Papers by Norbert Südland
Anomalous relaxation and diffusion processes in complex systemsa
Transport Theory and Statistical Physics, 2000
ABSTRACT
Physical review, Aug 1, 1999
The radial Fourier transform for the isotropic space with a fractal dimension is discussed. The m... more The radial Fourier transform for the isotropic space with a fractal dimension is discussed. The moments of diffusive displacements with non-Gaussian propagators arising as solutions of fractional diffusion equations are calculated. The Fourier propagator is applied to NMR correlation and spectral density functions in context with the orientational structure factor formalism. It is shown that the low-frequency molecular fluctuations of liquids in porous media with strong or forced adsorption at surfaces are due to reorientations mediated by translational displacements caused by surface diffusion of the adsorbate molecules. In terms of this formalism, field-cycling NMR experiments provide information on the static and dynamic fractal dimensions related to surface diffusion. The experimental results for liquids in porous silica glass can be explained by a surface fractal dimension d f ϭ2.5, where the mean squared displacement scales as ͗r 2 (t)͘ϰt 2/d w with d w ϭ1 ͑ballistic transport͒, if the surface population can exchange with the bulklike phase in the pores, and with d w ϭ2, if the bulklike phase is frozen. The former dynamics is interpreted in terms of bulk-mediated surface diffusion. ͓S1063-651X͑99͒08807-8͔
Fractal and fractional, Jan 28, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
A Fractional Calculus Approach to Adsorbate Dynamics in Nanoporous Materials
Birkhäuser Basel eBooks, 2002
ABSTRACT
Fractional Driftless Fokker-Planck Equation with Power Law Diffusion Coefficients
Springer eBooks, 2001
ABSTRACT
Integral Transforms and Special Functions, Dec 10, 2018
A simple ℵ-function and its dynamic equation is presented. An application to give an analytical s... more A simple ℵ-function and its dynamic equation is presented. An application to give an analytical solution to the Black Scholes equation is presented.
Annales Universitatis Paedagogicae Cracoviensis, 2022
Since Geronimo Cardano, algebraic equations of degree 4 have been solved analytically. Frequently... more Since Geronimo Cardano, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same resolvente, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that Carl Friedrich Gauss' (1799) fundamental theorem of algebra is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.
Fractal and fractional, Aug 18, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Axioms, Jan 3, 2023
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
Since Geronimo Cardano, algebraic equations of degree 4 have been solved analytically. Frequently... more Since Geronimo Cardano, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same resolvente, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that Carl Friedrich Gauss’ (1799) fundamental theorem of algebra is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.
Integral Transforms and Special Functions, 2018
A simple ℵ-function and its dynamic equation is presented. An application to give an analytical s... more A simple ℵ-function and its dynamic equation is presented. An application to give an analytical solution to the Black Scholes equation is presented.
Axioms
Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the sev... more Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Aleph-function. Using the q-analogue of the Leibniz rule for the fractional q-derivative of a product of two basic functions, we also provide a formula for the q-extended several variable Aleph-function, which is expressed in terms of an infinite series of the q-extended several variable Aleph-function. Since the three main formulas presented in this article are so general, they can be reduced to yield a number of identities involving q-extended simpler special functions. In this connection, we choose only one main formula to offer some of its particular instances involving diverse q-extended special functions, for example, the q-extended I-function, the q-extended H-function, and the q-extended Meijer’s G-function. The results presented he...
Applications to give an analytical solution to the Black Scholes equation
Integral Transforms and Special Functions, 2018
ABSTRACT A simple ℵ-function and its dynamic equation is presented. An application to give an ana... more ABSTRACT A simple ℵ-function and its dynamic equation is presented. An application to give an analytical solution to the Black Scholes equation is presented.
Analytic Theory on Probability, 2022
We translate the beginning of the central chapter of the original treatise [1814Lapl], chapter 3 ... more We translate the beginning of the central chapter of the original treatise [1814Lapl], chapter 3 in the 2 nd book, with its over 500 pages, and add some comments and annotations, in order that the interested reader of the 21 th century may understand what is meant. In adjustment to the English style of today we replace the future tense by the present, analogously to the expectation for a back-translation from a Hebrew text. This translation is kept closely in style to the original text, so that the reader, who has got no knowledge of the French language of the 19 th century, will get an impression of Laplace's style. In order to facilitate quotations, the page numbers of the French second edition of 1814 are placed within the translation at the places, where they are in the original, after the following full stop. The comments in the footnotes serve the subsequent explanation of the mathematical contents.
Fractal and Fractional
The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integra... more The purpose of this study is to offer a systematic, unified approach to the Mellin-Barnes integrals and associated special functions as Fox H, Aleph ℵ, and Saxena I function, encompassing the fundamental features and important conclusions under natural minimal assumptions on the functions in question. The approach’s pillars are the concept of a Mellin-Barnes integral and the Mellin representation of the given function. A Sinc quadrature is used in conjunction with a Sinc approximation of the function to achieve the numerical approximation of the Mellin-Barnes integral. The method converges exponentially and can handle endpoint singularities. We give numerical representations of the Aleph ℵ and Saxena I functions for the first time.
Fractional Calculus and Applied Analysis, 2004
We prove that Dirac's (symmetrical) delta function and the Hausdorff dimension function build up ... more We prove that Dirac's (symmetrical) delta function and the Hausdorff dimension function build up a pair of reciprocal functions. Our reasoning is based on the theorem by Mellin. Applications of the reciprocity relation demonstrate the merit of this approach.
Fractal and Fractional, 2022
In our work, we derived the fractional order q-integrals and q-derivatives concerning a basic ana... more In our work, we derived the fractional order q-integrals and q-derivatives concerning a basic analogue to the Aleph-function of two variables (AFTV). We discussed a related application and the q-extension of the corresponding Leibniz rule. Finally, we presented two corollaries concerning the basic analogue to the I-function of two variables and the basic analogue to the Aleph-function of one variable.
The paper discusses the application of MathLie in connection with Lie group analysis. The examine... more The paper discusses the application of MathLie in connection with Lie group analysis. The examined example is the (1 + 1)-dimensional case of the Doebner-Goldin equation after Madelung transform. The related Lie-algeras are calculated. We present the generators, commutator tables and adjoint representations from the algebras. Furthermore we discuss the reduction of an example to ordinary differential equations and solve it explicitly.
Gab es eine Vergletscherung der Alpen?, 1993
A geomorphologic–physical inquiry of ice–age is given, where a discrepancy between theory of ice–... more A geomorphologic–physical inquiry of ice–age is given, where a discrepancy between theory of ice–age and real existing glaciers is shown.
Solvo de algebraj ekvacioj
ABSTRACT
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Papers by Norbert Südland