2024-03-29T08:45:47Z
https://eprints.maths.manchester.ac.uk/cgi/oai2
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A note on vertices of simple modules
Eaton, Charles W.
20 Group theory and generalizations
We generalize the results of [Bessenrodt 1984], showing that vertices of
simple modules of blocks of the type studied in [Eaton 2005] are
radical. This means that in order to identify simple modules whose
vertices are not radical we must consider blocks 'involving' blocks
of positive defect of non-abelian simple groups.
2005-08-15
Article
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/2/1/notesonvertices.pdf
Eaton, Charles W. (2005) A note on vertices of simple modules. Preprint, submitted for publication. (Submitted)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:3
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A class of blocks behaving like blocks of p-solvable groups
Eaton, Charles W.
20 Group theory and generalizations
2006
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/3/1/a-class-of-blocks-behaving-like-blocks-of-p-solvable-groups.pdf
Eaton, Charles W. (2006) A class of blocks behaving like blocks of p-solvable groups. Journal of Algebra, 301. pp. 337-343. ISSN 0021-8693
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WH2-4GY89X8-4&_user=121749&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000024058&_version=1&_urlVersion=0&_userid=121749&md5=2001c116ec0fdec46fe049d12707fd3f
10.1016/j.jalgebra.2005.07.018
10.1016/j.jalgebra.2005.07.018
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:4
2017-11-07T22:38:43Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:5
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Characterisation of congenital nystagmus waveforms in terms of periodic orbits
Clement, R. A.
Whittle, J. P.
Muldoon, M. R.
Abadi, R. V.
Broomhead, D. S.
Akman, O.
92 Biology and other natural sciences
Because the oscillatory eye movements of congenital nystagmus vary from cycle to cycle, there is no clear relationship between the waveform produced and the underlying abnormality of the ocular motor system. We consider the durations of successive cycles of nystagmus which could be (1) completely determined by the lengths of the previous cycles, (2) completely independent of the lengths of the previous cycles or (3) a mixture of the two. The behaviour of a deterministic system can be characterised in terms of a collection of (unstable) oscillations, referred to as periodic orbits, which make up the system. By using a recently developed technique for identifying periodic orbits in noisy data, we find evidence for periodic orbits in nystagmus waveforms, eliminating the possibility that each cycle is independent of the previous cycles. The technique also enables us to identify the waveforms which correspond to the deterministic behaviour of the ocular motor system. These waveforms pose a challenge to our understanding of the ocular motor system because none of the current extensions to models of the normal behaviour of the ocular motor system can explain the range of identified waveforms.
2002-08
Article
PeerReviewed
application/pdf
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https://eprints.maths.manchester.ac.uk/5/1/Clement_VisRes42.pdf
Clement, R. A. and Whittle, J. P. and Muldoon, M. R. and Abadi, R. V. and Broomhead, D. S. and Akman, O. (2002) Characterisation of congenital nystagmus waveforms in terms of periodic orbits. Vision Research, 42 (17). pp. 2123-2130. ISSN 0042-6989
http://www.sciencedirect.com/science/article/B6T0W-46FG77W-2/2/1b2fe4afe04bda08c484f08085e26a7a
10.1016/S0042-6989(02)00124-4
10.1016/S0042-6989(02)00124-4
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:6
2017-10-20T14:12:02Z
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Dynamics and processivity of 40S ribosome scanning on mRNA in yeast
Berthelot, K.
Muldoon, M.
Rajkowitsch, L.
Hughes, J.
McCarthy, J. E. G.
92 Biology and other natural sciences
The eukaryotic 40S ribosomal subunit locates the translation initiation codon on an mRNA via the so-called scanning process that follows 40S binding to the capped 5' end. This key step in translation is required for the expression of almost all eukaryotic genes, yet the mechanism and dynamics of scanning are unknown. We have performed quantitative studies in vivo and in vitro of the movement of yeast 40S ribosomes along 5' untranslated regions (UTRs) of different lengths. 40S subunits perform cap-dependent scanning with high processivity for more than 1700 nucleotides in cells of Saccharomyces cerevisiae. Moreover, the observed rates of expression indicate that scanning is performed by an untethered 40S subunit that has been released from the 5' cap complex. Unexpectedly, the capability to maintain scanning competence on a long 5' UTR is more dependent on the Ded1/Dbp1 type of helicase than on eIF4A or eIF4B. In a yeast cell-free extract, scanning shows reduced processivity, with an estimated net 5'3' rate of approximately 10 nucleotides per second at 26°C. We have developed a biased bidirectional walking model of ribosomal scanning that provides a framework for understanding the above observations as well as other known quantitative and qualitative features of this process.
2004-02
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/6/1/MolMicrobiol.pdf
Berthelot, K. and Muldoon, M. and Rajkowitsch, L. and Hughes, J. and McCarthy, J. E. G. (2004) Dynamics and processivity of 40S ribosome scanning on mRNA in yeast. Molecular Microbiology, 51 (4). pp. 987-1002. ISSN 0950-382X
http://www.blackwell-synergy.com/doi/full/10.1046/j.1365-2958.2003.03898.x
10.1046/j.1365-2958.2003.03898.x
10.1046/j.1365-2958.2003.03898.x
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:7
2017-10-20T14:12:02Z
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Linear Filters and Non-linear Systems
Broomhead, D. S.
Huke, J.
Muldoon, M. R.
37 Dynamical systems and ergodic theory
94 Information and communication, circuits
It has been asserted in the literature that the low pass filtering of time series data may lead to erroneous results when calculating attractor dimensions. Here we prove that finite order, non-recursive filters do not have this effect. In fact, a generic, finite order, non-recursive filter leaves invariant all the quantities that can be estimated by using embedding techniques such as the method of delays.
1992-11-08
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/7/1/Broomhead_JRoYStatSocB54.pdf
Broomhead, D. S. and Huke, J. and Muldoon, M. R. (1992) Linear Filters and Non-linear Systems. Journal of the Royal Statistical Society. Series B, 54 (2). pp. 373-382. ISSN 1369-7412
http://www.jstor.org/stable/2346133
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:8
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oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:9
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https://eprints.maths.manchester.ac.uk/9/
The relation between local and global dual pairs
Montaldi, J
Ortega, J-P
Ratiu, TS
37 Dynamical systems and ergodic theory
53 Differential geometry
In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of the extra conditions.
2004
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/9/1/dual_pairs.pdf
Montaldi, J and Ortega, J-P and Ratiu, TS (2004) The relation between local and global dual pairs. Math Research Letters, 11. pp. 355-363. ISSN 1073-2780
http://www.mrlonline.org/mrl/2004-011-003/2004-011-003-007.pdf
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:11
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oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:12
2017-11-07T22:38:43Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:17
2017-11-07T22:38:43Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:18
2017-10-27T14:49:03Z
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Bifurcation and forced symmetry breaking in Hamiltonian systems
Grabsi, Fethi
Montaldi, James
Ortega, Juan-Pablo
37 Dynamical systems and ergodic theory
We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria .
2004
Article
PeerReviewed
text
en
https://eprints.maths.manchester.ac.uk/18/1/symbreak_cras.pdf
Grabsi, Fethi and Montaldi, James and Ortega, Juan-Pablo (2004) Bifurcation and forced symmetry breaking in Hamiltonian systems. C.R. Acad. Sci. Paris, Ser. I, 338 (7). pp. 565-570. ISSN 1631-073X
10.1016/j.crma.2004.01.029
10.1016/j.crma.2004.01.029
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:21
2017-11-08T18:18:28Z
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https://eprints.maths.manchester.ac.uk/21/
Nilpotent blocks of quasisimple groups for the prime two
An, Jianbei
Eaton, Charles W.
20 Group theory and generalizations
We investigate the nilpotent blocks of positive defect of the quasisimple groups for the prime 2. We show that every nilpotent block of a quasisimple group has abelian defect groups, and give explicit characterisations in many cases. A conjecture of Puig concerning the recognition of nilpotent blocks is also shown to hold for these groups.
2009-10-10
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/21/1/nilblkeven09final.pdf
An, Jianbei and Eaton, Charles W. (2009) Nilpotent blocks of quasisimple groups for the prime two. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:22
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Vertices for irreducible characters of a class of blocks
Eaton, Charles W.
20 Group theory and generalizations
We observe that Navarro's definition of a vertex for an
irreducible character of a $p$-solvable group may be extended to
irreducible characters in $p$-blocks with defect groups contained
in a normal $p$-solvable subgroup, and show that this definition
is independent of the choice of $N$. We show that the fundamental
properties of Navarro's vertices generalize, and as a corollary
show that the vertices of the irreducible Brauer characters in
blocks of the above form are radical and are intersections of
pairs of Sylow $p$-subgroups.
2005-04-15
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/22/1/verticesarticle.pdf
Eaton, Charles W. (2005) Vertices for irreducible characters of a class of blocks. Journal of Algebra, 286 (2). pp. 492-499. ISSN 0021-8693
http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WH2-4FK3P8F-1-1&_cdi=6838&_user=121749&_orig=search&_coverDate=04%2F15%2F2005&_qd=1&_sk=997139997&view=c&wchp=dGLbVlz-zSkzS&md5=5155f966e03f0f756d31e6b8d2b21655&ie=/sdarticle.pdf
doi:10.1016/j.jalgebra.2005.01.007
doi:10.1016/j.jalgebra.2005.01.007
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:24
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https://eprints.maths.manchester.ac.uk/24/
Iterated Function System Models of Digital Channels
Broomhead, D. S.
Huke, J. P.
Muldoon, M. R.
Stark, J.
37 Dynamical systems and ergodic theory
94 Information and communication, circuits
This paper introduces a new class of models of digital communications channels. Physically, these models take account of the digital nature of the input. Mathematically, they are iterated function systems. As a consequence of making explicit assumptions about the role of discreteness in the models, it is possible to make general statements about the behaviour of these channels without needing to assume that they are linear. We provide the mathematical background necessary to understand the behaviour of these models and prove a number of results about their observability. We also provide a number of examples intended to demonstrate their connection with linear state-space models, and to suggest how the nonlinear theory might be developed towards applications.
2004-11-08
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/24/1/RoySocIFS.pdf
Broomhead, D. S. and Huke, J. P. and Muldoon, M. R. and Stark, J. (2004) Iterated Function System Models of Digital Channels. Proceedings of the Royal Society of London, Series A, 460. pp. 3123-3142. ISSN 1364-5021
http://www.journals.royalsoc.ac.uk/openurl.asp?genre=article&id=doi:10.1098/rspa.2004.1336
10.1098/rspa.2004.1336
10.1098/rspa.2004.1336
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:30
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https://eprints.maths.manchester.ac.uk/30/
Geometric Mechanics and Symmetry:
The Peyresq Lectures
Montaldi, James
Ratiu, Tudor
37 Dynamical systems and ergodic theory
70 Mechanics of particles and systems
76 Fluid mechanics
This consists of lecture notes from 6 courses held at 2 summer schools in Peyresq, France in 2000 and 2001. The notes were written up by the lecturers together with some participants.
Cambridge University Press
Montaldi, J
Ratiu, TS
2005-06
Book
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/30/1/frontmatter.pdf
Montaldi, James and Ratiu, Tudor (2005) Geometric Mechanics and Symmetry: The Peyresq Lectures. London Mathematical Society Lecture Notes Series, 306 . Cambridge University Press, Cambridge etc. ISBN 0521539579
http://www.cambridge.org/9780521539579
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:32
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https://eprints.maths.manchester.ac.uk/32/
Symmetric Hamiltonian Bifurcations
Buono, Pietro-Luciano
Laurent-Polz, Frederic
Montaldi, James
37 Dynamical systems and ergodic theory
70 Mechanics of particles and systems
The purpose of these notes is to give a brief survey of bifurcation theory of Hamiltonian systems with symmetry; they are a slightly extended version of the 5 lectures given by JM on Hamiltonian Systems with Symmetry at the Peyresq Summer School. Attention is focussed on bifurcations near equilibrium solutions and relative equilibria. [Taken from introduction]
Cambridge University Press
Montaldi, James
Ratiu, Tudor
2005-05
Book Section
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/32/1/montaldi.pdf
Buono, Pietro-Luciano and Laurent-Polz, Frederic and Montaldi, James (2005) Symmetric Hamiltonian Bifurcations. In: Geometric Mechanics and Symmetry: The Peyresq Lectures. London Mathematical Society Lecture Notes Series (306). Cambridge University Press, Cambridge, UK (etc), pp. 357-402. ISBN 0-521-53957-9
http://www.cambridge.org/9780521539579
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:33
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oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:34
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Delay embedding in the presence of dynamical noise
Muldoon, M. R.
