Papers

In this article, we introduce and study a set of generalized Gaussian beam Radon transforms (GbRt) acting on tensor fields in $\mathbb{R}^2$. The operators considered include longitudinal, transverse, mixed GbRts, along with their integral moments. These operators extend the corresponding notions of the classical generalized Radon transforms for tensor fields. We establish reconstruction results for vector and symmetric 2-tensor fields using appropriate combinations of the defined transforms. This work extends a recent study on the recovery of scalar functions from their GbRt to the recovery of vector and tensor fields from analogously defined generalized GbRts.

We study the smallest singular value of nonuniform Fourier matrices in two settings: clustered nodes and perturbations of an equispaced grid. By reducing the problem to spectral norm estimates for periodic nonuniform interpolation matrices, we obtain nearly optimal bounds in both cases. For clustered nodes, we derive the first local separation condition in which each required gap depends only on the sizes of the two neighboring clusters. For perturbations with the bound \(1/4\leq L<1/2\), our result confirms the conjecture of Austin and Trefethen on the \(2\)-norm Lebesgue constant up to a logarithmic factor.

The dual $L_p$ John ellipsoids form a family that includes the origin-centered Löwner and Legendre ellipsoids. In this paper, we extend this family to general measures within the dual weighted $L_p$ Brunn–Minkowski theory. For measures with locally integrable densities that are continuous in a neighborhood of the boundary of the given star body, we establish a variational formula for $L_p$-harmonic radial combinations. The associated boundary integral leads to an optimization problem that remains meaningful under weaker assumptions on the boundary trace of the density. We prove the existence, uniqueness, and characterization of its solution. The resulting ellipsoids include the dual $L_p$ John ellipsoids for Gaussian measures. We also establish a monotonicity inequality, a measure inequality for homogeneous densities with its equality conditions on the weighted support, and an $L_p$ Löwner inclusion.

In this note we show that the image of the third Johnson homomorphism $\tau_3$ coincides with the kernel of Morita's trace map for the case of a surface of genus $g$ with one boundary component $\Sigma_{g,1}$, when $g \geq 6$. Moreover, as a consequence, we get that the image of $\tau_3$ on the handlebody subgroups coincides with the intersection of kernels of the Morita trace map and the Lagrangian trace maps.

Blackwells Bet (also known as the Cover Guessing Game) is an extremely clever use of a random variable to improve the probability of guessing which of two numbers is larger when you are only allowed to observe one of those numbers. This technique has hitherto been used to guess or predict outcomes with better-than-chance results. We show that this technique can also be used to construct subsets of unobserved items which have expected values below and above the expected value of the parent set. We discuss this in two situations: when the items in question can have arbitrary values, and when the items in question are Bernoulli trials. We also include simulations in an instance when the actual formulas require extensive.iteration of recursively-defined relations.

In 2002, Lins Neto introduced three remarkable one-parameter families of holomorphic foliations by curves on the complex projective plane, of degrees two, three, and four, whose properties established that the general form of the Poincaré Problem had no solution. These foliations are known to be birationally equivalent to certain quotients of linear foliations on abelian surfaces. We give explicit formulas for these birational equivalences, obtaining, in particular, parametrizations of the leaves of the foliations. For the families of degrees three and four, we determine explicit generators for the groups of birational transformations of the projective plane preserving them. For foliations in these families admitting a rational first integral, that is, for those whose parameter is an Eisenstein rational, we give a complete description of the nature and position of the singular points of a generic integral curve, and we present an algorithm that computes the rational first integral explicitly. Lins Neto's foliations can also be defined over algebraically closed fields of positive characteristic, and, in this setting, we characterize those that are algebraically integrable. Finally, we study the integrability of the reductions to fields of positive characteristic of some of the non-integrable foliations in the complex family.

We say that a finite group $G$ *forces* a finite group $H$ if every finite cover of $G$ contains a subgroup isomorphic to $H$, and we say $H$ is *forcible* if some finite group $G$ forces $H$. Complementing a classical result of Thompson–Mann, and answering a recent question of the third author, we show that a finite group is forcible if and only if it is abelian and its Sylow subgroups are elementary-by-cyclic. We also prove relative forcibility results for abelian $p$-groups in the settings of powerful $p$-groups and $p$-groups of bounded nilpotency class.

