scholarly journals Remarks on uniformly symmetrically continuous functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650069
Author(s):  
Tammatada Khemaratchatakumthorn ◽  
Prapanpong Pongsriiam
Keyword(s):  
Real Line ◽  
Nonempty Subset ◽  
Continuous Functions ◽  
The Real ◽  
Definition Of ◽  
Symmetrically Continuous Functions ◽  
Uniformly Continuous

We give the definition of uniform symmetric continuity for functions defined on a nonempty subset of the real line. Then we investigate the properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. We obtain some characterizations of uniformly symmetrically continuous functions. Several examples are also given.

A non-constant invariant function for certain ergodic flows

2008 ◽  
Vol 28 (3) ◽  
pp. 1031-1035
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Differentiable Manifold ◽  
Invariant Function ◽  
Continuous Functions ◽  
The Real ◽  
Almost All ◽  
Uniformly Continuous

AbstractLet U be the vector space of uniformly continuous real-valued functions on the real line $\mathbb {R}$ and let U0 denote the subspace of U consisting of all bounded uniformly continuous functions. If X is a compact differentiable manifold and we are given a flow on X, then we associate with the flow a function F:X→H1(X,U/U0) that is invariant under the flow. We give examples for which the flow on X is ergodic but there is no λ∈H1(X,U/U0) such that F(p)=λ for almost all points p.

scholarly journals ON A CONJECTURE REGARDING THE UPPER GRAPH BOX DIMENSION OF BOUNDED SUBSETS OF THE REAL LINE

Fractals ◽  
2013 ◽  
Vol 21 (03n04) ◽  
pp. 1350017
Author(s):  
GIORGOS KELGIANNIS ◽  
VAIOS LASCHOS
Keyword(s):  
Real Line ◽  
Continuous Functions ◽  
Box Dimension ◽  
The Real ◽  
Bounded Set ◽  
Isolated Points ◽  
Uniformly Continuous

Let X ⊂ ℝ be a bounded set; we introduce a formula that calculates the upper graph box dimension of X (i.e. the supremum of the upper box dimension of the graph over all uniformly continuous functions defined on X). We demonstrate the strength of the formula by proving various corollaries. We conclude by constructing a collection of sets X with infinitely many isolated points, having upper box dimension a taking values from zero to one while their graph box dimension takes any value in [ max {2a, 1}, a + 1], answering this way, negatively to a conjecture posed.1

scholarly journals Some remarks on incomplete gamma type function γ*(α, x_)

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 125-131 ◽  
Author(s):  
Emin Özcağ ◽  
İnci Egeb
Keyword(s):  
Recurrence Relation ◽  
Real Line ◽  
Summable Function ◽  
Type Function ◽  
The Real ◽  
Definition Of ◽  
Derivatives Of

The incomplete gamma type function ?*(?, x_) is defined as locally summable function on the real line for ?>0 by ?*(?,x_) = {?x0 |u|?-1 e-u du, x?0; 0, x > 0 = ?-x_0 |u|?-1 e-u du the integral divergining ? ? 0 and by using the recurrence relation ?*(? + 1,x_) = -??*(?,x_) - x?_ e-x the definition of ?*(?, x_) can be extended to the negative non-integer values of ?. Recently the authors [8] defined ?*(-m, x_) for m = 0, 1, 2,... . In this paper we define the derivatives of the incomplete gamma type function ?*(?, x_) as a distribution for all ? < 0.

Statistical σ approximation to Bögel-type continuous functions

2011 ◽  
Vol 48 (4) ◽  
pp. 475-488 ◽  
Author(s):  
Sevda Karakuş ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Result ◽  
The Real

In this paper, using the concept of statistical σ-convergence which is stronger than the statistical convergence, we obtain a statistical σ-approximation theorem for sequences of positive linear operators defined on the space of all real valued B-continuous functions on a compact subset of the real line. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. Also we compute the rate of statistical σ-convergence of sequence of positive linear operators.

scholarly journals Compactness in Metric Spaces

2016 ◽  
Vol 24 (3) ◽  
pp. 167-172
Author(s):  
Kazuhisa Nakasho ◽  
Keiko Narita ◽  
Yasunari Shidama
Keyword(s):  
Real Line ◽  
Metric Spaces ◽  
Topological Properties ◽  
The Real ◽  
The Third ◽  
Norm Space ◽  
Number Sequences ◽  
Sequential Compactness ◽  
Countable Compactness ◽  
Definition Of

Summary In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces. In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on [20], [5], [17], [14], and [4].

scholarly journals Convolutions with the Continuous Primitive Integral

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Erik Talvila
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Bounded Variation ◽  
Real Line ◽  
Fubini Theorem ◽  
Function Of Bounded Variation ◽  
Distributional Derivative ◽  
The Real ◽  
Integrable Distribution ◽  
Uniformly Continuous

IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ&#x1D4B1;, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈L1, the estimate is‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

The logic of absolute value inequalities

1964 ◽  
Vol 57 (2) ◽  
pp. 73-74
Author(s):  
Marvin L. Chachere
Keyword(s):  
Real Line ◽  
Two Dimensions ◽  
Elementary Algebra ◽  
Absolute Value ◽  
The Real ◽  
Definition Of ◽  
Absolute Value Inequalities

Teachers motivated by the need to express the distance of any point on the real line from zero find the definition of absolute value useful and interesting. It is used in elementary algebra to define addition and multiplication of “directed” numbers and to describe the graphs of broken lines in one and two dimensions. Recent emphasis on inequalities and the inclusion in newer textbooks of absolute value inequalities raise several questions that cannot be handled in a perfunctory manner.

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