Broomhead, D. S.
Huke, J. P.
Hegger, R.
37 Dynamical systems and ergodic theory
We present a new embedding theorem for time series, in the spirit of
Takens's theorem, but requiring multivariate signals. Our result is part
of a growing body of work that extends the
domain of geometric time series analysis to some genuinely
stochastic systems---including such natural examples as
x_{j+1} = \phi(x_j) + \eta_j
where \phi is some fixed map and the \eta_j
are i.i.d. random displacements.
1998-06
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/34/1/NoisyEmbed.pdf
Muldoon, M. R. and Broomhead, D. S. and Huke, J. P. and Hegger, R. (1998) Delay embedding in the presence of dynamical noise. Dynamics and Stability of Systems, 13 (2). pp. 175-186. ISSN 0268-1110
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:40
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https://eprints.maths.manchester.ac.uk/40/
Codes for spread spectrum applications generated using chaotic dynamical systems
Broomhead, D. S.
Huke, J. P.
Muldoon, M. R.
37 Dynamical systems and ergodic theory
94 Information and communication, circuits
An approach to finding codes for use in direct sequence
spread spectrum communications systems is described. It
is based upon an analogy between codes having auto- and
cross-correlation properties desirable for spread
spectrum systems, and certain dynamical systems
encountered in ergodic theory called systems with Lebesgue spectrum. Such systems are associated with
collections of orthogonal functions and these functions
can be used to generate collections of time series with
zero cross-correlation functions. To generate codewords
we must use truncated versions of these time series,
for which the cross-correlations are no longer
precisely zero: these truncated sequences correspond to
periodic orbits of the dynamical system. The method for
finding a code from a suitable periodic orbit is
described, and an example, using a simple dynamical
system known as the doubling map, is worked through in
some detail.
1999-03
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/40/1/Coding.pdf
Broomhead, D. S. and Huke, J. P. and Muldoon, M. R. (1999) Codes for spread spectrum applications generated using chaotic dynamical systems. Dynamics and Stability of Systems, 14 (1). pp. 95-105. ISSN 0268-1110
10.1080/026811199282100
10.1080/026811199282100
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:43
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https://eprints.maths.manchester.ac.uk/43/
Golden gaskets: variations on the Sierpinski sieve
Broomhead, Dave
Montaldi, James
Sidorov, Nikita
11 Number theory
28 Measure and integration
37 Dynamical systems and ergodic theory
We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, with a common contraction factor λ in (0, 1).
As is well known, for λ = 1/2 the attractor, S_λ, is a fractal called the Sierpinski sieve and for λ < 1/2 it is also a fractal. Our goal is to study S_λ for this IFS for 1/2 < λ < 2/3 , i.e. when there are ‘overlaps’ in S_λ as well as
‘holes’. In this introductory paper we show that despite the overlaps (i.e. the breaking down of the open set condition (OSC)), the attractor can still be a totally self-similar fractal, although this happens only for a very special family
of algebraic λ (so-called multinacci numbers). We evaluate the ausdorff dimension of S_λ for these special values by showing that S_λ is essentially the attractor for an infinite IFS that does satisfy the OSC. We also show that the set of points in the attractor with a unique ‘address’ is self-similar and compute its dimension.
For non-multinacci values of λ we show that if λ is close to 2/3 , then S_λ has a non-empty interior. Finally we discuss higher-dimensional analogues of the model in question.
2004-05-07
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/43/1/golden.pdf
Broomhead, Dave and Montaldi, James and Sidorov, Nikita (2004) Golden gaskets: variations on the Sierpinski sieve. Nonlinearity, 17. pp. 1455-1480. ISSN 0951-7715
http://www.iop.org/EJ/abstract/0951-7715/17/4/017
10.1088/0951-7715/17/4/017
10.1088/0951-7715/17/4/017
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:46
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Mass transfer from a finite strip near an oscillating stagnation point --- implications for atherogenesis
Heil, Matthias
Hazel, Andrew
76 Fluid mechanics
92 Biology and other natural sciences
We consider the mass transfer from a finite-length strip
near a two-dimensional, oscillating, stagnation-point flow in an incompressible, Newtonian fluid.
The problem is investigated using a combination of asymptotic and numerical methods. The aim of the study is to
determine the effect of the location of the strip, relative to the time-averaged position of the stagnation point,
on the mass transfer into the fluid. The study is motivated
by the problem of mass transfer from an injured region of the arterial wall into the blood, a process that may be of considerable importance in atherogenesis. For physiologically realistic parameter values,
we find that the fluid flow is quasi-steady, but the
mass transfer exhibits genuine time-dependence and
a high-frequency asymptotic solution provides an accurate
prediction of the time-average mass transfer. In this regime, there is a significant reduction in mass transfer when the centre of the strip is
located at the point of zero time-averaged wall shear rate, or equivalently wall shear stress, which
may serve to explain, at least partially, the correlation between arterial disease and regions of low wall shear stress.
2003
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/46/1/advect.pdf
Heil, Matthias and Hazel, Andrew (2003) Mass transfer from a finite strip near an oscillating stagnation point --- implications for atherogenesis. Journal of Engineering Mathematics, 47. pp. 315-334. ISSN 0022-0833
10.1023/B:ENGI.0000007976.65014.a9
10.1023/B:ENGI.0000007976.65014.a9
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:47
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https://eprints.maths.manchester.ac.uk/47/
Functions Preserving Matrix Groups and Iterations for the Matrix Square Root
Higham, Nicholas J.
Mackey, D. Steven
Mackey, Niloufer
Tisseur, Françoise
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
For any matrix automorphism group $\G$ associated with a bilinear
or sesquilinear form, Mackey, Mackey, and Tisseur have recently
shown that the matrix sign decomposition factors of $A\in\G$ also
lie in $\G$; moreover, the polar factors of $A$ lie in $\G$ if the
matrix of the underlying form is unitary. Groups satisfying the
latter condition include the complex orthogonal, real and complex
symplectic, and pseudo-orthogonal groups. This work is concerned
with exploiting the structure of $\G$ when computing the polar and
matrix sign decompositions of matrices in $\G$. We give sufficient
conditions for a matrix iteration to preserve the group structure
and show that a family of globally convergent rational
Pad\'e-based iterations of Kenney and Laub satisfy these
conditions. The well-known scaled Newton iteration for computing
the unitary polar factor does not preserve group structure, but we
show that the approach of the iterates to the group is precisely
tethered to the approach to unitarity, and that this forces a
different and exploitable structure in the iterates. A similar
relation holds for the Newton iteration for the matrix sign
function. We also prove that the number of iterations needed for
convergence of the structure-preserving methods can be precisely
predicted by running an associated scalar iteration. Numerical
experiments are given to compare the cubically and quintically
converging iterations with Newton's method and to test stopping
criteria. The overall conclusion is that the structure-preserving
iterations and the scaled Newton iteration are all of practical
interest, and which iteration is to be preferred is
problem-dependent.
2005
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/47/2/paper.pdf
Higham, Nicholas J. and Mackey, D. Steven and Mackey, Niloufer and Tisseur, Françoise (2005) Functions Preserving Matrix Groups and Iterations for the Matrix Square Root. SIAM Journal On Matrix Analysis and Applications, 26 (3). pp. 849-877. ISSN 1095-7162
10.1137/S0895479804442218
10.1137/S0895479804442218
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An Interview with Peter Lancaster
Higham, Nicholas J.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
This is the transcript of an interview with
Peter Lancaster recorded at the University of Manchester, March 15, 2005.
A shortened version of this interview appeared in SIAM News 38(6):5-6, Jul/Aug 2005.
See
http://www.siam.org/news/news.php?id=126
2005-11-25
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/48/1/Interview_Mar05_final.pdf
Higham, Nicholas J. (2005) An Interview with Peter Lancaster. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:49
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Computing the Polar Decomposition and the Matrix Sign Decomposition in Matrix Groups
Higham, Nicholas J.
Mackey, D. Steven
Mackey, Niloufer
Tisseur, Françoise
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
For any matrix automorphism group $\G$ associated with a
bilinear or sesquilinear form,
Mackey, Mackey, and Tisseur have recently shown that the
matrix sign decomposition factors of $A\in\G$ also lie in $\G$;
moreover, the polar factors of $A$ lie in $\G$ if the matrix of the underlying
form is unitary.
Groups satisfying the latter condition
include the complex orthogonal,
real and complex symplectic, and pseudo-orthogonal groups.
This work is concerned with exploiting the structure of $\G$ when computing the
polar and matrix sign decompositions of matrices in $\G$.
We give sufficient conditions for a matrix iteration to preserve the
group structure and show that a family of globally convergent
rational Pad\'e-based iterations of Kenney and Laub satisfy these conditions.
The well-known scaled Newton iteration for computing the unitary polar
factor does not preserve group structure,
but we show that the approach of the iterates to the group
is precisely tethered to the approach to unitarity,
and that this forces a different and exploitable structure in the iterates.
A similar relation holds for the Newton iteration for the matrix sign function.
We also prove that the number of iterations needed for convergence of the
structure-preserving methods
can be precisely predicted by running an associated scalar iteration.
Numerical experiments are given to compare the cubically and quintically
converging iterations with Newton's method and to test stopping criteria.
The overall conclusion is that the structure-preserving iterations
and the scaled Newton iteration are all of practical interest,
and which iteration is to be preferred is problem-dependent.
2004
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/49/1/paper.pdf
Higham, Nicholas J. and Mackey, D. Steven and Mackey, Niloufer and Tisseur, Françoise (2004) Computing the Polar Decomposition and the Matrix Sign Decomposition in Matrix Groups. SIAM Journal On Matrix Analysis and Applications, 25 (4). pp. 1178-1192. ISSN 1095-7162
10.1137/S0895479803426644
10.1137/S0895479803426644
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Blocks with trivial intersection defect groups
Eaton, Charles W
An, Jianbei
20 Group theory and generalizations
We show that each block whose defect groups intersect pairwise trivially either has cyclic or generalised quaternion defect groups, or is Morita equivalent to one of a given list of blocks of central extensions of automorphism groups of non-abelian simple groups. In particular we classify all blocks of automorphism groups of non-abelian simple groups whose defect groups are non-cyclic and intersect pairwise trivially. A consequence is that Donovans conjecture holds for blocks whose defect groups intersect pairwise trivially.
2004
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/51/1/pc09k1q7ac55wpyb.pdf
Eaton, Charles W and An, Jianbei (2004) Blocks with trivial intersection defect groups. Mathematisches Zeitschrift, 247 (3). pp. 461-486. ISSN 0025-5874
http://springerlink.metapress.com/(z0x1h045yqsvmh45hlcd03im)/app/home/contribution.asp?referrer=parent&backto=issue,2,12;journal,19,112;linkingpublicationresults,1:100443,1
10.1007/s00209-003-0545-8
10.1007/s00209-003-0545-8
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https://eprints.maths.manchester.ac.uk/65/
Functions of Matrices
Higham, Nicholas J.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
2005-11-25
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/65/1/fm_final.pdf
Higham, Nicholas J. (2005) Functions of Matrices. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:71
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https://eprints.maths.manchester.ac.uk/71/
The Conditioning of Linearizations of
Matrix Polynomials
Higham, Nicholas J.
Mackey, D. Steven
Tisseur, Françoise
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
The standard way of solving the polynomial eigenvalue problem of degree $m$
in $n\times n$ matrices
is to ``linearize'' to a pencil in $mn\times mn$ matrices
and solve the generalized eigenvalue problem.
For a given polynomial, $P$, infinitely many linearizations exist
and they can have widely varying eigenvalue condition numbers.
We investigate the conditioning of
linearizations from a vector space $\mathbb{DL}(P)$ of pencils
recently identified and studied by
Mackey, Mackey, Mehl, and Mehrmann.
We look for the best conditioned linearization and
compare the conditioning with that of the original polynomial.
Two particular pencils are shown always to be
almost optimal over linearizations in $\mathbb{DL}(P)$ for eigenvalues of
modulus greater than or less
than 1, respectively,
provided that the problem is not too badly scaled
and that the pencils are linearizations.
