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Ageing and rheology in soft materials
No full textWe study theoretically the role of aging in the rheology of soft materials. We define several generalized rheological response functions suited to aging samples (in which time translation invariance is lost). These are then used to study aging effects within a simple scalar model (the "soft glassy rheology" or SGR model) whose constitutive equations relate shear stress to shear strain among a set of elastic elements, with distributed yield thresholds, undergoing activated dynamics governed by a "noise temperature," x. (Between yields, each element follows affinely the applied shear.) For 1 < x < 2 there is a power-law fluid regime in which transients occur, but no aging. For x < 1, the model has a macroscopic yield stress. So long as this yield stress is not exceeded, aging occurs, with a sample's apparent relaxation time being of order its own age. The (age-dependent) linear viscoelastic loss modulus G[double-prime](omega,t) rises as frequency is lowered, but falls with age t, so as to always remain less than G[prime](omega,t) (which is nearly constant). Significant aging is also predicted for the stress overshoot in nonlinear shear startup and for the creep compliance. Though obviously oversimplified, the SGR model may provide a valuable paradigm for the experimental and theoretical study of rheological aging phenomena in soft solids. ©2000 Society of Rheology
Numerical analysis of a quadratic matrix equation
No full textThe quadratic matrix equation AX2+ BX + C = 0in n x nmatrices arises in applications and is of intrinsic interest as one of the simplest nonlinear matrix equations. We give a complete characterization of solutions in terms of the generalized Schur decomposition and describe and compare various numerical solution techniques. In particular, we give a thorough treatment of functional iteration methods based on Bernoulli’s method. Other methods considered include Newton’s method with exact line searches, symbolic solution and continued fractions. We show that functional iteration applied to the quadratic matrix equation can provide an efficient way to solve the associated quadratic eigenvalue problem ({lambda}2A + {lambda}B + C)x = 0
Monodromy of certain Painlevé-VI transcendents and reflection groups
Get PDFWe study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters #=%=0, '=]] and 2!=(27-1)2 with arbitrary 7, 27p]Ê. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space
Numerically stable generation of correlation matrices and their factors
No full textCorrelation matrices—symmetric positive semidefinite matrices with unit diagonal—are important in statistics and in numerical linear algebra. For simulation and testing it is desirable to be able to generate random correlation matrices with specified eigenvalues (which must be nonnegative and sum to the dimension of the matrix). A popular algorithm of Bendel and Mickey takes a matrix having the specified eigenvalues and uses a finite sequence of Givens rotations to introduce 1s on the diagonal. We give improved formulae for computing the rotations and prove that the resulting algorithm is numerically stable. We show by example that the formulae originally proposed, which are used in certain existing Fortran implementations, can lead to serious instability. We also show how to modify the algorithm to generate a rectangular matrix with columns of unit 2-norm. Such a matrix represents a correlation matrix in factored form, which can be preferable to representing the matrix itself, for example when the correlation matrix is nearly singular to working precision
Modelling of congenital nystagmus waveforms produced by saccadic system abnormalities
Get PDFModels of the mechanisms of normal eye movements are typically
described in terms of the block diagrams which are used in
control theory. An alternative approach to understanding the
mechanisms of normal eye movements involves describing the eye
movement behaviour in terms of smooth changes in state variables.
The latter approach captures the burst cell firing against motor error
(difference between target gaze angle and current gaze angle)
phase plane behaviour which is found experimentally and
facilitates the modelling of variations in burst cell behaviour.
