Cool a gas of atoms down enough and, under the right circumstances, they condense into a state known as a Bose-Einstein condensate. BECs are of significant interest because they display quantum mechanical behaviour on a macroscopic level. They also pose challenges to experimentalists and theorists alike.

The so-called Bosenova is the induced implosion of a BEC, causing a collapse followed by explosion (analogous to a star dying). More than seven years after this was first performed in the lab, theorists are still struggling to fully reproduce the experimental observations. We propose a new treatment of the scenario, with which we hope to produce much better quantitative agreement with experiment.

I will give a brief history of Bose-Einstein condensation, including some of the physics behind the phenomenon. I’ll then describe our application of the truncated Wigner method to the Bosenova, including the associated numerical techniques. I’ll round out the talk by showing movies of some simulations we have performed, and look at where we go from here.

080918102341

Prime numbers are arguably the oldest and most familiar non-trivial mathematical objects that humans have studied. In this expository talk we trace some of their history. We will focus on the difficult problem of deciding when a large integer is prime, a problem that is still with us, and still providing surprises, after three millennia. The going will be easy, and when we traverse the odd difficult patch we will not look down.

080911163958

Alexander Craig Aitken (1895-1967) is one of Dunedin’s glorious sons. He was a distinguished mathematician and statistician and received numerous honours and awards, among them FRSE (1925), FRS (1936) and the Gunning-Victoria Jubilee Prize (1948-52) of the Royal Society of Edinburgh, its highest award. He wrote a book about his experiences in the First World War, Gallipoli to the Somme (1963), for which he was elected a FRSL (1964). He was an excellent musician, had a prodigious memory – he could recite the decimal expansion of π to 1000 places – and an exceptional ability to calculate mentally. From the mid-1920s he suffered mental breakdowns almost as a matter of routine and disinterestedly recorded their effects. This talk takes a look at his life.

080904113124

With athletes sweating through the heat and humidity of the Beijing games, it’s timely to critique what we do and don’t know about hydration, athletic performance and health. I’ll present some of our hydration research, showing that the scientific community knows a lot but also relatively little, whereas – perhaps arguably – the media and thus public know nothing of use. Our research is common to most of the exercise physiology literature in using small samples (often <10 per group, single blinded) and thus repeated measures designs analysed with simple univariate statistics. I suspect that many of us rely on the statistics we’ve always known, and that we have little more stats knowledge than in our student days. It’s convenient to believe that such mediocrity of design is due to the high resource requirement of experimenting on humans under stress, whereas mediocrity of analysis is due to underpowered designs and the fact that it is ‘appropriate’ for what we want to know.

080826090759

For some fluid flow problems solution by conventional grid based methods present difficulties. While meshfree methods such as Smoothed Particle Hydrodynamics (SPH) offer potential benefits, typical implementations are not convergent.

I’ll show how convergent forms of the SPH equations may be derived.

080821145621

I will discuss how the work of Roy Kerr has profoundly affected the development of theoretical studies of the Einstein vacuum equations.

Note: I will not be discussing the Kerr solution as such.

080814155253

We consider the heat equation driven by additive noise. We discretize the equation in space by the standard continuous finite element method and derive two kinds of error estimates. The first kind of estimate measures the error in the mean square norm and shows the so called strong convergence. Here, error estimates for the deterministic problem give error estimates for the stochastic equation in a more or less straightforward fashion. In the second type of error analysis the error is measured in the weak sense of probability measures and implies the so called weak convergence. The analysis in this case is surprisingly complicated and uses more advanced results both from infinite dimensional stochastic analysis as well as from functional analysis. We prove that, similarly to stochastic ODEs, the rate of weak convergence is twice that of strong convergence. In both cases, the analysis is done in the operator semigroup framework for stochastic PDEs proposed by Da Prato and Zabczyk.

080804145425

Electrical impedance imaging is presently used for industrial process monitoring, particularly in harsh environments. Data consists of the relationship between voltage and current at electrodes placed around the region of interest. Data simulation requires solution of an elliptic partial differential equation, providing the primary computational cost of imaging via MCMC sampling. Fifteen years ago the required computation was considered intractable, whereas today ‘black box’ MCMC algorithms will adequately sample posterior distributions over pixel representations using GMRF type prior distributions, producing low-resolution images.

