Results 1  10
of
14
Perturbation of complex polynomials and normal operators
 Math. Nach
"... Abstract. We study the regularity of the roots of complex monic polynomials P(t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C ∞ curve P(t) (which always exists) is locally absolutely continuous. Generic means tha ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
Abstract. We study the regularity of the roots of complex monic polynomials P(t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C ∞ curve P(t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t0 there exists a positive integer N such that t ↦ → P(t0 ± (t − t0) N) admits smooth parameterizations of its roots near t0. We show that C n curves P(t) (where n = deg P) admit differentiable roots if and only if the order of contact of the roots is ≥ 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C ∞ curve of such operators can be arranged locally in an absolutely continuous way. 1.
CHOOSING ROOTS OF POLYNOMIALS WITH SYMMETRIES SMOOTHLY
, 2006
"... The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not C 1,α for any α> 0. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined s ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not C 1,α for any α> 0. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined sufficient conditions for smooth solvability if the polynomials have certain symmetries. In general a C 3n curve of hyperbolic polynomials of degree n admits twice differentiable parameterizations of its roots. If the polynomials have certain symmetries we are able to weaken the assumptions in that statement.
Lifting mappings over invariants of finite groups
"... Abstract. We characterize those regular, holomorphic or formal maps into the orbit space V/G of a complex representation of a finite group G which admit a regular, holomorphic or formal lift to the representation space V. In particular, the case of complex reflection groups is investigated. 1. ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. We characterize those regular, holomorphic or formal maps into the orbit space V/G of a complex representation of a finite group G which admit a regular, holomorphic or formal lift to the representation space V. In particular, the case of complex reflection groups is investigated. 1.
REFLECTION GROUPS ON RIEMANNIAN MANIFOLDS
, 2003
"... We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the the orbit space M/G is isometric to a Weyl chamber C which is a ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We investigate discrete groups G of isometries of a complete connected Riemannian manifold M which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the the orbit space M/G is isometric to a Weyl chamber C which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of istropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds.
ADDENDUM TO: “LIFTING SMOOTH CURVES OVER INVARIANTS FOR REPRESENTATIONS OF COMPACT LIE
"... Abstract. We improve the main results in [9] using a recent refinement of Bronshtein’s theorem [4] due to Colombini, Orrú, and Pernazza [5]. They are then in general best possible both in the hypothesis and in the outcome. A recent refinement of Bronshtein’s theorem [4] due to Colombini, Orrú, and P ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We improve the main results in [9] using a recent refinement of Bronshtein’s theorem [4] due to Colombini, Orrú, and Pernazza [5]. They are then in general best possible both in the hypothesis and in the outcome. A recent refinement of Bronshtein’s theorem [4] due to Colombini, Orrú, and Pernazza [5] (namely theorem 1(i) below) allows to essentially improve our main results in [9]. That means we conclude theorem 2 and corollary 3. The improvement consists in weakening the hypothesis considerably: In [9] we needed a curve c to be of class (i) C k in order to admit a differentiable lift with locally bounded derivative, (ii) C k+d in order to admit a C 1lift, and (iii) C k+2d in order to admit a twice differentiable lift. It turns out that theorem 2 and corollary 3 are in general best possible both in the hypothesis and in the outcome. Refinement of Bronshtein’s theorem. Bronshtein’s theorem [4] (see also Wakabayashi’s version [12]) states that, for a curve of monic hyperbolic polynomials
A GENERALIZATION OF PUISEUX’S THEOREM AND LIFTING CURVES OVER INVARIANTS
"... Abstract. Let ρ: G → GL(V) be a rational representation of a reductive linear algebraic group G defined over C on a finite dimensional complex vector space V. We show that, for any generic smooth (resp. C M) curve c: R → V /G in the categorical quotient V /G (viewed as affine variety in some C n) an ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. Let ρ: G → GL(V) be a rational representation of a reductive linear algebraic group G defined over C on a finite dimensional complex vector space V. We show that, for any generic smooth (resp. C M) curve c: R → V /G in the categorical quotient V /G (viewed as affine variety in some C n) and for any t0 ∈ R, there exists a positive integer N such that t ↦ → c(t0 ± (t − t0) N) allows a smooth (resp. C M) lift to the representation space near t0. (C M denotes the Denjoy–Carleman class associated with M = (Mk), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V /G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C ∞ curve which represents a lift of a germ of a quasianalytic C M curve in V /G is actually C M. There are applications to polar representations.
DIFFERENTIAL GEOMETRY AND LIE GROUPS APPLICATION FOR “INITIATIVKOLLEG DER UNIVERSITÄT WIEN”
"... Abstract. The disciplines covered by the proposed Initiativkolleg include differential geometry, groups of symmetries, (nonlinear) PDEs, singularities, and mathematical relativity. The local expertise together with the obvious synergies between these fields has the potential of creating a group of ..."
Abstract
 Add to MetaCart
Abstract. The disciplines covered by the proposed Initiativkolleg include differential geometry, groups of symmetries, (nonlinear) PDEs, singularities, and mathematical relativity. The local expertise together with the obvious synergies between these fields has the potential of creating a group of students who have close interaction both among themselves and with the existing research groups. The organizers ’ scientific connections allow international cosupervision of the PhD theses. The Initiativkolleg would also provide students with the opportunity to gain experience in scientific presentation, to visit international conferences and establish contacts with distinguished scientists. 1. Description of mathematical subfields and collaboration between them We describe several mathematical subfields, list the scientists working in them or interested, beginning with the main investigators of these subfields. Citations solely made of numbers ([1], [2], etc.) refer to the references at the end of this proposal. If they are preceeded by a letter ([Bu1], [C1], etc.) they refer to the literature in the corrsponding individual curriculum vitae (CV) attached to this document.
LIE THEORY AND APPLICATIONS. III
, 2008
"... This project is planned as a continuation of the project P 17108N04 which was finished in September 2007. The research in the new project will follow different lines which are explained below. Papers cited as [Mxy] can be found (fulltext) via the homepage of Peter ..."
Abstract
 Add to MetaCart
This project is planned as a continuation of the project P 17108N04 which was finished in September 2007. The research in the new project will follow different lines which are explained below. Papers cited as [Mxy] can be found (fulltext) via the homepage of Peter
Wirkungen von Lie Algebren und Lie Gruppen
"... The aim of this project was to study actions of Lie algebras and Lie groups on manifolds. In particular, I wanted to prove the following result. Conjecture. Let G be a compact Lie group which acts isometrically on a complete Riemannian manifold M, such that this action admits a section: There exists ..."
Abstract
 Add to MetaCart
The aim of this project was to study actions of Lie algebras and Lie groups on manifolds. In particular, I wanted to prove the following result. Conjecture. Let G be a compact Lie group which acts isometrically on a complete Riemannian manifold M, such that this action admits a section: There exists a closed submanifold Σ in M which meets each orbit orthogonally. Then the space Ωhor(M) G of all differential forms on M which are invariant under the Gaction and are horizontal in the sense that they kill each vector which is tangent to an orbit, is isomorphic to the space Ω(Σ) W (Σ) of all differential forms on the section Σ, which are invariant under the action of the Weyl group W (Σ) on Σ, where