Papers

For a graph $H$, $\langle H \rangle$ denotes the class of all subdivisions of $H$ and $tw(H)$ denotes the treewidth of $H$. In this paper, we prove the following. For $k\geq 1, R\geq 1$, let $G,H$ be two graphs such that strong isometric path complexities (Chakraborty et al. [Disc. Math., 2026]) of both $G$ and $\langle H \rangle$ are at most $k$, and $G$ admits an honest, “nicely rooted” $R$-bounded $H$-partition. Then, there is a graph $F$ with $tw(F)\leq tw(H)$ such that $G$ admits a $(1,33\cdot R\cdot k^2)$-quasi-isometry to $F$. Using results of Albrechtsen, Distel, and Georgakopoulos (2025), we also obtain that $K_{2,t}$-asymptotic minor-free graphs admit quasi-isometries with additive distortion to $K_{2,t}$-minor-free graphs. This answers an open question raised by the above authors. As part of our proof, we combine the graph-partition based method and the layering partition based method (Chepoi et al. [Discrete Comput. Geom. 2012]) to obtain additive quasi-isometry when both the source and all subdivisions of the target graph have bounded strong isometric path complexity.

The algebra of polysymmetric functions (PSym) is defined as the tensor product of copies of the algebra of symmetric functions (Sym) where the $i$th copy has variables scaled by $i$. One way to construct a basis of PSym is to start with a basis $\{f_\lambda\}$ of Sym and consider all pure tensors arising from this basis. Asvin G and Andrew O'Desky described four families of non-pure tensor bases, namely $H, E, E^+$ and $P$, which we call plethystic bases. In this paper, we study the expansions of the plethystic bases into the pure-tensor bases $\{h^\otimes_\tau\}$ and $\{e^\otimes_\tau\}$, and interpret the results via combinatorial objects called polywrapping block tabloids. The expansions of $H, E$ and $E^+$ in $h^\otimes$ and $e^\otimes$ are found via the $h$-expansion of the plethysm $h_n[p_r]$. We compute this $h$-expansion using a statistic on words and prove it through abacus methods.

We describe the Macaulay2 package MixedHodgeModules, which computes Hodge and weight filtrations on localizations S_f of a polynomial ring S at a non-constant reduced polynomial f, as well as on twisted localizations S_f f^{-\alpha}, where \alpha\in QQ is rational. The package includes functionality for computing Hodge ideals, weighted Hodge ideals, higher multiplier ideals, generation levels of Hodge filtrations, weight levels of elements, p-functions, and lengths of weight filtrations. It also provides routines for computing graded de Rham complexes, Du Bois complexes, and intersection Du Bois complexes of hypersurfaces.

The growing development of online educational video resources over the past two decades, has significantly reshaped university students' self-regulated study approaches and influenced diverse ways of learning. Within an environment of abundantly and publicly available online resources, educators cannot always control which materials students choose to supplement their study. Consequently, considerable deviations are likely to be present between the taught content by the educator and an arbitrary online educational video the student watches, in terms of terminology, notation and methods, especially in mathematical sciences. These deviations can be even more considerable when Transnational Education (TNE) students incorporate native language resources to comprehend the material. This paper presents the design, creation and implementation of interactive video resources produced in alignment with established pedagogical frameworks to support an undergraduate TNE mathematics module.

Tablet-based teaching and learning has become increasingly common in university mathematics courses, offering capabilities such as digital ink, dynamic visualisation, and real-time annotation. However, there is limited experiential evidence regarding how students perceive the specific benefits and drawbacks of tablet-supported teaching and learning, particularly in Transnational Education (TNE) settings. This paper presents results from a small-scale exploratory survey of undergraduate students ($n = 19$) attending mathematics courses within an applied sciences degree programme, who anonymously rated a set of statements regarding their experiences with tablet-based instruction on a five-point Likert scale. Initial analysis shows that students in this cohort perceived tablet teaching most positively in supporting step-by-step understanding and active engagement during lectures through real-time annotation and structured note-taking. In contrast, perceptions related to long-term recall, interest in mathematics, and preference over alternative teaching methods were more mixed. These findings suggest that students in this cohort perceived tablet-based teaching positively for supporting procedural understanding, while perceptions of other aspects of learning appeared more varied.

