LRA Collection:http://hdl.handle.net/2381/41372014-04-20T08:17:01Z2014-04-20T08:17:01ZOn the prescribed mean curvature problem on the standard n-dimensional ballSharaf, Khadijah Abdullah Mohammedhttp://hdl.handle.net/2381/284932013-12-07T02:02:05Z2013-12-06T13:36:38ZTitle: On the prescribed mean curvature problem on the standard n-dimensional ball
Authors: Sharaf, Khadijah Abdullah Mohammed
Abstract: In this thesis, we consider the problem of existence of conformal scalar flat metric with prescribed boundary mean curvature on the standard n-dimensional ball. Let B[superscript n] be the unit ball in R[superscript n], n ≥ 3, with Euclidean metric g[subscript 0]. Its boundary will be denoted by S[superscript n-1] and will be endowed with the standard metric still denoted by g[subscript 0]. Let H : S[superscript n-1] → R be a given function, we study the problem of finding a conformal metric g = u 4/n-2 g[subscript 0] such that R[subscript g] = 0 in B[superscript n] and h[subscript g] = H on S[superscript n-1]. Here R[subscript g] is the scalar curvature of the metric g in B[superscript n] and h[subscript g] is the mean curvature of g on S[superscript n-1]. This problem is equivalent to solving the following nonlinear boundary value equation: (see PDF for equation) where v is the outward unit vector with respect to the metric g[subscript 0]. In general there are several difficulties in facing this problem by means of variational methods. Indeed, in virtue of the non-compactness of the embedding H[superscript 1](B[superscript n]) → L 2(n-1)/n-2 (∂B[superscript n]), the Euler-Lagrange functional J associated to the problem, does not satisfy the Palais-Smale condition, and that leads to the failure of the standard critical point theory.
One part of this thesis deals with the case where H is a Morse function satisfying a non degeneracy condition. Using an algebraic topological method and the tools of the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the the n-dimensional balls.
The other part deals with the case where the non degeneracy condition is not satisfied and replaced by the so called β-flatness condition. In this case, we give precise estimates on the losses of the compactness and we identify the critical points at infinity of the variational problem. Then, we establish under generic boundary condition a Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem.2013-12-06T13:36:38ZThe effects of axial flow and surface mass-flux on the stability of the rotating-sphere boundary layerBarrow, Alistairhttp://hdl.handle.net/2381/284562013-11-28T02:02:01Z2013-11-27T09:39:53ZTitle: The effects of axial flow and surface mass-flux on the stability of the rotating-sphere boundary layer
Authors: Barrow, Alistair
Abstract: A theoretical investigation is carried out into the linear stability of the boundary-layer flow around a rotating sphere immersed in an incompressible viscous fluid. Two potentially stabilising mechanisms are considered: a forced uniform axial flow in the surrounding fluid, and the introduction of mass suction/injection through the surface of the sphere. The investigation is broadly split into a “local” analysis, where a parallel-flow assumption is made which limits the study to individual latitudinal positions; and a “global” analysis, where the entire streamwise extent of the flow is considered. In the local analysis, both stationary and travelling convective disturbances are considered. For a representative subset of the parameter space, critical Reynolds numbers are presented for the predicted onset of convective and absolute instabilities. Axial flow and surface suction are typically found to postpone the onset of all types of instability by raising the critical Reynolds number, whereas surface injection has the opposite effect. This is further demonstrated by a consideration of the convective and absolute growth rates at various parameter values.
The results of the global analysis suggest that the rotating sphere can support a self-sustained, linearly globally-unstable global mode for sufficiently large rotation rates. This is in contrast to the case of the rotating disk, where it is generally accepted that self-sustained linear global modes do not occur.2013-11-27T09:39:53ZApproximation on the complex sphereAlsaud, Huda Salehhttp://hdl.handle.net/2381/283682013-11-09T02:02:14Z2013-11-08T12:38:44ZTitle: Approximation on the complex sphere
Authors: Alsaud, Huda Saleh
Abstract: The aim of this thesis is to study approximation of multivariate functions on the complex sphere by spherical harmonic polynomials. Spherical harmonics arise naturally in many theoretical and practical applications. We consider different aspects of the approximation by spherical harmonic which play an important role in a wide range of topics. We study approximation on the spheres by spherical polynomials from the geometric point of view. In particular, we study and develop a generating function of Jacobi polynomials and its special cases which are of geometric nature and give a new representation for the left hand side of a well-known formulae for generating functions for Jacobi polynomials (of integer indices) in terms of associated Legendre functions. This representation arises as a consequence of the interpretation of projective spaces as quotient spaces of complex spheres. In addition, we develop new elements of harmonic analysis on the complex sphere, and use these to establish Jackson's and Kolmogorov's inequalities. We apply these results to get order sharp estimates for m-term approximation. The results obtained are a synthesis of new results on classical orthogonal polynomials geometric properties of Euclidean spaces. As another aspect of approximation, we consider interpolation by radial basis functions. In particular, we study interpolation on the spheres and its error estimate. We show that the improved error of convergence in n dimensional real sphere, given in [7], remain true in the case of the complex sphere.2013-11-08T12:38:44ZSolid-Liquid Interfacial Properties of Fe and Fe-C Alloys from Molecular Dynamics SimulationsMelnykov, Mykhailohttp://hdl.handle.net/2381/282692014-04-01T10:28:36Z2013-10-08T15:00:03ZTitle: Solid-Liquid Interfacial Properties of Fe and Fe-C Alloys from Molecular Dynamics Simulations
Authors: Melnykov, Mykhailo
Abstract: This project is devoted to the study of solid-liquid interfaces in pure Fe and Fe-C alloys using molecular simulation. It consists of three parts: first, we use the coexisting phases approach to calculate melting phase diagrams of several recent Fe-C interaction potentials, such as Embedded Atom Method (EAM) potential of Lau et al., EAM potential of Hepburn and Ackland, and Analytic Bond Order (ABOP) potential of Henriksson and Nordlund. Melting of both bcc (ferrite) and fcc (austenite) crystal structures is investigated with C concentrations up to 5 wt%. The results are compared with the experimental data and suggest that the potential of Hepburn and Ackland is the most accurate in reproducing the melting phase diagram of the ferrite but the austenite cannot be stabilised at any C concentration for this potential.
