Related papers: Shattering bounds for tuple systems

The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\theta$ starts with a random set of open…

Error-correcting codes resilient to synchronization errors such as insertions and deletions are known as insdel codes. Due to their important applications in DNA storage and computational biology, insdel codes have recently become a focal…

Let V be a smooth projective 3-fold of general type. Denote by $K^3$, a rational number, the self-intersection of the canonical sheaf of any minimal model of V. One defines $K^3$ as the canonical volume of $V$. Assume $p_g\ge 2$. We show…

A variational technique to describe the ground and scattering states below the break-up threshold for a three-nucleon system is developed. The method consists in expanding the wave function in terms of correlated Harmonic Hyperspherical…

In 1965, Bollob\'as proved that for a Bollob\'as set-pair system $\{(A_i,B_i)\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i|+|B_i|}{A_i}^{-1}$ is $1$. Heged\"{u}s and Frankl recently extended the concept of Bollob\'as…

In this paper we study the hard sphere packing problem in the Hamming space by the cavity method. We show that both the replica symmetric and the replica symmetry breaking approximations give maximum rates of packing that are asymptotically…

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $\mathbb Z^d$ from the limiting Wulff shape in arbitrary dimensions. As the number $n$ of elements diverges, we prove that the symmetric difference to…

We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of $n$ points from $[0,1]^d$ and cover-free families from the extremal set theory. This connection was discovered in a recent paper of…

A new computational method for solving the configuration-space Faddeev equations for the breakup scattering problem has been applied to nd scattering both below and above the two-body threshold.

Let $V$ be a norm-closed subset of the unit sphere of a Hilbert space $H$ that is stable under multiplication by scalars of absolute value 1. A {\em maximal vector} (for $V$) is a unit vector $\xi\in H$ whose distance to $V$ is maximum…

We estimate the size of the spectral gap at zero for some Hermitian block matrices. Included are quasi-definite matrices, quasi-semidefinite matrices (the closure of the set of the quasi-definite matrices) and some related block matrices…

We investigate the large scale chaotic, topological structure of the trajectories of an infinite sequence of dispersing, hence ergodic, $2D$ billiards with the configuration space $Q_n=\mathbb{T}^2 \setminus \bigcup_{i=0}^{n-1} D_i$, where…

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that the sumset $NA$ has predictable size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in some finite cone other than all…

We study the upper bounds for $A(n,d)$, the maximum size of codewords with length $n$ and Hamming distance at least $d$. Schrijver studied the Terwilliger algebra of the Hamming scheme and proposed a semidefinite program to bound $A(n, d)$.…

The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing…

We show that there are shape-independent upper bounds to the extinction cross section per unit volume of randomly oriented nanoparticles, given only material permittivity. Underlying the limits are restrictive sum rules that constrain the…

It is verified that the number of vertices in a $d$-dimensional cubical pseudomanifold is at least $2^{d+1}$. Using Adin's cubical $h$-vector, the generalized lower bound conjecture is established for all cubical 4-spheres, as well as for…

We consider a diffusion process $X_{t}$ and a skeleton curve $x_{t}(\phi)$ and we give a lower bound for $P(\sup_{t\leq T}d(X_{t},x_{t}(\phi))\leq R)$. This result is obtained under the hypothesis that the strong H\"{o}rmander condition of…

We show that if $\mathcal{E}$ is a subset of the $d$-dimensional vector space over a finite field $\mathbbm{F}_q$ ($d \geq 3$) of cardinality $|\mathcal{E}| \geq (d-1)q^{d - 1}$, then the set of volumes of $d$-dimensional parallelepipeds…

We consider the scalar Zakharov system in $\R^3$ for initial conditions $(\psi(0), n(0), n_t(0)) \in H^{\ell+1/2} \times H^\ell \times H^{\ell-1} $, $0\leq\ell \leq 1$. Assuming that the solution blows up in a finite time $t^* < \infty$, we…