Related papers: The Lonely Runner Conjecture
Satellite conjunctions involving "near misses" of space objects are becoming increasingly likely. One approach to risk analysis for them involves the computation of the collision probability, but this has been regarded as having some…
An old conjecture of Durfee 1978 bounds the ratio of two basic invariants of complex isolated complete intersection surface singularities: the Milnor number and the singularity (or geometric) genus. We give a counterexample for the case of…
Over the past few years it has been discovered that an "observable" can be set up on the lattice which obeys the discrete Cauchy-Riemann equations. The ensuing condition of discrete holomorphicity leads to a system of linear equations which…
Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. We establish a fully-inhomogeneous version of Gallagher's theorem, a diophantine fibre refinement, and a sharp and unexpected threshold for…
We study randomized variants of two classical algorithms: coordinate descent for systems of linear equations and iterated projections for systems of linear inequalities. Expanding on a recent randomized iterated projection algorithm of…
This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single…
In this paper we extend the notion of a Lorentz cone. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., monotone) with respect to the order…
We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of…
Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal…
An alternative approach towards quantum theory is described, and tentative attempts to connect his approach to special and general relativity are discussed. Important concepts are gauge groups and information/entropy connected to some…
The overall aim of this note is to initiate a "manifold" theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite…
I examine the results obtained so far in exploring the recent proposal of theories of the relativistic transformations between inertial observers that involve both an observer-independent velocity scale and an observer-independent…
In this survey, we give a short overview of the recent progress on the multidimensional L2 conjecture. It can also serve as an introduction to the subject.
The concept of distance covariance/correlation was introduced recently to characterize dependence among vectors of random variables. We review some statistical aspects of distance covariance/correlation function and we demonstrate its…
Loneliness does not only have emotional implications on a person but also on his/her well-being. The study of loneliness has been challenging and largely inconclusive in findings because of the several factors that might correlate to the…
In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most…
This brief survey of some singularity invariants related to Milnor fibers should serve as a quick guide to references. We attempt to place things into a wide geometric context while leaving technicalities aside. We focus on relations among…
This work proposes an algorithm to bound the minimum distance between points on trajectories of a dynamical system and points on an unsafe set. Prior work on certifying safety of trajectories includes barrier and density methods, which do…
The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in $R^n$ akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the…
We survey the classical results of the Dirichlet Approximation Theorem.