Journal of Advances in Mathematics and Computer Science 2020-06-03T11:57:33+00:00 Journal of Advances in Mathematics and Computer Science Open Journal Systems <p style="text-align: justify;"><strong>Journal of Advances in Mathematics and Computer Science (ISSN:&nbsp;2456-9968) </strong>aims to publish original research articles, review articles and short communications, in all areas of mathematics and computer science. Subject matters cover pure and applied mathematics, mathematical foundations, statistics and game theory, use of mathematics in natural science, engineering, medicine, and the social sciences, theoretical computer science, algorithms and data structures, computer elements and system architecture, programming languages and compilers, concurrent, parallel and distributed systems, telecommunication and networking, software engineering, computer graphics, scientific computing, database management, computational science, artificial Intelligence, human-computer interactions, etc. This is a quality controlled, OPEN peer reviewed, open access INTERNATIONAL journal.</p> Results on the Joint Essential Maximal Numerical Ranges 2020-06-03T09:42:46+00:00 O. S. Cyprian <p>In the present paper, we show the equivalent denitions of the joint essential maximal numerical range. In the current paper, we show that the properties of the classical numerical range such as compactness also hold for the joint essential maximal numerical range. Further, we show that the joint essential maximal numerical range is contained in the joint maximal numerical range.</p> 2020-05-24T00:00:00+00:00 ##submission.copyrightStatement## A Rigorous Homogenization for a Two-Scale Convergence Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil 2020-06-03T09:42:46+00:00 Adu Sakyi Peter Amoako-Yirenkyi Isaac Kwame Dontwi <p>We present a rigorous homogenization approach to modelling piping flow erosion in a spatially heterogeneous soil. The aim is to provide a justication to a formal homogenization approach to piping flow erosion with deposition in a spatially heterogeneous soil. Under the assumption that the soil domain is perforated periodically with cylindrical repeating microstructure, we begin by proving that a solution to the proposed set of microscopic equations exist. Two-scale convergence is then used to study the asymptotic behaviour of solutions to the microscopic problem as the microscopic length scale approaches zero(0). We thus derive rigorously a homogenized model or macro problem as well as explicit formula for the eective coecients. A strong observation from the numerical simulation was that, soil particle concentration in the water/soil mixture decreases but at a decreasing rate whereas soil particle deposition increases at regions with increasing amount of particle concentration in the flow causing a reduction in bare pore spaces across the soil domain.</p> 2020-05-27T00:00:00+00:00 ##submission.copyrightStatement## A Formal Homogenization Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil 2020-06-03T09:42:45+00:00 Adu Sakyi Peter Amoako-Yirenkyi Isaac Kwame Dontwi <p>We model and simulate piping erosion phenomena with deposition in a spatially heterogeneous soil mass motivated by seepage flow. The soil is considered to be a porous media with periodic positioning of pores spread around cylindrical structures or microstructures making the heterogeneities periodic in space.The period of the heterogeneities defines a microscopic length scale ϵ of the microscopic problem and this allows the use of periodic homogenization methods.<br>We studied the asymptotic behaviour of the solutions to the micro problem as ϵ ! 0 and obtained a homogenized model or macro problem with explicit formula for effective coefficients. Numerical simulations of the proposed model captures the expected behaviour of soil particle concentration and deposition as observed in piping flow erosion phenomena.</p> 2020-05-29T00:00:00+00:00 ##submission.copyrightStatement## A Unit Norm Conjecture for Some Real Quadratic Number Fields: A Preliminary Heuristic Investigation 2020-06-03T09:42:45+00:00 Elliot Benjamin <p>In this paper we make a conjecture about the norm of the fundamental unit, N(e), of some real quadratic number fields that have the form k = Q(√(p<sub>1</sub>.p<sub>2</sub>) where p<sub>1</sub> and p<sub>2</sub> are distinct primes such that p<sub>i</sub> = 2 or&nbsp; p<sub>i</sub> ≡ 1 mod 4, i = 1, 2. Our conjecture involves the case where the Kronecker symbol (p<sub>1</sub>/p<sub>2</sub>) = 1 and the biquadratic residue symbols (p<sub>1</sub>/p<sub>2</sub>)<sub>4</sub> = (p<sub>2</sub>/p<sub>1</sub>)<sub>4</sub> = 1, and is based upon Stevenhagen’s conjecture that if k = Q(√(p<sub>1</sub>.p<sub>2</sub>) is any real quadratic number field as above, then P(N(e) = -1)) = 2/3, i.e., the probability density that N(e) = -1 is 2/3. Given Stevenhagen’s conjecture and some theoretical assumptions about the probability density of the Kronecker symbols and biquadratic residue symbols, we establish that if k is as above with (p<sub>1</sub>/p<sub>2</sub>) = (p<sub>1</sub>/p<sub>2</sub>)<sub>4</sub> = (p<sub>2</sub>/p<sub>1</sub>)<sub>4</sub> = 1, then P(N(e) = -1)) = 1/3, and we support our conjecture with some preliminary heuristic data.</p> 2020-05-30T00:00:00+00:00 ##submission.copyrightStatement## A Predator-Prey Model with Logistic Growth for Constant Delayed Migration 2020-06-03T11:57:33+00:00 Apima Bong'ang'a Samuel <p>Predator prey models predict a broad range of results depending on characteristics of predators, prey and the environment in which they interact. The environment in which these species live in and interact is usually made up of many patches, and these patches are connected via migration. The instantaneous migration of these species from one patch to another may not be realistic since there may be barriers during migration such as a busy infrastructure through the natural habitat. A predator-prey model, with logistic growth for both species and constant delayed migration, is developed and analyzed in this paper. It is shown that these species will survive if they migrate at higher rates in search of sustaining resources. Thus, for the species to coexist, we recommend that factors that slow down migration rates should be addressed, for example, reducing human activities and settlement in natural habitat.</p> 2020-06-03T00:00:00+00:00 ##submission.copyrightStatement##