Verzweigung periodischer Lösungen bei rein nichtlinearen Differentialgleichungssystemen in der Ebene

Zusammenfassung:In dieser Arbeit werden die Abzweigung stationärer Punkte und periodischer Lösungen von isolierten stationären Punkten rein nichtlinearer Differentialgleichungen in der reellenEbene betrachtet.Das erste Kapitel enthält einige technische Hilfsmittel, während im zweiten ausführlich das Verhalten von Differentialgleichungen in der Ebene mit zwei homogenen Polynomen gleichen Grades als rechter Seite diskutiert wird.Im dritten Kapitel beginnt der Hauptteil der Arbeit. Hier wird eine Verallgemeinerung des Hopf'schen Verzweigungssatzes bewiesen, der den klassischen Satz als Spezialfall enthält.Im vierten Kapitel untersuchen wir die Abzweigung stationärer Punkte und im letzten Kapitel die Abzweigung periodischer Lösungen unter Störungen, deren Ordnung echt kleiner ist, als die erste nichtverschwindende Näherung der ungestörten Gleichung.Alle Voraussetzungen in dieser Arbeit sind leicht nachzurechnen und es werden zahlreiche Beispiele ausführlich diskutiert.

Mode coupling for precise measurements of the electronic g-factor of hydrogen-like ions in Penning traps

The g-factor is a constant which connects the magnetic moment $vec{mu}$ of a charged particle, of charge q and mass m, with its angular momentum $vec{J}$. Thus, the magnetic moment can be writen $ vec{mu}_J=g_Jfrac{q}{2m}vec{J}$. The g-factor for a free particle of spin s=1/2 should take the value g=2. But due to quantum electro-dynamical effects it deviates from this value by a small amount, the so called g-factor anomaly $a_e$, which is of the order of $10^{-3}$ for the free electron. This deviation is even bigger if the electron is exposed to high electric fields. Therefore highly charged ions, where electric field strength gets values on the order of $10^{13}-10^{16}$V/cm at the position of the bound electron, are an interesting field of investigations to test QED-calculations. In previous experiments [H"aff00,Ver04] using a single hydrogen-like ion confined in a Penning trap an accuracy of few parts in $10^{-9}$ was obtained. In the present work a new method for precise measurement of magnetic the electronic g-factor of hydrogen-like ions is discussed. Due to the unavoidable magnetic field inhomogeneity in a Penning trap, a very important contribution to the systematic uncertainty in the previous measurements arose from the elevated energy of the ion required for the measurement of its motional frequencies. Then it was necessary to extrapolate the result to vanishing energies. In the new method the energy in the cyclotron degree of freedom is reduced to the minimum attainable energy. This method consist in measuring the reduced cyclotron frequency $nu_{+}$ indirectly by coupling the axial to the reduced cyclotron motion by irradiation of the radio frequency $nu_{coup}=nu_{+}-nu_{ax}+delta$ where $delta$ is, in principle, an unknown detuning that can be obtained from the knowledge of the coupling process. Then the only unknown parameter is the desired value of $nu_+$. As a test, a measurement with, for simplicity, artificially increased axial energy was performed yielding the result $g_{exp}=2.000~047~020~8(24)(44)$. This is in perfect agreement with both the theoretical result $g_{theo}=2.000~047~020~2(6)$ and the previous experimental result $g_{exp1}=2.000~047~025~4(15)(44).$ In the experimental results the second error-bar is due to the uncertainty in the accepted value for the electron's mass. Thus, with the new method a higher accuracy in the g-factor could lead by comparison to the theoretical value to an improved value of the electron's mass. [H"af00] H. H"affner et al., Phys. Rev. Lett. 85 (2000) 5308 [Ver04] J. Verd'u et al., Phys. Rev. Lett. 92 (2004) 093002-1

