scholarly journals Properties of nano Δ* open sets

2021 ◽  
Vol 2070 (1) ◽  
pp. 012100
Author(s):  
K Meena ◽  
J Dhivya ◽  
R Kalaiselvi
Keyword(s):  
Equivalence Relation ◽  
Topological Spaces ◽  
Lower Approximation ◽  
Upper Approximation

Abstract This article aims at introducing nano Δ* open sets in Nano Topological Spaces (NTS). The NTS are nothing but the spaces created in terms of equivalence relation which is derived from lower approximation, upper approximation and boundaries of a subset of a universal set. An elaborate study on various properties and characterizations of Nano Δ* open sets in relation with nano δ open sets and nano δg-kemel operator are explained in this article.

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

scholarly journals On Some Types of Covering-Based ℐ , T -Fuzzy Rough Sets and Their Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Mohammed Atef ◽  
José Carlos R. Alcantud ◽  
Hussain AlSalman ◽  
Abdu Gumaei
Keyword(s):  
Rough Set ◽  
Rough Sets ◽  
Practical Application ◽  
Fuzzy Rough Sets ◽  
Fuzzy Rough Set ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Numerical Example ◽  
Variable Precision

The notions of the fuzzy β -minimal and maximal descriptions were established by Yang et al. (Yang and Hu, 2016 and 2019). Recently, Zhang et al. (Zhang et al. 2019) presented the fuzzy covering via ℐ , T -fuzzy rough set model ( FC ℐ T FRS ), and Jiang et al. (Jiang et al., in 2019) introduced the covering through variable precision ℐ , T -fuzzy rough sets ( CVP ℐ T FRS ). To generalize these models in (Jiang et al., 2019 and Zhang et al. 2019), that is, to improve the lower approximation and reduce the upper approximation, the present paper constructs eight novel models of an FC ℐ T FRS based on fuzzy β -minimal (maximal) descriptions. Characterizations of these models are discussed. Further, eight types of CVP ℐ T FRS are introduced, and we investigate the related properties. Relationships among these models are also proposed. Finally, we illustrate the above study with a numerical example that also describes its practical application.

scholarly journals On the Lattice of Intervals and Rough Sets

2009 ◽  
Vol 17 (4) ◽  
pp. 237-244 ◽  
Author(s):  
Adam Grabowski ◽  
Magdalena Jastrzębska
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Algebraic Theory ◽  
Algebraic Models ◽  
Lower Approximation ◽  
Definable Sets

On the Lattice of Intervals and Rough Sets 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].

Model of Covering Rough Sets in the Machine Intelligence

2012 ◽  
Vol 548 ◽  
pp. 735-739
Author(s):  
Hong Mei Nie ◽  
Jia Qing Zhou
Keyword(s):  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Machine Intelligence ◽  
Upper Approximation ◽  
Generalized Rough Sets

Rough set theory has been proposed by Pawlak as a useful tool for dealing with the vagueness and granularity in information systems. Classical rough set theory is based on equivalence relation. The covering rough sets are an improvement of Pawlak rough set to deal with complex practical problems which the latter one can not handle. This paper studies covering-based generalized rough sets. In this setting, we investigate common properties of classical lower and upper approximation operations hold for the covering-based lower and upper approximation operations and relationships among some type of covering rough sets.

The Rough Linear Approximate Space and Soft Linear Space

2016 ◽  
Vol 25 (2) ◽  
pp. 251-261
Author(s):  
Yingcang Ma ◽  
Shaoyang Li ◽  
Yamei Liu
Keyword(s):  
Boolean Algebra ◽  
Linear Space ◽  
Linear Algebra ◽  
Rough Sets ◽  
Linear Subspace ◽  
Real Life ◽  
Inclusion Relation ◽  
Soft Sets ◽  
Lower Approximation ◽  
Upper Approximation

AbstractThe studies of rough sets and soft sets, which can deal with uncertain problems in real life, have developed rapidly in recent years. We have known that linear space is a very important concept in linear algebra, so the aim of this paper was mainly focused on combining research in linear space, rough sets, and soft sets. First, according to the properties of upper (lower) approximation in rough linear space, the inclusion relation of the upper approximation’s union and the inclusion relation of the lower approximation’s intersection are improved. The equations of the upper approximation’s union and the lower approximation’s intersection are given. Secondly, the connection of linear space to rough sets is explored and the rough linear approximate space is proposed, which is proved to be a Boolean algebra under the intersection, union, and complementary operators. Thirdly, the combination of linear space and soft set is discussed, the definitions of soft linear space and soft linear subspace are proposed, and their properties are explored. Finally, the definitions of lower and upper approximation of a subspace X in soft linear space are given and their properties are studied. These investigations would enrich the studies of linear space, soft sets, and rough sets.

