T-sum of sigmoid-shaped fuzzy intervals

The usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of Zadeh’s extension principle based on a t-norm T. A t-norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation, if this arithmetic operation is closed for this class. It is important to know which t-norms are consistent with particular types of fuzzy intervals. Recently, Dombi and Gy?rbíró [J. Dombi, N. Gy?rbíró, Additions of sigmoid-shaped fuzzy intervals using the Dombi operator and infinite sum theorems, Fuzzy Sets and Systems 157 (2006) 952-963] proved that addition is closed if the Dombi t-norm is used with sigmoid-shaped fuzzy intervals. In this paper, we define a broader class of sigmoid-shaped fuzzy intervals. Then, we study t-norms that are consistent with these particular types of fuzzy intervals. Dombi and Gy?rbíró’s results are special cases of the results described in this paper.     

On fuzzy approximation operators in attribute reduction with fuzzy rough sets

Generally speaking, there are four fuzzy approximation operators defined on a general triangular norm (t-norm) framework in fuzzy rough sets. Different types of t-norms specify various approximation operators. One issue whether and how the different fuzzy approximation operators affect the result of attribute reduction is then arisen. This paper addresses this issue from the theoretical viewpoint by reviewing attribute reduction with fuzzy rough sets and then describing and proving some theorems which demonstrate the effects of the fuzzy approximation operators on the results of attribute reduction. First, we review some notions of attribute reduction with fuzzy rough sets, such as positive region, dependency degree and attribute reduction. We then present and prove some theorems which describe how and to what degree fuzzy approximation operators impact the performance of attribute reduction. Finally, we report some experimental simulation results which demonstrate the effectiveness and correctness of the theoretical contributions. One main contribution in this paper is that we have described and proven that each attribute reduction obtained using one type of fuzzy lower approximation operator always contains one reduction obtained using the other type of fuzzy lower approximation operator.     

Applications of interval valued t-norms (t-conorms) to fuzzy n-ary sub-hypergroups

In this paper, we study a generalization of group, hypergroup and n-ary group. Firstly, we define interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). We give a necessary and sufficient condition for, an interval-valued fuzzy subset to be an interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup are obtained with respect to infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Also, we obtain some results of T-product (S-product) of the interval-valued fuzzy subsets for infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Lastly, we investigate some properties of interval-valued fuzzy subsets of the fundamental n-ary group with infinitely ∨-distributive t-norms T (∧-distributive t-conorms S).     

Strong orientations of complete k-partite graphs achieving the strong diameter

In this paper, we consider each integer d between the lower and upper orientable strong diameter of the complete k-partite graph K(m₁,m₂,…,mk), and show that there does exist a strong orientation of K(m₁,m₂,…,mk) such that sdiam(K)=d.     

(?, T)-fuzzy rough approximation operators and the TL-fuzzy rough ideals on a ring

In this paper, we consider a ring as a universal set and study (?, T)-fuzzy rough approximation operators with respect to a TL-fuzzy ideal of a ring. First, some new properties of generalized (?, T)-fuzzy rough approximation operators are obtained. Then, a new fuzzy algebraic structure - TL-fuzzy rough ideal is defined and its properties investigated. And finally, the homomorphism of (?, T)-fuzzy rough approximation operators is studied.     

A parametric optimization approach to H∞ and H2 strong stabilization

This paper presents an approach for designing stable MIMO H_∞ and H₂ controllers by directly computing the norm-constrained stable transfer matrices Q in the H_∞ and H₂ suboptimal controller parameterizations. This is done by first converting the H₂ and H_∞ strong stabilization problems into some nonlinear unconstrained optimization problems through explicit parameterization of the norm-constrained Q's for any fixed order. Then, a two-stage numerical search is carried out by using a combination of a genetic algorithm and a quasi-Newton algorithm in order to reach an optimal solution. The effectiveness of the proposed algorithms is illustrated through some benchmark numerical examples.     

