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Recent developments in the asymptotic theory of separated flows. Leverhulme lectures - lecture notes.
Get PDFRobust Preconditioning for Second-Order Elliptic PDEs with Random Field Coefficients
No full textFluid flow and the transport of chemicals in flows in heterogeneous porous media are modelled mathematically using partial differential equations (PDEs). In deterministic modelling, material properties of
the porous medium are assumed to be known explicitly. This assumption leads to tractable computations. To tackle the more realistic stochastic groundwater flow problem, it is necessary to represent the unknown permeability coefficients as random fields with prescribed statistical properties. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions averaged to obtain statistical properties of the solution variables. Alternatively, the so-called stochastic finite element method (SFEM) discretises the probabilistic dimension of the PDE directly. However, this approach has not gained popularity with practitioners due to a perceived high computational cost. In this report we solve the stochastic Darcy flow problem via traditional and stochastic finite element techniques, in primal and mixed formulation where appropriate. Permeability coefficients are represented using Gaussian or lognormal random fields. We focus on fast and efficient linear algebra techniques for solving both the large numbers of deterministic problems required by the Monte Carlo approach and, in contrast, the single, structured, but extremely large linear system that arises as a consequence of the SFEM. To achieve optimal computational complexity, black-box algebraic multigrid is exploited in the design of fast solvers
Variétés abéliennes et invariants arithmétiques
No full textDans la continuité de nos travaux précédents, nous étudions un analogue, pour le modèle de Néron d´une variété abélienne semi-stable sur un corps de nombres, du class-invariant homomorphism introduit par M. J. Taylor, qui nous permet de mesurer la structure galoisienne de certains torseurs
Pseudo completions and completions in stages of o-minimal structures
Get PDFFor an o-minimal expansion R of a real closed field and a set \fancyscript{V} of -convex valuation rings, we construct a “pseudo completion” with respect to \fancyscript{V}. This is an elementary extension of generated by all completions of all the residue fields of the V \in \fancyscript{V}, when these completions are embedded into a big elementary extension of . It is shown that does not depend on the various embeddings up to an -isomorphism. For polynomially bounded we can iterate the construction of the pseudo completion in order to get a “completion in stages” of with respect to \fancyscript{V}. is the “smallest” extension of such that all residue fields of the unique extensions of all V \in \fancyscript{V} to are complete
Rheology of giant micelles
No full textGiant micelles are elongated, polymer-like objects created by the self-assembly of amphiphilic molecules (such as detergents) in solution. Giant micelles are typically flexible, and can become highly entangled even at modest concentrations. The resulting viscoelastic solutions show fascinating flow behaviour (rheology) which we address theoretically in this article at two levels. First, we summarize advances in understanding linear viscoelastic spectra and steady-state nonlinear flows, based on microscopic constitutive models that combine the physics of polymer entanglement with the reversible kinetics of self-assembly. Such models were first introduced two decades ago, and since then have been shown to explain robustly several distinctive features of the rheology in the strongly entangled regime, including extreme shear thinning. We then turn to more complex rheological phenomena, particularly involving spatial heterogeneity, spontaneous oscillation, instability and chaos. Recent understanding of these complex flows is based largely on grossly simplified models which capture in outline just a few pertinent microscopic features, such as coupling between stresses and other order parameters such as concentration. The role of 'structural memory' (the dependence of structural parameters such as the micellar length distribution on the flow history) in explaining these highly nonlinear phenomena is addressed. Structural memory also plays an intriguing role in the little-understood shear thickening regime, which occurs in a concentration regime close to but below the onset of strong entanglement, and which is marked by a shear-induced transformation from an inviscid to a gelatinous state
Absolute Conductivity Reconstruction in Magnetic Induction Tomography Using a Nonlinear Method
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Magnetic induction tomography (MIT) attempts to image the electrical and magnetic characteristics of a target using impedance measurement data from pairs of excitation and detection coils. This inverse eddy current problem is nonlinear and also severely ill posed so regularization is required for a stable solution. A regularized Gauss-Newton algorithm has been implemented as a nonlinear, iterative inverse solver. In this algorithm, one needs to solve the forward problem and recalculate the Jacobian matrix for each iteration. The forward problem has been solved using an edge based finite element method for magnetic vector potential A and electrical scalar potential V, a so called A, A-V formulation. A theoretical study of the general inverse eddy current problem and a derivation, paying special attention to the boundary conditions, of an adjoint field formula for the Jacobian is given. This efficient formula calculates the change in measured induced voltage due to a small perturbation of the conductivity in a region. This has the advantage that it involves only the inner product of the electric fields when two different coils are excited, and these are convenient computationally. This paper also shows that the sensitivity maps change significantly when the conductivity distribution changes, demonstrating the necessity for a nonlinear reconstruction algorithm. The performance of the inverse solver has been examined and results presented from simulated data with added nois
Variations on a theme of Timmesfeld: a finite group-theoretic analogue of the classification of groups of finite Morley rank and even type
No full textThe paper addresses a question whether there is a reasonable
self-contained theory of finite simple groups of even type which is closely parallel to the theory of groups and finite Morley rank
Isomorphism classes for Banach vector bundle structures of second tangents
No full textOn a smooth Banach manifold M, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle T^2M of M. This is a vector bundle in the presence of a linear connection on M and the corresponding local structure is heavily dependent on the choice of connection. In this paper we study the extent of this
dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on T^2M remains invariant under conjugate connections with respect to diffeomorphisms of M
Model A-EE : An Eulerian Two-Phase Gas-Solid Model
Get PDFThis paper investigates Model A (as discussed by Hudson &
Harris MIMS EPrint 2006.10) with the inclusion of a gas Energy equation. We present two formulations of the model and discretise them using the Lax-Wendroff, MacCormack and a high resolution scheme. Three different test cases are used to compare the results to determine the accuracy and robustness of the schemes. The results of Model A-EE are also compared to the isentropic model (Model A)
Nonlinear time series analysis of jerk congenital nystagmus
Get PDFNonlinear dynamics provides a complementary framework to control theory for the quantitative analysis of the oculomotor control system. This paper presents a number of findings relating to the aetiology and mechanics of the pathological ocular oscillation jerk congenital nystagmus (jerk CN). A range of time series analysis techniques were applied to both recorded jerk CN waveforms and simulated waveforms produced by an established model in which the oscillations are a consequence of an unstable neural integrator. The results of the time series analysis were then interpreted within the framework of a generalised model of the unforced oculomotor system.
This work suggests that for jerk oscillations, the origin of the instability lies in one of the five oculomotor subsystems, rather than in the final common pathway (the neu- ral integrator and muscle plant). Additionally, experimental estimates of the linearised foveation dynamics imply that a refixating fast phase induced by a near-homoclinic tra- jectory will result in periodic oscillations. Local dimension calculations show that the dimension of the experimental jerk CN data increases during the fast phase, indicating that the oscillations are not periodic, and hence that the refixation mechanism is of greater complexity than a homoclinic reinjection. The dimension increase is hypothe- sised to result either from a signal-dependent noise process in the saccadic system, or the activation of additional oculomotor components at the beginning of the fast phase. The modification of a recent saccadic system model to incorporate biologically realistic signal-dependent noise is suggested, in order to test the first of these hypotheses