Computing Algorithms for Solutions of Problems in Applied Mathematics and their Standard Program Realization 2 Stochastic Mathematics

366 Pages·2015·1.814 MB·english

by  Karlos J. Kachiashvili, D. Yu. Melikdzhanian, A. I. Prangishvili

MATHEMATICS RESEARCH DEVELOPMENTS

C A
OMPUTING LGORITHMS
S P
FOR OLUTIONS OF ROBLEMS
A M T
IN PPLIED ATHEMATICS AND HEIR
S P R
TANDARD ROGRAM EALIZATION

P 2
ART

S M
TOCHASTIC ATHEMATICS

K. J. KACHIASHVILI,
D. YU. MELIKDZHANIAN
AND
A. I. PRANGISHVILI

New York
Copyright © 2015 by Nova Science Publishers, Inc.
ISBN: (cid:28)(cid:26)(cid:27)(cid:16)(cid:20)(cid:16)(cid:25)(cid:22)(cid:23)(cid:25)(cid:22)(cid:16)(cid:26)(cid:20)(cid:24)(cid:16)(cid:21)(cid:3)(cid:11)(cid:72)(cid:16)(cid:37)(cid:82)(cid:82)(cid:78)(cid:12)
Contents
ListofFigures xiii
ListofTables xv
1 NumericalMethodsofProbabilityTheoryandMathematicalStatistics 1
1.1 MethodsofCombinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 MainConceptsandTheoremsofCombinatorics. . . . . . . . . . . 2
1.1.2 AlgorithmsfortheGenerationofNumericalSequences . . . . . . . 4
1.2 DiscreteProbabilityDistributions . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 SimplestDiscreteDistributions . . . . . . . . . . . . . . . . . . . 8
1.2.2 BinomialDistribution . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.3 GeometricandPascalDistributions . . . . . . . . . . . . . . . . . 12
1.2.4 HypergeometricDistribution . . . . . . . . . . . . . . . . . . . . . 13
1.2.5 PoissonDistribution . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.6 SeriesDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.7 ConnectionbetweenDifferentDiscreteDistributions . . . . . . . . 16
1.3 MajorContinuousProbabilityDistributions . . . . . . . . . . . . . . . . . 18
1.3.1 UniformDistribution . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.2 ExponentialDistribution . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.3 NormalDistribution . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.4 PropertiesoftheNormalDistributionFunction . . . . . . . . . . . 22
1.4 m-DimensionalNormalDistribution . . . . . . . . . . . . . . . . . . . . . 26
1.5 IrregularDistributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.1 TriangularDistribution . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.2 TrapezoidalDistribution . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.3 GeneralizedTrapezoidalDistribution . . . . . . . . . . . . . . . . 29
1.5.4 Antimodal-IDistribution . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.5 Antimodal-IIDistribution . . . . . . . . . . . . . . . . . . . . . . 30
1.6 BasicProbabilityDistributionsUsedinMathematicalStatistics . . . . . . . 31
1.6.1 Chi-SquareDistribution . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.2 PropertiesoftheChi-SquareDistributionFunction . . . . . . . . . 32
1.6.3 Student’sDistribution . . . . . . . . . . . . . . . . . . . . . . . . 34
1.6.4 PropertiesofStudent’sDistributionFunction . . . . . . . . . . . . 35
1.6.5 Fisher’sDistribution . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.6.6 PropertiesofFisher’sDistributionFunction . . . . . . . . . . . . . 39
1.6.7 ConnectionbetweenDifferentDistributions . . . . . . . . . . . . . 46
1.7 AdditionalProbabilityDistributionsUsedinMathematicalStatistics . . . . 47
