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The primary motivation for systematic bases in first principles electronic structure simulations is to derive physical and chemical properties of molecules and solids with predetermined accuracy. This requires a detailed understanding of the asymptotic behaviour of many-particle Coulomb systems near coalescence points of particles. Singular analysis provides a convenient framework to study the asymptotic behaviour of wavefunctions near these singularities. In the present work, we want to introduce the mathematical framework of singular analysis and discuss a novel asymptotic parametrix construction for Hamiltonians of many-particle Coulomb systems. This corresponds to the construction of an approximate inverse of a Hamiltonian operator with remainder given by a so-called Green operator. The Green operator encodes essential asymptotic information and we present as our main result an explicit asymptotic formula for this operator. First applications to many-particle models in quantum chemistry are presented in order to demonstrate the feasibility of our approach. The focus is on the asymptotic behaviour of ladder diagrams, which provide the dominant contribution to shortrange correlation in coupled cluster theory. Furthermore, we discuss possible consequences of our asymptotic analysis with respect to adaptive wavelet approximation.
The echo chamber model describes the development of groups in heterogeneous social networks. By heterogeneous social network we mean a set of individuals, each of whom represents exactly one opinion. The existing relationships between individuals can then be represented by a graph. The echo chamber model is a time-discrete model which, like a board game, is played in rounds. In each round, an existing relationship is randomly and uniformly selected from the network and the two connected individuals interact. If the opinions of the individuals involved are sufficiently similar, they continue to move closer together in their opinions, whereas in the case of opinions that are too far apart, they break off their relationship and one of the individuals seeks a new relationship. In this paper we examine the building blocks of this model. We start from the observation that changes in the structure of relationships in the network can be described by a system of interacting particles in a more abstract space.
These reflections lead to the definition of a new abstract graph that encompasses all possible relational configurations of the social network. This provides us with the geometric understanding necessary to analyse the dynamic components of the echo chamber model in Part III. As a first step, in Part 7, we leave aside the opinions of the inidividuals and assume that the position of the edges changes with each move as described above, in order to obtain a basic understanding of the underlying dynamics. Using Markov chain theory, we find upper bounds on the speed of convergence of an associated Markov chain to its unique stationary distribution and show that there are mutually identifiable networks that are not apparent in the dynamics under analysis, in the sense that the stationary distribution of the associated Markov chain gives equal weight to these networks.
In the reversible cases, we focus in particular on the explicit form of the stationary distribution as well as on the lower bounds of the Cheeger constant to describe the convergence speed.
The final result of Section 8, based on absorbing Markov chains, shows that in a reduced version of the echo chamber model, a hierarchical structure of the number of conflicting relations can be identified.
We can use this structure to determine an upper bound on the expected absorption time, using a quasi-stationary distribution. This hierarchy of structure also provides a bridge to classical theories of pure death processes. We conclude by showing how future research can exploit this link and by discussing the importance of the results as building blocks for a full theoretical understanding of the echo chamber model. Finally, Part IV presents a published paper on the birth-death process with partial catastrophe. The paper is based on the explicit calculation of the first moment of a catastrophe. This first part is entirely based on an analytical approach to second degree recurrences with linear coefficients. The convergence to 0 of the resulting sequence as well as the speed of convergence are proved. On the other hand, the determination of the upper bounds of the expected value of the population size as well as its variance and the difference between the determined upper bound and the actual value of the expected value. For these results we use almost exclusively the theory of ordinary nonlinear differential equations.
Hardy inequalities on graphs
(2024)
The dissertation deals with a central inequality of non-linear potential theory, the Hardy inequality. It states that the non-linear energy functional can be estimated from below by a pth power of a weighted p-norm, p>1. The energy functional consists of a divergence part and an arbitrary potential part. Locally summable infinite graphs were chosen as the underlying space. Previous publications on Hardy inequalities on graphs have mainly considered the special case p=2, or locally finite graphs without a potential part.
Two fundamental questions now arise quite naturally: For which graphs is there a Hardy inequality at all? And, if it exists, is there a way to obtain an optimal weight? Answers to these questions are given in Theorem 10.1 and Theorem 12.1. Theorem 10.1 gives a number of characterizations; among others, there is a Hardy inequality on a graph if and only if there is a Green's function. Theorem 12.1 gives an explicit formula to compute optimal Hardy weights for locally finite graphs under some additional technical assumptions. Examples show that Green's functions are good candidates to be used in the formula.
Emphasis is also placed on illustrating the theory with examples. The focus is on natural numbers, Euclidean lattices, trees and star graphs. Finally, a non-linear version of the Heisenberg uncertainty principle and a Rellich inequality are derived from the Hardy inequality.
We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.
For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.
The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.
In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators.
Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization.
We consider a G-invariant star-product algebra A on a symplectic manifold (M,ω) obtained by a canonical construction of deformation quantization. Under assumptions of the classical Marsden-Weinstein theorem we define a reduction of the algebra A with respect to the G-action. The reduced algebra turns out to be isomorphic to a canonical star-product algebra on the reduced phase space B. In other words, we show that the reduction commutes with the canonical G-invariant deformation quantization. A similar statement in the framework of geometric quantization is known as the Guillemin-Sternberg conjecture (by now completely proved).
We construct a deformation quantization on an infinite-dimensional symplectic space of regular connections on an SU(2)-bundle over a Riemannian surface of genus g ≥ 2. The construction is based on the normal form thoerem representing the space of connections as a fibration over a finite-dimensional moduli space of flat connections whose fibre is a cotangent bundle of the infinite-dimensional gauge group. We study the reduction with respect to the gauge groupe both for classical and quantum cases and show that our quantization commutes with reduction.
We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed.
The aim of this paper is to bring together two areas which are of great importance for the study of overdetermined boundary value problems. The first area is homological algebra which is the main tool in constructing the formal theory of overdetermined problems. And the second area is the global calculus of pseudodifferential operators which allows one to develop explicit analysis.
