This book, now in a second revised and enlarged edition, covers a course of mathematics designed primarily for physics and engineering students. It includes all the essential material on mathematical methods, presented in a form accessible to physics students and avoiding unnecessary mathematical jargon and proofs that are comprehensible only to mathematicians. Instead, all proofs are given in a form that is clear and sufficiently convincing for a physicist. Examples, where appropriate, are given from physics contexts. Both solved and unsolved problems are provided in each section of the book. The second edition includes more on advanced algebra, polynomials and algebraic equations in significantly extended first two chapters on elementary mathematics, numerical and functional series and ordinary differential equations. Improvements have been made in all other chapters, with inclusion of additional material, to make the presentation clearer, more rigorous and coherent, and the number of problems has been increased at least twofold. Mathematics for Natural Scientists: Fundamentals and Basics is the first of two volumes. Advanced topics and their applications in physics are covered in the second volume the second edition of which the author is currently being working on.
Quantum Physics of the Solid State: an Introduction" Draft foreword: 26/09/03 If only this book had been available when I was starting out in science! It would have saved me countless hours of struggle in trying to apply the general ideas of the standard solid-state text-books to solve real problems. The fact is that most of the texts stop at the point where the real difficulties begin. The great merit of this book is that it describes in an honest and detailed way what one really has to do in order to understand the multifarious properties of solids in terms of the fundamental physical theory of quantum mechanics. University students of the physical sciences are taught about the fundamental the ories, and know that quantum mechanics, together with relativity, is our basis for understanding the physical world. But the practical difficulties of using quantum mechanics to do anything useful are usually not very well explained. The truth is that the application of quantum theory to achieve our present detailed understand ing of solids has required the development of a large array of mathematical tech niques. This is closely analogous to the challenge faced long ago by theoretical astronomers in trying to apply Newton's equations of motion to the heavens -they too had to develop a battery of theoretical and computational techniques to do cal culations that could be compared with observation.
This book covers a course of mathematics designed primarily for physics and engineering students. It includes all the essential material on mathematical methods, presented in a form accessible to physics students, avoiding precise mathematical jargon and proofs which are comprehensible only to mathematicians. Instead, all proofs are given in a form that is clear and convincing enough for a physicist. Examples, where appropriate, are given from physics contexts. Both solved and unsolved problems are provided in each section of the book. Mathematics for Natural Scientists: Fundamentals and Basics is the first of two volumes. Advanced topics and their applications in physics are covered in the second volume.
This book covers the method of metric distances and its application in probability theory and other fields. The method is fundamental in the study of limit theorems and generally in assessing the quality of approximations to a given probabilistic model. The method of metric distances is developed to study stability problems and reduces to the selection of an ideal or the most appropriate metric for the problem under consideration and a comparison of probability metrics. After describing the basic structure of probability metrics and providing an analysis of the topologies in the space of probability measures generated by different types of probability metrics, the authors study stability problems by providing a characterization of the ideal metrics for a given problem and investigating the main relationships between different types of probability metrics. The presentation is provided in a general form, although specific cases are considered as they arise in the process of finding supplementary bounds or in applications to important special cases. Svetlozar T. Rachev is the Frey Family Foundation Chair of Quantitative Finance, Department of Applied Mathematics and Statistics, SUNY-Stony Brook and Chief Scientist of Finanlytica, USA. Lev B. Klebanov is a Professor in the Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic. Stoyan V. Stoyanov is a Professor at EDHEC Business School and Head of Research, EDHEC-Risk Institute—Asia (Singapore). Frank J. Fabozzi is a Professor at EDHEC Business School. (USA)
This volume is concerned with the problems in probability and statistics. Ill-posed problems are usually understood as those results where small changes in the assumptions lead to arbitrarily large changes in the conclusions. Such results are not very useful for practical applications where the presumptions usually hold only approximately (because even a slightest departure from the assumed model may produce an uncontrollable shift in the outcome). Often, the ill-posedness of certain practical problems is due to the lack of their precise mathematical formulation. Consequently, one can deal with such problems by replacing a given ill-posed problem with another, well-posed problem, which in some sense is 'close' to the original one. The goal in this book is to show that ill-posed problems are not just a mere curiosity in the contemporary theory of mathematical statistics and probability. On the contrary, such problems are quite common, and majority of classical results fall into this class. The objective of this book is to identify problems of this type, and re-formulate them more correctly. Thus, alternative (more precise in the above sense) versions are proposed of numerous classical theorems in the theory of probability and mathematical statistics. In addition, some non-standard problems are considered from this point of view.
This book consists of three major parts. The first two parts deal with general mathematical concepts and certain areas of operator theory. The third part is devoted to ill-posed problems. It can be read independently of the first two parts and presents a good example of applying the methods of calculus and functional analysis. The first part "Basic Concepts" briefly introduces the language of set theory and concepts of abstract, linear and multilinear algebra. Also introduced are the language of topology and fundamental concepts of calculus: the limit, the differential, and the integral. A special section is devoted to analysis on manifolds. The second part "Operators" describes the most important function spaces and operator classes for both linear and nonlinear operators. Different kinds of generalized functions and their transformations are considered. Elements of the theory of linear operators are presented. Spectral theory is given a special focus. The third part "Ill-Posed Problems" is devoted to problems of mathematical physics, integral and operator equations, evolution equations and problems of integral geometry. It also deals with problems of analytic continuation. Detailed coverage of the subjects and numerous examples and exercises make it possible to use the book as a textbook on some areas of calculus and functional analysis. It can also be used as a reference textbook because of the extensive scope and detailed references with comments.
The unique feature of this book is that it considers the theory of partial differential equations in mathematical physics as the language of continuous processes, that is, as an interdisciplinary science that treats the hierarchy of mathematical phenomena as reflections of their physical counterparts. Special attention is drawn to tracing the development of these mathematical phenomena in different natural sciences, with examples drawn from continuum mechanics, electrodynamics, transport phenomena, thermodynamics, and chemical kinetics. At the same time, the authors trace the interrelation between the different types of problems - elliptic, parabolic, and hyperbolic - as the mathematical counterparts of stationary and evolutionary processes. This combination of mathematical comprehensiveness and natural scientific motivation represents a step forward in the presentation of the classical theory of PDEs, one that will be appreciated by both students and researchers alike.
Quantum Physics of the Solid State: an Introduction" Draft foreword: 26/09/03 If only this book had been available when I was starting out in science! It would have saved me countless hours of struggle in trying to apply the general ideas of the standard solid-state text-books to solve real problems. The fact is that most of the texts stop at the point where the real difficulties begin. The great merit of this book is that it describes in an honest and detailed way what one really has to do in order to understand the multifarious properties of solids in terms of the fundamental physical theory of quantum mechanics. University students of the physical sciences are taught about the fundamental the ories, and know that quantum mechanics, together with relativity, is our basis for understanding the physical world. But the practical difficulties of using quantum mechanics to do anything useful are usually not very well explained. The truth is that the application of quantum theory to achieve our present detailed understand ing of solids has required the development of a large array of mathematical tech niques. This is closely analogous to the challenge faced long ago by theoretical astronomers in trying to apply Newton's equations of motion to the heavens -they too had to develop a battery of theoretical and computational techniques to do cal culations that could be compared with observation.
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