Slice theorem and orbit type stratification in infinite dimensions

We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification.     

Proper actions, fixed-point algebras and naturality in nonabelian duality

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C₀(X). We consider a category in which the objects are C^∗-dynamical systems (A,G,α) for which there is an equivariant homomorphism of (C₀(X),γ) into the multiplier algebra M(A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra Aα which is Morita equivalent to A_×α,rG. We show that the assignment (A,α)?Aα is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.     

A correspondence between a class of monoids and self-similar group actions I

We describe a correspondence between a class of left cancellative monoids and self-similar group actions in the sense of Nekrashevych et al. This correspondence originated in Perrot’s 1972 thesis, and developed the ideas to be found in Rees’ 1948 paper.     

On Orbits of Automorphism Groups

If G is a finite group and if A is a group of automorphisms of G whose fixed point subgroup is C _G (A) then every subgroup F of C _G (A) acts on the set of orbits of A in G. The peculiarities of this action are used here to derive several results on the number of orbits of A in an economical manner.Original Russian Text Copyright © 2005 Deaconescu M. and Walls G. L.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 533–537, May–June, 2005.     

Accessibility of the matrix Riccati differential equation

A known result on the classification of transitive Lie group actions on complex Grassmann manifolds is exploited to derive a necessary and sufficient accessibility criterion for the complex matrix differential Riccati equation. We treat both cases, the symmetric as well as the non-symmetric Riccati equation. Corresponding accessibility results for the real Riccati equation are also available, but not stated here. An application to the accessibility of generalized double bracket flows is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Locally Weyl Invariant Massless Bosonic and Fermionic Spin-1/2 Action in the (Y4, g) Space-Time

We search for a real bosonic and fermionic action in four dimensions which both remain invariant under local Weyl transformations in the presence of contortion and non-metricity tensor. In the presence of the non-metricity tensor the investigation is extended to (W _n, g) space-time while when the torsion is encountered we are restricted to the (U ₄, g) space-time. Our results hold in general for the (Y ₄, g) space-time and we also calculate extra contributions to the conformal gravity.     

Hamiltonian and Path Integral Quantization of the Conformally Gauge-Fixed Polyakov D1 Brane Action in the Presence of a Scalar Dilation Field

The conformally gauge-fixed Polyakov D1 brane action in the presence of a scalar dilaton field is seen to be a constrained system in the sense of Dirac. In the present work we study its Hamiltonian and path integral quantization in the instant-form of dynamics using the equal world-sheet time framework.     

Unsteady Flow to Wells Partially Penetrating in Two-Layered Aquifers

The unsteady flow of water to well in a layered aquifer with an interlayer flow is examined in this paper. The system studied comprises an aquifer consisting of two productive layers of finite thickness that are in a perfect hydraulic contact and a well which partially penetrates in one of the layers. Each layer is assumed to be homogeneous and isotropic and the water contained in the aquifer is of identical physical properties and small compressibility.The analytical solutions are derived for the case when the system is characterised by equality of hydraulic diffusivity of layers. These solutions give the results accurate enough for practical applications and allow to estimate the effects of partial penetration and contrast in parameters of formation on distribution of the flow potential both at large distances from wells and at the wells.The obtained solutions also provide a basis for predicting the interlayer flow arising from the performance of a pumping well as well as between a pair of wells which have an open interval located in different layers of the aquifer. Two cases have been analysed: (1) a pumping pair of wells which is used to prevent water inflow to the productive well, and (2) a pumping-injection pair of wells providing the demanded mixing of water from adjacent layers. Some examples of flow patterns and specially computed diagrams are given to illustrate the operation of such systems.