Master of Science in Advanced Mathematics and Mathematical Engineering (MAMME)
http://hdl.handle.net/2099.1/14224
2024-03-19T04:16:33ZSpectral gap of generalized MIT bag models
http://hdl.handle.net/2117/400748
Spectral gap of generalized MIT bag models
Duran Lamiel, Joaquim
We study some spectral properties of generalized MIT bag models. These are a family of Dirac operators $\{H_\tau\}_{\tau \in \mathbb R\cup \{-\infty, +\infty\}}$ used in the field of relativistic quantum mechanics to model confinement of quarks in hadrons, and their energies are related with the spectra of such operators. Their lowest positive eigenvalue is of special interest, and in \cite{Mas2022} it was conjectured that it is minimal for a ball among all domains of the same volume. In this work we prove that the conjecture holds true for corona domains of relatively small hole. Moreover, motivated by some open questions presented in \cite{Mas2022}, in this work we also study the convergence in several resolvent senses of $H_\tau$ as $\tau$ varies. More specifically, we show strong resolvent convergence of $H_\tau$ to $H_{\pm \infty}$ as $\tau \to \pm \infty$, we justify that one cannot improve this to norm resolvent convergence as $\tau \to \pm \infty$, and we show norm resolvent convergence of $H_\tau$ to $H_{\tau_0}$ as $\tau \to \tau_0$, for $\tau_0\in \mathbb R$. These results are new and will be sent for publication in an indexed journal.
2024-02-01T10:34:13ZDuran Lamiel, JoaquimWe study some spectral properties of generalized MIT bag models. These are a family of Dirac operators $\{H_\tau\}_{\tau \in \mathbb R\cup \{-\infty, +\infty\}}$ used in the field of relativistic quantum mechanics to model confinement of quarks in hadrons, and their energies are related with the spectra of such operators. Their lowest positive eigenvalue is of special interest, and in \cite{Mas2022} it was conjectured that it is minimal for a ball among all domains of the same volume. In this work we prove that the conjecture holds true for corona domains of relatively small hole. Moreover, motivated by some open questions presented in \cite{Mas2022}, in this work we also study the convergence in several resolvent senses of $H_\tau$ as $\tau$ varies. More specifically, we show strong resolvent convergence of $H_\tau$ to $H_{\pm \infty}$ as $\tau \to \pm \infty$, we justify that one cannot improve this to norm resolvent convergence as $\tau \to \pm \infty$, and we show norm resolvent convergence of $H_\tau$ to $H_{\tau_0}$ as $\tau \to \tau_0$, for $\tau_0\in \mathbb R$. These results are new and will be sent for publication in an indexed journal.Development of predictive flight fuel consumption models
http://hdl.handle.net/2117/400745
Development of predictive flight fuel consumption models
García Hernàndez, Sílvia
Fuel consumption is a crucial consideration in the aviation industry. This last one, integral to global connectivity, faces significant environmental concerns due to the escalating demand for air travel and its substantial contribution to greenhouse gas emissions.
In this thesis, we present predictive models calculating gate-to-gate fuel consumption, using simple variables such as flight distance, and taking into account the available number seats for each aircraft, in contrast with other flight consumption calculators. The main goal of this work is to construct an indicator that can be used to compare emissions with other travel alternatives. Specifically, we develop the theoretical framework for the presented models and showcase their results. Using the model with best accuracy, a LightGBM model, we demonstrate a real-world application by conducting a CO2 emission comparison between flight and rail routes.
2024-02-01T10:28:28ZGarcía Hernàndez, SílviaFuel consumption is a crucial consideration in the aviation industry. This last one, integral to global connectivity, faces significant environmental concerns due to the escalating demand for air travel and its substantial contribution to greenhouse gas emissions.
