Severi varieties are the parameter spaces for curves with prescribed homology class and genus on a smooth surface. We describe their limits along degenerations of surfaces, with a view towards the enumeration of curves. This includes a complete proof of the Caporaso-Harris recursive formula, with all the necessary background on deformations of curves and singularities.

We present a self-contained analysis of infinity from two mathematical perspectives: set theory and algebra. We begin with cardinal and ordinal numbers, examining deep questions such as the continuum hypothesis, along with foundational results such as the Schr\"oder-Bernstein theorem, multiple proofs of the well-ordering of cardinals, and various properties of infinite cardinals and ordinals. Transitioning to algebra, we analyze the interplay between finite and infinite algebraic structures, including groups, rings, and $R$-modules. Major results, such as the fundamental theorem of finitely generated abelian groups, Krull's Theorem, Hilbert's basis theorem, and the equivalence of free and projective modules over principal ideal domains, highlight the connections and differences between finite and infinite structures, as well as demonstrating the relationship between set-theoretic and algebraic treatments of infinity. Through this approach, we provide insights into how key results about infinity interact with and inform one another across set-theoretic and algebraic mathematics.

By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.

The author prove that there exists a function $\rho(n)$ which is a minorant for the prime indicator function $\mathbb{1}_{p}(n)$ and has distribution level $\frac{65}{123}$ in arithmetic progressions to smooth moduli. This refines the previous results of Baker--Irving and Stadlmann.

We investigate some classes of infinite series involving central binomial coefficients, particularly focusing on those arising from ratios such as $\binom{2n}{n}\binom{4n}{2n}^{-1}$,$\binom{4n}{2n}\binom{2n}{n}^{-1}$ and related expressions. We derive several new explicit identities and closed-form evaluations, building on and refining previous results by Bhandari (2022) and Adegoke et al. (2022).

The author proves that for $0.9985 < \gamma < 1$, there exist infinitely many primes $p$ such that $[p^{1/\gamma}]$ has at most 5 prime factors counted with multiplicity. This gives an improvement upon the previous results of Banks-Guo-Shparlinski and Xue-Li-Zhang.

Let $\chi(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring $\mathbb{Z}[x]/(\chi(x))$. We obtain formulas for the orders of these objects, and study their asymptotic behavior as the discriminant of $\chi(x)$ tends to infinity, in analogy with the Brauer-Siegel theorem. Finally, we describe the structure of the ideal class group when the degree of $\chi(x)$ is $2$ or $3$.

We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive expressions for these sums. In addition, we offer an alternative approach to express one class of sums and some related sums in closed form in terms of Stirling numbers and r-Stirling numbers of the second kind.

In this article, we obtain an explicit version of Heath-Brown's large sieve inequality for quadratic characters and discuss its applications to $L$-functions and quadratic fields.

For integers $k \geq 2$, the $k$-generalized Lucas sequence $\{L_n^{(k)}\}_{n \geq 2-k}$ is defined by the recurrence relation \[ L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \quad \text{for } n \geq 2, \] with initial terms given by $L_0^{(k)} = 2$, $L_1^{(k)} = 1$, and $L_{2-k}^{(k)} = \cdots = L_{-1}^{(k)} = 0$. In this paper, we extend work in \cite{Lucas} and show that the result in \cite{Lucas} still holds for $k\ge 3$, that is, we show that for $k\ge 3$, there is no $k$-generalized Lucas number appearing as a palindrome formed by concatenating two distinct repdigits.

In this survey, we studied the possibility of finding rational solutions to the equation $Ax^p+By^p+Cz^p=0$ via its attached hyperelliptic curve $Y^2=X^p+A^2(BC)^{p-1}/4$ and its rational points computed using computational tools.

Semantic communication (SemCom) aims to convey the intended meaning of messages rather than merely transmitting bits, thereby offering greater efficiency and robustness, particularly in resource-constrained or noisy environments. In this paper, we propose a novel framework which is referred to as joint source-channel noise adding with adaptive denoising (JSCNA-AD) for SemCom based on a diffusion model (DM). Unlike conventional encoder-decoder designs, our approach intentionally incorporates the channel noise during transmission, effectively transforming the harmful channel noise into a constructive component of the diffusion-based semantic reconstruction process. Besides, we introduce an attention-based adaptive denoising mechanism, in which transmitted images are divided into multiple regions, and the number of denoising steps is dynamically allocated based on the semantic importance of each region. This design effectively balances the reception quality and the inference latency by prioritizing the critical semantic information. Extensive experiments demonstrate that our method significantly outperforms existing SemCom schemes under various noise conditions, underscoring the potential of diffusion-based models in next-generation communication systems.

We establish a Tauberian theorem connecting the unknown asymptotic behavior of the partial sums $\sum_{n\le x}a_{n}$ to the known asymptotics of weighted sums $\sum_{n\le x}a_{n}g(n/x)$, as $x\rightarrow\infty$, where $g:(0,1]\to\mathbb{R}$ is a given function. Our approach relies on an identity relating a modified Mellin transform of $g$ to the Dirichlet series $\sum_{n\ge1}a_{n}n^{-s}$. As an application, we solve an open problem posed by Kalmynin and Kosenko regarding the "orthorecursive expansion of unity" associated with a sequence $(c_{n})_{n\geq0}$. Specifically, we improve their partial-sum bound $C_{N}=\sum_{0\leq n\leq N}c_{n}=\mathcal{O}(N^{-1/2})$, by obtaining the optimal estimate $C_{N}=\mathcal{O}(N^{-\alpha_{1}+\epsilon})$, where $\alpha_{1}\approx1.3465165$ is the smallest real part among the zeros of a transcendental function related to the digamma function.

We prove that if $A$ is a subset of those primes which are congruent to $1 \pmod{3}$ such that the relative density of $A$ in this residue class is larger than $\frac{1}{2},$ then every sufficiently large odd integer $n$ which satisfies $n \equiv 0 \pmod{3}$ can be written as a sum of three primes from $A.$ Moreover the threshold of $\frac{1}{2}$ for the relative density is best possible.

We investigate the validity of the synthetic Ricci curvature lower bound known as the measure contraction property (MCP) for sub-Riemannian structures beyond step two. We show that whenever the distance function is not Lipschitz in charts, the MCP may fail. This occurs already in fundamental examples such as the Martinet and Engel structures. Central to our analysis are new results on the stability of the local MCP under quotients by isometric group actions for general metric measure spaces, developed under a weaker variant of the essential non-branching condition which, in contrast with the classical one, is implied by the minimizing Sard property in sub-Riemannian geometry. Since the MCP is preserved under blow-ups, we focus on Carnot homogeneous spaces, proving that MCP descends to suitable quotients. As a byproduct, any structure whose tangent at some point admits a quotient to Martinet fails the MCP. We also obtain a computation-free proof that the Grushin plane shares the Heisenberg group's MCP. Applications include a detailed analysis of validity and failure of the MCP for Carnot groups of low dimension. Our results suggest a conjecture on the failure of the MCP in presence of Goh abnormal geodesics satisfying the strong generalized Legendre condition.

This paper investigates the symbolic powers of toric ideals. We first describe them in terms of the kernel of certain linear maps derived from the lattice structure of the toric ideal. Furthermore, we apply our results to show that symbolic powers of a toric ideal can also be expressed as saturations of regular powers with the monomial given by the product of all the variables. Finally, we conclude with a computationally significant result for computing symbolic powers of toric ideals.

The paper is devoted to the spreading of a message within the random graph evolving by the Norros-Reittu preferential attachment model. The latter model forms random Poissonian numbers of edges between newly added nodes and existing ones. For a pre-fixed time $T^*$, the probability mass functions of the number of nodes obtained the message and the total number of nodes in the graph, as well as the distribution function of their ratio are derived. To this end, the success probability to disseminate the message from the node with the message to the node without message is proved. The exposition is illustrated by the simulation study.

Let $G$ be a connected graph of genus $g$. The Picard group of degree $g$, $\text{Pic}^g(G)$, is the set of equivalence classes of divisors on $G$ of degree $g$, where two divisors are equivalent if one can be reached from the other through a sequence of chip-firing moves. We construct sets of representatives of the equivalence classes in $\text{Pic}^g(G)$ by defining a function $I_G$ on the spanning trees of $G$ from a triangulation of the Lawrence polytope of the cographic matroid $\mathcal{M}^\ast(G)$. Additionally, such sets of representatives correspond to stability conditions on the nodal curve dual to the graph $G$. We show that $I_G$ that are constructed from regular triangulations of Lawrence polytope correspond to classical stability conditions, which are induced by generic real-valued divisors on $G$.

We solve optimal stopping for multidimensional Brownian motion in a bounded domain, a question raised in Dynkin and Yushkevich (1967), where the one-dimensional case was presented. Taking a geometric approach, under regularity conditions we construct the optimal stopping free boundary in the multidimensional case. We characterise the value function as the pointwise infimum of potentials with recursive extensions dominating the gain function, and obtain its continuity. Explicit examples illustrate the result.

Fast Ewald summation is the most widely used approach for computing long-range Coulomb interactions in molecular dynamics (MD) simulations. While the asymptotic scaling is nearly optimal, its performance on parallel architectures is dominated by the global communication required for the underlying fast Fourier transform (FFT). Here, we develop a novel method, ESP - Ewald summation with prolate spheroidal wave functions (PSWFs) - that, for a fixed precision, sharply reduces the size of this transform by performing the Ewald split via a PSWF. In addition, PSWFs minimize the cost of spreading and interpolation steps that move information between the particles and the underlying uniform grid. We have integrated the ESP method into two widely-used open-source MD packages: LAMMPS and GROMACS. Detailed benchmarks show that this reduces the cost of computing far-field electrostatic interactions by an order of magnitude, leading to better strong scaling with respect to number of cores. The total execution time is reduced by a factor of 2 to 3 when using more than one thousand cores, even after optimally tuning the existing internal parameters in the native codes. We validate the accelerated codes in realistic long-time biological simulations.

Let $\sigma$ be a simplicial triangulation of the 2-sphere, $X$ the associated integral 2-cycle. A filling of $X$ is an integral 3-chain $Y$ with $\partial Y = X$; a taut filling is one with minimal $L_1$-norm. We show that any taut filling arises from an extension of $\sigma$ to a shellable simplicial triangulation of the 3-ball. The key to the proof is the general fact that any taut filling of an $n$-cycle splits under disjoint union, connected sum, and more generally what we call almost disjoint union, where summands are supported on sets that overlap in at most $n+1$ vertices. Despite the generality of this result, we have nothing to say about optimal fillings of spheres of dimension 3 or higher.

In this note we present a brief overview of connections between Chern-Simons theory and topological strings. A prominent role in this link has been played by large N dualities and holography. We demystify this by explaining why the Kahler form should be viewed as dual to the field strength associated with a 3-form gauge potential, sourced by Lagrangian D-branes. We explain how this leads to the computation of topological string amplitudes in terms of topological vertex for toric Calabi-Yau threefolds. Furthermore, applications of topological strings to a conceptual derivation of Skein relations for link invariants as well as some of its physical applications to black hole physics are also reviewed.

In this paper, we consider a nonlinear parabolic equation in the one dimensional interval. The equation was proposed as a mathematical model for moisture transport in porous media. Its features are that the diffusion coefficient depends on the unknown function and the boundary condition is not monotone. Namely, our equation is regarded as the pseudo monotone type. The aim of this paper is to prove the convergence of approximate solutions constructed by the finite volume method. Also, we establish the uniqueness of solutions by the dual equation method.

We show that a broad range of convex optimization algorithms, including alternating projection, operator splitting, and multiplier methods, can be systematically derived from the framework of subspace correction methods via convex duality. To formalize this connection, we introduce the notion of dualization, a process that transforms an iterative method for the dual problem into an equivalent method for the primal problem. This concept establishes new connections across these algorithmic classes, encompassing both well-known and new methods. In particular, we show that classical algorithms such as the von Neumann, Dykstra, Peaceman--Rachford, and Douglas--Rachford methods can be interpreted as dualizations of subspace correction methods applied to appropriate dual formulations. Beyond unifying existing methods, our framework enables the systematic development of new algorithms for convex optimization. For instance, we derive parallel variants of alternating projection and operator splitting methods, as dualizations of parallel subspace correction methods, that are well-suited for large-scale problems on modern computing architectures and offer straightforward convergence guarantees. We also propose new alternating direction method of multipliers-type algorithms, derived as dualizations of certain operator splitting methods. These algorithms naturally ensure convergence even in the multi-block setting, where the conventional method does not guarantee convergence when applied to more than two blocks. This unified perspective not only facilitates algorithm design and the transfer of theoretical results but also opens new avenues for research and innovation in convex optimization.

This work presents a methodology for reconstructing the spatial distribution of the neutron flux in a nuclear reactor, leveraging real-time measurements obtained from ex-core detectors. The Kirchhoff-Helmholtz (K-H) equation inherently defines the problem of estimating a scalar field within a domain based on boundary data, making it a natural mathematical framework for this task. The main challenge lies in deriving the Green's function specific to the domain and the neutron diffusion process. While analytical solutions for Green's functions exist for simplified geometries, their derivation of complex, heterogeneous domains-such as a nuclear reactor-requires a numerical approach. The objective of this work is to demonstrate the well-posedness of the data-driven Green's function approximation by formulating and solving the K-H equation as an inverse problem. After establishing the symmetry properties that the Green's function must satisfy, the K-H equation is derived from the one-speed neutron diffusion model. This is followed by a comprehensive description of the procedure for interpreting sensor readings and implementing the neutron flux reconstruction algorithm. Finally, the existence and uniqueness of the Green's function inferred from the sampled data are demonstrated, ensuring the reliability of the proposed method and its predictions.

