Papers

In this study, we analyze the number of pairs (i.e., two cards of the same rank) in a set of cards randomly selected from a deck of playing cards. While the standard deck of playing cards comprises 52 cards excluding the joker with 4 suits and 13 ranks, our analysis considers a generalized deck where the numbers of suits and ranks can be set arbitrarily. We derive the exact formulas for the mean and variance of the number of pairs in a given number of cards randomly selected from this generalized deck, expressed using the Gauss hypergeometric function. Moreover, we derive the asymptotic behavior of the mean and variance as the number of ranks tends to infinity.

We introduce a new class of magic figures defined on a finite triangular region of the trihexagonal tiling. The vertices of the region are labeled with the integers $1,2,\dots,3n(n+1)/2$, each used exactly once. The labeling is called \textbf{trihexagonal magic figure of order $n$} if all triangular faces have the same vertex-sum and all hexagonal faces have the same vertex-sum, with the latter equal to twice the former. We note three immediate structural features. First, the ratio between the number of independent constraints and the number of variables approaches $1$ as $n\to\infty$, indicating that the system is asymptotically tight. Second, a necessary parity condition arises: since the labeling uses the integers $1,2,\dots,3n(n+1)/2$, the total number of vertices must be odd, which holds if and only if $n \equiv 1$ or $2 \pmod{4}$. Third, the vertex set forms a perfectly regular triangular boundary, so the configuration has an exact polygonal symmetry without boundary irregularities.

Let $F$ be the fraction field of a discrete valuation ring. We give an explicit matrix in $\mathrm{SL}_2(F[X^{\pm1},Y^{\pm1}])\setminus \mathrm{E}_2(F[X^{\pm1},Y^{\pm1}])$. The proof of non-elementarity is based on techniques developed by Peter Abramenko in his preprint arXiv:0808.1095v1.

In this paper, we consider one-dimensional discrete Schrödinger operators \begin{align} Hu(n)=(\Delta+V)u(n)\nonumber \end{align} on $\ell^2(\mathbb{N})$ with a self-adjoint boundary condition at $n=0$, where $\Delta$ denotes the discrete Laplacian and $V(n)$ is a real-valued perturbation satisfying $$V(n)=\frac{O(1)}{1+n}.$$ We determine the sharp transition for the asymptotic coefficient of \(V\) governing the existence and nonexistence of embedded eigenvalues.

The Job Sequencing and Tool Switching Problem (SSP) is a well-known combinatorial optimization problem arising in the context of flexible manufacturing. Since the seminal work of Tang and Denardo (1988), the SSP has received significant attention in the literature, leading to the development of numerous exact and heuristic approaches. Despite these efforts, several benchmark instances proposed decades ago and containing only 20 jobs have remained unsolved to proven optimality. In this work, we propose an exact algorithm for the SSP, namely the Combinatorial Branch-and-Bound (C-B&B) algorithm, which combines two distinct branch-and-bound algorithms, each introducing novel features compared with the existing literature. The former relies on a new branching scheme designed to reduce the size of the implicit enumeration tree, together with a collection of new bounding functions. The latter builds on the branching scheme introduced by Laporte et al. (2004) and strengthens it with a new bounding function and two dominance rules. Within C-B&B, these exact algorithms are complemented by a preprocessing phase that incorporates a new branch-and-bound-based heuristic capable of rapidly generating a high-quality initial incumbent solution. Extensive computational experiments show that C-B&B represents a strong breakthrough over previously published approaches, proving optimality for more instances with significantly less computational effort and closing several benchmark instances that have remained open for decades.

