Papers

We characterize the pointwise notion of almost equicontinuity for families of real-valued measurable functions on subsets of $\mathbb R^n$ of finite measure. The characterization is given by means of an integral truncated translation condition. We also provide examples showing that the finite measure assumption and the truncation are essential.

In this paper, we study a class of quasilinear elliptic problems involving the $(1,q)-$Laplacian operator and a discontinuous superlinear nonlinearity governed by the Heaviside function. The main difficulty of the problem arises from the presence of the $1$-Laplacian operator, whose natural setting is the Space of Functions of Bounded Variation. Our approach is based on an approximation method involving $(p,q)-$Laplacian problems as $p\to1^+$. As a consequence, we prove the existence of a nontrivial and nonnegative solution belonging to $W^{1,p}_0(\Omega)$, in an appropriate weak sense. Moreover, we investigate the asymptotic behavior of the solutions as $\beta\to0^+$, showing that the family of solutions converges to a solution of the limit problem without discontinuity.

Let $G = \mathbf{GL}_d(\mathbb{F})$ be the general linear group over a field of cardinal $q$, and let $\mathbb{k}$ be a field of positive characteristic which does not divide $q(q-1)$. Building on the works of Cline, Parshall, and Scott, we show how to compute Ext-groups between $\mathbb{k}G$-modules using the quantum Schur algebra. The main novelty is our ability to compute these Ext-groups in higher degree than what was done before. More precisely, let $\ell$ be the order of $q$ in $\mathbb{k}$. In previous work, this method enabled the computation of the cohomology groups $H^*(\mathbf{GL}_d,M)$ in degree $*\leq \ell-1$. We show that for a lot of modules $M$, we can compute these cohomology groups in higher degree, with an example where we can compute until degree $3(\ell-1)$. We also show some new result on Ext-groups between modules over the quantum Schur algebra along the way.

In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(\rho_0,u_0)$ lying in $C^1 \times (L^2 \cap VMO^{-1})$, where $\rho_0$ has a positive lower bound. Furthermore, if $\rho_0 \in C^2$ and $||\rho_0-1||_{L^\infty}+||u_0||_{BMO^{-1}}$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.

Randomized sketching replaces large matrices with smaller surrogates that preserve geometry for downstream numerical linear algebra. Sparse sketches promise faster formation than dense projections, but irregular GPU atomic updates become costly when each input row contributes to several output rows, as in SparseStack. FP16 accumulation halves sketch storage and update traffic, relieving cache pressure. How much does this rounding affect sketch geometry and downstream accuracy? For SparseStack with four nonzeros per column, cuSPARSE's FP16 path is slower than its FP32 baseline. Our custom kernel instead pairs FP32 inputs and signed products with packed FP16 vector atomic accumulation, turning lower precision into substantial gains. On NVIDIA H200, it forms complete sketches up to $1.91\times$ faster than our FP32 baseline and up to $1.58\times$ faster than cuSPARSE FP32. Under stated independence and range assumptions, random sketch signs cancel round-to-nearest conversion-error cross terms, giving expected squared Frobenius error no greater than ideal stochastic rounding. We assess accumulation empirically. At a sketch dimension of twice the number of input columns, the least-squares pipeline is $1.44$-$1.47\times$ faster than FP32, with median paired residual increases below $0.4\%$. Rescaling exposes range limitations and cases where stochastic conversion degrades accuracy.

For a complex single variable polynomial $f$ of degree $d$, let $K$ be its filled Julia set, i.e., the union of all bounded orbits. Assume that $K$ has an invariant component $K^*$ on which $f$ acts as a degree $d_*<d$ map. This is a simplest instance of holomorphic polynomial-like renormalization (Douady-Hubbard). One can associate a certain Cantor-like subset $G'$ of the circle with $K^*$; it is defined as the set of arguments of all smooth or broken rays to $K^*$. We will describe a role the Hausdorff dimension of $G'$ and the respective Hausdorff measure play in geometry of $K^*$. In particular, we give upper and lower bounds on the modulus of renormalization in terms of the Hausdorff measure of $K^*$.

