# 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)

## research

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.

## news

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
· 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

(1) ￼

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

(2) ￼

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
http://flip.it/MtZjG

## 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.

## Mathdir

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.