This seminar will give an overview of the subject, but with emphasis on connectedness and separation.


Juice and muffins will be provided

061018114445

Low Copy Number DNA evidence arises in situations where there is very small amounts of genetic material - generally 100 picograms ($10^{-10}$g) or less. With “normal” PCR methods, this evidence would not be detected because it requires 250pg to 1ng to give a reliable signal. However Findlay et al. showed that by increasing the number of PCR cycles from 28 to 34 a signal may sometimes be obtained. As a consequence the effects of phenomena such as contamination, drop out and stutter are magnified due to the increase in relative sensitivity. In this talk I will describe how we alter the equations to evaluate LCN DNA evidence, how these equations are formalized (so that they may be implemented in an expert system) and the some of the current difficulties arising from inconsistencies in the formulation.

061010144556

Andrew Darlington:
Scheduling Factory Production

Caroline McLean):
Modelling Squat for Vessels

Brian Walters
American Option Valuation

Juice and muffins will be provided

061004082957

Difference equations arise not only in recursive discrete models but also from dimension reducing simplifications of continuous dynamical models. So the behaviour of difference equations has importance beyond their natural mathematical interest. Even simple difference equations exhibit bizarre and as yet ill understood chaotic behaviour. The prototypical case is the Collatz relation:

$x_{n+1} = x_n/2$ (if $x_n$ even), $3x_n +1$ (otherwise).

To shed light on this equation and to explore the phenomena of non-linear difference equations, simple piecewise linear extensions have been studied in a series of papers. We will show how simple convex geometric methods provide more general proofs and extensions of some previous results in the area

060908085811

After an absence of almost eight years I recently spent three weeks teaching at a local secondary school. I discovered that although a few things haven’t changed in the world of maths education, at least two paradigms have shifted so far across the horizon that if you don’t keep an eye on what’s going on you mightn’t be able to see where the horizon is any more.

These massive changes have been the introduction of NCEA and the proliferation of graphics calculators. This seminar represents an attempt to catch up with the second of these changes, although it may already be too late ...

We’ve all seen students who carry expensive graphics calculators around but can’t use them, but there are also an increasing number of students coming through to University who have been using their graphics calculators successfully in their maths classes for at least three years. They will want to keep on using them once they get here. Are we ready for them?

The seminar will begin with a couple of examples showing problems which may be readily solved using a graphics calculator. Then I’ll give everyone a chance to air their own wind.

And if you thought that 2007 was going to be bad, just you wait for 2014 ...

060904095307

Attend this fifty minute presentation with your colleagues and students to learn about Mathematica and the diverse applications it has for teaching and research.
Overview
Most numeric based professionals have heard of Mathematica, but many have yet to fully experience and appreciate the usefulness it has for a wide range of pure and applied mathematics purposes.
Dr Ray Hoare will discuss how easy it is to use Mathematica’s document-based user interface as a platform to develop and communicate mathematical ideas, whether you use standard mathematical, command line, or customised propriety notation. Ray will present Mathematica’s consistent environment for numeric and symbolic calculations; show off its unrivalled graphing power used to illustrate complex ideas; and discuss how programs can be created either from procedural, functional or rule-based approaches, to best achieve efficiency or clarity.
The demonstration will be followed by examples of Mathematica workbooks that show it being applied in a wide variety of disciplines and subject areas. In New Zealand we have as many economists using Mathematica as we have physicists.
Seminar highlights
* An introduction to the Mathematica notebook environment
* The benefits of Mathematica
* Examples of numerical and symbolic computation
* Graphing with Mathematica
* Programming methods available in Mathematica
* Extensions to the basic Mathematica
* Academic knowledge testing with Mathematica
* Application of Mathematica to a variety of real-world tasks
Show us your examples
You are invited to send Ray your own examples for special mention to your colleagues and students by emailing ray@hrs.co.nz at least a week before the seminar.
Open discussions
Hands on demonstrations will be available in Maths Lab A and Ray will be available after the seminars to chat with individuals or groups about the use of Mathematica for their research or teaching purposes. Please email ray@hrs.co.nz to schedule a meeting in advance.

Please Note: Please quote ref. 1777 when contacting us to register or to request your free info pack.

