Dipòsit Digital de la Universitat de Barcelona

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El Dipòsit Digital de la Universitat de Barcelona és el repositori institucional que conté en format digital els materials derivats de l'activitat docent, investigadora i institucional de la comunitat universitària.

Enviaments recents

Dinàmica de l’arquitectura Transformer: Del pas discret a la modelització contínua
(2026-06-10) Singh Kaur, Arshdeep; Director xxx
[eng] In this document we study the Transformer architecture from a mathematical viewpoint, interpreting it as a particle system, to model the trajectory of the tokens we derive a first order differential equation, and we construct an energy function to obtain a second order dynamics as the continuous limit of a method called Hamiltonian (Symplectic Euler). We relate this dynamics to Polyak’s heavy-ball method and analyze its behavior. Finally, we validate the results with numerical simulations and explore the clustering. [cat] En aquest treball estudiem matemàticament l’arquitectura Transformer que interpretem com un sistema de partícules, per modelar la trajectòria dels tokens derivem una equació diferencial de primer ordre i construïm una funció d’energia per obtenir una dinàmica de segon ordre a partir d’un esquema anomenat Hamiltonian (Euler simplèctic) i fer el seu límit continu. Relacionem aquesta dinàmica amb el mètode heavy-ball de Polyak i analitzem el comportament. Finalment, validem els resultats obtinguts amb simulacions numèriques i explorem el clustering.
On Time-Frequency Localization: Gabor Frames and the Balian-Low Theorem
(2026-06-09) Seguí Blanes, Gerard; Massaneda Clares, Francesc Xavier
[eng] This work studies time-frequency localization in the context of signal analysis. More precisely, it focuses on the Fourier transform, the short-time Fourier transform, and Gabor systems as tools to represent signals in both time and frequency. Special attention is given to the limitations imposed by the uncertainty principle and the Balian-Low theorem, which show that a signal representation cannot be simultaneously discrete, orthonormal, and well localized in time and frequency. Finally, the work presents Gabor frames as a natural alternative to orthonormal bases, emphasizing the role of Gaussian windows and the density condition for stable reconstruction. [cat] Aquest treball estudia la localització temps-freqüència en el context de l’anàlisi de senyals. Més concretament, se centra en la transformada de Fourier, la transformada de Fourier de temps curt i els sistemes de Gabor com a eines per representar senyals tant en el temps com en la freqüència. Es presta una atenció especial a les limitacions imposades pel principi d’incertesa i pel teorema de Balian-Low, que mostren que una representació de senyals no pot ser simultàniament discreta, ortonormal i ben localitzada en temps i freqüència. Finalment, el treball presenta els frames de Gabor com una alternativa natural a les bases ortonormals, destacant el paper de les finestres gaussianes i de la condició de densitat per a una reconstrucció estable.
Representation Theory and Structure of Semisimple Lie Algebras
(2026-06-10) Sancho Corsín, Irene; Marchesi, Simone
[eng] This work focuses on Lie algebras and representation theory, a field of mathematics that was born in the 19th century and remains relevant today in areas such as differential geometry, number theory, and physics. This dissertation covers several topics. We introduce matrix Lie groups, a specific type of Lie groups, as well as the concept of the Lie algebra associated with them and, more generally, the Lie algebra of an arbitrary Lie group. Next, we introduce the concept of representation. We study and prove the structure of finite-dimensional, complex, irreducible representations of the Lie algebra $sl(2;\mathbb{C})$, which is necessary for understanding the structure of semisimple Lie algebras. After introducing Cartan subalgebras, roots and root space decomposition, we state and prove the main theorem of this work. This theorem states that every finite-dimensional, complex, semisimple Lie algebra admits a simple system of roots, a fundamental result in the structure theory of semisimple Lie algebras. [cat] Aquest treball se centra en les àlgebres de Lie i la teoria de la representació, una àrea matemàtica nascuda