descent-ascent and the extragradient method are robust to such errors.

In this paper, we study a generalization of the D\'iaz-Saa inequality and its applications to nonlinear elliptic problems. We first present the necessary hypotheses and preliminary results before introducing an improved version of the inequality, which holds in a broader functional setting and allows applications to problems with homogeneous Neumann boundary conditions. The significance of cases where the inequality becomes an equality is also analyzed, leading to uniqueness results for certain classes of partial differential equations. Furthermore, we provide a detailed proof of a uniqueness theorem for strong positive solutions and illustrate our findings with two concrete applications: a multiple-phase problem and an elliptic quasilinear equation relevant to image processing. The paper concludes with possible directions for future research.

In our recent papers, we studied semiclassical spectral problems for the Bochner-Schr\"odinger operator on a manifold of bounded geometry. We survey some results of these papers in the setting of the magnetic Schr\"odinger operator in the Euclidean space and describe some ideas of the proofs.

In this paper, we study the following fractional nonlocal Sobolev-type inequality \begin{equation*} C_{HLS}\bigg(\int_{\mathbb{R}^n}\big(|x|^{-\mu} \ast |u|^{p_s}\big)|u|^{p_s} dx\bigg)^{\frac{1}{p_s}}\leq\|u\|_{\dot{H}^s(\mathbb{R}^n)}^2\quad \mbox{for all}~~u\in \dot{H}^s(\mathbb{R}^n), \end{equation*} induced by the classical fractional Sobolev inequality and Hardy-Littlewood-Sobolev inequality for $s\in(0,\frac{n}{2})$, $\mu\in(0,n)$ and where $p_{s}=\frac{2n-\mu}{n-2s}\geq2$ is energy-critical exponent. The $C_{HLS}>0$ is a constant depending on the dimension $n$, parameters $s$ and $\mu$, which can be achieved by $W(x)$, and up to translation and scaling, $W(x)$ is the unique positive and radially symmetric extremal function of the nonlocal Sobolev-type inequality. It is well-known that, up to a suitable scaling, \begin{equation*} (-\Delta)^{s}u=(|x|^{-\mu}\ast |u|^{p_s})|u|^{p_s-2}u\quad \mbox{for all}~~u\in\dot{H}^s(\mathbb{R}^n), \end{equation*} is the Euler-Lagrange equation corresponding to the associated minimization problem. In this paper, we first prove the non-degeneracy of positive solutions to the critical Hartree equation for all $s\in(0,\frac{n}{2})$, $\mu\in(0,n)$ with $0<\mu\leq4s$. Furthermore, we show the existence of a gradient type remainder term and, as a corollary, derive the existence of a remainder term in the weak $L^{\frac{n}{n-2s}}$-norm for functions supported in domains of finite measure, under the condition $s\in(0,\frac{n}{2})$. Finally, we establish a Struwe-type profile decomposition and quantitative stability estimates for critical points of the above inequality in the parameter region $s\in(0,\frac{n}{2})$ with the number of bubbles $\kappa\geq1$, and for $\mu\in(0,n)$ with $0<\mu\leq4s$. In particular, we provide an example to illustrate the sharpness of our result for $n=6s$ and $\mu=4s$.

Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as $f(x) = h(a x^q + b x + c) + u x^q + v x$, where $h(x) \in \mathbb{F}_{q^2}[x]$ and $a$, $b$, $c$, $u$, $v \in \mathbb{F}_{q^2}$. Using a commutative diagram satisfied by $f(x)$ and trace functions over finite fields, we reduce the problem whether $f(x)$ is a many-to-one mapping on $\mathbb{F}_{q^2}$ to another problem whether an associated polynomial $g(x)$ is a many-to-one mapping on the subfield $\mathbb{F}_{q}$. In particular, when $h(x) = x^{r}$ and $r$ satisfies certain conditions, we reduce $g(x)$ to polynomials of small degree or linearized polynomials. Then by employing the many-to-one properties of these low degree or linearized polynomials on $\mathbb{F}_{q}$, we derive a series of explicit characterization for $f(x)$ to be many-to-one on $\mathbb{F}_{q^2}$. On the other hand, for all $1$-to-$1$ mappings obtained in this paper, we determine the inverses of these permutation polynomials. Moreover, we also explicitly construct involutions from $2$-to-$1$ mappings of this form. Our findings generalize and unify many results in the literature.

In this paper, we study the asymptotic macroscopic behavior of the root sets of iterated, randomized derivatives of polynomials. The randomization depend on a parameter of inverse temperature $\beta \in (0, \infty]$, the case $\beta = \infty$ corresponding to the situation where one considers the derivative of polynomials, without randomization. Our constructions can be connected to random matrix theory: in particular, as detailed in Section 2, for $\beta = 2$ and roots on the real line, we get the distribution of the eigenvalues of minors of unitarily invariant random matrices. We prove that the asymptotic macroscopic behavior of the roots, i.e. the hydrodynamic limit, does not depend on $\beta$, and coincides with what we obtain for the non-randomized iterated derivatives, i.e. for $\beta = \infty$. Since recent results obtained for iterated derivations show that the limiting dynamics is governed by a non-local and non-linear PDE, we can transfer this information to the macroscopic behavior of the randomized setting. Our proof is completely explicit and relies on the analysis of increments in a triangular bivariate Markov chain.

This paper explores the emerging research direction of electromagnetic information theory (EIT), which aims to integrate traditional Shannon-based methodologies with physical consistency, particularly the electromagnetic properties of communication channels. We propose an EIT-based multiple-input multiple-output (MIMO) paradigm that enhances conventional spatially-discrete MIMO models by incorporating the concepts of electromagnetic (EM) precoding and EM combining. This approach aims to improve the modeling of next-generation systems while remaining consistent with Shannon's theoretical foundations. We explore typical EIT applications, such as densely spaced MIMO, near-field communications, and tri-polarized antennas, and analyze their channel characteristics through theoretical simulations and measured datasets. The paper also discusses critical research challenges and opportunities for EIT applications from an industrial perspective, emphasizing the field's potential for practical applications.

We prove a central limit error bound for convolution powers of laws with finite moments of order $r \in \mathopen]2,3\mathclose]$, taking a closeness of the laws to normality into account. Up to a universal constant, this generalises the case of $r=3$ of the sharpening of the Berry (1941) - Esseen (1942) theorem obtained by Mattner (2024), namely by sharpening here the Katz (1963) error bound for the i.i.d. case of Lyapunov's (1901) theorem. Our proof uses a partial generalisation of the theorem of Senatov and Zolotarev (1986) used for the earlier special case. A result more general than our main one could be obtained by using instead another theorem of Senatov (1980), but unfortunately an auxiliary inequality used in the latter's proof is wrong.

The generalized cyclotomic mappings over finite fields $\mathbb{F}_{q}$ are those mappings which induce monomial functions on all cosets of an index $\ell$ subgroup $C_0$ of the multiplicative group $\mathbb{F}_{q}^{*}$. Previous research has focused on the one-to-one property, the functional graphs, and their applications in constructing linear codes and bent functions. In this paper, we devote to study the many-to-one property of these mappings. We completely characterize many-to-one generalized cyclotomic mappings for $1 \le \ell \le 3$. Moreover, we completely classify $2$-to-$1$ generalized cyclotomic mappings for any divisor $\ell$ of $q-1$. In addition, we construct several classes of many-to-one binomials and trinomials of the form $x^r h(x^{q-1})$ on $\mathbb{F}_{q^2}$, where $h(x)^{q-1}$ induces monomial functions on the cosets of a subgroup of $U_{q+1}$.

Perfectly Matched Layers (PML) has become a very common method for the numerical approximation of wave and wave-like equations on unbounded domains. This technique allows one to obtain accurate solutions while working on a finite computational domain, and the technique is relatively simple to implement. Results concerning the accuracy of the PML method have been obtained, but mostly with regard problems at a fixed frequency. In this paper we provide very explicit time-domain bounds on the accuracy of PML for the two-dimensional wave equation and illustrate our conclusions with some numerical examples.

We extend the work of Galatos (2004) on generalised ordinal sums of residuated lattices. We show that the generalised ordinal sum of an odd quasi relation algebra (qRA) satisfying certain conditions and an arbitrary qRA is again a qRA. In a recent paper by Craig and Robinson (2024), the notion of representability for distributive quasi relation algebras (DqRAs) was developed. For certain pairs of representable DqRAs, we prove that their generalised ordinal sum is again representable. An important consequence of this result is that finite Sugihara chains are finitely representable.

An explicit first-order drift-randomized Milstein scheme for a regime switching stochastic differential equation is proposed and its bi-stability and rate of strong convergence are investigated for a non-differentiable drift coefficient. Precisely, drift is Lipschitz continuous while diffusion along with its derivative is Lipschitz continuous. Further, we explore the significance of evaluating Brownian trajectories at every switching time of the underlying Markov chain in achieving the convergence rate $1.0$ of the proposed scheme. In this context, possible variants of the scheme, namely modified randomized and reduced randomized schemes, are considered and their convergence rates are shown to be $1/2$. Numerical experiments are performed to illustrate the convergence rates of these schemes along with their corresponding non-randomized versions. Further, it is illustrated that the half-order non-randomized reduced and modified schemes outperforms the classical Euler scheme.

This paper studies a distributed algorithm for constrained consensus optimization that is obtained by fusing the Arrow-Hurwicz-Uzawa primal-dual gradient method for centralized constrained optimization and the Wang-Elia method for distributed unconstrained optimization. It is shown that the optimal primal-dual point is a semiglobally exponentially stable equilibrium for the algorithm, which implies linear convergence. The analysis is based on the separation between a slow centralized optimization dynamics describing the evolution of the average estimate toward the optimum, and a fast dynamics describing the evolution of the consensus error over the network. These two dynamics are mutually coupled, and the stability analysis builds on control theoretic tools such as time-scale separation, Lyapunov theory, and the small-gain principle. Our analysis approach highlights that the consensus dynamics can be seen as a fast, parasite one, and that stability of the distributed algorithm is obtained as a robustness consequence of the semiglobal exponential stability properties of the centralized method. This perspective can be used to enable other significant extensions, such as time-varying networks or delayed communication, that can be seen as ``perturbations" of the centralized algorithm.

We initiate systematic study of EZ-structures (and associated boundaries) of groups acting on spaces that admit consistent and conical (equivalently, consistent and convex) geodesic bicombings. Such spaces recently drew a lot of attention due to the fact that many classical groups act `nicely' on them. We rigorously construct EZ-structures, discuss their uniqueness (up to homeomorphism), provide examples, and prove some boundary-related features analogous to the ones exhibited by CAT(0) spaces and groups, which form a subclass of the discussed class of spaces and groups.

We consider vector and set optimization problems with respect to variable domination structures given by set-valued mappings acting between the preimage space and the image space of the objective mapping, as well as by set-valued mappings with the same input and output space, that coincides with the image space of the objective mapping. The aim of this paper is to derive necessary conditions for approximately nondominated points of problems with a single-valued objective function, employing an extension of Ekeland's Variational Principle for problems with respect to variable domination structures in terms of generalized differentiation in the sense of Mordukhovich. For set-valued objective mappings, we derive necessary conditions for approximately nondominated points of problems with variable domination structure taking into account the incompatibility between openness and optimality and a directional openness result for the sum of set-valued maps. We describe the necessary conditions for approximately nondominated points of set optimization problems with variable domination structure in terms of the limiting (Mordukhovich) generalized differentiation objects.

We present two limit theorems, a mean ergodic and a central limit theorem, for a specific class of one-dimensional diffusion processes that depend on a small-scale parameter $\varepsilon$ and converge weakly to a homogenized diffusion process in the limit $\varepsilon \rightarrow 0$. In these results, we allow for the time horizon to blow up such that $T_\varepsilon \rightarrow \infty$ as $\varepsilon \rightarrow 0$. The novelty of the results arises from the circumstance that many quantities are unbounded for $\varepsilon \rightarrow 0$, so that formerly established theory is not directly applicable here and a careful investigation of all relevant $\varepsilon$-dependent terms is required. As a mathematical application, we then use these limit theorems to prove asymptotic properties of a minimum distance estimator for parameters in a homogenized diffusion equation.

In this paper, we prove that uniformly bounded simple Lie conformal algebra must be finitely generated. Furthermore, we give a completely classification of simple uniformly bounded Lie conformal algebras with upper bound one.

We propose and analyze a sequential quadratic programming algorithm for minimizing a noisy nonlinear smooth function subject to noisy nonlinear smooth equality constraints. The algorithm uses a step decomposition strategy and, as a result, is robust to potential rank-deficiency in the constraints, allows for two different step size strategies, and has an early stopping mechanism. Under the linear independence constraint qualification, convergence is established to a neighborhood of a first-order stationary point, where the radius of the neighborhood is proportional to the noise levels in the objective function and constraints. Moreover, in the rank-deficient setting, the merit parameter may converge to zero, and convergence to a neighborhood of an infeasible stationary point is established. Numerical experiments demonstrate the efficiency and robustness of the proposed method.

This paper investigates a cell-free massive MIMO (multiple-input multiple-output) system where distributed access points (APs) perform integrated sensing and communications (ISAC) tasks, enabling simultaneous user communication and target detection/tracking. A unified framework and signal model are developed for the detection of potential targets and tracking of previously detected ones, even in arbitrary positions. Leveraging the Generalized Likelihood Ratio Test technique, novel detection/tracking algorithms are proposed to handle unknown target responses and interference. Scalable AP-user and AP-target association rules are evaluated, explicitly considering multi-zone sensing scenarios. Additionally, a scalable power control mechanism extends fractional power control principles to ISAC, balancing power allocation between communication and sensing tasks. For benchmarking, a non-scalable power control optimization problem is also formulated to maximize the minimum user data rate while ensuring a Quality of Service constraint for sensing, solved via successive convex approximation. Extensive numerical results validate the proposed framework, demonstrating its effectiveness in both communication and sensing, revealing the impact of interference from other targets, and highlighting fundamental trade-offs between sensing and communication performance.

We solve the Dry Ten Martini Problem for the unitary almost Mathieu operator with Diophantine frequencies in the non-critical regime.

This paper touches on several interaction points of semigroups and constructions from category theory: An adjunction is established between categories with selected arrows and semigroups. Regular semigroups are characterized by split epi - split mono factorization of the Karoubi envelope. We investigate how semigroupads (monads without requirement of unit transformation) map semigroups to semigroups and ensure certain properties provided they hold on meta level.

We prove the following: If $G$ is a finite solvable group and $H$ is another group whose subrack lattice is isomorphic to the subrack lattice of $G$, then $H$ is a solvable group and the derived length of $H$ coincides with the derived length of $G$.

We develop a technique to show the Morita equivalence of certain subrings of a ring with local units. We then apply this technique to develop conditions that are sufficient to show the Morita equivalence of subalgebras induced by partial subactions on generalized Boolean algebras and, subsequently, strongly $E^{\ast}$-unitary inverse subsemigroups. As an application, we prove that the Leavitt path algebra of a graph is Morita equivalent to the Leavitt path algebra of certain subgraphs and use this to calculate the Morita equivalence class of some Leavitt path algebras. Finally, as the main application, we prove a desingularization result for labelled Leavitt path algebras.

This paper investigates sharp stability estimates for the fractional Hardy-Sobolev inequality: $$\mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,{\rm d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-\Delta)^{\frac{s}{2}} u \right|^2 \,{\rm d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right),$$ where $N > 2s$, $s \in (0,1)$, $0 < t < 2s < N $, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $\mu_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality. The paper focuses on the quantitative stability results of the above inequality and the corresponding Euler-Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler-Lagrange equation, offering a thorough characterization of the Palais-Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell \cite{BE91} as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi \cite{CFM18} in the framework of fractional calculus.

Kuratowski's theorem says that the minimal (under subgraph containment) graphs that are not planar are the subdivisions of $K_5$ and of $K_{3,3}$. Here we study the minimal (under subdigraph containment) strongly-connected digraphs that are not planar. We also find the minimal strongly-connected non-outerplanar digraphs and the minimal strongly-connected non-series-parallel digraphs.

In this paper we explore the algebraic structure and combinatorial properties of eulerian magnitude homology. First, we analyze the diagonality conditions of eulerian magnitude homology, providing a characterization of complete graphs. Then, we construct the regular magnitude-path spectral sequence as the spectral sequence of the (filtered) injective nerve of the reachability category, and explore its consequences. Among others, we show that such spectral sequence converges to the complex of injective words on a digraph, and yields characterization results for the regular path homology of diagonal directed graphs.

In this paper we study the asymptotic behavior of a family of discrete functionals as the lattice size, $\varepsilon>0$, tends to zero. We consider pairwise interaction energies satisfying $p$-growth conditions, $p<d$, $d$ being the dimension of the reference configuration, defined on discrete functions subject to Dirichlet conditions on a $\delta$-periodic array of small squares of side $r_{\delta}\sim \delta^{d/d-p}$. Our analysis is performed in the framework of $\Gamma$-convergence and we prove that, in the regime $\varepsilon=o(r_{\delta})$, the discrete energy and their continuum counterpart share the same $\Gamma$-limit and the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals.

This paper addresses query scheduling for goal-oriented semantic communication in pull-based status update systems. We consider a system where multiple sensing agents (SAs) observe a source characterized by various attributes and provide updates to multiple actuation agents (AAs), which act upon the received information to fulfill their heterogeneous goals at the endpoint. A hub serves as an intermediary, querying the SAs for updates on observed attributes and maintaining a knowledge base, which is then broadcast to the AAs. The AAs leverage the knowledge to perform their actions effectively. To quantify the semantic value of updates, we introduce a grade of effectiveness (GoE) metric. Furthermore, we integrate cumulative perspective theory (CPT) into the long-term effectiveness analysis to account for risk awareness and loss aversion in the system. Leveraging this framework, we compute effect-aware scheduling policies aimed at maximizing the expected discounted sum of CPT-based total GoE provided by the transmitted updates while complying with a given query cost constraint. To achieve this, we propose a model-based solution based on dynamic programming and model-free solutions employing state-of-the-art deep reinforcement learning (DRL) algorithms. Our findings demonstrate that effect-aware scheduling significantly enhances the effectiveness of communicated updates compared to benchmark scheduling methods, particularly in settings with stringent cost constraints where optimal query scheduling is vital for system performance and overall effectiveness.

In their study of Levin-Wen models [Commun. Math. Phys. 313 (2012) 351-373], Kitaev and Kong proposed a weak Hopf algebra associated with a unitary fusion category $\mathcal{C}$ and a unitary left $\mathcal{C}$-module $\mathcal{M}$, and sketched a proof that its representation category is monoidally equivalent to the unitary $\mathcal{C}$-module functor category $\mathrm{Fun}^{\mathrm{u}}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^\mathrm{rev}$. We give an independent proof of this result without the unitarity conditions. In particular, viewing $\mathcal{C}$ as a left $\mathcal{C} \boxtimes \mathcal{C}^{\mathrm{rev}}$-module, we obtain a quasi-triangular weak Hopf algebra whose representation category is braided equivalent to the Drinfeld center $\mathcal{Z}(\mathcal{C})$. In the appendix, we also compare this quasi-triangular weak Hopf algebra with the tube algebra $\mathrm{Tube}_{\mathcal{C}}$ of $\mathcal{C}$ when $\mathcal{C}$ is pivotal. These two algebras are Morita equivalent by the well-known equivalence $\mathrm{Rep}(\mathrm{Tube}_{\mathcal{C}})\cong\mathcal{Z}(\mathcal{C})$. However, we show that in general there is no weak Hopf algebra structure on $\mathrm{Tube}_{\mathcal{C}}$ such that the above equivalence is monoidal.

We introduce and study $\mu$-elements, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several examples, we show that $\mu$-elements are indeed a genuine extension of essential elements. We study preservation of $\mu$-elements under contractions and extensions of quantale homomorphisms. We introduce $\mu$-complements and $\mu$-closedness and study their properties. We determine $\mu$-elements for several distinguished quantales, including ideals of $\mathbb{Z}_n$ and open subsets of topological spaces. Finally, we provide a complete characterization of $\mu$-elements in modular quantales.

