Granular Computing

The basic ideas and principles of granular computing (GrC) have been studied explicitly or implicitly in many fields in isolation. With the recent renewed and fast growing interest, it is time to extract the commonality from a diversity of fields and to study systematically and formally the domain independent principles of granular computing in a unified model. A framework of granular computing can be established by applying its own principles. We examine such a framework from two perspectives,granular computing as structured thinking and structured problem solving. From the philosophical perspective or the conceptual level,granular computing focuses on structured thinking based on multiple levels of granularity. The implementation of such a philosophy in the application level deals with structured problem solving.     

Preinvex fuzzy mappings

In this paper, we introduce the concepts of preinvex fuzzy mappings, quasi-preinvex fuzzy mappings, and invex sets by modifying the concepts proposed by Noor. We also give two characterization theorems for preinvex fuzzy mappings, and present some new results for preinvex fuzzy mappings, strictly preinvex fuzzy mappings, and quasi-preinvex fuzzy mappings.     

Data mappings again

Extracting data from a relational data base in order to respond to a request for information requires the use of a large number of algorithms. Some of these algorithms are responsible for validating data values, others support the relational operators while others still support the basic data structures used at the lowest levels of the system. This paper recognizes that the successful operation of a relational data base management system is dependent on the organization of these algorithms and emphasizes this point by describing a coherent architecture to embrace them.     

粒空间中基于粒计算的粒语义推理

在论域上定义公式,并因其与粒相对应,使得粒空间得以形成.在粒空间中由粒之间的关系确定出粒语义推理,并对此进行讨论.通过粒计算,证得粒语义推理满足命题逻辑自然推理系统的每一推理规则,因此粒空间上的可靠性成立,这表明形式推理蕴涵粒语义推理.尽管粒空间上的完备性不成立,但结果表明粒语义推理是各经典推理模式的扩充,所以粒语义推理既拓宽推理研究的范围,又引进粒计算的研究方法,促成粒计算与逻辑推理的进一步融合.     

Granular neural networks

Fuzzy neural networks (FNNs) and rough neural networks (RNNs) both have been hot research topics in the artificial intelligence in recent years. The former imitates the human brain in dealing with problems, the other takes advantage of rough set theory to process questions uncertainly. The aim of FNNs and RNNs is to process the massive volume of uncertain information, which is widespread applied in our life. This article summarizes the recent research development of FNNs and RNNs (together called granular neural networks). First the fuzzy neuron and rough neuron is introduced; next FNNs are analysed in two categories: normal FNNs and fuzzy logic neural networks; then the RNNs are analysed in the following four aspects: neural networks based on using rough sets in preprocessing information, neural networks based on rough logic, neural networks based on rough neuron and neural networks based on rough-granular; then we give a flow chart of the RNNs processing questions and an application of classical neural networks based on rough sets; next this is compared with FNNs and RNNs and the way to integrate is described; finally some advice is given on development of FNNs and RNNs in future.     

Semistrictly preinvex fuzzy mappings

In this paper, we introduce the concepts of a strictly preinvex fuzzy mapping, a semistrictly preinvex fuzzy mapping, a strictly prequasi-invex fuzzy mapping and a semistrictly prequasi-invex fuzzy mapping. Some properties of these mappings and relations among them are studied. A few interesting theorems are proved, including that: (1) a strictly prequasi-invex fuzzy mapping has at most one global minimum point; (2) the solution set is invex if the fuzzy mapping is prequasi-invex and (3) prequasi-invex fuzzy mappings satisfy most of the basic properties of the preinvex fuzzy mappings.     

Enriching Parikh matrix mappings

In this paper, we define the Super-Parikh (S-Parikh) matrix mapping as an extension of the Parikh matrix mapping introduced by Mateescu et al. Like the Parikh matrix, the extension revolves around a certain type of square matrices, but instead of non-negative integers, its matrix-mapped elements are non-negative rationals (fractions). We study the basic properties of the newly defined formalism and later on we investigate the injectivity of the mapping. Also, we begin a search for the reverse mapping – that is a method for obtaining a word, given the S-Parikh matrix.     