The use of mid- and high-level representations gives greatly improved results in specific applications. I will give an example of the use of boundary representations to quantify void fraction in oil pipelines. Computation time can also be slashed by adapting the Metropolis-Hastings step to cleverly use approximations to the forward map, such as the ‘delayed acceptance’ algorithm of Christen and Fox, or the ‘enhanced’ error model introduced by Kaipio and Somersalo.

080804102254

In the modeling of complex time-dependent physical phenomena the simultaneous effect of several different sub-processes has to be described. The operators describing the sub-processes are as a rule simpler than the whole spatial differential operator. Operator splitting is a widely used procedure for the numerical solution of such problems. The point in operator splitting is the replacement of the original model with one in which appropriately chosen groups of the sub-processes, described by the model, take place successively in time. This decoupling procedure allows us to solve a few simpler problems instead of the whole one.

In the talk several splitting methods will be constructed (sequential splitting, Strang splitting, weighted splitting, additive splitting, iterated splitting). We discuss the accuracy (local splitting error) of the methods. We also examine the effect of the choice of the numerical method chosen for the numerical solution of the sub-problems in the splitting procedure. We list the main benefits and drawbacks of this approach.

080722110126

The usual definition of an invertible matrix A assumes that A is square. This talk considers invertible matrices which are not square. There’s some elementary ring theory and graph theory involved.

080715082323

In the field of Einstein’s general relativity, promising successes have been achieved. This is the case both for the understanding of the nature of the theory, and for qualitative and quantitative aspects, since the formulation of Einstein’s field equations (EFE) as a Cauchy problem about 50 years ago. Nevertheless, there are several serious outstanding questions at the very heart of the theory. One of those issues is the following. In order to simplify the discussion, let us neglect all matter fields and restrict to ‘pure gravity’. Suppose that – in a well defined sense – one prescribes the ‘state’ of spacetime at some initial time, the initial data. Due to results by Choquet-Bruhet and others, there always exists a unique spacetime corresponding to these data in a local time neighborhood of the initial time which is a solution of EFE. Now, let us extend this spacetime in time as a solution of EFE such that the paths of all observers starting from the initial time are included completely. Is this extended spacetime always unique? The hope was that the answer is yes because otherwise Einstein's theory would have only limited predictive power. However, certain counter examples were found, one of them is the family of Taub-NUT solutions. Nevertheless, there are reasons to believe that all known examples of this kind are ‘non-generic’ in a certain sense, and the conjecture, called strong cosmic censorship conjecture, is that for ‘generic’ solutions, Einstein’s theory retrieves its ‘global’ predictive power. Analytical and numerical attempts to shed light on such aspects have been undertaken since many years. Although one was able to prove the strong cosmic censorship conjecture only in certain symmetry classes so far, these studies have revealed interesting new insides.

After having introduced the necessary background and given some comments on the current state of knowledge on strong cosmic censorship, I will discuss some of my studies of non-linear perturbations of Taub-NUTspacetimes with my own numerical code based on spectral methods.

080707102838

This seminar will look at how special relativity and quantum mechanics are combined in Dirac’s equation. After briefly reviewing the standard scenario, I shall indulge in a bit of subversive speculation about the role of time in quantum mechanics. Technicalities are unavoidable, but I shall try to reduce their impact by emphasizing what Dirac liked to call “pretty mathematics”.

080516153013

Oceanographic data sets are often sparse, with measurements from widely spaced vessel tracks or moorings. To assist with interpretation of the measurements, spatial interpolation is commonly used to span gaps between measurements to give more complete spatial patterns of oceanographic variables such as salinity, temperature or velocity. Typically the components of velocity are interpolated independently. Divergence free forms of Radial Basis Functions developed for interpolation of electromagnetic fields are shown to give more realistic values between sparse velocity data than interpolating velocity components independently. The interpolator enforces physical dependence between the velocity components by ensuring mass is conserved. The interaction of a rotating fluid with the frictional boundary layer induces weak secondary flows in stirred tea cups and curved ocean currents. RBF interpolation is used to extract weak helical secondary flows from ADCP measured velocities in a curved section of the Otago Harbour Channel. The spatial structure of these first detailed field observations of secondary flow exhibit remarkable agreement with existing models, but are 50% stronger than expected.

