# CEMAT’s Open Seminar

## Past sessions

### I. The Body of Least Resistance

Newton solves a variational problem about 10 years before the curve of quickest descent was found, a topic that is regarded to be the beginning of modern calculus of variations.

### II. The Force of Attraction of a Spherical Body

First the geometrical proof of this theorem is analyzed; then it is indicated that this result demonstrates the definitive role of mathematics in modern natural philosophy.

Poster
Slides I
Slides II

Introduced by Hugo Beirão da Veiga, Università di Pisa.

### Predicting the weather for the next week, outlook of the climate for the XXIst century

Current state of the art of weather forecasts has skill until day 6 to day 8. Nevertheless, it is possible to have reliable evolutions of the climate of the XXIst century; this apparent paradox will be explained in the presentation. Scenarios for the evolution of greenhouse gases throughout the century will be reviewed, as well as the climate impacts of each scenario. A climate portal is being finalized: a rich source of information for students, research, private and public sector, especially designed for guiding multi-decadal strategic decisions.

### Statistical Applications in Climatology.

The study of Climatology is based on the analysis and interpretation of weather data collected over many years. To analyze such data, knowledge of basic techniques and statistical methods is required. The climate analysis uses principles and techniques of meteorological analysis, numerical and statistical. Applied climatology makes the maximum use of meteorological and climatological knowledge and information for solving practical social, economic, and environmental problems.

Assessments of the effects of climate variability and climate change on human activities, as well as the effects of human activities on climate, are major factors in local, national, and global economic development, social programmes, and resource management.

The main focus of this presentation is to show statistical applications in climatology that are used at IPMA, for example, climatic normals, climatic indicators, trends, return periods and others.

### On Eigenvalues of the Transition Matrix of some Count Data Markov Chains

A stationary Markov chain is uniquely determined by its transition matrix, the eigenvalues of which play an important role for characterizing the stochastic properties of a Markov chain. Here, we consider the case where the monitored observations are counts, i.e., having values in either the full set of non-negative integers, or in a finite set of the form ${0,...,n}$ with a prespecified upper bound $n$. Examples of count data time series as well as a brief survey of some basic count data time series models is provided.

Then we analyze the eigenstructure of count data Markov chains. Our main focus is on so-called CLAR(1) models, which are characterized by having a linear conditional mean, and also on the case of a finite range, where the second largest eigenvalue determines the speed of convergence of the forecasting distributions. We derive a lower bound for the second largest eigenvalue, which often (but not always) even equals this eigenvalue. This becomes clear by deriving the complete set of eigenvalues for several specific cases of CLAR(1) models. Our method relies on the computation of appropriate conditional (factorial) moments.

### From Large Deviations to Statistical Mechanics: What Is the Most Likely Way for an Unlikely Event To Happen?

This talk is an introduction to the theory of large deviations, which studies the asymptotic behavior of probabilities of rare events. The talk is accessible to a general audience including graduate students in mathematics and physics. The theory of large deviations has its roots in the work of Ludwig Boltzmann, the founder of statistical mechanics. In 1877 he did the first large deviation calculation in science when he showed that large deviation probabilities of the empirical vector could be expressed in terms of the relative entropy function.

In this talk Boltzmann's insight is applied to prove a conditional limit theorem that addresses a basic issue arising in mathematics, statistical mechanics, and other applications. What is the most likely way for an unlikely event to happen? This question is answered in the context of $n$ tosses of a cubic die and other random experiments involving finitely many outcomes. Let $X_i$ denote the outcome of the $i$'th toss and define $S_n = X_1 + \ldots + X_n$. If the die were fair, then one would expect that for large $n$, $S_n/n$ should be close to the theoretical mean of $3.5$. Given that $S_n/n$ is close to a number $z$ not equal to $3.5$, the problem is to compute, in the limit $n \to \infty$, the distribution of $X_1$; i.e., the probability of obtaining $1$, $2$, $3$, $4$, $5$, $6$ on a single toss. Interestingly, this conditional limit theorem is intimately related to statistical mechanics because it gives a rigorous derivation, for a random ideal gas, of the equilibrium Maxwell-Boltzmann distribution of energy levels. A related conditional limit theorem for the distribution of $X_1$, $X_2$ illustrates the phenomenon of propagation of chaos.