In this paper, we establish a necessary and sufficient criterion for a finite-dimensional semisimple algebra over an algebraically closed field of characteristic $0$ to admit a regular quantum commutative decomposition. We apply this characterization to group algebras and show that a finite group algebra admits such a decomposition if and only if the underlying group is a peak group, that is, a finite group whose irreducible character degrees have a greatest element with respect to divisibility. We investigate the structure of peak groups and establish several criteria for their solvability in terms of the prime divisors of their largest irreducible character degree. In particular, we show that several important classes of groups are peak groups, while supersolvability alone does not imply the peak property. Finally, we show how to construct a peak group from any finite group.

We study a restricted form of Gross's finiteness problem for algebraic minimal Calabi–Yau threefolds: those fibred by quartic K3 surfaces and realised as anticanonical hypersurfaces in stacky scrolls $\PP^1\times[\PP^3/\ZZ_n]$. Beyond the ten straight-scroll families of \cite[Table~1]{MboyaSzendroi2023}, we introduce orbifold scrolls $(\PP^1\times\PP^3)/\ZZ_n$ and apply the Reid–Shepherd-Barron–Tai criterion to determine which admit canonical anticanonical hypersurfaces. Only finitely many weight vectors arise for each $n$; we classify $n=1,2,3$ completely and compute all Hodge numbers, giving fourteen deformation families in total. Engel, Filipazzi, Greer, Mauri and Svaldi \cite{EngelFilipazziGreerMauriSvaldi2025} have since established boundedness for fibred Calabi–Yau threefolds in general, settling the existence question this restricted case exemplifies; what remains, and what we supply here, is the explicit classification: weight vectors, singularity types, and Hodge data.

This paper studies exact controllability and L2-stabilizability of discrete-time backward-structured stochastic linear systems. For each prescribed finite horizon, equivalent characterizations of exact controllability are established in terms of the controllability Gramian, a Kalman-type rank condition, and the reachable subspace. Exact null controllability without a prescribed horizon is then characterized by a finite-dimensional rank condition, and a Hautus-type criterion is derived using positive semidefinite eigenmatrices of a positive operator. For L2-stabilizability, we establish a stabilizability decomposition and obtain a corresponding Hautus-type spectral criterion. As a byproduct, we show that exact null controllability implies L2-stabilizability. These results provide algebraic and spectral tests for the controllability and stabilizability of the considered systems.

We study interacting finite-color urn schemes on directed acyclic graphs, allowing the graph to be infinite. Each urn evolves through reinforcements driven by colors drawn from its in-neighboring urns via edge-dependent reinforcement matrices. Assuming that every vertex has only finitely many ancestors, we prove almost sure convergence of urn proportions and show that the limiting configuration is determined by vertices with no ancestors or self-loops. Under additional balance and irreducibility assumptions on reinforcement matrices, we also obtain second-order asymptotic results in all regimes of appropriately defined parameters.

In this paper, we propose and analyze a diffusion time-changed susceptible-infected-susceptible (SIS) epidemic model driven by time-changed Brownian motion. We prove that the proposed model admits a unique global positive solution for any initial value in $(0,N)$. The extinction and persistence of the disease are then investigated. To approximate the diffusion time-changed SIS model, we construct a positivity-preserving logarithmic Euler-Maruyama (LEM) method. Assuming that the time-changed is given by the inverse of a standard $\alpha$-stable subordinator with $\alpha\in(0,1)$, we prove that the numerical solution converges strongly to the exact solution with order $\alpha$. Finally, numerical experiments are provided to confirm the predicted convergence rates and illustrate the positivity-preserving property of the proposed method.

Let $A$ be the centralizer algebra of a matrix over an arbitrary field. We solve the fixed-source realization problem for matrix centralizers by proving that the matrix centralizer algebras derived equivalent to $A$ are precisely the opposite endomorphism algebras of tilting modules over $A$. We classify the basic tilting modules and determine their opposite endomorphism algebras. The tilting poset is a product of right weak orders on symmetric groups, with one factor for each primary block and degree equal to the number of distinct exponents in that block. Together with the center, this poset recovers the multiset of these numbers across all primary blocks, although it does not canonically match them with the local center factors. For each primary block, the target algebras are obtained by permuting the successive gaps between exponents, and their isomorphism classes are determined by the stabilizer of the gap word. Consequently, the quotient of the labeled mutation graph by target-algebra isomorphism is a Schreier multigraph. We also characterize when the weak-order orientation descends to its nonloop edges.