We present an efficient self-contained algorithm for computing the modified Bessel function of the first kind $I_{\nu}(z)$, implemented in a robust Fortran code supporting double and quadruple (quad) precision. The algorithm overcomes the limitations of Algorithm 644, which is restricted to double precision and applies overly conservative underflow and overflow thresholds, leading to failures in large parameter regions. Accuracy is validated against high-precision Maple calculations, and benchmarking shows execution time reductions to 54%-80% of Algorithm 644 (in double precision). Quad precision enhances numerical stability and broadens the domain of computations, making the implementation well suited for high-precision applications in physics and engineering. This work also provides a foundation for the development of efficient algorithms for other Bessel functions.

We discuss arithmetic questions related to the "poor man's ad\`ele ring" $\mathcal A$ whose elements are encoded by sequences $(t_p)_p$ indexed by prime numbers, with each $t_p$ viewed as an element in $\mathbb Z/p\mathbb Z$. Our main theorem is about the $\mathcal A$-transcendence of the element $(F_p(q))_p$, where $F_n(q)$ (Schur's $q$-Fibonacci numbers) are the $(1,1)$-entries of $2\times2$-matrices $$ \bigg(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q & 0 \end{matrix}\bigg) \bigg(\begin{matrix} 1 & 1 \\ q^2 & 0 \end{matrix}\bigg) \cdots \bigg(\begin{matrix} 1 & 1 \\ q^{n-2} & 0 \end{matrix}\bigg) $$ and $q>1$ is an integer. This result was previously known for $q>1$ square free under the GRH.

The bilevel variational inequality (BVI) problem is a general model that captures various optimization problems, including VI-constrained optimization and equilibrium problems with equilibrium constraints (EPECs). This paper introduces a first-order method for smooth or nonsmooth BVI with stochastic monotone operators at inner and outer levels. Our novel method, called Regularized Operator Extrapolation $(\texttt{R-OpEx})$, is a single-loop algorithm that combines Tikhonov's regularization with operator extrapolation. This method needs only one operator evaluation for each operator per iteration and tracks one sequence of iterates. We show that $\texttt{R-OpEx}$ gives $\mathcal{O}(\epsilon^{-4})$ complexity in nonsmooth stochastic monotone BVI, where $\epsilon$ is the error in the inner and outer levels. Using a mini-batching scheme, we improve the outer level complexity to $\mathcal{O}(\epsilon^{-2})$ while maintaining the $\mathcal{O}(\epsilon^{-4})$ complexity in the inner level when the inner level is smooth and stochastic. Moreover, if the inner level is smooth and deterministic, we show complexity of $\mathcal{O}(\epsilon^{-2})$. Finally, in case the outer level is strongly monotone, we improve to $\mathcal{O}(\epsilon^{-4/5})$ for general BVI and $\mathcal{O}(\epsilon^{-2/3})$ when the inner level is smooth and deterministic. To our knowledge, this is the first work that investigates nonsmooth stochastic BVI with the best-known convergence guarantees. We verify our theoretical results with numerical experiments.

Patient-specific computational modeling of the tricuspid valve (TV) is vital for the clinical assessment of heart valve diseases. However, this process is hindered by limitations inherent in the medical image data, such as noise and sparsity, as well as by complex valve dynamics. We present VALVEFIT, a novel GPU-accelerated and differentiable B-spline surface fitting framework that enables rapid reconstruction of smooth, analysis-suitable geometry from point clouds obtained via medical image segmentation. We start with an idealized TV B-spline template surface and optimize its control point positions to fit segmented point clouds via an innovative loss function, balancing shape fidelity and mesh regularization. Novel regularization terms are introduced to ensure that the surface remains smooth, regular, and intersection-free during large deformations. We demonstrate the robustness and validate the accuracy of the framework by first applying it to simulation-derived point clouds that serve as the ground truth. We further show its robustness across different point cloud densities and noise levels. Finally, we demonstrate the performance of the framework toward fitting point clouds obtained from real patients at different stages of valve motion. An isogeometric biomechanical valve simulation is then performed on the fitted surfaces to show their direct applicability toward analysis. VALVEFIT enables automated patient-specific modeling with minimal manual intervention, paving the way for the future development of direct image-to-analysis platforms for clinical applications.

In 1960, Ghouila-Houri proved that every strongly connected directed graph $G$ on $n$ vertices with minimum degree at least $n$ contains a directed Hamilton cycle. We asymptotically generalize this result by proving the following: every directed graph $G$ on $n$ vertices and with minimum degree at least $(1+o(1))n$ contains every orientation of a Hamilton cycle, except for the directed Hamilton cycle in the case when $G$ is not strongly connected. In fact, this minimum degree condition forces every orientation of a cycle in $G$ of every possible length, other than perhaps the directed cycles.

We study representations of $GL_{n}(\mathbb{F}_{q})$ that are distinguished with respect to a symmetric subgroup $H=GL_{n}(\mathbb{F}_{q})^{\sigma}$, where $\sigma$ is an involution. We prove that those representations satisfy $\pi \cong \pi ^{*,\sigma}$, thus positively answering a version of the Prasad-Lapid conjecture.

We study the arithmetic of Galois-invariant sets of points on algebraic curves with controlled reduction behavior. Let $C$ be a smooth projective curve with a smooth proper model $\mathcal{C}$ over $\mathcal{O}_{K,S}$. We define $\Omega_n$ as the set of $n$-element subsets of $C(\overline{K})$ that are invariant under $\text{Gal}(\overline{K}/K)$ and such that no two points in the set become identified modulo any prime $\mathfrak{p} \notin S$. Our main result establishes that $\Omega_n$ breaks into finitely many orbits under the action of $\text{Aut}_{\mathcal{O}_{K,S}}(\mathcal{C})$, generalizing finiteness theorems of Birch--Merriman, Siegel, and Faltings.

In this short note, we use flag algebras to prove that the number of colour alternating 6-cycles in a red/blue colouring of a large clique is asymptotically maximized by a uniformly random colouring. This settles the first open case of a problem of Basit, Granet, Horsley, K\"undgen and Staden.

A method is presented for the numerical solution of optimal boundary control problems governed by parabolic partial differential equations. The continuous space-time optimal control problem is transcribed into a sparse nonlinear programming problem through state and control parameterization. In particular, a multi-interval flipped Legendre-Gauss-Radau collocation method is implemented for temporal discretization alongside a Galerkin finite element spatial discretization. The finite element discretization allows for a reduction in problem size and avoids the redefinition of constraints required under a previous method. Further, a generalization of a Kirchoff transformation is performed to handle variational form nonlinearities in the context of numerical optimization. Due to the correspondence between the collocation points and the applied boundary conditions, the multi-interval flipped Legendre-Gauss-Radau collocation method is demonstrated to be preferable over the standard Legendre-Gauss-Radau collocation method for optimal control problems governed by parabolic partial differential equations. The details of the resulting transcription of the optimal control problem into a nonlinear programming problem are provided. Lastly, numerical examples demonstrate that the use of a multi-interval flipped Legendre-Gauss-Radau temporal discretization can lead to a reduction in the required number of collocation points to compute accurate values of the optimal objective in comparison to other methods.

We begin by deriving a number of combinatorial identities satisfied by the $q$-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the $q$-super Catalan numbers. Next, we introduce some $q$-convolution identities involving q-central binomial and q-Catalan numbers and derive a generating function for $q$-Catalan numbers. Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the $\gamma$-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural $q$-analogs, prove their $q$-$\gamma$-positivity and prove some identities they satisfy, generalizing identities of Kreweras and Le Jen-Shoo. Using yet another identity, we prove that these refinements are positive integer polynomials in $q$.

Cliquewidth is a dense analogue of treewidth. We further this analogy by showing that graphs of bounded cliquewidth admit a partition with `local, but dense' parts whose quotient has bounded treewidth. Specifically, each part is contained within the closed neighbourhood of some vertex. We use this to construct a quasi-isometry between graphs of bounded cliquewidth and bounded treewidth, and to show that graphs of bounded cliquewidth have Assouad-Nagata dimension 1.

The article offers a fresh perspective on Grzegorczyk logic Grz, introducing a simplified axiomatization and extending the analysis to its natural modal extensions, Grz.2 and Grz.3. I develop a control statement theory for these logics, utilizing concepts such as buttons, switches, and ratchets. Through this framework, I establish that Grz.2 is characterized by finite Boolean algebras.

We establish a density variant of the Frankl-R\"{o}dl theorem on the sphere $\mathbb{S}^{n-1}$, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish lower bounds on the probability that a randomly chosen pair of such vectors lies entirely within a measurable subset $A \subseteq \mathbb{S}^{n-1}$ of sufficiently large measure. Additionally, we prove a density version of spherical avoidance problems, which generalize from pairwise avoidance to broader configurations with prescribed pairwise inner products. Our framework encompasses a class of configurations we call inductive configurations, which include simplices with any prescribed inner product $-1 < r < 1$. As a consequence of our density statement, we show that all inductive configurations are sphere Ramsey.

We quantitatively relate the resonance sets of topologically finite infinite-area hyperbolic surfaces with no cusps to the resonance sets of certain metric graphs via the spine graph construction. In particular, we prove the existence of approximate resonance chains in resonance sets of these surfaces in the long-boundary-length regime. Our results are similar in spirit to those obtained in recent independent work by Li-Matheus-Pan-Tao, although our perspective and hypotheses are somewhat different. Our results also generalize older results obtained for three-funneled spheres by Weich. We primarily make use of transfer operators for holomorphic iterated function schemes, along with certain geometric bounds.

A valuation theory for superrings is developed, extending classical constructions from commutative algebra to the $\mathbb Z_2$-graded and supercommutative setting. We define valuations on superrings, investigate their fundamental properties, and explore the construction of Zariski-Riemann superspaces.

We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ over the ranges $0

This work introduces the High-Order Hermite Optimization (HOHO) method, an open-loop discrete adjoint method for quantum optimal control. Our method is the first of its kind to efficiently compute exact (discrete) gradients when using continuous, parameterized control pulses while solving the forward equations (e.g. Schrodinger's equation or the Linblad master equation) with an arbitrarily high-order Hermite Runge-Kutta method. The HOHO method is implemented in QuantumGateDesign.jl, an open-source software package for the Julia programming language, which we use to perform numerical experiments comparing the method to Juqbox.jl. For realistic model problems we observe speedups up to 775x.

The circular $\beta$ ensemble for $\beta =1,2$ and 4 corresponds to circular orthogonal, unitary and symplectic ensemble respectively as introduced by Dyson. The statistical state of the eigenvalues is then a determinantal point process ($\beta = 2$) and Pfaffian point process ($\beta = 1,4$). The explicit functional forms of the correlation kernels then imply that the general $n$-point correlation functions exhibit an asymptotic expansion in $1/N^2$, which moreover can be lifted to an asymptotic in $1/N^2$ for the spacing distributions and their generating function. We use $\sigma$-Painlev\'e characterisations to show that the functional form of the first correction is related to the leading term via a second derivative. Explicit functional forms are used to show that the spectral form factors for $\beta =1,2$ and 4 also admit an asymptotic expansion in $1/N^2$. Differential relations are identified expressing the first and second correction in terms of the limiting functional form, and evidence is presented that they hold for general $\beta$. For even $\beta$ it is proved that the two-point correlation function permits an asymptotic expansion in $1/N^2$, and moreover that the leading correction relates to the limiting functional form via a second derivative.

Recent work has shown that in certain settings, the Frank-Wolfe algorithm (FW) with open-loop step-sizes $\eta_t = \frac{\ell}{t+\ell}$ for a fixed parameter $\ell \in \mathbb{N},\, \ell \geq 2$, attains a convergence rate faster than the traditional $O(t^{-1})$ rate. In particular, when a strong growth property holds, the convergence rate attainable with open-loop step-sizes $\eta_t = \frac{\ell}{t+\ell}$ is $O(t^{-\ell})$. In this setting there is no single value of the parameter $\ell$ that prevails as superior. This paper shows that FW with log-adaptive open-loop step-sizes $\eta_t = \frac{2+\log(t+1)}{t+2+\log(t+1)}$ attains a convergence rate that is at least as fast as that attainable with fixed-parameter open-loop step-sizes $\eta_t = \frac{\ell}{t+\ell}$ for any value of $\ell \in \mathbb{N},\,\ell\geq 2$. To establish our main convergence results, we extend our previous affine-invariant accelerated convergence results for FW to more general open-loop step-sizes of the form $\eta_t = g(t)/(t+g(t))$, where $g:\mathbb{N}\to\mathbb{R}_{\geq 0}$ is any non-decreasing function such that the sequence of step-sizes $(\eta_t)$ is non-increasing. This covers in particular the fixed-parameter case by choosing $g(t) = \ell$ and the log-adaptive case by choosing $g(t) = 2+ \log(t+1)$. To facilitate adoption of log-adaptive open-loop step-sizes, we have incorporated this rule into the {\tt FrankWolfe.jl} software package.