1.7.1 KolmogorovDistribution . . . . . . . . . . . . . . . . . . . . . . . 47
1.7.2 Omega-SquareDistribution . . . . . . . . . . . . . . . . . . . . . 49
1.7.3 D-Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.8 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
1.8.1 SamplesandStatistics . . . . . . . . . . . . . . . . . . . . . . . . 53
1.8.2 VariationalSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.9 StatisticalEstimatesofMainCharacteristicsofaRandomVariable . . . . . 55
1.9.1 SampleMomentsandEmpiricProbabilities . . . . . . . . . . . . . 55
1.9.2 HistogramandtheConceptsConnectedwithIt . . . . . . . . . . . 56
1.9.3 EmpiricDistributionFunction . . . . . . . . . . . . . . . . . . . . 58
1.10 StatisticalEstimatesofDistributionParameters . . . . . . . . . . . . . . . 59
1.10.1 MethodsofObtainingofEstimators . . . . . . . . . . . . . . . . . 59
1.10.2 EstimatorsoftheParametersofSomeSpecialProbabilityDistribu-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.11 ConfidenceandToleranceIntervals . . . . . . . . . . . . . . . . . . . . . . 62
1.11.1 DeterminationofConfidenceIntervals . . . . . . . . . . . . . . . . 62
1.11.2 ToleranceInterval . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.11.3 Non-classical Method for Constructionof the Confidence Interval
forExpectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.11.4 MainPropertiesoftheFunctionsψ (h),Ψ (H),h (α),H (α) . 65
N N N N
1.12 TestingofStatisticalHypotheses . . . . . . . . . . . . . . . . . . . . . . . 70
1.12.1 SchemeofHypothesesTesting . . . . . . . . . . . . . . . . . . . . 70
1.12.2 TestofIdentityoftheDistributionsofTwoRandomVariables . . . 72
1.12.3 StatisticsofTwoSamplesfromNormalSets. . . . . . . . . . . . . 74
1.12.4 StatisticsoftheUnitedSamplesfromNormalSets . . . . . . . . . 75
1.12.5 TestingtheNormalityofProbabilityDistribution . . . . . . . . . . 77
1.12.6 IdentificationoftheDensityofProbabilityDistribution . . . . . . . 79
1.12.7 TestingofSomeOtherHypotheses. . . . . . . . . . . . . . . . . . 82
1.13 NonparametricMethodsofStatistics . . . . . . . . . . . . . . . . . . . . . 84
1.13.1 TestofIdentityoftheDistributionsofTwoRandomVariables . . . 84
1.13.2 CriteriaofSignsandSignRanks . . . . . . . . . . . . . . . . . . . 88
1.13.3 One-FactorAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 92
1.13.4 TwoFactorAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 95
1.13.5 TwoFactorAnalysisofaVariance . . . . . . . . . . . . . . . . . . 98
1.13.6 CorrelationAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 100
1.14 IdentificationofRegressionDependencies . . . . . . . . . . . . . . . . . . 104
1.15 RestorationofSomeSpecialTypesofNonlinearFunctionalDependencies . 107
1.15.1 GeometricRegression . . . . . . . . . . . . . . . . . . . . . . . . 108
1.15.2 ExponentialRegression . . . . . . . . . . . . . . . . . . . . . . . 109
1.15.3 LogarithmicRegression . . . . . . . . . . . . . . . . . . . . . . . 109
1.15.4 Geometric-ExponentialRegression . . . . . . . . . . . . . . . . . 109
1.15.5 ExponentialRegressionwithaFreeTerm . . . . . . . . . . . . . . 110
1.15.6 GeometricRegressionwithaFreeTerm . . . . . . . . . . . . . . . 110
1.15.7 InverseExponentialRegression . . . . . . . . . . . . . . . . . . . 111
1.15.8 Linear-ExponentialRegression . . . . . . . . . . . . . . . . . . . . 111