In this work, we develop a second-order optimization framework for physics-informed neural networks (PINNs) applied to high-dimensional parametric partial differential equations (PDEs). The framework is built on the Gauss–Newton pullback metric, which provides an operator-informed notion of curvature in parameter space and connects the method to the broader family of natural gradient schemes. We show that, for coordinate-separable neural architectures and linear differential operators (or linearized operators in the nonlinear case) admitting a finite separable representation, the residual Jacobian inherits a structured separable factorization. This yields an exact compressed formulation of the Gauss–Newton step in a reduced space, without assembling the full residual Jacobian on the exponentially large tensor-product collocation grid. The dimension of the reduced space (the effective compressed dimension) is determined by the local collocation grid sizes, the separable operator structure, and the contraction pattern of the architecture, thereby avoiding dependence on the full tensor-product grid size and replacing dense linear algebra in parameter space by a substantially smaller structured problem. Within our framework, we investigate canonical polyadic and tensor-train parametrizations and derive their full algebraic characterization relevant to the Gauss–Newton method, including the structure of the residual Jacobian, the resulting compressed system, and its effective compressed dimension. Numerical experiments on high-dimensional PDEs, including parametric problems, demonstrate the high efficiency of the proposed compressed Gauss–Newton method, which achieves substantially lower errors than tensor-compressed first-order baselines with orders of magnitude fewer iterations and only a fraction of the computing time.

We give a short proof of the following theorem of K. Ball: If a centrally symmetric convex body $K\subset \mathbb R^d$ is covered by finitely many planks, then the sum of their relative widths with respect to $K$ is at least $1$.

In this paper, we study the conditions under which covert communication at positive rates is feasible over discrete memoryless classical, classical-quantum, and quantum channels. For classical point-to-point channels, we show that if the dimension of the channel input probability simplex exceeds that of the channel output probability simplex, equivalently, if the input alphabet has larger cardinality than the output alphabet, then positive covert rates are achievable for certain classes of Discrete Memoryless Channels (DMCs). We further show that allowing the innocent symbol (i.e., the symbol transmitted in the no-communication mode) to be chosen appropriately can enlarge the class of DMCs for which positive covert rates are achievable. We also study covert communication over classical Multiple-Access Channels (MACs), where the additional transmitter effectively enlarges the set of available channel input pairs, and we show that the conditions required to satisfy the covertness constraint are less restrictive for MACs than for point-to-point channels. We extend these results to classical-quantum DMCs by showing that if the dimension of the input probability simplex exceeds the affine dimension of the set of output states that can be induced at the channel output, then positive covert rates are achievable for certain classes of classical-quantum DMCs. Finally, for quantum channels, we show that if the innocent state (i.e., the state transmitted in the no-communication mode) is mixed, then the covertness constraint can always be satisfied by a non-trivial input ensemble. Consequently, positive covert rates are achievable whenever the legitimate receiver can distinguish at least two states in a suitable such ensemble.

In this paper, we revisit the concept of entropy in the spatial context with the aim of deriving computable and interpretable metrics for point pattern analysis in domains such as histopathology. We discuss stochastic point process assumptions like Poisson homogeneity and Simple Sequential Inhibition (SSI) and review established results for Voronoi and Delaunay tilings of point sets for their potential to provide null distributions for rigorous empirical hypothesis testing. We present (1) a novel "dense basin entropy" defined in terms of the Ord distribution supported on Voronoi basins, which is shown to be sensitive to clustering, and (2) a related Markov chain designed as a simplification or approximation of Brownian motion in the underlying planar domain. We investigate bounds on the dense basin entropy in the SSI regime and in empirical data, and show that the entropy rate and spectral gap of the Markov chain lead to sharp discrimination metrics. Finally, we demonstrate a generalization to multiple-set analysis via aggregation of one set over the Markov invariant measure for another. Simulations and experiments in histopathology show that our proposed metrics offer insights complementary to other instantiations of entropy in spatial analysis. The code can be found on our SMProfiler GitHub: https://github.com/nadeemlab/SMProfiler.

