The Maximum Weight Independent Set problem is a fundamental NP-hard problem in combinatorial optimization with several real-world applications. Given an undirected vertex-weighted graph, the problem is to find a subset of the vertices with the highest possible weight under the constraint that no two vertices in the set can share an edge. An important part of solving this problem in both theory and practice is data reduction rules, which several state-of-the-art algorithms rely on. However, the most complicated rules are often not used in applications since the time needed to check them exhaustively becomes infeasible. In this work, we introduce three main results. First, we introduce several new data reduction rules and evaluate their effectiveness on real-world data. Second, we use a machine learning screening algorithm to speed up the reduction phase, thereby enabling more complicated rules to be applied. Our screening algorithm consults a Graph Neural Network oracle to decide if the probability of successfully reducing the graph is sufficiently large. For this task, we provide a dataset of labeled vertices for use in supervised learning. We also present the first results for this dataset using established Graph Neural Network architectures. Third, we present a new concurrent metaheuristic called Concurrent Difference-Core Heuristic. On the reduced instances, we use our new metaheuristic combined with iterated local search, called CHILS (Concurrent Hybrid Iterated Local Search). For this iterated local search, we provide a new implementation specifically designed to handle large graphs of varying densities. CHILS outperforms the current state-of-the-art on all commonly used benchmark instances, especially the largest ones.

We prove the Martingale Convergence Theorem by using the work of L. Dubins and I. Monroe about embedding a given discrete-time martingale in the sample paths of a Brownian motion.

We study the second-order asymptotics around the superdiffusive strong law~\cite{MMW} of a multidimensional driftless diffusion with oblique reflection from the boundary in a generalised parabolic domain. In the unbounded direction we prove the limit is Gaussian with the usual diffusive scaling, while in the appropriately scaled cross-sectional slice we establish convergence to the invariant law of a reflecting diffusion in a unit ball. Using the separation of time scales, we also show asymptotic independence between these two components. The parameters of the limit laws are explicit in the growth rate of the boundary and the asymptotic diffusion matrix and reflection vector field. A phase transition occurs when the domain becomes too narrow, in which case we prove that the central limit theorem for the unbounded component fails.

We present a so-called universal convergence theorem for inexact primal-dual penalty and augmented Lagrangian methods that can be applied to a large number of such methods and reduces their convergence analysis to verification of some simple conditions on sequences generated by these methods. If these conditions are verified, then both primal and dual convergence follow directly from the universal convergence theorem. This theorem allows one not only to derive standard convergence theorems for many existing primal-dual penalty and augmented Lagrangian methods in a unified and straightforward manner, but also to strengthen and generalize some of these theorems. In particular, we show how with the use of the universal convergence theorem one can significantly improve some existing results on convergence of a primal-dual rounded weighted $\ell_1$-penalty method, an augmented Lagrangian method for cone constrained optimization, and some other primal-dual methods.

Motivated by the critical need for unmanned aerial vehicles (UAVs) to patrol grid systems in hazardous and dynamically changing environments, this study addresses a routing problem aimed at minimizing the time-average Age of Information (AoI) for edges in general graphs. We establish a lower bound for all feasible patrol policies and demonstrate that this bound is tight when the graph contains an Eulerian cycle. For graphs without Eulerian cycles, it becomes challenging to identify the optimal patrol strategy due to the extensive range of feasible options. Our analysis shows that restricting the strategy to periodic sequences still results in an exponentially large number of possible strategies. To address this complexity, we introduce two polynomial-time approximation schemes, each involving a two-step process: constructing multigraphs first and then embedding Eulerian cycles within these multigraphs. We prove that both schemes achieve an approximation ratio of 2. Further, both analytical and numerical results suggest that evenly and sparsely distributing edge visits within a periodic route significantly reduces the average AoI compared to strategies that merely minimize the route travel distance. Building on this insight, we propose a heuristic method that not only maintains the approximation ratio of 2 but also ensures robust performance across varying random graphs.

We compute an analogue of Pascal's triangle enriched in bilinear forms over a finite field. This gives an arithmetically meaningful count of the ways to choose $j$ embeddings into an \'etale extension of degree $n$. We also compute a quadratic twist. These (twisted) enriched binomial coefficients are defined in joint work of Brugall\'e and the second-named author, building on work of Garibaldi, Merkurjev, and Serre. Such binomial coefficients support curve counting results over non-algebraically closed fields, using $A^1$-homotopy theory.

We consider the framework of methods for unconstrained minimization that are, in each iteration, restricted to a model that is only a valid approximation to the objective function on some affine subspace containing an incumbent point. These methods are of practical interest in computational settings where derivative information is either expensive or impossible to obtain. Recent attention has been paid in the literature to employing randomized matrix sketching for generating the affine subspaces within this framework. We consider a relatively straightforward, deterministic augmentation of such a generic subspace optimization method. In particular, we consider a sequential optimization framework where actions consist of one-dimensional linear subspaces and rewards consist of (approximations to) the magnitudes of directional derivatives computed in the direction of the action subspace. Reward maximization in this context is consistent with maximizing lower bounds on descent guaranteed by first-order Taylor models. This sequential optimization problem can be analyzed through the lens of dynamic regret. We modify an existing linear upper confidence bound (UCB) bandit method and prove sublinear dynamic regret in the subspace optimization setting. We demonstrate the efficacy of employing this linear UCB method in a setting where forward-mode algorithmic differentiation can provide directional derivatives in arbitrary directions and in a derivative-free setting. For the derivative-free setting, we propose SS-POUNDers, an extension of the derivative-free optimization method POUNDers that employs the linear UCB mechanism to identify promising subspaces. Our numerical experiments suggest a preference, in either computational setting, for employing a linear UCB mechanism within a subspace optimization method.

In any semigroup $S$ satisfying the Strong Folner Condition, there are three natural notions of density for a subset $A$ of $S$: Folner density $d(A)$, Banach density $d^*(A)$, and translation density $d_t(A)$. If $S$ is commutative or left cancellative, it is known that these three notions coincide. We shall show that these notions coincide for every semigroup $S$ which satisfies the Strong Folner Condition. Using this fact, we solve a problem that has been open for decades, showing that the set of ultrafilters every member of which has positive Folner density is a two sided ideal of $\beta S$. We also show that, if $S$ is a left amenable semigroup, then the set of ultrafilters every member of which has positive Banach density is a two sided ideal of $\beta S$. We investigate the density properties of subsets of $S$ in the case in which the minimal left ideals of the Stone-\v{C}ech compactification $\beta S$ are singletons. This occurs in many familiar examples, including all semilattices and all semigroups which have a right zero. We show that this is equivalent to the statement that $S$ satisfies the Strong Folner Condition and that, for every subset $A$ of $S$, $d(A)\in \{0,1\}$. We also examine the relation between the density properties of two semigroups when one is a quotient of the other. The Folner density of a subset of $S$ is always determined by some Folner net in $S$. We show that an arbitrary Folner net in $S$ determines the density of all of the subsets of $S$. And we prove that, if $S$ and $T$ are left amenable semigroups, then $d^*(A\times B)=d^*(A)d^*(B)$ for every subset $A$ of $S$ and every subset $B$ of $T$.

Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$, i.e., a subset containing no $3$ points on the same line. This problem was proposed by Erd\H{o}s under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n/s)$ when $n^{1/3}\leq s\leq n$ and $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=\Theta(\sqrt{n})$, our upper and lower bounds differ only by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erd\H{o}s, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $\Omega((n/\log{s})^{1/3})$ can always be found in $P$. When $s=\Theta(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

We prove congruences of Hecke eigenvalues between cuspidal Hilbert newforms $f_{79}$ and $h_{79}$ over $F=\mathbb Q(\sqrt{5})$, of weights (2,2) and (2,4) respectively, level of norm 79. In the main example, the modulus is a divisor of 5 in some coefficient field, in the secondary example a divisor of 2. The former allows us to prove that the 4-dimensional mod-5 representation of $\mathrm{Gal}(\overline{\mathbb Q} / \mathbb Q)$ on the 3rd cohomology of a certain Calabi-Yau threefold comes from a Siegel modular form $F_{79}$ of genus 2, weight 3 and paramodular level 79. This is a weak form of a conjecture of Golyshev and van Straten. In aid of this, we prove also a congruence of Hecke eigenvalues between $F_{79}$ and the Johnson-Leung-Roberts lift $\mathrm{JR}(h_{79})$, which has weight 3 and paramodular level $79\times 5^2$.

We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the "vanilla" PG method, achieving the best-known iteration complexity for finding an approximate stationary point of the problem. We then develop an "auto-conditioned" projected gradient (AC-PG) variant that achieves the same iteration complexity without requiring the input of the Lipschitz constant of the gradient or any line search procedure. The key idea is to estimate the Lipschitz constant using first-order information gathered from the previous iterations, and to show that the error caused by underestimating the Lipschitz constant can be properly controlled. We then generalize the PG methods to the stochastic setting, by proposing a stochastic projected gradient (SPG) method and a variance-reduced stochastic gradient (VR-SPG) method, achieving new complexity bounds in different oracle settings. We also present auto-conditioned stepsize policies for both stochastic PG methods and establish comparable convergence guarantees.

The transcendent part of the Drinfeld p-adic upper half plane is shown to be a Polish space. Using Radon measures associated with regular differential 1-forms invariant under Schottky groups allows to construct self-adjoint diffusion operators as Laplacian integral operators with kernel functions determined by the p-adic absolute value on the complex p-adic numbers. Their spectra are explicitly calculated and the corresponding Cauchy problems for their associated heat equations are found to be uniquely solvable and to determine Markov processes having paths which are cadlag. The heat kernels are shown to have explicitly given distribution functions, as well as boundary value problems associated with the heat equations under Dirchlet and von Neumann conditions are solved.

For manifolds equipped with group actions, we have the following natural question: To what extent does the equivariant cohomology determine the equivariant diffeotype? We resolve this question for Hamiltonian circle actions on compact, connected symplectic four-manifolds. They are equivariantly diffeomorphic if and only if their equivariant cohomology rings are isomorphic as algebras over the equivariant cohomology of a point. In fact, we prove a stronger claim: each isomorphism between their equivariant cohomology rings is induced by an equivariant diffeomorphism.

For a prime $\mathfrak{p} \subseteq \mathbb{F}_{q}[T]$ and a positive integer $r$, we consider the generalised Jacobian $J_{0}(\mathfrak{n})_{\mathbf{m}}$ of the Drinfeld modular curve $X_{0}(\mathfrak{n})$ of level $\mathfrak{n}=\mathfrak{p}^r$, with respect to the modulus~$\mathbf{m}$ consisting of all cusps on the modular curve. We show that the $\ell$-primary part of the group $J_{0}(\mathfrak{n})_{\mathbf{m}}(\mathbb{F}_{q}(T))_{\rm{tor}}[\ell^{\infty}]$ is trivial for all primes $\ell$ not dividing $q(q^{2}-1)$. Our results establish a function field analogue to those of Yamazaki--Yang for the classical case.

The aim of this article is to prove the derived equivalence for a local model of the simple flop of type $G_2^{\dagger}$, which was found by Kanemitsu. This flop is the only known simple flop that comes from a non-homogeneous roof. The proof of the derived equivalence is done by using tilting bundles, and also produces a noncommutative crepant resolution of the singularity that is derived equivalent to both sides of the flop.

We propose and analyse a novel, fully discrete numerical algorithm for the approximation of the generalised Stokes system forced by transport noise -- a prototype model for non-Newtonian fluids including turbulence. Utilising the Gradient Discretisation Method, we show that the algorithm is long-term stable for a broad class of particular Gradient Discretisations. Building on the long-term stability and the derived continuity of the algorithm's solution operator, we construct two sequences of approximate invariant measures. At the moment, each sequence lacks one important feature: either the existence of a limit measure, or the invariance with respect to the discrete semigroup. We derive an abstract condition that merges both properties, recovering the existence of an invariant measure. We provide an example for which invariance and existence hold simultaneously, and characterise the invariant measure completely. We close the article by conducting two numerical experiments that show the influence of transport noise on the dynamics of power-law fluids; in particular, we find that transport noise enhances the dissipation of kinetic energy, the mixing of particles, as well as the size of vortices.

Filtering is concerned with online estimation of the state of a dynamical system from partial and noisy observations. In applications where the state is high dimensional, ensemble Kalman filters are often the method of choice. This paper establishes long-time accuracy of ensemble Kalman filters. We introduce conditions on the dynamics and the observations under which the estimation error remains small in the long-time horizon. Our theory covers a wide class of partially-observed chaotic dynamical systems, which includes the Navier-Stokes equations and Lorenz models. In addition, we prove long-time accuracy of ensemble Kalman filters with surrogate dynamics, thus validating the use of machine-learned forecast models in ensemble data assimilation.

In this note, we define material-uniform hyperelastic bodies (in the sense of Noll) containing discrete disclinations and dislocations, and study their properties. We show in a rigorous way that the size of a disclination is limited by the symmetries of the constitutive relation; in particular, if the symmetry group of the body is discrete, it cannot admit arbitrarily small, yet non-zero, disclinations. We then discuss the application of these observations to the derivations of models of bodies with continuously-distributed defects.

Adapting a homotopy reconstruction theorem for general metric compacta, we show that every countable metric or ultrametric compact space can be topologically reconstructed as the inverse limit of a sequence of finite $T_0$ spaces which are finer approximations of the space.

We study the operator-valued partial derivative associated with covariance matrices on a von Neumann algebra B. We provide a cumulant characterization for the existence of conjugate variables and study some structure implications of their existence. Namely, we show that the center of the von Neumann algebra generated by B and its relative commutant is the center of B.

We give Euler-like recursive formulas for the $t$-colored partition function when $t=2$ or $t=3,$ as well as for all $t$-regular partition functions. In particular, we derive an infinite family of ``triangular number" recurrences for the $3$-colored partition function. Our proofs are inspired by the recent work of Gomez, Ono, Saad, and Singh on the ordinary partition function and make extensive use of $q$-series identities for $(q;q)_{\infty}$ and $(q;q)_{\infty}^3.$

Let $M$ be a closed surface, $q\geq 2$ and $n\geq 2$. In this paper, we analyze the Coxeter-type quotient group $B_n(M)(q)$ of the surface braid group $B_{n}(M)$ by the normal closure of the element $\sigma_1^q$, where $\sigma_1$ is the classic Artin generator of the Artin braid group $B_n$. Also, we study the Coxeter-type quotient groups obtained by taking the quotient of $B_n(M)$ by the commutator subgroup of the respective pure braid group $[P_n(M),P_n(M)]$ and adding the relation $\sigma_1^q=1$, when $M$ is a closed orientable surface or the disk.

One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where the approximating object is not constant, but a sequence as well. We extend Gaussian approximation results for the partial sum process by allowing each summand to be multiplied by a data-dependent matrix. The results allow for serial dependence of the data, and for high-dimensionality of both the data and the multipliers. In the finite-dimensional and locally-stationary setting, we obtain a functional central limit theorem as a direct consequence. An application to sequential testing in non-stationary environments is described.

We develop a class of functions Omega_N(x; mu, nu) in N-dimensional space concentrated around a spherical shell of the radius mu and such that, being convoluted with an isotropic Gaussian function, these functions do not change their expression but only a value of its 'width' parameter, nu. Isotropic Gaussian functions are a particular case of Omega_N(x; mu, nu) corresponding to mu = 0. Due to their features, these functions are an efficient tool to build approximations to smooth and continuous spherically-symmetric functions including oscillating ones. Atomic images in limited-resolution maps of the electron density, electrostatic scattering potential and other scalar fields studied in physics, chemistry, biology, and other natural sciences are examples of such functions. We give simple analytic expressions of Omega_N(x; mu, nu) for N = 1, 2, 3 and analyze properties of these functions. Representation of oscillating functions by a sum of Omega_N(x; mu, nu) allows calculating distorted maps for the same cost as the respective theoretical fields. We give practical examples of such representation for the interference functions of the uniform unit spheres for N = 1, 2, 3 that define the resolution of the respective images. Using the chain rule and analytic expressions of the Omega_N(x; mu, nu) derivatives makes simple refinement of parameters of the models which describe these fields.

In this paper, we consider nonparametric estimation over general Dirichlet metric measure spaces. Unlike the more commonly studied reproducing kernel Hilbert space, whose elements may be defined pointwise, a Dirichlet space typically only contain equivalence classes, i.e. its elements are only unique almost everywhere. This lack of pointwise definition presents significant challenges in the context of nonparametric estimation, for example the classical ridge regression problem is ill-posed. In this paper, we develop a new technique for renormalizing the ridge loss by replacing pointwise evaluations with certain \textit{local means} around the boundaries of obstacles centered at each data point. The resulting renormalized empirical risk functional is well-posed and even admits a representer theorem in terms of certain equilibrium potentials, which are truncated versions of the associated Green function, cut-off at a data-driven threshold. We study the global, out-of-sample consistency of the sample minimizer, and derive an adaptive upper bound on its convergence rate that highlights the interplay of the analytic, geometric, and probabilistic properties of the Dirichlet form. We also construct a simple regressogram type estimator that achieves the minimax optimal estimation rate over certain $L^p$ subsets of a Dirichlet ball with some knowledge of the geometry of the metric measure space. Our framework notably does not require the smoothness of the underlying space, and is applicable to both manifold and fractal settings. To the best of our knowledge, this is the first paper to obtain out-of-sample convergence guarantees in the framework of general metric measure Dirichlet spaces.

