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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.
This thesis is focused on the study and the exact simulation of two classes of real-valued Brownian diffusions: multi-skew Brownian motions with constant drift and Brownian diffusions whose drift admits a finite number of jumps.
The skew Brownian motion was introduced in the sixties by Itô and McKean, who constructed it from the reflected Brownian motion, flipping its excursions from the origin with a given probability. Such a process behaves as the original one except at the point 0, which plays the role of a semipermeable barrier. More generally, a skew diffusion with several semipermeable barriers, called multi-skew diffusion, is a diffusion everywhere except when it reaches one of the barriers, where it is partially reflected with a probability depending on that particular barrier. Clearly, a multi-skew diffusion can be characterized either as solution of a stochastic differential equation involving weighted local times (these terms providing the semi-permeability) or by its infinitesimal generator as Markov process.
In this thesis we first obtain a contour integral representation for the transition semigroup of the multiskew Brownian motion with constant drift, based on a fine analysis of its complex properties. Thanks to this representation we write explicitly the transition densities of the two-skew Brownian motion with constant drift as an infinite series involving, in particular, Gaussian functions and their tails.
Then we propose a new useful application of a generalization of the known rejection sampling method. Recall that this basic algorithm allows to sample from a density as soon as one finds an - easy to sample - instrumental density verifying that the ratio between the goal and the instrumental densities is a bounded function. The generalized rejection sampling method allows to sample exactly from densities for which indeed only an approximation is known. The originality of the algorithm lies in the fact that one finally samples directly from the law without any approximation, except the machine's.
As an application, we sample from the transition density of the two-skew Brownian motion with or without constant drift. The instrumental density is the transition density of the Brownian motion with constant drift, and we provide an useful uniform bound for the ratio of the densities. We also present numerical simulations to study the efficiency of the algorithm.
The second aim of this thesis is to develop an exact simulation algorithm for a Brownian diffusion whose drift admits several jumps. In the literature, so far only the case of a continuous drift (resp. of a drift with one finite jump) was treated. The theoretical method we give allows to deal with any finite number of discontinuities. Then we focus on the case of two jumps, using the transition densities of the two-skew Brownian motion obtained before. Various examples are presented and the efficiency of our approach is discussed.
In this paper we study the convergence of continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on a recent progress in the geometric theory of spirallike functions. We prove convergence theorems and illustrate them by numerical simulations.
By means of the cluster expansion method we show that a recent result of Poghosyan and Ueltschi (2009) combined with a result of Nehring (2012) yields a construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in R^d of Ginibre's Fermi-Dirac gas of such loops. The latter will be identified as a Gibbs perturbation of the ideal Fermi gas. On generalizing these considerations we will obtain the existence of a large class of Gibbs perturbations of the so-called KMM-processes as they were introduced by Nehring (2012). Moreover, it is shown that certain "limiting Gibbs processes" are Gibbs in the sense of Dobrushin, Lanford and Ruelle if the underlying potential is positive. And finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.
This article aims at the statistical assessment of time series with large fluctuations in short time, which are assumed to stem from a continuous process perturbed by a Lévy process exhibiting a heavy tail behavior. We propose an easily implementable procedure to estimate efficiently the statistical difference between the noisy behavior of the data and a given reference jump measure in terms of so-called coupling distances. After a short introduction to Lévy processes and coupling distances we recall basic statistical approximation results and derive rates of convergence. In the sequel the procedure is elaborated in detail in an abstract setting and eventually applied in a case study to simulated and paleoclimate data. It indicates the dominant presence of a non-stable heavy-tailed jump Lévy component for some tail index greater than 2.
We introduce the Volterra calculus of pseudodifferential operators with an anisotropic analytic parameter based on "twisted" operator-valued Volterra symbols. We establish the properties of the symbolic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side. In particular, we investigate the kernel cut-off operator via direct oscillatory integral techniques purely on symbolic level. We discuss the notion of parabolic for the calculus of Volterra operators, and construct Volterra parametrices for parabolic operators within the calculus.
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.
We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.
Linear and non-linear analogues of the Black-Scholes equation are derived when shocks can be described by a truncated Lévy process. A linear equation is derived under the perfect correlation assumption on returns for a derivative security and a stock, and its solutions for European put and call options are obtained. It is also shown that the solution violates the perfect correlation assumption unless a process is gaussian. Thus, for a family of truncated Lévy distributions, the perfect hedging is impossible even in the continuous time limit. A second linear analogue of the Black-Scholes equation is obtained by constructing a portfolio which eliminates fluctuations of the first order and assuming that the portfolio is risk-free; it is shown that this assumption fails unless a process is gaussian. It is shown that the di erence between solutions to the linear analogues of the Black-Scholes equations and solutions to the Black-Scholes equations are sizable. The equations and solutions can be written in a discretized approximate form which uses an observed probability distribution only. Non-linear analogues for the Black-Scholes equation are derived from the non-arbitrage condition, and approximate formulas for solutions of these equations are suggested. Assuming that a linear generalization of the Black-Scholes equation holds, we derive an explicit pricing formula for the perpetual American put option and produce numerical results which show that the difference between our result and the classical Merton's formula obtained for gaussian processes can be substantial. Our formula uses an observed distribution density, under very weak assumptions on the latter.
