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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.
Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations whose solutions describe the orbits of closed p-branes under the effect of a "generalized physical force". For the corresponding evolution equation, which is a system of parabolic nonlinear partial differential equations associated to the elliptic PDE, we can establish existence of short time solutions.
Boundary value problems for pseudodifferential operators (with or without the transmission property) are characterised as a substructure of the edge pseudodifferential calculus with constant discrete asymptotics. The boundary in this case is the edge and the inner normal the model cone of local wedges. Elliptic boundary value problems for non-integer powers of the Laplace symbol belong to the examples as well as problems for the identity in the interior with a prescribed number of trace and potential conditions. Transmission operators are characterised as smoothing Mellin and Green operators with meromorphic symbols.
Ellipticity of operators on manifolds with conical singularities or parabolicity on space-time cylinders are known to be linked to parameter-dependent operators (conormal symbols) on a corresponding base manifold. We introduce the conormal symbolic structure for the case of corner manifolds, where the base itself is a manifold with edges and boundary. The specific nature of parameter-dependence requires a systematic approach in terms of meromorphic functions with values in edge-boundary value problems. We develop here a corresponding calculus, and we construct inverses of elliptic elements.
We study the Cauchy problem for the oscillation equation of the couple-stress theory of elasticity in a bounded domain in R3. Both the displacement and stress are given on a part S of the boundary of the domain. This problem is densely solvable while data of compact support in the interior of S fail to belong to the range of the problem. Hence the problem is ill-posed which makes the standard calculi of Fourier integral operators inapplicable. If S is real analytic the Cauchy-Kovalevskaya theorem applies to guarantee the existence of a local solution. We invoke the special structure of the oscillation equation to derive explicit conditions of global solvability and an approximation solution.
The Cauchy problem for the linearised Einstein equation and the Goursat problem for wave equations
(2017)
In this thesis, we study two initial value problems arising in general relativity. The first is the Cauchy problem for the linearised Einstein equation on general globally hyperbolic spacetimes, with smooth and distributional initial data. We extend well-known results by showing that given a solution to the linearised constraint equations of arbitrary real Sobolev regularity, there is a globally defined solution, which is unique up to addition of gauge solutions. Two solutions are considered equivalent if they differ by a gauge solution. Our main result is that the equivalence class of solutions depends continuously on the corre- sponding equivalence class of initial data. We also solve the linearised constraint equations in certain cases and show that there exist arbitrarily irregular (non-gauge) solutions to the linearised Einstein equation on Minkowski spacetime and Kasner spacetime.
In the second part, we study the Goursat problem (the characteristic Cauchy problem) for wave equations. We specify initial data on a smooth compact Cauchy horizon, which is a lightlike hypersurface. This problem has not been studied much, since it is an initial value problem on a non-globally hyperbolic spacetime. Our main result is that given a smooth function on a non-empty, smooth, compact, totally geodesic and non-degenerate Cauchy horizon and a so called admissible linear wave equation, there exists a unique solution that is defined on the globally hyperbolic region and restricts to the given function on the Cauchy horizon. Moreover, the solution depends continuously on the initial data. A linear wave equation is called admissible if the first order part satisfies a certain condition on the Cauchy horizon, for example if it vanishes. Interestingly, both existence of solution and uniqueness are false for general wave equations, as examples show. If we drop the non-degeneracy assumption, examples show that existence of solution fails even for the simplest wave equation. The proof requires precise energy estimates for the wave equation close to the Cauchy horizon. In case the Ricci curvature vanishes on the Cauchy horizon, we show that the energy estimates are strong enough to prove local existence and uniqueness for a class of non-linear wave equations. Our results apply in particular to the Taub-NUT spacetime and the Misner spacetime. It has recently been shown that compact Cauchy horizons in spacetimes satisfying the null energy condition are necessarily smooth and totally geodesic. Our results therefore apply if the spacetime satisfies the null energy condition and the Cauchy horizon is compact and non-degenerate.
We introduce the calculus of Mellin pseudodifferential operators parameters based on "twisted" operator-valued Volterra symbols as well aas the abstract Mellin calclus with holomorphic symbols. We establish the properties of the symblic and operational calculi, and we give and make use of explicit oscillatory integral formulas on the symbolic side, e. g., for the Leibniz-product, kernel cut-off, and Mellin quantization. Moreover, we introduce the notion of parabolicity for the calculi of Volterra Mellin operators, and construct Volterra parametrices for parabolic operators within the calculi.
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.