1.15.9 Linear-ExponentialDependencewithaFreeTerm . . . . . . . . . 112
1.15.10ProductofGeometricDependencies . . . . . . . . . . . . . . . . . 112
1.15.11SumofExponentialDependencies . . . . . . . . . . . . . . . . . . 113
1.15.12SumofGeometricDependencies. . . . . . . . . . . . . . . . . . . 113
1.15.13SumofExponentialDependencieswithaFreeTerm . . . . . . . . 114
1.15.14SumofGeometricDependencieswithaFree Term . . . . . . . . . 114
1.15.15Exponential-SinusoidalRegression . . . . . . . . . . . . . . . . . 114
1.15.16Exponential-SinusoidalRegressionwithaFreeTerm . . . . . . . . 115
1.15.17PolynomialRegression . . . . . . . . . . . . . . . . . . . . . . . . 116
1.15.18Geometric-PolynomialRegression . . . . . . . . . . . . . . . . . . 116
1.15.19Exponential-PolynomialRegression . . . . . . . . . . . . . . . . . 117
1.15.20Logarithmic-PolynomialRegression . . . . . . . . . . . . . . . . . 118
1.15.21PeriodicRegression . . . . . . . . . . . . . . . . . . . . . . . . . 118
1.15.22LinearMultipleRegression. . . . . . . . . . . . . . . . . . . . . . 120
1.16 MainPropertiesofRestoredDependencies. . . . . . . . . . . . . . . . . . 121
2 SomeAdditionalProblems 135
2.1 AlgebraofIntegers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.2 PerformanceofArithmeticOperationswithRationalandComplexNumbers 138
2.3 SearchofWordsandPhrasesinaDictionary . . . . . . . . . . . . . . . . . 139
A DerivationofFormulaeandProofsofTheorems 143
A.1 NumericalMethodsofLinearAlgebra . . . . . . . . . . . . . . . . . . . . 143
A.1.1 SweepMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.1.2 SeidelMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.1.3 IterativeMethodsfortheDeterminationofEigenvaluesandEigen-
vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A.1.4 RotationMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
A.1.5 BasicPropertiesoftheEigenvaluesandEigenvectorsoftheOpera-
torsSˆ2andSˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
z
A.1.6 EigenvaluesoftheOperatorSˆ2. . . . . . . . . . . . . . . . . . . . 150
A.1.7 GeneralizedRecurrence Relationsfor theClebsch–GordanCoeffi-
cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
A.1.8 SpecialValuesoftheClebsch–GordanCoefficients . . . . . . . . . 152
A.2 NumericalAnalysisofaPowerSeriesanditsPolynomials . . . . . . . . . 153
A.2.1 CalculationofanExponent. . . . . . . . . . . . . . . . . . . . . . 153
A.2.2 CalculationofaPowerFunction . . . . . . . . . . . . . . . . . . . 154
A.2.3 FundamentalTheoremofAlgebra . . . . . . . . . . . . . . . . . . 154
A.2.4 EuclideanAlgorithmfortheFindingofGCD α(z),β(z) ) . . . . 155
{ }
A.2.5 LagrangeTheoremDeterminingtheBoundariesofRealZeros . . . 155
A.2.6 SturmTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A.2.7 InterpolationalFormulasforPolynomials . . . . . . . . . . . . . . 157
A.2.8 Gra¨ffe–LobatchevskyMethod . . . . . . . . . . . . . . . . . . . . 159
A.2.9 PropertiesofOrthogonalPolynomials . . . . . . . . . . . . . . . . 162
A.2.10 SumsofPartialFractions . . . . . . . . . . . . . . . . . . . . . . . 163
A.3 SolutionofNonlinearEquationsandtheDeterminationofExtremums . . . 164
A.3.1 PrincipleofContractionMappings . . . . . . . . . . . . . . . . . . 164
A.3.2 AuxiliaryTheoremsforthePrincipleofContractionMappings . . . 164
A.3.3 IterativeSequenceHavingaSquare-LawConvergence . . . . . . . 165