The potential of Lau et al. yields the best qualitative agreement with the real phase diagram in that the ferrite-liquid coexistence at low C concentrations is replaced by the austenite-liquid coexistence at higher C concentrations. However, the crossover C concentration is much larger and the ferrite melting temperature is much higher than in the real Fe-C alloy. The ABOP potential of Henriksson and Nordlund correctly predicts the relative stability of ferrite and austenite at melting, but significantly underestimates the solubility of C in the solid phases.
Second, we develop a new direct method for calculating the solid-liquid interfacial free energy using deformation of the solid-liquid coexistence system.
The deformation is designed to change the area of the interface, while preserving the volume of the system and crystal structure of the solid phase. The interfacial free energy is calculated as the deformation work divided by the change of the interfacial area. The method is applied to the bcc solid-liquid interface of pure Fe described by the Hepburn and Ackland potential. The obtained results are somewhat different from those calculated by the established methods so further development and analysis are required.
Third, we investigate the dependence on C concentration of the bcc solid-liquid interfacial free energy of Fe-C alloy described by the Hepburn and Ackland potential. We use the method proposed by Frolov and Mishin which is analogous to the Gibbs-Duhem integration along the solid-liquid coexistence line. The calculations are performed for three different crystal orientations (100), (110) and (111), allowing us to determine the anisotropy of the interfacial free energy and its dependence on C concentration along the coexistence line. Although the precision is somewhat limited by the high computational cost of such calculations.2013-10-08T15:00:03ZA Classification of Toral and Planar Attractors and Substitution Tiling SpacesMcCann, Sheila Margarethttp://hdl.handle.net/2381/282272013-09-27T01:03:33Z2013-09-26T09:27:20ZTitle: A Classification of Toral and Planar Attractors and Substitution Tiling Spaces
Authors: McCann, Sheila Margaret
Abstract: We focus on dynamical systems which are one-dimensional expanding
attractors with a local product structure of an arc times a Cantor set. We
define a class of Denjoy continua and show that each one of the class is homeomorphic to an orientable DA attractor with four complementary domains which in turn is homeomorphic to a tiling space consisting of aperiodic substitution tilings. The planar attractors are non-orientable as is the Plykin attractor in the 2-sphere which we describe.
We classify these attractors and tiling spaces up to homeomorphism and the
symmetries of the underlying spaces up to isomorphism. The criterion for
homeomorphism is the irrational slope of the expanding eigenvector of the
defining matrix from whence the attractor was formed whilst the criterion for
isomorphism is the matrix itself. We find that the permutation groups arising
from the 4 'special points' which serve as the repelling set of an attractor are isomorphic to subgroups of S[subscript 4]. Restricted to these 4 special points, we show that the isotopy class group of the self-homeomorphisms of an attractor, and likewise those of a tiling space, is isomorphic to Z ⊕ Z[subscript 2].2013-09-26T09:27:20ZThe Stability and Transition of the Compressible Boundary-Layer Flow over Broad Rotating ConesTowers, Paul Davidhttp://hdl.handle.net/2381/282192013-09-26T01:04:37Z2013-09-25T15:03:47ZTitle: The Stability and Transition of the Compressible Boundary-Layer Flow over Broad Rotating Cones
Authors: Towers, Paul David
Abstract: The subject of fluid flows over axisymmetric bodies has increased in recent
times, as they can be used to model flows over a swept wing, spinning projectiles and aeroengines amongst other things. A better mathematical understanding of the transition from laminar to turbulent flow within the boundary layer could lead to an improvement in the design of such applications.
We consider a compressible fluid flow over a rotating cone, defined by half-angle ψ. The mean flow boundary-layer equations are derived and we conduct a high Reynolds number asymptotic linear stability analysis. The flow is susceptible to instabilities caused by inviscid crossflow modes (type I ) and modes caused by a viscous-Coriolis balance force (type II ). Both are considered, along with the effects of changes in the cone half-angle, the magnitude of the local Mach number and the temperature at the cone wall. A surface suction along the cone wall is also analysed.2013-09-25T15:03:47ZAdaptive Radial Basis Function Interpolation for Time-Dependent Partial Differential EquationsNaqvi, Syeda Lailahttp://hdl.handle.net/2381/281842013-09-14T01:02:01Z2013-09-13T09:46:03ZTitle: Adaptive Radial Basis Function Interpolation for Time-Dependent Partial Differential Equations
Authors: Naqvi, Syeda Laila
Abstract: In this thesis we have proposed the meshless adaptive method by radial basis functions (RBFs) for the solution of the time-dependent partial differential equations (PDEs) where the approximate solution is obtained by the multiquadrics (MQ) and the local scattered data reconstruction has been done by polyharmonic splines. We choose MQ because of its exponential convergence for sufficiently smooth functions. The solution of partial differential equations arising in science and engineering, frequently have large variations occurring over small portion of the physical domain, the challenge then is to resolve the solution behaviour there. For the sake of efficiency we require a finer grid in those parts of the physical domain whereas a much coarser grid can be used otherwise.
During our journey, we come up with different ideas and have found many interesting results but the main motivation for the one-dimensional case was the Korteweg-de Vries (KdV) equation rather than the common test problems. The KdV equation is a nonlinear hyperbolic equation with smooth solutions at all times. Furthermore the methods available in the literature for solving this problem are rather fully implicit or limited literature can be found using explicit and semi-explicit methods. Our approach is to adaptively select the nodes, using the radial basis function interpolation.
We aimed in, the extension of our method in solving two-dimensional partial differential equations, however to get an insight of the method we developed the algorithms for one-dimensional PDEs and two-dimensional interpolation problem. The experiments show that the method is able to track the developing features of the profile of the solution. Furthermore this work is based on computations and not on proofs.2013-09-13T09:46:03ZPricing Discretely Monitored Barrier Options and Credit Default Swaps under Lévy Processesde Innocentis, Marcohttp://hdl.handle.net/2381/279122013-05-31T01:02:00Z2013-05-30T12:30:38ZTitle: Pricing Discretely Monitored Barrier Options and Credit Default Swaps under Lévy Processes
Authors: de Innocentis, Marco
Abstract: We introduce a new, fast and accurate method to calculate prices and sensitivities of European vanilla and digital options under the Variance Gamma model. For near at-the-money options of short maturity, our method is much faster than those based on discretization and truncation of the inverse Fourier transform integral (iFT method).