On the geometry of the spin-statistics connection in quantum mechanics

The Spin-Statistics theorem states that the statistics of a system of identical particles is determined by their spin: Particles of integer spin are Bosons (i.e. obey Bose-Einstein statistics), whereas particles of half-integer spin are Fermions (i.e. obey Fermi-Dirac statistics). Since the original proof by Fierz and Pauli, it has been known that the connection between Spin and Statistics follows from the general principles of relativistic Quantum Field Theory. In spite of this, there are different approaches to Spin-Statistics and it is not clear whether the theorem holds under assumptions that are different, and even less restrictive, than the usual ones (e.g. Lorentz-covariance). Additionally, in Quantum Mechanics there is a deep relation between indistinguishabilty and the geometry of the configuration space. This is clearly illustrated by Gibbs' paradox. Therefore, for many years efforts have been made in order to find a geometric proof of the connection between Spin and Statistics. Recently, various proposals have been put forward, in which an attempt is made to derive the Spin-Statistics connection from assumptions different from the ones used in the relativistic, quantum field theoretic proofs. Among these, there is the one due to Berry and Robbins (BR), based on the postulation of a certain single-valuedness condition, that has caused a renewed interest in the problem. In the present thesis, we consider the problem of indistinguishability in Quantum Mechanics from a geometric-algebraic point of view. An approach is developed to study configuration spaces Q having a finite fundamental group, that allows us to describe different geometric structures of Q in terms of spaces of functions on the universal cover of Q. In particular, it is shown that the space of complex continuous functions over the universal cover of Q admits a decomposition into C(Q)-submodules, labelled by the irreducible representations of the fundamental group of Q, that can be interpreted as the spaces of sections of certain flat vector bundles over Q. With this technique, various results pertaining to the problem of quantum indistinguishability are reproduced in a clear and systematic way. Our method is also used in order to give a global formulation of the BR construction. As a result of this analysis, it is found that the single-valuedness condition of BR is inconsistent. Additionally, a proposal aiming at establishing the Fermi-Bose alternative, within our approach, is made.

Mathematical aspects of Feynman integrals

In the present dissertation we consider Feynman integrals in the framework of dimensional regularization. As all such integrals can be expressed in terms of scalar integrals, we focus on this latter kind of integrals in their Feynman parametric representation and study their mathematical properties, partially applying graph theory, algebraic geometry and number theory. The three main topics are the graph theoretic properties of the Symanzik polynomials, the termination of the sector decomposition algorithm of Binoth and Heinrich and the arithmetic nature of the Laurent coefficients of Feynman integrals.rnrnThe integrand of an arbitrary dimensionally regularised, scalar Feynman integral can be expressed in terms of the two well-known Symanzik polynomials. We give a detailed review on the graph theoretic properties of these polynomials. Due to the matrix-tree-theorem the first of these polynomials can be constructed from the determinant of a minor of the generic Laplacian matrix of a graph. By use of a generalization of this theorem, the all-minors-matrix-tree theorem, we derive a new relation which furthermore relates the second Symanzik polynomial to the Laplacian matrix of a graph.rnrnStarting from the Feynman parametric parameterization, the sector decomposition algorithm of Binoth and Heinrich serves for the numerical evaluation of the Laurent coefficients of an arbitrary Feynman integral in the Euclidean momentum region. This widely used algorithm contains an iterated step, consisting of an appropriate decomposition of the domain of integration and the deformation of the resulting pieces. This procedure leads to a disentanglement of the overlapping singularities of the integral. By giving a counter-example we exhibit the problem, that this iterative step of the algorithm does not terminate for every possible case. We solve this problem by presenting an appropriate extension of the algorithm, which is guaranteed to terminate. This is achieved by mapping the iterative step to an abstract combinatorial problem, known as Hironaka's polyhedra game. We present a publicly available implementation of the improved algorithm. Furthermore we explain the relationship of the sector decomposition method with the resolution of singularities of a variety, given by a sequence of blow-ups, in algebraic geometry.rnrnMotivated by the connection between Feynman integrals and topics of algebraic geometry we consider the set of periods as defined by Kontsevich and Zagier. This special set of numbers contains the set of multiple zeta values and certain values of polylogarithms, which in turn are known to be present in results for Laurent coefficients of certain dimensionally regularized Feynman integrals. By use of the extended sector decomposition algorithm we prove a theorem which implies, that the Laurent coefficients of an arbitrary Feynman integral are periods if the masses and kinematical invariants take values in the Euclidean momentum region. The statement is formulated for an even more general class of integrals, allowing for an arbitrary number of polynomials in the integrand.