Radau IIA methods, specifically the adaptive order radau method in Fortran due to Hairer, are known to be state-of-the-art for the high-accuracy solution of highly stiff ordinary differential equations (ODEs). However, the traditional implementation was specialized to a specific range of tolerance, in particular only supporting 5th, 9th, and 13th order versions of the tableau and only derived in double precision floating point, thus limiting the ability to be truly general purpose for highly accurate scenarios. To alleviate these constraints, we implement an adaptive-time adaptive-order Radau method which can derive the coefficients for the Radau IIA embedded tableau to any order on the fly to any precision. Additionally, our Julia-based implementation includes many modernizations to improve performance, including improvements to the order adaptation scheme and improved linear algebra integrations. In a head-to-head benchmark against the classic Fortran implementation, we demonstrate our implementation is approximately 2x across a range of stiff ODEs. We benchmark our algorithm against several well-reputed numerical integrators for stiff ODEs and find state-of-the-art performance on several test problems, with a 1.5-times speed-up over common numerical integrators for stiff ODEs when low error tolerance is required. The newly implemented method is distributed in open source software for free usage on stiff ODEs.

We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 - 1$ is rigid in $\mathbb{R}^d$. For larger values of $d$, we achieve an approximate result: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $(n+2d)/2 - 1$ is rigid in $\mathbb{R}^d$. This bound is tight up to a factor of two in the coefficient of $d$. As a byproduct of our proof, we also obtain the following result, which may be of independent interest: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $d$ has pseudoachromatic number at least $d+1$; namely, the vertex set of such a graph can be partitioned into $d+1$ subsets such that there is at least one edge between each pair of subsets. This is tight.

Labourie raised the question of determining the possible asymptotics for the growth rate of compact $k$-surfaces, counted according to energy, in negatively curved $3$-manifolds, indicating the possibility of a theory of thermodynamical formalism for this class of surfaces. Motivated by this question and by analogous results for the geodesic flow, we prove a number of results concerning the asymptotic behavior of high energy $k$-surfaces, especially in relation to the curvature of the ambient space. First, we determine a rigid upper bound for the growth rate of quasi-Fuchsian $k$-surfaces, counted according to energy, and with asymptotically round limit set, subject to a lower bound on the sectional curvature of the ambient space. We also study the marked energy spectrum for $k$-surfaces, proving a number of domination and rigidity theorems in this context. Finally, we show that the marked area and energy spectra for $k$-surfaces in $3$-dimensional manifolds of negative curvature are asymptotic if and only if the sectional curvature is constant.

The contributions in this note begin with a new characterization of (positive) scalar multiples of minimal tripotents in a general JB$^*$-triple $E$, proving that a non-zero element $a\in E$ is a positive scalar multiple of a minimal tripotent in $E$ if, and only if, its inner quadratic annihilator (that is, the set $^{\perp_{q}}\!\{a\} = \{ b\in E: \{a,b,a\} =0\}$) is maximal among all inner quadratic annihilators of single elements in $E$. We subsequently apply this characterization to the study of surjective additive maps between atomic JBW$^*$-triples preserving truncations in both directions. Let $A: E\to F$ be a surjective additive mapping between atomic JBW$^*$-triples, where $E$ contains no one-dimensional Cartan factors as direct summands. We show that $A$ preserves truncations in both directions if, and only if, there exists a bijection $\sigma: \Gamma_1\to \Gamma_2$, a bounded family $(\gamma_k)_{k\in \Gamma_1}\subseteq \mathbb{R}^+$, and a family $(\Phi_k)_{k\in \Gamma_1},$ where each $\Phi_k$ is a (complex) linear or a conjugate-linear (isometric) triple isomorphism from $C_k$ onto $\widetilde{C}_{\sigma(k)}$ satisfying $\inf_{k} \{\gamma_k \} >0,$ and $$A(x) = \Big( \gamma_{k} \Phi_k \left(\pi_k(x)\right) \Big)_{k\in\Gamma_1},\ \hbox{ for all } x\in E,$$ where $\pi_k$ denotes the canonical projection of $E$ onto $C_k.$

We establish the existence of weak solutions for a nonlocal Klausmeier model within a small time interval $[0, T)$. The Klausmeier model is a coupled, nonlinear system of partial differential equations governing plant biomass and water dynamics in semiarid regions. The original model posits that plants disperse their seed according to classical diffusion. Instead, we opt for a nonlocal diffusive operator in alignment with ecological field data that validates long-range dispersive behaviors of plants and seeds. The equations, defined on a finite interval in $\mathbb{R}$, feature homogeneous Dirichlet boundary conditions for the water equation and nonlocal Dirichlet volume constraints for the plant biomass equation. The nonlocal operator involves convolution with a symmetric and spatially extended convolution kernel possessing mild integrability and regularity properties. We employ the Galerkin method to establish the existence of weak solutions. The key challenge comes from the nonlocal operator; we define it on a subspace of $L^{2}$ instead of $H^{1}$, precluding the use of Aubin's compactness theorem to prove the weak convergence of nonlinear terms. To overcome this, we modify the model and introduce two new equations for the spatial derivatives of plant biomass and water. This procedure allows us to recover enough regularity to establish compactness and complete the proof.

We define a fractional Ito stochastic integral with respect to a randomly scaled fractional Brownian motion via an $S$-transform approach. We investigate the properties of this stochastic integral, prove the Ito formula for functions of such stochastic integrals and apply this Ito formula for investigation of related generalized time-fractional evolution equations.

We prove that a variety of examples of minimal complex surfaces admit exotic diffeomorphisms, providing the first known instances of exotic diffeomorphisms of irreducible 4-manifolds. We also give sufficient conditions for the boundary Dehn twist on a spin 4-manifold with $S^3$ boundary to be non-trivial in the relative mapping class group. This gives many new examples of non-trivial boundary Dehn twists.

We show that the {\it full} mapping class group of any orientable closed surface with punctures admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension. This was proved for closed orientable surfaces and for {\it pure} mapping class groups by Aramayona and Mart\'inez P\'erez. As a consequence of our result we also obtain the proper geometric dimension of {\it full} spherical braid groups.

Microscopic behaviors of chemical reactions can be described by a random time-changed Poisson process, whose large-volume limit determines the macroscopic behaviors of species concentrations, including both typical and non-typical trajectories. When the reaction intensities (or fluxes) exhibit a separation of fast-slow scales, the macroscopic typical trajectory is governed by a system of $\varepsilon$-dependent nonlinear reaction rate equations (RRE), while the non-typical trajectories deviating from the typical ones are characterized by an $\varepsilon$-dependent exponentially nonlinear Hamilton-Jacobi equation (HJE). In this paper, for general chemical reactions, we study the fast-slow limit as $\varepsilon\to 0$ for the viscosity solutions of the associated HJE with a state-constrained boundary condition. We identify the limiting effective HJE on a slow manifold, along with an effective variational representation for the solution. Through the uniform convergence of the viscosity solutions and the $\Gamma$-convergence of the variational solution representations, we rigorously show that all non-typical (and also typical) trajectories are concentrated on the slow manifold and the effective macroscopic dynamics are described by the coarse-grained RRE and HJE, respectively. This approach for studying the fast-slow limit is applicable to, but not limited to, reversible chemical reactions described by gradient flows.

We prove that a $k$-regulous function defined on a non-singular affine variety can always be extended to the entire affine space.

We prove a structural theorem for generalized arithmetic progressions in $\F_p$ which contain a large product set of two other progressions.

This paper studies how the Elo rating system behaves when the underlying modelling assumptions are not met.

A rank $n$ Higgs bundle $(E,\theta)$ is called generically regular nilpotent if $\theta^n=0$ but $\theta^{n-1}\neq 0$. We show that for a generically regular nilpotent Higgs bundle, if it admits a harmonic metric, then its graded Higgs bundle admits a unique maximal harmonic metric. The proof relies on a generalization of Kalka-Yang's theorem for prescribed curvature equation over a non-compact hyperbolic surface to a coupled system. As an application, we show that the branched set of a branched minimal disk in $\mathbb{H}^3$ has to be the critical set of some holomorphic self-map of $\mathbb{D}$.

We consider model-based derivative-free optimization (DFO) for large-scale problems, based on iterative minimization in random subspaces. We provide the first worst-case complexity bound for such methods for convergence to approximate second-order critical points, and show that these bounds have significantly improved dimension dependence compared to standard full-space methods, provided low accuracy solutions are desired and/or the problem has low effective rank. We also introduce a practical subspace model-based method suitable for general objective minimization, based on iterative quadratic interpolation in subspaces, and show that it can solve significantly larger problems than state-of-the-art full-space methods, while also having comparable performance on medium-scale problems when allowed to use full-dimension subspaces.

A review of economic approaches showed the lack of a universal method for assessing management decisions in the face of an increasing volume of analyzed data and changing parameters of the external environment. The method of integral indicators is proposed. Integral indicators are one of the modern methods for researching the behavior of an enterprise. It provides an assessment of the impact of the external environment. It shows the ability of the enterprise to adapt to new conditions. The dynamics of the correlation indicator shows the reaction of the enterprise to the impact of external factors. The purpose of the scientific work was achieved: the optimal control of the enterprise was carried out in the conditions of changing the parameters of the external environment For this, the model of the economic object and the method of its analysis are formalized. The structure of an economic object (enterprise) is given. The characteristics of the parameters of the external environment are given. The state of an economic object (enterprise) is modeled taking into account the influence of the external environment. With the help of the software package created by the author, six optimal options for control decisions have been analyzed. The state of an economic object has been modeled depending on the state of the external environment by 5,000 parameters. The research showed significant changes in the values of the correlation of the parameters of the system and the intensity of business processes when the conditions for the functioning of the system change. The optimal control of an economic object (enterprise) is selected according to the integral indicator.

This paper is devoted to establishing the full scaling limit theorems for multivariate Hawkes processes. Under some mild conditions on the exciting kernels, we develop a new way to prove that after a suitable time-spatial scaling, the asymptotically critical multivariate Hawkes processes converge weakly to the unique solution of a multidimensional stochastic Volterra equation with convolution kernel being the potential measure associated to a matrix-valued extended Bernstein function. Also, based on the observation of their affine property and generalized branching property, we provide an exponential-affine representation of the Fourier-Laplace functional of scaling limits in terms of the unique solutions of multidimensional Riccati-Volterra equations with measure kernel. The regularity of limit processes and their alternate representations are also investigated by using the potential theory of L\'evy subordinators.

In this manuscript we present the tensor-train reduced basis method, a novel projection-based reduced-order model for the efficient solution of parameterized partial differential equations. Despite their popularity and considerable computational advantages with respect to their full order counterparts, reduced-order models are typically characterized by a considerable offline computational cost. The proposed approach addresses this issue by efficiently representing high dimensional finite element quantities with the tensor train format. This method entails numerous benefits, namely, the smaller number of operations required to compute the reduced subspaces, the cheaper hyper-reduction strategy employed to reduce the complexity of the PDE residual and Jacobian, and the decreased dimensionality of the projection subspaces for a fixed accuracy. We provide a posteriori estimates that demonstrate the accuracy of the proposed method, we test its computational performance for the heat equation and transient linear elasticity on three-dimensional Cartesian geometries.

A Toda flow is constructed on a space of bounded initial data through Sato-Segal-Wilson theory. The flow is described by the Weyl functions of the underlying Jacobi operators. This is a continuation of the previous work on the KdV flow.

Let $S$ be a $*$-monoid and let $a,b,c$ be elements of $S$. We say that $a$ is $(b,c)$-core-EP invertible if there exist some $x$ in $S$ and some nonnegative integer $k$ such that $cax(ca)^{k}c=(ca)^{k}c$, $x{\mathcal R}(ca)^{k}b$ and $x{\mathcal L}((ca)^{k}c)^{*}$. This terminology can be seen as an extension of the $w$-core-EP inverse and the $(b,c)$-core inverse. It is explored when $(b,c)$-core-EP invertibility implies $w$-core-EP invertibility. Another accomplishment of our work is to establish the criteria for the $(b,c)$-core-EP inverse of $a$ and to clarify the relations between the $(b,c)$-inverse, the core inverse, the core-EP inverse, the $w$-core inverse, the $(b,c)$-core inverse and the $(b,c)$-core-EP inverse. As an application, we improve a result in the literature focused on $(b,c)$-core inverses. We then establish the criterion for the $(B,C)$-core-EP inverse of $A$ in complex matrices, and give the solution to the system of matrix equations.

In this paper we consider an unconstrained stochastic optimization problem where the objective function exhibits a high order of smoothness. In particular, we propose a stochastic first-order method (SFOM) with multi-extrapolated momentum, in which multiple extrapolations are performed in each iteration, followed by a momentum step based on these extrapolations. We show that our proposed SFOM with multi-extrapolated momentum can accelerate optimization by exploiting the high-order smoothness of the objective function $f$. Specifically, assuming that the gradient and the $p$th-order derivative of $f$ are Lipschitz continuous for some $p\ge2$, and under some additional mild assumptions, we establish that our method achieves a sample complexity of $\widetilde{\mathcal{O}}(\epsilon^{-(3p+1)/p})$ for finding a point $x$ satisfying $\mathbb{E}[\|\nabla f(x)\|]\le\epsilon$. To the best of our knowledge, our method is the first SFOM to leverage arbitrary order smoothness of the objective function for acceleration, resulting in a sample complexity that strictly improves upon the best-known results without assuming the average smoothness condition. Finally, preliminary numerical experiments validate the practical performance of our method and corroborate our theoretical findings.

In this paper, we study the nonexistence of global weak solutions for a wave equation with nonlinear memory and damping terms. We give an answer to an open problem posed in [M. D'Abbicco, A wave equation with structural damping and nonlinear memory, Nonlinear Differ. Equ. Appl. 21 (2014), 751-773]. Moreover, comparing with the existing results, our results do not require any positivity condition of the initial values. The proof of our results is based on the asymptotic properties of solutions for an integral inequality.

We study non-convex delayed-noise online optimization problems by evaluating dynamic regret in the non-stationary setting when the loss functions are quasar-convex. In particular, we consider scenarios involving quasar-convex functions either with a Lipschitz gradient or weakly smooth and, for each case, we ensure bounded dynamic regret in terms of cumulative path variation achieving sub-linear regret rates. Furthermore, we illustrate the flexibility of our framework by applying it to both theoretical settings such as zeroth-order (bandit) and also to practical applications with quadratic fractional functions. Moreover, we provide new examples of non-convex functions that are quasar-convex by proving that the class of differentiable strongly quasiconvex functions (Polyak 1966) are strongly quasar-convex on convex compact sets. Finally, several numerical experiments validate our theoretical findings, illustrating the effectiveness of our approach.

We study a non-exchangeable multi-agent system and rigorously derive a strong form of the mean-field limit. The convergence of the connection weights and the initial data implies convergence of large-scale dynamics toward a deterministic limit given by the corresponding extended Vlasov PDE, at any later time and any realization of randomness. This is established on what we call a bi-coupling distance defined through a convex optimization problem, which is an interpolation of the optimal transport between measures and the fractional overlay between graphs. The proof relies on a quantitative stability estimate of the so-called observables, which are tensorizations of agent laws and graph homomorphism densities. This reveals a profound relationship between mean-field theory and graph limiting theory, intersecting in the study of non-exchangeable systems.

We study the structure of normal operators of double fibration transforms with conjugate points. Examples of double fibration transforms include Radon transforms, $d$-plane transforms on the Euclidean space, geodesic X-ray transforms, light-ray transforms, and ray transforms defined by null bicharacteristics associated with real principal type operators. We show that, under certain stable conditions on the distribution of conjugate points, the normal operator splits into an elliptic pseudodifferential operator and several Fourier integral operators, depending on the degree of the conjugate points. These results were first proved for geodesic X-ray transforms by Holman and Uhlmann (Journal of Differential Geometry, {\bf 108} (2018), pp.459--494).

We denote a path on $t$ vertices as $P_t$ and a cycle on $t$ vertices as $C_t$. For two vertex-disjoint graphs $G_1$ and $G_2$, the {\em union} $G_1\cup G_2$ is the graph with $V(G_1\cup G_2)=V(G_1)\cup V(G_2)$ and $E(G_1\cup G_2)=E(G_1)\cup E(G_2)$. A {\em diamond} (resp. {\em gem}) is a graph consisting of a $P_3$ (resp. $P_4$) and a new vertex adjacent to all vertices of the $P_3$ (resp. $P_4$), and a {\em butterfly} is a graph consisting of two triangles that share one vertex. In this paper, we show that $\chi(G)\le 3\omega(G)-2$ if $G$ is a ($P_2\cup P_4$, gem)-free graph, $\chi(G)\le \frac{\omega(G)^2+3\omega(G)-2}{2}$ if $G$ is a ($P_2\cup P_4$, butterfly)-free graph. We also study the class of ($P_2\cup P_4$, diamond)-free graphs, and show that, for such a graph $G$, $\chi(G)\leq4$ if $\omega(G)=2$, $\chi(G)\leq7$ if $\omega(G)=3$, $\chi(G)\leq9$ if $\omega(G)=4$, and $\chi(G)\leq2\omega(G)-1$ if $\omega(G)\ge 5$. Moreover, we prove that $G$ is perfect if $G$ is ($P_2\cup P_4$, diamond, $C_5$)-free with $\omega(G)\geq5$.

We study uniqueness and nondegeneracy of ground states for nonlinear scalar field equations in two dimensions with a point interaction at the origin. It is known that the all ground states are radial, positive, and decreasing functions. In this paper we prove the uniqueness of positive radial solutions by a method of Poho\v{z}aev identities. As a corollary, we obtain the uniqueness of ground states. Moreover, by a variational and ODE technique, we show that the ground state is a nondegenerate critical point of the action in the energy space.

The Milnor-Wood inequality states that if a (topological) oriented circle bundle over an orientable surface of genus $g$ has a smooth transverse foliation, then the Euler class of the bundle satisfies $$|\mathcal{E}|\leq 2-2g.$$ We give a new proof of the inequality based on a (previously proven by the authors) local formula which computes $\mathcal{E}$ from the singularities of a quasisection.

We consider the weakening of Ramsey's theorem that arises from considering only translation-invariant colourings of pairs, and show that this has the same strength (both from the viewpoint of Reverse Mathematics and from the viewpoint of Computability Theory) as the "adjacent Hindman's theorem", proposed by L. Carlucci (Arch. Math. Log. 57 (2018), 381--359). We also investigate some higher dimensional versions of both of these statements.

