Part I. Examples and Fundamental Concepts Introduction 1. First examples 2. Equivalence, classification, and invariants 3. Principle classes of asymptotic invariants 4. Statistical behavior of the orbits and introduction to ergodic theory 5. Smooth invariant measures and more examples Part II. Local Analysis and Orbit Growth 6. Local hyperbolic theory and its applications 7. Transversality and genericity 8. Orbit growth arising from topology 9. Variational aspects of dynamics Part III. Low-Dimensional Phenomena 10. Introduction: What is low dimensional dynamics 11. Homeomorphisms of the circle 12. Circle diffeomorphisms 13. Twist maps 14. Flows on surfaces and related dynamical systems 15. Continuous maps of the interval 16. Smooth maps of the interval Part IV. Hyperbolic Dynamical Systems 17. Survey of examples 18. Topological properties of hyperbolic sets 19. Metric structure of hyperbolic sets 20. Equilibrium states and smooth invariant measures Part V. Sopplement and Appendix 21. Dynamical systems with nonuniformly hyperbolic behavior Anatole Katok and Leonardo Mendoza.

Introduction to the Modern Theory of Dynamical Systems: EQUILIBRIUM STATES AND SMOOTH INVARIANT MEASURES

In this paper, we define soft ω -open sets and strongly soft ω -open sets as two new classes of soft sets. We study the natural properties of these types of soft sets and we study the validity of the exact versions of some known results in ordinary topological spaces regarding ω -open sets in soft topological spaces. Also, we study the relationships between the ω -open sets of a given indexed family of topological spaces and the soft ω -open sets (resp. strongly soft ω -open sets) of their generated soft topological space. These relationships form a biconditional logical connective which is a symmetry. As an application of strongly soft ω -open sets, we characterize soft Lindelof (resp. soft weakly Lindelof) soft topological spaces.

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On Two Classes of Soft Sets in Soft Topological Spaces

In this paper, we generalize three types of rough set models based on j-neighborhood space (i.e, type 1 j-neighborhood rough set, type 2 j-neighborhood rough set, and type 3 j-neighborhood rough set), and investigate some of their basic properties. Also, we present another three types of rough set models based on j-adhesion neighborhood space (i.e, type 4 j-adhesion neighborhood rough set, type 5 j-adhesion neighborhood rough set, and type 6 j-adhesion neighborhood rough set). The fundamental properties of approximation operators based on j-adhesion neighborhood space are established. The relationship between the properties of these types is explained. Finally, according to j-adhesion neighborhood space, we give a comparison between the Yao’s approach and our approach.

Comparison of six types of rough approximations based on j-neighborhood space and j-adhesion neighborhood space

Determination of entropy measures for the ordinal scale-based linguistic models

Soft rough set theory has been presented as a basic mathematical model for decision-making for many real-life data However, soft rough sets are based on a possible fusion of rough sets and soft sets which were proposed by Feng et al [20] The main contribution of the present article is to introduce a modification and a generalization for Feng's approximations, namely, soft β -rough approximations, and some of their properties will be studied A comparison between the suggested approximations and the previous one [20] will be discussed Some examples are prepared to display the validness of these proposals Finally, we put an actual example of the infections of coronavirus (COVID-19) based on soft β -rough sets This application aims to know the persons most likely to be infected with COVID-19 via soft β -rough approximations and soft β -rough topologies [ABSTRACT FROM AUTHOR] Copyright of Turkish Journal of Mathematics is the property of Scientific and Technical Research Council of Turkey and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use This abstract may be abridged No warranty is given about the accuracy of the copy Users should refer to the original published version of the material for the full abstract (Copyright applies to all Abstracts )

Soft β-rough sets and their application to determine COVID-19

The original model of rough sets was advanced by Pawlak, which was mainly involved with the approximation of things using an equivalence relation on the universal set of his approximation space. In this paper, two kinds of approximation operators via ideals which represent extensions of Pawlak’s approximation operator have been presented. In both kinds, the definitions of upper and lower approximations based on ideals have been given. Moreover, a new type of approximation spaces via two ideals which is called bi-ideal approximation spaces was introduced for the first time. This type of approximations was analyzed by two different methods, their properties are investigated, and the relationship between these methods is proposed. The importance of these methods was its dependent on ideals which were topological tools, and the two ideals represent two opinions instead of one opinion. At the end of the paper, an applied example had been introduced in the chemistry field by applying the current methods to illustrate the definitions in a friendly way.

Bi-ideal approximation spaces and their applications

In this paper, ideals in the context of multisets on the lattice of all submultisets with the order relation as the multiset inclusion have been introduced. Moreover, generalization of rough multiset model by defining new multiset approximation operators in more general setting via multiset ideal has been presented. The concepts of lower and upper multiset approxima- tions via multiset ideals have been mentioned. These new definitions decrease the upper multiset approximation and increase the lower multiset approximation and hence decreasing the boundary region and increasing the accuracy measure. Properties of these multiset approximations are studied and various examples are mentioned. Also, the comparison between the rough multiset approximations defined by Girish and John (11, 12) and the current multiset approximations has been presented. It's therefore shown that the current definitions are more generally. Finally, the multiset topology induced by the present methods is finer than the multiset topology induced by the old method (11, 12).

Sobhy A. El-Sheikh

Papers

Bi-ideal approximation spaces and their applications

Generalized rough multiset via multiset ideals

Some types of pairwise soft sets and the associated soft topologies

Strong and ultra separation axioms of fuzzy bitopological spaces

Pairwise soft separation axioms in soft bitopological spaces