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Model-theoretic imaginaries and coherent sheaves
Get PDFCategories of imaginaries (imaginary sorts are the objects and definable functions are the maps) defined using positive existential formulas are shown to be equivalent to categories of finitely presented / coherent functors on the category of models. Localised/relativised versions are also proved. This is linked with interpretation functors between categories of structures.
These results generalise what is already known in the additive case and include an alternative approach to an old result of Makkai and Reyes
Adaptive time-stepping for incompressible flow Part II: Navier-Stokes Equations
Get PDFWe outline a new class of robust and efficient methods for solving the Navier-Stokes equations. We describe a general solution strategy that has two basic building blocks; a fully implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system.
We present numerical experiments illustrating the potential of our approach
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
Definable Additive Categories (conference talk)
Get PDFThis is an account of a talk I gave, at a Conference on Modules and Representation Theory at Cluj in 2008, on definable additive
categories; I define these categories, say where they came from,
describe some of what is around them and then point out the
2-category which they form
Anomalous bubble propagation in elastic tubes
Get PDFAirway reopening is an important physiological event, as exemplified by the first breath of an infant
that inflates highly collapsed airways by driving a finger of air through its fluid-filled lungs. Whereas
fundamental models of airway reopening predict the steady propagation of only one type of bubble
with a characteristic rounded tip, our experiments reveal a surprising selection of novel bubbles with
counterintuitive shapes that reopen strongly collapsed, liquid-filled elastic tubes. Our multiple
bubbles are associated with a discontinuous relationship between bubble pressure and speed that sets
exciting challenges for modeler
Reconstruction algorithm for the polarization tomography problem with incomplete data.
Get PDFIn this paper we discuss some issues in polarization tomography. Specifically we describes a slice-by-slice reconstruction algorithm for the truncated transverse ray transform using data from rays normal to only a small number of directions. Specifically an unstable reconstruction procedure is given for three directions and a stable
reconstruction procedure for six directions. It is expected that these methods will prove useful in photoelastic tomography
First/second order transformation system. Tentative proof
Get PDFA tentative proof for the existence of a transformation from a first order system to a second order form
Symmetry reduction for stochastic hybrid systems
No full textThis paper is focused on adapting symmetry reduction, a technique that is highly successful in traditional model checking, to stochastic hybrid systems. We first show that performability analysis of stochastic hybrid systems can be reduced to a stochastic reachability analysis (SRA). Then, we generalize the notion of symmetry reduction as recently proposed for probabilistic model checking, to continuous probabilistic systems. We provide a rigorous mathematical foundation for the reduction technique in the continuous case and also investigate its observability perspective. For stochastic hybrid systems, characterizations for this reduction technique are provided, in terms of their infinitesimal generator
Reconstruction algorithm for the linearized polarization tomography problem with incomplete data.
Get PDFIn this paper we discuss some issues in polarization tomography. Specifically we describes a slice-by-slice reconstruction algorithm for the truncated transverse ray transform using data from rays normal to only a small number of directions. Specifically an unstable reconstruction procedure is given for three directions and a stable
reconstruction procedure for six directions. It is expected that these methods will prove useful in photoelastic tomography
Metric matroids and metric buildings
Get PDFThese notes represent fragments of an old joint work with I. M. Gelfand; I put in them some material which did not find its way in our book with Neil White "Coxeter Matroids", but, I believe, deserves to be documented.
In particular, I use the old term "W-matroid" for what later became a "Coxeter matroid for Coxeter group W".
We discuss realizations of W-matroids by retractions of buildings and their convexity properties. We define, in terms of metric properties of retractions, some class of geometric objects which includes,as partial cases, buildings and R-trees. For some of these new geometric structures `abstract' convexity properties of retractions have a natural interpretation in terms of usual Euclidean convexity