Using the Liu's method, we prove a new Witten rigidity theorem of elliptic genus of twisted Dirac operators in even dimensional spin manifolds under the circle action. Combined with the Han-Yu's method, we prove the Witten rigidity theorems of elliptic genus of twisted Toplitz operators of odd-dimensional spin manifolds under the circle action. Moreover, we have obtained several similar Witten rigidity theorems of elliptic genus.

Given a one-parameter family of flows over a parameter interval $\Lambda$, assuming there is a continuation of Morse decompositions over $\Lambda$, Reineck defined a singular transition matrix to show the existence of a connection orbit between some Morse sets at some parameter points in $\Lambda$. This paper aims to extend the definition of a singular transition matrix in cases where there is no continuation of Morse decompositions over the parameter interval. This extension will help study the bifurcation associated with the change of Morse decomposition from a topological dynamics viewpoint.

The umbral approach provides methods for comprehending and redefining special functions. This approach is employed efficiently in order to uncover intricacies and introduce new families of special functions. In this article, the umbral perspective is adopted to introduce a hybrid family of hypergeometric and Mittag-Leffler functions. The umbral-operational procedures are used to derive the generating functions, explicit representations, differential recurrence formulae, and specific integral formulae. Further, the Laplace and Sumudu transforms for the hypergeometric-Mittag-Leffler functions are established. The graphical representation and pattern for distribution of zeros for suitable values of parameters are also presented.

We show that certain radially symmetric steady states of compressible viscous fluids in domains with inflow/outflow boundary conditions are unconditionally stable. This means that any not necessarily radially symmetric solution of the associated evolutionary problem converges to a single radially symmetric steady state.

We perform a detailed Lie symmetry analysis for the hyperbolic system of partial differential equations that describe the one-dimensional Shallow Water magnetohydrodynamics equations within a rotating reference frame. We consider a relaxing condition $\mathbf{\mathbf{\nabla }}\left( h\mathbf{B} \right) \neq 0$ for the one-dimensional problem, which has been used to overcome unphysical behaviors. The hyperbolic system of partial differential equations depends on two parameters: the constant gravitational potential $g$ and the Coriolis term $f_{0}$, related to the constant rotation of the reference frame. For four different cases, namely $g=0,~f_{0}=0$; $g\neq 0\,,~f_{0}=0$; $g=0$, $f_{0}\neq 0$; and $g\neq 0$, $f_{0}\neq 0$ the admitted Lie symmetries for the hyperbolic system form different Lie algebras. Specifically the admitted Lie algebras are the $L^{10}=\left\{ A_{3,3}\rtimes A_{2,1}\right\} \otimes _{s}A_{5,34}^{a}$; $% L^{8}=A_{2,1}\rtimes A_{6,22}$; $L^{7}=A_{3,5}\rtimes\left\{ A_{2,1}\rtimes A_{2,1}\right\} $; and $L^{6}=A_{3,5}\rtimes A_{3,3}~$respectively, where we use the Morozov-Mubarakzyanov-Patera classification scheme. For the general case where $f_{0}g\neq 0$, we derive all the invariants for the Adjoint action of the Lie algebra $L^{6}$ and its subalgebras, and we calculate all the elements of the one-dimensional optimal system. These elements are then considered to define similarity transformations and construct analytic solutions for the hyperbolic system.

We present the first fixed-length elementary closed-form expressions for the prime-counting function, pi(n), and the n-th prime number, p(n). These expressions are represented as arithmetic terms, requiring only a fixed and finite number of elementary arithmetic operations from the set: addition, subtraction, multiplication, division with remainder, exponentiation. Mazzanti proved that every Kalmar function can be represented by arithmetic terms. We develop an arithmetic term representing the prime omega function, omega(n), which counts the number of distinct prime divisors of a positive integer n. From this term, we find immediately an arithmetic term for the prime-counting function, pi(n). We utilize these results, along with a new arithmetic term for binomial coefficients and new prime-related exponential Diophantine equations to construct an arithmetic term for the n-th prime number, p(n), thereby providing a constructive solution to a fundamental question in mathematics: Is there an order to the primes?

The study of interpolation nodes and their associated Lebesgue constants are central to numerical analysis, impacting the stability and accuracy of polynomial approximations. In this paper, we will explore the Morrow-Patterson points, a set of interpolation nodes introduced to construct cubature formulas of a minimum number of points in the square for a fixed degree $n$. We prove that their Lebesgue constant growth is ${\cal O}(n^2)$ as was conjectured based on numerical evidence about twenty years ago in the paper by Caliari, M., De Marchi, S., Vianello, M., {\it Bivariate polynomial interpolation on the square at new nodal sets}, Appl. Math. Comput. 165(2) (2005), 261--274.

We formulate a positivity conjecture relating the Verlinde ring associated with an untwisted affine Lie algebra at a positive integer level and a subcategory of finite-dimensional representations over the corresponding quantum affine algebra with a cluster algebra structure. Specifically, we consider a ring homomorphism from the Grothendieck ring of this representation category to the Verlinde ring and conjecture that every object in the category has a positive image under this map. We prove this conjecture in certain cases where the underlying simple Lie algebra is simply-laced with level 2 or of type $A_1$ at an arbitrary level. The proof employs the close connection between this category and cluster algebras of finite cluster type. As further evidence for the conjecture, we show that for any level, all objects have positive quantum dimensions under the assumption that some Kirillov-Reshetikhin modules have positive quantum dimensions.

In this paper, we {\color{black}study four kinds of polynomials orthogonal with the singularly perturbed Gaussian weight $w_{\rm SPG}(x)$, the deformed Freud weight $w_{\rm DF}(x)$, the jumpy Gaussian weight $w_{\rm JG}(x)$, and the Jacobi-type weight $w_{\rm {\color{black}JC}}(x)$. The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations from [J. Derezi\'{n}ski, A. Ishkhanyan, A. Latosi\'{n}ski, SIGMA 17 (2021), 056], we transform these Heun equations into Painlev\'{e} equations. It is interesting that the Painlev\'{e} equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, $w_{\rm SPG}(x)$, under a suitable double scalings for large $s$ and small $s$, where the Dyson's constant is recovered.}

We establish a bialgebra theory for averaging algebras, called averaging antisymmetric infinitesimal bialgebras by generalizing the study of antisymmetric infinitesimal bialgebras to the context of averaging algebras. They are characterized by double constructions of averaging Frobenius algebras as well as matched pairs of averaging algebras. Antisymmetric solutions of the Yang-Baxter equation in averaging algebras provide averaging antisymmetric infinitesimal bialgebras. The notions of an $\mathcal{O}$-operator of an averaging algebra and an averaging dendriform algebra are introduced to construct antisymmetric solutions of the Yang-Baxter equation in an averaging algebra and hence averaging antisymmetric infinitesimal bialgebras. Moreover, we introduce the notion of factorizable averaging antisymmetric infinitesimal bialgebras and show that a factorizable averaging antisymmetric infinitesimal bialgebra leads to a factorization of the underlying averaging algebra. We establish a one-to-one correspondence between factorizable averaging antisymmetric infinitesimal bialgebras and symmetric averaging Frobenius algebras with a Rota-Baxter operator of nonzero weight. Finally, we apply the study of averaging antisymmetric infinitesimal bialgebras to perm bialgebras, extending the construction of perm algebras from commutative averaging algebras to the context of bialgebras, which is consistent with the well constructed theory of perm bialgebras.

Additive codes may have better parameters than linear codes. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Here we show that a Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes.

In this paper, we study singular heat flows from a 3-dimensional complete bounded Riemannian manifold without boundary into the hyperbolic space with prescribe singularity along a closed curve. We prove the existence and regularity of the singular heat flows. Furthermore, we prove that the singular heat flows converge to a singular harmonic map at an exponential rate.

This survey is devoted to the geography problem of log-surfaces constructed as pairs consisting of a smooth projective surface and a reduced boundary divisor. In the first part we focus on the geography problem for log-surfaces associated with pairs of the form $(\mathbb{P}^{2}, C)$, where $C$ is an arrangement of smooth plane curves admitting ordinary singularities. In particular, we focus on the case where $C$ is an arrangement of smooth rational curves. In the second part, containing original new results, we study log surfaces constructed as pairs consisting of a $K3$ surface and a rational curve arrangement. In particular, we provide some combinatorial conditions for such pairs to have the log-Chern slope equal to $3$. Our survey is illustrated with many explicit examples of log-surfaces.

In this paper, we show that, for a solution to the stationary Fokker-Planck equation with general coefficients, defined as a measure with an $L^2$-density, this density not only exhibits $H^{1,2}$-regularity but also H\"{o}lder continuity. To achieve this, we first construct a reference measure $\mu=\rho dx$ by utilizing existence and elliptic regularity results for homogeneous boundary value problems, ensuring that the given divergence-type operator corresponds to a sectorial Dirichlet form. By employing elliptic regularity results for both divergence and non-divergence type equations, we demonstrate that the image of the resolvent operator associated with the sectorial Dirichlet form has $H^{2,2}$-regularity. Furthermore, through calculations based on the Dirichlet form and the $H^{2,2}$-regularity of the resolvent operator, we prove that the density of the solution measure for the stationary Fokker-Planck equation is, indeed, the weak limit of $H^{1,2}$-functions defined via the resolvent operator. Our results highlight the central role of Dirichlet form theory and resolvent approximations in establishing the regularity of solutions to stationary Fokker-Planck equations with general coefficients.

The classical Armijo backtracking algorithm achieves the optimal complexity for smooth functions like gradient descent but without any hyperparameter tuning. However, the smoothness assumption is not suitable for Deep Learning optimization. In this work, we show that some variants of the Armijo optimizer achieves acceleration and optimal complexities under assumptions more suited for Deep Learning: the (L 0 , L 1 ) smoothness condition and analyticity. New dependences on the smoothness constants and the initial gap are established. The results theoretically highlight the powerful efficiency of Armijo-like conditions for highly non-convex problems.

In a recent paper Zhang et al. constructed 17 families of permutation pentanomials of the form $x^t+x^{r_1(q-1)+t}+x^{r_2(q-1)+t}+x^{r_3(q-1)+t}+x^{r_4(q-1)+t}$ over $\mathbb{F}_{q^2}$ where $q=2^m$. In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over $\mathbb{F}_{q^2}$.

The regularity of solutions to the stochastic nonlinear wave equation plays a critical role in the accuracy and efficiency of numerical algorithms. Rough or discontinuous initial conditions pose significant challenges, often leading to a loss of accuracy and reduced computational efficiency in existing methods. In this study, we address these challenges by developing a novel and efficient numerical algorithm specifically designed for computing rough solutions of the stochastic nonlinear wave equation, while significantly relaxing the regularity requirements on the initial data. By leveraging the intrinsic structure of the stochastic nonlinear wave equation and employing advanced tools from harmonic analysis, we construct a time discretization method that achieves robust convergence for initial values \((u^{0}, v^{0}) \in H^{\gamma} \times H^{\gamma-1}\) for all \(\gamma > 0\). Notably, our method attains an improved error rate of \(O(\tau^{2\gamma-})\) in one and two dimensions for \(\gamma \in (0, \frac{1}{2}]\), and \(O(\tau^{\max(\gamma, 2\gamma - \frac{1}{2}-)})\) in three dimensions for \(\gamma \in (0, \frac{3}{4}]\), where \(\tau\) denotes the time step size. These convergence rates surpass those of existing numerical methods under the same regularity conditions, underscoring the advantage of our approach. To validate the performance of our method, we present extensive numerical experiments that demonstrate its superior accuracy and computational efficiency compared to state-of-the-art methods. These results highlight the potential of our approach to enable accurate and efficient simulations of stochastic wave phenomena even in the presence of challenging initial conditions.

We give a finite presentation of the cobordism symmetric monoidal bicategory of (smooth, oriented) closed manifolds, cobordisms and cobordisms with corners as an extension of the bicategory of closed manifolds, cobordisms and diffeomorphisms. The generators are the standard handle attachments, and the relations are handle cancellations and invariance under reversing the orientation of the attaching spheres. In other words, given a categorified TQFT and 2-morphisms associated to the standard handles satisfying our relations, we construct a once extended TQFT.

We study the dynamics of gradient flow in high dimensions for the multi-spiked tensor problem, where the goal is to estimate $r$ unknown signal vectors (spikes) from noisy Gaussian tensor observations. Specifically, we analyze the maximum likelihood estimation procedure, which involves optimizing a highly nonconvex random function. We determine the sample complexity required for gradient flow to efficiently recover all spikes, without imposing any assumptions on the separation of the signal-to-noise ratios (SNRs). More precisely, our results provide the sample complexity required to guarantee recovery of the spikes up to a permutation. Our work builds on our companion paper [Ben Arous, Gerbelot, Piccolo 2024], which studies Langevin dynamics and determines the sample complexity and separation conditions for the SNRs necessary for ensuring exact recovery of the spikes (where the recovered permutation matches the identity). During the recovery process, the correlations between the estimators and the hidden vectors increase in a sequential manner. The order in which these correlations become significant depends on their initial values and the corresponding SNRs, which ultimately determines the permutation of the recovered spikes.

Given a public transportation network, which and how many passenger routes can potentially be shortest paths, when all possible timetables are taken into account? This question leads to shortest path problems on graphs with interval costs on their arcs and is closely linked to multi-objective optimization. We introduce a Dijkstra algorithm based on polynomials over the tropical semiring that computes complete or minimal sets of efficient paths. We demonstrate that this approach is computationally feasible by employing it on the public transport network of the city of Wuppertal and instances of the benchmarking set TimPassLib, and we evaluate the resulting sets of passenger routes.

Extremely large-scale multiple-input multiple-output (XL-MIMO) communications, enabled by numerous antenna elements integrated into large antenna surfaces, can provide increased effective degree of freedom (EDoF) to achieve high diversity gain. However, it remains an open problem that how the EDoF is influenced by the directional radiation pattern of antenna elements. In this work, empowered by the wavenumber-domain channel representation, we analyze the EDoF in a general case where the directivity of antennas, determined by the antenna structure and element spacing, is considered. Specifically, we first reveal the uneven distribution of directivity-aware wavenumber-domain coupling coefficients, i.e., channel gain towards different directions, in the isotropic Rayleigh fading channel. EDoF is then calculated based on such distribution of coupling coefficients. A numerical method is also provided to obtain coupling coefficients via electromagnetic full-wave simulations. Due to the influence of antenna directivity, how EDoF and ergodic channel capacity vary with the element spacing are explored via simulations for different antenna types.

Preconditioned eigenvalue solvers offer the possibility to incorporate preconditioners for the solution of large-scale eigenvalue problems, as they arise from the discretization of partial differential equations. The convergence analysis of such methods is intricate. Even for the relatively simple preconditioned inverse iteration (PINVIT), which targets the smallest eigenvalue of a symmetric positive definite matrix, the celebrated analysis by Neymeyr is highly nontrivial and only yields convergence if the starting vector is fairly close to the desired eigenvector. In this work, we prove a new non-asymptotic convergence result for a variant of PINVIT. Our proof proceeds by analyzing an equivalent Riemannian steepest descent method and leveraging convexity-like properties. We show a convergence rate that nearly matches the one of PINVIT. As a major benefit, we require a condition on the starting vector that tends to be less stringent. This improved global convergence property is demonstrated for two classes of preconditioners with theoretical bounds and a range of numerical experiments.

A mathematical modeling process for phenomena with a single state variable that attempts to be realistic must be given by a scalar nonautonomous differential equation $x'=f(t,x)$ that is concave with respect to the state variable $x$ in some regions of its domain and convex in the complementary zones. This article takes the first step towards developing a theory to describe the corresponding dynamics: the case in which $f$ is concave on the region $x\ge b(t)$ and convex on $x\le b(t)$, where $b$ is a $C^1$ map, is considered. The different long-term dynamics that may appear are analyzed while describing the bifurcation diagram for $x'=f(t,x)+\lambda$. The results are used to establish conditions on a concave-convex map $h$ and a nonnegative map $k$ ensuring the existence of a value $\rho_0$ giving rise to the unique critical transition for the parametric family of equations $x'=h(t,x)-\rho\,k(t,x)$, which is assumed to approach $x'=h(t,x)$ as time decreases, but for which no conditions are assumed on the future dynamics. The developed theory is justified by showing that concave-convex models fit correctly some laboratory experimental data, and applied to describe a population dynamics model for which a large enough increase on the peak of a temporary higher predation causes extinction.

The modern algebra concepts are used to construct tables of algebraic spinors related to Clifford algebra multivectors with real and complex coefficients. The following data computed by Mathematica are presented in form of tables for individual Clifford geometric algebras: 1. Initial idempotent; 2. Two-sided ideal; 3. Left ideal basis (otherwise projector, or spinor basis); 4. Matrix representations (reps) for basis vectors in Clifford algebras in spinor basis; 5. General spinor; 6. Spinor in matrix form; 7. Squared hermitian norm of the spinor. Earlier in 1998, only the first four items computed by Maple were published by R. Ablamowicz.

We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. We illustrate the technique with a variety of applications, including tilings with tiles of irrational area, ordered integer factorizations, lattice walks enumerated by Euclidean length, and plane trees with vertices of irrational size. We also explore phase transitions in the asymptotics of families of irrational combinatorial classes.

We investigate the numerical solution of multiscale transport equations using Physics Informed Neural Networks (PINNs) with ReLU activation functions. Therefore, we study the analogy between PINNs and Least-Squares Finite Elements (LSFE) which lies in the shared approach to reformulate the PDE solution as a minimization of a quadratic functional. We prove that in the diffusive regime, the correct limit is not reached, in agreement with known results for first-order LSFE. A diffusive scaling is introduced that can be applied to overcome this, again in full agreement with theoretical results for LSFE. We provide numerical results in the case of slab geometry that support our theoretical findings.

