Related papers: Covering radius in the Hamming permutation space

For a graph $H$, the 1-subdivision of $H$, denoted by $H'$, is the graph obtained by replacing the edges of $H$ by internally disjoint paths of length 2. Recently, Conlon, Janzer and Lee (arXiv: 1903.10631) asked the following question: For…

Minimum numbers decide e.g. whether a given map f: S^m --> S^n/G from a sphere into a spherical space form can be deformed to a map f' such that f(x) not equal f'(x) for all x in S^m. In this paper we compare minimum numbers to…

Let $\mathbb{D}$ be a division ring and $\mathbb{F}$ be a subfield of the center of $\mathbb{D}$ over which $\mathbb{D}$ has finite dimension $d$. Let $n,p,r$ be positive integers and $\mathcal{V}$ be an affine subspace of the…

Let $z\in\mathbb C^n$ and $\|z\|$ be its Euclidean norm. Ebenfelt proposed a conjecture regarding the possible ranks of the Hermitian polynomials in $z,\bar z$ of the form $A(z,\bar z)\|z\|^2$, known as the SOS Conjecture, where SOS stands…

Given integers $ n \ge k >l \ge 1 $ and a $k$-graph $F$ with $|V(F)|$ divisible by $n$, define $t_l^k(n,F)$ to be the smallest integer $d$ such that every $k$-graph $H$ of order $n$ with minimum $l$-degree $\delta_l(H) \ge d $ contains an…

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations \pi, \sigma in S there is a point i in {1,...,n} such that \pi(i)=\sigma(i). Deza and Frankl \cite{MR0439648} proved that if S a…

We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f…

The Matrix Spencer Conjecture asks whether given $n$ symmetric matrices in $\mathbb{R}^{n \times n}$ with eigenvalues in $[-1,1]$ one can always find signs so that their signed sum has singular values bounded by $O(\sqrt{n})$. The standard…

Let $F_h(n)$ denote the minimum cardinality of an additive {\em $h$-fold basis} of $\{1,2,\cdots,n\}$: a set $S$ such that any integer in $\{1,2,\cdots, n\}$ can be written as a sum of at most $h$ elements from $S$. While the trivial bounds…

In this paper, we consider the limit $ \lim_{n \to \infty} \sum_{v\in S} \lambda_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) $ where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$ over a number field,…

We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes…

Let $Gr(k,n)$ be the Pl\"ucker embedding of the Grassmann variety of projective $k$-planes in $\P n$. For a projective variety $X$, let $\sigma_s(X)$ denote the variety of its $s-1$ secant planes. More precisely, $\sigma_s(X)$ denotes the…

Consider the linear space of functions on the binary hypercube and the linear operator $S_\delta$ acting by averaging a function over a Hamming sphere of radius $\delta n$ around every point. It is shown that this operator has a…

We show that the flag manifold $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$, with Grassmannian the special case $p=1$, has an $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding in an Euclidean space of dimension $(n-1)(n+2)/2$,…

What is the length of the shortest sequence $S$ of reals so that the set of consecutive $n$-words in $S$ form a covering code for permutations on $\{1,2, >..., n\}$ of radius $R$ ? (The distance between two $n$-words is the number of…

Let $X_1,X_2, \ldots $ and $Y_1, Y_2, \ldots$ be i.i.d. random uniform points in a bounded domain $A \subset \mathbb{R}^2$ with smooth or polygonal boundary. Given $n,m,k \in \mathbb{N}$, define the {\em two-sample $k$-coverage threshold}…

We study functions defined on the vertices of the Hamming graphs $H(n,q)$. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work, we…

We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $\mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian…

We seek an analog for the quantum permutation group $S_n^+$ of the normalized Hamming distance for permutations. We define three distances on the tracial state space of $C(S_n^+)$ that generalize the $L^1$-Wasserstein distance of…

We show how Lasry-Lions's result on regularization of functions defined on $\mathbb{R}^n$ or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds $M$ of…