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.

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 Lacunary Mean Ideal Convergence in Generalized Randomn-Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Normed Space ◽  
Complex Numbers ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Cauchy Sequences ◽  
Limit Points ◽  
Positive Integers ◽  
Cluster Points

An idealIis a hereditary and additive family of subsets of positive integersℕ. In this paper, we will introduce the concept of generalized randomn-normed space as an extension of randomn-normed space. Also, we study the concept of lacunary mean (L)-ideal convergence andL-ideal Cauchy for sequences of complex numbers in the generalized randomn-norm. We introduceIL-limit points andIL-cluster points. Furthermore, Cauchy andIL-Cauchy sequences in this construction are given. Finally, we find relations among these concepts.

scholarly journals Lacunary ward continuity in 2-normed spaces

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2257-2263 ◽  
Author(s):  
Huseyin Cakalli ◽  
Sibel Ersan
Keyword(s):  
Real Number ◽  
Normed Space ◽  
Positive Real Number ◽  
Normed Spaces ◽  
Positive Real ◽  
Cauchy Sequences ◽  
Positive Integers

In this paper, we introduce lacunary statistical ward continuity in a 2-normed space. A function f defined on a subset E of a 2-normed space X is lacunary statistically ward continuous if it preserves lacunary statistically quasi-Cauchy sequences of points in E where a sequence (xk) of points in X is lacunary statistically quasi-Cauchy if limr?1 1/hr |{k?Ir : ||xk+1 - xk, z||? ?}| = 0 for every positive real number ? and z ? X, and (kr) is an increasing sequence of positive integers such that k0 = 0 and hr = kr - kr-1 ? ? as r ? ?, Ir = (kr-1, kr]. We investigate not only lacunary statistical ward continuity, but also some other kinds of continuities in 2-normed spaces.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals Statistical Convergence and Ideal Convergence of Sequences of Functions in 2-Normed Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Saeed Sarabadan ◽  
Sorayya Talebi
Keyword(s):  
Statistical Convergence ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Convergence Of Sequences

We present various kinds of statistical convergence andℐ-convergence for sequences of functions with values in 2-normed spaces and obtain a criterion forℐ-convergence of sequences of functions in 2-normed spaces. We also define the notion ofℐ-equistatistically convergence and studyℐ-equi-statistically convergence of sequences of functions.

scholarly journals On Ideal Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

2014 ◽  
Vol 8 (5) ◽  
pp. 2307-2313
Author(s):  
Vatan KARAKAYA ◽  
Necip ŞİMŞEK ◽  
M�zeyyen ERTÜRK ◽  
Faik GÜRSOY
Keyword(s):  
Normed Spaces ◽  
Ideal Convergence ◽  
Intuitionistic Fuzzy ◽  
Convergence Of Sequences ◽  
Intuitionistic Fuzzy Normed Spaces

Rearrangements that Preserve Rates of Divergence

1982 ◽  
Vol 34 (4) ◽  
pp. 916-920
Author(s):  
Elgin H. Johnston
Keyword(s):  
Real Number ◽  
Infinite Series ◽  
Special Interest ◽  
Convergent Series ◽  
Divergent Series ◽  
Classical Theorem ◽  
Real Numbers ◽  
Conditionally Convergent Series ◽  
Positive Integers

Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak. A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.

scholarly journals Axioms for constructive fields

1973 ◽  
Vol 8 (2) ◽  
pp. 221-232 ◽  
Author(s):  
John Staples
Keyword(s):  
Real Number ◽  
Linear Algebra ◽  
General Notion ◽  
Constructive Mathematics ◽  
Real Numbers ◽  
Cauchy Sequences ◽  
Definition Of

In constructive mathematics the Dedekind cut definition of real number is not equivalent to the definition of real number by Cauchy sequences, and the Dedekind real numbers do not satisfy Heyting's axioms for constructive fields. A more general notion of constructive field is proposed which includes the Dedekind real numbers; some linear algebra is given which applies to such fields.

On Some Diophantine Inequalities Involving the Exponential Function

1965 ◽  
Vol 17 ◽  
pp. 616-626 ◽  
Author(s):  
A. Baker
Keyword(s):  
Real Number ◽  
Exponential Function ◽  
Diophantine Inequalities ◽  
Real Numbers ◽  
Image Position ◽  
Positive Integers

It is well known that for any real number θ there are infinitely many positive integers n such thatHere ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

The Arzelà–Ascoli theorem by means of ideal convergence

2018 ◽  
Vol 25 (3) ◽  
pp. 475-479
Author(s):  
Emre Taş ◽  
Tugba Yurdakadim
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Statistical Convergence ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Convergence Of Sequences ◽  
The Relationship

AbstractIn this paper, using the concept of ideal convergence, which extends the idea of ordinary convergence and statistical convergence, we are concerned with the I-uniform convergence and the I-pointwise convergence of sequences of functions defined on a set of real numbers D. We present the Arzelà–Ascoli theorem by means of ideal convergence and also the relationship between I-equicontinuity and I-continuity for a family of functions.

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