Some New Classes of Orthogonal Polynomials and Special Functions

In dieser Dissertation präsentieren wir zunächst eine Verallgemeinerung der üblichen Sturm-Liouville-Probleme mit symmetrischen Lösungen und erklären eine umfassendere Klasse. Dann führen wir einige neue Klassen orthogonaler Polynome und spezieller Funktionen ein, welche sich aus dieser symmetrischen Verallgemeinerung ableiten lassen. Als eine spezielle Konsequenz dieser Verallgemeinerung führen wir ein Polynomsystem mit vier freien Parametern ein und zeigen, dass in diesem System fast alle klassischen symmetrischen orthogonalen Polynome wie die Legendrepolynome, die Chebyshevpolynome erster und zweiter Art, die Gegenbauerpolynome, die verallgemeinerten Gegenbauerpolynome, die Hermitepolynome, die verallgemeinerten Hermitepolynome und zwei weitere neue endliche Systeme orthogonaler Polynome enthalten sind. All diese Polynome können direkt durch das neu eingeführte System ausgedrückt werden. Ferner bestimmen wir alle Standardeigenschaften des neuen Systems, insbesondere eine explizite Darstellung, eine Differentialgleichung zweiter Ordnung, eine generische Orthogonalitätsbeziehung sowie eine generische Dreitermrekursion. Außerdem benutzen wir diese Erweiterung, um die assoziierten Legendrefunktionen, welche viele Anwendungen in Physik und Ingenieurwissenschaften haben, zu verallgemeinern, und wir zeigen, dass diese Verallgemeinerung Orthogonalitätseigenschaft und -intervall erhält. In einem weiteren Kapitel der Dissertation studieren wir detailliert die Standardeigenschaften endlicher orthogonaler Polynomsysteme, welche sich aus der üblichen Sturm-Liouville-Theorie ergeben und wir zeigen, dass sie orthogonal bezüglich der Fisherschen F-Verteilung, der inversen Gammaverteilung und der verallgemeinerten t-Verteilung sind. Im nächsten Abschnitt der Dissertation betrachten wir eine vierparametrige Verallgemeinerung der Studentschen t-Verteilung. Wir zeigen, dass diese Verteilung gegen die Normalverteilung konvergiert, wenn die Anzahl der Stichprobe gegen Unendlich strebt. Eine ähnliche Verallgemeinerung der Fisherschen F-Verteilung konvergiert gegen die chi-Quadrat-Verteilung. Ferner führen wir im letzten Abschnitt der Dissertation einige neue Folgen spezieller Funktionen ein, welche Anwendungen bei der Lösung in Kugelkoordinaten der klassischen Potentialgleichung, der Wärmeleitungsgleichung und der Wellengleichung haben. Schließlich erklären wir zwei neue Klassen rationaler orthogonaler hypergeometrischer Funktionen, und wir zeigen unter Benutzung der Fouriertransformation und der Parsevalschen Gleichung, dass es sich um endliche Orthogonalsysteme mit Gewichtsfunktionen vom Gammatyp handelt.

Recycling urban waste as possible use for rooftop vegetable garden

Urban authorities in Europe are confronted with increasing demands by urban dwellers for allotment gardens, but vacant urban soil tends to be scarce and/or polluted by past industrial activities. A possible solution for local authorities could therefore be to promote rooftop gardening. However little technical information exists on certain forms of rooftop urban agriculture, called Z-Farming. In 2012, a pilot experiment was run in Paris (France). Simple and cheap systems of rooftop gardening were tested on a rooftop using as crop substrates only local urban organic waste so as to contribute to the urban metabolism. Production levels and heavy metal contents in cropping substrates and edible vegetables were measured. Available results show (i) high levels of crop production with limited inputs compared to land professional gardening, (ii) low levels of heavy metal pollutants in the edible parts of the crops, especially for Cd and Pb with respect to EU norms for vegetables and (iii) positive influence on yields on organizing the substrate in layers and enhancing the biological activity through earthworm inoculation. These encouraging results allow us to consider that rooftop gardening is feasible and seem to have a great potential to improve urban resiliency. It will nevertheless be necessary to identify more precisely the types of roof that can be used and to assess more fully the generic result of the low level of pollution, as well as the global sustainability of these cropping systems.

