Related papers: The number of orientable small covers over cubes
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
An orthogonal product basis (OPB) of a finite-dimensional Hilbert space $H=H_1\otimes H_2\otimes\cdots\otimes H_n$ is an orthonormal basis of $H$ consisting of product vectors $x_1\otimes x_2\otimes\cdots\otimes x_n$. We show that the…
Linear scaling methods, or O(N) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These…
We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of…
Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an…
A partial formula is provided to calculate the smallest number of vertices possible in a quadrangulation on the closed orientable 2-manifold of given genus. This extends the previously known partial formula due to N. Hartsfield and G.…
Choose a polynomial in three variables with not more than three or four monomials of moderate degree. Take simple coefficients as 1 and -1. Then draw a picture of the solution variety in real three space using a ray-tracing program like…
The growth rate function $r_N$ counts the number of irreducible representations of simple complex Lie groups of dimension $N$. While no explicit formula is known for this function, previous works have found bounds for $R_N=\sum_{i=1}^Nr_i$.…
We consider sequential random packing of cubes $z+[0,1]^n$ with $z\in \frac{1}{N}\ZZ^n$ into the cube $[0,2]^n$ and the torus $\QuotS{\RR^n}{2\ZZ^n}$ as $N\to\infty$. In the cube case $[0,2]^n$ as $N\to\infty$ the random cube packings thus…
Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials…
There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. The aim of this paper is to describe…
A set of segments in the plane may form a Euclidean TSP tour or a matching, among others. Optimal TSP tours as well as minimum weight perfect matchings have no crossing segments, but several heuristics and approximation algorithms may…
In this article, using the computer, are enumerated all locally-rigid packings by $N$ congruent circles (spherical caps) on the unit sphere ${\Bbb S}^2 $ with $N < 12.$ This is equivalent to the enumeration of irreducible spherical contact…
Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y' --> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y' respectively. We obtain apparently new estimates for the covering numbers of the…
In this paper, we obtain the limit formula of the observable diameter with non-Euclidean screen. In order to treat a sequence of observable diameters with varying screens, we define new types of observable diameters with errors.
A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of…
The main goal of this paper is to address the following problem: given a positive integer $n$, find the largest value $S(n)$ such that a square of edge length $S(n)$ in the Euclidean plane can be covered by $n$ unit squares. We investigate…
Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $\omega=(\omega_n)_{n\geq 1}$ uniformly distributed…
Each packing of R^d by translates of the unit cube [0,1)^d admits a decomposition into at most two parts such that if a translate of the unit cube is covered by one of them, then it also belongs to such a part.
It is well known that R^N has subspaces of dimension proportional to N on which the \ell_1 norm is equivalent to the \ell_2 norm; however, no explicit constructions are known. Extending earlier work by Artstein--Avidan and Milman, we prove…