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We derive fundamental asymptotic results for the expected covering radius $rho(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sphere $mathbb{S}^d subset mathbb{R}^{d+1}$, we obtain the precise asymptotic that $mathbb{E}rho(X_N)[N/log N]^{1/d}$ has limit $[(d+1)upsilon_{d+1}/upsilon_d]^{1/d}$ as $N to infty $, where $upsilon_d$ is the volume of the $d$-dimensional unit ball. This proves a recent conjecture of Brauchart et al. as well as extends a result previously known only for the circle. Likewise we obtain precise asymptotics for the expected covering radius of $N$ points randomly distributed on a $d$-dimensional ball, a $d$-dimensional cube, as well as on a 3-dimensional polyhedron (where the points are independently distributed with respect to volume measure). More generally, we deduce upper and lower bounds for the expected covering radius of $N$ points that are randomly and independently distributed on a metric measure space, provided the measure satisfies certain regularity assumptions.
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In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $operatorname{conv}(p)ge min{frac{1}{2}operatorname{inj}(p),operatorname{foc}(B_{operatorname{inj}(p)}(p))}$, where $operatorname{inj}(p)$ is the injectivity radius of $p$ and $operatorname{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $operatorname{inj}(q)ge min{operatorname{inj}(p), operatorname{conj}(q)}-d(p,q),$ where $operatorname{conj}(q)$ is the conjugate radius of $q$; 3) for any $0<r<min{operatorname{inj}(p),frac{1}{2}operatorname{conj}(B_{operatorname{inj}(p)}(p))}$, any (not necessarily minimizing) geodesic in $B_r(p)$ has length $le 2r$. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.
In the randomly-oriented Manhattan lattice, every line in $mathbb{Z}^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $mathbb{Z}^d$ and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.
A new family of binary linear completely transitive (and, therefore, completely regular) codes is constructed. The covering radius of these codes is growing with the length of the code. In particular, for any integer r > 1, there exist two codes with d=3, covering radius r and length 2r(4r-1) and (2r+1)(4r+1), respectively. These new completely transitive codes induce, as coset graphs, a family of distance-transitive graphs of growing diameter.
Let $mathcal{S}_n$ denote the set of permutations of ${1,2,dots,n}$. The function $f(n,s)$ is defined to be the minimum size of a subset $Ssubseteq mathcal{S}_n$ with the property that for any $rhoin mathcal{S}_n$ there exists some $sigmain S$ such that the Hamming distance between $rho$ and $sigma$ is at most $n-s$. The value of $f(n,2)$ is the subject of a conjecture by Kezdy and Snevily, which implies several famous conjectures about latin squares. We prove that the odd $n$ case of the Kezdy-Snevily Conjecture implies the whole conjecture. We also show that $f(n,2)>3n/4$ for all $n$, that $s!< f(n,s)< 3s!(n-s)log n$ for $1leq sleq n-2$ and that [f(n,s)>leftlfloor frac{2+sqrt{2s-2}}{2}rightrfloor frac{n}{2}] if $sgeq 3$.
We define kinetic Brownian motion on the diffeomorphism group of a closed Riemannian manifold, and prove that it provides an interpolation between the hydrodynamic flow of a fluid and a Brownian-like flow.
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