Papers

We investigate the deformable hypersurfaces of the conformally flat Riemannian products $\mathbb{H}^k\times \mathbb{S}^{n-k+1}$, $2\leq k\leq n-1$, of hyperbolic space and a sphere with constant sectional curvatures $-1$ and $1$, respectively.

We study the independence between an order statistic and its corresponding rank indicator for independent nonnegative random variables. A general characterization is established through a probability measure obtained by reweighting the distribution of a single observation according to its conditional probability of occupying a prescribed rank. This framework yields explicit expressions for the associated weighting functions and conditional distributions, as well as distribution-free measures of departure from independence based on Kolmogorov and Wasserstein distances. Special attention is devoted to the minimum and maximum order statistics, leading to new characterizations of independent right- and left-censoring under both single and multiple censoring mechanisms. We also establish sufficient conditions based on proportional hazards and proportional reversed hazards models and derive complete characterizations in the classical single-censoring setting. Discrete and continuous examples, together with numerical illustrations, are presented to demonstrate the applicability of the proposed methodology.

Starting from SU(N) on the lattice, we give a rigorous derivation of the Dyson–Schwinger equations in the continuum limit. We formulate the Dyson–Schwinger identities for the lattice Yang–Mills theory directly in terms of the link variables $U_\mu(m)\in SU(N)$, exploiting the invariance of the Haar measure under left group translations. This provides an exact lattice derivation of the corresponding master equation for the Wilson action, expressed through left-invariant Lie derivatives acting on individual links. Because the construction is carried out directly on the compact gauge group, it avoids the ambiguities associated with introducing Lie-algebra valued gauge potentials as primary integration variables at finite lattice spacing. For practical applications, in a second part we then break down the gauge degree of freedom by choosing Feynman gauge. We analyze the continuum-limit form of the resulting lattice identities and derive equations for the one- and two-point connected functions. Under a further simplifying reduction, these equations close to a tractable scalar system. Our results establish a direct bridge between exact lattice identities and the functional equations commonly used in continuum nonperturbative studies of Yang–Mills theory.

This study addresses the critical yet under researched area of supported evacuation for vulnerable populations during wildfires, such as hospital patients and long term care residents, by developing a two stage stochastic optimization model that optimizes facility location, fleet sizing, and vehicle routing under strict time windows. To overcome the problem NP hard complexity, the authors propose an innovative solution methodology leveraging Logic Based Benders Decomposition, featuring Combinatorial Benders Cuts and logic based inequalities. Extensive numerical experiments and real world data from a community wildfire drill in Roxborough Park, Colorado, demonstrate that the proposed approach yields high quality solutions, significantly improving shelter placement, vehicle utilization, and overall evacuation efficiency compared to alternative policies.

A Hider distributes two identical balls among three boxes whose search costs satisfy $a\ge b\ge c>0$. A Searcher opens boxes adaptively until both balls are found; every opening incurs the corresponding box cost and recovers at most one ball. We give an exact solution of this heterogeneous multi-look search-cost game. The value is the maximum of three explicit rational functions. In the three parameter regimes, an optimal Hider strategy is the product-form distribution restricted respectively to three, five, or all six placements. Every non-wasteful deterministic Searcher policy is payoff-equivalent to one of $72$ elementary decision trees, which induce only $42$ distinct opening-count profiles. An analytic argument settles the first regime. Two exact Searcher mixtures settle the second, and four settle the third. Feasibility of the parameterized mixtures over the full cost region is established by exact rational Bernstein-basis certificates. Independent implementations reproduce the policy set, the zero-sum linear-program values, and all $154$ dyadic nodes examined by the Bernstein verifier.

