Yazar "Cakalli, Huseyin" seçeneğine göre listele

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    Abel Statistical Quasi Cauchy Sequences
    (UNIV NIS, FAC SCI MATH, 2019) Cakalli, Huseyin
    In this paper, we investigate the concept of Abel statistical quasi Cauchy sequences. A real function f is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (alpha(k)) of point in R is called Abel statistically quasi Cauchy if lim(x -> 1)-(1 - x) Sigma(k:vertical bar Delta alpha k vertical bar >=epsilon) x(k) = 0 for every epsilon > 0, where Delta alpha(k) = alpha(k+1) - alpha(k) for every k is an element of N. Some other types of continuities are also studied and interesting results are obtained. It turns out that the set of Abel statistical ward continuous functions is a closed subset of the space of continuous functions.
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    AN APPROACH TO SOFT FUNCTIONS
    (UNIV PRISHTINES, 2017) Aras, Cigdem Gunduz; Sonmez, Ayse; Cakalli, Huseyin
    In this paper, using a more appreciate definition of a soft point, i.e. a soft point is a soft set (F, E) such that for the element e is an element of E, F(e) = {x} and F(e ') = (empty set) for all e ' is an element of E - {e}, we present a new approach to soft functions in a interesting way, and introduce the concepts of soft continuous, soft open, soft closed, and soft homeomorfic functions in a very different way from the source existing in the literature. In the investigation we prove theorems related to these concepts and provide with examples, and counterexamples.
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    Beyond Cauchy and Quasi-Cauchy Sequences
    (UNIV NIS, FAC SCI MATH, 2018) Cakalli, Huseyin
    In this paper, we investigate the concepts of downward continuity and upward continuity. A real valued function on a subset E of R, the set of real numbers, is downward continuous if it preserves downward quasi-Cauchy sequences; and is upward continuous if it preserves upward quasi-Cauchy sequences, where a sequence (x(k)) of points in R is called downward quasi-Cauchy if for every epsilon > 0 there exists an n(0) is an element of N such that x(n+1) xn < epsilon for n >= n(0), and called upward quasi-Cauchy if for every epsilon > 0 there exists an n(1) is an element of N such that xn x(n+1) < epsilon for n >= n(1). We investigate the notions of downward compactness and upward compactness and prove that downward compactness coincides with above boundedness. It turns out that not only the set of downward continuous functions, but also the set of upward continuous functions is a proper subset of the set of continuous functions.
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    Beyond the quasi-Cauchy sequences beyond the Cauchy sequences
    (AMER INST PHYSICS, 2016) Cakalli, Huseyin; Ashyralyev, A; Lukashov, A
    In this paper, we investigate the concept of upward continuity. A real valued function on a subset E of R, the set of real numbers is upward continuous if it preserves upward quasi Cauchy sequences in E, where a sequence (x(k)) of points in R is called upward quasi Cauchy if for every epsilon > 0 there exists a positive integer no such that x(n)-x(n+1) < epsilon for n >= n(0). It turns out that the set of upward continuous functions is a proper subset of the set of continuous functions.
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    FORWARD CONTINUITY
    (EUDOXUS PRESS, LLC, 2011) Cakalli, Huseyin
    A real function f is continuous if and only if (f(x(n))) is a convergent sequence whenever (x(n)) is convergent and a subset E of R is compact if any sequence x = (x(n)) of points in E has a convergent subsequence whose limit is in E where R is the set of real numbers. These well known results suggest us to introduce a concept of forward continuity in the sense that a function f is forward continuous if lim(n ->infinity)Delta(f) (x(n)) = 0 whenever lim(n ->infinity)Delta x(n) = 0 and a concept of forward compactness in the sense that a subset E of R is forward compact if any sequence x = (x.) of points in E has a subsequence z = (z(k)) = (x(nk)) of the sequence x such that lim(n ->infinity)Delta z(k)= 0 where Delta z(k) = z(k+1) z(k). We investigate forward continuity and forward compactness, and prove related theorems.
