Master of Science in Advanced Mathematics and Mathematical Engineering (MAMME)
http://hdl.handle.net/2099.1/14224
2019-12-07T04:31:32ZA minimal bound and divergence case on the elliptic curves over finite fields
http://hdl.handle.net/2117/171353
A minimal bound and divergence case on the elliptic curves over finite fields
Sota, Antonino
The main goal of this thesis is the study of elliptic curves over finite fields and the number of points of them. A study of interest is to analyze and find the cases when the difference #E(F_q^n )-#E(F_q) vanishes, where #E(F_q) denotes the number of points of an elliptic curve E over the finite field F_q. Moreover we can show that the sequence a_n=#E(F_q^n ) - #E(F_q)=q^n-q for n odd, q prime of the form q=4k+3 where k is a nonnegative integer and any elliptic curve of the form E : y^2=x^3+tx over F_q. Also a_n=#E(F_q^n ) - #E(F_q)=q^n-q for q prime of the form 3k+2 where k is a positive integer, n odd and any elliptic curve of the form E: y^2=x^3+b over F_q.
2019-10-31T13:13:57ZSota, AntoninoThe main goal of this thesis is the study of elliptic curves over finite fields and the number of points of them. A study of interest is to analyze and find the cases when the difference #E(F_q^n )-#E(F_q) vanishes, where #E(F_q) denotes the number of points of an elliptic curve E over the finite field F_q. Moreover we can show that the sequence a_n=#E(F_q^n ) - #E(F_q)=q^n-q for n odd, q prime of the form q=4k+3 where k is a nonnegative integer and any elliptic curve of the form E : y^2=x^3+tx over F_q. Also a_n=#E(F_q^n ) - #E(F_q)=q^n-q for q prime of the form 3k+2 where k is a positive integer, n odd and any elliptic curve of the form E: y^2=x^3+b over F_q.Numerical model of cardiac electromechanics
http://hdl.handle.net/2117/171347
Numerical model of cardiac electromechanics
Wieczorek I Masdeu, Nora
In this project, we develop a Finite Element Method (FEM) formulation that solves the cardiac electrophysiological problem of a three dimensional piece of tissue. This problem is modeled by an electromechanical model that includes the activation of a tension that depends on the cell's transmembrane potential and induces the contraction of the tissue. After applying an implicit time discretization, we end with a nonlinear system that depends on the position and potential at each node. This system is solved using the Newton-Raphson's method at each time iteration. Using this resolution methodology, we present a full implicit scheme. We also implement a faster and less accurate way of solving the coupled system with a staggered scheme: first computing the change of potential, and then actualizing the position of every node. Then, we simulate the electrophysiological model to observe the effect of the grid affects on the results. Finally, using the staggered algorithm, we simulate the propagation of a plane wave and the subsequent tissue contraction.
2019-10-31T12:57:18ZWieczorek I Masdeu, NoraIn this project, we develop a Finite Element Method (FEM) formulation that solves the cardiac electrophysiological problem of a three dimensional piece of tissue. This problem is modeled by an electromechanical model that includes the activation of a tension that depends on the cell's transmembrane potential and induces the contraction of the tissue. After applying an implicit time discretization, we end with a nonlinear system that depends on the position and potential at each node. This system is solved using the Newton-Raphson's method at each time iteration. Using this resolution methodology, we present a full implicit scheme. We also implement a faster and less accurate way of solving the coupled system with a staggered scheme: first computing the change of potential, and then actualizing the position of every node. Then, we simulate the electrophysiological model to observe the effect of the grid affects on the results. Finally, using the staggered algorithm, we simulate the propagation of a plane wave and the subsequent tissue contraction.An introduction to the Langlands Conjectures
http://hdl.handle.net/2117/171342
An introduction to the Langlands Conjectures
Felipe i Alsina, Marc
In this thesis, we take a look into a generalization of Local Class Field Theory (LCFT), called the Local Langlands Conjecture, which concerns about identifying special representations of the Weil group, a subgroup of the absolute Galois group of a field, with some representations of the general linear group with coefficients in that field. We study deeply the $n=1$ case, which corresponds to LCFT, and then we construct explicit elements of both sides of the bijection for the case $n=2$. Finally, we state the difficulties to formulate the analogous conjecture for global fields, the Global Langlands Conjecture.
