A Research Report submitted to the Department of Physics, Faculty of Science, University of the Witwatersrand, Johannesburg, in partial ful lment of the requirements for the degree of Master of Science. Signed on the 24th March 2016 in Johannesburg. / This research report uses cosmological N-body simulations to examine the the large scale mass distribution of the Universe, known as the cosmic web. The cosmic web can be classi ed into nodes, laments, sheets and voids - each with its own characteristic density and velocity elds. In this work, the author proposes a new Lagrangian cosmic web classi- cation algorithm, based on smoothed particle hydrodynamics. This scheme o ers adaptive resolution, resolves smaller substructure and obeys similar statistical properties with existing Eulerian methods. Using the new classi cation scheme, halo clustering dependence on cosmic web type is examined. The author nds halo clustering is signi cantly correlated with web type. Consequently, the mass dependence of halo clustering may be explained by the fractions of web types found for a particular halo mass. Finally, an analysis of dark matter halo spin, shape and fractional anisotropy is presented per web type to suggest avenues for future work.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulflment of the requirements for the degree of Master of Science. / The Method of Equivalent Lagrangians is used to find the solutions of a given differential equation by exploiting the possible existence of an isomorphic Lie point symmetry algebra and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as the Kummer Equation and the Combined Gravity-Inertial-Rossby Wave Equation and certain classes of partial differential equations related to the (1 + 1) linear wave equation. We also make generalisations to the (2 + 1) and (3 + 1) linear wave equations.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 85-91). Available also in a digital version from Dissertation Abstracts.

University Of The Witwatersrand Department Of Computational And Applied Mathematics Masters’ Dissertation 2015 / The ideal hitting location on a cricket bat, the ‘sweet-spot’, is taken to be defined in two parts: 1) the Location of Impact on a cricket bat that transfers the maximum amount of energy into the batted ball and 2) the Location of Impact that transfers the least amount of energy to the batsman’s hands post-impact with the ball; minimizing the unpleasant stinging sensation felt by the batsman in his hands. An analysis of di↵erent hitting locations on a cricket bat is presented with the cricket bat modelled as a one dimensional beam which is approximated by the Euler-Lagrange Beam Equation. The beam is assumed to have uniform density and constant flexural rigidity. These assumptions allow for the Euler-Lagrange Beam Equation to be simplified considerably and hence solved numerically. The solution is presented via both a Central Time, Central Space finite di↵erence scheme and a Crank-Nicolson scheme. Further, the simplified Euler-Lagrange Beam Equation is solved analytically using a Separation of Variables approach. Boundary conditions, initial conditions and the framework of various collision scenarios between the bat and ball are structured in such a way that the model approximates a batsman playing a defensive cricket shot in the first two collision scenarios and an aggressive shot in the third collision scenario. The first collision scenario models a point-like, impulsive, perpendicular collision between the bat and ball. A circular Hertzian pressure distribution is used to model an elastic, perpendicular collision between the bat and ball in the second collision scenario, and an elliptical Hertzian pressure distribution does similarly for an elastic, oblique collision in the third collision scenario. The pressure distributions are converted into initial velocity distributions through the use of the Lagrange Field Equation. The numerical solution via the Crank-Nicolson scheme and the analytical solution via the Separation of Variables approach are analysed. For di↵erent Locations of Impact along the length on a cricket bat, a post-impact analysis of the displacement of points along the bat and the strain energy in the bat is conducted. Further, through the use of a Fourier Transform, a post-impact frequency analysis of the signals travelling in the cricket bat is performed. Combining the results of these analyses and the two-part definition of a ‘sweet-spot’ allows for the conclusion to be drawn that a Location of Impact as close as possible to the fixed-end of the cricket bat (a point just below the handle of the bat) results in minimum amount of energy transferred to the hands of the batsman. This minimizes the ‘stinging’ sensation felt by the batsman in his hands and satisfies the second part of the definition of a sweet-spot. Due to the heavy emphasis of the frequency analysis in this study, the conclusion is drawn that bat manufacturers should consider the vibrational properties of bats more thoroughly in bat manufacturing. Further, it is concluded that the solutions from the numerical Crank-Nicolson scheme and the analytical Separation of Variables approach are in close agreement.

Thesis (Ph. D.)--Oregon State University, 1980. / Typescript (photocopy). Includes bibliographical references. Also available on the World Wide Web.

Thesis (Ph. D.)--Oregon State University, 1992. / Typescript (photocopy). Includes bibliographical references (leaves 68-70). Also available on the World Wide Web.

Thesis (Ph. D.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains x, 166 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 154-161).

