let $g$ be a finite group‎. ‎we denote by $psi(g)$ the integer $sum_{gin g}o(g)$‎, ‎where $o(g)$ denotes the order of $g in g$‎. ‎here we show that‎ ‎$psi(a_5)< psi(g)$ for every non-simple group $g$ of order $60$‎, ‎where $a_5$ is the alternating group of degree $5$‎. ‎also we prove that $psi(psl(2,7))

Although the alternating group is an index 2 subgroup of the symmetric group, there is no generating set that gives a Coxeter structure on it. Various generating sets were suggested and studied by Bourbaki, Mitsuhashi, Regev and Roichman, Vershik and Vserminov, and others. In a recent work of Brenti, Reiner and Roichman, it is explained that palindromes in Mitsuhashi’s generating set play a rol...

The group of alternating colored permutations is the natural analogue of the classical alternating group, inside the wreath product Zr ≀Sn. We present a ‘Coxeterlike’ presentation for this group and compute the length function with respect to that presentation. Then, we present this group as a covering of Z r 2 ≀ Sn and use this point of view to give another expression for the length function. ...

Recently, collaborative image filtering based on groupbased sparse representation has gained a popularity in image restoration. BM3D frame [1], one of the first example of such a representation, utilizes both local sparsity of small size image patches and group-sparsity of collections of selfsimilar image patches. As a sparsifying transforms in the spatial and similarity domains, fixed transfor...

Let An ⊆ Sn denote the alternating and the symmetric groups on 1, . . . , n. MacMahaon’s theorem [11], about the equi-distribution of the length and the major indices in Sn, has received far reaching refinements and generalizations, by Foata [5], Carlitz [3, 4], FoataSchützenberger [6], Garsia-Gessel [7] and followers. Our main goal is to find analogous statistics and identities for the alterna...

This paper proposes efficient algorithms for group sparse optimization with mixed `2,1-regularization, which arises from the reconstruction of group sparse signals in compressive sensing, and the group Lasso problem in statistics and machine learning. It is known that encoding the group information in addition to sparsity will lead to better signal recovery/feature selection. The `2,1-regulariz...

In this paper, a class of alternating group explicit iterative parallel method(AGI) is derived for solving diffusion equations based on a high order implicit scheme. Also the convergence analysis and the stability analysis are given. In order to verify the results, several numerical examples are presented in the end of the paper. Mathematics Subject Classification: 65M06, 74S20