The PI index of product graphs

The Padmakar–Ivan index of a graph G is the sum over all edges uv of G of number of edges which are not equidistant from u and v. In this work, an exact expression for the PI index of the Cartesian product of bipartite graphs is computed. Using this formula, the PI indices of C₄ nanotubes and nanotori are computed.     

The exact number of solutions for the second order nonlinear boundary value problem

A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA, P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained by the root function method as well. The paper was supported by the Grant VEGA No. 2/7140/27, Bratislava.     

Computing power indices: Multilinear extensions and new characterizations

Johnston [Johnston, R.J., 1978. On the measurement of power: some reactions to Laver. Environment and Planning A 10, 907–914], Deegan and Packel [Deegan, J., Packel, E.W., 1979. A new index of power for simple n-person games. International Journal of Game Theory 7, 113–123], and Holler [Holler, M.J., 1982. Forming coalitions and measuring voting power. Political Studies 30, 262–271] proposed three power indices for simple games: Johnston index, Deegan–Packel index, and the Public Good Index. In this paper, methods to compute these indices by means of the multilinear extension of the game are presented. Furthermore, a new characterization of the Public Good Index is given. Our methods are applied to two real-world examples taken from the political field.     

夏普里值在顾客满意度测评中的应用研究

利用夏普里值来识别导致顾客不满意的质量属性.因为夏普里值差异很小的质量属性具有高度的相关性,所以可以把具有相同或者类似的夏普里值的质量属性划分为同类属性,从而能够识别导致顾客不满意的质量属性.算例进一步验证该方法的可行性和正确性.     

一类二阶常微分方程边值问题的无穷多个解

利用变分原理和Z2不变群指标研究了二阶常微分方程边值问题{u″(t)-u(t) f(t,u(t))=0,0＜t＜1,u′(0)=0,α1u(1) u′(1)=0,(其中α1＞-1/2).得出了这类方程存在无穷个解的充分条件.     

g-循环矩阵的谱分解和Jordan块结构

首先证明了n级非奇异g-循环矩阵必定可以对角化，并且给出了它的谱分解。其次，当（n,g）=1时，给出了n级奇异g-循环矩阵相似于某些对角阵和某些幂零Jordan块的直和，进一步给出了其中幂零Jordan块的幂零指数的计算法。     

A successful algorithm for the undirected Hamiltonian path problem

In this paper a polynomial algorithm called the Minram algorithm is presented which finds a Hamiltonian Path in an undirected graph with high frequency of success for graphs up to 1000 nodes. It first reintroduces the concept described in [13] and then explains the algorithm. Computational comparison with the algorithm by Posa [10] is given.It is shown that a Hamiltonian Path is a spanning arborescence with zero ramification index. Given an undirected graph, the Minram algorithm starts by finding a spanning tree which defines a unique spanning arborescence. By suitable pivots it locates a locally minimal value of the ramification index. If this local minima corresponds to zero ramification index then the algorithm is considered to have ended successfully, else a failure is reported.Computational performance of the algorithm on randomly generated Hamiltonian graphs is given. The random graphs used as test problems were generated using the procedure explained in Section 6.1. Comparison with our version of the Posa algorithm which we call Posa-ran algorithm [10] is also made.     

Satisfactory solutions approach to parameter optimization of dynamic systems with vector performance index

A novel approach to parameter optimization of large dynamic systems using vector performance index is described. The approach entails characterizing and determining a set of satisfactory solutions to the multiobjective optimization problem. The satisfactory solutions are defined with respect to a prespecified and satisfactory set of bounds on the indices. A theoretical basis is provided to obtain a compact and connected set of satisfactory solutions in the parameter space. Compactness and connectedness are essential requirements, since they assure a range of values for the parameters. An expedient numerical technique for determining the range of satisfactory values for the parameters is illustrated with an example. The satisfactory solutions approach provides a basis for designing a system with multiple requirements when all of them cannot be formulated in the framework of a composite vector index problem.This work constitutes part of the first author's doctoral dissertation in the Department of Electrical Engineering and Computer Science, University of Illinois at Chicago, Chicago, Illinois. It was supported in part by the National Science Foundation, Grant No. ENG-76-09930. The first author is indebted to Drs. A. C. Raptis and T. P. Mulcahey of Argonne National Laboratory for their support and encouragement.