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Putnam and Beyond 2nd ed. 2017 [Minkštas viršelis]

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  • Formatas: Paperback / softback, 850 pages, aukštis x plotis: 254x178 mm, weight: 1628 g, 297 Illustrations, black and white; XVIII, 850 p. 297 illus., 1 Paperback / softback
  • Išleidimo metai: 03-Oct-2017
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319589865
  • ISBN-13: 9783319589862
Kitos knygos pagal šią temą:
Putnam and Beyond 2nd ed. 2017Didesnis paveikslėlis
  • Formatas: Paperback / softback, 850 pages, aukštis x plotis: 254x178 mm, weight: 1628 g, 297 Illustrations, black and white; XVIII, 850 p. 297 illus., 1 Paperback / softback
  • Išleidimo metai: 03-Oct-2017
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319589865
  • ISBN-13: 9783319589862
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319589862.html
Raktažodžiai:
This book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quad

ratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies.





Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and gradu

ate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons.
1 Methods of Proof
1(24)
1.1 Argument by Contradiction
1(2)
1.2 Mathematical Induction
3(8)
1.3 The Pigeonhole Principle
11(3)
1.4 Ordered Sets and Extremal Elements
14(4)
1.5 Invariants and Semi-Invariants
18(7)
2 Algebra
25(82)
2.1 Identities and Inequalities
25(22)
2.1.1 Algebraic Identities
25(3)
2.1.2 x2 ≥ 0
28(4)
2.1.3 The Cauchy-Schwarz Inequality
32(3)
2.1.4 The Triangle Inequality
35(3)
2.1.5 The Arithmetic Mean-Geometric Mean Inequality
38(5)
2.1.6 Sturm's Principle
43(3)
2.1.7 Other Inequalities
46(1)
2.2 Polynomials
47(24)
2.2.1 A Warmup in One-Variable Polynomials
47(3)
2.2.2 Polynomials in Several Variables
50(2)
2.2.3 Quadratic Polynomials
52(4)
2.2.4 Viete's Relations
56(5)
2.2.5 The Derivative of a Polynomial
61(3)
2.2.6 The Location of the Zeros of a Polynomial
64(2)
2.2.7 Irreducible Polynomials
66(2)
2.2.8 Chebyshev Polynomials
68(3)
2.3 Linear Algebra
71(25)
2.3.1 Operations with Matrices
71(1)
2.3.2 Determinants
72(6)
2.3.3 The Inverse of a Matrix
78(4)
2.3.4 Systems of Linear Equations
82(4)
2.3.5 Vector Spaces, Linear Combinations of Vectors, Bases
86(2)
2.3.6 Linear Transformations, Eigenvalues, Eigenvectors
88(3)
2.3.7 The Cayley-Hamilton and Perron-Frobenius Theorems
91(5)
2.4 Abstract Algebra
96(11)
2.4.1 Binary Operations
96(2)
2.4.2 Groups
98(5)
2.4.3 Rings
103(4)
3 Real Analysis
107(104)
3.1 Sequences and Series
108(28)
3.1.1 Search for a Pattern
108(2)
3.1.2 Linear Recursive Sequences
110(4)
3.1.3 Limits of Sequences
114(6)
3.1.4 More About Limits of Sequences
120(6)
3.1.5 Series
126(5)
3.1.6 Telescopic Series and Products
131(5)
3.2 Continuity, Derivatives, and Integrals
136(44)
3.2.1 Functions
136(2)
3.2.2 Limits of Functions
138(2)
3.2.3 Continuous Functions
140(3)
3.2.4 The Intermediate Value Property
143(3)
3.2.5 Derivatives and Their Applications
146(5)
3.2.6 The Mean Value Theorem
151(3)
3.2.7 Convex Functions
154(6)
3.2.8 Indefinite Integrals
160(3)
3.2.9 Definite Integrals
163(3)
3.2.10 Riemann Sums
166(2)
3.2.11 Inequalities for Integrals
168(4)
3.2.12 Taylor and Fourier Series
172(8)
3.3 Multivariate Differential and Integral Calculus
180(15)
3.3.1 Partial Derivatives and Their Applications
180(6)
3.3.2 Multivariate Integrals
186(4)
3.3.3 The Many Versions of Stokes' Theorem
190(5)
3.4 Equations with Functions as Unknowns
195(16)
3.4.1 Functional Equations
195(6)
3.4.2 Ordinary Differential Equations of the First Order
201(3)
