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In this paper, we introduce the concept of neutrosophic $G$-sequential continuity as a new tool to further studies presenting the definitions of neutrosophic soft sequence, neutrosophic soft quasi-coincidence, neutrosophic soft $q$-neighborhood, neutrosophic soft cluster point, neutrosophic soft boundary point, neutrosophic soft sequential closure, neutrosophic soft group, neutrosophic soft method, which constitute a base to define the concepts of neutrosophic soft $G$-sequential closure, neutrosophic soft $G$-sequential derived set, $G$-sequentially neutrosophic soft compactness of a subset of a neutrosophic soft topological space. Their characters are analyzed and some implications are given. A counterexample to each implication is also given.

In this paper we study the domain of generalized Riesz difference matrix RqΔ(α) of fractional order α in the classical sequence spaces c0 and c and introduced the sequence spaces r0 q (Δ(α) ) and rc q (Δ(α) ). We obtain the α−, β− and γ−duals of these spaces and using Hausdorff measure of noncompactness, we characterize certain classes of compact operators on the space r0 q (Δ(α) ). Keywords: Riesz difference sequence space, Fractional Difference operator Δ(α) , α−, β− and γ−duals , Compact operator, Hausdorff measure of non-compactness.

In this study, we introduce and examine the concepts of Δm−weighted statistical convergence and Δm−weighted N, pn −summability. Also some relations between Δm−weighted statistical convergence and Δm−weighted N, pn −summability are given.

In this paper, we introduce the concept of strong ρ−convergence of order β ( or Nβ p (ρ) −convergence ) of sequence of real numbers and give some inclusion relations between the set of all ρ−statistical convergence of order β and strong Nβ p (ρ)-convergence.

t. In this extended abstract, we introduce the concept of delta quasi Cauchy sequences in metric spaces. A function f defined on a subset of a metric space X to X is called delta ward continuous if it preserves delta quasi Cauchy sequences, where a sequence (xk) of points in X is called delta quasi Cauchy if limn→∞[d(xk+2, xk+1)−d(xk+1, xk)] = 0. A new type compactness in terms of δ-quasi Cauchy sequences, namely δ-ward compactness is also introduced, and some theorems related to δ-ward continuity and δ-ward compactness are obtained. Some other types of continuities are also discussed, and interesting results are obtained.

In this paper, we investigate the concept of Abel statistical ward continuity in 2-normed spaces. A function f defined on a 2-normed space X into X is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (xk) of points in X is called Abel statistically quasi Cauchy if limx→1− (1− x) k:||Δxk ,z||≥ε xk = 0 for every ε > 0 and z ∈ X, where Δxk = xk+1 − xk for every k ∈ N. Some other types of compactness and continuities are also studied and interesting results are obtained.

Açıkgöz, Ahu (Balikesir Author), Scientists have always adopted the concept of sequential continuity as an indispensable subject, not only in Topology but also in some other branches of Mathematics. Connor and Grosse-Erdmann gave this concept for real functions by using an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences instead of lim. Afterwards, this concept were adapted to a topological group X by replacing a linear functional G with an arbitrary additive function defined on a subgroup of the group of all X-valued sequences. Furthermore, alternative theorems in generalized setting were given and varied theorems that had not been achieved for real functions were presented. In this investigation, we offer neutrosophic soft G-continuity and analyze its nature in neutrosophic soft topological spaces.

In this paper, we investigate the concept of Abel statistical delta ward compactness and Abel statistical delta ward continuity in metric spaces. A function f defined on a metric space X into X is called Abel statistically delta ward continuous it preserves Abel statistical delta quasi Cauchy sequences, where a sequence (xk) of points in X is called Abel statistically delta quasi Cauchy if limx→1− (1 − x) k:|d(xk+2,xk+1)−d(xk+1,xk )|≥ε xk = 0 for every ε > 0, Some other types of compactnesses are also studied and interesting results are obtained.

It is well known 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 prove some results about G-compactness for topological group with operations which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.

Continuity, in particular sequential continuity, is an important subject of investigation not only in Topology, but also in some other branches of Mathematics. Connor and Grosse-Erdmann remodeled its definition for real functions by replacing {\sf lim} with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. Then, this definition was extended to a topological group $X$ by replacing a linear functional $G$ with an arbitrary additive function defined on a subgroup of the group of all $X$-valued sequences. Also, some new theorems in generalized setting were given and some other theorems that had not been obtained for real functions were presented. In this study, we introduce neutrosophic $G$-continuity and investigate its properties in neutrosophic topological spaces.

