In this thesis we study various aspects of the problem of finding rational linear spaces on hypersurfaces. This problem can be approached by the Hardy-Littlewood circle method, establishing a Local-Global Principle provided that the hypersurface is 'sufficiently non-singular' and the number of variables is large enough. However, the special structure of the linear spaces allows us to obtain some improvement over previous approaches. A generalised version is also addressed, which allows us to count linear spaces under somewhat more flexible conditions. We then investigate the local solubility. In particular, by adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity of the singular series. This improves on earlier approaches due to Davenport, Schmidt and others, which require the existence of non-singular p-adic solutions. Finally, we exhibit the strength of our methods by deriving unconditional results concerning the existence of linear spaces on systems of cubic and quintic equations.

This thesis contributes to the understanding of the use of topology in analysing 3D spatial data, focussing in particular on two aspects of the problem - what binary topological analysis functionality is required in a commercial 3D Geographical Information System (GIS), and how should this functionality be implemented to achieve the most efficient query performance. Topology is defined as the identification of spatial relationships between adjacent or neighbouring objects. The first stage of this research, a review of applications of topology, results in a generic list of requirements for topology in 3D. This was carried out in parallel with a review of topological frameworks and the relationships identified by one of the frameworks, Egenhofer and Herring's 9-Intersection, selected for implementation. Three generic binary relationship queries are identified (Find Objects with a Specific Relationship, Find Intersecting Objects and What Relationship is there Between These Objects) and a mechanism described to allow these to be adapted to specific application terminology. Approaches to the implementation of 3D binary topological queries include the use of data structures and an As-Required calculation, where computational geometry algorithms are run to determine relationships each time the user runs a query. The Three-Dimensional Formal Data Structure (3DFDS) was selected as a representative example of a Boundary-Representation (B- Rep) structure in GIS. Given the number of joins to be traversed when identifying binary relationships from a B-Rep structure, along with the requirement to query additional containment exception tables, an alternate structure, the Simplified Topological Structure (STS), was proposed to improve binary query performance. Binary relationship queries were developed and comparative performance tests carried out against 3DFDS, STS and a Proxy for the As-Required calculation, using a 1.08 million object test dataset. Results show that STS provides a significant performance improvement over 3DFDS. No definitive conclusion could be drawn when comparing STS with the Proxy for the As-Required approach.

This thesis is concerned with the anti-self-dual Yang-Mills equations and their reductions to Bogomol’nyi equations on constant curvature spaces. Chapters 1 and 2 contain introductory material. Chapter 1 discusses the origin of the equations in particle physics and their role in integrable systems. Chapter 2 describes the equations and the reduction process and outlines the construction of solutions via the twistor transform. In Chapter 3 we consider Bogomol’nyi equations on (2 + 1)-dimensional manifolds and show that for constant curvature space-times the equations are integrable and consider solutions in the negative scalar curvature case. In Chapter 4 we cover the negative scalar curvature case in more detail, constructing a number of soliton solutions including non-trivial scattering and consider the zero-curvature limit. In Chapter 5 we consider Bogomornyi equations in 3- diniensional hyperbolic space, derive an ansatz for solutions of the equation and use it to construct a number of new solutions. Chapter 6 contains concluding remarks.

No description available.

Spatial configurations tend to change. Dealing with spatial representations often means dealing with changing representations. Change in state for qualitative spatial representation languages has been analyzed through transition graphs in which relations form conceptual neighbourhoods via potential motion. Continuity has remained an implicitly assumed notion for any such understanding of motion. The work described in this thesis is concerned with formalizing an intuitive notion of spatio-temporal continuity for a qualitative theory of spatial change. Taking over a theory for spatial regions, I extend it for space-time. A mereotopological spatio-temporal theory based on space-time histories is developed. I formalize the intuitive notion of spatio-temporal continuity and christen it strong firm continuity. Continuous transitions in mereotopology for space-time histories are investigated. For strong firm continuity, transition rules for spatio-temporal histories are formulated. The conceptual neighbourhood for the spatial representation language RCC-8 specifies which transitions are continuous, and in its original presentation was simply posited without any proof of correctness. Formal proofs for the non-existence of transitions i.e., transitions absent from the RCC-8 conceptual neighbourhood are presented here.

