The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics

We consider the class of Levi nondegenerate hypersurfaces $M$ in \bC^{n+1} that admit a local (CR transversal) embedding, near a point $p\in M$, into a standard nondegenerate hyperquadric in $\Bbb C^{N+1}$ with codimension $k:=N-n$ small compared to the CR dimension $n$ of $M$. We show that, for hypersurfaces in this class, there is a normal form (which is closely related to the embedding) such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric. We also show that if the signature of $M$ and that of the standard hyperquadric in \bC^{N+1} are the same, then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric

Holomorphic extension of smooth CR-mappings between real-analytic and real-algebraic CR-manifolds

We establish results on holomorphic extension of CR-mappings of class $C^\infty$ between a real-analytic CR-submanifold of \C^N and a real-algebraic CR-submanifold of \C^{N'}

Modeling of gyrosynchrotron radio emission pulsations produced by MHD loop oscillations in solar flares

A quantitative study of the observable radio signatures of the sausage, kink, and torsional MHD oscillation modes in flaring coronal loops is performed. Considering first non-zero order effect of these various MHD oscillation modes on the radio source parameters such as magnetic field, line of sight, plasma density and temperature, electron distribution function, and the source dimensions, we compute time dependent radio emission (spectra and light curves). The radio light curves (of both flux density and degree of polarization) at all considered radio frequencies are than quantified in both time domain (via computation of the full modulation amplitude as a function of frequency) and in Fourier domain (oscillation spectra, phases, and partial modulation amplitude) to form the signatures specific to a particular oscillation mode and/or source parameter regime. We found that the parameter regime and the involved MHD mode can indeed be distinguished using the quantitative measures derived in the modeling. We apply the developed approach to analyze radio burst recorded by Owens Valley Solar Array and report possible detection of the sausage mode oscillation in one (partly occulted) flare and kink or torsional oscillations in another flare.Comment: ApJ, accepte

Deformation of generic submanifolds in a complex manifold

This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of the Thom transversality theorem, as well as the stratification of real-algebraic sets. Optimal results on the order of nondegeneracy are given.Comment: 24 page

Sausage oscillations of coronal loops

Aims. Analytical theory predicts the existence of trapped global (or fundamental) sausage fast magnetoacoustic modes in thick and dense coronal loops only, with the periods estimated as the ratio of double the loop length and the Alfvén speed outside the loop. We extend this study to the leaking regime, considering global sausage modes of long loops with small density contrasts. Methods. Anti-symmetric fast magnetoacoustic perturbations (sausage, or m = 0 modes) of a low β plasma slab with the symmetric Epstein profile of plasma density are modelled numerically. Results. It was found that long loops with sufficiently small density contrast can support global sausage leaky modes of detectable quality. The periods of the leaky modes are found to be approximately determined by the loop length and the external Alfvén speed. If the loop length can be estimated from imaging observations, the observed period of this mode provides us with the information about the Alfvén speed outside the loop. For typical flaring coronal loops, the estimated periods of the global sausage modes are about 5−60 s