Papers

Bracketing covers and $\delta$-covers provide finite discretizations of the anchored boxes that define the star discrepancy. Let $N_{[]}(d,\delta)$ and $N(d,\delta)$ denote the corresponding bracketing and covering numbers. We prove the lower bounds \[ N_{[]}(d,\delta)\ge \lceil \delta^{-d}\rceil, \qquad N(d,\delta)\ge \left\lceil \frac{d!}{d^d}\,\delta^{-d}\right\rceil. \] We give two explicit constructions of bracketing covers. For every fixed $d$, together with the lower bound they imply $N_{[]}(d,\delta)=(1+o_d(1))\delta^{-d}$ as $\delta\downarrow0$. A first construction uses box-dependent anisotropic local grids and gives simple explicit bounds. A second, homothetic logarithmic-shell construction again attains this coefficient and gives $\limsup_{d\to\infty}N_{[]}(d,\delta)^{1/d}\le\delta^{-1}+e+O(\delta)$ as $\delta\downarrow0$. Combining these finite estimates with Gnewuch's general bracketing bound and a Hoeffding–Bernstein chaining argument shows that, for every $d,n\in\mathbb N$, there exists an $n$-point set with star discrepancy at most $2.3463\sqrt{d/n}$. Consequently, $\lceil5.5052d\varepsilon^{-2}\rceil$ points suffice for star discrepancy at most $\varepsilon$.

Robust to intensity inhomogeneity, the local Chan–Vese (LCV) model extends the classical Chan–Vese (CV) image segmentation method by incorporating local statistical information around each pixel. Originally, the LCV model was solved using a finite difference scheme, following the approach used for the CV model. As an alternative to the finite difference scheme, a more efficient algorithm based on the Merriman-Bence-Osher (MBO) scheme was later developed for the CV model. In this paper, we derive a similar MBO-based algorithm to solve the LCV model and propose an efficient implementation. The algorithm is developed for both two-phase and multiphase segmentation, and an extension to color images is also discussed. To demonstrate the effectiveness of the proposed approach, we apply it to a variety of grayscale and color images, including medical and microscopy images.

A positive integer $k$ is called $Bn$-collinear if at least one lattice triangle with $n$ boundary points ($B(T)=n$) and $k$ interior lattice points exists, and every such triangle has all of its interior points collinear. Building on prior work on $B(T)=3$, we completely classify the $B4$- and $B5$-collinear integers. Using canonical lattice classifications together with arithmetic properties of Alder's generalized totient function $g(k)$, we prove that the only $B4$-collinear integers are $k\in\{1,2,5\}$. Furthermore, we show that no integer is $B5$-collinear. This establishes a structural contrast: while three and four boundary lattice points exhibit some collinearity constraints, five boundary points disrupt the pattern.

We address the prescribed Lorentzian mean curvature problem over a convex bounded domain $\Omega$ of $\mathbb R^m$ with bounded right-hand side and homogeneous capillary boundary condition. We prove that the problem has a unique $W^{2,2}$-regular weak solution $u$ with zero mean and that $|Du| \leq 1 - \theta$ for some $\theta\in(0,1)$ only depending on the data. Such $u$ is also the unique maximizer of an associated functional. A key step in proving that the maximizer is a weak solution consists in showing that it has no light segments, i.e. segments along which $|Du| = 1$. This holds for arbitrary bounded capillary boundary data and can thus be an interesting result on its own.