Moreover, under this scaling assumption,
these pencils are shown to be
about as well conditioned as the original polynomial.
For quadratic eigenvalue problems that are not too heavily damped,
a simple scaling is shown to convert the problem to one that is well scaled.
We also analyze the eigenvalue conditioning
of the widely used first and second companion linearizations.
The conditioning of the first companion linearization relative to that of
$P$ is shown to depend on the coefficient matrix norms,
the eigenvalue, and the left \ev s of the linearization and of $P$.
The companion form is found to be potentially much more ill conditioned than
$P$,
but if the 2-norms of the coefficient matrices are all approximately 1
then the companion form and $P$ are guaranteed to have similar
condition numbers.
Analogous results hold for the second companion form.
Our results are phrased in terms of both the standard relative condition number
and the condition number of Dedieu and Tisseur for the problem in
homogeneous form,
this latter condition number having the advantage of applying to zero and
infinite eigenvalues.
2005-11-25
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/71/1/paper8.pdf
Higham, Nicholas J. and Mackey, D. Steven and Tisseur, Françoise (2005) The Conditioning of Linearizations of Matrix Polynomials. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:72
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oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:73
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https://eprints.maths.manchester.ac.uk/73/
Topology from time series
Muldoon, M. R.
MacKay, R. S.
Huke, J. P.
Broomhead, D. S.
37 Dynamical systems and ergodic theory
54 General topology
89 Other areas of applied and interdisciplinary physics
We describe methods for the study of topological properties of the invariant manifolds of experimental dynamical systems. We explain how to compute such invariants as the Euler characteristic and Betti numbers using time series data, and suggest a number of potential applications.
1993-05-15
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/73/1/PhysicaD65.pdf
Muldoon, M. R. and MacKay, R. S. and Huke, J. P. and Broomhead, D. S. (1993) Topology from time series. Physica D, 65 (1-2). pp. 1-16. ISSN 0167-2789
10.1016/0167-2789(92)00026-U
10.1016/0167-2789(92)00026-U
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:74
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https://eprints.maths.manchester.ac.uk/74/
Symmetric Linearizations for Matrix Polynomials
Higham, Nicholas J.
Mackey, D. Steven
Mackey, Niloufer
Tisseur, Françoise
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
A standard way of treating the polynomial eigenvalue problem
$P(\l)x = 0$ is to convert it
into an equivalent matrix pencil---a process known as linearization.
Two vector spaces of pencils
$\Ell_1(P)$ and $\Ell_2(P)$, and their intersection $\DL(P)$,
have recently been defined and studied by
Mackey, Mackey, Mehl, and Mehrmann.
The aim of our work is to gain new insight into these spaces
and the extent to which their constituent pencils inherit
structure from $P$\@.
For arbitrary polynomials we show that every pencil in $\DL(P)$
is block symmetric
and we obtain a convenient basis for $\DL(P)$ built from block Hankel matrices.
This basis is then exploited to
prove that
the first $\deg(P)$ pencils in a sequence constructed by
Lancaster in the 1960s generate $\DL(P)$.
When $P$ is symmetric, we show that
the symmetric pencils in $\Ell_1(P)$ comprise
$\DL(P)$,
while for Hermitian $P$ the Hermitian pencils in $\Ell_1(P)$
form a proper subset of $\DL(P)$ that we explicitly characterize.
Almost all pencils in
each of these subsets are shown to be linearizations.
In addition to obtaining new results, this work provides a self-contained
treatment of some of the key properties of $\DL(P)$
together with some new, more concise proofs.
2005-11-29
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/74/1/paper5.pdf
Higham, Nicholas J. and Mackey, D. Steven and Mackey, Niloufer and Tisseur, Françoise (2005) Symmetric Linearizations for Matrix Polynomials. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:95
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Vector Spaces of Linearizations for Matrix Polynomials
Mackey, D. Steven
Mackey, Niloufer
Mehl, Christian
Mehrmann, Volker
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
The classical approach to investigating polynomial eigenvalue problems is linearization,
where the polynomial is converted into a larger matrix pencil with the same eigenvalues. For
any polynomial there are innitely many linearizations with widely varying properties, but in practice
the companion forms are typically used. However, these companion forms are not always entirely
satisfactory, and linearizations with special properties may sometimes be required.
Given a matrix polynomial P, we develop a systematic approach to generating large classes of
linearizations for P. We show how to simply construct two vector spaces of pencils that generalize the
companion forms of P, and prove that almost all of these pencils are linearizations for P. Eigenvectors
of these pencils are shown to be closely related to those of P. A distinguished subspace is then
isolated, and the special properties of these pencils are investigated. These spaces of pencils provide
a convenient arena in which to look for structured linearizations of structured polynomials, as well
as to try to optimize the conditioning of linearizations [7], [8], [12].
2005-12-01
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/95/1/genlin.pdf
Mackey, D. Steven and Mackey, Niloufer and Mehl, Christian and Mehrmann, Volker (2005) Vector Spaces of Linearizations for Matrix Polynomials. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:96
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Computing $f(A)b$ for Matrix Functions $f$
Davies, Philip I.
Higham, Nicholas J.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
For matrix functions $f$
we investigate how to compute a matrix-vector product $f(A)b$
without explicitly computing $f(A)$.
A general method is described that applies quadrature
to the matrix version of the Cauchy integral theorem.
Methods specific to the logarithm,
based on quadrature,
and fractional matrix powers,
based on solution of an ordinary differential equation initial value problem,
are also presented
Springer-Verlag
Borici, Artan
Frommer, Andreas
Joo, Balint
Kennedy, Anthony
Pendleton, Brian
2005
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/96/1/narep436.pdf
Davies, Philip I. and Higham, Nicholas J. (2005) Computing $f(A)b$ for Matrix Functions $f$. In: QCD and Numerical Analysis III. Lecture Notes in Computational Science and Engineering, 47 . Springer-Verlag, Berlin, pp. 15-24. ISBN 3-540-21257-4
http://www.springer.com/sgw/cda/frontpage/0,11855,4-10042-72-50493377-0,00.html
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:97
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oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:98
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https://eprints.maths.manchester.ac.uk/98/
Stability of Relative Equilibria of Point Vortices on the Sphere
Laurent-Polz, Frederic
Montaldi, James
Roberts, Mark
37 Dynamical systems and ergodic theory
76 Fluid mechanics
We describe the linear and nonlinear stability and instability of certain configurations of point vortices on the sphere forming relative equilibria. These configurations consist of up to two rings, with and without polar vortices. Such configurations have dihedral symmetry, and the symmetry
is used both to block diagonalize the relevant matrices and to distinguish the subspaces on which their eigenvalues need to be calculated.
2005-12-02
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/98/1/VorticesStability.pdf
Laurent-Polz, Frederic and Montaldi, James and Roberts, Mark (2005) Stability of Relative Equilibria of Point Vortices on the Sphere. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:99
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https://eprints.maths.manchester.ac.uk/99/
Longitudinal and Cross-Sectional Analysis of Cytotoxic T Lymphocyte (CTL) Responses and their Relationship to Vertical HIV Transmission
Jin, Xia
Roberts, Caroline G. P.
Nixon, Douglas F.
Cao, Yunzhen
Ho, David D.
Walker, Bruce D.
Muldoon, Mark
Korber, Bette T. M.
Koup, Richard A.
ARIEL Project Investigators, the
62 Statistics
92 Biology and other natural sciences
The ARIEL Project for the Prevention of HIV Transmission from Mother to Infant was established to evaluate virologic and immunologic parameters during vertical transmission. To determine the strength and breadth of the cytotoxic T lymphocyte (CTL) response and its correlation with human immunodeficiency virus (HIV) transmission, a cross-sectional study was done of 31 HIV-infected pregnant women, of whom 15 transmitted and 16 did not transmit HIV to their infants. The precursor frequencies of CTL specific for HIV-1 gag, pol, nef, and env from 5 different isolates of the clade B of HIV-1 were determined by limiting dilution analysis. Results showed that variable levels of HIV-specific CTL response were present in HIV-infected pregnant women during and after pregnancy. In addition, CTL precursor frequencies specific for pol and nef were higher during pregnancy in nontransmitters than in transmitters. Thus, CTL responding to different HIV antigens may not be contributing equally to the prevention of vertical transmission.
1998-11
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/99/1/journal/issues/v178n5/980240/980240.web.pdf
Jin, Xia and Roberts, Caroline G. P. and Nixon, Douglas F. and Cao, Yunzhen and Ho, David D. and Walker, Bruce D. and Muldoon, Mark and Korber, Bette T. M. and Koup, Richard A. and ARIEL Project Investigators, the (1998) Longitudinal and Cross-Sectional Analysis of Cytotoxic T Lymphocyte (CTL) Responses and their Relationship to Vertical HIV Transmission. Journal of Infectious Diseases, 178 (5). pp. 1317-1326. ISSN 0022-1899
http://www.journals.uchicago.edu/JID/journal/issues/v178n5/980240/980240.html
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:101
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An Introduction to the Quality of Computed Solutions
Hammarling, Sven
65 Numerical analysis
This report is concerned with the quality of the computed numerical solutions of mathematical problems. We give an introduction to ideas that are important in understanding and measuring the quality of computed solutions. In particular we
review the ideas of condition, stability and error analysis, and their realisation in numerical software. A number of illustrative examples are given.
SIAM
2005-07
Book Section
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/101/1/intro-qual.pdf
Hammarling, Sven (2005) An Introduction to the Quality of Computed Solutions. In: Accuracy and Reliability in Scientific Computing. SIAM, Philadelphia, USA, pp. 43-76. ISBN 0-89871-584-9
http://ec-securehost.com/SIAM/SE18.html
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Rationality as conformity
Hosni, Hykel
Paris, Jeff
03 Mathematical logic and foundations
We argue in favour of identifying one aspect of rational choice with the tendency to conform to the choice you expect another like-minded, but non-communicating, agent to make and study this idea in the very basic case where the choice is from a non-empty subset K of 2^A and no further structure or knowledge of A is assumed.
2005-03
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/102/1/rationality-as-conformity.pdf
Hosni, Hykel and Paris, Jeff (2005) Rationality as conformity. Knowledge Rationality and Action (Synthese), 144 (2). pp. 249-285. ISSN 0039-7857
http://www.springerlink.com/link.asp?id=w55w56h35553788j
10.1007/s11229-004-4684-1
10.1007/s11229-004-4684-1
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:103
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https://eprints.maths.manchester.ac.uk/103/
Interpretation, coordination and conformity
Hosni, Hykel
03 Mathematical logic and foundations
91 Game theory, economics, social and behavioral sciences
The aim of this note is to investigate a very general problem of (radical) interpretation in terms of a simple coordination game: the conformity game. We show how, within our mathematical framework, the solution concept for the conformity game does indeed provide an algorithmic procedure facilitating triangulation, in the sense of Davidson.
Springer
Pietarinen, Ahti-Veikko
Majer, Ondrej
Tulenheimo, Tero
2005-06
Book Section
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/103/1/interpretation_coordination_and_conformity.pdf
Hosni, Hykel (2005) Interpretation, coordination and conformity. In: Logic and Games: Foundational Perspectives. Springer. (In Press)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:105
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Vortex dynamics on cylinders
Montaldi, James
Souliere, Anik
Tokieda, Tadashi
37 Dynamical systems and ergodic theory
70 Mechanics of particles and systems
76 Fluid mechanics
Point vortices on a cylinder (periodic strip) are studied geometrically, using local integrals of motion. The Hamiltonian formalism is developed, a non-existence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging.
2003
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/105/1/cylinder.pdf
Montaldi, James and Souliere, Anik and Tokieda, Tadashi (2003) Vortex dynamics on cylinders. SIAM J. on Applied Dynamical Systems, 2. pp. 417-430. ISSN 1536-0040
http://epubs.siam.org/sam-bin/dbq/article/41556
10.1137/S1111111102415569
10.1137/S1111111102415569
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:106
2017-10-20T14:12:03Z
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A new bound for the smallest x with \pi(x) > \li(x)
Chao, Kuok Fai
Plymen, Roger
11 Number theory
We reduce the dominant term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays & Hudson. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [1.39792101 \times 10^316, 1.39847603 \times 10^316] for which \pi(x) > \li(x). This interval is strictly a sub-interval of the interval in Bays & Hudson [1], and is narrower by a factor of about 10.