本論文旨在討論複空間上的特殊拉格朗日子流行的各種建構方法。這些方法主要自R. Harvey, B. Lawson, D. Joyce, R. Bryanl 以及M. Haskins這幾位數學家發展及研究. 本文側重於擁有不同類型對稱位的特殊拉格朗日子流行的結構方法, 其中包指於n維環面群及特殊正交群不變的例子，以及直紋特殊拉格朗日子流行。除此之外,本論文也會討論以上建構方法所給出的具體制子。最後, 本文亦會討論一種可以建構擁有高虧格鏈的特殊拉格朗日子流行的方法。 / This thesis gives a survey on constructions of special Lagrangian submanifolds in C[superscript n]. These construction methods are mainly studied by Harvey and Lawson, D. Joyce, R. Bryant and M. Haskins. We mainly focus on special Lagrangian submanifolds with different kinds of symmetries. These include local constructions of T[superscript n] --and SO(n)--invariant examples, and ruled examples. We also discuss explicit examples arose from those constructions. Besides local constructions, a global gluing construction of special Lagrangian submanifolds with high genus links is also discussed. / Detailed summary in vernacular field only. / Lam, Yi Chun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 97-99). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Preliminaries --- p.17 / Chapter 2.1 --- Special Lagrangian Geometry --- p.17 / Chapter 2.2 --- Special Legendrian Links --- p.23 / Chapter 2.3 --- Harmonic Maps --- p.25 / Chapter 2.4 --- Moment Maps --- p.28 / Chapter 2.5 --- Evolution Equation --- p.30 / Chapter 2.6 --- Background Materials from Analysis --- p.33 / Chapter 3 --- T[superscript n] invariant Special Lagrangian Submanifolds --- p.36 / Chapter 3.1 --- Basic Example --- p.36 / Chapter 3.2 --- U(1)[superscript n-2] invariant Special Lagrangian Cones in C[superscript n] --- p.37 / Chapter 3.2.1 --- General Construction --- p.37 / Chapter 3.2.2 --- Reduction of O.D.E. System --- p.40 / Chapter 3.2.3 --- The 3--dimensional Case --- p.48 / Chapter 4 --- SO(n)--invariant Special Lagrangian Submanifolds --- p.52 / Chapter 4.1 --- Basic Example --- p.52 / Chapter 4.2 --- Special Lagrangian Submanifolds with Fixed Loci --- p.54 / Chapter 4.2.1 --- Reduction to P.D.E --- p.55 / Chapter 4.2.2 --- The 3--Dimensional Case --- p.63 / Chapter 5 --- Ruled Special Lagrangian Submanifolds --- p.65 / Chapter 5.1 --- Normal Bundle of a Submanifold --- p.65 / Chapter 5.2 --- Twisted Normal Bundles --- p.69 / Chapter 5.3 --- The 3Dimensional Case --- p.72 / Chapter 5.3.1 --- Construction of Special Lagrangian Ruled 3-folds --- p.71 / Chapter 5.3.2 --- Explicit Examples --- p.75 / Chapter 5.4 --- Twisted Special Lagrangian Cones --- p.78 / Chapter 6 --- Other Constructions --- p.81 / Chapter 6.1 --- Analysis on U(1)invariant Special Lagrangian submanifolds --- p.81 / Chapter 6.1.1 --- Non-singular Solutions --- p.83 / Chapter 6.1.2 --- Existence of Singular Solutions --- p.84 / Chapter 6.1.3 --- Properties of Singular Solutions --- p.86 / Chapter 6.2 --- Harmonic Maps and Minimal Immersions --- p.87 / Chapter 6.3 --- Construction of Special Lagrangian 3-folds with High Genus Links --- p.92 / Bibliography --- p.97

The application of Lagrange's equations to non-equilibrium reacting compressible thermofluid systems yields a modeling methodology for thermofluid dynamics compatible with the discrete energy methods used extensively in other energy domains; examples include mechanical systems simulations and molecular dynamics modeling. The introduction of internal energies as generalized coordinates leads to a thermomechanical model with a simple but general form. A finite element interpolation is used to formulate the ODE model in an ALE reference frame, without reference to any partial differential equations. The formulation is applied to highly nonlinear problems without the use of any time-splitting or shock-tracking methods. The method is verified via the solution of a set of example problems which incorporate a variety of reference frames, both open and closed control volumes, and moving boundaries. The example simulations include transient detonations with complex chemistry, piston-initiated detonations, canonical unstable overdriven detonations, high-resolution induction-zone species evolution within a pulsating hydrogen-air detonation, and the detonation of a solid explosive due to high-velocity impact. / text

Dissertation (PhD)--University of Stellenbosch, 2008. / Bibliography. Also available via the Internet.