Croissance et ensemble nodal de fonctions propres du laplacien sur des surfaces

Dans cette thèse, nous étudions les fonctions propres de l'opérateur de Laplace-Beltrami - ou simplement laplacien - sur une surface fermée, c'est-à-dire une variété riemannienne lisse, compacte et sans bord de dimension 2. Ces fonctions propres satisfont l'équation $\Delta_g \phi_\lambda + \lambda \phi_\lambda = 0$ et les valeurs propres forment une suite infinie. L'ensemble nodal d'une fonction propre du laplacien est celui de ses zéros et est d'intérêt depuis les expériences de plaques vibrantes de Chladni qui remontent au début du 19ème siècle et, plus récemment, dans le contexte de la mécanique quantique. La taille de cet ensemble nodal a été largement étudiée ces dernières années, notamment par Donnelly et Fefferman, Colding et Minicozzi, Hezari et Sogge, Mangoubi ainsi que Sogge et Zelditch. L'étude de la croissance de fonctions propres n'est pas en reste, avec entre autres les récents travaux de Donnelly et Fefferman, Sogge, Toth et Zelditch, pour ne nommer que ceux-là. Notre thèse s'inscrit dans la foulée du travail de Nazarov, Polterovich et Sodin et relie les propriétés de croissance des fonctions propres avec la taille de leur ensemble nodal dans l'asymptotique $\lambda \nearrow \infty$. Pour ce faire, nous considérons d'abord les exposants de croissance, qui mesurent la croissance locale de fonctions propres et qui sont obtenus à partir de la norme uniforme de celles-ci. Nous construisons ensuite la croissance locale moyenne d'une fonction propre en calculant la moyenne sur toute la surface de ces exposants de croissance, définis sur de petits disques de rayon comparable à la longueur d'onde. Nous montrons alors que la taille de l'ensemble nodal est contrôlée par le produit de cette croissance locale moyenne et de la fréquence $\sqrt{\lambda}$. Ce résultat permet une reformulation centrée sur les fonctions propres de la célèbre conjecture de Yau, qui prévoit que la mesure de l'ensemble nodal croît au rythme de la fréquence. Notre travail renforce également l'intuition répandue selon laquelle une fonction propre se comporte comme un polynôme de degré $\sqrt{\lambda}$. Nous généralisons ensuite nos résultats pour des exposants de croissance construits à partir de normes $L^q$. Nous sommes également amenés à étudier les fonctions appartenant au noyau d'opérateurs de Schrödinger avec petit potentiel dans le plan. Pour de telles fonctions, nous obtenons deux résultats qui relient croissance et taille de l'ensemble nodal.

Vacuum-ultraviolet photochemically initiated modification of polystyrene surfaces: morphological changes and mechanistic investigations

Vacuum-ultraviolet (VUV) irradiation (kexc: 172 ± 12 nm) of polystyrene films in the presence of oxygen produced not only oxidatively functionalized surfaces, but generated also morphological changes. Whereas OH- and C=O-functionalized surfaces might be used for e.g. secondary functionalization, enhanced aggregation or printing, processes leading to morphological changes open new possibilities of microstructurization. Series of experiments made under different experimental conditions brought evidence of two different reaction pathways: introduction of OH- and C=O-groups at the polystyrene pathways is mainly due to the reaction of reactive oxygen species (hydroxyl radicals, atomic oxygen, ozone) produced in the gas phase between the VUV-radiation source and the substrate. However, oxidative fragmentation leading to morphological changes, oxidation products of low molecular weight and eventually to mineralization of the organic substrate is initiated by electronic excitation of the polymer leading to C–C-bond homolysis and to a complex oxidation manifold after trapping of the C-centred radicals by molecular oxygen. The pathways of oxidative functionalization or fragmentation could be differentiated by FTIR-ATR analysis of irradiated polystyrene surfaces before and after washing with acetonitrile and microscopic fluorescence analysis of the surfaces secondarily functionalized with the N,N,N-tridodecyl-triaza-triangulenium (TATA) cation. Ozonization of the polystyrene leads to oxidative functionalization of the polymer surface but cannot initiate the fragmentation of the polymer backbone. Oxidative fragmentation is initiated by electronic excitation of the polymer (contact-mode AFM analysis), and evidence of the generation of intermediate C-centred radicals is given e.g. by experiments in the absence of oxygen leading to cross-linking (solubility effects, optical microscopy, friction-mode AFM) and disproportionation (fluorescence).

