Mar 26 2015

## Generalization of the fractional Poisson distribution

author: R. Herrmann

abstract:

A generalization of the Poisson distribution based on the generalized Mittag-Leffler function Eα,β(λ) is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter β is suggested.

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Jul 17 2014

## Reflection symmetric Erdelyi-Kober type operators – a quasi-particle interpretation

author: R. Herrmann

abstract:

Reflection symmetric Erdelyi-Kober type fractional integral operators are used to construct fractional quasi-particle generators. The eigenfunctions and eigenvalues of these operators are given analytically.
A set of fractional creation- and annihilation-operators is defined and the properties of the corresponding free Hamiltonian are investigated. Analogue to the classical approach for interacting multi-particle systems the results are interpreted as a fractional quantum model for a description of residual interactions of pairing type.

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Apr 07 2014

## A fractal approach to the dark silicon problem: a comparison of 3D computer architectures — standard slices versus fractal Menger sponge geometry

author: R. Herrmann

abstract:

The dark silicon problem, which limits the power-growth of future computer generations, is interpreted as a heat energy transport problem when increasing the energy emitting surface area within a given volume. A comparison of two 3D-configuration models, namely a standard slicing and a fractal surface generation within the Menger sponge geometry is presented. It is shown, that for iteration orders n>3 the fractal model shows increasingly better thermal behavior. As a consequence cooling problems may be minimized by using a fractal architecture.

Therefore the Menger sponge geometry is a good example for fractal architectures applicable not only in computer science, but also e.g. in chemistry when building chemical reactors, optimizing catalytic processes or in sensor construction technology building highly effective sensors for toxic gases or water analysis.

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Jan 22 2014

## Towards a geometric interpretation of generalized fractional integrals – Erdelyi-Kober type integrals on R^N as an example

author: R. Herrmann

abstract:

A family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on R^N . Based on this geometric view, several extensions are discussed.

reference: Fract. Calc. Appl. Anal. (2014) 17(2) 361-370

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Jul 24 2013

## On the origin of space

author: R. Herrmann

abstract:

Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a model is presented, which describes space generation as a dynamic process, where the dimension $d$ of space evolves smoothly with time in the range 0 <= d(t) <=3, where the lower and upper boundaries of dimension are derived from first principles. It is demonstrated, that a minimum threshold for the space dimension is necessary to establish an interaction with external probe particles. A possible application in cosmology is suggested.

reference: Cent. Eur. J. Phys. (2013) 11(10) 1212-1220

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May 03 2013

## Folded potentials in cluster physics – a comparison of Yukawa and Coulomb potentials with Riesz fractional integrals

author:
R. Herrmann

abstract:
In cluster physics a single particle potential to determine the microscopic part of the total energy of a collective configuration is necessary to calculate the shell- and pairing effects. In this paper we investigate the properties of the Riesz fractional integrals and compare their properties with the standard Coulomb and Yukawa potentials commonly used. It is demonstrated, that Riesz potentials may serve as a promising extension of standard potentials and may be reckoned as a smooth transition from Coulomb to Yukawa like potentials, depending of the fractional parameter $\alpha$. For the macroscopic part of the total energy the Riesz potentials treat the Coulomb-, symmetry- and pairing-contributions from a generalized point of view, since they turn out to be similar realizations of the same fractional integral at distinct $\alpha$ values.

reference: J. Phys. A.: Math. Theor. (2013) 46(40) 405203

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Apr 03 2013

## Uniqueness of the fractional derivative definition – The Riesz fractional derivative as an example

author:
R. Herrmann

abstract:
For the Riesz fractional derivative besides the well known integral representation two new differential representations are presented, which emphasize the local aspects of a fractional derivative. The consequences for a valid solution of the fractional Schroedinger equation are discussed.

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Mar 17 2013

## Numerical solution of the fractional quantum mechanical harmonic oscillator based on the Riemann and Caputo derivative

author:
R. Herrmann

abstract:
Based on the Riemann- and Caputo definition of the fractional derivative we tabulate the lowest n=31 energy levels and generated graphs of the occupation probability of the fractional quantum mechanical harmonic oscillator with a precision of 32 digits for 0.50 < \alpha < 2.00, which corresponds to the transition from U(1) to SO(3).

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Mar 15 2013

## The fractional Schroedinger equation and the infinite potential well – numerical results using the Riesz derivative

author:
R. Herrmann

abstract:
Based on the Riesz definition of the fractional derivative the fractional Schroedinger equation with an infinite well potential is investigated. First it is shown analytically, that the solutions of the free fractional Schroedinger equation are not eigenfunctions, but good approximations for large k and for $\alpha \approx 2$. The first lowest eigenfunctions are then calculated numerically and an approximate analytic formula for the level spectrum is derived.

reference: Gam. Ori. Chron. Phys. (2013) 1(1) 1-12

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Nov 14 2012