We investigate the class of completely regular codes in graphs with a distance partition C_0,..., C_\rho, where each set C_i, for 0<=i<=r-1, is an independent set. This work focuses on the existence problem for such codes in the n-dimensional infinite grid. We demonstrate that several parameter families of such codes necessarily arise from binary or ternary Hamming graphs or do not exist. Furthermore, employing binary linear programming techniques, we explore completely regular codes in infinite grids of dimensions 3 and 4 for the cases r=1 and r=2.

I introduce a novel mathematical framework integrating topological dynamics, operator algebras, and ergodic geometry to study lattices of asynchronous metric dynamical systems. Each node in the lattice carries an internal flow represented by a one-parameter family of operators, evolving on its own time scale. I formalize stratified state spaces capturing multiple levels of synchronized behavior, define an asynchronous evolution metric that quantifies phase-offset distances between subsystems, and characterize emergent coherent topologies arising when subsystems synchronize. Within this framework, I develop formal operators for the evolution of each subsystem and give precise conditions under which phase-aligned synchronization occurs across the lattice. The main results include: (1) the existence and uniqueness of coherent (synchronized) states under a contractive coupling condition, (2) stability of these coherent states and criteria for their emergence as a collective phase transition in a continuous operator topology, and (3) the influence of symmetries, with group-invariant coupling leading to flow-invariant synchrony subspaces and structured cluster dynamics. Proofs are given for each theorem, demonstrating full mathematical rigor. In a final section, I discuss hypothetical applications of this framework to symbolic lattice systems (e.g. subshifts), to invariant group actions on dynamical lattices, and to operator fields over stratified manifolds in the spirit of noncommutative geometry. Throughout, I write in the first person to emphasize the exploratory nature of this work. The paper avoids any reference to cosmology or observers, focusing instead on clean, formal mathematics suitable for a broad array of dynamical systems.

Let $X_\lambda^\mu := X_\lambda \cap X^\mu \subseteq G/P$ be a Richardson variety in a generalized partial flag manifold. We use equivariant stable map spaces to define a canonical resolution $\widetilde{X_\lambda^\mu}$ of singularities, albeit obtaining an orbifold not a manifold. The ``nodal curves'' boundary is an (orbifold) simple normal crossings divisor, and is conjecturally anticanonical. Its dual simplicial complex is the order complex of the open Bruhat interval $(\lambda,\mu) \subseteq W/W_P$, shown in [Bj\"orner-Wachs '82] to be a sphere or ball. In the case of $G/P$ a Grassmannian, the resolution $\widetilde{X_\lambda^\mu}$ is a GKM space, whose $T$-fixed points are indexed by rim-hook tableaux.

This paper aims to continue the studies initiated by Botha in [Linear Algebra Appl. 273 (1998), 65-82; Linear Algebra Appl. 286 (1999), 37-44; Linear Algebra Appl. 315 (2000), 1-23] by extending them to matrices over noncommutative division rings. In particular, we show that every such matrix can be written as either a sum or a product of two diagonalizable matrices. The number $2$ is not valid under mild conditions on the center, similar to those in Botha's work on fields. By applying this result and other results obtained so far, we latter establish some Waring-type results for matrices.

A feedforward neural network, including hidden layers, motivated by nonlinear functions (such as Tanh, ReLU, and Sigmoid functions), exhibits uniform approximation properties in Sobolev space, and discontinuous neural networks can reduce computational complexity. In this work, we present a discontinuous hybrid neural network method for solving the partial differential equations, construct a new hybrid loss functional that incorporates the variational of the approximation equation, interface jump stencil and boundary constraints. The RMSprop algorithm and discontinuous Galerkin method are employed to update the nonlinear parameters and linear parameters in neural networks, respectively. This approach guarantees the convergence of the loss functional and provides an approximate solution with high accuracy.

In this paper, it is proved, that for varieties with (m-1)-Du Bois singularities, the natural morphism from the Grothendieck dual of the m-th graded Du Bois complex to the Grothendieck dual of its zero-th cohomology sheaf is injective on cohomology. This confirms Conjecture G of Popa, Shen, and Vo [PSV24].

To circumvent the high path loss of mmWave propagation and reduce the hardware cost of massive multiple-input multiple-output antenna systems, full-dimensional hybrid beamforming is critical in 5G and beyond wireless communications. Concerning an uplink multi-cell system with a large-scale uniform planar antenna array, this paper designs an efficient hybrid beamformer using primitive Kronecker decomposition and dynamic factor allocation, where the analog beamformer applies to null the inter-cell interference and simultaneously enhances the desired signals. In contrast, the digital beamformer mitigates the intra-cell interference using the minimum mean square error (MMSE) criterion. Then, due to the low accuracy of phase shifters inherent in the analog beamformer, a low-complexity hybrid beamformer is developed to slow its adjustment speed. Next, an optimality analysis from a subspace perspective is performed, and a sufficient condition for optimal antenna configuration is established. Finally, simulation results demonstrate that the achievable sum rate of the proposed beamformer approaches that of the optimal pure digital MMSE scheme, yet with much lower computational complexity and hardware cost.

This work gives a computable formula for the average measure theoretic entropy of a family of expanding on average random Blaschke products, generalizing work by Pujals, Roberts and Shub [Expanding maps of the circle revisited: positive Lyapunov exponents in a rich family. $\textit{Ergodic Theory Dynam. Systems.}$ $\textbf{26}(6)$ $(2006),$ $1931$-$1937$] to the random setting. In doing so, we describe the random invariant measure and associated measure theoretic entropy for a class of admissible random Blaschke products, allowing for maps which are not necessarily expanding and may even have an attracting fixed point.

We establish an unfolding theorem for equivariant F-bundles (a variant of Frobenius manifolds), generalizing Hertling-Manin's universal unfolding of meromorphic connections. As an application, we obtain the mirror symmetry theorem for the big quantum cohomology of flag varieties, from the recent works on the small quantum cohomology mirror symmetry, via the equivariant unfolding theorem.

This paper introduces a novel class of topological spaces, termed SC*-regular spaces, which are defined using SC*-open sets. We explore their fundamental properties and examine their connections with existing regularity concepts, such as regular, almost, softly, weakly, alpha, zeta, and generalized-regular spaces respectively. Furthermore, we examined and define, analyze generalized SC*-closed sets and SC*-generalized closed functions, establishing key properties and preservation theorems. Several characterizations of SC*-regular spaces are also presented, providing new insights into generalized regularity in topology.

In this paper, we investigate the dynamics of a discrete-time phytoplankton-zooplankton model where the predator functional response and toxin distribution functions follow both Holling Type II and Holling Type III forms simultaneously. We analyze the types of fixed points and the global stability of the system. Additionally, we prove the occurrence of a Neimark-Sacker bifurcation at the positive fixed point. The theoretical findings are validated through numerical simulations

Let R be a commutative ring with identity and M be an R-module. A proper ideal I of R is said to be a $z^\circ$-ideal if for each $a \in I$ the intersection of all minimal prime ideals containing a is contained in I. The purpose of this paper is to introduce the notion of $z^\circ$-submodules of M as an extension of $z^\circ$-ideals of R. Moreover, we investigate some properties of this class of submodules when M is a reduced multiplication R-module.

We describe the singular fibers of a parabolic fibration $f:X\to Y$ whose moduli divisor $M_Y$ is numerically trivial and discriminant divisor $B_Y$ is zero.

A conjecture of J.M. Carnicer, E. Mainar and J.M. Pe\~{n}a states that the critical length of the space $P_{n}\odot C_{1}$ generated by the functions $x^{k}\sin x$ and $x^{k}\cos x$ for $k=0,...n$ is equal to the first positive zero $j_{n+\frac{1}{2},1}$ of the Bessel function $J_{n+\frac{1}{2}}$ of the first kind. It is known that the conjecture implies the following statement (D3): the determinant of the Hankel matrix \begin{equation} \left( \begin{array} [c]{ccc} f & f^{\prime} & f^{\prime\prime}\\ f^{\prime} & f^{\prime\prime} & f^{\left( 3\right) }\\ f^{\prime\prime} & f^{\prime\prime\prime} & f^{\left( 4\right) } \end{array} \right) \label{eqabstract} \end{equation} does not have a zero in the interval $(0,j_{n+\frac{1}{2},1})$ whenever $f=f_{n}$ is given by $f_{n}\left( x\right) =\sqrt{\frac{\pi}{2}} x^{n+\frac{1}{2}}J_{n+\frac{1}{2}}\left( x\right) .$ In this paper we shall prove (D3) and various generalizations.

Let R be a semiring. We say that a non-zero subsemimodule S of an R-semimodule M is second if for each a \in R, we have aS = S or aS = 0. The aim of this paper is to study the notion of second subsemimodules of semimodules over commutative semirings.

In this paper, the dynamics of a phytoplankton-zooplankton system with linear functional responses are examined. For the continuous-time model, the global asymptotic stability of the fixed points is demonstrated by constructing Lyapunov functions. For the discrete version of the model, both local and global dynamics are investigated using LaSalle's Invariance Principle. Furthermore, the occurrence of a Neimark-Sacker bifurcation at the positive fixed point is established, and it is proved that the resulting invariant closed curve is attracting.

We study congruences involving $p$-adic families of Hecke eigensystems of Yoshida lifts associated with two Hida families (say $\mathcal{F},\mathcal{G}$) of elliptic cusp forms. With appropriate hypotheses, we show that if a Hida family of genus two Siegel cusp forms admits a Yoshida lift at an appropriately chosen classical specialization, then all classical specializations are Yoshida lifts. Moreover, we prove that the characteristic ideal of the non-primitive Selmer group of (a self-dual twist of) the Rankin--Selberg convolution of $\mathcal{F}$ and $\mathcal{G}$ is divisible by the congruence ideal of the Yoshida lift associated with $\mathcal{F}$ and $\mathcal{G}$. Under an additional assumption inspired by pseudo-nullity conjectures in higher codimension Iwasawa theory, we establish the pseudo-cyclicity of the dual of the primitive Selmer group over the cyclotomic $\mathbb{Z}_p$-extension.

This note establishes that the opposite Gaussian product inequality (GPI) of the type proved by Russell & Sun (2022a) in two dimensions, and partially extended to higher dimensions by Zhou et al. (2024), continues to hold for an arbitrary mix of positive and negative exponents. A general quantitative lower bound is also obtained conditionally on the GPI conjecture being true.

To decode a short linear block code, ordered statics decoding (OSD) and/or the $A^*$ decoding are usually considered. Either OSD or the $A^*$ decoding utilizes the magnitudes of the received symbols to establish the most reliable and independent positions (MRIP) frame. A restricted searched space can be employed to achieve near-optimum decoding with reduced decoding complexity. For a low-rate code with large minimum distance, the restricted search space is still very huge. We propose to use concatenated coding to further restrict the search space by proposing an improved MRIP frame. The improved MRIP frame is founded according to magnitudes of log likelihood ratios (LLRs) obtained by the soft-in soft-out (SISO) decoder for the inner code. We focus on the construction and decoding of several $(n,k)$ = (128,36) binary linear block codes based on concatenated coding. We use the (128,36) extended BCH (eBCH) code as a benchmark for comparison. Simulation shows that there exist constructed concatenated codes which are much more efficient than the (128,36) eBCH code. Some other codes of length 128 or close to 128 are also constructed to demonstrate the efficiency of the proposed scheme.

Let $F$ be a $p$-adic field, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$, with $n \geq 0$. We prove that a smooth irreducible representation of good parity of $G$ is unitary if and only if it is of Arthur type. Combined with the algorithms of the first author or Hazeltine-Liu-Lo for detecting Arthur type representations, our result leads to an explicit algorithm for checking the unitarity of any given irreducible representation of good parity. Finally, we determine the set of unitary representations that may appear as local components of the discrete automorphic spectrum.

Let $R$ be a finite ring with unity. In general, the eigenvalues of the unitary Cayley graph $\text{Cay}(R, R^{\times})$ are not known when $R$ is a non-commutative. In this paper, we present an explicit formula for the eigenvalues of $\text{Cay}(R, R^{\times})$ for any finite ring $R$. However, our focus is on a more general case of the unitary Cayley graph. It is well known that the classical Ramanujan's sum represents the eigenvalues of $\text{Cay}(\mathbb{Z}_n, \mathbb{Z}_n^{\times})$. Consequently, the eigenvalues of $\text{Cay}(R, R^{\times})$ can be view as a generalization of classical Ramanujan's sum in the context of finite rings. Interestingly, the formula we derive for the eigenvalues of $\text{Cay}(R, R^{\times})$ extends the known formula of classical Ramanujan's sum to the context of finite rings.

We construct integral models and special affinoids of suitable tubular neighborhoods of local Shimura varieties at depth-zero. We show that the reductions of the special affinoids over suitable tamely ramified extensions are realized as parabolic Deligne-Lusztig varieties and explicitly compute part of the cohomology of local Shimura varieties at depth-zero. In the case of general linear groups, our constructions recover generalized semistable models of the Lubin-Tate spaces at depth-zero constructed by Yoshida.

Support vector machine (SVM) is a well known binary linear classification model in supervised learning. This paper proposes a globalized distributionally robust chance-constrained (GDRC) SVM model based on core sets to address uncertainties in the dataset and provide a robust classifier. The globalization means that we focus on the uncertainty in the sample population rather than the small perturbations around each sample point. The uncertainty is mainly specified by the confidence region of the first- and second-order moments. The core sets are constructed to capture some small regions near the potential classification hyperplane, which helps improve the classification quality via the expected distance constraint of the random vector to core sets. We obtain the equivalent semi-definite programming reformulation of the GDRC SVM model under some appropriate assumptions. To deal with the large-scale problem, an approximation approach based on principal component analysis is applied to the GDRC SVM. The numerical experiments are presented to illustrate the effectiveness and advantage of our model.

