In recent years, effort has been put into following the ideas of M. Ruzhansky and V. Turunen to construct a global pseudo-differential calculus on Lie groups. By this, we mean a collection of operators containing the left-invariant differential calculus with the additional requirement that it be stable under composition and adjunction. Moreover, we would like these operators to have adequate boundedness properties between Sobolev spaces. Our approach consists in using the group Fourier transform to defne a global, operator-valued symbol, yielding pseudo-differential operators via an analogue of the Euclidean Kohn-Nirenberg quantization. The present document treats the case of the Euclidean motion group, which is the smallest subset of Euclidean affne transformations containing translations and rotations. As our representations are infnite-dimensional, the proofs of the calculus properties are more naturally carried out on the kernel side, which means that particular care is required to treat the singularity at the origin. The key argument is a density result which allows us to approximate singular kernels via smooth ones and is proved herein via purely spectral arguments without using classical estimates on the heat kernel.

In this thesis we study bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium (with a one-dimensional unstable manifold) in flows with dimension four or higher. Particularly, we show that heterodimensional cycles can be born from such bifurcations. A heterodimensional cycle consists of two saddle periodic orbits having different indices (dimensions of unstable manifolds), and two heteroclinic connections between those orbits. We find heterodimensional cycles for the flow as the suspension of heterodimensional cycles for a Poincaré map around the homoclinic loops. Especially, those cycles are co-index 1, i.e. the difference between indices is 1. More specifically, each of those heterodimensional cycles are associated to periodic orbits of indices 2 and 3. As a partial result we mention a criterion for having index 3 for periodic orbits near a homoclinic loop to a saddle-focus equilibrium. Different types of perturbations are considered, where the original homoclinic loops can be either kept or split. In intermediate steps we find, in addition to the classical heterodimensional connection between two periodic orbits, two new types of heterodimensional connections: one is a heteroclinic between a homoclinic loop and a periodic orbit of index 2, and the other connects a saddle-focus equilibrium to a periodic orbit of index 3. Furthermore, we consider a symmetric case where the codimension of the bifurcations is minimised to 1. We prove that, by endowing the flow with a certain $\mathbb{Z}_2$ symmetry, a pair of heterodimensional cycles can be born from a one-parameter unfolding of the symmetric pair of homoclinic loops. Moreover, we show that the heterodimensional cycles obtained in either the general or the symmetric case can belong to a chain-transitive and volume-hyperbolic attractor of the flow, along with a persistent homoclinic tangency.

In this thesis we study the relationship between the existence of extremal Kähler metrics and stability. We introduce a space of symplectic potentials for toric manifolds, which we show gives metrics with mixed Poincaré type and cone angle singularities. We show uniqueness and that existence implies stability for extremal metrics arising from these potentials. For quadrilaterals, we give a computable criterion for stability in certain cases, giving a definite log-stable region for generic quadrilaterals. We use computational tools to find new examples of stable and unstable toric manifolds. For Poincaré type manifolds with an S1-action, we prove a version of the LeBrun-Simanca openness theorem and Arezzo-Pacard blow-up theorem.

In this thesis we study Dirac operators on the Euclidean Taub-NUT and Schwarzschild spaces coupled to abelian gauge fields, with the aim of computing the zero-modes and bound states. The work is motivated by recently proposed Geometric Models of Matter, where single particles are modelled by 4-manifolds and their quantum numbers realised as topological invariants of the model manifolds. In these models, the spin degrees of freedom are given by the zero-modes of the Dirac operator. In the case of the Taub-NUT manifold coupled to an U(1) gauged eld with selfdual curvature, which is the model for the electron, we are able to obtain explicit expressions for the zero modes of the Dirac operator. We show that they decompose into an irreducible representation of SU(2) and use this to interpret a known index theorem in this geometry first deduced by Pope. We also study the dynamical symmetry of this space in the classical and quantum cases, and show that the gauge eld allows the existence of classical bounded orbits and quantum bound states. We compute scattering cross sections and find a surprising electric-magnetic duality. Using twistor formalism we are able to show that the dynamical symmetry is preserved in the gauged case and that this makes possible to deduce the energy of the quantum bound states in an algebraic manner. We consider the Euclidean Schwarzschild manifold coupled to an U(1) gauge field as a neutron candidate. In this case the zero-modes of the Dirac operator also decompose into an irreducible representation of SU(2). Using the open code SLEIGN2, we compute the spectrum of the Laplace-Beltrami operator acting on scalar fields.

This thesis is concerned with the analysis and implementation of the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) on computational meshes consisting of general polygonal/polyhedral (polytopic) elements. Two model problems are considered: general advection-diffusion-reaction boundary value problems and time dependent parabolic problems. New hp-version a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration as well as an arbitrary number of faces and hanging nodes per element. The proposed method employs elemental polynomial bases of total degree p (Pp- bases) defined in the physical coordinate system, without requiring mapping from a given reference or canonical frame. A series of numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the p-version DGFEM employing a Pp-basis on both polytopic and tensor-product elements with a (standard) DGFEM and FEM employing a (mapped) Qp-basis. Moreover, a careful theoretical analysis of optimal convergence rate in p for Pp-basis is derived for several commonly used projectors, which leads to sharp bounds of exponential convergence with respect to degrees of freedom (dof) for the Pp-basis.

No description available.

No description available.

The primary aim of this thesis is to calculate the numerical solution of a given stiff system of ordinary differential equations. We deal with the implementation of the implicit Runge-Kutta methods, in particular for Radau IIA order 5 which is now a competitive method for solving stiff initial value problems. New software based on Radau IIA, called IRKMR5 written in MATLAB has been developed for fixed order (order 5) with variable stepsizes, which is quite efficient when it is used to solve stiff problems. The code is organized in a modular form so that it facilitates both the understanding of it and its modification whenever needed. The new software is not only more functional than its Fortran 77 Radau IIA counterpart but also more robust and better documented. When implicit methods are used to solve nonlinear problems it is necessary to solve systems of nonlinear algebraic equations. New investigations for a modified Newton iteration are undertaken. This new strategy manages the iterative solutions of nonlinear equations in the ODEs solver. It also involves when to re-evaluate the Jacobian and the iteration matrix. The strategy also significantly reduces the number of function evaluations and linear solves. We subsequently consider the mathematical analysis of the nonlinear algebraic equations that arise from using s-stage fully implicit Runge-Kutta methods. Results for uniqueness of solutions and an error bound was established. The termination criterion in the iterative solution of the nonlinear equations is also studied as well as two types of termination criterion (displacement and the residual test). The residual test has been compared with the displacement test on some test examples and the results are tabulated.

No description available.

No description available.