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Geometric constraint solving is a central topic in many fields such as parametric solid modeling, computer-aided design or chemical molecular docking. A geometric constraint problem consists of a set geometric objects on which a set of constraints is defined. Solving the geometric constraint problem means finding a placement for the geometric elements with respect to each other such that the set of constraints holds. Clearly, the primary goal of geometric constraint solving is to define rigid shapes. However an interesting problem arises when we ask whether allowing parameter constraint values to change with time makes sense. The answer is in the positive. Assuming a continuous change in the variant parameters, the result of the geometric constraint solving with variant parameters would result in the generation of families of different shapes built on top of the same geometric elements but governed by a fixed set of constraints. Considering the problem where several parameters change simultaneously would be a great accomplishment. However the potential combinatorial complexity make us to consider problems with just one variant parameter. Elaborating on work from other authors, we develop a new algorithm based on a new tool we have called h-graphs that properly solves the geometric constraint solving problem with one variant parameter. We offer a complete proof for the soundness of the approach which was missing in the original work. Dynamic geometry is a computer-based technology developed to teach geometry at secondary school, which provides the users with tools to define geometric constructions along with interaction tools such as drag-and-drop. The goal of the system is to show in the user's screen how the geometry changes in real time as the user interacts with the system. It is argued that this kind of interaction fosters students interest in experimenting and checking their ideas. The most important drawback of dynamic geometry is that it is the user who must know how the geometric problem is actually solved. Based on the fact that current user-computer interaction technology basically allows the user to drag just one geometric element at a time, we have developed a new dynamic geometry approach based on two ideas: 1) the underlying problem is just a geometric constraint problem with one variant parameter, which can be different for each drag-and-drop operation, and, 2) the burden of solving the geometric problem is left to the geometric constraint solver. Two classic and interesting problems in many computational models are the reachability and the tracing problems. Reachability consists in deciding whether a certain state of the system can be reached from a given initial state following a set of allowed transformations. This problem is paramount in many fields such as robotics, path finding, path planing, Petri Nets, etc. When translated to dynamic geometry two specific problems arise: 1) when intersecting geometric elements were at least one of them has degree two or higher, the solution is not unique and, 2) for given values assigned to constraint parameters, it may well be the case that the geometric problem is not realizable. For example computing the intersection of two parallel lines. Within our geometric constraint-based dynamic geometry system we have developed an specific approach that solves both the reachability and the tracing problems by properly applying tools from dynamic systems theory. Finally we consider Henneberg graphs, Laman graphs and tree-decomposable graphs which are fundamental tools in geometric constraint solving and its applications. We study which relationships can be established between them and show the conditions under which Henneberg constructions preserve graph tree-decomposability. Then we develop an algorithm to automatically generate tree-decomposable Laman graphs of a given order using Henneberg construction steps.

Tesis por compendio de publicaciones / It is well known that Takahashi's Theorem [7] characterizes the submanifolds in the Euclidean space whose coordinate functions are eigenfunctions of the Laplacian associated to the same nonzero eigenvalue: they are minimal submanifolds in a hypersphere. Later on, many authors have obtained different extensions of Takahashi's Theorem. One of these extensions is given by Dillen-Pas-Verstraelen in [2]. In that work, the authors study surfaces in the 3-dimensional space whose immersion ψ satisfy Δψ=Aψ+b, where Δ denotes the Laplacian operator, A is a 3x3 real matrix and b is a constant vector. They obtain that the only surfaces satisfying that equation are minimal ones, spheres and circular cylinders. After that different authors have studied this condition in the case of hypersurfaces Mn immersed in pseudo-Euclidean spaces Rn+1 for any index t≥0, and showed that Mn must be an open part of a minimal Rn+1 