Related papers: Mystery of point charges

We show that the maximum number of triples on $n$~points, if no three triples span at most five points, is $(1\pm o(1))n^2/5$. More generally, let $f^{(r)}(n;k,s)$ be the maximum number of edges of an $r$-uniform hypergraph on $n$~vertices…

By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system.…

For a dynamic system consisting of $n$ point vortices in an ideal plane fluid with a steady, incompressible and} irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under…

The goals of this paper are threefold. First, we show that a counterpart of the Newman bound related to the Chui conjecture is valid in the case where the gradient of Coulomb potential is generated by arbitrary positive charges placed at…

It is generally hard to count, or even estimate, how many integer points lie in a polytope P. Barvinok and Hartigan have approached the problem by way of information theory, showing how to efficiently compute a random vector which samples…

We consider supergravity theories with 8 supercharges in $d=6$. We show that all the proposed anomaly free theories with unbounded number of massless modes are restricted to a finite subset and thus argue that there is an upper bound on the…

We show that a point set of cardinality $n$ in the plane cannot be the vertex set of more than $59^n O(n^{-6})$ straight-edge triangulations of its convex hull. This improves the previous upper bound of $276.75^n$.

We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be…

In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…

We obtain upper bounds on the number of finite sets $\mathcal S$ of primes below a given bound for which various $2$ variable $\mathcal S$-unit equations have a solution.

We consider the Cartesian product X of n finite intervals of integers and a map F from X to itself. As main result, we establish an upper bound on the number of fixed points for F which only depends on X and on the topology of the positive…

We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.

Let $P$ be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions $n\leq N$ to $n! = P(x)$ which yields a power saving over the trivial bound. In particular, this applies…

We establish some upper and lower bounds for the number of rational points of Prym varieties over finite fields.

We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.

Maximal inequalities refer to bounds on expected values of the supremum of averages of random variables over a collection. They play a crucial role in the study of non-parametric and high-dimensional estimators, and especially in the study…

Let $M^m$ be an $m$-dimensional, smooth and closed manifold, equipped with a smooth involution $T\colon M^m \to M^m$ fixing submanifolds $F^n$ and $F^4$ of dimensions $n$ and $4$, respectively, where $4<n<m$ and $F^n\cup F^4$ does not…

In how many ways can $n$ queens be placed on an $n \times n$ chessboard so that no two queens attack each other? This is the famous $n$-queens problem. Let $Q(n)$ denote the number of such configurations, and let $T(n)$ be the number of…

We investigate the number of 4-edge paths in graphs with a fixed number of vertices and edges. An asymptotically sharp upper bound is given to this quantity. The extremal construction is the quasi-star or the quasi-clique graph, depending…

In the Newtonian 3-body problem, for any choice of the three masses, there are exactly three Euler configurations (also known as the three Euler points). In Helmholtz' problem of 3 point vortices in the plane, there are at most three…