This paper introduces a novel, efficient class of parallel direct and indirect iterative schemes to solve general obstacle and free boundary value problems. The uniqueness of the solution for the direct parallel method is established under the assumption that the model problem yields an $M$-matrix. The convergence analysis of the indirect approach is predicated on minimizing the discrete energy functional at each iterative stage of the linear approximation process. Under the aforementioned setting, we establish theoretical convergence results for both smooth and nonsmooth energy functionals defined over a convex set, in the sense of Ferris and Mangasarian \cite{ferris1994parallel}. For the numerical computation of the one-dimensional obstacle problem, we adopt a direct generalized parallel approach based on the SPIKE algorithm \cite{polizzi2006parallel}. To accelerate convergence and reduce computational complexity, a fast recursive version of the scheme is generalized to the nonsmooth case. For two- and three-dimensional obstacle problems, we employ a directional splitting method that treats each directional subproblem as a one-dimensional obstacle minimization problem. By framing the energy functional minimization as a parabolic time-dependent problem and utilizing an Armijo time-stepping rule during the solution update process, we successfully obtain results for higher-dimensional obstacles. Additionally, this study explores the possibility of extending the algorithm into a constrained quadratic programming optimization solver. We also evaluate the framework on image deblurring phenomena under the aforementioned setting, successfully recovering the original images. Finally, numerical illustrations are provided to validate the theoretical results.

We develop a probability geometric construction of generalized exponential families based on probability coordinate charts. A classical exponential family on the unit interval is transported to value space through a chart, producing what we call a G-exponential family. The construction changes the geometry in which exponential structure is represented while leaving the exponential function itself unchanged. This distinguishes it from the algebraic deformations used in the Tsallis and Kaniadakis frameworks. The transported families inherit the structure of the probability coordinate framework. The probability coordinate of each family member follows exactly the original exponential family on the unit interval. Initial Kolmogorov moments are therefore obtained by pulling classical coordinate moments back to value space. For the canonical family, the probability barycenter is the pullback of the mean parameter, while the Fisher information coincides with the Kolmogorov variance. Tail behaviour is inherited from the chart. Every family member is tail equivalent to the chart density, so heavy tails arise from geometry and remain unchanged in order under probability coordinate tilting. This is the opposite of classical exponential tilting. The sufficient statistic is bounded by construction, maximum likelihood reduces to moment matching in probability coordinates, and the maximum likelihood estimator has a bounded influence function. Thus robustness arises from geometric compactification. We also characterise G-exponential families as minimisers of relative entropy to the chart law under coordinate moment constraints and prove that transport preserves the information geometry of the family. A Cauchy chart example and a Monte Carlo study confirm efficient, calibrated, and bounded influence estimation even though every family member has an infinite mean.

Secret key generation from correlated observations at Alice and Bob, in the presence of an eavesdropper Eve, underpins physical-layer security. Classical pipelines quantize by hand, amplify privacy afterwards, and optimize no objective tied to a key rate. We propose Variational Probabilistic Quantization (VPQ): neural encoders that map the correlated sources directly into a discrete key alphabet, trained by a variational adversarial objective whose entropy, mismatch, and leakage terms match the three terms of the one-way secret key rate. A linear code-offset secure sketch then reconciles the encoder outputs into an identical key without a separate privacy amplification step. We prove that the VPQ losses lower-bound the one-way secret key capacity of the induced source, and derive in closed form the optimal worst-case key rate over the source class of a given alphabet size and mismatch probability, attained by finite-field linear sketches. On Gaussian fading channels, VPQ leaks less to a correlated eavesdropper than one classical and two recent learning-based baselines, and Reed–Solomon reconciliation operates within the predicted finite-blocklength gap.

This paper investigates over-the-air (OTA) computation enabled online federated learning (FL) in low-Earth orbit (LEO) satellite networks. Specifically, we consider a dual-layer OTA aggregation architecture, where ground devices upload analog model updates to serving satellites via uplink OTA aggregation, and satellites forward the aggregated signals to a data processing center through the second round OTA aggregation. Then, we formulate a long-term data-utilization maximization problem in which devices continuously collect new data and untrained samples gradually lose freshness. The problem is subject to the satellite beam budget, transmit-power limit, and global mean squared error (MSE) constraint that governs end-to-end aggregation distortion. This yields a coupled mixed-integer nonlinear programming (MINLP) problem, involving tightly coupled discrete beam-hopping decisions and continuous power control. Due to the combinatorial action space and nonconvex constraints, the problem is NP-hard and computationally intractable. Furthermore, the time-varying satellite topology and dynamic data generation render it a sequential decision-making problem, necessitating adaptive online scheduling. To address these issues, we cast the problem as a Markov decision process and develop a proximal policy optimization (PPO)-based deep reinforcement learning framework that jointly optimizes adaptive beam hopping and power control, using an MSE-aware reward to balance data utilization and aggregation accuracy. Numerical simulation results verify that the proposed algorithm consistently outperforms other benchmark schemes, achieving superior long-term data utilization and faster FL convergence while satisfying the MSE requirement.