4th International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM) -- MAY 11-15, 2017 -- Aydin, TURKEY, WOS: 000464504500018, 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.

In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function f on a subset E of R, the set of real numbers is down continuous if it preserves downward half Cauchy sequences. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions. © 2018 Elsevier B.V.

3rd International Conference on Analysis and Applied Mathematics (ICAAM) -- SEP 07-10, 2016 -- Almaty, KAZAKHSTAN, WOS: 000439421100034, 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., Inst Math & Math Modeling, Al Farabi Kazakh Natl Univ, L N Gumilyov Eurasian Natl Univ

The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of sequences of real numbers. Strongly ideal lacunary ward continuity is also investigated. Interesting results are obtained.

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.

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 $A$.

Açıkgöz, Ahu (Balikesir Author), Continuity is one of most important concepts in many mathematical disciplines. In some situations general notion of continuity is replaced by sequential continuity. Connor and Grosse-Erdmann replaced lim in the definition of sequential continuity of real functions by a linear functional G on a linear subspace of the vector space of all real sequences. Their definition was extended to topological group X by replacing a linear functional G with an additive function defined on a subgroup of the group of all X-valued sequences. In this paper we introduce neutrosophic G-continuity and investigate its properties in neutrosophic topological spaces.

WOS: 000386871300077, We introduce a new function space, namely the space of N-theta(alpha)(p)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions. A real valued function f defined on a subset A of R, the set of real numbers, is N-theta(alpha)(p)-ward continuous if it preserves N-theta(alpha)(p)-quasi-Cauchy sequences, that is, (f(x(n))) is an N-theta(alpha)(p)-quasi-Cauchy sequence whenever (x(n)) is N-theta(alpha)(p)-quasi-Cauchy sequence of points in A, where a sequence (x(k)) of points in R is called N-theta(alpha)(p)-quasi-Cauchy if lim(r ->infinity) 1/h(r)(alpha) Sigma(k is an element of lr) vertical bar Delta x(k)vertical bar(p) = 0, where Delta x(k) = x(k+1) - x(k) for each positive integer k, p is a constant positive integer, alpha is a constant in ]0,1], 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) not equal 0, and h(r) : k(r) - k(r-1) -> infinity. Some other function spaces are also investigated. (C) 2016 All rights reserved.

In this paper, the definitions of lacunary strong A−convergence of order α with respect to a modulus and lacunary A−statistical convergence of order α were given by adding α to the definitions of lacunary strong A−convergence with respect to a modulus and lacunary A−statistical convergence when A = (aik) is an infinite matrix of complex numbers and 0 < α ≤ 1. We study some connections between lacunary strong A−convergence of order α with respect to a modulus and lacunary A−statistical convergence of order α.

International Conference of Mathematical Sciences (ICMS) -- JUL 31-AUG 06, 2018 -- Maltepe Univ, Istanbul, TURKEY

In this extended abstract, we introduce a concept of statistically quasi-Cauchyness of a sequence in X in the sense that a sequence (xk) is statistically quasi-Cauchy in X if limn→∞ 1n |{k ≤ n : d(xk+1, xk) − c ∈ P}| for each c ∈P where (X, d) is a cone metric space, and P denotes interior of a cone P of X. It turns out that a function f from a totally bounded subset A of X into X is uniformly continuous if f preserves statistically quasi-Cauchy sequences.

International Conference of Mathematical Sciences (ICMS) -- JUL 31-AUG 06, 2018 -- Maltepe Univ, Istanbul, TURKEY, WOS: 000472950300048, In this paper, we investigate the concept of Abel statistical delta quasi Cauchy sequences. A real function! is called Abel statistically delta ward continuous it preserves Abel statistical delta quasi Cauchy sequences, where a sequence (alpha(k)) of point in R is called Abel statistically delta quasi Cauchy if lim(x -> 1)- (1 - x) Sigma(k:vertical bar Delta 2 alpha k vertical bar >=epsilon) x(k) = 0 for every epsilon > 0, where Delta(2)alpha(k) = alpha(k+2) - 2 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.

