Fermat’s Last Theorem and Related Problems

Darell Cox, Sourangshu Ghosh, Eldar Sultanow

Journal of Advances in Mathematics and Computer Science, Page 6-34
DOI: 10.9734/jamcs/2021/v36i530361

This paper is temporarily held for further investigation.

A Mathematical Model for the Transmission of HIV/AIDS with Early Treatment

T. O. Akinwumi, I. A. Olopade, A. O. Adesanya, M. O. Alabi

Journal of Advances in Mathematics and Computer Science, Page 35-51
DOI: 10.9734/jamcs/2021/v36i530362

In this paper, a mathematical model for the transmission of HIV/AIDS with early treatment is developed and analyzed to gain insight into early treatment of HIV/AIDS and other epidemiological features that cause the progression from HIV to full blown AIDS. We established the basic reproduction number which is the average number of new secondary infection generated by a single infected individual during infectious period. The analysis shows that the disease free equilibrium is locally and globally asymptotically stable whenever the threshold quantity   is less than unity i.e. Numerical analysis shows that the early treatment of latently infected individuals reduces the dynamical progression to full blown AIDS. The result also showed that immunity boosted substances increase the red blood cells, sensitivity analysis of basic reproduction number with respect to parameters showed that effective contact rate must not exceed 0.3 to avoid endemic stage.

The Analysis for the Recovery Cases of COVID-19 in Egypt using Odd Generalized Exponential Lomax Distribution

Hanem Mohamed, Amina E. Abo-Hussien, Salwa A. Mousa, Magda M. Ismail

Journal of Advances in Mathematics and Computer Science, Page 52-65
DOI: 10.9734/jamcs/2021/v36i530363

In this paper, an odd generalized exponential Lomax (OGEL, in short) distribution has been considered. Some mathematical properties of the distribution are studied. The methods of maximum likelihood and maximum product of spacing are used for estimating the model parameters.  Moreover, 95% asymptotic confidence intervals for the estimates of the parameters are derived. The Monte Carlo simulation is conducted for the two proposed methods of estimation to evaluate the performance of the various proposed estimators. The proposed methods are utilized to find estimates of the parameters of OGEL distribution for the daily recovery cases of COIVD-19 in Egypt from 12 May to 30 September 2020.The practical applications show that the proposed model provides better fits than the other models.

Spatial Mesh Refinement using Cubic Smoothing Spline Interpolation in Simulation of 2-D Elastic Wave Equation: Forward Modeling of Full-waveform Inversion

Amila Sudu Ambegedara, U. G. I. G. K. Udagedara, Erik M. Bollt

Journal of Advances in Mathematics and Computer Science, Page 66-83
DOI: 10.9734/jamcs/2021/v36i530364

Full-waveform inversion (FWI) is a non-destructive health monitoring technique that can be used to identify and quantify the embedded anomalies. The forward modeling of the FWI consists of a simulation of elastic wave equation to generate synthetic data. Thus the accuracy of the FWI method highly depends on the simulation method used in the forward modeling. Simulation of a 3-D seismic survey with small-scale heterogeneities is impossible with the classic finite difference approach even on modern super computers. In this work, we adopted a mesh refinement approach for simulation of the wave equation in the presence of small-scale heterogeneities. This approach uses cubic smoothing spline interpolation for spatial mesh refinement step in solving the wave equation. The simulation results for the 2-D elastic wave equation are presented and compared with the classic finite difference approach.