### Statistical Analysis of Compositional Data

Compositional data refer to data sets where not the values as such but rather the ratios between the variables contain the relevant information. Typical examples are chemical element concentrations, or any data where the unit reflects relative information (mg/kg, %, ppm, etc.). Usually, an inclease of the value in one variable has some effect on the values of other variables.

Compositional data are represented in the Aitchison geometry on the simplex, and for applying statistical methods designed for the Euclidean geometry they need to be transformed first. The isometric logratio (ilr) transformation has the best geometrical properties, but usually the results are difficult to interpret because the ilr coordinates are formed by non-linear combinations of the original variables. We show for different multivariate statistical methods how the ilr transformation can be sucessfully used for interpretation. Based on real data examples we compare results from a standard approach and from a compositional data approach.

### Can Mathematics Improve the Art of Roasting Coffee?

The goal of the R&D teams of Grupo Nabeiro Innovation Center is to research scientific and technological trends that may accelerate the innovation process of the company. The expected output is new inventions and new knowledge that may be converted into innovation. This strategy demands the application of statistical and mathematical technologies. There is a wide range of applications already in use, namely at the level of experimental design, in the analysis of differences or of the relationships between sensorial and consumer data, as well multivariate applications. The impact of this knowledge allows us to perfect our historical, traditional skills as masters of the art of roasting. Nonetheless, in spite of all the data available there is limited information available that directly links perceived sensory properties of coffee to specific compositional components and roasting profiles. We believe that mathematical technologies will be, at short-term, fundamental to truly understand the compositional drivers of coffee flavour.

The seminar will be organized in two parts, the first dedicated to Grupo Nabeiro history, organogram and future prospects. The second part will be dedicated to the needs of the organization with regard to R&D. Throughout the session, important contributions from mathematics will be pointed out, concluding with the trends for R&D defined in partnership with CEMAT.

### Editora Springer - Conteúdos de Matemática

Em primeiro lugar serão abordados os recursos de Matemática da Springer já incluídos na B-On (revistas, eBooks e a base de dados zbMATH), seguidos dos recursos não incluídos na B-On. Numa segunda parte será apresentado o acesso e opções de pesquisa da plataforma SpringerLink (www.springerlink.com). Na terceira e última parte serão apresentados serviços para autores e investigadores.

### Mathematical Finance in South Africa

I have been involved in Math Finance university education in South Africa since 1996. During this time I have produced numerous graduates & grown an extensive network of industry & academic partners. I'll talk about these experiences & take questions.

### Mathematics-A Catalyst for Innovation- Giving European Industry an Edge

We will discuss the role of Mathematics in Industry and in innovation processes. The focus will be European and we will look at good examples provided e.g. by the experiences of the network European Consortium for Mathematics in Industry (ECMI). I will also present the ongoing ESF Forward Look: “Mathematics and Industry” (see http://www.ceremade.dauphine.fr/FLMI/FLMI-frames-index.html) and discuss possible future developments on a European scale.

### Performance analysis of congestion control mechanisms with QoS constraints

With the rapid development of computer networks, particular in the Internet, the control of traffic congestion has become one of the fundamental and critical issues in present networks to accommodate the increasing diverse range of services and types of traffic that must be confronted by the users. It is also a major challenge to researchers in the field of performance modelling. Congestion control to enable different types of Internet traffic to satisfy specified QoS constraints is becoming significantly important. This talk will mainly present performance modelling and analysis of congestion control mechanisms under bursty and correlated traffic with QoS constraints and maintaining delay at a specified level with time-varying arrival rate.

### Quality of Experience assessment for network monitoring

The number of existing applications on Internet increases more and more. Their traffic characteristics and performance requirements are very different and evolve in time, and the users have diverse behaviours with these applications. An Internet Service Provider needs to satisfy the customers' requirements in spite of these heterogeneities. The Quality of Experience (QoE) seems a promising tool to manage a network taking into account this diversity. Classical methods to correlate this QoE from network performance criteria are based on long and expensive experiments with a panel of users. We propose then a fast and cheap method to assess up-to-date QoE by application from a passive traffic analysis.

### Electrostatic Imaging and Conformal Mapping

We present the solution of an inverse boundary value problem for harmonic functions arising in electrostatic imaging through conformal mapping techniques. The numerical method consists of two parts. In a first step, by successive approximations a nonlinear equation is solved to determine the boundary values of a holomorphic function on the outer boundary circle of an annulus. Then in a second step an ill-posed Cauchy problem is solved to determine the holomorphic function in the annulus. We establish convergence results for the iteration procedure and through numerical examples we illustrate the feasibility of the method.
This is joint work with I. Akduman, Istanbul, and H. Haddar, Paris.

