Papers

A recent SICOMP paper on classical and quantum algorithms for the shortest vector problem introduced a lattice-dependent parameter \(\gamma(L)\), bounded universally in the exponential sense by \(2^{0.402n+o(n)}\), and conjectured that this parameter is \(2^{o(n)}\) for most lattices. We prove the Haar–Siegel random-lattice version in a stronger, dimension-uniform form. Let \(X_n=\operatorname{SL}_n(\R)/\operatorname{SL}_n(\Z)\), let \(\mu_n\) be its invariant probability measure, and let \(\gamma(L)=\sup_{r\ge1} N_L(r\lambda_1(L))/r^n\), where \(N_L(R)\) counts nonzero vectors of \(L\) of Euclidean norm at most \(R\). For every \(n\ge3\) and every \(T>0\), \[ \mu_n\{L\in X_n:\gamma(L)>T\}\le C T^{-1} \] with an absolute constant \(C\). Consequently, for every sequence \(a_n\to\infty\), \(\gamma(L_n)\le a_n\) with \(\mu_n\)-probability tending to one; in particular \(\gamma(L_n)=2^{o(n)}\) with high probability. In the product model of independent Haar–Siegel lattices, \(\gamma(L_n)\le \exp(\sqrt n)\) eventually almost surely. The proof uses Rogers's second-moment estimate only through a dyadic self-normalization argument around the random scale \(\lambda_1(L)\).

Guo, Li, Shangguan, Tamo, and Wootters proposed a hypergraph analogue of the Nash–Williams–Tutte theorem, asserting that every $k$-weakly-partition-connected hypergraph admits a $k$-distinguishable tree assignment. We identify a sharp edge-count obstruction to the literal statement. A full tree assignment has $\rho(H)=\sum_{e\in E(H)}(|e|-1)$ labelled graph edges, whereas an ordered decomposition into $k$ spanning trees has exactly $k(t-1)$ edges. Since weak partition connectivity implies only $\rho(H)\ge k(t-1)$, every strict inequality rules out a full decomposition. In particular, for all $t\ge2$, $k\ge1$, and $q\ge1$, the hypergraph consisting of $k+q$ labelled copies of the full hyperedge is $k$-weakly-partition-connected but admits no $k$-distinguishable full tree assignment. We therefore isolate the critical regime $\rho(H)=k(t-1)$ and prove that every tree assignment of a critical $k$-weakly-partition-connected hypergraph admits a full ordered decomposition into $k$ spanning trees. Consequently, the critical conjecture reduces to the existence of some tree assignment having a decomposition whose signature fiber is a singleton. We establish this property for layer-contained certificates, without a star hypothesis or a rank restriction in interior layers. These certificates also imply weak partition connectivity by a quotient-rank argument and are stable under one-vertex sums. Finally, we derive a decomposition-indexed generalized-Laplace expansion for the relevant intersection-matrix minor, together with an exact formula for its collected monomial coefficients; we also give the required row-and-column perfect-shuffle normalization, identify criticality with the square full-assignment row count, and separate the critical conjecture from the additional overfull pruning problem.

Prescribed trace/norm estimates and Soto-Andrade-type sums control whole fibers or related global character sums. We prove a coset-refined trace theorem for cubic norm-one tori. Let $B/\mathbb{F}_q$ be finite étale cubic, $\operatorname{char}\mathbb{F}_q\ne2,3$, and let $T_B=\ker(\operatorname{N}_{B/\mathbb{F}_q}:\operatorname{Res}_{B/\mathbb{F}_q}\mathbb{G}_m\to\mathbb{G}_m)$. For every subgroup $H\subset T_B(\mathbb{F}_q)$ of index $m$, every coset $gH$, every $\gamma\in B^\times$, and every smooth fiber $\operatorname{Tr}(\gamma h)=s$, $s^3\ne27\operatorname{N}(\gamma)$, we prove $N_{gH,B}(s;\gamma)=m^{-1}N_B(s,\operatorname{N}\gamma)+E_{gH,B}(s;\gamma)$, with $|E_{gH,B}(s;\gamma)|\le3(1-1/m)\sqrt q$. The geometric input is a Picard-Kummer kernel calculation: no nontrivial torus character becomes geometrically constant on a smooth trace/norm curve, so nontrivial coset character sums have square-root cancellation. On the nodal boundary $s^3=27\operatorname{N}(\gamma)$, the kernel degenerates exactly to a cyclic cubic Kummer kernel. Its Frobenius-fixed part is the sole source of order-$q$ bias; after removing that explicit projection, remaining characters again have square-root cancellation up to bounded normalization/node correction. The same geometry gives local branch theory for $\operatorname{Tr}_A(\gamma\eta^n)=c$ over finite étale cubic $\mathbb{Z}_p$-algebras, $p\ge5$. The logarithmic tangent and trace-dual codifferent coordinates identify singular branches: nondegenerate classes have quadratic Hensel models, while the genuinely affine degenerate class has a cubic first-obstruction model; in full norm-fiber orbits singular branch counting reduces to one cubic norm equation.