Given a bounded open set $V\subset \mathbb{R}^2$, we consider the problem of slicing this set by a family of curves, whose average length is as small as possible. The first sections of the paper prove the existence of an optimal slicing and establish a connection with the problem of eradicating an invasive biological species from an island, considered in \cite{BMS}. Namely, an optimal slicing can be obtained as a limit of minimum time eradication strategies. The second part of the paper derives various necessary conditions for optimality, describing the behavior of an optimal slicing strategy in the interior of $V$ and near the boundary $\partial V$.

We study a dynamic integrated sensing-and-communication problem in which a multi-antenna transmitter communicates with a receiver while simultaneously tracking a moving target governed by a stable Gauss–Markov model. We formulate the sensing–communication trade-off as a rate–distortion problem, where the communication rate is defined through the Verdú–Han information spectrum and the sensing distortion is measured by the long-run minimum mean square tracking error in angle and distance. We construct a tractable outer region by lower bounding this distortion through the posterior Cramér–Rao bound (PCRB). The control problem is partially observed because the target state is not directly available to the transmitter. We therefore introduce a belief-augmented information state and show, under the stated regularity assumptions, that stationary randomized Markov policies suffice for the covariance-based scalarized problem used in the outer bound. We then derive, under the stated large-block conditions, a Gaussian covariance-control outer bound by showing that, in the large-block regime, the sensing belief transition becomes asymptotically covariance sufficient; consequently, Gaussian signaling with the same stationary covariance policy preserves the stationary PCRB asymptotically while maximizing the communication reward for each transmit covariance. Numerical experiments compare an extended Kalman filter-based achievable frontier with a PCRB-based Gaussian covariance-control outer frontier.

We propose a localized reduced basis method for parametrized parabolic partial differential equations based on an adaptive partitioning of the parameter domain using hierarchical binary trees. A POD-Greedy procedure is employed to construct reduced spaces associated with each parameter subdomain. A certified a posteriori error estimator is derived from an inf-sup stability analysis and shown to be equivalent to the underlying approximation error. This equivalence, together with suitable regularity assumptions on the parameter-to-solution map, enables a rigorous convergence analysis and is used to derive a strategy for selecting the number of POD modes to retain in each local reduced space. Numerical experiments for a parametrized convection–diffusion problem demonstrate the effectiveness of the proposed approach and confirm the predicted convergence behavior.

Sparse Code Multiple Access (SCMA) is a promising non-orthogonal multiplexing technique for enhancing connectivity and spectral efficiency in next-generation communication networks. This paper proposes near-optimal uplink SCMA codebook designs based on a novel deep neural network (DNN)-assisted framework. The DNN is trained in an unsupervised manner, with a loss function derived from the structural characteristics of the generated codebooks, thereby eliminating the need for labeled training data. Based on error-probability analysis, Rician-factor-dependent design criteria are derived for uplink Rician fading and used to formulate the loss function. The resulting criteria naturally encompass AWGN and Rayleigh fading as limiting cases, providing a unified framework for codebook design. Moreover, upper and lower bounds on the average mutual information (AMI) are derived, providing an information-theoretic interpretation of the PEP-based criterion and establishing its asymptotic consistency with the AMI-lower-bound-based criterion in the high-SNR regime. The analysis further reveals how the dominant design factors evolve as the channel transitions from Rayleigh fading to AWGN. Numerical results show that the proposed near-optimal codebooks achieve significant performance gains over representative benchmark schemes across a wide range of Rician factors and exhibit excellent error-rate performance in the error-floor region, making them particularly attractive for ultra-reliable applications.

