In what follows, V = {1, . . . , n} is the set of indices of given random variables X1, . . . , Xn, and B = {B1, . . . , Bm} is a set of subsets (possibly with repeat) of V . Furthermore, for S = {i1, . . . , il} ⊆ V , XS and H(XS) denote the random vector (Xi1 , . . . , Xil) and its Shannon entropy H(Xi1 , . . . , Xil) (H(X∅) = 0). The power set (the set of all subsets) and the set of all l-su...

We present a construction of long MDS codes which are not of the generalized Reed-Solomon (GRS) type. The construction employs subsets S, IS I = m, of a finite field F = GF(q) with the property that no t distinct elements of S add up to some fixed element of F . Large subsets of this kind are used to construct [n=m+2,k=t+l] non-GRS MDS codes over F. T ec hn io n C om pu te r Sc ie nc e D ep ar ...

Background. P and NP are two classes (sets) of languages in Computer Science. An open problem is whether NP P  . This paper tests a new idea to compare the two language sets and attempts to prove that these two language sets consist of same languages by elementary mathematical methods and basic knowledge of Turing machine. Methods. By introducing a filter function   w M C , that is the numbe...

One can check that there exists a lattice which is non trivial, Boolean, and strict. Let X be a set. Observe that there exists a family of subsets of X which is finite and non empty. Let S be a 1-sorted structure. One can check that there exists a family of subsets of S which is finite and non empty. Let L be a non trivial upper-bounded non empty poset. Note that there exists a filter of L whic...

We have initiated a study of catecholamine-containing neurons in Drosophila melanogaster because of the potential, with this organism, to perturb catecholamine metabolism using genetic tools. The major objectives of this study were (1) to define the pattern of catecholamine-containing neurons and (2) to determine the effect of the absence of dopa decarboxylase (DDC) enzyme activity on the catec...

The papers [18], [9], [21], [14], [23], [6], [24], [2], [3], [8], [10], [1], [22], [7], [11], [20], [16], [19], [4], [5], [13], [12], [17], and [15] provide the terminology and notation for this paper. For simplicity, we adopt the following convention: k, l, m, n, i, j denote natural numbers, K, N denote non empty subsets of N, K1, N1, M1 denote subsets of N, and X, Y denote sets. Let us consid...

Secret sharing scheme is a method to distribute secret information to a set P of participants so that only authorised subsets of P can reconstruct the secret. A set of subsets of P that can reconstruct the secret is called an access structure of the scheme. A simple undirected graph G is called a sum graph if there exists a labeling L of the vertices of G into distinct numbers, usually positive...

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by ErdSs and Rado, are given. These yield improvements over the known bounds for the arising Erd6s-Rado numbers ER(k; l), where the numbers ER(k; l) are defined as the least positive integer n such that for every partition of the k-element subsets of a totally ordered n-element set X into an arbitrary num...

We raise and investigate the algorithmic complexity of the following problem. Given a graph G = (V; E) and p-element sets L(v) for its vertices v 2 V such that jL(u) L(v)j p + r for all edges uv 2 E, do there exist q-element subsets C (v) L(v) with C (u) \ C (v) = ; for all uv 2 E ? Here p; q; r are positive integers, p q and p + r 2q. We characterize precisely which triples (p; q; r) admit a p...

we characterize the rational subsets of a finite group and discuss the relations to integral cayley graphs.