060809135102

ADInstruments designs and produces popular data analysis hardware and software systems (PowerLab and Chart) that are distributed internationally. All of their software is developed here in Dunedin by a small team, most of whom are PhD or Masters graduates from Otago University. For the past two years I’ve been working as a scientific programmer and software co-ordinator at ADI, and in this talk I’ll outline what products we produce, my role in the company, and discuss some of the interesting projects that I’ve been involved with. I’ll also talk about some of the challenges that a fresh graduate with a mathematics background can face in the workplace, and about some of the benefits of working for a small company like ADI.

060808140319

Superdiffusive differential equations are modelled by a fractional order diffusion term in the governing partial differential equation, and are used in applications such as groundwater hydrology, thermodynamics, and finance.

As in the classical diffusion PDEs, closed form solutions are elusive. The finite differences methods, used to solve the superdiffusive differential equations numerically, are discussed.

The consistency and the stability, and therefore the convergence of the explicit and implicit methods will be discussed.

The analysis of the finite difference methods for the fractional PDEs does not always parallel those for the classical PDEs. Moreover, the available methods for the fractional differential equations have a lower order convergence rate due to the "data history" that is needed for these problems. An approach to improve the low order spatial convergence rate of the numerical solutions, the computational complexities, some implementation details, and numerical examples are presented.

060810144711

Fractional derivatives have made a renaissance of late. They are used in parsimonious diffusion models in physics, chemistry, hydrology, and econophysics. We show how these models are derived from a simple random walk model. To lift the models to higher dimensions one can either use operator stable limit theorems or let dispersion happen along flow lines. We investigate the latter by subordinating the flow. This leads to a transport and diffusion model involving fractional powers of group generators with applications to hydrology as well as ecology. Along the way we prove a transference principle for subordinating groups and develop a functional calculus for generators of groups.

060802134731

The Maya rose from their origins in the ancient Olmec civilisation around 200AD and faded away around 900AD. During this time they used a complicated system of many calendars, being a people fond of the repeating cycles of the stars and spiritual matters. Legend holds that these calendars were passed on to them by the moon god, Itzamna, along with all their other knowledge and culture. The workings of at least three of these systems will be glossed over in some depth. In particular we shall discuss converting dates between these various calendars and our own Gregorian one, by some interesting and cunning modular arithmetic.

The so-called Long Count counts the number of days since some zero day (estimated to be Wed 12 Aug 3113BC) in the same way as we count our years from or until 1.AD. It does this in an almost base-20 system and conspiracy theorists and modern day soothsayers have been quick to jump on the fact that on (or around) Dec 21st 2012 the Long Count “ends”. This is true, as we will show, though only in the same way as our calendar ends on 31.12.9999, or perhaps ended on 31.12.999 AD.

060727095652

Fractional order differential equations (FODE) are useful tools for study of various complex processes. Although the theory of FODE has a long history, during the last three decades their applications in different areas of science have been discovered, as well as their linkage with other branches of mathematics. The CTRW theory has indicated that the governing equations of many non-classical (anomalous) sub- and super-diffusion processes can be described with the help of FODE. The contemporary development of FODE shows that these equations allow study of complex non-heterogeneous processes with changing modes. In the talk we give a brief survey of the current status of FODE theory, distributed and variable order differential equations, and some of their applications to diffusion in the cell membrane.

