Explicit class field theory via elliptic curves

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2017, Director: Xavier Guitart Morales[en] We present and describe a proof due to Deuring of the Hilbert’s 12th problem in the particular case of imaginary quadratic fields. This particular case consists in giving explicit generators of the maximal abelian extension of an imaginary quadratic field. To deal with abelian extensions of such fields, we study the classical formulation of Class Field Theory and present the main statements without proofs. The other important ingredient for the proof we will describe is the theory of elliptic curves with complex multiplication. We will combine Class Field Theory with the notions of elliptic curves to prove the desired construction

On Induced Hopf Galois Structures and their Local Hopf Galois Modules

The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as direct product of the corresponding regular groups of the inducing subextensions. We describe here the attached Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. We give a general matrix description of the Hopf action which is useful to compute bases of associated orders. In case of an induced Hopf Galois structures it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.Comment: Accepted to be published in Publicacions Matem\'atiques. We have included two minor change

Induced Hopf Galois structures and their local Hopf Galois modules

The regular subgroup determining an induced Hopf Galois structure for a Galois extension L/K is obtained as direct product of the corresponding regular groups of the inducing subextensions. We describe here the attached Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. We give a general matrix description of the Hopf action which is useful to compute bases of associated orders. In case of an induced Hopf Galois structures it allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.Peer ReviewedPreprin

Hopf-Galois module structure of quartic Galois extensions of Q

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Given a quartic Galois extension L/Q of number fields and a Hopf-Galois structure H on L/Q, we study the freeness of the ring of integers OL as module over the associated order AH in H. For the classical Galois structure Hc, we know by Leopoldt’s theorem that OL is AHc -free. If L/Q is cyclic, it admits a unique non-classical Hopf-Galois structure, whereas if it is biquadratic, it admits three such Hopf-Galois structures. In both cases, we obtain that freeness depends on the solvability in Z of certain generalized Pell equations. We shall translate some results on Pell equations into results on the AH-freeness of OL.Peer ReviewedPostprint (published version

Are crews empowered with all the resources needed to successfully address an inflight emergency? Checklists, a necessary but insufficient tool

[EN] Although safety in aviation has increased exponentially in recent decades, sadly, plane crashes will always be associated with the operation of the aviation world. The efforts should be directed towards the pursuit of “zero accidents”, providing aircrews with the necessary resources to minimize risks. One of the most important flight operation resources used in aviation is the checklist. However, when faced with an emergency, the crew cannot always make use of them, either because the huge diversity of potential emergencies does not allow to collect all of them in these manuals, or because the time available to react is so short that it makes it impossible to utilize them. This paper shows how aircraft checklists are a necessary but insufficient tool to solve complex, unpredictable, and novel emergencies. Furthermore, based on successfully solved critical events, the authors suggest the implementation of specific techniques for the learning and development of problem-solving programs focused on innovation and divergent thinking to decision making, as part of flight training, in order to increase the crews’ capability to face such emergencies.S

Inducing braces and Hopf Galois structures

Let $p$ be a prime number and let $n$ be an integer not divisible by $p$ and such that every group of order $np$ has a normal subgroup of order $p$. (This holds in particular for $p>n$.) We prove that left braces of size $np$ may be obtained as a semidirect product of the unique left brace of size $p$ and a left brace of size $n$. We give a method to determine all braces of size $np$ from the braces of size $n$ and certain classes of morphisms from the multiplicative group of these braces of size $n$ to $\mathrm{Z}_p^*$. From it we derive a formula giving the number of Hopf Galois structures of abelian type $\mathrm{Z}_p \times E$ on a Galois extension of degree $np$ in terms of the number of Hopf Galois structures of abelian type $E$ on a Galois extension of degree $n$. For a prime number $p\geq 7$, we apply the obtained results to describe all left braces of size $12p$ and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree $12p$.Comment: arXiv admin note: text overlap with arXiv:2205.0420

