We consider on the interval [0; a], rstly fourth-order di erential operators with eigenvalue parameter dependent boundary conditions and secondly a sixth-order di erential operator with eigenvalue parameter dependent boundary conditions. We associate to each of these problems a quadratic operator pencil with self-adjoint operators. We investigate the spectral proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst four terms of the eigenvalue asymptotics.

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Let G be a complex semisimple Lie Group with parabolic subgroup P, so that G/P is a generalized flag manifold. An algebraic construction of invariant differential operators between sections of homogeneous bundles over such spaces is given and it is shown how this leads to the classification of all such operators. As an example of a process which naturally generates such operators, the algebraic Penrose transform between generalized flag manifolds is given and computed for several cases, extending standard results in Twistor Theory to higher dimensions. It is then shown how to adapt the homogeneous construction to manifolds with a certain class of tangent bundle structure, including conformal manifolds. This leads to a natural definition of invariant differential operators on such manifolds, and an algebraic method for their construction. A curved analogue of the Penrose transform is given.

Originally presented as the author's thesis, Bonn. / Added t.p. with thesis statement inserted. Bibliography: p. 78-80.

Thesis--Bonn. / Includes bibliographical references (p. 58-60).

Thesis (doctoral)--Bonn, 1998. / Includes bibliographical references (p. 84-86).

The present paper deals with the calculus of Tk,x,y,z - operator. The operator is a three variable analogue of the operator given earlier by W. A. Al-Salam [1] and H. B. Mittal [10]. The operator is useful for finding operational representations and generating functions of polynomials of three variables and will be dealt in a separate communication.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in ful lment of the requirements for the degree of Masters of Science. May 27, 2015. / Abstract The unsteady three-dimensional ow of a thin slender rivulet of incompressible Newtonian uid down an inclined porous plane is investigated. The leak-o velocity is not speci ed in the model but is determined in the process of deriving the invariant solution. A second order nonlinear partial di erential equation in two spatial variables and time and containing the leak-o velocity is derived for the height of the thin slender rivulet. Using Lie group analysis it is found that the partial di erential equation can be reduced in two steps to an ordinary di erential equation provided the leak-o velocity satis es a rst order linear partial di erential equation in three variables. An exact analytical solution with a dry patch in the central region is derived for a special leak-o velocity. Two models are considered, one with the leak-o velocity proportional to the height of the rivulet and the other with leak-o velocity proportional to the cube of the height. Numerical solutions are obtained for the height of the rivulet using a shooting method which also determines the two-dimensional boundary of the rivulet on the inclined plane. The e ect of uid leak-o on the height and width of the rivulet is investigated numerically and compared in the two models. The conservation laws for the partial di erential equation with no uid leak-o are investigated. Two conserved vectors are derived, the elementary conserved vector and a new conserved vector. The Lie point symmetry of the partial di erential equation associated with each conserved vector is obtained. Each associated Lie point symmetry is used to perform a double reduction of the partial di erential equation, but the solutions obtained are not physically signi cant.

We develop a technique for identifying unknown coefficients in parabolic partial differential equations. The identification scheme is based on quasilinearization and is applied to both linear and nonlinear equations where the unknown coefficients may be spatially varying. Our investigation includes derivation, convergence, and numerical testing of the quasilinearization based identification scheme / Ph. D.

A (bounded linear) operator J on a Hilbert space is said to be jordan if J = S + N where S = S* and N² = 0. The operator T is subjordan if T is the restriction of a jordan operator to an invariant subspace, and pure subjordan if no nonzero restriction of T to an invariant subspace is jordan. The main operator theoretic result of the paper is that a compact subset of the real line is the spectrum of some pure subjordan operator if and only if it is the closure of its interior. The result depends on understanding when the operator D = θ + d/dx : L²(μ) —> L²(v) is closable. Here θ is an L²(μ) function, μ and v are two finite regular Borel measures with compact support on the real line, and the domain of D is taken to be the polynomials. Approximation questions more general than what is needed for the operator theory result are also discussed. Specifically, an explicit characterization of the closure of the graph of D for a large class of (θ, μ, v) is obtained, and the closure of the graph of D in other topologies is analyzed. More general results concerning spectral synthesis in a certain class of Banach algebras and extensions to the complex domain are also indicated. / Ph. D.