We study properties of graphs (or rather graph sequences) saying that some restricted count of subgraphs is approximatively what is expected in a random graph. It has been shown by several authors that many such properties characterize quasi-random graphs, but there are also some exceptions. We continue here the line of investigation in Janson and Sós (2013), and introduce some new versions of these properties, in order to better understand why many of these properties are quasi-random, and to understand the structure of the exceptions that are not. A new feature in the proofs is a simple decomposition of the subspace of symmetric functions in $L^2([0,1]^m)$ into subspaces that are irreducible for the action of measure-preserving transformations of $[0,1]$; this simplifies some arguments and gives structure to others.

We observe that the canonical map \(KK(A, \prod_{n \in \mathbb{N}} B_n) \to \prod_{n \in \mathbb{N}} KK(A,B_n)\) is an isomorphism of abelian groups whenever \(A\) enjoys the Universal Coefficient Theorem and \(B_n\) are unital, simple and purely infinite C*-algebras. This clarifies an aspect of previous work of Dadarlat–Eilers and Tikuisis–White–Winter.

This paper aims to explore the formation mechanism of probability distribution in situations where the differences among random experiments are distinguishable, and these differences continue to evolve along with the dynamic changes in conditions and their mechanisms of action. To this end, we are motivated to devise a new theoretical system – theory of uncertain prob ability (TUP) with Kolmogorov's system and nonlinear theories as special cases. TUP develops a novel model that integrates probability and uncertainty as well as the known and unknown to more accurately depict numerous typical random phenomena under more realistic assumptions, and thus provides appropriate tools for greater variety of real needs. It also allows for pioneering interpretation of the causal mechanisms underlying many important distributional characteristics and incorporation of pathwise property to distribution model.

The Haldane-Shastry spin chain is a prototypical integrable model with long-range interactions, notable for hosting quasiparticles with fractional statistics and serving as a discrete analogue of a conformal field theory. Its remarkable simplicity is closely tied to a full Yangian spin symmetry. While the highest-weight states for this symmetry are known explicitly, a systematic treatment of the descendant states, needed for the computation of various physical quantities, has remained incomplete. In this work, we provide a detailed construction of these descendants in terms of the algebraic Bethe ansatz following recent work of Ferrando et al. In the limit of extreme twist, it includes the Gelfand-Tsetlin basis. As an application, we derive product and determinant formulae for norms and overlaps of these states.

Semantic-aware communication has emerged as a transformative paradigm for next-generation communication systems, shifting the fundamental goal from transmitting bit-level symbols to reliably recovering and understanding the semantic meaning of information. Previous studies have demonstrated that representing the semantic content of source messages as graph-based structures can significantly improve communication efficiency and the accuracy of semantic inference at the receiver. However, existing solutions typically employ graphs that capture only pairwise relationships, thereby neglecting higher-order implicit correlations commonly observed in real-world scenarios, such as group interactions, multi-entity associations, and complex relational contexts. This limitation reduces semantic expressiveness and makes semantic inference susceptible to ambiguity and performance degradation, particularly under noisy or corrupted channel conditions. To address these issues, this paper proposes a novel hypergraph-based implicit semantic reasoning framework, HISR, which leverages hypergraphs to represent complex multi-entity relationships among semantic knowledge entities. In HISR, entities and their associated higher-order relations are mapped into dedicated semantic subspaces tailored to distinct relational contexts. This design not only disentangles diverse semantic interactions to mitigate the over-smoothing effects commonly found in traditional graph embedding methods but also enables robust semantic inference even when partial information loss occurs during transmission. Numerical results show that the proposed HISR achieves up to a 36.6% improvement in implicit semantic interpretation accuracy over the state-of-the-art benchmarks.

