A maximum principle for combinatorial Yamabe flow

This article studies a discrete geometric structure on triangulated manifolds and an associated curvature flow (combinatorial Yamabe flow). The associated evolution of curvature appears to be like a heat equation on graphs, but it can be shown to not satisfy the maximum principle. The notion of a parabolic-like operator is introduced as an operator which satisfies the maximum principle, but may not be parabolic in the usual sense of operators on graphs. A maximum principle is derived for the curvature of combinatorial Yamabe flow under certain assumptions on the triangulation, and hence the heat operator is shown to be parabolic-like. The maximum principle then allows a characterization of the curvature as well was a proof of long term existence of the flow.     

Unbounded quantum graphs with unbounded boundary conditions

We consider metric graphs with a uniform lower bound on the edge lengths but no further restrictions. We discuss how to describe every local self‐adjoint Laplace operator on such graphs by boundary conditions in the vertices given by projections and self‐adjoint operators. We then characterize the lower bounded self‐adjoint Laplacians and determine their associated quadratic form in terms of the operator families encoding the boundary conditions.     

Continuous interval-valued intuitionistic fuzzy aggregation operators and their applications to group decision making

In this paper, we introduce a new operator called the continuous interval-valued intuitionistic fuzzy ordered weighted averaging (C-IVIFOWA) operator for aggregating the interval-valued intuitionistic fuzzy values. It combines the intuitionistic fuzzy ordered weighted averaging (IFOWA) operator and the continuous ordered weighted averaging (C-OWA) operator by a controlling parameter, which can be employed to diminish fuzziness and improve the accuracy of decision making. We further apply the C-IVIFOWA operator to the aggregation of multiple interval-valued intuitionistic fuzzy values and obtain a wide range of aggregation operators including the weighted C-IVIFOWA (WC-IVIFOWA) operator, the ordered weighted (OWC-IVIFOWA) operator and the combined C-IVIFOWA (CC-IVIFOWA) operator. Some desirable properties of these operators are investigated. And finally, we give a numerical example to illustrate the applications of these operators to group decision making under interval-valued intuitionistic fuzzy environment.     

Multiple criteria decision making method based on normal interval‐valued intuitionistic fuzzy generalized aggregation operator

On the basis of the normal intuitionistic fuzzy numbers (NIFNs), we proposed the normal interval‐valued intuitionistic fuzzy numbers (NIVIFNs) in which the values of the membership and nonmembership were extended to interval numbers. First, the definition, the properties, the score function and accuracy function of the NIVIFNs are briefly introduced, and the operational laws are defined. Second, some aggregation operators based on the NIVIFNs are proposed, such as normal interval‐valued intuitionistic fuzzy weighted arithmetic averaging operator, normal interval‐valued intuitionistic fuzzy ordered weighted arithmetic averaging operator, normal interval‐valued intuitionistic fuzzy hybrid weighted arithmetic averaging operator, normal interval‐valued intuitionistic fuzzy weighted geometric averaging operator, normal interval‐valued intuitionistic fuzzy ordered weighted geometric averaging operator, normal interval‐valued intuitionistic fuzzy hybrid weighted geometric averaging operator, and normal interval‐valued intuitionistic fuzzy generalized weighted averaging operator, normal interval‐valued intuitionistic fuzzy generalized ordered weighted averaging operator, normal interval‐valued intuitionistic fuzzy generalized hybrid weighted averaging operator, and some properties of these operators, such as idempotency, monotonicity, boundedness, commutativity, are studied. Further, an approach to the decision making problems with the NIVIFNs is established. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness. © 2015 Wiley Periodicals, Inc. Complexity 21: 277–290, 2016     

Multi-criteria decision-making method based on normal intuitionistic fuzzy-induced generalized aggregation operator

Normal intuitionistic fuzzy numbers (NIFNs), which use normal fuzzy numbers to express their membership and non-membership functions, can reflect the evaluation information exactly in different dimensions. In this paper, we are committed to apply NIFNs to multi-criteria decision-making (MCDM) problems, and meanwhile some new aggregation operators are proposed, including normal intuitionistic fuzzy weighted arithmetic averaging operator, normal intuitionistic fuzzy weighted geometric averaging operator, normal intuitionistic fuzzy-induced ordered weighted averaging operator, normal intuitionistic fuzzy-induced ordered weighted geometric averaging operator and normal intuitionistic fuzzy-induced generalized ordered weighted averaging operator (NIFIGOWA). Based on the NIFIGOWA operator, an approach is introduced to solve MCDM problems where the criteria values are NIFNs and the criteria weight information is fixed. Finally, the proposed method is compared to the existing methods by virtue of a numerical example to verify its feasibility and rationality.     

