Papers

In this paper, we show that for any integer \(v \ge 8\) with \(v \ne 9,11,23\), a quantum Latin square of order $v$ exists for every cardinality \(c \in [v,\,v^{2}] \setminus \{\,v+1\,\}\).

We consider unconstrained minimization of smooth quasar-convex functions when only noisy function evaluations are accessible through a stochastic zeroth-order oracle. For these non-convex functions, the standard acceleration method relies on subspace-search mechanisms that require first-order information, being therefore unavailable in zeroth-order regimes. In contrast, the alternative and less conventional continuized method enable to avoid such mechanisms. In this work, we design a zeroth-order continuized algorithm, leading to accelerated convergence guarantees that parallel those of smooth convex optimization up to a quasar-convexity parameter. Our method incorporates a mirror step, improving the dimension dependence when there exists sparse solution.

A square-adjoining rectangle game generates the Fibonacci numbers and the Fibonacci spiral from a simple geometric rule. If one starts from a square, the four possible first moves are all equivalent by rotation. If one starts instead from a non-square rectangle, there are still four geometric placements for the first square, but they split into exactly two equivalence classes: long-side-first and short-side-first. We show that both classes are governed by the same Fibonacci-type recursion with different initial conditions, and that in both cases the successive aspect ratios converge to the golden ratio (phi). We then add a brief geometric remark: the Hypotenuse-Axis Intercept (HAI) construction from a right triangle produces a natural ordered seed whose outward and inward branches determine precisely those two first-move classes.

In this paper, we establish a sharp off-diagonal pointwise upper estimate for the Bergman reproducing kernel associated with a radial weight on the unit disc. Our proof is self-contained and assumes the weight satisfies a natural one-sided doubling condition on its moments. The standard kernels demonstrate that this estimate is sharp, up to a multiplicative constant, at every point. We also show that the estimate in fact characterizes the class of radial doubling weights under consideration. As applications, we first obtain optimal $L^p$-mean estimates for certain modified Bergman kernels. This approach recovers the key estimates in [Peláez et al., J. Math. Pures Appl. 105(2016), 102–130] and [Peláez et al., arxiv.org/pdf/2407.04645] via a novel and more direct proof. Second, we establish novel connections between the non-tangential maximal function of the Berezin transform, the Hörmander maximal function, and Carleson measures. Notably, these connections are new even in the setting of the standard weighted Bergman spaces. Finally, we extend our main results to higher dimensions, harmonic Bergman kernels, and two-weight fractional derivatives of kernels, the latter of which yields sharp estimates for Dirichlet reproducing kernels.

We investigate how nilpotency assumptions on the multiplicative group of a finite skew brace constrain its ideal structure and solvability. Our first main result shows that, if \(B=(B,+,\cdot)\) is finite and \((B,\cdot)\) is nilpotent, then the additive Fitting subgroup \(F(B,+)\) is a non-zero ideal of \(B\). As consequences, every finite simple skew brace with nilpotent multiplicative group is isomorphic to \(\Triv(C_p)\) for some prime \(p\), and every such skew brace admits an ideal of prime index. In particular, \[ B*B\neq B \qquad\text{and}\qquad \partial(B) \neq B. \] We also show that nilpotency of the multiplicative group does not, in general, imply either left nilpotency or solvability. Motivated by this obstruction, we then study the solvability of two-sided skew braces, both in the finite and in the general setting. We prove that, whenever \(I\) is an ideal of a two-sided skew brace \(B\), the internal commutator ideal \([I,I]_I\) is again an ideal of \(B\). This yields an extension theorem for solvability and implies that, for finite two-sided skew braces, solvability of the skew brace is equivalent to solvability of either the additive or the multiplicative group. In particular, every finite skew brace with abelian multiplicative group is solvable. Finally, we show that, although this equivalence fails in general for infinite two-sided skew braces, a residual form of it still survives: if the additive group is solvable, then every finite homomorphic image of the multiplicative group is solvable.

We present a third-order finite volume central WENO scheme for systems of nonlocal conservation laws in two spatial dimensions. The CWENO reconstruction of the conservative variable provides polynomials that can be evaluated in the entire domain, which is of advantage when approximating the nonlocal terms. Moreover, this method can be augmented with a limiter that preserves the maximum-principle and especially positivity of the solution.