Açıkgöz, Ahu (Balikesir Author), We adapt strong theta-precontinuity into fuzzy soft topology and investigate its properties. Also, the relations with the other types of continuities in fuzzy soft topological spaces are analized. Moreover, we give some new definitions.

In this paper, the concept of deferred statistical convergence of order α is generalized to topological groups, and some inclusion relations between the set of all statistically convergent sequences of order α in topological groups and the set of all deferred statistically convergent sequences of order α in topological groups are given.

International Conference of Mathematical Sciences (ICMS) -- JUL 31-AUG 06, 2018 -- Maltepe Univ, Istanbul, TURKEY, WOS: 000472950300015, A sequence (alpha(k)) of points in R, the set of real numbers, is called rho-statistically p quasi Cauchy if lim(n ->infinity )1/rho(n) vertical bar{k = epsilon}vertical bar = 0 for each epsilon > 0, where rho = (rho(n)) is a non-decreasing sequence of positive real numbers tending to infinity such that lim sup(n) rho n/n < infinity, Delta rho(n) = O(1), and Delta(p)alpha(k+p) = alpha(k+p) - alpha(k) for each positive integer k, p is a fixed positive integer. A real-valued function defined on a subset of R is called rho-statistically p-ward continuous if it preserves rho-statistical p-quasi Cauchy sequences. We obtain results related to rho-statistical p-ward continuity, rho-statistical p-ward compactness, p-ward continuity, ward continuity, and uniform continuity.

International Conference of Mathematical Sciences (ICMS) -- JUL 31-AUG 06, 2018 -- Maltepe Univ, Istanbul, TURKEY, WOS: 000472950300017, For a fixed positive integer p. a sequence (x(n)) in a metric space X is called p-quasi-Cauchy if (Delta(p)x(n)) is a null sequence where Delta(p)x(n) = d(x(n+p), x(n)) for each positive integer n. A subset E of X is called p-ward compact if any sequence (x(n)) of points in E has a p-quasi-Cauchy subsequence. A subset of X is totally bounded if and only if it is p-ward compact. A function f from a subset E of X into a metric space Y is called p-ward continuous if it preserves p-quasi Cauchy sequences, i.e. (f(x(n))) is a p-quasi Cauchy sequence in Y whenever (x(n)) is a p-quasi Cauchy sequence of points of E. A function f from a totally bounded subset of X into Y preserves p-quasi Cauchy sequences if and only if it is uniformly continuous. If a function is uniformly continuous on a subset E of X into Y, then (f(x(n))) is p-quasi Cauchy in Y whenever (x(n)) is a quasi cauchy sequence of points in E.

2nd International Conference on Recent Advances in Pure and Applied Mathematics (ICRAPAM) -- JUN 03-06, 2015 -- Istanbul Commerce Univ, Fac Arts & Sci, Istanbul, TURKEY, WOS: 000376574100003, 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.

WOS: 000461182200028, A sequence (alpha(k)) of real numbers is called lambda-statistically upward quasi-Cauchy if for every epsilon > 0 lim(n ->infinity)1/lambda(n)vertical bar{K is an element of I-n : alpha(k) - alpha(k+1) >= epsilon}vertical bar = 0, where (lambda(n)) is a non decreasing sequence of positive numbers tending to infinity such that lambda(n+1)

International Conference of Numerical Analysis and Applied Mathematics (ICNAAM) -- SEP 25-30, 2017 -- Thessaloniki, GREECE, WOS: 000445105400302, In this paper, we give a generalization of absolutely almost convergence, and prove interesting results.

3rd International Conference on Analysis and Applied Mathematics (ICAAM) -- SEP 07-10, 2016 -- Almaty, KAZAKHSTAN, WOS: 000439421100038, 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., Inst Math & Math Modeling, Al Farabi Kazakh Natl Univ, L N Gumilyov Eurasian Natl Univ

International Conference of Numerical Analysis and Applied Mathematics (ICNAAM) -- SEP 25-30, 2017 -- Thessaloniki, GREECE, WOS: 000445105400301, A sequence (alpha(k)) of real numbers is called lambda-statistically upward quasi-Cauchy if for every epsilon > 0 lim(n ->infinity 1/)lambda(n)vertical bar{k is an element of I-n : alpha(k) - alpha(k+1) >= epsilon}vertical bar = 0, where (lambda(n)) is a non-decreasing sequence of positive numbers tending to so such that lambda(n+1)

WOS: 000355844100004, 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 (x(n)) is called p-quasi-Cauchy if the sequence of differences between p-successive terms tends to 0.