We show that the results calculated with our method agree with those obtained with the iFT algorithm using very long and fine grids. Taking the results of our method as a benchmark, we show that the parabolic modification of the iFT method (Boyarchenko and Levendorskiĭ, 2012) is much more efficient than the standard (flat) version. Based on this conclusion, we consider an approach which uses a combination of backward induction and parabolic iFT to price discretely monitored barrier options, as well as credit default swaps, under wide classes of Lévy models. At each step of backward induction, we use piece-wise polynomial interpolation and parabolic iFT, which allows for efficient error control. We derive accurate recommendations for the choice of parameters of the numerical scheme, and produce numerical examples showing that oversimplified prescriptions in other methods can result in large errors.2013-05-30T12:30:38ZInner Ideals of Simple Locally Finite Lie AlgebrasRowley, Jamie Robert Derekhttp://hdl.handle.net/2381/278282013-03-28T02:02:15Z2013-03-27T12:16:08ZTitle: Inner Ideals of Simple Locally Finite Lie Algebras
Authors: Rowley, Jamie Robert Derek
Abstract: Inner ideals of simple locally finite dimensional Lie algebras over an algebraically closed field of characteristic 0 are described. In particular, it is shown that a simple locally finite dimensional Lie algebra has a non-zero proper inner ideal if and only if it is of diagonal type. Regular inner ideals of diagonal type Lie algebras are characterized in terms of left and right ideals of the enveloping algebra. Regular inner ideals of finitary simple Lie algebras are described. Inner ideals of some finite dimensional Lie algebras are studied. Maximal inner ideals of simple plain locally finite dimensional Lie algebras are classified.2013-03-27T12:16:08ZKernel Approximation on Compact Homogeneous SpacesOdell, Carl Richardhttp://hdl.handle.net/2381/275982013-03-14T15:23:06Z2012-11-28T10:06:45ZTitle: Kernel Approximation on Compact Homogeneous Spaces
Authors: Odell, Carl Richard
Abstract: This thesis is concerned with approximation on compact homogeneous spaces.
The first part of the research involves a particular kind of compact homogeneous space, the hypersphere, S ͩˉ¹ embedded in R ͩ. It is a calculation of three integrals associated with approximation using radial basis functions, calculating the Fourier-Gegenbauer coefficients for two such functions. The latter part of the research is a calculation of an error bound for compact homogeneous spaces when interpolating with a G-invariant kernel, a generalisation of a result already known for spheres.2012-11-28T10:06:45ZThe Convective Instability of the Boundary-Layer Flow over Families of Rotating SpheroidsSamad, Abdulhttp://hdl.handle.net/2381/275762013-03-14T15:22:49Z2012-11-07T11:58:20ZTitle: The Convective Instability of the Boundary-Layer Flow over Families of Rotating Spheroids
Authors: Samad, Abdul
Abstract: The majority of this work is concerned with the local-linear convective instability analysis of the incompressible boundary-layer flows over prolate spheroids and oblate spheroids rotating in otherwise still fluid. The laminar boundary layer and the perturbation equations have been formulated by introducing two distinct orthogonal coordinate systems. A cross-sectional eccentricity parameter e is introduced to identify each spheroid within its family. Both systems of equations reduce exactly to those already established for the rotating sphere boundary layer. The effects of viscosity and streamline-curvature are included in each analysis.
We predict that for prolate spheroids at low to moderate latitudes, increasing eccentricity has a strong stabilizing effect. However, at high latitudes of ϴ ≥ 60, increasing eccentricity is seen to have a destabilizing effect. For oblate spheroids, increasing eccentricity has a stabilizing effect at all latitudes. Near the pole of both types of spheroids, the critical Reynolds numbers approach that for the rotating disk boundary layer. However, in prolate spheroid case near the pole for very large values of e, the critical Reynolds numbers exceed that for the rotating disk. We show that high curvature near the pole of prolate spheroids is responsible for the increase in critical Reynolds number with increasing eccentricity.
For both types of spheroids at moderate eccentricity, we predict that the most amplified modes travel at approximately 76% of the surface speed at all latitudes. This is consistent with the existing studies of boundary-layer flows over the related rotating-disk, -sphere and -cone geometries. However, for large values of eccentricity, the traveling speed of the most amplified modes increases up to approximately 90% of the surface speed of oblate spheroids and up to 100% in the prolate spheroid case.2012-11-07T11:58:20ZPricing of Discretely Sampled Asian Options under Lévy ProcessesXie, Jiayaohttp://hdl.handle.net/2381/110372013-03-14T15:23:56Z2012-09-28T14:04:47ZTitle: Pricing of Discretely Sampled Asian Options under Lévy Processes
Authors: Xie, Jiayao
Abstract: We develop a new method for pricing options on discretely sampled arithmetic average in exponential Lévy models. The main idea is the reduction to a backward induction procedure for the difference Wn between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This allows for an efficient truncation of the state space. At each step of backward induction, Wn is calculated accurately and fast using a piece-wise interpolation or splines, fast convolution and either flat iFT and (refined) iFFT or the parabolic iFT. Numerical results demonstrate the advantages of the method.2012-09-28T14:04:47ZNon-Equilibrium Dynamics of Discrete Time Boltzmann SystemsPackwood, Davidhttp://hdl.handle.net/2381/109412013-03-14T15:23:17Z2012-08-16T10:09:34ZTitle: Non-Equilibrium Dynamics of Discrete Time Boltzmann Systems
Authors: Packwood, David
Abstract: Lattice Boltzmann methods are a fully discrete model and numerical method for simulating fluid dynamics, historically they have been developed as a continuation of lattice gas systems. Another route to a lattice Boltzmann system is a discrete approximation to the Boltzmann equation. An analysis of lattice Boltzmann systems is usually performed from one of these directions.
