Let m > 1; s 1 be xed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p(x) 0; q = q(x) 0;n = n(x) 0; r = r(x) 0 such that either x p x n ; y]x q = x r x; y m ]y s or x p x n ;y]x q = y s x; y m ]x r for all y 2 R. In the present paper it is shown that R is commutative if it satisses the property Q(m) (i.e. for all x; y 2 R;mx; y] = 0 implie...

In a recent paper [9], the authors introduced the extended Q-bipartite double of an almost dual bipartite cometric association scheme. Since the association schemes arising from linked systems of symmetric designs are almost dual bipartite, this gives rise to a new infinite family of 4-class cometric schemes which are both Q-bipartite and Q-antipodal. These schemes, the schemes arising from lin...

Based on the evaluation at t = −1 of the generating polynomial for the hyperoctahedral group by the number of descents, an observation recently made by Hirzebruch, a new q-secant number is derived by working with the Chow-Gessel q-polynomial involving the flag major index. Using the doubloon combinatorial model we show that this new q-secant number is a polynomial with positive integral coeffic...

Given a finite family F of convex sets in R, we say that F has the (p, q)r property if for any p convex sets in F there are at least r q-tuples that have non empty intersection. The piercing number of F is the minimum number of points we need to intersect all the sets in F . In this paper we willl find some bounds for the piercing number of families of convex sets with (p, q)r properties.

We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the q-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in ]0, 1[. Numerous properties of the modified Bernstein Polynomials are extended to their q-analogues: simultaneous approximation, p...

A graph is König for a q-path if every its induced subgraph has the following property. The maximum number of pairwise vertex-disjoint induced paths each on q vertices is equal to the minimum number of vertices, such that removing all the vertices produces a graph having no an induced path on q vertices. In this paper, for every q ≥ 5, we describe all König graphs for a qpath obtained from fore...

For d > 2 and n > d + 1, let P = {p1, . . . , pn} be a set of points in Rd whose convex hull contains the origin 0 in its interior. We show that if P∪{0} is in general position, then there exists a d-tuple Q = {pi1 , . . . , pid} ⊂ P such that 0 is not contained in the convex hull of Q∪{p} for any p ∈ P \Q. A generalization of this property is also considered.

We establish finiteand infinite-dimensional versions of the following assertion. If M is a matrix with the property that whenever P and Q are diagonal projections with P ≤ Q, the spectrum of PMP (considered as an operator on the range of P ) is contained in that of QMQ (considered as an operator on the range of Q), then there is a permutation matrix U such that U−1MU is triangular.

Let u t = ∇ 2 u − q(x)u := Lu in D × [0, ∞), where D ⊂ R 3 is a bounded domain with a smooth connected boundary S, and q(x) ∈ L 2 (S) is a real-valued function. is the outer normal to S, one can find q(x) uniquely. A similar result is obtained for the heat equation u t = Lu := ∇ · (a∇u). These results are based on new versions of Property C.