### New algorithms in computational mathematics suited for next-generation supercomputers

Present and future supercomputers offer many opportunities and advantages to attack complex and demanding industrial and applied mathematical problems, but provide also new challenges. In the Peta-Flops regime, these concern both, the way to exploit the increasingly available power and the need of designing algorithms which are scalable and fault-tolerant at the same time. An example of a probabilistic domain decomposition method, which is indeed scalable and naturally fault-tolerant, is presented. In practice, Monte Carlo as well as quasi-Monte Carlo methods are used to generate only few interfacial values in two-dimensional domains where boundary-value elliptic problems are formulated. This allows for a fully domain decomposition of the problem.

### Eighty Years of Control Charts: Some New Developments

J.M. Juran described in his memoirs the beginning of statistical process control (SPC). We can find the remarkable sentence that "Shewhart invented the control chart on May 16, 1924". More than eighty years passed and in the meantime control charts have become one of the main important tools of SPC. In this talk some important new developments will be discussed. Starting with Alwan and Roberts (1989) the extension of control charts to time series has been the subject of many papers. In order to monitor the parameters of a time series it is necessary to determine the control design by taking into account the probability structure of the underlying stochastic process. Explicit formulas of the average run lengths (ARL) are unknown for time series. Inequalities for the ARL of a Shewhart type chart for time series were proposed by Schmid (1995). Moreover, a time series has further characteristic quantities. Control charts for the simultaneous control of the mean and the autocovariances of a stationary Gaussian process were introduced by Rosolowski and Schmid (2003). A very challenging topic is the control of the parameters of a high-dimensional process. Up to now there are only a few papers dealing with the surveillance of the parameters of a multivariate time series (e.g., Theodossiu (1993), Kramer and Schmid (1997)). Several new control charts for the mean are proposed in Bodnar and Schmid (2005). There are many important applications of SPC in finance. This development started with the papers of Yashchin, Stein and Philips (1997) and Lam and Yam (1997). An interesting problem is to make statements about the optimal time points for the adjustment of a portfolio. In this situation we have a high-dimensional series and we have to monitor the optimal portfolio weights. Recently several control charts for this problem were introduced by Golosnoy and Schmid (2005).

### Boundary value problems for the Stokes and Navier-Stokes systems in polyhedral domains

This is a survey of new results concerning solvability and regularity properties of boundary value problems for the Stokes and Navier-Stokes systems in a three dimensional polyhedral domain. I start with pointwise estimates for Green's tensors of the Stokes operator. Then I turn to estimates of solutions in weighted and nonweighted Sobolev spaces, first for linear and then for nonlinear problems. These results are applied to a free boundary problem for the Navier-Stokes system.

### Uniqueness and stability of a classical solution to a non cut off Boltzmann equation

An initial value problem for a classical spatially homogeneous Boltzmann equation describes the time evolution of a distribution density for a mono atomic dilute gas of particles, independent of the spatial variable. Existence of a solution to such kind of initial value problem with non cut off collision kernel in the weighted Lebesgue space was proved by L. Arkeryd in 1981 for soft and hard potentials and by C. Villani in 1998 for soft and very soft potentials. These results impose on an initial data the finite entropy and the finite energy conditions. In the recent paper R. Duduchava, R. Kirsch S. Rjasanow 2005 proved existence and uniqueness of a local solution dropping the finite entropy and the finite energy (dropping even the finite impulse) condition. We will report on the uniqueness and stability of a classical solution to Boltzmann equations with non cut off kernels, soft and hard potentials, without the finite entropy constraint. Results on stability known before G.D. Blasio 1974, T. Gustafsson 1988, L. Arkeryd 1988 and B. Wennberg 1994, considered a cut off case only and imposed more constraints on the initial data.

### Uniqueness in Inverse Obstacle Scattering

The inverse problem we consider in this survey is to determine the shape of an obstacle from the knowledge of the far field pattern for the scattering of time-harmonic waves. We will concentrate on uniqueness issues, i.e., we will investigate under what conditions an obstacle and its boundary condition can be identified from a knowledge of its far field patterns for incident plane waves. We will review some classical and some recent results and draw attention to open problems.

### Helmholtz Problems in a Halfspace with an Impedance Condition

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