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.

Relativistic effects in physics and chemistry of element 105. [Part] IV.

Results of relativistic (Dirac-Slater and Dirac-Fock) and nonrelativistic (Hartree-Fock-Slater) atomic and molecular calculations have been compared for the group 5 elements Nb, Ta, and Ha and their compounds MCl_5, to elucidate the influence of relativistic effects on their properties especially in going from the 5d element Ta to the 6d element Ha. The analysis of the radial distribution of the valence electrons of the metals for electronic configurations obtained as a result of the molecular calculations and their overlap with ligands show opposite trends in behavior for ns_1/2, np_l/2, and (n -1 )d_5/2 orbitals for Ta and Ha in the relativistic and nonrelativistic cases. Relativistic contraction and energetic stabilization of the ns_1/2 and np_l/2 wave functions and expansion and destabilization of the (n-1)d_5/2 orbitals make hahnium pentahalide more covalent than tantalum pentahalide and increase the bond strength. The nonrelativistic treatment of the wave functions results in an increase in ionicity of the MCl_5 molecules in going from Nb to Ha making element Ha an analog of V. Different trends for the relativistic and nonrelativistic cases are also found for ionization potentials, electronic affinities, and energies of charge-transfer transitions as well as the stability of the maximum oxidation state.

Characterization of Lipid Droplets and Functional Analysis of Lipid Droplet-Associated Proteins in Dictyostelium discoideum

Lipid droplets (LDs) are the universal storage form of fat as a reservoir of metabolic energy in animals, plants, bacteria and single celled eukaryotes. Dictyostelium LD formation was investigated in response to the addition of different nutrients to the growth medium. LDs were induced by adding exogenous cholesterol, palmitic acid (PA) as well as growth in bacterial suspension, while glucose addition fails to form LDs. Among these nutrients, PA addition is most effective to stimulate LD formation, and depletion of PA from the medium caused LD degradation. The neutral lipids incorporated into the LD-core are composed of triacylglycerol (TAG), steryl esters, and an unknown neutral lipid (UKL) species when the cells were loaded simultaneously with cholesterol and PA. In order to avoid the contamination with other cellular organelles, the LD-purification method was modified. The isolated LD fraction was analysed by mass spectrometry and 100 proteins were identified. Nineteen of these appear to be directly involved in lipid metabolism or function in regulating LD morphology. Together with a previous study, a total of 13 proteins from the LD-proteome were confirmed to localize to LDs after the induction with PA. Among the identified LD-proteins, the localization of Ldp (lipid droplet membrane protein), GPAT3 (glycerol-3-phosphate acyltransferase 3) and AGPAT3 (1-acylglycerol-3-phosphate-acyltransferase 3) were further verified by GFP-tagging at the N-termini or C-termini of the respective proteins. Fluorescence microscopy demonstrated that PA-treatment stimulated the translocation of the three proteins from the ER to LDs. In order to clarify DGAT (diacylglycerol acyltransferase) function in Dictyostelium, the localization of DGAT1, that is not present in LD-proteome, was also investigated. GFP-tagged DGAT1 localized to the ER both, in the presence and absence of PA, which is different from the previously observed localization of GFP-tagged DGAT2, which almost exclusively binds to LDs. The investigation of the cellular neutral lipid level helps to elucidate the mechanism responsible for LD-formation in Dictyostelium cells. Ldp and two short-chain dehydrogenases, ADH (alcohol dehydrogenase) and Ali (ADH-like protein), are not involved in neutral lipid biosynthesis. GPAT, AGPAT and DGAT are three transferases responsible for the three acylation steps of de novo TAG synthesis. Knock-out (KO) of AGPAT3 and DGAT2 did not affect storage-fat formation significantly, whereas cells lacking GPAT3 or DGAT1 decreased TAG and LD accumulation dramatically. Furthermore, DGAT1 is responsible for the accumulation of the unknown lipid UKL. Overexpression of DGAT2 can rescue the reduced TAG content of the DGAT1-KO mutant, but fails to restore UKL content in these cells, indicating that of DGAT1 and DGAT2 have overlapping functions in TAG synthesis, but the role in UKL formation is unique to DGAT1. Both GPAT3 and DGAT1 affect phagocytic activity. Mutation of GPAT3 increases it but a DGAT1-KO decreases phagocytosis. The double knockout of DGAT1 and 2 also impairs the ability to grow on a bacterial lawn, which again can be rescued by overexpression of DGAT2. These and other results are incorporated into a new model, which proposes that up-regulation of phagocytosis serves to replenish precursor molecules of membrane lipid synthesis, whereas phagocytosis is down-regulated when excess fatty acids are used for storage-fat formation.