scholarly journals -Ward Continuity

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Huseyin Cakalli
Keyword(s):  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Lacunary Sequence ◽  
Real Numbers ◽  
Cauchy Sequences ◽  
New Type ◽  
Positive Integers ◽  
Convergent Sequences

A function is continuous if and only if preserves convergent sequences; that is, is a convergent sequence whenever is convergent. The concept of -ward continuity is defined in the sense that a function is -ward continuous if it preserves -quasi-Cauchy sequences; that is, is an -quasi-Cauchy sequence whenever is -quasi-Cauchy. A sequence of points in , the set of real numbers, is -quasi-Cauchy if , where , and is a lacunary sequence, that is, an increasing sequence of positive integers such that and . A new type compactness, namely, -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.

Matrix Transformations in an Incomplete Space

1968 ◽  
Vol 20 ◽  
pp. 727-734 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Convergent Series ◽  
Matrix Transformations ◽  
Cauchy Sequences ◽  
Complex Linear ◽  
Incomplete Space ◽  
Bounded Sequences ◽  
Convergent Sequences

Let X = (X, p) be a seminormed complex linear space with zero θ. Natural definitions of convergent sequence, Cauchy sequence, absolutely convergent series, etc., can be given in terms of the seminorm p. Let us write C = C(X) for the set of all convergent sequences for the set of Cauchy sequences; and L∞ for the set of all bounded sequences.

scholarly journals A new type of difference I-convergent sequence in IFnNS

2021 ◽  
pp. 22-22
Author(s):  
Vakeel Khan ◽  
Izhar Khan ◽  
Mobeen Ahmad
Keyword(s):  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Normed Spaces ◽  
Intuitionistic Fuzzy ◽  
New Type

In this paper, we introduce the notion of a generalized difference I-convergent (i.e.?m-I-convergent) and difference I-Cauchy (i.e.?m-I-Cauchy) sequence in intuitionistic fuzzy n-normed spaces. Further, we prove some results related to this notion. Also, we study the concepts of a generalized difference I+-convergent (i.e.?m-I+-convergent) sequence in intuitionistic fuzzy n-normed spaces and show the relation between them.

scholarly journals New Type Continuities via Abel Convergence

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Huseyin Cakalli ◽  
Mehmet Albayrak
Keyword(s):  
Closed Subset ◽  
Continuous Functions ◽  
Real Numbers ◽  
Uniform Limit ◽  
New Type ◽  
Convergent Sequences

We investigate the concept of Abel continuity. A functionfdefined on a subset ofℝ, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences. Some other types of continuities are also studied and interesting result is obtained. It turned out that uniform limit of a sequence of Abel continuous functions is Abel continuous and the set of Abel continuous functions is a closed subset of continuous functions.

scholarly journals Generalized ideal convergence in intuitionistic fuzzy normed linear spaces

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 811-820 ◽  
Author(s):  
Bipan Hazarika ◽  
Vijay Kumar ◽  
Bernardo Lafuerza-Guilién
Keyword(s):  
Real Number ◽  
Normed Spaces ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Linear Spaces ◽  
Intuitionistic Fuzzy ◽  
Cauchy Sequences ◽  
Positive Integers ◽  
Intuitionistic Fuzzy Normed Spaces ◽  
Cluster Points

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In [19], Kostyrko et al. introduced the concept of ideal convergence as a sequence (xk) of real numbers is said to be I-convergent to a real number e, if for each ? > 0 the set {k ? N : |xk - e| ? ?} belongs to I. The aim of this paper is to introduce and study the notion of ?-ideal convergence in intuitionistic fuzzy normed spaces as a variant of the notion of ideal convergence. Also I? -limit points and I?-cluster points have been defined and the relation between them has been established. Furthermore, Cauchy and I?-Cauchy sequences are introduced and studied. .

scholarly journals On Weak Statistical Convergence

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Vinod K. Bhardwaj ◽  
Indu Bala
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Statistical Convergence ◽  
Reflexive Space ◽  
Cauchy Sequences ◽  
Convergent Sequences

The main object of this paper is to introduce a new concept of weak statistically Cauchy sequence in a normed space. It is shown that in a reflexive space, weak statistically Cauchy sequences are the same as weakly statistically convergent sequences. Finally, weak statistical convergence has been discussed inlpspaces.