Generative AI tools are rapidly becoming standard for coding and other business operations. However, while these tools can drive productivity, their unpredictable, and often significant, costs are putting severe financial strain on organizations. The leadership of such organizations must split a fixed budget between human headcount and generative-AI token capacity, yet standard deterministic planning ignores both the heavy-tailed volatility of token consumption and the cognitive cost of auditing AI-generated output. We model the joint allocation by intersecting a cognitive-friction-adjusted production function with a chance-constrained stochastic budget frontier. Per-engineer token usage is treated as i.i.d., non-negative, and right-skewed; no parametric family is assumed, as only its mean, variance, and skewness enter the analysis. The chance constraint is reduced to a deterministic equivalent via the Central Limit Theorem with a Cornish–Fisher skewness correction. Solving the resulting Lagrangian yields a closed-form optimal headcount, a transcendental condition for optimal per-capita token intensity, and a bordered-Hessian second-order condition that holds for any reasonable volatility-to-headcount ratio. Comparative statics show that rising individual token volatility raises optimal per-capita token intensity while contracting headcount: although human labor and tokens are complements in production, the stochastic budget makes them substitutes at the margin. Individual usage skewness acts as a pure deadweight tax that shrinks the budget without altering the substitution dynamics. Both conclusions presuppose that usage dispersion is invariant to the planned mean allocation. In the case where dispersion instead scales proportionally with the mean, the substitution reverses, so which regime applies is a sharp, empirically testable question.

We consider First-Order Logic (FOL)-based semantic communication for neuro-symbolic decision-making in collaborative environments such as autonomous driving networks. Each connected autonomous vehicle (CAV) converts its partial sensor observations into a natural-language scene description and corresponding grounded FOL evidence. Under an uplink budget, a semantic encoder at each car selects the observations most informative for evaluating traffic rules and transmit to a Road Side Unit (RSU). The RSU fuses all received evidence, evaluates collaborative rules, performs logical deduction for vehicle-specific safety and right-of-way information for constrained downlink transmission. Each CAV combines the received deductions with its local description, enabling a local LLM agent to select a high-level driving action. We develop a principled, verifiable semantic communication method using a random-support Dirichlet–Categorical model of inductive logical probability, providing a modern statistical reinterpretation of Carnap's and Hintikka's systems. From this model, we derive a goal-oriented semantic information-bottleneck formulation that prioritizes evidence transmission by its reduction of uncertainty over task goals. Using 152 traffic rules extracted from the California Driver Handbook, we evaluate the framework on MDrive simulator in CARLA. Under identical communication budgets, semantic evidence selection completes every scenario without safety hazards, whereas uniform evidence selection produces collisions, showcasing semantic communication's superiority.

We study long-memory continuous-time moving-average processes driven by a Levy process and observed at random renewal times. The sampling scheme introduces an additional source of randomness through irregular observation times. We establish the asymptotic behaviour of normalized partial sums under both finite- and infinite-mean renewal sampling.

We perturb one-dimensional Dirac operators on a bounded interval subject to Dirichlet boundary conditions by potentials with Fourier coefficients exhibiting power-decay. As a consequence of Paley-Zygmund theorem, this broad family of potentials comprises distributions that are neither integrable functions nor measures. We localize the spectrum of the perturbed operator and, as the main result, show that its eigensystem generates a Riesz basis.

Currently, there exist only three strongly invertible L-space knots that are known to admit no Khovanov thin surgery. In this article, we give the first infinite family of such knots.

This work is concerned with weighted Geometric Rigidity Estimates and Korn's first inequalities in bulk and thin domains. We consider weights, that are a nonnegative power of the distance function to part of the boundary of the domain. For the case of thin domains, the distance is taken to be from the thin face of the domain boundary, and the constants depend on the domain thickness and when the mid-surface contains a flat region, the constants are proven to have optimal scaling as the thickness approaches zero. We employed some covering techniques utilized by Acosta, Cejas, and Duran in [\ref{bib:Aco.Cej.Dur.}] to prove a weighted Poincaré inequality, and later by Conti and Zwicknagl in [\ref{bib:Con.Zwi.}] to prove weighted Poincaré and classical Geometric Rigidity inequalities in Lipschitz domains. However, because the distance is taken to be only from part of the boundary, the covering parts become more delicate, especially for thin domains and the analysis becomes non-straightforward.

The purpose of this paper is to contribute to the further study of Loday algebroids by introducing, first, the concept of action Loday algebroid and clarifying the notion of a (co)morphism of Loday algebroids. We then focus on the study of Loday algebroids nonlinear connections and their related theory of characteristic classes. These classes arise from multidifferential operators (on the module of smooth sections of the Loday algebroid) in the Loday algebroid cohomology. In particular, the Chern-Simons differential operators for nonlinear connections on Loday algebroids are considered and a generalized Chern- Simons formula for such connections is established. Using representations of Loday algebroids, we define their secondary characteristic classes.

