Dissertations / Theses on the topic 'Hyperbolic'
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1
Hawksley, Ruth. "Hyperbolic monopoles." Thesis, University of Edinburgh, 1998. http://hdl.handle.net/1842/14019.
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A Euclidean SU(2) monopole consists of a connection and Higgs field on an SU(2) bundle over π3, satisfying certain partial differential equations. Monopoles may equivalently be described in terms of holomorphic vector bundles on twistor space, algebraic curves in twistor space, rational maps, or solutions to Nahm's equations (a set of ODEs for matrix-valued functions), all satisfying some further conditions. Research by Atiyah, Donaldson, Hitchin, Nahm and others has provided a beautiful and relatively complete picture of these different viewpoints and the links between them. Monopoles have also been studied on hyperbolic space π3, although the corresponding picture in this case is less well understood. One difficulty is that the conditions which must be imposed in order for all the various correspondences to be valid have not yet been completely determined. A partial answer is given in Chapter 2, where it is proved that any hyperbolic monopole arising from a spectral curve satisfies a certain natural boundary condition. The proof uses the algebraic geometry of the spectral curve and is similar to Hurtubise's proof of the analogous result in the Euclidean case. A large part of this thesis concentrates on the "Braam-Austin" description of hyperbolic monopoles. This is the hyperbolic version of Nahm's description of Euclidean monopoles; a monopole corresponds to a pair of discrete matrix-valued functions satisfying some difference equations. Euclidean monopoles appear as limits of hyperbolic monopoles as the curvature of π3 tends to zero. This "Euclidean limit" is described geometrically and is studied in terms of Braam-Austin data. Explicit conditions are given for such a sequence to have a subsequence converging to a Euclidean monopole. The result depends on a conjecture (§ 4.5) about properties of Braam-Austin monopole solutions.
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Lê, Nguyên Khoa 1975. "Time-frequency analyses of the hyperbolic kernel and hyperbolic wavelet." Monash University, Dept. of Electrical and Computer Systems Engineering, 2002. http://arrow.monash.edu.au/hdl/1959.1/8299.
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Vlamis, Nicholas George. "Identities on hyperbolic manifolds and quasiconformal homogeneity of hyperbolic surfaces." Thesis, Boston College, 2015. http://hdl.handle.net/2345/bc-ir:104137.
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Thesis advisor: Martin J. BridgemanThesis advisor: Ian Biringer
The first part of this dissertation is on the quasiconformal homogeneity of surfaces. In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces. We then move on to identities on hyperbolic manifolds. We study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function
Thesis (PhD) — Boston College, 2015
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics
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Ray, Gourab. "Hyperbolic random maps." Thesis, University of British Columbia, 2014. http://hdl.handle.net/2429/48417.
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Random planar maps have been an object of utmost interest over the last
decade and half since the pioneering works of Benjamini and Schramm, Angel and Schramm and Chassaing and Schaeffer. These maps serve as models
of random surfaces, the study of which is very important with motivations
from physics, combinatorics and random geometry.
Uniform infinite planar maps, introduced by Angel and Schramm, which
are obtained as local limits of uniform finite maps embedded in the sphere,
serve as a very important discrete model of infinite random surfaces. Recently, there has been growing interest to create and understand hyperbolic
versions of such uniform infinite maps and several conjectures and proposed
models have been around for some time. In this thesis, we mainly address
these questions from several viewpoints and gather evidence of their existence and nature.
The thesis can be broadly divided into two parts. The first part is
concerned with half planar maps (maps embedded in the upper half plane)
which enjoy a certain domain Markov property. This is reminiscent of that
of the SLE curves. Chapters 2 and 3 are mainly concerned with classi cation
of such maps and their study, with a special focus on triangulations. The
second part concerns investigating unicellular maps or maps with one face
embedded in a high genus surface. Unicellular maps are generalizations of
trees in higher genera. The main motivation is that investigating such maps
will shed some light into understanding the local limit of general maps via
some well-known bijective techniques. We obtain certain information about
the large scale geometry of such maps in Chapter 4 and about the local limit
of such maps in Chapter 5.Science, Faculty of
Mathematics, Department of
Graduate
5
Moussong, Gabor. "Hyperbolic Coxeter groups." Connect to this title online, 1988. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1112044027.
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Bult, Fokko Joppe van de. "Hyperbolic hypergeometric functions." [Amsterdam] : Amsterdam : Thomas Stieltjes Institute for Mathematics ; Universiteit van Amsterdam [Host], 2007. http://dare.uva.nl/document/97725.
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Koundouros, Stilianos. "Hyperbolic 3-manifolds." Thesis, University of Cambridge, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.615624.
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Markham, Sarah. "Hypercomplex hyperbolic geometry." Thesis, Durham University, 2003. http://etheses.dur.ac.uk/3698/.
