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The goal of this monograph is to develop the theory of wavelet harmonic analysis on the sphere. By starting with orthogonal polynomials and functional Hilbert spaces on the sphere, the foundations are laid for the study of spherical harmonics such as zonal functions. The book also discusses the construction of wavelet bases using special functions, especially Bessel, Hermite, Tchebychev, and Gegenbauer polynomials.
Wavelets --- harmonic analysis --- spherical harmonics --- special functions --- zonal functions
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Historically, the notion of entropy emerged in conceptually very distinct contexts. This book deals with the connection between entropy, probability, and fractional dynamics as they appeared, for example, in solar neutrino astrophysics since the 1970's (Mathai and Rathie 1975, Mathai and Pederzoli 1977, Mathai and Saxena 1978, Mathai, Saxena, and Haubold 2010).The original solar neutrino problem, experimentally and theoretically, was resolved through the discovery of neutrino oscillations and was recently enriched by neutrino entanglement entropy. To reconsider possible new physics of solar neutrinos, diffusion entropy analysis, utilizing Boltzmann entropy, and standard deviation analysis was undertaken with Super-Kamiokande solar neutrino data. This analysis revealed a non-Gaussian signal with harmonic content. The Hurst exponent is different from the scaling exponent of the probability density function and both Hurst exponent and scaling exponent of the Super-Kamiokande data deviate considerably from the value of ½, which indicates that the statistics of the underlying phenomenon is anomalous. Here experiment may provide guidance about the generalization of theory of Boltzmann statistical mechanics. Arguments in the so-called Boltzmann-Planck-Einstein discussion related to Planck's discovery of the black-body radiation law are recapitulated mathematically and statistically and emphasize from this discussion is pursued that a meaningful implementation of the complex ‘entropy-probability-dynamics’ may offer two ways for explaining the results of diffusion entropy analysis and standard deviation analysis. One way is to consider an anomalous diffusion process that needs to use the fractional space-time diffusion equation (Gorenflo and Mainardi) and the other way is to consider a generalized Boltzmann entropy by assuming a power law probability density function. Here new mathematical framework, invented by sheer thought, may provide guidance for the generalization of Boltzmann statistical mechanics. In this book Boltzmann entropy, generalized by Tsallis and Mathai, is considered. The second one contains a varying parameter that is used to construct an entropic pathway covering generalized type-1 beta, type-2 beta, and gamma families of densities. Similarly, pathways for respective distributions and differential equations can be developed. Mathai's entropy is optimized under various conditions reproducing the well-known Boltzmann distribution, Raleigh distribution, and other distributions used in physics. Properties of the entropy measure for the generalized entropy are examined. In this process the role of special functions of mathematical physics, particularly the H-function, is highlighted.
special functions --- fractional calculus --- entropic functional --- mathematical physics --- applied analysis --- statistical distributions --- geometrical probabilities --- multivariate analysis
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Investigations involving the theory and applications of mathematical analytic tools and techniques are remarkably wide-spread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences. In this Special Issue, we invite and welcome review, expository and original research articles dealing with the recent advances in mathematical analysis and its multidisciplinary applications.
Mathematical (or Higher Transcendental) Functions and Their Applications --- Fractional Calculus and Its Applications --- q-Series and q-Polynomials --- Analytic Number Theory --- Special Functions of Mathematical Physics and Applied Mathematics --- Geometric Function Theory of Complex Analysis
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During the past four decades or so, various operators of fractional calculus, such as those named after Riemann–Liouville, Weyl, Hadamard, Grunwald–Letnikov, Riesz, Erdelyi–Kober, Liouville–Caputo, and so on, have been found to be remarkably popular and important due mainly to their demonstrated applications in numerous diverse and widespread fields of the mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional calculus operators provide several potentially useful tools for solving differential, integral, differintegral, and integro-differential equations, together with the fractional-calculus analogues and extensions of each of these equations, and various other problems involving special functions of mathematical physics, as well as their extensions and generalizations in one and more variables. In this Special Issue, we invite and welcome review, expository, and original research articles dealing with the recent advances in the theory of fractional calculus and its multidisciplinary applications.