surfaces, a totally umbilical hypersurface or a standard pseudo-Riemannian product. Recently, that equation has been extended to operators different to the Laplacian one. In fact, Alías and Gürbüz study in [2] hypersurfaces in the Euclidean space Rn+1 whose position vector ψ satisfies Lkψ=Aψ+b, where Lk is the linealized differential operator associated to the mean curvature of order k+1, for k=0, 1,..., n-1 (note that for k=0 we obtain the Laplacian operator). Those authors show that the only hypersurfaces satisfying the above condition are k-minimal hypersurfaces, hyperspheres and generalized cylinders (for appropriate radii and dimensions). In view of that result for operators Lk, we study the same condition but for hypersurfaces immersed in pseudo-Euclidean spaces Rn+1 for any index t≥0, and show (in papers [5] and [6]) that the only hypersurfaces in the pseudo-Euclidean spaces satisfying that condition are k-minimal hypersurfaces, hyperspheres and generalized cylinders (for appropriate radii and dimensions). After solving the problem for hypersurfaces in pseudo-Euclidean spaces, we study the condition Lkψ=Aψ+b for hypersurfaces immersed in pseudo-Riemannian space forms, for arbitrary index t≥0 and nonzero constant curvature. We show (in papers [3] and [4]), that the only hypersurfaces satisfying that condition are k-minimal hypersurfaces, totally umbilical hypersurfaces, standard pseudo-Riemannian products and some quadratic hypersurfaces. In conclusion, the results obtained in this Thesis extend completely to pseudo- Euclidean spaces and pseudo-Riemannian space forms of nonzero constant curvature the results previously obtained in [2]. References [1] L.J. Alías and N. Gürbüz. An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006), 113-127. [2] F. Dillen, J. Pas and L. Verstraelen. On surfaces of finite type in Euclidean 3-space, Kodai Math. J. 13 (1990), 10-21. [3] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in non-flat Lorentzian space forms satisfying Lkψ=Aψ+b , Taiwanese J. Math. 16 (2012), 1173-1203. [4] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in non-flat pseudo-Euclidean space form satisfying a linear condition in the linearized operator of a higher order mean curvatures, Taiwanese J. Math. 17 (2013), 15-45. [5] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ=Aψ+b , Geom. Dedicata 153 (2011), 151-175. [6] P. Lucas and H.F. Ramírez-Ospina. Hypersurfaces in pseudo-Euclidean space satisfying a linear condition on the linearized operator of a higher order mean curvatures, Diff. Geom. and its Appl. 13 (2013), 175-189. [7] T. Takahashi. Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385.

El principal objectiu d'aquesta tesi és l'estudi de l'evolució mitjançant el flux de Ricci de superfícies amb singularitats de tipus cònic. Un segon objectiu, sorgit de les tècniques que utilitzem, és l'estudi de famílies de solitons del flux de Ricci en dimensió 2 i 3. El flux de Ricci és una equació d'evolució per a varietats Riemannianes, introduïda per R. Hamilton el 1982. És des dels avenços assolits per G. Perelman amb aquesta tècnica el 2002 quan el flux de Ricci s'ha establert com a una disciplina pròpia, aixecant un gran interès per la comunitat. Aquesta tesi conté quatre resultats originals. El primer resultat és una classificació exhaustiva dels solitons en superfícies llises i còniques. Amb aquesta classificació completem els precedents trobats per Hamilton, Chow i Wu entre d'altres, i obtenim descripcions explícites de tots els solitons en dimensió 2. El segon resultat és una Geometrització de les superfícies còniques mitjançant el flux de Ricci. Aquest resultat, que utilitza el primer resultat ja esmentat, estén la teoria de Hamilton al cas singular. Aquest és el resultat més extens, per al qual fem servir i desenvolupem tècniques tant d'anàlisi i EDPs com de geometria de comparació . El tercer resultat és l'existència d'un flux de Ricci que elimina les singularitats còniques . Això exposa clarament la no unicitat de solucions al flux, en analogia als fluxos de Ricci amb cusps de P. Topping . El quart resultat és la construcció d'un nou solitó gradient expansiu en dimensió 3. De la mateixa manera que amb els solitons cònics, donem una construcció explícita utilitzant tècniques de retrats de fase. Demostrem també que és l'únic solitó amb la seva topologia i la seva cota inferior de la curvatura, i que és un cas crític entre tots els solitons expansius en dimensió 3 amb curvatura acotada inferiorment. A més, mostrem que l'evolució de la seva curvatura escalar no és monòtona. / El principal objetivo de esta tesis es el estudio de la evolución mediante el flujo de Ricci de superficies con singularidades de tipo cónico. Un segundo objetivo, surgido de las técnicas que utilizamos, es el estudio de familias de solitones del flujo de Ricci en dimensión 2 y 3. El flujo de Ricci es una ecuación de evolución para variedades