A.3.4 NewtonMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.3.5 NewtonMethodforAnalyticalFunctions . . . . . . . . . . . . . . 166
A.3.6 NewtonMethodforaSystemofEquations. . . . . . . . . . . . . . 166
A.4 InterpolationandApproximationofFunctions . . . . . . . . . . . . . . . . 167
A.4.1 InterpolationError . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A.4.2 OptimalChoiceofInterpolationNodes . . . . . . . . . . . . . . . 168
A.4.3 InterpolationErrorforPolynomialsExpressedinTermsofDivided
Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.4.4 Explicit Expression for Interpolational Polynomial at Optimal
ChoiceofNodes . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
A.4.5 Interpolation Formulas for One-Parametric Families of Functions
ofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.4.6 InterpolationoftheFunctionaxc (1 bx)d . . . . . . . . . . . . 171
· −
A.4.7 CubicSplineInterpolation . . . . . . . . . . . . . . . . . . . . . . 172
A.5 NumericalDifferentiationandIntegration . . . . . . . . . . . . . . . . . . 173
A.5.1 ConnectionofDividedDifferenceswithDerivatives . . . . . . . . 173
A.5.2 QuadratureFormulasofanInterpolationalType . . . . . . . . . . . 173
A.5.3 Newton–CotesQuadratureFormulas . . . . . . . . . . . . . . . . . 174
A.5.4 QuadratureFormulasofaGaussianType . . . . . . . . . . . . . . 178
A.5.5 ModificationoftheGaussQuadratureFormula . . . . . . . . . . . 179
A.6 CalculationoftheValuesofSomeFunctions . . . . . . . . . . . . . . . . . 180
A.6.1 AsymptoticBehavioroftheInverseFunctionofxex . . . . . . . . 180
A.6.2 Determinationof the Radiusof Convergence of a Taylor Series of
theFunction (s,z) . . . . . . . . . . . . . . . . . . . . . . . . . 181
P
A.6.3 MainInequalitiesfortheFunction (s,z). . . . . . . . . . . . . . 182
P
A.6.4 ValuesoftheFunctionΨ(x)fortheFractionalValuesofArgument 184
A.6.5 DerivationoftheBasicFunctionalEquationsforΠ(z) . . . . . . . 186
A.6.6 GaussMultiplicationFormulaforΠ(z) . . . . . . . . . . . . . . . 186
A.6.7 IntegralRepresentationforΠ(z) . . . . . . . . . . . . . . . . . . . 187
A.6.8 IntegralRepresentationforΨ(z) . . . . . . . . . . . . . . . . . . . 187
A.6.9 IntegralRepresentationforLnΠ(z) . . . . . . . . . . . . . . . . . 189
A.6.10 IntegralRepresentationsofaBeta-Function . . . . . . . . . . . . . 190
A.6.11 AsymptoticExpansionoftheFunctionΨ(z) . . . . . . . . . . . . 191
A.6.12 PropertiesoftheOperatorzd . . . . . . . . . . . . . . . . . . . . 192
z
A.6.13 Coefficients of the Expansion of a Hypergeometric Function in a
PowerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
A.6.14 ConvergenceofaHypergeometricSeries . . . . . . . . . . . . . . 193
A.6.15 Transformation of Argument for the Hypergeometric Function:
z 1/z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
→
A.6.16 TransformationofArgumentfortheKummerFunction . . . . . . . 196
A.6.17 Recurrence FormulasforHypergeometricFunctionsoneofthePa-
rametersofWhichVariesbyUnit . . . . . . . . . . . . . . . . . . 196
A.6.18 MainInequalitiesforHypergeometricFunctions . . . . . . . . . . 197
A.6.19 AsymptoticExpansionofHypergeometricFunction(1) . . . . . . . 198
A.6.20 AsymptoticExpansionofHypergeometricFunction(2) . . . . . . . 199
A.6.21 AsymptoticExpansionofHypergeometricFunction(3) . . . . . . . 200
A.6.22 ApproximationoftheFunctionsJ (x)byTrigonometricalSums . 202
m