We extend the Pila-Zannier strategy to Drinfeld modules: we prove analogues of the Manin-Mumford theorem for a product of two Drinfeld modules of equal rank, and of the André-Oort theorem for a product of two Drinfeld modular curves. In characteristic zero, several steps of this strategy rest on $o$-minimality, which has no counterpart over a function field; we replace the counting step by the rigid analytic Pila-Wilkie theorem of Binyamini-Kato, and this appears to be its first arithmetic application. The functional transcendence input, namely an analogue of the Ax-Lindemann theorem in both settings, is established here by an independent point counting argument.

We propose a new Galerkin discretization scheme for wave scattering problems that is based on microlocalised basis functions. We show that the proposed method can be made uniformly accurate for large wavenumbers $k$ with a number of degrees of freedom only scaling as $k^{d-1/2}$, while leading to an essentially sparse linear system. In contrast, finite element methods are known to require a number of degrees of freedom scaling at least as $k^d$ to achieve the same property. A similar method based on a Gabor frame was previously introduced by two of the authors, but it was suffering from severe conditioning issues. In the present work, by replacing the Gabor frame by a Wilson basis, we completely alleviate this problem. We rigorously establish error estimates and condition number bounds for the proposed method, and we provide one-dimensional numerical examples illustrating our theoretical findings.

In this note we construct colorings of Euclidean space $\mathbb{R}^n$ with finitely many colors such that any two points at unit distance have different colors and, in addition, each color class is dense in $\mathbb{R}^n$. In particular, 12 dense colors suffice to color $\mathbb{R}^2$. In arbitrary dimension, we show that $n\chi(\mathbb{R}^n)+1$ colors suffice, where $\chi(\mathbb{R}^n)$ denotes the chromatic number of $\mathbb{R}^n$ in the standard formulation.

This paper investigates tilting modules over gentle algebras and their endomorphism algebras within the framework of marked surfaces and tilings introduced by Baur and Simões. Faithful dissections of a tiling are shown to correspond to tilting modules. For a faithful dissection, we define an auxiliary algebra and prove that it is isomorphic to the endomorphism algebra of the corresponding tilting module. We also construct a new tiling realizing this endomorphism algebra and introduce a flip preserving tilting modules.

A two-stage mixed-integer linear programming framework is introduced for subsea pipeline incident response planning, jointly optimizing Subsea Docking Plate (SDP) placement and resident autonomous underwater vehicle allocation to minimize both maximum and average response times under spatial uncertainty. Phase 1 minimizes the maximum response time across all potential leak locations. Phase 2 reduces the average response time subject to the maximum bound. A case study based on the Johan Sverdrup oil and gas field in Norway, with 9 SDP options and 33 pipelines, shows that just a few well-placed SDPs are enough to achieve top performance. A cost versus time Pareto analysis reveals the trade-off between deployment expenditure and response efficiency, providing actionable guidance for designing resilient and cost-effective subsea inspection networks.

We investigate maximal Boolean sublattices of the partition lattice Part(U) of a finite universe U. First, any largest size Boolean sublattice of Part(U) can be formed using all partitions whose blocks are subtrees of a tree with vertex set U. It is shown that a maximal Boolean sublattice of Part(U) always contains the least and the largest elements of Part(U). Boolean sublattices B of PartU containing 0 are characterized by a certain condition imposed on the cycles of a linear hypergraph corresponding to the atoms of B on the set U. We show that B can be extended to a Boolean sublattice of Part(U) with a largest size, if and only if the hypergraph induced by its atoms is a hypertree. This is the case when B is formed by all the partitions whose blocks are intervals in a generalized sense. The main result states that all partition lattices of height at least three have maximal Boolean sublattices for all possible dimensions at least three

This work presents analytical tools for studying functions defined on spaces of measures, together with two applications: mean field game master equations and first order Hamilton-Jacobi-Bellman equations on the set of probability measures. The first part introduces several notions of differentiability for such functions - vertical (or flat) and horizontal derivatives - as well as associated notions of convexity, monotonicity and sub-differentiability. The second part is concerned with master equations, for which uniqueness and stability of monotone solutions are established, and existence is discussed. The third part studies mean field optimal control problems and characterizes their value functions as the unique viscosity solutions of the associated Hamilton-Jacobi-Bellman equations. This work is intended to be accessible to readers unfamiliar with spaces of measures.