In this paper, a global existence result of smooth solutions to the multidimen- sional nonisentropic hydrodynamic model for semiconductors is proved, under the assumption that the initial data is a perturbation of the stationary solutions for the thermal equilibrium state. The resulting evolutionary solutions converge to the stationary solutions in time asymptotically exponentially fast.
Die Erweiterung des natürlichen Zahlbereichs um die positiven Bruchzahlen und die negativen ganzen Zahlen geht für Schülerinnen und Schüler mit großen gedanklichen Hürden und einem Umbruch bis dahin aufgebauter Grundvorstellungen einher. Diese Masterarbeit trägt wesentliche Veränderungen auf der Vorstellungs- und Darstellungsebene für beide Zahlbereiche zusammen und setzt sich mit den kognitiven Herausforderungen für Lernende auseinander. Auf der Grundlage einer Diskussion traditioneller sowie alternativer Lehrgänge der Zahlbereichserweiterung wird eine Unterrichtskonzeption für den Mathematikunterricht entwickelt, die eine parallele Einführung der Bruchzahlen und der negativen Zahlen vorschlägt. Die Empfehlungen der Unterrichtkonzeption erstrecken sich über den Zeitraum von der ersten bis zur siebten Klassenstufe, was der behutsamen Weiterentwicklung und Modifikation des Zahlbegriffs viel Zeit einräumt, und enthalten auch didaktische Überlegungen sowie konkrete Hinweise zu möglichen Aufgabenformaten.
Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox.
Die Vielfältigkeit des Winkelbegriffs ist gleichermaßen spannend wie herausfordernd in Hinblick auf seine Zugänge im Mathematikunterricht der Schule. Ausgehend von verschiedenen Vorstellungen zum Winkelbegriff wird in dieser Arbeit ein Lehrgang zur Vermittlung des Winkelbegriffs entwickelt und letztlich in konkrete Umsetzungen für den Schulunterricht überführt.
Dabei erfolgt zunächst eine stoffdidaktische Auseinandersetzung mit dem Winkelbegriff, die von einer informationstheoretischen Winkeldefinition begleitet wird. In dieser wird eine Definition für den Winkelbegriff unter der Fragestellung entwickelt, welche Informationen man über einen Winkel benötigt, um ihn beschreiben zu können. So können die in der fachdidaktischen Literatur auftretenden Winkelvorstellungen aus fachmathematischer Perspektive erneut abgeleitet und validiert werden. Parallel dazu wird ein Verfahren beschrieben, wie Winkel – auch unter dynamischen Aspekten – informationstechnisch verarbeitet werden können, so dass Schlussfolgerungen aus der informationstheoretischen Winkeldefinition beispielsweise in dynamischen Geometriesystemen zur Verfügung stehen.
Unter dem Gesichtspunkt, wie eine Abstraktion des Winkelbegriffs im Mathematikunterricht vonstatten gehen kann, werden die Grundvorstellungsidee sowie die Lehrstrategie des Aufsteigens vom Abstrakten zum Konkreten miteinander in Beziehung gesetzt. Aus der Verknüpfung der beiden Theorien wird ein grundsätzlicher Weg abgeleitet, wie im Rahmen der Lehrstrategie eine Ausgangsabstraktion zu einzelnen Winkelaspekten aufgebaut werden kann, was die Generierung von Grundvorstellungen zu den Bestandteilen des jeweiligen Winkelaspekts und zum Operieren mit diesen Begriffsbestandteilen ermöglichen soll. Hierfür wird die Lehrstrategie angepasst, um insbesondere den Übergang von Winkelsituationen zu Winkelkontexten zu realisieren. Explizit für den Aspekt des Winkelfeldes werden, anhand der Untersuchung der Sichtfelder von Tieren, Lernhandlungen und Forderungen an ein Lernmodell beschrieben, die Schülerinnen und Schüler bei der Begriffsaneignung unterstützen.
Die Tätigkeitstheorie, der die genannte Lehrstrategie zuzuordnen ist, zieht sich als roter Faden durch die weitere Arbeit, wenn nun theoriebasiert Designprinzipien generiert werden, die in die Entwicklung einer interaktiven Lernumgebung münden. Hierzu wird u. a. das Modell der Artifact-Centric Activity Theory genutzt, das das Beziehungsgefüge aus Schülerinnen und Schülern, dem mathematischen Gegenstand und einer zu entwickelnden App als vermittelndes Medium beschreibt, wobei der Einsatz der App im Unterrichtskontext sowie deren regelgeleitete Entwicklung Bestandteil des Modells sind. Gemäß dem Ansatz der Fachdidaktischen Entwicklungsforschung wird die Lernumgebung anschließend in mehreren Zyklen erprobt, evaluiert und überarbeitet. Dabei wird ein qualitatives Setting angewandt, das sich der Semiotischen Vermittlung bedient und untersucht, inwiefern sich die Qualität der von den Schülerinnen und Schülern gezeigten Lernhandlungen durch die Designprinzipien und deren Umsetzung erklären lässt. Am Ende der Arbeit stehen eine finale Version der Designprinzipien und eine sich daraus ergebende Lernumgebung zur Einführung des Winkelfeldbegriffs in der vierten Klassenstufe.