This paper is concerned with extending the notion of monotone solution to the mean field game (MFG) master equation to situations in which the coefficients are displacement monotone, instead of the previously introduced notion in the flat monotone regime. To account for this new setting, we work directly on the equation satisfied by the controls of the MFG. Following previous works, we define an appropriate notion of solution under which uniqueness and stability results hold for solutions without any differentiability assumption with respect to probability measures. Thanks to those properties, we show the existence of a monotone solution to displacement monotone mean field games under local regularity assumptions on the coefficients and sufficiently strong monotonicity. Albeit they are not the focus of this article, results presented are also of interest for mean field games of control and general mean field forward backward systems. In order to account for this last setting, we use the notion of $L^2$-monotonicity instead of displacement monotonicity, those two notions being equivalent in the particular case of MFG.

In this note, we present a result established in [BGR24] where we prove that nonlinear Schrodinger equations on the circle, without external parameters, admit plenty of infinite dimensional non resonant invariant tori, or equivalently, plenty of almost periodic solutions. Our aim is to propose an extended sketch of the proof, emphasizing the new points which have enabled us to achieve this result.

We provide few remarks on nonlinear Landau damping that concerns decay of the electric field in the classical Vlasov-Poisson system near spatially homogenous equilibria. In particular, this includes the analyticity framework, \`a la Grenier-Nguyen-Rodnianski, for non specialists, treating the analytic case studied by Mouhot-Villani, among other remarks for plasmas confined on a torus and in the whole space.

We establish a general form of Wiener's lemma for measures on locally compact abelian (LCA) groups by using Fourier analysis and the theory of F{{\o}}lner sequences. Our approach provides a unified framework that that encompasses both the discrete and continuous cases. We also show a version of Wiener's lemma for Bochner-Riesz means on both R^d and T^d . Mathematics Subject Classification (2010). 43A25.

In this work, we study the componentwise (Schur) product of monomial-Cartesian codes by exploiting its correspondence with the Minkowski sum of their defining exponent sets. We show that $ J$-affine variety codes are well suited for such products, generalizing earlier results for cyclic, Reed-Muller, hyperbolic, and toric codes. Using this correspondence, we construct CSS-T quantum codes from weighted Reed-Muller codes and from binary subfield-subcodes of $ J$-affine variety codes, leading to codes with better parameters than previously known. Finally, we present Private Information Retrieval (PIR) constructions for multiple colluding servers based on hyperbolic codes and subfield-subcodes of $ J$-affine variety codes, and show that they outperform existing PIR schemes.

We prove an asymptotic formula for the number of integral points of bounded log-anticanonical height on split smooth quintic del Pezzo surfaces over number fields, with respect to one of the lines as the boundary divisor.

We consider the spectrum of the Laplace operator on 3D rod structures, with a small cross section depending on a small parameter $\varepsilon$. The boundary conditions are of Dirichlet type on the basis of this structure and Neumann on the lateral boundary. We focus on the low frequencies. We study the asymptotic behavior of the eigenvalues and associated eigenfunctions, which are approached as $\varepsilon\to 0$ by those of a 1D model with Dirichlet boundary conditions, but which takes into account the geometry of the domain. Explicit and numerical computations enlighten the interest of this study, when the parameter becomes smaller. At the same time they show that in order to capture oscillations in the transverse direction we need to deal with the high frequencies. For prism like domains, we show the different asymptotic behavior of the spectrum depending on the boundary conditions.

Let $\mathcal{A}$ denote by the operator class with some proper conditions. %satisfying that for any two operators $T,\tilde{T}$ in $\mathcal{A}$, the non-zero intertwining operator between $T$ and $\tilde{T}$ has dense range. Then by taking the operators in $\mathcal{A}$ as atoms and using the flag structure as bondings, we introduce a new operator class. For operators with certain properties in this new class, we prove that the operator matrix of the intertwining operator is of the upper-triangular form. According to this critical result, we firstly show that the strongly irreducible operators in the new class preserve strong irreducibility under quasi-similarity, which gives a partial answer to the question proposed by C.L. Jiang. Also, when $\mathcal{A}$ is backward weighted shift operators class, we prove that the quasi-similarity between operators in the new class implies the similarity relation, which partially answers the question proposed by D.A. Herrero. Finally, we describe some properties of intertwining operators in term of geometric language.

Let $G = (V, E)$ be a graph with the vertex set $V (G)$ and edge set $E(G)$. The Sombor index of $G$, $SO(G)$, is defined as $\sum_{uv\in E(G)} \sqrt{deg(u)^2 + deg(v)^2}$, where $deg(u)$ is the degree of vertex $u$ in $V (G)$. The clean graph of a ring R, denoted by $Cl(R)$, is a graph with vertex set $\{(e, u) : e \in Id(R), u \in U(R)\}$ and two distinct vertices $(e, u)$ and$(f, v)$ are adjacent if and only if $ef = 0$ or $uv = 1$ ($Id(R)$ and $U(R)$ are the sets of idempotents and unit elements of R, respectively). The induced subgraph on $\{(e, u) : e \in Id^{*}(R), u \in U(R)\}$ is denoted by $Cl_2(R)$. In this paper, $SO(Cl2(\mathbb{Z}_n))$, for different values of the positive integer $n$, is investigated.

Isogeometric analysis (IGA) enables exact representations of computational geometries and higher-order approximation of PDEs. In non-smooth domains, however, singularities near corners limit the effectiveness of IGA, since standard methods typically fail to achieve optimal convergence rates. These constraints can be addressed through local mesh refinement, but existing approaches require breaking the tensor-product structure of splines, which leads to increased implementation complexity. This work introduces a novel local refinement strategy based on a polar parameterization, in which one edge of the parametric square is collapsed into the corner. By grading the standard mesh toward the collapsing edge, the desired locality near the singularity is obtained while maintaining the tensor-product structure. A mathematical analysis and numerical tests show that the resulting isogeometric approximation achieves optimal convergence rates with suitable grading parameters. Polar parameterizations, however, suffer from a lack of regularity at the polar point, making existing standard isogeometric approximation theory inapplicable. To address this, a new framework is developed for deriving error estimates on polar domains with corners. This involves the construction of polar function spaces on the parametric domain and a modified projection operator onto the space of $C^0$-smooth polar splines. The theoretical results are verified by numerical experiments confirming both the accuracy and efficiency of the proposed approach.

A generalization of the four-color theorem, Hadwiger's conjecture is considered as one of the most important and challenging problems in graph theory, and odd Hadwiger's conjecture is a strengthening of Hadwiger's conjecture by way of signed graphs. In this paper, we prove that odd Hadwiger's conjecture is true for the complements $\overline{K}(n,k)$ of the Kneser graphs $K(n,k)$, where $n\geq 2k \ge 4$. This improves a result of G. Xu and S. Zhou (2017) which states that Hadwiger's conjecture is true for this family of graphs. Moreover, we prove that $\overline{K}(n,k)$ contains a 1-shallow complete minor of a special type with order no less than the chromatic number $\chi(\overline{K}(n,k))$, and in the case when $7 \le 2k+1 \le n \le 3k-1$ the gap between the odd Hadwiger number and chromatic number of $\overline{K}(n,k)$ is $\Omega(1.5^{k})$.

We realize infinitely many covering groups $2.A_n$ (where $A_n$ is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works investigating special cases or proving conditional results in this direction, these are the first unramified realizations of infinitely many of these groups.

We discuss a nonlocal version of the Generalized Aw-Rascle-Zhang model, a second-order vehicular traffic model where the empty road velocity is a Lagrangian marker governed by a transport equation. The evolution of the car density is described by a continuity equation where the drivers' velocity depends on both the empty road velocity and the convolution of the car density with an anisotropic kernel. We establish existence and uniqueness results. When the convolution kernel is replaced by a Dirac Delta, the nonlocal model formally boils down to the classical (local) Generalized Aw-Rascle-Zhang model, which consists of a conservation law coupled with a transport equation. In the case of exponential kernels, we establish convergence in the nonlocal-to-local limit by proving an Oleinik-type estimate for the convolution term. To the best of our knowledge, this is the first nonlocal-to-local limit result for a system of two non-decoupling equations with a nonlocal flux function.

We establish existence, uniqueness and stability results for the so-called Generalized Aw-Rascle-Zhang model, a second order traffic model introduced by Fan, Herty and Seibold. Our analysis is motivated by the companion paper 'Nonlocal Generalized Aw-Rascle-Zhang model: well-posedness and singular limit'.

In this paper, we propose a discontinuous Hamilton Monte Carlo (DHMC) to sample from dimensional varying distributions, and particularly the grand canonical ensemble. The DHMC was proposed in [Biometrika, 107(2)] for discontinuous potential where the variable has a fixed dimension. When the dimension changes, there is no clear explanation of the volume-preserving property, and the conservation of energy is also not necessary. We use a random sampling for the extra dimensions, which corresponds to a measure transform. We show that when the energy is corrected suitably for the trans-dimensional Hamiltonian dynamics, the detailed balance condition is then satisfied. For the grand canonical ensemble, such a procedure can be explained very naturally to be the extra free energy change brought by the newly added particles, which justifies the rationality of our approach. To sample the grand canonical ensemble for interacting particle systems, the DHMC is then combined with the random batch method to yield an efficient sampling method. In experiments, we show that the proposed DHMC combined with the random batch method generates samples with much less correlation when compared with the traditional Metropolis-Hastings method.

The paper deals with analysis and design sliding mode control systems modeled by integro-differential equations. Filippov method and equivalent control approach are extended to a class of nonlinear discontinuous integro-differential equations. Sliding mode control algorithm is designed for a control system with distributed input delay. The obtained results are illustrated by numerical example.

We extend several predecessor works on even sextic monogenic polynomials. In particular, we prove a conjecture of Lenny Jones, thereby classifying even sextic monogenic polynomials with cyclic Galois group. This result is key to completing previous partial results on existence or non-existence of infinite families of even sextic monogenic polynomials with a prescribed Galois group. Some of the underlying ideas are relevant for investigation of more general families of even polynomials $f(X^2)$, or power-compositional polynomials $f(X^\ell)$.

This paper deals with N-person nonzero-sum discrete-time Markov games under a probability criterion, in which the transition probabilities and reward functions are allowed to vary with time. Differing from the existing works on the expected reward criteria, our concern here is to maximize the probabilities that the accumulated rewards until the first passage time to any target set exceed a given goal, which represent the reliability of the players income. Under a mild condition, by developing a comparison theorem for the probability criterion, we prove the existence of a Nash equilibrium over history-dependent policies. Moreover, we provide an efficient algorithm for computing epsilon-Nash equilibria. Finally, we illustrate our main results by a nonstationary energy management model and take a numerical experiment.

We study the asymptotic behavior of small deviation probabilities for the critical Galton-Watson processes with infinite variance of the offspring sizes of particles and apply the obtained result to investigate the structure of a reduced critical Galton-Watson process.

We use the framework of Abstract Elementary Classes ($\mathrm{AEC}$s) to introduce a new Construction Principle $\mathrm{CP}(\mathbf{K},\ast)$, which strictly generalises the Construction Principle of Eklof, Mekler and Shelah and allows for many novel applications beyond the setting of universal algebra. In particular, we show that $\mathrm{CP}(\mathbf{K},\ast)$ holds in the classes of free products of cyclic groups of fixed order, direct sums of a fixed torsion-free abelian group of rank 1 which is not $\mathbb{Q}$, (infinite) free $(k,n)$-Steiner systems, and (infinite) free generalised $n$-gons. From this we derive, in ZFC, that these classes of structures are not axiomatisable in the logic $\mathfrak{L}_{\infty,\omega_1}$, and, under $V=L$, that they are not axiomatisable in $\mathfrak{L}_{\infty,\infty}$.

The optimal matching of point clouds in $\mathbb{R}^d$ is a combinatorial problem; applications in statistics motivate to consider random point clouds, like the Poisson point process. There is a crucial dependance on dimension $d$, with $d=2$ being the critical dimension. This is revealed by adopting an analytical perspective, connecting e.\,g.~to Optimal Transportation. These short notes provide an introduction to the subject. The material presented here is based on a series of lectures held at the International Max Planck Research School during the summer semester 2022. Recordings of the lectures are available at https://www.mis.mpg.de/events/event/imprs-ringvorlesung-summer-semester-2022.

We study Sobolev and BV spaces on local trees which are metric spaces locally isometric to real trees. Such spaces are equipped with a Radon measure satisfying a locally uniform volume growth condition. Using the intrinsic geodesic structure, we define weak gradients and develop from it a coherent theory of Sobolev and BV spaces. We provide two main characterizations: one via Korevaar-Schoen-type energy functionals and another via the heat kernel associated with the natural Dirichlet form. Applications include interpolation results for Besov-Lipschitz spaces, critical exponents computations, and a Nash inequality. In globally tree-like settings we also establish $L^p$ gradient bounds for the heat semigroup.

The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many applications in problems of functional analysis, approximation theory, combinatorics, etc. In this paper we propose a modification of the invariant polytope algorithm enlarging the class of problems to which it is applicable. Precisely, we introduce mixed numeric and symbolic computations. A further minor modification of augmenting the input set with additional matrices speeds up the algorithm in certain cases. With this modifications we are able to automatically prove the finiteness conjecture for all pairs of binary $3\times 3$ matrices and sign $2\times 2$ matrices.