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    G-Connectedness in Topological Groups with Operations
    (UNIV NIS, FAC SCI MATH, 2018) Mucuk, Osman; Cakalli, Huseyin
    It is a well known fact that for a Hausdorff topological group X, the limits of convergent sequences in X define a function denoted by lim from the set of all convergent sequences in X to X. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences. Recently some authors have extended the concept to the topological group setting and introduced the concepts of G-continuity, G-compactness and G-connectedness. In this paper we present some results about G-hulls, G-connectedness and G-fundamental systems of G-open neighbourhoods for a wide class of topological algebraic structures called groups with operations, which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.
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    G-sequentially connectedness for topological groups with operations
    (AMER INST PHYSICS, 2016) Mucuk, Osman; Cakalli, Huseyin; Ashyralyev, A; Lukashov, A
    It is a well-known fact that for a Hausdorff topological group X, the limits of convergent sequences in X define a function denoted by l i m from the set of all convergent sequences in X to X. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing l i m with an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences. Recently some authors have extended the concept to the topological group setting and introduced the concepts of G-sequential continuity, G-sequential compactness and G-sequential connectedness. In this work, we present some results about G-sequentially closures, G-sequentially connectedness and fundamental system of G-sequentially open neighbourhoods for topological group with operations which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.
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    Ideal quasi-Cauchy sequences
    (SPRINGER INTERNATIONAL PUBLISHING AG, 2012) Cakalli, Huseyin; Hazarika, Bipan
    An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence (x(n)) of real numbers is said to be I-convergent to a real number L if for each epsilon > 0, the set {n : vertical bar x(n) - L vertical bar >= epsilon} belongs to I. We introduce I-ward compactness of a subset of R, the set of real numbers, and I-ward continuity of a real function in the senses that a subset E of R is I-ward compact if any sequence (x(n)) of points in E has an I-quasi-Cauchy subsequence, and a real function is I-ward continuous if it preserves I-quasi-Cauchy sequences where a sequence (x(n)) is called to be I-quasi-Cauchy when (Delta x(n)) is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, delta-ward continuity, and slowly oscillating continuity.
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    Ideal statistically quasi Cauchy sequences
    (AMER INST PHYSICS, 2016) Savas, Ekrem; Cakalli, Huseyin; Ashyralyev, A; Lukashov, A
    An ideal I is a family of subsets of N, the set of positive integers which is closed under taking finite unions and subsets of its elements. A sequence (x(k)) of real numbers is said to be S(I)-statistically convergent to a real number L, if for each epsilon > 0 and for each delta > 0 the set {n is an element of N: 1/n {k <= n: vertical bar x(k) - L vertical bar >= epsilon}vertical bar >= delta} belongs to I. We introduce S(I)-statistically ward compactness of a subset of R, the set of real numbers, and S(I)-statistically ward continuity of a real function in the senses that a subset E of R is S(I)-statistically ward compact if any sequence of points in E has an S(I)-statistically quasi Cauchy subsequence, and a real function is S(I)-statistically ward continuous if it preserves S(I)-statistically quasi-Cauchy sequences where a sequence (x(k)) is called to be S(I)-statistically quasi-Cauchy when (Delta x(k)) is S(I)-statistically convergent to 0. We obtain results related to S(I)-statistically ward continuity, S(I)-statistically ward compactness, N-theta-ward continuity, and slowly oscillating continuity.
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    Lacunary statistical ward continuity
    (AMER INST PHYSICS, 2015) Cakalli, Huseyin; Aras, Cigdem Gunduz; Sonmez, Ayse; Ashyralyev, A; Malkowsky, E; Lukashov, A; Basar, F
    The main object of this paper is to investigate lacunary statistically ward continuity. We obtain some relations between this kind of continuity and some other kinds of continuities. It turns out that any lacunary statistically ward continuous real valued function on a lacunary statistically ward compact subset E subset of R is uniformly continuous.
