### Mathematisches Kolloquium 2014

**Prof. Dr. Róbert Rajkó**(Faculty of Engineering, University of Szeged, Hungary)**"Chemometrics: Chemistry, Mathematics or what?"**Abstract: Chemometrics is basicly the application of mathematical or statistical methods to chemical data. The International Chemometrics Society (ICS) offers the following definition ( http://www.chemometry.com/Index/Chemometrics.html ):

"Chemometrics is the science of relating measurements made on a chemical system or process to the state of the system via application of mathematical or statistical methods. Chemometric research spans a wide area of different methods which can be applied in chemistry. There are techniques for collecting good data (optimization of experimental parameters, design of experiments, calibration, signal processing) and for getting information from these data (statistics, pattern recognition, modeling, structure-property-relationship estimations). Chemometrics tries to build a bridge between the methods and their application in chemistry."

Possible new definition from the author:

"Roughly speaking chemometrics is the collection of some mathematical statistical, linear algebraic, convex geometrical, computer technological and formal logical methods using them to explore and to exploit optimal chemical information from analytical measurements."

Considering philosophical aspects, a science can be exact science if it has mathematical fundaments. Otherwise it is "just" descriptive science. E.g., chemistry was descriptive science in the time of alchemy. After Mendeleev, Cayley, Student etc. and quantum chemistry, chemistry got its mathematical basement, thus it became an exact science, i.e. chemists could and can briefly formulate laws which state general concepts in the language of mathematics. These laws help us to systematize and reduce the descriptive chemical information. Chemometrics can provide routine tools for data analyses and laws for analytical chemistry.

The lecture shows some episodes from the author's ordeal adventure between 'Scylla and Charybdis', i.e. his some theoretical chemometric aspects between chemistry and mathematics.**16.12.2014, 11:00 Uhr, Raum 427 (Ulmenstraße 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. K. Neymeyr**Prof. Dr. Toshihiro Abe**(Tokyo University of Science)**"Symmetric circular models generated by stereographic projection"**Abstract: In this talk, inverse stereographic projection, from the real line to the circle, is used as the motivation for a four-parameter family of symmetric unimodal distributions which extends both the Minh and Farnum (2003) and Jones and Pewsey (2005) families of circular distributions. The normalizing constant of the density can be expressed in terms of Appell's function or, equivalently, the Gauss hypergeometric function, both of which are available in symbolic mathematical computing packages such as Mathematica. Important special cases of the family are identified and expressions for its trigonometric moments are obtained. Parameter estimation based on method of maximum likelihood techniques is discussed, and the approach is used to fit the family of distributions to an illustrative data set. A further extension to a family of rotationally symmetric distributions on the sphere is briefly made.**11.12.2014, 11:15 - 12:45 Uhr, SR 416 (Ulmenstraße 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. W.-D.Richter**Prof. Dr. Matthias Beck**(San Francisco State University)**"Combinatorial Reciprocity Theorems"**Abstract: A common theme of enumerative combinatorics is formed by counting functions that are polynomials. For example, one proves in any introductory graph theory course that the number of proper k-colorings of a given graph G is a polynomial in k, the chromatic polynomial of G. Combinatorics is abundant with polynomials that count something when evaluated at positive integers, and many of these polynomials have a (completely different) interpretation when evaluated at negative integers: these instances go by the name of combinatorial reciprocity theorems. For example, when we evaluate the chromatic polynomial of G at -1, we obtain (up to a sign) the number of acyclic orientations of G, that is, those orientations of G that do not contain a coherently oriented cycle.

Combinatorial reciprocity theorems appear all over combinatorics. This talk will attempt to show some of the charm (and usefulness!) these theorems exhibit. Our goal is to weave a unifying thread through various combinatorial reciprocity theorems, by looking at them through the lens of geometry.**03.12.2014, 15:00 Uhr, HS 125, (Ulmenstraße 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. A. Schürmann**Prof. Dr. Thorsten Theobald**(Goethe-Universität Frankfurt)**"Polyeder, Spektraeder und die Frage des Enthaltensein"**Abstract: Polyeder (bzw. im beschränkten Fall Polytope) sind die Zulässigkeitsbereiche linearer Optimierungsprobleme. In Analogie hierzu bezeichnet man als Spektraeder die Zulässigkeitsbereiche semidefiniter Optimierungsprobleme. In dem Vortrag geben wir zunächst einige allgemeine Einblicke in die in den vergangenen Jahren intensiv studierte Welt der Spektraeder. Wir betrachten dann die algorithmische Frage, ob ein gegebenes Polyeder oder Spektraeder $S_A$ (gegeben als der positiv semidefinite Bereich eines linearen Matrixbüschels $A(x)$) in einem anderen $S_B$ enthalten ist.