In this thesis, we present predictive models calculating gate-to-gate fuel consumption, using simple variables such as flight distance, and taking into account the available number seats for each aircraft, in contrast with other flight consumption calculators. The main goal of this work is to construct an indicator that can be used to compare emissions with other travel alternatives. Specifically, we develop the theoretical framework for the presented models and showcase their results. Using the model with best accuracy, a LightGBM model, we demonstrate a real-world application by conducting a CO2 emission comparison between flight and rail routes.Network creation games: NE graphs for high-price links
http://hdl.handle.net/2117/400301
Network creation games: NE graphs for high-price links
Gómez Arias, Mario
Network Creation Games (NCGs) are a long-standing, well-studied setting in Algorithmic Game Theory. In this game, the players consist of nodes in a graph that purchase other nodes at cost α, a parameter of the game. In the SumGame variant of the game, players incur a cost that depends on the total number of purchased nodes and their distances to the rest of the nodes (or players) in the resulting network. Since their appearance, a large research effort has been made to understand the arising network topologies. Moreover, when the game is tied to a bound in the possible values of α, interesting topologies such as trees arise.
Despite the underlying conceptual simplicity of NCGs, different conjectures still remain open. In all cases, few topologies have been discovered to assert the validity of these conjectures. Thus this work seeks to extend the set of topologies that confirm the range of α for which the PoA is not known to be constant and contribute insights to the biconnected component conjecture. Purchasing links within this range of values for α is commonly known as high-price links.
In the last decade, Artificial Intelligence (AI) systems have demonstrated massive potential in tackling different combinatorial problems. As an example, AlphaGo learned to master the Chinese game of Go and it even defeated a human Go world champion back in 2016. At the core, AlphaGo combines deep neural networks with advanced search algorithms. The latter include the so-called Monte Carlo Tree Search (MCTS), an efficient search algorithm that combines traditional Monte Carlo simulations with ideas from reinforcement learning.
In this thesis, we apply this relatively modern AI approach, namely MCTS, to different long lasting problems in NCGs. More precisely, the MCTS framework is adapted to the problem of finding NE in NCGs, application which departs considerably from previous applications such as the game of Go or other board games.
2024-01-25T15:24:19ZGómez Arias, MarioNetwork Creation Games (NCGs) are a long-standing, well-studied setting in Algorithmic Game Theory. In this game, the players consist of nodes in a graph that purchase other nodes at cost α, a parameter of the game. In the SumGame variant of the game, players incur a cost that depends on the total number of purchased nodes and their distances to the rest of the nodes (or players) in the resulting network. Since their appearance, a large research effort has been made to understand the arising network topologies. Moreover, when the game is tied to a bound in the possible values of α, interesting topologies such as trees arise.
Despite the underlying conceptual simplicity of NCGs, different conjectures still remain open. In all cases, few topologies have been discovered to assert the validity of these conjectures. Thus this work seeks to extend the set of topologies that confirm the range of α for which the PoA is not known to be constant and contribute insights to the biconnected component conjecture. Purchasing links within this range of values for α is commonly known as high-price links.
In the last decade, Artificial Intelligence (AI) systems have demonstrated massive potential in tackling different combinatorial problems. As an example, AlphaGo learned to master the Chinese game of Go and it even defeated a human Go world champion back in 2016. At the core, AlphaGo combines deep neural networks with advanced search algorithms. The latter include the so-called Monte Carlo Tree Search (MCTS), an efficient search algorithm that combines traditional Monte Carlo simulations with ideas from reinforcement learning.
