Eugénio A.M. Rocha

University of Aveiro, Portugal

Affiliation / Current Positions

• Assistant Professor at the Dept. of Math of the UAveiro
• Member of Research Unit “Center for Research & Development in Mathematics and Applications (CIDMA)”
• Co-Coordinator of GEOMETRIX
• Board member of General Assembly of CIM (2008-2022)
• Board member of Portuguese Mathematical Society CIM (2014-2022)


This is a log book with some of the research activities I’m doing lately.

thoughts, opinions and logging

Brainstorming and some logging

other stuff

A very incomplete list of supervision, organization and other stuff.

academic (ensino)

Some of my academic and teaching activities (Algumas actividade docents).


General news I found relevant or useful

Research Interests

Main Research Interests
· Partial Differential Equations [pubs]
· Nonlinear Control Theory [pubs]
· Educational Software [pubs] Other Current Interests [pubs]
· Epidemiology of HIV/Tuberculosis [pubs]
· Firms Economical Efficiency
· Dynamic Logics and Hybrid Systems
· Underwater AUVs Navigation
· Water Levels Statistics and Prediction

Other Scientific Interests [pubs]

· Raman Propagation Equations in Fiber Optics
· Retinal Eye Structures and Lesions
· Mathematical Knowledge and Preservation
· Graph Th with Apps to Automobile Industry
· Tables de Multiplication Jeux en ligne
· Integral Equations with General Delay
· Nanoscience and Nanotechnology

Regular aggregation sequences and their (in)finite series

Motivated by the (renormalization) of some classical divergent series in String Theory, e.g.

(see here), I start thinking on the convergence meaning of a class of divergent series, trying to make sense of them without following some of the (standard) approaches as Hardy resummation or Zeta function regularization (e.g. wikipedia). The main idea was to see if it is possible to define a class of sequences and a equivalence operation, by aggregating terms, such that their “value” is determined by some constant sequence in the same equivalent class.

In the comments of this post, you see the precise notions and a preliminary result, which is quite open for discussion…

First comment:
NOTATION. I use  to denote a tupple (e.g. a 2-tupple), of real numbers, since  means a set (i.e. no order and no element repetition) and  is an open interval. We will also consider tupples of functions. The size of a tupple  is denoted by . Any sequence  (an infinite-tupple) can be extended by zeros on the left, so it is infinite in the two directions. Therefore the index of a sequence is an element of , where the first element of a sequence  is denoted by . Moreover, since  for all , the relevant set of indices is . For , we denote by  the integers that are congruent to , i.e. the set .

Let , , and . Define the conditional function  as


which, for convenience, we just write as 

DEFINITION. A regular aggregation sequence (r.a.s.) is a sequence  such that exist fixed tupples  and , with ,  and , such that


We call the pair  a realization of u, which is not unique for each sequence u. The set of all r.a.s. is denoted by .

DEFINITION. The aggregation operation , with  and , is the map such that,

LEMMA. For  and , the map  has the following properties:

(1) It is well-defined, i.e. .

(2) We have

where , , and

is a bijection.

(3) Fix  and define  by

The function  is an homomorphism.

(4) Let ,  and , then

where .

(5) We have . Moreover, if  is a decomposition into prime numbers, then

Latest Professional Activity

Seminar at DISIM – Univ. of L’Aquila
[3 Nov 2021]
NIM game for Android
[20 Oct 2021]
PAPER: On the Schrödinger-Poisson system with a general indefinite nonlinearity
[29 Sep 2021]
PAPER: The effect of immigrant communities coming from higher incidence tuberculosis regions to a host country
[19 Jun 2021]

Introducing: Blocks and Fuel – Frameworks for Deep Learning in Python
[9 Jul 2016]
Introducing: Blocks and Fuel – Frameworks for Deep Learning in Python

Conference – 26-27/Nov/2022 Florence

The “Differential Inclusions and Set Valued Maps” Conference will be held at the Department of Mathematics of the University of Firenze, from 26 to 27 of November of 2022, and will be organised in three sessions, each consisting of Keynote Presentations and several contributed talks.
Keynote Presentations will be given by Alberto Bressan, Lech Gorniewicz, Nikolaos S. Papageorgiou and Alexandr A. Tolstonogov.


The Mathdir system is a web system for open science discussion and collaboration. Anyone with a social login may participate by adding topics, replies or creating their on groups of discussion.

Math Project’s Container

The website where I host projects for which I’m the founder or I’m heavily involved, and they do not have their own webpage elsewhere.


Centro Internacional de Matemática (CIM)
Ciência Hoje
Fundação Calouste Gulbenkian
Fundação para a Ciência e a Tecnologia (FCT)
Sociedade Portuguesa de Matemática (SPM)

Tables de multiplication
Apprendre les multiplications
Tables multiplications à imprimer
Jeux tables de multiplication