Richard Beals, “Analysis: An Introduction”
Cambridge University Press (September 13, 2004) | ISBN: 0521600472 | 272 pages | PDF | 1,8 Mb
“Analysis: An Introduction is most appropriate for a undergraduate who has already grappled with the main ideas from real analysis, and who is looking for a succinct, well-written treatise that connects these concepts to some of their most powerful applications. Beals’ book has the potential to serve this audience very well indeed.”
MAA Reviews, Christopher Hammond, Connecticut College
This self-contained text, suitable for advanced undergraduates, provides an extensive introduction to mathematical analysis, from the fundamentals to more advanced material. It begins with the properties of the real numbers and continues with a rigorous treatment of sequences, series, metric spaces, and calculus in one variable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book includes a number of interesting applications of the subject matter to areas both within and outside the field of mathematics. The aim throughout is to strike a balance between being too austere or too sketchy, and being so detailed as to obscure the essential ideas. A large number of examples and 500 exercises allow the reader to test understanding, practise mathematical exposition and provide a window into further topics.
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This book is a compilation of approximately nine hundred problems, which have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph.D. program. Students who work through this book will develop problem solving skills in areas such as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. The problems are organized by subject and ordered in an increasing level of difficulty. This new edition contains approximately 120 new problems and 200 new solutions. It is an ideal means for students to strengthen their foundation in basic mathematics and to prepare for graduate studies.
History of the Theory of Numbers, Volume ll: Diophantine Analysis (History of the Theory of Numbers)
Leonard Eugene Dickson, “History of the Theory of Numbers, Volume ll: Diophantine Analysis (History of the Theory of Numbers)”
Publisher: Dover Publications | Pages: 832 | ISBN: 0486442330 | DjVu | 12 MB
This 2nd volume in the series History of the Theory of Numbers presents material related to Diophantine Analysis. This series is the work of a distinguished mathematician who taught at the University of Chicago for 4 decades and is celebrated for his many contributions to number theory and group theory. 1919 edition.
Steven G. Krantz, “Geometric Function Theory: Explorations in Complex Analysis”
Birkhäuser Boston; 1 edition (September 20, 2005) | ISBN: 0817643397 | 314 pages | PDF | 2 Mb
Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous CauchyRiemann equations, and the corona problem.
The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.
Alexander Mielk, “Analysis, Modeling and Simulation of Multiscale Problems”
Springer; 1 edition (October 19, 2006) | ISBN: 3540356568 | 697 pages | PDF | 10,4 Mb
This book reports recent mathematical developments in the DFG Priority Programme “Analysis, Modeling and Simulation of Multiscale Problems”, which started as a German research initiative in 2000. The field of multiscale problems occurs in many fields of science, such as microstructures in materials, sharp-interface models, many-particle systems and motions on different spatial and temporal scales in quantum mechanics or in molecular dynamics. Recently developed tools are described in a comprehensive manner. This book provides the state of the art on the mathematical foundations of the modeling and the efficient numerical treatment of such problems.
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