Summary

Abstract

This volume presented the fundamentals of probability, parts of information theory, differential geometry, and stochastic processes at a level that is connected with physical modeling. The emphasis has been on reporting results that can be readily implemented as simple computer programs, though detailed numerical analysis has not been addressed. In this way it is hoped that a potentially useful language for describing physical problems from various engineering and scientific fields has been made accessible to a wider audience. Not only the terminology and concepts, but also the results of the theorems presented serve the goal of efficient physical description. In this context, efficiency means that the essence of any stochastic phenomenon drawn from a broad set of such phenomena can be captured with relatively simple equations in few variables. And these equations can be solved either analytically or numerically in a way that requires minimal calculations (either by human or computer). This goal is somewhat different than that of most books on stochastic processes. A common goal in other books is to train students of mathematics to learn how to prove theorems. While the ability to prove a theorem is at the center of a pure mathematician’s skill set, the results that are spun off during that process sometimes need reinterpretation and restatement in less precise (but more accessible) language in order to be used by practitioners. In other words, rather than stating results in the classical definition–theorem–proof style aimed at pure mathematicians, this book is intended for mathematical modelers including engineers, computational biologists, physical scientists, numerical analysts, and applied and computational mathematicians.