Binod Chandra Tripathy

Tripura University · Department of Mathematics

The sequence spaces ru ℓ ∞ (𝒪, ∇ q ), ru ℓ p (𝒪, ∇ q ), ru c (𝒪, ∇ q ), ru c 0 (𝒪, ∇ q ), ru m ϕ (𝒪, ∇ q , p ), ru n ϕ ( 𝒪, ∇ q , p ), ru m ϕ ( 𝒪, ∇ q ), ru n ϕ ( 𝒪, ∇ q ) are defined by the Orlicz function in this article. We examine all of its characteristics, including symmetry, solidity, and completeness. A few geometric properties on convexity...

The aim of this research article is to introduce the concept of strongly lacunary ideal convergence of sequences of fuzzy variables in a credibility space in different directions. We define strongly lacunary ideal convergence via credibility measure, credibility distribution function and expected value of the fuzzy variables which are indeed elemen...

This paper explores various concepts of almost convergence for sequences of fuzzy variables within the framework of credibility theory. It introduces the notion of almost ideal convergence and examines its different forms—almost surely, in credibility, in mean, in distribution, and uniformly almost surely. Key results establish that almost \(\maths...

We develop and examine the pre-modular space of null variable exponent-weighted backward generalized difference gai sequences of fuzzy functions in this paper. These sequences of fuzzy functions are important contributions to the concept of modular spaces because they have exponent weighting. Using extended s−fuzzy functions as well as this sequenc...

This study addresses the complexity of assessing river water quality, a multifaceted process influenced by numerous water quality parameters (WQPs) characterized by inherent uncertainties and diverse judgment information from decision-makers. These uncertainties and diverse judgment information can be effectively represented and simulated using fuz...

In this article, we have introduced the notion of convergence of difference double sequences in Pringsheim's sense, difference null in Pringsheim's sense, bounded difference, bounded convergence difference, bounded null difference, regular convergence difference and regular null difference double sequences of bi-complex numbers. We have proved that...

The main purpose of this paper is to investigate different types of deferred convergent double sequences of fuzzy variables by using the notions of ideal convergence and μ-density in a given credibility space. We also establish a number of instances to illustrate the newly introduced notions in the same environment. In this study, we also present i...

In this article we introduce the notion of double sequences of complex uncertain variables associated with multiplier sequences. We study their different algebraic and topological properties like solidity, symmetricity, completeness etc.

The increasing pollution of river systems poses a significant challenge to water quality management and public health. This study was conducted to evaluate the water quality of the Gomati River in India, focusing on the impact of anthropogenic activities on pollution levels across different seasons. The primary objective was to develop a comprehens...

In this paper we present the notion of bounded, convergent in Pringsheim sense, null in Pringsheim sense, regular, regular null and absolutely p-sumable double sequences of bi-complex numbers. We have also introduced the concept of repeated limit of the double sequences of bi-complex numbers. We have established that every P-convergent double sequ...

In this article, we introduced the idea of lacunary weak convergence of sequences defined by the class of Orlicz function. Various topological and algebraic properties, such as solidity, symmetry, and monotonicity, have been studied. To validate the failure instances, we gave appropriate examples and detailed discussions. Additionally, we have esta...

In this chapter, we delve into the fascinating realm of interval-valued neutrosophic sets by introducing two novel concepts: interval-valued neutrosophic b-open sets and interval-valued neutrosophic b-closed sets. These sets bring forth intriguing properties which we thoroughly explore. Additionally, we delve into the concept of interval-valued neu...

In the present article, we present how M-connectedness differs from similar concepts in general topology, and the notion of closed subspace mixed M-topology in mixed M-topological space by extending the concept of closed subspace M-topology. We also define the notions of M-connectedness and M-compactness on mixed M-topological spaces. We investiga...

The notion of relative uniform convergence of a sequence of functions was introduced and investigated by Moore, followed by E. W. Chittenden in the twentieth century. The concept attracted the researcher at the beginning, of the twenty-first century. Jackson F. H. initiated the concept of the notion of the quantum difference operator. We introduced...

In this paper we investigate the concept of ℐ {\mathcal{I}} -monotonic convergence of sequences of bi-complex numbers. We also discuss some partial order relations ≼ i 2 {\preccurlyeq_{i_{2}}} and ≼ i id {\preccurlyeq_{i_{\rm id}}} between two bi-complex numbers and ℐ {\mathcal{I}} -convergence, ℐ {\mathcal{I}} -monotonic sequences with respect to...

