Related papers: Four-point semidefinite bound for equiangular line…

We give upper bounds on the number of exceptional radial projections of arbitrary subsets of vector spaces over finite fields. Our bounds do not depend on the dimension of the ambient space. Let $\mathbb{F}_q^d$ be the $d$-dimensional…

A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ sides is not known when $s \ge 4$. In this paper, we construct a family of convex equilateral small $n$-gons,…

Let $\mathscr{S}_n(q)$ denote the set of symmetric bilinear forms over an $n$-dimensional $\mathbb{F}_q$-vector space. A subset $\mathcal{C}$ of $\mathscr{S}_n(q)$ is called a $d$-code if the rank of $A-B$ is larger than or equal to $d$ for…

In this paper, we study a partially overdetermined mixed boundary value problem in a half ball. We prove that a domain in which this partially overdetermined problem admits a solution if and only if the domain is a spherical cap…

A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…

Let $\mathrm{R}$ be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of $\mathrm{R}^k$ in terms of the number and degrees of the defining polynomials has been an important problem in…

Erdos, Herzog and Piranian asked whether, for $n$ points in the plane with fixed diameter (maximum distance between points), an arrangement of a regular $n$-gon maximizes their product of all pairs of distances. Recently, it was discovered…

By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constructed within a diagonal-norm summation-by-parts (SBP)…

A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any distinct two elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a…

Solutions of the semilinear wave equation are found numerically in three spatial dimensions with no assumed symmetry using distributed adaptive mesh refinement. The threshold of singularity formation is studied for the two cases in which…

This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles…

We derive the lower bound for Frobenius number of symmetric (not complete intersection) semigroups generated by four elements.

We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…

We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the…

Let $P$ be a collection of $n$ points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of $O(n^{2+\epsilon})$, for any $\epsilon>0$, on the maximum number of discrete changes that the…

We prove a new upper bound for the number of incidences between points and lines in a plane over an arbitrary field $\mathbb{F}$, a problem first considered by Bourgain, Katz and Tao. Specifically, we show that $m$ points and $n$ lines in…

We use semidefinite programming to bound the fractional cut-cover parameter of graphs in association schemes in terms of their smallest eigenvalue. We also extend the equality cases of a primal-dual inequality involving the…

We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…

The Erd\H{o}s-Anning theorem states that every point set in the Euclidean plane with integer distances must be either collinear or finite. More strongly, for any (non-degenerate) triangle of diameter~$\delta$, at most $O(\delta^2)$ points…

Semiclassical asymptotics for linear Schr\"odinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few years, as a proxy for the treatment…