In this thesis, we apply this relatively modern AI approach, namely MCTS, to different long lasting problems in NCGs. More precisely, the MCTS framework is adapted to the problem of finding NE in NCGs, application which departs considerably from previous applications such as the game of Go or other board games.Interpolation and geometrical transition of minimal surfaces with differential adhesion
http://hdl.handle.net/2117/398210
Interpolation and geometrical transition of minimal surfaces with differential adhesion
Pascual Mellado, Roger
The observed shape of cells can be understood and simplified to a contractile surface with a volume
preserving constraint. Vertex models are a suitable approach to simulate such systems and attempt to
numerically simulate their behavior by imposing mechanical equilibrium over time. In parallel, the similarities
between cell geometry and minimal surfaces have been investigated in depth. However, among other
differences, the latter have a volume constraint and exhibit the capacity to regulate their adhesion to
different orientations and neighboring cells. In this project, we employ both approaches, coming from
seemingly disconnected areas, to deduce the equivalence between vertex models (minimization of the energy
functional) and constrained minimal surfaces (discrete Laplacian formulation). We provide problems and
parameters for both to produce the same single-cell shape configuration and highlight their similarities and
differences.
2023-12-18T12:02:15ZPascual Mellado, RogerThe observed shape of cells can be understood and simplified to a contractile surface with a volume
preserving constraint. Vertex models are a suitable approach to simulate such systems and attempt to
numerically simulate their behavior by imposing mechanical equilibrium over time. In parallel, the similarities
between cell geometry and minimal surfaces have been investigated in depth. However, among other
differences, the latter have a volume constraint and exhibit the capacity to regulate their adhesion to
different orientations and neighboring cells. In this project, we employ both approaches, coming from
seemingly disconnected areas, to deduce the equivalence between vertex models (minimization of the energy
functional) and constrained minimal surfaces (discrete Laplacian formulation). We provide problems and
parameters for both to produce the same single-cell shape configuration and highlight their similarities and
differences.Reconstructing a dynamic world: What is next?
http://hdl.handle.net/2117/398208
Reconstructing a dynamic world: What is next?
Gastón Codony, Fernando
In this thesis, we study the problem denoted as Non-Rigid Structure from Motion and tackle the
two distinct parts of the problem: motion and shape estimation together with the corresponding
temporal segmentation into actions of the body. For the motion estimation, we implement a
Single Rotation algorithm (SRA) based on the Weiszfeld algorithm for the median of points in Rn .
We use Brand’s method to obtain a full corrective matrix and several estimates for the rotation
matrices but find that the first column triplet obtains better results than any of the triplets found
by Brand’s method, making SRA pointless. For the shape factor, we make an assumption that
shapes lie in a temporal union of subspaces. We perform Sparse Subspace Clustering to jointly
reconstruct the shape matrix while computing an affinity matrix that can be used to cluster frames
of the input data according to which subspace they belong to.; En esta tesis, estudiamos el problema de Non-Rigid Structure from Motion y abordamos
dos partes distintas del problema: estimación de movimiento y forma junto con la correspondiente
segmentación temporal en acciones del cuerpo no-rígido. Para la estimación del movimiento, implementamos un
Single Rotation Averaging (SRA) basado en el algoritmo de Weiszfeld para la mediana de puntos en R^n.
Usamos el método de Brand para obtener una matriz correctiva completa y varias estimaciones para las matrices de rotación,
pero encontramos que la primera tripleta de columnas obtiene mejores resultados que cualquiera de las tripletas encontrados
por el método de Brand, haciendo que SRA sea inútil. Para el factor de forma, asumimos que
las formas se encuentran en una unión temporal de subespacios. Realizamos Sparse Subspace Clustering para conjuntamente
reconstruir la matriz de forma mientras se calcula una matriz de afinidad que se puede utilizar para agrupar los frames
de los datos de entrada según a qué subespacio pertenecen.; En aquesta tesi, estudiem el problema de Non-Rigid Structure from Motion i abordem
dues parts diferents del problema: estimació de moviment i forma juntament amb la corresponent
segmentació temporal en accions del cos no rígid. Per a l'estimació del moviment, implementem un
Single Rotation Averaging (SRA) basat en l'algorisme de Weiszfeld per a la mitjana de punts a R^n.