Functionalization of hyperbranched polycarbosilanes - block copolymers and electrochemistry

Polycarbosilanes are a class of polymers at the interface between organic and inorganic chemistry. They are characterized by a high thermal and chemical inertness and high flexibility, especially pronounced for branched structures. Linear polycarbosilanes are well known as precursors for the preparation of SiCx ceramics. Additionally, more sophisticated architectures like dendrimers, hyperbranched polymers or block copolymers have been the subject of research for more than a decade. The scope of this work was to expand the properties and fields of application for polycarbosilane-containing structures. Thus, the work is divided in two major parts. The first part covers the synthesis and characterization of hyperbranched polycarbosilanes containing organometallic moieties. Hyperbranched poly-carbosilanes were synthesized using hydrosilylation of diallylmethylsilane and methyldiundecenylsilane. The degree of branching for polydiallymethylsilane was determined using standard 1H-NMR spectroscopy. The functional building blocks ferrocenyldimethylsilane and diferrocenylmethylsilane were synthesized which contain an isolated ferrocene unit or two ferrocenes bridged by silicon, respectively. Hyperbranched polycarbosilanes functionalized with ferrocenyl moieties were synthesized by modification of preformed polymers or by copolymerization of AB2 carbosilane monomers with AX-type ferrocenylsilanes. Polymers with Mn = 2500-9000g/mol and ferrocene contents of up to 67wt% were obtained. Electrochemical characterization by cyclic voltammetry revealed that polymers functionalized with isolated ferrocene units showed a single reversible oxidation wave, while voltammograms for polymers functionalized with diferrocenyl silane exhibited two well-separated reversible oxidation-reduction waves. This shows that the polymer bound ferrocenes bridged by silicon are electronically communicating and thus oxidation of the first ferrocene shifts the oxidation potential for the adjacent one. The polymers were utilized successfully for the preparation of modified electrodes with persistent and reproducible electrochemical response in organic solvents as well as in aqueous solution. The presented work has proven that ferrocenyl-functionalized hyperbranched polymers exhibit similar electrochemical properties as the analogous dendrimers. In a further approach it was shown that hyperbranched polymers containing organometallic moieties can be synthesized by polymerization of a new ferrocene-containing AB2 monomer - diallylferrocenylsilane. The second part of this work is dedicated to the preparation of core-functional hyperbranched polycarbosilanes. Low molecular weight ambifunctional molecules were synthesized that contain double bonds for the attachment of a polycarbosilane polymer as well as a second functionality available for further reaction and modification. Reactive vinyl groups in the core molecule allow an efficient attachment of hyperbranched polycarbosilane which was proven by MALDI-ToF and GPC. In combination with slow monomer addition techniques molecular weight and polydispersity of the polymers were controlled successfully. Core-functional polymers were characterized by NMR-spectroscopy, MALDI-ToF and GPC. Polymers with polydispersities <2 and molecular weights up to 5300g/mol were obtained. Transformation of the double bonds of the carbosilane was demonstrated with various silanes using hydrosilylation reaction or hydrogenation. Additionally, the core-functionality was varied resulting in polymers with bromo-, phthalimide-, amine- or azide moieties. Thus, a versatile synthetic strategy was developed that allows the synthesis of tailor-made polymers.A promising approach is the application of the polymer building blocks in copolymer synthesis. Bisglycidolization of amine-functional polycarbosilanes produces macro-initiators that are suitable for the multibranching-ring opening polymerization of glycidol. This experiments lead to the first example of hyperbranched-hyperbranched amphiphilic block copolymers, hb-PG-b-hb-PCS. Furthermore, the implementation of copper-catalyzed cycloaddition between azide-functional polycarbosilane and alkyne-functional poly(ethoxyethyl glycidylether) resulted in linear-hyperbranched block copolymers. The facile removal of acetal protecting groups provided convenient access to lin-PG-b-hb-PCS.

