Papers

We record an explicit counterexample to a refined form of Purdy's inequality for essential hyperplane arrangements in projective three-space. Let $\mathcal{A}$ be an arrangement of $n$ hyperplanes in $\mathbb{P}^3_{\mathbb{C}}$. Let $\ell$ be the number of distinct intersection lines of $\mathcal{A}$, and let $p$ be the number of intersection points, where an intersection point means a point at which at least three hyperplanes meet. The expected inequality is \[ p-\ell+n+2\geq 0. \] The classical obstruction is the rank $2+2$ product arrangement, or dually a configuration of points contained in two skew lines. We explain this obstruction first, and then show that it is not the only one. The reflection-arrangement search leads naturally to a subarrangement of the monomial reflection arrangement of type $G(3,3,4)$. Looking dually, this configuration is not contained in two skew lines, and has \[ f_0(S)=12,\qquad f_1(S)=58,\qquad f_2(S)=43. \] Therefore its dual arrangement has \[ n=12,\qquad \ell=58,\qquad p=43, \] and hence \[ p-\ell+n+2=-1. \] Thus the refined statement excluding only the two-skew-lines obstruction is false.

We generalise the recent two-ends Furstenberg inequality due to Wang and the second author from lines to a family of transversal curves, and give a much simplified proof. As an application, we present a result pertaining to the Fourier decay of fractal measures on convex curves.

For $\ell\geq 2$, let $\overline{A}_\ell(n)$ count the number of overpartitions of $n$ with no parts divisible by $\ell$. In this article, we employ the circle method to derive a Rademacher-type formula for $\overline{A}_\ell(n)$, when $\ell$ is a squarefree odd integer. As an application, we derive higher order Tuŕan inequalities for the $\ell$-regular overpartition function using a result of Griffin, Ono, Rolen, and Zagier.

We develop confidence regions for Reeb graphs from finite samples using the interleaving distance. Given a point cloud equipped with a filter function, we construct a finite proximity graph, extend the filter linearly, and use the Reeb cosheaf of the resulting filtered graph as the primary estimator. Mapper graphs are then treated as controlled cover-based coarsenings of this estimator, separating the statistical approximation problem from the visualization problem. We prove stability bounds for the Reeb estimators obtained both using intrinsic and extrinsic metrics, the latter under positive-reach assumptions, and derive interleaving-distance confidence regions from either \((a,b)\)-standard sampling assumptions or subsampling-based Hausdorff scale estimates. We also compare this object-level metric viewpoint with persistence-based guarantees by showing that the extended-persistence pseudometric is bounded by twice the interleaving distance, with sharp constant \(1\) for the \(H_0\)-related components. Numerical experiments illustrate how statistically significant features can be identified and then projected to Mapper graphs for interpretation.

We associate to each proper discrete metric space $X$ a Lie algebra that acts by locally generated derivations on an infinite tensor product of matrix algebras indexed by the points of $X$. We compute the homology of this Lie algebra with trivial scalar coefficients when $X$ is the integer lattice in $n$-dimensional Euclidean space.

Let $G=(V,E)$ be a simple graph of order $n$ and let $\lambda_1(G)\ge \cdots \ge \lambda_n(G)$ be the eigenvalues of its Laplacian matrix. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k\lambda_i(G)\le |E|+\binom{k+1}{2}$, which was recently confirmed by Kothari and Tudose. Before Brouwer's conjecture was proved, Lew (JCT-B, 2026) established a weaker form of Brouwer's Laplacian eigenvalue inequality and proposed two conjectures for upper bounds on the sum of the $k$ largest Laplacian eigenvalues, one in terms of the matching number and the other in terms of the vertex-cover number. Using Brouwer's Laplacian inequality, we prove both conjectures.

We present the analytical, exact, explicit, and assumption free formulas for the evolution operators corresponding to four instances of time-dependent Hamiltonians relevant to quantum spin batteries including two stochastic cases. We demonstrate how to recover and go beyond existing expansions and approximations directly from the exact solutions giving, for example, an explicit exact formula for Floquet Hamiltonians at all orders. The exact solutions are obtained through a completely novel combination of three mathematical techniques, the $\star$-algebra, path-sums and Omega calculus, which we briefly overview. These are widely applicable to other non-autonomous differential systems.

