Related papers: A New Upper Bound on Rubik's Cube Group
Various theories of quantum gravity predict the existence of a minimum length scale, which implies the Planck-scale modifications of the Heisenberg uncertainty principle to a so-called generalized uncertainty principle (GUP). Previous…
We give an improved upper bound for the Gr\"unbaum--Hadwiger--Ramos problem: Let $d,n,k \in \mathbb{N}$ such that $d \geq 2^n(1+2^{k-1})$. Given $2^{n+1}$ masses on $\mathbb{R}^d$, there exist $k$ hyperplanes in $\mathbb{R}^d$ that…
The Rubik's cube was invented in 1974 by Erno Rubik, who had no idea of the incredible popularity and mathematical fascinations his toy would bring. Through the years of study on the mathematical properties of the cube, the Rubik's Cube…
Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups.…
Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of…
We show that a graph with $n$ vertices and vertex cover of size $k$ has at most $4^k + n$ potential maximal cliques. We also show that for each positive integer $k$, there exists a graph with vertex cover of size $k$, $O(k^2)$ vertices, and…
We study the irreducible representations of simple algebraic groups in which some non-central semisimple element has at most one eigenvalue of multiplicity greater than 1. We bound the multiplicity of this eigenvalue in terms of the rank of…
For prime powers q<100, we compute new upper and lower bounds on N_q(4), the maximal number of points on a genus-4 curve over a finite field with q elements. We determine the exact value of N_q(4) for 17 prime powers q for which the value…
We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of "close" abelian cubic number fields with class numbers as large as possible. We also give a first step toward an…
This paper continues arXiv.org:math.AG/0609256, arXiv:0708.3991 and arXiv:0710.0162 . Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups…
The existence of a uniform upper bound for the maximum number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line has been subject of interest of hundreds of papers. After more than 30…
Let $E$ be an elliptic curve defined over a number field $k$ and $\ell$ a prime integer. When $E$ has at least one $k$-rational point of exact order $\ell$, we derive a uniform upper bound $\exp(C \log B / \log \log B)$ for the number of…
We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing…
In this paper, we give a new upper bound on the minimum Euclidean weight of Type II $\ZZ_{2k}$-codes and the concept of extremality for the Euclidean weights when $k=3,4,5,6$. Together with the known result, we demonstrate that there is an…
We use a global version of Heath-Brown's $p-$adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most $B$ on non-singular cubic curves defined over $\mathbb{Q}$. The bounds are…
We obtain a bound on the girth g of a quaternion unit gain graph in terms of the rank r of its adjacency matrix. In particular, we show that g <= r + 2 and characterize all quaternion unit gain graphs for which g = r+2. This extends…
We answer a 15-year-old open question about the exact upper bound for bivariate copulas with a given diagonal section by giving an explicit formula for this bound. As an application, we determine the maximal asymmetry of bivariate copulas…
Based on the rules of magic cubes, a game of two-dimensional magic cube was deliberately designed. This essay will explore its properties with the assistance of group theory and computer programming. It will first elaborate the rules of…
In this paper, we establish the explicit lower bound estimates for the rank of universal quadratic forms in some certain families of real cubic fields under the condition of density one. The more general results that represent all multiples…
We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.