Riemannianas, introducida por R. Hamilton en 1982. Es desde los logros alcanzados por G. Perelman con esta técnica en 2002 cuando el flujo de Ricci se ha establecido en una disciplina propia, despertando un gran interés en la comunidad. Esta tesis contiene cuatro resultados originales. El primer resultado es una clasificación exhaustiva de los solitones en superficies lisas y cónicas. Con esta clasificación completamos los precedentes hallados por Hamilton, Chow y Wu entre otros, y obtenemos descripciones explícitas de todos los solitones en dimensión 2. El segundo resultado es una Geometrización de las superficies cónicas mediante el flujo de Ricci. Este resultado, que utiliza el primer resultado ya mencionado, extiende la teoría de Hamilton al caso singular. Este es el resultado más extenso, para el que usamos y desarrollamos técnicas tanto de análisis y EDPs como de geometría de comparación. El tercer resultado es la existencia de un flujo de Ricci que elimina las singularidades cónicas. Esto expone claramente la no unicidad de soluciones al flujo, en analogía a los flujos de Ricci con cúspides de P. Topping. El cuarto resultado es la construcción de un nuevo solitón gradiente expansivo en dimensión 3. Del mismo modo que con los solitones cónicos, damos una construcción explícita utilizando técnicas de retratos de fase. Demostramos también que es el único solitón con su topología y su cota inferior de la curvatura, y que es un caso crítico entre todos los solitones expansivos en dimensión 3 con curvatura acotada inferiormente. Además, mostramos que la evolución de su curvatura escalar no es monótona. / The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds, introduced by R. Hamilton in 1982. It is from the achievements made by G. Perelman with this technique in 2002 when the Ricci flow has been established in a discipline itself, generating a great interest in the community. This thesis contains four original results. First result is a complete classification of solitons in smooth and cone surfaces. This cllassification completes the preceding results found by Hamilton, Chow and Wu and others, and we obtain explicit descriptions of all solitons in dimension 2. Second result is a Geometrization of cone surfaces by Ricci flow. This result, which uses the aforementioned first result, extends the theory of Hamilton to the singular case. This is the most comprehensive result in the thesis, for which we use and develop analysis and PDE techniques, as well as comparison geometry techniques. Third result is the existence of a Ricci flow that removes cone singularities. This clearly exposes the non-uniqueness of solutions to the flow , in analogy to the Ricci flow with cusps of P. Topping. The fourth result is the construction of a new expanding gradient Ricci soliton in dimension 3. Just as we do with solitons on cone surfaces, we give an explicit construction using techniques of phase portraits. We also prove that this is the only soliton with its topology and its lower bound of the curvature, and besides this is a critical case amongst all expanding solitons in dimension 3 with curvature bounded below. Furthermore, we show that the evolution of its scalar curvature is not monotone.

Esta tesis aborda los sistemas más conocidos de la mecánica clásica de forma unificada. Nuestro objetivo principal es desarrollar un marco de trabajo común para el estudio de perturbaciones, dicha tarea se realiza desde un punto de vista geométrico. Hemos estructurado esta memoria en tres partes: Parte I. Preliminares en Mecánica Clásica y Geometría: En esta primera parte recogemos herramientas que serán usadas a lo largo de nuestro estudio. En el primer capítulo fijamos la notación y se presentan algunos resultados básicos. En el segundo estudiamos el Sistema Extendido de Euler, como un problema de valor inicial paramétrico. Este enfoque permite derivar las principales propiedades de las funciones elípticas. En concreto, las conocidas relaciones cuadráticas entre funciones elípticas y la transformación de Jacobi para el módulo elíptico se obtienen de nuestro análisis. Parte II. Reducción de tipo Hopf de un modelo cuártico: En el tercer capítulo estudiamos una generalización de la fibración de Hopf clásica. Seguiremos la misma metodología que en la fibración de Hopf clásica, pero el cuerpo complejo será reemplazado por cuaternios. En el cuarto capítulo usamos las componentes de la representación cuaterniónica de la aplicación de Hopf para proponer una familia de Hamiltonianos multiparamétrica. Para una elección apropiada de los parámetros y considerando una regularización de la variable independiente, cuando sea necesario, algunos modelos destacados de la mecánica clásica tales como el sistema de Kepler, el flujo geodésico, el oscilador isotrópico de cuatro dimensiones y el sólido rígido libre aparecen como casos particulares. El análisis del modelo cuártico se lleva