Die Bienaymé-Galton-Watson Prozesse können für die Untersuchung von speziellen und sich entwickelnden Populationen verwendet werden. Die Populationen umfassen Individuen, welche sich identisch, zufällig, selbstständig und unabhängig voneinander fortpflanzen und die jeweils nur eine Generation existieren. Die n-te Generation ergibt sich als zufällige Summe der Individuen der (n-1)-ten Generation. Die Relevanz dieser Prozesse begründet sich innerhalb der Historie und der inner- und außermathematischen Bedeutung. Die Geschichte der Bienaymé-Galton-Watson-Prozesse wird anhand der Entwicklung des Konzeptes bis heute dargestellt. Dabei werden die Wissenschaftler:innen verschiedener Disziplinen angeführt, die Erkenntnisse zu dem Themengebiet beigetragen und das Konzept in ihren Fachbereichen angeführt haben. Somit ergibt sich die außermathematische Signifikanz. Des Weiteren erhält man die innermathematische Bedeutsamkeit mittels des Konzeptes der Verzweigungsprozesse, welches auf die Bienaymé-Galton-Watson Prozesse zurückzuführen ist. Die Verzweigungsprozesse stellen eines der aussagekräftigsten Modelle für die Beschreibung des Populationswachstums dar. Darüber hinaus besteht die derzeitige Wichtigkeit durch die Anwendungsmöglichkeit der Verzweigungsprozesse und der Bienaymé-Galton-Watson Prozesse innerhalb der Epidemiologie. Es werden die Ebola- und die Corona-Pandemie als Anwendungsfelder angeführt. Die Prozesse dienen als Entscheidungsstütze für die Politik und ermöglichen Aussagen über die Auswirkungen von Maßnahmen bezüglich der Pandemien. Neben den Prozessen werden ebenfalls der bedingte Erwartungswert bezüglich diskreter Zufallsvariablen, die wahrscheinlichkeitserzeugende Funktion und die zufällige Summe eingeführt. Die Konzepte vereinfachen die Beschreibung der Prozesse und bilden somit die Grundlage der Betrachtungen. Außerdem werden die benötigten und weiterführenden Eigenschaften der grundlegenden Themengebiete und der Prozesse aufgeführt und bewiesen. Das Kapitel erreicht seinen Höhepunkt bei dem Beweis des Kritikalitätstheorems, wodurch eine Aussage über das Aussterben des Prozesses in verschiedenen Fällen und somit über die Aussterbewahrscheinlichkeit getätigt werden kann. Die Fälle werden anhand der zu erwartenden Anzahl an Nachkommen eines Individuums unterschieden. Es zeigt sich, dass ein Prozess bei einer zu erwartenden Anzahl kleiner gleich Eins mit Sicherheit ausstirbt und bei einer Anzahl größer als Eins, die Population nicht in jedem Fall aussterben muss. Danach werden einzelne Beispiele, wie der linear fractional case, die Population von Fibroblasten (Bindegewebszellen) von Mäusen und die Entstehungsfragestellung der Prozesse, angeführt. Diese werden mithilfe der erlangten Ergebnisse untersucht und einige ausgewählte zufällige Dynamiken werden im nachfolgenden Kapitel simuliert. Die Simulationen erfolgen durch ein in Python erstelltes Programm und werden mithilfe der Inversionsmethode realisiert. Die Simulationen stellen beispielhaft die Entwicklungen in den verschiedenen Kritikalitätsfällen der Prozesse dar. Zudem werden die Häufigkeiten der einzelnen Populationsgrößen in Form von Histogrammen angebracht. Dabei lässt sich der Unterschied zwischen den einzelnen Fällen bestätigen und es wird die Anwendungsmöglichkeit der Bienaymé-Galton-Watson Prozesse bei komplexeren Problemen deutlich. Histogramme bekräftigen, dass die einzelnen Populationsgrößen nur endlich oft vorkommen. Diese Aussage wurde von Galton aufgeworfen und in der Extinktions-Explosions-Dichotomie verwendet. Die dargestellten Erkenntnisse über das Themengebiet und die Betrachtung des Konzeptes werden mit einer didaktischen Analyse abgeschlossen. Die Untersuchung beinhaltet die Berücksichtigung der Fundamentalen Ideen, der Fundamentalen Ideen der Stochastik und der Leitidee „Daten und Zufall“. Dabei ergibt sich, dass in Abhängigkeit der gewählten Perspektive die Anwendung der Bienaymé-Galton-Watson Prozesse innerhalb der Schule plausibel ist und von Vorteil für die Schüler:innen sein kann. Für die Behandlung wird exemplarisch der Rahmenlehrplan für Berlin und Brandenburg analysiert und mit dem Kernlehrplan Nordrhein-Westfalens verglichen. Die Konzeption des Lehrplans aus Berlin und Brandenburg lässt nicht den Schluss zu, dass die Bienaymé-Galton-Watson Prozesse angewendet werden sollten. Es lässt sich feststellen, dass die zugrunde liegende Leitidee nicht vollumfänglich mit manchen Fundamentalen Ideen der Stochastik vereinbar ist. Somit würde eine Modifikation hinsichtlich einer stärkeren Orientierung des Lehrplans an den Fundamentalen Ideen die Anwendung der Prozesse ermöglichen. Die Aussage wird durch die Betrachtung und Übertragung eines nordrhein-westfälischen Unterrichtsentwurfes für stochastische Prozesse auf die Bienaymé-Galton-Watson Prozesse unterstützt. Darüber hinaus werden eine Concept Map und ein Vernetzungspentagraph nach von der Bank konzipiert um diesen Aspekt hervorzuheben.
Übungsbuch zur Stochastik
(2023)
Dieses Buch stellt Übungen zu den Grundbegriffen und Grundsätzen der Stochastik und ihre Lösungen zur Verfügung. So wie man Tonleitern in der Musik trainiert, so berechnet man Übungsaufgaben in der Mathematik. In diesem Sinne soll dieses Übungsbuch vor allem als Vorlage dienen für das eigenständige, eigenverantwortliche Lernen und Üben.
Die Schönheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Phänomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Glücksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen.
We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.
We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.
