A general method to construct invariant PDEs on homogeneous manifolds

Let [Formula: see text] be an [Formula: see text]-dimensional homogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula: see text] for hypersurfaces of [Formula: see text] (i.e., with [Formula: see text] independent variables and [Formula: see text] dependent one) is defined as a [Formula: see text]-invariant hypersurface [Formula: see text]. We describe a general method for constructing such invariant partial differential equations for [Formula: see text]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [Formula: see text] of the [Formula: see text]-prolonged action of [Formula: see text]. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space [Formula: see text] and in the conformal space [Formula: see text]. Our method works under some mild assumptions on the action of [Formula: see text], namely: A1) the group [Formula: see text] must have an open orbit in [Formula: see text], and A2) the stabilizer [Formula: see text] of the fiber [Formula: see text] must factorize via the group of translations of the fiber itself.

Additive actions on hyperquadrics of corank two

<abstract><p>For a projective variety $ X $ in $ {\mathbb{P}}^{m} $ of dimension $ n $, an additive action on $ X $ is an effective action of $ {\mathbb{G}}_{a}^{n} $ on $ {\mathbb{P}}^{m} $ such that $ X $ is $ {\mathbb{G}}_{a}^{n} $-invariant and the induced action on $ X $ has an open orbit. Arzhantsev and Popovskiy have classified additive actions on hyperquadrics of corank 0 or 1. In this paper, we give the classification of additive actions on hyperquadrics of corank 2 whose singularities are not fixed by the $ {\mathbb{G}}_{a}^{n} $-action.</p></abstract>

Additive actions on complete toric surfaces

By an additive action on an algebraic variety [Formula: see text] we mean a regular effective action [Formula: see text] with an open orbit of the commutative unipotent group [Formula: see text]. In this paper, we give a classification of additive actions on complete toric surfaces.

Photoionization microscopy of the Rydberg Rb atom under a continuous infrared radiation laser field

The photoionization microscopy of the Rydberg Rb atom exposed to a continuous infrared radiation laser field is investigated based on the semiclassical open orbit theory. In contrast to the photoionization of the Rydberg hydrogen atom, the ionic core-scattering effect plays an important role in the photoionization of the Rb atom. Due to the core-scattering effect and the laser field, the electron trajectories become chaotic. A huge number of ionization trajectories from the ionic source to the detector plane appear, which makes the oscillatory pattern in the electron probability distribution become much more complicated. The ρ–θ curve on the detector plane exhibits a self-similar fractal structure for the ionization trajectories of the Rydberg Rb atom in the laser field. Due to constructive and destructive quantum interference of different electron trajectories, a series of concentric rings appear in the photoionization microscopy interference patterns on the detector plane. The electron probability density distributions on the detector are found to be changed sensitively with the scaled electron energy and the laser wavelength. Even as the detector plane is located at a macroscopic distance from the photoionization source, the photoionization microscopy interference patterns can be observed clearly. These calculations may provide a valuable contribution to the actual experimental study of the photoionization microscopy of non-hydrogenic Rydberg atom in the laser field.

The Parameter Space of Orbits of a Maximal Compact Subgroup Acting on a Flag Manifold

The orbits of a real form G of a complex semisimple Lie group GC and those of the complexification KC of its maximal compact subgroup K acting on Z=GC/Q, a homogeneous, algebraic, GC-manifold, are finite. Consequently, there is an open G-orbit. Lower-dimensional orbits are on the boundary of the open orbit with the lowest dimensional one being closed. Induced action on the parameter space of certain compact geometric objects (cycles) related to the manifold in question has been characterized using duality relations between G- and KC-orbits in the case of an open G-orbit and more recently lower-dimensional G-orbits. We show that the parameter space associated with the unique closed G-orbit in Z agrees with that of the other orbits characterized as a certain explicitly defined universal domain.

Upper Half-plane in the Grassmanian Gr(n; 2n)

We investigate the complex geometry of a multidimensional generalization D(n) of the upper-half-plane, which is homogeneous relative the group G = SL(2n;R). For n > 1 it is the pseudo Hermitian symmet- ric space which is the open orbit of G = SL(2n;R) on the Grassmanian GrC(n; 2n) of n-dimensional subspaces of C2n. The basic element of the construction is a canonical covering of D(n) by maximal Stein submanifolds — horospherical tubes

The effect of the laser polarization direction on the detached electron flux of H− in external field near surface

By using the semi-classical open orbit theory, the effect of the laser polarization direction on the detached electron flux of H− in electric field near metal surface has been studied. The result suggests that the azimuth angle [Formula: see text] has little effect on the electron flux distribution. We find that the influence of polarization angle θL on electron flux distribution is more obvious than the influence of the azimuth angle [Formula: see text]. In other words, the effect of the polarization angle θL on electron flux distribution is more important. We hope that our study will provide a new understanding of the electron flux distribution of negative ion in external field and surfaces, and can be used to guide the future experiment research on the negative ion photo-detachment microscopy.

The local structure theorem for real spherical varieties

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q\times _{L}S$. Here $Q$ is a parabolic subgroup with Levi decomposition $L\ltimes U$, and $S$ is a homogeneous space for a quotient $D=L/L_{n}$ of $L$, where $L_{n}\subseteq L$ is normal, such that $D$ is compact modulo center.