Papers

For coprime integers $\ell,\mu\ge 2$, Alanazi, Munagi, and Saikia (2026) studied $\overline{R}_{\ell,\mu}(n)$, the number of overpartitions of $n$ in which no part is divisible by $\ell$ or by $\mu$, together with the single-modulus analogue $\overline{R}_{\ell (n)$. We record a simple combinatorial mechanism that determines both functions modulo every power of $2$ in terms of the number of distinct part sizes of the underlying ordinary partition. We also deduce a clean characterization of $\overline{R}_{\ell}(n)$ and $\overline{R}_{\ell,\mu}(n)$ modulo $4$ in terms of perfect squares.

Fried recently enumerated Dyck paths having equally many black and white cells below them, for the chessboard coloring (Narayana numbers) and the column-alternating coloring (Fuss–Catalan numbers). We prove a generalization here: for the coloring of columns modulo any $c\ge2$, the number of Dyck paths of semilength $n$ whose $c$ residue classes carry equal weight is the Raney number $\Raney_{c+1,r}(m)$, where $n=cm+r-1$.

We consider the problem of minimizing the $k$-th order partial derivative $f=\partial_j^k g$ of an unknown function $g$ along a fixed coordinate direction $j$, based on noisy queries of $g$. Assuming that $g$ has Hölder regularity ${\beta+k}$ for some $\beta\ge 2$, that $f$ is strongly convex on a compact convex set $\Theta\subset\mathbb{R}^d$ and that $g$ and $f$ satisfy mild boundedness and Lipschitz regularity conditions on $\Theta$, we propose a kernel-based estimator of $\nabla f$ and analyze the projected stochastic gradient algorithm driven by this estimator. We obtain a non-asymptotic upper bound on the optimization error of the order $d^{(2\beta+k-1)/(\beta+k)}\,N^{-(\beta-1)/(\beta+k)}$, where $N$ is the total number of queries. We also establish a minimax lower bound of the order $N^{-(\beta-1)/(\beta+k)}$ showing that this rate is optimal in $N$ over all sequential algorithms.

We propose the Randomized BH procedure, a generalized framework that introduces randomization into the Benjamini-Hochberg (BH) procedure for false discovery rate (FDR) control. We show that the classical BH procedure as well as two recently developed families of null-distribution-free methods, competition tests and conformal tests, share a common theoretical foundation and arise as special cases of this framework. The Randomized BH relaxes the classical conditions required for FDR control, frees conformal tests from the strict exchangeability assumption while allowing calibration sets to be constructed via pre-screening, and enables competition tests to be analyzed through the well-developed theory of p-values. We illustrate the framework through two applications. The first is a deeper analysis of compound p-values through the lens of randomization. The second is a general approach for integrating distributed multiple testing problems while maintaining global FDR control. Numerical simulations confirm the validity of our theoretical results.

This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables, and by virtue of these estimates, we establish conditional stability estimates for this problem for the first time. Building upon this, we propose a Tikhonov-type regularization functional and derive its associated adjoint system. Furthermore, leveraging the established conditional stability results, we derive the convergence rate of the proposed regularization approach. Finally, we validate the effectiveness of our theoretical findings through extensive numerical experiments.

We discuss noncommutative differential geometry from a vector field centric point of view. This is based on the notion of first order vector field calculus (FOVC), which has been previously introduced by Borowiec under the name Cartan pair. We define the universal FOVC and construct an adjunction between the categories of FOVC and that of first order differential calculi, showing that the vector field approach is dual, though not equivalent, to the differential form one. This correspondence is then extended to covariant vector field and differential calculi. On Hopf algebras, (bi)covariant FOVC are in bijection with (bicovariant) quantum tangent spaces. For Hopf–Galois extensions, quantum tangent spaces give rise to vertical vector fields and in this setup we further describe base vector fields and horizontal vector fields and show that they are related via a noncommutative Atiyah sequence. Multiple examples, based on braided derivations, finite groups and the quantum Hopf fibration, are given. The vector field approach is further enriched by a sheaf-theoretic treatment, which recovers the former as a local, or affine, picture.

