Fuzzy Soft Topological Groups

Notions of Lowen type fuzzy soft topological space are introduced and some of their properties are established in the present paper. Besides this, a combined structure of a fuzzy soft topological space and a fuzzy soft group, which is termed here as fuzzy soft topological group is introduced. Homomorphic images and preimages are also examined. Finally, some definitions and results on fuzzy soft set are studied.     

A Local Property of Hausdorff Centered Measure of Self-Similar Sets

We analyze the local behavior of the Hausdorff centered measure for selfsimilar sets. If E is a self-similar set satisfying the open set condition, then Cs(E∩B(x,r)) ≤(2r)s for all x ∈ E and r 0, where Csdenotes the s-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.     

On the solutions of a system of two Diophantine equations

We obtain all positive integer solutions(m1,m2,a,b)with ab,gcd(a,b)=1 to the system of Diophantine equations km21-lat1bt2a2r=C1,km22-lat1bt2b2r=C2,with C1,C2∈{-1,1,-2,2,-4,4},and k,l,t1,t2,r∈Z such that k0,l0,r0,t10,t2 0,gcd(k,l)=1,and k is square-free.     

Differentiability of type-2 fuzzy number-valued functions

In this paper, we define a differentiability of the type-2 fuzzy number-valued functions. The definition is based on type-2 Hukuhara difference which is defined in the paper as well. The related theorem of the differentiability of the type-2 fuzzy number-valued functions is derived. In addition, a parametric closed form of the perfect triangular quasi type-2 fuzzy numbers is introduced, and finally, the applicability and an approach to solving type-2 fuzzy differential equations are illustrated using some examples and cases.     

Interpretation of association rules in multi-tier structures

Dealing with the large amount of data resulting from association rule mining is a big challenge. The essential issue is how to provide efficient methods for summarizing and representing meaningful discovered knowledge from databases. This paper presents a new approach called multi-tier granule mining to improve the performance of association rule mining. Rather than using patterns, it uses granules to represent knowledge that is implicitly contained in relational databases. This approach also uses multi-tier structures and association mappings to interpret association rules in terms of granules. Consequently, association rules can be quickly assessed and meaningless association rules can be justified according to these association mappings. The experimental results indicate that the proposed approach is promising.     

Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets

Covering rough sets generalize traditional rough sets by considering coverings of the universe instead of partitions, and neighborhood-covering rough sets have been demonstrated to be a reasonable selection for attribute reduction with covering rough sets. In this paper, numerical algorithms of attribute reduction with neighborhood-covering rough sets are developed by using evidence theory. We firstly employ belief and plausibility functions to measure lower and upper approximations in neighborhood-covering rough sets, and then, the attribute reductions of covering information systems and decision systems are characterized by these respective functions. The concepts of the significance and the relative significance of coverings are also developed to design algorithms for finding reducts. Based on these discussions, connections between neighborhood-covering rough sets and evidence theory are set up to establish a basic framework of numerical characterizations of attribute reduction with these sets.     

Octahedral norms and convex combination of slices in Banach spaces

We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of w^?$w^{?}$-slices in the dual unit ball has diameter 2, which answers an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing _?1${?}_1$ and for every ε>0$\epsilon >0$ there is an equivalent norm so that every convex combination of w^?$w^{?}$-slices in the dual unit ball has diameter at least 2−ε$2-\epsilon$.     

Nil Bohr0-sets and polynomial recurrence

In Huang et al.  [17] it was proved that for any Nil_d  Bohr₀-set A$A$, there are a minimal system (X,T)$(X,T)$ and a non-empty open subset U$U$ of X$X$ with A⊃{n∈Z:U∩T⁻ⁿU∩?∩T^−dnU≠0?}$A\supset \{n\in \mathbb{Z}:U\cap T^{-n}U\cap ?\cap T^{-dn}U\ne 0?\}$, and for any minimal system (X,T)$(X,T)$ and any open non-empty U⊂X$U\subset X$, the set {n∈Z:U∩T⁻ⁿU∩?∩T^−dnU≠0?}$\{n\in \mathbb{Z}:U\cap T^{-n}U\cap ?\cap T^{-dn}U\ne 0?\}$ is an almost Nil_d Bohr₀-set. The polynomial form of this problem is considered in this paper. It is shown that the latter is still true in the polynomial case, while the former is not in general. We also consider the special case when the system is a nilsystem.     

Vague集TOPSIS方法在方案优选中的应用

指出Vague集理论中现有距离度量的不足之处以及Vague集距离度量的基本准则,建立新的距离公式,给出建立在该距离公式基础上的TOPSIS方法,并将其应用到方案选优的实例中,得到合理的结论.