4,350 research outputs found
Near approximations via general ordered topological spaces
Rough set theory is a new mathematical approach to imperfect knowledge. The
notion of rough sets is generalized by using an arbitrary binary relation on
attribute values in information systems, instead of the trivial equality
relation. The topology induced by binary relations is used to generalize the
basic rough set concepts. This paper studies near approximation via general
ordered topological approximation spaces which may be viewed as a
generalization of the study of near approximation from the topological view.
The basic concepts of some increasing (decreasing) near approximations,
increasing (decreasing) near boundary regions and increasing (decreasing) near
accuracy were introduced and sufficiently illustrated. Moreover, proved
results, implications and add examples
Dialectics of Counting and the Mathematics of Vagueness
New concepts of rough natural number systems are introduced in this research
paper from both formal and less formal perspectives. These are used to improve
most rough set-theoretical measures in general Rough Set theory (\textsf{RST})
and to represent rough semantics. The foundations of the theory also rely upon
the axiomatic approach to granularity for all types of general \textsf{RST}
recently developed by the present author. The latter theory is expanded upon in
this paper. It is also shown that algebraic semantics of classical \textsf{RST}
can be obtained from the developed dialectical counting procedures. Fuzzy set
theory is also shown to be representable in purely granule-theoretic terms in
the general perspective of solving the contamination problem that pervades this
research paper. All this constitutes a radically different approach to the
mathematics of vague phenomena and suggests new directions for a more realistic
extension of the foundations of mathematics of vagueness from both foundational
and application points of view. Algebras corresponding to a concept of
\emph{rough naturals} are also studied and variants are characterised in the
penultimate section.Comment: This paper includes my axiomatic approach to granules. arXiv admin
note: substantial text overlap with arXiv:1102.255
Recommended from our members
From on-line sketching to 2D and 3D geometry: A fuzzy knowledge based system
The paper describes the development of a fuzzy knowledge based prototype system for conceptual design. This real time system is designed to infer user’s sketching intentions, to segment sketched input and generate corresponding geometric primitives: straight lines, circles, arcs, ellipses, elliptical arcs, and B-spline curves. Topology information (connectivity, unitary constraints and pairwise constraints) is received dynamically from 2D sketched input and primitives. From the 2D topology information, a more accurate 2D geometry can be built up by applying a 2D geometric constraint solver. Subsequently, 3D geometry can be received feature by feature incrementally. Each feature can be recognised by inference knowledge in terms of matching its 2D primitive configurations and connection relationships. The system accepts not only sketched input, working as an automatic design tools, but also accepts user’s interactive input of both 2D primitives and special positional 3D primitives. This makes it easy and friendly to use. The system has been tested with a number of sketched inputs of 2D and 3D geometry
Soft Concept Analysis
In this chapter we discuss soft concept analysis, a study which identifies an
enriched notion of "conceptual scale" as developed in formal concept analysis
with an enriched notion of "linguistic variable" as discussed in fuzzy logic.
The identification "enriched conceptual scale" = "enriched linguistic variable"
was made in a previous paper (Enriched interpretation, Robert E. Kent). In this
chapter we offer further arguments for the importance of this identification by
discussing the philosophy, spirit, and practical application of conceptual
scaling to the discovery, conceptual analysis, interpretation, and
categorization of networked information resources. We argue that a linguistic
variable, which has been defined at just the right generalization of valuated
categories, provides a natural definition for the process of soft conceptual
scaling. This enrichment using valuated categories models the relation of
indiscernability, a notion of central importance in rough set theory. At a more
fundamental level for soft concept analysis, it also models the derivation of
formal concepts, a process of central importance in formal concept analysis.
Soft concept analysis is synonymous with enriched concept analysis. From one
viewpoint, the study of soft concept analysis that is initiated here extends
formal concept analysis to soft computational structures. From another
viewpoint, soft concept analysis provides a natural foundation for soft
computation by unifying and explaining notions from soft computation in terms
of suitably generalized notions from formal concept analysis, rough set theory
and fuzzy set theory.Comment: 16 pages, 5 figures, 6 table
When is the condition of order preservation met?
This article explores a relationship between inconsistency in the pairwise
comparisons method and conditions of order preservation. A pairwise comparisons
matrix with elements from an alo-group is investigated. This approach allows
for a generalization of previous results. Sufficient conditions for order
preservation based on the properties of elements of pairwise comparisons matrix
are derived. A numerical example is presented.Comment: 19 page
Condition for neighborhoods induced by a covering to be equal to the covering itself
It is a meaningful issue that under what condition neighborhoods induced by a
covering are equal to the covering itself. A necessary and sufficient condition
for this issue has been provided by some scholars. In this paper, through a
counter-example, we firstly point out the necessary and sufficient condition is
false. Second, we present a necessary and sufficient condition for this issue.
Third, we concentrate on the inverse issue of computing neighborhoods by a
covering, namely giving an arbitrary covering, whether or not there exists
another covering such that the neighborhoods induced by it is just the former
covering. We present a necessary and sufficient condition for this issue as
well. In a word, through the study on the two fundamental issues induced by
neighborhoods, we have gained a deeper understanding of the relationship
between neighborhoods and the covering which induce the neighborhoods.Comment: 11 page
A necessary and sufficient condition for two relations to induce the same definable set family
In Pawlak rough sets, the structure of the definable set families is simple
and clear, but in generalizing rough sets, the structure of the definable set
families is a bit more complex. There has been much research work focusing on
this topic. However, as a fundamental issue in relation based rough sets, under
what condition two relations induce the same definable set family has not been
discussed. In this paper, based on the concept of the closure of relations, we
present a necessary and sufficient condition for two relations to induce the
same definable set family.Comment: 13 page
Applications of repeat degree on coverings of neighborhoods
In covering based rough sets, the neighborhood of an element is the
intersection of all the covering blocks containing the element. All the
neighborhoods form a new covering called a covering of neighborhoods. In the
course of studying under what condition a covering of neighborhoods is a
partition, the concept of repeat degree is proposed, with the help of which the
issue is addressed. This paper studies further the application of repeat degree
on coverings of neighborhoods. First, we investigate under what condition a
covering of neighborhoods is the reduct of the covering inducing it. As a
preparation for addressing this issue, we give a necessary and sufficient
condition for a subset of a set family to be the reduct of the set family. Then
we study under what condition two coverings induce a same relation and a same
covering of neighborhoods. Finally, we give the method of calculating the
covering according to repeat degree.Comment: 1
Multi-granular Perspectives on Covering
Covering model provides a general framework for granular computing in that
overlapping among granules are almost indispensable. For any given covering,
both intersection and union of covering blocks containing an element are
exploited as granules to form granular worlds at different abstraction levels,
respectively, and transformations among these different granular worlds are
also discussed. As an application of the presented multi-granular perspective
on covering, relational interpretation and axiomization of four types of
covering based rough upper approximation operators are investigated, which can
be dually applied to lower ones.Comment: 12 page
Modelling microgels with controlled structure across the volume phase transition
Thermoresponsive microgels are soft colloids that find widespread use as
model systems for soft matter physics. Their complex internal architecture,
made of a disordered and heterogeneous polymer network, has been so far a major
challenge for computer simulations. In this work we put forward a
coarse-grained model of microgels whose structural properties are in
quantitative agreement with results obtained with small-angle X-ray scattering
experiments across a wide range of temperatures, encompassing the volume phase
transition. These results bridge the gap between experiments and simulations of
individual microgel particles, paving the way to theoretically address open
questions about their bulk properties with unprecedented nano and microscale
resolution
- …