Supramolecular Organization of Metal Nanoparticles in Solution and of Phenyleneethynylenes on Surfaces

Most of the procedures reported for the synthesis of metal nanoparticles involve the use of strong reducing agents or elevated temperatures. This limits the possibility of developing metal nanoparticle based sensors for the in situ detection of analytes. One of the objectives of the present investigations is to (i) develop newer methodologies for the synthesis of metal nanoparticles in aqueous medium at ambient conditions and (ii) their use in the detection of metal cations by taking advantage of the unique coordination ability. Ideally, biocompatible molecules which possess both the reducing and stabilizing groups are desirable for such applications. Formation of stable supramolecular assembly, by bringing metal nanoparticles close to each other, results in plasmon coupling and this strategy can be effectively utilized for the development of metal nanoparticle based sensors.Another objective of the present study is to understand the supramolecular organization of molecules on surfaces. Various noncovalent interactions between the molecules and with surface play a decisive role in their organizations. An in-depth understanding of these interactions is essential for device fabrications. Recent photophysical studies have revealed that phenyleneethynylene based molecular systems are ideal for device application. The second objective of the thesis focuses on understanding the (i) organization of phenyleneethynylenes on highly oriented pyrolytic graphite (HOPG) surface with atomic level precision and (ii) weak intermolecular interactions which drive their organization.

Salmonella cross contamination in retail chicken outlets and the efficacy of spice extracts on salmonella enteritidis growth inhibhition on various surfaces

The source of samonella cross contamination in 15 retail chicken outlets in aresidual area in coimbatore city ,sourthern India was studied. Chopping boards and the butchers hands were predominant followed by knives and the weighing balance tray. Serotyping of the salmonella strains revealed that all strains were salmonella enteritis, except one which was found to be salmonella cerro.The anti bacterial activity of commonly used spices were evaluated.

Comparative effectiveness of solar disinfection using small-scale batch reactors with reflective, absorptive and transmissive rear surfaces

This study investigated the enhancement of solar disinfection using custom-made batch reactors with reflective (foil-backed) or absorptive (black-backed) rear surfaces, under a range of weather conditions in India. Plate counts of Escherichia coli ATCC11775 were made under aerobic conditions and under conditions where reactive oxygen species (ROS) were neutralised, i.e. in growth medium supplemented with 0.05% w/v sodium pyruvate plus incubation under anaerobic conditions. While the addition of either an absorptive or a reflective backing enhanced reactor performance under strong sunlight, the reflective reactor was the only system to show consistent enhancement under low sunlight, where the process was slowest. Counts performed under ROS-neutralised conditions were slightly higher than those in air, indicating that a fraction of the cells become sub-lethally injured during exposure to sunlight to the extent that they were unable to grow aerobically. However, the influence of this phenomenon on the dynamics of inactivation was relatively small

Preferential exposure of certain crystallographic planes on the surface of spinel ferrites: a study by LEIS on polycrystalline spinel ferrite surfaces

Spinel ferrites are commercially important because of their excellent magnetic and catalytic properties. The study by Low Energy Ion Scattering (LEIS) can reveal atomic scale information on the surface. The surface of selected spinel ferrites was investigated by LEIS. It has been found that it is the octahedral sites which are preferentially exposed on the surface of the spinel ferrites. So the probable planes which are exposed on spinel ferrite surfaces are D(110) or B(111). This prediction using LEIS gives scope for tailor-making compounds with catalytically active ions on the surface for various catalytic reactions.

Unusual surface and edge morphologies, sp2 to sp3 hybridized transformation and electronic damage after Ar+ ion irradiation of few-layer graphene surfaces

Roughness and defects induced on few-layer graphene (FLG) irradiated by Ar+ ions at different energies were investigated using X-ray photoemission spectroscopy (XPS) and atomic force microscopy techniques. The results provide direct experimental evidence of ripple formation, sp2 to sp3 hybridized carbon transformation, electronic damage, Ar+ implantation, unusual defects and edge reconstructions in FLG, which depend on the irradiation energy. In addition, shadowing effects similar to those found in oblique-angle growth of thin films were seen. Reliable quantification of the transition from the sp2-bonding to sp3-hybridized state as a result of Ar+ ion irradiation is achieved from the deconvolution of the XPS C (1s) peak. Although the ion irradiation effect is demonstrated through the shape of the derivative of the Auger transition C KVV spectra, we show that the D parameter values obtained from these spectra which are normally used in the literature fail to account for the sp2 to sp3 hybridization transition. In contrast to what is known, it is revealed that using ion irradiation at large FLG sample tilt angles can lead to edge reconstructions. Furthermore, FLG irradiation by low energy of 0.25 keV can be a plausible way of peeling graphene layers without the need of Joule heating reported previously

Spacelike maximal surfaces in 3D Lorentz-Minkowski space

We investigate spacelike maximal surfaces in 3-dimensional Lorentz-Minkowski space, give an Enneper-Weierstrass representation of such surfaces and classify those with a Lorentzian or Euclidian rotation symmetry.

