Venue: TBA

Venue: TBA

Venue: TBA

Title: Bivariate models involving independent gamma distributed components

Speaker: Barry C. Arnold, Department of Statistics, University of California, Riverside, USA

Date | Time: September 23, 2026 | 14:30

Location: Building I, Samsung room (Auditório Manuel Laranjeiro)  

Abstract: Several multivariate models involving independent gamma distributed components (three of which are new) are described. The flexible bivariate beta(2) model introduced by Arnold and Ng (2011) provides the template for the other models. It involved ratios of sums of independent gamma variables. The other models involve differences, sums, products and minima rather than ratios.

Short Bio: Barry C. Arnold is Distinguished Professor Emeritus at the Department of Statistics, University of California, Riverside, USA. He received his Ph.D. in Statistics from Stanford University in 1965. He has authored 14 books and more than 300 research papers in reputed peer-reviewed journals and contributed volumes. Professor Arnold has guided 17 Ph.D. scholars and has been on editorial boards of renowned journals. His research interests include estimation theory, probability, stochastic processes, mathematical learning models, biological models, characterizations, income distributions, order statistics, inequality measurement, record values, conditionally specified distributions, and Bayesian inference. He has been invited for scholarly lectures from across the world and is recognized by American Statistical Association, USA (Fellow); American Association for the Advancement of Science, USA (Fellow); Institute of Mathematical Statistics, USA (Fellow); Royal Statistical Society, UK (Fellow); International Statistical Institute, the Netherlands (Elected Member). He also received, in December 2023 an honorary doctorate from the Colegio de Postgraduados, Montecillo, Mexico.

Organizers: Isabel Natário and Mina Norouzirad and Carlos A. Coelho

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This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UID/00297/2025 (\url{https://doi.org/10.54499/UID/00297/2025}) and UID/PRR/00297/2025 (\url{https://doi.org/10.54499/UID/PRR/00297/2025}) (Center for Mathematics and Applications)

Venue: TBA

More information is available here.

Course

Title: Beyond multivariate normality. A spectrum of alternative constructions of  bivariate and multivariate distributions

Speaker: Barry C. Arnold, Department of Statistics, University of California, Riverside, USA

Date | Time: September 24, 2026 | 10h-12h and 14h-16h

Location: Building VII, room 1.9  

Abstract: Part I: The classical multivariate normal distribution does constitute a good starting point for our discussion. Our goal is to document various methods for constructing more flexible families of multivariate distributions using well-known (or sometimes, little-known) models as components. We will give many examples of such augmented distributions, but we will not strive to provide an exhaustive listing of all examples that have appeared in the literature. The emphasis will be on the description of methods or algorithms of augmentation, with only a reasonably extensive sample of examples in which the techniques have been or could be used.Part II: In Chapter 2, we will review positive features of the classical multivariate normal model. However, partially to motivate the study of alternative multivariate models, we will next look carefully at the classical multivariate normal model, paying close attention to the ``baggage" that comes along with an assumption of the classical model as an acceptable descriptor of our data. If you accept classical multivariate normality, you must also accept an extensive list of items as being appropriate for your data modeling project.

Short Bio: Barry C. Arnold is Distinguished Professor Emeritus at the Department of Statistics, University of California, Riverside, USA. He received his Ph.D. in Statistics from Stanford University in 1965. He has authored 14 books and more than 300 research papers in reputed peer-reviewed journals and contributed volumes. Professor Arnold has guided 17 Ph.D. scholars and has been on editorial boards of renowned journals. His research interests include estimation theory, probability, stochastic processes, mathematical learning models, biological models, characterizations, income distributions, order statistics, inequality measurement, record values, conditionally specified distributions, and Bayesian inference. He has been invited for scholarly lectures from across the world and is recognized by American Statistical Association, USA (Fellow); American Association for the Advancement of Science, USA (Fellow); Institute of Mathematical Statistics, USA (Fellow); Royal Statistical Society, UK (Fellow); International Statistical Institute, the Netherlands (Elected Member). He also received, in December 2023 an honorary doctorate from the Colegio de Postgraduados, Montecillo, Mexico.