Online Learning of Non-stationary Sequences

We consider an online learning scenario in which the learner can make predictions on the basis of a fixed set of experts. The performance of each expert may change over time in a manner unknown to the learner. We formulate a class of universal learning algorithms for this problem by expressing them as simple Bayesian algorithms operating on models analogous to Hidden Markov Models (HMMs). We derive a new performance bound for such algorithms which is considerably simpler than existing bounds. The bound provides the basis for learning the rate at which the identity of the optimal expert switches over time. We find an analytic expression for the a priori resolution at which we need to learn the rate parameter. We extend our scalar switching-rate result to models of the switching-rate that are governed by a matrix of parameters, i.e. arbitrary homogeneous HMMs. We apply and examine our algorithm in the context of the problem of energy management in wireless networks. We analyze the new results in the framework of Information Theory.

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible consideration of densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended the Euclidean structure of the simplex to a Hilbert space structure of the set of densities within a bounded interval, and van den Boogaart (2005) generalized this to the set of densities bounded by an arbitrary reference density. From the many variations of the Hilbert structures available, we work with three cases. For bounded variables, a basis derived from Legendre polynomials is used. For variables with a lower bound, we standardize them with respect to an exponential distribution and express their densities as coordinates in a basis derived from Laguerre polynomials. Finally, for unbounded variables, a normal distribution is used as reference, and coordinates are obtained with respect to a Hermite-polynomials-based basis. To get the coordinates, several approaches can be considered. A numerical accuracy problem occurs if one estimates the coordinates directly by using discretized scalar products. Thus we propose to use a weighted linear regression approach, where all k- order polynomials are used as predictand variables and weights are proportional to the reference density. Finally, for the case of 2-order Hermite polinomials (normal reference) and 1-order Laguerre polinomials (exponential), one can also derive the coordinates from their relationships to the classical mean and variance. Apart of these theoretical issues, this contribution focuses on the application of this theory to two main problems in sedimentary geology: the comparison of several grain size distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock or sediment, like their composition

Exercise sheet 7 pdf

Exercises and solutions in PDF

Exam 1999 pdf

Exam and solutions in PDF

Exam 2000

Exam and solutions in LaTex

Exercise sheet 7

Exercises and solutions in LaTex

Exam 1999

Exam and solutions in LaTex

Exam 2000 pdf

Exam and solutions in PDF

PGR Tracker Demonstration for Students

This short video is a demonstration of the University of Southampton's Post Graduate Research (PGR) Tracker software that PGR students use for recording and tracking their post graduate progression milestones and training. Click on the 'Download' link to watch the video with a clearer image.

PGR Tracker Demonstration for Supervisors

This short video is a demonstration of the University of Southampton's Post Graduate Research (PGR) Tracker software that supervisors use for tracking and recording their post graduate student’s progression and viewing training records. Click on the 'Download' link to watch the video with a clearer image.

PGR Tracker Administration Manual

This is a user guide to support faculty administrators who need to use the PGR Tracker system to administer and maintain post graduate student records and related tasks. Use the 'Download' option underneath the main display to save a PDF copy of the manual locally.

Estrategias de tercerización en Colombia como centro de operaciones enfocado a los Call Centers en Barranquilla

El sector BPO&O es un sector con alto potencial en Colombia que aún se encuentra Incipiente. Dentro de este sector, los Call Centers juegan un papel importante para su desarrollo y Barranquilla posee grandes cualidades para permitir este crecimiento si se plantean las estrategias adecuadas.

COMP6037 - Readings

Attached are the readings from past and present COMP6037 lectures.