al segle XIX i amb rellevància en l’actualitat en àmbits com la geometria diferencial, la teoria de nombres i la física. Aquesta dissertació cobreix diversos temes. Introduïm els grups de Lie de matrius, una classe de grups de Lie, així com el concepte d’àlgebra de Lie associada a ells i, més en general, l’àlgebra de Lie d’un grup arbitrari. A continuació, introduïm el concepte de representació. Estudiem i demostrem l’estructura de les representacions de dimensió finita, complexes, irreductibles de l’àlgebra de Lie $sl(2;\mathbb{C})$, que és necessària per entendre l’estructura de les àlgebres de Lie semisimples. Un cop introduïdes les subàlgebres de Cartan, les arrels i la descomposició en espais d’arrels, enunciem i demostrem el teorema principal d’aquest treball. Aquest teorema afirma que tota àlgebra de Lie semisimple, complexa i de dimensió finita admet un sistema simple d’arrels, un resultat fonamental en la teoria de l’estructura de les àlgebres de Lie semisimples.
Magnetically-altered eg-orbital occupancy to boost the two-electron oxygen reduction electrocatalysis for faster water decontamination
(Elsevier B.V., 2025-02-11) Jing, Baojian; Zhang, Qingshan; Liu, Minghui; Yang, Shilin; Zhang, Jiayu; Qiu, Shan; Sirés Sadornil, Ignacio; Deng, Fengxia
Herein, we demonstrate how the magnetic modulation of the eg-orbital occupancy of Co ions can become a novel approach to enhance the two-electron oxygen reduction reaction (2e- ORR) for advanced wastewater treatment by electro-Fenton (EF) process. Nickel foam (NF) substrate coated with ZIF67 was prepared and thermally treated to obtain ferromagnetic NF-supported Co/Co3O4 cathodes, whose response to a small magnetic field was further evaluated in EF. The results show an increase of 62.3 % increase in H2O2 yield and 46.5 % in faradaic efficiency, improving the degradation rate of the antibiotic sulfamerazine from 77.2 % to 92.4 % in 120 min. This enhancement is attributed to improved electron transfer facilitated by the alteration of eg-orbital occupancy of Co ions caused by spin regulation under the magnetic field, rather than to mass transport effects. Therefore, magnetic modulation of the eg-orbital occupancy emerges as a promising strategy to boost EF process for more efficient water decontamination.
Five centuries of lake deoxygenation and microbial shifts revealed by sedimentary DNA
(Elsevier, 2026-03-01) Vegas Vilarrúbia, Teresa Elena; Sacristán Soriano, Oriol; Borrego, Carles M.; Corella, Juan Pablo; Errea, Paz; Buchaca, Teresa; Rull, Valentí
Expanding global deoxygenation is a major ecological crisis. Oxygen loss in aquatic systems is reshaping ecosystems, altering biodiversity, nutrient cycles, and ecosystem functioning. Microbial communities, as key biogeochemical mediators, are highly responsive to oxygen availability, yet their long-term trajectories under sustained oxygen loss remain poorly understood. This limits our ability to anticipate ecosystem responses to climate-driven deoxygenation. We reconstructed five centuries of microbial and oxygen dynamics in meromictic Lake Montcortès (southern Pyrenees) using sedimentary deoxyribonucleic acid (DNA) and X-ray fluorescence spectroscopy of varved sediments. Average volume of anoxic water (VAW) increased from 545,560 m³ in the preindustrial period (1500–1750 Common Era) to 558,692 m³ after 1905, with greater persistence from 1937 onward. Instrumental data (2013–2020) confirmed this trend, with VAW peaking at 1018,400 m³ and anoxia reaching 11.5 m and anoxia occurring below 11.5 m depth. Microbial communities gradually shifted from suboxic-adapted taxa to anaerobic assemblages dominated by sulfate-reducing bacteria and methanogenic archaea, reflecting historical anthropogenic pressures and recent warming-driven anoxia. Despite compositional change, core functional traits persisted, with no evidence of abrupt regime shifts indicating microbial resilience under long-term redox stratification. The timing of stratification intensification in Lake Montcortès coincides with the mid-20th century onset of the Anthropocene, as defined by the Anthropocene Working Group. Its annually resolved varved sediments, combined with clear geochemical and microbial signals of anthropogenic impact, support its proposal as a candidate Global Boundary Stratotype Section and Point Global (GSSP or “golden spike”).