-
The Cauchy Operator and the Homogeneous Hahn Polynomials
Qiuxia Hu,
Xinhao Huang,
Zhizheng Zhang
Issue:
Volume 9, Issue 3, June 2021
Pages:
64-69
Received:
31 March 2021
Accepted:
23 April 2021
Published:
27 May 2021
Abstract: The Cauchy operator plays important roles in the theory of basic hypergeometric series. As some applications, our purpose is mainly to show new proofs of the Mehler’s formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials Φ(α)n(x,y|q)) by making use of the Cauchy operator and its properties. In addition, some interesting results are also obtained, which include a formal extension of the generating function for Φ(α)n(x,y|q)).
Abstract: The Cauchy operator plays important roles in the theory of basic hypergeometric series. As some applications, our purpose is mainly to show new proofs of the Mehler’s formula, the Rogers formula and the generating function for the homogeneous Hahn polynomials Φ(α)n(x,y|q)) by making use of the Cauchy operator and its properties. In addition, some i...
Show More
-
The Problem of Setting Traffic Signal Cycles at Crossroads
Dimitra Alexiou,
Leonidas Bakouros
Issue:
Volume 9, Issue 3, June 2021
Pages:
70-74
Received:
17 April 2021
Accepted:
13 May 2021
Published:
7 June 2021
Abstract: In this paper, the problem of setting traffic light cycles at crossroads and intersections is considered in order to reduce traffic congestion by minimizing total vehicle waiting time. A method to determine the family ℘ of all discrete cycle phasing systems with the minimum number of phases is presented. The aim is to detect the most appropriate phasing sequence for traffic control corresponding to a current traffic situation from among all the components of ℘. The method is applied at a complex multi-cross intersection. The problem, dealing with traffic movements and the conflicting relations that arise, is stated within the framework of graph theory. There are several methods for setting traffic signal cycles at traffic light intersections. In this paper and in the context of graph theory, we develop a method which aims to determine the family of all discrete phases of phase systems with the smallest number of phases. The aim is to select from the elements of the the appropriate phase system that corresponds to the current traffic situation.
Abstract: In this paper, the problem of setting traffic light cycles at crossroads and intersections is considered in order to reduce traffic congestion by minimizing total vehicle waiting time. A method to determine the family ℘ of all discrete cycle phasing systems with the minimum number of phases is presented. The aim is to detect the most appropriate ph...
Show More
-
Modelling Chlamydia Trachomatis Infection Among Young Women in Ghana: A Case Study at Tarkwa Nsuaem Municipality
Christiana Cynthia Nyarko,
Nicholas Nicodemus Nana Nsowa-Nuamah,
Peter Kwesi Nyarko,
Eric Neebo Wiah,
Albert Buabeng
Issue:
Volume 9, Issue 3, June 2021
Pages:
75-85
Received:
8 May 2021
Accepted:
3 June 2021
Published:
16 June 2021
Abstract: Chlamydia Genital infection has been a global health issue especially among most developing countries. Although, a lot of researchers have modelled CT infection to determine the impact of different intervals between Chlamydia infection and the development of Pelvic Inflammatory Disease (PID) on the cost-effectiveness of screening and the use of Chlamydia vaccine. This paper seeks to model the dynamics of Chlamydia Trachomatis (CT) infection among females who were diagnosed of vaginal discharge and the likelihood of developing PID complications. The model was formulated using a sexual network to explore the relationship between Chlamydia infection through diagnosed vaginal infection and PID. A sample of 147 females were diagnosed and screened of Chlamydia related symptoms on a routine check-up in the Tarkwa Nsuaem Municipality in the Western part of Ghana. Lyapunov functions was used to prove the necessary and sufficient conditions for Stability State of the system while Next Generation Method was also used to calculate the basic reproduction number (R0). The Stability Analysis of the Modified SIRS model shows that the system is locally and asymptotically stable at the Disease-Free Equilibrium (DFE) E0, when R0<1, and when R0>1, the Endemic Equilibrium (EE) E*, was found to be locally and asymptotically stable at certain conditions. It was observed that, as the distribution increases sharply at a given contact rate (β) of 0.05, many of the patients were infected within the first three days as compared to when the contact rate was 0.001. Moreover, at contact rates (β) of 0.5, R0 was greater than one, this shows how CT infection spreads in the population using parameter values in Table 1. Thus, the effects of change in the various initial conditions of the parameters (λ) and (β) on vaginal discharge and PID infections, turn to increase sharply at a higher infection rate for the first ten days of infection especially with vaginal discharge and then become stable over a period of time. This confirms the incubation period which is usually 7 to 10 days of infection. The paper concludes that, young women aged 18-24 years are more at risk of Chlamydia Trachomatis infection if diagnosed of vaginal discharge or PID and suggest early medication which is highly subsidised will help curb the spread of CT infection in the Municipality.
Abstract: Chlamydia Genital infection has been a global health issue especially among most developing countries. Although, a lot of researchers have modelled CT infection to determine the impact of different intervals between Chlamydia infection and the development of Pelvic Inflammatory Disease (PID) on the cost-effectiveness of screening and the use of Chl...
Show More
-
A Maximization Problem Involving a Fractional Laplace Type Operator
Issue:
Volume 9, Issue 3, June 2021
Pages:
86-91
Received:
25 May 2021
Accepted:
4 June 2021
Published:
16 June 2021
Abstract: Fractional Laplacian is an important nonlocal operator which has many applications in different kinds of differential equations. Recently, optimization problems involving the fractional Laplacian have been studied a lot by many authors. However, most of these papers are focusing on the optimization problems related to the first eigenvalue of the equation. Optimization problems related to the energy functional of the equation have not been investigated well enough. In this paper, we are going to study a maximization problem related to the energy functional of an equation involving a fractional Laplace type operator. Firstly, by using suitable variational framework in a fractional Sobolev space, we can show that a fractional equation has a solution which is in fact the global minimum of the corresponding energy functional. Moreover, by using reduction to absurdity we can obtain the uniqueness of the solution of the fractional equation. Then, we focus on a maximization problem related to the equation which takes the energy functional as the objective functional. Finally, by carefully analysing the properties of an arbitrarily choosen minimizing sequence and the tools of the rearrangement theory, we can prove that the maximization problem is solvable.
Abstract: Fractional Laplacian is an important nonlocal operator which has many applications in different kinds of differential equations. Recently, optimization problems involving the fractional Laplacian have been studied a lot by many authors. However, most of these papers are focusing on the optimization problems related to the first eigenvalue of the eq...
Show More