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    LACUNARY STATISTICALLY UPWARD AND DOWNWARD HALF QUASI-CAUCHY SEQUENCES
    (UNIV PRISHTINES, 2016) Cakalli, Huseyin; Mucuk, Osman
    A real valued function defined on a subset E of R, the set of real numbers, is lacunary statistically upward continuous if it preserves lacunary statistically upward half quasi-Cauchy sequences where a sequence (x(k)) of points in R is called lacunary statistically upward half quasi Cauchy if lim(r infinity) 1/h(r) vertical bar{k is an element of I-r : x(k) - x(k+1) >= epsilon}vertical bar - 0 for every epsilon > 0; and (x(k)) is called lacunary statistically downward half quasi-Cauchy if lim(r ->infinity) 1/h(r) vertical bar {k is an element of I-r : x(k+1) - x(k) >= epsilon}vertical bar = 0 for every epsilon > 0, where theta = (k(r)) is an increasing sequence of non-negative integers such that k(0) = 1 and h(r) : k(r) - k(r-1) -> infinity. We investigate lacunary statistically upward continuity and lacunary statistically downward continuity and prove some interesting theorems. It turns out that not only a lacunary statistically upward continuous function on a below bounded subset, but also a lacunary statistically downward continuous function on an above bounded subset is uniformly continuous.
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    Lacunary Ward Continuity in 2-normed Spaces
    (UNIV NIS, FAC SCI MATH, 2015) Cakalli, Huseyin; Ersan, Sibel
    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 (x(k)) of points in X is lacunary statistically quasi-Cauchy if lim(r ->infinity) 1/h(r) vertical bar{k is an element of I-r : parallel to x(k+1) - x(k), z parallel to >= epsilon}vertical bar = 0 for every positive real number epsilon and z 1/4 X, and (k(r)) is an increasing sequence of positive integers such that k(0) = 0 and h(r) = k(r) - k(r-1) -> infinity as r -> infinity, I-r = (k(r-1), k(r)]. We investigate not only lacunary statistical ward continuity, but also some other kinds of continuities in 2-normed spaces.
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    N-alpha(beta)(theta, I)- Ward Continuity
    (AMER INST PHYSICS, 2019) Sengul, Hacer; Cakalli, Huseyin; Et, Mikail; Cakalli, H; Kocinac, LDR; Harte, R; Cao, J; Savas, E; Ersan, S; Yildiz, S
    The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of order (alpha, beta) of sequences of real numbers. Strongly ideal lacunary ward continuity of order (alpha, beta) is also investigated. Interesting results are obtained.
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    N-theta-Ward Continuity
    (HINDAWI LTD, 2012) Cakalli, Huseyin
    A function f is continuous if and only if f preserves convergent sequences; that is, (f(alpha(n))) is a convergent sequence whenever (alpha(n)) is convergent. The concept of N-theta-ward continuity is defined in the sense that a function f is N-theta-ward continuous if it preserves N-theta-quasi-Cauchy sequences; that is, (f(alpha(n))) is an N-theta-quasi-Cauchy sequence whenever (alpha(n)) is N-theta-quasi-Cauchy. A sequence (alpha(k)) of points in R, the set of real numbers, is N-theta-quasi-Cauchy if lim(r ->infinity) (1/h(r)) Sigma(k is an element of Ir) vertical bar Delta alpha(k)vertical bar = 0, where Delta alpha(k) = alpha(k+1) - alpha(k), I-r = (k(r-1), k(r)], and theta = (k(r)) is a lacunary sequence, that is, an increasing sequence of positive integers such that k(0) = 0 and h(r) : k(r) - k(r-1) -> infinity. A new type compactness, namely, N-theta-ward compactness, is also, defined and some new results related to this kind of compactness are obtained.
