Abstract: In this paper we introduce some basic properties of simple Lie algebras over algebraically closed fields. We furthermore introduce root systems and describe a family of directed multigraphs that can be used to characterize these algebras. We conclude with a brief introduction to weights and representations.

Abstract: Cryptography is formed from the best of two fields, mathematics and computer science. The algorithms themselves are pure mathematics, and they can be applied in everyday life through the work of computer scientists. The RSA (Rivest, Shamir, Adleman) algorithm is the first published public key cryptographic algorithm, and one of the most famous. The mathematics behind this algorithm is examined, and then applied into a chat program written in JAVA so that chat program is secure.

Abstract: In this project we study the mathematics involved with a satellite's motion orbiting the Earth and also discuss the main historical events that defined the development of this field. We investigate how the equations of motion and Kepler's Laws, combined with coordinate and time systems, help track satellites in their elliptical or circular orbits. We present a study of the two-body problem, coordinate and time system conversions, as well as the Mathematics of some algorithms that are used to predict a satellite's motion.

Abstract: Traditional time-series models such as Autoregressive (AR) and moving Average (MA) models are based on the homoscedasticity assumption, which translates into a constant variance for the errors of a model. This assumption has been shown to be inappropriate when dealing with some economic and financial market data. A new class of models - conditional heteroscedastic models - was developed to deal with data that does not exhibit constant variance of the errors. The most well known models in this class are the Autoregresssive Conditional Heteroscedastic model (ARCH) and its generalized version (GARCH). Stock market volatility, the square root of the variance of stock returns presents a very good application of this type of model. In finance, volatility is the expression of risk. Since we must take risks to achieve rewards, finding appropriate methods to forecast volatility is ncessary in order to optimize our behavior and, in particular, our portfolio. I will present the general properties of the ARCH and GARCH models and use both Monte Carlo simulations and known financial time series data to test their performance.

Abstract: Life threatening events including bradycardia and apnea in infants are a major health concern for families and physicians. Events further described as “extreme events” occurred at least once in 20.6% of asymptomatic infants and 33% of infants born prematurely. The motivation for this study is the Collaborative Home Infant Monitoring Evaluation (CHIME); a study formed and funded by the National Institute of Health (NIH). The CHIME database is the largest infant longitudinal physiologic dataset. The work proposed here is part of a larger study to use heart rate variability to study the prediction of life-threatening events in infants and the classification of infant sleep state. This work seeks to develop methods to examine the effect of age on heart rate variability measures in each sleep state. These methods include both numerical summaries and graphical displays.

Abstract: Assume every person in a group of people has a unique tidbit of gossip to share. How many conversations must occur before everyone in the group knows all the gossip? It depends on what we assume about the conversations. The gossip number assumes that conversations occur between two people who tell each other everything they know. The e-mail gossip number assumes that one person shares all the gossip that he or she knows with all his or her friends in a mass mailing, but information exchange is only one way. We discuss some interesting results about the gossip number and the email gossip number of a graph.

Abstract: Previous benchmarking of dynamic memory allocators focused on text-based programs. With the introduction graphics-intensive applications, the question is raised if the previous results of text-based programs still holds true. This paper re-visits Zorn’s earlier work and finds that in commercial games, the Windows XP allocator and the Lea allocator, both general-purpose allocators, out perform other allocators.