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Generalized eigenvectors for resonances in the Friedrichs model and their associated Gamov vectors
(2006)
A Gelfand triplet for the Hamiltonian H of the Priedrichs model on R with multiplicity space K, dim K < infinity, is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are (generalized) eigenvalues of H. The corresponding eigen(anti)linear forms are calculated explicitly. Using the wave matrices for the wave (Moller) operators the corresponding eigen(anti)linear forms on the Schwartz space S for the unperturbed Hamiltonian Ho are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector lambda -> k/(zeta(0)-lambda)(-1), zeta(0) resonance, k epsilon K, which is uniquely determined by restriction of S to S boolean AND H-+(2), where H-+(2) denotes the Hardy space of the upper half-plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup for t >= 0 of the Toeplitz type on H-+(2). That is: Exactly those pre-Gamov vectors a lambda -> k/(zeta-lambda)(-1), ( from the lower half-plane, k epsilon K., have an extension to a generalized eigenvector of H if zeta is a resonance and if k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinear form
Finding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new linear method to identify the '' non-Gaussian subspace '' within a very general semi-parametric framework. Our proposed method, called NGCA (non-Gaussian component analysis), is based on a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector belonging to the low dimensional non-Gaussian target subspace, up to an estimation error. By applying this operator to a family of different nonlinear functions, one obtains a family of different vectors lying in a vicinity of the target space. As a final step, the target space itself is estimated by applying PCA to this family of vectors. We show that this procedure is consistent in the sense that the estimaton error tends to zero at a parametric rate, uniformly over the family, Numerical examples demonstrate the usefulness of our method
We prove the existence of a class of local in time solutions, including static solutions, of the Einstein-Euler system. This result is the relativistic generalisation of a similar result for the Euler-Poisson system obtained by Gamblin [8]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article.
Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n − SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V −iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions.
In this paper we present an inversion algorithm for ill-posed problems arising in atmospheric remote sensing. The proposed method is an iterative Runge-Kutta type regularization method. Those methods are better well known for solving differential equations. We adapted them for solving inverse ill-posed problems. The numerical performances of the algorithm are studied by means of simulations concerning the retrieval of aerosol particle size distributions from lidar observations.
Hypersubstitutions were introduced in [3] as a way of making precise the concepts of hyperidentity and M- hyperidentity. The monoid of hypersubstitutions has been widely studied by many authors. Knowledge of the monoid of hypersubstitutions can be applied to the concept of M-hyperidentities. In this paper, we show that the order of hypersubstitutions of type tau = (3) is 1, 2, 3 or infinite
Elliptic equations on configurations W = W-1 boolean OR (. . .) boolean OR W-N with edge Y and components W-j of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here edges. Starting from edge-degenerate operators on Wj, j = 1, . . . , N, we construct an algebra with extra 'transmission' conditions on Y that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator- valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on WY. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics
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
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.
Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multihypersubstitutions and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice.
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 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.
Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still largely missing. In this paper, we study linear multisymplectic PDEs and their discretization by means of numerical dispersion relations. It is found that multisymplectic methods in the sense of Bridges and Reich [Phys. Lett. A, 284 ( 2001), pp. 184-193] and Reich [J. Comput. Phys., 157 (2000), pp. 473-499], such as Gauss-Legendre Runge-Kutta methods, possess a number of desirable properties such as nonexistence of spurious roots and conservation of the sign of the group velocity. A certain CFL-type restriction on Delta t/Delta x might be required for methods higher than second order in time. It is also demonstrated by means of the explicit midpoint method that multistep methods may exhibit spurious roots in the numerical dispersion relation for any value of Delta t/Delta x despite being multisymplectic in the sense of discrete variational mechanics [J. E. Marsden, G. P. Patrick, and S. Shkoller, Commun. Math. Phys., 199 (1999), pp. 351-395]
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.
Symplectic geometry, wigner-weyl-moyal calculus, and quantum mechanics, in phase space ; Part 1
(2006)
Symplectic geometry, wigner-weyl-moyal calculus, and quantum mechanics, in phase space ; Part 2
(2006)
Symplectic geometry, wigner-weyl-moyal calculus, and quantum mechanics, in phase space ; Part 3
(2006)
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
The aim of this paper is to express the Conley-Zehnder index of a symplectic path in terms of an index due to Leray and which has been studied by one of us in a previous work. This will allow us to prove a formula for the Conley-Zehnder index of the product of two symplectic paths in terms of a symplectic Cayley transform. We apply our results to a rigorous study of the Weyl representation of metaplectic operators, which plays a crucial role in the understanding of semiclassical quantization of Hamiltonian systems exhibiting chaotic behavior.
We study pseudo-differential operators on a cylinder R x B where B has conical singularities. Configurations of that kind are the local model of corner singularities with cross section B. Operators in our calculus are assumed to have symbols a which are meromorphic in the complex covariable with values in the algebra of all cone operators on B. We show an explicit formula for solutions of the homogeneous equation if a is independent of the axial variable t is an element of R. Each non-bijectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula
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 subset of 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 Bull. Sci. Math. ( to appear). 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.