Using a variation of Woodin's $\mathbb{P}_{\mathrm{max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $\omega_{2}$ consisting of functions from $\omega$ to $\omega$. We also show that there can be no such sequence of length $\omega_{4}$.

New non-perturbative results on the eigenvalues of the spheroidal equation are presented. The results, found using an all orders WKB analysis, include a perturbative/non-perturbative (P/NP) relation as well as the first exponential correction to the perturbative series which is valid in certain regions of parameters. The quantum periods are also computed.

The Ewens-Pitman model is a probability distribution for random partitions of the set $[n]=\{1,\ldots,n\}$, parameterized by $\alpha\in[0,1)$ and $\theta>-\alpha$, with $\alpha=0$ corresponding to the Ewens model in population genetics. The goal of this paper is to provide an alternative and concise proof of the Feng-Hoppe large deviation principle for the number $K_{n}$ of partition sets in the Ewens-Pitman model with $\alpha\in(0,1)$ and $\theta>-\alpha$. Our approach leverages an integral representation of the moment-generating function of $K_{n}$ in terms of the (one-parameter) Mittag-Leffler function, along with a sharp asymptotic expansion of it. This approach significantly simplifies the original proof of Feng-Hoppe large deviation principle, as it avoids all the technical difficulties arising from a continuity argument with respect to rational and non-rational values of $\alpha$. Beyond large deviations for $K_{n}$, our approach allows to establish a sharp concentration inequality for $K_n$ involving the rate function of the large deviation principle.

The Dirichlet eigenvalues of the Laplacian on a triangle that collapses into a line segment diverge to infinity. In this paper, to track the behavior of the eigenvalues during the collapsing process of a triangle, we establish a quantitative error estimate for the Dirichlet eigenvalues on collapsing triangles. As an application, we solve the open problem concerning the simplicity of the second Dirichlet eigenvalue for nearly degenerate triangles, offering a complete solution to Conjecture 6.47 posed by R. Laugesen and B. Siudeja in A. Henrot's book ``Shape Optimization and Spectral Theory".

We introduce a new notion of local inverse metric entropy along backward trajectories for ergodic measures preserved by endomorphisms (non-invertible maps) on a compact metric space. A second notion of inverse measure entropy is defined by using measurable partitions. Our notions have several useful applications. Inverse entropy can distinguish between isomorphism classes of endomorphisms on Lebesgue spaces, when they have the same forward measure-theoretic entropy. In a general setting we prove that the local inverse entropy of an ergodic measure \mu is equal to the forward entropy minus the folding entropy. The inverse entropy of hyperbolic measures on compact manifolds is explored, focusing on their negative Lyapunov exponents. We compute next the inverse entropy of the inverse SRB measure on a hyperbolic repellor. We prove an entropy rigidity result for special Anosov endomorphisms of \mathbb T^2, namely that they can be classified up to smooth conjugacy by knowing the entropy of their SRB measure and the inverse entropy of their inverse SRB measure. Next we study the relations between our inverse measure-theoretic entropy and the generalized topological inverse entropy on subsets of prehistories. In general we establish a Partial Variational Principle for inverse entropy. We obtain also a Full Variational Principle for inverse entropy in the case of special TA-covering maps on tori. In the end, several examples of endomorphisms are studied, such as fat baker transformations, fat solenoidal attractors, special Anosov endomorphisms, toral endomorphisms, and the local inverse entropy is computed for their SRB measures.

In this paper, we address a social planner's optimal control problem for a partially observable stochastic epidemic model. The control measures include social distancing, testing, and vaccination. Using a diffusion approximation for the state dynamics of the epidemic, we apply filtering arguments to transform the partially observable stochastic optimal control problem into an optimal control problem with complete information. This transformed problem is treated as a Markov decision process. The associated Bellman equation is solved numerically using optimal quantization methods for approximating the expectations involved to mitigate the curse of dimensionality. We implement two approaches, the first involves state discretization coupled with linear interpolation of the value function at non-grid points. The second utilizes a parametrization of the value function with educated ansatz functions. Extensive numerical experiments are presented to demonstrate the efficacy of both methods.

The work is devoted to the global well-posedness in W^{1, (4, 2)}(R\times R^{+}) of the integro-differential problem involving the square of the one dimensional Laplace operator along with the drift term. Our proof is based on a fixed point technique. Moreover, we provide the assumption leading to the existence of the nontrivial solution for the problem under the consideration. Such equation is relevant to the cell population dynamics in the Mathematical Biology.

Given $a,b\ge 0$ and $t>0$, let $\rho =\{ \rho _{s}\} _{0\le s\le t}$ be a three-dimensional Bessel bridge from $a$ to $b$ over $[0,t]$. In this paper, based on a conditional identity in law between Brownian bridges stemming from Pitman's theorem, we show in particular that the process given by \begin{align*} \rho _{s}+\Bigl| b-a+ \min _{0\le u\le s}\rho _{u}-\min _{s\le u\le t}\rho _{u} \Bigr| -\Bigl| \min _{0\le u\le s}\rho _{u}-\min _{s\le u\le t}\rho _{u} \Bigr| ,\quad 0\le s\le t, \end{align*} has the same law as the time reversal $\{ \rho _{t-s}\} _{0\le s\le t}$ of $\rho $. As an immediate application, letting $R=\{ R_{s}\} _{s\ge 0}$ be a three-dimensional Bessel process starting from $a$, we obtain the following time-reversal and time-inversion results on $R$: $\{ R_{t-s}\} _{0\le s\le t}$ is identical in law with the process given by \begin{align*} R_{s}+R_{t}-2\min _{s\le u\le t}R_{u},\quad 0\le s\le t, \end{align*} when $a=0$, and $\{ sR_{1/s}\} _{s>0}$ is identical in law with the process given by \begin{align*} R_{s}-2(1+s)\min _{0\le u\le s}\frac{R_{u}}{1+u}+a(1+s),\quad s>0, \end{align*} for every $a\ge 0$.

This paper discusses how to develop a high-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the general d(>=1)-dimensional diagonal-anisotropic diffusion equation. Such an MRT-LB model considers the transformation matrix constructed in a natural way and the DdQ(2d^2+1) lattice structure. A key step in developing the high-order MRT-LB model is to determine the adjustable relaxation parameters and weight coefficients, which are used to eliminate the truncation errors at certain orders of the MRT-LB model, while ensuring the stability of the MRT-LB model. In this work, we first present a unified MRT-LB model for the diagonal-anisotropic diffusion equation. Then, through the direct Taylor expansion, we analyze the macroscopic modified equations of the MRT-LB model up to fourth-order, and further derive the fourth-order consistent conditions of the MRT-LB model. Additionally, we also construct the fourth-order initialization scheme for the present LB method. After that, the condition which guarantees that the MRT-LB model can satisfy the stability structure is explicitly given, and from a numerical perspective, once the stability structure is satisfied, the MRT-LB model must be L^2 stable. In combination with the fourth-order consistent and L^2 stability conditions, the relaxation parameters and weight coefficients of the MRT-LB model can be automatically given by a simple computer code. Finally, we perform numerical simulations of several benchmark problems, and find that the numerical results can achieve a fourth-order convergence rate, which is in agreement with our theoretical analysis. In particular, for the isotropic diffusion equation, we also make a comparison between the fourth-order MRT-LB models with the DdQ(2d^2+1) and DdQ(2d+1) lattice structures, and the numerical results show that the MRT-LB model with the DdQ(2d^2+1) lattice structure is more general.

We construct a bijective correspondence between the set of rigid modules over a gentle algebra and the set of admissible arc systems on the associated coordinated-marked surface. In particular, a maximal rigid module aligns with an equivalence class of admissible $5$-partial triangulations, which is an (admissible) set of simple arcs dissecting the surface into $s$-gons with $3\leqslant s\leqslant 5$. Furthermore, the rank of the maximal rigid module is equal to the rank of the algebra plus the number of internal $4$-gons and $5$-gons in the associated $5$-partial triangulation. Subsequently, these results facilitate an exploration of the higher Auslander-Reiten theory for gentle algebras with global dimension $n$. The $\tau_m$-closures of injective modules are realized as admissible $(m+2)$-partial triangulations, where $\tau_m$ are higher Auslander-Reiten translations with $2\leqslant m \leqslant n$. Finally, we provide a complete classification of gentle algebras that are $\tau_n$-finite or $n$-complete introduced by Iyama [I11].

We study large minors in small-set expanders. More precisely, we consider graphs with $n$ vertices and the property that every set of size at most $\alpha n / t$ expands by a factor of $t$, for some (constant) $\alpha > 0$ and large $t = t(n)$. We obtain the following: * Improving results of Krivelevich and Sudakov, we show that a small-set expander contains a complete minor of order $\sqrt{n t / \log n}$. * We show that a small-set expander contains every graph $H$ with $O(n \log t / \log n)$ edges and vertices as a minor. We complement this with an upper bound showing that if an $n$-vertex graph $G$ has average degree $d$, then there exists a graph with $O(n \log d / \log n)$ edges and vertices which is not a minor of $G$. This has two consequences: (i) It implies the optimality of our result in the case $t = d^c$ for some constant $c > 0$, and (ii) it shows expanders are optimal minor-universal graphs of a given average degree.

In classical matroid theory, extensions are the inverse operation of restrictions for matroids. In particular, extensions to a ground set with one additional element are called single-element extensions. In this paper we introduce a $q$-analogue of single-element extensions for $q$-matroids, which we call one-dimensional extensions. To enumerate such extensions, we establish a $q$-analogue of modular cuts and define a certain function which we call a modular cut selector. It assigns each newly appearing one-dimensional subspace to a modular cut. By using these notion, we establish the one-to-one correspondence between one-dimensional extensions and modular cut selectors. As an application, we develop a classification algorithm for $q$-matroids. Furthermore, by computation, we classify all the $q$-matroids on ground spaces over $\mathbb{F}_2$ of dimension $4$.

Motivated by a work of Fu-So-Song, we associate a symmetric matrix $A$ to a plane vector sequence $v$ and give a formula to find the signature of $A$ in terms of the sequence $v$. When $A$ is nonsingular, we interpret the relation between $A$ and $A^{-1}$ from a topological viewpoint. Finally, we associate an omnioriented quasitoric orbifold $X$ of real dimension four to the sequence $v$ and show that $A^{-1}$ is the intersection matrix of the characteristic suborbifolds of $X$.

Suppose we are given two metric spaces and a family of continuous transformations from one to the other. Given a probability distribution on each of these two spaces - namely the source and the target measures - the Wasserstein alignment problem seeks the transformation that minimizes the optimal transport cost between its pushforward of the source distribution and the target distribution, ensuring the closest possible alignment in a probabilistic sense. Examples of interest include two distributions on two Euclidean spaces $\mathbb{R}^n$ and $\mathbb{R}^d$, and we want a spatial embedding of the $n$-dimensional source measure in $\mathbb{R}^d$ that is closest in some Wasserstein metric to the target distribution on $\mathbb{R}^d$. Similar data alignment problems also commonly arise in shape analysis and computer vision. In this paper we show that this nonconvex optimal transport projection problem admits a convex Kantorovich-type dual. This allows us to characterize the set of projections and devise a linear programming algorithm. For certain special examples, such as orthogonal transformations on Euclidean spaces of unequal dimensions and the $2$-Wasserstein cost, we characterize the covariance of the optimal projections. Our results also cover the generalization when we penalize each transformation by a function. An example is the inner product Gromov-Wasserstein distance minimization problem which has recently gained popularity.

Rate distortion dimension describes the theoretical limit of lossy data compression methods as the distortion bound goes to zero. It was originally introduced in the context of information theory, and recently it was discovered that it has an intimate connection to Gromov's theory of mean dimension of dynamical systems. This paper studies the behavior of rate distortion dimension of $\mathbb{R}^d$-actions under ergodic decomposition. Our main theorems provide natural convexity and concavity of upper and lower rate distortion dimensions under convex combination of invariant probability measures. We also present examples which clarify the validity and limitations of the theorems.

We consider the system of elastic waves with critical space dependent damping $V(x)$. We study the Cauchy problem for this model in the $2$-dimensional Euclidean space ${\bf R}^{2}$, and we obtain faster decay rates of the total energy as time goes to infinity. In the $2$-D case we do not have any suitable Hardy type inequality, so generally one has no idea to establish optimal energy decay. We develope a special type of multiplier method combined with some estimates brought by the $2$-D Newton potential belonging to the usual Laplacian $-\Delta$, not the operator $-a^2\Delta - (b^{2}-a^{2})\nabla {\rm div}$ itself. The property of finite speed propagation is important to get results for this system.

We show that the generator of a conservative IID random system which expands on average codimension 1 planes has an essential spectral radius strictly smaller than $1$ on Sobolev spaces of small negative index. Consequently, such a system has finitely many ergodic components. If there is only one component for each power of the random system, then the system enjoys multiple exponential mixing and the central limit theorem. Moreover, these properties are stable under small perturbations. As an application we show that many small perturbations of random homogeneous systems are exponentially mixing.

We study a sequential estimation problem for an unknown reward in the presence of a random horizon. The reward takes one of two predetermined values which can be inferred from the drift of a Wiener process, which serves as a signal. The objective is to use the information in the signal to estimate the reward which is made available until a stochastic deadline that depends on its value. The observer must therefore work quickly to determine if the reward is favorable and claim it before the deadline passes. Under general assumptions on the stochastic deadline, we provide a full characterization of the solution that includes an identification with the unique solution to a free-boundary problem. Our analysis derives regularity properties of the solution that imply its ``smooth fit'' with the boundary data, and we show that the free-boundary solves a particular integral equation. The continuity of the free-boundary is also established under additional structural assumptions that lead to its representation in terms of a continuous transformation of a monotone function. We provide illustrations for several examples of interest.

Given a quadratically convex compact connected oriented hypersurface $N$ of the complex hyperbolic plane, we prove that the characteristic rays of the symplectic form restricted to $N$ determine a double geodesic foliation of the exterior $U$ of $N$. This induces an outer billiard map $B$ on $U$. We prove that $B$ is a diffeomorphism (notice that weaker notions of strict convexity may allow the billiard map to be well-defined and invertible, but not smooth) and moreover, a symplectomorphism. These results generalize known geometric properties of the outer billiard maps in the hyperbolic plane and complex Euclidean space.

It is shown that a compact $n$-dimensional K\"ahler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric with its symmetric metric has $\frac{n}{2}$-nonnegative Calabi curvature operator, yet $b_n =2.$ Furthermore, the compact K\"ahler manifolds with $\frac{n}{2}$-nonnegative Calabi curvature operator are completely classified. In addition, the previously known results for the K\"ahler curvature operator are improved when the metric is K\"ahler-Einstein.

In applications, a substantial number of problems can be formulated as non-linear least squares problems over smooth varieties. Unlike the usual least squares problem over a Euclidean space, the non-linear least squares problem over a variety can be challenging to solve and analyze, even if the variety itself is simple. Geometrically, this problem is equivalent to projecting a point in the ambient Euclidean space onto the image of the given variety under a non-linear map. It is the singularities of the image that make both the computation and the analysis difficult. In this paper, we prove that under some mild assumptions, these troublesome singularities can always be avoided. This enables us to establish a linear convergence rate for iterative sequences generated by algorithms satisfying some standard assumptions. We apply our general results to the low-rank partially orthogonal tensor approximation problem. As a consequence, we obtain the linear convergence rate for a classical APD-ALS method applied to a generic tensor, without any further assumptions.

Let $G$ be a finite group and $p$ be a prime. We prove that if $G$ has three codegrees, then $G$ is an $M$-group. We prove for some prime $p$ that if every irreducible Brauer character of $G$ is a prime, then for every normal subgroup $N$ of $G$ either $G/N$ or $N$ is an $M_p$-group.

In this paper we survey known results of characterizations of reflexive Banach spaces and properties stronger than reflexivity. Characterizations of reflexivity are mainly based on precise descriptions of weakly compact sets, affine sets in a Banach space $X$ or in $X^*$, properties of a Schauder basis of $X$ or $X^*$ and a bi-orthogonal system in $X$. Characterizations of stronger properties are based on the von Neumann-Jordan constant, finite-tree property and (finite-dimensional) representability.

Community detection, which focuses on recovering the group structure within networks, is a crucial and fundamental task in network analysis. However, the detection process can be quite challenging and unstable when community signals are weak. Motivated by a newly collected large-scale academic network dataset from the Web of Science, which includes multi-layer network information, we propose a Bipartite Assisted Spectral-clustering approach for Identifying Communities (BASIC), which incorporates the bipartite network information into the community structure learning of the primary network. The accuracy and stability enhancement of BASIC is validated theoretically on the basis of the degree-corrected stochastic block model framework, as well as numerically through extensive simulation studies. We rigorously study the convergence rate of BASIC even under weak signal scenarios and prove that BASIC yields a tighter upper error bound than that based on the primary network information alone. We utilize the proposed BASIC method to analyze the newly collected large-scale academic network dataset from statistical papers. During the author collaboration network structure learning, we incorporate the bipartite network information from author-paper, author-institution, and author-region relationships. From both statistical and interpretative perspectives, these bipartite networks greatly aid in identifying communities within the primary collaboration network.

In 2023, H.\,Brezis published a list of his ``favorite open problems", which he described as challenges he had ``raised throughout his career and has resisted so far". We provide a complete resolution to the first one--Open Problem 1.1--in Brezis's favorite open problems list: the existence of solutions to the long-standing Brezis-Nirenberg problem on a three-dimensional ball. Furthermore, using the building blocks of Del Pino-Musso-Pacard-Pistoia sign-changing solutions to the Yamabe problem, we establish the existence of infinitely many sign-changing, nonradial solutions for the full range of the parameter.

The concept of Property (II) was introduced and characterized by Chen and Lin \cite{CL}. In this work, given a compatiable collection of bounded sets $\mathcal{A}$, we provide a complete characterization of two generalizations of Property (II) : (i) every closed convex set $A\in \mathcal{A}$ is an intersection of closed convex hulls of finitely many balls and (ii) for every closed convex set $A\in \mathcal{A}$ and $\beta \geqslant 0$, $\overline{A+\beta B_X}$ is an intersection of closed convex hulls of finitely many balls.

This paper concentrates on positive definite functions on finite abelian groups, which are central to harmonic analysis and related fields. By leveraging the group structure and employing Fourier analysis, we establish a lower bound for the second largest value of positive definite functions. For illustrative purposes, we present three applications of our lower bound: (a) We obtain both lower and upper bounds for arbitrary functions on finite abelian groups; (b) We derive lower bounds for the relaxation and mixing times of random walks on finite abelian groups. Notably, our bound for the relaxation time achieves a quadratic improvement over the previously known one; (c) We determine a new lower bound for the size of the sumset of two subsets of finite abelian groups.

We present and analyze a three-stage stochastic optimization model that integrates output from a geoscience-based flood model with a power flow model for transmission grid resilience planning against flooding. The proposed model coordinates the decisions made across multiple stages of resilience planning and recommends an optimal allocation of the overall resilience investment budget across short- and long-term measures. While doing so, the model balances the cost of investment in both short- and long-term measures against the cost of load shed that results from unmitigated flooding forcing grid components go out-of-service. We also present a case study for the Texas Gulf Coast region to demonstrate how the proposed model can provide insights into various grid resilience questions. Specifically, we demonstrate that for a comprehensive yet reasonable range of economic values assigned to load loss, we should make significant investments in the permanent hardening of substations such that we achieve near-zero load shed. We also show that not accounting for short-term measures while making decisions about long-term measures can lead to significant overspending. Furthermore, we demonstrate that a technological development enabling to protect substations on short notice before imminent hurricanes could vastly influence and reduce the total investment budget that would otherwise be allocated for more expensive substation hardening. Lastly, we also show that for a wide range of values associated with the cost of mitigative long-term measures, the proportion allocated to such measures dominates the overall resilience spending.

