We study the binary Ehrenfeucht Mycielski sequence seeking a balance between the number of occurrences of different binary strings. There have been numerous attempts to prove the balance conjecture of the sequence, which roughly states that 1 and 0 occur equally often in it. Our contribution is twofold. First, we study weaker forms of the conjecture proved in the past and lay out detailed proof...

Perhaps the most strikingly counterintuitive theorem in mathematics is the "Banach-Tarski paradox": A ball in R3 can be decomposed into finitely many pieces which can be rearranged by rigid motions and reassembled to form two balls of the same size as the original. The Axiom of Choice is used to construct the decomposition; the "paradox" is resolved by noting that the pieces cannot be Lebesgue ...

let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

The generalized Mycielskians are the generalization of the Mycielski graphs, which were introduced by Mycielski in 1955. A topological index is a numeric parameter mathematically derived from a graph and is invariant under automorphism of graphs. Topological indices are widely used for establishing correlations between the structure of a molecular compound and its different physico-chemical pro...

Let G = (V,E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V − S, there exists u ∈ S such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the ...

A vertex coloring of a graph G=(V,E) is called an exact square G if any pair vertices at distance 2 receive distinct colors. The minimum number colors required by the chromatic and denoted χ[#2](G). set clique are 2. G, ω[#2](G), maximum cardinality clearly, ω[#2](G)≤χ[#2](G). In this article, we give tight upper bounds various standard operations, including Cartesian product, strong lexicograp...

Christensen has defined a generalization of the property of being of Haar measure zero to subsets of (abelian) Polish groups which need not be locally compact; a recent paper of Hunt, Sauer, and Yorke defines the same property for Borel subsets of linear spaces, and gives a number of examples and applications. The latter authors use the term “shyness” for this property, and “prevalence” for the...

It has always been a large part of the task of philosophers of mathematics to explain our grasp of the infinite. In the end every credible attempt to do this involves a reduction, an account which shows our understanding of the infinite to be explicable in finite terms. Russell, for example, thought when he wrote The Principles of Mathematics that propositions about infinite classes contain as ...

Gaussons: Solitons of the logarithmic Schrodinger equation. Iwo Bialynicki-Birula (Institute of Theoretical Physics, Warsaw University, Warsaw, Poland and Department of Physics, University of Pittsburgh, Pittsburgh, U.S.A.); and J erzy Mycielski (Institute of Theoretical Physics, Warsaw University, Warsaw, Poland). Physica Scripta (Sweden) 20,539-544,1979. Properties of the Schrodinger equation...

The local chromatic number of a graph, introduced by Erdős et al. in [3], is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk, based on the papers [13, 14, 15], would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of...