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The rank one symmetric spaces of non-compact type are the real, complex, quaternionic and octonionic hyperbolic spaces. Real hyperbolic geometry is widely studied complex hyperbolic geometry less so, whilst quaternionic hyperbolic geometry is still in its infancy. The purpose of this thesis is to investigate the conditions for discrete group action in quaternionic and octonionic hyperbolic 2-spaces and their geometric consequences, in the octonionic case, in terms of lower bounds on the volumes of non-compact manifolds. We will also explore the eigenvalue problem for the 3 x 3 octonionic matrices germane to the Jordan algebra model of the octonionic hyperbolic plane. In Chapters One and Two we concentrate on discreteness conditions in quaternionic hyperbolic 2-space. In Chapter One we develop a quaternionic Jørgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic 2-space generated by two elements, one of which is either loxodromic or boundary elliptic. In Chapter Two we give a generalisation of Shimizu's Lemma to groups of isometries of quaternionic hyperbolic 2-space containing a screw-parabolic element. In Chapter Three we present the Jordan algebra model of the octonionic hyperbolic plane and develop a generalisation of Shimizu's Lemma to groups of isometries of octonionic hyperbolic 2-space containing a parabolic map. We use this result to determine estimates of lower bounds on the volumes of non-compact closed octonionic 2-manifolds. In Chapter Four we construct an octonionic Jørgensen's inequality for non-elementary groups of isometries of octonionic hyperbolic 2-space generated by two elements, one of which is loxodromic. In Chapter Five we solve the real eigenvalue problem Xv = λv, for the 3 x 3 ɸ-Hermitian matrices, X, of the Jordan algebra model of the octonionic hyperbolic plane. Finally, in Chapter Six we consider the embedding of collars about real geodesies in complex hyperbohc 2-space, quaternionic hyperbolic 2-space and octonionic hyperbolic 2-space.
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ALMEIDA, HELLEN ANGELICA DA SILVA. "HYPERBOLIC COXETER GROUPS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2009. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=32643@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
Grupos de Coxeter ou de reflexões são importantes no estudo de inúmeras áreas da matemática, incluindo grupos e álgebras de Lie. Nesta dissertação apresentaremos a teoria básica de grupos de reflexões e a classificação dos grupos hiperbólicos, i.e., daqueles que agem no espaço hiperbólico tendo como domínio fundamental um politopo compacto.
Groups of Coxeter or of reflections they are important in the study of countless areas of the mathematics, including groups and algebras of Lie. In this dissertation we will present the basic theory of groups of reflections and the classification of the hyperbolic groups, this is of those that act in the hyperbolic space tends as fundamental domain a compact politopo.
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Marshall, Joseph. "Computation in hyperbolic groups." Thesis, University of Warwick, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.369403.
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Laibson, David I. "Hyperbolic discounting and consumption." Thesis, Massachusetts Institute of Technology, 1994. http://hdl.handle.net/1721.1/11966.
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Thompson, James Matthew. "Complex hyperbolic triangle groups." Thesis, Durham University, 2010. http://etheses.dur.ac.uk/478/.
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We prove several discreteness and non-discreteness results about complex hyperbolic triangle groups and discover two new lattices. These results use geometric (explicit construction of a fundamental domain), group theoretic and arithmetic methods.
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McLeod, John Angus. "Arithmetic hyperbolic reflection groups." Thesis, Durham University, 2013. http://etheses.dur.ac.uk/7743/.
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This thesis uses Vinberg’s algorithm to study arithmetic hyperbolic reflection groups which are contained in the groups of units of quadratic forms. We study two families of quadratic forms: the diagonal forms −dx_0^2 + x_1^2 + ... + x_n^2 ; and the forms whose automorphism groups contain the Bianchi groups. In the first instance we classify over Q the pairs (d,n) for which such a group can be found, and in some cases we can compute the volumes of the fundamental polytopes. In the second instance we use a combination of the geometric and number theoretic information to classify the reflective Bianchi groups by first classifying the reflective extended Bianchi groups, namely the maximal discrete extension of the Bianchi groups in PSL(2,C). Finally we identify some quadratic forms in the first instance and completely classify those in the second which have a quasi-reflective structure.
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Howard, Tamani M. "Hyperbolic Monge-Ampère Equation." Thesis, University of North Texas, 2006. https://digital.library.unt.edu/ark:/67531/metadc5322/.
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In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.
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Monaghan, Andrew. "Complex hyperbolic triangle groups." Thesis, University of Liverpool, 2013. http://livrepository.liverpool.ac.uk/14033/.
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In this thesis we study the discreteness criteria for complex hyperbolic triangle groups, generated by reflections in the complex hyperbolic 2-space. A complex hyperbolic triangle group is a group of isometries of the complex hyperbolic plane generated by three complex reflections. We study discreteness of some of these groups using arithmetic and geometric methods. We show that certain complex hyperbolic triangle groups of signature (p,p,2p) and (p,q,pq/(q-p)) are not discrete. The arithmetic methods we use are those studied by Conway and Jones and Parker. We also extend these results further. We finally give an area of discreteness for complex hyperbolic triangle groups of signature [m,n,0] using the compression property.
