Related papers: An improvement on Furstenberg's intersection probl…

The geometry of non-smooth $A_{n>2}$ caustics in solutions of the Helmholtz equation is analyzed using a Fock-Schwinger proper-time formulation. In this description, $A_3$ or cusp caustics are intimately related to poles of a quantity…

The intersection index at a common point of two analytic varieties of complementary dimensions in $\Bbb C^n$ is positive. This observation, which has been called a ``cornerstone'' of algebraic geometry ([GH, p.~62]), is a simple consequence…

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the…

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic…

We introduce the notion of intersective polynomials having coefficients in the ring of integers $\mathscr{O}_K$ of a number field $K$, and define a notion of upper density of subsets of $\mathscr{O}_K$. We prove that given any intersective…

The article continues the study of the 'regular' arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification…

We prove that every local complete intersection curve in $Spec(A)$, where $A$ is a commutative Noetherian ring of dimension three, is a set-theoretic complete intersection. An analogous result is established for local complete intersection…

We consider 5d supersymmetric gauge theories with unitary groups in the $\Omega$-background and study codim-2/4 BPS defects supported on orthogonal planes intersecting at the origin along a circle. The intersecting defects arise upon…

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block…

Let $G=(V,E)$ be a graph and $P\subseteq V$ a set of points. Two points are mutually visible if there is a shortest path between them without further points. $P$ is a mutual-visibility set if its points are pairwise mutually visible. The…

Many problems in extremal combinatorics can be reduced to determining the independence number of a specific auxiliary hypergraph. We present two such problems, one from discrete geometry and one from hypergraph Tur\'an theory. Using results…

We extend the Nonconventional Ergodic Theorem for generic measures by Furstenberg, to several situations of interest arising from quantum dynamical systems. We deal with the diagonal state canonically associated to the product state (i.e.…

Congruences, or $2$-parameter families of lines in $3$-space are of interest in many situations, in particular in geometric optics. In this paper we consider elements of their geometry which are invariant under affine changes of…

We study identifiability for bilinear inverse problems under sparsity and subspace constraints. We show that, up to a global scaling ambiguity, almost all such maps are injective on the set of pairs of sparse vectors if the number of…

The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the…

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P^n, n>=3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and…

Let $\mathcal{T}$ be a set of $n$ flat (planar) semi-algebraic regions in $\mathbb{R}^3$ of constant complexity (e.g., triangles, disks), which we call plates. We wish to preprocess $\mathcal{T}$ into a data structure so that for a query…

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed in Li and Pott (arXiv:2003.06678v1), which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in…

We investigate a dynamic inverse problem using a regularization which implements the so-called Wasserstein-$1$ distance. It naturally extends well-known static problems such as lasso or total variation regularized problems to a (temporally)…

The Furstenberg recurrence theorem (or equivalently, Szemer\'edi's theorem) can be formulated in the language of von Neumann algebras as follows: given an integer $k \geq 2$, an abelian finite von Neumann algebra $(\M,\tau)$ with an…