For a regular semisimple Higgs bundle with a smooth spectral curve, we prove that, over the \etale\ locus, the Atiyah class of the underlying bundle is induced by the Atiyah class of the spectral line bundle and takes values in the centralizer of the Higgs field. Further, when the discriminant is reduced, we construct a logarithmic refinement across the branch divisor: the Atiyah class extends as a class with logarithmic poles and values in a natural regularized centralizer sheaf.

Let $f$ be a transcendental entire function, $V$ be a simply connected Fatou component of $f,$ and $U$ be a Fatou component with $f(U)\subset V.$ There is a natural way to associate $f|_U$ to an inner function, namely a function $g_f:=\psi^{-1}\circ f\circ\varphi,$ where $\varphi:\mathbb{D}\to U$ and $\psi:\mathbb{D}\to V$ are Riemann maps. Inner functions have been used as a tool in the study of the iterates of transcendental entire, and more recently meromorphic, functions. However, there are only a few examples where associated inner functions have been calculated explicitly, with the case where $f$ has infinite degree in $U$ being the least well understood and more complicated. In this paper, we introduce a general method for calculating associated inner functions to a wide class of entire functions arising as `lifts'. In particular, if $f$ is a lift of a transcendental entire function $h,$ we show that an inner function associated to $f|_U$ can be obtained by relating it to an inner function associated to $h|_G,$ where $G$ is the Fatou component that lifts to $U.$ This result significantly generalises the main part of a theorem by Evdoridou, Rempe and Sixmith, and can be applied to several functions that have been studied so far. In both finite- and infinite-degree settings, the results hold for forward-invariant Fatou components as well as for wandering domains.

Pinching Waveguide Antennas (PWAs) offer significant potential for simultaneous wireless information and power transfer (SWIPT) by enabling precise near-field energy focusing. However, existing optimization frameworks are largely point-based (targeting a single coordinate for maximum gain), and thus highly sensitive to positioning errors and mobility, as near-field signals fluctuate significantly even over small spatial displacements. In this paper, we propose a spatially robust design framework based on discrete antenna selection optimized for service area (SA) coverage. Unlike point-based approaches, our model guarantees quality of service within predefined SAs for both information decoding (ID) and energy harvesting (EH) receivers, thereby improving robustness to user displacements. We formulate the problem as a non-convex binary quadratic program aimed at maximizing harvested energy within the EH SA subject to robust rate constraints in the ID SA. To characterize fundamental performance limits, we develop a semidefinite relaxation (SDR) framework that provides an upper bound on the achievable rate-energy (R-E) region. For the lower bound, we employ a low-complexity swap-based local search algorithm enforcing binary hardware constraints. Numerical results demonstrate that the proposed coverage-oriented design yields a robust R-E tradeoff and maintains stable performance across service regions, highlighting the advantages of discrete antenna activation over point-based near-field optimization approaches.

Let $n$ be a positive integer, $n \not=1$, and $S_n$ the set of all $n \times n$ real symmetric matrices. A nonempty subset $\U \subset S_n$ is called a matrix domain if it is open and connected and a map $\phi : \U \to S_n$ is said to be an order emebedding if for every pair $X,Y \in \U$ we have $X \le Y \iff \phi (X) \le \phi(Y)$. We describe the general form of such maps.

This paper explores the relationship between weak and viscosity solutions to a nonhomogeneous mixed local and non-local $p$-Laplace equation in a bounded Lipschitz domain in $\mathbb{R}^N$. Under certain conditions, we derive the comparison principle for weak subsolutions and weak supersolutions to the problem. For $1<p<\infty$, we establish that continuous weak supersolutions to the problem are viscosity supersolutions, using the comparison principle. Furthermore, we show that bounded viscosity supersolutions are weak supersolutions for $p \geq 2$.