Dans cette thèse nous étudions d’un point de vue topologique deux espaces dont l’utilité et l’importance dépassent le cadre de la topologie algébrique.  Le premier espace est constitué de toutes les fonctions holomorphes de la sphère de Riemann dans une variété de Grassmann complexe. Cet espace se scinde en composantes connexes et nous identifions entièrement le type d’homotopie de la composante des applications de degré un. Nous en déduisons des calculs homologiques explicites. Dans le cas des applications pointées, nous explicitons une action de l’opérade des deux petits disques sur l’espace des fonctions rationnelles, simplifiant ainsi quelques travaux de Mann et Milgram. Nous étudions également les espaces de fonctions continues et dans le cas de la Grassmannienne des deux plans complexes dans C4, nous obtenons une décomposition homotopique de son espace de lacet. Finalement le second espace que nous étudions est l'espace des lacets libres sur les configurations de points distincts dans Rn. Dans le cas de 3 points, nous obtenons de façon simple et élégante un résultat de scindement homologique dû à Fadell et Husseini. / In this thesis we study a topological point of view two spaces whose usefulness and importance beyond the scope of algebraic topology. The first space consists of all holomorphic maps of the Riemann sphere in a complex Grassmannian manifold. This space is divided into connected components and we identify the entire homotopy type of the component of degree one. We deduce explicit homological calculations. In the case of based map, we explain an action of the operad of two little disks on the space of rational functions, simplifying some work of Mann and Milgram. We also study the spaces of continuous maps and in the case of the Grassmannian of two planes complex C4, we obtain a homotopy decomposition of the space of loops. Finally the second space that we study is the free loop space of configurations of distinct points in Rn. In the case of three points, we obtain a simple and elegant result of homological splitting belonging to Fadell and Husseini.

Στην εργασία αυτή μελετάμε το πρόβλημα της ύπαρξη ή μη καθολικών χώρων για διάφορες κλάσεις τοπολογικών χώρων. / In this project we study the problem of the existence or no of universal spaces for some classes of topological spaces

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Για ένα, μάλλον, μακρύ διάστημα, (1950-1975) οι Μαθηματικοί ασχολήθηκαν με την εμφύτευση μιας αντιμεταθετικής τοπολογικής ημιομάδας σε ομάδα. Είναι γνωστό ότι για ημιομάδα S έχουμε αλγεβρική εμφύτευση στο σχέση ισοδυναμίας = , όπου στοιχεία της καινούργιας ομάδας). Το νέο στοιχείο είναι ότι ενώ η συνθήκη εμφύτευσης αναφέρεται σε Ομοιόμορφο χώρο, έχει εισαχθεί ο Η- μι-Ομοιόμορφος χώρος. Οι διαφορές μεταξύ των δύο χώρων είναι τεράστιες και ακριβώς, εκεί έγκειται η δημιουργικότητα της νέας δομής. Έτσι, η πρώτη θεώρηση για τη διατριβή είναι η προσπάθεια επιστημόνων να βρούνε συνθήκες, ώστε να μπορεί μια τοπολογική αντιμεταθετική ημιομάδα ( S,.,τ) (με τη συνήθη έννοια των . και τ ) να εμφυτεύεται στη δομή η γνωστή ισοδυ- ναμία ad=bc αν ). Τα έξη πρώτα εδάφια είναι εισαγωγικά. Στη συνέχεια εκθέτουμε όλη τη μεθοδο- λογία του θέματος / --

Dans cette thèse nous étudions un super-analogue de l'espace de Teichmüller des surfaces à trous. Le but de notre étude est la construction sur cet espace de coordonnées analogues aux coordonnées de décalage de Thurston-Bonahon-Fock-Penner. Ces coordonnées dépendent du choix d'une triangulation idéale de la surface de départ. Nous étudions les changements de coordonnées lorsque l'on change cette triangulation de la surface. Nous démontrons également que cet espace possède une structure de Poisson canonique et que cette structure est indépendante du choix de la triangulation. / In this thesis we study a superanalogue of the Teichmüller space of surfaces with holes.The aim of our study is the construction of coordinates on this space which are analogousto the Thurston-Bonahon-Fock-Penner shear coordinates. These coordinates depend on a choice of an ideal triangulation of the given surface. We study the changes of coordinates when we modify the triangulation by elementary moves. We also show that this spaceadmits a canonical Poisson structure which is independent of the choice of a triangulation.