For positive integers $n\ge r$, let $\lambda(r,n)$ denote the maximal relative projection constant of $r$-dimensional subspaces of $\ell_\infty^n$ and $\lambda(r)$ denote the maximal absolute projection constant, respectively. It is known that for any fixed $r$, $\lambda(r,n)$ is a non-decreasing sequence with limit $\lambda(r)$ as $n\to \infty$. A natural question is whether $\lambda(r,n)$ stabilizes at $\lambda(r)$ for some $n>r$. We prove that for any fixed $r$, \[\lambda(r,n)=\lambda(r) \qquad \text{for every}\qquad n\ge 2^{r}\binom{r+1}{2}.\] This answers a question of Basso. The technique used is of independent interest.

The Trautman conjecture asserted that a smooth three-dimensional CR manifold admitting a nowhere-zero closed section of its canonical bundle must be locally embeddable in $\mathbb{C}^2$. We modify a standard construction of nonembeddable smooth strongly pseudoconvex CR 3-manifolds so that the condition of having a nowhere-zero closed section of the canonical bundle is preserved, thus providing a strongly pseudoconvex counterexample to the Trautman conjecture.

Symmetry reductions of partial differential equations (PDEs) inherit more geometric structures than other types of reductions: invariant conservation laws, variational structures, and, under suitable conditions, Hamiltonian-type structures of the original model descend to the reduced model through the mechanism of invariant reduction. This paper develops computational algorithms that carry out such reductions explicitly. We use the interpretation of variational $p$-forms as conservation laws of an enlarged system — the (degree-shifted) tangent system — consisting of the original equations together with their linearizations, in which the perturbation variables are treated as anticommuting. This allows us to formulate the algorithms using well-known concepts from the theory of conservation laws, naturally adapted to the graded-commutative setting. The reduction of conservation laws, variational $1$-forms, and presymplectic structures thereby becomes a single algorithmic procedure. We present (i) a homotopy-based reduction algorithm for systems of evolution equations, implemented in Maple; (ii) a descent reduction algorithm applicable to general $\ell$-normal systems for $p>0$; and (iii) a simple reduction algorithm available for point symmetries under suitable conditions, including the description of reductions in terms of systems involving fewer independent variables. Examples include nonlinear evolution equations in one and two spatial dimensions, the Laplace equation, the incompressible Euler equations, and the cotangent system of Pavlov's equation.

Using a bijection between the lattice points in a lecture hall polytope and Lakshimibai-Seshadri (L-S) paths, we prove the Koszul property of lecture hall polytopes in complete generality, and give new proofs for their Integral Decomposition Property and for a criterion on Gorenstein property.

For a prime $p$, a pair $(s,m)\in\mathbb{N}^2$ with $m$ relatively prime to $p$, a homomorphism $\chi:C_m\rightarrow\operatorname{Aut}(C_{p^s})$, and an algebraically closed field $k$ of characteristic $p$, we consider the semidirect product $G=C_{p^s}\rtimes_{\chi} C_m$, the $p$-Sylow subgroup $H\cong C_{p^s}$ of $G$, and a $k[G]$-module $V$. Let $R$ be a complete discrete valuation ring of residue field $k$ and mixed characteristic $(0,p)$ that contains a primitive $p^s$-th root of unity. If $\chi$ is injective, we present two necessary and sufficient criteria for lifting $V$ to an $R[G]$-module $\widetilde{V}$ which is a free $R$-module: (i) when no extra requirement is made on $\widetilde{V}$ and (ii) when we require $\widetilde{V}^H=\{0\}$. The criteria correct several results in the literature and we use them to prove that, if $\chi$ is injective and $G$ acts faithfully on a connected smooth projective curve $X$ over $k$, then, under mild hypotheses satisfied if $X\rightarrow X/G$ is a Harbater–Katz–Gabber cover, the $k[G]$-module $H^0(X,\Omega_X)$ has a lift $\widetilde{V}$ to $R$ with $\widetilde{V}^H=\{0\}$. Also, with $B:=R[\frac{1}{p}]$, we prove the following obstruction when $p$ is odd, $m$ is even, $\chi$ is injective, and $X/C_{p^s}\cong\mathbb{P}^1_k$: if no such lift $\widetilde{V}$ exists with the $B[H]$-module $\widetilde{V}\otimes_R B$ defined over $\mathbb{Q}$, then the action of $G$ on $X$ does not lift to $R$.