A family of subsets of $[n]$ is called intersecting if it contains no pair of disjoint sets. It is called trivial if all its members contain a common element. Frankl and Kupavskii, and independently Balogh, Das, Liu, Sharifzadeh, and Tran, proved that there is a constant $c>0$ such that, whenever $n \geq 2k+2+c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are trivial. Balogh, Garcia, Li, and Wagner later improved this range to $n \geq 2k+100\ln k$. In this paper, we prove that the same conclusion holds for every $n\geq 2k+3$. This verifies the conjectured conclusion of Balogh, Garcia, Li, and Wagner throughout this range.

The dispersion of reactive solutes in shear flows is governed by the interplay between advective stretching, transverse diffusion, and boundary exchange kinetics. While classical analytical methods and grid-based numerical solvers have extensively characterised these transport mechanisms, accurately resolving the spatiotemporal evolution of solute plumes in asymmetric reactive environments remains computationally demanding. In this study, we introduce a physics-informed neural network (PINN) framework to simulate two-dimensional wall-reactive solute dispersion in canonical shear flows (Couette, Poiseuille, and Couette-Poiseuille) bounded by absorbing walls. By embedding the governing convection-diffusion equation and Robin boundary conditions into a unified loss function, the mesh-free PINN reconstructs the spatiotemporal concentration field. The network predictions are validated against an alternating-direction implicit (ADI) finite-difference benchmark, showing close agreement across non-reactive, symmetric, and asymmetric reactive regimes. The computations are carried out at $\mathrm{Pe}=10$ for impermeable walls, symmetric absorption $(\beta_1,\beta_2)=(1,1)$, and tenfold asymmetric wall-reactivity contrasts $(\beta_1,\beta_2)=(0.2,2)$ and (2,0.2). Leveraging the differentiable nature of the trained PINN, we extract wall-resolved transport diagnostics, including the apparent axial dispersion coefficient, cumulative wall-removal dynamics, and localised uptake fluxes. The results show that the imposed shear profile governs the streamwise organisation of reactive uptake, while unequal wall reactivities induce transverse asymmetry that modifies the macroscopic spreading rate. Overall, this framework establishes PINNs as an interpretable mesh-free tool for analysing boundary-coupled reactive transport in shear flows.

Uncrewed aerial vehicle (UAV)-enabled federated learning (FL) can provide flexible, on-demand edge intelligence for large-scale IoT deployments, but operating in shared unlicensed bands makes uplink update delivery interference-coupled and unreliable. In this paper, we develop a packet-level transmission framework that captures buffer overflow, delay violations, and transmission errors, and uses the resulting packet delivery ratio (PDR) to represent partial-update reception through a packetized, Bernoulli-masked FL aggregation process. We then formulate a fairness-consensus bilevel (FCB) optimization that jointly controls (i) transmission thresholds to maximize the average PDR while reaching consensus under partial observability and (ii) transmission powers to improve the worst PDR and enforce fairness across IoT learners. To solve this problem, we propose an alternating FCB optimizer composed of a consensus-based threshold controller (CTC), which drives the IoT learners toward a PDR-efficient consensus on transmission thresholds, and a fairness-based power controller (FPC), which updates transmission powers to improve the worst PDR and ensure fairness under the resulting consensus thresholds. Numerical results on CNN-based FL tasks show that the FCB optimizer improves FL aggregation and training performance by enhancing packet-level update delivery, consistently outperforming baseline transmission policies.

We prove that the largest Orlicz space into which the Hardy averaging operator maps coincides with the largest rearrangement-invariant (r.i.) space mapping into that space; in other words, the optimal Orlicz domain is automatically the optimal r.i.\ domain. More generally, we show that every Luxemburg–Orlicz norm is equivalent to a sublinear functional which makes more computable a certain expression for a dual norm arising in the $\K$-theory of interpolation. As applications we obtain Orlicz-space mapping properties for the Hardy–Littlewood maximal function, approximate identities and Calderón–Zygmund singular integral operators. These mapping properties are shown to be optimal for the Hardy-Littlewood maximal function and the approximate identities.

