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Der Beitrag setzt sich mit der halachischen Bedeutung von Dtn. 6,18 im Kontext der heutigen Zeit auseinander.
Given a system of entire functions in Cn with at most countable set of common zeros, we introduce the concept of zeta-function associated with the system. Under reasonable assumptions on the system, the zeta-function is well defined for all s ∈ Zn with sufficiently large components. Using residue theory we get an integral representation for the zeta-function which allows us to construct an analytic extension of the zeta-function to an infinite cone in Cn.
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.
Contents: 1 Introduction 1.1 Tikhanov-Phillips Regularization of Ill-Posed Problems 1.2 A Compact Course to Wavelets 2 A Multilevel Iteration for Tikhonov-Phillips Regularization 2.1 Multilevel Splitting 2.2 The Multilevel Iteration 2.3 Multilevel Approach to Cone Beam Reconstuction 3 The use of approximating operators 3.1 Computing approximating families {Ah}
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.
In this paper, we discuss the viscosity solutions of the weakly coupled systems of fully nonlinear second order degenerate parabolic equations and their Cauchy-Dirichlet problem. We prove the existence, uniqueness and continuity of viscosity solution by combining Perron's method with the technique of coupled solutions. The results here generalize those in [2] and [3].
In this paper we establish the regularity, exponential stability of global (weak) solutions and existence of uniform compact attractors of semiprocesses, which are generated by the global solutions, of a two-parameter family of operators for the nonlinear 1-d non-autonomous viscoelasticity. We employ the properties of the analytic semigroup to show the compactness for the semiprocess generated by the global solutions.
Toeplitz operators, and ellipticity of boundary value problems with global projection conditions
(2003)
Ellipticity of (pseudo-) differential operators A on a compact manifold X with boundary (or with edges) Y is connected with boundary (or edge) conditions of trace and potential type, formulated in terms of global projections on Y together with an additional symbolic structure. This gives rise to operator block matrices A with A in the upper left corner. We study an algebra of such operators, where ellipticity is equivalent to the Fredhom property in suitable scales of spaces: Sobolev spaces on X plus closed subspaces of Sobolev spaces on Y which are the range of corresponding pseudo-differential projections. Moreover, we express parametrices of elliptic elements within our algebra and discuss spectral boundary value problems for differential operators.
This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved.
Studying the influence of the updating scheme for MCMC algorithm on spatially extended models is a well known problem. For discrete-time interacting particle systems we study through simulations the effectiveness of a synchronous updating scheme versus the usual sequential one. We compare the speed of convergence of the associated Markov chains from the point of view of the time-to-coalescence arising in the coupling-from-the-past algorithm. Unlike the intuition, the synchronous updating scheme is not always the best one. The distribution of the time-to-coalescence for these spatially extended models is studied too.
We study mixed boundary value problems for an elliptic operator A on a manifold X with boundary Y , i.e., Au = f in int X, T±u = g± on int Y±, where Y is subdivided into subsets Y± with an interface Z and boundary conditions T± on Y± that are Shapiro-Lopatinskij elliptic up to Z from the respective sides. We assume that Z ⊂ Y is a manifold with conical singularity v. As an example we consider the Zaremba problem, where A is the Laplacian and T− Dirichlet, T+ Neumann conditions. The problem is treated as a corner boundary value problem near v which is the new point and the main difficulty in this paper. Outside v the problem belongs to the edge calculus as is shown in [3]. With a mixed problem we associate Fredholm operators in weighted corner Sobolev spaces with double weights, under suitable edge conditions along Z \ {v} of trace and potential type. We construct parametrices within the calculus and establish the regularity of solutions.
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.
We have numerically studied the bifurcations and transition to chaos in a two-dimensional fluid for varying values of the Reynolds number. These investigations have been motivated by experiments in fluids, where an array of vortices was driven by an electromotive force. In these experiments, successive changes leading to a complex motion of the vortices, due to increased forcing, have been explored [Tabeling, Perrin, and Fauve, J. Fluid Mech. 213, 511 (1990)]. We model this experiment by means of two-dimensional Navier-Stokes equations with a special external forcing, driving a linear chain of eight counter-rotating vortices, imposing stress-free boundary conditions in the vertical direction and periodic boundary conditions in the horizontal direction. As the strength of the forcing or the Reynolds number is raised, the original stationary vortex array becomes unstable and a complex sequence of bifurcations is observed. Several steady states and periodic branches and a period doubling cascade appear on the route to chaos. For increasing values of the Reynolds number, shear flow develops, for which the spatial scale is large compared to the scale of the forcing. Furthermore, we have investigated the influence of the aspect ratio of the container as well as the effect of no-slip boundary conditions at the top and bottom, on the bifurcation scenario.
