using a generalized spherical mean operator, we obtain the generalizationof titchmarsh's theorem for the dunkl transform for functions satisfyingthe lipschitz condition in l2(rd;wk), where wk is a weight function invariantunder the action of an associated reection groups.

in this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)d^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the banach algebras. this proof is given in two cases of the continuous and discontinuous function $g$, under the generalized lipschitz and caratheodory conditions.

Classical automated theorem provers can prove non-trivial mathematical theorems in highly specific settings. However they are generally unable to cope with even moderately difficult theorems in mainstream mathematics. While there are many reasons for the failure of the classical search-based paradigm, it is apparent that mathematicians can cope with long and complex proofs and have strategies t...

In this paper we show how to use the theorem prover Isabelle to provide tool support for Focus a speci cation and veri cation frame work for the stepwise development of distributed systems Focus is em bedded into Isabelle by modeling the basic notion of stream processing functions and by formalizing an appropriate set of assumption commit ment re nement rules Moreover the re nement calculus is ...