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BL
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| Record Number
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889827
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| Main Entry
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Mathai, A. M.
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| Title & Author
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Erdélyi-Kober fractional calculus : : from a statistical perspective, inspired by solar neutrino physics /\ A.M. Mathai, H.J. Haubold.
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| Publication Statement
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Singapore :: Springer,, 2018.
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| Series Statement
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SpringerBriefs in mathematical physics,; volume 31
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| Page. NO
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1 online resource (xii, 122 pages) :: illustrations (some color)
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| ISBN
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9789811311598
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: 9789811311604
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: 9811311595
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: 9811311609
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9789811311581
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9811311587
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Contents; Acronyms; 1 Solar Neutrinos, Diffusion, Entropy, Fractional Calculus; References; 2 Erdélyi-Kober Fractional Integrals in the Real Scalar Variable Case; 2.1 Introduction; 2.2 Some Notations; 2.2.1 Some of the Fractional Integrals and the Notations in the Literature; 2.3 Fractional Integrals of the First Kind in the Real Scalar Variable Case; 2.4 A Pathway Generalization of Erdélyi-Kober Fractional Integral Operator of the First Kind; 2.5 Some Special Cases; 2.6 Erdélyi-Kober Fractional Integrals of the First Kind and Hypergeometric Series.
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2.14.1 An Interpretation in Terms of Densities of Sum and Difference2.14.2 Fractional Integrals as Fractions of Total Probabilities; 2.14.3 A Geometrical Interpretation; 2.15 A General Definition of Fractional Integrals; 2.15.1 Mellin Convolution of Product and Second Kind Integrals; 2.15.2 Mellin Convolution of a Ratio and First Kind Fractional Integrals; References; 3 Erdélyi-Kober Fractional Integrals in the Real Matrix-Variante Case; 3.1 Explicit Evaluations of Matrix-Variate Gammaand Beta Integrals; 3.1.1 Explicit Evaluation of Real Matrix-Variate Gamma Integral.
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2.7 Mellin Transform of the Generalized Erdélyi-Kober Fractional Integral of the First Kind2.8 Riemann-Liouville Operators as Mellin Convolution; 2.9 Distribution of a Product and Erdélyi-Kober Operators of the Second Kind; 2.10 A Pathway Extension of Erdélyi-Kober Operator of the Second Kind; 2.11 Special Cases; 2.12 Another Form of Generalization of Erdélyi-Kober Operators of the Second Kind; 2.13 Mellin Transform of the Generalized Erdélyi-Kober Operator of the Second Kind; 2.14 A Geometrical and Some Physical Interpretations of Fractional Integrals.
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3.1.2 Evaluation of Matrix-Variate Type-1 Beta Integral in the Real Case3.1.3 General Partitions; 3.1.4 A Method of Avoiding Integration Over the Stiefel Manifold; 3.2 Erdélyi-Kober Fractional Integral Operator of the Second Kind for the Real Matrix-Variate Case; 3.3 A Pathway Generalization of Erdélyi-Kober Fractional Integral Operator of the Second Kind in the Real Matrix-Variate Case; 3.4 M-Transforms of Erdélyi-Kober Fractional Integral of the Second Kind in the Real Matrix-Variate Case.
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3.5 Generalization in Terms of Hypergeometric Series for Erdélyi-Kober Fractional Integral of the Second Kind in the Real Matrix-Variate Case3.6 Erdélyi-Kober Fractional Integral of the First Kind in the Real Matrix-Variate Case; 3.7 Pathway Extension of Erdélyi-Kober Fractional Integral of the First Kind in the Real Matrix-Variate Case; 3.8 A General Definition; 3.8.1 Special Cases; 3.8.2 Special Cases of First Kind Fractional Integrals; References; 4 Erdélyi-Kober Fractional Integrals in the Many Real Scalar Variables Case.
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| Abstract
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This book focuses on Erdelyi-Kober fractional calculus from a statistical perspective inspired by solar neutrino physics. Results of diffusion entropy analysis and standard deviation analysis of data from the Super-Kamiokande solar neutrino experiment lead to the development of anomalous diffusion and reaction in terms of fractional calculus. The new statistical perspective of Erdelyi-Kober fractional operators outlined in this book will have fundamental applications in the theory of anomalous reaction and diffusion processes dealt with in physics. A major mathematical objective of this book is specifically to examine a new definition for fractional integrals in terms of the distributions of products and ratios of statistically independently distributed positive scalar random variables or in terms of Mellin convolutions of products and ratios in the case of real scalar variables. The idea will be generalized to cover multivariable cases as well as matrix variable cases. In the matrix variable case, M-convolutions of products and ratios will be used to extend the ideas. We then give a definition for the case of real-valued scalar functions of several matrices.
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| Subject
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Fractional calculus.
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Fractional calculus.
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Functional analysis transforms.
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| Subject
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Mathematical physics.
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Mathematics-- Functional Analysis.
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Mathematics-- Mathematical Analysis.
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| Subject
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Science-- Mathematical Physics.
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| Dewey Classification
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515/.83
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| LC Classification
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QA314
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| Added Entry
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Haubold, H. J.
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