Computational mathematics : theory, methods and applications / Peter G. Chareton, editor.

Format:

Book

Contributor:

Chareton, Peter G.

Series:

Computational mathematics and analysis series.

Computational mathematics and analysis series

Language:

English

Subjects (All):

Numerical analysis--Data processing.

Numerical analysis.

Physical Description:

1 online resource (459 p.)

Edition:

1st ed.

Place of Publication:

New York : Nova Science Publishers, c2010.

Language Note:

English

Summary:

Chareton gathers the latest research in this study of computational mathematics. He highlights such topics as coherence-homotopies of higher order, Vandermonde systems, numerical conformal mappings for waveguides, commutativity formulas for fundamental group entropy, the completion of fuzzy metric spaces and more.

Contents:

Intro

COMPUTATIONAL MATHEMATICS: THEORY, METHODS AND APPLICATIONS

CONTENTS

PREFACE

ANALYTICAL AND NUMERICAL METHODS IN THE LINEAR STABILITY STUDY OF IDEAL FLOWS ON A SPHERE

ABSTRACT

1. INTRODUCTION

2. HILBERT SPACES AND GEOMETRIC STRUCTURE OF SMOOTH FUNCTIONS ON A SPHERE

3. INTEGRAL FORMULAS RELATED TO THE JACOBIAN

Lemma 1

4. STEADY BVE SOLUTIONS ON A ROTATING SPHERE

5. CONSERVATION LAW FOR PERTURBATIONS TO LP FLOWS AND RH WAVES

Theorem 1

6. CONSERVATION LAW FOR INFINITESIMAL PERTURBATIONS TO WV WAVES AND MODONS

Theorem 2

7. UNIFIED CONSERVATION LAW FOR DISTURBANCES OF BE SOLUTIONS

8. INSTABILITY CONDITIONS FOR LP FLOWS, RH WAVES, WV WAVES AND MODONS

Theorem 3

Example 1

Example 2

Example 3

Theorem 4

9. PECULIARITIES OF INSTABILITY CONDITIONS FOR WV WAVES AND MODONS

10. ESTIMATES OF THE MAXIMUM GROWTH RATE OF UNSTABLE MODES

Theorem 5

Theorem 6

11. ORTHOGONALITY OF UNSTABLE MODES TO THE BASIC FLOW (BVE SOLUTION)

Theorem 7

Corollary 1

Corollary 2

12. NUMERICAL EXPERIMENTS

Experiment 1

Experiment 2

Experiment 3

Experiment 4

13. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

REVIEWED BY

PURE AND MIXED MATHEMATICS IN THE WORK OF LEONHARD EULER

2. THE RISE OF THE CONCEPT OF FUNCTIONS

3. ORDINARY DIFFERENTIAL EQUATIONS

4. DIFFERENTIATION AND INTEGRATION OF FUNCTIONS OF TWO VARIABLES

5. PARTIAL DIFFERENTIAL EQUATION

6. INFINITE POLYNOMIALS AND SERIES

7. MECHANICS

8. THE CALCULUS OF VARIATIONS AND THE PRINCIPLE OF THE LEAST ACTION

9. OTHER RESULTS IN MIXED MATHEMATICS

APPLICATIONS OF COMPUTATIONAL GEOMETRY TO PROBLEMS OF POLITICAL COMPETITION

1. INTRODUCTION.

2. GEOMETRICAL SEARCH FOR OPTIMUM POSITIONS IN THE GAME WITH RESTRICTIONS: USE OF THE OPINION SURVEYS

2.1. The Opinion Surveys

2.1.1. Public Opinion and Politics Fiscal Survey Nº 2615 of the CIS

2.2. The Algorithm and the Simulation

2.2.1. A Graphic Approximation of the Algorithm

2.3. Simulation with an Example of the National Politics (Spain)

2.3.1. Algorithm Implementation

2.3.2. Results

2.4. Conclusions

3. GEOMETRICAL STUDY OF EQUILIBRIUM POSITIONS IN THE GAME WITH RESTRICTIONS

3.1. The Model

3.2. Equilibrium with Restrictions

3.2.1. Existence Conditions

3.2.2. Examples

3.3. Conclusions

APPENDIX A: DEVELOPMENT OF THE ALGORITHM

COHERENCE - HOMOTOPIES OF HIGHER ORDER

INTRODUCTION

1. COHERENT SYSTEMS AND COHERENT MAPS

Theorem 1.1

Theorem 1.2

Theorem 1.3

Theorem 1.4

2. LEVEL COHERENT CATEGORY

Theorem 2.1

Theorem 2.2

Theorem 2.3

Theorem 2.4

Theorem 2.5

Theorem 2.6

Theorem 2.7

Theorem 2.8

3. COHERENT SHIFT AND COHERENT CATEGORY

Proposition 3.1

Proposition 3.2.

