Approximate Approximations and a Boundary Point Method for the Linearized Stokes System

The method of approximate approximations, introduced by Maz'ya [1], can also be used for the numerical solution of boundary integral equations. In this case, the matrix of the resulting algebraic system to compute an approximate source density depends only on the position of a finite number of boundary points and on the direction of the normal vector in these points (Boundary Point Method). We investigate this approach for the Stokes problem in the whole space and for the Stokes boundary value problem in a bounded convex domain G subset R^2, where the second part consists of three steps: In a first step the unknown potential density is replaced by a linear combination of exponentially decreasing basis functions concentrated near the boundary points. In a second step, integration over the boundary partial G is replaced by integration over the tangents at the boundary points such that even analytical expressions for the potential approximations can be obtained. In a third step, finally, the linear algebraic system is solved to determine an approximate density function and the resulting solution of the Stokes boundary value problem. Even not convergent the method leads to an efficient approximation of the form O(h^2) + epsilon, where epsilon can be chosen arbitrarily small.

Quasi-Interpolation von Funktionen und Anwendungen auf Potentialprobleme

Ausgangspunkt der Dissertation ist ein von V. Maz'ya entwickeltes Verfahren, eine gegebene Funktion f : Rn ! R durch eine Linearkombination fh radialer glatter exponentiell fallender Basisfunktionen zu approximieren, die im Gegensatz zu den Splines lediglich eine näherungsweise Zerlegung der Eins bilden und somit ein für h ! 0 nicht konvergentes Verfahren definieren. Dieses Verfahren wurde unter dem Namen Approximate Approximations bekannt. Es zeigt sich jedoch, dass diese fehlende Konvergenz für die Praxis nicht relevant ist, da der Fehler zwischen f und der Approximation fh über gewisse Parameter unterhalb der Maschinengenauigkeit heutiger Rechner eingestellt werden kann. Darüber hinaus besitzt das Verfahren große Vorteile bei der numerischen Lösung von Cauchy-Problemen der Form Lu = f mit einem geeigneten linearen partiellen Differentialoperator L im Rn. Approximiert man die rechte Seite f durch fh, so lassen sich in vielen Fällen explizite Formeln für die entsprechenden approximativen Volumenpotentiale uh angeben, die nur noch eine eindimensionale Integration (z.B. die Errorfunktion) enthalten. Zur numerischen Lösung von Randwertproblemen ist das von Maz'ya entwickelte Verfahren bisher noch nicht genutzt worden, mit Ausnahme heuristischer bzw. experimenteller Betrachtungen zur sogenannten Randpunktmethode. Hier setzt die Dissertation ein. Auf der Grundlage radialer Basisfunktionen wird ein neues Approximationsverfahren entwickelt, welches die Vorzüge der von Maz'ya für Cauchy-Probleme entwickelten Methode auf die numerische Lösung von Randwertproblemen überträgt. Dabei werden stellvertretend das innere Dirichlet-Problem für die Laplace-Gleichung und für die Stokes-Gleichungen im R2 behandelt, wobei für jeden der einzelnen Approximationsschritte Konvergenzuntersuchungen durchgeführt und Fehlerabschätzungen angegeben werden.

Zur Konvergenz der Randpunktmethode

Das von Maz'ya eingeführte Approximationsverfahren, die Methode der näherungsweisen Näherungen (Approximate Approximations), kann auch zur numerischen Lösung von Randintegralgleichungen verwendet werden (Randpunktmethode). In diesem Fall hängen die Komponenten der Matrix des resultierenden Gleichungssystems zur Berechnung der Näherung für die Dichte nur von der Position der Randpunkte und der Richtung der äußeren Einheitsnormalen in diesen Punkten ab. Dieses numerisches Verfahren wird am Beispiel des Dirichlet Problems für die Laplace Gleichung und die Stokes Gleichungen in einem beschränkten zweidimensionalem Gebiet untersucht. Die Randpunktmethode umfasst drei Schritte: Im ersten Schritt wird die unbekannte Dichte durch eine Linearkombination von radialen, exponentiell abklingenden Basisfunktionen approximiert. Im zweiten Schritt wird die Integration über den Rand durch die Integration über die Tangenten in Randpunkten ersetzt. Für die auftretende Näherungspotentiale können sogar analytische Ausdrücke gewonnen werden. Im dritten Schritt wird das lineare Gleichungssystem gelöst, und eine Näherung für die unbekannte Dichte und damit auch für die Lösung der Randwertaufgabe konstruiert. Die Konvergenz dieses Verfahrens wird für glatte konvexe Gebiete nachgewiesen.

An approximation method using approximate approximations

The aim of this paper is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Computation of relativistic symmetry orbitals for finite double point groups

A program is presented for the construction of relativistic symmetry-adapted molecular basis functions. It is applicable to 36 finite double point groups. The algorithm, based on the projection operator method, automatically generates linearly independent basis sets. Time reversal invariance is included in the program, leading to additional selection rules in the non-relativistic limit.

