The slice condition and the more general weak slice conditions are geometric conditions on Euclidean space domains which have evolved over the last several years as a tool in various areas of analysis. This paper explores some of the finer distinctive properties of the various weak slice conditions.

Some locally finite simple Lie algebras are graded by finite (possibly nonreduced) root systems. Many more algebras are sufficiently close to being root graded that they still can be handled by the techniques from that area. In this paper we single out such Lie algebras, describe them, and suggest some applications of such descriptions.

For a quasidifferentiable function f defined on R 2 , it is proved, in the sense of Demyanov and Rubinov, that the following assertion [∂f (x),∂f (x)]∈Df (x) (∂f (x) + ∂f (x)), [∂f (x),∂f (x)]∈Df (x) (∂f (x) − ∂f (x)) ∈ Df (x) in this paper, where Df (x) denotes the set of all quasidifferentials of f at x. It is shown that this way can be viewed as an approach to determining or choosing a repre...

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing...

The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. In this paper, we investigate the concept of restricted positive definite functions and their relation with restricted representations of an inverse semigroup. We also introduce the restricted Fourier ...

We study a class of positively graded Lie algebras with a pattern of homogeneous components similar to that of the graded Lie algebra associated to the Nottingham group with respect to its lower central series.

In the present work the properties of Cartan subalgebras and their connection with regular elements in finite dimensional Lie algebras are extended to the case of Leibniz algebras. It is shown that Cartan subalgebras and regular elements of a Leibniz algebra correspond to Cartan subalgebras and regular elements of a Lie algebra by a natural homomorphism. Conjugacy of Cartan subalgebras of Leibn...

1. INTRODUCTION. Some years ago W. Plesken told the first author of a simple but interesting construction of a Lie algebra from a finite group. The authors posed themselves the question as to what the structure of this Lie algebra might be. In particular, for which groups does the construction produce a simple Lie algebra? The answer is given in the present paper; it uses some textbook results ...