In a graph, a set $D$ is $k$-dominating if every vertex in $V(G) \setminus D$ has at least $k$ neighbors in $D$. Jafari, Alikhani, and Bakhshesh introduced the concept of a $k$-coalition, which is a pair of disjoint sets $X_1$ and $X_2$ of vertices such that neither is a $k$-dominating set but $X_1 \cup X_2$ is a $k$-dominating set. A $k$-coalition partition is a vertex partition in which each set either forms a $k$-coalition with some other set or is itself a $k$-dominating set with exactly $k$ vertices. The $k$-coalition number $\operatorname{CO_k}(G)$ is the maximum number of sets in a $k$-coalition partition. We compute the $k$-coalition number for several families and bound the $k$-coalition number under disjoint union and graph join. We show that the set of possible sizes of $k$-coalition partitions forms an interval. Finally, we investigate $k$-coalition graphs and prove that every graph is a $k$-coalition graph.

We present a computer assisted proof of the existence of the oscillatory motion to collision and to infinity in the Earth-Moon planar circular restricted three body problem. In particular, we show that all types of motion (hyperbolic, parabolic, oscillatory to infinity, oscillatory to collision and terminating in collision) can be combined. We also prove the existence of arbitrarily large ejection/collision orbits. The results are achieved with the use of the Levi-Civita regularization, the McGehee regularization, topological tools and rigorous numerical computations.

Accurate prediction of tritium behaviour in molten salt breeding blankets is essential for the design and safe operation of ARC-class fusion reactors. This work presents a fully open-source, component-scale multiphysics framework for modelling tritium transport in an ARC liquid immersion blanket. Neutron transport, thermal hydraulics, and hydrogen isotope transport are coupled using OpenMC, OpenFOAM, and FESTIM, leveraging dedicated tools enabling direct transfer of spatially resolved fields between solvers. Assuming a zero inlet concentration, steady-state simulations predict a total tritium inventory of approximately 243 mg, with the blanket reaching steady-state tritium throughput within approximately 30 min, which is of a similar order to previous system-level estimates. The results show that tritium transport is dominated by turbulence-enhanced diffusion, with strong localisation in flow stagnation regions and reduced accumulation in highly turbulent zones. Sensitivity analyses indicate that predicted inventories are governed primarily by the numerical stabilisation scheme, with only a modest dependence on the turbulent Schmidt number. The proposed workflow provides a transparent and extensible basis for high-fidelity analysis of tritium transport in ARC-class breeding blankets.

Let $\rho$ be an absolutely irreducible 2-dimensional $p$-adic representation of the absolute Galois group of ${\mathbb Q}_p$, and let $\Pi(\rho)$ be the unitary Banach space representation of $G = {\rm GL}_2({\mathbb Q}_p)$ associated to $\rho$ by the $p$-adic Langlands correspondence. We deduce from results due to Colmez, Dospinescu, Paškūnas, Emerton, and others that if the locally analytic representation $\Pi(\rho)^\rm{la}$ has a subquotient isomorphic to a non-zero subquotient of a locally analytic principal series representation of $G$, then $\rho$ is trianguline.

We show that the weak Markus–Yamabe conjecture fails in every dimension $n\geq14$. We first prove a chain realization theorem: every polynomial Keller map $F=\I+H$ of $\R^n$ with component degrees $d_1,\ldots,d_n$ yields an explicit polynomial vector field on $\R^N$, with $N=\sum_i\max(d_i,2)-n$, whose Jacobian matrix has spectrum $\{-1\}$ at every point and whose singularities are in bijection with any prescribed fiber of $F$. Applied to the recent counterexample to the Jacobian conjecture, this gives a Hurwitz vector field of degree seven on $\R^{14}$ with three rational singularities. A rank-reduced dehomogenization of the associated cubic stabilization gives, independently, a Hurwitz field of degree three on $\R^{18}$. The dimensions $3\leq n\leq13$ remain open.

Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $\alpha_\gamma(n)$ be the largest number for which \[ \gamma_n(\lambda K+(1-\lambda)L)^{\alpha_\gamma(n)} \ge \lambda\gamma_n(K)^{\alpha_\gamma(n)} +(1-\lambda)\gamma_n(L)^{\alpha_\gamma(n)} \] holds for all convex bodies $K,L\subset\mathbb{R}^n$ containing the origin and all $\lambda\in[0,1]$. In this paper, we prove that \[ \alpha_\gamma(n) =1-\frac{2}{n-1} \frac{\Gamma(\frac n2)^2}{\Gamma(\frac{n-1}{2})^2}. \] The core of the proof is a raywise radial–tangential localization of the Hessian energy of a solution of a Neumann problem, which reduces source selection of the Neumann problem to a one-dimensional optimization. Monotonicity in the segment length and Laguerre spectral analysis determine the sharp one-dimensional value, whereas the planar endpoint is treated separately.

We establish sharp parameter thresholds governing $H^2$ -regularity of weak solutions in $H_0^1$ for two classes of elliptic problems with critical unbounded perturbations, in dimensions $N\geq 5$. More precisely, we consider two distinct $\lambda$-parametric elliptic problems $ - \Delta v + \lambda \frac{x\cdot \nabla v}{|x|^{2}} =f$ and $-\Delta v + \lambda \frac{v}{|x|^2}=f$ posed in a bounded $C^2$-domain $\Omega\subset \mathbb{R}^N$ containing the origin $x=0$. We observe that the singular perturbations $\frac{x\cdot \nabla v}{|x|^{2}}$ and $\frac{v}{|x|^2}$ are homogeneous operators of order 2 consistent with the scaling of the Laplacian. In view of the Hardy inequality the problems are well-posed in $H_0^1(\Omega)$ for $\lambda<\frac{N-2}{2}$ and $\lambda>-\frac{(N-2)^2}{4}$ respectively. The main results are as follows. For the first problem we show that any solution $v\in H_0^1(\Omega)$ belongs to $H^2(\Omega)$ for any $\lambda< \frac{N-2}{2}$ provided $f\in L^2(\Omega)$. This fully extends the previous $H^2$ regularity properties obtained by Kim and Tsai in \cite{Kim-Tsai} for $\lambda\leq 0$. For the second problem we show that $H^2$ regularity holds for any $\lambda>- \frac{N(N-4)}{4}$ and fails for any $\lambda\in \left(-\frac{(N-2)^2}{4},-\frac{N(N-4)}{4}\right]$. This extends sharply the range of $\lambda \in \left(-\frac{N(N-4)}{4}, \frac{N(N-4)}{4}\right)$ obtained when applying the Kato perturbation theory in \cite{Kato}. In addition, we develop sharp second order Hardy-Rellich type inequalities for the involved elliptic operators which are essential in the above proofs.

We extend the $\U$-fixed-point and $\U$-startpoint framework for quasi-uniform spaces. We reformulate the original existence framework through the filter axioms and supply a two-sided contraction condition on a generating family of quasi-pseudometrics. For this condition we prove a Banach-style fixed-point theorem on $T_0$ bicomplete quasi-uniform spaces, with a Picard-type error bound; a Boyd–Wong companion and a stability estimate follow. The conjugate quasi-uniformity yields an endpoint version, a selector argument extends the result to multivalued maps and to common startpoints of commuting families, a Knaster–Tarski variant covers monotone selectors on order-complete spaces, and a direct quasi-Hausdorff route handles weakly contractive multivalued maps. Moreover, the $T_0$-quotient removes the $T_0$ assumption from the corresponding involution-based fixed-point theorem.

We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains. Assuming that the linearized equation admits an exponential dichotomy with a reasonably large spectral gap relative to the maximal Lyapunov exponent of the background steady flow $v_0$, we construct the local stable and unstable manifolds of $(\rho_0,v_0)$. The proof is based on the Lyapunov–Perron method after reformulating the Euler equation as an ODE on the infinite-dimensional manifold of volume-preserving Lagrangian maps, with the density treated as a frozen Lagrangian parameter as well as the weight in the $L^2$ metric. We also discuss some applications to two-dimensional steady flows.