Current digital semantic communication systems have primarily focused on maintaining compatibility with conventional constellation-based modulation. In contrast, index modulation (IM) represents a more spectrally and energy-efficient alternative by exploiting additional dimensions for information conveyance. Recognizing this potential, this paper bridges the gap between IM and semantic communications by proposing a novel spatial semantic communication (SSC) system leveraging cutting-edge fluid antenna-IM (FA-IM) technology. Compatible with existing joint source-channel coding (JSCC) architectures, the proposed SSC system employs the residual quantization (RQ) approach to discretize analog semantic features for subsequent digital IM transmission. Notably, the proposed SSC system synergizes RQ and IM via a semantic-aware stream splitting scheme, which ensures that critical semantic information undergoes less severe channel fading, thereby further optimizing semantic transmission performance. Simulation results validate that the proposed SSC system effectively integrates the high fidelity of RQ, the reliability of semantic-aware splitting, and the spatial efficiency of FA-IM, thereby providing a robust solution for future digital semantic transmission. The open source code is available at: https://github.com/gxh1106/SSC.

We study entropy-regularized exploratory control in finite $N$-player stochastic differential games under a response model in which each player conditions on the opponents' currently realized actions and evaluates continuation with that profile frozen. The resulting Gibbs best responses form a system of full conditional densities, which need not admit a common joint law. We characterize joint realizability by a cross-partial condition on the entropy-scaled Hamiltonian gradients and, on simply connected action domains, by an equivalent entropy-weighted potential structure. When compatibility fails, a coordinate-path construction yields a joint density whose full conditionals satisfy explicit quadratic Kullback–Leibler bounds. We extend the analysis to stationary discounted problems and derive martingale and policy-improvement characterizations for learning the frozen response maps. A two-player linear-quadratic example illustrates the compatibility criterion and the associated learning procedure.

In this paper, we show that if the non-constant rational functions $f_1, \ldots, f_n\in K(x)$ over a number field $K$ cannot multiplicatively generate a power of a linear fractional function, then there are only finitely many elements $\alpha \in K$ such that $f_1(\alpha),\ldots,f_n(\alpha)$ are multiplicatively dependent modulo some subset `close' (with respect to the Weil height) to the division group of a finitely generated multiplicative subgroup of $K$. This improves some previous results.

We study higher cluster tilting objects through covering functors from derived categories of hereditary algebras. The covering formalism reduces the existence problem to the equivariant problem of finding \(G\)-stable \(d\)-cluster tilting objects in \(d\)-cluster categories. For triangulated categories with finitely many indecomposable objects this gives a complete ADE existence criterion and explicit counting formulas. We also prove that, in finite Frobenius models, the endomorphism algebras of all \(d\)-cluster tilting objects are derived equivalent. Applications are given to Cohen–Macaulay finite categories, finite noncommutative crepant resolutions, and rigidity dimensions of representation-finite self-injective algebras.

We construct ”Néron models” of Gm over any base scheme S.

Hyperderivative Reed-Solomon (HRS) codes form a class of maximum-distance-separable codes under the Niederreiter-Rosenbloom-Tsfasman metric and may be viewed as a derivative-evaluation extension of classical Reed-Solomon codes. For generalized Reed-Solomon codes, the Euclidean dual is again a generalized Reed-Solomon code. In this paper, we investigate the corresponding duality problem for HRS codes. Using a residue-theoretic argument, we derive an explicit component-wise representation for the Euclidean dual of an HRS code. The formula shows that, in general, the Euclidean dual is not an HRS code. Instead, it is blockwise upper-triangularly equivalent to a reverse-order HRS evaluation code, where the reversal occurs in the hyperderivative orders within each evaluation block. In particular, for full-domain HRS codes with low multiplicity, the triangular transformations reduce to diagonal scalings, and the Euclidean dual is obtained as the row reversal of an HRS code. Based on this reverse-order dual structure, we further study reverse self-dual HRS codes. We establish explicit criteria for reverse self-duality and construct several families from additive and multiplicative coset structures.