This paper explores the use of generative models to synthesize high-quality, site-specific multiple-input multiple-output (MIMO) channel data, addressing the high cost of the extensive measurement campaigns required to acquire real-world data for AI-native wireless networks. Two location-conditioned generative paradigms are compared: a conditional denoising diffusion implicit model (cDDIM), and a conditional flow matching model (cFMM). Both these models generate MIMO channel matrices conditioned on user coordinates, to preserve the spatial structure of the deployment site. The approaches are evaluated across three dimensions: statistical fidelity (including beam consistency and effective rank), generation efficiency, and utility in downstream tasks such as channel-state information compression and beam alignment. Results across diverse propagation scenarios (28 GHz and 3.5 GHz, both line-of-sight and non-line-of-sight) demonstrate that both models accurately capture site-specific characteristics, even when trained on scarce ground-truth data. Notably, cFMM achieves a quality comparable to cDDIM with roughly an order of magnitude less inference time. Augmenting scarce site-specific datasets with these synthetic channels yields hefty performance gains in downstream physical layer tasks compared to using scarce data alone or stochastic channels.

We describe a certified arbitrary-precision framework for evaluating a family of generalized multiple zeta functions. The family includes strict and weak-star chain sums, ordinary and colored multiple zeta values, affine-base and polynomial-base variants, and composite levels containing several affine or polynomial letters with complex coefficients. The numerical strategy combines finite-prefix recurrences with two complementary analytic-tail mechanisms: recursive Euler-Maclaurin expansion of one-variable tails and direct absolute tail majorants. The Euler-Maclaurin branch is fast when the relevant suffix expansions are regular, while the direct-tail branch gives robust certificates for multi-letter, weak-star, complex-coefficient, and branch-sensitive inputs. A computation is called certified only when its reported radius is obtained from a proved analytic bound for the omitted infinite tail. Strict-disk colored sums and boundary-color cases with summable absolute majorants are therefore within the certified scope; conditionally convergent colored cases whose convergence relies only on non-one unit-modulus oscillation are kept separate and reported as explicitly non-certified diagnostic outputs unless an independent analytic remainder bound is available.

The study of intersecting families, initiated by Erdős, Ko, and Rado, is a central topic in extremal combinatorics. A classical stability result of Hilton and Milner determines the largest non-trivial intersecting family, and in subsequent works researchers developed structural stability results via the notion of diversity. In this paper, we study cross-intersecting families. We establish a structural theorem for large cross-intersecting pairs, extending Kupavskii's theorem from intersecting families to the cross-intersecting setting. Our result characterizes extremal cross-intersecting pairs in terms of their diversity parts and maximal cross-intersecting extensions. As corollaries, we obtain cross-intersecting analogues of several classical theorems, including those of Han–Kohayakawa and Huang–Peng. A key ingredient in the proof is a new shifting method, called the $S_{U,V}^{Q}$-shift, which not only preserves global intersection properties but also maintains certain local substructures after shifting. We expect this method to be useful elsewhere, and it is already one of the key tools in establishing a product analogue of the Hilton–Milner theorem.

The John ellipsoid of a symmetric polytope $P=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{A}\mathbf{x}\|_\infty\le1\}$, $\mathbf{A}\in\mathbb{R}^{n\times d}$, is computed by a long line of leverage-score algorithms, from Cohen, Cousins, Lee and Yang (COLT 2019) to its successors [WY24, CLS+25], all reaching a $(1+\varepsilon)$-approximation in $\Theta(\varepsilon^{-1}\log(n/d))$ iterations. We separate this complexity into three costs the modern line conflates (certification, identification, and accuracy) and locate the historical $\varepsilon^{-1}$ in the first alone. In the equivalent D-optimal-design form $\min_{\mathbf{p}\in\Delta_n}-\log\det(\sum_i p_i\mathbf{a}_i\mathbf{a}_i^\top)$, the leverage-score oracle is exactly the first-order oracle and the $(1+\varepsilon)$-John guarantee the Frank-Wolfe gap $g(\mathbf{p})\le\varepsilon d$; through this dictionary the costs come apart. The $\varepsilon^{-1}$ is a certification artifact: the uniform average of the iterates, the certificate used throughout the line, has gap exactly $\Theta(1/T)$, however cheap each iteration is made. Pointed instead at the last iterate the same oracle is fast: a warm-started accelerated method reaches the guarantee in $C(\mathbf{A})+O(\sqrt{\kappa}\log(1/\varepsilon))$ queries after an $\varepsilon$-independent setup $C(\mathbf{A})$, and once the optimal face is identified the facial problem is an unconstrained self-concordant minimization whose Hessian the oracle recovers exactly, so damped Newton needs only $O(\log\log(1/\varepsilon))$ steps, for a total of $C(\mathbf{A})+O(d^2\log\log(1/\varepsilon))$ queries. The accuracy dependence is thus doubly logarithmic after an $\varepsilon$-independent, condition-dependent setup; the open problem is the remaining identification cost (a condition-free bound on reaching the optimal face) and lower bounds. Accuracy is not the obstruction.

