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A Lemon is not a Monstar: visualization of singularities of symmetric second rank tensor fields in the plane.
Get PDFIn the visualization of the topology of second rank symmetric tensor fields in the plane one can extract some key
points (degenerate points), and curves (separatrices) that characterize the qualitative behaviour of the whole
tensor field. This can provide a global structure of the whole tensor field, and effectively reduce the complexity of
the original data. To construct this global structure it is important to classify those degenerate points accurately.
However, in existing visualization techniques, a degenerate point is only classified into two types: trisector and
wedge types. In this work, we will apply the theory from the analysis of binary differential equations and demonstrate that, topologically, a simple degenerate point should be classified into three types: star (trisector), lemon
and monstar. The later two types were mistakenly regarded as a single type in the existing visualization techniques
Reconstruction of Grounded Objects from Cauchy Data on a Plane
Get PDFThis thesis explores the use of capacitance measurements made between electrodes embedded in or around a display surface, to detect the position, orientation and shape
of hands and fingers. This is of interest for unobtrusive 3D gesture input for interactive displays, so called touch-less interaction. The hand is assumed to be grounded.
The forward problem is solved using Green’s theorem and an appropriate Green’s function. This leads to an operator factorisation for the forward Dirichlet to Neumann
map Lambda_D : L^2(boundary H) -> L^2(boundary H). The foward map is demonstrated to be compact, injective and depends uniquely on the object. An alternative factorisation based on double layer potentials and involving a Fredholm equation of the second kind is also presented. These operator expressions are used in numerical calculations in two and three space dimensions using the Boundary Element Method for discretization.
Four methods are presented for the solution of the inverse problem of recovering the object from a measured forward map. The first uses modified Gauss-Newton optimization.
The method is successful if the degrees of freedom are limited to object position, size and orientation, but is unpractical for shape reconstruction. The second method recovers the zero potential contour of a solution to Laplace’s equation from Cauchy data on part of the boundary of a domain. An algorithm is used where at each iteration there is an approximation boundary D_k to boundary D on which approximate Cauchy data are calculated by solving a Tikhonov regularised linear system. This
data is used to modify boundary D_k by extrapolation towards the zero-surface giving the next
approximation boundary D_{k+1}
In the third method the problem is solved with the so-called Factorisation Method.
A test function g_z is used to characterise points z in D iff g_z in Range(Lambda^{1/2}_D). Implicit
regularisation due to the finite aperture of the measurement electrode results in a level
set P(z) that is finite and differentiable everywhere. The level representing the object
boundary D is found through minimization of the cost function.
The fourth method uses a monotonicity property of the forward map to test if
a probe object is contained within the unknown object. For an infinitesimal probe
object and finite aperture measurements the method is shown to be identical to the
factorisation method.
The thesis closes with conclusions on the relative merits of these methods
Definable additive categories: purity and model theory
Get PDFDefinable additive categories and their model theory are the topic of this paper. We begin with background and preliminary results on additive categories. Then definable subcategories, their properties and the morphisms between them are investigated, as are certain associated topological spaces (``spectra"). It was in the model theory of modules that these categories were first considered and model theory provides some of the tools for exploring them. Some general model-theoretic background is presented, then various aspects of the model theory of definable categories are considered
Bounded super real closed rings
Get PDFThis note is a complement to the paper "M. Tressl, Super real closed rings", where super real closed rings are introduced and studied. A bounded super real closed ring A is a commutative unital ring A together with an operation FA:An -> A for every bounded continuous map F:ℝn -> ℝ, so that all term equalities between the F's remain valid for the FA's. We show that bounded super real closed rings are precisely the convex subrings of super real closed rings: for every bounded super real closed ring there is a largest super real closed subring B contained in A and a smallest super real closed ring C containing A. Moreover B is convex in C. The assignment A↦C is an idempotent mono-reflector from bounded to arbitrary super real closed rings which allows to transfer many of the algebraic results from the unbounded to the bounded situatio
Directional effect of a magnetic field on oscillatory low-Prandtl-number convection
Get PDFThe directional effect of a magnetic field on the onset of oscillatory convection is studied
numerically in a confined three-dimensional cavity of relative dimensions 4:2:1
length:width:height filled with mercury and subject to a horizontal temperature gradient. The
magnetic field suppresses the oscillations most effectively when it is applied in the vertical direction,
and is the least efficient when applied in the longitudinal direction parallel to the temperature
gradient. In all cases, however, exponential growths of the critical Grashof number, Grc Gr, ratio
of buoyancy to viscous dissipation forces with the Hartmann number Ha, ratio of magnetic to
viscous dissipation forces are obtained. Insight into the damping mechanism is gained from the
fluctuating kinetic energy budget associated with the time-periodic disturbances at threshold. The
