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Non-axiomatizability of real spectra in L<sub>∞λ</sub>
Get PDFWe show that the property of a spectral space, to be a spectral subspace
of the real spectrum of a commutative ring, is not expressible in the
infinitary first order language L∞λ of its defining lattice.
This generalises a result of Delzell and Madden which says that not every
completely normal spectral space is a real spectru
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators
No full textA new algorithm is developed for computing , where is an matrix and is with . The algorithm works for any , its computational cost is dominated by the formation of products of with matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix or a sequence on an equally spaced grid of points . It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the recent analysis of Al-Mohy and Higham [\emph{SIAM J. Matrix Anal.\ Appl.} 31 (2009), pp.\ 970--989], which provides sharp truncation error bounds expressed in terms of the quantities for a few values of , where the norms are estimated using a matrix norm estimator. Shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm. Numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems, and analysis of rounding errors and of the conditioning of the problem provides theoretical support. Experimental comparisons with MATLAB codes based on Krylov subspace, Chebyshev polynomial, and Laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form that arise in exponential integrators, where the are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension built by augmenting with additional rows and columns, and the algorithm of this paper can therefore be employed
iGen: A program for the automated generation of models and parameterisations
No full textComplex physical systems can often be simulated using very high-resolution models but this is not always practical because of computational restrictions. In this case the model must be simplified or parameterised, but this is a notoriously difficult process that often requires the introduction of `model assumptions' that are hard or impossible to justify. Here we introduce a new approach to parameterising models. The approach makes use of a newly developed computer program, which we call iGen, that analyses the source code of a high-resolution model and formally derives a much faster parameterised model that closely approximates the original, reporting bounds on the error introduced by any approximations. These error bounds can be used to formally justify use of the parameterised model in subsequent numerical experiments. Using increasingly complex physical systems as examples we illustrate that iGen has the ability to produce parameterisations that run typically orders of magnitude faster than the underlying, high-resolution models from which they are derived and show that iGen has the potential to become an important tool in model development
Attractors near grazing-sliding bifurcations
No full textIn this paper we prove, for the first time, that multistability can occur in 3-dimensional
Fillipov type flows due to grazing-sliding bifurcations. We do this by reducing the study of the
dynamics of Filippov type flows around a grazing-sliding bifurcation to the study of appropri-
ately defined one-dimensional maps. In particular, we prove the presence of three qualitatively
different types of multiple attractors born in grazing-sliding bifurcations. Namely, a period-two
orbit with a sliding segment may coexsist with a chaotic attractor, two stable, period-two and
period-three orbits with a segment of sliding each may coexist, or a non-sliding and period-three
orbit with two sliding segments may coexist
Commuting Involution Graphs of Certain Finite Simple Classical Groups
No full textFor a group G and X a subset of G, the commuting graph of G on X, denoted by C(G,X), is the graph whose vertex set is X with x, y in X joined by an edge if x is not equal to y and x and y commute. If the elements in X are involutions, then C(G,X) is called a commuting involution graph. This thesis studies C(G,X) when G is either a 4-dimensional projective symplectic group; a 3-dimensional unitary group; 4-dimensional unitary group over a field of characteristic 2; a 2-dimensional projective general linear group; or a 4-dimensional affine orthogonal group, and X a G-conjugacy class of involutions. We determine the diameters and structure of the discs of these graphs
Stochastic production trees as products of i.i.d. componentwise exponential max-plus matrices
Get PDFWe introduce a class of stochastic production tree model, based on Petri nets, which admit a random matrix product description in the Max-plus algebra. With a kind of combinatorial change of variables we are able to simplify the form of the matrices arising from these models. For this class of \emph{Componentwise exponential} matrix we prove a new result relating the (Max-plus) spectrum of the product to the principal (classical) eigenvalue of an associated adjacency matrix by means of a sandwich inequality. This theorem highlights several important theoretical factors in the dynamics of Max-plus linear systems generally and gives us some neat insight into the different production tree models
Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices
Get PDFWe derive new perturbation bounds for eigenvalues of Hermitian matrices with block tridiagonal structure.
