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Organizers |
Issues in multivariate polynomial interpolation
by
Carl de Boor
University of Wisconsin
While univariate polynomial interpolation has been a basic tool of scientific computing for hundreds of years, multivariate polynomial interpolation is much less understood. Already the question from which polynomial space to choose an interpolant to given data has no obvious answer.
The talk presents, in some detail, one answer to this basic question, namely the "least interpolant" of Amos Ron and the speaker which, among other nice properties, is degree-reducing, then seeks some remedy for the resulting discontinuity of the interpolant as a function of the interpolation sites, then addresses the problem of a suitable representation of the interpolation error and the nature of possible limits of interpolants as some of the interpolation sites coalesce.
The last part of the talk is devoted to a more traditional setting, the complementary problem of finding correct interpolation sites for a given polynomial space, chiefly the space of polynomials of degree less than or equal to k for some k, and ends with a particular recipe for good interpolation sites in the square, the Padua points.
References: http://pages.cs.wisc.edu/ deboor/multiint/
Paper reference: http://pages.cs.wisc.edu/~deboor/multiint/
Date received: November 9, 2008
Some of my favourite convex functions
by
Jonathan Borwein
Newcastle and Dalhousie
I shall describe various examples of convex functions appearing (often unexpectedly) over the years in my research. Each example illustrates either the power of convexity, of modern symbolic computation, or of both.
Date received: October 7, 2008
Random matrices and von Neumann algebras
by
Vaughan Jones
Berkeley
In the diagrammatic calculus for computing expected values of words for N×N random matrices, only those that are planar survive in the limit as N→∞. Voiculescu used this limit to define a trace on the algebra of non-commutative polynomials and thus obtain remarkable results about the von Neumann algebras of free groups. In joint work with Shlyakhtenko and Guionnet we generalised this to certain systems of random matrices built on graphs. In this way we obtain subalgebras of considerable interest of the von Neumann algebras.
Date received: November 16, 2008
Conformally invariant random fractals
by
Gregory F. Lawler
University of Chicago
A number of interesting fractals arise as scaling limits of lattice statistical mechanical models at criticality. In two dimensions these limits also exhibit conformal invariance. I will give an introduction to some of the discrete models (self-avoiding walk, loop-erased walk, percolation), the continuous models (especially, SLE, the Schramm-Loewner evolution), and then discuss some more recent work concerning the fractal nature of the curves.
Date received: November 11, 2008
Computing in matrix groups over finite fields
by
C.R. Leedham-Green
Queen Mary, University of London
As a result of a major effort by many people over many years we are coming to a point where we can compute effectively with matrix groups defined over a finite field. I shall outline some of the principal ideas that have allowed us to reach this happy state.
Date received: November 4, 2008
The Poincaré Conjecture and the Geometrization Conjecture for 3-manifolds
by
John W. Morgan
Columbia
We will present an overview of the arguments due to Perelman proving these conjectures. The idea, which goes back to R. Hamilton, is to use the Ricci flow equation to deform any initial Riemannian metric on the given 3-manifold to one that is locally modeled on homogeneous metrics on 3-dimensional homogeneous spaces. The relation of this notion of geometrization to the Poincaré Conjecture is that in the special case that the 3-manifold is simply connected, the only possibility for the homogeneous metric is that it be round. It then follows immediately that the simply connected 3-manifold must be diffeomorphic to the 3-sphere, which is the statement of the Poincaré Conjecture.
There are many issues to be addressed in turning Hamilton's original idea into a mathematical argument proving results: short-time existence and uniqueness of solutions to the Ricci flow equation with arbitrary (smooth) initial conditions; analysis of the singularity development in finite time and the introduction of geometric surgeries to `cut away' the singularity regions and replace them by smooth manifolds; prolongation of the resulting Ricci flow with surgery and analysis of the geometric behavior of the solution for large time; and finally, application of all the geometric results as time goes to infinity together with the nature of the surgeries to prove the Geometrization Conjecture and as a special case the Poincaré Conjecture for the initial 3-manifold.
Date received: October 30, 2008
4000 years of algebra: a
whirlwind historical tour from BM 13901 to modern algebra
by
Karen Parshall
University of Virginia
To the high school student encountering it for the first time, algebra is a new and largely unintuitive abstract language of x's and y's, a's and b's together with rules for manipulating them and pictures for representing equations in terms of them like y = ax + b. If that same high school student goes on to university and happens into a course on algebra there, essentially gone are the by now familiar x's, y's, a's, and b's; essentially gone are the nice graphs that provide a way to picture what is going on. The university course reflects some brave new world in which algebra has somehow become "modern." This modern algebra involves abstract structures-groups, rings, fields, and other so-called "objects"-that are defined in terms of relatively small numbers of axioms and built up of substructures like subgroups, subrings, subfields, and others like ideals. How is it that these two endeavors-the high school analysis of polynomial equations and the modern algebra of the research mathematician-so seemingly different in their objectives, in their tools, and in their philosophical outlooks are both called "algebra"? Are they even related? In fact, they are, and it is the story of how they are that this talk will attempt to illuminate.
Date received: November 11, 2008
Estimation in finite groups
by
Cheryl E Praeger
University of Western Australia
Coauthors: Alice Niemeyer (and for some, Frank Lubeck and Tomasz Popiel)
When and how can we estimate the number of elements of a certain type in a finite group, for example, a finite group of Lie type? Such estimates are important for design and verification of randomised (Monte Carlo) algorithms, as well as of intrinsic interest in asymptotic group theory and applications to group actions.
Date received: October 29, 2008
Calcium, smooth muscle, and asthma
by
James Sneyd
University of Auckland
Contraction of airway smooth muscle is not, as far as we know, a good thing; its only known function is to cause asthma. It also happens that contraction of airway smooth muscle appears to be controlled by the frequency of oscillation of the intracellular free calcium concentration. Thus, this cell type provides an elegant bridge between the study of nonlinear dynamical systems, and the study of disease. In my talk I'll describe some experimental results, a mathematical model of those results, the questions that model has raised, and the questions we hope the model will answer.
Date received: November 11, 2008
Random planar graphs
by
Angelika Steger
ETH Zurich
The Gn, p model of random graphs, introduced by Erdös and Renyi in the 60's, has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in Gn, p appear independently. This situation changes dramatically if one considers graph classes with structural side constraints. For example, in a random planar graph Rn (a graph drawn uniformly at random from the class of all labelled planar graphs on n vertices) the edges are far from being independent. In this talk we survey progress on the study of properties of random planar graphs and the methods that were used and developed in order to achieve this progress.
Date received: November 18, 2008
On the number of composites less than a given number
by
Rob Akscyn
University of Waikato
We present a sieve of the composites based on their greatest proper factors, not the integers themselves (in contradistinction to the Sieve of Erasthothenes). By abandoning the traditional table of integers "problem space" (in which each integer is represented only once), we instead make each row a separate number line (going off the right margin towards infinity). This shift in problem space enables counting all the composites under n implicitly, by counting all the instances of their greatest proper factors instead (a one-to-one mapping under this multi-row representation).
This shift in viewpoint highlights additional structure in the integers: (a) the hyperbolic of n/p being an upper bound for greatest proper factors (strongly suggesting ln(n) is involved, as we already know from the Prime Number Theorem) and (b) that sieving of a finite n is a function of two factors (not just one!): a "thinning force" (sieving as we know it), and a "trimming force" (the hyperbolic serving as upper bound). In effect, each column integer is monotonically reused as a greatest proper factor over and over, all the way up to the hyperbolic curve, unless stopped before that limit because it has itself been sieved.
What this "Venetian Blind" model demonstrates is that "thinning" and "trimming" are not independent forces, and thus not 'compoundable', as use of Mertens product formula for estimating P(n) would imply. Instead, like sibling rivals, these two factors compete with one another; the trimming force eventually gaining the upper hand when further 'thinning' of the greatest proper factors completely ceases once the n/p hyperbolic trims inward to reach p squared.
Thereafter the remaining unsieved greatest proper factors (now all prime) are reduced solely by the trimming force of the hyperbolic (which continues until it reaches the square root of p).
In effect, separating these two forces for independent examination reveals that the thinning aspect of sieving is strongly-recursive, and consequently rapidly induce a pattern to the composites (and by duality, the primes) that is highly regular.
We believe this elementary approach represents a promising assault on key number theory problems such as Goldbach's Conjecture (as well as Mertens Paradox) by unveiling additional exploitable structure, and eventually could lead to stronger series formulas for upper and lower bounds on P(n) (on par with Riemann's formula for P(n)).
Date received: September 17, 2008
Mazur intersection property for Asplund spaces
by
Miroslav Bacak
University of Newcastle
The aim of this talk is to show that even purely geometric properties of Banach spaces can strongly depend on choice of set-theoretic axioms. Namely, we show that (under Martin's Maximum axiom) every Asplund space of density character w1 has a renorming with the Mazur intersection property. Combined with the previous result of Jim\' enez and Moreno, who (under CH) proved the negation of this statement, we obtain that the MIP renormability of Asplund spaces of density w1 is undecidable in ZFC. This result is contained in a recent joint paper with P. Hájek.
Date received: November 26, 2008
Monotone relaxation iterates and applications to singularly perturbed problems
by
Igor Boglaev
Massey University, Palmerston North
This talk deals with monotone relaxation iterative methods for solving nonlinear monotone difference schemes of elliptic type. The monotone w-Jacobi and SUR (Successive Under-Relaxation) methods are constructed. The monotone methods solve only linear discrete systems at each iterative step and converge monotonically to the exact solution of the nonlinear monotone difference schemes. Convergent rates of the monotone methods are estimated. The proposed methods are applied to solving singularly perturbed reaction-diffusion problems. Uniform convergence of the monotone methods is proved. Numerical experiments complement the theoretical results.
Date received: October 19, 2008
Groups in absolute algebraic geometry
by
James Borger
Australian National University
It is possible to build an enriched version of algebraic geometry based not on commutative rings but on lambda-rings. (Lambda-rings are commutative rings with certain extra operations. They originally arose in K-theory.) In a precise sense, this lambda-algebraic geometry is an algebraic geometry over a deeper base than the ring of integers, the usual base in arithmetic algebraic geometry.
I will make some remarks and raise some questions about the analogues of algebraic groups in lambda-algebraic geometry.
Date received: November 3, 2008
Testing irreducibility of sparse polynomials over GF(2)
by
Richard Brent
Australian National University
Coauthors: Paul Zimmermann
We consider three algorithms for testing irreducibility of sparse polynomials (for example, trinomials) over finite fields of small characteristic. In fact, we restrict our attention to characteristic two, though the algorithms generalise.
The first algorithm is straightforward and involves squaring of polynomials modulo the polynomial to be tested.
The second algorithm replaces squaring by modular composition, using an old algorithm of Brent and Kung which reduces the problem to matrix multiplication. In fact, there are several variations on this algorithm, depending on how the matrix multiplications are performed.
The third algorithm involves the recent "fast" modular composition of Kedlaya and Umans (2008).
We show that the theoretical (asymptotic) complexity analysis of the algorithms may be misleading in practice, and suggest a combination of the first two algorithms that may be faster than either.
Date received: September 21, 2008
The stability of Padé and generalized Padé approximations
by
John Butcher
University of Auckland
Numerical methods for the solution of ordinary differential equations possess an associated stability matrix whose characteristic polynomial determines stable behaviour of the method. If the principal eigenvalue is a maximally high-order approximation to the exponential function for given degrees of the coefficients in the characteristic polynomial, then it is a generalized Padé approximation. For the classical Padé approximations, the cases when A-stabilty is exhibited follows from a result of Hairer, Norsett and Wanner. This has recently been extended to generalized Padé approximations and aspects of this new result will be presented.
Date received: October 30, 2008
On the homology of finite dimensional Lie algebras
by
Grant Cairns
La Trobe University
This talk summarizes results in recent papers coauthored with Sebastian Jambor and Barry Jessup. This work is part of a program that seeks to better understand Lie algebra homology and its dependence on the characteristic of the underlying field.
Date received: October 30, 2008
Algebraic properties of chromatic roots
by
Peter J. Cameron
Queen Mary, University of London
The chromatic polynomial of a graph is the monic integer polynomial P(q) whose evaluation at a positive integer k is the number of proper k-colourings of the graph. A lot is known about the location of chromatic roots (roots of chromatic polynomials), but rather less about their algebraic properties.
A working group at the recent Newton Institute programme on Combinatorics and Statistical Mechanics looked at this. We made two conjectures:
Date received: August 6, 2008
Singularities of orders on surfaces
by
Kenneth Chan
University of New South Wales
Canonical orders are examples of noncommutative surfaces which have "mild" singularities. We show that such orders satisfy a numerical criterion which is analogous to the numerical criterion satisfied by commutative rational singularities.
Date received: November 18, 2008
Lean induced cycles
by
Yury Chebiryak
ETH, Zurich
Coauthors: Daniel Kroening, Igor Zinovik, Thomas Wahl
Covering, dominating, and induced paths in binary hypercubes are well-studied notions in combinatorics. For example, Blass et al. investigate lower bounds on the length of cube-dominating paths and cycles. In this talk, I will introduce a combinatorial problem of constructing lean induced cycles, which is defined to be longest chord-free cycles that span a minimum number of hypercube nodes. This problem is important in modeling gene networks in Systems Biology, as lean induced cycles correspond to stable network models. I will demonstrate how to use a SAT solver to compute lean induced cycles for hypercubes up to dimension 7 and classify the cycles with respect to the number of nodes they span. The classification is obtained using a custom-made All-SAT solver with blocking clauses. Efficient filtering of these clauses allows to reduce their number by two orders of magnitude for the 6-cube and thus to compute the classification in reasonable time.
Date received: October 31, 2008
Components in random planar graphs with n vertices and m edges
by
Chris Dowden
University of Canterbury
Let Pn, m denote a graph taken uniformly at random from the set of all labelled planar graphs with n vertices and m(n) edges. We shall use elementary counting arguments to investigate the probability that Pn, m has a component isomorphic to H, for various fixed H, as n →∞. We will provide a complete picture of exactly when the probability is bounded away from 0 and/or 1, showing that there is different behaviour depending on both the graph H and the ratio m/n.
Date received: November 2, 2008
Counting irreducible representations
by
Shannon Ezzat
University of Canterbury
Representation growth is quite a new area of mathematics. It concerns itself with counting the number of irreducible representations from a finitely generated group to matrices over a field, usually the complex numbers. This talk will give a brief introduction to the field, as well as compare it with its sister field, subgroup growth. Also, we will look at explicitly counting representations of the Heisenberg group over the rational integers and the Gaussian integers.
Date received: October 30, 2008
Combinatorial challenges in conservation biology
by
Beáta Faller
University of Canterbury
There is a diverse range of interesting mathematical questions that arise in evolutionary biology, including many of a combinatorial and probabilistic nature. This talk will present a few of the challenges that we have been facing when studying future biodiversity using extinction models and optimization methods. Its aim is to convince the audience how exciting applied mathematics can be.
Date received: November 1, 2008
Curious properties of Maximum Parsimony in estimating evolutionary trees and ancestral sequence states
by
Mareike Fischer
Allan Wilson Centre for Molecular Ecology and Evolution, and Biomathematics Research Centre, University of Canterbury
Coauthors: Bhalchandra D. Thatte
Maximum Parsimony (MP) and Maximum Likelihood (ML) are two of the most freqently used methods for inferring phylogenetic trees and for estimating ancestral root states. Both methods have been frequently discussed, and many scenarios are to-date well understood. For instance, it is well known that MP and ML can lead to different tree estimations (e.g. in the so-called Felsenstein zone) but that under a simple model of substitution, they always choose the same set of trees for sequences that developed ``under no common mechanism'' (as shown by Tuffley and Steel, 1997).
But some surprising properties of MP and ML have only recently been investigated: I will present examples for MP and ML favoring different sets of trees even under ``no common mechanism'' when the underlying model is changed slightly, for example, when substitution probabilities are subject to an upper bound or when a molecular clock condition is imposed. Additionally, I will show that the probability of MP choosing the correct ancestral state can, unlike ML, sometimes be increased by ignoring parts of the tree which may even be close to the root.
Thus, I will show that even 35 years after Fitch's parsimony algorithm was first introduced and 30 years after the discovery of the Felsenstein zone, there are still properties of MP and ML which are surprising, and that therefore both methods are still worth further investigation.
Date received: October 29, 2008
The finite volume method and Riemann problem for a mathematical model of a hydrothermal eruption
by
Luke Fullard
Massey University, Palmerston North
I will introduce the concept of the Finite Volume Method for solving a set of hyperbolic PDEs, (such as our current hydrothermal eruption model), and discuss its various advantages over finite difference methods in certain cases. Also, I will discuss the Riemann Problem for the situation of two fluids separated at an interface with a discontinuous pressure profile over the interface. A mathematical model for the initialization of a hydrothermal eruption will then be presented, making use of the previously mentioned methodologies.
Date received: November 2, 2008
Mathematical comparative analysis of syntax and semantic search engines for end user performance
by
Apakuki Gavoka
University of the South Pacific, Suva
This research investigates the effectiveness versus efficiency of search engines with respect to rankings of searches. It focuses on the mathematical formalization of where syntax searches and semantic search differ in effective and efficient discovery times of information on the web. The Page Rank algorithm drives the generic syntax search engine Google and semantic rank algorithm steers generic semantic search engine Hakia, these are the two search engines chosen for this research. Search cases were contrived and run through the two search engines and mathematically analysed. Interestingly, it is shown that a vector relationship exists in both the searches. The findings show that the popular belief of Google being the most efficient is highly questionable. However, the analyses establish that Google is more consistent in searches in contrast to Hakia.
Date received: October 23, 2008
The cyclic sliding operation in Garside groups
by
Volker Gebhardt
University of Western Sydney
Coauthors: Juan González-Meseses (University of Seville, Spain)
Garside groups are generalisations of the well-known Artin braid groups. Basically, the notion of Garside groups captures the fundamental algebraic properties of braid groups and separates them from properties arising from a specific geometric or topological context. The most fundamental characteristic is the existence of the greedy normal form.
I will recall some well-known invariants of conjugacy classes which were introduced to solve certain computational problems in Garside groups. We will see that the theoretical properties of these established invariants are in some sense unsatisfactory. This will lead us to the definition of what appears to be a more natural structure.
Date received: October 29, 2008
Method for calculating the spectra of self-adjoint extensions of simple symmetric operators
by
Yufang Hao
Department of Applied Mathematics, University of Waterloo, Canada
Coauthors: Achim Kempf
By the Cayley transform, a simple symmetric operator T with deficiency indices (1, 1) has a U(1)-family of self-adjoint extensions, which can be parameterized as T(a) with 0 ≤ a < 1. Under the assumption that one of these self-adjoint extensions, say at a = 0, has only a discrete set of eigenvalues with no accumulation point, all other self-adjoint extensions T(a) have only point spectra with no accumulation point as well, and their eigenvalues increase simultaneously in a continuous manner as a increases. Together these eigenvalues cover the real line exactly once. In addition with the knowledge of the derivatives of these eigenvalues with respect to a at a = 0, we provide an explicit formula for computing the eigenvalues of all other self-adjoint extensions T(a). This gives a computational realization of the abstract spectral theory of self-adjoint extensions of symmetric operators. As an application, we present a new generalized sampling theorem, in which samples are taken at a time-varying rate adjusted to the behaviour of the signal and the signal is stably reconstructed.
Date received: August 18, 2008
The Toeplitz-Hausdorff theorem in a constructive setting
by
Robin Havea
School of Computing, Information and Mathematical Sciences, University of the South Pacific, Suva
Coauthors: Douglas Bridges
The Toeplitz-Hausdorff theorem simply says that the numerical range of a Hilbert space operator is always convex. Most of the classical proof of this theorem is said to be `computational'. However, in Bishop's constructive mathematics (i.e. mathematics with intuitionistic logic this is not the case. We look at a proof given by Halmos where we identify and fix the nonconstructivity in it. By means of limiting examples, we also show that our result is the best we can hope for in a constructive setting.
Date received: October 19, 2008
Stability of variable stepsize BDF methods for initial value problems
by
Allison Heard
Dept of Mathematics, University of Auckland
Coauthors: John Butcher
The stability of a method is dependent on the formulation of a method as well as the method itself. Theoretical analysis of the underlying one-step method in Nordsieck form is considered and the ``scale and modify'' approach is applied to second and third order BDF methods.
Date received: October 28, 2008
Nesting polynomials in infinite radicals
by
Peter Humphries
University of Canterbury
A well-known problem of Ramanujan's asks for the evaluation
of the infinite nested radical
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Date received: November 25, 2008
Periodic initial value problems for (integrable) partial difference equations
by
Peter van der Kamp
La Trobe
For partial difference equations defined on a square, initial values can be given on staircases. By taking periodic initial conditions the equation reduces to a system of ordinary difference equations, or a mapping/correspondence. For integrable equations, integrals for these mappings/correspondences are obtained by taking the trace of the monodromy matrix. We show how to generalize the construction to more general (systems of) equations.