The importance of visual and olfactory stimuli during flower visits in Apis mellifera

Flowers attract honeybees using colour and scent signals. Bimodality (having both scent and colour) in flowers leads to increased visitation rates, but how the signals influence each other in a foraging situation is still quite controversial. We studied four basic questions: When faced with conflicting scent and colour information, will bees choose by scent and ignore the “wrong” colour, or vice versa? To get to the bottom of this question, we trained bees on scent-colour combination AX (rewarded) versus BY (unrewarded) and tested them on AY (previously rewarded colour and unrewarded scent) versus BX (previously rewarded scent and unrewarded colour). It turned out that the result depends on stimulus quality: if the colours are very similar (unsaturated blue and blue-green), bees choose by scent. If they are very different (saturated blue and yellow), bees choose by colour. We used the same scents, lavender and rosemary, in both cases. Our second question was: Are individual bees hardwired to use colour and ignore scent (or vice versa), or can this behaviour be modified, depending on which cue is more readily available in the current foraging context? To study this question, we picked colour-preferring bees and gave them extra training on scent-only stimuli. Afterwards, we tested if their preference had changed, and if they still remembered the scent stimulus they had originally used as their main cue. We came to the conclusion that a colour preference can be reversed through scent-only training. We also gave scent-preferring bees extra training on colour-only stimuli, and tested for a change in their preference. The number of animals tested was too small for statistical tests (n = 4), but a common tendency suggested that colour-only training leads to a preference for colour. A preference to forage by a certain sensory modality therefore appears to be not fixed but flexible, and adapted to the bee’s surroundings. Our third question was: Do bees learn bimodal stimuli as the sum of their parts (elemental learning), or as a new stimulus which is different from the sum of the components’ parts (configural learning)? We trained bees on bimodal stimuli, then tested them on the colour components only, and the scent components only. We performed this experiment with a similar colour set (unsaturated blue and blue-green, as above), and a very different colour set (saturated blue and yellow), but used lavender and rosemary for scent stimuli in both cases. Our experiment yielded unexpected results: with the different colours, the results were best explained by elemental learning, but with the similar colour set, bees exhibited configural learning. Still, their memory of the bimodal compound was excellent. Finally, we looked at reverse-learning. We reverse-trained bees with bimodal stimuli to find out whether bimodality leads to better reverse-learning compared to monomodal stimuli. We trained bees on AX (rewarded) versus BY (unrewarded), then on AX (unrewarded) versus BY (rewarded), and finally on AX (rewarded) and BY (unrewarded) again. We performed this experiment with both colour sets, always using the same two scents (lavender and rosemary). It turned out that bimodality does not help bees “see the pattern” and anticipate the switch. Generally, bees trained on the different colour set performed better than bees trained on the similar colour set, indicating that stimulus salience influences reverse-learning.

Non-parametric estimation of the diffusion coefficient of a branching diffusion with immigration

Wir betrachten Systeme von endlich vielen Partikeln, wobei die Partikel sich unabhängig voneinander gemäß eindimensionaler Diffusionen [dX_t = b(X_t),dt + sigma(X_t),dW_t] bewegen. Die Partikel sterben mit positionsabhängigen Raten und hinterlassen eine zufällige Anzahl an Nachkommen, die sich gemäß eines Übergangskerns im Raum verteilen. Zudem immigrieren neue Partikel mit einer konstanten Rate. Ein Prozess mit diesen Eigenschaften wird Verzweigungsprozess mit Immigration genannt. Beobachten wir einen solchen Prozess zu diskreten Zeitpunkten, so ist zunächst nicht offensichtlich, welche diskret beobachteten Punkte zu welchem Pfad gehören. Daher entwickeln wir einen Algorithmus, um den zugrundeliegenden Pfad zu rekonstruieren. Mit Hilfe dieses Algorithmus konstruieren wir einen nichtparametrischen Schätzer für den quadrierten Diffusionskoeffizienten $sigma^2(cdot),$ wobei die Konstruktion im Wesentlichen auf dem Auffüllen eines klassischen Regressionsschemas beruht. Wir beweisen Konsistenz und einen zentralen Grenzwertsatz.