The boundary value problem for the Laplace equation outside several cracks in a plane is studied. The jump of the solution of the Laplace equation and the boundary condition containing the jump of its normal derivative are specified on the cracks. The problem has unique classical solution under certain conditions. The new integral representation for the unique solution of this problem is obtain...

One can prove the following proposition (1) For all topological spaces S, T holds Ω[: S, T :] = [:ΩS , ΩT :]. Let X be a set and let Y be an empty set. Note that [:X, Y :] is empty. Let X be an empty set and let Y be a set. Observe that [:X, Y :] is empty. We now state the proposition (2) Let X, Y be non empty topological spaces and x be a point of X. Then Y 7−→ x is a continuous map from Y int...

Models of all the gradings on the exceptional simple Lie algebras induced by Jordan subgroups of their groups of automorphisms are provided.

We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application, we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which can serve as counterexamples for various Burnside type questions.

This paper studies the relationship between representations of a Lie group and representations of its Lie algebra. We will make the correspondence in two steps: First we shall prove that a given representation of a Lie group will provide us with a corresponding representation of its Lie algebra. Second, we shall go backwards and see whether a given representation of a Lie algebra will have a co...

The concept of (α, β)-fuzzy Lie algebras over an (α, β)-fuzzy field is introduced. We provide characterizations of an (∈,∈ ∨q)-fuzzy Lie algebra over an (∈,∈ ∨q)-fuzzy field. 2000 MSC: 17B99, 08A72

(Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems ([BRW], [SA])...

We study the roles of quark-pair correlations for baryon properties, in particular on non-leptonic weak decay of hyperons. We construct the quark wave function of baryons by solving the three body problem explicitly with confinement force and the short range attraction for a pair of quarks with their total spin being 0. We show that the existence of the strong quark-quark correlations enhances ...