Fem servir el mètode de Brand per obtenir una matriu correctiva completa i diverses estimacions per a les matrius de rotació,
però trobem que la primera tripleta de columnes obté millors resultats que qualsevol de les tripletes trobades
pel mètode de Brand, fent que SRA sigui inútil. Per al factor de forma, assumim que
les formes es troben en una unió temporal de subespais. Realitzem Sparse Subspace Clustering per conjuntament
reconstruir la matriu, mentre es calcula una matriu d'afinitat que es pot utilitzar per agrupar els frames
dades d'entrada segons a quin subespai pertanyen.
2023-12-18T11:49:19ZGastón Codony, FernandoIn this thesis, we study the problem denoted as Non-Rigid Structure from Motion and tackle the
two distinct parts of the problem: motion and shape estimation together with the corresponding
temporal segmentation into actions of the body. For the motion estimation, we implement a
Single Rotation algorithm (SRA) based on the Weiszfeld algorithm for the median of points in Rn .
We use Brand’s method to obtain a full corrective matrix and several estimates for the rotation
matrices but find that the first column triplet obtains better results than any of the triplets found
by Brand’s method, making SRA pointless. For the shape factor, we make an assumption that
shapes lie in a temporal union of subspaces. We perform Sparse Subspace Clustering to jointly
reconstruct the shape matrix while computing an affinity matrix that can be used to cluster frames
of the input data according to which subspace they belong to.
En esta tesis, estudiamos el problema de Non-Rigid Structure from Motion y abordamos
dos partes distintas del problema: estimación de movimiento y forma junto con la correspondiente
segmentación temporal en acciones del cuerpo no-rígido. Para la estimación del movimiento, implementamos un
Single Rotation Averaging (SRA) basado en el algoritmo de Weiszfeld para la mediana de puntos en R^n.
Usamos el método de Brand para obtener una matriz correctiva completa y varias estimaciones para las matrices de rotación,
pero encontramos que la primera tripleta de columnas obtiene mejores resultados que cualquiera de las tripletas encontrados
por el método de Brand, haciendo que SRA sea inútil. Para el factor de forma, asumimos que
las formas se encuentran en una unión temporal de subespacios. Realizamos Sparse Subspace Clustering para conjuntamente
reconstruir la matriz de forma mientras se calcula una matriz de afinidad que se puede utilizar para agrupar los frames
de los datos de entrada según a qué subespacio pertenecen.
En aquesta tesi, estudiem el problema de Non-Rigid Structure from Motion i abordem
dues parts diferents del problema: estimació de moviment i forma juntament amb la corresponent
segmentació temporal en accions del cos no rígid. Per a l'estimació del moviment, implementem un
Single Rotation Averaging (SRA) basat en l'algorisme de Weiszfeld per a la mitjana de punts a R^n.
Fem servir el mètode de Brand per obtenir una matriu correctiva completa i diverses estimacions per a les matrius de rotació,
però trobem que la primera tripleta de columnes obté millors resultats que qualsevol de les tripletes trobades
pel mètode de Brand, fent que SRA sigui inútil. Per al factor de forma, assumim que
les formes es troben en una unió temporal de subespais. Realitzem Sparse Subspace Clustering per conjuntament
reconstruir la matriu, mentre es calcula una matriu d'afinitat que es pot utilitzar per agrupar els frames
dades d'entrada segons a quin subespai pertanyen.Succinct encodings of graphs and other combinatorial structures
http://hdl.handle.net/2117/398207
Succinct encodings of graphs and other combinatorial structures
Lumbreras Navarro, Raúl
Considerem el següent problema. Donada una família de grafs amb un cert nombre de grafs amb n vèrtexs, volem trobar una representació per a qualsevol graf dins la família fent servir un nombre òptim de bits. És a dir, sigui t el nombre de grafs amb n vèrtexs a la família, volem codificar qualsevol d'aquests grafs fent servir log(t) + o(log(t)) bits, que és el mínim necessari asimptòticament. El primer tipus de famílies que considerem per codificar són diferents tipus de famílies d'arbres, per exemple, arbres ordenats, arbres k-aris i arbres no ordenats. Expliquem dos mètodes per codificar famílies d'arbres. El primer és per a famílies específiques d'arbres i el segon és per a famílies generals