In this thesis we consider diverse aspects of existence and correctness of asymptotic solutions to elliptic differential and pseudodifferential equations. We begin our studies with the case of a general elliptic boundary value problem in partial derivatives. A small parameter enters the coefficients of the main equation as well as into the boundary conditions. Such equations have already been investigated satisfactory, but there still exist certain theoretical deficiencies. Our aim is to present the general theory of elliptic problems with a small parameter. For this purpose we examine in detail the case of a bounded domain with a smooth boundary. First of all, we construct formal solutions as power series in the small parameter. Then we examine their asymptotic properties. It suffices to carry out sharp two-sided \emph{a priori} estimates for the operators of boundary value problems which are uniform in the small parameter. Such estimates failed to hold in functional spaces used in classical elliptic theory. To circumvent this limitation we exploit norms depending on the small parameter for the functions defined on a bounded domain. Similar norms are widely used in literature, but their properties have not been investigated extensively. Our theoretical investigation shows that the usual elliptic technique can be correctly carried out in these norms. The obtained results also allow one to extend the norms to compact manifolds with boundaries. We complete our investigation by formulating algebraic conditions on the operators and showing their equivalence to the existence of a priori estimates. In the second step, we extend the concept of ellipticity with a small parameter to more general classes of operators. Firstly, we want to compare the difference in asymptotic patterns between the obtained series and expansions for similar differential problems. Therefore we investigate the heat equation in a bounded domain with a small parameter near the time derivative. In this case the characteristics touch the boundary at a finite number of points. It is known that the solutions are not regular in a neighbourhood of such points in advance. We suppose moreover that the boundary at such points can be non-smooth but have cuspidal singularities. We find a formal asymptotic expansion and show that when a set of parameters comes through a threshold value, the expansions fail to be asymptotic. The last part of the work is devoted to general concept of ellipticity with a small parameter. Several theoretical extensions to pseudodifferential operators have already been suggested in previous studies. As a new contribution we involve the analysis on manifolds with edge singularities which allows us to consider wider classes of perturbed elliptic operators. We examine that introduced classes possess a priori estimates of elliptic type. As a further application we demonstrate how developed tools can be used to reduce singularly perturbed problems to regular ones.
The Green formula is proved for boundary value problems (BVPs), when "basic" operator is arbitrary partial differential operator with variable matrix coefficients and "boundary" operators are quasi-normal with vector-coeficients. If the system possesses the fundamental solution, representation formula for a solution is derived and boundedness properties of participating layer potentials from function spaces on the boundary (Besov, Zygmund spaces) into appropriate weighted function spaces on the inner and the outer domains are established. Some related problems are discussed in conclusion: traces of functions from weighted spaces, traces of potential-type functions, Plemelji formulae,Calderón projections, restricted smoothness of the underlying surface and coefficients. The results have essential applications in investigations of BVPs by the potential method, in apriori estimates and in asymptotics of solutions.
We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator-valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator As in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of As as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices Ps of As, interpreted as Mellin-edge representations of P.
The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy σ = (σψ, σ∧), where the second component takes value in operators on the infinite model cone of the local wedges. In general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the elliptcity of the principal edge symbol σ∧ which includes the (in general not explicitly known) number of additional conditions on the edge of trace and potential type. We focus here on these queations and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet-Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich-Dynin formula for edge boundary value problems, and we establish relations of elliptic operators for different weights, via the spectral flow of the underlying conormal symbols.
Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge Sobolev spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. Our methods from the calculus of boundary value problems on a manifold with edges will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator.
In single photon emission computed tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. The data gathered suffer from attenuation due to the tissue density µ. Each imaged slice incorporates noisy sample values of the nonlinear attenuated Radon transform (formular at this place in the original abstract) Traditional theory for SPECT reconstruction treats µ as a known parameter. In practical applications, however, µ is not known, but either crudely estimated, determined in costly additional measurements or plainly neglected. We demonstrate that an approximation of both f and µ from SPECT data alone is feasible, leading to quantitatively more accurate SPECT images. The result is based on nonlinear Tikhonov regularization techniques for parameter estimation problems in differential equations combined with Gauss-Newton-CG minimization.
A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice.
We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.
We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice $Z^{d} : X = (X_{i}(t), i ∈ Z^{d}, t ∈ [0, T], 0 < T < +∞)$. In a first part, these processes are characterized as Gibbs states on path spaces of the form $C([0, T],R)Z^{d}$. In a second part, we study the Gibbsian character on $R^{Z}^{d}$ of $v^{t}$, the law at time t of the infinite-dimensional diffusion X(t), when the initial law $v = v^{0}$ is Gibbsian.
Using an algorithm based on a retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical difficulty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.
It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (HT ) and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with R-bounded symbols, yielding by an iteration argument the R-boundedness of λ(A − λ)−1 in R(λ)≥ τ for some τ ∈ IR. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on Rd with operator valued coefficients.
In this thesis we mainly generalize two theorems from Mackaay-Picken and Picken (2002, 2004). In the first paper, Mackaay and Picken show that there is a bijective correspondence between Deligne 2-classes $\xi \in \check{H}^2(M,\mathcal{D}^2)$ and holonomy maps from the second thin-homotopy group $\pi_2^2(M)$ to $U(1)$. In the second one, a generalization of this theorem to manifolds with boundaries is given: Picken shows that there is a bijection between Deligne 2-cocycles and a certain variant of 2-dimensional topological quantum field theories. In this thesis we show that these two theorems hold in every dimension. We consider first the holonomy case, and by using simplicial methods we can prove that the group of smooth Deligne $d$-classes is isomorphic to the group of smooth holonomy maps from the $d^{th}$ thin-homotopy group $\pi_d^d(M)$ to $U(1)$, if $M$ is $(d-1)$-connected. We contrast this with a result of Gajer (1999). Gajer showed that Deligne $d$-classes can be reconstructed by a different class of holonomy maps, which not only include holonomies along spheres, but also along general $d$-manifolds in $M$. This approach does not require the manifold $M$ to be $(d-1)$-connected. We show that in the case of flat Deligne $d$-classes, our result differs from Gajers, if $M$ is not $(d-1)$-connected, but only $(d-2)$-connected. Stiefel manifolds do have this property, and if one applies our theorem to these and compare the result with that of Gajers theorem, it is revealed that our theorem reconstructs too many Deligne classes. This means, that our reconstruction theorem cannot live without the extra assumption on the manifold $M$, that is our reconstruction needs less informations about the holonomy of $d$-manifolds in $M$ at the price of assuming $M$ to be $(d-1)$-connected. We continue to show, that also the second theorem can be generalized: By introducing the concept of Picken-type topological quantum field theory in arbitrary dimensions, we can show that every Deligne $d$-cocycle induces such a $d$-dimensional field theory with two special properties, namely thin-invariance and smoothness. We show that any $d$-dimensional topological quantum field theory with these two properties gives rise to a Deligne $d$-cocycle and verify that this construction is surjective and injective, that is both groups are isomorphic.
