Thesis (doctoral)--Universität Greifswald, 1913. / Vita.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1893. / Vita.

In the first part of this work, two nonlinear partial differential equations, namely, a modified Camassa-Holm-Degasperis-Procesi equation and the generalized Kortewegde Vries equation with two power law nonlinearities are studied. The Lie symmetry method along with the simplest equation method is used to construct exact Solutions for these two equations. The second part looks at two systems of partial differential equations, namely, the generalized Boussinesq-Burgers equations and the (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method and the travelling wave hypothesis approach are utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+ 1 )-dimensional Davey-Stewartson equations. / Thesis (Msc. in Applied Mathematics) North-West University, Mafikeng Campus, 2012

published_or_final_version / Mathematics / Master / Master of Science

Let M denote a 2n-dimensional globally defined orientable manifold from which is constructed the product space N = M x R. It is assumed that N is endowed with a set of 2n independent smooth 1-forms {π(h),πʰ:h = 1,..,n}. Certain conditions are imposed on {π(h),πʰ} which imply the existence of local coordinates {qʰ,p(h)} on M and a function H(qʰ,p(h),t) on N, where t is the single coordinate on R, such that dπ = π(h) ∧ πʰ, where π has the structure of a Cartan form on N. The assumption that the function h = p(h)∂H/∂p(h)-H is non-zero on a region D ⊂ N, implies that π has maximal class on D. This construction gives rise to a local adapted contact structure on N and a local symplectic structure on M. A vector field X on N is said to be a contact field if there exists a smooth function σ : N → R such that ₤ₓπ = σπ. A vector field Z on N is called a canonical vector field if it admits the representation Z = ∂/∂t + (H, ) where (,) denotes the Poisson bracket on M. For a given function σ, a prescription is given for the construction of the space c(σ)(N) of contact fields in terms of solutions F of the p.d.e. Z<F> = σh. The vector space (UNFORMATTED EQUATION FOLLOWS) c(N) = ∪ (σ∊C)(∞)c(σ)(N) (END UNFORMATTED EQUATION) is shown to have the structure of a Lie sub-algebra of the Lie algebra of vector fields on N. It is shown that the associated subspace A(π) = {X:X˩π = 0} is such that c(σ)(N) ∩ A(π) = {0}. This implies that there is an X in c(σ)(N) such that X˩π ≠ 0. Thus, if the function H that appears in the Cartan form π is such that H = X˩π for any X ∊ c(σ)(N) it is possible to deduce that ∂H/∂t ≠ 0, which shows that such vector fields may be of relevance to the theory of non-conservative systems. A different set of 2n 1-forms {π(h),πʰ} is introduced on N that are subject to analogous conditions which ensure the existence of local coordinates (qʰ,p(h)) on M and a function K(qʰ,p(h),t) that gives rise to a new Cartan form π on N such that dπ= π(h) ∧ πʰ. By definition, the fundamental invariant of a parameter-dependent canonical transformation on N is dπ = dπ. In this new setting a contact field X satisfies the ₤ₓπ = σπ for some function σ: N to R. The relationship between the contact vector fields X and X is investigated in depth.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Master of Science. Johannesburg, March 2015. / The primary purpose of this study is to investigate the asymptotic distribution of the eigenvalues of self-adjoint second order di erential operators. We rst analyse the problem where the functions g and h are equal to zero. To improve on the terms of the eigenvalue problem for g; h = 0, we consider the eigenvalue problem for general functions g and h. Here we calculate explicitly the rst four terms of the eigenvalue asymptotics problem.

Thesis (Ph. D.)--Oregon State University, 1970. / Typescript (photocopy). Includes bibliographical references (leaves 28). Also available on the World Wide Web.

Thesis (Ph. D.)--Oregon State University, 1970. / Typescript (photocopy). Includes bibliographical references (leaf 44). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Chicago, 1915. / Vita. "A Private edition distributed by the University of Chicago Libraries." "Reprinted from Rendiconti del Circolo matematico di Palermo, Vol. XLI, 1916." Includes bibliographical references. Also available on the Internet.

Differential pairs may gradually replace single trace as the media transmitting high-speed signal on high-speed digital circuit. This dissertation investigates the impact on electromagnetic radiation from high-speed circuit board by the distance between differential traces and edge effect .At the same time, we use the result of measurement and simulation to make sure the measurement method correct .The result shows that,the shorter the distance is,the lower the EMI of differential pairs have .The edge effect will has the ability of destroying the suppression of EMI. Based on the consideration of measuring correctly, we use a solid kind of measurement method to search some non-ideal effects before studying EMI from differential pairs .These effect may exist on real measurement or ideal simulation. We hope to understand the environment of measurement as clearly as possible. In addition, we address one simple and fast method to investigate radiation mechanism that resulted from edge effect of differential pairs.By this method to calculate radiation, we will understand more and more the radiation mechanism.