Date received: November 17, 2008
Unbounded functional calculus for bounded groups with applications
by
Mihály Kovács
Department of Mathematics and Statistics, University of Otago
Coauthors: Boris Baeumer and Markus Haase
We develop the unbounded extension of the Hille-Phillips functional calculus for generators of bounded groups. Mathematical applications include the generalised Lévy-Khintchine formula for subordinate semigroups, the analyticity of semigroups generated by non-odd fractional powers of group generators and a shifted abstract Grünwald formula. We also give an application of the theory to subsurface hydrology, modeling solute transport on a regional scale using fractional dispersion along flow lines.
Date received: October 13, 2008
Scoring Bayes
by
Gerrard Liddell
University of Otago
If a utility value, or `truth score' is attached to correct statistical inferences, then it has been proved that every scientific experiment increases the truth score under the assumption of `reciprocal convexity'[1]. The relevant and invariant scoring functions that have been used by economists are not reciprocally convex, but reciprocal convexity is a necessary condition for proving that experimentation increases truth scores [2]. Other questions about the `credibility' of inferences also involve scoring functions. This talk will show how the analytic results needed can be reduced to decidable algebraic problems.
Date received: November 3, 2008
Glueing continuous functions constructively
by
Iris Loeb
University of Canterbury
Coauthors: Douglas Bridges
The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect, not just to the existence of their intersection.
Date received: October 21, 2008
Külshammer's second problem
by
Daniel Lond
University of Canterbury
Let G be a linear algebraic group over an algebraically closed field k, G an arbitrary finite group and Gp ⊂ G a Sylow p-subgroup of G, where p=char(k). It is known that there may be infinitely many equivalence classes of representations of G into G. Külshammer asks the following:
Given an equivalence class of representations of Gp into G, are there only finitely many representations r: G→ G, up to equivalence, such that the restriction of r to Gp belongs to that given class?
The aim of this elementary talk will be to describe the problem to a general audience and to show how a cohomology argument may provide some answers.
Date received: November 12, 2008
Rank dominations in matroids
by
Arun Mani
Monash University
A well known property of a matroid is its rank submodularity. This states that for any two subsets A, B of the ground set, E, of a matroid with rank function r:2E → N, r(A) + r(B) ≥ r(A ∪B) + r(A ∩B). We recast the submodularity of matroid ranks as a bijective map property between appropriately defined sets whose members are pairs of subsets of E. The bijection maintains an additive rank inequality across these subset pairs. We call such bijections rank dominating bijections of matroids. In this talk, we will present some results on the existence of rank dominating maps for certain special cases, along with a conjecture on their existence for the general case. We will also discuss some consequences of the presence of such rank dominating maps in matroids.
Date received: November 2, 2008
Frequency parameters of non-symmetric box-type structures using the Rayleigh-Ritz method and penalty functions
by
Luis Monterrubio
University of Waikato
Coauthors: Sinniah Ilanko
In this work, frequency parameters of non-symmetric box-type structures with several combinations of classical boundary conditions are obtained using the Rayleigh-Ritz method and penalty parameters. In this work, two different types of penalty parameters were used to model constraints. These penalty functions represent either stiffness or inertia, and as demonstrated in recent publications in both cases the penalty parameters can be either positive or negative. In all cases, the same set of admissible functions is used in the Rayleigh-Ritz method, which can be used to model a completely free plate. All geometric boundary conditions, as well as joints between plates are modelled using penalty parameters. Results compare well with those in the existing literature when available or with results obtained from finite element analysis using commercial software.
Date received: October 29, 2008
Reich theorem and mappings with fixed points theorem on G-metric spaces
by
Zead Mustafa
Department of Mathematics, Hashemite University, Jordan
Coauthors: Hamed Obiedat
In 2006, Zead Mustafa and Brailey Sims introduced a more robust concept of a generalized metric spaces, called G-metric space, and they developed a topological structure in such spaces. Also they discussed the fixed point theory of contractive mappings, and mappings satisfying various related conditions in complete G-metric spaces.
In this talk we prove some fixed point results for mappings satisfying sufficient contractive conditions on a complete G-metric space, moreover we show that if the G-metric space (X, G) is symmetric, then the existence and uniqueness of these fixed point results follows from Reich theorems in the usual metric space (X, dG), where (X, dG) is the metric induced by the G-metric space (X, G).
Date received: September 28, 2008
A review of the internet congestion control via rate control algorithms with examples and simulations
by
Salsabil Nusair
School of Computing, Information & Mathematical Sciences, University of the South Pacific, Suva
Coauthors: Jito Vanualailai
One of the most recent and exciting areas of research in computing science and information systems deals with the need to control traffic in the ever-growing Internet in a more systematic, rigorous and efficient manner. The world started experiencing a series of ``Internet congestion collapses'' in the early days of the Internet. The work of Jacobson in 1988 (V. Jacobson, Congestion avoidance and control, ACM Communication Review, 18 (1988), pp.314-329) first recorded this phenomenon and proposed a congestion control and avoidance algorithm. This algorithm, which is implemented in the transport layer protocol called TCP (Transmission Control Protocol), has served the Internet well during a time of unprecedented growth. However, as Srikant noted (R. Srikant, The Mathematics of Internet Congestion Control, Birkhauser, Boston, 2003), it was designed during a time when the Internet was a relatively small network compared to its size today. Therefore, there has been much interest in re-examining the role of congestion control in the Internet with the goal of enhancing TCP to make it scalable to large networks.
In this presentation, for non-computer science audience, we provide a simple explanation of the Internet congestion control problem, viewed as a resource allocation problem, and then review a convex optimization algorithm for providing a solution. Furthermore, we review a technique by Kelly (F. Kelly, Mathematical modeling of the Internet, Mathematics Unlimited --- 2001 and Beyond, Springer-Verlag, 2001, pp.685-702), who uses the Direct Method of Lyapunov to design a model of the Internet governed by ODEs, the solutions of which are an approximation to the solutions obtained via convex optimization. Throughout the presentation, we will use simple examples of the Internet to illustrate concepts and applications of convex optimization and the Direct Method of Lyapunov.
Date received: October 27, 2008
Space of test functions for (w1, w2)-tempered ultradistributions via Fourier transform
by
Hamed M. Obiedat
Hashemite University, Jordan)
Coauthors: Wasfi Shatanawi and Mohd Yasin
We introduce the space Sw1, w2 of test functions for (w1, w2)-tempered ultradistributions where w1 and w2 are two weights satisfying the classical Beurling conditions. Moreover, we give a topological characterization of the space Sw1, w2 without conditions on the derivatives. For functional in the dual space Sw1, w2, we prove a structure theorem by using the classical F. Riesz representation theorem.
Date received: September 28, 2008
An algebraic approach to quantum and classical information theory
by
Manas K Patra
Department of Computer Science, University of York, UK
Coauthors: Samuel L Braunstein
The aim of this paper is to give a unified description of classical and quantum information in the language of C* algebras. We then interpret and analyze several important notions from information theory (both classical and quantum) in this context. Quantum information theory combines quantum and classical information. Classical information theory treats information as sets of random variables and studies their statistical properties under various transformations. Some of the most important quantities are expectations (e.g. Shannon entropy), conditional expectations, or correlations of functions of random variables. We treat the random variables as classical observables burrowing from the language of classical statistical mechanics: further the set of observables can be given the structure of a commutative C* algebra. Quantum observables on the other hand form a noncommutative C* algebra. Hence, we regard a quantum information system as a non-commutative C* algebra with the classical components lying in its centre. The notion of statistical correlations and independence are shown to be connected with entanglement and separability. Moreover, the notion of subsystems has a natural generalization which avoids the use of tensor product, although in many (but not all) situations our formulation is equivalent to a tensor structure. Interestingly, we can prove important theorems (no broadcasting, no cloning) in this general formulation. Further, our formalism covers situations (e.g. indistinguishable particles, infinite dimensions) where the standard formulation cannot applied without drastic modifications. Finally, we also explore connections with free probability theory.
Date received: October 29, 2008
The notion of "recursive" subset in Euclidean space and related questions
by
Petrus H Potgieter
Department of Decision Sciences, University of South Africa
Roger Penrose, in his The Emperor's New Mind (1989), challenged the mathematical community to find a definition of ``recursive'' subsets of Euclidean space that would allow one to determined whether the Mandelbrot set is ``recursive'' or not. The talk discusses the properties that a ``nice'' definition should have and the elementary observation that it is impossible to simultaneously realize all of these properties. We consider several notions of recursiveness that have been introduced for sets in Euclidean space and counter-examples separating these notions. Finally, the relation to computational geometry is briefly discussed.
Date received: November 10, 2008
New algorithm for motion planning and posture control of 3-trailer systems.
by
Krishna Sami Raghuwaiya
University of the South Pacific, Suva
Coauthors: Bibhya Nand Sharma and Prof Jito Vanualailai
This paper utilizes a new Lyapunov-based control scheme to extract an algorithm that improves upon, in general, the motion planning and posture control of 3-trailer systems. The control scheme inherently guarantees point and posture stabilities, convergence and collision avoidance properties of the articulated systems in a priori known environment. We employ the concepts of ghost walls and minimum distance technique (MDT) to attain point and posture stabilities, in the sense of Lyapunov, of our kinodynamical model. The effectiveness of the control scheme and its control laws are demonstrated via simulations of two traffic-like scenarios.
Date received: October 23, 2008
CARTopt: a random search method for non-smooth optimization
by
B. L. Robertson
University of Canterbury
Coauthors: C. J. Price and M. Reale
A random search optimization method for finding optima of unconstrained optimization problems is described. The method operates by using a batch of random points at each iteration. These points are used to partition the optimization region into sub-regions using Classification and Regression Trees (CART). Each sub-region is classified as either high or low with respect to function value. The next batch of points has an increased probability distribution in sub-regions which are classified as low. Although the method focuses on regions where objective function is relatively low, points in the high sub-regions are still sampled reducing the risk of missing the global optima. The method requires no gradient information and thus can be applied to non-smooth problems. Numerical results will be presented to show the performance of the algorithm on a selection of non-smooth test functions.
Date received: October 28, 2008
Formation control of a swarm of mobile manipulators
by
Bibhya Sharma
University of the South Pacific, Suva
Coauthors: Jito Vanualailai and Avinesh Prasad
This paper presents a new Lyapunov-based centralized formation control planner for a swarm of 2-link mobile manipulators in a priori known environment. To ensure a significant degree of formation stiffness along the flight-path, information on moving ghost targets, inter-robot bounds for aggregation, and heading for the mobile manipulators are captured in the control planner. The final desired orientation of the formation is by observing a minimum distance between every member of the swarm and ghost walls. The nonlinear control laws extracted from the Lyapunov-based control scheme is utilized to obtain collision-free trajectories of the swarm in a low-degree formation, whilst ensuring the stability of the kinodynamic system governing the swarm. The effectiveness of the control scheme and its controllers are demonstrated by simulating interesting traffic-like situations.
Date received: October 14, 2008
Orbital forcing over the Cenozoic Era
by
Philip Sharp
University of Auckland
Long-term changes in Earth's orbit about the Sun can cause long-term changes in Earth's climate through a process known as orbital forcing. Establishing a link between the orbit and climate requires the orbit be known accurately. A suitable orbit is found by specifying a detailed model of the gravitational forces acting on the Earth and then using an accurate numerical integration method on a computer to calculate the orbit from the model. An orbit of the required accuracy for the Neogene Period (0 - 23 million years ago) has been calculated and used to calibrate the timing of geological events over the Neogene Period.
I will describe the challenges in calculating an accurate orbit for the Cenozoic Era (0 - 65 million years ago).
Date received: October 28, 2008
Global stability of a mathematical model of the internet
by
Ronal Singh
School of Computing, Information & Mathematical Sciences, University of the South Pacific, Suva
Coauthors: Jito Vanualailai
Utilizing the Direct Method of Lyapunov (DML), we study the global stability of a mathematical model of the Internet. We follow the pioneering work of Kelly who, in 2001 (F. Kelly, Mathematical modeling of the Internet, Mathematics Unlimited---2001 and Beyond, Springer-Verlag, 2001, pp.685-702) began a new direction in enhancing TCP (Transmission Control Protocol) with a novel technique that uses a type of distributed control algorithm derived from the DML and convex optimization. Kelly showed that it was possible to optimize the benefits for a user (e.g. bandwidth, cost) while meeting constraints (e.g. capacity of a link) and maintaining the stability of a network regardless of its size. In other words, the network would not experience congestion problems.
In this presentation, firstly, we review one of Kelly's results on the primal algorithm and secondly, we will attempt to create a new Lyapunov-based congestion algorithm by generalizing Kelly’s result.
Date received: October 25, 2008
Conditionally invariant measures and sets with low escape rates.
by
Ognjen Stancevic
School of Mathematics and Statistics, University of New South Wales
Consider a dynamical system T: X → X, and let H ⊂ X be a "hole" such that almost every point eventually enters the hole. Such systems were first studied by Yorke and Pianigiani in 1979. An important quantity associated with these open systems is the "escape rate": how fast, asymptotically, do points enter the hole. In some cases, escape rates can be related to conditionally invariant measures as well as to eigenvalues of the Perron-Frobenius operator. In this talk I will give a brief review of escape rates and possible applications in detection of "almost invariant" sets.
Date received: October 29, 2008
3-unisolvent sets, 4-dimensional Laguerre planes and generalized quadrangles
by
Gunter Steinke
Dept. of Mathematics and Statistics, University of Canterbury
Geometrically 3-unisolvent sets of functions correspond to Laguerre type geometries. In this talk we look at 3-unisolvent sets of functions from the 2-sphere to 2-dimensional Euclidean space that solve the Hermite interpolation problem and their corresponding geometric objects, 4-dimensional Laguerre planes. We give a construction via associated generalized quadrangles of such Laguerre planes whose automorphism group is not transitive on the set of circles (the graphs of the function in the 3-unisolvent set).
Date received: October 15, 2008
The best approximation of a function by a sum of radial functions
by
Steve Taylor
University of Auckland
We consider the best (in the least squares sense) approximation of a function on R3 by a finite sum of radial functions, each with a different centre.
A calculus of variations approach leads to a pretty system of integral equations for the unknown functions. Traditional methods for solving these integral equations have problems. However the system has a surprising connection to an initial value problem which serves as a useful tool for solving the system.
This problem arose in quantum chemistry from a desire to approximate the electronic charge density of a molecule by a sum of atomic-like charge densities.
Date received: November 2, 2008
Multiple solutions for systems of differential equations with nonlinear boundary conditions
by
Bevan Thompson
University of Queensland
Coauthors: Jutarat Kongson and Yongwimon Lenbury
We discuss existence results for three solutions of systems of nonlinear differential equations of the form y"=f(x, y, y') satisfying fully nonlinear boundary conditions g((y(0), y(1));(y'(0), y'(1)))=0. The Dirichlet, Periodic and Sturm-Liouville boundary conditions are included as special cases. Here assumptions on f guarentee solutions in an admissible bounding region of (x, y) space are bounded in C1 and g is compatible with this region. Our results extend those of Frigon and Montoki, Schmitt and Thompson, and Agarwal, Thompson and Tisdell.
Date received: October 31, 2008
An interactive window for split/rejoin maneuvers of swarms
by
Rajneel Totaram
University of the South Pacific, Suva
Coauthors: Bibhya Sharma, Jito Vanualailai
In this paper we develop a Java-based interactive window that illustrates the split/rejoin maneuvers of non-holonomic car-like robot swarm fixed in a prescribed formation. The split/rejoin maneuvers are desired so that the swarm is able to avoid obstacles in its path. The application allows the user to choose the size of the swarm and the number of obstacles in a constrained workspace. The shape of the formation will, by default of the prototype, depend on the size of the swarm provided by the user. The underlying controls of the swarm will be governed by the Lyapunov-based control scheme. Upon receiving the necessary details, the control scheme will design a set of suitable continuous acceleration controllers that produces the desired split/rejoin maneuvers of the swarm.
Date received: October 15, 2008
Cell structures for finite subset spaces
by
Christopher Tuffley
Massey University, Palmerston North
The kth finite subset space of a topological space X consists of the nonempty subsets of X of size at most k. The most famous of these is the 3rd finite subset space of the circle, which Bott proved in 1952 is homeomorphic to the 3-sphere. We will look at methods for studying these spaces when the underlying space X is "nice"; in particular, we will look at methods for constructing cell structures for finite subset spaces from a cell structure for the underlying space.
Date received: November 2, 2008
Developement of a 3D numerical model for salinity intrusion in Brisbane
by
Gurudeo Anand Tularam
Griffith University, Brisbane
Coauthors: Roshan Singh
A three-dimensional (3D) density dependent seawater intrusion field scale model is developed in this paper for Pine Rivers Shire (Australia). The 3D model not only allows for complex flow behaviour and boundary conditions but also takes into account the influence of tidal behaviour that is modelled using Cartwright's (2001) approach. The new model was solved using the finite element FEMWATER package. Along with various longer term salinity profiles for the region, the results show that regions within a distance of 1300m from the seawater boundary will be affected by higher levels of salinity concentration after 100 years. For longer simulations, deviation of the concentration contours from the normal behaviour in the vadose zone is observed at the end of the diffusive zone i.e. at 1300m from seaward boundary for the Pine Rivers Shire aquifer. A diffusive zone rather than a sharp interface was observed and the intrusion process was dominated by diffusion, rather that advection as expected. Parts of Australia have been under drought conditions with significant water shortages and lack of rainfall generally. The simulations suggest that irrigation for farming, pumping for industrial and other uses of groundwater should carefully monitored to avoid salinization of coastal aquifers through excessive extraction in the longer term.
Date received: October 13, 2008
Industrial mathematics initiatives: an (inter)national need?
by
Graeme Wake
Massey University, Albany
There is a world-wide trend to introduce (mostly post-graduate) developments to produce graduates attuned to industry and society need. Further, Universities are enjoying an increasing opportunity for industrial partnerships with industry in sponsored research and consulting. The OECD Global Science Forum surveyed initiatives in this area and made some important recommendations to member countries: see their report <http://www.oecd.org/dataoecd/47/1/41019441.pdf>. The intention of this informal discussion session is to survey what cooperative opportunities exist and any response we as member countries of the OECD should or could make.
Date received: October 31, 2008
Some properties of conformal transformations of symmetrical spinors
by
Graham Weir
Industrial Research Ltd, New Zealand
A general linear transformation of symmetrical spinors corresponds to a conformal transformation. The corresponding matrices are n ×n, whose elements are homogeneous polynomials of degree n-1. The determinants and inverses of these matrices have remarkable properties. In addition, these n ×n matrices can be built up successively by recursion, using the 2 ×2 and (n-1) ×(n-1) matrices, but in doing this, we encounter a diagonal matrix whose entries are the binomial coefficients, C(n, i). We briefly describe the determinant of this diagonal matrix (the product over i). Finally, we show that the fundamental system of differential equations arising from this system of symmetrical spinors is Baba's equation.
Date received: September 2, 2008
Singularities arising from the stretching of threads with viscous heating
by
Jonathan Wylie
Dept of Mathematics, City University of Hong Kong
Coauthors: Huaxiong Huang
We investigate the role played by viscous heating in extensional flows of viscous threads with temperature-dependent viscosity. We develop a formal asymptotic theory based on a long-wavelength approximation of the Navier-Stokes equations to describe such flows. Using this framework we show that there exists an interesting interplay between the effects of viscous heating, which accelerates thinning, and inertia, which prevents pinch-off. We first consider steady drawing of a thread that is fed through a fixed aperture at given speed and pulled with a constant force at a fixed downstream location. For pulling forces above a critical value, we show that inertialess solutions cannot exist and inertia is crucial in controlling the dynamics. We also consider the unsteady stretching of a thread that is fixed at one end and pulled with a constant force at the other end. In contrast to the case in which inertia is neglected, the thread will always pinch at the end where the force is applied. Our results show that viscous heating can have a profound effect on the final shape and total extension at pinching.
Date received: October 28, 2008
Idempotents in Stone-Cech compactifications and homogeneous maximal spaces
by
Yevhen Zelenyuk
University of the Witwatersrand, South Africa
Let G be an infinite group and let bG be the Stone-Cech compactification of G as a discrete semigroup. We take the points of bG to be the ultrafilters on G. Being a compact right topological semigroup, bG has idempotents. Every idempotent p ∈ bG determines a left translation invariant Hausdorff topology Tp on G with a neighborhood base at the identity e ∈ G consisting of subsets A∪{e} where A ∈ p. An idempotent p ∈ bG is regular if p is uniform (= for every A ∈ p, |A|=|G|) and the topology Tp is regular. We show that for every infinite group G, there exists a regular idempotent in bG. As a consequence we obtain that for every infinite cardinal k, there exists a homogeneous regular maximal space of dispersion character k, which is the answer to an old difficult question. Another consequence tells us that there exists a translation invariant regular maximal topology on the real line of dispersion character c stronger than the natural topology.
Date received: August 8, 2008
Nonlinear theories for shock-induced fingering instabilities
by
Qiang Zhang
Department of Mathematics, City University of Hong Kong
It is well known that, when a shock propagates through a material interface that separates fluids of different densities, the interface becomes unstable and fingers develop. It is a long-standing problem to develop a theory to predict the size of these fingers. In this talk, we present an analytical nonlinear theory which provides predictions for the growth rate and the size of the fingers. Our theoretical approach is based on the methods of Padé approximation and asymptotic matching. The theoretical predictions are in remarkably good agreement with the results from full-scale numerical simulations.