a cabo a través de una doble reducción. Por un lado, el sistema es geométricamente reducido, este modelo es un ejemplo detallado de reducción singular, en la cual la correspondiente reconstrucción es también proporcionada. Por otro lado, la reducción simpléctica llevada a cabo a través del uso de nuevas coordenadas canónicas es analizada. En concreto, se muestra la relación entre la reducción geométrica y simpléctica y se proporciona la formulación explícita para todos los cambios de variables que son usados. Parte III. Aplicaciones a la dinámica Roto-Orbital: Esta parte está dedicada al estudio de la dinámica de actitud y el movimiento orbital de modelos que aproximan un asteroide o un satélite con una triaxialidad genérica, bajo los efectos de una perturbación gravitacional. Este problema, denominado problema completo de los dos cuerpos, es un sistema dinámico Hamiltoniano no integrable, que requiere el uso de teorías de perturbaciones para su análisis. Dentro del contexto de Poincaré y Arnold, una teoría de perturbación debería ser desarrollada a partir un orden cero integrable y no degenerado. Nosotros exploraremos nuevos candidatos para el orden cero llamados intermediarios. La idea de los intermediarios consiste en definir un sistema integrable simplificado del problema en cuestión. En el quinto capítulo recordamos el concepto de intermediario, presentamos cinco modelos y establecemos una metodología común para su estudio. Es en este contexto donde el marco desarrollado para el modelo polinómico cuártico es completamente explotado. El sistema simplificado incluye parte del potencial donde el acoplamiento roto-orbital esta presente de tal manera, que el sistema definido por el orden cero es integrable. Los capítulos seis y siete aprovechan el marco de trabajo desarrollado en el estudio de dos intermediarios definidos en el capítulo anterior. Se asume que estos intermediarios tienen orbitas circulares y elípticas respectivamente. En el capítulo seis estudiamos equilibrios relativos y bifurcaciones del intermediario circular. Este modelo de intermediario define un flujo Poisson sobre espacio multiparamétrico. En el caso de un cuerpo de rotación lenta, identificamos condiciones bajo las cuales aparecen bifurcaciones de las trayectorias inestables clásicas, siendo dichos escenarios de gran interés en relación a la estabilización y control. Por otro lado, también se pone de manifiesto y se estudia en detalle el papel jugado por la triaxialidad del cuerpo. En el último capítulo la perturbación contiene al radio y como consecuencia las órbitas obtenidas serán de tipo roseta. Este modelo se asocia a dos tipos de aplicaciones, asteroides y satélites, es decir, en nuestra última aplicación consideramos órbitas elípticas en general; también analizamos las condiciones para que este modelo admita circulares. El objetivo de este estudio es encontrar un modelo suficientemente simplificado para ser considerado un orden cero, pero que incorpore parte del efecto perturbativo gravitatorio. Conclusión En esta tesis se aborda una generalización del sistema clásico de Euler, la solución general conecta con las doce funciones elípticas de Jacobi. Usando esta generalización y la fibración tipo Hopf cuaterniónica, se define y estudia en detalle una familia polinómica paramétrica de Hamiltonianos. Sobre dicha familia se llevan a cabo reducciones de tipo geométrico y simpléctico y se muestra que algunos modelos de la mecánica clásica están incluidos para ciertas elecciones de los parámetros. En este sentido, la familia propuesta proporciona un marco de trabajo común para abordar estos modelos clásicos. En las aplicaciones nos centramos en la modelización de problemas roto-orbitales. El modelo completo requiere el desarrollo de teorías perturbativas para obtener soluciones aproximadas. En este trabajo consideramos algunos candidatos para el orden cero. Presentamos un detallado análisis para el caso en el que el satélite presenta rotación lenta. Para cada misión concreta, el valor de los modelos dependerá de las comparaciones con experimentos numéricos. Para el caso de radio no constante aparecen un buen número de técnicas de la mecánica clásica a investigar. En este sentido, el capítulo siete es un primer paso que requiere más investigación. Como ejemplo valga la comparación de nuestro enfoque con la eliminación de la paralaje como punto de partida. Un segundo aspecto es la producción de las correspondientes variables de ángulo-acción. / This thesis addresses some of the very well known systems in classical mechanics in a uniform manner. Our main target is to develop a common framework to deal with perturbations. As such, the structure of this memoir comprises three parts: Part I. Preliminaries on Classical Mechanics and Geometry: In the first part of this memoir we gather some tools that will be used along our study. The first Chapter sets notation and presents some basic