Mathematics

We isolate three structural conditions on trust-region radius update rules for smooth unconstrained nonlinear optimisation, and study the class of mechanisms they define. The conditions act on the radius directly: a lower bound relative to…

We extend the construction of Steinberg algebras of ample groupoids to \'etale semicategories. We also relate ample semicategories to Boolean restriction semigroups via a representation result extending previously known results for…

Let $G$ be a locally compact second countable amenable group acting on a finite von Neumann algebra $(\mathcal{M},\tau)$ by trace-preserving automorphisms. In this article, we establish a Jacobs-de Leeuw-Glicksberg decomposition for this…

In 1934, Romanoff proved that the set of positive integers representable as the sum of a prime and a power of two has positive lower density. Erd\H{o}s later constructed an infinite arithmetic progression of odd integers none of which…

A classical extremal problem on progression free sets is to determine the maximum size of a $3$-term arithmetic progression free set in algebraic structures, for instance in intervals of integers or in finite vector spaces. To determine the…

Can a finite set of lattice points determine many rectangles and few isosceles triangles? This turns out to be a surprisingly interesting question in combinatorial geometry that we answer using basic analytic number theory combined with a…

Owing to the effectiveness of Tensor Train (TT) decomposition in managing high-order tensors, low-rank tensor completion within the TT-format has emerged as a prominent research focus. In this paper, we leverage the left-orthogonal property…

We construct a nested version of the non-commutative Hilbert scheme and embed the nested Hilbert scheme of points on $\mathbb{C}^n$ as the commutativity locus. In the $\mathbb{C}^2$-case, we exhibit this locus as the zero locus of two…

The real log canonical threshold (RLCT) is a central invariant in birational geometry and singularity theory, measuring the complexity of a singularity through discrepancy and valuation data on a log resolution. Beyond this…

We introduce and develop the theory of braided cogroupoids, a class of algebraic structures generalizing cogroupoids in a braided setting. We show that braided cogroupoids induce monoidal equivalences between the associated comodule…

We establish a criterion for Property $N_p$ for line bundles on a class of smooth projective toric varieties. More precisely, we prove that if a smooth projective toric variety $X$ of dimension $n\ge2$ satisfies the uniform unimodularity…

To our knowledge, there is no rigorous mathematical justification of the Reynolds equation for a spherical bearing. In this article, we demonstrate that the solution of the Stokes problem in a domain between two closely spaced spheres…

We introduce the class of relatively weakly convex functions, which extends the classical notion of weak convexity by measuring nonconvexity relative to a distance-generating function. We investigate the fundamental properties of this…

Let $k\geq 2$ be a positive integer. It is known that the set of visible lattice points from the origin in $\mathbb{Z}^k$ has a translation bounded pure point diffraction spectrum. We investigate these properties for sets of points…

In this article, we present a self-contained proof of Gromov's dihedral rigidity conjecture on scalar curvature in the three-dimensional case. The proof avoids many of the technical complications that arise in higher dimensions, while still…

In the study of energy-preserving methods for Hamiltonian systems, polynomial continuous-stage Runge--Kutta methods play an important role. Necessary and sufficient conditions for such methods to be energy-preserving have already been…

Let $E$ be a stable holomorphic vector bundle over a compact K\"ahler (or Gauduchon) manifold $(M,\omega_g)$. We show that for any real number $\mu>0$ and any initial Hermitian metric $h_0$ on $E$, there exists a unique iteration sequence…

Let $\Omega$ be a bounded domain in $R^d$. Denote by $\lambda_k$ (resp. $\mu_k$) the eigenvalues of the Laplace operator in $\Omega$ with Dirichlet (resp. Neumann) boundary conditions. Denote by $\Psi = \Psi (d,k,\Omega)$ the shift of…

We study a Mean Field Control system arising in the management of fisheries with a special emphasis on non-uniqueness issues. Namely, we focus on a situation where a group of players coordinate in order to harvest a fishery in the most…

This contribution studies the Boltzmann scheme on a ``D2T4''grid constructed on meshes using equilateral triangles. The center of each triangle is connected to itself and to three other triangles via the edges of the mesh. We adopt the…