A class of pseudo-parabolic partial differential equations is considered. We derive a posteriori error bounds for approximations obtained by FEMs in space and a BDF formula in time. The analysis is based on the $C_0$ semigroup theory and an adaptation of the concept of elliptic reconstruction to pseudo-parabolic problems. The analysis is complemented with numerical experiments.

We introduce optimal coarse correlated equilibria for continuous-time mean field games. A coarse correlated equilibrium is a randomized recommendation scheme from which no player can gain by ignoring the recommendation and switching to an alternative strategy. The problem is as follows: a moderator selects, among all mean-field coarse correlated equilibria, one that optimizes a prescribed performance criterion, which may differ from the representative player's objective. After formulating the problem, we develop a linear programming (LP) formulation, prove the existence of optimal LP coarse correlated equilibria, and relate the LP characterization to the original probabilistic setting. Building on this characterization, we design a no-regret primal-dual algorithm, based on an equivalent Lagrangian formulation of the external-regret constraint, for learning such equilibria. We provide explicit convergence rates for the learning algorithm, and numerical examples illustrate the method.

We develop a geometric framework for constrained optimization problems with inequality constraints through the introduction of quadratic slack variables. This formulation makes it possible to employ the language of Riemannian geometry and to solve the problem via the embedded gradient vector field method. We lift the feasible set to a smooth submanifold of an extended ambient space. The stratified structure of the resulting constraint manifold is analyzed in detail, yielding a natural partition according to which constraints are active. Using the embedded gradient vector field formalism, we derive explicit, determinantal formulas for the Lagrange multiplier functions directly from the geometry of the constraint manifold, recovering and re-framing the classical Karush-Kuhn-Tucker first-order necessary conditions without invoking the classical Lagrange multiplier method. Second-order optimality conditions are obtained by computing the restricted Hessian on each stratum, and a complete sign condition on the Lagrange multipliers is identified as the geometric counterpart of the classical complementary slackness condition. The theory is illustrated on the double ice-cream cone example, where the geometry of the problem determines the nature and number of local minima.

Consider the arithmetic function of two variables $f(a,b)= \gcd(ab,a+b)/\gcd(a,b)$, investigated in a recent preprint. We deduce asymptotic formulas for sums of the form $\sum_{a,b\le x} h(f(a,b))$, where $h$ belongs to a certain class of arithmetic functions. In particular, we obtain an asymptotic formula for the number of ordered pairs $(a,b)\in {\Bbb N}^2$ such that $a,b\le x$ and $f(a,b)=m$, where $m\in {\Bbb N}$ is fixed. This shows that in the case $m=1$ the corresponding density is the quadratic class number constant $C= \prod_p (1-1/(p^2(p+1))) \doteq 0.881513$.

The paper analyzes to what extent the dynamics of a nonautonomous $n$-dimensional dynamical system with two time scales, formulated in the slow time as $dx/dt=f(t/\varepsilon, t, x)$, can be approximated for small values of $\varepsilon$ by the dynamics of the averaged system $dz/dt=\hat f(t,z)$. Assuming that the skewproduct flow associated with the averaged system admits a local attractor $\mathcal{A}$, we prove that the solutions of the original system whose initial data lie in the basin of attraction of $\mathcal{A}$ track the fibers of the inflated attractor for all positive times. If the fiber map of $\mathcal{A}$ is continuous, inflation is no longer required. Alternative tracking results with a more classical formulation are also presented, under assumptions involving uniformly asymptotically stable solutions or uniform local attractors for the nonautonomous process, rather than for the skewproduct flow. Several examples illustrate the scope and applicability of the results. The twofold extension of the classical averaging results (to the doubly nonautonomous setting and to the whole positive halfline) is expected to be relevant to a broad range of application.