The conception of single-valued quadripartitioned neutrosophic d-ideal (SVQN-d-I) of single-valued quadripartitioned neutrosophic d-algebra (SVQN-d-A) as an expansion of neutrosophic d-Ideal and neutrosophic d-Algebra has been attempted to be introduced in this article. Additionally, we identify various characteristics of them. Additionally, SVQN-d...

In this article we have introduced the notion of almost convergence, strongly almost convergence, almost null and strongly almost null of double sequences in Pringsheim sense of bi-complex numbers. We have proved that these are linear spaces and with the help of the Euclidean norm defined on bi-complex numbers, we have proved their different algebr...

This paper discusses relative uniform convergence of triple sequence of functions that are defined on a compact domain. Another central idea that is discussed is the regular relative uniform convergence and the Cauchy relative uniform convergence of triple sequence of functions. The idea that a continuous triple sequence defined on a compact domain...

The purpose of this article is to define fuzzy (b,$\theta$)-quasi neighbourhood using the concept of fuzzy (b,$\theta$)-open sets introduced by Dutta and Tripathy with some structural properties by defining the (b,$\theta$)-cluster point in a different way. Moreover, our second objective is to introduce the notion of fuzzy (b,$\theta$)-continuous f...

In this research article, we introduce ℐ {\mathcal{I}} -statistically pre-Cauchy sequences of complex uncertain variables in five different aspects of uncertainty, namely: in mean, in measure, in distribution, in almost sure, and in uniformly almost sure. We also explore the connection between ℐ {\mathcal{I}} -statistically pre-Cauchy sequences and...

In this article we introduce the notion of relative uniform convergence of p-absolutely summable sequence of functions. We have investigated its different properties such as solidness, symmetric, convergence-free, monotonic etc. We have established some inclusion relations involving this sequence space.

In this article we have introduced the concept of Ces?ro convergence, Ces?ro null and Ces?ro bounded sequences of bi-complex numbers defined by BC-Orlicz function having hyperbolic norm. we have investigated some of their algebraic and topological properties by defining a D-norm on these spaces. Also inclusion results involving these sequence space...

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.

Let us consider X to be a linear space, the set of all bounded linear operators defined on X into itself. We denote it by B(X). Let \(T\in B(X)\) and X is a Banach space then the adjoint operator \(T^*\) of T is defined as a bounded linear operator on the dual of X which is denoted by \(X^*\) and defined by \((T^*\varPhi )(x) = \varPhi (Tx)\) for a...

The Orlicz function-defined sequence spaces of functions by relative uniform convergence of sequences related to p-absolutely summable spaces are a new concept that is introduced in this article. We look at its various attributes, such as solidity, completeness, and symmetry. We also look at a few insertional connections involving these spaces.

In this paper, we introduce the notions of I 3-convergent, I ⋆ 3-convergent and I 3-Cauchy triple sequences in a fuzzy normed linear space. Besides, we establish some basic results related to those notions. Furthermore, we define the concepts of I ⋆ 3-Cauchy sequence in fuzzy normed space and some relations between I 3-Cauchy sequence in a fuzzy no...

This research paper aims to introduce the concept of lacunary ideal Cauchy sequences of fuzzy variables in a credibility space. We establish the interrelationships between this notion with lacunary ideal convergent sequences in the same structure from several aspects of credibility. Furthermore, we explore the concepts of strongly lacunary Cauchy,...

In this article, we look into the concept of ideal relative uniform convergence of a double sequence of functions. In addition, we define ideal relative uniform Cauchy and ideal regular relative uniform convergence of a double sequence of positive linear functions defined on a compact domain D with respect to the scale function σ(x) defined on D. We a...

The concept of mixed multiset topology was introduced and investigated by different researchers from different aspects. In this paper, we introduce the notion of mixed multiset ideal topological space. Further, we define the concepts of τ 1 (τ 2)-pre-I-open mset, τ 1 (τ 2)-semi-I-open mset, τ 1 (τ 2)-α-I-open mset and τ 1 (τ 2)-δ-I-open mset in mix...

