An interesting 2-adic property of the Stirling numbers of the second kind S(n, k) was conjectured by the author in 1994 and proved by De Wannemacker in 2005: ν2(S(2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n. It was later generalized to ν2(S(c2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n, c ≥ 1 by the author in 2009. Here we provide full and two partial alternative proofs of the generalized version. The proofs are based on...

Given a graph G = (V, E) with nonnegative costs defined on edges, a positive integer k, and a collection of q terminal sets D = {S1, S2, . . . , Sq}, where each Si is a subset of V (G), the Generalized k-Multicut problem asks to find a set of edges C ⊆ E(G) at the minimum cost such that its removal from G cuts at least k terminal sets in D. A terminal subset Si is cut by C if all terminals in S...

Determining the size of minimum vertex cover of a graph G, denoted by β(G), is an NP-complete problem. Also, for only few families of graphs, β(G) is known. We study the size of minimum vertex cover in generalized Petersen graphs. For each n and k (n > 2k), a generalized Petersen graph P (n, k), is defined by vertex set {ui, vi} and edge set {uiui+1, uivi, vivi+k}; where i = 1, 2, . . . , n and...

given a mapping $f:xto x$ we naturally associate to it a monotonic map $gg_f:exp xto exp x$ from the power set of $x$ into itself, thus inducing a generalized topology on $x$. in this paper we investigate some properties of generalized topologies which are defined by such a procedure.

We present an efficient erasure decoding algorithm for generalized Reed Solomon codes constructed by utilizing the structure of the inverse of the VanderMonde matrices. Given an [n, k] generalized Reed Solomon code, decoding for r erasures, where r = n − k, requires rk + n Galois field multiplications and r(k − 1) Galois field additions after setting up the decoding structures for an erasure pa...

A generalized-theta-graph is a graph consisting of a pair of end vertices joined by k (k ≥ 3) internally disjoint paths. We denote the family of all the n-vertex generalized-theta-graphs with k paths between end vertices by Θnk . In this paper, we determine the sharp lower bound and the sharp upper bound for the total number of matchings of generalized-theta-graphs in Θnk . In addition, we char...

We discuss the symmetric homogeneous polynomial solutions of the generalized Laplace’s equation which arises in the context of the Calogero-Sutherland model on a line. The solutions are expressed as linear combinations of Jack polynomials and the constraints on the coefficients of expansion are derived. These constraints involve generalized binomial coefficients defined through Jack polynomials...

Let g be a reductive Lie algebra over C. We say that a g-module M is a generalized Harish-Chandra module if, for some subalgebra k ⊂ g, M is locally k-finite and has finite k-multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when k is a Cartan subalgebra. We ...

Let X be a non-negative random variable (r.v.) with cumulative distribution function (cdf) F. Suppose X(r, n,m, k), (r = 1, 2, . . . , n), m is a real number and k ≥ 1, is the rth smallest generalized order statistic. Several distributional properties of the generalized order statistics from exponential distribution are given. Some characterizations of the exponential distribution based on the ...

We develop various aspects of the nite model theory of L k (9) and L k 1! (9). We establish the optimality of normal forms for L k 1! (9) over the class of nite structures and demonstrate separations among descriptive complexity classes within L k 1! (9). We establish negative results concerning preservation theorems for L k (9) and L k 1! (9). We introduce a generalized notion of preservation ...