We propose a new method for retrieving the algebraic structure of a generic alternant code given an arbitrary generator matrix, provided certain conditions are met. We then discuss how this challenges the security of the McEliece cryptosystem instantiated with this family of codes. The central object of our work is the quadratic hull related to a linear code, defined as the intersection of all quadrics passing through the columns of a given generator or parity-check matrix, where the columns are considered as points in the affine or projective space. The geometric properties of this object reveal important information about the internal algebraic structure of the code. This is particularly evident in the case of generalized Reed-Solomon codes, whose quadratic hull is deeply linked to a well-known algebraic variety called the rational normal curve. By utilizing the concept of Weil restriction of affine varieties, we demonstrate that the quadratic hull of a generic dual alternant code inherits many interesting features from the rational normal curve, on account of the fact that alternant codes are subfield-subcodes of generalized Reed-Solomon codes. If the rate of the generic alternant code is sufficiently high, this allows us to construct a polynomial-time algorithm for retrieving the underlying generalized Reed-Solomon code from which the alternant code is defined, which leads to an efficient key-recovery attack against the McEliece cryptosystem when instantiated with this class of codes. Finally, we discuss the generalization of this approach to Algebraic-Geometry codes and Goppa codes.

In this paper, we deal with the construction of holomorphic functions on a simply connected domain satisfying that all its derivatives and antiderivatives under a composition operator have a dense orbit. Such functions will be called Luh hypercyclic vectors for the respective composition operator. We show that there is a dense linear manifold of Luh hypercyclic vectors. Moreover, we study the dynamics of cosine operator function generated by weighted composition operators on solid Banach function spaces, in particular on Orlicz and Morrey spaces, and we give sufficient conditions for supercyclicity of such cosine operator functions in terms of the corresponding weight function. Also, we give concrete examples of weighted translations satisfying these sufficient conditions.

We understand quantum principal bundle as faithfully flat Hopf--Galois extensions, with a structure Hopf algebra coacting on a total space algebra and with base algebra given by the coinvariant elements. To endow such bundles with a compatible differential structure, one requires the coaction to extend as a morphism of differential graded algebras. This leads to an exact noncommutative Atiyah sequence, a graded Hopf--Galois extension of differential forms and a canonical braiding on total space forms such that the latter are graded-braided commutative. We recall this approach to noncommutative differential geometry and further discuss the extension of quantum gauge transformations, in the sense of Brzezi\'nski, to differential forms. In this way we obtain an action of quantum gauge transformations on connections of the quantum principal bundle and their curvature. Explicit examples, such as the noncommutative 2-torus, the quantum Hopf fibration and smash product algebras are discussed.

When a circular symmetric piston suddenly expands into a still gas, a leading shock wave is generated. This paper investigates an inverse problem of reconstructing the trajectory of the piston from the given leading shock front and the given initial flow conditions. We observe that in piston models, as the initial density goes to zero, the piston approaches the shock front; however, in the region between the piston and the shock front, the strict hyperbolicity of the system degenerates. By applying asymptotic analysis, we provide quantitative characterizations of the distance between the piston and the shock front, and the degeneration of strict hyperbolicity. Consequently, by designing appropriate a priori assumptions to balance the benefits and drawbacks arising as the initial density approaches zero, we employ the method of characteristics to prove the global-in-time existence of the piecewise smooth solution for this inverse problem. In particular, the resulting flow structure exhibits significant velocity variations.

We first establish strong convergence rates for multiscale systems driven by $\alpha$-stable processes, with analyses constructed in two distinct scaling regimes. When addressing weak convergence rates of this system, we derive four averaged equations with respect to four scaling regimes. Notably, under sufficient H\"{o}lder regularity conditions on the time-dependent drifts of slow process, the strong convergence orders are related to the known optimal strong convergence order $1-\frac{1}{\alpha}$, and the weak convergence orders are 1. Our primary approach involves employing nonlocal Poisson equations to construct ``corrector equations" that effectively eliminate inhomogeneous terms.

We investigate the explicit convex relaxation of the ideal magnetohydrodynamics equations. We provide a non-trivial lower estimate on the lamination hull and an upper estimate on the $\Lambda$-convex hull, the latter providing inequalities which will be satisfied by weak limits of weak solution of the ideal MHD equations, which serve as a model of averaged turbulent magnetohydrodynamical flows.

Studying Manin's program for a family of spherical log Fano threefolds, we determine the asymptotic number of integral points whose height associated with an arbitrary ample line bundle is bounded. This confirms a recent conjecture by Santens and sheds new light on the logarithmic analogue of Iitaka fibrations, which have not yet been adequately formulated.

We consider a purely harmonic chain of oscillators which is perturbed by a stochastic noise. Under this perturbation, the system exhibits two conserved quantities: the volume and the energy. At the level of the hydrodynamic limit, under diffusive scaling, we show that depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or a non-linear system of coupled parabolic equations. Moreover, for general initial measures, under diffusive scaling, we can characterize the non-equilibrium volume fluctuations. The proofs are based on precise bounds on the two-point volume correlation function and a uniform fourth-moment estimate.

In previous work (arXiv:1908.09589), we studied rational generating functions ("ask zeta functions") associated with graphs and hypergraphs. These functions encode average sizes of kernels of generic matrices with support constraints determined by the graph or hypergraph in question, with applications to the enumeration of linear orbits and conjugacy classes of unipotent groups. In the present article, we turn to the effect of a natural graph-theoretic operation on associated ask zeta functions. Specifically, we show that two instances of rational functions, $W^-_\Gamma(X,T)$ and $W^\sharp_\Gamma(X,T)$, associated with a graph $\Gamma$ are both well-behaved under taking joins of graphs. In the former case, this has applications to zeta functions enumerating conjugacy classes associated with so-called graphical groups.

In this paper, the authors establish the existence and boundedness of multilinear Littlewood--Paley operators on products of BMO spaces, including the multilinear $g$-function, multilinear Lusin's area integral and multilinear $g^{\ast}_{\lambda}$-function. The authors prove that if the above multilinear operators are finite for a single point, then they are finite almost everywhere. Moreover, it is shown that these multilinear operators are bounded from $\mathrm{BMO}(\mathbb R^n)\times\cdots\times \mathrm{BMO}(\mathbb R^n)$ into $\mathrm{BLO}(\mathbb R^n)$ (the space of functions with bounded lower oscillation), which is a proper subspace of $\mathrm{BMO}(\mathbb R^n)$ (the space of functions with bounded mean oscillation). The corresponding estimates for multilinear Littlewood--Paley operators with non-convolution type kernels are also discussed.

In this paper, we derive a Pogorelov type interior $C^2$ estimate for the Hessian quotient equation $\frac{\sigma _n}{\sigma _k}\left( D^2u\right) =f$. As an application, we show that convex viscosity solutions are regular for $k\leq n-3$ if $u\in C^{1,\alpha}$ with $\alpha>1-\frac{2}{n-k}$ or $u\in W^{2,p}$ with $p\geq\frac{(n-1)(n-k)}{2}$. Both exponents are sharp in view of the example in arXiv:2401.12229.

We use the methods introduced by Lue Pan to study the locally analytic vectors in the completed cohomology of unitary Shimura curves. As an application, we prove a classicality result on two-dimensional regular $\sigma$-de Rham representations of $\text{Gal}(\bar L/L)$ appearing in the locally $\sigma$-analytic vectors of the completed cohomology, where $L$ is a finite extension of $\mathbb{Q}_p$ and $\sigma:L\hookrightarrow E$ is an embedding of $L$ into a sufficiently large finite extension $E$ of $\mathbb{Q}_p$. We also prove that if a two-dimensional representation of $\text{Gal}(\bar L/L)$ appears in the locally $\sigma$-algebraic vectors of the completed cohomology then it is $\sigma$-de Rham. Finally, we give a geometric realization of some locally $\sigma$-analytic representations of $\mathrm{GL}_2(L)$. This realization has some applications to the $p$-adic local Langlands program, including a locality theorem for Galois representations arising from classical automorphic forms, an admissibility result for coherent cohomology of Drinfeld curves, and some special cases of the Breuil's locally analytic Ext$^1$-conjecture for $\mathrm{GL}_2(L)$.

The Riemannian manifold of curves with a Sobolev metric is an important and frequently studied model in the theory of shape spaces. Various numerical approaches have been proposed to compute geodesics, but so far elude a rigorous convergence theory. By a slick modification of a temporal Galerkin discretization we manage to preserve coercivity and compactness properties of the continuous model and thereby are able to prove convergence for the geodesic boundary value problem. Likewise, for the numerical analysis of the geodesic initial value problem we are able to exploit the geodesic completeness of the underlying continuous model for the error control of a time-stepping approximation. In fact, we develop a convergent discretization of a comprehensive Riemannian calculus that in addition includes parallel transport, covariant differentiation, the Riemann curvature tensor, and sectional curvature, all important tools to explore the geometry of the space of curves. Selected numerical examples confirm the theoretical findings and show the qualitative behaviour. To this end, a low-dimensional submanifold of Sobolev curves with explicit formulas for ground truth covariant derivatives and curvatures are considered.

We study the representation theory of the quantum queer superalgebra ${U_{\lcase{v}}(\mathfrak{\lcase{q}}_{n})}$ and obtain some properties of the highest weight modules. Furthermore, based on the realization of ${U_{\lcase{v}}(\mathfrak{\lcase{q}}_{n})}$, we study the representation theory of the twisted queer $q$-Schur superalgebra ${{\widetilde{\mathcal{Q}}}_{\lcase{v}}(\lcase{n},\lcase{r})}$, and obtain the decomposition of its regular module as a direct sum of irreducible submodules, which also means ${{\widetilde{\mathcal{Q}}}_{\lcase{v}}(\lcase{n},\lcase{r})}$ is semisimple.

Alternating parity automata (APAs) provide a robust formalism for modelling infinite behaviours and play a central role in formal verification. Despite their widespread use, the algebraic theory underlying APAs has remained largely unexplored. In recent work, a notation for non-deterministic finite automata (NFAs) was introduced, along with a sound and complete axiomatisation of their equational theory via right-linear algebras. In this paper, we extend that line of work, in particular to the setting of infinite words. We present a dualised syntax, yielding a notation for APAs based on right-linear lattice expressions, and provide a natural axiomatisation of their equational theory with respect to the standard language model of {\omega}-regular languages. The design of this axiomatisation is guided by the theory of fixed point logics; in fact, the completeness factors cleanly through the completeness of the linear-time {\mu}-calculus.

An elementary combinatorial technique for computing lower bounds for the Yang indices of real Stiefel manifolds and oriented real Grassmannians is described. It is used to show that the Yang index of $St(n,k)$, and consequently $G(n,k)$, is at least $n - k$. For odd $n$, the bound for $G(n,2)$ can be improved to $n-1$. These results lead to generalized Borsuk-Ulam theorems.

We prove that for any countable finite dimensional CAT(0) cube complex, the Borel median graph on its Roller compactification has the Borel asymptotic dimension bounded from above by its dimension.

We derive an asymptotic formula for the divisor function $\tau(k)$ in an arithmetic progression $k\equiv a(\bmod \ q)$, uniformly for $q\leq X^{\Delta_{n,l}}$ with $(q,a)=1$. The parameter $\Delta_{n,l}$ is defined as $$ \Delta_{n,l}=\frac{1-\frac{3}{2^{2^l+2l-3}}}{1-\frac{1}{n2^{l-1}}}. $$ Specifically, by setting $l=2$, we achieve $\Delta_{n,l}>3/4+5/32$, which surpasses the result obtained by Liu, Shparlinski, and Zhang (2018). Meanwhile, this has also improved upon the result of Wu and Xi (2021). Notably, Hooley, Linnik, and Selberg independently established that the asymptotic formula holds for $q\leq X^{2/3-\varepsilon}$. Irving was the first to surpass the $2/3-$barrier for certain special moduli. We break the classical $3/4-$barrier in the case of prime power moduli and extend the range of $q$. Our main ingredients borrow from Mangerel's adaptation of Mili\'{c}evi\'{c} and Zhang's methodology in dealing with a specific class of weighted Kloosterman sums, rather than adopting Korobov's technique employed by Liu, Shparlinski, and Zhang.

This paper focuses on the well-posedness (or lack thereof) of three-dimensional time-harmonic wave propagation problems modeled by the Helmholtz equation. It is well-known that if the problem is set in bounded domain with Dirichlet boundary conditions, then the Helmholtz problem is well-posed for all (real-valued) frequencies except for a sequence of countably many resonant frequencies that accumulate at infinity. In fact, if the domain is sufficiently smooth, this can be quantified further and Weyl's law states that the number of resonant frequencies less than a given $\omega > 0$ scales as $\omega^3$. On the other hand, scattering problems set in $\mathbb{R}^3$ with a radiation condition at infinity and a bounded obstacle modeled by variations in the PDE coefficients are well-posed for all frequencies under mild regularity assumption on such coefficients. In 2001, Filinov provided a counter example of a rough coefficient such that the scattering problem is not well-posed for (at least) a single frequency $\omega$. In this contribution, we use this result to show that for all $\varepsilon > 0$ we can design a rough coefficient corresponding to a compactly supported obstacle such that the scattering problem is ill-posed for a countable sequence of frequencies accumulating at infinity, and such that the number of such frequencies less than any given $\omega > 0$ scales as $\omega^{3-\varepsilon}$.

