Mathematical principles of the Internet. Vol.2 Mathematical concepts

724 Pages·2019·3.715 MB·English

by  Bhatnagar N

Mathematical Principles
of the Internet
Volume 2
Mathematical Concepts
Mathematical Principles
of the Internet
Volume 2: Mathematical Concepts
Nirdosh Bhatnagar
CRC Press
Taylor & Francis Group
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International Standard Book Number Volume II: 978-1-1385-05513 (Hardback)
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...tothememoryofmyparents:
Smt.ShakuntlaBhatnagar&ShriRaiChandulalBhatnagar

Contents
Preface ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: xv
ListofSymbols :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: xxv
GreekSymbols ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::xxxiii
1. NumberTheory:::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1
1.1 Introduction ......................................................... 3
1.2 Sets ................................................................ 3
1.2.1 SetOperations ............................................... 5
1.2.2 BoundedSets ................................................ 7
1.2.3 IntervalNotation ............................................. 7
1.3 Functions ........................................................... 8
1.3.1 Sequences................................................... 8
1.3.2 PermutationMappings ........................................ 9
1.3.3 PermutationMatrices ......................................... 10
1.3.4 UnaryandBinaryOperations................................... 11
1.3.5 LogicalOperations ........................................... 11
1.4 BasicNumber-TheoreticConcepts ...................................... 12
1.4.1 Countability ................................................. 12
1.4.2 Divisibility .................................................. 12
1.4.3 PrimeNumbers .............................................. 12
1.4.4 GreatestCommonDivisor ..................................... 13
1.4.5 ContinuedFractions .......................................... 17
1.5 CongruenceArithmetic................................................ 20
1.5.1 ChineseRemainderTheorem................................... 23
1.5.2 MoebiusFunction ............................................ 24
1.5.3 Euler’sPhi-Function .......................................... 26
1.5.4 ModularArithmetic........................................... 28
1.5.5 QuadraticResidues ........................................... 30
1.5.6 JacobiSymbol ............................................... 32
viii Contents
1.6 CyclotomicPolynomials............................................... 33
1.7 SomeCombinatorics.................................................. 35
1.7.1 PrincipleofInclusionandExclusion............................. 35
1.7.2 StirlingNumbers ............................................. 36
ReferenceNotes ........................................................... 37
Problems ................................................................. 37
References................................................................ 42
2. AbstractAlgebra::::::::::::::::::::::::::::::::::::::::::::::::::::::: 45
2.1 Introduction ......................................................... 47
2.2 AlgebraicStructures .................................................. 47
2.2.1 Groups ..................................................... 48
2.2.2 Rings....................................................... 52
2.2.3 SubringsandIdeals ........................................... 53
2.2.4 Fields....................................................... 55
2.2.5 PolynomialRings ............................................ 57
2.2.6 BooleanAlgebra ............................................. 62
2.3 MoreGroupTheory .................................................. 63
2.4 VectorSpacesoverFields.............................................. 66
2.5 LinearMappings ..................................................... 70
2.6 StructureofFiniteFields .............................................. 71
2.6.1 Construction................................................. 73
2.6.2 MinimalPolynomials ......................................... 76
2.6.3 IrreduciblePolynomials ....................................... 79
2.6.4 FactoringPolynomials......................................... 80
2.6.5 Examples ................................................... 81
2.7 RootsofUnityinFiniteField .......................................... 86
2.8 EllipticCurves....................................................... 87
2.8.1 EllipticCurvesoverRealFields................................. 90
2.8.2 EllipticCurvesoverFiniteFields................................ 95
2.8.3 EllipticCurvesoverZp;p>3.................................. 96
2.8.4 EllipticCurvesoverGF ..................................... 99
2n
2.9 HyperellipticCurves.................................................. 100
2.9.1 BasicsofHyperellipticCurves.................................. 100
2.9.2 Polynomials,RationalFunctions,Zeros,andPoles................. 102
2.9.3 Divisors..................................................... 105
2.9.4 MumfordRepresentationofDivisors ............................ 111
2.9.5 OrderoftheJacobian ......................................... 117
ReferenceNotes ........................................................... 117
Problems ................................................................. 118
References................................................................ 132
3. MatricesandDeterminants :::::::::::::::::::::::::::::::::::::::::::::: 135
3.1 Introduction ......................................................... 137
3.2 BasicMatrixTheory.................................................. 137
3.2.1 BasicMatrixOperations....................................... 139
3.2.2 DifferentTypesofMatrices .................................... 140
Contents ix
3.2.3 MatrixNorm ................................................ 142
3.3 Determinants ........................................................ 144
3.3.1 Definitions .................................................. 144
3.3.2 VandermondeDeterminant..................................... 146
