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Trees and Valuation Rings
(2000)
We continue the investigation of the calculus of Fourier Integral Operators (FIOs) in the class of symbols with exit behaviour (SG symbols). Here we analyse what happens when one restricts the choice of amplitude and phase functions to the subclass of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG classes, has a "classical" counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1, 1); for such operators we refine the known results about the analogous problem for general SG hyperbolic operators. The material contained here will be used in a forthcoming paper to obtain a Weyl formula for a class of operators defined on manifolds with cylindrical ends, improving the results obtained in [9].
Solid varieties of semirings
(2000)
Hyperidentities and clones
(2000)
The theory of hyperidentities generalises the equational theory of universal algebras and is applicable in several fields of science, especially in computer sciences. This book presents the theory of hyperidentities and its relation to clone identities. The basic concept of hypersubstitution is used to introduce the monoid of hypersubstitutions, hyperidentities, M-hyperidentities, solid and M-solid varieties. This work integrates into a coherent framework many results scattered throughout the literature over the last eighteen years. In addition, the book contains some applications of hyperidentities to the functional completenes problem in multiple-valued logic. The general theory is also extended to partial algberas. The last chapter contains a list of exercises and open problems with suggestions of future work in this area of research.
Flux Tubes in Weyl Gravity
(2000)
Die vierte Dimension
(2000)
Approximation numbers of linear operators are a very useful tool in order to understand the structure and the numerical behaviour of the operators. In this paper, this concept is extended to polynomials on Banach spaces and the approximation numbers of diagonal polynomials are estimated. As a main tool the rank of polynomials as a graduation of finite type polynomials is introduced and studied.
We construct a new calculus of boundary value problems with the transmission property on a non-compact smooth manifold with boundary and conical exits to infinity. The symbols are classical both in covariables and variables. The operators are determined by principal symbol tuples modulo operators of lower orders and weights (such remainders are compact in weighted Sobolev spaces). We develop the concept of ellipticity, construct parametrices within the algebra and obtain the Fredholm property. For the existence of Shapiro-Lopatinskij elliptic boundary conditions to a given elliptic operator we prove an analogue of the Atiyah-Bott condition.
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.
Existence and semiclassical analysis of the total scattering cross-section for atom-ion collisaions
(2000)
In diesem Beitrag zum Sammelband MATHEMATIK -INTERDISZIPLINÄR wird zunächst der lange Weg von den frühen Bedürfnissen nach Messung über das Eudoxos-Archimedische Axiom bis hin zu HIBERTs Axiomen der Stetigkeit skizziert. Neben der Präzisierung der Euklidischen Raumvorstellung muss man sich in diesem Zusammenhang mit den Zweifeln an ihrer ausschließlichen Nutzung in den Anwendungen auseinandersetzen: Über die Begriffe des Hausdorffschen und des topologischen Raumes werden die Begriffe der C^r -Mannigfaltigkeit und des Riemannschen bzw. des pseudo-Riemannschen Raumes vorgestellt; somit sind die mathematischen Grundlagen der Speziellen und der Allgemeinen Relativitätstheorie von EINSTEIN begründet, wobei der Anlass - Konstanz der (Vakuum-) Lichtgeschwindigkeit nach MICHELSON - und der Beitrag von MINKOWSKI zur Geometrisierung der Physik gestreift wird. Die klassische nichteuklidische Geometrie von GAUSS, LOBACEVSKIJ und J. BOLYAI wird ebenso erwähnt wie die didaktisch begründete späte Behandlung der Stetigkeit in der Schule. Die schon für die klassische Differentialgeometrie wichtige dreimal stetige Differenzierbarkeit der betrachteten Funktionen ist Anlaß, das 5. Hilbertsche Problem "LIEs Begriff der kontinuierlichen Transformationsgruppe ohne die Annahme der Differenzierbarkeit der die Gruppe definierenden Funktionen" mit seiner positiven Lösung im 20. Jh. ebenso wie die Theorie der diskontinuierlichen oder gar schwach diskontinuierlichen Gruppen zu reflektieren.