Normalized graph Laplacians for directed graphs

We consider the normalized Laplace operator for directed graphs with positive and negative edge weights. This generalization of the normalized Laplace operator for undirected graphs is used to characterize directed acyclic graphs. Moreover, we identify certain structural properties of the underlying graph with extremal eigenvalues of the normalized Laplace operator. We prove comparison theorems that establish a relationship between the eigenvalues of directed graphs and certain undirected graphs. This relationship is used to derive eigenvalue estimates for directed graphs. Finally we introduce the concept of neighborhood graphs for directed graphs and use it to obtain further eigenvalue estimates.     

The OWA-based consensus operator under linguistic representation models using position indexes

When using linguistic approaches to solve decision problems, we need linguistic representation models. The symbolic model, the 2-tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes. Together with these three linguistic models, the corresponding ordered weighted averaging operators, such as the linguistic ordered weighted averaging operator, the 2-tuple ordered weighted averaging operator and the extended ordered weighted averaging operator, have been developed, respectively. In this paper, we analyze the internal relationship among these operators, and propose a consensus operator under the continuous linguistic model (or the 2-tuple fuzzy linguistic representation model). The proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures. Some desired properties of the consensus operator are also presented. In particular, the consensus operator provides an alternative consensus model for group decision making. This consensus model preserves the original preference information given by the decision makers as much as possible, and supports consensus process automatically, without moderator.     

The orness measures for two compound quasi-arithmetic mean aggregation operators

The paper first summarizes the orness measures and their common characteristics of some averaging operators: the quasi-arithmetic mean, the ordered weighted averaging (OWA) operator, the regular increasing monotone (RIM) quantifier and the weighted function average operator, respectively. Then it focuses on the aggregation properties and operator determination methods for two kinds of quasi-arithmetic mean-based compound aggregation operators: the quasi-OWA (ordered weighted averaging) operator and the Bajraktarevi? mean. The former is the combination of the quasi-arithmetic mean and the OWA operator, while the latter is the combination of the quasi-arithmetic mean and the weighted function average operator. Two quasi-OWA operator forms are given, where the OWA operator is assigned directly or generated from a RIM (regular increasing monotone) quantifier indirectly. The orness indexes to reflect the or-like level of the quasi-OWA operator and Bajraktarevi? mean are proposed. With generating function techniques, the properties of the quasi-OWA operator and Bajraktarevi? mean are discussed to show the rationality of these orness definitions. Based on these properties, two families of parameterized quasi-OWA operator and Bajraktarevi? mean with exponential and power function generators are proposed and compared. It shows that the method of this paper can also be applied to other function-based aggregation operators.     

Explicit Construction and Uniqueness for Universal Operator Algebras of Directed Graphs

Given a directed graph, there exist a universal operator algebraand universal C*-algebra associated to the directed graph. Inthis paper we give intrinsic constructions for these objects.We also provide an explicit construction for the maximal C*-algebraof an operator algebra. We discuss uniqueness of the universalalgebras for finite graphs, showing that for finite graphs thegraph is an isomorphism invariant for the universal operatoralgebra of a directed graph. We show that the underlying undirectedgraph is a Banach algebra isomorphism invariant for the universalC*-algebra of a directed graph.     

Universal Graphs with Forbidden Subgraphs and Algebraic Closure

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated “algebraic closure” operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases.     

Periodic solutions of nonlinear hyperbolic evolution systems

We shall deal with the periodic problem for nonlinear perturbations of abstract hyperbolic evolution equations generating an evolution system of contractions. We prove an averaging principle for the translation operator associated with the nonlinear evolution system, expressed in terms of the topological degree. The abstract results shall be applied to the damped hyperbolic partial differential equation.     

Mass Operator and Dynamical Implementation of Mass Superselection Rule

We start reviewing Giulini’s dynamical approach to Bargmann superselection rule proposing some improvements. First of all we discuss some general features of the central extensions of the Galilean group used in Giulini’s programme, in particular focussing on the interplay of classical and quantum picture, without making any particular choice for the multipliers. Preserving other features of Giulini’s approach, we modify the mass operator of a Galilei invariant quantum system to obtain a mass spectrum that is (i) positive and (ii) discrete, so giving rise to a standard (non-continuous) superselection rule. The model results to be invariant under time reversal but a further degree of freedom appears that can be interpreted as describing an internal conserved charge of the system (however, adopting a POVM approach, the unobservable degrees of freedom can be pictured as a generalized observable automatically gaining a positive mass operator without assuming the existence of such a charge). The effectiveness of Bargmann rule is shown to be equivalent to an averaging procedure over the unobservable degrees of freedom of the central extension of Galileian group. Moreover, viewing the Galileian invariant quantum mechanics as a non-relativistic limit, we prove that the above-mentioned averaging procedure giving rise to Bargmann superselection rule is nothing but an effective de-coherence phenomenon due to time evolution if assuming that real measurements includes a temporal averaging procedure. It happens when the added term Mc ² is taken in due account in the Hamiltonian operator since, in the dynamical approach, the mass M is an operator and cannot be trivially neglected as in classical mechanics. The presented results are quite general and rely upon the only hypothesis that the mass operator has point spectrum. These results explicitly show the interplay of the period of time of the averaging procedure, the energy content of the considered states, and the minimal difference of the mass operator eigenvalues.     