The spectral sensitivity $\lambda(f)$ of a Boolean function is the largest eigenvalue of the adjacency matrix of its sensitivity graph. It lower-bounds every standard measure of query complexity, and Aaronson, Ben-David, Kothari, Rao and Tal, who introduced it, asked whether block sensitivity is at most quadratic in it: is $bs(f)=O(\lambda(f)^{2})$? We show that it is not. We construct a total Boolean function on $2017584$ variables with $bs(f)\ge 14011$ and $\lambda(f)\le 89.0162$, so that $bs(f)\ge\lambda(f)^{2.127}$, and hence by composition a family with $\lambda(f_n)\to\infty$ and $bs(f_n)=\Omega(\lambda(f_n)^{2.127})$. The function is the indicator of a union of $k$ subcubes indexed by the vertices of a doubly regular tournament, and the freedom left in the construction is fixed by the Lovász local lemma. The main result has been formally verified in Lean. We also give numerical evidence that a member of the same family on $1255$ variables reaches an exponent near $2.20$, and exhibit a member on $30$ variables whose exponent already exceeds $2$ and whose spectral sensitivity can be computed exactly.

Let $K_\infty/K$ be a $p$-adic Lie extension of a $p$-adic field $K$. We study the subring of pro-analytic vectors in the de Rham period ring $\mathbf{B}_{\mathrm{dR}}^+(K_\infty)$. We show that the pro-analytic subring admits a Galois-equivariant isomorphism with a formal power series ring $\widehat{K}_{\infty}^{\mathrm{la}} [[t_{K_\infty}]]$ if and only if $K_\infty$ satisfies a certain orientability condition, which says that the $\widehat{K}_\infty$-level Sen operator admits a Galois-equivariant $\mathbf{B}_{\mathrm{dR}}^+$-lift. A key input is the vanishing of higher locally analytic vectors of $\widehat{K}_\infty$-representations. As an application, we show that the lifted Sen operator induces regular connections on pro-analytic vectors of $\mathbf{B}_{\mathrm{dR}}^+$-representations, and can be used to compute Galois cohomology.

The Galerkin finite element formulation of the incompressible Navier-Stokes equations presents two principal challenges: maintaining stable velocity-pressure coupling and controlling instability in advection-dominated regimes. Moreover, the monolithic formulation produces a coupled nonlinear saddle-point problem. In this work, we present a residual-based variational multiscale (VMS) stabilization of an incremental Helmholtz-Leray projection method that replaces this coupled saddle-point problem with a nonlinear velocity predictor, a pressure Poisson equation, and a velocity projection. The multiscale decomposition is applied only to the predicted velocity; neither the pressure nor the corrected, weakly divergence-free velocity is decomposed into coarse and fine scales. The modeled velocity fine scale contributes consistently to all three subproblems, introducing SUPG-like stabilization in the momentum predictor and a PSPG-like residual contribution in the pressure Poisson equation. We provide a formal error decomposition that separates the BDF2 time-discretization, projection-splitting, and spatial-VMS errors and, under stated stability and spatial-approximation assumptions, yields a combined velocity error estimate with second-order temporal accuracy. Numerical results for manufactured solutions, lid-driven cavity flow, flow past a cylinder, and the Taylor-Green vortex agree closely with established reference data. Comparisons with monolithic VMS indicate that omitting the pressure fine scale reduces drag overprediction and excess modeled dissipation, at the cost of increased divergence error. In the Taylor-Green tests, the projection formulation also reduces the average solution time per step by factors ranging from approximately $1.3\times$ to $2.7\times$ under identical solver settings.

