Results on the Joint Essential Maximal Numerical Ranges

O. S. Cyprian

Journal of Advances in Mathematics and Computer Science, Page 1-8
DOI: 10.9734/jamcs/2020/v35i330255

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.

A Rigorous Homogenization for a Two-Scale Convergence Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil

Adu Sakyi, Peter Amoako-Yirenkyi, Isaac Kwame Dontwi

Journal of Advances in Mathematics and Computer Science, Page 9-25
DOI: 10.9734/jamcs/2020/v35i330256

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.

A Formal Homogenization Approach to Piping Flow Erosion with Deposition in a Spatially Heterogeneous Soil

Adu Sakyi, Peter Amoako-Yirenkyi, Isaac Kwame Dontwi

Journal of Advances in Mathematics and Computer Science, Page 26-45
DOI: 10.9734/jamcs/2020/v35i330257

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.
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.

A Unit Norm Conjecture for Some Real Quadratic Number Fields: A Preliminary Heuristic Investigation

Elliot Benjamin

Journal of Advances in Mathematics and Computer Science, Page 46-50
DOI: 10.9734/jamcs/2020/v35i330258

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(√(p1.p2) where p1 and p2 are distinct primes such that pi = 2 or  pi ≡ 1 mod 4, i = 1, 2. Our conjecture involves the case where the Kronecker symbol (p1/p2) = 1 and the biquadratic residue symbols (p1/p2)4 = (p2/p1)4 = 1, and is based upon Stevenhagen’s conjecture that if k = Q(√(p1.p2) 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 (p1/p2) = (p1/p2)4 = (p2/p1)4 = 1, then P(N(e) = -1)) = 1/3, and we support our conjecture with some preliminary heuristic data.

The Decomposition for the α-level Fuzzy Soft Sets

Li Fu, Luo Tai Qie, Qiao Yun Liu

Journal of Advances in Mathematics and Computer Science, Page 62-73
DOI: 10.9734/jamcs/2020/v35i330260

In this paper, the properties of α-level fuzzy soft sets and α-level fuzzy soft lattices are discussed. Firstly, some soft operations between α-level fuzzy soft sets (lattices) are dened, such as the soft union and intersection operations, and illustrates them by the examples. Secondly, the relation between the α-level fuzzy soft lattices and the α-level fuzzy soft sets are studied, we mainly verify the properties not valid in the case discussed, but tenable in the classical soft set theory.
Thirdly, the decomposition theorem for the α-level fuzzy soft sets is proved. Lastly, a simple application in skin disease diagnosis is illustrated.

On the Computation of the Lagrange Multiplier for the Variational Iteration Method (VIM) for Solving Differential Equations

N. Okiotor, F. Ogunfiditimi, M. O. Durojaye

Journal of Advances in Mathematics and Computer Science, Page 74-92,
DOI: 10.9734/jamcs/2020/v35i330261

In this study, the Variational Iteration Method (VIM) is applied in finding the solution of differential equations with emphasis laid on the choice of the Lagrange multiplier used while employing VIM. Building on existing methods and variational theories, the operator D-Method and integrating factor are employed in certain aspects in the determination of exact Lagrange multiplier for VIM. When results of the computed exact Lagrange multiplier were compared with results of approximate Lagrange multiplier, it was observed that the computed exact Lagrange multiplier reduced significantly the number of iterations required to get a good approximate result, and in some cases the result converged to the exact solution after a single iteration. Evaluations are carried out using MAPLE Software.

Hopf Bifurcation Analysis for a Two Species Periodic Chemostat Model with Discrete Delays

Jane Ireri, Ganesh Pokhariyal, Stephene Moindi

Journal of Advances in Mathematics and Computer Science, Page 93-105
DOI: 10.9734/jamcs/2020/v35i330262

In this paper we analyze a Chemostat model of two species competing for a single limiting nutrient input varied periodically using a Fourier series with discrete delays. To understand global aspects of the dynamics we use an extension of the Hopf bifurcation theorem, a method that rigorously establishes existence of a periodic solution. We show that the interior equilibrium point changes its stability and due to the delay parameter it undergoes a Hopf bifurcation.
Numerical results shows that coexistence is possible when delays are introduced and Fourier series produces the required seasonal variations. We also show that for small delays periodic variations of nutrients has more influence on species density variations than the delay.

A Class of Explicit Integrators with o-grid Interpolation for Solving Non-linear Systems of First Order ODEs

U. W. Sirisena, S. I. Luka, S. Y. Yakubu

Journal of Advances in Mathematics and Computer Science, Page 106-118
DOI: 10.9734/jamcs/2020/v35i330263

This research work is aimed at constructing a class of explicit integrators with improved stability and accuracy by incorporating an off-gird interpolation point for the purpose of making them effcient for solving stiff initial value problems. Accordingly, continuous formulations of a class of hybrid explicit integrators are derived using multi-step collocation method through matrix inversion technique, for step numbers k = 2; 3; 4: The discrete schemes were deduced from their respective continuous formulations. The stability and convergence analysis were carried out and shown to be A(α)-stable and convergent respectively. The discrete schemes when implemented as block integrators to solve some non-linear problems, it was observed that the results obtained compete favorably with the MATLAB ode23 solver.

Performance Modelling of Health-care Service Delivery in Adekunle Ajasin University, Akungba-Akoko, Nigeria Using Queuing Theory

Orimoloye Segun Michael

Journal of Advances in Mathematics and Computer Science, Page 119-127
DOI: 10.9734/jamcs/2020/v35i330264

The queuing theory is the mathematical approach to the analysis of waiting lines in any setting where arrivals rate of the subject is faster than the system can handle. It is applicable to the health care setting where the systems have excess capacity to accommodate random variation. Therefore, the purpose of this study was to determine the waiting, arrival and service times of patients at AAUA Health- setting and to model a suitable queuing system by using simulation technique to validate the model. This study was conducted at AAUA Health- Centre Akungba Akoko. It employed analytical and simulation methods to develop a suitable model. The collection of waiting time for this study was based on the arrival rate and service rate of patients at the Outpatient Centre. The data was calculated and analyzed using Microsoft Excel. Based on the analyzed data, the queuing system of the patient current situation was modelled and simulated using the PYTHON software. The result obtained from the simulation model showed that the mean arrival rate of patients on Friday week1 was lesser than the mean service rate of patients (i.e. 5.33> 5.625 (λ > µ). What this means is that the waiting line would be formed which would increase indefinitely; the service facility would always be busy. The analysis of the entire system of the AAUA health centre showed that queue length increases when the system is very busy. This work therefore evaluated and predicted the system performance of AAUA Health-Centre in terms of service delivery and propose solutions on needed resources to improve the quality of service offered to the patients visiting this health centre.

A Predator-Prey Model with Logistic Growth for Constant Delayed Migration

Apima Bong'ang'a Samuel

Journal of Advances in Mathematics and Computer Science, Page 51-61
DOI: 10.9734/jamcs/2020/v35i330259

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.