Many biomedical challenges can be posed as tensor completion problems where the observed entries of a multidimensional array (a tensor) are used to impute the missing values. In such settings, incorporating side information about the modes of the tensor, such as gene-gene similarity, can significantly enhance the solutions of the completion problem. Most existing tensor completion methods can only incorporate side information in the form of matrices. In this study, we introduce a novel framework to incorporate side information in the form of tensors. Our new approach, called Coupled Tensor-Tensor Completion (CTTC), leverages the hidden connections among multimodal tensors to improve tensor completion performance. In addition to practical utility, CTTC has theoretical foundations in distance metric learning and group theory. We derive an alternating algorithm to solve the CTTC optimization problem and establish its convergence to a stationary point. Finally, we show that CTTC outperforms state-of-the-art tensor completion methods at predicting drug effects. Results: Compared with other tensor completion methods, including HaLRTC, CTRC, Cell, and NTDDR, CTTC demonstrates superior run-time and RSE tensor completion accuracy on two benchmark datasets, DTD and LINCS.

For a prime $p\geq 5$, let $\mathscr S_g$ be the moduli space over $\overline{\mathbb F}_p$ of $g$-dimensional principally polarized supersingular abelian varieties. We show that each of the following loci contains an open dense subscheme on which the principally polarized abelian varieties have automorphism group $\{\pm1\}$: (i) certain supersingular Ekedahl–Oort strata when $g$ is even, (ii) the loci in $\mathscr S_g$ with non-supersingular Ekedahl–Oort invariants of positive Coxeter type when $g\geq 3$, and (iii) the locus in $\mathscr S_g$ with $a$-number at least $2$ when $g\geq 4$. Consequently, for $g\geq 4$, the complement in $\mathscr S_g$ of the open locus where the automorphism group is $\{\pm1\}$ has codimension at least $2$. These results confirm Oort's conjecture for $p\geq 5$. We reduce them to statements about affine Deligne–Lusztig varieties for $\operatorname{GSp}_{2g}$ and prove analogues of (ii) for $\operatorname{GL}_{2g}$ and $\operatorname{GSO}_{4m}$.

We consider a weak martingale optimal transport problem related to the martingale analogue of the Benamou–Brenier formula. In contrast to the classical setting, the second marginal is not given, but specified only through a finite number of moment constraints of a particular form. For this problem, we derive a finite-dimensional dual formulation and prove the absence of a duality gap under the specified interiority condition on the constraint vector. In addition, we describe the structure of the optimal martingale coupling and its relation to Bass martingales.

We prove uniqueness of tangent flows for mean curvature flow with free boundary at singularities modeled on half-cylindrical self-shrinkers. More precisely, if a tangent flow is of the form $\mathbb{S}^{n-k} \times \mathbb{R}^k_+$ or $\mathbb{S}^{n-k}_+ \times \mathbb{R}^k $, then it is unique. This provides the free-boundary analogue of the uniqueness theory for cylindrical singularities.

