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Ideal convergence of sequences in neutrosophic normed spaces is defined by Omer Ki¸si [12]. This ¨ paper defines new sequence spaces using the Zweier matrix and neutrosophic norm. We explore (λ, μ)-Zweier convergence, (λ, μ) ideal convergence of double sequences in neutrosophic norm. We show that a double sequence that is (λ, μ)-Zweier convergent or ideal convergent has a unique limit with respect to neutrosophic norm. Additionally, we prove that (λ, μ)-Zweier ideal convergence is equivalent to (λ, μ)-Zweier ideal Cauchy for a double sequence in neutrosophic normed space. [ABSTRACT FROM AUTHOR]

In this article we introduce the notion of statistical convergence difference sequences of bi-complex numbers. Some properties of these sequence spaces like BC-module, Banach BC-module, BC-balanced set, BC-convex set, solidness, and symetricity are studied. [ABSTRACT FROM AUTHOR]

The central objective of this paper is to investigate the lacunary statistical convergence of order γ and strong lacunary summability of order γ for sequences of fuzzy functions by using the modulus function and a regular matrix. We examine the relevant inclusion relations between these spaces under specific conditions. Additionally, we study properties related to these spaces and establish several interesting results. [ABSTRACT FROM AUTHOR]

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In classical set theory, elements of the set are written once but the sets in which the same item is repeated several times in daily life are in all areas of our lives. These sets are called multisets and are studied in many fields such as Mathematics, Physics, Chemistry, and Computer Sciences. Sequences consisting of elements of these sets are called multiset sequences. In this paper, we study the concept of I-lacunary statistical convergence of multiset sequences and investigate some important results. [ABSTRACT FROM AUTHOR]

In this paper, we have presented rough statistical convergence of sequence on neutrosophic normed spaces as an important convergence criterion. As neutrosophication can handle partially dependent components, partially independent components and even independent components involved in real-world problems. By examining some properties related to rough convergence in these spaces we provide some useful functional tools in the situations of inconsistency and indeterminacy in the real world. Further, we have established some equivalent conditions on the set of statistical limit points as well as on the set of cluster points in these spaces for rough statistically convergent sequences. [ABSTRACT FROM AUTHOR]

With the advent of quantum computing, traditional cryptography algorithms are at risk of being broken. Post-quantum encryption algorithms, developed to include mathematical challenges to make it impossible for quantum computers to solve problems, are constantly being updated to ensure that sensitive information is protected from potential threats. In this study, a hybrid examination of a (p,q)-Bernstein-type polynomial, which is an argument that can be used for encryption algorithms with a post-quantum approach, was made from a mathematical and cryptography perspective. In addition, we have aimed to present a new useful operator that approximates functions and can be used in cases where it is not possible to work with functions in the fields of technology, medicine, and engineering. Based on this idea, a new version of the (p,q)-Bernstein-Schurer operator was introduced in our study on a variable interval and the convergence rate was calculated with two different methods. At the same time, the applications of the theoretical situation in the study were presented with the help of visual illustrations and tables related to the approach. Additionally, our operator satisfied the statistical-type Korovkin theorem and is suitable for variable interval approximation. This is the first paper to study the statistical convergence properties of (p,q)-Bernstein-Schurer operators defined on a variable bounded interval, to obtain special matrices with the help of (p,q)-basis functions, and to give an application of (p,q)-type operators for encrypted image transmission.

Many studies have been conducted on statistical convergence, which remains an area of active research. Since its introduction, statistical convergence has found applications in many fields. Nevertheless, there is a shortage of studies related to operator theory; especially of those on continuous, bounded, and compact operators. We explore the notions of statistical boundedness, continuity, and compactness of operators between normed spaces, establishing connections between these concepts and their counterparts in the traditional normed space theory. Additionally, we provide some examples and results that demonstrate the behavior and implications of statistical convergence in operator theory. [ABSTRACT FROM AUTHOR]

