68 pages | University of Oxford (2008) | ISBN: N/A | PDF | 1.1 MB

Our aim is to understand Atomic Physics, not just to illustrate the mathematics of Quantum Mechanics. This is both interesting and important, for Atomic Physics is the foundation for a wide range of basic science and practical technology. The structure and properties of atoms are the basis of Chemistry, and hence of Biology. Atomic Physics underlies the study of Astrophysics and Solid State Physics.

436 pages | The Trillia Group (December 18, 2009) | ISBN: 1931705038 | PDF | 2.1 MB

This final text in the Zakon Series on Mathematics Analysis follows the release of the author's Basic Concepts of Mathematics and Mathematical Analysis I and completes the material on Real Analysis that is the foundation for later courses in functional analysis, harmonic analysis, probability theory, etc. The first chapter extends calculus to n-dimensional Euclidean space and, more generally, Banach spaces, covering the inverse function theorem, the implicit function theorem, Taylor expansions, etc. Some basic theorems in functional analysis, including the open mapping theorem and the Banach-Steinhaus uniform boundedness principle, are also proved. The text then moves to measure theory, with a complete discussion of outer measures, Lebesgue measure, Lebesgue-Stieltjes measures, and differentiation of set functions. The discussion of measurable functions and integration in the following chapter follows an innovative approach, carefully choosing one of the equivalent definitions of measurable functions that allows the most intuitive development of the material. Fubini's theorem, the Radon-Nikodym theorem, and the basic convergence theorems (Fatou's lemma, the monotone convergence theorem, dominated convergence theorem) are covered. Finally, a chapter relates antidifferentiation to Lebesgue theory, Cauchy integrals, and convergence of parametrized integrals. Nearly 500 exercises allow students to develop their skills in the area.

367 pages | The Trillia Group (March 10, 2009) | ISBN: 193170502X | PDF | 2 MB

This text carefully leads the student through the basic topics of Real Analysis. Topics include metric spaces, open and closed sets, convergent sequences, function limits and continuity, compact sets, sequences and series of functions, power series, differentiation and integration, Taylor's theorem, total variation, rectifiable arcs, and sufficient conditions of integrability. Well over 500 exercises (many with extensive hints) assist students through the material. For students who need a review of basic mathematical concepts before beginning "epsilon-delta"-style proofs, the text begins with material on set theory (sets, quantifiers, relations and mappings, countable sets), the real numbers (axioms, natural numbers, induction, consequences of the completeness axiom), and Euclidean and vector spaces; this material is condensed from the author's Basic Concepts of Mathematics, the complete version of which can be used as supplementary background material for the present text.

208 pages | The Trillia Group (February 12, 2007) | ISBN: 1931705003 | PDF | 1.5 MB

This book helps the student complete the transition from purely manipulative to rigorous mathematics. The clear exposition covers many topics that are assumed by later courses but are often not covered with any depth or organization: basic set theory, induction, quantifiers, functions and relations, equivalence relations, properties of the real numbers (including consequences of the completeness axiom), fields, and basic properties of n-dimensional Euclidean spaces. The many exercises and optional topics (isomorphism of complete ordered fields, construction of the real numbers through Dedekind cuts, introduction to normed linear spaces, etc.) allow the instructor to adapt this book to many environments and levels of students. Extensive hypertextual cross-references and hyperlinked indexes of terms and notation add truly interactive elements to the text.

212 pages | Springer-Verlag (August 1999) | ISBN: 0817641092 | PDF | 1.1 MB

The Radon transform is an important topic in integral geometry which deals with the problem of expressing a function on a manifold in terms of its integrals over certain submanifolds. Solutions to such problems have a wide range of applications, namely to partial differential equations, group representations X-ray technology, nuclear magnetic resonance scanning, and tomography.