Abstract: Publication date: Sep 2018 Source:Mathematics and Statistics Volume 6 Number 3 Samuel Bertrand Liyimbeme Mouchili Since Galois rings are the generalization of Galois fields, the question we tried to answer is: How to move from the discrete logarithm in Galois fields to the one in Galois rings? That concept of the discrete logarithm in Galois rings is a little bit different from the one in Galois fields. Here, the discrete logarithm of an element is the tuple, which is not the case in Galois fields. However, thanks to the multiplicative representation of elements in Galois rings, each element can be uniquely represented in the form: ; where k is a nonnegative integer, is a generator of the Galois ring (the definition of a generator in a Galois ring will be given later on). Then the tuple will be called: the discrete logarithm of . The notion of generators in Galois rings comes from the one in the group theory. The Knowledge of the generators in multiplicative groups allows, as well to determine the generators in Galois rings ; p is a prime number and m is a nonnegative integer greater than or equal to two. These new concepts of discrete logarithm and generators in Galois rings will help to securely share common information and to perform ElGamal encryption in Galois rings. PubDate: Sep 2018

Abstract: Publication date: Sep 2018 Source:Mathematics and Statistics Volume 6 Number 3 Md. Jahurul Islam Md. Shahidul Islam and Md. Shafiqul Islam In this paper, we discuss Hausdorff measure and Hausdorff dimension. We also discuss iterated function systems (IFS) of the generalized Cantor sets and higher dimensional fractals such as the square fractal, the Menger sponge and the Sierpinski tetrahedron and show the Hausdorff measures and Hausdorff dimensions of the invariant sets for IFS of these fractals. PubDate: Sep 2018

Abstract: Publication date: Jan 2018 Source:Mathematics and Statistics Volume 6 Number 1 Pokutnyi Oleksandr Sufficient conditions for the existence of solutions for a weakly linear perturbed boundary value problem are obtained in the so called resonance (critical) case. Iterative process for finding solutions has been presented. Necessary and sufficient conditions of the existence of solutions, bounded solutions, generalized solutions and quasi solutions are obtained. PubDate: Jan 2018

Abstract: Publication date: Jan 2018 Source:Mathematics and Statistics Volume 6 Number 1 Siloko, I. U. Ishiekwene, C. C. and Oyegue, F. O. The bivariate kernel density estimator is fundamental in data smoothing methods especially for data exploration and visualization purposes due to its ease of graphical interpretation of results. The crucial factor which determines its performance is the bandwidth. We present new methods for bandwidth selection in bivariate kernel density estimation based on the principle of gradient method and compare the result with the biased cross-validation method. The results show that the new methods are reliable and they provide improved methods for a choice of smoothing parameter. The asymptotic mean integrated squared error is used as the measure of performance of the new methods. PubDate: Jan 2018

Abstract: Publication date: Apr 2018 Source:Mathematics and Statistics Volume 6 Number 2 Marian Anton and Landon Renzullo The field of computational topology is evolving rapidly and new algorithms are updated and released at a rapid pace. A good reference for currently available opensource libraries with peer-review publication can be found in [7]. In this paper we examine the descriptive potential of a combinatorial data structure known as Generating Set in constructing the boundary maps of a simplicial complex. By refining the approach of [1] in generating these maps, we provide algorithms that allow for relations among simplices to be easily accounted for. In this way we explicitly generate faces of a complex only once, even if a face is shared among multiple simplices. The result is a useful interface for constructing complexes with many relations and for extending our algorithms to ∆-complexes. Once we efficiently retrieve the representatives of "living" simplices i.e., of those that have not been related away, the construction of the boundary maps scales well with the number of relations and provides a simpler alternative to JavaPlex [8]. We note that the generating data of a complex is equivalent in information to its incidence matrix and provide efficient algorithms for converting from an incidence matrix to a Generating Set. PubDate: Apr 2018

Abstract: Publication date: Apr 2018 Source:Mathematics and Statistics Volume 6 Number 2 Chun P.B Ibrahim A.A and Kamoh N.M The use of the adjacency matrix of a graph as a generator matrix for some classes of binary codes had been reported and studied. This paper concerns the utilization of the stable variety of Cayley regular graphs of odd order for efficient interconnection networks as studied, in the area of Codes Generation and Analysis. The Use of some succession scheme in the construction of a stable variety of the Cayley regular graph had been considered. We shall enumerate the adjacency matrices of the regular Cayley graphs so constructed which are of odd order (2m+1), for m≥3 as in [1]. Next, we would show that the Matrices are cyclic and can be used in the generation of cyclic codes of odd lengths. PubDate: Apr 2018