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 ...

Linear recursion relations for the instanton corrections to the effective prepotential are derived for N=2 supersymmetric gauge theories with one hypermultiplet in the adjoint representation of SU(N) using the Calogero-Moser parameterization of the Seiberg-Witten spectral curves. S-duality properties of the Calogero-Moser parameterization and conjectures on the Seiberg-Witten spectral curves ge...

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

The celebrated Erdős-Hajnal conjecture states that for every n-vertex undirected graph H there exists ε(H) > 0 such that every graph G that does not contain H as an induced subgraph contains a clique or an independent set of size at least n. A weaker version of the conjecture states that the polynomial-size clique/independent set phenomenon occurs if one excludes both H and its complement H. We...

we investigate the classical h.~zassenhaus conjecture‎ ‎for integral group rings of alternating groups $a_9$ and $a_{10}$ of degree‎ ‎$9$ and $10$‎, ‎respectively‎. ‎as a consequence of our‎ ‎previous results we confirm the prime graph‎ ‎conjecture for integral group rings of $a_n$ for all $n leq 10$‎.

A famous conjecture (usually called Ryser’s conjecture), appeared in the Ph.D thesis of his student, J. R. Henderson [9], states that for an r-uniform r-partite hypergraph H, the inequality τ(H) ≤ (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem [16], and in the case of r = 3, which was proved by Aharoni in 2001 [2]...

This note is about spherical classes in H∗Q0S 0. A conjecture, due to Ed. Curtis, predicts that only Hopf invariant one and Kervaire invariant one elements will give rise to spherical classes in H∗Q0S 0. Yet, there has been no proof of this conjecture around. Assuming that this conjecture fails, there must exist some other spherical classes in H∗Q0S 0. This note determines the form of these pot...

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, i...

The Erdős-Hajnal Conjecture states that for every given H there exists a constant c(H) > 0 such that every graph G that does not contain H as an induced subgraph contains a clique or a stable set of size at least |V (G)|c(H). The conjecture is still open. However some time ago its directed version was proved to be equivalent to the original one. In the directed version graphs are replaced by to...