##### Set Theory INC# ∞# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions

Jaykov Foukzon

Journal of Advances in Mathematics and Computer Science, Page 90-112
DOI: 10.9734/jamcs/2021/v36i430359

In this paper intuitionistic set theory INC# ∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any
references to Catalan conjecture.

##### Modeling the Effect of Inpatient Rehabilitation of Tobacco Smokers on Smoking Dynamics

Amos Mwendwa Nyamai, Winifred N. Mutuku

Journal of Advances in Mathematics and Computer Science, Page 1-14
DOI: 10.9734/jamcs/2021/v36i430352

Aims:

1. Develop and analyze a mathematical model of the effect of inpatient rehabilitation of tobacco smokers on tobacco smoking using Kenya as a case study.
2. Perform stability analysis on the smoking free equilibrium point and endemic equilibrium point of the model.
3. Use numerical simulation to investigate the impact of inpatient rehabilitation of tobacco smokers on smoking.

Place and Duration of Study: Department of Mathematics and Actuarial Science, School of Pure and Applied Sciences (SPAS), Kenyatta University, Kenya, between May 2019 and September 2020.

Tobacco smoking is a serious burden in Kenya and the world at large. Smoking harms nearly every organ of the body and affects the overall health of a person. Despite the overwhelming facts about the consequences of tobacco smoking, it remains a bad wont which is socially accepted and widely spread. In this research we numerically analyze the dynamics of smoking incorporating the impact of inpatient rehabilitation to curb the smoking habit. We first present a three-compartment model incorporating inpatient rehabilitation, then develop the system of ordinary differential equations governing the smoking dynamics. The basic reproduction number is determined using next generation matrix method. The model equilibria were computed and the stability analysis carried out. The results of stability analysis indicate that the disease-free equilibrium (DFE) is both locally and globally asymptotically stable for RS < 1 while the endemic equilibrium is both locally and globally asymptotically stable for RS > 1. Numerical simulations of model carried out with the help of MATLAB shows that, when rehabilitation is implemented effectively, it helps in minimization of smoking in the community.

##### The Solutions of Generalized Euler Function Equation φ_2(n-φ_2 (n))=2ω(n)

Xu Yifan, Shen Zhongyan

Journal of Advances in Mathematics and Computer Science, Page 15-22
DOI: 10.9734/jamcs/2021/v36i430353

By using the properties of Euler function, an upper bound of solutions of Euler function equation  is given, where  is a positive integer. By using the classification discussion and the upper bound we obtained, all positive integer solutions of the generalized Euler function equation  are given, where is the number of distinct prime factors of n.

##### Covering a Regular Tetrahedron with Diminished Copies

Fangyu Zhang, Yuqin Zhang, Mei Han

Journal of Advances in Mathematics and Computer Science, Page 23-29
DOI: 10.9734/jamcs/2021/v36i430354

Let T be a unit regular tetrahedron. A diminished copy of T is the image of T under a homothety with positive ratio smaller than 1. Let m be a positive integer and let γm(T) be the smallest positive number r such that T can be covered by m translates of rT. Zong gave the results of γ4(T) = 3/4and γ5(T) = 9/13. However, the values of γ6(T) , γ7(T) and γ8(T) were not given then. In this article we give the upper bounds of γ6(T), γ7(T) and γ8(T).

##### Bayesian Prediction for Exponentiated Generalized Xgamma Distribution Based on Dual Generalized Order Statistics with Application to Poverty and COVID-19 Mortality Rates

R. E. Abd EL-Kader, A. M. Abd AL-Fattah, G. R. AL-Dayian, A. A. EL-Helbawy

Journal of Advances in Mathematics and Computer Science, Page 30-53
DOI: 10.9734/jamcs/2021/v36i430355

Statistical prediction is one of the most important problems in life testing; it has been applied in medicine, engineering, business and other areas as well. In this paper, the exponentiated generalized xgamma distribution is introduced as an application on the exponentiated generalized general class of distributions. Bayesian point and interval prediction of exponentiated generalized xgamma distribution based on dual generalized order statistics are considered. All results are specialized to lower records. The results are verified using simulation study as well as applications to real data sets to demonstrate the flexibility and potential applications of the distribution.

##### A Fractional Discrete Grey Model with Particle Swarm Optimizer and Its Applications in Forecasting the Gasoline Consumption in Chongqing China

Lanxi Zhang, Yubin Cai

Journal of Advances in Mathematics and Computer Science, Page 54-61
DOI: 10.9734/jamcs/2021/v36i430356

Forecasting gasoline consumption is of great significance for formulating oil production, foreign trade policies, and ensuring the balance of domestic refined oil supply. Based on grey system theory, a fractional accumulation operator is constructed to optimize the accumulation method of the traditional discrete grey model, and the Particle Swarm Optimization algorithm is used to solve the fractional nonlinear parameters. This model was used in the prediction of gasoline consumption in Chongqing, China, and compared with the existing 7 models. The results show that the fractional discrete grey model optimized by PSO has better prediction accuracy. The fractional discrete grey model optimized by PSO can be used as a quantitative method in the field of energy forecasting.

##### A Polygonal Finite Element Method for Stokes Equations

Xinjiang Chen

Journal of Advances in Mathematics and Computer Science, Page 62-78
DOI: 10.9734/jamcs/2021/v36i430357

In this paper, we extend the Bernardi-Raugel element  to convex polygonal meshes by using the generalized barycentric coordinates. Comparing to traditional discretizations defined on triangular and rectangular meshes, polygonal meshes can be more flexible when dealing with complicated domains or domains with curved boundaries. Theoretical analysis of the new element follows the standard mixed finite element theory for Stokes equations, i.e., we shall prove the discrete inf-sup condition (LBB condition) by constructing a Fortin operator. Because there is no scaling argument on polygonal meshes and the generalized barycentric coordinates are in general not polynomials, special treatments are required in the analysis. We prove that the extended Bernardi-Raugel element has optimal convergence rates. Supporting numerical results are also presented.

##### Approximation of Subsurface Seepage using 2- Dimensional Boussineq Equation

Chhaya K. Lande

Journal of Advances in Mathematics and Computer Science, Page 79-89
DOI: 10.9734/jamcs/2021/v36i430358

Groundwater is the main source of fresh water available for human beings. The surface water groundwater interaction affects the quantity and quality of groundwater. Hence the study of surfacewater-groundwater interaction is the emerging topic in this new era. In this paper, the analytical approximation of water table fluctuation in the aquifer is presented. The aquifer is subjected to the recharge and withdrawal activity through multiple basins and wells in the domain. The time dependent multiple recharge is considered. The flow is approximated by a non linear partial differential equation called Boussineq equation. The solution of Boussineq equation is developed using Finite Fourier cosine transform. Response of the solution to using numerical examples has been tested. Effect of aquifer parameters on the fluctuation of water table formation mainly water mound and cone of depression due to recharge and withdrawal are presented. The effect of permeability of aquifer base on the water table is also discussed.