This study introduces the concept of statistically pre-Cauchy sequences of fuzzy variables in five directions of credibility theory: almost surely, in measure, in mean, in distribution, and uniformly almost surely. However, the main focus is kept on statistically pre-Cauchy sequences in measure, in mean, and in distribution. Furthermore, a correlat...

This paper presents several statistical convergence concepts of complex uncertain sequences based on a regular matrix of Fibonacci numbers: statistical Fibonacci convergence almost surely, statistical Fibonacci convergence in measure, statistical Fibonacci convergence in mean and statistical Fibonacci convergence uniformly almost surely. Furthermor...

The study investigates the existence and uniqueness of solutions for Fisher's equations in disease propagation models by utilizing the principles of fixed point theorem, Morse theorem, and G-metric spaces. Fisher's equations, which describe the spatial-temporal dynamics of populations, are widely employed in modelling disease spread. By formulating...

In this article, we use a completely new technique to formulate the definitions of $(\tau_1,\tau_2)^*$-pre-open mset, $(\tau_1,\tau_2)^*$-semi-open mset, $(\tau_1,\tau_2)^*$-$\alpha$-open mset and $(\tau_1,\tau_2)^*$-$\beta$-open mset via two multiset topologies. We prove that every $\tau_2$-open mset is $(\tau_1,\tau_2)^*$-pre-open mset, $(\tau_1,...

In this paper, we study some basic properties of bicomplex numbers. We introduce two different types of partial order relations on bicomplex numbers, discuss bicomplex valued metric spaces with respect to two different partial orders, and compare them. We also define a hyperbolic valued metric space, the density of natural numbers, the statistical...

In the present article, we present the notions of fuzzy multiset interior and closure operator and study their basic characterizations and properties. Furthermore, using the fuzzy multiset closure operator we introduce the notion of fuzzy multiset closure space and present its related properties. Finally, the notion of connectedness in the context...

In this article, we introduce some M-topological operators called multiset kernel and multiset shell operators. Thereafter, we define a new separation axiom termed as multiset TD- spaces and investigate some of its basic properties. It is observed that this space precisely lies between multiset T0 and multiset T1-spaces. Also, we characterize multi...

In this treatise, we define statistically pre-Cauchy sequences of complex uncertain variable for five cases of uncertainty viz., in mean, in measure, in distribution, in almost surely and in uniformly almost surely and we confine our study to statistically pre-Cauchy sequence in mean, in measure and in distribution only. Furthermore, we establish t...

In this paper, we introduce a new class of set in ideal tritopological spaces which is called (i, j, k)-semi-I-open set. Besides, we study some of its properties. Moreover, we define the notion of (i, j, k)-semi-I-continuous functions and we show some of their properties and characterizations.

In this article an attempt has been made to procure the concept of pairwise neutrosophic simply open set, pairwise neutrosophic simply continuous mapping, pairwise neutrosophic simply open mapping, pairwise neutrosophic simply compactness, pairwise neutrosophic simply b-open set, pairwise neutrosophic simply b-continuous mapping, pairwise neutrosop...

This paper aims to establish the notion of Hamming distance under the possible environment and propose a multi-attribute decision-making (MADM) strategy based on grey relational analysis (GRA) under the Possibility environment. This study identified the most critical parameter or factors affected by climate change, urbanization, and machine failure...

Recently, El-Sharkasy et al., El-Sharkasy and Badr have shown that to find the mutations of deoxyribonucleic acid (DNA) and ribonucleic acid (RNA), multiset topology is very much essential. In this article we give the definitions of mixed (1,2)-pre-open (resp. mixed (1,2)-semi-open, mixed (1,2)-α-open, mixed (1,2)β-open) mset via two multiset topol...

The main aim of the paper is to characterize matrix transformation of complex uncertain sequences upto some extent. We introduce the spaces $\ell_p (\Gamma_{u.a.s})$ and $[\ell_\infty(p)]_{\Gamma_{u.a.s}}$ and then derive necessary and sufficient condition for an infinite real matrix $A = (a_{nk})$ to transform sequences from $\ell_1(\Gamma_{u.a.s}...

In this article an attempt is made to introduce and study the notion of single-valued neutrosophic rough continuous mapping, single-valued neutrosophic rough compactness via single-valued neutrosophic rough topological spaces (SVNRTS). By defining the concept of single-valued neutrosophic rough continuous function, single-valued neutrosophic rough...

