Mathematics and Statistics

This book is devoted to real analysis methods for the problem of constructing Markov processes with boundary conditions in probability theory. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called the Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called the Ventcel (Wentzell) boundary condition, on the boundary of the domain. Most likely, a Markovian particle moves both by continuous paths and by jumps in the state space and obeys the Ventcel boundary condition, which consists of six terms corresponding to diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon, and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first-order Ventcel boundary value problems for second-order elliptic Waldenfels integro-differential operators. By using the Calderón–Zygmund theory of singular integrals, we prove the existence and uniqueness of theorems in the framework of the Sobolev and Besov spaces, which extend earlier theorems due to Bony–Courrège–Priouret to the vanishing mean oscillation (VMO) case. Our proof is based on various maximum principles for second-order elliptic differential operators with discontinuous coefficients in the framework of Sobolev spaces.

My approach is distinguished by the extensive use of the ideas and techniques characteristic of recent developments in the theory of singular integral operators due to Calderón and Zygmund. Moreover, we make use of an Lp variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups by me. The present book is amply illustrated; 119 figures and 12 tables are provided in such a fashion that a broad spectrum of readers understand our problem and main results.

This book explains the basic principles of pattern recognition (PR) and machine learning (ML) in an easy-to-understand manner for beginners who are trying to learn these principles on their own. Readers with a basic knowledge of linear algebra and probability theory will find it easy to follow.

Many excellent books in this field have been published in the past.  However, these books are not necessarily intended for self-study by beginners.

This book limits the topics to the minimum essential themes that beginners should learn, and explains them in detail. This book focuses on supervised learning, first introducing classical but important methods that have contributed to the development of the field. It then explains various methods that have since attracted attention. In explaining these methods, the book also provides a historical account of how new technologies were created as a result of combining classical ideas. The book emphasizes that Bayes decision rule is a fundamental concept in PR and ML.

The following points make this book suitable for self-study by beginners.
(1) The book is self-contained, so that the reader does not need to refer to other books or literature.  
(2) To deepen the reader's understanding, exercises are provided at the end of each chapter with detailed solutions available online.
(3) To promote the reader's intuitive understanding, the book presents as many concrete examples as possible.
(4) ‘Coffee Break’ columns introduce knowledge and know-how from the author's experience.

Unsupervised learning will be discussed in a sequel.