A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset $E$ of $\textbf{R}$, is statistically upward compact if any sequence of points in $E$ has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in $E$ has a statistically downward half quasi-Cauchy subsequence where a sequence $(x_{n})$ of points in $\textbf{R}$ is called statistically upward half quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k}-x_{k+1}\geq \varepsilon\}|=0 \] is statistically downward half quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+1}-x_{k}\geq \varepsilon\}|=0 \] for every $\varepsilon>0$. We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that uniform limit of a sequence of statistically upward continuous functions is statistically upward continuous, and uniform limit of a sequence of statistically downward continuous functions is statistically downward continuous., 25 pages. arXiv admin note: substantial text overlap with arXiv:1205.3674, arXiv:1103.1230, arXiv:1102.1531, arXiv:1305.0697

The concept of I{convergence is an important generaliza- tion of statistical convergence which depends on the notion of an ideal I of subsets of the set N of positive integers. In this paper we introduce the ideas of I{Cauchy and I {Cauchy sequences in cone metric spaces and study their properties. We also investigate the relation between this new Cauchy type condition and the property of completeness.

WOS: 000412926000019, A sequence (alpha(k)) of points in R, the set of real numbers, is called rho-statistically convergent to an element l of R if lim (n ->infinity) 1/rho n |{k = epsilon}| = 0 for each epsilon > 0, where rho = (rho n) is a non-decreasing sequence of positive real numbers tending to 8 such that lim sup(n) rho n/n < infinity, Delta rho n = O(1), and Delta alpha(n) = alpha(n+ 1) - alpha(n) for each positive integer n. A real-valued function defined on a subset of R is called rho-statistically ward continuous if it preserves rho-statistical quasi-Cauchy sequences where a sequence (alpha(k)) is defined to be rho-statistically quasi-Cauchy if the sequence (Delta alpha(k)) is rho-statistically convergent to 0. We obtain results related to rho-statistical ward continuity, rho-statistical ward compactness, ward continuity, continuity, and uniform continuity. It turns out that the set of uniformly continuous functions coincides with the set of rho-statistically ward continuous functions not only on a bounded subset of R, but also on an interval.

WOS: 000407117100007, In this paper, the concept of a lacunary statistically delta-quasi-Cauchy sequence is investigated. In this investigation, we proved interesting theorems related to lacunary statistically delta-ward continuity, and some other kinds of continuities. A real valued function f defined on a subset A of R, the set of real numbers, is called lacunary statistically S ward continuous on A if it preserves lacunary statistically delta quasi-Cauchy sequences of points in A, i.e. (f (alpha(k))) is a lacunary statistically delta quasi-Cauchy sequence whenever (alpha(k)) is a lacunary statistically delta quasi-Cauchy sequence of points in A, where a sequence (alpha(k)) is called lacunary statistically delta quasi-Cauchy if (Delta alpha(k)) is a lacunary statistically quasi-Cauchy sequence. It turns out that the set of lacunary statistically delta ward continuous functions is a closed subset of the set of continuous functions.

In this paper, we study (Pn, s)-absolutely almost convergence where npn = 0(Pn) as n › ?,Pn = Pn ?n v=0 pv and s > 0. We also give necessary and sufficient conditions for a summability matrix to be (?n, s)-absolutely almost conservative, and obtain some inclusion results. © 2017, Sciendo. All rights reserved.

The main object of this paper is to investigate λ-statistically ward continuity. We obtain some relations between this kind of continuity and some other kinds of continuities. It turns out that any λ-statistically ward continuous real valued function on a λ-statistically ward compact subset E ⊂ R is uniformly continuous.

WOS: 000322038400021, In this paper, we investigate slowly oscillating continuity in cone metric spaces. It turns out that the set of slowly oscillating continuous functions is equal to the set of uniformly continuous functions on a slowly oscillating compact subset of a topological vector space valued cone metric space.