We explain how to compute simple perverse sheaves on the stack of $G$-zips and do these computations in several examples.

A causal manifold $(M,\gamma)$ is a manifold $M$ endowed with a closed proper cone $\gamma$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of $\gamma$. Let $\lambda$ be the antipodal of the polar cone of $\gamma$. An open set $U$ of $M$ is called $\gamma$-open if its Whitney normal cone contains the interior of $\gamma$. Similarly, $U$ is called $\lambda$-open if the micro-support of the constant sheaf on $U$ is contained in $\lambda$. We begin by proving that the two notions coincide. Next, we prove that if $(M,\gamma)$ admits a ``future time function'' the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $\lambda$ and the derived category of sheaves on the manifold $M$ endowed with the $\gamma$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space.

Numerical optimization (solving optimization problems using digital computers) currently dominates, but has three major drawbacks: high energy consumption, poor scalability, and lack of an execution time certificate. To address these challenges, this article explores the recent resurgence of analog computers, proposing a novel paradigm of arbitrarily small execution-time-certified analog optimization (solving optimization problems via analog computers). To achieve ultra-low energy consumption, this paradigm transforms optimization problems into ordinary differential equations (ODEs) and leverages the ability of analog computers to naturally solve ODEs (no need for time-discretization) in physically real time. However, this transformation can fail if the optimization problem, such as the general convex nonlinear programs (NLPs) considered in this article, has no feasible solution. To avoid transformation failure and enable infeasibility detection, this paradigm introduces the homogeneous monotone complementarity problem formulation for convex NLPs. To achieve scalability and execution time certificate, this paradigm introduces the Newton-based fixed-time-stable scheme for the transformed ODE, whose equilibrium time $T_p$ can be prescribed by choosing the ODE's time coefficient as $k=\frac{\pi}{2T_p}$. This equation certifies that the equilibrium time (execution time) is independent of the dimension of optimization problems and can be arbitrarily small if the analog computer allows.

We prove that the bounded and bounded below derived categories of (all) modules over the dual numbers have strongly unique (dg) enhancements. To this end we relate those categories to the category of sequences of vector spaces, which allows a complete classification of indecomposable objects. Along the way we also prove that all the derived categories of any hereditary category have strongly unique enhancements.

In this paper, we study the Chern-Hamilton energy functional on compact cosymplectic manifolds, fully classifying in dimension 3 those manifolds admitting a critical compatible metric for this functional. This is the case if and only if either the manifold is co-K\"ahler or if it is a mapping torus of the 2-torus by a hyperbolic toral automorphism and equipped with a suspension cosymplectic structure. Moreover, any critical metric has minimal energy among all compatible metrics. We also exhibit examples of manifolds with first Betti number $b_1 \geq 2$ admitting cosymplectic structures, but such that no cosymplectic structure admits a critical compatible metric.

Using Mackey functors, we provide a general framework for classifying torsors of algebraic tori in terms of Brauer groups of finite field extensions of the base field. This generalizes Blunk's description of the tori associated to del Pezzo surfaces of degree 6 to all retract rational tori, essentially the largest class for which this is possible.

The Piatetski-Shapiro sequences are of the form $\mathcal{N}_{c} := (\lfloor n^{c} \rfloor)_{n=1}^\infty$, where $\lfloor \cdot \rfloor$ is the integer part. Piatetski-Shapiro proved there are infinitely many primes in a Piatetski-Shapiro sequence for $1 < c < 12/11 = 1.0909\dots$ in 1953 and the best admissible range of $c$ for this result is by Rivat and Wu for $1 < c < 243/205 = 1.1853\dots$ in 2001. In this article, we prove there are infinitely many Piatetski-Shapiro prime numbers for $1 < c < 6/5 = 1.2$ with an asymptotic formula. Moreover, we also prove an asymptotic formula for Piatetski-Shapiro primes in arithmetic progressions with $1 < c < 6/5 = 1.2$.

A general regularization strategy is considered for the efficient iterative solution of the lowest-order weak Galerkin approximation of singular Stokes problems. The strategy adds a rank-one regularization term to the zero (2,2) block of the underlying singular saddle point system. This strategy includes the existing pressure pinning and mean-zero enforcement regularization as special examples. It is shown that the numerical error maintains the optimal-order convergence provided that the nonzero Dirichlet boundary datum is approximated numerically with sufficient accuracy. Inexact block diagonal and triangular Schur complement preconditioners are considered for the regularized system. The convergence analysis for MINRES and GMRES with corresponding block preconditioners is provided for different choices of the regularization term. Numerical experiments in two and three dimensions are presented to verify the theoretical findings and the effectiveness of the preconditioning for solving the regularized system.

Given an affine toric variety $X$ embedded in a smooth variety, we prove a general result about the mixed Hodge module structure on the local cohomology sheaves of $X$. As a consequence, we prove that the singular cohomology of a proper toric variety is mixed of Hodge-Tate type. Additionally, using these Hodge module techniques, we derive a purely combinatorial result on rational polyhedral cones that has consequences regarding the depth of reflexive differentials on a toric variety. We then study in detail two important subclasses of toric varieties: those corresponding to cones over simplicial polytopes and those corresponding to cones over simple polytopes. Here, we give a comprehensive description of the local cohomology in terms of the combinatorics of the associated cones, and calculate the Betti numbers (or more precisely, the Hodge-Du Bois diamond) of a projective toric variety associated to a simple polytope.

We present an efficient 140 line MATLAB code for topology optimization problems that include probabilistic parameters. It is built from the top99neo code by Ferrari and Sigmund and incorporates a stochastic sample-based approach. Old gradient samples are adaptively recombined during the optimization process to obtain a gradient approximation with vanishing approximation error. The method's performance is thoroughly analyzed for several numerical examples. While we focus on applications in which stochastic parameters describe local material failure, we also present extensions of the code to other settings, such as uncertain load positions or dynamic forces of unknown frequency. The complete code is included in the Appendix and can be downloaded from www.topopt.dtu.dk.

Given a bounded planar domain $\Omega \subset \mathbb{R}^2$, we show that any singular harmonic map into the circle $\mathbb{S}^1$ corresponding to a topologically nondegenerate critical point of the renormalised energy in the sense of Bethuel, Brezis and H\'elein is a limit of stationary $p$-harmonic maps for $p < 2$ as $p \to 2$

In this chapter, we investigate directional entropy for semigroup actions generated by one-dimensional linear cellular automata (LCAs) and the shift transformation on the compact metric space $\mathbb{Z}_m^{\mathbb{N}}$. This work provides a systematic study of both \emph{topological directional entropy} (TDE) within Milnor's geometric framework and \emph{measure-theoretic directional entropy} via the Kolmogorov--Sinai formalism.

In this paper, we extend Feigin-Frenkel duality at the critical level to the setting of complex rank. This is accomplished by considering the center of a vertex algebra in Deligne's interpolating categories, along with Feigin's Lie algebras of complex rank, $\mathfrak{gl}_{\lambda}$ and $\mathfrak{po}_{\lambda}$. More precisely, we define the universal affine vertex algebras associated with Lie algebras in $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{GL}_{\alpha},\mathbb{F})$, $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{O}_{\alpha},\mathbb{F})$ and $\underline{\mathrm{Re}}\mathrm{p}(\mathrm{Sp}_{\alpha},\mathbb{F})$, and describe their centers at the critical level explicitly by interpolating Molev's construction of Segal-Sugawara vectors. Using the formalism of Poisson vertex algebras, we identify a natural set of generators for the Drinfeld-Sokolov reduction of $\mathfrak{gl}_{\lambda}$ and $\mathfrak{po}_{\lambda}$, denoted by $\mathcal{W}(\mathfrak{gl}_{\lambda})$ and $\mathcal{W}(\mathfrak{po}_{\lambda})$, respectively. Finally, we show that the interpolated Feigin-Frenkel isomorphism maps the interpolated Segal-Sugawara vectors to these generators.

We extend the previously established zesting techniques from fusion categories to general tensor categories. In particular we consider the category of comodules over a Hopf algebra, providing a detailed translation of the categorical zesting construction into explicit Hopf algebraic terms: we show that the associative zesting of the category of comodules yields a coquasi-Hopf algebra whose comodule category is precisely the zested category. We explicitly write the modified multiplication and the associator, as well as the structures involved in the braided case. For pointed Hopf algebras, we derive concrete formulas for constructing zestings and establish a systematic approach for cyclic group gradings, providing explicit parameterizations of the zesting data.

This article deals with (1) the construction of a general non-linear fractal interpolation function on PCF self-similar sets, (2) the energy and normal derivatives of uniform non-linear fractal functions, (3) estimation of the bound of box dimension of the proposed fractal functions on the Sierpinski gasket and the von-Koch curve. Here, we present a more general framework to construct the attractor and the functions on the PCF self-similar sets using the Edelstein contraction, which broadens the class of functions. En route, we calculate the upper and lower box dimensions of the graph of non-linear interpolant. Finally, we provide several graphical and numerical examples for illustration of the construction and estimate the dimensions for different data sets.

In this paper, we study Borel probability measures of maximal entropy for analytic subsets in a dynamical system. It is well known that higher smoothness of the map over smooth space plays important role in the study of invariant measures of maximal entropy. A famous theorem of Newhouse states that smooth diffeomorphisms on compact manifolds without boundary have invariant measures of maximal entropy. However, we show that the situation becomes completely different when we study measures of maximal entropy for analytic subsets. Namely, we construct a smooth surface system which contains a smooth curve having no Borel probability measure of maximal entropy. Another evidence to show this difference is that, once an analytic set has one measure of maximal entropy, then the set has many measures of maximal entropy (no matter if we consider packing or Bowen entropy). For a general dynamical system with positive entropy $h_\mathrm{top}(T)$, we shall show that the system contains not only a Borel subset which has Borel probability measures of maximal entropy and has entropy sufficiently close to $h_\mathrm{top}(T)$, but also a Borel subset which has no Borel probability measures of maximal entropy and has entropy equal to the arbitrarily given positive real number which is at most $h_\mathrm{top}(T)$. We also provide in all $h$-expansive systems a full characterization for analytic subsets which have Borel probability measures of maximal entropy. Consequently, if let $Z\subset \mathbb{R}^n$ be any analytic subset with positive Hausdorff dimension in Euclidean space, then the set $Z$ either has a measure of full lower Hausdorff dimension, or it can be partitioned into a union of countably many analytic sets $\{Z_i\}_{i\in \mathbb{N}}$ with $\dim_{\mathcal{H}}(Z_i) < \dim_{\mathcal{H}}(Z)$ for each $i$.

Persistent homology is a fundamental tool in topological data analysis; however, it lacks methods to quantify the fragility or fineness of cycles, anticipate their formation or disappearance, or evaluate their stability beyond persistence. Furthermore, classical Betti numbers fail to capture key structural properties such as simplicial dimensions and higher-order adjacencies. In this work, we investigate the robustness of simplicial networks by analyzing cycle thickness and their resilience to failures or attacks. To address these limitations, we draw inspiration from persistent homology to introduce filtrations that model distinct simplicial elimination rules, leading to the definition of two novel Betti number families: thick and cohesive Betti numbers. These improved invariants capture richer structural information, enabling the measurement of the thickness of the links in the homology cycle and the assessment of the strength of their connections. This enhances and refines classical topological descriptors and our approach provides deeper insights into the structural dynamics of simplicial complexes and establishes a theoretical framework for assessing robustness in higher-order networks. Finally, we establish that the resilience of topological features to simplicial attacks can be systematically examined through biparameter persistence modules, wherein one parameter encodes the progression of the attack, and the other captures structural refinements informed by thickness or cohesiveness.

We derive exact expressions for the probabilities that partly random hyperplanes separate two Euclidean balls. The probability that a fully random hyperplane separates two balls turns out to be significantly smaller than the corresponding probabilities for hyperplanes which are not fully random in certain cases. Our results motivate studying partially random neural networks and provide a first step in this direction.

Fulton's matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen-Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

We consider the inverse problem of recovering a continuous-domain function from a finite number of noisy linear measurements. The unknown signal is modeled as the sum of a slowly varying trend and a periodic or quasi-periodic seasonal component. We formulate a variational framework for their joint recovery by introducing convex regularizations based on generalized total variation, which promote sparsity in spline-like representations. Our analysis is conducted in an infinite-dimensional setting and leads to a representer theorem showing that minimizers are splines in both components. To make the approach numerically feasible, we introduce a family of discrete approximations and prove their convergence to the original problem in the sense of $\Gamma$-convergence. This further ensures the uniform convergence of discrete solutions to their continuous counterparts. The proposed framework offers a principled approach to seasonal-trend decomposition in the presence of noise and limited measurements, with theoretical guarantees on both representation and discretization.

Generalizing work from the 1970s on the determinants of distance hypermatrices of trees, we consider the hyperdeterminants of order-$k$ Steiner distance hypermatrices of trees on $n$ vertices. We show that they can be nearly diagonalized as $k$-forms, generalizing a result of Graham-Lov\'{a}sz, implying a tensor version of ``conditional negative definiteness'', providing new proofs of previous results of the authors and Tauscheck, and resolving the conjecture that these hyperdeterminants depend only on $k$ and $n$ -- as Graham-Pollak showed for $k=2$. We conclude with some open questions.