With the advent of quantum computing, traditional cryptography algorithms are at risk of being broken. Post-quantum encryption algorithms, developed to include mathematical challenges to make it impossible for quantum computers to solve problems, are constantly being updated to ensure that sensitive information is protected from potential threats. In this study, a hybrid examination of a (p,q)-Bernstein-type polynomial, which is an argument that can be used for encryption algorithms with a post-quantum approach, was made from a mathematical and cryptography perspective. In addition, we have aimed to present a new useful operator that approximates functions and can be used in cases where it is not possible to work with functions in the fields of technology, medicine, and engineering. Based on this idea, a new version of the (p,q)-Bernstein-Schurer operator was introduced in our study on a variable interval and the convergence rate was calculated with two different methods. At the same time, the applications of the theoretical situation in the study were presented with the help of visual illustrations and tables related to the approach. Additionally, our operator satisfied the statistical-type Korovkin theorem and is suitable for variable interval approximation. This is the first paper to study the statistical convergence properties of (p,q)-Bernstein-Schurer operators defined on a variable bounded interval, to obtain special matrices with the help of (p,q)-basis functions, and to give an application of (p,q)-type operators for encrypted image transmission. [ABSTRACT FROM AUTHOR]

In the present paper, we mostly focus on Pp2-statistical convergence. We will look into the uniform integrability via the power series method and its characterizations for double sequences. Also, the notions of Pp2-statistically Cauchy sequence, Pp2-statistical boundedness and core for double sequences will be described in addition to these findings. [ABSTRACT FROM AUTHOR]

The present study introduces the notions of statistical convergence of order α and strong p-Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces. [ABSTRACT FROM AUTHOR]

The applications of a Fibonacci sequence in mathematics extend far beyond their initial discovery and theoretical significance. The Fibonacci sequence proves to be a versatile tool with real-world implications and the practical utility of manifests in various fields, including optimization algorithms, computer science and finance. In this research paper, we introduce new versions of convergence and summability of sequences in normed spaces with the help of the Fibonacci sequence called weak Fibonacci φ-lacunary statistical convergence and weak Fibonacci φ-lacunary summability, where φ is a modulus function under certain conditions. Furthermore, we obtain some relations related to these concepts in normed spaces. [ABSTRACT FROM AUTHOR]

Here in this paper, we have an idea of statistical boundedness for a partial metric space (X, φ) where φ is partial metric on X and establish the relationship between statistical convergence and statistical boundedness. Besides this, statistical analogue of monotone convergence theorem of reals has been established. [ABSTRACT FROM AUTHOR]

In this study, a different generalization of q-Bernstein operators with the parameter λ ∈ [ − 1 , 1 ] is created. The moments and central moments of these operators are calculated, a statistical approximation result for this new type of (λ , q) -Bernstein operators is obtained, and the convergence properties are analyzed using the Peetre K-functional and the modulus of continuity for this new operator. Finally, a numerical example is given to illustrate the convergence behavior of the newly defined operators. [ABSTRACT FROM AUTHOR]

This paper deals with the approximations of the functions by generalized Durrmeyer operators of Szász–Mirakjan, which are linear positive operators. Several approximation results are presented well, and we estimate the approximation properties along with the order of approximation and the convergence theorem of the proposed operators. For an explicit explanation of the operators, we determine the properties using the weight function. A quantitative approach is discussed for the operators; quantitative Voronovskaya type and Grüss type theorems are established, showing the operators' more efficient work. We investigate the A$$ A $$‐statistical convergence properties for the said operators, including the rate of approximation in a statistical sense. An important property for the rate of convergence of the operators is obtained in terms of the function with a derivative of the bounded variation. At last, the graphical representations and numerical analysis are discussed and shown to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study lacunary A-statistical convergence of complex uncertain sequences via Orlicz functions. We investigate some aspects of uncertainty such as mean, measure, almost surely, uniformly almost surely and distribution. Additionally, we make an effort to provide illustrative examples that elucidate the interrelations within the constructed sequence spaces. [ABSTRACT FROM AUTHOR]

In this study, we present statistical convergence, statistical limit points, and statistical cluster points of double sequences in neutrosophic fuzzy G-metric space with order q, extending the notion of neutrosophic fuzzy metric space. We support our assertions with relevant theorems and elucidate them through illustrative examples. Following the establishment of statistical convergence and the scrutiny of its properties within these spaces, we explore the concepts of lacunary statistical convergence and strongly lacunary convergence of double sequences, while also investigating the relationships among them. [ABSTRACT FROM AUTHOR]

