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Problem Based Journey From Elementary Number Theory To An Introduction To Matrix Theory, A: The President Problems [Hardback]

(Technion-israel Inst Of Tech, Israel)
  • Formāts: Hardback, 164 pages
  • Izdošanas datums: 11-Nov-2021
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811234876
  • ISBN-13: 9789811234873
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  • Hardback
  • Cena: 48,21 €
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Problem Based Journey From Elementary Number Theory To An Introduction To Matrix Theory, A: The President ProblemsPalielināt
  • Formāts: Hardback, 164 pages
  • Izdošanas datums: 11-Nov-2021
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • ISBN-10: 9811234876
  • ISBN-13: 9789811234873
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9789811234873.html
"The book is based on lecture notes of a course "from elementary number theory to an introduction to matrix theory" given at the Technion to gifted high school students. It is problem based, and covers topics in undergraduate mathematics that can be introduced in high school through solving challenging problems. These topics include Number theory, Set Theory, Group Theory, Matrix Theory, and applications to cryptography and search engines"--
Introduction xi
Introduction to the Course xi
1 Algebraic Structures
1(10)
1.1 Groups, Fields and Rings
2(3)
1.2 Polynomials
5(1)
1.3 Hints
6(1)
1.4 Solutions
7(1)
1.5 Notes
7(4)
1.5.1 Prestigious prizes in mathematics
7(4)
2 The Natural Numbers
11(20)
2.1 What is Induction?
11(2)
2.2 Mathematical Induction
13(3)
2.3 Easy to State Open Problems
16(1)
2.4 Tiling and Geometry Problems
17(1)
2.5 Hints
18(2)
2.6 Solutions
20(6)
2.7 Notes
26(5)
2.7.1 Fermat's Last Theorem
26(1)
2.7.2 The Catalan Conjecture
26(1)
2.7.3 Euler
27(1)
2.7.4 Euclid
27(1)
2.7.5 Gauss
28(1)
2.7.6 Newton and Leibniz
28(2)
2.7.7 Collatz and Tau
30(1)
2.7.8 Sylvester--Gallai theorem
30(1)
3 The Integers
31(10)
3.1 The Greatest Common Divisor
31(2)
3.2 Congruence
33(4)
3.3 Hints
37(1)
3.4 Solutions
38(3)
4 The Real Numbers
41(8)
4.1 Sequences and Rational Numbers
41(2)
4.2 Irrational Numbers
43(1)
4.3 Hints
44(1)
4.4 Solutions
44(3)
4.5 Notes
47(2)
5 Introduction to Set Theory
49(14)
5.1 Countable Sets
49(5)
5.2 Uncountable Sets
54(2)
5.3 Hints
56(1)
5.4 Solutions
57(2)
5.5 Notes
59(4)
5.5.1 Cantor, Fraenkel, Russel and Zermelo
59(1)
5.5.2 Hilbert's 23 Problems
59(2)
5.5.3 Godel and Cohen
61(1)
5.5.4 Bernstein and Schroder
61(2)
6 The Pigeonhole Principle and the Base 2 Number System
63(8)
6.1 The Pigeonhole Principle
63(1)
6.2 The Base 2 Number System
63(2)
6.3 Hints
65(1)
6.4 Solutions
66(2)
6.5 Notes
68(3)
6.5.1 Dirichlet
68(1)
6.5.2 Ask Marilyn
68(1)
6.5.3 The Erdos--Szekeres Theorem
68(3)
7 Introduction to Group Theory
71(16)
7.1 Subgroups
71(2)
7.2 Lagrange's, Euler's and Fermat's Theorems
73(3)
7.3 The RSA Public Key Cybersystem
76(2)
7.4 Permutations
78(4)
7.5 Hints
82(1)
7.6 Solutions
82(5)
8 Introduction to Matrix Theory
87(22)
8.1 Matrices
87(14)
8.2 Graphs and Matrices
101(2)
8.3 Hints
103(1)
8.4 Solutions
103(6)
9 Fibonacci Numbers, Determinants and Eigenvalues
109(24)
9.1 The Fibonacci Sequence
109(3)
9.2 Determinants
112(5)
9.3 Eigenvalues and Eigenvectors
117(4)
9.4 The Zeckendorf Representation of the Natural Numbers
121(1)
9.5 Hints
122(1)
9.6 Solutions
123(7)
9.7 Notes
130(3)
9.7.1 Fibonacci
130(1)
9.7.2 The golden ratio
130(1)
9.7.3 Cayley and Hamilton
131(1)
9.7.4 The Friendship Theorem
131(2)
10 The Mathematics Behind Google's Page Rank and a Game of Numbers
133(12)
10.1 Page Rank
133(7)
10.2 Back to the Numbers on the Pentagon Problem
140(1)
10.3 Notes
141(4)
10.3.1 Perron and Frobenius
141(1)
10.3.2 Brin and Page
141(1)
10.3.3 Alon, Peres, Mozes and Eriksson
141(4)
Bibliography 145(2)
Index 147