2019-10-31T12:51:02ZFelipe i Alsina, MarcIn this thesis, we take a look into a generalization of Local Class Field Theory (LCFT), called the Local Langlands Conjecture, which concerns about identifying special representations of the Weil group, a subgroup of the absolute Galois group of a field, with some representations of the general linear group with coefficients in that field. We study deeply the $n=1$ case, which corresponds to LCFT, and then we construct explicit elements of both sides of the bijection for the case $n=2$. Finally, we state the difficulties to formulate the analogous conjecture for global fields, the Global Langlands Conjecture.Heegner's lemma
http://hdl.handle.net/2117/171338
Heegner's lemma
Jofre Senciales, Jordi
Es un treball que explica una mica com Heegner va fer per trobar una familia de nombre congruents. Tambe explico les eienes que es necessiten de class field theory i modular functions. També explica el metode de descents de Fermat que és el que Heegner fa servir per doemostrar els seu teorema
2019-10-31T12:47:39ZJofre Senciales, JordiEs un treball que explica una mica com Heegner va fer per trobar una familia de nombre congruents. Tambe explico les eienes que es necessiten de class field theory i modular functions. També explica el metode de descents de Fermat que és el que Heegner fa servir per doemostrar els seu teoremaIsogeny-Based Post-Quantum Cryptography
http://hdl.handle.net/2117/171336
Isogeny-Based Post-Quantum Cryptography
Khandpur Singh, Ashneet
The present thesis focus on one of the post-quantum cryptosystems, in particular, the isogenybased cryptography. Because of its certain properties like the hard problem of computing isogenies between two elliptic curves, it makes it to be one of the few candidates for post-quantum cryptography. We will study the algorithms based on this isogeny-based cryptography and why they are useful against quantum computers. One of these algorithms is the Supersingular Isogeny Diffie Hellman (SIDH), a candidate for substituing ECDH in the Signal protocol to make it quantum safe.
2019-10-31T12:43:35ZKhandpur Singh, AshneetThe present thesis focus on one of the post-quantum cryptosystems, in particular, the isogenybased cryptography. Because of its certain properties like the hard problem of computing isogenies between two elliptic curves, it makes it to be one of the few candidates for post-quantum cryptography. We will study the algorithms based on this isogeny-based cryptography and why they are useful against quantum computers. One of these algorithms is the Supersingular Isogeny Diffie Hellman (SIDH), a candidate for substituing ECDH in the Signal protocol to make it quantum safe.Optimal coordinated motions for two square robots
http://hdl.handle.net/2117/171329
Optimal coordinated motions for two square robots
Ruiz Herrero, Víctor
We find the coordinated motion for two square robots that minimizes the sum of the length of the path of every robot, for every initial and final positions of both robots. We study it in an obstacle-free plane, so the unique constraint is that the robots can not collide.
2019-10-31T12:30:33ZRuiz Herrero, VíctorWe find the coordinated motion for two square robots that minimizes the sum of the length of the path of every robot, for every initial and final positions of both robots. We study it in an obstacle-free plane, so the unique constraint is that the robots can not collide.An Introduction to Polytope Theory through Ehrhart's Theorem
http://hdl.handle.net/2117/171328
An Introduction to Polytope Theory through Ehrhart's Theorem
Cano Córdoba, Filip
A classic introduction to polytope theory is presented, serving as the foundation to develop more advanced theoretical tools, namely the algebra of polyhedra and the use of valuations. The main theoretical objective is the construction of the so called Berline-Vergne valuation. Most of the theoretical development is aimed towards this goal. A little survey on Ehrhart positivity is presented, as well as some calculations that lead to conjecture that generalized permutohedra have positive coefficients in their Ehrhart polynomials. Throughout the thesis three different proofs of Ehrhart's theorem are presented, as an application of the new techniques developed.