We study sublinear time sampling methods for approximating the outlying eigenvectors of large matrices. Our main result is an algorithm that uniformly samples just $\tilde{O}(\log n/\epsilon^4)$ columns of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, and, for any eigenvalue $\lambda$ of $A$ with $|\lambda| \ge \epsilon n$, outputs an approximate eigenvector $v$ satisfying $\|Av - \lambda v\|_2 \le \epsilon n$. For approximating just the eigenvector of the largest magnitude eigenvalue, our algorithm samples only $\tilde{O}(\log n/\epsilon^2)$ columns. Given the ability to sample rows and columns of $A$ proportional to their squared norms, we give a similar result with an improved error bound of $\epsilon \|A\|_F$. For top eigenvector approximation, we show our bound is tight up to logarithmic terms. A key feature of our algorithms is that the output eigenvectors are spanned by a small number of $A$'s columns, and individual entries can be computed rapidly, in poly(log n, 1/epsilon) time per entry. This makes them applicable in the quantum-inspired algorithms framework of [Tang, STOC 2019], where we give the first sublinear time classical algorithms for eigenvector approximation with additive error $\epsilon \|A\|_F$. Finally, we present an alternative approach, based on a truncated Nystrom method, that, while not allowing poly(log n, 1/epsilon) time entrywise computation of the approximate eigenvectors, achieves near optimal sample complexity for general symmetric matrices, and improved bounds for positive semidefinite matrices. Technically, our bounds build on recent work on approximating the outlying eigenvalues of symmetric matrices via random sampling in [Bhattacharjee et al. '22] and [Swartworth and Woodruff '25]. We demonstrate for the first time that these approaches extend to the problem of eigenvector estimation.

We study a spectral regularization of the Dean–Kawasaki equation and quantify how the failure of positivity preservation affects its weak approximation of the empirical measure of independent Brownian particles. For initial densities bounded away from zero, we prove a uniform-in-time weak error measured through the Laplace transform and prove a superpolynomial convergence rate for smooth test functions. When the initial density is allowed to vanish, the negative part of the regularized solution leads to weaker upper bounds, and a one-dimensional example gives a lower error bound ruling out superpolynomial convergence. We also present numerical experiments that confirm the theoretical results and illustrate main observations.

We introduce Schur-Agler type class for the symmetrized polydisc $\mathbb{G}_d$ and establish a realization theorem for functions in this class. We state and prove an interpolation theorem on $\mathbb{G}_d$ with interpolating functions belonging to the associated Schur-Agler type class. Moreover, Toeplitz corona and extension theorems are established for $\mathbb{G}_d$. We also extend the realization, interpolation, Toeplitz corona and extension theorems to a more general symmetrized family of domains $\Theta_d$.

We address three questions of Spiro [Europ. J. Combin., 133:Paper No. 104307, 2026] on smallest $q$-kernels in digraphs. All three questions are on the existence of small $q$-kernels in digraphs, where 'smallness' is measured in different ways. We answer one of these questions in the affirmative, and the other two in the negative. We further study the latter two questions in the context of bipartite digraphs. In particular, we show that in the bipartite setting, the answer to one of the two questions is affirmative if and only if $q$ is odd. For the other question, we show that the answer is positive when the digraph is bipartite and $q$ is odd (the case when the digraph is bipartite and $q$ is even remains open).

The realization, interpolation, extension and Toeplitz corona problems are amongst the central themes in function theory of a domain in $\mathbb{C}^d$. The present work addresses these four problems for the hexablock $\mathbb{H}$, a domain in $\mathbb{C}^4$ arising in connection with a special case of $\mu$-synthesis in $H^{\infty}$ control theory. We determine Schur-Agler class $SA(\mathbb{H})$ for $\mathbb{H}$ and find a realization formula for $\mathbb H$. With the help of this realization formula we state and prove interpolation, extension and Toeplitz corona theorems for the hexablock. As an application, we obtain analogous theorems for the Euclidean unit ball in $\mathbb{C}^2$. Moreover, we recover the existing same results for the tetrablock $\mathbb E$, another domain associated with the $\mu$-synthesis, as consequences of the hexablock theory and in terms of the Schur-Agler class $SA(\mathbb{H})$ and admissible kernels on $\mathbb{H}$.

We establish a convex order comparison for products of averages of exchangeable nonnegative random variables. As a consequence, we solve a conjecture of Lamkin and Tkocz (2022) on the log-convexity of moments of sample means, as well as its pointwise version, in a more general form involving arbitrary integer partitions.

