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Affine Weyl groups and Langlands duality
Get PDFLet G be a compact connected semisimple Lie group. We show that, as well as the duality between K-theory and K-homology,
there is also a Langlands duality in the Baum-Connes correspondence for the (extended) affine Weyl group attached to G
Predicate Exchangeability and Language Invariance in Pure Inductive Logic
Get PDFIn Pure Inductive Logic, the rational principle of Predicate Exchangeability states that permuting the predicates
in a given language and replacing each occurrence of a predicate in an -sentence according to this
permutation should not change our belief in the truth of . In this paper we study when a prior probability
function on a purely unary language satisfying Predicate Exchangeability also satisfies the principle of
Unary Language Invariance
Mobius Transformations of Matrix Polynomials
No full textWe discuss Mobius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant,
constructs such as compound matrices,
and on structural features including sparsity and symmetry.
Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented.
The effect on canonical forms such as Smith forms
and local Smith forms,
on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated.
We show that many important transformations
are special instances of Mobius transformations,
and analyze a Mobius connection between alternating and
palindromic matrix polynomials.
Finally, the use of Mobius transformations in solving
polynomial inverse eigenproblems is illustrated
When is a Hamiltonian matrix the commutator of two skew-Hamiltonian matrices?
No full textThe mapping , where the matrices are skew-Hamiltonian with respect to transposition,
is studied. Let be the range of : we give an implicit characterization of , obtaining results that find an application in algebraic geometry. Namely, they are used in [R. Abuaf and A. Boralevi, Orthogonal bundles and skew-Hamiltonian matrices, In Preparation] to study orthogonal vector bundles.
We also give alternative and more explicit characterizations of for
. Moreover, we prove that for the complement of is nowhere dense in the set of -dimensional Hamiltonian matrices, denoted by , implying that almost all matrices in are in for . Finally, we show that is never surjective as a mapping from
to , where is the set
of -dimensional skew-Hamiltonian matrices. Along the way, we discuss the connections of this problem with several existing results in matrix theory
Tropical roots as approximations to eigenvalues of matrix polynomials
No full textThe tropical roots of
are points at which the maximum is attained at least twice.
These roots, which can be computed in only operations, can be good approximations to the moduli of
the eigenvalues of the matrix polynomial , in particular when the norms of the matrices vary widely.
Our aim is to investigate this observation and its applications.
We start by providing annuli defined in terms of the tropical roots of that contain the eigenvalues of .
Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of .
Our tropical localization of eigenvalues are less tight than
eigenvalue localization results derived from a generalized matrix version of Pellet's theorem but they are easier to interpret.
Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the Ehrlich-Aberth method
and our results further justify this choice.
Our results provide the basis for analyzing the effect of Gaubert and Sharify's tropical scalings for
on (a) the conditioning of linearizations of tropically scaled and
(b) the backward stability of eigensolvers based on linearizations of tropically scaled .
We anticipate that the tropical roots of , on which
the tropical scalings are based, will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems
such as that implemented in the MATLAB function \texttt{polyeig}
Linearizations of Matrix Polynomials in Bernstein Basis
Get PDFWe discuss matrix polynomials expressed
in a Bernstein basis,
and the associated polynomial eigenvalue problems.
Using Mobius transformations of matrix polynomials,
large new families of strong linearizations are generated.
We also investigate matrix polynomials
that are structured with respect to a Bernstein basis,
together with their associated spectral symmetries.
The results in this paper apply equally well
to scalar polynomials,
and include the development of new companion pencils
for polynomials expressed in a Bernstein basis
On the stability of computing polynomial roots via confederate linearizations
No full textA common way of computing the roots of a polynomial is to nd the eigenvalues of
a linearization, such as the companion (when the polynomial is expressed in the monomial basis),
colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the
monomial case, many studies exist on the stability of linearization-based rootnding algorithms. By
contrast, little seems to be known for other polynomial bases. This paper studies the stability of
algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First
we prove its normwise stability when the polynomial is properly scaled and the QZ algorithm (as
opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with
a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a rst-order
expansion of the backward error when the eigenvalues are computed via QR, which shows that the
method can be unstable. Finally, we focus on the special case of Chebyshev basis, in particular the
Chebfun rootnder: we discuss its stability and describe an optional functionality, made available
for improved stability, for computing the roots of a general continuous function f(x), implemented
in the recently updated version 5. The main message is that to guarantee backward stability QZ
applied to a properly scaled pencil is necessar
Automorphisms of finite soluble groups. Preliminary version
Get PDFThis preprint was distributed by Brian Hartley amongst his colleagues in 1994
Backward error analysis of the shift-and-invert Arnoldi algorithm
Get PDFWe perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm.
We consider inexactness in the solution of the arising linear systems, as well as in the orthonormalization steps, and
take the non-orthonormality of the computed Krylov basis into account.
We show that the computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed matrix, and we give bounds for the perturbation.
We show that the shift-and-invert Arnoldi algorithm is backward stable if the condition number of the small Hessenberg matrix is not too large.
This condition is then relaxed using implicit restarts.
Moreover we give notes on the Hermitian case, considering Hermitian backward errors, and
finally, we use our analysis to derive a sensible breakdown condition