We consider the following three conditions for polynomials in several variables. Condition (1) holds for a polynomial if, for every affine integral domain, the following is true: whenever the result of substituting algebraically independent elements into the polynomial lies in the kernel of a locally nilpotent derivation, the elements themselves must lie in that kernel. A polynomial satisfies condition (2) if it does not belong to the kernel of any nonzero LND of the polynomial algebra. It satisfies condition (3) if the quotient algebra modulo the ideal generated by the polynomial is rigid, that is, it admits no nontrivial LNDs. For irreducible binomials, we prove that all three conditions are equivalent. For reducible binomials, we establish all existing implications between these conditions.

The paper considers a class of optimal set motion problems, where the control determines the velocity of boundary points in the normal direction. Results and counterexamples are given, on the continuous dependence of the minimum cost from the initial configuration. The last sections of the paper study the slicing cost of bounded open sets. For strongly Lipschitz domains, it is proved that the slicing cost varies continuously w.r.t. the Hausdorff distance of the boundaries. We conjecture that the slicing cost for a disc is maximal among all sets with the same area.

Understanding what limits many-body quantum entanglement is a central problem in physics. Spatial locality has long provided a fundamental mechanism: correlations between a region and its complement must be mediated through a small spatial interface, thereby constraining their entanglement. Here we show that universal constraints on entanglement can persist even when interactions are strongly nonlocal, through collectivity: many weak interactions generate fluctuations controlled by their square-summed, rather than total, strength. We establish this mechanism rigorously for generic gapped Hamiltonians with Kac-normalized power-law interactions $r^{-\alpha}$ on a $D$-dimensional lattice. For arbitrary bipartitions, we prove that the ground-state entanglement scales at most logarithmically with system size for $\alpha<D/2$ and subextensively for $D/2<\alpha<D$, due to suppressed collective fluctuations around individual sites. For spatially regular bipartitions with codimension-one boundaries, we use a renormalization-group construction to extend this suppression to larger length scales, yielding polylogarithmic scaling for $D/2 < \alpha < (D+1) / 2$ and parametrically stronger subvolume bounds for $(D+1)/2<\alpha<D$. We also establish the corresponding optimality results: for arbitrary bipartitions, the logarithmic scaling for $\alpha<D/2$ is optimal and $\alpha=D/2$ marks the optimal threshold for universal logarithmic bounds, while for regular bipartitions the threshold at $\alpha=(D+1)/2$ is likewise optimal for universal polylogarithmic bounds when $D\ge2$. Together, these results reveal collectivity as a fundamental mechanism for constraining many-body entanglement alongside spatial locality.

In a recent paper arXiv:2503.20727, Ichino and Prasanna proved a supercongruence for binomial coefficients and related it to representations of $\mathrm{GL}_2$ over $p$-adic rings. We prove a $q$-analog of this supercongruence using the $q$-Lucas theorem, from which a strengthening of the main result of loc. cit. can easily be deduced. Our proof also substantially simplifies the original argument of arXiv:2503.20727.

Let $f$ be a Hénon map of $\mathbb C^2$. We provide quantitative versions of several results involving dynamical objects associated to $f$. Our results can be interpreted as a quantitative version of Pesin theory with geometric control. As an application we show that the perdiodic points of period $n$ of $f$ equidistribute towards its equilibrium measure exponentially fast as $n$ tends to infinity.

Let G be a linear group with Kazhdan property (T) generated by d elements. We show that there are generating sets of size d+2 with arbitrarily small Kazhdan constants. We also prove the same conclusion for every finitely generated group admitting a non-elementary convergence action on a compact metrizable space. This includes non-elementary word-hyperbolic and relatively hyperbolic groups, as well as finitely generated non-elementary subgroups of word-hyperbolic groups.

Cao and Wang recently classified a class of locally strongly convex centroaffine hypersurfaces whose non-vanishing Tchebychev vector field $T$ satisfies $K(T,X)=\alpha X$ and $X(\alpha)=0$ for every $X\perp T$, together with $K(T,T)=\beta T$. In the degenerate case $\beta=2\alpha$, their higher-dimensional argument assumes in addition that the $T^\perp$-component of the difference tensor $K$ vanishes identically. We show that this extra assumption is unnecessary and the theorem can be strengthened.