基于Theil不等系数的IOWHA算子组合预测模型及其应用

Theil不等系数是一种度量预测精度的另一种相关性指标,将Theil不等系数和IOWHA算子的概念相结合,提出一种新的基于Theil不等系数的IOWHA算子最优组合预测模型,并定义了优性组合预测的概念.利用该模型对税收收入进行了组合预测,结果表明该组合预测方法是可行和有效的,且该模型是一种优性组合预测.     

Development of some linguistic aggregation operators with conservation of interaction between criteria and their application in multiple attribute group decision problems

Decision making problems with fuzzy linguistic information and Choquet integral is investigated in this paper, to introduce three new aggregation operators: 2-tuple choquet integral averaging operator (TCIA), 2-tuple ordered choquet integral averaging operator (TOCIA) and combined 2-tuple choquet integral averaging operator which can be used to aggregate preference information that takes not only the form of linguistic variables but also study some of its desirable properties. Also, we developed a practical method based on TCIA as well as TOCIA operators for multiple attribute group decision making. Furthermore, in this approach alternative assessments are computed by the aggregation of 2-tuple linguistic information. Therefore, ranking alternatives or selecting the most desirable ones will be the outcome of the comparison between the 2-tuple linguistic information. Ultimately, the demonstration of the developed approaches, its practicality and its effectiveness is proved by a numerical example and a comparison with results issued from Wan method (Knowl-Based Syst 45:31–40, 2013).     

基于对偶犹豫模糊语言变量的多属性决策方法

首先定义了对偶犹豫模糊语言变量,然后给出其运算规则、得分值函数、精确值函数、比较规则以及对偶犹豫模糊语言变量的加权算术平均算子、有序加权算术平均算子和混合平均算子。针对属性值为对偶犹豫模糊语言变量的多属性决策问题,提出了一种基于对偶犹豫模糊语言变量集结算子的多属性决策方法。最后,结合国家电网公司合作单位选择问题,验证了该方法的有效性和可行性。     

Operator Algebraic Quotient Structures Induced by Direct Graphs II

The main purpose of this paper is to investigate the operator algebraic quotient structures induced by directed graphs. We enlarge our study of Cho (Compl Anal Oper Theory, 2008) to the general case. This can be done by constructing new graphs from given graphs called the pull-back graphs. We consider the corresponding groupoids, and von Neumann algebras of pull-back graphs.     

Operator Algebraic Quotient Structures Induced by Directed Graphs

Our result is about inclusions for (finite or infinite) countable directed graphs. In the proof, we use Free Probability Theory, groupoids, and algebras of operators (von Neumann algebras). We show that inclusions of directed graphs induce quotients for associated groupoid actions. With the use of operator thechniques, we then establish a duality between inclusions of graphs on the one hand and quotients of algebras on the other. Our main result is that each connected graph induces a quotient generated by a free group. This is a generalization of the notion of induced representations in the context of unitary representations of groups, i.e., the induction from the representations of a subgroup of an ambient group. The analogue is to systems of imprimitivity based on the homogeneous space. The parallel of this is the more general context of graphs (extending from groups to groupoids): We first prove that inclusions for connected graphs correspond to free group quotients, and we then build up the general case via connected components of given graphs.     

The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator

We show that the critical Kac–Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac–Ward operator on any planar graph is given by what we call a fermionic generating function. We also present a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac–Ward operators on isoradial graphs.     

A Scoring Criterion For Learning Chain Graphs

A chain graph allows both directed and undirected edges, and contains the underlying mathematical properties of the two. An important method of learning graphical models is to use scoring criteria to measure how well the graph structures fit the data. In this paper, we present a scoring criterion for learning chain graphs based on the Kullback Leibler distance. It is score equivalent, that is, equivalent chain graphs obtain the same score, so it can be used to perform model selection and model averaging.     

Group decision making with expertons and uncertain generalized probabilistic weighted aggregation operators

Expertons and uncertain aggregation operators are tools for dealing with imprecise information that can be assessed with interval numbers. This paper introduces the uncertain generalized probabilistic weighted averaging (UGPWA) operator. It is an aggregation operator that unifies the probability and the weighted average in the same formulation considering the degree of importance that each concept has in the aggregation. Moreover, it is able to assess uncertain environments that cannot be assessed with exact numbers but it is possible to use interval numbers. Thus, we can analyze imprecise information considering the minimum and the maximum result that may occur. Further extensions to this approach are presented including the quasi-arithmetic uncertain probabilistic weighted averaging operator and the uncertain generalized probabilistic weighted moving average. We analyze the applicability of this new approach in a group decision making problem by using the theory of expertons in strategic management.