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Characterising the shape and material properties of hidden targets from magnetic induction data
Get PDFThe aim of this paper is to show that, for the eddy current model, the leading order term for the perturbation in the magnetic field, due to the presence of a small conducting magnetic inclusion, can be expressed in terms of a symmetric rank 2 polarization tensor. This tensor contains information about the shape and material properties of the object and is independent of position. We apply a recently derived asymptotic formula for the perturbed magnetic field, due to the presence of a conducting inclusion, which is expressed in terms of a new class of rank 4 polarization tensors (H. Ammari, J. Chen, Z. Chen, J. Garnier and D. Volkov (2014) Target detection and characterization from electromagnetic induction data, Journal de Mathématiques Pures et Appliquées 101, 54-75.) and show that their result can be written in an alternative form involving a symmetric rank 2 tensor involving 6 instead of 81 complex components in an orthonormal coordinate frame. For objects with rotational and mirror symmetries we show that the number of coefficients is still smaller. We include numerical examples to demonstrate that the new polarization tensors can be accurately computed by solving a vector valued transmission problem by hp�finite elements and include examples to illustrate the agreement between the asymptotic formula describing the perturbed fields and the numerical predictions
A sufficient condition for the existence of Hamiltonian bifurcations with continuous isotropy
Get PDFWe present a framework for the study of the local qualitative dynamics of equivariant Hamiltonian flows specially designed for points in phase space with nontrivial isotropy. This is based on the classical construction of structure-preserving tubular neighborhoods for Hamiltonian Lie group actions on symplectic manifolds. This framework is applied to obtaining concrete and testable conditions guaranteeing the existence of bifurcations from symmetric branches of Hamiltonian relative equilibria
Spectral equivalence of matrix polynomials and the index sum theorem
Get PDFThe concept of linearization is fundamental for theory, applications, and spectral computations related to matrix polynomials.
However, recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations
to develop structure-preserving numerical algorithms
that compute the eigenvalues of structured matrix polynomials can be too restrictive,
because some structured polynomials do not have any linearization with the same structure.
This phenomenon strongly suggests that linearizations
should sometimes be replaced by other low degree matrix polynomials in applied numerical computations.
Motivated by this fact, we introduce equivalence relations
that allow the possibility of matrix polynomials
(with coefficients in an arbitrary field) to be equivalent,
with the same spectral structure, but have different sizes and degrees.
These equivalence relations are directly modeled
on the notion of linearization,
and consequently inherit the simplicity, applicability,
and most relevant properties of linearizations;
simultaneously, though, they are much more flexible
in the possible degrees of equivalent polynomials.
This flexibility allows us to define in a unified way
the notions of quadratification and -ification,
to introduce the concept of companion form of arbitrary degree, and to provide concrete and simple examples of these notions that generalize in a natural and smooth way
the classical first and second Frobenius companion forms.
The properties of -ifications are studied in depth;
in this process a fundamental result on matrix polynomials,
the ``Index Sum Theorem'', is recovered and extended to arbitrary fields.
Although this result is known in the systems theory literature for real matrix polynomials,
it has remained unnoticed by many researchers.
It establishes that the sum of the (finite and infinite) partial multiplicities,
together with the (left and right) minimal indices
of any matrix polynomial
is equal to the rank times the degree of the polynomial.
The ``Index Sum Theorem'' turns out to be a key tool
for obtaining a number of significant results:
on the possible sizes and degrees of -ifications and companion forms,
on the minimal index preservation properties of companion forms of arbitrary degree,
as well as on obstructions to the existence of structured companion forms
for structured matrix polynomials of even degree.
This paper presents many new results,
blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields.
Therefore we have written the paper in an expository
and self-contained style that makes it accessible to a wide variety of readers
An Improved Schur--Pade Algorithm for Fractional Powers of a Matrix and their Frechet Derivatives
Get PDFThe Schur--Pad���© algorithm [N. J. Higham and L. Lin, A Schur--Pad���© algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. Appl., 32(3):1056--1078, 2011] computes arbitrary real powers of a matrix using the building blocks of Schur decomposition, matrix square roots, and Pad���© approximants. We improve the algorithm by basing the underlying error analysis on the quantities , for several small , instead of . We extend the algorithm so that it computes along with one or more Fr���©chet derivatives, with reuse of information when more than one Fr���©chet derivative is required, as is the case in condition number estimation. We also derive a version of the extended algorithm that works entirely in real arithmetic when the data is real. Our numerical experiments show the new algorithms to be superior in accuracy to, and often faster than, the original Schur--Pad���© algorithm for computing matrix powers and more accurate than several alternative methods for computing the Fr���©chet derivative. They also show that reliable estimates of the condition number of are obtained by combining the algorithms with a matrix norm estimator
Growth-induced blisters in a circular tube
No full textThe growth of an elastic film adhered to a confining substrate might lead to the formation of delimitation blisters. Many results have been derived when the substrate is flat. The equilibrium shapes, beyond small deformations, are determined by the interplay between the sheet elastic energy and the adhesive potential due to capillarity. Here, we study a non-trivial generalization to this problem and consider the adhesion of a growing elastic loop to a confining circular substrate. The fundamental equations, i.e., the Euler Elastica equation, the boundary conditions and the transversality condition, are derived from a variational procedure. In contrast to the planar case, the curvature of the delimiting wall appears in the transversality condition, thus acting as a further source of adhesion. We provide the analytic solution to the problem under study in terms of elliptic integrals and perform the numerical and the asymptotic analysis of the characteristic lengths of the blister. Finally, and in contrast to previous studies, we also discuss the mechanics and the internal stresses in the case of vanishing adhesion. Specifically, we give a theoretical explanation to the observed divergence of the mean pressure exerted by the strip on the container in the limit of small excess-length
K-theory and limit-of-discrete-series for the universal cover of SL_2(R)
No full textThe unitary principal series of the universal cover of SL_2(R) admits a limit-of-discrete-series. We show how this representation leads to an explicit K-cycle which generates K_1 of the reduced C*-algebra
Some observations on weighted GMRES
No full textWe investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present a new alternative implementation of the weighted Arnoldi algorithm which under known circumstances will be favourable in terms of computational complexity. These implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used
Efficient and stable Arnoldi restarts for matrix functions based on quadrature
No full textWhen using the Arnoldi method for approximating f(A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a-priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numerical stability by various numerical experiments
Efficient and stable Arnoldi restarts for matrix functions based on quadrature
No full textWhen using the Arnoldi method for approximating f(A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We utilize an integral representation for the error of the iterates in the Arnoldi method which then allows us to develop an efficient quadrature-based restarting algorithm suitable for a large class of functions, including the so-called Stieltjes functions and the exponential function. Our method is applicable for functions of Hermitian and non-Hermitian matrices, requires no a-priori spectral information, and runs with essentially constant computational work per restart cycle. We comment on the relation of this new restarting approach to other existing algorithms and illustrate its efficiency and numerical stability by various numerical experiments
Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms
Get PDFSuppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point
subgroup C_G(H) of the complement is nilpotent of class c. It is proved that G has a nilpotent characteristic subgroup of index bounded in terms of c, |C_G(F)|, and |F|
whose nilpotency class is bounded in terms of c and |H| only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of C_G(F)=1
the whole group was proved to be nilpotent of (c,|H|)-bounded class. Examples show that the condition of F being cyclic is essential. B. Hartley's theorem based on
the classification provides reduction to soluble groups. Then representation theory arguments are used to bound the index of the Fitting subgroup. Lie ring methods are used for nilpotent groups. A similar theorem on Lie rings with a metacyclic Frobenius group of automorphisms FH is also proved