This paper solves ``The Dry Ten Martini Problem'' for $C^2$ cosine-type quasiperiodic Schr\"odinger operators with large coupling constants and Diophantine frequencies, a model originally introduced by Sinai in 1987 \cite{sinai}. This shows that the analyticity assumption on the potential is not essential for obtaining a dry Cantor spectrum and can be replaced by a certain geometric condition in the low regularity case. In addition, we prove the homogeneity of the spectrum and the absolute continuity of the integrated density of states (IDS).

Let $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ with $\operatorname{Im}\tau>0$ be a flat torus and $G(z;\tau)$ be the Green function on $E_{\tau}$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_{\tau})^{n}$: \[ G_{n}(z_{1},\cdots,z_{n};\tau):=\sum_{i<j}G(z_{i}-z_{j};\tau)-n\sum_{i=1}% ^{n}G(z_{i};\tau). \] Recently, Lin (J. Differ. Geom. to appear) proved that there are at least countably many analytic curves in $\mathbb H=\{\tau : \operatorname{Im}\tau>0\}$ such that $G_n(\cdot;\tau)$ has degenerate critical points for any $\tau$ on the union of these curves. In this paper, we prove that there is a measure zero subset $\mathcal{O}_n\subset \mathbb H$ (containing these curves) such that for any $\tau\in \mathbb H\setminus\mathcal{O}_n$, all critical points of $G_n(\cdot;\tau)$ are non-degenerate. Applications to counting the exact number of solutions of the curvature equation $\Delta u+e^{u}=\rho \delta_{0}$ on $E_{\tau}$ will be given.

Reconfigurable intelligent surfaces (RIS) can reshape the characteristics of wireless channels by intelligently regulating the phase shifts of reflecting elements. Recently, various codebook schemes have been utilized to optimize the reflection coefficients (RCs); however, the selection of the optimal codeword is usually obtained by evaluating a metric of interest. In this letter, we propose a novel weighted design on the discrete Fourier transform (DFT) codebook to obtain the optimal RCs for RIS-assisted point-to-point multiple-input multiple-output (MIMO) systems. Specifically, we first introduce a channel training protocol where we configure the RIS RCs using the DFT codebook to obtain a set of observations through the uplink training process. Secondly, based on these observed samples, the Lagrange multiplier method is utilized to optimize the weights in an iterative manner, which could result in a higher channel capacity for assisting in the downlink data transmission. Thirdly, we investigate the effect of different codeword configuration orders on system performance and design an efficient codeword configuration method based on statistical channel state information (CSI). Finally, numerical simulations are provided to demonstrate the performance of the proposed scheme.

Volterra integral and integro-differential equations have been extensively studied in both pure mathematics and applied science. In one direction, developments in analysis have yielded far-ranging existence, uniqueness, and regularity results. In the other, applications in science have inspired a substantial library of practical techniques to deal with such equations. The present work connects these research areas by examining five large classes of linear Volterra equations: integral and integro-differential equations with completely monotone (CM) kernels, corresponding to linear viscoelastic models; those with positive definite (PD) kernels, corresponding to partially-observed quantum systems; difference equations with PD kernels; a class of generalized delay differential equations; and a class of generalized fractional differential equations. We develop a system of correspondences between these problems, showing that all five can be understood within the same, spectral theory. We leverage this theory to recover practical, closed-form solutions of all five classes, and we show that interconversion yields a natural, continuous involution within each class. Our work unifies several results from science: the interconversion formula of Gross, recent results in viscoelasticity and operator theory for integral equations of the second type, classical formulas for Prony series and fractional differential equations, and the convergence of Prony series to CM kernels. Finally, our theory yields a novel, geometric construction of the regularized Hilbert transform, extends it to a wide class of infinite measures, and reveals a natural connection to delay and fractional differential equations. We leverage our theory to develop a powerful, spectral method to handle scalar Volterra equations numerically, and illustrate it with a number of practical examples.

We investigate the stability of maximizing measures for a penalty function of a two-dimensional subshift of finite type, building on the work of Gonschorowski et al. \cite{GQS}. In the one-dimensional case, such measures remain stable under Lipschitz perturbations for any subshift of finite type. However, instability arises in a penalty function of the Robinson tiling, which is a two-dimensional subshift of finite type with no periodic point and zero entropy. This raises the question of whether stability persists in two-dimensional subshifts of finite type with positive topological entropy. In this paper, we address this question by studying the $\mathbb{Z}^2$-hard square shift, a well-known example of a two-dimensional subshift with positive entropy. Our main theorem establishes that, in contrast to previous results, a penalty function of the hard square shift remains stable under Lipschitz perturbations.

In this work, we study the intersection of positive closed currents on domains. We use the theory of tangent currents in connection with King's residue formula. We find a sufficient condition for the local existence of tangent currents, and express the shadow of tangent currents and the $h$-dimension of tangent currents in terms of the complex Monge-Amp\`ere type current. Further, a reasonable integrability condition for the existence of the unique tangent current with minimal $h$-dimension is introduced. We apply it to the study of the intersection of positive closed currents, find a sufficient condition for the intersection of positive closed currents on domains and describe the intersection in terms of the complex Monge-Amp\`ere type current. At the same time, we obtain regularizations of positive closed currents that work well with the suggested intersection of positive closed currents. In particular, the standard regularization of currents by convolution actually produces the convergence towards the intersection of positive closed currents. In this sense, our approach generalizes King's work on currents defined by analytic varieties, which is obtained from Federer's slicing theory. Some classical examples are computed. Our work is applicable to general complex manifolds not necessarily compact or K\"ahler.

We consider a continuum of carbon-emitting firms who seek to maximise their stock price, and a regulator (e.g., Government) who wishes for the economy to flourish, whilst simultaneously punishing firms who behave non-green. Interpreting the regulator as a major player and the firms as the minor players, we model this setting through a mean field game with major and minor players. We extend the stochastic maximum principle derived by Carmona & Zhu [A probabilistic approach to mean field games with major and minor players. Annals of Applied Probability, 2016, 94, 745--788] by relaxing the assumptions on the forms of the minimisers for the Hamiltonians, allowing them to depend on more arguments. This allows the major and representative minor player to interact in a more natural fashion, thereby permitting us to consider more realistic models for our green and sustainable finance problem. Through our stochastic maximum principle, we derive explicit Nash equilibria for a number of examples.

The `sectional category' of a map has been studied in two ways: on the one hand with open covers, and on the other hand with Ganea and Whitehead style constructions. For these approaches to be equivalent, the normality of the codomain is necessary. We show that this is the only condition. The `Lusternik-Schnirelmann category' of a space, as well as Farber's `topological complexity', are particular cases of sectional categories. In addition, Mac\'ias-Virg\'os and Mosquera-Lois introduced the notion of `homotopic distance' between two maps, showing that it is a distance on the homotopy class of maps with same domain and codomain provided the domain is normal. We introduce the notion of `lifting category' which unifies the notions of sectional category, topological complexity, and homotopic distance, all these notions being special cases of lifting category. A special attention is also paid to the difference between the pointed and unpointed cases.

Given two affine permutations, some results of Lascoux and Deodhar, and independently Jacon-Lecouvey, allow to decide if they are comparable for the strong Bruhat order. These permutations are associated with tuples of core partitions, and the preceding problem is equivalent to compare the Young diagrams in each components for the inclusion. Using abaci, we give an easy rule to compute these Young diagrams one another. We deduce a procedure to compare, for the Bruhat order, two affine permutations in the window notation.

In this paper, we prove that, for any $\{2,3\}$-generated perfect group $P$, there exists a 2-arc-transitive graph which has full automorphism group isomorphic to $P\wr A_5=P^5{:} A_5$ and is a cover of the Petersen graph. This particularly shows that there are infinitely many $2$-arc-transitive normal covers of the Petersen graph. To the best of our knowledge, these are the first known examples of 2-arc-transitive non-trivial covers of 2-arc-transitive graphs with non-solvable transformation groups. Moreover, these graphs are non-Cayley graphs, and have girth 10.

We give a strengthening of the classical Khintchine inequality between the second and the $p$-th moment for $p \ge 3$ with optimal constant by adding a deficit depending on the vector of coefficients of the Rademacher sum.

We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\geq 2$ has $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq 0$ for some $\alpha<\frac{4}{n-1}$ and mean-convex boundary, then it is either isometric to $\Sigma\times \mathbb{R}_{\geq 0}$ for a closed manifold $\Sigma$ with nonnegative Ricci curvature or it has no interior ends.

In traditional facility location problems, a set of points is provided, and the objective is to determine the best location for a new facility based on criteria such as minimizing cost, time, and distances between clients and facilities. Conversely, inverse single facility location problems focus on adjusting the problem's parameters at minimal cost to make a specific point optimal. In this paper, we present an algorithm for the general case of the inverse single facility location problem with variable coordinates in a two-dimensional space. We outline the optimality conditions of this algorithm. Additionally, we examine the specific case namely the inverse minisum single facility location problem and test the algorithm on various instances. The results demonstrate the algorithm's effectiveness in these scenarios.

For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $\rho_0$, we prove in the infill asymptotics that the MLE is $n$-consistent, where $n$ denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in $\rho_0$, the analysis is highly non-standard. Therefore, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument $\rho$ is to $\rho_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neigborhood of $\rho_0$ asymptotically. The crucial argument to derive the stable convergence is to exploit the semimartingale structure of the sequential suitably rescaled local log-likelihood function (as a process in time). Both sequentially and as a process in $\rho$, it exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a multiple of the local time at $\rho_0$.

It is well-known that a function on an open set in $\mathbb R^d$ is smooth if and only if it is arc-smooth, i.e., its composites with all smooth curves are smooth. In recent work, we extended this and related results (for instance, a real analytic version) to suitable closed sets, notably, sets with H\"older boundary and fat subanalytic sets satisfying a necessary topological condition. In this paper, we prove that the resulting set-theoretic identities of function spaces are bornological isomorphisms with respect to their natural locally convex topologies. Extending the results to maps with values in convenient vector spaces, we obtain corresponding exponential laws. Additionally, we show analogous results for special ultradifferentiable Braun-Meise-Taylor classes.

A characteristic Galerkin-type semi-Lagrangian discontinuous finite element scheme (CSLDG) is investigated, which directly discretizes an integral invariant model derived from the coupling of a transport equation and its dual equation. Despite extensive research on the numerical implementation of this method, no studies have yet explored the well-posedness of the integral invariant model itself. To address this gap, a weak solution theory for CSLDG is developed: A precise definition of the weak solution for the integral invariant model is formulated. Utilizing the slice method, which is frequently employed in existence proofs for parabolic equations, the existence of the weak solution is established through the application of the Riesz Representation Theorem and mollifier techniques. The stability of the integral invariant weak solution is subsequently demonstrated by the strategic selection of the test function Psi, leading to the proof of its uniqueness.

Online Feedback Optimization uses optimization algorithms as dynamic systems to design optimal control inputs. The results obtained from Online Feedback Optimization depend on the setup of the chosen optimization algorithm. In this work we analyse the sensitivity of Online Feedback Optimization to the parameters of projected gradient descent as the algorithm of choice. We derive closed-form expressions for sensitivities of the objective function with respect to the parameters of the projected gradient and to time-varying model mismatch. The formulas are then used for analysis of model mismatch in a gas lift optimization problem. The results of the case study indicate that the sensitivity of Online Feedback Optimization to the model mismatch depends on how long the controller has been running, with decreasing sensitivity to mismatch in individual timesteps for long operation times.

This paper is devoted to an optimal control problem of fully coupled forward-backward stochastic differential equations driven by sub-diffusion, whose solutions are not Markov processes. The stochastic maximum principle is obtained, where the control domain may not be convex and the diffusion term is independent of the control variable. Additionally, problem with state constraint is researched by using Ekeland's variational principle. The theoretical results obtained are applied to a cash management optimization problem in bear market, and the optimal strategy is derived.

We consider a compact submanifold $M$ of a Riemannian manifold $N$ and we use the second variation formula as a tool to drive some geometric results on reach$(M, N)$ the reach of $M$ in $N$, including some useful relations between the extrinsic geometry of $M$ in $N$ and reach$(M, N)$. Our results generalize some theorems previously proved for the special case where $N$ is Euclidean space.

We prove that, among all radial subsets $\Omega\subset \mathbb{C}$ of prescribed measure, the ball is the only maximizer of the sum of the first $K$ eigenvalues ($K\geq 1$) of the corresponding Toeplitz operator $T_\Omega$ on the Fock space $\mathcal{F}^2(\mathbb{C})$. As a byproduct, we prove that balls maximize any Schatten $p$-norm of $T_\Omega$ for $p>1$ (and minimize the corresponding quasinorm for $p<1$), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in $L^p(\mathbb{C})$, with $p > 1$, characterizing those that maximize the sum of the first $K$ eigenvalues.

The problem of determining the maximum number of copies of $T$ in an $H$-free graph, for any graphs $T$ and $H$, was considered by Alon and Shikhelman. This is a variant of Tur\'{a}n's classical extremal problem. We show lower and upper bounds for the maximum number of $s$-cliques in a graph with no disjoint copies of arbitrary graph. We also determine the maximum number of $s$-cliques in an $n$-vertex graph that does not contain a disjoint union of $k$ paths of length two when $k=2,3$, or $s\geqslant k+2$, or $n$ is sufficiently large, this partly confirms a conjecture posed by Chen, Yang, Yuan, and Zhang \cite{2024Chen113974}.

In the setting of the Grushin differential operator $G=-\Delta_{x'}-|x'|^2\Delta_{x''}$ with domain ${\rm Dom}\,G=C^\infty_c(\mathbb{R}^d)\subset L^2(\mathbb{R}^d)$, we define a scalar transform which is a mixture of the partial Fourier transform and a transform based on the scaled Hermite functions. This transform unitarily intertwines $G$ with a multiplication operator by a nonnegative real-valued function on an appropriately associated `dual' space $L^2(\Gamma)$. This allows to construct a self-adjoint extension $\mathbb G$ of $G$ as a simple realization of this multiplication operator. Another self-adjoint extensions of $G$ are defined in terms of sesquilinear forms and then these extensions are compared. Aditionally, a closed formula for the heat kernel that corresponds to the heat semigroup $\{\exp(-t\mathbb G)\}_{t>0}$ is established.

We construct a family of estimators for a regression function based on a sample following a qdistribution. Our approach is nonparametric, using kernel methods built from operations that leverage the properties of q-calculus. Furthermore, under appropriate assumptions, we establish the weak convergence and strong consistency of this family of estimators.

In this paper, we propose a cross subcarrier precoder design (CSPD) for massive multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. The aim is to maximize the weighted sum-rate (WSR) performance while considering the smoothness of the frequency domain effective channel. To quantify the smoothness of the effective channel, we introduce a delay indicator function to measure the large delay components of the effective channel. An optimization problem is then formulated to balance the WSR performance and the delay indicator function. By appropriately selecting the weight factors in the objective function and the parameters in the delay indicator function, the delay spread of the effective channel can be reduced, thereby enhancing the smoothness of the effective channel. To solve the optimization problem, we apply the symplectic optimization, which achieves faster convergence compared to the gradient descent methods. Simulation results indicate that the proposed algorithm achieves satisfying WSR performance while maintaining the smoothness of the effective channel.

This paper studies data-driven stabilization of a class of unknown polynomial systems using data corrupted by bounded noise. Existing work addressing this problem has focused on designing a controller and a Lyapunov function so that a certain state-dependent matrix is negative definite, which ensures asymptotic stability of all closed-loop systems compatible with the data. However, as we demonstrate in this paper, considering the negative definiteness of this matrix introduces conservatism, which limits the applicability of current approaches. To tackle this issue, we develop a new method for the data-driven stabilization of polynomial systems using the concept of density functions. The control design consists of two steps. Firstly, a dual Lyapunov theorem is used to formulate a sum of squares program that allows us to compute a rational state feedback controller for all systems compatible with the data. By the dual Lyapunov theorem, this controller ensures that the trajectories of the closed-loop system converge to zero for almost all initial states. Secondly, we propose a method to verify whether the designed controller achieves asymptotic stability of all closed-loop systems compatible with the data. Apart from reducing conservatism of existing methods, the proposed approach can also readily take into account prior knowledge on the system parameters. A key technical result developed in this paper is a new type of S-lemma for a specific class of matrices that, in contrast to the classical S-lemma, avoids the use of multipliers.

The article examines a set of irreducible germs $f_P:U_P\to V_p$ of %finite generic morphisms $f:S\to\mathbb P^2$ to the projective plane whose branch curve germs $B_P\subset V_p$ have singularities equisingular deformation equivalent to singularities given by equations $x^{k_1}-y^{k_2}=0$ with coprime $k_1,k_2\in\mathbb N$.

We address optimal control problems on the space of measures for an objective containing a smooth functional and an optimal transport regularization. That is, the quadratic Monge-Kantorovich distance between a given prior measure and the control is penalized in the objective. We consider optimality conditions and reparametrize the problem using the celebrated structure theorem by Brenier. The optimality conditions can be formulated as a piecewise differentiable equation. This is utilized to formulate solution algorithms and to analyze their local convergence properties. We present a numerical example to illustrate the theoretical findings.

We investigate the stability of solutions to the Vlasov-Poisson system using the unifying framework of the kinetic Wasserstein distance, introduced by Iacobelli in (Section 4 in Arch. Ration. Mech. Anal. 244 (2022), no. 1, 27-50). This allows us to treat both macroscopic densities that lie in a Yudovich space, as recently considered by Crippa et al. (Theorem 1.6 in Nonlinearity 37 (2024), no. 9, 095015) for the $1$-Wasserstein distance, and higher order Wasserstein distances, for which only bounded macroscopic densities were treated by Iacobelli and the first author (Theorem 1.11 in Bull. Lond. Math. Soc. 56 (2024), 2250-2267). First, we establish an $L^p$-estimate on the difference between two force fields in terms of a suitable nonlinear quantity that controls the kinetic Wasserstein distance between their macroscopic densities. Second, we use this estimate in order to derive a closable Osgood-type inequality for the kinetic Wasserstein distance between two solutions. This enables us to prove our main theorem; for $1 \le p < +\infty$ we show the $p$-Wasserstein stability of solutions to the Vlasov-Poisson system with macroscopic densities belonging to a Yudovich space.

In this paper, we study the large time behaviour of solutions of multistable reaction-diffusion equations in $\mathbb{R}^N$, with a spatially periodic heterogeneity. By multistable, we mean that the problem admits a finite -- but arbitrarily large -- number of stable, periodic steady states. In contrast with the more classical monostable and bistable frameworks, which exhibit the emergence of a single travelling front in the long run, in the present case the large time dynamics is governed by a family of stacked travelling fronts, involving intermediate steady states, called propagating terrace. Their existence in the multidimensional case has been established in our previous work [13]. The first result of the present paper is their uniqueness. Next, we show that the speeds of the propagating terraces in different directions dictate the spreading speeds of solutions of the Cauchy problem, for both planar-like and compactly supported initial data. The latter case turns out to be much more intricate than the former, due to the fact that the propagating terraces in distinct directions may involve different sets of intermediate steady states. Another source of difficulty is that the Wulff shape of the speeds of travelling fronts can be non-smooth, as we show in the bistable case using a result of [4].

Let $n,q,t,s,p$ be non-negative integers where $n\geq s$ and $q\geq 1$. For $\mathbf{x}\in A_{q}^{n}\triangleq\{ 0,1,\ldots,q-1 \}^{n}$, let the $t$-insertion $s$-deletion $p$-substitution ball of $\mathbf{x}$, denoted by $\mathcal{B}_{t,s,p}(\mathbf{x})$, be the set of sequences in $A_{q}^{n+t-s}$ which can be obtained from $\mathbf{x}$ by performing $t$ insertions, $s$ deletions, and at most $p$ substitutions. We establish that for any $\mathbf{x}\in A_{q}^{n}$, $|\mathcal{B}_{t,s,p}(\mathbf{x})|\geq\sum_{i=0}^{t+p}\binom{n+t-s}{i}(q-1)^{i}$, with equality holding if and only if $t=s=0\vee s=p=0\vee s+p\geq n\vee r(\mathbf{x})=1$. Here, $r(\mathbf{x})$ denotes the number of runs in $\mathbf{x}$, and a run in $\mathbf{x}$ is a maximum continuous subsequence of identical symbols.

