Bieberbach's Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality

The Bieberbach conjecture about the coefficients of univalent functions of the unit disk was formulated by Ludwig Bieberbach in 1916 [Bieberbach1916]. The conjecture states that the coefficients of univalent functions are majorized by those of the Koebe function which maps the unit disk onto a radially slit plane. The Bieberbach conjecture was quite a difficult problem, and it was surprisingly proved by Louis de Branges in 1984 [deBranges1985] when some experts were rather trying to disprove it. It turned out that an inequality of Askey and Gasper [AskeyGasper1976] about certain hypergeometric functions played a crucial role in de Branges' proof. In this article I describe the historical development of the conjecture and the main ideas that led to the proof. The proof of Lenard Weinstein (1991) [Weinstein1991] follows, and it is shown how the two proofs are interrelated. Both proofs depend on polynomial systems that are directly related with the Koebe function. At this point algorithms of computer algebra come into the play, and computer demonstrations are given that show how important parts of the proofs can be automated.

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.

Collaborative design procedures for architects and engineers

This book argues for novel strategies to integrate engineering design procedures and structural analysis data into architectural design. Algorithmic procedures that recently migrated into the architectural practice are utilized to improve the interface of both disciplines. Architectural design is predominately conducted as a negotiation process of various factors but often lacks rigor and data structures to link it to quantitative procedures. Numerical structural design on the other hand could act as a role model for handling data and robust optimization but it often lacks the complexity of architectural design. The goal of this research is to bring together robust methods from structural design and complex dependency networks from architectural design processes. The book presents three case studies of tools and methods that are developed to exemplify, analyze and evaluate a collaborative work flow.

Characterization of DnmA, the Dnmt2 homolog in Dictyostelium discoideum

Eukaryotic DNA m5C methyltransferases (MTases) play a major role in many epigenetic regulatory processes like genomic imprinting, X-chromosome inactivation, silencing of transposons and gene expression. Members of the two DNA m5C MTase families, Dnmt1 and Dnmt3, are relatively well studied and many details of their biological functions, biochemical properties as well as interaction partners are known. In contrast, the biological functions of the highly conserved Dnmt2 family, which appear to have non-canonical dual substrate specificity, remain enigmatic despite the efforts of many researchers. The genome of the social amoeba Dictyostelium encodes Dnmt2-homolog, the DnmA, as the only DNA m5C MTase which allowed us to study Dnmt2 function in this organism without interference by the other enzymes. The dnmA gene can be easily disrupted but the knock-out clones did not show obvious phenotypes under normal lab conditions, suggesting that the function of DnmA is not vital for the organism. It appears that the dnmA gene has a low expression profile during vegetative growth and is only 5-fold upregulated during development. Fluorescence microscopy indicated that DnmA-GFP fusions were distributed between both the nucleus and cytoplasm with some enrichment in nuclei. Interestingly, the experiments showed specific dynamics of DnmA-GFP distribution during the cell cycle. The proteins colocalized with DNA in the interphase and were mainly removed from nuclei during mitosis. DnmA functions as an active DNA m5C MTase in vivo and is responsible for weak but detectable DNA methylation of several regions in the Dictyostelium genome. Nevertheless, gel retardation assays showed only slightly higher affinity of the enzyme to dsDNA compared to ssDNA and no specificity towards various sequence contexts, although weak but detectable specificity towards AT-rich sequences was observed. This could be due to intrinsic curvature of such sequences. Furthermore, DnmA did not show denaturant-resistant covalent complexes with dsDNA in vitro, although it could form covalent adducts with ssDNA. Low binding and methyltransfer activity in vitro suggest the necessity of additional factor in DnmA function. Nevertheless, no candidates could be identified in affinity purification experiments with different tagged DnmA fusions. In this respect, it should be noted that tagged DnmA fusion preparations from Dictyostelium showed somewhat higher activity in both covalent adduct formation and methylation assays than DnmA expressed in E.coli. Thus, the presence of co-purified factors cannot be excluded. The low efficiency of complex formation by the recombinant enzyme and the failure to define interacting proteins that could be required for DNA methylation in vivo, brought up the assumption that post-translational modifications could influence target recognition and enzymatic activity. Indeed, sites of phosphorylation, methylation and acetylation were identified within the target recognition domain (TRD) of DnmA by mass spectrometry. For phosphorylation, the combination of MS data and bioinformatic analysis revealed that some of the sites could well be targets for specific kinases in vivo. Preliminary 3D modeling of DnmA protein based on homology with hDNMT2 allowed us to show that several identified phosphorylation sites located on the surface of the molecule, where they would be available for kinases. The presence of modifications almost solely within the TRD domain of DnmA could potentially modulate the mode of its interaction with the target nucleic acids. DnmA was able to form denaturant-resistant covalent intermediates with several Dictyostelium tRNAs, using as a target C38 in the anticodon loop. The formation of complexes not always correlated with the data from methylation assays, and seemed to be dependent on both sequence and structure of the tRNA substrate. The pattern, previously suggested by the Helm group for optimal methyltransferase activity of hDNMT2, appeared to contribute significantly in the formation of covalent adducts but was not the only feature of the substrate required for DnmA and hDNMT2 functions. Both enzymes required Mg2+ to form covalent complexes, which indicated that the specific structure of the target tRNA was indispensable. The dynamics of covalent adduct accumulation was different for DnmA and different tRNAs. Interestingly, the profiles of covalent adduct accumulation for different tRNAs were somewhat similar for DnmA and hDNMT2 enzymes. According to the proposed catalytic mechanism for DNA m5C MTases, the observed denaturant-resistant complexes corresponded to covalent enamine intermediates. The apparent discrepancies in the data from covalent complex formation and methylation assays may be interpreted by the possibility of alternative pathways of the catalytic mechanism, leading not to methylation but to exchange or demethylation reactions. The reversibility of enamine intermediate formation should also be considered. Curiously, native gel retardation assays showed no or little difference in binding affinities of DnmA to different RNA substrates and thus the absence of specificity in the initial enzyme binding. The meaning of the tRNA methylation as well as identification of novel RNA substrates in vivo should be the aim of further experiments.