To address the overdispersion problem in matrix-variate integer-valued time series data arising in sociology, medicine, and related fields, this paper proposes a matrix integer-valued autoregressive model based on the negative binomial thinning operator. By introducing left and right matricial negative binomial thinning operators, the proposed model not only preserves the integer-valued nature of the data but also effectively handles overdispersion. The probabilistic and statistical properties of the proposed model are systematically investigated. Two estimation methods, namely projection estimation and iterative conditional least squares estimation are developed, and the corresponding asymptotic theories are established. Simulation studies provide concrete numerical results to evaluate the finite-sample performance of the estimators. Real data analysis demonstrates that the proposed model outperforms both the continuous matrix autoregressive model and the multivariate integer-valued autoregressive model in fitting matrix-variate integer-valued time series, while also proving effective in accommodating overdispersed count data.

We classify the homotheties of the complete Cheng–Yau Kähler–Einstein metrics on the two-dimensional Thullen domains $M(\mu)$, $\mu>0$, that admit Kähler immersions into $\mathbb{CH}^{\infty}$. With $\Ric(g_{\mathrm{CY}})=-3g_{\mathrm{CY}}$, an immersion of $c\,g_{\mathrm{CY}}$ exists exactly when $0<\mu\leq\mu_+$ and $0<c\leq3\min\{1,\mu\}/(\mu+2)$. The critical parameter $\mu_+>1$ is characterized by an explicit equation, and its endpoint is included. Every admissible metric has a proper global Kähler embedding. The same parameter interval classifies immersibility into flat and projective Hilbert space, where every positive homothety admits a proper global embedding. Beyond the wall, all three targets are excluded at every positive scale. The proof combines an algebraic normalization of the Cheng–Yau equation, all-order coefficient estimates, and complex continuation of a scalar flow. For $\mu\neq1$, the resulting complete nonhomogeneous examples disprove proposed Kähler–Einstein rigidity conjectures in the complex-hyperbolic and flat Hilbert settings. We also prove a complementary rigidity theorem for Bergman–Hartogs domains and construct complete $\eta$-Einstein Sasakian manifolds with nonhomogeneous Riemannian metrics and proper global Sasakian embeddings into the infinite-dimensional Heisenberg space form.

In this paper, we consider a time-periodic nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with saturated incidence and Neumann boundary conditions in a spatiotemporally heterogeneous environment. First, we define the basic reproduction number for the model, which depends on dispersal rates, total population size and saturation parameters, and is quite different from those of models incorporating standard or bilinear incidence. Then, we establish its variational characterization and investigate the impacts of those parameters on it. Next, we explore the existence, uniqueness and global attractivity of the equilibria. We also analyze the asymptotic behaviors of endemic steady states for small saturation parameters, and for both small and large diffusion rates. Our results show that the saturation effect enables the total population size to have a significant impact on the disease dynamics. Furthermore, as the saturation parameter tends to zero, the basic reproduction number and the endemic equilibrium reduce to those of the standard-incidence model, respectively. Finally, we support our findings by numerical simulations.

The paper studies the ruin probability for an insurer's surplus process when claim arrivals are governed by a Multiply Iterated Poisson Process (MIPP). This framework generalizes the classical Cramér-Lundberg model by permitting claims to arrive in clusters, which makes it well suited for representing catastrophic insurance events. The main result is the development of explicit conditions for ultimate ruin, demonstrating that the ruin probability is driven chiefly by the jump intensity and the number of iterations. In addition, we derive integro-differential equations for both survival and ruin probabilities and obtain their Laplace transforms. These transforms are connected through recursive relations that tie ruin probabilities at successive iteration levels. We further propose a Cramér-Lundberg-type approximation, yielding asymptotic formulas for ruin probabilities together with a Lundberg-style bound. Finally, numerical experiments - along with comparisons to the classical model - are included to support the theory and to illustrate how claim clustering affects an insurer's likelihood of ruin.

In this paper, we give a partial answer to Brezis' Open Problem~2.1, which concerns the uniqueness of solutions to the Ginzburg–Landau equation in the unit disc with the degree-one boundary condition. Let $\lambda_1$ be the first Dirichlet eigenvalue of $-\Delta$ in the unit disc and set $\varepsilon_*:=\lambda_1^{-1/2}$. We prove that there exists $\delta>0$ such that the radial solution is the unique weak solution for every $\varepsilon\in(\varepsilon_*-\delta,\infty)$. More generally, we establish the analogous uniqueness result for the Ginzburg–Landau system on bounded connected $C^{1,1}$ domains in $\mathbb R^N$, $2\leq N\leq4$, with nontrivial boundary data. In particular, uniqueness persists slightly below the convexity threshold $\varepsilon_*$, where the strict convexity argument is no longer available. The proof combines strict convexity for $\varepsilon\geq\varepsilon_*$ with a compactness argument, nondegeneracy of the solution at $\varepsilon=\varepsilon_*$ and the implicit function theorem.

