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.     

On a new class of analytic functions associated with conic domain

The aim of this paper is to generalize the conic domain defined by Kanas and Wisniowska, and define the class of functions which map the open unit disk E onto this generalized conic domain. A brief comparison between these conic domains is the main motivation of this paper. A correction is made in selecting the range interval of order of conic domain.     

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.     

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).     

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.     

On global solutions to fractional functional differential equations with infinite delay in Fréchet spaces

In this paper, we investigate global uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. We shall rely on a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. The results are obtained by using the α-resolvent family (S_α(t))_t≥0 on a complex Banach space X combined with the above-mentioned fixed point theorem. As an application, a controllability result with one parameter is also provided to illustrate the theory.     

A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces

In this paper, we introduce a modified new hybrid projection method for finding the set of solutions of the generalized mixed equilibrium problems and the convex feasibility problems for an infinite family of closed and uniformly quasi-?-asymptotically nonexpansive mappings. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Our results improve and extend the corresponding results announced by Qin et al. (2010) and many authors.     

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.     

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.     

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.     

On coefficient inequalities of functions associated with conic domains

In this paper, the concepts of Janowski functions and the conic regions are combined to define a new domain which represents the conic type regions. Different views of this modified conic domain for specific values are shown graphically for better understanding of the behavior of this domain. The class of such functions which map the open unit disk E onto this modified conic domain is defined. Also the classes of k-uniformly Janowski convex and k-Janowski starlike functions are defined and their coefficient inequalities are formulated. The coefficient bound for a certain class of analytic functions, proved by Owa et al. (2006) in [16], has also been improved.     

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.     

DNA library screening, pooling design and unitary spaces

Pooling design is an important research topic in bio-informatics due to its wide applications in molecular biology, especially DNA library screening. In this paper, with unitary spaces over finite fields, we present two new constructions whose efficiency ratio, i.e., the ratio between the number of tests and the number of items, is smaller than some of the existing constructions.     

Step sequence planning for a biped robot by means of a cylindrical shape model and a high-resolution 2.5D map

A novel step sequence planning (SSP) method for biped-walking robots is presented. The method adopts a free space representation custom-designed for efficient biped robot motion planning. The method rests upon the approximation of the robot shape by a set of 3D cylindrical solids. This feature allows efficient determination of feasible paths in a 2.5D map, comprising stepping over obstacles and stair climbing. A SSP algorithm based on A^∗-search is proposed which uses the advantages of the aforementioned environment representation. The efficiency of the proposed approach is evaluated by a series of simulations performed for eight walking scenarios.