Nach gemeinsamer Arbeit mit Kai Kellner und Christian Trabandt.**12.11.2014, 15:00 Uhr, HS 125 (Ulmenstraße 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. A. Schürmann**Maître d’Enseignement, Dr. Christophe Ley**(Université libre de Bruxelles)**"Skew-rotsymmetric distributions and related inferential procedures"**Abstract: Directional data on unit spheres appear more and more frequently in diverse fields, the most modern being machine learning, bioinformatics and cosmology. Most classical distributions on the sphere, like the Fisher-von Mises-Langevin and the axial Watson distribution, assume that the data are rotationally symmetric about some location on the sphere. While this is an appealing assumption in some situations (e.g., when the earth rotation prevents us from distinguishing between two points of the same latitude), it often turns out to be an unrealistic assumption.

While many more flexible models have been proposed in the simpler circular case, the existence of such models for data on higher-dimensional spheres is rather scarce (except for some complicated mixture models).

To palliate this need, I will present in this talk a new class of flexible distributions, termed skew-rotsymmetric distributions. I will show their shape with some nice pictures, describe their properties (one very interesting property is that they keep some of the properties of rotationally symmetric distributions, for instance there is no need to calculate a normalizing constant, which is one of the heaviest tasks in spherical settings) as well as a simple generating mechanism, and establish optimal tests for rotational symmetry against skew-rotsymmetric alternatives. Of particular interest is the skew-rotsymmetric-FvML case, which suffers from a singular information matrix and hence is subject to the same criticism as the famous skew-normal distribution of Azzalini... to be continued!

(10.11.2014, 11:15 - 12:45 Uhr, HS 326/327 (Ulmenstraße 69, Haus 3))**[Das Kolloquium wurde abgesagt.]**

Kolloquiumsleiter: Prof. Dr. W.-D.Richter**Prof. Dr. Arthur Pewsey**(Mathematics Department, University of Extremadura, Cáceres, Spain)**"A Class of Circulas for Use with Toroidal Data"**Abstract: We consider ‘circulas’; the circular analogues of copulas. More specifically, we concentrate on one particular class of bivariate circulas which is pre-existing but has not been studied in such explicit form or detail before. This class is appealing in many ways but does not necessarily result in especially attractive bivariate circular models when the marginals are not circular uniform. A major exception is an elegant bivariate wrapped Cauchy distribution proposed and developed by Kato & Pewsey (2013). We will consider properties of the circulas themselves, as well as those of distributions generated using circulas with marginals that are not circular uniform. Likelihood based inference for the latter distributions will be considered and applied in the modelling of wind directions at a Texan weather station and data on the pre-earthquake direction of steepest descent and post-earthquake direction of lateral ground movement before and after, respectively, an earthquake in Noshiro, Japan.

This is joint work with Chris Jones (OU, UK) and Shogo Kato (ISM, Japan).**16.10.2014, 11:15-12:45 Uhr, HS 421 (Ulmenstraße 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. W.-D.Richter**Associate Professor Stefan Hoderlein**(Boston College (USA))**"Nonparametric Identification of Endogenous and Heterogeneous Aggregate Demand Models: Complements, Bundles and the Market Level"**Abstract: This paper studies nonparametric identification in market level demand models for differentiated products. We generalize common models by allowing for the distribution of heterogeneity parameters (random coefficients) to have a nonparametric distribution across the population and give conditions under which the density of the random coefficients is identified. We show that key identifying restrictions are provided by (i) a set of moment conditions generated by instrumental variables together with an inversion of aggregate demand in unobserved product characteristics; and (ii) an integral transform (Radon transform) that maps the random coefficient density to the aggregate demand. This feature is shown to be common across a wide class of models, and we illustrate this by studying leading demand models. Our examples include demand models based on the multinomial choice (Berry, Levinsohn, Pakes, 1995), the choice of bundles of goods that can be substitutes or complements, and the choice of goods consumed in multiple units.**13.10.2014, 16:00 Uhr, HS 125 (Ulmenstr. 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. A. Meister**M. Sc. Sebastian Kühnert**(Universität Siegen)**"Parameterschätzung für Exponentially tempered Power Law - Verteilungen"**23.06.2014, 15:00 Uhr, HS 125 (Ulmenstraße 69, Haus 3)

Kolloquiumsleiter: Prof. Dr. A. Meister**Prof. Dr. Michael Joswig**(TU Berlin)**"Infinite games, linear programs and tropical geometry"**Abstract : Tropical geometry provides a framework for making available concepts from algebraic geometry and valuation theory in discrete mathematics. This way a number of maybe unexpected links appear. As an example we will discuss mean-payoff games, a certain class of infinite games, which are relevant from a computational complexity point of view. We will relate them to classical linear programming by means of tropical geometry.**15.01.2014, 15:00 Uhr, HS 125 (Ulmenstraße 69, Haus 3)**Kolloquiumsleiter: Prof. Dr. A. Schürmann