To understand a document with multiple events, event-event relation extraction (ERE) emerges as a crucial task, aiming to discern how natural events temporally or structurally associate with each other. To achieve this goal, our work addresses the problems of temporal event relation extraction (TRE) and subevent relation extraction (SRE). The latest methods for such problems have commonly built document-level event graphs for global reasoning across sentences. However, the edges between events are usually derived from external tools heuristically, which are not always reliable and may introduce noise. Moreover, they are not capable of preserving logical constraints among event relations, e.g., coreference constraint, symmetry constraint and conjunction constraint. These constraints guarantee coherence between different relation types,enabling the generation of a uniffed event evolution graph. In this work, we propose a novel method named LogicERE, which performs high-order event relation reasoning through modeling logic constraints. Speciffcally, different from conventional event graphs, we design a logic constraint induced graph (LCG) without any external tools. LCG involves event nodes where the interactions among them can model the coreference constraint, and event pairs nodes where the interactions among them can retain the symmetry constraint and conjunction constraint. Then we perform high-order reasoning on LCG with relational graph transformer to obtain enhanced event and event pair embeddings. Finally, we further incorporate logic constraint information via a joint logic learning module. Extensive experiments demonstrate the effectiveness of the proposed method with state-of-the-art performance on benchmark datasets.

Serre famously showed that almost all plane conics over $\mathbb{Q}$ have no rational point. We investigate versions of this over global function fields, focusing on a specific family of conics over $\mathbb{F}_2(t)$ which illustrates new behaviour. We obtain an asymptotic formula using harmonic analysis, which requires a new Tauberian theorem over function fields for Dirichlet series with branch point singularities.

We show rigidity results for the operator equations T(f.g) = Tf.Tg, T(f*g) = Tf.Tg and T(f.g) = Tf*Tg for bijective operators T acting on sufficently large spaces of smooth functions. Typically a condition like |T(f.g) - Tf.Tg| < a for all f, g with a fixed function a will imply T(f.g) = Tf.Tg. Theorems of Alesker, Artstein-Avidan, Faifman and Milman then yield characterizations (up to diffeomorphisms) of the Fourier transform by mapping products into convolutions and vice-versa on the Schwartz space.

In this paper we construct a parametrix for the fractional Helmholtz equation $((-\Delta)^s - \tau^{2s} r(x)^{2s} + q(x))u=0$ making use of geometrical optics solutions. We show that the associated eikonal equation is the same as in the classical case, while in the first transport equation the effect of nonlocality is only visible in the zero-th order term, which depends on $s$. Moreover, we show that the approximate geometrical optics solutions present different behaviors in the regimes $s\in(0,\frac 12)$ and $s\in [\frac 12,1)$. While the latter case is quite similar to the classical one, which corresponds to $s=1$, in the former case we find that the potential is a strong perturbation, which changes the propagation of singularities. As an application, we study the inverse problem consisting in recovering the potential $q$ from Cauchy data when the refraction index $r$ is fixed and simple. Using our parametrix based on the construction of approximate geometrical optics solutions, we prove that H\"older stability holds for this problem. This is a substantial improvement over the state of the art for fractional wave equations, for which the usual Runge approximation argument can provide only logarithmic stability. Besides its mathematical novelty, this study is motivated by envisioned applications in nonlocal elasticity models emerging from the geophysical sciences.

Radiation heat transfer in a graded-index medium often suffers accuracy problems due to the gradual changes in the refractive index. The finite element method, meshfree, and other numerical methods often struggle to maintain accuracy when applied to this medium. To address this issue, we apply physics-informed neural networks (PINNs)-based machine learning algorithms to simulate forward and inverse problems for this medium. We also provide the theoretical upper bounds. This theoretical framework is validated through numerical experiments of predefined and newly developed models that demonstrate the accuracy and robustness of the algorithms in solving radiation transport problems in the medium. The simulations show that the novel algorithm goes on with numerical stability and effectively mitigates oscillatory errors, even in cases with more pronounced variations in the refractive index.

Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. We also show that our new framework easily extends to non-commuting flows corresponding to nonabelian Lie groups. Thus Hamiltonian Lie group actions can be derived from a variational principle.

Let $F$ be a local field and let $R$ be its ring of integers. For a positive integer $n$, an integral quadratic form defined over $R$ is called primitively $n$-universal if it primitively represents all quadratic forms of rank $n$. It was proved in arXiv:2005.11268 that the minimal rank of primitively $1$-universal quadratic forms over the $p$-adic integer ring $\mathbb{Z}_p$ is $2$ if $p$ is odd, and $3$ otherwise. In this article, we completely determine the minimal rank of primitively $n$-universal quadratic forms over $R$ for any positive integer $n$ and any local ring $R$ such that $2$ is a unit or a prime.

Most state-of-the-art branch-and-bound solvers for mixed-integer linear programming rely on limited-precision floating-point arithmetic and use numerical tolerances when reasoning about feasibility and optimality during their search. While the practical success of floating-point MIP solvers bears witness to their overall numerical robustness, it is well-known that numerically challenging input can lead them to produce incorrect results. Even when their final answer is correct, one critical question remains: Were the individual decisions taken during branch-and-bound justified, i.e., can they be verified in exact arithmetic? In this paper, we attempt a first such a posteriori analysis of a pure LP-based branch-and-bound solver by checking all intermediate decisions critical to the correctness of the result: accepting solutions as integer feasible, declaring the LP relaxation infeasible, and pruning subtrees as subopti mal. Our computational study in the academic MIP solver SCIP confirms the expectation that in the overwhelming majority of cases, all decisions are correct. When errors do occur on numerically challenging instances, they typically affect only a small, typically single-digit, amount of leaf nodes that would require further processing.

This paper describes directed polymer on general time-correlated random field. Law of large numbers, existence and smoothness of limiting free energies are proved at all temperature. We also display the delocalized-localized transition, via separating techniques for entanglement of the random field.

We show that the centre of the walled Brauer algebra $B_{r,1}(\delta)$ over the complex field $\mathbb{C}$, for any parameter $\delta\in \mathbb{C}$, is generated by the supersymmetric polynomials evaluated at the Jucys-Murphy elements. Moreover, we prove that its dimension is independent of the parameter $\delta$.

This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and their applications in areas such as stochastic differential equations (SDEs) and financial modeling are pointed out. In particular, exploiting the rough paths view, solutions of SDEs continuously depend on the lift of the driver. Such continuous mappings can be approximated using (truncated) signatures, which are solutions of high-dimensional linear systems. In order to lower the complexity of these models, this paper presents methods for reducing the order of high-dimensional truncated signature models while retaining essential characteristics. The derivation of reduced models and the universal approximation property of (truncated) signatures are treated in detail. Numerical examples, including applications to the (rough) Bergomi model in financial markets, illustrate the proposed reduction techniques and highlight their effectiveness.

The term material with memory is generally used to indicate materials whose mechanical and/or thermodynamical behaviour depends not only on the process at the present time but also on the history of the process itself. Crucial in heat conductors with memory is the heat relaxation function which models the thermal response of the material. The present study is concerned about a thermodynamical problem with memory "aging"; that is, we analyze the temperature evolution within a rigid heat conductor with memory whose relaxation function takes into account the aging of the material. In particular, we account for variations of the relaxation function due to a possible deterioration of the thermal response of the material related to its age.

We study the recovery of one-dimensional semipermeable barriers for a stochastic process in a planar domain. The considered process acts like Brownian motion when away from the barriers and is reflected upon contact until a sufficient but random amount of interaction has occurred, determined by the permeability, after which it passes through. Given a sequence of samples, we wonder when one can determine the location and shape of the barriers. This paper identifies several different recovery regimes, determined by the available observation period and the time between samples, with qualitatively different behavior. The observation period $T$ dictates if the full barriers or only certain pieces can be recovered, and the sampling rate significantly influences the convergence rate as $T\to \infty$. This rate turns out polynomial for fixed-frequency data, but exponentially fast in a high-frequency regime. Further, the environment's impact on the difficulty of the problem is quantified using interpretable parameters in the recovery guarantees, and is found to also be regime-dependent. For instance, the curvature of the barriers affects the convergence rate for fixed-frequency data, but becomes irrelevant when $T\to \infty$ with high-frequency data. The results are accompanied by explicit algorithms, and we conclude by illustrating the application to real-life data.

Bilinear Fourier multipliers of the form $e^{i (|\xi| + |\eta|+ |\xi + \eta|)} \sigma (\xi, \eta)$ are considered. It is proved that if $\sigma (\xi, \eta)$ is in the H\"ormander class $S^{m}_{1,0} (\mathbb{R}^{2n})$ with $m=-(n+1)/2$ then the corresponding bilinear operator is bounded in $L^{\infty} \times L^{\infty} \to bmo$, $h^{1} \times L^{\infty} \to L^{1}$, and $L^{\infty} \times h^{1} \to L^{1}$. This improves a result given by Rodr\'iguez-L\'opez, Rule and Staubach.

In this paper, we solve unbalanced optimal transport (UOT) problem on surfaces represented by point clouds. Based on alternating direction method of multipliers algorithm, the original UOT problem can be solved by an iteration consists of three steps. The key ingredient is to solve a Poisson equation on point cloud which is solved by tangent radial basis function (TRBF) method. The proposed TRBF method requires only the point cloud and normal vectors to discretize the Poisson equation which simplify the computation significantly. Numerical experiments conducted on point clouds with varying geometry and topology demonstrate the effectiveness of the proposed method.

These notes are a brief summary of the main results from the book `Discriminants, Resultants and Multidimensional Determinants' by Gelfand-Kapranov-Zelevinsky. We sketch the key ideas involved in the proofs, using as little technical background as possible.

We prove a conjecture posted recently by by Horst Alzer and Janusz Matkowski concerning a bilinearity property of the Cauchy exponential difference for real-to-real functions. Moreover, we provide a generalization of their result for mappings defined on a linear space.

This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in $\mathbb{R}^3$. The method is built upon the principles of the closest point method, without the use of a grid or a closest point mapping. We show that the combination of local embedded stencils with these principles can be employed to approximate surface derivatives using polyharmonic spline kernels and polynomials (PHS+Poly) RBF-FD. Specifically, we show that it is enough to consider a constant extension along the normal direction only at a single node to overcome the rank deficiency of the polynomial basis. An extensive parameter analysis is presented to test the dependence of the approach. We demonstrate high-order convergence rates on problems involving surface advection and surface diffusion, and solve Turing pattern formations on surfaces defined either implicitly or by point clouds. Moreover, a simple coupling approach with a particle tracking method demonstrates the potential of the proposed method in solving PDEs on evolving surfaces in the normal direction. Our numerical results confirm the stability, flexibility, and high-order algebraic convergence of the approach.

Recently, Eduard Schesler and the second author constructed examples of finitely generated residually finite, hereditarily just infinite groups with positive first $L^2$-Betti number. In contrast to their result, we show that a finitely generated residually-$p$ just infinite group has trivial first $L^2$-Betti number. Moreover, we prove that the normal homology rank gradient of a finitely generated, residually finite, just infinite group vanishes.

In this article, we study geometric and analytical features of complete noncompact $\rho$-Einstein solitons, which are self-similar solutions of the Ricci-Bourguignon flow. We study the spectrum of the drifted Laplacian operator for complete gradient shrinking $\rho$-Einstein soliton. Moreover, similar to classical results due to Calabi-Yau and Bishop for complete Riemannian manifolds with nonnegative Ricci curvature, we prove new volume growth estimates for geodesic balls of complete noncompact $\rho$-Einstein solitons. In particular, the rigidity case is discussed. In addition, we establish weighted volume growth estimates for geodesic balls of such manifolds.

Given different speeds $c_1$ , ... , $c_K$, in the present paper we establish the existence of a solution to the Zakharov system in dimension 1 that behaves asymptotically like a $K$-solitary wave, each wave travelling with speed $c_k$. The proof is adapted from previous results for the NLS and gKdV equations.

We are interested in stochastic processes satisfying a nonlinear recurrence relation of the form $$X_{n + k} = \Phi_0 (X_n, ..., X_{n + k - 1}) + \Theta_n$$ where $\Theta$ is a noise term. We establish the existence of an invariant measure for this process under given sufficient conditions on $\Phi_0.$

Double circuits were introduced by Lov\'{a}sz in 1980 as a fundamental tool in his derivation of a min-max formula for the size of a maximum matching in certain families of matroids. This formula was extended to all matroids satisfying the so-called `double circuit property' by Dress and Lov\'{a}sz in 1987. We extend these notions to $k$-fold circuits for all natural numbers $k$ and derive foundational results about these $k$-fold circuits. Our results imply, in particular, that certain families of matroids which are known to satisfy the double circuit property, satisfy the $k$-fold circuit property for all natural numbers $k$. These families include all pseudomodular matroids (such as full linear, algebraic and transversal matroids) and count matroids.

In this paper, we introduce and study the following question. Let $\mathcal G$ be a family of graphs and let $k\geq 3$ be an integer. What is the largest value $f_k(n)$ such that every $n$-vertex graph in $\mathcal G$ has an induced subgraph with degree at most $k$ and with $f_k(n)$ vertices? Similar questions, in which one seeks a large induced forest, or a large induced linear forest, or a large induced $d$-degenerate graph, rather than a large induced graph of bounded degree, have been studied for decades and have given rise to some of the most fascinating and elusive conjectures in Graph Theory. We tackle our problem when $\mathcal G$ is the class of the outerplanar graphs, or the class of the planar graphs, or the class of the graphs whose degree is bounded by a value $d>k$. In all cases, we provide upper and lower bounds on the value of $f_k(n)$. For example, we prove that every $n$-vertex planar graph has an induced subgraph with degree at most $3$ and with $\frac{5n}{13}>0.384n$ vertices, and that there exist $n$-vertex planar graphs whose largest induced subgraph with degree at most $3$ has $\frac{4n}{7}+O(1)<0.572n+O(1)$ vertices.

For strongly continous semigroups on Hilbert spaces, we investigate admissibility properties of control and observation operators shifted along continuous scales of spaces built by means of either interpolation and extrapolation or functional calculus. Our results show equivalence of admissibility in, on the one hand, a fractional domain of the generator and, on the other hand, a (different, in general) quadratic interpolation space of the same "Sobolev order". Furthermore, such properties imply quantified resolvent bounds in the original state space topology. When the semigroup is a group, the resulting frequency-domain estimates are in fact equivalent to the aforementioned time-domain properties. In the case of systems with both control and observation, we are able to translate input-output regularity properties into high-frequency growth rates of operator-valued transfer functions. As an application, based on results by Lasiecka, Triggiani and Tataru on interior and boundary regularity of the wave equation under Neumann control, we derive optimal asymptotics for the Neumann-to-Dirichlet wave transfer function. With that in hand, we establish non-uniform energy decay rates for the wave equation posed in a rectangle and subject to Neumann damping on an arbitrary open subset of the boundary.

In this paper, we introduce orbit matrices of directed strongly regular graphs (DSRGs). Further, we propose a method of constructing directed strongly regular graphs with prescribed automorphism group using genetic algorithm. In the construction, we use orbit matrices, i.e. quotient matrices related to equitable partitions of adjacency matrices of putative directed strongly regular graphs induced by an action of a prescribed automorphism group. Further, we apply this method to construct directed strongly regular graphs with parameters $(36,10,5,2,3)$, $(52,12,3,2,3)$, $(52,15,6,5,6)$, $(55,20,8,6,8)$ and $(55,24,12,11,10)$.

Let $M$ be an $n\times n$ real symmetric matrix. The spread of $M$ is defined as the difference between its largest and smallest eigenvalue. When considering the spread of graph-related matrices, this topic has attracted significant attention, resulting in a substantial collection of findings. In this paper, we study a general spread problem regarding $A_{\alpha}$-matrix of graphs. The $A_{\alpha}$-matrix of a graph $G$, introduced by Nikiforov in 2017, is a convex combinations of its diagonal degree matrix $D(G)$ and adjacency matrix $A(G)$, defined as $A_{\alpha} (G) = \alpha D(G) + (1-\alpha) A(G)$. Let $\lambda_1^{(\alpha)} (G)$ and $\lambda_n^{(\alpha)} (G)$ denote the largest and least eigenvalues of $A_{\alpha} (G)$, respectively. We determined the unique graph that maximizes $\lambda^{(\alpha)}_1 (G) - \beta\cdot\lambda^{(\gamma)}_n (G)$ among all connected $n$-vertex graphs for sufficiently large $n$, where $0 \leq \alpha < 1$, $1/2\leq \gamma < 1$ and $0<\beta\gamma\leq 1$. As applications, we confirm a conjecture proposed by Lin, Miao, and Guo [Linear Algebra Appl. 606 (2020) 1--22], and derive a new finding.

We consider multivariate approximation problems in the average case setting with a zero mean Gaussian measure whose covariance kernel is a periodic Gevrey kernel. We investigate various notions of algebraic tractability and exponential tractability, and obtain necessary and sufficient conditions in terms of the parameters of the problem.

We study the asymptotic estimation of prime ideals that satisfy certain congruence and argument conditions in imaginary quadratic fields. We also discuss the phenomenon of Chebyshev's bias in the distribution of prime ideals among different residue classes.

The Bologna Process has affected almost every field and every aspect of higher education in European countries which participate in it. The strife towards unification of the degrees, quality assurance and expanded access to higher education brought a lot of positive adjustments in multiple aspects, however, at the cost of new challenges to undertake. One of these obstacles include the problem of student retention. Over the years, the percentage of Bachelors who remain at the university for Master's degree programme has dropped significantly. There is a myriad of possible reasons for this phenomenon and the aim of this paper is to provide at least a partial answer that question using the data we have collected from the students of multiple universities in Poland.