Instead of the sequential plant–control–estimator design pipeline, which is in general not optimal, we propose the Control and Estimation Co-Design (ContEst) framework, which poses the entire system design as a single two-stage problem solved by first-order methods. A key enabler is that the gradient of each inner control/estimation optimal cost with respect to the design parameters is available from the dual variables the inner solver already returns, by the envelope theorem: exactly under mild regularity (a unique inner minimizer), and as a subgradient under more relaxed conditions. That is, no differentiation through the optimizer or its optimality conditions is needed. We first present the framework in a linear quadratic Gaussian (LQG) regime, where the control and estimation costs become two semidefinite programs (SDPs) whose design gradients are read from the duals of the Lyapunov constraints. We extend the results to robust ($H_\infty$) formulation to cover worst-case control and filtering problems. We also present an extension to constrained and nonlinear systems, using an extended Kalman filter (eKF) and a local linearization of the information-state dynamics to yield an approximate but convex model predictive control (MPC) problem whose design gradients are read directly from the costates. We apply our methods to a variety of realistic design examples from different fields and report significant design improvements.

Let $R$ be a ring and $M$ a left $R$-module. We prove that $M$ is an $\omega_1$-indexed inverse limit of injective modules with split epimorphic connecting maps, and $M$ is also an $\omega$-indexed inverse limit of a stationary tower defined by a split monic endomorphism of one injective module. These constructions answer two questions of Bergman. We then study the class of direct summands of $\omega$-indexed inverse limit of injective modules with epimorphic connecting maps. Over a commutative Noetherian ring $R$, they are precisely the Enochs cotorsion modules if and only if $\operatorname{gldim}R\leq1$ and $\operatorname{Spec}R$ is countable.

We study Verlinde sums $V_{n}(N,m)$, which for our purposes are finite sums of trigonometric functions $\{r_{x}\}$, associated to $\textrm{ SU}(N)$. In this article we prove the following results: (i) the polynomial $D_N(w):=\prod_x(w-r_x)$ is equal to, up to an explicit and computable factor, the iterated resultant of $Q(u)=u^m-1$ with an explicit elementary polynomial $W_N$ which is independent of $n$ and $m$; (ii) for fixed $N$ and $m$, $V_n(N,m)$ satisfies a linear recursion of order at most $\binom{m-1}{N-1}$, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of $D_{N}$; (iii) for fixed $N$ and $m$, a Binet-type closed form expressing $V_n(N,m)$ as an explicit finite sum of $n$-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of $D_N(w)$ is derived. Several explicit examples are provided for $N=2,3,4,5,6$ and $8$.

Building on the work of Coskun and Ross, we investigate which cohomology classes of the orthogonal Grassmannian $OG(k,n)$ can be realized as irreducible subvarieties of $OG(k,n)$. We give a criterion for when the realizable classes in the orthogonal Grassmannian agree with those in the Grassmannian. We give a characterization for realizability in low dimension and codimension and give necessary conditions for realizability in certain orthogonal Grassmannians.

A positive integer $n$ is said to be strongly pseudoperfect if there is a subset $S$ of the positive divisors of $n$ such that $d \in S$ if and only if $n/d \in S$, and $\sum_{d \in S} d = 2n$. We construct new infinite families of strongly pseudoperfect numbers, and also prove new restrictions on when a positive integer $n$ can be strongly pseudoperfect.