On a new class of fuzzy implications: h-Implications and generalizations

A new class of fuzzy implications called the h-implications is introduced. They are implications generated from an additive generator of a representable uninorm in a similar way of Yager’s f- and g-implications which are generated from additive generators of continuous Archimedean t-norms and t-conorms. Basic properties of these implications are studied in detail. Modifications and generalizations of the initial definition are presented and their properties studied and compared between them. One of the modifications, called (h, e)-implications, is another example of a fuzzy implication satisfying the exchange principle but not the law of importation for any t-norm, in fact for any function F : [0, 1]² → [0, 1].     

H2 and H∞ norm computations of linear continuous-time periodic systems via the skew analysis of frequency response operators

The H₂ and H_∞ norm computations of finite-dimensional linear continuous-time periodic (FDLCP) systems through the frequency response operators defined by steady-state analysis are discussed. By the skew truncation, the H₂ norm can be reached to any degree of accuracy by that of an asymptotically equivalent linear time-invariant (LTI) continuous-time system. The H_∞ norm can be approximated by the maximum singular value of the frequency response of an asymptotically equivalent LTI continuous-time system over a certain frequency range via the modified skew truncation. By the latter result, a Hamiltonian test is proved for FDLCP systems in an LTI fashion, based on which a modified bisection algorithm is developed.     

On v-filters and normal v-filters of a residuated lattice with a weak vt-operator

In this paper, we study further the filter theory of residuated lattices. First, we discuss the concepts of filters and normal filters of residuated lattices and propose some equivalent conditions for them. Then we introduce and investigate the notions of v-filter and normal v-filter of a residuated lattice with a weak vt-operator and lay bare the formulas for calculating the v-filters and the normal v-filters generated by subsets. Finally we show that the lattices of v-filters and normal v-filters of a residuated lattice with a vt-operator are both complete Brouwerian lattices.     

On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension

In this work, we propose a new class of distance functions called weighted t-cost distances. This function maximizes the weighted contribution of different t-cost norms in n-dimensional space. With proper weight assignment, this class of function also generalizes m-neighbor and octagonal distances. A non-strict upper bound (denoted as R_u in this work) of its relative error with respect to Euclidean norm is derived and an optimal weight assignment by minimizing R_u is obtained. However, it is observed that the strict upper bound of weighted t-cost norm may be significantly lower than R_u. For example, an inverse square root weight assignment leads to a good approximation of Euclidean norm in arbitrary dimension.     

Lattice structure of e-implications on L

In this paper, we consider the set of all e-implications on L^∗ and define two operations on the set of all representable e-implications on L^∗, thus endowing it with monoid structure which is also a distributive complete lattice.     

On fuzzy relational equations and the covering problem

Previous studies have shown that fuzzy relational equations (FREs) based on either the max-continuous Archimedean t-norm or the max-arithmetic mean composition can be transformed into the covering problem, which is an NP-hard problem. Exploiting the properties common to the continuous Archimedean t-norm and the arithmetic mean, this study proposes a generalization of them as the “u-norm”, enabling FREs that are based on the max-continuous u-norm composition also to be transformed into the covering problem. This study also proposes a procedure for transforming the covering problem into max-product FREs. Consequently, max-continuous u-norm FREs can be solved by extending any procedure for solving either the covering problem or max-product FREs.     

Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution

Mixtures of factor analyzers enable model-based density estimation to be undertaken for high-dimensional data, where the number of observations n is small relative to their dimension p. However, this approach is sensitive to outliers as it is based on a mixture model in which the multivariate normal family of distributions is assumed for the component error and factor distributions. An extension to mixtures of t-factor analyzers is considered, whereby the multivariate t-family is adopted for the component error and factor distributions. An EM-based algorithm is developed for the fitting of mixtures of t-factor analyzers. Its application is demonstrated in the clustering of some microarray gene-expression data.     

灰色GM(1,1,t α)模型与自忆性原理的耦合及应用

针对实际工程应用中传统GM(1,1)模型预测的局限性,以含时间幂次项的灰色GM(1,1,tα)模型为基础,构建了灰色GM(1,1,tα)与自忆性原理的耦合预测模型;用动力系统自忆性原理来克服传统灰色模型对初值比较敏感的弱点;将灰色GM(1,1,t2)自忆性模型应用于某沿海高速软土地基沉降的模拟和预测,获得了满意的模拟和预测精度.实验算例表明,所提出的新模型显著地改善了传统灰色预测模型的模拟预测精度.     