d'arbres. A continuació, també considerem les famílies més generals de grafs que siguin tancades per subgrafs induïts i que tinguin separadors petits. Aquestes famílies inclouen grafs planars i qualsevol família de grafs tancada per menors. També presentem una aplicació detallada d'aquesta codificació.; Consideramos el siguiente problema. Dada una familia de grafos con un cierto número de grafos de n vértices, queremos encontrar una representación para cualquier grafo dentro de la familia utilizando un número de bits optimo. Es decir, sea t el número de grafos de n vértices en la familia, queremos codificar cualquiera de dichos grafos usando log(t) + o(log(t)) bits, que es el mínimo necesario asintóticamente. El primer tipo de familias que consideramos codificar son distintos tipos de familias de árboles, por ejemplo, árboles ordenados, árboles k-arios y árboles no ordenados. Explicamos dos métodos para codificar familias de árboles. El primero es para familias específicas de árboles y el segundo es para familias generales de árboles. A continuación, también consideramos las familias más generales de grafos que son cerradas por subgrafos inducidos y tienen separadores pequeños. Estas familias incluyen grafos planares y cualquier familia de grafos cerrada por menores. También presentamos una aplicación detallada de esta codificación.; We consider the following problem. Given a family of graphs with a certain number of n-vertex graphs, we want to find a representation for any graph within the family using an information-theoretical number of bits. That is to say, if there are a certain number of n-vertex graphs t in the family, then, we want to encode any of such graphs using log(t) + o(log(t)) bits, which is the asymptotically minimum needed. The first type of families we consider encoding are different types of tree families, for instance, ordered trees, k-ary trees and unordered trees. We explain two methods for encoding such families. The first one for specific families of trees and the second one for general families of trees. Then, we also consider the more general families of graphs which are closed under induced sub-graphs and have small separators. These families include planar graphs and any minor-closed family of graphs. We also present a detailed implementation of such encoding.
2023-12-18T11:45:44ZLumbreras Navarro, RaúlConsiderem el següent problema. Donada una família de grafs amb un cert nombre de grafs amb n vèrtexs, volem trobar una representació per a qualsevol graf dins la família fent servir un nombre òptim de bits. És a dir, sigui t el nombre de grafs amb n vèrtexs a la família, volem codificar qualsevol d'aquests grafs fent servir log(t) + o(log(t)) bits, que és el mínim necessari asimptòticament. El primer tipus de famílies que considerem per codificar són diferents tipus de famílies d'arbres, per exemple, arbres ordenats, arbres k-aris i arbres no ordenats. Expliquem dos mètodes per codificar famílies d'arbres. El primer és per a famílies específiques d'arbres i el segon és per a famílies generals d'arbres. A continuació, també considerem les famílies més generals de grafs que siguin tancades per subgrafs induïts i que tinguin separadors petits. Aquestes famílies inclouen grafs planars i qualsevol família de grafs tancada per menors. També presentem una aplicació detallada d'aquesta codificació.
Consideramos el siguiente problema. Dada una familia de grafos con un cierto número de grafos de n vértices, queremos encontrar una representación para cualquier grafo dentro de la familia utilizando un número de bits optimo. Es decir, sea t el número de grafos de n vértices en la familia, queremos codificar cualquiera de dichos grafos usando log(t) + o(log(t)) bits, que es el mínimo necesario asintóticamente. El primer tipo de familias que consideramos codificar son distintos tipos de familias de árboles, por ejemplo, árboles ordenados, árboles k-arios y árboles no ordenados. Explicamos dos métodos para codificar familias de árboles. El primero es para familias específicas de árboles y el segundo es para familias generales de árboles. A continuación, también consideramos las familias más generales de grafos que son cerradas por subgrafos inducidos y tienen separadores pequeños. Estas familias incluyen grafos planares y cualquier familia de grafos cerrada por menores. También presentamos una aplicación detallada de esta codificación.
