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Magnetohydrodynamic damping of convective flows in molten gallium
Get PDFWe report the results of an experimental study of magnetohydrodynamic damping of sidewall convection in a rectangular enclosure filled with gallium. In particular we investigate the suppression of convection when a steady magnetic field is applied separately in each of the three principal directions of the flow. The strongest damping of the steady flow is found for a vertical magnetic field, which is in agreement with theory. However, we observe that the application of a field transverse to the flow provides greater damping than a longitudinal one, which seems to contradict available theory. We provide a possible resolution of this apparent dichotomy in terms of the length scale of the experiment
Weak convergence of a numerical method for a stochastic heat equations
No full textWeak convergence with respect to a space of twice continuously differentiable test functions is established for a discretisation of a heat equation with homogeneous Dirichlet boundary conditions in one dimension, forced by a space-time Brownian motion. The discretisation is based on finite differences in space and time, incorporating a spectral approximation in space to the Brownian motion
Posets of matrices, and permutations with forbidden subsequences
Get PDFAbstract. The enumeration of permutations with specific forbidden subsequences has applications in areas ranging from algebraic geometry to the study of sorting algorithms. We consider a ranked poset of permutation matrices whose global structure incorporates the solution to the equivalent problem of enumerating permutations which contain a required subsequence. We describe this structure completely for saturated chains of lengths one and two, so settling several new and general instances of the original problem, and conclude with a superficial asymptotic investigation of arbitrary chains whose length is small by comparison with the rank of its constituent permutations. The value of this approach is reflected in the appearance of closed polynomial formulae (related to the Robinson-Schensted correspondence) and of a framework for the systematic analysis of associated combinatorial questions; indeed, we begin by studying a simpler poset of 0-1 sequences as the natural environment in which to introduce our insertion and deletion operators
Simultaneous tridiagonalization of two symmetric matrices
No full textWe show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the non-singularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder reflectors. Two algorithms are proposed, one using non-orthogonal rank-one modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and non-orthogonal rank-one modifications of the identity matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n3) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments
The emergence of reasons conjecture
No full textIn this paper we show how the maxent paradigm may be
used to produce an inductive method (in the sense of Carnap)
applicable to a wide class of problems in inductive logic. We then
state a conjecture about the answers given by this method and
prove the conjecture in a particular case which extends all
previous partial confirmations
Quasianalytic Denjoy-Carleman classes and o-minimality
No full textWe show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure conditions. Some of these structures do not admit analytic cell decomposition, and they show that there is no largest o-minimal expansion of the real field
Perturbation theory for homogeneous polynomial eigenvalue problems
No full textWe consider polynomial eigenvalue problems P(A,alpha,beta)x=0 in which the matrix polynomial is homogeneous in the eigenvalue (alpha,beta)&unknown;C2. In this framework infinite eigenvalues are on the same footing as finite eigenvalues. We view the problem in projective spaces to avoid normalization of the eigenpairs. We show that a polynomial eigenvalue problem is well-posed when its eigenvalues are simple. We define the condition numbers of a simple eigenvalue (alpha,beta) and a corresponding eigenvector x and show that the distance to the nearest ill-posed problem is equal to the reciprocal of the condition number of the eigenvector x. We describe a bihomogeneous Newton method for the solution of the homogeneous polynomial eigenvalue problem (homogeneous PEP
Model theory for algebra
Get PDFThe purpose of this article is to give a general introduction to the basic ideas and techniques from model theory. I begin with some general remarks concerning model theory and its relationship with algebra. There follows a “mini-course” on first order languages, structures and basic ideas in model theory. Then there is a series of subsections which describe briefly some topics from model theory
Kinetics of the shear banding instability in startup flows
No full textMotivated by recent light scattering experiments on semidilute wormlike micelles, we study the early stages of the shear banding instability using the nonlocal Johnson-Segalman model with a "two-fluid" coupling of flow to micellar concentration. We perform a linear stability analysis for coupled fluctuations in shear rate gamma, micellar strain (W) double under bar, and concentration phi about an initially homogeneous state. This resembles the Cahn-Hilliard (CH) analysis of fluid-fluid demixing (although we discuss important differences). First, assuming the initial state to lie on the intrinsic constitutive curve, we calculate the "spinodal" onset of instability in sweeps along this curve. We then consider start-up "quenches" into the unstable region. Here the instability in general occurs before the intrinsic constitutive curve can be attained, so we analyze the fluctuations with respect to the time-dependent start-up flow. We calculate the selected length and time scales at which inhomogeneity first emerges. When the coupling between flow and concentration is switched off, fluctuations in the "mechanical variables" gamma and (W) double under bar are independent of those in phi, and are unstable when the intrinsic constitutive curve has negative slope; but no length scale is selected. Coupling to the concentration enhances this instability at short length scales, thereby selecting a length scale, consistent with the recent light scattering experiments. The spinodal region is then broadened by an extent that increases with proximity to an underlying (zero-shear) CH fluid-fluid (phi) demixing instability. Far from demixing, the broadening is slight and the instability is still mechanically dominated (by delta(gamma) over dot and delta(W) double under bar) with only small deltaphi. Close to demixing, instability sets in at a very low shear rate, where it is dominated instead by deltaphi. In this way, the model captures a smooth crossover from shear banding instabilities that are perturbed by concentration coupling to demixing instabilities that are induced by shear