Improved multiplicity-editing of HMBC NMR spectra

A new improved multiplicity-edited HMBC experiment is introduced that leads to better J cross-talk suppression in the even (i.e. C + CH2 groups) and odd (i.e. CH + CH3 groups) subspectra. By combining data recorded with three different pulse sequences J cross-talk becomes a second-order effect in Delta1J, i.e. the deviation of an actual 1J coupling constant from the value 1J0 used in setting delays tau = (1J0)(-1/2), which is adequate for most applications. As for the original multiplicity-edited HMBC experiment, the improved experiment can be performed with a single excitation delay or implemented in a broadband version similar to broadband HMBC.     

具有转移条件的向量型Sturm-Liouville问题的特征值重数及Ambarzumyan定理

研究了定义在有限区间内具有转移条件的m维向量型Sturm-Liouville问题.主要得到了该问题特征值重数的若干结论.证明了当矩阵值势函数Q满足一定的条件时,只能有有限个重数为m的特征值.作为重数结果的应用,证明了该问题的Ambarzumyan定理.     

关于纤维锥的深度和Hilbert级数

Let (R,m) be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ? 1 and λ( JKKIIkk-1 ) = 1. We show that if depth G(I) ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK(I) assuming that depth G(I) ≥ d - 1.     

B型和C型Weyl模的一个新重数公式

A monomial basis and a filtration of subalgebras for the universal enveloping algebra U(gl) of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V (λ) as a U(gl)-module into a direct sum of Weyl modules V (μ)’s as U(gl-1)modules is described. In particular, a new multiplicity formula for the Weyl module V (λ) is obtained in this note.     

Multiple forced rotational solutions for the planar <Emphasis Type="Italic">N</Emphasis>-pendulum equation

This paper presents a theorem of existence of \((N - N_{0} + 1)2^{N_{0}}\) noncontractible periodic orbits of any given rotational vector with zero components for the planar forced N-pendulum equation, where N₀ denotes the number of zero components of given rotational vector. We make special arrangements for masses and length instead of the nondegenerate assumption. Moreover, for rotational vectors with identical components, similar results is also obtained.     

Effect of crosslinker multiplicity on the gel point in ATRP

The experimental gelation was studied in atom transfer radical polymerization (ATRP) of methyl acrylate (MA) with various branching reagents that structurally differ by the number of vinyl groups (multiplicity). MA was copolymerized with branching reagents containing 2, 3, 4, 5, or 6 acrylate moieties per molecule, respectively. Reactions with a constant concentration of branching vinyl groups (the same molar ratio of [Vinyl_branch]₀/[Initiator]₀) revealed a different gelation behavior when ethylene glycol diacrylate (2A) and trimethylolpropane triacrylate (3A) were used as crosslinkers, whereas the reactions using pentaerythritol tetraacrylate (4A) showed similar gelation behavior as compared to reactions using 3A. Additional reactions with dipentaerythritol pentaacrylate (5A) and hexafunctional acrylate crosslinkers (6A) revealed the presence of a concentration‐dependant gelation behavior. On the other hand, in reactions with the same molar concentration of various crosslinkers, gelation occurred at progressively lower MA conversions for reactions with an increasing number of vinyl groups per crosslinker. In addition, the number of unreacted pendant vinyl groups in the sols was compared for reactions with different multiplicity of the crosslinker and various ratios of [Vinyl_branch]₀/[Initiator]₀. Finally, a linear oligomeric crosslinker containing multiple branching vinyl groups along the side chain was used as the branching reagent for gelation. The gelation behavior during the ATRP of MA with the linear crosslinker was different as compared to the use of the aforementioned crosslinkers with a star‐like architecture. © 2010 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 2016–2023, 2010     

M／M／1算子的另一个特征值

证明2√λμ-λ-μ是偏微分方程形式的M／M／1排队模型主算子的几何重数为1的特征值．     

On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equation and some related spectral problems

We prove that under certain conditions, a vectorial Sturm-Liouville differential equation of dimension can only possess finitely many eigenvalues which have multiplicity . For the case , we find a sufficient condition on the potential function , and a bound depending on , such that the eigenvalues of the equation with index exceeding are all simple. These results are applied to find some sufficient conditions which imply that the spectra of two potential equations, or two string equations, have finitely many elements in common, and an estimate of the number of elements in the intersection of two spectra is provided.

     

Some interesting C<Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>C<Emphasis Type="Italic">H</Emphasis><Subscript>3</Subscript> spin systems in several diethyl malonate indene derivatives studied with NMR and molecular modeling

While studying the electrophilic cyclization of acetylenic malonates in synthesizing malonate indene derivatives with high regio and stereo-selectivity, we found that the CH₂ and CH₃ protons of two COOCH₂CH₃ groups in some diethyl malonate indene derivatives showed very interesting spin-spin coupling and signal multiplicity. To explain those phenomena, we synthesized several diethyl malonate indene derivatives, and then used ¹H NMR spectra and molecular modeling to investigate the CH ₂CH ₃ spin system of these two COOCH₂CH₃ groups in detail. The results showed that the spin systems of CH ₃CH ₂ of the two COOCH₂CH₃ groups in those compounds may exist in different froms (AMX ₃, ABX ₃ and A ₂ X ₃), and the effect of the aromatic ring induced by the phenyl group at the end of the triple bond is the main reason responsible for those phenomena.     

Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations

In this paper, we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation. We also obtain a precise global bifurcation diagram of the solution set. As a result, an open problem presented by C.-H. Hsu and Y.-W. Shih [C.-H. Hsu, Y.-W. Shih, Solutions of semilinear elliptic equations with asymptotic linear nonlinearity, Nonlinear Anal. 50 (2002) 275-283] is completely solved. Our argument is mainly based on bifurcation theory and continuation method.