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Mathematical Principles of the Internet, Volume 2: Mathematics [Pehme köide]

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This two-volume set on Mathematical Principles of the Internet provides a comprehensive overview of the mathematical principles of Internet engineering. The books do not aim to provide all of the mathematical foundations upon which the Internet is based. Instead, they cover a partial panorama and the key principles.





Volume 1 explores Internet engineering, while the supporting mathematics is covered in Volume 2. The chapters on mathematics complement those on the engineering episodes, and an effort has been made to make this work succinct, yet self-contained. Elements of information theory, algebraic coding theory, cryptography, Internet traffic, dynamics and control of Internet congestion, and queueing theory are discussed. In addition, stochastic networks, graph-theoretic algorithms, application of game theory to the Internet, Internet economics, data mining and knowledge discovery, and quantum computation, communication, and cryptography are also discussed.





In order to study the structure and function of the Internet, only a basic knowledge of number theory, abstract algebra, matrices and determinants, graph theory, geometry, analysis, optimization theory, probability theory, and stochastic processes, is required. These mathematical disciplines are defined and developed in the books to the extent that is needed to develop and justify their application to Internet engineering.



Preface xv
List of Symbols
xxv
Greek Symbols xxxiii
1 Number Theory
1(44)
1.1 Introduction
3(1)
1.2 Sets
3(5)
1.2.1 Set Operations
5(2)
1.2.2 Bounded Sets
7(1)
1.2.3 Interval Notation
7(1)
1.3 Functions
8(4)
1.3.1 Sequences
8(1)
1.3.2 Permutation Mappings
9(1)
1.3.3 Permutation Matrices
10(1)
1.3.4 Unary and Binary Operations
11(1)
1.3.5 Logical Operations
11(1)
1.4 Basic Number-Theoretic Concepts
12(8)
1.4.1 Countability
12(1)
1.4.2 Divisibility
12(1)
1.4.3 Prime Numbers
12(1)
1.4.4 Greatest Common Divisor
13(4)
1.4.5 Continued Fractions
17(3)
1.5 Congruence Arithmetic
20(13)
1.5.1 Chinese Remainder Theorem
23(1)
1.5.2 Moebius Function
24(2)
1.5.3 Euler's Phi-Function
26(2)
1.5.4 Modular Arithmetic
28(2)
1.5.5 Quadratic Residues
30(2)
1.5.6 Jacobi Symbol
32(1)
1.6 Cyclotomic Polynomials
33(2)
1.7 Some Combinatorics
35(2)
1.7.1 Principle of Inclusion and Exclusion
35(1)
1.7.2 Stirling Numbers
36(1)
Reference Notes
37(1)
Problems
37(5)
References
42(3)
2 Abstract Algebra
45(90)
2.1 Introduction
47(1)
2.2 Algebraic Structures
47(16)
2.2.1 Groups
48(4)
2.2.2 Rings
52(1)
2.2.3 Subrings and Ideals
53(2)
2.2.4 Fields
55(2)
2.2.5 Polynomial Rings
57(5)
2.2.6 Boolean Algebra
62(1)
2.3 More Group Theory
63(3)
2.4 Vector Spaces over Fields
66(4)
2.5 Linear Mappings
70(1)
2.6 Structure of Finite Fields
71(15)
2.6.1 Construction
73(3)
2.6.2 Minimal Polynomials
76(3)
2.6.3 Irreducible Polynomials
79(1)
2.6.4 Factoring Polynomials
80(1)
2.6.5 Examples
81(5)
2.7 Roots of Unity in Finite Field
86(1)
2.8 Elliptic Curves
87(13)
2.8.1 Elliptic Curves over Real Fields
90(5)
2.8.2 Elliptic Curves over Finite Fields
95(1)
2.8.3 Elliptic Curves over Zp, p > 3
96(3)
2.8.4 Elliptic Curves over GF2n
99(1)
2.9 Hyperelliptic Curves
100(17)
2.9.1 Basics of Hyperelliptic Curves
100(2)
2.9.2 Polynomials, Rational Functions, Zeros, and Poles
102(3)
2.9.3 Divisors
105(6)
2.9.4 Mumford Representation of Divisors
111(6)
2.9.5 Order of the Jacobian
117(1)
Reference Notes
