We reason that it is the gravitation of all ordinary matter, extending from dense distant past to sparse present, rather than dark shows up in galaxy rotation and velocity dispersion. Likewise, we argue this gradient gravitational energy due expansion, energy, explains Type 1a supernovae brightness vs. redshift data. Our conclusions follow statistical mechanics, thermodynamic theory based on at...

let $mathcal {n}_g$ denote the set of all proper‎ ‎normal subgroups of a group $g$ and $a$ be an element of $mathcal‎ ‎{n}_g$‎. ‎we use the notation $ncc(a)$ to denote the number of‎ ‎distinct $g$-conjugacy classes contained in $a$ and also $mathcal‎ ‎{k}_g$ for the set ${ncc(a) | ain mathcal {n}_g}$‎. ‎let $x$ be‎ ‎a non-empty set of positive integers‎. ‎a group $g$ is said to be‎ ‎$x$-d...

BACKGROUND Using appropriate terminology in nutrition education programs and behaviour change campaigns is important to optimise the effectiveness of these efforts. To inform future communications on the topic of healthy eating, this study explored adults' perceptions of the meaning of four terms used to describe unhealthy foods: junk food, snack food, party food, and discretionary food. METH...

‎for a finite group $g$ let $nu(g)$ denote the number of conjugacy classes of non-normal subgroups of $g$‎. ‎the aim of this paper is to classify all the non-nilpotent groups with $nu(g)=3$‎.

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fai...

Let $G = {rm Res}_{F/mathbb{Q}}(GL_n)$ where $F$ is a number field‎. ‎Let $S^G_{K_f}$ denote an ad`elic locally symmetric space for some level structure $K_f.$ Let ${mathcal M}_{mu,{mathbb C}}$ be an algebraic irreducible representation of $G({mathbb R})$ and we let $widetilde{mathcal{M}}_{mu,{mathbb C}}$ denote the associated sheaf on $S^G_{K_f}.$ The aim of this paper is to classify the data ...