In this paper, by a new constructive method, the authors reprove the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate fields. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C¹ functions. In particular, the authors give the optimal control time of the system. Finally, a new application is also given.
The genus-dependence of multi-loop superstring ams is estimated at large orders in perturbation theory using the super-Schottky group parameterization of supermoduli space. Restriction of the integration region to a subset of supermoduli space and a single fundamental domain of the super-modular group suggests an exponential dependence on the genus. Upper bounds for these estimates are obtained for arbitrary N-point superstring scattering amplitudes and are shown to be consistent with exact results obtained for special type II string amplitudes for orbifold or Calabi-Yau compactifications. The genus-dependence is then obtained by considering the effect of the remaining contribution to the superstring amplitudes after the coefficients of the formally divergent parts of the integrals vanish as a result of a sum over spin structures. The introduction of supersymmetry therefore leads to the elimination of large-order divergences in string pertubation theory, a result which is based only on the supersymmetric generalization of the polyakov measure and not the gauge group of the string model.
The derivation of the standard model from a higher-dimensional action suggests a further study of the fibre bundle formulation of gauge theories to determine the variations in the choice of structure group that are allowed in this geometrical setting. The action of transformations on the projection of fibres to their submanifolds are characteristic of theories with fewer gauge vector bosons, and specific examples are given, which may have phenomenological relevance. The spinor space for the three generations of fermions in the standard model is described algebraically.
On the existence of a non-zero lower bound for the number of Goldbach partitions of an even integer
(2002)
The Goldbach partitions of an even number greater than 2, given by the sums of two prime addends, form the non-empty set for all integers 2n with 2 ≤ n ≤ 2 × 1014. It will be shown how to determine by the method of induction the existence of a non-zero lower bound for the number of Goldbach partitions of all even integers greater than or equal to 4. The proof depends on contour arguments for complex functions in the unit disk.
The quantum cosmological wavefunction for a quadratic gravity theory derived from the heterotic string effective action is obtained near the inflationary epoch and during the initial Planck era. Neglecting derivatives with respect to the scalar field, the wavefunction would satisfy a third-order differential equation near the inflationary epoch which has a solution that is singular in the scale factor limit a(t) → 0. When scalar field derivatives are included, a sixth-order differential equation is obtained for the wavefunction and the solution by Mellin transform is regular in the a → 0 limit. It follows that inclusion of the scalar field in the quadratic gravity action is necessary for consistency of the quantum cosmology of the theory at very early times.
We analyze the notions of monotonicity and complete monotonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which monotonicity and complete monotonicity coincide in continuous time but not in discrete-time.
Zahlen in den Fingern
(2023)
Die Debatte über den Einsatz von digitalen Werkzeugen in der mathematischen Frühförderung ist hoch aktuell. Lernspiele werden konstruiert, mit dem Ziel, mathematisches, informelles Wissen aufzubauen und so einen besseren Schulstart zu ermöglichen. Doch allein die digitale und spielerische Aufarbeitung führt nicht zwingend zu einem Lernerfolg. Daher ist es umso wichtiger, die konkrete Implementation der theoretischen Konstrukte und Interaktionsmöglichkeiten mit den Werkzeugen zu analysieren und passend aufzubereiten.
In dieser Masterarbeit wird dazu exemplarisch ein mathematisches Lernspiel namens „Fingu“ für den Einsatz im vorschulischen Bereich theoretisch und empirisch im Rahmen der Artifact-Centric Activity Theory (ACAT) untersucht. Dazu werden zunächst die theoretischen Hintergründe zum Zahlensinn, Zahlbegriffserwerb, Teil-Ganze-Verständnis, der Anzahlwahrnehmung und -bestimmung, den Anzahlvergleichen und der Anzahldarstellung mithilfe von Fingern gemäß der Embodied Cognition sowie der Verwendung von digitalen Werkzeugen und Multi-Touch-Geräten umfassend beschrieben. Anschließend wird die App Fingu erklärt und dann theoretisch entlang des ACAT-Review-Guides analysiert. Zuletzt wird die selbstständig durchgeführte Studie mit zehn Vorschulkindern erläutert und darauf aufbauend Verbesserungs- und Entwicklungsmöglichkeiten der App auf wissenschaftlicher Grundlage beigetragen. Für Fingu lässt sich abschließend festhalten, dass viele Prozesse wie die (Quasi-)Simultanerfassung oder das Zählen gefördert werden können, für andere wie das Teil-Ganze-Verständnis aber noch Anpassungen und/oder die Begleitung durch Erwachsene nötig ist.
Spiele und spieltypische Elemente wie das Sammeln von Treuepunkten sind aus dem Alltag kaum wegzudenken. Zudem werden sie zunehmend in Unternehmen oder in Lernumgebungen eingesetzt. Allerdings ist die Methode Gamification bisher für den pädagogischen Kontext wenig klassifiziert und für Lehrende kaum zugänglich gemacht worden.