Theorem 3.1

Theorem 3.2

Theorem 3.3

Theorem 3.4

4. RELATIONS OF COHERENT CATEGORIES

Theorem 4.1

Theorem 4.2

APPENDIX: STRICT ORDERING VS ORDERING FOR DIRECTED SETS

STABLE MFS-BASED SOLUTION TO SINGULAR AND NON-SINGULAR INVERSE PROBLEMS FOR TWO-DIMENSIONAL HELMHOLTZ-TYPE EQUATIONS

Abstract

1Introduction

2MathematicalFormulationoftheInverseProblems

3SingularSolutionsfortheTwo-DimensionalHelmholtz-TypeOperator

4SingularInverseProblem:SingularitySubtractionTechnique

5StandardandModifiedMethodsofFundamentalSolutions

6Regularization

6.1TheTikhonovRegularizationMethod

6.2TheL-CurveMethod

7NumericalResults

7.1AccuracyErrors

7.2Non-SingularInverseProblems

7.2.1Examples.

7.2.2EffectoftheTRM

7.2.3ChoiceoftheOptimalRegularizationParameter

7.2.4NumericalStabilityoftheMethod

7.2.5NumericalConvergenceoftheMethod

7.3SingularInverseProblems

7.3.1Examples

7.3.2EffectoftheSST

7.3.3EffectoftheTRM

7.3.4ChoiceoftheOptimalRegularizationParameter

7.3.5NumericalStabilityoftheMethod

8Conclusion

References

VANDERMONDE SYSTEMS: THEORY AND APPLICATIONS

2PolynomialInterpolation

2.1LagrangeandNewtonform

2.2Lebesgueconstant

2.2.1Lebesgueconstantforequidistantnodes

2.3TheBj¨orckandPereyraalgorithm

3APropertyoftheElementarySymmetricFunctions

4PolynomialApproximationwithGauss-LobattoPoints

4.0.1Theinterpolationproblem

4.0.2Theleast-squaresproblem:explicitMoore-Penrosepseudo-inverseformula

4.0.3Theleast-squaresproblem:discreteorthogonalpolynomials

4.0.4Numericalproperties

A COMPARATIVE STUDY OF DIFFERENT SEMILOCAL CONVERGENCE RESULTS APPLIED TO KEPLER'S EQUATION

2Kantorovich'sTheoryAppliedtoKepler'sEquation

3Smale'sa-TheoryAppliedtoKepler'sEquation

4Conclusion

DISCRETE MAXIMUM PRINCIPLES FOR FEM SOLUTIONS OF NONLINEAR ELLIPTIC SYSTEMS

2DiscreteMaximumPrinciplesinDifferentSettings

2.1Algebraicbackgroundandthe'matrixmaximumprinciple'

2.2SomemotivationfortheDMP

2.2.1Linearequationsandcontinuousmaximumprinciples

2.2.2TheDMPforasinglenonlinearellipticequation

2.3GeometricpropertiestoensuretheDMP

2.4AnalgebraicDMPinHilbertspace

2.4.1Formulationoftheoperatorequation

2.4.2Galerkintypediscretization

2.4.3Maximumprinciplefortheabstractdiscretizedproblem

3DiscreteMaximumPrinciplesforEllipticReaction-DiffusionTypeSystems

3.1Systemswithnonlinearcoefficients

3.1.1Formulationoftheproblem

3.1.2Finiteelementdiscretization.