3.3.3 Binet-CauchyTheorem........................................ 146
3.4 MoreMatrixTheory .................................................. 148
3.4.1 RankofaMatrix ............................................. 148
3.4.2 AdjointofaSquareMatrix..................................... 149
3.4.3 NullityofaMatrix............................................ 149
3.4.4 SystemofLinearEquations .................................... 150
3.4.5 MatrixInversionLemma....................................... 151
3.4.6 TensorProductofMatrices..................................... 151
3.5 MatricesasLinearTransformations ..................................... 152
3.6 SpectralAnalysisofMatrices .......................................... 155
3.7 HermitianMatricesandTheirEigenstructures............................. 158
3.8 Perron-FrobeniusTheory .............................................. 161
3.8.1 PositiveMatrices ............................................. 162
3.8.2 NonnegativeMatrices ......................................... 163
3.8.3 StochasticMatrices ........................................... 165
3.9 SingularValueDecomposition.......................................... 165
3.10 MatrixCalculus...................................................... 168
3.11 RandomMatrices .................................................... 171
3.11.1 GaussianOrthogonalEnsemble................................. 171
3.11.2 Wigner’sSemicircleLaw ...................................... 173
ReferenceNotes ........................................................... 177
Problems ................................................................. 177
References................................................................ 201
4. GraphTheory ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 203
4.1 Introduction ......................................................... 205
4.2 UndirectedandDirectedGraphs ........................................ 205
4.2.1 UndirectedGraphs............................................ 206
4.2.2 DirectedGraphs.............................................. 207
4.3 SpecialGraphs....................................................... 209
4.4 GraphOperations,Representations,andTransformations ................... 211
4.4.1 GraphOperations............................................. 211
4.4.2 GraphRepresentations ........................................ 212
4.4.3 GraphTransformations ........................................ 214
4.5 PlaneandPlanarGraphs............................................... 215
4.6 SomeUsefulObservations............................................. 218
4.7 SpanningTrees ...................................................... 220
4.7.1 Matrix-TreeTheorem ......................................... 220
4.7.2 NumericalAlgorithm ......................................... 222
4.7.3 NumberofLabeledTrees ...................................... 224
4.7.4 ComputationofNumberofSpanningTrees ....................... 225
4.7.5 GenerationofSpanningTreesofaGraph......................... 225
4.8 The -core, -crust,and -shellofaGraph.............................. 226
K K K
x Contents
4.9 Matroids............................................................ 228
4.10 SpectralAnalysisofGraphs............................................ 232
4.10.1 SpectralAnalysisviaAdjacencyMatrix.......................... 232
4.10.2 LaplacianSpectralAnalysis .................................... 235
ReferenceNotes ........................................................... 235
Problems ................................................................. 236
References................................................................ 241
5. Geometry ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 243
5.1 Introduction ......................................................... 245
5.2 EuclideanGeometry .................................................. 246
5.2.1 RequirementsforanAxiomaticSystem .......................... 246
5.2.2 AxiomaticFoundationofEuclideanGeometry .................... 247
5.2.3 BasicDefinitionsandConstructions ............................. 249
5.3 CircleInversion ...................................................... 251
5.4 ElementaryDifferentialGeometry ...................................... 254
5.4.1 MathematicalPreliminaries .................................... 254
5.4.2 LinesandPlanes ............................................. 256
5.4.3 CurvesinPlaneandSpace ..................................... 257
5.5 BasicsofSurfaceGeometry............................................ 263
5.5.1 Preliminaries ................................................ 263
5.5.2 FirstFundamentalForm ....................................... 265
5.5.3 ConformalMappingofSurfaces ................................ 267
5.5.4 SecondFundamentalForm..................................... 268
5.6 PropertiesofSurfaces................................................. 271
5.6.1 CurvesonaSurface........................................... 272
5.6.2 LocalIsometryofSurfaces..................................... 278
5.6.3 GeodesicsonaSurface ........................................ 279
5.7 PreludetoHyperbolicGeometry........................................ 284
5.7.1 SurfacesofRevolution ........................................ 285
5.7.2 ConstantGaussianCurvatureSurfaces ........................... 287
5.7.3 IsotropicCurves.............................................. 288
5.7.4 AConformalMappingPerspective .............................. 289
5.8 HyperbolicGeometry ................................................. 292
5.8.1 UpperHalf-PlaneModel....................................... 293
5.8.2 IsometriesofUpperHalf-PlaneModel ........................... 295
5.8.3 PoincaréDiscModel.......................................... 297
5.8.4 SurfaceofDifferentConstantCurvature.......................... 301
5.8.5 Tessellations................................................. 301
5.8.6 GeometricConstructions ...................................... 302
ReferenceNotes ........................................................... 304
Problems ................................................................. 304
References................................................................ 346