We discuss the notions of resurgence, formalizability, and formation of singularities in the context of partial differential equations. The results show that Écalle’s how analyzability theory extends naturally to PDEs.

Let G be a split simple group of type G2 over a field k, and let g be its Lie algebra. Answering a question of J.-L. Colliot-Thélène, B. Kunyavskĭi, V. L. Popov, and Z. Reichstein, we show that the function field k(g) is generated by algebraically independent elements over the field of adjoint invariants k(g). Résumé. Soit G un groupe algébrique simple et déployé de type G2 sur un corps k. Soit...

Although contextual weak independence (CWI) has shown promise in leading to more efficient probabilistic inference, no investigation has examined how CWIs can be obtained. In this paper, we suggest and analyze two methods for obtaining this kind of independence.

ii ACKNOWLEDGMENTS I would like to express my sincere and deepest appreciation to my advisor, Dr. Donal F. Day for his advice, motivation and support as well as his encouragement and understanding throughout this work. who are my best friends in my life. Finally, I would like to dedicate this work to my wife, Kyung-Ran, our daughter, Lauren, my parents, and my families. A special thanks is exte...

It’s perhaps something of a surprise that almost none of the so-called New Atheists has anything to say about arguments for God’s existence. Instead, they do tend to focus on the social effects of religion and question whether religious belief is good for society. One might justifiably doubt that the social impact of an idea for good or ill is an adequate measure of its truth, especially when t...

We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for Fq [θ] and provide complete algebraic independence results for them.

let x1;x2;...;xn have a jointly multivariate exchangeable normal distribution. in this work we investigate another proof of the independence of x and s2 using order statistics. we also assume that (xi ; yi); i =1; 2;...; n; jointly distributed in bivariate normal and establish the independence of the mean and the variance of concomitants of order statistics.