We study black-box auditing for machine learning algorithms that claim R \ 'enyi differential privacy (RDP) guarantees. We introduce an auditing framework, based on hypothesis testing, that directly estimates Rényi divergence between neighboring executions using the Donsker-Varadhan (DV) variational estimator. Our analysis yields explicit and non-asymptotic confidence intervals for RDP auditing via class-restricted DV estimators, separating statistical estimation error from algorithmic privacy leakage. We prove matching minimax lower bounds showing that, up to logarithmic factors, our sample-complexity guarantees are information-theoretically optimal, thereby establishing the first optimal guarantees for auditing RDP via DV estimators. Empirically, we instantiate our framework for auditing DP-SGD in a fully black-box setting. Across MNIST and CIFAR-10, and over a wide range of privacy regimes, our auditors produce a strong overall improvement on empirical RDP lower bounds compared to prior state-of-the-art black-box methods especially at small and moderate Rényi orders where accurate auditing is most challenging.

The global existence of classical solutions to the 3D compressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain was recently established by Feng, Hong, and Zhu [Sci. China Math., 2024]. While Lin and Xiang [Sci. China Math., 2020] established global well-posedness for the 2D incompressible counterpart, the global well-posedness for the 3D incompressible case remains open. The analysis is rendered difficult by three major obstacles which are further compounded in the 3D case: the degeneracy induced by the lack of magnetic diffusion and spin viscosity; the coupling between micro-rotation and velocity fields characterized by a non-dissipative anti-symmetric structure; and the interaction between the velocity, magnetic field, and pressure, where the pressure acts as a non-state variable. In this paper, by adapting the two-layer energy method of Guo and Tice [Arch. Ration. Mech. Anal., 2013] and the techniques employed in Feng et al., together with refined trace estimates, we overcome these difficulties and establish the global well-posedness of classical solutions to the 3D incompressible magneto-micropolar fluid system without resistivity and spin viscosity in a strip domain. Moreover, we demonstrate the algebraic time-decay of solutions toward the equilibrium.

We study fixed-cardinality maximization of the inverse-matrix Solow–Polasky diversity, equivalently finite metric magnitude for the exponential kernel, on one-dimensional and ordered metric sets. The analysis starts from the known finite-line gap formula for the exponential kernel, which writes the excess inverse-matrix diversity as a sum of functions of consecutive gaps. Building on this formula, the main interval theorem proves that, for every $k\geq 2$, the unique maximizing $k$-point subset of $[0,1]$ is the equally spaced set. Thus the objective selects a uniform gap representation on the real line. A converse kernel proposition shows that, among normalized non-increasing distance kernels, requiring the corresponding adjacent-gap additive structure forces the exponential family. Further results transfer the interval theorem to ordered $\ell_1$ (L1, or Manhattan) curves by isometry: the maximizing sets are uniform in accumulated $\ell_1$ length. As a consequence, monotone biobjective Pareto fronts admit Solow–Polasky optimal finite approximations that are uniformly spaced in accumulated objective-space change, a natural representation when all parts of a continuous front should be covered. Examples, including a dense connected front and a finite disconnected ZDT3 front, illustrate how the continuous uniform-gap result appears on discrete candidate sets. Solow-Polasky diversity; diversity measures; finite metric magnitude; L1 distance; uniform spacing; Pareto-front approximation; multiobjective optimization; fixed-cardinality subset selection