Date received: October 28, 2008
L. E. J. Brouwer: solipsist - why?
by
Dr Philip Catton
University of Canterbury (Philosophy / History and Philosophy of Science)
Consider (A1) intuitionistic logic, (A2) the Kantian pure intuition of time, (A3) constructive mathematics, (A4) experience as a state of the subject. Consider furthermore, towards making four parallel comparisons, (B1) classical logic, (B2) the Kantian pure intuition of space [with a modification], (B3) classical mathematics, (B4) experience as the mode of a subject’s connecting to its public, intersubjective world. Note that: 1. Intuitionistic logic, as Gödel firmly established, deserves our respect if classical logic does; but the converse is, as Gödel again firmly established, also true — classical logic deserves our respect if intuitionistic logic does. 2. Time is, in various ways that Kant points out to us, a deeper condition on experience than space; and yet in ways that Kant also points out the converse is also true — space is in various ways a deeper condition on experience than time is (a thesis that Kant establishes with arguments that are not touched by the usual criticism against Kant that his philosophy is embarrassed by non-Euclidean geometries). [My modification on B2 is to weaken Kant's assessment of the pure intuition of space, at least to the extent of its not encompassing Euclideanness of the metric, and perhaps so far as its not encompassing any more than topological structure.] 3. Constructive mathematics commands special respect as mathematics but the notion that it is uniquely worthy cannot be sustained; classical mathematics too has its place. 4. The private, what-it-is-like-for-the-subject aspect of experience commands special attention; but Kant established as well the converse point, that our right to view experience as a mode of the subject’s connecting to what it is not actually precedes our right to view experience as a state of the subject. I argue that the parallelism between my various A1-B1, A2-B2, A3-B3 and A4-B4 comparisons is both perfectly strong, and significant - it explains the inevitability of Brouwer's solipsistic tendency, identifies the special significance of mathematics that proceeds (according to Brouwer's prescription) out of constructions within the pure intuition of time, and at the same time identifies deep reasons to conclude that such a constructive approach cannot even potentially complete the whole of mathematics.
Date received: October 30, 2008
The platypus and the mathematician.
by
Hannes Diener
University of Canterbury
Mathematicians generally like to insist that their field is not science, since its method of gaining knowledge is deductive rather than inductive. In this talk we argue that, nevertheless, there is a healthy dose of inductive reasoning in the way mathematicians think.
Date received: October 30, 2008
Nature's drawing
by
Ofer Gal
Unit for History and Philosophy of Science, University of Sydney
The challenge of assigning mathematics an explanatory role in natural philosophy forced 17th century savants to accept a difficult conversion: turning local motion---the paradigm of change---into the carrier of order. Kepler and Galileo met the challenge by treating geometrical curves not as ideal representations of motion, but as traces left by nature itself, which the mathematician is called upon to analyse. In contrast to their Renaissance predecessors, for whom mathematical order was eternal and static, Kepler and Galileo construed nature in terms of pure motion, essentially mathematical. Their successors no longer conceived of the
application of mathematics to nature as requiring difficult metaphysical legitimation. For Huygens and Hooke the mathematical structure of
nature resided in its malleability to the mathematician.
Date received: November 16, 2008
Rules of engagement: conventions for medieval recreational problems
by
John Hannah
University of Canterbury
Ancient and medieval mathematical texts often discuss highly impractical problems disguised by realistic-sounding contexts. For example, men with unknown amounts of money might exchange known fractions of their holdings in order to buy a horse of unknown value, and you would have to find all the unknowns. In this talk I shall discuss some of the conventions which seem to have determined which problems you were allowed to set, and which solutions were deemed valid. Examples from the work of Leonardo of Pisa (also known as Fibonacci) will be used to illustrate these conventions.
Date received: October 29, 2008
A symbolic history of the derivative
by
Clemency Montelle
University of Canterbury
How many ways to symbolically represent the derivative can you call to mind? Are they really equivalent? Among the many disagreements Newton and Leibniz are remembered for, they had a big one over notation. They both independently developed distinctive notation for the derivative when they published their results in calculus. In turn, allegiance in both British and European mathematical communities to strictly one or the other persisted for almost half a century until Leibniz's notation finally prevailed. We will look at this scuffle and its consequences and reflect upon the ways in which notational considerations can affect mathematics.
Date received: October 30, 2008
Euclid and Aristotle, in Persian and in Sanskrit
by
Kim Plofker
Union College, Schenectady NY, USA
The Indo-Persian empires of the mid-second millennium CE in northern India fostered, both deliberately and accidentally, a great number of intellectual exchanges between Greco-Islamic science and the indigenous Sanskrit tradition. The anonymous Hayatagrantha ("Book on Spherical Astronomy"), a Sanskrit translation of the Ris¯ala dar hay'a ("Treatise on Spherical Astronomy") by the fifteenth-century Samarqand astronomer `Ali al-Q¯ushj¯, bears witness to some of the philosophical adjustments that were required to fit the Persian version of traditional Euclidean geometry and Aristotelian cosmology into the intellectual framework of Indian mathematics.
Date received: November 3, 2008
Pappus of Alexandria: analysis/synthesis outside of Book 7 in the Mathematical Collection
by
Bronwyn Rideout
University of Canterbury
Although Pappus of Alexandria is often identified with inspiring the early modern mathematics of Descartes, Desargues, and, to a lesser extent, Newton, it has only been in the past century that he has been the focus of interest and textual criticism. However, translations of his text into English have been few and far between and consequently certain books have risen to prominence while others, unfairly receive nominal attention. In this talk, I will discuss Pappus' infamous work on Analysis and Synthesis in Book 7 of the Mathematical Collection and the implications it has on Books 2 and 3 of the Collection. One will find that Book 7 is far from the final word Pappus has to say on the subject as I show how Pappus bring analysis and synthesis into play numerically and in different scenarios.
Date received: November 2, 2008
Asymptotics of Null Lie Quadratics in Tension
by
Shreya Bhattarai
University of Western Australia
A null Riemannian cubic in tension is a curve that arises as a solution to a variational problem on a Riemannian manifold. If the manifold is a bi-invariant Lie group, there is an associated curve V(t) in the Lie algebra called a null Lie quadratic in tension which satisfies the equation
|
In "Asymptotics of Null Lie Quadratics in E3", Noakes determines accurate asymptotics for the case k = 0. I will talk about extending these results to the case k > 0.
Date received: November 3, 2008
Riemannian cubics and Lie quadratics: an introduction
by
Brian Corr
University of Western Australia
Riemannian cubics solve a variational problem for curves in Riemannian manifolds. When the manifold is a bi-invariant Lie group their study reduces to that of a class of curves in the Lie algebra (Lie quadratics). Some basic properties of Lie quadratics will be reviewed, especially in Euclidean 3-space.
Date received: November 16, 2008
Determining polynomial invariants of SE(3)
by
Deborah Crook
Victoria University of Wellington
This talk looks at the problem of determining polynomial invariants of the special Euclidean group SE(3) in its adjoint action on its Lie algebra, se(3). Both the action on a single element of se(3), and on a pair of elements, are examined.
Date received: November 2, 2008
Serial manipulators, screw systems and singularities
by
Peter Donelan
Victoria University of Wellington
Serial manipulators consist of a finite sequence of rigid links connected by a corresponding sequence of joints, starting from a base and terminating at the manipulator's end-effector. Each joint can be represented by a twist-an element of the Lie algebra of the Euclidean group SE(3)-or, more properly, a screw which is an element of the corresponding projective space. The forward kinematics of the manipulator are then represented by a product of exponentials in the Euclidean group. However, as the manipulator moves the screws and their span, the instantaneous screw system, vary. Selig obtained explicit formulae for the exponential map which are exploited to analyse these kinematics, with a view to understanding how the screw system varies for a given joint sequence and, in particular, how the given joints determine the manipulator's singularities. This has implications for the classification of over-constrained manipulators which exhibit unexpected self-motion.
Date received: November 2, 2008
Exterior - A Maple 10/11/12 library for computations in exterior calculus
by
Mark Hickman
Department of Mathematics & Statistics, University of Canterbury
Exterior is a package for Maple 10/11/12 that implements the exterior calculus. It allows the construction of jet bundles and exterior differential systems. The user interface is designed to mimic (as much as possible) standard mathematical notation both for the user input and the output. The package allows the user to compute, for example, symmetries of partial differential equations and exterior differential systems, characteristic vectors, Maurer-Cartan forms, torsion of lifted coframes and invariants that arise in the Cartan method of equivalence.
This talk will give concrete examples of computations using this package.
Date received: October 22, 2008
Representation of mechanical constraints on the Euclidean motion group
by
Manfred Husty
University Innsbruck, Austria
Using Study's representation as a convenient model we will discuss the representation of different mechanical systems on the motion group. We will show how mechanical constraints that represent serial robots, parallel robots and other mechanical systems map to different sets or varieties on the group. Especially we will discuss how these representations can be used to solve either direct or inverse kinematics of the systems. In the last part we will show how robot singularities fit into this framework.
Date received: November 2, 2008
Lie group approximation and quantum control
by
Wayne Lawton
Department of Mathematics, National University of Singapore
Approximation of trajectories in unitary groups by (trigonometric) polynomials has applications in classical control (polarization mode dispersion compensators, wavelet design, integrable systems), and the approximation methods are based on operator splitting formuli developed for quantum mechanics. This talk discusses potential applications to quantum control including a program to extend the recent solution of the Ten Martini Problem for the Almost Mathieu Operator to its time dependent analogue, the Kicked-Harper Operator, that provides a model for quantum chaos.
Date received: October 31, 2008
Geometry of Riemannian cubics
by
Gerrard Liddell
Maths, University of Otago
This talk will describe some of the geometry of Riemannian cubics for SO(3).
Date received: November 3, 2008
Riemannian cubics and friends
by
Lyle Noakes
University of Western Australia
A Riemannian cubic is a curve in a manifold solving a particular second order variational problem. Riemannian cubics are higher order geodesics, with applications in mechanical engineering, classical mechanics, and approximation theory with non-affine constraints. They reduce to cubic polynomials in the case of Euclidean space.
When the manifold is a bi-invariant Lie group, Riemannian cubics are studied in terms of an associated curve (the Lie quadratic) in the corresponding Lie algebra. The theory of Lie quadratics is quite rich (and still incomplete), even in SO(3) and SL(2, R).
The talk will review some central results about Riemannian cubics and (time permitting) some connections with the theory of Riemannian Bezier curves.
Date received: October 15, 2008
Quadratures for null Lie quadratics in sl(2) and so(3)
by
Michael Pauley
University of Western Australia
A Lie quadratic in a Lie algebra is a solution to a differential equation which arises in trajectory planning problems, and in computer graphics. A subclass called null Lie quadratics have special meaning, with regard to the applications, as well as the methods by which we study them. I will talk about how, in the Lie algebras sl(2) and so(3), it is possible to write quadrature formulae for null Lie quadratics.
Date received: October 29, 2008
The Riemannian Cox-de Boor algorithm
by
Tomasz Popiel
University of Western Australia
Coauthors: Lyle Noakes
The well-known Cox-de Boor algorithm for constructing polynomial curves generalises in a natural way to a Riemannian manifold M: line segments are replaced by minimal geodesics. The resulting curves can be used to solve interpolation problems in M, which arise in applications including robotics and computer animation, in which M is the Lie group SO(3) of rotations of Euclidean 3-space. Although these curves are straightforward to construct, information about their derivatives, which is needed for applications where the degree of smoothness of an interpolant is important, is usually difficult to establish. We present some recent developments.
Date received: October 27, 2008
Rational interpolation of rigid-body motions
by
J.M. Selig
London South Bank University
The group manifold of the group of rigid-body motions can be considered as an open set in a six-dimensional non-singular quadric, known as the Study quadric. Rational motions are rational curves in this quadric. Using a birational map the quadric can be transformed to six-dimensional projective space P6. These birational maps are simply derived from Cayley maps from the Lie algebra of the group to the group itself. In this way interpolation problems in the group can be transformed into interpolation problems in the Lie algebra. If only rotations about a fixed point are considered, this procedure restricts to well known methods in Computer Graphics and Computer Aided Design. Velocities can also be considered in a straightforward way and this leads to rational approximations for motions determined by variational principles, such as motions with stationary acceleration.
Date received: October 28, 2008
Modeling the invasion of Hawthorn at Porters Pass, NZ
by
Boris Baeumer
University of Otago
Coauthors: Agnes Radl
Hawthorn was introduced to Porters Pass in 1924. In 1983 and 2006 all Hawthorn trees in the area had their age estimated. We use this unique data set to test different (non-local) invasion of species models.
Date received: October 30, 2008
Empirical challenges in the evolution of the human genome
by
Gill Bejerano
Stanford University
I will outline briefly a modern view of the Human Genome highlighting the following points of potential interest: Extreme genomic sequence conservation---how surprising is it really? The discrepancy between observed short term and inferred long term dispensability of mammalian genomic DNA; Our growing appreciation for the potential complexity of the vertebrate gene regulatory code.
Date received: November 3, 2008
Poisson methods for modeling extinction and mutation
by
Peter D Drummond
Swinburne University of Technology
Coauthors: Alexei J. Drummond, T. G. Vaughan
We present an explicit unified stochastic model of fluctuations in population size due to random birth, death, density-dependent competition, mutation and environmental fluctuations. Stochastic dynamics provide insight into small populations, including processes such as extinction, that cannot be correctly treated by deterministic methods. We give exact analytical and simulation-based results for extinction times of our stochastic model without mutation with comparisons of the effects of environmental noise and intrinsic demographic stochasticity. Several methods - the discrete master equation approach, an exact mapping to a Fokker-Planck equation (the Poisson method), and stochastic equations are employed - showing they are precisely equivalent. We also calculate approximate extinction times using a steepest descent method, and demonstrate the ecological survival merit of using `unselfish' reduced birth rates, instead of `selfish' competition to control population size. This model can readily be extended to accommodate metapopulation structure and genetic variation, for example in viral populations. It thus represents a step towards a microscopic synthesis of population dynamics and population genetics.
Date received: November 3, 2008
Robust consensus methods for summarising phylogenetic trees
by
Barbara Holland
Allan Wilson Centre, Massey University
Coauthors: Barbara Keil
In phylogenetics consensus methods take as input a set of phylogenetic trees (on identical label sets) and attempt to identify where these trees agree. One popular method, majority-rule consensus displays all edges that appear in more than half the trees. Another popular approach, Adams consensus, preserves all the rooted triples that are displayed by all the input trees. For ``noisy'' data sets with large numbers of taxa both these methods can produce unresolved trees. This talk describes an attempt to modify the Adams consensus method by using ideas from the majority-rule approach to create a consensus method that is more robust to noisy data.
Date received: October 7, 2008
Reconstructing the evolutionary past of polyploid species: new combinatorial results
by
Katharina Huber
School of Computing Sciences, University of East Anglia, UK.
Coauthors: Martin Lott, Vincent Moulton, and Andreas Spillner.
Polyploid organisms are very common within plants but are also well documented within some animal groups. Essentially, such organisms arise when two organisms from different species hybridize giving rise to progeny with (possibly multiple) copies of their parents genome. The importance of polyploids for e.g. food production makes the development of methodology and algorithmic tools for reconstructing their evolutionary past an important albeit challenging task.
Recently a phylogenetic network reconstruction technique was introduced with this in mind. Its starting point is some kind of consensus over a set of multiply labeled trees (essentially rooted graph-theoretical trees in which every vertex of degree at most 2 is labeled but two distinct labeled vertices may have the same label) each supported by e.g. some gene. In this talk we will present recent results concerning the construction of such trees.
Date received: October 23, 2008
Fast phylogeny reconstruction through learning of ancestral sequences
by
Radu Mihaescu
UC Berkeley
Coauthors: Cameron Hill, Satish Rao, Alex Jaffe
Phylogenetic tree reconstruction is the task of recovering the topology of an evolutionary tree T from the evolved sequences at its leaves X. We present an algorithm which recovers the full phylogenetic tree T from logarithmic sized leaf sequences under the Cavender-Farris model of evolution with edge lengths under the Ising model phase transition: the probability of mutation along each edge is smaller than p0=(√(2)-1)/(2 √(2)).
Our work builds on recent progress by Daskalakis, Mossel, and Roch (DMR) who settled a conjecture of M. Steel by providing an O(n10) worst case running time algorithm achieving the same asymptotic results. The main advantages of our approach reside in the asymptotically optimal running time O(n2 log(n)) and the ability to provide partial topological information when some edges violate the length restriction.
We are able to circumvent the need for a-priori knowledge on lower and upper bounds on the edge lengths. Rather, we infer an edge length reliability interval from the size of the available sequences and proceed to recover the extremal components of the sub-forest of T given by edges falling in this reliability interval. In the case of trees with edges under the phase transition, the sequence length required for total reconstruction matches that of DMR.
Our methods are motivated by an intuitive minimum spanning tree framework. Similarly to DMR, we rely heavily on a method of Mossel for reconstructing sequences at the ancestral nodes of the tree with a bounded probability of error, in itself a very important problem in computational biology.
Date received: September 26, 2008
Testing hypotheses of treelikeness in genomic datasets
by
Vincent Moulton
School of Computing Sciences, University of East Anglia
Coauthors: Jo Dicks, Katharina Huber, George Savva
A common assumption of a phylogenetic analysis is that the evolutionary history of a dataset is best represented by a bifurcating tree. However, evolutionary processes including genetic recombination, horizontal gene transfer and allopolyploidy have been identified that give rise to datasets that cannot be represented adequately in this way. Our growing understanding of the importance of these events has led to the development of several mathematical and graphical representations of non-treelike evolution, phylogenetic networks. These approaches identify, within a dataset, conflicts with the hypothesis of treelike evolution. However, it is also known that apparent conflicts can arise from random variation in a dataset or through an incorrectly specified model of evolution rather than any underlying reticulate evolutionary process, so it is often difficult to intepret the results of a phylogenetic network analysis. In this talk, we address the problem of deciding whether or not a tree is adequate to describe the evolutionary signal a dataset contains. In particular, we introduce and discuss a simple test to assess the statistical significance of a non-treelike signal identified by a particular phylogenetic network tool, the NeighborNet.
Date received: October 15, 2008
Finding the trees in Darwin's forest
by
Lior Pachter
UC Berkeley
Coauthors: Robert Bradley, Nicolas Bray, Colin Dewey and Ariel Schwartz
The problem of determining homology among multiple related biological sequences, known as the alignment problem, is arguably the fundamental problem in comparative genomics. Accurate alignment is essential for both functional and evolutionary genomics studies. We explain how the problem of determining homology at the nucleotide level can be interpreted as finding the trees in "Darwin's forest", and focus on the tractability of the problem. In this letter, we argue that many recent negative results emphasizing uncertainty in alignment are misleading in that they confound uncertainty in the choice of model, uncertainty in alignment given a model, and error due to heuristics used for inference. We explain how hidden Markov models for pairwise alignment can be extended to provide effective models for multiple alignment, and show that these models indicate little uncertainty in alignment of both unrelated sequences and of orthologous sequences from related species. Moreover, we discuss an algorithm that provides an efficient approach to finding the alignments with highest expected accuracy. Together, these results provide a path to the removal of lingering doubts about the accuracy of multiple alignments.
Date received: September 2, 2008
Decision theory for saving species
by
Hugh Possingham
The University of Queensland, Maths and Ecology
Conservation science is booming, however much of its progress is hampered by a lack of quantitative thinking and tools. In this talk I will pose and solve three problems in conservation science that involve economic constraints. First we look at the problem of allocating resources between threatened species, some which may, or may not, be interdependent. Second we look at the problem of allocating resources to different actions across the globe. Third, with time, I will look at a problem of managing a threatened or harvested population where one of the objectives is learning.
Date received: October 21, 2008
Understanding frequency-dependent selection using the pairwise-interaction model
by
Hamish G Spencer
Allan Wilson Centre for Molecular Ecology & Evolution, University of Otago
Frequency-dependent selection (FDS) occurs when the fitnesses of the different genetic types of organisms in a population depend on their relative frequencies. In predator-prey situations, for instance, a rare prey type may be overlooked because it does not fit the predator’s search image. Hence, being rare confers selective advantage over commoner types and, clearly, by favouring rare genetic types, this sort of FDS has the potential to preserve genetic variability in a population. Observations of natural populations show that genetic variation is, in fact, ubiquitous. Why this is so is a central, unanswered problem in population-genetic theory, although FDS is often invoked as a plausible heuristic.
This talk examines the ability of a general model of FDS, the pairwise-interaction model, to maintain genetic variation. We also explore some mathematical properties of this model, showing, for example, why selection does not generally lead to populations with their mean fitness at a maximum.
