Abstract:
La integración de los avances en computación orientada a servicios, con aquellos alcanzados por la tecnología de la web y la inteligencia computacional, facilita el desarrollo de aplicaciones complejas para la solución de problemas de manera transversa en los diversos ámbitos de la investigación científica y tecnológica. Desde el campo de la inteligencia computacional la teoría de los conjuntos aproximados aporta un sólido fundamento teórico al razonamiento cualitativo exigido por el análisis de datos, caracterizados por la incertidumbre generada por la vaguedad e imprecisión asociada a éstos. El presente trabajo describe el desarrollo de un servicio web inteligente aplicado al procesamiento digital de imágenes, ilustrando las bondades de los conjuntos aproximados, para abordar de modo flexible y supervisado la clasificación de los pixeles asociados a ellas. Integrating recent developments in service-orientated computing, Web technologies and computational intelligence has facilitated the development of applications for solving complex problems in several fields of scientific and technological research. Rough sets theory provides a solid theoretical background within the computational intelligence (CI) field for the qualitative reasoning required for analysing datasets loaded with uncertainties due to the vagueness and lack of precision associated with them. This paper describes the development of an intelligent Web service to process digital imagery, demonstrating the benefits of rough sets theory in dealing with the flexible supervised classification of the pixels associated with them.

Abstract:
Rough sets, developed by Pawlak [6], are an important tool to describe a situation of incomplete or partially unknown information. One of the algebraic models deals with the pair of the upper and the lower approximation. Although usually the tolerance or the equivalence relation is taken into account when considering a rough set, here we rather concentrate on the model with the pair of two definable sets, hence we are close to the notion of an interval set. In this article, the lattices of rough sets and intervals are formalized. This paper, being essentially the continuation of [3], is also a step towards the formalization of the algebraic theory of rough sets, as in [4] or [9].

Abstract:
Digital topology was first studied by the computer image analysis researcher Azriel Rosenfeld [12]. The concept of *gα-closed sets in a topological spaces was introduced by M. Vigneshwaran and R. Devi[14]. In this paper, we study the properties of *gα-closed and *gα-open sets in the digital plane (Z2, k2). Also proved that the family of all *gα-open sets of (Z2, k2), say *GαO(Z2, k2), forms an alternative topology of Z2. Also we derive the properties of *gα-closed and *gα-open sets in the digital plane via the singletons points. Keywords: *gα-closed sets, *gα-open sets,digital plane, digital topology

Abstract:
Rough sets are efficient for data pre-processing in data mining. As a generalization of the linear independence in vector spaces, matroids provide well-established platforms for greedy algorithms. In this paper, we apply rough sets to matroids and study the contraction of the dual of the corresponding matroid. First, for an equivalence relation on a universe, a matroidal structure of the rough set is established through the lower approximation operator. Second, the dual of the matroid and its properties such as independent sets, bases and rank function are investigated. Finally, the relationships between the contraction of the dual matroid to the complement of a single point set and the contraction of the dual matroid to the complement of the equivalence class of this point are studied.

Abstract:
Rough set theory provides a useful mathematical foundation for developing automated computational systems that can help understand and make use of imperfect knowledge.Despite its recency,the theory and its extensions have been widely applied to many problems,including decision analysis,data mining,intelligent control and pattern recognition.This paper presents an outline of the basic concepts of rough sets and their major extensions,covering variable precision,tolerance and fuzzy rough sets.It also shows the diversity of successful applications these theories have entailed,ranging from financial and business,through biological and medicine,to physical,art,and meteorological.

Abstract:
The topology of the Internet at the Autonomous System (AS) level is not yet fully discovered despite significant research activity. The community still does not know how many links are missing, where these links are and finally, whether the missing links will change the conceptual model of the Internet topology. An accurate and complete model of the topology would be important for protocol design, performance evaluation and analyses. The goal of the work is to develop methodologies and tools to identify and validate such missing links between ASes. In this work, to develop several methods and identify a significant number of missing links, particularly of the peer-to-peer type. Interestingly, most of the missing AS links that to find exist as peer-to-peer links at the Internet Exchange Points (IXPs). First, in more detail, to provide a large-scale comprehensive synthesis of the available sources of information. To cross-validate and compare BGP routing tables, Internet Routing Registries, and traceroute data, while to extract significant new information from theless-studied Internet Exchange Points (IXPs). To identify 40% more edges and approximately 300% more peerto- peer edges compared to commonly used data sets. All of these edges have been verified by either BGP tables or traceroute. Second, to identify properties of the new edges and quantify their effects on important topologicalproperties. Given the new peer-to-peer edges, to find that for some ASes more than 50% of their paths stop going through their ISPs assuming policy-aware routing. A surprising observation is that the degree of an AS may be a poor indicator of which ASes it will peer with. IXPs(Internet Exchange Points) have not received attention in terms of Internet topology discovery, although they play a major role in the Internet connectivity.