We consider a nonparametric model $\mathcal{E}^{n},$ generated by independent observations $X_{i},$ $i=1,...,n,$ with densities $p(x,\theta_{i}),$ $i=1,...,n,$ the parameters of which $\theta _{i}=f(i/n)\in \Theta $ are driven by the values of an unknown function $f:[0,1]\rightarrow \Theta $ in a smoothness class. The main result of the paper is that, under regularity assumptions, this model can be approximated, in the sense of the Le Cam deficiency pseudodistance, by a nonparametric Gaussian shift model $Y_{i}=\Gamma (f(i/n))+\varepsilon _{i},$ where $\varepsilon_{1},...,\varepsilon _{n}$ are i.i.d. standard normal r.v.'s, the function $\Gamma (\theta ):\Theta \rightarrow \mathrm{R}$ satisfies $\Gamma ^{\prime}(\theta )=\sqrt{I(\theta )}$ and $I(\theta )$ is the Fisher information corresponding to the density $p(x,\theta ).$

Through the establishment of several extension theorems, we provide explicit expressions for norm one projections and 1-complemented subspaces in the Hardy space $H^p(\mathbb{T})$ for $1\leq p<\infty$, $p\neq 2$. Our characterization leads to two significant corollaries: first, all 1-complemented subspaces of $H^p(\mathbb{T})$ with dimension greater than one are isometric to $H^p(\mathbb{T})$; second, all norm one projections on $H^p(\mathbb{T})$ are restrictions of norm one projections on $L^p(\mathbb{T})$ that leave $H^p(\mathbb{T})$ invariant. The second corollary answers an open problem posed by P. Wojtaszczyk in 2003. Moreover, this is the first time that norm one projections and 1-complemented subspaces in an analytic function space have been completely understood.

We consider the cyclic representations $\Omega_{rs}$ of $ U_q(\widehat{\mathfrak{sl}}_2)$ at $q^N=1$ that depend upon two points $r,s$ in the chiral Potts algebraic curve. We show how $\Omega_{rs}$ is related to the tensor product $\rho_r\otimes \bar{\rho}_s$ of two representations of the upper Borel subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$. This result is analogous to the factorization property of the Verma module of $U_q(\widehat{\mathfrak{sl}}_2)$ at generic-$q$ in terms of two q-oscillator representation of the Borel subalgebra - a key step in the construction of the Q-operator. We construct short exact sequences of the different representations and use the results to construct Q operators that satisfy TQ relations for $q^N=1$ for both the 6-vertex and $\tau_2$ models.

We study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals.

We study an inverse boundary value problem in corrosion detection. The model is based on a conductivity equation with nonlinear Robin boundary condition. We prove that the nonlinear Robin term can be identified locally from Cauchy data measurements on a subset of the boundary. A possible strategy for turning a local identification result into a global one is suggested, and a partial result is proved in this direction. The inversion method is an adaptation to this nonlinear Robin problem of a method originally developed for semilinear elliptic equations. The strategy is based on linearization and relies on parametrizing solutions of the nonlinear equation on solutions of the linearized equation.

Hyperspectral anomaly detection refers to identifying pixels in the hyperspectral images that have spectral characteristics significantly different from the background. In this paper, we introduce a novel model that represents the background information using a low-rank representation. We integrate an implicit proximal denoiser prior, associated with a deep learning based denoiser, within a plug-and-play (PnP) framework to effectively remove noise from the eigenimages linked to the low-rank representation. Anomalies are characterized using a generalized group sparsity measure, denoted as $\|\cdot\|_{2,\psi}$. To solve the resulting orthogonal constrained nonconvex nonsmooth optimization problem, we develop a PnP-proximal block coordinate descent (PnP-PBCD) method, where the eigenimages are updated using a proximal denoiser within the PnP framework. We prove that any accumulation point of the sequence generated by the PnP-PBCD method is a stationary point. We evaluate the effectiveness of the PnP-PBCD method on hyperspectral anomaly detection in scenarios with and without Gaussian noise contamination. The results demonstrate that the proposed method can effectively detect anomalous objects, outperforming the competing methods that may mistakenly identify noise as anomalies or misidentify the anomalous objects due to noise interference.

The asymptotic behaviour of the Hartree equation is studied for short-range interaction potentials near translation-invariant steady states satisfying the Penrose stability condition. Phase-mixing estimates in finite regularity are derived, demonstrating density decay and scattering of solutions in weighted quantum Sobolev spaces. These results provide a quantum analogue of Landau damping in classical plasma physics and remain uniform in the semiclassical limit, bridging the quantum and classical regimes.

The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been extensively used for graphs, surprisingly little is known about analogous questions in the context of hypergraphs. In this work, we apply this method to asymptotically determine the number of independent sets in regular $k$-partite $k$-uniform hypergraphs which satisfy natural expansion properties. The resulting formula depends only on the local structure of the hypergraph, making it computationally efficient. In particular, we provide a simple closed-form expression for linear hypergraphs.

We introduce a category of non-symmetric operads with distinguished curvature term, and determine the right adjoint to the functor forgetting the distinction. As a result we give a universal characterization of the procedure of "twisting by a Maurer-Cartan element", and also of the curved A-infinity operad itself.

The optimization of large-scale multibody systems is a numerically challenging task, in particular when considering multiple conflicting criteria at the same time. In this situation, we need to approximate the Pareto set of optimal compromises, which is significantly more expensive than finding a single optimum in single-objective optimization. To prevent large costs, the usage of surrogate models, constructed from a small but informative number of expensive model evaluations, is a very popular and widely studied approach. The central challenge then is to ensure a high quality (that is, near-optimality) of the solutions that were obtained using the surrogate model, which can be hard to guarantee with a single pre-computed surrogate. We present a back-and-forth approach between surrogate modeling and multi-objective optimization to improve the quality of the obtained solutions. Using the example of an expensive-to-evaluate multibody system, we compare different strategies regarding multi-objective optimization, sampling and also surrogate modeling, to identify the most promising approach in terms of computational efficiency and solution quality.

Corti defined the notion of standard models of del Pezzo fibrations, and studied their existence over $\mathbb{C}$ with a fixed generic fibre. In this paper, we prove the existence of standard models of del Pezzo fibrations of degree $4$ in characteristic $>2$. To show this, we use the notion of Koll\'ar stability, which was introduced by Koll\'ar and Abban-Fedorchuk-Krylov.

We study canonical quotients in model theory, mainly stable quotients of type-definable groups and invariant types in NIP theories. We extend the modelling property to continuous theories and use it to study $n$-dependence in hyperdefinable sets. Furthermore, we study maximal WAP and tame quotients of $S_X(\mathfrak{C})$, where $\mathfrak{C}$ is a monster model of a complete theory $T$ and $X$ is an $\emptyset$-type-definable set and show that the Ellis groups of the maximal WAP quotient flow and the maximal tame quotient flow do not depend on the choice of the monster model $\mathfrak{C}$.

We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form \[ \|P - \Pi_{\mathbb{B}} \Pi_{\ker\mathscr{A}} P\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} \le c \, (\|\mathscr{A}[P]\|_{\dot{\mathrm{W}}^{k-1, p^*}(\mathbb{R}^n)} + \|\mathbb{B} P\|_{\mathrm{L}^p(\mathbb{R}^n)}) \] for all $ P \in \mathrm{C}_c^\infty(\mathbb{R}^n; V) $, where $ V $ is a finite-dimensional vector space, $ \mathscr{A} $ is a linear mapping, and $ \mathbb{B} $ is a constant coefficient homogeneous differential operator of order $ k $. In particular, we can treat the combination $(p,\mathscr{A},\mathbb{B},k)=(1,\operatorname{tr},\operatorname{Curl},1)$. Our results generalize the techniques from Gmeineder et al. (Math.Mod.Met.Appl.Sci,34:03,2024; arXiv:2405.10349), which exclusively dealt with reduced elliptic operators. In contrast to the reduced ellipticity case, however, the reduced constant rank case necessitates to introduce a correction, namely the projection $\Pi_\mathbb{B}$ on the left-hand side of the inequality.

The study of the homology of diagram algebras has emerged as an interesting and important field. In many cases, the homology of a diagram algebra can be identified with the homology of a group. In this paper we have two main aims. Firstly, we study the (co)homology of new families of diagram algebras such as the blob algebras and the walled Brauer algebras, both of which exhibit new phenomena in the field. Secondly, we show that in the cases where the homology of a diagram algebra can be identified with group homology one can also identify the cohomology of the algebra with the cohomology of a group. We use this to establish an integer-graded cohomology theory for these diagram algebras and identify this with the Tate cohomology of a group.

A cornerstone of extremal graph theory due to Erd\H{o}s and Stone states that the edge density which guarantees a fixed graph $F$ as subgraph also asymptotically guarantees a blow-up of $F$ as subgraph. It is natural to ask whether this phenomenon generalises to vertex-spanning structures such as Hamilton cycles. This was confirmed by B\"ottcher, Schacht and Taraz for graphs in the form of the Bandwidth Theorem. Our main result extends the phenomenon to hypergraphs. A graph on $n$ vertices that robustly contains a Hamilton cycle must satisfy certain conditions on space, connectivity and aperiodicity. Conversely, we show that if these properties are robustly satisfied, then all blow-ups of cycles on $n$ vertices with clusters of size at most $\operatorname{poly}(\log \log n)$ are guaranteed as subgraphs. This generalises to powers of cycles and to the hypergraph setting. As an application, we recover a series of classic results and recent breakthroughs on Hamiltonicity under degree conditions, which are then immediately upgraded to blown up versions. The proofs are based on a new setup for embedding large substructures into dense hypergraphs, which is of independent interest and does not rely on the Regularity Lemma or the Absorption Method.

We derive the effective transmission condition for a cluster of acoustic subwavelength resonators, modeled as small-scaled bubbles distributed not necessarily periodically along a smooth, bounded hypersurface, which need not be flat. The transmission condition specifies that the jump in the normal derivative of the acoustic field is proportional to its second time derivative, convoluted in time with a sinusoidal kernel. This kernel has a period determined by the common subwavelength resonance (specifically, the Minnaert resonance in this case). This dispersive transmission condition can also be interpreted as a Dirac-like surface potential that is convoluted in the time domain and spatially supported on the specified hypersurface. We highlight the following features: 1. High resonance regime: When the common resonance is large, the surface behaves as fully transparent, permitting complete transmission of the acoustic field. 2. Moderate resonance regime: For moderate resonance values, the surface acts as a screen with memory effects, capturing the dispersive behavior induced by the resonance. 3. Low resonance regime: When the common resonance is small, the surface functions as a partial reflective (or partial transmissive) screen with no memory effect.

The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure, in terms of the Fourier and Frostman dimensions of the measure. We generalise this result by using the Fourier spectrum; a family of dimensions that interpolate between the Fourier and Sobolev dimensions for measures. This gives us a continuum of Stein--Tomas type estimates, and optimising over this continuum gives a new $L^q\to L^2$ restriction theorem which often outperforms the Stein--Tomas result. We also provide results in the other direction by giving a range of $q$ in terms of the Fourier spectrum for which $L^q\to L^2$ restriction estimates fail, generalising an observation of Hambrook and {\L}aba. We illustrate our results with several examples, including the surface measure on the cone, the moment curve, and several fractal measures.

This paper is devoted to the study of the long time behavior of Nash equilibria in Mean Field Games within the framework of displacement monotonicity. We first show that any two equilibria defined on the time horizon $[0,T]$ must be close as $T \to \infty$, in a suitable sense, independently of initial/terminal conditions. The way this stability property is made quantitative involves the $L^2$ distance between solutions of the associated Pontryagin system of FBSDEs that characterizes the equilibria. Therefore, this implies in particular the stability in the 2-Wasserstein distance for the two flows of probability measures describing the agent population density and the $L^2$ distance between the co-states of agents, that are related to the optimal feedback controls. We then prove that the value function of a typical agent converges as $T \to \infty$, and we describe this limit via an infinite horizon MFG system, involving an ergodic constant. All of our convergence results hold true in a unified way for deterministic and idiosyncratic noise driven Mean Field Games, in the case of strongly displacement monotone non-separable Hamiltonians. All these are quantitative at exponential rates.

Let $I$ be a monomial ideal in a polynomial ring. In this paper, we study the asymptotic behavior of the set of associated radical ideals of the (symbolic) powers of $I$. We show that both $\asr(I^s)$ and $\asr(I^{(s)})$ need not stabilize for large value of $s$. In the case $I$ is a square-free monomial ideal, we prove that $\asr(I^{(s)})$ is constant for $s$ large enough. Finally, if $I$ is the cover ideal of a balanced hypergraph, then $\asr(I^s)$ monotonically increases in $s$.

The aim of this article is to study quadratic maps, in the sense of Leibman, between non-abelian groups. As a key result, we provide a complete characterization of quadratic maps through a universal construction that can be computed explicitly. Applications include the classification of quadratic maps on finite abelian groups, proving the non-existence of polynomial maps of higher degree on perfect groups, establishing stability results for polynomial maps, and presenting a general inverse theorem for the Gowers norm with non-abelian coefficients on perfect groups with bounded commutator width.

We survey decades of research identifying the (co)homology of configuration spaces with Lie algebra (co)homology. The different routes to this one proto-theorem offer genuinely different explanations of its truth, and we attempt to convey some sense of the conceptual core of each perspective. We close with a list of problems.

The proper quasivariety BCA of Bochvar algebras, which serves as the equivalent algebraic semantics of Bochvar's external logic, was introduced by Finn and Grigolia in and extensively studied in a recent work by two of these authors. In this paper, we show that the algebraic category of Bochvar algebras is equivalent to a category whose objects are pairs consisting of a Boolean algebra and a meet-subsemilattice (with unit) of the same. Furthermore, we provide an axiomatisation of the variety $V(BCA) generated by Bochvar algebras. Finally, we axiomatise the join of Boolean algebras and semilattices within the lattice of subvarieties of V(BCA).

We show that several sets of interest arising from the study of partition regularity and density Ramsey theory of polynomial equations over integral domains are undecidable. In particular, we show that the set of homogeneous polynomials $p \in \mathbb{Z}[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{Z}\setminus\{0\}$ is undecidable conditional on Hilbert's tenth problem for $\mathbb{Q}$. For other integral domains, we get the analogous result unconditionally. More generally, we determine the exact lightface complexity of the various sets of interest. For example, we show that the set of homogeneous polynomials $p \in \mathbb{F}_q(t)[x_1,\cdots,x_n]$ for which the equation $p(x_1,\cdots,x_n) = 0$ is partition regular over $\mathbb{F}_q(t)\setminus\{0\}$ is $\Pi_2^0$-complete. We also prove several other results of independent interest. These include a compactness principle and a uniformity principle for density Ramsey theory on countable cancellative left amenable semigroups, as well as the existence of the natural extension for measure preserving systems of countable cancellative left reversible semigroups.

Cutting planes are frequently used for solving integer programs. A common strategy is to derive cutting planes from building blocks or a substructure of the integer program. In this paper, we focus on knapsack constraints that arise from single row relaxations. Among the most popular classes derived from knapsack constraints are lifted minimal cover inequalities. The separation problem for these inequalities is NP-hard though, and one usually separates them heuristically, therefore not fully exploiting their potential. For many benchmarking instances however, it turns out that many knapsack constraints only have few different coefficients. This motivates the concept of sparse knapsacks where the number of different coefficients is a small constant, independent of the number of variables present. For such knapsacks, we observe that there are only polynomially many different classes of structurally equivalent minimal covers. This opens the door to specialized techniques for using lifted minimal cover inequalities. In this article we will discuss two such techniques, which are based on specialized sorting methods. On the one hand, we present new separation routines that separate equivalence classes of inequalities rather than individual inequalities. On the other hand, we derive compact extended formulations that express all lifted minimal cover inequalities by means of a polynomial number of constraints. These extended formulations are based on tailored sorting networks that express our separation algorithm by linear inequalities. We conclude the article by a numerical investigation of the different techniques for popular benchmarking instances.

We study the singularities of the moduli space of degree $e$ maps from smooth genus $g$ curves to an arbitrary smooth hypersurface of low degree. For $e$ large compared to $g$, we show that these moduli spaces have at worst terminal singularities. Our main approach is to study the jet schemes of these moduli spaces by developing a suitable form of the circle method.

This note initiates the study of the Fatou\,--\,Julia sets of a complex harmonic mapping. Along with some fundamental properties of the Fatou and the Julia sets, we observe some contrasting behaviour of these sets as those with in case of a holomorphic function. The existence of harmonic mapping with a wandering domain is also shown.

We give a counterexample to the Arithmetic Puncturing Conjecture and Geometric Puncturing Conjecture of Hassett-Tschinkel using symmetric powers of uniruled surfaces, and propose a corrected conjecture inspired by Campana's conjectures on special varieties. We verify Campana's conjecture on potential density for symmetric powers of products of curves. As a by-product, we obtain an example of a surface without a potentially dense set of rational points, but for which some symmetric power does have a dense set of rational points, and even satisfies Corvaja-Zannier's version of the Hilbert property.

In the first part of this paper, we prove that, under some natural non-degeneracy assumptions, the Greedy Parabolic Target-Following Method, based on {\em universal tangent direction} has a favorable local behavior. In view of its global complexity bound of the order $O(\sqrt{n} \ln {1 \over \epsilon})$, this fact proves that the functional proximity measure, used for controlling the closeness to Greedy Central Path, is large enough for ensuring a local super-linear rate of convergence, provided that the proximity to the path is gradually reduced. This requirement is eliminated in our second algorithm based on a new auto-correcting predictor direction. This method, besides the best-known polynomial-time complexity bound, ensures an automatic switching onto the local quadratic convergence in a small neighborhood of solution. Our third algorithm approximates the path by quadratic curves. On the top of the best-known global complexity bound, this method benefits from an unusual local cubic rate of convergence. This amelioration needs no serious increase in the cost of one iteration. We compare the advantages of these local accelerations with possibilities of finite termination. The conditions allowing the optimal basis detection sometimes are even weaker than those required for the local superlinear convergence. Hence, it is important to endow the practical optimization schemes with both abilities. The proposed methods have a very interesting combination of favorable properties, which can be hardly found in the most of existing Interior-Point schemes. As all other parabolic target-following schemes, the new methods can start from an arbitrary strictly feasible primal-dual pair and go directly towards the optimal solution of the problem in a single phase. The preliminary computational experiments confirm the advantage of the second-order prediction.