On null quadrature domains
(2006)
The characterization of null quadrature domains in Rn (n ≥ 3) has been an open problem throughout the past two and a half decades. A substantial contribution was done by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellip- soids. The first result with unbounded complements appeared in [15], there it is assumed the complement is contained in an infinitely cylinder. The aim of this paper is to show the relation between null quadrature domains and Newton's theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also succeed to make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a strip and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a strip. In addition, we present a Phragm¶en-Lindelöf type theorem which seems to be forgotten in the literature.
We prove the existence of Hp(D)-limit of iterations of double layer potentials constructed with the use of Hodge parametrix on a smooth compact manifold X, D being an open connected subset of X. This limit gives us an orthogonal projection from Sobolev space Hp(D) to a closed subspace of Hp(D)-solutions of an elliptic operator P of order p ≥ 1. Using this result we obtain formulae for Sobolev solutions to the equation Pu = f in D whenever these solutions exist. This representation involves the sum of a series whose terms are iterations of double layer potentials. Similar regularization is constructed also for a P-Neumann problem in D.
We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.
The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. AMS Classifications: 60G15 , 60G60 , 60H10 , 60J60
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.
We construct a theory of general boundary value problems for differential operators whose symbols do not necessarily satisfy the Atiyah-Bott condition [3] of vanishing of the corresponding obstruction. A condition for these problems to be Fredholm is introduced and the corresponding finiteness theorems are proved.
On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators
(2012)
We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.
In this paper, we discuss the global existence of solutions for Chemotaxis models with saturation growth. If the coe±cients of the equations are all positive smooth T-periodic functions, then the problem has a positive T-periodic solution, and meanwhile we discuss here the stability problems for the T-periodic solutions.
We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C n with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology.
An existence and uniqueness theory is developed for general nonlinear and nonautonomous differential-algebraic equations (DAEs) by exploiting their underlying differential-geometric structure. A DAE is called regular if there is a unique nonautonomous vector field such that the solutions of the DAE and the solutions of the vector field are in one-to-one correspondence. Sufficient conditions for regularity of a DAE are derived in terms of constrained manifolds. Based on this differential-geometric characterization, existence and uniqueness results are stated for regular DAEs. Furthermore, our not ons are compared with techniques frequently used in the literature such as index and solvability. The results are illustrated in detail by means of a simple circuit example.
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.
Contents: 1 Introduction 2 Statement of the problem and definitions 3 The main results 4 Proof of theorem 2 4.1 Reduction of problem (2) to functional - integral equations 4.2 The uniqueness of a solution of equation (2) 4.3 The existence of a solution of equation (2) 5 Proof of theorem 1 6 Proof of theorem 3 7 First boundary problem for hyperbolic differential equations 7.1 Statement of the problem 7.2 The formulation of the result and a sketch of the proof
On a method for solution of the ordinary differential equations connected with Huygens' equations
(2010)
Generalizing an idea of I. Vekua [1] who, in order to construct theory of plates and shells, fields of displacements, strains and stresses of threedimensional theory of linear elasticity expands into the orthogonal Fourier-series by Legendre Polynomials with respect to the variable along thickness, and then leaves only first N + 1, N = 0, 1, ..., terms, in the bar model under consideration all above quantities have been expanded into orthogonal double Fourier-series by Legendre Polynomials with respect to the variables along thickness, and width of the bar, and then first (Nsub(3) + 1)(Nsub(2) + 1), Nsub(3), Nsub(2) = 0, 1,..., terms have been left. This case will be called (Nsub(3), Nsub(2)) approximation. Both in general (Nsub(3), Nsub(2)) and in particular (0,0) (1,0) cases of approximation, the question of wellposedness of initial and boundary value problems, existence and uniqueness of solutions have been investigated. The cases when variable cross-section turns into segments of straight line, and points have been also considered. Such bars will be called cusped bars (see also [2]).
The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.
We study a boundary value problem for an overdetermined elliptic system of nonlinear first order differential equations with linear boundary operators. Such a problem is solvable for a small set of data, and so we pass to its variational formulation which consists in minimising the discrepancy. The Euler-Lagrange equations for the variational problem are far-reaching analogues of the classical Laplace equation. Within the framework of Euler-Lagrange equations we specify an operator on the boundary whose zero set consists precisely of those boundary data for which the initial problem is solvable. The construction of such operator has much in common with that of the familiar Dirichlet to Neumann operator. In the case of linear problems we establish complete results.
The aim of this paper is to demonstrate that nonparametric smoothing methods for estimating functions can be an useful tool in the analysis of life time data. After stating some basic notations we will present a data example. Applying standard parametric methods to these data we will see that this approach fails - basic features of the underlying functions are not reflected by their estimates. Our proposal is to use nonparametric estimation methods. These methods are explained in section 2. Nonparametric approaches are better in the sense that they are more flexible, and misspecifications of the model are avoided. But, parametric models have the advantage that the parameters can be interpreted. So, finally, we will formulate a test procedure to check whether a parametric or a nonparametric model is appropriate.
Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary
(2023)
Non-local boundary conditions – for example the Atiyah–Patodi–Singer (APS) conditions – for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, Bär and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While Bär and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions.
In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed.
We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems.
Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions – the analogue of the Atiyah–Patodi–Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.
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).
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.
We continue our study of invariant forms of the classical equations of mathematical physics,
such as the Maxwell equations or the Lamé system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded
in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations.
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.
In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL(n).
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.