Organizers: Isabel Natário and Mina Norouzirad and Carlos A. Coelho

LogosTodos.JPG

This work is funded by national funds through the FCT – Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UID/00297/2025 (\url{https://doi.org/10.54499/UID/00297/2025}) and UID/PRR/00297/2025 (\url{https://doi.org/10.54499/UID/PRR/00297/2025}) (Center for Mathematics and Applications)

Venue: TBA

More information is available here.

Venue: TBA

Venue: TBA

Venue: TBA

Venue: TBA

Room:  1.19 do VII 
 
Abstract: The study and characterization of the subvarieties of W (variety of semigroups whose square is completely regular) when W is a E-semigroup (the set of its idempotents, E(S), is a subsemigroup[1]) has been proposed by Borralho and Kinyon[3] following a question from Araújo et al.[2].
Although in its formulation this problem had been limited to semigroups where the idempotents commute, called E-commutative semigroups[1], ∀e, f ∈ E(S), e.f = f.e, in this work, I will also present results for the E-semigroups in W. These can be defined by the additional condition ∀e, f ∈ E(S), f.e ∈ W(e.f), where W(a) represents the set of weak inverses of a, W(a) = {x : x.a.x = x}.
When idempotents form a subsemigroup, we call it a band. If they commute, E(S) is
a semilattice of idempotents; otherwise, it is a semilattice of rectangular bands. Although the former is a particular case of the latter, this presentation treats both situations distinctly, as they are different structures with different properties.

The point of view we adopt to study the structure of a relatively complex semigroup is to first decompose it into a collection of subsemigroups, each of which has a somewhat
simpler structure, and then compose a complicated and “bigger” semigroup from simpler
ones.
In this work, this is accomplished in 3 steps:
1. Dividing the semigroup S in disjunct closed classes Sα of known structure.
2. Defining a “gross” structure by studying the relation between these classes induced
by the semigroup operation, Sα.Sβ → Sγ.
3. Studying a “fine” structure that relates the positions of two elements from different
classes to the position of their product. For any S ∈ W and any a ∈ S, a2 lies in a group; thus S is an epigroup, also called group-bound in Howie[4]. Then S splits into unipotency classes, Ke, defined by elements whose square lies in Ge, the largest subgroup whose idempotent is e[5]. In W, these unipotency classes are disjoint subsemigroups of S and can be indexed by the elements of the semilattice E(S). Starting from S 2 and taking into account that E(S2) = E(S), we refer to the conditions for being able to define homomorphisms in S, φα,β : Sα → Sβ for all α ≥ β of the semilattice E(S), and so define a multiplication of the elements of the
different unipotency classes. This way, we can study the “fine” structure of S ∈ W when
their idempotents form a subsemigroup.
Except as stated otherwise, the basic concepts used here can be found in Howie[4] and
Shevrin[5].
References
[1] J. Almeida, J.-E. Pin, and P. Weil, Semigroups whose idempotents form a subsemigroup, Math.
Proc. Camb. Phi. Soc. 111 (1992), 241-253.
[2] J. Ara ́ujo, M Kinyon, J. Konieczny, and A. Malheiro, Four notions of conjugacy for abstract
semigroups, in: Proceedings of the Royal Society of Edinburgh Section A: Mathematics 147(6) (2017),
1169-1214.
[3] M. Borralho and M. Kinyon, Variants of epigroups and primary conjugacy, Commun. Algebra
48(12) (2020), 5465-5473.
[4] J.M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs, New Series 12, (Oxford University Press, 1995).
[5] L.N. Shevrin, Epigroups, in Structural theory of automata, semigroups and universal algebra (Springer, 2005) 331-380.