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 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2019 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number Volume I: 978-1-1385-05483 (Hardback) International Standard Book Number Volume II: 978-1-1385-05513 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. 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CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com ...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