Abstract:
We show that for any tolerance $R$ on $U$, the ordered sets of lower and upper rough approximations determined by $R$ form ortholattices. These ortholattices are completely distributive, thus forming atomistic Boolean lattices, if and only if $R$ is induced by an irredundant covering of $U$, and in such a case, the atoms of these Boolean lattices are described. We prove that the ordered set $\mathit{RS}$ of rough sets determined by a tolerance $R$ on $U$ is a complete lattice if and only if it is a complete subdirect product of the complete lattices of lower and upper rough approximations. We show that $R$ is a tolerance induced by an irredundant covering of $U$ if and only if $\mathit{RS}$ is an algebraic completely distributive lattice, and in such a situation a quasi-Nelson algebra can be defined on $\mathit{RS}$. We present necessary and sufficient conditions which guarantee that for a tolerance $R$ on $U$, the ordered set $\mathit{RS}_X$ is a lattice for all $X \subseteq U$, where $R_X$ denotes the restriction of $R$ to the set $X$ and $\mathit{RS}_X$ is the corresponding set of rough sets. We introduce the disjoint representation and the formal concept representation of rough sets, and show that they are Dedekind--MacNeille completions of $\mathit{RS}$.

Abstract:
A point of a digital space is called simple if it can be deleted from the space without altering topology. This paper introduces the notion simple set of points of a digital space. The definition is based on contractible spaces and contractible transformations. A set of points in a digital space is called simple if it can be contracted to a point without changing topology of the space. It is shown that contracting a simple set of points does not change the homotopy type of a digital space, and the number of points in a digital space without simple points can be reduces by contracting simple sets. Using the process of contracting, we can substantially compress a digital space while preserving the topology. The paper proposes a method for thinning a digital space which shows that this approach can contribute to computer science such as medical imaging, computer graphics and pattern analysis.

Abstract:
We define and study a new class of regular sets called -regular sets. Properties of these sets are investigated for topological spaces and generalized topological spaces. Decompositions of regular open sets and regular closed sets are provided using -regular sets. Semiconnectedness is characterized by using -regular sets. -continuity and almost -continuity are introduced and investigated. 1. Introduction In general topology, repeated applications of interior and closure operators give rise to several different new classes of sets. Some of them are generalized form of open sets while few others are the so-called regular sets. These classes are found to have applications not only in mathematics but even in diverse fields outside the realm of mathematics [1–3]. Due to this, investigations of these sets have gained momentum in the recent days. Császár has already provided an umbrella study for generalized open sets in his latest papers [4–7]. In this paper, we introduce and study a new class of sets, called -regular sets, using semi-interior and semiclosure operators. Initially, we define them for a broader class, that is, for generalized topological spaces and discuss their various properties. Interrelationship of -regular sets with other existing classes such as semiopen sets, regular open sets, -sets, -sets, -sets, and -sets has been studied. A characterization of semiconnectedness is also provided using -regular sets. Moreover, -regular sets, where , of a generalized topological space are studied using -regular sets. In the last two sections, -regularity is studied in the domain of general topological spaces. Here several decompositions of regular open sets and regular closed sets are provided using -regular sets. In the last section, -continuity and almost -continuity are defined and interrelationship of almost -continuity with other existing mappings such as -map, graph mapping, almost precontinuity, and almost -continuity is investigated. 2. Preliminaries First we recall some definitions and results to be used in the paper. Definition 1 (see [6]). Let be a nonempty set. A collection of subsets of is called a generalized topology (in brief, ) on if it is closed under arbitrary unions. The ordered pair is called generalized topological space (in brief, ). Since an empty union amounts to the empty set, always belongs to . However, need not be a member of . The members of are called -open while the complements of -open sets are called -closed. The largest -open set contained in a set is called the interior of and is denoted by , whereas the smallest

Abstract:
In this paper, the ordered set of rough sets determined by a quasiorder relation $R$ is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomykala and J. A. Pomykala (1988) and M. Gehrke and E. Walker (1992) in case $R$ is an equivalence.