Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications

Tesis por compendioMathematical models have extensively been used in problems related to engineering, computer sciences, economics, social, natural and medical sciences etc. It has become very common to use mathematical tools to solve, study the behavior and different aspects of a system and its different subsystems. Because of various uncertainties arising in real world situations, methods of classical mathematics may not be successfully applied to solve them. Thus, new mathematical theories such as probability theory and fuzzy set theory have been introduced by mathematicians and computer scientists to handle the problems associated with the uncertainties of a model. But there are certain deficiencies pertaining to the parametrization in fuzzy set theory. Soft set theory aims to provide enough tools in the form of parameters to deal with the uncertainty in a data and to represent it in a useful way. The distinguishing attribute of soft set theory is that unlike probability theory and fuzzy set theory, it does not uphold a precise quantity. This attribute has facilitated applications in decision making, demand analysis, forecasting, information sciences, mathematics and other disciplines. In this thesis we will discuss several algebraic and topological properties of soft sets and fuzzy soft sets. Since soft sets can be considered as setvalued maps, the study of fixed point theory for multivalued maps on soft topological spaces and on other related structures will be also explored. The contributions of the study carried out in this thesis can be summarized as follows: i) Revisit of basic operations in soft set theory and proving some new results based on these modifications which would certainly set a new dimension to explore this theory further and would help to extend its limits further in different directions. Our findings can be applied to develop and modify the existing literature on soft topological spaces ii) Defining some new classes of mappings and then proving the existence and uniqueness of such mappings which can be viewed as a positive contribution towards an advancement of metric fixed point theory iii) Initiative of soft fixed point theory in framework of soft metric spaces and proving the results lying at the intersection of soft set theory and fixed point theory which would help in establishing a bridge between these two flourishing areas of research. iv) This study is also a starting point for the future research in the area of fuzzy soft fixed point theory.Abbas, M. (2014). Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48470TESISCompendi

Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology

Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted. The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing

Higher Dimensional Geometries from Matrix Brane constructions

Matrix descriptions of even dimensional fuzzy spherical branes $S^{2k}$ in Matrix Theory and other contexts in Type II superstring theory reveal, in the large $N$ limit, higher dimensional geometries $SO(2k+1)/U(k)$, which have an interesting spectrum of $SO(2k+1)$ harmonics and can be up to 20 dimensional, while the spheres are restricted to be of dimension less than 10. In the case $k=2$, the matrix description has two dual field theory formulations. One involves a field theory living on the non-commutative coset $SO(5)/U(2)$ which is a fuzzy $S^2$ fibre bundle over a fuzzy $S^4$. In the other, there is a U(n) gauge theory on a fuzzy $S^4$ with ${\cal O}(n^3)$ instantons. The two descriptions can be related by exploiting the usual relation between the fuzzy two-sphere and U(n) Lie algebra. We discuss the analogous phenomena in the higher dimensional cases, developing a relation between fuzzy $SO(2k)/U(k)$ cosets and unitary Lie algebras.Comment: 28 pages (Harvmac big) ; version 2 : minor typos fixed and ref. adde

Designing Software Architectures As a Composition of Specializations of Knowledge Domains

This paper summarizes our experimental research and software development activities in designing robust, adaptable and reusable software architectures. Several years ago, based on our previous experiences in object-oriented software development, we made the following assumption: ‘A software architecture should be a composition of specializations of knowledge domains’. To verify this assumption we carried out three pilot projects. In addition to the application of some popular domain analysis techniques such as use cases, we identified the invariant compositional structures of the software architectures and the related knowledge domains. Knowledge domains define the boundaries of the adaptability and reusability capabilities of software systems. Next, knowledge domains were mapped to object-oriented concepts. We experienced that some aspects of knowledge could not be directly modeled in terms of object-oriented concepts. In this paper we describe our approach, the pilot projects, the experienced problems and the adopted solutions for realizing the software architectures. We conclude the paper with the lessons that we learned from this experience

Comments on Multiple M2-branes

Recently a three-dimensional field theory was derived that is consistent with all the symmetries expected of the worldvolume action for multiple M2-branes. In this note we examine several physical predictions of this model and show that they are in agreement with expected M2-brane dynamics. In particular, we discuss the quantization of the Chern-Simons coefficient, the vacuum moduli space, a massive deformation leading to fuzzy three-sphere vacua, and a possible large n limit. In this large n limit, the fuzzy funnel solution correctly reproduces the mass of an M5-brane.Comment: 18 pages. Published versio