117(1)
Problems
118(14)
References
132(3)
3 Matrices and Determinants
135(68)
3.1 Introduction
137(1)
3.2 Basic Matrix Theory
137(7)
3.2.1 Basic Matrix Operations
139(1)
3.2.2 Different Types of Matrices
140(2)
3.2.3 Matrix Norm
142(2)
3.3 Determinants
144(4)
3.3.1 Definitions
144(2)
3.3.2 Vandermonde Determinant
146(1)
3.3.3 Binet-Cauchy Theorem
146(2)
3.4 More Matrix Theory
148(4)
3.4.1 Rank of a Matrix
148(1)
3.4.2 Adjoint of a Square Matrix
149(1)
3.4.3 Nullity of a Matrix
149(1)
3.4.4 System of Linear Equations
150(1)
3.4.5 Matrix Inversion Lemma
151(1)
3.4.6 Tensor Product of Matrices
151(1)
3.5 Matrices as Linear Transformations
152(3)
3.6 Spectral Analysis of Matrices
155(3)
3.7 Hermitian Matrices and Their Eigenstructures
158(3)
3.8 Perron-Frobenius Theory
161(4)
3.8.1 Positive Matrices
162(1)
3.8.2 Nonnegative Matrices
163(2)
3.8.3 Stochastic Matrices
165(1)
3.9 Singular Value Decomposition
165(3)
3.10 Matrix Calculus
168(3)
3.11 Random Matrices
171(6)
3.11.1 Gaussian Orthogonal Ensemble
171(2)
3.11.2 Wigner's Semicircle Law
173(4)
Reference Notes
177(1)
Problems
177(24)
References
201(2)
4 Graph Theory
203(40)
4.1 Introduction
205(1)
4.2 Undirected and Directed Graphs
205(4)
4.2.1 Undirected Graphs
206(1)
4.2.2 Directed Graphs
207(2)
4.3 Special Graphs
209(2)
4.4 Graph Operations, Representations, and Transformations
211(4)
4.4.1 Graph Operations
211(1)
4.4.2 Graph Representations
212(2)
4.4.3 Graph Transformations
214(1)
4.5 Plane and Planar Graphs
215(3)
4.6 Some Useful Observations
218(2)
4.7 Spanning Trees
220(6)
4.7.1 Matrix-Tree Theorem
220(2)
4.7.2 Numerical Algorithm
222(2)
4.7.3 Number of Labeled Trees
224(1)
4.7.4 Computation of Number of Spanning Trees
225(1)
4.7.5 Generation of Spanning Trees of a Graph
225(1)
4.8 The K-core, K-crust, and K-shell of a Graph
226(2)
4.9 Matroids
228(4)
4.10 Spectral Analysis of Graphs
232(3)
4.10.1 Spectral Analysis via Adjacency Matrix
232(3)
4.10.2 Laplacian Spectral Analysis
235(1)
Reference Notes
235(1)
Problems
236(5)
References
241(2)
5 Geometry
243
5.1 Introduction
245(1)
5.2 Euclidean Geometry
246(5)
5.2.1 Requirements for an Axiomatic System
246(1)
5.2.2 Axiomatic Foundation of Euclidean Geometry
247(2)
5.2.3 Basic Definitions and Constructions
249(2)
5.3 Circle Inversion
251(3)
5.4 Elementary Differential Geometry
254(9)
5.4.1 Mathematical Preliminaries
254(2)
5.4.2 Lines and Planes
256(1)
5.4.3 Curves in Plane and Space
257(6)
5.5 Basics of Surface Geometry
263(8)
5.5.1 Preliminaries
263(2)
5.5.2 First Fundamental Form
265(2)
5.5.3 Conformal Mapping of Surfaces
267(1)
5.5.4 Second Fundamental Form
268(3)
5.6 Properties of Surfaces
271(13)
5.6.1 Curves on a Surface
272(6)
5.6.2 Local Isometry of Surfaces
278(1)
5.6.3 Geodesies on a Surface
279(5)
5.7 Prelude to Hyperbolic Geometry
284(8)
5.7.1 Surfaces of Revolution
285(2)
5.7.2 Constant Gaussian Curvature Surfaces
287(1)
5.7.3 Isotropic Curves
288(1)
5.7.4 A Conformal Mapping Perspective
289(3)
5.8 Hyperbolic Geometry
292(12)
5.8.1 Upper Half-Plane Model
293(2)
5.8.2 Isometries of Upper Half-Plane Model
295(2)
5.8.3 Poincare Disc Model
297(4)
5.8.4 Surface of Different Constant Curvature
301(1)
5.8.5 Tessellations
301(1)
5.8.6 Geometric Constructions
302(2)
Reference Notes
304(1)
Problems
304(42)
References
346
Nirdosh Bhatnagar works, both in the academia and industry in Silicon Valley, California, USA. He is the author of several papers and reports. Nirdosh earned an MS in operations research, and MS and PhD in electrical engineering, all from Stanford University, Stanford, California.