operators of fractional calculus --- chaos and fractional dynamics --- fractional differential --- fractional differintegral equations --- fractional integro-differential equations --- fractional integrals --- fractional derivatives associated with special functions of mathematical physics --- applied mathematics --- identities and inequalities involving fractional integrals --- fractional derivatives
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For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation. Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier–Plancherel formula for non-commutative harmonic analysis, applied to locally-compact, non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences. The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties. One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory. The content of this Special Issue will highlight papers exploring non-commutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.
Weyl-Heisenberg group --- affine group --- Weyl quantization --- Wigner function --- covariant integral quantization --- Fourier analysis --- special functions --- rigged Hilbert spaces --- quantum mechanics --- signal processing --- non-Fourier heat conduction --- thermal expansion --- heat pulse experiments --- pseudo-temperature --- Guyer-Krumhansl equation --- higher order thermodynamics --- Lie groups thermodynamics --- homogeneous manifold --- poly-symplectic manifold --- dynamical systems --- non-equivariant cohomology --- Lie group machine learning --- Souriau-Fisher metric --- Born–Jordan quantization --- short-time propagators --- time-slicing --- Van Vleck determinant --- thermodynamics --- symplectization --- metrics --- non-equilibrium processes --- interconnection --- discrete multivariate sine transforms --- orthogonal polynomials --- cubature formulas --- nonequilibrium thermodynamics --- variational formulation --- nonholonomic constraints --- irreversible processes --- discrete thermodynamic systems --- continuum thermodynamic systems --- fourier transform --- rigid body motions --- partial differential equations --- Lévy processes --- Lie Groups --- homogeneous spaces --- stochastic differential equations --- harmonic analysis on abstract space --- heat equation on manifolds and Lie Groups
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This issue is a continuation of the previous successful Special Issue “Mathematical Analysis and Applications”
common fixed point --- metric-like space --- ?-Geraghty contraction --- triangular ?-admissible mapping --- fixed circle --- common fixed circle --- fixed-circle theorem --- extended partial Sb-metric spaces --- Sb-metric spaces --- fixed point --- generalized hypergeometric functions --- Gauss and confluent hypergeometric functions --- summation theorems of hypergeometric functions --- partial symmetric --- fixed point --- contraction and weak contraction --- Nadler’s theorem --- linear elastostatics --- simple layer potentials --- displacement problem --- existence and uniqueness theorems --- Fredholm alternative --- singular data --- differential equations --- Sheffer polynomial sets --- generating functions --- monomiality principle --- quasi metric space --- Suzuki contractions --- fixed point theorems --- modified ?-distance --- almost perfect functions --- generating function --- series transformation --- gamma function --- Hankel contour --- Fermi–Dirac function --- Bose–Einstein function --- Weyl transform --- series representation --- Hermite–Hadamard inequalities --- (p, q)-derivative --- (p, q)-integral --- convex functions --- fixed point --- Reich contraction --- Hardy–Rogers contraction --- almost b-metric space --- additive (Cauchy) equation --- additive mapping --- Hyers–Ulam stability --- generalized Hyers–Ulam stability --- hyperstability --- bounded index --- bounded L-index in direction --- slice function --- entire function --- bounded l-index --- generalized hypergeometric functions --- classical summation theorems --- generalization --- laplace transforms --- gamma and beta functions --- Szász-Mirakjan operators --- Szász-Mirakjan Beta type operators --- extended Gamma and Beta functions --- confluent hypergeometric function --- Modulus of smoothness --- modulus of continuity --- Lipschitz class --- local approximation --- Voronovskaja type approximation theorem --- operators theory 44A99, 47B99, 47A62 --- special functions 33C52, 33C65, 33C99, 33B10, 33B15 --- Stirling numbers and Touchard polynomials 11B73 --- n/a
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