We study elliptic boundary value problems in a wedge with additional edge conditions of trace and potential type. We compute the (difference of the) number of such conditions in terms of the Fredholm index of the principal edge symbol. The task will be reduced to the case of special opening angles, together with a homotopy argument.
Semiclassical asymptotics for the scattering amplitude in the presence of focal points at infinity
(2006)
Semiclassical asymptotics for the scattering amplitude in the presence of focal points at infinity
(2006)
We consider scattering in $\R^n$, $n\ge 2$, described by the Schr\"odinger operator $P(h)=-h^2\Delta+V$, where $V$ is a short-range potential. With the aid of Maslov theory, we give a geometrical formula for the semiclassical asymptotics as $h\to 0$ of the scattering amplitude $f(\omega_-,\omega_+;\lambda,h)$ $\omega_+\neq\omega_-$) which remains valid in the presence of focal points at infinity (caustics). Crucial for this analysis are precise estimates on the asymptotics of the classical phase trajectories and the relationship between caustics in euclidean phase space and caustics at infinity.
This paper is concerned with localization properties of coherent states. Instead of classical uncertainty relations we consider "generalized" localization quantities. This is done by introducing measures on the reproducing kernel. In this context we may prove the existence of optimally localized states. Moreover, we provide a numerical scheme for deriving them.
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.
On null quadrature domains
(2006)
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.
In this study we present iterative regularization methods using rational approximations, in particular, Pade approximants, which work well for ill-posed problems. We prove that the (k,j)-Pade method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Pade method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Bockmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385-397] where this method was treated to solve ill-conditioned equation systems. (c) 2006 Elsevier Inc. All rights reserved.
We consider quasicomplexes of Boutet de Monvel operators in Sobolev spaces on a smooth compact manifold with boundary. To each quasicomplex we associate two complexes of symbols. One complex is defined on the cotangent bundle of the manifold and the other on that of the boundary. The quasicomplex is elliptic if these symbol complexes are exact away from the zero sections. We prove that elliptic quasicomplexes are Fredholm. As a consequence of this result we deduce that a compatibility complex for an overdetermined elliptic boundary problem operator is also Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes of Boutet de Monvel operators.
Since 1971, the Freudenthal Institute has developed an approach to mathematics education named Realistic Mathematics Education (RME). The philosophy of RME is based on Hans Freudenthal’s concept of ‘mathematics as a human activity’. Prof. Hans Freudenthal (1905-1990), a mathematician and educator, believes that ‘ready-made mathematics’ should not be taught in school. By contrast, he urges that students should be offered ‘realistic situations’ so that they can rediscover from informal to formal mathematics. Although mathematics education in Vietnam has some achievements, it still encounters several challenges. Recently, the reform of teaching methods has become an urgent task in Vietnam. It appears that Vietnamese mathematics education lacks necessary theoretical frameworks. At first sight, the philosophy of RME is suitable for the orientation of the teaching method reform in Vietnam. However, the potential of RME for mathematics education as well as the ability of applying RME to teaching mathematics is still questionable in Vietnam. The primary aim of this dissertation is to research into abilities of applying RME to teaching and learning mathematics in Vietnam and to answer the question “how could RME enrich Vietnamese mathematics education?”. This research will emphasize teaching geometry in Vietnamese middle school. More specifically, the dissertation will implement the following research tasks: • Analyzing the characteristics of Vietnamese mathematics education in the ‘reformed’ period (from the early 1980s to the early 2000s) and at present; • Implementing a survey of 152 middle school teachers’ ideas from several Vietnamese provinces and cities about Vietnamese mathematics education; • Analyzing RME, including Freudenthal’s viewpoints for RME and the characteristics of RME; • Discussing how to design RME-based lessons and how to apply these lessons to teaching and learning in Vietnam; • Experimenting RME-based lessons in a Vietnamese middle school; • Analyzing the feedback from the students’ worksheets and the teachers’ reports, including the potentials of RME-based lessons for Vietnamese middle school and the difficulties the teachers and their students encountered with RME-based lessons; • Discussing proposals for applying RME-based lessons to teaching and learning mathematics in Vietnam, including making suggestions for teachers who will apply these lessons to their teaching and designing courses for in-service teachers and teachers-in training. This research reveals that although teachers and students may encounter some obstacles while teaching and learning with RME-based lesson, RME could become a potential approach for mathematics education and could be effectively applied to teaching and learning mathematics in Vietnamese school.
In this paper, we give a complete classification of all finite simple groups with maximal subgroups of index n, where n = 2(a)center dot 3b for a, b >= 1. As a consequence, for such n, all primitive permutation groups of degree n are given. The motivation of this work comes also from a study of Cayley graphs of certain valency on a finite simple group
We consider the problem of testing whether the density of a mul- tivariate random variable can be expressed by a prespecified copula function and the marginal densities. The proposed test procedure is based on the asymptotic normality of the properly standardized integrated squared distance between a multivariate kernel density estimator and an estimator of its expectation under the hypothesis. The test of independence is a special case of this 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.