We are concerned with positive radial solutions of the inhomogeneous elliptic equation $\Delta u+K(|x|)u^p+\mu f(|x|)=0$ on $\mathbb{R}^N$, where $N\ge 3$, $\mu>0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^{\alpha}$, $\alpha>-2$, near $r=0$, $K(r)\sim r^{\beta}$, $\beta>-2$, near $r=\infty$ and certain assumptions on $f$ are imposed, then the problem has a unique positive radial singular solution for a certain range of $\mu$. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if $p$ is between the critical Sobolev exponent $p_S(\alpha)$ and Joseph-Lundgren exponent $p_{JL}(\alpha)$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(\alpha)-2$.

This paper provides a rigorous study of tropicalizations of locally symmetric varieties. We give applications beyond tropical geometry, to the cohomology of moduli spaces as well as to the cohomology of arithmetic groups. We study two cases in detail: the case of level structures on the moduli space $\mathcal{A}_{g}$ of abelian varieties, and the special unitary case.

In this paper, we study an interaction problem between an elastic plate and a compressible viscous fluid located between the rigid bottom $z=0$ and the plate. First, by utilizing the vertical fluid dissipation, we show that $\ln\eta(t) \in L^1$ for any $t>0$ provided that $\ln\eta_0\in L^1$, ensuring that additional plate contact can form only on a set of a measure zero. Then, by utilizing the expanding capability of compressible fluid pressure, we show that all contact has to detach in finite time provided that the source force acting onto the plate is not pushing down excessively. Finally, we show that contact at any point can be detached in any given time with a strong enough source force localized around that point which is pulling the plate up. This is the first result where detachment of contact is proven in fluid-structure interaction.

In this short note, we classify pairs of conjugacy classes of the symmetric group such that any non-linear irreducible character of the symmetric group vanishes on at least one of them.

We investigate the optimal pricing strategy in a service-providing framework, where customers can become impatient and leave the system prior to service completion. In this setting, a price is quoted to an incoming customer based on the current number of customers in the system. When the quoted price is lower than the price the incoming customer is willing to pay (which follows a fixed probability distribution), then the customer joins the system and a reward equal to the quoted price is earned. A cost is incurred upon abandonment and a holding cost is incurred for customers waiting to be served. Our goal is to determine the pricing policy that maximizes the long-run average profit. Unlike traditional queueing systems without abandonments, we show that the optimal quoted prices do not always increase with the queue length in this setting. In particular, we prove that the optimal pricing policy is always uni-modal and provide conditions guaranteeing that the optimal policy is increasing in the number of customers in the system. Moreover, we introduce two heuristics that simplify the optimal dynamic pricing policy. Both heuristics admit customers until the number of customers in the system reaches a certain threshold. The cutoff static policy charges all admitted customers a fixed price while the two price policy charges one price when the arriving customer can enter service immediately and another price if the customer needs to wait. By selecting the price(s) and threshold that maximize the long-run average profit, both heuristics achieve near optimality and the two price policy provides more robustness compared to the cutoff static policy.

It is shown that the integral of the scalar curvature on a geodesic ball of radius $R$ in a three-dimensional complete manifold with nonnegative Ricci curvature is bounded above by $8\pi R$ asymptotically for large $R$ provided that the scalar curvature is bounded between two positive constants.

Given two subgroups H,K of a finite group G, the probability that a pair of random elements from H and K commutes is denoted by \pr(H,K). We address the following question. Let P be a p-subgroup of a finite group G and assume that \pr(P,P^x)\geq\e>0 for every x\in G. Is the order of P modulo O_p(G) bounded in terms of e only? With respect to this question, we establish several positive results but show that in general the answer is negative. In particular, we prove that if the composition factors of G which are isomorphic to simple groups of Lie type in characteristic p, have Lie rank at most n, then the order of P modulo O_p(G) is bounded in terms of n and e only. If P is a Sylow p-subgroup of G, then the order of P modulo O_p(G) is bounded in terms e only. Some other results of similar flavour are established. We also show that if \pr(P_1,P_2)>0 for every two Sylow p-subgroups P_1,P_2 of a profinite group G, then O_{p,p'}(G) is open in G.

We study the distortion of intermediate dimension under supercritical Sobolev mappings and also under quasiconformal or quasisymmetric homeomorphisms. In particular, we extend to the setting of intermediate dimensions both the Gehring--V\"ais\"al\"a theorem on dilatation-dependent quasiconformal distortion of dimension and Kovalev's theorem on the nonexistence of metric spaces with conformal dimension strictly between zero and one. Applications include new contributions to the quasiconformal classification of Euclidean sets and a new sufficient condition for the vanishing of conformal box-counting dimension. We illustrate our conclusions with specific consequences for Bedford--McMullen carpets, samples of Mandelbrot percolation, and product sets containing a polynomially convergent sequence factor.

We use semidefinite programming to bound the fractional cut-cover parameter of graphs in association schemes in terms of their smallest eigenvalue. We also extend the equality cases of a primal-dual inequality involving the Goemans-Williamson semidefinite program, which approximates \textsc{maxcut}, to graphs in certain coherent configurations. Moreover, we obtain spectral bounds for \textsc{max 2-sat} when the underlying graphs belong to a symmetric association scheme by means of a certain semidefinite program used to approximate quadratic programs, and we further develop this technique in order to explicitly compute the optimum value of its gauge dual in the case of distance-regular graphs.

In this article, we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of order $r$ and dimension $n$ drawn from the density \[ f(\mathcal{G}) \propto \exp\bigg(-\frac{1}{2r}\|\mathcal{G}\|^2_{\mathrm{F}}\bigg). \] For $\mathbf{w} \in \mathbb{S}^{n - 1}$, the unit-sphere in $\mathbb{R}^n$, we consider the matrix-valued contraction $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ when both $r$ and $n$ go to infinity such that $r / n \to c \in [0, \infty]$. We obtain semi-circle bulk-limits in all regimes, generalising the works of Goulart et al. (2022); Au and Garza-Vargas (2023); Bonnin (2024) in the fixed-$r$ setting. We also study the edge-spectrum. We obtain a Baik-Ben Arous-P\'{e}ch\'{e} phase-transition for the largest and the smallest eigenvalues at $r = 4$, generalising a result of Mukherjee et al. (2024) in the context of adjacency matrices of random hypergraphs. We also show that the extreme eigenvectors of $\mathcal{G} \cdot \mathbf{w}^{\otimes (r - 2)}$ contain non-trivial information about the contraction direction $\mathbf{w}$. Finally, we report some results, in the case $r = 4$, on mixed contractions $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$, $\mathbf{u}, \mathbf{v} \in \mathbb{S}^{n - 1}$. While the total variation distance between the joint distribution of the entries of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ and that of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{u}$ goes to $0$ when $\|\mathbf{u} - \mathbf{v}\| = o(n^{-1})$, the bulk and the largest eigenvalues of these two matrices have the same limit profile as long as $\|\mathbf{u} - \mathbf{v}\| = o(1)$. Furthermore, it turns out that there are no outlier eigenvalues in the spectrum of $\mathcal{G} \cdot \mathbf{u} \otimes \mathbf{v}$ when $\langle \mathbf{u}, \mathbf{v} \rangle = o(1)$.

We study the out-of-sample performance of multi-pass stochastic gradient descent (SGD) in the fundamental stochastic convex optimization (SCO) model. While one-pass SGD is known to achieve an optimal $\Theta(1/\sqrt{n})$ excess population loss given a sample of size $n$, much less is understood about the multi-pass version of the algorithm which is widely used in practice. Somewhat surprisingly, we show that in the general non-smooth case of SCO, just a few epochs of SGD can already hurt its out-of-sample performance significantly and lead to overfitting. In particular, using a step size $\eta = \Theta(1/\sqrt{n})$, which gives the optimal rate after one pass, can lead to population loss as large as $\Omega(1)$ after just one additional pass. More generally, we show that the population loss from the second pass onward is of the order $\Theta(1/(\eta T) + \eta \sqrt{T})$, where $T$ is the total number of steps. These results reveal a certain phase-transition in the out-of-sample behavior of SGD after the first epoch, as well as a sharp separation between the rates of overfitting in the smooth and non-smooth cases of SCO. Additionally, we extend our results to with-replacement SGD, proving that the same asymptotic bounds hold after $O(n \log n)$ steps. Finally, we also prove a lower bound of $\Omega(\eta \sqrt{n})$ on the generalization gap of one-pass SGD in dimension $d = \smash{\widetilde O}(n)$, improving on recent results of Koren et al.(2022) and Schliserman et al.(2024).

In this paper, we study the connected components of an obscuration-free planar polygonal light beam space modeling light propagation in optical systems composed of reflective surfaces and a focal plane. Through homotopy construction, we demonstrate that the connected components of this space are in bijection with the connected components of the reflective polygonal chains space, whose elements are the polygonal chains with their respective mirrors' orientations taken into account. In order to prove this, we introduce a topological invariant that provides an intelligible way for opticians to name homotopy-equivalent obscuration-free optical configurations thanks to previous work with polygonal chains.

A superpermutation is a sequence that contains every permutation of $n$ distinct symbols as a contiguous substring. For instance, a valid example for three symbols is a sequence that contains all six permutations. This paper introduces a new algorithm that constructs such sequences more efficiently than existing recursive and graph-theoretic methods. Unlike traditional techniques that suffer from scalability and factorial memory demands, the proposed approach builds superpermutations directly and compactly. This improves memory usage, enabling the construction of larger sequences previously considered impractical.

We describe an algorithm for sampling a low-rank random matrix $Q$ that best approximates a fixed target matrix $P\in\mathbb{C}^{n\times m}$ in the following sense: $Q$ is unbiased, i.e., $\mathbb{E}[Q] = P$; $\mathsf{rank}(Q)\leq r$; and $Q$ minimizes the expected Frobenius norm error $\mathbb{E}\|P-Q\|_F^2$. Our algorithm mirrors the solution to the efficient unbiased sparsification problem for vectors, except applied to the singular components of the matrix $P$. Optimality is proven by showing that our algorithm matches the error from an existing lower bound.

This paper presents a risk-aware safe reinforcement learning (RL) control design for stochastic discrete-time linear systems. Rather than using a safety certifier to myopically intervene with the RL controller, a risk-informed safe controller is also learned besides the RL controller, and the RL and safe controllers are combined together. Several advantages come along with this approach: 1) High-confidence safety can be certified without relying on a high-fidelity system model and using limited data available, 2) Myopic interventions and convergence to an undesired equilibrium can be avoided by deciding on the contribution of two stabilizing controllers, and 3) highly efficient and computationally tractable solutions can be provided by optimizing over a scalar decision variable and linear programming polyhedral sets. To learn safe controllers with a large invariant set, piecewise affine controllers are learned instead of linear controllers. To this end, the closed-loop system is first represented using collected data, a decision variable, and noise. The effect of the decision variable on the variance of the safe violation of the closed-loop system is formalized. The decision variable is then designed such that the probability of safety violation for the learned closed-loop system is minimized. It is shown that this control-oriented approach reduces the data requirements and can also reduce the variance of safety violations. Finally, to integrate the safe and RL controllers, a new data-driven interpolation technique is introduced. This method aims to maintain the RL agent's optimal implementation while ensuring its safety within environments characterized by noise. The study concludes with a simulation example that serves to validate the theoretical results.

We propose a new framework for multi-agent reinforcement learning (MARL), where the agents cooperate in a time-evolving network with latent community structures and mixed memberships. Unlike traditional neighbor-based or fixed interaction graphs, our community-based framework captures flexible and abstract coordination patterns by allowing each agent to belong to multiple overlapping communities. Each community maintains shared policy and value functions, which are aggregated by individual agents according to personalized membership weights. We also design actor-critic algorithms that exploit this structure: agents inherit community-level estimates for policy updates and value learning, enabling structured information sharing without requiring access to other agents' policies. Importantly, our approach supports both transfer learning by adapting to new agents or tasks via membership estimation, and active learning by prioritizing uncertain communities during exploration. Theoretically, we establish convergence guarantees under linear function approximation for both actor and critic updates. To our knowledge, this is the first MARL framework that integrates community structure, transferability, and active learning with provable guarantees.

The negative order KdV (nKdV) and the modified KdV (nmKdV) equations have two different formulations based on different hierarchy operators. Both equations can be written in terms of a nonlinear differential equation for a field $u(x,t)$ which we call the ``Lou form" of the equation. We find that for moving solutions of the nKdV equation and the nmKdV equation written in the ``Lou form" with $u(x,t) \rightarrow u (x-ct)= u(\xi) $, the equation for $u(\xi)$ can be mapped to the equation for the static solutions of the 1+1 dimensional $\phi^4$ field theory. Using this mapping we obtain a large number of solutions of the nKdV and the nmKdV equation, most of which are new. We also show that the nKdV equation can be derived from an Action Principle for both of its formulations. Furthermore, for both forms of the nmKdV equations as well as for both focusing and defocusing cases, we show that with a suitable ansatz one can decouple the $x$ and $t$ dependence of the nmKdV field $u(x,t)$ and obtain novel solutions in all the cases. We also obtain novel rational solutions of both the nKdV and the nmKdV equations.

We propose a constructive and dynamical redefinition of spatial structure, grounded in the interplay between mechanical evolution and observational acts. Rather than presupposing space as a static background, we interpret space as an emergent entity that arises through observational acts. Using the framework of pre-topologies, measurable structures, and the GNS construction, we analyze how the choice of observables and the system's time evolution dynamically determine the topological and measure-theoretic features of space. This approach highlights the observer-dependent and context-sensitive nature of spatial concepts in both classical and quantum domains.

