# Past Seminars

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Stochastic Geometry and Wireless Network Modeling

*Speaker:* François Baccelli

*Institution:* INRIA - Ecole Normale Superieure, France

*Date:* Fri 30 Nov 2007

*Time:* 3:15 pm

*Location:* Russel Love Theatre, Richard Berry Bldg, Parkville*Abstract:* The geometry of the location of mobiles and/or base stations plays a key role in several classes of wireless communication networks where it determines the signal to interference ratio for each potential channel and hence the possibility of establishing simultaneously some set of communications at a given bit rate. Stochastic geometry provides a natural way of defining (and computing) macroscopic properties of such networks, by some averaging over all potential geometrical patterns for example of the mobiles. The talk will survey recent results obtained by this approach for analyzing key properties of wireless networks such as coverage or
connectivity, and for evaluating the performance of a variety of protocols used in this context such as medium access control or routing.

Random thinnings and point processes under cover

*Speaker:* Dominic Schumacher

*Institution:* The University of Western Australia and Swiss National Science Foundation

*Date:* Fri 9 Nov 2007

*Time:* 3:15 pm

*Location:* Room 213, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne*Abstract:* A thinning of a point process (a "random point pattern") on R^D is obtained by deleting points independently according to probabilities supplied by a random function R^D -> [0,1]. In this talk I present a general estimate of the total variation distance between the distribution of a thinned point process and a Poisson process distribution and apply the result to a point process that is covered by i.i.d. balls. No previous knowledge about point processes or distances between probability distributions is required.

Pair-copula constructions of multiple dependence

*Speaker:* Claudia Czado

*Institution:* Technische Universität, München, Germany

*Date:* Mon 1 Oct 2007

*Time:* 3:15 pm

*Location:* Theatre 2, Ground Floor, 111 Barry Street, Carlton*Abstract:* Building on the work of Bedford, Cooke and Joe, we show how multivariate data, which exhibit complex patterns of dependence in the tails, can be modelled using a cascade of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of a general multivariate distribution and propose a method to perform inference. The model
construction is hierarchical in nature, the various levels corresponding to the incorporation of more variables in the conditioning sets, using pair-copulae as simple building blocks. Pair-copula decomposed models also represent a very flexible way to construct higher-dimensional copulae. We apply the methodology to a financial data set. This is joint
work with K. Aas, A. Frigessi, H. Bakken and A. Min.

Student Processes

*Speaker:* Nikolai Leonenko

*Institution:* Cardiff University, UK

*Date:* Fri 14 Sep 2007

*Time:* 3:15 pm

*Location:* Theatre 1, Ground Floor, 111 Barry Street, Carlton*Abstract:* With the aim of modelling key stylized features of observational series from economic and finance a number of stochastic processes with Student marginals and various types of dependence structures are discussed. In the field of finance, distributions of logarithmic asset returns can often be fitted extremely well by Student t-distribution with "?" degrees
of freedom, typically 3 = ? = 5. This implies infinite k-th moments for k =?. Another issue in modelling economic and financial time series is that their sample autocorrelation functions (acf) may decay quickly, but their absolute or squared increments may have acfs with non-negligible values for large lags. These ubiquitous phenomena call for on effort to develop reasonable models which can be integrated into the economic and financial theory as well as theories of turbulence. This approach has a long history since Mandebrot, who advocated to use heavy tails distributions (stable or Poreto type) and multifractals.

Alternatively, Barndorff-Nielsen has proposed to use different type of hyperbolic distributions in financial econometrics. In fact, the symmetric scaled t-distribution can be considered as limiting case of generalized hyperbolic distribution However, some properties of t-distribution can not be obtained from the corresponding limiting procedure. For instance, the characteristic function of the symmetric scaled t-distribution can not be obtained from the expression for haracteristic function of generalized hyperbolic distribution, based on the Laplace transform technique, which does not exist for the symmetric scaled t-distribution. Moreover, most of hyperbolic distributions are semiheavy tailed, while the symmetric scaled t-distributions heavy tailed. However, the basic ideas and constructions similar to symmetric scaled t-distribution is and more large class of the generalized hyperbolic distributions.

In this talk we propose a number of stochastic processes with Student marginal and various types of dependence structures, which are, in our opinion, interesting for economic and finance. This talk is based on joint work with Chris Heyde (Australian National University and Columbia University)

Path to extinction

*Speaker:* Fima Klebaner

*Institution:* Monash University, Melbourne

*Date:* Tue 11 Sep 2007

*Time:* 1:15 pm

*Location:* Russell Love Theatre, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne*Abstract:* We consider the size of a branching population at the time uT, where 0<u<1 and T is the time of extinction. For example, for u=0.5, it gives us size half-way to extinction, and when u varies between 0 and 1 we recover the path to extinction. We give an approximation when the population starts with a large number of individuals x, and show that the population is of order x^{1-u}. This approximation is not so precise just before extinction when u is close 1. For this case we consider random time T-a for small a, and give an approximation for this case. It is joint work with P. Jagers and S. Sagitov, Chalmers, Sweden.

