scholarly journals On λ statistical upward compactness and continuity

Filomat ◽  
2018 ◽  
Vol 32 (12) ◽  
pp. 4435-4443
Author(s):  
Huseyin Cakalli
Keyword(s):  
Positive Integer ◽  
Compact Subset ◽  
Real Numbers ◽  
Cauchy Sequences ◽  
Uniformly Continuous

A sequence (?k) of real numbers is called ?-statistically upward quasi-Cauchy if for every ? > 0 limn?? 1/?n |{k?In:?k-?k+1 ? ?}| = 0, where (?n) is a non-decreasing sequence of positive numbers tending to 1 such that ?n+1 ? ?n + 1, ?1 = 1, and In = [n-?n+1,n] for any positive integer n. A real valued function f defined on a subset of R, the set of real numbers is ?-statistically upward continuous if it preserves ?-statistical upward quasi-Cauchy sequences. ?-statistically upward compactness of a subset in real numbers is also introduced and some properties of functions preserving such quasi Cauchy sequences are investigated. It turns out that a function is uniformly continuous if it is ?-statistical upward continuous on a ?-statistical upward compact subset of R.

scholarly journals Variations on statistical quasi Cauchy sequences

2019 ◽  
Vol 38 (3) ◽  
pp. 141-150
Author(s):  
Huseyin Cakalli
Keyword(s):  
Positive Integer ◽  
Cauchy Sequence ◽  
Real Sequence ◽  
Bounded Subset ◽  
Cauchy Sequences ◽  
Uniformly Continuous

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $(\alpha_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward continuous on a subset $A$ of the set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy sequences, i.e. the sequence $f(\textbf{x})=(f(\alpha_{n}))$ is statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ is a statistically $p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued function $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves statistically $p$-quasi-Cauchy sequences of points in $A$.

scholarly journals Upward and downward statistical continuities

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2265-2273 ◽  
Author(s):  
Hüseyin Çakallı
Keyword(s):  
Continuous Function ◽  
Bounded Subset ◽  
Real Numbers ◽  
Cauchy Sequences ◽  
Uniformly Continuous

A real valued function f defined on a subset E of R, the set of real numbers, is statistically upward (resp. downward) continuous if it preserves statistically upward (resp. downward) half quasi-Cauchy sequences; A subset E of R, is statistically upward (resp. downward) compact if any sequence of points in E has a statistically upward (resp. downward) half quasi-Cauchy subsequence, where a sequence (xn) of points in R is called statistically upward half quasi-Cauchy if lim n?? 1/n |{k ? n : xk- xk+1 ? ?}| = 0, and statistically downward half quasi-Cauchy if lim n??1/n |{k ? n : xk+1 - xk ? ?}| = 0 for every ? > 0. We investigate statistically upward and downward continuity, statistically upward and downward half compactness and prove interesting theorems. It turns out that any statistically upward continuous function on a below bounded subset of R is uniformly continuous, and any statistically downward continuous function on an above bounded subset of R is uniformly continuous.

scholarly journals Variations on quasi-Cauchy sequences

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Hüseyin Çakallı
Keyword(s):  
Positive Integer ◽  
Cauchy Sequences ◽  
Uniformly Continuous

In this paper, we introduce and study new kinds of continuities. It turns out that a function f defined on an interval is uniformly continuous if and only if there exists a positive integer p such that f preserves p-quasi-Cauchy sequences where a sequence (xn) is called p-quasi-Cauchy if the sequence of differences between p-successive terms tends to 0.

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 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 Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
G. M. Moremedi ◽  
I. P. Stavroulakis
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Delay Difference ◽  
Constant Argument

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

scholarly journals ON EXISTENCE OF POSITIVE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATIONS

1994 ◽  
Vol 25 (3) ◽  
pp. 257-265
Author(s):  
J. H. SHEN ◽  
Z. C. WANG ◽  
X. Z. QIAN
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Difference Equations ◽  
Real Numbers ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Existence Of Positive Solutions

Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.

scholarly journals Dynamics of a Higher-Order System of Difference Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Qi Wang ◽  
Qinqin Zhang ◽  
Qirui Li
Keyword(s):  
Positive Integer ◽  
Positive Solutions ◽  
Difference Equations ◽  
Initial Conditions ◽  
Higher Order ◽  
Order System ◽  
Real Numbers ◽  
Precise Description ◽  
System Of Difference Equations ◽  
Positive Real

Consider the following system of difference equations:xn+1(i)=xn-m+1(i)/Ai∏j=0m-1xn-j(i+j+1)+αi,xn+1(i+m)=xn+1(i),x1-l(i+l)=ai,l,Ai+m=Ai,αi+m=αi,i,l=1,2,…,m;n=0,1,2,…,wheremis a positive integer,Ai,αi,i=1,2,…,m, and the initial conditionsai,l,i,l=1,2,…,m,are positive real numbers. We obtain the expressions of the positive solutions of the system and then give a precise description of the convergence of the positive solutions. Finally, we give some numerical results.

scholarly journals Generators for locally compact groups

1993 ◽  
Vol 36 (3) ◽  
pp. 463-467 ◽  
Author(s):  
Joan Cleary ◽  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Positive Integer ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Numbers ◽  
Locally Compact ◽  
Hausdorff Group

It is proved that if G is any compact connected Hausdorff group with weight w(G)≦c, ℝ is the topological group of all real numbers and n is a positive integer, then the topological group G × ℝn can be topologically generated by n + 1 elements, and no fewer elements will suffice.

Two Inequalities

1953 ◽  
Vol 37 (322) ◽  
pp. 244-246 ◽  
Author(s):  
G.N. Watson
Keyword(s):  
Positive Integer ◽  
Real Numbers ◽  
Intrinsic Interest ◽  
Elementary Analysis ◽  
Order Of Magnitude

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant. Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable. Let the determinant

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