Reciprocal complementary Wiener numbers of trees, unicyclic graphs and bicyclic graphs

The reciprocal complementary Wiener number of a connected graph G is defined as

where V(G) is the vertex set, d(u,v|G) is the distance between vertices u and v, d is the diameter of G. We determine the trees with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers, and the unicyclic and bicyclic graphs with the smallest and the second smallest reciprocal complementary Wiener numbers.     

On the nearest parametric approximation of a fuzzy number—Revisited

The problem of finding the nearest parametric approximation of a fuzzy number with respect to the average Euclidean distance is completely solved. Properties of translation invariance, scale invariance, additivity, preservation of expected value, value and ambiguity of this new approximation operator are studied and an algorithm for computing it is provided.     

The hierarchical product of graphs

A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some well-known properties of the cartesian product, such as reduced mean distance and diameter, simple routing algorithms and some optimal communication protocols are inherited by the hierarchical product. We also address the study of some algebraic properties of the hierarchical product of two or more graphs. In particular, the spectrum of the binary hypertree T_m (which is the hierarchical product of several copies of the complete graph on two vertices) is fully characterized; turning out to be an interesting example of graph with all its eigenvalues distinct. Finally, some natural generalizations of the hierarchic product are proposed.     

Generalized derivatives of distance functions and the existence of nearest points

The relationships between the generalized directional derivative of the distance function and the existence of nearest points as well as some geometry properties in Banach spaces are studied. It is proved in the present paper that the condition that for each closed subset G$G$ of X$X$ and x∈X?G$x\in X?G$, the Clarke, Michel-Penot, Dini or modified Dini directional derivative of the distance function is 1 or −1 implying the existence of the nearest points to x$x$ from G$G$ is equivalent to X$X$ being compactly locally uniformly convex. Similar results for uniqueness of the nearest point are also established.     

Abstract Voronoi diagrams revisited

Abstract Voronoi diagrams [R. Klein, Concrete and Abstract Voronoi Diagrams, Lecture Notes in Computer Science, vol. 400, Springer-Verlag, 1987] were designed as a unifying concept that should include as many concrete types of diagrams as possible. To ensure that abstract Voronoi diagrams, built from given sets of bisecting curves, are finite graphs, it was required that any two bisecting curves intersect only finitely often; this axiom was a cornerstone of the theory. In [A.G. Corbalan, M. Mazon, T. Recio, Geometry of bisectors for strictly convex distance functions, International Journal of Computational Geometry and Applications 6 (1) (1996) 45–58], Corbalan et al. gave an example of a smooth convex distance function whose bisectors have infinitely many intersections, so that it was not covered by the existing AVD theory. In this paper we give a new axiomatic foundation of abstract Voronoi diagrams that works without the finite intersection property.     

The nature of singularities in mean curvature flow of mean-convex sets

This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact mean-convex region of or of an -dimensional riemannian manifold. If , the moving boundary is shown to be very nearly convex in a spacetime neighborhood of any singularity. In particular, the tangent flows at singular points are all shrinking spheres or shrinking cylinders. If , the same results are shown up to the first time that singularities occur.

     

Inhomogeneous mean field models

The infinite set of coupled mean field equations for a classical inhomogeneous Ising ferromagnet is studied with respect to existence and uniqueness of its solutions.     

A Hidden-Markov estimation method for mean-shift detection of fraction defective in production process control

This paper considers a statistical method of estimating mean shift for a fraction defective of population. One traditional method for this estimation problem has been known as the CUSUM (cumulative sum) method, and it provides a method of estimating the occurrence of a shift in the mean from the observed data. We consider this estimation problem of shift occurrence in a production process. It is assumed that the process has two states, one is good (fraction defective low) and the other bad (fraction defective high), and starts in good state with probability one. We are interested in judging when the state has moved to the bad state by analyzing the observed data.In this paper, we model such a phenomenon as a hidden-Markov model. The states which are unobservable in a hidden-Markov model can be analyzed from the sequence of observed results. Hence, the advantage of this modeling technique is that the unknown parameters which are included in the hidden states can be estimated. We compare the performance of this hidden-Markov model with the CUSUM method based on several simulation data sets.     

线性回归诊断的若干问题

本文对于线性回归诊断提出了几种新的模型和方法。我们首次研究了方差加权和均值漂移的混合模型,得到了相应的诊断统计量。本文还引入了罚函数方法,并以此为工具,讨论了若干有偏估计的影响度量,最后,本文提出了基于重心的诊断统计量,对于识别异常点有较好的效果。     

模糊数值函数的强可导性、连续性、局部平均McShane可积性

通过所提出的模糊数值函数的强可导性和平均 Mc Shane可积性等概念 ,对模糊数值函数的连续性从导数和积分两方面进行了讨论和刻画 ,也回答了强可导性与强导数可积之间的关系 ,给出两个充分必要条件。