Let $C_q$ denote the group of the $q$th roots of unity. A question arising from the work of Becker, Klein, Slote, Volberg and Zhang is whether the dimension-free Bohnenblust–Hille constants for functions on $C_q^N$ grow subexponentially with the degree. We answer this question affirmatively. In fact, we prove a stronger estimate for functions whose Fourier characters involve at most $d$ coordinates. If $\BHint{d}{q}$ is the optimal constant for this larger class, then, for every fixed $q\geq2$, \[ \BHint{d}{q}\leq \exp\left(c_q\sqrt{d\log d} +O_q\left(\sqrt{\frac d{\log d}}\log\log d\right)\right), \] where $c_2=2$ and $c_q=\sqrt{2q\log(q-1)/(q-2)}$ for $q\geq3$. As an application, we obtain two-sided estimates for the Bohr radius of the Fourier layer formed by characters involving exactly $d$ coordinates, and we determine its asymptotic behaviour in natural joint regimes of $d$ and $N$.

This paper introduces a contour integral method (CIM) for efficiently computing outputs of parametric linear systems in control form over specified time intervals and to a user-prescribed accuracy. The CIM approximates the inverse Laplace transform via a quadrature rule applied along a modified integration contour. For parametric systems, we show how CIM integrates effectively with projection-based model order reduction (MOR) where a greedy algorithm builds the projection spaces following an error estimate we derive for this setting. We additionally demonstrate that the developed projection framework naturally enforces Hermite interpolation conditions. This combination substantially lowers the cost of evaluating the input-output relations across the parameter domain, for a wide range of input functions, and for initial conditions well captured by a low-dimensional subspace.. We demonstrate the accuracy and efficiency of the approach on benchmark non-parametric and parametric control systems, comparing against state-of-the-art projection-based MOR methods.

We prove that a Kazhdan group admitting a sofic embedding into a metric ultraproduct of symmetric groups with a centralizer that acts ergodically on the associated Loeb probability space is locally embeddable in finite groups (LEF). In particular, every finitely presented Kazhdan group admitting such an embedding is residually finite. The main technical theorem says that the centralizer of a sofic embedding of a Kazhdan group is itself a metric ultraproduct of permutation groups.

We study utility maximization for high-frequency trading in fractional Brownian motion models. We first consider exponential utility in the discretized arithmetic model. Using spectral methods for stationary Gaussian sequences, we derive the asymptotic growth rate of the optimal certainty equivalent and the scaling limit of the optimal positions. We then consider power and logarithmic utility in a positive-price model with positive wealth and investment fractions in $[0,1]$. In this case, the logarithm of the optimal certainty equivalent grows at rate $n^{1-H}$ for $H\neq1/2$, with a coefficient determined by the one-sided prediction error of fractional Gaussian noise. We also give a simple binary strategy which attains this leading rate, without claiming exact finite-horizon optimality.

We are interested in modelling superfluid helium-4, the common isotope of helium, of atomic number 4. We present a mathematically solvable toy model of the phase transition between the normal liquid and superfluid phases and use it to show how an order parameter can be obtained for the superfluid.

We study ruled minimal surfaces in a family of three-dimensional solvable Lie groups depending on a real parameter $p$ and endowed with left-invariant metrics. For $p\neq 0$, we classify all orthogonal ruled surfaces and construct new examples of minimal surfaces. For $p=0$, we parametrize all ruled surfaces and obtain additional examples of minimal surfaces.