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Bowen, Lewis Phylip. "Density in hyperbolic spaces." Access restricted to users with UT Austin EID Full text (PDF) from UMI/Dissertation Abstracts International, 2002. http://wwwlib.umi.com/cr/utexas/fullcit?p3077409.
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Billington, Stephen. "Topics in hyperbolic groups." Thesis, University of Warwick, 1999. http://wrap.warwick.ac.uk/110993/.
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Hyperbolic groups are a class of groups introduced by Gromov in 1987, which form an important part of geometric group theory. In Chapter 1, we give an introduction to this subject. In Chapter 2, we use the theory of complexes of groups to show that the integral homology and cohomology groups of a hyperbolic group are computable by a Turing machine. In Chapter 3, we present the boundary of a hyperbolic group as an inverse limit of topological spaces and use this to give computable estimates for properties of the boundary. In Chapter 4, we investigate symbolic dynamic properties concerning hyperbolic groups. In paricular, we give symbolic codings for the actions on the boundary of a hyperbolic and actions on the geodesic flow on a hyperbolic group. In Chapter 5 we investigate the problem of determining when graphs are Cayley graphs. The graphs which we are concerned with are regular and semi-regular planar graphs.
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Dincgez, Umut Can. "Three Dimensional Hyperbolic Grid Generation." Master's thesis, METU, 2006. http://etd.lib.metu.edu.tr/upload/2/12607147/index.pdf.
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This thesis analyzes procedure of generation of hyperbolic grids formulated by two constraints, which specify grid orthogonality and cell volume. The procedure was applied on a wide range of geometries and high quality two and three dimensional hyperbolic grids were generated by using grid control and smoothing procedures, which supply grid clustering in all directions and prevent grid deformation (grid shock), respectively.
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Thomson, Scott Andrew. "Short geodesics in hyperbolic manifolds." Thesis, Durham University, 2012. http://etheses.dur.ac.uk/3604/.
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Given a closed Riemannian n-manifold M, its shortest closed geodesic is called its systole and the length of this geodesic is denoted syst_1(M). For any ε > 0 and any n at least 2 one may construct a closed hyperbolic n-manifold M with syst_1(M) at most equal to ε. Constructions are detailed herein. The volume of M is bounded from below, by A_n/syst_1(M)^(n−2) where A_n is a positive constant depending only on n. There also exist sequences of n-manifolds M_i with syst_1(M_i) → 0 as i → ∞, such that vol(M_i) may be bounded above by a polynomial in 1/syst_1(M_i). When ε is sufficiently small, the manifold M is non-arithmetic, so that its fundamental group is an example of a non-arithmetic lattice in PO(n,1). The lattices arising from this construction are also exhibited as examples of non-coherent groups in PO(n,1). Also presented herein is an overview of existing results in this vein, alongside the prerequisite theory for the constructions given.
20
Bowditch, B. H. "Geometrical finiteness for hyperbolic groups." Thesis, University of Warwick, 1988. http://wrap.warwick.ac.uk/99188/.
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In this paper, we describe various definitions of geometrical finiteness for discrete hyperbolic groups in any dimension, and prove their equivalence. This generalises what has been worked out in two and three dimensions by Marden. Beardon, Maskit, Thurston and others. We also discuss the nature of convex fundamental domains for such groups. We begin the paper with a discussion of results related to the Margulls Lemma and Bieberbach Theorems.
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Marshall, T. H. (Timothy Hamilton). "Hyperbolic Geometry and Reflection Groups." Thesis, University of Auckland, 1994. http://hdl.handle.net/2292/2140.
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The n-dimensional pseudospheres are the surfaces in Rn+l given by the equations x12+x22+...+xk2-xk+12-...-xn+12=1(1 ≤ k ≤ n+1). The cases k=l, n+1 give, respectively a pair of hyperboloids, and the ordinary n-sphere. In the first chapter we consider the pseudospheres as surfaces h En+1,k, where Em,k=Rk x (iR)m-k, and investigate their geometry in terms of the linear algebra of these spaces. The main objects of investigation are finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices. In the second chapter we derive integral formulae for the volume of an n-dimensional spherical or hyperbolic simplex, both in terms of its dihedral angles and its edge lengths. For the regular simplex with common edge length γ we then derive power series for the volume, both in u = sinγ/2, and in γ itself, and discuss some of the properties of the coefficients. In obtaining these series we encounter an interesting family of entire functions, Rn(p) (n a nonnegative integer and pεC). We derive a functional equation relating Rn(p) and Rn-1(p). Finally we classify, up to isometry, all tetrahedra with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncatons. are all π/2, and the remaining dihedral angles are all sub-multiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups. For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary In particular, we find a sequence of manifolds with totally geodesic boundary of genus, g≥2, which we conjecture to be of least volume among such manifolds.