080508133329

Analytic techniques allow explicit solution of wave propagation and scattering in simple geometries, or for composite geometries typically limited to asymptotic regimes of 'large' or 'small' lengths. These solutions provide scaling laws that aid engineering and design, and explicit formulas for the inverse problem. Semi-analytic methods extend these tools to complex composite geometries by augmenting analytic spectral methods with numerical calculations that a computer can perform essentially exactly. We examine spectral methods in the setting of ocean wave scattering. There, removal of exponentials allows exact evaluation of solutions, while application of Liouville’s theorem reduces the Dirichlet-to-Neumann map to an operator between low-dimensional spaces. A (finite dimensional) computer can then exactly characterize scattering in a composite. These methods are applied to determining low-frequency sound transmission through light-weight timber-framed construction that is typical in New Zealand buildings. Those solutions agree closely with measurements, and were recently used in the design of a timber floor with excellent sound insulation properties.

Joint work with Hyuck Chung.

080428105639

At first sight these three topics seem to have no relationship to each other. However, as will be discussed in this talk, there is a deep connection between them which is based on the notion of an integrable system. Some systems in Nature are described by equations which turn out to be “completely integrable” so that their solutions can be expressed in terms of special meromorphic functions on appropriate Riemann surfaces. In this talk these connections will be discussed and it is shown how to make use of the theory of Riemann surfaces to obtain explicit numerical solutions.

080414131320

The current studies are related with the origin of life in general and on the origin of bacterial chromosomes in particular. I will start by presenting the biological background required to follow the presentation. Then I will introduce some "mathematical properties" that have been found on DNA sequences, and how this properties can be either inherited or not by their corresponding protein sequences. A main result is the relative closeness between the standard genetic code and one proposed for the RNA World. Finally, some comparisons will be shown regarding whether or not RNA putative sequences from the RNA world preserve the mathematical properties observed nowadays.

080408154759

Mathematical software is changing and extending the way maths is used, how we do maths, how we teach it, and the very nature of mathematics itself.
Mathematica heralded a new paradigm in programming and is still the only major package to use the simplicity and power of rewriting. Mathematical statements and models can be expressed with simple natural expressions akin to programming by pictures. Mathematica 3 in 1995 broke the barrier between programs and documents. For the first time you could use the program itself to make and analyse its own fully typeset scientific documents. The recent release of Mathematica 6 is a quantum leap. It integrates both graphics and the user interface with the power of mathematics and there are already thousands of interactive demonstrations of concepts across the disciplines.
The seminar will introduce the basic ideas, and present examples of automating the curriculum and mathematical modelling in the sciences and arts.

080409105424

We consider a simple transport process along the edges of an infinite network and are mainly interested in the long term behavior of the system. We describe this “flow” by a strongly continuous semigroup on an appropriate Banach space. Combining methods from functional analysis, graph theory and stochastic processes, we are able to characterize those networks for which the flow behaves asymptotically periodic in the strong or even in the uniform sense.

080401100725

As a mathematician, I have always been something of a dabbler; my best and most cited papers are those which establish unexpected connections between seemingly different fields rather than those which go deeply into a single topic. I will describe some of these connections and how they came about.

One of the greatest benefits of being a mathematician is the opportunity to visit many countries and work in close contact with mathematicians from other cultures. These contacts very often lead to the kind of unexpected connections just referred to.

080401100500

A statistician introduced to Sudoku for the first time might recognise it as a combination of two ideas from experimental design: gerechte designs and critical sets.

Suppose that a certain number of treatments (such as fertilisers) are being tested on plots in a square field. The standard arrangement, allowing for the fact that fertility might vary in a systematic way across the field, is to use a Latin square, in which each treatment occurs once in each row and column. If in addition there are certain regions of the field which are different in their properties (stony or boggy, for example), then we also require that each treatment occurs once in each such region. This is a gerechte design, introduced by W. U. Behrens.

