1.2 Relating the inverse and standard shuffles We now need to relate ∆(t) to ∆(t). We will do this by taking a more general view of shuffles. We will think of a shuffle as a random walk on the symmetric group Sn. The walk is defined by a set of generators G = {g1, . . . , gk} and a probability distribution P over G. (At each step of the walk, we pick a random gi according to P and apply gi to t...

A Smarandache quasigroup(loop) is shown to be universal if all its f, g-principal isotopes are Smarandache f, g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f, g-principal isotopes are Smarandache f, gprincipal isotopes. Conversely, it is shown that if these weak Smaranda...

The generalized inverse A T ,S of a matrix A is a {2}-inverse of A with the prescribed range T and null space S. A representation for the generalized inverse A T ,S has been recently developed with the condition σ(GA|T ) ⊂ (0,∞), where G is a matrix with R(G) = T and N(G) = S. In this note, we remove the above condition. Three types of iterative methods for A T ,S are presented if σ(GA|T ) is a...

The aim of this paper is to establish an explicit representation the generalized Drazin inverse $(a+b)^d$ under condition $$ab^2=0, ba^2=0, a^{\pi}b^{\pi}(ba)^2=0.$$ Furthermore, we apply our results give some for a $2\times 2$ block operator matrix. These extend on Bu, Feng and Bai [Appl. Math. Comput. 218, 10226-10237, 2012] Dopazo Martinez-Serano [Linear Algebra Appl. 432, 1896-1904, 2010].

The inverse problem of determining the unknown coefficient of the non-linear differential equation of torsional creep is studied. The unknown coefficient g = g(ξ2) depends on the gradient ξ := |∇u| of the solution u(x), x ∈ Ω ⊂ Rn , of the direct problem. It is proved that this gradient is bounded in C-norm. This permits one to choose the natural class of admissible coefficients for the conside...

Many important C∗-algebras, such as AF-algebras, Cuntz-Krieger algebras, graph algebras and foliation C∗-algebras, are the C∗-algebras of r-discrete groupoids. These C∗-algebras are often associated with inverse semigroups through the C∗-algebra of the inverse semigroup [HR90] or through a crossed product construction as in Kumjian’s localization [Kum84]. Nica [Nic94] connects groupoid C∗-algeb...