We study the problem of sampling weighted partial triangulations of a convex polygon with $n+2$ sides. We consider the distribution $\pi_{n,\lambda}$ under which each partial triangulation $\sigma$ is assigned probability proportional to $\lambda^{|\sigma|}$, where $\lambda>0$ is a model parameter and $|\sigma| \in \{0,\dots,n-1\}$ denotes the number of diagonals in $\sigma$. This model belongs to a broad class of weighted geometric partition problems that include lattice triangulations and dyadic tilings, and is closely related to several classical combinatorial structures, including the full triangulations of a convex polygon and the associated Catalan structures. Our main result is a simple exact sampling algorithm for $\pi_{n,\lambda}$ with expected running time $O\big((\min\{n,n\sqrt{\lambda}\}+1)\log n\big)$, which is optimal up to the logarithmic factor.

For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$.

Multiproposal MCMC (MP-MCMC) algorithms use clouds of proposals to efficiently traverse state spaces and overcome complex target geometries. While MCMC methods are embarrassingly parallel by nature, the non-trivial forms of parallelism provided by the MP-MCMC formalism sometimes leads to significant improvements over a naive approach. Here, one important tuning parameter is the number of proposals p used by a single MP-MCMC iteration. While a number of computational strategies have been proposed to efficiently leverage large numbers of proposals within the MP-MCMC paradigm, much remains unknown about these algorithms, particularly in the large p-regime. In this contribution, we discover surprising results by identifying and studying several promising new methods (Algorithm 1.1, Algorithm 3.3, Algorithm 3.4), ruling out other extant approaches and discovering new relationships between different MP-MCMC methodologies. Our analysis is centered on a general state space multiproposal involutive theory recently constructed by the authors combined with the consideration of the large p-limit kernels for MP-MCMC algorithms within a variety of different classes of proposal and acceptance structures.

We study the Lawn Mowing Problem restricted to periodic billiard paths in the unit square. Given the combinatorial data of a trajectory, we determine the optimal covering radius, and identify the shortest path that covers the square for any fixed blade radius.

We demonstrate the existence of quasiconformal mappings on closed manifolds (dimension at least 5) that cannot be decomposed as a composition of mappings with arbitrarily small conformal distortion. This answers in the negative the global version of a classical question of Gehring.

Orbital debris is a pressing problem which presents a danger to global space operations and a barrier to continued development of the space economy and space infrastructure. As research continues regarding orbital debris, there is a need for tools to understand the system-level implications of orbital debris solutions. This research considers the orbital debris problem as a dynamic process. Based on dynamic system theories, time-series variables of the numbers of orbital debris, orbital objects, and object launches should be causally linked, which means they share a common system attractor manifold. We propose a data-driven method based on complexity science to reconstruct a shadow attractor of the dynamic system using limited observable variables. The reconstructed shadow attractor helps us to understand the fundamental system dynamics for orbital debris and enables us to simulate the future of the orbital debris system based on changes to policy. These findings represent a significant advancement in our ability to understand high level impacts of space system policy with limited data available.

In this work we studied a generalized Fisher equation in cylindrical coordinate using Lie symmetry method. We have determined for what type of source function the generalized Fisher equation has Lie Symmetries other than time translation symmetry when the diffusion function is given by an exponential function. Also the reduced ordinary differential equations are obtained corresponding to Lie symmetries of the generalized Fisher equation.

We address the optimal control problems arising from partial differential equations with large discrete dimensional control systems. To obtain reduced order models, we find basis elements from the canonical polyadic (CP) decomposition. Tensor datasets are from snapshots of the large models. Our method to reduce the control system is to use dimensionality reduction approaches through sparse optimization and flexible hybrid methods is to obtain low rank CP tensor basis elements. The reduced optimal control problem leads to reduced state-dependent Riccati Equations which can be solved efficiently.