We provide highly accurate estimations of the 10th through 15th Dedekind Numbers, to a precision of 4 digits for $D(10)$, to 2 digits for $D(15)$. These estimates were obtained using three methods, including pair matching on large quantities of 9-dimensional monotone Boolean functions for $D(10)$, Reference Subsets for $D(10)$, $D(11)$, and $D(12)$. And our best method "Weight Layer Branching" which provided accurate estimates for all $D(10)$ through $D(15)$, strongly improving on the previous best known estimates by Korshunov and Tian-Shun Chen et al. arXiv:2606.09795

We study several interrelated problems arising from the interplay between extreme point theory, Grothendieck-type inequalities, and tensor product norms. We develop a general framework for characterizing the extreme points of the set of positive contractions $\mathcal{A}_{X\to Y}$ between finite-dimensional Banach spaces, with explicit results for $X=\ell_1^n$, $Y=\ell_\infty^n$ and vice versa. These characterizations are applied to evaluate several constants exactly. We show that the positive Grothendieck constant $K_G^{+,\mathbb{R}}(3)$ equals $9/8$ and that the smallest constant $\rho^{+}(X)$ for which $\|A\|_\pi \leqslant \rho^{+}(X)\|A\|_\epsilon$ holds for all $A \geqslant 0$ equals $5/4$ when $X=\ell^3_\infty(\mathbb{R})$. We also prove that $\rho^+(X)=1$ when $X=\ell_\infty^n(\mathbb{C})$ and $n\leqslant 3$. Finally, we prove that $\rho^+(X) = 1$ for every 2-dimensional subspace $X$ of $\ell^3_\infty(\mathbb{C})$; since this is stronger than the 2-summing property, it recovers Proposition~4.4 of \cite{AFJS95}.

We create examples of smooth, compact surfaces in $R^3$ for which the geodesic flow is Anosov. We determine their genus, thereby giving a (non-sharp) upper bound for the minimal genus of an embedded surface with Anosov geodesic flow. These examples are explicit physically realizable Anosov systems.

This article concerns interacting particle systems with singular kernels, driven either by degenerate deterministic controls or by degenerate decomposable noise. In the deterministic setting, we establish global exact controllability and a topologically robust property called solid controllability. Moreover, we prove a result that guarantees global approximate controllability with prescribed trajectories. For stochastic dynamics, we obtain ergodicity and exponential mixing by utilizing coupling and recurrence mechanisms based on controllability. Our approach exploits the singularity and applies to a broad class of models, including Biot-Savart, Coulomb, Riesz, and Yukawa interactions, as well as heterogeneous multi-species systems.

The SIGEST paper Nielsen and Strakoš (2024) characterized the spectrum of the preconditioned operator $\Delta^{-1}[\nabla \cdot (K\nabla u)]$ in a bounded open two-dimensional domain $\Omega$, where $\Delta$ denotes the Laplacian and $K(x,y)$ is a continuous symmetric matrix-valued function. An important part of the analysis states that for a diagonal tensor $K$ constant in an open subdomain $S \subset \Omega$, the closed interval defined by its diagonal elements belongs to the spectrum of the preconditioned operator. This result is correct, but the proof in Nielsen and Strakoš (2024) must be refined. This paper presents a refined proof and extends the previous work. As shown in the cited papers, for any point $\lambda$ in the open interval defined by the elements of the diagonal tensor constant in $S$ and any point $(x_0,y_0)\in S$, a rectangular subdomain $\Sigma_l\subset S$ can be constructed such that the generalized eigenvalue problem associated with the preconditioned operator restricted to $\Sigma_l$, of arbitrarily small size, has the eigenvalue $\lambda$ and infinitely many eigenfunctions. These are given by solutions of a locally defined wave equation. However, such solutions of the locally restricted generalized eigenvalue problem cannot be extended to the whole domain $\Omega$. Using instead rectangular subdomains whose size shrinks to zero, the present paper constructs a Weyl singular sequence of approximate eigenfunctions associated with $\lambda$, proving that $\lambda$ belongs to the spectrum of the preconditioned operator. Since self-adjoint operators in a separable Hilbert space can have at most a countable set of eigenvalues, this shows that the eigenvalues of the locally defined operator converge to points of the continuous spectrum of the preconditioned operator on the entire domain.