Daher zielt diese Bachelorarbeit darauf ab, eine systematische Strukturierung und Aufarbeitung von Gamification sowie innovative Ansätze für die Verwendung spieltypischer Elemente im Unterricht, konkret dem Mathematikunterricht, zu präsentieren. Dies kann eine Grundlage für andere Fachgebiete, aber auch andere Lehrformen bieten und so die Umsetzbarkeit von Gamification in eigenen Lehrveranstaltungen aufzeigen.
In der Arbeit wird begründet, weshalb und mithilfe welcher Elemente Gamification die Motivation und Leistungsbereitschaft der Lernenden langfristig erhöhen, die Sozial- und Personalkompetenzen fördern sowie die Lernenden zu mehr Aktivität anregen kann. Zudem wird Gamification explizit mit grundlegenden mathematikdidaktischen Prinzipien in Verbindung gesetzt und somit die Relevanz für den Mathematikunterricht hervorgehoben.
Anschließend werden die einzelnen Elemente von Gamification wie Punkte, Level, Abzeichen, Charaktere und Rahmengeschichte entlang einer eigens für den pädagogischen Kontext entwickelten Klassifikation „FUN“ (Feedback – User specific elements – Neutral elements) schematisch beschrieben, ihre Funktionen und Wirkung dargestellt sowie Einsatzmöglichkeiten im Unterricht aufgezeigt. Dies beinhaltet Ideen zu lernförderlichem Feedback, Differenzierungsmöglichkeiten und Unterrichtsrahmengestaltung, die in Lehrveranstaltungen aller Art umsetzbar sein können. Die Bachelorarbeit umfasst zudem ein spezifisches Beispiel, einen Unterrichtsentwurf einer gamifizierten Mathematikstunde inklusive des zugehörigen Arbeitsmaterials, anhand dessen die Verwendung von Gamification deutlich wird.
Gamification offeriert oftmals Vorteile gegenüber dem traditionellen Unterricht, muss jedoch wie jede Methode an den Inhalt und die Zielgruppe angepasst werden. Weiterführende Forschung könnte sich mit konkreten motivationalen Strukturen, personenspezifischen Unterschieden sowie mit mathematischen Inhalten wie dem Problemlösen oder dem Wechsel zwischen verschiedenen Darstellungen hinsichtlich gamifizierter Lehrformen beschäftigen.
Elliptic equations on configurations W = W1 ∪ ... ∪ Wn with edge Y and components Wj of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here, edges. Starting from edge-degenerate operators on Wj, j = 1, ..., N, we construct an algebra with extra "transmission" conditions on Y that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator-valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on W\Y. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics.
We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted Lp-spaces over B,1 < p < ∞. Under suitable ellipticity assumptions we can define a family of complex powers A up(z), z ∈ C. We also obtain sufficient information on the resolvent of A to show the boundedness of the pure imaginary powers. Examples concern unique solvability and maximal regularity of the solution of the Cauchy problem u' - Δu = f, u(0) = 0, for the Laplacian on conical manifolds.
We continue the investigation of the calculus of Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour (SG symbols). Here we analyse what happens when one restricts the choice of amplitude and phase functions to the subclass of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a "classical" counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1); for such operators we refine the known results about the analogous problem for general SG hyperbolic operators. The material contained here will be used in a forthcoming paper to obtain a Weyl formula for a class of operators defined on manifolds with cylindrical ends, improving the results obtained in [9].
Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.
Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set A in R^d. We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of A plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.
In this thesis we study reciprocal classes of Markov chains. Given a continuous time Markov chain on a countable state space, acting as reference dynamics, the associated reciprocal class is the set of all probability measures on path space that can be written as a mixture of its bridges. These processes possess a conditional independence property that generalizes the Markov property, and evolved from an idea of Schrödinger, who wanted to obtain a probabilistic interpretation of quantum mechanics.
Associated to a reciprocal class is a set of reciprocal characteristics, which are space-time functions that determine the reciprocal class. We compute explicitly these characteristics, and divide them into two main families: arc characteristics and cycle characteristics. As a byproduct, we obtain an explicit criterion to check when two different Markov chains share their bridges.
Starting from the characteristics we offer two different descriptions of the reciprocal class, including its non-Markov probabilities.
The first one is based on a pathwise approach and the second one on short time asymptotic. With the first approach one produces a family of functional equations whose only solutions are precisely the elements of the reciprocal class. These equations are integration by parts on path space associated with derivative operators which perturb the paths by mean of the addition of random loops. Several geometrical tools are employed to construct such formulas. The problem of obtaining sharp characterizations is also considered, showing some interesting connections with discrete geometry. Examples of such formulas are given in the framework of counting processes and random walks on Abelian groups, where the set of loops has a group structure.
In addition to this global description, we propose a second approach by looking at the short time behavior of a reciprocal process. In the same way as the Markov property and short time expansions of transition probabilities characterize Markov chains, we show that a reciprocal class is characterized by imposing the reciprocal property and two families of short time expansions for the bridges. Such local approach is suitable to study reciprocal processes on general countable graphs. As application of our characterization, we considered several interesting graphs, such as lattices, planar
graphs, the complete graph, and the hypercube.
Finally, we obtain some first results about concentration of measure implied by lower bounds on the reciprocal characteristics.
In this work we study reciprocal classes of Markov walks on graphs. Given a continuous time reference Markov chain on a graph, its reciprocal class is the set of all probability measures which can be represented as a mixture of the bridges of the reference walks. We characterize reciprocal classes with two different approaches. With the first approach we found it as the set of solutions to duality formulae on path space, where the differential operators have the interpretation of the addition of infinitesimal random loops to the paths of the canonical process. With the second approach we look at short time asymptotics of bridges. Both approaches allow an explicit computation of reciprocal characteristics, which are divided into two families, the loop characteristics and the arc characteristics. They are those specific functionals of the generator of the reference chain which determine its reciprocal class. We look at the specific examples such as Cayley graphs, the hypercube and planar graphs. Finally we establish the first concentration of measure results for the bridges of a continuous time Markov chain based on the reciprocal characteristics.