This paper is dedicated to the elementary proof of Pontryagins maximum principle for problems with free right end point. The proof for the standard problem is taken from the monography of Ioffe and Tichomirov. We assume piecewise continuous controls and the proof turns out to be very simple. We generalize the concept to the problem of optimal multiprocesses, to control problems with delays and to the control of Volterra integral equations. Furthermore, we discuss the problem on infinite horizon. Moreover, we state Arrow type sufficiency conditions. The optimality conditions are demonstrated on illustrative examples.

We point out a relationship between the norm residue isomorphism theorem of Suslin-Voevodsky-Rost and the theory of birational motives, as well as its generalisation to "higher jets".

The floating wind turbines sector has great energy potential. However, minimizing the movement of the structure under the combined effect of wind and waves while ensuring maximum power extraction over a wide operating range is one of the main challenges for the control of these turbines. This paper presents a review of control methods for floating wind turbines from the recent literature. The limitations of these controllers are discussed, before introducing a presentation of several promising data-based methods. In particular, this paper focuses on artificial intelligence techniques associated with data-based control methods. Finally, the CREATIF project dealing with real-time simulation of floating wind turbines and their intelligent controls is presented.

In this letter, we investigate a coordinated multiple point (CoMP)-aided integrated sensing and communication (ISAC) system that supports multiple users and targets. Multiple base stations (BSs) employ a coordinated power allocation strategy to serve their associated single-antenna communication users (CUs) while utilizing the echo signals for joint radar target (RT) detection. The probability of detection (PoD) of the CoMP-ISAC system is then proposed for assessing the sensing performance. To maximize the sum rate while ensuring the PoD for each RT and adhering to the total transmit power budget across all BSs, we introduce an efficient power allocation strategy. Finally, simulation results are provided to validate the analytical findings, demonstrating that the proposed power allocation scheme effectively enhances the sum rate while satisfying the sensing requirements.

Tournaments are competitions between a number of teams, the outcome of which determines the relative strength or rank of each team. In many cases, the strength of a team in the tournament is given by a score. Perhaps, the most striking mathematical result on the tournament is Moon's theorem, which provides a necessary and sufficient condition for a feasible score sequence via majorization. To give a probabilistic interpretation of Moon's result, Aldous and Kolesnik introduced the soccer model,the existence of which gives a short proof of Moon's theorem. However, the existence proof of Aldous and Kolesnik is nonconstructive, leading to the question of a ``canonical'' construction of the soccer model. The purpose of this paper is to provide explicit constructions of the soccer model with an additional stochastic ordering constraint, which can be formulated by martingale transport. Two solutions are given: one is by solving an entropy optimization problem via Sinkhorn's algorithm, and the other relies on the idea of shadow couplings. It turns out that both constructions yield the property of strong stochastic transitivity. The nontransitive situations of the soccer model are also considered.

We develop novel methods for constructing nearly Hamilton cycles in sublinear expanders with good regularity properties, as well as new techniques for finding such expanders in general graphs. These methods are of independent interest due to their potential for various applications to embedding problems in sparse graphs. In particular, using these tools, we make substantial progress towards a twenty-year-old conjecture of Verstra\"ete, which asserts that for any given graph $F$, nearly all vertices of every $d$-regular graph $G$ can be covered by vertex-disjoint $F$-subdivisions. This significantly extends previous work on the conjecture by Kelmans, Mubayi and Sudakov, Alon, and K\"uhn and Osthus. Additionally, we present applications of our methods to two other problems.

Within the context of renewable energy communities, this paper focuses on optimal operation of producers equipped with energy storage systems in the presence of demand response. A novel strategy for optimal scheduling of the storage systems of the community members under price-volume demand response programs, is devised. The underlying optimization problem is designed as a low-complexity mixed-integer linear program that scales well with the community size. Once the optimal solution is found, an algorithm for distributing the demand response rewards is introduced in order to guarantee fairness among participants. The proposed approach ensures increased benefits for producers joining a community compared to standalone operation.

Let n be any odd natural number other than a perfect square, in this article it is demonstrated that this new factorization algorithm is much more efficient than the implementation technique [2,3 p.1470], described in this article, of the Fermat's factorization algorithm [1 p.6,3 p.1470], implementation technique which I call the Fermat's factorization method (like the title, translated into English, of the reference document [2] published in Italian) and which is, among the implementation techniques [1 pp.6-8,2,3 pp.1470-1471] of the Fermat's factorization algorithm, the one with which a smaller iterations number occurs to identify the factors, trivial or non-trivial, of n (except for the circumstance in which two factors, trivial or non-trivial, of n are so close to each other that they are identified at the 1st iteration with each of the implementation techniques of the Fermat's factorization algorithm). In fact, through the way in which the Euler's function [4] is applied to the Fermat's factorization method, we arrive at this new factorization algorithm with which we obtain the certain reduction in the iterations number (except for the cases in which two factors of n are so close to each other that they are identified at the 1st iteration with the Fermat's factorization method) compared to the iterations number that occurs with the Fermat's factorization method. Furthermore, in this article I represent the hypotheses field according to which it will eventually be possible to further reduce the iterations number. Finally and always in relation to this new factorization algorithm, in this article I represent in detail the limit iterations number, which is smaller than the iterations number that occurs to reach the condition x - y = 1 which characterizes the pair of trivial factors of n, beyond which it is no longer possible for pairs of non-trivial factors of n to occur.

The purpose of this paper is to investigate the emergence of cross-diffusion terms in the time evolution of two slow-fast species in competition for resources. A class of triangular cross-diffusion systems is obtained as the singular limit of a fast reaction-diffusion system, as the time scale $\epsilon$ of the fast reaction goes to 0. We prove that the classical solution of the fast reaction-diffusion system converges towards a weak solution of the cross-diffusion system and we show a convergence rate for bounded solutions. The main tool used is a family of energy functionals, giving suitable a priori estimates, uniformly on $\epsilon$.

We introduce near triple arrays as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near triple arrays form a common generalization of such well-studied classes of designs as triple arrays, (near) Youden rectangles and Latin squares. We enumerate near triple arrays for a range of small parameter sets and show that they exist in the vast majority of the cases considered. As a byproduct, we obtain the first complete enumerations of $6 \times 10$ triple arrays on $15$ symbols, $7 \times 8$ triple arrays on $14$ symbols and $5 \times 16$ triple arrays on $20$ symbols. Next, we give several constructions for families of near triple arrays, and e.g. show that near triple arrays with 3 rows and at least 6 columns exist for any number of symbols. Finally, we investigate a duality between row and column intersection sizes of a row-column design, and covering numbers for pairs of symbols by rows and columns. These duality results are used to obtain necessary conditions for the existence of near triple arrays. This duality also provides a new unified approach to earlier results on triple arrays and balanced grids.

This paper presents a novel split parallel algorithm for solving quasi-static multiple-network poroelasticity (MPET) equations. By introducing a total pressure variable, the MPET system can be reformulated into a coupled Stokes-parabolic system. To efficiently solve this system, we propose a split parallel approach. In the first time step, a monolithic solver is used to solve all variables simultaneously. For subsequent time steps, the system is split into a Stokes subproblem and a parabolic subproblem. These subproblems are then solved in parallel using a stabilization technique. This split parallel approach differs from sequential or iterative decoupling, significantly reducing computational time. The algorithm is proven to be unconditionally stable, optimally convergent, and robust across various parameter settings. These theoretical results are confirmed by numerical experiments. We also apply this parallel algorithm to simulate fluid-tissue interactions within the physiological environment of the human brain.

We prove local H\"older regularity for bounded and sign-changing weak solutions to nonlocal Trudinger equations of the form \[ (|u|^{p-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} = 0, \] in the range $1< p<\infty$ and $s \in (0,1)$. One of the main difficulties in extending the local theory to the nonlocal Trudinger equation is that when $0 \ll u \ll \infty$ locally, a crucial change of variable is unavailable in the nonlocal case due to the presence of the Tail term. We adapt several new ideas developed in the past few years to prove the required H\"older regularity.

The main goal of this note is to show that subalgebras of regular evolution algebras are themselves evolution algebras. This allows us to assume, without loss of generality, that every subalgebra in the regular setting has a basis consisting of vectors with disjoint supports. Finally, we use this result to characterise the existence of codimension-one subalgebras in regular evolution algebras.

Reconfigurable intelligent surface (RIS)-aided cell-free (CF) massive multiple-input multiple-output (mMIMO) is a promising architecture for further improving spectral efficiency (SE) with low cost and power consumption. However, conventional RIS has inevitable limitations due to its capability of only reflecting signals. In contrast, beyond-diagonal RIS (BD-RIS), with its ability to both reflect and transmit signals, has gained great attention. This correspondence focuses on using BD-RIS to improve the sum SE of CF mMIMO systems. This requires completing the beamforming design under the transmit power constraints and unitary constraints of the BD-RIS, by optimizing active and passive beamformer simultaneously. To tackle this issue, we introduce an alternating optimization algorithm that decomposes it using fractional programming and solves the subproblems alternatively. Moreover, to address the challenge introduced by the unitary constraint on the beamforming matrix of the BD-RIS, a manifold optimization algorithm is proposed to solve the problem optimally. Simulation results show that BD-RISs outperform RISs comprehensively, especially in the case of the full connected architecture which achieves the best performance, enhancing the sum SE by around 40% compared to ideal RISs.

It is shown that any localisation of triangulated categories induces (up to an equivalence) a localisation of abelian categories when one passes to their abelianisations. From this one obtains for any enlargement of Grothendieck universes an example of an abelian category and a Serre subcategory within the smaller universe such that the corresponding quotient does only exist within the bigger universe. The second part of this note provides an analogue for the abelian hull of an arbitrary category.

A common observation in data-driven applications is that high-dimensional data have a low intrinsic dimension, at least locally. In this work, we consider the problem of point estimation for manifold-valued data. Namely, given a finite set of noisy samples of $\mathcal{M}$, a $d$ dimensional submanifold of $\mathbb{R}^D$, and a point $r$ near the manifold we aim to project $r$ onto the manifold. Assuming that the data was sampled uniformly from a tubular neighborhood of a $k$-times smooth boundaryless and compact manifold, we present an algorithm that takes $r$ from this neighborhood and outputs $\hat p_n\in \mathbb{R}^D$, and $\widehat{T_{\hat p_n}\mathcal{M}}$ an element in the Grassmannian $Gr(d, D)$. We prove that as the number of samples $n\to\infty$, the point $\hat p_n$ converges to $\mathbf{p}\in \mathcal{M}$, the projection of $r$ onto $\mathcal{M}$, and $\widehat{T_{\hat p_n}\mathcal{M}}$ converges to $T_{\mathbf{p}}\mathcal{M}$ (the tangent space at that point) with high probability. Furthermore, we show that $\hat p_n$ approaches the manifold with an asymptotic rate of $n^{-\frac{k}{2k + d}}$, and that $\hat p_n, \widehat{T_{\hat p_n}\mathcal{M}}$ approach $\mathbf{p}$ and $T_{\mathbf{p}}\mathcal{M}$ correspondingly with asymptotic rates of $n^{-\frac{k-1}{2k + d}}$.

The concept of parity due to Fitzpatrick, Pejsachowicz and Rabier is a central tool in the abstract bifurcation theory of nonlinear Fredholm operators. In this paper, we relate the parity to the Evans function, which is widely used in the stability analysis for traveling wave solutions to evolutionary PDEs. As application we obtain a flexible and general condition yielding local bifurcations of specific bounded entire solutions to (Carath{\'e}odory) differential equations. These bifurcations are intrinsically nonautonomous in the sense that the assumptions implying them cannot be fulfilled for autonomous or periodic temporal forcings. In addition, we demonstrate that Evans functions are strictly related to the dichotomy spectrum and hyperbolicity, which play a crucial role in studying the existence of bounded solutions on the whole real line and therefore the recent field of nonautonomous bifurcation theory. Finally, by means of non-trivial examples we illustrate the applicability of our methods.

This is the first part of our work which is devoted to the uniqueness sets for spaces of entire functions. In this part we consider a set $\Lambda$ with angular density with respect to the order $\rho>0,$ satisfying the Lindel\"of condition. We find the value of the critical zero set type for $\Lambda$ in geometrical terms. We give a necessary and sufficient condition for the coincidence of the critical zero set type and the critical uniqueness set type. At the end of the paper we present an application of our results to random zero sets in Fock-type spaces.

The present paper deals with the study of the fluence rate over both healthy and tumor tissues in the presence of focal laser ablation (FLA). We propose new analytical solutions for the coupled partial differential equations (PDE) system, which includes the transport equation modeling the light penetration into biological tissue, the bioheat equation modeling the heat transfer and its respective damage. The present building could be the first step to the knowledge of the mathematical framework for biothermophysical problems, as well as the main key to simplify the numerical calculation due to its no cost. We derive exact solutions and simulate results from them. We discuss the potential physical contributions and present respective conclusions about (1) the validness of the diffusion approximation of the radiative transfer equation; (2) the local behavior of the source of scattered photons; (3) the unsteady-state of the fluence rate; and (4) the boundedness of the critical time of the thermal damage to the cancerous tissue. We also discuss some controversial and diverging hypotheses.

Let $P$ be a set of points and $L$ a set of lines in the (extended) Euclidean plane, and $I \subseteq P\times L$, where $i =(p,l) \in I$ means that point $p$ and line $l$ are incident. The incidences can be interpreted as quadratic constraints on the homogeneous coordinates of the points and lines. We study the space of incidence preserving motions of the given incidence structure by linearizing the system of quadratic equations. The Jacobian of the quadratic system, our projective rigidity matrix, leads to the notion of independence/dependence of incidences. Column dependencies correspond to infinitesimal motions. Row dependencies or self-stresses allow for new interpretations of classical geometric incidence theorems. We show that self-stresses are characterized by a 3-fold balance. As expected, infinitesimal (first order) projective rigidity as well as second order projective rigidity imply projective rigidity but not conversely. Several open problems and possible generalizations are indicated.

On a connected surface $N$ with negative Euler characteristic, the free homotopy class of a loop obtained by smoothing an intersection of two closed geodesics may wind around a puncture. Chas and Kabiraj showed that this phenomenon does not occur when the surface $N$ is orientable. In this paper, we prove that it occurs when $N$ is non-orientable and both geodesics involved in the smoothing are actually one-sided. In particular, we study a loop obtained by traversing a one-sided closed geodesic and the $m$-th power of another one-sided closed geodesic for odd $m$. Then we show that its free homotopy class may wind aroud a puncture at most two values of $m$. Furthermore, if two such $m$'s exist, they are consecutive odd integers.

A module $M$ is said to be stable if it has no nonzero projective direct summand. For a ring $ R $, we study the conditions under which every $R$-module $M$ within a specific class can be decomposed into a direct sum of a projective module and a stable module, focusing on identifying the types of rings and the class of $R$-modules where this property holds. Some well-known classes of rings over which every finitely presented module can be decomposed into a direct sum of a projective submodule and a stable submodule are semiperfect rings or semilocal rings or rings satisfying some finiteness conditions like having finite uniform dimension or hollow dimension or being Noetherian or Artinian. By using the Auslander-Bridger transpose of modules, we prove that every finitely presented right $R$-module over a left semihereditary ring $R$ has such a decomposition; note that the semihereditary condition is on the opposite side. Our main focus in this article is to give examples where such a decomposition fails. We give some ring examples over which there exists an infinitely generated or finitely generated or cyclic module or finitely presented module or cyclically presented module where such a decomposition fails. Our main example is a cyclically presented module $M$ over a commutative ring such that~$M$ has no such decomposition and $M$ is not projectively equivalent to a stable module.

In this paper we consider Toeplitz operators with anti-analytic symbols on $H^1(\mathbb{C}^+)$. It is well known that there are no bounded Toeplitz operators $T_{\overline{\Theta}}\colon H^1(\mathbb{C}^+) \to H^1(\mathbb{C}^+)$, where $\Theta \in H^\infty(\mathbb{C}^+)$. We consider the subspace $H^1_{\Theta}=\left\lbrace f \in H^1(\mathbb{C}^+)\colon \int_{\mathbb{R}}f \overline{\Theta}=0\right\rbrace$ and show that it is natural to study the boundedness of $T_{\overline{\Theta}}\colon H^1_\Theta \to H^1(\mathbb{C}^+)$. We provide several different conditions equivalent to such boundedness. We prove that when $\Theta=e^{i\tau (\cdot)}$, with $\tau>0$ $T_{\overline{\Theta}}\colon H^1_\Theta \to H^1(\mathbb{C}^+)$ is bounded. Finally, we discuss a number of related open questions.

In this paper we study structure-preserving numerical methods for low Mach number barotropic Euler equations. Besides their asymptotic preserving properties that are crucial in order to obtain uniformly consistent and stable approximations of the Euler equations in their singular limit as the Mach number approaches zero, our aim is also to preserve discrete entropy stability. Suitable acoustic/advection splitting approach combined with time implicit-explicit approximations are used to achieve the asymptotic preserving property. The entropy stability of different space discretisation strategies is studied for different values of Mach number and is validated by the numerical experiments.

Multifractional processes extend the concept of fractional Brownian motion by replacing the constant Hurst parameter with a time-varying Hurst function. This extension allows for modulation of the roughness of sample paths over time. The paper introduces a new class of multifractional processes, the Gaussian Haar-based multifractional processes (GHBMP), which is based on the Haar wavelet series representations. The resulting processes cover a significantly broader set of Hurst functions compared to the existing literature, enhancing their suitability for both practical applications and theoretical studies. The theoretical properties of these processes are investigated. Simulation studies conducted for various Hurst functions validate the proposed model and demonstrate its applicability, even for Hurst functions exhibiting discontinuous behaviour.

A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance assumptions, an analytic counterpart of the moment vector is characterized alongside a new epi-translation invariant valuation. The former arises as the top-degree operator in a family of functional intrinsic moments, which are linked to functional intrinsic volumes through translations. The latter represents the top-degree operator in a class of Minkowski vectors, which are introduced in this article and which lack classical counterparts on convex bodies, as they vanish due to the Minkowski relations. Additional classification results are obtained for homogeneous valuations of extremal degrees.

We correct here two errors in our earlier paper "An algebraic model for finite loop spaces" [arXiv:1212.2033]

Let $\varphi:V\times V\to W$ be a bilinear map of finite vector spaces $V$ and $W$ over a finite field $\mathbb{F}_q$. We present asymptotic bounds on the number of isomorphism classes of bilinear maps under the natural action of $\mathrm{GL}(V)$ and $\mathrm{GL}(W)$, when $\dim(V)$ and $\dim(W)$ are linearly related. As motivations and applications of the results, we present almost tight upper bounds on the number of $p$-groups of Frattini class $2$ as first studied by Higman (Proc. Lond. Math. Soc., 1960). Such bounds lead to answers for some open questions by Blackburn, Neumann, and Venkataraman (Cambridge Tracts in Mathematics, 2007). Further applications include sampling matrix spaces with the trivial automorphism group, and asymptotic bounds on the number of isomorphism classes of finite cube-zero commutative algebras.

In this article we prove that $E(n)$-coactions over a finite-dimensional algebra $A$ are classified by tuples $(\varphi, d_1, ... , d_n)$ consisting of an involution $\varphi$ and a family $(d_i)_{i=1,...,n}$ of $\varphi$-derivations satisfying appropriate conditions. Tuples of maps can be replaced by tuples of suitable elements $(c, u_1, . . . , u_n)$, whenever $A$ is a semisimple Clifford algebra.