Variational inequalities as an effective tool for solving applied problems, including machine learning tasks, have been attracting more and more attention from researchers in recent years. The use of variational inequalities covers a wide range of areas - from reinforcement learning and generative models to traditional applications in economics and game theory. At the same time, it is impossible to imagine the modern world of machine learning without distributed optimization approaches that can significantly speed up the training process on large amounts of data. However, faced with the high costs of communication between devices in a computing network, the scientific community is striving to develop approaches that make computations cheap and stable. In this paper, we investigate the compression technique of transmitted information and its application to the distributed variational inequalities problem. In particular, we present a method based on advanced techniques originally developed for minimization problems. For the new method, we provide an exhaustive theoretical convergence analysis for cocoersive strongly monotone variational inequalities. We conduct experiments that emphasize the high performance of the presented technique and confirm its practical applicability.

For a finite, simple, and undirected graph $G$ with $n$ vertices and average degree $d$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-d\right|$. Provided that $G$ has largest eigenvalue $\lambda$, minimum degree at least $\delta$, and maximum degree at most $\Delta$, where $0\leq\delta<d<\Delta<n$, we show $$s\leq \frac{2n(\Delta-d)(d-\delta)}{\Delta-\delta} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \lambda \geq \begin{cases} \frac{d^2n}{\sqrt{d^2n^2-s^2}} & \mbox{, if } s\leq \frac{dn}{\sqrt{2}},\\[3mm] \frac{2s}{n} & \mbox{, if } s> \frac{dn}{\sqrt{2}}. \end{cases}$$ Our results are based on a smoothing technique relating the degree deviation and the largest eigenvalue to low-dimensional non-linear optimization problems.

In the present work we shall consider the existence and multiplicity of solutions for nonlocal elliptic singular problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: \begin{equation*} \left\{\begin{array}{lll} -\Delta u+V(x)u=\lambda(I_{\alpha_1}*a|u|^q)a(x)|u|^{q-2}u+\mu(I_{\alpha_2}*|u|^p)|u|^{p-2}u u\in H^1(\mathbb{R}^N) \end{array}\right. \end{equation*} where $\alpha_2<\alpha_1$; $\alpha_1,\alpha_2\in(0,N)$ and $0<q<1$; $p\in\left(2_{\alpha_2},2^*_{\alpha_2} \right)$. Recall also that $2_{\alpha_j}=(N+\alpha_j)/N$ and $2^*_{\alpha_j}=(N+\alpha_j)/(N-2), j=1,2$. Furthermore, for each $q\in(0,1)$, by using the Hardy-Littlewood-Sobolev inequality we can find a sharp parameter $\lambda^*> 0$ such that our main problem has at least two solutions using the Nehari method. Here we also use the Rayleigh quotient for the following scenarios $\lambda \in (0, \lambda^*)$ and $\lambda = \lambda^*$. Moreover, we consider some decay estimates ensuring a non-existence result for the Choquard type problems in the whole space.

It is established existence of solution with prescribed $L^p$ norm for the following nonlocal elliptic problem: \begin{equation*} \left\{\begin{array}{cc} \displaystyle (-\Delta)^s_p u\ +\ V (x) |u|^{p-2}u\ = \lambda |u|^{p - 2}u + \beta\left|u\right|^{q-2}u\ \hbox{in}\ \mathbb{R}^N, \displaystyle \|u\|_p^p = m^p,\ u \in W^{s, p}(\mathbb{R}^N). \end{array}\right. \end{equation*} where $s \in (0,1), sp < N, \beta > 0 \text{ and } q \in (p, \overline{p}_s]$ where $\overline{p}_s =p+ sp^2/N$. The main feature here is to consider $L^p$-subcritical and $L^p$-critical cases. Furthermore, we work with a huge class of potentials $V$ taking into account periodic potentials, asymptotically periodic potentials, and coercive potentials. More precisely, we ensure the existence of a solution of the prescribed norm for the periodic and asymptotically periodic potential $V$ in the $L^p$-subcritical regime. Furthermore, for the $L^p$ critical case, our main problem admits also a solution with a prescribed norm for each $\beta > 0$ small enough.

Motivated by a financial valuation problem on an asset-pricing model with dependent trade duration and returns, in this paper we study coupled fully non-local equations, where a linear non-local operator jointly acts on the time and space variables. We prove existence and uniqueness of the solution. Existence is established by providing a stochastic representation based on anomalous processes constructed as a time change via the undershooting of an independent subordinator. This leads to general non-stepped processes with intervals of constancy representing a sticky or trapping effect (i.e., constant price in financial applications). Our theory allows these intervals to be dependent on the immediately subsequent jump. A maximum principle is then proved and used to derive uniqueness. Based on these general results, we consider a particular case: a non-local analog of the Black and Scholes equation, addressing the problem of determining the seasoned price of a derivative security.

There is a well established theory that links semi-Markov chains having Mittag-Leffler waiting times to time-fractional equations. We here go beyond the semi-Markov setting, by defining some non-Markovian chains whose waiting times, although marginally Mittag-Leffler, are assumed to be stochastically dependent. This creates a long memory tail in the evolution, unlike what happens for semi-Markov processes. As a special case of our chains, we study a particular counting process which extends the well-known fractional Poisson process, the last one having independent, Mittag-Leffler waiting times.

We consider cubic forms $\phi_{a,b}(x,y,z) = ax^3 + by^3 - z^3$ with coefficients $a,b \in \mathbb{Z}$. We give an asymptotic formula for how many of these forms are locally soluble everywhere, i.e. we give an asymptotic formula for the number of pairs of integers $(a, b)$ that satisfy $1 \leq a \leq A$, $1 \leq b \leq B$ and some mild conditions, such that $\phi_{a,b}$ has a non-zero solution in $\mathbb{Q}_p$ for all primes $p$.

We study a version of first passage percolation on $\mathbb{Z}^d$ where the random passage times on the edges are replaced by contact times represented by random closed sets on $\mathbb{R}$. Similarly to the contact process without recovery, an infection can spread into the system along increasing sequences of contact times. In case of stationary contact times, we can identify associated first passage percolation models, which in turn establish shape theorems also for first contact percolation. In case of periodic contact times that reflect some reoccurring daily pattern, we also present shape theorems with limiting shapes that are universal with respect to the within-one-day contact distribution. In this case, we also prove a Poisson approximation for increasing numbers of within-one-day contacts. Finally, we present a comparison of the limiting speeds of three models -- all calibrated to have one expected contact per day -- that suggests that less randomness is beneficial for the speed of the infection. The proofs rest on coupling and subergodicity arguments.

In this work, we present sufficient cancellation conditions for the boundedness of inhomogeneous Calder\'on-Zygmund type operators on local Hardy spaces defined over spaces of homogeneous type in the sense of Coifman & Weiss for $ 0<p\leq 1 $. A new approach to atoms and molecules for local Hardy spaces in this setting are introduced {with special moment conditions}.

We investigate regularity properties of minimizers for non-autonomous convex variational integrands $F(x, \mathrm{D} u)$ with linear growth, defined on bounded Lipschitz domains $\Omega \subset \mathbb{R}^n$. Assuming appropriate ellipticity conditions and H\"older continuity of $\mathrm{D}_zF(x,z)$ with respect to the first variable, we establish higher integrability of the gradient of minimizers and provide bounds on the Hausdorff dimension of the singular set of minimizers.

We introduce a two-dimensional discrete-time dynamical system which represents the evolution of an angle and angular velocity. While the angle evolves by a fixed amount in every step, the evolution of the angular velocity is governed by a nonlinear map. We study the periodicity and stability of solutions to the system for a range of parameter values and initial conditions. The coupled system is shown to be periodic for certain parameter choices and initial conditions. In the limit, when the change in the angle tends to zero, the map is equivalent to the dynamics of a simple pendulum. Based on the integral of motion of the pendulum, an approximate invariant for the system is obtained. Simulations showing the behavior of the system for different parameter values and initial conditions are presented.

To find crystals of $\mathfrak{sl}_2$ representations of the form $\Lambda^n\text{Sym}^r\mathbb{C}^2$ it suffices to solve the combinatorial problem of decomposing Young's lattice into symmetric, saturated chains. We review the literature on this latter problem, and present a strategy to solve it. For $n \le 4$, the strategy recovers recently discovered solutions. We obtain (i) counting formulas for plethystic coefficients, (ii) new recursive formulas for plethysms of Schur functions, and (iii) formulas for the number of constituents of $\Lambda^n\text{Sym}^r\mathbb{C}^2$.

This work considers the nonconvex, nonsmooth problem of minimizing a composite objective of the form $f(g(x))+h(x)$ where the inner mapping $g$ is a smooth finite summation or expectation amenable to variance reduction. In such settings, prox-linear methods can enjoy variance-reduced speed-ups despite the existence of nonsmoothness. We provide a unified convergence theory applicable to a wide range of common variance-reduced vector and Jacobian constructions. Our theory (i) only requires operator norm bounds on Jacobians (whereas prior works used potentially much larger Frobenius norms), (ii) provides state-of-the-art high probability guarantees, and (iii) allows inexactness in proximal computations.

This work introduces a method for preprocessing measurements of electrical impedance tomography to considerably reduce the effect uncertainties in the electrode contacts have on the reconstruction quality, without a need to explicitly estimate the contacts. The idea is to compute the Jacobian matrix of the forward map with respect to the contact strengths and project the electrode measurements and the forward map onto the orthogonal complement of the range of this Jacobian. Using the smoothened complete electrode model as the forward model, it is demonstrated that inverting the resulting projected equation with respect to only the internal conductivity of the examined body results in good quality reconstructions both when resorting to a single step linearization with a smoothness prior and when combining lagged diffusivity iteration with total variation regularization. The quality of the reconstructions is further improved if the range of the employed projection is also orthogonal to that of the Jacobian with respect to the electrode positions. These results hold even if the projections are formed at internal and contact conductivities that significantly differ from the true ones; it is numerically demonstrated that the orthogonal complement of the range of the contact Jacobian is almost independent of the conductivity parameters at which it is evaluated. In particular, our observations introduce a numerical technique for inferring whether a change in the electrode measurements is caused by a change in the internal conductivity or alterations in the electrode contacts, which has potential applications, e.g., in bedside monitoring of stroke patients. The ideas are tested both on simulated data and on real-world water tank measurements with adjustable contact resistances.

We construct functions $f \colon [0,1] \to [0,1]$ whose graph as a subset of $\mathbb{R}^2$ has Hausdorff dimension greater than any given value $\alpha \in (1,2)$ but conformal dimension $1$. These functions have the property that a positive proportion of level sets have positive codimension-$1$ measure. This result gives a negative answer to a question of Binder--Hakobyan--Li. We also give a function whose graph has Hausdorff dimension $2$ but conformal dimension $1$. The construction is based on the author's previous solution to the inverse absolute continuity problem for quasisymmetric mappings.

In arXiv:2211.04917, it was shown that, over an algebraically closed field of characteristic zero, every fusion 2-category is Morita equivalent to a connected fusion 2-category, that is, one arising from a braided fusion 1-category. We extend this result to compact semisimple tensor 2-categories over an arbitrary field of characteristic zero. In order to do so, we generalize to an arbitrary field of characteristic zero many well-known results about braided fusion 1-categories over an algebraically closed field of characteristic zero. Most notably, we prove that the Picard group of any braided fusion 1-category is indfinite, generalizing the classical fact that the Brauer group of a field is torsion. As an application of our main result, we derive the existence of braided fusion 1-categories indexed by the fourth Galois cohomology group of the absolute Galois group that represent interesting classes in the appropriate Witt groups.

In this work we fully characterize the classes of matrix weights for which multilinear Calder\'on-Zygmund operators extend to bounded operators on matrix weighted Lebesgue spaces. To this end, we develop the theory of multilinear singular integrals taking values in tensor products of finite dimensional Hilbert spaces. On the one hand, we establish quantitative bounds in terms of multilinear Muckenhoupt matrix weight characteristics and scalar Fujii-Wilson conditions of a tensor product analogue of the convex body sparse operator, of a convex-set valued tensor product analogue of the Hardy-Littlewood maximal operator, and of a multilinear analogue of the Christ-Goldberg maximal operator. These bounds recover the sharpest known bounds in the linear case. Moreover, we define a notion of directional nondegeneracy for multilinear Calder\'on-Zygmund operators, which is new even in the scalar case. The noncommutavity of matrix multiplication, the absence of duality, and the natural presence of quasinorms in the multilinear setting present several new difficulties in comparison to previous works in the scalar or in the linear case. To overcome them, we use techniques inspired from convex combinatorics and differential geometry.

A long standing problem in abstract harmonic analysis concerns the strong Arens irregularity (sAir, for short) of the Fourier algebra $A(G)$ of a locally compact group $G.$ The groups for which $A(G)$ is known to be sAir are all amenable. So far this class includes the abelian groups, the discrete amenable groups, the second countable amenable groups $G$ such that $\overline{[G, G]}$ is not open in $G,$ the groups of the form $\prod_{i=0}^\infty G_i$ where each $G_i$, $i\ge1$, is a non-trivial metrizable compact group and $G_0$ is an amenable second countable locally compact group, the groups of the form $G_0\times G$, where $G$ is a compact group whose local weight $w(G)$ has uncountable cofinality and $G_0$ is any locally compact amenable group with $w(G_0)\le w(G)$, and the compact group $SU(2).$ We were primarily concerned with the groups for which $A(G)$ is sAir. We introduce a new class of $\ell^1$-bases in Banach algebras. These new $\ell^1$-bases enable us, among other results, to unify most of the results related to Arens products proved in the past seventy years since Arens defined his products. This includes the strong Arens irregularity of algebras in harmonic analysis, and in particular almost all the cases mentioned above for the Fourier algebras. In addition, we also show that $A(G)$ is sAir for compact connected groups with an infinite dual rank. The $\ell^1$-bases for the Fourier algebra are constructed with coefficients of certain irreducible representations of the group. With this new approach using $\ell^1$-bases, the rich algebraic structure of the algebras and semigroups under study such as the second dual of Banach algebras with an Arens product or certain semigroup compactifications (the Stone-\v Cech compactification of an infinite discrete group, for instance) is also unveiled

We study the orthospectrum and the simple orthospectrum of compact hyperbolic surfaces with geodesic boundary. We show that there are finitely many hyperbolic surfaces sharing the same simple orthospectrum and finitely many hyperbolic surfaces sharing the same orthospectrum. Then, we show that generic surfaces are determined by their orthospectrum and by their simple orthospectrum. We conclude with the example of the one-holed torus which is determined by its simple orthospectrum.

Let $k\geq 2$ and fix a $k$-uniform hypergraph $\mathcal{F}$. Consider the random process that, starting from a $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices, repeatedly deletes the edges of a copy of $\mathcal{F}$ chosen uniformly at random and terminates when no copies of $\mathcal{F}$ remain. Let $R(\mathcal{H},\mathcal{F})$ denote the number of edges that are left after termination. We show that $R(\mathcal{H},\mathcal{F})=n^{k-1/\rho\pm o(1)}$, where $\rho:=(\lvert E(\mathcal{F})\rvert-1)/(\lvert V(\mathcal{F})\rvert -k)$, holds with high probability provided that $\mathcal{F}$ is strictly $k$-balanced and $\mathcal{H}$ is sufficiently dense with pseudorandom properties. Since we may in particular choose $\mathcal{F}$ and $\mathcal{H}$ to be complete graphs, this confirms the major folklore conjecture in the area in a very strong form.

We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions $f$ from a class $\mathcal{H}$, but the supremum over $f\in $ $\mathcal{H}$ is taken outside the probability. This form is a prerequisite for the Koml\'{o}s-Major-Tusn\'{a}dy inequality in the space of bounded functionals $l^{\infty }(\mathcal{H})$, but contrary to the latter it essentially preserves the classical $n^{-1/2}\log n$ approximation rate over large functional classes $\mathcal{H}$ such as the H\"{o}lder ball of smoothness $1/2$. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.

We provide the final step in the resolution of Bourgain's slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq \mathbb{R}^n$ such that $$ Vol_{n-1}(K \cap H) > c, $$ where $c > 0$ is a universal constant. Our proof combines Milman's theory of $M$-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.

We show that induction along a Frobenius extension of Hopf algebras is a Frobenius monoidal functor in great generality, in particular, for all finite-dimensional and all pointed Hopf algebras. As an application, we show that induction functors from unimodular Hopf subalgebras to small quantum groups at roots of unity are Frobenius monoidal functors and classify such unimodular Hopf subalgebras. Moreover, we present stronger conditions on Frobenius extensions under which the induction functor extends to a braided Frobenius monoidal functor on categories of Yetter--Drinfeld modules. We show that these stronger conditions hold for any extension of finite-dimensional semisimple and co-semisimple (or, more generally, unimodular and dual unimodular) Hopf algebras.

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance $\Delta $; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value $f(t_i)$ of a regression function $f$ at a grid point $t_i$ (nonparametric GLM). When $f$ is in a H\"{o}lder ball with exponent $\beta >\frac 12 ,$ we establish global asymptotic equivalence to observations of a signal $\Gamma (f(t))$ in Gaussian white noise, where $\Gamma $ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.

In infectious disease modelling, the expected time from endemicity to extinction (of infection) may be analysed via WKB approximation, a method with origins in mathematical physics. The method is very general, but its uptake to date may have been limited by the practical difficulties of implementation. It is necessary to compute a trajectory of a (high dimensional) dynamical system, the `extinction path', and this trajectory is maximally sensitive to small perturbations, making numerical computation challenging. Our objective here is to make this methodology more accessible by presenting four computational algorithms, with associated Matlab code, together with discussion of various ways in which the algorithms may be tuned to achieve satisfactory convergence. We illustrate our methods using three standard infectious disease models. For each such model, we demonstrate that our algorithms are able to improve upon previously available results.