We consider the following problem. Given a family of graphs with a certain number of n-vertex graphs, we want to find a representation for any graph within the family using an information-theoretical number of bits. That is to say, if there are a certain number of n-vertex graphs t in the family, then, we want to encode any of such graphs using log(t) + o(log(t)) bits, which is the asymptotically minimum needed. The first type of families we consider encoding are different types of tree families, for instance, ordered trees, k-ary trees and unordered trees. We explain two methods for encoding such families. The first one for specific families of trees and the second one for general families of trees. Then, we also consider the more general families of graphs which are closed under induced sub-graphs and have small separators. These families include planar graphs and any minor-closed family of graphs. We also present a detailed implementation of such encoding.Variants and applications of Gehring's lemma
http://hdl.handle.net/2117/398206
Variants and applications of Gehring's lemma
Navarro Arroyo, Vicent
In this thesis, we present and prove the celebrated Gehring's Lemma in $\mathbb{R}^n$, that unveils a self-improving property of reverse Hölder inequalities, considering inhomogeneity. Subsequently, we apply the former lemma to demonstrate the Meyers' estimate, which provides insight into the self-improving regularity of solutions of elliptic PDEs.
2023-12-18T11:28:18ZNavarro Arroyo, VicentIn this thesis, we present and prove the celebrated Gehring's Lemma in $\mathbb{R}^n$, that unveils a self-improving property of reverse Hölder inequalities, considering inhomogeneity. Subsequently, we apply the former lemma to demonstrate the Meyers' estimate, which provides insight into the self-improving regularity of solutions of elliptic PDEs.On the number of drawings of a combinatorial triangulations
http://hdl.handle.net/2117/396459
On the number of drawings of a combinatorial triangulations
Cruces Mateo, Belen
Aquesta tesi explora la relació entre triangulacions combinatòries i geomètriques en geometria discreta i combinatòria. Per triangulacions combinatòries ens referim a grafs, mentre que per triangulacions geomètriques ens referim a dibuixos de grafs com a plans maximals amb línies rectes com a arestes sobre un conjunt de punts fixat al pla. Estudiem de quantes maneres es pot traçar una triangulació combinatòria com a triangulació geomètrica sobre un conjunt de punts donat. La nostra contribució central és demostrar que una triangulació combinatòria fixa amb n vèrtexs es pot dibuixar d'almenys 1,31^n maneres en un conjunt de n punts com a diferents triangulacions geomètriques. També analitzem els límits superiors i una versió acolorida daquest problema. L'enfocament suggerit pot ajudar a avançar en la resolució del problema obert per limitar el nombre de triangulacions geomètriques.
A més, aprofundim en fonaments històrics, com el treball de Tutte, que proporciona el nombre exacte de triangulacions combinatòries amb n vèrtexs.; Esta tesis explora la relación entre triangulaciones combinatorias y geométricas en geometría discreta y combinatoria. Con triangulaciones combinatorias nos referimos a grafos, mientras que con triangulaciones geométricas nos referimos a dibujos de grafos como planos maximales con líneas rectas como aristas sobre un conjunto de puntos fijado en el plano. Estudiamos de cuántas maneras se puede trazar una triangulación combinatoria como triangulación geométrica sobre un conjunto de puntos dado. Nuestra contribución central es demostrar que una triangulación combinatoria fija con n vértices se puede dibujar de al menos 1,31^n maneras en un conjunto de n puntos como diferentes triangulaciones geométricas. También analizamos los límites superiores y una versión coloreada de este problema. El enfoque sugerido puede ayudar a avanzar en la resolución del problema abierto para limitar el número de triangulaciones geométricas.
