Representation Theorem and Green’s Function (2)

compactifications and function spaces on weighted semigruops

chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...

An ~2-Theorem for Ramanujan's z-Function

This function was first studied by Ramanujan [6]. He wrote, for every prime p, z(p) = 2p I */2 COS Op and conjectured that 0v is real. This was proved by Deligne [2]. it is known that r(p~) =p11~/2 sin(a+ 1) 0p sin 0p If d(n) denotes the number of divisors of n, then it follows that It(n)[ _-< n 11/2 d(n), as r is a multiplicative function. Therefore, for some constant cl >0, { (CI 1ON F/.~ ~ "...

Genes, greens, and homocysteine.

A Representation Theorem and Applications

We introduce a set of transformations on the set of all probability distributions over a finite state space, and show that these transformations are the only ones that preserve certain elementary probabilistic relationships. This result provides a new perspective on a variety of probabilistic inference problems in which invariance considerations play a role. Two particular applications we consi...

Representation Theorem for Stacks

In this paper i is a natural number and x is a set. Let A be a set and let s1, s2 be finite sequences of elements of A. Then s1s2 is an element of A∗. Let A be a set, let i be a natural number, and let s be a finite sequence of elements of A. Then s i is an element of A∗. The following two propositions are true: (1) ∅ i = ∅. (2) Let D be a non empty set and s be a finite sequence of elements of...