The coherence conjecture of Pappas and Rapoport, proved by Zhu, asserts the equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a certain union of Schubert varieties in a partial affine flag variety. In this paper, we upgrade this equality of dimensions to an isomorphism of representations. This has interesting consequences in the setting of affine Demazure modules. Zhu's proof of coherence conjcture and our comparison theorem are established by introducing a parahoric Bruhat-Tits group scheme $\mathcal{G}$ over the affine line that is ramified at $0$. We further strengthen this comparison theorem by equipping any line bundle on the global affine Grassmannian of $\mathcal{G}$ with a unique equivariant structure of the global jet group scheme of $\mathcal{G}$.

In this paper, we present the numerical analysis and simulations of a multi-dimensional memristive device model. Memristive devices and memtransistors based on two-dimensional (2D) materials have demonstrated promising potential as components for next-generation artificial intelligence (AI) hardware and information technology. Our charge transport model describes the drift-diffusion of electrons, holes, and ionic defects self-consistently in an electric field. We incorporate two types of boundary models: ohmic and Schottky contacts. The coupled drift-diffusion partial differential equations are discretized using a physics-preserving Voronoi finite volume method. It relies on an implicit time-stepping scheme and the excess chemical potential flux approximation. We demonstrate that the fully discrete nonlinear scheme is unconditionally stable, preserving the free-energy structure of the continuous system and ensuring the non-negativity of carrier densities. Novel discrete entropy-dissipation inequalities for both boundary condition types in multiple dimensions allow us to prove the existence of discrete solutions. We perform multi-dimensional simulations to understand the impact of electrode configurations and device geometries, focusing on the hysteresis behavior in lateral 2D memristive devices. Three electrode configurations -- side, top, and mixed contacts -- are compared numerically for different geometries and boundary conditions. These simulations reveal the conditions under which a simplified one-dimensional electrode geometry can well represent the three electrode configurations. This work lays the foundations for developing accurate, efficient simulation tools for 2D memristive devices and memtransistors, offering tools and guidelines for their design and optimization in future applications.

For the non-monotone Hamilton-Jacobi equations of contact type, the associated Lax-Oleinik semiflow $(T_t, C(M))$ is expansive. In this paper, we provide qualitative estimates for both the lower and upper bounds of the topological entropy of the semiflow.

Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any $k \ge 2$, there exist a family of $k$ closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent.

Let $\Gamma \subset \mathbb{P}^n$ be a nondegenerate finite subscheme of degree $d$. Then the Castelnuovo-Mumford regularity ${\rm reg} ({\Gamma})$ of $\Gamma$ is at most $\left\lceil \frac{d-n-1}{t(\Gamma)} \right\rceil +2$ where $t(\Gamma)$ is the smallest integer such that $\Gamma$ admits a $(t+2)$-secant $t$-plane. In this paper, we show that ${\rm reg} ({\Gamma})$ is close to this upper bound if and only if there exists a unique rational normal curve $C$ of degree $t(\Gamma)$ such that the ${\rm reg} (\Gamma \cap C) = {\rm reg} (\Gamma)$.

We prove that the set of limit points of the set of all spectral gaps of closed arithmetic hyperbolic surfaces equals $[0,\frac{1}{4}]$.

We study homological invariants of the Steinberg algebra $\mathcal{A}_k(\mathcal{G})$ of an ample groupoid $\mathcal{G}$ over a commutative ring $k$. For $\mathcal{G}$ principal or Hausdorff with ${\mathcal{G}}^{\rm{Iso}}\setminus{\mathcal{G}}^{(0)}$ discrete, we compute Hochschild and cyclic homology of $\mathcal{A}_k(\mathcal{G})$ in terms of groupoid homology. For any ample Hausdorff groupoid $\mathcal{G}$, we find that $H_*(\mathcal{G})$ is a direct summand of $HH_*(\mathcal{A}_k(\mathcal{G}))$; using this and the Dennis trace we obtain a map $\overline{D}_*:K_*(\mathcal{A}_k(\mathcal{G}))\to H_n(\mathcal{G},k)$. We study this map when $\mathcal{G}$ is the (twisted) Exel-Pardo groupoid associated to a self-similar action of a group $G$ on a graph, and compute $HH_*(\mathcal{A}_k(\mathcal{G}))$ and $H_*(\mathcal{G},k)$ in terms of the homology of $G$, and the $K$-theory of $\mathcal{A}_k(\mathcal{G})$ in terms of that of $k[G]$.

We prove log-concavity of the lengths of the top rows of Young diagrams under Poissonized Plancherel measure. This is the first known positive result towards a conjecture of Chen that the length of the top row of a Young diagram under the Plancherel measure is log-concave. This is done by showing that the ordered elements of several discrete ensembles have log-concave distributions. In particular, we show the log-concavity of passage times in last passage percolation with geometric weights, using their connection to Meixner ensembles. In the continuous setting, distributions of the maximal elements of beta ensembles with convex potentials on the real line are shown to be log-concave. As a result, log-concavity of the $\beta$ versions of Tracy-Widom distributions follows; in fact, we also obtain log-concavity and positive association for the joint distribution of the $k$ smallest eigenvalues of the stochastic Airy operator. Our methods also show the log-concavity of finite dimensional distributions of the Airy-2 process and the Airy distribution. A log-concave distribution with full-dimensional support must have density, a fact that was apparently not known for some of these examples.

The Kuramoto model (KM) of $n$ coupled phase-oscillators is analyzed in this work. The KM on a Cayley graph possesses a family of steady state solutions called twisted states. Topologically distinct twisted states are distinguished by the winding number $q\in\mathbb{Z}$. These states are known to be stable for small enough $q$. In the presence of small noise, the KM exhibits metastable transitions between $q$-twisted states: A typical trajectory remains in the basin of attraction of a given $q$-twisted state for an exponentially long time, but eventually transitions to the vicinity of another such state. In the course of this transition, it passes in close proximity of a saddle of Morse index $1$, called a relevant saddle. In this work, we provide an exhaustive analysis of metastable transitions in the stochastic KM with nearest-neighbor coupling. We start by analyzing the equilibria and their stability. First, we identify all equilibria in this model. Using the discrete Fourier transform and eigenvalue estimates for rank-1 perturbations of symmetric matrices, we classify the equilibria by their Morse indices. In particular, we identify all stable equilibria and all relevant saddles involved in the metastable transitions. Further, we use Freidlin-Wentzell theory and the potential-theoretic approach to metastability to establish the metastable hierarchy and sharp estimates of Eyring-Kramers type for the transition times. The former determines the precise order, in which the metastable transitions occur, while the latter characterizes the times between successive transitions. The theoretical estimates are complemented by numerical simulations and a careful numerical verification of the transition times. Finally, we discuss the implications of this work for the KM with other coupling types including nonlocal coupling and the continuum limit as $n$ tends to infinity.

Recently, the authors have proved the finiteness of common zeros of two iterated rational maps under some compositional independence assumptions. In this article, we advance towards a question of Hsia and Tucker on a Zariski non-density of common zeros of iterated morphisms on a variety. More precisely, we provide an affirmative answer in the case of H\'{e}non type maps on $\mathbb{A}^2$, endomorphisms on $(\mathbb{P}^1)^n$, and polynomial skew products on $\mathbb{A}^2$ defined over $\overline{\mathbb{Q}}$. As a by-product, we prove a Tits' alternative analogy for semigroups generated by two regular polynomial skew products.

High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade multiderivative time-stepping have been used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become increasingly relevant. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the SSP theory for explicit and implicit two-derivative Runge--Kutta schemes, and discuss a special condition on the second derivative under which these implicit methods may be unconditionally SSP. This condition is then used in the context of implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes, where the time-step restriction is independent of the stiff term. Finally, we present the SSP theory for implicit-explicit (IMEX) multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.

We study ancient solutions to discrete heat equations on some weighted graphs. On a graph of the form of a product with $\bb Z,$ we show that there are no non-trivial ancient solutions with polynomial growth. This result is parallel to the case of finite graphs, which is also discussed. Along the way, we prove a backward uniqueness result for solutions with appropriate decaying rate based on a monotonicity formula of parabolic frequency.

We study relationships between asymptotic geometry of submanifolds in the hyperbolic space and their regularity properties near the ideal boundary, revisiting some of the related results in the literature. In particular, we discuss hypotheses when minimal submanifolds meet the ideal boundary orthogonally, and compute the essential spectrum of the Laplace operator on submanifolds that are asymptotically close to minimal submanifolds.

Twisted generalized Reed-Solomon (TGRS) codes constitute an interesting family of evaluation codes, containing a large class of maximum distance separable codes non-equivalent to generalized Reed-Solomon (GRS) ones. Moreover, the Schur squares of TGRS codes may be much larger than those of GRS codes with same dimension. Exploiting these structural differences, in 2018, Beelen, Bossert, Puchinger and Rosenkilde proposed a subfamily of Maximum Distance Separable (MDS) Twisted Reed-Solomon (TRS) codes over $\mathbb{F}_q$ with $\ell$ twists $q \approx n^{2^{\ell}}$ for McEliece encryption, claiming their resistance to both Sidelnikov Shestakov attack and Schur products--based attacks. In short, they claimed these codes to resist to classical key recovery attacks on McEliece encryption scheme instantiated with Reed-Solomon (RS) or GRS codes. In 2020, Lavauzelle and Renner presented an original attack on this system based on the computation of the subfield subcode of the public TRS code. In this paper, we show that the original claim on the resistance of TRS and TGRS codes to Schur products based--attacks is wrong. We identify a broad class of codes including TRS and TGRS ones that is distinguishable from random by computing the Schur square of some shortening of the code. Then, we focus on the case of single twist (i.e., $\ell = 1$), which is the most efficient one in terms of decryption complexity, to derive an attack. The technique is similar to the distinguisher-based attacks of RS code-based systems given by Couvreur, Gaborit, Gauthier-Uma\~na, Otmani, Tillich in 2014.

In this paper we obtain inequalities for the geometric mean of elements in the Grassmannians. These inequalities reflect the elliptic geometry of the Grassmannians as Riemannian manifolds. These include Semi-Parallelogram Law, Law of Cosines and geodesic triangle inequalities.

We identify two existing approaches to the derived equivalence for the stratified Mukai flop of cotangent bundles of Grassmannians -- one induced by the geometric categorical sl(2) action, and the other through the magic window category of graded matrix factorizations on the gauged Landau--Ginzburg model -- via the Kn\"orrer periodicity.

In 2019, Fox, Tidor and Zhao (arXiv:1911.03427) proved an induced arithmetic removal lemma for linear patterns of complexity 1 in vector spaces over a fixed finite field. With no further assumptions on the pattern, this induced removal lemma cannot guarantee a fully pattern-free recolouring of the space, as some `non-generic' instances must necessarily remain. On the other hand, Bhattacharyya et al. (arXiv:1212.3849) showed that in the case of translation-invariant patterns, it is possible to obtain recolourings that eliminate the given pattern completely, with no exceptions left behind. This paper demonstrates that such complete removal can be achieved for all partition-regular arithmetic patterns of complexity 1.

A casino offers the following game. There are three cups each containing a die. You are being told that the dice in the cups are all the same, but possibly nonstandard. For a bet of \$1, the game master shakes all three cups and lets you choose one of them. You win \$2 if the die in your cup displays at least the average of the other two, and you lose otherwise. Is this game fair? If not, how should the casino design the dice to maximize their profit? This problem is a special case of the following more general question: given a measurable space $X$ and a bounded measurable function $f : X^n \to \mathbb{R}$, how large can the expectation of $f$ under probability measures of the form $\mu^{\otimes n}$ be? We develop a general method to answer this kind of question. As an example application that is harder than the casino problem, we show that the maximal probability of the event $X_1 + X_2 + X_3 < 2 X_4$ for nonnegative iid random variables lies between $0.4$ and $0.422$, where the upper bound is obtained by mixed integer linear programming. We conjecture the lower bound to be the exact value.

We study a class of Gaussian random band matrices of dimension $N \times N$ and band-width $W$. We show that delocalization holds for bulk eigenvectors and that quantum diffusion holds for the resolvent, all under the assumption that $W \gg N^{8/11}$. Our analysis is based on a flow method, and a refinement of it may lead to an improvement on the condition $W \gg N^{8/11}$.

Topological zero modes in topological insulators or superconductors are exponentially localized at the phase transition between a topologically trivial and nontrivial phase. These modes are solutions of a Jackiw-Rebbi equation modified with an additional term which is quadratic in the momentum. Moreover, localized fermionic modes can also be induced by harmonic potentials in superfluids and superconductors or in atomic nuclei. Here, by using inverse methods, we consider in the same framework exponentially-localized zero modes, as well as gaussian modes induced by harmonic potentials (with superexponential decay) and polynomially decaying modes (with subexponential decay), and derive the explicit and analytical form of the modified Jackiw-Rebbi equation (and of the Schr\"odinger equation) which admits these modes as solutions. We find that the asymptotic behavior of the mass term is crucial in determining the decay properties of the modes. Furthermore, these considerations naturally extend to the nonhermitian regime. These findings allow us to classify and understand topological and nontopological boundary modes in topological insulators and superconductors.

We apply the Kosambi-Cartan-Chern theory to perform an extensive examination of Jacobi stability of geodesics around rotating black hole solutions to dynamical Chern-Simons gravity, a theory that introduces modifications to General Relativity via a scalar field non-minimally coupled to curvature scalars. We present a comparative study between Jacobi and Liapunov stability, pointing out the advantages of the more geometrical method over the usual Liapunov approach.

In this paper, we propose a novel approach to minimize the inference delay in semantic segmentation using split learning (SL), tailored to the needs of real-time computer vision (CV) applications for resource-constrained devices. Semantic segmentation is essential for applications such as autonomous vehicles and smart city infrastructure, but faces significant latency challenges due to high computational and communication loads. Traditional centralized processing methods are inefficient for such scenarios, often resulting in unacceptable inference delays. SL offers a promising alternative by partitioning deep neural networks (DNNs) between edge devices and a central server, enabling localized data processing and reducing the amount of data required for transmission. Our contribution includes the joint optimization of bandwidth allocation, cut layer selection of the edge devices' DNN, and the central server's processing resource allocation. We investigate both parallel and serial data processing scenarios and propose low-complexity heuristic solutions that maintain near-optimal performance while reducing computational requirements. Numerical results show that our approach effectively reduces inference delay, demonstrating the potential of SL for improving real-time CV applications in dynamic, resource-constrained environments.

Three-dimensional supergravity in the Batalin--Vilkovisky formalism is constructed by showing that the theory including the Rarita--Schwinger term is equivalent to an AKSZ theory.

There are now many examples of gapped fracton models, which are defined by the presence of restricted-mobility excitations above the quantum ground state. However, the theory of fracton orders remains in its early stages, and the complex landscape of examples is far from being mapped out. Here we introduce the class of planon-modular (p-modular) fracton orders, a relatively simple yet still rich class of quantum orders that encompasses several well-known examples of type I fracton order. The defining property is that any non-trivial point-like excitation can be detected by braiding with planons. From this definition, we uncover a significant amount of general structure, including the assignment of a natural number (dubbed the weight) to each excitation of a p-modular fracton order. We identify simple new phase invariants, some of which are based on weight, which can easily be used to compare and distinguish different fracton orders. We also study entanglement renormalization group (RG) flows of p-modular fracton orders, establishing a close connection with foliated RG. We illustrate our general results with an analysis of several exactly solvable fracton models that we show to realize p-modular fracton orders, including Z_n versions of the X-cube, anisotropic, checkerboard, 4-planar X-cube and four color cube (FCC) models. We show that each of these models is p-modular and compute its phase invariants. We also show that each example admits a foliated RG at the level of its non-trivial excitations, which is a new result for the 4-planar X-cube and FCC models. We show that the Z_2 FCC model is not a stack of other better-studied models, but predict that the Z_n FCC model with n odd is a stack of 10 4-planar X-cubes, possibly plus decoupled layers of 2d toric code. We also show that the Z_n checkerboard model for n odd is a stack of three anisotropic models.

Mixed-Integer Programming (MIP) is a powerful paradigm for modeling and solving various important combinatorial optimization problems. Recently, learning-based approaches have shown potential to speed up MIP solving via offline training that then guides important design decisions during search. However, a significant drawback of these methods is their heavy reliance on offline training, which requires collecting training datasets and computationally costly training epochs yet offering only limited generalization to unseen (larger) instances. In this paper, we propose Balans, an adaptive meta-solver for MIPs with online learning capability that does not require any supervision or apriori training. At its core, Balans is based on adaptive large-neighborhood search, operating on top of a MIP solver by successive applications of destroy and repair neighborhood operators. During the search, the selection among different neighborhood definitions is guided on the fly for the instance at hand via multi-armed bandit algorithms. Our extensive experiments on hard optimization instances show that Balans offers significant performance gains over the default MIP solver, is better than committing to any single best neighborhood, and improves over the state-of-the-art large-neighborhood search for MIPs. Finally, we release Balans as a highly configurable, MIP solver agnostic, open-source software.

Symmetry plays a central role in the sciences, machine learning, and statistics. While statistical tests for the presence of distributional invariance with respect to groups have a long history, tests for conditional symmetry in the form of equivariance or conditional invariance are absent from the literature. This work initiates the study of nonparametric randomization tests for symmetry (invariance or equivariance) of a conditional distribution under the action of a specified locally compact group. We develop a general framework for randomization tests with finite-sample Type I error control and, using kernel methods, implement tests with finite-sample power lower bounds. We also describe and implement approximate versions of the tests, which are asymptotically consistent. We study their properties empirically on synthetic examples, and on applications to testing for symmetry in two problems from high-energy particle physics.

We propose a short-term wind forecasting framework for predicting real-time variations in atmospheric turbulence based on nacelle-mounted anemometer and ground-level air-pressure measurements. Our approach combines linear stochastic estimation and Kalman filtering algorithms to assimilate and process real-time field measurements with the predictions of a stochastic reduced-order model that is confined to a two-dimensional plane at the hub height of turbines. We bridge the vertical gap between the computational plane of the model at hub height and the measurement plane on the ground using a projection technique that allows us to infer the pressure in one plane from the other. Depending on the quality of this inference, we show that customized variants of the extended and ensemble Kalman filters can be tuned to balance estimation quality and computational speed 1-1.5 diameters ahead and behind leading turbines. In particular, we show how synchronizing the sign of estimates with that of velocity fluctuations recorded at the nacelle can significantly improve the ability to follow temporal variations upwind of the leading turbine. We also propose a convex optimization-based framework for selecting a subset of pressure sensors that achieve a desired level of accuracy relative to the optimal Kalman filter that uses all sensing capabilities.