Low-Cost Piezoelectric Sensors for Time Domain Load Monitoring of Metallic Structures During Operational and Maintenance Processes

The versatility of piezoelectric sensors in measurement techniques and their performance in applications has given rise to an increased interest in their use for structural and manufacturing component monitoring. They enable wireless and sensor network solutions to be developed in order to directly integrate the sensors into machines, fixtures and tools. Piezoelectric sensors increasingly compete with strain-gauges due to their wide operational temperature range, load and strain sensing accuracy, low power consumption and low cost. This research sets out the use of piezoelectric sensors for real-time monitoring of mechanical strength in metallic structures in the ongoing operational control of machinery components. The behaviour of aluminium and steel structures under flexural strength was studied using piezoelectric sensors. Variations in structural behaviour and geometry were measured, and the load and μstrains during operational conditions were quantified in the time domain at a specific frequency. The lead zirconium titanate (PZT) sensors were able to distinguish between material types and thicknesses. Moreover, this work covers frequency selection and optimisation from 20 Hz to 300 kHz. Significant differences in terms of optimal operating frequencies and sensitivity were found in both structures. The influence of the PZT voltage applied was assessed to reduce power consumption without signal loss, and calibration to μstrains and loads was performed.This research was funded by Basque Government, grant number KK-2019/00051-SMARTRESNAK and by the European Commission, grant number 869884- RECLAIM

Using a RGB-D camera for 6DoF SLAM

This paper presents a method for fast calculation of the egomotion done by a robot using visual features. The method is part of a complete system for automatic map building and Simultaneous Localization and Mapping (SLAM). The method uses optical flow in order to determine if the robot has done a movement. If so, some visual features which do not accomplish several criteria (like intersection, unicity, etc,) are deleted, and then the egomotion is calculated. We use a state-of-the-art algorithm (TORO) in order to rectify the map and solve the SLAM problem. The proposed method provides better efficiency that other current methods.These authors want to express their gratitude to Spanish Ministry of Science and Technology (MYCIT) and the Research and Innovation Vice-president Office of the University of Alicante for their financial support through the projects DPI2009-07144 and GRE10-16, respectively

Inducing braces and Hopf Galois structures

© 2023 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Let p be a prime number and let n be an integer not divisible by p and such that every group of order np has a normal subgroup of order p. (This holds in particular for .) Under these hypotheses, we obtain a one-to-one correspondence between the isomorphism classes of braces of size np and the set of pairs , where runs over the isomorphism classes of braces of size n and runs over the classes of group morphisms from the multiplicative group of to ⁎ under a certain equivalence relation. This correspondence gives the classification of braces of size np from the one of braces of size n. From this result we derive a formula giving the number of Hopf Galois structures of abelian type on a Galois extension of degree np in terms of the number of Hopf Galois structures of abelian type E on a Galois extension of degree n. For a prime number , we apply the obtained results to describe all left braces of size 12p and determine the number of Hopf Galois structures of abelian type on a Galois extension of degree 12p.Peer ReviewedPostprint (published version

Are crews empowered with all the resources needed to successfully address an inflight emergency? Checklists, a necessary but insufficient tool

Although safety in aviation has increased exponentially in recent decades, sadly, plane crashes will always be associated with the operation of the aviation world. The efforts should be directed towards the pursuit of “zero accidents”, providing aircrews with the necessary resources to minimize risks. One of the most important flight operation resources used in aviation is the checklist. However, when faced with an emergency, the crew cannot always make use of them, either because the huge diversity of potential emergencies does not allow to collect all of them in these manuals, or because the time available to react is so short that it makes it impossible to utilize them. This paper shows how aircraft checklists are a necessary but insufficient tool to solve complex, unpredictable, and novel emergencies. Furthermore, based on successfully solved critical events, the authors suggest the implementation of specific techniques for the learning and development of problem-solving programs focused on innovation and divergent thinking to decision making, as part of flight training, in order to increase the crews’ capability to face such emergencies