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.     

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.     

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.     

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.     

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],?).     

Interval valued (\in,\in\!\vee\,q) -fuzzy filters of pseudo BL-algebras

We introduce the concept of quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set. By using this new idea, we introduce the notions of interval valued -fuzzy filters of pseudo BL-algebras and investigate some of their related properties. Some characterization theorems of these generalized interval valued fuzzy filters are derived. The relationship among these generalized interval valued fuzzy filters of pseudo BL-algebras is considered. Finally, we consider the concept of implication-based interval valued fuzzy implicative filters of pseudo BL-algebras, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

Approximation capabilities of multilayer fuzzy neural networks on the set of fuzzy-valued functions

This paper first introduces a piecewise linear interpolation method for fuzzy-valued functions. Based on this, we present a concrete approximation procedure to show the capability of four-layer regular fuzzy neural networks to perform approximation on the set of all d_p continuous fuzzy-valued functions. This approach can also be used to approximate d_∞ continuous fuzzy-valued functions. An example is given to illustrate the approximation procedure.     

Relationship between the eccentric connectivity index and Zagreb indices

For a (molecular) graph, the first Zagreb index M₁ is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M₂ is equal to the sum of the products of the degrees of pairs of adjacent vertices. If G is a connected graph with vertex set V(G), then the eccentric connectivity index of G, ξ^C(G), is defined as, ∑_vi∈V(G)d_ie_i, where d_i is the degree of a vertex v_i and e_i is its eccentricity. In this report we compare the eccentric connectivity index (ξ^C) and the Zagreb indices (M₁ and M₂) for chemical trees. Moreover, we compare the eccentric connectivity index (ξ^C) and the first Zagreb index (M₁) for molecular graphs.     

Three-dimensional beta shapes

The Voronoi diagram of a point set has been extensively used in various disciplines ever since it was first proposed. Its application realms have been even further extended to estimate the shape of point clouds when Edelsbrunner and Mücke introduced the concept of α-shape based on the Delaunay triangulation of a point set.In this paper, we present the theory of β-shape for a set of three-dimensional spheres as the generalization of the well-known α-shape for a set of points. The proposed β-shape fully accounts for the size differences among spheres and therefore it is more appropriate for the efficient and correct solution for applications in biological systems such as proteins.Once the Voronoi diagram of spheres is given, the corresponding β-shape can be efficiently constructed and various geometric computations on the sphere complex can be efficiently and correctly performed. It turns out that many important problems in biological systems such as proteins can be easily solved via the Voronoi diagram of atoms in proteins and β-shapes transformed from the Voronoi diagram.     

A note on soft topological spaces

Shabir and Naz (2011) [12] introduced and studied the notions of soft topological spaces, soft interior, soft closure and soft separation axioms. But we found that some results are incorrect (see their Remark 3.23). So the purpose of this note is, first, to point out some errors in Remark 4 and Example 9 of Shabir and Naz (2011) [12], and second, to investigate properties of soft separation axioms defined in Shabir and Naz (2011) [12]. In particular, we investigate the soft regular spaces and some properties of them. We show that if a soft topological space (X,τ,E) is soft T₁ and soft regular (i.e. a soft T₃-space), then (x,E) is soft closed for each x∈X (their Theorem 3.21).     

Distinct squares in run-length encoded strings

Squares are strings of the form ww where w is any nonempty string. Two squares ww and w^′w^′ are of different types if and only if w≠w^′. Fraenkel and Simpson [Avieri S. Fraenkel, Jamie Simpson, How many squares can a string contain? Journal of Combinatorial Theory, Series A 82 (1998) 112-120] proved that the number of square types contained in a string of length n is bounded by O(n). The set of all different square types contained in a string is called the vocabulary of the string. If a square can be obtained by a series of successive right-rotations from another square, then we say the latter covers the former. A square is called a c-square if no square with a smaller index can cover it and it is not a trivial square. The set containing all c-squares is called the covering set. Note that every string has a unique covering set. Furthermore, the vocabulary of the covering set are called c-vocabulary. In this paper, we prove that the cardinality of c-vocabulary in a string is less than , where N is the number of runs in this string.     

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.     

Experimental investigation on adaptive robust controller designs applied to a free-floating space manipulator

This paper aims to formulate and investigate the application of various nonlinear H_∞ control methods to a free-floating space manipulator subject to parametric uncertainties and external disturbances. From a tutorial perspective, a model-based approach and adaptive procedures based on linear parametrization, neural networks and fuzzy systems are covered by this work. A comparative study is conducted based on experimental implementations performed with an actual underactuated fixed-base planar manipulator which is, following the DEM concept, dynamically equivalent to a free-floating space manipulator.     

Unified approach to coefficient-based regularized regression

In this paper, we consider the coefficient-based regularized least-squares regression problem with the l^q-regularizer (1≤q≤2) and data dependent hypothesis spaces. Algorithms in data dependent hypothesis spaces perform well with the property of flexibility. We conduct a unified error analysis by a stepping stone technique. An empirical covering number technique is also employed in our study to improve sample error. Comparing with existing results, we make a few improvements: First, we obtain a significantly sharper learning rate that can be arbitrarily close to O(m⁻¹) under reasonable conditions, which is regarded as the best learning rate in learning theory. Second, our results cover the case q=1, which is novel. Finally, our results hold under very general conditions.     

Online computation with advice

We consider a model for online computation in which the online algorithm receives, together with each request, some information regarding the future, referred to as advice. The advice is a function, defined by the online algorithm, of the whole request sequence. The advice provided to the online algorithm may allow an improvement in its performance, compared to the classical model of complete lack of information regarding the future. We are interested in the impact of such advice on the competitive ratio, and in particular, in the relation between the size b of the advice, measured in terms of bits of information per request, and the (improved) competitive ratio. Since b=0 corresponds to the classical online model, and b=⌈log∣A∣⌉, where A is the algorithm’s action space, corresponds to the optimal (offline) one, our model spans a spectrum of settings ranging from classical online algorithms to offline ones.In this paper we propose the above model and illustrate its applicability by considering two of the most extensively studied online problems, namely, metrical task systems (MTS) and the k-server problem. For MTS we establish tight (up to constant factors) upper and lower bounds on the competitive ratio of deterministic and randomized online algorithms with advice for any choice of 1≤b≤Θ(logn), where n is the number of states in the system: we prove that any randomized online algorithm for MTS has competitive ratio Ω(log(n)/b) and we present a deterministic online algorithm for MTS with competitive ratio O(log(n)/b). For the k-server problem we construct a deterministic online algorithm for general metric spaces with competitive ratio k^O(1/b) for any choice of Θ(1)≤b≤logk.     

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}.     

Negative selection algorithms on strings with efficient training and linear-time classification

A string-based negative selection algorithm is an immune-inspired classifier that infers a partitioning of a string space Σ^? into “normal” and “anomalous” partitions from a training set S containing only samples from the “normal” partition. The algorithm generates a set of patterns, called “detectors”, to cover regions of the string space containing none of the training samples. Strings that match at least one of these detectors are then classified as “anomalous”. A major problem with existing implementations of this approach is that the detector generating step needs exponential time in the worst case. Here we show that for the two most widely used kinds of detectors, the r-chunk and r-contiguous detectors based on partial matching to substrings of length r, negative selection can be implemented more efficiently by avoiding generating detectors altogether: for each detector type, training set S⊆Σ^? and parameter r≤? one can construct an automaton whose acceptance behaviour is equivalent to the algorithm’s classification outcome. The resulting runtime is O(|S|?r|Σ|) for constructing the automaton in the training phase and O(?) for classifying a string.