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O-minimal structures
Get PDFThe notion of an o-minimal expansion of the ordered field of real numbers was
invented by L van den Dries [vdD1] as a framework for investigating the model theory
of the real exponential function exp : R -> R : x -> exp(x), and thereby settle an old
problem of Tarski. More on this later, but for the moment it is best motivated as being
a candidate for Grothendieck�s idea of �tame topology� as expounded in his Esquisse
d�un Programme [Gr]. In this lecture I shall explain these remarks
Convolutions of Cantor measures without resonance
Get PDFDenote by the distribution of the random sum , where and all the choices are independent. For , the measure is supported on , the central Cantor set obtained by starting with the closed united interval, removing an open central interval of length , and iterating this process inductively on each of the remaining intervals.
We investigate the convolutions , where is a rescaling map. We prove that if the ratio is irrational and , then where denotes any of correlation, Hausdorff or packing dimension of a measure.
We also show that, perhaps surprisingly, for uncountably many values of the convolution is a singular measure, although and is irrational
On solubility of groups with bounded centralizer chains
Get PDFThe -dimension of a group is the maximum length of a chain of
nested centralizers. It is proved that a periodic locally soluble
group of finite -dimension is soluble of derived length
bounded in terms of~, and the rank of its quotient by the
Hirsch--Plotkin radical is bounded in terms of~. Corollary: a
pseudo-(finite soluble) group of finite -dimension is
soluble of derived length bounded in terms of~
A New Scaling and Squaring Algorithm for the Matrix Exponential
Get PDFThe scaling and squaring method for the matrix exponential is based on the
approximation
,
where is the Pad\'e approximant to and the integers and
are to be chosen.
Several authors have identified a weakness of existing
scaling and squaring algorithms termed overscaling,
in which a value of much larger than necessary is chosen,
causing a loss of accuracy in floating point arithmetic.
Building on the scaling and squaring algorithm of Higham
[{\em SIAM J. Matrix Anal. Appl.}, 26\penalty0 (4):\penalty0
1179--1193, 2005],
which is used by MATLAB's \texttt{expm},
we derive a new algorithm that alleviates the overscaling problem.
Two key ideas are employed.
The first, specific to triangular matrices,
is to compute the diagonal elements in the squaring phase as exponentials
instead of from powers of .
The second idea is to base the backward error analysis that underlies the
algorithm
on members of the sequence
instead of ,
since for non-normal matrices it is possible
that is much smaller than ,
and indeed this is likely when overscaling occurs in existing algorithms.
The terms are estimated
without computing powers of by using a matrix 1-norm estimator
in conjunction with a bound of the form
that holds for certain fixed and less than .
The improvements to the truncation error bounds
have to be balanced by the potential for a large to cause inaccurate
evaluation of in floating point arithmetic.
We employ rigorous error bounds along with some heuristics to ensure that
rounding errors are kept under control.
Our numerical experiments show that the new algorithm
generally provides accuracy at least as good
as the existing algorithm of Higham at no higher cost,
while for matrices that are triangular or cause overscaling it usually
yields significant improvements in accuracy, cost, or both
A posteriori error bounds for discrete balanced truncation
Get PDFBalanced truncation of discrete linear time-invariant systems is an automatic method once an error tolerance is specified and yields an a priori error bound, which is why it is widely used in engineering for simulation and control. We present some new insight into this method. We derive a discrete version of Antoulas's -norm error formula \cite[p.218]{Ant05} and show how to adapt it to some special cases. This error bound is an a posteriori computable upper bound for the -norm of the error system defined as the system whose transfer function corresponds to the difference between the transfer function of the original system and the transfer function of the reduced system. The main advantage of our results is that we use the information already available in the balanced truncation algorithm in order to compute the -norm instead of computing one gramian of the corresponding error system. There is always a computational restriction on solving high-dimensional Stein equations for gramians. The a posteriori bound gives insight into the quality of the reduced system and can be used to solve many problems accompanying the order reduction operation
Bifurcations of Snap-back Repellers with application to Border-Collision Bifurcations
Get PDFThe bifurcation theory of snap-back repellers in hybrid dynamical systems is developed. Infinite sequences of bifurcations are shown to arise due to the creation of snap-back repellers in non-invertible maps. These are
analogous to the cascades of bifurcations known to occur close to homoclinic tangencies for diffeomorphisms. The theoretical results are illustrated with reference to bifurcations in the normal form for border-collision
bifurcations
Definite Matrix Polynomials and their Linearization by Definite Pencils
Get PDFHyperbolic matrix polynomials
are an important class of Hermitian matrix polynomials
that contain overdamped quadratics as a special case.
They share with definite pencils the spectral property that their eigenvalues
are real and semisimple.
We extend the definition of hyperbolic matrix polynomial
in a way that relaxes the requirement of definiteness of
the leading coefficient matrix,
yielding what we call definite polynomials.
We show that this class of polynomials has an elegant characterization in terms
of definiteness intervals on the extended real line,
and that it includes definite pencils as a special case.
A fundamental question is whether a definite
matrix polynomial can be linearized
in a structure-preserving way.
We show that the answer to this question is affirmative:
is definite if and only if it has a
definite linearization in ,
a certain vector space of Hermitian pencils;
and for definite we give a complete characterization of all the
linearizations in that are definite. For the important
special case of quadratics, we show how a definite quadratic
polynomial can be transformed into a definite linearization with a
positive definite leading coefficient matrix---a form that is
particularly attractive numerically
The Complex Step Approximation to the Fréchet Derivative of a Matrix Function
Get PDFWe show that the Fréchet derivative of a
matrix function at in the direction ,
where and are real matrices,
can be approximated by
for some suitably small .
This approximation,
requiring a single function evaluation at a complex argument,
generalizes the complex step approximation known in the
scalar case.
The approximation is proved to be of second order in
for analytic functions and also for
the matrix sign function.
It is shown that it does not suffer the inherent cancellation
that limits the accuracy of finite difference
approximations in floating point arithmetic.
However,
cancellation does nevertheless vitiate the approximation
when the underlying method for evaluating employs
complex arithmetic.
The ease of implementation of the approximation,
and its superiority over finite differences,
make it attractive when specialized methods for
evaluating the Fréchet derivative are not available,
and in particular for condition number estimation
when used in conjunction with a block 1-norm estimation algorithm
Nilpotent blocks of quasisimple groups for odd primes
Get PDFWe investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes. In particular, it is shown that every nilpotent block of a quasisimple group has abelian defect groups. A conjecture of Puig concerning the recognition of nilpotent blocks is also shown for these groups