The implantable cardioverter defibrillator (lCD) is the most effective treatment available for terminating potentially life-threatening ventricular fibrillation and ventricular tachycardia. The lCD detects and attempts to correct these arrhythmias by pacing, cardioversion, and defibrillation thereby providing lifesaving therapy to patients at risk for sudden cardiac death. Currently, 150,000 Americans receive ICDs each year. Although most lCD recipients are men, more women are now qualifying for insertion (Stutts, Cross, Conti, & Sears, 2007). Despite its established health benefits, lCD implantation is accompanied by psychological factors which merit research attention. This study investigated the experiences of women who live an lCD. The homogenous, purposeful sample consisted of 15 women who had an lCD that was implanted within the last three years and were receiving follow-up treatment at the same north Florida clinic. Data collection was accomplished through a semi-structured interview specific to the areas of pre-implantation, immediate post-implantation, and discharge home. Results were transcribed verbatim and then analyzed. Five core themes emerged from the transcripts along with multiple subcategories. The main themes included: Psychological Reactions, Physical Comfort, Procedural Issues, Body Image, and Feelings Regarding a Shock. Information obtained from this research is beneficial to nurses providing care to women with ICDs and to primary care advanced nurse practitioners in order to improve the overall health outcome and ongoing care of these women.

The advent of powerful computers has brought about the randomization technique for testing statistical hypotheses. Randomization tests are based on shuffles or rearrangements of the (combined) sample. Putting each of the I samples "in a bowl" forms the combined sample. Drawing samples "from the bowl" forms a shuffle. Shuffles can be made with or without replacement. In this thesis, analysis of means type randomization tests will be presented to solve the homogeneity of variance problem. An advantage of these tests is that they allow the user to graphically present the results via a decision chart similar to a Shewhart control chart. The focus is on finding tests that are robust to departures from normality. The proposed tests will be compared against commonly used nonrandomization tests. The type I error stability across several nonnormal distributions and the power of each test will be studied via Monte Carlo simulation.

The terminology and ideas involved in the statistical analysis of survival data are explained including the survival function, the probability density function, the hazard function, censored observations, parametric and nonparametric estimations of these functions, the product limit estimation of the survival function, and the proportional hazards estimation of the hazard function with explanatory variables. In Appendix A these ideas are applied to the actual analysis of the survival data for 54 cervical cancer patients.

Stochastic Calculus has been applied to the problem of pricing financial derivatives since 1973 when Black and Scholes published their famous paper "The Pricing of Options and Corporate Liabilities" in the Joumal of Political Economy. The purpose of this thesis is to show the mathematical principles underlying the methods applied to finance and to present a new model of the stock price process. As part of this paper, we present proofs of Ito's Formula and Girsanov's Theorem which are frequently used in financial applications. We demonstrate the application of these theorems to calculating the fair price of a European call option. There are two methods that result in the same price: the risk neutral valuation and the Black-Scholes partial differential equation. A new model of the stock price process is presented in Section 4. This model was inspired by the model of Cox and Ross published in 1976. We develop the model such that a martingale measure will exist for the present value of the stock price. We fit data to the traditional geometric Brownian motion model and the new model and compare the resulting prices. The data fit some stocks well, but in some cases the new model provided a better fit. The price of a European call is calculated for both models for several different stocks.

Many samples in the real world are very small in size and often do not follow a normal distribution. Existing tests for correlation have restrictions on the distribution of data and sample sizes, therefore the current tests cannot be used in some real world situations. In this thesis, two tests are considered to test hypotheses about the population correlation coefficient. The tests are based on statistics transformed by a saddlepoint approximation and by Fisher's Z-transformation. The tests are conducted on small samples of bivariate nonnormal data and found to perfom well. Simulations were run in order to compare the type I error rates and power of the new test with other commonly used tests. The new tests controlled type I error rates well, and have reasonable power performance.

Many distributions of multivariate data in the real world follow a non-normal model with distributions being skewed and/or heavy tailed. In studies in which multivariate non-normal distributions are needed, it is important for simulations of those variables to provide data that is close to the desired parameters while also being fast and easy to perform. Three algorithms for generating multivariate non-normal distributions are reviewed for accuracy, speed and simplicity. They are the Fleishman Power Method, the Fifth-Order Polynomial Transformation Method, and the Generalized Lambda Distribution Method. Simulations were run in order to compare the three methods by how well they generate bivariate distributions with the desired means, variances, skewness, kurtoses, and correlation, simplicity of the algorithms, and how quickly the desired distributions were calculated.

When the variance of a single population needs to be assessed, the well-known chi-squared test of variance is often used but relies heavily on its normality assumption. For non-normal populations, few alternative tests have been developed to conduct left tailed hypothesis tests of variance. This thesis outlines a method for generating new test statistics using a saddlepoint approximation. Several novel test statistics are proposed. The type-I error rates and power of each test are evaluated using a Monte Carlo simulation study. One of the proposed test statistics, R_gamma2, controls type-I error rates better than existing tests, while having comparable power. The only observed limitation is for populations that are highly skewed with heavy-tails, for which all tests under consideration performed poorly.

This thesis presents the Weil pairing on elliptic curves as a tool to implement a tripartite Diffie-Helman key exchange. Elliptic curves are introduced, as well as the addition operation that creates a group structure on its points. In leading to the definition of the Weil pairing, divisors of rational functions are studied, as well as the Weierstrass }-function, which shows the complex lattice as isomorphic to elliptic curves. Several important qualities of the Weil pairing are proved, and Miller's algorithm for quick calculation is shown. Next, the bipartite Diffie-Helman key exchange is discussed over finite fields and elliptic curves. Finally an example of a modifed Weil pairing is defined, which leads to the tripartite Diffie-Helman key exchange.

In this paper, we present and implement a method to assess the mathematical synergy of two-drug combinations based on a stochastic model. The drugs in question are two isomers that are applied to the human eye via a liquid eye drop. Techniques applied to the data in this paper can be applied to other two-drug combination studies. We derive the mean and the variance terms of the drug combination "effects" in closed form using Ito's method of stochastic differential equations. The model fit of the data to the individual subject is examined by both statistical and graphical methods. Two estimation methods in SAS, PROC NUN and PROC NLMIXED, are used to estimate model parameters. We perform simulation and power studies using R software to show the strengths of the proposed approach in estimating the model parameters. From this research, we find that the combination of drugs under study is synergistic in nature. We also confirm that the proposed stochastic model is appropriate.

This paper presents a detailed discussion of the Kronecker product of matrices. It begins with the definition and some basic properties of the Kronecker product. Statements will be proven that reveal information concerning the eigenvalues, singular values, rank, trace, and determinant of the Kronecker product of two matrices. The Kronecker product will then be employed to solve linear matrix equations. An investigation of the commutativity of the Kronecker product will be carried out using permutation matrices. The Jordan - Canonical form of a Kronecker product will be examined. Variations such as the Kronecker sum and generalized Kronecker product will be introduced. The paper concludes with an application of the Kronecker product to large least squares approximations.