The main focus of this paper is to introduce some aggregation operators namely, Rough-Bipolar Neutrosophic Arithmetic Mean (RBNAM) operator and Rough-Bipolar Neutrosophic Geometric Mean (RBNGM) operator under Rough-Bipolar Neutrosophic Set (RBNS) environment. Besides, we present the concept of score and accuracy functions under the RBNS environment...

The main aim of this investigation is to introduce rough I-statistical convergence in probabilistic n-normed spaces (briefly Pr-n-spaces). We establish some results on roughI-statistical convergence and also we introduce the notion of rough I-statistical limit set in Pr-n-spaces and discuss some topological aspects on this set. Moreover, we define...

In this article, we introduce the notion of relative uniform convergence of double sequence of positive linear functions defined by using Orlicz function. We also introduce different classes of relative uniform convergence sequence of functions and discuss their algebraic and topological properties.

In this paper, we define the expansive mapping in $G$-metric space and we prove some fixed point theorems in generalized M-fuzzy (GM-fuzzy) Metric Space.

In this paper, we introduce the notion of lacunary statistical convergence for the sequences of complex uncertain variables for almost sure, mean, measure and distribution. We investigate some of the basic properties of the notion. We have established relation between these notions.

The aim of this paper is to study the concept of matrix transformation between complex uncertain sequences in mean. The characterization of the matrix transformation has been made by applying the concept of convergence of complex uncertain series. Moreover, in this context, some well-known theorems of real sequence spaces have been established by c...

In this paper, a comprehensive literature review on the sequence of uncertain variables, complex uncertain variables defined by Orlicz function is realized. We procure different results and developments. In the recent years, ordinary sequences have been extended to new types and these extensions have been used in uncertain environment too, differen...

The intention of this article is to introduce and characterise the concept of preopen sets and prelocally closed sets in Generalised Topology and Minimal structure spaces.

The primary goal of this treatise is to introduce a new kind of generalized closed sets termed as Lgfs-closed sets in the light of fuzzy soft local function in a mixed fuzzy soft ideal topological space. We also define the notion of fuzzy soft *-separated set. In addition, we procure the idea of fuzzy soft regularity, normality and fuzzy soft compa...

Statistical convergence of complex uncertain triple sequence is already introduced in four aspects of uncertainty. In this article, we initiate the notion of statistical convergence of complex uncertain triple sequence with respect to uniformly almost surely. We prove that statistical convergence with respect to uniformly almost surely of a complex...

The main aim of this treatise is to introduce the notion of $\lambda^2$-statistical convergence in a given uncertainty space by considering sequences of complex uncertain variable. Also, we study the interrelationship between $\lambda^2$-statistical convergence with strongly $\lambda^2$-summable sequence in the same environment. Further, we extend...

The purpose of this paper is to introduce a novel similarity measure, the single-valued pentapartitioned neutrosophic exponential similarity measure (SVPNESM), and the single-valued pentapartitioned neutrosophic weighted exponential similarity measure (SVPNWESM) under the single-valued pentapartitioned neutrosophic set (SVPNS) environment for selec...

In this article, we procure the concept of single-valued pentapartitioned neutrosophic Lie (in short SVPN-Lie) algebra under single-valued pentapartitioned neutrosophic set (in short SVPN-set) environment. Besides, we study the notion of SVPN-Lie ideal of SVPN-Lie algebra, and produce several interesting results on SVPN-Lie algebra and SVPN-Lie ide...

In this article we initiate the concept of matrix transformation through uncertain measure of a complex uncertain sequence. We establish a necessary and sufficient condition under which an infinite matrix transforms a convergent sequence into another. As an application of this notion we prove the well-known Silverman-Toeplitz theorem and Kojima-Sch...

In this article an attempt is made to introduce and study the concept of single-valued quadripartitioned neutrosophic graph (SVQN-graph) by extending the idea of single-valued neutrosophic graph (SVN-graph) theory. Besides, we introduce the notion of degree, order and size of SVQN-graphs. Further, we furnish few illustrative examples to justify the...

In this paper, we propose a Variable Neighborhood Search (VNS) algorithm based on Multi-AttributeDecision-Making (MADM) strategy under possibility environment. Further, we provide a numerical example to show the applicability and rationality of the proposed MADM strategy.