We introduce a hierarchical sparse-grid (HSG) particle method for the numerical solution of the Vlasov–Poisson system. Sparse-grid PIC methods have so far been formulated within finite-difference frameworks, most notably through the sparse-grid combination technique (SGCT), which ties them to tensor-product Cartesian grids and globally defined component grids. This paper brings sparse-grid particle methods into the Galerkin setting: the field equation is solved in variational form on a hierarchical sparse-grid space spanned by B-splines of arbitrary degree, and the traditional charge deposition step is replaced by a direct Galerkin projection of the raw Monte Carlo density estimator onto this space. Beyond preserving the mesh-complexity and noise-reduction benefits of sparse grids, this reformulation substantially extends the versatility of the approach, opening the way to spatial adaptivity and to non-rectangular geometries, which are notoriously difficult to accommodate within the SGCT framework. We carry out a probabilistic error analysis decomposing the numerical error into a grid-based bias and a statistical noise component. Under mixed-derivative regularity assumptions on the particle distribution, the bias of the charge density in the $\mathrm{L}^2$-norm is shown to scale as $\mathcal{O}(h^{p+1}|\log h|^{d-1})$, where $p$ is the B-spline degree and $d$ the spatial dimension, and the statistical error in the $\mathrm{L}^1$-norm as $\mathcal{O}(|\log h|^{(d-1)/2}(Nh)^{-1/2})$, matching the accuracy of high-order SGCT-PIC methods. Corresponding bounds are derived for the electric field. The theoretical estimates are validated on classical kinetic plasma benchmarks, including configurations with limited regularity and strong anisotropies that are known to be challenging for sparse-PIC approximations.

In a recent paper we have analyzed a new formulation of the coupling of finite and boundary element methods (FEM-BEM) for Helmholtz problems, involving several heterogeneous bounded subdomains and one homogeneous unbounded subdomain. This formulation, called Generalized Optimized Schwarz Method (GOSM), is substructured, that is, its unknowns are associated with the subdomains interfaces. To derive the GOSM, the first step was to prove that a solution to the Helmholtz problem satisfies a specific multi-domain variational formulation, which involves one operator for each subdomain, each operator being independent of the others. In the present contribution, we design a variational formulation of that type for a more general geometrical and material configuration: several heterogeneous bounded subdomains, impenetrable obstacles and homogeneous subdomains are allowed. Note that, like in our recent paper, we assume that only one subdomain is unbounded, and that its boundary is bounded. The domain partition can have cross-points, that is, points where at least three subdomains are adjacent. We also prove that a solution to the initial Helmholtz problem can be recovered from a solution to the multi-domain variational formulation. This shows that the GOSM is a general and flexible framework to model acoustic wave propagation, as it can handle both multi-domain FEM-BEM coupling and (weakly imposed) boundary conditions on several obstacles.

We show that the support genus of the contact connected sum of two contact $3$-manifolds is at most the maximum of the support genera of the summands. In particular, iterated connected sums of contact manifolds of support genus one still have support genus at most one, and therefore cannot provide candidates for contact manifolds of higher support genus.

In this paper we analyze the Kuramoto-Sivashinsky equation (KSE), a model of flame-front propagation, on a two-dimensional square torus of arbitrary size $2 L$ with $L > \pi$. In this case, the linearized equation at the origin admits a finite number of growing modes, which corresponds to the positive eigenvalues of the linear operator $- \Delta^2 - \Delta$. The problem of analyzing the long-time behavior of solutions of the 2D KSE in two spatial dimensions remains largely open; the only global existence results are for sufficiently small tori, or for sufficiently anisotropic and thin domains, due to the lack of good a priori estimates. The main purpose of this paper is to analyze the instability around growing modes at the nonlinear level. More precisely, we consider the maximal growing mode $\lambda_0$ and we show that there is a finite dimensional manifold of initial data of size $\varepsilon$ arbitrarily small such that the corresponding solutions become of size $O(1)$ over a time-scale of order ${\rm log}(\varepsilon^{- 1})$. The proof is based on several ingredients such as a sharp quantitative construction of an approximate solution bifurcating from the maximal linearly growing mode, a fixed point argument with exponential weights to construct local in time solutions, and a continuation argument based on sharp energy estimates and para-differential calculus.