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Friel, Karren Janet. "Decision problems in hyperbolic groups." Thesis, University of Newcastle Upon Tyne, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.391970.
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Young, R. A. W. "The Uniform Hyperbolic Umbilic Approximation." Thesis, University of Manchester, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.376158.
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Woodward, J. M. "Integral lattices and hyperbolic manifolds." Thesis, University of York, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.441042.
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Yaman, Asli. "Boundaries of relatively hyperbolic groups." Thesis, University of Southampton, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.432635.
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Crampton, Benedict. "Hyperbolic braneworld backgrounds in supergravity." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/27229.
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The manifolds Hp,q [Symbol appears here. To view, please open pdf attachment] are a family of non-compact hyperboloids carrying inhomogeneous Euclidean metrics. In supergravity they appear as an interesting class of dimensional reductions, related to the well known sphere reductions by a simple analytic continuation. The spectrum of lower dimensional modes in these backgrounds is still poorly understood. In this thesis, we construct the complete Pauli reduction of type IIA supergravity on H2,2 [Symbol appears here. To view, please open pdf attachment] . We carefully analyse the spectrum of gravitational waves in the resulting Salam-Sezgin background, and identify the boundary conditions needed to render these modes normalisable. We give these boundary conditions a codimension-2 braneworld interpretation. We then exhibit a supersymmetric braneworld geometry based on the NS5-brane. In the remainder of this thesis we apply holographic methods to the problem of the fractionalisation transition in condensed matter theory. We exhibit a phase transition between a superconducting and a fractionalised phase in a bottom-up Einstein-Maxwell-Dilaton theory, and discuss the importance of entropy scaling in achieving this.
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Alabdullatif, Amal. "Hyperbolic variants of Poncelet's theorem." Thesis, University of Southampton, 2016. https://eprints.soton.ac.uk/415515/.
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In 1813, J. Poncelet proved his beautiful theorem in projective geometry, Poncelet's Closure Theorem, which states that: if C and D are two smooth conics in general position, and there is an n-gon inscribed in C and circumscribed around D, then for any point of C, there exists an n-gon, also inscribed in C and circumscribed around D, which has this point for one of its vertices. There are some formulae related to Poncelet's Theorem, in which introduce relations between two circles' data (their radii and the distance between their centres), when there is a bicentric n-gon between them. In Euclidean geometry, for example, we have Chapple's and Fuss's Formulae. We introduce a proof that Poncelet's Theorem holds in hyperbolic geometry. Also, we present hyperbolic Chapple's and Fuss's Formulae, and more general, we prove a Euclidean general formula, and two version of hyperbolic general formulae, which connect two circles' data, when there is an embedded bicentric n-gon between them. We formulate a conjecture that the Euclidean formulae should appear as a factor of the lowest order terms of a particular series expansion of the hyperbolic formulae. Moreover, we dene a three-manifold X, constructed from n = 3 case of Poncelet's Theorem, and prove that X can be represented as the union of two disjoint solid tori, we also prove that X is Seifert fibre space.
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Mondal, Sugata. "Small eigenvalues of hyperbolic surfaces." Toulouse 3, 2013. http://thesesups.ups-tlse.fr/2233/.