We study an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted Lp-space. Under suitable conditions we show that the resolvent (λ - A )-¹ exists in a sector of the complex plane and decays like 1/|λ| as |λ| -> ∞. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with striaght conical degeneracy and establish maximal regularity for the Cauchy problem u - Δu = f, u(0) = 0.
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 consider a mixed problem for a degenerate differentialoperator equation of higher order. We establish some embedding theorems in weighted Sobolev spaces and show existence and uniqueness of the generalized solution of this problem. We also give a description of the spectrum for the corresponding operator.
We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27 - 31, 2008, at the MSRI, University of Berkeley.
The quantization of contact transformations of the cosphere bundle over a manifold with conical singularities is described. The index of Fredholm operators given by this quantization is calculated. The answer is given in terms of the Epstein-Melrose contact degree and the conormal symbol of the corresponding operator.
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.
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 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.
In this paper we consider the hypo-ellipticity of differential forms on a closed manifold.The main results show that there are some topological obstruct for the existence of the differential forms with hypoellipticity.
The homotopy classification and the index of boundary value problems for general elliptic operators
(1999)
We give the homotopy classification and compute the index of boundary value problems for elliptic equations. The classical case of operators that satisfy the Atiyah-Bott condition is studied first. We also consider the general case of boundary value problems for operators that do not necessarily satisfy the Atiyah-Bott condition.
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.
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.
We construct a special asymptotic solution for the forced KdV equation. In the frame of the shallow water model this kind of the external driving force is related to the atmospheric disturbance. The perturbation slowly passes through a resonance and it leads to the solution exchange. The detailed asymptotic description of the process is presented.
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.
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.
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.
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.
We investigate the cognitive control in polyrhythmic hand movements as a model paradigm for bimanual coordination. Using a symbolic coding of the recorded time series, we demonstrate the existence of qualitative transitions induced by experimental manipulation of the tempo. A nonlinear model with delayed feedback control is proposed, which accounts for these dynamical transitions in terms of bifurcations resulting from variation of the external control parameter. Furthermore, it is shown that transitions can also be observed due to fluctuations in the timing control level. We conclude that the complexity of coordinated bimanual movements results from interactions between nonlinear control mechanisms with delayed feedback and stochastic timing components.
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
We have studied the bifurcations in a three-dimensional incompressible magnetofluid with periodic boundary conditions and an external forcing of the Arnold-Beltrami-Childress (ABC) type. Bifurcation-analysis techniques have been applied to explore the qualitative behavior of solution branches. Due to the symmetry of the forcing, the equations are equivariant with respect to a group of transformations isomorphic to the octahedral group, and we have paid special attention to symmetry-breaking effects. As the Reynolds number is increased, the primary nonmagnetic steady state, the ABC flow, loses its stability to a periodic magnetic state, showing the appearance of a generic dynamo effect; the critical value of the Reynolds number for the instability of the ABC flow is decreased compared to the purely hydrodynamic case. The bifurcating magnetic branch in turn is subject to secondary, symmetry-breaking bifurcations. We have traced periodic and quasi- periodic branches until they end up in chaotic states. In particular detail we have analyzed the subgroup symmetries of the bifurcating periodic branches, which are closely related to the spatial structure of the magnetic field.
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 have shown that the two-dimensional complex Ginzburg-Landau equation exhibits supertransient chaos in a certain parameter range. Using numerical methods this behavior is found near the transition line separating frozen spiral solutions from turbulence. Supertransient chaos seems to be a common phenomenon in extended spatiotemporal systems. These supertransients are characterized by an average transient lifetime which depends exponentially on the size of the system and are due to an underlying nonattracting chaotic set.
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.
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.
The stability of the quiescent ground state of an incompressible viscous fluid sheet bounded by two parallel planes, with an electrical conductivity varying across the sheet, and driven by an external electric field tangential to the boundaries is considered. It is demonstrated that irrespective of the conductivity profile, as magnetic and kinetic Reynolds numbers (based on the Alfvén velocity) are raised from small values, two-dimensional perturbations become unstable first.
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.
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.
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.
We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent.
