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Contents: Chapter 4: Pseudodifferential Operators 4.1. Preliminary Remarks 4.1.1. Why are pseudodifferential operators needed? 4.1.2. What is a pseudodifferential operator? 4.1.3. What properties should the pseudodifferential calculus possess? 4.2. Classical Pseudodifferential Operators on Smooth Manifolds 4.2.1. Definition of pseudodifferential operators on a manifold 4.2.2. Hörmander’s definition of pseudodifferential operators 4.2.3. Basic properties of pseudodifferential operators 4.3. Pseudodifferential Operators in Sections of Hilbert Bundles 4.3.1. Hilbert bundles 4.3.2. Operator-valued symbols. Specific features of the infinite-dimensional case 4.3.3. Symbols of compact fiber variation 4.3.4. Definition of pseudodifferential operators 4.3.5. The composition theorem 4.3.6. Ellipticity 4.3.7. The finiteness theorem 4.4. The Index Theorem 4.4.1. The Atiyah–Singer index theorem 4.4.2. The index theorem for pseudodifferential operators in sections of Hilbert bundles 4.4.3. Proof of the index theorem 4.5. Bibliographical Remarks
Contents: Chapter 3: Eta Invariant and the Spectral Flow 3.1. Introduction 3.2. The Classical Spectral Flow 3.2.1. Definition and main properties 3.2.2. The spectral flow formula for periodic families 3.3. The Atiyah–Patodi–Singer Eta Invariant 3.3.1. Definition of the eta invariant 3.3.2. Variation under deformations of the operator 3.3.3. Homotopy invariance. Examples 3.4. The Eta Invariant of Families with Parameter (Melrose’s Theory) 3.4.1. A trace on the algebra of parameter-dependent operators 3.4.2. Definition of the Melrose eta invariant 3.4.3. Relationship with the Atiyah–Patodi–Singer eta invariant 3.4.4. Locality of the derivative of the eta invariant. Examples 3.5. The Spectral Flow of Families of Parameter-Dependent Operators 3.5.1. Meromorphic operator functions. Multiplicities of singular points 3.5.2. Definition of the spectral flow 3.6. Higher Spectral Flows 3.6.1. Spectral sections 3.6.2. Spectral flow of homotopies of families of self-adjoint operators 3.6.3. Spectral flow of homotopies of families of parameter-dependent operators 3.7. Bibliographical Remarks
We investigate crack problems, where the crack boundary has conical singularities. Elliptic operators with two-sided elliptc boundary conditions on the plus and minus sides of the crack will be interpreted as elements of a corner algebra of boundary value problems. The corresponding operators will be completed by extra edge conditions on the crack boundary to Fredholm operators in corner Sobolev spaces with double weights, and there are parametrices within the calculus.
We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator-valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator As in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of As as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices Ps of As, interpreted as Mellin-edge representations of P.
Green formulae for elliptic cone differential operators are established. This is achieved by an accurate description of the maximal domain of an elliptic cone differential operator and its formal adjoint; thereby utilizing the concept of a discrete asymptotic type. From this description, the singular coefficients replacing the boundary traces in classical Green formulas are deduced.
Contents: Chapter 5: Manifolds with Isolated Singularities 5.1. Differential Operators and the Geometry of Singularities 5.1.1. How do isolated singularities arise? Examples 5.1.2. Definition and methods for the description of manifolds with isolated singularities 5.1.3. Bundles. The cotangent bundle 5.2. Asymptotics of Solutions, Function Spaces,Conormal Symbols 5.2.1. Conical singularities 5.2.2. Cuspidal singularities 5.3. A Universal Representation of Degenerate Operators and the Finiteness Theorem 5.3.1. The cylindrical representation 5.3.2. Continuity and compactness 5.3.3. Ellipticity and the finiteness theorem 5.4. Calculus of ΨDO 5.4.1. General ΨDO 5.4.2. The subalgebra of stabilizing ΨDO 5.4.3. Ellipticity and the finiteness theorem
Acclimatization
(2003)
Together with their wives Otto and Richard Schomburgk arrived in Port Adelaide (South Australia) on August 16th 1849. The essay looks at how these two brothers, who had received their scientific training and promotion in the circle surrounding Alexander von Humboldt, reacted to the unfamiliar conditions in the young British colony. Some indication will be given as to the differences between the Schomburgk brothers treatment of the natural resources of the new colony and that of the English colonists of the time.
Results of an inter-laboratory round-robin study of the application of time-resolved emission spectroscopy (TRES) to the speciation of uranium(VI) in aqueous media are presented. The round-robin study involved 13 independent laboratories, using various instrumentation and data analysis methods. Samples were prepared based on appropriate speciation diagrams and, in general, were found to be chemically stable for at least six months. Four different types of aqueous uranyl solutions were studied: (1) acidic medium where UO22+aq is the single emitting species, (2) uranyl in the presence of fluoride ions, (3) uranyl in the presence of sulfate ions, and (4) uranyl in aqueous solutions at different pH, promoting the formation of hydrolyzed species. Results between the laboratories are compared in terms of the number of decay components, luminescence lifetimes, and spectral band positions. The successes and limitations of TRES in uranyl analysis and speciation in aqueous solutions are discussed.
