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Structured Mapping Problems for Matrices Associated with Scalar Products Part I: Lie and Jordan Algebras
Get PDFGiven a class of structured matrices \Sb,
we identify pairs of vectors for which there exists
a matrix A\in\Sb such that ,
and also characterize the set of all matrices A\in\Sb
mapping to .
The structured classes we consider are the Lie and Jordan algebras
associated with orthosymmetric scalar products.
These include (skew-)symmetric,
(skew-)Hamiltonian,
pseudo (skew-)Hermitian,
persymmetric and perskew-symmetric matrices.
Structured mappings
with extremal properties are also investigated.
In particular, structured mappings of
minimal rank are identified and
shown to be unique when rank one is achieved.
The structured mapping of minimal Frobenius norm is always
unique and explicit formulas for it and its norm are obtained.
Finally the set of all structured mappings of minimal 2-norm
is characterized.
Our results generalize and unify existing work, answer a number of open questions,
and provide useful tools for structured backward error investigations
Local Characteristic p Completions of Weak BN-Pairs
No full textThis paper establishes recognition theorems for the finite Lie-type groups defined in odd characteristic . The hypotheses of these theorems are couched in terms of certain local data in the form of an amalgam which is called a weak -pair of rank 2. The groups to be recognized are completions of these amalgams which satisfy the condition of being of local characteristic (a condition satisfied by the finite Lie-type groups defined in characteristic ). The results presented will find application in one of the ongoing revisions of the finite simple group classification
A Schur-Newton Method for the Matrix p'th Root and its Inverse
Get PDFNewton's method for the inverse matrix th root,
, has the attraction that it
involves only matrix multiplication.
We show that if the starting matrix is for then
the iteration converges quadratically to
if the eigenvalues of lie in
a wedge-shaped convex set containing the disc
.
We derive an optimal choice of for the case where has real, positive
eigenvalues.
An application is described to roots of transition matrices from Markov models,
in which for certain problems the convergence condition is satisfied with
.
Although the basic Newton iteration is numerically unstable,
a coupled version is stable and a simple modification of it provides a new
coupled iteration for the matrix th root.
For general matrices we develop a hybrid algorithm that computes a Schur
decomposition, takes square roots of the upper (quasi)triangular factor,
and applies the coupled Newton iteration to a matrix for which fast convergence
is guaranteed.
The new algorithm can be used to compute either
or , and
for large that are not highly composite
it is more efficient than the method of Smith based entirely on the Schur
decomposition
The Small World of Corporate Boards
No full textWe demonstrate the importance of graph theory for understanding boards of
directors. Specifically, we focus on the ‘small world’ phenomenon. Our empirical results show
that a random graph model is remarkably good at explaining board structure and connectedness
in the United States, the United Kingdom and Germany. Although there are small-world traits
such as ‘clustering’ and ‘short-paths’ in the corporate world, they are no more pronounced than
would be expected by chance in a statistically similar, but randomly assembled corporate universe.
In short, boards of directors, especially in the United States, are no more ‘clubby’ than expected.
Finally, our results show the existence of positive degree correlation: directors who sit on many
boards do so in the company of other directors who sit on many boards. Board members whose
services are in high demand, serve on boards with similar directors
The spectra of lamplighter groups and Cayley machines
No full textWe calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr , with G a finite group, by calculating the Kesten—von Neumann—Serre spectral measures for the random walks on Schreier graphs of certain groups generated by automata. This generalises the work of Grigorchuk and Żuk on the lamplighter group. In the process we characterise when the usual spectral measure for a group generated by an automaton coincides with the Kesten—von Neumann—Serre spectral measure
On the ratio X/Y for some elliptically symmetric distributions
No full textThe distributions of the ratio X/Y are derived when (X,Y) has the elliptically symmetric Pearson-type II distribution, elliptically symmetric Pearson-type VII distribution and the elliptically symmetric Kotz-type distribution
Cycles in the chamber homology of GL(3)
No full textLet F be a nonarchimedean local field and let GL(N) = GL(N,F). We prove the existence of parahoric types for GL(N). We construct representative cycles in all the homology classes of the chamber homology of GL(3)
The rational points of a definable set
Get PDFLet be a set that is definable in an o-minimal structure over . This article shows that in a suitable sense, there are very few rational points of which do not lie on some connected semialgebraic subset of of positive dimensio
Uses and abuses of EIDORS: An extensible software base for EIT
Get PDFEIDORS is an open source software suite for image reconstruction in electrical impedance tomography and diffuse optical tomography, designed to facilitate
collaboration, testing and new research in these fields. This paper describes recent work to redesign the software structure in order to simplify its use and provide a
uniform interface, permitting easier modification and customization. We describe the key features of this software, followed by examples of its use. One general issue with inverse problem software is the difficulty of correctly implementing algorithms, and the consequent ease with which subtle numerical bugs can be inadvertently introduced.
EIDORS helps with this issue, by allowing sharing and reuse of well documented and debugged software. On the other hand, since EIDORS is designed to facilitate use by non-specialists, its use may inadvertently result in such numerical errors. In order to address this issue, we develop a list of ways in which such errors with inverse problems
(which we refer to as "cheats") may occur. Our hope is that
such an overview may assist authors of software to avoid such implementation issues
Boundary layers in a dilute particle suspension
Get PDFThe general problem of a boundary-layer flow carrying a dilute, mono-disperse suspension of small particles (together with gravitational effects) is considered. The problem is modelled using the �dusty-gas� equations, which are a coupled equation set linking the fluid motion to that of the particle motion (both of which are modelled as continua). A number of qualitatively distinct potential scenarios are predicted. These include a variety of boundary-layer breakdowns, and the formation of shock transitions in the distribution of the particulate phase (together with the development of particle-free zones). Numerical results predicting these differing behaviours are confirmed through local asymptotic analyses of the governing equations. Although we consider a general class of boundary layer, our results are compared and contrasted with previous studies of specific cases, most notably the constant freestream fluid velocity case (akin to the �clean� Blasius boundary layer). In the case of a boundary-layer flow driven by a linearly retarding free stream (the analogue of the classical �Howarth� boundary-layer problem), the effects of the particle phase are surprisingly seen to (slightly) delay the separation of the boundary layer