Además, profundizamos en fundamentos históricos, como el trabajo de Tutte, que proporciona el número exacto de triangulaciones combinatorias con n vértices.; This thesis explores the intricate relationship between combinatorial and geometric triangulations in discrete and combinatorial geometry. With combinatorial triangulations we refer to graphs, while with geometric triangulations we refer to maximal planar straight-line drawings on a point set in the plane. We study in how many ways can a combinatorial triangulation be drawn as geometric triangulation on a given point set. Our central contribution is proving that a fixed combinatorial triangulation with n vertices can be drawn in at least 1.31^n ways in a set of n points as different geometric triangulations. We also discuss upper bounds and a colored version of this problem. The suggested approach may help to advance the resolution of the open problem to bound the number of geometric triangulations.
Additionally, we delve into historical foundations, such as Tutte's work, which provides the exact number of combinatorial triangulations with n vertices.
2023-11-15T14:16:28ZCruces Mateo, BelenAquesta tesi explora la relació entre triangulacions combinatòries i geomètriques en geometria discreta i combinatòria. Per triangulacions combinatòries ens referim a grafs, mentre que per triangulacions geomètriques ens referim a dibuixos de grafs com a plans maximals amb línies rectes com a arestes sobre un conjunt de punts fixat al pla. Estudiem de quantes maneres es pot traçar una triangulació combinatòria com a triangulació geomètrica sobre un conjunt de punts donat. La nostra contribució central és demostrar que una triangulació combinatòria fixa amb n vèrtexs es pot dibuixar d'almenys 1,31^n maneres en un conjunt de n punts com a diferents triangulacions geomètriques. També analitzem els límits superiors i una versió acolorida daquest problema. L'enfocament suggerit pot ajudar a avançar en la resolució del problema obert per limitar el nombre de triangulacions geomètriques.
A més, aprofundim en fonaments històrics, com el treball de Tutte, que proporciona el nombre exacte de triangulacions combinatòries amb n vèrtexs.
Esta tesis explora la relación entre triangulaciones combinatorias y geométricas en geometría discreta y combinatoria. Con triangulaciones combinatorias nos referimos a grafos, mientras que con triangulaciones geométricas nos referimos a dibujos de grafos como planos maximales con líneas rectas como aristas sobre un conjunto de puntos fijado en el plano. Estudiamos de cuántas maneras se puede trazar una triangulación combinatoria como triangulación geométrica sobre un conjunto de puntos dado. Nuestra contribución central es demostrar que una triangulación combinatoria fija con n vértices se puede dibujar de al menos 1,31^n maneras en un conjunto de n puntos como diferentes triangulaciones geométricas. También analizamos los límites superiores y una versión coloreada de este problema. El enfoque sugerido puede ayudar a avanzar en la resolución del problema abierto para limitar el número de triangulaciones geométricas.
Además, profundizamos en fundamentos históricos, como el trabajo de Tutte, que proporciona el número exacto de triangulaciones combinatorias con n vértices.
This thesis explores the intricate relationship between combinatorial and geometric triangulations in discrete and combinatorial geometry. With combinatorial triangulations we refer to graphs, while with geometric triangulations we refer to maximal planar straight-line drawings on a point set in the plane. We study in how many ways can a combinatorial triangulation be drawn as geometric triangulation on a given point set. Our central contribution is proving that a fixed combinatorial triangulation with n vertices can be drawn in at least 1.31^n ways in a set of n points as different geometric triangulations. We also discuss upper bounds and a colored version of this problem. The suggested approach may help to advance the resolution of the open problem to bound the number of geometric triangulations.
Additionally, we delve into historical foundations, such as Tutte's work, which provides the exact number of combinatorial triangulations with n vertices.Tècniques geomètriques en monogeneïcitat
http://hdl.handle.net/2117/395997
Tècniques geomètriques en monogeneïcitat
Pedret Martínez, Francesc
By the primitive element theorem, any number field K of degree n can be written as Q(α) for some α in K. However, the analogous affirmation is not always true in the case of the ring of integers. When the ring of integers of K is Z[α], we say K is monogenic. Every cubic number field determines a non-trivial F3-orbit in H^1(Q,E[φ]), where E is the elliptic curve and φ is a certain 3-isogeny. In this work, we review the proof of this fact and use it to obtain bounds on the number of monogenic cubic number fields of discriminant D in terms of the Mordell-Weil group of E^D : Y^2 = 4X^3+D. We also compute a general expression for the cocycle associated to any pure cubic number field of Dedekind type I, which we use to characterize the sum of two such cocycles.