In this paper, we derive a common �xed point result for a class of contractive mappings which is a generalization of the contractive condition given in [1]. A convergence result for an iteration scheme is also obtained for such mappings. Suitable examples have been provided in support of the results obtained.

The main focus of this article is to introduce the notion of rough pentapartitioned neutrosophic set and rough pentapartitioned neutrosophic topology by using rough pentapartitioned neutrosophic lower approximation, rough pentapartitioned neutrosophic upper approximation, and rough pentapartitioned neutrosophic boundary region. Then, we provide som...

En este artículo se introduce y estudia el concepto de gráfico neutrosófico cuadriparticionado de un solo valor (SVQN-grafo) extendiendo la idea de la teoría del gráfico neutrosófico de un solo valor (SVN-grafo). Además, se define la noción de grado, orden y tamaño de los SVQN-grafos. Además, proporcionamos algunos ejemplos ilustrativos para justif...

The concept of statistical convergence plays a very prominent role in the study of sequence spaces. In this treatise, we extend the research on different types of statistical convergence viz., statistical convergence in mean, in measure, in distribution, in almost surely and with respect to uniformly almost surely of complex uncertain triple sequen...

Unlike traditional algebraic structures, where all operations are well-defined and all axioms are completely true, NeutroAlgebras and AntiAlgebras allow operations to be partially well-defined and axioms to be partially true or fully outer-defined, and axioms to be completely false. These NeutroAlgebras and AntiAlgebras represent a new research sub...

In this paper, we extend the study of statistical convergence of complex uncertain sequences in a given uncertainty space. We produce the relation between convergence and statistical convergence in an uncertain environment. We also initiate statistically Cauchy complex uncertain sequence to prove that a complex uncertain sequence is statistically c...

In this article, we introduce the notion of neutrosophic ideal bitopological space (NIBTS) by generalizing the idea of neutrosophic ideal topological space (NITS). Besides, we procure the notion of local function on neutrosophic bitopological space (NBTS) and studied its different properties. Further, we procure the concept of neutrosophic pre-I-op...

The main aim of this paper is to introduce several forms of statistical convergence of difference sequences of complex uncertain variables: [Formula: see text]-statistically convergence almost surely, [Formula: see text]-statistically convergence in measure, [Formula: see text]-statistically convergence in mean and [Formula: see text]-statistically...

In this paper, the concept of convergence of complex uncertain series is applied to study matrix transformation of complex uncertain sequences in terms of almost surely. We establish a necessary and sufficient condition under which an infinite matrix operator transforms a null complex uncertain sequence in almost surely into another null sequence a...

The geographical information system has been generally used for analyzing different types of data. The notions and results of topology have been applied in this connection, known as a spatial topological relation. In this article, we have studied the different layers of geographical data and their intersection property, separation axioms on spatial...

In this article, we introduce the class of sequence of functions C 1 ðΔ, ruÞ of Cesaŕo summable relative uniform difference sequence of functions. We have studied the topological properties of C 1 ðΔ, ruÞ. We also obtain the necessary and sufficient condition to characterize the matrix classes ðC 1 ðΔ, ruÞ, ℓ ∞ ðruÞÞ, ðC 1 ðΔ, ruÞ, cðruÞ, PÞ.

In this paper, we introduce a new type of almost convergent complex uncertain sequence with respect to uniformly almost surely. We characterize the notion of almost convergence of sequences of complex uncertain variables further. We establish the interconnection between convergent complex uncertain sequence, bounded complex uncertain sequence and a...

Uncertain theory which was initially introduced by Liu [2016. Unceratinty Theory. 5th ed. Berlin: Springer-Verlag] is significantly speared in various fields of engineering and scientific computing such as probability theory, statistics, fuzzy set theory, measure theory, summability theories, etc. As a part of this theory, the problems of convergen...

In this article an attempt is made to introduce the notion of neutrosophic infi-topological space as an extension of infi-topological space and fuzzy infi-topological space. Besides, we define some open sets, namely, neutrosophic infi-open set, neutrosophic infi-semi-open set, neutrosophic infi-pre-open set, neutrosophic infi-b-open set. Then, we d...

In this study, we give another generalization of second order backward difference operator ∇2 by introducing its quantum analog ∇q2. The operator ∇q2 represents the third band infinite matrix. We construct its domains c0(∇q2) and c(∇q2) in the spaces c0 and c of null and convergent sequences, respectively, and establish that the domains c0(∇q2) and...