Investigations with frequency domain photon density waves allow elucidation of absorption and scattering properties of turbid media. The temporal and spatial propagation of intensity modulated light with frequencies up to more than 1 GHz can be described by the P1 approximation to the Boltzmann transport equation. In this study, we establish requirements for the appropriate choice of turbid model media and characterize mixtures of isosulfan blue as absorber and polystyrene beads as scatterer. For these model media, the independent determination of absorption and reduced scattering coefficients over large absorber and scatterer concentration ranges is demonstrated with a frequency domain photon density wave spectrometer employing intensity and phase measurements at various modulation frequencies.
Content: I. The nature and form of international law 1. The acceptance of the existence of an international legal order 2. The legal position of the individual in international law II. Obligations of states in the protection of international human rights 1. Treaty-based human rights obligations 2. The nature of treaty-based human rights obligations 3. The ”absolute” and ”objective” character of human rights treaty obligations 4. Human rights conventions as self-contained regimes 5. The problem of characterisation of human rights obligations of states III. Human rights obligations arising from general principles of international law 1. Obligations erga omnes and human rights norms 2. The outlawing of genocide as obligation erga omnes 3. Protection from slavery as obligation erga omnes 4. The outlawing of acts of aggression as obligation erga omnes 5. Protection from racial discrimination as obligation erga omnes 6. The basic rights of the human person as obligation erga omnes 7. Jus Cogens and the search for peremptory norms of human rights 8. International crimes and human rights norms 9. The relationship between the concepts: erga omnes, jus cogens, international crime and human rights IV. International instruments for the coercive enforcement of state obligations to ‘respect and ensure’ human rights 1. Countermeasures as consequences of breach of treaties in international law 2. Application of reprisals for the enforcement of treaty-based human rights obligations 3. Intervention for the protection of human rights in international law 4. Intervention by the Security Council for the protection of human rights: the situation before the East-West détente 5. Humanitarian intervention after the end of the Cold War 6. The legal nature of ECOWAS intervention in the Liberian Civil War 7. The legality of NATO’s intervention in Kosovo 8. Some instances of intervention with mixed motives V. Non-forceful measures for the enforcement of states’ human rights obligations 1. Economic and financial pressure as means of enforcing states’ obligation to respect and observe human rights 2. The application of the clausula rebus sic stantibus for the protection of human rights 3. The enforcement of human rights through the World Bank 4. The enforcement of human rights through the ILO 5. Diplomatic recognition as an instrument for securing a state's respect and promotion of human rights 6. Refusal to comply with an extradition agreement as a means of enforcing a state’s human rights obligations 7. Denial of immunity as a means of enforcing a state’s human rights obligations 8. Publicity as an instrument for the enforcement of human rights VI. Judicial enforcement of state obligations to ‘respect and ensure’ human rights 1. Enforcement of human rights through International Criminal Tribunals 2. The International Criminal Tribunal for Yugoslavia 3. The International Criminal Tribunal for Rwanda 4. The International Special Court of Sierra Leone Résumé
We give a survey on procedures for testing functions which are based on quadratic deviation measures. The following problems are considered: Testing whether a density function lies in a parametric class of functions, whether continuous random variables are independent; testing cell probabilities and independence in sparse data sets; testing the parametric fit of a regression homoscedasticity in a regression model and testing the hazard rate in survival models with censoring and with and without covariates.
We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernal of this domain. The Lefschetz number is proved to be the sum of usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.
For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.
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.
Boundary value problems on manifolds with conical singularities or edges contain potential operators as well as trace and Green operators which play a similar role as the corresponding operators in (pseudo-differential) boundary value problems on a smooth manifold. There is then a specific asymptotic behaviour of these operators close to the singularities. We characterise potential operators in terms of actions of cone or edge pseudo-differential operators (in the neighbouring space) on densities supported by sbmanifolds which also have conical or edge singularities. As a byproduct we show the continuity of such potentials as continuous perators between cone or edge Sobolev spaces and subspaces with asymptotics.
Contents: Chapter 1: Localization (Surgery) in Elliptic Theory 1.1. The Index Locality Principle 1.1.1. What is locality? 1.1.2. A pilot example 1.1.3. Collar spaces 1.1.4. Elliptic operators 1.1.5. Surgery and the relative index theorem 1.2. Surgery in Index Theory on Smooth Manifolds 1.2.1. The Booß–Wojciechowski theorem 1.2.2. The Gromov–Lawson theorem 1.3. Surgery for Boundary Value Problems 1.3.1. Notation 1.3.2. General boundary value problems 1.3.3. A model boundary value problem on a cylinder 1.3.4. The Agranovich–Dynin theorem 1.3.5. The Agranovich theorem 1.3.6. Bojarski’s theorem and its generalizations 1.4. (Micro)localization in Lefschetz theory 1.4.1. The Lefschetz number 1.4.2. Localization and the contributions of singular points 1.4.3. The semiclassical method and microlocalization 1.4.4. The classical Atiyah–Bott–Lefschetz theorem
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.