We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schrödinger equation that models a one-dimensional quantum lattice. Using a recently developed exact solution for the discrete system, we derive the Dirichlet-to-Neumann maps analytically via Laplace transforms. This yields a convolution-type boundary condition governed by Bessel functions. We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit. Additionally, we present an efficient time-discretization scheme based on the trapezoidal rule for practical implementation. Numerical experiments using a Crank-Nicolson solver verify that our proposed TBCs eliminate spurious backscattering entirely and preserve reflectionless propagation of a Gaussian wave packet exiting the computational domain

A latin square of order $n$ can be viewed as a partition of the $n \times n$ all-ones matrix into permutation matrix summands. Here, we consider a relaxation in which the matrix summands are allowed to be induced from shorter permutations. For $\sigma \in S_k$, the `fuzzy permutation matrix' $P_\sigma^{\uparrow n}$ arises from combining all $\binom{n}{k}^2$ order-preserving embeddings of the $k \times k$ permutation matrix $P_\sigma$ into an $n \times n$ matrix. We define a fuzzy latin square as a linear combination of $n \times n$ fuzzy permutation matrices $P_\sigma^{\uparrow n}$ equaling a constant matrix. We study various aspects of these objects, including certain relevant vector space dimensions and a census of fuzzy latin squares with a small number of terms. In particular, we determine strong conditions on four-term fuzzy latin squares in the `vanishing' case (when the constant matrix is all zeros). We also report on a computer-assisted classification of six-term fuzzy latin squares in the non-vanishing case.

This paper is devoted to a new two-dimensional nonlinear dispersive wave model named as the Camassa-Holm-Zakharov-Kuznetsov (CH-ZK) equation which combines the nonlinear structure of the Camassa-Holm equation with the transverse Laplacian dispersion of the Zakharov-Kuznetsov equation. We first establish the local well-posedness of its Cauchy problem in a suitable Sobolev space and derive a blow-up criterion for strong solutions. Then the finite-time blow-up strong solutions for the CH-ZK equation have been constructed. Moreover, we prove a unique continuation property for the solutions to the CH-ZK equation. Finally, we investigate the existence of both peaked and smooth solitary waves, and obtain a rigidity theorem for the traveling wave solutions according to the magnitude of wave speed.

The year 2026 marks the 150th anniversary of a remarkable achievement in mathematics: in 1876, the French mathematician Édouard Lucas showed that the 39-digit number $M_{127} := 2^{127} -1$ is prime. This stood for 75 years as the largest known prime, and it remains the largest prime number discovered and certified without the aid of a mechanical device. We review the history behind this discovery and give a modern proof of the closely related Lucas-Lehmer test, which is still the main engine behind the certification of large prime numbers.

We develop an infinite-dimensional Bayesian framework for recovering spatially varying log-permeability from pressure observations in single-phase Darcy flow with a moving boundary, motivated by resin transfer moulding. The main contribution is an observation-wise representer formulation linking sensitivities of the parameter-to-observable map to the moving-boundary state and adjoint systems. This yields a reduced Levenberg–Marquardt method for maximum a posteriori (MAP) estimation on the Cameron–Martin space of a Gaussian prior and an explicit finite-rank covariance for the linearised MAP (LMAP) approximation. Once the representers are computed, the LM update and covariance assembly reduce to linear algebra in observation space, independently of the discretisation dimension. In 1D, the explicit moving-boundary solution is used to establish posterior well-posedness and existence of MAP estimators, and to derive closed-form Fréchet derivatives and representers. In 2D, we introduce a very weak mixed formulation coupling pressure, interface velocity and domain evolution. A formal one-sided directional shape linearisation yields the corresponding linearised state and adjoint systems, with observation-wise representers computed through independent, parallel adjoint problems. Numerical experiments show that the 1D LMAP approximation closely matches a reference MCMC posterior while requiring \(\mathcal{O}(10^1)\) rather than \(\mathcal{O}(10^5\text{--}10^6)\) forward simulations. In 2D, the method produces reconstructions and uncertainty estimates comparable to ensemble Kalman inversion with large ensembles, at substantially lower computational cost. The workflow is demonstrated across several geometries and experimental configurations without reformulating the inversion methodology.