We present a framework for generating nonconforming triangular meshes with multiple discretization layers. The framework exploits characteristic structural properties of meshes produced by a frontal Delaunay algorithm with uniform element size. The bulk region of such meshes exhibits a structured pattern resembling a regular triangular lattice. Owing to this structure, the bulk region can be coarsened by grouping elements into connected subsets of larger composite elements. When applied repeatedly, this procedure produces a mesh with multiple discretization layers. For complex geometries, the framework can be used to create composite multidomain meshes, where multiple discretization layers are generated within each subdomain. We present examples of meshes generated by the proposed framework and discuss postprocessing strategies for their refinement and coarsening. The resulting meshes are well suited for the application of adaptive mesh refinement techniques. The proposed framework can be readily integrated with existing mesh generators and finite-element solvers that support nonconforming triangular meshes.

Agraph G is minimally t-tough if the toughness of G is t and the deletion of any edge from G decreases its toughness, where t is a positive real number. It is conjectured that every $(K_1\cup P_4)$-free 1-tough graph is hamiltonian. In this paper, we characterize the structure of minimally 1-tough $(K_1\cup P_4)$-free graphs, and thus show that the above conjecture is true for minimally 1-tough graphs. Furthermore, it is also proved that the Kriesell's conjecture which states that each minimally 1-tough graph has a vertex of degree 2 holds for minimally 1-tough $(K_1\cup P_4)$-free graphs.

We prove global higher integrability of the gradient for solutions to nonlinear PDEs with homogeneous Dirichlet condition. We work with general open sets (i.e. not necessarily bounded), under a minimal thickness assumption of the complement. The main focus is on improving the global summability of the gradient, by preserving the “zero trace" Sobolev class. We pay due attention to providing precise a priori estimates. As an application, we get universal global Hölder estimates for solutions of the super-homogeneous Lane-Emden equation for the $N-$Laplacian, in contractible open sets with finite inradius.

We give a complete characterization of the (purely infinite) simplicity of Katsura algebras and the associated Steinberg algebras. This is achieved by characterizing when the singular ideals vanish via the self-similar groupoid model derived from Exel and Pardo. Analogous results are given for the algebras arising from the faithful quotient of the self-similar action. Finally, we describe polynomial-space algorithms to determine if each singular ideal vanishes and provide the first non-Hausdorff examples of non-contracting self-similar groupoids for which simplicity is algorithmically decidable.

We consider the first eigenvalue, $\lambda_\beta(\Omega,A)$, of a two-phase eigenvalue problem for the Laplacian with Robin boundary conditions, where the two phases are characterised by different ellipticity constants. We characterise the conditions under which a ball $B_R$ is a local minimum under a volume constraint for the minimisation problem $A\mapsto\lambda_\beta(B_r,A)$, in terms of the principal Neumann eigenvalue of the fixed inner ball $B_r$.

The paper studies bifurcation for additively parametrized scalar nonautonomous ODEs of the form $x'=f(t,x)+\lambda$, where $f$ is d-concave and coercive in the state variable. By formulating the problem through the hull of $f$ and the associated skewproduct flow, the autonomous notion of bifurcation can be extended in terms of loss of hyperbolicity of copies of the base. A fairly complete classification of the possible bifurcation diagrams is provided. The notion of a jump bifurcation, linked to abrupt changes in the global attractor, is introduced and related to the occurrence of critical transitions. Several nontrivial, genuinely nonautonomous examples are presented, illustrating the different bifurcation diagrams that may arise from the lack of unique ergodicity on the hull and showing how critical transitions can result from an underlying jump bifurcation point. The framework has potential applications in climate dynamics, ecology, circuits, optics, biology, and other models involving abrupt transitions.