A method is presented to carry out a transient simulation of eddy current problems with nonlinear materials. Coils are voltage-driven. The magnetic field due to currents in coils are considered by their Biot-Savart-fields. The magnetic vector potential is used in the finite element formulation. The time stepping method is based on implicit Euler. The arising nonlinear equation system is split into two parts, the common finite element system and a circuit equation. Each part is solved separately by Newton's method. Additionally, a line search is used to solve the nonlinear field equations. Inrush currents and average magnetic flux densities through cross sections of laminates of a nonlinear benchmark problem consisting of a laminated iron core inserted in a cylindrical coil are studied. All details of the numerical benchmark are given to evaluate the presented results. Numerical data describing the performance of the presented method are provided.

We develop a novel complete flux finite volume method for convection-diffusion equations on arbitrary subdivisions in two and three dimensions. Unlike standard finite volume discretizations, where the numerical flux is directly approximated from the flux definition, we derive the exact normal flux across each control volume edge/face from the underlying PDE. This exact flux splits naturally into a homogeneous part (the classical Scharfetter–Gummel flux) and an inhomogeneous part based on a Green's function that incorporates the tangential flux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without using correction or stabilization strategies. From this framework, we develop concrete numerical schemes on arbitrary grids using Lagrange finite element spaces and B-spline spaces, together with their companion dual meshes (control volume partitions). Numerical experiments in two and three dimensions confirm the optimal convergence and positivity preservation of the proposed schemes.

Multi-antenna unmanned aerial vehicle (UAV)-aided communication presents a promising solution to increase the system capacity and improve the quality of service (QoS) of the future wireless networks. In this paper, we equip a UAV platform with a rotatable antenna array (RAA), which can be rotated flexibly in three-dimensional (3D) space via an onboard gimbal, enabling additional spatial degrees of freedom (DoFs) for improving multiuser signal transmission and interference management. Compared with a conventional fixed antenna array (FAA), the RAA can proactively align users with the high-gain region of its antenna elements and reduce the spatial channel correlations among users. To demonstrate the advantages of RAA, we jointly design the RAA orientation and beamforming to maximize the sum-rate of multiple users subject to per-user QoS constraints. The formulated problem is highly nonconvex and exhibits strong coupling between the RAA orientation and beamforming variables. To solve this challenging problem, we propose first an optimization framework based on the penalty dual decomposition (PDD) method to iteratively optimize RAA orientation and beamforming. While the optimization framework yields high reliability in QoS satisfaction and favorable sum-rate performance, its iterative nature may hinder real-time deployment. To accelerate the joint design and preserve a high-quality solution, we further propose a deep learning (DL) framework based on graph neural networks (GNNs). Simulation results demonstrate that RAAs significantly outperform FAAs in UAV-aided communication. Additionally, the proposed optimization framework is capable of satisfying stringent QoS requirements with high reliability, while the proposed DL framework attains comparable sum-rate performance with substantially reduced computation time and exhibits robustness to user position information errors.

Let $X=(X_1,\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\ldots,D_n)\sim\operatorname{Dir}(1,\ldots,1)$ be independent of $X$, and define $K(x)=\mathbb{P}\{\sum_{i=1}^n x_iD_i\le1\}$. We prove that, for every $n\ge1$, whenever $\mathbb{E} X_i\le1$ for every $i$, $\mathbb{P}\{K(X)\le\alpha\}\le\alpha$ for all $0\le\alpha\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\mathbb{E}X_i \le 1$ for all $i$. This proves a conjecture of Gaffke (2005).

