An Advanced Galerkin Approach to Solve the Nonlinear \\[6pt]Reaction-Diffusion Equations With Different Boundary Conditions

This study proposed a scheme originated from the Galerkin finite element method (GFEM) for solving nonlinear parabolic partial differential equations (PDEs) numerically with initial and different types of boundary conditions. The scheme is applied generally handling the nonlinear terms in a simple way and throwing over restrictive assumptions. The convergence and stability analysis of the method are derived. The error of the method is estimated. In the series, eminent problems are solved, such as  Fisher's equation, Newell-Whitehead-Segel equation, Burger's equation, and  Burgers-Huxley equation to demonstrate the validity, efficiency, accuracy, simplicity and applicability of this scheme. In each example, the comparison results are presented both numerically and graphically

Step Pyramid Distribution for Prime Numbers

In this document, we will present a new way to visualize the distribution of Prime Numbers in the number system to spot Prime numbers in a subset of numbers using a simpler algorithm. Then we will look throw a classification algorithm to check if a number is prime using only 7 simple arithmetic operations with an algorithm complexity less than or equal to O (7) operations for any number.

The Infimum Norm of Completely Positive Maps

Let A  be a unital C* -algebra, let L: A→B(H)  be a linear map, and let ∅: A→B(H)  be a completely positive linear map. We prove the property in the following:  is completely positive}=inf {||T*T+TT*||1/2:  L= V*TπV  which is a minimal commutant representation with isometry} . Moreover, if L=L* , then  is completely positive  . In the paper we also extend the result  is completely positive}=inf{||T||: L=V*TπV}  [3 , Corollary 3.12].

Relationship between Event Prevalence Rate and Gini Coefficient of Predictive Model

Predictive models are currently used for early intervention to help identify patients with a high risk of adverse events. Assessing the accuracy of such models is a crucial part of the development process. To measure the predictive performance of a scoring model, quantitative indices such as the K-S statistic and C-statistic are used. This paper discusses the relationship between Gini coefficients and event prevalence rates. The main contribution of the paper is the theoretical proof of the relationship between the Gini coefficient and event prevalence rate.

Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero. Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= λ(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function λ satisfies the equation R(∂/∂z（λ(c∆^-1λ_zz+a))= 0 with some complex constants a and c≠0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function λ. If the functions λ and λ + λ_0 satisfy the equation for a real constant λ0, 0, then there exists a non-zero Killing vector field on the torus.

Reviewer Acknowledgements for Journal of Mathematics Research, Vol. 13, No. 6

Reviewer Acknowledgements for Journal of Mathematics Research, Vol. 13, No. 6

Extended Pythagoras Theorem Using Hexagons

This article provides the geometric and algebraic proof of the variant equation of the Pythagorean theorem x^2-xy+y2=z^2 . The hypothesis that will be proven is that just as squares govern the original version x^2+y^2=z^2 , hexagons are found to govern x^2-xy+y^2=z^2 . Both the special case x=y  and general case of x≠y  are examined.

Analysis of Social-Economic Factors on Population Change in Heilongjiang Province

The population is a constituent element of human society and is one of the indicators of a country's comprehensive strength. The quantity and quality of population directly determine the level of development of a country or region. Too small a population makes a country or region lack the motivation to develop, and too large a population strains local resources. Therefore, the state introduces relevant policies to regulate the population quantity in China. The analysis of the factors influencing the change in population size helps assess the current state of development and is essential for planning its future prosperity. This paper analyzes the impact of social-economics factors on population size change in Heilongjiang province since China's reform and opening up using double-logarithmic model (DLM) estimated by Elastic Net estimation (ENE). Meanwhile, this paper provides some policy recommendations to promote the growth of population size in Heilongjiang Province.

Determinants of Capital Adequacy Ratio of Banks in Botswana

Capital Adequacy Ratio (CAR) plays a very important role in the financial success of banks and acts as a buffer to prevent and absorb any unexpected losses. This study examines explanatory variables that influence CAR for nine banks in Botswana. Multiple linear regression was used for analysis, with CAR as the dependent variable and thirteen financial ratios as the independent variables. The study period is 2015-2019. Based on the data for this period, it was established that out of the thirteen financial ratios utilised, only four were found to have significant impact on the CAR of the nine banks under study, which are: Asset to Equity Ratio (A E), Return on Equity (ROE), Non-Performing Loans Ratio (NPL RATIO) and the Cost-to-Income Ratio (C I). The A E Ratio was found to be the most influential driver of the CAR and the NPL Ratio was found to be the least influential driver of the CAR for the banks under study.

N(2,2,0) Algebras and Related Topic

In this paper, we investigate the elementary properties of the N(2,2,0)-algebra. Especially, some properties of nilpotent N(2,2,0)-algebras are presented. Also some relationships between nilpotent N(2,2,0)-algebra and other algebras with the type of (2,0) are obtained.