Exercises in Applied Mathematics With a View toward Information Theory, Machine Learning, Wavelets, and Statistical Physics

This text presents a collection of mathematical exercises with the aim of guiding readers to study topics in statistical physics, equilibrium thermodynamics, information theory, and their various connections.  It explores essential tools from linear algebra, elementary functional analysis, and probability theory in detail and demonstrates their applications in topics such as entropy, machine learning, error-correcting codes, and quantum channels.  The theory of communication and signal theory are also in the background, and many exercises have been chosen from the theory of wavelets and machine learning.  Exercises are selected from a number of different domains, both theoretical and more applied.  Notes and other remarks provide motivation for the exercises, and hints and full solutions are given for many.  For senior undergraduate and beginning graduate students majoring in mathematics, physics, or engineering, this text will serve as a valuable guide as theymove on to more advanced work.

Prologue.- Part I: Algebra.- Linear Algebra.- Positive Matrices.- Algebra and Error Correcting Codes.- Part II: Analysis.- Complements in Real and Complex Analysis.- Complements in Functional Analysis.- Part III: Probability and Applications.- Probability Theory.- Entropy: Discrete Case.- Thermodynamics.

Daniel Alpay was born in Paris (France) and has a double formation of electrical engineer (Telecom Paris) and theoretical mathematics (Weizmann Institute, Rehovot, Israel). His research interests are in hypercomplex analysis, operator theory, stochastic processes (in particular in the setting of infinite dimensional analysis) and mathematical physics. He wrote a number of research books and more than 300 papers. Building on his research, he wrote two exercises books on complex analysis. He was a chaired professor at Ben-Gurion University (Beer-Sheva, Israel) and is now Professor at Chapman University (Orange, California), where he holds the Foster G. and Mary McGaw Professorship in Mathematical Sciences.