We derive upper bounds for the Lagrangian capacities of Liouville domains with finite Gutt–Hutchings capacities and show that the Lagrangian capacity of a convex or concave toric domain of arbitrary dimension equals its diagonal. In particular, this completely settles the conjecture of Cieliebak-Mohnke on the Lagrangian capacity of ellipsoids. Our proof is based on an $S^1$-equivariant variant of the techniques of Fukaya and Irie, and does not use holomorphic curves with local tangency constraints, which would inevitably cause transversality issues. Moreover, we show that any extremal Lagrangian torus in an $2n$-dimensional ellipsoid must lie on the boundary. Applications of our results and techniques include include a proof of the fact that every monotone Lagrangian torus in the complex projective space has non-trivial intersections with the Clifford torus and the first computations of the Lagrangian capacities for many non-subcritical Weinstein domains in dimensions 4 and 6.

In this paper, we study the arboreal Galois representations for polynomials of twisted Carlitz type, whose first iterated Galois group is linked to the torsion of a twisted Carlitz module. We prove two explicit families of polynomials having iterated Galois groups isomorphic to full iterated cyclic wreath product at every level. We then compare the arboreal Galois image of a polynomial of twisted Carlitz type with the adelic Galois image of its corresponding twisted Carlitz module, and show that arboreal maximality and adelic surjectivity are logically independent, except for a one-way local implication.

For a family $\mathcal{F}$, it is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in\mathcal{F}$. We use $\mathcal{SD}(\mathcal{F}) = \{F \triangle G : F, G \in \mathcal{F}\}$ to denote the family of symmetric differences of $\mathcal{F}$. In 2023, Frankl, Kiselev and Kupavskii conjectured that for any intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n > 10k$, the inequality $|\mathcal{SD}(\mathcal{F})| \le \sum_{\ell=0}^{k-1} \binom{n-1}{2\ell}$ holds. They further observed that a proof for the range $n>3k^2$ could likely be obtained via arguments similar to those in their earlier work, though no detailed derivation was given. In this paper, we establish the conjecture under the conditions $n\ge 100k\ln k$ and $k\ge 50$. We also determine the extremal families, which are precisely a certain class of stars. A concentration inequality plays a central role in the proof.

We employ the vector field method to establish a Liouville-type theorem for a class of \((p,q)\)-Laplace equations in the Euclidean space \(\mathbb{R}^n\). By modifying the exponents in the differential identity, we prove nonexistence in the subcritical range \(p-1<\alpha<q^*-1\), where \(q^*=nq/(n-q)\). The approach relies on constructing a suitable differential identity, carrying out precise integral estimates with cutoff functions, and combining sign control and decay of the cutoff errors.

Poisson and Jacobi structures play a fundamental role in the geometric description of many systems arising in classical mechanics. In most cases, the corresponding bivector field is induced by a non-degenerate 2-covariant tensor field. In this paper, we present a unified framework for constructing the associated brackets by studying the Poisson and Jacobi structures induced by these tensors. More specifically, under suitable assumptions on the tensor, we derive a formula for computing the Schouten–Nijenhuis bracket of the associated bivector field in terms of the curvature of a certain distribution and the exterior derivative of a differential form. This formula provides the obstruction to the existence of a Poisson or Jacobi structure. To illustrate the theory, we recover the classical brackets associated with symplectic, locally conformally symplectic, contact, cosymplectic, and cocontact geometries. Finally, we characterise the conditions under which fat bundles and almost cosymplectic structures of order $p$ determine a Jacobi bracket.