In this thesis the lattice Boltzmann method is presented ab initio as a fully discrete system in its own right. Using the Invariant Manifold hypothesis the microscopic and macroscopic fluid dynamics arising from such a model are found. In particular this analysis represents a validation for lattice Boltzmann methods far from equilibrium.
Far from equilibrium, at high Reynolds or Mach numbers, lattice Boltzmann methods can exhibit stability problems. In this work a conditional stability theorem for lattice Boltzmann methods is established. Furthermore several practical numerical techniques for stabilizing lattice Boltzmann schemes are tested.2012-08-16T10:09:34ZOperads and Moduli SpacesBraun, Christopher Davidhttp://hdl.handle.net/2381/109162013-03-14T15:23:41Z2012-08-02T11:31:46ZTitle: Operads and Moduli Spaces
Authors: Braun, Christopher David
Abstract: This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of ‘quantum invariants’ of manifolds inspired by ideas originating from physics.
We consider the extension of classical 2–dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We generalise open topological conformal field theories to open Klein topological conformal field theories and consider various related moduli spaces, in particular deducing a Möbius graph decomposition of the moduli spaces of
Klein surfaces, analogous to the ribbon graph decomposition of the moduli spaces of Riemann surfaces.
We also begin a study, in generality, of quantum homotopy algebras, which arise as ‘higher genus’ versions of classical homotopy algebras. In particular we study the problem of quantum lifting. We consider applications to understanding invariants of manifolds arising in the quantisation of Chern–Simons field theory.2012-08-02T11:31:46ZNonequilibrium Entropic Filters for Lattice Boltzmann Methods and Shock Tube Case StudiesZhang, Jianxiahttp://hdl.handle.net/2381/108552013-03-14T15:23:25Z2012-06-25T10:26:38ZTitle: Nonequilibrium Entropic Filters for Lattice Boltzmann Methods and Shock Tube Case Studies
Authors: Zhang, Jianxia
Abstract: The Lattice Boltzmann Method (LBM) is a discrete velocity method which involves a single particle distribution function with two repeating procedures propagation and collision. When the Bhatnagar-Gross-Krook operator is applied as the collision operator for LBM, this is called lattice Bhatnagar-Gross-Krook method (LBGK). In comparison with the traditional computation methods, LBM appears as an efficient alternative computational approach for simulating complex fluid systems. However, LBM suffers numerical stability deficiencies when applied in low-viscosity fluid flow, such as local blow-ups and spurious oscillations where sharp gradients appear. The development of LBM has taken a further step to resolve the stability problem with applying a discrete entropy H-theorem. However, the stability and accuracy problems are not completely dealt with by the entropic lattice Boltzmann method. One of the remedies for the stability deficiencies is to construct nonequilibrium entropy limiters for LBM. The original concepts with the construction of nonequilibrium entropy limiters are based on flux filters (also called flux-corrected transport) by Boris and Book. The principal idea of the nonequilibrium entropy limiters is to control a scalar quantity, the nonequilibrium entropy. In this thesis, there are 6 limiters are developed and tested in 1D athermal shock tubes in uniformed discretized space lattice sites. Among these limiters, two new nonequilibrium limiters are constructed. All the median entropy limiters are tested with different stencils, which also have an effect on removing spurious oscillations. Apart from the test on a three-velocity set, we use five-velocity sets for the applications of nonequilibrium entropy limiters of LBM. The five-velocity sets are {-3, -1, 0, 1, 3}, {-5, -2, 0, 2, 5} and {-7, -3, 0, 3, 7}. The performance of LBGK without limiters provides a frame of reference for comparison with the performance of LBGK which uses the nonequilibrium entropy limiters. The computations of the LBGK on different velocity sets have shown that the nonequilibrium entropy limiters are able to efficiently remove spurious oscillations for both post-shock and shock regions for high Reynolds number. Among the suggested limiters, we recommend the median nonequilibrium entropy limiter.2012-06-25T10:26:38ZNumerical Methods for Heath-Jarrow-Morton Model of Interest RatesKrivko, Mariahttp://hdl.handle.net/2381/108432013-03-14T15:23:33Z2012-06-14T08:36:59ZTitle: Numerical Methods for Heath-Jarrow-Morton Model of Interest Rates
Authors: Krivko, Maria
Abstract: The celebrated HJM framework models the evolution of the term structure of interest rates through the dynamics of the forward rate curve. These dynamics are described by a multifactor infinite-dimensional stochastic equation with the entire forward rate curve as state variable. Under no-arbitrage conditions, the HJM model is fully characterized by specifying forward rate volatility functions and the initial forward curve. In short, it can be described as a unifying framework with one of its most striking features being the generality: any arbitrage-free interest rate model driven by Brownian motion can be described as a special case of the HJM model. The HJM model has closed-form solutions only for some special cases of volatility, and valuations under the HJM framework usually require a numerical approximation. We propose and analyze numerical methods for the HJM model. To construct the methods, we first discretize the infinite-dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite-dimensional system of stochastic differential equations (SDEs) which we approximate in the weak and mean-square sense. The proposed numerical algorithms are highly computationally efficient due to the use of high-order quadrature rules which allow us to take relatively large discretization steps in the maturity time without affecting overall accuracy of the algorithms. They also have a high degree of flexibility and allow to choose appropriate approximations in maturity and calendar times separately. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.2012-06-14T08:36:59ZTilings Generated by Non-Parallel Projection SchemesPeden, Andrew Jameshttp://hdl.handle.net/2381/101942013-03-14T15:24:05Z2012-03-14T13:56:19ZTitle: Tilings Generated by Non-Parallel Projection Schemes
Authors: Peden, Andrew James
Abstract: This thesis defines and investigates rational and irrational 2:1 X-projection schemes and non-parallel projection schemes with strips at rational gradients.
Both irrational 2:1 X-projection schemes and non-parallel projection schemes with strips at rational gradients are shown to produce tilings with infinitely many prototiles, with the tilings produced by the second of these schemes nonetheless shown to display a property similar to repetitivity.
Rational 2:1 X-projection schemes are shown to produce tilings with a finite number of prototiles, with a subset of these filings shown to be repetitive. The points in the fundamental domain of our lattice L that correspond to translates of these filings are also investigated, with these points shown to be either dense in a finite number of lines or dense in the fundamental domain. This also leads to a proof of repetitivity in all rational 2:1 X-projection tangs and aperiodicity in a subset of these tilings. The tiling spaces of such filings are also investigated.