In this paper, we define new notions of (f, I)-statistical convergence for triple sequences of order ᾶ and strong (f, I)-Cesàro summability for triple sequences of order ᾶ. Furthermore, we examine the relationships among the spaces w³ᾶ,0(f, I), w³ᾶ(f, I) and w³ᾶ,∞(f, I). Additionally, we present the concepts of (f, I)-lacunary statistical convergence of order α and strong (f, I)-lacunary summability of order α for triple sequences, providing inclusion relations among these new concepts. [ABSTRACT FROM AUTHOR]

In the study done here regarding the theory of summability, we introduce some new concepts in fuzzy normed spaces. First, at the beginning of the original part of our study, we define the lacunary invariant statistical convergence. Then, we examine some characteristic features like uniqueness, linearity of this new notion and give its important relation with pre-given concepts.

Abstract In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity.

In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp-statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces. [ABSTRACT FROM AUTHOR]

In this article, we aim to define ℐ 2 ⁢ (풮) -summability and ℐ 2 ⁢ (풮 θ p , r ) -summability, and obtain interesting relationships among these by imposing certain conditions on p and r. Finally, we show that the space ℐ 2 ⁢ (풮 θ p , r ⁢ (G , B , Y)) ∩ l ∞ 2 ⁢ (V) is a closed subspace of l ∞ 2 ⁢ (V) . [ABSTRACT FROM AUTHOR]

In the study done here regarding the theory of summability, we introduce some new concepts in fuzzy normed spaces. First, at the beginning of the original part of our study, we define the lacunary invariant statistical convergence. Then, we examine some characteristic features like uniqueness, linearity of this new notion and give its important relation with pre-given concepts. [ABSTRACT FROM AUTHOR]

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity. [ABSTRACT FROM AUTHOR]

The aim of this article is to construct a (p , q) -analogue of wavelets Kantorovich–Baskakov operators and investigate some statistical approximation properties. We study weighted statistical approximation by means of a Bohman–Korovkin-type theorem, and statistical rate of convergence by means of the weighted modulus of smoothness ω ρ α associated to the space B ρ ⁢ α ⁢ (ℝ +) and Lipschitz-type maximal functions. [ABSTRACT FROM AUTHOR]

This paper's main purpose is to present λ²-statistically convergent and condition of being λ²-statistically Cauchy for double sequences in fuzzy nnormed spaces. At the same time, in fuzzy n-normed space, we have introduced the concept of λ²-summability and Cesaro summability for double sequences in fuzzy n normed spaces. Then, we studied the relation between these concepts and statistical convergence in fuzzy n-normed space. [ABSTRACT FROM AUTHOR]

In this study, the concept of ρ-statistical convergence with respect to the neutrosophic norm in the neutrosophic normed spaces is introduced. Some properties and some inclusion theorems related to this concept are investigated. [ABSTRACT FROM AUTHOR]

We first present a notion of a double lacunary sequence on product time scales. Using this notion, we define the notions of the double lacunary statistical convergence and double lacunary strongly p-Cesàro summability of 2-multiple functions on product time scales and we study some fundamental properties of both notions. We also present a theorem that connects the above-mentioned two concepts. Furthermore, we define a refinement of a double lacunary sequence on product time scales and provide some fundamental properties as well as inclusion theorems for a refined and a non-refined double lacunary sequence on product time scales. [ABSTRACT FROM AUTHOR]

In present paper, we aim to define a new summability method using σ-mean called σ-statistical summability in an intuitionistic fuzzy 2-normed space (briefly IF 2N S). We also define σ-statistical Cauchy sequence and Sσ-completeness in an IF 2NS and study some of their properties. We display an example that shows our method of summability is more stronger in these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we establish the concept -rung orthopair fuzzy normed space and explore its connections with intuitionistic fuzzy normed spaces. We introduced a novel type of statistical convergence and define the concept of statistical Cauchy sequences in -rung orthopair fuzzy normed spaces. Investigating the relationship between these concepts, we provide insights into their interplay. Furthermore, we contribute a criterion for statistical convergence and completeness in -rung orthopair fuzzy normed spaces. This criterion enhances our understanding of the convergence behavior in this unique mathematical setting, offering a comprehensive exploration of the analytical properties within the framework. [ABSTRACT FROM AUTHOR]

