An existence result for a quadrature surface free boundary problem

The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existences is given in the radial case.     

The Equivalence of Several Basic Theorems for Subdifferentials

Several different basic tools are used for studying subdifferentials. They are a nonlocal fuzzy sum rule in (Borwein et al., 1996; Zhu, 1996), a multidirectional mean value theorem in (Clarke and Ledyaev, 1994; Clarke et al., 1998), local fuzzy sum rules in (Ioffe, 1984, 1990) and an extremal principle in (Kruger and Mordukhovich, 1980; Mordukhovich, 1976, 1980, 1994). We show that all these basic results are equivalent and discuss some interesting consequences of this equivalence.     

Spectral distributions in nuclei: General principles and applications

The subject of spectral distribution methods where one derives and applies the locally smoothed forms of observables in nuclei is briefly reviewed. It is well understood that the local forms (with respect to energy) of the level density function, expectation values and strength densities are Gaussian, linear (or ratio of Gaussians) and a bivariate Gaussian respectively. To accomodate symmetries in the above forms, one has to deal with multivariate distributions in general; for example the angular-momentum (J) decomposition leads to a bivariate Gaussian form for the level density. These results extend to indefinitely large spaces by method of partitioning and they generate convolution forms. The origin of these remarkable spectral properties is discussed and shell model examples are given to substantiate their applicability to nuclear systems. Spectral distribution theory is a practical, usable theory because the smoothed forms are defined in terms of traces of low particle-rank operators, and the trace information propagates. Finally we discuss the application of the spectral methods for a wide range of nuclear problems; these include binding energies, orbit occupancies, electromagnetic andβ-decay sum rule quantities, analysis of operators, symmetry breaking, numerical level densities, and determination of bounds on time-reversal non-invariant part of nucleon-nucleon interaction.     

Interpolatory integration rules and orthogonal polynomials with varying weights

We investigate which types of asymptotic distributions can be generated by the knots of convergent sequences of interpolatory integration rules. It will turn out that the class of all possible distributions can be described exactly, and it will be shown that the zeros of polynomials that are orthogonal with respect to varying weight functions are good candidates for knots of integration rules with a prescribed asymptotic distribution.Research supported by the Deutsche Forschungsgemeinschaft (AZ: Sta 299/4-2).     

D→Klv～l衰变过程的研究

用光锥QCD求和规则研究D→Klv～_l衰变过程，首先计算D→K跃迁形状因子，通过构造新的关联函数，消除了twist-3波函数的不确定性给计算结果所带来的影响，从而使计算结果更加精确. 计算得到的分支比与最近的实验数据相一致. 关键词： QCD光锥求和规则 D介子半轻衰变 分支比 形状因子     

A note on generalized averaged Gaussian formulas

We have recently proposed a very simple numerical method for constructing the averaged Gaussian quadrature formulas. These formulas exist in many more cases than the real positive Gauss–Kronrod formulas. In this note we try to answer whether the averaged Gaussian formulas are an adequate alternative to the corresponding Gauss–Kronrod quadrature formulas, to estimate the remainder term of a Gaussian rule.     

Rhodathiaboranes with `anomalous' electron counts: synthesis, structure and reactivity

Analysis of the structures of 8,8-(PPh₃)₂-8,7-nido-RhSB₉H₁₀ and 9,9-(PPh₃)₂-9,7,8-nido-RhC₂B₈H₁₁ by RMS misfit calculations has confirmed that these rhodaheteroboranes possess nido 11-vertex cluster geometries in apparent contravention of Wade's rules. However, examination of the molecular structures of both species shows that the {RhP₂} planes are inclined by ca. 66° with respect to the metal-bonded SB₃ or CB₃ faces, and that two weak ortho-CHRh agostic interactions occupy the vacant co-ordination position thereby created. As a consequence of these agostic bonds the Rh atom, and hence the overall cluster, is provided with an additional electron pair, meaning that their nido structures are now fully consistent with Wade's rules. The chelated diphosphine compound 8,8-(dppe)-8,7-nido-RhSB₉H₁₀ is similar to the PPh₃ compound in showing the same agostic bonding. Attempts to prepare a bis-P(OMe)₃ analogue result in ligand scavenging and the formation of 8,8,8-{P(OMe)₃}₃-8,7-nido-RhSB₉H₁₀. Similarly, reaction between Cs[6-arachno-SB₉H₁₂] and RhCl(dmpe)CO does not result in CO loss but in formation of 8,8-(dmpe)-8-(CO)-8,7-nido-RhSB₉H₁₀, shown to exist as a mixture of two of three possible rotamers. Deprotonation of 8,8-(PPh₃)₂-8,7-nido-RhSB₉H₁₀ and 8,8-(dppe)-8,7-nido-RhSB₉H₁₀ with MeLi yields the anions [1,1-(PPh₃)₂-1,2-closo-RhSB₉H₉]⁻ and [1,1-dppe-1,2-closo-RhSB₉H₉]⁻, respectively, with octadecahedral cage structures. It is argued that anion formation causes the agostic bonding to be `switched-off' and results in the cluster adopting the closo architecture predicted by Wade's rules. This structural change is fully reversible on reprotonation, and if reprotonation of [1,1-(dppe)-1,2-closo-RhSB₉H₉]⁻ is carried out in MeCN, the product 8,8-(dppe)-8-(MeCN)-8,7-nido-RhSB₉H₁₀ forms. Interestingly, 8,8-(dppe)-8-(MeCN)-8,7-nido-RhSB₉H₁₀ reconverts to 8,8-(dppe)-8,7-nido-RhSB₉H₁₀ on standing in CDCl₃, suggesting that the agostic bonding is sufficiently strong to displace co-ordinated MeCN. All new compounds are fully characterised by multinuclear NMR spectroscopy and, in many cases, by single crystal X-ray diffraction.     

VA族元素对阳极铅(Ⅱ)氧化物膜半导体性质的影响(Ⅱ)

用光电化学电流法研究了铅、铅砷、铅锑和铅铋合金在4．5mol·L－1H2SO4溶液（22℃）中，以0．9V（vs．Hg／Hg2SO4）极化7h而形成的阳极膜中的氧化铅的半导体性质，合金添加剂砷、锑和铋对t－PbO（四方氧化铝）和o-PbO（斜方氧化铝）的禁带宽度没有影响，从量子效率和电位的关系可求Pb，Pb－lat％As（at％表示原子百分比，全文同），Pb-lat％Sb和Sb－lat％Bi上膜中t－Pbo的施主密度（ND）分别为9．3×1015，1．0×1016，3．1×1016和1．3×1017cm-3，平带位分别为-0．20，－0．22，－0．28和-0．08V（vs．Hg／Hg2SO4）．比较VA元素砷、锑和铋对上述膜中t－Pbo的ND（从而自由电子密度）和膜中t－Pbo的生长速率的影响，可认为法添加剂砷、锑和铋对阳极膜中t-Pbo的作用符合Hauffe规则．     

A six-dimensional quadrature procedure

An account is given of the use of Gaussian quadrature product formulae in the evaluation of certain six-dimensional, two-centre integrals involving one-electron Green's functions. These integrals occur in a new molecular variational principle recently proposed by Hall, Hyslop and Rees [1] from which an approximate energy may be derived which can be shown to be at least as good as that obtained from the Rayleigh-Ritz principle. Reductions in computing time are realized by removing certain singularities using a subtraction technique and also by using an empirically determined Richardson-type extrapolation formula.This paper was presented during the session on numerical integration methods for molecules of the 1970 Quantum Theory Conference in Nottingham. It has been revised in the light of the interesting discussion which followed.