In addition, the proportions in which the prototiles in a rational 2:1 X-projection tiling appear are also looked at, and a possible explanation of the values observed is provided.2012-03-14T13:56:19ZHochschild Cohomology and Periodicity of Tame Weakly Symmetric AlgebrasFallatah, Ahlam Omarhttp://hdl.handle.net/2381/101722013-03-14T15:22:59Z2012-03-09T16:16:06ZTitle: Hochschild Cohomology and Periodicity of Tame Weakly Symmetric Algebras
Authors: Fallatah, Ahlam Omar
Abstract: In this thesis we study the second Hochschild cohomology group of all tame weakly symmetric algebras having simply connected Galois coverings and only periodic modules. These algebras have been determined up to Morita equivalence by Białkowski, Holm and Skowroński in [4] where they give finite dimensional algebras A1(λ),A2(λ),A3,...A16 which are a full set of representatives of the equivalence classes. Hochschild cohomology is invariant under Morita equivalence, and this thesis describes HH²(Λ) for each algebra Λ = A1(λ),A2(λ),A3,…,A16 in this list. We also find the periodicity of the simple modules for each of these algebras. Moreover, for the algebra A1(λ) we find the minimal projective bimodule resolution of A1(λ) and discuss the periodicity of this resolution.2012-03-09T16:16:06ZAdaptive Discontinuous Galerkin Methods for Fourth Order ProblemsVirtanen, Juha Mikaelhttp://hdl.handle.net/2381/100912013-12-01T02:45:11Z2012-02-09T09:59:56ZTitle: Adaptive Discontinuous Galerkin Methods for Fourth Order Problems
Authors: Virtanen, Juha Mikael
Abstract: This work is concerned with the derivation of adaptive methods for discontinuous Galerkin approximations of linear fourth order elliptic and parabolic partial differential equations.
Adaptive methods are usually based on a posteriori error estimates. To this end, a new residual-based a posteriori error estimator for discontinuous Galerkin approximations to the biharmonic equation with essential boundary conditions is presented. The estimator is shown to be both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm, under minimal regularity assumptions. The reliability bound is based on a new recovery operator, which maps discontinuous finite element spaces to conforming finite element spaces (of two polynomial degrees higher), consisting of triangular or quadrilateral Hsieh-Clough-Tocher macroelements. The efficiency bound is based on bubble function techniques. The performance of the estimator within an h-adaptive mesh refinement procedure is validated through a series of numerical examples, verifying also its asymptotic exactness. Some remarks on the question of proof of convergence of adaptive algorithms for discontinuous Galerkin for fourth order elliptic problems are also presented.
Furthermore, we derive a new energy-norm a posteriori error bound for an implicit Euler time-stepping method combined with spatial discontinuous Galerkin scheme for linear fourth order parabolic problems. A key tool in the analysis is the elliptic reconstruction technique. A new challenge, compared to the case of conforming finite element methods for parabolic problems, is the control of the evolution of the error due to non-conformity. Based on the error estimators, we derive an adaptive numerical method and discuss its practical implementation and illustrate its performance in a series of numerical experiments.2012-02-09T09:59:56ZMultilevel Sparse Kernel-Based InterpolationSubhan, Fazlihttp://hdl.handle.net/2381/98942011-11-19T02:02:08Z2011-11-18T11:52:05ZTitle: Multilevel Sparse Kernel-Based Interpolation
Authors: Subhan, Fazli
Abstract: Radial basis functions (RBFs) have been successfully applied for the last four decades for fitting scattered data in Rd, due to their simple implementation for any d. However, RBF interpolation faces the challenge of keeping a balance between convergence performance and numerical stability. Moreover, to ensure good convergence rates in high dimensions, one has to deal with the difficulty of exponential growth of the degrees of freedom with respect to the dimension d of the interpolation problem. This makes the application of RBFs limited to few thousands of data sites and/or low dimensions in practice.
In this work, we propose a hierarchical multilevel scheme, termed sparse kernel-based interpolation (SKI) algorithm, for the solution of interpolation problem in high dimensions. The new scheme uses direction-wise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The new SKI algorithm can be viewed as an extension of the idea of sparse grids/hyperbolic cross to kernel-based functions.
To achieve accelerated convergence, we propose a multilevel version of the SKI algorithm.
The SKI and multilevel SKI (MLSKI) algorithms admit good reproduction properties: they are numerically stable and efficient for the reconstruction of large data in Rd, for d = 2, 3, 4, with several thousand data. SKI is generally superior over classical RBF methods in terms of complexity, run time, and convergence at least for large data sets. The MLSKI algorithm accelerates the convergence of SKI and has also generally faster convergence than the classical multilevel RBF scheme.2011-11-18T11:52:05ZConstruction operations to create new aperiodic tilings: local isomorphism classes and simplified matching rulesFletcher, Davidhttp://hdl.handle.net/2381/95342011-07-19T01:01:41Z2011-07-18T10:56:10ZTitle: Construction operations to create new aperiodic tilings: local isomorphism classes and simplified matching rules
Authors: Fletcher, David
Abstract: This thesis studies several constructions to produce aperiodic tilings with particular
properties. The first chapter of this thesis gives a constructive method, exchanging
edge to edge matching rules for a small atlas of permitted patches, that can decrease
the number of prototiles needed to tile a space. We present a single prototile that
can only tile R3 aperiodically, and a pair of square prototiles that can only tile R2
aperiodically.
The thesis then details a construction that superimposes two unit square tilings
to create new aperiodic tilings. We show with this method that tiling spaces can
be constructed with any desired number of local isomorphism classes, up to (and
including) an infinite value. Hyperbolic variants are also detailed.
