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E-grāmata: Mathematical Principles of the Internet, Volume 2: Mathematics

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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 Z, 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(106)
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 Geodesics 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(3)
6 Applied Analysis 349(108)
6.1 Introduction
351(1)
6.2 Basic Analysis
351(11)
6.2.1 Point Sets
352(1)
6.2.2 Limits, Continuity, Derivatives, and Monotonicity
352(4)
6.2.3 Partial Derivatives
356(2)
6.2.4 Jacobians, Singularity, and Related Topics
358(1)
6.2.5 Hyperbolic Functions
359(1)
6.2.6 Ordinary Differential Equations
360(2)
6.3 Complex Analysis
362(9)
6.3.1 Coordinate Representation
363(1)
6.3.2 De Moivre's and Euler's Identities
364(1)
6.3.3 Limits, Continuity, Derivatives, and Analyticity
365(1)
6.3.4 Contours or Curves
366(1)
6.3.5 Conformal Mapping
367(1)
6.3.6 Integration
367(1)
6.3.7 Infinite Series
368(1)
6.3.8 Lagrange's Series Expansion
369(1)
6.3.9 Saddle Point Technique of Integration
369(2)
6.4 Cartesian Geometry
371(2)
6.4.1 Straight Line and Circle
371(1)
6.4.2 Conic Sections
372(1)
6.5 Moebius Transformation
373(5)
6.6 Polynomial Properties
378(10)
6.6.1 Roots of a Polynomial
378(1)
6.6.2 Resultant of Two Polynomials
378(4)
6.6.3 Discriminant of a Polynomial
382(1)
6.6.4 Bezout's Theorem
383(5)
6.7 Asymptotic Behavior and Algorithmic-Complexity Classes
388(5)
6.7.1 Asymptotic Behavior
388(4)
6.7.2 Algorithmic-Complexity Classes
392(1)
6.8 Vector Algebra
393(2)
6.8.1 Dot Product
393(1)
6.8.2 Vector Product
394(1)
6.9 Vector Spaces Revisited
395(8)
6.9.1 Normed Vector Space
396(1)
6.9.2 Complete Vector Space
396(2)
6.9.3 Compactness
398(1)
6.9.4 Inner Product Space
399(1)
6.9.5 Orthogonality
400(2)
6.9.6 Gram-Schmidt Orthogonalization Process
402(1)
6.9.7 Projections
402(1)
6.9.8 Isometry
403(1)
6.10 Fourier Series
403(6)
6.10.1 Generalized Functions
404(1)
6.10.2 Conditions for the Existence of Fourier Series
405(1)
6.10.3 Complex Fourier Series
406(1)
6.10.4 Trigonometric Fourier Series
407(2)
6.11 Transform Techniques
409(20)
6.11.1 Fourier Transform
409(4)
6.11.2 Laplace Transform
413(3)
6.11.3 Mellin Transform
416(2)
6.11.4 Wigner-Ville Transform
418(1)
6.11.5 Wavelet Transform
419(3)
6.11.6 Hadamard Transform
422(1)
6.11.7 Discrete Fourier Transform
423(6)
6.12 Faa di Bruno's Formula
429(2)
6.13 Special Mathematical Functions
431(3)
6.13.1 Gamma and Incomplete Gamma Functions
431(1)
6.13.2 Beta Function
432(1)
6.13.3 Riemann's Zeta Function
432(1)
6.13.4 Polylogarithm Function
433(1)
6.13.5 Bessel Function
433(1)
6.13.6 Exponential Integral
433(1)
6.13.7 Error Function
434(1)
Reference Notes
434(1)
Problems
435(18)
References
453(4)
7 Optimization, Stability, and Chaos Theory 457(82)
7.1 Introduction
459(1)
7.2 Basics of Optimization Theory
460(5)
7.2.1 Convex and Concave Functions
460(3)
7.2.2 Convex Sets
463(2)
7.3 Inequalities
465(3)
7.4 Elements of Linear Programming
468(5)
7.4.1 Hyperplanes
468(1)
7.4.2 Farkas' Alternative
469(1)
7.4.3 Primal and Dual Problems
469(4)
7.5 Optimization Techniques
473(18)