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Une surface hyperbolique est une variete complete S de dimension 2 de courbure Sectionnelle egale a -1. Dans cette these on considere l'action du Laplacien de cette metrique. On appelle petite valeur propre toute valeur propre inferieure ou egale a 1/4. Notre theme general de recherche est de borner le nombre de valeurs propres en fonction de la topologie de S lorsque S est d'aire finie. Un theoreme d'Otal-Rosas qui dit que le nombre de petites valeurs propres d'une surface hyperbolique de genre g est au plus 2g-2, confirmant une conjecture de Buser. Nous donnons une version quantitative de ce resultat en donnant la minoration {\lambda_{2g-2}}(S)> 1/4 +{\epsilon_0}(S) pour une fonction {\epsilon_0}(S) > 0 explicite qui ne depend que de la geometrie de S. Notre demonstration utilise des inegalites geometriques comme celle de Faber-Krahn ou celle de Cheeger. Il est conjecture d'autre part que pour une surface hyperbolique non compacte de type (g, n), le nombre de petites valeurs propres paraboliques est ><=2g- 3. Nous montrons que sur un ouvert non-vide de l'espace modulaire Mg;n, ce nombre de valeurs propres est <= 2g- 2. Notre demonstration est basee sur un theoreme decrivant le comportement d'une suite de petites fonctions propres paraboliques sur des surfaces Sm qui tendent vers le bord de l'espace modulaire, et qui est motive par des des resultats de Lizhen Ji et de Scott Wolpert. Nous utilisons aussi ce theoreme pour donner une demonstration nouvelle et elementaire d'un resultat de D. Hejhal. Dans le dernier chapitre, nous etudions le maximum de {\lambda_1} vue comme fonction sur Mg, plus precisement nous nous demandons si ce maximum est superieur a 1/4. En utilisant des arguments topologiques, nous montrons que c'est bien le cas en genre 2 : il y a des surfaces dans M2 pour lesquelles {\lambda_1} > 1/4A hyperbolic surface S is a complete two dimensional manifold of sectional curvature -1. In this thesis we consider the Laplace operator associated to this metric (acting on functions). Any eigenvalue below 1/4 is called a small eigenvalue. The general theme of our research is to bound the number of small eigenvalues of S in terms of the topology of S when S has finite area. A theorem of Otal-Rosas says that the number of small eigenvalues of a closed hyperbolic surface of genus g is not more than 2g -2, confirming a conjecture of P. Buser. We prove a quantitative version of this result by giving the lower bound for the (2g- 2)-th eigenvalue : {\lambda_{2g-2}}(S) > 1/4 +{\epsilon_0}(S) where {\epsilon_0}(S) > 0 is an explicit function that depends only on the geometry of S. Our proof uses geometric inequalities of Faber-Krahn and of Cheeger. For a hyperbolic surface of finite area and type (g, n) it is a conjecture that the number of small cuspidal eigenvalues is <= 2g- 3. We show that on a non-empty open unbounded subset of the moduli space Mg;n, this number of eigenvalues is <= 2g -2. The proof is based on a theorem, motivated by results of Lizhen Ji and Scott Wolpert, that describes the behavior of small cuspidal eigenfunctions of surfaces Sm when the sequence (Sm) tends to the boundary of the moduli space. We use this theorem to give a new and elementary proof of a result of D. Hejhal also. In the last chapter, we study the maximum of {\lambda_1} viewed as a function on Mg. More precisely, we ask if the maximum is more than 1/4. Using topological arguments, we prove that in the case for genus two : there exist surfaces in Mg for which {\lambda_1} > 1/4
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Murray, Marilee Anne. "Hyperbolic Geometry and Coxeter Groups." Bowling Green State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1343040882.
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Steinberg, Daniel Howard. "Elastic curves in hyperbolic space." Case Western Reserve University School of Graduate Studies / OhioLINK, 1995. http://rave.ohiolink.edu/etdc/view?acc_num=case1058277066.
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Walker, Mairi. "Continued fractions and hyperbolic geometry." Thesis, Open University, 2016. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.700134.
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This thesis uses hyperbolic geometry to study various classes of both real and complex continued fractions. This intuitive approach gives insight into the theory of continued fractions that is not so easy to obtain from traditional algebraic methods. Using it, we provide a more extensive study of both Rosen continued fractions and even-integer continued fractions than many previous works, yielding new results, and revisiting classical theorems. We also study two types of complex continued fractions, namely Gaussian integer continued fractions and Bianchi continued fractions. As well as providing a more elegant and simple theory of continued fractions, our approach leads to a natural generalisation of continued fractions that has not been explored, before.
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Agol, Ian. "Topology of hyperbolic 3-manifolds /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1998. http://wwwlib.umi.com/cr/ucsd/fullcit?p9906477.
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Naeve, Trent Phillip. "Conics in the hyperbolic plane." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3075.
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An affine transformation such as T(P)=Q is a locus of an affine conic. Any affine conic can be produced from this incidence construction. The affine type of conic (ellipse, parabola, hyperbola) is determined by the invariants of T, the determinant and trace of its linear part. The purpose of this thesis is to obtain a corresponding classification in the hyperbolic plane of conics defined by this construction.
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Heard, Damian. "Computation of hyperbolic structures on 3 dimensional orbifolds /." Connect to theis, 2005. http://eprints.unimelb.edu.au/archive/00001577.
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Canadell, Cano Marta. "Computation of Normally Hyperbolic Invariant Manifolds." Doctoral thesis, Universitat de Barcelona, 2014. http://hdl.handle.net/10803/277215.
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The subject of the theory of Dynamical Systems is the evolution of systems with respect to time. Hence, it has many applications to other areas of science, such as Physics, Biology, Economics, etc. and it also has interactions with other parts of Mathematics.
The global behavior of a dynamical system is organized by its invariant objects, the simplest ones are equilibria and periodic orbits (and related invariant manifolds). Normally hyperbolic invariant manifolds (NHIM for short) are some of these invariant objects. They have the property to persist under small perturbations of the system. These NHIM are characterized by the fact that the directions on the points of the manifold split into stable, unstable and tangent components. The growth rate of stable directions (for which forward evolution of the system goes to zero) and unstable directions (for which backward evolution goes to zero) dominate the growth rate of the tangent directions. The robustness of normally hyperbolic invariant manifolds makes them very useful to understand the global dynamics. Both the theory and the computation of these objects are important for the general understanding of a dynamical system.