We resolve two open problems regarding uniquely colorable Cayley graphs posed by Klotz and Sander (2017). First, we construct an infinite family of uniquely 3-colorable integral circulant graphs with a clique number of 2. This provides a negative answer to Problem 3.6, which asks whether every uniquely colorable circulant graph satisfies $\chi(G) = \omega(G)$. Because verifying unique colorability inherently relies on the exact independence number, we demonstrate that traditional spectral bounds fail to tightly capture this parameter, necessitating a rigorous combinatorial proof based on exact structural isomorphisms. Second, we establish a general algebraic construction proving the existence of uniquely colorable Cayley graphs over nonabelian groups whose color classes are left cosets of strictly distinct subgroups. By utilizing right-coset partitions of non-normal subgroups, this result provides a definitive affirmative answer to Problem 2.4.

Motivated by Schoof-Pila-type point-counting algorithms, we study the fields of definition and factorization patterns of Galois orbits of $\ell$-torsion points on the Jacobians of genus-3 hyperelliptic curves over finite fields. We show that the degree of the field of definition of the $\ell$-torsion points can be bounded by $O(\ell^4)$, improving the previously expected $O(\ell^6)$ bound (which reflects the size of the $\ell$-torsion subgroup, of order $\ell^6$ for a genus 3 Jacobian). Moreover, we establish a precise correspondence between the rank of the $\ell$-torsion subgroup and the Galois orbits of $\ell$-torsion divisors. Our approach relies on a detailed analysis of the Jordan decomposition of the Frobenius action on $J$ and its nilpotent part.

We consider the task of optimizing smooth, possibly non-convex functions of a matrix variable given access only to directional derivatives rather than full gradients. This setting arises when fine-tuning large neural networks on consumer-grade hardware: the network's weights are matrices, memory constraints rule out backward-mode automatic differentiation, but directional derivatives remain available through forward mode. We frame gradient estimation in this setting as a structured recovery problem, in the spirit of signal processing. From this perspective we provide three contributions. First, we introduce an alternative to the standard random gradient estimator; the difference corresponds to replacing the adjoint of the sampling operator with its pseudoinverse. Second, when the gradient satisfies an approximate low-rank condition, techniques from matrix sensing yield a family of highly accurate gradient estimators that drop into any first-order method. Third, we note that while the computational cost of such estimators is high, this can be amortized by combining them with a matrix-aware optimizer such as spectral descent. Specifically, the gradient estimator computes a factorization of the gradient, allowing for the projection step of spectral descent to be done at no extra cost. We demonstrate our findings with two careful numerical experiments on synthetic functions with approximately low-rank gradients. We show that by exploiting this low-rank property one obtains much faster convergence to good approximate solutions.

For a given function $f$ on a discrete subspace $\Gamma$ of a noncompact Riemannian symmetric space $G/K$, we construct analytic, noncompactly supported functions $W$ on $G/K$ with $W|_{\Gamma}=f$ that, in addition, satisfy an invariant differential equation on $G/K$. The key step is to regard $W$ as an orthogonal projection onto a space of coherent states in a representation of the Lie group $G$ in a Hilbert space that admits a strictly positive definite reproducing kernel. The techniques are implemented for the heat kernel on $G/K$, and $W$ is expressed in terms of certain closed formulas for linearly independent coherent states.

Energy-based analysis of the three-dimensional incompressible Navier–Stokes equations with spatially localized dissipation is developed at the level of kinetic energy and Leray–Hopf weak solutions. A localized coercivity property linking viscosity and damping to global $L^2$ control is introduced and related to observability and spectral inequalities for the associated Stokes operator. Under this condition, exponential energy decay is established without regularity or linearization assumptions, and time-dependent damping is treated within a nonautonomous framework. A finite-dimensional spectral quantity is proposed as a computable proxy for coercivity, yielding explicit decay estimates for reduced-order models.

