Holomorphic Deformations of Real-Analytic CR Maps and Analytic Regularity of CR Mappings

Let \(M\subset {\mathbb {C}}^N\) and \(M'\subset {\mathbb {C}}^{N'}\) be real-analytic CR submanifolds, with M minimal. We provide a new sufficient condition, that happens to be also essentially necessary, for all sufficiently smooth CR maps \(h:U\rightarrow M'\) defined on a connected open subset of M and of rank larger than a prescribed integer r to be real-analytic on a dense open subset of U. This condition corresponds to the nonexistence of nontrivial holomorphic deformations of germs of real-analytic CR mappings whose rank is larger than r. As a consequence, we obtain several new results about analyticity of CR mappings that, at the same time, generalize and unify a number of previous existing ones.     

Experimental evidence for multiple oxidation pathways in the (salen) Mn-catalyzed epoxidation of alkenes

The substrate electronic effects on the selectivity in the catalytic epoxidation of para-substituted cis stilbenes 2a-i were investigated by using (R,R)-[N,N'-bis(3,5-di-tBu-salicylidene)-1,2-cyclohexanediamine]manganese(III) chloride 1 in benzene as the catalyst with iodosobenzene as the terminal oxidant. A Hammett study of the selectivity results reveals a stronger electrophilic character than previously assumed in the (salen)Mn-catalyzed reaction. In general, the best correlations with the experimental values were obtained by using the Hammett sigma + values, which gave rho = -1.37 for the rate of cis-epoxide formation and rho = -0.43 for the rate of the stepwise process leading to the corresponding trans product. The reaction involves two separate pathways as indicated also by the competitive breakdown of the intermediate on the path to trans epoxide for methoxy-substituted substrates. The asynchronicity in the concerted pathway leading to cis epoxide is apparent for 4-methoxy-4'-nitrostilbene, which yields cis epoxide with 75% ee entirely as a result of electronic effects.     

Push-pull structures with a pyrazine core and hexatriene chain: synthesis and light-emitting properties

In this paper, we describe the synthesis of various push-pull molecules with a central pyrazine unit connected to a hexatriene chain terminated by various p-substituted phenyl groups. The key steps involve metallation and subsequent transmetallation of 2-chloro and 2,6-dichloropyrazine followed by a Negishi cross-coupling reaction of the intermediate organozinc derivative with (2E,4E)-5-bromopentadienal. The aldehydes are then submitted to a Wittig reaction with the appropriate phosphonium salts readily obtained from various substituted benzyl alcohols. The light-emitting properties of the so obtained molecules are then investigated in terms of absorption and emission spectra and non-linear optics experiments have been carried out in two cases.     

Parametrization of local CR automorphisms by finite jets and applications

For any real-analytic hypersurface , which does not contain any complex-analytic subvariety of positive dimension, we show that for every point the local real-analytic CR automorphisms of fixing can be parametrized real-analytically by their jets at . As a direct application, we derive a Lie group structure for the topological group . Furthermore, we also show that the order of the jet space in which the group embeds can be chosen to depend upper-semicontinuously on . As a first consequence, it follows that given any compact real-analytic hypersurface in , there exists an integer depending only on such that for every point germs at of CR diffeomorphisms mapping into another real-analytic hypersurface in are uniquely determined by their -jet at that point. Another consequence is the following boundary version of H. Cartan's uniqueness theorem: given any bounded domain with smooth real-analytic boundary, there exists an integer depending only on such that if is a proper holomorphic mapping extending smoothly up to near some point with the same -jet at with that of the identity mapping, then necessarily .

Our parametrization theorem also holds for the stability group of any essentially finite minimal real-analytic CR manifold of arbitrary codimension. One of the new main tools developed in the paper, which may be of independent interest, is a parametrization theorem for invertible solutions of a certain kind of singular analytic equations, which roughly speaking consists of inverting certain families of parametrized maps with singularities.

     

Remarks on the rank properties of formal CR maps

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Silicon fibers produced by high-pressure LCVD

Using a unique fiber-growth control mechanism and high-reaction pressures (>1 bar), silicon fibers were grown by laser-assisted chemical vapor deposition (LCVD) from silane near the focal point of a cw Nd:YAG laser beam. Fiber-growth rates ranged from <1 to 500 m/s and fiber tip temperatures from 525 to 1412° C. At low fibertip temperatures (<600° C) silicon fibers yielding glassy fracture were obtained. Some crystallinity was observed by X-ray diffraction. Polycrystalline silicon fibers were formed at intermediate temperatures, single-crystal silicon fibers at high-laser intensities and high-tip temperatures. The single crystal LCVD silicon fibers were formed by a vapor-liquid-solid (VLS) mechanism. Single-crystal VLS-LCVD silicon fibers were also obtained from liquid silicon-metal alloys by initiating fiber growth from the end of thin palladium, gold and platinum wires.     

Structure of liquid Y3Al5O12 (YAG)

The total structure factor S(Q) and the radial distribution function G(r) of liquid Y3Al5O12 (YAG) were measured at 1770-2230 K by x-ray scattering from samples under containerless conditions in Ar and O2. Nominal coordination numbers are 4 for Al3+ and 6 for Y3+ ions. The G(r) has peaks at r approximately 1.8 A for Al-O, r approximately 2. 25 A for Y-O, and r approximately 3.3-3.6 A assigned to metal ions in adjacent AlO5-4 and YO9-6 polyhedral ions. Relative to the pure oxides, G(r) for molten YAG has smaller half-widths for the Al-O and Y-O peaks, and an increased sensitivity to temperature and the ambient gas composition.     

A new series of rod-like conjugated molecules with a pyrazine or a bipyrazine core. Synthesis and light emitting properties

In this paper, we describe the synthesis of a wide range of new rod-like conjugated molecules with a pyrazine or a bipyrazine core connected to electron acceptor (A) or donor (D) groups through π-conjugated bridges as transmitters for the internal charge transfer (ICT). The key steps of the synthesis involve metallation and subsequent transmetallation of 2-chloropyrazine derivatives followed by Sonogashira or Negishi cross-coupling reactions. The bipyrazine core was obtained with a Stille cross-coupling reaction between the 2-chloro-6-tributylstannylpyrazine and the 2-chloro-6-iodopyrazine. Functionalization of the 6,6′-dichloro-2,2′-bipyrazine was performed by metallation, transmetallation and cross-coupling reactions. The light emitting properties of the so obtained molecules are then investigated in terms of absorption and emission spectra.     

Finite jet determination of CR mappings

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M⊂CN, N?2, that is essentially finite and of finite type at each of its points, for every point p∈M there exists an integer ?p, depending upper-semicontinuously on p, such that for every smooth generic submanifold M^′⊂CN of the same dimension as M, if are two germs of smooth finite CR mappings with the same ?p jet at p, then necessarily for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in CN of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω,Ω^′⊂CN are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂Ω, such that if are two proper holomorphic mappings extending smoothly up to ∂Ω near some point p∈∂Ω and agreeing up to order k at p, then necessarily H₁=H₂.