2006
Article
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/106/1/new.pdf
Chao, Kuok Fai and Plymen, Roger (2006) A new bound for the smallest x with \pi(x) > \li(x). math.NT/0509312. pp. 1-16. (Submitted)
http://arxiv.org/abs/math.NT/0509312
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Homotopy Decompositions and K-theory of Bott Towers
Civan, Yusuf
Ray, Nigel
55 Algebraic topology
57 Manifolds and cell complexes
We describe Bott towers as sequences of toric manifolds $M^k$, and identify the omniorientations which correspond
to their original construction as complex varieties. We
show that the suspension of M^k is homotopy equivalent to
a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples,
and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky's analysis of the Adams Spectral Sequence. By
way of application we consider the enumeration of stably complex structures on M^k, obtaining estimates for those which arise from omniorientations and those which are
almost complex. We conclude with observations on the role
of Bott towers in complex cobordism theory.
2004-08-10
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/107/1/newhdrkbt.pdf
Civan, Yusuf and Ray, Nigel (2004) Homotopy Decompositions and K-theory of Bott Towers. K-theory, 34 (1). pp. 1-33. ISSN 0920-3036
http://dx.doi.org/10.1007/s10977-005-1551-x
10.1007/s10977-005-1551-x
10.1007/s10977-005-1551-x
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:108
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Circle actions on toric manifolds and their applications
Buchstaber, Victor M
55 Algebraic topology
57 Manifolds and cell complexes
2005-11-17
Conference or Workshop Item
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/108/1/vmbfinalslides05.pdf
Buchstaber, Victor M (2005) Circle actions on toric manifolds and their applications. In: Pure Mathematics Colloquium, University of Leicester, 17 November 2005, Leicester, England. (Unpublished)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:110
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Persistence and stability of relative equilibria
Montaldi, James
37 Dynamical systems and ergodic theory
58 Global analysis, analysis on manifolds
70 Mechanics of particles and systems
We consider relative equilibria in symmetric Hamiltonian systems, and their persistence or bifurcation as the momentum is varied. In particular, we extend a classical result about persistence of relative equilibria from values of the momentum map that are regular for the coadjoint action, to arbitrary values, provided that either (i) the relative equilibrium is at a local extremum of the reduced Hamiltonian or (ii) the action on the phase space is (locally) free. The first case uses just point-set topology, while in the second we rely on the local normal form for (free) symplectic group actions, and then apply the splitting lemma. We also consider the Lyapunov stability of extremal relative equilibria. The group of symmetries is assumed to be compact.
1997
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/110/1/persistence.pdf
Montaldi, James (1997) Persistence and stability of relative equilibria. Nonlinearity, 10. pp. 449-466. ISSN 1361-6544
http://www.iop.org/EJ/abstract/0951-7715/10/2/009
10.1088/0951-7715/10/2/009
10.1088/0951-7715/10/2/009
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:111
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Variations on a theme of Timmesfeld: a finite group-theoretic analogue of the classification of groups of finite Morley rank and even type
Borovik, Alexandre
20 Group theory and generalizations
The paper addresses a question whether there is a reasonable
self-contained theory of finite simple groups of even type which is closely parallel to the theory of groups and finite Morley rank.
2006-08-25
MIMS Preprint
NonPeerReviewed
application/pdf
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https://eprints.maths.manchester.ac.uk/111/1/atomic15.pdf
Borovik, Alexandre (2006) Variations on a theme of Timmesfeld: a finite group-theoretic analogue of the classification of groups of finite Morley rank and even type. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:113
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The Conjugacy Problem in Amalgamated Products I: Regular Elements and Black Holes
Borovik, Alexandre
Myasnikov, Alexei G.
Remeslennikov, Vladimir V.
20 Group theory and generalizations
We discuss the time complexity of the word and conjugacy problems for free products of two groups with
amalgamation over a subgroup. We stratify the set of elements of the fre product with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable assumptions about the groups groups involved. Moreover, the decision algorithm is fast on
the generic stratum.
2006-02-16
MIMS Preprint
NonPeerReviewed
application/pdf
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https://eprints.maths.manchester.ac.uk/113/1/amalgam_Ie.pdf
Borovik, Alexandre and Myasnikov, Alexei G. and Remeslennikov, Vladimir V. (2006) The Conjugacy Problem in Amalgamated Products I: Regular Elements and Black Holes. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:114
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Isomorphism classes for Banach vector bundle structures of second tangents
Dodson, CTJ
Galanis, GN
Vassiliou, E
58 Global analysis, analysis on manifolds
On a smooth Banach manifold M, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle T^2M of M. This is a vector bundle in the presence of a linear connection on M and the corresponding local structure is heavily dependent on the choice of connection. In this paper we study the extent of this
dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on T^2M remains invariant under conjugate connections with respect to diffeomorphisms of M.
2006-01-23
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/114/1/isomt2m.pdf
Dodson, CTJ and Galanis, GN and Vassiliou, E (2006) Isomorphism classes for Banach vector bundle structures of second tangents. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:115
2017-11-07T22:38:44Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:116
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A generalized second order frame bundle for Fréchet manifolds
Dodson, CTJ
Galanis, GN
Vassiliou, E
53 Differential geometry
58 Global analysis, analysis on manifolds
Working within the framework of Fréchet modelled infinite dimensional manifolds, we propose a generalized notion of second order frame bundle. We revise in this way the classical notion of bundles of linear frames of order
two, the direct definition and study of which is problematic due to intrinsic difficulties of the space models. However, this new structure keeps all the fundamental characteristics of a frame bundle: It is a principal Frechet bundle associated (differentially and geometrically) with the corresponding second order tangent bundle.
2005
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/116/1/FL2M.pdf
Dodson, CTJ and Galanis, GN and Vassiliou, E (2005) A generalized second order frame bundle for Fréchet manifolds. J. Geometry Physics, 55 (3). pp. 291-305. ISSN 0393-0440
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:117
2017-10-20T14:12:03Z
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Rationality as conformity
Hosni, Hykel
03 Mathematical logic and foundations
We address the problem of characterizing the choice processes of two like-minded yet
non-communicating agents who intend to select, from a finite set of options, the same
possible world. Hence, we call the resulting framework “Rationality-as-conformity”
Within the scope of our formalisation, in which a choice problem is defined on
non-empty subset of maps from a finite set A to the binary set {0, 1}, we introduce
and investigate three distinct logico-mathematical characterisations of Rationality-as-
conformity.
Finally, we discuss the applicability of our framework to problems such as pure
coordination games and radical interpretation which are traditionally related to “rationality”.
2005-11
Thesis
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/117/1/hh-thesis.pdf
Hosni, Hykel (2005) Rationality as conformity. Doctoral thesis, University of Manchester.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:118
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Openness of momentum maps and persistence of extremal relative equilibria
Montaldi, James
Tokieda, Tadashi
37 Dynamical systems and ergodic theory
53 Differential geometry
We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence.
2003
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/118/1/openMM.pdf
Montaldi, James and Tokieda, Tadashi (2003) Openness of momentum maps and persistence of extremal relative equilibria. Topology, 42. pp. 833-844. ISSN 0040-9383
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V1J-47HS1FW-2&_coverDate=07%2F31%2F2003&_alid=345214096&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=5676&_sort=d&view=c&_acct=C000010021&_version=1&_urlVersion=0&_userid=121749&md5=42d0a9f9d22e87b61137d
10.1016/S0040-9383(02)00047-2
10.1016/S0040-9383(02)00047-2
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:120
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Universal connection and curvature for statistical manifold geometry
Arwini, Khadiga
Del Riego, L
Dodson, CTJ
53 Differential geometry
62 Statistics
Statistical manifolds are representations of smooth families of
probability density functions
that allow differential geometric methods to be applied to
problems in stochastic processes, mathematical statistics and
information theory. It is common to have to consider a number of
linear connections on a given statistical manifold and so it is
important to know the corresponding universal connection and
curvature; then all linear connections and their curvatures are
pullbacks. An important class of statistical manifolds is that
arising from the exponential families and one particular family is
that of gamma distributions, which we showed recently to have
important uniqueness properties in stochastic processes. Here we
provide formulae for universal connections and curvatures on
exponential families and give an explicit example for the manifold
of gamma distributions.
2005
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/120/1/univconexpf.pdf
Arwini, Khadiga and Del Riego, L and Dodson, CTJ (2005) Universal connection and curvature for statistical manifold geometry. Houston Journal of Mathematics, In pre. ISSN 0362-1588 (In Press)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:121
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Neighbourhoods of independence and associated geometry
Arwini, Khadiga
Dodson, CTJ
53 Differential geometry
60 Probability theory and stochastic processes
We provide explicit information geometric tubular neighbourhoods containing all bivariate
processes sufficiently close to the cases of independent Poisson or Gaussian processes.
This is achieved via affine immersions of the 4-manifold of Freund bivariate distributions
and of the 5-manifold of bivariate Gaussians. We provide also the alpha-geometry for both
manifolds. The Central Limit Theorem makes our neighbourhoods of independence limiting cases
for a wide range of bivariate processes; the topological character of the results makes
them stable under small perturbations, which is important for applications.
2005-12-13
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/121/1/nhdindep.pdf
Arwini, Khadiga and Dodson, CTJ (2005) Neighbourhoods of independence and associated geometry. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:122
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Bundles of acceleration on Banach manifolds
Dodson, CTJ
Galanis, GN
53 Differential geometry
58 Global analysis, analysis on manifolds
We consider an infinite dimensional manifold M modelled on a
Banach space E and we construct smooth fiber bundle
structures on the tangent bundle of order two T^2M, which
consists of all smooth curves of M that agree up to their
acceleration, as well as on the corresponding second order frame
bundle L^2M. These bundles prove to be associated with
respect to the identity representation of the general linear group GL(E}) that serves as the structure group of both of them. Moreover, a
bijective correspondence between linear connections on T^2M and
connection forms of L^2M is revealed.
2005
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/122/1/dodson_galanis_WCNA.pdf
Dodson, CTJ and Galanis, GN (2005) Bundles of acceleration on Banach manifolds. Nonlinear Analysis, 63 (5-7). pp. 465-471. ISSN 0362-546X
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:123
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The n-valued groups: results and prospectives
Buchstaber, Victor M.
22 Topological groups, Lie groups
2005-11-24
Conference or Workshop Item
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/123/1/Multivalued_groups_Leeds--2005.pdf
Buchstaber, Victor M. (2005) The n-valued groups: results and prospectives. In: Pure Mathematics Colloqium, Leeds University, 24 Nov 2005, Leeds. (Unpublished)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:124
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Efficient Algorithms for the Matrix Cosine and Sine
Hargreaves, Gareth I.
Higham, Nicholas J.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
Several improvements are made to an algorithm of Higham and Smith for
computing the matrix cosine.
The original algorithm scales the matrix by a power of 2 to bring the
$\infty$-norm to 1 or less,
evaluates the [8/8] Pad\'e approximant,
then uses the double-angle formula $\cos(2A) = 2\cos^2A - I$ to recover the
cosine of the original matrix.
The first improvement is to phrase truncation error bounds in terms of
$\norm{A^2}^{1/2}$ instead of the (no smaller and potentially much larger
quantity) $\norm{A}$.
The second is to choose the
degree of the Pad\'e approximant to minimize the computational cost
subject to achieving a desired truncation error.
A third improvement is to use an absolute, rather than relative,
error criterion in the choice of Pad\'e approximant;
this allows the use of higher degree approximants without
worsening an a priori error bound.
Our theory and experiments show that each of these modifications brings a
reduction in computational cost.
Moreover, because the modifications tend to reduce the number of double-angle
steps they usually result in a more accurate computed cosine
in floating point arithmetic.
We also derive an algorithm for computing
both $\cos(A)$ and $\sin(A)$,
by adapting the ideas developed for the
cosine and intertwining the cosine and sine double angle recurrences.
