Sometimes slow growing is fast growing

The slow growing hierarchy is commonly defined as follows: G₀(x) = 0, G_x−1(x) := G_x(x) + 1 and G_λ(x) := G_λ[x](x) where λ<₀ is a limit and ·[·]:₀∩ Lim × ω → ₀ is a given assignment of fundamental sequences for the limits below ₀. The first obvious question which is encountered when one looks at this definition is: How does this hierarchy depend on the choice of the underlying system of fundamental sequences? Of course, it is well known and easy to prove that for the standard assignment of fundamental sequence the hierarchy (G_x)_x<0 is slow growing, i.e. each G_x is majorized by a Kalmar elementary recursive function.

It is shown in this paper that the slow growing hierarchy (G_x)_x<0 — when it is defined with respect to the norm-based assignment of fundamental sequences which is defined in the article by Cichon (1992, pp. 173–193) — is actually fast growing, i.e. each PA-provably recursive function is eventually dominated by G_x for some <₀. The exact classification of this hierarchy, i.e. the problem whether it is slow or fast growing, has been unsolved since 1992. The somewhat unexpected result of this paper shows that the slow growing hierarchy is extremely sensitive with respect to the choice of the underlying system of fundamental sequences.

The paper is essentially self-contained. Only little knowledge about ordinals less than ₀ — like the existence of Cantor normal forms, etc. and the beginnings of subrecursive hierarchy theory as presented, for example, in the 1984 textbook of Rose — is assumed.     

The closure approximation in the hierarchy equations

The expectation of the solution process in a stochastic operator equation can be obtained from averaged equations only under very special circumstances. Conditions for validity are given and the significance and validity of the approximation in widely used hierarchy methods and the self-consistent field approximation in nonequilibrium statistical mechanics are clarified. The error at any level of the hierarchy can be given and can be avoided by the use of the iterative method.Supported by the National Aeronautics and Space Administration (Grant NGR 11-003-020) and partially supported by the Office of Naval Research (Contract N 00014-69-A-0423 Themis).     

新的Liouville完全可积的广义Burgers方程族及其Hamilton结构

基于屠格式,从一个新的等谱问题,本文获得了一族广义Ｂｕｒｇｅｒｓ 方程及其Ｈａｍ ｉｌｔｏｎ 结构．最后证明了该族方程是Ｌｉｏｕｖｉｌｌｅ 完全可积的,并且有无穷多个彼此对合的公共守恒密度     

纳米多级结构枣核型多孔氧化亚铜的合成及拉曼性质

通过简单温和的水相还原反应, 在表面活性剂十二烷基苯磺酸钠(SDBS)的协同下合成了亚微米级枣核型多孔氧化亚铜纳米多级结构. 通过改变盐酸的加入量可以实现纳米粒子尺寸在300-900 nm范围可调. X射线衍射(XRD)和透射电子显微镜(TEM)分析表明, 亚微米粒子由小于10 nm的晶粒构成. 基于实验结果, 我们提出生长-刻蚀竞争生长模型来解释枣核型空心多孔结构的形成机制, 同时说明了尺寸调节机理. 利用激光拉曼光谱仪研究了样品的光学性质, 与相似尺寸的亚微米实心多面体的拉曼响应信号对比发现, 枣核型空心多孔结构的拉曼光谱表现出新颖的特性. 这些结果补充了氧化亚铜的拉曼光谱样本, 对文物表面颜料的无损拉曼检测具有一定的指导意义.     

草酸铕颗粒微米层状颗粒的合成及其光学性能

Hierarchical europium oxalate Eu₂(C₂O₄)₃·10H₂O micro-particles were synthesized through a simple precipitation method at room temperature in present of trisodium citrate. The prod-ucts were characterized by X-ray diffraction, X-ray photoelectron spectroscopy, field-emission scanning electron microscopy, and photoluminescence. The possible formation mechanism of the hierarchical europium oxalate Eu₂(C₂O₄)₃·10H₂O micro-particles was discussed.     

基于层次分析法的课程表的模糊综合评价

建立了高校课程表的评价指标体系,在此基础上提出了基于层次分析法的课程表的模糊综合评价方法,并用方根法简化模糊评价的权重集求解过程.通过对具体课程表的模糊综合评价,验证了该方法是科学的、可行的.     

Uncertainty index based interval assignment by Interval AHP

In a multi-attribute decision making problem, indigenous values are assigned to attributes based on a decision maker’s subjective judgments. The given judgments are often uncertain, because of the uncertainty of situations and intuitiveness of human judgments. In order to reflect the uncertainty in the assigned values, they are denoted as intervals whose widths represent the possibilities of attributes. Since it is difficult for a decision maker to assign values directly to attributes in case of more than two attributes, he/she gives a pairwise comparison matrix by comparing two attributes at one occasion. The given matrix contains two kinds of uncertainty, one is inconsistency among comparisons and the other is incompleteness of comparisons. This paper proposes the models to obtain intervals of attributes from the given uncertain pairwise comparison matrix. At first, the uncertainty indexes of a set of intervals are defined from the viewpoints of entropy in probability, sum or maximum of widths, or ignorance. Then, considering that too uncertain information is not useful, the intervals of attributes are obtained by minimizing their uncertainty indexes.     

A hierarchy of Turing degrees of divergence bounded computable real numbers

A real number x is f-bounded computable (f-bc, for short) for a function f if there is a computable sequence (x_s) of rational numbers which converges to x f-bounded effectively in the sense that, for any natural number n, the sequence (x_s) has at most f(n) non-overlapping jumps of size larger than 2^-n. f-bc reals are called divergence bounded computable if f is computable. In this paper we give a hierarchy theorem for Turing degrees of different classes of f-bc reals. More precisely, we will show that, for any computable functions f and g, if there exists a constant γ>1 such that, for any constant c, f(nγ)+n+cg(n) holds for almost all n, then the classes of Turing degrees given by f-bc and g-bc reals are different. As a corollary this implies immediately the result of [R. Rettinger, X. Zheng, On the Turing degrees of the divergence bounded computable reals, in: CiE 2005, June 8–15, Amsterdam, The Netherlands, Lecture Notes in Computer Science, vol. 3526, 2005, Springer, Berlin, pp. 418–428.] that the classes of Turing degrees of d-c.e. reals and divergence bounded computable reals are different.     

基于模糊层次分析的虚拟企业风险评价

在分析虚拟企业风险评价应该考虑的主要因素的基础上,建立了一套适用于虚拟企业风险评价的指标体系,提出了一种基于模糊集合理论与层次分析法相结合的虚拟企业风险评价方法。在多专家指标权重评价的基础上,运用模糊D elph i法对上层准则形成一系列的权重集合,得到三角模糊数表示的准则“重要性”评价。结合以模糊语言变量表示的备选方案偏好等级的“满意度”和下层指标的模糊权重,通过分层结合,得到虚拟企业风险的模糊评分,最后采用重心法和Chang Jing-rong的双系数法对所有方案的三角模糊数进行排序,得到虚拟企业方案综合选择的优先顺序。实例说明了虚拟企业风险评价的具体过程。