A novel explanation of several types of congenital nystagmus
waveforms is given in terms of a saccadic termination abnormality
Timing the Ancestor of the HIV-1 Pandemic Strains
No full textHIV-1 sequences were analyzed to estimate the timing of the ancestral sequence of the main group of HIV-1, the strains responsible for the AIDS pandemic. Using parallel supercomputers and assuming a constant rate of evolution, we applied maximum-likelihood phylogenetic methods to unprecedented amounts of data for this calculation. We validated our approach by correctly estimating the timing of two historically documented points. Using a comprehensive full-length envelope sequence alignment, we estimated the date of the last common ancestor of the main group of HIV-1 to be 1931 (1915-41). Analysis of a gag gene alignment, subregions of envelope including additional sequences, and a method that relaxed the assumption of a strict molecular clock also supported these results
Non-axisymmetric rotating-disk flows: nonlinear travelling-wave states
No full textWe consider the classical problem of the laminar flow of an incompressible rotating
fluid above a rotating, impermeable, infinite disk. There is a well-known class of
solutions to this configuration in the form of an exact axisymmetric solution to the
Navier�Stokes equations. However, the radial self-similarity that leads to the �rotating-
disk equations� can also be used to obtain solutions that are non-axisymmetric
in nature, although (in general) this requires a boundary-layer approximation. In
this manner, we locate several new solution branches, which are non-axisymmetric
travelling-wave states that satisfy axisymmetric boundary conditions at infinity and at
the disk. These states are shown to appear as symmetry-breaking bifurcations of the
well-known axisymmetric solution branches of the rotating-disk equations. Numerical
results are presented, which suggest that an infinity of such travelling states exist in
some parameter regimes. The numerical results are also presented in a manner that
allows their application to the analogous flow in a conical geometry.
Two of the many states described are of particular interest. The first is an exact,
nonlinear, non-axisymmetric, stationary state for a rotating disk in a counter-rotating
fluid; this solution was first presented by Hewitt, Duck & Foster (1999) and here we
provide further details. The second state corresponds to a new boundary-layer-type
approximation to the Navier�Stokes equations in the form of azimuthally propagating
waves in a rotating fluid above a stationary disk. This second state is a new non-
axisymmetric alternative to the classical axisymmetric Bodewadt solution
QR factorization with complete pivoting and accurate computation of the SVD
No full textA new algorithm of Demmel et al. for computing the singular value decomposition (SVD) to high relative accuracy begins by computing a rank-revealing decomposition (RRD). Demmel et al. analyse the use of Gaussian elimination with complete pivoting (GECP) for computing the RRD. We investigate the use of QR factorization with complete pivoting (that is, column pivoting together with row sorting or row pivoting) as an alternative to GECP, since this leads to a faster SVD algorithm. We derive a new componentwise backward error result for Householder QR factorization and combine it with the theory of Demmel et al. to show that high relative accuracy in the computed SVD can be expected for matrices that are diagonal scalings of a well-conditioned matrix. An a posteriori error bound is derived that gives useful estimates of the relative accuracy of the computed singular values. Numerical experiments confirm the theoretical predictions
Relative equilibria and conserved quantities in symmetric Hamiltonian systems
No full textThese are lectures given at a summer school in 1998. They begin with recalling the basic structure of Hamiltonian systems, and then proceed to discuss the effects of symmetry and the geometry of conservation laws
Steady boundary-layer solutions for a swirling stratified fluid in a rotating cone
No full textWe consider a set of nonlinear boundary-layer equations that have been derived by
Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified
fluid in a conical container. In contrast to the unsteady analysis of DFH, we re-
strict attention to steady solutions and extend the previous discussion further by
allowing the container to both co-rotate and counter-rotate relative to the contained
swirling fluid. The system is governed by three parameters, which are essentially non-
dimensional measures of the rotation, stratification and a Schmidt number. Some of
the properties of this system are related (in some cases rather subtly) to those found
in the swirling flow of a homogeneous fluid above an infinite rotating disk; however,
the introduction of buoyancy effects with a sloping boundary leads to other (new)
behaviours. A general description of the steady solutions to this system proves to be
rather complicated and shows many interesting features, including non-uniqueness,
singular solutions and bifurcation phenomena.
We present a broad description of the steady states with particular emphasis on
boundaries in parameter space beyond which steady states cannot be continued.
A natural extension of this work (motivated by recent experimental results) is to
investigate the possibility of solution branches corresponding to non-axisymmetric
boundary-layer states appearing as bifurcations of the axisymmetric solutions. In
an Appendix we give details of an exact, non-axisymmetric solution to the Navier�
Stokes equations (with axisymmetric boundary conditions) corresponding to the flow
of homogeneous fluid above a rotating disk