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Multidimensional Statistical Convergence in Credibility Theory
Abstract
The purpose of this article is to present the concepts of double statistical convergence in credibility and [Formula: see text]-double statistical convergence in credibility in pringsheim sense. By using these definitions we present a natural multidimensional extension of Credibility theory via Summability methods.
In this study, we introduce the concepts of φλ,μ-double statistically convergence of order β in fuzzy sequences and strongly λ- double Cesaro summable of order β for sequences of fuzzy numbers. Also we give some inclusion theorems.
Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of complex uncertain sequences: convergence almost surely (a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely. In addition, relationships among them are discussed.
The idea of statistical convergence was first introduced by H. Fast [Colloq. Math. 2, 241-244 (1951; Zbl 0044.33605)] but the rapid developments were started after the papers of J. A. Šalát [Math. Slovaca 30, 139-150 (1980; Zbl 0437.40003)] and J. A. Fridy [Analysis 5, 301-313 (1985; Zbl 0588.40001)]. Nowadays it has become one of the most active areas of research in the field of summability. In this paper we define and study statistical analogue of convergence and Cauchy for double sequences. We also establish the relation between statistical convergence and strongly Cesàro summable double sequences.
In this article we define the notion of statistically convergent difference double sequence spaces. We examine the spaces
_2 \bar \ell _\infty (\Delta ,q), _2^ - c(\Delta ,q), _2^ - c^B (\Delta ,q), _2^ - c^R (\Delta ,q), _2^ - c^{BR} (\Delta ,q)
etc. being symmetric, solid, monotone, etc. We prove some inclusion results too.
In this article we study different properties of convergent, null and bounded sequence spaces of fuzzy real numbers defined
by an Orlicz function, like completeness, solidness, symmetricity, convergence free etc. We prove some inclusion results,
too.
In this article we introduce the vector valued paranormed sequence spaces
2[`(c)] (q,p), 2[`(c)] 0 (q,p), (2[`(c)] )B (q,p), (2[`(c)] 0 )B (q,p), (2[`(c)] )R (q,p){}_2\overline c (q,p), {}_2\overline c _0 (q,p), ({}_2\overline c )^B (q,p), ({}_2\overline c _0 )^B (q,p), ({}_2\overline c )^R (q,p)
and
(2[`(c)] 0 )R (q,p)({}_2\overline c _0 )^R (q,p)
defined over a seminormed space (X,q). We study their different properties like completeness, solidness, symmetry, convergence freeness etc. We prove some inclusion
results.
In this article we introduce some vector valued double sequence spaces defined by Orlicz function. We study some of their
properties like solidness, symmetricity, completeness etc. and prove some inclusion results.
In this paper we extend the notion of λ-statistical convergence to the (λ, µ)statistical convergence for double sequences x = (x
jk
). We also determine some matrix transformations and establish some core theorems related to our new space of double sequences
S
λ,µ.
KeywordsDouble sequence-statistical convergence-matrix transformation-core
MR(2000) Subject Classification40C05-40H05
Possibility measures and credibility measures are widely used in fuzzy set theory. Compared with possibility measures, the advantage of credibility measures is the self-duality property. This paper gives a relation between possibility measures and credibility measures, and proves a sufficient and necessary condition for credibility measures. Finally, the credibility extension theorem is shown.
This paper will present a novel concept of expected values of fuzzy variables, which is essentially a type of Choquet integral and coincides with that of random variables. In order to calculate the expected value of general fuzzy variable, a fuzzy simulation technique is also designed. Finally, we construct a spectrum of fuzzy expected value models, and integrate fuzzy simulation, neural network, and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models.
The main aim of this article is to study strongly almost λ-convergence and statistically almost λ-convergence of complex uncertain sequences in two aspects. At first we introduce these notions via Orlicz function and then we do the same by following usual path of convergence. In general, these types of convergences of sequences can be done in five directions in the environment of uncertainty, namely strongly almost λ-convergence (respectively, statistically almost λ-convergence) in almost surely, in mean, in measure, in distribution and in uniformly almost surely. In this research article, we define the said notions and then characterize those in the direction of almost surely.
In this paper, we introduce a new type of convergence for a sequence of function, namely, lambda-statistically convergent sequences of functions in fuzzy 2-normed space, which is a natural generalization of convergence in fuzzy 2-normed space. In particular, we introduce the concepts of uniform lambda-statistical convergence and pointwise lambda-statistical convergence in the topology induced by fuzzy 2-normed spaces, and give some basic properties of these concepts.
In this article, we introduce some double sequence spaces of fuzzy real numbers defined by Orlicz function, study some of their properties like solidness, symmetricity, completeness etc, and prove some inclusion results.
The idea of almost convergence for double sequences was introduced by Moricz and Rhoades [Math. Proc. Cambridge Philos. Soc. 104 (1988) 283–294] and they also characterized the four dimensional strong regular matrices. In this paper we define and characterize the almost strongly regular matrices for double sequences and apply these matrices to establish a core theorem.
It is well-known that Markov inequality, Chebyshev inequality, Hlder''s inequality, and Minkowski inequality are important and useful results in probability theory. This paper presents the analogous inequalities in fuzzy set theory and rough set theory. In addition, sequence convergence plays an extremely important role in the fundamental theory of mathematics. This paper presents four types of convergence concept of fuzzy/rough sequence: convergence almost surely, convergence in credibility/trust, convergence in mean, and convergence in distribution. Some mathematical properties of those new convergence concepts are also given.
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
Recently, Patterson and Savaş (Math. Commun. 10:55-61, 2005), defined the lacunary statistical analog for double sequences x=(x(k,l)) as follows: A real double sequences x=(x(k,l)) is said to be P-lacunary statistically convergent to L provided that for each ε>0, P-limr,s1hr,s|{(k,l)∈Ir,s:|x(k,l)−L|≥ε}|=0. In this case write stθ2-limx=L or x(k,l)→L(stθ2).
In this paper we introduce and study lacunary statistical convergence for double sequences in topological groups and we shall also present some inclusion theorems.
MSC:
42B15, 40C05.