This paper presents a novel algorithm integrating global and robust optimisation methods to solve continuous non-convex quadratic problems under convex uncertainty sets. The proposed Robust spatial branch-and-bound (RsBB) algorithm combines the principles of spatial branch-and-bound (sBB) with robust cutting planes. We apply the RsBB algorithm to quadratically constrained quadratic programming (QCQP) pooling problems, utilising McCormick envelopes to obtain convex lower bounds. The performance of the RsBB algorithm is compared with state-of-the-art methods that rely on global solvers. As computational test bed for our proposed approach we focus on pooling problems under different types and sizes of uncertainty sets. The findings of our work highlight the efficiency of the RsBB algorithm in terms of computational time and optimality convergence and provide insights to the advantages of combining robustness and optimality search.

The global analytic hypoellipticity is proved for a class of second order partial differential equations with non-negative characteristic form globally defined on the torus. The class considered in this work generalizes at some degree the class of sum of squares considered by Bove-Chinni and also by Cordaro-Himonas.

We study the Gelfand-Fuks cohomology of smooth vector fields on $S^d$ relative to $\mathrm{SO}(d+1)$ following a method by Haefliger that uses tools from rational homotopy theory. In particular, we show that $H^*(\mathrm{BSO}(4);\mathbb{R})$ injects into the relative Gelfand-Fuks cohomology which corrects a claim by Haefliger. Moreover, for $S^3$ the relative Gelfand-Fuks cohomology agrees with the smooth cohomology of $\text{Diff}^+(S^3)$ and we provide a computation in low degrees.

Distribution shifts have long been regarded as troublesome external forces that a decision-maker should either counteract or conform to. An intriguing feedback phenomenon termed decision dependence arises when the deployed decision affects the environment and alters the data-generating distribution. In the realm of performative prediction, this is encoded by distribution maps parameterized by decisions due to strategic behaviors. In contrast, we formalize an endogenous distribution shift as a feedback process featuring nonlinear dynamics that couple the evolving distribution with the decision. Stochastic optimization in this dynamic regime provides a fertile ground to examine the various roles played by dynamics in the composite problem structure. To this end, we develop an online algorithm that achieves optimal decision-making by both adapting to and shaping the dynamic distribution. Throughout the paper, we adopt a distributional perspective and demonstrate how this view facilitates characterizations of distribution dynamics and the optimality and generalization performance of the proposed algorithm. We showcase the theoretical results in an opinion dynamics context, where an opportunistic party maximizes the affinity of a dynamic polarized population, and in a recommender system scenario, featuring performance optimization with discrete distributions in the probability simplex.

The Sudoku number $s(G)$ of graph $G$ with chromatic number $\chi(G)$ is the smallest partial $\chi(G)$-colouring of $G$ that determines a unique $\chi(G)$-colouring of the entire graph. We show that the Sudoku number of the random $3$-regular graph $\mathcal{G}_{n,3}$ satisfies $s(\mathcal{G}_{n,3}) \leq (1+o(1))\frac{n}{3}$ asymptotically almost surely. We prove this by analyzing an algorithm which $3$-colours $\mathcal{G}_{n,3}$ in a way that produces many locally forced vertices, i.e., vertices which see two distinct colours among their neighbours. The intricacies of the algorithm present some challenges for the analysis, and to overcome these we use a non-standard application of Wormald's differential equations method that incorporates tools from finite Markov chains.

In this paper, we focus on the famous Talenti's symmetrization inequality, more precisely its $L^p$ corollary asserting that the $L^p$-norm of the solution to $-\Delta v=f^\sharp$ is higher than the $L^p$-norm of the solution to $-\Delta u=f$ (we are considering Dirichlet boundary conditions, and $f^\sharp$ denotes the Schwarz symmetrization of $f:\Omega\to\mathbb{R}_+$). We focus on the particular case where functions $f$ are defined on the unit ball, and are characteristic functions of a subset of this unit ball. We show in this case that stability occurs for the $L^p$-Talenti inequality with the sharp exponent 2.

We study the evolution of the Gross-Pitaevskii equation linearized around the Ginzburg-Landau vortex of degree one under equivariant symmetry. Among the main results of this work, we determine the spectrum of the linearized operator, uncover a remarkable $L^2$-norm growth phenomenon related to a zero-energy resonance, and provide a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.

This paper is concerned with the energy decay of a viscoelastic variable coefficient wave equation with nonlocality in time as well as nonlinear damping and polynomial nonlinear terms. Using the Lyapunov method, we establish a polynomial energy decay for the solution under relatively weak assumptions regarding the kernel of the nonlocal term. More specifically, we improve the decay rate of the energy by additionally imposing a certain convexity assumption on the kernel. Several examples are provided to confirm such improvements to faster polynomial or even exponential decays.

This paper contains a review of the theoretical foundations of Clifford algebras, spinors and spinor bundles in the so-called co-frame formalism. A compact index-free notation is introduced, along with a series of identities useful for computations in supergravity theories.

We present a class of abelian groups that exhibit a high degree of freeness while possessing no non-trivial homomorphisms to a canonical free object. Unlike prior investigations, which primarily focused on torsion-free groups, our work broadens the scope to include groups with torsion. Our main focus is on reduced separable p-groups, for which we formulate and prove the Trivial Duality Conjecture. Key tools in our analysis include the multi black-box method and the application of specific homological properties of relative trees. These techniques allow us to navigate the intricate interplay between freeness and duality in a broader class of abelian groups

Graphcodes were recently introduced as a technique to employ two-parameter persistence modules in machine learning tasks (Kerber and Russold, NeurIPS 2024). We show in this work that a compressed version of graphcodes yields a description of a two-parameter module that is equivalent to a presentation of the module. This alternative representation as a graph allows for a simple translation between combinatorics and algebra: connected components of the graphcode correspond to summands of the module and isolated paths correspond to intervals. We demonstrate that graphcodes are useful in practice by speeding-up the task of decomposing a module into indecomposable summands. Also, the graphcode viewpoint allows to devise a simple algorithm to decide whether a persistence module is interval-decomposable in $O(n^4)$ time, which improves on the previous bound of $O(n^{2\omega+1})$.

This note examines the BV formulation of $N=1$, $D=4$ supergravity in the first-order Palatini--Cartan framework. Challenges in achieving an off-shell formulation are addressed by introducing corrections to the rank 2 BV action, offering in addition a solid foundation for the study of the theory on manifolds with boundary.

We study the log version of the prismatic Dieudonn\'{e} theory established by Ansch\"{u}tz-Le Bras. By applying this result to the integral toroidal compactification of a Shimura variety of Hodge type, we extend the prismatic realization, originally constructed by Imai-Kato-Youcis, to the compactification. This extension enables us to prove Lovering's conjecture on $p$-adic comparison isomorphisms for Shimura varieties.

Let \( \mathcal{F} \) be a family of graphs. The generalized Tur\'an number \( \operatorname{ex}(n, K_r, \mathcal{F}) \) is the maximum number of $K_r$ in an \( n \)-vertex graph that does not contain any member of \( \mathcal{F} \) as a subgraph. Recently, Alon and Frankl initiated the study of Tur\'an problems with bounded matching number. In this paper, we determine the generalized Tur\'an number of \( C_{\geq k} \) with bounded matching number.

We demonstrate that any full and faithful $*$-functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such $*$-functors between approximable categories and prove that the functor associating each locally finite coarse space with its approximable category is full and faithful.

A graph $G$ is cyclically $c$-edge-connected if there is no set of fewer than $c$ edges that disconnects $G$ into at least two cyclic components. We prove that if a $(k, g)$-cage $G$ has at most $2M(k, g) - g^2$ vertices, where $M(k, g)$ is the Moore bound, then $G$ is cyclically $(k - 2)g$-edge-connected, which equals the number of edges separating a $g$-cycle, and every cycle-separating $(k - 2)g$-edge-cut in $G$ separates a cycle of length $g$. In particular, this is true for unknown cages with $(k, g) \in \{(3, 13), (3, 14), (3, 15), (4, 9), (4, 10)$, $(4, 11),$ $(5, 7), (5, 9), (5, 10), (5, 11), (6, 7), (9, 7)\}$ and also the potential missing Moore graph with degree $57$ and diameter $2$. Keywords: cage, cyclic connectivity, girth, lower bound

This paper presents a distinctive prime detection approach. This method use GM-(n+1) sequences to effectively eliminate complex numbers. The sequences, which consist of odd a number of (n+1), exclude all components except for the initial prime integer. Only the first prime number is presented. This research proposes an approach using this model to identify exceptional candidates and examine their distribution. This study examines the interconnections among the laws of division, basic gaps, and their applications in analytical procedures. Computer studies may provide a novel perspective on the theory of prime numbers, demonstrating the effectiveness of this approach in refining the search space for primes.

Given a finite set of $2$-edge-coloured graphs $\mathcal F$ and a hereditary property of graphs $\mathcal{P}$, we say that $\mathcal F$ expresses $\mathcal{P}$ if a graph $G$ has the property $\mathcal{P}$ if and only if it admits a $2$-edge-colouring not having any graph in $\mathcal F$ as an induced $2$-edge-coloured subgraph. We show that certain classic hereditary classes are expressible by some set of $2$-edge-coloured graphs on three vertices. We then initiate a systematic study of the following problem. Given a finite set of $2$-edge-coloured graphs $\mathcal F$, structurally characterize the hereditary property expressed by $\mathcal F$. In our main results we describe all hereditary properties expressed by $\mathcal F$ when $\mathcal F$ consists of 2-edge-coloured graphs on three vertices and (1) patterns have at most two edges, or (2) $\mathcal F$ consists of both monochromatic paths and a set of coloured triangles. On the algorithmic side, we consider the $\mathcal F$-free colouring problem, i.e., deciding if an input graph admits an $\mathcal F$-free $2$-edge-colouring. It follows from our structural characterizations, that for all sets considered in (1) and (2) the $\mathcal F$-free colouring problem is solvable in polynomial time. We complement these tractability results with a uniform reduction to boolean constraint satisfaction problems which yield polynomial-time algorithms that recognize most graph classes expressible by a set $\mathcal F$ of $2$-edge-coloured graphs on at most three vertices. Finally, we exhibit some sets $\mathcal F$ such that the $\mathcal F$-free colouring problem is NP-complete.

We survey large value problems, including the large value problem for Dirichlet polynomials, the restriction problem, and problems from computer science. We describe known techniques and open problems, drawing on perspectives from all three fields.

Thurston's triangulation conjecture asserts that every hyperbolic 3-manifold admits a geometric triangulation into hyper-ideal hyperbolic tetrahedra. So far, this conjecture had only been proven for a few special 3-manifolds. In this article, we confirm this conjecture for a class of 3-manifolds. To be precise, let $M$ be an oriented compact 3-manifold with boundary, no component of which is a 2-sphere, and $\mathcal{T}$ is an ideal triangulation of $M$. If $\mathcal{T}$ satisfies properly gluing condition, and the valence is at least 6 at each ideal edge and 11 at each hyper-ideal edge, then $M$ admits an unique complete hyperbolic metric with totally geodesic boundary, so that $\mathcal{T}$ is isotopic to a geometric ideal triangulation of $M$. We use analytical tools such as combinatorial Ricci flow (CRF, abbr.) to derive the conclusions. There are intrinsic difficulties in dealing with CRF. First, the CRF may collapse in a finite time, second, most of the smooth curvature flow methods are no longer applicable since there is no local coordinates in $\mathcal{T}$, and third, the evolution of CRF is affected by certain combinatorial obstacles in addition to topology. To this end, we introduce the ideas as ``extending CRF", ``tetrahedral comparison principles", and ``control CRF with edge valence" to solve the above difficulties. In addition, the presence of torus boundary adds substantial difficulties in this article, which we have solved by introducing the properly gluing conditions on $\mathcal{T}$ and reducing the ECRF to a flow relatively easy to handle.

We establish Kronecker-type first and second limit formulas for "non-holomorphic" and "Jacobi-type" Eisenstein series over global function fields in the several-variable setting. Our main theorem demonstrates that the derivatives of these Eisenstein series can be understood as averaged integrals of certain period quantities along the associated "Heegner cycles" on Drinfeld modular varieties. A key innovation lies in our use of the Berkovich analytic structure of the Drinfeld period domains, which enables the parametrization of the Heegner cycles in question by Euclidean "parallelepiped" regions. This approach also facilitates a unified and streamlined formulation and proof of our results. Finally, we apply these formulas to provide period interpretations of the "Kronecker terms" of Dedekind-Weil zeta functions and Dirichlet $L$-functions associated with ring and ray class characters.

We give a construction of ``quantum Maslov characteristic classes'', generalizing to higher dimensional cycles the Hu-Lalonde-Seidel morphism. We also state a conjecture extending this to an $A _{\infty}$ functor from the exact path category of the space of monotone Lagrangian branes to the Fukaya category. Quantum Maslov classes are used here for the study of Hofer geometry of Lagrangian equators in $S ^{2}$, giving a rigidity phenomenon for the Hofer metric 2-systole, which stands in contrast to the flexibility phenomenon of the closely related Hofer metric girth studied by Rauch ~\cite{cite_Itamar}, in the same context of Lagrangian equators of $S ^{2}$. More applications appear in ~\cite{cite_SavelyevGlobalFukayacategoryII}.

In this paper, we introduce the concept of a relative Heffter space which simultaneously generalizes those of relative Heffter arrays and Heffter spaces. Given a subgroup $J$ of an abelian group $G$, a relative Heffter space is a resolvable configuration whose points form a half-set of $G\setminus{J}$ and whose blocks are all zero-sum in $G$. Here we present two infinite families of relative Heffter spaces satisfying the additional condition of being simple. As a consequence, we get new results on globally simple relative Heffter arrays, on mutually orthogonal cycle decompositions and on biembeddings of cyclic cycle decompositions of the complete multipartite graph into an orientable surface.

Majority dynamics is a process on a simple, undirected graph $G$ with an initial Red/Blue color for every vertex of $G$. Each day, each vertex updates its color following the majority among its neighbors, using its previous color for tie-breaking. The dynamics achieves \textit{unanimity} if every vertex has the same color after finitely many days, and such color is said to \textit{win}. When $G$ is a $G(n,p)$ random graph, L. Tran and Vu (2019) found a codition in terms of $p$ and the initial difference $2\Delta$ beteween the sizes of the Red and Blue camps, such that unanimity is achieved with probability arbitrarily close to 1. They showed that if $p\Delta^2 \gg1 $, $p\Delta \geq 100$, and $p\geq (1+\varepsilon) n^{-1}\log n$ for a positive constant $\varepsilon$, then unanimity occurs with probability $1 - o(1)$. If $p$ is not extremely small, namely $p > \log^{-1/16} n $, then Sah and Sawhney (2022) showed that the condition $p\Delta^2 \gg 1$ is sufficient. If $n^{-1}\log^2 n \ll p \ll n^{-1/2}\log^{1/4} n$, we show that $p^{3/2}\Delta \gg n^{-1/2}\log n$ is enough. Since this condition holds if $p\Delta \geq 100$ for $p$ in this range, this is an improvement of Tran's and Vu's result. For the closely related problem of finding the optimal condition for $p$ to achieve unanimity when the initial coloring is chosen uniformly at random among all possible Red/Blue assignments, our result implies a new lower bound $p \gg n^{-2/3}\log^{2/3} n$, which improves upon the previous bound of $n^{-3/5}\log n$ by Chakraborti, Kim, Lee and T. Tran (2021).

This paper investigates quasi-isometries between graphs with variable edge lengths. A quasi-isometry is a mapping between metric spaces that approximately preserves distances, allowing for a bounded amount of additive and multiplicative distortion. Recently, Nguyen, Scott, and Seymour conjectured that, by appropriately adjusting the edge lengths of the target graph along with modifying the additive distortion constant, the multiplicative distortion factor could be eliminated. We disprove this conjecture.

In the vicinity of the liquid--vapor critical point, supercritical fluids behave strongly compressibly and, in parallel, thermophysical properties have strong state dependence. These lead to various peculiar phenomena, one of which being the piston effect where a sudden heating induces a mechanical pulse. The coupling between thermal and mechanical processes, in the linear approximation, yields a non-trivially rich thermoacoustics. The numerous applications of supercritical fluids raise the need for reliable yet fast and efficient numerical solution for thermoacoustic time and space dependence in this sensitive domain. Here, we present a second-order accurate, fully explicit staggered space-time grid finite difference method for such coupled linear thermoacoustic problems. Time integration is based on the splitting of the state space vector field representing the interactions that affect the dynamics into reversible and irreversible parts, which splitting procedure leads to decoupled wave and heat equations. The former is a hyperbolic partial differential equation, while the latter is a parabolic one, therefore, different time integration algorithms must be amalgamated to obtain a reliable, dispersion error-free, and dissipation error-free numerical solution. Finally, the thermoacoustic approximation of the supercritical piston effect is investigated via the developed method.

We study the following Liouville system defined on a flat torus \begin{equation} \left\{ \begin{array}{lr} -\Delta u_i=\sum_{j=1}^n a_{ij}\rho_j\Big(\frac{h_j e^{u_j}}{\int_\Omega h_j e^{u_j}}-1\Big),\nonumber u_j\in H_{per}^1(\Omega)\mbox{ for }i\in I=\{1,\cdots,n\}\nonumber, \end{array} \right. \end{equation} where $h_j\in C^3(\Omega)$, $h_j>0$, $\rho_j>0$ and $u=(u_1,..,u_n)$ is doubly periodic on $\partial\Omega$. The matrix $A=(a_{ij})_{n\times n}$ satisfies certain properties. One central problem about Liouville systems is whether multi-bubble solutions do exist. In this work we present a comprehensive construction of multi-bubble solutions in the most general setting.

We find necessary and sufficient conditions for high-order persistence of resonant caustics in perturbed circular billiards. The main tool is a perturbation theory based on the Bialy-Mironov generating function for convex billiards. All resonant caustics with period $q$ persist up to order $\lceil q/n \rceil -1$ under any polynomial deformation of the circle of degree $n$.

This paper relates to the Fourier decay properties of images of self-similar measures $\mu$ on $\mathbb{R}^k$ under nonlinear smooth maps $f \colon \mathbb{R}^k \to \mathbb{R}$. For example, we prove that if the linear parts of the similarities defining $\mu$ commute and the graph of $f$ has nonvanishing Gaussian curvature, then the Fourier dimension of the image measure is at least $\max\left\{ \frac{2(2\kappa_2 - k)}{4 + 2\kappa_* - k} , 0 \right\}$, where $\kappa_2$ is the lower correlation dimension of $\mu$ and $\kappa_*$ is the Assouad dimension of the support of $\mu$. Under some additional assumptions on $\mu$, we use recent breakthroughs in the fractal uncertainty principle to obtain further improvements for the decay exponents. We give several applications to nonlinear arithmetic of self-similar sets $F$ in the line. For example, we prove that if $\dim_{\mathrm H} F > (\sqrt{65} - 5)/4 = 0.765\dots$ then the arithmetic product set $F \cdot F = \{ xy : x,y \in F \}$ has positive Lebesgue measure, while if $\dim_{\mathrm H} F > (-3 + \sqrt{41})/4 = 0.850\dots$ then $F \cdot F \cdot F$ has non-empty interior. One feature of the above results is that they do not require any separation conditions on the self-similar sets.

Non-orthogonal multiple access (NOMA) has gained significant attention as a potential next-generation multiple access technique. However, its implementation with finite-alphabet inputs faces challenges. Particularly, due to inter-user interference, superimposed constellations may have overlapping symbols leading to high bit error rates when successive interference cancellation (SIC) is applied. To tackle the issue, this paper employs autoencoders to design interference-aware super-constellations. Unlike conventional methods where superimposed constellation may have overlapping symbols, the proposed autoencoder-based NOMA (AE-NOMA) is trained to design super-constellations with distinguishable symbols at receivers, regardless of channel gains. The proposed architecture removes the need for SIC, allowing maximum likelihood-based approaches to be used instead. The paper presents the conceptual architecture, loss functions, and training strategies for AE-NOMA. Various test results are provided to demonstrate the effectiveness of interference-aware constellations in improving the bit error rate, indicating the adaptability of AE-NOMA to different channel scenarios and its promising potential for implementing NOMA systems

In this paper, we consider optimal control problems of stochastic Volterra equations (SVEs) with singular kernels, where the control domain is not necessarily convex. We establish a global maximum principle by means of the spike variation technique. To do so, we first show a Taylor type expansion of the controlled SVE with respect to the spike variation, where the convergence rates of the remainder terms are characterized by the singularity of the kernels. Next, assuming additional structure conditions for the kernels, we convert the variational SVEs appearing in the expansion to their infinite dimensional lifts. Then, we derive first and second order adjoint equations in form of infinite dimensional backward stochastic evolution equations (BSEEs) on weighted $L^2$ spaces. Moreover, we show the well-posedness of the new class of BSEEs on weighted $L^2$ spaces in a general setting.

