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The Weak Euler Scheme for Stochastic Differential Delay Equations
Get PDFWe develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has
order of convergence 1, as in the case of stochastic ordinary
differential equations (SODEs) (i.e., without delay).The result
holds for SDDEs with multiple finite fixed delays in the drift and
diffusion terms. Although the set-up is non-anticipating,
our approach uses the Malliavin calculus and the
anticipating stochastic analysis techniques of Nualart
and Pardoux
Post processing for stochastic parabolic partial differential equations
Get PDFWe investigate the strong approximation of stochastic parabolic
partial differential equations with additive noise. We introduce
post-processing in the context of a standard Galerkin approximation,
although other spatial discretisations are possible. In time, we
use an
exponential integrator. We prove strong error estimates
and discuss the best number of post-processing terms to take.
Numerically, we evaluate the efficiency of the methods and observe
rates of convergence. Some experiments with the implicit
Euler--Maruyama method are describe
How to Check for Short Cycles in Large Nonlinear Feedback Shift Registers
No full textIn \cite{Ly} it is claimed that for a given large nonlinear feedback shift register it is hard to check whether short cycles have been embedded by the given method. The purpose of the present note is to show how to check for short cycles in general using our proposed algorithm. To implement the algorithm we introduce the notion of nonlinear complexity and try to classify sequences into nonlinear complexity classes
On the Spectral Density Estimation of Periodically Correlated (Cyclostationary) Time Series
Get PDFWe consider the estimation of the spectral density matrix of a periodically
correlated (PC) time series (also known as cyclostationary time series).
We use the well known relation between the spectral density matrix
of a periodically correlated time series and a stationary vector time series
(Gladyshev, 1961). The spectral matrix of the stationary vector time
series is estimated using the eigenvalue decomposition of block Toeplitz
matrices. The method of estimation is illustrated with simulated and real
time series
Nonlinear dynamics of an interface between shear bands
No full textWe study numerically the nonlinear dynamics of a shear banding interface in two-dimensional planar shear flow, within the nonlocal Johnson-Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for a sufficiently small ratio of the interfacial width center dot to cell length L-x. The instability saturates in finite amplitude interfacial fluctuations. For decreasing center dot/L-x these undergo a nonequilibrium transition from simple traveling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos
Embedding Theorems for Non-uniformly Sampled Dynamical Systems
Get PDFThe embedding theorem of Takens, and its extensions, have provided the theoretical underpinning for a wide range of investigations of time series derived from nonlinear dynamical systems. The theorem applies when the dynamical system is sampled uniformly in time. There has, however, been increasing interest in situations where observations on the system are not uniform in time, and in particular where the consist of a series of inter-event (`interspike') intervals. Sauer has provided an embedding theorem for the case where these intervals are generated by an integrate-and-fire mechanism. Here we prove several embedding theorems pertaining to non-uniform sampling. We consider two situations: in the first, observations consist of the values of some function on the state space of the system, with the times between the successive observations being given by another function—the sampling interval function; in the second, sampling times are generated in the same way, but now the observations consist only of the intersample intervals. We prove embedding theorems both when the sampling interval function is allowed to be rather general, and when it corresponds to integrate-and-fire sampling. We point out that non-uniform sampling might lead to better reconstructions than uniform sampling, for certain kinds of time series
A Study of the Matrix Exponential
Get PDFThis report brings together a wide variety of facts concerning the
matrix exponential. Against a background of familiar results, we
present an analysis of matrix functions (the exponential in
particular) which exploits the Schur decomposition theorem. This
helps us explore the behavior of a function of a matrix whose
eigensystem is poorly conditioned. Finally, we investigate Pade
approximation of the matrix exponential and feature in the discussion
a potentially useful inverse error analysis
The Bessel ratio distribution
No full textLet X and Y be two random variables; then the exact distribution of the ratio X / Y is derived when X and Y are independent Bessel function random variable
Sklyanin algebras and Hilbert schemes of points
Get PDFWe construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P^2.
The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E ⊂ P^2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P^2 \ E