<p> The object of this dissertation is to investigate the properties, associated boundary conditions and generalized resolvents of symmetric ordinary differential operators associated with formally self-adjoint nth order ordinary differential expressions with complex coefficients. </p> <p> While symmetric differential operators with equal deficiency indices have been studied in some detail, expecially the particular case when the underlying differential expression has real coefficients, little research has been done on the properties of symmetric differential operators with unequal deficiency indices which are associated with a differential expression with complex coefficients. </p> <p> By extending the symmetric differential operators with unequal deficiency indices to suitable operators with equal deficiency indices in larger Hilbert spaces and introducing a new type of boundary conditions to these extensions, we obtain important information about the original symmetric differential operators with unequal deficiency indices. We are able to generate some well-known theorems of I. M. Glazman (1950) and E. A. Coddington (1954) dealing with the characterization of self-adjoint extensions of symmetric operators in terms of boundary conditions, the relation between the deficiency indices of operators on the whole real line and on the half-line, and the resolvent of self-adjoint extensions, from the theory of symmetric, in particular real, differential operators with equal deficiency indices. We also generalize the result of W. N. Everitt (1959) concerning the number of integrable-square solutions of differential equations with one particular and one singular end-point to the case in which both end-points are singular. Finally, under certain assumptions, we extend some of the fundamental results of K. Kodaira (1950) based upon the methods of algebraic geometry, concerning Green's functions and the minimal symmetric differential operator associated with an even-order formally self-adjoint ordinary differential expansion with real coefficients to the case of Green's functions and the minimal symmetric differential operator associated with an even-order formally self-adjoint ordinary differential expression with complex coefficients. </p> / Thesis / Doctor of Philosophy (PhD)

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-08T12:20:00Z No. of bitstreams: 2 Dissertação - Mario Jonas da Silva Santos - 2014.pdf: 1833814 bytes, checksum: 949d82c89c9bfa0c9361693ee94784ec (MD5) license_rdf: 21267 bytes, checksum: 73e23c2acaaf13389e092bd813e3223d (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-08T12:46:39Z (GMT) No. of bitstreams: 2 Dissertação - Mario Jonas da Silva Santos - 2014.pdf: 1833814 bytes, checksum: 949d82c89c9bfa0c9361693ee94784ec (MD5) license_rdf: 21267 bytes, checksum: 73e23c2acaaf13389e092bd813e3223d (MD5) / Made available in DSpace on 2015-10-08T12:46:40Z (GMT). No. of bitstreams: 2 Dissertação - Mario Jonas da Silva Santos - 2014.pdf: 1833814 bytes, checksum: 949d82c89c9bfa0c9361693ee94784ec (MD5) license_rdf: 21267 bytes, checksum: 73e23c2acaaf13389e092bd813e3223d (MD5) Previous issue date: 2014-09-26 / Let's start making an explanation of some important content. Starting with the set of complex numbers, polynomials, ellipse, derivative of a function in the complex variable and congruence triangles then we enunciate three lemma and demonstrates for them then enunciate and prove Theorem Marden. At the end we will have a proposal to class in the form of math workshop, students applied for the 3rd Series of high school. / Vamos começar fazendo uma explanação de alguns conteúdos importantes. Come çando com o conjunto dos números complexos, polinômios, elipse, derivada de uma função na variável complexa e conguência de triângulos em seguida vamos enuciar três lemas e demonstrá los para então enunciar e demonstrar o Teorema de Marden. Ao nal teremos uma proposta de aula em forma de o cina matemática, aplicada para alunos da 3a Série do ensino médio.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-10T14:58:57Z No. of bitstreams: 2 Dissertação - Pollyana Gomes Volpato - 2016.pdf: 1172027 bytes, checksum: 6e8ebfc67549380e690db90bc68b9104 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-01-11T09:42:09Z (GMT) No. of bitstreams: 2 Dissertação - Pollyana Gomes Volpato - 2016.pdf: 1172027 bytes, checksum: 6e8ebfc67549380e690db90bc68b9104 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-01-11T09:42:09Z (GMT). No. of bitstreams: 2 Dissertação - Pollyana Gomes Volpato - 2016.pdf: 1172027 bytes, checksum: 6e8ebfc67549380e690db90bc68b9104 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-12-09 / The main objective of this work is to demonstrate Marden’s Theorem, which tells us that given a third-degree polynomial with complex coefficients, the roots of this polynomial are not collinear and form a triangle T in the complex plane. There is a unique ellipse inscribed in T and tangent to the sides at their midpoints. The foci of this ellipse are the roots of the derivative of the polynomial. We show that such an ellipse is Steiner’s Ellipse. We make a generalization of the Marden Theorem using degree n polynomial. / Temos como objetivo central neste trabalho demonstrar o Teorema de Marden, que nos diz que, dado um polinômio de terceiro grau com coeficientes complexos, as raízes desse polinômio não são colineares e formam um triângulo T no plano complexo. Há uma única elipse inscrita em T e tangente aos lados nos seus pontos médios. Os focos dessa elipse são as raízes da derivada do polinômio. Mostramos que tal elipse é a Elipse de Steiner. Fazemos uma generalização do Teorema de Marden utilizando polinômio de grau n.

This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.