This paper explores the history and properties of the Gamma function with some analytical applications. Specifically, the Gamma function is employed to prove the legitimacy of the Standard Normal Distribution and for evaluation of some integrals involving the Laplace and Fourier Transforms using very simple techniques. Moreover, this paper demonstrates that the Gamma function is not a mere formula and proof in itself but rather an essential tool for applications in evaluating integrals that occur in practice and also in simplifying proofs of some other important identities and theorems in mathematics.
In this paper, we compare the performances of several models for fitting over-dispersed binary data. The distribution models considered in this study include the binomial (BN), the beta- binomial (BB), the multiplicative binomial (MBM), the Com-Poisson binomial (CPB) and the double binomial (DBM) models. Applications of these models to several well known data sets exhibiting under-dispersion and over-dispersion were considered in this paper. We applied these models to two frequency data sets and two data sets with covariates that have been variously analysed in the literature. The first relates to the Portuguese version of Duke Religiosity Index in a sample of 273 (202 women, 71 Male) postgraduate students of the faculty of Medicine of University of Sao Paulo. The second set that employs the Generalize Linear Model (GLM) is the correlated binary data which studies the cardiotoxic effects of doxorubicin chemoteraphy on the treatment of acute lymphoblastic leukemia in childhood. In the first data set, we have a single covariate, Sex (0,1) and two covariates in the second data set (dose and time).
Our results indicate that all the models considered here (excluding the binomial) behave reasonably well in modeling over-dispersed binary data with or without covariates, although both the multiplicative binomial and the double binomial models slightly behave better for these specific data sets. While this result may not be necessarily generalized to other variety of over and under-dispersed data, we would however, encourage the investigation of all possible models so that the right applicable model can be employed for a given data set under consideration. All analyses were carried out using PROC NLMIXED in SAS.
Dengue disease is a mosquito-borne infectious tropical disease caused by the dengue viruses of four serotypes, DEN 1 - DEN 4. It is transmitted between people by the bite of female adult Aedes mosquitoes. In the present work, we study a vector host epidemic model of dengue disease by considering control measures of the disease. The aim of the study is to observe the effects of control measures on the dengue disease development. Explicit formula for the metric, basic reproduction number R0 is obtained using Next Generation Matrix method. Stability of the disease free equilibrium and sensitivity analysis of model's parameters are discussed in terms of basic reproduction number. It is observed that the disease free equilibrium is locally and globally stable when R0 < 1 and unstable when R0> 1. Numerical results are carried out to illustrate the impact of control measures in the disease transmission.
In this paper, we prove some results for compatible and weakly compatible maps in non-Newtonian metric spaces. We also introduce E.A. and CLRT property in the context of non-Newtonian metric space and prove the corresponding fixed point results. We also provide examples to illustrate the concepts.
There exist several natural language processing systems that focus on checking the grammaticality (grammatical correctness or incorrectness) of natural language texts. Studies however showed that most existing systems do not assign specific scores to the grammaticality of the analysed text. Such scores would for instance prove very useful to second language learners and tutors, for judging the progress made in the learning process and assigning performance scores respectively. The current study was embarked upon to address this problem. A grammaticality grading model which comprised of 6 equations was developed using a vector space approach. The model was implemented in a natural language processing system. Correlation analysis showed that the grading (in %) performed using the developed model correlated at a coefficient of determination (R2) value of 0.9985 with the percentage of grammatical sentences in evaluated texts. The developed model is therefore deemed suitable for grammaticality grading in natural language texts. The developed model would readily find use in computer aided language learning and automated essay scoring.