J. A. Nelder invented the idea of a critical set in a Latin square, a set of entries from which the whole Latin square can be recovered uniquely. The notion immediately extends to gerechte designs, and we see that a Sudoku puzzle is a critical set for a gerechte design where the Latin square is 9 by 9 and the regions are the 3 by 3 subsquares.

There are many other connections between gerechte designs and other areas of mathematics such as finite geometry, perfect codes and orthogonal Latin squares, and many questions about gerechte designs themselves, for example: given a partition of a square n by n grid into n regions with n cells in each, how do we decide whether a gerechte design exists?

080401095948

Wheels are such a simple, efficient solution to the problem of moving around on land that it would seem unkind of an intelligent designer, and insufficiently selfish of our genes, to encumber us with legs. The usual excuse is that wheels would entail twisted blood vessels and nerves; but there is a simple engineering solution to this problem. It turns out that the wheel is a limiting case – actually one of the least interesting and least useful examples – of a class of mechanisms that are able to move over a horizontal plane without doing any work, and without any actuation or control. Tetrapods (this means you, and your dog) and arthropods (your cockroaches and spiders) appear to exploit these mechanisms. With a small amount of intelligence and a modicum of effort, they remain efficient on irregular terrain. We should stop thinking of brains and muscles in terms of classical control, forcing the body along a specified trajectory. Instead we can think of neuromuscular activity as adjusting the dynamics of the body to reflect an irregular ground constraint, such that the constrained system has wheel-like symmetries in space and time. According to this view, the brain does not over-ride the physics of the body, but couples it to the environment to get a free lunch. Some simple examples can be analyzed and more complex cases are being investigated using computational models. We are also doing experiments on the kinematics and dynamics of spiders and dogs.

080320102738

There are several good reasons to study general relativity in higher dimensions. Recently, there has been much interest in the idea that our (3+1) universe is only a submanifold (brane) on which the standard model fields are confined inside a higher dimensional space (bulk). Fundamental strings and branes are also basic objects in string theory and black holes (as well as other black objects) form an important class of solutions in this theory. On the other hand cosmic strings and domain walls are topological defects which can be naturally created during phase transitions in the early universe. Their interactions with astrophysical black holes may result in interesting observational effects. An important example is an interaction of a bulk black hole with a brane representing our world in the brane world models. A stationary test brane interacting with a bulk black hole can be used as a toy model for the study of topology change in merger black hole transitions and also have far going interesting consequences for the study of phase transitions in quantum chromodynamics trough the gauge/gravity (AdS/CFT) correspondence.

In my talk I will consider the specific problem of static brane configurations on a spherically symmetric black hole background in the case for which the thickness of the brane is small but not exactly zero.

080319110611

There is concern in the maths education community that enrolments in undergraduate mathematics courses is declining. At a conference in Mexico in July I have been asked to look at this from an international perspective and report on it and matters surrounding it. In the talk I’ll say what I have been able to do so far and ask if there was a decline in maths student numbers would it matter.

By the way, ‘mathematicians’ here equals mathematicians plus statisticians, … sometimes.

080311120029

We consider various methods, for example algebraic, geometric, metric, via differential equations, etc., for defining or characterizing the classical geometric mean (ab)^{1/2} of two positive numbers a,b and show that these methods generalize to the case of positive definite matrices. In the latter context these generalizations have a number of interesting mathematical connections and applications.

080305095727

A perfect matching (or 1-factor) in a graph G is a subset of independent edges which together cover all of the vertices in G. A graph is said to be n-extendable if for each subset of n independent edges N, we can find a perfect matching of G containing N. There are readily recognized bounds on the values of n for which certain classes of graphs can be n-extendable. We will mention some of these bounds and some situations where we can do a little better if we require that the n edges chosen to extend are “suitably separated”.

All terms will be explained, with examples and the talk should be accessible to senior students as well as academics.

080225134607

Let $f_1, f_2$ be bounded analytic functions in the unit disk. The
polynomial algebra ${\mathbb C}[f_1,f_2]$ belongs to the Hardy space
$H^p$ for all $p>0$. This algebra is naturally associated with the
range of the composition operator acting from the space of holomorphic
functions in the bidisk into holomorphic functions in the unit disk.
Among natural questions about this algebra are such as "when it is dense in the Hardy space", "when it has finite codimension", etc. These questions in general are far from being completely answered. In the talk we will discuss some results related to the case when both generators are inner functions.

