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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.
On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.
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.
Soit (A, H, F) un module de Fredholm p-sommable, où l'algèbre A = CT est engendrée par un groupe discret Gamma d'éléments unitaires de L(H) qui est de croissance polynomiale r. On construit alors un triplet spectral (A, H, D) sommabilité q pour tout q > p + r + 1 avec F = signD. Dans le cas où (A, H, F) est (p, infini)-sommable on obtient la (q, infini)-sommabilité de (A, H, D)pour tout q > p + r + 1.
We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C n with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology.
We consider a homogeneous pseudodifferential equation on a cylinder C = IR x X over a smooth compact closed manifold X whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators over X. When assuming the symbol to be independent on the variable t element IR, we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points since C can be thought of as a 'stretched' manifold with conical points at t = -infinite and t = infinite.
Linear and non-linear analogues of the Black-Scholes equation are derived when shocks can be described by a truncated Lévy process. A linear equation is derived under the perfect correlation assumption on returns for a derivative security and a stock, and its solutions for European put and call options are obtained. It is also shown that the solution violates the perfect correlation assumption unless a process is gaussian. Thus, for a family of truncated Lévy distributions, the perfect hedging is impossible even in the continuous time limit. A second linear analogue of the Black-Scholes equation is obtained by constructing a portfolio which eliminates fluctuations of the first order and assuming that the portfolio is risk-free; it is shown that this assumption fails unless a process is gaussian. It is shown that the di erence between solutions to the linear analogues of the Black-Scholes equations and solutions to the Black-Scholes equations are sizable. The equations and solutions can be written in a discretized approximate form which uses an observed probability distribution only. Non-linear analogues for the Black-Scholes equation are derived from the non-arbitrage condition, and approximate formulas for solutions of these equations are suggested. Assuming that a linear generalization of the Black-Scholes equation holds, we derive an explicit pricing formula for the perpetual American put option and produce numerical results which show that the difference between our result and the classical Merton's formula obtained for gaussian processes can be substantial. Our formula uses an observed distribution density, under very weak assumptions on the latter.
We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.
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.
Aller Anfang ist schwer
(1998)
Inhalt: Leitartikel Kongreß für Jüdische Studien - Sektion beim Deutschen Orientalistentag / Selbstkritik, Selbstverpflichtung oder Selbstzweck? / 16. DGfE-Kongreß, Der osteuropäische Chassidismus - neuere Forschungen / Schließung der Israelwissenschaft in der Humboldt-Universität / Einführung in das Fach Jüdische Studien / Notizen / Aus der jüdischen Welt / Veröffentlichungen unserer Mitglieder / Verbandsnachrichten
Disziplinäre Quergänge : (Un-)Möglichkeiten transdisziplinärer Frauen- und Geschlechterforschung
(1998)
Heft 2/1998 hat den thematischen Schwerpunkt der Transdisziplinarität in der Frauen- und Geschlechterforschung und dokumentiert die Beiträge sowie die Diskussion eines Workshops, der im Sommersemester 1998 an der Universität Potsdam im Rahmen der Planungen zu einem Magisternebenfach „Frauen- und Geschlechterstudien" stattgefunden hat. Beiträgerinnen sind u.a. Sabine Hark, Maike Baader, Beate Neumeier, Axeli Knapp, Silke Wenk, Ulrike Teubner. Frauen- und Geschlechterforschung hat wiederholt Interdisziplinarität reklamiert, um das Zugleich von Monotonie und Heterogenität der Reproduktion der Geschlechterhierarchie verstehen zu können. Aus den Einzeldisziplinen heraus waren Grenzgänge in andere Disziplinen geradezu notwendig, um das Dickicht der Geschlechterordnung, die Verknüpfungen zwischen symbolischen, strukturellen und individuellen Dimensionen von Geschlecht zu durchdringen. Der „Beziehungssinn" zwischen den Disziplinen wurde dabei allerdings selten gepflegt. Wie etwa die moderne Geschlechterordnung selbst zum Ordnungsprinzip und zur Modalität der Produktion wissenschaftlichen Wissens wurde, war allenfalls eine Randfrage. Der reflexive Blick auf die Prozesse der wechselseitigen Konstitution von Disziplingrenzen gerade durch interdisziplinäre Herangehensweisen blieb bisher weitgehend aus. In einer transdisziplinären Orientierung von Frauen- und Geschlechterstudien würde daher gerade die je fachspezifische Konstitution von Gegenständen, Methoden und disziplinären Grenzen sowie die durch sie bestimmten bzw. beschränkten Perspektiven zum Gegenstand, wenn es darum gehen soll, die überschneidenden Problemfelder, die sich aus der Perspektive der Geschlechterdifferenz als relevant erweisen, zwischen den Disziplinen zu bearbeiten. In einer transdisziplinären Perspektive also stünden die Disziplingrenzen selbst zur Disposition, Teil der Lehr- und Forschungspraxis wäre die Frage, wie verschiedene disziplinäre Zugänge die Objekte des Wissens konstruieren und was das für die möglichen Erkenntnisse bedeutet.