The stability of the quiescent ground state of an incompressible, viscous and electrically conducting fluid sheet, bounded by stress-free parallel planes and driven by an external electric field tangential to the boundaries, is studied numerically. The electrical conductivity varies as cosh–2(x1/a), where x1 is the cross-sheet coordinate and a is the half width of a current layer centered about the midplane of the sheet. For a <~ 0.4L, where L is the distance between the boundary planes, the ground state is unstable to disturbances whose wavelengths parallel to the sheet lie between lower and upper bounds depending on the value of a and on the Hartmann number. Asymmetry of the configuration with respect to the midplane of the sheet, modelled by the addition of an externally imposed constant magnetic field to a symmetric equilibrium field, acts as a stabilizing factor.
The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature o singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied.
Relative elliptic theory
(2002)
This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms.
Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We consider smooth symbols and ellipticity without additional boundary conditions which is the relevant case on a manifold with boundary. Starting from a class of symbols that has been investigated before for integer orders in boundary value problems with the transmission property we study operators of arbitrary real orders that play a similar role for operators without the transmission property. Moreover, we show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We finally investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary.
Using a special technique of data analysis, we have found out 34 grand minima of solar activity obtained from a 7,700 years long Δ14C record. The method used rests on a proper filtering of the Δ14C record and the extrapolation of verifiable results for the later history back in time. Additionally, we use a method of nonlinear dynamics, the recurrence rate, to back up the results. Our findings are not contradictory to the record of solar maxima resp. minima by Eddy [5], but constitute a considerable extension. Hence, it has become possible to look closer at the validity of models. This way, we have tested several models for solar activity, esp. the model of Barnes et al. [1]. There are hints for that the grand minima might solely be driven by the 209 year period found in the Δ14C record.
Reciprocal processes, whose concept can be traced back to E. Schrödinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. Léonard.
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.
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.
Content: 0.1 Preliminary Remarks Chapter 1: Noncommutative Analysis: Main Ideas, Definitions, and Theorems 1.1 Functions of One Operator (Functional Calculi) 1.2 Functions of Several Operators 1.3 Main Formulas of Operator Calculus 1.4 Main Tools of Noncommutative Analysis 1.5 Composition Laws and Ordered Representations
Quantization methods in differential equations : Chapter 2: Quantization of Lagrangian modules
(1999)
In this chapter we use the wave packet transform described in Chapter 1 to quantize extended classical states represented by so-called Lagrangian sumbanifolds of the phase space. Functions on a Lagrangian manifold form a module over the ring of classical Hamiltonian functions on the phase space (with respect to pointwise multiplication). The quantization procedure intertwines this multiplication with the action of the corresponding quantum Hamiltonians; hence we speak of quantization of Lagrangian modules. The semiclassical states obtained by this quantization procedure provide asymptotic solutions to differential equations with a small parameter. Locally, such solutions can be represented by WKB elements. Global solutions are given by Maslov's canonical operator [2]; also see, e.g., [3] and the references therein. Here the canonical operator is obtained in the framework of the universal quantization procedure provided by the wave packet transform. This procedure was suggested in [4] (see also the references there) and further developed in [5]; our exposition is in the spirit of these papers. Some further bibliographical remarks can be found in the beginning of Chapter 1.
The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah–Patodi–Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces.
Contents: Introduction 1 Operators with the transmission property 1.1 Operators on a manifold with boundary 1.2 Conditions with pseudodifferential projections 1.3 Projections and Fredholm families 2 Boundary value problems not requiring the transmission property 2.1 Interior operators 2.2 Edge amplitude functions 2.3 Boundary value problems 3 Operators with global projection conditions 3.1 Construction for boundary symbols 3.2 Ellipticity of boundary value problems with projection data 3.3 Operators of order zero
Pseudodifferential analysis on manifolds with boundary - a comparison of b-calculus and cone algebra
(1999)
We establish a relation between two different approaches to a complete pseudodifferential analysis of totally characteristic or Fuchs type operators on compact manifolds with boundary respectively conical singularities: Melrose's (overblown) b-calculus and Schulze's cone algebra. Though quite different in their definition, we show that these two pseudodifferential calculi basically contain the same 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.
Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges.
Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order of the singularity and σk operator-valued for k ≥ 1. The symbols determine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give examples and also comment on new challenges and interesting problems of the recent development.
We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.
In this paper, we study the existence of positive solutions of a one-parameter family of logistic equations on R+ or on R. These equations are stationary versions of the Fisher equations and the KPP equations. We also study the blow up region of a sequence of the solutions when the parameter approachs a critical value and the nonexistence of positive solutions beyond the critical value. We use the direct method and the sub and super solution method.
