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A lubrication-flow model for a free film in a corner is presented. The model, written in the hyperbolic coordinate system ξ = x² – y², η = 2xy, applies to films that are thin in the η direction. The lubrication approximation yields two coupled evolution equations for the film thickness and the velocity field which, to lowest order, describes plug flow in the hyperbolic coordinates. A free film in a corner evolving under surface tension and gravity is investigated. The rate of thinning of a free film is compared to that of a film evolving over a solid substrate. Viscous shear and normal stresses are both captured in the model and are computed for the entire flow domain. It is shown that normal stress dominates over shear stress in the far field, while shear stress dominates close to the corner.

Solutions to differential equations describing the behavior of physical quantities (e.g., displacement, temperature, electric field strength) often only have finite range of validity over a subdomain. Interest beyond the subdomain often arises. As a result, the problem of making the solution compatible across the connecting subdomain interfaces must be dealt with. Four different compatibility methods are examined here for hyperbolic (time varying) second-order differential equations. These methods are used to match two different solutions, one in each subdomain along the connecting interface. The entire domain that is examined here is a unit square in the Cartesian plane. The four compatibility methods examined are: point collocation; optimal least square fit; penalty function; Ritz-Galerkin weak form. Discretized L2 convergence is used to examine and compare the effectiveness of each method.

Baseilhac and Benedetti have created a quantum hyperbolic knot invariant similar to the colored Jones polynomial. Their invariant is based on the polyhedral decomposition of the knot complement into ideal tetrahedra. The edges of the tetrahedra are assigned cross ratios based on their interior angles. Additionally, these edges are decorated with charges and flattenings which can be determined by assigning weights to the longitude and meridian of the boundary torus of a neighborhood of the knot. Baseilhac and Benedetti then use a summation of matrix dilogarithms to get their invariants. This thesis investigates these invariants for the figure eight knot. In fact, it will be shown that the volume of the complete hyperbolic structure of the knot serves as an upper bound for the growth of the invariants.

Thesis advisor: Ian Biringer / We are interested in proving the following statement: Given a 3-manifold M with boundary and a homeomorphism of the boundary f : ∂M → ∂M such that there is some power that extends to M, there is some k depending only on the genus g(∂M) and some l < k such that ƒᶩ extends to M. We will prove that the power needed to extend is not uniformly bounded with some examples, we will prove the statement is true if M is boundary incompressible and we will show that the general statement reduces to effectivising some technical results about pure homeomorphisms extending to compression bodies. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Hyperbolic conservation laws form a special class of partial differential equations. They describe phenomena that involve conserved quantities and their solutions show discontinuities which reflect the formation of shock waves. We consider one-dimensional systems of hyperbolic conservation laws and produce approximations using finite difference, finite volume and finite element methods. Due to stability issues of classical finite element methods for hyperbolic conservation laws, we study the discontinuous Galerkin method, which was recently introduced. The method involves completely discontinuous basis functions across each element and it can be considered as a combination of finite volume and finite element methods. We illustrate the implementation of discontinuous Galerkin method using Legendre polynomials, in case of scalar equations and in case of quasi-linear systems, and we review important theoretical results about stability and convergence of the method. The applications of finite volume and discontinuous Galerkin methods to linear and non-linear scalar equations, as well as to the system of elastodynamics, are exhibited.

