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Recent studies (1) have indicated problems with our ability to use the genetic distances between species to estimate the time since their divergence (so called molecular clocks). An exponential decay curve has been detected in comparisons of closely related taxa in mammal and bird groups, and rough approximations suggest that molecular clock calculations may be problematic for the recent past (eg <1 million years). Unfortunately, this period encompasses a number of key evolutionary events where estimates of timing are critical such as modern human evolutionary history, the domestication of animals and plants, and most issues involved in conservation biology. A solution (formulated at UA) will be briefly outlined. A second area of active interest is the recent suggestion (2) that mitochondrial DNA diversity does not track population size in several groups, in contrast to standard thinking. This finding has been interpreted as showing that mtDNA may not be evolving neutrally, as has long been assumed.
Large ancient DNA datasets provide a means to examine these issues, by revealing evolutionary processes in real time (3). The data also provide a rich area for mathematical investigation as temporal information provides information about several parameters that are unknown in serial coalescent calculations (4).
References:

1. Ho SYW et al. Time dependency of molecular rate estimates and systematic overestimation of recent divergence times. Mol. Biol. Evol. 22, 1561-1568 (2005);
   Penny D, Nature 436, 183-184 (2005).
2. Bazin E., et al. Population size does not influence mitochondrial genetic diversity in animals. Science 312, 570 (2006);
   Eyre-Walker A. Size does not matter for mitochondrial DNA, Science 312, 537 (2006).
3. Shapiro B, et al. Rise and fall of the Beringian steppe bison. Science 306: 1561-1565 (2004);
   Chan et al. Bayesian estimation of the timing and severity of a population bottleneck from ancient DNA. PLoS Genetics, 2 e59 (2006).
4. Drummond et al. Measurably evolving populations, Trends in Ecol. Evol. 18, 481-488 (2003);
   Drummond et al. Bayesian coalescent inference of past population dynamics from molecular sequences. Molecular Biology Evolution 22, 1185-92 (2005).

The pricing of electricity involves several interesting features. Apart from daily, weekly and seasonal fluctuations, power prices often exhibit large spikes. To some extent this is because electricity cannot be stored. We propose a model for power prices in the Alberta market. This involves a diffusion process modified by a factor related to a Markov chain which describes the number of large generators on line. The model is calibrated and future contracts priced.

The breakup of a mass of fluid into drops is a ubiquitous phenomenon in daily life, the natural environment and technology, with common examples including a dripping tap, ocean spray and ink-jet printing. It is a feature of many generic industrial processes such as spraying, emulsification, aeration, mixing and atomisation, and is an undesirable feature in coating and fibre spinning. Surface-tension driven pinch-off and the subsequent recoil are examples of finite-time singularities in which the interfacial curvature becomes infinite at the point of disconnection. As a result, the flow near the point of disconnection becomes self-similar and independent of initial and far-field conditions. Similarity solutions will be presented for the cases of inviscid and very viscous flow, along with comparison to experiments. In each case, a boundary-integral representation can be used both to examine the time-dependent behaviour and as the basis of a modified Newton scheme for direct solution of the similarity equations.

To model real data sets using second order stochastic processes imposes that the data sets verify the second order stationarity condition. This stationarity condition concerns the unconditional moments of the process. It is in that context that most of models developed from the sixties' have been studied; We refer to the ARMA processes (Brockwell and Davis, 1988), the ARCH, GARCH and EGARCH models (Engle, 1982, Bollerslev, 1986, Nelson, 1990), the SETAR process (Lim and Tong, 1980 and Tong, 1990), the bilinear model (Granger and Andersen, 1978, Guégan, 1994), the EXPAR model (Haggan and Ozaki, 1980), the long memory process (Granger and Joyeux, 1980, Hosking, 1981, Gray, Zang and Woodward, 1989, Beran, 1994, Giraitis and Leipus, 1995, Guégan, 2000), the switching process (Hamilton, 1988). For all these models, we get an invertible causal solution under specific conditions on the parameters, then the forecast points and the forecast intervals are available.

Thus, the stationarity assumption is the basis for a general asymptotic theory for identification, estimation and forecasting. It guarantees that the increase of the sample size leads to more and more information of the same kind which is basic for an asymptotic theory to make sense.

Now non-stationarity modelling has also a long tradition in econometrics. This one is based on the conditional moments of the data generating process. It appears mainly in the heteroscedastic and volatility models, like the GARCH and related models, and stochastic volatility processes (Ghysels, Harvey and Renault 1997). This non-stationarity appears also in a different way with structural changes models like the switching models (Hamilton, 1988), the stopbreak model (Diebold and Inoue, 2001, Breidt and Hsu, 2002, Granger and Hyung, 2004) and the SETAR models, for instance. It can also be observed from linear models with time varying coefficients (Nicholls and Quinn, 1982, Tsay, 1987).

