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Unsteady flow in a rotating torus after a sudden change in rotation rate
No full textWe consider the temporal evolution of a viscous incompressible fluid in a torus of
finite curvature; a problem first investigated by Madden & Mullin (J. Fluid Mech.,
vol. 265, 1994, pp. 265�217). The system is initially in a state of rigid-body rotation
(about the axis of rotational symmetry) and the container�s rotation rate is then
changed impulsively. We describe the transient flow that is induced at small values
of the Ekman number, over a time scale that is comparable to one complete rotation
of the container. We show that (rotationally symmetric) eruptive singularities (of the
boundary layer) occur at the inner or outer bend of the pipe for a decrease or an
increase in rotation rate respectively. Moreover, on allowing for a change in direction
of rotation, there is a (negative) ratio of initial-to-final rotation frequencies for which
eruptive singularities can occur at both the inner and outer bend simultaneously.
We also demonstrate that the flow is susceptible to a combination of axisymmetric
centrifugal and non-axisymmetric inflectional instabilities. The inflectional instability
arises as a consequence of the developing eruption and is shown to be in
qualitative agreement with the experimental observations of Madden & Mullin (1994).
Throughout our work, detailed quantitative comparisons are made between asymptotic
predictions and finite- (but small-) Ekman-number Navier�Stokes computations using
a finite-element method. We find that the boundary-layer results correctly capture the
(finite-Ekman-number) rotationally symmetric flow and its global stability to linearised
perturbations
A constrained approach to multiscale stochastic simulation of chemically reacting systems
No full textStochastic simulation of coupled chemical reactions is often computationally intensive, especially if a chemical system contains reactions occurring on different time scales. In this paper, we introduce a multiscale methodology suitable to address this problem, assuming that the evolution of the slow species in the system is well approximated by a Langevin process. It is based on the conditional stochastic simulation algorithm (CSSA) which samples from the conditional distribution of the suitably defined fast variables, given values for the slow variables. In the constrained multiscale algorithm (CMA) a single realization of the CSSA is then used for each value of the slow variable to approximate the effective drift and diffusion terms, in a similar manner to the constrained mean-force computations in other applications such as molecular dynamics. We then show how using the ensuing Fokker-Planck equation approximation, we can in turn approximate average switching times in stochastic chemical systems
Dynamics of poles with position-dependent strengths and its optical analogues
No full textDynamics of point vortices is generalized in two ways: first by making the strengths complex, which allows for sources and sinks in superposition with the usual vortices, second by making them functions of position. These generalizations lead to a rich dynamical system, which is nonlinear and yet has enough conservation laws coming from a Hamiltonian-like formalism. We then discover that in this system the motion of a pair mimics the behavior of rays in geometric optics. We describe several exact solutions with optical analogues, notably Snell's law and the law of reflection off a mirror, and perform numerical experiments illustrating some striking behavior
Reachability in one-dimensional controlled polynomial dynamical systems
No full textIn this paper we address the reachability problem in controlled o-minimal dynamical systems.
This problem can be formulated as follows.
Given a controlled o-minimal dynamical system and
a target set, starting at an initial point, is there a
finite choice of time points and control parameters applied at
these points such that the target set is reachable.
We prove the existence of a finite control stategy and construct an algorithm which provides finite control
strategies for one-dimensional controlled polynomial dynamical systems.We also provide a complexity result which shows upper bounds on the numbers of switches in finite control strategies
Algorithmic Properties of Sigma--definability over Positive Predicate Structures
No full textIn this paper we propose a generalisation of the authors p results on semantical characterisation of --definability.
We prove that over every positive predicate structure a set is -definable if and only if it is definable by a disjunction of a recursively enumerable set of
existential formulas
Some recent work in Frechet geometry
No full textSome recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Frechet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Frechet spaces
HydraMP: Exploiting shared memory parallelism in HYDRA with OpenMP
Get PDFMulticore CPUs are now found in desktops, servers and supercomputers but many existing parallel performance analysis tools were designed for the single-core distributed-memory world. We investigate the practicality of taking an existing tool, namely the HYDRA response time analyser, and parallelising it with OpenMP to produce a multithreaded implementation suitable for execution on multicore shared-memory machines. We discuss the amount of software engineering work required and show that only a small number of lines of code need to be added to achieve dramatic speed-ups over the serial version. We also compare the run-times of our OpenMP-parallelised version with existing MPI-parallelised code on the same hardware
Products of random Max-plus matrices
Get PDFMax-plus stochastic linear systems describe a wide variety of non-linear queueing processes. The dynamics of these systems are dominated by a Max-plus analogue of the Lyupanov exponent the value of which depends on the structure of the underlying support graphs as well as the properties of the waiting-time distributions. For matrices whose associated weighted graphs have identically distributed edge weights (componentwise homogeneity) we are able to decouple these two effects and provide a sandwich of bounds for the Max-plus Lyupanov exponent relating it to some classical properties of the support graph and some extreme value expectations of the waiting-time distributions. This sandwich inequality is then applied to products of componentwise exponential, Gaussian and uniform matrices
An Algorithm for the Complete Solution of Quadratic Eigenvalue Problems
Get PDFWe develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials.
It incorporates scaling of
the problem parameters prior to the computation of eigenvalues, a choice of linearization with favorable
conditioning and backward stability properties,
and a preprocessing step that
reveals and deflates the zero and infinite eigenvalues contributed by singular
leading and trailing matrix coefficients.
The algorithm is backward stable for quadratics that are not too heavily damped.
Numerical experiments show that
our MATLAB implementation of the algorithm,
quadeig, outperforms the MATLAB
function polyeig in terms of both stability and efficiency
Isomonodromic deformations and twisted Yangians arising in Teichmüller theory
No full textIn this paper we build a link between the Teichmüller theory of hyperbolic Riemann surfaces and isomonodromic
deformations of linear systems whose monodromy group is the Fuchsian group associated to the
given hyperbolic Riemann surface by the Poincaré uniformization. In the case of a one-sheeted hyperboloid
with n orbifold points we show that the Poisson algebra Dn of geodesic length functions is the semiclassical
limit of the twisted q-Yangian Y
q (on) for the orthogonal Lie algebra on defined by Molev, Ragoucy and
Sorba. We give a representation of the braid-group action on Dn in terms of an adjoint matrix action. We
characterize two types of finite-dimensional Poissonian reductions and give an explicit expression for the
generating function of their central elements. Finally, we interpret the algebra Dn as the Poisson algebra of
monodromy data of a Frobenius manifold in the vicinity of a non-semi-simple point