Fuzzy approximately cubic mappings

We establish some stability results concerning the cubic functional equation

f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)     

On pseudo-fuzzy linear mappings

In this paper we study a more general class of linear maps on fuzzy subsets of vector spaces than those on fuzzy subspaces. We obtain a version of the fundamental theorem of homomorphisms which applies to these pseudo-fuzzy linear maps. In the case of fuzzy subspaces, the results become those for fuzzy linear maps. We introduce the notion of magnification of pseudo-fuzzy linear mappings. It is shown that this notion behaves well with respect to the fundamental theorem and with respect to the composition of mapping. We note that the class of distribution functions of random variables naturally produces pseudo-fuzzy linear mappings, which are not fuzzy linear mappings. We observe that pseudo-fuzzy linear mappings are examples of gain relations. We consider general properties of these relations as well as many other examples. From these results additional properties of pseudo-fuzzy linear mappings are obtained.     

Point-to-line mappings as Hough transforms

In 1962 [US Patent 3069654], Hough used a linear point-to-line mapping (PTLM) to detect large sets of collinear points in an image, by mapping the points into concurrent lines and detecting peaks where many lines intersect. In 1972, Duda and Hart [Commun. ACM 15 (1972) 11] pointed out that Hough's method is not practical, because the peaks need not lie in a bounded region. They (and others after them) therefore developed methods of detecting sets of collinear points using nonlinear point-to-curve mappings that map collinear points into concurrent curves whose intersections do lie in a bounded range. In this paper we show that any PTLM that maps collinear points into concurrent lines must be linear, and that no such PTLM can map all the sets of collinear points in an image into peaks that lie in a bounded region; thus Duda and Hart's objection applies to any PTLM-based Hough transform.     

Efficient mappings of pyramid networks

We consider primarily the simulation of large networks by smaller ones-an important consideration, because interconnection networks are typically of a fixed size, and yet applications may employ networks of a larger size. Current research (Dingle and Sudborough, 1993) describes methods to simulate common data structures and network architectures on the pyramid. However, these simulations assume that the pyramid grows with the size of the network or data structure. Because unbounded growth is not feasible, we address the issue of mapping several points of the guest data structure or network to a single host processor. We determine how a small pyramid may efficiently simulate the computation of a larger pyramid as well as that of tree networks     

Computation of area preserving mappings

The study of particle motion in accelerators of alternating gradient type, and in proton storage rings, is frequently undertaken by means of numerical computation of certain area preserving mappings. In this paper we give two methods for evaluating the uncertainty introduced by the use of finite precision arithmetics.     

Retract mappings of projectable MV-algebras

For MV-algebras (algebras of multivalued Lukasiewicz logics) we apply the same terminology and notation as in [3] and [8]. Retracts and retract mappings of abelian lattice ordered groups were studied in [4], cf. also [6], [7]; for the case of multilattice groups and cyclically ordered groups cf. [1] and [5]. To each MV-algebra ? there corresponds an abelian lattice ordered group G with a strong unit u such that (under the notation as in [8]), ? = ?₀(G,u) (cf. also Section 1 below). In [2], a different (but equivalent) system of axioms for defining the notion of MV-algebra was applied; instead of ?₀(G,u), the notation Γ(G,u) was used. In the present paper we investigate the relations between retract mappings of a projectable MV-algebra ? and the retract mappings of the corresponding lattice ordered group G.     

Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings

In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite (a finite) family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We obtain some weak convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known weak convergence theorems in the literature.     

Coloring periodicities of two-dimensional mappings

We show how the parameter plane for two-dimensional mappings can be colored according to periodicity to reveal patterns of finite attracting orbits and chaos.