Thus, using stationary unconditional moments suggest a global stationarity for the model, but using non-stationary unconditional moments or non-stationary conditional moments or assuming existence of states suggest that this global stationarity fails and that we only observe a local stationary behavior.

The growing evidence of instability in the stochastic behavior of stocks, of exchange rates, of some economic data sets like growth rates for instance, characterized by existence of volatility or existence of jumps in the variance or on the levels of the prices imposes to discuss the assumption of global stationarity and its consequence in modelling, particularly in forecasting. Thus we can address several questions with respect to these remarks.

1. What kinds of non-stationarity affect the major financial and economic data sets? How to detect them?

2. Local and global stationarities: How are they defined?

3. What is the impact of evidence of non-stationarity on the statistics computed from the global non stationary data sets?

4. How can we analyze data sets in the non-stationary global framework? Does the asymptotic theory work in non-stationary framework?

5. What kind of models create local stationarity instead of global stationarity? How can we use them to develop a modelling and a forecasting strategy?

These questions began to be discussed in some papers in the economic literature. For some of these questions, the answers are known, for others, very few works exist. In this talk I will discuss all these problems and will propose 2 new stategies and modelling to solve them. Several interesting topics in empirical finance awaiting future research will also be discussed.

The standard diffusion equation is obtained by Einstein's method and its generalisation, Fokker-Plank-Kolmogorov-Feller theory. The time between jumps in Einstein derivation is constant.

We discuss random walks with residence time distribution, which occurs for flows of solutes and suspensions/colloids in porous media, CO2 sequestration in coal mines, several processes in chemical, petroleum and environmental engineering. The rigorous application of the Einstein's method results in new equation, containing the time and the mixed dispersion terms expressing the dispersion of the particle time steps.

Usually, adding the second time derivative results in additional initial data. For the equation derived, the condition of limited solution when time tends to infinity provides with uniqueness of the Caushy problem solution.

The solution of the pulse injection problem describing a common tracer injection experiment is studied in greater detail. The new theory predicts delay of the maximum of the tracer, compared to the velocity of the flow, while its forward "tail" contains much more particles than in the solution of the classical parabolic (advection-dispersion) equation. This is in agreement with the experimental observations and predictions of the direct simulation.

Over the last 15 years a fairly substantial psychological literature has developed in which human reasoning and decision-making is viewed as the solution to a variety of statistical problems posed by the environments in which we operate. In this talk, I briefly outline the general approach to cognitive modelling that is adopted in this literature, which relies heavily on Bayesian statistics, and introduce a little of the current research in this field. In particular, I will discuss work by myself and others on the statistical basis of how people make simple inductive leaps and generalisations, and the links between these generalisations and how people acquire word meanings and learn new concepts. If time permits, the extensions of the work in which complex concepts may be characterised with the aid of nonparametric Bayesian tools such as Dirichlet processes will be briefly mentioned.

In this talk, we will survey a number of different free boundary problems involving slow viscous (Stokes) flows in which surface tension is active on the free boundary. Both steady and unsteady flows will be considered. Motivating applications range from industrial processes such as viscous sintering (where end-products are formed as a result of the surface-tension-driven densification of a compact of smaller particles that are heated in order that they coalesce) to biological phenomena such as understanding how organisms swim (i.e. propel themselves) at low Reynolds numbers. Common to our approach to all these problems will be an analytical/theoretical treatment of model problems via complex variable methods -- techniques well-known at infinite Reynolds numbers but used much less often in the Stokes regime. These model problems can give helpful insights into the behaviour of the true physical systems.

Nanomechanical sensors are often used to measure environmental changes with extreme sensitivity. Controlling the effects of surfaces and fluid dissipation presents significant challenges to achieving the ultimate sensitivity in these devices. In this talk, I will give an overview of theoretical/experimental work we are undertaking to explore the underlying physical processes in these systems. The talk will be general and aimed at introducing some recent developments in the field of nanomechanical sensors.

Partial differential equations are used to model populations (such as cells, animals or molecules) consisting of individuals that undergo two important processes: dispersal and settling. I will describe some general characteristics of these systems, as well as some of our recent projects.

The growth processes of human babies have been studied sufficiently in scientific fields, but there have still been many issues about the developments of human fetus which are not clarified. The aim of this research is to investigate the developing process of systemic organs of human fetuses based on the data set of measurements of fetus's bodies and organs. Specifically, this talk is concerned with giving a mathematical understanding for the harmonized developments of the organs of human fetuses. The method to evaluate such harmonies is proposed by the use of the maximal dilatation appeared in the theory of quasi-conformal mapping.