2005
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/124/1/paper2.pdf
Hargreaves, Gareth I. and Higham, Nicholas J. (2005) Efficient Algorithms for the Matrix Cosine and Sine. Numerical Algorithms, 40 (4). pp. 383-400. ISSN 1572-9265
10.1007/s11075-005-8141-0
10.1007/s11075-005-8141-0
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:125
2017-11-07T22:38:44Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:126
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https://eprints.maths.manchester.ac.uk/126/
Infinite dimensional second order differential equations via $T^2M$
Aghasi, M
Dodson, CTJ
Galanis, GN
Suri, A
53 Differential geometry
58 Global analysis, analysis on manifolds
The vector bundle structure obtained on the second order
(acceleration) tangent bundle T^2M of a smooth manifold M by
means of a linear connection on the base provides an alternative
way for the study of second order differential equations on
manifolds of finite and infinite dimension. Second order vector
fields and their integral curves provide a new way of solving a
wide class of second order differential equations on Frechet
manifolds and may be used also to describe geodesic curves on a
Riemannian manifold. The new technique proposed is illustrated by
concrete examples within the framework of Banach and Frechet
spaces as well as on Lie groups.
2005-12-16
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/126/1/t2de.pdf
Aghasi, M and Dodson, CTJ and Galanis, GN and Suri, A (2005) Infinite dimensional second order differential equations via $T^2M$. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:127
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Isomorphism classes for Banach vector bundle structures of second
tangents
Dodson, CTJ
Galanis, GN
Vassiliou, E
53 Differential geometry
58 Global analysis, analysis on manifolds
On a smooth Banach manifold M,$the equivalence classes of curves
that agree up to acceleration form the second order tangent bundle
T^2M of M.
This is a vector bundle in the presence of a linear connection on
M and the corresponding local structure is heavily dependent on
the choice of connection. In this paper we study the extent of this
dependence and we prove that it is closely related to the notions
of conjugate connections and
second order differentials. In particular, the vector bundle structure on
T^2M remains invariant under conjugate connections with respect
to diffeomorphisms of M.
2005
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/127/1/isomt2m.pdf
Dodson, CTJ and Galanis, GN and Vassiliou, E (2005) Isomorphism classes for Banach vector bundle structures of second tangents. Math. Proc. Camb. Phil. Soc.. ISSN 0305-0041 (In Press)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:129
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Using human immunodeficiency virus type 1 sequences to infer historical features of the acquired immune deficiency syndrome epidemic and human immunodeficiency virus evolution
Yusim, Karina
Peters, Martine
Pybus, Oliver
Bhattacharya, Tanmoy
Delaporte, Eric
Mulanaga, Claire
Muldoon, Mark
Theiler, James
Korber, Bette
92 Biology and other natural sciences
In earlier work, human immunodeficiency virus type 1 (HIV-1) sequences were analysed to estimate the timing of the ancestral sequence of the main group of HIV-1, the virus that is responsible for the acquired immune deficiency syndrome pandemic, yielding a best estimate of 1931 (95% confidence interval of 1915-1941). That work will be briefly reviewed, outlining how phylogenetic tools were extended to incorporate improved evolutionary models, how the molecular clock model was adapted to incorporate variable periods of latency, and how the approach was validated by correctly estimating the timing of two historically documented dates. The advantages, limitations, and assumptions of the approach will be summarized, with particular consideration of the implications of branch length uncertainty and recombination. We have recently undertaken new phylogenetic analysis of an extremely diverse set of human immunodeficiency virus envelope sequences from the Democratic Republic of the Congo (the DRC, formerly Zaire). This analysis both corroborates and extends the conclusions of our original study. Coalescent methods were used to infer the demographic history of the HIV-1 epidemic in the DRC, and the results suggest an increase in the exponential growth rate of the infected population through time.
Manchester Institute for the Mathematical Sciences
2001-06-29
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/129/1/PhilTransRoySocB356.pdf
Yusim, Karina and Peters, Martine and Pybus, Oliver and Bhattacharya, Tanmoy and Delaporte, Eric and Mulanaga, Claire and Muldoon, Mark and Theiler, James and Korber, Bette (2001) Using human immunodeficiency virus type 1 sequences to infer historical features of the acquired immune deficiency syndrome epidemic and human immunodeficiency virus evolution. Philosophical Transactions of the Royal Society of London, B, 356 (1410). pp. 855-866. ISSN 0962-8436
10.1098/rstb.2001.0859
10.1098/rstb.2001.0859
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:130
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https://eprints.maths.manchester.ac.uk/130/
Iterative Solution of a Nonsymmetric Algebraic Riccati Equation
Guo, Chun-Hua
Higham, Nicholas J.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
We study the nonsymmetric algebraic Riccati equation whose four
coefficient matrices are the blocks of
a nonsingular $M$-matrix or an irreducible singular
$M$-matrix $M$. The solution of practical interest is the minimal nonnegative
solution. We show that Newton's method with zero initial guess can be used to
find this solution without any further assumptions. We also present a
qualitative perturbation analysis for the minimal solution, which is
instructive in designing algorithms for finding more accurate approximations.
For the most practically important case, in
which $M$ is an irreducible singular $M$-matrix with zero row sums,
the minimal solution is either stochastic or substochastic and
the Riccati equation can be tranformed into a unilateral matrix equation by
a procedure of Ramaswami. The minimal solution of the Riccati equation can
then be found by computing the minimal nonnegative solution of
the unilateral equation using
the Latouche--Ramaswami algorithm.
We show that the Latouche--Ramawami algorithm, combined with a shift
technique suggested by He, Mini, and Rhee,
is breakdown-free in all cases and is able to find the minimal solution
more efficiently and more accurately than the algorithm without a shift.
Our approach is to find a proper
stochastic solution using the
shift technique even if it is not the minimal solution. We show how we can
easily recover the minimal solution when it is not the computed stochastic
solution.
2005-12-16
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/130/1/paper.pdf
Guo, Chun-Hua and Higham, Nicholas J. (2005) Iterative Solution of a Nonsymmetric Algebraic Riccati Equation. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:131
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Topics in Information Geometry
Dodson, CTJ
53 Differential geometry
60 Probability theory and stochastic processes
We introduce first some of the background ideas on information theory and its role in studying
analytic models for stochastic processes and the geometrization of families of measure functions.
This is then used to present the geometry of important examples of the Riemannian manifolds that
arise. Next, we obtain the proof of two theorems that characterise the metric neighbourhoods of the
two distinguished fundamental states: randomness and independence. These methods have had
applications in modelling cryptographic attacks, cosmological void distributions, porous media,
clustering of: galaxies, communications, and amino acids along protein chains in genomes.
2005
Conference or Workshop Item
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/131/1/InfoGeom.pdf
Dodson, CTJ (2005) Topics in Information Geometry. In: Workshop on Recent Results in Information Geometry, 13-19 Dec 2005, Santiago de Compostela, Spain. (Unpublished)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:132
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Krichever-Novikov continuous basis for plane algebraic curves
Buchstaber, Victor M.
Leykin, Dmitry
14 Algebraic geometry
2005-12-12
Conference or Workshop Item
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/132/1/KN-basis-Liverpool-2005.pdf
Buchstaber, Victor M. and Leykin, Dmitry (2005) Krichever-Novikov continuous basis for plane algebraic curves. In: Singularities seminar, 12 Dec 2005, Department of Mathematics Sciences, University of Liverpool. (Unpublished)
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:133
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https://eprints.maths.manchester.ac.uk/133/
An Iterative Method for Electrostatic Object Reconstruction in a Half Space
van Berkel, Cees
Lionheart, William R. B.
35 Partial differential equations
41 Electromagnetism; electron and ion optics
Sensing electrodes arranged in or around a display can provide input function for interactive displays. Commercially this is interesting because the sensing electrodes and electronics can be made in the same manufacturing process as that of the display itself thus reducing cost. In engineering terms the electrodes measure capacitance changes resulting from the presence and movement of objects such as hands and fingers in front of the display. At the quasi static frequencies used (100kHz) the human body is conductive and the hands or fingers provide a screen between the capacitive electrodes. There is no need to touch the actual display and the overall system constitutes a touchless gesture input system. Determining the shape of the hand or fingers is a boundary condition reconstruction problem of finding the boundary of an earthed conductive object D from electrostatic measurements. This is the ill-posed problem of recovering the zero-surface of a solution to Laplace s equation from Cauchy data on part of the boundary of a domain. The problem has similarities with object reconstruction in EIT or inverse scattering but is complicated because only a partial Dirichelet-Neumannn map is available as experimental data. We suggest an algorithm where at each iteration we have an approximation on which we calculate approximate Cauchy data by solving a Tikhonov regularized linear system. This data is used to modify the approximation by extrapolation towards the zero-surface giving the next approximation. We implemented the algorithm in two and three space dimensions using the Boundary Element Method for discretization. Numerical results using simulated data with added noise show that simply connected but not necessarily convex objects can be reconstructed with reasonable positional accuracy and approximate shape, but as might be expected the shape is more accurately determined near the plane of measurements.
2005-12-20
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/133/1/vanberkel_MST.pdf
van Berkel, Cees and Lionheart, William R. B. (2005) An Iterative Method for Electrostatic Object Reconstruction in a Half Space. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:134
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https://eprints.maths.manchester.ac.uk/134/
SDELab: stochastic differential equations with MATLAB
Gilsing, Hagen
Shardlow, Tony
34 Ordinary differential equations
60 Probability theory and stochastic processes
65 Numerical analysis
We introduce SDELab, a package for solving stochastic differential equations (SDEs) within MATLAB. SDELab features explicit and implicit integrators for a general class of Ito and Stratonovich SDEs, including Milstein's method, sophisticated algorithms for iterated stochastic integrals, and flexible plotting facilities.
2006-01-05
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/134/1/sdelab_paper.pdf
Gilsing, Hagen and Shardlow, Tony (2006) SDELab: stochastic differential equations with MATLAB. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:136
2017-10-20T14:12:03Z
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74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/136/
The Weak Euler Scheme for Stochastic
Differential Delay Equations
Buckwar, Evelyn
Kuske, Rachel
Mohammed, Salah-Eldin
Shardlow, Tony
39 Difference and functional equations
60 Probability theory and stochastic processes
65 Numerical analysis
We develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has
order of convergence 1, as in the case of stochastic ordinary
differential equations (SODEs) (i.e., without delay).The result
holds for SDDEs with multiple finite fixed delays in the drift and
diffusion terms. Although the set-up is non-anticipating,
our approach uses the Malliavin calculus and the
anticipating stochastic analysis techniques of Nualart
and Pardoux.
2006-01-05
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/136/1/weak_mal_web.pdf
Buckwar, Evelyn and Kuske, Rachel and Mohammed, Salah-Eldin and Shardlow, Tony (2006) The Weak Euler Scheme for Stochastic Differential Delay Equations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:137
2017-10-20T14:12:03Z
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74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/137/
Post processing for stochastic parabolic partial differential
equations
Lord, Gabriel
Shardlow, Tony
35 Partial differential equations
60 Probability theory and stochastic processes
65 Numerical analysis
We investigate the strong approximation of stochastic parabolic
partial differential equations with additive noise. We introduce
post-processing in the context of a standard Galerkin approximation,
although other spatial discretisations are possible. In time, we
use an
exponential integrator. We prove strong error estimates
and discuss the best number of post-processing terms to take.
Numerically, we evaluate the efficiency of the methods and observe
rates of convergence. Some experiments with the implicit
Euler--Maruyama method are described
2006-01-05
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/137/1/pp2005.pdf
Lord, Gabriel and Shardlow, Tony (2006) Post processing for stochastic parabolic partial differential equations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:138
2017-10-20T14:12:03Z
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https://eprints.maths.manchester.ac.uk/138/
Computing the Geodesic Interpolating Spline
Mills, Anna
Marsland, Stephen
Shardlow, Tony
65 Numerical analysis
68 Computer science
92 Biology and other natural sciences
We examine non-rigid image registration by knotpoint
matching. We consider registering two images, each with a set of
knotpoints marked, where one of the images is to be registered to the
other by a nonlinear warp so that the knotpoints on the template image
are exactly aligned with the corresponding knotpoints on the reference
image.
We explore two approaches to computing the Geodesic Interpolating
Spline registration. First, we describe a method which exploits the
structure of the objective function and constraints to permit
efficient optimisation and second, we outline an approach using the
framework of classical mechanics.
