A Study of Fuzzy Sequence Spaces

The purpose of this chapter is to introduce and study some new ideal convergence sequence spaces FSJθT, FS0JθT and FS∞JθT on a fuzzy real number F defined by a compact operator T. We investigate algebraic properties like linearity, solidness and monotinicity with some important examples. Further, we also analyze closedness of the subspace and inclusion relations on the said spaces.

Assessment of the convergence angle of teeth prepared for full crown by preclinical dental students

The objective of this study was to measure the buccolingual and mesiodistal convergence angles of six typodont teeth (# 26, 36, 45, 15, 21, and 13), prepared by preclinical dental students at Ajman University, for porcelain fused to a metal full crown and to compare them with the recommended convergence angle (6.5°). Additionally, we sought to compare the angles recorded for the six sets of teeth and relate the results according to the tooth position and surface and to know which one shows the greater tendency of straying from the normal convergence angle. Materials and methods: The angle of convergence of one hundred ninety-eight typodont teeth preparations was measured both buccolingually and mesiodistally by using a Dino-lite pro digital microscope (AM-413ZT Taiwan) with a Dinocapture (2.0 version 1.5.27.A, AnMo Electronics Corporation). All the results were recorded, and the data were analyzed by means of a one-sample t-test and one-way ANOVA. Results: The mean total convergence angle for this study was 11.29°± 6.66° from both surfaces, which is greater than the recommended value of 6.5° and statistically significant (p<0.000). Only 7.07% of teeth met the ideal convergence angle from both surfaces, and the one-sample test showed a statistically significant difference (p<0.057) from the recommended convergence angle, except for the mesiodistal convergence angle of the lower-right second premolar, which revealed no significant difference. The mean convergence angle for the buccolingual surface was 12.42°± 6.16°, which was higher than that of the mesiodistal surface (10.16°± 7°). One-way ANOVA showed a significant difference between all selected teeth (p<0.000), and a paired samples t-test showed a significant difference within two teeth only, the lower-right second premolar and upper-right canine (p<0.000), in which the mesiodistal measurement showed a lower convergence angle than the buccolingual angle. Conclusions: Preclinical students prepared teeth with a convergence angle higher than the recommended convergence angle. However, all the recorded angles were within the range of previous studies. It was concluded that the recommended convergence angle was difficult to achieve in preclinical practice.

Ideals, Nonnegative Summability Matrices and Corresponding Convergence Notions: A Short Survey of Recent Advancements

In this survey article, we look into some recent results concerning summability matrices, both regular as well as those which are not regular (called semi-regular) and generated matrix ideals as the overall view of the inter relationship between the notions of ideal convergence and summability methods by regular summability matrices.

Lacunary ℐ -Invariant Convergence of Sequence of Sets in Intuitionistic Fuzzy Metric Spaces

The concepts of invariant convergence, invariant statistical convergence, lacunary invariant convergence, and lacunary invariant statistical convergence for set sequences were introduced by Pancaroğlu and Nuray (2013). We know that ideal convergence is more general than statistical convergence for sequences. This has motivated us to study the lacunary ℐ -invariant convergence of sequence of sets in intuitionistic fuzzy metric spaces (briefly, IFMS). In this study, we examine the notions of lacunary ℐ -invariant convergence W ℐ σ θ η , ν (Wijsman sense), lacunary ℐ ∗ -invariant convergence W ℐ σ θ ∗ η , ν (Wijsman sense), and q -strongly lacunary invariant convergence W N σ θ η , ν q (Wijsman sense) of sequences of sets in IFMS. Also, we give the relationships among Wijsman lacunary invariant convergence, W N σ θ η , ν q , W ℐ σ θ η , ν , and W ℐ σ θ ∗ η , ν in IFMS. Furthermore, we define the concepts of W ℐ σ θ η , ν -Cauchy sequence and W ℐ σ θ ∗ η , ν -Cauchy sequence of sets in IFMS. Furthermore, we obtain some features of the new type of convergences in IFMS.

IDEAL CONVERGENCE OF DOUBLE SEQUENCES OF CLOSED SETS

In the present paper, we introduce the concepts of ideal inner and ideal outer limits which always exist even if empty sets for double sequences of closed sets in Pringsheim's sense. Next, we give some formulas for finding ideal inner and outer limits in a metric space. After then, we define Kuratowski ideal convergence of double sequences of closed sets by means of the ideal inner and ideal outer limits of a double sequence of closed sets. Additionally, we give some examples that our result is more general than the results obtained before.

Further aspects of I K-convergence in topological spaces

In this paper, we obtain some results on the relationships between different ideal convergence modes namely, I K, I K∗ , I, K, I ∪ K and (I ∪K) ∗ . We introduce a topological space namely I K-sequential space and show that the class of I K-sequential spaces contain the sequential spaces. Further I K-notions of cluster points and limit points of a function are also introduced here. For a given sequence in a topological space X, we characterize the set of I K-cluster points of the sequence as closed subsets of X.

on ideal convergence of triple sequences in intuitionistic Fuzzy normed space defined by compact operator

The main purpose of this article is to introduce and study some new spaces of I-convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operator i.e 3SI (μ,ν)(T ) and 3SI0(μ,ν)(T ) and examine some fundamental properties, fuzzy topology and verify inclusion relations lying under these spaces.

Olivier’s theorem: ideal convergence, algebrability and Borel classification

The classical Olivier’s theorem says that for any nonincreasing summable sequence (a(n)) the sequence (na(n)) tends to zero. This result was generalized by many authors. We propose its further generalization which implies known results. Next we consider the subset $${\mathcal {AOS}}$$ AOS of $$ \ell _{1} $$ ℓ 1 consisting of sequences for which the assertion of Olivier’s theorem is false. We study how large and good algebraic structures are contained in $${\mathcal {AOS}}$$ AOS and its subsets; this kind of study is known as lineability. Finally we show that $${\mathcal {AOS}}$$ AOS is a residual $$ \mathcal {G}_{\delta \sigma } $$ G δ σ but not an $$ {\mathcal {F}}_{\sigma \delta } \text {-set} $$ F σ δ -set .