Let G be a Lie group acting on a homogeneous space G/K. The center of the universal enveloping algebra of the Lie algebra of G maps homomorphically into the center of the algebra of differential operators on G/K invariant under the action of G. In the case that G is a Jacobi Lie group of rank 2, we prove that this homomorphism is surjective and hence that the center of the invariant differential operator algebra is the image of the center of the universal enveloping algebra. This is an extension of work of Bringmann, Conley, and Richter in the rank 1case.

<p>In this work, we apply $L^2$-index theory to compute the index of a non-Fredholm elliptic operator. The operator arises in Type I' string theory, and the index is found to be non-zero, thus implying existence of bound states.</p> / Dissertation

We study the integral equation related to the three and higher dimensional superradiance problem. Collective radiation phenomena has attracted the attention of many physicists and chemists since the pioneering work of R. H. Dicke in 1954. We first consider the three-dimensional superradiance problem and find a differential operator that commutes with the integral operator related to the problem. We find all the eigenfunctions of the differential operator and obtain a complete set of eigensolutions for the three-dimensional superradiance problem. Generalization of the three-dimensional superradiance integral equation is provided. A commuting differential operator is found for this generalized problem. For the three dimensional superradiance problem, an alternative set of complete eigenfunctions is also provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak. The uniqueness of the differential operator commuting with that kernel is indicated. Finally, a concentration problem for the signals which are bandlimited in disjoint frequency-intervals is considered. The problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted in a given time interval. A numerical algorithm for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L2(−∞,∞) are also proved. Numerical computations are carried out in support of the theory.

The Differential Operator D is a linear operator from C1[0,1] onto C[0,1]. Its domain C1[0,1] is thoroughly studied as a meager subspace of C[0,1]. This is analogous to the status of the set of all rational numbers Q in the set of the real numbers R. On the polynomial vector space Pn the Differential Operator D is a nilpotent operator. Using the invariant subspace and reducing subspace technique an appropriate basis for the underlying vector space can be found so that the nilpotent operator admits its Jordan Canonical form. The study of D on Pn is completely carried out. Finally, the solution space V of the nth order differential equation with leading coefficient one is studied. The behavior of D on V is explored using some notions from linear algebra and linear operators. NOTE- Due to the limitation of the above being in "text only form" , further details of this abstract can be viewed in the pdf file.

The concept of pseudo-differential operators allows one to study stochastic processes through their symbol. This approach has generated many new insights in recent years. However, most results are based on the assumption of bounded coefficients. In this thesis, we study Levy-type processes with unbounded coefficients and, especially, affine processes. In particular, we establish a connection between pseudo-differential operators and affine processes which are well-known from mathematical finance. Affine processes are an interesting example in this field since they have linearly growing and hence unbounded coefficients. New techniques and tools are developed to handle the affine case and then expanded to general Levy-type processes. In this way, the convergence of a simulation scheme based on a Markov chain approximation, results on path properties, and necessary conditions for the symmetry of operators were proven.

Neste trabalho mostramos uma caracterização para os espaços de potência fracionária associados ao operador 1 - &Delta;p, em que &Delta;p representa o fecho do operador laplaciano em Lp(Rn), usando o fato de que o mesmo pode ser visto como um operador pseudodiferencial com símbolo a(&xi;) = 1+4&pi;2|&xi;|2. No processo para obter essa caracterização representamos de maneira concreta a solução abstrata u : [0;+ &infin;) &rarr; Lp(Rn), obtida através da teoria de operadores setoriais e semigrupos analíticos, da equação u - &Delta;pu = 0 em (0;+&infin;) com condição inicial u(0) = f &isin; Lp(Rn). / In this work we show a characterization for the fractional power spaces associated with the operator 1 - &Delta;p, where &Delta;p, represents the closure of the Laplacian operator in Lp(Rn), using the fact that the operator may be seen as a pseudo-differential operator with symbol a(&xi;) = 1+4&pi;2|&xi;|2. In the process for this characterization we represent of concrete way the abstract solution u : [0;+&infin;) Lp(Rn), obtained through the theory of sector operators and analytic semigroups, of the equation u - &Delta;pu = 0 in (0;+&infin;) with initial condition u(0) = f &isin; Lp(Rn).