3.4.3 Ordinary Differential Equations of Higher Order
204(3)
3.4.4 Problems Solved with Techniques of Differential Equations
207(4)
4 Geometry and Trigonometry
211(46)
4.1 Geometry
211(32)
4.1.1 Vectors
211(5)
4.1.2 The Coordinate Geometry of Lines and Circles
216(5)
4.1.3 Quadratic and Cubic Curves in the Plane
221(9)
4.1.4 Some Famous Curves in the Plane
230(2)
4.1.5 Coordinate Geometry in Three and More Dimensions
232(5)
4.1.6 Integrals in Geometry
237(3)
4.1.7 Other Geometry Problems
240(3)
4.2 Trigonometry
243(14)
4.2.1 Trigonometric Identities
243(3)
4.2.2 Euler's Formula
246(3)
4.2.3 Trigonometric Substitutions
249(4)
4.2.4 Telescopic Sums and Products in Trigonometry
253(4)
5 Number Theory
257(34)
5.1 Integer-Valued Sequences and Functions
257(7)
5.1.1 Some General Problems
257(3)
5.1.2 Fermat's Infinite Descent Principle
260(1)
5.1.3 The Greatest Integer Function
261(3)
5.2 Arithmetic
264(18)
5.2.1 Factorization and Divisibility
264(1)
5.2.2 Prime Numbers
265(4)
5.2.3 Modular Arithmetic
269(2)
5.2.4 Fermat's Little Theorem
271(3)
5.2.5 Wilson's Theorem
274(1)
5.2.6 Euler's Totient Function
275(3)
5.2.7 The Chinese Remainder Theorem
278(2)
5.2.8 Quadratic Integer Rings
280(2)
5.3 Diophantine Equations
282(9)
5.3.1 Linear Diophantine Equations
282(3)
5.3.2 The Equation of Pythagoras
285(2)
5.3.3 Pell's Equation
287(2)
5.3.4 Other Diophantine Equations
289(2)
6 Combinatorics and Probability
291(554)
6.1 Combinatorial Arguments in Set Theory
291(7)
6.1.1 Combinatorics of Sets
291(2)
6.1.2 Combinatorics of Numbers
293(2)
6.1.3 Permutations
295(3)
6.2 Combinatorial Geometry
298(7)
6.2.1 Tessellations
298(4)
6.2.2 Miscellaneous Combinatorial Geometry Problems
302(3)
6.3 Graphs
305(8)
6.3.1 Some Basic Graph Theory
305(4)
6.3.2 Euler's Formula for Planar Graphs
309(2)
6.3.3 Ramsey Theory
311(2)
6.4 Binomial Coefficients and Counting Methods
313(17)
6.4.1 Combinatorial Identities
313(5)
6.4.2 Generating Functions
318(3)
6.4.3 Counting Strategies
321(6)
6.4.4 The Inclusion-Exclusion Principle
327(3)
6.5 Probability
330(515)
6.5.1 Equally Likely Cases
330(3)
6.5.2 Establishing Relations Among Probabilities
333(4)
6.5.3 Geometric Probabilities
337(4)
Methods of Proof
341(34)
Algebra
375(118)
Real Analysis
493(150)
Geometry and Trigonometry
643(74)
Number Theory
717(54)
Combinatorics and Probability
771(74)
Index of Notation 845(2)
Index 847
Rzvan Gelca, Texas Tech University, works in Chern-Simons theory, a   field of mathematics that blends low dimensional topology, mathematical physics, geometry, and the theory of group representations. He is also involved in mathematics competitions such as the mathematical Olympiads and the W.L. Putnam Mathematical Competition. He is co-author of 2 published books (with Titu Andreescu), namely Mathematical Olympiad Challenges and the first edition of Putnam and Beyond. In 2015 Gelca and Andreescu will also publish a monograph on Pells Equations.



Titu Andreescu, University of Texas-Dallas,  is highly involved with mathematics contests and olym

piads. He was the Director of AMC (as appointed by the Mathematical Association of America), Director of MOP, Head Coach of the USA IMO Team and Chairman of the USAMO. He has also authored a large number of books on the topic of problem solving and olympiad-style mathematics including the first edition of Putnam and Beyond (with Razvan Gelca), Mathematical Olympiad Treasures and Mathematical Olympiad Challenges (with Razvan Gelca). Additional Springer publications include Mathematical Bridges, Complex Numbers from A to Z, Number Theory and a new monograph on Pells Equations to be published in 2015.