The final chapters of the thesis apply the concept of Toeplitz arrays to this
construction, allowing it to be iterated. This gives a general method to produce
new aperiodic tilings, from a set of unit square tilings. Infinite iterations of the
construction are then studied. We show that infinite superimpositions of periodic
tilings are describable as substitution tilings, and also that most Robinson tilings
can be constructed by infinite superimpositions of given periodic tilings. Possible
applications of the thesis are then briefly considered.2011-07-18T10:56:10ZQuasichemical Models of Multicomponent Nonlinear DiffusionWahab, Hafiz Abdulhttp://hdl.handle.net/2381/92912011-05-07T01:02:02Z2011-05-06T09:14:58ZTitle: Quasichemical Models of Multicomponent Nonlinear Diffusion
Authors: Wahab, Hafiz Abdul
Abstract: Diffusion preserves positivity of concentrations, therefore, multicomponent diffusion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicomponent diffusion equations should be ordered and special tools are necessary to provide systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant interaction between components. We develop an approach to nonlinear multicomponent diffusion based on the idea of reaction mechanism borrowed from chemical kinetics.
Chemical kinetics gave rise to the very seminal tools for the modelling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion.
We demonstrate, how the mechanism based approach to multicomponent diffusion can be included into the general thermodynamic framework, and prove the corresponding dissipation inequalities. To satisfy the thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell-jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation.2011-05-06T09:14:58ZSlow Invariant Manifold and its approximations in kinetics of catalytic reactionsShahzad, Muhammadhttp://hdl.handle.net/2381/92882011-05-07T01:01:52Z2011-05-06T08:39:57ZTitle: Slow Invariant Manifold and its approximations in kinetics of catalytic reactions
Authors: Shahzad, Muhammad
Abstract: Equations of chemical kinetics typically include several distinct time scales. There exist many methods which allow to exclude fast variables and reduce equations to the slow manifold. In this thesis, we start by studying the background of the quasi equilibrium approximation, main approaches to this approximation, its consequences and other related topics.
We present the general formalism of the quasi equilibrium (QE) approximation with the proof of the persistence of entropy production in the QE approximation. We demonstrate how to apply this formalism to chemical kinetics and describe the difference between QE and quasi steady state (QSS) approximations. In 1913 Michaelis and Menten used the QE assumption that all intermediate complexes are in fast equilibrium with free substrates and enzymes. Similar approach was developed by Stuekelberg (1952) for the Boltzmann kinetics. Following them, we combine the QE (fast equilibria) and the QSS (small amounts) approaches and study the general kinetics with fast intermediates present in small amounts. We prove the representation of the rate of an elementary reaction as a product of the Boltzmann factor (purely thermodynamic) and the kinetic factor, and find the basic relations between kinetic factors. In the practice of modeling, a kinetic model may initially not respect thermodynamic conditions. For these cases, we solved a problem: is it possible to deform (linearly) the entropy and provide agreement with the given kinetic model and deformed thermodynamics ?
We demonstrate how to modify the QE approximation for stiffness removal in an example of the CO oxidation on Pt. QSSA was applied in order to get an approximation to the One dimensional Invariant Grid for oxidation of CO over Pt. The method of intrinsic low dimension manifold (ILDM) was implemented over the same example (CO oxidation on Pt) in order to automate the process of reduction and provide more accurate simplified mechanism (for one-dimension), yet at the cost of a significantly more complicated implementation.2011-05-06T08:39:57ZStabilizing Lattice Boltzmann Simulation of Flows Past Bluff Bodies by Introduction of Ehrenfests' LimitersKhan, Tahir Saeedhttp://hdl.handle.net/2381/91732013-04-23T08:48:30Z2011-03-16T16:45:10ZTitle: Stabilizing Lattice Boltzmann Simulation of Flows Past Bluff Bodies by Introduction of Ehrenfests' Limiters
Authors: Khan, Tahir Saeed
Abstract: The lattice Boltzmann method (LBM) have emerged as an alternative computational approach to the conventional computational fluid dynamics (CFD). Despite being computationally efficient and popular numerical method for simulation of complex fluid flow, the LBM exhibits severe instabilities in near-grid scale hydrodynamics where sharp gradients are present. Further, since the LBM often uses uniform cartesian lattices in space, the curved boundaries are usually approximated by a series of stairs that also causes computational inaccuracy in the method. An interpolation-based treatment is introduced for the curved boundaries by Mei et al. One of the recipe to stabilize the LBM is the introduction of Ehrenfests' step. The objective of this work is to investigate the efficiency of the LBM with Ehrenfests' steps for the flows around curved bluff bodies. For this purpose, we have combined the curved boundary treatment of Mei et al. and the LBM with Ehrenfests' steps and developed an efficient numerical scheme. To test the validity of our numerical scheme we have simulated the two-dimensional flow around a circular cylinder and an airfoil for a wide range of low to high Reynolds numbers (Re ≤ 30, 000). We will show that the LBM with Ehrenfests' steps can quantitatively capture the Strouhal-Reynolds number relationship and the drag coefficient without any need for explicit sub-grid scale modeling. Comparisons with the experimental and numerical results show that this model is a good candidate for the turbulence modeling of fluids around bluff bodies.2011-03-16T16:45:10ZDegrees of UnsolvabilityCooper, Stuart Barryhttp://hdl.handle.net/2381/90372011-03-14T16:51:12Z2011-02-07T10:04:21ZTitle: Degrees of Unsolvability
Authors: Cooper, Stuart Barry
Abstract: Thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at the University of Leicester Awarded 1971.2011-02-07T10:04:21ZThe cohomology of λ-rings and ψ-ringsRobinson, Michaelhttp://hdl.handle.net/2381/89272011-03-14T16:43:21Z2011-01-07T12:37:05ZTitle: The cohomology of λ-rings and ψ-rings
Authors: Robinson, Michael
Abstract: In this thesis we develop the cohomology of diagrams of algebras and then apply this to the cases of the λ-rings and the ψ-rings. A diagram of algebras is a functor from a small category to some category of algebras. For an appropriate category of algebras we get a diagram of groups, a diagram of Lie algebras, a diagram of commutative rings, etc.
We define the cohomology of diagrams of algebras using comonads. The cohomology of diagrams of algebras classifies extensions in the category of functors. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of algebras to the cohomology of the members of the diagram.
ψ-rings can be thought of as functors from the category with one object associated to the multiplicative monoid of the natural numbers to the category of commutative rings. So we can apply the theory we developed for the diagrams of algebras to the case of ψ-rings. Our main result tells us that there is a spectral sequence connecting the cohomology of the ψ-rings to the André-Quillen cohomology of the underlying commutative ring.