I-COMPACT RIGHT CHAIN DOMAINS

2011 ◽  
Vol 10 (06) ◽  
pp. 1107-1139
Author(s):  
H. H. BRUNGS ◽  
G. TÖRNER
Keyword(s):  
Valuation Ring ◽  
Convergent Sequence ◽  
Ideal Theory ◽  
Prime Ideals ◽  
Real Numbers ◽  
Minimal Set ◽  
Valuation Rings ◽  
Rank One ◽  
Special Case ◽  
Convergent Sequences

Generalizing the concept of convergency to valued fields, Ostrowski in the 1930s introduced pseudo-convergent sequences. In the present paper we classify pseudo-convergent sequences in right chain domains R according to the prime ideal P associated to the breadth I of the sequence using an ideal theory developed for right cones in groups. The ring R is I-compact if every pseudo-convergent sequence in R with breadth I has a limit in R, and we construct right chain domains R which are I-compact only for right ideals I in particular subsets [Formula: see text] of the set of all right ideals of R. Krull's perfect valuation rings and then Ribenboim's notion of a valuation ring complete par étages, where [Formula: see text] is the minimal set containing the completely prime ideals in a commutative valuation ring, is a special case. For a non-discrete right invariant rank-one right chain domain R there are exactly two possibilities for the set [Formula: see text] if the value group of R is the group of real numbers under addition, and there are infinitely many possibilities for [Formula: see text] in all other cases.

scholarly journals A variation on arithmetic continuity

2017 ◽  
Vol 35 (3) ◽  
pp. 195 ◽  
Author(s):  
Huseyin Cakalli
Keyword(s):  
Compact Subset ◽  
Convergent Sequence ◽  
Convergent Subsequence ◽  
Continuous Functions ◽  
Greatest Common Divisor ◽  
Real Numbers ◽  
Uniformly Continuous ◽  
Convergent Sequences

A sequence $(x_{k})$ of points in $\R$, the set of real numbers, is called \textit{arithmetically convergent} if  for each $\varepsilon > 0$ there is an integer $n$ such that for every integer $m$ we have $|x_{m} - x_{<m,n>}|<\varepsilon$, where $k|n$ means that $k$ divides $n$ or $n$ is a multiple of $k$, and the symbol $< m, n >$ denotes the greatest common divisor of the integers $m$ and $n$. We prove that a subset of $\R$ is bounded if and only if it is arithmetically compact, where a subset $E$ of $\R$ is arithmetically compact if any sequence of point in $E$ has an arithmetically convergent subsequence. It turns out that the set of arithmetically continuous functions on an arithmetically compact subset of $\R$ coincides with the set of uniformly continuous functions where a function $f$ defined on a subset $E$ of $\R$ is arithmetically continuous if it preserves arithmetically convergent sequences, i.e., $(f(x_{n})$ is arithmetically convergent whenever $(x_{n})$ is an arithmetic convergent sequence of points in $E$.

Lifting convergent sequences with networks

1971 ◽  
Vol 70 (1) ◽  
pp. 27-30
Author(s):  
J. H. Webb
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Real Numbers ◽  
Positive Real ◽  
Positive Integers ◽  
Locally Convex ◽  
Convergent Sequences

Definition (Moukoko Priso(2)). A locally convex spaceE[T] is said to have a strict absorbent network of type Σ if there exists in E a familyof absolutely convex absorbent sets such that(1) if {nk} is a sequence of positive integers andfor each k, then the seriesconverges in E[T](2) for each sequence {nk} there is a sequence {λk} of positive real numbers such that, ifand 0 ≤ μk ≤ λkfor each k, then(i) converges in E[T], and(ii) for each p.

scholarly journals On (λ,μ)- Zweier ideal convergence in intuitionistic fuzzy normed space

2020 ◽  
Vol 30 (4) ◽  
pp. 413-427
Author(s):  
Vakeel Khan ◽  
Mobeen Ahmad
Keyword(s):  
Normed Space ◽  
Double Sequences ◽  
Real Numbers ◽  
Fuzzy Normed Space ◽  
Positive Real ◽  
Ideal Convergence ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Fuzzy Normed Space ◽  
Cauchy Sequences ◽  
New Type

In this paper, we study and introduce a new type of convergence, namely (?,?)- Zweier convergence and (?,?)- Zweier ideal convergence of double sequences x = (xij) in intuitionistic fuzzy normed space (IFNS), where ? = (?n) and ?= (?m) are two non-decreasing sequences of positive real numbers such that each tending to infinity. Furthermore, we studied (?,?)- Zweier Cauchy and (?,?)- Zweier ideal Cauchy sequences on the said space and established a relation between them.

scholarly journals On generalized difference ideal convergence in random 2-normed spaces

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1273-1282 ◽  
Author(s):  
Bipan Hazarika
Keyword(s):  
Real Number ◽  
Normed Spaces ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Cauchy Sequences ◽  
Convergence Of Sequences ◽  
Positive Integers

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. In [17], Kostyrko et. al introduced the concept of ideal convergence as a sequence (xk ) of real numbers is said to be I-convergent to a real number ?, if for each ? > 0 the set {k ? N : |xk ? ?| ? ?} belongs to I. In [28], Mursaleen and Alotaibi introduced the concept of I-convergence of sequences in random 2-normed spaces. In this paper, we define and study the notion of ?n -ideal convergence and ?n -ideal Cauchy sequences in random 2-normed spaces, and prove some interesting theorems.

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