080213102314

“Doppler tracking” is a common method to track spacecraft trajectories in space and time. It rests on monitoring frequency ratios of electromagnetic signals which are exchanged between an earthbound station and the spacecraft. This is usually assumed to take place in flat (Minkowski) spacetime.

In my talk I will discusss generalisations to cosmological spacetimes. Here new features arise due to their time-dependent geometries. I show that an exact Doppler-tracking formula can be derived for standard cosmological spacetimes. This allows us to reliably establish upper bounds for a possible influence of global cosmic expansion on such measurements, which have sometimes in the past been claimed to be relevant for the analysis of the so-called “Pioneer Anomaly”. I will also stress some non-trivial conceptual points that are relevant in this context.

080212105812

Recent observations not only support the present epoch of an accelerated expansion of the universe but they also provide strong and growing evidence for inflation (as well as big bang) in the distant past. While the former is commonly attributed to some mysterious form of gravitational repulsion (so called dark energy), the latter is believed to be linked with a large quantum gravity effect at very short scales, or the dynamics of one or several scalar fields present in a fundamental theory of gravity and fields. Understanding dark energy (or cosmological constant problem) is essentially the most important aspect of current cosmology and gravity research. Although it may be quite nontrivial to obtain a natural realization of inflationary theory even in the context of string theory, one main goal has been to develop string based models to the point where one can incorporate reasonable models of inflation and dark energy into a fundamental theory of quantum gravity, and to see how the details of Cosmic Microwave Background (CMB), or other cosmological measurements, provide constraints on new model building ideas that the string theory or supergravity constructions suggest. In this talk, I will briefly review my recent works in the subject, focusing on recent attempts to implement inflation and dark energy in string theory. I will discuss, in general terms, some recent approaches to obtaining accelerating universes from higher-dimensional gravities.

080209120559

An overview of theoretical quantum computation will be provided, including reversible computation, quantum circuits and quantum algorithms and the hidden subgroup problem as well as important concepts such as entanglement.

080208104035

1. Iain Dangerfield:
Simple Groups and Solvable Groups
- adventures in abstract algebra.

2. Hugo Norton:
Fun with Alice and Bob
- public key cryptography and associated number theory.

3. Alex Young:
Go with the Flow
- diffusion and the numerics of subordination in a complex flow field.

071004143658

COMO
(Computational Modelling) Project Presentations

1. Chyou, Te-Yuan
Arachnophobia Conquered
The passive dynamics of the spider body, what
makes it tick.. or at least kick


2. Brimble, Hamish
On the ball
Automated tracking of players in court games.


3. Wang, Yikun
Monte Carlo simulation for fractional reaction diffusion equation.
… Monte Carlo simulated data is compared with the fractional reaction diffusion model (FDR) in both the linear and non-linear case. A preliminary experimental analysis of the relationship between fractal dimension and growth rate will be presented.

070928083225

A direct method will be presented for the symbolic computation of conservation laws of nonlinear PDEs in multi-spatial dimensions.

The method computes densities and fluxes based on two key tools: the Euler operator to test exactness and the homotopy operator to invert the total divergence.

The method has been implemented in Mathematica.

Using the (2+1)-dimensional shallow-water wave equations as an example, a computer package will be demonstrated that symbolically computes conservation laws of nonlinear PDEs.

070911102751

Since the Ministry of Education modified one of the National Administration Guidelines (NAGs) to specifically include gifted students, there has been increasing interest in this field of education and also increased pressure placed upon schools to identify and adequately cater for such students. The current method of assessing senior secondary school students, the National Certificate in Educational Achievement (NCEA), attempts to cater for different ability levels with its grade structure of Achieved, Merit and Excellence. For many subjects, including mathematics, the requirements for achieving at the excellence level include skills such as explaining processes and justifying decisions, both requiring high levels of linguistic competence. Teachers often stumble upon students who clearly possess the inherent mathematical ability to solve such problems, but who lack the linguistic ability to adequately communicate their solution. All too often, these students are prevented from reaching the excellence level (and scholarship level) of NCEA. This raises two interesting questions:
1) Could language be an indicator of giftedness?
2) What is it that gifted maths students who can explain their reasoning do differently to those who are gifted but cannot explain their reasoning?
If language can be used as a tool for thinking, then there would be clear benefits to two groups: those who can do the maths but cannot explain it, and those who cannot do the maths quite so well, but know that they can use their language skills to compensate.
This study aims to investigate the language characteristics of a group of gifted secondary school mathematicians within the context of the selection and coaching of the New Zealand team to the International Mathematics Olympiad in 2007. Student responses, verbal and written, are being examined from multiple viewpoints, including O'Halloran's systemic functional multi-modal discourse analysis of mathematical text, Bills' linguistic indicators, Krummheuer's components of argumentation, and gender differences.
The seminar will present a summary of the research project to date, an overview of the methodologies used and results from preliminary data analysis.