The main example of a λ-ring or a ψ-ring is the K-theory of a topological space.
We look at the example of the K-theory of spheres and use its cohomology to give a proof of the classical result of Adams. We show that there are natural transformations connecting the cohomology of the K-theory of spheres to the homotopy groups of spheres. There is a very close connection between the cohomology of the K-theory of the 4n-dimensional spheres and the homotopy groups of the (4n-1)- dimensional spheres.2011-01-07T12:37:05ZError Estimates for Interpolation of Rough and Smooth Functions using Radial Basis FunctionsBrownlee, Robert Alexanderhttp://hdl.handle.net/2381/88252011-03-14T16:44:54Z2010-12-06T11:38:49ZTitle: Error Estimates for Interpolation of Rough and Smooth Functions using Radial Basis Functions
Authors: Brownlee, Robert Alexander
Abstract: In this thesis we are concerned with the approximation of functions by radial basis function
interpolants. There is a plethora of results about the asymptotic behaviour of the error between appropriately smooth functions and their interpolants, as the interpolation points fill out a bounded domain in Euclidean space. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function - the native space.
This work establishes Lp-error estimates, for 1 ≤ p ≤ ∞, when the function being interpolated fails to have the required smoothness to lie in the corresponding native space; therefore, providing error estimates for a class of rougher functions than previously known.
Such estimates have application in the numerical analysis of solving partial differential equations using radial basis function collocation methods. At first our discussion focuses on the popular polyharmonic splines. A more general class of radial basis functions is admitted into exposition later on, this class being characterised by the algebraic decay of the Fourier transform of the radial basis function. The new estimates presented here offer some improvement on recent contributions from other authors by having wider applicability and a more satisfactory form. The method of proof employed is not restricted to interpolation alone. Rather, the technique provides error estimates for the approximation of rough functions for a variety of related approximation schemes as well.
For the previously mentioned class of radial basis functions, this work also gives error estimates when the function being interpolated has some additional smoothness. We find that the usual Lp-error estimate, for 1 ≤ p ≤ ∞, where the approximand belongs to the corresponding native space, can be doubled. Furthermore, error estimates are established for functions with smoothness intermediate to that of the native space and the subspace of the native space where double the error is observed.2010-12-06T11:38:49ZCalculating Ice–Water Interfacial Free Energy by Molecular SimulationHandel, Richard Jameshttp://hdl.handle.net/2381/86342010-10-25T13:18:21Z2010-10-18T13:12:12ZTitle: Calculating Ice–Water Interfacial Free Energy by Molecular Simulation
Authors: Handel, Richard James
Abstract: This study presents a calculation of the free energy of the ice–water interface using molecular simulation. The method used is an adaptation of the cleaving method, introduced by Broughton and Gilmer, and subsequently enhanced by Davidchack and Laird.
The calculation is direct in the sense that an interface is formed during the simulation: isolated ice and water systems are transformed, via a sequence of reversible steps, into a single system of ice and water in contact. The method is essentially computational, that is, it does not correspond to any possible physical experiment, since non-physical potential energies are introduced (and subsequently removed) during the transformation process.
The adaptation of the method to water presented significant challenges, notably the avoidance of hysteresis during the transformation, and the devising of an ‘external’ energy potential to control the position and orientation of water molecules.
The results represent the first direct calculation by simulation of the solid–liquid interfacial free energy for a model of a molecular (as opposed to atomic) system.2010-10-18T13:12:12ZAncient Egyptian astronomy: timekeeping and cosmography in the new kingdomSymons, Sarahhttp://hdl.handle.net/2381/85462010-10-02T01:02:25Z2010-10-01T08:26:54ZTitle: Ancient Egyptian astronomy: timekeeping and cosmography in the new kingdom
Authors: Symons, Sarah
Abstract: The first part of this study analyses and discusses astronomical timekeeping methods used in
the New Kingdom. Diagonal star clocks are examined first, looking at classification of
sources, decan lists, and the updating of the tables over time. The date list in the Osireion at
Abydos is discussed, and issues concerning its place in the history of astronomical
timekeeping are raised. The final stellar timekeeping method, the Ramesside star clock, is
then examined. The conventional interpretation of the observational method behind the tables
is challenged by a new theory, and a system of analysing the tables is introduced. The
conclusions of the previous sections are then gathered together in a discussion of the
development of stellar timekeeping methods.
The small instruments known as shadow clocks, and their later relatives the sloping sundials,
are also examined. The established hypothesis that the shadow clock was completed by the
addition of a crossbar is challenged and refuted.
The second part of this study is based on New Kingdom representations of the sky. Two
major texts and several celestial diagrams are discussed in detail, beginning with the Book of
Nut, which describes the motions of the sun and stars. New translations of the vignette and
dramatic text are presented and discussed. Portions of the Book of the Day describing the
behaviour of the sun and circumpolar group of stars are analysed.
Finally, celestial diagrams dating from the New Kingdom are described. Their composition
and significance is discussed and the conceptual framework behind the diagrams is recreated.
By introducing new theories and analysis methods, and using a modem but sympathetic
approach to the original sources, this study attempts to update and extend our knowledge of
these areas of ancient astronomy.2010-10-01T08:26:54ZOn Conditional Wiener Integrals and a Novel Approach to the Fermion Sign ProblemDumas, Warwick Michaelhttp://hdl.handle.net/2381/83302010-09-10T11:09:21Z2010-07-29T14:11:02ZTitle: On Conditional Wiener Integrals and a Novel Approach to the Fermion Sign Problem
Authors: Dumas, Warwick Michael
Abstract: The path-integral formulation of nonrelativistic quantum mechanics was introduced by Feynman in 1948. The use of Path Integral Monte Carlo can be put on a rigorous footing using conditional Wiener integrals. This thesis addresses the topics both of numerical error and of Monte Carlo error.
A piecewise constant numerical method which is of second order of accuracy for computing conditional Wiener integrals for a rather general class of sufficiently smooth functional is proposed. The method is based on simulation of Brownian bridges via the corresponding stochastic differential equations (SDEs) and on ideas of the weak-sense numerical integration of SDEs. A convergence theorem is proved. Special attention is paid to integral-type functionals. Results of some numerical experiments are presented.