The two and k-sample tests of equality of the survival distributions against the alternatives including cross-effects of survival functions, proportional and monotone hazard ratios, are given for the right censored data. The asymptotic power against approaching alternatives is investigated. The tests are applied to the well known chemio and radio therapy data of the Gastrointestinal Tumor Study Group. The P-values for both proposed tests are much smaller then in the case of other known tests. Differently from the test of Stablein and Koutrouvelis the new tests can be applied not only for singly but also to randomly censored data.
We consider a nonparametric survival model with random censoring. To test whether the hazard rate has a parametric form the unknown hazard rate is estimated by a kernel estimator. Based on a limit theorem stating the asymptotic normality of the quadratic distance of this estimator from the smoothed hypothesis an asymptotic ®-test is proposed. Since the test statistic depends on the maximum likelihood estimator for the unknown parameter in the hypothetical model properties of this parameter estimator are investigated. Power considerations complete the approach.
The accelerated life time model is considered. First, test procedures for testing the parameter of a parametric acceleration function is investigated; this is done under the assumption of parametric and nonparametric baseline distribution. Further, based on nonparametric estimators for regression functions tests are proposed for checking whether a parametric acceleration function is appropriate to model the influence of the covariates. Resampling procedures are discussed for the realization of these methods. Simulations complete the considerations.
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.
In this article we analyse the structure of Markov processes and reciprocal processes to underline their time symmetrical properties, and to compare them. Our originality consists in adopting a unifying approach of reciprocal processes, independently of special frameworks in which the theory was developped till now (diffusions, or pure jump processes). This leads to some new results, too.
A Hamiltonian system in potential form (formula in the original abstract) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in Rn. In this paper methods which reduce "Hamiltonian differential algebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraint invariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically.
Contents: Part I: Symplectic Geometry Chapter 1: Symplectic Spaces and Lagrangian Planes Chapter 2: The Symplectic Group Chapter 3: Multi-Oriented Symplectic Geometry Chapter 4: Intersection Indices in Lag(n) and Sp(n) Part II: Heisenberg Group, Weyl Calculus, and Metaplectic Representation Chapter 5: Lagrangian Manifolds and Quantization Chapter 6: Heisenberg Group and Weyl Operators Chapter 7: The Metaplectic Group Part III: Quantum Mechanics in Phase Space Chapter 8: The Uncertainty Principle Chapter 9: The Density Operator Chapter 10: A Phase Space Weyl Calculus
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.
Boundary value problems for (pseudo-) differential operators on a manifold with edges can be characterised by a hierarchy of symbols. The symbol structure is responsible or ellipicity and for the nature of parametrices within an algebra of "edge-degenerate" pseudo-differential operators. The edge symbol component of that hierarchy takes values in boundary value problems on an infinite model cone, with edge variables and covariables as parameters. Edge symbols play a crucial role in this theory, in particular, the contribution with holomorphic operatot-valued Mellin symbols. We establish a calculus in s framework of "twisted homogenity" that refers to strongly continuous groups of isomorphisms on weighted cone Sobolev spaces. We then derive an equivalent representation with a particularly transparent composition behaviour.
We prove a theorem describing the behaviour of the relative index of families of Fredholm operators under surgery performed on spaces where the operators act. In connection with additional conditions (like symmetry conditions) this theorem results in index formulas for given operator families. By way of an example, we give an application to index theory of families of boundary value problems.
We prove a general theorem on the local property of the relative index 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 as well as for elliptic boundary value problems with a symmetry condition for the conormal symbol.
We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain for a second order elliptic differential operator A. The differential operator is assumed to be of divergent form and the boundary operator B is of Robin type. The boundary is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset of the boundary and control the growth of solutions near this set. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set. Moreover, we prove the completeness of root functions related to L.
Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces.
Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness.
In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate.
We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase.
The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy.
The interdisciplinary workshop STOCHASTIC PROCESSES WITH APPLICATIONS IN THE NATURAL SCIENCES was held in Bogotá, at Universidad de los Andes from December 5 to December 9, 2016. It brought together researchers from Colombia, Germany, France, Italy, Ukraine, who communicated recent progress in the mathematical research related to stochastic processes with application in biophysics.
The present volume collects three of the four courses held at this meeting by Angelo Valleriani, Sylvie Rœlly and Alexei Kulik.
A particular aim of this collection is to inspire young scientists in setting up research goals within the wide scope of fields represented in this volume.
Angelo Valleriani, PhD in high energy physics, is group leader of the team "Stochastic processes in complex and biological systems" from the Max-Planck-Institute of Colloids and Interfaces, Potsdam.
Sylvie Rœlly, Docteur en Mathématiques, is the head of the chair of Probability at the University of Potsdam.
Alexei Kulik, Doctor of Sciences, is a Leading researcher at the Institute of Mathematics of Ukrainian National Academy of Sciences.