2023-11-08T12:07:35ZPedret Martínez, FrancescBy the primitive element theorem, any number field K of degree n can be written as Q(α) for some α in K. However, the analogous affirmation is not always true in the case of the ring of integers. When the ring of integers of K is Z[α], we say K is monogenic. Every cubic number field determines a non-trivial F3-orbit in H^1(Q,E[φ]), where E is the elliptic curve and φ is a certain 3-isogeny. In this work, we review the proof of this fact and use it to obtain bounds on the number of monogenic cubic number fields of discriminant D in terms of the Mordell-Weil group of E^D : Y^2 = 4X^3+D. We also compute a general expression for the cocycle associated to any pure cubic number field of Dedekind type I, which we use to characterize the sum of two such cocycles.Limit distribution of Hodge spectral exponents of plane curve singularities
http://hdl.handle.net/2117/394605
Limit distribution of Hodge spectral exponents of plane curve singularities
Gómez López, Roger
K. Saito va formular la pregunta de si una distribució continua és el límit de la distribució dels exponents espectrals de Hodge a l'interval [0,1) d'una hipersuperfície quan aquesta es mou en un sentit que s'ha de precisar. Ell ho va demostrar per a corbes irreductibles amb un límit molt específic. Aquest projecte es centra en el cas de singularitats de corbes planes. S'exploraran les diferents formes d'arribar a una distribució límit d'aquests invariants.; K. Saito formuló la pregunta de si una distribución continua es el límite de la distribución de los exponentes espectrales de Hodge en el intervalo [0,1) de una hipersuperficie cuando esta se mueve en un sentido que se ha de precisar. Él lo demostró para curvas irreducibles con un límite muy específico. Este proyecto se centra en el caso de singularidades de curvas planas. Se explorarán las diferentes formas de llegar a una distribución límite de estos invariantes.; K. Saito formulated the question whether a continuous distribution is the limit of the distribution of the Hodge spectral exponents in the interval [0,1) of a hypersurface as this hypersurface moves in a sense that has to be made precise. He proved it for irreducible plane curves with a very specific limit formulation. This project will focus on the case of irreducible plane curve singularities. Different formulations of achieving the limit of the distribution of these invariants will be explored.
2023-10-04T12:36:23ZGómez López, RogerK. Saito va formular la pregunta de si una distribució continua és el límit de la distribució dels exponents espectrals de Hodge a l'interval [0,1) d'una hipersuperfície quan aquesta es mou en un sentit que s'ha de precisar. Ell ho va demostrar per a corbes irreductibles amb un límit molt específic. Aquest projecte es centra en el cas de singularitats de corbes planes. S'exploraran les diferents formes d'arribar a una distribució límit d'aquests invariants.
K. Saito formuló la pregunta de si una distribución continua es el límite de la distribución de los exponentes espectrales de Hodge en el intervalo [0,1) de una hipersuperficie cuando esta se mueve en un sentido que se ha de precisar. Él lo demostró para curvas irreducibles con un límite muy específico. Este proyecto se centra en el caso de singularidades de curvas planas. Se explorarán las diferentes formas de llegar a una distribución límite de estos invariantes.
K. Saito formulated the question whether a continuous distribution is the limit of the distribution of the Hodge spectral exponents in the interval [0,1) of a hypersurface as this hypersurface moves in a sense that has to be made precise. He proved it for irreducible plane curves with a very specific limit formulation. This project will focus on the case of irreducible plane curve singularities. Different formulations of achieving the limit of the distribution of these invariants will be explored.