070913154906

This talk will be presented remotely from Auckland using the EVO system.
You may be interested to see a working demonstration of the system's capabilities even if the subject matter of the talk is not of direct interest to you.


Steinitz’s Theorem states that a graph is the 1-skeleton of a convex polyhedron if and only if it is 3-connected and planar. The polyhedron is called a geometric realization of the embedded graph. Its faces are bounded by convex polygons whose points are coplanar.

A map on the torus does not necessarily have such a geometric realization. In this paper, we relax the condition that faces are the convex hulls of coplanar points. We require instead that the convex hull of the points on a face can be projected onto a plane so that the boundary of the convex hull of the projected points is the image of the boundary of the face. We also require that the interiors of the convex hulls of different faces do not intersect. Call this an exhibition of the map. A map is polyhedral if the intersection of any two closed faces is simply connected. Our main result is that every polyhedral toroidal map can be exhibited. As a corollary, every toroidal triangulation has a geometric realization.

This talk is suitable for a general mathematical audience.
(joint work with D. Archdeacon and Jo Ellis-Monaghan)

070903110027

We consider the transport of particles in a network. In the vertices of the network the particles are distributed to the outgoing edges of the vertices by Kirchhoff rules. This will be formulated as an abstract Cauchy problem and then studied using semigroup methods. Particular attention is paid to the asymptotic behaviour of the system.

070815084325

In this talk I will show how ordinary differential equations (ODEs) are used to model the relationship between the dose of a drug and its effect on the body. I will then look at several important examples of linear and nonlinear ODEs and techniques of finding exact and approximate solutions.

070801103833

A huge range of morphological diversity is present in animals. These myriad morphological forms are produced through the activity of developmental genes and pathways that act, in the embryo, to sculpt the adult form.

Research in developmental genetics has begun to identify and untangle the developmental pathways that make an animal in a few species, but it is still unclear how these pathways evolve to give different morphologies.

We have been using insect segmentation as a model system to understand how developmental pathways evolve. In this seminar I will present some of our studies of honeybee segmentation, discuss their implications for our understanding of the evolution of development and indicate how computer modelling can help.

070808140118

As we grow more scale in our research projects we recognize that the complex of IT and communications systems required to support them are beyond the scope of a single project to sustain. A new model of IT support for research has emerged that focuses on defining layered services that can be used to support multiple projects, all dependent on advanced research and education networks. This model is inherently collaborative, requiring a level of transparency not previously considered. These layered services fall in three themes: Data management; Collaboration tools; High Performance Computing.

With the arrival of KAREN in December 2006 New Zealand has quickly moved to embrace this new approach, led by the TEC IDF funded BeSTGRID National Grid programme. Within BeSTGRID we’re evolving a nationally coordinated collaborative approach to research modeled on international best practice but adapted to the unique scope of New Zealand research, science and technology. This means establishing resources and supporting capability easily accessible from any NZ tertiary institution or CRI. While this programme is initially funded establish this capability at Canterbury, Massey, and Auckland Universities, the model can be more widely applied. The model describes how investment can best be apportioned into national, institutional, and project specific resources and capabilities. Paul will be describing the BeSTGRID programme and demonstrating it’s mechanisms and tools, live on the emerging national research grid, BeSTGRID.

070726113106

Biological and medical sciences are two of the main drivers for mathematical research in the 21st century. Applying mathematics to biology has got a long history, but recently there has been an explosion of interest in this field. The Human Genomic Project, the control of infectious disease, understanding global warming and etc. are already driving the research of hundreds of mathematicians.