Compatibility of t-norms with the concept of ?-partition

In this paper we study the behaviour of a kind of partitions formed by fuzzy sets, the ?-partitions, with respect to three important operations: refinement, union and product of partitions. In the crisp set theory, the previous operations lead to new partitions: every refinement of a partition is also a partition; the union of partitions of disjoint sets is a partition of the union set; the product of two partitions of two sets is a partition of the intersection of the partitioned sets. It has been proven that ?-partitions extend the three previous properties when the intersection of fuzzy sets is defined by the minimum t-norm and the union by the maximum t-conorm. In this paper we consider any t-norm defining the intersection of fuzzy sets and we characterize those t-norms for which refinements, unions and products of ?-partitions are ?-partitions. We pay special attention to these characterizations in the case of continuous t-norms.     

On the additive constant of the k-server Work Function Algorithm

We consider the Work Function Algorithm for the k-server problem (Chrobak and Larmore, 1991; Koutsoupias and Papadimitriou, 1995) [2] and [4]. We show that if the Work Function Algorithm is c-competitive, then it is also strictly(2c)-competitive. As a consequence of (Koutsoupias and Papadimitriou, 1995) [4] this also shows that the Work Function Algorithm is strictly (4k−2)-competitive.     

A direct boosting algorithm for the k-nearest neighbor classifier via local warping of the distance metric

Though the k-nearest neighbor (k-NN) pattern classifier is an effective learning algorithm, it can result in large model sizes. To compensate, a number of variant algorithms have been developed that condense the model size of the k-NN classifier at the expense of accuracy. To increase the accuracy of these condensed models, we present a direct boosting algorithm for the k-NN classifier that creates an ensemble of models with locally modified distance weighting. An empirical study conducted on 10 standard databases from the UCI repository shows that this new Boosted k-NN algorithm has increased generalization accuracy in the majority of the datasets and never performs worse than standard k-NN.     

Upper bounds on the queuenumber of k-ary n-cubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn(G), is called the queuenumber of G. Heath and Rosenberg [SIAM J. Comput. 21 (1992) 927-958] showed that boolean n-cube (i.e., the n-dimensional hypercube) can be laid out using at most n−1 queues. Heath et al. [SIAM J. Discrete Math. 5 (1992) 398-412] showed that the ternary n-cube can be laid out using at most 2n−2 queues. Recently, Hasunuma and Hirota [Inform. Process. Lett. 104 (2007) 41-44] improved the upper bound on queuenumber to n−2 for hypercubes. In this paper, we deal with the upper bound on queuenumber of a wider class of graphs called k-ary n-cubes, which contains hypercubes and ternary n-cubes as subclasses. Our result improves the previous bound in the case of ternary n-cubes. Let denote the n-dimensional k-ary cube. This paper contributes three main results as follows:

(1)

if n?3.

(2)

if n?2 and 4?k?8.

(3)

if n?1 and k?9.

     

On the V-stability of complex dynamical networks

This paper introduces the concept of Lyapunov V-stability for complex dynamical networks. Under the new framework, each dynamical node is associated with a passivity degree, which indicates to what extent an effort is required for stabilizing the node. From this approach, the network stability problem is converted to measuring the negative definiteness of one simple matrix that characterizes the topology of the network. Pinning control is then suggested and investigated based on the new V-stability formulation. As an illustrative example, a network with different node dynamics and non-uniform coupling strengths is simulated to verify the analytic results. Moreover, a comparison study for three different kinds of networks is provided to further illustrate the novelty and efficacy of the proposed V-stability criterion and stabilization scheme.     

Characterizing r-perfect codes in direct products of two and three cycles

An r-perfect code of a graph G=(V,E) is a set C⊆V such that the r-balls centered at vertices of C form a partition of V. It is proved that the direct product of Cm and Cn (r?1, m,n?2r+1) contains an r-perfect code if and only if m and n are each a multiple of ²(r+1)+r² and that the direct product of Cm, Cn, and C? (r?1, m,n,??2r+1) contains an r-perfect code if and only if m, n, and ? are each a multiple of r³+³(r+1). The corresponding r-codes are essentially unique. Also, r-perfect codes in C2r×Cn (r?2, n?2r) are characterized.