This user-friendly book is designed for a first, one-semester course in statistical signal analysis for a broad audience of students in engineering and the physical sciences. A bibliography is included for readers who wish to pursue things in greater depth.
This book stems from a course within the NSF-funded Gateway Coalition Initiative in Engineering Education, aimed at modernizing the engineering curriculum and attracting more students. It emphasizes the integration of technology, particularly through Mathematica, to enhance learning experiences in statistics courses across various universities.
Linear and Nonlinear Dynamics in Continuous Media
The book delves into the application of distributions in various fields of physical and engineering sciences, emphasizing their significance in modeling and analysis. It covers foundational concepts and advanced topics, providing a comprehensive understanding of how distributions can be utilized to solve complex problems. The text includes practical examples and applications, making it a valuable resource for students and professionals seeking to enhance their knowledge in mathematical methods relevant to physical phenomena and engineering challenges.
This comprehensive exposition on analytic methods for solving problems in science and engineering is grounded in distribution theory and features many modern topics relevant to practitioners and researchers. It aims to provide both specialists and non-specialists with practical mathematical tools for research and analysis. Volume 1 focuses on asymptotic methods, including stationary phase and steepest descent techniques for Fourier and other integral transforms, alongside topics such as fractional calculus, the uncertainty principle, wavelets, and multiresolution analysis. Volume 2 analyzes the three basic types of linear PDEs—elliptic, parabolic, and hyperbolic—and includes discussions on first-order nonlinear PDEs and conservation laws, along with nonlinear waves, Burger's equations, KdV equations, and gas dynamics. Volume 3 explores distributional tools in generalized stochastic processes and fields, covering probability distributions, Brownian motion, stochastic differential equations, and multiscale anomalous fractional dynamics. With clear explanations, an accessible writing style, and numerous illustrations/examples, this work serves as a valuable self-study reference for anyone looking to enhance their understanding and proficiency in these problem-solving methods. It is tailored for a broad scientific and engineering audience while maintaining mathematical precision.
Focusing on the interrelations between diffusion stochastic processes, stochastic differential equations (SDEs), and fractional infinitesimal operators, the book offers a concise yet thorough exposition of these concepts. It has been classroom tested at Case Western Reserve University, ensuring its effectiveness for both senior and graduate students. This practical approach enhances understanding of complex mathematical theories through real-world applications and examples.
These lecture notes are woven around the subject of Burgers' turbulence/KPZ model of interface growth, a study of the nonlinear parabolic equation with random initial data. The analysis is conducted mostly in the space-time domain, with less attention paid to the frequency-domain picture. However, the bibliography contains a more complete information about other directions in the field which over the last decade enjoyed a vigorous expansion. The notes are addressed to a diverse audience, including mathematicians, statisticians, physicists, fluid dynamicists and engineers, and contain both rigorous and heuristic arguments. Because of the multidisciplinary audience, the notes also include a concise exposition of some classical topics in probability theory, such as Brownian motion, Wiener polynomial chaos, etc.