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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 the relationship between statistically pre-Cauchy and statistically convergent sequence by using complex uncertain variables. Finally, we initiate the study of statistically pre-Cauchy sequences of complex uncertain variables via Orlicz functions also.

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 sequences in a given uncertainty space. We emphasize to focus on characterizing statistical convergence of complex uncertain triple sequence in some extent. Moreover, we initiate the notion of complex uncertain Cauchy triple sequence and establish the interconnection of it with a statistically convergent triple sequence.

In this paper, we explore the existence of operation approach on open sets in a generalized fuzzy topological space. We introduce the concept of fuzzy γµ-open set and study some basic properties of it. We obtain an interesting result that the intersection of two fuzzy γµ-open sets may not be a fuzzy γµ-open set, but if the operation is regular then the intersection becomes a fuzzy γµ-open set. We also initiate the notions of fuzzy minimal γµ-open set and fuzzy γµ-locally finite space and establish various results related to these.

The main aim of this paper is to introduce the notion of [Formula: see text]-statistical convergence in a given uncertainty space by considering sequences of complex uncertain variable. Also, we study the interrelationship between [Formula: see text]-statistical convergence with strongly [Formula: see text]-summable sequence in the same environment. Further, we extend the study by initiating the notions of [Formula: see text]-statistically Cauchy complex uncertain sequence.

The main aim of this article is to introduce the notion of statistical convergence of a complex uncertain triple sequence. Four types of statistically convergent complex uncertain triple sequences ...

Convergence of real sequences, as well as complex sequences are studied by B. Liu and X. Chen respectively in uncertain environment. In this treatise, we extend the study of almost convergence by introducing double sequences of complex uncertain variable. Almost convergence with respect to almost surely, mean, measure, distribution and uniformly almost surely are presented and interrelationships among them are studied and depicted in the form of a diagram. We also define almost Cauchy sequence in the same format and establish some results. Conventionally we have, every convergent sequence is a Cauchy sequence and the converse case is not true in general. But taking complex uncertain variable in a double sequence, we find that a complex uncertain double sequence is a almost Cauchy sequence if and only if it is almost convergent. Some suitable examples and counter examples are properly placed to make the paper self sufficient.

In this paper, we present the concept of complex uncertain triple sequences and study almost convergence therein. Almost convergence with respect to all five aspects in uncertain space, viz., almost convergence in mean, measure, distribution, almost surely and uniformly almost surely, are initiated and interrelationships among them are established. We have also studied almost Cauchy triple sequence of complex uncertain variables and established some results. It is known that every convergent sequence is a Cauchy sequence but the converse is not true in general. But taking complex uncertain variables in triple sequences, we find that a complex uncertain triple sequence is an almost Cauchy sequence if and only if it is almost convergent.

The paper presents the notion of $$(i,j)^{*}~\gamma $$ -generalized fuzzy closed set in a fuzzy bitopological space considering a parallel form of (i, j) fuzzy $$\gamma $$ -open set. We introduce the idea of $$\gamma $$ -generalized fuzzy q-neighbourhood of a fuzzy point. Moreover, we study various related properties and cite some interesting results on this set. Also, we show that generalized fuzzy closed sets and $$\gamma $$ -generalized fuzzy closed sets are completely independent of each other in the same environment. Furthermore, we introduce the notion of $$\gamma $$ -generalized fuzzy continuous function in a fuzzy bitopological space. Finally, we propose a new type of closure operator and establish important results based on this concept.

In this paper, we introduce the concept of convergence of complex uncertain series. We initiate matrix transformation of complex uncertain sequence and extend the study via linearity and boundedness. In this context, we prove Silverman-Toeplitz theorem and Kojima-Schur theorem considering complex uncertain sequences. Finally, we establish some results on co-regular matrices .

In this paper, we apply continuous random distributions such as the exponential distribution, gamma distribution, beta distribution and uniform distribution and discontinuous random distribution such as Poisson distribution by using neutrosophic random variables. This study opens a new way for dealing with issues that follow the classical distributions which appear in classical random variables and at the same time contain data not specified accurately.

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 of matrix transformation of complex uncertain sequences are also presented.

This paper aims to present the notion of (i, j)*-fuzzy γ-open set in a fuzzy bitopological space as a parallel form of (i, j)-fuzzy γ-open set due to Tripathy and Debnath (2013) [17] and show that both of them are independent concepts. Then we extend our study to (i, j)*-generalized fuzzy γ-closed set and (i, j)*-γ-generalized fuzzy closed set. We show that (i, j)*-γ-generalized fuzzy closed set and (i, j)*-generalized fuzzy γ-closed set are also independent of each other in nature. Though every (i, j)*-fuzzy γ-closed set is a (i, j)*-generalized fuzzy γ-closed set but with (i, j)*-γ-generalized fuzzy closed set, the same relation is not linear. Similarly though every (i, j)*-fuzzy closed set is (i, j)*-γ-generalized fuzzy closed set but it is independent to (i, j)*-generalized fuzzy γ-closed set. Various properties related to (i, j)*-generalized fuzzy γ-closed set are also studied. Finally, (i, j)*-generalized fuzzy γ-continuous function and (i, j)*-generalized fuzzy γ-irresolute functions are introduced and interrelationships among them are established. We characterized these functions in different directions for different applications.