2006-01-05
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/138/1/comp_gis.pdf
Mills, Anna and Marsland, Stephen and Shardlow, Tony (2006) Computing the Geodesic Interpolating Spline. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:141
2017-10-27T16:38:37Z
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https://eprints.maths.manchester.ac.uk/141/
A note on the geometry of linear Hamiltonian systems of signature 0 in R4
Montaldi, James
37 Dynamical systems and ergodic theory
53 Differential geometry
It is shown that a linear Hamiltonian system on R4 is elliptic or hyperbolic according to the number of Lagrangian planes in the null-cone H^−1(0), or equivalently the number of invariant Lagrangian planes. Some extension to higher dimensions is described.
Elsevier
2007-02
Article
PeerReviewed
text
en
https://eprints.maths.manchester.ac.uk/141/1/linear_systems.pdf
Montaldi, James (2007) A note on the geometry of linear Hamiltonian systems of signature 0 in R4. J. Differential Geometry and its Applications, 25. pp. 344-350. ISSN 1749-9097
10.1016/j.difgeo.2007.02.003
10.1016/j.difgeo.2007.02.003
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:142
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74797065733D61727469636C65
https://eprints.maths.manchester.ac.uk/142/
Monodromy of certain Painlevé-VI transcendents and reflection groups
Dubrovin, Boris
Mazzocco, Marta
20 Group theory and generalizations
33 Special functions (properties of functions as functions)
34 Ordinary differential equations
We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters #=%=0, '=$\frac{1}{2}$]] and 2!=(27-1)2 with arbitrary 7, 27p]Ê. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space.
2000-02-21
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/142/1/9806056-2.pdf
Dubrovin, Boris and Mazzocco, Marta (2000) Monodromy of certain Painlevé-VI transcendents and reflection groups. Inventiones Mathematicae, 141 (1). pp. 55-147. ISSN 0020-9910
http://www.springerlink.com/openurl.asp?genre=article&issn=0020-9910&volume=141&issue=1&spage=55
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:143
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https://eprints.maths.manchester.ac.uk/143/
Tomographic reconstruction of stress from photoelastic measurements using elastic regularization.
Szotten, David
Lionheart, William RB
Tomlinson, Rachel A
53 Differential geometry
74 Mechanics of deformable solids
78 Optics, electromagnetic theory
In this paper we consider the problem of recovering the stress tensor field of a three dimensional object from
measurements of the polarization state of transmitted light. In contrast to the ray transform approach suggested
by Sharafutdinov, which uses the inversion of planar Radon transforms to recover a single component of the
deviatoric stress normal to a plane, we study the simultaneous reconstruction of all components of the deviatoric
stress in each voxel using a matrix approximation to the truncated transverse ray transform. This approach allows us to employ partial differential operators related to linear elasticity in a regularizing penalty term resulting in a well posed problem. We note that the hydrostatic stress is determined by the deviatoric stress (up to
an additive constant) from the equilibrium equation, and that our numerical results confirm that the full stress tensor can be recovered using elastic regularization.
2006-01-11
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/143/1/matrix_photoelast.pdf
Szotten, David and Lionheart, William RB and Tomlinson, Rachel A (2006) Tomographic reconstruction of stress from photoelastic measurements using elastic regularization. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:144
2017-11-07T22:38:44Z
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:146
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https://eprints.maths.manchester.ac.uk/146/
Conjugate connections and differential equations on infinite dimensional manifolds
Aghasi, M.
Dodson, C.T.J.
Galanis, G.N.
Suri, A.
53 Differential geometry
58 Global analysis, analysis on manifolds
On a smooth manifold M, the vector bundle structures of the
second order tangent bundle, T^2M, bijectively correspond to
linear connections. In this paper we classify such structures for
those Frechet manifolds which can be considered as projective
limits of Banach manifolds. We investigate also the relation
between ordinary differential equations on Frechet spaces and
the linear connections on their trivial bundle. Such equations
arise in theoretical physics.
2006-01-23
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/146/1/ccode.pdf
Aghasi, M. and Dodson, C.T.J. and Galanis, G.N. and Suri, A. (2006) Conjugate connections and differential equations on infinite dimensional manifolds. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:149
2017-10-20T14:12:04Z
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74797065733D4D494D535F7072657072696E74
https://eprints.maths.manchester.ac.uk/149/
The Weak Euler Scheme for Stochastic
Differential Delay Equations
Buckwar, Evelyn
Kuske, Rachel
Mohammed, Salah-Eldin
Shardlow, Tony
39 Difference and functional equations
60 Probability theory and stochastic processes
65 Numerical analysis
We develop a weak numerical Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The weak Euler scheme has
order of convergence 1, as in the case of stochastic ordinary
differential equations (SODEs) (i.e., without delay).The result
holds for SDDEs with multiple finite fixed delays in the drift and
diffusion terms. Although the set-up is non-anticipating,
our approach uses the Malliavin calculus and the
anticipating stochastic analysis techniques of Nualart
and Pardoux.
2006-01-24
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/149/1/weak_mal_web.pdf
Buckwar, Evelyn and Kuske, Rachel and Mohammed, Salah-Eldin and Shardlow, Tony (2006) The Weak Euler Scheme for Stochastic Differential Delay Equations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:150
2017-10-20T14:12:04Z
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https://eprints.maths.manchester.ac.uk/150/
A Review on the Numerical Solution of the 1D Euler Equations
Hudson, Justin
35 Partial differential equations
65 Numerical analysis
76 Fluid mechanics
This paper presents a review on the numerical solution of the Euler equations. Three different high resolution versions of the following schemes are considered: Roe's scheme, the HLLE scheme and the AUSM+ scheme. We present a variety of test cases, each designed to test the robustness of each scheme and compare the results to determine which scheme was the most accurate.
2006-01-26
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/150/1/JuH-EE.pdf
application/postscript
en
https://eprints.maths.manchester.ac.uk/150/2/JuH-EE.ps
Hudson, Justin (2006) A Review on the Numerical Solution of the 1D Euler Equations. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:151
2017-11-08T18:18:29Z
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https://eprints.maths.manchester.ac.uk/151/
Numerical Modelling of Eulerian Two-Phase Gas-Solid Flow
Hudson, Justin
Harris, David
35 Partial differential equations
65 Numerical analysis
76 Fluid mechanics
This paper investigates the numerical solution of the equations governing two-phase flow of a solid granular material dispersed in a gas. We consider two different models in both of which the dispersed and continuous phases are treated as continua. An Eulerian description of the flow is adopted. Four different formulations of the two models are derived and a high resolution scheme is presented to obtain numerical solutions of the equations in each of the formulations. We investigate whether the chosen numerical scheme is suitable for the equations governing the models and use the numerical results to obtain quantitative and qualitative insight into the predictions of each of the models. Three test cases, new to the literature, are considered, and the numerical results compared.
2006-01-26
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/151/1/JuH-1Dnd0.pdf
application/postscript
en
https://eprints.maths.manchester.ac.uk/151/2/JuH-1Dnd0.ps
Hudson, Justin and Harris, David (2006) Numerical Modelling of Eulerian Two-Phase Gas-Solid Flow. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:152
2017-11-08T18:18:29Z
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https://eprints.maths.manchester.ac.uk/152/
Symmetric Linearizations for Matrix Polynomials
Higham, Nicholas J.
Mackey, D. Steven
Mackey, Niloufer
Tisseur, Françoise
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
A standard way of treating the polynomial eigenvalue problem
$P(\l)x = 0$ is to convert it
into an equivalent matrix pencil---a process known as linearization.
Two vector spaces of pencils
$\Ell_1(P)$ and $\Ell_2(P)$, and their intersection $\DL(P)$,
have recently been defined and studied by
Mackey, Mackey, Mehl, and Mehrmann.
The aim of our work is to gain new insight into these spaces
and the extent to which their constituent pencils inherit
structure from $P$\@.
For arbitrary polynomials we show that every pencil in $\DL(P)$
is block symmetric
and we obtain a convenient basis for $\DL(P)$ built from block Hankel matrices.
This basis is then exploited to
prove that
the first $\deg(P)$ pencils in a sequence constructed by
Lancaster in the 1960s generate $\DL(P)$.
When $P$ is symmetric, we show that
the symmetric pencils in $\Ell_1(P)$ comprise
$\DL(P)$,
while for Hermitian $P$ the Hermitian pencils in $\Ell_1(P)$
form a proper subset of $\DL(P)$ that we explicitly characterize.
Almost all pencils in
each of these subsets are shown to be linearizations.
In addition to obtaining new results, this work provides a self-contained
treatment of some of the key properties of $\DL(P)$
together with some new, more concise proofs.
2006-01-24
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/152/2/paper6.pdf
Higham, Nicholas J. and Mackey, D. Steven and Mackey, Niloufer and Tisseur, Françoise (2006) Symmetric Linearizations for Matrix Polynomials. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:153
2017-11-08T18:18:29Z
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https://eprints.maths.manchester.ac.uk/153/
Model A-EE : An Eulerian Two-Phase Gas-Solid Model
Hudson, Justin
Harris, David
35 Partial differential equations
65 Numerical analysis
76 Fluid mechanics
This paper investigates Model A (as discussed by Hudson &
Harris MIMS EPrint 2006.10) with the inclusion of a gas Energy equation. We present two formulations of the model and discretise them using the Lax-Wendroff, MacCormack and a high resolution scheme. Three different test cases are used to compare the results to determine the accuracy and robustness of the schemes. The results of Model A-EE are also compared to the isentropic model (Model A).
2006-01-30
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/153/1/AEEpaper.pdf
application/postscript
en
https://eprints.maths.manchester.ac.uk/153/2/AEEpaper.ps
Hudson, Justin and Harris, David (2006) Model A-EE : An Eulerian Two-Phase Gas-Solid Model. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:154
2017-10-20T14:12:04Z
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https://eprints.maths.manchester.ac.uk/154/
A Schur-Newton Method for the Matrix p'th Root and its Inverse
Guo, Chun-Hua
Higham, Nicholas J.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
Newton's method for the inverse matrix $p$th root, $A^{-1/p}$, has
the attraction that it involves only matrix multiplication. We
show that if the starting matrix is $c^{-1}I$ for $c\in\R^+$ then
the iteration converges quadratically to $A^{-1/p}$ if the
eigenvalues of $A$ lie in a wedge-shaped convex set containing the
disc $\{\, z: |z-c^p| < c^p\,\}$. We derive an optimal choice of
$c$ for the case where $A$ 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 $c=1$. 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 $p$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
$A^{1/p}$ or $A^{-1/p}$, and for large $p$ that are not highly
composite it is more efficient than the method of Smith based
entirely on the Schur decomposition.
2006-02-16
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/154/2/paper6.pdf
Guo, Chun-Hua and Higham, Nicholas J. (2006) A Schur-Newton Method for the Matrix p'th Root and its Inverse. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:156
2017-10-20T14:12:04Z
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A Schur--Parlett Algorithm for Computing Matrix Functions
Davies, Philip I.
Higham, Nicholas J.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
An algorithm for computing matrix functions is presented. It employs a Schur
decomposition with reordering and blocking followed by the block form of a
recurrence of Parlett, with functions of the nontrivial diagonal blocks
evaluated via a Taylor series. A parameter is used to balance the conflicting
requirements of producing small diagonal blocks and keeping the separations of
the blocks large. The algorithm is intended primarily for functions having a
Taylor series with an infinite radius of convergence, but it can be adapted for
certain other functions, such as the logarithm. Novel features introduced here
include a convergence test that avoids premature termination of the Taylor
series evaluation and an algorithm for reordering and blocking the Schur form.
Numerical experiments show that the algorithm is competitive with existing
special-purpose algorithms for the matrix exponential, logarithm, and cosine.
Nevertheless, the algorithm can be numerically unstable with the default choice
of its blocking parameter (or in certain cases for all choices), and we explain
why determining the optimal parameter appears to be a very difficult problem. A
MATLAB implementation is available that is much more reliable than the function
\texttt{funm} in MATLAB~6.5 (R13).
Springer-Verlag
Borici, Artan
Frommer, Andreas
Joo, Balint
Kennedy, Anthony
Pendleton, Brian
2003
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/156/1/paper.pdf
Davies, Philip I. and Higham, Nicholas J. (2003) A Schur--Parlett Algorithm for Computing Matrix Functions. SIAM Journal On Matrix Analysis and Applications, 25 (2). pp. 464-485. ISSN 1095-7162
10.1137/S0895479802410815
10.1137/S0895479802410815
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Computing the Condition Number of Tridiagonal and
Diagonal-Plus-Semiseparable Matrices in Linear Time
Hargreaves, Gareth I.
