Generalized fuzzy h-bi-ideals and h-quasi-ideals of hemirings

A new kind of generalized fuzzy h-ideals of a hemiring, namely, the (∈,∈∨q)-fuzzy h-bi-ideal (resp., h-quasi-ideal) is studied and the relationships between these generalized fuzzy h-ideals are described. Some characterization theorems of prime and semiprime (∈,∈∨q)-fuzzy h-bi-ideals (resp., h-quasi-ideals) of a hemiring are also given. In particular, we show that the h-hemiregular hemirings and h-intra-hemiregular hemirings can be described by using some of their generalized fuzzy h-ideals. Finally, the implication-based fuzzy h-bi-ideals (resp., h-quasi-ideals) of a hemiring are considered.     

A new type of fuzzy n-ary hyperstructures

The aim of this paper is to consider and study a new kind of fuzzy n-ary hyperstructures, such as fuzzy n-ary hypergroups and fuzzy (m,n)-ary hyperrings.     

Fuzzy and possibilistic clustering for fuzzy data

The Fuzzy k-Means clustering model (FkM) is a powerful tool for classifying objects into a set of k homogeneous clusters by means of the membership degrees of an object in a cluster. In FkM, for each object, the sum of the membership degrees in the clusters must be equal to one. Such a constraint may cause meaningless results, especially when noise is present. To avoid this drawback, it is possible to relax the constraint, leading to the so-called Possibilistic k-Means clustering model (PkM). In particular, attention is paid to the case in which the empirical information is affected by imprecision or vagueness. This is handled by means of LR fuzzy numbers. An FkM model for LR fuzzy data is firstly developed and a PkM model for the same type of data is then proposed. The results of a simulation experiment and of two applications to real world fuzzy data confirm the validity of both models, while providing indications as to some advantages connected with the use of the possibilistic approach.     

Geometric visualization of clusters obtained from fuzzy clustering algorithms

Fuzzy-clustering methods, such as fuzzy k-means and expectation maximization, allow an object to be assigned to multiple clusters with different degrees of membership. However, the memberships that result from fuzzy-clustering algorithms are difficult to be analyzed and visualized. The memberships, usually converted to 0-1 values, are visualized using parallel coordinates or different color shades. In this paper, we propose a new approach to visualize fuzzy-clustered data. The scheme is based on a geometric visualization, and works by grouping the objects with similar cluster memberships towards the vertices of a hyper-tetrahedron. The proposed method shows clear advantages over the existing methods, demonstrating its capabilities for viewing and navigating inter-cluster relationships in a spatial manner.     

State feedback control of continuous-time T-S fuzzy systems via switched fuzzy controllers

This paper studies the problem of state feedback control of continuous-time T-S fuzzy systems. Switched fuzzy controllers are exploited in the control design, which are switched based on the values of membership functions, and the control scheme is an extension of the parallel distributed compensation (PDC) scheme. Sufficient conditions for designing switched state feedback controllers are obtained with meeting an H_∞ norm bound requirement and quadratic D stability constraints. It is shown that the new control design method provides less conservative results than the corresponding ones via the parallel distributed compensation (PDC) scheme. A numerical example is given to illustrate the effectiveness of the proposed method.     

Self-learning fuzzy logic controllers for pursuit-evasion differential games

This paper addresses the problem of tuning the input and the output parameters of a fuzzy logic controller. The system learns autonomously without supervision or a priori training data. Two novel techniques are proposed. The first technique combines Q(λ)-learning with function approximation (fuzzy inference system) to tune the parameters of a fuzzy logic controller operating in continuous state and action spaces. The second technique combines Q(λ)-learning with genetic algorithms to tune the parameters of a fuzzy logic controller in the discrete state and action spaces. The proposed techniques are applied to different pursuit-evasion differential games. The proposed techniques are compared with the classical control strategy, Q(λ)-learning only, reward-based genetic algorithms learning, and with the technique proposed by Dai et al. (2005) [19] in which a neural network is used as a function approximation for Q-learning. Computer simulations show the usefulness of the proposed techniques.     

Vague soft hemirings

Xu et al. introduced the concept of vague soft sets, which is an extension to the soft set and the vague set. In this paper, we apply the concept of vague soft sets to hemiring theory. The notion of (∈,∈∨q)-vague (soft) left h-ideals of a hemiring is introduced and some related properties are investigated.     

Necessary optimality conditions for fuzzy variational problems

The main aims of this study are to derive the fuzzy Euler-Lagrange conditions for both fuzzy unconstrained and constrained variational problems based on the concepts of differentiability and integrability of a fuzzy mapping that may be parameterized by the left and right-hand functions of its α-level sets.     

Arithmetic operators in interval-valued fuzzy set theory

We introduce the addition, subtraction, multiplication and division on L^I, where L^I is the underlying lattice of both interval-valued fuzzy set theory [R. Sambuc, Fonctions Φ-floues. Application à l’aide au diagnostic en pathologie thyroidienne, Ph.D. Thesis, Université de Marseille, France, 1975] and intuitionistic fuzzy set theory [K.T. Atanassov, Intuitionistic fuzzy sets, 1983, VII ITKR’s Session, Sofia (deposed in Central Sci. Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)]. We investigate some algebraic properties of these operators. We show that using these operators the pseudo-t-representable extensions of the ?ukasiewicz t-norm and the product t-norm on the unit interval to L^I and some related operators can be written in a similar way as their counterparts on ([0,1],?).     

Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings

This paper generalizes a classical result about the space of bounded closed sets with the Hausdorff metric, and establishes the completeness of CB(X) with respect to the completeness of the metric space X, where CB(X) is the class of fuzzy sets with nonempty bounded closed α-cut sets, equipped with the supremum metric d_∞ which takes the supremum on the Hausdorff distances between the corresponding α-cut sets. In addition, some common fixed point theorems for fuzzy mappings are proved and two examples are given to illustrate the validity of the main results in fixed point theory.     