In a further part of the research, the goal is to develop Monte Carlo methods for fermion simulations that are resistant to the explosion of variance which happens due to the fermion sign problem. A novel approach is developed which represents a radical departure from the current approaches. This is based on the principle of using a geometrical interpretation of the problem in order to find ways to maximize the negative covariance between the countersigned functional contributions. The fundamental connection between quantum exchange and the fermion sign problem is exploited. It is shown that this leads to a mathematical proof of the well-known exact solution to the sign problem for 1-dimensional fermion systems, and also to a novel exact solution in the case of a pair of 2-dimensional fermions.2010-07-29T14:11:02ZSelected topics in Dirichlet problems for linear parabolic stochastic partial differential equationsStanciulescu, Vasile Nicolaehttp://hdl.handle.net/2381/82712010-09-10T13:14:05Z2010-07-26T13:57:38ZTitle: Selected topics in Dirichlet problems for linear parabolic stochastic partial differential equations
Authors: Stanciulescu, Vasile Nicolae
Abstract: This thesis is devoted to the study of Dirichlet problems for some linear parabolic SPDEs. Our aim in it is twofold. First, we consider SPDEs with deterministic coefficients which are smooth up to some order of regularity. We establish some theoretical results in terms of existence, uniqueness and regularity of the classical solution to the considered problem. Then, we provide the probabilistic representations (the averaging-over-characteristic formulas of its solution. We, thereafter, construct numerical methods for it. The methods are based on the averaging-over-characteristic formula and the weak-sense numerical integration of ordinary stochastic differential equations in bounded domains. Their orders of convergence in the mean-square sense and in the sense of almost sure convergence are obtained. The Monte Carlo technique is used for practical realization of the methods. Results of some numerical experiments are presented. These results are in agreement with the theoretical findings.
Second, we construct the solution of a class of one dimensional stochastic linear heat equations with drift in the first Wiener chaos, deterministic initial condition and which are driven by a space-time white noise and the white noise. This is done by giving explicitly its Wiener chaos decomposition. We also prove its uniqueness in the weak sense. Then we use the chaos expansion in order to show that the unique weak solution is an analytic functional with finite moments of all orders. The chaos decomposition is also utilized as a very useful tool for obtaining a continuity property of the solution.2010-07-26T13:57:38ZStyles of effective heads of mathematics in secondary schoolsEales, Alanhttp://hdl.handle.net/2381/76422010-03-06T02:01:13Z2010-03-05T14:56:04ZTitle: Styles of effective heads of mathematics in secondary schools
Authors: Eales, Alan
Abstract: Heads of department are in pivotal positions in secondary schools. Their tasks are well documented but it is not the tasks themselves that are critical for effectiveness but rather HOW the tasks are carried out. This research investigated the styles of effective heads of department and how these styles can be developed.
For the first study a panel of judges was used to select seven effective heads of mathematics and, following interviews with each one, pen-pictures of their style were drawn up.
The second study used four heads of mathematics who were by reputation regarded as effective and were well known to the researcher. Each head of mathematics nominated four colleagues and from interviews with these and the heads of mathematics themselves, extensive pen-pictures of the professional life of each of the heads of mathematics were drawn up.
Comparison was made to some leadership theories and it is possible to suggest some generalisations, for example that potential styles are limited by behavioural or affective aspects which suggests that the professional lives of middle-managers are crucially influenced by their personal lives.
There was no 'style for all seasons' but the pen-pictures showed the heads of mathematics to be:
- 'people-centered'
- supporting teachers and pupils through highlypersonalised, professional help;
- efficient administrators;
- open about their aims - idealistic but pragmatic;
- extremely hard working
- accepting responsibility for their subject;
- members of personal and professional support networks
- and involved in the mathematics education debate, both locally and nationally.
INSET and appraisal are increasingly school-focussed and school-based.
This research trialled aspects of appraisal such as the use of a colleague who can listen, reflect and draw up a written record. Processes such as these are crucial for the development of middle-managers in secondary schools.2010-03-05T14:56:04ZNon-oscillatory finite volume methods for conservation laws on unstructured grids.Aboiyar, Terhemenhttp://hdl.handle.net/2381/41382011-11-24T12:54:58Z2009-01-07T15:12:48ZTitle: Non-oscillatory finite volume methods for conservation laws on unstructured grids.
Authors: Aboiyar, Terhemen
Abstract: This work focuses on the use of polyharmonic splines, a class of radial basis functions,
in the reconstruction step of finite volume methods.
We first establish the theory of radial basis functions as a powerful tool for scattered data approximation. We thereafter provide existing and new results on the approximation order and numerical stability of local interpolation by polyharmonic splines. These results provide the tools needed in the design of the Runge KuttaWeighted Essentially Non-Oscillatory (RK-WENO) method and the Arbitrary high order using high order DERivatives-WENO (ADER-WENO) method. In the RK-WENO method, a WENO reconstruction based on polyharmonic splines is coupled with Strong Stability Preserving (SSP) Runge-Kutta time stepping.
The polyharmonic spline WENO reconstruction is also used in the spatial discretisation of the ADER-WENO method. Here, the time discretisation is based on a Taylor
expansion in time where the time derivatives are replaced by space derivatives using the
Cauchy-Kowalewski procedure. The high order °ux evaluation of the ADER-WENO method is achieved by solving generalized Riemann problems for the spatial derivatives across cell interfaces.
Adaptive formulations of the RK-WENO and ADER-WENO methods are used to solve advection problems on unstructured triangulations. An a posteriori error indicator is used to design the adaptation rules for the dynamic modification of the triangular mesh during the simulation. In addition, the flexibility of the stencil selection strategy for polyharmonic spline reconstruction is utilised in developing a WENO reconstruction method with stencil adaptivity.
Finally, order variation procedures are combined with mesh adaptation in order to
handle regions of the computational domain where the solution is smooth in a different
fashion from the vicinity of singularities and steep gradients with the goal of delivering accurate solutions with less computational effort and fewer degrees of freedom when compared to adaptive methods with fixed order of reconstruction.2009-01-07T15:12:48Z