Analyzing Gene Centres with the Help of the Checklist Method – the Case of Syria

The present survey of species diversity of cultivated plants is the first for Syria. Some cultivated species will be added in the future, because due to the civil war in Syria, it was not possible to visit the country in the frame of the present work, as initially planned. Checklists proved to be a useful tool for overviewing the cultivated plants of selected areas and allow a characterization of the state of plant genetic resources of Syria. Syria has experienced several civilizations. Man settled in this productive land since ancient times and used its resources. However, such use has led to changes in vegetation and decline of wildlife through the country, in seashore areas, interior, mountains, and grassland. Plant domestication and growing started more than 10,000 years ago in West Asia. Since then, plentiful of economic plant species were present and used by man and his domesticated animals. Forming a part of the Fertile Crescent, where many of the world’s agricultural plants have evolved, Syria is extremely rich in agrobiodiversity. Wild progenitors of wheat and barley and wild relatives of many fruit trees such as almonds and pistachio as well as forage species are still found in marginal lands and less disturbed areas. These are threatened by a wide range of human activities, notably modern, extensive agriculture, overgrazing, overcutting and urban expansion. Syria is also considered as part of one of the main centres of origin, according to Vavilov, who had collected in Syria in 1926. The first expeditions to crop fields showed the exclusive nature of cultivated plants in Syria with a high number of endemic forms. Furthermore, Syria is a part of a biodiversity hotspot. Several studies have been performed to study agrobiodiversity in different parts of Syria, but usually on wild species. Many collections have been carried out; however, they focussed preferably on cereals and pulses, and particularly on wheat, like Vavilov’s expedition. Only 30 crops make up the major part of the conserved Syrian crop plant material in the genebank, indicating that most of the remaining 7,000 species of cultivated plants and many other valuable genetic resources species have only been included on a limited scale in the genebank collections. Although a small country (185,180 km2), Syria accommodates numerous ecosystems that allow for a large diversity of plant genetic resources for agriculture ranging from cold-requiring to subtropical crops to live and thrive. Only few references are available in this respect. The aim of the present study was to complete a checklist of Syria’s cultivated plants of agriculture and horticulture excluding plants only grown as ornamental or for forestry. Furthermore, plants taken for reforestation have not been included, if they do not have also agricultural or horticultural uses. Therefore, the inclusion of plants into the checklist follows the same principles as “Mansfeld’s Encyclopedia”. Main sources of information were published literature, floras of Syria, Lebanon and the Mediterranean, as well as Syrian printed sources in Arabic and/or English, reports from FAO on agricultural statistics in Syria, and data from ICARDA and Bioversity International. In addition, personal observations gathered during professional work in the General Commission for Scientific Agricultural Research (GCSAR) in Syria (since 1989) and participation in projects were taken into account. These were: (1) A project on “Conservation and Sustainable Use of Dry Land Agrobiodiversity in the Near East” with participation of Jordan, Lebanon, Syria, and the Palestinian Authority, focussing on landraces and wild relatives of barley, wheat, lentil, alliums, feed legumes, and fruit trees (1999–2005). (2) A project for vegetable landraces (1993–1995) in collaboration with the former International Plant Genetic Resources Institute and the UN Development Programme, in which 380 local vegetable accessions were evaluated. For medicinal plants and fruit trees I was in personal contact with departments of GCSAR and the Ministry of Agriculture and Agrarian Reform, as well as with private organizations. The resulting checklist was compared with the catalogues of crop plants of Italy and a checklist of cultivated plants of Iraq. The cultivated plant species are presented in alphabetical order according to their accepted scientific names. Each entry consists of a nomenclatural part, folk names, details of plant uses, the distribution in Syria (by provinces), a textual description, and references to literature. In total, 262 species belonging to 146 genera and 57 families were identified. Within-species (intraspecific) diversity is a significant measure of the biodiversity. Intraspecific diversity for wild plants has been and remains to be well studied, but for crop plants there are only few results. Mansfeld’s method is an actual logical contribution to such studies. Among the families, the following have the highest number of crop species: Leguminosae (34 spp.), Rosaceae (24), Gramineae (18), Labiatae (18), Compositae (14), Cruciferae (14), Cucurbitaceae (11), Rutaceae (10), Malvaceae (9), Alliaceae (7), and Anacardiaceae (7). The establishment of an effective programme for the maintenance of plant genetic resources in Syria started in the mid-1970s. This programme considered ex situ and in situ collection of the genetic resources of various field crops, fruit trees and vegetables. From a plant genetic resources viewpoint, it is clear that the homegarden is an important location for the cultivation of so-called neglected and underutilized species (neglected from a research side and underutilized from a larger economic side). Such species have so far not received much care from ecologists, botanists and agronomists, and they are considerably under-represented in genebanks.