Mixed elliptic boundary value problems are characterised by conditions which have a jump along an interface of codimension 1 on the boundary. We study such problems in weighted edge Sobolev spaces and show the Fredholm property and the existence of parametrices under additional conditions of trace and potential type on the interface. Our methods from the calculus of boundary value problems on a manifold with edges will be illustrated by the Zaremba problem and other mixed problems for the Laplace operator.
In this paper, by a new constructive method, the authors reprove the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate fields. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C¹ functions. In particular, the authors give the optimal control time of the system. Finally, a new application is also given.
Do we know the answer?
(2003)
In the recent literature there is a hypothesis that the human parser uses number and case information in different ways to resolve an initially incorrect case assignment. This paper investigates what role morphological case information plays during the parser’s detection of an ungrammaticality or its recognition that a reanalysis is necessary. First, we compare double nominative with double accusative ungrammaticalities in a word by word, speeded grammaticality task and in this way show that only double nominatives lead to a so-called ”illusion of grammaticality” (a low rate of ungrammaticality detection). This illusion was found to disappear when the second argument was realized by a pronoun rather than by a full definite determiner phrase, i.e. when the saliency of the second argument was increased. Thus, the accuracy in recognizing an ungrammaticality induced by the case feature of the second argument is dependent on the type of this argument. Furthermore, we found that the accuracy in detecting such case ungrammaticalities is distance sensitive insofar as a shorter distance leads to a higher accuracy. The results are taken as support for an ”expectationdriven” parse strategy in which the way the parser uses the information of a current input item depends on the expectation resulting from the parse carried out so far. By contrast, ”input-driven” parse strategies, such as the diagnosis model (Fodor & Inoue, 1999) are unable to explain the data presented here.
The present paper addresses a current view in the psycholinguistic literature that case exhibits processing properties distinct from those of other morphological features such as number (cf. Fodor & Inoue, 2000; Meng & Bader, 2000a/b). In a speeded-acceptability judgement experiment, we show that the low performance previously found for case in contrast to number violations is limited to nominative case, whereas violations involving accusative and dative are judged more accurately. The data thus do not support the proposal that case per se is associated with special properties (in contrast to other features such as number) in reanalysis processes. Rather, there are significant judgement differences between the object cases accusative and dative on the one hand and the subject nominative case on the other. This may be explained by the fact that nominative has a specific status in German (and many other languages) as a default case.
The present paper addresses a current view in the psycholinguistic literature that case exhibits processing properties distinct from those of other morphological features such as number (cf. Fodor & Inoue, 2000; Meng & Bader, 2000a/b). In a speeded-acceptability judgement experiment, we show that the low performance previously found for case in contrast to number violations is limited to nominative case, whereas violations involving accusative and dative are judged more accurately. The data thus do not support the proposal that case per se is associated with special properties (in contrast to other features such as number) in reanalysis processes. Rather, there are significant judgement differences between the object cases accusative and dative on the one hand and the subject nominative case on the other. This may be explained by the fact that nominative has a specific status in German (and many other languages) as a default case.
Counting Markedness
(2003)
This paper reports the results of a corpus investigation on case conflicts in German argument free relative constructions. We investigate how corpus frequencies reflect the relative markedness of free relative and correlative constructions, the relative markedness of different case conflict configurations, and the relative markedness of different conflict resolution strategies. Section 1 introduces the conception of markedness as used in Optimality Theory. Section 2 introduces the facts about German free relative clauses, and section 3 presents the results of the corpus study. By and large, markedness and frequency go hand in hand. However, configurations at the highest end of the markedness scale rarely show up in corpus data, and for the configuration at the lowest end we found an unexpected outcome: the more marked structure is preferred.
Computational models such as E-Z Reader and SWIFT are ideal theoretical tools to test quantitatively our current understanding of eye-movement control in reading. Here we present a mathematical analysis of word skipping in the E-Z Reader model by semianalytic methods, to highlight the differences in current modeling approaches. In E-Z Reader, the word identification system must outperform the oculomotor system to induce word skipping. In SWIFT, there is competition among words to be selected as a saccade target. We conclude that it is the question of competitors in the “game” of word skipping that must be solved in eye movement research.
We question the assumption of serial attention shifts and the assumption that saccade programs are initiated or canceled only after stage one of word identification. Evidence: (1) Fixation durations prior to skipped words are not consistently higher compared to those prior to nonskipped words. (2) Attentional modulation of microsaccade rate might occur after early visual processing. Saccades are probably triggered by attentional selection.
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 dependence between survival times and covariates is described e.g. by proportional hazard models. We consider partly parametric Cox models and discuss here the estimation of interesting parameters. We represent the ma- ximum likelihood approach and extend the results of Huang (1999) from linear to nonlinear parameters. Then we investigate the least squares esti- mation and formulate conditions for the a.s. boundedness and consistency of these estimators.