A solution of a Riemann problem for a strictly hyperbolic system of conservation laws is traditionally expected to consist of rarefaction waves, jump discontinuities, and constant states. In this paper, we investigate whether a Riemann solution has this structure when the solution is only assumed to be measurable and essentially bounded. To discriminate continuous and discontinuous features in an $L^\infty$ solution, we introduce one-sided accumulation sets based on local essential images. Supposing that throughout a bounded open interval a solution is continuous in the essential image (ess-im) sense, we prove that it is a rarefaction wave if it is resonant (the characteristic speed equals $x/t$), and otherwise it is constant. Although an ess-im discontinuity might not be a jump discontinuity, we show that all ess-im accumulation states lie on a common Hugoniot locus and have the same speed. Anomalies are possible if there are limit points of ess-im discontinuities, but if the set of ess-im discontinuities is finite, then an $L^\infty$ Riemann solution has bounded variation and is composed of finitely many rarefaction waves, jump discontinuities, and constant states.

The stability number of a forbidden family measures how many different structures are needed to approximate all near-extremal constructions avoiding it. An infinite stability number means that no finite list of structures suffices. We construct a simple explicit $3$-graph whose stability number is infinite. This extends the infinite-stability phenomenon for finite forbidden families, established by Hou–Li–Liu–Mubayi–Zhang, to the single-forbidden setting, and further develops the single-$3$-graph direction of Balogh–Clemen–Luo, in which exponentially many exact extremal constructions coexist with stability.

We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at representations of integers as the sum of primes and reversed primes. We show that: (1) Every large odd integer is the sum of a prime and two reversed primes ($N=p_1+\overleftarrow{p_2}+\overleftarrow{p_3}$). (2) Every large odd integer is the sum of two primes and a reversed prime ($N=p_1+p_2+\overleftarrow{p_3}$). (3) Almost all even integers are the sum of a prime and a reversed prime ($N=p_1+\overleftarrow{p_2}$). (4) All large integers are the sum of a reversed prime and a square-free number ($N=\overleftarrow{p}+\eta$, $\mu^2(\eta)=1$). To obtain our results, along with associated asymptotics, we apply the Hardy–Littlewood circle method and a novel refinement of the “Zsiflaw–Legeis" theorem on the distribution of reversed primes in arithmetic progressions. Notably, our variant of the Zsiflaw–Legeis theorem does not require one to fix the digit length, unlike previous versions.

We prove, for any W$^*$-probability space $(M,\varphi)$ where $M$ is a type $\mathrm{III}_1$ factor, any nontrivial, proper closed $F\subseteq \mathbb{R}$, and any nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, that the ultrapower $M(\sigma^\varphi,F)^{\mathcal{U}}$ of the spectral subspace $M(\sigma^\varphi,F)$ is a proper subset of the spectral subspace $M^{\mathcal{U}}(\sigma^{\varphi^{\mathcal{U}}},F)$. We discuss the model-theoretic implications of this result.

We prove that there are globally hyperbolic spacetimes which are refocusing but not strongly refocusing. In fact, every globally hyperbolic strongly refocusing spacetime of dimension at least $3$ admits globally hyperbolic metrics which are refocusing but not strongly refocusing. This answers a question by Chernov, Kinlaw, and Sadykov. We then prove that globally hyperbolic spacetimes which are Legendrian refocusing (a notion introduced in this paper) admit globally hyperbolic strongly refocusing metrics. As a corollary, a contact Bott-Samelson type result by Frauenfelder, Labrousse, and Schlenk can be applied to Legendrian refocusing spacetimes to show that the Cauchy surface of a globally hyperbolic Legendrian refocusing spacetime of dimension at least $3$ is compact, that its fundamental group is finite, and that its universal cover has the integral cohomology ring of a compact rank one symmetric space (CROSS).