This article concerns the structure of $\mathrm{C}^{\ast}$-algebraic group actions induced on corona algebras from a given $\sigma$-unital $\mathrm{C}^{\ast}$-dynamical system over a locally compact group $G$. We prove that such actions satisfy the so-called dynamical folding property, which generalizes a fundamental property observed for corona algebras in works of Manuilov--Thomsen and Phillips--Weaver. We then focus on corona actions induced from $G$-$\mathrm{C}^{\ast}$-dynamics that are assumed to absorb a given strongly self-absorbing and unitarily regular $G$-action $\gamma$. It is proved that these corona actions are $\gamma$-saturated, which is a stronger property than being separably $\gamma$-stable. Conversely, if one assumes that the underlying $\mathrm{C}^{\ast}$-dynamics absorbs the trivial action on the compact operators, then $\gamma$-saturation of the corona action is equivalent to the original action being $\gamma$-absorbing. These results are a dynamical version of recent work by Farah and the third-named author.

The goal of this note is to study the geometry of large size-conditioned Bienaym\'e trees whose offspring distribution is subcritical, belongs to the domain of attraction of a stable law of index $\alpha=1$ and satisfies a local regularity assumption. We show that a condensation phenomenon occurs: one unique vertex of macroscopic degree emerges, and its height converges in distribution to a geometric random variable. Furthermore, the height of such trees grows logarithmically in their size. Interestingly, the behavior of subcritical Bienaym\'ee trees with $\alpha=1$ is quite similar to the case $\alpha \in( 1,2]$, in contrast with the critical case. This completes the study of the height of heavy-tailed size-conditioned Bienaym\'e trees. Our approach is to check that a random-walk one-big-jump principle due to Armend\'ariz & Loulakis holds, by using local estimates due to Berger, combined with the previous approach to study subcritical Bienaym\'e trees with $\alpha>1$.

This is the last of a three part work about relative free splitting complexes $\mathcal{FS}(\Gamma,\mathscr{A})$ and their actions by relative outer automorphism groups $\text{Out}(\Gamma;\mathscr{A})$. We obtain quantitative relations between the stable translation length $\tau_\phi$ and the relative train track dynamics of~$\phi \in \Out(\Gamma;\A)$. First, if $\phi$ has an orbit with diameter bounded below by a certain constant $\Omega(\Gamma;\mathscr{A}) \ge 1$ then $\phi$ has a filling attracting lamination. Also, there is a positive lower bound $\tau_\phi \ge A(\Gamma;\mathscr{A}) > 0$ amongst all $\phi$ which have a filling attracting lamination. Both proofs rely on a study of \emph{filling paths} in a free splitting. These results are all new even for $\text{Out}(F_n)$.

Adaptive therapy is a recent paradigm in cancer treatment aiming at indefinite, safe containment of the disease when cure is judged unattainable. In modeling this approach, inherent limitations arise due to the structure of the vector fields and the bounds imposed by toxic side-effects of the drug. In this work we analyze these limitations in a minimal class of models describing a cancer population with a slowly co-evolving drug resistance trait. Chemotherapeutic treatment is introduced as any bounded time-varying input, forcing the cells to adapt to a changing environment. We leverage the affine structure and low dimension of the system to explicitly construct controllable subsets of the state space enclosing sets of equilibria. We show that these controllable sets entirely determine the asymptotic behavior of all trajectories that cannot lead to a cure.

The aim of this paper is to compare singularities of closed positive currents whose non-pluripolar complex Monge--Amp\`ere masses equal. We also provide a short alternative proof for the monotonicity of non-pluripolar complex Monge--Amp\`ere masses, generalizing results of Witt-Nystr\"om, Darvas--Di Nezza--Lu, Lu--Nguy\^en and Vu.

The effect of multiplicative white noise on the resonance capture in non-isochronous systems with time-decaying pumping is investigated. It is assumed that the intensity of perturbations decays with time, and its frequency is asymptotically constant. The occurrence of attractive solutions with an amplitude close to the resonant value and a phase synchronized with the excitation are considered. The persistence of such a regime in a stochastically perturbed system is analyzed. By combining the averaging method and the construction of suitable stochastic Lyapunov functions, conditions are derived that guarantee the stochastic stability of the resonant modes on infinite or asymptotically large time intervals. The proposed theory is applied to the Duffing oscillator with decaying parametric excitation and noise.

Our main objective in this article is to study factorization problems for twisted triple product Galois representation over quadratic totally real fields coming from families of Hilbert cuspforms. We study this problem in two cases namely rank (1,1) and rank (0,2) depending on the order of vanishing of $ L $-functions at central critical values. This work is a quadratic totally real generalization of (algebraic) factorization result obtained by B\"{u}y\"{u}kboduk et al. in the spirit of work by Greenberg and Palvannan. Notably, this is the first result in the higher rank (0,2) case.

Given a smooth closed $n$-manifold $M$ and a $\kappa$-tuple of basepoints $\boldsymbol{q}\subset M$, we define a Morse-type $A_\infty$-algebra $CM_{-*}(\Omega(M,\boldsymbol{q}))$, called the based multiloop $A_\infty$-algebra, as a graded generalization of the braid skein algebra due to Morton and Samuelson. For example, when $M=T^2$ the braid skein algebra is the Type A double affine Hecke algebra (DAHA). The $A_\infty$-operations couple Morse gradient trees on a based loop space with Chas-Sullivan type string operations. We show that, after a certain "base change", $CM_{-*}(\Omega(M,\boldsymbol{q}))$ is $A_\infty$-equivalent to the wrapped higher-dimensional Heegaard Floer $A_\infty$-algebra of $\kappa$ disjoint cotangent fibers which was studied in the work of Honda, Colin, and Tian. We also compute the based multiloop $A_\infty$-algebra for $M=S^2$, which we can regard as a derived Hecke algebra of the $2$-sphere.

We study a variational model in nonlinear elasticity allowing for cavitation which penalizes both the volume and the perimeter of the cavities. Specifically, we investigate the approximation (in the sense of {\Gamma}-convergence) of the energy by means of functionals defined on perforated domains. Perforations are introduced at flaw points where singularities are expected and, hence, the corresponding deformations do not exhibit cavitation. Notably, those points are not prescribed but rather selected by the variational principle. Our analysis is motivated by the numerical simulation of cavitation and extends previous results on models which solely accounted for elastic energy but neglected contributions related to the formation of cavities.

Inspired by certain topics in local deformation theory, we primarily utilize F. Campana's methods to investigate how far a fiberwise Fujiki family is from being locally Moishezon and how far a fiberwise K\"ahler family is from being locally projective. We investigate these questions from two main perspectives: cohomological data and global semi-positivity data on the total space.

This article presents the $L^p$-Heisenberg-Pauli-Weyl uncertainty inequality for the group Fourier transform on a broad class of two-step nilpotent Lie groups, specifically the two-step MW groups. This inequality quantitatively demonstrates that on two-step MW groups, a nonzero function and its group Fourier transform cannot both be sharply localized. The proof primarily relies on utilizing the dilation structure inherent to two-step nilpotent Lie groups and estimating the Schatten class norms of the group Fourier transform. The inequality we establish is new even in the simplest case of Heisenberg groups. Our result significantly sharpens all previously known $L^p$-Heisenberg-Pauli-Weyl uncertainty inequalities for $1 \leq p < 2$ within the realm of two-step nilpotent Lie groups.

This paper proposes a universal, optimal algorithm for convex minimization problems of the composite form $g_0(x)+h(g_1(x),\dots, g_m(x)) + u(x)$. We allow each $g_j$ to independently range from being nonsmooth Lipschitz to smooth, from convex to strongly convex, described by notions of H\"older continuous gradients and uniform convexity. Note that, although the objective is built from a heterogeneous combination of such structured components, it does not necessarily possess smoothness, Lipschitzness, or any favorable structure overall other than convexity. Regardless, we provide a universal optimal method in terms of oracle access to (sub)gradients of each $g_j$. The key insight enabling our optimal universal analysis is the construction of two new constants, the Approximate Dualized Aggregate smoothness and strong convexity, which combine the benefits of each heterogeneous structure into single quantities amenable to analysis. As a key application, fixing $h$ as the nonpositive indicator function, this model readily captures functionally constrained minimization $g_0(x)+u(x)$ subject to $g_j(x)\leq 0$. In particular, our algorithm and analysis are directly inspired by the smooth constrained minimization method of Zhang and Lan and consequently recover and generalize their accelerated guarantees.

Quantum error correction (QEC) is critical for practical realization of fault-tolerant quantum computing, and recently proposed families of quantum low-density parity-check (QLDPC) code are prime candidates for advanced QEC hardware architectures and implementations. This paper focuses on the finite-length QLDPC code design criteria, specifically aimed at constructing degenerate quasi-cyclic symmetric lifted-product (LP-QLDPC) codes. We describe the necessary conditions such that the designed LP-QLDPC codes are guaranteed to have a minimum distance strictly greater than the minimum weight stabilizer generators, ensuring superior error correction performance on quantum channels. The focus is on LP-QLDPC codes built from quasi-cyclic base codes belonging to the class of type-I protographs, and the necessary constraints are efficiently expressed in terms of the row and column indices of the base code. Specifically, we characterize the combinatorial constraints on the classical quasi-cyclic base matrices that guarantee construction of degenerate LP-QLDPC codes. Minimal examples and illustrations are provided to demonstrate the usefulness and effectiveness of the code construction approach. The row and column partition constraints derived in the paper simplify the design of degenerate LP-QLDPC codes and can be incorporated into existing classical and quantum code design approaches.

In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial $A_2$-weight. We prove that having a generalized fundamental solution with upper Gaussian bounds is equivalent to Moser's $L^2$-$L^\infty$ estimates for local weak solutions. In the special case of real coefficients, Moser's $L^2$-$L^\infty$ estimates are known, which provide an easier proof of Gaussian upper bounds, and a known Harnack inequality is then used to derive Gaussian lower bounds.

It is a folklore belief that metastable wells in low-temperature statistical mechanics models exhibit high-temperature behavior. We prove a rigorous version of this phenomenon in the setting of the exponential random graph model (ERGM) through the lens of concentration of measure. To do this, we first present a new general result deriving concentration inequalities in a metastable well from the metastable mixing of a Markov chain with the appropriate stationary distribution, extending a result of Chatterjee [Cha05] which is suited for more traditional forms of global mixing. We then apply this result to the supercritical (low-temperature) ERGM which was recently proven to exhibit metastable mixing by Bresler, Nagaraj, and Nichani [BNN24], and obtain a novel concentration inequality for Lipschitz observables of the supercritical ERGM conditioned on a large metastable well, answering a question posed by [BNN24]. This extends a result of Ganguly and Nam [GN24] from the subcritical (high-temperature) regime to a metastable well in the supercritical regime, and we are also able to extend the applications of their concentration inequality to these metastable wells. Namely, we obtain an upper bound on the Wasserstein distance between the ERGM conditioned on a metastable well and an appropriate Erd\H{o}s-R\'enyi model, as well as derive a central limit theorem for the count of edges in certain small subcollections of possible edges. Finally, to supplement the mathematical content of the article, we also discuss the results of what appears to be the first simulation study of a metastable well in the supercritical ERGM.

We prove an inversion formula for the exterior $k$-plane transform. As a consequence, we show that if $m < k$ then an $m$-current in $\mathbf R^n$ can be reconstructed from its projections onto $\mathbf R^k$, which proves a conjecture of Solomon.

Let ${p > 2}$ be an odd prime and ${G = SL_2(\mathbb{F}_p)}$. Denote the subgroup of upper triangular matrices as $B$. Finally, let ${\mathbb{F}}$ be an algebraically closed field of characteristic ${p}$. The Green correspondence gives a bijection between the non-projective indecomposable ${\mathbb{F}[G]}$ modules and non-projective indecomposable ${\mathbb{F}[B]}$ modules, realised by restriction and induction. In this paper, we start by recalling a suitable description of the non-projective indecomposable modules for these group algebras. Next, we explicitly describe the Green correspondence bijection by pinpointing the modules' position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of these descriptions: formulae for lifting the ${\mathbb{F}[B]}$ module decomposition of an ${\mathbb{F}[G]}$ module, and a complete description of ${\text{Ind}_B^G}$ and ${\text{Res}^G_B}$.

We study a random band matrix $H=(H_{xy})_{x,y}$ of dimension $N\times N$ with mean-zero complex Gaussian entries, where $x,y$ belong to the discrete torus $(\mathbb{Z}/\sqrt{N}\mathbb{Z})^{2}$. The variance profile $\mathbb{E}|H_{xy}|^{2}=S_{xy}$ vanishes when the distance between $x,y$ is larger than some band-width parameter $W$ depending on $N$. We show that if the band-width satisfies $W\geq N^{\mathfrak{c}}$ for some $\mathfrak{c}>0$, then in the large-$N$ limit, we have the following results. The first result is a local semicircle law in the bulk down to scales $N^{-1+\varepsilon}$. The second is delocalization of bulk eigenvectors. The third is a quantum unique ergodicity for bulk eigenvectors. The fourth is universality of local bulk eigenvalue statistics. The fifth is a quantum diffusion profile for the associated $T$ matrix. Our method is based on embedding $H$ inside a matrix Brownian motion $H_{t}$ as done in [Dubova-Yang '24] and [Yau-Yin '25] for band matrices on the one-dimensional torus. In this paper, the key additional ingredient in our analysis of $H_{t}$ is a new CLT-type estimate for polynomials in the entries of the resolvent of $H_{t}$.

Patterns and nonlinear waves, such as spots, stripes, and rotating spirals, arise prominently in many natural processes and in reaction-diffusion models. Our goal is to compute boundaries between parameter regions with different prevailing patterns and waves. We accomplish this by evolving randomized initial data to full patterns and evaluate feature functions, such as the number of connected components or their area distribution, on their sublevel sets. The resulting probability measure on the feature space, which we refer to as pattern statistics, can then be compared at different parameter values using the Wasserstein distance. We show that arclength predictor-corrector continuation can be used to trace out transition and bifurcation curves in parameter space by maximizing the distance of the pattern statistics. The utility of this approach is demonstrated through a range of examples involving homogeneous states, spots, stripes, and spiral waves.

The program of internal type theory seeks to develop the categorical model theory of dependent type theory using the language of dependent type theory itself. In the present work we study internal homotopical type theory by relaxing the notion of a category with families (cwf) to that of a wild, or precoherent higher cwf, and determine coherence conditions that suffice to recover properties expected of models of dependent type theory. The result is a definition of a split 2-coherent wild cwf, which admits as instances both the syntax and the "standard model" given by a universe type. This allows us to give a straightforward internalization of the notion of a 2-coherent reflection of homotopical type theory in itself: namely as a 2-coherent wild cwf morphism from the syntax to the standard model. Our theory also easily specializes to give definitions of "low-dimensional" higher cwfs, and conjecturally includes the container higher model as a further instance.

Traditional control methods often show limitations in dealing with complex nonlinear systems, especially when it is difficult to accurately obtain the exact system model, and the control accuracy and stability are difficult to guarantee. To solve this problem, the Koopman operator theory provides an effective method to linearise nonlinear systems, which simplifies the analysis and control of the system by mapping the nonlinear dynamics into a high-dimensional space. However, the existing extended dynamical mode decomposition (EDMD) methods suffer from randomness in the selection of basis functions, which leads to bias in the finite-dimensional approximation to the Koopman operator, thus affecting the accuracy of model prediction. To solve this problem, this paper proposes a BLS-EDMD method based on the Broad learning system (BLS) network. The method achieves a high-precision approximation to the Koopman operator by learning more accurate basis functions, which significantly improves the prediction ability of the model. Building on this, we further develop a model predictive controller (MPC) called BE-MPC. This controller directly utilises the high-dimensional and high-precision predictors generated by BLS-EDMD to predict the system state more accurately, thus achieving precise control of the underwater unmanned vehicle (UUV), and its effectiveness is verified by simulation.

We study the non-modal stability of black hole spacetimes under linear perturbations. We show that large-amplitude growth can occur at finite time, despite asymptotic decay of linear perturbations. In the example presented, the physical mechanism is a transient form of superradiance, and is qualitatively similar to the transition to turbulence in Navier-Stokes shear flows. As part of the construction we provide a theorem for the positivity of QNM energies, and introduce a truncated-Hamiltonian approach to black hole pseudospectra which does not suffer from convergence issues.

Quantum counterparts of Schrodinger's classical bridge problem have been around for the better part of half a century. During that time, several quantum approaches to this multifaceted classical problem have been introduced. In the present work, we unify, extend, and interpret several such approaches through a classical large deviations perspective. To this end, we consider time-symmetric ensembles that are pre- and post-selected before and after a Markovian experiment is performed. The Schrodinger bridge problem is that of finding the most likely joint distribution of initial and final outcomes that is consistent with obtained endpoint results. The derived distribution provides quantum Markovian dynamics that bridge the observed endpoint states in the form of density matrices. The solution retains its classical structure in that density matrices can be expressed as the product of forward-evolving and backward-evolving matrices. In addition, the quantum Schrodinger bridge allows inference of the most likely distribution of outcomes of an intervening measurement with unknown results. This distribution may be written as a product of forward- and backward-evolving expressions, in close analogy to the classical setting, and in a time-symmetric way. The derived results are illustrated through a two-level amplitude damping example.

This review paper explores the Riccati-type pseudo-potential formulation applied to the quasi-integrable sine-Gordon, KdV, and NLS models. The proposed framework provides a unified methodology for analyzing quasi-integrability properties across various integrable systems, including deformations of the sine-Gordon, Bullough-Dodd, Toda, KdV, pKdV, NLS and SUSY sine-Gordon models. Key findings include the emergence of infinite towers of anomalous conservation laws within the Riccati-type approach and the identification of exact non-local conservation laws in the linear formulations of deformed models. As modified integrable models play a crucial role in diverse fields of non-linear physics-such as Bose-Einstein condensation, superconductivity, gravity models, optics, and soliton turbulence-these results may have far-reaching applications.

A mesh-based parametrization is a parametrization of a geometric object that is defined solely from a mesh of the object, e.g., without an analytical expression or computer-aided design (CAD) representation of the object. In this work, we propose a mesh-based parametrization of an arbitrary $d'$-dimensional object embedded in a $d$-dimensional space using tools from high-order finite elements. Using mesh-based parametrizations, we construct a boundary-preserving parametrization of the nodal coordinates of a computational mesh that ensures all nodes remain on all their original boundaries. These boundary-preseving parametrizations allow the nodes of the mesh to move only in ways that will not change the computational domain. They also ensure nodes will not move between boundaries, which would cause issues assigning boundary conditions for partial differential equation simulations and lead to inaccurate geometry representations for non-smooth boundary transitions. Finally, we integrate boundary-preserving, mesh-based parametrizations into high-order implicit shock tracking, an optimization-based discontinuous Galerkin method that moves nodes to align mesh faces with non-smooth flow features to represent them perfectly with inter-element jumps, leaving the intra-element polynomial basis to represent smooth regions of the flow with high-order accuracy. Mesh-based parametrizations enable implicit shock tracking simulations of shock-dominated flows over geometries without simple analytical parametrizations. Several demonstrations of mesh-based parametrizations are provided.