Building on recent studies of 0-Auslander categories, we establish a connection between $d$-Auslander extriangulated categories and categories of $(d+2)$-term complexes up to homotopy. We give a precise homological condition under which an algebraic extriangulated category admits an extriangulated ideal quotient equivalent to $\mathcal{K}^{[-d-1,0]}(\mathcal{A})$. We then demonstrate that $d$-cluster-tilting subcategories in triangulated categories serve as a key source of $d$-Auslander extriangulated categories. Using these structural results, we answer a question posed by Iyama in the Appendix of arXiv:2509.08246 by proving that $\mathcal{K}^{[-d-1,0]}(\mathcal{N})$ admits a triangulated structure when $\mathcal{N}$ is a weakly idempotent complete algebraic $(d+4)$-angulated category.

Optimizer experiments observe responses to algorithmic configurations without uniquely revealing hidden mechanisms. We develop a causal optimizer interaction calculus that separates pathwise realization, Mobius decomposition, and experimental identification. Under a fixed innovation coupling, every finite-horizon innovation-driven optimizer admits a behaviorally minimal pathwise realization. For any finite effect support and intervention design, an incidence operator gives the complete observational gauge, exact identifiability, sharp quotient stability, held-out predictions, and exact noiseless configuration complexity. Smooth hidden relaxation generates interactions through inverse hidden-state stiffness. Building on this structural law, we prove an observable-readout transfer theorem: arbitrary smooth update or trace readouts inherit an explicit five-term interaction through first and second hidden responses. Unlike the reduced optimal value, a general readout has no universal interaction sign. Its Boolean effects remain exact integrals of continuous interaction curvature and can therefore be identified by factorial interventions. We also derive Gaussian quotient minimax risk, exact confidence sets and tests, misspecification decomposition, certified downstream decisions, and optimal replication. A controlled real-data experiment on a 65-dimensional strongly convex logistic model validates the complete reduced-value chain. Boolean effects and independently integrated curvature agree within 4.21e-11, while nine held-out continuous intensities agree within 8.88e-13. Gaussian campaigns attain the predicted coverage and power, and 4,500 real-minibatch observations reject an order-two interaction model. Neural trace audits provide complementary evidence that the declared response classes remain informative in nonconvex training.

Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric. We formulate this effort as a path-space value problem. Restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path whose endpoint reaches a Hessian-relative generalized-eigenvalue condition target. Path elimination gives an exact min-plus semigroup and Bellman principle, while fixed-horizon kinetic energy is exactly squared complexity divided by twice the horizon. The main response result is global on a Hadamard state space: geodesic convexity produces a smooth intervention-cube path branch and a uniformly coercive Jacobi form, while one Green inverse generates the value Hessian, two-sided force-to-curvature bounds, exact Möbius effects, and arbitrary prescribed finite-order responses. For the hard condition target, a bordered Jacobi–KKT theorem differentiates the moving projection endpoint and multiplier on every regular active spectral stratum; its indefinite inverse also explains why hard-target interactions need not share the unconstrained sign. The theory specializes to affine-invariant positive-definite geometry. A determinant-one two-dimensional diagonal model has an exact target interval, a closed-form forced path, and a strictly negative-definite interaction matrix. A moving diagonal Hessian gives a closed-form hard-target projection, multiplier, and pair effects of either sign, while a coordinate-sequential three-dimensional protocol yields an exact path metric strictly larger than the ambient projection distance. Thus the global Green and bordered hard-target responses are explicit laws of restricted metric-path elimination built on Bellman composition.

Since self-orthogonal codes, few-weight codes and linear complementary dual codes(LCD codes, for short) have nice applications in coding theory and cryptography, they have received continuous attention. In 2024, by introducing the notion of the augment code, Heng et al.[16] constructed several classes of few-weight self-orthogonal codes basing on defining sets, which are introduced by Ding et al.[7] in 2007. In this manuscript, for two classes of defining sets, we consider the corresponding augmented codes, construct three classes of four-weight self-orthogonal projective codes and determine the parameters of their dual codes. And then, we construct three classes of LCD codes. In particular, we prove that there exists a class of these LCD codes whose dual codes are almost optimal LCD codes according to the sphere packing bound.

