172,343 research outputs found
Spectral methods for bivariate Markov processes with diffusion and discrete components and a variant of the Wright-Fisher model
The aim of this paper is to study differential and spectral properties of the
infinitesimal operator of two dimensional Markov processes with diffusion and
discrete components. The infinitesimal operator is now a second-order
differential operator with matrix-valued coefficients, from which we can derive
backward and forward equations, a spectral representation of the probability
density, study recurrence of the process and the corresponding invariant
distribution. All these results are applied to an example coming from group
representation theory which can be viewed as a variant of the Wright-Fisher
model involving only mutation effects.Comment: 6 figure
Some examples of matrix-valued orthogonal functions having a differential and an integral operator as eigenfunctions
The aim of this paper is to show some examples of matrix-valued orthogonal
functions on the real line which are simultaneously eigenfunctions of a
second-order differential operator of Schr\"{o}dinger type and an integral
operator of Fourier type. As a consequence we derive integral representations
of these functions as well as other useful structural formulas. Some of these
functions are plotted to show the relationship with the Hermite or wave
functions
IFSM representation of Brownian motion with applications to simulation
Several methods are currently available to simulate paths of the Brownian
motion. In particular, paths of the BM can be simulated using the properties of
the increments of the process like in the Euler scheme, or as the limit of a
random walk or via L2 decomposition like the Kac-Siegert/Karnounen-Loeve
series.
In this paper we first propose a IFSM (Iterated Function Systems with Maps)
operator whose fixed point is the trajectory of the BM. We then use this
representation of the process to simulate its trajectories. The resulting
simulated trajectories are self-affine, continuous and fractal by construction.
This fact produces more realistic trajectories than other schemes in the sense
that their geometry is closer to the one of the true BM's trajectories. The
IFSM trajectory of the BM can then be used to generate more realistic solutions
of stochastic differential equations
Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials
We study double integral representations of Christoffel-Darboux kernels
associated with two examples of Hermite-type matrix orthogonal polynomials. We
show that the Fredholm determinants connected with these kernels are related
through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain
Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax
pair whose compatibility conditions lead to a non-commutative version of the
Painleve' IV differential equation for each family.Comment: Final version, accepted for publication on CMP: Communications in
Mathematical Physics. 24 pages, 1 figur
Condensation and Slow Dynamics of Polar Nanoregions in Lead Relaxors
It is now well established that the unique properties of relaxor
ferroelectrics are due to the presence of polar nanoregions (PNR's). We present
recent results from Neutron and Raman scattering of single crystals of PZN,
PZN-xPT, and PMN. Both sets of measurements provide information on the
condensation of the PNR's and on their slow dynamics, directly through the
central peak and, indirectly, through their coupling to transverse phonons. A
comparative analysis of these results allows identification of three stages in
the evolution of the PNR's with decreasing temperature: a purely dynamic stage,
a quasi-static stage with reorientational motion and a frozen stage. A model is
proposed, based on a prior study of KTN, which explains the special behavior of
the transverse phonons (TO and TA) in terms of their mutual coupling through
the rotations of the PNR's.Comment: AIP 6x9 style files, 10 pages, 4 figures, Conference-Fundamental
Physics of Ferroelectrics 200
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