Traditional entropy-based methods - such as cross-entropy loss in classification problems - have long been essential tools for quantifying uncertainty and disorder in data and developing artificial intelligence algorithms. However, the rapid growth of data across various domains has introduced new challenges, particularly the integration of heterogeneous datasets with intrinsic disparities. In this paper, we extend zentropy theory into the data science domain by introducing intrinsic entropy, enabling more effective learning from heterogeneous data sources. We propose a zentropy-enhanced neural network (ZENN) that simultaneously learns both energy and intrinsic entropy components, capturing the underlying structure of multi-source data. To support this, we redesign the neural network architecture to better reflect the intrinsic properties and variability inherent in diverse datasets. We demonstrate the effectiveness of ZENN on classification tasks and energy landscape reconstructions, showing its superior generalization capabilities and robustness-particularly in predicting high-order derivatives. As a practical application, we employ ZENN to reconstruct the Helmholtz energy landscape of Fe3Pt using data generated from DFT and capture key material behaviors, including negative thermal expansion and the critical point in the temperature-pressure space. Overall, our study introduces a novel approach for data-driven machine learning grounded in zentropy theory, highlighting ZENN as a versatile and robust deep learning framework for scientific problems involving complex, heterogeneous datasets.

This paper proposes a robust and computationally efficient estimation framework for fitting parametric distributions based on trimmed L-moments. Trimmed L-moments extend classical L-moment theory by downweighting or excluding extreme order statistics, resulting in estimators that are less sensitive to outliers and heavy tails. We construct estimators for both location-scale and shape parameters using asymmetric trimming schemes tailored to different moments, and establish their asymptotic properties for inferential justification using the general structural theory of L-statistics, deriving simplified single-integration expressions to ensure numerical stability. State-of-the-art algorithms are developed to resolve the sign ambiguity in estimating the scale parameter for location-scale models and the tail index for the Frechet model. The proposed estimators offer improved efficiency over traditional robust alternatives for selected asymmetric trimming configurations, while retaining closed-form expressions for a wide range of common distributions, facilitating fast and stable computation. Simulation studies demonstrate strong finite-sample performance. An application to financial claim severity modeling highlights the practical relevance and flexibility of the approach.

The Batalin-Vilkovisky formulation of a general local gauge theory can be encoded in the structure of a so-called presymplectic gauge PDE -- an almost-$Q$ bundle over the spacetime exterior algebra, equipped with a compatible presymplectic structure. In the case of a trivial bundle and an invertible presymplectic structure, this reduces to the well-known AKSZ sigma model construction. We develop an extension of the presympletic BV-AKSZ approach to describe local gauge theories with background fields. It turns out that such theories correspond to presymplectic gauge PDEs whose base spaces are again gauge PDEs describing background fields. As such, the geometric structure is that of a bundle over a bundle over a given spacetime. Gauge PDEs over backgrounds arise naturally when studying linearisation, coupling (gauge) fields to background geometry, gauging global symmetries, etc. Less obvious examples involve parameterised systems, Fedosov equations, and the so-called homogeneous (presymplectic) gauge PDEs. The latter are the gauge-invariant generalisations of the familiar homogeneous PDEs and they provide a very concise description of gauge fields on homogeneous spaces such as higher spin gauge fields on Minkowski, (A)dS, and conformal spaces. Finally, we briefly discuss how the higher-form symmetries and their gauging fit into the framework using the simplest example of the Maxwell field.

We describe a family of 1+1 classical integrable space-discrete models of the Landau-Lifshitz type through the usage of ansatz for $U$-$V$ (Lax) pair with spectral parameter satisfying the semi-discrete Zakharov-Shabat equation. The ansatz for $U$-$V$ pair is based on $R$-matrices satisfying the associative Yang-Baxter equation and certain additional properties. Equations of motion are obtained using a set of $R$-matrix identities. In the continuous limit we reproduce the previously known family of the higher rank Landau-Lifshitz equations.

Many multi-variate time series obtained in the natural sciences and engineering possess a repetitive behavior, as for instance state-space trajectories of industrial machines in discrete automation. Recovering the times of recurrence from such a multi-variate time series is of a fundamental importance for many monitoring and control tasks. For a periodic time series this is equivalent to determining its period length. In this work we present a persistent homology framework to estimate recurrence times in multi-variate time series with different generalizations of cyclic behavior (periodic, repetitive, and recurring). To this end, we provide three specialized methods within our framework that are provably stable and validate them using real-world data, including a new benchmark dataset from an injection molding machine.

Motivated by practical applications where stable long-term performance is critical-such as robotics, operations research, and healthcare-we study the problem of distributionally robust (DR) average-reward reinforcement learning. We propose two algorithms that achieve near-optimal sample complexity. The first reduces the problem to a DR discounted Markov decision process (MDP), while the second, Anchored DR Average-Reward MDP, introduces an anchoring state to stabilize the controlled transition kernels within the uncertainty set. Assuming the nominal MDP is uniformly ergodic, we prove that both algorithms attain a sample complexity of $\widetilde{O}\left(|\mathbf{S}||\mathbf{A}| t_{\mathrm{mix}}^2\varepsilon^{-2}\right)$ for estimating the optimal policy as well as the robust average reward under KL and $f_k$-divergence-based uncertainty sets, provided the uncertainty radius is sufficiently small. Here, $\varepsilon$ is the target accuracy, $|\mathbf{S}|$ and $|\mathbf{A}|$ denote the sizes of the state and action spaces, and $t_{\mathrm{mix}}$ is the mixing time of the nominal MDP. This represents the first finite-sample convergence guarantee for DR average-reward reinforcement learning. We further validate the convergence rates of our algorithms through numerical experiments.

In the Lighthill-Whitham-Richards (LWR) model for single-lane traffic, vehicle trajectories follow the first-in-first-out (FIFO) principle and can be represented by a monotone three-dimensional surface of cumulative vehicle count. In contrast, the generalized bathtub model, which describes congestion dynamics in transportation networks using relative space, typically violates the FIFO principle, making its representation more challenging. Building on the characteristic distance ordering concept, we observe that trips in the generalized bathtub model can be ordered by their characteristic distances (remaining trip distance plus network travel distance). We define a new cumulative number of trips ahead of a trip with a given remaining distance at a time instant, showing it forms a monotone three-dimensional surface despite FIFO violations. Using the inverse function theorem, we derive equivalent formulations with different coordinates and dependent variables, including special cases for Vickrey's bathtub model and the basic bathtub model. We demonstrate numerical methods based on these formulations and discuss trip-based approaches for discrete demand patterns. This study enhances understanding of the generalized bathtub model's properties, facilitating its application in network traffic flow modeling, congestion pricing, and transportation planning.

Portfolio optimization involves selecting asset weights to minimize a risk-reward objective, such as the portfolio variance in the classical minimum-variance framework. Sparse portfolio selection extends this by imposing a cardinality constraint: only $k$ assets from a universe of $p$ may be included. The standard approach models this problem as a mixed-integer quadratic program and relies on commercial solvers to find the optimal solution. However, the computational costs of such methods increase exponentially with $k$ and $p$, making them too slow for problems of even moderate size. We propose a fast and scalable gradient-based approach that transforms the combinatorial sparse selection problem into a constrained continuous optimization task via Boolean relaxation, while preserving equivalence with the original problem on the set of binary points. Our algorithm employs a tunable parameter that transmutes the auxiliary objective from a convex to a concave function. This allows a stable convex starting point, followed by a controlled path toward a sparse binary solution as the tuning parameter increases and the objective moves toward concavity. In practice, our method matches commercial solvers in asset selection for most instances and, in rare instances, the solution differs by a few assets whilst showing a negligible error in portfolio variance.

Lunar mean-motion resonances (MMRs) significantly shape cislunar dynamics beyond GEO, forming stable-unstable orbit pairs with corresponding intermingled chaotic and regular regions. The resonance zone is rigorously defined using the separatrix of unstable resonant periodic orbits surrounding stable quasi-periodic regions. Our study leverages the planar, circular, restricted three-body problem (PCR3BP) to estimate the (stable) resonance widths and (unstable) chaotic resonance zones of influence of the 2:1 and 3:1 MMRs across various Jacobi constants, employing a Poincar\'e map at perigee and presenting findings in easily interpretable geocentric orbital elements. An analysis of the semi-major axis versus eccentricity plane reveals broader regions of resonance influence than those predicted by semi-analytical models based on the perturbed Kepler problem. A comparison with high-fidelity 3-dimensional ephemeris propagation of several spacecraft - TESS, IBEX, and Spektr-R - in these regions is made, which shows good agreement with the simplified CR3BP model.

Many first-order equational theories, such as the theory of groups or boolean algebras, can be presented by a smaller set of axioms than the original one. Recent studies showed that a homological approach to equational theories gives us inequalities to obtain lower bounds on the number of axioms. In this paper, we extend this result to higher-order equational theories. More precisely, we consider simply typed lambda calculus with product and unit types and study sets of equations between lambda terms. Then, we define homology groups of the given equational theory and show that a lower bound on the number of equations can be computed from the homology groups.

This paper explores multiparameter quantum metrology using Greenberger-Horne-Zeilinger (GHZ)-type photon-added coherent states (PACS) and investigates both independent and simultaneous parameter estimation with linear and non-linear protocols, highlighting the significant potential of quantum resources to enhance precision in multiparameter scenarios. To provide a comprehensive analysis, we explicitly derive analytical expressions for the quantum Cram\'er-Rao bound (QCRB) for each protocol. Additionally, we compare the two estimation strategies, examining the behavior of their QCRBs and offering insights into the advantages and limitations of these quantum states in various contexts. Our results show that simultaneous estimation generally outperforms independent estimation, particularly in non-linear protocols. Furthermore, we analyze how the QCRB varies with the coherent state amplitude $|\alpha|^2$, the number of estimated parameters $d$, and the photon excitation order $n$ across three protocols. The results indicate that increasing $|\alpha|^2$ and decreasing $d$ improves estimation precision. For low $n$, the variation in the QCRB is similar for both symmetric and antisymmetric cases; however, at higher $n$, the antisymmetric case exhibits slightly better precision. The dependence on $d$ is comparable for both types of states. We also compare PACS-based GHZ states with NOON states and entangled coherent states, demonstrating the relative performance of each. Finally, we conclude with an analysis of homodyne detection in the context of a linear protocol, discussing its impact on estimation accuracy.

Hebbian learning is a key principle underlying learning in biological neural networks. It postulates that synaptic changes occur locally, depending on the activities of pre- and postsynaptic neurons. While Hebbian learning based on neuronal firing rates is well explored, much less is known about learning rules that account for precise spike-timing. We relate a Hebbian spike-timing-dependent plasticity rule to noisy gradient descent with respect to a natural loss function on the probability simplex. This connection allows us to prove that the learning rule eventually identifies the presynaptic neuron with the highest activity. We also discover an intrinsic connection to noisy mirror descent.

We study classical continuous automorphisms of the torus (CAT maps) from the viewpoint of the Koopman theory. We find analytical formulae for Koopman modes defined coherently on the whole of the torus, and their decompositions associated with the partition of the torus into ergodic components. The spectrum of the Koopman operator is studied in four cases of CAT maps: cyclic, quasi-cyclic, critical (transition from quasi-cyclic to chaotic behaviour) and chaotic. We generalise these results to quantum CAT maps defined onto a noncommutative torus (and on its dual space). Finally, we study usual quantum chaos indicators onto quantum CAT maps from the viewpoint of the Koopman picture. The analogy with the classical case suggests that couples of these indicators are in fact necessary to certify a quantum chaotic behaviour.

Decentralized optimization has become vital for leveraging distributed data without central control, enhancing scalability and privacy. However, practical deployments face fundamental challenges due to heterogeneous computation speeds and unpredictable communication delays. This paper introduces a refined model of Asynchronous Decentralized Stochastic Gradient Descent (ADSGD) under practical assumptions of bounded computation and communication times. To understand the convergence of ADSGD, we first analyze Asynchronous Stochastic Block Coordinate Descent (ASBCD) as a tool, and then show that ADSGD converges under computation-delay-independent step sizes. The convergence result is established without assuming bounded data heterogeneity. Empirical experiments reveal that ADSGD outperforms existing methods in wall-clock convergence time across various scenarios. With its simplicity, efficiency in memory and communication, and resilience to communication and computation delays, ADSGD is well-suited for real-world decentralized learning tasks.

We study sequential decision-making in batched nonparametric contextual bandits, where actions are selected over a finite horizon divided into a small number of batches. Motivated by constraints in domains such as medicine and marketing -- where online feedback is limited -- we propose a nonparametric algorithm that combines adaptive k-nearest neighbor (k-NN) regression with the upper confidence bound (UCB) principle. Our method, BaNk-UCB, is fully nonparametric, adapts to the context dimension, and is simple to implement. Unlike prior work relying on parametric or binning-based estimators, BaNk-UCB uses local geometry to estimate rewards and adaptively balances exploration and exploitation. We provide near-optimal regret guarantees under standard Lipschitz smoothness and margin assumptions, using a theoretically motivated batch schedule that balances regret across batches and achieves minimax-optimal rates. Empirical evaluations on synthetic and real-world datasets demonstrate that BaNk-UCB consistently outperforms binning-based baselines.