Statistical convergence is more extensive than the classical convergence and has recently drawn the recognition of many researchers. The Korovkin-type approximation theorems are usually based on the convergence analysis of sequences of positive linear operators. Gradually, such approximation theorems are extended over more general sequence spaces with several settings via different kinds of statistical summability techniques. In this paper, we introduce presumably a new statistical Riesz-Euler product summability technique to prove a Korovkin-type approximation theorem. Moreover, we demonstrate another result for the rate of statistical convergence under our proposed summability technique. [ABSTRACT FROM AUTHOR]

In this work, a new generalization of statistical boundedness is provided for difference sequences in regard to lacunary α-density and lacunary statistical sense. Apart from examining some inclusion theorems on related sequence spaces, we show that... does not form a sequence algebra unlike .... [ABSTRACT FROM AUTHOR]

We generalized the concepts in the probability of rough lacunary statistical analysis of order η by introducing the difference operator A of fractional order, where a is a proper fraction and y = (Ymnk) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. The main focus of the present paper is to generalize rough lacunary statistical analysis of order n of triple difference sequence spaces and give the relations between statistical analysis and lacunary statistical analysis of order n. [ABSTRACT FROM AUTHOR]

This paper aims to explore the concepts of statistical convergence sequence and statistical summability in non-Archimedean fields (NAF). Statistical convergence has been studied in various mathematical fields such as measure theory, probability theory, and number theory, and plays a significant role in summability theory and functional analysis. The goal of this study is to provide characterizations for triple sequences using the (Mλt,u,v) method of summability over NAF and to prove inclusion relations between statistical convergence and statistical (Mλt,u,v) summability for triple sequences over such fields. Furthermore, statistics At,u,v-summability for triple sequence has been examined in non-Archimedean fields. The non-trivially valued, complete, non-Archimedean fields are denoted by K throughout the article. [ABSTRACT FROM AUTHOR]

Recently, Devaiya and Srivastava [3] studied the T r -strong convergence of numerical sequences and Fourier series using a lower triangular matrix T = (b m , n) , and generalized the results of Kórus [8]. The main objective of this paper is to introduce [ T r , G , u , q ] -strongly convergent sequence spaces for r ∈ N , and defined by a sequence of modulus functions. We also provide a relationship between [ T , G , u , q ] and [ T r , G , u , q ] -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for T r -strong convergent sequences, statistical convergence and Fourier series using the idea of [ T r , G , u , q ] -strongly convergent sequence spaces. [ABSTRACT FROM AUTHOR]

Tomographic absorption spectroscopy (TAS) is important for the combustion diagnostics of hydrogen and hydrocarbon fuels in various flow fields. In this study, we proposed and conducted time-averaged TAS for the reconstructions of mean H 2 O mole fraction X H2O in a dry, axisymmetric, turbulent jet flow with a Reynolds number of 5670 to reveal the effect of turbulence on absorption spectroscopy measurement. The absorption line of H 2 O near 1368.598 nm was selected and scanned at 1 kHz. The effect of turbulence on TAS, which is positively related to the turbulent intensity along the optical path, has been confirmed and can be effectively alleviated by conducting time averaging. It is shown that the proper selections of scan rate and the number of time steps for statistical convergence are required for time-averaged TAS. With time averaging and smoothing operations on the integrated absorbance, the two-dimensional mean X H2O distributions were reconstructed with the present method and the result showed a good agreement with CFD simulation. The mean deviation of the measured results can be reduced by over 25% when the number of time steps for average is 30000. The present study highlights the potential of TAS for the visualization and diagnostics of complex flows such as turbulent non-reactive flow and turbulent combustion. • Time-averaged tomographic absorption spectroscopy for isothermal flow is proposed and conducted in a turbulent jet. • The effect of turbulence on TDLAS is confirmed to be positively related to local turbulent intensity. • Time-averaging and smoothing on the integrated absorbance improves the tomographic reconstruction. [ABSTRACT FROM AUTHOR]

