Related papers: A New Upper Bound on Rubik's Cube Group
We give explicit upper bounds for the entropy in the cusp for one-parameter diagonal flows on $SL_d(\mathbb{R})/SL_{d}(\mathbb{Z})$. These results include bounds for the entropy of the cusp as a whole, as well as for the cusp regions…
A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets $A, B\subset\mathbb R$, each of size $n^{1/2}$, the convex grid…
Rubik's Revenge, a 4x4x4 variant of the Rubik's puzzles, remains to date as an unsolved puzzle. That is to say, we do not have a method or successful categorization to optimally solve every one of its approximately $7.401 \times 10^{45}$…
This paper focuses on greedy expansions, one possible representation of numbers, and on arithmetical operations with them. Performing addition or multiplication some additional digits can appear. We study bounds on the number of such digits…
Various authors have calculated how many pairwise incomparable points can be selected from a partially ordered set. We tackle this question for the family of subsets of a finite set obtained by removing or adding a bounded number of…
We propose a method for computing upper bounds for the Heilbronn problem for triangles.
Let $F$ be a finite field of characteristic $p>0$ with $q = p^{n}$ elements. In this paper, a complete characterization of the unit groups $U(FG)$ of group algebras $FG$ for the abelian groups of order $32$, over finite field of…
We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our…
We improve the upper bound on the Ramsey number R(3,10) from 42 to 41. Hence R(3,10) is equal to 40 or 41.
We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy…
In this paper, we introduce new general frameworks for estimating the maximal dimension of Hilbert cubes contained in finite truncations of arbitrary sets. As applications, we investigate Hilbert cubes in a range of arithmetic sets,…
Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use their method to produce a refined bound on the number of…
Let $U_n(q)$ be the upper triangular group of degree $n$ over the finite field $\F_q$ with $q$ elements. In this paper, we present constructions of large degree ordinary irreducible representations of $U_n(q)$ where $n\geq 7$, and then…
Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions.…
Linckelmann showed in 2011 that a group algebra is separably equivalent to the group algebra of its Sylow p-subgroups. In this article we use this relationship, together with Mackey decomposition, to demonstrate that a group algebra of a…
We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb{F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a…
A dominating set on an $n $-dimensional hypercube is equivalent to a binary covering code of length $n $ and covering radius 1. It is still an open problem to determine the domination number $\gamma(Q_n)$ for $ n\geq10$ and $…
Several upper bounds on the size of quantum codes are derived using the linear programming approach. These bounds are strengthened for the linear quantum codes.
Let $N$ denote the maximum number of congruent infinite cylinders that can be arranged in $\mathbb{R}^3$ so that every pair of cylinders touches each other. Littlewood posed the question of whether $N=7$, which remains unsolved. In this…
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane.…