The main goal of my thesis is to develop efficient algorithms for the computation of normally hyperbolic invariant manifolds, give a rigorous mathematical theory and implement them to explore new mathematical phenomena. For simplicity, we consider the problem for discrete dynamical systems, since it is known that the discrete case implies the continuous case using time one flow. We consider a diffeomorphism F : Rm → Rm and a d-torus parameterized by K : Td → Rm which is invariant under F. This means that there exists a diffeomorphism f : Td → Td (the internal dynamics) such that it satisfies F ◦ K = K ◦ f, (0.3) called the invariance equation.
Our goal is to solve this invariance equation considering two different scenarios: one in which we do not know the internal dynamics of the invariant torus (where K and f are our unknowns), see Chapter 4, and the other in which we impose that the internal dynamics is a rigid rotation with a quasi-periodic frequency (where K is the unknown and f is the rigid rotation), for which we also need to add an adjusting parameter to equation (0.3), see Chapters 2 and 3. Additionally, in both cases we are also interested in computing the invariant tangent and normal bundles.L’objecte d’estudi dels Sistemes Dinàmics és l’evolució dels sistemes respecte del temps. Per aquesta raó, els Sistemes Dinàmics presenten moltes aplicacions en altres àrees de la Ciència, com ara la Física, Biologia, Economia, etc. i tenen nombroses interaccions amb altres parts de les Matemàtiques. Els objectes invariants organitzen el comportament global d’un sistema dinàmic, els més simples dels quals són els punts fixos i les òrbites periòdiques (així com les seves corresponents varietats invariants). Les Varietats Invariants Normalment Hiperbòliques (NHIM forma abreviada provinent de l’anglès) són alguns d’aquests objectes invariants. Aquests objectes posseeixen la propietat de persistir sota petites pertorbacions del sistema. Les NHIM estan caracteritzades pel fet que les direccions en els punts de la varietat presenten una divisió en components tangent, estable i inestable. L’índex de creixement de les direccions estables (per les quals la iteració endavant del sistema tendeix cap a zero) i inestables (per les quals la iteració enrere del sistema tendeix cap a zero) domina l’índex de creixement de les direccions tangents. La robustesa de les varietats invariants normalment hiperbòliques les fa de gran utilitat a l’hora d’estudiar la dinàmica global. Per aquesta raó, tant la teoria com el càlcul d’aquests objectes sós molt importants per al coneixement general d’un sistema dinàmic. L’objectiu principal d’aquesta tesi és desenvolupar algoritmes eficients pel càlcul de varietats invariants normalment hiperbòliques, donar-ne resultats teòrics rigorosos i implementar-los per a explorar nous fenòmens matemàtics. Per simplicitat, considerarem el problema per a sistemes dinàmics discrets, ja que és ben conegut que el cas discret implica el cas continu usant operadors d’evolució. Considerem així difeomorfismes donats per F : Rm → Rm i un d-tor F-invariant parametritzat per K : Td → Rm. És a dir, existeix un difeomorfisme f : Td → Td (la dinàmica interna) tal que satisfà l’equació F ◦ K = K ◦ f, (0.1) anomenada equació d’invariància. La nostra finalitat és solucionar aquesta equació d’invariància considerant dos possibles escenaris: un en el qual no coneixem quina és la dinàmica interna del tor (on K i f són les nostres incògnites), veure Capítol 4, i un altre en el qual imposem que la dinàmica interna sigui una rotació rígida amb freqüència quasi-periòdica (on K és una incògnita i f és la rotació rígida), pel qual necessitarem, a més a més, afegir un paràmetre ajustador a l’equació (0.1), veure Capítols 2 i 3. En ambdós casos també estarem interessats en el càlcul dels fibrats invariants tangent i normals.
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Leclerc, Marc-Antoine. "The Hyperbolic Formal Affine Demazure Algebra." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/35218.
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In this thesis, we extend the construction of the formal (affine) Demazure algebra due to Hoffnung, Malagón-López, Savage and Zainoulline in two directions. First, we introduce and study the notion of formal Demazure lattices of a Kac-Moody root system and show that the definitions and properties of the formal (affine) Demazure operators and algebras hold for such lattices. Second, we show that for the hyperbolic formal group law the formal Demazure algebra is isomorphic (after extending the coefficients) to the Hecke algebra.
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Al-Nayef, Anwar Ali Bayer, and mikewood@deakin edu au. "Semi-hyperbolic mappings in Banach spaces." Deakin University. School of Computing and Mathematics, 1997. http://tux.lib.deakin.edu.au./adt-VDU/public/adt-VDU20051208.110247.
