Unusual Structure,Fluxionality, and Reaction Mechanism of Carbonyl Hydrosilylation by Silyl Hydride Complex [(ArN)Mo(H)(SiH2Ph)(PMe3)3]

The reactions of bis(borohydride) complexes [(RN?)Mo(BH₄)₂(PMe₃)₂] ( 4 : R=2,6‐Me₂C₆H₃; 5 : R=2,6‐iPr₂C₆H₃) with hydrosilanes afford new silyl hydride derivatives [(RN?)Mo(H)(SiR′₃)(PMe₃)₃] ( 3 : R=Ar, R′₃=H₂Ph; 8 : R=Ar′, R′₃=H₂Ph; 9 : R=Ar, R′₃=(OEt)₃; 10 : R=Ar, R′₃=HMePh). These compounds can also be conveniently prepared by reacting [(RN?)Mo(H)(Cl)(PMe₃)₃] with one equivalent of LiBH₄ in the presence of a silane. Complex 3 undergoes intramolecular and intermolecular phosphine exchange, as well as exchange between the silyl ligand and the free silane. Kinetic and DFT studies show that the intermolecular phosphine exchange occurs through the predissociation of a PMe₃ group, which, surprisingly, is facilitated by the silane. The intramolecular exchange proceeds through a new non‐Bailar‐twist pathway. The silyl/silane exchange proceeds through an unusual Mo^VI intermediate, [(ArN?)Mo(H)₂(SiH₂Ph)₂(PMe₃)₂] ( 19 ). Complex 3 was found to be the catalyst of a variety of hydrosilylation reactions of carbonyl compounds (aldehydes and ketones) and nitriles, as well as of silane alcoholysis. Stoichiometric mechanistic studies of the hydrosilylation of acetone, supported by DFT calculations, suggest the operation of an unexpected mechanism, in that the silyl ligand of compound 3 plays an unusual role as a spectator ligand. The addition of acetone to compound 3 leads to the formation of [trans‐(ArN)Mo(OiPr)(SiH₂Ph)(PMe₃)₂] ( 18 ). This latter species does not undergo the elimination of a Si? O group (which corresponds to the conventional Ojima′s mechanism of hydrosilylation). Rather, complex 18 undergoes unusual reversible β‐CH activation of the isopropoxy ligand. In the hydrosilylation of benzaldehyde, the reaction proceeds through the formation of a new intermediate bis(benzaldehyde) adduct, [(ArN?)Mo(η²‐PhC(O)H)₂(PMe₃)], which reacts further with hydrosilane through a η¹‐silane complex, as studied by DFT calculations.     

Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, 3j Symbols, and Character Localization

In this paper, we employ a technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index J) of generic Wigner matrix elements D^J_MM￠(g){D^{J}_{MM'}(g)} . We use this result to derive asymptotic formulae for the character χ ^J (g) of an SU(2) group element and for Wigner’s 3j symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for χ ^J (g) is in fact exact. The result hints at a “Duistermaat-Heckman like” localization property for discrete sums.     

A dispersive regularization of the modulational instability of stratified gravity waves

We derive high-order corrections to a modulation theory for the propagation of internal gravity waves in a density-stratified fluid with coupling to the mean flow. The methodology we use allows for strong modulations of wavenumber and mean flow, extending previous approaches developed for the quasi-monochromatic regime. The wave mean flow modulation equations consist of a system of nonlinear conservation laws that may be hyperbolic, elliptic or of mixed type. We investigate the regularizing properties of the asymptotic correction terms in the case when the system becomes unstable and ill-posed due to a change of type (loss of hyperbolicity). A linear analysis reveals that the regularization by the added correction terms does so by introducing a short-wave cut-off of the unstable wavenumbers. We perform various numerical experiments that confirm the regularizing properties of the correction terms, and show that the growth of unstable modes is tempered by nonlinearity. We also find an excellent agreement between the solution of the corrected modulation system and the modulation variables extracted from the numerical solution of the nonlinear Boussinesq equations.     

Colored Group Field Theory

Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to higher dimensional topological spaces. The perturbative development of the usual GFT’s is rather involved combinatorially and plagued by topological singularities (which we discuss in great detail in this paper), thus very difficult to control and unsatisfactory.     

Global smoothness and uniform convergence of smooth Gauss–Weierstrass singular operators

In this article we continue with the study of smooth Gauss–Weierstrass singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L_p norm, 1≤p≤∞, by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.     

Radial equivalence for the two basic nonlinear degenerate diffusion equations

In this paper we prove that there exists an explicit correspondence between the radially symmetric solutions of two well-known models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. We establish exact correspondence formulas between these solutions. We also study in detail the application of the results in the important case of self-similar solutions. In particular, we derive the existence of new self-similar solutions for the evolution p-Laplacian equation.     

Tree Quantum Field Theory

We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e., it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define non-perturbatively differential renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse–Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a non-perturbative definition of field theory in non-integer dimension.     

飞秒激光光刻波导偏振器

飞秒激光诱导折射率变化提供了一种灵活的三维光子器件制作手段.飞秒激光光刻的II类波导具有偏振导光特性,可以作为波导偏振器,但是对于要保留的偏振分量损耗太大.本文阐述了一种利用飞秒激光在熔融石英中制作的新型低损耗波导偏振器.它由中间的一根I类波导及两侧的两根II类纳米光栅轨迹构成.基于飞秒激光诱导的纳米光栅的偏振依赖散射特性,II类纳米光栅轨迹能够对I类波导的倏逝场进行调制.偏振方向垂直于纳米光栅的模式相对于偏振方向平行于纳米光栅的模式有更大的散射损耗,因此导通的是偏振方向平行于纳米光栅的模式.研究了消光比随I类波导与II类纳米光栅轨迹之间的间距的变化关系,选择一个最佳间距来进一步研究消光比随II类纳米光栅轨迹长度的变化关系.在间距6 μm,II类纳米光栅轨迹扫描长度6 mm处实现了最大15.91 dB的消光比.通过增加II类轨迹的长度或者数量,很容易得到更高的消光比.