Date received: August 24, 2008
A new method for tackling the stochastic dynamics of viral infection
by
Timothy G. Vaughan
Swinburne University of Technology
Coauthors: Peter D. Drummond and Alexei J. Drummond
The dynamics of viral infection are intrinsically stochastic in nature, and inter-dependent processes such as the infection of cells and the subsequent production of virions by those cells lead to the development of statistical correlations between the various sub-populations involved. Such correlations form an integral part of the population dynamics, but are often difficult to calculate for realistic numbers of cells and virions using standard Monte Carlo techniques. In this talk, we will demonstrate how stochastic models of viral infection can be tackled using the Poisson representation - an exact means of expressing a discrete birth/death master equation in terms of a diffusion process. By comparing the numerical results thus obtained with those calculated using Gillespie's exact algorithm, we will assess the validity and effectiveness of this approach, and discuss its potential application to the investigation of the mutation-driven evolutionary dynamics of single-host viral infections.
Date received: November 3, 2008
Symplectic methods for the simulation of Hamiltonian systems
by
John Butcher
The University of Auckland
For the faithful simulation of gravitational and other Hamiltonian problems, symplectic (or canonical) numerical methods have a crucial role. A typical method of this type is the 2-stage implicit Runge-Kutta method based on Gaussian quadrature. While preserving quadratic invariants, this method has the disadvantage of being fully implicit and therefore expensive to implement. In comparison, a general linear method with similar accuracy exists, which is diagonally implicit and therefore less expensive. Although it is only G-symplectic it preserves many invariants for millions of time steps and apparently forever.
Date received: October 30, 2008
Resonance, chaos and stability in the general three-body problem
by
Rosemary Mardling
School of Mathematical Sciences, Monash University
The quest to understand and predict the stability of general three-body configurations has existed since Newton formulated his equations of motion and his law of gravity. Poincaré modernized the quest with his prize-winning work in the early twentieth century, effectively inventing the theory of chaos in the process. Interest waned until the work of Kolmogorov, Arnold and Moser in the 1950s and 60s when the famous KAM tori entered the mathematical lexicon, although nothing specific could be said about three-body stability until the work of Wisdom in 1980 who studied the planar circular restricted problem (one body is massless and the other two have small mass ratio).
Here we present a new formulation which uses the chaos concept of resonance overlap to determine stability in the general three-body problem. It involves no free parameters and is entirely general and robust, with no restrictions on the masses or orbital elements.
Paper reference: http://adsabs.harvard.edu/abs/2008LNP...760...59M
Date received: November 2, 2008
Jupiter: shield or sniper?
by
Philip Sharp
University of Auckland
Coauthors: K. R. Grazier, W. I. Newman
After the formation of the planets in the solar system, a large number of small bodies orbiting the Sun remained. Over the following half a billion years, most of the larger bodies inside the orbit of Saturn were removed. The obvious mechanisms for removal are accretion by a planet, ejection from the solar system or incorporation into the clouds or belts of objects beyond Neptune.
We simulated the trajectories of four sets of 10, 000 massless particles over 100 million years using a simulation method that achieved the theoretical lower bound on the integration error. The particles in the first set were initially between Jupiter and Saturn, those in the second between Saturn and Uranus, the third between Uranus and Neptune and the fourth in the Kuiper Belt.
We found, contrary to previous analysis, that Jupiter does not protect the terrestrial planets from bombardment by bodies orbiting the Sun.
Date received: October 28, 2008
Binary star scattering encounters resulting in single stars
by
Winston Sweatman
Massey University, Albany
Binary stars can be separated into their component stars through interaction with other binary or single stars. It is important to understand this process as the dynamics of much larger stellar N-body systems can be driven by few-body encounters.
Approximating stars by point masses, there exist theoretical approaches for this process at extremes of high and low energy. These have been used to estimate the ionisation cross-section, which is the measure of likelihood of separation into single stars during an encounter.
Date received: October 21, 2008
Future challenges for variational analysis
by
Jonathan Borwein
Newcastle and Dalhousie
Modern nonsmooth analysis is now roughly thirty-five years old. In this talk I shall briefly assess where the subject stands today from the perspective of both theory and applications. I will also discuss some open problems and current challenges for the subject.
Date received: November 2, 2008
Outer approximation schemes for generalized semi-infinite variational inequality problems
by
Regina Sandra Burachik
University of South Australia
Coauthors: Lopes, J. O. (Universidade Federal do Piaui, Brazil)
We introduce and analyze outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems Generalized Semi-Infinite Variational Inequality Problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.
Date received: October 23, 2008
Direction-set updates for derivative-free optimization
by
Ian Coope
University of Canterbury
The updating of direction sets in direct search methods for unconstrained optimization is examined. Both weak and strong quasi-Newton updates are considered together with other simple quadratic interpolation conditions. Efficient and numerically stable techniques are described for implementing the appropriate updates. The updating schemes are applicable to both line search and trust region algorithms as well as some newer grid-based methods for derivative-free optimization and the updates considered can usually be calculated in O(n2) arithmetic operations.
Date received: November 2, 2008
Kinds of vector invex
by
Bruce Craven
University of Melbourne
Necessary Lagrangian conditions for a constrained minimum become come sufficient under generalized convex assumptions, in particular invex, and duality results follow. Many classes of vector functions with properties related to invex have been studied, but it has not been clear how far these classes are distinct. Various inclusions between these classes are now established. Some modifications of invex can be regarded as perturbations of invex. There is a stability criterion for when the invex property is preserved under small perturbations. Some results extend to nondifferentiable (Lipschitz) functions.
Date received: October 27, 2008
A new least squares best fit problem for utility estimation with application to the fitting of elasticities to data in CGE modelling
by
Andrew Eberhard
RMIT University
Coauthors: A. Eberhard, S. Schreider, L. Stojkov and D. Ralph
We consider the problem of fitting of a utility to a finite sample of demand data. Theory is developed to justify this process which shows that this process provides a partial positive answer to the problem of revealed preference when no sampling errors are present. It is showed that even when the underlying utility is not consistent with a concavity assumption one can still form approximations involving concave utilities. Conditions are given in terms of the boundedness of parameters fitted in the approximate Afriat utilities that ensure the approximations converge to a concave utility. When sampling errors are present one must solve a nonlinear best fit problem of unique character. Application is then made to the estimation of elasticities that are used in economic models. Some numerical simulations are provided.
Date received: October 29, 2008
A multistage stochastic programming approach to open pit mine production scheduling with uncertain geology
by
Gary Froyland
University of New South Wales
Coauthors: Natashia Boland, University of Newcastle;
Irina Dumitrescu, University of New South Wales
The Open Pit Mine Production Scheduling Problem (OPMPSP) studied in recent years is usually based on a single geological estimate of material to be excavated and processed over a number of decades. However techniques have now been developed to generate multiple stochastic geological estimates that more accurately describe the uncertain geology. While some attempts have been made to use such multiple estimates in mine production scheduling, none of these allow mining and processing decisions to flexibly adapt over time, in response to observation of the geological properties of the material mined. In this paper, we use multiple geological estimates in a mixed integer multistage stochastic programming approach, in which decisions made in later time periods can depend on observations of the geological properties of the material mined in earlier periods. Since the material mined in earlier periods is determined by our decisions, the information received about uncertain properties, and when that information is available, is decision-dependent. Thus we tackle the difficult case of stochastic programming with endogeneous uncertainty. We extend a successful mixed integer programming formulation of the OPMPSP to this stochastic case, and show that non-anticipativity can be modelled with linear constraints involving variables already present in the model. We extend this observation to the general class of endogenous stochastic programs, and exploit the special structure of our model to show that in some cases we can omit a significant proportion of these constraints. Using data supplied by our industry partner, (a multinational mining company), we show that this approach is reasonably tractable, and demonstrate the improvements that can be made to mine schedules through the explicit use of multiple geological estimates.
Date received: October 27, 2008
Necessary and sufficient conditions for inversion of perturbed linear operators on Banach space.
by
Phil Howlett
University of South Australia
Coauthors: Amie Albrecht, Charles Pearce
In this paper we find necessary and sufficient conditions for the existence of a Laurent series expansion with a finite order pole at the origin for a perturbed bounded linear operator on Banach space.
Date received: October 30, 2008
Runge-Kutta discretization and inexact restoration for optimal control
by
C Yalcin Kaya
University of South Australia
Coauthors: Bulent Karasozen
A computational technique for a class of optimal control problems is presented. First Runge-Kutta discretization is carried out to obtain a finite-dimensional approximation of the continuous-time problem. Then an inexact restoration (IR) method is applied to the discretized problem to find an approximate solution. Convergence of the technique to a solution of the continuous-time problem is facilitated, under some general conditions, by convergence of the IR method and convergence of the discrete (approximate) solution as finer subdivisions are taken. Numerical experiments are presented for a discussion of the technique.
Date received: October 29, 2008
SIP approach to continuously constrained LQ optimal control problems via piecewise polynomial control parameterization
by
Yanqun Liu
Department of Mathematics and Statistics, RMIT University
In this paper, we consider the class of LQ optimal control problems with continuous constraints involving both the state and control. We extend an existing SIP method for the cases where the constraints involves only the system states. The existing method employs piecewise constant control to reduce the optimal control problem to an SIP problem. Here we use piecewise polynomial control instead. We provide convergence results with treatment of the control term in the constraint function. We present a number of illustrative examples demonstrating an improved accuracy over piecewise constant controls without increasing the size of the resulted SIP problem.
Date received: October 24, 2008
Necessary optimality conditions for some control problems of elliptic equations with venttsel boundary conditions
by
Yousong Luo
RMIT University
In this paper we derive a necessary optimality condition for a local optimal solution of some control problems. These optimal control problems are governed by a semi-linear Vettsel boundary value problem of a linear elliptic equation. The control is applied to the state equation via the boundary of the domain and a functional of the control together with the solution of the state equation under such a control will be minimized. A constraint on the solution of the state equation is also considered.
Date received: October 23, 2008
A stochastic direct method for bound constrained non-smooth global optimization
by
Chris Price
Maths and Stats, University of Canterbury
Coauthors: M. Reale and B. L. Robertson
A stochastic algorithm for global optimization subject to simple bounds is described. The algorithm is in the spirit of the direct algorithm of Jones, Perttunen, and Stuckman. Like direct it generates succcessively finer covers of the feasible region. Each cover consists of a finite number of boxes, where each box is defined by simple bounds on each variable. Its principal differences are that it subdivides each box into two rather than three smaller boxes, and that it calculates the objective function at a randomly selected point in each box, rather than the box's centre. The stochastic nature of the method permits a limited memory version to be developed. The sequence of best known function values is shown to converge to the essential global minimum with probability one on non-smooth functions. Numerical results are presented.
Date received: October 27, 2008
Optimal attitude control of an accompanying satellite rotating around the space station
by
Fang Wang
RMIT University
Coauthors: Pavel M. Trivailo,Honghua Zhang
This paper deals with the optimal control of an accompanying satellite rotating around the space station in the presence of sinusoidal disturbances. The concept of accompanying satellites (AS) around the space station (SS) is introduced. Both the AS and the SS are modeled as rigid bodies with the reference coordinate frames described for the AS pointing to the SS. There are many functions for the AS of the SS such as navigation, relaying communication data and inspecting the SS. Since all of these functions require attitude control of the AS, it is necessary to study the optimal control for the AS rotating around the SS.
Untill now there have been many studies about the optimal control of satellites. However, all of them did not consider the case when the satellite is underactuated, i.e., actuators in one or two dimensions are failed, though the fact is that the AS has a probability of disabled (or damaged or malfunctioned) actuators during its running period. Hence it is also necessary to study the problem when AS is underactuated, which is the novelty of the paper. Without loss of generality, the underactuated axis is assumed to be the third axis of the AS.
The purpose of this paper can be stated as designing an optimal control law so that the underactuated AS can achieve suitable attitude in accordance with the expected thrust direction before orbit maneuvering, and then attain reorientation towards the desired direction (e.g., the SS) after the orbit maneuvering with the least thrust. Based on this purpose, the paper first defines the reference coordinate, frames for the AS pointing to the SS according to different missions. Then by using unit quaternion, the full set of nonlinear equations of motion is derived. These equations are solved numerically using direct transcription method to obtain optimal solution for the underactuated satellite. The direct method seeks to transform the continuous optimal control problem into a discrete mathematical programming problem, which in turn is solved using a non-linear programming algorithm. By discretizing the state and control variables at a series of nodes, the integration of the dynamical equations of motion is not required. The state equations are enforced as constraints by using interpolating polynomials and an implicit integration scheme is used in each discrete segment, which ensures fast computational times. The Chebyshev-pseudospectral method, due to its ease of implementation, high accuracy and fast computation times, was chosen as the direct optimization method to be employed to solve the problem. The analytical and simulation results show that the proposed control law is effective in the case of failed actuators.
Further research in this area is discussed. It involves cases when the AS has flexible attachments; the AS is under disturbances; and the inertia matrix is unknown or poorly known, of which the controller for the underactuated AS will become more complicated.
Date received: September 28, 2008
Overshoot characterisation for continuous-time systems
by
Rob Wenczel
RMIT University
Coauthors: Robin Hill (RMIT)
This talk is concerned with the design of feedback controllers to optimally track a step input. For this problem the amount of overshoot in the system response is an issue of considerable engineering significance. For a given open-loop system, there are fundamental limitations on the extent to which a linear time-invariant controller can reduce the overshoot response to a step. Existing results for discrete-time systems use convex optimisation tools, with density arguments to achieve a tight bound.
We show that the same techniques yield results of identical structure, for the case of continuous-time systems
Date received: October 16, 2008
Solving large scale highly nonlinear systems of equations and spherical designs
by
Rob Womersley
School of Mathematics and Statistics, University of New South Wales
Spherical L-designs are sets of N points on the unit sphere such that the average of the function values at the points gives the integral over the sphere for any spherical polynomial of degree at most L. Thus they provide the nodes of an equal weight quadrature rule of precision L for the sphere.
Building on a new characterization of spherical designs by Sloan and Womersley, spherical designs can be characterized by finding a global minimum of zero for an objective function or solving a system of nonlinear equations. This talk concentrates on issues related to solving the resulting large scale (up to degree 100 with 10, 000 variables and equations) highly nonlinear system of equations and difficulties with being sure that there is a true solution near to the computed solution.
Date received: October 29, 2008
The generalized superelliptic equation
by
Michael Bennett
University of British Columbia
Coauthors: Sander Dahmen
Given an irreducible binary form F(x, y) of degree at least three, one might expect that the form represents at most finitely many kth powers, for k > 3 variable. While such a conclusion does follow from a suitable number field generalization of the ABC-conjecture, it has not been previously possible to exhibit, for example, a single cubic form for which we can prove it to hold. In this talk, I will sketch recent joint work with Sander Dahmen which establishes this result for a large class of forms. We rely upon new ideas from the theory of Frey curves and their associated Galois representations.
Date received: October 29, 2008
Rational solutions of y2=x6+k
by
Andrew Bremner
Arizona State University
Coauthors: Nikos Tzanakis
We discuss techniques for finding all rational points on the genus 2 curves y2=x6+k, and apply these techniques to find all solutions of the title equation for |k| ≤ 50, with the exception of k=-47, -39.
Date received: October 16, 2008
On the missing values of n! mod p
by
Kevin Broughan
University of Waikato
Coauthors: A. Ross Barnett
Consider the question of the values of n! mod p for odd primes p and 1 ≤ n ≤ p-1. Numerical evidence shows that about p/e of the residue classes are missing, but there has been little progress in explaining this phenomenon. That is except for a result of Cobeli, Vajaitu and Zharescu [2] who show that the 1/e missing values proportion arises as an average when the set of all sequences is considered.
Here we show how the 1/e proportion arises, and then give some details of the recent result that when sequences are considered which obey a "no identical neighbors" condition, and so better model the factorial, the mysterious 1/e proportion is maintained.
[1] Broughan, K. A. and Barnett, A.R, On the missing values of n! mod p, (submitted 2008).
[2] Cobeli, C., Vâjâitu, M., and Zaharescu, A. The sequence n! ( mod ) p, J. Ramanujan Math. Soc. 15 (2000), p135-154.
Date received: September 7, 2008
Series and iterations for 1/p
by
Shaun Cooper
Massey University
I will show how the Rogers-Ramanujan continued fraction and four other similar functions can be used to derive series and iterations for 1/p. This extends recent work of H. H. Chan, W.-C. Liaw, K. P. Loo and the author.
Date received: October 28, 2008
Local and global zeros of ternary quadratic forms
by
John Friedlander
University of Toronto
Coauthors: Henryk Iwaniec
We study a problem of Serre and variations thereof concerning the existence of non-trivial zeros of the Legendre quadratic form ax2 + by2 - z2.
Date received: October 28, 2008
Computing level one Hecke eigensystems (mod p)
by
Alexandru Ghitza
University of Melbourne
Coauthors: Craig Citro
We describe an algorithm for computing the complete list of systems of Hecke eigenvalues coming from modular forms (mod p) of level one. The focus will be on the theoretical underpinnings, but we will also touch upon the implementation of the algorithm in Sage.
Date received: October 28, 2008
Iwasawa theory of elliptic curves for supersingular primes
by
Byoung Du Kim
Victoria University of Wellington
I will introduce Iwasawa theory for elliptic curves. The focus will be on the new theory, ``the plus/minus Iwasawa theory'' for supersingular primes, and I will explain how it is used to solve some problems including the parity conjecture.
Date received: November 3, 2008
A fundamental system of units for a family of algebraic number fields of degree 12
by
Claude Levesque
U. Laval, Quebec
Let a and b be the two real roots of the
(assumed irreducile) polynomial X2+DX +d where
D, d are integers such that d divides D and D2 -4d > 0.
Let K be the sextic field Q( w) where
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Date received: October 28, 2008
On the spacings between C-nomial coefficients
by
Florian Luca
Mathematical Institute, UNAM, Morelia
Coauthors: Pantelimon Stanica, Diego Marques
Let (Cn)n ≥ 0 be the Lucas sequence Cn+2=aCn+1+bCn for all n ≥ 0, where C0=0 and C1=1. For 1 ≤ k ≤ m-1 let
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Date received: October 28, 2008
Rational-derived polynomials
by
Jim MacDougall
University of Newcastle
A polynomial in Q[x] is called rational-derived if all of its roots are rational and all the roots of all of its derivatives are rational. All such polynomials of degree 3 are easily characterised. Surprisingly, already at degree 4 we do not know how to describe all such polynomials. In particular, no rational-derived quartic with 4 distinct roots has been found. This talk will survey what is known about this and some related problems.
Date received: October 24, 2008
Inequities in the Shanks-Rényi prime number race
by
Greg Martin
University of British Columbia
Coauthors: Daniel Fiorilli (Université de Montréal)
Let p(x;q, a) denote the number of primes up to x that are congruent to a (mod q). It has been well-observed that an inequality of the type p(x;q, a) > p(x;q, b) is more likely to hold if a is a nonsquare modulo q and b is a square modulo q (the so-called “Chebyshev bias” in comparative prime number theory). However, the tendencies of the various p(x;q, a) (for nonsquares a) to dominate p(x;q, b) have different strengths: the asymptotic (logarithmic) density of the real numbers x for which p(x;q, a) > p(x;q, b) can depend on a and b as well as on q. Some of these densities have been computed, but only by using laborious numerical integration of functions involving zeros of the appropriate Dirichlet L-functions. We present an asymptotic formula for these densities, which in its most explicit form explains which nonsquares a are most dominant for a given square b.
Date received: October 21, 2008
Modular curves of genus 3
by
Roger Oyono
University of French Polynesia
Coauthors: Enrique Gonzalez Jimenez
We present a method for computing equations of non-hyperelliptic modular curves (defined over the rationals) of genus 3.
Date received: October 30, 2008
Perfect powers expressible as sums of two cubes
by
Samir Siksek
University of Warwick
Coauthors: Imin Chen
We attack the equation a3+b3=cn (a, b, c coprime integers) using a combination of the modular approach (via Galois representations and modular forms) together with an obstruction to solutions which is of Brauer-Manin type. We solve this equation for a set of prime exponents n having Dirichlet density 0.628.
Date received: October 21, 2008
Cubic Thue equations with many solutions
by
Cameron Stewart
University of Waterloo
Let F be a cubic binary form with non-zero discriminant and integer coefficients. We shall show how that there is a positive number c, which depends on F, such that the Thue equation F(x, y)=m has at least c(logm)(1/2) solutions in integers x and y for infinitely many positive integers m.
Date received: November 2, 2008
Application of a theorem of Akhtari to families of quartic diophantine equations.
by
Gary Walsh
University of Ottawa
We elaborate on a recent theorem due to Shabnam Akhtari, and describe the applicability of this theorem to solving certain families of classical diophantine equations, and extensions thereof.
