Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup X(S) of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/X(S...

Professor S. Luo in [Phys. Rev. A 82, 052122(2010)] proposed two conjectures on the classical correlation and quantum correlation in a bipartite state ρAB, respectively. In this paper, we prove the conjecture on the classical correlation completely. Moreover, we show that Q(ρAB) 6 S(ρB) is always valid, and the conjecture on quantum correlation is true if S(ρB) 6 S(ρA) or ρAB is separable. We o...

In 1916, Schur introduced the Ramsey number r(3; m), which is minimum integer n > 1 such that for any m-coloring of edges complete graph Kn, there a monochromatic copy K3. He showed m) ? O(m!), and simple construction demonstrates ? 2?(m). An old conjecture Erd?s states = 2?(m). this note, we prove m-colorings with bounded VC-dimension, is, property set system induced by neighborhoods vertices ...

in the present paper, among other results, a decomposition formula is given for the w-bounded continuous negative definite functions of a topological *-semigroup s with a weight function w into a proper h*-algebra a in terms of w-bounded continuous positive definite a-valued functions on s. a generalization of a well-known result of k. harzallah is obtained. an earlier conjecture of the author ...

Let $E$ be an elliptic curve over $Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let $E^{(D)}(Bbb{Q})$ be the group of $Bbb{Q}$-rational points of $E^{(D)}$. It is conjectured by J. Silverman that there are infinitely many primes $p$ for which $...