Converse of a Well Known Theorem on Integral Means

Some Consequences of a Well Known Theorem on Conics

On the converse of Wolstenholme’s Theorem

The problem of distinguishing prime numbers from composite numbers (. . .) is known to be one of the most important and useful in arithmetic. (. . .) The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. Wilson’s Theorem states that if p is prime then (p− 1)! ≡ −1 (mod p). It is easy to see that the c...

A Converse to Dye’s Theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of F2 on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not univ...

Weil Converse Theorem

Hecke generalized this equivalence, showing that an integral form has an associated Dirichlet series which can be analytically continued to C and satisfies a functional equation. Conversely, Weil showed that, if a Dirichlet series satisfies certain functional equations, then it must be associated to some integral form. Our goal in this paper is to describe this work. In the first three sections...

Closed Means Continuous Iff Polyhedral: a Converse of the Gkr Theorem

Given x0, a point of a convex subset C of an Euclidean space, the two following statements are proven to be equivalent: (i) any convex function f : C → R is upper semi-continuous at x0, and (ii) C is polyhedral at x0. In the particular setting of closed convex mappings and Fσ domains, we prove that any closed convex function f : C → R is continuous at x0 if and only if C is polyhedral at x0. Th...