In this thesis we provide several different systematic methods for constructing complex root spaces that remain invariant under an antilinear transformation. The first method is based on any element of the Weyl group, which is extended to factorizations of the Coxeter element and a reduced Coxeter element thereafter. An antilinear deformation method for the longest element of the Weyl group is given as well. Our last construction method leads to an alternative construction for q-deformed roots. For each of these construction methods we provide examples. In addition, we show a method of construction that for some special cases leads to rotations in the dual space and vice versa, starting from a rotation we find the root space involved. We then continue to apply these deformations to a generalized Calogero model and Affine Toda field theory. We provide a general solution for the ground state wave function of the Calogero model that is independent of a root representation and we extend this to the deformed case. An important property of this deformed Calogero model is that the amount of singularities in its potential is significantly reduced. We find that the exchange of particles in this model then leads to anyonic exchange factors. Following this we solve the model and find the ground state eigenvalues and eigenfunctions for the deformed Calogero model. We apply the q-deformed roots to an Affine Toda field theory and find that one may formulate a classical theory respecting the mass renormalisation of the quantum case.

The thesis starts with a short introduction to smooth dynamical systems and Hamiltonian dynamical systems. The aim of the introductory chapter is to collect basic results and concepts used in the thesis to make it self–contained. The second chapter of the thesis deals with the interaction of two charges moving in R2 in a magnetic field B. This problem can be formulated as a Hamiltonian system with four degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reduce this Hamiltonian system to one with two degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitable regime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non–integrability for this system with a Coulomb type interaction. Explicit knowledge of the reconstruction map and a dynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types of dynamical behaviours in this system: from periodic to quasiperiodic and chaotic orbits, from bounded to unbounded motion. In the third chapter of the thesis we study the interaction of two charges moving in R3 in a magnetic field B. This problem can also be formulated as a Hamiltonian system, but one with six degrees of freedom. We keep the assumption that the magnetic field is uniform and the interaction potential has rotational symmetry and reduce this Hamiltonian system to one with three degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain a system with two degrees of freedom. Furthermore, when the interaction potential is chosen to be Coulomb we prove the existence of an invariant submanifold where the system can be reduced by a further degree of freedom. The reductions simplify the analysis of some properties of this system: we use the reconstruction map to obtain a classification for the dynamics in terms of boundedness of the motion and the existence of collisions. Moreover, we study the scattering map associated with this system in the limit of widely separated trajectories. In this limit we prove that the norms of the gyroradii of the particles are conserved during an interaction and that the interaction of the two particles is responsible for a rotation of the guiding centres around a fixed centre in the case of two charges whose sum is not zero and a drift of the guiding centres in the case of two charges whose sum is zero.

This thesis consists of two parts, connected by one central theme: the dynamics of the "shape of space". To give the reader some inkling of what we mean by "shape of space", consider the fact that the shape of a triangle is given solely by its three internal angles; its position and size in ambient space are irrelevant for this ultimately intrinsic description. Analogously, the shape of a 3 dimensional space is given by a metric up to coordinate and conformal changes. Considerations of a relational nature strongly support the development of such dynamical theories of shape. The first part of the thesis concerns the construction of a theory of gravity dynamically equivalent to general relativity (GR) in 3+1 form (ADM). What is special about this theory is that it does not possess foliation invariance, as does ADM. It replaces that "symmetry" by another: local conformal invariance. In so doing it more accurately reflects a theory of the "shape of space", giving us reason to call it shape dynamics. (SD). Being a very recent development, the consequences of this radical change of perspective on gravity are still largely unexplored. In the first part we will try to present some of the highlights of results so far, and indicate what we can and cannot do with shape dynamics. Because this is a young, rapidly moving field, we have necessarily left out some interesting new results which are not yet in print and were developed alongside the writing of the thesis. The second part of the thesis will develop a gauge theory for "shape of space"--theories. To be more precise, if one admits that the physically relevant bservables are given by shape, our descriptions of Nature carry a lot of redundancy, namely absolute local size and absolute spatial position. This redundancy is related to the action of the infinite-dimensional conformal and diffeomorphism groups on the geometry of space. We will show that the action of these groups can be put into a language of infinite-dimensional gauge theory, taking place in the configuration space of 3+1 gravity. In this context gauge connections acquire new and interesting meanings, and can be used as "relational tools".

