A New Method for Approximations in Probability and Operator Theories

The aim of this work is to introduce a new approach in normal approximations, approximations of semigroups of operators, and approximations by accompanying laws. We describe the method and provide several examples showing how the method works. Loosely speaking, the approach is based on multiplicative representations of differences to estimate. The method considerably reduces the technical part of proofs compared to the traditional approaches. In probability theory, it nicely applies to the nonidentically distributed case; another advantage of the method is its extendability to the case of dependent random variables.     

Solving the Sum-of-Ratios Problem by an Interior-Point Method

We consider the problem of minimizing the sum of a convex function and of p1 fractions subject to convex constraints. The numerators of the fractions are positive convex functions, and the denominators are positive concave functions. Thus, each fraction is quasi-convex. We give a brief discussion of the problem and prove that in spite of its special structure, the problem is IE1.gif" alt=" $$\mathcal{N}\mathcal{P}$$ " align="middle" border="0"> -complete even when only p=1 fraction is involved. We then show how the problem can be reduced to the minimization of a function of p variables where the function values are given by the solution of certain convex subproblems. Based on this reduction, we propose an algorithm for computing the global minimum of the problem by means of an interior-point method for convex programs.     

Li_2O-(LiCl)_2-B_2O_3-Al_2O_3系玻璃的红外光谱

本工作系统地测量和分析了Li_O-(LiCl)_2-B_2O_3-Al_2O_3系玻璃的红外光谱,据此研究了玻璃中硼的配位数和硼酸盐结构基团随组成的变化以及Al~(3+)和Cl~-离子在结构中所起的作用,并对光谱中某些吸收峰机理作了定性探讨。     

Level structures of 95,97Mo — A comparative study

High-spin states of ^95,97Mo (Z=42, N=53,55) nuclei have been investigated through ⁸²Se(¹⁸O, xn) reaction at Eb=60 MeV. The level scheme in ⁹⁵Mo has been observed upto ≏ 10 MeV in the present experiment. The level structure shows mainly single particle character. In ⁹⁷Mo, the ground state level sequence has been extended to ≏ 4.5 MeV while the previous information had been up to 2.4 MeV. A negative parity band built on 1437 keV (11/2⁻) excited state has been extended to 5.5 MeV. The structure seems to show a coexistence of single particle and collective modes of excitation. Properties of both the nuclei have been compared with shell model calculations using OXBASH.     

Anomalous dimensions of Wilson operators in the N = 4\mathcal{N} = 4 supersymmetric Yang-Mills theory

We present results for the universal anomalous dimension γ_un(j) of Wilson twist-2 operators in the IE2.gif" alt=" $$\mathcal{N} = 4$$ " align="middle" border="0"> supersymmetric Yang-Mills theory in the first three orders of the perturbation theory. We obtain these expressions by extracting the most complicated terms from the corresponding anomalous dimensions in QCD. The result obtained agrees with the hypothesis of the integrability of the IE3.gif" alt=" $$\mathcal{N} = 4$$ " align="middle" border="0"> supersymmetric Yang-Mills theory in the context of the AdS/CFT correspondence. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 249–262, February, 2007.     

Strong laws of large numbers for weighted sums of $$\tilde \rho $$ -mixing random variables

Strong laws are established for linear statistics that are weighted sums of a IE2.gif" alt=" $$\tilde \rho $$ " align="middle" border="0"> -mixing random sample. The results obtained generalize the results of Baxter et al. [SLLN for weighted independent indentically distributed random variables, J. Theoret. Probab. 17 (2004), 165–181] to IE3.gif" alt=" $$\tilde \rho $$ " align="middle" border="0"> -mixing random variables. This paper is supported by Key discipline of Zhejiang Province (Key discipline of Statistics of Zhejiang Gongshang University) and National Natural Science Foundation of China.     

The Abelian category of quotients of $$\mathcal{L}\mathfrak{F}$$ -spaces

We construct the category of quotients of IE2.gif" alt=" $$\mathcal{L}\mathfrak{F}$$ " align="middle" border="0"> -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.     

Comparative analysis of real and \mathcal{P}\mathcal{T}-symmetric Scarf potentials

The Scarf I and Scarf II potentials are discussed within a common mathematical framework, which is then specified to handle the two potentials separately both in the conventional Hermitian and in the IE2.gif" alt=" $$\mathcal{P}\mathcal{T}$$ " align="middle" border="0"> -symmetric setting. The physically admissible solutions are identified in each case together with the corresponding energy eigenvalues. Several main differences between the IE3.gif" alt=" $$\mathcal{P}\mathcal{T}$$ " align="middle" border="0"> -symmetric Scarf I and II potentials are pointed out. These include the presence and absence of the quasi-parity quantum number, the sign of the pseudo-norm, the mechanism of the spontaneous breakdown of IE4.gif" alt=" $$\mathcal{P}\mathcal{T}$$ " align="middle" border="0"> symmetry and the non- IE5.gif" alt=" $$\mathcal{P}\mathcal{T}$$ " align="middle" border="0"> orthogonality of otherwise admissible solutions in the Scarf I potential. Similarities and differences with respect to the corresponding Hermitian systems are also pointed out.     

Synthesis of fulleropyrrolidine dicarboxylic acid and its scavenging activity to superoxide radicals (O_2 ^{\bar \cdot } )

Photochemical reaction of C60 with iminodiacetic ester (NH(CH₂COOR)₂, R=Me, Et, t-Bu) produced fulleropyrrolidine derivatives 2 in a 55%–36% yield (based on consumed C60). The reaction activity correlated with the steric hindrance of R groups. The order of the reaction rates decreased from Me to t-Bu (R=Me>Et>t-Bu). Fulleropyrrolidine derivative 2 (R=Me) were hydrolyzed with NaH and methanol in toluene, and then acidified with HCl to result in the corresponding fulleropyrrolidine dicarboxylic acid 3 in a 65% yield (relative to fulleropyrrolidine derivative 2). C60 derivatives 2 and 3 were structurally characterized by ¹H NMR, ¹³C NMR, IR, and elementary analysis, MS. A chemiluminescence technique was applied to study the effects of their scavenging superoxide radicals IE2.gif" alt=" $$(O_2 ^{\bar \cdot } )$$ " align="middle" border="0"> generated by pyrogallol autoxidation. The results show fulleropyrrolidine dicarboxylic acid 3 had scavenging activity and the efficiencies were dependent on their concentrations. At the concentration of 3.0×10⁻⁴ mol·L⁻¹, fulleropyrrolidine dicarboxylic acid 3 showed a radical scavenging efficiency of approximately 70%. Finally, the influence of structure factor on the scavenging activity was discussed. The results show that the monoadduct of C60, owing to keeping almost intact double bands in C60 moiety, had obvious scavenging activity for superoxide radicals IE3.gif" alt=" $$(O_2 ^{\bar \cdot } )$$ " align="middle" border="0"> . __________ Translated from Chinese Journal of Applied Chemistry, 2006, 23(1) (in Chinese)     

Variational measures related to local systems and the Ward property of \mathcal{P}-adic path bases

Some properties of absolutely continuous variational measures associated with local systems of sets are established. The classes of functions generating such measures are described. It is shown by constructing an example that there exists a IE2.gif" alt=" $$\mathcal{P}$$ " align="middle" border="0"> -adic path system that defines a differentiation basis which does not possess Ward property.