Harmonische Funktionen auf dem Bruhat-Tits-Gebäude der PGL_3 über Funktionenkörpern

Harmonische Funktionen auf dem Bruhat-Tits-Gebäude der PGL(3) über Funktionenkörpern lassen sich als ein Analogon zu den auf der oberen Halbebene definierten klassischen Spitzenformen verstehen. An die Stelle des starken Abklingens der Spitzenformen tritt hier die Endlichkeit des Trägers modulo einer gewissen Untergruppe. Der erste Teil der vorliegenden Arbeit befaßt sich mit der Untersuchung und Charakterisierung dieses Trägers. Im weiteren Verlauf werden gewisse Konzepte der klassischen Theorie auf harmonische Funktionen übertragen. So wird gezeigt, daß diese sich ebenfalls als Fourierreihe darstellen lassen und es werden explizite Formeln für die Fourierkoeffizienten hergeleitet. Es stellt sich heraus, daß sich die Harmonizität in gewissen Relationen zwischen den Fourierkoeffizienten widerspiegelt und sich umgekehrt aus einem Satz passender Koeffizienten eine harmonische Funktion erzeugen läßt. Dies wird zur expliziten Konstruktion zweier quasi-harmonischer Funktionen genutzt, die ein Pendant zu klassischen Poincaré-Reihen darstellen. Abschließend werden Hecke-Operatoren definiert und Formeln für die Fourierkoeffizienten der Hecke-Transformierten einer harmonischen Funktion hergeleitet.

Lecture notes pdf

Lecture notes in PDF

MA17xexam01

Exam questions and solutions in LaTex. Diagrams for the questions are all together in the support.zip file, as .eps files

MA17xexam00pdf

Exam questions and solutions in PDF

Lecture notes

Lecture notes in LaTex

Exercise sheet 2

Exercises and solutions in LaTex

MA17xexam00

Exam questions and solutions in LaTex. Diagrams for the questions are all together in the support.zip file, as .eps files

Lecture notes

Lecture notes in LaTex

MA17xexam01pdf

Exam questions and solutions in PDF

Exercise sheet 2 pdf

Exercises and solutions in PDF

Lecture notes pdf

Lecture notes in PDF

MATH2038 Notes, Lectures and handouts

The MATH2038 (Partial Differential Equations) course, as given in semester 2 2008/9. Syllabus has changed slightly from previous years, as has coursework weighting.

Sparse shape representation using the Laplace-Beltrami eigenfunctions and its application to modeling subcortical structures

We present a new sparse shape modeling framework on the Laplace-Beltrami (LB) eigenfunctions. Traditionally, the LB-eigenfunctions are used as a basis for intrinsically representing surface shapes by forming a Fourier series expansion. To reduce high frequency noise, only the first few terms are used in the expansion and higher frequency terms are simply thrown away. However, some lower frequency terms may not necessarily contribute significantly in reconstructing the surfaces. Motivated by this idea, we propose to filter out only the significant eigenfunctions by imposing l1-penalty. The new sparse framework can further avoid additional surface-based smoothing often used in the field. The proposed approach is applied in investigating the influence of age (38-79 years) and gender on amygdala and hippocampus shapes in the normal population. In addition, we show how the emotional response is related to the anatomy of the subcortical structures.

Gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type

We study the Gevrey solvability of a class of complex vector fields, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), given by L = partial derivative/partial derivative t + (a(x) + ib(x))partial derivative/partial derivative x, b not equivalent to 0, near the characteristic set Sigma = {0} x S(1). We show that the interplay between the order of vanishing of the functions a and b at x = 0 plays a role in the Gevrey solvability. (C) 2008 Elsevier Inc. All rights reserved.

Motion Detection for Video Surveillance

This thesis is related to the broad subject of automatic motion detection and analysis in videosurveillance image sequence. Besides, proposing the new unique solution, some of the previousalgorithms are evaluated, where some of the approaches are noticeably complementary sometimes.In real time surveillance, detecting and tracking multiple objects and monitoring their activities inboth outdoor and indoor environment are challenging task for the video surveillance system. Inpresence of a good number of real time problems limits scope for this work since the beginning. Theproblems are namely, illumination changes, moving background and shadow detection.An improved background subtraction method has been followed by foreground segmentation, dataevaluation, shadow detection in the scene and finally the motion detection method. The algorithm isapplied on to a number of practical problems to observe whether it leads us to the expected solution.Several experiments are done under different challenging problem environment. Test result showsthat under most of the problematic environment, the proposed algorithm shows the better qualityresult.