We present a high-order multiphysics solver for the simulation of microwave-heated flows. The solver couples a discontinuous Petrov-Galerkin (DPG) finite element method for the time-harmonic Maxwell equations with continuous Galerkin finite element methods for the heat equation and the incompressible Navier-Stokes equations. We validate the electromagnetic solver against multiple benchmark problems: wave propagation in a rectangular waveguide, a cavity problem with a singular solution, and a microwave-heated obstacle problem, comparing our results against numerical and experimental data from the literature. The results confirm the validity of the implementation and demonstrate its ability to perform adaptive mesh refinement using the DPG method's built-in error estimator. The final part of the study showcases the capabilities of the multiphysics framework through simulations of microwave-heated flow around obstacles with singular geometric features. These results highlight the potential of the proposed framework for the simulation and optimization of microwave-assisted chemical processes. Finally, the developed high-order multiphysics solver has a low memory footprint, since the electromagnetic solver relies on a Conjugate Gradient (CG) iterative solver and the fluid solver is implemented in a matrix-free fashion, making the overall approach scalable and well-suited for large-scale parallel simulations.

Motivated by the conjectural existence of M-theory, we investigate a web of correspondences between enumerative invariants of a Calabi-Yau fivefold $Z$ with a torus action. This includes a correspondence between the fivefold Gromov-Witten invariants and K-theoretic Pandharipande-Thomas invariants of a threefold $X \subset Z$, mediated by so-called membrane indices, which generalise Gopakumar-Vafa invariants. We establish the correspondence for strip geometries for restricted torus actions and test it numerically for general torus actions. Further numerical evidence is provided for the closed vertex, local surfaces and local projective spaces. For the latter we present conjectural formulae for their low-degree membrane indices. When the fivefold is the product of the affine plane with a suitable toric variety, we prove geometric engineering and equate the generating series of Pandharipande-Thomas invariants with the corresponding instanton partition function while equality with the Gromov-Witten series is probed numerically.

We use tools from nonlinear algebra to study the equilibria of small linear translational spring networks. Specifically we use the techniques of homotopy continuation, monodromy, and parameter homotopy (a.k.a. cheater homotopy) to solve all rigid linear translational spring networks up to $5$ nodes in both $2$ and $3$ dimensions. We describe a method of implementing parameter homotopy that arises naturally from the physical structure of the system. We give precise total degree bounds on the maximum number of solutions for general planar spring networks. We discuss further efficiency gains obtained from polyhedral homotopy methods. We compare the computation efficiency of these techniques against a baseline of Newton's method.

Modern feedback design for distributed parameter systems presupposes that the closed-loop dynamics define a well-posed evolution problem. This presupposition becomes nontrivial for temporally degenerate parabolic systems, where temporal degeneracy affects not only the analytical properties of the evolution equation but also the mathematical formulation of the feedback interconnection itself. It is shown that admissible feedback interconnections for temporally degenerate parabolic systems are completely characterized by an operator compatibility condition linking the singular reaction operator with the actuator and observation operators. This characterization removes the singular component of the closed-loop dynamics and reduces the degenerate evolution equation to a regular evolution equation. Building upon this regularized formulation, a critical–residual decomposition yields a uniform exponential stability certificate, which is subsequently extended to the original infinite-dimensional evolution through a finite-to-infinite lifting theorem. A constructive static output feedback synthesis is finally obtained as a consequence of these results. Numerical experiments illustrate the regularization mechanism, validate the stability certificate, and confirm the finite-to-infinite lifting.

We prove the existence of multi-peak solutions for singularly perturbed elliptic problems in the so-called zero-mass case by applying the classical Lyapunov-Schmidt reduction method. The goal is to construct solutions concentrating around several distinct interior points of the domain, identified as nondegenerate critical points of a potential.

We prove Chui's conjecture on sums of Cauchy kernels. It states that among all configurations of Cauchy kernels on the unit disk, the equispaced one minimizes $L^1$ norm of the sum in the unit disk. Moreover, we obtain a generalization to a wider set of functionals, including weighted $L^p$ norms for $p > 1$ with radially symmetric weights.

This work concerns ideals invariant under the action of the group of linear base changes on a generic matrix; we call these GL-invariant ideals. We determine the ordinary powers, their saturations with respect to determinantal ideals, and the symbolic powers of GL-invariant ideals. We give explicit formulas for asymptotic invariants known as (skew) Waldschmidt constants and asymptotic resurgence, which measure the growth of these families and compare the ordinary and symbolic topologies. We also prove that the generalized symbolic Rees algebras associated with these ideals are Noetherian.