2,825 research outputs found
Random-time processes governed by differential equations of fractional distributed order
We analyze here different types of fractional differential equations, under
the assumption that their fractional order is random\ with
probability density We start by considering the fractional extension
of the recursive equation governing the homogeneous Poisson process
\ We prove that, for a particular (discrete) choice of , it
leads to a process with random time, defined as The distribution of the
random time argument can be
expressed, for any fixed , in terms of convolutions of stable-laws. The new
process is itself a renewal and
can be shown to be a Cox process. Moreover we prove that the survival
probability of , as well as its
probability generating function, are solution to the so-called fractional
relaxation equation of distributed order (see \cite{Vib}%).
In view of the previous results it is natural to consider diffusion-type
fractional equations of distributed order. We present here an approach to their
solutions in terms of composition of the Brownian motion with the
random time . We thus provide an
alternative to the constructions presented in Mainardi and Pagnini
\cite{mapagn} and in Chechkin et al. \cite{che1}, at least in the double-order
case.Comment: 26 page
TEMPERED RELAXATION EQUATION AND RELATED GENERALIZED STABLE PROCESSES
Fractional relaxation equations, as well as relaxation functions time-changed by independent stochastic processes have been widely studied (see, for example, MAI, STAW and GAR). We start here by proving that the upper-incomplete Gamma function satisfies the tempered-relaxation equation (of index Ďâ(0,1)); thanks to this explicit form of the solution, we can then derive its spectral distribution, which extends the stable law. Accordingly, we define a new class of selfsimilar processes (by means of the n-times Laplace transform of its density) which is indexed by the parameter Ď: in the special case where Ď=1, it reduces to the stable subordinator. Therefore the parameter Ď can be seen as a measure of the local deviation from the temporal dependence structure displayed in the standard stable case
Probabilistic analysis of the telegrapher's process with drift by means of relativistic transformations
The telegrapher's process with drift is here examined and its distribution is
obtained by applying the Lorentz transformation. The related characteristic function as well as the distribution are also derived by solving an initial
value problem for the generalized telegraph equation
On random flights with non-uniformly distributed directions
This paper deals with a new class of random flights defined in the real space characterized
by non-uniform probability distributions on the multidimensional sphere. These
random motions differ from similar models appeared in literature which take
directions according to the uniform law. The family of angular probability
distributions introduced in this paper depends on a parameter which
gives the level of drift of the motion. Furthermore, we assume that the number
of changes of direction performed by the random flight is fixed. The time
lengths between two consecutive changes of orientation have joint probability
distribution given by a Dirichlet density function.
The analysis of is not an easy task, because it
involves the calculation of integrals which are not always solvable. Therefore,
we analyze the random flight obtained as
projection onto the lower spaces of the original random
motion in . Then we get the probability distribution of
Although, in its general framework, the analysis of is very complicated, for some values of , we can provide
some results on the process. Indeed, for , we obtain the characteristic
function of the random flight moving in . Furthermore, by
inverting the characteristic function, we are able to give the analytic form
(up to some constants) of the probability distribution of Comment: 28 pages, 3 figure
Diffusion in multiscale spacetimes
We study diffusion processes in anomalous spacetimes regarded as models of
quantum geometry. Several types of diffusion equation and their solutions are
presented and the associated stochastic processes are identified. These results
are partly based on the literature in probability and percolation theory but
their physical interpretation here is different since they apply to quantum
spacetime itself. The case of multiscale (in particular, multifractal)
spacetimes is then considered through a number of examples and the most general
spectral-dimension profile of multifractional spaces is constructed.Comment: 23 pages, 5 figures. v2: discussion improved, typos corrected,
references adde
Large deviations for a damped telegraph process
In this paper we consider a slight generalization of the damped telegraph
process in Di Crescenzo and Martinucci (2010). We prove a large deviation
principle for this process and an asymptotic result for its level crossing
probabilities (as the level goes to infinity). Finally we compare our results
with the analogous well-known results for the standard telegraph process
Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes
Different initial and boundary value problems for the equation of vibrations
of rods (also called Fresnel equation) are solved by exploiting the connection
with Brownian motion and the heat equation. The analysis of the fractional
version (of order ) of the Fresnel equation is also performed and, in
detail, some specific cases, like , 1/3, 2/3, are analyzed. By means
of the fundamental solution of the Fresnel equation, a pseudo-process ,
with real sign-varying density is constructed and some of its properties
examined. The equation of vibrations of plates is considered and the case of
circular vibrating disks is investigated by applying the methods of
planar orthogonally reflecting Brownian motion within . The composition of
F with reflecting Brownian motion yields the law of biquadratic heat
equation while the composition of with the first passage time of
produces a genuine probability law strictly connected with the Cauchy process.Comment: 33 pages,8 figure
A qualitative study of independent fast food vendors near secondary schools in disadvantaged Scottish neighbourhoods
Background:
Preventing and reducing childhood and adolescent obesity is a growing priority in many countries. Recent UK data suggest that children in more deprived areas have higher rates of obesity and poorer diet quality than those in less deprived areas. As adolescents spend a large proportion of time in school, interventions to improve the food environment in and around schools are being considered. Nutrient standards for school meals are mandatory in the UK, but many secondary pupils purchase foods outside schools at break or lunchtime that may not meet these standards.
Methods:
Qualitative interviews were conducted with fast food shop managers to explore barriers to offering healthier menu options. Recruitment targeted independently-owned shops near secondary schools (pupils aged c.12-17) in low-income areas of three Scottish cities. Ten interviews were completed, recorded, and transcribed for analysis. An inductive qualitative approach was used to analyse the data in NVivo 10.
Results:
Five themes emerged from the data: pride in what is sold; individual autonomy and responsibility; customer demand; profit margin; and neighbourhood context. Interviewees consistently expressed pride in the foods they sold, most of which were homemade. They felt that healthy eating and general wellbeing are the responsibility of the individual and that offering what customers want to eat, not necessarily what they should eat, was the only way to stay in business. Most vendors felt they were struggling to maintain a profit, and that many aspects of the low-income neighbourhood context would make change difficult or impossible.
Conclusions:
Independent food shops in low-income areas face barriers to offering healthy food choices, and interventions and policies that target the food environment around schools should take the neighbourhood context into consideration
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