Perron Theorem in the Monotone Iteration Method for Traveling Waves in Delayed Reaction-Diffusion Equations

In this paper we revisit the existence of traveling waves for delayed reaction diffusion equations by the monotone iteration method. We show that Perron Theorem on existence of bounded solution provides a rigorous and constructive framework to find traveling wave solutions of reaction diffusion systems with time delay. The method is tried out on two classical examples with delay: the predator-prey and Belousov-Zhabotinskii models.Comment: 17 pages. To appear in Journal of Differential Equation

Method of Recruitment Produces a Group of Persons Who Look Different from the Groups with HIV in Most Studies: A Case-Control Study

Abstract Introduction: More than 1.1 million citizens are infected with HIV and 40,000 new cases are reported annually. Between the years 2004-2007, the Centers for Disease Control (CDC) reported a 15% increase of HIV/AIDS incidence rate in 34 states with established HIV reporting. Although the incidence of HIV infection caused by illicit drugs injection and heterosexual activities have decreased, the CDC reported 26% of new HIV cases among men who have sex with men (MSM). Objectives: The aim of this study is to reveal if the method of recruitment will produce a group of persons who look different from the groups with HIV in most studies. Also to examine the risk factors such as individual’s behavior and characteristics that lead to the increase of HIV incidence despite the existence of different treatments that was claimed to be effective and the increased awareness of HIV infection. The risk factors examined in the study were alcohol intake, drug use, education level, marital status, and the socioeconomic status among people affected with HIV. Methods: The data was obtained from the National health and Nutrition Examination Survey (NHANES) for the years 1999-2012, and it was retrieved from The Centers of Disease Control and prevention (CDC) website. A descriptive analysis was run on the sought variables to be examined, then cases and controls were matched for age +5 years, gender, and race/ethnic groups. The study is a case control study using a sample of 238 participants (N=238) in the age range of 0-80 years and residing in all 50 states. Demographic characteristics were examined for cases and controls. A conditional logistic regression was conducted to examine the risk factors associated with HIV status. Lastly, a comparison of cases and controls and the original data were conducted. Results: The results reveal that the method of recruitment of participants did have an impact on the result of the study compared to other studies. It was also found that individuals who drink more than five alcoholic drinks per day increase the odds of being infected with HIV than those who do not. Conclusion: The method of recruitment used by NHANES produced a group of persons who look different from the groups with HIV in other studies. The results of the study strongly suggests the association between risky behavior taken by individuals, such as alcohol drinking and HIV status. These findings also suggest that public health professionals have to increase the awareness of alcohol consumption among the public to reduce HIV transmission

Sampling for multi-parameter spectral theory

In this talk we extend the sampling method to deal with the multiparameter spectral theory of Sturm-Liouville systems. To do so, we use the 2-dimensional version of the Whittaker-Shannon-Kotelnikov sampling theorem to find a representation for the characteristic function which leads to the computation of the eigencurves of the system

Computing the spectrum of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions

This paper deals with the computation of the eigenvalues of non self-adjoint Sturm-Liouville problems with parameter dependent boundary conditions using the \textit{regularized sampling method}. A few numerical examples among which singular ones will be presented to illustrate the merit of the method and comparison made with the exact eigenvalues when they are available

Transmutations for Strings

We investigate the existence and representation of transmutations, also known as transformation operators, for strings. Using measure theory and functional analytic methods we prove their existence and study their representation. We show that in general they are not close to unity since their representation does not involve a Volterra operator but rather the eigenvalue parameter. We also obtain conditions under which the transmutation is either a bounded or a compact operator. Explicit examples show that they cannot be reduced to Volterra type operators.

Continued fraction solution of Krein's inverse problem

The spectral data of a vibrating string are encoded in its so-called characteristic function. We consider the problem of recovering the distribution of mass along the string from its characteristic function. It is well-known that Stieltjes' continued fraction provides a solution of this inverse problem in the particular case where the distribution of mass is purely discrete. We show how to adapt Stieltjes' method to solve the inverse problem for a related class of strings. An application to the excursion theory of diffusion processes is presented.Comment: 18 pages, 2 figure

Traveling waves for a model of the Belousov-Zhabotinsky reaction

Following J.D. Murray, we consider a system of two differential equations that models traveling fronts in the Noyes-Field theory of the Belousov-Zhabotinsky (BZ) chemical reaction. We are also interested in the situation when the system incorporates a delay $h\geq 0$. As we show, the BZ system has a dual character: it is monostable when its key parameter $r \in (0,1]$ and it is bistable when $r >1$. For $h=0, r\not=1$, and for each admissible wave speed, we prove the uniqueness of monotone wavefronts. Next, a concept of regular super-solutions is introduced as a main tool for generating new comparison solutions for the BZ system. This allows to improve all previously known upper estimations for the minimal speed of propagation in the BZ system, independently whether it is monostable, bistable, delayed or not. Special attention is given to the critical case $r=1$ which to some extent resembles to the Zeldovich equation.Comment: 23 pages, to appear in the Journal of Differential Equation