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NLEVP: A Collection of Nonlinear Eigenvalue Problems. Users' Guide
Get PDFThis is the Users' Guide for NLEVP: a collection of nonlinear eigenvalue problems provided in the form of a MATLAB toolbox. A separate paper describes the collection and its organization
-Harmonic Cohomology and Equivariant Cohomology on Riemannian Manifolds With Boundary
Get PDFGiven a Riemannian manifold with boundary and a
torus which acts by isometries on and let be in the Lie
algebra of and corresponding vector field on , we
consider Witten's coboundary operator \d_{X_M} = \d+\iota_{X_M} on
invariant forms on . In \cite{Our paper} we introduce the
absolute -cohomology H^*_{X_M}(M)=
H^*(\Omega^{*}_G,\,\d_{X_M}) and the relative -cohomology
H^*_{X_M}(M,\,\partial M)= H^*(\Omega^{*}_{G,D},\,\d_{X_M}) where
the is for Dirichlet boundary condition and is
the invariant forms on M. Let be the adjoint of
and the resulting \emph{Witten-Hodge-Laplacian} is
\Delta_{X_M}= \d_{X_M}\delta_{X_M} + \delta_{X_M}\d_{X_M} where
the space is called the -harmonic forms. In
this paper, we prove that the (even/odd) -harmonic cohomology
which is the -cohomology of the subcomplex
(\ker\Delta_{X_M},\d_{X_M}) of the complex
(\Omega^{*}_G,\d_{X_M}) is enough to determine the total absolute
and relative -cohomology. As conclusion, we infer that the free
part of the absolute and relative equivariant cohomology groups are
determined by the (even/odd) -harmonic cohomology when the set
of zeros of the corresponding vector field is equal to the
fixed point set for the -action
Hermitian Quadratic Matrix Polynomials: Solvents and Inverse Problems
Get PDFA monic quadratic Hermitian matrix polynomial can be factorized into a product of two linear matrix polynomials, say
. For the inverse problem of finding a
quadratic matrix polynomial with prescribed spectral data (eigenvalues and
eigenvectors) it is natural to prescribe a right solvent and then determine
compatible left solvents . This problem is explored in the present paper. The
splitting of the spectrum between real eigenvalues and nonreal conjugate pairs
plays an important role. Special attention is paid to the case of real-symmetric
quadratic polynomials
The Social Entropy Process: Axiomatising the Aggregation of Probabilistic Beliefs
Get PDFThe present work stems from a desire to combine ideas arising from two historically different schemes of probabilistic reasoning, each having its own axiomatic traditions, into a single broader axiomatic framework, capable of providing general new insights into the nature of probabilistic inference in a multiagent context.
In the present sketch of our work we first describe briefly the background context, and we then present a set of natural principles to be satisfied by any general method of aggregating the partially defined probabilistic beliefs of several agents into a single probabilistic belief function. We will call such a general method of aggregation a social inference process. Finally we define a particular social inference process, the Social Entropy Process (abbreviated to SEP), which satisfies the principles formulated earlier. SEP has a natural justification in terms
of information theory, and is closely related to the maximum entropy inference process: indeed it can be regarded as a natural extension of that inference process to the multiagent context
On sheafification of modules
Get PDFWe consider the localisation of the category of presheaves of modules over a ringed space to the category of sheaves of modules. With a suitable finiteness condition (being "elementary") on the torsion theory, we deduce that over noetherian spaces the latter category is locally finitely presented.
This account is some of the background to our preprint "M. Prest and A. Ralph, Locally finitely presented categories of sheaves of modules" and to that of Philip Bridge, "Local presentability of categories of sheaves of modules"
Shape corrections for 3D EIT
Get PDFMovement of the boundary in biomedical Electrical Impedance Tomography (EIT) has been always a source of error in image reconstruction. In the case of pulmonary EIT, where the patient's chest shape changes during respiration, this is inevitable, so it is essential to be able to correct for shape changes and consequently avoid artifacts. Assuming that the conductivity is isotropic, an assumption that is reasonable for lung tissue but admittedly violated for muscle, the boundary shape up to a Mobius transformation (conformal mapping) as well as the conductivity can theoretically be determined by 3D EIT data. While in two dimensions the space of conformal mappings are infinite dimensional, in the three dimensional case the Mobius transformations are given by a finite number of parameters. In this paper, we concentrate on the three dimensional case and take a linear approximation. We will give results of numerical studies analogous to the two dimensional work of Boyle et al. on the effect of electrode movement and shape error in 3D EIT
Σ_K–constraints for Hybrid Systems
Get PDFIn this paper we introduce and study computational aspects
of Σ_K-constraints which are powerful enough to represent computable continuous data, but also simple enough to be an approach to approximate constraint solving for a large class of quantified continuous constraints.
We illustrate how Σ_K-constraints can be used for reasoning
about hybrid systems
Dynamics of poles with position-dependent strengths and its optical analogues
Get PDFDynamics of point vortices is generalized in two ways. Firstly by allowing complex strengths which allows for sources and sinks in combination with the the usual vorticity, and secondly by allowing the strengths to be functions of position. We describe several exact solutions with optical analogues, notably Snell's law and law of reflection off a mirror
Infinitely divisible cylindrical measures on Banach spaces
Get PDFIn this work infinitely divisible cylindrical probability measures on arbitrary Banach
spaces are introduced. The class of infinitely divisible cylindrical probability measures
is described in terms of their characteristics, a characterisation which is not known in
general for infinitely divisible Radon measures on Banach spaces. Furthermore, continuity
properties and the relation to infinitely divisible Radon measures of infinitely
divisible cylindrical probability measures are considered
Modelling infection spread using location tracking
Get PDFThe precision of location tracking technology has improved greatly over the last few decades. We aim to show that by tracking the locations of individuals in a closed environment, it is now possible to record the nature and frequency of interactions between them. Further, that it is possible to use such data to predict the way in which an infection will spread throughout such a population, given parameters such as transmission and recovery rates. We accordingly present a software package that is capable of recording and then replaying location data provided by a high-precision location tracking system. The software then employs a combination of SIR modelling and the epidemiological technique of contact tracing in order to predict the spread of an infection. We use this software to conduct a number of experiments using a sample data set, and compare the SIR graphs generated from these to similar graphs generated using the traditional SIR differential equations