kinetic energy produced by the vertical shear of the longitudinal basic flow dominates the oscillatory
transition, and when a magnetic field is applied, it increases in order to balance the stabilizing
magnetic energy. Moreover, subtle changes in the spatial distribution of this shear energy are at the
origin of the exponential growth of Grc. The destabilizing effect of the velocity fluctuations strongly
decreases when Ha is increased due to the decay of the velocity fluctuations in the bulk
accompanied by the appearance of steep gradients localized in the Hartmann layers, so that an
increase of the shear of the basic flow at Grc is required in order to sustain the instability. This yields
an increase in Grc, which is reinforced by the fact that the shear of the basic flow naturally decreases
at constant Gr with the increase of Ha, particularly when the magnetic field is applied in the vertical
direction. For transverse and longitudinal fields, the decay of the velocity fluctuations is combined
with an increase of the shear energy term due to a sustained growth in stabilizing magnetic energy
with H
Nonlinear vortex development in rotating flows
Get PDFWe present the results of a combined experimental and numerical investigation into steady secondary vortex flows confined between two concentric right circular cylinders. When the flow is driven by the symmetric rotation of both end walls and the inner cylinder, toroidal vortex structures arise through the creation of stagnation points (in the meridional plane) at the inner bounding cylinder or on the mid-plane of symmetry. A detailed description of the flow regimes is presented, suggesting that a cascade of such vortices can be created. Experimental results are reported, which visualize some of the new states and confirm the prediction that they are stable to (mid-plane) symmetry-breaking perturbations.
We also present some brief results for the flows driven by the rotation of a single end wall. Vortex structures may also be observed at low Reynolds numbers in this geometry. We show that standard flow visualization methods lead to some interesting non-axisymmetric particle paths in this case
A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices
Get PDFQuantum counterparts of certain simple classical systems can exhibit chaotic behaviour
through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices.
We use known bounds on the distribution
function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have an important uniqueness property, can yield an approximation to the GOE distribution. That has
the advantage that then both chaotic and non chaotic cases fit in the
information geometric framework of the manifold of gamma distributions, which has been
the subject of recent work on neighbourhoods of randomness for general stochastic systems.
Additionally, gamma
distributions give approximations, to eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite random hermitian matrices with real quaternionic elements, except near the origin. Gamma distributions do not
precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes
Symplectic, BVD, and Palindromic Approaches to Discrete-Time Control Problems
Get PDFWe give several different formulations for the discrete-time linear-quadratic control
problem in terms of structured eigenvalue problems, and discuss the relationships among
the associated structured objects: symplectic matrices and pencils, BVD-pencils and
polynomials, and the recently introduced classes of palindromic pencils and matrix polynomials.
We show how these structured objects can be transformed into each other, and
also how their eigenvalues, eigenvectors and invariant/deflating subspaces are related
Phylogenetic Trees Predicted by an Irreversible Markov Process
Get PDFA new Markovian method for the prediction of phylogenetic trees has been developed earlier by the authors. Here, the method is illustrated and applied using mitochondrial DNA data for vertebrate species (and technicalities of the
theory are avoided). Discussions include the sensitivity of the results to DNA
alignment techniques and the inclusion of polytomy in tree structures. Several
comparisons are made with tree structures in the literature which have been
predicted using statistical techniques
Information Geometry: Near Randomness and Near Independence
Get PDFThe main motivation for this book lies in the breadth of applications
in which a statistical model is used to represent small departures
from, for example, a Poisson process. Our approach uses information geometry to provide
a common context but we need only rather elementary material from differential geometry, information theory and mathematical statistics. Introductory
sections serve together to help those
interested from the applications side in making use of our methods
and results.
Reported in this monograph is a body of results, and
computer-algebraic methods that seem to have quite general
applicability to statistical models admitting representation
through parametric families of probability density functions. Some
illustrations are given from a variety of contexts for geometric
characterization of statistical states near to the three important standard
basic reference states: (Poisson) randomness, uniformity, independence. The
individual applications are somewhat heuristic
models from various fields and we incline more to terminology and notation
from the applications rather than from formal statistics. However, a
common thread is a geometrical representation for statistical perturbations of the
basic standard states, and hence results gain qualitative stability.
Moreover, the geometry is controlled by a metric structure that
owes its heritage through maximum likelihood to information theory
so the quantitative features---lengths of curves, geodesics,
scalar curvatures etc.---have some respectable authority. We
see in the applications simple models for galactic
void distributions and galaxy clustering, amino acid clustering
along protein chains, cryptographic protection, stochastic fibre
networks, coupled geometric features in hydrology and quantum chaotic behaviour