3.1.3Discretemaximumprincipleforsystemswithnonlinearcoefficients

3.2Systemswithgeneralreactiontermsofsublineargrowth

3.3Systemswithgeneralreactiontermsofsuperlineargrowth

3.4Sufficientconditionsandtheirgeometricmeaning

4DiscreteMaximumPrinciplesforEllipticSystemsIncludingFirstOrderTerms

4.1Nonsymmetricsystemswithnonlinearreactioncoefficients

4.2Nonsymmetricsystemswithsublinearreactionterms

4.3Nonsymmetricsystemswithsuperlinearreactionterms

4.4Nonsymmetricsystemswithnonlinearconvectioncoefficients

5Somereal-lifeexamples

5.1Reaction-diffusionsystemsinchemistry

5.2Linearellipticsystems

5.3Nonsymmetrictransportsystems

Acknowledgments

NUMERICAL CONFORMAL MAPPINGS FOR WAVEGUIDES

2WaveScatteringinTwo-DimensionalWaveguides

2.1Theoriginalproblem

2.2TheBuildingBlockMethod

2.4Solvingtheresultingproblem

3ConformalMappingMethods

3.1ModifiedSchwarz-Christoffelmappingsforpolygonswithroundedcor-ners

3.2Approximatecurvefactors

3.3TheOuterPolygonMethod

3.4Usingthegeodesicalgorithmforchannels

COMPUTATIONAL STUDY OF THE 3D AFFINE TRANSFORMATION

2Definitionofthe3-PointProblem

3NumericalSolutions

3.1GeneralpolynomialsolverbasedonnumericalGroebnerbasisandeigen-systemmethod

3.2Globalminimization

3.3HomotopySolution

4SymbolicSolutions

4.1Dixon'sResultant:BasicConcepts

4.2ConstructionofDixonResultant

Cayley'sformulationofB´ezout'smethod

Example

Dixon'sgeneralizationoftheCayley-B´ezout'smethod

4.3ImprovedDixonresultant-Kapur,SaxenaandYangmethod

4.4HeuristicmethodstoacceleratetheDixonresultant

4.5Earlydiscoveryoffactors:theEDFmethod

4.6ApplicationoftheEDFmethod

4.7ApplicationofReducedGroebnerbasis

4.8Computationofotherparameters.

5DefinitionoftheN-PointProblem

6SolutionoftheOverdeterminedModel

6.1DirectNumericalSolutionviaGlobalMinimization

6.2Newton-RaphsonwithDeflation

6.3ExtendedgeneralProcrustesalgorithm

7TheDeterminedModel

8NumericalSolutionoftheDeterminedModel

9ComplexityStudyoftheAlgorithms

10TheProperSelectionofthe3PointsforInitialGuessValues

11Conclusions

DISTANCES BASED ON NEIGHBORHOOD SEQUENCES IN THE TRIANGULAR GRID

1.1Abriefhistoryofdigitalgeometry

2BasicDefinitionsandNotations

3TheShortestPaths

4ConditionforMetricDistances

5ComputingtheDistance

6DigitalCircles

7Conclusion

A STREAM IN THE STUDY ON NORMALITY OF S-PRODUCTS

I.S-ProductsandInfiniteProducts

2ProductsofCompactFactors

3TheDefinitionofS-Products

4S-ProductsofMetricSpaces

II.S-ProductswithCountableTightness

5TightnessandProducts

6S-ProductsofParacompactp-Spaces

7GeneralizedMetricSpaces

8S-ProductsofGeneralizedMetricSpaces

9S-ProductsofParacompactC-ScatteredSpaces

10Non-NormalityofS-Products

III.S-ProductswithoutCountableTightness

11CollectionwiseNormalityofS-Products

12CountableParacompactnessofS-Products

13TheShrinkingPropertyofS-Products

14Non-NormalityofS-Products,Revisited

IV.RectangularProducts

15RectangularProductsandCoveringDimension

16ProductsofMetricSpaces

17ProductsofGeneralizedMetricSpaces

18NormalCoversofProducts

THE COMPLETION OF FUZZY METRIC SPACES AND OF OTHER RELATED STRUCTURES

1IntroductionandPreliminaries

2TheCompletionofFuzzyMetricSpaces

3TheCompletionofStrongFuzzyMetricSpacesandofNon-ArchimedeanFuzzyMetricSpaces

4TheCompletionofFuzzyMetricGroups

5TheCompletionofIntuitionisticFuzzyMetricSpaces

Acknowledgments.

References.

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-62417-078-1

OCLC:

923666746