Extended Ensemble Filter for High-dimensional Nonlinear State Space Models

Oryiema Robert, David Angwenyi, Kevin Midenyo

Journal of Advances in Mathematics and Computer Science, Page 84-97
DOI: 10.9734/jamcs/2021/v36i530365

There are several functional forms for non-linear dynamical filters. Extended Kalman filters are algorithms that are used to estimate more accurate values of unknown quantities of internal dynamical systems from a sequence of noisy observation measured over a period of time. This filtering process becomes computationally expensive when subjected to high dimensional data which consequently has a negative impact on the filter performance in real time. This is because integration of the equation of evolution of covariances is extremely costly, especially when the dimension of the problem is huge which is the case in numerical weather prediction.
This study has developed a new filter, the First order Extended Ensemble Filter (FoEEF), with a new extended innovation process to improve on the measurement and be able to estimate the state value of high dimensional data. We propose to estimate the covariances empirically, which lends the filter amenable to large dimensional models. The new filter is derived from stochastic state-space models and its performance is tested using Lorenz 63 system of ordinary differential equations and Matlab software.
The performance of the newly developed filter is then compared with the performances of three other filters, that is, Bootstrap particle Filter (BPF), First order Extended Kalman Bucy Filter (FoEKBF) and Second order Extended Kalman Bucy Filter (SoEKBF).
The performance of the FoEEF improves with the increase in ensemble size. Even with as low number of ensembles as 40, the FoEEF performs as good as the FoEKBF and SoEKBF. This shows, that the proposed filter can register a good performance when used in high-dimensional state-space models.

A Meta-Heuristic Search Algorithm based on Infrasonic Mating Displays in Peafowls

Kenekayoro Patrick

Journal of Advances in Mathematics and Computer Science, Page 98-106
DOI: 10.9734/jamcs/2021/v36i530366

Meta-heuristic techniques are important as they are used to find solutions to computationally intractable problems. Simplistic methods such as exhaustive search become computationally expensive and unreliable as the solution space for search algorithms increase. As no method is guaranteed to perform better than all others in all classes of optimization search problems, there is a need to constantly find new and/or adapt old search algorithms. This research proposes an Infrasonic Search Algorithm, inspired from the Gravitational Search Algorithm and the mating behaviour in peafowls. The Infrasonic Search Algorithm identified competitive solutions to 23 benchmark unimodal and multimodal test functions compared to the Genetic Algorithm, Particle Swarm Optimization Algorithm and the Gravitational Search Algorithm.

Some Geometric Properties of a Non-Strict Eight Dimensional Walker Manifold

Silas Longwap, Gukat G. Bitrus, Chibuisi Chigozie

Journal of Advances in Mathematics and Computer Science, Page 107-119
DOI: 10.9734/jamcs/2021/v36i530367

An 8 dimensional Walker manifold (M; g) is a strict walker manifold if we can choose a coordinate system fx1; x2; x3; x4; x5; x6; x7; x8g on (M,g) such that any function f on the manfold (M,g), f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x5; x6; x7; x8): In this work, we dene a Non-strict eight dimensional walker manifold as the one that we can choose the coordinate system such that for any f in (M; g); f(x1; x2; x3; x4; x5; x6; x7; x8) = f(x1; x2; x3; x4): We derive cononical form of the Levi-Civita connection, curvature operator, (0; 4)-curvature tansor, the Ricci tensor, Weyl tensorand study some of the properties associated with the class of Non-strict 8 dimensionalWalker manifold. We investigate the Einstein property and establish a theorem for the metric to be locally conformally at.

On the Ideal Based Zero Divisor Graphs of Unital Commutative Rings and Galois Ring Module Idealizations

Owino Maurice Oduor

Journal of Advances in Mathematics and Computer Science, Page 1-5
DOI: 10.9734/jamcs/2021/v36i530360

Let R be a commutative ring with identity 1 and I is an ideal of R. The zero divisor graph of the ring with respect to ideal has vertices defined as follows: {u ∈ Ic | uv ∈ I for some v ∈ Ic}, where Ic is the complement of I and two distinct vertices are adjacent if and only if their product lies in the ideal. In this note, we investigate the conditions under which the zero divisor graph of the ring with respect to the ideal coincides with the zero divisor graph of the ring modulo the ideal. We also consider a case of Galois ring module idealization and investigate its ideal based zero divisor graph.