We describe injective endomorphisms of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^4}$ with a four-element family $\mathscr{F}^4$ of inductive non-empty subsets of $\omega$. In particular we proved that every injective monoid endomorphism of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}^4}$ is the identity transformation. Also we describe all (not necessary monoid) injective endomorphism of $\boldsymbol{B}_{\omega}^{\mathscr{F}^4}$.

We construct surface diffusion flows $f:\mathbb{T}^2\times[0,T)\to\mathbb{R}^3$ that drive smooth closed embedded tori to pinchoff in finite time. The flow remains embedded for $t\in[0,T)$ and develops a curvature singularity only at a distinguished point $p$ as $t\nearrow T$. Away from $(p,T)$ the flow converges smoothly as $t\nearrow T$. We characterise the singularity profile: If $A(t)$ denotes the radius of the waist, and the surface is given locally near the waist by the rotation of a radial graph $(z,t)\mapsto r(z,t)$, then there exists a constant $\mu>0$ and smooth function $U:\mathbb R\to\mathbb R$ such that \[ A(t)=\{4\mu(T-t)\}^{1/4}(1+o(1)), \qquad A(t)^{-1}r(A(t)\zeta,t)\stackrel{C^\infty_{loc}}{\longrightarrow} U(\zeta). \] Here $U$ is a rigorous realisation of the classical fundamental positive even conical similarity profile first computed numerically by Wong, Miksis, Voorhees and Davis and subsequently analysed by Bernoff, Bertozzi and Witelski.

We introduce the centered weak discrete Riemannian gradient (c-wDRG) framework for the unified analysis of optimization methods on Riemannian manifolds. The framework uses a center point to represent the relevant logarithmic differences in a common tangent space and covers Riemannian steepest descent, proximal point, proximal gradient, implicit midpoint, geodesic average-vector-field, Gonzalez, and Itoh–Abe methods. We derive c-wDRG certificates for these schemes and establish convergence bounds with explicit curvature-dependent step-size conditions. Under suitable c-wDRG parameter and step-size conditions, nonaccelerated schemes achieve \(O(k^{-1})\) or linear convergence of the objective gap. We further develop accelerated c-wDRG schemes with \(O(k^{-2})\) or accelerated linear convergence under the respective parameter conditions. Numerical experiments support the theoretical convergence results.

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,\ldots,v_n\}$, and let $\omega:V(G)\to \mathbb R^+$ be a positive vertex-weight function satisfying $\omega(v_i)=x_i$ for each $v_i \in V(G)$. The weighted Kirchhoff index of $G$ is defined by $K(G;x_1,x_2,\ldots,x_n)=\sum_{1\le i<j\le n}x_i x_j r_G(v_i,v_j)$, where $r_G(v_i,v_j)$ denotes the resistance distance between $v_i$ and $v_j$. In this paper, we give a combinatorial interpretation of the weighted Kirchhoff index of an arbitrary connected graph. More precisely, we express $K(G;x_1,x_2,\ldots,x_n)$ in terms of the sums of weights of matchings in an appropriately weighted subdivision graph of $G$, and in the subgraphs obtained from this weighted subdivision graph by deleting the subdivision graphs corresponding to \(2\)-regular subgraphs of $G$. This gives an affirmative answer to a question posed by Li, Li and Yan [Discrete Math. 345 (2022) 113109] concerning a combinatorial explanation of the weighted Kirchhoff index of a general graph by using matchings in weighted subdivision graphs and their subgraphs. As special cases, our formula recovers the known formulas for the weighted Kirchhoff index of trees and unicyclic graphs, as well as the known formula for the ordinary Kirchhoff index of an arbitrary connected graph.

We generalize the notion of the Hopf link to the context of spatial graphs and handlebody-links, and study their symmetry and classification problems.

Fox's conjecture predicts that the absolute values of the coefficients of the Alexander polynomial of an alternating link form a trapezoidal sequence. Kálmán, Mészáros, and Postnikov gave a new proof of trapezoidality for special alternating links using a polynomial associated with flat vector arrangements. We prove that the flat arrangement polynomial has trapezoidal coefficients for every full-dimensional flat vector arrangement, answering the corresponding open question. Our proof gives a geometric interpretation of this polynomial in terms of convex polytopes. As an application, we prove that the Murasugi–Stoimenow polynomial of every connected Eulerian digraph has trapezoidal coefficients.