A.7 NumericalMethodsforSolvingDifferentialEquations . . . . . . . . . . . 203
A.7.1 ErrorofApproximationoftheRunge–KuttaMethods . . . . . . . . 203
A.7.2 ErrorofApproximationofm-StepDifferenceMethods . . . . . . . 208
A.7.3 DerivationofFormulasfortheResidualsofaDiffusionEquation . 210
A.8 NumericalMethodsUsedinGeometry . . . . . . . . . . . . . . . . . . . . 211
A.8.1 ExpressionoftheMatrixAinTermsoftheRotationAngleandthe
AxisofRotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
A.9 NumericalMethodsofProbabilityTheoryandMathematicalStatistics . . . 212
A.9.1 Asymptotic Expansionsfor the Function of Probabilitiesof Bino-
mialDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.9.2 AsymptoticExpansionofFisher’sDistributionFunction . . . . . . 213
A.9.3 RemaindersoftheExpansionsofaKolmogorovDistributionFunc-
tionUsedforSmallValuesoftheArgument . . . . . . . . . . . . . 215
A.9.4 NonclassicalMethodfor theConstructionoftheConfidenceInter-
valfortheExpectationofaRandomVariable . . . . . . . . . . . . 216
A.9.5 SpecialValuesoftheFunctionψ (h) . . . . . . . . . . . . . . . . 217
N
A.9.6 Limitofh (α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
N
A.10 SomeAdditionalProblems . . . . . . . . . . . . . . . . . . . . . . . . . . 219
A.10.1 DecompositionofNaturalNumbersonPrimeFactors . . . . . . . . 219
A.10.2 EuclideanAlgorithmforIntegers . . . . . . . . . . . . . . . . . . 220
A.10.3 SolvingofLinearEquationsinIntegers . . . . . . . . . . . . . . . 220
B ProgramRealizationofFormulasandAlgorithms 223
B.1 NumericalMethodsofLinearAlgebra . . . . . . . . . . . . . . . . . . . . 223
B.1.1 ProblemswhichdonotUseIterativeAlgorithms . . . . . . . . . . 223
B.1.2 IterativeMethodsforSolutionofLinearEquations . . . . . . . . . 225
B.1.3 CalculationoftheEigenvaluesandEigenvectorsofMatrices . . . . 227
B.1.4 AdditionalProblemsofLinearAlgebra . . . . . . . . . . . . . . . 228
B.2 NumericalAnalysisofaPowerSeriesanditsPolynomials . . . . . . . . . 228
B.2.1 ActionsOveraPowerSeries . . . . . . . . . . . . . . . . . . . . . 228
B.2.2 ActionswithPolynomials . . . . . . . . . . . . . . . . . . . . . . 231
B.2.3 CalculationofSomeSpecialPolynomialsandTheirCoefficients . . 235
B.3 SolutionofNonlinearEquationsandDeterminationofExtremums . . . . . 237
B.4 InterpolationandApproximationofFunctions . . . . . . . . . . . . . . . . 239
B.4.1 Spline-Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 239
B.4.2 BaseClassesfortheApproximationofFunctions . . . . . . . . . . 241
B.4.3 InterpolationoftheNonlinearFunctionsofaCertainClass . . . . . 246
B.4.4 DeterminationoftheBoundariesfortheSearchoftheApproxima-
tionParameters fortheNonlinearFunctionsofaCertainClass . . . 249
B.4.5 RestorationofSomeSpecialTypesofFunctionalDependencies . . 250
B.4.6 InterpolationandApproximationofFunctionsbyMeans
ofPolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
B.4.7 RestorationofPeriodicDependence . . . . . . . . . . . . . . . . . 259
B.4.8 RestorationofDependencethatisLinearwithRespecttoParameters261
B.5 NumericalIntegrationofFunctions . . . . . . . . . . . . . . . . . . . . . . 262
B.6 CalculationoftheValuesofSomeFunctions . . . . . . . . . . . . . . . . . 265
B.6.1 Calculationof theValuesof ElementaryTranscendentalFunctions
andGamma-Functions . . . . . . . . . . . . . . . . . . . . . . . . 265
B.6.2 Calculation of the Values of the Hypergeometric and Cylindrical