In this paper, an attempt has been made to introduce a new similarity measure namely single-valued pentapartitioned neutrosophic dice similarity measure (SVPNDSM) under the single-valued pentapartitioned neutrosophic set (SVPNS) environment, and to formulate several interesting results on SVPNDSM and SVPNWDSM. In this present work, the SVPNDSM unde...

The aim of this article is to introduce the characterization of strongly almost convergence in complex uncertain double sequences. We investigate the concept of strongly almost convergence almost surely, convergence in measure, convergence in mean, convergence in distribution, and convergence uniformly almost surely of complex uncertain double sequ...

The contemporary situation of the world is very pathetic due to the spread of COVID-19. In this article, we have prepared a decision making model on COVID-19 pandemic patients with the help of the neutrosophic similarity measures. The model is to predict the COVID-19 patents for testing positive and testing negative. The decision making is based on...

In this paper we have studied the regular relative uniform convergence and relative uniform Cauchy double sequences of functions defined on a compact domain D. We have established that for the double sequences of functions in the complete subset of the set of continuous functions on D, a double sequence is relative uniform convergent if and only if...

In this paper we have examined the approximate point spectrum, defect spectrum and compression spectrum of the operator D(r,0,s,0,t) on the sequence spaces c0, c, and $bv_p (1<p<\infty)$.

In this article we established the concept of multi-continuity in minimal structure spaces (in short $\mathscr{M}$ space) and the notion of product minimal space in multiset topological space. Continuity between $\mathscr{M}$-space, generalized multiset topology and multiset ideal topological spaces. We have investigated some basic properties of $\...

In this paper, we procure the notions of neutrosophic simply b-open set, neutrosophic simply b-open cover, and neutrosophic simply b-compactness via neutrosophic topological spaces. Then, we establish some remarks, propositions, and theorems on neutrosophic simply b-compactness. Further, we furnish some counter examples where the result fails.

In this paper, we introduce the difference double sequence of complex uncertain variables defined by Orlicz function. We study some of their properties like solidness, symmetricity, and completeness and prove some inclusion results.

The main aim of this treatise is to study the concept of strongly almost convergence of complex uncertain triple sequences and to investigate the interrelationship among different aspect of this concept on the complex uncertain triple sequences.

The main aim of this article is to study strongly almost λ-convergence and statistically almost λ-convergence of complex uncertain sequences in two aspects. At first we introduce these notions via Orlicz function and then we do the same by following usual path of convergence. In general, these types of convergences of sequences can be done in five...

In this article, we procure the idea of single-valued pentapartitioned neutrosophic cosine similarity measure (SVPNCSM) and single-valued pentapartitioned neutrosophic weighted cosine similarity measure (SVPNWCSM) under the single-valued pentapartitioned neutrosophic set (SVPNS) environment. Besides, we formulate several interesting results on SVPN...

The study of uncertainty theory evolved and developed largely in the last decade. In this paper, we introduce the concept of summability and absolutely summability with respect to almost surely through matrix transformation of complex uncertain sequences and establish the interrelationship between these two concepts. In this context, applications o...

In this paper, we introduce the notion of lacunary convergence for double sequences of complex uncertain variables. We have established the relation between lacunary convergence and strong Cesàro convergence. Also, we have established the relation between different concepts of lacunary convergence of double sequences of complex uncertain variables.

The main aim of this paper is to introduce I-st limit points and I-st cluster points of a sequence of fuzzy numbers and also study some of its basic properties. Conditions for a I-st limit point of a I-st cluster point are investigated.

In this article we define matrix maps between complex uncertain sequences. We introduce the notion of bounded sequences of complex uncertain sequence for almost sure, mean, measure and distribution. We introduce the limitation method for different notion of boundedness of sequence of complex uncertain variables and establish relation between the dif...

In this article we introduce the notion of relative uniform convergence of difference double sequence of positive linear functions. We define the difference double sequence spaces \(_2\ell _\infty (\varDelta , ru),~ _2c(\varDelta , ru),~ _2c_0(\varDelta , ru), _2{c_0}^B(\varDelta , ru),~_2c^B(\varDelta , ru), _2c^R(\varDelta , ru), ~_2{c_0}^R(\varD...