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 A Study on -Quasi-Cauchy Sequences

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Hüseyin Çakalli ◽  
Huseyin Kaplan
Keyword(s):  
Real Function ◽  
Cauchy Sequence ◽  
Closed Subset ◽  
Continuous Functions ◽  
Uniform Limit ◽  
Cauchy Sequences ◽  
Uniformly Continuous

Recently, the concept of -ward continuity was introduced and studied. In this paper, we prove that the uniform limit of -ward continuous functions is -ward continuous, and the set of all -ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function defined on an interval is uniformly continuous if and only if (()) is -quasi-Cauchy whenever () is a quasi-Cauchy sequence of points in .

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 $E$.

scholarly journals Ideal and some applications of simply open sets

2017 ◽  
Vol 13 (3) ◽  
pp. 7264-7271
Author(s):  
Arafa A Nasefa ◽  
R Mareay
Keyword(s):  
Closed Subset ◽  
Continuous Functions ◽  
Logical Space ◽  
Fundamental Properties ◽  
New Type ◽  
Nowhere Dense Set ◽  
Dense Set ◽  
The Ideal

Recently there has been some interest in the notion of a locally closed subset of a topo- logical space. In this paper, we introduce a useful characterizations of simply open sets in terms of the ideal of nowhere dense set. Also, we study a new notion of functions in topo- logical spaces known as dual simply-continuous functions and some of their fundamental properties are investigated. Finally, a new type of simply open sets is introduced.

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.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

An Extremum Result

1962 ◽  
Vol 14 ◽  
pp. 597-601 ◽  
Author(s):  
J. Kiefer
Keyword(s):  
Probability Measure ◽  
Compact Space ◽  
Ad Hoc ◽  
Continuous Functions ◽  
Orthogonality Condition ◽  
Technical Detail ◽  
Real Numbers ◽  
Discrete Probability ◽  
Linearly Independent ◽  
Image Position

The main object of this paper is to prove the following:Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;(b)The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).

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