Šiame magistro baigiamajame darbe nagrinėjamas paprastasis diferencialinis operatorius su viena klasikine (pirmo arba antro tipo) sąlyga kairiajame intervalo krašte ir kita nelokaliąja (integraline, Samarskio ir Bitsadzės ar antro tipo) sąlyga dešiniajame intervalo gale. Mokslinėje literatūroje nemažai rašoma apie tokio uždavinio realiojo spektro struktūrą, tačiau kompleksinis spektras yra pakankamai mažai nagrinėjamas. Magistro baigiamajame darbe aprašyta šio uždavinio realiojo ir kompleksinio spektro struktūra, ištirtos kompleksinių tikrinių reikšmių teigiamųjų realiųjų dalių egzistavimo sąlygos, bei jų priklausomybė nuo nelokaliųjų kraštinių sąlygų parametrų. / In this master thesis there is investigated ordinary differential operator with one classical (first or second type) boundary condition in the left side of the interval and other nonlocal (integral, Samarski – Bitsadze or second type) boundary condition in the right side of the interval. The structure of the real spectrum of this problem is quite wide described in the scientific literature, but the complex spectrum is investigated not enough. There is described the real and complex spectrum structure of this problem. Also in the master thesis there are analyzed existence conditions of positive real parts of complex eigenvalues, and their dependence on nonlocal boundary condition parameters.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / No primeiro capÃtulo da dissertaÃÃo, à apresentada uma breve introduÃÃo do trabalho. Em seguida, no segundo capÃtulo, sÃo demonstradas noÃÃes e propriedades de espaÃos vetoriais topolÃgicos. Dando seguimento ao presente estudo, no terceiro capÃtulo, efetua-se a abordagem da teoria das distribuiÃÃes, onde se proporciona, como exemplo a distribuiÃÃo delta de Dirac, na qual, por conseguinte, sÃo definidas ainda operaÃÃes com distribuiÃÃes, entre elas a convoluÃÃo de uma distribuiÃÃo com uma funÃÃo teste, e por fim, ainda no mesmo capitulo à feito uma anÃlise das distribuiÃÃes com suporte compacto. No capÃtulo quatro, por sua vez, explana-se a transformada de Fourier e suas propriedades, bem como, propriedades de funÃÃes que pertencem ao espaÃo de Schwartz e ainda, à feito um estudo das distribuiÃÃes temperadas. Finalmente, no quinto e Ãltimo capÃtulo à demonstrado o teorema de Malgrange-Ehrenpreis, que à a temÃtica principal do trabalho elaborado, o qual afirma que todo operador diferencial com coeficientes constantes tem uma soluÃÃo fundamental. Destarte, à implementado um estudo de alguns exemplos afins ao teorema. / In the first chapter of the dissertation, is a brief introduction. Then in the second chapter, are shown notions and properties of topological vector spaces. Following the present study, the third chapter, is effected the approach to the theory of distributions, which provides, as an example the Dirac delta distribution, in which, therefore, are dened further distribution operations, including the convolution of a distribution with a test function, and finally, still in same chapter an analysis is made of distributions with compact support. In chapter four, in turn, explains to the Fourier transform and its properties, as well as properties of functions belonging to Schwartz space and also a study is made of tempered distributions. Finally, the fifth and final chapter is shown the Malgrange-Ehrenpreis theorem, which is the main theme of the work done,which states that any differential operator with constant coecients has a fundamental solution. Thus, it implemented a study of some examples related to the theorem.

Neste trabalho mostramos uma caracterização para os espaços de potência fracionária associados ao operador 1 - &Delta;p, em que &Delta;p representa o fecho do operador laplaciano em Lp(Rn), usando o fato de que o mesmo pode ser visto como um operador pseudodiferencial com símbolo a(&xi;) = 1+4&pi;2|&xi;|2. No processo para obter essa caracterização representamos de maneira concreta a solução abstrata u : [0;+ &infin;) &rarr; Lp(Rn), obtida através da teoria de operadores setoriais e semigrupos analíticos, da equação u - &Delta;pu = 0 em (0;+&infin;) com condição inicial u(0) = f &isin; Lp(Rn). / In this work we show a characterization for the fractional power spaces associated with the operator 1 - &Delta;p, where &Delta;p, represents the closure of the Laplacian operator in Lp(Rn), using the fact that the operator may be seen as a pseudo-differential operator with symbol a(&xi;) = 1+4&pi;2|&xi;|2. In the process for this characterization we represent of concrete way the abstract solution u : [0;+&infin;) Lp(Rn), obtained through the theory of sector operators and analytic semigroups, of the equation u - &Delta;pu = 0 in (0;+&infin;) with initial condition u(0) = f &isin; Lp(Rn).

This report presents and discusses proofs of the Malgrange-Ehrenpreis theorem, which states that every non-zero linear partial differential operator with constant coefficients has a fundamental solution. The main topic is explicit formulae, and more specifically, how they can be used to prove the theorem. Two different formulas will be considered in detail and the aim is to provide a fundamental and elementary description of how to prove the Malgrange-Ehrenpreis theorem using those formulas. In addition to the proofs, an example of how to use one of the formulas for the Cauchy-Riemann operator is shown. Finally, the report also contains a chapter discussing a few different notable methods of proof and their historical signifance.