Periodic orbits in Hamiltonian systems with involutory symmetries

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetries. In both classes, involutions reverse the sign of the Hamiltonian function. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R and there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian. In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R

A Joint Model of Longitudinal Data and Informative Time with Time-Dependent Covariate

In analysis of longitudinal data, a number of methods have been proposed. Most of the traditional longitudinal methods assume that the independent variables are not dependent on time and are the same across study. However, one of the main advantages of longitudinal studies is the ability to observe outcomes and covariates at the same time, and a researcher can define whether changes in a covariate lead to changes in the outcome of interest. In addition, the methods focused on a predetermined observation time that does not carry information about the response variable. Moreover, it is possible in real research to have time-varying covariates, unbalanced observation time, and the observation times may be informative. The usual longitudinal statistical analysis might be biased if their assumptions are not valid. The purpose of this study was to develop a joint model of a longitudinal outcome and informative time with time-dependent covariates. In this study, a joint model and analysis of longitudinal data with possibly informative observation times and time-dependent covariates via joint probability distributions has been proposed. The maximum likelihood parameter estimates of the proposed model were obtained from Monte Carlo simulated data by employing a nonlinear optimization in R. Furthermore, the model selection criteria and likelihood ratio test statistic were computed to select the best fitting model and for comparing nested models. Additionally, the R codes were developed for the proposed model and an application is presented on the bladder cancer data used for explanation purposes. In the application, the results show that the time-dependent covariate appear to be important predictor in the longitudinal data

A First-Order Autoregressive Hurdle Poisson Model

Count regression models are used when the response variable takes count or non-negative values. Poisson and negative binomial distributions are commonly used to model count data. A frequent matter with the count data is to have an excess number of zeros that can result in overdispersed data when using Poisson or negative binomial distributions. Appropriate approaches to use when modeling excess-zero data is to use either a hurdle or a zero-inated Poisson (ZIP) distribution. Recently, the hurdle models are commonly used in fields such as medicine, epidemiology, genetics, and marketing. Excess-zero data occur frequently as a series of data that are repeatedly measured over time as well. In this dissertation, the hurdle distribution is used to model time series data that are counts with a high frequency of zeros. Particularly, a first order autoregressive hurdle process is formulated to model excess-zero time series data. Comparisons with two existing zero-inate time series models are presented and the models are evaluated based on their prediction capabilities. It is concluded that the developed hurdle autoregressive model provides better prediction of future observations compared to the other zero-inated Poisson models. The three models are used to analyze the crime data and the results show that the three models do not provide good prediction of future observations

Collegiate Women in Saudi Arabia: Leading Collectively for the Development of Self, Others, and Society

This is a constructivist grounded theory study that explored and investigated the leadership understandings of collegiate women in Saudi Arabia’s private non-profit universities, the opportunities they have to develop leadership, and how and why they develop leadership. The researcher engaged in semi-structured interviews with 25 collegiate women who have experiences in student leadership in one or more cocurricular program at their respective university. The findings revealed that collegiate women: (a) have a collective sense of the importance in developing their leadership potential to better themselves, to better each other, and for the betterment of the Saudi Arabian society; (b) they are interested in and motivated to develop their leadership potential; (c) develop leadership in inconsistent and informal ways; and (d) understand leadership as a relational practice. This is an unprecedented study in the field of college student leadership development within the context of Saudi Arabia. The findings have a number of important implications for action and future research in Saudi Arabia, as well as in neighboring countries that share similar complexities pertaining to women’s role and status in society