Volume 9, Issue 4 - Special Issue: Proceedings of the 22nd Meeting of the Tunisian Mathematical Society, Mahdia, Tunisia, 2017

This note is a contribution to the Proceedings of the Conference of the Tunisian Mathematical Society CSMT 2017. After briefly revisiting the case of the standard Hardy–Littlewood maximal operator, we will discuss the behavior of the Dunkl maximal operator in both the scalar and vector-valued cases.

Let G be a type 1 connected and simply connected solvable Lie group. The generalized moment map for π in G^{\widehat{G}}, the unitary dual of G , sends smooth vectors of the representation space of π to 𝒰⁢(𝔤)*{{\mathcal{U}(\mathfrak{g})}^{*}}, the dual vector space of 𝒰⁢(𝔤){\mathcal{U}(\mathfrak{g})}. The convex hull of the image of the generalized moment map for π is called its generalized moment set , denoted by J⁢(π){J(\pi)}. We say that G^{\widehat{G}} is generalized moment separable when the generalized moment sets differ for any pair of distinct irreducible unitary representations. Our main result in this paper provides a second proof of the generalized moment separability theorem for G .

Alexandroff spaces are the topological spaces in which the intersection of arbitrary many open sets is open. Let T be an indecomposable tournament. In this paper, first, we associate a trivial topology to T . Then we define another topology on T , called the graphic topology of T , and we show that it is an Alexandroff topology. Our motivation is to investigate some properties of this topology.

We give an introduction to superalgebra, founded on the difference between even (commuting) and odd (anti-commuting) variables. We give a sketch of Graßmann’s work, and show how derivations of those structures induce various superalgebra structures, Lie superalgebras of Cartan type being obtained with even derivations, while odd derivations induce Jordan-type superalgebras.