In this talk, I will focus on the modelling of spread of epidemics and describing the conditions under which an epidemic is likely to occur. First, I will look at simple epidemics (in closed populations) and derive the corresponding model, known as SIR together with two real examples of epidemics. The last part of my talk will be about the geographic spread of epidemics that is the centre of interest of epidemiologists these days. At the end I will look at the data of the fascinating plague of Europe, known as the Black Death.

070717100354

1. Iain Dangerfield
Simple Groups and Solvable Groups

2. Hugo Norton
Public Secrets: Cryptography and Number Theory

3. Alex Young
Numerics of Subordinated Flow


Refreshments will be provided

070524082100

1. Chyou, Te-Yuan
Arachnophobia Conquered
The inside story of the spider, what makes it tick.. or at least kick; with animations

2. Brimble, Hamish
On the ball
Automated tracking of players in court games.

3. Wang, Yikun
Monte Carlo
… simulation for fractional reaction diffusion equation.

4. Walters, Brian
Stopping time problems

070517140954

There are many reasons that researchers categorise continuous variables, including ease of modelling and interpretation, facilitation of clinical decision making, and concern about the appropriateness of assuming a linear functional form. However, there are more compelling reasons NOT to categorise (in particular not to dichotomize). These include loss of power, problems with choosing cutpoints and an increase in type I error rates. When the relationships are not linear, fractional polynomials offer an alternative to categorisation while maintaining a parametric form (unlike splines, for example). This seminar will introduce the method of fractional polynomials and show their (fairly) simple implementation in the software package Stata (and give references for their implementation in SAS and R).

070515112541

A cover charge of $5 will apply to cover drinks and nibbles before the seminar. Jeff will speak at 6.30 p.m.

For those interested we plan to go out for a late dinner afterwards.


The time to stationarity in a Markov chain is an important concept, especially in the application of Markov chain Monte Carlo methods. The time to stationarity can be defined in a variety of ways. In this talk we explore two possibilities – the “time to mixing” (as given by the presenter in a paper on “Mixing times with applications to perturbed Markov chains” in Linear Algebra Appl. 417, 108-123, (2006)) and the “time to coupling”. Both these related concepts are explored through the presentation of some general results, without detailed proofs, for expected times to mixing and coupling in finite state space Markov chains. A collection of special cases are explored in order to illustrate some general comparisons between the two expectations. For those not overly familiar with Markov chains an overview of the basic essential concepts will be included in the presentation.

070420164701

In this talk I will introduce the normal-Laplace (NL) and the generalized normal-Laplace (GNL) distributions and discuss some of their applications. These include: fitting size distributions; option pricing for financial assets; directional statistics and survival analysis.

The four-parameter NL distribution provides a good model for size distributions. It can also be used to provide a flexible family of hazard rate functions (including a ‘bath-tub’ shaped hazard) for use in survival analysis.

The five-parameter GNL distribution is used in the creation of a Lévy process (Brownian-Laplace motion) whose increments can exhibit skewness and excess kurtosis (as seen in empirical logarithmic returns on stocks and other financial assets). An option pricing formula for assets following Brownian-Laplace motion is derived.

Finally wrapped versions of both the NL and GNL distributions provide attractive parametric models for directional data. They can exhibit both skewness and kurtosis.

070417104606

Let a0(ω), a1(ω), a2(ω), • • • , an(ω) be a sequence of independent random variables defined on a fixed probability space (Ω,Pr,A). There are many known results for the expected number of real zeros of a polynomial

a0(ω) ψ0(x) + a1(ω) ψ1(x) + a2(ω) ψ2(x) + • • • + an(ω) ψn(x)

where ψj(x), j = 0, 1, 2, • • • n is a specific function of x. In this talk we highlight different characteristics arising for the random polynomial dictated by assuming different values for ψj(x). Although, we are mainly concerned with the number of real roots we also study the density of complex roots generated by assuming complex random coefficients for polynomials.