Let $q$ be an odd prime power, let $n\ge 2$, and let $V\subsetneq \mathbb F_{q^n}$ be a proper $\mathbb F_q$-vector subspace. Given a nonzero quadratic form $Q(X,Y)\in \mathbb F_{q^n}[X,Y]$, we consider the graph $\Gamma(Q,V)$ that naturally arises from the condition $Q(X,Y)\in V$. We determine all quadratic forms $Q$ for which $\Gamma(Q,V)$ is undirected for every $V$. Besides the case $Q(x,y)=XY$, studied earlier by the second author, this essentially leads to the forms $X^2\pm Y^2$ and the family $Q_b(X, Y):=X^2+bXY+Y^2, b\ne 0$. We then study connectedness and clique number for the corresponding graphs. Our results reveal a clear contrast between these cases. The graphs $\Gamma(X^2\pm Y^2, V)$ are well structured, disconnected and their clique number can be as large as $\# V$. On the other hand, the family $Q_b$ seems to yield less structured graphs: the graphs are connected (in fact, of diameter $2$) if $\# V\ge q^{3n/4}$ and, in many cases, their clique number is $o(\# V)$. Our proofs are mainly based on character sums, while requiring a few algebraic and combinatorial ideas. We end the paper with some open problems and remarks, including a short discussion of the complementary case where $q$ is even.

We introduce a new measure of robustness for statistical estimators, which we call empirical sensitivity. An estimator $\hat \theta$ has bounded empirical sensitivity if, with high probability over a dataset $X = (X_1, \dots, X_n) \sim \mathcal{D}^{\otimes n}$, for any dataset $Y$ obtained by modifying at most $\eta n$ points in $X$, we have that $\hat \theta(Y)$ is close to $\hat \theta(X)$. We study bounds on this quantity for the prototypical problem of Gaussian mean estimation. We prove new lower bounds, showing that for any estimator $\hat \mu$ which achieves an optimal $\ell_2$-error bound of $O\left(\sqrt{d/n}\right)$, the empirical sensitivity is at least $\Omega\left(\eta + \sqrt{\eta d/n}\right)$. The two terms arise due to obstructions on the mean and variance (via an Efron-Stein argument) of such an estimator. We show that this bound is tight up to logarithmic factors, by employing recent results for robust empirical mean estimation.

Reinforced random walks are random walks on graphs whose transition probabilities along edges from a vertex are proportional to the weights of those edges, but where the weight of an edge evolves in a way that depends on the past traversals across it. In an edge-reinforced random walk (ERRW), the weight of an edge increases by $1$ whenever that edge is traversed, in either direction. On a finite graph, an ERRW admits a remarkable representation as a random walk in a random environment. The law of the environment is given by the so-called magic formula, with this law depending on the initial edge weights. This representation provides a natural route for studying statistical properties of ERRWs. This work focuses on various information-theoretic quantities associated with ERRWs on finite graphs, motivated in part by the problem of statistically distinguishing between different ERRW models from observed trajectories. In particular, we study the entropy rate of an ERRW. We also study the Kullback–Leibler divergence (KL divergence) between two ERRW environment laws, and the KL divergence between the corresponding finite-trajectory distributions. Leveraging structural properties of the underlying random environment, we derive an annealed representation of the entropy rate, a closed-form formula for the environment-level KL divergence, and quantitative bounds on the convergence of trajectory-level KL divergence toward environment-level KL divergence. These information-theoretic quantities are motivated by the two-point hypothesis testing problem for ERRW trajectories, and in particular by the associated Stein exponent. We also expect them to play a fundamental role in the study of other testing problems for ERRWs, including identity testing and closeness testing.

We analyze the Jacobian of the logit mapping for stochastic user equilibrium (SUE) and use it to develop two improved algorithms for path-based SUE. We show that the Jacobian decomposes into two matrices: one that annihilates differences of feasible path flow vectors, and another whose eigenvalues are all non-positive reals, provided link costs are monotone non-decreasing and separable. Using these properties, we first show that the method of successive averages (MSA) with a small constant step-size $s$ converges linearly at a rate $1-s$, with the largest admissible step-size depending on the eigenvalues of the Jacobian of the logit mapping. Building on this result, we develop an adaptive constant step-size rule that retains the global convergence of MSA while achieving asymptotic linear convergence. Our second algorithm is a Newton-based method using a reformulation of SUE as a root-finding problem. Unlike gradient-projection approaches that operate on the Hessian of the SUE objective function (a dense matrix), our method exploits the structure of the Jacobian of the logit mapping, making computations tractable and removing the need for manifold optimization. Numerical experiments show superlinear convergence on most tested networks, with our methods outperforming existing approaches on large networks or when demand is high. To our knowledge, this article is the first to report runtimes for logit-based SUE on networks as large as Chicago Regional and Philadelphia, providing a benchmark for future algorithmic development.