In this paper, we have introduced three new types of convergence concepts, namely convergence in p-distance, completely convergence and convergence in metric by considering triple sequences of complex uncertain variable. We have established the interrelationships among these notions and also with the existing ones. In this process, we have proven that the notions of convergence in metric and convergence in almost surely are equivalent in nature. Overall, this study presents a more complete scenario of interconnections between the notions of convergences initiated in different directions.

Generalized closed sets play an important role in the study of topological spaces. In the literature we have seen the relation between closed set and generalized closed set is linear in nature, both in crisp and fuzzy environment. In particular, every closed set is a generalized closed set. In this present treatise, a new type of generalized closed set via $$(i,j)^{*}$$ -fuzzy $$\gamma $$ -open set, namely $$(i,j)^{*}$$ - $$\gamma gf\gamma $$ -closed set have been introduced in a fuzzy bitopological space and interrelationships of it with the existing ones have been made. It has been proved that this newly defined generalized fuzzy closed set is totally independent with $$(i,j)^{*}$$ -fuzzy closed set. Various characterizations of this set have been established and some interesting results are cited. As applications of this set various types of continuities are studied in the same environment. More applications are proposed by introducing a new kind of closure operator namely $$(i,j)^{*}$$ - $$\gamma $$ - $$cl_{\gamma }^{*}$$ -operator and important results have been reported based on this concept.

The main aim of this paper is to introduce statistical convergence of a complex uncertain double sequence. Characterization of statistical convergence is given via uncertain measure, uncertain expected value operator, uncertain distribution function and also with respect to almost surely, uniformly almost surely. The concept of boundedness is introduced to establish some results. Finally, the notion of statistically complex uncertain double Cauchy sequence is proposed to obtain interrelationship with statistically convergent complex uncertain double sequence.

The generalized closed set plays an important role in the study of fuzzy topological spaces, and this paper is a prolongation of the idea of \(g^*\)-fuzzy closed set in the light of fuzzy \(\gamma ^*\)-open set in the same environment. In particular, this paper presents three different types of \(g^*\)-fuzzy closed sets, namely \(\gamma ^*\)-\(g^*\)-fuzzy closed set, \(g^*\)-\(\gamma ^*\)-fuzzy closed set, and \(\gamma ^*\)-\(g^*\)-\(\gamma ^*\) fuzzy closed set via fuzzy \(\gamma ^*\)-open set in fuzzy topological spaces. Also, we establish the interrelationships among these newly defined fuzzy sets with the existing ones. As an application, we introduce and study some new classes of spaces called fuzzy \(\gamma ^*\)-\({T_{1/2}}^*\) spaces, fuzzy \({T_{1/2}}^*\)-\(\gamma ^*\) space and fuzzy \(\gamma ^*\)-\({T_{1/2}}^*\)-\(\gamma ^*\) space.

In this treatise, we propose a new type of closed set in fuzzy topological spaces called \(g^*\) fuzzy closed set, which is lying in between the fuzzy closed set and the generalized fuzzy closed set. Also, we study another class of fuzzy sets called \(\theta \)-\(g^*\) fuzzy closed sets which is weaker than \(\theta \)- fuzzy closed sets but stronger than \(\theta \)-g fuzzy closed sets and an interrelationships among these newly defined fuzzy closed sets along with the existing generalized fuzzy closed sets are established. Furthermore, the idea of fuzzy \(g^*\)-connectedness is introduced in the light of \(g^*\)-fuzzy closed sets. Finally, we define \({T_{1/2}}^*\)-space, \(^*T_{1/2}\)-space, \({_\theta T_{1/2}}^*\)-space and \({^*}_\theta T_{1/2}\)-space and some applications of these newly defined spaces are discussed.

In this present treatise, we introduce the notion of (i, j) generalized fuzzy \(\gamma \)-closed (\((i,j)~ gf\gamma \)-closed) set and \((i,j)~ \gamma \)-generalized fuzzy closed (\((i,j) ~\gamma \)-gf closed) set in a fuzzy bitopological space. We show that \((i,j)~ \gamma \)-generalized fuzzy closed set and (i, j) generalized fuzzy \(\gamma \)-closed set are totally independent of each other. Different properties related to those sets are studied. Also, (i, j) generalized fuzzy \(\gamma \)-continuous functions and (i, j) generalized fuzzy \(\gamma \)-irresolute functions based on \((i,j)~ gf\gamma \)-closed set are introduced and interrelationship among them are established. Finally, we characterize these functions in various directions for different purposes.