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The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional.
Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations.
Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation.
It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations.
Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is chaotic.
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Kuhlmann, Sally Malinda. "Geodesic knots in hyperbolic 3 manifolds." Connect to thesis, 2005. http://repository.unimelb.edu.au/10187/916.
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This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds.Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded.
The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially.
We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.
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Bahuaud, Eric. "Intrinsic characterization of asymptotically hyperbolic metrics /." Thesis, Connect to this title online; UW restricted, 2007. http://hdl.handle.net/1773/5781.
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Hardin, Douglas Patten. "Hyperbolic iterated function systems and applications." Diss., Georgia Institute of Technology, 1985. http://hdl.handle.net/1853/30864.
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Mansouri, Abdol-Reza 1962. "The variational bicomplex for hyperbolic equations /." Thesis, McGill University, 2001. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=33805.
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This thesis presents the geometric investigation of hyperbolic partial differential equations in the plane as carried out by Niky Kamran, Ian Anderson, and Martin Juras. In particular, the relation between the Darboux integrability of an arbitrary hyperbolic equation and the Laplace invariants of the linearization of this equation is established. This extends to non-linear hyperbolic equations in the plane a classical result of Goursat for linear hyperbolic equations. The formal setting for this geometric investigation is afforded by the constrained variational bicomplex, which allows the solution to a partial differential equation to be viewed as a manifold on which standard differential geometric operations such as exterior differentiation and Lie differentiation can be performed. The key element in this investigation is the judicious construction and use of appropriate moving coframes which will reflect the properties of the equations under investigation.
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Cheng, Kan. "Hyperbolic conservation laws with source terms." Thesis, University of Oxford, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.343042.
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Thorgeirsson, Sverrir. "Hyperbolic geometry: history, models, and axioms." Thesis, Uppsala universitet, Algebra och geometri, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227503.
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Velani, Sanju Lalji. "Metric diophantine approximation in hyperbolic space." Thesis, University of York, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.304351.
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Bode, Michel. "Random graphs on the hyperbolic plane." Thesis, University of Birmingham, 2016. http://etheses.bham.ac.uk//id/eprint/6526/.
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In this thesis, we study a recently proposed model of random graphs that exhibit properties which are present in a wide range of networks arising in real world settings. The model creates random geometric graphs on the hyperbolic plane, where vertices are connected if they are within a certain threshold distance. We study typical properties of these graphs. We identify two critical values for one of the parameters that act as sharp thresholds. The three resulting intervals of the parameters that correspond to three possible phases of the random structure: A.a.s., the graph is connected; A.a.s., the graph is not connected, yet there is a giant component; A.a.s., every component is of sublinear size. Furthermore, we determine the behaviour at the critical values. We also consider typical distances between vertices and show that the ultra-small world phenomenon is present. Our results imply that most pairs of vertices that belong to the giant component are within doubly logarithmic distance.
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Cockburn, Alexander Hugh. "Aspects of vortices and hyperbolic monopoles." Thesis, Durham University, 2015. http://etheses.dur.ac.uk/11107/.
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This thesis discusses BPS monopoles in hyperbolic space and BPS vortices in the presence of magnetic impurities. We prove explicit formulae for the spectral curve and rational map of a JNR-type hyperbolic monopole, and we use these to study some Platonic examples as well as some new 1-parameter families analogous to Euclidean monopole scattering. Explicit fields and Braam-Austin data for axial hyperbolic monopoles of a particular mass are derived using a correspondence to 1-monopoles, and this data is deformed to give new 1-parameter families. Numerical techniques are used to study the effect of magnetic impurities on vortices on a flat background. Analytic results for vortices with magnetic impurities are found by adapting previous results on vortices on the hyperbolic plane and the 2-sphere.
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Braam, Peter J. "Magnetic monopoles and hyperbolic three-manifolds." Thesis, University of Oxford, 1987. http://ora.ox.ac.uk/objects/uuid:daa73d43-6d58-404c-9926-ebf23f59cfc6.
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Let M = H3/Γ be a complete, non-compact, oriented geometrically finite hyperbolic 3-manifold without cusps. By constructing a conformal compactification of M x S1 we functorially associate to M an oriented, conformally flat, compact 4-manifold X (without boundary) with an S1-action. X determines M as a hyperbolic manifold. Using our functor and the differential geometry of conformally flat 4-manifolds we prove that any Γ as above with a limit set of Hausdorff dimension ≤ 1 is Schottky, Fuchsian or extended Fuchsian. Furthermore, the Hodge theory for H2 (X;R) carries over to H1(M, δM;R) and H2(M;R) which correspond to the spaces of harmonic L2-forms of degree 1 and 2 on M. Comparison of lattices through the Hodge star gives an invariant h(M) ε GL(H2(M;R)/GL(H2(M;Z)) of the hyperbolic structure. Secondly we pay attention to magnetic monopoles on M which correspond to S1invariant solutions of the anti-self-duality equations on X. The basic result is that we associate to M an infinite collection of moduli spaces of monopoles , labelled by boundary conditions. We prove that the moduli spaces are not empty (under reasonable conditions), compute their dimension , prove orientability , the existence of a compactification and smoothness for generic S1-invariant conformal structures on X. For these results one doesn't need a hyperbolic structure on M , the existence of a conformal compactification X suffices. A twistor description for monopoles on a hyperbolic M can be given through the twistor space of X , and monopoles turn out to correspond to invariant holomorphic bundles on twistor space. We analyse these bundles. Explicit formulas for monopoles can be found on handlebodies M , and for M = surface x R we describe the moduli spaces in some detail.