Construction of fractal interpolation surfaces on rectangular grids

We present a general method of generating continuous fractal interpolation surfaces by iterated function systems on an arbitrary data set over rectangular grids and estimate their Box-counting dimension.

Construction of recurrent fractal interpolation surfaces(RFISs) on rectangular grids

A recurrent iterated function system (RIFS) is a genaralization of an IFS and provides nonself-affine fractal sets which are closer to natural objects. In general, it's attractor is not a continuous surface in R3. A recurrent fractal interpolation surface (RFIS) is an attractor of RIFS which is a graph of bivariate continuous interpolation function. We introduce a general method of generating recurrent interpolation surface which are at- tractors of RIFSs about any data set on a grid.

Dipodal Ferrocene Derivatives for Nanostructured Redox-Functionalised Surfaces

Self-assembled monolayers (SAMs) on solid surfaces are of great current interest in science and nanotechnology. This thesis describes the preparation of several symmetrically 1,1’-substituted ferrocene derivatives that contain anchoring groups suitable for chemisorption on gold and may give rise to SAMs with electrochemically switchable properties. The binding groups are isocyano (-NC), isothiocyanato (-NCS), phosphanyl (-PPh2), thioether (-SR) and thienyl. In the context of SAM fabrication, isothiocyanates and phosphanes are adsorbate systems which, surprisingly, have remained essentially unexplored. SAMs on gold have been fabricated with the adsorbates from solution and investigated primarily by X-ray photoelectron spectroscopy and near-edge X-ray absorption fine structure spectroscopy. The results of these analytical investigations are presented and discussed in matters of the film quality and possible binding modes. The quality of self-assembled monolayers fabricated from 1,1’-diisocyanoferrocene and 1,1’-diisothiocyanatoferrocene turned out to be superior to that of films based on the other adsorbate species investigated. Films of those absorbates as well as of dppf afforded well-defined SAMs of good quality. All other films of this study based on sulfur containing anchoring groups exhibit chemical inhomogeneity and low orientational order of the film constituents and therefore failed to give rise to well-defined SAMs. Surface coordination chemistry is naturally related to molecular coordination chemistry. Since all SAMs described in this thesis were prepared on gold (111) surfaces, the ferrocene-based ligands of this study have been investigated in their ability for complexation towards gold(I). The sulfur-based ferrocene ligands [fc(SR)2] failed to give stable gold(I) complexes. In contrast, 1,1’-diisocyanoferrocene (1) proved to be an excellent ligand for the complexation of gold(I). Several complexes were prepared and characterised utilising a series of gold(I) acetylides. These complexes show interesting structural motifs in the solid state, since intramolecular aurophilic interactions lead to a parallel orientation of the isocyano moieties, combined with an antiparallel alignment of neighbouring units. The reaction of 1 with the gold(I) acetylide [Au(C≡C–Fc)]n turned out to be very unusual, since the two chemically equivalent isocyano groups undergo a different reaction. One group shows an ordinary coordination and the other one undergoes an extraordinary 1,1-insertion into the Au-C bond. As a sideline of the research of this thesis several ferrocene derivatives have been tested for their suitability for potential surface reactions. Copper(I) mediated 1,3-dipolar cycloadditions of azidoferrocene derivatives with terminal alkynes appeared very promising in this context, but failed to a certain extent in terms of ‘click’ chemistry, since the formation of the triazoles depended on the strict exclusion of oxygen and moisture and yields were only moderate. Staudinger reactions between dppf and azidoferrocene derivatives were also tested. The nucleophilic additions of secondary amines to 1,1’-diisothiocyanatoferrocene led to the respective thiourea derivatives in quantitative yields.

