Hypha consists of two regions; cap (apex) and cylindrical shaft (subapex and mature combined). The hyphal-cap is the most critical part due to its dominant role in the hyphal-wall growth and mor- phogenesis. Just how the hyphal-wall growth is regulated in order to maintain its tubular shape has been the subject of much research for over 100 years. Here, we derived a formula of approximate idealized hyphal-contour based on gradients of secant lines joining a fixed coor- dinate at the starting hyphal-shaft to any coordinates on the contour. The formula is capable of being a control for experimental analysis in which it is not limited to one specific shape of the hyphal-like cell. Also, it potentially can play a role as built-in or ready-made hyphal-fitting profile that “best fits” microscopic images of various actual hyphal- like cells. In other words, given a microscopic image of hyphal-like cell, mycologists and microbiologists would not have to wonder about mathematical representation of its contour since the formula has pre- pared for it.

Benford’s law is widely applied in testing anomalies in various dataset, including accounting fraud detection and population numbers. It is a statistical regularity, which is said that it works better with larger datasets that span large orders of magnitude distributed in a non-uniform way. In this study, we examine the potential metrics in small-sized Quran dataset that are applicable for the Benford’s law. Against our expectations, we find that the Quran dataset conforms to the Benford’s law. We provide evidence that metrics such as total paragraph per chapter and total verse per chapter conform to Benford’s distribution. However, total verse is closer to Benford’s law prediction compared to total paragraph.

Several different approaches are implemented and used to solve higher order non-linear boundary value problems (BVPs). Galerkin weighted residual technique (GWRT) are commonly used to solve linear and non-linear BVPs. In this paper, we have proposed GWRT for the numerical computations of general third order three-point non-linear BVPs. Modified Legendre and Bezier Polynomials, over the interval [0, 1], are chosen separately as a basis functions. The main advantage of this method is its efficiency and simple applicability. Numerical result is presented to illustrate the performance of the proposed method. The results clearly show that the proposed method is suitable for solving third order nonlinear BVPs

In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a prime near ring. If there exist p,q ϵ M and two sided nonzero β-derivation f on M, where β:M→M is a homomorphism, satisfying the following conditions:

f([s,t])=s^p [β(s),β(t)]s^q ∀ s,t ϵ M
f([s,t])=-s^p [β(s),β(t)]s^q ∀ s,t ϵ M

In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a prime near ring. If there exist  and two sided generalized β-derivation G associated with the non-zero two sided β-derivation  on M, where  is a homomorphism, satisfying the following conditions:

1. G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M
2. G([p_1,q_1 ])=〖p_1〗^(u_1 ) [β(p_1 ),β(q_1)]〖p_1〗^(v_1 ) ∀ p_1,q_1 ϵ M

Then M is a commutative ring.

In this paper, we prove commutativity of prime near rings by using the notion of β-derivations. Let M be a zero symmetric prime near ring. If there exist p ≥ 0, q ≥ 0 and a nonzero two sided β-derivation d on M, where β : M → M is a homomorphism, such that d satisfy one of the following conditions:

• [β(s),d(t)] = s^p(β(s)oβ(t))s^q ∀ s, t ∈ M
• [β(s),d(t)] = −s^p(β(s)oβ(t))s^q ∀ s, t ∈ M
• [d(s),β(t)] = t^p(β(s)oβ(t))t^q ∀ s, t ∈ M
• [d(s),β(t)] = −t^p(β(s)oβ(t)t^q ∀ s, t ∈ M

In this study, we proposed a malaria-hygiene mathematical model using non-linear differential equation. The model equations are divided into seven compartments consisting of five human compartments (Hygienic Susceptible, Unhygienic Susceptible, Hygienic Infected, Unhygienic Infected, and Recovered) and two vector compartments (Non-Disease Carrier vector and Disease carrier vector). Differential Transformation Method (DTM) is applied to solve the mathematical model. The solutions obtained by DTM are compared with Runge-Kutta order 4th method (RK4). The graphical solutions illustrate similarity between DTM and RK4. It therefore imply that DTM can be consider a reliable alternative solution method.

We establish the existence of continuous solutions to initial value problem for first order functional integro-differential inclusion. The study holds in the case when the set-valued function has Lipschitz selection, also we discuss the existence of maximal and minimal solutions. The continuous dependence and uniqueness of the solution will be proved. As an application, the initial value problem for the arbitrary-order differential inclusion will be studied.

The present study is related to the numerical solution of the human immunodeficiency virus (HIV) infection model with full logistic proliferation and variable source term (depending on the viral load) used for the supply of new CD4+ T-cells from thymus instead of using simple logistic proliferation and constant source term. In simple logistic proliferation term only the healthy or infected CD4+ T-cells proliferation are considered while in full logistic proliferation term both the proliferation of healthy and infected are considered. Consequently, the variable source term is used for the supply of new healthy CD4+ T-cells from thymus, which is a decreasing function depending on the concentration of viral load. Continuous Galerkin-Petrov method, in particular cGP(2)-method has been invoked for finding the approximate solution of the model. For cGP(2)-method, we have two unknowns on each time interval which have to be calculated by solving 2 × 2 block system. This method is an accurate of order three in the whole time interval and shows the convergence of order four in the discrete time points. We examined the impact of various clinical parameters and study the existence of the infected state. Additionally, the Runge Kutta method of order four briefly RK4-method has also been used to verify and strengthen the results obtained by cGP(2)-method. Obtained results are displayed both graphically and in tabular form. The results obtained in this study confirm the idea that the cGP(2)-method is a powerful technique which can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

In this paper, we analyze a new form of non-local boundary conditions for a two-dimensional elliptic partial differential equation model. Some relations for the existence of the different types of eigenvalues and their corresponding eigenfunctions are proved. The figures of the relations are also dragged to show the effect of the non-locality in boundary conditions on the eigenvalue problem.