Determinants of crop diversity and composition in Enset-coffee agroforestry homegardens of Southern Ethiopia

Households in much of the tropics depend for their livelihoods on the variety and continued production of food and other products that are provided by their own farms. In such systems, maintenance of agrobiodiversity and ensuring food security are important for the well being of the population. The enset-coffee agroforestry homegardens of Southern Ethiopia that are dominated by two native perennial crops, Coffee (Coffea arabica L.) and Enset (Enset ventricosum Welw. Cheesman), are examples of such agricultural systems. This study was conducted in Sidama administrative zone of Southern Ethiopia to determine the factors that influence the diversity and composition of crops in the systems. Data were collected from 144 sample homegardens selected from four districts. Stepwise multiple regression analysis was used to relate indices of crop diversity and area share of major crops with the physical and socioeconomic factors. The study revealed that socioeconomic factors, mainly proximity to markets, affected negatively crop species richness. The production area of the main crops enset and coffee decreased with increasing proximity to market and road while that of maize and khat increased. At household level, farm size had a significant effect on area share of enset and coffee. As farm size increased the share of the cash crop, coffee increased but that of the staple, enset declined. Enset, which is the backbone of the system in terms of food security, is declining on small farms and the share of monoculture maize system is increasing. The trend towards declining agrobiodiversity, and reduction in the production area of the main perennial crops and their gradual replacement with monoculture fields could make the systems liable to instability and collapse. As these sites are high potential agricultural areas, intensification can be achieved by integrating high-value and more productive crops, such as fruits, spices and vegetables, while maintaining the integrated and complex nature of the systems.

Water Resource Pollution and Impacts on the local livelihood: A case study of Beas River in Kullu District, India

The rivers are considered as the life line of any country since they make water available for our domestic, industrial and recreational functions. The quality of river water signifies the health status and hygienic aspects of a particular region, but the quality of these life lines is continuously deteriorating due to discharge of sewage, garbage and industrial effluents into them. Thrust on water demand has increased manifolds due to the increased population, therefore tangible efforts to make the water sources free from pollution is catching attention all across the globe. This paper attempts to highlight the trends in water quality change of River Beas, right from Manali to Larji in India. This is an important river in the state of Himachal Pradesh and caters to the need of water for Manali and Kullu townships, besides other surrounding rural areas. The Manali-Larji Beas river stretch is exposed to the flow of sewage, garbage and muck resulting from various project activities, thereby making it vulnerable to pollution. In addition, the influx of thousands of tourists to these towns also contributes to the pollution load by their recreational and other tourist related activities. Pollution of this river has ultimately affected the livelihood of local population in this region. Hence, water quality monitoring was carried out for the said stretch between January, 2010 and January, 2012 at 15 various locations on quarterly basis, right from the upstream of Manali town and up to downstream of Larji dam. Temperature, color, odor, D.O. , pH, BOD, TSS, TC and FC has been the parameters that were studied. This study gives the broad idea about the characteristics of water at locations in the said river stretch, and suggestions for improving water quality and livelihood of local population in this particular domain.