results. In the second Chapter we study the extended Euler systems as an initial value problem with parameters. Particular realizations of this system lead to several Lie-Poisson structures. The twelve Jacobi elliptic functions are shown in a unified way. Part II. Hopf Reduction on a Quartic Polynomial Model: In the third Chapter we study a four dimensional generalization of the classical Hopf fibration. We follow the same methodology as in the classical Hopf fibration, but instead of complex numbers the generalization of the classic Hopf map is defined in terms of quaternions. The fourth Chapter uses the components of the quaternionic Hopf map to propose a parametric Hamiltonian function, which is an homogeneous quartic polynomial with six parameters, defining an integrable family of Hamiltonian systems. For suitable choices of the parameters, adding an appropriate regularization when needed, some remarkable classical models such as the Kepler, geodesic flow, 4-D isotropic oscillator and free rigid body systems appear as particular cases. The analysis of the quartic model is performed through a twofold reduction. On the one hand, the system is geometrically reduced. On the other hand, symplectic reduction is examined. Moreover, we show the relation between the geometric reduction and the reduction carried out by the Projective Andoyer variables. Part III. Applications to Roto-Orbital Dynamics: This part is devoted to the study of the attitude dynamics and the orbital motion of models approximating a generic triaxial spacecraft under gravity-gradient torque perturbation. The full problem is a non-integrable Hamiltonian dynamical system. Within the context of Poincaré and Arnold, a perturbation theory should be developed upon an integrable and non-degenerate zero order. We study alternative candidates for the zero order, the intermediaries. The idea of the intermediary is to define a simplified integrable system of the problem at stake. In the fifth Chapter we recall the concept of intermediary, present five of them and we set a common methodology. Sixth and seventh Chapters take advantage of the previous framework considering two intermediary models. Those intermediaries are assumed to be in circular and elliptic orbits respectively. We study relative equilibria and bifurcations of the circular intermediary in chapter six. This intermediary model defines a Poisson flow over a large parameter space. In the case of slow rotational motion we identify conditions under which different bifurcations of the classical unstable trajectories occur, being those scenarios of great interest in relation to stabilization and control purposes. The role played by the triaxiality is also shown. The final chapter examines a body moving in a rosette-like orbit. More precisely we are thinking about two types of applications, namely to artificial satellites or asteroids around a planet. In other words, we consider perturbed elliptic orbits in general; we also investigate conditions for which this model admits the circular ones. This scenario leads to medium orbits rather than to the low type of orbits studied in the preceding chapter. The intention of this study is to analyze a model simply enough to be considered as an alternative zero order, but incorporating partially the effects of the gravity torque perturbation. Conclusions The main conclusion of this Memoire may be summarized as follows. A generalized study of the classical Euler system is presented, connecting its solutions with the twelve Jacobi elliptic functions. Using that and the quaternionic Hopf fibration a quartic homogeneous polynomial parametric family is proposed and studied in detail. Geometric and symplectic reductions are performed in the family. It is shown that, for suitable choices of parameters, several classical mechanical systems arise as family realizations and we provide a common framework to study them. In the application we focus on modeling problems in the roto-orbital dynamics. The full model is a non-integrable problem which requires the development of perturbation theories in order to obtain approximate solutions. Several candidates for the zero order term, on which the whole theory relies, are considered in this context. We analyze the role played by the integrals and the relation with the physical parameters involved. In particular, we present a fairly complete analysis of the case when the satellite has slow rotation. For each mission, the relative value of each model will finally depend on numerical experiments. When the radius is not constant, there is a number of techniques of classical mechanics to be considered and the last chapter is just a preliminary step to do more research. As an example we mention the comparison of our approach with the elimination of the parallax as the starting point. A second aspect could be the production of the corresponding action-angle variables.