2019-10-31T12:19:49ZCano Córdoba, FilipA classic introduction to polytope theory is presented, serving as the foundation to develop more advanced theoretical tools, namely the algebra of polyhedra and the use of valuations. The main theoretical objective is the construction of the so called Berline-Vergne valuation. Most of the theoretical development is aimed towards this goal. A little survey on Ehrhart positivity is presented, as well as some calculations that lead to conjecture that generalized permutohedra have positive coefficients in their Ehrhart polynomials. Throughout the thesis three different proofs of Ehrhart's theorem are presented, as an application of the new techniques developed.Toric Orbifolds
http://hdl.handle.net/2117/171325
Toric Orbifolds
Elkoroaristizabal Peleteiro, Ander
The objective of this essay is to introduce some of the broad theory involving toric varieties, and fit into it some recent advances about toric orbifolds. We first introduce the geometric-combinatorial objects the theory of toric varieties is built upon, and pursue the toric variety of a lattice polytope. Once we are familiar with the basics about toric varieties, we introduce topological orbifolds, a generalization of manifolds that appear naturally when studying toric varieties. Finally, we alter the construction of the toric variety defined by a fan in order to create new orbifolds from non-orbifold toric varieties.
2019-10-31T12:15:07ZElkoroaristizabal Peleteiro, AnderThe objective of this essay is to introduce some of the broad theory involving toric varieties, and fit into it some recent advances about toric orbifolds. We first introduce the geometric-combinatorial objects the theory of toric varieties is built upon, and pursue the toric variety of a lattice polytope. Once we are familiar with the basics about toric varieties, we introduce topological orbifolds, a generalization of manifolds that appear naturally when studying toric varieties. Finally, we alter the construction of the toric variety defined by a fan in order to create new orbifolds from non-orbifold toric varieties.Approximation schemes for randomly sampling colorings
http://hdl.handle.net/2117/171323
Approximation schemes for randomly sampling colorings
González I Sentís, Marta
Graph colouring is arguably one of the most important issues in Graph Theory. However, many of the questions that arise in the area such as the chromatic number problem or counting the number of proper colorings of a graph are known to be hard. This is the reason why approximation schemes are considered. In this thesis we consider the problem of approximate sampling a proper coloring at random. Among others, approximate samplers yield approximation schemes for the problem of counting the number of colourings of a graph. These samplers are based in Markov chains, and the main requirement of these chains is to mix rapidly, namely in time polynomial in the number of vertices. Two main examples are the Glauber and the flip dynamics. In the project we study under which conditions these chains mix rapidly and hence under which conditions there exist efficient samplers.
2019-10-31T12:10:34ZGonzález I Sentís, MartaGraph colouring is arguably one of the most important issues in Graph Theory. However, many of the questions that arise in the area such as the chromatic number problem or counting the number of proper colorings of a graph are known to be hard. This is the reason why approximation schemes are considered. In this thesis we consider the problem of approximate sampling a proper coloring at random. Among others, approximate samplers yield approximation schemes for the problem of counting the number of colourings of a graph. These samplers are based in Markov chains, and the main requirement of these chains is to mix rapidly, namely in time polynomial in the number of vertices. Two main examples are the Glauber and the flip dynamics. In the project we study under which conditions these chains mix rapidly and hence under which conditions there exist efficient samplers.Amenability and Thompson’s group F
http://hdl.handle.net/2117/171322
Amenability and Thompson’s group F
Acedo Moscoso, Diego
Amenability is a group theoretical property consisting in the existence of a finite measure defined on all subsets of the group. The concept is motivated and introduced, and some criteria, characterizations and generalizations are presented. Then, this property is studied in a particular group of homeomorphisms of the interval [0,1], Thompson's group F. This group is introduced, along with its most relevant properties, and some possible Folner sequences are proposed and studied.
2019-10-31T12:06:27ZAcedo Moscoso, DiegoAmenability is a group theoretical property consisting in the existence of a finite measure defined on all subsets of the group. The concept is motivated and introduced, and some criteria, characterizations and generalizations are presented. Then, this property is studied in a particular group of homeomorphisms of the interval [0,1], Thompson's group F. This group is introduced, along with its most relevant properties, and some possible Folner sequences are proposed and studied.