Maxmoments and mixing in stochastic systems

*Speaker:* Daniel Tokarev

*Institution:* MASCOS

*Date:* Thu 6 Sep 2007

*Time:* 1:15 pm

*Location:* Russell Love Theatre, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne*Abstract:* We consider the expectations of the maxima of independent non-negativerandom variables. In the case of identically distributed random variables, when such expectations are referred to as maxmoments, we solve an analogue of the moment problem and present the asymptotics of the maxmoments as the number of random variables increases to infinity, and give an application to simple branching processes. In the case of non-identically distributed independent random variables, we derive tight upper and lower bounds for these expectations in terms of the maxmoments (which, incidentally, is equivalent to deriving tight bounds for the L_1-distance between the arithmetic and geometric means of the respective distribution functions). We also investigate a related optimization problem in the case when the random variables can follow one of two fixed different distributions. [This talk is based on joint research with K. Hamza, A. Sudbury, F. Klebaner and P. Jagers.]

Multilevel clustering of extremes

*Speaker:* Dr S.Yu.Novak

*Institution:* Middlesex University, UK

*Date:* Fri 13 Jul 2007

*Time:* 3:15 pm

*Location:* Room 213, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne*Abstract:* A sample element is considered “extreme’’ if it exceeds a certain level. Various quantities of interest in Extreme Value Theory can be expressed in terms of the number
of such exceedances. For example, the distribution of the sample maximum is closely related to that of the number of such exceedances.
This topic is of interest to insurers as an insurance company may be interested in a number of claims varying between certain levels. The empirical point process of exceedances (EPPE) takes into account heights as well as locations of extremes.
EPPE’s play a central role in Extreme Value theory. We describe the class of limit laws for EPPE’s, and present necessary and sufficient conditions for the weak convergence of an EPPE to a given element of that class.

On the connections between chaos theory and statistical mechanics

*Speaker:* Henk van Beijeren

*Institution:* Institute for Theoretical Physics, Utrecht University

*Date:* Thu 7 Jun 2007

*Time:* 3:15 pm

*Location:* Russell Love Theatre, Richard Berry Building, Department of Mathematics and Statistics, The University of Melbourne*Abstract:* The past years have seen a surge of activity on the connections between
chaos theory and statistical mechanics. Among the connections known I want to mention:
1) the Gaussian thermostat formalism, developed by Hoover, Evans et al.
Here the irreversible entropy production in a stationary non-equilibrium system is related to the sum of all of its Lyapunov exponents.
2) the escape-rate formalism of Gaspard and and Nicolis, in which transport coefficients determining the rate of escape of systems from phase space through an open boundary are related tot the Kolmogorov-Sinai entropy and the sum of all positive Lyapunov exponents on a small subset of phase space.
3) Ruelle's thermodynamic formalism, in which chaotic as well as transport properties can be obtained from a single dynamical partition function. This is even more ambitious, but for the majority of many-particle systems calculation of the dynamical partition
function is a very hard task.
Here I will briefly introduce dynamical systems and discuss their characteristic properties. I will show how quantities like Lyapunov
exponents, Kolmogorov-Sinai entropies and topological pressures may be calculated for a dilute
Lorentz gas (disordered billiard), which is a system with fixed scatterers on random positions, with which a point particle makes elastic collisions.
Comparisons of the results with computer simulation results show a very good agreement.
For a dilute hard sphere gas in equilibrium both the KS entropy (equal
to the sum of all positive Lyapunov exponents) and the largest Lyapunov exponent can be calculated analytically to leading orders in the density. Again,
comparisons to computer simulations show good agreement. The smallest positive Lyapunov exponents for these
systems show very interesting collective behavior, which can also be explained through kinetic theory calculations.
Finally I wil discuss some outstanding open problems.

COSNet/MASCOS Seminar - Heat Conduction in Nonlinear Systems

*Speaker:* Professor Bambi Hu

*Institution:* Department of Physics, Hong Kong Baptist University, and Department of Physics, University of Houston

*Date:* Fri 20 Apr 2007

*Time:* 3:15 pm

*Location:* Theatre 1, Ground Floor, 111 Barry Street, Carlton*Abstract:* Heat conduction is an old yet important problem. Since Fourier introduced the law bearing his name two hundred years ago, a first-principle derivation of this law from statistical mechanics is still lacking. Worse still, the validity of this law in low dimensions, and the necessary and sufficient conditions for its validity are still far from clear. In this talk I'll review recent works done on this subject. I'll also report our latest work on asymmetric heat conduction. The study of heat conduction is not only of theoretical interest but also of practical interest. It'll shed light on the thermal properties of carbon nanotubes.
The study of electric current has led to the invention of electric diodes and transistors. The study of heat conduction may also lead to the invention of thermal diodes and transistors in the future.

Adjoint based Optimization Methods

*Speaker:* Andreas Griewank

*Institution:* Humboldt Universitaet zu Berlin and DFG Research Center Matheon

*Date:* Thu 19 Apr 2007

*Time:* 3:15 pm

*Location:* Theatre 1, Ground Floor, 111 Barry Street, Carlton*Abstract:* As recently confirmed by an NP completeness result of Uwe Naumann the evaluation of Jacobians, Hessians, and other derivative matrices is a potentially costly computational task. In contrast, not only tangents representing Jacobian-vector products but also first and second order adjoints representing vector-Jacobian and Hessian vector products, can be obtained at a cost similar to the underlying vector or scalar function.
After reviewing the above results we discuss their impact on the design of optimization algorithms. I particular we present a new class of adjoint based quasi-Newton updates and a matrix-free one-shot approach to optimizing very large systems like those arising through the discretization of partial differential equations in aerodynamics and geophysics.

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