Invited Speakers

Wolf-Dieter Richter holds a Ph.D. and a Habilitation in mathematics by Tech-nical University of Dresden where his research focus was on multivariate limit theorems for large deviations, an asymptotic study of the Laplace-Gauss integral and a generalization of Cavalieri’s method of integration. At the University of Rostock he became a lecturer in 1986 and a professor in 1992. He applied large deviation results to statistical decision theory and then focused his work on exact multivariate distribution theory and its applications in probability and statistics. In doing so, he identified normalization constants of density generating functions as two- and multidimensional generalizations of the circle number π and correspondingly introduced and studied the ball number function. Wolf-Dieter Richter became a Fellow of the Alexander von Humboldt-Stiftung at the University of Marburg in 1991, an ISI-Elected Member in 2012, was the supervisor of ten Ph.D. students, has Erdos number three and is an Emeritus Professor since 2017. Based on his consequent vector space approach, he introduces new generalizations of complex numbers in his current work, gives an exact geo-metric explanation for the occurring single or multiple imaginary numbers and combines his generalizations of Euler’s formula with the theory of directional probability laws.

Speech Title: "Generalized Complex Numbers"

Abstract: The generalizations of complex numbers considered here differ from well-known generalizations such as quaternions, octonions, bicomplex and multicomplex numbers and Clifford algebras, to name some of the well-known ones, in at least two fundamental respects. First of all, products of elements of such traditional algebraic structures are explained by the fact that certain expressions in brackets are formally treated as when multiplying expressions in brackets of real numbers, with additional assumptions being made for the multiplication of so-called basic elements. In the present work, on the other hand, suitable vector-valued vector products are introduced and used which are geometrically motivated as rotations and stretches. Furthermore, traditional generalizations of complex numbers do not provide any information about which concrete mathematical objects fulfill the formulated wishes with regard to multiplication and whether the fulfilment of these wishes is unequivocal or ambiguous, while in concrete objects they are always specified in the present work. The new vector products allow the introduction of vector-valued vector-division, vector powers and exponential functions as well as a corresponding generalization of the Euler formula. The latter applies to the theory of directional probability laws. The new vector division opens new perspectives of differentiability and function theory.

Filipe J. Marques is Associate Professor at the Mathematics Department of NOVA School of Science and Technology | FCT NOVA and he is a research member of Nova Math – Center for Mathematics and Applications | FCT NOVA. He holds a PhD in Mathematics with a specialization in Statistics by NOVA School of Science and Technology | FCT NOVA. His main areas of research are Multivariate Statistics, Mathematical Statistics, Computational Statistics, Distribution Theory, and also Nonparametric statistics.

Speech Title: "How to Simultaneously Test Different Block Diagonal Covariance Structures"

Abstract: The analysis of the covariance structure is a crucial topic in multivariate statistics. Not only does the covariance matrix contain valuable information about the dependence structure between variables, enabling informed conclusions and optimal decision-making, but it also helps in calibrating statistical models. With the increasing use of more complex models, it has become essential to verify assumptions about the structure of covariance matrices. In this study, we review several significant covariance structures and demonstrate how it is possible to simultaneously test the presence of different structures in the diagonal blocks of a covariance matrix. We examine the distribution of the likelihood ratio test statistic and derive the expression for its h-th null moment. To ensure practical usability, we develop near-exact approximations for the likelihood ratio statistic. We provide a practical application using real data, along with numerical studies and simulations, to illustrate the applicability of the test and assess the precision of the developed near-exact approximations. Finally, we demonstrate how the proposed methodology can be extended to the study of more complex covariance structures.

Assoc. Prof. María del Carmen Segovia García
University of Granada, Spain