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    New kinds of continuities
    (PERGAMON-ELSEVIER SCIENCE LTD, 2011) Cakalli, Huseyin
    A sequence (x(n)) of points in a topological group is slowly oscillating if for any given neighborhood U of 0, there exist delta = delta(U) > 0 and N = N(U) such that x(m)-x(n) epsilon U if n >= N(U) and n <= m <= (1+delta)n. It is well known that in a first countable Hausdorff topological space, a function f is continuous if and only if (f(x(n))) is convergent whenever (x(n)) is. Applying this idea to slowly oscillating sequences one gets slowly oscillating continuity, i.e. a function f defined on a subset of a topological group is slowly oscillating continuous if (f (x(n))) is slowly oscillating whenever (x(n)) is slowly oscillating. We study the concept of slowly oscillating continuity and investigate relations with statistical continuity, lacunary statistical continuity, and some other kinds of continuities in metrizable topological groups. (C) 2010 Elsevier Ltd. All rights reserved.
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    New results in quasi cone metric spaces
    (JOURNAL MATHEMATICS & COMPUTER SCIENCE-JMCS, 2016) Yaying, Teja; Hazarika, Bipan; Cakalli, Huseyin
    In this paper, we prove some interesting results using forward and backward convergence in quasi cone metric spaces. We study forward and backward sequential compactness, sequential countably compactness, and sequential continuity property in quasi cone metric spaces and give some interesting results. (C) 2016 all rights reserved.
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    A New Study on the Strongly Lacunary Quasi Cauchy Sequences
    (AMER INST PHYSICS, 2018) Cakalli, Huseyin; Kaplan, Huseyin
    In this paper, the concept of a strongly lacunary delta(2) quasi-Cauchy sequence is introduced. We proved interesting theorems related to strongly lacunary delta(2) -quasi-Cauchy sequences. A real valued function f defined on a subset A of the set of real numbers, is strongly lacunary delta(2) ward continuous on A if it preserves strongly lacunary delta(2) quasi-Cauchy sequences of points in A, i.e. (f(alpha(k))) is a strongly lacunary delta(2) quasi-Cauchy sequence whenever (alpha(k)) is a strongly lacunary delta(2) quasi-Cauchy sequences of points in A, where a sequence (alpha(k)) is called strongly lacunary delta(2) quasi-Cauchy if (Delta(2)alpha(k)) is a strongly lacunary delta(2) quasi-Cauchy sequence where Delta(2)alpha(k) = alpha(k+2)-2 alpha(k+1)+ alpha(k) for each positive integer k.
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    New Type Continuities via Abel Convergence
    (HINDAWI LTD, 2014) Cakalli, Huseyin; Albayrak, Mehmet
    We investigate the concept of Abel continuity. A function f defined on a subset of R, 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.
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    A NEW TYPE CONTINUITY FOR REAL FUNCTIONS
    (UNIV PRISHTINES, 2016) Braha, Naim L.; Cakalli, Huseyin
    A real valued function f de fined on a subset of R is delta(2)-ward continuous if lim(n ->infinity) Delta(3)f(x(n)) = 0 whenever lim(n ->infinity) Delta(3)x(n) = 0, where Delta(3) z(n) = z(n+3) - 3z(n+2) + 3z(n+1) - z(n) for each positive integer n, R denotes the set of real numbers, and a subset E of R is delta(2)-ward compact if any sequence of points in E has a delta(2)-quasi Cauchy subsequence where a sequence (x(n)) is delta(2)-quasi Cauchy if lim(n ->infinity) Delta(3) z(n)=0. It turns out that the uniform limit process preserves this kind of continuity, and the set of ffi 2 - ward continuous functions is a closed subset of the set of continuous functions.
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    New Types of Continuity in 2-Normed Spaces
    (UNIV NIS, FAC SCI MATH, 2016) Cakalli, Huseyin; Ersan, Sibel
    A sequence (chi(n)) of points in a 2-normed space X is statistically quasi-Cauchy if the sequence of difference between successive terms statistically converges to 0. In this paper we mainly study statistical ward continuity, where a function defined on a subset E of X is statistically ward continuous if it preserves statistically quasi-Cauchy sequences of points in E. Some other types of continuity are also discussed, and interesting results related to these kinds of continuity are obtained in 2-normed space setting.