The main message of this paper is that an eigenvalue is insensitive to blockwise perturbation, if it is well-separated from the spectrum of the diagonal blocks nearby the perturbed blocks. Our bound is particularly effective when the matrix is block-diagonally dominant and graded. Our approach is to obtain eigenvalue bounds via bounding eigenvector components, which is based on the observation that an eigenvalue is insensitive to componentwise perturbation if the corresponding eigenvector components are small. We use the same idea to explain two well-known phenomena, one concerning aggressive early deflation used in the symmetric tridiagonal QR algorithm and the other concerning the extremal eigenvalues of Wilkinson matrices
Shape Deformation in Two-Dimensional Electrical Impedance Tomography
No full textElectrical Impedance Tomography (EIT) uses mea- surements from surface electrodes to reconstruct an image of the conductivity of the contained medium. However, changes in measurements result from both changes in internal conductivity and changes in the shape of the medium relative to the electrode positions. Failure to account for shape changes results in a conductivity image with significant artifacts. Previous work to address shape changes in EIT has shown that in some cases boundary shape and electrode location can be uniquely deter- mined for isotropic conductivities; however, for geometrically conformal changes, this is not possible. This prior work has shown that the shape change problem can be partially addressed. In this paper, we explore the limits of compensation for boundary movement in EIT, using three approaches: first, a theoretical model was developed to separate a deformation vector field into conformal and non-conformal components, from which the reconstruction limits may be determined; next, finite element models were used to simulate EIT measurements from a domain whose boundary has been deformed; finally, an experimental phantom was constructed from which boundary deformation measurements were acquired. Results, both in simulation and with experimental data, suggest that some electrode movement and boundary distortions can be reconstructed based on conduc- tivity changes alone while reducing image artifacts in the process
Kinetic modelling of metabolic pathways
No full textWe show how to build a kinetic model of a metabolic pathway. We provide the example of building a model of glycerol synthesis using Copasi. However, the techniques required remain the same for any pathway, using any software
Algebraic Topology of PDEs
Get PDFWe consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of
G and corresponding vector field X_M on M, one defines Witten’s inhomogeneous coboundary operator (even/odd invariant forms on M) and its adjoint .
First,Witten [35] showed that the resulting cohomology classes have X_M-harmonic representatives
(forms in the null space of ), and the cohomology groups
are isomorphic to the ordinary de Rham cohomology groups of the set N(X_M) of zeros of X_M. The first principal purpose is to extend Witten’s results to manifolds with boundary.
In particular, we define relative (to the boundary) and absolute versions of the X_M-cohomology and show the classes have representative X_M-harmonic fields with appropriate boundary conditions. To do this we present the relevant version of the Hodge-Morrey-Friedrichs decomposition theorem for invariant forms in terms of the operators d_{X_M} and \deta_{X_M}; the proof involves showing that certain boundary value problems are elliptic. We also elucidate the connection between the X_M-cohomology groups and the relative and absolute equivariant cohomology, following work of Atiyah and Bott. This connection is then exploited to show that every harmonic field with appropriate boundary conditions on N(X_M)
has a unique corresponding an X_M-harmonic field on M to it, with corresponding boundary conditions. Finally, we define the interior and boundary portion of X_M-cohomology
and then we define the X_M-Poincar´e duality angles between the interior subspaces of X_M-harmonic fields on M with appropriate boundary conditions.
Second, in 2008, Belishev and Sharafutdinov [9] showed that the Dirichlet-to-Neumann (DN) operator \Lambda inscribes into the list of objects of algebraic topology by proving that the de Rham cohomology groups are determined by \Lambda.
In the second part of this thesis, we investigate to what extent is the equivariant topology of a manifold determined by a variant of the DN map?. Based on the results in the first part above, we define an operator \Lambda_{X_M} on invariant forms on the boundary ¶M which we call
the X_M-DN map and using this we recover the long exact X_M-cohomology sequence of the topological pair (M;\partial M) from an isomorphism with the long exact sequence formed from the generalized boundary data. Consequently, This shows that for a Zariski-open subset of the Lie algebra, \Lambda_{X_M} determines the free part of the relative and absolute equivariant cohomology groups of M. In addition, we partially determine the mixed cup product of X_M-cohomology groups from \Lambda_{X_M}. This shows that \Lambda_{X_M} encodes more information about the equivariant algebraic topology of M than does the operator \Lambda on the boundary. Finally, we elucidate
the connection between Belishev-Sharafutdinov’s boundary data on N(X_M) and ours on \partial M.
Third, based on the first part above, we present the (even/odd) X_M-harmonic cohomology which is the cohomology of certain subcomplex of the complex (\Omega_G^*, d_{X_M}) and we prove that it is isomorphic to the total absolute and relative X_M-cohomology groups