48,189 research outputs found
A description based on Schubert classes of cohomology of flag manifolds
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
We address some global solvability issues for classes of smooth nonsingular
vector fields in the plane related to cohomological equations in
geometry and dynamical systems. The first main result is that is not
surjective in iff the geometrical condition -- the existence
of separatrix strips -- holds. Next, for nonsurjective vector fields, we
demonstrate that if the RHS has at most infra-exponential growth in the
separatrix strips we can find a global weak solution 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
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,
is a self-adjoint operator, and 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
In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp
estimates for were obtained for self-adjoint operators and
and for various classes of functions on the real line . In this paper
we extend those results to the case of functions of normal operators. We show
that if a function belongs to the H\"older class \L_\a(\R^2), 0<\a<1,
of functions of two variables, and and 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 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 in the case when f\in\L_\a(\R^2) and
belongs to the Schatten-von Neuman class \bS_p.Comment: 32 page
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