Mixed Integer Linear Programs (MILPs) are highly flexible and powerful tools for modeling and solving complex real-world combinatorial optimization problems. Recently, machine learning (ML)-guided approaches have demonstrated significant potential in improving MILP-solving efficiency. However, these methods typically rely on separate offline data collection and training processes, which limits their scalability and adaptability. This paper introduces the first multi-task learning framework for ML-guided MILP solving. The proposed framework provides MILP embeddings helpful in guiding MILP solving across solvers (e.g., Gurobi and SCIP) and across tasks (e.g., Branching and Solver configuration). Through extensive experiments on three widely used MILP benchmarks, we demonstrate that our multi-task learning model performs similarly to specialized models within the same distribution. Moreover, it significantly outperforms them in generalization across problem sizes and tasks.

We introduce a method for performing cross-validation without sample splitting. The method is well-suited for problems where traditional sample splitting is infeasible, such as when data are not assumed to be independently and identically distributed. Even in scenarios where sample splitting is possible, our method offers a computationally efficient alternative for estimating prediction error, achieving comparable or even lower error than standard cross-validation at a significantly reduced computational cost. Our approach constructs train-test data pairs using externally generated Gaussian randomization variables, drawing inspiration from recent randomization techniques such as data-fission and data-thinning. The key innovation lies in a carefully designed correlation structure among these randomization variables, referred to as antithetic Gaussian randomization. This correlation is crucial in maintaining a bounded variance while allowing the bias to vanish, offering an additional advantage over standard cross-validation, whose performance depends heavily on the bias-variance tradeoff dictated by the number of folds. We provide a theoretical analysis of the mean squared error of the proposed estimator, proving that as the level of randomization decreases to zero, the bias converges to zero, while the variance remains bounded and decays linearly with the number of repetitions. This analysis highlights the benefits of the antithetic Gaussian randomization over independent randomization. Simulation studies corroborate our theoretical findings, illustrating the robust performance of our cross-validated estimator across various data types and loss functions.

We discuss a large class of non-relativistic continuum field theories where the Euclidean spatial symmetry of the classical theory is violated in the quantum theory by an Adler-Bell-Jackiw-like anomaly. In particular, the continuous translation symmetry of the classical theory is broken in the quantum theory to a discrete symmetry. Furthermore, that discrete symmetry is extended by an internal symmetry, making it non-Abelian. This presentation streamlines and extends the discussion in [1]. In an Appendix, we present an elementary introduction to 't Hooft and Adler-Bell-Jackiw anomalies using a well-known system.

The survival of the domestic cat (Felis catus) in various ecosystems has become increasingly relevant due to its impact on wildlife, public health, and society. In countries like Mexico, social factors such as abandonment have led to the feralization of the species and an unexpected increase in its population in urban areas. To design and implement effective population control methods, a thorough analysis of the species' population dynamics, along with the social factors influencing it, is necessary. We propose a reaction-diffusion model to simulate the natural dispersal of the population within a bounded domain. After exploring the species' spreading ability, we construct a complex dynamical system based on the biological characteristics of cats and their intraspecific and interspecific interactions, which we explain and study in detail. Both deterministic and stochastic parameters are considered to enhance the realism of the simulations. Our results indicate that the population reaches equilibrium, highlighting the need for control methods combined with social regulations to achieve sustainability in the system.

We study the electrical distribution network reconfiguration problem, defined as follows. We are given an undirected graph with a root vertex, demand at each non-root vertex, and resistance on each edge. Then, we want to find a spanning tree of the graph that specifies the routing of power from the root to each vertex so that all the demands are satisfied and the energy loss is minimized. This problem is known to be NP-hard in general. When restricted to grids with uniform resistance and the root located at a corner, Gupta, Khodabaksh, Mortagy and Nikolova [Mathematical Programming 2022] invented the so-called Min-Min algorithm whose approximation factor is theoretically guaranteed. Our contributions are twofold. First, we prove that the problem is NP-hard even for grids; this resolves the open problem posed by Gupta et al. Second, we give a refined analysis of the Min-Min algorithm and improve its approximation factor under the same setup. In the analysis, we formulate the problem of giving an upper bound for the approximation factor as a non-linear optimization problem that maximizes a convex function over a polytope, which is less commonly employed in the analysis of approximation algorithms than linear optimization problems.

Longitudinal imaging allows for the study of structural changes over time. One approach to detecting such changes is by non-linear image registration. This study introduces Multi-Session Temporal Registration (MUSTER), a novel method that facilitates longitudinal analysis of changes in extended series of medical images. MUSTER improves upon conventional pairwise registration by incorporating more than two imaging sessions to recover longitudinal deformations. Longitudinal analysis at a voxel-level is challenging due to effects of a changing image contrast as well as instrumental and environmental sources of bias between sessions. We show that local normalized cross-correlation as an image similarity metric leads to biased results and propose a robust alternative. We test the performance of MUSTER on a synthetic multi-site, multi-session neuroimaging dataset and show that, in various scenarios, using MUSTER significantly enhances the estimated deformations relative to pairwise registration. Additionally, we apply MUSTER on a sample of older adults from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study. The results show that MUSTER can effectively identify patterns of neuro-degeneration from T1-weighted images and that these changes correlate with changes in cognition, matching the performance of state of the art segmentation methods. By leveraging GPU acceleration, MUSTER efficiently handles large datasets, making it feasible also in situations with limited computational resources.

An adult human body is made up of some 30 to 40 trillion cells, all of which stem from a single fertilized egg cell. The process by which the right cells appear to arrive in their right numbers at the right time at the right place -- development -- is only understood in the roughest of outlines. This process does not happen in isolation: the egg, the embryo, the developing foetus, and the adult organism all interact intricately with their changing environments. Conceptual and, increasingly, mathematical approaches to modelling development have centred around Waddington's concept of an epigenetic landscape. This perspective enables us to talk about the molecular and cellular factors that contribute to cells reaching their terminally differentiated state: their fate. The landscape metaphor is however only a simplification of the complex process of development; it for instance does not consider environmental influences, a context which we argue needs to be explicitly taken into account and from the outset. When delving into the literature, it also quickly becomes clear that there is a lack of consistency and agreement on even fundamental concepts; for example, the precise meaning of what we refer to when talking about a `cell type' or `cell state.' Here we engage with previous theoretical and mathematical approaches to modelling cell fate -- focused on trees, networks, and landscape descriptions -- and argue that they require a level of simplification that can be problematic. We introduce random dynamical systems as one natural alternative. These provide a flexible conceptual and mathematical framework that is free of extraneous assumptions. We develop some of the basic concepts and discuss them in relation to now `classical' depictions of cell fate dynamics, in particular Waddington's landscape.

The development of logic has largely been through the 'deductive' paradigm: conclusions are inferred from established premisses. However, the use of logic in the context of both human and machine reasoning is typically through the dual 'reductive' perspective: collections of sufficient premisses are generated from putative conclusions. We call this paradigm, 'reductive logic'. This expression of logic encompass as diverse reasoning activities as proving a formula in a formal system to seeking to meet a friend before noon on Saturday. This paper is a semantical analysis of reductive logic. In particular, we provide mathematical foundations for representing and reasoning about 'reduction operators'. Heuristically, reduction operators may be thought of as `backwards' inference rules. In this paper, we address their mathematical representation, how they are used in the context of reductive reasoning, and, crucially, what makes them 'valid'.

We consider an interacting particle system on star graphs. As in the case of the Kdv equation, we have infinitely many invariants ( martingale invariants). It enables us to obtain the limiting distribution of the Markov chain. Each of the martingale invariants is a homogeneous polynomial with coefficients of Narayana numbers.The identity for the enumeration of plane unlabeled trees, which gives Narayana numbers, becomes the key identity to obtain the probability of death states by a change of variables.

Many natural computational problems, including e.g. Max Weight Independent Set, Feedback Vertex Set, or Vertex Planarization, can be unified under an umbrella of finding the largest sparse induced subgraph, that satisfies some property definable in CMSO$_2$ logic. It is believed that each problem expressible with this formalism can be solved in polynomial time in graphs that exclude a fixed path as an induced subgraph. This belief is supported by the existence of a quasipolynomial-time algorithm by Gartland, Lokshtanov, Pilipczuk, Pilipczuk, and Rz\k{a}\.zewski [STOC 2021], and a recent polynomial-time algorithm for $P_6$-free graphs by Chudnovsky, McCarty, Pilipczuk, Pilipczuk, and Rz\k{a}\.zewski [SODA 2024]. In this work we extend polynomial-time tractability of all such problems to $P_7$-free graphs of bounded clique number.

Consider a graph $G$ with a long path $P$. When is it the case that $G$ also contains a long induced path? This question has been investigated in general as well as within a number of different graph classes since the 80s. We have recently observed in a companion paper (Long induced paths in sparse graphs and graphs with forbidden patterns, arXiv:2411.08685, 2024) that most existing results can recovered in a simple way by considering forbidden ordered patterns of edges along the path $P$. In particular we proved that if we forbid some fixed ordered matching along a path of order $n$ in a graph $G$, then $G$ must contain an induced path of order $(\log n)^{\Omega(1)}$. Moreover, we completely characterized the forbidden ordered patterns forcing the existence of an induced path of polynomial size. The purpose of the present paper is to completely characterize the ordered patterns $H$ such that forbidding $H$ along a path $P$ of order $n$ implies the existence of an induced path of order $(\log n)^{\Omega(1)}$. These patterns are star forests with some specific ordering, which we called constellations. As a direct consequence of our result, we show that if a graph $G$ has a path of length $n$ and does not contain $K_t$ as a topological minor, then $G$ contains an induced path of order $(\log n)^{\Omega(1/t \log^2 t)}$. The previously best known bound was $(\log n)^{f(t)}$ for some unspecified function $f$ depending on the Topological Minor Structure Theorem of Grohe and Marx (2015).

We study the evolution of a system of many point particles initially concentrated in a small region in $d$ dimensions. Particles undergo overdamped motion caused by pairwise interactions through the long-ranged repulsive $r^{-s}$ potential; each particle is also subject to Brownian noise. When $s<d$, the expansion is governed by non-local hydrodynamic equations. In the one-dimensional case, we deduce self-similar solutions for all $s\in (-2,1)$. The expansion of Coulomb gases remains well-defined in the infinite-particle limit: The density is spatially uniform and inversely proportional to time independent of the spatial dimension.

We propose a mirror derivation of the quiver description of open topological strings known as the knots-quivers correspondence, based on enumerative invariants of augmentation curves encoded by exponential networks. Quivers are obtained by studying M2 branes wrapping holomorphic disks with Lagrangian boundary conditions on an M5 brane, through their identification with a distinguished sector of BPS kinky vortices in the 3d-3d dual QFT. Our proposal suggests that holomorphic disks with Lagrangian boundary conditions are mirror to calibrated 1-chains on the associated augmentation curve, whose intersections encode the linking of boundaries.

We propose a novel and systematic recurrence method for the energy spectra of non-Hermitian systems under open boundary conditions based on the recurrence relations of their characteristic polynomials. Our formalism exhibits better accuracy and performance on multi-band non-Hermitian systems than numerical diagonalization or the non-Bloch band theory. It also provides a targeted and efficient formulation for the non-Hermitian edge spectra. As demonstrations, we derive general expressions for both the bulk and edge spectra of multi-band non-Hermitian models with nearest-neighbor hopping and under open boundary conditions, such as the non-Hermitian Su-Schrieffer-Heeger and Rice-Mele models and the non-Hermitian Hofstadter butterfly - 2D lattice models in the presence of non-reciprocity and perpendicular magnetic fields, which is only made possible by the significantly lower complexity of the recurrence method. In addition, we use the recurrence method to study non-Hermitian edge physics, including the size-parity effect and the stability of the topological edge modes against boundary perturbations. Our recurrence method offers a novel and favorable formalism to the intriguing physics of non-Hermitian systems under open boundary conditions.

We explore the relation between a classical periodic Hamiltonian system and an associated discrete quantum system on a torus in phase space. The model is a sinusoidally perturbed Harper model and is similar to the sinusoidally perturbed pendulum. Separatrices connecting hyperbolic fixed points in the unperturbed classical system become chaotic under sinusoidal perturbation. We numerically compute eigenstates of the Floquet propagator for the associated quantum system. For each Floquet eigenstate we compute a Husimi distribution in phase space and an energy and energy dispersion from the expectation value of the unperturbed Hamiltonian operator. The Husimi distribution of each Floquet eigenstate resembles a classical orbit with a similar energy and similar energy dispersion. Chaotic orbits in the classical system are related to Floquet eigenstates that appear ergodic. For a hybrid regular and chaotic system, we use the energy dispersion to separate the Floquet eigenstates into ergodic and integrable subspaces. The distribution of quasi-energies in the ergodic subspace resembles that of a random matrix model. The width of a chaotic region in the classical system is estimated by integrating the perturbation along a separatrix orbit. We derive a related expression for the associated quantum system from the averaged perturbation in the interaction representation evaluated at states with energy close to the separatrix.

This article provides a tutorial on over-the-air electromagnetic signal processing (ESP) for next-generation wireless networks, addressing the limitations of digital processing to enhance the efficiency and sustainability of future 6th Generation (6G) systems. It explores the integration of electromagnetism and signal processing (SP) highlighting how their convergence can drive innovations for 6G technologies. Key topics include electromagnetic (EM) wave-based processing, the application of metamaterials and advanced antennas to optimize EM field manipulation with a reduced number of radiofrequency chains, and their applications in holographic multiple-input multiple-output systems. By showcasing enabling technologies and use cases, the article demonstrates how wave-based processing can minimize energy consumption, complexity, and latency, offering an effective framework for more sustainable and efficient wireless systems. This article aims to assist researchers and professionals in integrating advanced EM technologies with conventional SP methods.

$A$-type Little String Theories (LSTs) are engineered from parallel M5-branes on a circle $\mathbb{S}_\perp^1$, probing a transverse $\mathbb{R}^4/\mathbb{Z}_M$ background. Below the scale of the radius of $\mathbb{S}_\perp^1$, these theories resemble a circular quiver gauge theory with $M$ nodes of gauge group $U(N)$ and matter in the bifundamental representation (or adjoint in the case of $M=1$). In this paper, we study these LSTs in the presence of a surface defect, which is introduced through the action of a $\mathbb{Z}_N$ orbifold that breaks the gauge groups into $[U(1)]^N$. We provide a combinatoric expression for the non-perturbative BPS partition function for this system. This form allows us to argue that a number of non-perturbative symmetries, that have previously been established for the LSTs, are preserved in the presence of the defect. Furthermore, we discuss the Nekrasov-Shatashvili (NS) limit of the defect partition function: focusing in detail on the case $(M,N)=(1,2)$, we analyse two distinct proposals made in the literature. We unravel an algebraic structure that is responsible for the cancellation of singular terms in the NS limit, which we generalise to generic $(M,N)$. In view of the dualities of higher dimensional gauge theories to quantum many-body systems, we provide indications that our combinatoric expression for the defect partition are useful in constructing and analysing quantum integrable systems in the future.

This paper introduces a new objective measure for assessing treatment response in asthmatic patients using computed tomography (CT) imaging data. For each patient, CT scans were obtained before and after one year of monoclonal antibody treatment. Following image segmentation, the Hounsfield unit (HU) values of the voxels were encoded through quantile functions. It is hypothesized that patients with improved conditions after treatment will exhibit better expiration, reflected in higher HU values and an upward shift in the quantile curve. To objectively measure treatment response, a novel linear regression model on quantile functions is developed, drawing inspiration from Verde and Irpino (2010). Unlike their framework, the proposed model is parametric and incorporates distributional assumptions on the errors, enabling statistical inference. The model allows for the explicit calculation of regression coefficient estimators and confidence intervals, similar to conventional linear regression. The corresponding data and R code are available on GitHub to facilitate the reproducibility of the analyses presented.

Some recent papers have extended the concept of finite-time stability (FTS) to the context of 2D linear systems, where it has been referred to as finite-region stability (FRS). FRS methodologies make even more sense than the classical FTS approach developed for 1D-systems, since, typically, at least one of the state variables of 2D-systems is a space coordinate, rather than a time variable. Since space coordinates clearly belong to finite intervals, FRS techniques are much more effective than the classical Lyapunov approach, which looks to the asymptotic behavior of the system over an infinite interval. To this regard, the novel contribution of this paper goes in several directions. First, we provide a novel sufficient condition for the FRS of linear time-varying (LTV) discrete-time 2D-systems, which turns out to be less conservative than those ones provided in the existing literature. Then, an interesting application of FRS to the context of iterative learning control (ILC) is investigated, by exploiting the previously developed theory. In particular, a new procedure is proposed so that the tracking errors of the ILC law converges within the desired bound in a finite number of iterations. Finally, a sufficient condition to solve the finite-region stabilization problem is proposed. All the results provided in the paper lead to optimization problems constrained by linear matrix inequalities (LMIs), that can be solved via widely available software. Numerical examples illustrate and validate the effectiveness of the proposed technique.

Based on the Newtonian mechanics, in this article, we present a heuristic derivation of the Friedmann equations, providing an intuitive foundation for these fundamental relations in cosmology. Additionally, using the first law of thermodynamics and Euler's equation, we derive a set of equations that, at linear order, coincide with those obtained from the conservation of the stress-energy tensor in General Relativity. This approach not only highlights the consistency between Newtonian and relativistic frameworks in certain limits but also serves as a pedagogical bridge, offering insights into the physical principles underlying the dynamics of the universe.