A Golomb ruler of order~$n$ is an integer set $\{a_0<a_1<\cdots<a_{n-1}\}$ whose $\binom{n}{2}$ pairwise differences $a_j-a_i$ ($i<j$) are all distinct. The optimal Golomb ruler problem asks for $\mathrm{OGR}(n)=\min\{a_{n-1}-a_0\}$ and is a classical combinatorial benchmark. The values $\mathrm{OGR}(2),\dots,\mathrm{OGR}(28)$ are settled through distributed volunteer search (the Distributed.net OGR project); $\mathrm{OGR}(29)$ is in active computation, with verification expected in late 2026 or early 2027. The recent upper bound $\mathrm{OGR}(29)\le 757$ of Lee, Park, and Kim (arXiv:2510.0122, October~2025) tightens the search window. This note describes a compact CNF encoding of the decision problem $\mathrm{GR}(n,L)$ (“is there a Golomb ruler of order~$n$ with length exactly~$L$?”) with $O(n^2 L)$ clauses, together with a small-case verification sweep that emits machine-checkable LRAT certificates of optimality for $\mathrm{OGR}(n)$ at $n\le 12$. We then give closed-form encoding-size estimates for $\mathrm{OGR}(29, L=757)$ and propose a cube-and-conquer decomposition aimed at a Mallob-style parallel run on commodity multi-core hardware. The certificate pipeline (CaDiCaL with –lrat=true, a structural sanity-check, then formal validation by drat-trim or cake\_lpr) is end-to-end. A formally verified optimality proof for any single $\mathrm{OGR}(n)$ value beyond the trivial $n\le 5$ would be a first in the field.

This study focuses on dead-core solutions for a two-dimensional isothermal reaction-diffusion problem involving a single chemical reaction described by power-law kinetics within a bounded reactor domain. The steady-state nonlinear diffusion-reaction boundary value problem is solved numerically by applying a suitable time-stepping technique. The lumped finite element technique with piecewise linear shape functions is utilized to discretize the spatial domain. Dead zones, defined as regions inside a reactor where no reaction occurs because the reactant has been completely consumed, present a major difficulty in the design and operation of chemical reactors. The impact of the reaction exponent and the Thiele modulus on the concentration fields and on the size of such inactive regions is numerically investigated. The findings indicate that particular reactor shapes can lead to the formation of several separate dead zones.

The eigenfunctions of the Laplacian are a natural basis of functions for many tasks in computational mathematics. On the circle and sphere, the eigenfunctions are given by complex periodic exponentials and spherical harmonics, respectively, and much work has been done to develop fast algorithms for analyzing and synthesizing data in these bases. In this work, we generalize these special-case transforms to Laplace-Beltrami eigenfunctions of arbitrary surfaces, referred to as manifold harmonics. The resulting fast algorithm for computing linear combinations of the manifold harmonics is based on a butterfly factorization, which hierarchically compresses the transform matrix by constructing nested low-rank approximations of carefully selected submatrices. Several numerical examples are provided which demonstrate the speedups and reduction in memory requirements achieved by our algorithm for a variety of geometries, discretizations, and applications. In addition, a detailed analysis of the algorithm is given in the case that the underlying manifold is the flat periodic square.

Given a positive integer $k$ and graph $G$, the $\mathbb{Z}_k$-Ramsey number $R(G,\mathbb{Z}_k)$ is the least $N$ (if it exists) such that every coloring $f:E(K_N)\rightarrow \mathbb{Z}_k$ contains a copy $G'$ of $G$ such that $\sum_{e\in E(G')}f(e)=0$. Motivated by a question of Caro and Mifsud, we study the $\mathbb{Z}_k$-Ramsey number of graphs with a sufficiently large 2-packing, i.e. a set of vertices $S\subseteq V(G)$ such that $N[u]\cap N[v]=\emptyset$ for all distinct $u,v\in S$. In particular, we prove that $R(G,\mathbb{Z}_p)\leq n+6p-9$ for all $n$-vertex graphs $G$ and all primes $p$ such that $p$ divides $e(G)$, the minimum degree of $G$ is at least $1$, and there exists a $2$-packing of $G$ with size $p-1$. This upper bound improves depending on vertex degrees in the $2$-packing, with equality in certain cases. The result also implies an upper bound of the form $R(G,\mathbb{Z}_p)\leq n+C$ for $n$-vertex graphs $G$ of bounded maximum degree.