Date received: September 11, 2008
A fundamental cryptographic(?) algorithm
by
Hugh Williams
University of Calgary, Canada
Cryptosystems which rely for their security on the presumed difficulty of solving the discrete logarithm problem in quadratic number fields execute somewhat more slowly than the standard Diffie-Hellman or RSA techniques. Although this gap has narrowed somewhat in the last several years, in order to narrow it further, there is still a fundamental difficulty that must be addressed. This is the fast implementation of the operation of finding a reduced ideal equivalent to the product of two given ideals, the operation analogous to that of modular multiplication in rational number theory. In 1988 Daniel Shanks described an algorithm, which he called NUCOMP, for performing this operation. The beauty of this algorithm is that it does not require the large intermediate numbers that are needed after the usual ideal multiplication, which is subsequently followed by a reduction procedure. Although Shanks’ version of NUCOMP was developed for imaginary quadratic fields, van der Poorten was able to show that it could also be used for real quadratic fields. In this talk, I will describe the most recent version of NUCOMP, and present an analysis of why and how well it works.
Date received: October 30, 2008
On finiteness of odd superperfect numbers
by
Tomohiro Yamada
Department of Mathematics, Kyoto University
Some new results concerning the equation s(N)=aM, s(M)=bN are proved, which implies that there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
Date received: October 28, 2008
On Hecke eigenvalues at Piatetski-Shapiro primes
by
Liangyi Zhao
Nanyang Technological University
Coauthors: Stephan Baier
Let l(n) be the normalized n-th Fourier coefficient of holomorphic cusp form for the full modular group. We show that for some constant C > 0 depending on the cusp form and every fixed c in 1 < c < 8/7, the mean value of l(p) is O ( exp( -C √{logN} )) as p runs over all (Piatetski-Shapiro) primes of the form [nc] with some natural number n ≤ N.
Date received: September 8, 2008
An intrinsic description of the trace spaces Hs(∂G) and classes of harmonic functions on G.
by
Giles Auchmuty
University of Houston
This talk will outline an characterization of the Hilbert trace spaces Hs(∂G) on bounded regions G in Rn with minimal boundary regularity. The description uses a spectral characterization in terms of the Steklov eigenfunctions of the Laplacian on the region. This leads to corresponding Hilbert spaces of real harmonic functions on the regions. It is shown that these spaces are reproducing kernel Hilbert spaces with respect to a natural inner product. Representation theorems for the solutions of Dirichlet, Robin and Neumann boundary value problems in these spaces are described.
Date received: October 31, 2008
Inverse positivity for general Robin problems on Lipschitz domains
by
Daniel Daners
The University of Sydney
We prove that elliptic boundary value problems in divergence form can be written in many equivalent forms. This is used to prove regularity properties and maximum principles for problems with Robin boundary conditions with negative or indefinite boundary coefficient on Lipschitz domains. We do this by rewriting such problem as a problem with positive boundary coefficient. We finally show that such a result cannot be proved for domains with an outward pointing cusp.
Date received: November 2, 2008
Convergence and thresholds in nonlinear diffusion problems
by
Yihong Du
University of New England
Coauthors: Hiroshi Matano
We study the Cauchy problem
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Date received: October 19, 2008
Sectorial forms and degenerate operators
by
Tom ter Elst
University of Auckland
Coauthors: Wolfgang Arendt
In the theory of sectorial forms and holomorphic semigroups a basic assumption is that the form is closed, or at least closable. This is a nasty difficult condition. In a recent paper with Wolfgang Arendt we proved that one can associate in a natural way a holomorphic semigroup generator to any sectorial form, even if it is not closable. Thus one can forget closability. This opens the door to consider complex degenerate elliptic differential operators without demanding that they are symmetric or strongly elliptic. In the talk we present several examples and applications.
Date received: October 23, 2008
Direct " Delay" reductions of the Toda equation
by
Nalini Joshi
School of Mathematics and Statistics, The University of Sydney
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painlevé equations. The Lax pair associated to this equation is obtained, also by reduction.
Paper reference: arXiv:0810.5581
Date received: November 1, 2008
Fully nonlinear curvature flows of nonconvex hypersurfaces
by
James McCoy
University of Wollongong
In this talk I consider a natural class of fully nonlinear curvature flows with closed, compact initial data that is not necessarily convex. I will show that some fundamental behaviour of solutions to these curvature flows is analogous to the case of the mean curvature flow, including the result that the only smooth, compact self-similar shrinking solutions of positive speed are shrinking spheres.
Date received: September 24, 2008
An abstract approach to domain perturbation for parabolic equations
by
Parinya Sa Ngiamsunthorn
University of Sydney
Let V be a reflexive Banach space. Suppose Kn, n ≥ 1 and K are closed and convex subsets of V. We show that Mosco convergence of Kn to K is equivalent to Mosco convergence of L2((0, T), Kn) to L2((0, T), K), where L2((0, T), Kn) consists of all function u ∈ L2((0, T), V) with u(t) ∈ Kn a.e. t ∈ (0, T). An application in domain perturbation for parabolic equations will be discussed.
Date received: October 22, 2008
Analysis of degenerate elliptic operators of Grusin type
by
Adam Sikora
Australian National University
Coauthors: Joint work with Derek W. Robinson
We analyze degenerate, second-order, elliptic operators H in divergence form on L2(Rn×Rm).
We assume the coefficients are real symmetric and a1Hd ≥ H ≥ a2Hd for some
a1, a2 > 0 where
|
Our principal results state that the submarkovian semigroup St=e-tH is conservative and its
kernel Kt satisfies bounds
|
Paper reference: math.AP/0607584
Date received: November 3, 2008
Bifurcation for some non-Fréchet differentiable problems
by
Charles Stuart
EPFL, Lausanne, Switzerland
We consider some basic aspects of bifurcation theory in the context of maps that are differentiable in the sense of Hadamard. In finite dimensions this property is equivalent to Fréchet differentiability, but in infinite dimensions it is a weaker condition. The stationary nonlinear Schrödinger equation will be used to illustrate the general results.
Date received: October 3, 2008
Hypersurface Lp estimates for approximate eigenfunctions of a differential operator
by
Melissa Tacy
Australian National University
In this talk I will present Lp estimates for approximate eigenfunctions of a differential operator restricted to a hypersurface of a compact manifold. This proof, similar to Koch, Tataru and Zworski's proof of eigenfunction estimates over the whole manifold, exploits locality to transform the problem into one concerning evolution equations. Strichartz estimates are then used, with one spatial variable taking the place of time, to achieve the required estimates.
Date received: October 28, 2008
Convergence of anisotropically decaying solutions of a semilinear parabolic equation
by
Eiji Yanagida
Mathematical Institute, Tohoku University
We consider the Cauchy problem for a semilinear parabolic equation with a supercritical power nonlinearity. It is known that the asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular, if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In this talk we consider solutions whose initial value decays in an anisotropic way. Then we show that the solution converges to a steady state which is explicitly determined by an average formula. The proof is given by using previous results on the global stability and quasi-convergence of solutions, self-similar solutions of the linearized equation around a singular steady state, and a comparison technique. This is a joint work with Peter Polacik of the University of Minnesota.
Date received: October 30, 2008
More mathematical graduates than ever: behind the figures from UOA
by
Bill Barton
Department of Mathematics, University of Auckland
Coauthors: Louise Sherryn
The Pipeline Project is an international project looking at mathematics graduates from secondary and tertiary systems. The New Zealand data, like other countries, has been hard to mine--but we now have a comprehensive time-series for The University of Auckland.
This talk will examine this data and ask questions about some of the trends. We will also relate it to known data from other universities and data overseas.
Date received: October 29, 2008
Annotations and digital ink in teaching electronically and via the access grid ... in Australia and the UK ... and NZ?
by
Bill Blyth
Australian Mathematical Sciences Institute
In recent years it has become common for mathematics lecturers to use computer projection in their lectures and seminar presentations. Often some handwriting on a whiteboard is used for asides: for clarification, for worked examples and sketches. If the handwriting is done electronically, as is necessary over the Access Grid, it is referred to as Digital Ink.
We will give a brief overview of e-teaching approaches. This will include using the beamer class in LaTeX to produce pdf slides (with stepped uncovering of a slide) and annotation of any pdf file using PDF Annotator (and jarnal). We will demonstrate using a TabletPC to produce highlighting (note that a laser pointer is not effective in an AGR) and annotations of pdf "notes" or slides. When using a TabletPC, WindowsXP provides a very good (and fully integrated) Digital Ink with Word, PowerPoint and Excel. Digital Ink within Maple and also with an interactive whiteboard will be demonstrated. We discuss an example of marking remote student work (as a pdf file and using PDF Annotator).
We'll make a few preliminary comments about the Australian national program of collaborative teaching of Honours mathematics and statistics via the Access Grid (AG); and also comment about the taught courses for PhD students in the mathematical sciences in the UK using Access Grid. A comprehensive seminar series has been given in Canada and has begun in Australia. Since NZ has KAREN, a network of AG Rooms, opportunities for Australia & New Zealand collaborations abound.
Date received: October 30, 2008
eLearning and automated assessment using Maple
by
Bill Blyth
Australian Mathematical Sciences Institute & RMIT University
Coauthors: Alexandra Labovic (RMIT University)
In the first semester of a traditional calculus course, the weekly Maple lab sessions are not used to directly support the lectures ... nearly half of the work is at school level! A major aim is for the students to enjoy the experience of using Maple.
Students work in groups of size 2 to 4. After Maple introductions, they complete an Introduction to Animation session and then choose an extended animation project from a list of five problems. They have to demonstrate their animations in the lab for assessment. Students enjoy the animation project.
Following the animation projects, Spot the Curve uses plots and animations to understand horizontal and vertical translation of curves: students identify the randomly generated translations used and appreciate automatic marking within Maple. Student feedback has been very positive.
Our trapezoidal rule assignment is now more fun: it's disguised as a Fish Pond (a trout farm). Trapezoidal rule is used to approximate the cross-sectional area (and hence the number of trout) ... the students download a template for individualized fish ponds, with automatic marking. These projects are enjoyable deep learning activities.
The Fish Pond assignment could be set and automatically marked with standard Computer Aided Assessment, CAA, systems such as MapleTA. However in second semester, we have introduced a pendulum assignment for which students obtain immediate automatic marking of numeric and symbolic (exact) answers and save their comments and plots in their Maple file. A tutor marks these elements and generates a full marking report (all in Maple) for students. This can’t be implemented with CAA systems currently available, such as MapleTA.
Developing innovative eLearning and eAssessment materials is resource intensive and so collaboration, using collaborative environments such as the Access Grid, is particularly appealing.
Date received: October 30, 2008
Digitally assisted discovery and proof in mathematics
by
Jonathan Borwein
Newcastle and Dalhousie
I will argue that the mathematical community (appropriately defined) is facing a great challenge to re-evaluate the role of proof in light of the power of current computer systems, of modern mathematical computing packages and of the growing capacity to data-mine on the internet. With great challenges come great opportunities.
I intend to illustrate the current challenges and opportunities for the learning and doing of mathematics. For example, the knowledgeable user of Maple and Google if presented with the unremarkable number
a=1.4331274267223117583?
can "discover" within seconds-via various pathways-that it has the more remarkable continued fraction
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 , ?]
The same user may then quickly discover (from, say, JSTOR, MathWorld, Wikipedia or elsewhere) that such arithmetic continued fractions only arise as ratios of Bessel functions (of which they may never have heard, but in the new order, so what?). Indeed, a = I0(2)/I1(2) where I0 and I1 are the Bessel functions of the first kind of order zero and one respectively. Armed with this knowledge a proof is easy.
The continued fraction-a concept which a lamentably small number of university mathematics students meet, perhaps because hand-computation is difficult-itself affords a fine illustration of the power of computers to make concepts more accessible. It provides one of many fine illustrations of the value of seeking different representations for the same object and so to provide for better motivated learning of new concepts. Above all the growing richness of this matrix of tools places an extra onus on us to understand and explicate the appropriate role of proof. As Jacques Hadamard put it "The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it".
I shall also explore some of the impediments to the assimilation of these new techniques into our parole. These include inertia, organizational and technical bottlenecks, under-prepared or mis-prepared teachers of mathematics, and the lack of material from which to train them.
Reference: Jonathan Borwein and Keith Devlin, The Computer as Crucible: an Introduction to Experimental Mathematics, AK Peters, 2008.
Date received: September 29, 2008
Distance education with discrete mathematics
by
Graham Clarke
RMIT University
Discrete Mathematics provides opportunities and challenges as a subject for distance education. The nature of the challenges has changed with new developments in technology, but many problems remain. We review some difficulties and successes in the provision of Discrete Mathematics to students in remote and distant locations.
Date received: October 30, 2008
An urgent mathematical mustering: calling all king Terry's horses & all king Tao's men
by
Patricia Cretchley
USQ and QUT
Financial imperatives and managerial ambitions are dictating funding foci in Australian Universities and elsewhere. Some managers use a numbers cut-off to justify withdrawing of majors with relatively small numbers, typically mathematics. The damaging effects to mathematics programs and staffing are certainly but not only being felt in regional universities in Australia, as the Flinders and USQ stories tell.
To promote discussion on what more we can be do to protect mathematics programs from destructive education policies and management, I outline the Flinders/USQ stories and the people inexorably linked by the damage done in these two universities.
The Cast reads like that of a Fairy Tale: The Villian Dean, the Hero Fields Medalist, and the delightful Talented Child whose urgent plea to Terry Tao sets the scene for a mathematics support campaign second to none, internationally. And a Cast of Thousands of aspiring mathematics students, not all of them Regional Peasants!
But the synopsis is not a Fairy Tale. What unfolds is a large-scale Tragedy of damage to mathematics students and academics, and the destruction of decades of careful mathematics program-building. As one of those closely affected, I report the agonies of academics and students sadly... but determinedly.
This talk is designed to help set the scene for a Round Table that focuses on issues of the educational funding and management of mathematics in Australian and New Zealand. What we can learn from these events, each other's experiences, and practices elsewhere? How do we best steer our mathematical futures?
Date received: November 20, 2008
Fermat's Last Theorem in romantic mathematics
by
Miroslav Haviar
M Bel University, Slovakia
Coauthors: Pavel Klenovcan
Several years ago we established a course of Romantic Mathematics offered for all students at M Bel University in Slovakia with a particular target group being the future teachers of Mathematics. The course has been based on S. Singh's book Fermat's Last Theorem and its aim has been to achieve that students would become: (i) better aware of the history and current developments of Mathematics and (ii) more enthusiastic about the beauty and the challenges of Mathematics via the thrilling stories like the one of Andrew Wiles.
We share our experience about teaching (or rather performing) this course and illustrate the work of students.
Date received: October 30, 2008
Challenging pre-service primary teachers
by
Carolyn Kennett
Macquarie University
Coauthors: Dilshara Hill
Pre-service primary teachers bring to their studies a wide range of mathematical backgrounds, beliefs and experiences. It is in the interests of mathematics as well as the improved education of our children that these students come out of university with positive attitudes and experiences in mathematics.
As teachers of mathematics we should be aware of the need to challenge some traditional erroneous beliefs about mathematics as well as facilitating successful experiences in mathematics. We look at some of the things that work and some that don’t in a first year course specifically designed for pre-service primary and early childhood teachers.
Date received: November 3, 2008
American Indian participation in mathematics in the U.S. - obstacles and opportunities
by
Bob Megginson
University of Michigan
This talk will focus on some of the barriers that have prevented the greater participation of U.S. American Indians in the mathematical sciences. A bit about American Indian educational history in the U.S. will be presented, where it is relevant to participation in mathematics, along with a discussion of some perceptions about American Indian ability to do mathematics that have been damaging. Though the audience will likely not need convincing, the talk will end with some evidence that the ability of American Indians to do mathematics should certainly not be in question.
Date received: November 20, 2008
Advanced features in mathematics typesetting and presentation
by
Ross Moore
Macquarie University
As methods of electronic communication have been developed, so also has the (La)TeX software been evolving to take advantage of newly emerging technologies. The typesetting of mathematics has always presented challenges that are much greater than for normal prose, whatever the language. With the adoption of Unicode on all modern computing platforms, the trend will be toward electronic documents in which all manner of content, including mathematics and/or exotic scripts, can coexist and remain easily searchable and copyable (if not editable) with standard software tools. In this talk I will present some techniques which should be of particular interest for handling mathematical content, whether by a publisher, researcher, teacher or student. Examples will be shown that are readily available on the AustMS website and course unit sites at Macquarie University, and elsewhere.
Date received: October 24, 2008
Demise of the "back of envelope" sketch?
by
Ross Moore
Macquarie University
We teachers, and our students, see sophisticated technical graphics all the time in movies and on TV shows, whether for entertainment or enlightenment. General purpose mathematical software tools, such as Mathematica, Maple and others, provide the capability to produce detailed graphics which can better present some mathematical ideas than the "back of envelope" or "chalk-board" sketches that we all grew up with. It cannot be leaving a good impression or creating adequate comprehension, when we try to present complex geometrical ideas using just roughly drawn sketches.
In this talk I'll show some of the graphics that I have been using in a course on Vector Calculus, to illustrate functions and vector fields in 2 and 3 dimensions, and their integrals over curves and surfaces. That is, "div, curl, and all that...", visualised with accurately-drawn 2D and 3D graphics, employing colours and transparency (i.e., opacity), and animations, to help develop a better understanding of what the integrals mean. Furthermore, at least with Mathematica, the presentations used in lectures can be saved as well-presented PDFs for the lecture "notes". These PDFs can retain the full quality of the graphics and include the animations.
It is not my contention that all lecturers need to become graphic artists; but that some of us should be gaining good experience with these software tools, and sharing the fruits with our colleagues and students.
Date received: October 24, 2008
The use of reflective journals in a first year mathematics unit
by
Leanne Rylands
University of Western Sydney
Coauthors: Carmel Coady
``I hate maths'', ``I can't do maths''. Anecdotal evidence suggests that more and more students are entering university with very negative feelings towards maths. Most such students avoid maths if at all possible.
In 2008 our we introduced a new subject specifically designed to help students develop strategies to lessen the effects of maths anxiety and test phobia, as well as to revise basic maths and build their confidence. A reflective journal was part of the assessment in this mathematics subject.
I will talk about our experiences with this so far.
Date received: October 30, 2008
The decline of Australian mathematical sciences capability
by
Jan Thomas
Australian Mathematical Sciences Institute
In 1995 a review of Australian mathematical sciences found them to be facing challenges but to be in reasonably good health. In 2006 another review found a very different situation with difficulties at every level from inadequate primary teacher education to a narrowing research base. In short, the mathematical sciences in Australia are in crisis. This is having a profound effect on opportunities for students in schools to access a quality mathematics education. Australia does not have enough graduates in mathematics and statistics and this affects teacher supply. Increasingly access to a quality mathematics education equates with being able to pay private school fees. In the late 1980s, Australia came close to recognising, and was beginning to cater for, all students’ mathematics education needs. In 2008 we have managed to turn this around so there is now an impossible gateway for many. Data behind this situation will be presented and discussed
Date received: October 29, 2008
Computing permutations with stacks and deques
by
Mike Atkinson
University of Otago
Coauthors: M. H. Albert, S. A. Linton
Upper and lower estimates are given for the number of permutations that can be computed using a deque, two stacks in parallel, or two stacks in series
Date received: October 22, 2008
Pathwidth and caterpillar algorithms
by
Michael J. Dinneen
University of Auckland
I will present some old and new results related to the problems of computing pathwidth and finding caterpillars (or their offsprings) in graphs. I will first illustrate a simple linear-time algorithm that determines, for fixed k, whether a graph has pathwidth at most k. I end the talk with some applications and open algorithmic problems in the area. This talk is intended to be self-contained and should be understandable by anyone with a little background in graph theory.
Date received: October 27, 2008
Kernelization lower bounds
by
Rod Downey
Victoria University of Wellington
Kernelization is a method of algorithmic pre-processing which tends to be basis of many practical algorithms, particularly in parameterized complexity. We discuss recent work on demonstrating lower bounds for these problems modulo complexity assumptions as well as applications to density hard instances.
Date received: October 31, 2008
The Maximum Induced Planar Subgraph problem
by
Graham Farr
Monash University
Coauthors: Keith Edwards, Kerri Morgan
The Maximum Induced Planar Subgraph problem asks for the largest set of vertices in a given input graph G that induces a planar subgraph of G. Equivalently, we may ask for the smallest set of vertices in G whose removal leaves behind a planar subgraph. This problem has been linked by Edwards and Farr to the problem of fragmentability of graphs, where we seek the smallest proportion of vertices in a graph whose removal breaks the graph into small (bounded size) pieces. It is also related to the classical Maximum Planar Subgraph problem which is of central importance in graph layout algorithms. This talk describes some algorithms developed for this problem, together with theoretical and experimental results on their performance. Most of the algorithms presented are joint work either with Keith Edwards (Dundee) or Kerri Morgan. The experimental analysis was done by Morgan.
Date received: October 30, 2008
Detecting regular visit patterns
by
Joachim Gudmundsson
NICTA
Coauthors: Anh Pham, Bojan Djordjevic and Thomas Wolle
We are given a trajectory T and an area A. T might intersect A several times, and our aim is to detect whether T visits A with some regularity, e.g. what is the longest time span that a GPS-GSM equipped elephant visited a specific lake on a daily (weekly or yearly) basis, where the elephant has to visit the lake most of the days (weeks or years), but not necessarily on every day (week or year).