Quantum locally recoverable codes (qLRCs), introduced recently by Golowich and Guruswami, allow any single-qudit erasure to be recovered from a small set of other qudits. Most known constructions require a large alphabet. We systematically investigate qLRCs obtained, via the CSS construction, from classical constacyclic codes over small fields $\Ff_q$ with $q\in\{2,3,4,5,7\}$. First, we prove that a nonzero dual-containing $\lambda$-constacyclic code exists only when $\lambda^2=1$, so that negacyclic and (repeated-root) cyclic codes exhaust the constacyclic route to qLRCs. Second, we show that the locality of a constacyclic code equals the minimum distance of its dual minus one, and we give a simple purity criterion for the resulting quantum codes. Third, we show that odd-like duadic codes whose splitting is given by $\mu_{-1}$ yield pure qLRCs; specializing to $q$-ary quadratic residue codes of prime length $p\equiv 3 \pmod 4$ gives an infinite family of pure qLRCs with unbounded minimum distance and certified locality. Finally, by means of concrete computations, we obtain a classification of qLRCs from cyclic, negacyclic, and repeated-root cyclic codes of moderate lengths, which contains the first binary qLRCs from repeated-root cyclic codes and many parameter sets that cyclic codes cannot attain.

The spectral characterization of graphs is a central problem in spectral graph theory. In this paper we study when a tree is determined, among trees, by its generalized spectrum. We use the equivalent formulation given by the adjacency spectrum together with the total-walk sequence $W_k(G)=\mathbf 1^{\mathsf T}A(G)^k\mathbf 1$. We determine the exact matching-number threshold for this tree-level reconstruction problem. If $T$ and $T'$ are trees with matching number at most 4 and have the same adjacency spectrum and the same total-walk sequence, then $T\cong T'$. Moreover, in this range it is enough to require equality of $W_k$ for $3\le k\le8$. The bound is sharp: for every positive integer $m$ we construct a pair of non-isomorphic trees with matching number 5 having the same adjacency spectrum and identical total-walk sequences. The proof of the positive result is based on a finite-core reduction and an algebraic reconstruction of the possible pendant attachments.

A $t$-wise balanced design on the edge set of a complete graph is graphical if its block multiset is invariant under the induced action of the symmetric group on the vertices. Chouinard conjectured that, for each fixed index $\lambda$, there are only finitely many nontrivial simple graphical $t$-wise balanced designs with $t>1$. We prove the conjecture for $t$-designs, which are the $t$-wise balanced designs whose blocks all have the same size. Our theorem does not require simplicity, and we bound all parameters of the designs by explicit polynomials in $\lambda$.

We introduce weighted Laplacian flow (WLF), a deterministic particle-flow framework for sampling from a target distribution known only up to normalization. The proposed method evolves the logarithmic density ratio through a transport equation and determines the particle velocity from a target-weighted Poisson problem, resulting in a nonlocal and kernel-free mechanism for redistributing mass. For bounded smooth domains, we establish global well-posedness of the flow for Lipschitz initial data and prove that the flow contracts the discrepancy between the evolving and target densities at an exact exponential rate in terms of the oscillation of the log-density ratio, as well as in the associated projective metric. The relative entropy also satisfies a precise dissipation relation involving the two directional Kullback–Leibler divergences. In addition, we identify a variational interpretation of WLF. The method can be viewed as a gradient flow of the reverse Kullback–Leibler divergence under a target-anchored metric. Consequently, the method achieves an explicit relaxation scale without requiring log-concavity or spectral-gap assumptions on the target distribution. Numerical experiments on multimodal, heavy-tailed, and high-dimensional targets demonstrate the effectiveness of the proposed approach in capturing long-range mass transport.

We establish the orthonormal Strichartz estimates in the abstract Keel–Tao framework, under precisely the same hypotheses as in their original theorem. As an important consequence, we resolve the endpoint conjecture for the Schrödinger equation in dimensions $d\ge 2$, which was first raised by Frank, Lewin, Lieb, and Seiringer. This consequence also yields an affirmative answer to the conjecture for the free transport equation, which was stated in dual form by Bennett, Bez, Gutiérrez, and Lee. As a further application, we also establish a refinement of the Strichartz estimate for a single function.

Let $1\leq m<n$, and let $\gamma:[0,1]\to\mathbb R^n$ be a smooth non-degenerate curve. For $\theta\in[0,1]$, set \[ V_\theta := \operatorname{span} \{\gamma'(\theta),\ldots,\gamma^{(m)}(\theta)\}, \] and let $\pi_\theta$ denote the orthogonal projection onto $V_\theta$. We prove that if $E\subset\mathbb R^n$ is a Borel set with $\dim_{\mathrm H}E>m$, then \[ \mathcal H^m(\pi_\theta(E))>0 \] for almost every $\theta\in[0,1]$.