Near-field beamfocusing enabled by extremely large-aperture arrays (ELAA) is a promising 6G technique for massive connectivity and high spectrum efficiency. While beamfocusing concentrates energy at an intended user, the radiated field outside the focal point exhibits a structured leakage that varies with the focal-point coordinates. This paper shows that this leakage enables a new form of passive user localization in which distributed far-field sensors measuring only received power can infer the user's location by exploiting this location-dependent power signature. Using the induced noncentral chi-square statistics, we derive a Bayesian Cramér-Rao lower bound (BCRLB) that establishes the fundamental limits of this inference problem. We then evaluate a model-based grid-search estimator and an attention-based permutation-invariant deep learning regressor (DeepSet). Results under both line-of-sight (LoS) and multipath propagation confirm that reliable location inference is feasible, with accuracy improving as more sensors and snapshots are used.

The classical Gleason–Kahane–Zelazko theorem characterizes multiplicative linear functionals on a unital Banach algebra through the scalar identity $\Lambda(x^{2})=\Lambda(x)^{2}$. We develop analogues of this theorem for bounded linear functionals on reproducing kernel Hilbert spaces of holomorphic functions on domains in $\mathbb{C}^{d}$, replacing conditions on the whole space by tractable conditions involving only kernel functions. Our first main result shows that if $k$ is a diagonal holomorphic kernel on a domain $\Omega\subseteq\mathbb{C}^{d}$ containing the origin, and if $k_{w}^{2}\in\mathcal H(k)$ for every $w\in\Omega$, then a bounded linear functional $\Lambda$ on $\mathcal H(k)$ satisfying $\Lambda(1)=1$ is multiplicative if and only if $\Lambda(k_{w}^{2})=\Lambda(k_{w})^{2}$ for all $w\in\Omega$. Kernels satisfying $2$-point Pick property and their powers furnish a natural class of examples. When $k$ arises as a Schur product or a tensor product of complete diagonal Pick kernels, we obtain a further, more algebraic characterization of multiplicativity, expressed in terms of the values of $\Lambda$ on kernel functions and their reciprocals. This framework subsumes the weighted Bergman and Dirichlet-type spaces, as well as the Hardy space on the polydisc. We extend the analysis to Hilbert spaces associated with diagonal Dirichlet series kernels on half-planes, encompassing in particular the Hardy space of Dirichlet series and its Riemann zeta reproducing kernel. Explicit examples demonstrate that the boundedness hypothesis on $\Lambda$ cannot be omitted. Finally, our characterization of multiplicative linear functionals leads to characterizations of weighted composition operators on a reproducing kernel Hilbert space associated with a diagonal holomorphic kernel.

We investigate the emergence of synchronization in a network of coupled chaotic macroeconomic systems. Each node represents an economy characterized by three key variables savings, gross domestic product (GDP), and foreign capital inflows. These economies interact or are connected through a fitness-based probability that depends on the potential GDP of each node. This formulation allows both structural heterogeneity, arising from uneven network connectivity, and dynamical heterogeneity, due to differences in local parameters, to be explored within a unified framework. Using both numerical simulations and a mean-field approximation, by varying the coupling strength and the degree of heterogeneity of both network topology and dynamical behavior of the nodes, we analyze synchronization transitions. Our results show that the mean-field approach accurately captures the collective dynamics in homogeneous and fully connected networks even with heterogeneity within the intrinsic dynamic of the nodes but fails when strong heterogeneity in the structure of the network is introduced. In heterogeneous networks, the system exhibits partial synchronization and on–off intermittency, where coherent phases of global synchronization alternate with abrupt desynchronization bursts. The distribution of laminar phase durations follows a power-law scaling, consistent with theoretical predictions for intermittent synchronization. From an economic perspective, these results suggest that global business cycle synchronization is inherently fragile: strong integration can promote temporary coordination among economies, but structural and dynamical disparities inevitably lead to intermittent breakdowns of collective behavior.