In this thesis, stochastic dynamics modelling collective motions of populations, one of the most mysterious type of biological phenomena, are considered. For a system of N particle-like individuals, two kinds of asymptotic behaviours are studied : ergodicity and flocking properties, in long time, and propagation of chaos, when the number N of agents goes to infinity. Cucker and Smale, deterministic, mean-field kinetic model for a population without a hierarchical structure is the starting point of our journey : the first two chapters are dedicated to the understanding of various stochastic dynamics it inspires, with random noise added in different ways. The third chapter, an attempt to improve those results, is built upon the cluster expansion method, a technique from statistical mechanics. Exponential ergodicity is obtained for a class of non-Markovian process with non-regular drift. In the final part, the focus shifts onto a stochastic system of interacting particles derived from Keller and Segel 2-D parabolicelliptic model for chemotaxis. Existence and weak uniqueness are proven.
Geomagnetic field modeling using spherical harmonics requires the inversion for hundreds to thousands of parameters. This large-scale problem can always be formulated as an optimization problem, where a global minimum of a certain cost function has to be calculated. A variety of approaches is known in order to solve this inverse problem, e.g. derivative-based methods or least-squares methods and their variants. Each of these methods has its own advantages and disadvantages, which affect for example the applicability to non-differentiable functions or the runtime of the corresponding algorithm.
In this work, we pursue the goal to find an algorithm which is faster than the established methods and which is applicable to non-linear problems. Such non-linear problems occur for example when estimating Euler angles or when the more robust L_1 norm is applied. Therefore, we will investigate the usability of stochastic optimization methods from the CMAES family for modeling the geomagnetic field of Earth's core. On one hand, basics of core field modeling and their parameterization are discussed using some examples from the literature. On the other hand, the theoretical background of the stochastic methods are provided. A specific CMAES algorithm was successfully applied in order to invert data of the Swarm satellite mission and to derive the core field model EvoMag. The EvoMag model agrees well with established models and observatory data from Niemegk. Finally, we present some observed difficulties and discuss the results of our model.
Die vorliegende Diplomarbeit beschäftigt sich mit zwei Aspekten der statistischen Eigenschaften von Clusterverfahren. Zum einen geht die Arbeit auf die Frage der Existenz von unterschiedlichen Clusteranalysemethoden zur Strukturfindung und deren unterschiedlichen Vorgehensweisen ein. Die Methode des Abstandes zwischen Mannigfaltigkeiten und die K-means Methode liefern ausgehend von gleichen Daten unterschiedliche Endclusterungen. Der zweite Teil dieser Arbeit beschäftigt sich näher mit den asymptotischen Eigenschaften des K-means Verfahrens. Hierbei ist die Menge der optimalen Clusterzentren konsistent. Bei Vergrößerung des Stichprobenumfangs gegen Unendlich konvergiert diese in Wahrscheinlichkeit gegen die Menge der Clusterzentren, die das Varianzkriterium minimiert. Ebenfalls konvergiert die Menge der optimalen Clusterzentren für n gegen Unendlich gegen eine Normalverteilung. Es hat sich dabei ergeben, dass die einzelnen Clusterzentren voneinander abhängen.
Estimation and testing of distributions in metric spaces are well known. R.A. Fisher, J. Neyman, W. Cochran and M. Bartlett achieved essential results on the statistical analysis of categorical data. In the last 40 years many other statisticians found important results in this field. Often data sets contain categorical data, e.g. levels of factors or names. There does not exist any ordering or any distance between these categories. At each level there are measured some metric or categorical values. We introduce a new method of scaling based on statistical decisions. For this we define empirical probabilities for the original observations and find a class of distributions in a metric space where these empirical probabilities can be found as approximations for equivalently defined probabilities. With this method we identify probabilities connected with the categorical data and probabilities in metric spaces. Here we get a mapping from the levels of factors or names into points of a metric space. This mapping yields the scale for the categorical data. From the statistical point of view we use multivariate statistical methods, we calculate maximum likelihood estimations and compare different approaches for scaling.
An expansion for a class of functions is called stable if the partial sums are bounded uniformly in the class. Stable expansions are of key importance in numerical analysis where functions are given up to certain error. We show that expansions in homogeneous functions are always stable on a small ball around the origin, and evaluate the radius of the largest ball with this property.
Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.