We prove a persistence theorem for attracting weakly normally hyperbolic invariant tori under small time-periodic perturbations. The theorem extends recent continuation results for weakly normally hyperbolic limit cycles to invariant tori of arbitrary dimension, providing a general analytical framework for their detection. As an application, we establish an averaging principle showing that attracting normally hyperbolic invariant $d$-tori of the averaged system give rise to attracting normally hyperbolic invariant $(d+1)$-tori of the original non-autonomous system. We further introduce a polynomiality-preserving construction that simultaneously lifts the dimensions of the phase space and the attracting normally hyperbolic invariant tori, yielding recursive lower bounds for the maximal number of codimension-$1$ normally hyperbolic invariant tori of polynomial vector fields of a given degree, thereby extending the counting aspect of Hilbert's sixteenth problem to higher-dimensional invariant tori. In particular, we prove that this number grows at least polynomially with the degree and becomes unbounded for invariant tori of higher codimension.

In earlier works, we studied natural examples of stable-like random walks on finitely generated nilpotent groups and, more generally, on groups of polynomial volume growth. These walks are driven by measures of radial type, coordinate-wise type, or convex combinations of such measures. In the present work, we explore when two such random walks have comparable behavior, as measured by the equivalence of their associated Dirichlet forms. We develop criteria for this equivalence in terms of both geometric and algebraic features of the driving measures and the group.

The Service Rate Region (SRR) serves as a critical metric for evaluating the concurrent service capacity of distributed storage systems. While several works have characterized the SRR for MDS codes and first order Reed-Muller codes, for high-order Reed-Muller codes the problem becomes way more complicated and only partial results were given by Ly, Soljanin, and Lalitha [IEEE ISIT 2025]. In this paper, we refine the SRR analysis by explicitly characterizing the intersection patterns of recovery sets for high-order Reed-Muller codes, deriving the exact region for the case m=r+1 and providing new types of strictly tighter constraints to bridge the gap between existing approximations and the exact SRR polytope.

In this paper, we establish an explicit formula for automorphic periods which will be used in a sequel to study special values of $p$-adic Rankin-Selberg $L$-functions. Our motivation is to extend the Bertolini-Darmon-Prasanna formula to the case where the archimedean local sign is opposite to that in the original setting. To this end, we prove a formula for the $\mathrm{GU}(1,1)$-period of a theta lift from $\mathrm{GSO}(2)$ to $\mathrm{GSp}_4$, which is adapted to $p$-adic interpolation. We also introduce a $p$-depletion Hecke operator for this theta lift, which will play a crucial role in relating the automorphic period to the image of the $p$-adic Abel-Jacobi map.

In every dimension three and higher we construct a nonzero function that satisfies the stationary Schrödinger equation with bounded, real-valued potential and decays like $\exp(-c|x|^{4/3})$. This gives a counterexample to the Landis conjecture in dimension three and higher.

A $(k,d)$-edge coloring of a loopless multigraph $G$ is an edge coloring using at most $k$ colors such that the subgraph formed by each color class has maximum degree at most $d$. The least such $k$ is denoted by $\chi'_d(G)$. Let $G$ be a loopless non-bipartite multigraph with maximum degree $\Delta(G)$ and odd girth $g_0(G)$, and let $d\ge1$ be odd. We prove that \[ \chi'_d(G)\le\left\lceil\frac{g_0(G)\Delta(G)-1}{dg_0(G)-1}\right\rceil. \] For $d=1$, this is Goldberg's odd-girth refinement of Shannon's theorem, while for $g_0(G)=3$ it is the defective Shannon bound of Aboulker, Aubian, and Huang. For every odd $d>1$, every odd $g_0\ge3$, and every $\Delta>d$, an almost full ring multigraph $R(\Delta,g_0)$, an odd cycle with edge multiplicities alternating between $\lfloor\Delta/2\rfloor$ and $\lceil\Delta/2\rceil$, except that two consecutive edges have multiplicity $\lfloor\Delta/2\rfloor$, attains equality. We also derive a range in which the defective Goldberg–Seymour conjecture holds.