We extend the Schanuel integration framework, originally introduced for finite unions of convex sets, to arbitrary semi-algebraic sets. We prove that the resulting Schanuel integral of indicator functions is independent of the choice of ordered linear bases and therefore defines a well-defined Euler characteristic in the semi-algebraic category. We further show that this Schanuel–Euler characteristic coincides with the classical Euler characteristic defined via Borel–Moore homology and cylindrical algebraic decomposition. The recursive fiberwise structure of Schanuel integration provides an elementary and geometric interpretation of Euler characteristic and yields simplified proofs of several classical properties, including invariance under semi-algebraic isomorphisms.

A multivariate transformation of the unit cube with component transformations that are piecewise continuously differentiable and uniform distribution preserving (udp) is considered. A stochastic inverse transformation is defined using randomization to overcome the non-injective nature of the udp transformations. The inverse transformation preserves the uniform margins of a random vector distributed according to a copula and yields different copulas for different randomizations. A copula density transformation result for the multivariate stochastic inverse is proved and illustrated in the bivariate case.

For radial initial data, we construct explicit higher-order \(L^p(\mathbb R^N)\)-asymptotic profiles for the heat equation with Hardy potential. These profiles, denoted $A_n$ are obtained from the small-argument expansion, up to an arbitrary order \(n\), of the modified Bessel function appearing in the radial Hardy heat kernel. If $u$ is the mild solution generated by this kernel, we prove that the corresponding remainder $u(x,t)-A_n(x,t)$ admits a polynomial decay depending on $n$ in \(L^p(\mathbb R^N)\) as \(t\to\infty\). We also treat the non-radial case through spherical harmonics: each angular mode evolves according to a radial Hardy heat equation with a modified parameter, leading to finite and infinite angular expansion versions of the asymptotic profile under suitable summability assumptions.

In this paper we investigate the existence and qualitative properties of weak solutions for a class of nonlinear fractional equations driven by the spectral fractional Laplacian under mixed Dirichlet–Neumann boundary conditions. More precisely, we consider subcritical problems of the form \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^s u=\lambda u + \mu f(x,u) & \text{ in } \Omega\smallskip, \\ \displaystyle u\chi_{\SD} +\frac{\partial u}{\partial\nu}\chi_{\SN}=0 & \text{ on } \partial\Omega, \end{array} \right. \end{equation*} where \(s\in(1/2,1)\), \(\Omega\subset\mathbb R^N\) is a bounded smooth domain, and the boundary \(\partial\Omega\) is endowed with a mixed Dirichlet–Neumann configuration satisfying suitable geometric and analytic assumptions. Working in the natural spectral fractional framework associated with the mixed boundary operator, we establish the existence of nontrivial weak solutions by means of suitable variational methods. The analysis is based on a topological and variational principle combined with a localization argument involving certain compactly supported cutoff functions. More precisely, the proof relies on a delicate local comparison argument involving localized plateau-type test functions and explicit asymptotic estimates near the origin. Such an approach is sufficiently flexible and may potentially be adapted to several different settings. As applications, we establish simplified existence criteria in the autonomous case through the use of the Chebyshev radius of the domain and present concrete examples illustrating the applicability of the abstract theory.

We propose an optimized stencil strategy for the Generalized Finite Difference Method (GFDM) applied to non-linear problems. We take advantage of the flexibility of GFDM to engineer specific stencils by agglomerating nodes and balancing size with numerical accuracy. Previous work, focusing on the stencil construction for the linear convection-diffusion problem, showed that optimizing stencils and scaling parameters improves the scheme. The present study aims to investigate similar benefits for non-linear problems such as the Burgers' equations and the weakly compressible Navier-Stokes system.