Aus dem Inhalt: Inhaltsverzeichnis Abbildungsverzeichnis Tabellenverzeichnis 1 Einleitung und Motivation 2 Multivariate Copulafunktionen 2.1 Einleitung 2.2 Satz von Sklar 2.3 Eigenschaften von Copulafunktionen 3 Abhängigkeitskonzepte 3.1 Lineare Korrelation 3.2 Copulabasierte Abhängigkeitsmaße 3.2.1 Konkordanz 3.2.2 Kendall’s und Spearman’s 3.2.3 Asymptotische Randabhängigkeit 4 Elliptische Copulaklasse 4.1 Sphärische und elliptische Verteilungen 4.2 Normal-Copula 4.3 t-Copula 5 Parametrische Schätzverfahren 5.1 Maximum-Likelihood-Methode 5.1.1 ExakteMaximum-Likelihood-Methode 5.1.2 2-stufige parametrische Maximum-Likelihood-Methode 5.1.3 2-stufige semiparametrische Maximum-Likelihood-Methode 5.2 Momentenmethode 5.3 Kendall’s -Momentenmethode 6 Parameterschätzungen für Normal- und t-Copula 6.1 Normal-Copula 6.1.1 Maximum-Likelihood-Methode 6.1.2 Momentenmethode 6.1.3 Kendall’s Momentenmethode 6.1.4 Spearman’s Momentenmethode 6.2 t-Copula 6.2.1 Verfahren 1 (exakte ML-Methode) 6.2.2 Verfahren 2 (2-stufige rekursive ML-Methode) 6.2.3 Verfahren 3 (2-stufige KM-ML-Methode) 6.2.4 Verfahren 4 (3-stufige M-ML-Methode) 7 Simulationen 7.1 Grundlagen 7.2 Parametrischer Fall 7.3 Nichtparametrischer Fall 7.4 Fazit A Programmausschnitt Literaturverzeichnis
As is known from Kondratyev's work, boundary value problems for elliptic operators on a manifold with conical singularities and boundary are controlled by a principal symbolic hierarchy, where the conormal symbols belong to the typical new components, compared with the smooth case, with interior and boundary symbols. A similar picture may be expected on manifolds with corners when the base of the cone itself is a manifold with conical or edge singularities. This is a natural situation in a number of applications, though with essential new difficulties. We investigate here corresponding conormal symbols in terms of a calculus of holomorphic parameter-dependent edge boundary value problems on the base. We show that a certain kernel cut-off procedure generates all such holomorphic families, modulo smoothing elements, and we establish conormal symbols as an algebra as is necessary for a parametrix constructions in the elliptic case.
We develop an approach to the problem of optimal recovery of continuous linear functionals in Banach spaces through information on a finite number of given functionals. The results obtained are applied to the problem of the best analytic continuation from a finite set in the complex space Cn, n ≥ 1, for classes of entire functions of exponential type which belong to the space Lp, 1 < p < 1, on the real subspace of Cn. These latter are known as Wiener classes.
Peter Jones' theorem on the factorization of Ap weights is sharpened for weights with bounds near 1, allowing the factorization to be performed continuously near the limiting, unweighted case. When 1 < p < infinite and omega is an Ap weight with bound Ap(omega) = 1 + epsilon, it is shown that there exist Asub1 weights u, v such that both the formula omega = uv(1-p) and the estimates A1 (u), A1 (v) = 1 + Omikron (√epsilon) hold. The square root in these estimates is also proven to be the correct asymptotic power as epsilon -> 0.
Contents: Introduction 1 Edge calculus with parameters 1.1 Cone asymptotics and Green symbols 1.2 Mellin edge symbols 1.3 The edge symbol algebra 1.4 Operators on a manifold with edges 2 Corner symbols and iterated asymptotics 2.1 Holomorphic corner symbols 2.2 Meromorphic corner symbols and ellipicity 2.3 Weighted corner Sobolev spaces 2.4 Iterated asymptotics 3 The edge corner algebra with trace and potential conditions 3.1 Green corner operators 3.2 Smoothing Mellin corner operators 3.3 The edge corner algebra 3.4 Ellipicity and regularity with asymptotics 3.5 Examples and remarks
We establish a new calculus of pseudodifferential operators on a manifold with smooth edges and study ellipticity with extra trace and potential conditions (as well as Green operators) at the edge. In contrast to the known scenario with conditions of that kind in integral form we admit in this paper ‘singular’ trace, potential and Green operators, which are related to the corresponding operators of positive type in Boutet de Monvel’s calculus for boundary value problems.
We establish elements of a new approch to ellipticity and parametrices within operator algebras on a manifold with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaces. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The “full” calculus is voluminous; so we content ourselves here with some typical aspects such as symbols in terms of order reducing families, classes of relevant examples, and operators near the conical exit to infinity.
We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.