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Saleh, Ibrahim A. "Cluster automorphisms and hyperbolic cluster algebras." Diss., Kansas State University, 2012. http://hdl.handle.net/2097/14195.
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Doctor of PhilosophyDepartment of Mathematics
Zongzhu Lin
Let A[subscript]n(S) be a coefficient free commutative cluster algebra over a field K. A cluster automorphism is an element of Aut.[subscript]KK(t[subscript]1,[dot, dot, dot],t[subscript]n) which leaves the set of all cluster variables, [chi][subscript]s invariant. In Chapter 2, the group of all such automorphisms is studied in terms of the orbits of the symmetric group action on the set of all seeds of the field K(t[subscript]1,[dot,dot, dot],t[subscript]n). In Chapter 3, we set up for a new class of non-commutative algebras that carry a non-commutative cluster structure. This structure is related naturally to some hyperbolic algebras such as, Weyl Algebras, classical and quantized universal enveloping algebras of sl[subscript]2 and the quantum coordinate algebra of SL(2). The cluster structure gives rise to some combinatorial data, called cluster strings, which are used to introduce a class of representations of Weyl algebras. Irreducible and indecomposable representations are also introduced from the same data. The last section of Chapter 3 is devoted to introduce a class of categories that carry a hyperbolic cluster structure. Examples of these categories are the categories of representations of certain algebras such as Weyl algebras, the coordinate algebra of the Lie algebra sl[subscript]2, and the quantum coordinate algebra of SL(2).
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Schrecker, Matthew. "Hyperbolic problems in fluids and relativity." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:2dec2eb9-4253-4625-a071-0c19d0c1f76d.
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In this thesis, we present a collection of newly obtained results concerning the existence of vanishing viscosity solutions to the one-dimensional compressible Euler equations of gas dynamics, with and without geometric structure. We demonstrate the existence of such vanishing viscosity solutions, which we show to be entropy solutions, to the transonic nozzle problem and spherically symmetric Euler equations in Chapter 4, in both cases under the simple and natural assumption of relative finite-energy. In Chapter 5, we show that the viscous solutions of the one-dimensional compressible Navier-Stokes equations converge, as the viscosity tends to zero, to an entropy solution of the Euler equations, again under the assumption of relative finite-energy. In so doing, we develop a compactness framework for the solutions and approximate solutions to the Euler equations under the assumption of a physical pressure law. Finally, in Chapter 6, we consider the Euler equations in special relativity, and show the existence of bounded entropy solutions to these equations. In the process, we also construct fundamental solutions to the entropy equations and develop a compactness framework for the solutions and approximate solutions to the relativistic Euler equations.
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De, Capua Antonio. "Hyperbolic volume estimates via train tracks." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:426f7186-e881-482b-90d8-5cbb9b9a38b7.
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In this thesis we describe how to estimate the distance spanned in the pants graph by a train track splitting sequence on a surface, up to multiplicative and additive constants. If some moderate assumptions on a splitting sequence are satisfied, each vertex set of a train track in it will represent a vertex of a graph which is naturally quasi-isometric to the pants graph; moreover the splitting sequence gives an edge-path in this graph so, more precisely, our distance estimate holds between the extreme points of this path. The present distance estimate is inspired by a result of Masur, Mosher and Schleimer for distances in the marking graph. However, we can apply their line of proof only after some manipulation of the splitting sequence: a rearrangement, changing the order the elementary moves are performed in, so that the ones producing Dehn twists are brought together; and then an untwisting, which suppresses the majority of these latter moves to give a new sequence, which does not end with the same track as before, but does not include any portion that is almost stationary in the pants graph. The required distance is then, up to constants, the number of splits occurring in the untwisted sequence. A consequence of our main theorem together with a result of Brock is that, given a pseudo-Anosov self-diffeomorphism Ï of a surface S, the maximal splitting sequence introduced by Agol gives us an estimate for the hyperbolic volume of the mapping torus built from S and Ï. There are also some interesting consequences for the hyperbolic volume of a solid torus minus a closed braid, via a machinery employed by Dynnikov and Wiest.