Quickly evolving technologies, such as Interactive Theorem Provers (ITPs), Automated Theorem Provers (ATPs), and Large Language Models (LLMs), all falling under the general heading 'AI for mathematics,' are transforming mathematical practice in profound ways. This chapter explores the implications of these innovations, focusing on their impact on how mathematical knowledge is created and shared. It also discusses how they are reshaping the social dimension of mathematics, altering collaboration dynamics, trust relationships, and the collective production of knowledge. For instance, tools like ITPs facilitate large-scale collaborations and make new types of teamwork possible, where trust is not a necessary ingredient. ITPs also help us mitigate our human fallibility, yet they raise questions about the nature of formalization and the relationship between traditional and formal mathematics. Technologies such as LLMs are reshaping the division of epistemic labour between humans and machines and urge philosophers of mathematics to ask questions about the value of their work.

Given a graph $H$, the minimum feedback vertex number of $H$ is the minimum number of vertices whose removal results in an acyclic graph. In this paper, we investigate Turán-type extremal problems for bipartite graphs in terms of their feedback vertex number. Our first result concerns bipartite graphs $H$ with minimum feedback vertex number one. Such graphs can be obtained from a forest by identifying a specified collection of leaves into a single vertex. For these graphs, we show that $\text{ex}(n, H)$ is upper bounded by $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the shortest cycle contained in $H$. In addition, we consider a family of bipartite graphs with minimum feedback vertex number three. Let $E_{k,t}$ be the graph obtained from the theta graph $\theta_{k,t}$ by joining a new vertex $x$ to one side of the bipartition and another vertex $y$ to the other. Let $E^+_{k,t}$ denote the graph obtained by adding the edge $xy$ to $E_{k,t}$. We prove that for any $k\geq 2$ and sufficiently large $t$, $\text{ex}(n, E^+_{k,t})= \Theta(n^{\frac{3k-1}{2k-1}}).$

We study the quadratic nonlinear Schrödinger equation in energy-critical and energy-supercritical regimes and establish unconditional uniqueness at critical regularity on both $\mathbb{T}^{d}$ and $\mathbb{R}^{d}$. We introduce a new infinite quadratic hierarchy that featuring a linear structure for tensor product forms, instead of marginal densities. Consequently, this newly constructed hierarchy differs from the quantum Gross-Pitaevskii hierarchy, and its structure is in fact closer to that of the classical Boltzmann hierarchy. We prove this quadratic hierarchy admits combinatorial structures that are compatible with the bilinear $U$-$V$ estimates we prove for the quadratic nonlinearity. These tools enable us to establish unconditional uniqueness at the critical regularity via a quadratic hierarchy approach, and thus to provide an affirmative answer to Bourgain's uniqueness concern [3,p.152] for the quadratic nonlinearity in the weak-form bilinear case.

We study the fractional Plateau problem for $s$-minimal sets with prescribed exterior datum. For fixed exterior data, minimizers need not be unique, and singularities may occur beyond the critical dimension. We first prove a generic uniqueness theorem: along any strictly increasing family of exterior data, nonuniqueness occurs for at most countably many parameters. We then show that one can make arbitrarily small perturbations for which the interior regularity theory improves by one dimension.

Computationally efficient models for multimodal traffic flows with inter-modal interactions are foundational for coordinated traffic management. However, such models are lacking in the current literature. This study introduces a multimodal link transmission model (M-LTM) accommodating both continuous road traffic and discrete tramway traffic at the network level. M-LTM builds on a link model that captures the inter-modal interactions through the moving bottleneck theory. We further develop node models describing the flow transfer, tram operations, and mode interactions at tram stops and intersections. Specifically, the tram dwell process and dwell-induced congestion of road traffic are simulated at the stop node, and the intersection node model can reproduce queue spillback and tram priority. Case studies are conducted on two synthetic one-node networks, a synthetic arterial network, and the real network of Dresden to demonstrate the predictive power and the applicability of the proposed model in network traffic analysis and management. The GEH statistics of road traffic are below 4, and the average tram arrival time absolute errors are less than 10 seconds.

