Problems and Solutions in Mathematics, Chen Ji-Xiu, Jiang Guo-Ying, 2011

Problems and Solutions in Mathematics, Chen Ji-Xiu, Jiang Guo-Ying, 2011.
   
   This book covers six aspects of graduate school mathematics: Algebra, Topology, Differential Geometry, Real Analysis, Complex Analysis and Partial Differential Equations. It contains a selection of more than 500 problems and solutions based on the Ph.D. qualifying test papers of a decade of influential universities in North America. The mathematical problems under discussion are kept within the scope of the textbooks for graduate students. Finding solutions to these problems, however, involves a deep understanding of mathematical principles as well as an acquisition of skills in analysis and computation. As a supplement to textbooks, this book may prove to be of some help to the students in taking relevant courses. It may also serve as a reference book for the teachers concerned.




Примеры.
Let A be an 77 x 77 matrix with entries in C. Show that A has 77 distinct eigenvalues in C if and only if A commutes with no nonzero nilpotent matrix.

Let A be an n x n real matrix, all of whose (complex) eigenvalues are real and positive. Show that for any integer m > I, there exist at least one n x n real matrix B with Bm = A.
(Hint. Make use of the Jordan form S + TV of a conjugate of A1 where S is diagonal and N is strictly triangular.)

Contents.
Preface.
Preface to the Second Edition.
Part 1 Algebra.
Section 1 Linear Algebra.
Section 2 Group Theory.
Section 3 Ring Theory.
Section 4 Field and Galois Theory.
Part 2 Topology.
Section 1 Point Set Topology.
Section 2 Homotopy Theory.
Section 3 Homology Theory.
Part 3 Differential Geometry.
Section 1 Differential Geometry of Curves.
Section 2 Differential Geometry of Surfaces.
Section 3 Differential Geometry of Manifolds.
Part 4 Real Analysis.
Section 1 Measurability and Measure.
Section 2 Integral.
Section 3 Space of Integrable Functions.
Section 4 Differential.
Section 5 Miscellaneous Problems.
Part 5 Complex Analysis.
Section 1 Analytic and Harmonic Functions.
Section 2 Geometry of Analytic Functions.
Section 3 Complex Integration.
Section 4 The Maximum Modulus and Argument Principles.
Section 5 Series and Normal Families.
Part 6 Partial Differential Equations.
Section 1 General Theory.
Section 2 Elliptic Equations.
Section 3 Parabolic Equations.
Section 4 Hyperbolic Equations.
Abbreviations of Universities in this Book.



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