Techniques that rigorously bound the overall rounding error exhibited by a numerical program are of significant interest for communities developing numerical software. However, there are few available tools today that can be used to rigorously bound errors in programs that employ conditional statements (a basic need) as well as mixed-precision arithmetic (a direction of significant future interest) employing global optimization in error analysis. In this paper, we present a new tool that fills this void while also employing an abstraction-guided optimization approach to allow designers to trade error-bound tightness for gains in analysis time -- useful when searching for design alternatives. We first present the basic rigorous analysis framework of Satire and then show how to extend it to incorporate abstractions, conditionals, and mixed-precision arithmetic. We begin by describing Satire's design and its performance on a collection of benchmark examples. We then describe these aspects of Satire: (1) how the error-bound and tool execution time vary with the abstraction level; (2) the additional machinery to handle conditional expression branches, including defining the concepts of instability jumps and instability window widths and measuring these quantities; and (3) how the error changes when a mix of precision values are used. To showcase how \satire can add value during design, we start with a Conjugate Gradient solver and demonstrate how its step size and search direction are affected by different precision settings. Satire is freely available for evaluation, and can be used during the design of numerical routines to effect design tradeoffs guided by rigorous empirical error guarantees.

A well-developed classification program for 4d $\mathcal{N}=2$ super conformal field theories (SCFTs) leverages Seiberg-Witten geometry on the Coulomb branch of vacua; theories are arranged by increasing $\mathfrak{rank}$, the complex dimension of their Coulomb branch. An alternative organizational scheme focusses on the associated vertex operator algebra (VOA), which is more closely related to the Higgs branch. From the VOA perspective, a natural way to arrange theories is by their ``index of nilpotency'', the smallest integer $\mathfrak{n}$ such that $T^\mathfrak{n} = 0$ in the $C_2$ algebra, where $T$ is the VOA stress tensor. It follows from the Higgs branch reconstruction conjecture that $\mathfrak{n} < \infty$ for any 4d ${\cal N}=2$ SCFT. Extrapolating from several examples, we conjecture that $\mathfrak{n}$ is an RG monotone, $\mathfrak{n}_{\rm IR} \leq \mathfrak{n}_{\rm UV}$. What's more, we find in all cases that $\mathfrak{rank} \leq \mathfrak{n}-1$. Theory ordering by $\mathfrak{n}$ appears thus more refined than ordering by $\mathfrak{rank}$. For example, in the list of $\mathfrak{rank}=1$ theories, the Kodaira SCFTs and $SU(2)$ ${\cal N}=4$ SYM have $\mathfrak{n} =2$, while all others have $\mathfrak{n} >2$.

This paper presents a data-driven methodology to estimate the storage function of a passive system. The methodology consists in parametrizing the storage function with a dictionary then running a linear program. Results on a benchmark are presented to illustrate its properties, including its robustness to noise. Various uses of the storage function that do not require knowledge of a model are also discussed.

High dose rate brachytherapy (HDR-BT) is an appealing treatment option for superficial cancers that permits the delivery of higher local doses than other radiation modalities without a significant increase in toxicity. In order for HDR-BT to be used in these situations, needles through which the radiation source is passed must be strategically placed in close proximity to the patient's body. Currently, this crucial step is performed manually by physicians or medical physicists. The use of 3D-printed masks customized for individual patients has been advocated as a way to more precisely and securely position these needles, with the potential of producing better and safer treatment plans. In this paper, we propose optimization approaches for positioning needles within 3D-printed masks for HDR-BT, focusing on skin cancers. We numerically show that the models we propose efficiently generate more homogeneous plans than those derived manually and provide an alternative to manual placement that can save planning time and enhance plan quality.

Hirota's discrete KdV (dKdV) equation is an integrable autonomous partial difference equation on $\mathbb{Z}^2$ that reduces to the Korteweg-de Vries (KdV) equation in a continuum limit. In this paper, we introduce a new non-autonomous version of the dKdV equation. Furthermore, we show that the new equation is integrable and admits discrete Painlev\'e transcendent solutions described by $q$-Painlev\'e equations of $A_J^{(1)}$-surface types ($J=3,4,5,6$).

Using a reaction-diffusion model with free boundaries in one space dimension for a single population species with density $u(t,x)$ and population range $[g(t), h(t)]$, we demonstrate that the Allee effects can be eliminated if the species maintains its population density at a suitable level at the range boundary by advancing or retreating the fronts. It is proved that with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density level at the range boundary is maintained at other levels. If the invading cane toads in Australia used this strategy at the range boundary to become a super invader, then our results may explain why toads near the invading front evolve to have longer legs and run faster.

This paper bridges optimization and control, and presents a novel closed-loop control framework based on natural gradient descent, offering a trajectory-oriented alternative to traditional cost-function tuning. By leveraging the Fisher Information Matrix, we formulate a preconditioned gradient descent update that explicitly shapes system trajectories. We show that, in sharp contrast to traditional controllers, our approach provides flexibility to shape the system's low-level behavior. To this end, the proposed method parameterizes closed-loop dynamics in terms of stationary covariance and an unknown cost function, providing a geometric interpretation of control adjustments. We establish theoretical stability conditions. The simulation results on a rotary inverted pendulum benchmark highlight the advantages of natural gradient descent in trajectory shaping.

Federated learning (FL) has emerged as a promising framework for distributed learning, enabling collaborative model training without sharing private data. Existing wireless FL works primarily adopt two communication strategies: (1) over-the-air (OTA) computation, which exploits wireless signal superposition for simultaneous gradient aggregation, and (2) digital communication, which allocates orthogonal resources for gradient uploads. Prior works on both schemes typically assume \emph{homogeneous} wireless conditions (equal path loss across devices) to enforce zero-bias updates or permit uncontrolled bias, resulting in suboptimal performance and high-variance model updates in \emph{heterogeneous} environments, where devices with poor channel conditions slow down convergence. This paper addresses FL over heterogeneous wireless networks by proposing novel OTA and digital FL updates that allow a structured, time-invariant model bias, thereby reducing variance in FL updates. We analyze their convergence under a unified framework and derive an upper bound on the model ``optimality error", which explicitly quantifies the effect of bias and variance in terms of design parameters. Next, to optimize this trade-off, we study a non-convex optimization problem and develop a successive convex approximation (SCA)-based framework to jointly optimize the design parameters. We perform extensive numerical evaluations with several related design variants and state-of-the-art OTA and digital FL schemes. Our results confirm that minimizing the bias-variance trade-off while allowing a structured bias provides better FL convergence performance than existing schemes.

Reinforced random walks (RRWs), including vertex-reinforced random walks (VRRWs) and edge-reinforced random walks (ERRWs), model random walks where the transition probabilities evolve based on prior visitation history~\cite{mgr, fmk, tarres, volkov}. These models have found applications in various areas, such as network representation learning~\cite{xzzs}, reinforced PageRank~\cite{gly}, and modeling animal behaviors~\cite{smouse}, among others. However, statistical estimation of the parameters governing RRWs remains underexplored. This work focuses on estimating the initial edge weights of ERRWs using observed trajectory data. Leveraging the connections between an ERRW and a random walk in a random environment (RWRE)~\cite{mr, mr2}, as given by the so-called "magic formula", we propose an estimator based on the generalized method of moments. To analyze the sample complexity of our estimator, we exploit the hyperbolic Gaussian structure embedded in the random environment to bound the fluctuations of the underlying random edge conductances.

This paper proposes an accelerated consensus-based distributed iterative algorithm for resource allocation and scheduling. The proposed gradient-tracking algorithm introduces an auxiliary variable to add momentum towards the optimal state. We prove that this solution is all-time feasible, implying that the coupling constraint always holds along the algorithm iterative procedure; therefore, the algorithm can be terminated at any time. This is in contrast to the ADMM-based solutions that meet constraint feasibility asymptotically. Further, we show that the proposed algorithm can handle possible link nonlinearity due to logarithmically-quantized data transmission (or any sign-preserving odd sector-bound nonlinear mapping). We prove convergence over uniformly-connected dynamic networks (i.e., a hybrid setup) that may occur in mobile and time-varying multi-agent networks. Further, the latency issue over the network is addressed by proposing delay-tolerant solutions. To our best knowledge, accelerated momentum-based convergence, nonlinear linking, all-time feasibility, uniform network connectivity, and handling (possible) time delays are not altogether addressed in the literature. These contributions make our solution practical in many real-world applications.

In strictly axisymmetric configurations of tokamaks, field-line tracing reduces from a three-dimensional ODE system to a two-dimensional one, where Poincar\'e-Bendixson theorem applies and guarantees the nonexistence of chaos. The formulae of functional perturbation theory (FPT) mostly simplify to compact closed-form expressions to allow the computation to finish instantly, which could improve and accelerate the existing plasma control systems by detangling the plasma dynamics from the magnetic topology change. FPT can conveniently calculate how the key geometric objects of magnetic topology: 1. the divertor X-point(s) and the magnetic axis, 2. the last closed flux surface (LCFS) 3. flux surfaces change under perturbation. For example, when the divertor X-point shifts outwards, the LCFS there must expand accordingly, but not necessarily for other places of the LCFS, which could also contract, depending on the perturbation. FPT can not only facilitate adaptive control of plasma, but also enable utilizing as much as possible space in the vacuum vessel by weakening the plasma-wall interaction (PWI) via tuning the eigenvalues of $\mathcal{DP}^m$ of the divertor X-point(s), such that the field line connection lengths in the scrape-off layer (SOL) are long enough to achieve detachment. Increasing flux expansion $f_x$ is another option for detachment and can also be facilitated by FPT. Apart from the edge, FPT can also benefit the understanding of the plasma core. Since the magnetic axis O-point would also shift under perturbation and the shift is known by FPT, the O-point can be controlled without full knowledge of the plasma response, which shall not significantly change the tendency.

This paper studies the linear quadratic regulation (LQR) problem of unknown discrete-time systems via dynamic output feedback learning control. In contrast to the state feedback, the optimality of the dynamic output feedback control for solving the LQR problem requires an implicit condition on the convergence of the state observer. Moreover, due to unknown system matrices and the existence of observer error, it is difficult to analyze the convergence and stability of most existing output feedback learning-based control methods. To tackle these issues, we propose a generalized dynamic output feedback learning control approach with guaranteed convergence, stability, and optimality performance for solving the LQR problem of unknown discrete-time linear systems. In particular, a dynamic output feedback controller is designed to be equivalent to a state feedback controller. This equivalence relationship is an inherent property without requiring convergence of the estimated state by the state observer, which plays a key role in establishing the off-policy learning control approaches. By value iteration and policy iteration schemes, the adaptive dynamic programming based learning control approaches are developed to estimate the optimal feedback control gain. In addition, a model-free stability criterion is provided by finding a nonsingular parameterization matrix, which contributes to establishing a switched iteration scheme. Furthermore, the convergence, stability, and optimality analyses of the proposed output feedback learning control approaches are given. Finally, the theoretical results are validated by two numerical examples.

Let $G=(V,E)$ be an undirected unweighted multi-graph and $S\subseteq V$ be a subset of vertices called the Steiner set. A set of edges with the least cardinality whose removal disconnects $S$, that is, there is no path between at least one pair of vertices from $S$, is called a Steiner mincut for $S$ or simply an $S$-mincut. Connectivity Carcass is a compact data structure storing all $S$-mincuts in $G$ announced by Dinitz and Vainshtein in an extended abstract by Dinitz and Vainshtein in 1994. The complete proof of various results of this data structure for the simpler case when the capacity of $S$-mincut is odd appeared in the year 2000 in SICOMP. Over the last couple of decades, there have been attempts towards the proof for the case when the capacity of $S$-mincut is even, but none of them met a logical end. We present the following results. - We present the first complete, self-contained exposition of the connectivity carcass which covers both even and odd cases of the capacity of $S$-mincut. - We derive the results using an alternate and much simpler approach. In particular, we derive the results using submodularity of cuts -- a well-known property of graphs expressed using a simple inequality. - We also show how the connectivity carcass can be helpful in efficiently answering some basic queries related to $S$-mincuts using some additional insights.

We study the properties of B-branes in a class of nonabelian GLSMs realizing the canonical line bundle $K_{Gr(2,N)}$ in their geometric phase. By analysing the hemisphere partition function, i.e. B-brane central charge, we propose a grade restriction rule and the corresponding window categories for a specific class of paths between phases. We find very striking differences between the cases of N even and N odd. In particular, for the case of N even, we suggest that more than one window category can be possible, for a fixed path. A detailed computation of the open Witten index and some monodromies provides evidence for our proposal for window categories. In addition, we make some remarks about B-branes on the the strongly coupled phase, for the case $N=4$, based on our window proposal.

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

In this paper, we address the Wigner distribution and the star exponential function for a time-dependent harmonic oscillator for which the mass and the frequency terms are considered explicitly depending on time. To such an end, we explore the connection between the star exponential, naturally emerging within the context of deformation quantization, and the propagators constructed through the path integral formalism. In particular, the Fourier-Dirichlet expansion of the star exponential implies a distinctive quantization of the Lewis-Riesenfeld invariant. Further, by introducing a judicious time variable, we recovered a time-dependent phase function associated with the Lewis-Riesenfeld construction of the standard Schr\"odinger picture. In particular, we applied our results to the cases of the Caldirola-Kanai and the time-dependent frequency harmonic oscillators, recovering relevant results previously reported in the literature.

Consider a pair of sparse correlated stochastic block models $\mathcal S(n,\tfrac{\lambda}{n},\epsilon;s)$ subsampled from a common parent stochastic block model with two symmetric communities, average degree $\lambda=O(1)$ and divergence parameter $\epsilon \in (0,1)$. For all $\epsilon\in(0,1)$, we construct a statistic based on the combination of two low-degree polynomials and show that there exists a sufficiently small constant $\delta=\delta(\epsilon)>0$ and a sufficiently large constant $\Delta=\Delta(\epsilon,\delta)$ such that when $\lambda>\Delta$ and $s>\sqrt{\alpha}-\delta$ where $\alpha\approx 0.338$ is Otter's constant, this statistic can distinguish this model and a pair of independent stochastic block models $\mathcal S(n,\tfrac{\lambda s}{n},\epsilon)$ with probability $1-o(1)$. We also provide an efficient algorithm that approximates this statistic in polynomial time. The crux of our statistic's construction lies in a carefully curated family of multigraphs called \emph{decorated trees}, which enables effective aggregation of the community signal and graph correlation from the counts of the same decorated tree while suppressing the undesirable correlations among counts of different decorated trees.

Reduced Rank Regression (RRR) is a widely used method for multi-response regression. However, RRR assumes a linear relationship between features and responses. While linear models are useful and often provide a good approximation, many real-world problems involve more complex relationships that cannot be adequately captured by simple linear interactions. One way to model such relationships is via multilinear transformations. This paper introduces Higher Order Reduced Rank Regression (HORRR), an extension of RRR that leverages multi-linear transformations, and as such is capable of capturing nonlinear interactions in multi-response regression. HORRR employs tensor representations for the coefficients and a Tucker decomposition to impose multilinear rank constraints as regularization akin to the rank constraints in RRR. Encoding these constraints as a manifold allows us to use Riemannian optimization to solve this HORRR problems. We theoretically and empirically analyze the use of Riemannian optimization for solving HORRR problems.

The behavior of extreme observations is well-understood for time series or spatial data, but little is known if the data generating process is a structural causal model (SCM). We study the behavior of extremes in this model class, both for the observational distribution and under extremal interventions. We show that under suitable regularity conditions on the structure functions, the extremal behavior is described by a multivariate Pareto distribution, which can be represented as a new SCM on an extremal graph. Importantly, the latter is a sub-graph of the graph in the original SCM, which means that causal links can disappear in the tails. We further introduce a directed version of extremal graphical models and show that an extremal SCM satisfies the corresponding Markov properties. Based on a new test of extremal conditional independence, we propose two algorithms for learning the extremal causal structure from data. The first is an extremal version of the PC-algorithm, and the second is a pruning algorithm that removes edges from the original graph to consistently recover the extremal graph. The methods are illustrated on river data with known causal ground truth.

We perform full integration of the stationary axisymmetric Einstein-Maxwell-dilaton-axion (EMDA) theory with and without potential using a recently proposed generalization of Carter's approach to spacetimes beyond type D, allowing the Killing tensor. Crucial to our construction is a new parametrization of the dilaton and axion fields based on the analyticity argument. The general solution in the ungauged case is asymptotically locally flat and contains two more parameters compared to EMDA black holes previously obtained using Harrison transformations. In the gauged case, the general solution is asymptotically AdS and includes flat and hyperbolic topological solutions, as well as generalization of the Kerr-Sen-AdS metric with three additional parameters. Our approach can be applied to more general four-dimensional ungauged and gauged supergravities.

Polynomial inequality proving is fundamental to many mathematical disciplines and finds wide applications in diverse fields. Current traditional algebraic methods are based on searching for a polynomial positive definite representation over a set of basis. However, these methods are limited by truncation degree. To address this issue, this paper proposes an approach based on reinforcement learning to find a {Krivine-basis} representation for proving polynomial inequalities. Specifically, we formulate the inequality proving problem as a linear programming (LP) problem and encode it as a basis selection problem using reinforcement learning (RL), achieving a non-negative {Krivine basis}. Moreover, a fast multivariate polynomial multiplication method based on Fast Fourier Transform (FFT) is employed to enhance the efficiency of action space search. Furthermore, we have implemented a tool called {APPIRL} (Automated Proof of Polynomial Inequalities via Reinforcement Learning). Experimental evaluation on benchmark problems demonstrates the feasibility and effectiveness of our approach. In addition, {APPIRL} has been successfully applied to solve the maximum stable set problem.

This paper explores the application of reinforcement learning to optimize the parameters of a Type-1 Takagi-Sugeno fuzzy controller, designed to operate as an artificial pancreas for Type 1 diabetes. The primary challenge in diabetes management is the dynamic nature of blood glucose levels, which are influenced by several factors such as meal intake and timing. Traditional controllers often struggle to adapt to these changes, leading to suboptimal insulin administration. To address this issue, we employ a reinforcement learning agent tasked with adjusting 27 parameters of the Takagi-Sugeno fuzzy controller at each time step, ensuring real-time adaptability. The study's findings demonstrate that this approach significantly enhances the robustness of the controller against variations in meal size and timing, while also stabilizing glucose levels with minimal exogenous insulin. This adaptive method holds promise for improving the quality of life and health outcomes for individuals with Type 1 diabetes by providing a more responsive and precise management tool. Simulation results are given to highlight the effectiveness of the proposed approach.

Near-field communication with large antenna arrays promises significant beamforming and multiplexing gains. These communication links, however, are very sensitive to user mobility as any small change in the user position may suddenly drop the signal power. This leads to critical challenges for the robustness of these near-field communication systems. In this paper, we propose \textit{sphere precoding}, which is a robust precoding design to address user mobility in near-field communications. To gain insights into the spatial correlation of near-field channels, we extend the one-ring channel model to what we call one-sphere channel model and derive the channel covariance considering user mobility. Based on the one-sphere channel model, a robust precoding design problem is defined to optimize the minimum signal-to-interference-plus-noise ratio (SINR) satisfaction probability among mobile users. By utilizing the eigen structure of channel covariance, we further design a relaxed convex problem to approximate the solution of the original non-convex problem. The low-complexity solution effectively shapes a sphere that maintains the signal power for the target user and also nulls its interference within spheres around the other users. Simulation results highlight the efficacy of the proposed solution in achieving robust precoding yet high achievable rates in near-field communication systems.

This paper describes recursive algorithms for state estimation of linear dynamical systems when measurements are noisy with unknown bias and/or outliers. For situations with noisy and biased measurements, algorithms are proposed that minimize $\epsilon$ insensitive loss function. In this approach which is often used in Support Vector Machines, small errors are ignored making the algorithm less sensitive to measurement bias. Apart from $\epsilon$ insensitive quadratic loss function, estimation algorithms are also presented for $\epsilon$ insensitive Huber M loss function which provides good performance in presence of both small noises as well as outliers. The advantage of Huber cost function based estimator in presence of outliers is due to the fact the error penalty function switches from quadratic to linear for errors beyond a certain threshold. For both objective functions, estimation algorithms are extended to cases when there are additional constraints on states and exogenous signals such as known range of some states or exogenous signals or measurement noises. Interestingly, the filtering algorithms are recursive and structurally similar to Kalman filter with the main difference being that the updates based on the new measurement ("innovation term") are based on solution of a quadratic optimization problem with linear constraints.