This paper is a sequel to the erratum by the authors to a paper by Crampon and Marquis (see arXiv:1202.5442). The main result of the Erratum was relating several notions of geometrical finiteness in round convex projective geometry and we prove here that our series of implications was sharp, by providing counterexamples to the implications that were not established. Our counterexamples are 4-dimensional convex domains $\Omega$ acted on by $\rho (\Gamma)$ where $\Gamma$ is a lattice of $\mathrm{SL}_2 (\mathbb R)$ and $\rho$ is the irreducible representation of $\mathrm{SL}_2 (\mathbb R)$ of dimension $5$. We give a description of all $\rho(\Gamma)$-invariant convex domains, and in particular we construct one which is "close enough" to the convex hull $\mathcal C$ of the limit set of $\rho(\Gamma)$ so that the Hilbert volume $\mathrm{Vol}_{\Omega/\Gamma}(\mathcal C/\Gamma)$ of the convex core is infinite. We include an appendix with a smoothing procedure in the spirit of Cooper, Long and Tillman (arXiv:1511.06206) and Danciger, Guéritaud and Kassel (arXiv:1704.08711).

Relative multisymplectic geometry replaces differential forms on a single manifold by cocycles in the mapping cone of a smooth map $F\colon M\to N$. Building on the relative Cartan calculus, the Lie $n$-algebras of relative observables, and the relative homotopy moment maps developed in companion work, we establish a range of applications showing that the framework is a working tool rather than a formal generalization. We first construct the integration pairing between relative differential forms and smooth relative chains, and prove two structural results that make it usable: a period criterion, reducing relative integrality to the periods of the target form together with a defect homomorphism on the classes killed by $F_*$, and a functoriality theorem for morphisms of arrows. The criterion yields a structure theorem for levels: the integers $k$ at which $k\varpi$ is relatively integral always form a cyclic group $k_0\mathbb{Z}$, with no hypothesis on $F$, and $k_0$ is computable from finitely many integrals whenever the relevant homology is finitely generated. Together these tools yield a characterization of homotopy-invariant bulk–boundary action functionals, hence a precise treatment of Wess–Zumino terms; a relative Weil–Kostant theorem, whose specialization to a Lagrangian submanifold is the Bohr–Sommerfeld condition of geometric quantization; a relative Noether identity, with a bulk–boundary splitting of the conserved charges; and a rigidity theorem making comoment maps unique, strict and equivariant, so that the Kostant–Souriau cocycle disappears. Two closing sections analyse the degenerate edges $M=\varnothing$ and $N=\mathrm{pt}$, at which the absolute theories are recovered, and separate weak from strong nondegeneracy, determining which results require which.

Official statistics in Greece started as early as 1828 with the first Census. A department of statistics was established in 1861, and the General Statistical Service of Greece was founded in 1925. It was reformed to become the National Statistical Service of Greece in 1953, which was then replaced by the independent Hellenic Statistical Authority in 2010. The science of probability and statistics was essentially introduced in the late 1960s and before 1982. Seven chairs, plus were established in probability and statistics, albeit with various names, at the four Greek universities and ASOEE, which existed then, and six aspirants were elected to the post of professor of the respective chair. Since 1982, when the law abolished the archaic system of chairs, more than 200 PhD holders in probability and statistics or related subjects have been elected or promoted to the new ranks of professor, associate professor, and assistant professor at the 25 public Greek universities, which exist today. Several of these professors or associate professors have been elected to the post of Rector or Vice Rector. Now, probability and statistics, as well as related subjects, are very well treated in Greece. Today, the Greek contribution to these specialties is internationally recognized

Recently, Andrews and El Bachraoui studied a partition function $s_1(n)$, which counts the number of two-color partitions into distinct parts of $n$ whose smallest part occurs in one prescribed color only, while every larger part may occur in either color or in both colors. They obtained a complete description modulo 4 for $s_1(n)$. They also considered a $q$-series $T_{o}(q)$ which is the odd companion series of the generating function for $s_1(n)$. At the end of their paper, they presented a conjecture on the parity of the coefficients of $T_o(q)$. In this paper, we confirm this conjecture. Moreover, we establish an infinite family of congruences modulo 8 for the coefficients of $S_1(q)$ and prove that the set of integers satisfying $s_1(n)\equiv 0\pmod 8$ has natural density one.