40,249 research outputs found
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
Multiaspect soft sets and its generalizations / Nor Hashimah Sulaiman
The theory of soft sets introduced in 1999 by Molodtsov is an alternative mathematical tool for dealing with uncertainties. It basically deals with information representations of objects characterized by parameters which are defined over a single common universal set. Combinations of the theory with fuzzy sets and interval-valued fuzzy sets have resulted in the so-called fuzzy soft sets and interval-valued fuzzy soft sets. Various theoretical studies on these theories and the variants have been made, and applications of the theories in various areas particularly in the area of decision making are continuously explored. Soft sets, fuzzy soft sets and interval-valued fuzzy soft sets have greater potential in information representation should the universe sets of elements not be restricted to only a common universal set. Real life situations may involve descriptions of objects, situations or entities based on certain characteristics or attributes which may be associated with different sets of elements of different types of universal sets. In this thesis, we introduce the concepts of multiaspect soft set (MASS), multiaspect fuzzy soft set (MAFSS) and multiaspect interval-valued fuzzy soft set (MAIVFSS) which are generalizations of soft sets, fuzzy soft sets and intervalvalued fuzzy soft sets, respectively. These concepts provide platforms for information representations that allow elements from different universal sets be taken into consideration in the description of a particular object, item or entity. MASS is defined for crisp data representation while MAFSS and MAIVFSS are respectively defined for fuzzy data representation with single and interval-valued membership degrees. For each concept, the set operations are established and the algebraic properties are studied. The concepts of mapping for multiaspect soft classes, multiaspect fuzzy soft classes and multiaspect interval-valued fuzzy soft classes are presented. In addition, we put forward the axiomatic definitions of distance, distance-based similarity measures and entropy for MAFSS and MAIVFSS. We introduce weighted and nonweighted distances and similarity measures based on the Hamming distance and the Euclidean distance. Relationships between the three measures are investigated. In the final part of the thesis, we highlight the applicability of some of the introduced concepts in solving group decision making problem under MAFSS and MAIVFSS environment
Relations on FP-Soft Sets Applied to Decision Making Problems
In this work, we first define relations on the fuzzy parametrized soft sets
and study their properties. We also give a decision making method based on
these relations. In approximate reasoning, relations on the fuzzy parametrized
soft sets have shown to be of a primordial importance. Finally, the method is
successfully applied to a problems that contain uncertainties.Comment: soft application
Linear-algebraic list decoding of folded Reed-Solomon codes
Folded Reed-Solomon codes are an explicit family of codes that achieve the
optimal trade-off between rate and error-correction capability: specifically,
for any \eps > 0, the author and Rudra (2006,08) presented an n^{O(1/\eps)}
time algorithm to list decode appropriate folded RS codes of rate from a
fraction 1-R-\eps of errors. The algorithm is based on multivariate
polynomial interpolation and root-finding over extension fields. It was noted
by Vadhan that interpolating a linear polynomial suffices if one settles for a
smaller decoding radius (but still enough for a statement of the above form).
Here we give a simple linear-algebra based analysis of this variant that
eliminates the need for the computationally expensive root-finding step over
extension fields (and indeed any mention of extension fields). The entire list
decoding algorithm is linear-algebraic, solving one linear system for the
interpolation step, and another linear system to find a small subspace of
candidate solutions. Except for the step of pruning this subspace, the
algorithm can be implemented to run in {\em quadratic} time. The theoretical
drawback of folded RS codes are that both the decoding complexity and proven
worst-case list-size bound are n^{\Omega(1/\eps)}. By combining the above
idea with a pseudorandom subset of all polynomials as messages, we get a Monte
Carlo construction achieving a list size bound of O(1/\eps^2) which is quite
close to the existential O(1/\eps) bound (however, the decoding complexity
remains n^{\Omega(1/\eps)}). Our work highlights that constructing an
explicit {\em subspace-evasive} subset that has small intersection with
low-dimensional subspaces could lead to explicit codes with better
list-decoding guarantees.Comment: 16 pages. Extended abstract in Proc. of IEEE Conference on
Computational Complexity (CCC), 201
Tensor products and regularity properties of Cuntz semigroups
The Cuntz semigroup of a C*-algebra is an important invariant in the
structure and classification theory of C*-algebras. It captures more
information than K-theory but is often more delicate to handle. We
systematically study the lattice and category theoretic aspects of Cuntz
semigroups.
Given a C*-algebra , its (concrete) Cuntz semigroup is an object
in the category of (abstract) Cuntz semigroups, as introduced by Coward,
Elliott and Ivanescu. To clarify the distinction between concrete and abstract
Cuntz semigroups, we will call the latter -semigroups.
We establish the existence of tensor products in the category and study
the basic properties of this construction. We show that is a symmetric,
monoidal category and relate with for
certain classes of C*-algebras.
As a main tool for our approach we introduce the category of
pre-completed Cuntz semigroups. We show that is a full, reflective
subcategory of . One can then easily deduce properties of from
respective properties of , e.g. the existence of tensor products and
inductive limits. The advantage is that constructions in are much easier
since the objects are purely algebraic.
We also develop a theory of -semirings and their semimodules. The Cuntz
semigroup of a strongly self-absorbing C*-algebra has a natural product giving
it the structure of a -semiring. We give explicit characterizations of
-semimodules over such -semirings. For instance, we show that a
-semigroup tensorially absorbs the -semiring of the Jiang-Su
algebra if and only if is almost unperforated and almost divisible, thus
establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results
corrected, in particular added 5.2.3-5.2.
- …