Given a finite collection of stochastic alternatives, we study the problem of sequentially allocating a fixed sampling budget to identify the optimal alternative with a high probability, where the optimal alternative is defined as the one with the smallest value of extreme tail risk. We particularly consider a situation where these alternatives generate heavy-tailed losses whose probability distributions are unknown and may not admit any specific parametric representation. In this setup, we propose data-driven sequential sampling policies that maximize the rate at which the likelihood of falsely selecting suboptimal alternatives decays to zero. We rigorously demonstrate the superiority of the proposed methods over existing approaches, which is further validated via numerical studies.

Several important dynamical systems are in $\mathbb{R}^2$, defined by $(x',y')=(f(x,y),g(x,y))$. A question of fundamental importance is how such systems might behave quantum mechanically. In developing quantum theory, Dirac and others realized that classical Hamiltonian systems can be mapped to their quantum counterparts via canonical quantization. The resulting quantum dynamics is always physical, characterized by completely-positive and trace-preserving evolutions in the Schr\"odinger picture. However, whether non-Hamiltonian systems can be quantized systematically while respecting the same physical requirements has remained a long-standing problem. Here we resolve this question when $f(x,y)$ and $g(x,y)$ are arbitrary polynomials. By leveraging open-systems theory, we prove constructively that every polynomial system admits a physical generator of time evolution in the form of a Lindbladian. We refer to our method as cascade quantization, and demonstrate its power by analyzing several paradigmatic examples of nonlinear dynamics such as bifurcations, noise-activated spiking, and Li\'{e}nard systems. In effect, any classical system whose $f(x,y)$ and $g(x,y)$ are analytic functions can be quantized with arbitrary precision. Crucially, our method is exact. Being free from any approximations, cascade quantization dispenses with simplifying assumptions such as the weakly-nonlinear limit, or semiclassical dynamics in the quantized system -- both of which have been critical in facilitating quantization in the literature. We also highlight the advantages of cascade quantization over the existing proposals, by weighing it against examples from the variational paradigm using Lagrangians, as well as non-variational approaches.

Motivated by the paradigm of a super-Maltusian population catastrophe, we study a simple stochastic population model which exhibits a finite-time blowup of the population size and is strongly affected by intrinsic noise. We focus on the fluctuations of the blowup time $T$ in the asexual binary reproduction model $2A \to 3A$, where two identical individuals give birth to a third one. We determine exactly the average blowup time as well as the probability distribution $\mathcal{P}(T)$ of the blowup time and its moments. In particular, we show that the long-time tail $\mathcal{P}(T\to \infty)$ is purely exponential. The short-time tail $\mathcal{P}(T\to 0)$ exhibits an essential singularity at $T=0$, and it is dominated by a single (the most likely) population trajectory which we determine analytically.

We propose a novel approach to the analysis of programmable geometrically exact shear deformable beam systems made of shape memory polymers. The proposed method combines the viscoelastic Generalized Maxwell model with the Williams, Landel and Ferry relaxation principle, enabling the reproduction of the shape memory effect of structural systems featuring complex geometry and topology. Very high efficiency is pursued by discretizing the differential problem in space through the isogeometric collocation (IGA-C) method. The method, in addition to the desirable attributes of isogeometric analysis (IGA), such as exactness of the geometric reconstruction of complex shapes and high-order accuracy, circumvents the need for numerical integration since it discretizes the problem in the strong form. Other distinguishing features of the proposed formulation are: i) ${\rm SO}(3)$-consistency for the linearization of the problem and for the time stepping; ii) minimal (finite) rotation parametrization, that means only three rotational unknowns are used; iii) no additional unknowns are needed to account for the rate-dependent material compared to the purely elastic case. Through different numerical applications involving challenging initial geometries, we show that the proposed formulation possesses all the sought attributes in terms of programmability of complex systems, geometric flexibility, and high order accuracy.

In the Subset Feedback Arc Set in Tournaments, Subset-FAST problem we are given as input a tournament $T$ with a vertex set $V(T)$ and an arc set $A(T)$, along with a terminal set $S \subseteq V(T)$, and an integer $ k$. The objective is to determine whether there exists a set $ F \subseteq A(T) $ with $|F| \leq k$ such that the resulting graph $T-F $ contains no cycle that includes any vertex of $S$. When $S=V(T)$ this is the classic Feedback Arc Set in Tournaments (FAST) problem. We obtain the first polynomial kernel for this problem parameterized by the solution size. More precisely, we obtain an algorithm that, given an input instance $(T, S, k)$, produces an equivalent instance $(T',S',k')$ with $k'\leq k$ and $V(T')=O(k^2)$. It was known that FAST admits a simple quadratic vertex kernel and a non-trivial linear vertex kernel. However, no such kernel was previously known for Subset-FAST. Our kernel employs variants of the most well-known reduction rules for FAST and introduces two new reduction rules to identify irrelevant vertices. As a result of our kernelization, we also obtain the first sub-exponential time FPT algorithm for Subset-FAST.

Mathematical models of epidemics often use compartmental models dividing the population into several compartments. Based on a microscopic setting describing the temporal evolution of the subpopulation sizes in the compartments by stochastic counting processes one can derive macroscopic models for large populations describing the average behavior by associated ordinary differential equations such as the celebrated SIR model. Further, diffusion approximations allow to address fluctuations from the average and to describe the state dynamics also for smaller populations by stochastic differential equations. In general, not all state variables are directly observable, and we face the so-called "dark figure" problem, which concerns, for example, the unknown number of asymptomatic and undetected infections. The present study addresses this problem by developing stochastic epidemic models that incorporate partial information about the current state of the epidemic, also known as nowcast uncertainty. Examples include a simple extension of the SIR model, a model for a disease with lifelong immunity after infection or vaccination, and a Covid-19 model. For the latter, we propose a ``cascade state approach'' that allows to exploit the information contained in formally hidden compartments with observable inflow but unobservable outflow. Furthermore, parameter estimation and calibration are performed using ridge regression for the Covid-19 model. The results of the numerical simulations illustrate the theoretical findings.

We apply statistical analysis to search for processes responsible for turbulence in physical systems. In our previous studies, we have shown that solar wind turbulence in the inertial range of large magnetohydrodynamic scales exhibits Markov properties. We have recently extended this approach on much smaller kinetic scales. Here we are testing for the Markovian character of stochastic processes in a kinetic regime based on magnetic field and velocity fluctuations in the solar wind, measured onboard the Magnetospheric Multiscale (MMS) mission: behind the bow shock, inside the magnetosheath, and near the magnetopause. We have verified that the Chapman-Kolmogorov necessary conditions for Markov processes is satisfied for local transfer of energy between the magnetic and velocity fields also on kinetic scales. We have confirmed that for magnetic fluctuations, the first Kramers-Moyal coefficient is linear, while the second term is quadratic, corresponding to drift and diffusion processes in the resulting Fokker-Planck equation. It means that magnetic self-similar turbulence is described by generalized Ornstein-Uhlenbeck processes. We show that for the magnetic case, the Fokker-Planck equation leads to the probability density functions of the kappa distributions, which exhibit global universal scale invariance with a linear scaling and lack of intermittency. On the contrary, for velocity fluctuations, higher order Kramers-Moyal coefficients should be taken into account and hence scale invariance is not observed. However, the nonextensity parameter in Tsallis entropy provides a robust measure of the departure of the system from equilibrium. The obtained results are important for a better understanding of the physical mechanism governing turbulent systems in space and laboratory.

We investigate the evolution of anisotropies in Einstein-Gauss-Bonnet theory with a scalar field coupled to the Gauss-Bonnet term. Specifically, we examine the simplest scenario in which the scalar field lacks a kinetic term, and its kinetic contribution arises from an integration by parts of the Gauss-Bonnet scalar. We consider four- and five-dimensional anisotropic spacetimes, focusing on Bianchi I and extended Bianchi I geometries. Our study reveals that the asymptotic solutions correspond to locally symmetric spacetimes where at least two scale factors exhibit analogous behavior or, alternatively, to isotropic configurations where all scale factors evolve identically. Additionally, we discuss the effects of a cosmological constant, finding that the presence of the cosmological constant does not lead to an isotropic universe.

The complexity of solving equations over finite groups has been an active area of research over the last two decades, starting with Goldmann and Russell, \emph{The complexity of solving equations over finite groups} from 1999. One important case of a group with unknown complexity is the symmetric group $S_4.$ In 2023, Idziak, Kawa{\l}ek, and Krzaczkowski published $\exp(\Omega(\log^2 n))$ lower bounds for the satisfiability and equivalence problems over $S_4$ under the Exponential Time Hypothesis. In the present note, we prove that the satisfiability problem $\textsc{PolSat}(S_4)$ can be reduced to the equivalence problem $\textsc{PolEqv}(S_4)$ and thus, the two problems have the same complexity. We provide several equivalent formulations of the problem. In particular, we prove that $\textsc{PolEqv}(S_4)$ is equivalent to the circuit equivalence problem for $\operatorname{CC}[2,3,2]$-circuits, which were introduced by Idziak, Kawe{\l}ek and Krzaczkowski. Under their strong exponential size hypothesis, such circuits cannot compute $\operatorname{AND}_n$ in size $\exp(o(\sqrt{n})).$ Our results provide an upper bound on the complexity of $\textsc{PolEqv}(S_4)$ that is based on the minimal size of $\operatorname{AND}_n$ over $\operatorname{CC}[2,3,2]$-circuits.

The theory of argumentation frameworks ($AF$s) has been a useful tool for artificial intelligence. The research of the connection between $AF$s and logic is an important branch. This paper generalizes the encoding method by encoding $AF$s as logical formulas in different propositional logic systems. It studies the relationship between models of an AF by argumentation semantics, including Dung's classical semantics and Gabbay's equational semantics, and models of the encoded formulas by semantics of propositional logic systems. Firstly, we supplement the proof of the regular encoding function in the case of encoding $AF$s to the 2-valued propositional logic system. Then we encode $AF$s to 3-valued propositional logic systems and fuzzy propositional logic systems and explore the model relationship. This paper enhances the connection between $AF$s and propositional logic systems. It also provides a new way to construct new equational semantics by choosing different fuzzy logic operations.

A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every long edge is strictly longer than every short edge. We call a graph $G$ pandichotomous in $X$ if $G$ admits a geometric realization in $X$ for every partition of its edge set into short and long edges. We exhibit a very close relationship between the degeneracy of a graph $G$ and its pandichotomic Euclidean or spherical dimension, that is, the smallest dimension $k$ such that $G$ is pandichotomous in $\mathbb{R}^k$ or the sphere $\mathbb{S}^k$, respectively. First, every $d$-degenerate graph is pandichotomous in $\mathbb{R}^{d}$ and $\mathbb{S}^{d-1}$ and these bounds are tight for the sphere and for $\mathbb{R}^2$ and almost tight for $\mathbb{R}^d$, for $d\ge 3$. Second, every $n$-vertex graph that is pandichotomous in $\mathbb{R}^k$ has at most $\mu kn$ edges, for some absolute constant $\mu<7.23$. This shows that the pandichotomic Euclidean dimension of any graph is linearly tied to its degeneracy and in the special cases $k\in \{1,2\}$ resolves open problems posed by Alam, Kobourov, Pupyrev, and Toeniskoetter. Further, we characterize which complete bipartite graphs are pandichotomous in $\mathbb{R}^2$: These are exactly the $K_{m,n}$ with $m\le 3$ or $m=4$ and $n\le 6$. For general bipartite graphs, we can guarantee realizations in $\mathbb{R}^2$ if the short or the long subgraph is constrained: namely if the short subgraph is outerplanar or a subgraph of a rectangular grid, or if the long subgraph forms a caterpillar.

A seminal result of [Fleischer et al. and Karakostas and Kolliopulos, both FOCS 2004] states that system optimal multi-commodity static network flows are always implementable as tolled Wardrop equilibrium flows even if users have heterogeneous value-of-time sensitivities. Their proof uses LP-duality to characterize the general implementability of network flows by tolls. For the much more complex setting of $\textit{dynamic flows}$, [Graf et al., SODA 2025] identified necessary and sufficient conditions for a dynamic $s$-$d$ flow to be implementable as a tolled dynamic equilibrium. They used the machinery of (infinite-dimensional) strong duality to obtain their characterizations. Their work, however, does not answer the question of whether system optimal dynamic network flows are implementable by tolls. We consider this question for a general dynamic flow model involving multiple commodities with individual source-destination pairs, fixed inflow rates and heterogeneous valuations of travel time and money spent. We present both a positive and a, perhaps surprising, negative result: For the negative result, we provide a network with multiple source and destination pairs in which under the Vickrey queuing model no system optimal flow is implementable -- even if all users value travel times and spent money the same. Our counter-example even shows that the ratio of the achievable equilibrium travel times by using tolls and of the system optimal travel times can be unbounded. For the single-source, single-destination case, we show that if the traversal time functions are suitably well-behaved (as is the case, for example, in the Vickrey queuing model), any system optimal flow is implementable.

Autonomous navigation in intelligent mobile systems represents a core research focus within artificial intelligence-driven robotics. Contemporary path planning approaches face constraints in dynamic environmental responsiveness and multi-objective task scalability, limiting their capacity to address growing intelligent operation requirements. Decision-centric reinforcement learning frameworks, capitalizing on their unique strengths in adaptive environmental interaction and self-optimization, have gained prominence in advanced control system research. This investigation introduces methodological improvements to address sample homogeneity challenges in reinforcement learning experience replay mechanisms. By incorporating determinant point processes (DPP) for diversity assessment, we develop a dual-criteria sampling framework with adaptive selection protocols. This approach resolves representation bias in conventional prioritized experience replay (PER) systems while preserving algorithmic interoperability, offering improved decision optimization for dynamic operational scenarios. Key contributions comprise: Develop a hybrid sampling paradigm (PER-DPP) combining priority sequencing with diversity maximization.Based on this,create an integrated optimization scheme (PER-DPP-Elastic DQN) merging diversity-aware sampling with adaptive step-size regulation. Comparative simulations in 2D navigation scenarios demonstrate that the elastic step-size component temporarily delays initial convergence speed but synergistically enhances final-stage optimization with PER-DPP integration. The synthesized method generates navigation paths with optimized length efficiency and directional stability.

Knowing the sea surface velocity field is essential for various applications, such as search and rescue operations and oil spill monitoring, where understanding the movement of objects or substances is critical. However, obtaining an accurate approximation of these advection processes is challenging, even with modern measuring equipment, such as high-frequency radar or advanced simulations based on oceanic flow models. Therefore this paper presents a data-driven framework to approximate sea surface velocity from spatially distributed observations, thus enabling efficient probability advection modeling across submesoscale domains. The system approximates transient flows by leveraging quasi-steady flow assumptions. To overcome the limitations of point measurements in capturing domain-wide circulation, the method employs a fusion of two simplified 2D flow models to approximate submesoscale dynamics, enabling complete velocity field reconstruction from scattered data. To ensure reliable flow dynamics, the approach iteratively adjusts boundary conditions in numerical simulations to align the simulated flow with observations. Experimental validation in Kvarner Bay using GPS-tracked drifters confirmed the system's ability to replace computationally intensive transient simulations by approximating flow fields based on model simplifications. The results demonstrate its efficiency across domains, making it a practical tool for real-world submesoscale applications requiring swift passive scalar advection.

Language model alignment (or, reinforcement learning) techniques that leverage active exploration -- deliberately encouraging the model to produce diverse, informative responses -- offer the promise of super-human capabilities. However, current understanding of algorithm design primitives for computationally efficient exploration with language models is limited. To better understand how to leverage access to powerful pre-trained generative models to improve the efficiency of exploration, we introduce a new computational framework for RL with language models, in which the learner interacts with the model through a sampling oracle. Focusing on the linear softmax model parameterization, we provide new results that reveal the computational-statistical tradeoffs of efficient exploration: 1. Necessity of coverage: Coverage refers to the extent to which the pre-trained model covers near-optimal responses -- a form of hidden knowledge. We show that coverage, while not necessary for data efficiency, lower bounds the runtime of any algorithm in our framework. 2. Inference-time exploration: We introduce a new algorithm, SpannerSampling, which obtains optimal data efficiency and is computationally efficient whenever the pre-trained model enjoys sufficient coverage, matching our lower bound. SpannerSampling leverages inference-time computation with the pre-trained model to reduce the effective search space for exploration. 3. Insufficiency of training-time interventions: We contrast the result above by showing that training-time interventions that produce proper policies cannot achieve similar guarantees in polynomial time. 4. Computational benefits of multi-turn exploration: Finally, we show that under additional representational assumptions, one can achieve improved runtime (replacing sequence-level coverage with token-level coverage) through multi-turn exploration.

The precision, stability, and performance of lightweight high-strength steel structures in heavy machinery is affected by their highly nonlinear dynamics. This, in turn, makes control more difficult, simulation more computationally intensive, and achieving real-time autonomy, using standard approaches, impossible. Machine learning through data-driven, physics-informed and physics-inspired networks, however, promises more computationally efficient and accurate solutions to nonlinear dynamic problems. This study proposes a novel framework that has been developed to estimate real-time structural deflection in hydraulically actuated three-dimensional systems. It is based on SLIDE, a machine-learning-based method to estimate dynamic responses of mechanical systems subjected to forced excitations.~Further, an algorithm is introduced for the data acquisition from a hydraulically actuated system using randomized initial configurations and hydraulic pressures.~The new framework was tested on a hydraulically actuated flexible boom with various sensor combinations and lifting various payloads. The neural network was successfully trained in less time using standard parameters from PyTorch, ADAM optimizer, the various sensor inputs, and minimal output data. The SLIDE-trained neural network accelerated deflection estimation solutions by a factor of $10^7$ in reference to flexible multibody simulation batches and provided reasonable accuracy. These results support the studies goal of providing robust, real-time solutions for control, robotic manipulators, structural health monitoring, and automation problems.

We develop methods for systematic construction of superintegrable polynomials in matrix/eigenvalue models. Our consideration is based on a tight connection of superintegrable property of Gaussian Hermitian model and $W_{1 + \infty}$ algebra in Fock representation. Motivated by this example, we propose a set of assumptions that may allow one to recover superintegrable polynomials. The main two assumptions are box adding/removing rule (Pierri rule) and existence of Hamiltonian for superintegrable polynomials. We detail our method in case of the Gaussian Hermitian model, and then apply it to the cubic Kontsevich model.

The triad refers to embedding of two systems of polynomials, symmetric ones and those of the Baker-Akhiezer type into a power series of the Noumi-Shiraishi type. It provides an alternative definition of Macdonald theory and its extensions. The basic triad is associated with the vector representation of the Ding-Iohara-Miki (DIM) algebra. We discuss lifting this triad to two elliptic generalizations and further to the bi-elliptic triad. At the algebraic level, it corresponds to elliptic and bi-elliptic DIM algebras. This completes the list of polynomials associated with Seiberg-Witten theory with adjoint matter in various dimensions.

This paper proposes a novel analysis for the Scaffold algorithm, a popular method for dealing with data heterogeneity in federated learning. While its convergence in deterministic settings--where local control variates mitigate client drift--is well established, the impact of stochastic gradient updates on its performance is less understood. To address this problem, we first show that its global parameters and control variates define a Markov chain that converges to a stationary distribution in the Wasserstein distance. Leveraging this result, we prove that Scaffold achieves linear speed-up in the number of clients up to higher-order terms in the step size. Nevertheless, our analysis reveals that Scaffold retains a higher-order bias, similar to FedAvg, that does not decrease as the number of clients increases. This highlights opportunities for developing improved stochastic federated learning algorithms