7.5.1 Taylor's Series of Several Variables
473(1)
7.5.2 Minimization and Maximization of a Function
474(3)
7.5.3 Nonlinear Constrained Optimization
477(1)
7.5.4 Optimization Problem with Equality Constraints
478(3)
7.5.5 Optimization Problem with Inequality Constraints
481(6)
7.5.6 Nonlinear Optimization via Duality Theory
487(4)
7.6 Calculus of Variations
491(3)
7.7 Stability Theory
494(14)
7.7.1 Notions Used in Describing a System
494(5)
7.7.2 Stability Concepts
499(9)
7.8 Chaos Theory
508(8)
7.8.1 Preliminaries
509(2)
7.8.2 Characterization of Chaotic Dynamics
511(3)
7.8.3 Examples of Chaotic Maps
514(2)
Reference Notes
516(1)
Problems
517(18)
References
535(4)
8 Probability Theory 539(70)
8.1 Introduction
541(1)
8.2 Axioms of Probability Theory
542(2)
8.3 Random Variables
544(2)
8.4 Average Measures
546(2)
8.4.1 Expectation
547(1)
8.4.2 Common Second Order Expectations
548(1)
8.5 Independent Random Variables
548(1)
8.6 Transforms and Moment Generating Functions
549(2)
8.6.1 z-Transform
549(1)
8.6.2 Moment Generating Function
550(1)
8.7 Examples of Some Distributions
551(8)
8.7.1 Discrete Distributions
552(1)
8.7.2 Continuous Distributions
553(4)
8.7.3 Multivariate Gaussian Distribution
557(2)
8.8 Some Well-Known Results
559(5)
8.8.1 Well-Known Inequalities
559(2)
8.8.2 Law of Large Numbers
561(1)
8.8.3 Gaussian Central Limit Theorem
562(2)
8.9 Generalized Central Limit Theorem
564(9)
8.9.1 Characteristic Function
564(1)
8.9.2 Infinitely Divisible Distributions
565(4)
8.9.3 Stable Distributions
569(4)
8.10 Range Distribution
573(2)
8.10.1 Joint Distribution of U and V
573(1)
8.10.2 Distribution of Range
574(1)
8.10.3 A Property of the Average Value of Range
574(1)
8.11 Large Deviation Theory
575(5)
8.11.1 A Prelude via Saddle Point Technique
576(1)
8.11.2 Cramer and Bahadur-Rao Theorems
577(3)
Reference Notes
580(1)
Problems
580(26)
References
606(3)
9 Stochastic Processes 609(70)
9.1 Introduction
611(1)
9.2 Terminology and Definitions
611(2)
9.3 Measure Theory
613(3)
9.3.1 Sigma-Algebra, Measurable Sets, and Measurable Space
613(1)
9.3.2 Borel Sets
614(1)
9.3.3 Measure
614(1)
9.3.4 Measurable Function
615(1)
9.4 Examples of Stochastic Processes
616(9)
9.4.1 Poisson Process
616(3)
9.4.2 Shot Noise Process
619(3)
9.4.3 Gaussian Process
622(1)
9.4.4 Brownian Motion Process
622(1)
9.4.5 Gaussian White Noise Process
623(1)
9.4.6 Brownian Motion and Gaussian White Noise
624(1)
9.5 Point Process
625(10)
9.5.1 Poisson Point Process
628(2)
9.5.2 Laplace Functional
630(2)
9.5.3 Marked Point Process
632(2)
9.5.4 Transformation of Poisson Point Processes
634(1)
9.6 Renewal Theory
635(7)
9.6.1 Ordinary Renewal Process
635(2)
9.6.2 Modified Renewal Process
637(1)
9.6.3 Alternating Renewal Process
638(2)
9.6.4 Backward and Forward Recurrence-Times
640(1)
9.6.5 Equilibrium Renewal Process
641(1)
9.6.6 Equilibrium Alternating Renewal Process
641(1)
9.7 Markov Processes
642(19)
9.7.1 Discrete-Time Markov Chains
643(7)
9.7.2 Continuous-Time Markov Chains
650(8)
9.7.3 Continuous-Time Markov Processes
658(3)
Reference Notes
661(1)
Problems
662(15)
References
677(2)
Index 679
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.
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