Modification and Analysis of Group-13-Nitride Semiconductor Surfaces: Influence of Adsorbates on Device Characteristics and Processing

Im Rahmen dieser interdisziplinären Doktorarbeit wird eine (Al)GaN Halbleiteroberflächenmodifikation untersucht, mit dem Ziel eine verbesserte Grenzfläche zwischen dem Material und dem Dielektrikum zu erzeugen. Aufgrund von Oberflächenzuständen zeigen GaN basierte HEMT Strukturen üblicherweise große Einsatzspannungsverschiebungen. Bisher wurden zur Grenzflächenmodifikation besonders die Entfernung von Verunreinigungen wie Sauerstoff oder Kohlenstoff analysiert. Die nasschemischen Oberflächenbehandlungen werden vor der Abscheidung des Dielektrikums durchgeführt, wobei die Kontaminationen jedoch nicht vollständig entfernt werden können. In dieser Arbeit werden Modifikationen der Oberfläche in wässrigen Lösungen, in Gasen sowie in Plasma analysiert. Detaillierte Untersuchungen zeigen, dass die inerte (0001) c-Ebene der Oberfläche kaum reagiert, sondern hauptsächlich die weniger polaren r- und m- Ebenen. Dies kann deutlich beim Defektätzen sowie bei der thermischen Oxidation beobachtet werden. Einen weiteren Ansatz zur Oberflächenmodifikation stellen Plasmabehandlungen dar. Hierbei wird die Oberflächenterminierung durch eine nukleophile Substitution mit Lewis Basen, wie Fluorid, Chlorid oder Oxid verändert, wodurch sich die Elektronegativitätsdifferenz zwischen dem Metall und dem Anion im Vergleich zur Metall-Stickstoff Bindung erhöht. Dies führt gleichzeitig zu einer Erhöhung der Potentialdifferenz des Schottky Kontakts. Sauerstoff oder Fluor besitzen die nötige thermische Stabilität um während einer Silicium-nitridabscheidung an der (Al)GaN Oberfläche zu bleiben. Sauerstoffvariationen an der Oberfläche werden in NH3 bei 700°C, welches die nötigen Bedingungen für die Abscheidung darstellen, immer zu etwa 6-8% reduziert – solche Grenzflächen zeigen deswegen auch keine veränderten Ergebnisse in Einsatzspannungsuntersuchungen. Im Gegensatz dazu zeigt die fluorierte Oberfläche ein völlig neues elektrisches Verhalten: ein neuer dominanter Oberflächendonator mit einem schnellen Trapping und Detrapping Verhalten wird gefunden. Das Energieniveau dieses neuen, stabilen Donators liegt um ca. 0,5 eV tiefer in der Bandlücke als die ursprünglichen Energieniveaus der Oberflächenzustände. Physikalisch-chemische Oberflächen- und Grenzflächenuntersuchung mit XPS, AES oder SIMS erlauben keine eindeutige Schlussfolgerung, ob das Fluor nach der Si3N4 Abscheidung tatsächlich noch an der Grenzfläche vorhanden ist, oder einfach eine stabilere Oberflächenrekonstruktion induziert wurde, bei welcher es selbst nicht beteiligt ist. In beiden Fällen ist der neue Donator in einer Konzentration von 4x1013 at/cm-2 vorhanden. Diese Dichte entspricht einer Oberflächenkonzentration von etwa 1%, was genau an der Nachweisgrenze der spektroskopischen Methoden liegt. Jedoch werden die elektrischen Oberflächeneigenschaften durch die Oberflächenmodifikation deutlich verändert und ermöglichen eine potentiell weiter optimierbare Grenzfläche.

The Perception of Subjective Surfaces

It is proposed that subjective contours are an artifact of the perception of natural three-dimensional surfaces. A recent theory of surface interpolation implies that "subjective surfaces" are constructed in the visual system by interpolation between three-dimensional values arising from interpretation of a variety of surface cues. We show that subjective surfaces can take any form, including singly and doubly curved surfaces, as well as the commonly discussed fronto-parallel planes. In addition, it is necessary in the context of computational vision to make explicit the discontinuities, both in depth and in surface orientation, in the surfaces constructed by interpolation. It is proposed that subjective surfaces and subjective contours are demonstrated. The role played by figure completion and enhanced brightness contrast in the determination of subjective surfaces is discussed. All considerations of surface perception apply equally to subjective surfaces.

Computing Distance Functions from Generalized Sources on Weighted Polyhedral Surfaces

We present algorithms for computing approximate distance functions and shortest paths from a generalized source (point, segment, polygonal chain or polygonal region) on a weighted non-convex polyhedral surface in which obstacles (represented by polygonal chains or polygons) are allowed. We also describe an algorithm for discretizing, by using graphics hardware capabilities, distance functions. Finally, we present algorithms for computing discrete k-order Voronoi diagrams