Estudamos a álgebra de comparação do círculo, descrevemos o espaço de Gelfand do seu quociente pelos compactos e damos uma fórmula para o índice dos seus operadores de Fredholm. Depois generalizamos o resultado para as matrizes com elementos na álgebra de comparação e damos uma aplicação para operadores diferenciais no círculo. / We study the comparison algebra on the circle, we describe the Gelfand space of its quotient by the compacts and we give a formula to compute the index of its Fredholm operators. After that, we generalize the result to the matrices with entries in the comparison algebra and give an application to differential opperators in the circle.

Estudamos a álgebra de comparação do círculo, descrevemos o espaço de Gelfand do seu quociente pelos compactos e damos uma fórmula para o índice dos seus operadores de Fredholm. Depois generalizamos o resultado para as matrizes com elementos na álgebra de comparação e damos uma aplicação para operadores diferenciais no círculo. / We study the comparison algebra on the circle, we describe the Gelfand space of its quotient by the compacts and we give a formula to compute the index of its Fredholm operators. After that, we generalize the result to the matrices with entries in the comparison algebra and give an application to differential opperators in the circle.

by Chi-Fai Chan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 53-54). / Abstract also in Chinese. / Chapter Chapter 1 --- INTRODUCTION --- p.1 / Chapter §1.1 --- Polynomial Interpolation --- p.1 / Chapter §1.2 --- Conjugate Gradient Type Methods --- p.6 / Chapter §1.3 --- Outline of the Thesis --- p.10 / Chapter Chapter 2 --- INTEGRAL EQUATIONS --- p.11 / Chapter §2.1 --- Integral Equations --- p.11 / Chapter §2.2 --- Numerical Treatments of Second Kind Integral Equations --- p.15 / Chapter Chapter 3 --- FAST ALGORITHM FOR SECOND KIND INTEGRAL EQUATIONS --- p.20 / Chapter §3.1 --- Introduction --- p.20 / Chapter §3.2 --- The Approximation --- p.24 / Chapter §3.3 --- Error Analysis --- p.35 / Chapter §3.4 --- Numerical Examples --- p.40 / Chapter §3.5 --- Concluding Remarks --- p.51 / References --- p.53

Errata inserted / Bibliography: leaves 170-175 / iii, 175 leaves : ill ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, 1987

Thesis (Ph. D. in Mathematics)--Vanderbilt University, May 2007. / Title from title screen. Includes bibliographical references.

Graduation date: 1991

Die Dissertation untersucht die Invertierbarkeit im Unendlichen fuer Normgrenzwerte von Bandoperatoren - sogenannte band-dominierte Operatoren. Das dazu verwendete Instrument ist die Methode der Limitoperatoren. Es werden grundlegende Eigenschaften von Limitoperatoren bewiesen, Zusammenhaenge zur Invertierbarkeit im Unendlichen hergeleitet, sowie darueber hinaus gehende Anwendungen, z.B. zur Konvergenz von Projektionsverfahren, studiert.

This text is concerned with the Fredholm theory and stable approximation of bounded linear operators generated by a class of infinite matrices $(a_{ij})$ that are either banded or have certain decay properties as one goes away from the main diagonal. The operators are studied on $\ell^p$ spaces of functions $\Z^N\to X$, where $p\in[1,\infty]$, $N\in\N$ and $X$ is a complex Banach space. The latter means that our matrix entries $a_{ij}$ are indexed by multiindices $i,j\in\Z^N$ and that every $a_{ij}$ is itself a bounded linear operator on $X$. Our main focus lies on the case $p=\infty$, where new results are derived, and it is demonstrated in both general theory and concrete operator equations from mathematical physics how advantage can be taken of these new $p=\infty$ results in the general case $p\in[1,\infty]$.

The two-dimensional, viscous flow around an elliptic cylinder undergoing prescribed unsteady motions is analyzed. The fluid is taken to be incompressible. Departing from the conventional vorticity-stream function approach, the Biot-Savart law of induced velocities is utilized to account for the contribution to the velocity field of the different vorticity fields comprising the flow. These include the internal vorticity due to the rotation of the body, the free vorticity in the fluid surrounding the body, and the bound vorticity distributed along the body contour. In order to apply the method, the body must be assumed to be replaced by fluid of the same density as the undisturbed surroundings. The replacement fluid must have a rigid motion exactly the same as the actual body motion. This can be achieved by placing suitable distributed vorticity fields within and on the surface of the body. The bound vorticity on the body surface is in the form of a vortex sheet, and its distribution is governed by a Fredholm integral equation of the second kind. The equation is derived in detail. It is solved numerically. The motion of the free vorticity in the flow field is governed by the Navier-Stokes equations written in terms of vorticity. The descretized vorticity transport equation is derived for a control volume and is solved numerically using an explicit method with a forward-difference for the time derivative, and a central-difference for the diffusive terms. An upwind method is used for convection terms. The results obtained using the present method are compared with a number of special cases available in the literature. Viscous flows around a circular cylinder rotating in any arbitrary fashion possess an exact solution, as presented in Chapter 2. Two cases of this flow are chosen for comparison. In the first case the circular cylinder is initially given an impulsive twist such that it rotates with a constant velocity about its axis. In the second case, the angular velocity of the circular cylinder increases with time exponentially. For a Reynolds number of 100, based on the cylinder radius and the internal vorticity, the exact solutions are compared with the numerical results. Viscous flow around an elliptic cylinder of .0996 aspect ratio rotating with a constant angular velocity is another special case, available in the literature, which is chosen for comparison. For this case the Reynolds number, based on the cylinder semi-major-axis and internal vorticity is 202. The agreement in all above-mentioned cases is excellent. Finally, viscous flow around an elliptic cylinder of .25 aspect ratio undergoing a combined translation and pitching oscillation is presented. A Reynolds number of 500, based on the semi-major-axis and body translational velocity, is chosen for this case. No similar case has been reported until now. This case, however, is only one of the many cases that can be handled by the present method.

Universiẗat, Diss., 2001--Dortmund.