Difference Sets and Recursion Theory

There is a recursive set of natural numbers which is the difference set of some recursively enumerable set but which is not the difference set of any recursive set.     

On the use of the Galerkin method for 3D numerical modelling of the general circulation: the South Atlantic eperiment

A linear three‐dimensional hydrodynamical numerical model, with the application of the Galerkin Method for the vertical dependence, is here presented. The spherical coordinate system is used, in order to allow large‐scale simulations. The equations and mathematical development of the model are shown in detail, together with the boundary and initial conditions, and the sequence of equations' solution. The model is applied to the South Atlantic Ocean, for estimating typical seasonal circulations, and the results are summarized in maps of currents at surface and 1000 m depth, and in transport values of the Brazil Current between 30°S and 40°S. Copyright © 2002 John Wiley & Sons, Ltd.     

关于Menger图的几种运算

设Ｇ是一个简单连通图，若分离Ｇ的余一独立集Ｓ的最小点数等于连接Ｓ的点之间的内部不相交路的最大个数，则称Ｇ是Ｍｅｎｇｅｒ图。我们考虑了图的几种运算并给出了运算后的图是Ｍｅｎｇｅｒ图的条件。     

The Maximal Closed Classes of Unary Functions in p-Valued Logic

In many-valued logic the decision of functional completeness is a basic and important problem, and the thorough solution to this problem depends on determining all maximal closed sets in the set of many-valued logic functions. It includes three famous problems, i.e., to determine all maximal closed sets in the set of the total, of the partial and of the unary many-valued logic functions, respectively. The first two problems have been completely solved ([1], [2], [8]), and the solution to the third problem boils down to determining all maximal subgroups in the k-degree symmetric group S_k, which is an open problem in the finite group theory. In this paper, all maximal closed sets in the set of unary p-valued logic functions are determined, where p is a prime. Mathematics Subject Classification: 03B50, 20B35.     

Essential spectrum and -solutions of one-dimensional Schrödinger operators

In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of -solutions. The proof of this theorem is based on inverse spectral theory.

     

Basis Set Orbital Relaxation in Atomic and Molecular Hydrogen Systems

Orbital relaxation (OR) amounts to variation of the orbital exponents in hydrogen molecules and ions relative to the exponents of the isolated atom; it is represented as the sum of the one- and two-center contributions depending on the effective atomic charge and on the presence of other atoms in the molecule. The procedure for isolating the contributions of the exponent includes treatment of the OR of hydrogen in a special set of neutral and charged atoms and molecules with certain multiplicities of their electronic states. Within the framework of the spin-unrestricted Hartree-Fock method, we found and discussed the optimal values of the exponents of the basis orbitals of hydrogen atoms and molecules using the minimal split valence-shell basis set, the basis set that includes the polarization function, and the expanded set of grouped natural orbitals. A simple energy model is suggested for OR. Expressions are derived for evaluating the exponents of the relaxed orbitals in hydrogen-containing systems.Original Russian Text Copyright © 2004 by A. I. Ermakov, A. E. Merkulov, A. A. Svechnikova, and V. V. Belousov__________Translated from Zhurnal Strukturnoi Khimii, Vol. 45, No. 6, pp. 973–978, November–December, 2004.     

From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns

Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.     

Three-Way Decision Models Based on Ideal Relations in Multi-Attribute Decision-Making

In recent years, research on applications of three-way decision (e.g., TWD) has attracted the attention of many scholars. In this paper, we combine TWD with multi-attribute decision-making (MADM). First, we utilize the essential idea of TOPSIS in MADM theory to propose a pair of new ideal relation models based on TWD, namely, the three-way ideal superiority model and the three-way ideal inferiority model. Second, in order to reduce errors caused by the subjectivity of decision-makers, we develop two new methods to calculate the state sets for the two proposed ideal relation models. Third, we employ aggregate relative loss functions to calculate the thresholds of each object, divide all objects into three different territories and sort all objects. Then, we use a concrete example of building appearance selection to verify the rationality and feasibility of our proposed models. Furthermore, we apply comparative analysis, Spearman’s rank correlation analysis and experiment analysis to illustrate the consistency and superiority of our methods.     

概率语言多属性群决策方法及其在新型智慧城市市民获得感评价中的应用

针对公众参与的语言信息多属性群决策问题,研究了考虑参与者满意度的概率语言多属性群决策方法。首先,根据参与者的语言评价信息确定并规范化概率语言决策矩阵。然后,对大群体进行共识分析,由最大化参与者群体的满意度构建线性规划模型,确定参与者群组的权重;构造正、负理想方案的评价向量,构建多目标规划模型,用拉格朗日乘子法求解属性权重;定义各方案的加权贴近度,并以此对方案进行排序和优选。最后,通过新型智慧城市市民获得感评价案例验证了模型的可行性和有效性。     

Hydrophobic effect for anionic dye - cationic surfactant interaction in aqueous and mixed solvent

Abstract

The interaction between anionic dyes [Reactive Orange 122 (R.O 122), Reactive Blue 19 (R.B 19), Reactive Violet 5 (R.V 5) and Acid Green 20 (A.G 20)] with cationic surfactant cetyltrimethylammoniun bromide (CTAB) has been investigated by spectrophotometry and conductance technique. The used dyes are characterized by tautomeric behavior which affects the mechanism of the interaction. Various parameters such as dye structure, surfactant composition, solvent composition, temperature and pH of the medium were studied. The spectral data were applied for calculating the binding constant between dye and surfactant (K_b), fraction of micellization (?_mic), and standard free energy change of binding (ΔG°_b) in 0,10,20 and 30 v/v % acetonitile (AN). Conductance technique was constructed to estimate the ion pairing constant (K_a) at different temperatures and v/v % AN. Thermodynamic parameters (ΔG°, ΔH° and ΔS°) for ion pair formation were evaluated. The role of hydrophobic and electrostatic effect on dye-surfactant interaction was discussed.