arXiv preprints from January 1, 2026 through September 6, 2026 — 03:20:07 EST
The goal of this paper is to establish a local rigidity result for the integrability of standard-like maps. The main focus of the paper is the remarkable integrable potential discovered by Suris in the 80's. We show that locally, the integrability of this potential is rigid. The proof relies on a similar strategy that was used for...
It is well-known that Leonard pairs have a close connection with bispectral orthogonal polynomials of the Askey scheme. In this paper, we introduce the notion of a Leonard trio $(V,\oV,Z)$, an algebraic structure extending Leonard pairs, for which the overlap coefficients of eigenfunctions of $V$ and $\oV$ are biorthogonal rational...
We prove the following isoperimetric-type inequality: for every convex body $K$ in $\mathbb R^n$ and some $σ\subset[n]:=\{1,\dots,n\}$ there exists a suitable Hanner polytope $B_K$ with the same volume as $K$ and such that the volume of each of its orthogonal projections onto every subspace whose basis is formed by the canonical...
In this paper, we prove an extension of the noncompact version of Llarull's theorem due to Zhang and Li-Su-Wang-Zhang, giving an upper bound for the infimum of scalar curvature in terms of the bottom spectrum of the Laplacian. Moreover, we extend the theorem to manifolds with boundary, relaxing the strict positivity condition on the...
The paper investigates quantitative weak mixing of Salem substitutions flows. We prove that for a substitution whose substitution matrix is irreducible over the rationals and the dominant eigenvalue is a Salem number, for almost every suspension flow with a piecewise constant roof function, quantitative weak mixing holds with a rate...
We obtain the generic real Jordan canonical forms for $n\times n$ matrices with real entries. More precisely, we prove that the set of $n\times n$ real matrices is the union of the closures of $\lfloor n/2\rfloor+1$ sets, which are called generic bundles, as they are particular "bundles". In general, a bundle is the set of $n\times n$...
We set up and solve the initial value problem for equivariant harmonic maps of cohomogeneity one manifolds, i.e. we show the local existence of a harmonic map in the neighborhood of a singular orbit. Furthermore, we present some theory of regular-singular systems of first order.
We study bounded domains $Ω\subset\mathbb{C}^n$ whose Bergman metric is locally symmetric, i.e. its Riemannian curvature tensor is parallel with respect to the Levi-Civita connection. Following the strategy developed in \cite{UnifThm2}, we obtain two rigidity results. If the Bergman metric of $Ω$ is complete, then $Ω$ is (globally)...
We consider the free-boundary relativistic Euler equations in Minkowski spacetime $\mathbb{M}^{1+3}$ equipped with a physical vacuum boundary, which models the motion of a relativistic gas. We concern ourselves with the family of isentropic, barotropic, and polytropic gas, with an equation of state $p = ρ^{1+κ}, κ\in (0,\frac23]$. We...
In this paper, we have found some features of $\ast$- Ricci Bourguignon Soliton on Kenmotsu manifold. We estimated the conditions for $\ast$-Ricci Bourguignon on Kenmotsu manifold to be compressing, balancing or enlarging accordingly. We have found some curvature properties of Kenmotsu manifold admitting $\ast$-Ricci Bourguignon...
We investigate the conditions under which pseudo-Riemannian inner products induce pseudo-Riemannian algebraic Ricci solitons on four-dimensional Lie algebras. By analyzing the algebraic Ricci soliton equation for each four-dimensional Lie algebra, we obtain a complete description of when such pseudo-Riemannian algebraic Ricci solitons...
We are interested in the multiplicity of solutions to the following scalar field equation $$ -Δu - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\}. $$ We establish the existence of infinitely many radial and non-radial solutions.
This paper is concerned with the axiomatic basis of structures within Hypercompositional Algebra. It is proven that the axioms employed in the definition of numerous hypercompositional structures lack independence. Accordingly, novel definitions are introduced in this work which minimize the established definitions by reducing the...
The recent discovery of a family of aperiodic monotiles, which includes David Smith's famous Hat, has shaken the field of plane tessellations. Music composers have already utilised the visual representation of plane tilings in their artwork (Tom Johnson through his use of Vuza's canons, talea and color in isorhythmic motets...)....
In this paper, we study a combinatorial Ricci flow on closed pseudo $3$-manifolds $(M,\mathcal{T})$. We prove that if every edge in the triangulation $\mathcal{T}$ has valence at least $9$, then the combinatorial Ricci flow converges exponentially fast to a hyperbolic metric. As a consequence, for any compact $3$-manifold $N$ with...
We give two new upper bounds on the covering minima of convex bodies, depending on covering minima of certain projections and intersections with linear subspaces. We show one bound to be sharp for direct sums of two convex bodies, generalizing previous results on the covering radius and lattice width of direct sums. We apply our...
We consider infinite-dimensional Bayesian linear inverse problems governed by time-dependent partial differential equations (PDEs) and develop a mathematical and computational framework for optimal design of mobile sensor paths in this setting. The proposed path optimal experimental design (path-OED) framework is established...
The first non-zero Laplace eigenvalue of a hyperbolic surface, or its spectral gap, measures how well-connected the surface is: surfaces with a large spectral gap are hard to cut in pieces, have a small diameter and fast mixing times. For large hyperbolic surfaces (of large area or large genus $g$, equivalently), we know that the...
We establish foundational properties of fractional operators on Lie groups of homogeneous type. We prove embedding theorems for the associated Sobolev-type spaces.
We prove that homological filling functions over a ring $R$ equipped with the discrete norm are quasi-isometry invariants for all groups of type $\mathrm{FP}_n$. This confirms a conjecture of Bader-Kropholler-Vankov in the case of discrete norms. The proof uses a technique of equipping free chain complexes with a geometric structure,...
In this paper we present, for any integers $0\leq ν\leq n$, a set of inequalities satisfied by the Chern classes of any minimal complex projective variety of dimension $n$ and numerical dimension $ν$. In the cases where $ν$ is either very small or very large compared with $n$, this recovers many previously known results. We...
We introduce a new measure of coarseness for characterizing phase separation processes such as those described by Cahn--Hilliard equations. An advantage of our measure is that it remains consistent throughout the evolution, including for solutions with no periodic structure. We use our measure to compare two previous models of...
We study determinantal varieties from conditional independence models with hidden variables, focusing on their irreducible decompositions, dimensions, degrees, and Gröbner bases. Each variety encodes a collection of matroids, whose flats capture algebraic dependencies among variables. Using this approach, we provide a systematic...
We frame the Painlevè mechanical system, which has been extensively studied because of the paradox it generates, within the class of Regular Geometric Impulsive Mechanical Systems (RGIMS), by modeling it as a mechanical system subject to a rough unilateral positional constraint $\cal{S}$, where friction is represented by an...
Continuing our previous study \cite{DJKZ} on the monostable reaction-diffusion-convection equation, we analyze the bistable case under weak regularity assumptions. Our approach applies monostable results on the subintervals where the reaction term $g$ has constant sign, thereby establishing both existence and nonexistence of bistable...