1,599 research outputs found
On the fractional Poisson process and the discretized stable subordinator
The fractional Poisson process and the Wright process (as discretization of
the stable subordinator) along with their diffusion limits play eminent roles
in theory and simulation of fractional diffusion processes. Here we have
analyzed these two processes, concretely the corresponding counting number and
Erlang processes, the latter being the processes inverse to the former.
Furthermore we have obtained the diffusion limits of all these processes by
well-scaled refinement of waiting times and jumpsComment: 30 pages, 4 figures. A preliminary version of this paper was an
invited talk given by R. Gorenflo at the Conference ICMS2011, held at the
International Centre of Mathematical Sciences, Pala-Kerala (India) 3-5
January 2011, devoted to Prof Mathai on the occasion of his 75 birthda
Laplace-Laplace analysis of the fractional Poisson process
We generate the fractional Poisson process by subordinating the standard
Poisson process to the inverse stable subordinator. Our analysis is based on
application of the Laplace transform with respect to both arguments of the
evolving probability densities.Comment: 20 pages. Some text may overlap with our E-prints: arXiv:1305.3074,
arXiv:1210.8414, arXiv:1104.404
Continuous time random walk, Mittag-Leffler waiting time and fractional diffusion: mathematical aspects
We show the asymptotic long-time equivalence of a generic power law waiting
time distribution to the Mittag-Leffler waiting time distribution,
characteristic for a time fractional CTRW. This asymptotic equivalence is
effected by a combination of "rescaling" time and "respeeding" the relevant
renewal process followed by a passage to a limit for which we need a suitable
relation between the parameters of rescaling and respeeding. Turning our
attention to spatially 1-D CTRWs with a generic power law jump distribution,
"rescaling" space can be interpreted as a second kind of "respeeding" which
then, again under a proper relation between the relevant parameters leads in
the limit to the space-time fractional diffusion equation. Finally, we treat
the `time fractional drift" process as a properly scaled limit of the counting
number of a Mittag-Leffler renewal process.Comment: 36 pages, 3 figures (5 files eps). Invited lecture by R. Gorenflo at
the 373. WE-Heraeus-Seminar on Anomalous Transport: Experimental Results and
Theoretical Challenges, Physikzentrum Bad-Honnef (Germany), 12-16 July 2006;
Chairmen: R. Klages, G. Radons and I.M. Sokolo
On an explicit finite difference method for fractional diffusion equations
A numerical method to solve the fractional diffusion equation, which could
also be easily extended to many other fractional dynamics equations, is
considered. These fractional equations have been proposed in order to describe
anomalous transport characterized by non-Markovian kinetics and the breakdown
of Fick's law. In this paper we combine the forward time centered space (FTCS)
method, well known for the numerical integration of ordinary diffusion
equations, with the Grunwald-Letnikov definition of the fractional derivative
operator to obtain an explicit fractional FTCS scheme for solving the
fractional diffusion equation. The resulting method is amenable to a stability
analysis a la von Neumann. We show that the analytical stability bounds are in
excellent agreement with numerical tests. Comparison between exact analytical
solutions and numerical predictions are made.Comment: 22 pages, 6 figure
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Special opportunities for conserving cultural and biological diversity: The co-occurrence of Indigenous languages and UNESCO Natural World Heritage Sites
Recent research indicates that speakers of Indigenous languages often live in or near United Nations Educational, Scientific, and Cultural Organization (UNESCO) Natural World Heritage Sites (WHSs). Because language is a key index of cultural diversity, examining global patterns of co-occurrence between languages and these sites provides a means of identifying opportunities to conserve both culture and nature, especially where languages, WHSs, or both are recognized as endangered. This paper summarizes instances when Indigenous languages share at least part of their geographic extent with Natural WHSs. We consider how this co-occurrence introduces the potential to coordinate conservation of nature and sociocultural systems at these localities, particularly with respect to the recently issued UNESCO policy on engaging Indigenous people and the forthcoming International Year of Indigenous Languages. The paper concludes by discussing how the presence of Indigenous people at UNESCO Natural WHSs introduces important opportunities for co-management that enable resident Indigenous people to help conserve their language and culture along with the natural settings where they occur. We discuss briefly the example of Australia as a nation exploring opportunities for employing and strengthening such coordinated conservation efforts
Fractional calculus and continuous-time finance II: the waiting-time distribution
We complement the theory of tick-by-tick dynamics of financial markets based
on a Continuous-Time Random Walk (CTRW) model recently proposed by Scalas et
al., and we point out its consistency with the behaviour observed in the
waiting-time distribution for BUND future prices traded at LIFFE, London.Comment: Revised version, 17 pages, 4 figures. Physica A, Vol. 287, No 3-4,
468--481 (2000). Proceedings of the International Workshop on "Economic
Dynamics from the Physics Point of View", Bad-Honnef (Germany), 27-30 March
200
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