060710135046

Two examples of mathematics applied to industry and biology are presented. The first is the three-dimensional flow in a slender pipe of simple cross-section which gradually bends the rapid flow through a substantial angle. The main focus is the turbulent wall layer flow in the pipe, when the ratio of the relative radius of curvature to the magnitude of the turbulent fluctuations becomes a crucial factor; there are three different downstream developments depending on the magnitude of that ratio. One is when turbulence dominates, for which novel points are that although the physical situation arises commonly in industrial settings, it has been little studied previously by theory or experiments, and that, as a most surprising feature, the fully developed flow far downstream is not unique, being found to depend instead on the global flow behaviour. Further, a quite accurate predictive tool based on approximation is suggested for the downstream flow.
The lipid bilayer membrane which surrounds red blood cells is the biological example. The membranes are two molecules thick, and so measured in nanometres, but extend to several micrometres laterally: they are on a mesoscale at which both molecular and continuum descriptions are valid. Moreover, in both descriptions membranes display both fluid and solid characteristics. Existing continuum models require searches of their parameter spaces for values giving physical properties which agree with experiment or other simulations. Beyond a conceptual link between low-level uncoupling and high-level coupling is a demand for multiscale numerical simulations of the human body. Presently, the passing of data between levels of such simulations is often ad hoc and performed offline, and a model which can “handshake” any two layers is coveted. With these motivations, a model of the lipid bilayer in which molecular information is retained in a continuum description is presented. Crucially, this model has the potential for a direct link between molecular and continuum properties. The model minimizes a free energy functional of a lipid-water mix with the minimization corresponding to a time scale separation which ensures that continuum variables represent their discrete molecular counterparts. By considering the origins of the hydrophobic effect, the model is rendered more physical, and membrane-like numerical solutions in both one and two dimensions are shown, before a discussion of the many projects which this work suggests.

060710135219

In talk I will give an introduction to some ideas underlying exciting developments in equation and differential equation solving. The treatment is geometric and the talk will be illustrated with pictures to make the ideas accessible.

Newton’s introduction of calculus provided tools for solving equations approximately. Later modern algebra through Buchberger’s Algorithm, replaced the earlier analysis ideas, and gave algorithmic approaches to many important questions involving general polynomial systems. It has become apparent that the poor complexity of these methods severely limit their applicability.

Interestingly, analysis and Newton’s ideas are making a comeback in the new area of Numerical Algebraic Geometry. In that area, components (or solution manifolds) of polynomial systems, are represented by random “witness” points lying on the components. These points are computed with good complexity by homotopy continuation methods (i.e. global Newton methods). The first book, on this area, by Sommese and Wampler appeared in 2005. In will also discuss the extension of these ideas to systems of partial differential equations.

060609151519

The students and their topics are -

Andrew Darlington (COMO):
“Optimizing Soup Production”

Josh Howie (MATH):
“Topology and Modern Analysis”

Caroline McLean (COMO):
“Squat of Vessels”

060531151558

Many data sets in finance and geophysics exhibit heavy tails, with numerous outliers and significant skewness. Classical statistical and stochastic models in these areas need to be extended to allow a reasonable fit to the data. Scientists and engineers understand the problem, and are agreeable to new ideas and models. Some elegant approaches are available using extreme value theory, distributions with power-law or exponential tails, generalized central limit theory, and subordination. The new models motivate connections with diverse areas of research including fractals and fractional calculus. In this talk, I will survey some of these developments starting with a careful look at the data, and a discussion of the underlying problems from a practical point of view.

This is joint work with colleagues in statistics, probability, applied mathematics, physics, geology, and finance.

060529141610

In this talk we define torsion theories of modules over a ring in three different ways:

1. by using exact sequences of homomorphisms;

2. by using the concept of complementary classes;

3. by using Gabriel filters.


The prototype example of a torsion theory is given by the class of all abelian torsion groups (modules over the ring of integers).

Given time, we will look at some of the current research problems I am working on.

The usual pattern of research in torsion theory involves the following steps:

1. Take a concept from the theory of modules and rings;

2. create a torsion theoretic analogue of that concept;

3. try to get torsion analogues of theorems from the theory of modules and rings, so that

4. by restricting ourselves to one of the two most trivial torsion theories, we recover the original theorem.

060531152046

This seminar will introduce the wedge product and the exterior derivative, and look at some of their simplest applications. These include:
Generating Jacobians from differential forms.
Unifying the integral theorems of Green, Gauss and Stokes.
Testing differential equations for exactness and integrability.

060522082545

In his memoir written in 1774, P.S. Laplace introduced an error distribution that now bears his name. Since then, for many years the popularity of the Laplace distribution in stochastic modeling has been by far less than that of its four-years-older “sibling” - the second law of Laplace, better known as the Gaussian (normal) distribution. It is only in recent years that this distribution, together with its various generalizations, has been revived, and is now being used in a variety of fields, including archaeology, biology, economics, environmental science, finance, and physics.

