MIMS EPrints
Not a member yet
2151 research outputs found
Sort by
K-theory and the connection index
No full textLet G denote a split simply connected almost simple p-adic
group. The classical example is the special linear group SL(n). We study the spherical C*-algebra of G and prove that the rank of the K-theory group K_0 is the connection index f(G). We relate this result to the recent conjecture in http://eprints.ma.man.ac.uk/1553 and to a recent result of Solleveld in http://arxiv.org/abs/1008.0177
Dierential operators and Cherednik algebras
Get PDFWe establish a link betweentwo geometric approaches to the representation theory of rationalCherednik algebras of type A: one based on anoncommutative Proj construction \cite{GS}; the other
involving quantum hamiltonian reduction of an algebra of differential operators \cite{GG}. In this paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra.That enables us to give a direct and relatively short proof of the key result \cite[Theorem~1.4]{GS} without recourse to Haiman's deep results on the n! theorem \cite{Ha1}. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from \cite{GG}
The Problem with the Linpack Benchmark 1.0 Matrix Generator
Get PDFWe characterize the matrix sizes for which the Linpack Benchmark 1.0 matrix generator constructs
a matrix with identical columns
Computing the Fréchet Derivative of the Matrix Exponential, with an application to Condition Number Estimation
Get PDFThe matrix exponential is a much-studied matrix function having many
applications.
The Fr\'echet derivative of the matrix exponential describes the first order
sensitivity of to perturbations in and its norm determines a condition
number for .
Among the numerous methods for computing the
scaling and squaring method is the most widely used.
We show that the implementation of the method in
[N.~J. Higham.
The scaling and squaring method for the matrix exponential revisited.
{\em SIAM J. Matrix Anal. Appl.}, 26(4):1179--1193, 2005]
can be extended to compute both and the
Fr\'echet derivative
at in the direction ,
denoted by ,
at a cost about three
times that for computing alone.
The algorithm is derived from the scaling and squaring method
by differentiating the Pad\'e approximants
and the squaring recurrence,
re-using quantities computed during the evaluation of the
Pad\'e approximant,
and intertwining the recurrences in the squaring phase.
To guide the choice of algorithmic parameters an extension of the
existing backward error analysis for the
scaling and squaring method is developed which shows that,
modulo rounding errors,
the approximations obtained are
and ,
with the same in both cases,
and with computable bounds on
and .
The algorithm for is used to develop an algorithm
that computes together with an estimate of its condition number.
In addition to results specific to the exponential,
we develop some results and techniques for arbitrary functions.
We show how a matrix iteration for yields an iteration for
the Fr\'echet derivative and
show how to efficiently compute the Fr\'echet derivative of a power
series.
We also show that a matrix polynomial
and its Fr\'echet derivative can be
evaluated at a cost at most three times that of computing the polynomial itself
and give a general framework for evaluating a matrix function and its
Fr\'echet derivative via Pad\'e approximation
A 195,747,435 vertex graph related to the Fischer group Fi23, part I
Get PDFThis is the first of a series of three papers
devoted to the study of G, the point-line collinearity
graph of the maximal 2-local geometry for Fischer's second largest simple group Fi23. In this paper the first two discs of an arbitrary vertex of G are described and a start is made on properties of the third disc
A 195,747,435 vertex graph related to the Fischer group Fi23, part III
Get PDFThis is the final part of the three part series in which the
structure of G, the point-line collinearity graph of the
maximal 2-local geometry for Fischer's second largest simple group Fi23, is analysed. Using the results obtained in parts I and II,
here we are largely concerned with the fourth disc of an arbitrary
vertex of G, giving a detailed description of
G
Deflating Quadratic Matrix Polynomials with Structure Preserving Transformations
Get PDFGiven a pair of distinct \e s (\l_1,\l_2) of an \nbyn quadratic matrix polynomial Q(\l)
with nonsingular leading coefficient and their corresponding \ev s,
we show how to transform Q(\l) into a quadratic of the form
\twobytwoa{\Qd(\l)}{0}{0}{q(\l)} having the same \e s as Q(\l), with
\Qd(\l) an quadratic matrix \py\ and
q(\l) a scalar quadratic \py\ with roots \l_1 and \l_2.
This block diagonalization cannot be achieved by a similarity
transformation applied directly to Q(\l) unless the \ev s corresponding to
\l_1 and \l_2
are parallel.
We identify conditions under which we can construct a family of
elementary similarity transformations that
(a) are rank-two modifications of the identity matrix,
(b) act on linearizations of Q(\l),
(c) preserve the block structure of a large class of block symmetric
linearizations of Q(\l),
thereby defining new quadratic matrix polynomials
Q_1(\l) that have the same \e s as Q(\l),
(d) yield quadratics Q_1(\l) with the property that their \ev s
associated with \l_1 and \l_2 are parallel and hence
can subsequently be deflated by a similarity applied directly to Q_1(\l).
This is the first attempt at building elementary
transformations that preserve the block structure of widely used linearizations
and which have a specific action
The Complex Step Approximation to the Fréchet Derivative of a Matrix Function
Get PDFWe show that the Fr\'echet 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 Fr\'echet derivative are not available,
and in particular for condition number estimation
when used in conjunction with a block 1-norm estimation algorithm
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 note on the least informative model of a theory
Get PDFWe consider one possible interpretation of the `least informative model' of a relational and finite theory and show that it is well defined for a particular class of theories. We conjecture that it is always defined for theories