During the modelling of such applications, we encountered an elementary problem on bitstrings, that we call LDS (Longest Dense Substring). The bits of the bitstring correspond to a sequence of regular time points, in which a bit is set to 1 iff the trajectory T intersects the area A at the corresponding time point. For the LDS problem, we are given a string s as input and want to output a longest substring of s, such that the ratio of 1s in the substring is at least a certain threshold.
In our model, LDS is a core problem for many applications that aim at detecting regularity of T intersecting A. We propose an optimal algorithm to solve LDS, and also for related problems that are closer to applications, we provide efficient algorithms for detecting regularity.
Date received: October 21, 2008
What is the largest real flow root for a graph?
by
Gordon Royle
University of Western Australia
The flow polynomial of a graph G is the polynomial FG(k) that counts the number of nowhere zero k-flows on G, and the flow roots of G are the real and complex zeros of the flow polynomial. Although the flow polynomial is dual to the chromatic polynomial, much less is known about flow roots than chromatic roots, including the fundamental question of whether there is an absolute upper bound on real flow roots.
For integer flow roots, it is well known that every bridgeless graph has a nowhere-zero 6-flow and it is a difficult unsolved problem whether this extends to nowhere-zero 5-flows. However there is no such upper bound known for real flow roots, although Welsh conjectured that 4 was a possible value for such an upper bound.
In this talk, I will describe a computational attack on this problem that has shown that Welsh's conjecture is not true, and discuss possible replacements for the conjecture.
Date received: November 2, 2008
Algorithmic problems in conservation biology
by
Charles Semple
University of Canterbury
Coauthors: Magnus Bordewich and Andreas Spillner
A central task in conservation biology is measuring, predicting, and preserving biological diversity as species face extinction. Dating back to 1992, phylogenetic diversity is a prominent notion for measuring the biodiversity of a collection of species. This talk gives a flavour of some the combinatorial and algorithmic problems and recent solutions associated with computing this measure.
Date received: November 2, 2008
Symmetry in search
by
Toby Walsh
NICTA and University of New South Wales
Symmetry turns up in many problems. When solving combinatorial problems using search, symmetry may be inherent in the problem (e.g. interchangeable machines to assign to a job) or may arise through modelling it (e.g. naming elements in a set). In either case, symmetry can lead to redundant search, as many symmetrically equivalent blind alleys may be explored wastefully. To avoid this, symmetry-breaking constraints can be added, to exclude all but one of each equivalence class of solutions. Alternatively, the search method can be modified to exclude symmetric parts of the search space. I will describe recent results in this area and outline some of the open problems in the field.
Date received: October 23, 2008
Holomorphic Solutions for a Class of Functional Differential Equations
by
Bruce van Brunt
Massey University, Palmerston North
In this talk we study special cases of initial-value problems of
the form
|
We are concerned with the existence and continuation of nonconstant solutions that are holomorphic at a fixed point. Results from complex dynamics are used to show that, in general, the nonlinear functional term precludes solutions holomorphic at repelling fixed points and produces a natural boundary for holomorphic solutions at the attracting fixed point. For simple cases we investigate the behaviour of solutions near the natural boundary.
Date received: October 29, 2008
Mathematical modelling and parameter identification methods in systems
by
Christopher Eric Hann
Department of Mechanical Engineering, University of Canterbury
Coauthors: J. Geoffrey Chase, Geoffrey M. Shaw, Thomas Desaive, Paul Docherty, Christina Starfinger, Katherine Kok, Richard Brown, Sam Houghton.
The combination of mathematical modelling and parameter identification is a powerful tool for understanding and/or controlling real systems, natural or artificially made. However, modelling and identification are usually viewed as two separate entities and are often in direct conflict with each other. For example as the level of detail in a mathematical model increases, the amount of physical measurements required for validation increases as well as the number of parameters. Therefore, parameter identification can become infeasible with many combinations of parameters providing equally good model fits to the measured data. Parameter identification can also become computationally intractable with excessively large numbers of simulations required to avoid local minima's or false solutions.
This paper takes a minimal modelling approach where only the essential dynamics of a system are captured with the emphasis on the specific application or outcome required. Furthermore, a new concept of parameter identification is presented where for a given differential equation model, the inverse problem and forward problem are treated as one without any requirement of numerical DE solvers. Specifically, the model equations are first reformulated in terms of integrals of measured data. The inverse problem is then transformed into the context of a semi-discrete dynamical system, where the steady state solution corresponds precisely to the "best fit" model solution in the least squares sense. The integral formulation is critical for stability, results in fast convergence, and overall the method is more accurate and 104-106 times faster than current methods, depending on the application.
To demonstrate the concepts, the minimal modelling and parameter identification methodologies are implemented on several case studies in biomedical engineering. However, note, that the methods are general, and could be equally applied in any engineering field, particularly where lumped parameter or compartmental type non-linear differential equation models are used. Examples include the glucose-insulin and cardiovascular systems, with application to diagnosis and therapy in the Christchurch Intensive Care Unit.
A third example concentrates on results of a new technology being developed for non-invasive breast cancer detection. This technology is called Digital Image-based Elasto-tomography or "DIET". Minimal modelling and parameter identification play a major role in both the computer vision side and tissue stiffness reconstruction of the DIET system. Results of the computer vision algorithms applied to silicon breast phantoms are presented. Finally, some initial results of applying the "minimal modelling" approach and integral-based parameter identification to simulated 2D tissue displacement data using the Navier equations are given.
Date received: October 28, 2008
A solvable model for chimera states in heterogeneous networks
by
Carlo Laing
Massey University, Auckland
Chimera states in networks of coupled oscillators occur when some fraction of the oscillators synchronise with one another, while the remaining oscillators are incoherent. Several groups have studied chimerae in networks of identical oscillators, but I will discuss these states in a heterogeneous model for which the natural frequencies of the oscillators are chosen from a distribution. We obtain exact results by reduction to a finite set of differential equations and find that heterogeneity can destroy chimerae, destroy all states except chimerae, or destabilise chimerae in Hopf bifurcations, depending on the form of the heterogeneity.
Date received: October 27, 2008
Through Fourier-tinted spectacles: advection-dispersion equations transformed to a dynamical system
by
Robert McKibbin
Massey University, Albany
A system of partial differential equations that may be used to describe the transport of a pollutant in a groundwater aquifer that is composed of several different parallel sedimentary layers is constructed. The groundwater flow throughout is assumed to be under the influence of a uniform pressure gradient parallel to the layers, and the layer thicknesses are small compared to the lateral extent of the aquifer. Transfer of the pollutant across the layer interfaces may occur when the concentrations within adjoining layers are not equal.
The result is a set of coupled linear time-dependent one-dimensional advection-dispersion equations. A Fourier transform (FT) is applied with respect to the spatial variable; the resulting set of coupled linear first-order ordinary differential equations for the (complex-valued) transformed pollutant concentrations (in time, and carrying the FT parameter) may then be solved using standard linear dynamical system techniques. The predicted pollutant concentrations are retrieved by numerically evaluating the inverse transforms. The complex FT of the concentration in each layer may also be examined using a phase-plane (complex plane). Some illustrative results are presented.
Date received: September 21, 2008
Determining a safe path through a dynamic threat environment in real time
by
G.N. Mercer
UNSW at ADFA, Canberra
Coauthors: H.S. Sidhu and M.P. Rowe
The problem of minimising the risk to a vehicle when travelling through a threat environment is considered. In a military application the threat environment could be a minefield (land or sea) or a radar network. A safest path route is required in real time but often there is limited computational power on the battlefield which restricts the type of methods that can be used. In addition there can be ``pop up'' threats that can dramatically change the safest path with little prior warning. We apply an energy based springs and masses model to determine the safest route. This model reduces down to finding the steady state of a large system of ODEs. The model is very robust and capable of producing good enough real time safe paths even under a rapidly changing threat environment.
Date received: November 6, 2008
A fundamental analysis of a membrane bioreactor containing a sludge disintegration system
by
Mark Nelson
School of Mathematics & Applied Statistics, University of Wollongong
Coauthors: T.C.L. Yue
We analyze the steady-state operation of a membrane bioreactor system (MR) incorporating a sludge disintegration unit (SD). The latter is used to prevent excess sludge production. The relationship between process control parameters and the performance of the MR-SD is determined by finding the steady-states of the model and determining their stability as a function of the residence time. Asymptotic solutions for the steady-state solutions in the limit of high residence times are obtained. These show that at sufficiently high residence times the mixed liquor suspended solids (MLSS) content of the bioreactor is independent of the operation of the sludge disintegration unit. Thus the main role played by the sludge disintegration unit is to improve the performance at `low' residence times. For a specified MLSS concentration the values of the dimensionless residence time and the sludge disintegration factor are determined that ensure zero excess sludge production. If the sludge disintegration factor is sufficiently high then the MLSS content is guaranteed to be below the target value (`negative' excess sludge production) provided that the residence time is higher than the washout value. It is shown that zero excess sludge production can be achieved with a slight decrease in effluent quality.
Date received: October 27, 2008
Multi-scaling analysis of a predator-prey model
by
John J Shepherd
RMIT University
Coauthors: A Stacey, T Grozdanovski
The general Lotka-Volterra equations are the simplest differential equation system for modelling the results of a two-species interaction in which one species is preyed upon by the other. Although, in their simplest form, they are too idealized to accurately model real-world communities, they display features that make them worthy of continued study.
When competition within the species (intraspecies competition) is incorporated, the resulting system moves closer to a realistic representation of real predator-prey populations.
In this talk, we consider such a system, where intraspecies competition occurs in the prey, but not the predator; while the growth rate for the predator is much slower than that for the prey. Such a situation arises in a range of predator-prey communities – for example, foxes and rabbits; lions and antelope.
We exploit this differential in growth rates by applying a multi-scaling technique to obtain approximate expressions for the evolving predator and prey populations, valid over all time. These are then used to describe predator and prey behaviour.
Date received: October 29, 2008
Classical and pulsating combustion waves in a chain-branching reaction model
by
Harvinder Sidhu
Applied and Industrial Research Group, School of Physical, Environmental and Mathematical Sciences, UNSW at ADFA, Canberra
Any scientist who has gazed into a campfire will appreciate the complexity of combustion and the difficulty in constructing a theoretical model of the process. In most investigations of these phenomena only the simplest models have been utilized. These simple models have all featured one-step chemistry, where the reaction is assumed to be well modelled by a single step of fuel and oxidant combining to produce products and heat. In the past these models have been comprehensively analysed using the classical methods of matched asymptotics and linearised stability analysis, yielding now familiar results for planar flame propagation The one-step, large activation energy model has led to many useful qualitatively correct predictions such as: ignition, extinction and stability of diffusion flames, propagation and stability of premixed flames; structure and stability of flame balls. In particular, it has been possible to show many qualitative features of flame stability that can, in a generic sense, be observed in experimental work, but it has not been possible to make general quantitative comparisons to experimental work. This is mainly because real flames do not arise as a result of one-step chemistry. Some researchers also claim that many simple kinetic schemes do not give results which correspond to experimental observations and can produce erroneous conclusions.
At the other extreme, several groups are involved in the study of flame behaviour using full numerical solutions of the equations of energy and mass transfer for all of the species involved with detailed chemistry. Although such investigations are useful in providing quantitative prediction for observed phenomena, there is still a great deal of uncertainty about the reliability of these complex models when applied to the prediction of stable combustion regimes and particularly the onset of exotic combustion phenomena such as pulsating and cellular combustion. It is in these very regimes when the reactions begin to change rapidly in space and/or time, that any numerical method is tested to its extreme limits and the conclusions drawn from these numerical results must be viewed carefully.
The main aim of our current work is to systematically investigate the stability of the flame solutions with complex kinetics. We will progress from the one-step scheme which has been a dominating paradigm in combustion theory, to the two-step scheme and beyond. In today's talk I will present results of our work thus far.
Date received: October 27, 2008
Modelling of coexistence of endophyte-free and endophyte-infected grasses in New Zealand grazing system.
by
Tanya Soboleva
AgResearch Limited, New Zealand
Coauthors: Anthony J. Parsons
While endophyte contributes positively grasslands productivity, it has well-known adverse effects on livestock. New strains of endophytes with desirable properties are cultivated in the laboratory conditions then artificially infected into ryegrasses, which later could be introduced into the pasture.
The presented model is aimed to design best strategies for releasing into environment those ryegrasses artificially infected with new endophytes. The model represents a system of differential equations for two competing types of the ryegrass with different attractiveness to insects and grazers. The system has two non-trivial stable nodes located in distant regions of the phase space. The separatrix is separating the phase trajectories reaching the state with strong prevalence of endophyte-infected grass from the trajectories tending to the state with strong prevalence of endophyte-free grass. Such topology of the phase diagram usually suggests strong sensitivity to initial conditions around the separatrix. However, this is not the case for a feasible (from the point of view of the considered application) time scale. The variables of the system evolve quickly toward some universal relation (a line at the phase diagram) between them and then very slowly approach fixed points. The points along this line, especially those close to the separatrix , can be considered as quasi-steady states for this system. This phenomenon of rapid evolution of dynamical variables toward a universal relationship is the essence of the so-called ``large-river effect'', which was previously studied in the reference to the theory of phase transitions.
Date received: October 20, 2008
An asymptotically stable collision-avoidance system
by
Jito Vanualailai
School of Computing, Information & Mathematical Sciences, University of the South Pacific, Suva, Fiji
Coauthors: Bibhya Sharma and Shin-ichi Nakagiri
Artificial potential fields, which are widely used in robotics for path planning and collision avoidance, are normally beset by difficulties arising from the existence of local minima. This presentation proposes a solution that involves an asymptotically stable point-mass system governed by differential equations. The system represents a planar point robot moving from its initial position to the desired goal whilst avoiding a static obstacle. Because the system is asymptotically stable, its Lyapunov function, which produces artificial potential fields around the goal and the obstacle, has no local minima other than the goal configuration in the pathwise-connected proper subset of free space which contains the goal configuration. As an application, we consider the point stabilization of a planar mobile car-like robot moving in the presence of a static obstacle.
Date received: October 25, 2008
Burning issues: critical storage and assembly.
by
Graeme Wake
Centre for Mathematics in Industry, Massey University, Albany
Coauthors: Weiwei Luo, University of Alabama Huntsville.
The storage problem for solid combustible materials is recast into a dynamical systems framework so as to provide an easily accessible platform for fire investigators to use as a decision-support tool. This is easier than the usual path-following methods and will be the basis for commercially-developed software which is under development. This, and some new work on critical assembly conditions, were developed initially as an aid to a recent marine fire investigation. Some interesting but simple non-local steady-state equations provide useful bounds for the determination of the critical conditions for the storage problem.
Date received: September 15, 2008
A multi-compartment, two population, age-distribution model of cancer cell growth; transfer from in vivo to in vitro
by
David JN Wall
Department of Mathematics & Statistics, University of Canterbury
Coauthors: Liene Daukste and Britta Basse
Human cancers have been shown to contain a population of relatively slower growing cells with a cell cycle time, depending on the tumour, between three days and several weeks. When cells from tumours removed at surgery are grown in culture, they initially grow at this slow rate. However, loss of the slower growing cell population over time results in the emergence of a population of more rapidly growing cells (a cell line) with doubling times of 1-3 days. It has been postulated that the more slowly growing population is maintained by a small population of more rapidly growing population. In this talk we extend a previously developed and analysed model to describe the behaviour of a complex system with two cell populations with different kinetic characteristics. The aim of this model is to provide a framework for understanding the difference in behaviour of cancer cell lines and the human tumours from which they were derived.
We discuss results regarding the stability of age-distributions displaying balanced exponential growth, and then consider the existence of steady age-distributions displaying balanced exponential growth.
Date received: November 3, 2008
A Bayesian take on RBFs
by
Colin Fox
University of Otago
Radial basis functions (RBFs) provide a convenient representation of (implicity defined) boundaries and explicit functions, and are therefore useful as mid-level representations of unknowns in statistical inference. In the presence of measurement noise, or other uncertainty, these high-dimensional representations necessarily induce a bias in quantities of interest, or `statistics', and often provide fickle estimates. These problems are easy to demonstrate, and just as easy to fix through the design of the Bayesian's `prior' distribution.
Date received: November 3, 2008
On some aggregation/disaggregation based approximations
by
Markus Hegland
Australian National University
Originally, aggregation and disaggregation were considered as acceleration techniques similar to multigrid methods for the solution of linear systems of equations. In some recent work we have demonstrated that these methods can also be used for the numerical solution of the chemical master equations.
In the simplest case, one would like to approximate smooth summable sequences by piecewise polynomials. I will show how the chemical master equations provide a notion of smoothness and will derive approximation error bounds using convolution and sampling theorems. If time permits, I will discuss some issues arising in the multi- and high-dimensional case.
Date received: October 21, 2008
Approximating functions in Clifford algebras
by
Paul Leopardi
Australian National University
As is well known, the Clifford algebras over the real field R can be represented as matrix algebras over R or the complex field C. This means that various functions defined over matrices, such as square root and logarithm, can also be defined over the Clifford algebras. The talk will discuss the similarities and differences between matrix and Clifford functions, and look at specific approximation algorithms, notably the Denman-Beavers square root and the Cheng-Higham-Kenney-Laub logarithm.
Date received: October 30, 2008
Geodesic interpolating splines
by
Stephen Marsland
Massey University, Palmerston North
Spline interpolation is a common problem in many areas of data analysis, including statistic sand image and signal processing. There has been some interest in constructing splines that are guaranteed diffeomorphic, e.g., that distort space in a smooth, invertible way. I will demonstrate different methods of computing such splines and show that the ability to reach a metric on the diffeomorphism group from this provides some benefits for data analysis.
Date received: October 29, 2008
Preserving Energy resp. dissipation in numerical partial differential equations, using the "Average Vector Field" method.
by
David McLaren
La Trobe University
Coauthors: E. Celledoni, V. Grimm, R.I. McLachlan, B. Owren, G.R.W. Quispel
We give a method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated in many examples. In the Hamiltonian case they are: the sine-Gordon, Korteweg-de Vries, nonlinear Schrödinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and Heat equations.
Date received: October 16, 2008
Fast algorithms for integral transforms based on domain decomposition and composition equations
by
Garry Newsam
Defence Science & Technology Organisation, Australia
The standard decomposition of the unit interval I ≡ [-1/2, 1/2] into the union of two equal subintervals induces a corresponding decomposition of a function f on I as:
| (1) |
| (2) |
The algorithm is readily generalised to any other operator (e.g. the Radon transform) that commutes in a simple way with the shift operator, and to nonuniform domain decompositions. Its main advantage over existing fast algorithms is that in the course of computation it evaluates the transform not just on the full interval but also on a complete pyramid of subintervals: this allows operations such as feature detection to be carried out at all length scales rather than just a single scale.
Date received: October 14, 2008
High order of convergence using lattice sequences for numerical integration
by
Dirk Nuyens
University of New South Wales
Coauthors: Fred J. Hickernell, Peter Kritzer and Frances Y. Kuo
We study the worst case integration error of combinations of quadrature rules in a reproducing kernel Hilbert space. We show that the error, with respect to the total number N of function evaluations used, cannot decrease faster than O(N-1) in the case where several quasi-Monte Carlo rules are combined to a compound quasi-Monte Carlo rule. However, if the errors of the quadrature rules constituting the compound rule have an order of convergence O(N-a) for a > 1 then, by introducing weights, this order of convergence can be shown to be recovered for the compound rule. We apply our results to the case of lattice sequences.
Date received: November 24, 2008
On the construction of equiangular tight frames
by
Shayne Waldron
University of Auckland
Tight frames are optimal for signal reconstruction when there is one erasure, and equiangular tight frames are optimal when there are two erasures. We give a survey of the known methods for constructing real and complex equiangular tight frames. This includes the identification of real equiangular frames with graphs, and Zauner's conjecture on the existence of an equiangular tight frame of n=d2 vectors for Cd.
Date received: November 10, 2008
Hungry? Go ballistic! Or what should you do to find your dinner?
by
Richard Brown
University of Canterbury
Coauthors: Alex James, Mike Plank
An important question in foraging theory is "What is the optimal search strategy for a predator to maximise its efficiency in finding its prey?" Going hand in hand is the question of whether real-life predators search in an optimal way.
These question have been the focus of a significant amount of research over the last decade with many articles published, both theoretical and based on field work. In particular, there is a significant body of research pointing to certain heavy-tailed Levy random walks as being optimal predator search strategies in a variety of scenarios.
This talk will address certain recent advances in the area, focusing on the case of search for moving targets.
Date received: October 30, 2008
Passive dynamics of animal locomotion
by
Te-yuan Chyou
Department of Mathematics and Statistics, University of Otago
Coauthors: Gerrard Liddell, Mike Paulin
For decades biologists believed that animals walk because the brains calculates the motion trajectories for the limbs. More recently, there is a new hypothesis about animal locomotion that suggests the contrary. Animal framework is built for walking in the first place. It has the correct dynamics so that they can work without relying on controls. Instead, the walking gait is generated simply by interaction of gravity and inertia, in a stable, naturally emerged limit-cycle, known as passive dynamic walking. The feasibility of passive dynamic walking had been demonstrated for biped system that consists of only a pair of legs. In this talk we will look into full-body passive dynamic walking models, by showing that some simple and animal-like mechanical linkages can generate walking gait by using only gravity, and can recover from small perturbation without the need of controller input. The contribution of a torso on the stability and efficiency of passive biped walking will also be addressed.
