Related papers: The Width of a Ball in a Hypercube

We derive formulas for the number of polycubes of size $n$ and perimeter $t$ that are proper in $n-1$ and $n-2$ dimensions. These formulas complement computer based enumerations of perimeter polynomials in percolation problems. We…

We present a deterministic polynomial-time algorithm for estimating the volume of a hypercube intersected by a fixed number of constraints of the type $f(x) \leq b$, where $f$ is the sum of univariate functions that are each nonnegative,…

In this paper we study the problem of maximizing the distance to a given point $C_0$ over a polytope $\mathcal{P}$. Assuming that the polytope is circumscribed by a known ball we construct an intersection of balls which preserves the…

This report presents an expression for the number of a multiset's sub-multisets of a given cardinality as a function of the multiplicity of its elements. This is also the number of distinct samples of a given size that may be produced by…

We prove that at least $\Omega(n^{0.51})$ hyperplanes are needed to slice all edges of the $n$-dimensional hypercube. We provide a couple of applications: lower bounds on the computational complexity of parity, and a lower bound on the…

We present a necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball of initial states at some time, of an ordinary differential equation to be convex. In particular, convexity is guaranteed…

It is possible to have a packing by translates of a cube that is maximal (i.e.\ no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness…

For a given undirected graph $G$, an \emph{ordered} subset $S = {s_1,s_2,...,s_k} \subseteq V$ of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in…

In tri-partite systems, there are three basic biseparability, $A$-$BC$, $B$-$CA$ and $C$-$AB$ biseparability according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the…

Some of the most remarkable tilings and discrete quasiperiodic sets used in quasicrystal physics can be obtained by using strip projection method in a superspace of dimension four, five or six, and the projection of a unit hypercube as a…

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…

We consider the problem of determining whether a set of primes, or, more generally, prime ideals in a number field, can be realized as a finite union of residue classes, or of Frobenius conjugacy classes. We give criteria for a set to be…

Let $V \subset \mathbb{R}$ be a finite set with $|V| = n $ and suppose we are given each pairwise distance independently with probability $p$. We show that if $p = (1+\epsilon)/n$, for some fixed $\epsilon >0$, then we can reconstruct a…

In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower…

Equations over linearly ordered semilattices are studied. For any equation $t(X)=s(X)$ we find irreducible components of its solution set and compute the average number of irreducible components of all equations in $n$ variables.

This article studies the number of ways of selecting $k$ objects arranged in $p$ circles of sizes $n_1,\ldots,n_p$ such that no two selected ones have less than $s$ objects between them. If $n_i\geq sk+1$ for all $1\leq i \leq p$, this…

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined…

For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior…

We obtain a new lower bound on the radius of the largest ball of separable unnormalized states around the identity matrix for a 3-qubit system. This also enables us to improve the corresponding lower bounds for multi-qubit systems. These…

This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…