Outer operators for the noncommutative symmetric hardy spaces associated with finite subdiagonal algebra

In this article, we extended main results on outer operators of [6] to the symmetric Hardy spaces, when associated subdiagonal algebra is finite.     

Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups

Stochastic equations indexed by negative integers and taking values in compact groups are studied. Extremal solutions of the equations are characterized in terms of infinite products of independent random variables. This result is applied to characterize several properties of the set of all solutions in terms of the law of the driving noise.     

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m$m$ fermions (or bosons) in N$N$ single particle states and interacting via k$k$-body interactions, we have EGUE(k$k$) [embedded GUE of k$k$-body interactions] with GUE embedding and the embedding algebra is U(N)$U\left(N\right)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k$k$) representation for a Hamiltonian that is k$k$-body and an independent EGUE(t$t$) representation for a transition operator that is t$t$-body and employing the embedding U(N)$U\left(N\right)$ algebra, finite-N$N$ formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k₀$k_0$ number of particles from a system of m$m$ spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French (1975). Numerical results obtained using the exact formulas for two-body (k=2$k=2$) Hamiltonians (in some examples for k=3$k=3$ and 4$4$) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.     

Dirac方程的特征函数的迹公式

本文研究具有周期有限带位势的Dirac算子, 利用Dirac算子与单值算子的交换性,定义Bloch函数和乘子曲线, 进而获得揭示Bloch函数与位势间内在关系的Dubrovin-Novikov型公式; 通过复球面上的留数计算, 最终分别得到相应于谱带左端点、右端点以及双侧端点的特征函数的迹公式.     

BIS(DIAZASILAPHOSPHETIDINES) AND THEIR METAL COMPLEXES

The syntheses and solid-state structures of bis(diazasilaphosphetidines) of the type [Me ₂ Si(μ-N ^t Bu) ₂ P] ₂ NR, R = Ph, ⁱ Pr, ^t Bu, of the P-chiral analogue [Me ₂ Si(μ-N ^t Bu)(μ-N-2,6- ⁱ PrPh)P] ₂ (C ₂ H ₄ ), and of some of their metal complexes are reported. The highly electron-rich, modular bis(phosphines) are easy to synthesize and may have applications in catalysis.     

One-Pot Stereoselective Synthesis of Alkyl (Z)-2-[4-Oxo-3-phenyl-2-(phenylimino)-1,3-thiazolan-5-yliden]acetates from Acetylenic Esters and N,N′ -Diphenylthiourea

N, N′ -diphenylthiourea reacts with dialkyl acetylenedicarboxylates in acetone to form 1:1 adducts, which undergo a cyclization reaction to produce alkyl (Z)-2-[4-oxo-3-phenyl-2-(phenylimino)-1,3-thiazolan-5-yliden]acetates in fairly good yields. NMR spectra indicated that the reaction is completely stereoselective.     

N,S-Substituted Dienes from Mono(arylthio)substituted- and S-, S,S-Substituted Dienes

Mono(thio)substituted dienes 1a-1b gave compounds 3a-c and 5d-g with piperazine and piperidine derivatives in dichloromethane. Compounds 8 , 9 , and 10 were obtained from the reactions of perchlorobutadiene ( 6 ) with 1,4-butanedithiol ( 7 ) in ethanol in the presence of sodium hydroxide. Compounds 12a-b , 13a-b were obtained from the reactions of perchlorobutadiene ( 6 ) with allylmercaptan (CH 2 =CH--CH 2 --SH) and mercaptoethanol (HO--CH 2 --CH 2 --SH).     

Einige Neue N,S-Substituierte 2-Nitrodienverbindungen Aus Reaktionen Von Mono(p-fluorarylthio)substituierten Dienen Mit Aminen

2-Nitrodiene compound 1 was stirred with p-fluorothiophenol for a long time and compound 3 was obtained. Compound 1 gave bis(thio)substituted 2-nitrodiene compound 4 and tris(thio)substituted compound 5 with 2 moles of p-fluorothiophenol in the presence of NaOH in ethanol. The compounds 9a–g have been prepared from 8a–g and 3. Compound 7 was obtained from the reaction of mono(thio)substituted 2-nitrodiene with morpholine. Compound 3 gives 11a–d in the reaction with piperidines in CH₂Cl₂ (or ether). Compound 13a–b have been obtained from the reaction of compound 3 with primary amines 12a–b. Compound 3 gives 15 and 16 in the reaction with 2,5-dimethylpiperazine in CH₂Cl₂.