In DNA-based storage systems, data are retrieved by sequencing molecules sampled from a DNA pool. We study how the way coding redundancy is shared between files affects their expected retrieval times. Our focus lies on the case of two files that are encoded by a systematic linear code over an arbitrary finite field. We consider the conjecture that the sum obtained by dividing each file dimension by its expected retrieval time is at most one whenever the dimension of at least one file is more than one. For this, we develop a geometric view of the retrieval process. As molecules are sampled, we follow the growing span of the corresponding columns of the generator matrix and track how much of this span comes from each file. Among the samples that enlarge the overall span, this lets us compare those that make progress toward recovering both files with those that make progress toward neither. We call a column mixed if the corresponding encoded symbol combines information from both files. In this work, we sharpen a projection bound and use it to control the effect of mixed columns. We prove the conjecture whenever the total information dimension is at least twice the number of mixed columns plus two. For equal-sized files, this allows up to one fewer mixed column than the dimension of either file and extends the previous result for codes with no mixed columns.

In this paper, we consider Markov additive processes (MAPs) whose additive components are $\bR$-valued and have no positive jumps. Specifically, we assume that the modulators are Feller processes with only finitely many jumps on each bounded time interval and that the additive components also have only finitely many jumps on each bounded time interval. For such MAPs, we characterize the associated scale operators and use them to solve the two-sided exit problem, which concerns the first entrance time into one half-line on the event that it is reached before the other, and to characterize the potential measures of the processes killed when their additive components exit an interval. Our proofs rely on properties of analogues of local times, properties of the exit systems obtained by pairing them with suitable kernels, and results on the extension of $C_0$-semigroups to $C_0$-groups.

We present a high-order finite element arbitrary Lagrangian-Eulerian (ALE) method for hydrodynamics with slip-wall boundary conditions on analytically curved domains. The method combines distinct Lagrangian, remesh, and remap phases while preserving the geometric features of curved boundaries and allowing tangential motion of boundary nodes. In the Lagrangian phase, we use weak boundary condition enforcement to impose the slip-wall condition without freezing tangential motion. In the mesh optimization phase, we adapt TMOP (target-matrix optimization paradigm) to analytic walls by replacing selected boundary coordinate degrees of freedom with curve or surface parameters. The Newton solve then moves boundary nodes tangentially with respect to the prescribed geometry while carrying the drift introduced by the Lagrangian phase. In the remap phase, we transfer the solution from the old mesh to the optimized mesh by a conservative advection-based remap over pseudo-time. We present numerical results in two and three dimensions demonstrating the method on high-order curved meshes and showing that it preserves boundary fidelity while producing the expected hydrodynamic solution features.

We study the computational complexity of hypothesis testing in the spiked Wigner model, a prototypical model for detecting low-rank structure in a large random matrix. Below the "BBP" eigenvalue transition, it is expected that strong detection — with both type I and II errors vanishing — requires exponential time. Assuming this as a conjecture, we determine the limits of polynomial-time weak detection, exactly characterizing the possible tradeoffs between type I and II errors. Specifically, the optimal tradeoff is achieved by a particular linear spectral statistic. Thus, the question of weak detection is entirely reduced to that of strong detection. The proof builds on ideas of Nagda-Raghavendra (2025) and Moitra-Wein (2025). The low-degree likelihood ratio (LDLR) plays a key role: any test that slightly beats the LDLR can be boosted to have an even higher success probability. This leads us to establish a computational analogue of the Neyman-Pearson lemma for a subclass of additive Gaussian models: for a given super-polynomial runtime, the best possible tradeoff between type I and II errors is either the one achieved by thresholding the LDLR, or the trivial tradeoff that results from strong detection.

In a restricted untwisted sector, Diaconescu, Moore & Witten (2000, 2003) gave a quantum M-theoretic consistency check, at the level of partition functions, of what one may call "Hypothesis K", namely that magnetic and electric type IIA brane charges are jointly quantized in K-theory. We show that the more recently discussed "Hypothesis H" — that magnetic and electric charges in 11D are jointly quantized in coHomotopy — together with a topological smallness condition on the M-theory circle, systematically implies that IIA charges are quantized in a nonabelian deformation of twisted K-theory by the previously neglected non-linear NS1-brane charge. This is essentially a derivation of twisted K-theory from IR-completed 11D SuGra, and conversely a further consistency check on Hypothesis H from Hypothesis K. We motivate and precisely state this result, discuss its structural implications, and outline the elaborate algebro-topological proof.