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
For an $n \times n$ tridiagonal matrix we exploit the structure of
its QR factorization to devise two new algorithms for computing the 1-norm
condition number in $O(n)$ operations. The algorithms avoid
underflow and overflow, and are simpler than existing algorithms since
tests are not required for degenerate cases. An error analysis of the
first algorithm is given, while the second algorithm is shown to be
competitive in speed with existing algorithms. We then turn our
attention to an $n \times n$ diagonal-plus-semiseparable matrix, $A$,
for which several algorithms have recently been developed to solve
$Ax=b$ in $O(n)$ operations. We again exploit the QR factorization of
the matrix to present an algorithm that computes the 1-norm condition
number in $O(n)$ operations.
2006
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/157/2/narep447%5B1%5D.pdf
Hargreaves, Gareth I. (2006) Computing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time. SIAM Journal On Matrix Analysis and Applications, 27 (3). pp. 801-820. ISSN 1095-7162
10.1137/040606636
10.1137/040606636
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:158
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Permutation groups of finite Morley rank
Borovik, Alexandre
Cherlin, Gregory
20 Group theory and generalizations
The paper bounds the Morley rank of a definably primitive permutation group of finite Morley rank in terms of the rank of the set on which it acts.
2007-09-03
MIMS Preprint
NonPeerReviewed
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https://eprints.maths.manchester.ac.uk/158/1/BorovikCherlin-B.pdf
Borovik, Alexandre and Cherlin, Gregory (2007) Permutation groups of finite Morley rank. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:159
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Structured Condition Numbers and Backward Errors in Scalar Product Spaces
Tisseur, Françoise
Graillat, Stef
15 Linear and multilinear algebra; matrix theory
65 Numerical analysis
We investigate the effect of structure-preserving perturbations on the solution
to a linear system, matrix inversion, and distance to singularity.
Particular attention is paid to linear and nonlinear
structures that form Lie algebras,
Jordan algebras and automorphism groups of a scalar product.
These include complex symmetric, pseudo-symmetric, persymmetric, skew-symmetric, Hamiltonian,
unitary, complex orthogonal and symplectic matrices.
We show that under reasonable assumptions on the scalar product,
there is little or no difference between structured and unstructured condition
numbers and distance to singularity for matrices in Lie and Jordan algebras.
Hence, for these classes of matrices, the usual unstructured perturbation
analysis is sufficient.
We show this is not true in general for structures in automorphism groups.
Bounds and computable expressions for the structured condition numbers
for a linear system and matrix inversion are derived for these nonlinear
structures.
Structured backward errors for the approximate solution of linear systems
are also considered.
Conditions are given for the structured backward error to be finite.
We prove that for Lie and Jordan algebras, whenever
the structured backward error is finite, it is within a small factor of or equal
to the unstructured one.
The same conclusion holds for orthogonal and unitary structures but cannot
easily be extended to other matrix groups.
This work extends and unifies earlier analyses.
2006-02-17
MIMS Preprint
NonPeerReviewed
application/pdf
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https://eprints.maths.manchester.ac.uk/159/1/ftsg-final.pdf
Tisseur, Françoise and Graillat, Stef (2006) Structured Condition Numbers and Backward Errors in Scalar Product Spaces. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:163
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Topological dimension and local coordinates from time series data
Broomhead, D. S.
Jones, R.
King, Gregory P.
37 Dynamical systems and ergodic theory
53 Differential geometry
94 Information and communication, circuits
A method for the estimation of the topological dimension of a manifold from time series data is presented. It is based on the approximation of the manifold near a point chi by its tangent space at chi. The dimension of the tangent space is estimated by constructing a maximal set of linearly independent vectors from the data near chi using the method of singular value decomposition. The method is used to analyse experimental data obtained from a nonlinear electronic oscillator in a chaotic state.
1987-06-21
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/163/1/DSB_JPhysA20.pdf
Broomhead, D. S. and Jones, R. and King, Gregory P. (1987) Topological dimension and local coordinates from time series data. Journal of Physics A, 20 (9). L563-L569. ISSN 0305-4470
http://www.iop.org/EJ/abstract/0305-4470/20/9/003/
10.1088/0305-4470/20/9/003
10.1088/0305-4470/20/9/003
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:164
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Phytoplankton biomass and residual
nitrate in the pelagic ecosystem
Platt, Trevor
Broomhead, David S.
Sathyendranath, Shubha
Edwards, Andrew M.
Murphy, Eugene J.
37 Dynamical systems and ergodic theory
92 Biology and other natural sciences
We develop and analyse a simple, two-compartment (chlorophyll and nitrate) model of the surface mixed layer of the ocean. The mixed-layer depth is modulated intermittently to simulate the effects of storms. The optical properties of the water column are linked to changes in the chlorophyll biomass. The model can be treated analytically. Mathematical bounds are found for the autotrophic biomass and the residual nitrate in terms of the intensity and frequency of storms and the bio-optical properties of the phytoplankton. The results are discussed in the context of the high-nutrient, low-chlorophyll regimes, where unconsumed nitrate is a persistent occurrence.
2003-02-12
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/164/1/2003ProcRoySoc.pdf
Platt, Trevor and Broomhead, David S. and Sathyendranath, Shubha and Edwards, Andrew M. and Murphy, Eugene J. (2003) Phytoplankton biomass and residual nitrate in the pelagic ecosystem. Proceedings of the Royal Society of London, Series A, 459 (2033). pp. 1063-1073. ISSN 1471-2946
10.1098/rspa.2002.1079
10.1098/rspa.2002.1079
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:165
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Delay Embeddings for Forced Systems. II.
Stochastic Forcing
Stark, J.
Broomhead, D. S.
Davies, M. E.
Huke, J.
37 Dynamical systems and ergodic theory
53 Differential geometry
60 Probability theory and stochastic processes
94 Information and communication, circuits
Takens’ Embedding Theorem forms the basis of virtually all approaches
to the analysis of time series generated by nonlinear deterministic dynamical systems.
It typically allows us to reconstruct an unknown dynamical system which gave rise
to a given observed scalar time series simply by constructing a new state space out
of successive values of the time series. This provides the theoretical foundation for
many popular techniques, including those for the measurement of fractal dimensions
and Liapunov exponents, for the prediction of future behaviour, for noise reduction
and signal separation, and most recently for control and targeting. Current versions of
Takens’ Theorem assume that the underlying system is autonomous (and noise-free).
Unfortunately this is not the case for many real systems. In a previous paper, one of us
showed how to extend Takens’ Theorem to deterministically forced systems. Here, we
use similar techniques to prove a number of delay embedding theorems for arbitrarily
and stochastically forced systems. As a special case, we obtain embedding results for
Iterated Functions Systems, and we also briefly consider noisy observations.
2003-12-03
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/165/1/2003jnonlinsci.pdf
Stark, J. and Broomhead, D. S. and Davies, M. E. and Huke, J. (2003) Delay Embeddings for Forced Systems. II. Stochastic Forcing. Journal of Nonlinear Science, 13 (6). pp. 519-577. ISSN 1432-1467
10.1007/s00332-003-0534-4
10.1007/s00332-003-0534-4
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:166
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Mutual information and capacity of a linear digital channel
Broomhead, D. S.
Sidorov, Nikita
60 Probability theory and stochastic processes
68 Computer science
94 Information and communication, circuits
In this paper we analyse a simple model of a digital communications channel. This model proves to be closely related to an iterated function system (IFS) related to the well-known Bernoulli convolution. We derive it from a randomly forced first-order ordinary differential equation. This allows the parameter of the Bernoulli convolution—the contraction rate, λ—to be related to the rate at which symbols are input to the channel. It is shown that for a channel with equiprobable binary inputs the mutual information between input and output distributions is the stationary measure of the complement of the overlap region of the IFS. We show that the mutual information is Hölder continuous with respect to λ and decreases hyper-exponentially as λ → 1. We also study the case of non-equiprobable binary inputs and show that the maximum of the mutual information—the channel capacity—does not always correspond to equiprobable inputs.
2004-08-20
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/166/1/2004Nonlinearity17.pdf
Broomhead, D. S. and Sidorov, Nikita (2004) Mutual information and capacity of a linear digital channel. Nonlinearity, 17 (6). pp. 2203-2223. ISSN 1361-6544
10.1088/0951-7715/17/6/010
10.1088/0951-7715/17/6/010
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Eye movement instabilities and nystagmus can be
predicted by a nonlinear dynamics model of the
saccadic system
Akman, O. E.
Broomhead, D. S.
Abadi, R. V.
Clement, R. A.
37 Dynamical systems and ergodic theory
92 Biology and other natural sciences
The study of eye movements and oculomotor disorders has, for four decades, greatly benefitted from the application of control theoretic concepts. This paper is an example of a complementary approach based on the theory of nonlinear dynamical systems. Recently, a nonlinear dynamics model of the saccadic system was developed, comprising a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and the oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations were produced, including jerk, bidirectional jerk and pendular nystagmus.
The aim of this study was to investigate the bifurcations and attractors of the model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addition to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth gluing bifurcations. Evidence was also found for the existence of Hopf-initiated canards.
The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist in regions of parameter space where the equations are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of the corresponding slow manifold. The simulated bidirectional jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. In contrast to control models of the oculomotor system, the bifurcation analysis places clear restrictions on which kinds of behaviour are likely to be associated with each other in parameter space, enabling predictions to be made regarding the possible changes in the oscillation type that may be observed upon changing the model parameters. The analysis suggests that CN is one of a range of oculomotor disorders associated with a pathological saccadic braking signal, and that jerk and pendular nystagmus are the most probable oscillatory instabilities. Additionally, the transition from jerk CN to bidirectional jerk and pendular nystagmus observed experimentally when the gaze angle or attention level is changed is attributed to a gluing bifurcation. This suggests the possibility of manipulating the waveforms of subjects with jerk CN experimentally to produce waveforms with an extended foveation period, thereby improving visual resolution.
2005-06-06
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/167/1/Akmanetal-JMB05.pdf
Akman, O. E. and Broomhead, D. S. and Abadi, R. V. and Clement, R. A. (2005) Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system. Journal of Mathematical Biology, 51 (6). pp. 661-694. ISSN 1432-1416
10.1007/s00285-005-0336-4
10.1007/s00285-005-0336-4
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Dimensionality Reduction Using Secant-Based Projection Methods: The Induced Dynamics in Projected Systems
Broomhead, D. S.
Kirby, M. J.
37 Dynamical systems and ergodic theory
53 Differential geometry
65 Numerical analysis
In previous papers we have developed an approach to the data reduction problem which is based on a well-known, constructive proof of Whitney’s embedding theorem [Broomhead, D. S. and Kirby, M., SIAM Journal of Applied Mathematics 60(6), 2000, 2114–2142; Broomhead, D. S. and Kirby, M., Neural Computation 13, 2001, 2595–2616]. This approach involves picking projections of the high-dimensional system which are optimised in the sense that they are easy to invert. This is done by considering the effect of the projections on the set of unit secants constructed from the data. In the present paper we discuss the implications of this idea in the case that the high-dimensional data is generated by a dynamical system. We ask if the existence of an easily invertible projection leads to practical methods for the construction of an equivalent, low-dimensional dynamical system. The paper consists of a review of the secant-based projection method and simple methods for finding good representations of the (nonlinear) inverse of the projections. We then discuss two variants of a way to find the dynamical system induced by a projection which lead to quite distinct numerical approximations. One of these is developed further as we describe various ways in which knowledge of the full dynamical system can be incorporated into the approximate projected system. The ideas of the paper are illustrated in some more or less simple examples, which range from a simple system of nonlinear ODEs which have an attracting limit cycle, to low-dimensional solutions of the Kuramoto–Sivashinsky equation which need many Galerkin modes for their description.
2005-08
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/168/1/2005NonlinearDyn41.pdf
Broomhead, D. S. and Kirby, M. J. (2005) Dimensionality Reduction Using Secant-Based Projection Methods: The Induced Dynamics in Projected Systems. Nonlinear Dynamics, 41 (1-3). pp. 47-67. ISSN 1573-269X
10.1007/s11071-005-2792-1
10.1007/s11071-005-2792-1
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Multitype randomised Reed-Frost epidemics and epidemics upon random graphs
Neal, Peter
60 Probability theory and stochastic processes
92 Biology and other natural sciences
We consider a multitype epidemic model which is a natural exten-
sion of the randomised Reed-Frost epidemic model. The main result
is the derivation of an asympotic Gaussian limit theorem for the ¯nal
size of the epidemic. The method of proof is simpler, and more direct,
than is used for similar results elsewhere in the epidemics literature.