Many methods have been proposed for the simulation of constrained mechanical systems. The most obvious of these have mild instabilities and drift problems. Consequently, stabilization techniques have been proposed A popular stabilization method is Baumgarte's technique, but the choice of parameters to make it robust has been unclear in practice. Some of the simulation methods that have been proposed and used in computations are reviewed here, from a stability point of view. This involves concepts of differential-algebraic equation (DAE) and ordinary differential equation (ODE) invariants. An explanation of the difficulties that may be encountered using Baumgarte's method is given, and a discussion of why a further quest for better parameter values for this method will always remain frustrating is presented. It is then shown how Baumgarte's method can be improved. An efficient stabilization technique is proposed, which may employ explicit ODE solvers in case of nonstiff or highly oscillatory problems and which relates to coordinate projection methods. Examples of a two-link planar robotic arm and a squeezing mechanism illustrate the effectiveness of this new stabilization method.
This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.
The propagation of test fields, such as electromagnetic, Dirac or linearized gravity, on a fixed spacetime manifold is often studied by using the geometrical optics approximation. In the limit of infinitely high frequencies, the geometrical optics approximation provides a conceptual transition between the test field and an effective point-particle description. The corresponding point-particles, or wave rays, coincide with the geodesics of the underlying spacetime. For most astrophysical applications of interest, such as the observation of celestial bodies, gravitational lensing, or the observation of cosmic rays, the geometrical optics approximation and the effective point-particle description represent a satisfactory theoretical model. However, the geometrical optics approximation gradually breaks down as test fields of finite frequency are considered.
In this thesis, we consider the propagation of test fields on spacetime, beyond the leading-order geometrical optics approximation. By performing a covariant Wentzel-Kramers-Brillouin analysis for test fields, we show how higher-order corrections to the geometrical optics approximation can be considered. The higher-order corrections are related to the dynamics of the spin internal degree of freedom of the considered test field. We obtain an effective point-particle description, which contains spin-dependent corrections to the geodesic motion obtained using geometrical optics. This represents a covariant generalization of the well-known spin Hall effect, usually encountered in condensed matter physics and in optics. Our analysis is applied to electromagnetic and massive Dirac test fields, but it can easily be extended to other fields, such as linearized gravity. In the electromagnetic case, we present several examples where the gravitational spin Hall effect of light plays an important role. These include the propagation of polarized light rays on black hole spacetimes and cosmological spacetimes, as well as polarization-dependent effects on the shape of black hole shadows. Furthermore, we show that our effective point-particle equations for polarized light rays reproduce well-known results, such as the spin Hall effect of light in an inhomogeneous medium, and the relativistic Hall effect of polarized electromagnetic wave packets encountered in Minkowski spacetime.
For elliptic operators on manifolds with boundary, we define spectral boundary value problems, which generalize the Atiyah-Patodi-Singer problem to the case of nonhomogeneous boundary conditions, operators of arbitrary order, and nonself-adjoint conormal symbols. The Fredholm property is proved and equivalence with certain elliptic equations on manifolds with conical singularities is established.
We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.
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.
Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville Problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular and some singular SLPs of even orders (tested up to order eight), with a mix of boundary conditions (including non-separable and finite singular endpoints), accurately and efficiently.
The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved.
Next, a concrete implementation to the inverse Sturm–Liouville problem
algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of order n=2,4 is verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied.
In conclusion, this work provides methods that can be adapted successfully for solving a direct (regular/singular) or inverse SLP of an arbitrary order with arbitrary boundary conditions.
In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.
We consider the numerical treatment of Hamiltonian systems that contain a potential which grows large when the system deviates from the equilibrium value of the potential. Such systems arise, e.g., in molecular dynamics simulations and the spatial discretization of Hamiltonian partial differential equations. Since the presence of highly oscillatory terms in the solutions forces any explicit integrator to use very small step size, the numerical integration of such systems provides a challenging task. It has been suggested before to replace the strong potential by a holonomic constraint that forces the solutions to stay at the equilibrium value of the potential. This approach has, e.g., been successfully applied to the bond stretching in molecular dynamics simulations. In other cases, such as the bond-angle bending, this methods fails due to the introduced rigidity. Here we give a careful analysis of the analytical problem by means of a smoothing operator. This will lead us to the notion of the smoothed dynamics of a highly oscillatory Hamiltonian system. Based on our analysis, we suggest a new constrained formulation that maintains the flexibility of the system while at the same time suppressing the high-frequency components in the solutions and thus allowing for larger time steps. The new constrained formulation is Hamiltonian and can be discretized by the well-known SHAKE method.
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'.
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.
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.
We consider a class of infinite-dimensional diffusions where the interaction between the components is both spatial and temporal. We start the system from a Gibbs measure with finiterange uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exists t0 > 0 such that the distribution at time t = t0 is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion of both the initial interaction and certain time-reversed Girsanov factors coming from the dynamics.