Lifting quasianalytic mappings over invariants

Let $\rho : G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb{C}[V]^G$. We study the problem of lifting mappings $f : \mathbb{R}^q \supseteq U \to \sigma(V) \subseteq \mathbb{C}^n$ over the mapping of invariants $\sigma=(\sigma_1,\sigma_n) : V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that, if $f$ belongs to a quasianalytic subclass $\mathcal{C} \subseteq C^\infty$ satisfying some mild closedness properties which guarantee resolution of singularities in $\mathcal{C}$ (e.g.\ the real analytic class), then $f$ admits a lift of the same class $\mathcal{C}$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift which belongs to $SBV_{\operatorname{loc}}$ (i.e.\ special functions of bounded variation). If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.Comment: 17 pages, 1 table, minor corrections, to appear in Canad. J. Mat

Differentiable perturbation of unbounded operators

If $A(t)$ is a C^{1,\al}-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized $C^1$ in $t$. If $A$ is $C^\infty$ then the eigenvalues can be parameterized twice differentiable.Comment: amstex 9 pages. Some misprints correcte

Tensor fields and connections on holomorphic orbit spaces of finite groups

For a representation of a finite group $G$ on a complex vector space $V$ we determine when a holomorphic $\binom{p}{q}$-tensor field on the principle stratum of the orbit space $V/G$ can be lifted to a holomorphic $G$-invariant tensor field on $V$. This extends also to connections. As a consequence we determine those holomorphic diffeomorphisms on $V/G$ which can be lifted to orbit preserving holomorphic diffeomorphisms on $V$. This in turn is applied to characterize complex orbifolds.Comment: 15 pages, LaTeX, some arguments rearrange

The Convenient Setting for Denjoy--Carleman Differentiable Mappings of Beurling and Roumieu Type

We prove in a uniform way that all Denjoy--Carleman differentiable function classes of Beurling type $C^{(M)}$ and of Roumieu type $C^{\{M\}}$, admit a convenient setting if the weight sequence $M=(M_k)$ is log-convex and of moderate growth: For $\mathcal C$ denoting either $C^{(M)}$ or $C^{\{M\}}$, the category of $\mathcal C$-mappings is cartesian closed in the sense that $\mathcal C(E,\mathcal C(F,G))\cong \mathcal C(E\times F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $\mathcal C$-diffeomorphisms is a regular $\mathcal C$-Lie group if $\mathcal C \supseteq C^\omega$, but not better.Comment: 41 pages; numbering changed in order to make it consistent with the published version; to appear in Rev. Mat. Complut. arXiv admin note: text overlap with arXiv:0909.563

Regular infinite dimensional Lie groups

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply connected and the image group is regular.Comment: AmSTeX, using diag.tex with fonts lams?.ps, 38 page

PocketGraph : graph representation of binding site volumes

The representation of small molecules as molecular graphs is a common technique in various fields of cheminformatics. This approach employs abstract descriptions of topology and properties for rapid analyses and comparison. Receptor-based methods in contrast mostly depend on more complex representations impeding simplified analysis and limiting the possibilities of property assignment. In this study we demonstrate that ligand-based methods can be applied to receptor-derived binding site analysis. We introduce the new method PocketGraph that translates representations of binding site volumes into linear graphs and enables the application of graph-based methods to the world of protein pockets. The method uses the PocketPicker algorithm for characterization of binding site volumes and employs a Growing Neural Gas procedure to derive graph representations of pocket topologies. Self-organizing map (SOM) projections revealed a limited number of pocket topologies. We argue that there is only a small set of pocket shapes realized in the known ligand-receptor complexes