FunctionsofRealVariablesbyElementaryMethods . . . . . . . . 267
B.7 NumericalSolutionofDifferentialEquations . . . . . . . . . . . . . . . . 270
B.7.1 Numerical Solution of Ordinary Differential Equations by the
Runge–KuttaMethod . . . . . . . . . . . . . . . . . . . . . . . . . 270
B.7.2 Numerical Solving of One-Dimensional, Two-Dimensional, and
Three-DimensionalBoundaryProblems . . . . . . . . . . . . . . . 275
B.7.3 Numerical Solving of One-Dimensional, Two-Dimensional, and
Three-Dimensional Diffusion Equations and Wave Equations by
MeansofExplicitSchemes . . . . . . . . . . . . . . . . . . . . . . 283
B.7.4 Numerical Solving of One-Dimensional, Two-Dimensional and
Three-Dimensional Diffusion Equations and Wave Equations by
MeansofImplicitSchemes . . . . . . . . . . . . . . . . . . . . . . 289
B.8 NumericalMethodsUsedinGeometry . . . . . . . . . . . . . . . . . . . . 291
B.8.1 Three-DimensionalRotationMatrices . . . . . . . . . . . . . . . . 291
B.8.2 DescriptionofPlaneCurvesbySplines . . . . . . . . . . . . . . . 293
B.9 NumericalMethodsofProbabilityTheoryandMathematicalStatistics . . . 295
B.9.1 MethodsofCombinatorics . . . . . . . . . . . . . . . . . . . . . . 295
B.9.2 Functions of Statistical Distributions, Density Functions, and
Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
B.9.3 StatisticalEstimatesofParameters . . . . . . . . . . . . . . . . . . 301
B.9.4 TestingofStatisticalHypotheses . . . . . . . . . . . . . . . . . . . 303
B.9.5 NonparametricMethodsofStatistics. . . . . . . . . . . . . . . . . 306
B.10 SomeAdditionalProblems . . . . . . . . . . . . . . . . . . . . . . . . . . 316
B.10.1 Performance of Arithmeticaland Algebraic OperationsOver Inte-
gers,RationalandComplexNumbers . . . . . . . . . . . . . . . . 316
B.10.2 SearchofWordsandPhrasesinaDictionary . . . . . . . . . . . . 317
C UsedDesignations 319
C.1 ObjectsofMathematicalLogicandGeneralAlgebra . . . . . . . . . . . . 319
C.2 ObjectsofLinearAlgebraandFunctionalAnalysis . . . . . . . . . . . . . 323
C.3 OperationsofMathematicalAnalysis . . . . . . . . . . . . . . . . . . . . 325
C.4 DesignationsofMathematicalFunctionsandConstants . . . . . . . . . . . 327
C.5 GeometricalObjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
C.6 DesignationsofProbabilityTheoryandMathematicalStatistics. . . . . . . 335
References 337
About the Authors 355
Index 357
Figures
1.1 Plotsofthegeometricdependence . . . . . . . . . . . . . . . . . . . . . . 121
1.2 Plotsofexponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
1.3 Plotsoflogarithmicfunctions. . . . . . . . . . . . . . . . . . . . . . . . . 122
1.4 Plotsoftheproductofgeometricandexponentialdependencies. . . . . . . 124
1.5 Plotsofinverse-exponentialdependence . . . . . . . . . . . . . . . . . . . 125
1.6 Plotsoftheproductoflinearandexponentialdependencies . . . . . . . . . 126
1.7 Plotsofproductofgeometricdependencies . . . . . . . . . . . . . . . . . 127
1.8 Plotsofsumofexponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
1.9 Plotsofsumofgeometricdependencies . . . . . . . . . . . . . . . . . . . 131
1.10 Plotsofexponential-sinusoidaldependence . . . . . . . . . . . . . . . . . 132
A.1 CriticalareasD ,D , andthehypothesis-acceptanceregionD ;tanϕ =
1 2 0 1
h/(1+h);tanϕ = H/(1+H). . . . . . . . . . . . . . . . . . . . . . . 217
2