An Intramolecular Silylene Borane Capable of Facile Activation of Small Molecules,Including Metal‐Free Dehydrogenation of Water

The first single‐component N‐heterocyclic silylene borane 1 (LSi‐R‐BMes₂; L=PhC(N^t Bu)₂; R=1,12‐xanthendiyl spacer; Mes=2,4,6‐Me₃C₆H₂), acting as a frustrated Lewis pair (FLP) in small‐molecule activation, can be synthesized in 65 % yields. Its HOMO is largely localized at the silicon(II) atom and the LUMO has mainly boron 2p character. In small‐molecule activation 1 allows access to the intramolecular silanone–borane 3 featuring a Si=O→B interaction through reaction with O₂, N₂O, or CO₂, and formation of silanethione borane 4 from reaction with S₈. The Si^II center in 1 undergoes immediate hydrogenation if exposed to H₂ at 1 atm pressure in benzene, affording the silane borane 5‐H₂ , L(H₂)Si‐R‐BMes₂. Remarkably, no H₂ activation occurs if the single silylene LSiPh and Mes₃B intermolecularly separated are exposed to dihydrogen. Unexpectedly, the pre‐organized Si–B separation in 1 enables a metal‐free dehydrogenation of H₂O to give the silanone–borane 3 as reactive intermediate.     

Insertion Reactions of Neutral Phosphidozirconocene Complexes as a Convenient Entry into Frustrated Lewis Pair Territory

Neutral phosphidozirconocene complexes [Cp₂Zr(PR₂)Me] (Cp=cyclopentadienyl; 1a : R=cyclohexyl (Cy); 1b : R=mesityl (Mes); 1c : R=tBu) undergo insertion into the Zr?P bond by non‐enolisable carbonyl building blocks (O=CR′R′′), such as benzophenone, aldehydes, paraformaldehyde or CO₂, to give [Cp₂Zr(OCR′R′′PR₂)Me] ( 3 – 7 ). Depending on the steric bulk around P, complexes 3 – 7 react with B(C₆F₅)₃ to give O‐bridged cationic zirconocene dimers that display typical frustrated Lewis pair (FLP)/ambiphilic ligand behaviour. Thus, the reaction of {[Cp₂Zr(μ‐OCHPhPCy₂)][MeB(C₆F₅)₃]}₂ ( 10a ) with chalcone results in 1,4 addition of the Zr⁺/P FLP, whereas the reaction of {[Cp₂Zr(μ‐OCHFcPCy₂)][MeB(C₆F₅)₃]}₂ ( 11a ; Fc=(C₅H₄)CpFe) with [Pd(η³‐C₃H₅)Cl]₂ yields the unique Zr?Fe?Pd trimetallic complex 13a , which has been characterised by XRD analysis.     

Families of multisums as mock theta functions

In view of the Bailey lemma and the relations between Hecke-type sums and Appell–Lerch sums given by Hickerson and Mortenson, we find that many Bailey pairs given by Slater can be used to deduce mock theta functions. Therefore, by constructing generalized Bailey pairs with more parameters, we derive some new families of mock theta functions. Meanwhile, some identities between new mock theta functions and classical ones are established. Furthermore, based on the proofs of the main theorems, many q-hypergeometric transformations are obtained.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited, II

It is shown that the admissible solutions of the continuity and Bernoulli or Burgers' equations of a perfect one-dimensional liquid are conditioned by a relation established in 1949–1950 by Pauli, Morette, and Van Hove, apparently, overlooked so far, which, in our case, stipulates that the mass density is proportional to the second derivative of the velocity potential. Positivity of the density implies convexity of the potential, i.e., smooth solutions, no shock. Non-elementary and symmetric solutions of the above equations are given in analytical and numerical form. Analytically, these solutions are derived from the original Ansatz proposed in Ref. 1 and from the ensuing operations which show that they represent a particular case of the general implicit solutions of Burgers' equation. Numerically and with the help of an ad hoc computer program, these solutions are simulated for a variety of initial conditions called compatible if they satisfy the Morette–Van Hove formula and anti-compatible if the sign of the initial velocity field is reversed. In the latter case, singular behaviour is observed. Part of the theoretical development presented here is rephrased in the context of the Hopf–Lax formula whose domain of applicability for the solution of the Cauchy problem of the homogeneous Hamilton–Jacobi equation has recently been enlarged.     

Structure and Photoelectric Characteristics of Functional Polymers and of Compositions Based on Them

The electronic processes in disordered organic solids were analyzed, classified, and summarized. They include the photogeneration of charge pairs, electron transport, geminal and bimolecular recombination, and the capture of charge carriers in local electronic states. General criteria were formulated for the targeted search for and the creation of new prospective polymers and their compositions with organic pigments and related compounds having specific photoelectric characteristics for electrographic and holographic media, photoelectric converters, and electroluminescent and photorefractive devices.     

多波长系统孤子耦合方程的可积性

多波长系统孤子耦合方程存在Lax对，具有可积性，利用Hirota双线性方法求出了孤子耦合方程的单孤子解和双孤子解. 关键词： Lax对 Hirota双线性变换 可积性