This paper interprets the relationship between the statistical Cesàro summability method and statistical convergence for sequences in intuitionistic fuzzy normed spaces. There are several studies in the literature on the Cesàro summability method and its statistical version, in classical, fuzzy, and intuitionistic fuzzy normed spaces. However, what distinguishes this study from others, besides the techniques used, is that it is a more comprehensive research in terms of both the space being studied and the path to the statistical convergence from the statistical Cesàro summability method. Our primary goal in this work is to investigate whether conditions designated as O-type like q-boundedness and statistical slow oscillation require statistical Cesàro summable sequences be statistical convergent with respect to the intuitionistic fuzzy norm (μ , ν). Moreover, we inquire about the existence of a more inclusive condition than these conditions, from which we can derive a similar inference. Our second aim is to present a series of auxiliary results designed to be utilized in the proofs of our main results. To go into a little detail, we discuss some algebraic properties of statistical limits, including additivity and homogeneity, as well as the existence of a statistical limit of subsequences and moving averages of a sequence. Finally, we give some examples of the application of the intuitionistic fuzzy normed space in engineering, in order to extend the area of influence of the relevant space and to support the studies carried out in the context of this space. [ABSTRACT FROM AUTHOR]

This paper presents and examines lacunary A-statistical convergence for fuzzy variable sequences utilising Orlicz functions. We delve into various facets of credibility theory including mean, measure, almost surely, uniform almost surely and distribution. We aim to illustrate the interrelationships of the constructed sequence spaces through relevant examples. [ABSTRACT FROM AUTHOR]

In this study, we define lacunary Δm-statistical convergence in the framework of intuitionistic fuzzy normed spaces (IFNS) for triple sequences. We prove several results for lacunary Δm-statistical convergence of triple sequence in IFNS. We further established lacunary Δm-statistical Cauchy sequences and provided the Cauchy convergence criterion for this novel idea of convergence. [ABSTRACT FROM AUTHOR]

In recent years, a variety of work has been done in the field of single, double and triple sequences. Study on n-tuple sequence is new in this field. The main interest of this paper is to explore the idea of n-tuple sequences x = (xi1,i2,...,in) in metric spaces. We introduce the concept of n-sequence space of interval number and discussed its arithmetic properties. Furthermore, we combined the concept of Orlicz function, statistical convergence, interval number and n-sequence to construct some new n-sequence spaces and discussed their properties. Some suitable examples for these spaces have been constructed. [ABSTRACT FROM AUTHOR]

The usual convergence of sequences has many generalizations with the aim of providing deeper insights into the summability theory. In this paper, following a very recent and new approach, we introduce the notion of I3 and I*3-convergence of triple sequences in neutrosophic normed spaces mainly as a generalization of statistical convergence of triple sequences. We investigate a few fundamental properties and study the relationship between the two notions. We also introduce and investigate the concept of I3 and I*3-Cauchy sequence of triple sequences and show that the condition (AP3) plays a crucial role to study the interrelationship between them. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and study the strong Cesaro summability of difference sequence spaces through fusion of modulus function. On the newly established sequence space, linear structure is imposed and a paranorm is established. Apart from various inclusion relations, a new variant of statistical convergence is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we expand upon different concepts of convergence, including convergence in distribution, in the mean, and in measure, while also exploring uniform almost sure convergence of uncertain double sequences of fuzzy numbers in statistical context. Additionally, we present intriguing findings concerning these convergence concepts. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of statistical limit superior and statistical limit inferior in neutrosophic normed space. We study some of its basic properties and establish some implication relations. [ABSTRACT FROM AUTHOR]

In this study, a different generalization of q-Bernstein operators with the parameter λ∈[−1,1] is created. The moments and central moments of these operators are calculated, a statistical approximation result for this new type of (λ,q)-Bernstein operators is obtained, and the convergence properties are analyzed using the Peetre K-functional and the modulus of continuity for this new operator. Finally, a numerical example is given to illustrate the convergence behavior of the newly defined operators.

Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of ℐ -statistical supremum and ℐ -statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of ℐ -statistical monotonicity and establish the condition under which an ℐ -statistical monotonic sequence is ℐ -statistical convergent. We end up giving a necessary and a sufficient condition for the ℐ -statistical convergence of a real-valued sequence. [ABSTRACT FROM AUTHOR]