Cubic structures applied to ideals of BCI-algebras

The notions of cubic a-ideals and cubic p-ideals are introduced, and several related properties are investigated. Characterizations of a cubic a-ideal are established. Relations between cubic p-ideals, cubic a-ideals and cubic q-ideals are discussed. The cubic extension property of a cubic a-ideal is discussed.     

The solitary solution of the Liouville equation produced by the Exp-function method does not hold for all initial conditions

In this paper we argue that the solitary solution of the Liouville equation produced by the Exp-function method does not satisfy the original differential equation for all initial conditions. Moreover, the region where the solution is correct is located entirely on a curve in the parameter plane of initial conditions. We derive the explicit equation for this curve and argue that classical Exp-function type methods cannot identify such constraints related to initial conditions.     

Some common fixed point theorems for generalized nonlinear contractive mappings

Sufficient conditions for the existence of a common fixed point for generalized nonlinear contractive mappings are derived. As applications, some results on the set of best approximation for this class of mappings are also obtained. The proved results generalize and extend various known results in the literature.     

Some new sequence spaces derived by the domain of the triple band matrix

Let λ denote any of the classical spaces ?_∞,c,c₀, and ?_p of bounded, convergent, null, and absolutely p-summable sequences, respectively, and let λ(B) also be the domain of the triple band matrix B(r,s,t) in the sequence space λ, where 1<p<∞. The present paper is devoted to studying the sequence space λ(B). Furthermore, the β- and γ-duals of the space λ(B) are determined, the Schauder bases for the spaces c(B), c₀(B), and ?_p(B) are given, and some topological properties of the spaces c₀(B), ?₁(B), and ?_p(B) are examined. Finally, the classes (λ₁(B):λ₂) and (λ₁(B):λ₂(B)) of infinite matrices are characterized, where λ₁∈{?_∞,c,c₀,?_p,?₁} and λ₂∈{?_∞,c,c₀,?₁}.     

Verification of reactive systems via instantiation of Parameterised Boolean Equation Systems

Verification problems for finite- and infinite-state processes, like model checking and equivalence checking, can effectively be encoded in Parameterised Boolean Equation Systems (PBESs). Solving the PBES then solves the encoded problem. The decidability of solving a PBES depends on the data sorts that occur in the PBES. We describe a pragmatic methodology for solving PBESs, viz., by attempting to instantiate them to the sub-fragment of Boolean Equation Systems (BESs). Unlike solving PBESs, solving BESs is a decidable problem. Based on instantiation, verification using PBESs can effectively be done fully automatically in most practical cases. We demonstrate this by solving several complex verification problems using a prototype implementation of our instantiation technique. In addition, practical issues concerning this implementation are addressed. Furthermore, we illustrate the effectiveness of instantiation as a transformation on PBESs when solving verification problems involving systems of infinite size.     

Sample-weighted clustering methods

Although there have been many researches on cluster analysis considering feature (or variable) weights, little effort has been made regarding sample weights in clustering. In practice, not every sample in a data set has the same importance in cluster analysis. Therefore, it is interesting to obtain the proper sample weights for clustering a data set. In this paper, we consider a probability distribution over a data set to represent its sample weights. We then apply the maximum entropy principle to automatically compute these sample weights for clustering. Such method can generate the sample-weighted versions of most clustering algorithms, such as k-means, fuzzy c-means (FCM) and expectation & maximization (EM), etc. The proposed sample-weighted clustering algorithms will be robust for data sets with noise and outliers. Furthermore, we also analyze the convergence properties of the proposed algorithms. This study also uses some numerical data and real data sets for demonstration and comparison. Experimental results and comparisons actually demonstrate that the proposed sample-weighted clustering algorithms are effective and robust clustering methods.     

Some new fixed point theorems of generalized nonlinear contractive multivalued maps in complete metric spaces

In this paper, we present some new fixed point theorems for generalized nonlinear contractive multivalued maps and these results generalize or improve the corresponding recent fixed point results in the literature.     

Optimal convex error estimators for classification

A cross-validation error estimator is obtained by repeatedly leaving out some data points, deriving classifiers on the remaining points, computing errors for these classifiers on the left-out points, and then averaging these errors. The 0.632 bootstrap estimator is obtained by averaging the errors of classifiers designed from points drawn with replacement and then taking a convex combination of this “zero bootstrap” error with the resubstitution error for the designed classifier. This gives a convex combination of the low-biased resubstitution and the high-biased zero bootstrap. Another convex error estimator suggested in the literature is the unweighted average of resubstitution and cross-validation. This paper treats the following question: Given a feature-label distribution and classification rule, what is the optimal convex combination of two error estimators, i.e. what are the optimal weights for the convex combination. This problem is considered by finding the weights to minimize the MSE of a convex estimator. It also considers optimality under the constraint that the resulting estimator be unbiased. Owing to the large amount of results coming from the various feature-label models and error estimators, a portion of the results are presented herein and the main body of results appears on a companion website. In the tabulated results, each table treats the classification rules considered for the model, various Bayes errors, and various sample sizes. Each table includes the optimal weights, mean errors and standard deviations for the relevant error measures, and the MSE and MAE for the optimal convex estimator. Many observations can be made by considering the full set of experiments. Some general trends are outlined in the paper. The general conclusion is that optimizing the weights of a convex estimator can provide substantial improvement, depending on the classification rule, data model, sample size and component estimators. Optimal convex bootstrap estimators are applied to feature-set ranking to illustrate their potential advantage over non-optimized convex estimators.