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Involutions in Janko's Simple Group J4
Get PDFIn this paper we determine the suborbits of Janko's largest simple group in its
conjugation action on each of its two conjugacy classes of involutions. We also
provide matrix representatives of these suborbits in an accompanying computer file
Modelling acidosis and the cell cycle in multicellular tumour spheroids
No full textA partial differential equation model is developed to understand the effect that nutrient and acidosis have on the distribution of proliferating and quiescent cells within a multicellular tumour spheroid. The rates of cell quiescence and necrosis are assumed to depend upon the local nutrient and acid concentrations. Quiescent cells are assumed to consume less nutrient and produce less acid than proliferating cells and a description of anaerobic metabolism by the cells is included. Parameterised with data from the literature, our model predicts that the tumour size is reduced in the presence of acid. Analysis of the differences in nutrient consumption and acid production by quiescent and proliferating cells shows low nutrient levels do not necessarily lead to increased acid concentration via anaerobic metabolism. Instead it is the balance between proliferating and quiescent cells within the tumour which is important; decreased nutrient levels lead to more quiescent cells, which produce less acid than proliferating cells. We examine this effect via a sensitivity analysis which also includes a quantification of the effect that nutrient and acid concentrations have on the rates of cell quiescence and necrosis
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
Computing a Nearest Correlation Matrix with Factor Structure
Get PDFAn correlation matrix has factor structure if its
off-diagonal agrees with that of a rank matrix.
Such correlation matrices arise, for example,
in factor models of
collateralized debt obligations (CDOs)
and multivariate time series.
We analyze the properties of these matrices and,
in particular, obtain an explicit formula for the rank in
the one factor case.
Our main focus is on the nearness problem of finding the nearest
factor correlation matrix C(X) = \diag(I-XX^T) + XX^T to a given
symmetric matrix, subject to natural nonlinear constraints on the
elements of the matrix ,
where distance is measured in the Frobenius norm.
For a special one parameter case we obtain an explicit solution.
For the general factor case we obtain the gradient and Hessian of
the objective function and derive an instructive result on the
positive definiteness of the Hessian when .
We investigate several numerical methods for solving the nearness
problem: the alternating directions method;
a principal factors method used by
Anderson, Sidenius, and Basu in the CDO application,
which we show is equivalent to the alternating projections method
and lacks convergence results;
the spectral projected gradient method of
Birgin, Mart{\'\i}nez, and Raydan;
and Newton and sequential quadratic programming methods.
The methods differ in whether or not they can take account of the
nonlinear constraints and in their convergence properties.
Our numerical experiments show that the performance of the methods
depends strongly on the problem,
but that the spectral projected gradient method is the clear winner
Rule Systems for Runtime Verification: A Short Tutorial
Get PDFIn this tutorial, we introduce two rule-based systems for on and off-line trace analysis, RuleR and LogScope. RuleR is a conditional rule-based system, which has a simple and easily implemented algorithm for effective runtime verification, and into which one can compile a wide range of temporal logics and other specification formalisms used for runtime verification. Specifications can be parameterized with data, or even with specifications, allowing for temporal logic combinators to be defined. We outline a number of simple syntactic extensions of core RuleR that can lead to further conciseness of specification but still enabling easy and efficient implementation. RuleR is implemented in Java and we will demonstrate its ease of use in monitoring Java programs. LogScope is a derivation of RuleR adding a simple very user-friendly temporal logic. It was developed in Python, specifically for supporting testing of spacecraft flight software for NASA’s next 2011 Mars mission MSL (Mars Science Laboratory). The system has been applied by test engineers to analysis of log files generated by running the flight software. Detailed logging is already part of the system design approach, and hence there is no added instrumentation overhead caused by this approach. While post-mortem log analysis prevents the autonomous reaction to problems possible with traditional runtime verification, it provides a powerful tool for test automation. A new system is being developed that integrates features from both RuleR and LogScope
Model category structures arising from Drinfeld vector bundles
Get PDFWe present a general construction of model category structures on the category
\Ch(\Qco(X)) of unbounded chain complexes
of quasi-coherent sheaves on a semi-separated scheme . The construction is based
on making compatible the filtrations of individual
modules of sections at open affine subsets of . It does not require closure under
direct limits as previous methods. We apply it to describe the
derived category \mathbb D (\Qco(X))
via various model structures on \Ch(\Qco(X)). As particular instances, we recover
recent results on the flat model
structure for quasi-coherent sheaves. Our approach also includes the case of
(infinite-dimensional) vector bundles, and of
restricted flat Mittag-Leffler quasi-coherent sheaves, as introduced by Drinfeld.
Finally, we prove that the unrestricted case does not induce a model category
structure as above in general
Fiedler Companion Linearizations and the Recovery of Minimal Indices
Get PDFA standard way of dealing with a matrix polynomial
is to convert it into an equivalent matrix pencil --
a process known as linearization.
For any regular matrix polynomial,
a new family of linearizations generalizing
the classical first and second Frobenius companion forms
has recently been introduced by Antoniou and Vologiannidis,
extending some linearizations previously defined
by Fiedler for scalar polynomials.
We prove that these pencils are linearizations
even when
is a singular square matrix polynomial,
and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial
from the minimal indices and bases of these linearizations.
In addition, we provide a simple way to recover the eigenvectors of a regular polynomial
from those of any of these linearizations,
without any computational cost.
The existence of an eigenvector recovery procedure
is essential for a linearization
to be relevant for applications
Products of homogeneous subspaces in free Lie algebras
No full textLet be a free Lie algebra of rank over a field . If
then we let denote the homogeneous subspace of
spanned by Lie products of weight in the free generators of .
We obtain formulae for the dimensions of the subspaces
for all and . This answers a question raised
by Eamonn O'Brien in private communication
Structured Polynomial Eigenproblems Related to Time-Delay Systems
Get PDFA new class of structured polynomial eigenproblems arising in the stability analysis
of time-delay systems is identified and analyzed together with new types of closely related structured
polynomials. Relationships between these polynomials are established via the Cayley transformation.
Their spectral symmetries are revealed, and structure-preserving linearizations constructed. A
structured Schur decomposition for the class of structured pencils associated with time-delay systems
is derived, and an algorithm for its computation that compares favorably with the QZ algorithm is
presented along with numerical experiment
Model reduction of switched dynamical systems
Get PDFA hybrid dynamical system is a system described by both differential equations (continuous
flows) and difference equations (discrete transitions). It has the benefit of allowing more flexible
modeling of dynamic phenomena, including physical systems with impact such as the bouncing ball,
switched systems such as the thermostat, and even the internet congestion as examples. Hybrid dynamical
systems pose a challenge since almost all reduction methods cannot be directly applied. Here
we show some recent developments in the area of model reduction of switched dynamical systems