A brief observation on recent research of routing problems shows that most of the methods used to tackle the problems are using heuristics and metaheuristics; and they often use problem specific knowledge to build or improve solutions. In the last few years, research on hyper-heuristic has been investigated which aims to raise the generality of optimisation systems. This thesis is concerned with the investigation of ant-based hyper-heuristic. Ant algorithms have been applied to vehicle routing problems and have produced competitive results. Therefore, it is assumed that there is a reasonable possibility that ant-based hyperheuristic could perform well for the problem. The thesis first surveys the literature for some common solution methodologies for optimisation problems and explores in some detail the ant algorithms and ant algorithm hyperheuristic methods. Furthermore, the literature specifically concerns with routing problems; the capacitated routing problem (CVRP) and the travelling salesman problem (TSP). The thesis studies the ant system algorithm and further proposes the ant algorithm hyper-heuristic, which introduces a new pheromone update rule in order to improve its performance. The proposed approach, called the ant-based hyper-heuristic is tested to two routing problems; the CVRP and TSP. Although it does not produce any best known results, the experimental results have shown that it is competitive with other methods. Most importantly, it demonstrates how simple and easy to implement low level heuristics, with no extensive parameter tuning. Further analysis shows that the approach possesses learning mechanism when compared to random hyper-heuristic. The approach investigates the number of low level heuristics appropriate and found out that the more low level heuristics used, the better solution is generated. In addition an ACO hyper-heuristic which has two categories of pheromone updates is developed. However, ant-based hyper-heuristic performs better and this is inconsistent with the performance of ACO algorithm in the literature. In TSP, we utilise two different categories of low level heuristics, the TSP heuristics and the CVRP heuristics that were previously used for the CVRP. From the observation, it can be seen that by using any heuristics for the same class of problems, ant-based hyper-heuristic is seen to be able to produce competitive results. This has demonstrated that the ant-based hyper-heuristic is a reusable method. One major advantage of this work is the usage of the same parameter for all problem instances with simple moves and swap procedures. It is hoped that in the future, results obtained will be better than current results by using better intelligent low level heuristics.

Στην παρούσα διπλωματική εργασία παρουσιάζεται η μεθοδολογία εξαγωγής διαφορικών εξισώσεων κατά Hamilton. Οι προκύπτουσες διαφορικές εξισώσεις είναι n το πλήθος (όπου n είναι το πλήθος των βαθμών ελυθερίας του συστήματος) διαφορικές εξισώσεις πρώτου βαθμού και περιγράφουν την δυναμική συμπεριφορά ενός μηχανικού συστήματος. Παρουσιάζεται το εκτενές θεωρητικό πλαίσιο που απορρέει απο τις εξισώσεις του Ηamilton και αξιοποιούνται οι θεωρητικές αρχές κατα την περιγραφή και τον έλεγχο του συστήματος σφαίρας-ράβδου. / In the diploma thesis we present the Hamiltonian formalism in order to derive first order differential equations which represent the dynamical behaviour of a mechanical system. The derived differential equations are n first order equations (where n is the number of system's degrees of freedom). We present the extensive theoretical framework that is derived from the Hamilton's equations which we use so that to describe and control the ball and beam system.

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The integrability of an m-component system of hydrodynamic type, Ut = v(u)ux, by the generalized hodograph method requires the diagonalizability of the m x m matrix v(u). The diagonalizability is known to be equivalent to the vanishing of the corresponding Haantjes tensor. This idea is applied to hydrodynamic chains - infinite-component systems of hydrodynamic type for which the 00 x 00 matrix v(u) is 'sufficiently sparse'. For such 'sparse' systems the Haantjes tensor is well-defined, and the calculation of its components involves only a finite number of summations. The calculation of the Haantjes tensor is done by using Mathematica to perform symbolic calculations. Certain conservative and Hamiltonian hydrodynamic chains are classified by setting Haantjes tensor equal to zero and solving the resulting system of equations. It is shown that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability of such sysyems. In the cases of the Hamiltonian hydrodynamic chains we were able to first construct one extra conservation law and later a generating function for conservation laws, thus establishing the integrability.

Cette thèse contient essentiellement deux chapitres principaux qui ont en commun de porter sur ce que l'on appelle en géométrie de Poisson les chemins cotangents. Dans le premier chapitre, nous introduisons pour chaque hamiltonien, un lagrangien sur les chemins à valeurs dans l'espace cotangent dont les points stationnaires indiquent si le champ de bivecteur est de Poisson ou au moins définit une distribution intégrable - une classe de champs de bivecteurs qui généralise les structures de Poisson tordus que nous étudions en détail. Nous traitons dans le deuxième chapitre d'un autre résultat classique à propos des chemins cotangents, dû à Klimčík, Strobl et étudiée par Cattaneo et Felder. Un bivecteur sur une variété $M$ est de Poisson si et seulement si l'ensemble $C_pi$ des chemins cotangents pour $pi$ est co"{i}sotrope dans la variété symplectique des chemins à valeurs dans $T^*M$. Notre but dans le deuxième chapitre est de reprendre la caractérisation des bivecteurs de Poisson, en travaillant avec des fonctions locales sur l'ensemble des chemins lisses, pour lesquels l'utilisation d'une variété de Banach peut être évitée. Ceci permet d'étendre au cas périodique / In this thesis, we study cotangents paths. In chapter 1 we introduce for every Hamiltonian a Lagrangian on paths valued in the cotangent space whose stationary points projects onto Hamiltonian vector fields. We show that the remaining components of those stationary points tell whether the bivector field is Poisson or at least defines an integrable distribution - a class of bivector fields generalizing twisted Poisson structures that we study in detail. In chapter 2, we establish a local function version of a result due to Klimčík and Strobl then Cattaneo and Felder claiming that a bivector field on a manifold $M$ is Poisson if and only if cotangent paths form a coisotropic submabifold of the infinite dimensional symplectic manifold of paths valued in $T^*M$. Our purpose in chapter 2 is to prove this result without using the Banach manifold setting used by Cattaneo and Felder, which fails in the periodic case because cotangent loops do not form a Banach sub-manifold. Instead, we use local functions on the path space, a point of view that allows to speak of a coisotropic set

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