Problems and Solutions in Mathematics

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World Scientific, 1998 - Mathematics - 539 pages
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This book contains a selection of more than 500 mathematical problems and their solutions from the PhD qualifying examination papers of more than ten famous American universities. The problems cover six aspects of graduate school mathematics: Algebra, Differential Geometry, Topology, Real Analysis, Complex Analysis and Partial Differential Equations. The depth of knowledge involved is not beyond the contents of the textbooks for graduate students, while solution of the problems requires deep understanding of the mathematical principles and skilled techniques. For students this book is a valuable complement to textbooks; for lecturers teaching graduate school mathematics, a helpful reference.
 

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Contents

Linear Algebra
3
Group Theory 26
26
Ring Theory 44
44
Field and Galois Theory 59
59
Point Set Topology 83
83
Homotopy Theory 99
99
Homology Theory 118
118
Differential Geometry of Curves 153
153
Differential 302
302
Miscellaneous Problems 322
322
Analytic and Harmonic Functions 335
335
Geometry of Analytic Functions 360
360
Complex Integration 377
377
The Maximum Modulus and Argument Principles 413
413
Series and Normal Families 433
433
General Theory 457
457

Differential Geometry of Surfaces 171
171
Differential Geometry of Manifold 194
194
Measurablity and Measure 231
231
Integral 256
256
Space of Integrable Functions 283
283
Elliptic Equations 472
472
Parabolic Equations 496
496
Hyperbolic Equations 513
513
Abbreviations of Universities in This Book 539
539
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