Two characterizations of geometric finiteness in terms of weaker conditions are known for actions on Gromov hyperbolic spaces. In this paper, we establish analogous characterizations for general convergence group actions. To this end, we introduce the notion of an orbit uniform metric. We prove that a point is either conical or bounded parabolic if and only if its orbit is discrete with respect to an orbit uniform metric. As a consequence, we characterize geometric infiniteness in terms of the uncountability of the set of non-conical limit points. We further characterize geometric infiniteness by the existence of escaping sequences of hyperbolic elements, thereby extending the corresponding result for actions on Gromov hyperbolic spaces.

Edge open packing is a relaxation of induced matching in which the selected edges may induce disjoint stars. We study the Maximum Edge Open Packing problem on permutation graphs, interval graphs, and well-partitioned chordal graphs. For the first two classes, we introduce an oriented star-conflict graph whose vertices are ordered edges. We prove that its compatibility graph admits a natural transitive orientation: a product-order orientation for permutation graphs and a left-to-right orientation for interval graphs. In each case, a maximum edge open packing is obtained from a maximum clique, equivalently a longest directed path, in the compatibility graph. Given the corresponding representation, both algorithms run in \(O(n^2+m^2)\leq O(n^4)\) time, where \(n=|V(G)|\) and \(m=|E(G)|\). For well-partitioned chordal graphs, we give a dynamic program over a partition tree. Its states use the fact that the endpoint set of an edge open packing meets each clique bag in at most two vertices. Given a partition-tree representation, the edge open packing number is computed in \(O(n^4)\) time, and an optimal packing can be reconstructed within the same time bound.

We present several explicit classes of algebraic integers $\alpha>1$ such that $\lfloor \alpha^n\rfloor$ is composite for all but finitely many $n$.

For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets. Let $X_1(n)^\circ$ denote the open subscheme of the modular curve $X_1(n)$ obtained by removing the cusps corresponding to reducible Néron polygons. For $n \geq 10$, we give a purely algebraic and incidence-theoretic proof that, for every field $k$ with $\mathrm{char}(k)$ not dividing $n$, there is a natural bijection between $X_1(n)^\circ(k)$ and rescaling classes of $k$-realizations of $T_n$. For $k = \mathbb{C}$, this recovers a theorem of Borisov and Roulleau. We then upgrade the field-valued correspondence to an isomorphism of schemes over $\mathbb{Z}[1/n]$. The main new ingredient is a deformation-theoretic argument which allows us to verify the isomorphism on points valued in Artinian local rings. As consequences, the modular curve $X_1(n)^\circ$ acquires a natural model over $\mathbb{Z}[1/n]$ as a matroid realization space, and, for primes $p \geq 11$, the non-representability of $T_p$ over $\mathbb{Q}$ is equivalent to the prime-order case of Mazur's celebrated theorem on rational torsion points of elliptic curves. In an appendix, we explain how to upgrade the realization space of a matroid from an affine scheme over $\mathbb{Z}$ to an affine band scheme (in the sense of Baker-Jin-Lorscheid) over $\mathbb{F}_1^{\pm}$.

Motivated by the study of the level lines of the $(2+1)$D Solid-On-Solid (SOS) model above a floor, near the corners of the box, we derive the limit law of an ensemble of $K$ curves from a $(1+1)$D SOS model, with an area tilt, and above a wedge-shaped floor in $\{-N,\ldots,N\}$. We show that there exist explicit critical points $\alpha_0=1>\alpha_1>\ldots>\alpha_K>0$ such that, for each $r \geq 1$, along the intervals $\pm(\alpha_{r} N,\alpha_{r-1} N)$, the bottom $K+1-r$ curves, rescaled by $(N^{2/3},N^{1/3})$, tend to the law of a Geometrically-Area-Tilted Ensemble of non-crossing Brownian paths (Brownian GATE), independently across those $2K$ intervals. All other curves, centered and rescaled by $(N,\sqrt{N})$, tend to a product of $K$ suitable Brownian bridges.