Hankel used his principle of the permanence of formal laws (PFL) as a guide for the extension of number systems and as a necessary condition for the legitimacy of their formal theories. He acknowledged that these applications have important limitations, evidenced by the extension to hypercomplex numbers and by what he saw as the unavoidable inconsistency of a formal theory of irrational numbers. Yet, intriguingly enough, he remained fully committed to the PFL. I argue that this was due to his understanding it as an expression of a conservative strategy, inherited from Peacock and Hamilton, which permits the revision of the basic laws of arithmetic if there are reasons for revision that are found, upon deliberation, to outweigh the reasons for their preservation. Then I discuss criticisms by Schubert and Pringsheim, who reformulated the PFL to align it with their own anti-revisionary conservative strategy, at the cost of relinquishing parts of modern mathematics. I conclude by emphasizing the deep philosophical difference between these kinds of conservatism in mathematics.

We initiate the study of Lyapunov-type inequalities for Dirichlet problems driven by the discrete p-Laplacian on weighted graphs. The approach is capacitary and is based on point p-capacities and the associated capacitary radii. First, we prove general Lyapunov-type inequalities on arbitrary connected locally finite weighted graphs. These inequalities provide intrinsic lower bounds, expressed in terms of the capacitary radii, for the positive part of the potential whenever the corresponding Dirichlet problem admits a nontrivial solution. Next, we estimate these capacitary radii in several geometric settings and prove the sharpness of the resulting Lyapunov-type inequalities. As an application, we derive lower bounds for the first weighted Dirichlet eigenvalue of the discrete p-Laplacian.

We investigate the stability and asymptotic behavior of spatially periodic cnoidal waves in the Korteweg-de Vries equation subject to localized perturbations. Standard stability arguments in Hamiltonian systems break down in this setting, since localized perturbations preclude a characterization of stable periodic waves as strict minimizers of a suitable energy functional subject to finitely many constraints. As a result, the nonlinear stability of periodic waves under localized perturbations has remained a long-standing open problem in Hamiltonian systems, with previous results only addressing plane waves that can be reduced to constant states by passing to polar coordinates. In this paper, we develop a novel method that resolves this obstruction by combining variational arguments, Floquet-Bloch theory, and Duhamel-based estimates with spatiotemporal modulation. Our framework applies to general periodic waves in Hamiltonian systems with symmetry and reduces the nonlinear stability problem to verifying diffusive spectral stability conditions for the second variation of a suitable conserved energy. Applying our approach to cnoidal waves in the Korteweg-de Vries equation, we obtain the first nonlinear stability result for periodic waves in Hamiltonian systems under localized perturbations that cannot be reduced to constant states.

In this paper, we introduce the notion of $q$-pre-Lie algebras from the perspective of representations of Lie algebras, providing a parametrized generalization that unifies pre-Lie algebras and anti-pre-Lie algebras. For a $q$-pre-Lie algebra $(A,\circ)$, the commutator of $\circ$ is a Lie bracket and the left multiplication operator scaled by $q$ gives a representation of the associated commutator Lie algebra. We also introduce the notions of $q$-$\mathcal{O}$-operators and $q$-Novikov algebras, and investigate their relationships with $q$-pre-Lie algebras. Several explicit constructions of $q$-pre-Lie algebras are provided. Moreover, we give a complete classification of graded $q$-pre-Lie algebra structures on the Witt algebra and prove the existence of such structures on the Virasoro algebra only when $q=1$ and $q=2$. Finally, we classify compatible root-graded $q$-pre-Lie algebra structures on finite-dimensional complex simple Lie algebras.

