593,713 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
On an origin of numerical diffusion: Violation of invariance under space-time inversion
The invariant properties of the convection equation du/dt + adu/dx = 0 (where d is the partial differential operator) with respect to spatial reflection, time reversal, and space-time inversion are studied. Generally, a finite-difference analog of this equation may possess some or none of these properties. It is shown that, under certain conditions, the von Neumann amplification factor of an analog satisfies a special relation for each invariant property this analog possesses. Particularly, an analog is neutrally stable and thus free of numerical diffusion if it possesses the invariant property related to space-time inversion. It is also explained why generally (1) an upwind scheme possesses neither the invariant property related to spatial reflection nor that related to space-time inversion, and (2) an explicit scheme possesses neither the invariant property related to time reversal nor that related to space-time inversion. Extension to the viscous case and a remarkable connection between the current work and a new numerical framework for solving conservation laws are also discussed
Laws of the iterated logarithm for a class of iterated processes
Let be a Brownian motion or a spectrally negative
stable process of index 1<\a<2. Let be the hitting time
of a stable subordinator of index independent of . We use a
connection between and the stable subordinator of index \beta/\a to
derive information on the path behavior of . This is an extension of
the connection of iterated Brownian motion and (1/4)-stable subordinator due to
Bertoin \cite{bertoin}.
Using this connection, we obtain various laws of the iterated logarithm for
. In particular, we establish law of the iterated logarithm for local
time Brownian motion, , where is a Brownian motion (the case
\a=2) and is the local time at zero of a stable process of index
independent of . In this case with
for some constant . This establishes the lower bound
in the law of the iterated logarithm which we could not prove with the
techniques of our paper \cite{MNX}. We also obtain exact small ball probability
for using ideas from \cite{aurzada}.Comment: 13 page
On a new concept of stochastic domination and the laws of large numbers
Consider a sequence of positive integers , and an array of
nonnegative real numbers satisfying
This paper introduces
the concept of -stochastic domination. We develop some techniques
concerning this concept and apply them to remove an assumption in a strong law
of large numbers of Chandra and Ghosal [Acta. Math. Hungarica, 1996]. As a
by-product, a considerable extension of a recent result of Boukhari [J.
Theoret. Probab., 2021] is established and proved by a different method. The
results on laws of large numbers are new even when the summands are
independent. Relationships between the concept of -stochastic
domination and the concept of -uniform integrability are
presented. Two open problems are also discussed.Comment: 26 page
Elasticity, Shape Fluctuations and Phase Transitions in the New Tubule Phase of Anisotropic Tethered Membranes
We study the shape, elasticity and fluctuations of the recently predicted
(cond-mat/9510172) and subsequently observed (in numerical simulations)
(cond-mat/9705059) tubule phase of anisotropic membranes, as well as the phase
transitions into and out of it. This novel phase lies between the previously
predicted flat and crumpled phases, both in temperature and in its physical
properties: it is crumpled in one direction, and extended in the other. Its
shape and elastic properties are characterized by a radius of gyration exponent
and an anisotropy exponent . We derive scaling laws for the radius of
gyration (i.e. the average thickness) of the tubule about a
spontaneously selected straight axis and for the tubule undulations
transverse to its average extension. For phantom (i.e.
non-self-avoiding) membranes, we predict , and
, exactly, in excellent agreement with simulations. For
membranes embedded in the space of dimension , self-avoidance greatly
swells the tubule and suppresses its wild transverse undulations, changing its
shape exponents and . We give detailed scaling results for the shape
of the tubule of an arbitrary aspect ratio and compute a variety of correlation
functions, as well as the anomalous elasticity of the tubules. Finally we
present a scaling theory for the shape of the membrane and its specific heat
near the continuous transitions into and out of the tubule phase and perform
detailed renormalization group calculations for the crumpled-to-tubule
transition for phantom membranes.Comment: 34 PRE pages, RevTex and 11 postscript figures, also available at
http://lulu.colorado.edu/~radzihov/ version to appear in Phys. Rev. E, 57, 1
(1998); minor change
N-extended Chern-Simons Carrollian supergravities in 2+1 spacetime dimensions
In this work we present the ultra-relativistic -extended AdS
Chern-Simons supergravity theories in three spacetime dimensions invariant
under -extended AdS Carroll superalgebras. We first consider the
and cases; subsequently, we generalize our analysis to
, with even, and to
, with . The -extended AdS Carroll
superalgebras are obtained through the Carrollian (i.e., ultra-relativistic)
contraction applied to an extension of , to , to an
extension of , and to the direct sum of an algebra and ,
respectively. We also analyze the flat limit (, being
the length parameter) of the aforementioned -extended
Chern-Simons AdS Carroll supergravities, in which we recover the
ultra-relativistic -extended (flat) Chern-Simons supergravity
theories invariant under -extended super-Carroll algebras. The
flat limit is applied at the level of the superalgebras, Chern-Simons actions,
supersymmetry transformation laws, and field equations.Comment: 48 pages. Version accepted for publication in Journal of High Energy
Physic
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