36,077 research outputs found
Fixed point theorems for nonlinear contractions with applications to iterated function systems
[EN] We introduce a new type of nonlinear contraction and present some fixed point results without using continuity or semi-continuity. Our result complement, extend and generalize a number of fixed point theorems including the the well-known Boyd and Wong theorem [On nonlinear contractions, Proc. Amer. Math. Soc. 20(1969)]. Also we discuss an application to iterated function systems.Pant, R. (2018). Fixed point theorems for nonlinear contractions with applications to iterated function systems. Applied General Topology. 19(1):163-172. doi:10.4995/agt.2018.7918SWORD16317219
Fractal and chaotic solutions of the discrete nonlinear Schr\"odinger equation in classical and quantum systems
We discuss stationary solutions of the discrete nonlinear Schr\"odinger
equation (DNSE) with a potential of the type which is generically
applicable to several quantum spin, electron and classical lattice systems. We
show that there may arise chaotic spatial structures in the form of
incommensurate or irregular quantum states. As a first (typical) example we
consider a single electron which is strongly coupled with phonons on a
chain of atoms --- the (Rashba)--Holstein polaron model. In the adiabatic
approximation this system is conventionally described by the DNSE. Another
relevant example is that of superconducting states in layered superconductors
described by the same DNSE. Amongst many other applications the typical example
for a classical lattice is a system of coupled nonlinear oscillators. We
present the exact energy spectrum of this model in the strong coupling limit
and the corresponding wave function. Using this as a starting point we go on to
calculate the wave function for moderate coupling and find that the energy
eigenvalue of these structures of the wave function is in exquisite agreement
with the exact strong coupling result. This procedure allows us to obtain
(numerically) exact solutions of the DNSE directly. When applied to our typical
example we find that the wave function of an electron on a deformable lattice
(and other quantum or classical discrete systems) may exhibit incommensurate
and irregular structures. These states are analogous to the periodic,
quasiperiodic and chaotic structures found in classical chaotic dynamics
Statistical Inference for Partially Observed Markov Processes via the R Package pomp
Partially observed Markov process (POMP) models, also known as hidden Markov
models or state space models, are ubiquitous tools for time series analysis.
The R package pomp provides a very flexible framework for Monte Carlo
statistical investigations using nonlinear, non-Gaussian POMP models. A range
of modern statistical methods for POMP models have been implemented in this
framework including sequential Monte Carlo, iterated filtering, particle Markov
chain Monte Carlo, approximate Bayesian computation, maximum synthetic
likelihood estimation, nonlinear forecasting, and trajectory matching. In this
paper, we demonstrate the application of these methodologies using some simple
toy problems. We also illustrate the specification of more complex POMP models,
using a nonlinear epidemiological model with a discrete population,
seasonality, and extra-demographic stochasticity. We discuss the specification
of user-defined models and the development of additional methods within the
programming environment provided by pomp.Comment: In press at the Journal of Statistical Software. A version of this
paper is provided at the pomp package website: http://kingaa.github.io/pom
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
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