Separability is an important structural assumption often placed on the covariance when working with matrix-variate data, because it greatly simplifies both interpretation and computation of subsequent covariance-based statistical tasks. Yet testing the separability assumption is difficult in the high-dimensional regime. We propose to test separability by recasting the problem as a sphericity test after whitening the data using the separable maximum likelihood estimate of the covariance. The test is calibrated by Monte Carlo simulation, yielding finite-sample level control. Furthermore, we prove the test's high-dimensional consistency under dense alternatives. To reduce its reliance on distributional assumptions, we introduce an angular version of the test based on radial normalization after whitening. We demonstrate the practical utility, empirical power, and computational efficiency of the prop

A technique is provided in this paper to integrate nonlinear Schrödinger equation with time-dependent Dirichlet boundary conditions with high-order Yoshida splittings which are based on Strang method. For that, a modification of Strang method is required in which the linear and stiff part of the equation is integrated with a rational-like version of midpoint rule for which the required boundary values can be calculated without resorting to any differentiation of data. Although Yoshida splitting (with real coefficients) cannot be applied to parabolic problems to obtain order higher than two because of stability, the modified Strang method is also applicable to such type of problems and local order $3$ and global order $2$ are also obtained without differentiation of data.

We investigate pathwise well-posedness of the stochastic modulated Korteweg-de Vries equation (KdV) on the circle with a multiplicative noise, where a time non-homogeneous modulation acts on the linear dispersion term. (i) In the Young case (= fractional-in-time case with Hurst parameter greater than $\frac 12$), we establish a new regularization-by-noise phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that, given any $s \in \mathbb R$ and any multiplicative Young noise, however rough it is in space, the stochastic modulated KdV is pathwise locally well-posed in $H^s(\mathbb T)$, provided that the modulation is sufficiently irregular. (ii) In the rough case (= white-in-time case), irregularity of the modulation does not induce any smoothing on the stochastic convolution, and in fact, there is a slight loss in the spatial regularity. In this case, by slightly regularizing the multiplicative noise term, we prove pathwise local well-posedness in $H^s(\mathbb T)$ for any given $s \in \mathbb R$, provided that the noise is sufficiently smooth in space. We achieve these goals by combining (i) the sewing lemma approach to the nonlinear Young integration theory, introduced by Chouk and the second author (2014), and (ii) the pathwise construction of stochastic convolutions as Young or rough integrals via the random tensor estimate and the sewing lemma, introduced by the first, fourth, and fifth authors (2026). In the appendix, we also present an example of regularization by noise for a stochastic modulated Schrödinger equation with a multiplicative Young noise.

We study the existence, multiplicity, and concentration of normalized semiclassical states for a magnetic Schrödinger–Poisson system in $\mathbb{R}^3$ featuring both the Sobolev-critical local nonlinearity $|u|^4u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint $\int_{\mathbb{R}^3}|u|^2\,dx=a^2\varepsilon^3,$ where $a>0$ denotes the prescribed mass and $\varepsilon>0$ is the semiclassical parameter. By combining constrained variational methods, a suitable penalization scheme, concentration–compactness arguments, and Ljusternik–Schnirelmann theory, we first prove the existence of a normalized semiclassical solution for sufficiently small $a$ and $\varepsilon$. We then establish a multiplicity result showing that, for every sufficiently small $\varepsilon>0$, the number of distinct normalized solutions is bounded from below by the Ljusternik–Schnirelmann category of the minimum set \[ \mathcal M = \{x\in\mathbb{R}^3:V(x)=\min_{\mathbb{R}^3}V\}. \] Finally, we describe the semiclassical concentration phenomenon by showing that the maximum points of the resulting solutions approach $\mathcal M$ as $\varepsilon\to0$.

An important quantity in the theory of gradient descent (GD) is the sharpness, defined as the largest eigenvalue of the objective Hessian. Classical analyses typically require the step size to be uniformly smaller than twice the reciprocal of the sharpness, but this condition is frequently violated in the training of deep neural networks. Recent work bridges this gap in the setting of overparametrised least-squares with a single scalar output, providing a normal form for large-step GD in a neighbourhood of an isolated flat minimum and establishing three corresponding convergence results. In this paper, we extend this theory in two directions: (1) to overparametrised least-squares with vector-valued outputs (including regression with arbitrarily many observations), and (2) to a neighbourhood of a manifold of flat minima (which we show is essential for applications such as matrix factorisation). We generalise both the normal form and all three convergence theorems of \cite{macdonaldeos} to this broader setting, overcoming several technical challenges. We further show that our framework applies to deep matrix factorisation under mild assumptions, yielding several new structural results. In particular, we prove that the set of flat minima forms a fibre bundle over a product of spheres, and that the sharpness is Morse-Bott along this manifold.