This thesis considers a number of problems which are related to the study of random fractals. We define a class of iterations (which we call random hierarchical systems) of probability distributions, which are defined by applying a random map to a set of k independent and identically distributed random variables. Classical examples of this sort of iteration include the Strong Law of Large Numbers, Galton-Watson branching processes, and the construction of random self-similar sets. In Chapter 2, we consider random hierarchical systems on ℝ, under the condition that the random map is bounded above by a random weighted mean, and that the initial distribution is bounded below. Under moment conditions on the initial distribution we show that there exists almost sure convergence to a constant. In Chapters 3 to 5 we consider the asymptotics of some examples of random hierarchical systems, some of which arise when considering certain properties of random fractal graphs. In one example, which is related to first-passage percolation on a random hierarchical lattice, we show the existence of a family of non-degenerate fixed points and show that the sequence of distributions will converge to one of these. The results of some simulations are reported in Chapter 7. Part III investigates the spectral properties of random fractal graphs. In Chapter 8 we look at one example in detail, showing that there exist localised eigenfunctions which lead to certain eigenvalues having very high multiplicity. We also investigate the behaviour of the Cheeger constants of this example. We then consider eigenvalues of homogeneous random fractal graphs, which preserve some of the symmetry of deterministic fractals. We then use relationships between homogeneous graphs and more general random fractal graphs to obtain results on the eigenvalues of the latter. Finally, we consider a few further examples.

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We investigate almost-dual-like solutions of the WDVV equations for which the metric, under the standard definition, is degenerate. Such solutions have previously been considered in [21] as complex Euclidean v-systems with zero canonical form but were not regarded as solutions since a non-degenerate metric is required for a solution. We have found that, in every case we considered, we can impose a metric and hence recover a solution. We also found that for the deformed A_n(c) family (first appearing in [8]) with the choice of parameters that renders the metric singular we can also recover a solution. The generalised root system A(n-1,n) (as it appears in our notation) has zero canonical form but we found that by restricting the covectors we can again recover a solution which we generalise to a family with (n+1) parameters which we denote as P_n. We next look at extended v-systems. These are root-systems which possess the small orbit property (as defined in [36]) which we then extend into a dimension perpendicular to the original system. We then impose the v-conditions onto these systems and obtain 1-parameter infinite families of v-systems. We also find that for the B_n family we can extend into two perpendicular directions. We then go on to look at a generalisation of the Legendre transformations (which originally appeared in [13])which map solutions to WDVV to other solutions. We find that such transformations are generated not only by constant vector fields but by functional vector fields too and we find a very simple rule which such vector fields must obey. Finally we link our work on extended v-systems and on generalised Legendre transformations to that on extended affine Weyl groups found in [16] and [17].

We consider a real analytic family of area-preserving maps on C2, fµ, depending analytically on the parameter, such that f0 is a map at 1:3 resonance. Such maps can be formally embedded in an one degree of freedom Hamiltonian system, called the normal form of the map. We denote the third iterate of the map by Fµ = f3µ. We show that given a certain non degeneracy condition on the map F0, there exists a Stokes constant, θ, that when it does not vanish, it describes the splitting of the separatrices that the normal form predicts. We show that this constant can be approximated numerically for any non-degenerate map F0. For a non-vanishing and small enough µ, we show that if the Stokes constant does not vanish the separatrices split. Moreover, let Ω be the area of the parallelogram defined by the 2 vectors tangent at the two separatrices at a homoclinic point. For any M ε N we have the estimate. In this equation λµ is the largest eigenvalue of the saddle points around the origin and θn's are real constants with θ0 = 4π|θ|.

This thesis discusses subgroups of mapping class groups of particular surfaces. First, we study the Torelli group, that is, the subgroup of the mapping class group that acts trivially on the first homology. We investigate generators of the Torelli group, and we give an algorithm that factorizes elements of the Torelli group into products of particular generators. Furthermore, we investigate normal closures of powers of standard generators of the mapping class group of a punctured sphere. By using the Jones representation, we prove that in most cases these normal closures have infinite index in the mapping class group. We prove a similar result for the hyperelliptic mapping class group, that is, the group that consists of mapping classes that commute with a fixed hyperelliptic involution. As a corollary, we recover an older theorem of Coxeter (with 2 exceptional cases), which states that the normal closure of the m-th power of standard generators of the braid group has infinite index in the braid group. Finally, we study finite index subgroups of braid groups, namely, congruence subgroups of braid groups. We discuss presentations of these groups and we provide a topological interpretation of their generating sets.