Spectral shift function and trace formula

The complete proofs of Krein’s theorem on the spectral shift function and the trace formula are given for a pair of self-adjoint operators such that either (i) their difference is trace-class or (ii) the difference of their resolvents is trace-class. The proofs, essentially due to Krein, is based on Herglotz’s theorem on the boundary value of the analytic functions whose imaginary part is non-negative on the upper half plane, and an almost optimal class of functions are obtained for which the trace formula is valid. Also an alternative method based on Weyl-von Neumann’s theorem for self-adjoint operators, avoiding the complex function theory and inspired by Voiculescu’s work, is given for the first case. Furthermore, some applications of the spectral shift function have been discussed.     

On a problem of Zygmund

It is proved that there exists an integrable function on [0, 1]² whose integral is nondifferentiable in each direction belonging to a set everywhere dense in [0, 2π] but is strongly differentiable. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 749–762, November, 1998.     

Decomposition of multiattribute expected-utility functions

This paper integrates and extends the theory of the decomposition of multiattribute expected-utility functions based on utility independence. In a preliminary section, the standard decision model of expected utility is briefly discussed, including the fact that the decision maker's preference forlotteries with two outcomes determines the utility function uniquely. The decomposition possibilities of a utility function are captured by the concept ofautonomous sets of attributes, an affine separability of some kind known as generalized utility independence.Overlapping autonomous sets lead to biaffine-associative, i.e.multiplicative oradditive decompositions. The multiplicative representation shows that autonomy has strongerclosure properties than utility independence, for instance with respect to set-theoretic difference. Autonomy is also a concept with a wider scope since it applies to the decomposition of Boolean functions, games and a number of other topics in combinatorial optimization. This relationship to the well-known theory ofsubstitution decomposition in discrete mathematics also reveals a kind of discrete core behind the decomposition of utility functions. The entirety of autonomous sets can be represented by a compact data structure, the so-calledcomposition tree, which frequently corresponds to a natural hierarchy of attributes. Multiplicative/additive ormulti-affine functions correspond to the hierarchy steps. The known representation of multi-affine functions is shown to be given by aMoebius inversion formula. The entire approach has the advantage that it allows the application of more sophisticated representation methods on a detailed level, whereas it employs onlyfinite set theory andarithmetic on the main levels in the hierarchy.     

新型重整催化剂的研究—Pt／BaKL型沸石中钡的作用

本文采用XRD,NH_3-TPD,IR以及TEM等方法考察了Pt/BaKL沸石中Ba~(2+)的作用。NH_3-TPD结果表明,交换Ba~(2+)后的L型沸石的酸性,由于阳离子的作用仅在弱酸范围内有些改变,但与载铂的沸石的催化活性无明显的对应关系。IR及XRD谱线表明,BaKL沸石中Ba~(2+)在最佳交换度(26.3%)时,随预培烧温度的提高,使Ba~(2+)处于沸石孔道中B、C、D位置,接近于最佳分布,此时IR谱线上1421cm~(-1)处有一新峰。当沸石负载铂后,因铂与Ba~(2+)的相互作用,新峰消失,导致铂粒在沸石上分布更加均匀;铂粒在1.5—3.0nm之间占有最大比例,使得Pt/BaKL沸石呈现芳构化活性及选择性高于Pt/KL沸石,分别为～98%和～95%。     

用θ-2θ型衍射仪测定溶液结构的新方法

叙述了用θ－２θ型Ｘ射线衍射仪精确测定电解质溶液结构的新实验技术。设计制作了具有恒温功能的超厚液体样品池，并建立了样品池窗口强度的校正方法。优化了液体Ｘ射线衍射数据和结构参数精细化的计算机程序，获得了非常令人满意的实验结果。由θ－２θ型Ｘ射线衍射仪精确测定的径向分布函数与θ－θ型衍射仪自由散射比较，表明ＤＲＦ分辨率有所提高。     

The Mathematical Theory of Ito Diffusions on Hypersurfaces, with Applications to NMR Relaxation Problems

A mathematical framework for translational Brownian motion on hypersurfaces is presented, using an imbedding of the surface and Ito diffusions in the ambient space. This includes a survey of Ito calculus and differential geometry. Computational methods for time correlation functions relevant to spin relaxation studies on curved interfaces are given, and explicit calculations of time correlation functions and order parameters for a Rippled surface are presented.     

Molecular modeling of cytochrome P450 3A4

The three-dimensional structure of human cytochrome P450 3A4 was modeled based on crystallographic coordinates of four bacterial P450s: P450 BM-3, P450cam, P450terp, and P450eryF. The P450 3A4 sequence was aligned to those of the known proteins using a structure-based alignment of P450 BM-3, P450cam, P450terp, and P450eryF. The coordinates of the model were then calculated using a consensus strategy, and the final structure was optimized in the presence of water. The P450 3A4 model resembles P450 BM-3 the most, but the B helix is similar to that of P450eryF, which leads to an enlarged active site when compared with P450 BM-3, P450cam, and P450terp. The 3A4 residues equivalent to known substrate contact residues of the bacterial proteins and key residues of rat P450 2B1 are located in the active site or the substrate access channel. Docking of progesterone into the P450 3A4 model demonstrated that the substrate bound in a 6-orientation can interact with a number of active site residues, such as 114, 119, 301, 304, 305, 309, 370, 373, and 479, through hydrophobic interactions. The active site of the enzyme can also accommodate erythromycin, which, in addition to the residues listed for progesterone, also contacts residues 101, 104, 105, 214, 215, 217, 218, 374, and 478. The majority of 3A4 residues which interact with progesterone and/or erythromycin possess their equivalents in key residues of P450 2B enzymes, except for residues 297, 480 and 482, which do not contact either substrate in P450 3A4. The results from docking of progesterone and erythromycin into the enzyme model make it possible to pinpoint residues which may be important for 3A4 function and to target them for site-directed mutagenesis.     

Dipolar relaxations in an epoxy-amine system

Dipolar relaxations in a reactive epoxy-amine system based on diglycidyl ether of bisphenol A with 4,4^′-diaminodiphenylsulfone were studied with the Havriliak-Negami function. The system was cured isothermally at 140 °C using simultaneous kinetic and microdielectric studies.The relaxation time was calculated from the frequency f_max of the peak of the loss factor ε^′′ versus frequency f. A linear relation exists between the logarithm of the relaxation time τ and the glass transition temperature T_g. Then, logτ follows the Di Benedetto equation revisited by Pascault and Williams allowing the prediction of the relaxation time τ during cure.The unrelaxed permittivity at high frequency ε_u, the relaxed or static permittivity ε_s and the skewness parameter β were found independent on the conversion or the curing time. The distribution parameter α decreases as curing time t increases. The difficulty of orientation of dipoles in the electric field due to vitrification is responsible of these behaviours.