Related papers: Lines Missing Every Random Point

We consider weighted geodesic random walks in a complete Riemannian manifold $(M,g)$. We show that for almost all sequences of weights (with respect to a suitable measure), these weighted geodesic random walks satisfy, when suitably scaled,…

An oblivious point on a translation surface is a point with no closed geodesic passing through it. Nguyen, Pan, and Su (2017) showed that there are at most finitely many oblivious points on any given translation surface and constructed a…

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and…

We consider the problem of choosing Euclidean points to maximize the sum of their weighted pairwise distances, when each point is constrained to a ball centered at the origin. We derive a dual minimization problem and show strong duality…

We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In…

In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent…

We give examples of real enumerative problems without real solutions. Most of the examples concern rational curves in ${\mathbb C}{\mathbb P}^3$ passing through a real set of points and lines.

Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large,…

Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…

We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of…

We show that, for two-dimensional space-periodic incompressible flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy achieved in standard Eulerian spectral methods. This allows the determination…

Given a computable probability measure P over natural numbers or infinite binary sequences, there is no computable, randomized method that can produce an arbitrarily large sample such that none of its members are outliers of P.

We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.

We establish the weak large deviations principle for empirical measures of Markov chains on $\mathbb R^d$ under mild assumptions. In particular, no irreducibility is assumed and the initial measure may be arbitrary. The proof is entirely…

We identify complete monotonicity properties underlying a variety of well-known sharp Strichartz inequalities in euclidean space.

We assume that the points in volumes smaller than an elementary volume (which may have a Planck size) are indistinguishable in any physical experiment. This naturally leads to a picture of a discrete space with a finite number of degrees of…

The motion of a compact body in space and time is commonly described by the world line of a point representing the instantaneous position of the body. In General Relativity such a world-line formalism is not quite straightforward because of…

Let $L_{a,b}$ be a line in the Euclidean plane with slope $a$ and intercept $b$. The dimension spectrum $\spec(L_{a,b})$ is the set of all effective dimensions of individual points on $L_{a,b}$. The dimension spectrum conjecture states…

Suppose that 2d-2 tangent lines to the rational normal curve z\mapsto (1 : z : ... : z^d) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always…