scholarly journals Non Homogeneous Rough Finite State Automaton

2021 ◽  
Vol 11 (2) ◽  
pp. 629-641
Author(s):  
B. Praba ◽  
R. Saranya
Keyword(s):  
Artificial Intelligence ◽  
Machine Learning ◽  
Dynamical Behaviour ◽  
Finite State Automaton ◽  
Finite State Automata ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Finite State ◽  
Transition Map

Objective: The study of finite state automaton is an essential tool in machine learning and artificial intelligence. The class of rough finite state automaton captures the uncertainty using the rough transition map. The need to generalize this concept arises to adhere the dynamical behaviour of the system. Hence this paper focuses on defining non-homogeneous rough finite state automaton. Methodology: With the aid of Rough finite state automata we define the concept of non-homogeneous rough finite state automata. Findings: Non homogeneous Rough Finite State Automata (NRFSA) Mt is defined by a tuple (Q,Σ,δt,q0 (t),F(t)) The dynamical behaviour of any system can be expressed in terms of an information system at time t. This leads us to define non-homogeneous rough finite state automaton. For each time ‘t’ we generate lower approximation rough finite state automaton Mt_ and the upper approximation rough finite state automaton Mt- and the defined concepts are elaborated with suitable examples. The ordered pair , Mt=(M(t)-,M(t)-) is called as the non-homogeneous rough finite state automaton. Conclusion: Over all our study reveals the characterization of the system which changes its behaviour dynamically over a time ‘t’. Novelty: The novelty of the proposed article is that it clearly immense the system behaviour over a time ‘t’. Using this concept the possible and the definite transitions in the system can be calculated in any given time ‘t’.

scholarly journals The Lattice Structures of Approximation Operators Based on L-Fuzzy Generalized Neighborhood Systems

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Qiao-Ling Song ◽  
Hu Zhao ◽  
Juan-Juan Zhang ◽  
A. A. Ramadan ◽  
Hong-Ying Zhang ◽  
...  
Keyword(s):  
Complete Lattice ◽  
Lattice Isomorphism ◽  
Double Negative ◽  
Lattice Structures ◽  
Fuzzy Relations ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Approximation Operators ◽  
Neighborhood System ◽  
Generalized Neighborhood System

Following the idea of L -fuzzy generalized neighborhood systems as introduced by Zhao et al., we will give the join-complete lattice structures of lower and upper approximation operators based on L -fuzzy generalized neighborhood systems. In particular, as special approximation operators based on L -fuzzy generalized neighborhood systems, we will give the complete lattice structures of lower and upper approximation operators based on L -fuzzy relations. Furthermore, if L satisfies the double negative law, then there exists an order isomorphic mapping between upper and lower approximation operators based on L -fuzzy generalized neighborhood systems; when L -fuzzy generalized neighborhood system is serial, reflexive, and transitive, there still exists an order isomorphic mapping between upper and lower approximation operators, respectively, and both lower and upper approximation operators based on L -fuzzy relations are complete lattice isomorphism.

scholarly journals Error bounds for approximations with deep ReLU neural networks in Ws,p norms

2019 ◽  
Vol 18 (05) ◽  
pp. 803-859 ◽  
Author(s):  
Ingo Gühring ◽  
Gitta Kutyniok ◽  
Philipp Petersen
Keyword(s):  
Neural Networks ◽  
Approximation Error ◽  
Upper And Lower Bounds ◽  
Trade Off ◽  
Lower Approximation ◽  
Upper Approximation ◽  
Regular Functions ◽  
Function Classes ◽  
The Neural Network ◽  
Sobolev Norms

We analyze to what extent deep Rectified Linear Unit (ReLU) neural networks can efficiently approximate Sobolev regular functions if the approximation error is measured with respect to weaker Sobolev norms. In this context, we first establish upper approximation bounds by ReLU neural networks for Sobolev regular functions by explicitly constructing the approximate ReLU neural networks. Then, we establish lower approximation bounds for the same type of function classes. A trade-off between the regularity used in the approximation norm and the complexity of the neural network can be observed in upper and lower bounds. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.

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