735 research outputs found
Perturbation theory for normal operators
Let be a -mapping with values unbounded
normal operators with common domain of definition and compact resolvent. Here
stands for , (real analytic),
(Denjoy--Carleman of Beurling or Roumieu type), (locally Lipschitz),
or . The parameter domain is either or or an infinite dimensional convenient vector space. We completely describe
the -dependence on of the eigenvalues and the eigenvectors of
. Thereby we extend previously known results for self-adjoint operators
to normal operators, partly improve them, and show that they are best possible.
For normal matrices we obtain partly stronger results.Comment: 32 pages, Remark 7.5 on m-sectorial operators added, accepted for
publication in Trans. Amer. Math. So
Non-commutative desingularization of determinantal varieties, I
We show that determinantal varieties defined by maximal minors of a generic
matrix have a non-commutative desingularization, in that we construct a maximal
Cohen-Macaulay module over such a variety whose endomorphism ring is
Cohen-Macaulay and has finite global dimension. In the case of the determinant
of a square matrix, this gives a non-commutative crepant resolution.Comment: 52 pages, 3 figures, all comments welcom
Desingularization of Implicit Analytic Differential Equations
The question of finding solutions to given implicit differential equations
(IDE) has been answered by several authors in the last few years, using
different approaches, in an algebraic and also a geometric setting. Many of
those results assume in one way or another that the subimmersion theorem can be
applied at several stages of the reduction algorithm, which, roughly speaking,
allows to reduce a given IDE to a collection of ODE depending on parameters.
The main purpose of the present paper is to improve some of the known results
by introducing at each stage of the reduction algorithm a desingularization of
the manifolds with singularities that may appear when the subimmersion theorem
cannot be applied. This can be done for analytic IDE by using some fundamental
results on subanalytic subsets and desingularization of closed subanalytic
subsets due mainly to Lojasiewicz, Hironaka, Gabrielov, Hardt, Bierstone,
Milman and Sussmann, among others. We will show how this approach helps to
understand the dynamics given by the Lagrange-D'Alembert-Poincare equations for
the symmetric elastic sphere.Comment: 50 page
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