In this third part we study the group $Aut_{\mathbb{Q}}(S)$ of numerically trivial automorphisms of a non-algebraic properly elliptic surface $S$. We treat completely the Kähler case.

This work studies the linear Jin-Xin relaxation model by means of the relative entropy method, both at the continuous and fully discrete levels. Under the strict subcharacteristic condition, we obtain an O($\epsilon$) estimate for the convergence toward the equilibrium transport equation. We then develop discrete relative entropy estimates for a Lie splitting scheme and an asymptotic preserving staggered scheme. A fixed-time analysis further explains the O($\epsilon$^2 ) behavior observed in the strongly stiff regime and the transition occurring when $\epsilon$ becomes comparable to the time step.

We prove a local limit theorem for the quantity $\| g_n \cdots g_1 \|$, where $g_1, g_2, \ldots$ is a sequence of independent and identically distributed random invertible square matrices. The norm $\| \cdot \|$ is arbitrary and we make no proximality assumption.

Shan~[J. Graph Theory (2026)] proved that every 23-tough $(P_4\cup P_1)$-free graph on at least three vertices is Hamiltonian. We improve this bound to 7 by replacing the final cut analysis in Shan's framework with an asymmetric separation criterion and a cograph covering lemma.

We prove that every finite spectral pair \((A,\Gamma)\) with \(A\subset\mathbb Z\), \(\Gamma\subset\mathbb R/\mathbb Z\), and \(0\in\Gamma\) has a rational spectrum, that is, \(\Gamma\subset\mathbb Q/\mathbb Z\). Our argument relies on a modulus rigidity theorem for generalizedVandermonde systems satisfying inverse-orthogonality relations, constructed by hyperbolic positive-definite kernels. Combined with Galois conjugation and Kronecker's theorem, this rigidity forces the associated exponential nodes to be roots of unity. As an application, using the periodicity and fiberization of one-dimensional spectra, we show that every spectrum \(\Lambda\) of a bounded measurable spectral set \(\Omega\subset\mathbb R\) with \(|\Omega|=1\) and \(0\in\Lambda\) is contained in \(\mathbb Q\). This rational-spectrum result completes a chain of equivalences proven by Dutkay and Lai, which reduces the full one-dimensional Fuglede's conjecture to its finite cyclic analogues over \(\mathbb{Z}_n\).

In this paper, we provide a categorical framework for understanding coarse subspaces of coarse spaces. First, we introduce the notion of a controlled total relation between coarse spaces and show that the category whose morphisms are closeness classes of controlled total relations is isomorphic to the conventional category of coarse spaces defined using closeness classes of controlled maps. Next, we show that the assignment associating to each coarse space the finite-join partially ordered set of its coarse subspaces is functorial, and prove that this partially ordered set is naturally isomorphic to the poset of subobjects in the category of coarse spaces. Furthermore, we formulate asymptotic disjointness between coarse subspaces and show that mono-morphisms preserve this relation. These results provide a categorical interpretation of the framework of coarse subspaces introduced by Leitner–Vigolo [Lecture Notes in Math.~(2023)] and characterize coarse subspaces as objects intrinsic to the category of coarse spaces. They also provide a foundation for a coarse-geometric interpretation of the properness criterion established by Kobayashi [Math.~Ann.~(1989); J.~Lie Theory (1996)] and Benoist [Ann.~of Math.~(1996)] (cf.~Nagaya–Ogawa–Okuda [Proc.~Japan Acad.~Ser.~A (2025)]).

We study binomial probabilities at a fixed integer distance from the upper mode, in the regime where the fractional part of the mean remains visible. The complete asymptotic expansion is obtained to all orders, uniformly for the success probability in compact subintervals of $(0,1)$ and for bounded shifts from the mode. The main point is structural. A logarithmic tail in the standard local expansion is exactly the binomial size-bias factor, and removing it leaves a pure Appell expansion. A second, symmetric normalisation is governed by an even Appell sequence and recovers the classical Stirling prefactor. The resulting coefficients explain the exact binomial mode rule, recover the central-binomial expansions as a special case, and give closed forms for averaged oscillating coefficients.