Date received: October 24, 2008
A multiscale, spatially-distributed model of airway hyper-responsiveness
by
Graham Donovan
Auckland Bioengineering Institute
Coauthors: Antonio Politi, James Sneyd, Merryn Tawhai
Airway hyper-responsiveness (AHR), along with airway hyper-sensitivity, is a defining feature of asthma, and greater understanding of this emergent phenomenon may lead to better insight into and treatment of the condition. Our model couples together the organ scale with the tissue scale in the lung in a multiscale approach to the problem. At the organ level, parenchymal tissue is modeled as a compressible Blatz-Ko material in three dimensions, with expansion and recoil of lung tissue due to tidal breathing. The governing equations of finite elasticity deformation are solved using a finite element method. An airway tree is embedded in this tissue, with airway smooth muscle behavior described by a modified Hai-Murphy cross-bridge model (Wang et al., Biophys. J. 94:2008). Each airway segment is initially assumed to be radially symmetric and longitudinally stiff, and thus the embedded airway tree is essentially 1D. Preliminary results from the integrated model indicate potential use in the study of many phenomena associated with asthmatic AHR, including spatial distribution of ventilation defects, patchiness, and effects of deep inspirations.
Date received: October 28, 2008
Mathematical modeling of the HER2-receptor overexpression in breast cancer
by
Amina Eladdadi
Mathematical Sciences Department, Rensselaer Polytechnic Institute, New York
Coauthors: David Isaacson
Overexpression of the HER2 receptor due to the neu gene amplification contributes to the development of human breast cancers. The carcinogenic effects of HER2 protein overexpression on cell growth and cell proliferation have been observed in a variety of experimental systems. These observations suggest that HER2 overexpression provides tumor cells with a growth advantage leading to a more aggressive phenotype. To investigate the effects of HER2 receptor overexpression on cell proliferation, we have developed two mathematical models that describe the proliferative behavior of HER2-overexpressing cells.
We address by means of mathematical models and numerical simulation the following major questions:
1. How does the cell proliferation rate depend on the number (expression level of the HER2 receptor?
2. How do changes in the number of HER2 and EGFR receptors during the cell-cycle affects the cell proliferation rate?
The cell proliferation models enable us to simulate the proliferative behavior of the HER2-overexpressing cells with various HER2 and EGFR expression levels at various ligand concentrations. Both mathematical models predict a growth advantage associated with excess in cell surface HER2 receptors.
Date received: September 25, 2008
The countercurrent mechanism
by
Scott Graybill
Department of Mathematics and Statistics, University of Canterbury
Coauthors: Alex James, Mike Plank, Tim David, Zoltan Endre
A concentration gradient exists in the kidney, with the higest concentration deep within the medulla. This gradient can be explained by the countercurrent mechanism, where a modest gradient is increased by the interaction of two parallel tubules.
This talk will discuss how the water and solute transport properties of the two parallel tubules can establish and maintain this concentraion gradient and present some results.
Date received: October 29, 2008
Understanding complicated oscillations in intracellular calcium dynamics
by
Emily Harvey
University of Auckland
Coauthors: Vivien Kirk, James Sneyd
Calcium controls a huge range of crucial cellular processes. It is thought that oscillations in intracellular calcium act as signals, with the message being encoded in the frequency and form of the oscillations. However, the mechanisms underlying these oscillations are not well known. In this talk I will show how the analysis of mathematical models of intracellular calcium can give us insight into these underlying mechanisms, particularly for the complicated oscillations known as mixed mode oscillations.
Date received: November 2, 2008
Existence of solutions in a model of chondrogenesis
by
Bogdan Kazmierczak
Institute of Fundamental Technological Research, Poland
Coauthors: M.Alber, St.Newman, G. Hentschel
The paper considers conditions sufficient for the existence of global classical solutions to a new model of chondrogenesis during the vertebrate limb formation. We assume that the diffusion coefficient of the fibronectin is positive and that the function describing the interaction between the fibronectin and cells satisfies some additional properties.
Date received: September 1, 2008
Modeling autoregulation in the rat kidney
by
Nicole Kleinstreuer
Centre for Bioengineering, University of Canterbury
Coauthors: Tim David, Mike Plank, Zoltan Endre
A transient mathematical model of whole-organ renal autoregulation in the rat is presented, incorporating the myogenic response throughout the renal vasculature and the tubuloglomerular feedback response at the level of the nephrons. The myogenic response to a change in circumferential wall tension is modeled with vessel size-specific parameters, including effects of in-vivo viscosity variation and flow-induced dilation. This myogenic model is coupled with a system representing change in concentration of the tubular filtrate and corresponding resistance changes of the afferent arteriole via the TGF mechanism. Computer simulation results of the steady state and transient autoregulatory response to pressure perturbations are examined, as well as the modulatory influences of metabolic and hemodynamic factors. A comprehensive model of autoregulation allows for the examination of both normal and pathological states, such as the altered NO production in chronic kidney disease or the inhibited tubular reabsorption of water seen in diabetes.
Date received: October 20, 2008
Reducing the load of spatial scales in a calcium model
by
Shawn Means
University of Auckland
Multiscale models of calcium dynamics often rely crucially on accurate
representations of the calcium concentration in restricted spaces. However, connecting a local model of a microdomain to the model of calcium dynamics in the bulk of the cell is often computationally expensive. We present a method which reduces this expense on a model of calcium dynamics for a cardiac cell which resolves the multiple spatial scales of the dyadic cleft (order 10 nm) and the bulk interior (order 1 um). The method involves using a combined analytical / numerical solution scheme coupled via appropriate boundary conditions.
Date received: November 16, 2008
Computational neural models embedded in virtual animals
by
Mike Paulin
University of Otago
It is now possible to build complex integrative models of animals, with brains and bodies, which can act autonomously in virtual environments. These models extend scientists’ ability to visualize the relevant physics and biology, and to develop and test theories despite the essential complexity of neural systems. Wrapping the detailed physics, engineering, mathematical abstraction and numerical complexity in a form that looks and behaves like an animal, and can be measured and explored like the real thing, provides an interface not only between experimental biologists and mathematical/computational modelers, but also between scientists and the lay public. I will demonstrate a model of the electrosensory system of the spiny dogfish, Squalus acanthias. The dogfish’s ability to isolate weak signals in a noisy environment is due to a brainstem structure called the dorsal octavolateral nucleus. The virtual dogfish has a computational model of this neural structure, receiving realistic sense data from a realistic physical model of the environment. The human auditory system contains an analogous brainstem structure called the dorsal cochlear nucleus, albeit much less accessible to experimentation than the dogfish’s dorsal nucleus. Understanding the dogfish electrosensory hindbrain may lead to new designs for signal filters, in particular for applications in human hearing.
Date received: October 27, 2008
Who eats whom? Population dynamics in the ocean
by
Mike Plank
University of Canterbury
Marine ecosystems have a very high degree of size structure, meaning that body size (which typically spans several orders of magnitude from plankton to large fish) is the most important determinant of who eats whom. Many marine ecosystems have the remarkable property that total biomass in logarithmically sized bins is almost invariant with respect to body size. In this talk, I will present a stochastic, individual-based model of predation, growth and death, and show how this scales up to a size-structured population model. Depending on certain ecological parameters, this model can have a stable steady state (which is related to the invariance of biomass property), or can exhibit travelling wave solutions.
Date received: November 23, 2008
Optimal movement in the prey capture behaviour of weakly electric fish
by
Claire Postlethwaite
University of Auckland
Coauthors: Malcolm MacIver, Mary Silber, Tiffany Psemeneki, Jangir Selmanikov
Animal behaviour arises through a complex mixture of biomechanical, neuronal, sensory, and control constraints. By focusing on a simple, stereotyped movement, the prey capture strike of a weakly electric fish, we show that the trajectory of a strike is one which minimises effort. Specifically, we model the fish as a rigid ellipsoid moving through a fluid with no viscosity, governed by Kirchhoff's equations, and generate trajectories which are optimal with respect to a mechanical cost function. We compare these to measured prey capture strikes of weakly electric fish. The fish has certain movement limitations which are not incorporated in the mathematical model, such as not being able to move sideways. Nonetheless, we show quantitatively that the computed least-cost trajectories are remarkably similar to the measured trajectories.
Date received: October 19, 2008
Development of a model-based tracking algorithm for reconstruction of 3D spider motion.
by
Kiri Pullar
University of Otago
Coauthors: Mike Paulin
An algorithm that recovers 3D body pose from video sequences has numerous applications such as motion capture, gesture recognition, surveillance of people or animals and animation for movies or computer games. The aim of this work is to develop a tracking approach that exploits our knowledge of physics to enable makerless motion capture of a spider during locomotion.
The basic elements of our tracking approach are an articulated body model, extracted features from video frames and Bayesian filtering. Articulated objects like the spider present a number of difficulties for successful tracking. The three key challenges are: large number of degrees of freedom (a complete model requires ~64 DOF), fast, non-linear motion and self occlusion of limbs. I intend to overcome these complications by beginning with a simple inverted pendulum based model, then adding complexity such as multiple links, springs and constraints until I have a physics based model that can be applied to the tracking of 3D spider motion.
Date received: October 27, 2008
Multitype contact processes: stochastic spatial competition models
by
Joseph Stover
University of Canterbury
The contact process is the basic interacting particle system used for creating stochastic spatial biological models. A new type of particle is introduced for each species or even for stages of growth within a species. Analytical studies have proven extremely difficult for these types of models, but mean field analysis and computer simulations can provide some insight into their behavior.
Date received: November 2, 2008
Modelling the interaction of prolactin and LIF signalling in the bovine mammary gland
by
Kumar Vetharaniam
AgResearch Limited, New Zealand
Coauthors: Kuljeet Singh
The hormone prolactin plays a major part in controlling the synthesis of all the major milk components and is important for maintaining lactation. A key pathway through which prolactin acts involves the activation of STAT-5 proteins which then translocate to the nucleus and mediate the transcription of prolactin’s target genes. A potential inhibitor of PRL action is activated STAT-3, which up-regulates the SOCS-3 protein which in turn blocks STAT-5 activation. Experiments have lead to speculation that increased levels of STAT-3 may be responsible for turning off milk production, and suggest a possible role for LIF, a factor known to activate STAT-3.
We have constructed a mathematical model, consisting of a set of coupled, delay-differential equations, which describes prolactin and LIF signalling along the STAT pathways. This model allows us to investigate under what conditions the appearance of LIF could stop milk synthesis by blocking prolactin, and also to investigate possible interventions to ameliorate the effect of STAT-3.
Date received: October 30, 2008
Modelling the calcium dynamics in airway smooth muscle cells
by
Inga Wang
University of Auckland
Asthma is a condition characterized by airway hyper-responsiveness, which results in reversible increases in airway smooth muscle (ASM) contraction, and variable amounts of inflammation of the bronchial mucosa. Hence the understanding of the regulation and mechanics of ASM contraction and the surrounding lung tissue is crucial for medical research. The ASM contraction is regulated by the changes in intracellular calcium concentration and the responsiveness of the ASM to this calcium. In this talk, I will present a model of calcium dynamics in ASM cells.
Date received: November 15, 2008
Dynamics in reaction-diffusion excitable systems
by
Wenjun Zhang
University of Auckland
Excitable systems of reaction-diffusion equations are used to model many biophysical processes, including changes of calcium concentration in various cell types. Understanding the interaction between soliton and periodic waves is important in these systems. By moving to travelling waves coordinates and using ideas from geometric singular perturbation theory, we show how complicated bifurcations associated with this interaction arise from the singular limit. We illustrate the method with examples of calcium models and the FitzHugh-Nagumo model.
Date received: November 11, 2008
The dynamics of finite data: hyperbolicity and distortion
by
Arno Berger
University of Alberta
Numerical data generated by real-world dynamical systems such as oceanographic models are typically finite. It is often not clear what qualitative and quantitative dynamical information can be deduced from such data in a rigorous manner. This talk will discuss two concepts that address this question: finite-time hyperbolicity and (scale) distortion of data sets.
Date received: November 3, 2008
Second largest eigenvalues and almost invariant structures for hyperbolic systems
by
Chris Bose
Department of Mathematics and Statistics, University of Victoria, Canada
Coauthors: Gary Froyland, University of New South Wales
We discuss the problem of constructing eigenfunctions with eigenvalues inside the unit disc for measurable, nonsingular and invertible transformations. For expanding one-dimensional maps the problem is well-understood in terms of spectral properties of the associated transfer operator restricted to a suitable subspace of L1 consisting of regular functions. For invertible maps we will show that one must instead EXPAND the domain of the transfer operator to obtain similar results. A construction using the so-called generalized baker's map shows this to be a feasible program and reveals some of the structure that will hold for such 'almost invariant functions' in the hyperbolic setting.
Date received: October 30, 2008
Ergodicity for partially hyperbolic diffeomorphisms
by
Keith Burns
Northwestern University
I will survey recent results which extend Hopf's method for proving ergodicity to a large class of partially hyperbolic diffeomorphisms.
Date received: November 3, 2008
Critical dimension of non-singular actions
by
Anthony Dooley
University of New South Wales
Let (X, B, m, T) be a non-singular ergodic dynamical system. We define the lower critical dimension of T to be the largest value of a ∈ [0, 1] such that [1/(na)] liminfSk=1n [(dm○T)/(dm)] > 0 a.e. and the upper critical dimension b in a similar way. These are invariants of metric equivalence. There is a natural weaker notion of equivalence, which we call Hurewicz equivalence which also preserves a and b. We shall discuss the classification of dynamical systems using these notions.
Date received: November 3, 2008
Coherent sets in flows and Perron-Frobenius cocycles
by
Gary Froyland
University of New South Wales
Coauthors: Simon Lloyd, Naratip Santitissadeekorn (UNSW), Anthony Quas (UVic)
We present an analysis of one-dimensional models of dynamical systems that possess ``coherent sets''; global structures that disperse more slowly than local trajectory separation. We introduce the notion of a Perron-Frobenius cocycle, give a characterisation of the Lyapunov spectrum of this cocycle, and present a strengthened version of the Multiplicative Ergodic Theorem. We describe applications to flows, including those arising from ocean models.
Date received: November 3, 2008
A cone exchange map
by
Arek Goetz
San Francisco State University
Coauthors: Peter Ashwin and Anthony Quas
A basic system to rotations on cones presents surprising challenges in determining the long term behavior of the orbits. We will report of our recent joint work with Ashwin and Quas. In particular we explain that a piecewise rotation with two half planes that is invertible has bounded rings and recurrent points. This is all despite the existence of an obvious finite invariant measure.
Date received: November 3, 2008
The geometric dimension of an equivalence relation and finite extensions of finite groups
by
Valentyn Golodets
University of New South Wales
Coauthors: A.H. Dooley
We say that the geometric dimension of a countable group G is equal to an integer n if any free Borel action of G on a standard Borel probability space (X, m) induces an equivalence relation EXG of geometrical dimension n in the sense of Gaboriau, or briefly geom-dim(EXG)=n.
Let G be as above and geom-dim(G)=n and let K be a finite extension of G. Does geom-dim(K)=n?
We prove that for any integer n, n ≥ 1, there exists a big enough class of group An such that if G belong to An then geom-dim(G)=n, and any finite extension K of G belongs to An too.
The important case n = 1 is considered more explicitly. We prove that A1 contains a big enough set of free products amenable groups. In particular, all free groups and all free finite products of finite groups belongs to A1.
We use some results and constructions from combinatorial group theory belonging to Karrass, Hanna Neumann, John Stallings and others, in a combination with methods of orbital equivalence theory.
Date received: November 3, 2008
Extreme behaviour in chaotic dynamical systems with suspension flow applications.
by
Mark Holland
University of Exeter, UK
Coauthors: M. Nicol, A. Torok.
We study extreme statistics for chaotic dynamical systems. That is, we investigate the long time range distribution of the maxima of observations on a typical solution trajectory. As applications we consider suspension flows over chaotic maps and mention preliminary results on extremal properties of the Lorenz equations.
Date received: October 30, 2008
Fast approximation of long term dynamical behaviour for continuous-time systems
by
Péter Koltai
Technische Universität München, Germany
Coauthors: Gary Froyland and Oliver Junge
Transfer operator methods are widely used in applications to determine long term dynamical behaviour. They are based on simulations of the dynamical system. However, for systems arising from ODEs, simulation is computationally very expensive. Instead of the associated transfer operator we propose to analyze the infinitesimal generator of the system. We develope theory and show numerical examples in the talk.
Date received: October 31, 2008
Large deviations and moderate deviations for slowly mixing dynamical systems
by
Ian Melbourne
University of Surrey, UK
We obtain results on large deviations for a large class of nonuniformly hyperbolic dynamical systems with polynomial decay of correlations 1/nb, b > 0. This includes systems modelled by Young towers with polynomial tails, extending recent work of M. Nicol and the author which assumed b > 1. As a byproduct of the proof, we obtain slightly stronger results even when b > 1. The results are sharp in the sense that there exist examples (such as Pomeau-Manneville intermittency maps) for which the obtained rates are best possible. In addition, we obtain results on moderate deviations.
Date received: October 23, 2008
Extreme value theory for dynamical systems
by
Matthew Nicol
University of Houston
Coauthors: Mark Holland (University of Exeter) and Andrew Torok (University of Houston)
We present results on extreme value theory (which concerns the distribution of successive maxima of a time-series of observations) for dynamical systems. The main result is that a broad class of observations on certain non-uniformly hyperbolic systems exhibit the same extreme value statistics as iid processes with the same distribution function.
Date received: October 30, 2008
Feedback control of unstable periodic orbits
by
Claire Postlethwaite
University of Auckland
It is often the case that the desired output from a system (be it experimental or numerical) is a periodic orbit or pattern which is unstable. In 1992 Pyragas introduced a method of time-delayed feedback control which can be used to stabilise such unstable solutions. This method has attracted much attention, as it has the advantages of being both non-invasive, and also requiring only a knowledge of the period of the orbit a priori.
I will present results on two examples. The first example involves stabilising unstable periodic orbits resulting from a subcritical Hopf bifurcation in the Lorenz equations. The second involves the stabilisation of periodic orbits with arbitrarily large period which have originated from a heteroclinic bifurcation.
Date received: October 19, 2008
Motion Estimation of Image Sequence Data in the Framework of the Frobenius-Perron Operator
by
Naratip Santitissadeekorn
University of New South Wales
In many areas of science, main sources of data are given in a form of image sequences or movies, which capture the underlying dynamics. To understand the transport behavior of these dynamical systems, it is mandatory to extract the velocity fields from image sequences.
Traditionally, the Particle Image Velocimetry (PIV) is used in fluid imagery to locally approximate velocity filed based on a cross-correlation technique. Suppose that an image seguence depicts a flow in such a way that the essential characteristics of the flow are captured. In particular, an image sequence may be viewed as the evolution of image pixel intensities under the Frobenius-Perron operator.
Then, a variational model based on the principle the infinitesimal generator of the Frobenius-Perron operator can be developed to globally estimate the velocity field.
Date received: October 30, 2008
Folded saddle-nodes: where canards meet hopf
by
Martin Wechselberger
School of Mathematics & Statistics, University of Sydney
Coauthors: Martin Krupa (Radboud Universiteit, Nijmegen, The Netherlands)
Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organising centres for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory -- the blow-up technique -- and from delayed Hopf bifurcation theory -- complex time paths analysis -- to analyse the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. Furthermore, we define a way in/way out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity reminiscent of classical delayed Hopf bifurcation theory.
Date received: October 20, 2008
Calabi-Yau equations and 4 dimensional lattice Green functions
by
Tony Guttmann
Department of Mathematics and Statistics, University of Melbourne
Lattice Green functions in two dimensions are straightforward, and have now largely been solved for most three-dimensional lattices. In four dimensions there has been no complete solution. However recently a large family of 4th order ordinary differential equations, possessing maximal unipotent monodromy and satisfying the Calabi-Yau condition have been solved. We show that this enables us to obtain 4-dimensional lattice Green functions for the first time.
Date received: October 31, 2008
Harmonic deformations of hyperbolic 3-manifolds
by
Craig Hodgson
University of Melbourne
Coauthors: Steve Kerckhoff (Stanford)
This talk will give an introduction to our work on harmonic deformations of hyperbolic 3-manifolds, and describe some topological applications. In particular, this work gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem, including precise estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling.
Paper reference: math.GT/0301226
Date received: October 30, 2008
The p-HH-norm on the Cartesian product of n copies of a normed space
by
Eder Kikianty
Victoria University, Australia
Coauthors: Gord Sinnamon (University of Western Ontario, London, Canada)
The n-fold Cartesian product Xn of a normed space X is again a normed space when it is equipped with any one of the well-known p-norms, p ∈ [1, ∞]. These norms are equivalent but are not equal for different p. In 2008, Kikianty and Dragomir introduced the p-HH-norms on the vector space X2 of pairs of elements from X. These norms are equivalent to the p-norms but, unlike the p-norms, they do not depend only on the norms of the two elements in the pair, but also reflect the relative position of the two elements within the original space X. In this talk, I will discuss recent work in which the p-HH-norms are defined on the space Xn for n > 2.
