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Berkeley Problems in Mathematics : Problem Books in Mathematics - Paulo Ney De Souza

Berkeley Problems in Mathematics

Problem Books in Mathematics

Hardcover

Published: 8th January 2004
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In 1977 the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first major requirements toward the Ph. D. degree in Mathematics. Its purpose was to determine whether first-year students in the Ph. D. program had successfully mastered basic mathematics in order to continue in the program with the likelihood of success. Since its inception, the exam has become a major hurdle to overcome in the pursuit of the degree. The purpose of this book is to publicize the material and aid in the preparation for the examination during the undergraduate years. The book is a compilation of over 1,250 problems which have appeared on the preliminary exams in Berkeley over the last twenty-five 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. Tags with the exact exam year provide the opportunity to rehearse complete examinations. The appendix includes instructions on accessing electronic versions of the exams as well as a syllabus and statistics of passing scores. This new edition has been updated with the most recent exams, including exams given during the Fall 2003 semester. There are numerous new problems and solutions which were not included in previous editions.

From the reviews of the third edition:

"This new edition has been updated with the most recent exams ... . There are numerous new problems and solutions which were not included in previous editions. It is an invaluable source of problems and solutions for every mathematics student who plans to enter a Ph. D program. ... 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. ... Tags with the exact exam year provide the opportunity to rehearse complete examinations. ... This new edition has been updated with the most recent exams ... ." (Zentralblatt fur Didaktik der Mathematik, November 2004)

"The Mathematics department of the University of California, Berkeley, has set a written preliminary examination to determine whether first year Ph.D. students have mastered enough basic mathematics to succeed in the doctoral program. Berkeley Problems in Mathematics is a compilation of all the ... questions, together with worked solutions ... . All the solutions I looked at are complete ... . Some of the solutions are very elegant. ... This is an impressive piece of work and a welcome addition to any mathematician's bookshelf." (Chris Good, The Mathematical Gazette, 90:518, 2006)

"During the last twenty-five years problems from written preliminary examinations that are required for the Ph.D. degree at the Mathematics Department of the University of California, Berkeley, have been assembled. ... The book is suited for students in mathematics, physics or engineering. Solutions are well explained, making the book valuable for self-study. The problems have a satisfactory high level, so the book is a rich resource of examples for lecturers as well, who need exercises ... . This book certainly is to be recommended." (Paula Bruggen, Bulletin of the Belgian Mathematical Society, 12:4, 2005)

Contents
Preface
Problems
Real Analysis
Elementary Calculus
Limitsand Continuity
Sequences, Series, and Products
Differential Calculus
Integral Calculus
Sequences of Functions
Fourier Series
Convex Functions
Multivariable Calculus
Limitsand Continuity
Differential Calculus
Integral Calculus
Differential Equations
First Order Equations
SecondOrder Equations
Higher Order Equations
Systems of Differential Equations
Metric Spaces
Topology of Rn
General Theory
Fixed Point Theorem
Complex Analysis
Complex Numbers
Series and Sequences of Functions
Conformal Mappings
Functions on the Unit Disc
Growth Conditions
Analytic and Meromorphic Functions
Cauchy's Theorem
Zeros and Singularities
Harmonic Functions
Residue Theory
Integrals Along the Real Axis
Algebra
Examples of Groups and General Theory
Homomorphisms and Subgroups
Cyclic Groups
Normality, Quotients, and Homomorphisms
Sn, An , Dn
Direct Products
Free Groups, Generators, and Relations
Finite Groups
Ringsand Their Homomorphisms
Ideals
Polynomials
Fields and Their Extensions
Elementary Number Theory
Linear Algebra
Vector Spaces
Rankand Determinants
Systems of Equations
Linear Transformations
Eigenvalues and Eigenvectors
Canonical Forms
Similarity
Bilinear, Quadratic Forms, and Inner Product Spaces
General Theory of Matrices
Solutions
Real Analysis
Elementary Calculus
Limits and Continuity
Sequences, Series, and Products
Differential Calculus
Integral Calculus
Sequences of Functions
Fourier Series
Convex Functions
Multivariable Calculus
Limitsand Continuity
Differential Calculus
Integral Calculus
Differential Equations
First Order Equations
Second Order Equations
Higher Order Equations
Systems of Differential Equations
Metric Spaces
Topology of Rn
General Theory
Fixed Point Theorem
Complex Analysis
Complex Numbers
Series and Sequences of Functions
Conformal Mappings
Functions on the Unit Disc
Growth Conditions
Analytic and Meromorphic Functions
Cauchy's Theorem
Zeros and Singularities
Harmonic Functions
Residue Theory
Integrals Along the Real Axis
Algebra
Examples of Groups and General Theory
Homomorphisms and Subgroups
Cyclic Groups
Normality, Quotients, and Homomorphisms
Sn, An , Dn
Direct Products
Free Groups, Generators, and Relations
Finite Groups
Rings and Their Homomorphisms
Ideals
Polynomials
Fields and Their Extensions
Elementary Number Theory
Linear Algebra
Vector Spaces
Rankand Determinants
Systems of Equations
Linear Transformations
Eigenvalues and Eigenvectors
Canonical Forms
Similarity
Bilinear, Quadratic Forms, and Inner Product Spaces
General Theory of Matrices III
Appendices
How to Get the Exams
On-line
Off-line, the Last Resort
Passing Scores
The Syllabus
References
Index
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780387204291
ISBN-10: 0387204296
Series: Problem Books in Mathematics
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 593
Published: 8th January 2004
Publisher: SPRINGER VERLAG GMBH
Country of Publication: US
Dimensions (cm): 24.4 x 16.6  x 3.8
Weight (kg): 1.07
Edition Number: 3
Edition Type: Revised