The paper deals with a non-linear singular partial differential equation: (E) t∂/∂t = F(t, x, u, ∂u/∂x) in the holomorphic category. When (E) is of Fuchsian type, the existence of the unique holomorphic solution was established by Gérard-Tahara [2]. In this paper, under the assumption that (E) is of totally characteristic type, the authors give a sufficient condition for (E) to have a unique holomorphic solution. The result is extended to higher order case.
A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too .
A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every nite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. The explicit form of the Laplace functional of the conditioned process is used to obtain several results on the long time behaviour of the mass of the conditioned and unconditioned processes. The general case is considered first, where the mutation matrix which modelizes the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are also analysed.
We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of difusion processes. Under suitable ergodicity assumptions on the underlying difusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense.
We study (pseudo-)differential operators on a manifold with edge Z, locally modelled on a wedge with model cone that has itself a base manifold W with smooth edge Y . The typical operators A are corner degenerate in a specific way. They are described (modulo ‘lower order terms’) by a principal symbolic hierarchy σ(A) = (σ ψ(A), σ ^(A), σ ^(A)), where σ ψ is the interior symbol and σ ^(A)(y, η), (y, η) 2 T*Y \ 0, the (operator-valued) edge symbol of ‘first generation’, cf. [15]. The novelty here is the edge symbol σ^ of ‘second generation’, parametrised by (z, Ϛ) 2 T*Z \ 0, acting on weighted Sobolev spaces on the infinite cone with base W. Since such a cone has edges with exit to infinity, the calculus has the problem to understand the behaviour of operators on a manifold of that kind. We show the continuity of corner-degenerate operators in weighted edge Sobolev spaces, and we investigate the ellipticity of edge symbols of second generation. Starting from parameter-dependent elliptic families of edge operators of first generation, we obtain the Fredholm property of higher edge symbols on the corresponding singular infinite model cone.
Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions.
In this article we study the geometry associated with the sub-elliptic operator ½ (X²1 +X²2), where X1 = ∂x and X2 = x²/2 ∂y are vector fields on R². We show that any point can be connected with the origin by at least one geodesic and we provide an approximate formula for the number of the geodesics between the origin and the points situated outside of the y-axis. We show there are in¯nitely many geodesics between the origin and the points on the y-axis.
The ill-posed inversion of multiwavelength lidar data by a hybrid method of variable projection
(1999)
The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements.
The determination of the atmospheric aerosol size distribution is an inverse illposed problem. The shape and the material composition of the air-carried particles are two substantial model parameters. Present evaluation algorithms only used an approximation with spherical homogeneous particles. In this paper we propose a new numerically efficient recursive algorithm for inhomogeneous multilayered coated and absorbing particles. Numerical results of real existing particles show that the influence of the two parameters on the model is very important and therefore cannot be ignored.
The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed.
This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schrödinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.
We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.
A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p solution must vanish.
Europa Universalis IV
(2020)
Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n − SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V −iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions.
Parabolic equations on manifolds with singularities require a new calculus of anisotropic pseudo-differential operators with operator-valued symbols. The paper develops this theory along the lines of sn abstract wedge calculus with strongly continuous groups of isomorphisms on the involved Banach spaces. The corresponding pseodo-diferential operators are continuous in anisotropic wedge Sobolev spaces, and they form an alegbra. There is then introduced the concept of anisotropic parameter-dependent ellipticity, based on an order reduction variant of the pseudo-differential calculus. The theory is appled to a class of parabolic differential operators, and it is proved the invertibility in Sobolev spaces with exponential weights at infinity in time direction.
We prove the existence of a class of local in time solutions, including static solutions, of the Einstein-Euler system. This result is the relativistic generalisation of a similar result for the Euler-Poisson system obtained by Gamblin [8]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article.
We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein{Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for hyperbolic and elliptic equations in these spaces. The well posedness is obtained in these spaces.
This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case.
We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set.
The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which takes into account both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration this modification is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.
We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true
regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.
We introduce a theoretical framework for performing statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses. This extends the standard statistical setting for multiple hypotheses testing, which is restricted to a finite set. This work is motivated by numerous modern applications where the observed signal is modeled by a stochastic process over a continuum. As a measure of type I error, we extend the concept of false discovery rate (FDR) to this setting. The FDR is defined as the average ratio of the measure of two random sets, so that its study presents some challenge and is of some intrinsic mathematical interest. Our main result shows how to use the p-value process to control the FDR at a nominal level, either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting, the latter one leading to a less conservative procedure. The interest of this approach is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables. Conceptually, an interesting feature of the setting advocated here is that it focuses directly on the intrinsic hypothesis space associated with a testing model on a random process, without referring to an arbitrary discretization.
Microsaccades
(2015)
The first thing we do upon waking is open our eyes. Rotating them in our eye sockets, we scan our surroundings and collect the information into a picture in our head. Eye movements can be split into saccades and fixational eye movements, which occur when we attempt to fixate our gaze. The latter consists of microsaccades, drift and tremor. Before we even lift our eye lids, eye movements – such as saccades and microsaccades that let the eyes jump from one to another position – have partially been prepared in the brain stem. Saccades and microsaccades are often assumed to be generated by the same mechanisms. But how saccades and microsaccades can be classified according to shape has not yet been reported in a statistical manner. Research has put more effort into the investigations of microsaccades’ properties and generation only since the last decade. Consequently, we are only beginning to understand the dynamic processes governing microsaccadic eye movements. Within this thesis, the dynamics governing the generation of microsaccades is assessed and the development of a model for the underlying processes. Eye movement trajectories from different experiments are used, recorded with a video-based eye tracking technique, and a novel method is proposed for the scale-invariant detection of saccades (events of large amplitude) and microsaccades (events of small amplitude). Using a time-frequency approach, the method is examined with different experiments and validated against simulated data. A shape model is suggested that allows for a simple estimation of saccade- and microsaccade related properties. For sequences of microsaccades, in this thesis a time-dynamic Markov model is proposed, with a memory horizon that changes over time and which can best describe sequences of microsaccades.