Distribution goodness-of-fit testing has developed a powerful rate-level theory: we often know how the required sample size scales with the alphabet size, the separation from the null, and the target error probability. Uniformity testing is the canonical example. One can distinguish the uniform distribution on $N$ categories from alternatives at total-variation distance at least $\epsilon$ with far fewer than $N$ samples, and the optimal scaling is now well understood. But rate-level theory leaves an important question unresolved: among several tests with the same sample-complexity order, which one actually gives the best risk or power? This is a constant-level question. It is especially relevant in modern applications where distribution testing is used not merely as an asymptotic abstraction, but as a practical design tool. This note argues that sharp constants in distribution testing play a role analogous to Fisher information in parametric estimation and Pinsker's constant in nonparametric estimation. First, they distinguish between tests that are all rate-optimal but not equally powerful. Second, they reveal the effective signal-to-noise ratio governing the testing problem. Third, they can guide tuning-parameter choices in downstream applications. We illustrate this perspective through large-alphabet uniformity testing and then explain why the same logic matters for choosing the number of bins in calibration testing.

We derive bounds for the volume of tubular neighbourhoods of smooth Pfaffian hypersurfaces, generalising known results for algebraic varieties. The bounds are given in terms of the Pfaffian format of the defining functions. As an application, we obtain tail bounds on the probability distribution of a condition number measuring the robustness of neural network classifiers with Pfaffian activation functions, in both the uniform and Gaussian settings. In the special case of single-hidden-layer sigmoid networks with rational weights, we derive polynomial-in-width bounds for tubular neighbourhoods of the decision boundary.

In this paper, we prove a conjecture of Andrews and El Bachraoui concerning the parity of certain two-color partitions. Precisely, we show that if the Fourier coefficient $t_o(n)$ of the corresponding $q$-series is odd, then $8n+9$ is represented by the binary quadratic form $x^2+2y^2$.

We introduce strongly exposing Lipschitz maps, a vector-valued extension of Weaver's peaking functions and a nonlinear analogue of absolutely strongly exposing operators. Our main result shows that a Lipschitz map is strongly exposing if and only if its canonical linearization is absolutely strongly exposing. This equivalence serves as a bridge between the linear and Lipschitz settings and enables us to transfer several results from the former to the latter. As applications, we establish norm-denseness and residuality results for strongly exposing Lipschitz maps, obtain an isomorphic characterization related to the denseness of strongly norm-attaining Lipschitz maps. We also investigate weak sequential denseness of strongly exposing Lipschitz maps. In particular, we prove that this property holds whenever the derived set of the underlying metric space is finite, while further examples show that, unlike for strongly norm-attaining Lipschitz maps, weak sequential denseness may fail beyond trivial cases.

Fix n >= 2. A set A = {a_0 < a_1 < ... < a_{n-1}} of n residues in Z_N is "valid mod N" if the all-ones multiset is the only size-n multiset drawn from A whose sum is p := sum_i a_i (mod N). For the super-increasing set A = {2^k - 1 : 0 <= k <= n-1} we determine the least valid modulus exactly: N_min(n) = 2^n - 2^floor(log_2 n) for all n >= 2. Both directions of the proof are elementary, resting on a sharp minimal-digit-sum estimate for representations by binary coins, and the full theorem has been machine-checked in Lean 4/Mathlib for all n (https://github.com/jarfo/min-modulus). We conjecture that no size-n residue set admits a smaller valid modulus. This validity condition is exactly what makes the permanent of an n x n matrix equal to a single coefficient of a row-product polynomial modulo x^N - 1, extractable by a size-N discrete Fourier (or number-theoretic) transform; the theorem thus identifies the smallest transform, N ~ 2^n, for which this evaluation is exact. That application – and the resulting common framework for the classical formulas of Ryser and Glynn and this transform – is developed in a companion paper (arXiv:2607.09949).