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This article is essentially an English translation of a paper of mine, published in \emph{La Gaceta de la RSME}. Its aim is to present, for a broad mathematical audience, a research programme in local representation theory that goes beyond the classical restrictions to characters of $p'$-degree, characters of height zero, and blocks of abelian defect. The final and most recent part of this programme concerns Alperin's main problem of block theory: the search for local rules for character values. In that direction I describe the conjectures on picky elements and subnormalizers, which suggest that the sets ${\rm Irr}^x(G)$ and the subgroups ${\rm Sub}_G(x)$ are the natural objects attached to a $p$-element $x$.
The objective of Emergency Medical Services (EMSs) is to promptly respond to calls from citizens for first aid, providing pre-hospital care and, if necessary, to transfer patients to an appropriate Emergency Department (ED) by ambulance. The efficiency of such a system strongly depends on the deployment of ambulance home bases, i.e., locations where ambulances and their crews are strategically positioned, ready to respond to emergency calls. This paper presents a general Discrete Event Simulation (DES) model designed to capture the stochastic behaviour and workflow of regional ambulance emergency systems. The proposed model incorporates and integrates information collected from different sources, reproducing very accurately the operation of the ambulance system, thus allowing a more comprehensive and realistic analysis. To show the applicability and reliability of the proposed general model, a case study provided by the Azienda Regionale Emergenza Sanitaria - ARES 118 (an Italian Regional Emergency Medical Services Authority - ARES~118}) is presented. It concerns a territory within the Lazio region of Italy, including a medium-size city along with sparsely populated areas. The repo
This article provides an accessible introduction to fractional derivatives, a concept that extends classical calculus by allowing derivatives of non-integer order. It explores both the fundamental definitions and some of the most relevant properties and applications of this mathematical tool. It was originally published in Spanish in the Gaceta de la Real Sociedad Española and automatically translated using Github copilot.
Stone duality establishes a contravariant equivalence between the category of Boolean algebras and the category of compact, Hausdorff, totally disconnected topological spaces (Stone spaces). These spaces are precisely the profinite spaces and form the natural environment for arithmetic objects such as the $p$-adic integers, absolute Galois groups, and profinite completions. In this article we give a rigorous and self-contained development of the construction of the Stone space associated to a Boolean algebra, prove Stone's representation theorem, and describe profinite spaces as inverse limits of finite systems. From there we introduce the Stone--Čech compactification via its universal property and discuss central arithmetic examples (such as $β\mathbb{N}$ and profinite completions), emphasizing their structural role in arithmetic geometry and in the framework of Condensed Mathematics of Clausen--Scholze. The text is aimed at advanced undergraduate and beginning graduate students, as well as readers interested in the interface between topology, number theory, and new categorical formulations of topology.
In this expository article, we study the relation between the boolean functions and the hypercontractivity theorems of Aline Bonami. We focus on the social choice theory, and present some of the most important results in the area, such as the Friedgut-Kalai-Naor (FKN) and the Kahn-Kalai-Linial (KKL) theorems, and the famous Fourier Entropy/Influence conjecture. -- En este artículo expositivo estudiamos la relación entre las funciones booleanas y los teoremas de hipercontractividad de Aline Bonami. Nos concentramos en la teoría de la elección social, y presentamos algunos de los resultados más importantes en el área como los teoremas de Friedgut-Kalai-Naor (FKN) y de Kahn-Kalai-Linial (KKL), y la famosa conjetura Entropíıa de Fourier/Influencia.
This document is intended to present in detail the processing criteria and the analysis techniques used for the production of the Vulnerability Map Sanitary based on the use of public and open data sources. The paper makes use of statistical analysis techniques (MCA, PCA, etc.) and machine learning (autoencoders) for the processing and analysis of information. The final product is a map at the census track level that seeks to quantify the population's access to basic health benefits.
In this paper, we review data mining approaches for health applications. Our focus is on hardware-centric approaches. Modern computers consist of multiple processors, each equipped with multiple cores, each with a set of arithmetic/logical units. Thus, a modern computer may be composed of several thousand units capable of doing arithmetic operations like addition and multiplication. Graphic processors, in addition may offer some thousand such units. In both cases, single instruction multiple data and multiple instruction multiple data parallelism must be exploited. We review the principles of algorithms which exploit this parallelism and focus also on the memory issues when multiple processing units access main memory through caches. This is important for many applications of health, such as ECG, EEG, CT, SPECT, fMRI, DTI, ultrasound, microscopy, dermascopy, etc.
Proof assistants are computer softwares that allow us to write mathematical proofs so as to assess their correctness. In November 2021, I started the project of checking the simplicity of the alternating groups within the Lean theorem prover and its mathlib library. This text aims at reviewing this experiment. -- (French) Les assistants de preuves sont des logiciels qui permettent de rédiger des démonstrations mathématiques et d'en garantir leur correction. En novembre 2021, j'ai débuté un projet de vérification de la simplicité des groupes alternés au sein de l'assistant de preuve Lean, et de sa librairie mathlib. Ce texte est un essai de compte rendu de cette expérience. -- Published version in Spanish
This is the English version of the paper: "Complejidad de los números naturales", Gaceta de la Real Sociedad Matemática Española 3 (2000) 230--250. In this paper, several conjectures about the complexity of natural numbers are proposed. In a recent joint paper with H. Altman "Integer Complexity, stability, and self-similarity" arXiv:2111.00671, we resolve the last of these conjectures. This is why I consider its translation useful now as a reference.
Metal-rich asteroids could one day become hardware stores for Mars, supplying the materials needed to build and repair a growing colony。 A new analysis shows that carefully selected asteroids may be reachable with current spacecraft technology。 Some could even provide the ingredients for making rocket fuel in space, dramatically reducing the burden
A supermassive black hole has been found lurking about 30,000 light-years from the center of its galaxy—the first dormant black hole detected so far from a galactic core。 It revealed itself only after shredding a passing star and producing a brief, brilliant flare。 Astronomers think it may be the remnant of a devoured galaxy or a black hole flung o
NASA’s Psyche spacecraft aced its Mars flyby, using the planet’s gravity to speed toward its 2029 encounter with a metal-rich asteroid。 The spacecraft tested its cameras, magnetometer, and particle-detecting instruments, capturing unusual views of Mars and measuring its magnetic environment。 It also detected neutrons from the planet and spotted the
Engineers have transformed a notoriously brittle cobalt-aluminum compound into a material that is both extremely strong and capable of bending without breaking。 Their nanoscale design produced a yield strength about six to 10 times greater than high-strength structural steel while sustaining substantial deformation at room temperature。 The approach
The solar wind may be stripping away Mars’ atmosphere by creating giant rolling waves along its outer edge。 Data from MAVEN and Tianwen-1 show that these waves form plasma clouds that carry atmospheric particles into space。 The process is concentrated on one side of Mars and may have played a major role in drying out the once potentially habitable
NASA’s Curiosity rover has entered a Martian valley covered by an astonishing “sea” of tiny polygon-shaped fractures。 The honeycomb patterns, each only a few inches wide, stretch across the landscape and even wrap around a nearby 20-foot-tall butte
Researchers have found a way to build much larger “twisted” oxide materials while precisely controlling how their atomic layers line up。 Because these materials can be made over large areas and transferred onto different surfaces, the technique could help turn twistronics from a laboratory curiosity into a practical platform for next-generation ele