As in the case of X2, the p-HH-norms on Xn depend on the relative positions in X of the elements in the n-tuple. They preserve the completeness of the space X. Also, they preserve geometric properties of the space X such as completeness, smoothness, Fréchet smoothness, strict convexity, uniform convexity, when 1 < p < ∞; and reflexivity, when 1 ≤ p < ∞. When the underlying space is the field of real numbers, the p-HH-norm on Rn is the hypergeometric mean of a positive n-tuple. Unlike the case of X2, the embeddings that establish the equivalence of the p-HH-norms to the p-norms in Xn have norms that depend on the space X, specifically, on the convexity of the unit ball in X.
Although the p-norms are all equivalent on the finite product Xn, they are all inequivalent as n goes to infinity and, consequently, they each determine a different normed subspace of X∞. When X=R these are the familiar lp sequence spaces. Similarly, the p-HH-norms give rise to new normed spaces of sequences in X and, when X=R, to new sequence spaces of real numbers. Comparing these new sequence spaces with those arising from the p-norms is an important area for future work.
Date received: September 10, 2008
Regularization of some equivariant Euler classes
by
Rongmin Lu
University of Adelaide
The theory of zeta-function regularization has grown from its appearance in the Ray-Singer definition of analytic torsion into a useful tool in mathematical physics. We propose a variant of zeta-function regularization - W-regularization - and apply this to some S1- and S1×S1-equivariant Euler classes, which are defined for certain infinite-dimensional vector bundles using an approximation technique. We find that we obtain new multiplicative genera and recover some familiar ones.
Date received: October 31, 2008
Minimal triangulations of 3-manifolds
by
J.H. Rubinstein
Department of Mathematics and Statistics, University of Melbourne
Coauthors: S. Tillmann, (Melbourne) and W. Jaco (Oklahoma State)
Using some new techniques involving labelling edges and tetrahedra, via Z2 torsion elements in homology, we have been able to classify the minimal triangulations of some infinite classes of spherical 3-dimensional manifolds. We are currently working on extending this to other types of geometric structures.
Paper reference: arXiv:0805.2425
Date received: October 29, 2008
Analytic torsion for twisted de Rham complexes
by
Mathai Varghese
School of Mathematical Sciences, University of Adelaide
Coauthors: Siye Wu (U. Colorado, Boulder)
We define analytic torsion for the twisted de Rham complex, consisting of differential forms on a compact Riemannian manifold X with coefficients in a flat vector bundle E, with a differential given by a flat connection on E plus a closed odd degree differential form on X. The definition in our case is more complicated than in the case discussed by Ray-Singer, as it uses pseudodifferential operators. We show that this analytic torsion is independent of the choice of metrics on X and E, establish some basic functorial properties, and compute it in many examples. We also establish the relationship of an invariant version of analytic torsion for T-dual circle bundles with closed 3-form flux. This is joint work with Siye Wu.
Paper reference: arXiv:0810.4204
Date received: October 29, 2008
String Structures and Characteristic Classes for Loop Group Bundles
by
Raymond Vozzo
University of Adelaide
The string class of a loop group bundle P is the obstruction to lifting the structure group to the central extension of the loop group. The string class is related to the first Pontrjagyn class of a certain G-bundle associated to P. In this talk we will review the known results regarding this class and develop a notion of higher string classes for loop group bundles, which are associated to characteristic classes of certain G-bundles.
Date received: October 27, 2008
A new approach to hyperbolic geometry
by
Norman Wildberger
University of New South Wales
Hyperbolic geometry can be introduced in a simpler and more general way by using the framework of rational trigonometry and universal geometry within the Cayley Beltrami Klein model.
This way the subject extends to DeSitter space, incorporates a fundamental duality between points and lines, emphasises the importance of the null cone, and works over a general field. Many theorems of Euclidean geomety now can be given their proper Hyperbolic formulation, and the connections with Special Relativity become much clearer.
This talk will have lots of pictures.
Date received: October 31, 2008
Generalised dihedral quotients of finitely-presented groups
by
Matthew Auger
University of Auckland
A generalised dihedral group is a split extension of an abelian group A by the cyclic group C2 where the non-trivial element of C2 acts on A by inverting elements. I will discuss a method of determining which generalised dihedral groups are quotients of a given finitely-presented group.
Date received: October 30, 2008
On Generalised Inflations of Algebras
by
Graham Clarke
RMIT University
An easy way to enlarge a semigroup is to adjoin elements which mimic the behaviour of existing elements. This process creates an inflation of the semigroup, and associativity is preserved. More recently, Bob Monzo and I introduced the idea of a generalised inflation of a semigroup. Subsequently several papers have investigated this construction in various different contexts. I will discuss the different directions in which the research in this area has gone.
Date received: October 30, 2008
Recent progress in the study of regular maps on surfaces
by
Marston Conder
University of Auckland
A regular map is a 2-cell embedding of a connected graph (or multigraph) on a surface, such that the group of all its incidence-preserving automorphisms has a single orbit on flags (incident vertex-edge pairs). Following the computer-assisted determination of all regular and orientably-regular maps of characteristic -1 to -200 two years ago, a lot of new things have been discovered. We now have theorems about the genus spectra of such maps that are chiral, and such maps that have simple underlying graph. Also more is now known about regular Cayley maps (that is, orientably-regular maps whose underlying graph is a Cayley graph); for example, the curious result that a map for a cyclic group is reflexible if and only if it is anti-balanced. A detailed classification of all regular Cayley maps for cyclic groups is now in sight. I will report on many of these developments, some of which were obtained in joint work with Young Soo Kwon, Jozef Sirán and Tom Tucker.
Date received: October 30, 2008
Multiplicative structure in the centre of the Iwahori-Hecke algebra of type A.
by
Andrew Francis
University of Western Sydney
Coauthors: John Graham (Babcock & Brown), Weiqiang Wang (University of Virginia), Lenny Jones (Shippensburg University)
The connection between the centre of the Hecke algebra and the symmetric polynomials in Jucys-Murphy elements allows one to define an integral basis for the centre in terms of symmetric polynomials. This has a side effect of allowing one to show that the "diagonal" structure constants with respect to the minimal basis for the centre are independent of n. This in turn gives a filtration on the centre.
In general it would be nice if there were an integral basis for the centre that is multiplicative, but it turns out that this is not possible.
Date received: November 1, 2008
Fixed point free elements of prime order in primitive permutation groups
by
Michael Giudici
University of Western Australia
It is an easy consequence of the Orbit-Counting Lemma that every transitive permutation group of degree at least 2 has a fixed point free element. Using the Classification of Finite Simple Groups, Fein, Kantor and Schacher showed that there is actually one of prime power order. It has been proved that for primitive groups, except for a family involving M11 acting on 12 points, prime power order can be replaced with prime order. These two results do not give any information about which prime. This talk will discuss some recent work with Tim Burness which aims to provide such information for fixed point free elements of prime order in primitive groups.
Date received: October 29, 2008
Enhancing the nilpotent cone
by
Anthony Henderson
University of Sydney
Coauthors: Pramod N. Achar (Louisiana State University), Benjamin F. Jones (University of Georgia)
Many features of an algebraic group are controlled by the geometry of its nilpotent cone, which in the case of GLn(C) is merely the variety N of n×n nilpotent matrices. The study of the orbits of the group in its nilpotent cone leads to combinatorial data relating to the representations of the Weyl group, via the famous Springer correspondence. In the case of GLn(C), the basic manifestation of this correspondence is the fact that conjugacy classes of nilpotent matrices and irreducible representations of the symmetric group are both parametrized by partitions of n.
Pramod Achar and I have shown that studying the orbits of GLn(C) in the enhanced nilpotent cone Cn×N leads to exotic combinatorial data of type B/C (previously defined by Shoji under the name of "limit symbols"). This is closely related to Syu Kato's exotic Springer correspondence for the symplectic group.
I will review this story and report on more recent joint work with Achar and Ben Jones, in which we consider the question of whether the orbit closures in the enhanced nilpotent cone are normal varieties, as is known to be the case for the ordinary nilpotent cone N.
Date received: September 25, 2008
Equivalence classes of highly nonlinear functions between groups
by
Kathy Horadam
Mathematics, RMIT University
For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine (EA) equivalence and Carlet-Charpin-Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ-equivalent functions are EA-inequivalent because the equivalences are defined by different properties of functions.
I have recently solved the corresponding problem for functions between arbitrary finite groups, by relating graphs of functions to transversals and appealing to the theory of group extensions. There is a formula for all the functions in the generalised CCZ equivalence class of a given function, in terms directly comparable to the formula for all functions in its generalised EA equivalence class. As the EA classes are orbits under a group action, it is likely the former are too.
I will outline these results for the elementary abelian case, which is most useful for applications.
Date received: October 25, 2008
Monodromy groups of polytopes and self-invariance
by
Isabel Hubard
University of Auckland
Coauthors: Alen Orbanic and Asia Weiss
For every polytope P there is the universal regular polytope of the same rank as P corresponding to the Coxeter group C = [∞, . . . , ∞]. For a given automorphism d of C , using monodromy groups, we construct a combinatorial structure Pd. When Pd is a polytope isomorphic to P we say that P is self-invariant with respect to d, or d-invariant. We develop algebraic tools for investigating these operations on polytopes, and in particular give a criterion on the existence of a polarity for self-dual polytopes.
Date received: October 30, 2008
Representation growth and alternating quotients
by
Ben Martin
University of Canterbury
Let G be a group. Given a positive integer n, we define rn(G) to be the number of isomorphism classes of complex n-dimensional representations of G (note that rn(G) can be infinite). We say that G has polynomial representation growth if there exist a, b ≥ 0 such that rn(G) ≤ anb for every n. In this talk I will discuss a question of Brent Everitt: does there exist a finitely generated group G such that
(1) G has polynomial representation growth; and
(2) G has the alternating group Am as a quotient for infinitely many m?
Date received: November 2, 2008
Blocks of generalized q-Schur algebras of type A
by
Andrew Mathas
University of Sydney
Coauthors: Marcos Soriano (Hannover)
Donkin introduced an analogue of the q-Schur algebras indexed by an arbitrary saturated set of weights. We classify the blocks of these algebras in type A. Quite surprisingly, these blocks are just the restrictions of the blocks of the corresponding (quantized) enveloping algebra. The proof is a slick combinatorial application of the Jantzen sum formula which gives new information even in the cases where this result was previously known.
Date received: November 6, 2008
Dualizing complexes via flat modules
by
Amnon Neeman
Australian National University
I will describe recent work, culminating in the PhD thesis of Daniel Murfet, giving a completely new approach to Grothendieck's dualizing complexes.
Date received: September 17, 2008
The Ore Conjecture
by
Eamonn O'Brien
University of Auckland
The Ore conjecture, posed in 1951, states that every element of every finite non-abelian simple group is a commutator. Recently, Liebeck, Shalev, Tiep and I used a combination of character-theoretic methods and computation to establish it. Here we summarise some of the key ideas.
Date received: October 29, 2008
Some results on the cohomology and representation theory of quantum groups
by
Brian Parshall
University of Virginia
Coauthors: L. Scott; C. Bendel, D. Nakano, C. Pillen
Let G be a semisimple, simply connected algebraic group over a field k of positive characteristic p. A family of rational representations of G can be constructed from the irreducible modules for the associated quantum enveloping algebra at a p-th root of unity. (They were first defined by Lusztig.) This talk will discuss some of the homological properties of these modules, applications, and open questions. We will also discuss recent results on the cohomology of quantum enveloping algebras at mth roots of unity for small values of m.
Date received: October 29, 2008
Beads on runners
by
Arun Ram
University of Melbourne
Coauthors: Alexander Kleshchev
Khovanov-Lauda algebras are a family of algebras whose representation theory provides a categorification of quantum groups. In this work we classify and construct homogeneous representations of these algebras. The construction generalises the construction of irreducible representations of the symmetric groups and the notions of partitions, skew shapes, and abaci.
Date received: September 22, 2008
Embeddings of finite unitary reflection groups
by
Don Taylor
University of Sydney
A unitary reflection is a linear transformation, of finite order, of a complex vector space, which fixes a hyperplane pointwise. The finite groups generated by unitary reflections were classified long ago: Mitchell (1914), Shephard and Todd (1954).
Given a reflection group one can ask for all subgroups generated by reflections and, more generally, for all subgroups that act as a reflection group on a subspace (as in recent work of Lehrer and Springer). In this talk I describe embeddings that are not covered by these methods. One consequence is that all `exceptional groups' of rank at least three are embedded in K6 or E8.
Date received: October 20, 2008
Projective modules over quantum symmetric algebras
by
Ruibin Zhang
University of Sydney
We study finitely generated projective modules for quantum analogues of symmetric algebras which arise from the theory of quantum groups. It is shown that finitely generated projective modules for quantum symmetric algebras over the function field C(q) are stably free. It is also shown that every finitely generated projective module for a quantum symmetric algebra over the power series ring C[[t]] is free. The latter case rather resembles the Quillen-Suslin theorem on Serre's problem for polynomial algebras.
Date received: November 9, 2008
Retractions and projections for Chebyshev subsets of function and sequence spaces
by
Sergey Ajiev
University of New South Wales
Along with Lebesgue and sequence spaces with mixed norms, anisotropic Besov, Lebesgue, Lizorkin-Triebel and Sobolev spaces of differentiable functions defined on a domain and endowed with various norms are considered. We estimate the constants and determine the exponents for the local Hölder regularity of the Chebyshev centres, metric projections and some retractions for the closed convex subsets of these spaces. Attention is paid to the sharpness of some results.
Date received: October 30, 2008
Sobolev Spaces and Fractional Smoothness Spaces on Irregular Domains
by
Oleg V. Besov
Steklov Mathematical Institute
Function spaces on irregular domains of certain type of multidimensional Euclidean spaces are studied. Such domains may have, for instance, the shape of the external peak with a power-like degeneracy in the neighborhood of some boundary point. The embedding theorems for these spaces are established.
Date received: October 30, 2008
Characterization of function spaces via non-smooth kernels
by
Qui Bui
University of Canterbury
Coauthors: Tim Candy
The characterization of function spaces via a kernel in the Schwartz class S was established by Bui, Paluszinski and Taibleson in the mid-1990's. In this talk, using the concept of a bounded distribution introduced by E. Stein, I will present an extension of this characterization to the case where the kernel is not in S. This is joint work with Tim Candy.
Date received: October 30, 2008
Hardy's uncertainty principle
by
Michael Cowling
University of Birmingham (UK)
Hardy showed that if |f(x)| ≤ C exp(-a|x|2) and |[^f](x)| ≤ C exp(-a|x|2), where a, a > 0, then f=0 if aa is big enough, and if aa is the critical value, then f is a gaussian.
This has recently been extended in various ways, for instance to deal with operators (by Cowling, Demange, and Sundari; to appear). This talk surveys the inequality and some new extensions.
Date received: October 31, 2008
Contractions of Lie Groups and the Cowling-Haagerup Theorem
by
Anthony Dooley
University of New South Wales
Inönü and Wigner introduced the notion of a contraction, or continuous deformation, of one Lie group into another (generally non-isomorphic) Lie group. I discuss here the contraction of compact Lie groups arising as the K in the Iwasawa decomposition of rank one semi-simple group G, into the associated generalised Heisenberg group. It turns out that the techniques developed can be used to describe the relationship between the K-picture and the N-picture of the representation theory of G, and this has consequences for the Cowling-Haagerup constants.
Date received: November 3, 2008
Multilinear operators with non-smooth kernels and commutators of singular integrals
by
Xuan Thinh Duong
Macquarie university
Coauthors: Loukas Grafakos and Lixin Yan
We obtain endpoint estimates for multilinear singular integrals operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calderón-Zygmund kernels. As a consequence, we deduce endpoint L1 ×... ×L1 to weak L1/m estimates for the mth order commutator of Calderón. Our results reproduce known estimates for m = 1, 2 but are new for m ≥ 3. We also explore connections between the 2nd order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.
Date received: October 30, 2008
Invariant subspaces of submarkovian semigroups
by
Tom ter Elst
University of Auckland
Coauthors: Derek Robinson
If S is a submarkovian semigroup acting on a Hilbert space L2(X) and W is a measurable subset of X then we characterize the invariance of L2(W) by capacity conditions on the boundary of W.
Date received: October 27, 2008
Frequency-scale frames and the solution of the Mexican hat problem
by
Richard S. Laugesen
University of Illinois
Coauthors: H.-Q. Bui
Yves Meyer revealed the depth of our ignorance of non-orthogonal wavelets when he remarked, in his monograph on wavelets and operators, that we do not even know whether the Mexican hat wavelet system is complete in Lp for 1 < p < ∞. Completeness has been known only for p=2, by the sufficient frame condition of Daubechies. I have proved completeness for p > 2, in joint work with H.-Q. Bui. The talk will describe this wavelet spanning problem and the tools we develop to solve it, which include approximately dual frames in the Fourier domain.
Date received: October 26, 2008
Duality of Hardy space on product spaces of homogeneous type
by
Ji Li
Mathematics Department, Macquarie University
Coauthors: Yongsheng Han, Guozhen Lu
In this paper, we introduce the Carleson measure space CMOp on product spaces of homogeneous type in the sense of Coifman and Weiss, and prove that it is the dual space of the product Hardy space Hp of two homogeneous spaces. Our results thus extend the duality theory of Chang and R. Fefferman on the bi-disc, which was established using bi-Hilbert transform. Our method is to use discrete Littlewood-Paley-Stein theory in product spaces.
Date received: September 24, 2008
An inequality for bi-orthogonal pairs
by
Christopher Meaney
Macquarie University
We use ideas from Salem's proof of the Rademacher-Menshov Theorem, combined with a result of Menshov, to give a logarithmic lower bounds on sums of vectors in bi-orthogonal pairs. This is applied to estimates on Lebesgue functions for orthogonal expansions.
Date received: October 23, 2008
Using solid angles to get approximate measures of volume for polytopes
by
Sinai Robins
Nanyang Technological University
Coauthors: David Desario
Using lattices in Rd, we compute local contributions (solid angles) at each lattice point and find an interesting new Fourier series for their global sum over any real polytope, extending a few known results of I.G. Macdonald and A. Barvinok.
The global sums that we encounter form a different sort of discrete measure for the volume for any given real polytope. When p=2, we retrieve the classical solid angle polynomial of Macdonald, which turns out to be an integral of a classical theta function over a polytope. For other p's we have new variations on the theme of discrete volumes. This variation on a theme also extend the notion of a spherically symmetric solid angle to an Lp-ball solid angle, computed locally at each integer point inside a convex polytope The methods involve Fourier analysis. I'll define inasmuch as it is possible much all of the notions we encounter.
Date received: October 21, 2008
Bochner-Riesz analysis on on asymptotically conic manifolds
by
Adam Sikora
Australian National University
Coauthors: Joint work with Colin Guillarmou and Andrew Hassell
Let (M, g) be a complete noncompact manifold with the Riemannian metric g which is asymptotically conic in the sense that M compactifies to a manifold with boundary M' in such a way that g becomes a scattering metric on M'. Let D be the positive Laplacian associated to g, and L = D+ V, where V is a potential function obeying certain conditions. We analyze the spectral measure dEL(l) = [1/(2pi)] R(l+i0) - R(l- i0), where R(l) = (L - l2)-1. We obtain L1 → L∞ estimates on derivatives (in l) of the spectral measure. Hence, under assumption that the manifolds M is nontrapping, we obtain restriction theorems, i.e. Lp → Lp' mapping properties of the spectral projections, which are as good as those currently known for flat Euclidean space. As an immediate application, we prove spectral multiplier and Bochner Riesz summability results for L, similar to those known for the standard Laplace operator.
Date received: November 2, 2008
Symmetric norms and spaces of operators
by
F. Sukochev
University of New South Wales
Coauthors: N. Kalton
In 1937, von Neumann showed that if ∥·∥E is a symmetric norm
on Rn then one can define a norm on the space of
n×n matrices by
|
Date received: October 24, 2008
Harmonic measure distribution functions for sequences of planar domains
by
Lesley A. Ward
University of South Australia
Coauthors: Marie A. Snipes
The harmonic measure distribution function h(r), for r ≥ 0, of a planar domain D specifies the harmonic measure of the part of the boundary of D that lies within distance r of a fixed basepoint in D. It thus relates the geometry of the domain to the behaviour of Brownian motion in the domain. We establish sufficient conditions under which these functions hn for a sequence of domains Dn converge pointwise to the function h for a limiting domain D, at all points of continuity of h. We establish this convergence for a model example.
Date received: October 23, 2008
Comparison of solutions of the heat and Laplace equations
by
Neil Watson
University of Canterbury
Solutions of the heat equation have sometimes been used in harmonic analysis, in preference to harmonic functions. There are some advantages in doing this, as larger functions can be handled, and the kernel for the infinite strip is the same as that for the half-space. In this talk, I will present some comparisons between the solutions of the heat and Laplace equations, particularly for the case of the half-space.
Date received: November 2, 2008