Change points in time series are perceived as heterogeneities in the statistical or dynamical characteristics of the observations. Unraveling such transitions yields essential information for the understanding of the observed system’s intrinsic evolution and potential external influences. A precise detection of multiple changes is therefore of great importance for various research disciplines, such as environmental sciences, bioinformatics and economics. The primary purpose of the detection approach introduced in this thesis is the investigation of transitions underlying direct or indirect climate observations. In order to develop a diagnostic approach capable to capture such a variety of natural processes, the generic statistical features in terms of central tendency and dispersion are employed in the light of Bayesian inversion. In contrast to established Bayesian approaches to multiple changes, the generic approach proposed in this thesis is not formulated in the framework of specialized partition models of high dimensionality requiring prior specification, but as a robust kernel-based approach of low dimensionality employing least informative prior distributions.
First of all, a local Bayesian inversion approach is developed to robustly infer on the location and the generic patterns of a single transition. The analysis of synthetic time series comprising changes of different observational evidence, data loss and outliers validates the performance, consistency and sensitivity of the inference algorithm. To systematically investigate time series for multiple changes, the Bayesian inversion is extended to a kernel-based inference approach. By introducing basic kernel measures, the weighted kernel inference results are composed into a proxy probability to a posterior distribution of multiple transitions. The detection approach is applied to environmental time series from the Nile river in Aswan and the weather station Tuscaloosa, Alabama comprising documented changes. The method’s performance confirms the approach as a powerful diagnostic tool to decipher multiple changes underlying direct climate observations.
Finally, the kernel-based Bayesian inference approach is used to investigate a set of complex terrigenous dust records interpreted as climate indicators of the African region of the Plio-Pleistocene period. A detailed inference unravels multiple transitions underlying the indirect climate observations, that are interpreted as conjoint changes. The identified conjoint changes coincide with established global climate events. In particular, the two-step transition associated to the establishment of the modern Walker-Circulation contributes to the current discussion about the influence of paleoclimate changes on the environmental conditions in tropical and subtropical Africa at around two million years ago.
We study the dynamics of four wave interactions in a nonlinear quantum chain of oscillators under the "narrow packet" approximation. We determine the set of times for which the evolution of decay processes is essentially specified by quantum effects. Moreover, we highlight the quantum increment of instability.
In this thesis, we give two constructions for Riemannian metrics on Seiberg-Witten moduli spaces. Both these constructions are naturally induced from the L2-metric on the configuration space. The construction of the so called quotient L2-metric is very similar to the one construction of an L2-metric on Yang-Mills moduli spaces as given by Groisser and Parker. To construct a Riemannian metric on the total space of the Seiberg-Witten bundle in a similar way, we define the reduced gauge group as a subgroup of the gauge group. We show, that the quotient of the premoduli space by the reduced gauge group is isomorphic as a U(1)-bundle to the quotient of the premoduli space by the based gauge group. The total space of this new representation of the Seiberg-Witten bundle carries a natural quotient L2-metric, and the bundle projection is a Riemannian submersion with respect to these metrics. We compute explicit formulae for the sectional curvature of the moduli space in terms of Green operators of the elliptic complex associated with a monopole. Further, we construct a Riemannian metric on the cobordism between moduli spaces for different perturbations. The second construction of a Riemannian metric on the moduli space uses a canonical global gauge fixing, which represents the total space of the Seiberg-Witten bundle as a finite dimensional submanifold of the configuration space. We consider the Seiberg-Witten moduli space on a simply connected Käuhler surface. We show that the moduli space (when nonempty) is a complex projective space, if the perturbation does not admit reducible monpoles, and that the moduli space consists of a single point otherwise. The Seiberg-Witten bundle can then be identified with the Hopf fibration. On the complex projective plane with a special Spin-C structure, our Riemannian metrics on the moduli space are Fubini-Study metrics. Correspondingly, the metrics on the total space of the Seiberg-Witten bundle are Berger metrics. We show that the diameter of the moduli space shrinks to 0 when the perturbation approaches the wall of reducible perturbations. Finally we show, that the quotient L2-metric on the Seiberg-Witten moduli space on a Kähler surface is a Kähler metric.
Als Grundlage vieler statistischer Verfahren wird der Prozess der Entstehung von Daten modelliert, um dann weitere Schätz- und Testverfahren anzuwenden. Diese Arbeit befasst sich mit der Frage, wie diese Spezifikation für parametrische Modelle selbst getestet werden kann. In Erweiterung bestehender Verfahren werden Tests mit festem Kern eingeführt und ihre asymptotischen Eigenschaften werden analysiert. Es wird gezeigt, dass die Bestimmung der kritischen Werte mit mehreren Stichprobenwiederholungsverfahren möglich ist. Von diesen ist eine neue Monte-Carlo-Approximation besonders wichtig, da sie die Komplexität der Berechnung deutlich verringern kann. Ein bedingter Kleinste-Quadrate-Schätzer für nichtlineare parametrische Modelle wird definiert und seine wesentlichen asymptotischen Eigenschaften werden hergeleitet. Sämtliche Versionen der Tests und alle neuen Konzepte wurden in Simulationsstudien untersucht, deren wichtigste Resultate präsentiert werden. Die praktische Anwendbarkeit der Testverfahren wird an einem Datensatz zur Produktwahl dargelegt, der mit multinomialen Logit-Modellen analysiert werden soll.