Mathematical analysis and Functions of real variables
A Guide to Advanced Real Analysis is perfect for graduate students preparing for qualifying exams, or for faculty who would like an overview of the subject. This book is an outline of the core material in the standard graduate-level real analysis course. It is intended as a resource for students in such a course as well as others who wish to learn or review the subject. On the abstract level, it covers the theory of measure and integration and the basics of point set topology, functional analysis, and the most important types of function spaces. On the more concrete level, it also deals with the applications of these general theories to analysis on Euclidean space: the Lebesgue integral, Hausdorff measure, convolutions, Fourier series and transforms, and distributions. The relevant definitions and major theorems are stated in detail. Proofs, however, are generally presented only as sketches, in such a way that the key ideas are explained but the technical details are omitted. In this way a large amount of material is presented in a concise and readable form.
MATHEMATICS -- Calculus, MATHEMATICS -- Mathematical Analysis, Functions of real variables, Mathematical analysis, Mathematical analysis, and Functions of real variables
Includes bibliographical references (p. 101-102) and index Prologue: notation, terminology, and set theory -- Topology -- Measure and integration: general theory -- Measure and integration: constructions and special examples -- Rudiments of functional analysis -- Function spaces -- Topics in analysis on Euclidean Space
Number theory, Number theory, and Elementare Zahlentheorie
1. Greatest common divisors -- 2. Unique factorization -- 3. Linear Diophantine equations -- 4. Congruences -- 5. Linear congruences -- 6. The Chinese remainder theorem -- 7. Fermat's theorem -- 8. Wilson's theorem -- 9. The number of divisors of an integer -- 10. The sum of the divisors of an integer -- 11. Amicable numbers -- 12. Perfect numbers -- 13. Euler's theorem and function -- 14. Primitive roots and orders -- 15. Decimals -- 16. Quadratic congruences -- 17. Gauss's lemma -- 18. The quadratic reciprocity theorem -- 19. The Jacobi symbol -- 20. Pythagorean triangles -- 21. x⁴ + y⁴ [not equal] z⁴ -- 22. Sums of two squares -- 23. Sums of three squares -- 24. Sums of four squares -- 25. Waring's problem -- 26. Pell's equation -- 27. Continued fractions -- 28. Multigrades -- 29. Carmichael numbers -- 30. Sophie Germain primes -- 31. The group of multiplicative functions -- 32. Bounds for [pi](x) -- 33. The sum of the reciprocals of the primes -- 34. The Riemann hypothesis -- 35. The prime number theorem -- 36. The abc conjecture -- 37. Factorization and testing for primes -- 38. Algebraic and transcendental numbers -- 39. Unsolved problems
This Guide is a friendly introduction to plane algebraic curves. It emphasizes geometry and intuition, and the presentation is kept concrete. You'll find an abundance of pictures and examples to help develop your intuition about the subject, which is so basic to understanding and asking fruitful questions. Highlights of the elementary theory are covered, which for some could be an end in itself, and for others an invitation to investigate further. Proofs, when given, are mostly sketched, some in more detail, but typically with less. References to texts that provide further discussion are often included. Computer algebra software has made getting around in algebraic geometry much easier. Algebraic curves and geometry are now being applied to areas such as cryptography, complexity and coding theory, robotics, biological networks, and coupled dynamical systems. Algebraic curves were used in Andrew Wiles'proof of Fermat's Last Theorem, and to understand string theory, you need to know some algebraic geometry. There are other areas on the horizon for which the concepts and tools of algebraic curves and geometry hold tantalizing promise. This introduction to algebraic curves will be appropriate for a wide segment of scientists and engineers wanting an entrance to this burgeoning subject.