Le présent travail est une contribution à la K-théorie bivariante des C*-algèbres au sens de Kasparov, et en particulier à sa version équivariante. Un rôle clé dans cette théorie est joué par l'élément"gamma" de Kasparov, une sorte de classe fondamentale équivariante d'un groupe localement compact. On s'intéresse à la représenter par desK-cycles (modules de Fredholm) possédant de bonnes propriétés.Dans cette thèse on donne une nouvelle construction de tels K-cyclespour les groupes hyperboliques au sens de Gromov. Les modules de Fredholm obtenus sont finiment sommables, i.e. ils possèdent une propriété de régularité particulièrement forte. On donne aussi une majoration de leur degré minimal de sommabilité.On s'inspire des travaux de V. Lafforgue: les K-cycles considérés sontsimilaires à ceux utilisés par Lafforgue dans sa démonstration de la Conjecture de Baum-Connes à coefficients pour les groupes hyperboliques. Leur construction est basée sur les idées de Mineyev sur les "bicombings homologiques" des groupes hyperboliques et procède par récurrence sur les squelettes d'un complexe de Rips associé au groupe.Une preuve non-constructive de la sommabilité finie d'un élément "gamma"a été obtenue par Emerson et Nica pour les groupes hyperboliques decaractéristique d'Euler-Poincaré zéro. Des constructions explicites deK-cycles représentant l'élément "gamma" d'un groupe hyperbolique ont étédonnées par Kasparov-Skandalis et V. Lafforgue, mais on ne sait passi leurs modules sont finiment sommables. En général, on ne peut pasespérer trouver des éléments "gamma" finiment sommables pour d'autresclasses de groupes discrets. / This work is a contribution to the bivariant K-theory of C*-algebras in the sense of Kasparov and in particular to its equivariant version. In this theory, a key role is played by Kasparov’s “gamma”-element, a kind of equivariant fundamental equivariant class for a locally compact group. It is of interest to find particularly well behaved K-cycles (Fredholm modules) representing this class.We present a new construction of K-cycles representing a "gamma"-element for hyperbolic groups in the sens of Gromov. The Fredholm modules obtained are finitely summable i.e. they possess particularly strong regularity properties. We also obtain an upper bound of their minimal degree of summability.Our approach is inspired by the work of V. Lafforgue: the K-cycles under consideration are similar to those used by Lafforgue in his demonstration of Baum-Connes conjecture with coefficients for hyperbolic groups. Their construction is based on Mineyev’s ideas on homological bicombings and proceeds by induction over the skeleta of a Rips complex associated to the group.A non-constructive proof of the finite summablity of a “gamma” element was obtained by Emerson and Nica for the hyperbolic groups of Euler-Poincaré characteristic zero. Explicit constructions of K-cycles representing the “gamma”-element of hyperbolic groups were given by Kasparov-Skandalis and V. Lafforgue, but it is not known whether their modules are finitely summable. In general one cannot hope to find finitely summable “gamma” elements for other classes of discrete groups.

This paper will investigate the geometry that results when Euclid's fifth postulate is replaced by the Lobachevskian assumption. The investigations will be limited to the plane.

The goal of this thesis is to present a comprehensive study of the parabolic and hyperbolic Anderson models with constant initial condition, driven by a Gaussian noise which is fractional in space with index H > 1/2 or H < 1/2, and is either white in time, or fractional in time with index H_0 > 1/2. As a preliminary step, we study the linear stochastic heat and wave equations with the same type of noise. In the case H_0 > 1/2 and H < 1/2, we present a new result, regarding the solution of the parabolic Anderson model with general initial condition given by a measure.

This dissertation presents a study of hyperbolic turbulence modelling for the Gaussian ten-moment equations. In gaskinetic theory, moment closures offer the possibility of deriving a series of gas-dynamic governing equations from the Boltzmann equation. One typical example, the Gaussian ten-moment model, which takes the form of hyperbolic-relaxation equations, is considered as a competitive model for viscous gas flow when heat transfer effects are negligible. The hyperbolic nature of this model gives it several numerical advantages, compared to the Navier-Stokes equations. However, until this study, the application of the ten-moment equations has been limited to laminar flows, due to the lack of appropriate turbulence models. In this work, the ten-moment equations are, for the first time, Reynolds-averaged. The resulting equations inherit the hyperbolic balance-law form from the original equations with new unknowns, which require approximation by turbulence models. Most of the traditional turbulence models for the Reynolds-averaged Navier-Stokes equations are not perfectly well-suited for the Reynolds-averaged ten-moment equations, because the second-order derivatives presented in these models can break the pure hyperbolic nature of the original model. The relaxation methods are therefore proposed in this project to reform the existing turbulence models. Two relaxation methods, the Chen-Levermore-Liu p-system and Cattaneo-Vernotte models, are used to hyperbolize the Prandtl’s one-equation model, standard k-ε model and Wilcox k-ω model. The hyperbolic versions of these turbulence models are first shown to be equivalent to their original forms. They are then coupled to the Reynolds-averaged ten-moment equations to build the overall hyperbolic governing equations for turbulence flows. An axisymmetric version of Reynolds-averaged ten-moment equations is also derived. A dispersion analysis is conducted for the resulting governing equations, which shows the corresponding dispersive behaviour and stability. The effect of the relaxation parameters is investigated through several numerical tests. All derived turbulence models are applied to solve canonical validation test problems, including two-dimensional planar mixing-layer, free-jet and circular free-jet. The numerical evaluations are analysed and compared against existing experimental measurements.