48,189 research outputs found

    A description based on Schubert classes of cohomology of flag manifolds

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    We describe the integral cohomology rings of the flag manifolds of types B_n, D_n, G_2 and F_4 in terms of their Schubert classes. The main tool is the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure. As an application, we compute the Chow rings of the corresponding complex algebraic groups, recovering thereby the results of R. Marlin.Comment: 25 pages, AMS-LaTeX; typos corrected, an error concerning the Schubert classes corrected, Remarks 4.4 and 4.8 adde

    Global Solvability in Functional Spaces for Smooth Nonsingular Vector Fields in the Plane

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    We address some global solvability issues for classes of smooth nonsingular vector fields LL in the plane related to cohomological equations Lu=fLu=f in geometry and dynamical systems. The first main result is that LL is not surjective in C∞(R2)C^\infty(\R^2) iff the geometrical condition -- the existence of separatrix strips -- holds. Next, for nonsurjective vector fields, we demonstrate that if the RHS ff has at most infra-exponential growth in the separatrix strips we can find a global weak solution Lloc1L^1_{loc} near the boundaries of the separatrix strips. Finally we investigate the global solvability for perturbations with zero order p.d.o. We provide examples showing that our estimates are sharp.Comment: 22 pages, 2 figures, submitted to the PDE volume of the proceedings of the ISAAC2009 conferenc

    Multiple operator integrals and higher operator derivatives

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    In this paper we consider the problem of the existence of higher derivatives of the function t\mapsto\f(A+tK), where \f is a function on the real line, AA is a self-adjoint operator, and KK is a bounded self-adjoint operator. We improve earlier results by Sten'kin. In order to do this, we give a new approach to multiple operator integrals. This approach improves the earlier approach given by Sten'kin. We also consider a similar problem for unitary operators.Comment: 24 page

    Functions of normal operators under perturbations

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    In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for f(A)−f(B)f(A)-f(B) were obtained for self-adjoint operators AA and BB and for various classes of functions ff on the real line R\R. In this paper we extend those results to the case of functions of normal operators. We show that if a function ff belongs to the H\"older class \L_\a(\R^2), 0<\a<1, of functions of two variables, and N1N_1 and N2N_2 are normal operators, then \|f(N_1)-f(N_2)\|\le\const\|f\|_{\L_\a}\|N_1-N_2\|^\a. We obtain a more general result for functions in the space \L_\o(\R^2)=\big\{f:~|f(\z_1)-f(\z_2)|\le\const\o(|\z_1-\z_2|)\big\} for an arbitrary modulus of continuity \o. We prove that if ff belongs to the Besov class B_{\be1}^1(\R^2), then it is operator Lipschitz, i.e., \|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|. We also study properties of f(N1)−f(N2)f(N_1)-f(N_2) in the case when f\in\L_\a(\R^2) and N1−N2N_1-N_2 belongs to the Schatten-von Neuman class \bS_p.Comment: 32 page
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