In this talk, we will review fundamental properties of the Laplace and related distributions, discuss their applications, and present some recent developments in this area.

060426132926

This pre NCEA study investigated the impact of student employment on academic achievement within the senior high school (N=223). External national examinations for all year 11, year 12, and year 13 students were analyzed against a survey of the students working status. The incidence of student employment during the school term (82-88%) exceeded than official statistics (25-30%). While the total hours worked during the school term was negatively related to respective subject examination outcomes, the problem appeared more detrimental for year 12 and year 13 students, and in particular Mathematics.

060303104133

Mathematical models of tumor growth in tissue, the immune response, and the administration of therapies can suggest treatment strategies that optimize treatment efficacy and minimize negative side-effects. In this talk I will describe two types of models: systems of differential equations and high-dimensional, cellular automata models. The advantages and challenges of each model will be discussed. In particular, we will see how optimal control theory and results from nonlinear dynamics can be applied to the simpler models, while more complex models might serve as an in silico laboratory test-bed.
The goal of this talk is to highlight some recent results, and to give some insight into the potential role of mathematics in studying biological processes; no advanced knowledge of biology or optimal control theory will be assumed.
This is joint work with Prof. L. De Pillis of Harvey Mudd College.

060405151426

John Rossi and I recently showed that for certain meromorphic functions f(z), min |z|=r |f(z)| is reasonably large compared with max |z|=r |f(z)| on a sequence of r going to infinity. The question is: how much thicker than a sequence is the set of r on which this holds? I will try to put these results into context as painlessly as possible. Estimating the size of the set involves only elementary calculus.

060329141428

Sport is an international phenomenon attracting well-paid, high-performing athletes. We are bouyed by the success of our national teams at major events like the Commonwealth Games. But unfortunately sport has its "dark side" - a perspective best illustrated by the intrusion of performance-enhancing drugs and various methods employed to avoid their detection.

This seminar will highlight something of the history of drug misuse in sport, describe the current international opinion on standards and sanctions and look at the work of the New Zealand Sports Drug Agency. It will serve to remind us that the vast majority of our athletes are drug-free.

060320111133

Dr. Kovacs is a candidate for a staff position in Applied Mathematics

Reaction-diffusion equations are commonly used to model the growth and spreading of biological species. The classical diffusion term implies a Gaussian dispersal kernel in the corresponding integro-difference equation, which is often unrealistic in practice. We propose a fractional reaction-diffusion equation where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. The resulting model captures the faster spreading rates and power law invasion profiles observed in many applications, and is strongly motivated by a generalized central limit theorem for random movements with power-law probability tails. We develop practical numerical methods to solve the fractional reaction-diffusion equation by time discretization and operator splitting, along with some existing methods from the literature on anomalous super-diffusion. In the process, we establish the mathematical relationship between the discrete time integro-difference and continuous time reaction-diffusion analogues of the model, along with error bounds. Our general approach also applies to other alternative non-Gaussian dispersal kernels, and it identifies the analogous continuous time evolution equations for those models.

Joint work with Mark Meerschaert and Boris Baumer

060303141741

It has been conjectured that the scattering of ice-coupled waves could be used for remotely predicting the average thickness of an ice sheet, a task which is time-consuming and difficult by other means. I have used a model for simulating the scattering of waves, a model in which the ice sheet has a section of arbitrarily varying thickness, to examine the response of ice sheets to waves. I will give some background to the model, describe a potential method for predicting the ice thickness, and present our latest results.

060306154503

This talk describes a solution method for the scattering of ice-coupled waves by a sea-ice/ice shelf transition, such as occurs in the Ross Sea, Antarctica. Sea ice is usually about 1-2m thick, while ice shelves may be from 10-20m thick. The transition zone between them is essentially a ramp, which may be walked up.

The ramp is taken to be a linear increase in the ice thickness, although an arbitrary thickness profile is possible.

Mathematically, the problem is to solve a 2D partial differential equation over an infinite strip. A Green's function is used to transform the problem from two dimensions to one dimension, reducing it to a pair of coupled integral equations. One of this is over a finite interval, while the other over a semi-infinite interval. The latter may be solved analytically using the Wiener-Hopf technique, and the former is solved using numerical quadrature.

060303094020