In particular, the results are specialised to epidemics upon extensions
of the Bernoulli random graph.
2006-03-06
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/171/1/psrr09-2005.pdf
Neal, Peter (2006) Multitype randomised Reed-Frost epidemics and epidemics upon random graphs. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:173
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How to Check for Short Cycles in Large Nonlinear Feedback Shift Registers
Petrides, George
Mykkeltveit, Johannes
40 Sequences, series, summability
68 Computer science
In \cite{Ly} it is claimed that for a given large nonlinear feedback shift register it is hard to check whether short cycles have been embedded by the given method. The purpose of the present note is to show how to check for short cycles in general using our proposed algorithm. To implement the algorithm we introduce the notion of nonlinear complexity and try to classify sequences into nonlinear complexity classes.
2006-03-08
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/173/1/nonlin3.pdf
Petrides, George and Mykkeltveit, Johannes (2006) How to Check for Short Cycles in Large Nonlinear Feedback Shift Registers. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:175
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Embedding Nonlinear Dynamical Systems: A Guide to Takens' Theorem
Huke, J. P.
37 Dynamical systems and ergodic theory
53 Differential geometry
The embedding theorem forms a bridge between the theory of nonlinear dynamical systems and the analysis of experimental time series. This memorandum describes the theorem and gives a detailed account of its proof. The necessary differential topology is briefly reviewed, and then a proof of the theorem is presented; this proof follows broadly the argument of Takens, although it differs in some details. Some extensions to the theorem, which facilitate its use in applications, are described. The memo concludes with a brief discussion of what the theorem implies about time series, viewed as the raw material for signal processing algorithms.
2006-03-09
MIMS Preprint
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/175/1/embed.pdf
Huke, J. P. (2006) Embedding Nonlinear Dynamical Systems: A Guide to Takens' Theorem. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:176
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Analysis of a nonlinear dynamics model of the saccadic system
Akman, Ozgur
34 Ordinary differential equations
37 Dynamical systems and ergodic theory
92 Biology and other natural sciences
Models of the mechanisms of saccadic eye movements are typically described in terms of the block diagrams used in control theory. Recently, a nonlinear dynamics model of the saccadic system was developed. The model comprises a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations, which were obtained by combining parts of the existing control models with data from experimental observations of saccadic dynamics. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and an oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations could be produced, including jerk, bilateral jerk and pendular nystagmus.
The aim of this study was to investigate the bifurcations and attractors of the nonlinear dynamics model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard local and global stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addi- tion to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth glu- ing bifurcations. These nonsmooth bifurcations were observed to result from simultaneous transcritical and homoclinic bifurcations in a pair of related smooth systems. Evidence was also found for the existence of Hopf-initiated canards, and for a global bifurcation involving the catastrophic destruction of a symmetry-invariant limit cycle. Unlike the pitchfork and gluing bifurcations, this bifurcation could not be explained in terms of the related smooth systems.
The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist in regions of parameter space where the equa- tions are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of an underlying slow manifold. This provides an alternative explanation for the shape of the jerk oscillation, which contrasts with the prevalent control model view that CN is caused by structural abnormalities. The simulated bilateral jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles.
The bifurcation structure of the model suggests the possibility of moving between the differ- ent simulated behaviours by varying the parameters of the model. This was in agreement with experimental evidence showing that sub jects can exhibit several different types of behaviour in a single recording period. In addition, the bifurcation analysis places restric- tions on which kinds of behaviour are likely to be associated with each other in parameter space. On the basis of these restrictions, several experiments were suggested to assess the validity of the model as a predictor of saccadic behaviour. In particular, it was proposed that reducing the level of attention of a sub ject in a controlled way could induce a change from a jerk to a pendular oscillation.
2003
Thesis
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/176/1/Akman-PhD.pdf
Akman, Ozgur (2003) Analysis of a nonlinear dynamics model of the saccadic system. Doctoral thesis, University of Manchester Institute of Science and Technology (UMIST).
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Nonlinear time series analysis of jerk congenital nystagmus
Akman, O. E.
Broomhead, D. S.
Clement, R. A.
Abadi, R. V.
37 Dynamical systems and ergodic theory
92 Biology and other natural sciences
Nonlinear dynamics provides a complementary framework to control theory for the quantitative analysis of the oculomotor control system. This paper presents a number of findings relating to the aetiology and mechanics of the pathological ocular oscillation jerk congenital nystagmus (jerk CN). A range of time series analysis techniques were applied to both recorded jerk CN waveforms and simulated waveforms produced by an established model in which the oscillations are a consequence of an unstable neural integrator. The results of the time series analysis were then interpreted within the framework of a generalised model of the unforced oculomotor system.
This work suggests that for jerk oscillations, the origin of the instability lies in one of the five oculomotor subsystems, rather than in the final common pathway (the neu- ral integrator and muscle plant). Additionally, experimental estimates of the linearised foveation dynamics imply that a refixating fast phase induced by a near-homoclinic tra- jectory will result in periodic oscillations. Local dimension calculations show that the dimension of the experimental jerk CN data increases during the fast phase, indicating that the oscillations are not periodic, and hence that the refixation mechanism is of greater complexity than a homoclinic reinjection. The dimension increase is hypothe- sised to result either from a signal-dependent noise process in the saccadic system, or the activation of additional oculomotor components at the beginning of the fast phase. The modification of a recent saccadic system model to incorporate biologically realistic signal-dependent noise is suggested, in order to test the first of these hypotheses.
2006-03-10
MIMS Preprint
NonPeerReviewed
application/pdf
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https://eprints.maths.manchester.ac.uk/177/1/Akmanetal-JCNS06.pdf
Akman, O. E. and Broomhead, D. S. and Clement, R. A. and Abadi, R. V. (2006) Nonlinear time series analysis of jerk congenital nystagmus. [MIMS Preprint]
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:179
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https://eprints.maths.manchester.ac.uk/179/
Timing the Ancestor of the HIV-1 Pandemic Strains
Korber, B.
Muldoon, M.
Theiler, J.
Gao, F.
Gupta, R.
Lapedes, A.
Hahn, B. H.
Wolinsky, S.
Bhattacharya, T.
92 Biology and other natural sciences
HIV-1 sequences were analyzed to estimate the timing of the ancestral sequence of the main group of HIV-1, the strains responsible for the AIDS pandemic. Using parallel supercomputers and assuming a constant rate of evolution, we applied maximum-likelihood phylogenetic methods to unprecedented amounts of data for this calculation. We validated our approach by correctly estimating the timing of two historically documented points. Using a comprehensive full-length envelope sequence alignment, we estimated the date of the last common ancestor of the main group of HIV-1 to be 1931 (1915-41). Analysis of a gag gene alignment, subregions of envelope including additional sequences, and a method that relaxed the assumption of a strict molecular clock also supported these results.
Manchester Institute for the Mathematical Sciences
2000-06-09
Article
PeerReviewed
application/pdf
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https://eprints.maths.manchester.ac.uk/179/1/1789.pdf
Korber, B. and Muldoon, M. and Theiler, J. and Gao, F. and Gupta, R. and Lapedes, A. and Hahn, B. H. and Wolinsky, S. and Bhattacharya, T. (2000) Timing the Ancestor of the HIV-1 Pandemic Strains. Science, 288 (5472). pp. 1786-1796. ISSN 0036-8075
10.1126/science.288.5472.1789
10.1126/science.288.5472.1789
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:181
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https://eprints.maths.manchester.ac.uk/181/
The Trap of Complacency in Predicting the Maximum
du Toit, Jacques
Peskir, Goran
35 Partial differential equations
45 Integral equations
60 Probability theory and stochastic processes
62 Statistics
2007
Article
PeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/181/1/2006.30.pdf
du Toit, Jacques and Peskir, Goran (2007) The Trap of Complacency in Predicting the Maximum. Ann. Probability, 35. pp. 340-365.
10.1214/009117906000000638
10.1214/009117906000000638
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:182
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https://eprints.maths.manchester.ac.uk/182/
On Reflecting Brownian Motion with Drift
Peskir, Goran
60 Probability theory and stochastic processes
2006
Conference or Workshop Item
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/182/1/2006.31.pdf
Peskir, Goran (2006) On Reflecting Brownian Motion with Drift. In: Symp. Stoch. Syst., 2005, Osaka.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:183
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https://eprints.maths.manchester.ac.uk/183/
The Effects of State Dependent and State Independent Probabilistic Updating on Boolean Network Dynamics
Savage, Natasha Saint
37 Dynamical systems and ergodic theory
We study semi-synchronous Boolean networks with robabilistic updating schemes and various topologies (tree, loop, and random). As well as state independent probabilistic updating we investigate a state dependent scheme which allows us to control the `accuracy'
of nodes. A node is accurate at $n$ if it has been updated at $n$, or if its state is as it would be if it had updated.
The state dependent re-evaluation probabilities are determined by the `accuracy heuristic': a stochastic equation which depends on the estimation of a distribution; we look at ways of estimating this distribution and derive variance expressions for the estimators.
Through our work on random Boolean trees we observe that (in
general) the output of a Boolean function with correlated inputs, becomes less correlated as the number of inputs is increased. We also discover that the correlation of a Boolean function's output directly affects the ability of the heuristic to achieve the node's target accuracy.
Deterministic random Boolean network dynamics are viewed in a new way, via the distribution of node output distributions (the probability a node's state is $1$ or $0$). This view shows the `activity' of nodes across the network. We find that as in-degree is increased the topology has less effect on the activity and the distribution of the Boolean functions dominates. We present a theoretical result to support this theory.
To understand the dynamics of probabilistically updating Boolean networks we use a numerical approximation to Flyvbjerg's frozen component.
The concept of stability in probabilistically updating Boolean networks is addressed and investigated. For the loop topology the dynamics of active loops fall into two categories: those with an odd number of inversion nodes and those with an even number. We discuss the stability of a fixed point in both cases. For the random topology we derive an annealed approximation which indicates a phase transition similar to that previously found in the deterministic networks.
2005
Thesis
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/183/1/thesis_reports.pdf
Savage, Natasha Saint (2005) The Effects of State Dependent and State Independent Probabilistic Updating on Boolean Network Dynamics. Doctoral thesis, University of Manchester.
oai:eprints.maths.manchester.ac.uk.MIMS.EPrints:184
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7375626A656374733D4D5343:4D53435F3730
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https://eprints.maths.manchester.ac.uk/184/
Symplectic integrators and optimal control
Cross, Mathew I.
49 Calculus of variations and optimal control; optimization
65 Numerical analysis
70 Mechanics of particles and systems
When selecting a numerical method to integrate an ODE system, it is intuitively clear that preservation of geometric properties is desirable. The particular subclasses of ODE systems we will consider are Lagrangian and Hamiltonian systems. The dynamical equations for these derive from variational principles, and we obtain structure preserving integrators by discretizing the principles rather than the ODEs they generate. We demonstrate some advantages that these symplectic integrators have over methods that are more rudimentary by looking at some examples from optimal control theory.
Our major motivation for considering symplectic integrators is solving an image registration problem, where, using the least effort, we associate a set of landmark points on one image to a corresponding set of points on another. A mathematical formulation of this problem is as a Hamiltonian system; this becomes apparent once we realize that we are computing the motion of particles (the landmark points) under some appropriate potential function. We
investigate the performance of symplectic methods on this, more complex, problem. We show that by formulating the problem as a system of nonlinear equations rather than one of optimal control, the explicit Euler method performs better than the symplectic integrators, especially on a set of data points generated by a real experiment. We give some
evidence that the higher-order methods in Matlab's ODE suite
may be better still, but we do not pursue this line of investigation in any detail.
2005-05-01
Thesis
NonPeerReviewed
application/pdf
en
https://eprints.maths.manchester.ac.uk/184/1/MSc2005.pdf
Cross, Mathew I. (2005) Symplectic integrators and optimal control. Masters thesis, University of Manchester.
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