070402095408

Decision problems in general, and conservation decisions in particular, involve the assessment of which of several alternative decisions is most likely to meet the objectives of the decision maker. Decision making under uncertainty must consider both the probability of particular outcomes following a decision, and the value of these outcomes to the decision maker. Decision theory allows for the explicit consideration of both uncertainty and value, and leads to decisions that are coherent (Lindley 1985).
Most conservation problems involve sequential decisions in dynamic, stochastic systems. Decision makers must account both for the immediate consequences of actions and the potential consequence of today's actions on tomorrow's decision opportunities. Thus, in harvest management the objective function includes both current harvest yield and future, anticipated harvest; strategies that fulfill this objective are by definition sustainable. The Principle of Optimality (Bellman 1957) leads to a globally optimal solution to a broad class of Markov decision problems via dynamic programming (DP). In contrast to 'open loop' procedures such as simulation-gaming and genetic algorithms, DP is 'closed loop', explicitly providing for feedback of anticipated future states to present decisions.
In the above, uncertainty exists (e.g., due to demographic or environmental stochasticity) but under a single model of the underlying biological and physical processes. By contrast, there often is profound uncertainty about the processes relating candidate decisions to outcomes. This uncertainty degrades the ability to make an optimal decision. Under sequential decision making and monitoring uncertainty can potentially be reduced, via adaptation. Decision making not only changes the natural system state; in doing so, it influences the relative belief in alternative models; this is known as dual control. Thus, a key premise of adaptive management, which depends on dual control, is that decision making and 'learning' are synergistic, not competitive processes.
Challenges remain for the implementation of adaptive management under dual control. Some of these are technical, including limitations on the dimensionality of decision models under DP, and the most effective ways of incorporating information into decision making. Advances in computing efficiency and algorithms, and Bayesian methods, have reduced or eliminated the technical limitation to adaptive management in conservation. The remaining challenges are principally social or political, and include disagreement over fundamental objectives, confusion of values issues with technical issues, and definitions of adaptive management that confuse rather than clarify.

070315135121

In this seminar I will present a general overview of the mathematical and computational issues which appear in the theory of gravitation. Special emphasis is put on the interplay between geometry and the development of numerical methods.

070315140027

Small particles suspended in a turbulent gas can cluster together. It is widely believed that this is due to particles being ‘centrifuged’ away from vortices. It has also been proposed that this clustering effect results in an increased rate of collision of particles.

I will describe recent results which quantify the clustering of particles, by means of a mapping to a perturbation of a nine-dimensional quantum harmonic oscillator. The centrifugal effect plays no role in my model, and the results are in good agreement with simulations of particles in turbulent flows.

I also argue that the increased rate of collision of particles in turbulent flows is primarily caused by the generation of caustics in the velocity field of the particles, rather than spatial clustering.

These results are relevant to the initiation of rainfall from cumulus clouds, or the formation of planets from dust around a young star.

070315140510

Photonic Crystal Fibres (PCFs) are silicon or glass fibres designed with a special 2D structure that allows them to guide certain frequencies of light. For a fixed frequency we solve Maxwell's equations for the z-component of the wave vector to determine the modes of light permissible in the fibre. For the simplest PCF, the problem is to calculate the spectrum of a 2nd order operator that has a periodic piece-wise constant coefficient. After transforming the problem onto the period cell, the spectral Galerkin method (also known as the plane-wave expansion method) is applied. As well as presenting the error analysis of this method we have considered solving a modified problem where the piece-wise constant coefficient function is replaced with a smooth coefficient function. The error analysis for the smooth problem is also presented and I answer the question: Is smoothing worth it?

070129091903

The mathematics is sometimes taught via procedures based in a single representation without consideration of the role of the underlying concepts. This approach easily transfers to GC use, with consequent loss of mathematical thinking. This workshop will briefly discuss some key theoretical ideas, such as processes and objects in mathematics and the role of representations, and apply them to algebra and calculus to see how the GC might be a useful teaching tool. Most of the time will then be taken up with a hands-on session implementing some of these ideas on the GC. We will consider areas such as: graphing, solving equations; transformation and symmetry of functions; differentiability and continuity; differentiation and integration; etc. This workshop is aimed at those who are beginning to use graphic calculator (GC) technology, and will use the Casio 9850G and the TI-84plus calculators with occasional reference to CAS.

061115134507